EXTENSION OF THE YAKHOT–ORSZAG RENORMALIZATION GROUP-BASED CLOSURE SCHEME TO ISOTROPIC AND ANISOTROPIC TWO-DIMENSIONAL TURBULENCE

Abstract

The renormalization group based closure developed by Yakhot and Orszag for the analysis of three-dimensional turbulence is extended to the two-dimensional case. The logarithmically corrected Ek ∝ ( log k/k1)–1/3 k–3 energy spectrum of the enstrophy inertial range and the k−5/3 spectrum of the energy range are derived self-consistently by the method. The values for the Kolmogorov constants C′ = 2.29 in the enstrophy range and C = 6.45 in the energy range are obtained without the need of adjustable parameters. These solutions are proven to be linearly stable against anisotropic perturbations and the decay of anisotropy at small scales given anisotropic large scales is shown to be ∝ ( In k/k1)−1/12 in the enstrophy range and ∝ k−1/6 in the energy range. The difficulty in observing a k−3 spectrum in two-dimensional turbulence has a counterpart in the renormalization group closure, in the fact that such a spectrum would require an enstrophy input from the stochastic force exceeding the contribution of the flux from larger scales. Application of the method to the problem of the passive scalar leads to Prandtl numbers equal to one and Batchelor constants equal to the Kolmogorov constant of the respective range. Calculation of the two-dimensional skewness s2 = <∂xvx (∂xq)2>/[<(∂xvx)2>1/2<(∂xq)2>] leads to the result s2 ≃ −.265 in the enstrophy range.