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New conservation laws for helically symmetric flows and their importance for tubulence theory

Our present understanding of statistical 3D turbulence dynamics for reasonable large wave numbers (or small scales) largelyrelies on the dissipation of kinetic energy a quantity which is invariant under all symmetry groups of Navier-Stokes equations except

some special combination of the scaling groups. On the other hand in 2D turbulence, which is translational invariant in one direction,

the transfer mechanism among scales is rather different since the vortex stretching is non-existing. Instead, the transfer and scale

determining key invariant is enstrophy: an area integral of the vorticity squared which is one of the infinite many integral invariants

of 2D inviscid fluid mechanics (see 4 below). Hence the basic mechanism between and 2D and 3D turbulence is very different. To

close this gap we consider flows with a helical symmetry which is a twist of translational and rotational symmetry. The resulting equa-

tions are "2 1 D" which means they have three independent velocity components though only two independent spatial variables. We 2

presently show that the helically symmetric equations of motion admit an infinite number of new non-trivial conservation laws (CL). It is to be expected that the new CLs may give some deeper insight into turbulence dynamics and hence bridging 2D and 3D turbulence.

Author(s):

Olga Kelbin

Chair of Fluid Dynamics, Department of Mechanical Engineering, Technische Universität Darmstadt

Germany

Alexei Cheviakov

Department of Mathematics and Statistics, University of Saskatchewan

Canada

Martin Oberlack

Chair of Fluid Dynamics, Department of Mechanical Engineering, Technische Universität Darmstadt

Germany

Ivan Delbende

Université Pierre et Marie Curie-Paris 6, LIMSI-CNRS

France