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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Middletown, CT, USA
Biferale et al, Phys. Fluids 20:065103 (2008)
Gulitski et al, JFM 2007
Xu et al, NJP 2008
Ott and Mann, JFM (2000)
So we would like 4000x4000 pixels
so we would like frame
rates in the kHz range.
Designed for charged particle detection at high energy physics
512 light sensitive strips with integrated amplifiers
Reads out a 1D projection of the light intensity
Up to 70,000 images per second
447 silicon detectors
arranged in 4 concentric
Outer diameter: 30 cm
Inner diameter: 8cm
~100Hz event rate
4 MB/sec data rate from
~100 MB/sec from all
Richard Kass, Ohio State University
Tracer particles in the flow are optically imaged onto the detectors.
Imaging volume is (2mm)3
35 MB/sec data rate per detector
Two crossed imagers give resolution equal to
512x512 pixel imager:
50 micron polystyrene spheres.
70,000 images per second on 4 strip detectors
Histogram of accelerations has higher probabilities of rare events than either the scalar gradient or the velocity gradients.
Mordant, Leveque and Pinton, PRL 2001, NJP 2004
Zeff et al, Nature 2003
Luthi et al, JFM 2005
Hoyer et al, Exp. In Fluids 2005
Mean eignevalues of the Cauchy Green Strain Tensor (Rl=50)
Ouellette et al, NJP 2005
Phantom v7 cameras
27 kHz at 256 x 256
Ayyalasomayajula et al, PRL 2006
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.
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.
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:
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-2/3What 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).
Applications that require an understanding of rod dynamics include:
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!
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.
The Jeffery Equation is:
and for isotropic turbulence
So the rotation rate variance for randomly oriented thin rods in isotropic turbulence is
After advection by the flow partially aligns rods with the strain rate
Factors to consider:
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-2/3Accelerations of Large Neutrally Buoyant Particles