Finite element methods for viscous incompressible BiGlobal instability analysis on unstructured meshes

Viscous linear 3-D BiGlobal instability analyses of incompressible flows have been performed using finite element numerical methods, with a view to extend the scope of application of this analysis methodology to flows over complex geometries. The initial value problem (IVP), based on the linearized Navier--Stokes equations (LNSE), as well as the real and the complex partial-differential-equation-based generalized eigenvalue problems (EVP), have been solved. A mixed [P.sub.2][P.sub.1] finite element spatial discretization on unstructured meshes for both the LNSE and the EVP approaches has been used. For the time-discretization of the LNSE a characteristics method has been used for the first time in the context of flow stability analysis; the complex BiGlobal EVP has also been solved for the first time in the context of a finite element numerical discretization. In either its real or its complex form, the EVP has been solved without the need to introduce pseudocompressibility into the incompressible equations, which has simplified the systems to be solved without sacrificing accuracy. An Arnoldi approach has been used to recover the most significant eigenvalues. In this context, the associated solutions to the resulting linear systems were obtained by iterative methods: generalized minimal residual with incomplete lower-upper preconditioning or conjugate gradient with I-Cholesky preconditioning, depending on whether the coefficient matrix was symmetric or not. The 3-D instability of the classic 2-D lid-driven cavity flow and that of the rectangular duct flow were used as validation cases for the real and complex EVP, respectively. New results have been obtained fur the 3-D BiGlobal instability of two closed and one open flow, namely, the regularized lid-driven cavity of rectangular and triangular shape and flow in the wake of a model bluff body. DOI: 10.2514/1.25410