Velocity and thermal boundary layer equations for turbulent Rayleigh-Bénard convection

In turbulent Rayleigh-Bénard convection, the boundary layers are nonsteady with fluctuations, the time-averaged large-scale circulating velocity vanishes far away from the top and bottom plates, and the motion arises from buoyancy. In this paper, we derive the full set of boundary layer equations for both the temperature and velocity fields from the Boussinesq equations for a quasi-two-dimensional flow above a heated plate, taking into account all the above effects. By solving these boundary layer equations, both the time-averaged temperature and velocity boundary layer profiles are obtained.

Figure 1

Values of $B$ obtained in the DNS [8] (circles) and [26] (crosses). The data points for the same $\text{Pr}$, which ranges from 0.005 to 300, are connected by a solid line.

Figure 2

Comparison of the DNS temperature profile (circles), averaged in time along the axis of a cylindrical cell, at $\text{Pr}=0.1$ and $\text{Ra}={10}^8$ with the theoretical result $\theta \left(\zeta \right)$ (solid line). We show also the results from the PBP theory for $\text{Pr}=0.1$ (dashed lines). Here $\zeta \propto \xi$ is chosen such that ${\theta }_{\zeta }\left(0\right)=1$.

Figure 3

Comparison (similar to Fig. 2) of the DNS profile of the velocity component in the LSC direction near the hot plate (circles), averaged in time and along the axis of the cylindrical cell and normalized by its maximum value, with the theoretical result ${\psi }_{\zeta }\left(\zeta \right)/{\psi }_{\zeta }\left({\zeta }_{\text{m}}\right)$ (solid line). We show also the results from the PBP theory for $\text{Pr}=0.1$ (dashed lines). Here $\zeta \propto \xi$ is chosen such that ${\theta }_{\zeta }\left(0\right)=1$.