Bounds on dissipation for Navier-Stokes flow with Kolmogorov forcing

In this paper, bounds on average viscous dissipation are derived for Kolmogorov flow in a spatially periodic domain with steady and unsteady forcing, at arbitrarily large Grash of number G. For a force of the form F-0 sin mzi or F-0 sin mz cos omega ti, we derive various bounds on the total dissipation in the flow, D-u,, as well as on the dissipation D-m obtained from the x-velocity averaged over the x, y plane (the mean velocity of the flow). We derive upper bounds on D-u and D-v = D-u - D-m, as well as lower bounds on D-m and D-m /D-u, adopting constraints of the kind introduced by Howard and Busse and assuming a steady force. The background flow method introduced by Doering and Constantin is used to obtain an improved lower bound on D-m/D-u of O(G(-1)), and a lower bound on D-u, of O(G(-1/2)) where G := F0L3/nu (2) is the Grashof number. Some of these results are then generalized to time-periodic forcing. Direct numerical simulation of the flow indicates that these bounds leave substantial gaps at large Grashof number G, the calculated D-m (G) and D-u (G) being 0 (G(-1/2)) and O(1), respectively, as G --> infinity. Our theoretical bounds on D-m, D-u are shown to be attained by steady laminar-type flows for neighboring forcing functions, which seems to indicate that these bounds cannot be improved by adding further dynamical constraints. However, our elementary upper bound on D-v can probably be improved by placing more constraints on the flows. These results serve to emphasize the difference between boundary-driven turbulence and body-force driven turbulence where the appropriate dissipation bound is believed saturated at least up to logarithms. (C) 2001 Elsevier Science B.V. All rights reserved.