1. Introduction to Laser Doppler Velocimetry
Ken Kiger
Burgers Program For Fluid Dynamics
Turbulence School
College Park, Maryland, May 24-27
2. 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
3. Phase Doppler Anemometry (PDA)
• Single point particle sizing/velocimetry method
Drop Size and Velocity
measurements in an atomized
Stream of Moleten Metal
Droplet Size Distributions
Measured in a Kerosene
Spray Produced by a Fuel Injector
4. Laser Doppler Anemometry
• LDA
– A high resolution - single point technique for velocity measurements in turbulent flows
– 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
A Back Scatter LDA System for One Velocity Component Measurement (Dantec Dynamics)
5. 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
6. • General wave propagation
Review of Wave Characteristics
tx
Atx 2cos,
x
A
x,t Re Aei kx t
A = Amplitude
k = wavenumber
x = spatial coordinate
t = time
= angular frequency
= phase
2
k
c
c
22
7. Electromagnetic waves: coherence
• Light is emitted in “wavetrains”
– Short duration, t
– Corresponding phase shift, (t); where may vary on scale t> t
• Light is coherent when the phase remains constant for
a sufficiently long time
– Typical duration ( tc) and equivalent propagation length ( lc) over
which some sources remain coherent are:
– Interferometry is only practical with coherent light sources
E Eo exp i kx t t
Source nom (nm) lc
White light 550 8 m
Mercury Arc 546 0.3 mm
Kr86 discharge lamp 606 0.3 m
Stabilized He-Ne laser 633 ≤ 400 m
8. Electromagnetic waves: irradiance
• Instantaneous power density given by Poynting vector
– Units of Energy/(Area-Time)
• More useful: average over times longer than light freq.
S c2
oE B S c oE2
I S T
c o E2
T
c o
2
E E
c o
2
E0
2
tdtf
T
f
T
T
t
t
T
2
2
1
Frequency Range
6.10 x 1014
5.20 x 1014
3.80 x 1014
9. 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
ˆb
k
e
k
u
ˆse
10. 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 t, as though the
waves were stationary…
ˆb
k
e
k
Δtbeu
Δtbeu ˆ
u
Number of wavefronts particle passes
during t due to particle velocity:
11. 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, t…
k
k
eb
ˆ
tc
tc
Number of wavefronts that pass a
stationary particle during t due to the
wavefront velocity:
12. LDA: Doppler shift, effect I
• The net effect due to a moving observer w/ a
stationary source is then the difference:
Δttc beu ˆ
Number of wavefronts that pass a
moving particle during t due to
combined velocity (same as using
relative velocity in particle frame):
Net frequency observed by
moving particle
c
f
c
c
t
f
b
b
p
eu
eu
ˆ
1
ˆ
1
swavefrontof#
0
13. 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 detector (classic train whistle
problem)
u
tseu
se
tc
receiver
lens
u
Distance a scattered wave front would travel
during t in the direction of detector, if u
were 0:
tc
p
s
p
s
s
f
c
Δtf
Δttc eueu ˆˆ
emittedwavesofnumber
by wavetraveleddistancenet
Due to source motion, the distance is
changed by an amount:
Therefore, the effective scattered
wavelength is:
Δtseu ˆ
14. LDA: Doppler shift, I & II combined
• Combining the two effects gives:
• For u << c, we can approximate
c
c
f
c
f
c
cfc
f
s
b
s
p
s
p
s
obs
eu
eu
eueu ˆ
1
ˆ
1
ˆ
1
ˆ 0
bs
bs
ssb
sb
obs
c
f
f
c
f
ccc
f
cc
ff
eeu
eeu
eueueu
eueu
ˆˆ
ˆˆ
1
1
ˆˆ
1
ˆ
1
ˆ
1
ˆ
1
0
0
0
2
0
1
0
15. LDA: problem with single source/detector
• Single beam frequency shift depends on:
– velocity magnitude
– Velocity direction
– observation angle
• 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.
– Need to repeat for optical waves
bsobs
c
f
ff eeu ˆˆ0
0
to 1111 cos rkEE
to 2222 cos rkEE
P
cos 1t cos 2t
1
2
cos 1 2 t cos 1 2 t
16. • Repeat, but allow for different frequencies…
Optical Heterodyne
I
c o
2
E1 E2 E1 E2
I
c o
2
Eo1
2
Eo2
2
2Eo1Eo2 cos k1 k2 r 1 2 t 1 2
1
2
Io1 Io2 2 Io1Io2 cos k1 k2 r 1 2 t 1 2
ACIPEDI
E1 E01 exp i k1x 1t 1 E01 exp i 1
E2 E02 exp i k2x 2t 2 E02 exp i 2
I
c o
2
Eo1
2
Eo2
2
2E01E02
exp i 1 2 exp i 1 2
2
I
c o
2
Eo1
2
Eo2
2
E01 exp i 1 E02 exp i 2 E01 exp i 1 E02 exp i 2
I
c o
2
Eo1
2
Eo2
2
2E01E02 cos 1 2
17. How do you get different scatter
frequencies?
• For a single beam
– 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)
bss
c
f
ff eeu ˆˆ0
0
18. Dual beam method
Real MV formed by two beams
Beam crossing angle
Scattering angle
„Forward‟ Scatter
Configuration
19. Dual beam method (cont)
Note that ,1 ,2
ˆ ˆ ˆ( ) 2sin( / 2)b b ge e x
so:
ˆgMeasure the component of in the directionu x1,2,
0
2,2,
0
02,
1,1,
0
01,
ˆˆ
ˆˆ
ˆˆ
bb
bss
bss
c
f
f
c
f
ff
c
f
ff
eeu
eeu
eeu
Dg f
2sin2
xu
21212121
2sin4
cos2
2
1
tIIIItI oooo gxurkk
20. Fringe Interference description
• Interference “fringes” seen as standing waves
– Particles passing through fringes scatter light in regions of
constructive interference
– Adequate explanation for particles smaller than individual fringes
f
sxu
2
sin2
21. Gaussian beam effects
-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
A single laser beam profile
Figures from Albrecht et. al., 2003
22. Non-uniform beam effects
Centered Off Center
Particle Trajectory
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
24. Sign ambiguity…
• Change in sign of velocity has no effect on frequency
Ds f
2sin2
xu
uxg> 0
uxg< 0
beam 2
beam 1
Xg
2121 2cos22 tfIII Doo rkk
25. 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
XgBragg Cell
fb = 5.8 e14
fb2 = fbragg + fb
fb1 = fb
,1 ,1 ,1
,2 ,2 ,2
,1 ,2 0
ˆ ˆ( )
ˆ ˆ( ) ( )
ˆ ˆ( )
b
s b s b
b
s b bragg s b
b
D bragg b b bragg D
f
f f
c
f
f f f
c
f
f f f f
c
u e e
u e e
u e e
Hypothetical shift
Without Bragg Cell
2
01 02 cos( 2 { } )D braggI E f f t
If fD < fbragg then u < 0
New Signal
26. Frequency shift: Fringe description
• Different frequency causes an apparent velocity in
fringes
– Effect result of interference of two traveling waves as slightly
different frequency
27. Directional ambiguity (cont)
nm, fbragg = 40 MHz and = 20°
Upper limit on positive velocity limited only by
time response of detector
0( )
2sin( / 2)
D bragg
xg
f f
u
fbragg
uxg (m/s)
fDs-1
28. 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:
– 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
8.6
20
2*48*16
5
20
20*5
29. Laser Doppler Anemometry
Velocity Measurement Bias
, ,
n1 1
x
1 1
U
N N n
xx i i x i i
i i
x N N
i i
i i
U U U
U
Mean Velocity nth moment
- 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
Bias Compensation Formulas
30. Phase Doppler Anemometry
The overall phase difference
is proportional to particle diameter
The geometric factor,
- 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
Multiple Detector
Implementation
Figures from Dantec
2 niD
, , ,np,ni
