Introduction to Laser Doppler Velocimetry

Ken Kiger

Burgers Program For Fluid DynamicsTurbulence School

College Park, Maryland, May 24-27

Laser Doppler Anemometry (LDA)• Single-point optical velocimetry method

Study of the flow between rotatingimpeller blades of a pump

3-D LDA Measurements ona 1:5 Mercedes-Benz

E-class model car in wind tunnel

Phase Doppler Anemometry (PDA)• Single point particle sizing/velocimetry method

Drop Size and Velocitymeasurements in an atomized

Stream of Moleten Metal

Droplet Size DistributionsMeasured in a Kerosene

Spray Produced by a Fuel Injector

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)

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

• General wave propagation

Review of Wave Characteristics

tx

Atx 2cos,

x

A

x,t Re Aei kx t A = Amplitudek = wavenumberx = spatial coordinatet = time = angular frequency = phase

2

k

c

c

22

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) lcWhite light 550 8 mMercury Arc 546 0.3 mmKr86 discharge lamp 606 0.3 mStabilized He-Ne laser 633 ≤ 400 m

Electromagnetic waves: irradiance• Instantaneous power density given by Poynting vector

– Units of Energy/(Area-Time)

• More useful: average over times longer than light freq.

Sc 2oEB

S coE2

I STco E 2

Tco2

EE co2

E02

tdtfT

fT

T

t

tT

2

2

1Frequency Range

6.10 x 1014

5.20 x 1014

3.80 x 1014

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

ek

u

ˆse

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

ek

Δtbeu

Δtbeu

u

Number of wavefronts particle passes during t due to particle velocity:

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

ke bˆ

tc

tcNumber of wavefronts that pass a

stationary particle during t due to the wavefront velocity:

LDA: Doppler shift, effect I

• The net effect due to a moving observer w/ a stationary source is then the difference:

Δttc beu

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

cf

c

ct

f

b

b

p

eu

eu

ˆ1

ˆ1

s wavefrontof #

0

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

tcreceiver

lens

u

Distance a scattered wave front would travel during t in the direction of detector, if u were 0:

tc

p

s

p

ss f

c

Δtf

Δttc eueu ˆˆ

emitted wavesofnumber

by wave traveleddistancenet

Due to source motion, the distance is changed by an amount:

Therefore, the effective scattered wavelength is:

Δtseu

LDA: Doppler shift, I & II combined• Combining the two effects gives:

• For u << c, we can approximate

c

cf

c

f

c

cfcf

s

b

s

p

s

p

sobs eu

eu

eueu ˆ1

ˆ1

ˆ1

ˆ 0

bs

bs

ssb

sbobs

c

ff

cf

cccf

ccff

eeu

eeu

eueueu

eueu

ˆˆ

ˆˆ1

1

ˆˆ1

ˆ1

ˆ1

ˆ1

00

0

2

0

1

0

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

fff eeu ˆˆ0

0

to 1111 cos rkEE

to 2222 cos rkEEP

cos 1t cos 2t 1

2cos 1 2 t cos 1 2 t

• Repeat, but allow for different frequencies…

Optical Heterodyne

I co2

E1 E2 E1 E2

I co2

Eo12 Eo2

2 2Eo1Eo2 cos k1 k2 r 1 2 t 1 2

1

2Io1 Io2 2 Io1Io2 cos k1 k2 r 1 2 t 1 2

ACIPEDI

E1 E01 exp i k1x 1t 1 E01 exp i1

E2 E02 exp i k2x 2t 2 E02 exp i2

I co2

Eo12 Eo2

2 2E01E02

exp i 1 2 exp i 1 2 2

I co2

Eo12 Eo2

2 E01 exp i1 E02 exp i2 E01 exp i1 E02 exp i2

I co2

Eo12 Eo2

2 2E01E02 cos 1 2

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

fff eeu ˆˆ0

0

Dual beam method

Real MV formed by two beamsBeam crossing angle Scattering angle

‘Forward’ Scatter Configuration

Dual beam method (cont)

Note that ,1 ,2ˆ ˆ ˆ( ) 2sin( / 2) b b g e e x

so:

ˆ gMeasure the component of in the directionu x

1,2,0

2,2,0

02,

1,1,0

01,

ˆˆ

ˆˆ

ˆˆ

bb

bss

bss

c

ff

c

fff

c

fff

eeu

eeu

eeu

Dg f2sin2

xu

21212121

2sin4cos2

2

1

tIIIItI oooo gxurkk

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

fs

xu

2sin2

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

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

Multi-component dual beam

Three independent directions

xg

xb

Two – Component Probe LookingToward the Transmitter

Sign ambiguity…

• Change in sign of velocity has no effect on frequency

Ds f2sin2

xu

uxg> 0

uxg< 0

beam 2

beam 1

Xg

2121 2cos22 tfIII Doo rkk

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

beam 2

beam 1XgBragg 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( )

bs b s b

bs b bragg s b

bD bragg b b bragg D

ff f

cf

f f fc

ff f f f

c

u e e

u e e

u e e

Hypothetical shift Without Bragg Cell

201 02 cos( 2 { } )D braggI E f f t

If fD < fbragg then u < 0

New Signal

Frequency shift: Fringe description

• Different frequency causes an apparent velocity in fringes– Effect result of interference of two traveling waves as

slightly different frequency

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 fu

fbragg

uxg (m/s)

fD

s-1

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

8.620

2*48*16

520

20*5

Laser Doppler AnemometryVelocity Measurement Bias

, ,n1 1x

1 1

U

N N n

xx i i x i ii ix N N

i ii i

U U UU

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

Phase Doppler Anemometry

The overall phase differenceis 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 DetectorImplementation

Figures from Dantec

2niD

,,,np,ni