We carried out high resolution direct numerical simulations of two-dimensional Rayleigh-Bénard convection for which the heat flux at the boundary is fixed. This models the scenario when the boundary plates are poor conductors compared to the fluid. The nondimensional control parameter in the Boussinesq equations is the forcing scale imposed by the heat flux . The Nusselt number —the enhancement of the heat flux beyond pure conduction—is determined by measuring the Rayleigh number, Ra, from long time averages of the temperature drop across the convection cell.
Simulations of a fluid with Prandtl number Pr=1 in an aspect ratio 2 cell were performed using a Fourier-Chebyshev spectral method. Figure 1 shows snapshots of the temperature over a range of extending up to , exhibiting dynamics from steady plumes in panel (a) to fully developed turbulence in panel (c).
(Color) Snapshots of the temperature: (a) ; (b) ; (c) ; (d) close-up of (c). (Enhanced online.) [URL: http://dx.doi.org/10.1063/1.2786003.1]
(Color) Snapshots of the temperature: (a) ; (b) ; (c) ; (d) close-up of (c). (Enhanced online.) [URL: http://dx.doi.org/10.1063/1.2786003.1]
One way that the fixed-flux Rayleigh-Bénard problem differs qualitatively from the fixed-temperature problem is that the bifurcation to convection occurs at the wavelength given by horizontal—rather than the vertical—scale of the domain. It is seen that the largest scale persists and organizes the flow even in the turbulent regime.
Figure 2 shows the data along with those produced by a fourth-order finite difference method1 for cells of aspect ratio 2 (FD-2) and 4 (FD-4). A least-squares fit of the eight highest data points to the power law yields an exponent of . This is well below the best known rigorous analytic upper bound with ,2 but it is close to experimental measurements of convection in liquid mercury with a conjectured exponent 3