Hostname: page-component-7dd5485656-frp75 Total loading time: 0 Render date: 2025-10-31T12:01:39.193Z Has data issue: false hasContentIssue false
  • English
  • Français

Revisiting Batchelor's theory of two-dimensional turbulence

Published online by Cambridge University Press:  30 October 2007

DAVID G. DRITSCHEL ,
CHUONG V. TRAN  and
RICHARD K. SCOTT
DAVID G. DRITSCHEL
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
CHUONG V. TRAN
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
RICHARD K. SCOTT
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a χ2/3k−1 enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation χ in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes.

We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing χ in the limit Re → ∞. Our proposal is supported by high Reynolds number simulations which confirm that χ decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy 〈ω2〉/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing χ, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum χ2/3 k−1 with 〈ω2〉 k−1 (ln Re)−1).

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 591 , 25 November 2007 , pp. 379 - 391
Copyright
Copyright © Cambridge University Press 2007

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Bartello, P. & Warn, T. 1996 Self-similarity of decaying two-dimensional turbulence. J. Fluid Mech. 326, 357–372.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12, 233–239.Google Scholar
Benzi, R., Patarnello, S. & Santangelo, P. 1988 Self-similar coherent structures in two-dimensional decaying turbulence. J. Phys. A: Math. Gen. 21, 1221–1237.Google Scholar
Chasnov, J. R. 1997 On the decay of two-dimensional homogeneous turbulence. Phys. Fluids 9, 171–180.CrossRefGoogle Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
DiPerna, R. J. & Lions, P.-L. 1989 Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547.CrossRefGoogle Scholar
DiPerna, R. J. & Majda, A. J. 1987 Concentrations in regularizations for 2-D incompressible flow. Commun. Pure Appl. Maths 40, 301–345.Google Scholar
Dmitruk, P. & Montgomery, D. C. 2005 Numerical study of the decay of enstrophy in a two-dimensional Navier–Stokes fluid in the limit of very small viscosities. Phys. Fluids 17, 035114.Google Scholar
Dritschel, D. G. 1989 Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Computer Phys. Rep. 10, 77–146.Google Scholar
Eyink, G. L. 2001 Dissipation in turbulence solutions of 2D Euler equations. Nonlinearity 14, 787–802.CrossRefGoogle Scholar
Hou, T. Y. & Li, R. 2006 Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. J. Nonlinear Sci. 16, 639–664.Google Scholar
Kida, S., Yamada, M. & Ohkitani, K. 1988 The energy spectrum in the universal range of two-dimensional turbulence. Fluid Dyn. Res. 4, 271–301.Google Scholar
Lopes Filho, M. C., Mazzucato, A. L. & Nussenzveig Lopes, H. J. 2006 Weak solutions, renormalized solutions, and enstrophy defects in 2D turbulence. Arch. Rat. Mech. Anal. 179, 353–387.Google Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.Google Scholar
Santangelo, P., Benzi, R. & Legras, B. 1989 The generation of vortices in high-resolution, two-dimensional decaying turbulence and the influence of initial conditions on the breaking of self-similarity. Phys. Fluids A 1, 1027–1034.CrossRefGoogle Scholar
Tran, C. V. & Shepherd, T. G. 2002 Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence. Physica D 165, 199–212.Google Scholar
Tran, C. V. & Dritschel, D. G. 2006 Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit. J. Fluid Mech. 559, 107–116.Google Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in 2 dimensions. J. Comput. Phys. 30, 96–106.Google Scholar