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Euler and Navier–Stokes equations in a new time-dependent helically symmetric system: derivation of the fundamental system and new conservation laws

Published online by Cambridge University Press:  31 March 2017

Dominik Dierkes Open the ORCID record for Dominik Dierkes [Opens in a new window]  and
Martin Oberlack
Dominik Dierkes*
Affiliation:
Chair of Fluid Dynamics, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany Graduate School of Excellence Computational Engineering, TU Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany
Martin Oberlack
Affiliation:
Chair of Fluid Dynamics, TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany Graduate School of Excellence Computational Engineering, TU Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany
*
†Email address for correspondence: [email protected]
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Abstract

The present contribution is a significant extension of the work by Kelbin et al. (J. Fluid Mech., vol. 721, 2013, pp. 340–366) as a new time-dependent helical coordinate system has been introduced. For this, Lie symmetry methods have been employed such that the spatial dependence of the originally three independent variables is reduced by one and the remaining variables are: the cylindrical radius $r$ and the time-dependent helical variable $\unicode[STIX]{x1D709}=(z/\unicode[STIX]{x1D6FC}(t))+b\unicode[STIX]{x1D711}$, $b=\text{const.}$ and time $t$. The variables $z$ and $\unicode[STIX]{x1D711}$ are the usual cylindrical coordinates and $\unicode[STIX]{x1D6FC}(t)$ is an arbitrary function of time $t$. Assuming $\unicode[STIX]{x1D6FC}=\text{const.}$, we retain the classical helically symmetric case. Using this, and imposing helical invariance onto the equation of motion, leads to a helically symmetric system of Euler and Navier–Stokes equations with a time-dependent pitch $\unicode[STIX]{x1D6FC}(t)$, which may be varied arbitrarily and which is explicitly contained in all of the latter equations. This has been conducted both for primitive variables as well as for the vorticity formulation. Hence a significantly extended set of helically invariant flows may be considered, which may be altered by an external time-dependent strain along the axis of the helix. Finally, we sought new conservation laws which can be found from the helically invariant Euler and Navier–Stokes equations derived herein. Most of these new conservation laws are considerable extensions of existing conservation laws for helical flows at a constant pitch. Interestingly enough, certain classical conservation laws do not admit extensions in the new time-dependent coordinate system.

JFM classification

Mathematical Foundations: General fluid mechanics Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 818 , 10 May 2017 , pp. 344 - 365
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Copyright
© 2017 Cambridge University Press 

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