A SOLUTION OF THE NAVIER–STOKES EQUATIONS ILLUSTRATING THE RESPONSE OF A LAMINAR BOUNDARY LAYER TO A GIVEN CHANGE IN THE EXTERNAL STREAM VELOCITY

Abstract

Exact solutions of the Navier-Stokes equations are derived by a Laplace-transform technique for two-dimensional, incompressible flow past an infinite plane porous wall. It is assumed that the flow is independent of the distance parallel to the wall and that the velocity component normal to the wall is constant. A general formula is derived for the velocity distribution as a function of the given free-stream velocity and, from this, general formulae for the skin friction and displacement thickness are obtained. Among the particular cases considered are (i) flow in which the free-stream velocity is changed impulsively from one value to another, (ii) uniform acceleration of the main stream from a given value of the velocity, and (iii) flow in which the free-stream velocity consists of a mean value with a superimposed decaying oscillation. In the first case a secondary boundary layer of a Rayleigh impulsive flow type is created immediately after the impulsive increase in free-stream velocity and this secondary layer increases in thickness with time until the final steady state is reached. In the second case also, a secondary boundary layer is formed at the instant when the free-stream velocity begins to accelerate and ultimately the skin friction consists of a quasi-steady part and a part proportional to the product of the free-stream acceleration and a ‘virtual mass’ equal to the mass of fluid in the displacement area. In the final case the fluctuating skin friction is found to have a phase advance over the fluctuating free-stream velocity. For sufficiently large frequencies this phase advance is ¼π and the amplitude of the skin friction fluctuation is proportional to the square root of the frequency and to the amplitude of the free-stream velocity fluctuation.