Lagrangian Velocity Structure Functions Research Articles

About Universality and Thermodynamics of Turbulence.

This paper investigates the universality of the Eulerian velocity structure functions using velocity fields obtained from the stereoscopic particle image velocimetry (SPIV) technique in experiments and direct numerical simulations (DNS) of the Navier-Stokes equations. It shows that the numerical and experimental velocity structure functions up to order 9 follow a log-universality (Castaing et al. Phys. D Nonlinear Phenom. 1993); this leads to a collapse on a universal curve, when units including a logarithmic dependence on the Reynolds number are used. This paper then investigates the meaning and consequences of such log-universality, and shows that it is connected with the properties of a “multifractal free energy”, based on an analogy between multifractal and thermodynamics. It shows that in such a framework, the existence of a fluctuating dissipation scale is associated with a phase transition describing the relaminarisation of rough velocity fields with different Hölder exponents. Such a phase transition has been already observed using the Lagrangian velocity structure functions, but was so far believed to be out of reach for the Eulerian data.

Kolmogorov’s Lagrangian similarity law revisited

Kolmogorov’s similarity turbulence theory in a Lagrangian frame is assessed with new direct numerical simulations of isotropic turbulence with and without hyperviscosity, which attain higher Reynolds numbers than previously available. It is demonstrated that hyperviscous simulations can be used to accurately predict the second order Lagrangian velocity structure function (LVSF-2) in the inertial range, by using an original new procedure. The results strongly support Kolmogorov’s Lagrangian similarity assumption and allow the universal constant of LVSF-2 to be computed with a new level of confidence with C0=7.4±0.2.

Concentration effects on turbulence in dilute polymer solutions far from walls.

We report measurements of the modification of turbulence far from any walls by small concentrations of long-chain polymers. We consider a range of statistical properties of the flow, including Eulerian and Lagrangian velocity structure functions, Eulerian acceleration correlation functions, and the relative dispersion of particle pairs. In all cases, we find that the polymer concentration has a strong effect on the extent to which the statistical properties are changed compared to their values in pure water. These effects can be captured by the recently proposed energy flux-balance model (when suitably extended into the time domain for Lagrangian statistics). However, unlike previous measurements, which found that the concentration effect could be completely scaled out, we consistently find that our data collapse onto two different master curves, one for small concentration and one for larger concentration. We suggest that the difference between the two may be related to the onset of interactions among polymer chains, which is likely to be more easily observed at the small Weissenberg numbers we consider here.

A multifractal model for linking Lagrangian and Eulerian velocity structure functions

A multifractal model is developed to connect the Lagrangian multifractal dimensions with their Eulerian counterparts. We propose that the characteristic time scale of a Lagrangian quantity should be the Lagrangian time scale, and it should not be the Eulerian time scale which was widely used in previous studies on Lagrangian statistics. Using the present model, we can obtain the scaling exponents of Lagrangian velocity structure functions from the existing data or models of scaling exponents of Eulerian velocity structure functions. This model is validated by comparing its prediction with the results of experiments, direct numerical simulations, and the previous theoretical models. The comparison shows that the proposed model can better predict the scaling exponents of Lagrangian velocity structure functions, especially for orders larger than 6.

A Lagrangian stochastic model for tetrad dispersion

We present results for tetrad (four-particle) dispersion in homogeneous isotropic turbulence by means of a simple Lagrangian stochastic model with a focus on the inertial subrange. We show that for appropriate values of C 0, the constant of proportionality in the second-order Lagrangian velocity structure function, the shape statistics agree well with equivalent results from a direct numerical simulation (DNS) of turbulence. Moreover, we show that the shape statistics are independent of C 0 for a wide range of C 0-values. We also show that the parameters which characterise the shape of the tetrads can be approximately related to appropriate ratios of the growth rates of the mean square separation, the mean square area and the mean square volume of the tetrads. By means of exit times, we are able to estimate the equivalent values for the DNS data. We also consider the statistics of four-point velocity differences (via a diffusion tensor) which agree well with the DNS. We show that the nature of the velocity field experienced by the tetrad varies significantly with C 0.

Velocity Statistics in Quantum Turbulence

Quantum turbulence in thermal counterflow of superfluid 4He is studied at length scales comparable to the average distance R between quantized vortices. The Lagrangian velocities of micrometer-sized deuterium particles, smaller here than R, are calculated in a planar section of the experimental volume by using the particle tracking velocimetry technique. It is confirmed that the Lagrangian velocity normalized distributions are strongly non-Gaussian, with power-law tails, in contrast to the nearly Gaussian distributions obtained in classical turbulent flows. The probability density functions of the velocity differences along the particle trajectories, that is, the Lagrangian velocity structure functions of order 1, are also computed. Their shape changes from a strongly non-Gaussian form to a nearly Gaussian one as the corresponding Lagrangian time delay increases, showing thus an unexpected classical-like behavior.

High order lagrangian velocity statistics in a turbulent channel flow with Reτ = 80

The scaling properties of high-order Lagrangian velocity structure functions are investigated numerically for a turbulent flow with a friction-velocity based Reynolds number Reτ = 80. The Lagrangian particles are released from locations of different distances to the wall. The relative scaling exponents ξLU(q) of the longitudinal velocity component are found to increase with the released distance to the wall and to approach asymptotically to theoretical predictions. However, the scaling exponents ξLV(q) of the transverse velocity component are smaller than ξLU(q), indicating a more intermittent nature. Specifically for the release locations in the center region, the relative scaling exponents ξLU(q) agree very well with theoretical predictions.

Pair reversal in homogeneous isotropic turbulence

We show that the separation of particle pairs in the inertial subrange of homogeneous isotropic turbulence is strongly influenced by the pairs that separate quasi-diffusively. We quantify the influence of the diffusive separators by considering the probability that a pair will 'reverse' direction across a given separation i.e. its separation will decrease (before eventually increasing) and derive an analytical expression for the expected number of reversals across this separation for a quasi-one-dimensional model of relative dispersion in the inertial subrange with Gaussian turbulence. We compare this theoretical result with three different Lagrangian stochastic models in which the influence of the diffusive and ballistic separators (the latter dominated by velocity memory) can be varied by means of the value of C0, the constant of proportionality in the Lagrangian velocity structure function, which appears explicitly in Lagrangian stochastic models. We also compare these results with data from a direct numerical simulation of turbulence. The results indicate the importance of the transverse relative velocity component (i.e. the ability of pairs to rotate), which is absent in Q1D models, in determining the correct quantitative relative dispersion statistics.

Kolmogorov similarity scaling for one-particle Lagrangian statistics

We use direct numerical simulation data up to a Taylor scale Reynolds number Rλ = 1000 to investigate Kolmogorov similarity scaling in the inertial sub-range for one-particle Lagrangian statistics. Although similarity scaling is not achieved at these Reynolds numbers for the Lagrangian velocity structure function, we show clearly that it is achieved for the Lagrangian acceleration frequency spectrum and the scaling range becomes wider with increasing Reynolds number. Stochastic and heuristic model calculations suggest that the difference in behavior observed for the structure function and spectrum is simply a consequence of different rates of convergence to scaling behavior with increasing Reynolds number. Our estimate C0 ≈ 6.9 ± 0.2 for the Lagrangian structure function constant is close to earlier estimates based on extrapolation of the peak value of the compensated structure function. The results presented here suggest prospects for studying Kolmogorov similarity for Lagrangian statistics using the latest innovations in simulation, and measurement techniques are more hopeful than previously suggested in the literature.

Anomalous scaling for Lagrangian velocity structure functions in fully developed turbulence

A hierarchical structure model is developed for anomalous scaling of the Lagrangian velocity structure functions in fully developed turbulence. This model is an extension of the Eulerian hierarchical structure model of She and Leveque [Phys. Rev. Lett. 72, 366 (1994)] to the Lagrangian velocity structure functions, where the straining and sweeping hypotheses are used to build up the relationship between the singular scalings of Lagrangian and Eulerian intermittent structures. The Lagrangian scaling exponents obtained from the straining hypothesis are in good agreement with the experimental results of the Bodenschatz group.

Numerical study of high-order Lagrangian structure functions in a turbulent channel flow with low Reynolds number

The scaling exponents of Lagrangian velocity structure functions from orders 1 to 10 in a low Reynolds number turbulent channel flow are investigated by using direct numerical simulation. The Reynolds number Reτ is 80 (based on friction velocity on the wall). The Lagrangian velocity structure functions are shown to obey the scaling relations <Δvq(τ)>∼τζL(q). The scaling exponents are normalized by ζL(2) (so-called ESS procedure). The coincidence between the theoretical predictions and numerical calculations is very good for the longitudinal scaling exponent in the channel center. It is also found that the high-order longitudinal scaling exponents agree with theoretical values better than those for the transverse direction.

Estimation of the Kolmogorov constant for the Lagrangian velocity spectrum and structure function under different PBL stability regimes generated by LES

The Lagrangian description of turbulence is very important in studies of mixing and dispersion. The following study focuses on the main aspects using data from high resolution Large Eddy Simulations and investigating basic results in Kolmogorov similarity giving an estimation of inertial subrange universal constants. It was possible to evaluate the velocity structure functions and the Lagrangian spectra in the inertial subrange, under different stability conditions, by tracking an ensemble of Lagrangian particles in the PBL simulated with a Large Eddy Simulation model. This allows a direct estimation of the inertial subrange constants and their dependence on the Planetary Boundary Layer stability conditions.

Lagrangian and Eulerian Velocity Structure Functions in Hydrodynamic Turbulence

The Lagrangian and Eulerian transverse velocity structure functions of fully developed fluid turbulence are found based on the inviscid Navier-Stokes equation. The structure functions are shown to obey the scaling relations S(n)(L)(tau) proportional to tau(xi(n)) and S(n)(E)(l) proportional to l(zeta(n)) inside the inertial range. The scaling exponents zeta(n) and xi(n) are calculated analytically without using dimensional considerations. This result is a significant step toward understanding the nature of hydrodynamic turbulence. The obtained values are in agreement with recent numerical and experimental data.

The effects of parametric uncertainties in simulations of a reactive plume using a Lagrangian stochastic model

A combined Lagrangian stochastic model with micro-mixing and chemical sub-models is used to investigate a reactive plume of nitrogen oxides (NO x) released into a turbulent grid flow doped with ozone (O 3). Sensitivities to the model input parameters are explored for different source NO x scenarios. The wind tunnel experiments of Brown and Bilger (1996) provide the simulation conditions for the first case study where photolysis reactions are not included and the main uncertainties occur in parameters defining the turbulence scales, source size and reaction rate of NO with O 3. Using nominal values of the parameters from previous studies, the model gives a good representation of the radial profile of the conserved mean scalar Γ ¯ NO x although slightly over predicts peak mean NO 2 concentrations Γ ¯ NO 2 compared to the experiments. The high dimensional model representation (HDMR) method is used to investigate the effects of uncertainties in model inputs on the simulation of chemical species concentrations. For this scenario, the Lagrangian velocity structure function coefficient has the largest impact on simulated Γ ¯ NO x profiles. Photolysis reactions are then included in a chemical scheme consisting of eight reactions between species NO, O, O 3 and NO 2. Independent and interactive effects of 22 input parameters are studied for two source NO x scenarios using HDMR, including turbulence parameters, temperature dependant rate parameters, photolysis rates, temperature, fraction of NO in total NO x at the source and background ozone concentration [O 3]. For this reactive case, the variance in the predicted mean plume centre Γ ¯ O 3 is caused by parameters describing both physical (mixing time-scale coefficient) and chemical processes (activation energy for the reaction O 3+NO). The variance in predicted plume centre Γ ¯ NO 2 and root mean square NO 2 concentration γ NO 2 ′ , is strongly influenced by the fraction of NO in the source NO x, and to a lesser extent the mixing time-scale coefficient. Adjusting the latter gives improved agreement with the Brown and Bilger experiment. Some weak parameter interactions are observed.

Estimation of the Kolmogorov constant by large-eddy simulation in the stable PBL

Abstract In this paper is evaluated the inertial subrange Kolmogorov constant C 0 in a stable boundary-layer. The importance of the constant C 0 is well known as predictions of turbulent dispersion by means of Lagrangian stochastic models depend upon its value. Different values of the C 0 constant has been proposed along the years, most of them determined at low Reynolds number and/or under different techniques. Here we estimate the constant C 0 by tracking an ensemble of Lagrangian particles injected in a stable planetary boundary-layer simulated with a large-eddy simulation model and analyzing the ensemble-averaged Lagrangian velocity structure function in the inertial subrange. Our estimative of C 0 , w is 3.7, which is consistent with values found in literature. The evaluation of C 0 , u and C 0 , v cannot be easily accomplished since it is difficult to identify an inertial subrange for the wind field horizontal components.

Lagrangian Properties of Particles in Turbulence

The Lagrangian description of turbulence is characterized by a unique conceptual simplicity and by an immediate connection with the physics of dispersion and mixing. In this article, we report some motivations behind the Lagrangian description of turbulence and focus on the statistical properties of particles when advected by fully developed turbulent flows. By means of a detailed comparison between experimental and numerical results, we review the physics of particle acceleration, Lagrangian velocity structure functions, and pairs and shapes evolution. Recent results for nonideal particles are discussed, providing an outlook on future directions.

A conditionally cubic-Gaussian stochastic Lagrangian model for acceleration in isotropic turbulence

The modelling of fluid-particle acceleration in homogeneous isotropic turbulence in terms of stochastic models for the Lagrangian velocity, acceleration and a dissipation rate variable is considered. The basis for the Reynolds model (A. M. Reynolds, Phys. Rev. Lett. vol. 91, 2003, 084503) is reviewed and examined by reference to direct numerical simulations (DNS) of isotropic turbulence at Taylor-scale Reynolds number (Rλ) up to about 650. In particular, we show DNS data that support stochastic modelling of the logarithm of pseudo-dissipation as an Ornstein–Uhlenbeck process and reveal non-Gaussianity of the acceleration conditioned on fluctuations of the pseudo-dissipation rate. The DNS data are used to construct a new stochastic model that is exactly consistent with Gaussian velocity and conditionally cubic-Gaussian acceleration statistics. This model captures the effects of small-scale intermittency on acceleration and the conditional dependence of acceleration on pseudo-dissipation (which differs from that predicted by the refined Kolmogorov hypotheses). Non-Gaussianity of the conditionally standardized acceleration probability density function (PDF) is accounted for in terms of model nonlinearity. The large-time behaviour of the new model is that of a velocity-dissipation model that can be matched with DNS data for conditional second-order Lagrangian velocity structure functions. As a result, the diffusion coefficient for the new model incorporates two-time information and its Reynolds-number dependence as observed in DNS. The resulting model predictions for conditional and unconditional velocity autocorrelations and time scales are shown to be in very good agreement with DNS.

Effects of vortex filaments on the velocity of tracers and heavy particles in turbulence

The Lagrangian statistics of heavy particles and of fluid tracers transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. The Lagrangian velocity structure functions are measured in a time range spanning about three decades, from a tenth of the Kolmogorov time scale, τη, up to a few large-scale eddy turnover times. Strong evidence is obtained that fluid tracer statistics are contaminated in the time range τ∊[1:10]τη by a bottleneck effect due to vortex filament. This effect is found to be significantly reduced for heavy particles which are expelled from vortices by inertia. These findings help in clarifying the results of a recent study by H. Xu et al. [Phys. Rev. Lett. 96, 024503 (2006)], where differences between experimental and numerical results on scaling properties of fluid tracers were reported.

Multifractal Dimension of Lagrangian Turbulence

We report experimental measurements of the Lagrangian multifractal dimension spectrum in an intensely turbulent laboratory water flow by the optical tracking of tracer particles. The Legendre transform of the measured spectrum is compared with measurements of the scaling exponents of the Lagrangian velocity structure functions, and excellent agreement between the two measurements is found, in support of the multifractal picture of turbulence. These measurements are compared with three model dimension spectra. When the nonexistence of structure functions of order less than -1 is accounted for, the models are shown to agree well with the measured spectrum.

Estimation of the Lagrangian Velocity Structure Function Constant C0 by Large-Eddy Simulation

The inertial subrange Kolmogorov constant C0, which determines the effective tur- bulent diffusion in velocity space, plays an important role in the Lagrangian modelling of pollutants. A wide range of values of the constant are found in the literature, most of them determined at low Reynolds number and/or under different assumptions. Here we estimate the constant C0 by tracking an ensemble of Lagrangian particles in a planetary boundary layer simulated with a large-eddy simulation model and analysing the Lagrangian velocity structure function in the inertial subrange. The advantage of this technique is that it easily allows Reynolds numbers to be achieved typical of convective turbulent flows. Our estimates of C0 is C0 = 4.3 ± 0.3 consistent with values found in the literature.