The Lagrangian description of turbulence is characterized by a unique conceptual simplicity and by an immediate connection with the physics of dispersion and mixing. In this article, we report some motivations behind the Lagrangian description of turbulence and focus on the statistical properties of particles when advected by fully developed turbulent flows. By means of a detailed comparison between experimental and numerical results, we review the physics of particle acceleration, Lagrangian velocity structure functions, and pairs and shapes evolution. Recent results for nonideal particles are discussed, providing an outlook on future directions.

Lagrangian Properties of Particles in Turbulence

▪ Abstract Turbulence is ubiquitous in atmospheric clouds, which have enormous turbulence Reynolds numbers owing to the large range of spatial scales present. Indeed, the ratio of energy-containing and dissipative length scales is on the order of 105 for a typical convective cloud, with a corresponding large-eddy Reynolds number on the order of 106 to 107. A characteristic trait of high-Reynolds-number turbulence is strong intermittency in energy dissipation, Lagrangian acceleration, and scalar gradients at small scales. Microscale properties of clouds are determined to a great extent by thermodynamic and fluid-mechanical interactions between droplets and the surrounding air, all of which take place at small spatial scales. Furthermore, these microscale properties of clouds affect the efficiency with which clouds produce rain as well as the nature of their interaction with atmospheric radiation and chemical species. It is expected, therefore, that fine-scale turbulence is of direct importance to the evolu...

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Particle-turbulence interactions in atmospheric clouds

We review studies of the statistics of isotropic turbulence in an incompressible fluid at high Reynolds numbers using direct numerical simulation (DNS) from the viewpoint of fundamental physics. The Reynolds number achieved by the largest DNS, with 4096 3 grid points, is comparable with the largest Reynolds number in laboratory experiments. The high-quality DNS data in the inertial subrange and the dissipative range enable the examination of detailed statistics at small scales, such as the normalized energy-dissipation rate, energy and energy-flux spectra, the intermittency of the velocity gradients and increments, scaling exponents, and flow-field structure. We emphasize basic questions of turbulence, universality in the sense of Kolmogorov’s theory, and the dependence of the statistics on the Reynolds number and scale.

http://www.o3d.org/abracco/annual_rev_3dnumerical.pdf

Study of High-Reynolds Number Isotropic Turbulence by Direct Numerical Simulation

We use silicon strip detectors (originally developed for the CLEO III high-energy particle physics experiment) to measure fluid particle trajectories in turbulence with temporal resolution of up to 70000 frames per second. This high frame rate allows the Kolmogorov time scale of a turbulent water flow to be fully resolved for 140 [ges ] Rλ [ges ] 970. Particle trajectories exhibiting accelerations up to 16000 m s
 −2 (40
 times the r.m.s. value) are routinely observed. The probability density function of the acceleration is found to have Reynolds-number-dependent stretched exponential tails. The moments of the acceleration distribution are calculated. The scaling of the acceleration component variance with the energy dissipation is found to be consistent with the results for low-Reynolds-number direct numerical simulations, and with the K41-based Heisenberg–Yaglom prediction for Rλ [ges ] 500. The acceleration flatness is found to increase with Reynolds number, and to exceed 60 at Rλ = 970. The coupling of the acceleration to the large-scale anisotropy is found to be large at low Reynolds number and to decrease as the Reynolds number increases, but to persist at all Reynolds numbers measured. The dependence of the acceleration variance on the size and density of the tracer particles is measured. The autocorrelation function of an acceleration component is measured, and is found to scale with the Kolmogorov time τη.

Measurement of particle accelerations in fully developed turbulence

A Lagrangian tracking method is introduced, which uses a prediction of the particle distribution for the subsequent time-step as a mean to seize the temporal domain. Errors introduced by the prediction process are corrected by an image matching technique (‘shaking’ the particle in space), followed by an iterative triangulation of particles newly entering the measurement domain. The scheme was termed ‘Shake-The-Box’ and previously characterized as ‘4D-PTV’ due to the strong interaction with the temporal dimension. Trajectories of tracer particles are identified at high spatial accuracy due to a nearly complete suppression of ghost particles; a temporal filtering scheme further improves on accuracy and allows for the extraction of local velocity and acceleration as derivatives of a continuous function. Exploiting the temporal information enables the processing of densely seeded flows (beyond 0.1 particles per pixel, ppp), which were previously reserved for tomographic PIV evaluations. While TOMO-PIV uses statistical means to evaluate the flow (building an ‘anonymous’ voxel space with subsequent spatial averaging of the velocity information using correlation), the Shake-The-Box approach is able to identify and track individual particles at numbers of tens or even hundreds of thousands per time-step. The method is outlined in detail, followed by descriptions of applications to synthetic and experimental data. The synthetic data evaluation reveals that STB is able to capture virtually all true particles, while effectively suppressing the formation of ghost particles. For the examined four-camera set-up particle image densities N
 I up to 0.125 ppp could be processed. For noise-free images, the attained accuracy is very high. The addition of synthetic noise reduces usable particle image density (N
 I ≤ 0.075 ppp for highly noisy images) and accuracy (still being significantly higher compared to tomographic reconstruction). The solutions remain virtually free of ghost particles. Processing an experimental data set on a transitional jet in water demonstrates the benefits of advanced Lagrangian evaluation in describing flow details—both on small scales (by the individual tracks) and on larger structures (using an interpolation onto an Eulerian grid). Comparisons to standard TOMO-PIV processing for synthetic and experimental evaluations show distinct benefits in local accuracy, completeness of the solution, ghost particle occurrence, spatial resolution, temporal coherence and computational effort.

Shake-The-Box: Lagrangian particle tracking at high particle image densities

The motion of fluid particles as they are pushed along erratic trajectories by fluctuating pressure gradients is fundamental to transport and mixing in turbulence. It is essential in cloud formation and atmospheric transport, processes in stirred chemical reactors and combustion systems, and in the industrial production of nanoparticles. The concept of particle trajectories has been used successfully to describe mixing and transport in turbulence, but issues of fundamental importance remain unresolved. One such issue is the Heisenberg-Yaglom prediction of fluid particle accelerations, based on the 1941 scaling theory of Kolmogorov. Here we report acceleration measurements using a detector adapted from high-energy physics to track particles in a laboratory water flow at Reynolds numbers up to 63,000. We find that, within experimental errors, Kolmogorov scaling of the acceleration variance is attained at high Reynolds numbers. Our data indicate that the acceleration is an extremely intermittent variable--particles are observed with accelerations of up to 1,500 times the acceleration of gravity (equivalent to 40 times the root mean square acceleration). We find that the acceleration data reflect the anisotropy of the large-scale flow at all Reynolds numbers studied.

Fluid particle accelerations in fully developed turbulence

We use silicon strip detectors (originally developed for the CLEO III high energy particle physics experiment) to measure fluid particle trajectories in turbulence with temporal resolution of up to 70,000 frames per second. This high frame rate allows the Kolmogorov time scale of a turbulent water flow to be fully resolved for 140 = 500. The acceleration flatness is found to increase with Reynolds number, and to exceed 60 at R_lambda = 970. The coupling of the acceleration to the large scale anisotropy is found to be large at low Reynolds number and to decrease as the Reynolds number increases, but to persist at all Reynolds numbers measured. The dependence of the acceleration variance on the size and density of the tracer particles is measured. The autocorrelation function of an acceleration component is measured, and is found to scale with the Kolmogorov time tau_eta.

Measurement of Particle Accelerations in Fully Developed Turbulence

Experimental Lagrangian acceleration probability density function measurement

We report experimental results on the three-dimensional Lagrangian acceleration in highly turbulent flows. Tracer particles are tracked optically using four silicon strip detectors from high energy physics that provide high temporal and spatial resolution. The components of the acceleration are shown to be statistically dependent. The probability density function of the acceleration magnitude is comparable to a log-normal distribution. Assuming isotropy, a log-normal distribution of the magnitude can account for the observed dependency of the components. The time dynamics of the acceleration components is found to be typical of the dissipation scales, whereas the magnitude evolves over longer times, possibly close to the integral time scale.

Alice M. Crawford

Papers

Fluid particle accelerations in fully developed turbulence

Measurement of particle accelerations in fully developed turbulence

Measurement of Particle Accelerations in Fully Developed Turbulence

Experimental Lagrangian acceleration probability density function measurement

Three-Dimensional Structure of the Lagrangian Acceleration in Turbulent Flows