(1+1)-dimensional turbulence

A class of dynamical models of turbulence living on a one-dimensional dyadic-tree structure is introduced and studied. The models are obtained as a natural generalization of the popular GOY shell model of turbulence. These models are found to be chaotic and intermittent. They represent the first example of (1+1)-dimensional dynamical systems possessing non trivial multifractal properties. The dyadic structure allows to study spatial and temporal fluctuations. Energy dissipation statistics and its scaling properties are studied. Refined Kolmogorov Hypothesis is found to hold.Comment: 18 pages, 9 figures, submitted to Phys.of Fluid

PDF of Velocity Fluctuation in Turbulence by a Statistics based on Generalized Entropy

An analytical formula for the probability density function (PDF) of the velocity fluctuation in fully-developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the R\'{e}nyi entropy or the Tsallis-Havrda-Charvat (THC) entropy. The parameters appeared in the PDF, including the index $q$ which appears in the measures of the R\'{e}nyi entropy or of the THC entropy are determined self-consistently with the help of the observed value $\mu$ of the intermittency exponent. The derived PDF explains quite well the experimentally observed density functions.Comment: 10 pages, 2 figure

Inertial range scaling of the scalar flux spectrum in two-dimensional turbulence

Two-dimensional statistically stationary isotropic turbulence with an imposed uniform scalar gradient is investigated. Dimensional arguments are presented to predict the inertial range scaling of the turbulent scalar flux spectrum in both the inverse cascade range and the enstrophy cascade range for small and unity Schmidt numbers. The scaling predictions are checked by direct numerical simulations and good agreement is observed

Harmonious Representation of PDF's reflecting Large Deviations

The framework of multifractal analysis (MFA) is distilled to the most sophisticated one. Within this transparent framework, it is shown that the harmonious representation of MFA utilizing two distinct Tsallis distribution functions, one for the tail part of probability density function (PDF) and the other for its center part, explains the recently observed PDF's of turbulence in the highest accuracy superior to the analyses based on other models such as the log-normal model and the $p$ model.Comment: 11 pages, 2 figure

Kolmogorov Similarity Hypotheses for Scalar Fields: Sampling Intermittent Turbulent Mixing in the Ocean and Galaxy

Kolmogorov's three universal similarity hypotheses are extrapolated to describe scalar fields like temperature mixed by turbulence. By the analogous Kolmogorov third hypothesis for scalars, temperature dissipation rates chi averaged over lengths r > L_K should be lognormally distributed with intermittency factors I that increase with increasing turbulence energy length scales L_O as I_chi-r = m_T ln(L_O/r). Tests of Kolmogorovian velocity and scalar universal similarity hypotheses for very large ranges of turbulence length and time scales are provided by data from the ocean and the Galactic interstellar medium. The universal constant for turbulent mixing intermittency m_T is estimated from oceanic data to be 0.44+-0.01, which is remarkably close to estimates for Kolmogorov's turbulence intermittency constant m_u of 0.45+-0.05 from Galactic as well as atmospheric data. Extreme intermittency complicates the oceanic sampling problem, and may lead to quantitative and qualitative undersampling errors in estimates of mean oceanic dissipation rates and fluxes. Intermittency of turbulence and mixing in the interstellar medium may be a factor in the formation of stars.Comment: 23 pages original of Proc. Roy. Soc. article, 8 figures; in "Turbulence and Stochastic Processes: Kolmogorov's ideas 50 years on", London The Royal Society, 1991, J.C.R. Hunt, O.M. Phillips, D. Williams Eds., pages 1-240, vol. 434 (no. 1890) Proc. Roy. Soc. Lond. A, PDF fil

On an alternative explanation of anomalous scaling and how well-defined is the concept of inertial range

The main point of this communication is that there is a small non-negligible amount of eddies-outliers/very strong events (comprising a significant subset of the tails of the PDF of velocity increments in the nominally-defined inertial range) for which viscosity/dissipation is of utmost importance at whatever high Reynolds number. These events contribute significantly to the values of higher-order structure functions and their anomalous scaling. Thus the anomalous scaling is not an attribute of the conventionally-defined inertial range, and the latter is not a well-defined concept. The claim above is supported by an analysis of high-Reynolds-number flows in which among other things it was possible to evaluate the instantaneous rate of energy dissipation.Comment: 7 pages, 6 figure

On the universality of small scale turbulence

The proposed universality of small scale turbulence is investigated for a set of measurements in a cryogenic free jet with a variation of the Reynolds number (Re) from 8500 to 10^6. The traditional analysis of the statistics of velocity increments by means of structure functions or probability density functions is replaced by a new method which is based on the theory of stochastic Markovian processes. It gives access to a more complete characterization by means of joint probabilities of finding velocity increments at several scales. Based on this more precise method our results call in question the concept of universality.Comment: 4 pages, 4 figure

Local properties of extended self-similarity in 3D turbulence

Using a generalization of extended self-similarity we have studied local scaling properties of 3D turbulence in a direct numerical simulation. We have found that these properties are consistent with lognormal-like behavior of energy dissipation fluctuations with moderate amplitudes for space scales $r$ beginning from Kolmogorov length $\eta$ up to the largest scales, and in the whole range of the Reynolds numbers: $50 \leq R_{\lambda} \leq 459$. The locally determined intermittency exponent $\mu(r)$ varies with $r$; it has a maximum at scale $r=14 \eta$, independent of $R_{\lambda}$.Comment: 4 pages, 5 figure

Analysis of Velocity Fluctuation in Turbulence based on Generalized Statistics

The numerical experiments of turbulence conducted by Gotoh et al. are analyzed precisely with the help of the formulae for the scaling exponents of velocity structure function and for the probability density function (PDF) of velocity fluctuations. These formulae are derived by the present authors with the multifractal aspect based on the statistics that are constructed on the generalized measures of entropy, i.e., the extensive R\'{e}nyi's or the non-extensive Tsallis' entropy. It is revealed that there exist two scaling regions separated by a crossover length, i.e., a definite length approximately of the order of the Taylor microscale. It indicates that the multifractal distribution of singularities in velocity gradient in turbulent flow is robust enough to produce scaling behaviors even for the phenomena out side the inertial range.Comment: 10 Pages, 5 figure

The effect of turbulent intermittency on the deflagration to detonation transition in SN Ia explosions

We examine the effects of turbulent intermittency on the deflagration to detonation transition (DDT) in Type Ia supernovae. The Zel'dovich mechanism for DDT requires the formation of a nearly isothermal region of mixed ash and fuel that is larger than a critical size. We primarily consider the hypothesis by Khokhlov et al. and Niemeyer and Woosley that the nearly isothermal, mixed region is produced when the flame makes the transition to the distributed regime. We use two models for the distribution of the turbulent velocity fluctuations to estimate the probability as a function of the density in the exploding white dwarf that a given region of critical size is in the distributed regime due to strong local turbulent stretching of the flame structure. We also estimate lower limits on the number of such regions as a function of density. We find that the distributed regime, and hence perhaps DDT, occurs in a local region of critical size at a density at least a factor of 2-3 larger than predicted for mean conditions that neglect intermittency. This factor brings the transition density to be much larger than the empirical value from observations in most situations. We also consider the intermittency effect on the more stringent conditions for DDT by Lisewski et al. and Woosley. We find that a turbulent velocity of $10^8$ cm/s in a region of size $10^6$ cm, required by Lisewski et al., is rare. We expect that intermittency gives a weaker effect on the Woosley model with stronger criterion. The predicted transition density from this criterion remains below $10^7$ g/cm$^3$ after accounting for intermittency using our intermittency models.Comment: 31 pages, accepted for publication in Ap