Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

CSIC Building (#406), Seminar Room 4122.
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Incompressible Fluid Turbulence & Generalized Solutions of Euler Equations

              Gregory Eyink

                      Department of Applied Mathematics & Statistics, Johns Hopkins University

Abstract:   Experiments and numerical simulations show that energy dissipation in incompressible fluid turbulence tends to a positive value in the inviscid limit (infinite Reynolds number). Lars Onsager (1949) proposed an explanation for this phenomenon in terms of energy cascade for certain singular solutions of Euler equations.

This proposal can be fruitful both for turbulence theory, by suggesting novel results that can be tested empirically, and also for theory of partial differential equations, by giving important hints on the character of Euler solutions in the zero-viscosity limit. We shall review some of the classical ideas on turbulent energy cascade and their current status within rigorous theory of PDE's. In particular, we shall discuss old ideas of Geoffrey Taylor (1937) on the role of vortex line-stretching in generating turbulent energy dissipation. Taylor's argument was based on a statistical hypothesis that material lines in a turbulent flow will tend to elongate, on average, and appealed to the Kelvin Theorem (1869) on conservation of circulations. For smooth solutions the Kelvin Theorem for all loops is equivalent to the Euler equations of motion, but we shall present rigorous results which suggest that the theorem breaks down in turbulent flow due to nonlinear effects. This turbulent "cascade of circulations" has been verified by high-Reynolds-number numerical simulations. We propose another conjecture, that circulations on material loops may be martingales of a generalized Euler flow (in the sense of Brenier and Shnirelman).

This hypothesis introduces an arrow of time, according to whether the martingale property holds in the past or in the future. We shall show that this property has a close analogue in the "Kraichnan model" of random advection, which accounts for anomalous scalar dissipation in that model. The "Kraichnan model" is also known to probabilists as a generalized stochastic flow and its basic features have been put on a rigorous footing by Le Jan and Raimond (2002, 2004). We propose a geometric treatment of this model, formally as a diffusion process on an infinite-dimensional semi-group of volume-preserving maps. We expect that this model has great potential to provide further insight into the nature of incompressible, high Reynolds number flows.
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