Condition number analysis of flow fields arising from CFD simulations

Abstract

Condition numbers of pressure and velocity matrices obtained from an unstructured finite volume discretization of the three dimensional Navier–Stokes equations are studied for two distinct flow configurations. The exact condition number is calculated based on singular values. The continuous estimation of condition number of the velocity matrices with respect to the time varying velocity field is achieved through the Gershgorin theoretical bounds for the eigenvalues. The focus of the present study is on the analysis of flow dynamics with respect to the degree of ill-conditioning of the linear systems. For the two benchmark problems, condition numbers of pressure and velocity matrices are presented. The pressure matrix is shown to be more ill-conditioned than velocity and requires strong preconditioning. For a 3D lid-driven cavity, the appearance and disappearance of corner vortices at higher Reynolds number is clearly reflected in the condition number variation with time. In a second application of flow past a circular cylinder, condition number variation with time during the initial phase clearly reveals the onset of vortex shedding. The condition number based on Gershgorin bounds is also used to switch the SGS and ILU preconditioners for velocity matrices when they are well conditioned. Consequently, an overall reduction in the simulation time is demonstrated.