Investigation of turbulent convection under a rotational constraint - Chan, Shu-Kwan (Journal of fluid mechanics, 1974)
Turbulent convection for a rotating layer of fluid heated from below is studied in this paper. The boundaries of the fluid layer are taken to be free. The underlying principle, used is the Malkus hypothesis that the flow tends to transport the maximum amount of heat possible, subject to certain constraints. By taking the Prandtl number to be infinite, a linear differential constraint and an integral constraint are used. The variational problem that follows then depends on two dimensionless parameters, the Taylor number T and the Rayleigh number R. Asymptotic analysis for the turbulent regime shows that the flow arranges itself so as to tend to offset the stabilizing effect of the rotational constraint, at least in so far as the heat flux is concerned. The dimensionless heat flux, or the Nusselt number, has in general different dependence on T and R, depending on the particular region in the parameter space. For T [les ] O(R), the flow is essentially non-rotating. For O(R) [les ]T [les ] O(R4/3), the flow will always have finitely many horizontal wavenumbers, though the total number of modes increases as T increases in this region. For O(R4/3) [les ] T [les ] O (R3/2), the Nusselt number has a functional dependence proportional to R3/T2, having essentially infinitely many horizontal modes as both R and T increase indefinitely in this region. The last expression is particularly interesting, as it agrees qualitatively with results in finite-amplitude laminar convection. It is also linearly dependent on the layer thickness, as one might expect from dimensional argument. It is suggested that, in the context of the maximum principle, the result in this region of the parameter space may be applicable as well to the same fluid layer with rigid boundaries through the existence of an Ekman layer that is thinner than the thermal layer.