Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

Abstract

This paper is concerned with the derivation of asymptotic approximations to the solutions of the Orr‐Sommerfeld equation and its adjoint which are uniformly valid in a full neighborhood of the critical point. In the case of plane Couette flow, it is shown that the expansions can be expressed to all orders in terms of a restricted class of generalized Airy functions and that the slowly varying coefficients in these expansions can be determined from simple recurrence relations. The expansions can easily be related to the Laplace integral representation of the solutions but this is not essential in their derivation. The structure of the theory also suggests how it can be immediately generalized to provide “first approximations” for a more general class of velocity profiles. Detailed results are given in the special case when the wave number α → 0. In this limit, the characteristic equation can be reduced to an especially simple form which permits a convenient global description of the modal structure.

An Exact Solution of the Orr‐Sommerfeld Equation for Plane Couette Flow

“…Thus, we let (9) where K is a constant to be determined. An integration by parts then shows that (10) and hence (8) (20) and these approximations are in agreement with those given previously [6].…”

“…Integrals of this type have been discussed briefly by Reid [6] + e 2 2 (7/, e)A~<K + aV), (6) where, as indicated, Cl' -is independent of 'J]. On differentiating (6) we obtain the two equations W. H. Reid (7) and on then eliminating <ffi we have (8) The form of this equation suggests that we look for a solution in terms of a Laplace integral with finite limits of integration.…”

On the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group-invariant solutions

“…The most famous and thoroughly studied approach, the spectral/modal approach (spectral expansion of disturbances in time and space, followed by the eigenfunction analysis), was derived by Orr. 1 A large collection of literature exists on the application of this approach to homogeneous incompressible [2][3][4][5][6] and compressible shear flows, [7][8][9] as well as inhomogeneous stratified or magnetized shear flows. [10][11][12][13][14] A comprehensive introduction to this field can be found, for instance, in the book by Schmid and Henningson.…”

Uniform asymptotic approximations to the solutions of the Dunn-Lin equations

“…cannot, of course, be linearly independent, but must satisfy four exact connection formulae. Although these connection formulae are not essential to the development of the present theory, as has been shown by Reid [12,13] in his discussion of the Orr-Sommerfeld equation, nevertheless their use can lead to a more systematic derivation of the coefficients in the expansions of balanced type. Thus, for example, by using the connection formula…”

“…Moreover, lest the present theory be obscured by unnecessary technical complexities, we have restricted our attention to the case where the Prandtl number is equal to unity. The present approach is closely related to some methods developed by Olver [9] whereby he obtained uniform approximations to a certain class of second-order equations, and can be regarded as a direct extension of the work of Reid [12,13] on the Orr-Sommerfeld equation to higher-order systems with a simple turning point. The techniques employed here also represent a complete departure from the usual procedure of seeking approximations in terms of viscous and non-viscous types (see, for example, [1] and [14]).…”