Multiscaling in Strong Turbulence Driven by a Random Force (original) (raw)
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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 −
Anomalous scaling in a model of hydrodynamic turbulence with a small parameter
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.
Anomalous scaling of low-order structure functions of turbulent velocity
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.
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.
Anomalous scaling in fluid mechanics: The case of the passive scalar
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.
Anomalous Scaling and Generic Structure Function in Turbulence
Journal de Physique II, 1996
We discuss on an example a general mechanism of apparition of anomalous scaling in scale invariant systems via zero modes of a scale invariant operator. We discuss the relevance of such mechanism in turbulence, and point out a peculiarity of turbulent flows, due to the existence of both forcing and dissipation. Following these considerations, we show that if this mechanism of anomalous scaling is operating in turbulence, the structure functions can be constructed by simple symmetry considerations. We find that the generical scale behavior of structure functions in the inertial range is not self-similar Sn( ) ∝ ζn but includes an "exponential self-similar" behavior Sn( ) ∝ exp[ζnα −1 α ] where α is a parameter proportional to the inverse of the logarithm of the Reynolds number. The solution also follows exact General Scaling and approximate Extended Self-Similarity.
Anomalous and dimensional scaling in anisotropic turbulence
Physical Review E, 2002
We present a numerical study of anisotropic statistical fluctuations in homogeneous turbulent flows. We give an argument to predict the dimensional scaling exponents, ζ j d (p) = (p + j)/3, for the projections of p-th order structure function in the j-th sector of the rotational group. We show that measured exponents are anomalous, showing a clear deviation from the dimensional prediction. Dimensional scaling is subleading and it is recovered only after a random reshuffling of all velocity phases, in the stationary ensemble. This supports the idea that anomalous scaling is the result of a genuine inertial evolution, independent of large-scale behavior.
Viscous Lengths in Hydrodynamic Turbulence are Anomalous Scaling Functions
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.
Asymptotic exponents from low-Reynolds-number flows
New Journal of Physics, 2007
The high-order statistics of fluctuations in velocity gradients in the crossover range from the inertial to the Kolmogorov and sub-Kolmogorov scales are studied by direct numerical simulations (DNS) of homogeneous isotropic turbulence with vastly improved resolution. The derivative moments for orders 0 n 8 are represented well as powers of the Reynolds number, Re, in the range 380 Re 5275, where Re is based on the periodic box length L x . These low-Reynolds-number flows give no hint of scaling in the inertial range even when extended self-similarity is applied. Yet, the DNS scaling exponents of velocity gradients agree well with those deduced, using a recent theory of anomalous scaling, from the scaling exponents of the longitudinal structure functions at infinitely high Reynolds numbers. This suggests that the asymptotic state of turbulence is attained for the velocity gradients at far lower Reynolds numbers than those required for the inertial range to appear. We discuss these findings in the light of multifractal formalism. Our numerical studies also resolve the crossover of the velocity gradient statistics from Gaussian to non-Gaussian behaviour that occurs as the Reynolds number is increased.