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

Abstract

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 $S_n(R)$ (${\zeta }_n\ne n/3$) 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, $z_n={\zeta }_2$.

PRE submitted, 22 pages + 11 figures, REVTeX. The Eps files of figures will be FTPed by request to [email protected]

References in corpus (4)

• Turbulence without pressure
• Anomalous Scaling in a Model of Passive Scalar Advection: Exact Results
• Exact Resummations in the Theory of Hydrodynamic Turbulence: I The Ball of Locality and Normal Scaling
• Exact Resummations in the Theory of Hydrodynamic Turbulence: II A Ladder to Anomalous Scaling

Cited by in corpus (15)

• Fully developed isotropic turbulence: nonperturbative renormalization group formalism and fixed point solution
• Anomalous Scaling in a Model of Passive Scalar Advection: Exact Results
• Towards a Nonperturbative Theory of Hydrodynamic Turbulence:Fusion Rules, Exact Bridge Relations and Anomalous Viscous Scaling Functions
• Variable energy flux in turbulence
• Analytic Calculation of the Anomalous Exponents in Turbulence: Using the Fusion Rules to Flush Out a Small Parameter
• Recent Developments in Understanding Two-dimensional Turbulence and the Nastrom-Gage Spectrum
• The Viscous Lengths in Hydrodynamic Turbulence are Anomalous Scaling Functions
• The Universal Scaling Exponents of Anisotropy in Turbulence and their Measurement
• Computing the Scaling Exponents in Fluid Turbulence from First Principles: the Formal Setup
• On the elimination of the sweeping interactions from theories of hydrodynamic turbulence
• Locality and stability of the cascades of two-dimensional turbulence
• Hydrodynamic Turbulence: a 19th Century Problem with a challenge for the 21st Century
• Dissipation scales and anomalous sinks in steady two-dimensional turbulence
• Dissipative Scaling Functions in Navier-Stokes Turbulence: Experimental Tests
• Multi-locality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence