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New conservation laws of helically symmetric, plane and rotationally symmetric viscous and inviscid flows

Published online by Cambridge University Press:  13 March 2013

Olga Kelbin ,
Alexei F. Cheviakov  and
Martin Oberlack
Olga Kelbin
Affiliation:
Chair of Fluid Dynamics, TU Darmstadt, Petersenstraße 30, 64287 Darmstadt, Germany
Alexei F. Cheviakov*
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada S7N 5E6
Martin Oberlack
Affiliation:
Chair of Fluid Dynamics, TU Darmstadt, Petersenstraße 30, 64287 Darmstadt, Germany Center of Smart Interfaces, TU Darmstadt, Petersenstraße 32, 64287 Darmstadt, Germany GS Computational Engineering, TU Darmstadt, Dolivostraße 15, 64293 Darmstadt, Germany
*
†Email address for correspondence: cheviakov@math.usask.ca
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Abstract

Helically invariant reductions due to a reduced set of independent variables $(t, r, \xi )$ with $\xi = az+ b\varphi $ emerging from a cylindrical coordinate system of viscous and inviscid time-dependent fluid flow equations, with all three velocity components generally non-zero, are considered in primitive variables and in the vorticity formulation. Full sets of equations are derived. Local conservation laws of helically invariant systems are systematically sought through the direct construction method. Various new sets of conservation laws for both inviscid and viscous flows, including families that involve arbitrary functions, are derived. For both Euler and Navier–Stokes flows, infinite sets of vorticity-related conservation laws are derived. In particular, for Euler flows, we obtain a family of conserved quantities that generalize helicity. The special case of two-component flows, with zero velocity component in the invariant direction, is additionally considered, and special conserved quantities that hold for such flows are computed. In particular, it is shown that the well-known infinite set of generalized enstrophy conservation laws that holds for plane flows also holds for the general two-component helically invariant flows and for axisymmetric two-component flows.

JFM classification

Mathematical Foundations: General fluid mechanics Mathematical Foundations: Mathematical Foundations

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Type
Papers
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Journal of Fluid Mechanics , Volume 721 , 25 April 2013 , pp. 340 - 366
Copyright
©2013 Cambridge University Press

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