Review of Jackson Herring’s Early Work on Thermal Convection

Abstract

Jack Herring had three mid-1960s numerical papers on Rayleigh-Bénard thermal convection that might seem primitive by today’s standards, but already encapsulated many of the questions that are still being asked. All of them use severely truncated versions of the incompressible Navier–Stokes–Boussinesq equations with only one, or just a few, horizontal Fourier modes. In the first two papers, 1963 and 1964, the presented results used only one Fourier mode α and three variables. The single mode’s variables are its vertical velocity profile wα(z,t), its temperature profile θα(z,t) and the horizontally uniform vertical profile of the background temperature ψ(z,t). All of the second- and third-order terms are ignored except the convective heat flux wθ¯. The objective was to find asymptotic steady-state solutions. Each paper found evidence for the one-third Nusselt versus Rayleigh scaling of Nu∼Ra1/3, originally derived from Malkus’ maximum flux principle. The 1963 paper uses free-slip upper and lower boundaries, with magnitudes of Nu that are a factor of three larger than the experiments. In the 1964 paper, by introducing no-slip/rigid boundary conditions, the magnitude of Nu dropped to within 20% of the experimental values. Both Nu(Ra) relations are in good agreement with circa-1990 direct numerical simulations (DNS). This dependence upon the boundary condition at the walls suggests that to obtain physically realistic scaling, no-slip boundary conditions are necessary. The third paper is discussed only in terms of what it might have been aiming to accomplish and its relation to the earlier free-slip results.