A dynamical model for turbulence .4. Buoyancy-driven flows

We apply a recent model of turbulence to turbulent convection at high Rayleigh number Ra and compare the results with new laboratory and DNS data. Mie derive a closed set of equations for the total turbulent kinetic energy, turbulent kinetic energy in the,z direction, temperature variance, and convective flux. The equations are coupled, time dependent, and nonlocal. We solve the equations both analytically and numerically. In the first case, wie neglect diffusion and derive the relation Nu=Nu(sigma,Ra), where Nu is the Nusselt number and sigma is the Prandtl number. For sigma much greater than 1, Nu becomes independent of sigma; for sigma much less than 1, Nu is proportional to sigma(1/3); for 0.025 (mercury)less than or equal to sigma less than or equal to 0.7 (helium), Nu is proportional to,sigma(2/7). The numerical solution (with diffusion) yields (a) Nusselt number Nu, <theta(2) >(w), <theta(2) >(c) (temperature variance near the wall and at the center). lambda(T) (thermal boundary layer thickness). and Pe (Peclet number) versus Ra, (b) the z profile of mean temperature T, <theta(2) >, horizontal, and vertical Peclet numbers, (c) spectra versus k(h) (horizontal wave number) of total kinetic energy, vertical kinetic energy. temperature variance, and temperature flux; (d) dependence of the Nu vs Ra relation on the Prandtl number. For large aspect ratios, the agreement with both laboratory and DNS data is satisfactory. The model contains no free parameters. (C) 1997 American Institute of Physics.