Turbulent Transport in Benard Convection

Stochastic evolution equations for turbulent Benard convection are derived from the Boussinesq equations by transforming the inertial forces into a sum of systematic linear transport terms and random nonlinear fluctuating forces by means of the projection operator method. Then the heat flux and the velocity fluxes of turbulent Benard convection are formulated in terms of the temperature gradient and the velocity gradient explicitly with turbulent transport coefficients. It is found that turbulence produces interference between the velocity flux of the vertical velocity component and the heat flux that is similar to the interference between the electric current and the heat flux in the thermoelectric phenomena of metals, so that the heat flux is generated not only by the temperature gradient but also by the velocity gradient. The large-scale flows of turbulent Benard convection are characterized by this interference effect. It is also shown that a simple scaling law holds for the Rayleigh number and the Prandtl number dependence of the turbulent transport coefficients in the hard turbulence region, as in the case of the scaling law of the Nusselt number discovered by Castaing et al. (1989).