Searcharxiv⌕ Search

arXiv subjects

Itamar Procaccia

Publications and source records attributed to Itamar Procaccia.

At least 271 records · Page 15Linked to original sources
arXiv1997

Computing the Scaling Exponents in Fluid Turbulence from First Principles: Demonstration of Multi-scaling

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 $ζ_n$ of the $n$th 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 $ζ_n$. This hierarchy was discussed in detail in a recent publication [V.S. L'vov and I. Procaccia, Phys. Rev. E, submitted, chao-dyn/9707015]. 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 i necessary for any anomalous scaling appears at this point. The Holder inequalities on the scaling exponents select the renormalizaiton scale as the outer scale of turbulence $L$.

chao-dyn↗
arXiv1997

Computing the Scaling Exponents in Fluid Turbulence from First Principles: the Formal Setup

We propose a scheme for the calculation from the NS equations of the scaling exponents $ζ_n$ of the $n$th order correlators in fully developed hydrodynamic turbulence. The scheme is nonperturbative and constructed to respect the fundamental rescaling symmetry of the Euler equation. It constitutes an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of $ζ_n$. As a consequence $ζ_n$ are determined by solvability conditions and not from power counting. It is arguedthat in order to achieve such a formulation one must recognize that the many-point space-time correlators are not scale invariant in time arguments. The assumption of full scale invariance leads unavoidably to K41 exponents. It is argued the determination of of $ζ_n$ requires equations for infinitely many renormalized objects. One ca n however proceed in controlled successive approximations by truncations of the infinite hierarchy of equations. Clues as to how to truncate without reintroducing power counting can be obtained from renormalized perturbation theory. To this aim we show that the fully resummed perturbation theory is equivalent in its contents to the exact hierarchy of equations obeyed by the $n$th order correlators and Green's function. In light of this important result we can safely use finite resummations to construct successive closures. This paper presents the conceptual and technical details of the scheme. The analysis of the high-order closure procedures which do not destroy the rescaling symmetry and the actual calculations for truncated models will be presented in a forthcoming paper in collaboration with V. Belinicher.

chao-dyn↗
arXiv1997

Direct Numerical Simulations of the Kraichnan Model: Scaling Exponents and Fusion Rules

We present results from direct numerical simulations of the Kraichnan model for passive scalar advection by a rapidly-varying random scaling velocity field for intermediate values of the velocity scaling exponent. These results are compared with the scaling exponents predicted for this model by Kraichnan. Further, we test the recently proposed fusion rules which govern the scaling properties of multi-point correlations, and present results on the linearity of the conditional statistics of the Laplacian operator on the scalar field.

chao-dyn↗
arXiv1997

Invariants for Correlations of Velocity Differences in Turbulent Fields

The phenomenology of the scaling behavior of higher order structure functions of velocity differences across a scale $R$ in turbulence should be built around the irreducible representations of the rotation symmetry group. Every irreducible representation is associated with a scalar function of $R$ which may exhibit different scaling exponents. The common practice of using moments of longitudinal and transverse fluctuations mixes different scalar functions and therefore may mix different scaling exponents. It is shown explicitly how to extract pure scaling exponents for correlations functions of arbitrary orders.

chao-dyn↗
arXiv1997

Exact Result for the 3rd Order Correlations of Velocity in Turbulence with Helicity

All statistical models of turbulence take into account Kolmogorov's exact result known as the "4/5 law" which stems from energy conservation. This law states that the energy flux expressed as a spatial derivative of the 3rd order velocity correlator equals the rate of energy dissipation. We have found an additional exact result which stems from the conservation of helicity in turbulence without inversion symmetry. It equates the flux of helicity expressed as a second spatial derivative of the 3rd order velocity correlator with the rate of helicity dissipation. This exact result must be incorporated to all statistical theories of turbulence with helicity. After submitting this paper for publication we learned that the main result was independently found by Otto Chkhetiani in JETP Lett . 63, 808 (1996).

chao-dyn↗
arXiv1997

Perturbative and Non-Perturbative Analysis of the 3'rd Order Zero Modes

The anomalous scaling behavior of the n-th order correlation functions $F_n$ of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero-modes) of the Kraichnan equation $\hat{B}_n F_n=0$. In this paper we present an extensive analysis of the simplest (non-trivial) case of n=3 in the isotropic sector. The main parameter of the model, denoted as $ζ_h$, characterizes the eddy diffusivity and can take values in the interval $0\le ζ_h \le 2$. After choosing appropriate variables we can present computer-assisted non-perturbative calculations of the zero modes in a projective two dimensional circle. In this presentation it is also very easy to perform perturbative calculations of the scaling exponent $ζ_3$ of the zero modes in the limit $ζ_h\to 0$, and we display quantitative agreement with the non-perturbative calculations in this limit. Another interesting limit is $ζ_h\to 2$. This second limit is singular, and calls for a study of a boundary layer using techniques of singular perturbation theory. Our analysis of this limit shows that the scaling exponent $ζ_3$ vanishes like $\sqrt{ζ_2/log{ζ_2}}$. In this limit as well, perturbative calculations are consistent with the non-perturbative calculations.

chao-dyn↗
arXiv1997

Fusion Rules in Navier-Stokes Turbulence: First Experimental Tests

We present the first experimental tests of the recently derived fusion rules for Navier-Stokes (N-S) turbulence. The fusion rules address the asymptotic properties of many-point correlation functions as some of the coordinates coalesce, and form an important ingredient of the nonperturbative statistical theory of turbulence. Here we test the fusion rules when the spatial separations lie within the inertial range, and find good agreement between experiment and theory. An unexpected result is a simple linear law for the Laplacian of the velocity fluctuation conditioned on velocity increments across large separations.

chao-dyn↗
arXiv1996

Non-Perturbative Zero Modes in the Kraichnan Model for Turbulent Advection

The anomalous scaling behavior of the n-th order correlation functions ${\cal F}_n$ of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero-modes) of the Kraichnan equation $\hat{\cal B}_n {\cal F}_n=0$. Previous analysis found zero-modes in perturbation theory in a small parameter. We present non-perturbative analysis of the simplest (non-trivial) case of n=3 and compare the results with the perturbative predictions.

chao-dyn↗
arXiv1996

On Conditional Statistics in Scalar Turbulence: Theory vs. Experiment

We consider turbulent advection of a scalar field $T(\B.r)$, passive or active, and focus on the statistics of gradient fields conditioned on scalar differences $ΔT(R)$ across a scale $R$. In particular we focus on two conditional averages $\langle\nabla^2 T\big|ΔT(R)\rangle$ and $\langle|\nabla T|^2\big|ΔT(R) \rangle$. We find exact relations between these averages, and with the help of the fusion rules we propose a general representation for these objects in terms of the probability density function $P(ΔT,R)$ of $ΔT(R)$. These results offer a new way to analyze experimental data that is presented in this paper. The main question that we ask is whether the conditional average $\langle\nabla^2 T\big| ΔT(R)\rangle$ is linear in $ΔT$. We show that there exists a dimensionless parameter which governs the deviation from linearity. The data analysis indicates that this parameter is very small for passive scalar advection, and is generally a decreasing function of the Rayleigh number for the convection data.

chao-dyn↗
arXiv1996

Hydrodynamic Turbulence Has Infinitely Many Anomalous Dynamical Exponents

On the basis of the Navier-Stokes equations we develop the statistical theory of many space-time correlation functions of velocity differences. Their time dependence is {\em not} scale invariant: $n$-order correlations functions exhibit $n-1$ distinct decorrelation times that are characterized by $n-1$ anomalous dynamical scaling exponents. We derive exact scaling relations that bridge all these dynamical exponents to the static anomalous exponents $ζ_n$ of the standard structure functions.

chao-dyn↗
arXiv1996

Towards a Nonperturbative Theory of Hydrodynamic Turbulence:Fusion Rules, Exact Bridge Relations and Anomalous Viscous Scaling Functions

In this paper we derive here, on the basis of the NS eqs. a set of fusion rules for correlations of velocity differences when all the separation are in the inertial interval. Using this we consider the standard hierarchy of equations relating the $n$-th order correlations (originating from the viscous term in the NS eq.) to $n+1$'th order (originating from the nonlinear term) and demonstrate that for fully unfused correlations the viscous term is negligible. Consequently the hierarchic chain is decoupled in the sense that the correlations of $n+1$'th order satisfy a homogeneous equation that may exhibit anomalous scaling solutions. Using the same hierarchy of eqs. when some separations go to zero we derive a second set of fusion rules for correlations with differences in the viscous range. The latter includes gradient fields. We demonstrate that every n'th order correlation function of velocity differences ${\cal F}_n(\B.R_1,\B.R_2,\dots)$ exhibits its own cross-over length $η_{n}$ to dissipative behavior as a function of, say, $R_1$. This length depends on $n$ {and on the remaining separations} $R_2,R_3,\dots$. When all these separations are of the same order $R$ this length scales like $η_n(R)\sim η(R/L)^{x_n}$ with $x_n=(ζ_n-ζ_{n+1}+ζ_3-ζ_2)/(2-ζ_2)$, with $ζ_n$ being the scaling exponent of the $n$'th order structure function. We derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents $ζ_n$ of the $n$'th order structure functions. One of these relations is the well known ``bridge relation" for the scaling exponent of dissipation fluctuations $μ=2-ζ_6$.

chao-dyn↗
arXiv1996

The Viscous Lengths in Hydrodynamic Turbulence are Anomalous Scaling Functions

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 $\bbox{\cal F}_n(\B.R_1,\B.R_2,\dots)$ exhibits its own cross-over length $η_{n}$ to dissipative behavior as a function of, say, $R_1$. This length depends on $n$ {and on the remaining separations} $R_2,R_3,\dots$. One result of this Letter is that when all these separations are of the same order $R$ this length scales like $η_n(R)\sim η(R/L)^{x_n}$ with $x_n=(ζ_n-ζ_{n+1}+ζ_3-ζ_2)/(2-ζ_2)$, with $ζ_n$ 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 $μ=2-ζ_6$.

chao-dyn↗
arXiv1996

Fusion Rules and Conditional Statistics in Turbulent Advection

Fusion rules in turbulence address the asymptotic properties of many-point correlation functions when some of the coordinates are very close to each other. Here we put to experimental test some non-trivial consequences of the fusion rules for scalar correlations in turbulence. To this aim we examine passive turbulent advection as well as convective turbulence. Adding one assumption to the fusion rules one obtains a prediction for universal conditional statistics of gradient fields. We examine the conditional average of the scalar dissipation field $\left<\nabla^2 T(\B.r)|T(\B.r+\B.R)-T(\B.r)\right>$ for $R$ in the inertial range, and find that it is linear in $T(\B.r+\B.R)-T(\B.r)$ with a fully determined proportionality constant. The implications of these findings for the general scaling theory of scalar turbulence are discussed.

chao-dyn↗
arXiv1996

The Universal Scaling Exponents of Anisotropy in Turbulence and their Measurement

The scaling properties of correlation functions of non-scalar fields (constructed from velocity derivatives) in isotropic hydrodynamic turbulence are characterized by a set of universal exponents. It is explained that these exponents also characterize the rate of decay of the effects of anisotropic forcing in developed turbulence. This set has never been measured in either numerical or laboratory experiments. These exponents are important for the general theory of turbulence, but also for modeling anisotropic flows. We propose in this letter how to measure these exponents using existing data bases of direct numerical simulations and by designing new laboratory experiments.

chao-dyn↗
arXiv1996

Comment on ``Multicomponent turbulence, the spherical limit, and non-Kolmogorov spectra''

It is shown that the generalization of the Navier-Stokes equations to a theory with $N$ ``internal state" copies of the velocity fields is a step in a wrong direction: the $N\to\infty$ limit has no physical sense and produces wrong results, whereas the treatment of the first order terms in $1/N$ is even more complicated than the initial problem of description of turbulence in the frame of the Navier-Stokes equation.

chao-dyn↗
arXiv1996

Extended Self-Similarity in Turbulent Systems: an Analytically Soluble Example

In turbulent flows the $n$'th order structure functions $S_n(R)$ scale like $R^{ζ_n}$ when $R$ is in the "inertial range". Extended Self-Similarity refers to the substantial increase in the range of power law behaviour of $S_n(R)$ when they are plotted as a function of $S_2(R)$ or $S_3(R)$. In this Letter we demonstrate this phenomenon analytically in the context of the ``multiscaling" turbulent advection of a passive scalar. This model gives rise to a series of differential equations for the structure functions $S_n(R)$ which can be solved and shown to exhibit extended self similarity. The phenomenon is understood by comparing the equations for $S_n(R)$ to those for $S_n(S_2)$.

chao-dyn↗
arXiv1996

Scaling Behavior in Turbulence is Doubly Anomalous

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 {\em two ends} of this range.

chao-dyn↗