On the two-dimensional steady flow of a viscous fluid behind a solid body.―I

Abstract

1.1. The purpose of this paper is to exhibit, for reasons given below, calculations of the velocity distribution some distance downstream behind any symmetrical obstacle in a stream of viscous fluid, but particularly behind an infinitely thin plate parallel to the stream, the motion being two-dimensional. For a slightly viscous fluid, Blasius worked out the velocity distribution in the boundary layer from the front to the downstream end of the plate; and in a previous paper, I calculated the velocity in the wake for a distance varying from 0.3645 to 0.5 of the length of the plate from its downstream end (according to distance from its plane). In these calculations the fluid was supposed unlimited, and the undisturbed velocity in front of the plate was taken as constant. The viscosity being assumed small, the work was carried out on the basis of Pranstl's boundary layer theory, with zero pressure gradient in the direction of the stream. The velocity is then constant everywhere expect within a thin layer near the plate, and in a wake which must gradually broaden out downstream. (The broadening of the wake just behind the plate is so gradual that it could not be shown by calculations of the accuracy obtained in I). Pressure variations in a direction at right angles to the stream are negligible, and so is the velocity in that direction. Later, Tollmien attcked the problem from the other end, and found a first asymptotic approximation for the velocity distribution in the wake at a considerable distance downstream. He simplified the Prandtl equations by assuming that the departure from the constant velocity, U 0 , of the main stream is small, and neglecting terms quadratic in this departure. In other words, he applied the notion of the Oseen approximation to the Prandtl equations. His result for the velocity is U = U 0 {1 - a X -½ exp (-U 0 Y 2 /4νX)}.