Variational bounds on energy dissipation in incompressible flows: Shear flow

A variational principle for upper bounds on the time averaged rate of viscous energy dissipation for Newtonian fluid flows is derived from the incompressible Navier-Stokes equations. When supplied with appropriate test "background" flow fields, the variational formulation produces explicit estimates for the energy dissipation rate. This dissipation rate is related to the drag of the fluid on the boundaries, and so these estimates translate into bounds on the drag. We analyze the problem of boundary-driven shear flow in detail, comparing the rigorous estimates obtained from the variational method with both recent experimental results and predictions of a conventional closure approximation from statistical turbulence theory.