Download 2 Theory (2 pages)

1 Introduction
Laser Doppler Anemometry is a remote sensing method that can be used to measure
velocities in fluids. Apart from the obvious advantages of remote sensing methods
(for instance application to corrosive fluids) a second advantage of LDA is the high
spatial resolution.
2 Theory (2 pages)
2.1 Laser Doppler Anemometry (1 page)
The theoretical background of LDA can be based on either relativistic or classical
arguments. From a relativistic point of view one uses the frequency shift that light
waves undergo when scattered by moving particles, i.e. the Doppler shift. Assume a
particle is moving along with the flow in a fluid with velocity U ( see fig….). If the
particle moves through a beam 1 (with positive y) of monochromatic light with
frequency 0, it receives the light at a shifted frequency given by:
 e U
(2.1)
 O1   0 1  L1 
c 

where c is the speed of light and eL1 is a unit vector in the direction of the light
beam. The light emitted by a moving particle or the scattered light received by an
observer at rest will undergo an additional frequency shift given by:
1
1
eS  U 
 e  U  e S  U 
(2.2)
   0 1  L1

1 
c 
c 
c 


This equation relates the particle velocity to a measurable frequency shift S1. It
does not however allow to distinguish between particles traversing the beam at
different locations, so spatial resolution will be poor.
If one takes the first beam to be coherent and monochromatic and combines this
beam with a second equivalent beam, this draw back can be eliminated. To this end
one creates an intersection of the two beams. The scattered light originating from
this intersection, also called the measurement volume, will form a linear
combination of the scattered light resulting from beam 1 and that of beam 2. The
addition of waves of equal intensity and differing frequency yields a light wave
with an amplitude modulated with frequency (1-2)/2 and an irradiance modulated
with a frequency (1-2), the so-called beat frequency. One finds for the scattered
light that it is frequency is given by:
1
Uy
U  eS  U 
(2.3)
 d   S 2   S1   0 e L1  e L 2   1 
  2 sin 
c
c 
0
Here it is assumed that U  es so that:
1
 eS  U 
1 
 1
c 

This establishes the linear relationship between the so-called Doppler frequency d
and the particle velocity component Uy perpendicular to the optical axis of the
system. The relation is not sensitive for the direction of flow however. For the
situation where this information is necessary one may apply two beams of differing
frequency. Assuming that the frequency of the second beam is S2 = 0 + 2, this
results in;
1
Uy
U

 e S  U 
 d   S 2   S1   0 e L1  e L 2     2 1 
 2
  2 sin 
c
c 
0


This variation can also be applied for the detection of more velocity components.

 S1   O1 1 
From a classical point of view the two beams will interfere at the intersection or
within the measurement volume, resulting in a series of light and dark surfaces or
fringes, parallel to the optical axis of the system. A particle moving through these
surfaces will scatter light with a frequency equal to that of the illumination, i.e.:
Uy
(2.4)
d 
y
Where y is the spatial period of the intensity in the direction of flow. This spatial
period is the spatial period of the intensity of the combined beams, i.e. of the beat
signal. Since the spatial pattern is independent of time this yields:
cos((k 1  k 2 )  r )  cos(( k 1x  k 2 x ) x  (k 1 y  k 2 y ) y )  cos( 2k 0 y sin  )
(2.5)
The spatial period of this cosine-term then is given by;
0
2 (n  (n  1))
(2.6)
y  y n  y n1 

2k 0 sin 
2 sin 
Combining eq. (2.4) and (2.6) this finally yields the same result as eq. (2.3).
2.2 Liquid flow through a pipe (1 page)
The characteristics of flows are studied in the field of fluid dynamics. The most
general description of flows is known as the Navier-Stokes equation:
v
   v   v  v  v     τ  p  g
(2.1)
t
The equation may basically be considered as a balance of forces over some cubic
volume element dV = dxdydz. In this respect the Navier-Stokes equation can be
interpreted as a formulation of the second law of Newton, with on the left hand side
the change of impulse in the volume dV due to the forces exerted on this volume,
summed on the right hand side. The first two terms on the right quantify the inertia
forces. The third term stands for the viscous forces that may be interpreted as the
frictional forces acting on and within the volume. The forth term and the fifth term
represent the pressure forces and the gravity force respectively.
In a first approximation the Navier-Stokes equation can be simplified through the
specification of the fluid that is dealt with. Assuming incompressibility, one may
take the mass balance into consideration;

   v
(2.2)
t
Furthermore if the equation is applied to a Newtonian fluid, the viscous forces are
linearly related to the spatial derivative of the velocity through the dynamic
viscosity. For other classes of fluids different and generally more complicated
relationships between viscous forces and the spatial derivative of the velocity have
been established. For the case at hand however the resulting equation is;
v

   v   v   2 v  p  g
(2.3)
t
For relatively slow moving fluids one may assume a laminar, or layered, flow
pattern. In this model it is assumed that the flow pattern consists of a number of
layers parallel to the velocity, that differ in velocity. The velocity is virtually zero at
some wall and increases with distance from this wall. Assuming a stationary
laminar flow in a cylindrical pipe, eq. (2.3) can be solved analytically.
A stationary flow is assumed so that the time dependence on the left hand side of
the equation is zero. Furthermore the acceleration g is a vector perpendicular to the
direction of flow. It can therefore only have an influence on velocities at different
vertical positions. The measurements will however only be performed at the height
of the axis of the pipe so that the last term can be left out of the equation as well.
Apart from specification of the fluid, specification of the geometry helps.
Finally the gradient operator and the Laplace operator for cylindrical coordinates
can be applied, where the angular dependence can be left out for reasons of axial
symmetry. Since the velocity is independent of the axial variable y, its derivative
with respect to y is zero. The same goes for the pressure with respect to the radius.
Furthermore the velocity component vr in the direction of the radius is zero so that
the final result is;
 1   v y    p
 r
 e y   e y
(2.4)
0   

r

r

r

y

  

Integrating eq. (2.4) with boundary conditions r = 0 ; vy/r = 0 and r = R ; v = 0
this yields;

R 2  p 
r2 
r2 
  1  2   U y (0)1  2 
(2.5)
U y (r ) 
4  y  R 
 R 
This flow pattern is also known as Poiseuille flow and shows that the velocity
distribution for a laminar flow in a horizontal cylindrical pipe is parabolic. The
volumetric throughput then is found trough integration over the cross section of the
flow pipe (Hagen-Poiseuille law):
R 4  p 
R 2
    U y (0)
(2.6)
V 
8  y 
2
The former derivation is applicable as long as the laminar model can be assumed.
For higher velocities however a new regime is entered, known as turbulence.
Characteristic for turbulence is that viscous forces are small compared to the inertia
forces. The inertia forces may in this context be taken to be a measure of the kinetic
energy contained in a flow. The viscous forces on the other hand, are responsible
for the distribution of the kinetic energy in the flow. For a laminar flow the kinetic
energy is passed from layer to layer through the viscous forces. If the viscous forces
are too low compared to the kinetic energy present, another mechanism arises. This
mechanism distributes the energy more efficiently. A spatially and temporally
unstable pattern of eddies arises where large eddies transfer their kinetic energy to
smaller ones and these again to even smaller ones. This yields an ever changing
flow pattern that spreads the kinetic energy in a very effective manner. The
direction and magnitude of the velocity vector then are no longer unambiguous and
therefore eq. (2.3) can no longer be solved analytically.
To describe the regime of the flow use is made of the Reynolds number Re defined
as the dimensionless ratio of the inertia forces and the viscous forces;
U 2 Ul
(2.7)
Re 

U l

where l is some length characteristic for the dimensions of the flow, which in this
case is the diameter of the flow pipe.
Roughly speaking, for a straight cylindrical flow pipe the flow is laminar for Re <
2000. If 2000 < Re < 2500 the regime may change to the turbulent. A range is given
because other phenomena then the viscous and inertia forces may be of influence.
Here one may think for instance of oscillations introduced by the pump since
influences like these promote the instability of the laminar flow.
2.3 Other relevant theory ( 0,5 page)
For the determination of the position of the measurement volume in the flow pipe
the propagation of the incident laser beams is to be corrected for the refraction of
the beams. In fig. … the situation is sketched for a horizontally positioned flow pipe
with incoming laser beams in the plane of the sketch. With Snellius’ law one may
therefore rewrite eq. … as:
Uy 
 fluid d air d

2 sin 
2 sin 
(2.8)
Furthermore one may extend the trajectory of the beams at either side of the flow
pipe as if there was no fluid. The two intersections then form the virtual
measurement volumes for an observer at the position of lens 1 and that of lens 2
respectively. These two positions are fixed with respect to the lenses, irrespective of
the location of the flow pipe. This will turn out to come in handy in the design of
the system.
3 System design and measurement procedure (2 – 3
pages)
3.1 System Design
Many varieties of LDA systems are possible. A common and general set up will be
discussed here that is also used in the physics lab at the VU (see fig.1). This LDA
system is a single component LDA system with forward-scattered detection. In
other words the system allows for the measurement of one velocity component at a
time, by measurement of the forward scattering. Some remarks on other options
will be made in the following discussion as well.
In the most general sense the system is composed of three compartments. The first
consists of the illuminating compartment that is used to create a measurement
volume. The second part of the system consist of the optical detection apparatus
with the objective of measuring the scattered light and the last part is the signal
processing compartment that is used for the processing of the output of the
photodiode.
3.1.1 Illuminating System
The first optical element in this system is the light source. As discussed in the
theory, a coherent and monochromatic beam is required, so that a laser is used. The
beam is split with a beam splitter. The resulting beams should be of equivalent or
nearly equivalent intensity. This yields the highest degree of contrast between the
fringes in the measurement volume and thus the highest amplitude for the signal so
that the signal-to-noise ratio is maximized. If the amplitudes of the two beams differ
less than or up to some 5% the contrast is virtually maximal so that one can allow
for deviations of that order. An OD-filter may be used to correct for any larger
differences. Furthermore a Bragg-cell may be implemented after the beam splitter
to create a frequency difference as suggested in the theory.
After the beam splitter, and possibly the other optical elements mentioned, the two
beams should be aligned in parallel, so that a lens 1 is can be used to let the beams
intersect. The beams are to pass the lens equidistant from the optical axis of the lens
so that the angle  is equal for both of the beams. Hereby the angle  is determined
by the distance between the beams and the focal length of lens 1.
An appropriate choice for this angle is not unambiguous. In the first place the
choice for some angle  will have consequences for the intensity of the signal.
Since the intensity for forward scattering is generally high enough for a good
signal-to-noise ratio, these considerations will not be discussed here. The choice for
a specific angle  however also has consequences for the size and formation of the
measurement volume.
In the first place a smaller angle  results in a larger size of the measurement
volume in the direction of the optical axis. Since the velocity gradient in this
direction is non zero this will result in a certain distribution in the velocities
measured and thus in a broadening of the Doppler frequency. A smaller angle 
may therefore yield a larger uncertainty in the Doppler frequency, so a large angle
may be preferable. It should be noted however that the size of the measurement
volume in the direction of the optical axis also depends on the width of the beams at
the point of intersection. Due to the focusing of the beams very small waists may be
feasible but spherical aberration (see 6) may result in a displacement of the waists
and therefore a larger beam width at the point of intersection. To minimize
spherical aberration the beams should not be to far apart compared to the focal
length. This implies that the angle  should not be too large, such that the paraxial
region of lens 1 is approached.
Furthermore a the wavefronts of the light waves emerging from lens 1 are spherical
due to the focusing, except for the region around the waists where they approximate
planar wavefronts yielding a nice and regular fringe pattern. Again a problem may
arise due to spherical aberration and the possible resulting displacement of the
waists. It then no longer can be assumed that the wavefronts are planar at the point
of intersection. The result would be that the fringes fan out so that a gradient in the
fringe pattern or a fringe gradient arises (see par. 6). Again a relatively small angle
 should be chosen, such that the paraxial region of lens 1 is approached.
These considerations imply that one should look for a large angle  without causing
too much spherical aberration.
Apart from choosing a small angle , the spherical aberration is minimized by
choosing the ‘flat’ side of lens 1 to face the measurement volume.
3.1.2 Optical Detection System
The aim of the optical detection system is to measure the scattered light waves
originating from the measurement volume. The light waves that impinge on the
particles are scattered through Mie scattering. This type of scattering occurs when
the size of the scattering particles is comparable to or larger then the wavelength of
the illuminating light waves. The detection system can be set up to measure the
scattered light in any direction. Generally the intensity of the scattered light is the
highest in the forward direction, so that often forward scattering is detected as in the
design discussed here. If backward scattering is measured however, high power
lasers are needed since the intensity in this direction is relatively low. The
advantage of this set up is that the illuminating system and the detection system can
be integrated.
For forward-scattered detection a simple arrangement of two converging lenses may
be used to increase the light gathering power of the system. Furthermore the signal
is detected using a photodiode. A diaphragm may be positioned between the
measurement volume and the second lens so that mainly the scattered light
originating from the measurement volume enters the system.
On may choose for a set up as suggested in fig. … Here the second virtual
measurement volume is located at the focal length of lens 2. This yields an image
located at infinity that on its turn is projected on the focal plane of lens 3. This
configuration has the advantage that it is relatively easy to align.
In the selection of lenses for this system, the first point to note is that the étendue,
also called speed or light-gathering capability, of the photodiode should be fully
used. To this end the ratio of the diameter effective aperture of lens 3 end its focal
length should be equal to this quantity. To be more exact, the ratio between the
diameter of the exit pupil of the optical detection system and the distance between
this exit pupil and the entrance of the photodiode should be equal to the étendue of
the photodiode.
Finally it may be convenient to choose f2 to be larger then f3 in order to minimize
both transverse and longitudinal magnification, so that the system is less sensitive to
inaccuracies in the position of the virtual measurement volume.
3.1.3 Signal Processing System
The output signal of the photodiode may be processed using an amplifier, an A/Dconverter and a digital processor. It is useful to apply an amplifier that allows for
correction of bias in the signal since intensity of the scattered light may vary
notably for different circumstances. Furthermore the digital processor should have a
high sampling rate and an option to perform Fourier transforms so that spectral
measurements can be made.
3.1.4 Flow circuit
In general the object of measurement of the LDA system is some flow. In a first
approach however it is useful to measure a flow with controlled flow parameters.
This offers the opportunity to calibrate the system.
To this end a relatively simple option is to apply the LDA measurements to water,
or any other fluid, in some flow circuit. A glass pipe may offer a measurement
window and if a valve and flowmeter are implemented in addition to a stable pump,
most relevant parameters are controlled. In addition it may also be useful to have a
cooling circuit (a simple cooling circuit connected to a running tap will do) and a
thermometer at one’s disposal, since the dynamic viscosity  can be highly
temperature dependent.
The fluid should contain particles that have sizes comparable to the spatial period of
the fringe pattern as given by eq. (2.6). In plain tap water particles with sizes in the
order of m are present. When using a fluid that does not contain such particles
seeding may be needed. In that case one may add some flower, milk or Detol.
4 Optimization and Measurement Procedure
4.1 Optimization (0,5 page)
Optimization of the illuminating system comes down to optimization of the
amplitude, volume and the regularity of the fringe pattern. Various points of interest
for signal optimization have been discussed in the paragraph on system design
already:
- For an optimal signal-to-noise-ratio the intensity of the two beams should be
as close as possible or at any rate less than some 5 %.
- For a small measurement volume the angle  should not be to small, but
preferable such that the two beams fall within the paraxial region.
- For a regular fringe pattern the beams should fall within the paraxial region
of lens 1, resulting in a relatively small angle .
Apart from these constraints, the alignment of the two beams should be optimized.
For the optical detection system it comes down to maximizing the amount of
scattered light that falls into the photodiode. Again some conditions for
optimization of the optical detection system have been discussed in the paragraph
on system design;
-
For maximization of the intensity of the signal the étendue of the photodiode
should be matched.
- For ease of alignment it is useful to configure the two lenses in such a way
that the focal length of second lens equals the distance between the second
virtual measurement volume and the second lens. Furthermore the focal
length of the third lens should equal the distance between the photodiode
and the third lens.
One may add that the first is not absolutely true. If the signal is to strong a smaller
diaphragm diameter may be needed, since the photodiode may be overloaded.
The optimization of the signal data processing system is primarily related to the
optimization of the signal-to-noise-ratio. This can be done by adjusting the bias and
amplification, on the basis of the signal in the time domain. In this dimension the
signal should clearly and directly show ‘bursts’ of light with the Doppler frequency
as discussed in the paragraph on data analysis.
4.2 Procedure (0,5 page)
The procedure to perform a calibration measurement can be stated as follows:
- Optimize the general parameters of instrument as described above
- Optimize the instrument for the specific measurement to be carried out
- Make an estimate of all the relevant variables and their inaccuracies
- Perform a series of spectral measurements for various flow velocities
- Estimate the center frequencies and the standard deviation of each of the
spectral peaks
- Calculate the measured flow velocities/volumetric throughput
- Calculate the uncertainties
- Plot the volumetric throughput based on LDA measurement versus that
based on the flow meter data, with their respective uncertainties
5 Signal processing and data analysis (0,5 page)
The signal is composed of low frequency background noise, the Doppler frequency
and high frequency components in the order of the frequency of the laser beam.
The low frequency noise is the result of complex interferences that are typical for
Mie scattering. Light scattered by different parts of the illuminated particle
interferes and since the particle has some roughness it will lead to other frequencies
than the Doppler frequency as well. The intensity of this noise decreases with
increasing frequency.
The Doppler signal has been discussed shortly in the theory. The addition of waves
of equal intensity and differing frequency yields a light wave with an irradiance that
is modulated beat frequency. There is however not a continuously a scattering
particle present. The Doppler signal increases in intensity when a particle enters the
measurement volume up to the center and decreases again upon leaving the
measurement volume. This is the result of the Gaussian distribution of the intensity
of the laser beams, that reflects on the intensity distribution of the measurement
volume. For a single particle this results in an additional modulation with a
‘frequency’ that corresponds to
p 
c

Uy
 p 2y mv
Where ymv is the width of the measurement volume in the direction of he flow.
This signal is called the pedestal signal. The result is that a particle traversing the
measurement volume manifests itself through a ‘signal burst’, a short intense burst.
As more particles traverse the measurement volume, every now and than such
signal bursts occur and these contribute to the Doppler signal.
This effect can also be interpreted as a –random – modulation of the Doppler signal
with a frequency that depends on the
maximu result is that even this signal is modulated In other words the One finds
for the scattered
originating from the measurement volume due to complex interference of Mie
scattering. Since the
The light waves that impinge on the particles are scattered through Mie scattering.
This type of scattering occurs when the size of the scattering particles is comparable
to or larger then the wavelength of the illuminating light waves.
The signal consists of components with vd, vaverage, v1 or v2, and vpedestal.
As shortly described in par. 2, the signal processing after photodiode comes down
to spectral analysis in the range where the Doppler frequency is expected. The
spectrum consists of a intense peek around the Doppler frequency and a noise signal
that decreases exponentially with increasing frequency. The background noise is the
result low frequency influences of other light sources present.
It is useful to look at the signal in the time dimension as well since this allows for
the recognition of distortions like overexposure of the photodiode, leading to boxcar
terms that in a Fourier transformation yield high frequency terms. Furthermore it
allows for adjustment of the diaphragm that may be to tight (only high frequency
terms) or to wide (boxcar), and the bias and amplification (which may yield to small
amplification … low signal to noise ratio, or to large … boxcar)
The spectral measurement is to be averaged over a temporal period that is
substantially larger than the period of the Doppler frequency, since it is highly
dependent of the frequency of passing particles and the regularity of the fringe
pattern.
One may fit the spectra to obtain the center frequency of the peek and the standard
deviation. The next step is an error analysis, including the propagation errors that
result from the calculation of velocity U. This should include all relevant
parameters like the distance between the beams, before passing lens 1, the focal
length of lens 1, the position of the measurement volume etc.
The result is a set of velocities with there respective standard deviations that may be
compared, i.e. plotted versus the velocities that were measured with the flow meter.
6 Example (1 page)
6.1 Experimental setup
The experimental set up for this experiment differs little from the system discussed
on par. 2. The specifications of the instrumentation was not mentioned however so
this will be done here.













Laser: He-Ne laser with 632,816 nm wavelength
Lens 1: Diameter … and back focal length 17.8 cm
Flow pipe: External Diameter 3,4 cm Internal Diameter 3,04 cm
Lens 2: Diameter 3 and focal length 100 mm
Lens 3: Diameter 3 cm and focal length 80 mm
Photodiode
Signal Amplifier
Digital Storage Oscilloscope (DSO)
Digital Spectrum Analyzer (DSA)
Pump
Flow Meter
Beam Splitter
Mirrors
Expected range of Doppler frequencies
Check background noise by taking one beam out
6.2 Measured data
6.3 Error Analysis
7 Complications (1,5 pages)
In principle the error analysis allows for the estimation of uncertainties in the
relevant parameters. Here the relevant parameters follow from the theory described
in par. 2. In the derivation of the equations however same assumptions have been
made that can lead to complications that do not follow from the error analysis.
Mass/volume of particles (Brownian movement and is vparticle = vflow?)
Very influential is the assumption that first order optics can be applied. This point
was made already in the paragraph on system design.
In the discussion on the system design the optics was described based on first order
or paraxial theory. The first order approximation however may introduce serious
errors. Particularly the first lens can be subject to this problem to its diameter.
Spherical aberration is the phenomenon that the focal length depends on the point of
incidence of the beam. For a converging lens the focal length decreases with
increasing distance between the optical axis and the point of incidence. This is also
the case for the two sides of just one beam. This implies that the waist of the beam
may be slightly of the optical axis. As mentioned in the before this may have two
consequences for the measurement volume:
 the width of the beams at the point of intersection will be relatively large
 the wave fronts at intersection will have bending with limited radius
Especially the second point may lead to problems. The result may be that the fringe
pattern fans out and that a fringe gradient is introduced. This may contribute heavily
to the width of the spectral peek at the Doppler frequency.
To measure the fringe bias qualitatively one may use a wire on a rotating disc. This
results in a nice pedestal wave with the signal with the Doppler frequency
superposed on it. By passing the wire on different positions through the
measurement volume on may be able to see variations in the Doppler frequency
One could try to calibrate the system through measurement of this fringe gradient
with the wire and one could try to estimate the spherical aberration of each of the
beams resulting in a possibility to estimate the gradient.
Another problem that may arise is the sampling bias. If the period of the sampling is
larger than the period of the signal, and more specifically a
Sampling bias (0,5 page)
Sampling bias is introduced with the digitalization of the signal.
8 Suggestions for other experiments (0,5)
-
Profile measurements for laminar or turbulent flow
Velocity measurement of rotating disc with wire
Fringe gradient
Two methods (reference and pedestrian crossing)
9 Literature
References
-
Wellink
Bibliography
- Hecht
- Environmental physics
- Fysische Transport verschijnselen
Links
- nasa
-