Filtering the eigenvalues at infinite from the linear stability analysis of incompressible flows

Steady state, two-dimensional flows may become unstable under two and three-dimensional disturbances if the flow parameters exceed some critical values. In many practical situations, determining the parameters at which the flow becomes unstable is essential. Linear hydrodynamic stability of a laminar flow leads to a generalized eigenvalue problem (GEVP) where the eigenvalues correspond to the rate of growth of the disturbances and the eigenfunctions to the amplitude of the perturbation. Solving GEVP's is challenging, because the incompressibility of the liquid gives rise to singularities leading to non-physical eigenvalues at infinity that require substantial care. The high computational cost of solving the GEVP has probably discouraged the use of linear stability analysis of incompressible flows as a general engineering tool for design and optimization. In this work, we propose a new procedure to eliminate the eigenvalues at infinity from the GEVP associated to the linear stability analysis of incompressible flow. The procedure takes advantage of the structure of the matrices involved and avoids part of the computational effort of the standard mapping techniques used to compute the spectrum of incompressible flows. As an example, the method is applied in the solution of linear stability analysis of plane Couette flow.