Heat Flux and Wall Shear Stress in Large Aspect-Ratio Turbulent Vertical Convection

We present a theoretical analysis of large aspect-ratio turbulent vertical convection that yields two relationships between heat flux and wall shear stress, measured respectively by the Nusselt number ($\text{Nu}$) and shear Reynolds number (${\text{Re}}_{\tau }$), in terms of the Rayleigh ($\text{Ra}$) and Prandtl numbers ($\text{Pr}$): ${\text{Re}}_{\tau }^2\text{Nu}=f(\text{Pr}){\text{Pr}}^{-1}\text{Ra}$ in the high-$\text{Ra}$ limit and $\text{Nu}\approx C{\text{Pr}}^{\epsilon }{\text{Re}}_{\tau }$ with $\epsilon =1\mathrm{/}3$ for $\text{Pr}\gg 1$ and $\epsilon =1$ for $\text{Pr}\ll 1$, where $f(\text{Pr})$ is not a power law of $\text{Pr}$ and $C$ is a constant. These relationships imply $\text{Nu}\approx [C^{2}f(\text{Pr}){]}^{1\mathrm{/}3}{\text{Pr}}^{-(1-2\epsilon )\mathrm{/}3}{\text{Ra}}^{1\mathrm{/}3}$ and ${\text{Re}}_{\tau }\approx [f(\text{Pr})\mathrm{/}C{]}^{1\mathrm{/}3}{\text{Pr}}^{-(1+\epsilon )\mathrm{/}3}{\text{Ra}}^{1\mathrm{/}3}$ for high $\text{Ra}$.