Uniform estimates on the velocity in Rayleigh–Bénard convection

Abstract

The kinetic energy of a fluid located between two plates at different temperatures is easily bounded by classical inequalities. However, experiments and numerical simulations indicate that when the convection is turbulent, the volume of the domains in which the speed is large, is rather small. This could imply that the maximum of the speed, in contrast with its quadratic mean, does not admit an a priori upper bound. It is proved that, provided the pressure remains bounded, a uniform estimate for the speed maximum does indeed exist, and that it depends on the maxima of certain ratios between temperature, pressure, and velocity.