# Low-Mach Number Flow and the Discontinuous Galerkin Method

A. Hennink

Research output: ThesisDissertation (TU Delft)

## Abstract

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.
Original language English Doctor of Philosophy Delft University of Technology Kloosterman, J.L., SupervisorLathouwers, D., Supervisor 22 Feb 2022 https://doi.org/10.4233/uuid:9b35b35a-0789-4883-a471-f8df0d7939ad Published - 2022

### Funding

Part of this work was funded by the sCO2 ­HeRo project that has received funding from the European research and training program 2014–2018 (grant agreement ID 662116).

Part of this work was supported by the ENEN+ project that has received funding from the Euratom research and training Work Programme 2016–2017 — 1 #755576.

## Keywords

• Low-Mach
• Pressure correction
• Discontinuous Galerkin

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