Scaling in Rayleigh-Benard convection

We consider the Nusselt-Rayleigh number problem of Rayleigh-Benard convection and make the hypothesis that the velocity and thermal boundary layer widths, delta(u) and delta(T), in the absence of a strong mean flow are controlled by the dissipation scales of the turbulence outside the boundary layers and, therefore, are of the order of the Kolmogorov and Batchelor scales, respectively. Under this assumption, we derive Nu similar to Ra-1/3 in the high Ra limit, independent of the Prandtl number, delta(T)/L similar to Ra-1/3 and delta(u)/L similar to Ra-Pr-1/3(1/2), where L is the height of the convection cell. The scaling relations are valid as long as the Prandtl number is not too far from unity. For Pr similar to 1, we make a more general ansatz, delta(u) similar to nu(alpha), where nu is the kinematic viscosity and assume that the dissipation scales as similar to u(3)/L, where u is a characteristic turbulent velocity. Under these assumptions we show that Nu similar to Ra-alpha/(3-alpha), implying that Nu similar to Ra-1/5 if delta(u) were scaling as in a Blasius boundary layer and Nu similar to Ra-1/2 (with some logarithmic correction) if it were scaling as in a standard turbulent shear boundary layer. It is argued that the boundary layers will retain the intermediate scaling alpha = 3/4 in the limit of high Ra.

Automated summary

In this study, we investigate the Nusselt-Rayleigh number problem of Rayleigh-Benard convection and propose that the velocity and thermal boundary layer widths, delta(u) and delta(T), are controlled by the dissipation scales of turbulence outside the boundary layers. We derive the scaling relations for Nu, delta(T)/L, and delta(u)/L in the high Ra limit and find that they are independent of the Prandtl number. For Pr close to 1, we make a more general assumption and show that Nu scales with Ra-alpha/(3-alpha), where alpha is the intermediate scaling coefficient. We argue that the boundary layers will retain the intermediate scaling alpha = 3/4 in the limit of high Ra.