Hostname: page-component-669899f699-g7b4s Total loading time: 0 Render date: 2025-05-04T12:59:06.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Extended self-similarity works for the Burgers equation and why

Published online by Cambridge University Press:  13 April 2010

SAGAR CHAKRABORTY ,
URIEL FRISCH  and
SAMRIDDHI SANKAR RAY
SAGAR CHAKRABORTY
Affiliation:
NBIA, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark Theoretical Sciences, SNBNCBS, Kolkata-98, India
URIEL FRISCH*
Affiliation:
UNS, CNRS, Laboratoire Cassiopée, OCA, B.P. 4229, 06304 Nice Cedex 4, France
SAMRIDDHI SANKAR RAY
Affiliation:
Department of Physics, Indian Institute of Science, Bangalore, India
*
†Email address for correspondence: uriel@obs-nice.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Extended self-similarity (ESS), a procedure that remarkably extends the range of scaling for structure functions in Navier–Stokes turbulence and thus allows improved determination of intermittency exponents, has never been fully explained. We show that ESS applies to Burgers turbulence at high Reynolds numbers and we give the theoretical explanation of the numerically observed improved scaling at both the IR and UV end, in total a gain of about three quarters of a decade: there is a reduction of subdominant contributions to scaling when going from the standard structure function representation to the ESS representation. We conjecture that a similar situation holds for three-dimensional incompressible turbulence and suggest ways of capturing subdominant contributions to scaling.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 649 , 25 April 2010 , pp. 275 - 285
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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Arad, I., Dhruva, B., Kurien, S. L'vov, V. S., Procaccia, I. & Sreenivasan, K. R. 1998 Extraction of anisotropic contributions in turbulent flows. Phys. Rev. Lett. 81, 5330–5333.CrossRefGoogle Scholar
Aurell, E., Frisch, U., Lutsko, J. & Vergassola, M. 1992 On the multifractal properties of the energy dissipation derived from turbulence data. J. Fluid Mech. 238, 467–486.CrossRefGoogle Scholar
Bardos, C., Frisch, U., Pauls, W., Ray, S. S. & Titi, E. S. 2010 Entire solutions of hydrodynamical equations with exponential dissipation. Commun. Math. Phys. 293, 519–543.CrossRefGoogle Scholar
Bec, J., Frisch, U. & Khanin, K. 2000 Kicked Burgers turbulence. J. Fluid Mech. 416, 239–267, arXiv:chao-dyn/991000.CrossRefGoogle Scholar
Benzi, R., Biferale, L., Fisher, R., Lamb, D. Q. & Toschi, F. 2009 Eulerian and Lagrangian statistics from high resolution numerical simulations of weakly compressible turbulence. J. Fluid Mech. (in press). arXiv:0905.0082 [physics.flu-dyn].Google Scholar
Benzi, R., Ciliberto, S., Baudet, C. & Chavarria, G. R. 1995 On the scaling of three-dimensional homogeneous and isotropic turbulence. Physica D 80, 385–398.CrossRefGoogle Scholar
Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F. & Succi, S. 1993 Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29–R32.CrossRefGoogle ScholarPubMed
Bhattacharjee, J. K. & Sain, A. 1999 Homogeneous isotropic turbulence: large momentum expansion. Physica A 270, 165–172.Google Scholar
Biferale, L. & Procaccia, I. 2005 Anisotropy in turbulent flows and in turbulent transport. Phys. Rep. 414, 43–164.CrossRefGoogle Scholar
Burgers, J. M. 1974 The Nonlinear Diffusion Equation. D. Reidel.Google Scholar
Cole, J. D. 1951 On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236.CrossRefGoogle Scholar
Cox, S. M. & Matthews, P. C. 2002 Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455.Google Scholar
Falkovich, G., Gawedzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975.Google Scholar
Frisch, U. 1995 Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press.Google Scholar
Frisch, U., Afonso, M. M., Mazzino, A. & Yakhot, V. 2005 Does multifractal theory of turbulence have logarithms in the scaling relations? J. Fluid Mech. 542, 97–103.CrossRefGoogle Scholar
Frisch, U. & Vergassola, M. 1991 A prediction of the multifractal model: the intermediate dissipation range. Europhys. Lett. 14, 439–444.CrossRefGoogle Scholar
Fujisaka, H. & Grossman, S. 2001 Scaling hypothesis leading to extended self-similarity in turbulence. Phys. Rev. E 63, 026305.CrossRefGoogle ScholarPubMed
Gurbatov, S. N., Simdyankin, S. I., Aurell, E., Frisch, U. & Toth, G. 1997 On the decay of Burgers turbulence. J. Fluid Mech. 344, 339–374.CrossRefGoogle Scholar
van der Hoeven, J. 2009 On asymptotic extrapolation. J. Symb. Comput. 44, 1000–1016.CrossRefGoogle Scholar
Hopf, E. 1950 The partial differential equation (u t+uu x = u xx. Commun. Pure Appl. Math. 3, 201–230.CrossRefGoogle Scholar
Kassam, A. K. & Trefethen, L. N. 2005 Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214–1233.Google Scholar
Kolmogorov, A. N. 1941 Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18.Google Scholar
Kraichnan, R. 1968 Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–953.CrossRefGoogle Scholar
Kraichnan, R. 1994 Anomalous scaling of a randomly advected passive scalar. Phys. Rev. Lett. 72, 1016–1019.CrossRefGoogle ScholarPubMed
Meneveau, C. 1996 Transition between viscous and inertial-range scaling of turbulence structure functions. Phys. Rev. E 54, 3657–3663.CrossRefGoogle ScholarPubMed
Mitra, D., Bec, J., Pandit, R. & Frisch, U. 2005 Is multiscaling an artifact in the stochastically forced Burgers equation? Phys. Rev. Lett. 94, 194501.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, vol. 2 (ed. Lumley, J.). MIT Press.Google Scholar
Paladin, G. & Vulpiani, A. 1987 Anomalous scaling and generalized Lyapunov exponents of the one-dimensional Anderson model. Phys. Rev. B 35, 2015–2020.CrossRefGoogle ScholarPubMed
Pauls, W. & Frisch, U. 2007 A Borel transform method for locating singularities of Taylor and Fourier series. J. Stat. Phys. 127, 1095–1119.CrossRefGoogle Scholar
Sain, A. & Bhattacharjee, J. K. 1999 Extended self-similarity and dissipation range dynamics of three-dimensional turbulence. Phys. Rev. E 60, 571–577.CrossRefGoogle ScholarPubMed
Schumacher, J., Sreenivasan, K. R. & Yakhot, V. 2007 Asymptotic exponents from low-Reynolds-number flows. New J. Phys. 9, 89.CrossRefGoogle Scholar
Segel, D., L'vov, V. & Procaccia, I. 1996 Extended self-similarity in turbulent systems: an analytically soluble example. Phys. Rev. Lett. 76, 1828–1831.CrossRefGoogle ScholarPubMed
Vainshtein, S. I. & Sreenivasan, K. R. 1994 Kolmogorov's (4/5)th law and intermittency in turbulence. Phys. Rev. Lett. 73, 3085–3088.CrossRefGoogle Scholar
Yakhot, V. 2001 Mean-field approximation and extended self-similarity in turbulence. Phys. Rev. Lett. 87, 234501.Google Scholar