Velocity Structure Functions Research Articles

On the radar estimation of turbulence parameters in a stably stratified atmosphere

Four estimators of the rate of dissipation of turbulent energy based on the velocity structure function, rms turbulent velocities, Brunt‐Väisälä frequencies, and ground diffraction pattern fading rates are discussed and compared using Arecibo Initiative on the Dynamics of the Atmosphere MF radar data in both the imaging Doppler interferometry (IDI) and spaced antenna modes. A consistent set of empirical equations is developed which define and describe the relationships between the turbulent velocity, the Brunt‐Väisälä frequency, the buoyancy length scale, the rate of dissipation of turbulent energy, the eddy diffusivity, and the time constant of the gravity wave generated, intermittent, decaying, coherent structures responsible for the scatterers detected by the IDI technique. It should be noted that the constants of proportionality are significantly different from the currently accepted values.

Study of Spatial Spectra of Horizontal Turbulence in the Ocean Using Drifter Data

The statistics of a pair of Lagrangian particles offer, in principle, a possibility to estimate the structure functions of velocity, then the spatial autocorrelations, and finally the spatial spectra. On the basis of this strategy, the authors have developed an approach to estimate spatial spectra of mesoscale horizontal turbulence in the ocean using data of satellite-tracked drifters. The approach was applied to the data of 19 drifters deployed in the California Current System in 1993. It was found that the shape of both those spectra and this spectra calculated using drifterborne longitudinal and transverse correlations estimated by other authors are qualitatively in good accordance with theoretical predictions for 2D isotropic nondivergent turbulent flow. To relate obtained spectra to some physical parameters, kinematic stochastic models were developed that consisted of a population of randomly spaced, 2D axisymmetric eddies of a given shape. Numerical experiments with different eddy shapes showed that the model spectra obey a self-similarity; that is, at a given eddy shape they depend on the variance of stochastic process and a length scale of the eddy only. A model with the exponential eddy shape was found to fit drifterborne spectra better than other models. The best agreement between the drifterborne and model spectra was achieved when the radius of an exponentially shaped model eddy was taken equal to the internal Rossby radius.

On the Large Time Asymptotics¶of Decaying Burgers Turbulence

The decay of Burgers turbulence with compactly supported Gaussian "white noise" initial conditions is studied in the limit of vanishing viscosity and large time. Probability distribution functions and moments for both velocities and velocity differences are computed exactly, together with the "time-like" structure functions . The analysis of the answers reveals both well known features of Burgers turbulence, such as the presence of dissipative anomaly, the extreme anomalous scaling of the velocity structure functions and self similarity of the statistics of the velocity field, and new features such as the extreme anomalous scaling of the "time-like" structure functions and the non-existence of a global inertial scale due to multiscaling of the Burgers velocity field. We also observe that all the results can be recovered using the one point probability distribution function of the shock strength and discuss the implications of this fact for Burgers turbulence in general.

Tsallis statistics and fully developed turbulence

An analysis of fully developed turbulence is developed based on the assumption that the underlying statistics of the system is that of the Tsallis ensemble. The multifractal spectrum fT(α) corresponding to the Tsallis-type distribution function is determined self-consistently in the sense that all parameters can be obtained through the observed value of the intermittency exponent. It is shown that the scaling exponents ζm of the velocity structure function derived with the help of the multifractal spectrum fit very well with experimental data. It is revealed that the asymptotic expression of ζm for m>>1 has a log term. The present self-consistent approach narrowed down the value of intermittency exponent µ for the fully developed turbulence to µ = 0.235±0.015.

Scaling properties of the streamwise component of velocity in a turbulent boundary layer

Experimental results concerning the statistics of the streamwise component of velocity in a turbulent boundary layer near a solid wall are presented. Energy spectra, probability density functions and velocity structure functions are investigated as a function of the distance to the wall. Energy spectra and structure functions display power-law scalings which depend on this distance. On the other hand, the relative behavior of structure functions remains constant in the logarithmic sublayer. This behavior exhibits two distinct power-laws. At small scales, relative scaling exponents are relatively close to those usually reported in isotropic and homogeneous turbulence. At larger scales, where shear effects are dominant, relative exponents display larger intermittency corrections.

The kinetic energy spectrum of the two-dimensional enstrophy turbulence cascade

A direct numerical simulation of forced two-dimensional turbulence with hyperviscosity is performed at resolution 40962. A stage is reached at which the flux of enstrophy from large to small scales is approximately constant in time. The cubic and quintic relations for the third-order velocity structure function derived by Lindborg [J. Fluid Mech. 388, 259 (1999)] are verified. The calculated kinetic energy spectrum in the constant enstrophy flux range has the form E(k)=Kεω2/3k−3, where εω is the enstrophy dissipation. This is in accordance with the prediction of Kraichnan [Phys. Fluids 10, 1417 (1970)] and Batchelor [Phys. Fluids 12, II233 (1969)]. The logarithmic correction, suggested by Kraichnan [J. Fluid Mech. 47, 525 (1970)], is not present in the calculated spectrum.

Anomalous scaling exponents and coherent structures in high Re fluid turbulence

The scaling exponents of the velocity structure functions are predicted by a simple model based on the probability density function (PDF) of the local turbulent kinetic energy fluctuations with no hypotheses on the shape of the most intermittent coherent structures. The energy fluctuations are presumed to be induced by statistically independent coherent structures and, therefore, are characterized by exponential-like PDFs. Accounting also for the scaling of the energy with the scale, a recursive formula is obtained which permits the calculation of the scaling exponents of the velocity structure functions. The predicted exponents are checked to be in good agreement with previous theoretical and experimental results. The main hypotheses behind the present approach are also validated by analyzing experimental velocity time series acquired in a jet flow at moderately high Reλ. It is also shown that present approach allows the calculation of the scaling exponents also in cases of nonhomogeneous turbulence.

Decaying magnetohydrodynamics: effects of initial conditions

We study the effects of homogenous and isotropic initial conditions on decaying magnetohydrodynamics (MHD). We show that for an initial distribution of velocity and magnetic-field fluctuations, appropriately defined structure functions decay as a power law in time. We also show that for a suitable choice of initial cross correlations between velocity and magnetic fields even-order structure functions acquire anomalous scaling in time where as scaling exponents of the odd-order structure functions remain unchanged. We discuss our results in the context of fully developed MHD turbulence.

Small scale intermittency and bursting in a turbulent channel flow

The statistical properties of the streamwise velocity fluctuations in a fully developed turbulent channel flow are studied experimentally by means of single hot wire measurements. The intermittency features, studied through the scaling of the moments of the velocity structure function computed using the extended self- similarity and through the probability density function of the wavelet coefficients, are found to be dependent on the distance from the wall. The maximum intermittency effects are observed in the region between the buffer layer and the inner part of the logarithmic region where it is known that the bursting phenomenon, related to coherent structures such as low speed streaks and streamwise vortices, is the dominant dynamical feature. An eduction technique based on wavelet transform for identification of organized motion is developed and used to analyze the turbulent signals. Streamwise velocity conditional averages computed on events educed with the proposed method are reported. Events responsible for intermittency are found to consist of regions of high velocity gradients and are directly correlated with the observed increase of intermittency close to the wall.

Relations between intermittency and structure function exponents in turbulence

We derive from the Navier–Stokes equation an exact equation satisfied by the dissipation rate correlation function, 〈ε(x+r,t+τ)ε(x,t)〉, which we study in the equal time limit, for homogeneous, isotropic turbulence. We exploit its mathematical similarity to the corresponding equation derived from the one-dimensional stochastic Burgers equation to show that the main intermittency exponents are μ1=2−ζ6 and μ2=2z̃4−ζ4, where the ζ’s are exponents of velocity structure functions and z̃4 is a dynamical exponent characterizing the fourth order structure function. We discuss the role of sweeping and Galilean invariance in determining the intermittency exponents.

LES computations and comparison with Kolmogorov theory for two-point pressure–velocity correlations and structure functions for globally anisotropic turbulence

A new extension of the Kolmogorov theory, for the two-point pressure–velocity correlation, is studied by LES of homogeneous turbulence with a large inertial subrange in order to capture the high Reynolds number nonlinear dynamics of the flow. Simulations of both decaying and forced anisotropic homogeneous turbulence were performed. The forcing allows the study of higher Reynolds numbers for the same number of modes compared with simulations of decaying turbulence. The forced simulations give statistically stationary turbulence, with a substantial inertial subrange, well suited to test the Kolmogorov theory for turbulence that is locally isotropic but has significant anisotropy of the total energy distribution. This has been investigated in the recent theoretical studies of Lindborg (1996) and Hill (1997) where the role of the pressure terms was given particular attention. On the surface the two somewhat different approaches taken in these two studies may seem to lead to contradictory conclusions, but are here reconciled and (numerically) shown to yield an interesting extension of the traditional Kolmogorov theory. The results from the simulations indeed show that the two-point pressure–velocity correlation closely adheres to the predicted linear relation in the inertial subrange where also the pressure-related term in the general Kolmogorov equation is shown to vanish. Also, second- and third-order structure functions are shown to exhibit the expected dependences on separation.

Structure function of passive scalars in two-dimensional turbulence

The structure function of a scalar theta(x,t), passively advected in a two-dimensional turbulent flow u(x,t), is discussed by means of the fractal dimension delta1g of the passive-scalar graph. A relation between delta1g, the scaling exponent zeta(theta)(1) of the scalar structure function D1(theta)r, and the structure function D2(r) of the underlying flow field is derived. Different from the three-dimensional (3D) case, the 2D structure function also depends on an additional parameter, characteristic of the driving of the passive scalar. In the enstrophy inertial subrange a mean-field approximation for the velocity structure function gives a scaling of the passive scalar graph with delta1g<2 for intermediate and large values of the Prandtl number Pr. In the energy inertial subrange a model for the energy spectrum and thus D2(r) gives a passive-scalar graph scaling with exponent delta1g=5/3. Finally, we discuss an application to recent observations of scalar dispersion in nonuniversal 2D flows.

Soap film flows: Statistics of two-dimensional turbulence

Soap film flows provide a very convenient laboratory model for studies of two-dimensional (2-D) hydrodynamics including turbulence. For a gravity-driven soap film channel with a grid of equally spaced cylinders inserted in the flow, we have measured the simultaneous velocity and thickness fields in the irregular flow downstream from the cylinders. The velocity field is determined by a modified digital particle image velocimetry method and the thickness from the light scattered by the particles in the film. From these measurements, we compute the decay of mean energy, enstrophy, and thickness fluctuations with downstream distance, and the structure functions of velocity, vorticity, thickness fluctuation, and vorticity flux. From these quantities we determine the microscale Reynolds number of the flow Rλ≈100 and the integral and dissipation scales of 2D turbulence. We also obtain quantitative measures of the degree to which our flow can be considered incompressible and isotropic as a function of downstream distance. We find coarsening of characteristic spatial scales, qualitative correspondence of the decay of energy and enstrophy with the Batchelor model, scaling of energy in k space consistent with the k−3 spectrum of the Kraichnan–Batchelor enstrophy-scaling picture, and power-law scalings of the structure functions of velocity, vorticity, vorticity flux, and thickness. These results are compared with models of 2-D turbulence and with numerical simulations.

Measurements of Turbulent Energy Dissipation Rate with a CW Doppler Lidar in the Atmospheric Boundary Layer

Abstract The results of a theoretical and experimental study of the feasibility of the turbulent energy dissipation rate ϵT measurements with a continuous wave (CW) CO2 Doppler lidar in the atmospheric boundary layer are presented. Three methods of probing ϵT are considered: 1) Doppler spectrum width, 2) the temporal spectrum (temporal structure function) of wind velocity measured by the Doppler lidar, and 3) spatial structure function. In these methods, information on the dissipation rate is extracted by means of analysis of the corresponding statistical characteristics of wind velocity in the inertial subrange of the turbulence, taking into account the spatial averaging of the measured wind velocity fluctuations over sounded volume. In the first and third methods, the spatial structure of the turbulence is analyzed directly. In the second method, to determine ϵT from the measured temporal characteristics, it is necessary to use a model for the spatiotemporal correlation function of wind velocity. As a r...

A second‐order Lagrangian stochastic model for particle trajectories in inhomogeneous turbulence

AbstractThe second‐order Lagrangian stochastic model of Du et al. for particle trajectories in homogeneous turbulence is extended to inhomogeneous turbulence. the model is shown to be in good agreement with experimental dispersion data for laboratory‐scale convective boundary layers and for a neutrally stable wind‐tunnel boundary layer. the results of numerical simulations suggest that finite Reynolds number effects are not important in determining the mean concentrations of tracer particles dispersing in these flows. Estimates for the value of the Lagrangian velocity structure function constant, obtained by optimizing model agreement with experimental dispersion data, are found to be flow‐dependent.

Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence

The scaling properties of one- and two-point statistics of the acceleration, pressure, and pressure gradient are studied in incompressible isotropic turbulence by direct numerical simulation. Ensemble-averaged Taylor-scale Reynolds numbers (Rλ) are up to about 230 on grids from 323 to 5123. From about Rλ 40 onwards the acceleration variance normalized by Kolmogorov variables is found to increase as Rλ1/2. This nonuniversal behavior is traced to the dominant irrotational pressure gradient contributions to the acceleration (whereas the much weaker solenoidal viscous part is universal). Longitudinal and transverse two-point correlations of the pressure gradient differ according to kinematic constraints, but both (especially the latter) extend over distances of intermediate scale size large compared to the Kolmogorov scale. These extended-range properties essentially provide the Eulerian mechanism whereby (as found in recent work) the accelerations of a pair of fluid particles can remain significantly correlated for relatively long periods of time even as they move apart from each other. Although a limited inertial range is attained in the energy spectrum, little evidence for classical inertial scaling is found in acceleration correlations and pressure structure functions. The probability density function (PDF) of pressure fluctuations has negatively skewed tails that exhibit a stretched-exponential form. Pressure gradient statistics show a rapid increase in intermittency with Reynolds number, characterized by widening tails in the PDF and large flatness factors. The practicality of computing acceleration correlations from velocity structure functions is also assessed using direct numerical simulations (DNS); within some resolution limitations good agreement is obtained with experimental data in grid turbulence at comparable Reynolds number.

Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow.

Flow between concentric cylinders with a rotating inner cylinder is studied for Reynolds numbers in the range 2x10(3)<R<10(6) (Taylor Reynolds numbers, 10 < R(lambda)< 290) for a system with radius ratio eta=0.724. Even at the highest Reynolds number studied, the energy spectra do not show power law scaling (i.e., there is no inertial range), and the dissipation length scale is surprisingly large. Nevertheless, the velocity structure functions calculated using extended self-similarity exhibit clear power-law scaling. The structure function exponents zeta(p) fit Kolmogorov's log-normal model within the experimental uncertainty, zeta(p)=(p/3) [1+(mu/6)(3-p)] (for p < or =10) with mu=0.27. These zeta(p) values are close to those found in other flows. Measurements of torque scaling are presented that are an order of magnitude more accurate than those previously reported [Lathrop et al., Phys Rev. A 46, 6390 (1992)]. Measurements of velocity in the fluid core reveal the presence of azimuthal traveling waves up to the highest Reynolds numbers examined. These waves show evidence of a transition at R(T)=1.3 x 10(4); this transition was observed previously in measurements of torque, but our wave velocity and wall shear stress measurements provide the first evidence from local quantities of the transition at R(T). Velocity measurements indicate that at R(T) there is a change in the coherent structures of the core flow; this is consistent with our analyses of the scaling of the torque. Our measurements were made at two aspect ratios, and no significant dependence on aspect ratio was observable for R > R(T).

Statistical study of anisotropic anomalous behavior of a cylinder wake

An experimental investigation of the statistical properties of the velocity difference in an anisotropic turbulent cylinder wake at moderate Reλ is conducted by a triple hot-wire anemometer (HWA) probe. The energy spectra show a clear large scale anisotropy due to the presence of the Von Karman vortices. In spite of the low Reλ and the large scale anisotropy, the extended-self-similarity (ESS) allows location of broad scaling ranges and the calculation of the scaling exponents of the three velocity structure function components in order to examine the intermittency anomalies. More intermittent behavior of the transverse velocity components with respect to the longitudinal one is found both by ESS and by comparison of the longitudinal and transverse velocity difference probability distribution functions (PDF).

Scaling and exotic regimes in decaying Burgers turbulence

We analyze the stochastic scaling laws arising in the invicid limit of the decaying solutions of the Burgers equation. The linear scaling of the velocity structure functions is shown to reflect the domination by shocks of the long-time asymptotics. We exhibit new self-similar statistics of solutions describing phases with diluted shocks. Some speculations are included on the nature of systems whose large time behavior is described by the new statistics.

Reply To 'Comments on the 'Universality of the Lagrangian velocity structure function constant (C0) across different kinds of turbulence’' by A. M. Reynolds

Reply To 'Comments on the 'Universality of the Lagrangian velocity structure function constant (C0) across different kinds of turbulence’' by A. M. Reynolds