Turbulent heat transport: upper bounds by a priori estimates
Topic
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.
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.
Speakers
Additional Information
PIMS Distinguished Chair Lectures 2007
Felix Otto (University of Bonn)
Felix Otto (University of Bonn)