Approximations to the eigenvalue relation for the Orr-Sommerfeld problem

Abstract
A comprehensive study is made of the eigenvalue relation for the Orr-Sommerfeld problem. One of the major results obtained is a 'first approximation’ to the eigenvalue relation which is uniformly valid along the entire curve of marginal stability. Two derivations of this approximation are given, one based on the use of uniform approximations to the solutions of the Orr-Sommerfeld equation and the other based on the differential equation satisfied by the eigenvalue relation itself. The theory is developed in detail for symmetrical flows in a channel but it is also applicable, with minor modifications, to flows of the boundary-layer type. Near the nose of the marginal curve the error associated with the approximation is of the order of e 3 , where e = (ixRU c )- 1/3 , and as R → ∞ along the upper and lower branches of the marginal curve the errors are of the order of e 12/5 and e 2 ln e respectively. A comparison is also made with four heuristic approximations to the eigenvalue relation, two of which have been widely used in the past, and detailed calculations for plane Poiseuille flow clearly demonstrate the superiority of the uniform approximation.

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