Lagrangian Velocity Increments Research Articles

Lagrangian turbulence in the woods

Abstract

Lagrangian statistics in isotropic turbulent flows with deterministic and stochastic forcing schemes

Artificial input of energy into the flow is necessary to create and maintain a statistically stationary isotropic turbulence for sampling in studying the statistics. Due to the nonlinear coupling among different Fourier modes through the triadic interaction, whether or not various forcing schemes affect the statistics in turbulence is an important and open question. We present detailed comparison of Lagrangian statistics of fluids particles in forced isotropic turbulent flows in 1283, 2563, and 5123 simulations, with Taylor-scale Reynolds numbers in the range of 64–171, using a deterministic and a stochastic forcing scheme, respectively. Several Lagrangian statistics are compared, such as velocity and acceleration autocorrelations, and moments of Lagrangian velocity increments. The differences in the Lagrangian statistics obtained from the two forcing schemes are shown to be small, indicating that the isotropic forcing schemes used have little effects on the Lagrangian statistics in the isotropic turbulence.

Introduction of longitudinal and transverse Lagrangian velocity increments in homogeneous and isotropic turbulence

Based on geometric considerations, longitudinal and transverse Lagrangian velocity increments are introduced as components along, and perpendicular to, the displacement of fluid particles during a time scale τ. It is argued that these two increments probe preferentially the stretching and spinning of material fluid elements, respectively. This property is confirmed (in the limit of vanishing τ) by examining the variances of these increments conditioned on the local topology of the flow. These longitudinal and transverse Lagrangian increments are found to share some qualitative features with their Eulerian counterparts. In particular, direct numerical simulations at up to 300 show that the distributions of are negatively skewed at all τ, which is a signature of time irreversibility in the Lagrangian framework. Transverse increments are found more intermittent than longitudinal increments, as quantified by the comparison of their respective flatnesses and scaling laws. Although different in nature, standard Cartesian Lagrangian increments (projected on fixed axis) exhibit scaling properties that are very close to transverse Lagrangian increments.

A new assessment of the second-order moment of Lagrangian velocity increments in turbulence

The behaviour of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov 4/5-law into a relation predicting the linear (in time) scaling for the second-order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterisation of the Lagrangian structure functions based on Batchelor model is introduced and tested on data for 3d turbulence, and for 2d turbulence in the inverse cascade regime. Such parameterisation supports the idea, previously suggested, that both Eulerian and Lagrangian data are consistent with a linear scaling plus finite-Reynolds number effects affecting the small- and large timescales. When large-time saturation effects are properly accounted for, compensated plots show a detectable plateau already at the available Reynolds number. Furthermore, this parameterisation allows us to make quantitative predictions on the Reynolds number value for which Lagrangian structure functions are expected to display a scaling region. Finally, we show that this is also sufficient to predict the anomalous dependency of the normalised root mean squared acceleration as a function of the Reynolds number, without fitting parameters.

Generation of Lagrangian intermittency in turbulence by a self-similar mechanism

Intermittency, i.e., extreme fluctuations at small scales, causes the deviation of turbulence statistics from Kolmogorov's 1941 theoretical predictions. Intermittency effects are especially strong for Lagrangian statistics. Our understanding of how Lagrangian intermittency manifests, however, is still elusive. Here, we study the Lagrangian intermittency in the framework of an exact, yet unclosed probability density function (PDF) equation. Combining this theoretical approach with data from experiments and simulations, no a priori phenomenological assumptions about the structure or properties of the flow have to be made. In this description, the non-self-similar evolution of the velocity increment PDF is determined at all scales by a single function, which is accessible through data from experiments and simulations. This ‘intermittency generating function’ arises from the dependence of the acceleration of a fluid element on its velocity history, thereby coupling different scales of turbulent motion. Empirically, we find that the intermittency generating function has a simple, approximately self-similar form, which has the surprising implication that Lagrangian intermittency—the absence of self-similarity in the Lagrangian velocity increment statistics—is driven by a self-similar mechanism. The simple form of the intermittency generating function furthermore allows us to formulate a simple model parametrization of the velocity increment PDFs.

A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows

A phenomenological theory of the fluctuations of velocity occurring in a fully developed homogeneous and isotropic turbulent flow is presented. The focus is made on the fluctuations of the spatial (Eulerian) and temporal (Lagrangian) velocity increments. The universal nature of the intermittency phenomenon as observed in experimental measurements and numerical simulations is shown to be fully taken into account by the multiscale picture proposed by the multifractal formalism, and its extensions to the dissipative scales and to the Lagrangian framework. The article is devoted to the presentation of these arguments and to their comparisons against empirical data. In particular, explicit predictions of the statistics, such as probability density functions and high order moments, of the velocity gradients and acceleration are derived. In the Eulerian framework, at a given Reynolds number, they are shown to depend on a single parameter function called the singularity spectrum and to a universal constant governing the transition between the inertial and dissipative ranges. The Lagrangian singularity spectrum compares well with its Eulerian counterpart by a transformation based on incompressibility, homogeneity and isotropy and the remaining constant is shown to be difficult to estimate on empirical data. It is finally underlined the limitations of the increment to quantify accurately the singular nature of Lagrangian velocity. This is confirmed using higher order increments unbiased by the presence of linear trends, as they are observed on velocity along a trajectory.

Influence of waves on Lagrangian acceleration in two-dimensional turbulent flows

The Lagrangian statistics in rotating Saint-Venant turbulence are studied by means of direct numerical simulation using a pseudo-spectral discretization fully resolving, both in time and space, all the inertio-gravity waves present in the system. To understand the influence of waves, three initial conditions are considered, one which is dominated by waves, one which is dominated by vortices, and one which is intermediate between these two extreme cases. It is found that Lagrangian acceleration is strongly influenced by the presence of waves, as are the Lagrangian velocity increments, for incremements that are sufficiently short, and that vortexmotion influences mostly the component of Lagrangian acceleration which is perpendicular to the local velocity.

Turbulence in fluid and plasma: search for a new paradigm

A first principle explanation of the origin of intermittency and nonlinear structure formation in the Lagrangian velocity increments of a turbulent flow is presented in the context of a scale-invariant analytical formalism that is being developed recently. The copious generation of power laws and nonlinear exponents in the structure functions are shown to follow quite naturally in the present formalism.

The influence of walls on Lagrangian statistics in two-dimensional turbulence

The influence of solid walls on the Lagrangian statistics of statistically stationary two-dimensional turbulence is investigated by comparing the flow in a circular wall-bounded and in an unbounded periodic domain. A Fourier pseudo-spectral method is used, which is combined in the wall bounded case with a volume penalization technique to impose no-slip conditions. A particular emphasis is put on the acceleration of fluid particles. It is investigated to what extent the impact of the boundaries influences the shape of the probability density functions of Lagrangian velocity increments. It is shown that the influence of walls is not confined to a small near-wall region but alters the statistics in the entire flow domain. This can be explained by the vorticity generation in the turbulent boundary layer which destabilizes and leads to the formation of vortices that subsequently detach and travel into the bulk flow. The enstrophy level is thus increased with respect to the one in the unbounded periodic domain.

Conditional Eulerian and Lagrangian velocity increment statistics of fully developed turbulent flow

Conditional statistics of homogeneous isotropic turbulent flow is investigated by means of high Reynolds number direct numerical simulations performed with 20483 collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data, longitudinal probability density functions P(δl||u|εl) conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also transverse increments conditioned on either the dissipation rate or the square of the vorticity have quasi-Gaussian probability density functions (PDFs). Concerning Lagrangian statistics we found that conditioning on a trajectory averaged energy dissipation rate ετ significantly reduces the scale dependence of the increment PDFs P(δτui|ετ). By means of dimensional arguments we propose a novel condition for Lagrangian increments which is shown to reduce even more the flatness of the corresponding PDFs and thus intermittency in the inertial range of scales. The conditioned Lagrangian PDF corresponding to the smallest increment considered is reasonably well described by a prediction for the PDF of acceleration of a non-intermittent velocity field. Conditioned structure functions show approximately K41 scaling with a larger scaling range than the unconditioned ones.

Exact relation between Eulerian and Lagrangian velocity increment statistics

We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments.

Statistics of a mixed Eulerian–Lagrangian velocity increment in fully developed turbulence

We investigate the relationship between Eulerian and Lagrangian probability density functions obtained from numerical simulations of two-dimensional as well as three-dimensional turbulence. We show that in contrast to the structure functions of the Lagrangian velocity increment δτv(y)=u(x(y, τ), τ)- u(y, 0), where u(x, t) denotes the Eulerian velocity and x(y, t) the particle path initially starting at x(y, 0)=y, the structure functions of the velocity increment δτw(y)=u(x(y, τ), τ)- u(y, τ) exhibit a wide range of scaling behavior. Similar scaling indices are detected for the structure functions for particles diffusing in frozen turbulent fields. Furthermore, we discuss a connection to the scaling of Eulerian transversal structure functions.

Lagrangian structure functions in turbulence: A quantitative comparison between experiment and direct numerical simulation

A detailed comparison between data from experimental measurements and numerical simulations of Lagrangian velocity structure functions in turbulence is presented. Experimental data, at Reynolds number ranging from Rλ=350 to Rλ=815, are obtained in a swirling water flow between counter-rotating baffled disks. Direct numerical simulations (DNS) data, up to Rλ=284, are obtained from a statistically homogeneous and isotropic turbulent flow. By integrating information from experiments and numerics, a quantitative understanding of the velocity scaling properties over a wide range of time scales and Reynolds numbers is achieved. To this purpose, we discuss in detail the importance of statistical errors, anisotropy effects, and finite volume and filter effects, finite trajectory lengths. The local scaling properties of the Lagrangian velocity increments in the two data sets are in good quantitative agreement for all time lags, showing a degree of intermittency that changes if measured close to the Kolmogorov time scales or at larger time lags. This systematic study resolves apparent disagreement between observed experimental and numerical scaling properties.

Particle acceleration in turbulent flows: A class of nonlinear stochastic models for intermittency

The problem of modeling the velocity and acceleration of inertial particles in turbulent flows is discussed. Particular attention is focused on the modeling of the particle Lagrangian velocity increment, especially, but not exclusively, in the case in which only the low frequencies of the carrier turbulent flow field are available. The need for suitable models arises in the simulation of particle laden flows by the means of new computational techniques such as large-eddy simulation. For this, stochastic differential equations, sde, are often introduced, though there is a lack of clarity in how such models should deal with the experimental observed far from Gaussian statistics, intermittency, and heavy tailed probability density function for particle acceleration. It is well known that Langevin-type equations are not capable of reproducing such features. It is first shown how the stochastic model for the particle Lagrangian velocity increments is far from being a Langevin equation, and it is characterized by nonlinear drift and diffusion; the statistical characteristics of this first model are shown to be in qualitative agreement with experimental findings. These results suggest an improved model for the particle dynamics based upon a more general family of nonlinear sde; the family, which is generated by a single parameter, includes both the Langevin equation and the first model as special cases. An analysis of the statistical properties of the new sde shows that the model is capable of accurately reproducing the strong deviations from Gaussianity observed in recent experiments.

Lagrangian velocity structure functions in Bolgiano turbulence

Single-particle Lagrangian velocity statistics in the Bolgiano–Obukhov regime of two-dimensional turbulent convection is investigated. At variance with flows displaying the classical K41 phenomenology, here the leading contribution to the Lagrangian velocity fluctuations is given by the largest eddies. This implies a linear behavior in time for a typical velocity fluctuation in the time interval t. The contribution to the Lagrangian velocity fluctuations of local eddies (i.e., with a characteristic time of order t), whose space/time scalings are ruled by the Bolgiano–Obukhov theory, is thus not detectable by standard Lagrangian statistical observables. To disentangle contributions arising from the large eddies from those of local eddies, a strategy based on exit-time statistics has successfully been exploited. Lagrangian velocity increments in Bolgiano convection thus provide a physically relevant example of a signal with more than smooth fluctuations.

Stochastic simulation of Lagrangian trajectories in near-wall turbulence

A dissipation-conditioned stochastic model for the simulation of marked fluid-particle trajectories is formulated that reproduces the extreme intermittency of Lagrangian accelerations and Lagrangian velocity increments observed in recent direct numerical simulations (DNS) of near-wall turbulence [Lee, Yeo, and Choi, Phys. Rev. Lett. 92, 144502 (2004); Choi, Yeo, and Lee, Phys. Fluids 16, 779 (2004)]. Model agreement with data from the DNS for the mean-square displacements of marked fluid particles is good and significantly better than that which can be obtained with stochastic models formulated in terms of the mean rather than the fluctuating rate of dissipation of turbulent kinetic energy. The model accounts, in part, for the observed extreme intermittency of rotations of the Lagrangian velocity vector.

Lagrangian statistics from direct numerical simulations of isotropic turbulence

A comprehensive study is reported of the Lagrangian statistics of velocity, acceleration, dissipation and related quantities, in isotropic turbulence. High-resolution direct numerical simulations are performed on 643and 1283grids, resulting in Taylor-scale Reynolds numbers Rλin the range 38-93. The low-wavenumber modes of the velocity field are forced so that the turbulence is statistically stationary. Using an accurate numerical scheme, of order 4000 fluid particles are tracked through the computed flow field, and hence time series of Lagrangian velocity and velocity gradients are obtained.The results reported include: velocity and acceleration autocorrelations and spectra; probability density functions (p.d.f.'s) and moments of Lagrangian velocity increments; and p.d.f.'s, correlation functions and spectra of dissipation and other velocity-gradient invariants. It is found that the acceleration variance (normalized by the Kolmogorov scales) increases as R½λ- a much stronger dependence than predicted by the refined Kolmogorov hypotheses. At small time lags, the Lagrangian velocity increments are distinctly non-Gaussian with, for example, flatness factors in excess of 10. The enstrophy (vorticity squared) is found to be more intermittent than dissipation, having a standard-deviation-to-mean ratio of about 1.5 (compared to 1.0 for dissipation). The acceleration vector rotates on a timescale about twice the Kolmogorov scale, while the timescales of acceleration magnitude, dissipation and enstrophy appear to scale with the Lagrangian velocity timescale.