Simple asymptotics for the temporal spectrum of an Orr-Sommerfeld problem
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.
third-order ordinary differential equation
scientific article; zbMATH DE number 1540349