Development of a high-order interior penalty discontinuous Galerkin method for compressible turbulent flows. Application to Reynolds-averaged Navier-Stokes and large eddy simulations

Over the last decade several studies and workgroups came to the conclusion that high-order discretisation schemes are the most suited to overcome the limitations of today’s flow solvers. The latter are at most second-order accurate and therefore often deliver poor results on currently used meshes. Within this context, discontinuous Galerkin (DG) methods have recently been identified as having the biggest potential for future flow solvers. Further research is however needed to improve the computational efficiency and the robustness of these schemes. Both issues are faced in the subsequent chapters of the thesis.

The first part of the thesis is devoted to the solution of the Reynolds-averaged Navier-Stokes (RANS) equations using an interior penalty (IP) formulation. Whereas the intention is not to develop new turbulence models, the focus lies on the adaptation of existing models to the particular features of high-order discretisation schemes. In contrast to classical low-order methods, high-order schemes are prone to Gibbs oscillations which – if no care is taken – can lead to the breakdown of the algorithm. Given the stability issues caused by negative values of the eddy viscosity, different changes of the one-equation Spalart-Allmaras (S-A) model are discussed.
Whilst having identical sparsity patterns as the popular second Bassi-Rebay scheme (BR2), the simpler IP formulation is up to 30% less time-consuming. Unfortunately, its stability strongly depends on a seemingly more or less arbitrary penalty parameter. As this parameter also effects the conditioning of the resulting system of discretised equations, its choice is of utmost importance. This might explain why the IP formulation has received only little attention over the past decades. The thesis analyses for the first time the use of the interior penalty discretisation to solve a system of RANS equations. One of the principal contributions of this work is the generalisation of the penalty parameter to anisotropic meshes and a highly variable viscosity. Several numerical examples are presented to illustrate the effect of the newly proposed definition of the penalty parameter.

The focus of the second part lies on turbulent boundary layer resolution. Based on detailed grid convergence analyses as a function of interpolation order, element type and grid size, clear guidelines for the choice of boundary layer meshes for practical applications are provided. Besides straight boundaries, the effect of boundary curvature or a hybrid discretisation are investigated. Moreover, the concept of “quasi-straight” elements is introduced. Following this innovative approach, the accuracy of the calculated shear friction can further be improved by a proper choice of the position of the inner-element nodes, without increasing the total number of unknowns. Finally, several numerical applications clearly demonstrate the potential of the method to solve industrial problems.

In the last part, large eddy simulations of the Taylor-Green vortex and the decay of homogeneous isotropic turbulence are presented. This last chapter constitutes a first step towards the use of discontinuous Galerkin methods for large eddy simulations. Besides a validation study for the Smagorinsky and the WALE subgrid scale model, the resolution requirements of resolved LES are determined. Furthermore different subgrid filters are compared and the choice of the Smagorinsky constant in the case of filtered LES is briefly examined.