Energy dissipation in body-forced plane shear flow

We study the problem of body-force-driven shear flows in a plane channel of width l with free-slip boundaries. A mini-max variational problem for upper bounds on the bulk time-averaged energy dissipation rate epsilon is derived from the incompressible Navier-Stokes equations with no secondary assumptions. This produces rigorous limits on the power consumption that are valid for laminar or turbulent solutions. The mini-max problem is solved exactly at high Reynolds numbers Re = Ul/v, where U is the r.m.s. velocity and v is the kinematic viscosity, yielding an explicit bound on the dimensionless asymptotic dissipation factor beta = epsilonl/U-3 that depends only on the 'shape' of the shearing body force. For a simple half-cosine force profile, for example, the high Reynolds number bound is beta less than or equal to pi(2)/root216 = 0.6715.... We also report extensive direct numerical simulations for this particular force shape up to Re approximate to 400; the observed dissipation rates are about a factor 3 below the rigorous high-Re bound. Interestingly, the high-Re optimal solution of the variational problem bears some qualitative resemblance to the observed mean flow profiles in the simulations. These results extend and refine the recent analysis for body-forced turbulence in Doering Folas (2002).