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× 10⁷ ≤ Raw ≤ 5× 10¹⁴, 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 VF. 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 Γ=1 and 0.5, we obtain the variation of the local BL thicknesses with the longitudinal distance for various Raw. We find that the average shear acting on the edges of these local BLs (u|̄z=δ) increases as u|̄z=δ ~ Raw1/3 for 8× 10⁷≤ Raw ≤ 10¹² at Γ=1, and as u|̄z=δ ~ Raw0.38 for 1× 10¹¹≤ Raw ≤ 5× 10¹⁴ at Γ=0.5. We then estimate the average local thermal BL thickness to find the global Nusselt number $Nu$.We find that Nu~ Rawᵐ, where m≈ 0.327 for 8× 10⁷ ≤ Raw ≤ 1× 10¹² at Γ=1, and $m=0.33$ for 1×10¹¹≤ Raw ≤ 5×10¹⁴ at Γ=0.5. Inspite of the increasing shear on these BLs with increasing Raw, 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 (Vbl) within these BLs, upto Raw≤ 5×10¹⁴.