Lagrangian measurements using Doppler techniques:
Laser and Ultrasound
Lagrangian measurements using Doppler techniques:
Laser and Ultrasound
Nicolas Mordant (Ecole Normale Supérieure de Paris)
Romain Volk, Artyom Petrosyan,
Jean-François Pinton (Ecole Normale Supérieure de Lyon)
FRANCE
Nicolas Mordant (Ecole Normale Supérieure de Paris)
Romain Volk, Artyom Petrosyan,
Jean-François Pinton (Ecole Normale Supérieure de Lyon)
FRANCEQuickTime™ et undécompresseur TIFF (non compressé)sont requis pour visionner cette image.
QuickTime™ et undécompresseur TIFF (non compressé)sont requis pour visionner cette image.QuickTime™ et undécompresseur TIFF (non compressé)sont requis pour visionner cette image.QuickTime™ et undécompresseur TIFF (non compressé)sont requis pour visionner cette image.
Experimental goals in Lagrangian measurementsExperimental goals in Lagrangian measurements
Track individual particles along their trajectories
- single out individual particles
- field of view as wide as possible
Measure trajectories for long enough to have information about the dynamics
- acceleration time scale: Kolmogorov time
- velocity: integral time scale TL
Track individual particles along their trajectories
- single out individual particles
- field of view as wide as possible
Measure trajectories for long enough to have information about the dynamics
- acceleration time scale: Kolmogorov time
- velocity: integral time scale TL
Experimental Issues for Lagrangian measurements (I)Experimental Issues for Lagrangian measurements (I)
Temporal resolution
Lab experiment in water:
Spatial resolution
in water:
Integral scales
time: (3<C0<7)
space: several centimeters
Temporal resolution
Lab experiment in water:
Spatial resolution
in water:
Integral scales
time: (3<C0<7)
space: several centimeters
values for =20 W/kg, =1 m/s, =10-6m2/s
Experimental Issues for Lagrangian measurements (II)Experimental Issues for Lagrangian measurements (II)
for Doppler measurements: no direct spatial sampling
spatial localization: imposed by the measurement volume
for Doppler measurements: no direct spatial sampling
spatial localization: imposed by the measurement volume
the whole difficulty lies in the temporal resolution
either small measurement volumeor homogeneous turbulenceeither small measurement volumeor homogeneous turbulence
Experiments in Lyon:KLAC & KLOP
Experiments in Lyon:KLAC & KLOP
“old” ultrasound Doppler experiment (KLAC) preliminary laser Doppler experiment (KLOP)
same physical principle same flow: French Washing Machine
different time resolutionDoppler frequency shift for 1 m/s
acoustics: 2.5 kHz laser: 50 kHz (+ “big” particles 250 m)
different measurement volumestypical size
acoustics: 10 cm laser: 3 mm
“old” ultrasound Doppler experiment (KLAC) preliminary laser Doppler experiment (KLOP)
same physical principle same flow: French Washing Machine
different time resolutionDoppler frequency shift for 1 m/s
acoustics: 2.5 kHz laser: 50 kHz (+ “big” particles 250 m)
different measurement volumestypical size
acoustics: 10 cm laser: 3 mm
Experiments in Lyon:KLAC
Experiments in Lyon:KLAC
emission at 2.5 MHz angle between beams
45 degreesequivalent fringe length 0.85
mm direct sampling of the acoustic wave
(heterodyne detection)
10 liters of water, 2x 1kW motors
250 m particles typical size of the measurement
volume: 10 cm a the center
emission at 2.5 MHz angle between beams
45 degreesequivalent fringe length 0.85
mm direct sampling of the acoustic wave
(heterodyne detection)
10 liters of water, 2x 1kW motors
250 m particles typical size of the measurement
volume: 10 cm a the center
Mordant, Lévêque & Pinton, NJP 2004
Experiments in Lyon:KLOP
Experiments in Lyon:KLOP
laser 1W splitted into two beams angle between beams
1.5 degrees, fringe length: 20m sampling of the light intensity
5 liters of water, 2x 600W motors
10 m fluorescent particles typical size of the measurement
volume: 3 mm a the center
so far: measurement of the absolute value of the velocity only
(addition of acousto-optic modulators soon)
laser 1W splitted into two beams angle between beams
1.5 degrees, fringe length: 20m sampling of the light intensity
5 liters of water, 2x 600W motors
10 m fluorescent particles typical size of the measurement
volume: 3 mm a the center
so far: measurement of the absolute value of the velocity only
(addition of acousto-optic modulators soon)
telescopes to increase the beam size
PM
the frequency demodulation (1)the frequency demodulation (1)goal: extract the spectral component with the best time resolution
example: time-frequency picture of a laser signal
high acceleration(~200 g)
the frequency demodulation (2)the frequency demodulation (2)
Fourier analysis: blind approach (no a priori information on the signal)
uncertainty principle:
for an accuracy of 0.05 m/s and 0.2 ms in our configuration
for ultrasound: for laser:
for an accuracy of 0.05 m/s and 0.2 ms in our configuration
for ultrasound: for laser:
parametric approach:add a priori information on the structure of the signal
parametric approach:add a priori information on the structure of the signal
the frequency demodulation (3)the frequency demodulation (3)
the noise is assumed to be white gaussian of variance b2
the likelihood of the parameter set {An, fn , b2, N} is thus
one has to maximize the likelihood to get the optimal parametersBUT not possible analytically
one can maximize analytically in respect with the amplitudes at fixed frequencies and noise variance
one has to maximize the likelihood to get the optimal parametersBUT not possible analytically
one can maximize analytically in respect with the amplitudes at fixed frequencies and noise variance
Mordant, Pinton & Michel, JASA 2002
the frequency demodulation (4)the frequency demodulation (4)
• b is estimated separately
• N is postulated (N=1 in general)
• {An(t)} are estimated from the frequencies
• {fn(t)} are estimated from a second order approximation of the likelihood in the vicinity of its maximum (requires a first estimate)
• the overall estimator is embedded in a Kalman filter (prediction/correction scheme) to get a tracking algorithm
• the algorithm outputs the Hessian of the likelihood which gives the confidence in the estimation
• b is estimated separately
• N is postulated (N=1 in general)
• {An(t)} are estimated from the frequencies
• {fn(t)} are estimated from a second order approximation of the likelihood in the vicinity of its maximum (requires a first estimate)
• the overall estimator is embedded in a Kalman filter (prediction/correction scheme) to get a tracking algorithm
• the algorithm outputs the Hessian of the likelihood which gives the confidence in the estimation
the frequency demodulation: results (1)the frequency demodulation: results (1)
the frequency demodulation: results (2)the frequency demodulation: results (2)
the frequency demodulation: results (3)the frequency demodulation: results (3)
distribution of recorded events
the frequency demodulation: results (4)the frequency demodulation: results (4)
no dependence of the acceleration varianceon the length of the recorded events
(less bias than for the velocity?)
Velocity distribution (KLOP)Velocity distribution (KLOP)
preliminary results from the Laser experiment (only 2.105 data points)
(only the absolute value of the velocity so far)Gaussian distribution of the velocity with vrms~0.3 m/s
~25 W/kg (?), R~100(?)
acceleration PDF (KLOP)acceleration PDF (KLOP)
arms~400 m/s2 compatible with Heisenberg-Yaglom
with a0~2 (Vedula & Yeung) and ~25 W/kg
solid line: Bodenschatz data R=285
acceleration correlation (KLOP)acceleration correlation (KLOP)
zero crossing at 1.6
Yeung & Pope report 2.2 at R=90)
zero crossing at 1.6
Yeung & Pope report 2.2 at R=90)
perspectives of the KLOP experimentperspectives of the KLOP experiment
• improve the signal over noise ratio(larger fluo. particles or higher reflectivity particles)
• increase the amount of data • increase the Reynolds number
• more powerful laser (larger measurement volume)
• other kind of particles (inertial, different sizes)
• improve the signal over noise ratio(larger fluo. particles or higher reflectivity particles)
• increase the amount of data • increase the Reynolds number
• more powerful laser (larger measurement volume)
• other kind of particles (inertial, different sizes)
results from the KLAC experiment (1)results from the KLAC experiment (1)
large measurement volume: velocity autocorrelation
Mordant, Metz, Michel & Pinton PRL 2001
results from the KLAC experiment (2)results from the KLAC experiment (2)
Kolmogorov constant C0 :
with
then
here C0~4 at R=800
important for stochastic modelling of dispersion:
results from the KLAC experiment (3)results from the KLAC experiment (3)
|ai| surrogate for the acceleration magnitude
long time decorrelation: integral time
Mordant, Lévêque & Pinton, NJP 2004
results from the KLAC experiment (4)results from the KLAC experiment (4)
velocity time increments
intermittency
Mordant, Metz, Michel & Pinton PRL 2001
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