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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 R l of 815 the Eulerian structure functions would have well defined scaling ranges.

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lagrangian descriptions of turbulence

Lagrangian Descriptions of Turbulence

Greg Voth

Wesleyan University,

Middletown, CT, USA

second order lagrangian structure function
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 function1
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


  • 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 with more recent data: intermittency corrections


Gulitski et al, JFM 2007


Lagrangian Tetrad Dynamics

Xu et al, NJP 2008

early lagrangian experiments
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:
3d particle tracking velocimetry
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?

Ott and Mann, JFM (2000)

imaging requirements to resolve particle trajectories in intense turbulence
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
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
Silicon Vertex Detector during assembly of CLEO III at Cornell

447 silicon detectors

arranged in 4 concentric


Outer diameter: 30 cm

Inner diameter: 8cm

43,000 channels

~100Hz event rate

4 MB/sec data rate from


~100 MB/sec from all


LHC produces


Richard Kass, Ohio State University

strip detectors for particle tracking in turbulence
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


35 MB/sec data rate per detector

Two crossed imagers give resolution equal to

512x512 pixel imager:

5122*70kHz=18 Gb/sec


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.


acoustic particle tracking
Acoustic Particle Tracking

Mordant, Leveque and Pinton, PRL 2001, NJP 2004

2.5 MHz

probability distribution of energy dissipation strain rate squared and enstrophy vorticity squared
Probability Distribution of Energy Dissipation (strain rate squared)and Enstrophy (vorticity squared)
3d particle tracking velocimetry1
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
Cameras that move with the mean velocity

Ayyalasomayajula et al, PRL 2006

experimental problems
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
Recent Developments
  • Instrumented 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.
commercial systems
Commercial Systems
  • TSI V3V
  • La Vision FlowMasterTomoPIV
recent developments1
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…
real time image compression
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.




Brightness & location

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
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
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
Motion of Non-tracer particles in Turbulence

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
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:

what happens when the particles are large and neutrally buoyant

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

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.


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).

Modeling particle size dependence:

RNN and RLL are the transverse and longitudinal acceleration correlation functions.

RNN and RLL are measured in Xu et al, PRL (2007).

tracking rods in 3d turbulence
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

Sader 2008

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

Tracking Rods allows a single particle measurement that contains information about the velocity gradient tensor!

raw data
Raw Data

Rods are made of Nylon fibers d=300 μm (2h) L=1.5 mm (10h)a=5density=1.15 g/cm3

Rods are stained using fluorescent dye.

Nd:YAG laser with 50 W average power is used for illumination.

rotation rate variance and the energy dissipation rate shin and koch jfm2005
Rotation rate variance and the energy dissipation rate (Shin and Koch JFM2005)

The Jeffery Equation is:


By definition

and for isotropic turbulence

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.
  • 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

Accelerations of Large Neutrally Buoyant Particles