Skip to main content
×
×
Home

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

  • Dominik Dierkes (a1) (a2) and Martin Oberlack (a1) (a2)
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.

Copyright
Corresponding author
Email address for correspondence: dierkes@fdy.tu-darmstadt.de
References
Hide All
Anco S. C. & Bluman G. W. 2002a Direct construction method for conservation laws of partial differential equations part ii: General treatment. Eur. J. Appl. Maths 13 (05), 567585.
Anco S. C. & Bluman G. W. 2002b Symmetry and Integration Methods for Differential Equations. Springer.
Bluman G. W., Cheviakov A. F. & Anco S. C. 2010 Applications of Symmetry Methods to Partial Differential Equations. Springer.
Bogoyavlenskij O. I. 2000 Helically symmetric astrophysical jets. Phys. Rev. E 62, 86168627.
Cheviakov A. F. 2007 Gem software package for computation of symmetries and conservation laws of differential equations. Comput. Phys. Commun. 176 (1), 4861.
Cheviakov A. F. & Oberlack M. 2014 Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations. J. Fluid Mech. 760, 368386.
Delbende I., Rossi M. & Daube O. 2012 DNS of flows with helical symmetry. Theor. Comput. Fluid Dyn. 26 (1–4), 141160.
Dritschel D. G. 1991 Generalized helical Beltrami flows in hydrodynamics and magnetohydrodynamics. J. Fluid Mech. 222, 525541.
Jamil M. & Fetecau C. 2010 Helical flows of Maxwell fluid between coaxial cylinders with given shear stresses on the boundary. Nonlinear Anal. Real World Appl. 11 (5), 43024311.
Johnson J. L., Oberman C. R., Kulsrud R. M. & Frieman E. A.1958 Some stable hydromagnetic equilibria. 1 (4), 281–296.
Kelbin O., Cheviakov A. F. & Oberlack M. 2013 New conservation laws of helically symmetric, plane and rotationally symmetric viscous and inviscid flows. J. Fluid Mech. 721, 340366.
Mitchell A., Morton S. & Forsythe J.1997 Wind turbine wake aerodynamics Rep. ADA425027. Air Force Academy Colorado Springs, Department of Aeronautics.
Oberlack M. 2000 Symmetrie, Invarianz und Selbstähnlichkeit in der Turbulenz. Shaker Aachen.
Rosenhaus V. & Shankar R.2015 Sub-symmetries and infinite conservation laws for the Euler equations. arXiv:1509.05101.
Sarpkaya T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45 (03), 545559.
Schnack D. D., Caramana E. J. & Nebel R. A. 1985 Three-dimensional magnetohydrodynamic studies of the reversed-field pinch. Phys. Fluids 28 (1), 321333.
Vermeer L. J., Sorensen J. N. & Crespo A. 2003 Wind turbine wake aerodynamics. Prog. Aerosp. Sci. 39 (6–7), 467510.
Zienkiewicz O. C., Taylor R. L., Sherwin S. J. & Peiro J. 2003 On discontinuous Galerkin methods. Int. J. Numer. Meth. Engng 58 (8), 11191148.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 5
Total number of PDF views: 157 *
Loading metrics...

Abstract views

Total abstract views: 344 *
Loading metrics...

* Views captured on Cambridge Core between 31st March 2017 - 16th January 2018. This data will be updated every 24 hours.