A Contribution to the Theory of Turbulence

Abstract

S of the problem of Wind-Tunnel Turbulence have recently been published by Taylor, von Karman, and Dryden. These studies are concerned with the decay of homogeneous isotropic turbulence. The scale and intensity of the turbulence is assumed to be known at a given instant throughout the fluid and the decay with respect to time is asked for. In this note the authors investigate the turbulence produced at a constant rate in the boundary, and decaying in a direction normal to the surface of generation. This problem, so far as the authors are aware, has not been treated before. The problem becomes especially simple if one considers a fluid in a semiinfinite space bounded by a plane along which the turbulence is produced. The mean motion is assumed to be constant throughout and parallel to the plane. This investigation is not only of theoretical interest but should be useful in the study of the behavior of turbulent boundary layers, and might be of help in the study of the influence of surface roughness. The stated problem can be treated by means of the equation of the statistical theory of turbulence recently proposed by von Karman. It furthermore might serve as the basis for an experiment in which the consequences of the theory can be tested. The considered turbulent motion has statistically an isotropic character due to the absence of mean shear stresses, i.e., the correlations of the velocity components at a certain point in the space with respect to a set of orthogonal axes are independent of the rotation of this system. The turbulence, however, is not homogeneous, since the correlations are functions of the distance from the boundary. Consider a space with the orthogonal axes xiy x2} x3, in which the fluid is bounded by a (xix3) plane and has a, constant mean velocity U in the X direction. The