Lagrangian Acceleration Research Articles

Mechanism to explore lamination rate

A mechanism amenable to laminate and fold flows is identified and quantified. This laminating mechanism follows from a physical and experimental approach relying on the interlaced structure of velocity and Lagrangian acceleration. The Lagrangian acceleration being the resultant of the forces applied on particle fluids, the component of acceleration perpendicular to the velocity vector allows the quantification of a rate of change of the velocity's direction, i.e., the local angular Lagrangian velocity, theta. The spatial variation in theta is then used to predict and measure the lamination and folding rate. To support and illustrate this approach, three basic experimental flows, driven by electromagnetic forces, are discussed and compared. Folding rate intensities are extracted for different characteristic length scales. Also, good agreement is found between grid deformation and the prediction of lamination rate. This quantification of lamination rate opens new avenues for the design of mixers, in particular at low Reynolds numbers.

Wave breaking and the surface velocity field for three-dimensional water waves

It is shown that there is an exact reduction from the full three-dimensional inviscid water wave problem with vorticity to a set of hyperbolic equations for the free surface horizontal velocity field. The key term in the reduced equations is g + (Dw/Dt)|S where g > 0 is the gravitational constant and (Dw/Dt)|S is the Lagrangian vertical acceleration at the surface. Vertical accelerations—both Eulerian and Lagrangian—at the surface are widely used as a diagnostic for wave breaking and this reduction gives a precise theoretical argument for how accelerations drive the surface dynamics. When the acceleration term is small, the surface equations reduce to the model equation proposed in Pomeau et al (2008 Nonlinearity 21 T61–T79, Proc. R. Soc. Lond. A 464 1851–66) which provides a universal law for spreading of the crest during wave breaking, and when the acceleration term is negative the equations can switch type (hyperbolic to elliptic) and solutions can fail to exist. Addition of surface tension leads to dispersive regularization of the surface equations.

Intermittency and scale-dependent statistics in fully developed turbulence

Fully developed homogeneous isotropic turbulent fields, computed by direct numerical simulation, are compared to divergence-free random fields having the same energy spectrum and either the same helicity spectrum as that of the turbulent data, or vanishing helicity. We show that the scale-dependent velocity flatness quantifies the spatial variability of the energy spectrum. The flatness exhibits a substantial increase at small scales for the turbulent field, but remains constant for the random fields. A diagnostic, the scale-dependent helicity, is proposed to quantify the geometrical statistics of the flow, which shows that only the turbulent flow is intermittent. Finally, statistical scale-dependent analyses of both Eulerian and Lagrangian accelerations confirm the inherently different dynamics of turbulent and random flows.

Lagrangian measurements of inertial particle accelerations in a turbulent boundary layer

Two-dimensional Lagrangian acceleration statistics of inertial particles in a turbulent boundary layer with free-stream turbulence are determined by means of a particle tracking technique using a high-speed camera moving along the side of the wind tunnel at the mean flow speed. The boundary layer is formed above a flat plate placed horizontally in the tunnel, and water droplets are fed into the flow using two different methods: sprays placed downstream from an active grid, and tubes fed into the boundary layer from humidifiers. For the flow conditions studied, the sprays produce Stokes numbers varying from 0.47 to 1.2, and the humidifiers produce Stokes numbers varying from 0.035 to 0.25, where the low and high values refer to the outer boundary layer edge and the near-wall region, respectively. The Froude number is approximately 1.0 for the sprays and 0.25 for the humidifiers, with a small variation within the boundary layer. The free-stream turbulence is varied by operating the grid in the active mode as well as a passive mode (the latter behaves as a conventional grid). The boundary layer momentum-thickness Reynolds numbers are 840 and 725 for the active and passive grid respectively. At the outer edge of the boundary layer, where the shear is weak, the acceleration probability density functions are similar to those previously observed in isotropic turbulence for inertial particles. As the boundary layer plate is approached, the tails of the probability density functions narrow, become negatively skewed, and their peak occurs at negative accelerations (decelerations in the streamwise direction). The mean deceleration and its root mean square (r.m.s.) increase to large values close to the plate. These effects are more pronounced at higher Stokes number. In the vertical direction, there is a slight downward mean deceleration and its r.m.s., which is lower in magnitude than that of the streamwise component, peaks in the buffer region. Although there are free-stream turbulence effects, and the complex boundary layer structure plays an important role, a simple model suggests that the acceleration behaviour is dominated by shear, gravity and inertia. The results are contrasted with inertial particles in isotropic turbulence and with fluid particle acceleration statistics in a boundary layer. The background velocity field is documented by means of hot-wire anemometry and laser Doppler velocimetry measurements. These appear to be the first Lagrangian acceleration measurements of inertial particles in a shear flow.

Fluid acceleration in the bulk of turbulent dilute polymer solutions

We studied the effects of long-chain polymers on the small scales of turbulence by experimental measurements of Lagrangian accelerations in the bulk of turbulent flows of dilute polymer solutions. Lagrangian accelerations were measured by following tracer particles with a high-speed optical tracking system. We observed a significant decrease in the acceleration variance in dilute polymer solutions as compared with in pure water. The shape of the normalized acceleration probability density functions, however, remained the same as in Newtonian water flows. We also observed an increase in the turbulent Lagrangian acceleration autocorrelation time with polymer concentration. The decrease of acceleration variance and the increase of acceleration autocorrelation time are consistent with a suppression of viscous dissipation, and cannot be explained by a mere increase of effective viscosity due to the polymers.

Lagrangian particle tracking in three dimensions via single-camera in-line digital holography

Lagrangian particle trajectories are measured in three spatial dimensions with a single camera using the method of digital in-line holography. Lagrangian trajectories of 60–120 μm diameter droplets in turbulent air obtained with data from one camera compare favorably with tracks obtained from a simultaneous dual-camera data set, the latter having high spatial resolution in all three dimensions. Using the single-camera system, particle motion along the optical axis is successfully tracked, allowing for long, continuous 3D tracks, but the depth resolution based on standard reconstruction methods is not sufficient to obtain accurate acceleration measurements for that component. Lagrangian velocity distributions for all three spatial components agree within reasonable sampling uncertainties and Lagrangian acceleration distributions agree for the two lateral components. An equivalent single-camera, imaging-based 2D tracking system would be challenged by the particle densities tested, but the holographic configuration allows for 3D tracking in the dilute limit. The method also allows particle size, shape and orientation to be measured along the trajectory. Lagrangian measurements of particle size provide a direct measure of particle size uncertainty under realistic conditions sampled from the entire measurement volume.

Modeling inertial particle acceleration statistics in isotropic turbulence

Our objective is to explain recent Lagrangian acceleration measurements of inertial particles in decaying, nearly isotropic turbulence [Ayyalasomayajula et al., Phys. Rev. Lett. 97, 144507 (2006)]. These experiments showed that as particle inertial effects increased, the variance in the particle acceleration fluctuations was reduced, and the tails of the normalized particle acceleration probability density function (PDF) became systematically attenuated. We model this phenomenon using a base flow that consists of a two-dimensional array of evenly spaced vortices with signs and intensities that vary randomly in time. We simulate a large sample of inertial particles moving through the fluid without disturbing the flow (one-way coupling). Consistent with Bec et al. [J. Fluid Mech. 550, 349 (2006)], we find that our model exhibits preferential concentration or clustering of particles in regions located away from the vortex centers. That is, inertial particles selectively sample the flow field, oversampling regions with high strains and undersampling regions with high vorticities. At low Stokes numbers, this biased “sampling” of the flow is responsible for the reduction in the acceleration variance and partially explains the attenuation of the tails of the acceleration PDF. However, contrary to previous findings, we show that the tails of the PDF are also diminished by “filtering” induced by the attenuated response of the inertial particles to temporal variations in the fluid acceleration: Inertial particles do not respond to fluctuations with frequencies much higher than the inverse of the particle stopping time. We show that larger fluid acceleration events have higher frequencies and hence experience greater filtering by particle inertia. We contrast the vortex model with previous Lagrangian acceleration models by Sawford [Phys. Fluids A 3, 1577 (1991)] and Reynolds [Phys. Fluids 15, L1 (2003)] and show that although these models capture some aspects of the inertial particle behavior, it is necessary to employ a model of the flow with spatial structure to capture the effect of sampling on the inertial particle dynamics.

Motion of inertial particles with size larger than Kolmogorov scale in turbulent flows

We report experimental results on the motion of tracer and non-tracer particles in intense turbulent water flows between counter-rotating disks measured by three-dimensional Lagrangian particle tracking. The sizes of the non-tracer particles were in the range of η < d p ≪ L , where η is the Kolmogorov length scale and L is the integral scale. We propose a modified Stokes number that takes into account the effects from finite particle size and inertia. We compare results from tracers and from two types of particles (heavy +small, approx. neutrally bouyant +large) for which the conventional Stokes numbers differ by a factor of ≈8% and the modified Stokes numbers by ≈60%. The conventional Stokes numbers of the particles investigated were in the range of 0.7 and 1.5, while the modified Stokes numbers were smaller between 0.1 to 0.3. We observed that the tails of the measured acceleration PDFs were slightly narrower compared to tracer particles with the heavier+smaller particles showing a larger effect. The measured Lagrangian acceleration correlations of the large particles were approximately the same as that of the tracer particles. This suggests that trajectories of large particles are not biased towards the low-vorticity, high-straining region as was observed previously in the case of small, very heavy particles. These findings are also supported by the measurements of the local slopes of the fourth order Lagrangian structure functions.

Extreme Lagrangian Acceleration in Confined Turbulent Flow

A Lagrangian study of two-dimensional turbulence for two different geometries, a periodic and a confined circular geometry, is presented to investigate the influence of solid boundaries on the Lagrangian dynamics. It is found that the Lagrangian acceleration is even more intermittent in the confined domain than in the periodic domain. The flatness of the Lagrangian acceleration as a function of the radius shows that the influence of the wall on the Lagrangian dynamics becomes negligible in the center of the domain, and it also reveals that the wall is responsible for the increased intermittency. The transition in the Lagrangian statistics between this region, not directly influenced by the walls, and a critical radius which defines a Lagrangian boundary layer is shown to be very sharp with a sudden increase of the acceleration flatness from about 5 to about 20.

Measurement of acceleration and multiple velocity components using a laser Doppler velocity profile sensor

For the investigation of turbulent flows, the measurement of Lagrangian acceleration is of great interest as it represents a direct component of the Navier–Stokes equations. The presented sensor is based on the common laser Doppler technique, but offers in addition combined high spatial resolution in the micrometre range and the possibility of measuring the velocity component along the optical axis. As a result of the sensor setup, signals of particles with inclined trajectories show frequency modulation within a single burst similar to the signal of an accelerated particle. A model-based approach to distinguish between both quantities is presented, and a signal processing technique based on the Hilbert transform has been developed. The processing is comparatively fast and showed good agreement with preset values, even for signals of poor quality. The variance of velocity and acceleration measurements nearly reaches the Cramér–Rao lower bound. Experimental verification is done by the measurement of a harmonic oscillator with known parameters and a stagnation flow within a free jet.

Lagrangian investigation of two-dimensional decaying turbulence

We present a numerical investigation of two-dimensional decaying turbulence in the Lagrangian framework. Focusing on single particle statistics, we investigate Lagrangian trajectories in a freely evolving turbulent velocity field. The dynamical evolution of the tracer particles is strongly dominated by the emergence and evolution of coherent structures. For a statistical analysis we focus on the Lagrangian acceleration as a central quantity. For more geometrical aspects we investigate the curvature along the trajectories. We find strong signatures for the self-similar universal behavior.

Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ~ 1 (Series 1) and the other is with kmaxη ~ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ~ −0.32Rλ0.11 and F ~ 1.14Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ &gt; 100, in contrast to Rλ &lt; 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and βL ≈ 3.0.

Collisions of Cloud Droplets in a Turbulent Flow. Part IV: Droplet Hydrodynamic Interaction

Abstract The paper presents a computationally accurate and efficient method for calculation of cloud droplets’ collision efficiency in a turbulent flow with the properties typical of atmospheric clouds. According to Part III, the statistical properties of a turbulent flow are represented by a set of noncorrelated samples of turbulent velocity gradients and Lagrangian accelerations. Long series of these samples were generated for turbulent parameters typical of different atmospheric clouds. Each sample can be assigned to a certain point of the turbulent flow. Each such point can be surrounded by a small elementary volume with the linear length scale of the Kolmogorov length scale, in which the Lagrangian acceleration and the velocity gradient tensor can be considered uniform in space and invariable in time. For each sample (or an elementary volume), fluxes of droplets of one size onto droplets of another size are calculated both in the presence and absence of hydrodynamical droplet interaction (HDI). In each elementary volume, the collision efficiency is calculated as the ratio of these fluxes. Using a set of the collision efficiency and kernels, the probability distribution functions (PDFs) and the mean values of collision efficiency and collision kernels are calculated under different dissipation rates and Reynolds numbers. It is shown that turbulence significantly increases the collision efficiency, especially for droplets of close sizes and droplet pairs containing a few-microns-radius droplet. The results suggest that the main mechanism by means of which turbulence increases the rate of cloud droplets’ collisions is its influence on HDI.

Impact of the floating-point precision and interpolation scheme on the results of DNS of turbulence by pseudo-spectral codes

In this paper we investigate the impact of the floating-point precision and interpolation scheme on the results of direct numerical simulations (DNS) of turbulence by pseudo-spectral codes. Three different types of floating-point precision configurations show no differences in the statistical results. This implies that single precision computations allow for increased Reynolds numbers due to the reduced amount of memory needed. The interpolation scheme for obtaining velocity values at particle positions has a noticeable impact on the Lagrangian acceleration statistics. A tri-cubic scheme results in a slightly broader acceleration probability density function than a tri-linear scheme. Furthermore the scaling behavior obtained by the cubic interpolation scheme exhibits a tendency towards a slightly increased degree of intermittency compared to the linear one.

Lagrangian conditional statistics, acceleration and local relative motion in numerically simulated isotropic turbulence

Lagrangian statistics of fluid-particle velocity and acceleration conditioned on fluctuations of dissipation, enstrophy and pseudo-dissipation representing different characteristics of local relative motion are extracted from a direct numerical simulation database of stationary (forced) homogeneous isotropic turbulence. The grid resolution in the simulations is up to 20483, and the Taylor-scale Reynolds number ranges from about 40 to 650, where characteristics of small-scale intermittency in the Eulerian flow field are well developed. A key joint statistic of the conditioning variables is the dissipation-enstrophy cross-correlation, which is asymmetric, but becomes less so at high Reynolds number. Conditional velocity autocorrelations are consistent with rapid changes in the velocity of fluid particles moving in regions of large velocity gradients. Examination of statistics conditioned upon enstrophy, especially in a local coordinate frame moving with the vorticity vector, and of the centripetal acceleration suggests the presence of vortex-trapping effects which persist for several Kolmogorov time scales. Further results on acceleration statistics and joint velocity-acceleration autocorrelations are also presented to help characterize in detail the properties of a joint stochastic process of velocity, acceleration and the pseudo-dissipation. Together with recent work on Eulerian conditional acceleration and Reynolds-number dependence of basic Lagrangian quantities, the present results are directly useful for the development of a new stochastic model formulated to account for intermittency and Reynolds-number effects as described in detail in a companion paper.

Collisions of Small Drops in a Turbulent Flow. Part III: Relative Droplet Fluxes and Swept Volumes

Abstract Swept volumes of cloud droplets with radii below 20 μm are calculated under conditions typical of atmospheric cloud turbulence characterized by enormous values of Reynolds numbers, high turbulent intermittency, and characteristic values of the dissipation rate. To perform the calculations, the motion equation for small droplets proposed by Maxey is generalized for Stokes numbers St &amp;gt; 0.1, which allows one to simulate relative droplet motion even for very high turbulence intensities typical of deep cumulus clouds. Analytical considerations show that droplet motion is fully determined by turbulent shears and the Lagrangian accelerations. A new statistical representation of a turbulent flow has been proposed based on the results of the scale analysis of turbulence characteristics and those related to the droplet motion. According to the method proposed, statistical properties of turbulent flow are represented by a set of noncorrelated samples of turbulent shears and Lagrangian accelerations. Each sample can be assigned to a certain point of the turbulent flow. Each such point can be surrounded by a small “elementary” volume with linear length scales of the Kolmogorov length scale, in which the Lagrangian acceleration and turbulent shears can be considered as uniform in space and invariable in time. This present study (Part III) investigates the droplet collisions in a turbulent flow when hydrodynamic droplet interaction (HDI) is disregarded. Using a statistical model, long series of turbulent shears and accelerations were generated, reproducing probability distribution functions (PDF) at high Reynolds numbers, as they were obtained in recent laboratory and theoretical studies. Swept volumes of droplets are calculated for each sample of an acceleration–shear pair, and the PDF of swept volumes is calculated for turbulent parameters typical of cloud turbulence. The effect of turbulent flow intermittency manifests itself in two aspects: 1) an increase of swept volume variance with increasing Reynolds number, and 2) formation of the swept volume PDF that has a sharp maximum and an elongated tail. In spite of the fact that the magnitude of the mean swept volume increases significantly with Reynolds number and the dissipation rate, this increase does not exceed ∼60% of pure gravity values even under turbulent conditions typical of strong cumulus clouds. A comparison with the classical results of Saffman and Turner is presented and discussed.

Measurement of Turbulent Kinetic Energy Dissipation Rate with a Lagrangian Float

Abstract This study tests the ability of a neutrally buoyant float to estimate the dissipation rate of turbulent kinetic energy ɛ from its vertical acceleration spectrum using an inertial subrange method. A Lagrangian float was equipped with a SonTek acoustic Doppler velocimeter (ADV), which measured the vector velocity 1 m below the float's center, and a pressure sensor, which measured the float's depth. Measurements were taken in flows where estimates of ɛ varied from 10−8 to 10−3 W kg−1. Previous observational and theoretical studies conclude that the Lagrangian acceleration spectrum is white within the inertial subrange with a level proportional to ɛ. The size of the Lagrangian float introduces a highly reproducible spectral attenuation at high frequencies. Estimates of the dissipation rate of turbulent kinetic energy using float measurements ɛfloat were obtained by fitting the observed spectra to a model spectrum that included the attenuation effect. The ADV velocity measurements were converted to a wavenumber spectrum using a variant of Taylor's hypothesis. The spectrum exhibited the expected −5/3 slope within an inertial subrange. The turbulent kinetic energy dissipation rate ɛADV was computed from the level of this spectrum. These two independent estimates, ɛADV and ɛfloat, were highly correlated. The ratio ɛfloat/ɛADV deviated from one by less than a factor of 2 over the five decades of ɛ measured. This analysis confirms that ɛ can be estimated reliably from Lagrangian float acceleration spectra in turbulent flows. For the meter-sized floats used here, the size of the float and the noise level of the pressure measurements sets a lower limit of ɛfloat &amp;gt; 10−8 W kg−1.

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.

On chirality and turbulent dispersion

Turbulent dispersion in the presence of chiral flow structures is investigated using Lagrangian stochastic models. Attention is focused upon modeled conditional Lagrangian accelerations that have quadratic dependencies upon Lagrangian velocities; a scenario that is consistent with mean accelerations derived from the Navier–Stokes equations and with the results of direct numerical simulations (i.e. with solutions of the Navier–Stokes equations). Chirality endows fluid-particle trajectories with a velocity-dependent preferred sense of rotation and is closely associated with the widely-utilized Eulerian diagnostic, turbulence-helicity. A non-vanishing mean chirality statistic is shown to be associated with oscillating Lagrangian velocity autocorrelation functions, oscillatory growth of dispersion, a significant reduction in dispersion for fixed turbulence kinetic energy and dissipation, vanishing mean angular momentum and the production of bi-modal plume structures. It is shown that two-point mean helicity statistics, characterizing the correlation between the velocity of one particle and the rotation of a second particle, can discriminate between two-particle Lagrangian stochastic models in the well-mixed class. A model is constructed for which this non-physical correlation vanishes.

On upscaling operator-stable Lévy motions in fractal porous media

The dynamics of motile particles, such as microbes, in random porous media are modeled with a hierarchical set of stochastic differential equations which correspond to micro, meso and macro scales. On the microscale the motile particle is modeled as an operator stable Lévy process with stationary, ergodic, Markov drift. The micro to meso and meso to macro scale homogenization is handled with generalized central limit theorems. On the mesoscale the Lagrangian drift (or the Lagrangian acceleration) is assumed Lévy to account for the fractal character of many natural porous systems. Diffusion on the mesoscale is a result of the microscale asymptotics while diffusion on the macroscale results from the mesoscale asymptotics. Renormalized Fokker–Planck equations with time dependent dispersion tensors and fractional derivatives are presented at the macro scale.