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The role of angular momentum conservation in homogeneous turbulence

Published online by Cambridge University Press:  27 July 2009

P. A. DAVIDSON
P. A. DAVIDSON*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

Loitsyanky's integral I = − ∫ r2〈u ⋅u′〉dr is known to be approximately conserved in certain types of fully developed, isotropic turbulence, and its near conservation controls the rate of decay of kinetic energy. Landau suggested that this integral is related to the angular momentum H = ∫ (x × u)dV of some large volume V of the turbulence, according to the expression I = 〈H2〉/V. He also suggested that the approximate conservation of I is related to the principle of conservation of angular momentum. However, Landau's analysis can be criticized because, formally, it applies only to inhomogeneous turbulence evolving in a closed domain. So how are we to interpret the near conservation of I? And what is its relationship, if any, to angular momentum conservation? We show that the key to extending Landau's analysis to strictly homogeneous turbulence is to rewrite Loitsyansky's integral in terms of the vector potential of the velocity field, i.e. I = 6 ∫〈A ⋅ A′〉dr, where ∇ × A = u. This yields I = 6〈[∫VAdV]2〉/V for any large spherical volume V of radius R. Crucially, J = 3∫VAdV can be rewritten as the weighted integral of the angular momentum density throughout all space. This fundamentally changes the way in which we interpret the dynamical behaviour of I. For example, we show that the conservation of 〈J2〉/V, and hence of I, which occurs when the long-range correlations are weak, is a direct consequence of the decorrelation of the flux of angular momentum out through a spherical control surface S and the local angular momentum in the vicinity of S. Thus, within the framework of strictly homogeneous turbulence, we provide the first self-consistent interpretation of Loitsyanky's integral in terms of angular momentum conservation. We also show that essentially the same ideas carry over to certain types of anisotropic turbulence, such as magnetohydrodynamic (MHD), rotating and stratified turbulence. This is important because conservation of angular momentum, which manifests itself in the form of a Loitsyansky-like invariant, places a fundamental restriction on the way in which the integral scales can evolve in such turbulence. This, in turn, controls the rate of decay of energy. We illustrate this by deriving new decay laws for MHD and stratified turbulence. The MHD decay laws are consistent with the available numerical evidence, but further study is required to verify, or otherwise, the predictions for stratified turbulence.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 632 , 10 August 2009 , pp. 329 - 358
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Batchelor, G. K. & Proudman, I. 1956 The large-scale structure of homogenous turbulence. Phil. Trans. R. Soc. Lond. A 248, 369–405.Google Scholar
Brethouwer, G., Billant, P., Lindborg, E., Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech 585, 343–368.CrossRefGoogle Scholar
Chasnov, J. R. 1993 Computation of the Loitsyansky integral in decaying isotropic turbulence. Phys. Fluids A 5 (11), 2579–2581.CrossRefGoogle Scholar
Cho, J. & Vishniac, E. T. 2000 The Anisotropy of magnetohydrodynamic Alfvenic turbulence, Astrophys. J. 539, 273–282.CrossRefGoogle Scholar
Davidson, P. A. 1997 The role of angular momentum in the magnetic damping of turbulence. J. Fluid Mech 336, 123–150.CrossRefGoogle Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Davidson, P. A. 2007 On the large-scale structure of homogeneous two-dimensional turbulence. J. Fluid Mech. 580, 431.CrossRefGoogle Scholar
Davidson, P. A., Ishida, T. & Kaneda, Y. 2006 Linear and angular momentum invariants in homogenous turbulence. In IUTAM Symposium on Computational Physics and New Perspectives in Turbulence (ed. Kaneda, Y.). pp. 19–26, Springer.Google Scholar
Eyink, G. L. & Thomson, D. J. 2000 Free decay of turbulence and the breakdown of self-similarity. Phys. Fluids 12 (3), 477–479.Google Scholar
Godeferd, F. S. & Staquet, C. 2003 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 2. Large scale and small scale anisotropy. J. Fluid Mech 486, 115–159.CrossRefGoogle Scholar
Herring, J. R. Kimura, Y., James, R., Clyne, J., & Davidson, P.A. 2006 Statistical and dynamical questions in stratified turbulence. In Mathematical and Physical Theory of Turbulence (ed. Cannon, J. & Shivamoggi, S.), pp. 101–114. Chapman and Hall.Google Scholar
Ishida, T., Davidson, P. A., & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455.Google Scholar
Jackson, J. D. 1975 Classical Electrodynamics, 2nd edn. Wiley.Google Scholar
Kolmogorov, A. N. 1941 On the degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl. Akad. Nauk. SSSR 31 (6), 538–541.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics, 1st edn. Pergamon.Google Scholar
Lesieur, M. 1990, Turbulence in Fluids, 2nd edn. Kluwer.CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in strongly stratified turbulence. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Llor, A. 2006 Invariants of free turbulent decay. arXiv:physics/0612220v1.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2. MIT PressGoogle Scholar
Oberlack, M. 2002 On the decay exponent of isotropic turbulence. Proc. Appl. Math. Mech. 1 (1), 294–297Google Scholar
Okamoto, N., Davidson, P. A. & Kaneda, Y. 2008 The decay of low magnetic Reynolds number turbulence in an imposed magnetic field. In Isaac Newton Institute Workshop on Inertial Range Dynamics and Mixing, Cambridge, UK.Google Scholar
Ossia, S. & Lesieur, M. 2000 Energy backscatter in large-eddy simulations of three-dimensional incompressible isotropic turbulence. J. Turbul. 1, 010.Google Scholar
Proudman, I. & Reid, W. H. 1954 On the decay of a normally distributed and homogeneous turbulent velocity field. Phil. Trans. R. Soc. Lond. A 247, 163–189.Google Scholar
Riley, J. J. & deBruynkops, S. M. 2004 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15 (7), 2047–2059.CrossRefGoogle Scholar
Ruelle, D. 1990 Is there screening in turbulence? J. Stat. Phys. 61, 865–868.CrossRefGoogle Scholar
Saffman, P. G. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27 (3), 581–593.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics Cambridge University Press.Google Scholar
Sagaut, P. & Cambon, C. 2008 Homogenous Turbulence Dynamics, Cambridge University PressCrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2006 Stratified turbulence dominated by vortical motion. J. Fluid Mech 517, 281–308.Google Scholar