ON THE STATISTICAL VORTICITY STRUCTURE THEORY OF HOMOGENEOUS ISOTROPIC TURBULENCE

Abstract

The present paper is a further development, after the Great Proletarian Cultural Revolu-tion, of our statistical point of view that the element of homogeneous isotropic turbulenceis a small axially symmetrical vortex from the solution of the Navier-Stokes equations ofmotion. After long years of observation and analysis in practice, we recognize that thestructure of vortices, as elements of turbulence, does not obey the simple law of similarityor self-preservation. The structure undergoes a kind of stretching phenomena in the processof the decay of vortices. Therefore, we first bring out a vortex scale—vortex Reynoldsnumber relation, and then introduce a pseudo-similarity or psoudo-self-preservation conceptinto the theory. We solve the case of small vortex Reynolds number flows. The zerothorder approximation of the solution can be expressed in terms of confluent hypergeometricfunctions. The law of the decay of turbulent energy and the expression for the microscaleof turbulence from the initial to the final period of decay together with the double velocitycorrelation thus calculated all agree very well with the experiments of Batchelor and Town-send. We have also computed the energy spectrum transfer function corresponding to thetriple velocity correlation. But in this range of small Reynolds number flows there arestill no experimental measurements available for its verification. In the Appendix by using the method of multi-Fourier transform, we have derived theFourier transforms of the triple and n-tiple velocity correlations and the simpler form ofthe energy spectrum transfer function corresponding to the triple velocity correlation.