This thesis describes a numerical method for computational fluid dynamics. Special attention is paid to low­Mach number flows. The spatial discretization is a discontinuous Galerkin method, based on modal basis functions. The convection is discretized with the local Lax­Friedrichs flux. The diffusion in the enthalpy equation is discretized with the symmetric interior penalty method, which is generalized in a straightforward manner for the viscous stress in the momentum equation. The numerical method does not deviate fundamentally from previous literature. The temporal derivatives in the enthalpy and momentum equations are dis­ cretized with a second­order backward finite difference method. An algorithmic pressure correction scheme is used decouple the momentum and the continuity equations, giving rise to explicit artificial boundary conditions. If the pressure and the momentum are discretized with an equal­order polynomial space, then the pres­sure equation is stabilized with an extra penalty term to suppress the discontinuities in the solution, as explained in chapter 2. Using a time­splitting method is far more difficult when the flow is compressible, due the variable density. Low­Mach number flows also do not lend themselves well to solving the coupled transport equations, because the density is a function of the enthalpy, not the pressure. This differs from high­Mach number flows, where one can solve a transport equation for the density. Chapter 4 describes in great detail how a non­constant density can be incorporated into a time­splitting scheme for low­Mach number flows. Chapter 4 also discusses the best form of the enthalpy transport equation to solve (primitive or conservative), and for which variable (primitive or conserved). This question arises in low­Mach number flows, because the density is a function of the temperature. Here the conservative transport equation is solved for the specific enthalpy. The main difficulty with this approach is that the temporal enthalpy derivative is nonlinear due to the variable density. This can be addressed with an easily implemented adjustment of the finite difference scheme (‘method #2’ in sections 4.3–4.4). The resulting discretization displays second­order temporal accuracy (as measured in the spatial 𝐿2 norm) for the enthalpy and the mass flux, without having to iterate within a time step. Furthermore, the enthalpy transport equation needs to be stabilized with a sim­ple change of variables, in which the specific enthalpy is ‘offset’ by a constant. Though it may be counter­intuitive, the enthalpy offset is critical to the stability and the accuracy of the temporal discretization. This would also be true if one were to solve for the volumetric enthalpy, because the enthalpy offset determines whether there is a one­to­one mapping between the volumetric enthalpy and the density. The spatial and temporal discretizations and their implementations are exten­sively verified and validated with the test cases at the end of the chapters. In particular, sections 3.3.1, 3.3.2, and 4.5.1 feature exhaustive tests with manufac­tured solutions with nontrivial fluid properties. Sections 2.7, 3.4, and 4.5.2 contain validations for laminar flows. Chapter 5 also shows simulations of turbulent flows. @en

Over the past two decades, there has been much development in discontinuous Galerkin methods for incompressible flows and for compressible flows with a positive Mach number, but almost no attention has been paid to variable-density flows at low speeds. This paper presents a pressure-based discontinuous Galerkin method for flow in the low-Mach number limit. We use a variable-density pressure correction method, which is simplified by solving for the mass flux instead of the velocity. The fluid properties do not depend significantly on the pressure, but may vary strongly in space and time as a function of the temperature. We pay particular attention to the temporal discretization of the enthalpy equation, and show that the specific enthalpy needs to be ‘offset’ with a constant in order for the temporal finite difference method to be stable. We also show how one can solve for the specific enthalpy from the conservative enthalpy transport equation without needing a predictor step for the density. These findings do not depend on the spatial discretization. A series of manufactured solutions with variable fluid properties demonstrate full second-order temporal accuracy, without iterating the transport equations within a time step. We also simulate a Von Kármán vortex street in the wake of a heated circular cylinder, and show good agreement between our numerical results and experimental data.

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Accurate methods to solve the Reynolds-Averaged Navier-Stokes (RANS) equations coupled to turbulence models are still of great interest, as this is often the only computationally feasible approach to simulate complex turbulent flows in large engineering applications. In this work, we present a novel discontinuous Galerkin (DG) solver for the RANS equations coupled to the k−ϵ model (in logarithmic form, to ensure positivity of the turbulence quantities). We investigate the possibility of modeling walls with a wall function approach in combination with DG. The solver features an algebraic pressure correction scheme to solve the coupled RANS system, implicit backward differentiation formulae for time discretization, and adopts the Symmetric Interior Penalty method and the Lax-Friedrichs flux to discretize diffusive and convective terms respectively. We pay special attention to the choice of polynomial order for any transported scalar quantity and show it has to be the same as the pressure order to avoid numerical instability. A manufactured solution is used to verify that the solution converges with the expected order of accuracy in space and time. We then simulate a stationary flow over a backward-facing step and a Von Kármán vortex street in the wake of a square cylinder to validate our approach.

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We present a new discretization of the mono-energetic Fokker–Planck equation. We build on previous work (Kópházi and Lathouwers, 2015) where we devised an angular discretization for the Boltzmann equation, allowing for both heterogeneous and anisotropic angular refinement. The angular discretization is based on a discontinuous finite element method on the unit sphere. Here we extend the methodology to include the effect of the Fokker–Planck scatter operator describing small angle particle scatter. We describe the construction of an interior penalty method on the sphere surface. Results are provided for a variety of test cases, ranging from purely angular to fully three-dimensional. The results show that the scheme can resolve highly forward-peaked flux distributions with forward-peaked scatter.

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