Lagrangian Velocity Statistics Research Articles

On small-scale and large-scale intermittency of Lagrangian statistics in canopy flow

Abstract

Roughness, inertia, and diffusion effects on anomalous transport in rough channel flows

We study how the complex interplay between roughness, inertia, and diffusion controls the tracer transport in rough channels. We first simulate the flow and tracer transport over wide ranges of channel roughness, Reynolds number ($Re$), and P\'eclet number ($Pe$) observable in nature. $Pe$ exerts a first-order control on first-passage time distributions, and the effect of roughness on the tracer transport becomes evident with the increase in $Re$. The interplay between the roughness and $Re$ causes eddy flows, which intensify or suppress anomalous transport depending on $Pe$. At infinite $Pe$, the late-time scaling follows a universal power-law scaling, which is explained by conducting a scaling analysis. With extensive numerical simulations and stochastic modeling, we show that the roughness, inertia, and diffusion effects are encoded in Lagrangian velocity statistics represented by velocity distribution and correlation. We finally predict the anomalous transport using a stochastic model that considers the Lagrangian velocity statistics.

Single-particle Lagrangian statistics from direct numerical simulations of rotating-stratified turbulence

Geophysical fluid flows are predominantly turbulent and often strongly affected by the Earth's rotation, as well as by stable density stratification. Using direct numerical simulations of forced Boussinesq equations, we study the influence of these effects on the motion of fluid particles, focusing on cases where the frequencies associated with rotation and stratification (RaS), $N$ and $f$ respectively, are held at a fixed ratio $N/f=5$. As the intensity of RaS increases, a sharp transition is observed between a regime dominated by eddies to a regime dominated by waves, which can also be seemingly described by simply comparing the time scale $1/N$ and $\tau_\eta$ (the Kolmogorov time scale). We perform a detailed study of Lagrangian statistics of acceleration, velocity and related quantities in the two regimes. The flow anisotropy induces a clear difference between particle motion in the horizontal and vertical directions. In the regime $N\tau_\eta <1$, acceleration statistics in both horizontal and vertical directions, exhibit well known characteristics of isotropic turbulence. In contrast for $N\tau_\eta >1$, they are directly influenced by imposed RaS. The Lagrangian velocity statistics exhibit visible anisotropy for all runs; nevertheless the degree of anisotropy becomes very strong in the regime $N\tau_\eta >1$. We find that in the regime $N\tau_\eta <1$, rotation enhances the mean displacement of particles in horizontal planes at short times, but inhibits them at longer times. This inhibition of horizontal displacement becomes stronger for $N\tau_\eta >1$, with no clear diffusive behavior. Displacements in the vertical direction are always inhibited. The inhibition becomes extremely strong when $N\tau_\eta >1$, with the particles almost being trapped horizontally.

Recirculating flow-induced anomalous transport in meandering open-channel flows

Channel meanders in rivers induce complex three-dimensional (3D) flow characteristics such as secondary flows and flow recirculation. Helical secondary flows promote transverse dispersion, and flow recirculation zones trap tracers. As a result, these meander-driven flows may cause anomalous transport manifested by unusually elevated levels of tracer concentration at early and late times of breakthrough curves (BTCs). In this study, we perform 3D numerical simulations in meandering channels across a wide range of channel sinuosity to investigate the impact of meander geometry on flow and transport. We solve 3D Reynolds-averaged Navier-Stoke equations blended with a SST k − ω turbulence model to obtain velocity and turbulence fields. We then incorporate the obtained flow fields into a Lagrangian particle tracking model to simulate solute transport. Sinuosity higher than 1.5 leads to the onset of horizontal recirculating flows along the apex outer banks, and these recirculation zones expand as sinuosity increases. The analysis of the transport simulations elucidates that the interplay between the secondary flows and recirculation zones induces anomalous transport. The helical secondary flows bring particles into the recirculation zones by promoting transverse dispersion, and the recirculating flows delay particle transport by the trapping effect. We show that the tail power-law slope and truncated time of BTCs as well as Lagrangian tortuosity distributions change dramatically with the emergence of recirculation zones. These analyses demonstrate that the recirculation zones are acting as the primary driver of anomalous transport. Finally, we successfully predict the observed anomalous transport with a Spatial Markov Model (SMM), which is an upscaled transport model that incorporates Lagrangian velocity distribution and spatial velocity correlation. The successful predictions show that the Lagrangian velocity statistics effectively capture the underlying mechanisms of anomalous transport in meandering open-channel 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.

Eulerian and Lagrangian statistics in an exactly solvable turbulent shear model with a random background mean

We study the Lagrangian statistics of passively advected particles in an elementary velocity model for turbulent shear. The stochastic velocity model is exactly solvable and includes features that highlight the important differences between Lagrangian and Eulerian velocity statistics, which are not equal in the present context. A major element of the velocity model is the presence of a random, spatially uniform background mean, which is superimposed on a turbulent shear with a spectrum that typically follows a power law. We directly solve for the Eulerian and Lagrangian statistics and show how the sweeping motion of the background mean affects the Lagrangian velocity statistics with faster decaying correlations that oscillate more rapidly compared to the Eulerian velocity. This arises due to interaction of the cross-sweeps of the mean flow with the shear component, which determines Lagrangian tracer transport rates. We derive explicit expressions for the tracer dispersion that demonstrate how the dispersion rate depends on model parameters. We validate the predictions with numerical experiments in various test regimes that also highlight the behavior of Lagrangian particles in space. The proposed exactly solvable model serves as a test problem for Eulerian spectral recovery via Lagrangian data assimilation and parameter estimation methods.

Upscaling and Prediction of Lagrangian Velocity Dynamics in Heterogeneous Porous Media

AbstractThe understanding of the dynamics of Lagrangian velocities is key for the understanding and upscaling of solute transport in heterogeneous porous media. The prediction of large‐scale particle motion in a stochastic framework implies identifying the relation between the Lagrangian velocity statistics and the statistical characteristics of the Eulerian flow field and the hydraulic medium properties. In this paper, we approach both challenges from numerical and theoretical points of view. Direct numerical simulations of Darcy‐scale flow and particle motion give detailed information on the evolution of the statistics of particle velocities both as a function of travel time and distance along streamlines. Both statistics evolve from a given initial distribution to different steady‐state distributions, which are related to the Eulerian velocity probability density function. Furthermore, we find that Lagrangian velocities measured isochronally as a function of travel time show intermittency dominated by low velocities, which is removed when measured equidistantly as a function of travel distance. This observation gives insight into the stochastic dynamics of the particle velocity series. As the equidistant particle velocities show a regular random pattern that fluctuates on a characteristic length scale, it is represented by two stationary Markov processes, which are parametrized by the distribution of flow velocities and a correlation distance. The velocity Markov models capture the evolution of the Lagrangian velocity statistics in terms of the Eulerian flow properties and a characteristics length scale and shed light on the role of the initial conditions and flow statistics on large‐scale particle motion.

Stochastic Dynamics of Lagrangian Pore‐Scale Velocities in Three‐Dimensional Porous Media

AbstractUpscaling dispersion, mixing, and reaction processes from the pore to the Darcy scale is directly related to the understanding of the dynamics of pore‐scale particle velocities, which are at the origin of hydrodynamic dispersion and non‐Fickian transport behaviors. With the aim of deriving a framework for the systematic upscaling of these processes from the pore to the Darcy scale, we present a detailed analysis of the evolution of Lagrangian and Eulerian statistics and their dependence on the injection condition. The study is based on velocity data obtained from computational fluid dynamics simulations of Stokes flow and advective particle tracking in the three‐dimensional pore structure obtained from high‐resolution X‐ray microtomography of a Berea sandstone sample. While isochronously sampled velocity series show intermittent behavior, equidistant series vary in a regular random pattern. Both statistics evolve toward stationary states, which are related to the Eulerian velocity statistics. The equidistantly sampled Lagrangian velocity distribution converges on only a few pore lengths. These findings indicate that the equidistant velocity series can be represented by an ergodic Markov process. A stochastic Markov model for the equidistant velocity magnitude captures the evolution of the Lagrangian velocity statistics. The model is parameterized by the Eulerian velocity distribution and a relaxation length scale, which can be related to hydraulic properties and the medium geometry. These findings lay the basis for a predictive stochastic approach to upscale solute dispersion in complex porous media from the pore to the Darcy scale.

Velocity and acceleration statistics in rapidly rotating Rayleigh-Bénard convection.

Background rotation causes different flow structures and heat transfer efficiencies in Rayleigh-Bénard convection (RBC). Three main regimes are known: rotation-unaffected, rotation-affected and rotation-dominated. It has been shown that the transition between rotation-unaffected and rotation-affected regimes is driven by the boundary layers. However, the physics behind the transition between rotation-affected and rotation-dominated regimes are still unresolved. In this study, we employ the experimentally obtained Lagrangian velocity and acceleration statistics of neutrally buoyant immersed particles to study the rotation-affected and rotation-dominated regimes and the transition between them. We have found that the transition to the rotation-dominated regime coincides with three phenomena; suppressed vertical motions, strong penetration of vortical plumes deep into the bulk and reduced interaction of vortical plumes with their surroundings. The first two phenomena are used as confirmations for the available hypotheses on the transition to the rotation-dominated regime while the last phenomenon is a new argument to describe the regime transition. These findings allow us to better understand the rotation-dominated regime and the transition to this regime.

Parameterizing the Spatial Markov Model From Breakthrough Curve Data Alone

AbstractThe spatial Markov model (SMM) is an upscaled Lagrangian model that effectively captures anomalous transport across a diverse range of hydrologic systems. The distinct feature of the SMM relative to other random walk models is that successive steps are correlated. To date, with some notable exceptions, the model has primarily been applied to data from high‐resolution numerical simulations and correlation effects have been measured from simulated particle trajectories. In real systems such knowledge is practically unattainable and the best one might hope for is breakthrough curves (BTCs) at successive downstream locations. We introduce a novel methodology to quantify velocity correlation from BTC data alone. By discretizing two measured BTCs into a set of arrival times and developing an inverse model, we estimate velocity correlation, thereby enabling parameterization of the SMM in studies where detailed Lagrangian velocity statistics are unavailable. The proposed methodology is applied to two synthetic numerical problems, where we measure all details and thus test the veracity of the approach by comparison of estimated parameters with known simulated values. Our results suggest that our estimated transition probabilities agree with simulated values and using the SMM with this estimated parameterization accurately predicts BTCs downstream. Our methodology naturally allows for estimates of uncertainty by calculating lower and upper bounds of velocity correlation, enabling prediction of a range of BTCs. The measured BTCs fall within the range of predicted BTCs. This novel method to parameterize the SMM from BTC data alone is quite parsimonious, thereby widening the SMM's practical applicability.

A multi-time-step noise reduction method for measuring velocity statistics from particle tracking velocimetry

We present a method to improve the accuracy of velocity measurements for fluid flow or particles immersed in it, based on a multi-time-step approach that allows for cancellation of noise in the velocity measurements. Improved velocity statistics, a critical element in turbulent flow measurements, can be computed from the combination of the velocity moments computed using standard particle tracking velocimetry (PTV) or particle image velocimetry (PIV) techniques for data sets that have been collected over different values of time intervals between images. This method produces Eulerian velocity fields and Lagrangian velocity statistics with much lower noise levels compared to standard PIV or PTV measurements, without the need of filtering and/or windowing. Particle displacement between two frames is computed for multiple different time-step values between frames in a canonical experiment of homogeneous isotropic turbulence. The second order velocity structure function of the flow is computed with the new method and compared to results from traditional measurement techniques in the literature. Increased accuracy is also demonstrated by comparing the dissipation rate of turbulent kinetic energy measured from this function against previously validated measurements.

Continuous time random walks for the evolution of Lagrangian velocities

We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. This approach describes persistence of velocities over a characteristic spatial scale, unlike classical random walk methods, which model persistence over a characteristic time scale. We first establish the relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop a space continuous CTRW approach for the spatial and temporal dynamics of Lagrangian velocities. While classical CTRW formulations have non-stationary Lagrangian velocity statistics, the proposed approach quantifies the evolution of the Lagrangian velocity statistics under both stationary and non-stationary conditions. We provide explicit expressions for the Lagrangian velocity statistics, and determine the behaviors of the mean particle velocity, velocity covariance and particle dispersion. We find strong Lagrangian correlation and anomalous dispersion for velocity distributions which are tailed toward low velocities as well as marked differences depending on the initial conditions. The developed CTRW approach predicts the Lagrangian particle dynamics from an arbitrary initial condition based on the Eulerian velocity distribution and a characteristic correlation scale.

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.

Time irreversibility of the statistics of a single particle in compressible turbulence.

We investigate time irreversibility from the point of view of a single particle in Burgers turbulence. Inspired by the recent work for incompressible flows [Xu et al., Proc. Natl. Acad. Sci. USA 111, 7558 (2014)], we analyze the evolution of the kinetic energy for fluid markers and use the fluctuations of the instantaneous power as a measure of time irreversibility. For short times, starting from a uniform distribution of markers, we find the scaling 〈[E(t)-E(0)](n)〉∝t and 〈p(n)〉∝Re(n-1) for the power as a function of the Reynolds number. Both observations can be explained using the "flight-crash" model, suggested by Xu et al. Furthermore, we use a simple model for shocks that reproduces the moments of the energy difference, including the pre-factor for 〈E(t)-E(0)〉. To complete the single-particle picture for Burgers we compute the moments of the Lagrangian velocity difference and show that they are bifractal. This arises in a similar manner to the bifractality of Eulerian velocity differences. In the above setting, time irreversibility is directly manifest as particles eventually end up in shocks. We additionally investigate time irreversibility in the long-time limit when all particles are located inside shocks and the Lagrangian velocity statistics are stationary. We find the same scalings for the power and energy differences as at short times and argue that this is also a consequence of rare "flight-crash" events related to shock collisions.

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.

Lagrangian statistics of light particles in turbulence

We study the Lagrangian velocity and acceleration statistics of light particles (micro-bubbles in water) in homogeneous isotropic turbulence. Micro-bubbles with a diameter db = 340 μm and Stokes number from 0.02 to 0.09 are dispersed in a turbulent water tunnel operated at Taylor-Reynolds numbers (Reλ) ranging from 160 to 265. We reconstruct the bubble trajectories by employing three-dimensional particle tracking velocimetry. It is found that the probability density functions (PDFs) of the micro-bubble acceleration show a highly non-Gaussian behavior with flatness values in the range 23 to 30. The acceleration flatness values show an increasing trend with Reλ, consistent with previous experiments [G. Voth, A. La Porta, A. M. Crawford, J. Alexander, and E. Bodenschatz, “Measurement of particle accelerations in fully developed turbulence,” J. Fluid Mech. 469, 121 (2002)]10.1017/S0022112002001842 and numerics [T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura, and A. Uno, “Small-scale statistics in highresolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics,” J. Fluid Mech. 592, 335 (2007)]10.1017/S0022112007008531. These acceleration PDFs show a higher intermittency compared to tracers [S. Ayyalasomayajula, Z. Warhaft, and L. R. Collins, “Modeling inertial particle acceleration statistics in isotropic turbulence,” Phys. Fluids. 20, 095104 (2008)]10.1063/1.2976174 and heavy particles [S. Ayyalasomayajula, A. Gylfason, L. R. Collins, E. Bodenschatz, and Z. Warhaft, “Lagrangian measurements of inertial particle accelerations in grid generated wind tunnel turbulence,” Phys. Rev. Lett. 97, 144507 (2006)]10.1103/PhysRevLett.97.144507 in wind tunnel experiments. In addition, the micro-bubble acceleration autocorrelation function decorrelates slower with increasing Reλ. We also compare our results with experiments in von Kármán flows and point-particle direct numerical simulations with periodic boundary conditions.

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.

Direct numerical simulation on strained turbulent flows and particles within

We present results from direct numerical simulations of strained turbulent flows. Our focus is twofold. One is to improve our understanding of the interactions of large and small scale dynamics in strained turbulent flow; another focus is to understand the influence of the straining on the motions of passive and inertial particles of varied Stokes numbers. We seek to emphasize the effects of the strain geometry and strain rate on the particles' behaviors. Eulerian flow field results and the Lagrangian particle velocity and acceleration statistics will be discussed. The Rogallo algorithm is applied for simulating the flow field in a non-cubical domain.

A joint velocity‐concentration PDF method for tracer flow in heterogeneous porous media

The probability density function (PDF) of the local concentration of a contaminant, or tracer, is an important component of risk assessment in applications that involve flow in heterogeneous subsurface formations. In this paper, a novel joint velocity‐concentration PDF method for tracer flow in highly heterogeneous porous media is introduced. The PDF formalism accounts for advective transport, pore‐scale dispersion (PSD), and molecular diffusion. Low‐order approximations (LOAs), which are usually obtained using a perturbation expansion, typically lead to Gaussian one‐point velocity PDFs. Moreover, LOAs provide reasonable approximations for small log conductivity variances (i.e., σY2 &lt; 1). For large σY2, however, the one‐point velocity PDFs deviate significantly from the Gaussian distribution as demonstrated convincingly by several Monte Carlo (MC) simulation studies. Furthermore, the Lagrangian velocity statistics exhibit complex correlations that span a wide range of scales, including long‐range correlations due to the formation of preferential flow paths. Both non‐Gaussian PDFs and complex long‐range correlations are accurately represented using Markovian velocity processes (MVPs) in the proposed joint PDF method. LOA methods can be generalized to some extent by presuming a certain shape for the concentration PDF (e.g., a β PDF fully characterized by the concentration mean and variance). The joint velocity‐concentration PDF method proposed here does not require any closure assumptions on the shape of the marginal concentration PDF. The Eulerian joint PDF transport equation is solved numerically using a computationally efficient particle‐based approach. The PDF method is validated with high‐resolution MC reference data from Caroni and Fiorotto (2005) for saturated transport in velocity fields, which are stationary in space and time, for domains with σY2 = 0.05, 1, and 2 and Péclet numbers ranging from 100 to 10,000. PSD is modeled using constant anisotropic dispersion coefficients in both the reference MC computations and our PDF method.

Origin of Lagrangian Intermittency in Drift-Wave Turbulence

The Lagrangian velocity statistics of dissipative drift-wave turbulence are investigated. For large values of the adiabaticity (or small collisionality), the probability density function of the Lagrangian acceleration shows exponential tails, as opposed to the stretched exponential or algebraic tails, generally observed for the highly intermittent acceleration of Navier-Stokes turbulence. This exponential distribution is shown to be a robust feature independent of the Reynolds number. For small adiabaticity, algebraic tails are observed, suggesting the strong influence of point-vortex-like dynamics on the acceleration. A causal connection is found between the shape of the probability density function and the autocorrelation of the norm of the acceleration.