Investigation of selfsimilar solutions describing flows in mixing layers

Abstract

A complete investigation is made of the selfsimilar solutions of the boundary layer equation for the stream function with zero pressure gradient. They are a good description of the flow pattern in mixing layers since far from the separation point the latter is formed mainly under the effect of the boundary conditions and depends slightly on the initial conditions. The selfsimilar function φ ( ζ; m) (ζ is the selfsimilar variable, and m the selfsimilarity parameter) satisfies a well-known third-order non-linear differential equation. It is successfully reduced to a first-order equation /1/, which enables us to investigate the behaviour of all the integral curves of φ ( ζ; m) and, in particular, the examination of the question of the existence and uniqueness of the solutions of the two- and three-point problems that occur in the theory of displacement layers. For m=1 these are classical problems /2–4/ and the Blasius boundary layer problem and for m = 2 the Goldstein problem for the wake /5/. The mixing layer encountered in the theory separations /6–11/ refers to the case m ε (1, 2]. The case m = ∞ occurs in the theory of non-stationary separation /12/. From the viewpoint of the behaviour of the integral curves, the cases m>1 and 0< m⩽1 differ substantially. For 0< m⩽1 their pattern is reformed in such a manner that solutions describing the flows in mixing layers with reverse velocities do not occur. Examples of the latter are given in /13, 14/.