Efficient unsteady high Reynolds number flow computations on unstructured grids

Despite the advances in computer power and numerical algorithms over the last decades, solutions to unsteady flow problems remain computing time intensive. Especially for high Reynolds number flows, nonlinear multigrid, which is commonly used to solve the nonlinear systems of equations, converges slowly. The stiffness induced by the high-aspect ratio cells and turbulence is not tackled well by this solution method.In this paper, it is investigated if a Jacobian-free Newton-Krylov (jfnk) solution method can speed up unsteady flow computations at high Reynolds numbers. Preconditioning of the linear systems that arise after Newton linearization is commonly performed with matrix-free preconditioners or approximate factorizations based on crude approximations of the Jacobian. Approximate factorizations based on a Jacobian that matches the target residual operator are unpopular because these preconditioners consume a large amount of memory and can suffer from robustness issues. However, these preconditioners remain appealing because they closely resemble A^-1.In this paper, it is shown that a jfnk solution method with an approximate factorization preconditioner based on a Jacobian that approximately matches the target residual operator enables a speed up of a factor 2.5-12 over nonlinear multigrid for two-dimensional high Reynolds number flows. The solution method performs equally well as nonlinear multigrid for three-dimensional laminar problems. A modest memory consumption is achieved with partly lumping the Jacobian before constructing the approximate factorization preconditioner, whereas robustness is ensured with enhanced diagonal dominance.