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Conference on Turbulence and Interactions TI2006 May June Porquerolles France

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12 pages
Niveau: Supérieur, Doctorat, Bac+8
Conference on Turbulence and Interactions TI2006, May 29 - June 2, 2006, Porquerolles, France EXACTCOHERENT STRUCTURES IN TURBULENT SHEAR FLOWS Fabian Waleffe Departments of Mathematics and Engineering Physics University of Wisconsin, Madison, WI 53706, USA Email: ABSTRACT Exact coherent structures are three-dimensional, nonlinear traveling wave solutions of the Navier-Stokes equations. These solutions are typically unstable from onset, yet they capture the basic statistical and structural features of low Reynolds number turbulent shear flows remarkably well. These exact coherent structures have now been found in all canonical shear flows: plane Couette, Poiseuille and pipe flow. They are generic for shear flows and exist for both no-slip and stress boundary conditions. Their discovery opens up new avenues for turbulence research and forces a fundamental rethinking of the true nature of turbulence. INTRODUCTION What is ‘Turbulence'? Is it the random interaction of ‘eddies'? That is indeed the prevailing view, motivated on the one hand by the kinetic theory of gases where gases are modeled as the random collisions of point molecules, and on the other hand by one's first impression of turbulent flows: they do look very disordered and ‘random.' And they do help mixing milk and coffee. So the basic model of turbulence is that it merely enhances molecular diffusion. The molecular vis- cosity ? is augmented by an eddy or turbulent viscosity ?T = TvT which is the product of a characteristic or ‘mixing length' T and a charac- teristic velocity vT .

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Conference on Turbulence and Interactions TI2006, May 29 - June 2, 2006, Porquerolles, France
EXACT COHERENT STRUCTURES IN TURBULENT SHEAR FLOWS
Fabian Waleffe Departments of Mathematics and Engineering Physics University of Wisconsin, Madison, WI 53706, USA Email: waleffe@math.wisc.edu
ABSTRACT Exact coherent structures are three-dimensional, nonlinear traveling wave solutions of the Navier-Stokes equations. These solutions are typically unstable from onset, yet they capture the basic statistical and structural features of low Reynolds number turbulent shear flows remarkably well. These exact coherent structures have now been found in all canonical shear flows: plane Couette, Poiseuille and pipe flow. They are generic for shear flows and exist for both no-slip and stress boundary conditions. Their discovery opens up new avenues for turbulence research and forces a fundamental rethinking of the true nature of turbulence.
I NTRODUCTION
poorly, if not catastrophically, in strongly inho-mogeneous flows, and in particular near walls. Other models abandon the eddy viscosity concept What is ‘Turbulence’? Is it the random interaction and seek to model directly the Reynolds stresses. of ‘eddies’? That is indeed the prevailing view, But the basic point of view is unchanged. In-motivated on the one hand by the kinetic theory deed the basic model in that class is the ‘return to of gases where gases are modeled as the random isotropy’. Again, not surprisingly, these models collisions of point molecules, and on the other have big troubles near walls.[16,15] hand by one’s first impression of turbulent flows: they do look very disordered and ‘random.’ And There is a fundamentally different view of tur-they do help mixing milk and coffee. bulence that may go back to Hopf, but has been developed in more recent times by Predrag Cvi-So the basic model of turbulence is that it merely tanovic and co-workers, primarily in the context enhances molecular diffusion. The molecular vis- of low-order dynamical systems. In this view, tur-cosity ν is augmented by an eddy or turbulent bulence is not the random interaction of ‘eddies’ viscosity ν T = ` T v T which is the product of a but rather the random ‘switching’ from one unsta-characteristic or ‘mixing length’ ` T and a charac- ble periodic solution to another. Cvitanovic and teristic velocity v T . Various models specify how collaborators have developed a quantitative cycle to prescribe v T and ` T . In that point of view, ho- expansion method to calculate average properties mogeneous, isotropic turbulence appears as the of a chaotic system in terms unstable periodic or-fundamental problem and the local turbulent ki- bits, with the short period solutions dominating netic energy K ( x , t ) and energy dissipation rate the expansion [5]. We now have some evidence ε ( x , t ) as the fundamental quantities of interests. that this point of view may also apply to turbu-Not surprisingly these models typically perform lence in fluids.
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