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

Abstract

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. We 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, we neglect diffusion and derive the relation Nu=Nu(σ,Ra), where Nu is the Nusselt number and σ is the Prandtl number. For σ≫1, Nu becomes independent of σ; for σ≪1, Nu is proportional to σ1/3; for 0.025 (mercury)⩽σ⩽0.7 (helium), Nu is proportional to σ2/7. The numerical solution (with diffusion) yields (a) Nusselt number Nu, 〈θ2〉w, 〈θ2〉c (temperature variance near the wall and at the center), λT (thermal boundary layer thickness), and Pe (Peclet number) versus Ra, (b) the z profile of mean temperature T, 〈θ2〉, horizontal, and vertical Peclet numbers, (c) spectra versus kh (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.