High-order methods for the numerical solution of the BiGlobal linear stability eigenvalue problem in complex geometries.

Abstract

A high-order computational tool based on spectral and spectral/hp elements (J. Fluid. Mech. 2009; to appear) discretizations is employed for the analysis of BiGlobal fluid instability problems. Unlike other implementations of this type, which use a time-stepping-based formulation (J. Comput. Phys. 1994; 110(1):82–102; J. Fluid Mech. 1996; 322:215–241), a formulation is considered here in which the discretized matrix is constructed and stored prior to applying an iterative shift-and-invert Arnoldi algorithm for the solution of the generalized eigenvalue problem. In contrast to the time-stepping-based formulations, the matrix-based approach permits searching anywhere in the eigenspace using shifting. Hybrid and fully unstructured meshes are used in conjunction with the spatial discretization. This permits analysis of flow instability on arbitrarily complex 2-D geometries, homogeneous in the third spatial direction and allows both mesh (h)-refinement as well as polynomial (p)-refinement. A series of validation cases has been defined, using well-known stability results in confined geometries. In addition new results are presented for ducts of curvilinear cross-sections with rounded corners.