Coherent heat transport in two-dimensional penetrative Rayleigh-Benard convection

摘要

This paper investigates the steady coherent solutions, bifurcated from the linear stability of a stationary flow, in two-dimensional (2-D) penetrative convection. The results show that the thickness of the upper stably stratified layer, which is measured by theta(M) (the dimensionless temperature at which the density is maximal, with 0 <= theta(M) <= 1), plays an important role in the linear and nonlinear dynamics. First, we investigate steady solutions of fixed aspect ratio L = 2 pi/alpha(c) (where alpha(c) is the critical wavenumber). The results show that the instability is supercritical when theta(M) < 0.4 and is subcritical when theta(M) > 0.4. When theta(M) > 0.4, the results show that the type of solution depends on the Prandtl number (Pr). For instance, when Pr less than or similar to 2.4 at theta(M) = 0.5, the solution in one type of pair of convection cells does not exist, as the Rayleigh number Ra exceeds a critical value due to a saddle-node bifurcation. When Pr > 2.4, steady solutions can be found up to Ra = 108 for all theta(M), which exhibit the scaling of heat transfer (characterized by the Nusselt number Nu): Nu similar to Ra-1/4. Then, the optimal 2-D steady solutions are tracked up to Ra = 10(9) by varying the aspect ratio L, which shows that heat transfer roughly follows the Nu similar to Ra-gamma (gamma approximate to 1/3) scaling in the regime of 10(7) < Ra < 10(9). It is interesting that the optimal temperature field has an arm-like horizontal structure when Pr < 10, while it has no significant horizontal structures when Pr > 10. Thus, the mean temperature in the mixing region is higher at large Pr. The steady solutions show that Nu similar to Pr-1/12 for theta(M) = 0 in a certain range of Pr by fixing the Rayleigh numbers, e.g. 1 < Pr < 10 for 2-D optimal steady solutions at Ra = 10(8) and 2 < Pr < 30 for 2-D steady solutions of fixed aspect ratio at Ra = 10(7). But when the Prandtl number is large or the upper stably stratified layer is thick, both the steady solutions of fixed aspect ratio and the 2-D optimal steady solutions are very weakly dependent on Pr.