We report the results of high resolution direct numerical simulations of two-dimensional RayleighBénard convection for Rayleigh numbers up to Ra = 10 10 in order to study the influence of temperature boundary conditions on turbulent heat transport. Specifically, we considered the extreme cases of fixed heat flux (where the top and bottom boundaries are poor thermal conductors) and fixed temperature (perfectly conducting boundaries). Both cases display identical heat transport at high Rayleigh numbers fitting a power law Nu ≈ 0.138 × Ra .285 with a scaling exponent indistinguishable from 2/7 = .2857 . . . above Ra = 10 7 . The overall flow dynamics for both scenarios, in particular the time averaged temperature profiles, are also indistinguishable at the highest Rayleigh numbers. The findings are compared and contrasted with results of recent three-dimensional simulations.

We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh-Bénard convection, focusing on its ability to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers (P r) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number (Ra) sufficiently, and we explore the resulting convection for Ra up to 10 10 . When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as Ra → ∞. The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with Ra. When the large-scale shear is present with P r 2, the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with Ra for P r = 1. When the shear is present with P r 3, the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of Ra, but the growth rates are slower than any previously reported for Rayleigh-Bénard convection without large-scale shear. We find the Nusselt numbers grow proportionally to Ra 0.077 when P r = 3 and to Ra 0.19 when P r = 10. Analogies with tokamak plasmas are described.

The Darcy-Boussinesq equations at infinite Darcy-Prandtl number are used to study convection and heat transport in a basic model of porous-medium convection over a broad range of Rayleigh number Ra. High-resolution direct numerical simulations are performed to explore the modes of convection and measure the heat transport, i.e. the Nusselt number Nu, from onset at Ra = 4π 2 up to Ra = 10 4. Over an intermediate range of increasing Rayleigh numbers, the simulations display the 'classical' heat transport Nu ∼ Ra scaling. As the Rayleigh number is increased beyond Ra = 1255, we observe a sharp crossover to a form fitted by Nu ≈ 0.0174 × Ra 0.9 over nearly a decade up to the highest Ra. New rigorous upper bounds on the high-Rayleighnumber heat transport are derived, quantitatively improving the most recent available results. The upper bounds are of the classical scaling form with an explicit prefactor: Nu 6 0.0297 × Ra. The bounds are compared directly to the results of the simulations. We also report various dynamical transitions for intermediate values of Ra, including hysteretic effects observed in the simulations as the Rayleigh number is decreased from 1255 back down to onset.

No abstract

No abstract