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Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations

Published online by Cambridge University Press:  07 November 2014

Alexei F. Cheviakov  and
Martin Oberlack
Alexei F. Cheviakov*
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, S7N 5E6, Canada
Martin Oberlack
Affiliation:
Chair of Fluid Dynamics, Technische Universität Darmstadt, Otto-Berndt-Strasse 2, 64287 Darmstadt, Germany
*
†Email address for correspondence: cheviakov@math.usask.ca
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Abstract

Local conservation laws are systematically constructed for three-dimensional time-dependent viscous and inviscid incompressible fluid flows, in primitive variables and vorticity formulation, using the direct construction method. Complete sets of local conservation laws in primitive variables are derived for the case of conservation law multipliers depending on derivatives up to the second order. In the vorticity formulation, there exists an infinite family of vorticity-dependent conservation laws involving an arbitrary differentiable function of space and time, holding for both viscous and inviscid cases. The infinite conservation law family is used to generate further independent hierarchies of conservation laws that essentially involve vorticity and arbitrary flow parameters, which are determined by known evolution equations such as those for momentum, energy or helicity, though not necessarily in the form of a conservation law. The new conservation laws are not restricted to any reduced flow geometry such as planar or axisymmetric limits. Examples are considered.

JFM classification

Mathematical Foundations: General fluid mechanics Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 760 , 10 December 2014 , pp. 368 - 386
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Copyright
© 2014 Cambridge University Press 

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