Browsing by Author "Munz, Claus-Dieter (Prof. Dr.)"

Now showing 1 - 13 of 13

Open Access
Compressible multi-phase simulation at extreme conditions using a discontinuous Galerkin scheme
(2015) Fechter, Stefan; Munz, Claus-Dieter (Prof. Dr.)
This work provides a contribution to the approximation of compressible multi-phase flows using a high-order discontinuous Galerkin spectral element method. Compressibility effects have to be considered for operating conditions close to the critical point. Important examples for such extreme ambient conditions include fuel injection systems of aeronautical, automotive and rocket engines. The simulation of compressible multi-phase flows at these ambient conditions imposes high demands on the numerical treatment as well as the numerical method. On the one hand, due to the compressible treatment of both fluid phases and their corresponding numerical methods, especially regarding the numerical resolution of the phase interface. On the other hand, the evaluation of equation of states, that are valid in the vicinity of the critical point, is expensive. As additional challenge are hydrodynamics and thermodynamics coupled closely by the compressible flow equations. This implies that a thermodynamically consistent numerical method has to be chosen. The building blocks of the described numerical method for compressible multi-phase flows include a compressible flow solver for the bulk phases, a level-set based interface tracking method, a comprehensive description of the equation of state and a model for the interface approximation. The interaction of these parts within the solution algorithm is described and validated in the thesis.
Open Access
Divergenzkorrekturen und asymptotische Untersuchungen bei der numerischen Simulation idealer magnetohydrodynamischer Strömungen
(2006) Kemm, Friedemann; Munz, Claus-Dieter (Prof. Dr.)
Es werden Lösungsansätze für zwei wesentliche Problembereiche bei der Simulation kompressibler idealer magnetohydrodynamischer Strömungen entwickelt: - Schwach kompressible Strömungen, - Auftreten magnetischer Monopole in der numerischen Lösung. Für den erstgenannten Problembereich wird ein Ansatz aus der schwach kompressiblen Gasdynamik für die MHD adaptiert. Es werden Methoden der asymptotischen Analyse verwendet, um den Übergang von der Kompressibilität zur Inkompressibilität zu untersuchen. Aufgrund der Ergebnisse der Analyse wurden Vorschläge für numerische Verfahren entwickelt. In dieser Arbeit wird nachgewiesen, daß sich im Fall kleiner Machzahlen die Konstruktion eines Verfahrens aus der schwach kompressiblen Gasdynamik für die MHD direkt übernehmen läßt. Nun wird eine MHD-Strömung aber auch dann schwach kompressibel, wenn die Alfvenzahl klein wird. In dieser Arbeit wird nun gezeigt, daß auch für den Fall kleiner Alfvenzahlen sowohl für den eindimensionalen Fall als auch für zweidimensionale Spezialfälle mithilfe der Mehrskalenanalyse der Übergang zur Inkompressibilität untersucht werden kann. Hieraus wird jeweils die Konstruktion eines numerischen Verfahrens abgeleitet. Dieses kann auch als Vorlage für Verfahren dienen, welche allgemeine dreidimensionale Strömungen kleiner Alfvenzahl approximieren, aber der gewählten Analysis nicht zugänglich sind. Für das Problem magnetischer Monopole werden in der vorliegenden Arbeit Methoden konstruiert, welche die Divergenzbedingung an das Magnetfeld zu einem inhärenten Teil des Evolutionsoperators machen. Hierfür wird der GLM-Ansatz für das elektrische Feld bei den Maxwellgleichungen auf die MHD übertragen und um eine neue, die gemischte GLM-Korrektur, erweitert. Es gelingt, ein allgemeines Modell aufzustellen, das neben der GLM-Korrektur auch die Powell-Korrektur als Spezialfall enthält. Überdies lassen sich verbesserte Versionen der Powell-Korrektur angeben. Der Ansatz wird anhand eines reduzierten Testsystems entwickelt und ist so allgemein, daß er wiederum auf die Maxwellgleichungen angewandt werden kann. Für die zu wählenden Parameter werden auf analytischem Wege Schätzungen hergeleitet, welche die Wirksamkeit der jeweiligen Divergenzkorrekturmethode optimieren. Die theoretischen Vorhersagen werden mittels numerischer Tests sowohl für das Testsystem als auch für die volle MHD verifiziert.
Open Access
A domain decomposition method for the efficient direct simulation of aeroacoustic problems
(2008) Utzmann, Jens; Munz, Claus-Dieter (Prof. Dr.)
A novel domain decomposition approach is developed in this thesis, which significantly accelerates the direct simulation of aeroacoustic problems. All relevant scales must be resolved with high accuracy, from the small, noise generating flow features (e.g., vortices) to the sound with small pressure amplitudes and large wavelengths. Furthermore, the acoustic waves must be propagated over great distances and without dissipation and dispersion errors. In order to keep the computational effort within reasonable and feasible limits, the calculation domain is divided into subregions with respect to the local physical requirements. In these domains, the numerical method which is most suitable and optimized for the considered subproblem is employed. The proposed method differs from established approaches, e.g. the grid coupling is not limited to Chimera techniques but presents a consistent way for the space-time coupling of high order methods. Various domain decomposition options are examined and implemented in a common code framework. In the subdomains, the Navier-Stokes, Euler and linearized Euler equations are solved, for which methods from the discontinuous Galerkin (DG), finite volume (FV) and finite difference (FD) class are available with their respective special properties. For example, DG methods are very suitable for highly accurate solutions on unstructured grids due to their locality, while FD methods are very efficient on Cartesian grids for the simulation of linear wave propagation. In turn, FV methods are very robust in the presence of strong gradients, e.g. shocks. All implemented methods have in common, that they are explicit one-step time integration schemes and thus are especially applicable for unsteady calculations. Furthermore, their order of accuracy in space and time may be chosen arbitrarily. A newly developed numerical solver, the STE-FV method on Cartesian grids, closes the gaps in the repertoire of numerical schemes in the coupling framework. It forms a fast high order method that features great robustness also at nonlinearities by employing a WENO algorithm. For validation purposes, convergence studies and benchmark tests, e.g. the popular double Mach reflection in 2D and an explosion in 3D, are performed for the STE-FV method with orders in space and time up to six and beyond. The coupling of different grids is based on high order interpolations and the data exchange over the ghost elements of the calculation domains. The Gauss integration points in the cells are used here in order to find a source domain for the interpolation and for providing high order boundary conditions afterwards. The grids are not required to be matching or overlapping. Furthermore, arbitrary constellations of structured and unstructured grids are possible. The optimal time steps, which can be different of each other, are allowed in the subregions. This is made possible by employing the Cauchy-Kovalevskaja procedure, which delivers a Taylor series that provides boundary information for the intermediate points of time for domains with a smaller time step. The implementation structure inside the code framework is largely modular. The fluid and acoustics solvers can be used as stand-alone codes, and also new ones can be easily added. Furthermore, external programs, which may run on separate computer systems, can be linked to the framework. The distribution to different system architectures is also possible for the internal solvers. Hence, the respective properties of the numerical methods regarding vectorization and parallelization can be exploited in an optimal way. It is shown on the basis of convergence studies for different constellations of grids, equations and methods, that the domain decomposition approach is capable of maintaining high order of accuracy globally. An examination regarding high-frequency perturbations reveals a natural filtering process if perturbations cannot be resolved on a coarse mesh anymore. Hence, a spatial filtering operator is not a necessity. Another study shows, that the magnitude of reflections occurring at the domain boundaries are in good accordance with theoretical estimations. Besides the change from nonlinear to linear equations, also the jump in resolution matters in this context. However, the reflections are negligible in general. The accuracy and efficiency of the proposed domain decomposition method is illustrated for benchmark examples like the acoustic scattering at a sphere or at multiple cylinders and for the Von Karman vortex street. Here, especially the method's potential for efficient far field calculations becomes clear, but also the advantages in the presence of complex geometries are emphasized. Finally, the simulation of a nozzle flow with a supersonic free jet and the associated noise underlines the practical applicability of the domain decomposition approach.
Open Access
High order discontinuous Galerkin methods for the simulation of multiscale problems
(2015) Beck, Andrea; Munz, Claus-Dieter (Prof. Dr.)
This work provides a contribution to the accurate, stable and efficient numerical simulation of hydrodynamic non-linear multiscale problems with high order discretizations. Due to their wide range of spatial and temporal scale, these types of problems demand not only highly accurate and efficient numerical discretization schemes, but also careful code design with regards to supercomputing architectures. Still, as a rule, even for the most sophisticated algorithms and hardware, a full resolution of all occurring scales remains infeasible. Thus, an approximate solution with drastically reduced number of degrees of freedom is sought, which retains the most important characteristics of the full solution. This solution is obtained by solving a truncated multiscale problem, supplemented by a suitable modeling strategy for the omitted scales and their interaction with the truncated solution. This approach is only meaningful if the resolvable scales determine the mean solution features accurately, and if the non-resolved scales show some form of universality behavior, which allows the derivation of meaningful models. Hydrodynamic turbulence is one example of these types of problems. In this work, two frameworks for the numerical solution of the compressible Navier-Stokes equations are presented: A self-developed Fourier pseudo-spectral solver, and a co-developed framework based on the Discontinuous Galerkin Spectral Element Method (DGSEM). Both discretization schemes are highly efficient for the resolution of multiscale problems as they – due to their spectral character – exhibit very low approximation errors over a wide range of scales, and thus return a very high resolution capability per invested degree of freedom. Since DGSEM is based on the variational form of the governing equations, it allows an element-based discretization of the computational domain, which in turn leads to superior parallelization and the possibility for flexible, unstructured meshes. These features make it attractive for the full resolution of turbulence in a Direct Numerical Simulation (DNS) approach and – as demonstrated in this work – highly competitive when compared to other discretization strategies. These favorable discretization properties carry over into the under-resolved situation (Large Eddy Simulation, LES), where a lower-dimensional version of the problem is solved numerically. However, depending on the discretization of the scale-producing mechanism, its truncation can introduce a self-feeding error into the solution, that can lead to a global instability. The source and effects of these aliasing errors are investigate in this work. Strategies for countering or avoiding it are presented, and the code framework is extended accordingly. These strategies are compared and evaluated, showing that only the exact quadrature of the non-linear terms recovers the favorable approximation properties and thus the efficiency of the spectral approach. With this discretization strategy, it is shown that high order DGSEM can outperform established, lower-order LES formulations in terms of accuracy per invested degree of freedom for challenging test cases at moderate Reynolds number turbulence. Extension to higher Reynolds numbers necessitates the introduction of some form of closure for the un-resolved scales, due to the increase in the truncation error. Aspects of two modeling approaches are discussed: An implicit modeling strategy for DGSEM can be based on the modification of the dissipation introduced by the inter-cell fluxes. The addition of an explicit modeling term which provides a subgrid dissipation mechanism raises the question whether de-aliasing remains essential in that situation. The de-aliasing strategy is revisited, and its interactions with an explicit closure model are examined. It is shown that only through a proper de-aliasing mechanism, the superior scale-resolving capabilities of the scheme can be recovered, and that a decoupling of explicit model and numerics is imperative. Through these investigations, a consistent strategy for stable and accurate DNS and LES of turbulent flows with high order DGSEM has been established. As an outlook, further research strategies into LES modeling should take full advantage of the spectral character of DGSEM, and the associated scale range resolved within each element can be exploited in both an implicit as well as explicit closure approach.
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.
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.
Open Access
Ein konservatives MPV-Verfahren zur Simulation der Strömungen in allen Machzahlbereichen
(2003) Park, Jea-Ho; Munz, Claus-Dieter (Prof. Dr.)
Zur Simulation sowohl inkompressibler als auch kompressibler Strömungen wird ein neues numerisches Verfahren eingeführt. Dabei wird zunächst die konservative Form der Navier-Stokes-Gleichungen aufgestellt, da sie das numerische Erhaltungsprinzip für Strömungen höherer Machzahl gewährleistet. Auf den Druck wird noch der sogenannte MPV-Ansatz angewandt. Dies ist eine Art Druckzerlegungsmethode. Damit kann die beim inkompressiblen Grenzfall auftretende Singularität beseitigt werden. Eigentlich beruht sie auf der sich sehr schnell ausbreitenden Druckstörung bei Strömungen niedriger Machzahl. Für die Zeitintegration der Grundgleichungen wird deshalb ein semi-implizites Verfahren verwendet. Durch die implizite Behandlung der Schallterme wird die Stabilitätsbedingung unseres Verfahrens unabhängig von der Machzahl. Damit kann man auch höhere Ordnung in der Zeit für Strömungen in allen Machzahlbereichen erreichen. Zum Lösen der Druck-Geschwindigkeits-Kopplung untersucht man das iterative SIMPLE-Typ Verfahren und auch das nicht-iterative Druck-Korrektur Verfahren. In Hinsicht auf die Reduktion der Rechenzeit ist die Entwicklung eines solchen nicht-iterativen Verfahrens sehr wichtig. Im kartesischen Koodinatensystem wird ein versetztes Gitter verwendet. Auf allgemeinen Koordinaten wird zusätzlich die zellzentrierte Variablenanordnung ausführlich untersucht. Dazu werden einige typische Testbeispiele verschiedener Machzahl in der Stömungsmechanik berechnet und mit in der Literatur bekannten Musterergebnissen verglichen.
Open Access
Mesh curving techniques for high order parallel simulations on unstructured meshes
(2014) Hindenlang, Florian; Munz, Claus-Dieter (Prof. Dr.)
In this work, the generation of high order curved three-dimensional hybrid meshes and its application are presented. Meshes with linear edges are the standard of today's state-of-the-art meshing software. Industrial applications typically imply geometrically complex domains, mostly described by curved domain boundaries. To apply high order methods in this context, the geometry - in contrast to classical low order methods - has to be represented with a high order approximation, too. Therefore, a high order element mapping has to be used for the discretization of curved domain boundaries. The main idea here is to rely on existing linear mesh generators and provide additional information to produce high order curved elements. A very promising candidate for future numerical solvers in computational fluid dynamics is the family of high order discontinuous Galerkin (DG) schemes. They are locally conservative schemes, with a continuous polynomial representation within each element and allow a discontinuous solution across element faces. Elements couple only to direct face neighbors, and the discontinuity is resolved via numerical flux functions. As the main focus of this work are curved elements, the different formulations and possible implementations of the DG scheme with non-linear element mappings are discussed in detail. Especially, a highly efficient variant of the DG scheme for hexahedra, namely the discontinuous Galerkin spectral element method (DG-SEM), is presented. The main focus of this thesis is the generation of high order meshes. Several techniques to generate curved elements are described and their applicability to complex geometries is demonstrated. Starting from a linear mesh, the first step curves the element faces representing the curved geometry. Two approaches are presented, the first based on continuity conditions using surface normal vectors and the second based on interpolation of additionally generated surface points. The high order mapping of the volumetric element is computed as a blending of the curved element faces. In the case of boundary layer meshes, the blending may lead to inverted elements. As a remedy to this problem, an additional mesh deformation approach is proposed and validated. Independent thereof, another approach is presented, allowing one to directly generate curved volume mappings from the agglomeration of block-structured meshes. One of the reasons making high order DG schemes attractive for the simulation of fluid dynamics is their parallel efficiency. As future applications in fluid dynamics comprise the resolution of three-dimensional unsteady effects and are increasingly complex, the simulations require more and more computing resources, and weak and strong scalability of the numerical method becomes extremely important. In the last part of this thesis, the parallelization concept of the DG-SEM code Flexi is described in detail. A new domain decomposition strategy based on space filling curves is introduced, and is shown to be simple and flexible. A thorough parallel performance analysis confirms that the overall implementation scales perfectly. Ideal speed-up is maintained for high polynomial degrees, up to the limit of one element per core. As the DG scheme only communicates with direct neighbors, the same parallel efficiency is found on both cartesian meshes as well as fully unstructured meshes. The findings underline that the proposed Discontinuous Galerkin scheme exhibit a great potential for highly resolved simulations on current and future large scale parallel computer systems.
Open Access
Modelling of intra- and inter species charged particle collisions for flow simulation in pulsed plasma thrusters
(2008) D'Andrea, Danilo; Munz, Claus-Dieter (Prof. Dr.)
A better physical understanding of electrical space propulsion systems like Pulsed Plasma Thrusters requires the numerical modelling and simulation of highly rarefied plasma flows. Mathematically, such phenomena demand a kinetic description which is established by the complete, time-dependent Boltzmann equation. An attractive numerical approach to tackle this complex non-linear problem consists of a combination of the well-known Particle-in-Cell (PIC) and Monte Carlo methods extended by a PIC-based Fokker-Planck solver, on which we focus our attention in the following. This numerical model accommodates the physics of interaction of charged particles with electromagnetic fields, inelastic electron-neutral scattering as well as intra- and inter-species charged particle Coulomb collisions. To describe elastic intra- and inter-species charged particle Coulomb collisions it is convenient to start from the Boltzmann collision integral with the classical Rutherford differential cross section. A Taylor series expansion up to second order in velocity of the post-collision distribution functions and the adoption of a cut-off value for the impact parameter permits the final integration of the Boltzmann integral to obtain the Fokker-Planck equation. The central quantities appearing in the Fokker-Planck equation are the friction force vector and the diffusion tensor. The keys to compute the friction and diffusion coefficients are the Rosenbluth potentials which are in turn complicate integrals of the field particle distribution function and the relative velocity between test and field particles. Usually, strong assumptions like isotropic velocity distribution of the scatterer, are made to evaluate the Rosenbluth potentials. Observing that the Rosenbluth potentials are convolution intergrals addresses the use of fast Fourier transform techniques to calculate these quantities and their derivatives rapidly with the advantage of being free of any additional assumption. Furthermore, such a determination the Rosenbluth potentials is the basis to model collisional relaxation in a complete self-consistent manner. In order to fit the three-dimensional Fokker-Planck equation of the scattered distribution function into a particle-based method framework, the equivalence with the stochastic differential equation (SDE) is exploit. The stochastic variable C(t) which obeys the SDE is later identified with the charged particle velocity. Also in this context the friction force vector and a matrix derived from the diffusion tensor play the central role. By means of Ito-Taylor expansion and Ito calculus the stochastic differential equation is discretised and numerical schemes are derived. In this work, explicit weak schemes up to approximation order two have been applied to update the particles velocity. These weak Ito-Taylor schemes together with the Fourier transform method and particle-mesh interface techniques form a remarkable simulation tool to study collisional relaxation processes from first principles. For instance by means of this tool, a more realistic evaluation of the time scales can be provided since the classical test-particle approach is not necessary anymore thanks to self-consistency. The introduced intra-species charged particle modelling can be easily adapted for inter-species electron-ion particle collisions. Finally, the structure of the developed PIC-based method to solve the Fokker-Planck equation also allows to combine intra- and inter-species collisions to perform coupled simulations.
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.
Open Access
On the efficiency of implicit discontinuous Galerkin spectral element methods for the unsteady compressible Navier-Stokes equations
(2020) Vangelatos, Serena; Munz, Claus-Dieter (Prof. Dr.)
In this work we investigate the implicit time integration for the Discontinuous Galerkin Spectral Element Method (DGSEM) with respect to its efficiency and accuracy compared to explicit time integration. Considering unsteady simulations of laminar and turbulent flows described by the compressible Navier-Stokes equations, not only the spatial but also the temporal discretization is of high importance. Due to the explicit time step restriction caused by the stability condition, explicit schemes are widely used for time accurate simulations. In the case of very stiff problems this condition becomes severe leading to the consideration of implicit time integration schemes, which can be constructed as unconditionally stable. The implicit time step is contrastingly only driven by accuracy requirements. However, solving the arising (non-) linear equation systems within the DGSEM context is still a challenging issue. We consider Explicit first stage Singly Diagonally Implicit Runge-Kutta (ESDIRK) schemes in combination with a Jacobian-Free Newton-Krylov solver. In the first part, we introduce a novel strategy for solving the non-linear and linear systems with adaptive tolerances in order to avoid over- and under-solving. These tolerances are automatically adjusted to temporal accuracy requirements. This strategy leads not only to an user friendly handling but also to an highly efficient solver. In a second step, we employ the block-Jacobi preconditioner neglecting all the off-diagonals blocks, so that the resulting linear system is able to be solved element-locally. The advantages of this preconditioner are low storage requirements, parallel scalability and simple implementation. The results highlight that implicit DGSEM can be competitive with an explicit Runge-Kutta scheme in terms of computational time and accuracy. The implicit solver can outperform the explicit scheme in the case of very severe time step restrictions with the same spatial accuracy.
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.
Open Access
Simulation of multiphase flows with multiparameter equations of state and the discontinuous Galerkin method
(2017) Hempert, Fabian; Munz, Claus-Dieter (Prof. Dr.)
Numerical simulations of multiphase flows for industrial applications have become increasingly complex. The demand on the resolution of temporal and spatial scales has increased and more complex and numerically demanding thermodynamic states of the fluid are required. The aim of this study is to demonstrate the applicability of a high order method, i.e., the discontinuous Galerkin spectral element method, with an accurate equation of state, valid for a wide range of pressures and temperatures, e.g., the Helmholtz energy formulation. Although these two aspects have been intensely investigated separately, a combination of both in an efficient manner remains challenging for complex applications, e.g., cavitational flows or real gas jets. The present work presents the application of a novel approach, which uses a dense gas approach with a discontinuous Galerkin method with a tabulated equation of state including the gaseous, liquid and two-phase states of the fluid. This new approach allows for detailed investigations of flow phenomena, which require accurate fluid properties and have been unfeasible to simulate in the past. The investigated cases include supersonic real gas jets and cavitational flows. Riemann-problems are investigated to demonstrate the differences between ideal and real equation of state approximations. The results show on one hand that at high pressures the ideal approximation of the equation of state shows large differences. On the other hand, a very good agreement of the applied method compared to analytical results is shown. The simulation results for the supersonic real gas jet suggest large differences for the applied cases between the real gas and ideal gas approximation. A difference are the shock structures which might lead to differences in acoustics and mixing. Further, the mass flow rates show significant differences. For the cavitational flow a detailed parameter study for single vapor bubble collapses in a liquid is executed. The presented results demonstrate difference influence quantities for such collapses, e.g., the influence of the grid resolution to the maximum collapse pressure. Subsequently, a micro channel flow simulation is conducted for water for many known effects could be reproduced by the simulation. An example is the shock propagation within the wet steam area, which is very slow compared to the mean flow velocities and is traveling in the upstream direction. For both the real gas and cavitating flow, using the low dissipation discontinuous Galerkin scheme shows superior results compared to a second order finite volume scheme used in this work. The proposed framework shows great potential for the simulation of flows, that require an accurate representation of small spatial and temporal scales and multiparameter equation of states. First simulation results of industrially relevant flows are presented for both single and multiphase application. However, to fully exploit the potential of the combination high order methods with accurate equation of states further development is necessary, e.g., stability and sub-grid scale models.