Probability Densities in Strong Turbulence

Abstract

According to modern developments in turbulence theory, the "dissipation" scales (u.v. cut-offs) $\eta$ form a random field related to velocity increments ${\delta }_{\eta }u$. In this work we, using Mellin's transform combined with the Gaussain large -scale boundary condition, calculate probability densities (PDFs) of velocity increments $P({\delta }_{r}u,r)$ and the PDF of the dissipation scales $Q(\eta ,Re)$, where $Re$ is the large-scale Reynolds number. The resulting expressions strongly deviate from the Log-normal PDF $P_L({\delta }_{r}u,r)$ often quoted in the literature. It is shown that the probability density of the small-scale velocity fluctuations includes information about the large (integral) scale dynamics which is responsible for deviation of $P({\delta }_{r}u,r)$ from $P_L({\delta }_{r}u,r)$. A framework for evaluation of the PDFs of various turbulence characteristics involving spatial derivatives is developed. The exact relation, free of spurious Logarithms recently discussed in Frisch et al (J. Fluid Mech. {\bf 542}, 97 (2005)), for the multifractal probability density of velocity increments, not based on the steepest descent evaluation of the integrals is obtained and the calculated function $D(h)$ is close to experimental data. A novel derivation (Polyakov, 2005), of a well-known result of the multi-fractal theory [Frisch, "Turbulence. {\it Legacy of A.N.Kolmogorov}", Cambridge University Press, 1995)), based on the concepts described in this paper, is also presented.

arXiv comment25 pages and 9 figures

References in corpus (9)

• Particles and fields in fluid turbulence
• The multifractal nature of turbulent energy dissipation
• ANOMALOUS SCALING OF THE PASSIVE SCALAR
• Turbulence without pressure
• Exact Second-Order Structure-Function Relationships
• Mean- Field Approximation and a Small Parameter in Turbulence Theory
• Local 4/5-Law and Energy Dissipation Anomaly in Turbulence
• Dynamical equations for high-order structure functions, and a comparison of a mean field theory with experiments in three-dimensional turbulence
• Does multifractal theory of turbulence have logarithms in the scaling relations?

Cited by in corpus (26)

• Asymptotic Exponents from Low-Reynolds-Number Flows
• Extreme velocity gradients in turbulent flows
• Resolving the fine-scale structure in turbulent Rayleigh-Benard convection
• Sub-Kolmogorov-Scale Fluctuations in Fluid Turbulence
• Measurement of local dissipation scales in turbulent pipe flow
• An exact relation between Eulerian and Lagrangian velocity increment statistics
• Fully developed turbulence and the multifractal conjecture
• Multiscale velocity correlations in turbulence and Burgers turbulence: Fusion rules, Markov processes in scale, and multifractal predictions
• Intermittency in the relative separations of tracers and of heavy particles in turbulent flows
• Stochastic interpolation of sparsely sampled time series via multi-point fractional Brownian bridges
• Stochastic Dynamical Model of Intermittency in Fully Developed Turbulence
• Emergence of skewed non-Gaussian distributions of velocity increments in isotropic turbulence
• Generation of small-scale structures in the developed turbulence
• Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space
• Transition to turbulence scaling in Rayleigh-Bénard convection
• Quantum Carleman linearisation efficiency in nonlinear fluid dynamics
• Beyond scaling and locality in turbulence
• Law-of-the-wall for streamwise energy spectra in high-Reynolds-number turbulent boundary layers
• Connecting boundary layer dynamics with extreme bulk dissipation events in Rayleigh-Bénard flow
• Generalized description of intermittency in turbulence via stochastic methods
• Superstatistical wind fields from point-wise atmospheric turbulence measurements
• Shot noise multifractal model for turbulent pseudo-dissipation
• Scale-resolved turbulent Prandtl number for Rayleigh-Bénard convection at $\mathbf{Pr}=10^{-3}$
• Skin Friction in Simple Wall - Bounded Shear Flows in Large Reynolds Number Limit
• Probability Density Functions in Homogeneous and Isotropic Magneto-Hydrodynamic Turbulence
• Transition to dilatation-dominated isothermal compressible turbulence