An ideal seeding device is proposed generating particles that exactly follow the flow out are still a major source of error, i.e., with a particle counting bias wherein the probability of measuring velocity is a function of velocity. The error in the measured mean can be as much as 25%. Many schemes have been put forward to correct for this error, but there is not universal agreement as to the acceptability of any one method. In particular it is sometimes difficult to know if the assumptions required in the analysis are fulfilled by any particular flow measurement system. To check various correction mechanisms in an ideal way and to gain some insight into how to correct with the fewest initial assumptions, a computer simulation is constructed to simulate laser anemometer measurements in a turbulent flow. That simulator and the results of its use are discussed

Baratuci, W.

Edwards, R. V.

English

NASA Technical Reports Server

CORMCTING FOR PARTICLE COUNTING BIAS ERROR IN TURBULENT FLOW 
Robert V. Bdwardr and Williao Baratuci 
Cheaical Engineering Department 
Caee Center for Complex Flow Measurements 
Cleveland, Ohio 44106 
https://ntrs.nasa.gov/search.jsp?R=19860001972 2020-03-20T16:14:02+00:00Z
RATIONALE 
Even i f  t he  r e s u l t  of t h i s  meeting is an idea l  reeding device t h a t  gen- 
e r a t e s  p a r t i c l e s  t ha t  exact ly  follow the  flow and a r e  of s u f f i c i e n t  major source 
of e r r o r ,  I r e f e r  t o  p a r t i c l e  counting bias  wherein the  probabi l i ty  of 
measuring ve loc i ty  i f  i t  occurs is a funct ion of vC?locity. The e r r o r  i n  the  
measured mean can be as much a s  25% ( r e f .  1). 
Many schemes have been put forward t o  cor rec t  f o r  t h i s  e r r o r ,  but t he re  
is not  universal  agreement a s  t o  t h e  accep tab i l i t y  of anycone method. I n  par- 
t i c u l a r  it  is sometimes d i f f i c u l t  t o  know i f  the  assumptions required i n  t h e  ana- 
l y s i s  a r e  f u l f i l l e d  .by any p a r t i c u l a r  flow aeaeurement system. 
I n  an e f f o r t  t o  check various cor rec t ion  mechanism i n  an i dea l  way and t o  
gain some in s igh t  i n t o  how t o  cor rec t  wlth the  fewest i n i t i a l  assumptions, a 
computer simulation was constructed t o  s imulate  l a s e r  anemometer measurement8 i n  
a turbulent  flow. That simulator and the  r e s u l t s  of i t e  use a r e  the  t op i c  of 
t h i s  paper. 
INTRODUCTION 
A l l  measurements of mean quan t i t i e s  i n  a sparse ly  seeded turbulent  flow 
using a l a s e r  anemometer generate a measured ve loc i ty  probabi l i ty  function 
Pm(v) t ha t  diffe ' rs  from the  t rue  Eulerlan probabi l i ty  of i n t e r e s t  P(v). The 
r e l a t i on  between the  two is  given by 
Any cor rec t ion  scheme '8 function is t o  e l iminate  the e f f e c t  of r,(v). 
- 
1 
rm(v) 
P,,,(v) = - P(v) , 
*m> 
where r,,,(v) i s  the measurement ra te  - i f
the velocity i s  v. < > denotes 
average val ue. 
s 
First a pseudo-continuour r i g ~ l ,  the "hot wire" signal, is generated by 
passing a digital white noire eignal through a digital filter. Shown here is a 
typical segment of the "hot wire" rignai. 
Valoci t y  
1.4 
Somple Hot Wire Velocity Signal 
40 
T i n a  
The " l a r e r  anemometer" r i gna l  is generated by a random rampling of the  hot 
wire r ignal .  The simulator is conrtructed 00 tha t  t he  average mearurement r a t e  
a s  a funct ion of ve loc i ty  ran be r a t  t o  ba any der i red  funct ion of the veloci ty .  
I n  t h i s  utudy the  mearureunt  p robabi l i ty ,  rm(v), wr r a t  t o  be e i t h e r  a l i n e a r  
or  quadrat ic  funct ion of the  ve loc i ty  ugn i tude .  Shown here ir  a block diagram 
of the p a r t i c l e  sampling rect ion.  
Flow Diagram of Part lc le  Arr+val Slm~lst lon 
"Hot Wire" Laser 
Signal SIMPLER PROCESSING Output 
, A L 
RANDO)(( 
NOISE 
GENERATOR 
9' 
c 
* 
f(?.v,) 
Shown here Is a hietogram of the hot wire elgnal and laeer anemometer 
elgnal for a typical elmulation. They are cloee i n  ehspe, but there are 
d i f  f erencee. 
Hot Wire and Laear Ve:ocity Hietograme 
Protdll  l t y  
a 4  
If one takes the rat io  of the laser  anemometer hietogram to the hot wire 
hietogram, the result  should be the normalized rate corresponding t o  the velo- 
c i t y  (see the f i r s t  figure).  That ra t io  is shown here. Note the l iuzar depen- 
dence of the rate on velocity.  
VELOCITY 
CORRECTION SCHEMES 
Three cor rec t ion  schemes from the  l i t e r a t u r e  were checked using the  
s i a r l a t  or. 
J 
1 )  Sample and Hold. A continuous s igna l  is generated by holding t h e  l a s t  j 
measured ve loc i ty  u n t i  1 a new ooeasurement arr ives .  That new one is the held.. . i 
.? It has been predicted t h a t  i n  the  l i m i t  of many Dleasurements per flow correla-  :: 
t i o n  time, the continuous s igna l  generates unbiased s t a t i s t i c s .  ! 
i 
2 )  P e r i d i c  Ueasurement. Time is divided i n t o  i n t e rva l s  of constant f 
length. I f  only the  f i r s t  measure i n  each i n t e r v a l  i s  recorded, a per iod ica l ly  
sampled s igna l  is generated. It has been predicted t ha t  a l l  b ias  vanishes when s 
i t  is highly probable t ha t  there  is  a measurerent i n  each in t e rva l .  
3)  Dead Time. Any data  recording device has a r e se t  time during which no 
new measurement is recorded. I f  t he  dead time is  small compared t o  the  flow 
co r r e l a t i on  time and i f  the p a r t i c l e  r a t e  is high enough, t he  bias  should 
vanish. 
This f i gu re  shows each scheme graphical ly .  
Sample Hot Wire Velocity Signal 
4 7  - 
ae- I I I 
0 26 40 W 60 
T i l e  
& - .- . * - -  , -:. &d:f%\\ - % . .  - . - -  
-1 
1 
' I  
' f  The alas i n  the mean de f in i t e ly  does decrease as the man rate per flow 
I correlation time increases. This  is shown i n  the next figure. 
Meon Heacured Velocity vs. Sampling Rote 
V = 1.0. T I  - 0.30 
I Sample and Hold Analysis 
5 [ 
f 
ban Sapling Rate Per Correlation T i w  
If the  aean p a r t i c l e  a r r i v a l  r a t e  is held  constant  and i f  the  per iod ic  
aaapl ing  r a t e  is varied ,  the b i a s  i n  the  measured mean does decrease  but not t o  
-
zero.  See  below. 
Measured Mean Vc:loci ty vs. Q 
V-. 997+/-. 008. TI-. 300+/-. 003. soql ing rat&-1. fl 
Periodic Sampling Anulysi~ 
0 - ratio o f  -ling period t o  correlation tin# 
If the dead time is kept smaller t ~ a n  the f l o w  correlat ion time, the bias 
is supposed t o  decrease t o  t e ro  ae the pa r t i c l e  a r r i v a l  r a t e  increases. This is 
borne out by the following figure. Although the  f igure  doesn't show the  b ias  
goiag t o  tero,  the measurements closely f i t  the theory that  does go to  zero. We 
haven't had enough computer ti= t o  check the high pa r t i c l e  ra te  limlt. 
Weeuured Hem Velocity vs. RT 
V = 1.0, T I  = 0.30 
Dead Time Analysts 
RT - Maan nukr oC -.#nk per [kad T i m  
Recently Edwards and Meyers put f o r t h  a cor rec t ion  scheue t h a t  involved 
d i r e c t  measurement of rm(v) ( r e f .  2 ) .  S i m ~ l y ,  one measured tha t  r a t e  f o r  each 
in te rva l  i n  t h e  ve loc i ty  histogram by counting t h e  average number of measure- 
ments occurring i n  a small i n t e r v a l  A t  a f t e r  the  appearance of a ve loc i ty  i n  a 
given in t e rva l .  It can be shown t h a t  t he  procedure's averaging r e s u l t  is exact 
i n  the  l i m i t  of A t  going t o  zero. Unfortunately, , t  can a l s o  be show. t h e t  the  
r e l a t i v e  measurement e r r o r  goes t o  i n f i n i t y  a s  C t  goes t o  zero. The f i gu re  below 
shows the measured r a t e  as a function of A t .  Note t h a t  as  A t  increases,  the  
measured r a t e  tends to~rard the  mean value, independent of uhich i n t e r v a l  one 
s t a r t s  in. 
Rato by Time o f  Arrival vs. Delta - t 
Edworde Lineor Method 
V.loolty Hlotogra 
I l n  0 I 
7 
VIloolty H i o t o g a  
I l n  I 2 
------ 
am 
Delta - t 
Edwards and Baratuci have invented a scheme that computes the l i m i t  as A t  
goes t o  zero start ing with re lat ive ly  large values of A t  (ref.  3 ) .  A l i n e  i s  
f i t  t o  the re su l t s  for various A t  and the intercept i s  taken as the "ccrrect" 
value. 
Rate o f  Particle Arrival vs. Delta - t 
Velocity Histogram Bin U 5 
Edwards Method ( Linear) 
Ymlghtd L . a t  
Sqrrrtm F l t  
------ 
The foLlowing figure shows the measured rate versus ve loc i ty  for a typical  
simulation with a quadratic dependence on velocfty magnitude. The measured 
rates  are very close to the values set by the aimulatlon. 
Hoto 
Rate by Histogram vs. Velocity 
Edwards - Baratuci Method 
Uelghtod L e a t  
The foll.owing figure from reference 3 glves a summary of the results of 
our correction schemes for a set of simulations. McLaughlrn and Tiederman (ref. 
1) and the hietogram corrections are included only as a self consistency check 
of the simulator. The two Edwards correction schemes are here because in general 
one does not know the functional form of rm(v). Edwards (quadratic) is a general 
quadratic fit to r,(v). 
Edwards - Bara tuc i  Cor rec t ion  Method. 
Ratm dependence on v e l o c i t y  i s  l i n e a r .  
Mem Turbul m c e  
V e l o c i t y  I n t e n s i  t y  
Eul m r  i an 8.998 2 .a08 8.302 1?: ,084 
HcLaughlin & 1.000 2 .a12 0.302 2 ,817 
T i  ederman 
Hi stogram 1.002 2 .a10 0.302 2 .006 
( l i n e a r )  
Hi stogram 1.033 *_ .a12 0.300 2 .009 
(quadrat ic)  
Edwards 1.000 5 .a27 0.304 +_ .a14 
( l i n e a r )  
Edwards 1.010 2 .a23 0.290 2 ,018 
(quadrat ic1 
Note : Vbar a 1.0, T I  = 0.30, HE; - 1.0. 
Note : Edwards ( l i n e a r )  means a l i n e a r  f i t  was 
done on rm (At ,v .) and Edwards (quadra t ic )  
)Z 
means'a f i t  was done t o  a second order 
polynomial. The same explanat ion app l i es  
t o  the  Histogram Methods. 
Note r Error  bars are the  standard dev ia t i on  
of t h e  mean o f  t h e  mean v e l o c i t y  obtainmd 
from twenty data e t s  o f  2700 p o i n t s  each. 
This table (from ref ,  3) is the same as the previous except that r,(v) i s  
generated ae a quadratic function of v. 
Edwards - Bara tuc i  C o r r r c t i o n  Hrthod. 
Rate dopondenco on v o l o c i  t y  i s  quadratic. 
Mean Turbulence 
Vo loc i t y  In tons i  t y  
-------.-----------------------------.------------- 
Eul e r  i an 1.8B0 2 .810 8.301 2 .a04 
McL8ughlln 1 1.893 = ,813 , 8.429 2 .a19 
T i  odormar? 
H i  rtopram 1.047 2 .029 ' 0.296 2 .a30 
( l i n e a r )  
Histogram 1.010 +_ .019 0.297 2 .01l 
(quadrat ic)  
Edwards 0.968 2 .a97 0.289 +_ .053 
' ( l i n e a r )  
Edwards 1.016 +_ .a45 8.274 2 .028 
(quadrat ic)  
Note r Vbar = 1.0, T I  = 0.30, k% = 1.0, N = 2700. 
Note : Edwards ( l i n e a r )  moans a l i n e a r  f l t  warn 
dona on r (4t ,v ) and Edwardo (quadra t ic )  
m k 
means a + i t  was done t o  r second order 
pol yncrnrial . Thr same explanat ion app l ies  
: ; ;ii,togram Methods. 
Note : Er ro r  bars are the  stand;r~ dev ia t ion  
ox the  mean o+ t h e  mes,l v e l o c i t y  o b t a ~ n e d  
f r o m  twenty data  i e t s  of 2700 p o l n t s  each. 
27 
- 
f a -  -- - * € 9  -;y ". . -r31C- 
CONCLUSIONS , 
Sample and Hold eliminates biasfor high 
particle densities. 
Dead Timo reduces bias for sampling ratas tested 
C Prediction not checked due to large run time I 
Periodic Sampling with long periods reduces but 
does not eliminate velocity bias 
New correction is excellent when the form of 
rate as a function of velocity is known 
Further work using "unknown" functional forms is 
in process 
REFERENCES 
1. McLaughlin, D.K. and Tiederman, W.G.: Biasing Correct:,ms for Individual 
Realization of Laser Anemometer Measurements in Turbulent Flow. Phys. Fluids 
16, 2082, 1973. 
2. Edwards, R.V. and Meyere, J.: An Overview of Particle Sampling Bias. 
Presented at the Second International Symposium on Applications of Larzr 
Anemometry to Fluid Mechanics, July 2-4, 1984, Sisbon, Portugal. 
3. Baratuci, W.: Particle Arrival Statistics in Laser Anemometry. M.S. Thesis, 
Case Western Reserve University, i985. 


Correcting for particle counting bias error in turbulent flow

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