Lagrangian Velocity Correlation Research Articles

Statistics of the Inertial Energy Transfer Range in d-Dimensional Turbulence (2 ≤ d ≤ 3) in a Lagrangian Renormalized Approximation

Statistics in the inertial energy transfer range (IETR) of d-dimensional turbulence ( 2≤d≤3) are studied using a Lagrangian renormalized approximation (LRA). The LRA suggests that the energy spectrum in the IETR is given by Kd|ε¯|2/3k−5/3, where Kd is a constant and ε¯ is the energy flux across wave-number k; the energy transfer is forward for dc<d≤3 but inverse for 2≤d<dc, where dc≈2.065; at d=dc, Kd diverges and the skewness of the longitudinal velocity difference vanishes; and the d-dependence of the two-time Lagrangian velocity correlation spectra under appropriate normalization is weak in the IETR.

Two-time Lagrangian velocity correlation function for particle pairs in two-dimensional inverse energy-cascade turbulence

We numerically investigate a two-time Lagrangian velocity correlation function (TTLVCF) for particle pairs in two-dimensional energy inverse-cascade turbulence. We consider self similarity of the correlation function by means of incomplete similarity. In this framework, we propose a self-similar form of the correlation function, whose scaling exponents cannot be determined by only using the dimensional analysis based on the Kolmogorov's phenomenology. As a result, the scaling laws of the correlation function can depend on the initial separation. This initial-separation dependency is frequently observed in laboratory experiments and direct numerical simulations of the relative dispersion, which is directly related to the correlation function, at moderate Reynolds numbers. We numerically verify the self-similar form by direct numerical simulations of two-dimensional energy inverse-cascade turbulence. The involved scaling exponents and the dependencies on finite Reynolds number effects are determined empirically. Then, we consider implication of the scaling laws of the correlation function on the relative dispersion, i.e. the Richardson-Obukhov $t^3$ law. Our results suggest a possibility not to recover the Richardson-Obukhov $t^3$ law at infinite Reynolds number.

Scale-similar structures of homogeneous isotropic non-mirror-symmetric turbulence based on the Lagrangian closure theory

We investigate the effect of helicity on the scale-similar structures of homogeneous isotropic and non-mirror-symmetric turbulence based on the Lagrangian renormalised approximation (LRA), which is a self-consistent closure theory proposed by Kaneda (J. Fluid Mech., vol. 107, 1981, pp. 131–145). In this study, we focus on the time scale representing the scale-similar range. For the LRA, the Lagrangian two-time velocity correlation and response function determine the representative time scale. The LRA predicts that both the Lagrangian two-time velocity correlation and response function equation do not explicitly depend on helicity. We assume the extended scale-similar spectra and time scale by considering the helicity dissipation rate. Considering the small-scale structures, the requirements for the energy and helicity fluxes and response function equation to be scale similar, yield the conventional inertial-range power laws and provide the energy and helicity spectra $\propto k^{-5/3}$ and the time scale $\propto \varepsilon ^{-1/3} k^{-2/3}$ , where $\varepsilon$ and $k$ denote the energy dissipation rate and wavenumber, respectively. Notably, energy flux can be scale similar only when $k^H /k \ll 1$ , where $k^H = \varepsilon ^H/\varepsilon$ and $\varepsilon ^H$ denotes the helicity dissipation rate. This condition makes the energy cascade process in the scale-similar range completely independent of helicity. We also investigate the localness of the interscale interaction in the energy and helicity cascades for the LRA. We demonstrate that the helicity cascade is slightly non-local in scales compared with the energy cascade. This study provides a foundation on the modelling of non-mirror-symmetric turbulent flows.

Anomalous transport through free-flow-porous media interface: Pore-scale simulation and predictive modeling

Pore-scale velocity and turbulence structures near streambeds may control solute transport and dispersion in streams. In this study, pore-scale flow and transport simulations are performed to investigate the effects of pore-scale processes on anomalous transport, which is often manifested by remarkably long residence times of solute particles, in coupled free-flow and porous media systems. By solving the 2D Reynolds-averaged Navier–Stokes equations integrated with a renormalization group k − ε turbulence model, we resolve complex pore-scale flows featured by vortices and preferential flows. Then, we incorporate the simulated velocity and turbulence fields into a Lagrangian particle tracking model that considers advection, turbulent diffusion, and molecular diffusion. Simulation results reveal that the interplay between pore-scale vortices and turbulence structures near the free-flow-permeable bed interface controls anomalous transport. High porosity induces strong turbulence penetration and preferential flow paths within permeable beds. The enhanced subsurface turbulence facilitates the escape of solute particles from recirculation zones via turbulent diffusion, causing steep power-law slopes in breakthrough curves (BTCs). In contrast, low porosity introduces heavy tailing in BTCs from particles that are trapped in the near-interface recirculation zones characterized by low velocities and limited turbulence. We upscale and predict particle transport via a Spatial Markov model (SMM) honoring the interplay between Lagrangian velocity distribution and velocity correlation. The SMM reproduces anomalous transport behaviors obtained from the numerical simulations. These results demonstrate that Lagrangian velocity statistics effectively encode anomalous transport mechanisms in the coupled systems.

Effects of the Reynolds number on a scale-similarity model of Lagrangian velocity correlations in isotropic turbulent flows

A scale-similarity model of a two-point two-time Lagrangian velocity correlation (LVC) was originally developed for the relative dispersion of tracer particles in isotropic turbulent flows (HE, G. W., JIN, G. D., and ZHAO, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E, 80, 066313 (2009)). The model can be expressed as a two-point Eulerian space correlation and the dispersion velocity V. The dispersion velocity denotes the rate at which one moving particle departs from another fixed particle. This paper numerically validates the robustness of the scale-similarity model at high Taylor micro-scale Reynolds numbers up to 373, which are much higher than the original values (Rλ = 66, 102). The effect of the Reynolds number on the dispersion velocity in the scale-similarity model is carefully investigated. The results show that the scale-similarity model is more accurate at higher Reynolds numbers because the two-point Lagrangian velocity correlations with different initial spatial separations collapse into a universal form compared with a combination of the initial separation and the temporal separation via the dispersion velocity. Moreover, the dispersion velocity V normalized by the Kolmogorov velocity Vη ≡ η/τη in which η and τη are the Kolmogorov space and time scales, respectively, scales with the Reynolds number Rλ as $$V/V_\eta\propto{R_\lambda^{1.39}}$$ obtained from the numerical data.

Lagrangian Statistics and Intermittency in Gulf of Mexico

Due to the nonlinear interaction between different flow patterns, for instance, ocean current, meso-scale eddies, waves, etc, the movement of ocean is extremely complex, where a multiscale statistics is then relevant. In this work, a high time-resolution velocity with a time step 15 minutes obtained by the Lagrangian drifter deployed in the Gulf of Mexico (GoM) from July 2012 to October 2012 is considered. The measured Lagrangian velocity correlation function shows a strong daily cycle due to the diurnal tidal cycle. The estimated Fourier power spectrum E(f) implies a dual-power-law behavior which is separated by the daily cycle. The corresponding scaling exponents are close to −1.75 and −2.75 respectively for the time scale larger (resp. 0.1 ≤ f ≤ 0.4 day−1) and smaller (resp. 2 ≤ f ≤ 8 day−1) than 1 day. A Hilbert-based approach is then applied to this data set to identify the possible multifractal property of the cascade process. The results show an intermittent dynamics for the time scale larger than 1 day, while a less intermittent dynamics for the time scale smaller than 1 day. It is speculated that the energy is partially injected via the diurnal tidal movement and then transferred to larger and small scales through a complex cascade process, which needs more studies in the near future.

Prediction of Lagrangian dispersion of fluid particles in isotropic turbulent flows using large-eddy simulation method

The single-, two- and multi-particle dispersions in isotropic turbulent flows are investigated using the direct numerical simulation (DNS), filtered DNS (FDNS) and large-eddy simulation (LES) with a spectral eddy viscosity subgrid scale (SGS) model. The contributions of filtering operation and SGS model to the dispersions are separately studied by comparing the statistics obtained from the three methods. The missing SGS motions in LES can be observed to significantly hinder two-particle and four-particle dispersions if the initial separations are less than or comparable to the filter width. A theoretical analysis of the non-monotonic behavior at short time of the one-time, two-point Lagrangian velocity correlation functions with large initial separations based on the Taylor expansion and the Kolmogorov similarity theory is derived, and the Reynolds number effect on the performance of the spectral eddy viscosity SGS model is also investigated. The results show that the SGS model used performs better with increasing Reynolds numbers. It is concluded that the particle SGS model is needed to be developed to correctly capture the Lagrangian two- and multi-point dispersion statistics of fluid particles.

Pore-scale dispersion: Bridging the gap between microscopic pore structure and the emerging macroscopic transport behavior.

We devise an efficient methodology to provide a universal statistical description of advection-dominated dispersion (Péclet→∞) in natural porous media including carbonates. First, we investigate the dispersion of tracer particles by direct numerical simulation (DNS). The transverse dispersion is found to be essentially determined by the tortuosity and it approaches a Fickian limit within a dozen characteristic scales. Longitudinal dispersion was found to be Fickian in the limit for bead packs and superdiffusive for all other natural media inspected. We demonstrate that the Lagrangian velocity correlation length is a quantity that characterizes the spatial variability for transport. Finally, a statistical transport model is presented that sheds light on the connection between pore-scale characteristics and the resulting macroscopic transport behavior. Our computationally efficient model accurately reproduces the transport behavior in longitudinal direction and approaches the Fickian limit in transverse direction.

Lagrangian velocity and acceleration correlations of large inertial particles in a closed turbulent flow

We investigate the response of large inertial particle to turbulent fluctuations in an inhomogeneous and anisotropic flow. We conduct a Lagrangian study using particles both heavier and lighter than the surrounding fluid, and whose diameters are comparable to the flow integral scale. Both velocity and acceleration correlation functions are analyzed to compute the Lagrangian integral time and the acceleration time scale of such particles. The knowledge of how size and density affect these time scales is crucial in understanding particle dynamics and may permit stochastic process modelization using two-time models (for instance, Sawford’s). As particles are tracked over long times in the quasi-totality of a closed flow, the mean flow influences their behaviour and also biases the velocity time statistics, in particular the velocity correlation functions. By using a method that allows for the computation of turbulent velocity trajectories, we can obtain unbiased Lagrangian integral time. This is particularly useful in accessing the scale separation for such particles and to comparing it to the case of fluid particles in a similar configuration.

Anomalous transport on regular fracture networks: Impact of conductivity heterogeneity and mixing at fracture intersections.

We investigate transport on regular fracture networks that are characterized by heterogeneity in hydraulic conductivity. We discuss the impact of conductivity heterogeneity and mixing within fracture intersections on particle spreading. We show the emergence of non-Fickian transport due to the interplay between the network conductivity heterogeneity and the degree of mixing at nodes. Specifically, lack of mixing at fracture intersections leads to subdiffusive scaling of transverse spreading but has negligible impact on longitudinal spreading. An increase in network conductivity heterogeneity enhances both longitudinal and transverse spreading and leads to non-Fickian transport in longitudinal direction. Based on the observed Lagrangian velocity statistics, we develop an effective stochastic model that incorporates the interplay between Lagrangian velocity correlation and velocity distribution. The model is parameterized with a few physical parameters and is able to capture the full particle transition dynamics.

Lagrangian velocity and acceleration statistics of fluid and inertial particles measured in pipe flow with 3D particle tracking velocimetry

Three-dimensional particle tracking velocimetry (3D-PTV) has been applied to particle-laden pipe flow at Reynolds number 10,300, based on the bulk velocity and the pipe diameter. The volume fraction of the inertial particles was equal to 1.4×10−5. Lagrangian velocity and acceleration statistics were determined both for tracers and for inertial particles with Stokes number equal to 2.3, based on the particle relaxation time and the viscous time scale. The decay of Lagrangian velocity and acceleration correlation functions was measured both for the fluid and for the dispersed phase at various radial positions. The decay of Lagrangian velocity correlations is faster for inertial particles than for flow tracers, whereas the decay of Lagrangian acceleration correlations is about 25% slower for inertial particles than for flow tracers. Further differences between inertial and tracer particles are found in velocity fluctuations evaluated for both positive and negative time lags. The asymmetry in time of velocity cross-correlations is more pronounced for inertial particles. Quadrant analysis revealed another difference still near the wall: ejection and sweep events are less frequent for inertial particles than for tracers.

On the anisotropy of the turbulent passive scalar in the presence of a mean scalar gradient

AbstractWe investigate the origin of the scalar gradient skewness in isotropic turbulence on which a mean scalar gradient is imposed. The problem of the advection of an anisotropic scalar field is reformulated in terms of the advection of an isotropic vector field. For this field, triadic closure equations are derived. It is shown how the scaling of the scalar gradient skewness depends on the choice of the time scale used for the Lagrangian decorrelation of the vector field. The persistent anisotropy in the small scales for the third-order statistics is shown to be perfectly compatible with Corrsin–Obukhov scaling for second-order quantities, since second- and third-order scalar quantities are governed by a different triad correlation time scale. Whereas the inertial range dynamics of second-order scalar quantities is governed by the Lagrangian velocity correlation time, the third-order quantities remain correlated over a time related to the large-scale dynamics of the scalar field. It is argued that this time is determined by the average time it takes for a fluid particle to travel between ramp-cliff scalar structures.

Étude numérique de la diffusion turbulente dʼune turbulence homogène stablement stratifie à lʼaide dʼune analyse linéaire

In this Note, we study numerically turbulent diffusion in the case of homogeneous stably stratified turbulence. This study is based on a linear analysis in spectral space. To express the Lagrangian velocity correlations using Eulerian correlations, we adapt the Corrsin assumption and then we calculate the tensor components of the mean square displacement. We begin by writing the basic linear equations governing the evolution of the spectra of double correlations then we present the Corrsin hypothesis as a model of turbulent diffusion. Subsequently, we describe the numerical method used and analyze the numerical results. Clearly, the Corrsin assumption can give the proper behavior, at large times, of the horizontal diffusion and improves the prediction of vertical diffusion. As an application of the numerical code elaborated, we study the influence of Prandtl number on the vertical diffusion which results in an accentuation of the suppression of vertical diffusion when the Prandtl number (less than 1) diminishes.

Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence

A scale-similarity model for Lagrangian two-point, two-time velocity correlations (LVCs) in isotropic turbulence is developed from the Kolmogorov similarity hypothesis. It is a second approximation to the isocontours of LVCs, while the Smith-Hay model is only a first approximation. This model expresses the LVC by its space correlation and a dispersion velocity. We derive the analytical expression for the dispersion velocity from the Navier-Stokes equations using the quasinormality assumption. The dispersion velocity is dependent on enstrophy spectra and shown to be smaller than the sweeping velocity for the Eulerian velocity correlation. Therefore, the Lagrangian decorrelation process is slower than the Eulerian decorrelation process. The data from direct numerical simulation of isotropic turbulence support the scale-similarity model: the LVCs for different space separations collapse into a universal form when plotted against the separation axis defined by the model.

Subgrid scale fluid velocity timescales seen by inertial particles in large-eddy simulation of particle-laden turbulence

Large-eddy simulation (LES) has emerged as a promising tool for simulating turbulent flows in general and, in recent years,has also been applied to the particle-laden turbulence with some success (Kassinos et al., 2007). The motion of inertial particles is much more complicated than fluid elements, and therefore, LES of turbulent flow laden with inertial particles encounters new challenges. In the conventional LES, only large-scale eddies are explicitly resolved and the effects of unresolved, small or subgrid scale (SGS) eddies on the large-scale eddies are modeled. The SGS turbulent flow field is not available. The effects of SGS turbulent velocity field on particle motion have been studied by Wang and Squires (1996), Armenio et al. (1999), Yamamoto et al. (2001), Shotorban and Mashayek (2006a,b), Fede and Simonin (2006), Berrouk et al. (2007), Bini and Jones (2008), and Pozorski and Apte (2009), amongst others. One contemporary method to include the effects of SGS eddies on inertial particle motions is to introduce a stochastic differential equation (SDE), that is, a Langevin stochastic equation to model the SGS fluid velocity seen by inertial particles (Fede et al., 2006; Shotorban and Mashayek, 2006a; Shotorban and Mashayek, 2006b; Berrouk et al., 2007; Bini and Jones, 2008; Pozorski and Apte, 2009).However, the accuracy of such a Langevin equation model depends primarily on the prescription of the SGS fluid velocity autocorrelation time seen by an inertial particle or the inertial particle–SGS eddy interaction timescale (denoted by $\delt T_{Lp}$ and a second model constant in the diffusion term which controls the intensity of the random force received by an inertial particle (denoted by C_0, see Eq. (7)). From the theoretical point of view, dTLp differs significantly from the Lagrangian fluid velocity correlation time (Reeks, 1977; Wang and Stock, 1993), and this carries the essential nonlinearity in the statistical modeling of particle motion. dTLp and C0 may depend on the filter width and particle Stokes number even for a given turbulent flow. In previous studies, dTLp is modeled either by the fluid SGS Lagrangian timescale (Fede et al., 2006; Shotorban and Mashayek, 2006b; Pozorski and Apte, 2009; Bini and Jones, 2008) or by a simple extension of the timescale obtained from the full flow field (Berrouk et al., 2007). In this work, we shall study the subtle and on-monotonic dependence of $\delt T_{Lp}$ on the filter width and particle Stokes number using a flow field obtained from Direct Numerical Simulation (DNS). We then propose an empirical closure model for $\delta T_{Lp}$. Finally, the model is validated against LES of particle-laden turbulence in predicting single-particle statistics such as particle kinetic energy. As a first step, we consider the particle motion under the one-way coupling assumption in isotropic turbulent flow and neglect the gravitational settling effect. The one-way coupling assumption is only valid for low particle mass loading.

Simulation of concentrated suspensions using the force-coupling method

Abstract The force-coupling method (FCM) represents the dynamics of low Reynolds number suspension flows through a distributed, low-order, finite force-multipole expansion and provides an efficient, matrix-free method to solve the mobility problem for the particle motion. In concentrated suspensions, strong short-range lubrication forces are generated between particles in close proximity as fluid in the intervening gap is displaced by the relative motion of the particles. These forces, together with near-surface contact forces, play an important role in the suspension rheology and self-diffusion of particles. However these forces lead to ill-conditioned problems for determining the particle stresses and particle motion in large systems of particles at higher volume fractions. A robust and effective iteration scheme for determining the particle stresslets is described together with a new scheme for including lubrication forces as near-field corrections to the FCM resistance problem. Both the lubrication and far-field interactions are solved as fully coupled systems in O ( N p log ( N p ) ) operations, for N p particles, using preconditioned conjugate gradient solvers. Numerical results for particles settling under gravity, particle pairs in linearly varying flows and in concentrated suspensions are compared with previous theoretical results and simulations. Numerical simulations with more than 4000 non-Brownian, spherical particles in a homogeneous shear flow provide results on the pair-distribution function and Lagrangian velocity correlations. The extension of the methods to simulate bidisperse systems or wall-bounded suspensions are discussed.

Lagrangian Statistical Model for Transport in Highly Heterogeneous Velocity Fields

We define an effective Lagrangian statistical model in phase space (x, t, v) for describing transport in highly heterogeneous velocity fields with complex spatial organizations. The spatial Markovian nature (and temporal non-Markovian nature) of Lagrangian velocities leads to an effective transport description that turns out to be a correlated continuous time random walk. This model correctly captures the Lagrangian velocity correlation properties and is demonstrated to represent a forward model for predicting transport in highly heterogeneous porous media for different types of velocity organizations.

Effective transport in random shear flows

We obtain an effective transport description for the superdiffusive motion of random walkers in stratified flow by projection of the process on the direction of stratification. The effective dimensionally reduced motion is shown to describe a correlated random walk characterized by the Lagrangian velocity correlation. We analyze the projected motion through exact analytical solutions for the distribution density for an arbitrary correlated Gaussian noise and derive an evolution equation for the one-point and conditional two-point displacement densities. The latter gives an explicit effective equation for superdiffusive transport in stratified random flow and demonstrates that the displacement density has a Gaussian scaling form for all times.

Effects of subgrid-scale modeling on Lagrangian statistics in large-eddy simulation

The application of large-eddy simulation (LES) to turbulent transport processes requires accurate prediction of the Lagrangian statistics of flow fields. However, in most existing SGS models, no explicit consideration is given to Lagrangian statistics. In this paper, we focus on the effects of SGS modeling on Lagrangian statistics in LES ranging from statistics determining single-particle dispersion to those of pair dispersion and multiparticle dispersion. Lagrangian statistics in homogeneous isotropic turbulence are extracted from direct numerical simulation (DNS) and the LES with a spectral eddy-viscosity model. For the case of longtime single-particle dispersion, it is shown that, compared to DNS, LES overpredicts the time scale of the Lagrangian velocity correlation but underpredicts the Lagrangian velocity fluctuation. These two effects tend to cancel one another leading to an accurate prediction of the longtime turbulent dispersion coefficient. Unlike the single-particle dispersion, LES tends to underestimate significantly the rate of relative dispersion of particle pairs and multiple-particles, when initial separation distances are less than the minimum resolved scale due to the lack of subgrid fluctuations. The overprediction of LES on the time scale of the Lagrangian velocity correlation is further confirmed by a theoretical analysis using a turbulence closure theory.

Set of measurement data from flume experiments on steady uniform debris flows

The paper documents a detailed set of measurements from laboratory experiments involving open–channel flows of granular–liquid mixtures. Polyvinyl Chloride pellets mixed with clear water are used, and represent simplified analogues of the mixtures of rocks and muddy water encountered in stony inertial debris flows. The flows are examined in steady uniform conditions, achieved by setting up a closed loop with an inclined channel and a high–speed conveyor belt. Such idealized conditions make it possible to estimate stresses within the flowing mixture, and to accumulate statistics of local granular configurations and motions. These are extracted from video sequences imaged through the sidewall, using algorithms based on the Voronoï diagram. Estimates of granular concentration and granular temperature are derived from local grain patterns and from a detailed analysis of Lagrangian and Eulerian velocity correlations. Depth profiles of kinematic and dynamic quantities are then obtained for various ratios of solid to liquid discharges. These data were earlier used to probe depth variations of rheological behaviour in granular–liquid flows. To make them available for other purposes, they are assembled here into a comprehensive dataset, and provided in digital form in the electronic supplement to this special issue.