Passive Scalar and Scalar Flux in Homogeneous Anisotropic Turbulence

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1 Passve Scalar and Scalar Flux n Homogeneous Ansotropc Turbulence A. BRIARD a, T. GOMEZ a, C. CAMBON b a. Alembert, boîte 162, 4, Place Jusseu Pars Cedex 5 b. LMFA, Ecole Centrale de Lyon, Unversté de Lyon, France Résumé : Le but de ce traval est l étude théorque et la modélsaton d un champ scalare passf dans une turbulence homogène ansotrope à l ade d une fermeture EDQNM (Eddy-Damped Quas-Normal Markovan) adaptée à un tel contexte. Cette fermeture est une extenson du modèle spectral pour le champ de vtesse de V. Mons et. al. [1] à un champ scalare passf et au flux scalare, qu naît de l nteracton de gradents de vtesse et de scalare. Des résultats généraux orgnaux de crossance et décrossance des énerges du scalare et du flux en présence de gradents moyens sont c présentés ans qu une valdaton du modèle. Abstract : Ths work ams at studyng both numercally and theoretcally a passve scalar feld n a homogeneous and ansotropc turbulent flow, thanks to an adapted EDQNM (Eddy-Damped Quas-Normal Markovan) closure. Ths closure s an extenson of the spectral model for the velocty feld developed by V. Mons et. al. [1] to the passve scalar and scalar flux cases. The latter s created thanks to both velocty and scalar mean gradents. New general results regardng decay and growth laws of passve scalar and scalar flux energes n the presence of mean gradents are presented, along wth the valdaton of the present model. Mots clefs : Passve scalar, Scalar flux, Ansotropy, Shear flow, Mean scalar gradent 1 Introducton The study of Homogeneous Ansotropc Turbulence (HAT) s of great nterest for the complete understandng of real problems, such as atmospherc flows where scalar (temperature) and velocty gradents strongly nteract. Velocty and temperature gradents can also be the consequence of boundary condtons (no-slp condton, adabatc wall,...). Shear flows have receved a partcular attenton [1, 2, 3, 4] so that some results are now well-known, such as the exponental growth of the knetc energy and the return to sotropy of small scales, accordng to Kolmogorov local sotropy hypothess [5]. The specfc case of Homogeneous Isotropc Turbulence wth a mean Scalar Gradent (HITSG) has also been wdely studed, n partcular by Bos [6]. However,

2 n hs spectral model based on EDQNM closure, there were several adjustable constants for the velocty feld and transfers kept sotropc for the passve scalar feld. Fnally, the case of a shear flow wth scalar gradent (HSTSG), or more generally, of HAT wth both velocty and scalar gradents, has not been fully nvestgated yet, both numercally and theoretcally. Important results are nevertheless reported by Bos [7], wth the ssues mentoned earler, and the workable expermental results date back to Tavoulars and Corrsn [8]. These dfferent ponts are the motvaton for the present work, whch s a frst step nto a complete HAT study. We am at mappng the behavor of a passve scalar feld under varous knds of ansotropy and for dfferent Prandtl numbers, very large or very small. Here, both HITSG and HSTSG are nvestgated numercally wth a fully consstent EDQNM model based on the poneerng work of Cambon [9]. Such a study of the scalar feld, for dfferent knds of ansotropc flows, has never been performed so far, at least wth an unque model that does not rely on adjustable constants (except the classcal one of EDQNM). Ths spectral modellng has been the object of a prevous work by Mons, Cambon and Sagaut [1] on the velocty feld exclusvely. Ths models reles on the ntrnsc decomposton of the spectral Reynolds stress tensors nto drectonal and polarzaton parts. The man equatons of the exstng model [1] for the velocty feld are recalled. Then, the new extenson to the passve scalar and scalar flux s presented. Fnally, the code s valdated n the HITSG and HSTSG frameworks and new results are presented n HITSG regardng the scalar varance and the cospectrum energy. 2 Equatons of the model In ths part, the three fnal evoluton equatons of the ansotropc spectral model of Mons, Cambon and Sagaut [1] are gven. The startng pont s the dynamcal equatons for the full spectral tensor of velocty correlatons ˆR, n terms of the complete (3D) wave vector k. These equatons nvolve terms, whch reflect the drect effect of the mean velocty gradents, that are lnear and closed, and call nto play unclosed contrbutons from thrd-order correlatons. The latter deal wth energy transfers and nonlnear pressure-stran correlatons, and are closed by a generalzed EDQNM procedure, as a nonlnear contrbuton. In the fnal step, the model does not retan the full spectral nformaton, but only an optmal set of sphercal-averaged descrptors, whch depend on the wave vector modulus k only. Ths set s extracted from the decomposton of the spectral tensor ˆR n terms of angular sphercal harmoncs at the frst non-trval degree (second degree here). The evoluton of the velocty feld s thus drven by three sets of equatons : the sotropc part of the knetc spectrum E, ts drectonal ansotropy part EH (dr) and ts polarzaton ansotropy part EH (pol) : ( t + 2νk 2 )E(k, t) = S L(so) (k, t) + S NL(so) (k, t), (1) ( t + 2νk 2 )E(k, t)h (dr) ( t + 2νk 2 )E(k, t)h (pol) ν s the knematc vscosty, S NL(so), S NL(dr) are non-lnear transfer terms fully computed wth EDQNM, and S L(so), S L(dr) gradents. H (dr) and H (pol) (k, t) = S L(dr) (k, t) + S NL(dr) (k, t), (2) (k, t) = S L(pol) and S NL(pol) (k, t) + S NL(pol) (k, t). (3) and S L(pol) are lnear ones, drectly dependng on the velocty are devatorc ansotropy ndcators tensors lnked to the spectral Reynolds

3 stress tensor ˆR through 2E(k, t)h () (k, t) = S k ˆR() (k, t)d2 k, (4) where S k denotes a sphere of radus k. The orgnal result here s the extenson to the passve scalar and scalax flux cases wth the same compact shape. Smlar calculatons gve the evoluton of the sotropc part of the scalar spectrum E T and ts pseudo-drectonal part E T H (T ) : ( t + 2ak 2 )E T (k, t) = S T,NL(so) (k, t) + S T,L(so) (k, t), (5) ( t + 2ak 2 )E T (k, t)h (T ) (k, t) = S T,NL(dr) (k, t) + S T,L(dr) (k, t). (6) a s the scalar dffusvty and H (T ) s a pseudo scalar drectonal ansotropy ndcator obtaned wth 2E T (k, t)h (T ) (k, t) = E (T,dr) (k, t)d 2 k, (7) S k where E (T,dr) s the drectonal part of the spectral scalar-scalar correlaton < ˆθ( p)ˆθ(k) >= E T (k)δ(k p). (8) The scalar spectrum s smply the sphercal average of E T, smlar to E for the velocty feld [1]. The scalar flux F (k), or the spectral velocty-scalar cross-correlaton, s drven only by one equaton, as ths quantty remans zero n sotropc turbulence. We defne the sphercal average of the scalar flux so that ( t + (a + ν)k 2 )E F H (F ) E F H (F ) (k, t) = F (k, t)d 2 k, S k (9) (k, t) = S F,NL (k, t) + S F,L (k, t). (1) The spectrum E F cannot exst on ts own and s always lnked to the ansotropy ndcator H (F ) as the scalar flux s zero n the sotropc framework. S T,NL(so), S T,NL(dr), S F,NL (k, t), S T,L(so), S T,L(dr) and S F,L (k, t) are the non-lnear and lnear passve scalar and scalar flux transfers. For the numercal ntegraton of these 2 ndependent equatons, a logarthmc dscretzaton n wave numbers s used. The knetc and scalar spectra are ntally sotropc and the scalar flux s zero. Re λ s the Reynolds number based on Taylor mcroscale, and n what follows, the Prandtl number s 1, so that ν = a. 3 Numercal results 3.1 Valdaton n HITSG Frstly, the code s valdated n the HITSG framework : the velocty feld s sotropc and decays accordng to Comte-Bellot and Corrsn (CBC) theory [11] whereas the mean scalar gradent Λ creates the cospectrum F = E F H (F ) 3, the only non-zero component of the scalar flux, and produces scalar energy K T (t) = E T (k, t)dk. The cospectrum energy evolves as dk F dt = P F (t) ɛ F (t) + Π F (t) (11)

4 1 1 DNS of Overholt and Pope EDQNM 1 ǫf/pf Re.769 λ Re.76 λ Re 1.72 λ E(k,t),F(k,t) 1 1 E(k, t) F(k, t) k 5/3 k 7/ Re λ k L k η k FIGURE 1 At left, comparson of the cospectrum dsspaton and producton rato wth DNS. At rght, the k 7/3 range of the cospectrum F. where Π F (t) = S F,NL 3 (k, t)dk s the cospectrum destructon, drven by pressure effects, ɛ F (t) ΛK(t) the cospectrum producton. the cospectrum dsspaton rate, and P F (t) = S F,L 3 (k, t)dk = 2 3 The rato of cospectrum dsspaton and producton s nvestgated and compared to the drect numercal smulaton (DNS) of Overholt and Pope [12]. The ntal Reynolds number s Re λ () = 233 so that nertal ranges can be clearly observed. Results are presented n Fg. 1 along wth the k 7/3 range of the cospectrum, predcted by dmensonal analyss by Lumley [13]. Moreover, the theoretcal Re 1 λ dependence at hgh Reynolds number, found by Bos, s recovered as well, usng a smple calculaton ɛ F (t) 3(ν + a) k 2 F(k, t)dk = P F (t) 2Λ E(k, t)dk kη k L ɛ 1/3 k 1/3 dk kη k L ɛ 2/3 k 5/3 dk Re 1 λ where 1/k L s the ntegral scale and k η the Kolmogorov wavenumber. The good agreement between theory, DNS and EDQNM smulatons valdates our code. 3.2 Decay and growth laws n HITSG In ths secton, new results regardng the decay of the cospectrum n the HITSG framework are presented. The emphass s put on the decay of the cospectrum energy K F (t) = F(k, t)dk. We propose theoretcal decay exponents, followng [11], Meld and Sagaut [14] n the sotropc case, and Brard, Gomez and Sagaut [15] for the passve scalar. Let s ntroduce the nfrared exponent σ so that E(k, t) k σ and the decay exponent α of the knetc energy K(t) = E(k, t)dk t α. Smulatons show that the cospectrum nfrared slope s exactly σ and does not depend at all on the scalar one σ T. In the hgh Reynolds numbers regme (Re λ 2), nertal effects are domnant and thus K F (t) = k L F(k, t) k 4/3 L ɛ 1/3. Usng the well-known decay exponent of the ntegral scale 1/k L, one fnds K F (t) t α F, α F = σ p F 1 σ p + 3 (12)

5 1 αf(t) CBC Cospectrum Hgh Re CBC Cospectrum Low Re 1 2 Re λ 1 σ = 1 σ = 2 σ = 3 σ = nǫf (t) Re λ 1 1 σ = 1 σ = 2 σ = 3 σ = FIGURE 2 At left, decay exponent α F of the cospectrum energy n hgh and low Reynolds numbers regmes for varous nfrared exponents σ, and at rght, decay exponent n ɛf of the cospectrum dsspaton n low Reynolds numbers regme. where p F s chosen to be (p + p T )/2 and accounts for backscatter n Batchelor turbulence (σ = 4) and s zero otherwse. In the low Reynolds numbers regme, there are no longer nertal ranges. Hence, producton effects lead the decay of the cospectrum so that dk F dt P F (t) whch drectly gves K F (t) t α F, α F = σ 1. (13) 2 The great agreement between these theoretcal decay exponents and numercal smulatons s llustrated n Fg. 2 and s one of the man result of ths paper. It s worth notng that these new results could not be obtaned n exstng DNS as the knetc feld was forced so that no decay could occur. As for the cospectrum dsspaton ɛ F, t s not a conserved quantty n the nertal range unlke the knetc and scalar ones ɛ and ɛ T. Hence, t s mpossble to derve a smlar power law n hgh Reynolds numbers regme. Nevertheless, n low Reynolds numbers regme, as nertal ranges dsappear, t s possble to express the decay exponent of ɛ F, smply as n ɛf = α F 1 accordng to (12). Ths s confrmed n Fg. 2 as well. A smlar result s now presented regardng the passve scalar tself. In the presence of a mean-gradent, there s a contnuous producton of scalar energy accordng to dk T dt = 2ΛK F (t) ɛ T (t), (14) where ɛ T (t) s the scalar energy dsspaton rate. Instead of decayng [15], the scalar energy K T grows both n hgh and low Reynolds numbers regmes, and scales as ΛK F. Usng the prevous results on the cospectrum exponent α F, we fnd the scalar growth exponent αt Λ n hgh Reynolds numbers regme K T (t) t αλ T, α Λ T = 1 p F p σ p + 3 (15) and n low Reynolds numbers regme K T (t) t αλ T, α Λ T = σ 3. (16) 2 The good agreement between these theoretcal exponents and numercal smulatons s revealed n Fg.

6 α Λ T (t) CBC Scalar wth Λ Hgh Re CBC Scalar wth Λ Low Re σ = 1 σ = 2 σ = 3 σ = 4 n Λ ǫt (t) CBC Scalar wth Λ Hgh Re CBC Scalar wth Λ Low Re σ = 1 σ = 2 σ = 3 σ = Re λ Re λ FIGURE 3 At left, decay exponent αt Λ of the scalar energy n hgh and low Reynolds numbers regmes for varous nfrared exponents σ, and at rght the decay exponent n Λ ɛ T of the scalar dsspaton rate..1 H (T) (k, t).5 H (T) 11 = H (T) 22 H (T) 33.5 k T k FIGURE 4 Scalar ansotropy tensor H (T ) at Re λ = It s nterestng to pont out that the growth exponent of the scalar energy wth a mean gradent was already found by Chasnov [16] n the partcular case of Saffman turbulence (σ = 2) where α Λ T = 4/5. Chasnov computed analytcally decay and growth exponents for passve and actve scalar felds for constant and non-constant mean gradents n Saffman turbulence. Hs results for the actve scalar were assessed by Large Eddy Smulaton (LES). Our result here for the passve scalar s more general as t s vald for all knd of ntal turbulence state, and can be seen as an a posteror valdaton of Chasnov s result. The last pont to explore s the behavor of the scalar dsspaton rate ɛ T whch was not studed n [16]. From the evoluton equaton of K T (16), t seems clear that ɛ T behaves lke K F and so one smply has n Λ ɛ T = α F, whch s also recovered n Fg. 3. A strkng result s that wth a mean scalar gradent, decay and growth exponents, both n hgh and low Reynolds numbers regmes, for the cospectrum and the scalar feld, do not depend at all on the nfrared scalar slope σ T, meanng that the knetc feld fully leads the dynamcs of the flow through the mean gradent Λ. We conclude ths part wth a remark on the return to sotropy mechansm of small scales. Scalar ansotropy tensors H (T ) are dsplayed n Fg. 4 : they clearly show that small scales of the flow have returned to sotropy as H (T ) there. Ths s n agreement wth Kolmogorov local sotropy hypothess [5] for second-order moments.

7 3.3 Shear flow and scalar gradent Here, the case of shear flow s addressed. Few results only exst regardng the passve scalar and scalar flux n presence of both velocty and scalar gradents. The shear rate s S = du 1 /dx 3 where U s the mean velocty feld. Scalar ndcators analogous to knetc ones are now defned. Frst, the scalar shear rapdty as and the scalar ansotropy tensors b T (t) = 1 K T (t) S T R = ɛ T K T S, E T (k, t)h (T ) (k, t)dk. In Fg. 5, the case of a shear flow wthout mean scalar gradent s presented. In such a framework, there s no scalar flux. Smulatons show that SR T and bt reach constant values for 3, as for the analogous knetc ansotropy ndcators [4, 1] e.34 b T,ǫ T/(KTS).4.2 b T 11 b T 22 b T 33 b T 13 ǫ T /(K T S) K (T) (t),ǫt(t) K(t) K T (t) ǫ T (t) e FIGURE 5 At left, scalar ansotropy tensors b T wth ST R. At rght, K(t), K T (t) and ɛ T. Both for σ = 2, Λ = and S = 1 2. Moreover, an exponental decrease of the scalar energy s observed, n agreement wth the evoluton equaton t K T = ɛ T. Let s replace K T and ɛ T n ths equaton by K T (t) = K T exp(γ T ) and ɛ T (t) = ɛ T exp(γ T ). An analytcal expresson for the scalar exponental rate γ T s then obtaned γ T = ɛ T SK T = S T R, K T (t) K T () exp(γ T ). (17) Such an expresson for γ T has already been obtaned wth dfferent arguments by Gonzalez [17]. The exponent γ T found by plottng K T s n very good agreement wth the asymptotc value of SR T, whch gves γ T.52. The mportant result s that the fnal value of the scalar decay rate γ T does not depend on the shear rate S nor on the nfrared exponents σ and σ T. When a scalar gradent s added, the scalar flux has two non-zero components. The second one, the streamwse flux F S, arses from shear effects, and ts energy s K S F (t). A short valdaton s proposed aganst Tavoulars and Corrsn [8] results n Fg. 6. Approprate characterstc parameters gve the dmensonless shear S = 6, 19 and scalar gradent Λ = Let s menton that we start from an ntal sotropc condton, whch s clearly not the case n the experment. Two fnal Reynolds numbers are gven n [8] : R λg = 16 scaled for an sotropc framework, and

8 2.5 ρuθ(t), ρvθ(t).5.5 ρ uθ ρ vθ ρ exp uθ ρ exp vθ PrT(t) EDQNM Experment FIGURE 6 Comparsons wth expermental results of Tavoulars and Corrsn, wth σ = 2, S = 6.19 and Λ = R λ11 = 266 for nhomogeneous flows, more approprate. We chose the second one. Data s avalable at three locatons : x 1 /h = 7.5, 9.5 and 11. Usng the approprate converson n dmensonless tme ths provdes expermental nformaton at = 8.63, 1.94 and There are good agreements n Fg. 6 for the cospectrum and streamwse flux correlatons ρ vθ and ρ uθ. Fnally, a dfference s observed for the turbulent Prandtl number whch s P r exp T P r T (t) = Λ S R 12 (t) K F (t), (18) 1.1, whereas P redqnm T.74. The value obtaned expermentally seems qute large : ndeed, Herrng et. al. [18] and Leseur [19] have both reported, from atmospherc data and theoretcal consderatons, that one should obtan.6 P r T.8, n agreement wth our result. Moreover, other values of the turbulent Prandtl numbers reported from DNS and experments agree better wth.7 than wth 1.1. Ths bref comparson valdates our model for the HSTSG case. In Fg. 7, the scalar energy grows exponentally wth the presence of the scalar gradent, wth the same growth rate as the knetc energy, and so do the energes of the scalar flux. Ths s an orgnal result, although t could have been deduced from expermental works and exstng DNS as ρ vθ and ρ uθ become always constant for suffcently hgh. One can see n Fg. 7 that the ratos ɛ F /(K F S) and ɛ S F /(KS F S) both tend to zero. Ths wll be used n an upcomng work to gve analytcal expressons for the scalar flux growth rates. 4 Conclusons Ths work s a step further nto the theoretcal and numercal study of the Homogeneous Ansotropc Turbulence. The present EDQNM spectral model reles on sx equatons, that can be seen as generalzed Ln equatons, ncludng three for the velocty feld (equatons (1), (2), and (3)) whch are part of the Mons, Cambon and Sagaut model [1]. The present study extends ths model to the cases of a passve scalar feld (equatons (5) and (6)) and the assocated scalar flux (equaton (1)). The model s assessed both n the frameworks of Homogeneous Isotropc Turbulence wth Scalar Gradent and of Homogeneous Shear Turbulence wth Scalar Gradent. New decay laws are found for the cospectrum energy K F (t) : In hgh Reynolds numbers regme, the decay s drven by nertal effects and the decrease of the knetc energy K(t).

9 K(t),KT(t),KF(t),K S F (t) K K T K F and ǫ F K S F and ǫs F e ǫ (S) F /(K(S) F S) ǫ S F /(KS F S) ǫ F /(K F S) FIGURE 7 At left, exponental growth of the knetc, scalar, cospectrum and streamwse flux energes. The cospectrum and streamwse flux dsspaton rates ɛ F and ɛ S F are dsplayed n grey. At rght, ratos ɛ F /(K F S) and ɛ S F /(KS F S). Both for σ = 2, Λ = 1 and S = 1 2. In the low Reynolds numbers regme, as nertal effects are neglgble, producton of cospectrum leads the decay. The decay exponent of the cospectrum dsspaton rate ɛ F (t) s found as well, only n the low Reynolds numbers regme, n agreement wth Bos study [6]. Smlarly, the scalar dsspaton rate ɛ T (t) s found to decay as K F (t), whereas the scalar energy K T (t) grows wth tme because of the producton term lnked to the cospectrum. The latter s n agreement wth the theoretcal work of Chasnov [16] n Saffman turbulence, and thus can be seen as a generalzaton and an addtonal valdaton. The strong result wth these new theoretcal exponents, assessed wth numercal smulatons, s that they do not depend on the ntensty of the scalar gradent Λ, nor on the scalar large scales ntal condtons σ T (E T (k, t) k σ T ). Therefore, we conclude that n HITSG, the passve scalar s fully domnated by the velocty feld, even ts large scales. Then, the cases of a passve scalar wth shear only, and shear combned wth scalar gradent, are brefly nvestgated. In the frst case, the exponental decrease of the scalar energy K T (t) s recovered both numercally and analytcally. In the second case, all energes are found to grow exponentally wth the same rate γ as the knetc feld, found n [1]. These two last cases deserve more nvestgaton and are at the center of an upcomng work, along wth the nfluence of a Prandtl number very dfferent from 1. Références [1] S. Tavoulars, U. Karnk, Further experments on the evoluton of turbulent stresses and scales n unformly sheared turbulence. J. Flud Mech. (24), [2] A. Pumr, B. I. Shraman, Persstent Small Scale Ansotropy n Homogeneous shear flows. Phys. Rev. Lett. (75), [3] F. A. De Souza, V. D. Nguyen, S. Tavoulars, The structure of hghly sheared turbulence. J. Flud Mech. (33), [4] P. Sagaut, C. Cambon, Homogeneous Turbulence Dynamcs. Cambrdge Unversty Press, 28. [5] A. N. Kolmogorov, The local structure of turbulence n ncompressble vscous flud for very large Reynolds numbers. Dokl. Akad. Nauk SSSR (3), 1941.

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