The bulk rate of energy dissipation is the power required to
maintain a flow state. We consider the idealized situation of
flow in a channel with slippery (no-stress) wall driven by an
imposed shearing body force in the streamwise direction. The
Navier-Stokes equations are used to derive a mini-max problem for
an upper limit to the long-time averaged bulk power consumption,
valid for laminar or turbulent flows. This variational problem
yields rigorous bounds in qualitative agreement with the
conventional cascade picture of turbulent dynamics. Moreover, it
can be solved exactly in the high Reynolds number limit.
Quantitative results are compared to the results of direct
numerical solutions of the Navier-Stokes equations for a
particular form of the driving force. Curiously, a component of
the high Reynolds number solution of the variational problem is
reminiscent of statistical aspects of the turbulent flow.
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