**Turbulent energy dissipation for forced flow in a slippery
channel**

The bulk rate of energy dissipation is the power required to
maintain a flow state. One of the fundamental concepts of
turbulence theory, to be reviewed in this talk, is that the
dissipation rate should become independent of viscosity at high
Reynolds numbers due to a "nonlinear cascade" of energy to small
scales. In order to investigate this in a rigorous mathematical
setting, we consider the idealized situation of flow in a channel
with slippery (no-stress) walls 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 upper bounds that
are in qualitative agreement with the conventional cascade picture
of turbulent dynamics. Moreover, mini-max problem can be solved
exactly in the infinite Reynolds number limit, and quantitative
results are compared to the results of direct numerical solutions of
the Navier-Stokes equations for a particular "shape" 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. This is joint work with Bruno Eckhardt and
Jörg Schumacher, University of Marburg.

*Charles Doering*

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