The power spectrum of phototube current from a laser Doppler velocimeter operating in the heterodyne mode has been computed. The spectrum is obtained in terms of the space time correlation function of the fluid. The spectral width and shape predicted by the theory are in agreement with experiment. For normal operating parameters the time average spectrum contains information only for times shorter than the Lagrangian integral time scale of the turbulence. To examine the long time behavior, one must use either extremely small scattering angles, much longer wavelength radiation or a different mode of signal analysis, e.g., FM detection

Angus, J. C.

Dunning, J. W., Jr.

Edwards, R. V.

English

NASA Technical Reports Server

N7 3- 16 67 5
NASA TECHNICAL NASA TM X- 68184
MEMO RAN DU M
CO
r—I
00
vO
I
X
<
CO
<
TURBULENT TRANSPORT MEASUREMENTS WITH A
LASER DOPPLER VELOCIMETER
by Robert V. Edwards and John C. Angus
Case Western Reserve University
Chemical Engineering Division
Cleveland, Ohio 44106
and John W. Dunning, Jr.
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
TECHNICAL PAPER presented at
First Pacific Chemical Engineering Meeting sponsored
by the Japanese Society of Chemical Engineers
Kyoto, Japan, October 10-14, 1972-
https://ntrs.nasa.gov/search.jsp?R=19730007948 2020-03-23T07:37:48+00:00Z
TURBULENT TRANSPORT MEASUREMENTS WITH A
LASER DOPPLER VELOCIMETER
Robert V. Edwards , JohnC. Angus ,
and John W. Dunning, Jr.
Lewis Research Center
ABSTRACT
The power spectrum of phototube current from a Laser Doppler
Velocimeter operating in the heterodyne mode has been computed.
The spectrum is obtained in term's of the space-time correlation func-
tion, G(Ar, r), of the fluid. The spectral width and shape predicted by
o> the theory are in agreement with experiment. For normal operating
c^ parameters the time-aver age spectrum contains information only for
W times shorter than the Lagrangian-integral time scale of the turbu-
lence. To examine the long-time behavior, one must use either ex-
tremely small scattering angles, much-longer-wavelength radiation
or a different mode of signal analysis, e.g., F. M. detection.
1. Introduction
In a Laser Doppler Velocimeter (LDV) laser light is scattered
from particles suspended in the fluid. The power spectrum of the
scattered light contains information on the motion of the particles
and hence information on molecular and turbulent transport processes.
A schematic view of the measurement region is shown in figure 1.
The light received at the detector is a linear sum of electric
fields scattered from each particle in the illuminated region. The sig-
nal at the detector can be written
Case Western Reserve University, Chemical Engineering Division,
Cleveland, Ohio, 44106.
f(t) = A cos [coQt + 0(t)J (1)
where A is an amplitude, o> is the laser frequency, and <£(t) is the re-
sultant phase.
0(t) is the phase that results from properly adding the phase received
from each of the particles. This phase is determined by the positions of
the particles and the scattering angle, 0. For the situation shown in fig-
ure 1, the phase of light scattered from the i particle is
. +c (2)
1
where X. is the X coordinate of the i particle, A the laser wave-
length and a is a constant determined by the optics: 01 = 2 sin (0/2).
The resultant phase, d>, and amplitude, E are obtained by summing thes
contributions, E. exp (i<fc) from the individual illuminated particles. This
is most easily seen graphically by a plot in the complex plane, figure 2 „
If the relative particle positions are fixed and if the suspending fluid
is convecting at some constant velocity v in the X direction, the signal
f (t) is
f(t) = A(t) cos u> + ±H v t + C (3)
The term (2ira/\) v is the frequency change usually interpreted as the
Doppler shift. It is indeed proportional to the velocity and if this were
all that happened, life would be simple indeed! However, as we men-
tioned, only the particles in the illuminated region contribute to the sig-
nal. If the particles in the illuminated region are removed and another
set substituted by convection, the phase of the new signal is different,
since the new particles do not occupy the same positions as the old set.
Further, since the suspended particles are randomly placed in any real
system, the new phase cannot be predicted solely with knowledge of the
old phase; i. e., the two phase terms are uncorrelated. Therefore, in
a real system with a finite illuminated volume, the signal (eq. (3)) only
stays correlated with itself at time t + T as long as some particles present
in it at t are still present at time t + T. If the particles maintain their rel-
ative positions, this time is on the order of a/v where a is the character-
istic dimension of the illuminated region in the flow direction.
The power density spectrum of a random signal is the Fourier transform
of the autocorrelation of the signal. In this case, the autocorrelation is zero
for all times greater than a/v so that the resulting spectrum has a width on
the order v/a. To a first (but good) approximation this spectrum is cen-
tered at 2ircxv/\, so that a measurement of the spectrum can yield v and a.
Until this point, the particles have been assumed not to change relative
position and that the flow was steady. If we now allow the particles to move
relative to each other (diffuse), another phenomenon appears. In undergoing
molecular or turbulent motion, the particles lose correlation with them-
selves, which means that the signal loses correlation with itself. This con-
tributes to the spectrum of the received signal. The interpretation of this
contribution of the spectrum in terms of the stochastic properties of tur-
bulent fluid motion is the purpose of the remainder of this paper.
2. Theory
The instantaneous A.C. heterodyne current from the photodetector for
a Laser Doppler Velocimeter may be written [1], assuming a constant ref-
erence beam £„
i(t)'oc Es(t)Er(t) = Vcpnst. Re/^ e ' " P[rn.(t)] (4)
where <x implies proportionality and where r f t ) is the time dependent
tivposition of the n particle in the system measured with respect to a
laboratory frame. The vector K is the scattering vector and is fixed by
the optics. P[r_(t)] is the weighting function for the E fields seen by the
photodetector. The weighting function, which can be complex, is deter-
mined by the optics and can be computed from first principles. It is es-
sentially the product of the E field amplitude weighting function for the
two beams that define the sample volume. The term "const. " contains
terms describing the scattering efficiency of the particles, the quantum
efficiency of the photodetector, etc., which are unimportant for the pres-
ent discussion.
By the Weiner -Khinchin theorem, the power density spectrum of the
photodetector current pan be written
(5)
where
R..(K, T) = const. i(t)i(t + T) (6)11
is the autocorrelation of the current. The oyerbar denotes a time average.
This now may be written in terms of the particle positions,
iK.[rn(o),r (T)]Rn = const. R e _ / _ e n m P[rn(o:)]P[rm(7)] (7)
m n
or
iK-[r (p)-r (T)]
lu = const. Re^/_e n n P[rn(o)]P[rn(r)]
n .
Statioharity has been assumed in eqs, (7) and (8). In all but fluid-particle sys-
tems of very high number density, the relative particle positions can be
taken as random. Therefore the second term in equation (8) is vanishingly
small compared to the first term. Note here, that because of the average,
the second term tends to zero even for the limiting case of very few par-
ticles in the sample volume at one time.
Equation (8) can now be rewritten,
r~ -iK.Arn(T) ~Ru = const. Re) e n P[rn(o)] P[rn(o) + Arn(T)] (9)
n
where
= rn(r) - r (o)1.1. n n
This expression can be formally evaluated in terms of two probability den-
sity functions:
(1) that there is a particle within the volume element r, r + dr,
£ dr (10)
N
(2) that a particle in the volume element r, r +. dr moves a distance
Ar, in time T,
G(Ar, r,r)d Ar (11)
The autocorrelation of the photodetector current is,from equations (9),
(10), and (11),
p C ~/ / \~K AT* ~ ~ ' 'v ~RU = const. Re p e^' - G(Ar, r;r)P(r)P[r + Ar(r)]dr d Ar
•
 J
 (12)
3. Laminar Flow
Consider a system in steady, straight laminar flow. In this case,
for the scattering particles normally used in LDV systems,
G(Ar, TJT) = .6[Ar -v(r)T] (13)
i.e., the particles track the flow. Equation (12) becomes
r*
R.. = const. Re p I e"iK'v(r)TP(r)P[r 4- v(r)r]dr (14)
The dependence of v on r takes account of gradients in the velocity.
Detailed calculations of the spectrum for laminar pipe flow have been
presented in a previous paper [1]. A typical spectrum is shown in fig-
ure 3. This spectrum demonstrates the effects of both the finite illu-
iminated region and of a gradient in the velocity. Here VQ is the veloc-
ity in the center of the sample volume and z /R is the fractional dis-
tance from the center of the pipe.
4. Turbulent Flow
We define the average velocity v by
v ( r ) = / £± G(Ar, T, r)d Ar (15)
T
For convenience, Ar will be defined in terms of the mean velocity v,
and Ar the deviation from the mean motion
Ar = Ar + vr (16)
The computations for this section will be done for v ± v(r). This is
not a necessary condition, but it aids in the clarity of the presentation.
Equation (12) now becomes
P* P . _
Rti = const. Re / ,/ G(Ar + v-fie'^' (Ar+YT)p(r)P(r + Ar +v"r)dr d(Ar + VT)
J J • (17)
In view of equation (15) we define
G(Ar, r)d Ar = G(Ar, r)d Ar (18)
where G is the probability that a particle moves Ar, Ar + d Ar with
respect to the mean motion v T in time T. This definition becomes clear
if one notes that the probability that a particle moves Ar in time T is
identical to the probability that it moves Ar with respect to the mean
flow in time T.
Equation (17) becomes
.. = const. Re / G(Ar, Ar e~ iK 'VTQ^(Ar + vr)d Ar (19)
where
Q0 = / P(r)P(r + Ar + vr)dr (20)
Equation (19) can always be rewritten using the Fourier convolution and
shift theorems as
* 1VI tr rr ~G(K',
 T)e • QQ(K - K')dK (21)
where Q is the Fourier space transform of Q and G is the Fourier
space transform of G.
R-. can be shown to tend to zero with increasing T for two reasons
(1)
lim e-iK'-VT f (K, )dK, = 0 (22)
for any square integrable f(K'),
(2) G(K', T) can be shown to be a decreasing function of r for any
K', in particular K, so that
lim
T-oo
-K)dK' = 0 (23)
For the first mechanism, we have demonstrated in a previous paper [1],
that a characteristic decay time is
TQ = 2o/v
8
o
where a is characteristic dimension of P in the mean flow direction.
For the second mechanism, the characteristic decay time can be taken to
be when
G(K, r)/G(K, 0) •« e ' / (24)
i.e., when the standard deviation (_AK (tu ' , of G is approximately
equal to K. But from the theory of Fourier transforms we know that
= 0(1) (25)
o 1/2where (Ar ) ' is the standard deviation of G(Ar, T).
The characteristic decay time, T , for the second mechanism
therefore occurs when
K[Ar2(T)]1/2 = Q(l) (26)
In other words, it is the time when the average scattering particle has
diffused a distance 1/|KJ with respect to the mean motion. Since |K
4 1for a typical LDV is on the order of 10 cm" . the second mechanism
2 1 / 2 4causes significant decay of R.. when (Ar ) ' & 10 cm. This length
is very small compared to the normally encountered Lagrangian integral
length scales, A^, which are on the order of the shear dimension of the
flow system. Therefore, no matter what value TQ takes, R.. decays in
at most the time it takes the average particle to wander a distance with
respect to the mean flow that is very small compared to the Lagrangian
integral length scale A, . This means that the integrand involved in the
evaluation of R.. is significantly different from zero only for times that
are short compared to the Lagrangian integral time scale, TV •
In this case, G(Ar, T) can be approximated by an asymptotic expres-
sion for small T/TT.
(27)5(Ar - v'r)F(v';r)dv', «!F (^5 r)
T \
 r I
where F(v') is the probability density function for a turbulent fluctuation
V.
In this section, several simplifying assumptions are made:
(1) G(Ar, T) is isotropic
(2) the flow is one dimensional
(3) the vector K lies in the mean flow direction
These assumptions are made only for clarity of presentation and do not
represent a restriction on the method of calculating the spectra.
If the G(Ar, T) from equation (27) is substituted into equation (19),
and that used in equation (5), we find after some rearrangement
I(K, co) = const. / —L-FfrOQ^ -"l^v dv' (28)
v' + v /
where v' = co/K' - v. The integration variable, v', in equation (28) can
now be interpreted as the fluctuating velocity. This becomes clear when
one recognizes that the term
K(v + vQ - co
v + v'
(29)
is the spectrum generated by a flow with a constant velocity v + v' (see
ref. 1). Equation (28) gives the spectrum as a weighted sum of spectra
generated by each velocity present in the sample region. Individual par-
ticles can change their velocities during their traverse of the sample vol-
ume; but since r « TV they do not change their velocity significantly
over, T , the decay time for R...
For P = 1, a very large sample volume,
I(K, w) = const. I F (" -K 'v ) (30)
K \ K /
In this case the spectrum has the shape of the probability density function
for the fluctuating velocity with a scale factor of K.
10
A
If K is sufficiently small, G(K, T) and hence R- for turbulent flow,
do not decay until times long compared to TV . In this case the observed
particle motion takes on a Brownian motion character and G(Ar, T) can be
approximated by (4)
G(Ar, T)
(4776 r)3/2
where e is the eddy diffusivity. Using equations (31), (21), and (5), the
spectrum of the LDV becomes
I(K, o>) = const. - - — — Q> (K ' -K)dK' (32)
•*• — ^
For equation (32) to be valid K must be less than I/A, . Since A, is
typically greater than 1 cm. , it appears infeasible to measure eddy dif-
fusivlfcies by the above technique. However, the theory presented here
also applies to ultrasonic scattering where small K vectors are possible.
A check of the small K limit predicted here is feasible by ultrasonic
techniques. For the Brownian motion of small particles, where the core-
relation length is much smaller than 1/K, the results have been confirmed
[2,3].
5. Calculation for Real Systems
Consider a three dimensional system with a sample volume having a
ssian intensity with RMS widths a , a , a in tl
x y z .
directions and with orthogonal turbulence intensities,
Gau he three orthogonal
 
(v,2)l/2 ( 2)l/2 (v,2)l/2
J — 5 —V V
11
The moments of the spectrum can be calculated from the three dimensional
form of equation (28). When the mean flow is in the X-direction, and when
K is in the X-direction, the mean and second central moment of the spec-
trum are
= K V (33a)
VAR = Kv'* +
X
v2 vi2 v;2+ + y
yl« A £i A 2
,'i1
5 ^2
(33b)
or
VAR = K2 vx2 + v
2
4(7X
L
_^2
f
Vs
>
/ :I- (33c)
These formulae are independent of the particular form of the fluctuating
_2 2
velocity probability density F(v'). The v /4a terms have been discussed
in reference [1] and are the finite sample volume contribution to the width of
the spectrum. The other terms are the broadening due to the turbulence.
One will notice that the variance is not simply the sum of the square of the
finite volume broadening and the turbulence broadening. It contains, in ad-/ g o \dition, turbulent finite volume broadening (e.g., v' /4a Jdue to the RMS
\ X X *
velocity fluctuations carrying the scattering centers across the sample vol-
ume. Typically, a_A_ < <r_/ff. « 1 and (v' /v r/ < 0.05, sotheef-x z x y
feet of y and z fluctuations is to increase the finite transit time term by
less than 3 percent. Since, in a well-designed experiment, the finite transit
time broadening is small compared to the turbulence broadening, the effect
of y and z fluctuations is negligible. Therefore, a good approximation to
the variance of the spectrum is
VAR = Kv2 2 (34)
12
If the K-vector is in the y direction
= 0 (35)
VAR = K2v' 2 +y (36)
4<T
The LDV is therefore capable of measuring each component of the fluctuating
intensity independently.
Near the centerline of turbulent channel flow, the gradient in the mean
velocity is small. Further F(v') can be approximated by
rt 2
F(v')dvf = exp - vy
^
"*•<
Idv-dv-dv^ (37)
Using equations (37) and (28)
const.
'27TV ,2
9 / 9
, -v'2 /2v'21
 e x/ x exp-,
v
x
+ v
x
(- 00'
,n2
,dv'
(38)
Since v » |v' | one has
A. Ai
-ao
2
exp
2f f x (Kv x + Kvx
r; (38b)
13
I(K,cb) const. exp -
2K v2,,,;
(co - Kvx)2 (39)
The heterodyne turbulence spectrum in this case is approximated by a
Gaussian with variance (v2/4a2 + K v' ). Figure 4 is a turbulence spectrum
X.' X
taken at the channel center. The solid line is a Gaussian fitted to the data.
Near the center of a channel, the dependence of the turbulence intensity
2~on position is not strong, so that over the sample volume, vf , can be con-
sidered constant and the gradient in the mean velocity can be handled as in
reference [1]. Near walls where large gradients in the mean velocity may
be present and a strong positional dependence of v', may be also present,
a more complex model for G(Ar, r) must be used. This will be treated in
a later paper.
6. Conclusion
The power spectrum of phototube current for a Laser Doppler Velo-
cimeter (LDV) operating in the heterodyne mode has been computed for the
case of turbulent flows. It is shown that for normal operating parameters
the spectrum contains information only on the short-time behavior of the
fluid motion. To examine the long-time behavior, i.e., times greater than
the Lagrangian-integral time scale of the turbulence, one must use ex-
tremely small scattering angles.
It should be noted that this paper describes the power spectrum of a
signal averaged over long times. This includes situations in which there
may be a very few or very many particles in the sample volume at one
time. We have not considered the case of F. M. detection. With F. M.
detection it is possible to extend the time and length scale of the LDV
measurement beyond those indicated in this paper.
14
7. Acknowledgements
Two of the authors, Robert V. Edwards and John C. Angus, acknowl-
edge the financial support of the Office of Naval Research, the National
Institute of Arthritis and Metabolic Diseases (NIH) and the National Science
Foundation. All of the authors wish to thank the NASA Lewis Research
Center, Cleveland, Ohio, for their generous cooperation.
REFERENCES
1. R. V. Edwards, J. C. Angus, M. J. French, and J. W. Dunning, Jr.,
J. Appl. Phys. 42 (1971) 837-50.
2. S. B. Dubin, J. H. Lunacek, and G. B. Benedek, Prop. Nat. Acad.
Sci. U.S. 57 (1967) 1164-71.
3. J. W. Dunning, Jr. and J. C. Angus, J. Appl. Phys. 39 (1968)
2479-80.
4. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1-44.
REFERENCE
BEAM
o
CONSTANT
PHASE LINES
BEAM TO BE
SCATTERED
TO PHOTO-
MULTIPLIER
TUBE
Figure 1. - Sample region.
Figure 2. - Summation of individual fields.
1.0r-
roz-0.04CM
' r al°0.l CM
in
Z
o
25.1 27.1 29.1
FREQUENCY,
31.1 33.1
Figure 3. - Spectra observed with laminar pipe flow showing
effects of large sample volume in radial (z) direction. In
each case the sample volume was centered at 0.1 cm from
centerline of 1 cm pipe, the characteristic dimension, ox,
in the flow direction was approximately 0.0045 cm and the
velocity at the tube center was 10 cm/sec. Note that K-VO
is identical in each case.
1.0,-
.8
.6
ISII
ee.O
o DATA
— EQ.(39)
1400 1800 2200
FREQUENCY, kHz
2600
Figure 4. - Spectrum obtained fromjurbulent
flow of water. Parameters were: v • 288
cm/sec, Re-T 85700, 9= 19.8°, ox • 10.4 urn.
Calculated percent turbulence was 2.95 per-
cent.
NASA-Lewii-Com'


Turbulent transport measurements with a laser Doppler velocimeter

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server