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) FRANCE

Experimental 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 T L  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 T L

Experimental Issues for Lagrangian measurements (I)  Temporal resolution Lab experiment in water:  Spatial resolution in water:  Integral scales time: (3<C 0 <7) space: several centimeters  Temporal resolution Lab experiment in water:  Spatial resolution in water:  Integral scales time: (3<C 0 <7) space: several centimeters values for  =20 W/kg,  =1 m/s, =10 -6 m 2 /s

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 volume or homogeneous turbulence either small measurement volume or homogeneous turbulence

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 resolution Doppler frequency shift for 1 m/s acoustics: 2.5 kHzlaser: 50 kHz (+ “big” particles 250  m)  different measurement volumes typical size acoustics: 10 cmlaser: 3 mm  “old” ultrasound Doppler experiment (KLAC)  preliminary laser Doppler experiment (KLOP)  same physical principle  same flow: French Washing Machine  different time resolution Doppler frequency shift for 1 m/s acoustics: 2.5 kHzlaser: 50 kHz (+ “big” particles 250  m)  different measurement volumes typical size acoustics: 10 cmlaser: 3 mm

Experiments in Lyon: KLAC  emission at 2.5 MHz  angle between beams 45 degrees equivalent 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 degrees equivalent 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  laser 1W splitted into two beams  angle between beams 1.5 degrees, fringe length: 20  m  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: 20  m  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) 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) 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 noise is assumed to be white gaussian of variance  b 2 the likelihood of the parameter set {A n, f n,  b 2, N} is thus one has to maximize the likelihood to get the optimal parameters BUT 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 parameters BUT 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)  b is estimated separately N is postulated (N=1 in general) {A n (t)} are estimated from the frequencies {f n (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) {A n (t)} are estimated from the frequencies {f n (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 (2)

the frequency demodulation: results (3) distribution of recorded events

the frequency demodulation: results (4) no dependence of the acceleration variance on the length of the recorded events (less bias than for the velocity?)

Velocity distribution (KLOP) preliminary results from the Laser experiment (only data points) (only the absolute value of the velocity so far) Gaussian distribution of the velocity with v rms ~0.3 m/s  ~25 W/kg (?), R ~100 (?)

acceleration PDF (KLOP) a rms ~400 m/s 2 compatible with Heisenberg-Yaglom with a 0 ~2 (Vedula & Yeung) and  ~25 W/kg solid line: Bodenschatz data R =285

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 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) large measurement volume: velocity autocorrelation Mordant, Metz, Michel & Pinton PRL 2001

results from the KLAC experiment (2) Kolmogorov constant C 0 : with then here C 0 ~4 at R =800 important for stochastic modelling of dispersion:

results from the KLAC experiment (3) |a i | surrogate for the acceleration magnitude long time decorrelation: integral time Mordant, Lévêque & Pinton, NJP 2004

results from the KLAC experiment (4) velocity time increments intermittency Mordant, Metz, Michel & Pinton PRL 2001