Lagrangian Descriptions of Turbulence Greg Voth Wesleyan University, Middletown, CT, USA

Second Order Lagrangian Structure Function A small Lagrangian inertial range may be visible. For Rl of 815 the Eulerian structure functions would have well defined scaling ranges. Biferale et al, Phys. Fluids 20:065103 (2008)

Second Order Lagrangian Structure Function Lagrangian structure functions require larger Reynolds numbers to observe a scaling range if at all. Despite the lack of clean scaling, the Lagrangian Kolmogorov constant C0~6 is very useful for stochastic modeling of turbulent mixing.

Higher Order Lagrangian Structure Functions K41 Error bars are large for available data, but intermittency appears and deviations from K41 are larger than for the Eulerian velocity. They are even somewhat stronger than deviations for the passive scalar.

Acceleration Statistics

Acceleration Statistics with more recent data: intermittency corrections Rl0.13 Gulitski et al, JFM 2007

Lagrangian Tetrad Dynamics Xu et al, NJP 2008

Experimental Measurements of Particle Trajectories in Turbulence

Early Lagrangian Experiments Richardson, 1922 Balloons were released from Brighton England with labels asking a discoverer to return them with the landing location marked. This data was used to justify Richardson’s law for two particle dispersion:

Sato and Yamamoto, JFM 175:183 (1987) Note that ‘solid state image sensor’ was worth labelling in 1987…vidicons and other vacuum tube based imaging systems had only recently been replaced for electronic imaging purposes.

3D Particle Tracking Velocimetry Developed by the Drakos group at ETH Zurich in the mid 1990s Early implementations were limited to low Reynolds numbers (Rl~100) by available cameras (50 Hz frame rate). Question: How many parameters are required to specify a camera’s viewpoint? 7 parameters minimum: 3 from vector to camera position. 3 Euler angles to give the camera’s viewing orientation. 1 Magnification to convert camera pixel coordinates to position in space. In addition many optical distortion parameters are sometimes included…we find that one radial distortion is all that is necessary for a reasonably good lens with care that the viewing is perpendicular to the windows. If the viewing is not perpendicular then you have additional refraction effects. Ott and Mann, JFM (2000)

Imaging requirements to resolve particle trajectories in intense turbulence Spatial resolution: Number of pixels needed in one direction is So we would like 4000x4000 pixels Temporal resolution: Frame period should be on the order of the Kolmogorov time so we would like frame rates in the kHz range. Total Data Rate: 16 GB/sec

Silicon Strip Detectors for optical particle tracking Designed for charged particle detection at high energy physics collider experiments. 512 light sensitive strips with integrated amplifiers Reads out a 1D projection of the light intensity Up to 70,000 images per second

Silicon Vertex Detector during assembly of CLEO III at Cornell 447 silicon detectors arranged in 4 concentric cylinders Outer diameter: 30 cm Inner diameter: 8cm 43,000 channels ~100Hz event rate 4 MB/sec data rate from Silicon ~100 MB/sec from all detectors LHC produces ~1.8Gb/sec Richard Kass, Ohio State University

Strip Detectors for particle tracking in turbulence Tracer particles in the flow are optically imaged onto the detectors. Imaging volume is (2mm)3 512 channels 70kHz 35 MB/sec data rate per detector Two crossed imagers give resolution equal to 512x512 pixel imager: 5122*70kHz=18 Gb/sec

Raw Data from a Strip Detector

Trajectory measured with silicon strip detectors Tracer particles: 50 micron polystyrene spheres. 70,000 images per second on 4 strip detectors 30s event

Histogram of accelerations has higher probabilities of rare events than either the scalar gradient or the velocity gradients. Fit:

Acoustic Particle Tracking Mordant, Leveque and Pinton, PRL 2001, NJP 2004 2.5 MHz Provides access to position (from relative phase across the array) and velocity (from the doppler shift). Impressive custom built electronic design to perform the ultra low noise heterodyne mixing from the phased array. Limited to relatively large tracer particles to get enough scattered acoustic intensity.

Intermittency of Lagrangian Velocity Increments

3D PIV to obtain the velocity gradient tensor Zeff et al, Nature 2003

Probability Distribution of Energy Dissipation (strain rate squared) and Enstrophy (vorticity squared)

3D Particle Tracking Velocimetry Luthi et al, JFM 2005 Hoyer et al, Exp. In Fluids 2005 Mean eignevalues of the Cauchy Green Strain Tensor (Rl=50)

3D Particle Tracking Velocimetry with high speed cameras Ouellette et al, NJP 2005 Phantom v7 cameras 27 kHz at 256 x 256

Cameras that move with the mean velocity Ayyalasomayajula et al, PRL 2006

Experimental Problems 3D particle tracking is always starved for light: Illumination beam must be expanded to cover the detection volume. Volume imaging requires small apertures. Particles should be small enough to passively follow the flow. High speed imaging means very little time to collect photons. Images are never nice clean gaussian spots on a dark background Stray reflections Multiple reflections from tracer particles Sometimes exactly focused particles only illuminate one pixel which greatly degrades position accuracy At high seeding densities, images of particles overlap.

Recent Developments Instrumented Particles Marked Particles Particles with sensors for acceleration, pressure, etc. and radios to transmit data out of the flow. Marked Particles Particles with patterns created on their surface can be imaged to extract orientation in addition to position. Index matched particles with tracers embedded allow standard 3DPTV to extract particle orientation.

Goettingen Turbulence Facility

Commercial Systems TSI V3V La Vision FlowMaster TomoPIV

Recent Developments Instrumented Particles Particle positions are found to a few microns from distances of 50 cm. Calibrations drift. A useful solution developed independently by several groups is dynamic calibration: Once you have an initial calibration of the position and orientation of the cameras, use real data with nonlinear optimization to adjust the calibration parameters to minimize mismatch between the rays from different cameras. There is a reason plumbers are paid well…

Raw Image

Real-time Image Compression Intermediate hardware between camera and computer Input pixel brightness array, output a “vector” of (brightness, x, y) Compression factors above 100 are routinely achieved. Compression Circuit Pixels Brightness & location 55 233 156 67 29 96 28 9 45 23 68 8 79 58 176 54 55 233 156 67 29 96 28 9 45 23 68 8 79 58 176 54 (233, 1, 0) (156, 2, 0) (96, 1,1) (176, 2, 3) Before, video ram limited data collection time to ~7 seconds=4Gb. Now with direct hard drive recording of compressed video, we can acquire data for ~7 days. Other compression schemes are possible, but the simple real-space basis seems to be nearly ideal for 3D particle tracking data.

Image Compression Circuit 650 MB/sec data rate from the cameras (Basler 1024X1280 pixels with 500 Hz frame rate). Field Programmable IC: Altera® FPGA Custom circuit board interfaces the FPGA with a camera ~1 ½ year development and construction time by a master’s student ~costs about $600

Summary of Experimental Particle Tracking Techniques The wide range of scales in 3D turbulence precludes complete measurement of the flow, so one must choose either an Eulerian (fixed in space) or Lagrangian (particle tracking) measurements. Rapid advances in imaging technology have led to rapid advances in capabilities of particle tracking systems.

Motion of Non-tracer particles in Turbulence What happens when particles do not follow the flow? This can happen either because the particles have a density difference with the fluid or because they have a size large compared with the Kolmogorov scale. Very important problem in a variety of Clouds are a major source of uncertainty in predictions of the rate of climate change expected. An unsolved problem is how water droplets in clouds rapidly grow from ~10 microns (where condensation becomes less important) to ~1mm (where they fall as rain). Droplet collisions are the growth mechanism and turbulence is believed to be an important ingredient.

Equation of motion for small particles at low particle Reynolds number For particles with a large density difference like water droplets in clouds, the external force (first term), fluid acceleration (second term) and Stokes drag (fourth term) dominate. In this case a single non-dimensional parameter defines the motion, the Stokes number:

Preferential Concentration

Accelerations of Heavy Particles

What happens when the particles are large and neutrally buoyant? Particles larger than the Kolmogorov scale start to average over scales smaller than the particle diameter. If particles have size in the inertial range, then the acceleration variance should be only a function of particle diameter (d) and energy dissipation rate (e). The only combination of these with units of acceleration squared is e4/3d-2/3

Modeling particle size dependence: The Faxen model developed by Calzavarini et al (JFM 2009) predicts the particle acceleration is the average of the fluid acceleration over the particle size (for neutrally buoyant particles). RNN and RLL are the transverse and longitudinal acceleration correlation functions. RNN and RLL are measured in Xu et al, PRL (2007).

Probability Distribution of Acceleration of Large Particles

Tracking Rods in 3D Turbulence Shima Parsa, Wesleyan University Nick Ouellette, Yale University Applications that require an understanding of rod dynamics include: fibers for paper making ice crystal dynamics in clouds drag reduction

Rod rotation rate is determined by the velocity gradients The rotation rate of an ellipsoid in Stokes flow was predicted by Jeffrey in 1922 =unit vector along the rod. =rod aspect ratio Sader 2008 Tracking Rods allows a single particle measurement that contains information about the velocity gradient tensor!

Flow between Two Oscillating Grids At 3 Hz grid frequency Blum et al Physics of Fluids 2010

Raw Data Rods are made of Nylon fibers d=300 μm (2h) L=1.5 mm (10h) a=5 density=1.15 g/cm3 Rods are stained using fluorescent dye. Nd:YAG laser with 50 W average power is used for illumination.

A Rod Trajectory

Rod Velocity

Rotation rate of rods in 3D

Rotation rate variance and the energy dissipation rate (Shin and Koch JFM2005) The Jeffery Equation is: so By definition and for isotropic turbulence XX Explain energy dissipation rate, equation & importance in turbulence E=2v <S_ij S_ij> It can be measured from rotation, rotation is coupled with the velocity gradient and can be used as a way to measure the strain rate, so energy dissipation rate Use the figure 10 and equation on page 160 Shin and koch Explain that fibers are adopting themselves with the direction in the fluid XX We introduced the rotation rate for asymptotically large aspect ratio as (lambda dot) which depends on the velocity gradient and initial orientation. Sij is the strain rate which is the symmetric part of the velocity gradient, Rotation rate variance is defined as (lambda dot times lambda dot) and by multiplying the above equation can be simply written as 3 averages, at time t=0 the orientation of fiber is not correlated with the velocity gradient ( strain and vorticity) so the averages can be decomposed to products and for a system of isotropic turbulence we can relate the moments of velocity gradinet to kolmogorov shear rate and we will have (lambda dot =4/15etta) Where etta is ... Epsilon is the energy dissipation rate and nu viscosity. This assumption is true for only t=0, but in real world experiment we know that we can not assume that fiber is at t=0 and we are dealing with particles that their orientation are correlated with the local axis of strain and vorticity. Shin and koch simulation results suggest that for short fibers the rotation rate variance is related to kolmogorov shear rate as... Measuring the rotation rate variance provides an interesting way to measure the energy dissipation rate in a turbulent flow which is somehow hard to measure. So the rotation rate variance for randomly oriented thin rods in isotropic turbulence is

From simulation of thin rods in turbulence by Shin and Koch (JFM 2005) randomly oriented After advection by the flow partially aligns rods with the strain rate Factors to consider: The simulation results are for one-way coupling of infinite aspect ratio rods. 15% density mismatch in the experiment The simulation is at much smaller The simulation is based on one-way coupling of rods with flow.

Parameter Space for Rigid Rods

Summary Lagrangian description is necessary to study the univerality of the temporal dynamics of turbulence. Particle tracking tools have helped a robust phenomenology from the Lagrangian viewpoint. The problem has inspired a wide range of innovative measurements. The tools developed have turned out to useful in addressing many new problems about motion of non-tracer particles in turbulence.

Accelerations of Large Neutrally Buoyant Particles If particles have size in the inertial range, then the acceleration variance should be only a function of particle diameter (d) and energy dissipation rate (e). The only combination of these with units of acceleration squared is e4/3d-2/3