Introduction to Laser Doppler Velocimetry Ken Kiger Burgers Program For Fluid Dynamics Turbulence School College Park, Maryland, May 24-27 Laser Doppler Anemometry (LDA) • Single-point optical velocimetry method Study of the flow between rotating impeller blades of a pump 3-D LDA Measurements on a 1:5 Mercedes-Benz E-class model car in wind tunnel Phase Doppler Anemometry (PDA) • Single point particle sizing/velocimetry method Droplet Size Distributions Drop Size and Velocity Measured in a Kerosene measurements in an atomized Stream of Moleten Metal Spray Produced by a Fuel Injector Laser Doppler Anemometry • LDA – A high resolution - single point technique for velocity measurements in turbulent flows A Back Scatter LDA System for One Velocity Component Measurement (Dantec Dynamics) – Basics • • • • Seed flow with small tracer particles Illuminate flow with one or more coherent, polarized laser beams to form a MV Receive scattered light from particles passing through MV and interfere with additional light sources Measurement of the resultant light intensity frequency is related to particle velocity LDA in a nutshell • Benefits – – – – – – Essentially non-intrusive Hostile environments Very accurate No calibration High data rates Good spatial & temporal resolution • Limitations – – – – Expensive equipment Flow must be seeded with particles if none naturally exist Single point measurement technique Can be difficult to collect data very near walls Review of Wave Characteristics A x • General wave propagation x t x,t ReAe i kxt x, t A cos2 A k x t = Amplitude = wavenumber = spatial coordinate = time = angular frequency = phase k 2 c 2 2c Electromagnetic waves: coherence • Light is emitted in “wavetrains” – Short duration, Dt – Corresponding phase shift, (t); where may vary on scale t>Dt E Eo exp ikx t t • Light is coherent when the phase remains constant for a sufficiently long time – Typical duration (Dtc) and equivalent propagation length (Dlc) over which some sources remain coherent are: Source White light Mercury Arc Kr86 discharge lamp Stabilized He-Ne laser nom (nm) 550 546 606 633 Dlc 8 mm 0.3 mm 0.3 m ≤ 400 m – Interferometry is only practical with coherent light sources Electromagnetic waves: irradiance • Instantaneous power density given by Poynting vector – Units of Energy/(Area-Time) S c 2oE B S co E 2 • More useful: average over times longer than light freq. Frequency Range t T 2 1 6.10 x 1014 f T f t dt T t T 2 5.20 x 1014 3.80 x 1014 I S T co E 2 T co co 2 E E E0 2 2 LDA: Doppler effect frequency shift • Overall Doppler shift due two separate changes – The particle ‘sees’ a shift in incident light frequency due to particle motion – Scattered light from particle to stationary detector is shifted due to particle motion u eˆ b eˆ s k k LDA: Doppler shift, effect I • Frequency Observed by Particle – The first shift can itself be split into two effects • (a) the number of wavefronts the particle passes in a time Dt, as though the waves were stationary… u eˆ b u eb Δt Number of wavefronts particle passes during Dt due to particle velocity: k k u eˆ b Δt LDA: Doppler shift, effect I • Frequency Observed by Particle – The first shift can itself be split into two effects • (b) the number of wavefronts passing a stationary particle position over the same duration, Dt… eˆ b cD t Number of wavefronts that pass a stationary particle during Dt due to the wavefront velocity: cDt k k LDA: Doppler shift, effect I • The net effect due to a moving observer w/ a stationary source is then the difference: Number of wavefronts that pass a moving particle during Dt due to combined velocity (same as using relative velocity in particle frame): Net frequency observed by moving particle cDt u eˆ b Δt # of wavefronts Dt c u eˆ b 1 c u eˆ b f 0 1 c fp LDA: Doppler shift, effect II • An additional shift happens when the light gets scattered by the particle and is observed by the detector – This is the case of a moving source and stationary u detector (classic train whistle problem) u e s Dt u es receiver lens cD t Distance a scattered wave front would travel during Dt in the direction of detector, if u were 0: Due to source motion, the distance is changed by an amount: cD t u eˆ s Δt Therefore, the effective scattered wavelength is: net distance traveledby wave s number of waves emitted cDt u eˆ s Δt c u eˆ s f p Δt fp LDA: Doppler shift, I & II combined • Combining the two effects gives: f obs u eˆ b 1 cf p fp c c f0 ˆ ˆ u e s c u e s u eˆ s s 1 1 c c • For u << c, we can approximate f obs u eˆ b u eˆ s f 0 1 1 c c 1 2 u eˆ b u eˆ s u eˆ s f 0 1 1 c c c 1 f 0 1 u eˆ s eˆ b c f f 0 0 u eˆ s eˆ b c LDA: problem with single source/detector • Single beam frequency shift depends on: – velocity magnitude – Velocity direction – observation angle f obs f0 f 0 u eˆ s eˆ b c • Additionally, base frequency is quite high… – O[1014] Hz, making direct detection quite difficult • Solution? – Optical heterodyne • Use interference of two beams or two detectors to create a “beating” effect, like two slightly out of tune guitar strings, e.g. 1 cos1t cos2t cos1 2 t cos1 2 t 2 – Need to repeat for optical waves E1 Eo1 cosk1 r 1t E2 Eo 2 cosk 2 r 2t P Optical Heterodyne • Repeat, but allow for different frequencies… I co E1 E2 E1 E2 2 E1 E01 expik1x 1t 1 E01 expi1 E2 E02 expik2 x 2t 2 E02 expi2 co 2 2 E o1 E o2 E 01 exp i1E02 expi2 E01 expi1E02 expi2 2 expi1 2 expi1 2 co 2 2 E o1 E o2 2E 01 E 02 I 2 2 I I co 2 2 E o1 E o2 2E o1E o2 cosk1 k2 r 1 2 t 1 2 2 1 Io1 Io2 2 Io1Io2 cosk1 k 2 r 1 2 t 1 2 2 I co 2 2 E o1 E o2 2E 01 E 02 cos1 2 2 I PED I AC How do you get different scatter frequencies? • For a single beam f0 f s f 0 u eˆ s eˆ b c – Frequency depends on directions of es and eb • Three common methods have been used – Reference beam mode (single scatter and single beam) – Single-beam, dual scatter (two observation angles) – Dual beam (two incident beams, single observation location) Dual beam method Real MV formed by two beams Beam crossing angle g Scattering angle q ‘Forward’ Scatter Configuration Dual beam method (cont) f0 u eˆ s ,1 eˆ b ,1 c f f s , 2 f 0 0 u eˆ s , 2 eˆ b , 2 c f Df 0 u eˆ b , 2 eˆ b ,1 c f s ,1 f 0 I t Note that so: (eˆ b,1 eˆ b,2 ) 2sin(g / 2) xˆ g u xg Df D 2 sin g 2 Measure the component of u in the xˆ g direction 1 4 sin g 2 I I 2 I I cos k k r u x t o1 o 2 o1 o 2 2 g 1 2 1 2 Fringe Interference description • Interference “fringes” seen as standing waves – Particles passing through fringes scatter light in regions of constructive interference g 2 sin 2 u xs Df – Adequate explanation for particles smaller than individual fringes Gaussian beam effects A single laser beam profile -Power distribution in MV will be Gaussian shaped -In the MV, true plane waves occur only at the focal point -Even for a perfect particle trajectory the strength of the Doppler ‘burst’ will vary with position Figures from Albrecht et. al., 2003 Non-uniform beam effects Particle Trajectory Centered Off Center DC AC DC+AC - Off-center trajectory results in weakened signal visibility -Pedestal (DC part of signal) is removed by a high pass filter after photomultiplier Figures from Albrecht et. al., 2003 Multi-component dual beam ^ xg ^ xb Three independent directions Two – Component Probe Looking Toward the Transmitter Sign ambiguity… • Change in sign of velocity has no effect on frequency I 2I o 2I o cosk1 k 2 r 2Df Dt 1 2 u xs Df D 2 sin g 2 Xg uxg> 0 beam 2 beam 1 uxg< 0 Velocity Ambiguity • Equal frequency beams – No difference with velocity direction… cannot detect reversed flow • Solution: Introduce a frequency shift into 1 of the two beams Bragg Cell fb2 = fbragg + fb Xg fb = 5.8 e14 fb1 = fb fb u (eˆ s ,1 eˆ b ,1 ) c f f s ,2 ( fb fbragg ) b u (eˆ s ,2 eˆ b,2 ) c f Df D fbragg b u (eˆ b,1 eˆ b,2 ) fbragg Df D 0 c f s ,1 fb Hypothetical shift Without Bragg Cell New Signal 2 I 2E01 cos(2{Df D0 fbragg }t ) If DfD < fbragg then u < 0 Frequency shift: Fringe description • Different frequency causes an apparent velocity in fringes – Effect result of interference of two traveling waves as slightly different frequency DfD s-1 Directional ambiguity (cont) fbragg uxg (m/s) uxg (Df D 0 fbragg ) 2sin(g / 2) 514nm, fbragg = 40 MHz and g = 20° Upper limit on positive velocity limited only by time response of detector Velocity bias sampling effects • LDA samples the flow based on – Rate at which particles pass through the detection volume – Inherently a flux-weighted measurement – Simple number weighted means are biased for unsteady flows and need to be corrected • Consider: – Uniform seeding density (# particles/volume) – Flow moves at steady speed of 5 units/sec for 4 seconds (giving 20 samples) would measure: 5 * 20 5 20 – Flow that moves at 8 units/sec for 2 sec (giving 16 samples), then 2 units/sec for 2 second (giving 4 samples) would give 16 * 8 4 * 2 6 .8 20 Laser Doppler Anemometry Velocity Measurement Bias N Ux U i 1 x ,i i U N U n x N i 1 Mean Velocity i 1 x ,i n U x i N i i 1 i nth moment Bias Compensation Formulas - The sampling rate of a volume of fluid containing particles increases with the velocity of that volume - Introduces a bias towards sampling higher velocity particles Phase Doppler Anemometry The overall phase difference is proportional to particle diameter D 2ni D b q,,g,n p ,ni Multiple Detector Implementation The geometric factor, b - Has closed form solution for p = 0 and 1 only - Absolute value increases with elevation angle relative to 0°) - Is independent of np for reflection Figures from Dantec