Discontinuous solutions of the boundary-layer equations

Abstract

Since 1904, when Prandtl formulated the boundary-layer equations, it has been presumed that due to the viscous nature of the boundary layers the solution of the Prandtl equations should be sought in the class of continuous functions. However, there are clear mathematical reasons for discontinuous solutions to exist. Moreover, under certain conditions they represent the only possible solutions of the boundary-layer equations.In this paper we consider, as an example, an unsteady analogue of the laminar jet problem first studied by Schlichting in 1933. In Schlichting's formulation the jet emerges from a narrow slit in a flat barrier and penetrates into a semi-infinite region filled with fluid which would remain at rest if the slit were closed. Assuming the flow steady, Schlichting was able to demonstrate that the corresponding solution to the Prandtl equations may be written in an explicit analytic form. Here our concern will be with unsteady flow that is initiated when the slit is opened and the jet starts penetrating into the stagnant fluid. To study this process we begin with the numerical solution of the unsteady boundary-layer equations. Since discontinuities were expected, the equations were written in conservative form before finite differencing. The solution shows that the jet has a well-established front representing a discontinuity in the velocity field, similar to the shock waves that form in supersonic gas flows.Then, in order to reveal the ‘internal structure’ of the shock we turn to the analysis of the flow in a small region surrounding the discontinuity. WithRedenoting the Reynolds number, the size of the inner region is estimated as an orderRe−1/2quantity in both longitudinal and lateral directions. We found that the fluid motion in this region is predominantly inviscid and may be treated as quasi-steady if considered in the coordinate frame moving with the jet front. These simplifications allow a simple formula for the front speed to be deduced, which proved to be in close agreement with experimental observation of Turner (J. Fluid Mech.vol. 13 (1962), p. 356).