Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-18T20:59:11.859Z Has data issue: false hasContentIssue false
  • English
  • Français

The hybrid Euler–Lagrange procedure using an extension of Moffatt's method

Published online by Cambridge University Press:  02 August 2010

A. M. SOWARD  and
P. H. ROBERTS
A. M. SOWARD*
Affiliation:
School of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, UK
P. H. ROBERTS
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The hybrid Euler–Lagrange (HEL) description of fluid mechanics, pioneered largely by Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646), has had to face the fact, in common with all Lagrangian descriptions of fluid motion, that the variables used do not describe conditions at the coordinate x, upon which they depend, but conditions elsewhere at some displaced position xL(x, t) = x + ξ(x, t), generally dependent on time t. To address this issue, we employ ‘Lie dragging’ techniques of general tensor calculus to extend a method introduced by Moffatt (J. Fluid Mech., vol. 166, 1986, pp. 359–378) in the fluid dynamic context, whereby the point x is dragged to xL(x, t) by a ‘fictitious steady flow’ η*(x, t) in a unit of ‘fictitious time’. Whereas ξ(x, t) is a Lagrangian concept intimately linked to the location xL(x, t), the ‘dragging velocity’ η*(x, t) has an essentially Eulerian character, because it describes the fictitious velocity at x itself. For the case of constant-density fluids, we show, using solenoidal η*(x, t) instead of solenoidal ξ(x, t), how the HEL theory can be cast into Eulerian form. A useful aspect of this Eulerian development is that the mean flow itself remains solenoidal, a feature that traditional HEL theories lack. Our method realizes the objective sought by Holm (Physica D, vol. 170, 2002, pp. 253–286) in his derivation of the Navier–Stokes–α equation, which is the basis of one of the methods currently employed to represent the sub-grid scales in large-eddy simulations. His derivation, based on expansion to second order in ξ, contained an error which, when corrected, implied a violation of Kelvin's theorem on the constancy of circulation in inviscid incompressible fluid. We show that this is rectified when the expansion is in η* rather than ξ, Kelvin's theorem then being satisfied to all orders for which the expansion converges. We discuss the implications of our approach using η* for the Navier–Stokes–α theory.

JFM classification

Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 661 , 25 October 2010 , pp. 45 - 72
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, D. G. & McIntyre, M. E. 1978 a An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978 b On wave action and its relatives. J. Fluid Mech. 89, 647–664 [and corrigendum, 95, 796 (1978)].CrossRefGoogle Scholar
Braginsky, S. I. 1964 a Self-excitation of magnetic field during the motion of a highly conducting fluid. JETP 47, 1084–1098. [Engl. transl. Sov. Phys. JETP 20, 726–735 (1965)].Google Scholar
Braginsky, S. I. 1964 b Theory of the hydromagnetic dynamo. JETP 47, 2178–2193. [Engl. transl. Sov. Phys. JETP 20, 1462–1471 (1965).]Google Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 2003 Remote recoil: a new wave–mean interaction effect. J. Fluid Mech. 492, 207–230.Google Scholar
Bühler, O. & McIntyre, M. E. 2005 Wave capture and wave–vortex duality. J. Fluid Mech. 534, 67–95.Google Scholar
Davidson, P. A. 2000 An energy criterion for the linear stability of conservative flows. J. Fluid Mech. 402, 329–348.CrossRefGoogle Scholar
D'Inverno, R. 1992 Introducing Einstein's Relativity. Clarendon.Google Scholar
Eckart, C. 1963 Some transformations of the hydrodynamic equations. Phys. Fluids 6, 1037–1041.Google Scholar
Frieman, E. & Rotenberg, M. 1960 On hydromagnetic stability of stationary equilibria. Rev. Mod. Phys. 32, 898–902.Google Scholar
Geurts, B. J., Kuczaj, A. K. & Titi, E. S. 2008 Regularization modeling for large-eddy simulation of homogeneous isotropic decaying turbulence, J. Phys. A: Math. Theor. 41 344008.Google Scholar
Holm, D. D. 2002 Averaged Lagrangians and mean effects of fluctuations in ideal fluid dynamics. Physica D 170, 253–286.Google Scholar
McIntyre, M. E. 1980 Towards a Lagrangian-mean description of stratospheric circulations and chemical transports. Phil. Trans. R. Soc. Lond. A 296, 129–148.Google Scholar
McIntyre, M. E. 1988 A note on the divergence effect and the Lagrangian-mean surface elevation in periodic water waves. J. Fluid Mech. 189, 235–242.Google Scholar
Moffatt, H. K. 1986 Magnetostatic equilibria and analogous Euler flows of arbitrary complex topology. Part 2. Stability considerations. J. Fluid Mech. 166, 359–378.Google Scholar
Roberts, P. H. & Soward, A. M. 2006 a The Navier–Stokes–alpha equations revisited. Eulerian and Lagrangian means in rotating, magnetohydrodynamic flows. I. General results. Geophys. Astrophys. Fluid Dyn. 100, 457–483.CrossRefGoogle Scholar
Roberts, P. H. & Soward, A. M. 2006 b Covariant description of non-relativistic magnetohydrodynamics. Geophys. Astrophys. Fluid Dyn. 100, 485–502.Google Scholar
Roberts, P. H. & Soward, A. M. 2009 The Navier–Stokes–alpha equations revisited. Geophys. Astrophys. Fluid Dyn. 103, 303–316.CrossRefGoogle Scholar
Schutz, B. 1980 Geometrical Methods of Mathematical Physics. Cambridge University Press.Google Scholar
Soward, A. M. 1971 a Nearly symmetric kinematic and hydromagnetic dynamos. J. Math. Phys. 12, 1900–1906.Google Scholar
Soward, A. M. 1971 b Nearly symmetric advection. J. Math. Phys. 12, 2052–2062.Google Scholar
Soward, A. M. 1972 A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. R. Soc. Lond. A 272, 431–462.Google Scholar
Soward, A. M. & Roberts, P. H. 2008 On the derivation of the Navier–Stokes–alpha equations from Hamilton's principle. J. Fluid Mech. 604, 297–323.Google Scholar
Tough, J. G. & Roberts, P. H. 1968 Nearly symmetric hydromagnetic dynamos. Phys. Earth Planet. Inter. 1, 288–296.Google Scholar