On the decay of a normally distributed and homogenous turbulent velocity field

Abstract

This paper examines the dynamical behaviour of a field of homogeneous turbulence in which the joint-probability distribution of the fluctuating velocity components at three points is approximately normal. In principle, the analysis is formulated entirely in terms of the mean values u i u' j and u i u j u'' k , where the number of primes denotes the point at which the velocity components are taken. First, the kinematical properties of the three-point correlation are obtained by techniques similar to those used in the well-known theory of the two-point correlation. In the particular case of isotropic turbulence, the necessary extensions to the existing invariant theory lead to the result that the three-point correlation is completely defined by two scalar functions. Two independent dynamical relations between these correlations are then derived from the Navier-Stokes equation, and the remainder of the paper is based on this (determinate) system of equations. These remarks refer only to the principle of the calculations; in fact, most of the results are obtained in terms of the Fourier transforms of the correlations defined above. The first set of deductions from the governing equations refer to the decay of isotropic turbulence at large Reynolds numbers. In particular, the exact solution of the inviscid equations for the vorticity is obtained, and it is shown to be consistent with the predictions of Kolmogoroff’s theory of local similarity after a sufficiently long time of decay. The distribution of energy transfer between eddies of different sizes is also examined for a special form of the energy spectrum of turbulence, and the general features of this distribution appear to be satisfactory in the main energy-containing range of the spectrum. The remaining results are concerned with energy transfer in the large eddies. It is shown, beyond all reasonable doubt, that the magnitude of this energy transfer is such that the large eddies are not permanent during decay. Immediate consequences of this result are that Loitsiansky’s integral is not an invariant of the motion, and that the usual triple correlation function k (f) is proportional to r -4 for large values of r . These conclusions are inconsistent with the theory initiated by Loitsiansky, and developed by Lin and Batchelor. The cause of this inconsistency is attributed to the dynamical unlikelihood of the basic assumptions made by these earlier authors.