Similarity, Self-Similarity and Intermediate Asymptotics

We fluid dynamicists live on intimate terms with dimensional analysis, working almost instinctiveIy in terms of force coefficients and Mach and Reynolds numbers, and are surprised to learn that earlier in this century our colleagues sometimes plotted experimental results in raw dimensional form. We are also familar with self-similar solutions, knowing for example how Sedov and Taylor deduced from simple dimensional considerations that the blast wave from an intense explosion spreads as the 2/5-power of time. In fact, the well known book of Sedov may have given us the erroneous impression that all self-similar solutions are revealed by dimensional analysis. However, we need only recall the Prandtl-Blasius solution for the laminar boundary layer on a flat plate to realize that self-similarity is often revealed only by additional group properties of the problem beyond dimensional invariance. In Russian these are both termed self-similar solutions of the first kind. We are less familiar with self-similar solutions of the second kind. Perhaps the best known example is Guderley’s local solution for a converging spherical shock wave at the instant it collapses onto the centre. The radius decreases (and then grows again) as the 0.717-power of time. However, that exponent cannot be found from dimensional or other properties of the problem, but only from a solvability condition in the course of integrating the solution numerically. We may have supposed that a self-similar solution of this kind is rare, or even exceptional. Not so, says Barenblatt. Self-similarity of the first kind is actually the exception, with the second kind providing a much richer set of solutions. In this pioneering book he introduces us to such important new concepts as incomplete self-similarity and intermediate asymptotics, and shows how they enlarge the scope of similitude even to shed new light on turbulence. A keg point is that although any self-similar solution is the exact solution of a degenerate problem, it is a t the same time the asymptotic solution of a non-degenerate and hence non-self-similar problem. For example, the result of Sedov and Taylor solves the idealized problem of a finite amount of energy released at a point, but it is used instead as an approximation to the solution of the non-self-similar motion produced by release of the energy within a finite volume, at such relatively large times that the flow tends toward self-similarity. In this case the transition from the non-self-similar motion to self-similarity is complete : the dimensionless parameter containing the initial size of the explosion, which spoils the self-similarity, can be disregarded at times when it becomes large (or small). Then the self-similar variables can be found by dimensional analysis. Likewise the non-self-similar boundary layer on a blunted plate approaches the Prandtl-Blasius self-similar solution at distances downstream large compared with the blunting. Then the self-similarity is revealed by invariance of the limiting problem under a simple atfine transformation. These self-similarities are of the first kind.