Turbulent Prandtl number effect on passive scalar advection

Transcription

1 Physica D 5 53 ) Tubulent Pandtl numbe effect on passive scala advection Weinan E, Eic Vanden-Eijnden Couant Institute of Mathematical Sciences, New Yok Univesity, New Yok, NY, USA Abstact A genealization of Kaichnan s model of passive scala advection is consideed. Physically motivated egulaizations of the model ae consideed which take into account both the effects of viscosity and molecula diffusion. The balance between these two effects on the inetial ange behavio fo the scala is shown to be paameteized by a new tubulent Pandtl numbe. Thee diffeent egimes ae identified in the paamete space depending on degees of compessibility. In the stongly and weakly compessible egimes, the inetial ange behavio of the scala does not depend on the tubulent Pandtl numbe. In the egime of intemediate compessibility, the inetial ange behavio does depend on the tubulent Pandtl numbe. Published by Elsevie Science B.V. Keywods: Passive scala advections; Tubulence; Kaichnan model; Regulaizations; Tubulent Pandtl numbe Kaichnan s model fo the passive scala advection [] has become a popula benchmak in the studies of intemittency in hydodynamic tubulence. In this model, one studies the behavio of a scala field, θx,t), passively advected by a tubulent velocity, u ν x, t), and subject to molecula diffusion θ t + uν x, t) )θ = κ θ. ) The velocity field is assumed to be a zeo mean Gaussian andom pocess, white-in-time, isotopic, and such that Eδu ν x,y,t) δu ν x,y,s))= D x y ξ δt s) fo l ν x y l, ) whee δu ν x,y,t)= u ν x, t) u ν y, t) and <ξ<. Hee l is the integal scale o coelation length) fo u ν, and l ν is the viscous length scale below which the velocity becomes smooth and dominated by viscous effects: Eδu ν x,y,t) δu ν x,y,s))= Dl ξ ν x y δt s) fo x y l ν. 3) The Gaussian natue of u ν and its white-in-time chaacte ae simplifying assumptions that make the Kaichnan model tactable. In contast, the spatial dependence of u ν is non-tivial and moe ealistic of eal tubulent velocity fields. Indeed, inteesting behavio fo the scala occus when <ξ<when u ν is non-smooth and is only Hölde continuous on the inetial ange of scales l ν x y l. This is pecisely the case of inteest fo fully developed tubulent velocity fields fo which Kolmogoov s agument suggests ξ = 3. In fact, the non-smoothness of u ν is esponsible fo intemittency coections in the behavio of the scala θ, as we explain below. Coesponding autho //$ see font matte Published by Elsevie Science B.V. PII: S67-789)96-8

2 W. E, E.Vanden-Eijnden / Physica D 5 53 ) Fo some of the egimes discussed below, the tanspot equation ) does not have a statistical steady state in the pesence of focing. Theefoe, we will focus on the decaying situation. Fo simplicity, we will assume that the initial condition, θx,) = θ x), is a zeo-mean, isotopic Gaussian andom pocess, independent of u ν, and with covaiance E θ x)θ y)) = B x y ), 4) whee B) is smooth and tends apidly to fo L. The length L is the integal scale fo the scala field θ and typically one has L l. As usual, we ae inteested in the behavio of the stuctue functions of abitay ode n, S n, t), defined as S n = x y,t)= E θx,t) θy,t) n 5) in the inetial ange of scales fo the passive scala defined as maxl ν,l κ ) L inetial ange). 6) Hee l κ is the diffusive length scale defined as l κ = κ/d) /ξ. The paametes ae ξ, which is dimensionless, D, with dimension [length] ξ [time],b = B), with dimension [tempeatue], and the vaious lengths l ν,l κ,l. Nomal scaling means that the behavio of the stuctue functions in the inetial ange is independent of l ν,l κ, and /L to leading ode in these paametes. Dimensional analysis then gives S n, t) = B n/ ξ ) f n nomal scaling), 7) Dt whee f n s ae dimensionless functions. Howeve, it was shown by seveal goups [ 5] that the nomal scaling in 7) does not hold fo the Kaichnan model. Moe pecisely, to leading ode in l ν,l κ, and /L, the stuctue functions depend on L in the inetial ange. 8) This makes the Kaichnan model a elevant example fo the study of intemittency. Notice that due to 8), the scaling of stuctue functions cannot be obtained by dimensional analysis. We will conside a genealization of the Kaichnan model due to Gawȩdzki and Vegassola [6] see also [7]) whee the velocity is allowed to be compessible. As in the oiginal Kaichnan model, u ν is assumed to be a zeo mean Gaussian andom pocess with covaiance Eu ν α x, t)uν β y, s) = C δ αβ c αβ x y))δt s). 9) Compessibility is now incopoated into the model by taking c αβ as c αβ x) = Ac P αβ x) + BcS αβ x), ) whee to leading ode cαβ P x) = D δ αβ + ξ x ) αx β x x ξ, cαβ S x) = D d + ξ )δ αβ ξ x ) αx β x x ξ ) In the so-called Batchelo egime, one typically assumes that l κ l ν and studies the behavio of the stuctue functions in the ange l κ l ν whee the velocity is smooth. We will not conside this case hee.

3 638 W. E, E.Vanden-Eijnden / Physica D 5 53 ) fo l ν x l. Hee d is the spatial dimension. The dimensionless paametes A and B measue the divegence and otation of the field u ν. A = coesponds to incompessible fields with u ν =. B = coesponds to iotational fields with u ν =. Following Ref. [6], we chaacteize compessibility by intoducing P = C S, S = A + d )B, C = A. ) P = when the velocity is incompessible and P = when it is iotational. Gawȩdzki and Vegassola [6] studied the situation when l ν = and identified two diffeent egimes which can aleady be seen at the level of S see also [7]):. When P d/ξ, coesponding to a egime of weak compessibility, one has, to leading ode in l κ, /L, S, t) = C B ξ Dt fo l ν =, l κ L. 3) Hee and in the fomulas below, C is a geneic numeical constant. ) is nomal scaling.. In contast, when P <d/ξ, coesponding to a egime of stonge compessibility, one has to leading ode in l κ, /L, S, t) = C B L ξ ) ζ Dt L fo l ν =, l κ L, 4) whee ζ = d ξ + ξp. + ξp 5) The scaling in ) is anomalous. The main pupose of the pesent pape is to study the effect of the simultaneous pesence of viscosity and molecula diffusion. We account fo the effect of viscosity by assuming that the tensos cαβ P x) and cs αβ x) enteing the covaiance of u ν behave fo x l ν as cαβ P x) = Dlξ ν δ αβ + x ) αx β x x, cαβ S x) = Dlξ ν d + )δ αβ x ) αx β x x. 6) Thus, Dl ξ ν can be identified as the dynamic viscosity ν, and taking the limit as l ν amounts to letting ν. The standad way to measue the elative stength of viscous and diffusive effects is though the Pandtl numbe. In the pesent model, the Pandtl numbe is given by P = ν ) ξ κ = Dlξ ν lν =. 7) κ l κ P is the only non-dimensional paamete one can constuct based on D, l ν and κ. Howeve, it tuns out fo the pesent model that the elative stength of viscous and diffusive effects must be chaacteized by a diffeent Pandtl numbe which we shall efe to as the tubulent Pandtl numbe: ) ξ+α P T = Dl ξ+α ν lν = P, 8) κl ξ+α L whee α = d + ξ ξp. 9) + ξp

4 W. E, E.Vanden-Eijnden / Physica D 5 53 ) As shown below, in the ange of paametes whee the behavio of S depends on P T,wehaveξ <α<, i.e. P T P since l ν L P T tends to P as α ξ ). The tubulent Pandtl numbe P T has the following intepetation. Let be the aveage time it takes fo two paticles to be sepaated by distance L if thei initial distance is zeo, and decompose as = +, whee esp. ) is the amount of time duing which the distance between the two paticles is less esp. moe) than the viscous length scale l ν duing the sepaation pocess. The elative stength of viscous and diffusive effects can be then chaacteized by the atio / : the latte tuns out to be popotional to the squae oot of P T. We identify thee diffeent egimes accoding to thei degee of compessibility:. In the weakly compessible egime when P d + ξ, ) ξ the scaling of S is, to leading ode in l ν,l κ, /L given by S, t) = C B ξ fo maxl ν,l κ ) L. ) Dt. In the stongly compessible egime when P d ξ, ) the scaling of S is, to leading ode in l ν,l κ, /L given by S, t) = C B L ξ ) ζ Dt L fo maxl ν,l κ ) L, 3) whee ζ is given by 5). 3. In the intemediate egime when d + ξ < P < d ξ ξ, the scaling of S depends on the tubulent Pandtl numbe in 8). Moe pecisely, if 4) P T, 5) S scales as in 3), wheeas if P T is of ode one, o P T, 6) S scales as in ), with C depending on the pecise value of P T. A moe pecise fomulation of this esult is given in Poposition. The coesponding phase diagams ae shown in Fig.. The diffeent scalings in ) o in 3) fo S ae elated to diffeent types of genealized flows that can be associated with the tanspot equation ). Genealized flows fo passive scala wee intoduced in Ref. [8] and will be discussed in moe details elsewhee. Essentially, a genealized flow is a family of pobability distibution functions fo the tajectoies of n test paticles advected by the velocity field u ν and subject to molecula diffusion, in the limit whee the egulaization paametes, l ν and l κ, ae both taken to zeo. In this limit, the family of pobability distibution functions exhibits popeties of banching o coalescence between the test paticle tajectoies, elated to the non-lipschitz chaacte of the velocity u ν in the limit as l ν and not obseved fo standad flows; these

5 64 W. E, E.Vanden-Eijnden / Physica D 5 53 ) Fig.. Phase diagams fo the thee egimes WC: weakly compessible egime; IR: intemediate egime; SC: stongly compessible egime). popeties ae fomulated in a moe pecise way in tems of the pai distance pobability density function in 9) and 3). Banching is associated with the scaling in ), wheeas coalescence yields the scaling in 3). The above classification is essentially equivalent to the popety that the limiting genealized flow depends on the way the egulaization paametes ae emoved, i.e. the way the limit as l ν,l κ is taken. Ou poof of the above classification fo S essentially amounts to studying the behavio of the pobability density function fo the pai distance between two paticles. We now tun to moe pecise statement fo the classification given ealie. It is easy to see that S, t) = B )P ν,κ,t) P ν,κ,t))d. 7) Hee P ν,κ ρ, t) d = Pob{ ϕ t x) ϕ t y), ]}, 8) whee x y =ρ>and ϕ t satisfies dϕ t x) = u ν ϕ t x), t) dt + κ dβt), ϕ x) = x, whee β is a Wiene pocess. In othe wods, P ν,κ ρ, t) is the pobability density function that the distance between two test paticles is at time t if it was ρ initially. Thus, to undestand the behavio of S in the inetial ange, it is cucial to undestand the behavio of P ν,κ in the limit as l ν,l κ o, equivalently, as ν, κ ). We will establish the following poposition. Poposition. Let P ν,κ ρ, t) be defined as in 8). We have:. In the weakly compessible egime when ) is satisfied, fo any fixed ρ,t >, lim P ν,κ ρ, t) d = Pρ, t) d, weakly as measues. The limiting measue is absolutely continuous with espect to the Lebesgue measue and P satisfies lim Pρ, t) > fo >, t>. 9) ρ +. In the stongly compessible egime when ) is satisfied, lim P ν,κ ρ, t) d = Pρ, t) d,

6 weakly as measues. Moeove, W. E, E.Vanden-Eijnden / Physica D 5 53 ) Pρ, t) d = Aρ, t)δ) + Pρ, s) d, 3) whee Aρ, s) = Pρ, s) d >. P is integable and satisfies lim ρ + Pρ, t) = fo >, t>. 3) 3. In the intemediate egime when 4) is satisfied, we must distinguish thee situations. Fo any fixed constant C>, lim P T lim P T C P ν,κ ρ, t) d = P ρ, t) d, 3) P ν,κ ρ, t) d = P ρ, t) d, 33) lim P ν,κ ρ, t) d = P 3 ρ, t) d, 34) P T weakly as measues. P satisfies 9), and the measue P d is absolutely continuous with espect to the Lebesgue measue. P depends on C, and satisfies both 9) and 3). P 3 satisfies both 3) and 3). The popeties in 9) and 3), espectively, eflect the banching o the coalescence behavios of the genealized flow. In contast, the popety in 3) eflects the absence of banching, and the absolute continuity of the distibution associated with P with espect to the Lebesgue measue eflects the absence of coalescence. Notice that, in the intemediate egime, if thee is coalescence in the sense of 3), then it dominates the scaling of S. This explains that the scaling is the same if lim P T = C, ) o lim P T =. Poof. Because the velocity field in Kaichnan model is an isotopic white-noise, the density P ν,κ satisfies a Fokke Planck equation given explicitly by P ν,κ t whee = bν,κ )P ν,κ ) + aν,κ )P ν,κ ), 35) a ν,κ ) = κ + a ν ), b ν,κ ) = d )κ + b ν ), 36) and a ν,b ν behave fo l ν l outside the viscous laye as ) )) ) a ν lν ) = a) + O + O, b ν ) = b) + O + O l l with lν )) 37) a) = DS + ξc ) ξ, b) = Dd + ξ)s ξc ) ξ, 38) and fo l ν inside the viscous laye as a ν ) = Dl ξ ν S + C ) + O l ν )), b ν ) = Dl ξ ν d + )S C ) + O l ν )). 39)

7 64 W. E, E.Vanden-Eijnden / Physica D 5 53 ) We notice now that if the limiting P is well defined it must satisfies the equation obtained by setting ν, κ in 35). Thus, P t = b)p ) + a)p ). 4) This equation makes sense fo, ) but is singula at =. Identifying P as a suitable limit of P ν,κ amounts to classifying the bounday = fo Eq. 4). This is a well-known poblem in pobability theoy [9] and a complete classification of the bounday = can be given as follows. Let A) = exp β bρ) aρ) dρ whee β> is abitay. We have ), 4). If A and Aa) ae integable at =, = isaegula bounday.. If A is not integable at =, but Aa) β A dρ is integable, = isanentance bounday. 3. If Aa) is not integable at =, but A β Aa) dρ is integable, = isanexit bounday. 4. In all the othe cases, = isanatual bounday. A bounday condition at = is equied and only allowed if = is a egula bounday. The condition may be an absobing condition fo which lim a)a)p =, + a eflecting condition fo which lim b)p + ) + a)p ) =, 43) o a linea combination of the two mixed condition). Fo Eq. 35), we obtain A) = C α, a)a)) = C α ξ, 44) whee α is given by 9). By analyzing the integability of the functions in 44) accoding to the classification given above,wehave. In the weakly compessible egime, = is an entance bounday whee no bounday condition is allowed. It follows then that lim P ν,κ = P is well defined on evey subsequences ν, κ i.e. P is independent of P T ). It also follows fom Felle s theoy [9] that the distibution associated with P is absolutely continuous with espect to the Lebesgue measue and P satisfies 9).. In the stongly compessible egime, = is an exit bounday whee no bounday condition is allowed. Again lim P ν,κ = P is well defined on evey subsequences. It also follows fom Felle s theoy [9] that P satisfies 3) and 3). 3. In the intemediate egime, = is a egula bounday whee a bounday condition is equied. In this case, P is well defined only if the limit as ν, κ is taken on subsequences ν, κ whee P T is appopiately constained, which specifies the effective bounday condition at = fo Eq. 4), as we show now. In the intemediate egime, we shall obtain the type of bounday condition at = fo the limiting equation fo P by studying the behavio as ν, κ of the aveage time,, it takes fo two paticles to be sepaated by a finite i.e. independent of l ν,l κ ) distance d if thei initial distance is zeo; since the integal scale L is the only 4)

8 W. E, E.Vanden-Eijnden / Physica D 5 53 ) length scale besides l ν,l κ in the model, it is natual to take d = L. We shall compae the limit of as ν, κ to the aveage time τ R it takes fo two paticles to be sepaated by L if thei initial distance is zeo fo the pocess associated with the limit equation 4) when the latte is solved with a eflecting bounday condition at =. By definition of the vaious types of bounday conditions at = that ae allowed it follows indeed that. If the atio /τ R tends to as ν, κ, we obtain a eflecting bounday condition at =.. If this atio tends to a finite constant C, ), we obtain a mixed bounday condition. 3. If this atio tends to infinity, we obtain an absobing bounday condition. We shall show now that the atio /τ R behaves as + OP T ) / ) as ν, κ consistent with ou above classification and the esults in 3) 34). It is a standad esult in pobability theoy that T ν,κ ), whee T ν,κ ) satisfies b ν,κ dt ν,κ ) + a ν,κ ) d T ν,κ d d = 45) fo the bounday conditions dt ν,κ d =, = T ν,κ L) =. 46) This gives L L = a ν,κ )A ν,κ )) A ν,κ ) d d, 47) whee A ν,κ is defined as in 4) fo convenience we take β = L) L A ν,κ b ν,κ ) ρ) ) = exp a ν,κ ρ) dρ. 48) The integals involved in 47) exist because fo l κ,wehave A ν,κ ) = C ν,κ d+ + o d+ ), a ν,κ )A ν,κ )) = C ν,κ d + o d ). 49) This implies that = is an entance bounday fo the egulaized equation 35) fo d>, and a egula bounday fo d =.) Let us decompose = +, whee = lν L a ν,κ )A ν,κ )) A ν,κ ) d d, L = a ν,κ )A ν,κ )) l ν L A ν,κ ) d d. The time esp. ) is the amount of time duing which the distance between the two paticles is less esp. moe) than the viscous length scale l ν duing the sepaation pocess. We now estimate the behavios of and as ν, κ. We denote by f ν,κ g ν,κ if f ν,κ /g ν,κ as ν, κ. Fom 5), we have L α l α lν lν ν ā ν,κ )) b ν,κ ) ) exp α ā ν,κ ) d d, 5) 5) 5)

9 644 W. E, E.Vanden-Eijnden / Physica D 5 53 ) whee ā ν,κ ) = κ + Dl ξ ν S + C ), b ν,κ ) = κd ) + Dl ξ ν d + )S C ). 53) It follows that Aν,κ L α l α ν α l ξ ) / ν 54) Dκ with A ν,κ = P / It is easy to check that + S + C )z ) exp P / z ) d ) + d + )S C )z dz + S + C )z z dz. 55) A ν,κ = OP / ) as P, lim P Aν,κ, ). 56) On the othe hand, it can be veified by diect calculation that in the limit as ν, κ, L ξ τ R = DS + ξc ) ξ + α) ξ), 57) whee α is given by 9). Combining the above expessions, we obtain that τ R with C ν,κ given by + C ν,κ Dl ξ+α ) / ν κl ξ+α = + C ν,κ P T ) / 58) C ν,κ = Aν,κ ξ + α) ξ)s + ξc ). 59) α Since P C [, ) if P T and P if P T, ], it follows using the popeties of A ν,κ that lim P T τ R =, lim P T C, ), lim τ R τ R =, 6) P T whee C, ). This concludes the poof. Acknowledgements We ae gateful to K. Gawȩdzki fo helpful discussions. Weinan E is patially suppoted by a Pesidential Faculty Fellowship fom NSF. Eic Vanden-Eijnden is patially suppoted by NSF Gant DMS

10 W. E, E.Vanden-Eijnden / Physica D 5 53 ) Refeences [] R.H. Kaichnan, Phys. Fluids 968) [] B. Shaiman, E. Siggia, Phys. Rev. E ) [3] M. Chetkov, G. Falkovich, I. Kolokolov, V. Lebedev, Phys. Rev. E 5 995) [4] K. Gawȩdzki, A. Kupiainen, Phys. Rev. Lett ) [5] E. Balkovsky, V. Lebedev, Phys. Rev. E ) [6] K. Gawȩdzki, M. Vegassola, Physica D 38 ) [7] Y. Le Jan, O. Raimond, 999. math. PR/ [8] W. E, E. Vanden-Eijnden, Poc. Natl. Acad. Sci. USA 97 ) [9] W. Felle, Ann. Math )