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 ($\text{Pr}$): ${\text{Re}}_{\tau }^2\text{Nu}=f\left(\text{Pr}\right){\text{Pr}}^{-1}\text{Ra}$ in the high-$\text{Ra}$ limit and $\text{Nu}\approx C{\text{Pr}}^{\varepsilon }{\text{Re}}_{\tau }$ with $\varepsilon =1/3$ for $\text{Pr}\gg 1$ and $\varepsilon =1$ for $\text{Pr}\ll 1$, where $f\left(\text{Pr}\right)$ is not a power law of $\text{Pr}$ and $C$ is a constant. These relationships imply $\text{Nu}\approx {\left[C^{2}f\left(\text{Pr}\right)\right]}^{1/3}{\text{Pr}}^{-(1-2\varepsilon )/3}{\text{Ra}}^{1/3}$ and ${\text{Re}}_{\tau }\approx {[f\left(\text{Pr}\right)/C]}^{1/3}{\text{Pr}}^{-(1+\varepsilon )/3}{\text{Ra}}^{1/3}$ for high $\text{Ra}$.

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• Turbulence
• Turbulent convection