Transcript Document

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
–
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Essentially non-intrusive
Hostile environments
Very accurate
No calibration
High data rates
Good spatial & temporal resolution
• Limitations
–
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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   ReAe i kxt   
 x, t   A cos2   
    
A
k
x
t


= Amplitude
= wavenumber
= spatial coordinate
= time
= angular frequency
= phase

k
2



c
 
2


2c

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 ikx  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 2oE  B
S  co 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
 co E
2
T
co
co 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
cos1t cos2t   cos1  2 t  cos1  2 t
2
– Need to repeat for optical waves
E1  Eo1 cosk1  r  1t 

E2  Eo 2 cosk 2  r  2t 
P
Optical Heterodyne
• Repeat, but allow for different frequencies…
I
co
E1  E2  E1  E2 
2
E1  E01 expik1x  1t  1 E01 expi1
E2  E02 expik2 x  2t  2  E02 expi2 
co 2
2
E o1  E o2
 E 01 exp
i1E02 expi2   E01 expi1E02 expi2 


2

expi1   2  expi1   2 

co  2
2
E o1  E o2  2E 01 E 02 
I



2 
2




I


I

co 2
2
E o1  E o2
 2E o1E o2 cosk1  k2  r  1  2 t  1  2 
2
1
 Io1  Io2  2 Io1Io2 cosk1  k 2  r  1  2 t  1  2 
2
I

co 2
2
E o1  E o2
 2E 01 E 02 cos1  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 cosk1  k 2  r  2Df 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)
514nm, 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 
2ni 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