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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