Variational bounds on shear-driven turbulence and turbulent Boussinesq convection

Abstract

Based on earlier studies by Hopf (1941), Doering and Constantin (1992, 1994, 1995) have recently formulated a new “background” technique for obtaining upper bounds on turbulent fluid flow quantities. This method produces upper bounds on the limit supremum of long time averages, making no statistical assumptions about the flow in contrast to the well-known Howard-Busse approach. The full optimisation problems posed by this method for the momentum transport in turbulent Couette flow and the heat transport (with zero background flow) in turbulent Boussinesq convection are solved here for the first time at asymptotically large Reynolds number and Rayleigh number within Busse's multiple boundary layer approximation to extract the best (lowest) possible upper bounds available. Intriguingly, the original bounds isolated by Busse (1969, 1970) within the confines of statistical stationarity are recovered exactly using this new formalism. The optimal background velocity profile for turbulent Couette flow is found to be shearless in the interior thus differing from Busse's “ 1 4 ” mean shear result. In the convective case, an interesting degeneracy in the formulation of the background variational problem leads to an indeterminacy in the optimal background temperature profile. Only for one special choice is the isothermal core feature of Busse's mean profile recovered.