The theory of decay process of incompressible isotropic turbulence

Abstract

One of the most serious difficulties in the theory of homogeneous turbulence is the indeterminacy of the equation for the velocity correlation function of any order, each involving the correlation of higher-by-one order. In the present paper this difficulty is resolved by treating the two dynamical equations for the second- and third-order velocity correlations, and by introducing the assumption of the zero fourth-order cumulant of the velocity field which yields a relationship between the fourth- and second-order velocity correlations. Actual calculation, however, is carried out in the wave-number space, and a pair of simultaneous equations for the energy spectrum function are derived in part I. Another difficulty of the subject arises from the present lack of knowledge about the initial state of turbulence. In part II, some probable initial conditions for the energy spectrum are examined, among which the initial spectrum of single-line type is chosen as the most suitable for the present problem and its dynamical consequences are fully discussed. The power-series solution for the initial spectrum as well as the energy decay law due to it are computed and compared with experimental data. It is found that the solution, in so far as the approximate expression calculated in the present paper is concerned, corresponds to the earlier initial period of decay. A solution which would be essentially in agreement with experiments is expected to be given by extending the present solution to the further developed stage of decay.