The Simulation of Turbomachinery Flows at
Arbitrary Mach Numbers and The Analysis of
Leakage Flows in Shrouded Axial Turbines
Von der Fakultät Energie-, Verfahrens- und Biotechnik der
Universität Stuttgart zur Erlangung der Würde eines
Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung
vorgelegt von
Jan Eric Anker
aus Karlsruhe
Hauptberichter: Prof. M. V. Casey, DPhil
Mitberichter: Prof. Dr.-Ing. E. Göde
Tag der mündlichen Prüfung: 22. April 2013
Institut für Thermische Strömungsmaschinen und
Maschinenlaboratorium der Universität Stuttgart
2013

Acknowledgment
The foundation for this thesis was laid during my position as a Research As-
sociate at the Institute of Thermal Turbomachinery and Machinery Labora-
tory (ITSM) of University of Stuttgart. The work presented herein related
to the study of leakage flow effects in shrouded axial turbines was funded
by Forschungsvereinigung Verbrennungskraftmaschinen (FVV) in the frame-
work of the projects Deckbandströmunseinfluss I-III, which is gratefully ac-
knowledged.
First of all I would like to thank Prof. Michael V. Casey, the Director of
ITSM, for supervising this thesis, many fruitful discussions, and for promot-
ing my scientific work. His continuous interest in my research and his ex-
cellent guidance on turbomachinery flow physics was an inspiration and con-
tributed significantly to the success of this thesis.
I would also like to express my gratitude to the late Prof. Heinz Stetter,
the former Director of ITSM, for hiring me as a Research Associate and for his
unwavering confidence in me. Furthermore, I would like to thank Dr. Joachim
Messner, the former Vice-Director of ITSM, for ensuring that there always
was sufficient funding for research and that the associates could present their
work at international conferences.
My gratitude is also extended to Prof. Eberhard Göde for his support as a
Commissary Director in the interim period before Prof. Casey was appointed
as Director of ITSM. In addition, I would like to thank Prof. Göde for co-
promoting this thesis.
Furthermore, I am obliged to the various members of the Supervising Com-
mittees of the aforementioned FVV projects for their constructive advice and
guidance. Special thanks is due to the Chairmen of the first two projects,
Dr. N. Vortmeyer and Dr. C. Mundo, for promoting the proposals for follow-on
projects on leakage flow effects. I am also obliged to Dr. A. Jung, the Chair-
man of the third FVV project, for giving demanding, but constructive feedback
and pushing for a very systematic study of the impact of the leakage flow in
shrouded turbines.
In the context of the projects on leakage flow effects, I would additionally
like to thank Dr. Jürgen F. Mayer, Head of the Section for Numerical Methods
at ITSM, for his support in general matters and for his excellent reviews of
the reports I wrote for FVV. His feedback, especially with regards to formal
aspects, was very valuable. Furthermore, I am thankful that he gave me the
freedom to develop numerical methods in the in-house flow solver ITSM3D
during the FVV projects.
For their help in visualizing the leakage flow effects in Virtual Reality
(VR), I would also like to thank Mr. Lars Frenzel and Dr. Uwe Wössner of
the High Performance Computing Center Stuttgart (HLRS). The use of VR-
visualization proved to be a very effective means to present results to the
members of the Supervising Committee in the aforementioned FVV projects.
At this place I would like to thank Dr. Rüdiger Merz, who brought me to
ITSM by hiring me as a Research Assistant (HiWi). His inspiring and opti-
i
mistic personality convinced me to carry out the MSc thesis in the Section
for Numerical Methods at ITSM, which was the starting point in my career
in Computational Fluid Dynamics (CFD). I learned a lot from him during his
supervision and I am grateful to have had his advice and mentorship.
Furthermore, I owe many thanks to my former colleagues, students, and
research assistants at the University of Stuttgart, who were supporting me
in my work and personal development in numerous ways. In particular I
would like to thank Mr. Holger Bauer, Mr. Francis Delhopital, Dr. Gerhard
Eyb, Mr. Klaus Findeisen, Mr. Frank Fuchs, Mr. Peter Gerkens, Mr. André
Gerlach, Mr. Wolfram Gerschütz, Mr. Michael Göhner, Mr. Steffen Kämmerer,
Mr. Peter Kraus, Dr. Xandra Margot, Mr. Michael Rath, Mr. Dietmar Reiß,
Mr. Markus Schatz, Mr. Andreas Seibold, Mr. Udo Seybold, Mr. Roland Sigg,
Mr. Lutz Völker, and Mr. Volker Wittstock for their valuable colleagual sup-
port, interesting discussions, productive collaborations and various memo-
rable moments.
Of the students I was supervising when I was a Research Associate at
ITSM, I would particularly thank Mr. Alexander Berreth, Mr. Jürgen Bosch,
Mr. Reinaldo A. Gomes, Mr. Perttu Jukola, Mrs. Birthe Schrader, Mr. Alex
Schulz, Mr. Senol Sög˘üt, and Mr. Christian Wölfel for the great collaboration
on various CFD related topics. I want also to express my gratitude to Mr. H. C.
Bauer for his help with interlibrary loans of literature and for sourcing vari-
ous scientific literature and references.
I am indebted to my parents for their support during my studies. Last but
not least, very special thanks go to my wife for her loving support and her
encouragement while I was writing down this thesis.
Brussels, in July 2013 Jan E. Anker
ii
Contents
Contents iii
Nomenclature ix
Abstract xvii
Kurzfassung xix
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Leakage flow effects in shrouded axial
turbomachinery . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Methods for the simulation of turbomachinery flows at
all speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Modeling of transitional and turbulent flows in turboma-
chinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Objectives of the present thesis . . . . . . . . . . . . . . . . . . . 15
1.4 Outline of the present thesis . . . . . . . . . . . . . . . . . . . . . 16
2 A baseline solution scheme 19
2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Discretization scheme and solution procedure . . . . . . . . . . . 20
2.2.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . 21
2.2.2 The necessity of adding artificial dissipation in a central
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 The artificial dissipation scheme of Jameson,
Schmidt and Turkel . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Time-stepping scheme . . . . . . . . . . . . . . . . . . . . 25
2.2.5 Convergence acceleration techniques . . . . . . . . . . . . 26
2.2.6 Normalization of the flow equations . . . . . . . . . . . . 26
2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Inlet, outlet and mixing planes . . . . . . . . . . . . . . . 27
2.3.2 Wall boundary conditions . . . . . . . . . . . . . . . . . . 27
2.3.3 Periodic boundary condition . . . . . . . . . . . . . . . . . 28
iii
3 A Mach number independent solution scheme 29
3.1 The time-derivative preconditioning scheme of Merkle . . . . . 29
3.1.1 Eigenvalue scaling for convergence enhancement . . . . 31
3.1.2 Perturbation analysis for low Mach number flow . . . . . 33
3.1.3 Robustness aspects . . . . . . . . . . . . . . . . . . . . . . 33
3.1.4 Scaling issues for low Reynolds number flow . . . . . . . 34
3.2 Preconditioned discretization and solution scheme . . . . . . . . 35
3.2.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . 36
3.2.2 Artificial dissipation . . . . . . . . . . . . . . . . . . . . . 37
3.2.3 Time-stepping scheme . . . . . . . . . . . . . . . . . . . . 39
3.2.4 Calculation of the time step . . . . . . . . . . . . . . . . . 39
3.2.5 Treatment of the initial and boundary conditions . . . . 39
3.2.6 The use of perturbed, normalized flow quantities . . . . . 40
3.2.7 Implicit residual smoothing technique for preconditioned
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Impact of preconditioning . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Enhancement of the convergence rates . . . . . . . . . . . 41
3.3.2 Accuracy preservation of the solution scheme for lowMach
number flow simulations . . . . . . . . . . . . . . . . . . . 42
3.4 Computational results . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.1 Inviscid flow over a bump . . . . . . . . . . . . . . . . . . 44
3.4.2 Flat plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.3 Supersonic flow through a row of wedges . . . . . . . . . 45
3.4.4 Low-speed turbine . . . . . . . . . . . . . . . . . . . . . . 46
4 Low-diffusive discretization schemes 51
4.1 Scalar JST-scheme with TVD properties . . . . . . . . . . . . . 51
4.2 JST-switched matrix dissipation . . . . . . . . . . . . . . . . . . 52
4.3 Strengths and weaknesses of JST-switched schemes . . . . . . . 53
4.4 Scalar SLIP scheme of Jameson . . . . . . . . . . . . . . . . . . 54
4.5 Characteristic upwind scheme of Roe . . . . . . . . . . . . . . . 57
4.5.1 Roe-averaging . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5.2 Method adopted in the current work . . . . . . . . . . . . 58
4.6 Advected upstream splitting techniques . . . . . . . . . . . . . . 59
4.6.1 The AUSM+ scheme . . . . . . . . . . . . . . . . . . . . . 60
4.6.2 Formulation of the AUSM+ scheme for a cell-vertex scheme 62
4.6.3 Preconditioned AUSM+-scheme . . . . . . . . . . . . . . . 65
4.6.4 Robustness issues . . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Computational results . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7.1 Ni-Bump test case . . . . . . . . . . . . . . . . . . . . . . . 70
4.7.2 Flat plate laminar boundary layer . . . . . . . . . . . . . 70
4.7.3 Supersonic flow through a row of wedges . . . . . . . . . 72
4.7.4 Low-speed turbine . . . . . . . . . . . . . . . . . . . . . . 74
iv
5 Development of a NRBC treatment for preconditioned systems 89
5.1 The Giles-Saxer approach . . . . . . . . . . . . . . . . . . . . . . 89
5.2 The Giles Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Quasi-three-dimensional steady-state NRBC . . . . . . . . . . . 91
5.4 1D characteristic theory for imposing BCs . . . . . . . . . . . . . 93
5.4.1 One-dimensional characteristic theory as a means to im-
pose compatible boundary conditions . . . . . . . . . . . . 93
5.4.2 Axially subsonic inlet . . . . . . . . . . . . . . . . . . . . 95
5.4.3 Axially subsonic outlet . . . . . . . . . . . . . . . . . . . . 97
5.4.4 Axially supersonic inlet . . . . . . . . . . . . . . . . . . . 98
5.4.5 Axially supersonic outlet . . . . . . . . . . . . . . . . . . 99
5.5 Implementation of the Q3D NRBC . . . . . . . . . . . . . . . . . 99
5.5.1 Subsonic inflow boundary . . . . . . . . . . . . . . . . . . 100
5.5.2 Subsonic outlet boundary . . . . . . . . . . . . . . . . . . 103
5.5.3 Axially subsonic inlet or outlet boundaries in an overall
supersonic flow . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5.4 Axially supersonic inlet and outlet boundaries . . . . . . 105
5.5.5 Adjacent solid walls . . . . . . . . . . . . . . . . . . . . . . 105
5.6 Non-reflecting mixing planes . . . . . . . . . . . . . . . . . . . . 105
5.6.1 Coupling condition . . . . . . . . . . . . . . . . . . . . . . 106
5.6.2 Numerical implementation . . . . . . . . . . . . . . . . . 106
5.7 Verification of the non-reflectivity . . . . . . . . . . . . . . . . . . 107
5.8 Comparison of developed and original NRBC . . . . . . . . . . . 109
6 Analysis of leakage flow effects 115
6.1 Description of the shrouded turbine . . . . . . . . . . . . . . . . 115
6.2 Modeling of the shrouded turbine . . . . . . . . . . . . . . . . . . 116
6.2.1 Grids and boundary conditions . . . . . . . . . . . . . . . 116
6.2.2 Numerical modeling . . . . . . . . . . . . . . . . . . . . . 118
6.3 Validation of the computational results . . . . . . . . . . . . . . 118
6.4 Importance of the geometrical parameters of the labyrinth seal 124
6.5 Main flow - leakage flow interaction . . . . . . . . . . . . . . . . 124
6.5.1 The impact of the leakage flow on the main flow . . . . . 124
6.5.2 Interaction between the potential field of the rotor and
the cavity flow . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5.3 Consequences of the leakage flow extraction on the sec-
ondary flow field of the rotor . . . . . . . . . . . . . . . . . 126
6.5.4 Consequences of the leakage flow re-injection on the sec-
ondary flow field of the rotor . . . . . . . . . . . . . . . . . 128
6.5.5 The impact of the presence of cavities on the formation
of the secondary flow field . . . . . . . . . . . . . . . . . . 129
6.5.6 Impact of the Leakage Flow on the Secondary Flow Field
in the Second Stator Row . . . . . . . . . . . . . . . . . . 130
6.6 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . 132
7 Summary and conclusions 135
v
A Perturbation analysis 139
A.1 A preconditioning method for inviscid, low Mach number flow . 139
A.2 A preconditioning method for low Mach number, viscous flows . 143
A.3 Summary of the implications for the preconditioning parameters 147
B Dispersion analysis 149
B.1 Governing equations and dispersion analysis . . . . . . . . . . . 149
B.1.1 Dispersion analysis for ideal gas flow . . . . . . . . . . . 151
B.1.2 Dispersion analysis for incompressible flow . . . . . . . . 151
B.2 Preconditioning for ideal gases . . . . . . . . . . . . . . . . . . . 153
B.2.1 Wave speeds when time-derivative preconditioning is not
applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.2.2 Wave speeds when low Mach number
time-derivative preconditioning is applied . . . . . . . . . 154
B.2.3 Wave speeds when low Reynolds number time-
derivative preconditioning is applied . . . . . . . . . . . . 155
B.3 Preconditioning for incompressible flow . . . . . . . . . . . . . . 157
B.3.1 Wave speeds when time-derivative preconditioning for
inviscid flows is applied . . . . . . . . . . . . . . . . . . . 158
B.3.2 Wave speeds when time-derivative preconditioning for
viscous dominated flows is applied . . . . . . . . . . . . . 158
B.4 Implications of the dispersion analysis . . . . . . . . . . . . . . . 160
C Scaling of the artificial dissipation 163
C.1 Inviscid flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
C.2 Viscous dominated flows . . . . . . . . . . . . . . . . . . . . . . . 166
D Issues in low Mach number flow simulations 171
D.1 Singularity of the flow equations . . . . . . . . . . . . . . . . . . 171
D.2 Avoidance of round-off errors at low Mach numbers . . . . . . . 172
D.3 Use of a perturbed temperature variable . . . . . . . . . . . . . . 174
E Flux limiters 177
E.1 Monotonicity conditions and limiters . . . . . . . . . . . . . . . . 177
E.2 Limiter functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
E.3 Discussion of Jameson’s SLIP and USLIP schemes . . . . . . . . 181
F The Roe scheme for preconditioned systems 183
F.1 Conservative and non-conservative formulation . . . . . . . . . 183
F.2 Use of conservative versus non-conservative variables . . . . . . 185
F.3 The principle of Roe-averaging . . . . . . . . . . . . . . . . . . . 187
F.4 Eigenvector and absolute Jacobian matrices . . . . . . . . . . . . 189
G Influence of geometry of the labyrinth seal 193
H Numerical modeling of labyrinth seals 197
vi
I Unsteadiness of leakage flow and necessary level of numerical
modelling 199
Bibliography 203
vii
viii
Nomenclature
Latin symbols
a Isentropic speed of sound
a0 Constant in the CUSP scheme
A Area, flux Jacobian
A0 Measurement plane
B Flux Jacobian
bp Factor in Roe’s characteristic upwind scheme
c Arbitrary wave speed
C Flux Jacobian
c, ci,
[u, v, w] Cartesian velocity vector
c11, c12,
c21 Factors in Roe’s characteristic upwind scheme
c52 Coefficient in the NRBC treatment
c Constant for determining the threshold value
cp Specific isobaric heat capacity
cv Specific isochore heat capacity
CFL Courant-Friedrichs-Lewy number
d Component of the artificial dissipation vector,
quantity related to preconditioning
dp Quantity related to preconditioning
D Determinant, mean diameter
D Vector of the artificial dissipation
|D| Absolute Mach number function (AUSM scheme)
e Specific energy
E Axial flux vector
E◦ Kinetic energy
f Weighting factor in the κ-scheme
F Circumferential flux vector, general flux vector
G Radial flux vector
h Specific enthalpy, spanwise coordinate
H Span, blade height, helicity
H Vector of source terms, general flux vector
i Grid index, summation index
ιˆ Imaginary unit, ιˆ2 = −1
ix
I Unity matrix
j Grid index, summation index
J Jacobian matrix
k Grid index, user-defined
scaling coefficient for the artificial dissipation,
proportionality constant, wave number, summation index
k First vector in the system k, l,m
of orthogonal vectors
kλ Thermal conductivity coefficient
l Characteristic length, wave number, summation index
l Left eigenvector, second vector in the system
k, l,m of orthogonal vectors
L Reference length, limiter function, matrix of left eigenvectors
m Scaled Mach number, wave number, summation index
m˙ Mass flux
m Third vector in the system k, l,m
M Mach number
M1,M2, Measurement plane
M3
MP Midplane
Mˇ Scaled Mach number to ensure pressure-velocity coupling
(AUSM scheme)
M Mach number function (AUSM scheme)
|∆M| Absolute Mach number function (AUSM scheme)
n Fourier mode, summation index
n˙ Rotational speed
~n Direction vector, surface vector
N Blade number, number of grid points in the circumferential direction
(periodic nodes are only counted once per pitch)
|N | Absolute Mach number function (AUSM scheme)
p Static pressure
pˆ State vector (frequency domain)
P Element of the limiter function of the SLIP scheme
P Element of the pressure sensor of the JST scheme,
effective limiter of the SLIP scheme
|P| Absolute Mach number function (AUSM scheme)
P Presssure flux vector (in the AUSM scheme)
Pe Peclet number
Pr Prandtl number
q State variable
qe Exponent in the SLIP scheme
qˆ State vector (frequency domain)
Q State vector (conservative variables)
Qp Vector of primitive state variables
Qv Vector of viscous, primitive state variables
r Radial coordinate, radius, ratio of successive
x
slopes/differences, reaction
r Flux vector, contribution to the residual
R Specific gas constant, matrix of right eigenvectors,
residual, matrix of diffusive terms
Re Reynolds number
s Specific entropy, stages in the Runge-Kutta scheme,
limiter, chord length, width of a gap/tip
scl Clearance height
S Surface, element of the limiter function of the SLIP scheme
t Time, summation index
T Absolute temperature
u First component of the Cartesian velocity vector c
V∞ Axial velocity of the far-field
vˆ Contravariant velocity component
Vˆ Volume weighted, contravariant velocity component
Vˇ Cell volume (control volume)
U General state vector
v Velocity, second component of the Cartesian velocity vector c
v, Velocity vector in an absolute frame
[vx, vφ, vr] of reference (Cylindrical coordinates)
vˆ, Contravariant velocity vector in an absolute
[vˆξ, vˆη, vˆζ ] frame of reference
V Volume, velocity magnitude (absolute frame of reference)
Vl Lower threshold for the eigenvalues corresponding
to advective (linear) transport velocities
Vn Lower threshold for the eigenvalues corresponding
to acoustic (non-linear) transport velocities
Vr Reference velocity
VNN Von Neumann Number
wx Axial velocity component
wφ Circumferential velocity component
wr Radial velocity component
w, Velocity vector in a relative frame
[wx, wφ, wr] of reference (Cylindrical coordinates)
wˆ, Contravariant velocity vector in a relative frame
[wξ, wη, wζ ] of reference
W Magnitude of the velocity vector in the relative system
Wˆ Volume weighted, contravariant velocity vector
in the relative system
x Axial distance, first Cartesian coordinate, length
y Second Cartesian coordinate
z Third Cartesian coordinate
xi
Greek symbols
α Thermal diffusivity coefficient, coefficient in the AUSM scheme,
coefficient in the JST scheme, coefficient in the Runge-Kutta scheme,
scaling factor for the dissipation
β Coefficient in the AUSM scheme, quantity in the NRBC treatment
γ Coefficient of the Runge-Kutta scheme, quantity in the NRBC
treatment, isentropic coefficient
Γ Preconditioning matrix
δpc Preconditioning parameter
 Accuracy, tolerance, preconditioning parameter,
perturbation parameter, threshold
ε Resulting coefficient for the dissipation in the JST scheme
ijk Permutation tensor, three-dimensional Levi-Civita symbol
m Machine accuracy
rnd Round-off error
ζ Third grid coordinate, loss coefficient
η Second grid coordinate, normalized wall distance, similarity variable
Θ Scaling factor in the JST scheme
κ Factor in the scheme with the same name
λ Eigenvalue
λ45 Eigenvalue-based factor in the dissipation of
the Roe scheme
Λ Diagonal matrix of the eigenvalues
µ Dynamic viscosity
ν Pressure sensor, kinematic viscosity
σ Condition number
σ1, σ2 Relaxation factors (NRBC)
σp Factor in AUSM scheme
Ξ Mach number related quantity
pi Number pi, ratio between the circumference
and the diameter of a circle
pi Dissipation vector (Roe scheme)
Π Computational grid
ρ Spectral radius
% Density
τ Stress tensor
Υ Mach number weighted conservative state vector
δΥ Difference of the surface weighted velocity vectors (Roe scheme)
φ General state variable, characteristic variable, flow coefficient
Φ State vector
χ Constant for the artificial dissipation
ξ First grid coordinate
ψ General state variable, stage loading coefficient
Ψ Limiter function
Ψ State vector
xii
ω Angular frequency
ω, ωi Vorticity vector
Ω Angular speed of the rotating frame of reference
ωAUSM Scaling parameter of the AUSM scheme
Latin sub- or superscripts
0 Total quantity, quantity that is vanishing
when preconditioning is shut off
(0) Zeroth order
1 First component, related to equation 1
(1) First order
(2) Second order
3 Third component, related to equation 3
(4) Fourth order
a Outlet
adv Advected part
apl Applied
atd Anti-diffusive
bc Boundary condition
c Convective
cal Calculated by the flow solver
CD Central difference term
cl Clearance
comp Compressible
d Diffusive
e Inlet related quantity, neutral (preconditioning matrix)
edi Essentially diffusive
ex Outgoing
F Related to convective fluxes
i Grid index, summation index, inner
in Incoming
inv Inviscid
incomp Incompressible
is Isentropic
j Grid index, summation index
k Grid index, summation index,
dimensionless (preconditioning matrix)
l Laminar, molecular, summation index, limiter
L Left state, left eigenvector
lim Limit
m Summation index
n Summation index
(n) State at time step n
xiii
o Outer
p Derivative with respect to the pressure at constant temperature,
isobaric, primitive, particular, pressure dissipation term
pc Preconditioning related quantity
pgr Pressure gradient
r Radial component, reference quantity
rot Rotor
R Right state, right eigenvector
Roe Roe average
s Isentropic, summation index, non-reflecting in steady-state
(s) State of the Runge-Kutta stage s
S Surface weighted, related to sources
sec Secondary
st Stator
t Turbulent, summation index
T Derivative with respect to the temperature at
constant pressure, transposed, temperature dissipation term
x Axial component
v Viscous
V Volume weighted
w Fin width
Greek sub- or superscripts
φ Circumferential
ξ Related to the first grid direction, derivative with respect to ξ
η Related to the second grid direction, derivative with respect to η
ζ Related to the third grid direction, derivative with respect to ζ
Symbolic superscripts and overbar symbols
+ Forward, positive part
− Backward, negative part
ˆ Quantity related to the dual mesh, viscous control volume,
contravariant
∗ Normalized, critical, equidistantly distributed, system
† Reference
′ Pseudoacoustic, scaled due to preconditioning,
perturbed variable, local variation
′′ Scaled due to preconditioning
¯ Averaged, scaled quantity, anisotropically scaled, average interface value
˜ Numerical value, perturbed quantity, normalized
ˆ Limited value, Fourier mode
xiv
Operators, functions and mathematical symbols
max( ) Maximum function
min( ) Minimum function
sign( ) Signum function
O( .. ) In the order of
d Total derivative
D Substantial derivative, dissipation operator
δ Variation
δ+ Forward difference
δ− Backward difference
 Much smaller than (one order in magnitude at least)
 Much greater than (one order in magnitude at least)
∆ Difference
∂ Partial derivative∮
Surfaceintegral∫
Integral∑
Sum
Real( ) The real part of a complex number
ln Natural logarithm
∞ Infinity
∀ For all
∈ Is element of
C Set of imaginary numbers
N Set of natural numbers
R Set of real numbers
Z Set of integer numbers√ Square root
· Inner product
∧ Logical and
∨ Logical or
| | Absolute value of a scalar, determinant of a matrix
‖ ‖ Euclidian norm of a vector
[ ]T Transposition operator
∇ Gradient operator
Diag( .. ) Diagonal matrix
∝ Proportional to
6= Not equal
:= Per definition equal to, set equal to
≈ Approximately equal to
≡ Identically equal to
× Times
xv
Conventions
The summation convention of Einstein
If not otherwise stated in the text, then the summation convention of Einstein
is used for all sub- and superscripts i, j, k, l, p, q. According to this convention,
a summation from 1 to 3 is to be performed if the same index appears twice in
the same term. Example:
∂φi
∂xi
=
3∑
j=1
∂φj
∂xj
=
∂φ1
∂x1
+
∂φ2
∂x2
+
∂φ3
∂x3
=
∂φx
∂x
+
∂φy
∂y
+
∂φz
∂z
Abbreviations
1D One-dimensional
1Dm One-dimensional theory applied on the averaged state
1Dp One-dimensional theory applied pointwise
3D Three-dimensional
AUSM+ Enhanced Advection Upstream Splitting Method
CFL Courant-Friedrichs-Lewy
CUSP Convective Upwind Split Pressure
JST Jameson-Schmidt-Turkel
JSTMatd Jameson-Schmidt-Turkel Matrix-dissipation
LED Local Extremum Diminishing
MUSCL Monotone Upstream-centered Schemes for Conservation Laws
nonPC Not preconditioned
NRPCBC Non-reflecting boundary condition treatment for
preconditioned systems
NRBC Non-reflecting boundary condition
PC Preconditioning
PCBC Boundary condition treatment for preconditioned systems
Q3D Quasi-three-dimensional
RPM Rotations per minute
SLIP Symmetric Limited Positive
TVD Total Variation Diminishing
USLIP Upstream Limited Positive
VNN Von Neumann Number
Other conventions
• Non-transposed vectors are always coloumn vectors and transposed vec-
tors are always row vectors
• Formulas marked in (bold face) represent always a vectorial quantity
xvi
Abstract
In the present thesis a preconditioned solution scheme for the simulation of
turbomachinery flow at arbitrary Mach numbers is presented. A time-deriva-
tive preconditioning technique is applied to an explicit, time-marching Navier-
Stokes code, originally concipated for compressible, high-speed turbomachin-
ery applications. Various aspects of the accurate simulation of incompressible
or low Mach number, compressible flows are addressed. The preconditioning
scheme is formulated for fluids with a general equation of state. As verified
by several test cases, the use of preconditioning allows the code to simulate
flows efficiently and accurately for all Mach numbers.
Four different dissipation schemes have been developed and adapted for
the use in combination with preconditioning. It is shown how the robustness
of Liou’s improved Advection Upwind Splitting Method (AUSM+) scheme can
be enhanced by controlling the anti-diffusive parts of the dissipative fluxes.
The different dissipation schemes are assessed on several test cases in terms
of accuracy, solution monotonicity, and robustness.
Since turbomachinery computations often are performed on truncated do-
mains, a non-reflecting boundary condition (NRBC) treatment should be used.
Because preconditioning alters the dynamics of the Navier-Stokes equations,
the state-of-the-art NRBC treatment of Giles and Saxer can not be applied.
For this reason, a new NRBC treatment for preconditioned systems and gen-
eral equations of state is developed. The effectiveness of the novel treatment
is demonstrated on two turbomachinery test cases.
After validation against available measurement data, the solution scheme
is used for a computational study of the interaction of labyrinth seal leakage
flow and main flow in an axial, low-speed turbine. The results are used to in-
vestigate leakage flow phenomena and their dependence on the geometrical
parameters of the sealing arrangements. The most influential geometrical pa-
rameter, the clearance height, is varied between a vanishing and a realistic
height to systematically identify the impact of the leakage flow on main and
secondary flow as well as on the loss generation mechanisms.
The computational results show that the ingress of shroud leakage flow
from the rotor significantly influences the main flow. The high swirl of the
leakage flow causes a suction side incidence onto the following stator row and
consequently leads to an amplification of the upper secondary channel vortex
and increased losses.
The results imply that in order to reduce the losses in shrouded, axial
turbines, the swirl of the reentering leakage flow should be corrected by con-
structive means. If this is technically infeasible, then the blades in the follow-
ing blade row should be twisted to ensure a correct incidence and avoid flow
separation. Since the computational results also show that not only the leak-
age flow but also the mere presence of cavities induce aerodynamic losses, the
geometry of the inlet and outlet cavities must be considered with care in the
design and optimization process of sealing arrangements for axial turbines.
xvii
xviii
Kurzfassung
In der vorliegenden Arbeit wird ein präkonditioniertes Lösungsverfahren zur
Simulation von Turbomaschinenströmungen bei beliebigen Machzahlen vor-
gestellt. Das Präkonditionierungsverfahren wird in einem Navier-Stokes Co-
de entwickelt, welcher ursprünglich für die Simulation von transsonischen
Turbomaschinenströmungen konzipiert wurde. Verschiedene Gesichtspunkte
für die genaue Berechnung von Strömungen mit geringen Geschwindigkeiten
werden behandelt. Das Präkonditionierungsverfahren wird für Fluide mit ei-
ner generellen Zustandsbeschreibung formuliert. Mehrere Testfälle bestäti-
gen, dass die Anwendung von Präkonditionierung eine genaue und effiziente
Strömungsberechnung bei beliebigen Machzahlen ermöglicht.
Vier unterschiedliche Dissipationsverfahren werden entwickelt und für
die Verwendung von Präkonditionierung angepasst. Dabei wird gezeigt, wie
man die Robustheit der modifizierten Advection Upwind Splitting Method
(AUSM+) von Liou durch eine gezielte Steuerung der antidiffusiven Terme
der dissipativen Flüsse verbessern kann. Die Dissipationsschemen werden
anhand mehrerer Testfälle hinsichtlich Robustheit, Genauigkeit und Monoto-
nie des resultierenden Lösungsschemas untersucht.
Da die Berechnung von Turbomaschinenströmungen oft auf Rechengebie-
ten mit kurzen Zu- und Abströmgebieten ausgeführt werden, sollte eine nicht-
reflektierende Randbehandlung (NRRB) benutzt werden. Weil Präkonditio-
nierung die Dynamik der zu integrierenden Erhaltungsgleichungen ändert,
wurde im Rahmen der vorliegenden Arbeit eine NRRB für präkonditionierte
Systeme und generelle Zustandsgleichungen entwickelt. Die Effektivität der
neuen NRRB wird anhand zweier Turbomaschinentestfälle verifiziert.
Nach einer Validierung mit vorhandenen Messdaten wird das entwickelte
Lösungsverfahren dazu benutzt, systematische numerische Untersuchungen
der Interaktion von Deckbandleckage- und Hauptströmung in einer Nieder-
geschwindigkeitsaxialturbine durchzuführen. Die Spaltweite der Labyrinth-
dichtung wird variiert, um den Einfluss der Deckbandströmung auf Haupt-
und Sekundärströmung sowie Verlustentstehung zu identifizieren.
Die Berechnungsergebnisse zeigen, dass die Leckageströmung der Rotor-
reihe einen signifikanten Einfluss auf die Hauptströmung hat. Der hohe Drall
der wiedereintretenden Leckageströmung führt zu einer saugseitigen Fehl-
anströmung der nachfolgenden Leitradreihe und infolgedessen zu einer Ver-
stärkung des oberen Kanalwirbels sowie zu einer Erhöhung der Verluste.
Die Ergebnisse legen nahe, dass der Drall der wiedereintretenden Lecka-
geströmung durch konstruktiveMaßnahmen korrigiert werden sollte, um Ver-
luste zu minimieren. Alternativ sollte die nachfolgende Schaufelreihe gewun-
den werden, um eine Fehlanströmung infolge der Leckageströmung zu ver-
meiden. Da die Ergebnisse zeigen, dass nicht allein die Deckbandleckageströ-
mung, sondern auch die bloße Präsenz von Kavitäten aerodynamische Verlu-
ste hervorrufen kann, sollte die Geometrie der Ein- und Austrittskammern
der Labyrinthdichtung in die Design- und Optimierungsüberlegungen von
Schaufelreihen mit Deckbandgeometrie einbezogen werden.
xix
xx
Chapter 1
Introduction
Turbomachines are the most common devices used in energy conversion pro-
cesses and in aircraft propulsion so that the research and development in this
discipline remains important. Current research objectives consist of develop-
ing propulsion and power systems having a higher efficiency and reliability,
which simultaneously lead to lower emissions of pollutants and noise. To this
end it is necessary to improve the understanding of the the interplay between
the various physical phenomena in turbomachinery and how they can be in-
fluenced to optimize the overall performance of such devices.
1.1 Motivation
The anergy generated in turbomachinery can be attributed to various sources.
Depending on the configuration the presence of wakes, secondary flow, bound-
ary layers on blades and endwalls, the appearance of shocks and the occur-
rence of leakage flow may all contribute to the generation of losses.
The losses due to tip leakage flow represent a substantial part (typically a
third) of the overall exergy losses generated in turbomachinery. The leakage
flow leads to a reduction in the performance since a part of the flow bypasses
the blade without exerting work and anergy is created through dissipation
in the tip region. The leakage may also induce losses in performance in sub-
sequent blade rows since it causes flow inhomogeneities after reentering the
main flow.
The most obvious way to minimize exergy losses caused by leakage flow
in turbomachinery, is to reduce the leakage mass flow by applying shrouds
on the rotor tips and labyrinth seals between the rotating and non-rotating
components. The reduction of the exergy losses through a minimization of
the leakage flow is however technically limited. To be able to minimize the
losses in efficiency incurred by the inevitable presence of leakage flow, it is
important to understand the governing mechanisms when the leakage flow
and the main flow are interacting.
The use of numerical simulations of fluid flows has become an important
tool in the process of optimizing fluid machinery and in achieving insight in
the flow physics of turbomachinery configurations. By employing Computa-
1
2 1. Introduction
tional Fluid Dynamics (CFD) in the development process, the number of ex-
pensive experiments in testing facilities can be reduced.
The requirements on the numerical schemes are increasing for several rea-
sons: The enhancement of the performance of computers, the competition in
the energy and propulsion sector, and the demand for efficient energy con-
version processes. The fluid dynamics software should deliver reliable and
accurate predictions of the flow field, be robust, computationally efficient and
easy to use. In this context it should be noted that the flow field encoun-
tered in turbomachinery is one of the most complicated in the field of fluid
dynamics practice. The Mach number of the flow in turbomachinery can vary
significantly within one single device. It would therefore be desirable to have
a Navier-Stokes solver which is capable of simulating flow in turbomachinery
accurately and efficiently at arbitrary Mach numbers.
In this thesis the Mach number independent simulation of turbomachinery
flows with an arbitrary state description of the fluid is addressed. To be able
to simulate incompressible, compressible, and transonic flows with one single
solution algorithm, a time-derivative preconditioning method, non-reflecting
boundary conditions, and low-diffusion discretization schemes are developed.
The developed solution scheme is used to simulate and to numerically study
the interaction between leakage and main flow in a 1 12 stage shrouded tur-
bine. From the analysis, general conclusions concerning the impact of leakage
flow on the main flow and the associated exergy loss mechanisms are drawn.
1.2 Literature survey and overview of
state-of-the-art
This thesis concerns both the development of methods for the simulation of
turbomachinery flows as well as the analysis of leakage flow and loss genera-
tion mechanisms in turbomachinery flows. In the following literature survey,
modeling aspects and numerical issues will be treated separately from the
physical analysis of turbomachinery flows.
1.2.1 Leakage flow effects in shrouded axial
turbomachinery
The impact of tip clearance flow on the main flow and the associated exergy
losses was extensively studied in the 1990s, which a plethora of publications
on both experimental (e.g., [37, 127, 293]) and numerical (e.g., [22, 147, 151,
192]) studies in this field testifies. The interaction between the tip clearance
flow and the main flow for turbomachinery rows with unshrouded blades is
not comparable to the interaction of leakage flow and main flow in shrouded
turbomachinery. In turbomachinery with free ending blade tips, the pressure
difference between the suction and the pressure side of the blades causes a
leakage flow over the blade tip whereas in shrouded turbomachinery the ax-
ial pressure difference over the blade row causes a part of the fluid to flow over
1.2 Literature survey 3
the shroud instead of passing through the blade channel. Except for effects
due to friction, the leakage flow will not change its flow direction when pass-
ing over the shroud and will thus reenter in the main flow with a flow angle
differing greatly from that of the main flow. The ingress of leakage flow in the
main flow can cause complex flow effects and will in most cases be responsible
for significant exergy losses.
The losses caused by leakage flow over shrouded blades can be minimized
by reducing the leakage and much effort has been undertaken in this direction
[223, 264]. Constraints on the design determine the minimum clearance that
can be achieved. To minimize the leakage losses optimised sealing arrange-
ments should be used. A further reduction of the losses due to shroud leakage
effects can be obtained by controlling the interaction between the leakage flow
and the main flow.
The first work on the impact of leakage flow on the main flow was con-
ducted by Traupel [275], Kacker and Okapuu [138] and Denton [73]. These
publications provide global correlations for losses caused by leakage flow, but
they do not take into account that leakage flow effects not only result in en-
tropy generation due to mixing but also can induce losses in subsequent blade
rows due to the changes in the main flow field.
In recent years there has been a series of publications on the interaction
of the leakage and the main flow in axial turbomachinery. The paper of Ko-
rschunov and Döhler [144] presents the effect of an artificially produced jet
(which simulates leakage flow) entering the main flow at different angles on
a following linear cascade. Pfau et al. [206] have examined the steady and
unsteady interactions within the inlet seal cavity with a three step labyrinth
seal on a rotor shroud both experimentally and numerically.
Attempts to optimize the geometry of the sealing arrangements, shroud,
and the casing to minimize the losses incurred by the interaction between the
leakage flow have been reported in several publications. Wallis et al. [292]
have tried to reduce the aerodynamic losses due to mixing when the leakage
flow reenters the main flow by applying turning bladelets on the shroud in
a four stage turbine. Özdemir [197] and Klein [143] investigate the effect of
shortening the shroud, so that the rotating blade row is only partially instead
of fully covered by the shroud. Schlienger et al. [244] present the effects of
labyrinth seal variation on multistage axial turbine flow but can not report
any advantageous effect on the overall turbine efficiency. In contrast to that,
by optimizing the shroud geometry, sealing arrangement, as well as the cavi-
ties and by additionally applying deflectors, Rosic et al. [232, 233] show that
the exergy losses due to the presence of leakage flow can be significantly re-
duced. Both Peters [202] and Adami [3] show that the mixing losses caused by
the ingress of leakage flow in the main flow can be reduced if the outer edge
of the downstream wall of the outlet cavity is chamfered. Mahle [176] shows
in a numerical study that a turning device close to the outlet of the labyrinth
seal can lead to an increase of the overall efficiency of the turbine.
Both Wallis [292] and Gier et al. [96] have examined the contributions
of different types of losses in a shrouded turbine. They found that not only
4 1. Introduction
the leakage flow itself but also the presence of cavities is responsible for a
substantial part of the generation of losses in an axial turbine. The effect of
the presence of cavities on the secondary flow field was also investigated in the
work of Anker et al. [15]. In the work of Rosic and Denton [231] and of Rosic
et al. [232, 233] the influence of both the presence of leakage flow and cavities
on the main flow and the losses in a 3-stage axial turbine is studied in detail
and various loss mechanisms are identified. Apart from these publications,
there are not many systematic studies on the influence of the leakage flow on
the aerodynamics of axial turbines.
Numerous articles exist on modelling labyrinth seals in their effect on the
rotordynamics of turbomachinery devices (e.g., [34, 55, 264, 272]) and on nu-
merical modelling of the losses caused by leakage flow in throughflow codes
or inviscid solvers (e.g., [77, 138, 162]). There are however only few studies on
the numerical modelling of the labyrinth seals in 3D, turbulent simulations
of multi-stage turbomachinery. The motivation for using a numerical model
would be to avoid a full discretization of the details of the cavities and the
labyrinth seals. Among others Hunter and Orkwis [122], Brown [45], Anker
et al. [15], Rosic et al. [234], Gier et al. [95] have formulated simple numerical
models to avoid a detailed discretization of labyrinth seals in their simula-
tions. Even though the developed models capture important effects and are
better used than not accounting for leakage flow at all, they need either to
be calibrated with experimental data or results from detailed simulations to
deliver accurate results. Anker et al. [15] furthermore show that the first and
last cavity in a sealing arrangement need to be discretized as the flow field
first is sufficiently homogeneous between the first and the last sealing fin to
be modelled by simplified 1D approaches. For detailed predictions of the leak-
age flow effects, a full, geometrically complete discretization of the turbine,
shroud and cavity geometry should be used.
In a series of projects funded by Forschungsvereiningung Verbrennungs-
kraftmaschinen (FVV), the Chair of Steam and Gas Turbines (DGT) of the
Ruhr-University Bochum and the Institute for Thermal Turbomachinery and
Machinery Laboratory (ITSM) at University of Stuttgart, have collaborated
to study the influence of the leakage flow in a 11
2
stage axial turbine with a
labyrinth seal on the periphery of the moving row [6, 7, 9, 10, 13, 14, 19, 92,
202, 207, 208, 310]. In these projects extensive measurements with pneu-
matic five-hole probes and hot-wire anemometry were carried out together
with steady-state and time-accurate simulations. Some of the earlier results
have been presented in Wolter et al. [311], Peters et al. [203, 204], Giboni et
al. [93, 94] and Anker et al. [11, 12, 15, 17].
The leakage flow interaction has only been studied in a few works using
time-accurate measurement and simulations. The numerical and experimen-
tal work carried at ITSM (Anker et al. [10, 13, 15, 19]) and DGT (Peters,
Giboni, Wolter, et al. [92–94, 204, 311]) in the aforementioned FVV projects
address the temporal effects of the interaction between the leakage and the
main flow. Using time-accurate measurements Schlienger et al. [244, 245]
study the interaction between re-entering leakage flow and main flow in a
1.2 Literature survey 5
two-stage turbine. Adami et al. [3] apply an unsteady CFD approach to study
the interaction between the leakage flow and the main flow.
The use of time-accurate experiments related to turbomachinery flow has
the advantage over the application of stationary measurement techniques
that both the temporal and circumferential variations in the flow field are
resolved; both local unsteady effects as well as effects attributed to a varying
flow field in a blade row that is moving relatively to another row can be traced.
Analogously, the use of unsteady flow simulations has the advantage that in-
teraction between the flow in the rotor and the stator can be simulated – the
potential fields of the blades, the wakes and the flow are tracked from one
blade row to the next and their temporal evolution can be simulated without
any loss in accuracy. In time-accurate simulations the flow field at the com-
putational interface between two adjacent blade rows are directly coupled as
opposed to steady-state simulations where the blade rows are connected via
mixing planes, where the flow field in adjacent blade rows are merely coupled
over circumferentially averaged quantities.
In the numerical studies of ITSM mentioned above the periodically un-
steady interaction between the leakage flow, secondary flow and the main
flow is studied. As shown by the work of Anker et al. [15] by comparing time-
averaged flow field of an unsteady simulation with the results of a steady-
state simulation where mixing planes were used, the essential effects are still
captured accurately even when steady-state simulations are carried out.
The results presented in this thesis contributes to the understanding of
the interaction of leakage flow and main flow in axial turbomachinery. In
this work a comprehensive study is carried out in which, by subtraction of
the numerical solution for a vanishing and a realistic clearance height, the
impact of the leakage flow on the main flow, the secondary flow and the loss
generating mechanisms are explicitly identified. Attention is drawn to the
effect that the leakage flow has on the flow field of the shrouded rotor row. In
particular effects introduced by the interaction of the cavity flow and the main
flow in the rotor are examined and different loss generating mechanisms are
discussed.
1.2.2 Methods for the simulation of turbomachinery flows
at all speeds
The rapid increase in computer performance has promoted the development
and use of numerical methods in all technical and scientific disciplines. The
rise in the computing power allows the use of models with increasing complex-
ity and the use of finer numerical resolution of the dynamics of the systems
modeled.
In this context, computational fluid dynamics (CFD) software has become
an indispensable tool in many industrial sectors in which fluid flow is of con-
cern. The application of CFD in the design process over the past three decades
has led to a significant improvement in the performance of aircraft engines
[119]. The design of modern turbines and compressors is unthinkable with-
6 1. Introduction
out the help of CFD software; the effective use of CFD simulations is a key
ingredient in the design of fluid machinery and aeronautic devices. The ben-
efits of applying CFD range from shorter design cycles to better performance
and reduced cost and weight [74, 134].
The flow in turbomachinery may be incompressible, subsonic, transonic,
or supersonic. Many aircraft engines have mixed flows, in which all of these
regimes are present in a single blade row. For this reason it would be advan-
tageous to have a Navier- Stokes solver which is capable of simulating flow in
turbomachinery accurately and efficiently over a broad Mach number range.
Historically, low speed flows were first solved using solution schemes that
solve the Euler or Navier-Stokes equations for incompressible fluids. In the
pseudo-compressibility approach introduced by Chorin [61], a pressure time
derivative is introduced in the continuity equation, which makes the system
of incompressible Navier-Stokes equations hyperbolic and provides a means
of updating the pressure field [205, 230]. Alternatively for the simulation of
incompressible or low Mach number flow, pressure-based methods [200] have
been formulated, which are frequently applied in both academic and indus-
trial flow solvers. In the pressure based methods, the momentum equations
are solved in a segregated fashion applying the most current iterate of the
pressure field. In a second step of each iteration, the pressure field is updated
ensuring that continuity is fulfilled [87].
Since the time-dependent compressible Euler equations are hyperbolic, re-
gardless whether the flow is sub- or supersonic, density-based time-marching
schemes were developed in the context of transonic, external aerodynamic
applications [32, 131] and later extended to viscous, internal turbomachin-
ery applications [56, 265]. These methods proved to be very effective for the
computation of high Reynolds number flows in the transonic and supersonic
regimes. However, these schemes become inefficient and inaccurate at low
Mach numbers due to the disparity of acoustic and the advective velocities for
low speed flows. For this reason, the computation of low Mach number flow
was dominated by solution procedures [61, 200] solving the incompressible
Navier-Stokes equations for many years.
The application of time-marching methods could be extended to low Mach
number compressible flow after it was realized that the difficulty in solving
the compressible equations for low Mach number flow was attributed to the
large disparity of the acoustic wave speed and the waves convected at parti-
cle speed. Low Mach number perturbation methods were the first techniques,
which were used to alleviate these problems and to calculate low speed com-
pressible flow [104, 187, 222].
Later, various local preconditioning procedures were developed in which
the time derivatives of the compressible Euler and Navier-Stokes equations
are altered to control the eigenvalues and to accelerate convergence [43, 59,
160, 276, 278, 290, 301]. D. Choi and Merkle [58, 59] were the first to formu-
late a preconditioning scheme for the full Euler equations for compressible,
ideal gas flow. The scheme was later extended by Y.-H. Choi and Merkle [60]
to the Navier-Stokes equations to also account for viscous effects. Weiss and
1.2 Literature survey 7
Smith adapted their method and introduced a pressure gradient sensor to in-
crease the robustness of the resulting solution scheme [301]. Subsequently,
Merkle et al. [188] extended the method for the simulation of fluid with gen-
eral equations of state. Turkel [276] generalized the Pseudo-Compressibility
approach of Chorin [61] and presented in his work a unified theory of precon-
ditioned methods that can both be used for compressible and incompressible
flows. The preconditioning methods of Merkle and Turkel are very similar
and these methods have been devised by applying one-dimensional eigen-
value, dispersion, and perturbation analysis to equilibrate the propagation
velocities to ensure a good conditioning of the flow equations [289].
Van Leer et al. [160, 173, 175] have developed an alternative precondi-
tioning method which is based on a two-dimensional wave-front analysis of
the Euler equations. In their approach the time-derivatives of the flow equa-
tions are not only scaled for low Mach number but also for supersonic flows.
Their preconditioning method accounts for many design considerations (opti-
mal eigenvector structure, flow angle insensitivities, clustering of numerical
eigenvalues, et cetera) for inviscid, ideal gas flows [69, 158]. Their method
was extended by Lee et al. [158, 159] to account for viscous effects. However,
compared to the preconditioning method of Merkle, in their method viscous
effects have been treated with less rigor. Furthermore, it is not evident how
to extend the preconditioning method of van Leer to fluids described by a gen-
eral thermal and caloric equation of state.
Common for all of the preconditioning methods mentioned above, is that
the temporal terms in the equations to be solved are altered to facilitate the
use of time-stepping methods and to enhance their convergence rates. The
modification of the temporal terms to equilibrate the propagation speeds in
the numerical solution process is not a restriction of the generality of precon-
ditioning method; in the limit of a converged solution, the time derivatives
disappear so that the changed dynamics of the equations does not affect the
solution accuracy. When unsteady simulations are to be performed, the pre-
conditioning method must be incorporated into a dual time-stepping scheme,
where preconditioning is applied at the inner pseudo-time level, thereby not
affecting the outer loop stepping through physical time [48, 286, 301]. Since
the preconditioned Navier-Stokes equations are valid at all speeds, they have
potential for providing uniform convergence over all Reynolds and Mach num-
ber regimes [159, 189].
Preconditioning is beneficial for multigrid schemes [69, 173] and improves
convergence because it removes the inherent stiffness in the governing equa-
tions caused by the disparity in wave propagation speeds. The elimination
of the stiffness is not only important in explicit schemes but also in implicit
ones. Due to the presence of nonlinearities and factorization errors, implicit
time-marching schemes require the use of a finite Courant-Friedrichs-Lewy
(CFL) number which also make the convergence dependent on the dynamics
of the flow equations.
For some solution schemes preconditioning also has the advantage that
it ensures that the dissipative terms resulting from the discretization of the
8 1. Introduction
inviscid fluxes are correctly scaling in the low Mach number limit. If applied
without the use of preconditioning both the popular Roe- and the Jameson-
Schmidt-Turkel schemes become not only inefficient but also inaccurate when
the Mach number approaches zero [105, 287, 289, 291].
While preconditioning methods have become popular and are also used for
the simulation of real gas [80, 188], inert multicomponent or multiphase [120,
121, 124, 164], reacting [33, 81, 91, 224, 241, 286, 300, 309], or cavitat-
ing flow [36, 54, 65, 84, 150, 195, 251], there are not many reports [51,
279, 298, 301] known to the author where preconditioning is used for sim-
ulating turbomachinery flow at low Mach numbers. None of the aforemen-
tioned papers discuss a proper treatment of boundary conditions in conjunc-
tion with turbomachinery applications. Moreover, only few authors have pre-
sented preconditioning schemes that can be used for the simulation of fluids
with arbitrary equations of state in the context of turbomachinery simula-
tions [24, 88, 108, 301].
In the last two decades much work in the CFD community has been at-
tributed to the development of improved discretization schemes [44, 128, 168,
169, 267, 268, 270]. However, first in recent years the attention has been
drawn to the development of classical dissipation [36, 196, 216, 279] and low
diffusion schemes in conjunction with preconditioning [79, 141, 171, 199, 236].
In the last years there has also been an increasing interest in the develop-
ment of discretization schemes that can be used for completely stagnating to
supersonic flows and yield a dissipation that scales correctly in the limit of a
vanishing Mach number [163, 167, 225, 237].
Whether the fluid flow is laminar, transitional, or turbulent, it is inher-
ently unsteady in multi-stage turbomachinery. To resolve the interaction be-
tween the potential fields of blade rows moving relative to each other and the
flow field in detail, time-accurate simulations have to be applied.
Even though computational resources have increased significantly over the
last decades, unsteady flow simulations, even in the RANS context, are still
very time consuming. Unsteady simulations, and the subsequent handling of
large data sets and the associated post-processing require significantly more
effort from the user than when a steady simulation is carried out. This is
especially true for the simulation of multi-stage turbomachinery applications
where the modelling of many subsequent blade rows is necessary. For these
reasons steady-state stage calculations remain common practice in the indus-
trial design process of turbomachinery.
When steady-state stage computations are to be performed, so-called mix-
ing planes are utilized to couple different frames of reference and to transfer
circumferentially averaged data from the computational domain of adjacent
blade rows.
Since blade rows in turbomachinery can be closely spaced it is often nec-
essary to impose inlet and outlet boundaries as well as mixing planes close
to the blading in a region where the flow is far from being uniform. In order
to ensure that physical waves can traverse the numerical boundaries without
interacting with the flow field in the interior of the domain, so-called non-
1.2 Literature survey 9
reflecting boundary conditions (NRBC) have been developed.
The idea of NRBC is to ensure that every wave can leave the computational
domain without causing reflections or spurious oscillations at the boundaries.
This means that the boundary condition is transmissive for outgoing waves
but suppresses all spatially varying distributions at the boundary that would
correspond to modes of incoming waves. The basic theoretical concepts for
constructing NRBC were developed in the 1970s [82, 83, 115, 145]. Since
then a large number of publications have been issued on the topic of NRBC in
many branches of engineering and applied mathematics [100]. Unifying and
extending some of these theories, Giles [98, 99] formulated a method for the
application of NRBC to the linearized 2D Euler equations.
In the literature many authors report the use of a boundary condition
treatment based solely on Riemann invariants (e.g., [126]) or equivalently [66]
a 1D characteristic boundary condition treatment based on the Euler equa-
tions (e.g., [211]). The principle of using characteristic theory is to separate
the solution updates at the boundary into contributions from incoming and
outgoing waves. This procedure is necessary to ensure that corrections of
the boundary values, which are undertaken to meet the specified conditions,
are compatible [117, 118] and that the corrections are propagated correctly
into the interior of the computational domain. Lele and Poinsot [211] have
extended the characteristic boundary condition treatment to also account for
viscous effects. Their method is often referred to as Navier-Stokes Character-
istic Boundary Condition (NSCBC) treatment in the literature.
Whether or not it accounts for viscous effects, a disadvantage associated
with the use of a one-dimensional characteristic theory for the treatment of
boundary conditions is that these methods rely on the assumption that the
flow at the boundaries can be regarded as locally one-dimensional. The single-
dimensionality assumption has proven to perform well when the flow is al-
most aligned with the boundary; however distortions of the flow fields and
reflections emerge when the flow does not cross the boundary perpendicularly
or the flow is non-uniform in the transverse directions [247].
A further problem associated with the boundary condition treatment based
solely on one-dimensional characteristic theory is related to the specification
of spatially uniform boundary conditions. Giles and Saxer [242, 243] show
that even if the characteristic variables at the boundary that are correspond-
ing to outgoing waves are extrapolated from the interior, reflections will not
be avoided if spatially constant distributions of flow variables are prescribed.
As shown by Seybold et al. [247, 248], the use of 1D characteristic treat-
ment does not lead to reflections if a distribution of the flow variables is im-
posed, which corresponds to the solution one would obtain at the position of
the current boundaries, if the computational domain would be extended up-
and downstream, respectively. As noted by Denton [72], a physically sound
boundary condition treatment for turbomachinery applications should impose
desired values for averaged quantities but should allow a distribution of the
flow variables in circumferential direction. Unfortunately, the use of a bound-
ary condition treatment based on one-dimensional characteristic theory does
10 1. Introduction
not allow an automatic prediction of a physical flow field distribution about a
specified averaged value.
Yoo et al. [314, 315] achieved an improvement of the NSCBC type of bound-
ary condition treatment by also accounting for transverse terms. This kind
of boundary condition was further enhanced by Lodato et al. [170], who de-
rived compatibility conditions at the edges and corners of the computational
domain, where walls meet inlet or outlet boundaries. The inclusion of trans-
verse terms allows for variations of the solution variables at the boundary and
the resulting numerical treatment leads to non-reflecting boundaries. This
method has been successfully applied in the context of Large Eddy Simula-
tions of reacting flows but has not been adapted and tested for simulations of
turbomachinery flow.
For steady-state simulations of turbomachinery flows Saxer and Giles [242,
243] devised a quasi-three-dimensional non-reflecting boundary condition treat-
ment. In their approach, the solution at the boundary is decomposed into
Fourier modes. The zeroth mode, which corresponds to the average solution,
is treated according to standard one-dimensional characteristic theory and
enables the user to specify the circumferential averages of certain flow field
parameters at the boundary. The remaining harmonics of the solution at the
boundary are altered to satisfy a condition derived by Giles basing on the lin-
earized, two-dimensional Euler equations. This ensures that the variations
corresponding to incoming (or reflected) waves are eliminated. In the quasi-
three-dimensional approach of Saxer, the grid layers in the radial direction
are treated independently, so that the Fourier decomposition of the state vari-
ables at the boundary is limited to the circumferential direction.
The quasi-3D (Q3D) steady-state NRBCs due to Saxer and Giles allows
variations of the flow variables around an average value. In general they
lead to a more physical prediction of the flow fields near the boundaries than
when boundary conditions based merely on 1D characteristic theory are used
[63, 247, 248].
In the derivation of the Q3D-NRBC it is assumed that the radial variations
of the flow field are small compared to those in the pitchwise direction, so that
this treatment may not always lead to a fully non-reflecting behavior of the
boundaries. In addition, the Q3D-NRBC treatment requires the specification
of the correct radial distribution of the boundary values. However, boundary
values are often only known as average values for the whole boundary surface.
Motivated by these limitations, the theory of Giles was extended to three di-
mensions by Anker and Seybold [21] and applied by Schrader [246]. In the
publication of Anker et al. [20] it is shown that the developed steady-state,
fully three-dimensional NRBC treatment outperforms the Q3D-NRBC in flow
configurations where the flow is three-dimensional and periodicity exists in
both lateral directions. The developed treatment applies a double Fourier ex-
pansion and when periodicity does not exist in both spanwise and circumfer-
ential direction, errors linked to Fourier decomposition are introduced, which
significantly degrade the performance of this kind of boundary condition treat-
ment.
1.2 Literature survey 11
It should be noted that Fatsis and van den Braembussche [86] also devel-
oped a three-dimensional non-reflecting boundary treatment for steady state
flow simulations. However, their boundary condition treatment is approxi-
mate, whereas the theory developed by Anker and Seybold is exact within the
linearized Euler equations. Imanari and Kodoma [125] have introduced a non-
reflecting treatment for three-dimensional Euler equations by expanding the
pressure field normal to the boundary into Fourier-Bessel series. They obtain
a non-reflecting property by suppressing the perturbations that correspond
to pressure waves leaving the domain. For the simulations of strut cascades
they achieve good results, but their treatment is not general as they neglect
the influence that vorticity and entropy waves have on the pressure pertur-
bations. Fan and Lakshminarayana [85] solve partial differential equations
in their three-dimensional boundary treatment for unsteady flows; however,
the underlying theory contains several approximations and from their work
it is not clear, whether or not their treatment leads to a fully non-reflecting
boundary condition treatment.
Given the limitations of the 3D-NRBC treatment mentioned above, the
Q3D-NRBCs of Saxer and Giles [242, 243] represent state-of-the-art for a
boundary condition treatment for the simulation of turbomachinery applica-
tions. This boundary condition treatment has attained popularity and has
been implemented in both research and commercial flow solvers [88, 123, 190,
210].
The treatment of one-dimensional boundary conditions in the context of
time-derivative preconditioning schemes has been subject to several studies.
Using a reflection analysis, Darmofal et al. [66, 67] have shown that a Rie-
mann/Characteristic boundary condition treatment based on the unprecon-
ditioned Euler equations are reflective when used with preconditioning. It
is evident that the changed dynamics of the flow equations due to the use
of time-derivative preconditioning must be accounted for when a characteris-
tic boundary condition treatment is applied. How to concretely accommodate
for the time-derivative preconditioning method in a boundary condition treat-
ment based on Riemann invariants, or equivalently, on characteristic theory,
has been addressed in various publications (e.g., [108, 277]), but to the knowl-
edge of the author there has not been published any work on multidimen-
sional non-reflecting boundary conditions for preconditioned systems and/or
flows with general equations of state.
1.2.3 Modeling of transitional and turbulent flows in tur-
bomachinery
The fluid flow encountered in turbomachinery devices is usually turbulent.
Turbulent flows occur when the Reynolds number is larger than the critical
one and a regular, laminar flow pattern collapses due to instability of the
flow. In turbulent flow vortices are formed, which in a cascade process break
up into successively smaller vortices until they are dissipated by viscous ef-
fects [116, 271]. The vortical motion in turbulent flows has a significant im-
12 1. Introduction
pact on the averaged flow field, on the loss generation, and on separation
and attachment of the flow over obstacles. Since the Navier-Stokes equations
describe general flows, they can be used to simulate turbulent flow directly.
Except for moderate low Reynolds numbers and simple geometries, a Direct
Numerical Simulation (DNS) is computationally very expensive and practi-
cally not feasible due to the presence of vastly different length scales of the
vortices [4, 212, 228]. An alternative approach, which demands less compu-
tational resources, is only to resolve the largest eddies and model the small-
est ones. The so-called Large Eddy Simulation (LES) including the Detached
Eddy Simulation (DES) techniques are increasing in popularity and have a
strong predictive capability [40, 107, 119, 178]. These methods are used by re-
search institutions for phenomenological studies [316] over aeronautical com-
ponents, for the simulation of elementary flames, or for the computation of
the reactive flow in combustors [181, 273].
Unfortunately, both a DES and a LES are computationally expensive since
they require an unsteady simulation on meshes fine enough to resolve all the
eddies in the main flow down to the inertial range, which for high Reynolds
numbers can represent a formidable task. While a DES/LES is practically
feasible for the aerodynamical study of isolated configurations, like the flow
over wings or single turbomachinery blades, it is still far too expensive for
being used for the simulation of complete turbomachinery stages.
To account for the influence of turbulence without resolving any of the tur-
bulent eddies, statistical turbulence models have been developed. In these
models the flow is simulated by solving the Reynolds-Averaged Navier-Stokes
(RANS) equations in conjunction with a model for the Reynolds stresses [227],
which describe the average influence of the turbulent eddies on the main
flow. There are two different principal approaches for modeling the Reynolds
stresses appearing in the RANS equations. In the most advanced Differen-
tial Reynolds Stress Modeling (DRSM) approaches [110–112, 135, 155, 259]
a transport equation for each of the six different elements of the Reynolds
stress tensor is solved as well as a transport equation for the dissipation rate
of the turbulent kinetic energy [4, 53, 90, 212]. A computationally less ex-
pensive and a numerically more robust approach consists in modelling the
Reynolds stresses as a function of the velocity gradients of the mean flow
field. In the so-called eddy viscosity models the Boussinesq hypothesis [41]
is applied, which assumes an analogy between the diffusive transport of mo-
mentum caused by the molecular and by the turbulent motion. In these mod-
els the influence of the turbulence on the main flow is effectively modelled
by adding an eddy viscosity term to the molecular viscosity coefficient. In
the simplest models the eddy viscosity is modelled via algebraic relations in
dependence of the normalized wall distance, mean flow quantities as well as
gradients of velocity and pressure. The oldest eddy-viscosity model is the mix-
ing length model of Prandtl [214], in which the eddy viscosity is assumed to be
proportional to the square of the mixing length and the norm of the velocity
gradient. As detailed in Wilcox [306], modern algebraic turbulence models,
e.g., the Cebeci-Smith [50, 255] and the Baldwin-Lomax [27] models repre-
1.2 Literature survey 13
sent extensions of this model by accounting for wall-damping [281], turbulent
intermittency [64, 142] and Clausers [62] formula for the eddy viscosity in
the defect layer. Algebraic turbulence models are robust and are for this rea-
son still used [76]. These models are suited for attached flows with positive
pressure gradients [306] or slightly negative pressure gradients as long as
the flow does not separate [52]; for separated flows and/or flows with adverse
pressure gradients only the Johnson-King [133] turbulence model is recom-
mended [52, 183], which solves an ordinary differential equation for the max-
imum value of the shear stress. For complex, 3D flow configurations or flows
with adverse pressure gradients, the use of so-called one- or two-equation tur-
bulence models is preferable [52] in which respectively one- or two partial
differential equations (PDE) are solved in addition to the RANS equations to
model the effect of turbulence. Frequently applied one-equation models are
the Baldwin-Barth [25, 26] and the Spalart-Allmaras turbulence model [258]
as well as variants of the latter [240, 252, 257], which include effects due
to rotation. Among the two-equation turbulence models, the k–ε type (Stan-
dard [156, 157], Realizable [250], RNG [313]) and k–ω type (Modified Wilcox
model [308]) are commonly applied in science and engineering. Contrary to
the k–ε turbulence models, the k–ω models do not need wall damping func-
tions to correctly resolve turbulent boundary layers. However, k–ω models are
very sensitive to the free stream values of ω [212]. In order to combine the
advantages of both models, Menter [184] developed a model, which employs
the original k–ω model of Wilcox [305] in the boundary layer and switches to
the standard k–ε model outside. Inspired by the Johnston-King model [133],
Menter included in his Shear-Stress Transport (SST) model a modification of
the definition of the eddy-viscosity, which accounts for the effect of the trans-
port of the turbulent shear stress. In a comparative study on standard aero-
dynamic test cases, Bardina et al. [28] conclude that Menter’s SST as well
as the Spalart-Allmaras model outperform both the Standard k–ε model of
Launder and Sharma [156, 157] as well as Wilcox’s original k–ω model. The
Spalart-Allmaras and Menter’s SST models are popular eddy-viscosity models
and can now be regarded as a standard choice for industrial use [49].
In the context of the simulation and analysis of the leakage flow effects
in shrouded axial turbines, Rosic et al. [232, 233] have been successful even
though only a mixing-length approach for modeling the turbulence was used.
Seybold [249] simulated the flow through the labyrinth seal in an axial tur-
bine using various standard eddy-viscosity models ranging from the algebraic
Baldwin-Lomax model to the k–ω SST model. Like in the work of Sög˘üt [256],
which concerned the simulation of a complete shrouded axial turbine, no con-
clusion could be made in favour of any of the turbulence models used.
In the FLOMANIA project the performance of various one-equation, lin-
ear and non-linear two-equation models, DRSM, DES and LES models have
been assessed by a large consortium on a range of different test cases includ-
ing turbomachinery configurations [106]. In that collaborative assessment of
turbulence models, it was found that the Spalart-Allmaras and the k-ω SST
models perform well for predominantly attached, high Reynolds number flow.
14 1. Introduction
For simpler, two-dimensional configurations non-linear two-equation models
as well as DRSM models may lead to a better reproduction of the flow physics
than the aforementioned models, even though not universally [49]. In the
FLOMANIA project, it was furthermore concluded that all RANS based mod-
els, including DRSM, fail for complex, three-dimensional separating-reattach-
ing flow. For such flows, acceptable results were only obtained when LES
or hybrid LES-RANS approaches are used. These findings show that unless
LES-based models are used, there is still considerable uncertainty associated
with turbulence modeling for general flow configurations.
In order to accurately predict the losses in turbomachinery, not only the
prediction of the effect of turbulence itself but also the accurate determina-
tion of the on-set and the extent of the laminar-turbulent transition is impor-
tant [154, 266]. The accurate prediction of the boundary layer development
and its interaction with the main flow is especially important for low-pressure
turbines, where the flow often is transitional [152]. Despite the importance
of transition modeling, compared to the development of statistical turbulence
models for fully turbulent flow, the efforts that have been invested on de-
veloping models that reliably determine the laminar-turbulent transition of
boundary layers are relatively low.
When turbomachinery flow is simulated using the RANS equations, a com-
mon method to account for the laminar-turbulent transition is to turn on the
turbulence model at a predefined, given chord length (e.g., [136, 238]). A more
advanced approach in the context of RANS simulations [213], consists in us-
ing empirical correlations [1, 180] for the determination of the transition onset
location. None of these methods are predictive and the coupling of empirical
correlations with RANSmodels can be challenging as some of these models re-
quire the determination of integral quantities like for instance the boundary
layer thickness [154]. Some of these modeling approaches assume instanta-
neous transition at the onset location and thereby neglect the importance of
the transitional region, which can lead to approximate results.
Transition can be predicted directly by applying low Reynolds models [307];
this method is however not regarded as a reliable means to predict the tran-
sition point or the extent of the transitional region [154, 239, 266, 295, 303].
The more general modeling approach is to solve for additional transport equa-
tions for states that are essential in the description of transitional flows (e.g.,
intermittency, laminar kinetic energy, or transition Reynolds number). Phe-
nomenological models [57, 80, 296, 297] as well as correlation based mod-
els [260, 261, 266] have been developed. Transition modeling is a topic of
active research [39] and current efforts are focused on developing models that
do not need integral quantities and solely rely on local flow quantities [153,
185, 295–297].
In brief, whereas LES based models are quite promising for the accurate
prediction of the fluid flow of aerodynamic configurations but are too expen-
sive in terms of computational resources to be applied in practical turbo-
machinery simulations, the application of the computationally more efficient
RANS models are connected with modeling uncertainties both for transitional
1.3 Objectives of the present thesis 15
and fully turbulent flows.
While the improvement of RANS based turbulence models is important,
a more pertinent issue for the simulation of low speed flows is to prevent
that a wrong scaling of the dissipation terms or a numerically inconsistent
boundary condition treatment are degrading the accuracy and the reliability
of the solution scheme [6, 312]. Rather than to concentrate on devising and
assessing RANS based turbulence models, which is a topic on its own, the
development of an accurate, efficient, robust, and Mach number independent
solution scheme is the focal point in the numerical part of this thesis. For
this reason, issues related to modeling of turbulence will not be given much
attention in later chapters of this thesis. The modeling method used in this
work will be based on the algebraic approach developed by Merz [190] and
Jung [136], which for the simulation of the attached flow through axial tur-
bines has proven to be sufficiently robust and accurate for engineering pur-
pose [149, 221] and can be used together with the time-inclination method of
Giles [294] without the need for modifications.
1.3 Objectives of the present thesis
The aim of the present thesis is to contribute to the simulation and the anal-
ysis of turbomachinery flows.
On the numerical side, the goal of the present work is to develop a numer-
ical scheme for the simulation of flows through multi-stage turbomachinery
at arbitrary Mach number levels. It is required, that the solution scheme
be robust, accurate and efficient and it be possible to simulate fluids obeying
arbitrary thermal and caloric state equations.
The developments are to be based on an already existing density-based
solution scheme for the simulation of high-speed flow developed byMerz [190],
Krückels [147], Raif [221], Jung [136] and Bauer [30, 31]. The solution scheme
should be low diffusive and be suited for the simulation of transonic flows
containing strong shocks as well as large lowMach number regions so that the
flow in arbitrary turbomachinery applications can be simulated accurately.
For the simulation of multi-stage turbomachinery the blade-rows can be
situated close to each other and in the simulation of such configurations the
inlet, outlet and the mixing planes may have to be positioned close to the
blade rows, where the flow is inhomogeneous. To avoid that the numerical
boundary treatment impairs the solution, a non-reflecting boundary condition
needs to be developed, that is both suitable for the simulation of lowMach and
transonic flows.
The second objective of the present thesis, is to use the developed solution
scheme to simulate and analyse the flow in a 1 1
2
-stage, shrouded low-speed
axial turbine situated at the University of Bochum [42]. The flow in this tur-
bine is to be studied for a realistic clearance height of the labyrinth seal of
the rotor row and the effect of the leakage flow on the main flow is to be in-
vestigated. Since the minimization of leakage flow is constructively limited
16 1. Introduction
and the presence of leakage flow is thus inevitable, it is for the future design
of turbomachinery components important to understand and identify the an-
ergy producing processes involved. The interaction between the cavity, leak-
age, secondary and main flow is a recent topic of interest and it is therefore
important that a systematic study, which contributes to the comprehension of
the physical phenomena and the loss generation taking place, be carried out.
1.4 Outline of the present thesis
The numerical methods developed in the current work are implemented in the
ITSM3D flow solver. For this reason, the baseline solution scheme of that flow
solver is described in Chapter 2. The chapter summarizes the discretization
and solution scheme, which has been developed and applied successfully by
Merz, Krückels and others for steady-state simulations of the flow through
transonic turbomachinery devices [148, 191].
In order to be able to simulate flows at arbitrary Mach numbers with
the density-based solution scheme used in ITSM3D, a time-derivative pre-
conditioning technique is introduced. Chapter 3 describes the precondition-
ing method applied, its theoretical basis and the modifications introduced to
make the method robust for practical simulations. Results from several ver-
ification and validation test cases confirm that the preconditioned solution
scheme leads to Mach number independent convergence rates. The presented
test cases show furthermore, that accurate solutions can be obtained for low
Mach number, transonic, and supersonic flow conditions.
In order to obtain a solution scheme which is low diffusive, monotonicity
preserving, and simultaneously leads to a sharp resolution of shocks, sev-
eral upwind-biased artificial dissipation schemes were developed in the cur-
rent work. These schemes are presented and assessed in Chapter 4. Since
they were developed in the context of Merkle’s time-derivative precondition-
ing scheme, they have been formulated for simulation of flows with a general
equation of state. Results from several test cases show that the developed dis-
cretization schemes lead to less diffusive solutions than the baseline scheme
and yield accurate and monotone solutions for both low and high Mach num-
ber flows.
In the current work, the quasi-three-dimensional non-reflecting boundary
condition treatment of Giles and Saxer [242, 243] was extended to precon-
ditioned systems and general equations of state. This extension was neces-
sary since the use of time-derivative preconditioning alters the characteristic
speeds of the governing equations. In addition, numerical tests showed that
the use of the original boundary condition treatment of Giles and Saxer was
neither robust nor did it lead to non-reflecting solutions when used in conjunc-
tion with the preconditioned solution scheme. The developed theory as well as
the implementation details are described in Chapter 5. The presented results
verify that the solutions obtained with the new boundary condition treatment
are free of reflections for incompressible flows as well as for ideal gas flow at
1.4 Outline of the present thesis 17
arbitrary Mach numbers.
In Chapter 6 the low-diffusive, preconditioned solution scheme together
with the developed non-reflecting boundary condition treatment is used to
simulate the flow in 1 1
2
-stage axial turbine with shroud on the rotor row. By
comparing the flow field for a realistic, a small and a vanishing clearance
height, the influence of the leakage flow on the main and secondary flow is
systematically investigated. The physical effects attributed to the interaction
of leakage and main flow are discussed. In particular, effects introduced by
the interaction of the cavity flow and the main flow in the rotor are examined
and different loss generating mechanisms are discussed.
18
Chapter 2
A baseline Finite-Volume
solution scheme for the
Navier-Stokes equations
In the current work the flow solver ITSM3D was used for developing new
schemes for the simulation of turbomachinery flows as well as for investigat-
ing the interaction between leakage and main flow in axial turbomachinery.
As the name suggests the flow solver ITSM3D was developed at the Institute
for Thermal Turbomachinery andMachinery Laboratory (ITSM) of University
of Stuttgart. The baseline scheme, which is outlined in this chapter, was de-
veloped by Bauer [29], Jung [136], Krückels [146], Raif [221], and Merz [190],
among others. The flow solver ITSM3D is used in the current work since it is
computationally efficient and offers the use of non-reflecting inlet, outlet and
mixing planes. With the time-inclination method of Giles, the flow solver can
be used to simulate the rotor-stator interaction in axial turbomachinery for
arbitrary pitch ratios [137].
2.1 Governing equations
For turbomachinery applications it is convenient to solve the Navier-Stokes
equations in a cylindrical coordinate system that is aligned with the the rota-
tional axis and rotates with the blade row. This choice of a coordinate system
is the basis of the flow equations solved in ITSM3D as it naturally accounts
for both the axis-symmetry as well as the angular motion of the turbomachin-
ery geometry in an elegant fashion. The Navier-Stokes equations solved in
ITSM3D are formulated for a x − φ − r coordinate-system, that is rotating
with an angular speed Ω in a mathematical negative sense around the x-axis.
These equations read as
∂Q
∂t
+
∂E
∂x
+
1
r
∂F
∂φ
+
1
r
∂(rG)
∂r
+H = 0 (2.1)
with
E = Ec − Ev, F = Fc − Fv, G = Gc −Gv, H = Hc −Hv,
19
20 2. A baseline solution scheme
and
Q =

%
%wx
%wφ
%wr
%e0
 , Ec =

%wx
%w2x + p
%wφwx
%wrwx
%h0wx
 , Ev =

0
τxx
τφx
τrx
τxiwi + kλ
∂T
∂x
 ,
Fc =

%wφ
%wxwφ
%w2φ + p
%wrwφ
%h0wφ
 , Fv =

0
τxφ
τφφ
τrφ
τφiwi + kλ
1
r
∂T
∂φ
 , Gc =

%wr
%wxwr
%wφwr
%w2r + p
%h0wr
 ,
Gv =

0
τxr
τφr
τrr
τriwi + kλ
∂T
∂r
 , Hc = 1r

0
0
%wr(wφ + 2Ωr)
−%(wφ + Ωr)2 − p
0
 , Hv = 1r

0
0
τrφ
−τφφ
0
 .
In the baseline scheme in ITSM3D it is assumed that the fluid behaves as a
caloric and thermal ideal gas, whereby the following state equations
p = %RT, %e0 =
p
γ − 1 +
1
2
%
[
w2x + w
2
φ + w
2
r − (Ωr)2
]
(2.2)
are used, where R represents the individual gas constant and γ the ratio of
specific heats. The shear stresses are calculated assuming that the fluid be-
haves like a Newtonian fluid. The laminar viscosity µl is determined using
Sutherland’s law and turbulence is modelled applying the Baldwin-Lomax
turbulence model, which is an algebraic model for the turbulent viscosity µt. It
is assumed that the laminar and the turbulent Prandtl numbers are constant,
i.e.,
Prl = 0.72, Prt = 0.9 (2.3)
respectively, so that the thermal conductivity kλ can be calculated via the
following relation:
kλ = cp
(
µl
Prl
+
µt
Prt
)
. (2.4)
2.2 Discretization scheme and solution proce-
dure
The discretization- and solution scheme, that is applied in the flow solver
ITSM3D is based on work of Jameson et al. [132]. In this scheme the spatial
and temporal discretization are treated separately according to the method
of lines. Correspondingly, the spatial and temporal discretization will in the
following be described separately.
In the flow solver ITSM3D the Finite-Volume method is used for the spa-
tial discretization of the flow equations. In this method, the physical space
2.2 Discretization scheme and solution procedure 21
is subdivided in finite volumes for which discrete balance equations are for-
mulated for the conservation of mass, momentum and energy. These discrete
equations approximate the integral average of the temporal change of the in-
dependent solution variables in each control volume. The use of an integral
form ensures a conservation of the fluxes and admits weak solutions of the
conservation equations; the latter property is important for capturing shocks
in the simulation of transonic flows.
2.2.1 Spatial discretization
The conservative three-dimensional Navier-Stokes equations in cylindrical co-
ordinates are discretized spatially using a cell-vertex Finite-Volume scheme.
The computational domain is subdivided in curved hexahedral elements and
the independent flow quantities are stored at the vertices of the elements.
The volumes for discretization of the convective fluxes is formed by the eight
hexahedral elements that share a node. With the assumption that the point
of gravity of the resulting control volume is coinciding with the common cor-
ner point (i, j, k) and that the flow quantities are linearly distributed over the
control volume Vˇi,j,k, the integral of the unsteady terms of the conservation
equations can be approximated to∫
Vˇi,j,k
∂Qi,j,k
∂t
dV ≈ Vˇi,j,k · ∂Qi,j,k
∂t
(2.5)
with a second order accuracy. If the aforementioned assumptions are not met,
the latter discretization is at worst of first order accuracy.
The surface integral of the convective fluxes and the volume integral of
the convective sources are approximated through a discrete sum of fluxes
and sources. The vector of the respective sums for each transport equation
is called the convective residual vector Rc. The corresponding sum result-
ing from the discretization of diffusive terms is called the diffusive residual
vector Rd. With the introduction of these residual vectors, the following semi-
discrete form of the conservation equation is obtained
∂Qi,j,k
∂t
= − 1
Vˇi,j,k
(Rc +Rd). (2.6)
In ITSM3D the convective residual vector Rc is calculated according to the
method of Hall [109]. With this scheme the convective residual vector is cal-
culated as
Rc,i,j,k =
24∑
m=1
(
E¯c,mSx,m + F¯c,mSφ,m + G¯c,mSr,m
)
i,j,k
+
8∑
l=1
(
H¯c,lVl
)
i,j,k
=
24∑
m=1
(rF,c,x,m + rF,c,φ,m + rF,c,r,m)i,j,k +
8∑
l=1
(rS,c,l)i,j,k
(2.7)
for each control volume Vˇi,j,k. The surface of the control volume Vˇi,j,k is formed
by the 24 outer surfaces Sm of the eight cells Vl that surrounds the central
node (i, j, k), cf. Fig. 2.1.
22 2. A baseline solution scheme
(i, j, k)
x
r
φ
Figure 2.1: Control volume Vi,j,k for the determination of the convective fluxes and
sources
Through the assumption of a linear distribution of the state variables, the
flux contribution of a face Sm is determined via an arithmetic averaging of
the flux vectors at its four corners; for instance the axial component of the
convective flux contribution rF,C,x,m of the surface Sm calculates as
rF,c,x,m = E¯c,mSx,m =
1
4
(Ec,m,1 + Ec,m,2 + Ec,m,3 + Ec,m,4)Sx,m, ∀m = 1, . . . , 24.
(2.8)
Similarly, the specific contribution rS,c,l of the convective source termsHc,l of a
hexahedral element Vl to the convective residual is calculated by an arithmetic
averaging of the source terms of the eight corners i = 1, . . . , 8:
rS,c,l =
1
8
Vl
8∑
i=1
Hc,l,i. (2.9)
A cell-vertex scheme has the advantage over a cell-centered scheme that the
fluxes can be determined directly without the need for interpolation. Fur-
thermore, for a cell-vertex scheme the boundary conditions can be applied
without extrapolation of the states. According to Rossow [235], cell-vertex
schemes have the advantage that they possess a second order accuracy on
smooth meshes and a first order accuracy on distorted meshes. On smooth
meshes cell-centered schemes also possess a second order accuracy, but they
exhibit a zero order accuracy on distorted meshes.
In order to determine the residuals RD of the diffusive contributions, it
is necessary to calculate the gradients of the velocities and the temperature.
In ITSM3D the diffusive residuals are calculated according to the approach
2.2 Discretization scheme and solution procedure 23
of Radespiel and Rossow [217], which exhibits a second order accuracy on
smooth grids and a first order accuracy on distorted meshes. This accuracy
is achieved by calculating the gradients on the dual mesh by simultaneously
accounting for the skewedness and curvilinearity of the grid. A detailed de-
scription of the scheme for the computation of the diffusive contributions can
be found in the work of Radespiel and Rossow [217] as well as in the thesis of
Merz [190].
2.2.2 The necessity of adding artificial dissipation in a
central scheme
The finite-volume scheme described above applies a central discretization.
Since numerical dissipation is not inherent to this type of discretization, an
odd-even decoupling of the states of successive nodes may occur if no artificial
dissipation is introduced in the scheme. In the baseline scheme in ITSM3D
artificial dissipation D is added to the convective and diffusive residuals in
the following manner:
∂Qi,j,k
∂t
= − 1
Vˇi,j,k
(Rc +Rd +D). (2.10)
The artificial dissipation terms are calculated in such a way, that they damp
high frequency oscillations but leave the waves that extend over many grid
nodes undamped. The damping of the high frequency oscillations in the so-
lution is important due to the non-linear terms in the momentum equations.
Through these terms energy can be transferred from long to short waves. If
no mechanism is dissipating the energy of the low frequency waves, these
may reach a sufficient amplitude that the scheme becomes unstable even if
it were stable for the linearized flow equations. The laminar and the turbu-
lent viscosity in the flow simulations damp the high frequency oscillations.
However, except in regions of low cell Reynolds numbers, this damping is only
effective for frequencies that are higher than those which can be resolved on
a practical mesh for RANS computations, wherefore artificial dissipation is
indispensable not only for inviscid but also for viscous flow computations.
2.2.3 The artificial dissipation scheme of Jameson,
Schmidt and Turkel
The artificial dissipation applied in ITSM3D is based on the work of Jameson,
Schmidt, and Turkel (JST) [132] and is constructed of a combination of second
and fourth order differences of the solution vector. Applying a grid conformal
ξ−η−ζ coordinate system, the artificial dissipationDi,j,k related to the control
volume Vˇi,j,k reads as
Di,j,k =
(
−D(2)ξ +D(4)ξ −D(2)η +D(4)η −D(2)ζ +D(4)ζ
)
Qi,j,k. (2.11)
24 2. A baseline solution scheme
The second and fourth order dissipation operators are defined as
D
(2)
ξ = δ
−
ξ
(
ρ¯Vξ ε
(2)
)
i+1/2,j,k
δ+ξ (2.12)
and
D
(4)
ξ = δ
−
ξ
(
ρ¯Vξ ε
(4)
)
i+1/2,j,k
δ+ξ δ
−
ξ δ
+
ξ , (2.13)
respectively, with the forward and backward difference operators δ+ξ and δ
−
ξ
for the ξ-direction given by
δ+ξ ψi,j,k := ψi+1,j,k − ψi,j,k, δ−ξ ψi,j,k := ψi,j,k − ψi−1,j,k. (2.14)
The scaling factor ρ¯Vξ can be determined isotropically by a simple addition of
the volume weighted spectral radii for all grid directions
ρ¯Vξ,i+1/2,j,k = ρ
V
ξ,i+1/2,j,k + ρ
V
η,i+1/2,j,k + ρ
V
ζ,i+1/2,j,k. (2.15)
The scalar formulation of the artificial dissipation in Eq. (2.11) with the
isotropic scaling factors in Eq. (2.15) was originally developed for the solution
of the Euler equations with meshes having an aspect ratio close to unity. For
the solution of the RANS equations for typical turbomachinery applications
meshes with large aspect ratios are used. To avoid that the stretching of the
grids does not lead to excess dissipation, the anisotropic scaling factors of
Radespiel et al. [218] are used in ITSM3D. For instance for the ξ direction,
the anisotropic scaling factor is calculated as
ρ¯Vξ,i+1/2,j,k =
(
Θξ · ρVξ
)
i+1/2,j,k
, (2.16)
where the weighting factor Θξ is defined as
Θξ = 1 +max
(
ρVη
ρVξ
,
ρVζ
ρVξ
)α
with α =
1
2
. (2.17)
The coefficients ε(2) and ε(4), which appear in the equations (2.12) and
(2.13), are dependent on the pressure gradient and control the amount of the
second and fourth order dissipation. These coefficients are determined by
ε
(2)
i+1/2,j,k = k
(2)max(νξ,i−1,j,k, νξ,i,j,k, νξ,i+1,j,k, νξ,i+2,j,k) (2.18)
and
ε(4) = max(0, k(4) − ε(2)i+1/2,j,k) (2.19)
with the pressure sensor
νξ,i,j,k =
∣∣∣∣pi−1,j,k − 2pi,j,k + pi+1,j,kpi−1,j,k + 2pi,j,k + pi+1,j,k
∣∣∣∣ . (2.20)
The fourth order dissipation ensures stability and convergence of the solution
in regions where the gradients are small. Since the fourth order dissipation
2.2 Discretization scheme and solution procedure 25
leads to overshoots in the solution near shocks, this part of the dissipation is
shut off gradually with increasing activation of the second order dissipation.
To avoid that the accuracy of the central discretization scheme is not glob-
ally degraded, the second order dissipation is only activated locally in regions
where the pressure sensor ν detects large variations in the pressure.
The constants k(2) and k(4) are defined by the user; typical values are 1/2
and 1/64, respectively. The dissipation operators for the η- and ζ-direction are
defined in analogous fashion as the operators defined in the equations (2.12)
and (2.13) for the ξ-direction.
2.2.4 Time-stepping scheme
The semi-discrete Navier-Stokes equations (2.10) represent a system of first
order ordinary differential equations with respect to time. In ITSM3D a five-
stage, explicit Runge-Kutta scheme is used for the time-integration of this
system of differential equations. This scheme has the property of being effec-
tive in damping high frequency errors and is according to Mavriplis [179] very
efficient in combination with a multigrid technique.
In the Runge-Kutta scheme that is used in ITSM3D the dissipative and the
diffusive terms are not calculated on each stage, which significantly reduces
the computational overhead. Furthermore, this so-called hybrid Runge-Kutta
scheme has the advantage over the classical one, that for the same maximal
Courant-Friedrichs-Lewy number CFLmax it possesses a higher value for the
maximal stable von Neumann number VNNmax, which adds stability to a solu-
tion scheme for viscous flow. For a detailed discussion of the performance and
the computational costs of various hybrid Runge-Kutta schemes, the reader
may refer to the work of Merz [190].
The hybrid Runge-Kutta that is used in ITSM3D was devised by Radespiel
and Swanson [219] and reads as
Q
(0)
i,j,k = Q
(n)
i,j,k,
Q
(s)
i,j,k = Q
(0)
i,j,k − αs
∆t
Vˇi,j,k
(
R(s−1)c +R
(0)
d +
s∑
t=1
γstD
(s−1)
)
i,j,k
, s = 1, 2, . . . , 5
Q
(n+1)
i,j,k = Q
(5)
i,j,k
(2.21)
with
R(s−1)c = Rc(Q
(s−1)), R(0)d = Rd(Q
(0)), D(s−1) = D(Q(s−1)) (2.22)
and the coefficients
γ11 = γ21 = 1, γ22 = 0, γ31 = γ41 =
11
25
, γ32 = γ42 = 0, γ33 = γ43 =
14
25
,
γ44 = 0, γ51 =
154
625
, γ52 = 0, γ53 =
196
625
, γ54 = 0, γ55 =
11
25
,
α1 =
1
4
, α2 =
1
6
, α3 =
3
8
, α4 =
1
2
, α5 = 1.
(2.23)
26 2. A baseline solution scheme
The state vector in the grid point (i, j, k) at the beginning and the end of a
time step n are thereby denoted by Q(n)i,j,k and Q
(n+1)
i,j,k , respectively.
It should be noted that when an explicit integration scheme is used, that
the differential equations (2.10) are physically decoupled from each other, if
the time step is chosen too large. This leads to an unstable integration; –
to ensure a stable integration it is necessary that the domain of dependence
of each node in the computational fields be larger than the physical one. The
maximum time step that according to linear theory allows a stable integration
of the Navier-Stokes equations calculates according to Merz [190] to
∆tmax =
CFLmax
ρξ + ρη + ρζ + 4 ·max(4/3 · ν, α) · CFLmaxVNNmax [(∇ξ)
2 + (∇η)2 + (∇ζ)2]
.
(2.24)
The maximum stable CFL and von Neumann number of the hybrid scheme
used in ITSM3D are given by
CFLmax = 4, VNNmax = 9, (2.25)
respectively.
With the aforementioned discretization and solution scheme the coupled
systems of ordinary differential equations (2.10) are integrated starting from
an initial guess of the solution and constant boundary conditions until a max-
imum number of time steps have been carried out or a steady state has been
reached, for which the temporal derivatives and consequently the residuals
vanish. The use of the unsteady Navier-Stokes equations for the simulation
of steady state flow problems, in which the time is only used as an itera-
tion variable, has the advantage over the solution of the corresponding steady
state equations, that it is possible to use a single solution scheme for the sim-
ulation of steady transonic flows, since the unsteady Navier-Stokes equations
always exhibit a hyperbolic-parabolic character irrespective whether the flow
is sub- or supersonic.
2.2.5 Convergence acceleration techniques
The convergence of the explicit solution scheme of the flow solver ITSM3D is
accelerated using a multiple-grid technique, the implicit residual smoothing
technique of Radespiel et al. [218], and a local time stepping procedure. A
detailed description of the various convergence enhancement techniques im-
plemented in ITSM3D can be found in the thesis of Merz [190].
2.2.6 Normalization of the flow equations
In order to minimize round-off errors in the unpreconditioned solution schemes
in ITSM3D the flow quantities are normalized with the inlet total pressure
p0,e, the total inlet temperature T0,e, a characteristic length lr (usually the
2.3 Boundary conditions 27
chord of the first stator) and the specific gas constant R. For a detailed de-
scription of the normalization procedure, the reader may refer to the work of
Merz [190].
2.3 Boundary conditions and their numerical
treatment
The solution of the Navier-Stokes equations (2.1) requires the specification
of conditions for the states at the boundaries of the computational domain.
In the following, the boundary conditions and their numerical treatment in
steady-state simulations of turbomachinery applications are described.
2.3.1 Inlet, outlet and mixing planes
At the inlet- and outlet boundaries as well as at the mixing planes where the
stator and rotor blocks are coupled via circumferential averaged quantities, a
boundary condition treatment based on the work of Giles [98] and Saxer [242]
is applied. With their non-reflecting boundary condition treatment, it is pos-
sible to carry out computations on truncated domains where the boundaries
of the computational domain are lying close to the blades without obtaining
spurious reflections at the boundaries.
In ITSM3D the user has to specify the radial distribution of the total pres-
sure, the total temperature, and the flow angles. At the outlet boundary the
radial distribution of the static pressure has to be given. If the flow is axially
supersonic, then the user has also to specify the Mach number distribution
at the inlet. In this case, the specification of the static pressure at the outlet
boundary can be omitted.
At the mixing planes, the coupling of the circumferential averaged flow
quantities is carried out such that the mass, the momentum and the energy
are conserved over the interface.
2.3.2 Wall boundary conditions
On solid walls the so-called no-slip condition is used, which implies that the
fluid at the wall is not moving relative to it. The walls in ITSM3D are either
treated as adiabatic or as isothermal. In the former case, the temperature in
the nodes on the walls are determined from the condition that the tempera-
ture gradient on the wall vanishes. For isothermal walls, the temperature in
the nodes at the walls follow from the boundary condition. In either of the
aforementioned cases, the pressure at the wall is variable and is determined
by the flow solver in dependence on the flow field in the interior.
28 2. A baseline solution scheme
2.3.3 Periodic boundary condition
The use of periodic boundary conditions for the simulation of turbomachinery
flow can lead to a drastic reduction of the computational time as well as the
memory requirements. In the case that the inflow in the configuration is ro-
tationally symmetric and mixing planes are used to couple the simulated flow
field between the rotor- and the stator rows, it is sufficient to discretize one
single blade channel in each row instead of the full annulus.
The periodic boundary conditions in ITSM3D are realised by a direct cou-
pling of the state variables via the use of ghost cells, which are constructed
by an angular extension of the grid. At the periodic boundaries, not only the
independent state variables but also the second order differences are copied
into the layer of phantom cells, such that the nodes on the periodic boundaries
can be treated in the same fashion as the nodes in the interior of the domain.
Chapter 3
Development of a Mach number
independent solution scheme
The baseline scheme presented in the previous chapter is efficient for tran-
sonic flow computations. For the simulation of turbomachinery with large
regions of low Mach number flow, the scheme becomes inefficient due to the
disparity in the speed of sound and the convection velocities. To ensure that
the density based solution scheme is efficient independently of the Mach num-
ber level of the flow, time-derivative preconditioning must be used.
In the current work the time-derivative preconditioning scheme of Merkle
and coworkers was introduced in the flow solver ITSM3D. This chapter de-
scribes theMerkle preconditioning scheme and its implementation in ITSM3D.
In addition, issues related to the simulation of flow in the low Mach and
low Reynolds number limit are addressed. The performance of the devel-
oped scheme for the simulation of turbomachinery flow is assessed and it is
shown that with this scheme accurate flow computations can be carried out
efficiently at arbitrary Mach numbers.
3.1 The time-derivative preconditioning
scheme of Merkle
The principle of preconditioning is to modify the dynamics of the Navier-
Stokes equations such that the disparities in the transport velocities are re-
duced and that the different physical processes can evolve at the same rate.
The propagation velocities of the various processes can in principle be al-
tered by either scaling the convective or the unsteady terms. However, in
order to avoid the introduction of unnecessary source terms and to obtain a
solution scheme that ensures global flux conservation, it is important to use
the original, unscaled conservative flux vectors as given in Eq. (2.1). Thus,
when preconditioning methods are used, the dynamics of the governing equa-
tions are always changed by multiplying the vector of the time-derivatives of
the independent variables by an appropriate matrix.
The idea of time-derivative preconditioning is to alter the dynamics of the
29
30 3. A Mach number independent solution scheme
time-stepping procedure but not necessarily to change the resulting solution.
If the scaling of the artificial dissipation is not altered, the residuals repre-
senting the sum of convective, diffusive, and dissipative fluxes are identical
for the preconditioned and the unpreconditioned scheme. Thus, in the limit
of a converged solution, the flow fields obtained with a preconditioned and an
unpreconditioned solution scheme are the same.
When preconditioning is applied, it would be possible to use the conserva-
tive state vector Q = [%, %wx, %wφ, %wr, %e0] as a set of independent variables.
In the preconditioning scheme of Merkle, the vector of the primitive, viscous
variablesQv = [p, wx, wφ, wr, T ]T is used instead of the conservative ones, since
this set of variables can be used for describing constant density flow. Fur-
thermore, for constant density flow this choice of solution variables results in
the artificial compressibility method of Chorin [61] if an appropriate precon-
ditioning matrix is used [301]. The choice of the viscous, primitive variables
is also convenient when fluids with arbitrary equations are to be simulated,
since it is easier to calculate the density % and the specific enthalpy h from Qv
than to calculate the temperature T and pressure p from Q [188]. The precon-
ditioned solution scheme developed in the current work, can handle all kinds
of thermal and caloric state equations, which can be written in the following
form
% = %(p, T ), h = h(p, T ). (3.1)
With the choice of Qv as solution vector, the derivation of the precondi-
tioning matrix begins by transforming the independent variables in Eq. (2.1)
from the vector of conservative variables Q to the vector of viscous, primitive
variables Qv = [p, wx, wφ, wr, T ]T as follows:
∂Q
∂Qv
∂Qv
∂t
+
∂E
∂x
+
1
r
∂F
∂φ
+
1
r
∂(rG)
∂r
= H, (3.2)
where the Jacobian ∂Q
∂Qv
is given by
∂Q
∂Qv
=

%p 0 0 0 %T
wx%p % 0 0 wx%T
wφ%p 0 % 0 wφ%T
wr%p 0 0 % wr%T
h0%p−
(1− %hp) %wx %wφ %wr
h0%T+
%hT

. (3.3)
The indices p and T represent the isothermal derivative with respect to the
pressure and the isobaric derivative with respect to the temperature, respec-
tively; i.e., hT is the specific heat capacity at constant pressure.
To ensure uniform, efficient convergence over all Mach numbers, the phys-
ical Jacobian is replaced with the preconditioning matrix Γv, so that the fol-
lowing equations
Γv
∂Qv
∂t
+
∂E
∂x
+
1
r
∂F
∂φ
+
1
r
∂(rG)
∂r
= H (3.4)
3.1 The time-derivative preconditioning scheme of Merkle 31
have to be solved. As first proposed byMerkle et al. [188], fluids with a general
equation of state can be dealt with when Γv is chosen as
Γv =

%′p 0 0 0 %
′
T
wx%
′
p % 0 0 wx%
′
T
wφ%
′
p 0 % 0 wφ%
′
T
wr%
′
p 0 0 % wr%
′
T
h0%
′
p − h′p %wx %wφ %wr h0%′T + %h′T
 (3.5)
in a form analogous to the physical Jacobian ∂Q
∂Qv
. The preconditioning matrix
Γv differs from the Jacobian ∂Q∂Qv only by the change of the partial derivatives
%p, %T , hT and 1 − %hp, which represent physical properties, by the respective
artificial terms %′p, %′T , h′T , and h′p. In the following sections, it will be shown
how these quantities can be chosen to ensure well-conditioned flow equations
at all Mach numbers.
3.1.1 Eigenvalue scaling for convergence enhancement
The eigenvalues of the linearized unsteady Euler equations for the ξ-direction
of a grid-conformal, curvilinear ξ-η-ζ coordinate system read as
λξ,1,2,3 = wˆξ,
λξ,4,5 =
1
2
wˆξ
(
1 + d
d′
)± 1
2
√
wˆ2ξ
(
1− d
d′
)2
+ 4%hT
d′ (∇ξ)2
(3.6)
with
d = %hT%p + %T (1− %hp), d′ = %hT%′p + %′T (1− %hp) (3.7)
if the parameters h′T and h′p with h′T := hT and h′p := hp have been set to their
respective unscaled counterpart [6, 9]. The term
wˆξ = wxξx + wφξφ + wrξr (3.8)
represents the contravariant velocity component along the grid direction ξ
with wx, wφ and wr representing the relative velocity components in the cylin-
drical system. When preconditioning is not used, d = d′, and the last two
eigenvalues should equal the propagation velocities of the pressure waves.
From this reasoning it follows that the isentropic speed of sound is given by(
∂p
∂%
)
s
≡ a2 = %hT
%T (1− %hp) + %%phT =
%hT
d
, (3.9)
as can also be derived directly by means of basic thermodynamic relations [19,
47]. Consequently, the ratio %hT/d′ may be interpreted as the square of the
artificial speed of sound a′2. In order to avoid eigenvalue stiffness caused by
the disparity between acoustic and convective velocities for subsonic flows,
this artificial speed of sound is reduced to the convective reference velocity Vr
according to
a′ =
√
%hT
d′
:= Vr. (3.10)
32 3. A Mach number independent solution scheme
The convective reference velocity is oriented after the norm of the local veloc-
ity vector according to
Vr = min(‖w‖, a), (3.11)
where w = [wx, wφ, wr] represents the relative velocity vector. By combining
Eqs. (3.7) and (3.10) the following prescription for the preconditioning param-
eter %′p results:
%′p =
1
V 2r
− 1− %hp
%hT
%′T . (3.12)
The convective reference velocity Vr is limited by the sonic speed, so if the
other preconditioning parameters are set to their corresponding physical val-
ues, at supersonic speed the preconditioned flow equations (3.4) revert to the
unscaled Navier-Stokes equations (2.1). Preconditioning is not necessary at
supersonic velocities, since density based schemes are efficient at these condi-
tions.
In the incompressible limit the energy equation of the unscaled Navier-
Stokes equations is decoupled from the continuity and momentum equations.
This property is also desirable for the preconditioned system of flow equations,
which is the reason why the preconditioning parameter %′T should be scaled
proportional to its physical counterpart %T . In the solution scheme presented
here %′T is set according to
%′T = δpc · %T , (3.13)
where the parameter δpc is defined by the user. For the choice δpc = 0 or for
the case of incompressible flow, the preconditioning matrix is regular for all
reference speeds (Vr > 0) as long as the density and the heat capacity take on
physical values (%, hT > 0), as shown in the work of Anker et al. [7, 9]. In the
case that the parameter δpc is set to a non-vanishing value, it should be noted
that the regularity of the preconditioning matrix for compressible fluids can
not longer be guaranteed for all finite reference speeds. However, for ideal
gases the preconditioning matrix first gets singular for supersonic values of
the reference speed Vr if the parameter δpc is bounded by unity. Thus, by
bounding the reference velocity Vr by Eq. (3.11) and limiting the parameter
δpc to the range δpc ∈ [0, 1] the regularity of the preconditioning matrix can be
ensured for ideal gas flow.
With the preconditioning parameter %′p determined according to Eq. (3.12)
it can be be shown by combining the Eqs. (3.6), (3.9), (3.10) that the condition
number σ for ideal gas flow tends to σ = |1+
√
5|
|1−√5| ≈ 2.62 for ‖w‖ → 0 indepen-
dently of the choice of the preconditioning parameter δpc.
For an incompressible fluid with % = const. the sonic speed is infinite and
the time derivatives in the continuity equation vanish. By preconditioning the
Navier-Stokes-Equations (2.1) using the matrix given in Eq. (3.5) and setting
the reference velocity to Vr = ‖w‖, temporal derivatives of the pressure are
added to the continuity equation. For the special case of Vr =
√
2 · ‖w‖ the pre-
conditioning results in the artificial compressibility approach of Chorin [61].
This choice leads to the condition number of σ = 2 independently of the mag-
nitude of the flow speed. The physical values %T = %p = 0 that cause the
3.1 The time-derivative preconditioning scheme of Merkle 33
sonic velocity in incompressible fluids to be infinite are replaced with the ar-
tificial properties %′p and %′T when preconditioning is used. This replacement
makes time-marching solution schemes practically feasible for compressible,
low Mach number as well as for incompressible flow.
3.1.2 Perturbation analysis for inviscid, low Mach num-
ber flows
Instead of performing an eigenvalue analysis for finding suitable values for
the preconditioning parameters, a perturbation analysis of the inviscid flow
equations for low Mach numbers may be carried out as demonstrated in Ap-
pendix A. The analysis presented there shows that in order to ensure a proper
balance between the temporal and the convective terms in the flow equations
for low Mach numbers, the physical quantity %p should be replaced with a
scaled quantity %′p given by
%′p = kinv/u
†2, (3.14)
with u† as a representative velocity scale (e.g., the norm of the local velocity
vector). Other physical properties (e.g., %T , hT , 1− %hp) need not to be changed
in order to achieve a well-conditioned system of inviscid flow equations.
3.1.3 Robustness aspects
In order to prevent singularities at solid walls, not ‖w‖, but the maximum of
‖w‖ and Vlim = lim · a is used as reference velocity for subsonic flows, where
lim is typically set to 10−5. As will be shown later in this chapter, precondi-
tioning yields a significant enhancement of the convergence rates at low Mach
numbers. However, as first discussed by Darmofal and Schmid [68], precon-
ditioned systems show an increased sensitivity to pressure fluctuations and
are unstable in the presence of pressure fluctuations that are large compared
to the dynamic pressure, especially in the initial phase of a simulation. One
approach devised by Weiss et al. [299] to handle this sensitivity is to introduce
the term Vpgr = pgr
√
δp/% and to limit the lower bound of the pseudo-acoustic
speed a′ in the case that the spatial pressure variations δp are larger than the
potential dynamic pressure variations. For the case of a compressible, inviscid
fluid the pseudo-acoustic velocity is calculated according to
a′ = Vr = min[a,max(‖w‖, Vlim, Vpgr)]. (3.15)
In the present work this modification was adopted with the pressure variation
δp taken as the maximum pressure variation over a control volume and pgr ∈
[0.1, 2]. To further increase the robustness, the reference velocity Vr at a node i
was in the current work determined as the maximum of the reference velocity
Vr,i of the local node i and the corresponding values Vr,j in the surrounding
nodes j = 1, . . . Nj:
Vr = min[a,max(Vr,i, Vr,n)], Vr,n = max
j=1,...,Nj
(‖w‖j, Vlim,j, Vpgr,j). (3.16)
34 3. A Mach number independent solution scheme
The procedure of looping over neighbouring nodes may be repeated; in the cur-
rent work it was found that accounting for the immediate neighbours already
leads to a very robust scheme. This result is confirmed by Venkateswaran et
al. [285], who also report a significant gain in the robustness by applying this
method.
Because the instability of a preconditioned scheme in the presence of large
pressure fluctuations is due to non-linear effects, Venkateswaran et al. [284,
288] suggest to embed the preconditioning scheme in an implicit-implicit dual
time-stepping procedure, in which preconditioning is only applied at the inner
level where the linearized flow equations are solved. They claim to achieve ro-
bustness of the solution scheme without employing the aforementioned pres-
sure switch. Since an explicit Runge-Kutta scheme was used in the current
work, this method of attaining robustness was not considered.
3.1.4 Scaling issues for low Reynolds number flow
With the preconditioned solution scheme presented so far, inviscid flows can
be calculated efficiently at arbitrary Mach numbers. However, as recognized
by Choi and Merkle [59] additional modifications are necessary for obtaining
an efficient scheme for viscous flows. Even though flows in turbomachinery
configurations usually have high Reynolds numbers where it could be antici-
pated that the dominant characteristics controlling the solution of the Navier-
Stokes equations are the same as those of the Euler equations, the Reynolds
numbers of the cross stream directions in the boundary layers may still be
small.
In the following the modifications necessary for calculating diffusion dom-
inated flow efficiently will be described. The extension of the preconditioning
method to viscous flows assumes that the Mach number as well as its square
is lower than the Reynolds number (i.e., M/Re  1 and M2/Re  1), which
is the case for most technical applications. In the case of high Mach numbers
and low Reynolds numbers, the viscous effects must be accounted for in a dif-
ferent manner in order to ensure a good conditioning of the flow equations
(cf. Appendix B).
Using perturbation analysis for viscous, ideal gas flows, it is shown in Ap-
pendix A that the preconditioning parameter %′T should be limited by
%′T ≤ %rRe/Tr (3.17)
in order to prevent the temporal derivative of the temperature from becoming
dominant in the continuity equation. Tr and %r represent a reference temper-
ature and reference density, respectively. In the present work this precondi-
tioning parameter was not scaled in dependence of the Reynolds number and
the inviscid prescription %′T = δpc%T according to Eq. (3.13) was also used for
the viscous case. A combination of Eqs. (3.13) and (3.17) leads to the following
condition for the preconditioning parameter δpc:
δpc ≤ %r
%TTr
Re. (3.18)
3.2 Preconditioned discretization and solution scheme 35
This restriction can be met for all flow conditions by setting the parameter δpc
to zero. A variation of this parameter from δpc = 0 to δpc = 1 did not prove to
have significant influence on the computation of high Reynolds, adiabatic flow
with the preconditioned solution scheme in ITSM3D. The results presented in
the current work were obtained with the preconditioning parameter δpc set to
unity.
According to Merkle [186], and as also shown in Appendix A, a scaling
of the energy equation by setting h′T = hT/Pr is appropriate if the Prandtl
number deviates largely from unity. Since most fluids encountered in tur-
bomachinery applications have Prandtl numbers in the order of unity, this
modification was not adopted in the present work.
As shown in Appendix A by employing a perturbation analysis, the tempo-
ral term in the continuity equation balances the convective term for viscous
dominated flows provided that
%′p = kvRe
2/V 2r (3.19)
holds [186], with kv as a proportionality constant having a magnitude in the
order of unity. Furthermore, as shown in Appendix B by means of a dispersion
analysis for low Reynolds number flow, this prescription of the precondition-
ing parameter %′p leads to a good conditioning of the flow equations both for
compressible and incompressible flows, when the other preconditioning pa-
rameters h′p and h′T are set to their physical values and the preconditioning
parameter %′T simultaneously is set to its physical counterpart or to zero.
Using a Fourier stability analysis [289], it can be shown that the proper
Reynolds number to use in Eq. (3.19) is the cell-Reynolds number, Re∆x =
v∆x/ν. The cell-Reynolds number can be incorporated and viscous effects can
be accounted for by limiting the reference velocity Vr by the velocity scale of
diffusion of momentum Vvis given by Vvis = ν/∆x in the following manner:
Vr = min(a,max(‖w‖, Vlim, Vpgr, Vvis)). (3.20)
The introduction of the cell-Reynolds number shows that whether or not low
Mach preconditioning is used, depends on both the flow field and the grid reso-
lution. An inspection of Eq. (3.20) shows that the introduction of the diffusion
velocity scale Vvis may reduce the inviscid preconditioning. When active, the
low Reynolds number scaling causes the convective terms to be stiff, but since
in this case the viscous terms dominate, the convective fluxes are negligible
and do not inhibit the convergence.
3.2 Discretization and solution scheme for the
preconditioned Navier-Stokes equations
In this section the method for solving the preconditioned Navier-Stokes equa-
tions (3.4) is described. Since the developed preconditioned solution scheme is
36 3. A Mach number independent solution scheme
based on the scheme that was described in Chapter 2 for solving of the origi-
nal, unpreconditioned Navier-Stokes equations, this description is kept short
and focuses on the differences between the schemes.
3.2.1 Spatial discretization
In the solution scheme used, the preconditioned equations (3.4) are solved in
the integral form∫
V
Γv
∂Qv
∂t
dV +
∮
A
(E r dφ dr + F dx dφ+G r dx dφ) +
∫
V
H dV = 0, (3.21)
where V represents the content of an arbitrary control volume and A rep-
resents its surface. The convective and the diffusive fluxes are in principle
determined as in the original scheme; minor differences are only due to the
use of a different set of independent solution variables. In the original scheme
the temperature and the pressure must be determined from the conservative
state vector Q = [%, %wx, %wφ, %wr, %e0]T ; due to the implemented functionality
in ITSM3D of simulating ideal gas flows with a temperature-dependent heat
capacity, the temperature is determined iteratively. In the preconditioned so-
lution scheme, the viscous, primitive state variables are used as independent
solution variables and are consequently directly available. In this scheme,
the elements of the conservative state vector Q must be determined from the
elements of the vector Qv = [p, wx, wφ, wr, T ] of primitive state variables. This
determination can be carried out directly even for general equations of state,
since Merkle’s preconditioning scheme requires the user to specify the density
and the enthalpy as explicit functions of the pressure and the temperature.
Similar to the original scheme, the temporal term in the preconditioned
Navier-Stokes equations is discretized assuming that the distribution of the
primitive viscous state variables are linearly distributed over the control vol-
umes. With the preconditioning matrix Γv,i,j,k calculated as function of the
state vector Qv,i,j,k in the central node (i, j, k) of the control volume, it follows
that the discretization∫
Vˇi,j,k
Γv,i,j,k
∂Qv,i,j,k
∂t
dV ≈ Vˇi,j,k · Γv,i,j,k ∂Qv,i,j,k
∂t
(3.22)
is of second order accuracy in cases where the central node of a control volume
Vˇi,j,k coincides with the center of gravity.
With the sum of the convective fluxes Rc, the sum of the diffusive fluxes
Rd and the artificial dissipation vector D, the semi-discrete, preconditioned
Navier-Stokes equations can be written as
Γv
∂Qv
∂t
∣∣∣∣
i,j,k
= − 1
Vˇi,j,k
(Rc +Rd +D)i,j,k. (3.23)
The convective residual vector Rc and the diffusive residual vector Rd are
calculated in the same manner for the preconditioned Navier-Stokes equa-
tions as for the unpreconditioned ones using the discretization techniques of
3.2 Preconditioned discretization and solution scheme 37
Radespiel and Rossow [217] and Hall [109], respectively. The techniques for
calculating the convective and the diffusive residuals were already described
in Chapter 2 and will be omitted here. In the following only the calculation of
the artificial dissipation is described.
3.2.2 Artificial dissipation
Like in the solution and discretization scheme for the unscaled Navier-Stokes
equations (cf. Chapter 2), the artificial dissipation vector D in the precondi-
tioned, baseline solution scheme is calculated according to the approach of
Jameson et al. [132] using a blend of second and fourth order differences:
Di,j,k =
(−D2ξ +D4ξ −D2η +D4η −D2ζ +D4ζ)Qv,i,j,k. (3.24)
When time-derivative preconditioning is used, the differences are construct-
ed in dependence of the primitive, viscous state variablesQv = [p, wx, wφ, wr, T ]T
instead of differences of the vector Q = [%, %wx, %wφ, %wr, %e0]T of conservative
state variables. The second and fourth order difference operators employ the
preconditioning matrix Γv; for the ξ-direction these are defined as
D
(2)
ξ = δ
−
ξ
(
ρ¯′Vξ ε
(2)Γv
)
i+1/2,j,k
δ+ξ , D
(4)
ξ = δ
−
ξ
(
ρ¯′Vξ ε
(4)Γv
)
i+1/2,j,k
δ+ξ δ
−
ξ δ
+
ξ . (3.25)
The respective difference operators Dη and Dζ for the η- and ζ-directions are
defined analogously.
For the simulation of low Mach number flows, it would also be possible to
construct the artificial dissipation using differences of the conservative vari-
ables. However, for low Mach number flows it easier to avoid round-off errors
(cf. Appendix D.2) and ensure smoothness of the simulated flow field if the
vector of primitive, viscous variables are used to construct artificial dissipa-
tion, as the gradient of these variables is the driving force for the convective
and diffusive fluxes in the flow [71]. In the case of incompressible flow, the
use of differences of the conservative variables to construct the artificial dis-
sipation would not be suitable, as the density differences vanish identically.
As shown in Appendix C, the incorporation of the preconditioning matrix and
the use of the spectral radii of the preconditioned system ensures that the ar-
tificial dissipation scales correctly in the limit of low Mach numbers both for
inviscid and viscous flow.
The formulation of the artificial dissipation used in the Eqs. (3.24) and
(3.25) above is conservative [278]; with this method of constructing the artifi-
cial dissipation, flows with an arbitrary Mach number level can be simulated.
As demonstrated in the work of Anker et al. [17] as well as in subsequent sec-
tions of this chapter, the resulting scheme can be used to simulate transonic
flows and accurately capture the shocks as well as reliably predict low Mach
number flow. It should be noted that the dissipation operators of the precondi-
tioning scheme in Eq. (3.25) through the presence of a matrix-vector product
are computationally more costly than the dissipation operators in Eqs. (2.12)
38 3. A Mach number independent solution scheme
and (2.13) of the original scheme; the computational overhead of the conser-
vatively formulated dissipation corresponds approximately to the additional
costs of a matrix-dissipation scheme.
If conservativity of the solution scheme can be sacrificed, which is some-
thing that could be done for purely subsonic flows, then the dissipation vector
D can be determined through Eq. (3.24) with the difference operators defined
as
D
(2)
ξ = δ
−
ξ
(
ρ¯′Vξ ε
(2)
)
i+1/2,j,k
δ+ξ , D
(4)
ξ = δ
−
ξ
(
ρ¯′Vξ ε
(4)
)
i+1/2,j,k
δ+ξ δ
−
ξ δ
+
ξ (3.26)
instead of the computationally more expensive operators defined in Eq. (3.25).
If these nonconservative dissipation operators are used, then the artificial
dissipation should first be added to the residuals Rc and Rd after they have
been multiplied with the inverse preconditioning matrix Γ−1v ; the use of a non-
conservative dissipation scheme leads to the following system of semi-discrete
equations
∂Qi,j,k
∂t
= − 1
Vˇi,j,k
[
Γ−1v (Rc +Rd) +D
]
i,j,k
, (3.27)
which are to be integrated forward in time.
Like in the solution scheme for the unscaled Navier-Stokes equations, the
scaling factors ρ¯′Vξi+1/2,j,k for the artificial dissipation are given as
ρ¯′Vξi+1/2,j,k = ρ
′V
ξi+1/2,j,k
1 + max
(ρ′Vηi+1/2,j,k
ρ′Vξi+1/2,j,k
)1/2
,
(
ρ′Vζi+1/2,j,k
ρ′Vξi+1/2,j,k
)1/2 (3.28)
in dependence of the volume weighted spectral radii ρ′Vξ , ρ′
V
η , and ρ′
V
ζ according
to a suggestion of Martinelli [177] and Radespiel et al. [218]; the correspond-
ing factors for the η- and ζ-direction are defined in an analogous fashion. In
contrast to the original scheme, the spectral radii ρ′V of the preconditioned
Euler equations are used rather than the volume weighted spectral radii ρV
of the unscaled Euler equations. For example, the spectral radius for the ξ-
direction, which is used for scaling of the artificial dissipation, is determined
as
ρ′Vξ =
1
2
‖w · Sξ‖
(
1 +
d
d′
)
+
1
2
√
‖w · Sξ‖2
(
1− d
d′
)2
+ 4
%hT
d′
(3.29)
instead of
ρVξ = ‖w · Sξ‖+ a · ‖Sξ‖ , (3.30)
which is the spectral radius used in the original scheme. As discussed in
Appendix C it is necessary to use the spectral radius of the preconditioned
equations to ensure a correct magnitude of the artificial dissipation in the
limit of low Mach and Reynolds numbers.
The factors ε(2) and ε(4) for controlling the artificial dissipation are calcu-
lated in the same way as in the original scheme, i.e., they are determined
according to Eqs. (2.18) and (2.19).
3.2 Preconditioned discretization and solution scheme 39
3.2.3 Time-stepping scheme
The semi-discrete system of preconditioned Navier-Stokes equations (3.23)
are integrated with the five-stage, hybrid, explicit Runge-Kutta scheme of
Radespiel and Swanson [219]
Qv
(0)
i,j,k = Qv
(n)
i,j,k,
Qv
(s)
i,j,k = Qv
(0)
i,j,k − αs
∆t
Vˇi,j,k
(
Γ(s−1)v
)−1 [
R(s−1)c +R
(0)
d +
s∑
t=1
γstD
(s−1)
]
i,j,k
,
Qv
(n+1)
i,j,k = Qv
(5)
i,j,k
(3.31)
where s = 1, . . . , 5 denotes the stage number. The coefficients αs, γst; s =
1, . . . , 5; t ≤ s are the same as those for the unpreconditioned flow equations
and are given in Eq. (2.23).
In contrast to the Runge-Kutta scheme given in Eq. (2.21) for the integra-
tion of the unscaled Navier-Stokes equations, the sum of the residuals in the
preconditioned integration scheme given in Eq. (3.31) is multiplied with the
inverse of the preconditioning matrix Γv. The inverse preconditioning ma-
trix Γ−1v is determined as a function of independent solution Q
(s−1)
v , which is
present at the beginning of Runge-Kutta stage s.
3.2.4 Calculation of the time step
The size of the maximum stable time-integration step ∆tmax for the system of
preconditioned Navier-Stokes equations (3.4) is calculated as
∆tmax =
CFLmax
ρ′ξ + ρ′η + ρ
′
ζ + 4 ·max(4/3 · ν, α) · CFLmaxVNNmax [(∇ξ)
2 + (∇η)2 + (∇ζ)2]
,
(3.32)
analogous to the relation (2.24) for the maximum allowable time step for the
unpreconditioned Navier-Stokes equations [18].
3.2.5 Treatment of the initial and boundary conditions
For the steady state numerical simulation of turbine flows, six different types
of boundaries have to be considered: Inlet, outlet, non-rotating and rotating
walls, periodicity and mixing planes.
Solid walls are treated as isothermal and the no-slip condition is applied
for viscous flows. When inviscid flows are simulated or a symmetry plane is
needed, slip-walls can be defined instead.
In all of the calculations presented in this paper, the wall temperature is
set to the total temperature at the inlet boundary. Periodicity in the pitch-
wise direction is ensured using phantom cells that keep copies of the periodic
values such that the points on these boundaries can be treated like interior
points.
40 3. A Mach number independent solution scheme
At the inlet and outlet boundaries and at the mixing planes a non-reflecting
post-correction method is used. The boundary conditions used represent an
extension of the work of Giles [98] and Saxer [242] to preconditioned systems
with arbitrary equations of state and the essential elements of the developed
theory are presented in Chapter 5.
3.2.6 The use of perturbed, normalized flow quantities
In order to minimize round-off errors in the unpreconditioned solution scheme
in ITSM3D the flow quantities are normalized as previously described in
Chapter 2.2.6. Since the preconditioned solution can handle incompressible
fluids and fluids with general state equations, the isobaric heat capacity cp
is used for normalization rather than the specific gas constant. As for the
unpreconditioned scheme, also the inlet total pressure temperature p0,e, the
total inlet temperature T0,e and a characteristic length lr are used as reference
quantities for the normalization.
When the preconditioned solution schemes in ITSM3D are used, the so-
called perturbed, normalized quantities
p˜∗=
p− p0,e
p0,e
= p∗ − 1, and T˜ ∗= T − T0,e
T0,e
= T ∗ − 1 (3.33)
are used as independent solution variables as this does not spoil the mantissa
for storage of the nearly constant part of the pressure and the temperature
(cf. Appendix D).
To also avoid round-off errors linked to the calculation and use of the en-
thalpy, the enthalpy at the inlet is used as reference condition.
Gradients and differences (artificial dissipation) are directly calculated by
means of the perturbed and normalized variables. However, state functions
as well as the preconditioning matrix are determined in function of the ab-
solute quantities p∗, T ∗, which are obtained from the perturbed variables by
incrementation, i.e., p∗ = p˜∗ + 1, T ∗ = T˜ ∗ + 1.
To address the singularity problem outlined in Section D.1, the pressure
should be normalized with a reference value for the dynamic pressure. This
was not done in the current work; however with the use the normalization
and perturbation procedure described in this section, low Mach number flow
(M = 10−4) could be accurately calculated using a single precision (4 Byte)
representation of floating numbers.
3.2.7 Implicit residual smoothing technique for precon-
ditioned systems
In order to accelerate the convergence of the unscaled Navier-Stokes equa-
tions to steady state, an implicit residual smoothing technique with vari-
able coefficients [177, 218] was implemented in the flow solver ITSM3D by
Merz [190]. The implicit residual smoothing technique used in conjunction
3.3 Impact of preconditioning 41
with preconditioning is the same as for the original scheme with the excep-
tion that the smoothing coefficients now are calculated as functions of the
spectral radii of the preconditioned system instead of those of the unscaled
Navier-Stokes equations.
3.3 Preconditioning and its impact on the effi-
ciency and accuracy of a solution scheme
The use of time-derivative preconditioning schemes can lead to a dramatic
enhancement in the efficiency and a significant improvement of the accuracy
of the solution scheme at low Mach numbers. In this section, both of these
beneficial aspects will be discussed.
3.3.1 Enhancement of the convergence rates
Preconditioning generally leads to enhanced convergence rates for low Mach
number flows. The reason for this can be found in the improved damping of
the errors as well as in the increase in the maximum allowable time-steps
due to eigenvalue clustering. In the following these effects will be discussed
separately.
By effectively setting the numerical speed of sound a′ to the norm ‖w‖ of
the relative velocity vector for convective dominated flows, the maximum al-
lowable time step for low Mach number flow can be increased substantially.
To show this, one-dimensional inviscid flow is assumed. In this case, the max-
imum allowable time step ∆tmax for an explicit unpreconditioned scheme can
be estimated as
∆tmax = CFLmax ·∆x/ρ (3.34)
where CFLmax represents the maximum stable CFL number of the integration
scheme and ρ = max
i
|λi| represents the spectral radius of the flux Jacobian of
the linearized flow equations. The slowest modes of the solution are trans-
ported with a speed that corresponds to the value of the smallest eigenvalue
of the system of linearized Euler equations. Thus, in explicit time-stepping
schemes for which the maximum stable integration time step is limited by the
CFL condition given in Eq. (3.34), the number of time steps needed to convect
the slowest modes through the computational domain is directly related to the
condition number σ = max
i
|λi|/min
i
|λi|. For the unscaled Euler-equations the
condition number is given by
σ = (a+ ‖w‖)/‖w‖ = (1 +M)/M, (3.35)
which for low Mach numbers can be approximated to σ ≈ 1/M . Thus, if pre-
conditioning is not used and if the effect of numerical damping is not strong,
42 3. A Mach number independent solution scheme
the number of iterations required to converge a simulation of the Euler equa-
tions goes beyond all bounds as the Mach number tends to zero.
If time-derivative preconditioning is used, then the numerical speed of
sound a′ is oriented after the convection speed. With a′ = ‖w‖ the maximum
allowable time step ∆t′max can be estimated as ∆t′max = CFLmax ·∆x/‖w‖. Con-
sequently, the ratio of the maximum allowable time step with and without the
use of preconditioning can be determined to
∆t′max/∆tmax = (a+ ‖w‖)/(a′ + ‖w‖) =
1
2
(a+ ‖w‖)/‖w‖, (3.36)
which for low Mach numbers can be approximated to ∆t′max/∆tmax ≈ 12 1M .
Thus, for low Mach number flows, the maximum allowable time step when
preconditioning is used is significantly larger than when preconditioning is
not used.
Implicit solution methods are known to be less sensitive to stiffness of the
governing equations than their explicit counterparts. For unconditionally sta-
ble integration schemes there is no limit in the maximum stable CFL number,
wherefore the stiffness nominally should not affect the number of iterations
needed for convergence of the flow equations. It should be noted that the
unconditional stability of an integration scheme usually refers to a property
given for the solution of linear flow equations. For non-linear problems, the
solution can only be found iteratively using finite CFL numbers. Thus, also
for implicit schemes, the stiffness and the disparity of the wave propagation
speeds of the governing equations matter.
The possibility to use a larger time step, enhances the convergence speed
assuming the damping of errors is not impaired by the use of larger time
steps. The use of time-derivative preconditioning has in fact a clearly positive
effect on the damping rates both on explicit and implicit schemes as shown
by Buelow [46], Venkateswaran and Merkle [289] by carrying out a von Neu-
mann analysis of the solution scheme. As furthermore shown by Lynn [173]
and van Leer [174] by analysing the Fourier footprints, preconditioning has
the effect of clustering the eigenvalues of the discretized equations. Numerical
tests show that the clustering of eigenvalues leads to a significant enhance-
ment of the error damping properties of a given multigrid scheme.
3.3.2 Accuracy preservation of the solution scheme for
low Mach number flow simulations
Several discretization schemes, among them the JST, Roe, and the AUSM+
discretization schemes degrade in accuracy for low Mach numbers if applied
in their original form. By the use of a perturbation analysis, it is shown in
Appendix C, that the original JST scheme for the solution of the unprecondi-
tioned Navier-Stokes equations leads to a wrong scaling of the artificial dissi-
pation for low Mach number inviscid or viscous flows. Using the normalized
flow equations as a basis for the analysis, it is shown that in the case of low
Mach number flow (under inviscid or high Reynolds number flow conditions),
3.3 Impact of preconditioning 43
the artificial dissipation D1 added to the continuity equations scales in the
order of the Mach number (D1 = O(M)) whereas the artificial dissipation
D2,3 added to the momentum and the energy equation scales inversely propor-
tional to the Mach number (D2,3 = O(1/M)). Ideally, the amount of artificial
dissipation added to each of the flow equations should scale in the order of
unity independently of the Mach and Reynolds number level. The wrong scal-
ing of the artificial dissipation leads for lowMach number flows to an odd-even
decoupling of the pressure field, which manifests itself in unphysical wiggles
in the pressure field of a computation, as can be seen in Fig. 3.4a, where the
simulated pressure field for a low speed test rig obtained with the original
JST scheme are shown. At the same time, the excess amount of artificial dis-
sipation that is added to the momentum and the energy equations degrades
the accuracy of the solution. The effect of this can be seen in Fig. 3.5, where
the circumferentially averaged flow velocities computed with (PC) and with-
out (NonPC) the use of preconditioning are compared for the aforementioned
low speed test rig.
However, if preconditioning is used and the scaling factor of the artificial
dissipation is based on the spectral radius of the preconditioned equations,
the artificial dissipation scales in the order of unity in terms of normalized
flow equations independently of the Mach number level of the simulated flow
(D1,2,3 = O(1)). The result of a correct scaling for the test case discussed above
can be seen in Fig. 3.4b. The use of time-derivative preconditioning leads to
a correct scaling of the artificial dissipation added to the continuity equation
and consequently to a smooth simulated pressure field. Furthermore, since
the addition of excess dissipation to the momentum and the energy equation
is avoided, the accuracy of the results are improved. This is shown in Fig. 3.5,
where it is visible that the preconditioned scheme is able to resolve secondary
flow effects, while the original scheme is not.
As demonstrated in Appendix C, for low Reynolds number flow both the
use of the original scheme as well as inviscid preconditioning leads to a wrong
scaling of the artificial dissipation. In the case of low Reynolds number flow,
the artificial dissipation terms first scale correctly when the preconditioning
method properly accounts for viscous effects.
44 3. A Mach number independent solution scheme
3.4 Computational results
3.4.1 Inviscid flow over a bump
To demonstrate that the preconditioned solution scheme is able to calculate
a configuration efficiently and accurately over a broad Mach number range,
results from the computation of the Ni-Bump [226] test case are presented
in the following. In this test case the internal inviscid, two-dimensional flow
through a parallel channel having a 4.2 % thick circular bump on the lower
wall is considered. The computational grid is shown in Fig. 3.1 and consists
of 177 × 3 × 21 grid points. In all simulations conducted, the scalar JST
dissipation scheme was used.
The calculations were initialized by a flow field which was gained using
the non-preconditioned solution scheme for an isentropic outlet Mach number
of Mais,a=0.59. All simulations were run until machine zero of the continuity
residual was reached. Table 3.1 shows the isentropic Mach numbers for which
the calculations were performed with (PC) and without (NonPC) precondition-
ing and the number of iterations N that were necessary for achieving a drop
in the residuals by 5 orders in magnitude.
In the case of an isentropic outlet Mach number of Mais = 0.85 the orig-
inal boundary values of the test case [226] were prescribed and the flow is
transonic. The solutions obtained with the original and the preconditioned
scheme are plotted in Figs. 3.2a and 3.2b, respectively. The calculated flow
fields are nearly identical, which shows that the shock capturing property of
the original scheme is retained when preconditioning is used. For this case,
the original scheme is faster than the preconditioned one.
When the preconditioned scheme was used at the other Mach number
cases, a drop in the residual by five orders of magnitude was always achieved
within 7650 iterations. As expected, the necessary number of iterations us-
ing the original scheme increases with decreasing Mach number. However,
a more serious problem that arises when using the original scheme, is the
deterioration of accuracy with decreasing Mach number.
In Figs. 3.3a and 3.3b the Mach number distributions obtained with the
different schemes for the case of an isentropic Mach number of Mais = 0.037
are compared; it is apparent that the original scheme is much more diffu-
sive than the preconditioned scheme, since the former produces a long tail
after the Bump with reduced Mach numbers in the solution. Furthermore,
in contrast to the nearly symmetrical flow solution obtained with the precon-
ditioned scheme the Mach number lines of the flow field simulated with the
non-preconditioned solution are tilted in streamwise direction. Obviously, the
accuracy of the solution scheme is only preserved at low Mach numbers when
preconditioning is used.
3.4 Computational results 45
Mais,a N (PC) N (NonPC)
0.013 6480 13290
0.037 7330 8930
0.084 7650 8930
0.850 5210 4090
Table 3.1: Number of iterations N needed for a residual drop by five orders of mag-
nitude in the Ni-Bump case in dependence of the isentropic Mach number
Mais,a
Figure 3.1: Computational grid for the Ni-Bump test case
3.4.2 Flat plate
In Chapter 4.7.2 a comparison of the performance of various low diffusion
schemes are shown for the case of a viscous flow over a flat plate (Mais,a =
0.136). A comparison of the results for the JST dissipation scheme with and
without preconditioning clearly shows that preconditioning improves the re-
sults (cf. Fig. 4.6).
The different dissipation schemes used in conjunction with precondition-
ing for the simulation of the flat plate test case presented in detail in Chap-
ter 4.7.2 lead to comparable convergence rates; for a CFL-number of CFL = 1
all schemes achieve a residual drop of 4 orders in magnitude within 4,000-
5,000 iterations. In contrast to this, the non-preconditioned solution scheme
needs 30,000 iterations for obtaining the same residual drop. Even though
one iteration with the preconditioned solution scheme can consume up to 25%
more CPU-time than the non-preconditioned one, it is clear that precondition-
ing not only enhances the accuracy of the solution but has also the potential
of substantially increasing the computational efficiency of a given solution
scheme at low Mach numbers.
3.4.3 Supersonic flow through a row of wedges
In Chapter 4.7.3 results of the simulation of the supersonic flow through a row
of wedges are shown. As shown by comparing the pressure field (cf. Fig. 4.11),
the difference in the results obtained using the JST scheme with and with-
out preconditioning are marginal; the use of time-derivative preconditioning
46 3. A Mach number independent solution scheme
(a) (NonPC) (b) (PC)
Figure 3.2: Calculated Mach number contours (∆Ma = 0.025) of the transonic Ni-
Bump test case (Mais=0.85) using the non-preconditioned (NonPC) and
the preconditioned (PC) JST-scheme
(a) (PC) (b) (NonPC)
Figure 3.3: Calculated Mach number contours (∆Ma = 0.0002) of the subsonic Ni-
Bump test case (Mais = 0.037) using the non-preconditioned (NonPC)
and the preconditioned (PC) JST-scheme
does not in any way impair the shock capturing properties of the JST-scheme.
Thus, when preconditioning is used, both low Mach number flow and super-
sonic flow can be simulated accurately.
3.4.4 Low-speed turbine
In order to verify that the presented scheme is able to simulate low-speed mul-
tistage turbomachinery efficiently, in this section results from the simulation
of the flow in a 1.5 stage axial air turbine are presented. In this low-speed
test case, the Mach number ranges from Ma = 0.05− 0.15 in the main flow. It
will be shown that on coarse grids the secondary flow effects are only captured
when preconditioning is used.
The turbine considered is operated in a test rig at the Ruhr-Universität
Bochumwhere extensive steady-state measurements have been carried out [6,
9]. For a detailed description of the test case, the reader is referred to Chap-
ter 6 of this thesis. In this section, the configuration is simulated for a clear-
ance height of scl = 1 mm and a rotor speed of n = 500 rpm.
The 1.5 stage axial turbine was discretized using a block-structured grid.
The numerical grids used consist of 327,000 grid points in total; the first stator
was discretized using 61×25×33 nodes.
All calculations were conducted with a CFL-number of CFL=1.5 and run
until the mass flow rates had reached constant values. The fluid was modelled
3.4 Computational results 47
as an ideal gas in the simulations applying the original, unpreconditioned
solution scheme. In the simulations using the preconditioned solution scheme
the fluid was modelled either as incompressible (% = const) or as an ideal gas.
Using preconditioning, three multiple-grid levels, and an impulsive start,
convergence of the mass flow rates was achieved within 4000 iterations in-
dependent of the fluid model used. However, when preconditioning was not
used, about 12000 iterations were needed in order to obtain convergence.
In Figs. 3.4a and 3.4b the static pressure contours of the solution computed
with the original (NonPC) and the preconditioned scheme (PC) are plotted in
the azimuthal plane at midspan, respectively. Since the simulated flow fields
obtained with the preconditioned scheme are qualitatively independent of the
fluid model applied, only the flow field where the fluid was modeled as ideal
gas is shown. The original, non-preconditioned solution scheme produces un-
physical wiggles in the solution, especially at the leading and trailing edges of
the blades. In addition, the isobars of the computed flow field are not perpen-
dicular to the boundary layers. Previous work showed that using finer grids
leads to smoother isobars in the main flow. The use of a finer grid does not
alleviate the problem of solution decoupling in the boundary layers. With a
finer discretization of the boundary layer, the Mach number of the nodes lying
next to the wall decreases, whereby the low Mach number deficiencies in the
neighbourhood of the wall become pronounced. In contrast to this, the precon-
ditioned scheme produces smooth and physical solutions in all regions of the
flow.
In Fig. 3.5 the calculated circumferentially averaged axial and circumfer-
ential velocities in the plane M2 (located 30 mm behind the rotor) are com-
pared to measurement data. As seen by the circumferentially averaged axial
velocities, both the preconditioned scheme and the original one underpredict
the axial velocities. While a mass flow of 13.04 kg/s was measured, the pre-
conditioned scheme predicts a mass flow of 12.53 kg/s and 12.61 kg/s for the
compressible and the incompressible case respectively, whereas in the simula-
tion with the original scheme a mass flow rate of 12.24 kg/s was obtained. The
measured circumferentially averaged velocities in M2, depicted in Fig. 3.5a,
reveal that there is a reduced mass flow in the hub region of the rotor, since
the hubside passage vortex causes an overturning of the main flow. While
the preconditioned solver captures the tendencies of the secondary flow, the
original scheme completely fails to reproduce the secondary flow effects. Fur-
thermore, in the casing region in M2, where the leakage flow enters the main
flow, the preconditioned solver predicts the averaged circumferential veloci-
ties fairly well, cf. 3.5b.
From the findings of Volpe [291] and from previous work of the present
author [16], it is known that by using enough grid points and lowering the
artificial damping parameter k(4) from 0.03 to k(4) = 0.015 compensates for
some of the deficiencies of using a non-preconditioned code for the simulation
of low Mach number flows. However, the use of 3.2 Mio. grid points as in a
previous work of the author [16] instead of the 327,000 grid points used here,
is not practicable when taking into account that the original scheme exhibits
48 3. A Mach number independent solution scheme
a slower convergence rate. The results presented here clearly show the need
for preconditioning when simulating lowMach number flow in multistage tur-
bomachinery.
3.4 Computational results 49
(a) NonPC, % = p/RT
(b) PC, % = p/RT
Figure 3.4: Lines of constant pressure of the flow field in the low-speed turbine com-
puted with the preconditioned (PC) and the original scheme (NonPC)
50 3. A Mach number independent solution scheme
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40 45 50
h/
H 
[-]
Axial velocity [m/s]
Measurement
PC(incomp)
PC(comp)
NonPC(comp)
(a)
0
0.2
0.4
0.6
0.8
1
-10 -5 0 5 10 15 20 25 30 35 40
h/
H 
[-]
Circumferential velocity [m/s]
Measurement
PC(incomp)
PC(comp)
NonPC(comp)
(b)
Figure 3.5: Circumferentially averaged velocities over the relative channel height
h/H in the measurement plane M2 (30 mm behind rotor) of the low-
speed turbine
Chapter 4
Development of low-diffusive
discretization schemes for
preconditioned systems
The baseline solution scheme in ITSM3D applies a central discretization of
the fluxes. To stabilize the solution scheme and to damp numerical oscil-
lations, the artificial dissipation technique of Jameson, Schmidt and Turkel
(JST) [131] is applied, which uses a blend of second and fourth order differ-
ences of the solution variables. This solution scheme is robust and suited
for the simulation of transonic flows through turbomachinery [147, 190]. To
improve the overall solution scheme and to ensure an accurate resolution of
the flow field at arbitrary Mach numbers without numerical overshoots of
the state variables, various alternative dissipation schemes have been imple-
mented in the flow solver ITSM3D as a part of the current work.
In this chapter the schemes developed for the construction of the artificial
dissipation are described and their extension to preconditioning are discussed.
Common for the formulated dissipation models is that they can be applied for
the simulation of fluids with arbitrary equations of state at all flow speeds.
4.1 Scalar JST-scheme with TVD properties
The scalar dissipation scheme of Jameson, Schmidt and Turkel (JST) is the
baseline scheme in ITSM3D and was described in the previous chapters. It
is a popular scheme since it is conceptually simple and very robust. How-
ever, in its orginal formulation it does not ensure monotonicity of the solution
and causes overshoots in the state variables in the vicinity of shocks, contact
discontinuities and boundaries layers. The overshoots of the flow solution
near shocks can be avoided using the improved pressure switch developed by
Swanson and Turkel [268], which reads as:
νξ,i,j,k =
|pi−1,j,k − 2pi,j,k + pi+1,j,k|
(1− χ)PJST + χPTVD + d , (4.1)
51
52 4. Low-diffusive discretization schemes
with
PJST = |pi+1,j,k + 2pi,j,k + pi−1,j,k|,
PTVD = |pi+1,j,k − pi,j,k|+ |pi,j,k − pi−1,j,k|, (4.2)
and d = 10−5, where the parameter χ ∈ [0, 1] is specified by the user. By
setting the parameter χ to χ=0 the original pressure switch is obtained, and
with χ=1 the solution scheme should, according to Swanson and Turkel [268],
obtain the property of being Total Variation Diminishing (TVD).
4.2 JST-switched matrix dissipation
Even if the use of preconditioning reduces the level of the artificial dissipa-
tion in the momentum equations of the Navier-Stokes equations at low Mach
numbers, the accuracy of the solution scheme can be improved further by ap-
plying a matrix formulation of the dissipation. While the matrix dissipation
formulation in conjunction with preconditioning proposed by other authors
are limited to incompressible flow [51] or ideal gas mixtures [263], the scheme
presented in the following is formulated without any limiting assumption re-
garding the state equation of the fluid.
In contrast to the scalar scheme of Jameson et al. [131], in which the spec-
tral radius of the flux Jacobian is used for scaling the dissipation terms of each
equation, in the matrix dissipation approach each component of the dissipa-
tion vector is in effect scaled individually. Formally, the matrix dissipation for-
mulation is obtained from the scalar scheme by simply replacing the volume
weighted spectral radii ρ¯′Vξ in Eq. (3.26) with the volume weighted absolute
Jacobian matrix |A¯VΓ,ξ|, so that the second and fourth difference operators are
calculated according to
D
(2)
ξ = δ
−
ξ
(
Γv|A¯VΓ,ξ|ε(2)
)
i+1/2,j,k
δ+ξ ,
D
(4)
ξ = δ
−
ξ
(
Γv|A¯VΓ,ξ|ε(4)
)
i+1/2,j,k
δ+ξ δ
−
ξ δ
+
ξ .
(4.3)
The volume-weighted absolute Jacobian is defined as
|A¯VΓ,ξ| = L−1Γ,ξ · |Λ¯
′V
ξ | · LΓ,ξ (4.4)
with Lξ as the matrix of left eigenvectors for the Jacobian of the inviscid flux
vector in ξ-direction and
|Λ¯′Vξ | = Diag. (|λ¯
′V
ξ,1|, . . . , |λ¯
′V
ξ,5|) (4.5)
as the corresponding diagonal matrix of the absolute, volume-weighted, anisotrop-
ically scaled eigenvalues. The applicable eigenvectors and the product of the
absolute Jacobian with the difference vector δQv are given in Appendix F.
As indicated by the use of overbar-quantities in the Eqs. (4.3) to (4.5), in
order to increase the robustness of the matrix dissipation scheme, the volume-
weighted eigenvalues are scaled anisotropically in the same way that the spec-
tral radii were scaled in the scalar scheme.
4.3 Strengths and weaknesses of JST-switched schemes 53
Near stagnation points and sonic lines, some of the eigenvalues of the Ja-
cobian |A¯VΓ,ξ| approach zero. To avoid vanishing dissipation terms for such
situations, the elements of the diagonal eigenvalue matrix are limited as sug-
gested by Swanson and Turkel [268] as follows:
|λ¯′Vξ,1,2,3| = Θξ ·max(VlσVξ , |λ′Vξ,1,2,3|),
|λ¯′Vξ,4,5| = Θξ ·max(VnσVξ , |λ′Vξ,4,5|),
(4.6)
with σVξ as the volume weighted spectral radius and Θξ as the coefficient for
anisotropic scaling, which is defined in Eq. (2.17). Brodersen [44] has tested
the influence of the parameters Vl, Vn for a matrix dissipation scheme for the
unpreconditioned Navier-Stokes equations; as a good compromise between
stability and robustness he suggests that these parameters should be set to
0.3 and 0.4, respectively. In the current work these proposed values were
adopted.
It should be noted that if all eigenvalues were equal, the corresponding Ja-
cobian matrix |A¯VΓ,ξ| would be a diagonal matrix. Since three eigenvalues are
equal and the remaining two only differ by the sign of the pseudoacoustic ve-
locity, the resulting product of the Jacobian matrix and the first or third order
differences is not very complex, cf. Appendix F where the product |A¯VΓ,ξ| · δQv
is given. In agreement with the findings of Brodersen, the matrix dissipa-
tion scheme only results in a negligible overhead in CPU-time per time step
compared to the scalar dissipation.
4.3 Strengths and weaknesses of JST-switched
schemes
The scalar and the matrix-dissipation JST-schemes are simple, they are ro-
bust [254] and have been implemented in a variety of flow solvers (e.g., [23,
108, 126, 201, 279]).
Both the scalar and the matrix JST-switched dissipation schemes exhibit
deficiencies due to the use of a blend of second and fourth order differences of
the solution variables for the dissipation terms. For instance, in the matrix
dissipation scheme the differences (δQv)i+1/2,j,k = Qv,i+1,j,k −Qv,i,j,k are scaled
with the absolute Jacobian |Aξ|i+1/2,j,k for building the second order difference
dissipation terms. If the absolute Jacobian is calculated from the Roe-average
states of the left "L" (i, j, k) and right "R" (i+1, j, k) side of the interface and an
appropriate central discretization of the convective fluxes is used, the result
is a first order upwind scheme. Thus, the second order difference dissipation
terms are physically motivated, but when activated, they lead to a degrada-
tion of the order of accuracy of the solution scheme. The fourth order differ-
ence dissipation terms on the other hand are only numerically motivated. The
idea of the fourth order difference dissipation is merely to prevent odd-even
decoupling and to stabilize the central difference scheme. The third order
differences δ3Qv,i+1/2 = Qv,i+2 − 3Qv,i+1 + 3Qv,i − Qv,i−1 for the construction of
54 4. Low-diffusive discretization schemes
the dissipation based on the fourth order differences are also scaled with the
Jacobian |Aξ|i+1/2,j,k. This Jacobian is calculated as a function of the state
variables in the nodes left and right of the interface at i + 1/2, j, k. Thus, not
every state contributing to the third order differences is taken into account
and these terms lack a clear physical motivation. However, these terms have
the advantage that for appropriate values of the scaling parameter k(4) they
do not degrade the second order accuracy of the central differencing scheme.
Jameson [129, 130] has proposed an alternative artificial dissipation model
which combines the advantages of the second and fourth order difference dis-
sipation used in the JST scheme, leads to second order accuracy in smooth
regions of the flow and in addition has the property of providing essentially
monotone solutions. Such a scheme can be obtained when the dissipative flux
is constructed by using only the left and right states of the cell interfaces. In
order to attain a scheme which exhibits a second order accuracy, the left and
right states of the intermediate points are extrapolated from nearby data. To
obtain monotonicity of the scheme, appropriate limiters are used. This scheme
is the topic of the following section.
4.4 Scalar SLIP scheme of Jameson
As an alternative to the JST-switched schemes Jameson [129] proposed the
scalar SLIP scheme, that is computationally and conceptually simple and yet
leads to monotone solutions. In the scalar SLIP scheme the dissipation vector
di+1/2 associated with the flux at the interface between the nodes i and i+1 is
calculated as follows
dξ,i+1/2 =
1
2
αVi+1/2 (Qi+1 −Qi) . (4.7)
Jameson suggests setting the coefficient αV to the absolute value of the mini-
mum eigenvalue of the linearized flux Jacobian A:
αV = min
k
λˆVk , and λˆ
V
k =
{ |λVk |, if λVk ≥ ,
1
2
(
+
λVk
2

)
, if |λk| < , (4.8)
with  as a positive threshold proportional to the speed of sound a that pre-
vents the scheme from admitting stationary expansion shocks that would vi-
olate the entropy condition. Tweedt et al. [279] suggest a similar formulation
for the dissipation in which they also account for cell stretching.
When preconditioning is used, the formulation for the dissipation in Eq. (4.7)
should be substituted with the following formulation
dξ,i+1/2,j,k =
1
2
(Γvα
′V )i+1/2,j,k (Qv,i+1,j,k −Qv,i,j,k) , (4.9)
where Γv represents the preconditioning matrix of Merkle, Qv the vector of
primitive, viscous variables, and the scaling factor α′ defined like α in Eq. (4.8)
4.4 Scalar SLIP scheme of Jameson 55
but based on the eigenvalues of the preconditioned system. The resulting
dissipation vector for the ξ-direction is given by
Dξ,i,j,k = dξ,i+1/2,j,k − dξ,i−1/2,j,k. (4.10)
The corresponding terms Dη and Dζ for the η- and ζ-direction are defined
analogously. The complete dissipation vector Di,j,k for the control volume Vˇi,j,k
is determined by
D = Dξ +Dη +Dζ . (4.11)
In the following, only the dissipation in the ξ-direction will be considered and
the j and k indices will be omitted in the notation for the sake of simplicity.
The discretization scheme formulated so far leads to a scheme which is
of first order accuracy. To reduce the amount of artificial dissipation and to
attain a scheme of second order accuracy, Jameson [129, 130] suggests us-
ing either a symmetric or a upstream limited positive (SLIP / USLIP) one-
dimensional reconstruction procedure. With this procedure, the diffusive flux
is formed by the reconstructed states Qi+1/2,L and Qi+1/2,R at the interfaces of
the dual mesh instead of the values in the neighbouring nodes, Qi and Qi+1,
respectively. The former states are obtained by interpolation of nearby data
from the left and the right side of the interface. This interpolation proce-
dure is subject to limiters to preserve monotonicity in a similar manner to
the reconstruction of the solution in Van Leer’s Monotone Upstream-centered
Schemes for Conservation Laws (MUSCL) [282].
When the SLIP-procedure is applied, each of component of the state vari-
able vector Qv = [q1v , . . . , q5v ]T at the interface i+ 1/2 is determined by
qkv,i+1/2,L = q
k
v,i +
1
2
L(∆qkv,i−1/2,∆q
k
v,i+3/2),
qkv,i+1/2,R = q
k
v,i+1 − 12L(∆qkv,i−1/2,∆qkv,i+3/2),
∀k = 1, . . . , 5 (4.12)
with ∆qkv,i+1/2 = q
k
v,i+1 − qkv,i and the SLIP limiter defined as
L(φ1, φ2) =
1
2
P (φ1, φ2)(φ1 + φ2), P (φ1, φ2) = 1−
∣∣∣∣ φ1 − φ2|φ1|+ |φ2|+ l
∣∣∣∣eq , (4.13)
where eq is a positive power. To obtain a higher order dissipation scheme, the
components of the extrapolated state vectorsQv,i+1/2,L andQv,i+1/2,R at the left
and the right side of the interface i+1/2 are calculated according to Eqs. (4.12)
and (4.13). Then the diffusive flux for the ξ-direction is determined as
dξ,i+1/2,j,k =
1
2
(Γvα
′V )i+1/2,j,k
(
Qv,i+1/2,j,k,R −Qv,i+1/2,j,k,L
)
(4.14)
instead of the prescription in Eq. (4.9). However, in the case that the limiter
function L returns zero, the last formulation in Eq. (4.14) becomes equivalent
to the first order formulation in Eq. (4.9).
In the current work the parameter eq of the SLIP limiter given in Eq. (4.13)
was set to eq = 2. Inspired by the work of Venkatakrishnan [283] on limiters
for unstructured grids, the threshold value l to prevent the denominator from
56 4. Low-diffusive discretization schemes
getting zero for vanishing differences φ1 and φ2 of the state variables was
defined as
l = c‖∆x‖3/2, c ∈ [1, 5], (4.15)
where ‖∆x‖ represents the grid spacing length. If φ1 and φ2 have opposite
signs (which is the case for an extremum), then P (φ1, φ2) vanishes and the
left and right states are simply taken without extrapolation, thereby locally
reducing accuracy of the scheme to first order and preventing overshoots of
the solution variables.
As discussed by Jameson the presence of the threshold quantity l in the
SLIP limiter (4.13) relaxes the Local Extremum Diminishing (LED) property
of the scheme. With the introduction of the threshold quantity the scheme
preserves its second order accuracy at smooth extremes, leads to a faster con-
vergence rates, and it decreases the likelihood of limit cycles in which the lim-
iter interacts unfavorably with the corrections of the solution scheme [129].
It should furthermore be noted that to achieve a local extremum diminish-
ing scheme other limiter-functions L(φ1, φ2) than the one given in Eq. (4.13)
can be used. Jameson [129] shows that any limiter fulfilling the properties
P1 : L(φ1, φ2) = L(φ2, φ1), P2 : L(αφ1, αφ2) = αL(φ1, φ2),
P3 : L(φ1, φ1) = φ1, P4 : L(φ1, φ2) = 0 if φ1 · φ2 < 0, (4.16)
leads to a LED scheme, if used in conjunction with the reconstruction and dis-
sipation scheme described above by Eqs. (4.12) and (4.14). Thus, as a substi-
tute for the SLIP limiter defined in Eq. (4.13) one of the following well-known
limiters
L(φ1, φ2) = S(φ1, φ2) ·min(|φ1|, |φ2|),
L(φ1, φ2) = S(φ1, φ2) · 2|φ1||φ2||φ1|+ |φ2| = max
[
0,
2φ1φ2
|φ1|+ |φ2|
]
,
L(φ1, φ2) = S(φ1, φ2) ·max [min(2|φ1|, |φ2|),min(|φ1|, 2|φ2|)]
(4.17)
with
S(φ1, φ2) =
1
2
[sign(φ1) + sign(φ2)] , (4.18)
i.e., the Min-mod, the van Albada, or Roe’s Superbee limiter could be used. For
a discussion and analysis of limiters, reconstruction procedures, and mono-
tonicity properties, the reader may refer to Appendix E.
It should be remarked that the dissipative fluxes of the the SLIP scheme
effectively consists of a blend of differences of second and fourth order of the
solution variables. From Eqs. (4.12), (4.13) and (4.14) it follows that the k-
th component dkξ,v,i+1/2 of the dissipative flux vector dξ,v,i+1/2 at the interface
i+ 1/2 calculates as
dkξ,v,i+1/2 =
1
2
(Γα′)i+1/2(1− Pki+1/2)∆qkv,i+1/2−
1
4
(Γα′)i+1/2Pki+1/2
(
∆qkv,i−1/2 − 2∆qkv,i+1/2 +∆qkv,i+3/2
)
,
(4.19)
∀ k = 1, . . . , 5 with Pki+1/2 = P (∆qkv,i−1/2,∆qkv,i+3/2). Thus, it is evident that the
dissipation vector Dξ = dξ,v,i+1/2 − dξ,v,i−1/2 is constructed merely by the use of
4.5 Characteristic upwind scheme of Roe 57
second order differences of the solution variables in cases where the limiter
is fully active (P =0), while it is only constructed by fourth order differences
in cases for which the limiter is fully shut off (P = 1) for each variable. A
major difference between the SLIP and the JST scheme, is that in the for-
mer an individual limiter for each solution variable is applied, whereas in the
JST scheme the switching between the dissipation terms based on second and
fourth order differences is only determined by a pressure sensor and is carried
out in the same fashion for all flow equations.
4.5 Characteristic upwind scheme of Roe
The scalar SLIP scheme can according to Jameson [129] resolve shock fronts
over just a few grid nodes. While the scheme is perfectly suited for the dis-
cretization of single equations for transported scalars, for a system of trans-
port equations like the Euler equations, the performance of the scheme can
be improved by carrying out a characteristic decomposition and applying the
SLIP reconstruction and limiting procedure individually for each transport
equation. As shown in Appendix F, if a first order upwind scheme, which is
based on a characteristic decomposition of the solution variables, is applied to
the preconditioned Euler-equations the dissipative fluxes are given by
dξ,v,i+1/2 =
1
2
Γv,i+1/2|AVΓ,ξ|i+1/2(Qv,i+1 −Qv,i) (4.20)
with the absolute Jacobian |AΓ,ξ| determined as
|AΓ,ξ| = L−1ξ,v|Λ
′V
ξ |Lξ,v, AΓ,ξ = Γ−1v
∂Eˆ
∂Qˆ
. (4.21)
Lξ,v represents thereby the matrix of left eigenvectors of the Jacobian matrix
AΓ,ξ for the fluxes in ξ-direction. The matrix |Λ′ξ| is defined as the diagonal ma-
trix of the absolute values of the volume weighted eigenvalues λ′Vξ,i; i = 1, . . . , 5
of the Jacobian AΓ,ξ, i.e., |Λ′Vξ | = Diag.(|λ′Vξ,1|, . . . , |λ′Vξ,5|). It should be noted that
the volume weighted eigenvalues (cf. Eq. (F.38)) and not the eigenvalues based
merely on contravariant velocities need to be taken in order to account for the
area of the interface of the dual cell at i+ 1/2.
A comparison of Eq. (4.20) with Eq. (4.9), shows that the dissipation term
of the first order characteristic upwind scheme only differs from the scalar
SLIP scheme in the sense that the absolute Jacobian matrix |AΓ,ξ| is used for
scaling the dissipative terms instead of the eigenvalue based norm α′.
To attain a second order scheme, the states Qv,i and Qv,i+1 in Eq. (4.20)
for the dissipative flux vector may be replaced with the extrapolated states
Qv,i+1/2,L andQv,i+1/2,R on the respective left and the right side of the interface
at i + 1/2. To determine these states, the SLIP reconstruction and limiting
procedure as described in Eqs. (4.12) and (4.13) can be used.
58 4. Low-diffusive discretization schemes
4.5.1 Roe-averaging
Whether a first or a second order scheme is used, to be able to capture a shock
front as accurately as possible, the absolute Jacobian should be calculated
from the Roe-averaged states. If the Jacobian A = ∂E
∂Q
at an interface i+1/2 is
based on the Roe-averaged [229] state vector Q¯Roev,i+1/2= f(Qv,i+1/2,R,Qv,i+1/2,L),
the following relation holds
E(Qv,i+1/2,R)− E(Qv,i+1/2,L) = A(Q¯Roev,i+1/2) · (Qv,i+1/2,R −Qv,i+1/2,L). (4.22)
If Roe-averages are used, the Hugoniot jump conditions are fulfilled [118] and
as shown in Appendix F a full upwind discretization results. As furthermore
shown theoretically by Jameson [130], if a characteristic upwind scheme is
used where the absolute Jacobian is based on Roe averaged states, shock
fronts can be resolved over three nodes (where only one node is contained
in the interior of the shock front).
4.5.2 Method adopted in the current work
In the current work, the absolute Jacobian |A|i+1/2 is calculated from the Roe-
average of the left Qv,i+1/2,L and right Qv,i+1/2,R states in the case of ideal
gas flow. For incompressible fluids the reference state is calculated by sim-
ple arithmetic averaging of the left and right states. The dissipation operator
for the ξ-direction can be written as
Dξ,i,j,k = δ
−
ξ
[(
Γv|A¯Vξ |
)∣∣
Q¯v,RL
· δ+ξ Qv,RL
]
i+1/2,j,k
(4.23)
with the difference δ+Qv,i+1/2,RL = Qv,i+1/2,R − Qv,i+1/2,L and Q¯v,i+1/2,RL as an
average (Roe or arithmetic) between the left and right states. The dissipation
for the η- and ζ-direction is determined analogously.
The characteristic upwind scheme was implemented in the flow solver
ITSM3D. There the user can choose between the second order SLIP-recon-
struction due to Jameson and a third order κ-extrapolation scheme [118] in
conjunction with a van Albada type of limiter. In the latter case, the proce-
dure used in the work of Raif [221] is followed, in which each of the compo-
nents of the state variable vector Qv = (q1v , . . . , q5v)T at the interface i + 1/2 is
determined by
qkv,i+1/2,L = q
k
v,i +
[
s
4
(f+∆+ + f−∆−)
]
k,i
,
qkv,i+1/2,R = q
k
v,i+1 −
[
s
4
(f−∆+ + f+∆−)
]
k,i+1
with
f−k,i = 1− κsk,i, f+k,i = 1 + κsk,i, κ = 13 ,
∆+k,i = q
k
v,i+1 − qkv,i, ∆−k,i = qkv,i − qkv,i−1,
(4.24)
where the van Albada limiter s at node i is defined as
sk,i = max
[
0,
2∆−k,i∆
+
k,i + l
(∆+k,i)
2 + (∆−k,i)2 + l
]
. (4.25)
4.6 Advected upstream splitting techniques 59
It should be noted that the limiter given in the last equation is a modified
version of the original limiter proposed by van Albada et al. [280]. As seen
in Fig. E.1 where the limiter function is plotted in a Ψ − r diagram (as "van
Albada 2"), the modified limiter is not as the original limiter contained in
Sweby’s second order TVD region. However, since Ψ(1) = 1, the limiter is still
leading to a third order accuracy of the resulting κ-scheme in smooth regions
of the flow. Kermani et al. [139, 140] recommend the use of the modified form,
albeit they refrain scaling the value of κ with the limiter function s.
Whether the SLIP reconstruction procedure or the κ-scheme with the mod-
ified van Albada limiter is used, the absolute Jacobian is calculated without
applying anisotropically scaled eigenvalues (Θξ=1). However, to increase the
robustness of the scheme and to avoid the presence of unphysical expansion
shocks in the numerical solution, the eigenvalues are limited according to
Eq. (4.6) but with the value of the parameters Vl, Vn now both lowered to the
range Vl, Vn ∈ [0.02, 0.1].
4.6 Dissipation scheme based on the Advection
Upstream Splitting Method (AUSM+)
With the characteristic upwind dissipation scheme presented in the last sec-
tion one can resolve stationary shocks with one single interior point [130]
and resolve boundary layers well. However, the performance of the upwind
characteristic splitting scheme can be attained by a less complex scheme that
is not based on a characteristic decomposition of the flow field. The use of
so-called low diffusion schemes such as the AUSM type of schemes devel-
oped by Liou and Steffen [169] have gained popularity and has been imple-
mented in both academic and commercial codes for a wide range of flow prob-
lems [88, 167, 172, 198].
As with other upwindmethods, the interface flux formulas for low-diffusion
flux-splitting schemes can be divided into a central flux plus a dissipation
term. Liou and Steffen [169] show how this can be done for the original
AUSM-scheme. By using a central discretization as a basis Radespiel and
Kroll [220] present a modified version of that scheme. Some of the drawbacks
of the AUSM-scheme have been eliminated by the work of Liou [165], where
he presents a revised scheme called AUSM+ and also makes suggestions how
to implement this scheme into a central differencing code. In a later publica-
tion Edwards and Liou [78] explain how this scheme can be extended for the
use in conjunction with time-derivate preconditioning.
In the current work the suggestions of Liou were followed and his scheme
has been implemented in the central differencing Navier-Stokes code ITSM3D.
As will be seen from results from various test cases, the performance of the
scheme is very encouraging.
In the following sections, the implementation of the AUSM+ scheme in a
central differencing framework is described. Beginning with the original for-
mulation for the sake of completeness, the fluxes of this upwind biased scheme
60 4. Low-diffusive discretization schemes
are then divided into a central plus a diffusive part. While Liou [165] showed
that an approximation of his scheme is equivalent to the CUSP-scheme of
Jameson [128], in this work it is shown in a more in-depth analysis that the
dissipation terms of the AUSM+-scheme can be splitted up in an anti-diffusive
and a diffusive part. Issues regarding preconditioning will be addressed be-
fore the explanation of how to control the anti-diffusive part for increasing
the robustness is given and the implementation details of the scheme are pre-
sented.
4.6.1 The AUSM+ scheme
In order to formulate the original AUSM+ splitting of Liou [166] the precon-
ditioned Euler equations
Γv
∂Qˆv
∂t
+
∂Eˆc
∂ξ
+
∂Fˆc
∂η
+
∂Gˆc
∂ζ
= 0 (4.26)
with
Qˆv = Vˇi,j,kQv, Eˆc = Sξ,xEc + Sξ,φFc + Sξ,rGc,
Fˆc = Sη,xEc + Sη,φFc + Sη,rGc, Gˆc = Sζ,xEc + Sζ,φFc + Sζ,rGc
(4.27)
for a curvilinear coordinate system for which the source terms have been ne-
glected (cf. Anker et al. [6]) are used as a basis. The quantities Sξ,Sη,Sζ denote
the cell surface vectors of the dual mesh in the ξ-,η- and ζ-direction, respec-
tively, and Vˇi,j,k represents the corresponding cell volume. The dual mesh is
defined by connecting the cell centers of the original mesh. In the follow-
ing, only the splitting of the flux vector Eˆc for the i-direction at the interface
i + 1/2, j, k will be considered, the splitting of the fluxes at the other faces
is treated analogously. The subscript "i + 1/2" (or simply "1/2") refers in the
following to this specific interface.
In the AUSM type of schemes, the inviscid flux is split up into a convective
flux and a pressure term
Eˆi+1/2 = m˙i+1/2Φi+1/2 +Pi+1/2 (4.28)
with the state vector Φ defined as Φ = [1, vˆξ, vˆη, vˆζ , h0]T with vˆξ, vˆη, and vˆζ rep-
resenting the contravariant velocity components in the absolute system in ξ-,
η- and ζ-direction of the grid, respectively. If relative systems are considered,
the absolute contravariant velocity vector vˆ = [vˆξ, vˆη, vˆζ ] has to be replaced
with the relative contravariant velocity vector wˆ = [wˆξ, wˆη, wˆζ ].
For the i-direction the state vector Φi+1/2 at the interface Si+1/2 between
node i and i+ 1 of the grid is taken as
Φi+1/2 =
{
Φi, mi+1/2 ≥ 0,
Φi+1, mi+1/2 < 0.
(4.29)
The interface mass flux m˙i+1/2 is given by
m˙i+1/2 = a
S
i+1/2mi+1/2%i+1/2 (4.30)
4.6 Advected upstream splitting techniques 61
where aSi+1/2 denotes the surface weighted speed of sound, which calculates as
aSi+1/2 = ai+1/2 · ‖Sξ‖i+1/2 (4.31)
with ‖Sξ‖i+1/2 representing the projection of the interface of the dual mesh
between the nodes i and i + 1 in the ξ-direction. The interface Mach number
mi+1/2 is defined as
mi+1/2 =M+(M1/2,i) +M−(M1/2,i+1) (4.32)
and the interface density reads as
%i+1/2 =
{
%i, mi+1/2 ≥ 0,
%i+1, mi+1/2 < 0.
(4.33)
The pressure flux vector Pξ,i+1/2 is determined by the following equation
Pξ,i+1/2 =
[
0, pi+1/2Sξ,i+1/2,x, pi+1/2Sξ,i+1/2,φ, pi+1/2Sξ,i+1/2,r, 0
]T (4.34)
with the interface pressure defined as
pi+1/2 = P+(pi)pi + P−(pi+1)pi+1. (4.35)
Before explaining how the Mach numbers Mi,Mi+1 and the splitting func-
tionsM and P are defined it should be noted that the resulting flux in Eq. (4.28)
also can be written in the following form
Eˆi+1/2 = a
S
i+1/2(%iΦim
+
i+1/2 + %i+1Φi+1m
−
i+1/2) +Pi+1/2,
= aSi+1/2(Ψim
+
i+1/2 +Ψi+1m
−
i+1/2) +Pi+1/2,
(4.36)
with the state vector Ψ defined as Ψ = [%, %vˆξ, %vˆη, %vˆζ , %h0]T and the positive
and negative interface Mach numbers m±i+1/2 given as
m±i+1/2 =
1
2
(mi+1/2 ± |mi+1/2|). (4.37)
The Mach numbers M1/2,i,M1/2,i+1 for determination of the interface Mach
number mi+1/2 according to Eq. (4.32) are defined as
M1/2,i =
vˆξ,1/2,i
ai+1/2
, vˆξ,1/2,i =
vi · Sξ,i+1/2
‖Sξ,i+1/2‖ , (4.38)
and
M1/2,i+1 =
vˆξ,1/2,i+1
ai+1/2
, vˆξ,1/2,i+1 =
vi+1 · Sξ,i+1/2
‖Sξ,i+1/2‖ , (4.39)
respectively, with v = [vx, vφ, vr] representing the velocity vector in the cylin-
drical x − φ − r coordinate system. The quantity Sξ,i+1/2 = [Sξ,x, Sξ,φ, Sξ,r]i+1/2
represents the surface vector of the cell interface at i + 1/2; its components
represent the projections in the x, φ, and r directions, respectively.
62 4. Low-diffusive discretization schemes
The Mach numberM± and pressure splitting functions P± are calculated
as
M± =
{
M±(1), |M | ≥ 1,
M±(4), |M | < 1,
(4.40)
with
M±(1) =
1
2
(M ± |M |), M±(4) = ±
1
4
(M ± 1)2 ± β(M2 − 1)2 (4.41)
and
P± =
{ 1
2
[1± sign(M)] , |M | ≥ 1,
1
4
(M ± 1)2(2∓M)±αM(M2−1)2, |M | ≤ 1, (4.42)
respectively. When monotonicity of the polynomials is required, the parame-
ters have to be chosen within the ranges α ∈ [−3/4, 3/16] and β ∈ [−1/16, 1/2]
as discussed by Liou [165]. As will be clear from the expressions that will be
derived for the diffusive fluxes, increasing the values of the parameters α and
β gives a larger inherent diffusion of the AUSM discretization.
Liou [166] recommends setting the parameters α and β to
α = 3/16, β = 1/8. (4.43)
For a discussion of the setting of the parameters and the properties of the
splitting functions the reader may refer to the work of Liou [166].
According to Liou the interface speed of sound a1/2 may be calculated as a
simple average of the right and left states:
ai+1/2 =
1
2
(ai + ai+1) . (4.44)
In order to be able resolve a stationary shock exactly, Liou [166] shows that
the speed of sound should be calculated as follows
ai+1/2 = min(a˜i, a˜i+1), a˜ = a
∗2/max(a∗, |vˆξ|), (4.45)
where a∗ denotes the critical speed of sound.
4.6.2 Formulation of the AUSM+ scheme for a cell-vertex
scheme
There are several ways to implement the AUSM+ scheme in a finite volume
code. When a cell-centered scheme is used, the implementation of the original
upwind formulation given by Eqs. (4.28) to (4.45) is straightforward. Alterna-
tively, if a cell vertex scheme is used for discretization of the inviscid fluxes,
then the original formulation of Liou [165] can also be applied directly if a
dual mesh with control volumes Vˇi,j,k is constructed. As a further option, the
AUSM+ interface fluxes can as be split up into a central and a diffusive con-
tribution. This is the approach that was used in the current work; the central
part of the fluxes is calculated as described in Section 2.2.1, i. e., the fluxes are
4.6 Advected upstream splitting techniques 63
evaluated at each individual of the four faces of each side of the control volume
Vi,j,k (cf. Fig. 2.1). Opposed to this, the diffusive fluxes are calculated using the
dual mesh using a cell-centered approach; the diffusive fluxes are calculated
from the reconstructed states located on the center of each face of the control
volumes Vˇi,j,k of the dual mesh. This corresponds to the way the central and
the diffusive fluxes of the JST scheme are implemented in ITSM3D [190].
The use of the dual-volume for the calculation of both the central and the
diffusive terms would have the advantage that an exact cancellation between
central and dissipation terms and full upwinding is obtained for supersonic
flows. The approach used here is however favorable when the flow and the
flow phenomena are not aligned with the grid. The separate calculation of
the diffusive and central terms has furthermore the advantage that a hybrid
Runge-Kutta scheme can be used, in which the former terms are not recalcu-
lated in every stage.
In the AUSM+ scheme the convective interface flux given by Eqs. (4.28) or
(4.36) can be split into central and diffusive terms according to
Eˆi+1/2 = Eˆ
CD
c,i+1/2 + Pˆ
CD
i+1/2 − dˆc,i+1/2 − dˆp,i+1/2, (4.46)
where the central terms are defined as
EˆCDc,i+1/2 =
1
2
aSi+1/2(MiΨi +Mi+1Ψi+1),
PCDi+1/2 = p
CD
1/2 ·T, T = [0, Sξ,x, Sξ,φ, Sξ,r, 0]T , pCD1/2 = 12(pi + pi+1).
(4.47)
It should be noted that the vector EˆCDc,i+1/2 represents only a fully central term if
the interface speed of sound ai+1/2 is determined in a central fashion according
to Eq. (4.44). With this definition of the central fluxes and the consistency
property P+(M)+P−(M) = 1 it follows that the diffusion terms related to the
discretization of the pressure term becomes
dˆp,1/2 =
[
1
2
(pi + pi+1)− P+(Mi)pi − P−(Mi+1)pi+1
]
T
= [∆P(Mi+1)pi+1 −∆P(Mi)pi] ·T (4.48)
with the function ∆P(M) = P+ − P− defined as
∆P =
{
sign(M), |M | ≥ 1,
1
2
M [3−M2 + 4α(M2 − 1)2], |M | ≤ 1, (4.49)
which is in agreement with the result of Liou [166].
If the resulting interface Mach number m1/2 is positive-semidefinite, then
m−1/2 is vanishing and the diffusive part of the convective terms is obtained as
dˆc,1/2 =
1
2
aSi+1/2 [MiΨi +Mi+1Ψi+1]− aSi+1/2m+i+1/2Ψi
= aSi+1/2
{
[(1
2
Mi −M+i )Ψi + (12Mi+1 −M−i+1)Ψi+1] +M−i+1(Ψi+1 −Ψi)
}
= aSi+1/2
{
1
2
[|∆M(Mi+1)|Ψi+1 − |∆M(Mi)|Ψi] +M−i+1(Ψi+1 −Ψi)
}
,
(4.50)
64 4. Low-diffusive discretization schemes
with M+i = M+(Mi), M−i+1 = M−(Mi+1). To obtain the last expression, the
consistency propertyM+(M)+M−(M) =M was exploited. The Mach number
function |∆M| is defined as
|∆M| :=M+ −M+ =
{ |M |, |M | ≥ 1,
1
2
(M2 + 1) + 2β(M2 − 1)2, |M | ≤ 1. (4.51)
Similarly, for negative semi-definite interface Mach numbers m1/2 the convec-
tive dissipation becomes
dˆc,1/2 =
1
2
aSi+1/2 [MiΨi +Mi+1Ψi+1]− aSi+1/2m−i+1/2Ψi+1
= aSi+1/2
{[(
1
2
Mi −M+i
)
Ψi +
(
1
2
Mi+1 −M−i+1
)
Ψi+1
]−M+i (Ψi+1 −Ψi)}
= 1
2
aSi+1/2
{
[∆M(Mi+1)Ψi+1 −∆M(Mi)Ψi]−M+i (Ψi+1 −Ψi)
}
.
(4.52)
Since the functionM+ is positive semi-definite and the functionM− is nega-
tive semi-definite for all Mach numbers, the last term in both Eq. (4.50) and
Eq. (4.52), represent an antidiffusive contribution. As will be shown later in
this chapter, it will be useful to divide the convective dissipation vector dˆc,1/2
into an essentially diffusive dˆc,edi,1/2 and an anti-diffusive part dˆc,atd,1/2:
dˆc,1/2 = dˆc,edi,1/2 + dˆc,atd,1/2, (4.53)
with the essentially diffusive part defined as
dˆc,edi,1/2 =
1
2
aSi+1/2 [∆M(Mi+1)Ψi+1 −∆M(Mi)Ψi] (4.54)
and the antidiffusive contribution dˆc,atd,1/2 given by
dˆc,atd,1/2 =
{
+aSi+1/2M−i+1(Ψi+1 −Ψi), ∀m1/2 ≥ 0,
−aSi+1/2M+i (Ψi+1 −Ψi), ∀m1/2 ≤ 0.
(4.55)
For a vanishing interface Mach number m1/2, the Mach numbersMi andMi+1
must be equal in magnitude but opposite in sign. This follows from the sym-
metry property −M−(−M) = M+(M) of the Mach number functionsM±. In
this case, both expressions of the anti-diffusive dissipation term in Eq. (4.55)
are equal. It should however also be noted that in the special case that
the Mach number Mi and Mi+1 are equal in magnitude but are opposite in
sign, the anti-diffusive dissipation fully cancels the dissipation term defined
in Eq. (4.54); however in this case also the central part EˆCDc of the convective
fluxes vanishes.
In order to analyze the convective dissipation term, the Mach number vari-
ation over the interface is neglected, i.e., the validity of M1/2 := Mi ≈ Mi+1 is
assumed. With this supposition, the following relation
dˆc,1/2 = −1
2
aS1/2
∣∣∆M(M1/2)∣∣∆Ψi+aS1/2|N |(M1/2)∆Ψi, ∆Ψi = Ψi+1−Ψi (4.56)
with
|N | = 1
4
(1−Mm)2 + β(M2m − 1)2, Mm = min(|M |, 1) (4.57)
4.6 Advected upstream splitting techniques 65
holds for both positive and negative Mach numbersM1/2.
The first part of the dissipation term corresponds to the convective part of
the dissipation of the CUSP scheme introduced by Jameson [129] if the con-
stant a0 in his scheme is set equal to a0 = 2β +1/2 and his modifications [130]
for resolving stationary shocks sharply and capturing contact discontinuities
are not taken into account. The second anti-diffusive part of the dissipation
vanishes for |M1/2| ≥ 1.
A simple addition of both dissipation terms in Eq. (4.56) results in the
following expression for the convective dissipation
dˆc,1/2 = −1
2
aS1/2
∣∣M1/2∣∣∆Ψi. (4.58)
Since the convective dissipation term for the original AUSM+ scheme van-
ishes for M1/2 = 0, measures must be taken in order to stabilize the scheme
near stagnation points and walls. This point will be addressed in a later sec-
tion.
4.6.3 Preconditioned AUSM+-scheme
In order for the AUSM+ dissipation to scale correctly at low Mach numbers
preconditioning must be used. As shown by Edwards and Liou [78] atten-
tion must thereby be paid to how the dissipative contributions of the fluxes
scale with the speed of sound. Furthermore, pressure-velocity coupling at low
speed flows must be ensured. In this section their work is reviewed and it
is explained which modifications were adopted in the preconditioned AUSM+
scheme that was implemented in the flow solver ITSM3D.
To attain a correct scaling of the flux terms in the low Mach number limit,
the physical Mach numbers Mi and Mi+1 used in the flux splitting must be
substituted with the effective numerical Mach number M ′i and M ′i+1of the
preconditioned Navier-Stokes equations, respectively. These numerical Mach
numbers are given by
M ′i =
vˆξ,i
a′i+1/2
, M ′i+1 =
vˆξ,i+1
a′i+1/2
, (4.59)
where a′ represents the numerical speed of sound and vˆξ the contravariant ve-
locity component in the ξ-direction of the grid. The numerical speed of sound
a′1/2 at the interface i+ 1/2 is defined to be
a′1/2 =
1
2
(a′i + a
′
i+1), a
′ =
√
v · Sξ
(
1− d
d′
)2
+ 4%hT
d′ ‖Sξ‖2
(1 + d
d′ )‖Sξ‖
. (4.60)
For M → 1 preconditioning is successively turned off and d′ → d so that the
numerical speed of sound a′ for |M | ≥ 1 equals the physical speed of sound
a. It is important to be aware that regardless of how a′ may be calculated,
the product of a′ and M ′ must give the physical velocity component vˆξ in ξ-
direction for consistency reasons.
66 4. Low-diffusive discretization schemes
The introduction of a numerical speed a′ and an accordingly defined Mach
number M ′ is not sufficient for using the AUSM+ scheme for the simulation
of low Mach number flow. This will be shown in the following by carrying
out an analysis of the dissipative terms according to Edwards and Liou [78].
Based on this analysis, appropriate measures to enforce the pressure-velocity
coupling will be derived.
To analyse the numerical dissipation resulting from the discretization of
the pressure with the numerical speed of sound a′ and the scaled Mach num-
ber M ′, the effective interface pressure given in Eq. (4.35) is in accordance
with Edwards and Liou [78] for subsonic flows written out as
pi+1/2 =
1
2
(pi + pi+1)︸ ︷︷ ︸
pCD
− 1
2
σp(M
′
i +M
′
i+1)(pi+1 − pi)︸ ︷︷ ︸
dp,1
−
1
2
σp(pi + pi+1)(M
′
i+1 −M ′i)]︸ ︷︷ ︸
dp,2
+O(δp · |∆P|)
(4.61)
with the factor
σp = 3 + 4α+O(M ′2). (4.62)
As long as α > −3/4 the second order terms in the last expression can be
neglected for low Mach numbers. To be able to analyze their scaling with
respect to the Mach number level, the dissipative terms are normalized with
the reference quantities u†, %†, p†. This yields the following expression for the
first dissipation term:
d∗p,1 =
1
2
σp(M
′
i +M
′
i+1)(p
∗
i+1 − p∗i ) = σpM¯ ′ · δp∗. (4.63)
For low Mach number, steady state flow it should be noted that the spatial
pressure difference δp∗ represents a difference of the second order pressure.
These kind of pressure differences are connected to the kinetic, incompress-
ible part of the velocity field. For steady state problems, these differences can
neither be linked to global pressure variations (zeroth order pressure) nor to
temporal pressure fluctuations related to the propagation of acoustic waves
(first order pressure). The reader may refer to Chapter D.1 or to the work of
Müller [194] for a discussion on the role of the zeroth, first and second order
pressure. As elaborated in Appendix A it should also be noted that the use of
time-derivative preconditioning also ensures that the temporal pressure vari-
ations scale in the same order as the advection terms for low Mach number
flow. Thus, the pressure differences δp∗ appearing in Eq. (4.63) are for low
Mach number flow proportional to the variation of the kinetic energy of the
flow field, i.e., δp∗ ∼ δ(%∗v∗2). In this case the dissipation term d∗p,1 obviously
scales correctly in the order of the convective scales:
d∗p,1 = σpM¯
′ · δp∗ ∼ M¯ ′ · δp∗ ∼ M¯ ′ · δ(%∗v∗2) = O(1). (4.64)
The term dp,2 in Eq. (4.61) can for ideal gas flow be written as
d∗p,2 =
1
2
σp(p
∗
i + p
∗
i+1)(M
′
i+1 −M ′i) ∼ σp%¯∗a¯∗2δM ′ = σp%¯∗/M¯2 · δM ′. (4.65)
4.6 Advected upstream splitting techniques 67
Since δM ′ scales in the order of unity, the dissipation term d∗p,2 scales inversely
proportional to the square of the Mach number, i.e., d∗p,2 = O(M¯−2).
In order to be able to treat each of the parts dp,1 and dp,2 of the pressure
diffusion separately and so alleviate the wrong scaling of the second part of
the pressure dissipation term, the following Mach numbers
Mˇ ′i =
1
2
[
(1 + M¯2r )M
′
i + (1− M¯2r )M ′i+1
]
,
Mˇ ′i+1 =
1
2
[
(1− M¯2r )M ′i + (1 + M¯2r )M ′i+1
] (4.66)
as suggested by Edwards and Liou [78] should be used for the flux splitting,
where M¯r represents a reference Mach number. Generally, the reference Mach
number has to be calculated locally as the ratio of the pseudoacoustic velocity
a′ to the physical sonic speed a:
M¯r = a¯
′/a¯. (4.67)
The use of a Mach number defined by Eq. (4.66) and (4.67) for the flux splitting
makes both parts of the pressure dissipation terms scale correctly for low
Mach number, ideal gas flow:
d∗p,1 =
1
2
σp(Mˇ
′
i + Mˇ
′
i+1)(p
∗
i+1 − p∗i ) = 12σp(M ′i +M ′i+1)(p∗i+1 − p∗i )
∼ M¯ ′ · δp∗ ∼ M¯ ′ · δ(%∗u∗2) = O(1),
d∗p,2 =
1
2
σp(p
∗
i + p
∗
i+1)(Mˇ
′
i+1 − Mˇ ′i) = 12σp(p∗i + p∗i+1)(M ′i+1 −M ′i) · M¯2r
∼ p¯∗M¯2r · δM ′ ∼ %¯∗a¯∗2M¯2r · δM ′ ∼ %¯∗M¯2r /M¯2 · δM ′ = O(1).
(4.68)
For an incompressible fluid or compressible flow in the low Mach num-
ber limit the reference Mach number goes to zero, hence the left and right
Mach numbers Mˇ
′
i and Mˇ
′
i+1 become identical. Thus, in this case the expres-
sion in Eq. (4.58) for the convective dissipation is exact, since the assumption
of a vanishing Mach number variation over the interfaces is fulfilled. This
means that the convective dissipation terms completely vanish as the velocity
magnitude approaches zero, which is a desirable property for resolving con-
tact discontinuities or boundary layers in special. However, this property is
responsible for raising the possibility of odd-even decoupling with vanishing
flow speed. To alleviate this problem Edwards and Liou [78] suggest adding
one element of the pressure diffusion used in the AUSMDV scheme to en-
hance the pressure-velocity coupling at low speeds. In this case the pressure
diffusion term m˙D,i+1/2 defined as
m˙D,i+1/2 = −aSi+1/2|∆D(Mi,Mi+1)|
pi+1 − pi
pi/%i + pi+1/%i+1
(4.69)
with
|∆D(Mi,Mi+1)| = D+ −D−, D+ =M+(4) −M+(1), D− =M−(4) −M−(1) (4.70)
and β = 1/8 is added to the interface mass flux defined in Eq. (4.30), so that
the resulting interface mass flux reads as
m˙i+1/2 = a
S
i+1/2(Ψim
+
i+1/2 +Ψi+1m
−
i+1/2) + m˙D,i+1/2. (4.71)
68 4. Low-diffusive discretization schemes
The function |∆D| for β = 1/8 becomes
|∆D| = 3/4− 1
2
(|Mm,i|+ |Mm,i+1|)+ 1
8
(M4m,i+M
4
m,i+1), Mm = min(1,M). (4.72)
The function |∆D| attains its maximum value of 3/4 for vanishing Mach num-
bersMi,Mi+1 and reduces to zero for supersonic flows.
In the low Mach number limit the normalized pressure diffusion term
scales as
m˙∗D,i+1/2 = −
3
4
a∗Si+1/2
p∗i+1 − p∗i
p∗i /%
∗
i + p
∗
i+1/%
∗
i+1
∼ 1
a¯∗
δp∗ ∼ 1
a¯∗
δ(%∗u∗2) = M¯δ(%∗u∗2),
(4.73)
which means that this term will not provide any effect for low Mach numbers
if not scaled differently.
When preconditioning is used, the interface speed of sound ai+1/2 is re-
placed with the numerical speed of sound a′i+1/2, which for low Mach number
flow scales in the order of the convective velocity v. Thus, when precondition-
ing is used for the simulation of low Mach number flow, the pressure diffusion
term scales as m˙∗D,i+1/2 ∼ v¯
∗
a¯∗2 δ(%
∗u∗2) = M¯2δ(%∗u∗2). To achieve a correct scal-
ing, the pressure diffusion term has to be scaled with 1/M2r with Mr defined
according to Eq. (4.67). To also ensure that the pressure diffusion terms are
shut off rapidly enough with an increasing Mach number level, Edwards and
Liou suggests scaling the pressure term with the factor dM = (1 −M2r )/M2r .
With this scaling the pressure diffusion term reads as follows
m˙D,i+1/2 = −a′Si+1/2
1−M2r,i+1/2
M2r,i+1/2
· |∆D(M ′i ,M ′i+1)| ·
pi+1 − pi
pi/%i + pi+1/%i+1
. (4.74)
For low Mach number flow the term m˙D,i+1/2 scales in dimensional form in the
order of the variations of the mass flux (O[δ(%v)]); in dimensionless form this
term scales in the order of unity, which is correct.
In the case of incompressible flow the scaling term (1−M2r )/M2r is not used
since there is no relation between the terms p/ρ in Eq. (4.69) and the sonic
speed, i. e., these terms must not be compensated for anymore in order to
obtain a correct scaling of the diffusion terms. Since an incompressible flow
field is not affected by the magnitude of the pressure but only by spatial or
temporal variations in the pressure, the pressure difference term in Eq. (4.69)
should be normalized with the local value for the dynamic pressure pdyni =
(%v2)i. Thus, for incompressible flow, the pressure diffusion term is calculated
as follows
m˙D,i+1/2 = −
a′i+1/2
v2i+1/2
· |∆D(M ′i ,M ′i+1)| · ‖Sξ‖ · (pi+1 − pi) . (4.75)
For incompressible flows, the Mach numberM ′ based on the numerical speed
of sound a′ is constant as long as the reference velocity Vr used for scaling
the preconditioning parameter %′p is based on the local velocity scale, i.e.,
4.6 Advected upstream splitting techniques 69
a′ =
√
5v2 and M ′ = 1/
√
5. In this case the Mach number function |∆D|
can be approximated to |∆D| ≈ 0.31. With a′i+1/2 scaling in the order of the
magnitude of convective velocity v, the pressure diffusion term, which is de-
fined in Eq. (4.75) for constant density flow, scales in the order of the mass
flux variations in the flow field, which is appropriate.
4.6.4 Robustness issues
With the original formulation of the anti-diffusive part of the convective dis-
sipation as given in Eq. (4.55), the net dissipation vanishes completely in the
low Mach number limit. From a theoretical viewpoint this is ideal but is
detrimental to the robustness of the scheme. In order to ensure stability of a
given solution scheme for practical problems, some dissipation should always
be added to the central terms. This is accomplished by a proper scaling of the
anti-diffusive term dˆc,atd,1/2.
To obtain the best compromise between robustness and low-diffusivity of
the solution scheme on stretched grids, the applied part dˆaplc,atd,1/2,ξ of the anti-
diffusive dissipation for the ξ-direction of the mesh is determined as
dˆaplc,atd,1/2,ξ = ω˜AUSM,ξ · dˆc,atd,1/2,ξ (4.76)
with the scaling parameter ω˜AUSM defined as
ω˜AUSM,ξ = 1− ωAUSM · γ1/2,ξ. (4.77)
The term ωAUSM is a user defined parameter (ωAUSM ∈ [0, 1]) and the second
quantity γ1/2,ξ is given by
γ1/2,ξ = min
[
1,max
(
ση
σξ
,
σζ
σξ
)]
(4.78)
as a function of the spectral radii σξ, ση and σζ of the inviscid flux Jacobians
in ξ-, η- and ζ-direction, respectively. The scaling of the anti-diffusive terms
in the η- and ζ-direction is given analogously. This approach is similar to that
one used by Radespiel and Kroll [216] for the regular AUSM-scheme with
the difference that in the current work the anti-diffusive term is controlled
while they limit the Mach number used as argument for the dissipation func-
tion. If the parameter ωAUSM is set to zero, the original AUSM+ dissipation
is recovered. For finite values of this parameter, the amount of anti-diffusion
added can be controlled, which improves the damping behaviour of the solu-
tion scheme but does not degrade the ability of the original dissipation scheme
for resolving boundary layers accurately.
With this scaling, the anti-diffusive terms are primarily contributing to
the dissipation at the largest faces of the cells. In viscous flow simulations,
the grid is usually clustered in the wall normal direction. If the ξ-direction
of the grid corresponds to this direction, the cell face area ‖Sξ‖ will typically
be larger than the cell face areas ‖Sη‖ and ‖Sζ‖. Since the variations of the
70 4. Low-diffusive discretization schemes
(a) (PC), χ = 0.01 (b) (PC), χ = 0.99
Figure 4.1: Calculated Mach number contours (∆M = 0.025) of the transonic Ni-
Bump test case (Mis = 0.85) using the preconditioned JST-scheme with
the original [131] and the modified pressure switch [268]
velocities within a cell are negligible, the corresponding spectral radius σξ
will consequently be larger than ση and σζ . Thus, anti-diffusive terms will be
reduced in grid directions parallel to the wall but will be retained in the wall-
normal direction so that the original non-smearing behaviour of the AUSM
scheme is preserved.
4.7 Computational results
4.7.1 Ni-Bump test case
To demonstrate the effect of the improved pressure switch of Swanson and
Turkel, which is described in Chapter 4.1, the inviscid flow in the Ni-Bump
test case [226] was simulated for an isentropic Mach number of Mis,a = 0.85.
A description of the grid and the boundary conditions used, can be found in
Chapter 3.4.1, where the test case was used to assess the preconditioning
scheme.
In Fig. 4.1 the calculated Mach number contours are shown for two differ-
ent values of the parameter χ of the pressure switch defined in Eq. (4.1). It
should be noted that when the parameter χ of the modified pressure switch is
set to unity the solution scheme should obtain TVD-properties [268]. The new
pressure switch reverts to the original one for a value χ=0. From Fig. 4.1a it
is apparent that a value of χ=0 for the pressure switch leads to overshoots of
the pressure around the shock. With a full activation of the modified form of
the pressure switch (χ=0.99), the wiggles in the vicinity of the shock are pre-
vented as can be seen from Fig. 4.1b. Obviously, the modified pressure switch
is a suitable means to ensure monotone pressure distributions in transonic,
inviscid flow simulations with the JST scheme.
4.7.2 Flat plate laminar boundary layer
As a test case for assessing the ability of the different dissipation schemes to
resolve viscous flows, a laminar boundary layer developing over a flat blade at
4.7 Computational results 71
zero incidence was chosen. In the investigated set-up of the case the compu-
tational domain is rectangular with the inflow and outflow boundary located
directly at the leading edge and the trailing edge of the flat plate, respectively.
Figure 4.2 provides an overview over the geometry modeled and the boundary
conditions applied. The grids used consist of 49×3×73 and 97×3×145 grid
points in x-, φ- and r-direction, respectively. The mesh points in the stream-
wise direction and in the direction normal to the wall were clustered with a
stretching factor of 1.015% in order to provide an adequate resolution of the
boundary layer and of the flow near the leading edge.
The results of the simulations in which the fluid was modeled as incom-
pressible and those in which the fluid was modeled as an ideal gas were
nearly identical. For the sake of brevity, only the latter ones are presented
in the following. The ratio of the total inlet pressure p0,e to the to the static
outlet pressure pa was chosen to give a low isentropic outlet Mach number
(Mis,a=0.136). Thus, the flow field can be considered as incompressible, which
makes a comparison of the computed variables with the theoretical Blasius
solution meaningful.
In Figs. 4.3 and 4.4 the simulated dimensionless tangential velocity u∗ =
u/U∞ is shown as a function of the normalized wall distance η = y
√
Rex/x
gained with the different dissipation schemes on the fine grid. The velocity
distributions are evaluated at the end of the plate; the numerical solution
obtained with the current grid is self-similar from the middle of the plate and
onwards in the downstream direction. Plots of the solutions obtained with the
AUSM type dissipation and Roe’s Characteristic Upwind Dissipation scheme
are given in Fig. 4.3a. The free parameters of these schemes were defined
such that these schemes yield low-diffusive solutions; the actual values of
the parameters are listed in Table 4.1. Evidently, these schemes are able to
reproduce the Blasius solution quite well.
In order to distinguish more clearly between different dissipation schemes
with regard to their resolution capability, simulations were also run on the
coarser grid. The corresponding results are displayed in Figs. 4.5 to 4.8.
Obviously, the AUSM dissipation and Roe’s Characteristic Upwind Dissi-
pation scheme exhibit the best performance of all dissipation models consid-
ered in the current work, when the parameters of these schemes are set such
that corresponding dissipation terms almost vanish as the Mach number ap-
proaches zero. In contrast to the results produced by the dissipation schemes
based on fourth order differences (JST and JSTMatd), the AUSM dissipation
scheme and the Characteristic Upwind Dissipation scheme do not result in an
overshoot of the normalized tangential velocities. However, even though the
Matrix Dissipation Scheme fails to provide a non-monotone solution, it clearly
outperforms the scalar JST-scheme.
From Fig. 4.6 it is seen that the preconditioned JST-scheme obviously ex-
hibits a less diffusive behaviour than the non-preconditioned one, even though
a higher value for k(4) was used in the preconditioned solution scheme.
In Figs. 4.8a and 4.8b the solutions obtained with the scalar JST scheme
for the extreme values χ = 0.01 and χ = 0.99 (TVD) of the pressure switch
72 4. Low-diffusive discretization schemes
x
φ
r, y
No-slip wall
Slip wall
H = 25mm
pa = 1000mbar
p0,e = 1013mbar
T0,e = 296.15K
∆φ = 0.36◦
Rw = 10295mm
l = 50mm
Figure 4.2: Geometry and boundary conditions of the flat plate laminar boundary
layer test case
Scheme Parameters Limiter
JSTLoD(NonPC) k(4)=0.015 -
JST(NonPC) k(4)=0.03 -
JSTLoD k(4)=0.03 / k(4)=0.015 (TVD) -
JST k(4)=0.05 -
JSTMatdLoD k(4)=0.015, Vl=0.3, Vn=0.4 -
JSTMatd k(4)=0.05, Vl=0.3, Vn=0.4 -
JSTMatdHiD k(4)=0.05, Vl=0.5, Vn=0.5 -
CharUpwDissLoD Vl = 0.02, Vn = 0.02 v. Albada
CharUpwDiss Vl = 0.1, Vn = 0.1 v. Albada
AUSM ωAUSM=0.5,α = 3/16,β=1/8 v. Albada
AUSMLoD ωAUSM=0.1,α=3/16,β=−1/16 v. Albada
Table 4.1: Values of the parameters used in the dissipation schemes
parameter are depicted. The pressure switch has only a small effect on the
solution since the pressure gradient normal to the wall is negligible. This
shows that a switch based on the pressure alone is not suitable for all kinds
of flows.
4.7.3 Supersonic flow through a row of wedges
To validate the preconditioned solution scheme for supersonic flow and to as-
sess the shock resolution capability of the different dissipation schemes, the
supersonic flow through a row of wedges was simulated.
The geometry and the boundary conditions of the test case are given in
Fig. 4.9. As indicated, air flow enters the configuration with a Mach number of
Me=2. With the assumption that the flow is inviscid, it can be shown by using
basic aerodynamical arguments, that an oblique shock starts at the tip of each
wedge, hits the lower wall of the neighbouring wedge and is reflected back to
the wedge from which it originated. Under the given boundary conditions the
4.7 Computational results 73
reflected shock hits the corner of the wedge and gets canceled there.
The computational grid is shown in Fig. 4.10 and it consists of 169×3×33
grid points. With the prescription of an inlet Mach number of Me = 2 the sim-
ulated shock impinged on the wall behind the corner, which led to additional
reflections of the shock in the channel between the wedges. To compensate for
numerical dissipation and to ensure that the shock actually hits the corner
and gets annihilated there, the inlet Mach number used in the simulations
was lowered to Me=1.985.
In Fig. 4.11 the simulated static pressure contours obtained by using the
JST scheme (χ= 0.01) with and without preconditioning are plotted for a ra-
dial height of r=1060mm in dependence of the axial position. The difference
between the results is marginal. Thus, it is once more shown that precondi-
tioning does not negatively affect the shock capturing properties of the JST-
scheme.
A comparison of the solutions obtained with the JST-scheme and the JST-
based matrix dissipation for the extreme values χ=0.01 and χ=0.99 (TVD) of
the pressure switch parameter is given in Fig. 4.12. Obviously, the solutions
obtained with the scalar JST-scheme and with the JST-switched Matrix Dis-
sipation scheme are monotone when the pressure switch parameter is set to
χ=0.99. The use of the original pressure switch (χ=0.01) leads to a less diffu-
sive behaviour and larger gradients in the solution, but results in unphysical
oscillations. It should be noted, that the robustness of the solution scheme
improves with an increasing value of this parameter; when the matrix dissi-
pation in this test case was used with χ=0.01 the other parameters Vl, Vn and
k(4) had to be increased in order to obtain a stable integration. Nevertheless,
independent of the setting of the pressure switch parameter χ, the matrix dis-
sipation scheme clearly leads to a better resolution of shocks than the scalar
scheme does.
In Fig. 4.13 the results obtained with the Characteristic Upwind Dissipa-
tion scheme of Roe, the AUSM dissipation scheme and the matrix dissipation
scheme are compared. Of the schemes compared, the JST-switched matrix
dissipation leads to the largest gradients, but at the cost that oscillations are
produced in the solution. The Roe as well as the AUSM dissipation scheme
lead to a nearly monotone solution; both schemes are less diffusive than the
matrix-dissipation scheme when the parameter of the pressure switch is set
to χ=0.99.
For this test case, the AUSM dissipation model led to the best compromise
between robustness, shock resolution ability and monotonicity of the solution
scheme. For a comparison of the most and the least diffusive scheme that
did not lead to unphysical oscillations in the vicinity of the shock, the static
pressure field obtained with the AUSM+ dissipation scheme and the solution
gained with the scalar JST-scheme (TVD) are plotted in Fig. 4.14.
74 4. Low-diffusive discretization schemes
4.7.4 Low-speed turbine
To further assess the performance of the dissipation schemes, the low-speed
multistage turbomachinery test case already presented in Chapter 3.4.4 was
simulated using the various dissipation schemes presented above.
In Figs. 4.15a and 4.15b the results for the JSTMatrix Dissipation scheme,
Roe’s Characteristic Upwind Dissipation scheme, and the AUSM+ dissipa-
tion are compared. These schemes have been used in conjunction with time-
derivative preconditioning where the fluid was modelled as an ideal gas. The
capability of these schemes to resolve the flow features are better than that of
the preconditioned scalar JST scheme; for the results obtained with the latter
scheme with and without the use of preconditioning, the reader may refer to
Chapter 3.4.4.
In this case, the ability to resolve secondary flow effects appears to be sim-
ilar for the JST-switched matrix dissipation (χ=0.99), the AUSM dissipation,
and the Characteristic Upwind Dissipation scheme. However, as can be seen
from Fig. 4.16, the Characteristic Upwind Dissipation scheme and the AUSM
dissipation scheme, which both are applying limiters, do not generate over-
shoots of the solution variables in the vicinity of the boundary layers like the
JST-type of schemes do. Thus, it can also for this test case be concluded that
the upwind-biased dissipation schemes are superior to the ad hoc JST dissi-
pation techniques.
4.7 Computational results 75
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8
u
/U
in
f [-
]
eta [-]
Blasius
AUSMLoD
CharUpwDissLoD
(a)
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8
u
/U
in
f [-
]
eta [-]
Blasius
JSTMatd
JST(NonPC)
JST
(b)
Figure 4.3: Laminar flow over a flat plate: Simulated normalized tangential velocity
u∗ = u/U∞ over the normalized wall distance η = y
√
Rex/x in depen-
dence of the dissipation scheme on the fine grid with 97×3×145 nodes
76 4. Low-diffusive discretization schemes
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5
u
/U
in
f [-
]
eta [-]
Blasius
JSTMatd
JST(NonPC)
JST
AUSMLoD
CharUpwDissLoD
(a) Close-up
Figure 4.4: Laminar flow over a flat plate: Simulated normalized tangential velocity
u∗ = u/U∞ over the normalized wall distance η = y
√
Rex/x in depen-
dence of the dissipation scheme on the fine grid with 97×3×145 nodes
4.7 Computational results 77
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14 16
u
/U
in
f [-
]
eta [-]
Blasius
AUSMLoD
AUSM
CharUpwDissLoD
CharUpwDiss
(a)
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5
u
/U
in
f [-
]
eta [-]
Blasius
AUSMLoD
AUSM
CharUpwDissLoD
CharUpwDiss
(b) Close-up
Figure 4.5: Laminar flow over a flat plate: Simulated normalized tangential velocity
u∗ = u/U∞ over the normalized wall distance η = y
√
Rex/x in depen-
dence of the dissipation scheme on the coarse grid with 49×3×73 nodes
78 4. Low-diffusive discretization schemes
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14 16
u
/U
in
f [-
]
eta [-]
Blasius
JST
JST(NonPC)
JSTMatd
(a)
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5
u
/U
in
f [-
]
eta [-]
Blasius
JST
JST(NonPC)
JSTMatd
(b) Close-up
Figure 4.6: Laminar flow over a flat plate: Simulated normalized tangential velocity
u∗ = u/U∞ over the normalized wall distance η = y
√
Rex/x in depen-
dence of the dissipation scheme on the coarse grid with 49×3×73 nodes
4.7 Computational results 79
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14 16
u
/U
in
f [-
]
eta [-]
Blasius
AUSM
CharUpwDiss
JSTMatd
(a)
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5
u
/U
in
f [-
]
eta [-]
Blasius
AUSM
CharUpwDiss
JSTMatd
(b) Close-up
Figure 4.7: Laminar flow over a flat plate: Simulated normalized tangential velocity
u∗ = u/U∞ over the normalized wall distance η = y
√
Rex/x in depen-
dence of the dissipation scheme on the coarse grid with 49×3×73 nodes
80 4. Low-diffusive discretization schemes
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14 16
u
/U
in
f [-
]
eta [-]
Blasius
JST(TVD)
JST
(a)
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5
u
/U
in
f [-
]
eta [-]
Blasius
JST(TVD)
JST
(b) Close-up
Figure 4.8: Laminar flow over a flat plate: Simulated normalized tangential velocity
u∗ = u/U∞ over the normalized wall distance η = y
√
Rex/x in depen-
dence of the dissipation scheme on the coarse grid with 49×3×73 nodes
4.7 Computational results 81
150mm 175.77mm 150mm
∆φ = 0.36◦
Periodic boundary Slip wall
10.0◦
φ pa=358.32mbar
p0,e=1000mbar
T0,e=518.15K
Mae=2 r
x
Rw,o=1087.54mm
Rw,u=1000mm
Air
Figure 4.9: Geometry and boundary conditions of the test case where the supersonic
flow through a row of wedges is simulated
Figure 4.10: Computational grid for the simulation of the supersonic flow through a
row of wedges
82 4. Low-diffusive discretization schemes
0.1
0.15
0.2
0.25
0.3
0.35
0.4
200 220 240 260 280 300 320
p 
[m
ba
r]
x [mm]
Analytical
JSTLoD(NonPC)
JSTLoD(PC)
(a)
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.21
0.22
0.23
200 205 210 215 220 225 230 235
p 
[m
ba
r]
x [mm]
Analytical
JSTLoD(NonPC)
JSTLoD(PC)
(b) Close-up
Figure 4.11: Supersonic flow through a row of wedges: Simulated static pressure at
r = 1060mm in dependence of the dissipation scheme
4.7 Computational results 83
0.1
0.15
0.2
0.25
0.3
0.35
0.4
200 220 240 260 280 300 320
p 
[m
ba
r]
x [mm]
Analytical
JSTLoD
JSTLoD(TVD)
JSTMatdHiD
JSTMatdLoD(TVD)
(a)
0.12
0.14
0.16
0.18
0.2
0.22
0.24
200 205 210 215 220 225 230 235
p 
[m
ba
r]
x [mm]
Analytical
JSTLoD
JSTLoD(TVD)
JSTMatdHiD
JSTMatdLoD(TVD)
(b) Close-up
Figure 4.12: Supersonic flow through a row of wedges: Simulated static pressure at
r = 1060mm in dependence of the dissipation scheme
84 4. Low-diffusive discretization schemes
0.1
0.15
0.2
0.25
0.3
0.35
0.4
200 220 240 260 280 300 320
p 
[m
ba
r]
x [mm]
Analytical
JSTMatdHiD
JSTMatdLoD(TVD)
AUSMLoD
CharUpwDissHiD
(a)
0.12
0.14
0.16
0.18
0.2
0.22
0.24
200 205 210 215 220 225 230 235
p 
[m
ba
r]
x [mm]
Analytical
JSTMatdHiD
JSTMatdLoD(TVD)
AUSMLoD
CharUpwDissHiD
(b) Close-up
Figure 4.13: Supersonic flow through a row of wedges: Simulated static pressure at
r = 1060mm in dependence of the dissipation scheme
4.7 Computational results 85
(a) JSTLoD(TVD)
(b) AUSMLoD
Figure 4.14: Static pressure contours (∆p = 0.01mbar) of the simulated supersonic
flow field through a row of wedges in dependence of the dissipation
scheme
86 4. Low-diffusive discretization schemes
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40 45 50
h/
H 
[-]
Axial velocity [m/s]
Measurement
JSTMatd
CharUpwDissLoD
AUSMLoD
(a)
0
0.2
0.4
0.6
0.8
1
-10 -5 0 5 10 15 20 25 30 35 40
h/
H 
[-]
Circumferential velocity [m/s]
Measurement
JSTMatd
CharUpwDissLoD
AUSMLoD
(b)
Figure 4.15: Circumferentially averaged velocities over the relative channel height
h/H in the measurement plane M2 (30 mm behind rotor) of the low-
speed turbine
4.7 Computational results 87
(a) JSTMatd (χ=0.99) (b) CharUpwDiss
Figure 4.16: Mach number contours of the simulated flow in the first stator of the
low-speed turbine in dependence of the discretization scheme
88
Chapter 5
Development of a non-reflecting
boundary condition treatment
In the numerical simulation of turbomachinery it is often necessary to impose
inlet and outlet boundaries near the blades where the flow is far from being
uniform. The correct distribution of the values at the boundaries, however,
is in general unknown and the boundary conditions can only be specified as
averaged values. In order to achieve a solution that does not contain unphys-
ical reflections at the boundaries and is practically independent of the size of
the computational domain, it is important to be able to apply a non-reflecting
boundary condition (NRBC) treatment.
The quasi-three-dimensional boundary condition treatment developed by
Saxer and Giles [242, 243] is state-of-the-art for steady-state simulations of
turbomachinery flow. These boundary conditions were developed for the un-
scaled Navier-Stokes equations for ideal gas flow. In order to be able to use
this kind of boundary condition treatment with a solution scheme for the sim-
ulation of both incompressible and compressible flows with arbitrary Mach
numbers, the boundary condition treatment of Saxer and Giles is extended to
preconditioned system and general equations of state.
5.1 The quasi-three-dimensional approach of
Giles and Saxer for a NRBC treatment
In the original boundary condition treatment devised by Giles and Saxer the
solution at the boundary is decomposed into Fourier modes, where the ze-
roth mode corresponds to the average solution. This average mode is treated
according to standard one-dimensional characteristic theory and enables the
user to specify the circumferential averages of certain flow field parameters at
the boundary. The remaining harmonics of the solution at the boundary are
altered to satisfy a condition derived by Giles [98] based on the linearized,
two-dimensional Euler equations, which ensures that any variation corre-
sponding to incoming (or reflected) waves are eliminated. Using the quasi-
three-dimensional approach of Saxer [242], the grid layers in the radial di-
89
90 5. Development of a NRBC treatment for preconditioned systems
rection are treated independently; the required Fourier decomposition of the
state variables at the boundary is consequently limited to the circumferential
direction. This method is motivated by the fact, that in axial turbomachinery
the flow field variations are usually larger in the pitch-wise than in the radial
direction.
5.2 The Giles-condition for a non-reflecting
boundary for preconditioned systems
In order to arrive at the Giles-condition for a non-reflecting boundary for pre-
conditioned systems, the Navier-Stokes equations (3.4) are linearized about
the average state Q¯v. Neglecting the diffusive fluxes and the source terms,
the following equations are obtained
Γv
∂Q˜v
∂t
+ A
∂Q˜v
∂x
+B
∂Q˜v
∂y
+ C
∂Q˜v
∂z
= 0 (5.1)
for the perturbed state variable Q˜v = Qv − Q¯v with the matrices A,B,C given
by A = ∂Ec
∂Qv
|Q¯v , B = ∂Fc∂Qv |Q¯v , C = ∂Gc∂Qv |Q¯v . To simplify the analysis, Cartesian
coordinates are now used; substituting (wx, wφ, wr)with (u, v, w) the convective
terms in Eq. (2.1) can otherwise be kept unchanged.
In order to be able to separate waves into incoming and outgoing ones, the
perturbed state variable is decomposed into Fourier modes
Q˜v(x, y, z, t) =
∞∑
`=−∞
qˆRv,` e
ıˆ(−ω`t+k`x+l`y+m`z). (5.2)
Applying one particular mode qˆRv,`eıˆ(−ω`t+k`x+l`y+m`z) to the linearized Euler equa-
tions leads to the following relation
0 = (−ωlΓv+klA+llB+mlC)qˆRv,` = (klI −ωlA−1Γv + llA−1B +mlA−1C︸ ︷︷ ︸
Ap
)qˆRv,`. (5.3)
Due to the linearity of the Equations (5.1), this relation holds for any mode;
for the sake of brevity the index ` is dropped in the following analysis. Non-
trivial solutions qˆRv of Eq. (5.3) exist when the dispersion relation
|Ap + kI| = 0 (5.4)
is satisfied; this equation relates the wave numbers k, l,m to the frequency ω
over a homogeneous fifth-order polynomial expression. As can be seen from
Eq. (5.3), qˆRv is a right eigenvector of the matrix Ap with eigenvalue −k. It
should be noted, that to discern between modes going in axial positive and
axial negative directions, it is not the eigenvectors in dependence of the fre-
quency ω that are of interest but the eigenvectors in dependence of the axial
wave number k.
5.3 Quasi-three-dimensional steady-state NRBC 91
To formulate a condition for a non-reflecting axial boundary, the perturbed
state vector Q˜v for a particular mode is decomposed into the sum of its eigen-
functions
Q˜v =
(
5∑
n=1
anqˆ
R
v,ne
ıˆknx
)
eıˆ(ly+mz−ωt), (5.5)
where ki, i = 1, . . . , 5 represent the roots of the dispersion relation (5.4). Re-
flections at the boundary can now be prevented by specifying ai = 0 for each i
in Eq. (5.5) corresponding to an incoming wave. An equivalent condition for a
non-reflecting boundary is
(pˆLv,i)
T · Q˜v = 0 (5.6)
for vectors pˆLv,i(ki) that belong to incoming waves and have the property of
being orthogonal to the right eigenvectors qˆRv,j(kj), i.e.,
(pˆLv,i)
T (ω, ki, l,m) · qˆRv,j(ω, kj, l,m) = 0, ∀i 6= j. (5.7)
It should be noted that since the relation in Eq. (5.6) is formulated for a par-
ticular wave-mode, it represents a condition in the frequency domain. The
vectors pˆLv,i with the desired orthogonality property are found as left eigenvec-
tors of the matrix Ap, i.e.,
0 = (pˆLv,i)
T (−ωA−1Γv + kiI + lA−1B +mA−1C), i = 1, 2, . . . , 5. (5.8)
5.3 Quasi-three-dimensional steady-state NRBC
treatment for preconditioned systems
In the last section the Giles-condition for a non-reflecting boundary was re-
viewed for the general three-dimensional, unsteady case and the necessary
left eigenvectors were defined for the case of preconditioning. Assuming that
the variations in the span-wise directions are small compared to the varia-
tions in the circumferential directions, the necessary Fourier transform for
implementing the Giles-condition given by Eq. (5.6) can be limited to the cir-
cumferential direction by setting the radial wave number m to zero. In this
way, radial flow variations are only considered in the average modes.
Since steady flows are considered, only a treatment that yields a non-
reflecting boundary in the limit of a converged solution for which the temporal
modes vanish is of interest, i.e., a boundary condition treatment for ω → 0 is
sought. In this case, at a boundary x = x0 the wave representation of the
perturbed state variable according to Eq. (5.2) can be simplified to
Q˜v(x0, z) = q¯(x0, z) +
∞∑
n=−∞, n6=0
qˆv,n(x0, z) · eıˆyln . (5.9)
q¯(x, z) corresponds to the zeroth Fourier mode and represents the pitch-wise
solution average at the boundary specified using the one-dimensional charac-
teristic boundary theory for preconditioned systems given in the next section.
92 5. Development of a NRBC treatment for preconditioned systems
At every span-wise location the Giles-condition given by Eq. (5.6) is applied
for every Fourier mode n 6= 0. In the limit of l/ω → ∞ and m = 0 the left
special eigenvectors defined in Eq. (5.8) are, after a considerable amount of
algebra, calculated to be
Lv,s,comp =
pˆT1. . .
pˆT5

comp
=

−1−%hp
%hT
0 0 0 1
−1
%
−u −v 0 0
0 0 0 1 0
−β %va −%ua 0 0
β %va −%ua 0 0
 (5.10)
for compressible fluids (%p 6= 0) with the general thermal % = %(p, T ) and caloric
h = h(p, T ) equation of state. For incompressible fluids with % = const. as the
thermal equation of state, the set of left special eigenvectors becomes
Lv,s,incomp =
pˆT1. . .
pˆT5

incomp
=

−1−%hp
%hT
0 0 0 1
−1
%
−u −v 0 0
0 0 0 1 0
γ −%v %u 0 0
−γ −%v %u 0 0
 . (5.11)
The parameters β and γ in Eqs. (5.10) and (5.11) are given by
γ = ıˆ · sign(l),
β =
{
ıˆ · sign(l)√a2 − (u2 + v2), u2 + v2 ≤ a2,
−sign(v)√(u2 + v2)− a2, u2 + v2 ≥ a2. (5.12)
For a detailed derivation of the eigenvectors, the reader may refer to a sepa-
rate report of Anker et al. [19]. The first left special eigenvector for the com-
pressible case presented here differs from the result obtained by Saxer [242]
since a different set of primitive variables and a general and not an ideal equa-
tion of state are used. However, the four last eigenvectors correspond to the
eigenvectors derived by Saxer, which can be explained by the fact that precon-
ditioning does not affect the steady state. The matrix of left special eigenvec-
tors Ls,comp for the primitive variable vectors Qp = [%, u, v, w, p]T (which Saxer
uses) and compressible flow can in fact be calculated from
Ls,comp = Lv,s,comp
[
∂Qv
∂Qp
]
. (5.13)
Because the physical speed of sound is infinite in incompressible fluids, it
is obvious that the eigenvectors [pˆ1, . . . , pˆ5]Tcomp that are formulated in depen-
dence of the speed of sound cannot be used for the incompressible case. The
eigenvectors [pˆ1, . . . , pˆ5]Tincomp were derived taking the property % = const. into
account.
5.4 1D characteristic theory for imposing BCs 93
5.4 Boundary condition treatment based on one-
dimensional characteristic theory
In the quasi-three-dimensional boundary condition treatment approach of Giles
and Saxer, the circumferential averages of flow variables at the boundary are
imposed by means of one-dimensional characteristic theory. In the follow-
ing subsections standard one-dimensional boundary condition treatment is
extended to preconditioned systems and general equations of state.
5.4.1 One-dimensional characteristic theory as a means
to impose compatible boundary conditions
When a time-stepping scheme is used for the simulation of fluid flow, the un-
steady Navier-Stokes equations are solved. Whether or not time-derivative
preconditioning is used, the governing equations are hyperbolic-parabolic in
time and the propagation speeds of the different physical modes will conse-
quently be of importance. As one immediately sees from a physical point of
reasoning, for the boundary condition treatment it is essential whether pres-
sure perturbations can propagate upstream or not.
The idea of using characteristic theory for imposing boundary conditions
is to decompose the solution at a boundary in contributions of waves entering
and leaving the computational domain. While the outgoing waves can not be
manipulated, the ingoing modes can be acted upon in order to impose certain
values at the boundaries. As the characteristic variables represent decoupled
state variables, the application of a characteristic boundary condition treat-
ment ensures that the individual corrections of the solution variables at the
boundary do not interfere with each other and that the corrections are com-
patible with the propagation of the waves in the interior of the computational
domain.
A boundary condition treatment based on one-dimensional boundary con-
dition treatment assumes that it is sufficient to only consider the propagation
of waves perpendicular to the boundary under consideration; it is assumed
that the flow is homogeneous in the directions transverse to the normal of the
boundary, that the flow is convection dominated and that the influence of the
diffusive terms on the wave propagation can be neglected.
For the sake of simplicity, it will in the following be assumed that the inlet
boundary is positioned at x = x0 = const. and the outlet boundary located
further downstream at x = x1 = const.. In this way, the fluxes in the y- and
z-direction do not need to be accounted for so that the one-dimensional, pre-
conditioned Euler equations, which have been linearized about the averaged
state Q¯v,
∂Q˜v
∂t
+ Γ−1v A︸ ︷︷ ︸
A∗
∂Q˜v
∂x
= 0 (5.14)
with the flux Jacobian A defined as A = ∂E
∂Qv
|Q¯v , are sufficient for a proper
94 5. Development of a NRBC treatment for preconditioned systems
description of the transport of information near the boundaries.
By multiplying the last equation with the matrix L of the left eigenvec-
tors of the system matrix A∗ = Γ−1v A, one obtains the decoupled system of
equations
L
∂Q˜v
∂t
+ LA∗L−1︸ ︷︷ ︸
ΛΓ
L
∂Q˜v
∂x
= 0. (5.15)
As shown in Anker et al. [19] the matrix L reads as
L =
l
T
1
...
lT5
 =

−1−%hp
%hT
0 0 0 1
0 0 1 0 0
0 0 0 1 0
1 %(u′′0 + a
′′) 0 0 0
1 %(u′′0 − a′′) 0 0 0
 (5.16)
and its inverse L−1, which equals the matrix of the right eigenvector R, be-
comes
L−1 = R = [r1, . . . , r5] =

0 0 0
a′′−u′′0
2a′′
a′′+u′′0
2a′′
0 0 0 1
2%a′′ − 12%a′′
0 1 0 0 0
0 0 1 0 0
1 0 0 1−%hp
%hT
a′′−u′′0
2a′′
1−%hp
%hT
a′′+u′′0
2a′′
 . (5.17)
The auxiliary quantities a′′, u′′ and u′′0 are defined as
u′′ =
1
2
u(1 +
d
d′
), u′′0 =
1
2
u(1− d
d′
), a′′ =
√
u′′0
2 + a′2. (5.18)
The terms d, d′ and pseudo-acoustic velocity a′ are defined by Eqs. (3.7) and
(3.10). It should be noted that these quantities reduce to
a′′ = a, u′′ = u, u′′0 = 0 (5.19)
in the case that preconditioning is switched off.
With the definition of the vector of the characteristic variables φ as φ =
LQv, the relation (5.15) can be transformed to the following system of trans-
port equations
∂φ˜
∂t
+ ΛΓ
∂φ˜
∂x
= 0. (5.20)
Since ΛΓ is a diagonal matrix,
ΛΓ = Diag (λ′1, . . . , λ
′
5) , with λ
′
1,2,3 = u
′′, λ′4,5 = u
′′ ± a′′, (5.21)
the perturbed characteristic variables φ˜ = LQ˜v represent decoupled quanti-
ties; from Eq. (5.20) it is evident that each characteristic variable φ˜i is trans-
ported at its own characteristic speed λ′i (i = 1, . . . , 5).
5.4 1D characteristic theory for imposing BCs 95
It should be noted, that in the preconditioning method used, the scaling of
the flow equations is shut off for supersonic speed. Furthermore, the precon-
ditioning matrix Γv is defined in such a way that the signs of the eigenvalues
of the system matrix A∗ are not altered. Consequently, for a flow along the
positive x-direction (u > 0), the signs of the first four eigenvalues are always
positive, whereas the fifth eigenvalue is dependent on whether or not the flow
is axially supersonic or not:
sign(λ′1,2,3,4) > 0, sign(λ
′
5) = sign(u− a). (5.22)
Only those characteristic variables that are entering the domain at a bound-
ary can and should be prescribed; the value of the outgoing characteristic
variables are determined by the evolution of the flow field in the interior of
the computational domain and can not be manipulated. Thus, at a subsonic
inlet boundary, the first four characteristic quantities (φ˜i, i = 1, . . . , 4) need
to be prescribed whereas the value of the fifth characteristic quantity φ˜5 is
determined by extrapolation. Vice versa at a subsonic outflow boundary, only
the fifth characteristic quantity φ˜5 is associated with a boundary condition,
the others are a mere consequence of the flow field upstream.
At a supersonic inflow boundary, the value of all characteristics follow from
the specification of the state variables at the boundary. This obviates the need
for a distinction between in- and outgoing wave modes and makes the imple-
mentation of a suitable boundary condition treatment easy. A proper numer-
ical treatment of a supersonic outflow is even simpler; since all characteristic
variables are determined by the dynamics of the interior of the flow field, the
states that are predicted at the boundary by the flow solver can be taken over
without any correction.
In the following the boundary condition treatment for each of the four
aforementioned cases is discussed.
5.4.2 Axially subsonic inlet
The average changes in the four incoming characteristics are calculated from
the requirement that the average flow field match the total temperature T0,bc(r),
the total pressure p0,bc(r), the relative flow angle βbc(r) in the n-φ-plane and
the pitch angle γbc(r) in the n-t-plane specified by the user. Accordingly, in
the following u, v, w represent the velocity component in the normal, circum-
ferential and tangential direction of the boundary, respectively. An equivalent
specification of the average inlet conditions is to drive the following boundary
residuals to zero:
R1 := %¯T¯ (s¯− sbc(r)), R2 := %¯a¯′′(v¯ − u¯β∗bc(r)),
R3 := %¯a¯′′(w¯ − u¯γ∗bc(r)), R4 := %¯(h¯0 − h0,bc(r))
(5.23)
with β∗bc = tan βbc, γ
∗
bc = tan γbc and the entropy and total enthalpy given by
sbc = s(p0,bc, T0,bc) and h0,bc = h(p0,bc, T0,bc), respectively. A bar “¯” denotes a cir-
cumferential average of the respective quantity. In coherence with the orig-
inal boundary condition treatment devised by Giles, the boundary residuals
96 5. Development of a NRBC treatment for preconditioned systems
are defined in dependence of the total enthalpy and the entropy to ensure a
uniform distribution of these quantities; as noted by Giles [97] if the bound-
ary residuals are directly specified in dependence of the total pressure and the
total temperature, a linear numerical treatment may introduce second order
effects that will cause variations in the total enthalpy and the entropy at the
boundary. The residuals have been scaled with %¯T¯ , %¯a¯′′, and %¯, respectively,
in order to obtain simpler expressions in the boundary condition treatment.
These reference quantities are assumed to be constant within a time-step.
For the definition of the first boundary residual Giles employs the entropy
related function s∗ with s∗ := ln(p) − γ ln %, which for ideal gases corresponds
to s∗ = s
cv
. Since specifying s∗ := const. does not enforce s = const. for general
equations of state for which the isochoric heat capacity cv may be temperature
and pressure dependent, the entropy function is used directly. This choice
is also more convenient, as the specific isochoric heat capacity can neither
be calculated directly from the thermal % = %(p, T ) and caloric h = h(p, T )
equation of state nor from the entropy function s = s(p, T ).
The nonlinear equations (5.23) are linearized about the circumferentially
averaged solution Q¯(n)v of the last time-step n
R(n+1) = R(n) +
∂(R1, R2, R3, R4)
∂(φ1, φ2, φ3, φ4)︸ ︷︷ ︸
J
∣∣∣∣∣∣∣∣
Q¯
(n)
v
·

δφ¯1
δφ¯2
δφ¯3
δφ¯4

(n)
:= 0. (5.24)
The Jacobian J is calculated as
J =
∂(R1, R2, R3, R4)
∂(p, u, v, w, T )
∂(p, u, v, w, T )
∂(φ1, φ2, φ3, φ4)︸ ︷︷ ︸
(r1,r2,r3,r4)
, (5.25)
where ri; i = 1, . . . , 4 represent the first four right of the eigenvectors given in
Eq. (5.17). When solving these linearized equations one obtains the necessary
variations of the average incoming characteristic variables δφ¯(n)i = φ¯
(n+1)
i −
φ¯
(n)
i ; i = 1, . . . , 4 for driving the boundary residuals with increasing time-step
to zero: 
δφ¯1
δφ¯2
δφ¯3
δφ¯4

(n)
= −J−1∣∣
Q¯
(n)
v
·

R1
R2
R3
R4

(n)
. (5.26)
By exploiting the identity
1− %hp
%hT
+
sp
sT
≡ 0, (5.27)
5.4 1D characteristic theory for imposing BCs 97
which holds for general equations of state, the inverse of the Jacobian becomes
J−1 =

1
%hT
0 0 0
− β∗bc
%a′′M ′′
M ′′−M ′′y β∗bc
%a′′M ′′ −
M ′′z β∗bc
%a′′M ′′
β∗bc
%a′′M ′′
− γ∗bc
%a′′M ′′ −
M ′′y γ∗bc
%a′′M ′′
M ′′−M ′′z γ∗bc
%a′′M ′′
γ∗bc
%a′′M ′′
− 2
M ′′ −
2M ′′y
M ′′ −2M
′′
z
M ′′
2
M ′′
 (5.28)
with the Mach numbersM ′′,M ′′y ,M ′′z and the abbreviations Ξ, given by
M ′′ = Ξ
a′′ , M
′′
y =
v
a′′ , M
′′
z =
w
a′′ ,
Ξ = a′′ + u′′ + vβ∗bc + wγ
∗
bc.
(5.29)
In the work of Anker et al. [19] details of the derivation of the inverse Jacobian
J−1 and the proof of the validity of the thermodynamic identity in Eq. (5.27)
are given.
The determinant D of the Jacobian J is given by
D =
1
2
%3a′′cpΞ. (5.30)
Since D is greater than zero for physical thermal state variables (p, T > 0
⇒ %, cp, a′′, u′′ > 0) and inflow conditions u > 0, the regularity of the Jacobian
J for physically meaningful applications is guaranteed.
The average variation of the outgoing fifth characteristic variable δφ¯5 is
given by
δφ¯
(n)
5 = l
T
5
∣∣
Q¯(n)
· (δQ¯v)(n)cal
= δp¯
(n)
cal − [%¯(a¯′′ − u¯′′0)](n) · δu¯(n)cal
(5.31)
where the subscript ”cal” denotes changes predicted by the flow solver, i.e.,
δp¯
(n)
cal = p¯
(n+1)
cal − p¯(n) where p¯(n+1)cal represents the predicted static pressure at the
actual time step n+ 1 before the boundary values are post-corrected. The left
eigenvectors for calculating the characteristic variables are given in Eq. (5.16)
above.
When the boundary treatment is simply based on the average one-dimen-
sional characteristic variables, the solution variables Qv,j at every point j =
1, . . . , N + 1 of the current circumferential grid line are post-corrected accord-
ing to
Q
(n+1)
v,j = Q¯
(n)
v + R|Q¯(n)v δφ¯
(n) (5.32)
with δφ¯ = [δφ¯1, δφ¯2, δφ¯3, δφ¯4, δφ¯5]T whereR represents the matrix of right eigen-
vectors given in Eq. (5.17).
5.4.3 Axially subsonic outlet
At an axially subsonic boundary four characteristics are leaving and one char-
acteristic is entering the computational domain. The implementation of the
boundary conditions at outflow is easier than at inflow, because only one
98 5. Development of a NRBC treatment for preconditioned systems
boundary condition has to be imposed. As outlet boundary condition for ax-
ially subsonic flow, the user has to specify the radial variation of the static
pressure pbc(r). Using the residual R5 = p¯(r)− pbc(r) and linearizing from the
current time level yields
R
(n+1)
5 = R
(n)
5 +
∂R5
∂φ5
∣∣∣∣
Q¯
(n)
v
δφ¯5
(n)
:= 0. (5.33)
Calculating the scalar Jacobian ∂R5
∂φ5
and solving the equation for the average
change in the fifth characteristic variable leads to
δφ¯
(n)
5 = −
2a′′
a′′ + u′′0
∣∣∣∣
Q¯
(n)
v
·R(n)5 . (5.34)
The variations of the four first characteristics are calculated according to
δφ¯1
δφ¯2
δφ¯3
δφ¯4

(n)
=

lT1
lT2
lT3
lT4

Q¯
(n)
v
· δQ¯(n)v,cal =

−1−%hp
%hT
0 0 0 1
0 0 1 0 0
0 0 0 1 0
1 %(u′′0 + a
′′) 0 0 0

Q¯
(n)
v
·

δp¯
δu¯
δv¯
δw¯
δT¯

(n)
cal
. (5.35)
If a boundary condition treatment based on one-dimensional characteristic
theory is applied, the characteristic changes determined in the last two equa-
tions are transformed back into corrections of the solution variables at the
boundary according to Eq. (5.32).
5.4.4 Axially supersonic inlet
In the case that the flow is axially supersonic at the inlet, all characteristics
are incoming so that five boundary conditions have to be given. In addition
to the boundary conditions for the axially subsonic case, the user has to spec-
ify the radial distribution of the axial Mach number Mx,bc(r). Since the flow
field at the boundary is independent of the downstream flow field, the cor-
rect boundary values can be calculated once and for all from the boundary
conditions specified by the user. When general equations of state are used, a
Newton-Raphson procedure must be applied in order to find the correct values
of the primitive, viscous variables from the set of nonlinear algebraic equa-
tions
h0(p0,bc, T0,bc) = h(p0,bc, T0,bc) = h(p, T ) +
1
2
(u2 + v2 + w2 − (Ωr)2),
s(p0,bc, T0,bc) = s(p, T ), tan βbc =
v
u
, tan γbc =
w
u
, Mx,bc = ua(p,T ) .
(5.36)
A concrete description of an algorithm to determine the boundary conditions
for general equation of state can be found in the work of Anker et al. [19]. For
ideal gases, the vector of primitive, viscous state variables can be determined
analytically from the boundary conditions:
M =
√
1 + tan2 βbc + tan
2 γbc ·Mx,bc, T = T0,bc1+ γ−1
2
M2
,
p = p0,bc ·
(
1 + γ−1
2
M2
)− γ
γ−1 , u = a ·Mx,bc =
√
γRT ·Mx,bc,
v = u tan βbc, w = u tan γbc.
(5.37)
5.5 Implementation of the Q3D NRBC 99
For the practical use of this boundary condition treatment in a numerical sim-
ulation, it should be noted that the flow field must be initialized to be super-
sonic in the vicinity of the boundary. If upstream running pressure waves can
reach the boundary, this boundary condition treatment will not be compatible
with the dynamics of the flow field in the computational domain.
5.4.5 Axially supersonic outlet
When the flow is axially supersonic at the outlet boundary, all characteristics
are outgoing, so that no boundary condition can be imposed. For practical
simulations it is recommended to apply a subsonic outlet boundary for the
case that the simulated flow field during the simulation should be subsonic. In
the case that the flow is supersonic, the presence of a subsonic outlet boundary
will not affect the interior of the flow field.
5.5 Implementation of the Q3D non-reflecting
boundary condition treatment
Except in the case that the flow is supersonic in the flow direction normal
to the boundary (which here for simplicity is assumed to be the axial direc-
tion), the Giles-condition must be applied in addition to a one-dimensional
characteristic theory to a obtain a non-reflecting boundary condition treat-
ment. At an axially supersonic inflow boundary, all characteristic variables
are incoming, which obviates the need for a non-reflecting boundary condition
treatment; if a proper discretization scheme is used, all wave modes are ex-
clusively transported downstream and reflections can not be produced. By the
same token, at an axially supersonic outflow boundary, where all characteris-
tics are are leaving the domain, the non-reflectivity of the boundary condition
treatment is not an issue.
If a quasi-three-dimensional non-reflecting boundary condition treatment
is applied, then the variations in the characteristic variables δφ(n)j =φ(n+1)j −φ(n)j at
each grid point j at a boundary are divided into the circumferential average
δφ¯
(n)
:=φ¯
(n+1)−φ¯(n) over one pitch and a local change δφ′(n)j according to
δφ
(n)
j (r) = δφ¯
(n)
(r) + δφ
′(n)
j , j = 1, 2, . . . , N, (5.38)
where N denotes the number of grid points in the circumferential direction
when only accounting for the periodic point once.
It should be noted that the reference state used Q¯(n)v for the calculation of
the perturbed state variable Q˜v and the perturbed characteristic variable vec-
tor φ˜ in the post-correction procedure of the boundary values between time-
step n and n + 1 is the same. It is thus not of any importance whether the
variations in the aforementioned quantities are determined using the respec-
tive absolute or the perturbed quantities; for the total variation of the charac-
100 5. Development of a NRBC treatment for preconditioned systems
teristic variables the following relation holds:
δφ
(n)
j := φ
(n+1)
j − φ(n)j = φ˜
(n+1)
j − φ˜
(n)
j = δφ˜
(n)
j ; j = 1, 2, . . . , N. (5.39)
As suggested by Giles [98], to ensure that the boundary condition treat-
ment does not cause instabilities in the solution process, the correction in the
incoming characteristic variables should be relaxed, i.e., the overall variations
of the incoming characteristic variables
δφ
(n)
j (r) = σ(δφ¯
(n)
(r) + δφ
′(n)
j (r)) (5.40)
are attenuated with a relaxation factor σ. Numerical investigations of Giles [98]
suggest that a suitable choice of the relaxation factor is given by
σ = 1/N. (5.41)
In the following subsections the implementation of the non-reflecting correc-
tions at the boundaries will be described in dependence of the prevailing flow
conditions at the boundaries.
5.5.1 Subsonic inflow boundary
At a subsonic inflow boundary, the changes in the average incoming charac-
teristic variables are determined using one-dimensional characteristic theory
as described in Section 5.4.2.
In order to obtain a non-reflecting inlet boundary, the Giles-condition[
pˆ1, pˆ2, pˆ3, pˆ4
]T · ˆ˜qk = 0 (5.42)
has to be satisfied for each Fourier mode ˆ˜qk (k 6= 0) of the perturbed viscous,
primitive variables Q˜v = Qv − Q¯v. By using the relation L−1 ˆ˜φk = ˆ˜qk, this
boundary condition can be expressed in terms of the spatial Fourier modes of
the characteristic variables. For compressible flows it is given by
1 0 0 0 0
0 −v 0 −a′′+u−u′′0
2%a′′ −a
′′−u+u′′0
2%a′′
0 0 1 0 0
0 −%ua 0 −β(a′′−u′′0 )−va
2a′′ −β(a
′′+u′′0 )+va
2a′′

Q¯
(n)
v
· ˆ˜φk = 0. (5.43)
For incompressible fluids the corresponding condition is determined analo-
gously; for a detailed description, the reader may refer to the work of Anker
et al. [19]. In the following only the compressible case is discussed for the
sake of brevity. The condition given in the last equation is satisfied trivially
when all Fourier components (except the average Fourier mode) of the char-
acteristic variables are zero, i.e., ˆ˜φk=0,∀ k 6=0. This shows that one-dimensional
characteristic theory alone can be applied to uniform flow fields, such as the
far-field of external aerodynamic applications. However, in the general case
5.5 Implementation of the Q3D NRBC 101
inhomogeneous flow fields are encountered which implies that the condition
stated in Eq. (5.43) needs to be fulfilled non-trivially.
Since the outgoing characteristics do not depend on the boundary condi-
tions but are only given by the interior flow field, they are not manipulable.
Thus, the general way to satisfy the condition given in Eq. (5.43) is to define
the incoming characteristics as functions of the outgoing ones:
ˆ˜φ1,k
ˆ˜φ2,k
ˆ˜φ3,k
ˆ˜φ4,k

(n)
:=

0
− βu+va
%[a(u2+v2)+(a′′−u′′0 )(ua−vβ)]
0
a(u2+v2)−(a′′+u′′0 )(ua−vβ)
a(u2+v2)+(a′′−u′′0 )(ua−vβ)

Q¯
(n)
v
· ˆ˜φ(n)5,k . (5.44)
Since the last condition is formulated in the frequency domain, the local
outgoing characteristics have to be calculated as
φ′5,j
(n) = lT5
∣∣
Q¯
(n)
v
· Q˜(n)v
= (pj − p¯)(n) + %¯(n)(u′′0 − a′′)(n) · (uj − u¯)(n)
(5.45)
for j = 1, 2, . . . , N . Afterwards a discrete Fourier transformation according to
ˆ˜φ
(n)
5,k =
1
N
N−1∑
j=0
φ
′∗
5,j
(n) · e2piıˆjk/N , k = 0, 1, . . . , N − 1 (5.46)
by assuming periodicity (φ′5,1 = φ′5,N+1) has to be carried out. The set of equidis-
tantly distributed characteristic variables φ′∗5,j; j = 0, . . . , N is obtained from
the characteristic variables on the computational grid φ′5,j; j = 1, . . . , N + 1
by using spline-interpolation where the endpoints of both sets match, i.e.,
φ
′∗
5,0 = φ
′
5,1 and φ
′∗
5,N = φ
′
5,N+1 = φ
′
5,1.
The Fourier coefficients of the second characteristic variable are calculated
from the Fourier coefficients of the fifth characteristic according to Eq. (5.44):
ˆ˜φ
(n)
2,ks = −
βu+ va
%[a(u2 + v2) + (a′′ − u′′0)(ua− vβ)]
∣∣∣∣
Q¯
(n)
v︸ ︷︷ ︸
c52
· ˆ˜φ(n)5,k = c52 · ˆ˜φ(n)5,k (5.47)
∀k = 0, 1, . . . , N −1. The correct steady state distribution of the second charac-
teristic variables φ′∗2,js is obtained from the Fourier coefficients φˆ2,ks by means
of an inverse discrete Fourier transform:
φ
′∗
2,js
(n)
=
N−1∑
k=0
φˆ
(n)
2,ks · e−2piıˆjk/N , j = 0, 1, . . . , N − 1. (5.48)
Since the Fourier coefficients ˆ˜φ5,k originate from a real function, they pos-
sess a conjugate even symmetry with respect to k=0. Due to the definition of
β in Eq. (5.12), where the wave number l corresponds to the Fourier-index k
102 5. Development of a NRBC treatment for preconditioned systems
used here, the factor c52 has the same property. The inverse discrete Fourier
transform (5.48) for determining the correct steady state distribution of the
second characteristics can, as shown in Anker et al. [19], then be simplified to
φ
′∗
2,js =

2 Real
[
N/2−1∑
k=1
ˆ˜φ2,ks · e−2piıˆjk/N
]
+ ˆ˜φ2,N/2s, for even N,
2 Real
[
(N−1)/2∑
k=1
ˆ˜φ2,ks · e−2piıˆjk/N
]
, for odd N.
(5.49)
In Eq. (5.49) it was taken into account that the Fourier coefficient ˆ˜φ2,0s equals
zero, since the characteristic variables are defined as perturbations from the
average state of the last time step n.
The distribution of the correct steady state second characteristic variables
on the computational grid φ′2,js are gained from φ
′∗
2,js, again using a cubic inter-
polation-spline. When the flow is supersonic, but axially subsonic, the factor
c52 is real-valued and independent of the circumferential wave number, which
is why the Giles-condition can be applied in the time domain and the discrete
Fourier transformations can be omitted.
For each node j along the boundary, the local correction of the second char-
acteristic variable δφ′2,j
(n) is determined as the difference between the correct
steady state value φ′2,js
(n) and the current value φ′2,j
(n):
δφ
′(n)
2,j = (φ
′
2,js − φ′2,j)(n) = φ′2,js(n) − lT2
∣∣
Q¯
(n)
v
· Q˜(n)v
= φ′2,js
(n) − (vj − v¯)(n).
(5.50)
Since the harmonics of the correct third steady state characteristics are zero,
the correction of the third characteristic variable is simply
δφ
′(n)
3,j = −φ′3,j(n) = − lT3
∣∣
Q¯
(n)
v
· Q˜(n)v = −(wj − w¯)(n). (5.51)
As noted by Giles [98], an implementation of the other two of the four con-
ditions in Eq. (5.44) would result in a flow field which to first-order would
have uniform entropy and stagnation enthalpy at each radius. However, the
neglected second order effects would introduce variations in entropy and stag-
nation enthalpy. To avoid this, the steady state corrections of the local first
and fourth characteristic variables are obtained from the conditions that the
local entropy sj and the local stagnation enthalpy h0j should match their av-
erage values, which is equivalent to drive the residuals
R1,j = %¯T¯ (sj − s¯), R4,j = %¯(h0j − h¯0) (5.52)
to zero. The use of the residuals defined by Eq. (5.52) together with a one-
step Newton-Raphson procedure results in the following local corrections of
the first and fourth characteristic variables:
δφ′1,j
(n) = − 1
%hT
∣∣∣
Q¯
(n)
v
·R(n)1,j ,
δφ′4,j
(n) = − 2a′′
a′′+u−u′′0
∣∣∣
Q¯
(n)
v
· [−R1,j + (%¯v¯) · δφ′2,j + (%¯w¯) · δφ′3,j +R4,j](n).
(5.53)
5.5 Implementation of the Q3D NRBC 103
Now that the local variations of the incoming characteristic variables have
been determined, they are added to the average changes according to
δφi,j = σ(δφ
′
i,j + δφ¯i); i = 1, . . . , 4 (5.54)
where the relaxation factor σ is chosen to be 1/N . The total variation of the
fifth characteristic variable δφ(n)5,j is calculated as
δφ
(n)
5,j = l
T
5
∣∣
Q¯
(n)
v
(Q
(n+1)
v,j −Q(n)v,j )
= p
(n+1)
j,cal − p(n)j + %¯(u′′0 − a′′)(n)(u(n+1)j,cal − u(n)j ).
(5.55)
Finally, the total characteristic changes δφ = [δφ1, . . . , δφ5]T are trans-
formed back into changes of the viscous, primitive variables and the state
vector is updated according to
Q
(n+1)
v,j = Q
(n)
v,j +R|Q¯(n)v · δφ
(n)
j . (5.56)
with the Matrix R of the right eigenvectors given in Eq. (5.16) above.
5.5.2 Subsonic outlet boundary
The Giles-condition for a non-reflecting spatial distribution of the perturbed
primitive, viscous variables at the outlet boundary reads as
[β, %va,−%ua, 0, 0]
Q¯
(n)
v
· ˆ˜qk = 0, (5.57)
which in terms of the characteristic variables can be expressed as[
0,−%ua, 0, β(a
′′ − u′′0) + va
2a′′
,
β(a′′ + u′′0)− va
2a′′
]
Q¯
(n)
v
· ˆ˜φk = 0 (5.58)
for all Fourier modes k 6= 0. The Fourier modes of the ingoing fifth charac-
teristic variable are defined as functions of the Fourier modes of the outgoing
second and fourth characteristics according to
ˆ˜φ
(n)
5,ks :=
2%uaa′′
β(a′′ + u′′0)− va
∣∣∣∣
Q¯
(n)
v
· ˆ˜φ(n)2,k −
β(a′′ − u′′0) + va
β(a′′ + u′′0)− va
∣∣∣∣
Q¯
(n)
v
· ˆ˜φ(n)4,k , (5.59)
for k = 0, 1, . . . , N − 1, where N again denotes the number of grid points in
circumferential direction when accounting for the periodic nodes only once.
The Fourier modes ˆ˜φ2,k and
ˆ˜φ4,k are computed by first calculating φ′2,j and φ′4,j
from the following relations:
φ′2,j
(n) = lT2
∣∣
Q¯
(n)
v
· (Qv,j − Q¯v)(n) = (vj − v¯)(n),
φ′4,j
(n) = lT4
∣∣
Q¯
(n)
v
· (Qv,j − Q¯v)(n)
= (pj − p¯)(n) + %¯(n)(u′′0 + a′′)(n) · (uj − u¯)(n),
(5.60)
104 5. Development of a NRBC treatment for preconditioned systems
∀j = 1, 2, . . . , N . Then these quantities are transferred onto an equidistant
grid and a discrete Fourier transformations of the spatially equidistantly dis-
tributed characteristic variables φ
′∗(n)
2,j and φ
′∗(n)
4,j are performed; for example
the Fourier coefficients of the second characteristic variables φ
′∗(n)
2,j are calcu-
lated as
ˆ˜φ
(n)
2,k =
1
N
N−1∑
j=0
φ
′∗(n)
2,j · e2piıˆjk/N , k = 0, 1, . . . , N − 1. (5.61)
Using the simplification due to conjugate even symmetry of the Fourier coeffi-
cients ˆ˜φ5,ks with respect to k = 0 allows the non-reflecting steady state values
φ
′∗
5,js for the incoming fifth characteristic variables to be written as
φ
′∗
5,js
(n)
=

2 Real
[
(N/2)−1∑
k=1
ˆ˜φ
(n)
5,ks · e−2piıˆjk/N
]
+ ˆ˜φ5,N/2s, for even N,
2 Real
[
(N−1)/2∑
k=1
ˆ˜φ
(n)
5,ks · e−2piıˆjk/N
]
, for odd N,
(5.62)
∀j = 1, 2, . . . , N . With φ′5,j(n) calculated as
φ′5,j
(n)
= lT5
∣∣
Q¯
(n)
v
Q˜
(n)
v,j = p
(n)
j − p¯(n) + %¯(n)(u′′0 − a′′)(n)(u(n)j − u¯(n)) (5.63)
and the average variation of the fifth characteristic variable δφ¯(n)5 given by
Eq. (5.34) one obtains the total variation as
δφ
(n)
5,j = σ(δφ¯
(n)
5 + φ
′
5,js
(n) − φ′5,j(n)). (5.64)
The total variations of the first four characteristic variables are determined
directly by

δφ1
δφ2
δφ3
δφ4

(n)
j
=

lT1
lT2
lT3
lT4

Q¯
(n)
v
·

δpj
δuj
δvj
δwj
δTj

(n)
j,cal
=

−1−%hp
%hT
0 0 0 1
0 0 1 0 0
0 0 0 1 0
1 %(u′′0 + a
′′) 0 0 0

Q¯
(n)
v
·

δpj
δuj
δvj
δwj
δTj

(n)
cal
.
(5.65)
Finally, the total characteristic changes are transformed back into changes of
the viscous, primitive variables and the state vector is updated according to
Eq. (5.56).
5.5.3 Axially subsonic inlet or outlet boundaries in an
overall supersonic flow
In the case that the flow is supersonic (M > 1) but that the axial flow velocity
is subsonic (Mx < 1), then the boundary condition treatment described in the
last sections only changes slightly. Under this condition the pressure waves
5.6 Non-reflecting mixing planes 105
are still able to propagate upstream in the axial direction, which is why the
in- and outgoing waves in the boundary condition treatment must be discrim-
inated. However, in this case, the Giles-condition (5.6) simplifies due to the
fact that the coefficient β, which is defined in Eq. (5.12), is only real-valued
and not dependent of the wave number l anymore. Thus, for supersonic flows,
a Giles-condition can be implemented directly without the need for applying
Fourier-transforms of the boundary values.
5.5.4 Axially supersonic inlet and outlet boundaries
Since all wave modes at an axially supersonic inflow boundary are exclusively
transported downstream, the boundary condition treatment can be imple-
mented independently of the flow field located downstream. Thus, the bound-
ary condition treatment described in Section 5.4.4 is sufficient to achieve a
non-reflecting boundary.
At a supersonic outflow boundary all wave modes are leaving the domain,
which makes it impossible to impose any boundary values. Consequently, any
kind of boundary condition treatment in this case is inapplicable.
5.5.5 Adjacent solid walls
In calculations of viscous flow, the boundary points that lie in one of the flow
boundary planes as well as on a solid wall require a special treatment, since
the non-reflecting boundary treatment in combination with the precondition-
ing is singular at no-slip walls. The singularity of the walls becomes evident
when the limit of coefficient c52 defined in Eq. (5.47) is taken for v = 0 and
u→ 0.
In the current work, the grid points of the inlet- or outlet boundaries which
simultaneously are located on a wall boundary are treated by first imposing
the no-slip condition by setting all components of the velocity vector to zero.
At the inlet boundary the static pressure and the temperature are set to the
user specified total values. At the outlet boundaries the pressure is set to the
corresponding user defined value, whereas the wall temperature is given by
the specified thermal boundary condition.
5.6 Non-reflectingmixing-planes for the precon-
ditioned Navier-Stokes equations
In steady multistage calculations of turbomachinery, mixing planes are used
for coupling different frames of reference and transferring circumferentially
averaged data from one row to another. The mixing planes developed in the
frame of this work apply one-dimensional characteristic theory to enforce the
condition that mass, impulse and energy are conserved across the interface.
Reflections at these boundaries are prevented applying the Giles-condition
given in Eq. (5.6). Since a mixing plane in principle represents an inlet
106 5. Development of a NRBC treatment for preconditioned systems
boundary coupled with an upstream outlet boundary, the extension of the
non-reflecting inlet and outlet boundaries to non-reflecting mixing planes is
straightforward. For this reason, in the following only a brief description of
the mixing-planes developed for the simulation of turbomachinery flows using
the preconditioned Navier-Stokes equations will be given.
5.6.1 Coupling condition
The developed mixing planes couple the flow in the rotor and stator rows by
imposing that the circumferentially averaged, viscous, primitive state vectors
Q¯v on each side i ∈ {1, 2} of a given interface take on identical values in an
absolute frame of reference for all radial position r, i.e.,
Qv(r)
∣∣
1,abs = Qv(r)
∣∣
2,abs , ∀r. (5.66)
To ensure that mass, momentum and energy fluxes are conserved, the latter
condition (5.66) is applied on the flux averaged quantities at the circumferen-
tial lines where the mass flow does not vanish, as suggested by Saxer [242].
At the walls, simple area averaged flow quantities are used for coupling the
stator and the rotor rows.
It should be noted that the use of circumferentially averaged quantities
will lead to artificially increased values of the entropy in the downstream
blade row as studied by Fritsch and Giles [89]. The use of mass-averaged
stagnation flow quantities for coupling would lead to lower mixing losses; for
a principal discussion of suitable averaging techniques in internal flows, the
reader may refer to the work of Greitzer et al. [103]. Michelassi et al. [193]
have compared various matching criteria and do not observe any noticeable
difference between the use of stagnation and regular flow variables. Since de-
terministic stresses [2] would anyhow need to be modelled to achieve a mix-
ing model that is as realistic as possible, in the current work the approach of
Saxer was adopted for the sake of simplicity.
5.6.2 Numerical implementation
For the imposition of the coupling condition (5.66) it is necessary to distin-
guish between incoming and outgoing wave modes. For this reason a bound-
ary condition treatment analogous to the procedure for inlet and outlet bound-
aries based on characteristic theory is used; in fact, the mixing planes can be
regarded as inlet boundaries that are linked via the aforementioned coupling
condition to an outflow boundary located upstream.
In a similar fashion as the treatment of the inlet and outlet boundaries, the
overall variations of outgoing characteristic variables δφ5,j,ex are accounted
for in their entirety whereas the corresponding variations of the incoming
variables are relaxed. The complete variations in the incoming characteristic
variables δφi,j,in, i = 1, . . . , 4 and δφ5,j,in that are used in the boundary condition
treatment of the downstream and the upstream side of each mixing plane,
5.7 Verification of the non-reflectivity 107
respectively, are determined as
δφi,j,in = σ1δφi,in + σ2δφ
′
i,j,in; ∀i = 1, . . . , 5; j = 1, . . . , N (5.67)
with the relaxation factors σ1 = 0.3 and σ2 = max(0.02, 1/N). The quantities
δφ¯i,in, i = 1, . . . , 5 are calculated as the necessary change of the average incom-
ing characteristic variables required to meet the coupling condition (5.66). For
a derivation of the latter quantities and for a detailed description of the the-
ory of the developed mixing planes treatment for preconditioned system, the
reader may refer to the report of Anker et al. [19]. As in the case of an inlet
or outlet boundary, the local variations in the incoming characteristic quan-
tities δφ′i,j,in, i = 1, . . . , 5 are calculated applying the Giles-condition (5.6) for a
non-reflecting distribution of the flow variables at the boundary.
5.7 Verification of the non-reflectivity of the
developed NRBC treatment
An ideal boundary condition treatment for the steady-state simulation of tur-
bomachinery flow should prevent unphysical reflections at the in- and outlet
boundaries as well as at the mixing planes. To verify that a boundary con-
dition treatment is non-reflecting, a single-blade simulation may be carried
out on a short and a long domain. If a non-reflecting boundary condition
treatment is used, pressure perturbations will only be generated by the tur-
bomachinery blade. For this reason, the solution obtained on a small com-
putational domain, should not differ from the solution obtained on a larger
domain, if the circumferentially averaged solution of the large domain has
been used as boundary condition for the smaller domain. To assess NRBC for
unsteady flow simulations, it needs to be proven that the boundary condition
treatment does not influence vortices, temperature streaks (or alternatively
entropy variations), and pressure waves that are superimposed on free stream
flow, when they are passing through the boundaries that are meant to be non-
reflecting; cf. for instance the work of Yoo and Im [314] for a description of an
appropriate set of test cases. For steady-state simulations such a verification
procedure is not necessary as it only needs to be shown that the boundary
condition treatment is non-reflecting in the limit of a converged solution; for
this reason the aforementioned procedure of assessing the independence of
the flow solution on the size of the domain is already a sufficient means to
test the non-reflectivity of the boundary condition treatment for steady-state
simulations.
In order to demonstrate that the developed NRBC treatment leads to so-
lutions that are independent of the size of the computational domain, the vis-
cous flow field in the linear VKI-1 cascade [253] was simulated at isentropic,
outlet Mach numbers of Mis,a = 1.21,0.81 and 0.1, respectively. The meshes
used contain 154×53×3 and 113×53×3 nodes, respectively, and were con-
structed such that the nodes of the smallest grid coincide with the nodes of the
108 5. Development of a NRBC treatment for preconditioned systems
largest mesh. By proceeding in this way, grid dependence and interpolation
errors can be avoided [247].
In Figs. 5.1 and 5.2 computational results are shown for the cases of an
isentropic Mach number of Mis,a= 0.81 and Mis,a= 1.21, respectively. The re-
sults were obtained with the preconditioned (PC) solution scheme described
in Chapter 3 and the NRBC treatment, that was developed for the precon-
ditioned Navier-Stokes equations (NRPCBC) and described in the preceeding
sections. The solid lines represent isobars of the solution, that were obtained
on the smallest domain, whereas the dotted lines represent the correspond-
ing pressure contours of solution calculated on the largest domain. In order
to have a reference and to show the consequences, which a reflecting bound-
ary treatment may have, solutions obtained by Seybold [247] using the non-
preconditioned solution scheme (NonPC) and boundary treatments based ei-
ther solely on one-dimensional characteristic theory (1DpBC) or on the orig-
inal quasi-three-dimensional non-reflecting theory (NRBC) due to Giles and
Saxer [98, 242] for the unscaled Navier-Stokes equations are depicted for com-
parison.
For the case of an isentropic outlet Mach number of Mis = 1.21 it can be
observed in Fig. 5.1 that the developed boundary treatment for the precondi-
tioned Navier-Stokes equations leads to a solution that is dependent on the
domain size of the solution in the vicinity of the outflow boundary. However,
by comparing this solution with the corresponding solution obtained with the
unscaled Navier- Stokes equations and the original, non-reflecting boundary
condition treatment (NRBC-NonPC), this non-ideal behaviour is not specific
for the developed boundary treatment. It is well known, that a boundary
treatment based on the work of Giles is not capable of fully preventing re-
flections when shocks are crossing the boundary. This is due to the nonlinear
nature of shock formation and second order effects not considered in the linear
theory of Giles [63, 209, 247].
By comparing the solutions that were obtained with a non-reflecting bound-
ary condition treatment (NRBC-NonPC or NRPCBC-PC) with the solutions
gained by a treatment where one-dimensional characteristic theory is applied
pointwise (1DpBC-NonPC) for the Mach numbers of Mis=1.21 and Mis=0.81,
it becomes evident that the use of a non-reflecting boundary treatment only
leads to relatively small deviations in the flow field when the size of the com-
putational domain is changed. When a boundary treatment based merely on
one-dimensional characteristic theory is applied, the solution obtained on a
small and a large domain differ significantly.
In Figs. 5.3 and 5.4 computational results are shown, where a pressure
ratio was prescribed, which for ideal gases corresponds to an isentropic outlet
Mach number of Mis = 0.10. The results depicted in Fig. 5.3 were obtained
by simulations, where the fluid was modelled as an ideal gas, whereas the
results shown in Fig. 5.4 were obtained by simulations were the fluid was
modeled as incompressible (% = const.). The boundary treatment that is sim-
ply based on one-dimensional characteristic theory for preconditioned systems
(1DpPCBC) clearly leads to reflections for compressible and for incompress-
5.8 Comparison of developed and original NRBC 109
Small domain
Large computational domain
Small computational domain
NRBC-NonPC
NRPCBC-PC
1DpBC-NonPC
Figure 5.1: Pressure contours of the simulated flow field (Mis = 1.21) in the VKI-1
cascade in dependence of the boundary treatment and the size of the
computational domain
ible fluids, whereas the solutions obtained with the developed non-reflecting
boundary treatment on a small and large computational domain agree so well,
that it is difficult to observe any difference between them.
It is thus verified that the developed boundary conditions are non-reflecting
for incompressible fluids and subsonic, ideal gas flow.
5.8 Comparison of the developed and the orig-
inal NRBC treatment
The preconditioning method described in Chapter 3 was first used with the
original boundary conditions of Giles and Saxer, which is based on the char-
acteristics of the unscaled Navier-Stokes equations. However this often led to
instabilities in the vicinity of the inlet boundaries, outlet boundaries, and the
mixing planes. In order to demonstrate the effectiveness of the novel bound-
110 5. Development of a NRBC treatment for preconditioned systems
Small domain
Large computational domain
Small computational domain
NRPCBC-PC
1DpBC-NonPC
NRBC-NonPC
Figure 5.2: Pressure contours of the simulated flow field (Mis = 0.81) in the VKI-1
cascade in dependence of the boundary treatment and the size of the
computational domain
ary treatment at low Mach numbers, the flow through the second stator row
of the Bochum low speed turbine [6, 42] was simulated using the precondi-
tioned scheme (PC) with the original quasi-three-dimensional, non-reflecting
boundary treatment (NRBC) and with the consistent boundary treatment
(NRPCBC) presented earlier in this chapter. The latter boundary conditions
were also used without applying the non-reflecting correction, thereby basing
this boundary treatment (1DmPCBC) on the averaged characteristics only.
As boundary conditions for the isolated vane an inlet total pressure of 1013
mbar and an outlet static pressure of 1000mbar were specified. The difference
of 13 mbar corresponds approximately to the actual pressure difference over
the second stator.
The calculations were carried out on a grid consisting of 49×25× 33 grid
points. The flow solver was run using the scalar JST-scheme with a CFL
number of CFL = 2.0 and the coefficient k(4) of the fourth order dissipation
set to k(4) = 0.05. The coefficient of the pressure sensor of the precondition-
ing scheme pgr was set to unity. The computations with a boundary condition
5.8 Comparison of developed and original NRBC 111
Small domain
Small computational domain
Large computational domain
NRPCBC-PC
1DpPCBC-PC
Figure 5.3: Pressure contours of the simulated flow field in the VKI-1 cascade for
the isentropic outlet Mach number of Mis = 0.10 in dependence of the
boundary treatment and the size of the computational domain. The fluid
was modelled as an ideal gas
treatment based merely on the one-dimensional characteristics of the precon-
ditioned scheme diverged on the finest grid level. In order to increase the
robustness of the scheme the parameter pgr was raised to pgr=10. With this
setting, divergence could be avoided, but the residual stagnated after a drop
by 2.5 orders. In Fig. 5.5a the calculated flow field is depicted. It reveals an
occurrence of unphysical reflections at the inlet and outlet boundaries.
The original boundary condition treatment of Giles and Saxer did not lead
to a divergence of the flow computations, but with the preconditioning param-
eter set to pgr=1 the residuals did not drop at all. First after increasing the
parameter pgr to pgr = 10 a residual drop by 1.5 orders was attained. From
the computed flow field in Fig. 5.5b the reason for the poor convergence can
be seen: The lines of constant pressure are wiggling at the boundary, reveal-
ing a poor boundary treatment. The deficiency in the boundary treatment
is dependent on the Mach number. By lowering the outlet pressure to 700
mbar and thereby increasing the isentropic Mach number Mis to a value of
Mis=0.6, the solution converged and the original boundary condition seemed
to be non-reflecting for this case, cf. Fig. 5.5c.
112 5. Development of a NRBC treatment for preconditioned systems
Small domain
Small computational domain
Large computational domain
1DpPCBC-PC
NRPCBC-PC
Figure 5.4: Pressure contours of the simulated flow field in the VKI-1 cascade for
the isentropic outlet Mach number of Mis = 0.10 in dependence of the
boundary treatment and the size of the computational domain. The fluid
was modelled as an incompressible fluid (% = const.)
When using the novel non-reflecting boundary conditions for precondi-
tioned systems (NRPCBC) the residuals converge well and a physical flow
field is obtained. In Fig. 5.5d the isolines of the calculated static pressure are
depicted.
The problems encountered when a reflective or a non-consistent boundary
treatment is used, are worsened when the Mach number decreases. Further-
more, since setting pgr to a value greater than unity is nothing other than re-
ducing the preconditioning, the presented results show that the new boundary
conditions are indispensable, when low Mach number flow is to be computed
with time-derivative preconditioning methods.
5.8 Comparison of developed and original NRBC 113
(a) 1DmPCBC-PC,
∆p=13 mbar
(b) NRBC-PC,
∆p=13 mbar
(c) NRBC-PC,
∆p=313 mbar
(d) NRPCBC-PC,
∆p=13 mbar
Figure 5.5: Lines of constant pressure of the computed flow fields using different
boundary condition treatments
114
Chapter 6
Analysis of leakage flow effects
in axial turbomachinery
The solution scheme developed in the current work is in this chapter applied
to study the interaction of the leakage and main flow in a 1 1
2
stage low-speed
axial turbine with a shroud on the rotor row. The use of a preconditioned
solution scheme with non-reflecting boundary conditions is important for ob-
taining reliable results as the Mach number level is low (M = 0.05− 0.15) and
there are three blade rows. The test turbine is situated at the Chair of Steam
and Gas Turbines (DGT) of the Ruhr-University Bochum and the aerodynam-
ics of this turbine was subject to investigation in several projects on leakage
flow effects („Deckbandströmunsgeinfluss I-III“) funded by Forschungsvere-
inigung Verbrennungskraftmaschinen (FVV) in which the present author was
involved in the numerical investigations [6, 7, 9, 10, 13, 14, 19, 92, 202, 207,
208, 310]. The analysis presented in the following originates from the afore-
mentioned projects and contributes to the understanding of the interaction of
leakage flow and main flow in axial turbomachinery.
6.1 Description of the shrouded, low-speed ax-
ial turbine and available experimental data
The turbine considered in the current work is a 1 1
2
stage axial low-speed air
turbine comprising of inlet guide vane, rotor and stator blade rows. It is op-
erated in a test rig at the Ruhr-University of Bochum. The turbine consists
of two identical stator blade rows and a rotor with a labyrinth seal on the
shroud in between them; see Fig. 6.1 for a three-dimensional plot of the tur-
bine geometry. The prismatic profile of the turbine blades was taken from the
tip-section of an industrial gas turbine rotor-blade.
The air through the turbine passes a filter and is straightened and aligned
before it enters the measurement plane M0 ahead of the first stator. It is
drawn through the turbine by a centrifugal blower located downstream of
last measurement plane A1. In order to examine the effect of the influence
115
116 6. Analysis of leakage flow effects
of the seal clearance height and to keep a constant cavity depth, the casing
surface and not the wall separating the adjacent cavities was cut back in the
experiments (Fig. 6.2).
Figure 6.3 shows also a meridional view of the turbine and its measure-
ment planes. The axial distance between two adjacent blade rows corresponds
to 59.2% of the axial chords, which is considerably larger than in normal
practice, to allow space for measurement instrumentation. The test rig is
operated at peak efficiency condition (design condition) with a rotor speed of
n=500 rpm. Additional operational data are listed in Table 6.1.
The flow fields in the planes M1, M2 and M3 were measured with a pneu-
matic 5-hole probe at defined inlet and outlet conditions in M0 and A1 for two
different clearance heights, scl=1mm (scl/D=0.07 %) and scl=3mm, respec-
tively. In these planes time-accurate measurements were also carried out for
the largest clearance height using special triple-hot-wire probes. A detailed
description of the test rig and the extensive steady-state experimental data
available can be found in the report of Pfost et al. [207].
Number of rotor
blades Nrot 36
Number of stator
blades Nst 37
Chord length s 163 mm
Axial chord length sx 101.2 mm
Blade height H 170 mm
Aspect ratio H/s 1.043
Mean diameter D 1490 mm
Hub/tip radius ra-
tio Di/Do 0.795
Clearance height scl 1mm/3mm
Reynolds
number Re 4 · 10
5
Stagger angle βBi 37.5◦
Rotational
speed n˙
500 rpm
Rotor velocity
(midspan) vφ,rot 39m/s
Flow
coefficient
φ =
vx/vφ,rot
0.35
Stage loading
coefficient
ψ =
2∆hs/v2φ,rot
2.4
Reaction r 0.46
Table 6.1: Test rig specifications [6]. Geometrical and operational data
6.2 Modeling of the shrouded turbine
6.2.1 Grids, geometrical modelling and boundary
conditions
The 1 1
2
stage axial turbine with a generic labyrinth seal on the rotor was
discretized using a block-structured grid. In Fig. 6.4 the block topology and
the positions of the mixing planes are shown. Each of the individual blocks is
an H-type grid with hexahedral cells. Except for the mixing planes, the block-
interfaces are conformal, i.e., the grid points at the intersection of adjoint
blocks overlap. Fig. 6.5 shows some views of the grid used for the simulation
of the turbine for a clearance height of scl = 3mm.
The numerical grids used for the discretization of the 1 1
2
stage turbine
consist of 700 000 grid points in total. These grids were obtained from fine
6.2 Modeling of the shrouded turbine 117
Figure 6.1: 3D-plot of the turbine geometry and indication of the measurement
planes
meshes with 5.4 million grid points that were used in a grid independence
study by deleting every other grid line. The clearance height of the stator seals
is 0.1 mm. Since the experimental data has shown the stator hub leakage
flows to be negligible, the stator seals were not modeled.
At the inlet plane (M0) the measured radial distribution of total pressure
and temperature and axial flow direction is specified. For the outlet boundary
in A1 the measured radial distribution of the static pressure is prescribed.
Solid surfaces are assumed to be isothermal, and the no-slip condition is ap-
plied. Periodicity in the pitch-wise direction is ensured using phantom cells
that keep copies of the periodic values such that the points on these bound-
aries can be treated like interior points.
In the current work the labyrinth geometry is completely discretized. In
the work of Anker et al. [14] a numerical model for labyrinth seals was de-
veloped, with which a discretization of the labyrinth seal can be avoided. The
results of this study are summarized in Appendix H. Even though good re-
sults were obtained in the aforementioned study, a simplified modelling is
not suitable when the computational results are to be used for studying phe-
nomenological effects.
118 6. Analysis of leakage flow effects
1-3
6
17
rotor 1
direction
stator 2stator 1
rotational
Figure 6.2: Turbine and labyrinth seal geometry, and the definition of flow angles
6.2.2 Numerical modeling
The equations solved are the three-dimensional Favre-averaged Navier-Stokes
equations. They are formulated in a cylindrical coordinate system that rotates
at a constant angular velocity. The fluid was modelled as a perfect gas and a
modified algebraic Baldwin-Lomax model is used to describe the effect of tur-
bulence.
Due to the presence of low Mach numbers flow in the turbine, the pre-
conditioned solution scheme described in Chapter 3 was used. In order to
prevent spurious reflections at the inlet and outlet boundaries and at the
mixing planes the developed non-reflecting boundary treatment described in
Chapter 5 is used. For the purpose of obtaining a good resolution of the flow
field and simultaneously avoiding overshoots of the solution variables near
the boundary layers, the Roe scheme, which was extended for the use in con-
junction with Merkle’s time derivative preconditioning method (Chapter 4),
was employed in combination with a third order variable extrapolation and
the van Albada limiter.
The interactions of the main flow, the leakage and the cavity flow due to
the presence of the upstream and downstream stators are in the present tur-
bine configuration relatively small due to the large axial gaps between the
blade rows. As demonstrated in the work of Anker et al. [13] and as shown
in a summarized fashion in Appendix I, it is sufficient to conduct steady state
simulations in order to capture the time-average of the impact of the leakage
flow on the main flow. For this reason, the analysis of the interaction between
leakage and main flow focuses on results obtained from steady state simula-
tions.
6.3 Validation of the computational results
To verify that the solver ITSM3D captures the important effects when leakage
flow interacts with the main flow, the calculated circumferentially averaged
6.3 Validation of the computational results 119
Figure 6.3: Location of the measurement planes
velocities at the measurement planes M2 and M3 are compared to the data
obtained experimentally in the Bochum test facility for a clearance height of
scl=3mm, see Figs. 6.6 and 6.7. With the exception of the computational re-
sults denoted as ”fine” (obtained on the fine grids with 5.6 Mio. grid points),
the results used in this study were computed on meshes with 700 000 grid
points. The results confirm that the solutions used in this study can be re-
garded as grid independent.
As with all other plots in this chapter, the velocities presented are nor-
malized with the circumferential velocity vφ,rot (vφ,rot = 39m/s) of the rotor at
midspan and denoted with an asterix ”∗”. The measurement planes M2 and
M3 are positioned at a distance of 30 mm behind the trailing edge of the rotor
and the second stator, respectively. Since the flow field in M1 (30 mm behind
the trailing edge of the first stator) is nearly invariant of the clearance height
(cf. [6]), those results are omitted for brevity.
The measured circumferentially averaged axial velocity in M2 depicted in
Fig. 6.6a reveals that there is a reduced throughput in the hub region of the
rotor. The slightly reduced axial velocities near the hub are due to the hub-
side passage vortex which causes an overturning of the main flow. Figure 6.6b
shows the occurrence of large circumferential velocities in the casing region
incurred by the injection of leakage flow with large components of swirl.
The measured and predicted circumferentially averaged axial and circum-
ferential velocities in M3 are depicted in Fig. 6.7. Obviously, the flow solver
overpredicts the under- and overturning of the flow caused by the upper sec-
ondary channel vortex (near the casing) and this may be due to a weakness of
the Baldwin-Lomax turbulence model.
The computational results for a clearance height of scl=1mm are in similar
120 6. Analysis of leakage flow effects
Figure 6.4: Block topology and the position of the mixing planes
good agreement with the measurement data and this justifies the use of the
numerical results for discussing different aspects of the interaction of leak-
age flow with the main flow in the turbine considered. Additionally, it should
be noted that the present results were compared with results computed with
CFX-5 in the work of Sög˘üt [256]. The good agreement between the simu-
lations of two different flow solvers furthermore confirms the validity of the
results used in the following study.
6.3 Validation of the computational results 121
A
B
(a) Meridional view of the grid in the middle of
the blade channel
(b) Azimuthal view of the grid at
mid-span
(c) Detail A: Modeling of the labyrinth tip (d) Detail B: Modeling of the outlet gap
(e) Detail C: Modeling of the rotor tip (f) Detail D: Modeling of the trailing edge
of the rotor
Figure 6.5: Computational grid (reduced) for a clearance height of scl = 3mm
122 6. Analysis of leakage flow effects
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
h/
H 
[-]
Normalized axial velocity [-]
Measurement
Sim. (fine)
Sim. (coarse)
(a)
0
0.2
0.4
0.6
0.8
1
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
h/
H 
[-]
Normalized circumferential velocity [-]
Measurement
Sim. (fine)
Sim. (coarse)
(b)
Figure 6.6: Measured and simulated circumferentially averaged normalized axial v∗x
and circumferential v∗φ velocity components in M2 at a clearance height
of scl = 3mm
6.3 Validation of the computational results 123
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
h/
H 
[-]
Normalized axial velocity [-]
Measurement
Sim. (fine)
Sim. (coarse)
(a)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
h/
H 
[-]
Normalized circumferential velocity [-]
Measurement
Sim. (fine)
Sim. (coarse)
(b)
Figure 6.7: Measured and simulated circumferentially averaged normalized axial v∗x
and circumferential v∗φ velocity components in M3 at a clearance height
of scl = 3mm
124 6. Analysis of leakage flow effects
6.4 Relative importance of the geometrical
parameters of the labyrinth seal
The influence of the various geometrical parameters of the labyrinth seal were
investigated in the work of Anker et al. [14]. A short summary of the main re-
sults of that study can be found in Appendix G. One finding presented in the
aforementioned publication was that the leakage flow effects are primarily de-
pendent on the labyrinth seal leakage height scl. For this reason the labyrinth
seal leakage height is the only geometrical parameter that is changed in the
following study of the interaction between the leakage and the main flow.
6.5 Analysis of the interaction between
leakage, main and secondary flow
In this section the computational results for four different configurations of
the sealing arrangement of the rotor row are analysed, compared and dis-
cussed. The first two simulations for a tip clearance gap of scl = 1mm and
scl = 3mm identify the effect of the leakage flow on the main flow and have
already been addressed above. The second two identify the interaction of the
cavity flow with the main flow. Both of these have no leakage flow (tip clear-
ance gap of scl = 0mm), whereby in one case a smooth rotor casing with no
cavity is considered and in the second case the cavity at inlet and exit of the
rotor shroud seal is included in the numerical model.
6.5.1 The impact of the leakage flow on the main flow
For the largest clearance height (scl = 3mm) the leakage flow over the rotor
shroud experiences nearly no net deflection. This is visible from Fig. 6.8 in
which the streamlines of the flow passing through the labyrinth seal of the
rotor row are visualized in the rotating frame of reference. When this leakage
flow re-enters the main flow it leads to an increase of tangential momentum
near the casing. This becomes evident from Fig. 6.9, where the difference
between the simulated circumferential velocities for a clearance height of scl=
3mm and scl=0mm are plotted in the measurement plane M2.
The ingress of leakage flow with a high component of swirl after the rotor
leads to high suction side incidence (negative incidence) in the tip-region of the
following stator row. This can be seen from Fig. 6.10 in which the streamlines
of the flow around the second stator are visualized. The leakage flow through
the seal experiences the same pressure drop over the rotor as the main flow
but as no work is extracted this leads to increased velocities in the re-injected
flow near the casing (Fig. 6.11).
The leakage flow re-enters the main flow from the downstream cavity pref-
erentially on the suction side of the blade (Fig. 6.12). Due to the slower mov-
ing fluid in the wake, the leakage jet penetrates more deeply there than in
6.5 Main flow - leakage flow interaction 125
the mid-channel. For a clearance height of scl = 3mm the leakage flow leads
also to a reverse flow on the casing between the reentering leakage jet and the
suction side of the rotor blade [12, 14]. The flow blockage that arises reduces
the flow velocity on the suction side of the rotor flow near the tip (Fig. 6.13).
Figure 6.8: Undeflected streamlines in the labyrinth seal
6.5.2 Interaction between the potential field of the rotor
and the cavity flow
In Fig. 6.14 the calculated radial velocities in the inlet and outlet planes of the
labyrinth seal are plotted for the simulations with different clearance heights
of the labyrinth seal. The location of the labyrinth inlet and outlet planes
is sketched in Fig. 6.15. The largest positive radial velocity components (ex-
traction from main flow) are found in the pressure region of the blades and
the largest negative components (re-injection of leakage flow) are found in the
vicinity of the suction side. For the smallest clearance heights of scl = 0mm
and scl=1mm the leakage flow re-enters the main flow at the labyrinth inlet
plane, whereas for all clearance heights reverse flow situations occur at the
outlet cavity.
The circumferential variation of the radial velocity components at the in-
let and outlet planes of the cavities are caused by the potential field of the
rotor. In the available time-accurate numerical results on this test case [8],
this inhomogeneous distribution is the steady part of the rotor flow field in
the rotating frame of reference. The time-accurate simulations show that un-
steady perturbations are superimposed on this distribution by the wake of the
upstream vane and the potential fields of both stator rows, but that these ef-
fects are small due to the large axial distance between the stator and rotor
126 6. Analysis of leakage flow effects
PHI
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
.5
.4
.3
.2
.1
.1∆v
∗
φ [-]
Second stator row
Rotor row (shroud not plotted)
Figure 6.9: Difference ∆v∗φ between the simulated normalized circumferential veloc-
ities in M2 (30 mm downstream of the rotor) for a clearance height of
scl = 3mm and scl = 0mm
rows. For a detailed discussion of the effect of the potential effects of the sta-
tor and the rotor row on the leakage flow, the reader is referred to the work of
Anker et al. [8, 14].
6.5.3 Consequences of the leakage flow extraction on the
secondary flow field of the rotor
The extraction of leakage flow has a negligible effect on the flow field in the
first stator. In the rotor row the extraction of leakage flow leads to a removal
of the slowly moving fluid on the shroud wall, such that there is a reduc-
tion of the boundary layer thickness at the casing wall of the rotor. This
affects the development of the upper secondary channel vortex, as can be
seen from Fig. 6.16 where the helicity H of the relative velocity vector field
wi = wi(x, y, z) defined as
H = wi · ωi, ωi = ijk ∂wk
∂xj
(6.1)
is plotted at an axial plane in the middle of the rotor as a function of the
clearance height. Since the norm of the vorticity ‖ωi‖ is a measure of the
strength of the secondary flow, the helicity represents the rate of transport of
secondary flow. Helicity has the advantage over vorticity that it identifies sec-
ondary flows more clearly than vorticity. Unlike the vorticity which is induced
by the no-slip walls and has its maximum there, the helicity vanishes at the
walls since the vorticity and velocity vector fields are orthogonal to each other
in sheared layers.
6.5 Main flow - leakage flow interaction 127
Figure 6.10: Calculated streamlines, showing a suction-sided incidence of the second
stator caused by leakage flow
A
0.85
0.75
0.65
0.55
0.45
0.35
0.25
0.15
0.05
scl=3mm
V ∗ [-]
scl=0mm
Figure 6.11: Magnitude V ∗ of the normalized absolute velocities in the measurement
plane M2 (30 mm behind the rotor)
An increase in the shroud clearance height reduces the strength of the up-
per channel vortex such that the relative secondary kinetic energy decreases
from the measurement plane M1 up to the axial midplane MP in the rotor, as
can be seen from Table 6.2. The secondary kinetic energy coefficient is thereby
defined as
ζ◦sec,MP-M1 =
E◦sec,MP − E◦sec,M1
E◦MP − E◦M1
(6.2)
where E◦ and E◦sec denote the kinetic energy of the main and secondary flow
of the rotor, respectively. The secondary flow vectors were calculated as de-
scribed in the report of Anker et al. [6]. The secondary kinetic energy co-
efficients have been used to identify changes in performance as they appear
to be more reliable than predicted losses [114]. This method determines the
strength of the secondary velocities at a point in the flow from the difference
128 6. Analysis of leakage flow effects
Figure 6.12: The leakage flow re-entering the main flow behind the rotor
Clearance height ζ◦sec,MP-M1 [%] ζ
◦
sec,M2-M1 [%] ζM3-M2 [%]
3 mm -0.53 4.94 4.99
1 mm -0.16 2.44 2.65
0 mm (w/o cavity) 0.83 1.65 2.84
0 mm (with cavity) 1.34 1.76 2.40
Table 6.2: Secondary kinetic energy coefficients for the rotor and the second stator
row
between the local velocity vector and the main flow vector. The direction of
the main flow is defined by the circumferential average of the pitch angle and
by the spanwise average of the yaw angle for this point.
It should be noted that for the clearance heights of scl=1mm and scl=3mm
the secondary kinetic energy coefficients ζ◦sec,MP-M1 have negative values, which
implies that the extraction of leakage flow leads to a homogenization of the
secondary flow field. Paradoxically, the secondary flow losses in the rotor tend
to decrease with increasing tip gap as the leakage flow removes more of the
endwall boundary layer.
6.5.4 Consequences of the leakage flow re-injection on
the secondary flow field of the rotor
The re-injection of leakage flow strongly influences the secondary flow field in
the casing region downstream of the rotor. This can be seen from Fig. 6.17a
and 6.17b where the helicity in M2 is plotted for a clearance height of scl =
0mm (with cavities) and scl = 3mm, respectively. A positive helicity cor-
responds to a clockwise rotation of the flow seen from the perspective of a
particle moving with the flow. Consequently, the white region near the cas-
6.5 Main flow - leakage flow interaction 129
C
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
W ∗ [-]
scl=0mm (w. cavities) scl=3mm
Figure 6.13: MagnitudeW ∗ of the normalized velocities in the rotor system near the
casing (h/H=0.95)
ing represents the upper secondary flow vortex. In Fig. 6.18a the simulated
helicity for scl = 0mm (with cavities) is subtracted from the simulated he-
licity for a clearance height of scl = 3mm. Obviously, the circumferentially
non-uniform leakage flow strengthens the upper channel vortex and leads to
a more pronounced neighbouring trailing edge vortex. This effect leads to in-
creased secondary kinetic energy coefficients in plane M2 (cf. Table 6.2). Thus,
the re-injection considerably outweighs the benefits of the extraction of leak-
age flow discussed above, such that leakage flow leads to a net increase in the
secondary losses over the rotor.
6.5.5 The impact of the presence of cavities on the for-
mation of the secondary flow field
The changes in the rotor secondary flow field are not only caused by the leak-
age flow but also by the mere presence of cavities. When cavities are present,
and the clearance height is of moderate size, the flow leaves the main flow
on the pressure side and re-enters on the suction side of the rotor at both the
inlet and the outlet cavity. This strengthens the secondary channel vortex in
this region leading to an increased kinetic energy of the secondary flow field
in the rotor. A comparison of the computed helicity in the axial midplane
MP in the rotor for a vanishing clearance height with and without cavities
(Figs. 6.16a and 6.16b) shows that the first cavity of the labyrinth seal is re-
sponsible for a strengthening of the upper channel vortex and for an increase
in the secondary kinetic energy coefficient (cf. Table 6.2).
130 6. Analysis of leakage flow effects
R
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
v∗r [-]
1 mm 3 mm0 mm
φ
0.2 · x
R
0.15
0.05
-0.05
-0.15
-0.25
-0.35
-0.45
-0.55
-0.65
-0.75
-0.85
v∗r [-]
0 mm 1 mm 3 mm
φ
0.2 · x
Figure 6.14: Normalized radial velocities v∗r . Labyrinth inlet plane (left), labyrinth
outlet plane (right)
Extracted leakage flow Re-injected leakage flow
M1 MP M2
Labyrinth inlet plane
Labyrinth outlet plane
First cavity
Main flow
Figure 6.15: Location of the axial midplane MP in the rotor and the inlet- and outlet
planes of the labyrinth seal
The presence of cavities does not lead to such a pronounced effect in the
secondary flow field behind the rotor. As can be seen from Fig. 6.18b, where
the difference in the simulated helicity for zero clearance height with and
without cavities is plotted in M2, the presence of cavities reinforces the upper
channel vortex. However, this effect only leads to a marginal change of the
secondary kinetic energy coefficient. (It should be noted, that the helicity
differences are plotted over a smaller range in Fig. 6.18b than in Fig. 6.18a.)
6.5.6 Impact of the Leakage Flow on the Secondary Flow
Field in the Second Stator Row
The formation of secondary flow in the second stator is strongly influenced by
the leakage flow. The helicity of the simulated flow fields in M3 (30 mm behind
the second stator) for a clearance height of scl=3mm and scl=0mm is depicted
in Fig. 6.19. A negative value for the helicity corresponds to a counterclock-
6.5 Main flow - leakage flow interaction 131
HELI
200
150
100
50
0
-50
-100
-150
-200
H [m · s−2]
(a) scl=0mm (with cavity)
HELI
200
150
100
50
0
-50
-100
-150
-200
H [m · s−2]
(b) scl=0mm (w/o cavity)
HELI
200
150
100
50
0
-50
-100
-150
-200
H [m · s−2]
(c) scl=1mm
HELI
200
150
100
50
0
-50
-100
-150
-200
H [m · s−2]
(d) scl=3mm
Figure 6.16: Helicity H in an axial plane in the middle of the rotor
wise rotation of the flow seen from the perspective of a particle moving with
the flow. Thus, the upper channel secondary vortex can be identified by dark
regions in the casing region. Apparently, the leakage flow gives rise to a mag-
nification of the upper channel vortex, which also leads to an increased sec-
ondary kinetic energy coefficient, cf. Table 6.2. There are two reasons for this.
Firstly, the leakage flow has a high swirl and causes a suction side incidence
on the upper part of the second stator row. This generates a positive radial
pressure gradient on the suction side and a negative radial pressure gradient
on the pressure side, and this strengthens the driving mechanisms respon-
sible for the development of the channel vortex. Secondly, the re-injection of
leakage flow into the main flow behind the rotor leads to a recirculation zone
which causes a thickening of the boundary layer ahead of the second stator.
This thickening of the inlet boundary layer of the second blade row also aug-
ments the upper channel secondary vortex with an increase in the clearance
height.
132 6. Analysis of leakage flow effects
75
45
15
-15
-45
-75
H [m/s2]
(a) scl=0mm (with cavity)
75
45
15
-15
-45
-75
H [m/s2]
(b) scl=3mm
Figure 6.17: Helicity H in M2 (30 mm behind the rotor)
75
45
15
-15
-45
-75
∆H [m/s2]
(a) Difference between scl = 3mm and
scl=0mm (with cavity)
HEL
20
12
4
-4
-12
-20
∆H [m/s2]
(b) Difference between scl = 0mm (with
cavity) and scl=0mm (w/o cavity)
Figure 6.18: Helicity differences ∆H in M2 (30 mm behind the rotor)
6.6 Discussion of the results and consequences
for the design of shrouded axial turbines
From the results presented above, several possible design improvements in
shrouded axial turbines can be identified. Firstly, the leakage flow from the
exit cavity should not be forced to enter radially into the main flow but should
have more inclination in the axial direction to avoid the effects of flow block-
age. Secondly, to avoid the negative effects incurred by the excess swirl in the
leakage flow upon reentry in the main flow, it could prove advantageous to in-
stall deflectors on the casing immediately after the outlet of the labyrinth seal.
Alternatively, for large clearance heights one might install turning devices di-
rectly on the shroud to lower the swirl of the leakage flow and simultaneously
to extract work. In either case, the inlet of the labyrinth seal should be de-
signed such that primarily near-wall fluid with a low level of swirl and exergy
is extracted from the main flow. Furthermore, since not only the leakage flow
but also the mere presence of cavities induces losses, the shape of the inlet
and outlet cavities of the labyrinth seals must be considered with care in the
design.
If the injection of leakage flow with a high level of swirl can not be avoided,
6.6 Discussion of results 133
(a) scl=0mm (b) scl=3mm
Figure 6.19: Helicity H in M3 (30 mm behind the second stator)
it would be important to twist the tip-section of the blade row following the
labyrinth seal. In this case suction side incidence on the following blade row
and subsequent increase in the aerodynamic losses could be avoided.
134
Chapter 7
Summary and conclusions
In this thesis a numerical scheme for the solution of the Navier-Stokes equa-
tions for the simulation of flows in turbomachinery at arbitrary Mach num-
bers is presented. The developed numerical methods are implemented in a
density based solution scheme, which was originally concipated for the sim-
ulation of transonic flows in rotating machines. To ensure that the solution
scheme is efficient and accurate at all Mach number levels, a time-derivative
preconditioning method is incorporated in the solution scheme. In the present
work the finite volume discretization and the applied time-stepping scheme
are described, details of the preconditioning method are given, and general
issues related to the simulation of low Mach number flows are addressed.
To ensure a sharp resolution of the flow features both for low-speed as well
as for transonic flows, several low dissipation schemes are developed for the
robust use in the context of time-derivative preconditioning methods. In the
current work, both the scalar and the matrix dissipation scheme due to Jame-
son, Schmidt and Turkel (JST) are adapted for the use in conjunction with
preconditioning methods. As demonstrated in the current work, while the im-
proved pressure switch due to Swanson and Turkel suppresses overshoots of
the solution near shocks, these schemes fail to produce monotone solutions for
boundary layers regardless which form of the pressure switch is used.
For this reason, also Roe’s characteristic upwind dissipation and Liou’s
AUSM+ scheme are formulated for the use in conjunction with the time-
derivative preconditioning scheme of Merkle. Applying a variable extrapola-
tion and limiters, these schemes yield monotone and bounded solutions both
over shocks and boundary layers. In the current work it is in particular shown
how the AUSM+ upwind scheme can be applied to a cell-vertex scheme. By
splitting up the convective fluxes in a central, a diffusive, and an anti-diffusive
part, an approach is developed for the implementation of this scheme in a
central-differencing code. To enhance the robustness of the AUSM+ scheme,
a method to control the anti-diffusive part of the dissipative fluxes is devised.
Common for all of the dissipation schemes implemented in the current work,
is that they have been formulated for fluids with arbitrary equations of state.
In order to avoid unphysical reflections at the inlet, outlet, and mixing
planes in turbomachinery simulations, a non-reflecting boundary condition
135
136 7. Summary and conclusions
treatment should be used. In the current work the quasi-three-dimensional
non-reflecting boundary condition treatment devised by Giles and Saxer was
extended to preconditioned systems and general equations of state; as demon-
strated in the current work, since preconditioning changes the dynamics of
the governing equations, a boundary condition treatment based on the charac-
teristics of the unscaled Navier-Stokes equations can not be used. The theory
of a new boundary treatment is derived and the main aspects of the imple-
mentation are presented.
To demonstrate the accuracy and efficiency of the developed preconditioned
solution scheme several test cases are presented. Varying the isentropic Mach
number in the Ni-Bump test case from 0.013 to 0.85 shows that with the use of
time-derivative preconditioning Mach number independent convergence rates
can be obtained. While preconditioning does not impair the shock capturing
property of the original solution scheme, the accuracy of density based solu-
tion methods is preserved at low Mach numbers only when preconditioning is
used.
In order to assess the ability of the presented dissipation schemes of re-
solving viscous flows, results from the simulation of a laminar boundary layer
developing over a flat plate are discussed. It is shown that the AUSM dis-
sipation and the Roe’s characteristic upwind dissipation model lead to a bet-
ter resolution than the dissipation schemes based on fourth order differences
(the scalar and the matrix JST dissipation schemes); furthermore the former
schemes do not result in overshoots in the solution variables. However, even
though the matrix dissipation scheme fails to provide a monotone solution, it
clearly represents an improvement of the scalar JST-scheme.
In a third test case the simulation of the supersonic flow through a row
of wedges is considered. It is shown that the scalar JST-scheme with the
original pressure switch resolves shocks very well but leads to oscillations in
the solution variables in the vicinity of the shock. As a remedy, a modified
pressure switch which gives the scheme TVD-properties may be used, but
then the increased diffusivity of the scheme reduces the crispness of the shock-
front. While the use of matrix dissipation lowers the numerical diffusion and
enhances the shock resolution, it is the AUSM dissipation scheme that leads
to the best compromise between shock resolution ability and monotonicity of
the solution scheme.
In a further test case the turbulent flow in a 1.5 stage low-speed test rig
was computed. Without the use of preconditioning unphysical wiggles in the
pressure field are observed. On a coarse grid the original scheme completely
fails to reproduce the secondary flow effects measured, whereas the precon-
ditioned solver captures the tendencies of the secondary flow effects and pro-
duces smooth solutions in all regions of the flow.
To validate the novel non-reflecting boundary condition treatment devel-
oped in the present work, two test cases are presented. In the first test case
the viscous flow through a linear cascade with VKI-1 turbine blades are sim-
ulated at different Mach numbers. By varying the size of the computational
domain it is verified that the developed boundary conditions are non-reflecting
137
for incompressible fluids and subsonic, ideal gas flows.
Furthermore, the flow through an isolated vane using different bound-
ary conditions in conjunction with the preconditioned scheme is simulated.
The original boundary condition treatment due to Giles and Saxer for the
unscaled Navier-Stokes equations can be used in the case of an isentropic
outlet Mach number of Mis=0.6. However, the use of these unmodified bound-
ary conditions leads to instabilities and an unphysical pressure distribution
in the vicinity of the boundaries at low Mach numbers. Obviously, the orig-
inal boundary treatment can only be used at compressible flow conditions,
where the preconditioned equations revert to the physical ones. The benefit
of the novel boundary conditions presented in this work is given by the fact,
that they are the only ones that lead to stable integration and non-reflecting
boundaries over a broad Mach number range.
The developed solution scheme is used to investigate the impact of rotor
labyrinth seal leakage flow on the main flow and on the loss generation mech-
anisms in an axial turbine. After a validation of the computational results
with available measurement data, a parameter study is carried out to study
the influence of the geometrical parameters on the leakage flow effects. Of all
parameters investigated, the labyrinth seal leakage gap has the most impor-
tant effect.
A variation of the latter parameter between a vanishing and a realistic
clearance height allows a systematic investigation of the impact of the leakage
flow on main and secondary flow. The computational results show that the
ingress of leakage flow downstream of the shrouded rotor not only leads to
excess swirl but also to an increased kinetic energy in the upper part of the
casing region. Furthermore, the circumferentially non-uniform re-injection of
the leakage flow leads to a blockage of the main flow on the suction side of the
rotor near the tip.
The influence of extraction and re-injection of the leakage flow on the sec-
ondary flow field in the rotor and downstream stator is studied. The egress of
leakage flow reduces the boundary layer thickness on the casing wall of the
rotor which weakens the upper secondary channel vortex in the rotor channel.
However, the non-uniform re-injection of leakage flow strengthens this vortex
and leads to an overall increase of the losses in the rotor. The high swirl of the
leakage flow causes a suction side incidence onto the following stator row and
thus gives rise to radial pressure gradients. The re-injection of leakage flow
also leads to a recirculation zone after the rotor, which results in a thickening
of the boundary layer ahead of the second stator. Both effects are responsible
for the amplification of the upper secondary channel vortex in the subsequent
stator.
The effect of the inlet and exit cavities on the rotor flow field is also studied.
It is shown that, when cavities are present and the clearance height is of
moderate size, the flow leaves the main flow on the pressure side and re-
enters on the suction side of the rotor both at the inlet and the exit cavity.
This effect is also responsible for a strengthening of the secondary channel
vortex in the rotor.
138 7. Summary and conclusions
Several possible design improvements are identified from the analysed
flow effects. The leakage flow from the exit cavity should not be forced to enter
radially into the main flow but should have more inclination in the axial di-
rection to avoid the effects of flow blockage. The excess swirl of the reentering
leakage flow could either be reduced via baffles mounted on the casing wall af-
ter the outlet of the labyrinth seal or by applying winglets or turning devices
in the labyrinth seal. If the flow velocity of the re-entering leakage flow can
not be adjusted by constructive means to avoid excess swirl, the blade of the
following stator row should be twisted in the casing region to avoid suction
side incidence and a subsequent increase in the aerodynamic losses. Since
the present results show that not only the leakage flow but also the mere
presence of cavities induce aerodynamic losses, the geometry of the inlet and
outlet cavities must be considered with care in the design and optimization
process of sealing arrangements for axial turbines.
Appendix A
Derivation of time-derivative
preconditioning methods by the
use of perturbation analysis
Time-derivative preconditioning methods can be designed as methods that
cluster the eigenvalues of the flow equations under consideration. With the
motivation of alleviating the disparity of the eigenvalues, these techniques
were introduced for the simulation of inviscid, low Mach number flows. It
is sufficient to analyse and modify the eigenvalue distribution to avoid stiff-
ness of the flow equations and so attain an efficient computation of inviscid
flows, but not for systems of transport equations like the Navier-Stokes equa-
tions that with the presence of both diffusive and convective terms describe
flows governed by multiple physical processes. A more general method of
achieving a good conditioning of the flow equations is to scale the various
physical terms such that all are of the same order even under limiting condi-
tions. In the following a perturbation analysis similar to that of Merkle and
Venkateswaran [186, 189] will be used to derive a preconditioning technique
that leads to well-conditioned flow equations in the limit of low Mach and/or
Reynolds numbers.
A.1 Derivation of a preconditioningmethod for
inviscid, low Mach number flows
In this section perturbation analysis applied on the Euler equations will be
used to demonstrate that the Merkle preconditioning technique for inviscid,
low speed flows can be derived by the criterion that all terms in the flow equa-
tions are of the same order in the limit of low Mach number flows. The follow-
ing derivation will show an alternative way of determining the precondition-
ing parameter %′p in Merkle’s preconditioning scheme.
The one-dimensional Navier-Stokes equations
∂Q
∂t
+
∂E
∂x
= 0 (A.1)
139
140 A. Perturbation analysis
with
Q = [%, %u, %e0]
T ,
∂E
∂x
= [%u, %u2 + p, %uh0]
T (A.2)
serve as a basis for the following analysis. By using the chain rule, the un-
steady term can be rewritten as ∂Q
∂t
= ∂Q
∂Qv
∂Qv
∂t
with
Qv = [p, u, T ]
T (A.3)
and the last equations can be transformed to
%p
∂p
∂t
+ %T
∂T
∂t
+
∂(%u)
∂x
= 0,
%
∂u
∂t
+ %u
∂u
∂x
+
∂p
∂x
= 0,
−(1− %hp)∂p
∂t
+ %hT
∂T
∂t
+ %u
∂h
∂x
− u∂p
∂x
= 0.
(A.4)
A normalization with the reference scales
L†, u†, p†, ρ†, T † (A.5)
leads to the following system [5]:
%∗p
∂p∗
∂t∗
+ %∗T
∂T ∗
∂t∗
+
∂(%∗u∗)
∂x∗
= 0,
%∗
∂u∗
∂t∗
+ %∗u∗
∂u∗
∂x∗
+
(
p†
%†u†2
)
∂p∗
∂x∗
= 0,
−(1− %∗h∗p)
∂p∗
∂t∗
+ %∗h∗T
∂T ∗
∂t∗
+ %∗u∗
∂h∗
∂x∗
− u∗ ∂p
∗
∂x∗
= 0
(A.6)
where the asterix "∗" denotes that a flow variable has been normalized, i.e.,
a state variable φ∗ is defined as φ∗ = φ/φ†, where φ† is a reference state. To
avoid the presence of the dimensionless characteristic numbers p†/(%†h†) and
L†/(t†u†) in the flow equations, the enthalpy scale h† and the time scale t† were
respectively related to the other dimensionless quantities according to
h† = p†/ρ†, t† = L†/u†. (A.7)
If the reference pressure p† is set to the thermodynamic pressure, then the
characteristic number p†/(%†h†) is in the order of unity for ideal gases. The
reference time scale t† could also have been defined using the speed of sound
a† as reference velocity. The use of the convective velocity u† as reference
speed is however more convenient than the use of the speed of sound as the
former term appears explicitly in the flow equations and consequently leads
to the simplest algebra. Also for incompressible flows, the application of the
convective velocity as reference velocity is a natural choice. For a rigorous
analysis of low Mach number flow, a multi-scale analysis should be carried
out in which both an acoustic reference time scale τ † based on the sonic speed
A.1 A preconditioning method for inviscid, low Mach number flow 141
and a convective time scale t† are used. However, as shown by Müller [194], if
the temporal resolution of acoustic effects is not of interest, a multiple-scale
analysis, in which both convective and acoustic velocity and time scales are
accounted for, leads to the same conclusions as a single scale analysis in the
study of the asymptotical behaviour of the Navier-Stokes equations at low
Mach numbers. In the work of Müller [194] it shown that the acoustically fil-
tered asymptotic flow equations derived by a multiple-scale analysis coincide
with the asymptotic flow equations derived by a single-scale analysis.
With the aforementioned choice of reference scales, only the ratio of the
dynamic pressure to the static pressure,
 = %†u†
2
/p†, (A.8)
appears in the flow equations. At low Mach numbers the reference velocity
u† approaches zero so that this ratio , which is scaling the pressure gradient
in the momentum equations, becomes small. For an ideal gas the ratio  is
proportional to the square of the Mach number, i.e.,  = γM2. In the following
the limiting form of the equations as this term goes to zero is studied.
Since the term  only appears before the pressure gradient in the momen-
tum equations, it is natural to expand the pressure in a power series of this
parameter
p = p(0) + p(2) + . . . . (A.9)
It would also have been possible to expand the pressure in dependence of the
Mach number and then also having a first order pressure p(1) in the the series
expansion:
p = p(0) +Mp(1) +M
2p(2) + . . . . (A.10)
As can be shown by a more thorough analysis [5, 194], the first order pressure
drops out of the flow equations and does not need to be considered, unless
time-accurate, acoustic effects are of importance. In a single-scale analysis all
the variables should ideally be perturbed; however as shown by Müller [194]
this does not change the result of the current perturbation analysis. When
applying the pressure expansion in Eq. (A.9) in the following equations, it
must be assumed that the Mach numbers are at least one order less than
unity (M  1) to ensure a separation of the zeroth, first and second order
pressure.
Using the expansion given in Eq. (A.9) the normalized momentum equa-
tion
%
∂u∗
∂t∗
+ %∗u∗
∂u∗
∂x∗
+
1
ε
∂(p∗(0) + p
∗
(2) + . . .)
∂x∗
= 0 (A.11)
is obtained. Since the term 1

∂p∗
(0)
∂x∗ cannot be matched by any other term in the
equation, it follows
∂p∗(0)
∂x∗
= 0, (A.12)
i.e., the zeroth order pressure is independent of the spatial coordinate and can
only vary temporally by being subject to a global change. This implies that the
142 A. Perturbation analysis
zeroth order pressure does not directly have an impact on the velocity field; it
only has an indirect impact on the flow field by a change in the thermodynamic
states like the density. By assuming steady boundary conditions, vanishing
temporal variations of the zeroth order pressure follows. For this reason only
the second order pressure p∗(2) and not the zeroth order pressure p
∗
(0) appears
in the following flow equations.
After separating out the zeroth order pressure, the momentum equation
can be written as
%∗
∂u∗
∂t∗
+ %∗u∗
∂u∗
∂x∗
+
∂p∗(2)
∂x∗
= 0. (A.13)
Since this equation only has a temporal term containing the velocity, it can
only be used in a time marching scheme to update the velocity but not the
second order pressure. With the expansion of the pressure in Eq. (A.9) and by
accounting for relation (A.12), the energy equation can be written as
−(1− %∗h∗p)
∂p∗(2)
∂t∗
+ %∗h∗T
∂T ∗
∂t∗
+ %∗u∗
∂h∗
∂x∗
− u∗∂p
∗
(2)
∂x∗
= 0. (A.14)
It should be noted that all the terms in the last equation involving the pres-
sure are of second order while the other terms are of zeroth order in the Mach
number. From this it follows that the energy equation is not an adequate
means to update the second order pressure p(2) unless one scales the quantity
h∗p. The last equation furthermore show that even the energy equation for in-
viscid, ideal gas flows can be represented by the following transport equation
for the pressure p
∂p
∂t
+ u
∂p
∂x
+ γp
∂u
∂x
= 0, (A.15)
the energy equation is better suited to update the temperature than the pres-
sure in the case of low Mach number flow. It should be noted that for incom-
pressible flows, the term 1− %hp is zero, from which it follows that the energy
field for constant density flow is independent of the temporal variations of the
second order pressure.
Having separated out the zeroth order pressure term, the continuity equa-
tion reads as follows
%∗p
∂p∗(2)
∂t∗
+ %∗T
∂T ∗
∂t∗
+
∂(%∗u∗)
∂x∗
= 0. (A.16)
In the last equation all terms are of zeroth order in  except the temporal
derivative of the pressure which is of first order and consequently will vanish
for low values of . Hence this equation is not suitable for updating the pres-
sure at low Mach numbers. To ensure that the time derivative of the pressure
is of the same order as the convective term irrespective of the Mach number
level of the flow, the physical property %∗p should be replaced with an artificial
quantity %′∗p defined as
%
′∗
p =
kp

=
kpp
†
%†u†2
(A.17)
A.2 A preconditioning method for low Mach number, viscous flows 143
with the constant kp in the order of unity. In dimensional form the last equa-
tion reads as
%
′
p = kp/u
†2. (A.18)
By replacing the property %p with the quantity %′p, the following system of flow
equations results
kp
∂p∗(2)
∂t∗
+ %∗T
∂T ∗
∂t∗
+
∂(%∗u∗)
∂x∗
= 0,
%∗
∂u∗
∂t∗
+ %∗u∗
∂u∗
∂x∗
+
∂p∗(2)
∂x∗
= 0,
%∗h∗T
∂T ∗
∂t∗
+ %∗u∗
∂h∗
∂x∗
= 0
(A.19)
for ideal gases in the limit of a vanishing Mach number. As all terms are
in the order of unity, the convective and the temporal terms are in balance
and the system is well-conditioned for time marching schemes. The property
%T may be changed to avoid the appearance of temporal derivatives of the
temperature in the continuity equation; with the aforementioned modifica-
tion of the parameter %p, the other properties hT and (1 − %hp) need not to be
scaled [5, 289] to achieve a good conditioning of the flow equations for inviscid
flow in the low Mach number limit.
A.2 Derivation of a preconditioningmethod for
low Mach number, viscous flows
In this section perturbation analysis applied on the Navier-Stokes equations
will be used to show that the Merkle preconditioning technique for viscous,
low speed flows can be derived by the criterion that all terms in the flow equa-
tions are of the same order in the limit of low Mach and low Reynolds number
flows. The derivation will show an alternative way of determining proper
values for the preconditioning parameters %′p, %′T , h′T , and h′p in Merkle’s time-
derivative preconditioning scheme.
The Navier-Stokes equations
%p
∂p
∂t
+ %T
∂T
∂t
+
∂(%u)
∂x
= 0,
%
∂u
∂t
+ %u
∂u
∂x
+
∂p
∂x
=
4
3
∂
∂x
(
µ
∂u
∂x
)
,
−(1− %hp)∂p
∂t
+ %hT
∂T
∂t
+ %u
∂h
∂x
= u
∂p
∂x
+
∂
∂x
(
kλ
∂T
∂x
)
+ µ
4
3
(
∂u
∂x
)2 (A.20)
are used as a starting point of the following analysis. By normalizing the flow
quantities with the following reference scales
L†, u†, p†, ρ†, T †, h†, t†, µ†, k†λ, (A.21)
144 A. Perturbation analysis
and by expanding the pressure according to Eq. (A.9), the Navier-Stokes equa-
tion reads as
(
L†
t†u†
)(
 %∗p
∂p∗(2)
∂t∗
+ %∗T
∂T ∗
∂t∗
)
+
∂(%∗u∗)
∂x∗
= 0,(
ReL†
t†u†
)
%∗
∂u∗
∂t∗
+ Re %∗u∗
∂u∗
∂x∗
+ Re
p†
%†u†2
∂p∗(2)
∂x∗
=
4
3
∂
∂x∗
(
µ∗
∂u∗
∂x∗
)
,(
PeL†
t†u†
)[
%∗h∗T
∂T ∗
∂t∗
− dp
∂p∗(2)
∂t∗
]
+ Pe %∗u∗
∂h∗
∂x∗
= u∗Pe
∂p∗(2)
∂x∗
+
∂
∂x∗
(
k∗λ
∂T ∗
∂x∗
)
+
Pr
u†2
h†
µ∗
(
∂u∗
∂x∗
)2
(A.22)
with the abbreviation dp given by dp = 1− %hp and the Reynolds, Prandtl, and
Peclet numbers defined as
Re =
%†u†L†
µ†
, Pr =
µ†h†
k†λT †
, Pe = RePr =
%†u†L†h†
k†λT †
. (A.23)
Here it is assumed that the Mach number level is low. It then follows that
the gradient of the zeroth order pressure vanishes, just as in the perturbation
analysis for the inviscid flow equations in the previous section. By assuming
steady boundary conditions, it follows that the temporal variations of the ze-
roth order pressure vanish. For this reason, only the second order pressure
p∗(2) and not the zeroth order pressure p
∗
(0) appears in the flow equations (A.22).
The reference time scale t† is determined by the condition that the coef-
ficient of the time-derivative term in the momentum equations should be in
the order of unity to balance the viscous term. This leads to the following
expression
t† =
ReL†
u†
=
L†2
ν†
. (A.24)
To ensure that the pressure gradient term in the momentum equations
balances the viscous forces, the term Re p†/(%†u†2) must be in the order of
unity. From this condition it follows that the perturbation parameter  has to
satisfy
 =
%†u†2
p†
1
Re†
, (A.25)
which for ideal gases leads to the requirement  = γM †2/Re†.
With the aforementioned definitions of the time scale and the perturbation
A.2 A preconditioning method for low Mach number, viscous flows 145
parameter , the normalized Navier-Stokes equations read as follows
%∗p
%†u†2
p†Re†
2
∂p∗(2)
∂t∗
+
%∗T
Re
∂T ∗
∂t∗
+
∂(%∗u∗)
∂x∗
= 0,
%∗
∂u∗
∂t∗
+ Re %∗u∗
∂u∗
∂x∗
+
∂p∗(2)
∂x∗
=
∂
∂x∗
(
µ∗
∂u∗
∂x∗
)
,
Pr
[
%∗h∗T
∂T ∗
∂t∗
− dp%
†u†2
p†
1
Re
∂p∗(2)
∂t∗
]
+
Pe %∗u∗
∂h∗
∂x∗
− Pru∗%
†u†2
p†
∂p∗(2)
∂x∗
=
∂
∂x∗
(
k∗λ
∂T ∗
∂x∗
)
+
Pr
u†2
h†
µ∗
(
∂u∗
∂x∗
)2
.
(A.26)
To make sure that the temporal pressure term in the continuity equation
scales correctly for low Mach number viscous flows, the physical quantity %∗p
needs to be replaced by the artificial quantity %′∗p defined as
%′∗p = kv
Re2p†
%†u†2
(A.27)
where kv is a constant in the order of one. Written in dimensional form the
last definition reads as follows
%′p = kv
Re2
u2
. (A.28)
For ideal gases the last relation can also be written as
%′p = kv
Re2
γM2
%p. (A.29)
To avoid that the temporal derivative of the temperature dominates the
continuity equation, the coefficient %∗T/Re should not exceed unity. In terms of
dimensional quantities, this is given when
%T ≤ %†Re/T †. (A.30)
This is achieved by replacing the physical property %T with the scaled prop-
erty %′T that fulfills the last condition. As already mentioned in previous
sections, the scaled property %′T should ideally be proportional to the physi-
cal property %T , such that the energy equation remains decoupled from the
continuity equation in the case of incompressible flows. With %′T defined as
%′T = δpc%T , δpc ∈ [0, 1] according to Eq. (3.13), the condition (A.30) is fulfilled
when the parameter δpc is limited by
δpc ≤ %
†Re
%TT †
. (A.31)
146 A. Perturbation analysis
An analysis of the energy equation shows that there are circumstances,
where the physical properties hT and dp = 1− %hp need to be changed in order
to ensure that the various physical terms are of the same order. Before dis-
cussing this, it should be noted that if the reference enthalpy is related to the
thermodynamic pressure according to h† = p†/%† then the term u†2/h† scales
with the square of the Mach number for the case of ideal gas flow. Hence, for
low Mach and moderate Prandtl numbers (i.e., PrM2  1) the viscous dissipa-
tion term Pr u†
2
h† µ
∗ (∂u∗
∂x∗
)2 can be neglected. In the case of low Mach, moderate
Prandtl number, ideal gas flow, the pressure gradient term Pru∗ %
†u†2
p†
∂p∗
(2)
∂x∗ can
be neglected as well. To ensure a good conditioning of the flow equations, the
temporal temperature derivative in the energy equation should scale in the
same order as the diffusive terms. The only diffusive term of significance for
low Mach number flow is the heat conduction term. From this it follows that
the factor Pr%∗h∗T of the temporal temperature derivative should scale in the
order of unity. Thus, for the simulation of flow with Prandtl numbers far from
unity, the normalized isobaric heat capacity h∗T may be replaced with a scaled
quantity h′∗T defined as h′∗T = 1/%∗Pr. In terms of dimensional quantities this
reads as
h′T =
p†
T †
1
%Pr
. (A.32)
For viscous, ideal gas, low Mach number flow, where the local value for the
density is close to the reference density, the scaled quantity h′T may be intro-
duced as h′T = R/Pr. Alternatively, one might, as suggested by Merkle, set the
scaled quantity h′T to h′T = hT/Pr.
The term in the energy equation containing the temporal pressure deriva-
tive, dp %
†u†2
p†
Pr
Re
∂p∗
(2)
∂t∗ , scales in the order of O(PrM2/Re) for ideal gas flow. To
avoid that this term dominates the energy equation for situations where the
Reynolds number is significantly lower than the square of the Mach number
(Re  M2) and the Prandtl number is in the order of unity, the coefficient
dp = 1− %hp needs to be replaced with an artificial coefficient h′p such that the
term h′p
%†u†2
p†
Pr
Re stays in the order of unity. From this condition it follows that
the coefficient h′p must be bounded by
h′p ≤
p†
%†u†2
Re
Pr
, (A.33)
which for ideal gas flow translates into the following condition h′p ≤ 1γM†2 RePr .
In flows encountered in engineering systems the Reynolds number is usually
many orders larger than the Mach number. For this reason the last condition
is practically always met and the scaling of the temporal pressure derivative
in the energy equation is not much of an issue.
A.3 Summary of the implications for the preconditioning parameters 147
A.3 Summary of the implications for the pre-
conditioning parameters
The conditions derived in the previous sections to ensure a proper balance
between the various physical terms in the flow equations in the limit of low
Mach and Reynolds numbers are summarized in Table A.1.
The main preconditioning parameter %′p in the limit of low Mach number,
inviscid flow should be determined as
%′p = kp/u
†2. (A.34)
Using eigenvalue analysis the following condition for the preconditioning pa-
rameter %′p was derived (Eq. 3.12):
%′p =
1
V 2r
− 1− %hp
%hT
%′T .
This relation gives the same result as Eq. (A.34) when the reference speed Vr
is set to the reference scale u† for the convective velocity and the parameter
%′T is set to zero.
For low Mach number viscous flow, it follows from Eq. (A.28) that the ref-
erence speed Vr used to determine %′p should be oriented after the diffusion
velocity, i.e., Vr is to be determined as Vr = kvν/L†.
The preconditioning parameter %′T should vanish for constant density flow
to avoid the presence of temporal derivatives of the temperature in the conti-
nuity equation. This can be achieved by setting this parameter proportional
to its physical counterpart. For low Reynolds number flow, the parameter %′T
should be constrained by the condition given in Eq. (A.30).
In the case of inviscid flow, the parameters h′T and h′p are not subject to
any restriction to attain a good conditioning of the flow equations. To achieve
simple expressions for the eigenvalues it is favorable to set both parameters
to their physical counterparts, i.e., h′T = hT and h′p := dp = 1− %hp. The former
choice ensures that three of the eigenvalues of the linearized Euler equations
corresponds to the convective velocity. For low Reynolds number flows both
parameters h′T and h′p are subject to restrictions if a good conditioning is to
be achieved. In the case of viscosity dominated flow, the parameter h′p should
be bounded by the constraint given in Eq. (A.33) to avoid that temporal pres-
sure variations dominate the evolution of the energy field. For low Reynolds
number flows where the Prandtl number is far from unity (e.g., flow of en-
gine oil) the parameter h′T should, according to Eq. (A.32), be scaled inversely
proportional to the Prandtl number.
148 A. Perturbation analysis
Preconditioning
parameter %
′
p %
′
T h
′
p h
′
T
Inviscid, low
Mach no. flow kp/u
†2 kT%T , Rem.
(i,iv)
No cond.,
Rem. (ii)
No cond.,
Rem. (iii)
Viscous, low
Mach no. flow kvRe
2/u†2 %′T ≤ %†Re /T † h′p ≤ p
†
%†u†2
Pr
Re p
†/(% T † Pr)
Table A.1: Summary of conditions imposed on the preconditioning parameters in the
low Mach number and low Reynolds number limit. Remarks and expla-
nations: i) %′T should vanish for incompressible flows - for this reason this
parameter should be formulated proportional to its physical counterpart;
ii) the choice h′p := dp = 1 − %hp is favorable as this gives simpler expres-
sions for the eigenvalues; iii) the choice h′T :=hT gives also simpler eigen-
values and ensures that the speed of the entropy wave is not altered; iv)
for compressible flow the parameter kT should not exceed unity to ensure
regularity of the preconditioning matrix for subsonic reference speeds.
Appendix B
Dispersion analysis of the
unscaled and the preconditioned
Navier-Stokes equations
In viscous flow various physical phenomena interplay; whether the transport
processes are governed by advection, acoustic waves and/or viscous diffusion
is dependent on the Mach and Reynolds number levels. With the purpose
of analysing the role of the various physical processes and to determine the
wave speeds under different limiting conditions, a dispersion analysis simi-
lar to that one of Buelow [46] is carried out in this chapter. In contrast to
the analysis conducted by Buelow, the dispersion analysis here is carried out
for two different values of the preconditioning parameter %′T , which steers the
amount of coupling between the energy and the continuity equation. In ad-
dition to ideal gas flow, the wave speeds of the preconditioned Navier-Stokes
equations are analysed for the case that the fluid is incompressible.
B.1 Governing equations and dispersion anal-
ysis employed for determiningwave speeds
To analyse the wave speeds of the unscaled and preconditioned Navier-Stokes
equations, a dispersion analysis is carried out using the one-dimensional Navier-
Stokes equations
∂Q
∂t
+
∂Ec
∂x
=
∂Ev
∂x
(B.1)
with
Q = [ρ, %u, %e0]
T , Ec = [ρu, %u
2 + p, %h0]
T
,
Ev = [0, τxx, τxxu− q˙x]T , τxx = 4/3µ∂u∂x , q˙x = −kλ ∂T∂x .
(B.2)
Changing from the vector Q of conservative variables to the vector Qv =
[p, u, T ]T of primitive, viscous variables and linearizing the equations about
an arbitrary, but fixed averaged state Q¯v yields
Γe
∂Q˜v
∂t
+ A
∂Q˜v
∂x
=
∂
∂x
Rxx
∂Q˜v
∂x
(B.3)
149
150 B. Dispersion analysis
with
Γe =
∂Q
∂Qv
∣∣∣∣
Q¯v
, Av =
∂E
∂Qv
∣∣∣∣
Q¯v
, Rxx =
0 0 00 4/3µ 0
0 4/3µu kλ

Q¯v
(B.4)
and the linearized state vector Q˜v defined as Q˜v = Qv − Q¯v. To simplify the
equations, a constant viscosity and a fixed thermal conductivity are assumed
in the current analysis. When Merkle’s preconditioning technique is applied,
the physical Jacobian Γe is replaced with the preconditioning matrix
Γv =
 %′p 0 %′T%′pu % %′Tu
%′ph0 + %hp − 1 %u %′Th0 + %hT
 (B.5)
for which the quantities %p and %T have been substituted with the artificial
quantities %′p and %′T , so that the numerical speed of sound a′ is in the same or-
der as the maximum of the convective and diffusive velocity scales. In the first
instance ideal gas flow will be assumed which is why the notion of Buelow will
be used, in which the pseudo-acoustic velocity a′ is related to the isentropic
speed of sound a according to
a′2 =  · a2. (B.6)
Hence, if  is set to the square of the Mach number M = u/a, the numerical
speed of sound a′ reduces to the convective velocity u. With the last equation
and Eqs. (3.10) and (3.12) the following relation for %′p is obtained:
%′p =
1
a2
− 1− %hp
%hT
%′T . (B.7)
The parameter %′T is determined according to Eq. (3.13), i.e., %′T = δpc%T where
δpc is a constant parameter in the range δpc = [0, 1]. It should be noted that if
both parameters  and δpc are set to unity, then the preconditioning matrix Γv
reverts to the Jacobian matrix ∂Q
∂Qv
.
A dispersion analysis consists of analysing for which values of the wave-
number k and angular frequency ω waves exist in the form
Q˜v(x, t) = Qˆv(x, t)e
ıˆ(kx−ωt) (B.8)
as a solution of the preconditioned Navier-Stokes equations. By substituting
this expression in Eq. (B.3) it follows that non-trivial solutions exist for∣∣−ıˆωI + ıˆkAΓ,v + k2RΓ,xx∣∣ = 0 (B.9)
with AΓ,v = Γ−1v Av and RΓ,xx = Γ−1v Rxx. The problem of finding solutions of the
dispersion relation in Eq. (B.9) can be recast as finding the roots ω/k of the
polynomial given by
|AΓ,v,xx − (ω/k) · I| = 0, (B.10)
with the matrix AΓ,v,xx given by
AΓ,v,xx = AΓ,v − ıˆ u
ν Rek
·Rxx,Γ (B.11)
B.1 Governing equations and dispersion analysis 151
and the wave number based Reynolds number Rek defined as
Rek =
u
νk
. (B.12)
The roots (ω/k)j, j = 1, . . . , 3 of the dispersion relation (B.9) correspond to the
phase velocity (i.e., the propagation velocity) [304] of the different wave modes
Qˆv,j(x, t)e
ıˆ(kjx−ωjt) governed by the Navier-Stokes equations (B.3).
B.1.1 Dispersion analysis for ideal gas flow
For an ideal gas the preconditioned inviscid flux Jacobian AΓ,v = Γ−1v Aˆ reads
as
AΓ,v = Γ
−1
v Aˆ =
u[γ − (γ − 1)δpc] %c2 −
(1−δpc)%uc2
T
1
%
u 0
u[(γ−(γ−1)δpc)−1]
%cp
(γ − 1)T u[1− γpc]
 (B.13)
with the abbreviation γpc = (1 − δpc)(γ − 1) while the preconditioned viscous
flux Jacobian RΓ,xx = Γ−1v Rxx becomes
Rxx,Γ = Γ
−1
v Rxx =
0 0 00 4/3ν 0
0 0 ν/Pr
 (B.14)
with the Prandtl number defined as Pr = cpµ/kλ. To simplify the expressions,
the Prandtl number is set to 3/4 in the following analyses.
The roots of the dispersion relation can be calculated as
(ω/k)1 = u(1− ıˆ43 1Rek ),
(ω/k)2,3 =
u
2
[
(1 + )− ıˆ4
3
1
Rek
]
± u
√
 1
M2
+
(
ıˆ2
3
1
Rek
+ −1
2
)2
+ ıˆ4
3
1
Rek
γpc.
(B.15)
In subsequent sections these roots will be analysed for different limits of the
Mach and Reynolds numbers.
B.1.2 Dispersion analysis for incompressible flow
For the unscaled, linearized Navier-Stokes equations for incompressible flows
for which the temporal derivative of the density vanishes (∂%
∂t
= 0), it is im-
mediately seen that waves of the form Q˜ = Qˆeıˆ(kx−ωt) with finite transport ve-
locities do not exist. For the constant density case, the flow equations exhibit
only then a hyperbolic character when the artificial compressibility method or
a preconditioning approach is used where temporal derivatives of the pressure
and the temperature are added to the continuity equation.
In the case of a convection dominated incompressible fluid flow, the pre-
scriptions for the preconditioning parameters %′p and %′T given by Eqs. (3.12)
and (3.13) reduce to
%′p =
1
V 2r
, %′T = 0 (B.16)
152 B. Dispersion analysis
and the reference velocity Vr might be set to the convection velocity u, i.e.,
Vr = u. With the other preconditioning parameters %′T , h′T and h′p set to their
physical counterparts, i.e.,
%′T = %T = 0, h
′
T = hT , h
′
p = 1− %hp = 0 (B.17)
the matrices AΓ,v and Rxx,Γ become
AΓ,v = Γ
−1
v Aˆ =
0 %u2 01
%
u 0
0 0 u
 , Rxx,Γ = Γ−1v Rxx =
0 0 00 4
3
ν 0
0 0 ν/Pr
 . (B.18)
As for the ideal gas flow case, the Prandtl number is set to 3/4 to simplify the
following analyses. With the preconditioned flux Jacobians in Eq. (B.18), the
solution of the dispersion relation (B.10) becomes
(ω/k)1 = u
(
1− 4
3
ıˆ 1Rek
)
,
(ω/k)2,3 =
1
2
u
(
1− 4
3
ıˆ 1Rek
)
± 1
2
u
√(
1− 4
3
ıˆ 1Rek
)2
+ 4.
(B.19)
For viscosity dominated flow, the preconditioning parameter %′p may be de-
termined as
%′p =
Renek
u2
(B.20)
with the exponent ne either set to 1 or 2. The preconditioned flux Jacobian
matrix AΓ,v in this case becomes
AΓ,v =
0 %u
2
Renek
0
1
%
u 0
0 0 u
 , (B.21)
if the other preconditioning parameters have not been scaled, i.e., they have
been set according to Eq. (B.17). Whether the preconditioning parameter %′p
is specified according to Eq. (B.16) or (B.20), the flux Jacobian RΓ,xx = Γ−1v Rxx
given in Eq. (B.18) remains the same. In the case of viscous preconditioning,
for which the parameter %′p has been set according to Eq. (B.20), the roots of
the dispersion relation (B.9) become
(ω/k)1 = u(1− 43 ıˆ 1Rek ),
(ω/k)2,3 =
1
2
u
(
1− 4
3
ıˆ 1Rek
)
± 1
2
u
√(
1− 4
3
ıˆ 1Rek
)2
+ 4Renek
.
(B.22)
Further below, the roots of the dispersion relation for inviscid and viscous
preconditioning will be examined in the low and high Reynolds number limit.
B.2 Preconditioning for ideal gases 153
B.2 Wave speeds of the preconditioned Navier-
Stokes equations for ideal gas flow
In the following, the wave speeds of the Navier-Stokes equations for ideal
gas flow will be determined using dispersion analysis. The effect of using low
Mach or low Reynolds number time-derivative preconditioning will be studied
by calculating the distribution of the wave speeds for various limiting flow
conditions.
B.2.1 Wave speeds when time-derivative preconditioning
is not applied
In this subsection, the wave speeds of the unscaled Navier-Stokes equations
will be analysed. Thus, the roots of Eq. (B.15) will be investigated for  = 1
and δpc = 1 for which the preconditioning matrix Γv given in Eq. (B.5) reverts
to the physical Jacobian Γe = ∂Q∂Qv .
In the limit of high Reynolds numbers, the roots of the dispersion relation
simplify to
(ω/k)1 = u, (ω/k)2,3 = u± a. (B.23)
These roots represent the speed of the particle wave (i.e., the convection of
entropy) and the speed of the two acoustic waves. The physical condition
number σ becomes σ = 1+ 1M . Obviously, the condition number goes beyond all
bounds as the Mach number tends to zero, which demonstrates the stiffness
of the inviscid flow equations as the Mach number goes to zero.
For viscosity dominated flows where the Reynolds number is low, differ-
ent cases appear depending on the ratio between the Mach and the Reynolds
number. The ratio Rek/M can be regarded as an acoustic Reynolds number
Rea,k based on the speed of sound, i. e.,
Rea,k = Rek/M = a/(νk). (B.24)
For viscous dominated flow (Rek  1) having a high acoustic Reynolds number
(Rek/M 1) the roots of the dispersion relation become
(ω/k)1 = −ıˆ4
3
u
Rek
, (ω/k)2,3 = u± a. (B.25)
This result is based on the assumption that the ratio Rek/M2 is at least one
order larger than unity. This ratio determines the ratio between the pressure
forces and viscous stresses in the momentum equations as well as the ratio
between the viscous dissipation and the pressure work term in the energy
equation [182].
The roots in Eq. (B.25) show that the pressure waves are not effectively
attenuated by the presence of viscosity and that only the entropy wave is
subject to viscous damping in the case of large acoustic Reynolds numbers
(Rea,k  1). The physical condition number σ in this case becomes σ = 34(1 +
154 B. Dispersion analysis
Limits ω1/k ω2/k ω3/k σ
Re 1 u u+ a u− a 1 + 1M
Re  1, Re/M  1,
Re/M2  1 −ıˆ
4
3
u
Rek u+ a u− a 34(1 + 1M)Rek
Re 1, Re/M 1 −ıˆ43 uRek u [γ · u] −ıˆ43 uRek 43 1Rek [43 1γRek ]
Table B.1: The roots of the dispersion relation and the condition number of the un-
scaled Navier-Stokes equations for ideal gas flow ( = 1, δpc = 1). The
values listed in brackets are the transport velocities for δpc = 0 if they
differ from those for δpc = 1.
1
M)Rek =
3
4
(Rek + Rea,k), which shows that there is a significant disparity in
the wave speeds for large acoustic Reynolds numbers.
For viscous dominated flow (Rek  1) and low acoustic Reynolds numbers
(Rea,k  1) the wave speeds are determined as
(ω/k)1,3 = −ıˆ4
3
u
Rek
, (ω/k)2 = u. (B.26)
Thus, in this case two wave modes are subject to viscous damping, while the
third mode represents an undamped particle velocity. The condition number
σ for this situation becomes σ = 4
3
1
Rek
. This shows that not only in the limit of
low Mach numbers but also in the limit of low Reynolds numbers, a disparity
of the wave speeds of the unscaled Navier-Stokes equations occurs.
Table B.1 summarizes the roots of the dispersion relation (B.10) for the
unscaled Navier-Stokes equations under various limiting conditions. In the
table the wave speeds are also listed for the case that the parameter δpc is set
to zero instead of unity. As can be seen from the table, the parameter δpc has
no impact other than for the low acoustic Reynolds number case. In that case,
the parameter leads only to an insignificant change in the condition number.
B.2.2 Wave speeds when low Mach number
time-derivative preconditioning is applied
In this section the roots of the dispersion relation (B.10) are determined and
analysed for various limiting conditions for the case that low Mach number
time-derivative preconditioning is used. To determine the wave speeds in this
case, the dispersion relation (B.10) is evaluated with the parameter  set to
 =M2. In the first instance, the roots will be determined for the parameter δpc
set to unity. With this choice of the parameters  and δpc, the preconditioning
parameters %′p and %′T become
%′p =
1
V 2r
− 1− %hp
%hT
%′T , Vr =
√
a = u, %′T = %T . (B.27)
In the limit of high Reynolds (Rek → ∞) and low Mach numbers (M 
1), the roots of the dispersion relation for the aforementioned choices of the
B.2 Preconditioning for ideal gases 155
parameters  and δpc become
ω1/k = u, ω2,3/k =
1
2
(1±
√
5)u. (B.28)
The condition number in this case is σ = |(1 + √5)/(1 − √5)| ≈ 2.62, which
shows that good conditioning is attained for this choice of preconditioning
parameters in the limit of low Mach number flow.
In the limit of low Reynolds number flow (Rek  1), the use of low Mach
number preconditioning exacerbates the disparity of the wave speeds. To
show this, the roots of the dispersion relation (B.10) are calculated in the limit
of low Reynolds numbers (Rek  1) for both high and low values of the acous-
tic Reynolds number Rea,k = Rek/M. In the former case, under the additional
assumption that the ratio Rek/M2 is also large (Rek/M2  1), the roots of the
dispersion relation (B.10) are
(ω/k)1 = −ıˆ4
3
u
Rek
, (ω/k)2 = −ıˆ3
4
Rek u, (ω/k)3 = −ıˆ4
3
u
Rek
, (B.29)
which leads to a condition number of σ = 16
9
1
Re2k
. For low acoustic Reynolds
numbers (Rea,k  1) the roots are determined as
(ω/k)1 = −ıˆ4
3
u
Rek
, (ω/k)2 = uM
2, (ω/k)3 = −ıˆ4
3
u
Rek
, (B.30)
which leads to a condition number of σ = 4
3
1
RekM2
.
In the limit of low Reynolds numbers, it can be concluded that the use
of low Mach number preconditioning does not lead to an equilibration of the
wave speeds.
Table B.2 summarizes the roots of the dispersion relation in the case that
low Mach preconditioning is used (ε = M2). In the table the wave speeds
for the case that the parameter δpc was set to zero instead of unity are also
given. As for the unscaled flow equations, a change of this parameter has only
a minor impact; the wave speeds are only altered insignificantly for the low
acoustic Reynolds number case.
B.2.3 Wave speeds when low Reynolds number time-
derivative preconditioning is applied
As suggested by Buelow [46] for low particle and high acoustic Reynolds num-
bers (Rek  1, Rea,k = Rek/M  1) the value for the preconditioning param-
eter  must be defined such that the numerical speed of sound a′ is oriented
after the viscous diffusion velocity. This is achieved by scaling the numerical
speed of sound inversely proportional to the reciprocal value of the acoustic
Reynolds number. With the parameter  set to
 = M2/Re2k (B.31)
156 B. Dispersion analysis
Limits ω1/k ω2/k ω3/k σ
Re 1 u 1
2
u(1+
√
5) 1
2
u(1−√5) 2.62
Re  1, Re/M  1,
Re/M2  1 −ıˆ
4
3
u
Rek −ıˆ34Rek u −ıˆ43 uRek
16
9
1
Re2k
Re 1, Re/M 1 −ıˆ43 uRek
uM2
[γuM2]
−ıˆ4
3
u
Rek
4
3
1
RekM2
[4
3
1
γRekM2
]
Table B.2: The roots of the dispersion relation and the condition number of the
preconditioned Navier-Stokes equations for ideal gas flow ( = M2 and
δpc = 1). The values listed in brackets are the transport velocities for
δpc = 0 if they differ from those for δpc = 1.
the magnitude of the numerical speed of sound a′ follows from Eq. (B.6)
a′2 = a2 = (νk)2. (B.32)
In the following, wave speeds for this particular choice of the parameter 
and with the parameter δpc set to unity will be analyzed for various limiting
conditions.
In the limit of high Reynolds numbers (Rek → ∞) and bounded values for
the Mach number, the roots of the dispersion become
(ω/k)1 = u, (ω/k)2,3 =
1
2
(1± 1)u. (B.33)
This leads to a condition number σ that goes beyond all limits, which shows
that the preconditioning choice according to Eq. (B.31) is not suited for high
Reynolds number flow.
In the limit of low Reynolds numbers (Rek  1) and large acoustic Reynolds
numbers (Rea,k = Rek/M  1) the roots of the dispersion relation (B.10) be-
come
(ω/k)1 = −ıˆ4
3
u
Rek
, (ω/k)2,3 =
u
6Rek
(−4ıˆ± 2
√
5). (B.34)
To arrive at these expressions, it was again necessary to assume that the ra-
tio Rek/M2 is large (Rek/M2  1). For the case of high acoustic Reynolds
numbers, setting the preconditioning parameter according to Eq. (B.31) is fa-
vorable, as this leads to a condition number of σ = 4/3.
For low acoustic Reynolds numbers (Rea,k/M  1) however, the precondi-
tioning parameter does not prevent Mach or Reynolds number stiffness, since
the roots in this case are given by
(ω/k)1 = −ıˆ4
3
u
Rek
, (ω/k)2 = u
M2
Re2k
, (ω/k)3 = −ıˆ4
3
u
Rek
, (B.35)
which leads to a spectral radius σ of σ = 3
4
M2
Rek
.
Table B.3 summarizes the roots of the dispersion relation (B.10) when the
preconditioning parameter  is set according to Eq. (B.31). As can be seen
B.3 Preconditioning for incompressible flow 157
Limits ω1/k ω2/k ω3/k σ
Re 1 u 0 u ∞
Re 1, Re/M 1,
Re/M2  1 −ıˆ
4
3
u
Rek
u
6Rek
(−4ıˆ+ 2√5) u
6Rek
(−4ıˆ− 2√5) 4
3
Rek  1, Rek/M 
1
−ıˆ4
3
u
Rek
u M
2
Re2k
[γu M
2
Re2k
]
−ıˆ4
3
u
Rek
3
4
M2
Rek
[3
4
γ M
2
Rek
]
Table B.3: The roots of the dispersion relation and the condition number of the pre-
conditioned Navier-Stokes equations for ideal gas flow and the precondi-
tioning parameters  and δpc set to  = M2/Re2k and δpc = 1, respectively.
The values listed in brackets are the wave speeds for δpc = 0 if they differ
from those for δpc = 1.
from the table, if the preconditioning parameter δpc is set to zero instead of
unity, this changes the spectral radius for the case of low acoustic Reynolds
numbers but only by an insignificant factor γ.
For low particle and low acoustic Reynolds numbers (Rek  1, Rea,k =
Rek/M 1) Buelow [46] suggests setting the preconditioning parameter  to
 = 1/Rek. (B.36)
The roots of the dispersion equation (B.10) for this choice of the precondition-
ing parameter are given in Table B.4 for various limiting conditions. While
a preconditioning parameter according to Eq. (B.36) does not relieve stiffness
in the low Mach number limit for inviscid flow (M  1, Rek → ∞) or elim-
inate Reynolds number stiffness at low particle but high acoustic Reynolds
numbers (Rek  1, Rek/M  1), it leads to a favorable condition number of
σ = 4/3 in the limit of low particle and acoustic Reynolds numbers (Rek  1,
Rek/M 1).
If the preconditioning parameter δpc is changed from unity to zero, the con-
dition numbers change only in the case of low particle and acoustic Reynolds
numbers. The change in the condition number in that case is not important;
however, in the case that the parameter δpc is set to zero, the second root
(ω/k)2 =
u
2Rek
(2.42 + ıˆ0.33) (B.37)
contains a positive imaginary part, which corresponds to a growing mode.
Thus, if the choice for the parameter  according to Eq. (B.36) should be used,
the parameter δpc should be set to unity.
B.3 Wave speeds of the preconditioned Navier-
Stokes equations for incompressible flow
In the following the wave speeds of the preconditioned Navier-Stokes equa-
tions for incompressible flow will be determined using dispersion analysis.
158 B. Dispersion analysis
Limits ω1/k ω2/k ω3/k σ
Re 1 u 0 u ∞
Re 1, Re/M 1,
Re/M2  1 −ıˆ
4
3
u
Rek
u√
RekM
ıˆ − u√
RekM
ıˆ 3
4
√
Rek
M
Re 1, Re/M 1 −ıˆ43 uRek
u
Rek
[ u
2Rek
(2.42+ ıˆ0.33)]
−ıˆ4
3
u
Rek
[− u
2Rek
(0.42+ˆı3.0)]
4
3
[1.23]
Table B.4: The roots of the dispersion relation and the condition number of the pre-
conditioned Navier-Stokes equations for the preconditioning parameters
set to  = 1Rek and δpc = 1. The values listed in brackets are the transport
velocities for δpc = 0 if they differ from those for δpc = 1.
The effect of using time-derivative preconditioning for inviscid or viscous flows
will be studied by calculating the wave speeds for various limiting flow condi-
tions.
B.3.1 Wave speeds when time-derivative preconditioning
for inviscid flows is applied
With the preconditioning parameter set to
%′p =
1
V 2r
with Vr = u (B.38)
and the other parameters %′T , h′T and h′p set their respective physical counter-
parts, the roots of the dispersion relation (B.10) of the Navier-Stokes equa-
tions (B.3) under the assumption of constant density flow (% = const.) reduce
to
(ω/k)1 = u, (ω/k)2,3 =
1
2
(1±
√
5)u. (B.39)
This leads to a condition number of σ =
∣∣∣1+√5
1−√5
∣∣∣ ≈ 2.62, which shows that the
wave speeds are properly clustered.
For low Reynolds number flow (Rek  1) the wave speeds become
λ1 = −4
3
ıˆ
u
Rek
, λ2 = −4
3
ıˆ
u
Rek
, λ3 = −3
4
ıˆuRek. (B.40)
This leads to a condition number of σ = 16
9
1
Re2k
, which shows that the prescrip-
tion in Eq. (B.38) for the preconditioning parameter %′p is not suited for low
Reynolds number flow. Table B.5 summarizes the roots of the dispersion rela-
tion for %′p = 1/u2 and incompressible flow under various limiting conditions.
B.3.2 Wave speeds when time-derivative preconditioning
for viscous dominated flows is applied
In the following, the effect of choosing the preconditioning parameter %′p such
that the numerical speed of sound a′ is oriented after the diffusion velocity
B.3 Preconditioning for incompressible flow 159
Limits ω1/k ω2/k ω3/k σ
Re 1 u 1
2
u(1 +
√
5) 1
2
u(1−√5) 2.62
Re 1 −ıˆ43 uRek −ıˆ34Rek u −ıˆ43 uRek
16
9
1
Re2k
Table B.5: The roots of the dispersion relation and the condition number of the
preconditioned Navier-Stokes equations for incompressible flows with
%′p = 1/u2
Limits ω1/k ω2/k ω3/k σ
Re 1 u − uRe2k u Re
2
k
Re 1 −ıˆ43 uRek u6Rek (−4ıˆ+ 2
√
5) u
6Rek
(−4ıˆ− 2√5) 4
3
Table B.6: The roots of the dispersion relation and the condition number σ of the
preconditioned Navier-Stokes equations for incompressible flows and %′p =
Re2k/u2
Vvis = νk will be examined. Thus, the wave speeds when the preconditioning
parameter %′p is set to
%′p =
Re2k
u2
=
1
V 2vis
= (νk)2 (B.41)
will be examined.
The roots (B.22) of the dispersion relation (B.10) of the Navier-Stokes equa-
tions (B.3) under the assumption of constant density flow (% = const.) become
(ω/k)1,3 = u, (ω/k)2 = − uRe2k
, (B.42)
for large Reynolds numbers (Rek  1), which leads to a condition number of
σ = Re2k. Thus, for high Reynolds numbers, the prescription in Eq. (B.41) for
the preconditioning parameter %′p is not suitable.
In the limit of low Reynolds numbers (Rek  1), a dispersion analysis leads
to the following values of the wave speeds
(ω/k)1 = −4
3
ıˆ
1
Rek
, (ω/k)2,3 =
u
6Rek
(−4ıˆ± 2
√
5). (B.43)
The condition number σ in this case reads as σ = 4
3
, which shows that a pre-
scription of the preconditioning parameter %′p according to Eq. (B.41) is suited
for low Reynolds number flow. Table B.6 summarizes the roots of the disper-
sion relation for incompressible flows when the preconditioning parameter %′p
is defined according to Eq. (B.41).
For the case of ideal gas flow the wave velocities were determined for the
case that the preconditioning parameter  was set to  = 1/Rek. For incom-
pressible flows, this choice corresponds to setting the preconditioning param-
eter %′p as
%′p = Rek/u
2. (B.44)
160 B. Dispersion analysis
Limits ω1/k ω2/k ω3/k σ
Rek  1 u − uRek u Rek
Rek  1 −ıˆ43 uRek −34 ıˆu −ıˆ43 uRek 169 1Rek
Table B.7: The roots of the dispersion relation and the condition number of the
preconditioned Navier-Stokes equations for incompressible flows with
%′p =
Rek
u2
For this choice of the preconditioning parameter, the roots of the dispersion
relation (B.10) reduce to
(ω/k)1,3 = u, (ω/k)2 = − uRek (B.45)
for high Reynolds number flow (Rek  1), which leads to a condition number
σ of σ = Rek. For low Reynolds numbers (Rek  1) the wave speeds become
(ω/k)1 = −4
3
ıˆ
u
Rek
, (ω/k)2 = −3
4
ıˆu, (ω/k)3 = −4
3
ıˆ
u
Rek
(B.46)
from which a condition number σ of σ = 16
9
1
Rek
follows. Thus, both for high
and low Reynolds numbers, the prescription according to Eq. (B.44) for the
preconditioning parameter %′p does not lead to favorable condition numbers.
Table B.7 summarizes the wave speeds of the preconditioned Navier-Stokes
equations under various limiting conditions when the preconditioning param-
eter %′p is set according to Eq. (B.44).
B.4 Conclusions and implications of the results
of the dispersion analysis
The dispersion analysis above shows that the use of low Mach preconditioning
with
 = c, c =M
2 (B.47)
or
%′p =
1
V 2r
− 1− %hp
%hT
%′T , Vr =
√
 · a = u (B.48)
with %′T = δpc%T , δpc ∈ [0, 1], h′T = hT , and h′p = 1 − %hp leads to a good condi-
tioning (σ ≈ 2.62) of the linearized Navier-Stokes in the case of high Reynolds
numbers, ideal gas flow. Analogously, by setting the parameters %′T , h′T , h′p to
their respective physical counterparts and the preconditioning parameter %′p
to
%′p =
1
V 2r
with Vr = u (B.49)
a good conditioning (σ ≈ 2.62) is also achieved for incompressible flows.
B.4 Implications of the dispersion analysis 161
For low particle Reynolds numbers, viscous effects must be accounted for
in the preconditioning formulation, since the use of time-derivative precondi-
tioning otherwise may aggravate the low Reynolds number stiffness. For low
particle (Rek  1) and high acoustic Reynolds numbers (Rea,k = Rek/M  1)
the following preconditioning strategy
 = max(c, v), v = M2/Re2k = (νk)
2/a2 (B.50)
or equivalently
%′p =
1
V 2r
− 1− %hp
%hT
%′T , Vr = max(u, νk) (B.51)
with %′T = δpc%T , δpc ∈ [0, 1] leads to a good conditioning (σ=4/3) for ideal gas
flow.
In cases with low particle (Rek  1) and low acoustic Reynolds numbers
(Rea,k  1) the prescription for the preconditioning parameter %′p given in
Eq. (B.51) is not suitable. A good conditioning (σ = 4/3) for low acoustic
Reynolds numbers and ideal gas flow is attained using the following set of
preconditioning parameter
 = max(c, v), v = 1/Rek (B.52)
or analogously
%′p =
1
V 2r
− 1− %hp
%hT
%′T , Vr = max(u, a/
√
Rek). (B.53)
For incompressible flows the Mach number is identically equal zero, so
that only the case analogous to the high acoustic Reynolds number case for
ideal gas flow can be considered. To accommodate for regions of low particle
Reynolds numbers Rek = u/(νk) 1 in incompressible flows, the prescription
(B.51) for the preconditioning parameter should be used. In the limit of low
particle Reynolds numbers this yields a condition number of σ = 4/3, which is
the same as for the case of ideal gas flow.
Whether the parameter δpc is set to unity or to zero for ideal gas flow this
only changes the condition number in the limit of low Reynolds number flow.
The value of δpc does not alter the condition number significantly, but it should
be noted that a value of δpc = 0 for the low particle and low acoustic Reynolds
number case leads to growing modes. Thus, if the prescription (B.52) for the
preconditioning parameters  should be used or the preconditioning param-
eter %′p should be defined according to Eq. (B.53), it is recommended to set
the parameter δpc to unity. For incompressible flows the parameter δpc has no
effect since %T vanishes identically.
As discussed by Buelow [46], to obtain a preconditioning method that fol-
lows the prescription (B.52) for low particle and acoustic Reynolds and the
prescription (B.51) for low particle and high acoustic Reynolds numbers, the
following definition for v could be applied
v =
M2(1− Rek)
Re2k +M
2Rek(1− Rek)
. (B.54)
162 B. Dispersion analysis
With this prescription, the parameter v goes to M2/Re2k for high acoustic
Reynolds numbers and to 1/Rek for low values of the same.
It should however be noted, that the Reynolds number is wave number
dependent. Thus, the desired behaviour of the preconditioning scheme under
limiting conditions can only be ensured for a given wave number. As suggested
by Choi and Merkle [60] the parameter v may be selected such that the von
Neumann Number of the solution scheme is controlled.
For most practical CFD simulations, the Reynolds number is much larger
than the Mach number; – cases for which the Mach number is high and the
Reynolds number simultaneously is low are seldom encountered, which is why
it was decided not to account for the case of low particle and low acoustic
Reynolds number flow in the current work.
For the case of low particle and high acoustic Reynolds number flow, the
Reynolds number Rek used in the definition of the preconditioning parameter
v is taken to be the cell Reynolds number. The Reynolds number is oriented
after the wave number k = 2pi/∆x of the shortest waves, i.e., the Reynolds
number is defined as
Rek =
u∆x
ν
(B.55)
with ∆x as the local distance between two grid nodes. The reason for this
choice is that the viscosity will predominantly act on the shortest wave lengths.
Stability analysis of the discretized flow equations [46] also shows that the
cell Reynolds number is a critical parameter in judging the influence of the
diffusive terms in the discretized flow equations. With this choice, the precon-
ditioning parameter  reads for ideal gases as
 = max(c, v), c =M
2, v = M2/Re2k, (B.56)
which in terms of the preconditioning parameters %′p and %′T translates into
%′p =
1
V 2r
− 1− %hp
%hT
%′T , Vr = max(u, ν/∆x), %
′
T = δpc%T (B.57)
with δpc ∈ [0, 1]. The formulation in Eq. (B.57) applies both for incompressible
fluid and for ideal gas flow.
Appendix C
Scaling of the artificial
dissipation for inviscid and
viscous, low Mach number flow
In order to avoid low frequency errors and to stabilize a numerical scheme for
convection equations, dissipative terms must be introduced in the discretiza-
tion. The dissipative terms may be introduced implicitly via the use of an
upwind discretization or as artificial terms that are explicitly added to cen-
trally discretized fluxes. Whichever type of scheme is used, it is essential for
the accuracy and stability of the solution scheme that the dissipative terms re-
main in the same order as the physical, non-dispersive part of the convective
fluxes.
In this chapter it will be shown that the artificial dissipation in the scheme
of Jameson, Schmidt and Turkel (JST) [131] scales wrongly in the low Mach
or low Reynolds number limit. If, however, a suitable preconditioning scheme
is applied and properly accounted for in the construction of the artificial dis-
sipation, a correct scaling of the dissipative terms can be ensured both in the
limit of inviscid and viscous low Mach number flow.
The analysis carried out in this chapter is based on the perturbation meth-
ods introduced in Appendix A and on the theoretical work of Venkateswaran
and Merkle [289].
C.1 Scaling of artificial dissipation of the JST-
scheme for inviscid flows
In order to analyse how the artificial dissipation of JST-scheme scales in the
low Mach number limit both with and without preconditioning, the linearized
Euler equations
∂Q
∂t
+ A
∂Q
∂x
= D, A =
∂E
∂Q
∣∣∣∣
Q¯
(C.1)
are used. The vector
D = δ+
[
k(4)
σ
∆x
(
δ−δ+δ−Q
)]
(C.2)
163
164 C. Scaling of the artificial dissipation
represents thereby the artificial dissipation with k(4) as a scaling coefficient
and σ = |u| + a as the spectral radius. The operators δ+ and δ− represent the
forward and backward difference operators; for an arbitrary state variable φ
these operators are defined as
δ+φi = φi+1 − φi, δ−φi = φi − φi−1, (C.3)
where the subscripts i− 1, i, and i+ 1 denote the index of successive nodes.
Changing from the vector of conservative variables Q = [%, %u, %e0]T to the
vectorQv = [p, u, T ]T of viscous, primitive variables by applying the chain rule,
the following formulation is obtained
Γe
∂Qv
∂t
+ Av
∂Qv
∂x
= D, (C.4)
with the Jacobians Γe and Av defined as
Γe =
(
∂Q
∂Qv
)
, Av =
(
∂E
∂Qv
)
, (C.5)
respectively. Assuming that the grid spacing, the Jacobian matrix Γe, and
the spectral radius are locally constant, the artificial dissipation vector can be
written as
Dv =
k(4) σ
∆x
Γe δ
(4)Qv (C.6)
where δ(4) = δ+δ−δ+δ− represents the fourth order central difference operator.
To analyse the size of the various terms, a non-dimensionalization of the
inviscid flow equations is carried out in the same manner as for the precondi-
tioned Euler equation in Appendix A.1 where the following reference scales
L†, u†, ρ†, T †, p†, t† = L†/u†, h† = p†/ρ† (C.7)
were used for normalization. With this kind of normalization the parameter
 with  = %†u†2/p† appears as the only dimensionless number in the flow
equations. This quantity represents the ratio of the dynamic to the static
pressure and for ideal gases is proportional to the square of Mach number,
i.e.,  = γM†
2
. By expanding the pressure in a series of this parameter
p = p(0) + p(2) +O(2) (C.8)
and afterwards separating out the zeroth order pressure assuming low Mach
numbers (M 1), one arrives at the following form of the non-dimensionalized
linearized Euler equations
Γ∗e
∂Q∗v
∂t∗
+ A∗v
∂Q∗v
∂x∗
= D∗v (C.9)
with
Γ∗e =
 %∗p 0 %∗T0 %∗ 0
−d∗p 0 %∗h∗T
 , A∗v =
 u∗%∗p % u∗%∗T1 %∗u∗ 0
−d∗pu∗ 0 %∗u∗h∗T
 (C.10)
C.1 Inviscid flows 165
and the artificial dissipation vector written as
Dv = k
(4) σ
∗
∆x∗

%∗p δ
(4)p∗(2) + %
∗
T δ
(4)T ∗
%∗δ(4)u∗
−d∗p δ(4)p∗(2) + %∗h∗T δ(4)T ∗
 (C.11)
with d∗p = 1− %∗h∗p. The asterix "∗" as a superscript on a flow variable denotes
that it has been normalized, i.e., a state variable φ∗ is defined as φ∗ = φ/φ†.
All terms in the temporal and spatial terms above scale in the order of unity
in magnitude and zeroth order in the Mach number except those involving the
parameter .
In the artificial dissipation terms the normalized spectral radius σ∗ = |u∗|+
a∗ can be written as M+1
M
, which for low Mach number yields the following
approximation |u∗|+ a∗ ≈ 1
M
. With the parameter  proportional to the square
of the Mach number, the artificial dissipation vector scales as
Dv = [Dv,1, Dv,2, Dv,3]
T = [O(M), O(1/M), O(1/M)]T . (C.12)
This result shows that if preconditioning is not used for the simulation of low
Mach number flow, the artificial dissipation added to the continuity equation
is too low while excess dissipation is added to the momentum and the energy
equations.
The result in Eq. (C.12) is attained by neglecting the temperature differ-
ence term that is added to the continuity equation and the pressure differ-
ence term that is added to the energy equation. In the limit of a vanishing
thermal expansion coefficient of the fluid, both of the neglected terms van-
ish identically. For low Mach number, ideal gas flow the pressure difference
term added to the energy equation is two orders in magnitude smaller the
temperature difference term. For ideal gas flow, the temperature difference
term Dv,1,T added to the continuity equation scales as Dv,1,T = O(1/M). How-
ever, for the stabilization of the continuity and momentum equation it is the
pressure and velocity difference terms that are important since it is the dif-
ferences in the total pressure that is the driving force for the kinetic motion of
the flow when gravity is not accounted for. Anyhow, whether the temperature
difference term is accounted for or not, the dissipation terms of the unpre-
conditioned JST scheme scale only correctly, i.e., in the same order as the
convective terms, for sonic flow conditions.
Ideally, the dissipative fluxes should scale in the order of unity for all Mach
number levels so that these terms match the convective terms. If inviscid
preconditioning is used, the parameter %∗p is replaced by the scaled quantity
%
′∗
p defined as
%
′∗
p =
p†
%†u†2
, (C.13)
which implies that all terms in the matrices Γ∗e and A∗v will scale in the same
order, except the terms −d∗p and −d∗pu∗, which can be omitted for low Mach
numbers. Additionally, if the spectral radius σ, which is used as a scaling
166 C. Scaling of the artificial dissipation
parameter of the artificial dissipation, is set to the spectral radius of the pre-
conditioned system, i.e., σ∗ = |u∗|, all dissipation terms will scale in the order
unity
Dv = [O(1), O(1), O(1)]T , (C.14)
i.e., they scale correctly independently of the Mach number level. This result
for the artificial dissipation of the preconditioned JST scheme is achieved ir-
respective of the value of the thermal expansion coefficient and regardless
whether the temperature difference term Dv,1,T is added to the continuity
equation or not.
C.2 Scaling of artificial dissipation of the JST-
scheme for viscous dominated flows
To investigate the scaling of artificial dissipation terms of the JST discretiza-
tion scheme, the linearized Navier-Stokes equations (B.3) to which the artifi-
cial dissipation vector Dv defined in Eq. (C.6) has been added
Γe
∂Qv
∂t
+ A
∂Qv
∂x
=
∂
∂x
(
Rxx
∂Qv
∂x
)
+Dv (C.15)
are normalized using the following reference quantities
L†, u†, p†, ρ†, T †, t†, µ†, k†λ, h
† = p†/ρ†. (C.16)
This yields
Γ∗e
∂Q∗v
∂t∗
+ A∗v
∂Q∗v
∂x∗
=
∂
∂x∗
(
R∗xx
∂Q∗v
∂x∗
)
+D∗v (C.17)
with
Γ∗e =
 %∗pRe 0 1Re%∗T0 %∗ 0
−d∗pPr 0 %∗h∗TPr
 , A∗v =
 u
∗%∗p % u
∗%∗T
Re p
†
%u†2
%∗u∗ 0
−d∗pu∗Pe 0 %∗u∗h∗TPe
 ,
R∗xx =
0 0 00 4/3µ∗ 0
0 4/3µ∗u∗ µ∗/Pr
 , D∗v = k(4)σ∗∆x∗

(
%∗p δ
(4)p∗ + %∗T δ
(4)T
)
1
Re
%∗δ(4)u∗(
%∗h∗T δ
(4)T ∗ − d∗p δ(4)p∗
)
Pr
 .
(C.18)
The Reynolds, Prandtl and Peclet numbers are defined according to Eq. (A.23).
To arrive at the normalized equations (C.17), the same procedure as applied
in Chapter A.2 has been used, which means that the reference time scale t† is
defined as
t† =
ReL†
u†
=
L†2
ν†
. (C.19)
This reference time scale has been chosen such that the temporal term bal-
ances the viscous forces in the momentum equations.
C.2 Viscous dominated flows 167
The coefficient Re p
†
%†u†2
of the pressure gradient term in the momentum
equation should scale in the same order as the other terms to ensure a balance
between temporal, acoustic, advective and diffusive terms. This coefficient is
used as perturbation variable
 =
%†u†2
p†
1
Re
, (C.20)
to expand the pressure into the following series
p = p(0) + p(2) +O(2). (C.21)
Applying this pressure expansion, Eq. (C.17) can be written as
Γ∗e
∂Q˜∗v
∂t∗
+ A∗v
∂Q˜∗v
∂x∗
=
∂
∂x∗
(
R∗xx
∂Q˜∗v
∂x∗
)
+D∗v (C.22)
with the perturbed state vector Q˜∗ =
[
p∗(2), u
∗, T ∗
]
and the matrices
Γ∗e =
 %∗pRe 0 1Re%∗T0 %∗ 0
−(1− %∗h∗p)Pr 0 %∗h∗TPr
 , A∗v =
 u∗%∗p %∗ u∗%∗T1 %∗u∗ 0
−d∗pu∗Pe 0 %∗u∗h∗TPe
 ,
R∗xx =
0 0 00 4/3µ∗ 0
0 4/3µ∗u∗ µ∗/Pr
 , D∗v = k(4) σ∗∆x∗

%∗p
1
Re δ
(4)p∗(2) +
%∗T
Re δ
(4)T
%∗δ(4)u∗
%∗h∗TPr δ(4)T ∗ − d∗pPr δ(4)p∗(2)
 .
(C.23)
From the perturbation analysis carried out in Appendix A.2 for low Reynolds
number flow, it was deduced that the spatial gradients of the zeroth order
pressure vanishes when the quantity  approaches zero. Assuming constant
boundary conditions, the temporal derivatives of the zeroth order pressure
also vanishes. For this reason only the second order pressure p∗(2) and not the
zeroth order pressure p∗(0) appears in the last equations.
If the Prandtl number scales in the order of unity, then the leading term of
the convective and the diffusive fluxes in the equations (C.22) above scales in
order of unity for low Reynolds number flow. As will be shown in the following,
if preconditioning is not used, the dissipative terms do not scale in the order
of unity and will consequently not be in proper balance with the convective
and diffusive fluxes.
With the perturbation variable  for viscous, lowMach number flow defined
by Eq. (C.20), which for an ideal gases can be expressed as  = kvγM2/Re, and
with the normalized spectral radius σ∗ = |u∗|+a∗, which can be approximated
as σ∗ ≈ 1
M
for low speed flows, the artificial dissipation vector scales as
Dv = [Dv,1, Dv,2, Dv,3]
T =
[O(M/Re2), O(1/M), O(1/M)]T (C.24)
168 C. Scaling of the artificial dissipation
when preconditioning is not used. To arrive at this result, it was assumed
that the Prandtl number is in the order of unity and that no dissipation based
on temperature differences are added to the continuity equation. It was also
assumed that the dissipation term, which is based on pressure differences and
is added to the energy equation, can be neglected. For incompressible flows,
the last two assumptions are fulfilled since both %T and dp vanish identically;
for ideal gas flow not. The neglected dissipation termDv,1,T = k(4)σ/∆x·%∗T/Re·
δ(4)T based on temperature differences and the neglected pressure dissipation
term Dv,3,p = −k(4)σ/∆x · d∗p%∗T · Prδ(4)p scale as
Dv,1,T = O(1/M · 1/Re), Dv,3,p = O(M/Re) (C.25)
for Prandtl numbers in the order of unity. Whether or not the last terms
are accounted for, the scaling of the artificial dissipation will in any case be
wrong unless both the Mach and the Reynolds number scale in the order of
unity. If preconditioning is not used then the unsteady terms in Eq. (C.22) will
neither scale in the order of unity. However, what matters for the accuracy of
the converged solution is that the dissipation terms scale in the order of the
leading flux term.
In the case that time-derivative preconditioning is used, the physical deri-
vatives %p and %T appearing in the matrix Γe and in the dissipation vector Dv
are replaced with the scaled quantities %′p and %′T , respectively. Furthermore,
the spectral radius σ∗ is oriented after the spectral radius of the precondi-
tioned system, i.e., σ∗ ≈ |u∗|. When preconditioning is used, the leading term
of the convective and diffusive fluxes in each of the equations (C.22) still scales
in the order of unity. If preconditioning for inviscid flow is used, the precondi-
tioning parameter %′∗p is set to %
′∗
p = kinvp
†/(%†u†2), which for ideal gas flow can
be written as %′∗p = kinvγ/M †
2. In the case the inviscid preconditioning is used
the artificial dissipation vector scales as
Dv =
[O(1/Re2), O(1), O(1)]T , (C.26)
if the terms Dv,1,T and Dv,3,p are neglected and it is assumed that the Prandtl
number scales in the order of unity. If the parameter %′T is set to its physical
counterpart %T the term Dv,1,T can, due to its scaling Dv,1,T = O(1/Re), be
neglected as the pressure diffusion term Dv,1,p with Dv,1,p = O(1/Re2) will
dominate for low Reynolds number flow. The temperature dissipation term
Dv,1,T vanishes identically in the case that the preconditioning parameter %′T
is set to zero. The second term Dv,3,p can also be neglected as it scales in
the order of the perturbation parameter , which in the present analysis is
assumed to be small.
If, however, viscous preconditioning is used, then the parameter %′∗p is set
according to Eq. (A.27), which for ideal gases yields
%
′∗
p = kv
Re2p†
%†u†2
= kv
Re2
γM†
2 . (C.27)
C.2 Viscous dominated flows 169
If in addition the preconditioning parameter %′∗T is set to zero, all components
of the artificial dissipation vector Dv will scale in the same order as the con-
vective and diffusive terms, i.e.,
Dv = [O(1), O(1), O(1)] . (C.28)
The pressure diffusion term Dv,3,p can be neglected for low values of the per-
turbation variable ; the temperature diffusion term Dv,1,T vanishes identi-
cally for %∗′T = 0. This result is valid for all Mach and Reynolds numbers for
which the perturbation parameter is at least one order less than unity, i.e.,
 = M2/Re 1.
If the preconditioning parameter %′T is set to be proportional to its physical
counterpart %′T = δpc%T , then also a dissipation term Dv,1,T based on temper-
ature differences is added to the continuity equation. This term then scales
as
D∗v,1,T = O(δpc%∗TT ∗/Re). (C.29)
If this term is added to the continuity equation, its magnitude should not
exceed unity. For the sake of avoiding that this term overwhelms the other
terms in the continuity equation the restriction
δpc ≤ Re/(%∗TT ∗) (C.30)
for the preconditioning parameter δpc follows. As discussed in Chapter A.2,
this condition has anyway to be met by the preconditioning scheme to ensure
that the temporal temperature derivative term added to the continuity equa-
tion is not dominating.
170
Appendix D
General issues in simulating low
Mach number flow
As elaborated in Chapter 3 as well as in the previous appendices, a major chal-
lenge in simulating lowMach number flows consists in accommodating for the
disparity of the advective and acoustic wave speeds. An additional matter of
general importance that needs to be dealt with is that in the limit of vanishing
Mach numbers the compressible Navier-Stokes equations do not necessarily
converge to the corresponding transport equations for incompressible flows.
Furthermore, several flow quantities, like the pressure and the temperature,
are nearly constant for low Mach number flow. If no measures are taken when
simulating low speed flows, the largest part of the mantissa is used on storing
the constant part of the solution. This may lead to an insufficient numerical
resolution of the variations in the solution variables, which in turn can cause
significant round-off errors.
D.1 Singularity of the flow equations
To discuss the importance of a proper normalization of the pressure, so that
the compressible flow equations coincide with the ones for incompressible
flows in the limit of a vanishing Mach number, an analysis similar to that
of Müller [194] is carried out in the following.
The starting point of the analysis are the Euler equations, which have
been normalized with the reference quantities p†, ρ†, u†, L†, and t† = L†/u†.
The momentum equations read as
∂(%∗c∗i )
∂t∗
+
∂(%∗c∗i c
∗
j)
∂x∗j
+
p†
%†u†2
∂p∗
∂x∗i
= 0, (D.1)
where the asterix ∗ as a superscript denotes that the respective quantity has
been normalized. The pressure gradient in the momentum equation for ideal
gases can be expressed in the form
p†
%†u†2
∂p∗
∂x∗i
=
1
γM †2
∂p∗
∂x∗i
. (D.2)
171
172 D. Issues in low Mach number flow simulations
Thus, in the limit of vanishing Mach numbers, the momentum equations read
as
∂p∗
∂x∗i
= 0i. (D.3)
A multi-scale perturbation analysis [5, 194] of the compressible Navier-
Stokes equations for low Mach number flows, in which the pressure p is ex-
panded in a series
p = p(0) +Mp(1) +M
2p(2) + . . . (D.4)
of the Mach number, reveals that the zeroth order pressure term p(0) repre-
sents the thermodynamic part of the pressure and only influences the global
compression of the flow field and the average value of the thermodynamic
state variables (like the density) but not the variation of the local flow proper-
ties. The first order pressure p(1), on the other hand, is linked to the acoustics
waves and the local compression of the flow field, whereas the second order
pressure p(2) is directly coupled to the advection of the flow and the incom-
pressible part of flow dynamics. To avoid a decoupling of the pressure and the
local velocity field, it must be ensured that the second order pressure term p(2)
remains effective in the flow equations in the limit of vanishing Mach num-
bers.
To avoid a so-called pressure-velocity decoupling at low Mach numbers,
Bijl [35] suggests to introduce a dimensionless pressure defined as
p∗ =
p− p(0)
%†u†i
2 , (D.5)
where p(0) represents the zeroth order pressure in the flow; for practical com-
putations the zeroth order pressure p(0) can be taken as the inlet or outlet
pressure of the domain. With this choice of normalization, the pressure p∗ is
in the order of unity and the normalized momentum equations can be written
as
∂(%∗c∗i )
∂t∗
+
∂(%∗c∗i c
∗
j)
∂x∗j
+
∂p∗
∂x∗i
= 0. (D.6)
This means that the pressure gradient term is of the same order as the con-
vective and the unsteady terms.
For the purpose of normalizing the pressure as well as the velocity compo-
nents, it is appropriate to orient the reference velocity u† after the maximum
norm of the velocity in the flow. This ensures that the normalized velocities
scale in the order of unity for low Mach number flow. The density at the inlet
may be used as reference density %† for the normalization of the pressure in
Eq. (D.5).
D.2 Avoidance of round-off errors at low Mach
numbers
In low Mach number flows some flow quantities like the pressure are nearly
constant. Thus, if no care is taken the major part of the mantissa will be
D.2 Avoidance of round-off errors at low Mach numbers 173
used for storing the constant part of the pressure and large round-off errors
may arise. To quantify round-off errors and to demonstrate how the use of a
perturbed pressure may alleviate the numerical accuracy issue, an analysis
similar to that one of Müller [194] will be carried out in the following.
In order to estimate the round-off errors associated with the numerical
calculation of a pressure difference ∆p∗ = p∗R − p∗L between the right and the
left state of a cell interface, we assume that the flow quantities have been
normalized by the reference quantities given in Eq. (A.21) such that their
magnitude is in the order of unity.
Due to finite precision the exact values for pressures p∗L and p∗R are trun-
cated; their numerical representations p˜∗L and p˜∗R can be written as
p˜∗i = (1 + m,i)p
∗
i , i ∈ {L,R} (D.7)
where m represents the machine accuracy, which in other words is the small-
est machine number that added to unity gives a numerical sum different from
unity [215]. The numerical representation of a pressure difference ∆p˜∗ can be
expressed in the following fashion
∆p˜∗ = p˜∗R − p˜∗L = ∆p∗
(
1 +
p∗R
∆p∗
m,R − p
∗
L
∆p∗
m,L
)
. (D.8)
The relative error of the pressure difference ∆p˜∗ due to numerical round-off
errors can be estimated as
rnd =
∣∣∣∣∆p˜∗ −∆p∗∆p∗
∣∣∣∣ ≤ p∗L + p∗R|∆p∗| m (D.9)
with m = max(|m,L|, |m,R|). For convection dominated, steady state flow it fol-
lows from the Bernoulli equation that the variations in the normalized pres-
sure scale in the order of O(M2):
δp∗ =
δp
p
= −%u
2
p
δu
u
= −γM2δu∗. (D.10)
Assuming the presence of low Mach number flow and that the variations in
the normalized velocity differences are in the order of unity, the round-off
error associated with the calculation of pressure differences is of the order
O(mM−2). With a 4 byte representation (single precision) of floating numbers
the machine accuracy on common computers is approximately 10−7. Already
at a Mach number of M = 10−3 the relative error becomes of the order of
O(10−1) for steady state flows, which is clearly unacceptable.
Round-off errors can be avoided if the nearly constant part is subtracted
from the pressure and this part is stored separately. Whether the pressure
variable itself p(xi, t) or a perturbed pressure variable p′(xi, t) defined as
p′(xi, t) = p(xi, t)− p0, (D.11)
is used, where p0 represents the pressure on some point of the computational
domain, does not matter in the calculation of gradients or differences:
∂p′
∂x
=
∂p
∂x
, ∆p = ∆p′. (D.12)
174 D. Issues in low Mach number flow simulations
When thermodynamical quantities have to be calculated, which depend on the
absolute pressure and the absolute temperature level, the perturbed thermo-
dynamic state variables can only be utilized after adding the constant refer-
ence quantity to them. When perturbed thermodynamic state variables are
used, the density is for ideal gas flows calculated from the following relation
% =
p′ + p0
R(T ′ + T0)
. (D.13)
The relative round-off errors in the computation of the differences ∆p′ of
the perturbed pressure variable can be determined as
rnd =
∣∣∣∣∆p˜′ −∆p∆p
∣∣∣∣ ≤ |p′L|+ |p′R||∆p| m. (D.14)
If the reference pressure p0 is chosen as
min(pL, pR) ≤ p0 ≤ max(pL, pR) (D.15)
then the round-off error is bounded by
rnd =
∣∣∣∣∆p˜′ −∆p∆p
∣∣∣∣ ≤ m. (D.16)
The relative error is independent of the Mach number and is bounded by the
machine accuracy. The validity of the analysis above is not compromised when
the static pressure variable is normalized by the dynamic pressure as devel-
oped in Section D.1.
D.3 Use of a perturbed temperature variable in
the simulations of low Mach number flow
In the preceeding sections it has been shown that it is important to use a pres-
sure that is both perturbed and normalized to ensure that the Navier-Stokes
equations for compressible flows coincide with the Navier-Stokes equations
for incompressible flows in the limit of a vanishing Mach number. The use
of a perturbed pressure has also the advantage that round-off errors can be
avoided in numerical simulations.
In this section it will be shown that the temperature needs to be perturbed
as well to ensure that the kinetic energy term is of comparable size as the
static enthalpy when applying the energy equation.
If the total enthalpy h0 for a caloric perfect gas is formulated as
h0(T, u) = cpT +
1
2
u2, (D.17)
then the contribution of the kinetic energy term may become negligible in
comparison to the inner energy. To make this apparent, the flow quantities
are normalized with a reference velocity u†, which yields
h∗0(T, u) =
γRT
(γ − 1)u†2 +
1
2
u∗2 =
1
(γ − 1)M2 +
1
2
u∗2. (D.18)
D.3 Use of a perturbed temperature variable 175
Thus, the total enthalpy should be formulated as
h0(T )− h0(T0) = cp(T − T0) + 1
2
u2, (D.19)
which in normalized variables yields
∆h∗0 = c
∗
p(T
∗ − T ∗0 ) +
1
2
u∗2. (D.20)
For adiabatic, inviscid flow, the temperature variations δT are directly related
to the velocity variations δu via the relation cp δT ∼ u δu, from which it follows,
that if a suitable reference temperature T0 is chosen, the static enthalpy term
will not dominate the kinetic energy term in magnitude. It should further-
more be noted that for adiabatic, inert, low Mach number ideal gas flow, the
variation in temperature or in enthalpy can be low. Thus, also for the sake of
generally avoiding round-off errors in the calculation of differences and gradi-
ents, not only a perturbed pressure variable but also a perturbed temperature
or enthalpy variable should be used.
If the flow is non-adiabatic, or if the flow is viscous and the Peclet numbers
are less than unity, additional measures must be applied to avoid singular-
ity of the flow equations and round-off errors in the low Mach number limit.
These issues will not be addressed here since in the current work only flows
with marginal heat losses are considered. The interested reader may refer
to the work of Müller [194] for details on how to accommodate for such flow
conditions.
176
Appendix E
Flux limiters to ensure bounded
and monotone solutions
For flow simulations it can be of large importance to have a numerical scheme
that leads to bounded and monotone solutions. The property of being mono-
tone means that the numerical scheme for the solution of linear transport
equations does not increase the total variation of the solution and that the so-
lution scheme does not introduce any new extremes in the state variables [113,
119]. It is important to avoid that the state variables that are naturally
bounded (e.g., the density, which has to be positive) take on unphysical values
at any stage in the solution process. Overshoots and undershoots in the solu-
tion may not only be a nuisance from a physical point of view, but may also
trigger instability of the solution scheme.
Gudunov’s order barrier theorem [101, 102, 302] states that a linear nu-
merical scheme that leads to monotone solutions in general can at most be
first-order accurate. In other words, there are no linear schemes of second or
higher order which will be able to prevent the generation of oscillations in the
solution in regions of large gradients. Since a first order scheme does not yield
a sufficient accuracy for practical computational meshes, non-linear schemes
have to be constructed, in order to simultaneously obtain a scheme that is
monotone and is essentially of higher order [119]. Such non-linear schemes
can be realized by the use of slope or flux limiters, which prevent oscillations
by locally reducing the order of the scheme in the vicinity of local extrema in
the solution.
E.1 Monotonicity conditions and limiters
To demonstrate the ideas of limiters, potential spatial discretizations of the
scalar convection equation
∂φ
∂t
+ c
∂φ
∂x
= 0 (E.1)
for the transport variable φ for a positive convection velocity c will be consid-
ered in the following. A discretization of this transport equation on a mesh
177
178 E. Flux limiters
Π : x = 0, . . . , i·∆x, . . . ; i ∈ N; ∆x > 0 leads to the following difference equation
∂φi
∂t
+
c
∆x
(
φi+1/2,L − φi−1/2,L
)
= 0. (E.2)
With the state variable φi+1/2 at the interface taken as
φi+1/2,L = φi +
1
2
Ψ(ri)(φi − φi−1) (E.3)
with ri as the ratio of successive gradients at point i given as ri = (φi+1 −
φi)/(φi − φi−1) various schemes can be defined in dependence on how the lim-
iter function Ψ is defined. For Ψ(r) = r a second order central scheme results,
whereas Ψ(r) = 1 yields a second order upwind scheme [70]. Obviously, choos-
ing Ψ = 2r leads to a first order downwind and setting Ψ to Ψ = 0 to a first
order upwind scheme.
The reconstruction procedure above applies for positive wave speeds c > 0;
for negative wave speeds, the state at the interface i+1/2 can be reconstructed
as
φi+1/2,R = φi+1 − 1
2
Ψ(r′i+1)(φi+2 − φi+1), (E.4)
with r′i+1 = (φi − φi+1)/(φi+1 − φi+2) = 1/ri+1. Thus, the interface state φi+1/2
is for c < 0 rather reconstructed from the right from the states in the nodes
i, i+1, i+2 than from left from the states in the nodes i−1, i, i+1. To distinguish
whether the states at the interface have been constructed from the left or the
right the subscripts ’L’ and ’R’ are used. For c < 0 the convection equation
reads as
∂φi
∂t
+
c
∆x
(
φi+1/2,R − φi−1/2,R
)
= 0, (E.5)
which is the same form as the semi-discrete equations (E.2) with the differ-
ence that the interface states have been reconstructed from the right rather
than from the left.
In order to formulate conditions for the limiter function, the explanations
of Hirsch [119] will be followed and the difference equation (E.2) is rewritten
in the following form
∂φi
∂t
+
c
∆x
[
1 +
1
2
Ψ(ri)− 1
2
Ψ(ri−1)
ri−1
]
(φi − φi−1) = 0, ∀c > 0. (E.6)
For negative wave speeds, the corresponding difference equation (E.5) can be
rewritten accordingly
∂φi
∂t
+
(−c)
∆x
[
1− 1
2
ri+1Ψ(1/ri+1) +
1
2
Ψ(1/ri)
]
(φi − φi+1) = 0, ∀c < 0. (E.7)
To yield a monotone scheme, the term in the square brackets in both dis-
cretizations (E.6) and (E.7) must be positive. The reason for the positivity con-
dition is that if the solution possesses a local maximum at node i (φi > φi±1),
the solution should decrease (∂φi
∂t
< 0), whereas the solution should increase
E.1 Monotonicity conditions and limiters 179
at node i if it has a local minimum there. The square bracket in Eq. (E.6) is
positive when
Ψ(ri−1)
ri−1
−Ψ(ri) ≤ 2 (E.8)
and the square in the brackets in Eq. (E.7) is positive when
ri+1Ψ(1/ri+1)−Ψ(1/ri) ≤ 2. (E.9)
In the following conditions will be derived under which Eq. (E.8) and (E.9) are
valid.
The limiter Ψ(r) should preserve the sign of the gradient used for the re-
construction. In addition, it would be natural to reduce the scheme to a first
order scheme, when there is an extremum of the solution at point i, i.e., when
ri is negative. Expressed mathematically, this leads to the following condi-
tions on the limiter:
Ψ(r) ≥ 0, ∀r ≥ 0; Ψ(r) = 0, ∀r < 0. (E.10)
To derive a further condition on the limiter function Ψ(r), it should be noted
that the state φi+1/2,L of the left side of the interface at i+1/2 could as well be
reconstructed via the following relation
φi+1/2,L = φi +
1
2
Ψ(1/ri)(φi+1 − φi) = φi + 1
2
Ψ(1/ri) ri (φi − φi−1) (E.11)
instead of Eq. (E.3). Since the reconstructed states should be independent
whether the limiter applies forward or backward ratios of successive slopes,
the relation Ψ(r) = rΨ(1/r) must be fulfilled. With this property and the
conditions given in Eq. (E.10), the following sufficient conditions for the ful-
fillment of Eq. (E.8) and (E.9) can be formulated:
Ψ(r) ≤ 2 ∧ Ψ(r) ≤ 2r. (E.12)
Summarizing these last two last conditions and the constraints formulated
in Eq. (E.10), the solution of the scalar convection equation will be monotone
with the discretization defined by either Eqs. (E.2) and (E.3) or Eqs. (E.4) and
(E.5) if the following condition is satisfied:
0 ≤ Ψ(r) ≤ min(2r, 2). (E.13)
In the case of a linear solution on an equidistant mesh the ratio r of successive
gradients is unity. In this case, it would be desirable that the limiter function
Ψ equal unity, so that the reconstruction given by Eq. (E.3) leads to a scheme
of second order accuracy. To make sure that a limiter function Ψ(r) fulfills this
property, the following restriction
1 ≥ Ψ(r) ≥ r ∨ r ≥ Ψ(r) ≥ 1 (E.14)
may be required from a limiter in addition to the condition given in Eq. (E.13).
The last condition also follow from Sweby’s suggestion [269] to formulate a
limiter that represents a convex combination of the Lax-Wendroff and Beam-
Warming scheme to avoid that a limiter be too compressive or be too diffusive.
The resulting admissible region of a limiter is referred to in the literature as
Sweby’s second order TVD region [161, 302].
180 E. Flux limiters
E.2 Limiter functions
Various limiter functions Ψ(r) can be defined, that with the discretization
(E.2), (E.3) leads to a second order TVD scheme. The limiter following the
lower bound of Sweby’s second order TVD region is the so-called Min-mod
limiter
Ψ(r) = max [0,min(r, 1)] . (E.15)
The limiter is formulated in a different form in Eq. (4.17). It takes the min-
imum of the consecutive slopes for the reconstruction and leads to the most
diffusive solutions of the limiter functions which fall in Sweby’s second order
TVD region. A limiter that is following the upper bound of the second order
TVD region is the Superbee limiter of Roe:
Ψ(r) = max [0,min(2r, 1),min(r, 2)] . (E.16)
For reconstruction, it takes the maximum of two consecutive slopes, as long
as their ratio is less than 2. In the other case, it takes the minimum of both
slopes. It is the least diffusive and the most compressive limiter that leads to
a second order TVD scheme. In Fig. E.1 both the Min-mod and the Superbee
limiter functions, as well as other classical limiters are shown. For a detailed
discussion of the performance of various limiters the reader may refer to the
book of Hirsch [119].
 0
 0.5
 1
 1.5
 2
 0  0.5  1  1.5  2  2.5  3
Ps
i(r)
 [-]
r [-]
Superbee
Minmod
van Leer
van Albada
van Albada 2
Figure E.1: Various limiter functions plotted in the Sweby-diagram. For 0 ≤ Ψ(r) ≤
min(2r, 2) a limiter yields a monotone scheme if applied with the recon-
struction procedure given in Eq. (E.3). For limiters lying between or
on the Min-mod and the Superbee limiters, a second order TVD scheme
results.
E.3 Discussion of Jameson’s SLIP and USLIP schemes 181
E.3 Discussion of Jameson’s SLIP and USLIP
schemes
The SLIP scheme of Jameson [129, 130] can not be analysed with the Sweby-
diagram since the respective limiter as defined in Eq. (4.13) used for control-
ling the reconstruction (4.12) of the left and right states at an interface i+1/2
is not using consecutive slopes. Furthermore, the method of reconstruction
used in the SLIP scheme is different from that one described in either of the
Eqs. (E.3) and (E.4). However, the conditions given in Eq. (4.16) on the limiter
function L(φ1, φ2) are sufficient to ensure that the resulting diffusive flux for-
mulation in Eq. (4.9), which is based on the reconstruction given in Eq. (4.12),
leads to a monotone scheme.
If Jameson’s Upstream Symmetric Limited Positive (USLIP) scheme is
used as discretization method, which is an upwind-biased form of the SLIP
scheme, then the condition given in Eq. (E.13) on a limiter function Ψ(r) can
be applied. In the USLIP scheme, the states at an interface i+ 1/2 are recon-
structed as follows
qkv,i+1/2,L = q
k
v,i +
1
2
L(∆qkv,i−1/2,∆q
k
v,i+1/2),
qkv,i+1/2,R = q
k
v,i+1 − 12L(∆qkv,i+1/2,∆qkv,i+3/2).
(E.17)
This is the same reconstruction as the one given in Eqs. (E.3) and (E.4),
wherefore the limiter function L(φ1, φ2) of Jameson given in Eq. (4.13) for
the use in the USLIP scheme can be analysed with the Sweby-diagram de-
scribed above. The limiter function L(φ1, φ2) of Jameson can be written as
L(φ1/φ2, 1) · φ2 = L(r, 1) · φ2 = L(1, r) · φ1. The limiter function L(φ1, φ2) is con-
structed in such a manner that it for φ1 · φ2 < 0 returns zero. For positive
ratios r of successive gradients and for the parameter eq set either to eq = 1 or
eq = 2, the limiter can be rewritten as
Ψ(r)|eq=1 = L(r, 1) =
1
2
(1 + r)− 1
2
|1− r| and Ψ(r)|eq=2 = L(r, 1) =
2r
r + 1
,
(E.18)
respectively. For a parameter eq set to eq = 1 the Jameson limiter corresponds
to the Min-mod limiter, whereas it for eq = 2 corresponds to the van Leer lim-
iter. Thus, for eq = 1 or eq = 2, the limiter of Jameson given in Eq. (4.13) to-
gether with the USLIP reconstruction procedure described in Eq. (E.17) leads
to a second order TVD scheme.
182
Appendix F
Roe’s Characteristic Upwind
Dissipation scheme for
preconditioned systems
In this chapter Roe’s characteristic upwind dissipation scheme is formulated
for preconditioned systems and the necessary absolute Jacobians for the flux
difference splitting are defined. Both a conservative and a non-conservative
scheme will be formulated and the choice of the solution variable will be dis-
cussed. The principles of Roe-averaging will be explained and finally the
necessary system-matrices, eigenvectors and resulting artificial dissipation
terms are derived.
F.1 Conservative and non-conservative
formulation
As a basis for the following analysis, the one-dimensional preconditioned Eu-
ler equations
Γv
∂Qˆv
∂t
+
∂Eˆc
∂ξ
= 0 (F.1)
for the ξ-direction are used. The vectors Qˆv and Eˆc are defined in Eq. (4.27)
and denote the vector of primitive viscous variables and the vector of fluxes in
the ξ-direction, respectively.
A linearisation around an average state ¯ˆQv leads to the following equa-
tions
∂Qˆv
∂t
+ AˆΓv ,ξ
∣∣∣ ¯ˆ
Qv
· ∂Qˆv
∂ξ
= 0, AˆΓv ,ξ = Γ
−1
v Aˆξ,v, Aˆξ,v =
∂Eˆc
∂Qˆv
(F.2)
and a successive multiplication with the left eigenvector matrix Lξ,v of the flux
Jacobian AˆΓv ,ξ yields the following system of decoupled equations
Lξ,v
∂Qˆv
∂t
+ Lξ,vΓ
−1
v Aˆξ,vL
−1
ξ,v︸ ︷︷ ︸
Λξ,v
Lξ,v
∂Qˆv
∂ξ
= 0, (F.3)
183
184 F. The Roe scheme for preconditioned systems
where Λξ,v = Lξ,vΓ−1v Aˆξ,vL
−1
ξ,v = Diag. (λ′1,v, . . . , λ′5,v) is a diagonal matrix of the
eigenvalues of the preconditioned flux Jacobian matrix. Defining the charac-
teristic variable φˆ as φˆ = Lξ,vQˆv, the last equation can be transformed to the
following system
∂φˆ
∂t
+ Λξ,v
∂φˆ
∂ξ
= 0 (F.4)
of decoupled characteristic variables.
Formulated as a semi-discrete first order upwind scheme the last equation
yields
∂φˆ
∂t
+ Λξ,v
φˆi+1 − φˆi−1
2∆ξ
= |Λξ,v|φˆi+1 − 2φˆi + φˆi−1
2∆ξ
, (F.5)
with the diagonal matrix |Λξ,v| = Diag.(|λ′ξ,v,1|, . . . , |λ′ξ,v,5|) of the absolute eigen-
values.
By multiplying the last equation from the left with the inverse of the ma-
trix of the left eigenvectors, the following system of semi-discrete equations
results
∂Qˆv,i
∂t
+ AΓv ,ξ
1
2∆ξ
(Qˆv,i+1 − Qˆv,i−1) = |AΓv ,ξ|
1
2∆ξ
(Qˆv,i+1 − 2Qˆv,i + Qˆv,i−1) (F.6)
with the absolute Jacobian |AΓv ,ξ| given by |AΓv ,ξ| = L−1ξ,v|Λξ,v|Lξ,v. Treating the
central and the dissipation terms nonlinearly yields the following formulation
∂Qˆv,i
∂t
+ Γ−1v,i
1
2∆ξ
(Eˆc,i+1 − Eˆc,i−1) = Dv,nc,i (F.7)
with the dissipation term defined as
Dv,nc,i =
1
2∆ξ
[
(|AΓ,ξ|i+1/2)
(
Qˆv,i+1 − Qˆv,i
)
− (|AΓ,ξ|i−1/2)
(
Qˆv,i − Qˆv,i−1
)]
. (F.8)
With the forward and backward difference operators δ+ and δ− defined as
δ+ψi = ψi+1 − ψi and δ−ψi = ψi − ψi−1, respectively, the dissipation term can in
compact form be expressed as
Dv,nc,i =
1
2∆ξ
δ−
(
|AΓv ,ξ|i+1/2δ+Qˆv,i
)
. (F.9)
If this formulation is used, the artifical dissipation terms have to be added
to the convective residual vector after the latter has been multiplied with the
inverse preconditioning matrix. The use of this formulation would be disad-
vantageous for flows with shocks since the dissipation terms are written in
a non-conservative form [277]. As can be seen from Eq. (F.7), the convective
fluxes are multiplied with the inverse of the preconditioning matrix, which
is varying from cell to cell, while the dissipation terms are not. To ensure
conservativity of the discretization and to achieve a cancellation of the dissi-
pation terms and the centrally discretized convective fluxes to yield full up-
winding, the dissipation terms must be formulated as a part of the convective
F.2 Use of conservative versus non-conservative variables 185
fluxes. For the sake of achieving a discretization that is accurate when the
flow variables are subject to strong variations, the formulation of the dissi-
pation scheme should be such that when preconditioning is shut off (at su-
personic conditions), the dissipation terms are effectively constructed from
differences of the conservative variables. The choice of the variables used for
the construction of the artificial dissipation terms will be discussed in the next
section.
To attain a conservative formulation, Eq. (F.6) is multiplied by the precon-
ditioning matrix Γv; afterwards both the central and the dissipation terms are
written in a non-linear representation to yield the following system
Γv,i
∂Qˆv,i
∂t
+
1
2∆ξ
(Eˆv,i+1 − Eˆv,i−1) = 1
2∆ξ
δ−
(
Γv,i+1/2|AΓ,ξ|i+1/2δ+Qˆv,i
)
. (F.10)
Summing up the contributions of the convective fluxes and the artificial dissi-
pation yields
Γv,i
∂Qˆv,i
∂t
+
1
2∆ξ
[
(Eˆv,i+1 − Eˆv,i−1)− δ−
(
Γv,i+1/2|AΓ,ξ|i+1/2δ+Qˆv,i
)]
= 0. (F.11)
Introducing the flux vector at the interface i + 1/2 of the dual mesh as the
contribution of both the convective and the dissipative fluxes as
Fˆv,i+1/2 = Eˆv,i+1/2 − 1
2
Γv,i+1/2|AΓ,ξ|i+1/2δ+Qˆv,i (F.12)
with Eˆv,i+1/2 = 12(Eˆv,i+1 + Eˆv,i), Eq. (F.11) can also be written as
Γv,i
∂Qˆv,i
∂t
+
1
∆ξ
(
Fˆv,i+1/2 − Fˆv,i−1/2
)
= 0. (F.13)
The fact that the dissipation term can be written as a part of the convective
fluxes, shows the conservativity of the formulation of the dissipative fluxes in
Eq. (F.10).
F.2 The use of conservative versus
non-conservative solution variables
In order to obtain a formulation, for which effectively flux differences of con-
served quantities are taken (when preconditioning is shut off), the Euler equa-
tions without preconditioning
∂Qˆ
∂t
+
∂Eˆc
∂ξ
= 0 (F.14)
are used as a starting point for the following analysis. Preconditioning is
introduced by multiplying the temporal term with the preconditioning matrix
Γk, which is defined as
Γk = ΓvΓ
−1
e , Γe =
∂Qˆ
∂Qˆv
. (F.15)
186 F. The Roe scheme for preconditioned systems
When preconditioning is shut off, Γv = Γe, and the matrix Γk equals the iden-
tity matrix. A linearization of these equations around an averaged state ¯ˆQ
yields the following equations
∂Qˆ
∂t
+ AˆΓk,ξ
∣∣∣ ¯ˆ
Q
· ∂Qˆ
∂ξ
= 0, AˆΓk,ξ = Γ
−1
k Aˆξ,c, Aˆξ,c =
∂Eˆc
∂Qˆ
(F.16)
and a successive multiplication with the left eigenvector matrix Lξ,k of the flux
Jacobian AˆΓk,ξ yields the following system of decoupled equations
Lξ,k
∂Qˆ
∂t
+ Lξ,kAˆΓk,ξL
−1
ξ,k︸ ︷︷ ︸
Λξ,k
Lξ,k
∂Qˆ
∂ξ
= 0, (F.17)
where Λξ,k = Lξ,kAˆΓk,ξL
−1
ξ,k = Diag. (λ′1,k, . . . , λ′5,k) represents a diagonal matrix
of the eigenvalues of the preconditioned, conservative flux Jacobian matrix.
With the characteristic variable φˆk now defined as φˆ = Lξ,kQˆ, the last system
of equations can be written as
∂φˆ
∂t
+ Λξ,k
∂φˆ
∂ξ
= 0, (F.18)
which in a semi-discrete upwind formulation yields
∂φˆi
∂t
+ Λξ,k
φˆi+1 − φˆi−1
2∆ξ
= |Λξ,k|φˆi+1 − 2φˆi + φˆi−1
2∆ξ
. (F.19)
Transformation back by multiplying with the inverse of the left eigenvector
matrix yields
∂Qˆi
∂t
+ AˆΓk,ξ ·
Qˆi+1 − Qˆi−1
2∆ξ
= |AˆΓk,ξ|
Qˆi+1 − 2Qˆi + Qˆi−1
2∆ξ
, (F.20)
where the absolute Jacobian |AˆΓk,ξ| is given by
|AˆΓk,ξ| = L−1ξ,k|Λξ,k|Lξ,k. (F.21)
Multiplying from the left with the preconditioning matrix Γk and and treating
the central term non-linearly yields
Γk
∂Qˆi
∂t
+
Eˆi+1 − Eˆi−1
2∆ξ
= Γk|AˆΓk,ξ|
Qˆi+1 − 2Qˆi + Qˆi−1
2∆ξ
. (F.22)
This formulation of the dissipation term is conservative but is has the dis-
advantage that it can not be used for the simulation of incompressible flows,
as in this case the density variations vanish identically and consequently no
dissipation would be added to the continuity equation. To be able to simulate
both compressible and incompressible flow it would be preferable to use the
F.3 The principle of Roe-averaging 187
vector Qˆv of primitive viscous variables to construct the artificial dissipation.
With δQ = ΓeδQv the last equation is transformed to
ΓkΓe
∂Qˆv,i
∂t
+
Eˆi+1 − Eˆi−1
2∆ξ
= Γk|AˆΓk,ξ|Γe
Qˆv,i+1 − 2Qˆv,i + Qˆv,i−1
2∆ξ
. (F.23)
This formulation is still conservative and has the advantage that the artifi-
cial dissipation is constructed using differences of the primitive, viscous state
variables. Thus, this formulation should be suited for the simulation of incom-
pressible or compressible, low Mach number flows as well as flows containing
shocks provided that the matrix Γk|AˆΓk,ξ|Γe scales correctly.
To show that the formulation of the artificial dissipation in either Eq. (F.22)
or (F.23) on the linear level is equivalent to the conservative formulation (F.10)
used in the previous section, the absolute Jacobian |AˆΓk,ξ| in the last equation
has to be related to the absolute Jacobian |AˆΓv ,ξ|. In Hirsch [118] the valid-
ity of Lξ,kΓe = Lξ,v is shown. If it is also exploited, that the eigenvalues of
the preconditioned flow equations do not change with a transformation of the
independent variables, i.e., Λξ,v = Λξ,k, then the following relation follows
Γ−1e |AˆΓk,ξ|Γe = Γ−1e L−1ξ,k|Λξ,k|Lξ,kΓe = L−1ξ,v|Λξ,k|Lξ,v = L−1ξ,v|Λξ,v|Lξ,v = |AˆΓv ,ξ|.
(F.24)
Using this relation to substitute the conservative absolute Jacobian |AˆΓk,ξ| in
Eq. (F.23) with the absolute Jacobian |AˆΓv ,ξ| together with the equality Γv =
ΓkΓe yields
Γv
∂Qˆv,i
∂t
+
Eˆi+1 − Eˆi−1
2∆ξ
= Γv|AΓv ,ξ|
Qˆv,i+1 − 2Qˆv,i + Qˆv,i−1
2∆ξ
. (F.25)
This equation corresponds on the linear level to Eq. (F.10), which was to be
demonstrated.
F.3 The principle of Roe-averaging
The absolute Jacobians, which are needed in the calculation of the dissipative
terms in the Roe scheme, may be based on the arithmetic average of the left
and right states of the interface for which the Jacobian is to be determined.
However, this procedure will not lead to a full upwinding. In this section it
will be briefly explained, which properties the averages must have in order to
obtain a scheme that leads to full upwinding.
The starting point of the analysis is the following system of transport equa-
tions
∂U
∂t
+
∂F(U)
∂x
= 0, (F.26)
where U represent an arbitrary state vector and F represents a convective
flux vector.
188 F. The Roe scheme for preconditioned systems
On a one-dimensional, equidistant grid Π : x1, . . . , xi, . . . , xN with xi = i ·
∆x; i ∈ N, ∆x > 0 the semi-discrete form of these equations can be written as
∂Ui
∂t
Vi +Hi+1/2 −Hi−1/2 = 0 (F.27)
where Hi+1/2 and Hi−1/2 represents the numerical flux terms at the interfaces
ni+1/2 and ni−1/2 of the volume Vi enclosing node i. The numerical flux can be
determined as the sum of a central and a dissipative part:
Hi+1/2 =
1
2
(Fi + Fi+1)ni+1/2 − 1
2
B (Ui+1 −Ui)ni+1/2. (F.28)
By introducing the interface flux Jacobian Ai+1/2 as
Ai+1/2 =
{ Fi+1−Fi
Ui+1−Ui , Ui+1 6= Ui
∂F
∂U
, Ui+1 = Ui
(F.29)
and setting the coefficient matrix B for the dissipative terms to the absolute
Jacobian B := |A|, the following equation is obtained
Hi+1/2 = Fini+1/2 +
1
2
(Fi+1 − Fi)ni+1/2 − 12B (Ui+1 −Ui)ni+1/2
= Fini+1/2 +
1
2
(A− |A|)i+1/2 (Ui+1 −Ui)ni+1/2. (F.30)
Using the definition of the absolute Jacobian |A| = L−1|Λ|L, where L repre-
sents the matrix of left eigenvectors of the Jacobian A, the last relation can be
expressed as
Hi+1/2 = Fini+1/2 +
1
2
[
L−1 (Λ− |Λ|)]
i+1/2
(
φi+1 − φi
)
ni+1/2. (F.31)
For positive values of the eigenvalues λj, j = 1, . . . , 5 the last relation reduces
to Hi+1/2 = Fini+1/2. In the opposite case that all eigenvalues are negative,
the last term on the right hand side in Eq. (F.31) equals (Fi+1 − Fi)ni+1/2 and
the flux vector Hi+1/2 becomes Hi+1/2 = Fi+1ni+1/2. In both of these cases
the flux vector Hi+1/2 is solely determined from the states lying upwind of
the interface. In the case that the eigenvalues have a mixed set of signs,
full upwinding is ensured for each of the individual characteristic variables
φj, j = 1, . . . , 5.
From this analysis it follows, that to ensure a full upwinding for the char-
acteristic upwind scheme, the flux Jacobian A(Ui,Ui+1) should possess the
following properties:
1. Ai+1/2(Ui+1 −Ui) = Fi+1 − Fi; ∀Ui,Ui+1.
2. For Ui = Ui+1 the flux Jacobian should correspond to the analytical
value: Ai+1/2(U,U) = ∂F(U)∂U
∣∣∣
U
.
3. The Jacobian A should only possess real eigenvalues and should have
linearly independent eigenvectors.
F.4 Eigenvector and absolute Jacobian matrices 189
If an averaging technique due to Roe is used, the aforementioned proper-
ties can be fulfilled for the unscaled Euler equations. If the Jacobian matrix
A is based on the Roe-averaged quantities
%˜i+1/2 =
√
%i%i+1,
v˜i+1/2 =
√
%ivi +
√
%i+1vi+1√
%
i
+
√
%i+1
,
h˜0,i+1/2 =
√
%ih0,i +
√
%i+1h0,i+1√
%
i
+
√
%
i+1
,
a˜i+1/2 = (γ − 1)
(
h˜0,i+1/2 − 1
2
v˜2i+1/2
)
(F.32)
then condition (1) above is fulfilled [274] for ideal gas flow. Condition (2) is
fulfilled with a consistent derivation of the flux Jacobian, and condition (3) is
given by the Euler equations for ideal gas flow [118].
In the current work the Roe-averaging procedure given in Eq. (F.32) was
applied in conjunction with preconditioning for the determination of the in-
terface states Qv,i+1/2 in the simulation of ideal gas flow. For the simulation
of incompressible fluids flow the interface states were determined by simple
arithmetic averaging of the left and right states of each cell interface.
F.4 Eigenvector and absolute Jacobian
matrices
The system matrix AˆΓv ,ξ = Γ−1v Aˆξ = Γ−1v ∂Eˆ∂Qˆv can be determined as [6]:
AˆΓv ,ξ =

[%′T (1− %hp)+
%%ph
′
T ]
wˆξ
d′
%2h′T ξx
d′
%2h′T ξφ
d′
%2h′T ξr
d′
%wˆξ
d′ (%Th
′
T
−hT%′T )
ξx
%
wˆ 0 0 0
ξφ
%
0 wˆ 0 0
ξr
%
0 0 wˆ 0
wˆξ
d′ [%ph
′
p−
%′p(1− %hp)]
%h′pξx
d′
%h′pξφ
d′
%h′pξz
d′
wˆξ
d′ (%%
′
phT+
%Th
′
p)

, (F.33)
where the contravariant velocity wˆξ is given by Eq. (3.8). With the assumption
that preconditioning variables %′T , h′p and h′T are set to their physical values,
(i. e., only %′p is used as an active preconditioning parameter), the matrix Lξ=
[l1(λξ,1), . . . , l5(λξ,5)] of left eigenvectors becomes
Lξ =

−1−%hp
%hT
0 0 0 1
0 l˜x l˜φ l˜r 0
0 m˜x m˜φ m˜r 0
1 %λ˜wξ,4ξ˜x %λ˜
w
ξ,4ξ˜φ %λ˜
w
ξ,4ξ˜r 0
1 %λ˜wξ,5ξ˜x %λ˜
w
ξ,5ξ˜φ %λ˜
w
ξ,5ξ˜r 0
 (F.34)
190 F. The Roe scheme for preconditioned systems
with λ˜wξ,4 = (λ˜ξ,4−w˜ dd′ ) and λ˜wξ,5 = (λ˜ξ,5−w˜ dd′ ). The left eigenvectors li, i = 1, . . . , 5
are defined as the vectors fulfilling (li)T (AˆΓ − λξ,iI) = 0 with the eigenvalues
λξ,i defined according to Eq. (3.6). The quantities denoted by a "˜" are nor-
malized with ‖∇ξ‖, for instance the normalized contravariant velocity w˜ξ is
defined as
w˜ξ =
wˆ√
ξ2x + ξ
2
φ + ξ
2
r
=
w · Sξ
‖Sξ‖ . (F.35)
The vectors [l˜x, l˜φ, l˜r] and [m˜x, m˜φ, m˜r] represent two mutually orthogonal unit
vectors which both are orthogonal to the normalized grid projection vector
[ξ˜x, ξ˜φ, ξ˜r]. Due to the orthogonality of the left li and right eigenvectors ri, the
matrix of right eigenvectors Rξ = [rR(λξ,1), . . . , rR(λξ,5)] equals the inverse of
the Matrix Lξ, i. e., Rξ = L−1ξ . Thus, this matrix reads as
Rξ =

0 0 0
λ˜ξ,5−w˜ dd′
λ˜ξ,5−λ˜ξ,4
λ˜ξ,4−w˜ dd′
λ˜ξ,4−λ˜ξ,5
0 l˜x m˜x
ξ˜x
%(λ˜ξ,4−λ˜ξ,5)
ξ˜x
%(λ˜ξ,5−λ˜ξ,4)
0 l˜φ m˜φ
ξ˜φ
%(λ˜ξ,4−λ˜ξ,5)
ξ˜φ
%(λ˜ξ,5−λ˜ξ,4)
0 l˜r m˜r
ξ˜r
%(λ˜ξ,4−λ˜ξ,5)
ξ˜r
%(λ˜ξ,5−λ˜ξ,4)
1 0 0 1−%hp
%hT
λ˜ξ,5−w˜ dd′
λ˜ξ,5−λ˜ξ,4
1−%hp
%hT
λ˜ξ,4−w˜ dd′
λ˜ξ,4−λ˜ξ,5

. (F.36)
By defining the volume weighted diagonal matrix to
|ΛVξ | = Diag(|λVξ,1|, |λVξ,2|, |λVξ,3|, |λVξ,4|, |λVξ,5|) (F.37)
with the volume weighted eigenvalues given by
λVξ,1,2,3 = w · Sξ = wxSξ,x + wφSξ,φ + wrSξ,r,
λVξ,4,5 =
1
2
w · Sξ
(
1 + d
d′
)± 1
2
√
(w · Sξ)2
(
1− d
d′
)2
+ 4%hT
d′
(F.38)
the product |AΓ,ξ| · δQv = Lξ · |Λξ| ·L−1ξ · δQv = δpi after some algebra reduces to
δpi1 = c11 · δp+ %c12 · (δΥ)ξ,
δpi2 = |λV1 | · δu+ Sξ,x
[
c21 · δp− |λ
V
1 |+ |λV45|
‖Sξ‖2 · (δΥ)ξ
]
,
δpi3 = |λV1 | · δv + Sξ,y
[
c21 · δp− |λ
V
1 |+ |λV45|
‖Sξ‖2 · (δΥ)ξ
]
,
δpi4 = |λV1 | · δw + Sξ,z
[
c21 · δp− |λ
V
1 |+ |λV45|
‖Sξ‖2 · (δΥ)ξ
]
,
δpi5 = bp · δpi1 + |λV1 | (δT − bp · δp)
(F.39)
F.4 Eigenvector and absolute Jacobian matrices 191
with δQv = [δp, δu, δv, δw, δT ]T and the terms (δΥ)ξ, bp, c11, c12, c21, λV45 given by
(δΥ)ξ = Sξ,xδu+ Sξ,yδv + Sξ,zδw, bp =
(1− %hp)
%hT
,
c11 =
|λV5 |(λV4 − UV dd′ )− |λV4 |(λV5 − UV dd′ )
λV4 − λV5
, c21 =
|λV4 | − |λV5 |
%(λV4 − λV5 )
,
c12 =
(|λV5 | − |λV4 |)(λV5 − UV dd′ )(λV4 − UV dd′ )
(λV4 − λV5 ) · ‖Sξ‖2
,
λV45 =
|λV5 |(λV5 − UV dd′ )− |λV4 |(λV4 − UV dd′ )
(λV4 − λV5 )
.
(F.40)
192
Appendix G
The influence of the geometrical
parameters of the labyrinth seal
In the work of Anker et al. [14], steady-state simulations were carried out us-
ing the preconditioned solution scheme to study the influence of the geomet-
rical parameters of the labyrinth seal on the leakage flow in the Bochum tur-
bine. In that study, using the same geometry and the same operating point as
described above (Ch. 6.1) as a basis (scl, sw=3mm, sa, se=6mm), the clearance
height (scl/mm=1, 2, 3, 5), the width of the fins of the labyrinth seals (sw/mm=
3, 6, 9), the width of the inlet gap of the labyrinth seal (se/mm = 3, 6, 12) and
the width of the labyrinth outlet gap (sa/mm=3, 6, 12) were changed indepen-
dently of each other. Figure G.1 illustrates the geometrical variations that
were subject to investigation. The purpose of the study was to calibrate a
numerical model of the labyrinth seal which replaces a discretization of the
details of the sealing arrangements.
The study showed that the clearance height is the most influential param-
eter of those investigated. For the geometrical configurations examined, the
simulations predicted a nearly linear increase in the leakage mass flow rate
with an in increase in the seal leakage height (Fig. G.2a). The effect of varying
this parameter from scl = 0mm to scl = 3mm has been described in previous
chapters and shall not be repeated here. In the aforementioned study [14] it
se sa
scl
sw
Figure G.1: Variations in the geometry of the labyrinth seal in the work of Anker et
al. [14]
193
194 G. Influence of geometry of the labyrinth seal
was however discovered that the difference∆vφ in the circumferential velocity
vφ of the leakage flow between its entry in the first cavity of the labyrinth seal
rotor to its re-injection in the main flow after the rotor is strongly dependent
on the clearance height of the seal. In Fig. G.2b the change in the circumfer-
ential velocity ∆vφ is displayed in dependence of the clearance height. The
swirl of the leakage flow increases from the entry in the seal to its re-injection
in the main flow for low values of the clearance height scl, for which the mass
flow rates are small. For low clearance heights more near-wall flow with a
low circumferential velocity enters the labyrinth seal. For large clearance
heights more fluid of the main flow, which has the circumferential velocity of
the rotor, enters the labyrinth seal. While the leakage flow for large clear-
ance heights passes through the labyrinth seal with a high swirl and without
much deflection, for low clearance heights it enters the labyrinth seal with a
low circumferential velocity and is accelerated to the half of the circumferen-
tial speed of the rotor through friction. This result is of importance since it
implies that as long as the leakage flow rate is low, it is principally possible to
extract fluid of the boundary layer with a low exergy and thus to avoid that
fluid with high values of the kinetic energy is flowing through the labyrinth
seal without doing work on the rotor blade.
The dependence of the leakage mass flow rate on the width of the inlet (se)
or the outlet (sa) gap of the labyrinth seal is much smaller than that of the
clearance height (scl). The leakage mass flow rate m˙cl and the difference in
the swirl of the leakage flow ∆vφ change only marginally with a variation in
the width of the inlet or the outlet gap of the labyrinth seal. A variation in the
size of the inlet gap does not qualitatively change the flow field in the inlet
plane of the labyrinth seal, whereas a variation in the size of the outlet gap
significantly changes the structure of the flow field at the outlet plane of the
labyrinth seal (cf. Fig. G.3). With a larger width of the outlet gap, the relative
area in which fluid enters the labyrinth seal increases significantly. However
this does not lead to a noticeable impact on the main or the secondary flow in
the measurement plane M2.
A change in the width of the labyrinth fins does not change the leakage
flow rate, but leads to a slight increase in the swirl of the reentering leakage
flow [14]. As with a change in the width of the inlet and the outlet gap, a varia-
tion of this parameter does not lead to a noticeable impact on the downstream
flow field.
195
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8
scl [mm]
m˙
/
m˙
H
[%
]
(Sim.,coarse)
(Sim.,fine)
(a)
-10
-5
0
5
10
20
25
30
0 1 2 3 4 5 6 7 8
∆
v φ
/u
[%
]
scl [mm]
(Sim., coarse)
(Sim., fine)
15
(b)
Figure G.2: (a) Percentage of the flow passing through the labyrinth seal, (b) change
in the circumferential velocity of the leakage flow ∆vφ in dependence of
the clearance height scl of the labyrinth seal
196 G. Influence of geometry of the labyrinth seal
CR
*
0.875
0.825
0.775
0.725
0.675
0.625
0.575
0.525
0.475
0.425
0.375
0.325
0.275
0.225
0.175
0.125
0.075
0.025
-0.025
6 mm (Ref.)3 mm 12 mm
(a)
CR
*
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
-0.4
-0.45
-0.5
-0.55
-0.6
6 mm (Ref.)3 mm 12 mm
(b)
Figure G.3: Simulated radial velocities at the inlet plane (a) and the outlet plane (b)
of the labyrinth seal in dependence of the width of the inlet and outlet
gap. The width was scaled in the pictures.
Appendix H
Development and use of
numerical labyrinth seal models
In the work of Anker et al. [14] a numerical model for labyrinth seals was
developed by which a discretization of the labyrinth seal can be avoided in the
simulation of shrouded turbines. The mass flow through the labyrinth seal
is determined applying an approach suggested by Denton and Johnson [75]
where it is assumed that the flow after each labyrinth chamber is fully mixed
out. The development of the circumferential velocity of the leakage flow was
determined using an approach suggested by Benckert [34], in which the laby-
rinth seal is subdivided in discrete segments and the change in the circumfer-
ential velocity is determined by bilancing the angular momentum in consecu-
tive segments through the labyrinth seal. The labyrinth model was calibrated
with the results of the parameter study described in Appendix G.
To avoid that the use of a homogeneous boundary conditions for the nu-
merical extraction and injection of leakage flow in the simplified model leads
to an unrealistic prediction of the flow in the adjacent blade channels, the
first and last cavity in a sealing arrangement need to be discretized, as shown
in the work of Anker et al. [14]. A comparison of the computational results
where the labyrinth seal was discretized and results which were obtained us-
ing the labyrinth model, showed that the labyrinth model qualitatively leads
to a correct prediction of the impact of the leakage flow.
Even though the model is not suited for the detailed simulation of leakage
flow effects and needs calibration to deliver accurate results, the approximate
model is better used than to completely neglecting the leakage flow. Since
the model only contains few parameters that need to be adapted to the spe-
cific turbine geometry, it is expected that the model delivers comparably good
results, if it is used to account for the leakage flow effects in simulations of
industrial turbines.
197
198
Appendix I
Unsteadiness of leakage flow
effects and necessary level of
numerical modelling
In a previous study of Anker et al. [14] it was investigated whether the time-
averaged solution of an unsteady simulation differs from the corresponding
solution of a steady-state simulation for a clearance height of scl = 3mm for
the shrouded Bochum turbine. While the rotor and stator systems are directly
coupled in an unsteady simulation, the rotor and stator rows are linked via
mixing planes in a steady-state simulation. The method of coupling stator
and rotor systems is of importance in the current case, since the reentering
leakage flow causes a recirculation behind the rotor row and the use of mix-
ing planes, which does not allow reverse flow, potentially can degrade the
accuracy of the predicted flow field. In the following a few results of the afore-
mentioned study of Anker et al. [14] will be shown.
Representative for other circumferential positions, in Fig. I.1a and I.1b the
velocity vectors of the steady-state simulations and the time-averaged velocity
vectors of the unsteady simulations are plotted in the meridional plane after
the trailing edge of the rotor, respectively. The solutions differ only slightly
through vanishing velocity vectors on the right border of the domain close to
the wall in the steady-state results due to the presence of a mixing plane.
Otherwise, there are only minor differences between both velocity fields visi-
ble and the position of the vortex core is the same in both simulations.
In Figs. I.2 and I.3 the magnitude of the main and secondary flow vectors
of the steady-state simulations and the time-averaged unsteady computations
are compared in the measurement plane M2. The comparison shows a good
agreement between the solutions.
From the presented excerpt of results of the work of Anker et al. [14] it
can for the Bochum turbine be concluded that the steady-state results can
be used to predict the time-averaged effects of the interaction between the
leakage and the main flow.
199
200 I. Unsteadiness of leakage flow and necessary level of numerical modelling
(a) Steady-state
(b) Unsteady (time-averaged)
Figure I.1: Computed velocity vectors in the meridional plane after the trailing edge
of the rotor for a clearance height of scl = 3mm (Reference vector u -
Circumferential velocity of the rotor at midspan). The results originate
from the report of Anker et al. [14].
201
CA
*
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(a) Steady-state
CA
*
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(b) Unsteady (time-averaged)
Figure I.2: Computed normalized magnitude of the velocity vectors in the absolute
system for a clearance height of scl = 3mm. The results originate from
the report of Anker et al. [14].
202 I. Unsteadiness of leakage flow and necessary level of numerical modelling
CASEK
*
0.5
0.4
0.3
0.2
0.1
0
(a) Steady-state
CASEK
*
0.5
0.4
0.3
0.2
0.1
0
(b) Unsteady (time-averaged)
Figure I.3: Normalized magnitude of the vectors of the secondary flow for a clear-
ance height of scl=3mm. The results originate from the report of Anker
et al. [14].
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