Renormalization group analysis of turbulence. I. Basic theory

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 spectrumC _K=1.617; turbulent Prandtl number for high-Reynolds-number heat transferP _t =0.7179; Batchelor constantBa=1.161; and skewness factor¯S ₃=0.4878. A differentialK- model is derived, which, in the high-Reynolds-number regions of the flow, gives the algebraic relationv=0.0837 K²/ , decay of isotropic turbulence asK=O(t ^–1.3307), and the von Karman constant=0.372. A differential transport model, based on differential relations betweenK, , and, is derived that is not divergent whenK 0 and is finite. This latter model is particularly useful near walls.