COLLOQUIUM
4:00 p.m., Monday (September 24)
MATH 105
Felix Otto
Universitat Bonn
Turbulent heat transport: upper bounds by a priori estimates
Abstract:
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 noslip 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 Nu\stackrel{\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 NavierStokes equation and used by Teman et. al. for
the KuramotoSivashinski equation. We propose a background temperature field
with nonmonotone boundary layers; direct numerical simulations show an
average temperature field with the same qualitative behavior. We obtain the
slightly improved bound Nu\stackrel{\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^2spaces.
This is joint work with Charles Doering and Maria Reznikoff.
Refreshments will be served at 3:45 p.m. in the Faculty Lounge,
Math Annex (Room 1115).
