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

TL;DR

This paper provides a theoretical framework linking heat flux and wall shear stress in large aspect-ratio turbulent vertical convection, revealing their dependence on Rayleigh and Prandtl numbers with specific scaling laws.

Contribution

It introduces new relationships between heat flux and wall shear stress in turbulent vertical convection, incorporating Prandtl number effects and high-Rayleigh number asymptotics.

Findings

Derived formulas relating Nu and Re_τ to Ra and Pr.

Identified Prandtl number regimes with different scaling exponents.

Provided asymptotic behaviors for high Ra.

Abstract

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 $\text{Nu} \approx C \text{Pr}^{\varepsilon} \text{Re}_\tau$ with $\varepsilon=1/3$ for $\text{Pr} \gg1$ and $\varepsilon=1$ 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 $\text{Nu} \approx [C^2f(\text{Pr})]^{1/3} \text{Pr}^{-(1-2\varepsilon)/3} \text{Ra}^{1/3}$ and $\text{Re}_\tau\approx[f(\text{Pr})/C]^{1/3}\text{Pr}^{-(1+\varepsilon)/3}\text{Ra}^{1/3}$ for high $\text{Ra}$.