High-Order Numerical Methods for BiGlobal Flow Instability Analysis and Control

Abstract

In recent years flow instability and flow control research has focused attention on two novel and promising areas, investigation of perturbations in the limit of short times after their introduction into the flow Schmid & Henningson (2001) and study of modal and non-modal perturbations in complex, essentially non–parallel flows. A rather complete, up to that time, account of the rapidly growing latter area was discussed in Collis et al. (2004); Theofilis (2003); the role of (especially global) instability analysis in flow control is discussed elsewhere Theofilis (2009b). The present article deals with the theory and numerical aspects underlying the recent rapid developments in global instability research, in an attempt to generate a self–contained account of the areas in which the authors have been working in the last decade, as opposed to producing a review-type article on themany developments which have recently taken place. When discussing numerical methods for the solution of the partial–derivative eigenvalue problem it is instructive to remind the reader of the process by which one arrives at the various large–scale eigenvalue problems solved in a global instability context; pictorially, this process is shown in the charts presented in figure 1. In the first of these decision trees, one is confronted with the temporal derivative in the linearized Navier–Stokes equations (LNSE) and the possibility to either discretize this term (in a so–called time-stepping approach), thereby solving for arbitrary temporal development of the perturbations (including transient growth) or work in frequency space by introducing eigenmodes, a procedure which is permissible by the separability of the temporal and spatial derivatives in the LNSE. Once this decision has been made, the next step is to deal with how to treat the LNSE matrix; there too, two paths may be followed, one (Time-steppers) along which the matrix describing the evolution of perturbations is not formed explicitly and tools analogous with those used in direct numerical simulations are employed, and another one (Generalized Eigenvalue Problem – GEVP) along which the matrix (of leading dimension potentially reaching 1 Tb Kitsios et al. (2008)) is stored in (shared or distributed) memory. From the outset the question 7