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