Was Loitsyansky correct? A review of the arguments

Loitsyansky's integral, I, is important because it controls the rate of dec ay of kinetic energy in freely-evolving, isotropic turbulence. Traditionall y it was assumed that I is conserved in decaying turbulence and this leads to Kolmogorov's decay law, u(2) similar to t(-10/7). However, the modern co nsensus is that I is not conserved, which is a little surprising since Kolm ogorov's law is reasonably in line with the experimental data. This discrep ancy led Davidson (2000 J. Fluid Mech. submitted) to reassess the entire pr oblem. He concluded that, for certain initial conditions, which are probabl y typical of wind tunnel turbulence, freely evolving turbulence reaches an asymptotic state in which the variation of I is negligible, a conclusion wh ich is at odds with the predictions of certain closure models. In this revi ew we revisit this debate. We explain why the widespread belief in the time dependence of I owes much to a misinterpretation of Batchelor and Proudman 's original analysis (1956 Phil. Trans. R. Soc. A 248 369). Indeed, a surve y of the experimental and numerical data shows that there is little evidenc e for significant long-range pressure forces of the type which underpin the supposed variation of I. Interestingly, Batchelor and Proudman reached the same conclusion almost half a century ago. We conclude by extending the id eas of Loitsyansky and Kolmogorov to MHD turbulence. We note that there exi sts a Loitsyansky integral for MHD turbulence (Davidson 1997 J. Fluid Mech. 336 123) and show that this leads to energy decay laws which coincide with the experimental data.