High-order Structure Functions Research Articles

Ordinary and Viscosity‐damped Magnetohydrodynamic Turbulence

We compare the properties of ordinary strong magnetohydrodynamic (MHD) turbulence in a strongly magnetized medium with the recently discovered viscosity-damped regime. We focus on energy spectra, anisotropy, and intermittency. Our most surprising conclusion is that in ordinary strong MHD turbulence, the velocity and magnetic fields show different high-order structure function scalings. Moreover, this scaling depends on whether the intermittency is viewed in a global or local system of reference. This reconciles seemingly contradictory earlier results. On the other hand, the intermittency scaling for viscosity-damped turbulence is very different and difficult to understand in terms of the usual phenomenological models for intermittency in turbulence. Our remaining results are in reasonable agreement with expectations. First, we find that our high-resolution simulations for ordinary MHD turbulence show that the energy spectra are compatible with a Kolmogorov spectrum, while viscosity-damped turbulence shows a shallow k-1 spectrum for the magnetic fluctuations. Second, a new numerical technique confirms that ordinary MHD turbulence exhibits Goldreich-Sridhar type anisotropy, while viscosity-damped MHD turbulence shows extremely anisotropic eddy structures. Finally, we show that many properties of incompressible turbulence for both the ordinary and viscosity-damped regimes carry over to the case of compressible turbulence.

Dynamical equations for high-order structure functions, and a comparison of a mean-field theory with experiments in three-dimensional turbulence.

Two recent papers [V. Yakhot, Phys. Rev. E 63, 026307, (2001) and R. J. Hill, J. Fluid Mech. 434, 379, (2001)] derive, through two different approaches that have the Navier-Stokes equations as the common starting point, a set of dynamic equations for structure functions of arbitrary order in turbulence. These equations are not closed. Yakhot proposed a "mean-field theory" to close the equations for locally isotropic turbulence, and obtained scaling exponents of structure functions and expressions for the peak in the probability density function of transverse velocity increments, and for its behavior for intermediate amplitudes. At high Reynolds numbers, some relevant experimental data on pressure gradient and dissipation terms are presented that are needed to provide closure, as well as on other aspects predicted by the theory. Comparison between the theory and the data shows varying levels of agreement, and reveals gaps inherent to the implementation of the theory.

Anomalous scaling, nonlocality, and anisotropy in a model of the passively advected vector field.

A model of the passive vector quantity advected by the Gaussian velocity field with the covariance approximately delta(t-t('))|x-x(')|(epsilon) is studied; the effects of pressure and large-scale anisotropy are discussed. The inertial-range behavior of the pair correlation function is described by an infinite family of scaling exponents, which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques. The exponents are organized in a hierarchical order according to their degree of anisotropy, with the spectrum unbounded from above and the leading (minimal) exponent coming from the isotropic sector. This picture extends to higher-order correlation functions. Like in the scalar model, the second-order structure function appears nonanomalous and is described by the simple dimensional exponent: S2 approximately r(2-epsilon). For the higher-order structure functions, S(2n)approximately r(n(2-epsilon)+ delta(n)), the anomalous scaling behavior is established as a consequence of the existence in the corresponding operator product expansions of "dangerous" composite operators, whose negative critical dimensions determine the anomalous exponents delta(n)<0. A close formal resemblance of the model with the stirred Navier-Stokes equation reveals itself in the mixing of relevant operators and is the main motivation of the paper. Using the renormalization group, the anomalous exponents are calculated in the O(epsilon) approximation, in large d dimensions, for the even structure functions up to the twelfth order.

Sweeping decorrelation hypothesis in a turbulent round jet

Higher-order spectra and structure functions are used to investigate the random sweeping decorrelation hypothesis. Measurements in a turbulent round jet at moderate Taylor microscale Reynolds numbers (Rλ = 230–545) indicate that higher-order (up to 4) velocity spectra and structure functions do not follow the Kolmogorov (1941a) scaling. The normal Reynolds stresses appear to influence the dynamics of the inertial range (IR). Spectra and structure functions of the transverse velocity fluctuation v exhibit different behaviours from those of the longitudinal velocity fluctuation u. The extent of the −5/3 and 2/3 scaling ranges is greater for u than v, and the compensated higher-order spectra of u collapse better than the corresponding v spectra. Statistical independence between large and small scales implies a zero-correlation coefficient between velocity fluctuations and velocity increments. In the present moderate Rλ turbulent jet, the correlation coefficient is significant throughout the whole IR. The correlation coefficient between v and the increment δv is larger than that between u and δu in the IR. Both correlation coefficients decrease as Rλ increases.

QUASI-CLOSURE AND SCALING OF TURBULENCE

A statistically stationary isotropic turbulence is of quasi-closure, i.e. its high-order statistical moments can be derived from its low-order moments. A workable quasi-closure scheme is developed for the structure functions of incompressible homogeneous isotropic turbulence based upon a non-Gaussian statistical model. The second order structure function is obtained by solving the spectral dynamic equation or by using an empirical formula such as the Batchelor fit, and then the high-order structure functions is calculated by the quasi-closure scheme. We study the absolute and relative scaling of the structure functions of isotropic turbulence in connection with Kolmogorovs' 1941 theory (K41) and his 1962 theory (K62). In contrast to K62 and various intermittency models, our results suggest a different picture of scaling of isotropic turbulence: the anomalous scaling of structure functions observed in experiments and numerical simulations is a finite Reynolds number effect, and the K41 normal scaling is valid in the real Kolmogorov inertial range corresponding to an infinite Reynolds number.

Estimating global and local scaling exponents in turbulent flows using discrete wavelet transformations

High frequency longitudinal velocity (u) measurements were performed in the atmospheric surface layer to investigate the inertial subrange structure of turbulence. The u measurements, collected over a wide range of atmospheric stability conditions, were used to investigate local and global intermittency buildup in the inertial subrange. Global scaling exponents and other statistical properties were derived using nondecimated (NDWT) and critically sampled orthonormal (OWT) wavelet transformations. These statistical measures were contrasted to similar statistical measures derived by applying NDWT and OWT to an ensemble of fractional Brownian motion (fBm) time series with Hurst exponent of 1/3. Such comparisons permit direct assessment as to whether discrepancies in observed intermittency corrections are artifacts of wavelet transformations or limitations in sample size. This study demonstrated that both NDWT and OWT were able to resolve intermittency-based departures from global power laws observed in higher-order structure functions of turbulence time series. Particularly, global power laws inferred from OWT and NDWT were consistent with new intermittency correction results derived from the dynamics of the fourth order structure functions. This study is the first to report on the ensemble behavior of such a power law for a wide range of surface boundary conditions (e.g., variable surface heating and friction velocity). The wavelet computed global intermittency departures from the classical Kolmogorov theory (or K41) were marginally smaller than those computed by the traditional structure function approach. In terms of local exponents, we found that the application of NDWT to fBm time series resulted in a wide empirical frequency distribution of local scaling exponents (α). The latter finding clearly demonstrates that previous and present α determined by wavelet analysis cannot be used as evidence for multifractality in turbulence. We also demonstrated that the classical local regression estimation of α is theoretically impaired by heteroscedascity when the local scale is finite. While the spread in α does not reflect any multifractal signatures, the modes of the local α frequency distribution support findings from global exponent analysis. We found that the modes of the local α distribution are well reproduced by global intermittency models for u and by K41 for the fBm.

Anisotropic scaling contributions to high-order structure functions in high-reynolds-number turbulence

We make an attempt at obtaining the scaling exponents for the anisotropic components of structure functions of order 2 through 6. We avoid mixing these components with their isotropic counterparts for each order by using tensor components that are entirely anisotropic. We do this by considering terms of the isotropic sector corresponding to j=0 in the SO(3) decomposition of each tensor, and then constructing components that are explicitly zero in the isotropic sector. We use an interpolation formula to compensate for the large-scale encroachment of inertial-range scales. This allows us to examine the lowest order anisotropic scaling behavior. The resulting anisotropic exponents for a given tensorial order are larger than those known for the corresponding isotropic part. One conclusion that emerges is that the anisotropy effects diminish with decreasing scale, although much more slowly than previously thought.

Statistics of filtered velocity in grid and wake turbulence

Data on spatially filtered turbulence are commonly needed for a priori sub-grid model studies and for a posteriori testing of large eddy simulation (LES) codes. In this paper, hot-wire anemometry is used to record very long records of such data, required for good convergence of high-order statistics. An array consisting of four X-wire probes placed transversal to the flow direction is built. Unlike previous single-probe hot-wire measurements, which only allowed stream-wise filtering using Taylor’s hypothesis, the array permits cross-stream filtering as well. Measurements which are spatially filtered at a length-scale Δ pertaining to the inertial-range of turbulence are performed in grid and wake turbulence. The data can be used directly to compare with results from LES. From the data, fundamental differences between filtered and unfiltered velocity fields are examined through probability density functions and the scaling behavior of high-order structure functions. A comparative study of probability density functions of filtered and unfiltered velocity increments shows that the tails of the distributions are affected by the filtering even at scales much larger than the filter scale. Significant differences are also observed in regard to the scaling of structure functions. It is shown that extended self-similarity, a recent technique for measuring inertial range scaling exponents, yields questionable results when applied to structure functions of filtered velocity.

Closure approach to high-order structure functions of turbulence

Let T(p) be the structure function of order p of turbulence. First T2 is determined, then T3 is derived from T2, and finally T(p) (p>3) are derived from T2 and T3. This closure scheme is realized by a non-Gaussian statistical model. We use it to study the scaling law of T(p), and we find that the available data on scaling exponents favor Kolmogorov's 1941 theory rather than his 1962 theory. We also predict the high-order universal constants of inertial-range scaling.

Spectral Reduction: A Statistical Description of Turbulence

A method is described for predicting statistical properties of turbulence. Collections of Fourier amplitudes are represented by nonuniformly spaced modes with enhanced coupling coefficients. The statistics of the full dynamics can be recovered from the time-averaged predictions of the reduced model. A Liouville theorem leads to inviscid equipartition solutions. Excellent agreement is obtained with two-dimensional forced-dissipative pseudospectral simulations. For the two-dimensional enstrophy cascade, logarithmic corrections to the high-order structure functions are observed.

High-order structure functions of turbulence

The scaling behaviour of high-order structure functions Gp(r) =〈(u(x+r)−u(x))p〉 is studied in a variety of laboratory turbulent flows. The statistical accuracy of the structure function benefits from novel instrumentation for its real-time measurement. The nature of statistical errors is discussed extensively. It is argued that integration times must increase for decreasing separations r. Based on the statistical properties of probability density functions we derive a simple estimate of the required integration time for moments of a given order. We further give a way for improving this accuracy through careful extrapolation of probability density functions of velocity differences.Structure functions are studied in two different kinematical situations. The (standard) longitudinal structure functions are measured using Taylor's hypothesis. In the transverse case an array of probes is used and no recourse to Taylor's hypothesis is needed. The measured scaling exponents deviate from Kolmogorov's (1941) prediction, more strongly so for the transverse exponents.The experimental results are discussed in the light of the multifractal model that explains intermittency in a geometrical framework. We discuss a prediction of this model for the form of the structure function at scales where viscosity becomes of importance.

Lagrangian dynamics and high-order moments intermittency in passive solar advection

A Lagrangian method for the numerical simulation of the Kraichnan passive scalar model is discussed. Two different applications are presented. First, the dependence of scalar anomalies on the roughness of the advecting velocity field is considered. Second, the anomalous behavior of high order structure functions is investigated.

Simple passive scalar advection-diffusion model

This paper presents a simple, one-dimensional model of a randomly advected passive scalar. The model exhibits anomalous inertial range scaling for the structure functions constructed from scalar differences. The model provides a simple computational test for recent ideas regarding closure and scaling for randomly advected passive scalars. Results suggest that high order structure function scaling depends on the largest velocity eddy size, and hence scaling exponents may be geometry dependent and nonuniversal.

Scaling laws and vanishing-viscosity limits for wall-bounded shear flows and for local structure in developed turbulence

Scaling laws for wall-bounded turbulence are derived and their properties are analyzed via vanishing-viscosity asymptotics; a comparison of the results with recent experiments shows that the observed scaling law differs significantly from the customary logarithmic law of the wall. The Izakson-Millikan-von Mises derivation of turbulence structure, properly interpreted, confirms this analysis. Analogous relations for the local structure of turbulence are given, including results on the scaling of the higher-order structure functions; these results suggest that there are no Reynolds-number-independent corrections to the Kolmogorov exponent and thus that the classical 1941 version of the Kolmogorov theory already gives the limiting behavior. The use of small-viscosity asymptotics is explained, and the consequences of the theory and of the experimental evidence for the Navier-Stokes equations and for the statistical theory of turbulence are discussed. © 1997 John Wiley & Sons, Inc.

生物に学ぶ高次構造制御と機能 羊毛の高次構造と機能 毛根の中のハイテク構造化機構

生物に学ぶ高次構造制御と機能 羊毛の高次構造と機能 毛根の中のハイテク構造化機構

Solar wind low-frequency magnetohydrodynamic turbulence: extended self-similarity and scaling laws

Abstract. In this paper we review some of the work done in investigating the scaling properties of Magnetohydrodynamic turbulence, by using velocity fluctuations measurements performed in the interplanetary space plasma by the Helios spacecraft. The set of scaling exponents ξq for the q-th order velocity structure functions, have been determined by using the Extended Self-Similarity hypothesis. We have found that the q-th order velocity structure function, when plotted vs. the 4-th order structure function, displays a range of self-similarity which extends over all the lengths covered by measurements, thus allowing for a very good determination of ξq. Moreover the results seem to show that the scaling exponents are the same regardless the various observation periods considered. The obtained scaling exponents have been compared with the results of some intermittency models for Kraichnan's turbulence, derived in the framework of infinitely divisible fragmentation processes, showing the good agreement between these models and our observations. Finally, on the basis of the actually available data sets, we show that scaling laws in Solar Wind turbulence seem to be different from turbulent scaling laws in the ordinary fluid flows. This is true for high-order velocity structure functions, while low-order velocity structure functions show the same scaling laws. Since our measurements involve length scales which extend over many order of magnitude where dissipation is practically absent, our results show that Solar Wind turbulence can be regarded as a testing bench for the investigation of general scaling behaviour in turbulent flows. In particular our results strongly support the point of view which attributes a key role to the inertial range dynamics in determining the intermittency characteristics in fluid flows, in contrast with the point of view which attributes intermittency to a finite Reynolds number effect.

Anomalous scaling and anisotropy in turbulence

Using realtime signal processing techniques it is possible to obtain precise values for the scaling exponents ζ(p) of high-order structure functions. The value of these exponents differs from the Kolmogorov 1941 prediction, that they are anomalous. This anomalous behavior has been firmly confirmed in our experiments where it has been seen in structure functions which show convincing scaling behavior and that have a high statistical accuracy.The advantage of performing experiments in laboratory windtunnel turbulent flows is that development of detector technology is relatively easy. As a result, we have been able to measure transverse structure functions using a line of velocity probes. Such a configuration obviates usage of Taylor's hypothesis. Also the scaling of the transverse structure functions is anomalous. However, the transverse exponent ζT(p) is significantly different from the standard longitudinal exponent ζL(p): scaling anomaly is non-isotropic. We speculate about the meaning of this result.

Gaussian nature of the COBE data from multipoint correlations.

Important information about the early universe can be obtained from a study of the cosmic background radiation (CBR) in the microwave range. We study high-order correlation and structure functions of temperature fluctuations collected from the first two years of Differential Microwave Radiometer (DMR) observations from the Cosmic Background Explorer (COBE) satellite. The intent is to determine whether the radiation data possess significant deviations from Gaussianity. The most difficult problem in drawing meaningful conclusions is the presence of instrumental noise, which is quite strong even after a two-year averaging has been performed. We have taken into account the noise in various ways. Within the limitations imposed by the noise, our study shows that the fluctuations are quite likely to be Gaussian-like. The results can, therefore, be said to favor the inflation scenario of the universe.

Anomalous scaling and anisotropy in turbulence

Using realtime signal processing techniques it is possible to obtain precise values for the scaling exponents ζ(p) of high-order structure functions. The value of these exponents differs from the Kolmogorov 1941 prediction, they are anomalous. This anomalous behavior has been firmly confirmed in our experiments where it has been seen in structure functions that show convincing scaling behavior and that have a high statistical accuracy.The advantage of performing experiments in laboratory windtunnel turbulent flows is that development of detector technology is relatively easy. As a result, we have been able to measure transverse structure functions using a line of velocity probes. Such a configuration obviates usage of Taylor's hypothesis. Also the scaling of the transverse structure functions is anomalous. However, the transverse exponent ζT(p) is significantly different from the standard longitudinal exponent ζL(p): scaling anomaly is non-isotropic. We speculate about the meaning of this result.

Experimental evidence for differences in the extended self-similarity scaling laws between fluid and magnetohydrodynamic turbulent flows.

It has been recently suggested that the various $q$th order velocity structure functions in turbulent flows are related to each other through well defined scaling laws, which extend outside the usual inertial domain. Even if theoretical models provide different scaling laws for fluid and magnetohydrodynamic turbulent flows, no attempt has been made up to now to furnish experimental evidence for these differences. By using measurements from the solar wind turbulence and from turbulence in ordinary fluid flows, we show that the differences can be observed only by looking at the high-order velocity structure functions.