In this paper, bounds on average viscous dissipation are derived for Kolmog
orov 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 dissipat
ion 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 an
d 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 t
ime-periodic forcing. Direct numerical simulation of the flow indicates tha
t these bounds leave substantial gaps at large Grashof number G, the calcul
ated 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 s
teady laminar-type flows for neighboring forcing functions, which seems to
indicate that these bounds cannot be improved by adding further dynamical c
onstraints. However, our elementary upper bound on D-v can probably be impr
oved by placing more constraints on the flows. These results serve to empha
size the difference between boundary-driven turbulence and body-force drive
n turbulence where the appropriate dissipation bound is believed saturated
at least up to logarithms. (C) 2001 Elsevier Science B.V. All rights reserv
ed.