Approximation of pressure perturbations by FEM

TL;DR

The paper addresses linear hydrodynamic stability of shear flows under Tollmien–Schlichting perturbations by replacing the velocity continuity equation with a Poisson equation for the pressure perturbation and formulating a variational problem. It then develops a finite element discretization of the resulting weak form, leading to a generalized eigenproblem $(K_h+L_h G_h^{-1}H_h)A_h = c_h S_h A_h$ with $B_h = G_h^{-1}H_h A_h$, solvable by QZ, QR, or LR methods. This framework enables computation of stability characteristics such as neutral curves and wave speeds, and is demonstrated as a basis for a Prandtl boundary layer stability program. The approach provides a robust, alternative computational path for predicting stability properties of shear flows in practical settings.

Abstract

In the mathematical problem of linear hydrodynamic stability for shear flows against Tollmien-Schlichting perturbations, the continuity equation for the perturbation of the velocity is replaced by a Poisson equation for the pressure perturbation. The resulting eigenvalue problem, an alternative form for the two-point eigenvalue problem for the Orr-Sommerfeld equation, is formulated in a variational form and this one is approximated by finite element method (FEM). Possible applications to concrete cases are revealed.