High-order Structure Functions Research Articles

Universal shape of scaling functions in turbulence.

Using a novel device that enables the real-time measurement of high-order structure functions in turbulence with superior statistical accuracy, we study the shape of the structure function at scales where it is influenced by viscosity. The experiments on a variety of laboratory turbulent flows demonstrate that this shape is universal. We argue that it reflects a fundamental property of multifractals: multiscaling. We present accurate experimental scaling functions $\ensuremath{\zeta}(p)$ and, finally, comment on the nature of the skewness of the distribution function of velocity increments by comparing transverse and longitudinal distribution functions.

Measurements of optical turbulence with higher-order structure functions

Higher-order structure functions have been used to extract atmospheric optical C(n)(2) profiles from a vertical sequence of temperature data collected by a single probe carried by a meteorological balloon. This technique circumvents trends and fluctuations in the atmospheric mean temperature and simplifies the equipment and complexity of measurement collection compared with traditional, horizontal differential-probe pair systems.

Numerical and analytical estimates for the structure functions in two-dimensional magnetohydrodynamic flows

In two-dimensional magnetohydrodynamic turbulence, the Kraichnan–Iroshnikov dimensional analysis suggests a linear scaling law for the exponents ζp=p/4 of the structure functions for the Elsässer variables z±=u±B. Numerical simulations are presented and higher order structure functions are calculated using the extended self-similarity hypothesis of Benzi et al. [Phys. Rev. E 48, R29 (1993)]. In addition, an estimate for the first structure function ζ1≥1/4 is derived using a geometric technique introduced by Constantin and Procaccia [Phys. Rev. E 47, 3307 (1993)] in the the context of the transport of a passive scalar in three-dimensional Navier–Stokes turbulence.

Scaling of high-order structure functions in magnetohydrodynamic turbulence

A phenomenological model for the description of intermittency corrections in magnetohydrodynamic flows is presented. The strength of the model lies in its lack of adjustable parameters. A comparison to measurements in the solar wind is presented.

Scaling exponents for turbulence and other random processes and their relationships with multifractal structure.

In the recent literature on high-Reynolds-number turbulence, several different types of scaling exponents---such as multifractal exponents for velocity increments, for energy and scalar dissipation, for the square of the local vorticity, and so forth---have been introduced. More recently, a new exponent called the cancellation exponent has been introduced for characterizing rapidly oscillating quantities. Not all of these exponents are independent; some of them are simply related to more familiar scaling for velocity and temperature structure functions either exactly or through plausible hypotheses familiar for turbulence. A primary purpose of this paper is to establish the interrelationships among the various exponents. In doing so, we obtain several additional relations. Much of the paper is relevant to general stochastic processes, although the discussion is heavily influenced by the turbulent context. We first examine the case of one-dimensional random processes and subsequently consider two- and three-dimensional processes. Special consideration is given to characteristic values appropriate to the geometry of turbulence, as well as the lifetimes of eddies of various scales. Finally, we discuss some properties of the tails of the probability density function to which the scaling properties of high-order structure functions are related and discuss the implications of multifractality on their structure.

The influence of coherent structures and microfronts on scaling laws using global and local transforms

This study examines the influence of coherent structures and attendant microfronts on scaling laws. Toward this goal, we analyse atmospheric observations of turbulence collected 45 m above a flat surface during the Lammefjord Experiment in Denmark. These observations represent more than 40 hours of nearly stationary strong wind conditions and include more than 1600 samples of the main coherent structures. These samples occupy about 40% of the total record and explain the majority of the Reynolds stress.To study the dependence of the scaling laws on the choice of basis set, the time series of velocity fluctuations are decomposed into Fourier modes, the local Haar basis set and eigenvectors of the lagged covariance matrix. The three decompositions are compared by formulating joint projections. The decompositions are first applied to the samples of phased-locked coherent structures centred about eddy microfronts. The eigenvector decomposition is able to partially separate the small-scale variances due to the coherent eddy microfronts from that due to the small-scale structure with random phase. In the Fourier spectrum, both of these contributions to the variance appear together at the higher wavenumbers and their individual contributions cannot be separated. This effect is relatively minor for the scale distribution of energy but exerts an important influence on higher-moment statistics. Deviations from the −$\frac53$ scaling are observed to be slight and depend on choice of basis set.The microfronts strongly influence the higher-order statistics such as the sixth-order structure function traditionally used to estimate the energy transfer variance. The intermittency of fine-scale structure, energy transfer variance and dissipation are not completely characterized by random phase, as often assumed, but are partly associated with microfronts characterized by systematic phase with respect to the main transporting eddies. These conclusions are supported by both the higher-order structure function and the higher-order Haar transform.The Fourier and Haar spectra are also computed for the entire record. The peak of the Haar energy spectrum occurs at smaller scales than those of the Fourier spectrum. The Haar transform is local and emphasizes the width of the events. The Fourier spectrum peaks at the scale of the main periodicity, if it exists, which includes the spacing between the events.

An identification of Energy Cascade in Turbulence by Orthonormal Wavelet Analysis

Orthonormal wavelet expansion method is applied to an analysis of atmospheric turbulence data which shows more than two decades of the inertial subrange spectrum. The result of the orthonormal wavelet analysis of the turbulence data is discussed in comparison with those of an artificial random noise. The local wavelet spectra of turbulence show a characteristic structure, which is absent in the artificial random noise and is identified with the trace of the energy cascade process. The higher-order structure function of velocity, obtained by the wavelet analysis, shows the intermittent structure of the flow field.

Physical model of intermittency in turbulence: Inertial-range non-Gaussian statistics.

We present a physical model to describe the equilibrium probability distribution function (PDF) of velocity differences across an inertial-range distance in 3D isotropic turbulence. The form of the non-Gaussian PDF agrees well with data from direct numerical simulations. It is shown that these PDF's obey a self-similar property, and the resulting inertial-range exponents of high-order velocity structure functions are in agreement with both experimental and numerical data. The model suggests a physical explanation for the phenomenon of intermittency and the nature of multifractality in fully developed turbulence, namely, local self-distortion of turbulent structures.

A microscopic vortical structure in fully developed turbulence

A microscopic vortical structure in fully developed three-dimensional turbulence is considered. The strain rates due to eddies of a certain scale stretch and shrink the vorticity field of smaller scales, whose strain rates excite the vorticity of even smaller scales. The excitation propagates toward small scales, the self-similar structure being anticipated in wavenumber space. It is indicated that a pair of oppositely oriented vortex sheets are a candidate for such a self-similar structure: the two sheets approach each other and the vorticity is enhanced during the process. The time evolution of the vorticity is expressed in the dynamical scaling form proposed previously by Nelkin and the present author [Phys. Rev. A 31, 1980(1985)] in which the higher-order structure functions and generalized dimensions in fully developed turbulence were interpreted in good agreement with experiment. Hence the derived vortical structure is expected to be responsible for the intermittent effects.

Correlation and structure functions for multimode-laser-beam incidence in atmospheric turbulence

Correlation and structure functions are formulated for multimode-laser-beam incidence in atmospheric turbulence. Laser modes are represented by Hermite–Gaussian beams. Higher-order single-mode correlation and structure functions are special cases of our results that correctly reduce to the known Gaussian-beam-wave (TEM00-mode) results. Since the Rytov approximation is employed in our formulation, the results presented are valid only under weak-scattering conditions.

Hot- and cold-wire sensitivity corrections for moments of the fine scale turbulence in heated flows

A method is presented for the correction of measured higher-order moments of hot-wire velocity and cold-wire temperature signals for the respective parasitic sensitivities to temperature and velocity fluctuations. The method is applied to third-order moments of time derivatives and higher-order structure functions, 〈 (Δu)p(Δϑ)q) 〉, measured in a marine boundary layer, where u is the velocity, ϑ is the temperature, and Δu=u (x+r)−u (x). Corrections are less than 5% for 〈 (dϑ/dt)3〉, 7% for 〈 (dϑ/dt)2 (du/dt) 〉, over 200% for 〈 (dϑ/dt)(du/dt)2〉, and less than 0.1% for 〈 (du/dt)3〉. The dominant correction to the third-order temperature moments arises from the velocity sensitivity of the cold wire whereas the dominant correction for the cross statistics was the temperature sensitivity of the hot wire.