Simple asymptotics for the temporal spectrum of an Orr-Sommerfeld problem
type: Item
instance of: scholarly article
title: Simple asymptotics for the temporal spectrum of an Orr-Sommerfeld problem
zbMATH Open document ID: 0991.76021
DOI: 10.1016/S0893-9659(99)00185-8
author: B. S. Ng
author: W. H. Reid
published in: Applied Mathematics Letters
publication date: 2000-12-04T00:00:00Z
review text: This paper deals with an asymptotic approximation to the eigenvalue problem resulting from Orr-Sommerfeld equation (OSE) in the limit of vanishing wave number \(\alpha\), \(\alpha \to 0\), while \(\alpha\operatorname {Re}\) (where Re is the Reynolds number) remains large but finite. In this case, the OSE reduces to a third-order ordinary differential equation, whose eigenvalue relation can be specified in terms of `inviscid' and `viscous' solutions. A perturbation-analysis based approach involving two scaling variables combined with the generalized Airy functions is used to approximate the viscous solutions up to 2nd order in the `slow' variable \(\varepsilon\). It is shown that the eigenvalue problem can then be reformulated in terms of the `usual' Airy functions, and that it is -- in limiting situations -- relatively easy to obtain solutions for the \(P\)- and \(A\)- modes. However, to arrive at the corresponding results for the \(S\)-modes, the Airy functions have to be replaced by the leading order terms of their complete asymptotic expansions. This implies that the eigenvalue relation becomes (for fixed values of \(\alpha \operatorname {Re}\)) a function of the wave speed only, so that the eigenvalues can be obtained from a simple numerical iteration process. Numerical calculations have been performed for \(\alpha \operatorname {Re}=10.000\), and two figures are presented. It is stated that the results are visually indistinguishable from those obtained by a direct numerical solution of the original third-order ordinary differential equation. The paper is well written and interesting, however, a few additional (and more recent) references would have helped the reader to see more clearly the link to complementary current research in the field of hydrodynamic instability.
reviewed by: Nina Kirchner
Mathematics Subject Classification ID: 76E05
Mathematics Subject Classification ID: 76M45
Mathematics Subject Classification ID: 34L16
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 1540349
zbMATH Keywords: hydrodynamic instability
zbMATH Keywords: Orr-Sommerfeld equation
zbMATH Keywords: even modes
zbMATH Keywords: asymptotic approximation
zbMATH Keywords: inviscid solution
zbMATH Keywords: visous solution
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: vanishing wave number
zbMATH Keywords: third-order ordinary differential equation
zbMATH Keywords: generalized Airy functions
zbMATH Keywords: numerical iteration
MaRDI profile type: Publication
cites work: Accurate solution of the Orr–Sommerfeld stability equation
cites work: A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer
cites work: Q3896709
cites work: Stability of the Laminar Parabolic Flow of a Viscous Fluid between Parallel Fixed Walls
description: scientific article; zbMATH DE number 1540349