Renormalization group analysis of turbulence. I. Basic theory

Abstract

We develop the dynamic renormalization group (RNG) method for hydrodynamic turbulence. This procedure, which uses dynamic scaling and invariance together with iterated perturbation methods, allows us to evaluate transport coefficients and transport equations for the large-scale (slow) modes. The RNG theory, which does not include any experimentally adjustable parameters, gives the following numerical values for important constants of turbulent flows: Kolmogorov constant for the inertial-range spectrum CK=1.617; turbulent Prandtl number for high-Reynolds-number heat transfer Pt=0.7179; Batchelor constant Ba=1.161; and skewness factor -S3=0.4878. A differential K- {Mathematical expression} model is derived, which, in the high-Reynolds-number regions of the flow, gives the algebraic relation v=0.0837 K2/ {Mathematical expression}, decay of isotropic turbulence as K=O(t-1.3307), and the von Karman constant κ=0.372. A differential transport model, based on differential relations between K, {Mathematical expression}, and ν, is derived that is not divergent when K→ 0 and {Mathematical expression} is finite. This latter model is particularly useful near walls. © 1986 Plenum Publishing Corporation.