29 The Long-Wave Approximation for Unbounded Flows

The stability characteristics of unbounded flows are substantially different from those of bounded or semi-bounded flows. These flows are of two basic types. For the shear-layer type, U( ? ?) ? U( ?), and without loss of generality we can choose the origin of velocity so that U( ? ?)= ? U( ?). This may be effected by a Galilean transformation which will change c _r, but not c _i, and hence not the growth rate of the disturbance. We also choose the characteristic velocity V so that U( ?)=1. For flows of the jet type, U( ? ?)= U( ?), and in this case we choose the origin of velocity so that U( ?)=0. The characteristic velocity V is then usually chosen so that max {( U(z)}=1.

In studying the stability of unbounded flows an important question that immediately arises is whether or not the basic flow can be treated as nearly parallel in the sense of equations (25.4). A general and precise answer to this question is not presently available but it is a matter of continuing concern and study, especially in connexion with flows of the boundary-layer type. The nature of the difficulties, however, can easily be illustrated by considering Bickley s (1937) solution of the boundary-layer equations for a two-dimensional jet. If we let M denote the (constant)...