Coordinate perturbation and multiple scale in gasdynamics

Abstract

Usually, the application of the coordinate perturbation technique consists in transforming the equations to perturbed coordinates, and determining from the transformed equations the amount of coordinate straining appropriate to obtain a uniformly valid expansion. However, the transformed equations may become unwieldy with increasing order of the system, number of variables, and order of the approximation. There exists a much simpler way of applying the technique, which bypasses the transformed equations and provides the appropriate coordinate stretching by simple algebraic manipulations on the nonuniformly valid expansion obtained by straightforward expansion from the original equations. Interesting results are obtained by applying the procedure to two gasdynamical problems. In the first the flow field around a supersonic two-dimensional wing is determined up to third order, including a uniformly valid representation of the front shock shape, valid even when the shock does not start at the leading edge. The second problem concerns the oscillations in a closed tube following an arbitrary initial disturbance, both when the two ends are closed, and when one of the two ends contains an oscillating piston (the inviscid Chester problem). In both problems the uniformly valid expansions are substantially simpler than the non-uniformly valid. But most interesting is the result that the uniformly valid expansions cannot be obtained without supplementing the coordinate perturbation technique by the multiple scale technique.