Дисертації з теми "High Resolution Shock Capturing"

With the advances in the computing world, computational fluid dynamics (CFD) is becoming more and more critical tool in the field of fluid dynamics. In the past few decades, a huge number of CFD models have been developed with ever improved performance. In this research a robust CFD model, called Riemann2D, is extended to model flow over a mobile bed and applied to a full scale dam break problem. Riemann2D, an object oriented hyperbolic solver that solves shallow water equations with an unstructured triangular mesh and using high resolution shock capturing methods, provides a generic framework for the solution of hyperbolic problems. The object-oriented design of Riemann2D has the flexibility to apply the model to any type of hyperbolic problem with the addition of new information and inheriting the common components from the generic part of the model. In a part of this work, this feature of Riemann2D is exploited to enhance the model capabilities to compute flow over mobile beds. This is achieved by incorporating the two dimensional version of the one dimensional non-capacity model for erodible bed hydraulics by Cao et al. (2004). A few novel and simple algorithms are included, to track the wet/dry and dry/wet fronts over abruptly varying topography and stabilize the solution while using high resolution shock capturing methods. The negative depths computed from the surface gradient by the limiters are algebraically adjusted to ensure depth positivity. The friction term contribution in the source term, that creates unphysical values near the wet/dry fronts, are resolved by the introduction of a limiting value for the friction term. The model is validated using an extensive variety of tests both on fixed and mobile beds. The results are compared with the analytical, numerical and experimental results available in the literature. The model is also tested against the actual field data of 1957 Malpasset dam break. Finally, the model is applied to simulate dam break flow of Warsak Dam in Pakistan. Remotely sensed topographic data of Warsak dam is used to improve the accuracy of the solution. The study reveals from the thorough testing and application of the model that the simulated results are in close agreement with the available analytical, numerical and experimental results. The high resolution shock capturing methods give far better results than the traditional numerical schemes. It is also concluded that the object oriented CFD model is very easy to adapt and extend without changing the generic part of the model.

The problem of modelling both the unsteady hydrodynamics and the bed morphological variations in natural channels is generally performed by solving the De Saint Venant balance equations for the liquid phase together with the Exner continuity equation for the sediments carried as bed-load. This thesis focuses on the development of an high-order accurate centred scheme of the finite volume type for the numerical solution of the coupled De Saint Venant-Exner system. A new scheme, called PRICE-C, is proposed. It solves the system of equations in a non-conservative form, however it has the important characteristic of reducing automatically to a conservative scheme if the underlying PDE system is a conservation law. It is applied to the shallow water equations in the presence of either a fix or a movable bed. The scheme is first introduced in a one-dimensional framework, and it is then extended to the two-dimensional case. The extension is not straightforward in the case of an unstructured mesh, since averages over suitable edge-based control volumes have to be performed. The scheme is extended to high order of accuracy in space and time via the ADER-WENO and MUSCL technique respectively for the one- and twodimensional case. The well-balanced property of the scheme is proven, i.e. the ability to reach steady states also in the presence of discontinuous water surface or discontinuous bottom profile. The scheme can deal with subcritical and supercritical flows, as well as transcritical situations. Moreover the proposed approach leads to a correct estimate of the celerity of surface discontinuities as well sediment bores and small bottom perturbations. The main characteristic of the scheme is its simplicity: it is based on a simple centred approach, that means that the knowledge of the eigenvalues of the matrix of the system is not required. This is important since the interaction between sediment transport and water flow not always admits detailed knowledge of the eigenstructure. Hence this scheme can be useful to engineers since they need simple numerical tools that can be easily used without entering in the mathematical detail of the differential hyperbolic system under consideration. Moreover the centred strategy gives generality to the scheme: in fact, it can be applied without modification to any kind of hyperbolic equations with non-conservative terms.
La modellazione dell’idrodinamica e delle variazioni orfologiche in canali naturali `e generalmente effettuata risolvendo numericamente le equazioni delle onde lunghe in acque basse, che regolano il moto della fase fluida, assieme all’equazione di Exner, che descrive l’evoluzione del fondo. L’argomento della presente tesi consiste nello sviluppo di un schema ai volumi finiti di tipo ”centrato” per la soluzione accoppiata di tale sistema di equazioni. Un nuovo schema, denominato PRICE-C, `e qui introdotto: esso risolve le equazioni in forma conconservativa, ma ha l’importante propriet`a di degenerare in uno schema conservativo se il sottostante sistema di equazioni ammette una forma conservativa. Lo schema `e applicato alle equazioni delle onde lunghe in acque basse sia nel caso di fondo fisso che di fondo mobile, dapprima in un ambito unidimensionale e successivamente in quello bidimensionale. L’estensione non `e immediata nel caso in cui il reticolo di calcolo sia non-strutturato, dal momento che le equazioni differenziali devono essere mediate su opportuni volumi di controllo. Lo schema `e poi esteso ad alti ordini di accuratezza nello spazio e nel tempo attraverso le procedure ADER-WENO e MUSCL rispettivamente per il caso unidimensionale e bidimensionale. Inoltre si dimostra come lo schema proposto verifichi la ”well-balanced property”, che consiste nella capacit`a di raggiungere soluzioni stazionarie, anche in presenza di discontinuit`a della superficie libera e del fondo. Condizioni di corrente lenta e rapida, come pure condizioni di tipo transcritico vengono correttamente risolte. Inoltre lo schema in grado di riprodurre le celerit`a di propagazione di discontinuit`a della superficie e fronti di sedimenti al fondo, cos`? come la celerit`a di propagazione di piccoli disturbi del fondo. Caratteristica principale dello schema `e la sua semplicit`a: `e basato su un semplice approccio di tipo centrato, cio`e non necessita la conoscenza degli autovalori della matrice del sistema. Questa `e un’importante caratteristica dal momento che non sempre autovalori e autovettori sono calcolabili analiticamente, in particolare nel caso di complesse formule di chiusura per il trasporto al fondo. Quindi questo schema pu`o rivelarsi utile per l’ingegnere che spesso necessita di un semplice strumento numerico che possa essere applicato ad un sistema di equazioni differenziali di tipo iperbolico senza dover entrare nel dettaglio delle propriet`a atematiche del sistema stesso. Data la sua generalit`a, infatti, lo schema pu`o essere applicato ad ogni tipo di sistema iperbolico contenente termini non-conservativi.

Many applications in engineering practice can be described by thehyperbolic partial differential equations (PDEs). Numerical modeling of this typeof equations often involves large gradients or shocks, which makes it achallenging task for conventional numerical methods to accurately simulate suchsystems. Thus developing accurate and efficient shock capturing numericalschemes becomes important for the study of hyperbolic equations.In this dissertation, a detailed study of the numerical methods for linearand nonlinear unsteady hyperbolic equations was carried out. A new finitedifference shock capturing scheme of finite volume style was developed. Thisscheme is based on the high order Pad?? type compact central finite differencemethod with the weighted essentially non-oscillatory (WENO) reconstruction toeliminate non-physical oscillations near the discontinuities while maintain stablesolution in the smooth areas. The unconditionally stable semi-implicit Crank-Nicolson (CN) scheme is used for time integration.The theoretical development was conducted based on one-dimensionalhomogeneous scalar equation and system equations. Discussions were alsoextended to include source terms and to deal with problems of higher dimension.For the treatment of source terms, Strang splitting was used. For multidimensionalequations, the ?? -form Douglas-Gunn alternating direction implicit(ADI) method was employed. To compare the performance of the scheme withENO type interpolation, the current numerical framework was also applied usingENO reconstruction.The numerical schemes were tested on 1-D and 2-D benchmark problems,as well as published experimental results. The simulated results show thecapability of the proposed scheme to resolve discontinuities while maintainingaccuracy in smooth regions. Comparisons with the experimental results validatethe method for dam break problems. It is concluded that the proposed scheme isa useful tool for solving hyperbolic equations in general, and from engineeringapplication perspective it provides a new way of modeling open channel flows.

Turbulent mixing evolving from the Richtmyer-Meshkov instability, also known as shock-induced turbulent mixing, is investigated using numerical simulations of fundamental test problems with high-resolution computational methods. An existing state-of-the-art implicit large eddy simulation algorithm for compressible multispecies flows is extended to include the effects of viscous dissipation, thermal conductivity and species diffusion by deriving a novel set of governing equations for binary mixtures. This allows for direct numerical simulations of shock-induced turbulent mixing to be performed for arbitrary gas mixtures cases where the ratio of specific heats may vary with mixture composition at much greater computational efficiency. Using direct numerical simulation, a detailed study is performed of the effects of Reynolds number on the transition to turbulence in shock-induced mixing evolving from narrowband initial conditions. Even though the turbulence in the highest Reynolds number case is not fully developed, a careful analysis shows that the high Reynolds number limit of several key quantities is able to be estimated from the present data. The mixing layer is also shown to be persistently anisotropic at all Reynolds numbers, which also has important consequences for modelling. At the time of writing, the highest Reynolds number case from this set of simulations is the highest achieved in any fully-resolved direct numerical simulations presented in the open literature for this class of problems. Implicit large eddy simulation is employed to investigate the influence of broadband initial conditions on the late-time evolution of a shock-induced turbulent mixing layer. Both the bandwidth of initial modes as well as their relative amplitudes are varied, showing that both the growth rate of the mixing layer width and the decay rate of fluctuating kinetic energy strongly depend on initial conditions. Finally, both implicit large eddy simulations and direct numerical simulations are performed of an idealised shock tube experiment to analyse the effects of additional long wavelength, low amplitude modes in the initial perturbation. These calculations represent the first direct numerical simulations performed of Richtmyer-Meshkov instability evolving from broadband initial conditions.

Thesis (Ph.D.)--Aerospace Engineering, Georgia Institute of Technology, 2009.
Committee Chair: Smith, Marilyn; Committee Co-Chair: Zhou, Hao-Min; Committee Member: Dieci, Luca; Committee Member: Menon, Suresh; Committee Member: Ruffin, Stephen

These thesis review some recent results on the construction of very high order multidimensional upwind schemes for the solution of steady and unsteady conservation laws on unstructured triangular grids.

We also consider the extension to the approximation of solutions to conservation laws containing second order dissipative terms. To build this high order schemes we use a subtriangulation of the triangular Pk elements where we apply the distribution used for a P1 element.

This manuscript is divided in two parts. The first part is dedicated to the design of the high order schemes for scalar equations and focus more on the theoretical design of the schemes. The second part deals with the extension to system of equations, in particular we will compare the performances of 2nd, 3rd and 4th order schemes.

The first part is subdivided in four chapters:

The aim of the second chapter is to present the multidimensional upwind residual distributive schemes and to explain what was the status of their development at the beginning of this work.

The third chapter is dedicated to the first contribution: the design of 3rd and 4th order quasi non-oscillatory schemes.

The fourth chapter is composed of two parts: we start by understanding the non-uniformity of the accuracy of the 2nd order schemes for advection-diffusion problem. To solve this issue we use a Finite Element hybridisation.

This deep study of the 2nd order scheme is used as a basis to design a 3rd order scheme for advection-diffusion.

Finally, in the fifth chapter we extend the high order quasi non-oscillatory schemes to unsteady problems.

In the second part, we extend the schemes of the first part to systems of equations as follows:

The sixth chapter deals with the extension to steady systems of hyperbolic equations. In particular, we discuss how to solve some issues such as boundary conditions and the discretisation of curved geometries.

Then, we look at the performance of 2nd and 3rd order schemes on viscous flow.

Finally, we test the space-time schemes on several test cases. In particular, we will test the monotonicity of the space-time non-oscillatory schemes and we apply residual distributive schemes to acoustic problems.
Doctorat en Sciences de l'ingénieur
info:eu-repo/semantics/nonPublished

L’objectif de ce travail de thèse est de proposer la Méthodes des éléments finis de Galerkin discontinus (DGFEM) à la discrétisation des équations compressibles de Navier-Stokes. Plusieurs challenges font l’objet de ce travail. Le premier aspect a consisté à montrer l’ordre de convergence optimal de la méthode DGFEM en utilisant les polynômes d’interpolation d’ordre élevé. Le deuxième aspect concerne l’implémentation de méthodes de ‘‘shock-catpuring’’ comme les limiteurs de pentes et les méthodes de viscosité artificielle pour supprimer les oscillations numériques engendrées par l’ordre élevé (lorsque des polynômes d’interpolation de degré p>0 sont utilisés) dans les écoulements transsoniques et supersoniques. Ensuite nous avons implémenté des estimateurs d’erreur a posteriori et des procédures d ’adaptation de maillages qui permettent d’augmenter la précision de la solution et la vitesse de convergence afin d’obtenir un gain de temps considérable. Finalement, nous avons montré la capacité de la méthode DG à donner des résultats corrects à faibles nombres de Mach. Lorsque le nombre de Mach est petit pour les écoulements compressibles à la limite de l’incompressible, la solution souffre généralement de convergence et de précision. Pour pallier ce problème généralement on procède au préconditionnement qui modifie les équations d’Euler. Dans notre cas, les équations ne sont pas modifiées. Dans ce travail, nous montrons la précision et la robustesse de méthode DG proposée avec un schéma en temps implicite de second ordre et des conditions de bords adéquats
The aim of this thesis is to deal with compressible Navier-Stokes flows discretized by Discontinuous Galerkin Finite Elements Methods. Several aspects has been considered. One is to show the optimal convergence of the DGFEM method when using high order polynomial. Second is to design shock-capturing methods such as slope limiters and artificial viscosity to suppress numerical oscillation occurring when p>0 schemes are used. Third aspect is to design an a posteriori error estimator for adaptive mesh refinement in order to optimize the mesh in the computational domain. And finally, we want to show the accuracy and the robustness of the DG method implemented when we reach very low mach numbers. Usually when simulating compressible flows at very low mach numbers at the limit of incompressible flows, there occurs many kind of problems such as accuracy and convergence of the solution. To be able to run low Mach number problems, there exists solution like preconditioning. This method usually modifies the Euler. Here the Euler equations are not modified and with a robust time scheme and good boundary conditions imposed one can have efficient and accurate results

In industry and research, CFD methods play an essential role in the study of compressible flows which occur, for example, around airplanes or in jet engines, and complement experiments as well as theoretical analysis. In transonic flows, the flow speed may already exceed the speed of sound locally, giving rise to discontinuous flow phenomena, such as shock waves. These phenomena are numerically challenging due to having a size of only a few mean free paths and featuring a large gradient in physical quantities. The application of traditional low-order approaches, such as the FEM or the FVM, is usually limited by their immense computational costs for large three-dimensional problems with complex geometries when aiming for highly accurate solutions. By contrast, high-order methods, such as the DG method, inherently enable a deep insight into complex fluid flows due to their high-order spatial convergence rate for smooth problems while requiring comparatively few. This work presents two different numerical approaches in the context of unfitted DG discretizations of the Euler equations for inviscid compressible flow. In these approaches, we employ a sharp interface description by means of the zero iso-contour of a level-set function for the treatment of immersed boundaries and shock fronts, respectively. Thus, we omit the elaborate and computationally expensive generation of boundary-fitted grids. The robustness, stability, and accuracy of the presented numerical approaches are tested against a variety of benchmarks. The presented shock-capturing approach makes use of a DG IBM. It features a cell-agglomeration strategy in order to avoid ill-conditioned system matrices and a severe explicit time-step restriction both caused by small and ill-shaped cut cells. In high Mach number flows, the polynomial approximation oscillates in the vicinity of discontinuous flow phenomena, degrading the accuracy and the stability of the numerical method. As a remedy, we adapt a two-step shock-capturing strategy consisting of a modal-decay detection and a smoothing based on artificial viscosity for the application on an agglomerated cut-cell grid. However, the second-order artificial viscosity term drastically restricts the globally admissible time-step size. We address this issue by means of an adaptive LTS scheme, which we extend by a dynamic rebuild of the cell clustering for an efficient application in unsteady flows. The presented shock-fitting approach employs an XDG method, which we enhance by verifying the implementation of two level-set functions. Their zero iso-contours describe a solid body and a shock front, respectively. We present a novel sub-cell accurate reconstruction procedure of the shock front. In particular, we show a one-dimensional proof of concept for a stationary normal shock wave by applying an implicit pseudo-time-stepping procedure in order to correct the interface position inside a cut background cell. Thus, this work builds a fundamental basis on the way towards a high-order XDG method for supersonic compressible flow in three dimensions.

碩士
國立成功大學
航空太空工程學系
86
Shock wave focusing is a phenomena of energy collection, and the phenomena was applied to the medical treatment of kidney stones in recent years. Extra-Corporeal Shock Wave Lithotripsy(ESWL) is based on the fact that the sound impedance of water and human tissue are nearly the same. In ESWL, a blast wave generated at the first focus of the ellipsoidal reflector moves toward the second focus, and the high pressure at the second focus will be used to strike calculi. In order to more accurately capturing the high pressure in focus,the high order weighted essentially non-oscillatory scheme(WENO) is employed to solve 2D/axisymmetric compressible inviscid Euler equations in conjunction with a finite volume approach, and the equation of state for water is described by the Tait equation. The computed results are compared with the result by Sommerfeld and Muller.It was found that the present results are reasonably accurate.In the aspect of shock wave focusing, we considered the focusing of plane shock wave over parabolic reflector and spherical blast waves over ellipsoidal reflectors in water. It was found that the maximum pressure at focus are higher than those obtained by TVD schemes.

Supersonic flows over both simple and complex geometries involve features over a wide spectrum of spatial and temporal scales, whose resolution in a numerical solution is of significant importance for accurate predictions in engineering applications. While CFD has been greatly developed in the last 30 years, the desire and necessity to perform more complex, high fidelity simulations still remains. The present thesis has introduced two major innovations regarding the fidelity of numerical solutions of the compressible \ns equations. The first one is the development of new a priori mesh quality measures for the Finite Volume (FV) method on mixed-type (quadrilateral/triangular) element meshes. Elementary types of mesh distortion were identified expressing grid distortion in terms of stretching, skewness, shearing and non-alignment of the mesh. Through a rigorous truncation error analysis, novel grid quality measures were derived by emphasizing on the direct relation between mesh distortion and the quality indicators. They were applied over several meshes and their ability was observed to identify faithfully irregularly-shaped small or large distortions in any direction. It was concluded that accuracy degradation occurs even for small mesh distortions and especially at mixed-type element mesh interfaces the formal order of the FV method is degraded no matter of the mesh geometry and local mesh size. Therefore, in the present work, the high-order Discontinuous Galerkin (DG) discretization of the compressible flow equations was adopted as a means of achieving and attaining high resolution of flow features on irregular mixed-type meshes for flows with strong moving shocks. During the course of the thesis a code was developed and named HoAc (standing for High Order Accuracy), which can perform via the domain decomposition method parallel $p$-adaptive computations for flows with strong shocks on mixed-type element meshes over arbitrary geometries at a predefined arbitrary order of accuracy. In HoAc in contrast to other DG developments, all the numerical operations are performed in the computational space, for all element types. This choice constitutes the key element for the ability to perform $p$-adaptive computations along with modal hierarchical basis for the solution expansion. The time marching of the DG discretized Navier-Stokes system is performed with the aid of explicit Runge-Kutta methods or with a matrix-free implicit approach. The second innovation of the present thesis, which is also based on the choice of implementing the DG method on the regular computational space, is the development of a new $p$-adaptive limiting procedure for shock capturing of the implemented DG discretization. The new limiting approach along with positivity preserving limiters is suitable for computations of high speed flows with strong shocks around complex geometries. The unified approach for $p$-adaptive limiting on mixed-type meshes is achieved by applying the limiters on the transformed canonical elements, and it is fully automated without the need of ad hoc specification of parameters as it has been done with standard limiting approaches and in the artificial dissipation method for shock capturing. Verification and validation studies have been performed, which prove the correctness of the implemented discretization method in cases where the linear elements are adequate for the tessellation of the computational domain both for subsonic and supersonic flows. At present HoAc can handle only linear elements since most grid generators do not provide meshes with curved elements. Furthermore, p-adaptive computations with the implemented DG method were performed for a number of standard test cases for shock capturing schemes to illustrate the outstanding performance of the proposed $p$-adaptive limiting approach. The obtained results are in excellent agreement with analytical solutions and with experimental data, proving the excellent efficiency of the developed shock capturing method for the DG discretization of the equations of gas dynamics.
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