Energy dissipation in body-forced plane shear flow

Abstract

We study the problem of body-force-driven shear flows in a plane channel of width $\ell$ 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 $Re = U\ell/\nu$, where $U$ is the r.m.s. velocity and $\nu$ is the kinematic viscosity, yielding an explicit bound on the dimensionless asymptotic dissipation factor $\beta=\epsilon \ell/U^3$ 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 $\beta \le \pi^2/\sqrt{216} = 0.6715\ldots$. We also report extensive direct numerical simulations for this particular force shape up to $Re \approx 400$; 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 & Foias (2002).