On the variational method of closure in the theory of turbulence

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Abstract
The variational approach to the closure problem of turbulence theory with particular attention to the perturbation–variation method of Qian [Phys. Fluids 26, 2098 (1983)] is studied. It is shown that although the method is based on a clear physical idea, it is not self-consistent. The procedure to obtain the equation for the dynamic damping coefficient does contain arbitrariness, which leads to the dependence of this equation on the choice of variables. This ambiguity is illustrated by numerical evaluations of the Kolmogorov constant in two-dimensional and three-dimensional cases. The equation for the dynamic damping coefficient, which is invariant, with respect to the change of variables, is obtained and analyzed. The principal inevitability of arbitrariness in closure methods is discussed.