Some comments on scaling exponents of turbulence (original) (raw)
Journal of Statistical Physics, 2000
This manuscript is a draft of work in progress, meant for network distribution only. It will be updated to a formal preprint when the numerical calculations will be accomplished. In this draft we develop a consistent closure procedure for the calculation of the scaling exponents ζn of the nth order correlation functions in fully developed hydrodynamic 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 ζn. This hierarchy was discussed in detail in a recent publication [V.S. L'vov and I. Procaccia, Phys. Rev. E, submitted, chao-dyn/970507015]. The scaling exponents in this set of equations cannot be found from power counting. In this draft we discuss in detail low order non-trivial closures of this infinite set of equations, and prove that these closures lead 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 re-normalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalizaiton scale as the outer scale of turbulence L.
A new approach to computing the scaling exponents in fluid turbulence from first principles
Physica A: Statistical Mechanics and its Applications, 1998
In this short paper we describe the essential ideas behind a new consistent closure procedure for the calculation of the scaling exponents n of the nth order correlation functions in fully developed hydrodynamic turbulence, starting from ÿrst principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an inÿnite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents n. This hierarchy was discussed in detail in a recent publication [V.S. L'vov and I. Procaccia, Physica A (1998), in press, chao-dyn=9707015]. The scaling exponents in this set of equations cannot be found from power counting. In this short paper we discuss in detail low order non-trivial closures of this inÿnite set of equations, and prove that these closures lead to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-di erential equations, re ecting 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 ÿtting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The H older inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L.
Logarithmic scaling in the near-dissipation range of turbulence
Pramana, 2005
A logarithmic scaling for structure functions, in the form Sp ∼ [ln(r/η)] ζp , where η is the Kolmogorov dissipation scale and ζp are the scaling exponents, is suggested for the statistical description of the near-dissipation range for which classical power-law scaling does not apply. From experimental data at moderate Reynolds numbers, it is shown that the logarithmic scaling, deduced from general considerations for the near-dissipation range, covers almost the entire range of scales (about two decades) of structure functions, for both velocity and passive scalar fields. This new scaling requires two empirical constants, just as the classical scaling does, and can be considered the basis for extended self-similarity.
Scaling Behavior in Turbulence is Doubly Anomalous
Physical Review Letters, 1996
It is shown that the description of anomalous scaling in turbulent systems requires the simultaneous use of two normalization scales. This phenomenon stems from the existence of two independent (infinite) sets of anomalous scaling exponents that appear in leading order, one set due to infrared anomalies, and the other due to ultraviolet anomalies. To expose this clearly we introduce here a set of local fields whose correlation functions depend simultaneously on the the two sets of exponents. Thus the Kolmogorov picture of "inertial range" scaling is shown to fail because of anomalies that are sensitive to the two ends of this range. PACS numbers 47.27.Gs, 47.27.Jv, 05.40.+j  0 = (ρ · ∇ ′ ) 0 ≡ 1 , 2 = (ρ · ∇ ′ ) 2 −
Linking Eulerian and Lagrangian structure functions’ scaling exponents in turbulence
Physica A: Statistical Mechanics and its Applications, 2006
In fully developed turbulence, intermittency is classically characterized by z E ðqÞ, the Eulerian scaling exponent of structure functions. The same approach can be used in a Lagrangian framework, using z L ðqÞ to characterize the temporal intermittency of the velocity of a particle advected by a turbulent intermittent field. An interesting question is then to know how to relate the scaling functions and explore the links between z L ðqÞ and z E ðqÞ. We first provide different transformations between these functions, associated to different transformations linking space and time scales. We test these relations using experimental estimates for the two functions. For small and medium order of moments they are both close to data. For larger moments, experimental estimates have still too much scatter to conclude. The present paper underlines the need of more precise estimates and provides a methodology for future comparisons of Eulerian and Lagrangian scaling exponents.
A new scaling property of turbulent flows
1995
We discuss a possible theoretical interpretation of the self scaling property of turbulent flows (Extended Self Similarity). Our interpretation predicts that, even in cases when ESS is not observed, a generalized self scaling, must be observed. This prediction is checked on a number of laboratory experiments and direct numerical simulations.
Scaling property of turbulent flows
Physical Review E, 1996
We discuss a possible theoretical interpretation of the self scaling property of turbulent flows (Extended Self Similarity). Our interpretation predicts that, even in cases when ESS is not observed, a generalized self scaling, must be observed. This prediction is checked on a number of laboratory experiments and direct numerical simulations.
Scaling Relations in Elastic Turbulence
Physical Review Letters, 2019
We report the scaling relations between the exponents of the power-law decays of kinetic and elastic energies, pressure, as well as torque fluctuations in elastic turbulence (ET). The relations are derived by estimating that the divergent part of the elastic stress is much larger than its vortical part, and its contribution into the full elastic stress is dominant in the range of the power spectrum amplitudes observed experimentally in ET. The estimate is in line with polymer stretching by flow: the polymers are stretched mostly by the divergent part associated with a strain rate, whereas a rotational, or vortical, flow plays a minor role in the polymer stretching. The scaling relations agree well with the exponent values obtained experimentally and numerically in the ET regime of a viscoelastic fluid in different flow geometries.
Scaling laws and dissipation scale of a passive scalar in fully developed turbulence
Physica D-nonlinear Phenomena, 1996
The scaling laws of the temperature structure functions and their relation with those of velocity have been experimentally studied. The relationship between the dissipative scales for velocity and the temperature is first investigated. In agreement with recent numerical simulations of Pumir (1994), it is found that, for Prandtl number close to 1, the dissipation scale for a scalar is smaller than that of velocity. Thus temperature structure functions present a larger scaling interval than that of the velocity. The intermittent corrections of scaling are then analyzed. It is shown that, as proposed in literature (R. Benzi et al., 1992), the second-order structure function is affected only by the velocity intermittency. This structure function is then used as the reference for testing the applicability of the extended self-similarity (ESS) to the passive scalar case. ESS holds, but in a narrower interval than that observed in velocity statistics. Finally, a hierarchy for the temperature structure functions, similar to that proposed by She and Leveque (1994) for the velocity field, is introduced and experimentally tested.