An Overview of Particle Sampling Bias

Transcription

1 An Oeiew of Paticle Sampling Bias Robet V. Edwads Case Westen Resee Uniesity Case Cente fo Complex Flow easuements Cleeland, Ohio and James F. eyes NASA - Langley Reseach Cente Hampton, Viginia 3665 Second ntenational Symposium on Applications of Lase Anemomety to Fluid echanics July -4, 1984 Lisbon, Potugal

2 An Oeiew of Paticle Sampling Bias Robet V. Edwads Case Westen Resee Uniesity Chemical Engineeing Depatment Cleeland, Ohio Case Cente fo Complex Flow easuements and James F. eyes NASA - Langley Reseach Cente ail Stop 35A Hampton, Viginia 3665 Abstact The complex elation between paticle aial statistics and the inteaial statistics is exploed. t is known that the mean inteaial time gien an initial elocity is geneally not the inese of the mean ate coesponding to that elocity. Necessay conditions fo the measuement of the conditional ate ae gien. ntoduction The poblem to be dealt with is that in a spasely seeded flow, the pobability of ecoding a elocity depends on: (1) the pobability of the elocity appeaing in the measuement olume. () the pobability of a paticle aiing at the measuement olume when the elocity is. (3) the pobability of detecting a elocity fom a paticle of elocity een if it passes though the olume. (4) the pobability of ecoding a measuement. n geneal, all of the latte effects depend on the elocity itself. Fo example, fo a flow with unifom spatial seeding, the highe speeds will cay moe paticles pe unit time though the measuement egion than will the slowe speeds. Also, fo aious easons, the electonics

3 ae less likely to ecod a highe speed paticle than a lowe speed one {Duao and Whitelaw (1979), Duao, Lake and Whitelaw (1980)}. f one ecods a histogam of measued elocities at a fixed point, p m (),it is elated to the Euleian elocity distibution by an equation of the fom P () N p() h() m whee N is the total numbe of measuements, and h() is the conditional pobability of ecoding a measuement if the elocity is. Anothe intepetation of the tem h() is the elatie measuement ate fo the elocity. t has been witten {Edwads and Jensen (1983)} h () () m m whee m () is the measuement ate fo elocity and <> denotes expected alue. Unde the best of cicumstances, h() is difficult to compute {Buchhae (1975)}. Wose, the assumptions one has to make to compute h() ae often not alid. any times the seeding paticle density is not unifom. Unde these cicumstances, any coection applied to eliminate the effect of h() may actually bias the data moe. Poblems of this type wee noted by {Hoesel and Rodi (1977)}. They pointed out that esidence time coection was inappopiate unless the paticle density was unifom. They went on to suggest that if the paticle density could not be assumed to be unifom, then esidence time weighting is not appopiate. One should attempt to measue h() by looking at the paticle inteaial times. Statistics of Paticle nteaial Time f the flow is steady and the seeding is unifom the pobability of the time t between paticles is of the fom P () () e ( ) whee () is the mean aial ate coesponding to the elocity. This follows eadily fom the fact that the olumetic distibution of the paticles follows a Poisson distibution. n geneal () can be witten () A( )

4 whee is the paticle density and A( ) is the effectie measuement olume coss section. t has the dimensions of aea and is positie definite. The coss section can be a function of flow angle. Fo this steady flow, the aeage inteaial time is gien by (()) 1 Fo a tubulent flow, the situation is moe complex since the eleant pobability is the conditional pobability of an inteaial time if the initial elocity is. t is not possible to gie a global computable expession fo the function, howee the asymptotic behaio is known. At times small compaed to the micoscale, the expected ate emains () and at times lage compaed to the micoscale, the flow is uncoelated with and thus the expected ate becomes <>. An appoximate fom fo P (,) the pobability of an inteaial time gien the initial elocity can be deied using the methods descibed in {Edwads and Jensen (1983)}. Howee, fo illustatie puposes, a much simple fom can be used, i.e., P (, ) exp ( ( ) ) exp( (() ) ) e xp ( ) ( ) ( 0 ) () () exp ( ( ) ) This fom ignoes the expected aiance in ate fo long times, but that effect is demonstably small. Figue 1 shows some foms of this function fo aious mean ates and initial elocities. Using this fomula, the mean inteaial time, gien the initial elocity is gien by () 1 ((() 1) ( 1 )) e () () Clealy, the mean inteaial time is not simply the inese of the ate. Fo instance, when the expected aial ate () appoaches zeo, the mean inteaial time does not appoach infinity. This eflects the fact that the elocity only stays nea any gien alue fo a time on the ode of. The asymptotes ae: 3

5 () 1 () 1, ( ) ( ) 1, ( ) 1 1 ( 1 ) 1 () Only in the second, high paticle density, case does appea to look like the inese of (). Fo the moe typical, low paticle density case, the mean inteaial time is a weak function of. See Table 1 fo a moe detailed examination of the aiation of with ate. The weak dependence of < >on is appaent in the measuements of {Duao, Lake and Whitelaw (1980)}. n thei measuements, they included a non-negligible eset time, but this complication does not negate the aboe explanation of the measuement esults. Effect of nstumentation As has been noted in many aticles {Duao, Lake and Whitelaw (1980), Edwads and Jensen (1983), and eyes and Clemmons (1979)} the paticle aial ate is not the same as the measuement ate. The use of Bagg cells, the existence of counte eset times, etc., can all alte the ecoded statistics. This esult is a measuement ate fo a elocity, m (), that is diffeent fom (). n the est of this pesentation, the ate m () will be used. A geneal deiation fo the statistics of any instumentation set up cannot be gien, howee as befoe a few geneal statements can be made: (1) The expected ate fo times shot compaed to, will be m (). () The expected ate fo times lage compaed to will be < m >. Recall that m () is a conditional pobability so that the times efeed to aboe mean the times afte the occuence of in the flow. The diffeence between m () and () can be illustated by examining some data taken by Steenson, Thompson, Bemme and Roesle. n that study, they make elocity measuements in a tubulent flow while aying the effectie paticle density. The data collection system had a eset time (dead time) that was shote than the flow coelation time. As the paticle density inceased past the point whee the poduct of the aial ate and the eset time exceeded one (satuated), the measued 4

6 means changed. See figue. This clealy indicated a change of the measued statistics fom those of the paticle aial statistics. {Edwads and Jensen (1983)} deied an appoximate fom fo h m () fo a system with a eset time T. iz. h m () () ( 1 T ) ( 1 ( T R ) ( ) R ) T T whee R R() d, R( ) is the elocity autocoelation function, and T T 0 is a nomalization constant. Again one can gain some insight into the expected behaio by consideing the asymptotic behaio of this expession. T << T c = R T=, The ntegal Coelation Time Unde this condition, RT T. Then hm() 1 ()( 1 T ) ( 1 ( ) T) (1) f ()T >> 1, many paticles aie duing the eset time. Then h m () = 1, thee is no effect of the paticle aial statistics. () f ()T << 1, few paticles pe eset time, then () h () The paticle aial statistics ae appaent in the measuement statistics. The solid line in Figue eflects this change in the statistics as the aial ate is aied. T >> T c This coesponds to a eset time longe than the flow coelation time. Then At this asymptote, R R T T c 5

7 h m () 1 () ( 1 T ) ( 1 T ( )) c Fo eithe ()T >> 1 o ()T << 1, the paticle aial statistics ae eflected in the measuement ate. Fo eset times longe than the flow integal time scale, the measued ates behaio is astly diffeent fom the behaio of the mean inteaial time. With a lage eset time, the inteaial pobability function only contains infomation about the mean flow. Using the assumptions used ealie, P () e fo T 0 othewise With this pobability function, /< > = 1. This is clealy shown in {Duao, Lake and Whitelaw (1980)} esults fo a eset time lage than the flow coelation time. The meaning of the aboe is that although a system with a eset time lage than the flow coelation time will hae a mean inteaial time that is independent of the initial elocity, the mean aial ate is not independent of the oiginal elocity. To undestand how this can happen, one must ealize that the inteaial time distibution is a eflection of the conditional pobability of anothe measuement if a measuement of is obtained. On the othe hand, the measued ate, m () is a eflection of the pobability of getting the initial measuement of. easuement of m () As was shown aboe, one cannot measue m () (and thus h()) by measuing the mean paticle inteaial times. Een wose, if thee is a dead time lage than the flow coelation, one cannot detemine m () fom the inteaial time statistics. Howee it is conceiable that estimates fo m () can be obtained by examining the data ate fo times that ae small compaed to the micoscale. Fo a steady flow, it is easy to measue m (). One simply picks a time inteal t and fo successie non-oelapping inteals, measue the numbe of measuements one gets. The mean numbe of measuements diided by the time t is an estimate of the ate m (). Roughly, if N is the total numbe of measuements, the elatie eo in the estimated ate is (N ) -1/. This follows if each measuement is independent. Fo 6

8 some cicumstances such as a satuated detecto, {Edwads and Jensen (1983)} o multiple measuements of the same paticle, each measuement is not independent and thus the elatie eo will be lage. Fo a tubulent flow, the elocity is not constant and the pobability of getting two o moe measuements of the same elocity with a lase anemomete is ey small. Howee if one makes a histogam of the measued elocities using a finite numbe (say K) of non-oelapping anges, one can aange the diisions so that each ange contains at least two measuements. The ate estimates can be pefomed fo each inteal of the histogam. Selection of Histogam nteals n the wose case, an estimate of the wose factional change of the ate oe a elocity ange H,is H /, whee is the aeage elocity in the inteal. This assumes a ate popotional to the elocity as exemplified by claughlin and Tiedeman's one dimensional models. ost othe models gie a weake dependence on elocity. Let R be the ange of measued elocities, o 4 standad deiations, whichee is lage. The change in ate in each inteal compaed to the change in ate acoss the entie elocity ange is oughly H / R. Fo a eal data set with a finite numbe of measuements, the aboe consideations place contadictoy equiements on the selection of H, the width of the histogam inteals. Accuate estimation of ate in each ange demands a lage numbe of sample measuements and thus as lage a as possible. On the othe hand, accuate esolution of the change of ate acoss the measued ange, R, equies a small H.We do not know of a pocedue fo optimizing, but expeimentally hae settled on H / R = 1/9. A detailed exposition of one pocedue fo estimating m () and thus of deiing the tue Euleian elocity pobability distibution, p(), fom the measued distibution is gien in a companion pape by J. eyes. Sample and Hold Pocessing the ecoded data by a sample and hold scheme is exactly equialent to estimating integals of the elocity by a fowad step integation algoithm. One holds the peious elocity alue until a new one is obtained. {Dimotakis (6)} had poposed a backwad step algoithm fo use in the situation of many measuements pe flow 7

9 coelation time. Thee is no essential diffeence in the esults obtained fo a fowad o backwads integation scheme. f the aeage of the measuement pe flow coelation time is small, then the appoximation to integation fails. Howee, when the measuements pe flow coelation time is lage, the appoximate integation schemes ae good appoximations to the continuous integals {Edwads and Jensen (1983)}. When these latte conditions ae obtained, sample and hold pocessing o the method suggested by Dimotakis is clealy the best method to use. The paticle statistics ae aoided. Conclusions The measued aial statistics fo lase anemometes in spasely seeded flow is indeed complex. No theoy can adequately pedict these statistics as many uncontollable and unmeasuable aiables in the system can influence the statistics to an impotant degee. Theefoe unless the athe specialized and ae conditions of many measuements pe flow coelation times ae obtained, one should not use any of the peiously poposed coections. The mean inteaial time between measuements gien an initial elocity is elated to the mean measuement ate in a complex manne. n some cases whee the measuement system cannot ecod measuements in a time shote than the flow coelation time, the mean measued inteaial time is a constant. This effect can be used to get an ode of magnitude estimate of the flow coelation time. f the measuement system is capable of measuing paticles sepaated in time by less than the flow micoscale time, it is possible to measue the equied coection function. This can be done by measuing the measuement ate fo each elocity in time anges that ae small compaed to the coelation time of the flow. Refeences 1. Duao, D. F. G.; and Whitelaw, J. H.: Relationship Between Velocity and Signal Quality in Lase-Dopple Anemomety, J. Phys. E: Sci. nstum., 1, 47 (1979).. Duao, D. F. G.; Lake, J.; and Whitelaw, J. H.: Bias Effects in Lase Dopple Anemomety, J. Phys. E: Sci. nstum., 13, 44 (1980). 8

10 3. Edwads, R. V.; and Jensen, A. S.: Paticle-Sampling Statistics in Lase Anemometes: Sample-and-Hold and Satuable Systems, J. Fluid ech., 133, 397 (1983). 4. Buchhae, P.: Biasing Eos in ndiidual Paticle easuements with the LDA Counte Signal Pocesso, Poc. LDA Symp. Copenhagen 58 (1975). 5. Hoesel, W.; and Rodi, W.: New Biasing Elimination ethod fo Lase-Dopple Velocimete Counte Pocessing, Re. Sci. nstum., 48, no. 7, 910 (1977). 6. Dimotakis, P.: Single-Scatteing Paticle Lase Dopple easuements of Tubulence, AGARD Confeence No. 193, Applications of Non-intusie nstumentation in Fluid Flow Reseach pape eyes, J. F.; and Clemmons, J.., J: Pocessing Lase Velocimete High-Speed Bust Counte Data, Lase Velocimety and Paticle Sizing, eds., H.D. Thompson and W.H. Steenson, Hemisphee Publishing Co. p. 300 (1979). 8. Steenson, W. H.; Thompson, H. D.; Bemme, R.; and Roesle, T.: Lase Velocimete easuements in Tubulent and ixing Flows - Pat, Ai Foce Tech. Rep. AFAPL-TR , Pat Table 1.- Vaiation of /< > with ate fo aious mean ates. 9

11 (Time) -1 ( A) ( ) 10, () 1 0 (Time) Time Time (A) (B)..1 0 ( B) ( ) 1, () (Time) (Time) ( D) ( ) 1, () 01. ( C) ( ) 01., () Time (C) Time (D) Figue 1.- Appoximate conditional pobabilities fo the inteaial time gien the initial elocity as a function of appaent paticle concentation. The flow coelation time in each figue is Tubulent Region Velocity s Data Rate Files: Dec. 13 X - Position: 5 mm (A) Velocity, m/s aximum Sampling Rate easued Data Coected Data fo Velocity Bias Data Rate Figue.- Velocity s Data Rate at Location A. 10