Scaling Behavior in Turbulence is Doubly Anomalous

Europhysics Letters (EPL), 2000

The major difficulty in developing theories for anomalous scaling in hydrodynamic turbulence is the lack of a small parameter. In this Letter we introduce a shell model of turbulence that exhibits anomalous scaling with a tunable small parameter. The small parameter ǫ represents the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. We show that in the limit N → ∞ anomalous scaling sets in proportional to ǫ 4 . Moreover we give strong evidences that the birth of anomalous scaling appears at a finite critical ǫ, being ǫc ≈ 0.6.

Physical Review E, 1994

A mechanism for anomalous scaling in turbulent advection of passive scalars is identified as being similar to a recently discovered mechanism in Navier-Stokes dynamics [V. V. Lebedev and V. S. L'vov, JETP Lett. 59, 577 (1994)l. This mechanism is demonstrated in the context of a passive scalar field that is driven by a rapidly varying velocity field. The mechanism is not perturbative, and its demonstration within renormalized perturbation theory calls for a resummation of infinite sets of diagrams. For the example studied here we make use of a small parameter, the ratio of the typical time scales of the passive scalar vs that of the velocity field, to classify the diagrams of renormalized perturbation theory such that the relevant ones can be resummed exactly. The main observation here, as in the Navier-Stokes counterpart, is that the dissipative terms lead to logarithmic divergences in the diagrammatic expansion, and these are resummed to an anomalous exponent. The anomalous exponent can be measured directly in the scaling behavior of the dissipation two-point correlation function, and it also affects the scaling laws of the structure functions. It is shown that when the structure functions exhibit anomalous scaling, the dissipation correlation function does not decay on length scales that are in the scaling range. The implication of our findings is that the concept of an "inertial range" in which the dissipative terms can be ignored is untenable. The consequences of this mechanism for other cases of possible anomalous scaling in turbulence are discussed.

Physical review, 1996

Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws Sn(R) ∼ R ζn , and the correlation of energy dissipation Kǫǫ(R) ∼ R -µ . The goal is to understand the exponents ζn and µ from first principles. In paper II of this series it was shown that the existence of an ultraviolet scale (the dissipation scale η) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted ∆. The exact resummation of ladder diagrams resulted in the calculation of ∆ which satisfies the scaling relation ∆ = 2 -ζ2. In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e. ζn not linear in n) with the renormalization scale being the infrared outer scale of turbulence L. It is shown that deviations from K41 scaling of Sn(R) (ζn = n/3) must appear if the correlation of dissipation is mixing (i.e. µ > 0). We derive an exact scaling relation µ = 2ζ2 -ζ4. We present analytic expressions for ζn for all n and discuss their relation to experimental data. One surprising prediction is that the time decay constant τn(R) ∝ R zn of Sn(R) scales independently of n: the dynamic scaling exponent zn is the same for all n-order quantities, zn = ζ2.

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.

Journal of Fluid Mechanics, 2005

It is now believed that the scaling exponents of moments of velocity increments are anomalous, or that the departures from Kolmogorov's (1941) self-similar scaling increase nonlinearly with the increasing order of the moment. This appears to be true whether one considers velocity increments themselves or their absolute values. However, moments of order lower than 2 of the absolute values of velocity increments have not been investigated thoroughly for anomaly. Here, we discuss the importance of the scaling of non-integer moments of order between +2 and −1, and obtain them from direct numerical simulations at moderate Reynolds numbers (Taylor microscale Reynolds numbers R λ 450) and experimental data at high Reynolds numbers (R λ ≈ 10,000). The relative difference between the measured exponents and Kolmogorov's prediction increases as the moment order decreases towards −1, thus showing that the anomaly that is manifest in high-order moments is present in low-order moments as well. This conclusion provides a motivation for seeking a theory of anomalous scaling as the order of the moment vanishes. Such a theory does not have to consider rare events-which may be affected by non-universal features such as shear-and so may be regarded as advantageous to consider and develop.

Communications on Pure and Applied Analysis, 2014

In fond memory of Mark I. Vishik.

Physical Review Letters, 2005

We analyze multi-point correlation functions of a tracer in an incompressible flow at scales far exceeding the scale L at which fluctuations are generated (quasi-equilibrium domain) and compare them with the correlation functions at scales smaller than L (turbulence domain). We demonstrate that the scale invariance can be broken in the equilibrium domain and trace this breakdown to the statistical integrals of motion (zero modes) as has been done before for turbulence. Employing

Physical Review Letters, 1999

We address scaling in inhomogeneous and anisotropic turbulent flows by decomposing structure functions into their irreducible representation of the SO(3) symmetry group which are designated by j, m indices. Employing simulations of channel flows with Re λ ≈ 70 we demonstrate that different components characterized by different j display different scaling exponents, but for a given j these remain the same at different distances from the wall. The j = 0 exponent agrees extremely well with high Re measurements of the scaling exponents, demonstrating the vitality of the SO(3) decomposition.

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.

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.