Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-10T05:54:59.137Z Has data issue: false hasContentIssue false
  • English
  • Français

Reynolds-number dependency in homogeneous, stationary two-dimensional turbulence

Published online by Cambridge University Press:  08 March 2010

ANNALISA BRACCO  and
JAMES C. MCWILLIAMS
ANNALISA BRACCO*
Affiliation:
EAS and CNS, Georgia Institute of Technology, Atlanta, GA 30332, USA
JAMES C. MCWILLIAMS
Affiliation:
Department of Atmospheric and Oceanic Sciences and IGPP, UCLA, Los Angeles, CA 90095, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Turbulent solutions of the two-dimensional Navier–Stokes equations are a paradigm for the chaotic space–time patterns and equilibrium distributions of turbulent geophysical and astrophysical ‘thin’ flows on large horizontal scales. Here we investigate how homogeneous, stationary two-dimensional turbulence varies with the Reynolds number (Re) in stationary solutions with large-scale, random forcing and viscous diffusion, also including hypoviscous diffusion to limit the inverse energy cascade. This survey is made over the computationally feasible range in Re ≫ 1, approximately between 1.5 × 103 and 5.6 × 106. For increasing Re, we witness the emergence of vorticity fine structure within the filaments and vortex cores. The energy spectrum shape approaches the forward-enstrophy inertial-range form k−3 at large Re, and the velocity structure function is independent of Re. All other statistical measures investigated in this study exhibit power-law scaling with Re, including energy, enstrophy, dissipation rates and the vorticity structure function. The scaling exponents depend on the forcing properties through their influences on large-scale coherent structures, whose particular distributions are non-universal. A striking result is the Re independence of the intermittency measures of the flow, in contrast with the known behaviour for three-dimensional homogeneous turbulence of asymptotically increasing intermittency. This is a consequence of the control of the tails of the distribution functions by large-scale coherent vortices. Our analysis allows extrapolation towards the asymptotic limit of Re → ∞, fundamental to geophysical and astrophysical regimes and their large-scale simulation models where turbulent transport and dissipation must be parameterized.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 646 , 10 March 2010 , pp. 517 - 526
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

Alexakis, A. & Doering, C. R. 2006 Energy and enstrophy dissipation in steady state two-dimensional turbulence. Phys. Lett. A 359, 652–657.CrossRefGoogle Scholar
Babiano, A. & Provenzale, A. 2007 Coherent vortices and tracer cascades in two-dimensional turbulence. J. Fluid Mech. 574, 429–448.CrossRefGoogle Scholar
Batchelor, G. K. 1969 Computation of energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. 12, 233–239.CrossRefGoogle Scholar
Benzi, R., Paladin, G. & Vulpiani, A. 1990 Power spectra in two-dimensional turbulence. Phys. Rev. A 42, 3654–3656.CrossRefGoogle ScholarPubMed
Boffetta, G. 2007 Energy and enstrophy fluxes in the double cascade of two-dimensional turbulence. J. Fluid Mech. 589, 253–260.CrossRefGoogle Scholar
Boffetta, G., Celani, A. & Vergassola, M. 2000 Inverse energy cascade in two-dimensional turbulence: deviations from Gaussian behaviour. Phys. Rev. E 61, R29–R32.CrossRefGoogle Scholar
Borue, V. 1993 Spectral exponents of enstrophy cascade in stationary two-dimensional homogeneous turbulence. Phys. Rev. Lett. 71, 3967–3970.CrossRefGoogle ScholarPubMed
Borue, V. 1994 Inverse energy cascade in stationary two-dimensional homogeneous turbulence. Phys. Rev. Lett. 72, 1475–1478.CrossRefGoogle ScholarPubMed
Bracco, A., LaCasce, J. H., Pasquero, C. & Provenzale, A. 2000 a The velocity distribution of barotropic turbulence. Phys. Fluids 12 (10), 2478–2488.CrossRefGoogle Scholar
Bracco, A., McWilliams, J. C., Murante, G., Provenzale, A. & Weiss, J. B. 2000 b Revisiting freely decaying two-dimensional turbulence at millennial resolution. Phys. Fluids 12 (11), 2931–2941.CrossRefGoogle Scholar
Charney, J. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1095.2.0.CO;2>CrossRefGoogle Scholar
Eyink, Gregory L. 1995 Exact results on scaling exponents in the two-dimensional enstrophy cascade. Phys. Rev. Lett. 74 (19), 3800–3803.CrossRefGoogle Scholar
Falkovich, G. & Lebedev, V. 1994 Universal direct cascade in two-dimensional turbulence. Phys. Rev. E 50, 3883–3899.CrossRefGoogle ScholarPubMed
Gotoh, T. 1998 Energy spectrum in the inertial and dissipation ranges of two-dimensional steady turbulence. Phys. Rev. E 57, 2984–2992.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 Local structure of turbulence in an incompressible fluid for very large Reynolds number. Dokl. Akad. Nauk. SSSR 30, 19–21.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.CrossRefGoogle Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Kraichnan, R. H. 1971 Inertial range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535.CrossRefGoogle Scholar
Ladyzhenskaya, O. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.Google Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.CrossRefGoogle Scholar
McWilliams, J. C. 1990 The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361–385.CrossRefGoogle Scholar
Paret, J. & Tabeling, P. 1997 Experimental observation of the two-dimensional inverse energy cascade. Phys. Rev. Lett. 79, 2486–2489.CrossRefGoogle Scholar
Pasquero, C. & Falkovich, G. 2002 Stationary spectrum of vorticity cascade in two-dimensional turbulence. Phys. Rev. E 65, 056305.CrossRefGoogle ScholarPubMed
Rose, H. A. & Sulem, P.-L. 1978 Fully developed turbulence and statistical mechanics. J. Phys. France 39, 441–484.CrossRefGoogle Scholar
Schorghofer, N. 2000 a Energy spectra of steady two-dimensional turbulent flows. Phys. Rev. E 61 (6), 6572–6577.CrossRefGoogle ScholarPubMed
Schorghofer, N. 2000 b Universality of probability distributions among two-dimensional turbulent flows. Phys. Rev. E 61 (6), 6568–6571.CrossRefGoogle ScholarPubMed
Sreenivasan, K. & Antonia, R. 1997 The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech. 29, 435–472.CrossRefGoogle Scholar