Two-Point Closure Study of Covariance Budgets for Turbulent Rayleigh-Benard Convection

Statistical properties of turbulent Rayleigh-Benard convection in a laterally-periodic domain between free -slip boundaries are investigated in the parameter range Pr(, )< O(1), R (TURN) (2-4)R(,c), and aspect ratio (eta) (TURN) 4(eta)(,c). Ensemble-mean dynamics of this system are studied by means of a non-Markovian two-point closure obtained from the Direct Interaction Approximation by application of a two- time perturbation formalism and non-equilibrium fluctuation-dissipation relation. The computations are economized by use of Pade approximation methods to estimate the modal response function and triad relaxation time. Comparisons at the level of second- and third -order moments are made between the closure predictions and averages obtained from Direct Spectral Simulation (DSS) solutions of the full Navier-Stokes-Boussinesq equations. The final-state convective heat flux and volume-integrated kinetic energy and thermal variance are predicted to O(5%) accuracy with the closure model, which contains no adjustable constants or functions. At Rayleigh numbers near the convective threshold (<2R(,c)),(, )characteristic correlation times for the vertical velocity and temperature fields grow to(, )>O(10) vertical thermal diffusion times. Evidence is given that the two-time perturbation formalism furnishes an efficient iteration scheme for estimating statistically -steady dynamics in this parameter regime. Vertical profiles are constructed of the final -state covariance budgets for R = 2R(,c); these agree at the 5-10% level with budgets computed from horizontal averages of the DSS solutions. In the kinetic energy budget, the term arising from correlation between buoyancy-induced pressure and vertical velocity dominates the total transport term at all depths in the convecting layer. In the other second-moment budgets, triple correlations arising from the eddy flux of thermal variance and vertical heat flux play essential roles at nearly all levels.