## Turbulent heat transport: upper bounds by a priori estimates

- Date: 09/24/2007

Felix Otto (University of Bonn)

University of British Columbia

We are interested in the transport of heat through a layer of viscous

fluid which is heated from below and cooled from above. Two mechanisms

are at work: Heat is transported by simple diffusion and by advection

through the flow. The transport by advection is triggered by buoyancy

(hotter parts have lower density) but is hindered by the no-slip

boundary condition for the fluid velocity at the bottom and top

surfaces.

Neglecting inertia, the equations contain a single dimensionless

parameter, the Rayleigh number $Ra$. It measures the relative strength

of advection with respect to diffusion. For $Ra gg 1$, the flow is

aperiodic and the heat transport is mediated by plumes. As a

consequence, the horizontally averaged temperature displays boundary

layers.

Inspired by the work of Constantin and Doering, we are interested in

rigorous bounds on the average heat transport (the Nusselt number $Nu$)

in terms of $Ra$. By PDE methods, Constantin and Doering prove

$Nustackrel{le}{sim} Ra^{1/3}log^{2/3}Ra$.

We use the conceptually intriguing method of the background

(temperature) field, introduced by Hopf for the Navier--Stokes equation

and used by Teman et. al. for the Kuramoto--Sivashinski equation. We

propose a background temperature field with non--monotone boundary

layers; direct numerical simulations show an average temperature field

with the same qualitative behavior. We obtain the slightly improved

bound $Nustackrel{le}{sim} Ra^{1/3}log^{1/3}Ra$. The crucial ingredient

is a maximal regularity statement for the Stokes operator in suitably

weighted $L^2$--spaces.

This is joint work with Charles Doering and Maria Reznikoff.

PIMS Distinguished Chair Lectures 2007