Approximating the rate of heat transport

Tuesday, April 13, 2010 - 11:00am - 11:45am
EE/CS 3-180
Keywords: convection, heat transport, Nusselt number,
Constantin-Doering-Hopf technique, long time statistical properties,
numerical approximation, convergence of long time statistical properties

Abstract: We survey a few recent results on estimating the long time averaged rate
of heat transport in the vertical direction (the Nusselt number) in
Rayleigh-Bénard convection. In the first half of the talk, we recall
rigorous upper bounds on the Nusselt number that are of the form of
Ra1/3 modulo logarithmic correction for both the infinite Prandtl
number model and the classical Boussinesq model for convection with large
but finite Prandtl number. The main technique is the
Constantin-Doering-Hopf approach. We also discuss the infinite Prandtl
number limit in the Boussinesq model for convection, and the formal
infinite Rayleigh number limit within the infinite Prandtl number model
for convection. In the second half of the talk, we discuss numerical
schemes (time discretization) that are able to capture the long time
statistical properties of the convection problems. We first recall that
the maximum long time averaged rate of heat transport in the vertical
direction (true maximum Nusselt number) is a long time statistical
property of the convection system. We then show that appropriate time
discretizations of the systems will be able to capture the true maximum
Nusselt number asymptotically. Several specific schemes that satisfy the
desired properties will be presented. This numerical approach complements
the Constantin-Doering-Hopf approach in the sense that it provides a
computational asymptotic lower bound. Noise effects will be mentioned.