Eddy thermal diffusivity model and mean temperature profiles in turbulent vertical convection

In this paper, we propose a space-dependent eddy thermal diffusivity model for turbulent vertical natural convection in a fluid between two infinite vertical walls at different temperatures. Using this model, we derive analytical results for the mean temperature profile. Our results reveal that mean temperature profiles for different Rayleigh and Prandtl numbers are described by two universal scaling functions in the inner region next to the walls and the outer region near the centerline between the two walls and the characteristic temperature scales in the inner and outer regions are expressed in terms of the two parameters of the model which determine the characteristic velocities for heat transfer in the two regions. We show that these results are in good agreement with direct numerical simulation data.

💡 Research Summary

The paper addresses turbulent vertical natural convection between two infinite vertical walls held at different temperatures. Recognizing that existing models either assume a constant eddy thermal diffusivity or rely on empirical scaling that fails across a broad range of Prandtl (Pr) and Rayleigh (Ra) numbers, the authors propose a spatially varying eddy diffusivity model, K(x), that satisfies the physical boundary conditions at the walls and the symmetry about the channel centreline.

The governing equations under the Oberbeck‑Boussinesq approximation are reduced to mean momentum and temperature equations. Closure is achieved by defining the turbulent heat flux as u′T′ = −K(x) dT/dx. K(x)/ν is modeled in three distinct zones: (i) an inner region adjacent to each wall where K/ν = A y³ (y = x/H), ensuring K and its first two derivatives vanish at the wall; (ii) an intermediate linear region where K/ν = c₁ + c₂ y, providing continuous matching between the inner and outer zones; (iii) an outer region near the centreline where K/ν = Cₘ