Approximation of pressure perturbations by FEM

TL;DR

This paper develops a finite element method to approximate pressure perturbations in linear hydrodynamic stability problems, providing a variational formulation of the eigenvalue problem related to Tollmien-Schlichting waves.

Contribution

It introduces a variational FEM approach for pressure perturbations in the Orr-Sommerfeld eigenvalue problem, offering a new computational framework.

Findings

FEM effectively approximates pressure perturbations in shear flows.

The variational formulation facilitates numerical analysis of stability problems.

Applications to specific flow cases demonstrate method viability.

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.