Recursive renormalization-group calculation for the eddy viscosity and thermal eddy diffusivity of incompressible turbulence

Abstract

The recursive renormalization-group method with an asymptotically free conditional average is conducted for calculating the effective eddy viscosity and thermal eddy diffusivity of incompressible turbulence. For different values of spatial rescaling factor Λ, the dependence of renormalized eddy viscosity on wave number is examined at the zeroth- and the first-order truncations of expansion of subgrid-scale energy spectrum density in Taylor series, respectively. A strong cusp-like behavior is observed near the supergrid–subgrid cutoff for the first order. This is in agreement with the result of testing-field model. However, no cusp behavior is found for the zeroth order. Calculation of Kolmogorov constant C K indicates a special region where C K is insensitive to Λ. The first order shows a range much larger than that for the zeroth order, and gives C K =1.56±0.04 in the range of 0.4⩽ Λ⩽0.8, consistent with the generally accepted experimental values (1.2–2.2). Furthermore, the wave-number dependence of renormalized thermal eddy diffusivity is investigated, and no cusp behavior is observed around the supergrid–subgrid cutoff at the first-order truncation of expansion of subgrid-scale energy spectrum density, different from that of renormalized eddy viscosity. The Batchelor constant C B is also found insensitive to Λ in the range of 0.4⩽ Λ⩽0.8, and is estimated to be 1.15±0.05, in good agreement with the result of ε-renormalization group approach.