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cofw%51f77-- I

TITLE. MONTECARLO SIMULAHONOF TWO-PARTICLE RELATIvE DIFFUSIONUSING EULERIAN STATISTICS

J. T. Lee and G. L. StoneLoa Alamos National LaboratoryLos khUtlOS, NM 87545

Robert E. Lawson, Jr.U.S. Environmental Protection Agency

AUTHCWS) Research Triangle Park, NC 27711

LA-UR--85-2745

DE85 015727

Michael ShipmanNorthrop Senices, Inc.Research Triangle Park, NC 27709

SU13MITTE0 TO: AMS Seventh Symposium on Turbulence md DiffueionNOV. 12-15, 1985Boulder, CO

DISCLAIMER

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LosMlmR

LOSAlmos National LaboratoryLos Almos,New Mexico 87545mmnlntm~ m TH18DrMJmT la WI t~lh (4

Notre CARW SIWJIATION OF TVO-PARTICLE RELATIVE DIFFUSION

USING WLERIM STATISTICS

by

Jo T. Lme ●nd G. L. Stoma

Los Alamo- M tiona 1 Iabora tory

Lo. Alnmoo, NH 87545

Robert C. Iawoon, Jr. ”

U.S. bvironmcntal Protection Agancy

R@aaarch Triangla Park, NC 27711

and

HichQal Shipm,~n

Northrop Sarvicas, Inc.

Rasaarch Triangl@ Park, NC 27709

●On ●esignmant from th Na tional Ocoanlc ●nd Atmospheric Administration,U, S. D9parbuant of Commorca

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1. INTRODUCTION

Plont@ Carlo techniques hava baan us~d ●xtenoivcly to simulata ●bsolute or

total diffusion in ● turbulent flow field. Less ●ffort has been devoted t.o

oimulati~ relative diffumion in spite of it. importmtce in analysing tha

statistics of pl~e mend-ring, instantaneous plume widths ●nd concentration

f Luc tua tions. Since meet models of relativa diffusion are baaed on the

●titietico of pairs of tracer particles, calculating the trajectorlta of

partich-pairs is ● key problem in Honta Mrlo simulations.

An ●pproximate method for the analyeia .If relative diffusion hae been

discusoed in several recent publications. In this approach the trajectories of

particles released as pairs or in cluators ara ●esumed to be otetistically

independent, but the initial particle velocities ● rc “conditioned” to *ccount

for the effects of interparticle velocity correlations. Due to the

independence of the trajectories, this ●pproach is raferred to ● e a one-

particle model. It vas recently ●ppliad to atmospheric diffusion from ●

continuous point oource by Cifford (1982) ●nd to d! ffusion from ● Einitt-size

finite-duration source by be ●nd Stone (1983a)~ herekfter referrad to am LSa,

It uas crltiaed by Smith (1983), defended by Cifford (1983) ●nd reviewed by

Sawford (1984), but it. validity ham not baan rasolvad in a deflnitiva mannar.

There is ●n obvious inconsistency in the model slnc~ it ●ccounts for

interpartjcle v~locity correlations at the source but ignoraa them in the

particle trajectory calculations downwind of tha source.

In this otudy we modify the one-particl, Monte Carlo slmulationo of

relative diffusion in LSa to ●ccount for lnt.arparticlm velocity correlations

both ●t the source and ●lon~ the particle trajectories. The intarparticlc

correlations ●re calculated directly from the Eu16rlan apace-tima wloc{ty

●utocorralatlon funationo Slnae tho particla trajectories are coupled, this 1.

,

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rcf,rmd toaaa two-par ticlo modol. Msults from thmca two-par tic 10

simulatias ● rQ coqtarad to our proviow umc-pmrticla r-ul a to dotaraino tlw

impor Unca of tlm htorpar tic lo corrc h tiocm do-wind of tho source. The

❑edals ● m ●ralua tad by comprison with now wind- tunnel dam on tho ●bsoluta

diffusion ●nd mcandming of a pi-a in ● field of grid-8 maratod turbulmcc.

2. THEORETICALANALYSIS

Ua conaid~r the ona-dimensional motion of ● pair of tracer particlas in ●

fhld of swtionary homogsncouo turbulanc~ ● s illustrated ●chomatically in

Flu. 1. Hara t is the tima aftar roloaaa of the particlcc, U is tha mtan wind

v~locity, y is tha cross-wind coordimta, v la tho croos-wind componont of th,

turbulmtt valocity ●nd 2 is the partlcl~ s~paration. ThQ valocity of ●ach

tracar particl~ 1. ●qual to tho local valua of tha turbulant field velocity

V(y,t) ●long it- trajectory. Tlmmforc, the particla vclocitias can ba

oimulatod using tho Eularian spsca- tima velocity ●utocormlation function for

tha turbul~nt field, RE. Ikwavor, it is mor~ co~~vcniant to USQ tho Eul@rlan

●utocorralation function RC in tho convactivs raforonco frama which movaa W1th

th~ mtan wind valocity U. W@hSSUIl10 that I lc an Qxponcntial function ofL

separation in spac~ c ●nd timo t, i.a. RC(C,T) = ●xF(-lcl/L)aXp(-T/tc) Whcrt

tc i. tha Eul@rlan lntqral tlmo scala in tha conv~ctiva raforanca frama ●nd L

is the Eul@rian lnt~gral l~n~th scala. Exponential autocorr~lations arc

physically unrealistic tn coma way., but they ● ro consistent with many of tht

wmll known proportion uf ●bsoluta and ral~tlva diffusion so diacuatad by

Tann@kcs (1979) ●nd Sawford (1984). rhay ●lso parmft the tlma ecala tC mnd tha

Lagranuian intqral tim scats tL to bc simply rclatad to tho fixnd-fram~

Eularian SCS1-S t~ and L ● o dccivcd by Lao and Ston@ (L983b), haraaftar

r~fcrrad to as LSb.

. ,1

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Ths dieplacamauts of the partichs from time t to t + 6t ●tc aasumcd to be

lilmar, i.a. Y2 - Y1 + vldt ●nd Y4 - Y3 + vs~t whgra at << tL~ w- ●lao •s~uma

ths t the valocitica a t tima t + 6 t ● re the sum of correlated atid random parte

●nd that tha correlated parte can ba rela tad to the veloci ties a t tima t in a

linaar manner ● s fellows:

‘2 - ●V1 + bv3 + CV4 + Vi (1)

V4 (2)= dv3 + 9V1 + fv2 + Vi,

wh~ra vi and v~ ● ra uncorralated random valoci ties, The coefficients in these

●qua tious and the variancas of the random velocities vi ●nd vi ● re determined

by taking momenta ●nd ●nsembla ●verages in which the ralativo positions of

points 1, 2, 3, ●nd 4 arc held fixed, i.a. Eulcrian en.smblc avurage~. The

valocfty correlations in tha resulting ●quationa ● re set equal to the value

dicta t-d by RC. For example, the term [V*v31/[V2] 1s se~ ●qual to

Rc(r=y2-y3, t=t2- t3) where [ ] denote Eulerian ●nsemble avarag~o. Thus

th~ coefficients hav~ different numerlc~l valu~a ● t ●ach tlmo ●tep ●nd for ●ach

particle-pair trajectory in the LaSran~ian ●nsemble. Oa tails of the ●valuation

of thaae coafficlants a-a prssontad in Lao, ●t ●l. (1985).

Use of tho Eulerian ●utocorrclation function RC introduces the parameter

a - uv~/L into the problam. This paramatcr ●ppear- in studies of l!ulerian-

LagranCian relationships in homoganaouc turbulence, ●.g. Baldwin and Johnson

(1972) ●nd LSb. Based upon thair theoretical studies •~[! analyois of wind

tunn~l datao Baldwin ●nd Johnson suggest chat a is t.i ‘Jrdtii untl,v ●nd wtll

probably ba Limitad to tha range of values 0.3 6 a 6 5. In this mtudy we

con. idarod tha ranga J,3 { a C - @lnc* the a = o limit corraspcnds to tha

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froqumttly uaod ‘lhylor hypothesis of frosan Wrbulanc@ in which L = UtE (LSb).

As ● practial matar, tha nu96rical solutions ● ra wry naar tha froson

turbulcnca limit for a > 5.

Tha r~latiwa diffusion UR ●nd tho ❑,andorlng UC ara dofinod roop~ctfmly

● m the stinda’ 1 davi~tion of tic particle displacamanta ralativa to the

cmtroid of the particla-pair and the smndard daviation of the cantroid

displacamanta ralativa to the mean wind. Tk abaoluta or :etal dispersion uT

is the a~ndard daviatlor of tha particla displacam~nts rolativa to tha maan

wind, 2and a; = a: + UC. They are prasmtad in the dimansionlass form of

‘2 = u2/2[#]~2 am a function of T - t/~.“ A comparison of ona-particla ●nd

tw-particlo rasults is presented in Pig. 2 for ● n initial particla 89paratlon

of to/L = 0.01 ●nd a = 1. Both aoiutions hava ●n initial rsgion in which ER

incrcaaac as T followtd by ●n accalaratod growth region in which XR incrcasaa

●pproxlmat~ly aa T3’2 in ●gra,mant with the lnartial rang~ scaling law of

Batchalor (1950). All of tha curves approach an ●symptotic growth rata of T1/2

with tR = EC = ~T/(2)L’2 ● s T approaches infinity which is ●lao in ●8racmQnt

with Batchalor-s th~ory. Tha affgct of ths lntcrparticla co:ralation tarma in

tha two-particle modal 1s to dalay tha onset of tha acc,l~rat,d ~cowth region.

Tha on~-particla ZR valu~a art, ● t moat, a factor of ●bout 3 largar t!tam the

tm-particle valuas. Tha affact of tho parameter a on tho solutions la

prastntod in Lao, tt al. (1985).

3, EXPERIMENTALSTUDIES

Tho thaoratical reaulta pr~aattcd in Saction 2 w~ra “valuat~d u~lng

grid-gancratad turbulanc~ in the Mtooroloaical Wind Tunnel ● t thg EPA Fluid

Hodallng Facility which is dcscribad by Snydar (1979), A schmatic of tha

cxparimantil ●rran~omcnt is shown in FIs, 3, Th@ standard grid waa modifiad by

,“ , .’

#

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th ●ddition of ● cbeckarboard pattarn of 20-cn-aquara Plamnita squaroo with a

40 cm horisontml ●nd vertioal spacing botwon ●quaras in order to produco more

•und-rin~ of L’hc plum.. Tho mean ●xial velocity, the turbul~nca intensities

in tho ●xial ●nd vertical directions, ●nd tha tmpoml ●nd epa tial corra la tions

of the ●xial ●nd vertical volocitiaa ware maaaurad using hot-wira and hot-film

X-array ●cnoora.

Ttm turbu lance intcnsi ty uaa naarly uniform in the p Lana norms 1 to the

flow. It dacayad slowly in tha downwind diraction from ● valua of 0.076 ● t tha

aourca location, x - 0, to 0.04 ● t the downs traam and of the exparimenml

region, x ~ 6 m. Tlw maan vcloci ty U was nearly uniform throughout tha

●xparimmttal rqion with a valua of 3. L fO. 1 ❑/a. h u on the ●utworrtlation

of the vartical v~locity w ●a ● function of tha timo delay at a fixed point in

apacs ware fit wi’h ●xponantial curves to datermina the integral time .aca la tE,

hta on tha autocorralation of tha ●xial valocity u and the vertical velocity w

● a ● function of tha vertical aopara tion z batwaen two probes were also fit

with ●xponential to datermina the raspactivc integral length aca lee Luz and

LwZ” Theta quantities ● re tabulated tn Table 1 for three axial locations

downwind of tha source. Valuaa of UtE/Lua arc alao tabulated in Table 1.

ThQs@ ra ties are vtry C1OSC to unity which indica tes frozen turbulence with u

●pproaching infinity,

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x Ow tE Lus

m m/s 8 m m

0.5 0.207 0.0224 0.109 0.0712 0.960

3.0 0.159 o.02@9 0.133 0.0858 1.002

6.0 0.130 0.0121 0.144 0.0995 0.984

Table 1 Eul@rian integral long th ●nd time scales a long the ●xis of the wind

tunnal (y = 0, z = O).

TIW ●bsolute or total diffusion UT was determined by injecting athyiane

(C2H4) continuously and ieokinatically into the flow field through ● small

diametgr tube as shown schematically in Fig. 3. Plume samples ware withdrawn

through ● sampling rake of four 1.6 mm diameter tubes, each tube being routed

to a aaparate flame ionization detactor. The rake was traversed acrosa the

plume vertically at various axial positions downs tream of the source. The

measured concentration profiles were fit to the Gaussian plums aquation to

data rmine the mean and tha eundard deviati~~ UT of tha diatributione. We

assumed that uv = uz = UT, and this warn varified by meaeuring both vertical and.

lataral profiles ● t a few axial Locatione.

Tlm m~andering of thg plume centroid UC was determined bv Mking

inawntanaoua photographs of a smoke plume which was injacted isokine tically

into the tunnel as ehown schematically in Fig. 3. The‘hrough Saa

plums was photographed at#’ intarvals ueing a shutter spacd of 1/500 a.A

Nagativss of the exposures were digitiaod ueing a scannina microdansitomttar to

obhin ~~rtical pro filas of optical deneityo and the s-coordinate of tha plum,

ccntroicl W-S obtained by taking the first mom~nt of thaoo pro fil~e. A typical

photograph, optical density pro fil~s ●nd centroid Locetlona art mhown in Fig. 4

,“

4

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for O<x<3m. Values of UC for an ensemble of 70 photographs are shown in

Fig. 3 ●long with the UT values discussed abovec

Reaulta of the Monte Qrlo simulatzoas for the wind tunnel conditions are

9 lao shown in Fig. 5. Values of aw ●nd \z measured as a function of dieurice

downwind of the source were used in these simulations, and the turbulence was

assumed to be frozen with a and ~ infinitely large. Thus there were no free

parameters to ad~uat in the model. ‘he agreement between the experimental

results and tha two-particle simulations is very good, but the one-particle

model undareatimatea UC aignificently. It is therefore concluded that the two-

particle

relativa

coneista

model provides a more ●ccurate description of plume meandering and

diffusion. This experimental evaluation is limited, however, eince it

of only one set of flow conditions which produced frozen turbulence.

ACKNOWLEDOHEN’TS

The participation of Los Alamos personnel in this proJect was eupported by

the U.S. Army Atmospheric Sciences bboratory and the U.S. tkpt. of Energy. We

gratefully ●cknowledge the contributions of William Ohmstede, Donald Hoard,

tirmen Neppo, Brian Templeman, William Snyder, and Roger Thompeon.

REFERENCES

Baldwin, L. V. ●nd Johnson, G, R. (1972) The estimation of turbulent

diffusivitlaa from ●nemometer measurements - 1, Fluid particles. Colorado

State University, CEP71-72LVB-0RJ42.

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Batchalor, G. IL (1950) The applicatioti of *a similarity theory of turbulence

to a~oapheric diffuoiom. Q.J1 R. met. Sot. 76, 133-146..—— —

Gifford, F. A. (1982) Horizontal dlffusiom in the a~osphere: a Iagrangiaa-

dyuamical theory. Atmospheric Environment ~, 505-512.

Gifford, F. A. (1983) Diacuasion of “Horizontal diffusion in the atioaphere: a

hgrangian-dynamical theory.” Atmospheric Environment~, 196-197.

!

Lee, J. T., Stone, G. L., Iawson, R. E. and Shipman, M. (1985) Monte Carlo

simultion of two-particle relative diffusion using Eulerian stitistice.

Los Alamos National tiboratory, LA-UR-85-2008, Los Alamos, NH 87545.

Lee, J. T. and Stone, G. L. (1983a) The use of Eulerian initiel conditions in

a bgrangian model of turbulent diffusion. Atmoepharic Environment 17,—

2477-2481.

Lee, J. T. and Stone, G. L. (1983b) Eularian-L~ngrangian ralationohips In

Monte-Carlo simulations of turbulent diffusion. Atmospheric Environment

17, 2483-2487.—

Sawford, B. L. (1984) The basis for, and some limiutiona of, the bngevin

equation in atmospheric relative dispersion modeling. Atmospheric

Environment~, 2405-2411.

t- ,.

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Smitb, F, B. (1983) ~fSCU081~ of “Horizontal diffusion in the atmosphere: A

hngranghn-dpmiul theory. ” Atmospheric Environment~, 194-196.

Snyder, U. H. (1979) The EPA meteorological wind tunnel: its design,

construction and operating characteristic. U.S. Environmental Protection

Agency, EPA-600/4-79-051. Hat. Tech. Info. Service, Springfield, VA

22161.

Tennekes, H. (1979) The exponential kgrangian correlation function and

turbulent diffusion in the inertial sub-range. Atmospheric Environment lJ,

1565-1567.

.“

4

t=o

u

t

Fig. 1 Schematic of a typical particle-pair trajectory.

10-1

~ ● ~ TOTAL .0!

---- ONE-PARTICLE /M ‘A

~ TWO-PAFiTICLE /b

#

flO/L = 0.01

(y=l A

/ .0’

I 1 1 m m 1 1 I I I I I I I I

100 101

T

Fig . 2 ‘~parlson of ore-particle and two-particle mdels of wandering abd

relative diffusion.

102

SOURCEGRID PLUME

I\

IAIR u

4.5 m

EXPERIMENTAL

REGION

c#14 OR

SMOKEA

X=o

3F&g./ Sche-tic of experi~ntal arrangement, side view.

—

6.0 m

6 !●

1)

9‘ifFig, # Instantaneous ?~+ photographs of the amokc plume with

vertical profiles of the optical deneity and centroid poeitions over

the region O < x < 3 m.

250

20.0

15.0

Ew

G 1~

5.0

0.0

I

0 UT CONCENTRATION DATA UT I. 0 Uc PHOTOGRAPHIC DATA

o~ TWO-PARTICLE MODEL

‘- - ONE-PARTICLE MODELq

/J+

f!!m– Uc 4

Cf#f(.

0

1 1 I 1 .

0.0 1.0 2.0 3.0 4.0 5.0 6.0J(, m~.—

—~ig. # @qarisom of concentratim and plwtographic data to ome~rticle ●d

,, ,.