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.
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 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  22 Feb 2022 
DOIs  
Publication status  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
 LowMach
 Pressure correction
 Discontinuous Galerkin