A shift-invert strategy for global flow instability analysis using matrix-free methods

A new time-stepping shift-invert algorithm for linear stability analysis of large-scale laminar flows in complex geometries is presented. This method, based on a Krylov subspace iteration, enables the solution of complex non-symmetric eigenvalue problems in a matrix- free framework. Compared with the classical exponential method, the new approach has the advantage of converging to specific parts of the full global spectrum. Validations and comparisons to the exponential power method have been performed in three different cases: (i) the stenotic flow, (ii) the backward-facing step and (iii) the two-dimensional swirl flow. It is shown that, although the exponential method remains the method of choice if leading eigenvalues are sought, the present method can be competitive when access to specific parts of the full global spectrum is required. In addition, as opposed to other methods, this strategy can be directly applied to any time-stepper, regardless of the temporal or spatial discretization of the latter.