Heat Rises: Energy transport bounds for Rayleigh-B\'enard convection in an infinite Prandtl number fluid

Charles Doering
University of Michigan
Departments of Mathematics and Physics

Rayleigh-B\'enard convection is the buoyancy-driven flow resulting from heating a fluid from below and cooling it from above sufficiently to destabilize the pure conduction state where the fluid is at rest. The Prandtl number $Pr$ of a fluid is the ratio of its kinematic viscosity, i.e., its momentum diffusivity, to its thermal diffusivity. (High Prandtl number fluids include lubricating oils with $Pr \sim 10^4$ and the earth's mantle with $Pr = {\cal O}(10^{21})$). The Nusselt number $Nu$ is the enhancement factor due to the convective fluid motion of the vertical heat flux over purely conductive thermal transport. The Rayleigh number $Ra$ is a nondimensional measure of the temperature drop across the layer. A key challenge for theory and experiment, the result of which is of significant importance for a wide variety of scientific and engineering applications, is to ascertain the functional dependence of $Nu$ on $Ra$ and $Pr$. Of particular interest is the high $Ra$ behavior of $Nu$, often corresponding to turbulent convection. Rayleigh-B\'enard convection is frequently modeled by the Boussinesq equations, the incompressible Navier-Stokes equations for the fluid velocity field with a temperature-dependent buoyancy force together with the advection-diffusion equation for the temperature field. The infinite Prandtl number limit of the Boussinesq system is the Stokes equations coupled to the advection-diffusion equation for the temperature.

In this work we obtain the following result: for the infinite Prandtl number limit of the Boussinesq equations, the Nusselt number is rigorously bounded in terms of the Rayleigh number according to $Nu \le
.644 \times Ra^{1/3} [\log{Ra}]^{1/3}$ as $Ra \rightarrow \infty$. This estimate follows from the utilization of a novel logarithmic profile in the background method for producing bounds on bulk transport together with new estimates for the bi-Laplacian in a weighted $L^{2}$ space.
It is a quantitative improvement over the best previously available analytic bound and it comes within a logarithmic factor of the pure $Ra^{1/3}$ scaling anticipated by the classical marginally stable boundary layer argument, numerical computations of the optimal bound using the background method, and some recent experiments and direct numerical simulations for finite Prandtl number fluids. This is joint work with Felix Otto (Bonn) and Maria Reznikoff (GaTech).

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