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Journal of Statistical Physics

, Volume 93, Issue 3–4, pp 797–832 | Cite as

Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multiscaling

  • Victor I. Belinicher
  • Victor S. L'vov
  • Anna Pomyalov
  • Itamar Procaccia
Article

Abstract

We develop a consistent closure procedure for the calculation of the scaling exponents ζ n of the nth-order correlation functions in fully developed hydro-dynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ n . This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest non-trivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier–Stokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to non-Kolmogorov values of the scaling exponents ζ n . Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.

Turbulence universal statistics anomalous scaling multi-fractals 

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Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • Victor I. Belinicher
    • 1
    • 2
  • Victor S. L'vov
    • 1
    • 3
  • Anna Pomyalov
    • 1
  • Itamar Procaccia
    • 1
  1. 1.Department of Chemical PhysicsWeizmann Institute of ScienceRehovotIsrael
  2. 2.Institute for Semiconductor PhysicsRussian Academy of ScienceNovosibirskRussia
  3. 3.Institute of Automatization and ElectrometryRusisan Academy of ScienceNovosibirskRussia

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