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 $\eta_{n}$ 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 $\eta_n(R)\sim \eta (R/L)^{x_n}$ with $x_n=(\zeta_n-\zeta_{n+1}+\zeta_3-\zeta_2)/(2-\zeta_2)$, with $\zeta_n$ 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 $\zeta_n$ of the $n$'th order structure functions. One of these relations is the well known ``bridge relation" for the scaling exponent of dissipation fluctuations $\mu=2-\zeta_6$.Comment: PRE, Submitted. REVTeX, 18 pages, 7 figures (not included) PS Source of the paper with figures avalable at http://lvov.weizmann.ac.il/onlinelist.htm

Exact Resummations in the Theory of Hydrodynamic Turbulence: II A Ladder to Anomalous Scaling

In paper I of this series on fluid turbulence we showed that exact resummations of the perturbative theory of the structure functions of velocity differences result in a finite (order by order) theory. These findings exclude any known perturbative mechanism for anomalous scaling of the velocity structure functions. In this paper we continue to build the theory of turbulence and commence the analysis of nonperturbative effects that form the analytic basis of anomalous scaling. Starting from the Navier-Stokes equations (at high Reynolds number Re) we discuss the simplest examples of the appearance of anomalous exponents in fluid mechanics. These examples are the nonlinear (four-point) Green's function and related quantities. We show that the renormalized perturbation theory for these functions contains ``ladder`` diagrams with (convergent!) logarithmic terms that sum up to anomalous exponents. Using a new sum rule which is derived here we calculate the leading anomalous exponent and show that it is critical in a sense made precise below. This result opens up the possibility of multiscaling of the structure functions with the outer scale of turbulence as the renormalization length. This possibility will be discussed in detail in the concluding paper III of this series.Comment: PRE in press, 15 pages + 21 figures, REVTeX, The Eps files of figures will be FTPed by request to [email protected]

Exact Resummations in the Theory of Hydrodynamic Turbulence: III. Scenarios for Anomalous Scaling and Intermittency

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 $S_n(R)\sim R^{\zeta_n}$, and the correlation of energy dissipation $K_{\epsilon\epsilon}(R) \sim R^{-\mu}$. The goal is to understand the exponents $\zeta_n$ and $\mu$ from first principles. In paper II of this series it was shown that the existence of an ultraviolet scale (the dissipation scale $\eta$) 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 $\Delta$. The exact resummation of ladder diagrams resulted in the calculation of $\Delta$ which satisfies the scaling relation $\Delta=2-\zeta_2$. In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e. $\zeta_n$ not being linear in $n$) with the renormalization scale being the infrared outer scale of turbulence $L$. It is shown that deviations from K41 scaling of $S_n(R)$ ($\zeta_n\neq n/3$) must appear if the correlation of dissipation is mixing (i.e. $\mu>0$). We derive an exact scaling relation $\mu = 2\zeta_2-\zeta_4$. We present analytic expressions for $\zeta_n$ for all $n$ and discuss their relation to experimental data. One surprising prediction is that the time decay constant $\tau_n(R)\propto R^{z_n}$ of $S_n(R)$ scales independently of $n$: the dynamic scaling exponent $z_n$ is the same for all $n$-order quantities, $z_n=\zeta_2$.Comment: PRE submitted, 22 pages + 11 figures, REVTeX. The Eps files of figures will be FTPed by request to [email protected]

Fusion Rules in Turbulent Systems with Flux Equilibrium

Fusion rules in turbulence specify the analytic structure of many-point correlation functions of the turbulent field when a group of coordinates coalesce. We show that the existence of flux equilibrium in fully developed turbulent systems combined with a direct cascade induces universal fusion rules. In certain examples these fusion rules suffice to compute the multiscaling exponents exactly, and in other examples they give rise to an infinite number of scaling relations that constrain enormously the structure of the allowed theory.Comment: Submitted to PRL on July 95, 4 pages, REVTe

Inertial- and Dissipation-Range Asymptotics in Fluid Turbulence

We propose and verify a wave-vector-space version of generalized extended self similarity and broaden its applicability to uncover intriguing, universal scaling in the far dissipation range by computing high-order (\leq 20\/) structure functions numerically for: (1) the three-dimensional, incompressible Navier Stokes equation (with and without hyperviscosity); and (2) the GOY shell model for turbulence. Also, in case (2), with Taylor-microscale Reynolds numbers 4 \times 10^{4} \leq Re_{\lambda} \leq 3 \times 10^{6}\/, we find that the inertial-range exponents (\zeta_{p}\/) of the order - p\/ structure functions do not approach their Kolmogorov value p/3\/ as Re_{\lambda}\/ increases.Comment: RevTeX file, with six postscript figures. epsf.tex macro is used for figure insertion. Packaged using the 'uufiles' utilit

Parametric Generation of Second Sound by First Sound in Superfluid Helium

We report the first experimental observation of parametric generation of second sound (SS) by first sound (FS) in superfluid helium in a narrow temperature range in the vicinity of $T_\lambda$. The temperature dependence of the threshold FS amplitude is found to be in a good quantitative agreement with the theory suggested long time ago and corrected for a finite geometry. Strong amplitude fluctuations and two types of the SS spectra are observed above the bifurcation. The latter effect is quantitatively explained by the discreteness of the wave vector space and the strong temperature dependence of the SS dissipation length.Comment: 4 pages, 4 postscript figures, REVTE

Universality in Turbulence: an Exactly Soluble Model

The present note contains the text of lectures discussing the problem of universality in fully developed turbulence. After a brief description of Kolmogorov's 1941 scaling theory of turbulence and a comparison between the statistical approach to turbulence and field theory, we discuss a simple model of turbulent advection which is exactly soluble but whose exact solution is still difficult to analyze. The model exhibits a restricted universality. Its correlation functions contain terms with universal but anomalous scaling but with non-universal amplitudes typically diverging with the growing size of the system. Strict universality applies only after such terms have been removed leaving renormalized correlators with normal scaling. We expect that the necessity of such an infrared renormalization is a characteristic feature of universality in turbulence.Comment: 31 pages, late

Finite size corrections to scaling in high Reynolds number turbulence

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 $\eta$, i.e., they are due to finite size effects as anisotropic forcing or boundary conditions at large scales. We find that the deviations \dzm from the classical Kolmogorov scaling $\zeta_m = m/3$ of the velocity moments \langle |\u(\k)|^m\rangle \propto k^{-\zeta_m} decrease like $\delta\zeta_m (Re) =c_m Re^{-3/10}$. 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)

Longitudinal Structure Functions in Decaying and Forced Turbulence

In order to reliably compute the longitudinal structure functions in decaying and forced turbulence, local isotropy is examined with the aid of the isotropic expression of the incompressible conditions for the second and third order structure functions. Furthermore, the Karman-Howarth-Kolmogorov relation is investigated to examine the effects of external forcing and temporally decreasing of the second order structure function. On the basis of these investigations, the scaling range and exponents $\zeta_n$ of the longitudinal structure functions are determined for decaying and forced turbulence with the aid of the extended-self-similarity (ESS) method. We find that $\zeta_n$'s are smaller, for $n \geq 4$, in decaying turbulence than in forced turbulence. The reasons for this discrepancy are discussed. Analysis of the local slopes of the structure functions is used to justify the ESS method.Comment: 15 pages, 16 figure