UCRL-JC-126353 PREPRINT
Comparison of Reflection Boundary Conditions for Langevin Equation Modeling of Convective Boundary Layer Dispersion J.S. Nasstrom D.L. Ermak
This paper was prepared for submittal to the American Meteorological Society 12th Symposium on Boundkry Layers and Turbulence Vancouver, B.C., Cana&a July 28-August 1,1997
April
.—
1997
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5.5
COMPARISON OF REFLECTKIN BOUNDARY CONDITIONS FORLANGEVIN EOUATKINMODELING OFCONVECTIVE BOUNDARY LAYERDISPERSION JohnS.Nasstrom’andDonaldL, Ermak LamnceaLivermo~NW%m.dXmtCMY . . , valodtias, but doesnot providethe relatbnsh~betweena specific
1. INTRODUCTION Lagrangian stochastic modeling based on the Langevin equation has been shown to be useful for simulatingvertical de of tram materialin theconvdvaboundwyfayer orcBL (e.g.,Luhar&Brittar,1989).ThismodeCng qqxoachm ~t for the effectsof the bng vafocitycorrelationtime scafea(asaockted with large-scab turbulent structures),skewed vertical vebcity d@riiions, and vartfcaflyinhomogenaousturbulentproperties foundin the CBL. It has bean recognizedthat Langevinequation modelsassumingskewedbut homogeneous vebcity statisticscan c+ture the importantaspedsof diffusionfromsourcesin the CBL, -[y ~~~ aOurCSS (tfurfayandPhysick1993), We comparethree reflectionboundaryconditionsusingtwo different Langevin-equatbn-basad numericalmodelsfor vertical dispersion in skewed, homogeneousturbulence. One model, dewibad by ErmakW Nasatrom(1995),is basedon a Langevin equationwith a skewedrandomforce and a Iiiaar deterministic force.The secondmodal,usedby Hurfeyand Physick(1993),is basadonaLan@vinaquatbnwitha Gaussianrandomforcaanda non-lineardetarministiiforce.Thsreffadionbwndarycoddions are all basedon the q.ymach describedby Thomson&Montgomery (1994). 2.
REFLECTION BOUNDARY CONDITIONS
Thomson ad Montgomery (1994)based their apprd 10 particfereflactbn at boundar”bson the conca@that a wall-mixed spatialand vebcity daributionawiffbe maintainedif eachparticle thatencounters,for exanp!e,the bwar bourdaryis reflactedwitha vabcity that is chosenso that the ensemble-average upwardand downwardparticleflux is the sameas if therewarean imaginary r~ofWdti*@~.Sn@ti@fixdtbh* mustbe zero,thiscriterii reducesto ‘jwipf(wi)tii
‘jwrpf(wr)dwr
(1)
Yf(j--
is thefluid v*O diatribuhn attheboundary,w is the “kwdant(downward)vabcityand w, is the refieciwl(upww$ vebcity (the same relationshipwith wi and w, interchanged appfies at the upper boundary.)The probabilitydistributionof rellactti (posihw)velocitii is proportionalto the flux 01particles withamPwardvafocdywandis
where
P+(w)= AwPJw),
W>0,
where A is a positive normalizationconstant.The analogous distributionforrnddant(negative)vebcitii is W< O. P_(w)= -AwPf(w), The criteriagivenby Eq. (1) providesa retationshpbetween the ensemble01incidentvebciiies andthe ensembleof reflected ‘ Correspondingauthoraddress JohnS. Nasstrom,L-103,LLNL, P.O. Box 808, Livermore, CA 94550, USA; e-mail:
[email protected]
Wi andthe resultantw,. Thus,anyralationahiibetweenWj md w, tha meals the Eq. (1) criteria wifl maintainthe welf-mixed conditii. (we wiffthen@y theserelatiish@a underwall-mixed andnon-wallmixedconditions.) One method(methodI) of i@ment@ Eq.(1),thatresultsin wiarldwr, kto apOSitiVe ~bSt~tfWm@tUdIBSOf choose wrsucfrthat ~~+(w)dw= o
~~-(w)dw.
Wfwn rf (w) is GauAan (orly
otherfunctionsymmetriiabout w = O), thismethodrealms to thewaif-knownreflerlii boundary W, = -wi. Andhw tihd (methodII) thd resultsin a cddii negativecwrelatibnbetweenths rM@tuds d ~i ~d W,, is to ChoeaWr suchthat w.
w,
p+(ww= p-(w)dw. o
-00
A thirdmethod(methodIII) is to randomtyselecta reflectedWocity valuefromthe d~tributbn f’+ at the lowerboundary( f’_ at the upperboundary). We implementedthesemethodsby constructingtablesof w versus cumulative probability using 256 bins from w = -120W to 120W withevenfyspacedintervalsof w, and linearlyinterpolating betweenvalues. 3.
COMPARISON OFREFLECTION METHODS
We firsttestedeachof the threeraflec+ii methodswitheach Langevinequationmodalto determinethe time step requiredto maintaina well-mixeddwributii. Wepdonned simufatibnswitha velocityskawnessofl, LagrangiantimescaJa t=o.5(h/c7w), and an initial uniform spatial distriition of particles between boundariiat z= Oandz=h. Dapariurestroma uniformspatial d~tributionof less than approximately5% wereobtainedusinga time step Ar=0.05r, and I%usingatims step Ar=O. OIT. werecalculatedusing500,000partidasand (Positbndistributions 40 binsbetweenthe top andbottomboundary,and wereaveraged from r/t=l to 4.) We then compared the three refledion methods using simulationsof dispamii horna continuouspointsourcein theCEIL. UsingdatafromW* and Daardodf’s(1981)laboratoryexperiment (saaFigc1) froma sourceat nondiianaional height
