Flux scaling in Rayleigh B\'enard convection: a local boundary layer analysis

We study the effect of shear due to the large scale flow (LSF) on the heat flux in Rayleigh B'enard convection for a range of near-plate Rayleigh numbers $8\times 10^7 \leq Ra_w \leq 5\times 10^{14}$, by studying its effect on the local boundary layers (BLs) on either sides of the plumes, which are much thinner than the global shear BL created by the LSF velocity $V_F$. Considering these local BLs forced externally by the LSF, we obtain a fifth order algebraic equation for the local boundary layer thicknesses. Solving these equations numerically using $Re$ relations for aspect ratios $\Gamma=1$ and 0.5, we obtain the variation of the local BL thicknesses with the longitudinal distance for various $Ra_w$. We find that the average shear acting on the edges of these local BLs ($\overline{u|}_{z=\delta}$) increases as $\overline{u|}_{z=\delta} \sim Ra_w^{1/3}$ for $8\times 10^7\leq Ra_w \leq 10^{12}$ at $\Gamma=1$, and as $\overline{u|}_{z=\delta} \sim Ra_w^{0.38}$ for $1\times 10^{11}\leq Ra_w \leq 5\times 10^{14}$ at $\Gamma=0.5$. We then estimate the average local thermal BL thickness to find the global Nusselt number $Nu$.We find that $Nu\sim Ra_w^m$, where $m\approx 0.327$ for $8\times 10^7 \leq Ra_w \leq 1\times 10^{12}$ at $\Gamma=1$, and $m=0.33$ for $1\times10^{11}\leq Ra_w \leq 5\times10^{14}$ at $\Gamma=0.5$. Inspite of the increasing shear on these BLs with increasing $Ra_w$, we then surprisingly obtain the classical 1/3 scaling of flux since the shear forcing acting on those BLs remains sub-dominant compared to the NCBL velocities ($V_{bl}$) within these BLs, upto $Ra_w\leq 5\times10^{14}$.