Lagrangian Acceleration Statistics Research Articles

Measurements Of The Energy Dissipation Rate In Homogeneous Turbulence Using Dense 3D Lagrangian Particle Tracking And FlowFit

We present measurements of the full velocity gradient tensor and all volumetric dissipation rate elements based on dense fields of fluid particle trajectories in homogeneous turbulence at Re_λ~270 and ~370 in a Kármán flow between two counter-rotating disks with impellers. Applying the Shake-The-Box (STB) Lagrangian Particle Tracking (LPT) algorithm, we are able to instantaneously track up to 80.000 particles in a volume of 40 x 40 x 15 mm³. The mean interparticle distance is lower than 7 Kolmogorov lengths for the Re_λ ~270 case. A data assimilation scheme (FlowFit) with continuity and Navier-Stokes- constraints is used to interpolate the scattered velocity and acceleration data by a continuous 3D B-Spline representation, enabling to recover (locally) the smallest flow scales. In the presentation, we show Lagrangian velocity and acceleration statistics, as well as the Eulerian counterparts on velocity gradients and pressure fields. We compute the energy dissipation rate directly by making use of quadruples of particle trajectories in close proximity (r < 3η) and compare it to indirect approaches using second-order velocity- and velocityacceleration structure functions in the inertial subrange.

Sharp transitions in rotating turbulent convection: Lagrangian acceleration statistics reveal a second critical Rossby number

In RB convection for fluids with Prandtl number $Pr\gtrsim 1$, rotation beyond a critical (small) rotation rate is known to cause a sudden enhancement of heat transfer which can be explained by a change in the character of the BL dynamics near the top and bottom plates of the convection cell. Namely, with increasing rotation rate, the BL signature suddenly changes from Prandtl--Blasius type to Ekman type. The transition from a constant heat transfer to an almost linearly increasing heat transfer with increasing rotation rate is known to be sharp and the critical Rossby number $Ro_{c}$ occurs typically in the range $2.3\lesssim Ro_{c}\lesssim 2.9$ (for Rayleigh number $Ra=1.3\times 10^9$, $Pr=6.7$, and a convection cell with aspect ratio $\Gamma=\frac{D}{H}=1$, with $D$ the diameter and $H$ the height of the cell). The explanation of the sharp transition in the heat transfer points to the change in the dominant flow structure. At $1/Ro\lesssim 1/Ro_c$ (slow rotation), the well-known LSC is found: a single domain-filling convection roll made up of many individual thermal plumes. At $1/Ro\gtrsim 1/Ro_c$ (rapid rotation), the LSC vanishes and is replaced with a collection of swirling plumes that align with the rotation axis. In this paper, by numerically studying Lagrangian acceleration statistics, related to the small-scale properties of the flow structures, we reveal that this transition between these different dominant flow structures happens at a second critical Rossby number, $Ro_{c_2}\approx 2.25$ (different from $Ro_{c_1}\approx 2.7$ for the sharp transition in the Nusselt number $Nu$; both values for the parameter settings of our present numerical study). When statistical data of Lagrangian tracers near the top plate are collected, it is found that the root-mean-square (rms) values and the kurtosis of the horizontal acceleration of these tracers show a sudden increase at $Ro_{c_2}$.

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.

Acceleration statistics of tracer particles in filtered turbulent fields

We present results from direct numerical simulations of tracer particles advected in filtered velocity fields to quantify the impact of the scales of turbulence on Lagrangian acceleration statistics. Systematically removing spatial scales reduces the frequency of extreme acceleration events, consistent with the notion that they are rooted in the small-scale structure of turbulence. We also find that acceleration variance and flatness as a function of filter scale closely resemble experimental results of neutrally buoyant, finite-sized particles, corroborating the picture that particle size determines the scale on which turbulent fluctuations are sampled.

Lagrangian acceleration statistics in a turbulent channel flow

Lagrangian acceleration statistics in a fully developed turbulent channel flow at $Re_\tau = 1440$ are investigated, based on tracer particle tracking in experiments and direct numerical simulations. The evolution with wall distance of the Lagrangian velocity and acceleration time scales is analyzed. Dependency between acceleration components in the near-wall region is described using cross-correlations and joint probability density functions. The strong streamwise coherent vortices typical of wall-bounded turbulent flows are shown to have a significant impact on the dynamics. This results in a strong anisotropy at small scales in the near-wall region that remains present in most of the channel. Such statistical properties may be used as constraints in building advanced Lagrangian stochastic models to predict the dispersion and mixing of chemical components for combustion or environmental studies.

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.

How gravity and size affect the acceleration statistics of bubbles in turbulence

We report the results of the first systematic Lagrangian experimental investigation in a previously unexplored regime of very light (air bubbles in water) and large (D/η ≫ 1) particles in turbulence. Using a traversing camera setup and particle tracking, we study the Lagrangian acceleration statistics of ∼3 mm diameter (D) bubbles in a water tunnel with nearly homogeneous and isotropic turbulence generated by an active grid. The Reynolds number (Reλ) is varied from 145 to 230, resulting in size ratios, D/η, in the range of 7.3–12.5, where η is the Kolmogorov length scale. The experiments reveal that gravity increases the acceleration variance and reduces the intermittency of the probability density function (PDF) in the vertical direction. Once the gravity offset has been subtracted, the variances of both the horizontal and vertical acceleration components are about 5 ± 2 times larger than those measured in the same flow for fluid tracers. Moreover, for these light particles, the experimental acceleration PDF shows a substantial reduction of intermittency at growing size ratios, in contrast with neutrally buoyant or heavy particles. All these results closely match numerical simulations of finite-sized bubbles with the Faxén corrections.

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.

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.

Long Time Memory of Lagrangian Acceleration Statistics in 2D and 3D Turbulence

In this paper we report on a comparison of Lagrangian acceleration statistics in the direct energy cascade of three-dimensional turbulence and the corresponding observables for the case of the inverse energy cascade in two dimensions. We focus on the time scales describing the memory of the acceleration statistics of a tracer particle. We show that for both systems the Markov time scale, which is an indicator for the length of the memory of a stochastic process, is in the order of magnitude of the Lagrangian integral time scale. We also show that the decorrelation time for the cross-correlation between the squared components of the acceleration is larger than the integral time scale.

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.

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.

STOCHASTIC MODELS OF LAGRANGIAN ACCELERATION OF FLUID PARTICLE IN DEVELOPED TURBULENCE

Modeling statistical properties of motion of a Lagrangian particle advected by a high-Reynolds-number flow is of much practical interest and complement traditional studies of turbulence made in Eulerian framework. The strong and nonlocal character of Lagrangian particle coupling due to pressure effects makes the main obstacle to derive turbulence statistics from the three-dimensional Navier–Stokes equation; motion of a single fluid-particle is strongly correlated to that of the other particles. Recent breakthrough Lagrangian experiments with high resolution of Kolmogorov scale have motivated growing interest to acceleration of a fluid particle. Experimental stationary statistics of Lagrangian acceleration conditioned on Lagrangian velocity reveals essential dependence of the acceleration variance upon the velocity. This is confirmed by direct numerical simulations. Lagrangian intermittency is considerably stronger than the Eulerian one. Statistics of Lagrangian acceleration depends on Reynolds number. In this review we present description of new simple models of Lagrangian acceleration that enable data analysis and some advance in phenomenological study of the Lagrangian single-particle dynamics. Simple Lagrangian stochastic modeling by Langevin-type dynamical equations is one the widely used tools. The models are aimed particularly to describe the observed highly non-Gaussian conditional and unconditional acceleration distributions. Stochastic one-dimensional toy models capture main features of the observed stationary statistics of acceleration. We review various models and focus in a more detail on the model which has some deductive support from the Navier–Stokes equation. Comparative analysis on the basis of the experimental data and direct numerical simulations is made.

Conditional Lagrangian acceleration statistics in turbulent flows with Gaussian-distributed velocities

The random intensity of noise approach to the one-dimensional Laval-Dubrulle-Nazarenko-type model having deductive support from the three-dimensional Navier-Stokes equation is used to describe Lagrangian acceleration statistics of a fluid particle in developed turbulent flows. Intensity of additive noise and cross correlation between multiplicative and additive noises entering a nonlinear Langevin equation are assumed to depend on random velocity fluctuations in an exponential way. We use an exact analytic result for the acceleration probability density function obtained as a stationary solution of the associated Fokker-Planck equation. We give a complete quantitative description of the available experimental data on conditional and unconditional acceleration statistics within the framework of a single model with a single set of fit parameters. The acceleration distribution and variance conditioned on Lagrangian velocity fluctuations and the marginal distribution calculated by using independent Gaussian velocity statistics are found to be in a good agreement with the recent high-Reynolds-number Lagrangian experimental data. The fitted conditional mean acceleration is very small, that is, in agreement with direct numerical simulations, and increases for higher velocities but it departs from the experimental data, which exhibit anisotropy of the studied flow.

Multifractal statistics of Lagrangian velocity and acceleration in turbulence.

The statistical properties of velocity and acceleration fields along the trajectories of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. We present results for Lagrangian velocity structure functions, the acceleration probability density function, and the acceleration variance conditioned on the instantaneous velocity. These are compared with predictions of the multifractal formalism, and its merits and limitations are discussed.

Superstatistical Lagrangian stochastic modeling

A simple superstatistical Lagrangian stochastic model (Phys. Fluids 15 (2003) L1; Phys. Rev. Lett. 91 (2003) 84503) that accounts explicitly for fluctuations in the rate of dissipation of turbulent kinetic energy has been shown to be in remarkably close agreement with recently acquired data for unconditional Lagrangian acceleration statistics. In this paper, a more elaborate version of the model is shown to predict correctly the observed conditional dependency of Lagrangian acceleration statistics on velocity. The modeling approach is then extended to the simulation of large/heavy-particle accelerations in turbulence. Model predictions for the distribution of accelerations of 450 μm diameter particles with near-neutral buoyancy are shown to be in excellent agreement with experimental data. Tsallis statistics are found to describe accurately model predictions for distributions of heavy-particle velocities and accelerations.

Reynolds number effects in Lagrangian stochastic models of turbulent dispersion

A second-order autoregressive equation is used to model the acceleration of fluid particles in turbulence in order to study the effect of Reynolds number on Lagrangian turbulence statistics. It is shown that this approach provides a good representation of dissipation subrange structure of Lagrangian velocity and acceleration statistics. The parameters of the model, two time scales representing the energy-containing and dissipation scales, are determined by matching the model velocity autocorrelation function to Kolmogorov similarity forms in the inertial subrange and the dissipation subrange. The model is tested against the Lagrangian statistics obtained by Yeung and Pope [J. Fluid Mech. 207, 531 (1989)] from direct numerical simulations of turbulence. Agreement between the model predictions and simulation data for second-order Lagrangian statistics such as the velocity structure function, the acceleration correlation function, and the dispersion of fluid particles is excellent, indicating that the main departures from Kolmogorov’s theory of local isotropy shown by the simulation data are due to low Reynolds number. For Reynolds numbers typical of laboratory experiments and direct numerical simulations of turbulence the root-mean-square dispersion of marked particles is changed from the Langevin equation (i.e., infinite Reynolds number) prediction by up to about 50% at large times. Most of this change can be accounted for by the change in the Lagrangian integral time scale. It is also shown that Reynolds number effects in laboratory dispersion or Lagrangian turbulence measurements can cause significant errors (typically of order 50%) when the value of the Kolmogorov Lagrangian structure function constant C0 is estimated by fitting the predictions of the Langevin equation to these data. A value C0 = 7 is obtained by fitting the new model to the direct simulation data.