A grid-transparent numerical method for compressible viscous flows on mixed unstructured grids

The goal of the present work is the development of a numerical method for compressible viscous flows on mixed unstructured grids. The discretisation is based on a vertex-centred finite-volume method. The concept of grid transparency is developed as a framework for the discretisation on mixed unstructured grids. A grid-transparent method does not require information on the cell types. For this reason, the numerical method developed in the present work can be applied to triangular, quadrilateral, and mixed grids without modification. The inviscid fluxes are discretised using the approximate Riemann solver of Roe. A limited linear-reconstruction method leads to monotonic capturing of shock waves and second-order accuracy in smooth regions of the flow. The discretisation of the viscous fluxes on triangular and quadrilateral grids is first studied by reference to Laplace's equation. A variety of schemes are evaluated against several criteria. The chosen discretisation is then extended to the viscous fluxes in the Navier-Stokes equations. A careful study of the various terms allows a form to be developed which may be regarded as a thin-shear-layer approximation. In contrast to previous implementations, however, the present approximation does not require knowledge of normal and tangential coordinate directions near solid surfaces. The effects of turbulence are modelled through the eddy-viscosity hypothesis and the one-equation model of Spalart and Allmaras. The discrete equations are marched to the steady-state solution by an explicit Runge-Kutta method with local time-stepping. The turbulence-model equation is solved by a point-implicit method. To accelerate the convergence rate, an agglomeration multigrid method is employed. In contrast to previous implementations, the governing equations are entirely rediscretised on the coarse grid levels. The solution method is applied to various inviscid, laminar, and turbulent flows. The performance of the multigrid method is compared for triangular and quadrilateral grids. Care is taken to assess numerical errors through grid-refinement studies or comparisons with analytical solutions or experimental data. The main contributions of the present work are the careful development of a solution method for compressible viscous flows on mixed unstructured grids and the comparison of the impact of triangular, quadrilateral, and mixed grids on convergence rates and solution quality.