Stability analysis of sheared thermal boundary layers and its implication for modelling turbulent Rayleigh–Bénard convection

Abstract

Predicting the heat flux through a horizontal layer of fluid confined between a hot bottom plate and a cold top one has always spurred theoretical, numerical and experimental work on Rayleigh–Bénard convection. Customarily, the Nusselt number (the heat flux in non-dimensional form) has been modelled in the form of one or several power-laws of three parameters, the Rayleigh, Prandtl and Reynolds numbers. Quantifying the large-scale flow that spontaneously develops in a turbulent Rayleigh–Bénard cell, the Reynolds number, unlike the Rayleigh and Prandtl numbers, is not a control parameter strictly speaking and, depending on the model, is sought as another power-law or introduced as an external input. Whereas balancing the different transport mechanisms can predict the exponents in these power laws, experimental and numerical results are required to adjust the various prefactors. The early and simple model of Malkus (1954) and Howard (1966) assumed that the value of the Nusselt number could be directly deduced from the marginal stability of the two sheared thermal boundary layers along the upper and lower plates, interacting via the large-scale flow. Maintaining this simplicity, this work shows that in the classical regime of turbulent convection, considering the linear critical conditions of absolute (as opposed to convective) thermo-convective instabilities alleviates the flaws of the original model. Revisiting available Direct Numerical Simulations from which a Reynolds number can be unambiguously extracted, the present approach then yields the Nusselt number as a function of the Rayleigh and Prandtl numbers agreeing well with the numerical results.