The large-scale structure of homogenous turbulence

Abstract The starting-point for this paper lies in some results obtained by Proudman which becomes of order r-8. This semi-irrotational property of the asymptotic form, which arises from the fact that pressure forces act only indirectly on the vorticity, allows the asymptotic form of to be determined explicitly. By methods that are new in turbulence theory and that involve a good deal of tensor manipulation, it is found that U'} = mnV2 “ V2 ~ drjdr) when r is large, where the coefficient Cpqmn is related to the fourth integral moment of in a known way. There is a corresponding expression for the leading term in the spectrum tensor at small wave-numbers, which is now not analytic. The spectrum function giving the distribution of energy with respect to wave-number magnitude k is in general of the form E (k) = a 4 + 0 (⁵ In k) 9 when k is small. Corresponding expressions are found for the asymptotic forms of the various terms occurring in the dynamical equation giving the rate of change of Both the inertia and pressure terms in this equation are found to be of order r~5, and as a consequence the coefficient Cpqmn (and likewise C) is not a dynamical invariant. It is shown that the integral J*2qzqrmrwdr (which exists, despite the apparent logarithmic divergence at large r) is uniquely related to CpqmnJ and it too varies during the decay, contrary to past belief. The final period of decay is examined afresh, and it is found that the energy then varies as which is also the result found experimentally; the power (— f) arises from the fact that the spectrum tensor is of order k2 when the wave-number k is small, and is unaffected by the non-analytic character of that leading term. The covariance tqw] does not have a simple form in the final period; it is determined by the parameter on the previous history of the decay in a complicated way. It is rather puzzling that measurements indicate that the longitudinal correlation coefficient has a simple Gaussian form in the final period of decay, as would be the case for an analytic spectrum. We suggest this observation may be true only for turbulence of very low initial Reynolds number, for which the non-analytic part of the spectrum tensor has little time to develop. Finally, the results are specialized to correspond to turbulence which is completely isotropic. For reasons related to the symmetry, is now no larger than 0 (r-6) when r is large (we have been unable to determine the exact order), and the leading term in the spectrum tensor, of order k2, is analytic. As suggested by Proudman _ -j-u2 r*f (r) dr = (w2) 1 lim rAk{r)