Was Loitsyansky correct? A review of the arguments

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Abstract
Loitsyansky's integral, I, is important because it controls the rate of decay of kinetic energy in freely-evolving, isotropic turbulence. Traditionally it was assumed that I is conserved in decaying turbulence and this leads to Kolmogorov's decay law, u 2 ∼ t −10/7. However, the modern consensus is that I is not conserved, which is a little surprising since Kolmogorov's law is reasonably in line with the experimental data. This discrepancy led Davidson (2000 J. Fluid Mech. submitted) to reassess the entire problem. He concluded that, for certain initial conditions, which are probably typical of wind tunnel turbulence, freely evolving turbulence reaches an asymptotic state in which the variation of I is negligible, a conclusion which is at odds with the predictions of certain closure models. In this review 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 survey of the experimental and numerical data shows that there is little evidence 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 ideas of Loitsyansky and Kolmogorov to MHD turbulence. We note that there exists 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.

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