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 Grashof number G. For a force of the form F0 sin mzi or F0 sin mz cos ωti, we derive various bounds on the total dissipation in the flow, Du, as well as on the dissipation Dm obtained from the x-velocity averaged over the x,y plane (the mean velocity of the flow). We derive upper bounds on Du and Dv=Du−Dm, as well as lower bounds on Dm and Dm/Du, 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 Dm/Du of O(G−1), and a lower bound on Du, of O(G−1/2) where G≔F0L3/ν2 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 Dm(G) and Du(G) being O(G−1/2) and O(1), respectively, as G→∞. Our theoretical bounds on Dm,Du 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 Dv 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.