Fourth order finite difference and multigrid methods for modeling instabilities in flat plate boundary layers—2-D and 3-D approaches

Abstract

An efficient spatial approach which includes a fourth-order finite difference technique on stretched and staggered grids, a fully implicit time-marching scheme, and a semi-coarsening multigrid method based on a modified form of distributive relaxation is developed for solving the transition problem. An improved outflow boundary treatment is also presented that needs only a very short buffer domain to damp all order-one wave reflections. As a first application, we investigate the instability of 2-D incompressible flows over a semi-infinite flat plate by tracking the growth of the stable mode for Tollmien-Schlichting waves. Numerical results show good agreement with the analytic solution obtained from linear stability theory (LST) for the parallel base with small disturbance case. For the non-parallel base flow case, the results are in good agreement with other direct numerical simulations (DNS). An extended 3-D code is then tested as the second applications. Again, good agreement is observed between our approach and LST in the case of small disturbance. Finally, the secondary instability is successfully simulated with moderate grid size and reasonable CPU costs.