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Extended self-similarity works for the Burgers equation and why

  • SAGAR CHAKRABORTY (a1) (a2), URIEL FRISCH (a3) and SAMRIDDHI SANKAR RAY (a4)
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.

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Corresponding author
Email address for correspondence: uriel@obs-nice.fr
References
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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, 53305333.
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, 467486.
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, 519543.
Bec J., Frisch U. & Khanin K. 2000 Kicked Burgers turbulence. J. Fluid Mech. 416, 239267, arXiv:chao-dyn/991000.
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].
Benzi R., Ciliberto S., Baudet C. & Chavarria G. R. 1995 On the scaling of three-dimensional homogeneous and isotropic turbulence. Physica D 80, 385398.
Benzi R., Ciliberto S., Tripiccione R., Baudet C., Massaioli F. & Succi S. 1993 Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29R32.
Bhattacharjee J. K. & Sain A. 1999 Homogeneous isotropic turbulence: large momentum expansion. Physica A 270, 165172.
Biferale L. & Procaccia I. 2005 Anisotropy in turbulent flows and in turbulent transport. Phys. Rep. 414, 43164.
Burgers J. M. 1974 The Nonlinear Diffusion Equation. D. Reidel.
Cole J. D. 1951 On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225236.
Cox S. M. & Matthews P. C. 2002 Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430455.
Falkovich G., Gawedzki K. & Vergassola M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913975.
Frisch U. 1995 Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press.
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, 97103.
Frisch U. & Vergassola M. 1991 A prediction of the multifractal model: the intermediate dissipation range. Europhys. Lett. 14, 439444.
Fujisaka H. & Grossman S. 2001 Scaling hypothesis leading to extended self-similarity in turbulence. Phys. Rev. E 63, 026305.
Gurbatov S. N., Simdyankin S. I., Aurell E., Frisch U. & Toth G. 1997 On the decay of Burgers turbulence. J. Fluid Mech. 344, 339374.
van der Hoeven J. 2009 On asymptotic extrapolation. J. Symb. Comput. 44, 10001016.
Hopf E. 1950 The partial differential equation (u t+uu x = u xx. Commun. Pure Appl. Math. 3, 201230.
Kassam A. K. & Trefethen L. N. 2005 Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 12141233.
Kolmogorov A. N. 1941 Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 1618.
Kraichnan R. 1968 Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945953.
Kraichnan R. 1994 Anomalous scaling of a randomly advected passive scalar. Phys. Rev. Lett. 72, 10161019.
Meneveau C. 1996 Transition between viscous and inertial-range scaling of turbulence structure functions. Phys. Rev. E 54, 36573663.
Mitra D., Bec J., Pandit R. & Frisch U. 2005 Is multiscaling an artifact in the stochastically forced Burgers equation? Phys. Rev. Lett. 94, 194501.
Monin A. S. & Yaglom A. M. 1971 Statistical Fluid Mechanics, vol. 2 (ed. Lumley J.). MIT Press.
Paladin G. & Vulpiani A. 1987 Anomalous scaling and generalized Lyapunov exponents of the one-dimensional Anderson model. Phys. Rev. B 35, 20152020.
Pauls W. & Frisch U. 2007 A Borel transform method for locating singularities of Taylor and Fourier series. J. Stat. Phys. 127, 10951119.
Sain A. & Bhattacharjee J. K. 1999 Extended self-similarity and dissipation range dynamics of three-dimensional turbulence. Phys. Rev. E 60, 571577.
Schumacher J., Sreenivasan K. R. & Yakhot V. 2007 Asymptotic exponents from low-Reynolds-number flows. New J. Phys. 9, 89.
Segel D., L'vov V. & Procaccia I. 1996 Extended self-similarity in turbulent systems: an analytically soluble example. Phys. Rev. Lett. 76, 18281831.
Vainshtein S. I. & Sreenivasan K. R. 1994 Kolmogorov's (4/5)th law and intermittency in turbulence. Phys. Rev. Lett. 73, 30853088.
Yakhot V. 2001 Mean-field approximation and extended self-similarity in turbulence. Phys. Rev. Lett. 87, 234501.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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