Richardson number-dependent bounds on buoyancy flux
Seminar Room 1, Newton Institute
Parameterizing the mixing of a stratified fluid subject to shear is a fundamental challenge for models of environmental and industrial flows. In particular, it is of great value to parameterize the efficiency of turbulent mixing, in the sense of the proportion of the kinetic energy converted into potential energy (through irreversible mixing of fluid of different density) compared to the total amount converted to both potential energy and internal energy (through viscous dissipation). Various competing models have been presented to relate the mixing efficiency to bulk properties of the flow, especially through different Richardson numbers, which quantify the relative importance of buoyancy and shear within the flow.
One promising approach is to construct rigorous bounds on the long-time average of the buoyancy flux (i.e. the mixing rate) within simple model stratified shear flows, imposing physically reasonable constraints on the model flow fields. In this talk, we apply this technique to stably stratified Couette flow, imposing as a constraint the mixing efficiency, and show that a bound on the long-time average of the buoyancy flux can only be found for a certain range of bulk Richardson numbers. For sufficiently strong stratifications, it appears that no bound exists, suggesting that it is not possible to find a statistically steady state. We discuss the implications of this result for various classical stratified shear turbulence models.