Two-dimensional turbulence in the inverse cascade range.
Numerical and physical experiments on forced two-dimensional Navier-Stokes equations show that transverse velocity differences are described by "normal" Kolmogorov scaling <(deltav)(2n)> proportional r(2n/3) and obey Gaussian statistics. Since nontrivial scaling is a sign of the strong nonlinearity of the problem, these two results seem to contradict each other. A theory explaining these observations is presented in this paper. The derived self-consistent expression for the pressure gradient contributions leads to the conclusion that small-scale transverse velocity differences are governed by a linear Langevin-like equation, stirred by a nonlocal, universal, solution-dependent Gaussian random force. This explains the experimentally observed Gaussian statistics of transverse velocity differences and their Kolmogorov scaling. The solution for the PDF of longitudinal velocity differences is based on the numerical smallness of the energy flux in two-dimensional turbulence. The theory makes a few quantitative predictions that can be tested experimentally.
Yakhot
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Department of Aerospace and Mechanical Engineering, Boston University, Boston, Massachusetts 02215, USA.