Abstract
The problem of a semi-infinite flat plate started initially from rest in an incompressible fluid with an arbitrary velocity without flow reversal is considered in this paper. In reference (1) the unsteady laminar boundary layer on a semi-infinite flat plate started to move with a velocity proportional to tn, where n is positive, is analyzed in detail. It is shown that a solution, constructed from the leading edge and properly continued into the far downstream region of the leading edge, possesses the proper limiting behavior that either at small t or in the far downstream region the flow field is substantially independent of x and is given by the Rayleigh-type solution. This solution also possesses the proper behavior at very large times. In view of the fact that serious difficulties have been encountered in trying to perturb the Rayleigh-type solution either to obtain solutions in the upstream region or to obtain solutions valid at large times, it is of interest to see whether the method of constructing the unsteady laminar boundary-layer solution as shown in reference (1) can be applied to arbitrary plate velocity and still lead to solutions with proper limiting behavior. This is found to be so if the plate was started initially from rest. In particular, the limiting form of the results for large t produces a solution for the motion of an oscillating plate in a uniform stream. This is of interest at least in so far as the theoretical results can be checked by experiment, not mentioning its connection with a number of practical problems. The effects of compressibility have been considered briefly with Howarth’s transformation in reference (2). Unfortunately, the simple energy integrals of Busemann and of Crocco when Pr = 1 cannot be extended to unsteady flow with physically sensible thermal condition on the plate. It seems that the temperature field would be better determined numerically for each specific example in question and is omitted from the present paper.
