Turbulent natural convection in a differentially heated vertical channel: A theoretical study

To understand thermally driven wall-bounded turbulent flows, the model system of turbulent natural convection in a differentially heated vertical channel is often studied. One important question is how the heat flux depends on the control parameters of the flow. We have carried out a theoretical study in the large aspect-ratio limit, where the height and width of the vertical walls of the channel are much larger than their separation. In this limit, the mean flow quantities depend only on the coordinate along the direction normal to the vertical walls. Our theoretical analysis, which is based on the mean momentum balance and mean energy balance equations and makes the minimal closure approximations needed, yields the dependence of the heat flux and wall shear stress, as measured by the Nusselt number (Nu) and the shear Reynolds number (Re_τ) on the Rayleigh (Ra) and Prandtl (Pr) numbers:

Nu≈[C²f(Pr)]^1/3Pr^-(1-2ε)/3Ra^1/3 and Re_τ≈[f(Pr)/C]^1/3Pr^-(1+ε)/3Ra^1/3 in the high-Ra limit. Here, C is a constant, f(Pr) is a function of Pr that is not in the form of a power law and ε = 1/3 and 1, respectively for Pr ≫ 1 and Pr ≪ 1. We have tested our theoretical results using available direct numerical simulation (DNS) data for Pr≥1 and an excellent agreement is found. In contrast, the DNS data do not support the asymptotic laws of heat transfer and the mean maximum velocity obtained by a scaling approach in previous theoretical studies.