Renormalization group analysis for thermal turbulent transport

In this study, we continue with our previous renormalization group analysis of incompressible turbulence, aiming at determination of various thermal transport properties. In particular, the temperature field T is considered a passive scalar. The quasinormal approximation is assumed for the statistical correlation between the velocity and temperature fields. A differential argument leads to derivation of the turbulent Prandtl number Pr(t) as a function of the turbulent Peclet Pe(t) number, which in turn depends on the turbulent eddy viscosity nu(t). The functional relationship between Pr(t) and Pe(t) is comparable to that of Yakhot et al. [Int. J. Heat Mass Transf. 30, 15 (1987)] and is in close consistency with direct-numerical-simulation results as well as measured data from experiments. The study proceeds further with limiting the operation of renormalization group analysis, yielding an inhomogeneous ordinary differential equation for an invariant thermal eddy diffusivity sigma. Simplicity of the equation renders itself a closed-form solution of sigma as a function of the wave number k, which, when combined with a modified Batchelor's energy spectrum for the passive temperature T, facilitates determination of the Batchelor constant C(B) and a parallel Smagorinsky model and the model constant C(P) for thermal turbulent energy transport.