Beyond Scaling and Locality in Turbulence

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.

Progress of Theoretical Physics Supplement, 1998

Thrbulence velocity measurements have been made in the surface layer of the atmosphere at Taylor microscale Reynolds numbers between 10,000 and 20,000. Even at these high Reynolds numbers, the structure functions do not scale unambiguously. It is shown that the scaling improves significantly by implementing a plausible correction due to the mean shear. For second and fourth order structure functions, the exponents for the corrected data are close to those determined by extended self-similarity (ESS). ESS improves scaling enormously for all orders, and is used to obtain exponents for moment orders between-0.08 and 10. Anomaly prevails even for very low orders. A major qualitative conclusion is that it is difficult to discuss the scaling effectively without first understanding quantitatively the effects of finite shear and finite Reynolds numbers.

Physical Review E, 2008

We analyze the data stemming from a forced incompressible hydrodynamic simulation on a grid of 2048 3 regularly spaced points, with a Taylor Reynolds number of R λ ∼ 1300. The forcing is given by the Taylor-Green flow, which shares similarities with the flow in several laboratory experiments, and the computation is run for ten turnover times in the turbulent steady state. At this Reynolds number the anisotropic large scale flow pattern, the inertial range, the bottleneck, and the dissipative range are clearly visible, thus providing a good test case for the study of turbulence as it appears in nature. Triadic interactions, the locality of energy fluxes, and structure functions of the velocity increments are computed. A comparison with runs at lower Reynolds numbers is performed, and shows the emergence of scaling laws for the relative amplitude of local and non-local interactions in spectral space. The scalings of the Kolmogorov constant, and of skewness and flatness of velocity increments, performed as well and are consistent with previous experimental results. Furthermore, the accumulation of energy in the small-scales associated with the bottleneck seems to occur on a span of wavenumbers that is independent of the Reynolds number, possibly ruling out an inertial range explanation for it. Finally, intermittency exponents seem to depart from standard models at high R λ , leaving the interpretation of intermittency an open problem.

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.

Physical Review Letters, 1994

We study analytically and numerically the corrections to scaling in turbulence which arise due to the finite ratio of the outer scale L of turbulence to the viscous scale η, i.e., they are due to finite size effects as anisotropic forcing or boundary conditions at large scales. We find that the deviations δζ m from the classical Kolmogorov scaling ζ m = m/3 of the velocity moments |u(k)| m ∝ k −ζm decrease like δζ m (Re) = c m Re −3/10 . Our numerics employ a reduced wave vector set approximation for which the small scale structures are not fully resolved. Within this approximation we do not find Re independent anomalous scaling within the inertial subrange. If anomalous scaling in the inertial subrange can be verified in the large Re limit, this supports the suggestion that small scale structures should be responsible, originating from viscosity either in the bulk (vortex tubes or sheets) or from the boundary layers (plumes or swirls).

Physics of Fluids, 2008

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 −

Physical Review Letters, 1996

It has been recently suggested that the various qth order velocity structure functions in turbulent flows are related to each other through well defined scaling laws, which extend outside the usual inertial domain. Even if theoretical models provide different scaling laws for fluid and magnetohydrodynamic turbulent flows, no attempt has been made up to now to furnish experimental evidence for these differences. By using measurements from the solar wind turbulence and from turbulence in ordinary fluid flows, we show that the differences can be observed only by looking at the high-order velocity structure functions. PACS numbers: 47.27.i, 47.65.+a, 95.30.Qd The most interesting aspect of fully developed turbulence is the existence of universal scaling behavior of small-scale Iluctuations [1]. Indeed, turbulent Ilows are characterized by the presence of self-similarity in the inertial range, that is, on the spatial scale A « f, « L (L is the energy injection scale and A is the dissipative scale). In this range fluctuations reach an equilibrium state characterized by a continuous flux of energy from the scale L up to A, which can be viewed in the real space as an energy cascade generated by the breakdown of eddies at different length scales (Richardson's energy cascade). The main statistical feature has been evidenced by Kolmogorov [2,3], who derived the well known relation aV, -. , '"r'I' (I) between the velocity differences AVE =~V (x + 8)-V(x)~across the distance f, =~Z~[ V(x) being the velocity field] and et, which represents the energy transfer rate per unit mass [4]. The same picture can be extended to magnetohydrodynamic (MHD) turbulence, even if, owing to the decorrelation effect of the large-scale magnetic field [5], the nonlinear interactions are lowered. This originates the Kraichnan scaling relatioñ V, -(c"«) i'~'I' (2) (c~= Bn/$47r p is the Alfven velocity of the large-scale magnetic field Bo and p is the constant plasma mass density), which is valid for hydromagnetic flows. The interesting measurable quantities are the qth order velocity structure function Se = (BVe) (brackets denoting space (q) q average) and the scaling exponents $~d efined through (q) (3) which must hold in the inertial range [2,3]. In the absence of intermittency the linear scaling laws hold, seq = q/m (I = 3 for the Kolmogorov scaling law and I = 4 for the Kraichnan scaling law). If intermittency is taken into account, anomalous scaling laws are obtained where the scaling exponents g~a re nonlinear functions of q both in fluid flows [6,7] and in MHD Ilows [8 -10]. Cascade models based on Richardson's picture have been developed in order to take into account intermittency corrections to the classical scaling laws, in both the fluid [11 -14] and the MHD case [10,15 -17]. Among others She and Leveque [14] developed a model based on the hypotheses that the Kolmogorov refined similarity (1) is verified and that the moments of the energy transfer rate obey a given hierarchy (q+ 1) E'g (4) (0 ( p ( 1). The quantity et, obtained from the ( ) hierarchy in the limit q~~, is associated with the most intermittent structures. It has been conjectured that this quantity has a divergent scaling e~-4 . On the basis (oo) of relation (2), and assuming the same hierarchy (4), the model has been extended to MHD by Grauer, Krug, and Marliani [16] and independently by Politano and Pouquet [17]. An expression for the scaling exponents valid in both cases can be derived as $q= -(1x)+C 1 -1 -ffl C

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.

Physical Review Letters, 1996

It is shown that the idea that scaling behavior in turbulence is limited by one outer length L and one inner length η is untenable. Every n'th order correlation function of velocity differences F n(R1, R2,. . .) exhibits its own cross-over length ηn to dissipative behavior as a function of, say, R1. This length depends on n and on the remaining separations R2, R3,. . .. One result of this Letter is that when all these separations are of the same order R this length scales like ηn(R) ∼ η(R/L) xn with xn = (ζn − ζn+1 + ζ3 − ζ2)/(2 − ζ2), with ζn being the scaling exponent of the n'th order structure function. We derive a class of scaling relations including the "bridge relation" for the scaling exponent of dissipation fluctuations µ = 2 − ζ6.