LAUSR

Abstract In Rayleigh Bénard convection, for a range of Prandtl numbers $4.69 \leqslant Pr \leqslant 5.88$ and Rayleigh numbers $5.52\times 10^5 \leqslant Ra \leqslant 1.21\times 10^9$, we study the effect of shear by the inherent large-scale flow (LSF) on the local boundary layers on the hot plate. The velocity distribution in a horizontal plane within the boundary layers at each $Ra$, at any instant, is (A) unimodal with a peak at approximately the natural convection boundary layer velocities $V_{bl}$; (B) bimodal with the first peak between $V_{bl}$ and $V_{L}$, the shear velocities created by the LSF close to the plate; or (C) unimodal with the peak at approximately $V_{L}$. Type A distributions occur more at lower $Ra$, while type C occur more at higher $Ra$, with type B occurring more at intermediate $Ra$. We show that the second peak of the bimodal type B distributions, and the peak of the unimodal type C distributions, scale as $V_{L}$ scales with $Ra$. We then show that the areas of such regions that have velocities of the order of $V_{L}$ increase exponentially with increase in $Ra$ and then saturate. The velocities in the remaining regions, which contribute to the first peak of the bimodal type B distributions and the single peak of type A distributions, are also affected by the shear. We show that the Reynolds number based on these velocities scale as $Re_{bs}$, the Reynolds number based on the boundary layer velocities forced externally by the shear due to the LSF, which we obtained as a perturbation solution of the scaling relations derived from integral boundary layer equations. For $Pr=1$ and aspect ratio $\varGamma =1$, $Re_{bs} \sim Ra^{0.375}$ for small shear, similar to the observed flux scaling in a possible ultimate regime. The velocity at the edge of the natural convection boundary layers was found to increase with $Ra$ as $Ra^{0.35}$; since $V_{bl}\sim Ra^{1/3}$, this suggests a possible shear domination of the boundary layers at high $Ra$. The effect of shear, however, decreases with increase in $Pr$ and with increase in $\varGamma$, and becomes negligible for $Pr\geqslant 100$ at $\varGamma =1$ or for $\varGamma \geqslant 20$ at $Pr=1$, causing $Re_{bs}\sim Ra_w^{1/3}$.