Asymptotic approximation of the eigenvalues and the eigenfunctions for the Orr-Sommerfeld equation on infinite intervals

TL;DR

This paper develops asymptotic methods to approximate eigenvalues and eigenfunctions of the Orr-Sommerfeld equation on infinite domains, using WKB and hypergeometric functions for different wave limits, aiding fluid flow stability analysis.

Contribution

It introduces novel asymptotic approximation techniques for the Orr-Sommerfeld eigenproblem on infinite domains, employing WKB and hypergeometric functions for short- and long-wave limits.

Findings

Eigenvalues estimated using WKB methods in short-wave limit

Eigenfunctions approximated via Green's functions

Solutions expressed with hypergeometric functions in long-wave limit

Abstract

Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two and three dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, WKB methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms Green's functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which eigenvalues and eigenfunctions can be approximated. The approximated eigenvalues can, for instance, be used as a starting point in predicting transitions in boundary layers with computer simulations (computational fluid dynamics). In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions.