Three-Dimensional Flow Stability Analysis Based on the Matrix-Forming Approach Made Affordable

Abstract

Theoretical developments for hydrodynamic instability analysis are often based on eigenvalue problems, the size of which depends on the dimensionality of the reference state (or base flow) and the number of coupled equations governing the fluid motion. The straightforward numerical approach consisting on spatial discretization of the linear operators, and numerical solution of the resulting matrix eigenvalue problem, can be applied today without restrictions to one-dimensional base flows. The most efficient implementations for one-dimensional problems feature spectral collocation discretizations which produce dense matrices. However, this combination of theoretical approach and numerics becomes computationally prohibitive when two-dimensional and three-dimensional flows are considered. This paper proposes a new methodology based on an optimized combination of high-order finite differences and sparse algebra, that leads to a substantial reduction of the computational cost. As a result, three-dimensional eigenvalue problems can be solved in a local workstation, while other related theoretical methods based on the WKB expansion, like global-oscillator instability or the Parabolized Stability Equations, can be extended to three-dimensional base flows and solved using a personal computer.