Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T04:53:21.715Z Has data issue: false hasContentIssue false
  • English
  • Français

On the decay of Saffman turbulence subject to rotation, stratification or an imposed magnetic field

Published online by Cambridge University Press:  23 August 2010

P. A. DAVIDSON
P. A. DAVIDSON*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
†Email address for correspondence: pad3@eng.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider freely decaying, statistically axisymmetric turbulence evolving in the presence of a background rotation, an imposed stratification or a uniform magnetic field. We focus on the case of Saffman turbulence in which E(k → 0) ~ k2 and show that, if the large scales evolve in a self-similar manner, then u⊥2 ℓ⊥2 ℓ∥ = constant in rotating, stratified and magnetohydrodynamic turbulence. This may be contrasted with E(k → 0) ~ k4 turbulence, in which u⊥2 ℓ⊥4 ℓ∥ ≈ constant. (Here the subscripts ⊥ and ∥ indicate directions perpendicular and parallel to the axis of symmetry, and ℓ⊥, ℓ∥ and u⊥ are suitably defined integral scales.) This constraint on the integral scales allows us to make simple, testable predictions for the temporal evolution of ℓ⊥, ℓ∥ and u⊥ in all three systems. There are only limited data sets against which to compare these predictions, but they are consistent with those data which are currently available.

JFM classification

Turbulent Flows: Homogeneous turbulence Turbulent Flows: Rotating turbulence Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 663 , 25 November 2010 , pp. 268 - 292
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

Batchelor, G. K. & Proudman, I. 1956 The large-scale structure of homogenous turbulence. Phil. Trans. R. Soc. Lond. A 248, 369–405.Google Scholar
Bennett, J. C. & Corrsin, S. 1978 Small Reynolds number nearly isotropic turbulence in a straight duct and a contraction. Phys. Fluids 21 (12), 2129–2140.CrossRefGoogle Scholar
Bigot, B., Galtier, S. & Politano, H. 2008 Energy decay laws in strongly anisotropic magnetohydrodynamic turbulence. Phys. Rev. Lett. 100, 074502.CrossRefGoogle ScholarPubMed
Bokhoven, L. J. A., Cambon, C., Liechtenstein, L., Godeferd, F. S. & Clercx, H. J. H. 2008 Refined vorticity statistics in decaying rotating three-dimensional turbulence. J. Turbulence 9 (6), 1–25.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
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, 431–450.CrossRefGoogle Scholar
Davidson, P. A. 2009 The role of angular momentum conservation in homogenous turbulence. J. Fluid Mech. 632, 329–358.CrossRefGoogle Scholar
Davidson, P. A., Staplehurst, P. J. & Dalziel, S. B. 2006 On the evolution of eddies in a rapidly rotating system. J. Fluid Mech. 557, 135–144.CrossRefGoogle Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68, 015301.CrossRefGoogle ScholarPubMed
Ishida, T., Davidson, P. A. & Kaneda, Y. 2006 On the decay of isotropic turbulence. J. Fluid Mech. 564, 455–475.CrossRefGoogle Scholar
Jacquin, L., Leuchter, O., Cambon, C. & Mathieu, J. 1990 Homogenous turbulence in the presence of rotation. J. Fluid Mech. 220, 1–52.CrossRefGoogle 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
Krogstad, P.-A. & Davidson, P. A. 2010 Is grid turbulence Saffman turbulence? J. Fluid Mech. 642, 373–394.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics, 1st edn. Pergamon.Google Scholar
Lavoie, P., Djenedi, L. & Antonia, R. A. 2007 Effects of initial conditions in decaying turbulence generated by passive grids. J. Fluid Mech. 585, 395–420.CrossRefGoogle Scholar
Moffatt, H. K. 1967 On the suppression of turbulence by a uniform magnetic field. J. Fluid Mech. 28, 571–592.CrossRefGoogle Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Morize, C. & Moisy, F. 2006 Energy decay of rotating turbulence with confinement effects. Phys. Fluids 18, 065107.CrossRefGoogle Scholar
Okamoto, N., Davidson, P. A. & Kaneda, Y. 2010 The decay of low magnetic Reynolds number turbulence in an imposed magnetic field. J. Fluid Mech. 651, 295–318.CrossRefGoogle Scholar
Saffman, P. G. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27 (3), 581–593.CrossRefGoogle Scholar
Squires, K. D., Chasnov, J. R., Mansour, N. N. & Cambon, C. 1994 The asymptotic state of rotating turbulence at high Reynolds number. AGARD-CP-551, pp. 4.1–4.9.Google Scholar
Staplehurst, P. J., Davidson, P. A. & Dalziel, S. B. 2008 Structure formation in homogeneous freely decaying rotating turbulence. J. Fluid Mech. 598, 81–105.CrossRefGoogle Scholar
Staquet, C. & Godeferd, F. S. 1998 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 1 Flow energetics. J. Fluid Mech. 360, 295–340.CrossRefGoogle Scholar
Teitelbaum, I. & Mininni, P. D. 2009 Effect of helicity and rotation on the free decay of turbulent flows. Phys. Rev. Lett. 103, 014501.CrossRefGoogle ScholarPubMed
Thiele, M. & Muller, W.-C. 2009 Structure and decay of rotating homogeneous turbulence. J. Fluid Mech. 637, 425–442.CrossRefGoogle Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd Edn. Cambridge University Press.Google Scholar
Zeman, O. 1994 A note on the spectra and decay of rotating homogeneous turbulence. Phys. Fluids 6, 3221–3223.CrossRefGoogle Scholar