Renormalization group analysis for thermal turbulent transport

Turbulent magnetic Prandtl number in kinematic magnetohydrodynamic turbulence: two-loop approximation

The turbulent magnetic Prandtl number in the framework of the kinematic magnetohydrodynamic (MHD) turbulence, where the magnetic field behaves as a passive vector field advected by the stochastic Navier-Stokes equation, is calculated by the field theoretic renormalization group technique in the two-loop approximation. It is shown that the two-loop corrections to the turbulent magnetic Prandtl number in the kinematic MHD turbulence are less than 2% of its leading order value (the one-loop value) and, at the same time, the two-loop turbulent magnetic Prandtl number is the same as the two-loop turbulent Prandtl number obtained in the corresponding model of a passively advected scalar field. The dependence of the turbulent magnetic Prandtl number on the spatial dimension d is investigated and the source of the smallness of the two-loop corrections for spatial dimension d=3 is identified and analyzed.

Heisenberg approximation in passive scalar turbulence

We use Heisenberg's approximation to derive analytic expressions for eddy viscosity and eddy diffusivity from the transfer integrals of energy and mean-square scalar arising from the Navier-Stokes and passive scalar dynamics. In the same scheme, we evaluate the flux integrals for the transports of energy and mean-square scalar. These procedures allow for the evaluation of relevant amplitude ratios, from which we calculate the universal numbers, namely, Batchelor constant B, Kolmogorov constant C, and turbulent Prandtl number σ, under two different schemes (with and without ε expansion). Our results are comparable with existing theoretical, numerical, and experimental values. As a byproduct, we obtain a relation between C, B, and σ, namely, B=σ C. To compare our results with the experimental values, we calculate Batchelor constant in one dimension (B'). Within the same framework, we also see that with increasing values of space dimension d, the Prandtl number σ increases and approaches unity, while the Kolmogorov constant C and Batchelor constant B approach very close to each other. For large space dimensions, we find the asymptotic B=B(0)d(1/3), and evaluate B(0).

Solvable model in renormalization group analysis for effective eddy viscosity

This study presents a solvable model in renormalization group analysis for the effective eddy viscosity. It is found fruitful to take a simple hypothesis that large-scale eddies are statistically independent of those of smaller scales. A limiting operation of renormalization group analysis yields an inhomogeneous ordinary differential equation for the invariant effective eddy viscosity. The closed-form solution of the equation facilitates derivations of an expression of the Kolmogorov constant C(K) and of the Smagorinsky model for large-eddy simulation of turbulent flow. The Smagorinsky constant C(S) is proportional to C(3/4)(K). In particular, we shall illustrate that the value of C(K) ranges from 1.35 to 2.06, which is in close agreement with the generally accepted experimental values (1.2 approximately 2.2).