06 Fakultät Luft- und Raumfahrttechnik und Geodäsie
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Item Open Access Shape derivatives and shock capturing for the Navier-Stokes equations in discontinuous Galerkin methods(2017) Sonntag, Matthias; Munz, Claus-Dieter (Prof. Dr.)This work addresses two different topics, the shape derivatives for the compressible Navier-Stokes equations on the one hand and, on the other hand, the treatment of shocks or other flow discontinuities in Discontinuous Galerkin methods. There is a strong demand for very efficient methods for shape optimization in the aerospace industry, for example drag reduction or lift maximization of an aircraft. The use of gradient based optimization schemes requires derivatives of the cost function with respect to the shape of an object. With the shape derivatives presented in this work, these derivatives can be calculated independent of the parametrization of the object's shape, and, since the derivation takes place in the continuous space, they can be applied to almost any discretization. Nevertheless, one has to take the numerical scheme, which is later applied, into account. For methods based on the variational formulation a difference in the shape derivative, compared to the pointwise approach, arises, which cannot be neglected. Hence, one objective of this work is to derive the shape derivatives of the drag- and lift-coefficient for the Navier-Stokes equations in variational formulation and compare it with the pointwise approach both analytically and numerically. A discrepancy has to be expected, especially for flow phenomena with high gradients or discontinuities which do not fulfill the strong form of the governing equations. These flow phenomena require a special treatment in numerical methods of high order. In the second part of this work, a shock capturing for the Discontinuous Galerkin method is developed which prevents the oscillations originating from the approximation of discontinuities with high order polynomials. Therefore a hybrid approach is presented, where the original DG scheme is coupled with a second order Finite Volume method. In all elements containing shocks or discontinuities the operator of the DG method is replaced by the Finite Volume scheme. This scheme is, due to the use of slope limiters, well known for its strengths in handling shocks. However, in regions where the flow is smooth the Finite Volume method requires a finer resolution for the same accuracy than the Discontinuous Galerkin scheme. Using the same mesh for the FV method as for the DG scheme would lead to a big reduction in resolution. Hence, to compensate this loss the original elements of the mesh are divided into logical sub-cells. By associating exactly one Finite Volume sub-cell to each degree of freedom of a DG element, the same data structures can be used. This enables an efficient implementation of the outlined shock capturing designated for high performance computations. Therefore, not only the basic properties of this hybrid DG/FV sub-cell approach are investigated with several examples, but also studies regarding the parallel efficiency are performed.Item Open Access Linear stability investigations of three-dimensional disturbances in the boundary layer over anisotropic compliant walls(2023) Zengl, Marcus; Rist, Ulrich (apl. Prof. Dr.-Ing.)In dieser Arbeit werden dreidimensionale Störungen in der Grenzschicht über anisotropen nachgiebigen Wänden mit linearer Stabilitätstheorie untersucht. Ein oberflächenbasiertes Modell wird verwendet, um die nachgiebige Wand abzubilden. Hierbei wird das anisotrope Wandmodell von Carpenter erweitert, um einen zusätzlichen Schiebewinkel der Wand bezüglich der Strömungsrichtung einzubringen. Basierend auf diesem Wandmodell wird eine Randbedingung für die Lineare Stabilitätstheorie hergeleitet. Aufgrund der Tatsache, dass diese Randbedingung die Orr-Sommerfeld- und Squire-Gleichung koppelt, wurden zwei neuartige Lösungsverfahren, ein Schießverfahren und ein Matrixlöser, für diesen besonderen Umstand entwickelt. Der Schießlöser transformiert das zugrunde gelegte Eigenwertproblem in ein Randwertproblem und verwendet ein klassisches Schießverfahren zur Lösung des Problems. Um das numerisch steife Problem mit seinem parasitärem Fehlerwachstum zu berücksichtigen beinhaltet das Lösungsverfahren eine Gram-Schmid Orthonormierungsroutine. Durch eine neuartige Skalierung der Phase des zu minimierenden Residuums wird das zeitliche und räumliche Modell robust und performant für gegebene Eigenmoden gelöst. Das durch die gekoppelte Orr-Sommerfeld- and Squire-Gleichung entstehende Eigenwertproblem wird auch mit einer Matrix-basierenden Methode gelöst. Das durch die nachgiebige Wand entstehende zeitliche quadratische Eigenwertproblem wird dabei berücksichtigt. Hierbei wird eine pseudospektrale Diskretisierung mit Chebyshev-Kollokation verwendet. Besonders betrachtet wird die Formulierung des diskretisierten Problems auf seine numerischen Fehler. Die numerische Genauigkeit der Lösungsverfahren wird genau überprüft, um die Gitterunabhängigkeit der Ergebnisse sicherzustellen. Um das Potenzial der nachgiebigen Wände zur Verzögerung des laminar-turbulenten Umschlags zu untersuchen, wurde die Vorgehensweise von Carpenter [15] übernommen. Carpenter optimierte die Parameter der nachgiebigen Wand so, dass Tollmien-Schlichting (TS) Moden so weit wie möglich abgeschwächt werden, während Fluid-Struktur (FISI) Moden grenzwertig stabil bleiben. Dieses Vorgehen wurde ausgewählt, weil Fluid-Struktur Moden absolut instabil sein können, was zu sofortigem Strömungsumschlag führen kann. Stabilitätsrechnungen wurden ausgeführt für zwei Sätze von Wandparametern, die Carpenter mit seinem zweidimensionalen Rahmenwerk optimiert hat. Hierbei wurden nicht nur dreidimensionale Störungen betrachtet, sondern es wurde auch der Einfluss des neu eingebrachten Schiebewinkels der nachgiebigen Wand untersucht. Die Ergebnisse wurden bezüglich der zeitlichen Anfachung der TS- und FISI-Moden, und bezüglich des mit N-Faktoren vorhergesagten Umschlagspunkts beurteilt. Es wird gezeigt, dass dreidimensionale Störungen bestimmte N-Faktoren vor ihren zweidimensionalen Pendants erreichen. Die vorhergesagte laminare Länge ist etwas kürzer als mit zweidimensionalen Verfahren vorhergesagt. Es scheint als ob der eingebrachte Schiebewinkel für die untersuchten Parametersätze keinen Vorteil bezüglich Laminarhaltung bringt. Schließlich wurden optimale Störungen berechnet, um das Transiente Energiewachstum für die anisotrope nachgiebige Wand zu untersuchen. Hierbei wurden die Anfangsverteilungen von Eigenmoden so optimiert, dass deren Überlagerung ein maximales Energiewachstum für eine vorgegebene Zeit erfährt. Die Einhüllende dieser optimalen Störungen wird dann für variierende Wellenzahlen in Strömungs- und Spannweitenrichtung, und variierende Wachstumszeit berechnet. Die Ergebnisse zeigen kein durch die nachgiebige Wand hervorgerufenes relevantes transientes Wachstum. Es wird gezeigt, dass der klassische Mechanismus für transientes Wachstum, der bei der steifen Wand dominiert, nicht verändert wird.Item Open Access Experimental and numerical aeroacoustic investigation of impinging flows at low Mach number(2007) Zucchini, Marco; Munz, Claus-Dieter (Prof. Dr. rer. nat. habil.)This work presents the development of methods for the experimental and numerical investigation of flow-induced noise. Moreover it offers a systematic validation and comparison of various numerical prediction techniques for small Mach number aeroacoustics. The work is motivated from the need to validate all the calculation steps of aeroacoustic simulations beyond pure analytical solutions. This encompasses the fluid dynamics and source calculations up to the propagation of sound for problems of technical interest. The configurations investigated are chosen to ensure geometrical simplicity, given the limitations associated with the numerical calculations of complex geometries, in addition to geometrical relevance with respect to technical problems. The investigation concentrates on the numerical calculations of a low subsonic jet impinging on flat inclined plates with various downstream obstacles. Particular focus is placed on broad band noise contributions.Item Open Access High-order methods for computational astrophysics(2015) Núñez-de la Rosa, Jonatan; Munz, Claus-Dieter (Prof. Dr.)In computational fluid dynamics, high-order numerical methods have gained quite popularity in the last years due to the need of high fidelity predictions in the simulations. High-order methods are suitable for unsteady flow problems and long-term simulations because they are more efficient when obtaining higher accuracy than low-order methods, and because of their outstanding dissipation and dispersion properties. In the present work, the development and application of three high-order numerical methods, namely, the conservative finite difference (FD) method, the finite volume (FV) method, and the discontinuous Galerkin spectral element method (DGSEM), is presented. These methods are used here for solving three equations systems arising in computational astrophysics on flat spacetimes, specifically, the ideal magnetohydrodynamics (MHD), relativistic hydrodynamics (SRHD) and relativistic magnetohydrodynamics (SRMHD). Our computational framework has been subject to the standard testbench in computational astrophysics. Numerical results of problems having smooth flows, and problems with shock-dominated flows, are also reported. Finite volume methods are numerical methods based on the weak solution of conservation laws in integral form. Unlike finite volume methods, where cell averages of the solution are evolved in time, in the conservative finite difference schemes only the solution at specific nodal points are considered. This difference offers a high efficiency of finite difference over finite volume methods in two and three dimensional high-order calculations because of the form of the utilized stencils in the reconstruction step. Recently, a lot of effort has been put into the development of efficient high-order accurate reconstruction procedures on structured and unstructured meshes. The most widely used procedure to achieve high-order spatial accuracy in finite volume and conservative finite difference methods is the WENO reconstruction. The basic idea of the WENO schemes is based on an adaptive reconstruction procedure to obtain a higher-order approximation on smooth regions while the scheme remains non-oscillatory near discontinuities. For this reason, the WENO formulation is particularly effective when solving conservation laws containing discontinuities. In this work, the FD and FV methods are extended to very high-order accuracy on regular Cartesian meshes by making use of the arbitrary high-order reconstruction WENO operator. The time discretization is carried out with a strong stability-preserving Runge-Kutta (SSPRK) method. The MHD, SRHD and SRMHD equations are then solved with these two methods for problems having strong shock configurations. The discontinuous Galerkin (DG) methods combine the ideas of the finite element (FE) and the finite volume methods. From the FE methods, the solution and test functions in the variational formulation of the conservation law are locally represented by polynomials, allowing to be discontinuous at element faces. In order to stabilize the scheme, from the FV methods are borrowed the ideas of using Riemann solvers, which permit to connect a given element with its direct neighboring ones. One special case in the family of DG methods is the DGSEM. In these methods, the domain is decomposed into quadrilateral/hexahedral elements, and the solution and the fluxes are represented by tensor-product basis functions (high-order Lagrangian interpolants). The integrals are approximated by quadrature, and the nodal points, where the solution is computed, are the Gauss-Legendre quadrature points. With these choices, the DG operator has a dimension-by-dimension splitting form, which yields more efficiency due to less operations and less memory consumption. In this work, the DGSEM has been also extended to the equations of computational astrophysics on flat spacetimes, but restricted only to the MHD and SRHD equations. Because discontinuous solutions form part of the nature of the hyperbolic conservation laws, shock capturing strategies have to be devised, especially for the discontinuous Galerkin method. For the DGSEM, a hybrid DG/FV shock capturing approach is used as the main building block for stabilization of the solution when shocks take place. The hybrid DGSEM/FV is constructed in such a way that, in regions of smooth flows, the DGSEM method is employed, and those parts of the flow having shocks, the DGSEM elements are interpreted as quadrilateral/hexahedral subdomains. In each of these subdomains, the nodal DG solution values are used to build a new local domain composed now of finite volume subcells, which are evolved with a robust finite volume method with third order WENO reconstruction. This new numerical framework for computational astrophysics based on the hybridization of high-order methods brings very promising results.Item Open Access Numerical prediction of flow induced noise in free jets of high Mach numbers(2009) Schönrock, Olaf; Munz, Claus-Dieter (Prof. Dr.)A direct aeroacoustic simulation methodology is developed on the basis of the numerical schemes implemented in the commercial tool ANSYS CFX. The focus lies upon the efficient and direct numerical prediction of the flow-induced noise generated by natural gas and pneumatic applications. The respective compressed gas related components are characterized by tiny supersonic gas jets, strong noise emissions, poor accessibility by measurement techniques and excessive simulation costs in particular. Highly resolved computational grids close to DNS requirements become necessary just in order to capture the time-averaged flow profile, tiny shocks and gradients correctly. Furthermore the coexistent supersonic flow velocity results in an exceptionally small timestepping in compliance with the CFL condition, e.g. for LES aeroacoustic simulations. Considering the assumably nonlinear noise propagation and the acoustic feedback within enclosed environments the well-established hybrid approaches cannot be employed here as well. The flow and acoustics of the whole domain rather have to be captured within a single tool instead. In fact, the corresponding simulation costs inhibit the numerical prediction and reduction of the emitted noise levels for those compressed gas components at the industrial scale. In this work the test subject is a dedicated natural gas injector in an open and a confined environment and with varying boundary conditions. Specific to the injector nozzle, four under-expanded supersonic gas jets (M=1.4, Re=30000) are formed and cause a strong flow three-dimensionality. Furthermore a turbulence cluster establishes between the jets driving jet fluctuations and aeroacoustics. To enable aeroacoustic simulations in the first place, ANSYS CFX is augmented by a transient inlet boundary condition and a non-reflective farfield boundary condition based on an implicit damping sponge layer. In order to reduce the simulation costs the scale-adaptive turbulence model (SAS-SST) recently implemented in ANSYS CFX is validated for the gas injection problem and especially for CFL numbers much larger than one. Since a degrading solution quality has to be expected then a timestep study is conducted in order to detect the limit for aeroacoustic simulations. Bottom line the different turbulence modeling allows a strongly increased global timestepping such that a net simulation costs reduction by a factor of 19 compared to LES is achieved. In spite of the generally lower solution quality the predicted noise levels, spectral distributions as well as noise sensitivities are in well agreement with own experimental data. In an alternative simulation approach the research code NSDG2D is applied to a simplified 2D setup with very promising results. The more sophisticated solver numerics based on an explicit Discontinuous Galerkin scheme allows local dynamic adaption to the problem, amongst others by local timestepping and locally adaptive element orders. These features prove to be feasible especially for locally varying unsteady compressible flows and the supersonic gas injection in particular. Considering these advantages a further reasonable simulation costs reduction compared to ANSYS CFX can be projected for the 3D application as well.Item Open Access Turbulenzmodellierung und Detached-Eddy-Simulationen mit einem Discontinuous-Galerkin-Verfahren hoher Ordnung(2011) Lübon, Christian; Wagner, Siegfried (Prof. Dr.-Ing.)In letzter Zeit wuchs die Bedeutung der numerischen Simulation von Strömungen im industriellen Umfeld. Meist ist die Strömung dreidimensional, turbulent, abgelöst und das bei komplexen Geometrien. Aufgrund der Komplexität solcher Probleme ist eine kontinuierliche Weiterentwicklung der numerischen Methoden und Lösungsverfahren zwingend erforderlich. In der vorliegenden Arbeit wird ein Discontinuous Galerkin Verfahren hoher Ordnung vorgestellt, um abgelöste turbulente Strömungen um komplexe Geometrien mit unstrukturierten Gittern und hoher Genauigkeit zu berechnen. Die Discontinuous Galerkin (DG) Verfahren verbinden Ideen aus den Finite Elemente (FE) und Finite Volumen (FV) Verfahren, beispielsweise hohe Genauigkeit durch einen polynomialen Ansatz innerhalb der Zellen, oder die Wellenausbreitung mit den damit verbundenen Riemann Problemen. Ursprünglich wurde die Methode f ur hyperbolische Erhaltungsgleichungen wie die Euler-Gleichungen entwickelt. In der realen Anwendung ist die Strömung meist aber turbulent, dreidimensional und instationär. Die ursprünglich entwickelten DG Verfahren für die Euler-Gleichungen enthielten nur Ableitung erster Ordnung. Der Durchbruch zur Lösung der Navier-Stokes Gleichungen mit ihrer Ableitung zweiter Ordnung gelang Bassi und Rebay. Danach musste ein weiterer großer Schritt getan werden, die Erweiterung der Algorithmen zur Behandlung von turbulenten Strömungen. Die instationären Reynolds gemittelten Navier-Stokes (URANS) Gleichungen mussten gelöst werden. Dafür wurden die Algorithmen von Bassi und Rebay sowie von Landmann um Turbulenzmodelle erweitert. Darauf aufbauend wurde in der vorliegenden Arbeit die Methode zuerst für dreidimensionale turbulente Strömungen und später auf eine Detached Eddy Simulation (DES) erweitert. Um die hohe Ordnung und den damit verbundenen Effizienzgewinn des Verfahrens nachzuweisen wurde zuerst eine Konvergenzanalyse durchgeführt, später dann die ausgezeichnete Skalierbarkeit auf massiv parallelen Höchstleistungsrechnern demonstriert. Danach werden einige ausgewählte Ergebnisse, wie die Strömung längs einer Platte oder die Strömung um eine Kugel, die nicht mit herkömmlichen URANS Methoden bestimmt werden kann, mit hoher Genauigkeit berechnet und mit der Theorie beziehungsweise dem Experiment verglichen.Item Open Access Laminar-to-turbulent transition in airfoil boundary layer flows at oscillating inflow conditions(2023) Ohno, Duncan; Rist, Ulrich (apl. Prof. Dr.-Ing.)This thesis numerically investigates the laminar-to-turbulent transition in boundary-layer flows on natural laminar flow airfoils under oscillating inflow conditions. Large-scale fluctuations in the form of periodic vertical gusts generate an oscillating pressure gradient, resulting in a complex transient behavior of the boundary layer. Under these conditions, two scenarios are investigated: an attached flow with natural Tollmien-Schlichting (TS) wave transition and a boundary-layer flow featuring a laminar separation bubble (LSB). The study aims to provide a deeper understanding of the transient mechanisms involved as well as the basis for new transition prediction methods for unsteady conditions. Direct numerical simulations (DNS) are performed where the gust disturbance is imposed on the fully-resolved transitional boundary layers via unsteady boundary conditions. In this novel approach, transient base flows are generated in advance with unsteady Reynolds-averaged Navier-Stokes (URANS) simulations of entire unsteady airfoil flows in conjunction with the disturbance velocity approach (DVA) to introduce sinusoidal gusts. The spatio-temporal evolution of the modal disturbances is analyzed using the continuous wavelet transform (CWT), which is then compared with linear stability theory (LST) by employing a trajectory-following method for transient flows. Several physical effects responsible for the transient characteristics of the studied flows are identified and observations from previous experimental studies are classified. The results demonstrate that the quasi-steady linear theory adequately predicts the transient behavior of the convective modal disturbances for a wide range of gust perturbations. The so-called convective-transition mode with a subsequent calmed region is found for cases with a high degree of unsteadiness. This study is the first to provide a physical explanation for the occurrence of this mechanism for natural transition.Item Open Access A parallel discontinuous Galerkin code for the Navier-Stokes and Reynolds-averaged Navier-Stokes equations(2008) Landmann, Björn; Wagner, Siegfried (Prof. Dr.-Ing.)The numerical simulation of flow problems has gained further importance during the recent years. This is-obviously next to the increase of computing power-due to the steady improvements of the numerical discretisation methods and the improvement of the efficiency of the associated solution algorithms. Even wider acceptance could be obtained, if the flexibility, the automatism or the efficiency of the flow simulation could be further improved. One promising and relatively new discretisation approach, which recently attracted attention, is the discontinuous Galerkin (DG) method. The DG approach seems to have the potential to solve some problems, which mainly have their origin in the presently used discretisation methods. In the present work, an attempt has been made to examine the qualities of the present state of the art discretisation methods based on the DG approach in space. Therefore, different DG methods, including some recently developed methods, are employed for the discretisation of the compressible Euler- and Navier-Stokes equations as well as for the Reynolds-averaged Navier-Stokes equations. The turbulence modeling is applied with a one-equation or a two equation model, namely the Spalart-Allmaras or the k-omega model. The temporal discretisation of the partial differential equations is either performed explicitely with the aid of classical Runge-Kutta methods or with an implicit discretisation approach.Item Open Access Aeroacoustic simulation of turbulent boundary layer induced automotive gap noise(2021) Erbig, Lars; Munz, Claus-Dieter (Prof. Dr. rer. nat.)Item Open Access High order large eddy simulation for the analysis of tonal noise generation via aeroacoustic feedback effects at a side mirror(2017) Frank, Hannes; Munz, Claus-Dieter (Prof. Dr.)In this work, the flow around a side mirror and the resulting tonal noise generation are investigated using highly accurate compressible large eddy simulations. Avoiding tonal noise, which can be perceived as disturbing whistling sound, is a crucial target in automotive aeroacoustics. However, the underlying mechanisms are not completely understood and can typically not be captured with state of the art computational aeroacoustics solvers used in industry. Acoustic feedback effects known from tonal airfoil self-noise are a possible cause at smooth mirror housings that exhibit laminar separation upstream of the trailing edge. Since this application demands high accuracy, a simulation code based on the high order discontinuous Galerkin spectral element method is employed. To enhance geometrical flexibility, it is augmented with an extension to non-conforming curved elements in three dimensions. In the first part of the investigation, the simulation framework is used to analyze an early development stage side mirror exhibiting tonal noise generation. Adopting the corresponding experimental configuration, the study considers an isolated side mirror mounted on the wind tunnel floor. The computational flow field is shown to agree remarkably well with the experimental one based on comparisons with static wall pressure, hotwire and PIV measurements. Discrete peaks are obtained in the computational acoustic spectrum, originating at the trailing edge of the mirror downstream of laminar separation. The identified tonal noise source regions match the experimental ones and quantitative agreement is achieved for one of the tonal peak frequencies. Perturbation simulations reveal global acoustic feedback instabilities selecting the same discrete frequencies observed in the developed flow. The feedback loop comprises convective disturbance growth in the separated shear layer, scattering at the trailing edge and reinforcement through receptivity to the emitted sound in the upstream boundary layer. In a second step, this mechanism is studied in more detail based on a specifically designed simplified two-dimensional model. A subdomain approach is introduced to exploit the two-dimensional shape and circumvent the computational cost associated with the bluff body wake of the model. Simulations of a range of free-stream velocities exhibit tonal frequencies varying similarly to the experimentally observed so-called 'ladder structure'. The tone frequencies are shown to evolve according to a theoretical feedback model based on linear stability theory. Finally, the efficacy of various modifications to the mirror contour to eliminate tonal noise generation is evaluated. The present work contributes to the understanding of tonal noise generation mechanisms and can guide future designs. Moreover, it corroborates the capacity of the present discontinuous Galerkin framework to accurately capture relevant but delicate aeroacoustic effects at complex geometries.