A Navier-Stokes equation solver using agglomerated multigrid featuring directional coarsening and line-implicit smoothing

An explicit agglomeration multigrid method is presented for the Navier-Stokes equations with the Spalart-Allmaras turbulence model and applied to turbulent aerodynamic flows. The use of agglomeration provides an automated method for producing a sequence of coarse grids for use with multigrid. The solver employs a finite-volume formulation with a multistage relaxation scheme to produce steady-state solutions. Artificial dissipation is provided through a blend of an undivided Laplacian and biharmonic operator. Block-Jacobi preconditioning with matrix dissipation has been implemented. This solver is capable of using structured and unstructured grids, including mixed-element grids, with the primary focus being the use of unstructured grids. Competitive convergence rates approaching the theoretical limit of 0.75 for a second-order method are demonstrated for inviscid flow problems. Validation of inviscid and viscous results relative to two reference solvers and experimental results are presented. The main focus of this research is the improvement of convergence rates for turbulent flow problems.; The performance of explicit multigrid methods applied to turbulent flow problems has typically been shown to be far slower than that for inviscid flow problems. Two techniques to improve the performance of explicit agglomeration multigrid for turbulent flow problems are presented. Directional-coarsening alters the agglomeration algorithm to directly alleviate numerical stiffness arising from grids containing high-aspect-ratio cells. Line-implicit smoothing alters the block-Jacobi preconditioner to introduce implicit terms along non-crossing lines constructed in the grid. Both methods are investigated separately and in combination to determine the best set of parameters and the limits of these methods. Substantial reductions in the cost of obtaining a solution are demonstrated for laminar and turbulent flow problems, with solutions converged over 10 orders of residual reduction being obtained in 25--33% of the time required for the base multigrid solver using isotropic coarsening.