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 ǫ 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 = U ℓ/ν, where U is the rms velocity and ν is the kinematic viscosity, yielding an explicit bound on the dimensionless asymptotic dissipation factor β = ǫℓ/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 β ≤ π 2 / √ 216 = .6715. . .. We also report extensive direct numerical simulations for this particular force shape up to Re ≈ 400; the observed dissipation rates are about a factor of three below the rigorous high-Re bound. Interestingly, the high-Re optimal solution of the variational problem bears some qualitative resemblence to the observed mean flow profiles in the simulations. These results extend and refine the recent analysis for body-forced turbulence in J.