A Lagrangian stochastic model for dispersion in stratified turbulence

Transcription

1 Iowa Sae Universiy From he SelecedWorks of Paul A. Durbin 2005 A Lagrangian sochasic model for dispersion in sraified urbulence S. K. Das, Sanford Universiy Paul A. Durbin, Sanford Universiy Available a: hps://works.bepress.com/paul_durbin/1/

2 A Lagrangian sochasic model for dispersion in sraified urbulence S. K. Das and P. A. Durbin Ciaion: Physics of Fluids 17, (2005); doi: / View online: hp://dx.doi.org/ / View Table of Conens: hp://sciaion.aip.org/conen/aip/journal/pof2/17/2?ver=pdfcov Published by he AIP Publishing Aricles you may be ineresed in A sochasic perurbaion mehod o generae inflow urbulence in large-eddy simulaion models: Applicaion o neurally sraified amospheric boundary layers Phys. Fluids 27, (2015); / Verical dispersion of ligh inerial paricles in sably sraified urbulence: The influence of he Basse force Phys. Fluids 22, (2010); / On Lagrangian ime scales and paricle dispersion modeling in equilibrium urbulen shear flows Phys. Fluids 16, 3374 (2004); / Lengh scales of urbulence in sably sraified mixing layers Phys. Fluids 12, 1327 (2000); / The effec of nonverical shear on urbulence in a sably sraified medium Phys. Fluids 10, 1158 (1998); /

3 PHYSICS OF FLUIDS 17, A Lagrangian sochasic model for dispersion in sraified urbulence S. K. Das and P. A. Durbin Deparmen of Mechanical Engineering, Sanford Universiy, Sanford, California Received 31 Augus 2004; acceped 19 November 2004; published online 24 January 2005 In his paper we discuss he developmen of a Lagrangian sochasic model LSM for urbulen dispersion of a scalar species. Given any ensorally linear second-momen closure SMC urbulence model we show how o derive a mahemaically equivalen se of sochasic differenial equaions SDEs, i.e., he second-momen equaions consruced from hese SDEs are exacly he same wihin a realizabiliy consrain as he given SMC. This se of equaions forms he LSM. Boh urbulence anisoropy and buoyancy effecs are incorporaed by his mehod. In order o achieve he correc criical Richardson number and o obain he simples Lagrangian formulaion, a revised se of consans for he isoropizaion of producion model is proposed. They improve agreemen wih experimens. The LSM is applied o homogeneous shear flow wih varying degrees of sraificaion. Our model is shown o capure imporan physics associaed wih buoyan flows and we also paramerize our resuls. Finally he form of he curren model is compared wih a few of he well-known Lagrangian sochasic models American Insiue of Physics. DOI: / I. INTRODUCTION Taylor 1 inroduced he Lagrangian sochasic model LSM for urbulen dispersion in he amosphere. His seminal model was valid only for homogeneous urbulence. Over ime, varians have been explored. Legg and Raupach 2 made an imporan revision by adding a erm ha correcs a spurious drif which occurred when he velociy variance was no homogeneous. Thomson 3 reviewed LSMs and formulaed a well mixed crierion as a furher mehod of assessing differen formulaions. He also described how o design LSMs o mee his crierion. Wilson and Sawford 4 provide a brief review of models ha mee his consrain. Mos of hese models did a fairly good job for unsraified flows bu hey were no designed for hermal sraificaion and proved unsaisfacory in srongly sably sraified flows. 4 Pearson e al. 5 proposed a heoreical model for dispersion in horizonal mean flow wih sably sraified, saionary urbulence. They added a separae equaion for he flucuaing densiy, conaining a parameer ha was described as a measure of molecular mixing processes. Wih no molecular mixing =0 he mean square displacemen of he released fluid paricles ceased o grow afer a cerain ime whereas nonzero, posiive values of produced a linear increase a large imes. Venkaram e al. 6 proposed a semiempirical model ha did no suppor he role of molecular mixing in urbulen dispersion in sably sraified amospheric flows. Comparison of boh hese models wih experimens was no conclusive abou he role of molecular mixing. Alhough hese models were designed for sraificaion, hey did no ake oher complexiies such as mean shear and urbulence inhomogeniy ino accoun. Lagrangian dispersion has largely been pursued in he conex of amospheric dispersion. By naure i is a saisical closure. The individual, random rajecories are an arifice. Their purpose is o produce saisics via Mone Carlo simulaion. In some sense here should be a parallel beween LSMs and Eulerian second-momen closure models, which is he heme of his paper. Second-momen closure models have been designed for complex engineering flows. They can rea complexiies such as urbulence anisoropy, inhomogeniy, mean shear, buoyancy, ec. This suggess ha such effecs migh be incorporaed ino Lagrangian formulaions by developing a correspondence beween he Lagrangian and Eulerian closures. Essenially, he aim is o incorporae sraificaion ino he LSM by making i consisen wih he Eulerian formulaion. The premise is ha he Eulerian closure provides accepable predicions or, perhaps, ha he laer is a vehicle for developing he empirical modeling, which hen can be exploied by he LSM. Pope 7 examined he relaion beween a generalized Langevin model and second-momen closures. In his approach, he Langevin model is essenially an Eulerian sochasic equaion ha implemens a probabiliy densiy funcion pdf closure. He showed how he second momen of he Langevin equaion was a Reynolds sress closure. Hence, he explored he relaion beween sochasic models and secondmomen closure models. Durbin and Speziale 8 consruced sochasic differenial equaions whose second momen would exacly reproduce a given Reynolds sress closure up o a realizabiliy consrain. They did no address sraified urbulence. The objecive of he presen paper is o exend he approach of Durbin and Speziale; 8 ha is, o develop a Lagrangian model ha accouns for sraificaion effecs by devising i o reproduce a given second-momen closure SMC model. This is an alernaive o previous mehods 5 for devising buoyan componens. We show how o consruc a se of sochasic differenial equaions SDEs whose second momens equal o a given, general form of SMC equaions for Reynolds sress, Reynolds emperaure flux, and emperaure /2005/17 2 /025109/10/$ , American Insiue of Physics

4 S. K. Das and P. A. Durbin Phys. Fluids 17, variance. Again, his is done up o a realizabiliy consrain: he sochasic model mus produce realizable saisics. Mos SMC models do no ensure realizabiliy; indeed, by deriving hem from a valid sochasic equaion hey can be revised o ensure realizabliy. 8 Heinz 9 derived sochasic models ha included an equaion for poenial emperaure and, hence, are suied for hermally sraified flows. However, his formulaion did no reproduce he well-known SMC models, such as ha of Gibson and Launder. 10 We choose he general linear formulaion as he SMC. This is widely used wih he Gibson Launder coefficiens. 10 Recenly, Ji and Durbin 11 have analyzed his model and found a few deficiencies. I produces a criical Richardson number ha is significanly oo large. We propose a differen se of values for consans for he Reynolds scalar flux and variance closures. The consans of he Reynolds sress equaions reain heir acceped values. The recalibraion of scalar ranspor is no essenial o our formulaion, however, a criical Richardson number of abou 0.25 is desirable for sraified flow applicaions. The presen consans improve agreemen wih experimenal and numerical daa. II. MATHEMATICAL FORMULATION A. Turbulence second-momen closure The general linear model forms he framework for he sochasic model. This closure of he Reynolds sress ranspor equaions is conained in he righ side of du i u j d u i = c 1 k u i u j 2 3 k ij c 2 P ij 2 3 P ij c 3 D ij 2 3 P ij c s ks ij + P ij 2 3 ij + 1 c 5 G ij c 5G ij, = c 1 k u i 1 c 2 u k U i x k + c 3 u k U k x i 1 c 4 u i u k x k 1 c 5 g i 2, d 2 = 2 u k R x k k 2. These are he equaions for homogeneous urbulence. In 1 S ij = 1 2 U i x j + U j x i, 1 P = 1 2 P kk, G = 1 2 G kk. U i and u i are he mean and flucuaing velociies, and are he mean and flucuaing emperaures, is he ime, x i is he spaial coordinae, g i is he acceleraion due o graviy, is he hermal expansion coefficien, k is he urbulen kineic energy, is is dissipaion rae, and R is a ime-scale raio of mechanical o scalar ime scales. Terms wih an overbar mean ha hey are ensemble averaged quaniies. R is no a universal consan, is value may vary for differen flows. According o Warhaf, 12 in he presence of a uniform mean scalar gradien, R is found o be very nearly consan and equal o 1.5. This model can be derived by expansion abou isoropy. The general linear expansion allows he consans c 1, c 2, c 3, c s, c 5, c 1, c 2, c 3, c 4, c 5, and R. Their values have been derived mosly by calibraion agains experimens and by heoreical reasoning. Furher deails can be found in Durbin and Peersson Reif. 13 B. Sochasic differenial equaions Our goal is o derive a se of Langevin sochasic differenial equaions ha will have 1 as heir second momen. Firs we review how o consruc he second-momen equaion of a given Io-ype sochasic differenial equaion. For more informaion he reader is referred o exs such as hose by Arnold 14 and Øksendal. 15 Le us ake he case of he simples Langevin equaion du i = u i T + c 0 1/2 dw i, where u i is he dependen variable a random funcion of T is a ime scale, c 0 is a consan, is a deerminisic funcion of, and W i is he Wiener sochasic process. For derivaion of he second-momen equaion we need he following properies of dw i : dw i =0, dw i dw j = ij, u j dw i =0, dw = O 1/2. Then he derivaion of he second-momen equaion proceeds as follows. Evaluae he differenial, keeping erms o order du 2 : d u i u j = u i + du i u j + du j u i u j = u i du j + u j du i + du i du j. Average he above using 3 o ge 2 3 P ij = u i u k U j x k u j u k U i x k, D ij = u i u k U k x j u j u k U k x i, G ij = g i u j + g j u i, du i u j = 2 u iu j T + c 0 ij. 4 This is he second-momen closure corresponding o 2. In essence his is he procedure used herein o derive secondmomen urbulence closure equaions from he sochasic differenial equaions of our Lagrangian model. Subsequenly, we will no deail he seps.

5 A Lagrangian sochasic model for dispersion Phys. Fluids 17, C. Lagrangian sochasic model If we choose he values of c 2, c 3, and c 5 in Eq. 1 as equal o c 2, c 3, and c 5, respecively, and se c s =0, hen he simples form of our model resuls. The sandard isoropizaion of producion IP model uses c s =0. The general form of our LSM will be described in he following secion where he moive for hese specificaions of he consans will become clear. The simplified form is du i = c 1 2 k u U i U k i + c 2 1 u k + c 3 u k x k x i 1 c 5 g i + c0 dw i, d = c 1 c 1 2 k 1 c 4 u k + c dw. x k The coefficiens c 0 and c are specified as c 0 = 3 c c 2 + c 3 P + c G 5, 6 c = 2c 4 u k + x k k 2c 1 c 1 R. 2 Realizabily akes he form of requiring hese o be nonnegaive, so ha he square roos in Eq. 5 are real valued. If c 1SMC is a sandard value of he consan c 1, he Langevin model will be well posed if he reurn o isoropy consans are replaced by 8 c 1 = max c 1SMC,1 c 2 + c 3 P c G 5, c 1 = max c 1 SMC,0.5 2 k c4 u k + c x k R. The mehod of he preceding secion shows how 5 wih 6 reproduces 1. Alhough 5 is conrived only o reproduce 1 mahemaically, i inheris cerain physical aribues. Oscillaory moion is produced by he resoring force g in he firs equaion. The naural frequency is less han he Brun Väisälä frequency. I equals 1 c 5 gs, S being he mean emperaure gradien in he negaive direcion of graviy. Pearson e al. 5 argue ha inernal wave radiaion can be modeled in his way. The frequency of radiaed inernal waves is he Brun Väisälä frequency imes he cosine of he angle o he direcion of graviy. The empirical consan 1 c 5 plays he role of he average over radiaion angles in he physical reasoning of Pearson e al. 5 The densiy flucuaions of fluid elemens relax oward he mean sae by mixing wih heir environmen. The second of equaions 5 can be characerized as describing his via linear damping of he emperaure flucuaion of he fluid paricle. I migh be quesioned wheher i is valid o assume ha he paricle relaxes oward he ensemble averaged environmen. Tha raionale was no invoked in 5, we only required i o reproduce he saisics of he SMC. However, his physical inerpreaion is available. 5 A calibraion of he consans and discussion of he properies of his model are presened in a laer secion. D. General case The Langevin equaion of he general linear model is somewha more involved han 5. The whie noise forcing is no longer isoropic. A device inroduced by Durbin and Speziale 8 can be used o ensure he desired second momen. We presen he equaion wihou derivaion. To a large exen i is a maer of working backward from he SMC o he sochasic equaion. However, he laer is no unique; he second momen does no uniquely deermine he sochasic process. The following is a form ha has 1 as is momen equaion: du i = c 1 2 k u U i U k i + c 2 1 u k + c 3 u k x k x i 1 c 5 g i + M ik dw k + P ik dw k + D ik dw k + G ik dw k iv + c 0 1/2 dw i, d = c 1 c 1 2 k 1 c 4 u k + c 1/2 x dw. k The marices muliplying he whie noise erms are devised o produce requisie erms of 1. In order o do so, hey mus saisfy he following: M ik M kj S ij = c s ks ij, P ik P kj P ij = c 2 c 2 P ij, D ik D kj D ij = c 3 c 3 D ij, G ik G kj G ij = c 5 c 5 G ij. The righ-hand sides of he above equaions are he erms ha appear in he SMC model equaions. In order o ge hose erms when second-momen equaions are generaed from he general form of he LSM, he marices given in he firs erms on he lef of Eqs. 8 are required. The s which are he maximum eigenvalues of he marices on he righ achieve he equaliy. The coefficiens c 0 and c are 1 now given by c 0 = 3 c c 2 + c 3 P + c G 5 1 S + P + D + G, 9 c = 2c 4 u k + x k k 2c 1 c 1 R 2. Realizabiliy requires ha each erm under he square roo in he SDEs be posiive. The urbulence consans are such ha his is saisfied in mos siuaions. Bu someimes c 0 and c, as calculaed by he above equaions, may become negaive. As previously noed, uncondiional realizabiliy is enforced by defining c 1 and c 1 as 7 8

6 S. K. Das and P. A. Durbin Phys. Fluids 17, c 1 = max c 1 SMC, 1.5 S + P + D + G c 2 + c 3 P c G 5 +1, 10 c 1 = max c 1 SMC,0.5 2 k c4 u k + c x k R, where c 1 SMC and c 1 SMC are he consans c 1 and c 1 of he original SMC model. The marices M, P, D, and G can be regarded as generalized square roos of he marices in he righ-hand side of 8. We show he procedure o calculae M. The res can be calculaed by he same mehod. Marix c s ks can be diagonalized by he orhogonal marix U of eigenvecors as follows: c s ks = U diag 1, 2, 3 U, 11 where I is no very difficul o show ha he generalized square roo M is hen given by M = U diag 0, 1 2 1/2, 1 3 1/2 U. 12 In saionary, nonhomogeneous flow, he mean Lagrangian ime derivaive is D u i = u iu k, D = u k. x k x k These erms mus be included in he Lagrangian equaion o make i consisen wih he Eulerian mean. 16 Legg and Raupach 2 derived such erms by aribuing hem o a mean pressure gradien. They are needed o avoid spurious drif and o saisfy he well mixed condiion. Afer adding hese erms, he final and mos general form of our sochasic equaion is du i = c 1 2 k u U i U k i + c 2 1 u k + c 3 u k x k x i 1 c 5 g i + u iu k x k + M ik dw k + P ik dw k + D ik dw k + G ik dw k iv + c 0 1/2 dw i, d = c 1 c 1 2 k 1 c 4 u k + u k x k x k + c 1/2 dw. 13 The second-momen equaions consruced from hese sochasic differenial equaions can be shown afer a fair bi of algebra o be similar o he Eulerian closure equaions. In inhomogeneous urbulence, he SMC model conains riple correlaion erms whereas in he Lagrangian approach hese erms are a naural par of he Lagrangian derivaive. So no exra erm is required. However, in he Eulerian models, pressure diffusion erms need o be modeled; usually hey are clubbed ogeher wih he urbulen ranspor erms and are modeled collecively. In he curren approach, he pressure diffusion is negleced. Also, as menioned before, he above equaion se 13 is no unique. SDEs, oher han hose given by 13, can be consruced ha can generae he same second momens. Indeed, a unique se of SDEs canno be consruced by consideraion of second momens alone. Heinz 9 also derived Lagrangian equaions for urbulen moion and buoyancy in inhomogeneous flows. He derived he corresponding Fokker Planck equaion for he pdf, generaed momen equaions from i, and evaluaed he unknown erms of his SDEs by comparing erms wih he conservaion equaions of momenum and poenial emperaure and ranspor equaions of second momens. Bu hese secondmomen equaions were no any of he well-known SMC models and hence his LSM could no reproduce he caegory of Eulerian closure models ha are considered here. III. MODEL CALIBRATION The IP model is one of he simples and mos widely esablished SMC urbulence models. Bu unil recenly i had no been esed properly for buoyan flows. Ji and Durbin, 11 while sudying he srucural equilibrium behavior of SMC models in sably sraified, spanwise roaing, homogeneous shear flows, found ha he IP model did no perform well for values of gradien Richardson number around This is he criical value of Ri g a which urbulence becomes saionary. Ji and Durbin 11 fel ha SMC urbulence models in general should be calibraed o have he correc sabilizaion poin, and proceeded o do so. In his secion we propose anoher se of consans differen from heirs o use wih our simplified model 5. Roaion is no aken ino accoun, as his is no our focus. Only hose consans ha are associaed wih acive scalar ranspor in he sandard IP model will be alered, leaving he oher consans unouched. Here, he choice is made wih he Lagrangian formulaion as a guideline. From he general form of he LSM given by Eqs. 13 and 8, i is clear ha if c s =0 and c 2, c 3, and c 5 are made equal o c 2, c 3, and c 5, respecively, hen he simplified form of he LSM is arrived a. The proposed values of he se of consans is given in Table I. The nex sep is o es hese coefficiens agains experimenal and numerical DNS and LES daa for sraified flows. These are experimens of Tavoularis and Corrsin 17 and Rohr e al., 18 direc numerical simulaions of Rogers e al., 19 TABLE I. Model consans for IP and his model. Consans c 1 c 2 c 3 c 5 c s c 1 c 2 c 3 c 4 c 5 R IP model Our model / /3 1.5

7 A Lagrangian sochasic model for dispersion Phys. Fluids 17, The models do no predic he iniial ransien S 4 which may be due o mismach beween he iniial condiions of he experimens and he model calculaions. Bu i is clearly seen ha our model performs much beer han he IP model afer he iniial ransiens. Figure 4 compares he evoluion of R uw and R w wih experimenal and numerical DNS and LES daa. The predicion of his model is given in he ime inerval 8 S 17. Since all he experimenal and numerical daa were aken during he ime 8 S 15, he performance of his model seems reasonable. FIG. 1. Schemaic for flow configuraion. Gerz and Schumann, 20 Hol e al., 21 and Shih e al., 22 and large eddy simulaions of Kalenbach e al. 23 They have all focused on homogeneous sraified shear flow cases. Homogeneous shear flow wih varying degrees of sable sraificaion characerized by differen values of Ri g is compued. The mean velociy is unidirecional, wih a specified consan value of du/ dz. The mean emperaure gradien d /dz is posiive and consan. A schemaic for he flow is given in Fig. 1. Ji and Durbin 11 have specified he problem, he governing differenial equaions, and he iniial condiions in greaer deail. We do no repea hose deails here, insead he reader is referred o heir work for more informaion. We show es resuls for boh he original IP model and he consans ha we are proposing. Figure 2 shows he urbulen kineic energy evoluion. Time is nondimensionalized as S where S du/dz, and urbulen kineic energy has been normalized by is iniial value k 0. Oher parameers appearing in he figures are defined as u u 2, R uw uw/ u w, and R w w / w, where 2. I is clear ha a Ri g =0.25 he IP model shows an increase in urbulen kineic energy wih ime, while i neiher grows nor decays for our model. In fac Ji and Durbin 11 repored ha he IP model predics he criical Ri g a 0.48 whereas in our model i is prediced a around 0.24 which is exremely close o he experimenally observed value of Figure 3 shows he evoluion of he rms velociy u. IV. DISPERSION IN HOMOGENEOUS STABLY STRATIFIED SHEAR FLOW In dispersion analysis, he Reynolds sress is given and he saisics of he dispersing paricles are o be compued. For he simplified model, only k,, P, and G are needed. They are obained by solving he ordinary differenial equaions of he SMC urbulence model given by Eqs. 1. The values of he consans are specified in Table I. For his problem he LSM reduces o du = c 1 2 k u+ c 2 1 w du dz + c 0 1/2 dw u, dw = c 1 2 k w+ 1 c 5 g + c 0 1/2 dw w, d = c 1 c 1 2 wd k dz + c 1/2 dw, where he consans c 0 and c are given by c 0 = 3 c 2 uw du 1 c 2 dz + c g 5 w 1, 15 c = k 2c 1 c 1 R Realizabiliy is ensured by calculaing c 1 and c 1 as follows: FIG. 2. Evoluion of urbulen kineic energy wih ime a differen Ri g. Lines: dash, 0; dash do, 0.13; solid, 0.25; hick dash, 0.37; hick dash do, 0.48; hick solid, 0.60.

8 S. K. Das and P. A. Durbin Phys. Fluids 17, FIG. 3. Evoluion of u rms velociy wih ime a differen Ri g. Lines model predicions : solid, 0; doed, 0.075; dash, 0.21; dash do, Symbols experimenal daa of Rohr e al :, 0;, 0.075;, 0.21;, uw du c 1 = max c 1 SMC,1 + c 2 dz c g 5 w, c 1 = max c 1 SMC,0.5 c 1 + R. 16 Iniially he focus is on he case where urbulence is saionary. As menioned before, he SMC urbulence model given by 1 predics saionary urbulence a Ri g =0.24. The advanage wih saionary urbulence is ha while solving he LSM we deal wih fixed values of urbulence saisics and do no have o worry abou hese values changing wih ime. I is o be noed hough ha he LSM in general is capable of handling nonsaionary urbulence, in ha case he Eulerian saisics ha are o be fed in our SDEs will be varying in ime. In his problem we are pariculary ineresed in he variaion of dispersion lengh squared, Z 2 wih ime. Z 2 is calculaed as he mean of he Z 2 over a large number of rajecories, where Z for each paricle is defined as Z w. = 0 For saisically saionary urbulence we have Z 2 evaluaed as Z 2 = 0 0 w w = w 0 2 R ww 0 =2w 2 0 R ww d. 17 So if he verical velociy auocorrelaion R ww is known hen he verical dispersion lengh can be readily calculaed. I urns ou ha for his problem an analyical closed form soluion of our model exiss. To proceed furher i is necessary o find ou an equaion for R ww. In order o achieve his he w equaion needs o be manipulaed. Subsiuing T L =2k/ c 1 and c 5 =1/3 in he w equaion i can be wrien as dw = w T L g + c 0 1/2 dw w. 18 Muliplying boh sides of his equaion by w, aking he mean and finally dividing by w 2 0 gives FIG. 4. R uw and R w vs Ri g. Lines model predicions : solid, S=8; dash do, S=11; do, S=14; dash, S=17. Symbols:, Rohr e al ;, Tavaoloris and Corrsin 1981 ;, Hol e al ; +, Gerz and Schumann 1991 ;, Kalenbach e al

9 A Lagrangian sochasic model for dispersion Phys. Fluids 17, dr ww = 1 R ww + 2 T L 3 gr w. 19 The derivaion invokes he nonanicipaing propery w dw =0 for 0. To find an equaion for R w w /w 2 0, a similar procedure is followed for he sochasic equaion, o find dr w = 1 T L R w S R ww, 20 where T L =2k/ 2c 1 c 1 and S =d /dz. In marix noaion he above wo equaions can be wrien as dr = AR, 21 where R = R ww R w, A = 1/T L The soluion of 21 is R = e A R 0. S I is no very difficul o show ha R ww = Ae A + Be B, where 2 3 g. 1/T L Z 2 =2w 2 0 A A + B B + A A 2 e A 1 + B B 2 e B 1, A = T L + 1 T L + 1 T L 1 T L N2, B = T L T L 1 1 T L T L N2, N 2 = gs, A = p sr ww 0 qr w 0 / ps qr, B = q pr w 0 rr ww 0 / ps qr, p = A 1/T L / A 1/T L 2 + S 2, q = B 1/T L / B 1/T L 2 + S 2, r = S / A 1/T L 2 + S 2, s = S / B 1/T L 2 + S The facor of 8/3 muliplying N 2 is 4 1 c 5. A and B are eigenvalues of he marix A which appears in Eq. 21. N is he Brun Väisälä, or buoyancy, frequency. So in addiion o he urbulen ime scale, anoher due o buoyancy has made an appearance in he soluion. This eners hrough he addiional SDE for. For moderae o srongly sable sraificaion NT L 1, so in general he eigenvalues A and B can become complex, especially for srongly sable sraificaion. As given in Eqs. 23, R ww and Z 2 conain erms such as e A and e B, so we expec o see wavelike oscillaions in he soluion; hese are effecs of inernal graviy waves in he flow field. Consider he shor ime and long ime behaviors of Z 2. As 0, Z 2 w from he soluion 23. This is he sandard shor ime behavior expeced from all Lagrangian models. As, Z 2 2w 2 0 A A + B B A A 2 + B B 2. Hence, he model predics a linear increase of Z 2 a large imes. Differeniaing 17 wih respec o gives dz 2 =2w 2 R ww d. 0 0 As he inegral becomes he Lagrangian ime scale of urbulence, which in cases of buoyan flows is expeced o have posiive values. Thus, a large imes, Z 2 is expeced o grow linearly, as he model predics. As menioned before, he soluion of Eqs. 1 gives he Reynolds sresses for saionary, sraified, homogenous shear flow. The value of he dissipaion rae is obained from DNS sudies, 22 which sae ha a high Reynolds number he value of he dimensionless shear rae Sk/ becomes consan a around 5.5. Figure 5 shows plos of normalized verical dispersion lengh squared, 2 Z 2 N 2 /w 2 0 and verical velociy auocorrelaion R ww wih normalized ime N a criical Richardson number. A shor imes some small oscillaions are seen which are due o inernal graviy waves bu hey ge damped ou prey quickly. In order o see heir effec more clearly he urbulen ime scale is increased by changing he value of Sk/ o 20. The resuls are ploed in Fig. 6 where he effec of he waves are clearly seen in boh he 2 and R ww plos as more pronounced damped oscillaions. Afer hey decay away, 2 shows a linear increase wih N as prediced by our analyical soluion. In order o furher explore our model, he LSM is solved for a range of values of Ri g oher han he criical one. I is well known ha for his problem he so-called moving equilibrium soluion o he SMC equaions exiss, i.e., he raios of all he Reynolds sresses, scalar fluxes, and variance o he urbulen kineic energy are seady whereas only he urbulen kineic energy iself varies wih ime. In Fig. 2 i is seen ha a values of Ri g greaer han Ri g cri he urbulen kineic energy ends o show much less variaion in ime han for cases where Ri g Ri g cri. Hence he Eulerian saisics for hese cases will be varying quie slowly wih ime. So jus for illusraion purposes i seems plausible o feed consan values of hese saisics ino our LSM for values of Ri g higher han criical. The LSM is solved numerically for values of Ri g =0.25, 0.30, 0.35, and The Eulerian saisics correspond o consan urbulen kineic energy which is he same for all

10 S. K. Das and P. A. Durbin Phys. Fluids 17, FIG. 5. a Normalized verical dispersion lengh squared 2 Z 2 N 2 /w 2 0 and b verical auocorrelaion R ww vs normalized ime N a criical Ri g wih Sk/ =5.5. cases. The problem specificaion is du/dz=1.0 s 1, d /dz =5.0 K/m, g=9.81 m/s 2, and he value of is adjused for differen Ri g. The value of k is fixed a m 2 /s 2 and Sk/ a 20. The values of he res of he urbulence saisics are obained from he moving equilibrium soluion of he SMC model. Figure 7 shows he resuls. Par a shows how he verical dispersion is reduced as sabiliy is increased. From par b i is apparen ha he frequency of he inernal graviy waves follows he 0.8N curve quie closely. Mahemaically he imaginary par of he complex eigenvalues A and B, given by Eq. 23, gives he frequency of he inernal graviy waves for saionary urbulence. Assuming he inegral ime scale of urbulence o be much larger han he buoyancy ime scale a very reasonable assumpion he imaginary par of he eigenvalues becomes 0.82N. I can be inerpreed o mean ha he inernal graviy waves are radiaed a an angle of abou arccos 0.8, i.e., 37 wih he horizonal. Pearson e al. 5 assumed in heir model ha inernal graviy waves were radiaed a an angle of arccos 0.8 for sably sraified flows. They explicily specified he value of 0.8 as a consan o capure he angle of radiaion; in our case he value comes from he specificaion of he consan c 5 in he SMC model. We also paramerize he dispersion curves. In his problem here are wo ime scales, viz. he inegral ime scale T L and he buoyancy ime scale 1/N. Figure 8 a shows a plo of he dimensionless dispersion lengh wih ime nondimensionalized as N. The curves do collapse a small imes bu a large imes hey separae. The frequencies of he oscillaions of he curves mach quie well. In Fig. 8 b he ime is nondimensionalized by he inegral ime scale T L. Overall he curves collapse ono each oher bu here is no mach wih he frequencies. So i is clear from he figures ha only as long as he inernal graviy waves have an effec on he dispersion, N is a good ime parameer; for very shor imes i does an excellen job. However, he inegral ime scale conrols he overall dispersion, especially a large imes. The same conclusion can also be arrived a mahemaically from he analyical soluion for Z 2 given by Eq. 23. The inernal graviy waves come only hrough he exponenial erms and a large imes hose erms become neglible. On he oher hand he real par of he eigenvalues conain he inegral ime scale and hough i has an effec hroughou he process, i becomes relaively much more imporan a large imes. V. COMPARISON WITH PREVIOUS MODELS The random force model of Legg and Raupach 2 consised of a single equaion for w which in differenial form can be wrien as dw = w + dw2 1/2 2w2 T L + dz T L dw w. 24 The w equaion of he curren LSM for one-dimensional neural flows reduces o FIG. 6. a Normalized verical dispersion lengh squared 2 and b verical auocorrelaion R ww vs normalized ime N a criical Ri g wih Sk/ =20.0.

11 A Lagrangian sochasic model for dispersion Phys. Fluids 17, FIG. 7. a Verical dispersion lengh Z rms Z 2 vs ime a differen Ri g. Lines: solid, Ri g =0.25; dash, Ri g =0.30; do, Ri g =0.35; dash do, Ri g =0.40. b Frequency of he inernal graviy waves vs Ri g. Symbol:, frequency from LSM. Lines: solid, N; dash, 0.8N. dw = c 1 dw2 w+ 2 k dz + c 0 1/2 dw w. 25 If we assume 2k/ c 1 =T L and c 0 =c 1 w 2 /k hen our w equaion becomes exacly 24. Furhermore if he urbulence inhomogeniy erm is dropped hen he LSM revers back o Taylor s model. 1 Pearson e al. 5 developed a model for horizonal mean flow wih sably sraified, saisically saionary urbulence ha consised of equaions for w and flucuaing densiy. Afer convering o emperaure from densiy heir equaions are given by dw = 2 Nw + g + H, 26 d = N w d dz, where and are consans and H is a saionary random funcion which represens he flucuaing pressure gradien. They claimed ha O 1 was dependen on molecular mixing process and he consan =0.8 se he angle of propagaion of he radiaed inernal graviy waves. Differen forms of he pressure gradien funcion were assumed in heir work. Is represenaion as whie noise is an assumpion of a fla frequency specrum. I is seen ha heir model conains only one ime scale, he Brun Väisälä, or buoyancy frequency, in heir deerminisic erms. However he form of H can be chosen o conain he urbulence ime scale in he random componen. Equaion 14 is he presen LSM for his case. Subsiuing T L, T L, and c 5 gives dw = w T L g + c 0 1/2 dw w, 27 d = wd T L dz + c 1/2 dw. Unlike 26 a urbulence ime scale T L or T L is explicily presen in he model. The analyical soluion given in Eq. 23 shows he presence of he Brun Väisälä frequency as well. In urbulen dispersion problems in a sraified environmen i is expeced o have ime scales boh due o urbulence and buoyancy presen in he soluion. The wo ime scales are relaed in he following way: T L = c 1 k 2 = c 1 S * 2 S = c 1/2 1 2 S*Ri g N where Sk = S*, since Ri g = N2 S 2. For saionary sraified urbulence S * approaches a consan value of abou 5.5. As menioned before, many researchers have prediced he criical value of Ri g =0.25. So T L and FIG. 8. a Normalized verical dispersion lengh Z 2 N/ w 2 0 vs dimensionless ime N a differen Ri g. b Normalized verical dispersion lengh vs dimensionless ime /T L a differen Ri g. Lines: solid, Ri g =0.25; dash, Ri g =0.30; do, Ri g =0.35; dash do, Ri g =0.40.

12 S. K. Das and P. A. Durbin Phys. Fluids 17, similarly T L 1/N under criical condiions. Aside from his, he presen model differs from he model of Pearson e al. 5 VI. CONCLUSION The main objecive of he paper is o devise a mehod o obain Lagrangian sochasic models for dispersion, given a SMC. The moive is ha predicion of urbulen dispersion in amospheric flows can be divorced from predicion of Eulerian saisics. Indeed, he laer are ofen prescribed as surface layer similariy profiles. Given he general, ensorally linear Reynolds sress model, he corresponding Lagrangian model is a se of sochasic differenial equaions reproducing i as heir second momen. A se of consans was proposed for he general linear model. They were seleced o provide a simple form of LSM, while providing he correc criical Richardson number, and reasonable agreemen o experimenal and simulaion daa. The sochasic formulaion also ensures realizabiliy, alhough ha is effeced by consrains on he reurn o isoropy coefficiens. The model was illusraed by compuaions of homogeneous shear flow wih varying degrees of sable sraificaion. Depending on he raio of buoyancy o urbulence ime scale, oscillaory dispersion can be observed. Tha migh a firs seem o be unphysical. A raionale is ha paricles released a a given heigh will move randomly, eiher upward, or downward. Those moving upward will experience negaive buoyancy. A a laer ime, hey will have a endency o move downward. Hence he correlaion funcion will oscillae: upward velociy ends o be correlaed wih downward velociy afer one buoyancy ime scale. The dispersion formula 17 connecs he correlaion funcion o he variance of dispersing paricles. Oscillaions of he correlaion funcion ranslae ino nonmonoonic dispersion. Alhough he curren model has been formulaed following a very differen approach is form has been shown o become very similar o oher well known LSMs when one or more of various simplifying assumpions such as neural condiions, urbulence homogeniy, and absence of mean shear are applied. So his LSM is expeced o give beer resuls for more complex flows. The proposed LSM has a srong physical and mahemaical basis bu furher ess, especially for unsably sraified flow, are needed. Even for sably sraified flows furher experimens are needed o es he model. One meri of he approach is ha i inheris he empiricism of he second momen model. ACKNOWLEDGMENTS This work was funded by NASA Ames Research Cener, wih R. Miraflor being he conrac monior. S.K.D. would like o hank M. Ji for his helpful discussion on calibraion of SMC models and also for providing his codes for doing he same. 1 G. I. Taylor, Diffusion by coninuous movemens, Proc. London Mah. Soc. 20, B. J. Legg and M. R. 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