Dynamics of line plumes on horizontal surfaces in turbulent convection

Abstract

We study the dynamics of line plumes on the bottom plate in turbulent convection over six decades of Rayleigh number $(10^5<Ra_w<10^{11})$ and two decades of Prandtl number $(0.7<Pr<600)$. We qualitatively identify the main dynamics as motion along the plumes, lateral merging of the plumes and initiation of the plumes; various other minor types of motion also occur along with these main dynamics. The mean velocity along the length of the plumes scales as the large scale flow velocity, with the fraction of the length of the plumes affected by shear increasing with $Ra_w$ as $L_{ps}/L_p\sim Ra_w^{0.04} Pr^{-0.1}$. In agreement with \cite{how}, the mean time of initiation of the plumes $\bar{t^*}$, scales as the diffusive time scale near the plate, $Z_w^2/\alpha$, where $Z_w$ is the appropriate length scale near the plate. The fraction of the length of the plumes that undergoes merging decreases with increase in $Ra_w$ as, $L_{pm}/L_p \sim Ra_w^{-0.04} Pr^{0.1}$. The plumes merge with a constant velocity during their merging cycle, however, the values of these constant velocities depend on the location and the time of measurement. The merging velocities at all $Ra_w$ and $Pr$ have a common lognormal distribution, but their mean and variance increased with increasing $Ra_w$ and decreasing $Pr$. $\bar{V_m}$, which are an order lower than the large scale velocities, scale as the average entrainment velocity at the sides of the plumes. This implies that $\bar{V_m}$ scales as the velocity scale near the plate $\nu/Z_w$. $Re_H$, the Reynolds number interms of $\bar{V_m}$ and the layer height $H$ scales as $Ra^{1/3}$, in the same way as the Nusselt number $Nu$; therefore $Re_{H}\sim Nu$. These relations imply that $Re_w=\bar{V_m}Z_w/\nu$ the Reynolds number near the plate is an invariant for a given fluid in turbulent convection.