A new turbulence model for Large Eddy Simulation

Adv. Studes Theor. Phys., Vol. 1, 2007, no. 6, 247-270 A new turbulence model for Large Eddy Smulaton Francesco Gallerano Dpartmento d Idraulca Trasport e Strade Unversta degl Stud d Roma La Sapenza Va Eudossana 18, Roma Italy Francesco.Gallerano@unroma1.t Govann Cannata Dpartmento d Idraulca Trasport e Strade Unversta degl Stud d Roma La Sapenza Va Eudossana 18, Roma Italy Abstract The present - day Large Eddy Smulaton models based on the Smagornsky assumpton and the drawbacks of the dynamc calculaton of the closure coeffcent for the generalsed subgrd scale turbulent stress tensor are presented. The relatons between numercal scheme conservaton property of mass, momentum and knetc energy and the drawbacks of the dynamc Smagornsky - type turbulence models are shown. A new turbulence model s proposed. The proposed model: a) s able to take nto account the ansotropy of the turbulence; b) remove any balance assumpton between the producton and dsspaton of subgrd scale turbulent knetc energy; c) s able to elmnate the numercal effects produced by the non conservaton a pror of the resolved knetc energy. New closure relatons for the unknown terms of the subgrd scale vscous dsspaton balance equaton are proposed. The fltered momentum equatons are solved by usng a sxth order fnte dfference scheme. The proposed model s tested for a turbulent channel flow at Reynolds numbers (based on frcton velocty and channel half-wdth) rangng from 395 to 2340. Keywords: LES, ansotropy, SGS turbulent knetc energy, two-equaton

248 F. Gallerano and G. Cannata 1 Introducton Among the most common LES models present n lterature are the dynamc Smagornsky-type SGS models (e.g., Dynamc Smagornsky Model [4], Dynamc Mxed Model [23], [13], Lagrangan Dynamc Model [11], Dynamc Twoparameter Model [15], n whch the generalzed SGS turbulent stress tensor, τ j, s related to the resolved stran-rate tensor by means of a scalar eddy vscosty. It s assumed n these models that the eddy vscosty s a scalar proportonal to the cubc root of the generalzed SGS turbulent knetc energy dsspaton and that such dsspaton s locally and nstantaneously balanced by the producton of the generalzed SGS turbulent knetc energy (.e., by the rate of knetc energy per unt of mass transferred from the large scales, larger than the flter sze, to the unresolved ones). Consequently, t s evdent that the dynamc Smagornsky-type SGS models are fraught wth four relevant drawbacks. The frst drawback s represented by the scalar defnton of the eddy vscosty; the second one concerns the local balance assumpton of the generalzed SGS turbulent knetc energy producton and dsspaton, the thrd drawback s related to the dynamc calculaton of the coeffcent used to model the eddy vscosty (Smagornsky coeffcent), whlst the fourth drawback s related to the problems arsng from the numercal scheme adopted for the smulatons of three-dmensonal unsteady flows (LES). The scalar defnton (frst nconsstency) of the eddy vscosty s equvalent to assumng that the prncpal axes of the generalzed SGS turbulent stress tensor, or the unresolved part of t (represented by the cross and Reynolds terms), are algned wth the prncpal axes of the resolved stran-rate tensor. Ths assumpton has been dsproved by many expermental tests and by DNS, whch demonstrate that there s no algnment between the generalzed SGS turbulent stress tensor, or the unresolved part of t, and the resolved stranrate tensor (Tao et al. [19], Meneveau & Katz [10]). Moreover, the eddy vscosty s proportonal to the product of two terms, of whch the dmensons are, respectvely, those of a length and a velocty (Tennekes & Lumley [20]). These terms, whch represent, respectvely, the turbulence length scales and turbulence velocty scales, are, more generally, second-order tensors of whch the product s a fourth-order tensor whch represents the eddy vscosty (Monn & Yaglom [12]). The scalar defnton of the eddy vscosty, used n the above-mentoned dynamc Smagornsky-type SGS models, presupposes the exstence of a sngle turbulence velocty scale and a sngle turbulence length scale. Ths s equvalent to assumng that the second-order tensors whch represent the turbulence length scales and the turbulence velocty scales are sotropc and that, therefore, the turbulence s sotropc. In ths manner, the turbulence ansotropy nduced by the contnuous transfer of energy from the mean flow towards

A new turbulence model for Large Eddy Smulaton 249 the turbulent fluctuatons, whch s generally extremely ansotropc, s not consdered. Even though the energy cascade process causes a reducton of the turbulence ansotropy, many authors (Spezale & Gatsk [17], Sreenvasan [18]) demonstrated that even n the dsspaton range of the smallest turbulent scales, where vscous dsspaton occurs, there s a hgh ansotropy level even at hgh Reynolds numbers. The second nconsstency of the Smagornsky dynamc models s related to the assumpton of a local and nstantaneous balance between producton and dsspaton of the generalzed SGS turbulent knetc energy, formulated n the above-mentoned models to obtan the turbulent vscosty expresson. Ths assumpton s confrmed statstcally and never nstantaneously, and only locally at the scales assocated wth wavenumbers wthn the nertal subrange, and the latter exsts only for sotropc turbulence and at hgh Reynolds numbers. Moreover, snce the dsspaton of the generalzed SGS turbulent knetc energy s, by defnton, postve, the assumpton of local balance mples that also the producton of generalzed SGS turbulent knetc energy s postve. However, the assumpton that the producton s always postve mples that the energy transfer always occurs from the largest to the smallest scales and prevents postve transfers of knetc energy from the subgrd scales to the resolved ones (backscatter). Snce the energy exchange processes between the resolved and unresolved scales generally occur n both drectons (forward scatter and back scatter), as has been observed by varous authors (Pomell et al. [14], Horut [7]), the assumpton that the producton of generalzed SGS turbulent knetc energy s always postve does not enable the complexty of the energy exchange processes whch characterze the turbulence to be adequately taken nto account. The current dynamc models clam to represent the energy transfer from the smaller to the larger scales (backscatter) by the change to negatve values of the Smagornsky coeffcent C S (whch appears n the defnton of the eddy vscosty) whch s dynamcally calculated by means of Germano s procedure [4]. It has, however, been found that when the coeffcent C S assumes negatve values the numercal calculaton becomes extremely unstable. Ths nstablty s due to the long autocorrelaton tme of the coeffcent C S whch, once t becomes negatve n some regon of the doman, t may reman negatve for excessvely long perods of tme durng whch the exponental growth of the local velocty felds causes a dvergence of the total knetc energy (Ghosal et al [6]). The thrd nconsstency of the dynamc models concerns the calculaton of the above mentoned Smagornsky coeffcent C S. It s calculated wth varatonal methods, (e.g. wth a least squares mnmzaton method [8] or Lagrangan method [11]). These methods dentfy a sngle value of the scalar coeffcent C S from a system of fve ndependent scalar equatons relatng the

250 F. Gallerano and G. Cannata components of the ansotropc part of the generalzed SGS turbulent stress tensor to the components of the resolved stran-rate tensor. Ths procedure does not provde completely acceptable results. Moreover, when smulatng confned flows at hgh Reynolds number, the results of the dynamc procedure are of doubtful relablty n the regon close to the wall ncludng both the vscous sublayer and the buffer layer (Sarghn et al. [16]). In ths regon, the flter wdth used n the dynamc procedure s larger than most eddes that govern the momentum and energy transfer. Consequently, the dynamc procedure used under these condtons for the calculaton of the coeffcent C S s not able to fully account for the local subgrd dsspatve processes that affect the entre doman. The fourth nconsstency of the dynamc mxed models, based on the Smagornsky closure relaton, s connected to the problems arsng from the numercal scheme adopted for the smulatons of three-dmensonal unsteady flows (LES). In the smulatons of three-dmensonal unsteady flows (that are realzed by a hgh-order fnte dfference scheme wth the dvergence form of the convectve terms) the resolved knetc energy s not perfectly conserved because the contnuty equaton s not perfectly satsfed. Consequently, wthout an adequate turbulence model, the resolved knetc energy s destned to rse n long tme smulatons. In the dynamc mxed models, based on the Smagornsky closure relaton, the calculaton of the closure coeffcent C S (by dynamc procedure) s not able to compensate the effects produced by the convectve terms (expressed n dvergence form n the resolved momentum equaton) that are not able to perfectly conserve resolved knetc energy. In ths paper, a new model for the generalsed SGS turbulent stress tensor s proposed n whch, n order to adequately account for the ansotropy of both, the turbulence length scales and the turbulence velocty scales, the eddy vscosty s defned as a symmetrc fourth order tensor. A mxed formulaton s adopted n the model, n whch the modfed Leonard tensor s calculated explctly, whlst the unresolved resdual part of the tensor τ j s obtaned by the contracton of the eddy vscosty tensor wth the resolved stran rate tensor. The prncpal axes of the generalsed SGS turbulent stress tensor τ j are assumed to be algned wth those of the modfed Leonard tensor, n agreement wth the assumpton of scale smlarty [1]. In order to overcome the nconsstences of the dynamc mxed Smagornsky - type models and n order to contan the ncrease of resolved knetc energy, the turbulent closure relaton for the generalsed SGS turbulent stress tensor s expressed as a functon of the generalsed SGS turbulent knetc energy Eand of the SGS vscous dsspaton ε. In the proposed model the closure coeffcent whch appears n the closure relaton s unquely determned wthout adoptng Germano s dynamc procedure. The closure relaton at the bass of the proposed Two-Equaton Model

A new turbulence model for Large Eddy Smulaton 251 (TEM): a) comples wth the rule of turbulent closure relatons; b) takes nto account both the ansotropy of the turbulence velocty scales and of the turbulence length scales; c) removes any assumpton of balance between the producton and dsspaton of the turbulent knetc energy; d) allows the use of a flter of whch the wdth s not necessarly assocated wth the wavenumbers lyng wthn the nertal subrange; e) assumes scale smlarty n the defnton of the second-order tensor representng the turbulence velocty scales; f) guarantees an adequate energy dran from the grd scales to the subgrd scales and guarantees backscatter; g) overcomes the nconsstences lnked to the dynamc calculaton of the closure coeffcent used n the modellng of the generalzed SGS turbulent stress tensor; h) s able to compensate dynamcally the ncrease of resolved knetc energy (produced by the numercal dscretzaton of the dvergence form of the convectve term n momentum equatons) and then allow the large eddy smulaton of three-dmensonal unsteady flows, also for long tme smulatons. A sxth order accurate scheme for a non-unform staggered grd wth good conservaton propertes s adopted: the proposed scheme conserves mass and momentum; the non-conservaton of knetc energy s weak. It s a functon of a commutaton error whch s very small for smoothly varyng meshes (Vaslyev [21]). 2 Energy conservaton property of the numercal scheme n Large Eddy Smulaton The fourth nconsstency of the dynamc mxed models (based on the Smagornsky closure relaton) s connected to the problems arsng from the numercal scheme adopted for the smulatons of three - dmensonal unsteady flows (LES). The three - dmensonal unsteady flows smulatons requre numercal schemes wth a hgh order of accuracy: a low order of accuracy of centred fnte dfference schemes ntroduces an ant-dsspatve factor, whch reduces the ablty of the generalsed SGS turbulent stress tensor to represent the knetc energy transfer from the resolved scales to the unresolved ones, wth an ncrease of the resolved knetc energy. The numercal scheme, besdes beng accurate, must fulfl the conservaton requrement. Conservaton propertes of the mass, the momentum and the knetc energy equatons, for ncompressble flows, are regarded as analytcal requrements for a proper set of dscrete equatons. Consder the followng governng equaton

252 F. Gallerano and G. Cannata for the scalar quantty φ: φ t + 1 Q (φ) + 2 Q (φ) + 3 Q (φ) +... = 0 (1) the term k Q (φ) s conservatve (conserves the volume ntegral of φ over the whole doman n perodc feld) f t can be wrtten n dvergence form k Q (φ) = (k F j ) x j (2) Note that mass s conserved a pror snce the contnuty equaton appears n dvergence form. The convectve term of the momentum equatons for an ncompressble flow can be expressed n dvergence and advectve form: (Dv.) = u ju x j ; (Adv.) = u j u x j (3) The relaton between the dvergence and the advectve form of the convectve term s gven by: u u j = u ju u j u (4) x j x j x j The advectve form of the convectve term conserves momentum (.e. the convectve term volume ntegral over the whole doman, n perodc feld, does not modfy the global volume ntegral of momentum) only f the contnuty equaton (for an ncompressble flow) s perfectly satsfed. The convectve term of the momentum equaton s conservatve a pror f t s wrtten n dvergence form. Ths defnton of the conservaton a pror ndcates the property of conservng momentum ndependently of the modaltes by whch the contnuty equaton s satsfed. The governng equaton for the knetc energy, K = u u /2, can be developed by takng the vector dot product of the velocty and the momentum equaton, ( u u t + u ju + p σ ) j = 0 (5) x j x x j where p s the pressure dvded by the constant densty, and σ j s the vscous stress. In the above equaton the convectve term can be rewrtten n the followng form, correspondng to that n the momentum equaton, u j u u = u ju u /2 + 1 x j x j 2 u u j u (6) x j

A new turbulence model for Large Eddy Smulaton 253 Ths term s composed by two parts: the frst s n conservatve form and the second nvolves the contnuty equaton. The convectve term (expressed n dvergence form n the momentum equaton) conserves a pror momentum but does not conserve a pror knetc energy: n fact equaton (6) shows how the contnuty equaton s nvolved n the knetc energy conservaton property of the convectve terms that are expressed n dvergence form. In other words, knetc energy volume ntegral s conserved, n a perodc feld, (by the dvergence form of the convectve term) only when the contnuty equaton s perfectly satsfed. The passage from the prevous analytcal consderatons to the effect that they produce on numercal smulatons mposes a reflecton on the followng statement: the contnuty equaton cannot be perfectly satsfed by numercal smulaton. In the smulatons of three-dmensonal unsteady flows (LES) (that are realzed by a hgh-order fnte dfference scheme wth the dvergence form of the convectve terms) the resolved knetc energy s not perfectly conserved because the contnuty equaton s not perfectly satsfed. Consequently, n LES, the numercal dscretzaton of the dvergence form of the convectve term n the momentum equaton ntroduce a numercal error assocated wth a producton of resolved knetc energy. Fgure 1: Profles of resolved knetc energy Reynolds averaged over successve ntervals of tme (T1, T2, T3, T4). Smulaton performed by usng the turbulence model of Zang et al. [23]. Channel flow, Re*=395. Wthout a turbulence model that s able to dsspate the above mentoned knetc energy producton, the resolved knetc energy s destned to rse n long tme smulatons. In the dynamc mxed models based on the Smagornsky closure relaton, the calculaton of the closure coeffcent C S (by dynamc procedure) s not

254 F. Gallerano and G. Cannata able to compensate the effects produced by the convectve terms (expressed n dvergence form n the resolved momentum equaton) that are not able to perfectly conserve resolved knetc energy. In order to verfy the above mentoned nconsstency of the dynamc mxed models based on the Smagornsky closure relaton, smulatons of a turbulent channel flow at Re*=395 (Re* s the frcton-velocty-based Reynolds number) have been performed, by a sxth-order staggered fnte dfference scheme proposed by Vaslyev [21]. The generalsed SGS turbulent stress tensor has been calculated by means of the mxed dynamc model of Zang et al. [23]. The resolved knetc energy has been averaged over tme ntervals greater than the ntegral turbulent tme scale. In Fgure 1 the over tme averaged resolved knetc energy profles are shown. From the fgure t s possble to deduce that the resolved knetc energy ncreases. In ths paper t s demonstrated that, n order to contan the ncrease of resolved knetc energy, the turbulent closure relaton for the generalsed SGS turbulent stress tensor must be expressed drectly as a functon of the generalsed SGS turbulent knetc energy E and of the SGS vscous dsspaton ε. The generalsed SGS turbulent knetc energy and the SGS vscous dsspaton are unknown quanttes that are calculated by solvng the relatve balance equatons. In these equatons there are unknown terms that are calculated by dynamc procedures: t s shown that the dynamc procedures for the calculaton of the closure coeffcents of the producton and dsspaton terms of the SGS vscous dsspaton balance equaton are able to compensate dynamcally the ncrease of resolved knetc energy (produced by the numercal dscretzaton of the dvergence form of the convectve term n the momentum equaton) and then allow the large eddy smulaton of three-dmensonal unsteady flows, also for long tme smulatons. 3 The turbulence model The generalsed SGS turbulent stress tensor, τ j, can be splt nto three tensors τ j = u u j u u j = L m j + C m j + R m j (7) where u s the th component of the nstantaneous velocty, the overbar represents the applcaton of the grd flterng operator G, L m j = u u j u u j, C m j = u u j u u j + u u j u u j, R m j = u u j u u j (8) are, respectvely, the so-called modfed Leonard tensor, modfed cross tensor and modfed Reynolds tensor and u = u u.

A new turbulence model for Large Eddy Smulaton 255 In dynamc Smagornsky-type mxed models the modfed cross tensor Cj m and the modfed Reynolds tensor Rj m are related to the resolved stran rate tensor S j by means of scalar eddy vscosty ν T. The algnment assumpton between the tensor Cj m + Rj m and S j s not expermentally verfed [10], [19]. In order to remove ths assumpton and take nto account the ansotropy of the unresolved scales of turbulence, we express the generalsed SGS turbulent stress tensor n the form: τ j = L m j 2ν jmn S mn (9) The eddy vscosty s expressed n the above equaton by a fourth order tensor proportonal to the product of a second-order tensor, b j, whch represents the turbulence length scales, and a and a second-order tensor, d mn, whch represents the turbulence length scales, accordng to the equaton ν jmn = Cb j d mn, where b j = b j and d mn = d nm (10) The expresson of the eddy vscosty n terms of a fourth order tensor enables the ansotropc character of the turbulence to be fully represented, snce t assume nether the exstence of a sngle turbulence velocty scale nor a sngle turbulence length scale, as s found n models n whch the vscosty s expressed as a scalar. The second order tensor that represents the turbulence velocty scales s defned as b j = EL m j / L m kk (11) In whch E s the generalsed SGS turbulent knetc energy. In ths equaton t s assumed that the second-order tensor b j s proportonal to the square root of the generalzed SGS turbulent knetc energy E and that t s algned wth the modfed Leonard tensor. In ths manner t s assumed that the ansotropy of the unresolved turbulence velocty scales, expressed by b j, s equal to the ansotropy of the smallest resolved scales, assocated wth the modfed Leonard tensor L m j. Ths assumpton s based on scale smlarty, accordng to whch the scales that are contguous n the wavenumber space have strct dynamc analoges related to the energy exchange processes, whch occur between them. In order to take nto account the complextes of the phenomena lnked to the ansotropy of the unresolved turbulence length scales, the ansotropy of the tensor d mn s related to the ansotropy of the flter used. Consequently, the second order tensor d mn s defned as / 3 d mn = m n 1 2 3 (12) In whch m s the vector of whch the components are the flters dmensons n the three coordnates drectons.

256 F. Gallerano and G. Cannata Wth (10), (11) and (12) the generalzed SGS turbulent stress tensor takes the form: τ j = L m j 2Cd mn Smn EL m j / L m kk (13) whch may be rewrtten n the more compact form τ j = (1 + r) L m j, where r = 2Cd mn Smn E / L m kk (14) The coeffcent r n (14) s unquely determned by usng the exstng relaton between the generalzed SGS turbulent knetc energy and the generalzed SGS turbulent stress tensor. By defnton, the generalzed SGS turbulent knetc energy s equal to half the trace of the generalzed SGS turbulent stress tensor Equaton (14) and (15) gves from whch we have Introducng (17) nto (14) gves: τ j = ( E = τ kk /2 (15) τ kk = (1 + r)l m kk, (16) r = (2E L m kk) /L m kk. (17) 1 + 2E Lm kk L m kk ) ( ) 2E L m j = L m j. (18) The generalzed SGS turbulent stress tensor s expressed n equatons (18) by means of a tensor of whch the prncpal axes are algned wth those of the modfed Leonard tensor L m j. The closure relaton expressed by equatons (18) s obtaned wthout any assumpton of local balance between producton and dsspaton of the generalzed SGS turbulent knetc energy. The assumpton of local balance between the producton and dsspaton of the generalzed SGS turbulent knetc energy s vald, n fact, (n a statstcal sense) only for homogeneous and sotropc turbulence and wthn the nertal subrange, whch only occurs at hgh Reynolds numbers and at hgh wavenumbers. Gven that the assumpton of local balance between producton and dsspaton of the generalzed SGS turbulent knetc energy only occurs n the nertal subrange (when ths exsts), the above-mentoned assumpton requres the use of spatal flters of whch the dmensons are assocated wth wavenumbers belongng to the nertal subrange tself. Wth the ncrease n the Reynolds number the nertal subrange occurs at ncreasngly hgh wavenumbers and, therefore, at ncreasngly small turbulence scales. Consequently, the assumpton of local balance between producton and dsspaton of the generalzed L m kk

A new turbulence model for Large Eddy Smulaton 257 SGS turbulent knetc energy requres the use of fner grds as the Reynolds number ncreases. The closure relaton expressed by (18), removng the abovementoned balance assumpton, does not requre the use of spatal flters (and, therefore, calculaton grds) of dmensons assocated wth wavenumbers fallng n the nertal subrange and may therefore be used wth coarser calculaton grds. The sequence of equatons (9)-(18) demonstrates that the defnton of the eddy vscosty n terms of a fourth-order tensor and the defnton of the tensor of the turbulence velocty scales algned wth the modfed Leonard tensor are equvalent to the assumpton of scale smlarty. Ths enables the formulaton of a closure relaton for the generalzed SGS turbulent stress tensor smlar to that of many scale smlarty models derved from the Bardna model [1]. It can, therefore, easly be seen that the exstence of scale smlarty between the turbulent structures assocated wth contguous scales s n lne wth the defnton of a fourth-order turbulent vscosty tensor, whch removes the assumpton of local sotropy. Consequently, the formulaton of the generalzed SGS turbulent stress tensor defned by the sequence of equatons (9)-(18) may be consdered applcable n LES wth flter wdth fallng nto the range of wavenumbers greater than the wavenumber correspondng to the maxmum turbulent knetc energy. On the other hand, n the range of wavenumbers that are below the latter, even though the strong ansotropy of the turbulent structures suggests the use of the fourth-order eddy vscosty tensor, the scale smlarty assumpton may not be reasonably formulated. In ths range, n fact, the part of the turbulent knetc energy produced by the largest unresolved eddes s hgh and consequently the above-mentoned turbulent structures may not entrely represent the energy dsspaton process (a basc assumpton of scale smlarty models). The generalsed turbulent knetc energy E s calculated by solvng ts balance equaton DE Dt = 1 τ(u k, u k, u m ) τ (u m, u k ) ū k τ (p, u m) 2 x m x m x m 2 ( E uk +ν ντ, u ) k x m x m x m x m (19) The symbols τ(f, g) and τ(f, g, h) represent the generalzed second and thrdorder central moments (Germano [5]) related to the generc quanttes f, g and h, namely τ(f, g) = f g f g (20) τ(f, g, h) = f g h f g h f τ(g, h) g τ(f, h) h τ(f, g) (21) Equaton (19) s form nvarant under Eucldean transformaton of the frame and frame ndfferent (Gallerano et al. [3]).

258 F. Gallerano and G. Cannata Consequently, the modelled balance equaton of E must be form-nvarant and frame-ndfferent, lke the exact balance equaton of E. The sum of the 1 st and 3 rd term of the rght-hand sde of equaton (19) expresses the turbulent dffuson of the generalsed SGS turbulent knetc energy, for whch the followng closure relaton s proposed: 1 2 τ (u, u, u k ) + τ (p, u k ) = D E E (22) x k The scalar coeffcent D s dynamcally calculated by means of the dentty: 1 1 2 T (u, u, u k ) + T (p, u k ) 2 τ (u, u, u k ) + τ (p, u k ) = + 1 2 u u u k 1 2 u u u k 1 2 u k T (u, u ) + 1 2 u kτ (u, u ) u T (u, u k ) + u τ (u, u k ) + pu k p u k (23) where the angular brackets represent the applcaton of a flterng operator F, wth characterstc dmenson, T, whch s double that the grd flter one and T (f, g) = fg f g (24) T (f, g, h) = fgh f g h f T (g, h) g T (f, h) h T (f, g) (25) are, respectvely, the generalsed second and thrd order central moments relatve to test-scale flterng operator G1=FG. The second-order generalsed central moments relatve to the test and s lnked to the second-order generalsed central moments relatve to the grd flter by means of the Germano dentty [4]: T (f, g) = τ (f, g) + f g f g (26) For the generalsed central moments, relatve to the test flter, on the left - hand sde of (23) we propose the followng closure relaton 1 2 T (u, u, u k ) + T (p, u k ) = D E T T E T x k (27) where E T s the generalsed SGS turbulent knetc energy relatve to the test flter. By ntroducng equatons (22) and (27) nto the left-hand sde of (23), by ntroducng equatons (15) and (18) nto the rght-hand sde of (23) and by usng (26), the closure coeffcent D s obtaned. By ntroducng (18) and (22) nto (19), the proposed modelled form of the generalsed SGS turbulent knetc energy s obtaned: DE Dt = x k ( D E E ) x k ( ) 2E L m qq L m mk ū k x m 2 E + ν ε (28) x m x m

A new turbulence model for Large Eddy Smulaton 259 The last term on the rght-hand sde of equaton (19) s the generalsed SGS vscous dsspaton ( uk ε = ντ, u ) k (29) x m x m In the proposed turbulence model the generalsed SGS vscous dsspaton s calculated by solvng ts balance equaton. The exact ε balance equaton, expressed n terms of the generalzed central moments, takes the form: ε t + u kε x k +2ν ( u τ x k x j 2ν ( u x k +2ν u k x j τ 2 ε ν + ν ( τ u k, u, u ) x k x k x k x j x j ( u k, u )) + 2ν ( uk τ, p ) x j x k x x ) ( τ (u, u k ) u + 2ντ, u k, u ) x j x j x k x j x j ( u, u ) + 2ν u ( uk τ, u ) x k x j x k x j x j ( u, u ) k + 2ν τ (u, u k ) 2 u x k x j x j x j x k ( +2ν 2 2 u 2 ) u τ, = 0 (30) x j x k x j x k +2ν u x j τ Ths equaton s obtaned from the Naver-Stokes equaton and the fltered Naver-Stokes equaton. In the Gallerano et al. model [3], the 5 th and 7 th terms of the above mentoned equaton are added together. It s easy to demonstrate that the 11 th term and the sum between the 5 th and the 7 th terms are the representatons, n an nertal frame, of two objectve but frame-dependent zero-order tensors. The 11 th term and the above sum are unknown quanttes that must be modelled. The modelled expresson of the sum of the two terms (5 th and 7 th ) proposed n [3] s not correct, snce they represent quanttes wth dfferent physcal meanngs; the 5 th s a convectve transport term, whle the 7 th represents a dspersve transport term. Furthermore, there s an evdent contradcton n [3]: the modelled expressons n [3] of the 11 th term and the sum between the 5 th and 7 th term are objectve quanttes and frame-ndfferent, whle the expressons of the correspondng terms that appear n the exact ε balance equaton are objectve quanttes but frame-dependent. In ths paper the above contradctons are overcome.

260 F. Gallerano and G. Cannata The sum of the 4 th and 6 th terms of equaton (30) are modelled as follow ν ( τ u k, u, u ) + 2ν ( uk τ, p ) E 2 L m kl ε = C Fε (31) x k x j x j x k x x ε L m jj x l The closure coeffcent C Fε s calculated dynamcally, by means of the dentty: [ ( ν T u k, u, u ) ( uk + 2T, p )] x j x j x x ( ν τ u k, u, u ) ( uk + 2τ, p ) = x j x j x x ( u u ur ν u k u k νt, u ) r x j x j x m x m ( u 2ν T u k, u ) u u + ν u k x j x j x j x j ( ur + u k τ, u ) ( r u + 2ν τ u k, u ) x m x m x j x j uk p uk p +2ν 2ν (32) x x x x On the rght-hand sde of equaton (32) the unknown second-order generalsed central moment, τ (u k, u / x j ), and the correspondng second-order generalsed central moment relatve to the test flter, T (u k, u / x j ), appear. By assumng the scale smlarty assumpton, the above mentoned generalsed central moment relatve to the grd flter becomes: τ (u k, u / x j ) = τ (u k, u / x j ) (τ (u, u / x )/τ (u m, u m / x m )) (33) where τ (u, u / x ) s an unknown second-order generalsed central moment that must be modelled. The followng closure relatons for ths quantty and for the correspondng second-order generalsed central moment relatve to the test flter are proposed: τ ( u, u ) x = C Tε E ε ε x q δ q, T ( u, u ) x = C Tε E T ε T ε T x q δ q (34) where ε T s the generalsed SGS vscous dsspaton relatve to the test flter. The closure coeffcent C Tε s calculated by means of the followng dentty: ( T u, u ) ( τ u, u ) u u = u u (35) x x x x By ntroducng (34) 1 and (34) 2 nto the left-hand sde of equaton (35), the closure coeffcent C Tε s obtaned.

A new turbulence model for Large Eddy Smulaton 261 For the generalsed central moments, relatve to the test flter, on the lefthand sde of (31) the proposed closure relaton s νt ( u k, u x j, u x j ) + 2νT ( uk, p ) ( ) 2 E T L m T kl ε T = C Fε (36) x x ε T L m jj T x l where L mt k s the modfed Leonard tensor relatve to the test flter. By ntroducng the closure relatons (31) and (36) nto the left-hand sde of equaton (32), by ntroducng (33) and (34) nto the rght-hand sde of equaton (32) and by usng (26), the closure coeffcent C Fε s obtaned. The 5 th term of equaton (30) s calculated by usng the scale smlarty hypothess, expressed by equaton (33), and by modellng only the unknown term on the rght-hand sde of (33) by means of the closure relaton (34). The closure coeffcent on the rght-hand sde of (34) s dynamcally calculated by means of equaton (35). The resultng closure relaton for the 5 th term of (30) s: 2ν ( ( u τ u k, u )) = 2ν x k x j x j x k u x j C Tε E ε ( ε τ uk, u δ q x q τ ( ) u n, u n x n ) x j (37) As t s easy to demonstrate, the proposed expresson for the 5 th term of equaton (30) results dependent on the frame of reference, under Eucldean transformatons of the frame, the same manner of the no-modelled term. The 7 th term of (30) s related exclusvely to the resolved velocty feld and the generalsed SGS turbulent stress tensor. It s calculated by means of the closure relaton (18) for the generalsed SGS turbulent stress tensor. The resultng closure relaton for the 7 th term of (30) s: 2ν ( ) u τ (u, u k ) = 2ν ( u x k x j x j x k x j ( )) 2E L m x j L m k qq (38) Under a Eucldean transformaton of the frame, the modelled expresson of the 7 th term of equaton (30) results dependent on the frame of reference the same manner of the no-modelled term. The 8 th term of equaton (30) represents a producton term of ε. For ths tensor the proposed closure relaton s ( u 2ντ, u k, u ) = C Pε ε ( )/ L m j S j L m x k x j x kk (39) j The closure coeffcent C Pε s dynamcally calculated by means of the dentty: ( u 2νT, u k, u ) ( u 2ν τ, u k, u ) = x k x j x j x k x j x j

262 F. Gallerano and G. Cannata ( ) ( ) u uk u uk u u 2ν τ + 2ν τ x k x j x j x j x k x j ( ) ( ) u u u k u uk u +2ν τ 2ν T x j x k x j x k x j x j ( ) ( ) uk u u u u u k 2ν T 2ν T x j x k x j x j x k x j u u k u u uk u +2ν 2ν x k x j x j x k x j x j (40) The unknown generalsed second-order central moments n (38) are expressed n terms of resolved varables by assumng the scale smlarty assumpton. In partcular, by assumng the scale smlarty assumpton, the unknown generalsed second order central moment n the 3 rd term of the rght-hand sde of (40) becomes: τ ( ) ( u u k u = τ, u ) ( ( )/ ( k um u m uq τ τ, u )) q x k x j x k x j x s x s x n x n (41) whch, for (29), s equvalent to τ ( ) u u k = ε ( ( u τ, u )/ ( k uq τ, u )) q x k x j ν x k x j x n x n (42) Analogously, the unknown generalsed second order central moments n the 1 st and 2 nd term of the rght-hand sde of (40) are calculated by means of expressons smlar to equaton (42). For the generalsed thrd-order central moment, relatve to the test flter, on the left-hand sde of (40) the proposed closure relaton s 2νT ( u, u k, u ) x k x j x j = C Pε ε ( T L mt j S T )/ j L mt kk (43) where S T j s the resolved stran rate tensor relatve to the test flter. By ntroducng the closure relatons (39) and (43) nto the left-hand sde of equaton (40), by ntroducng the closure relaton (42) nto the rght-hand sde of (40) and by usng (26), the closure coeffcent C Pε s obtaned. The unknown quanttes n the 9 th, 10 th and 11 th term of equaton (30) are calculated, by usng the scale smlarty assumpton, by means of equatons (41) and (42). In partcular, for the 11 th term of (30) we obtan: 2ν u ( u τ, u ) k = 2ε u ( ( u τ, u )/ ( k uq τ, u )) q x j x k x j x j x k x j x n x n (44)

A new turbulence model for Large Eddy Smulaton 263 As t s easy to demonstrate, the proposed expresson for the 11 th term of equaton (30) results dependent on the frame of reference, under a Eucldean transformaton of the frame, the same manner of the no-modelled term. The 12 th term of equaton (30) s calculated by means of the closure relaton (18) for the generalsed SGS turbulent stress tensor. The last term of equaton (30) represents the destructon of ε and s modelled as follow ( 2ν 2 2 u 2 ) u ε 2 τ, = C Dε (45) x j x k x j x k E where the closure coeffcent C Dε s calculated by the dentty: ( 2ν 2 2 u 2 ) ( u T, 2ν 2 2 u 2 ) u τ, = x j x k x j x k x j x k x j x k 2ν 2 2 u 2 u 2ν 2 2 u 2 u (46) x j x k x j x k x j x k x j x k The followng closure relaton s proposed for the 1 st term on the left of (46) ( 2ν 2 2 u 2 ) u ( T, = C ) Dε ε T 2 / E T (47) x j x k x j x k By ntroducng (45) and (47) nto (46) the closure coeffcent C Dε s obtaned. The fnal modelled form of equaton (30) s +2ν ( u x k x j 2ν x k ε t + u kε 2 ε ν + E 2 L m kl ε C Fε x k x k x k x k ε L m jj x l ( ( E ε C Tε δ q τ u k, u )/ ( τ u n, u ))) n ε x q x j x n ( ( )) u 2E ε ( ) L m L m j S j x j x j L m k C Pε qq L m kk +2ν u ( ε x j ν τ u, u )/ ( uq τ, u ) q x k x j x n x n +2ν u ( ε x k ν τ uk, u )/ ( uq τ, u ) q x j x j x n x n +2ν u ( ε x j ν τ u, u )/ ( k uq τ, u ) s x k x j x s x q +2ν ( 2E x j L m L m k nn ) 2 u x j x k + C Dε ε 2 E = 0 (48) The proposed modelled balance equaton of the SGS vscous dsspaton s form-nvarant under Eucldean transformatons of the frame and has the same dependence on the frame as the exact equaton (the demonstraton s omtted for the sake of brevty).

264 F. Gallerano and G. Cannata 4 The numercal scheme For the smulaton of the unsteady three-dmensonal turbulent flow t s very mportant to control the dsspaton produced by the numercal scheme. The numercal dsspaton removes energy from the dynamcally mportant smallscale eddes; for ths reason unsteady, three-dmensonal turbulent smulatons are much less tolerant of numercal dsspaton (Mornsh et al [13]), consequently, the numercal scheme must be accurate. Mornsh et al [13] derved the general famly of fully conservatve hgher order accurate fnte dfference schemes for unform staggered grds. The extenson of the scheme, suggested by Mornsh et al., to non unform meshes produces a fourth order accurate fnte dfference scheme that s not fully conservatve. Vaslyev [21] generalzed the hgh order schemes of Mornsh et al. to non-unform meshes by preservng the symmetres of the unform mesh case, obtanng numercal schemes wth good conservaton propertes. The schemes of Vaslyev do not smultaneously conserve mass, momentum and knetc energy. However, dependng on the form of the convectve term, conservaton of ether momentum or energy n addton to mass can be acheved. In ths paper we produce a sxth order accurate scheme for a non-unform staggered grd wth good conservaton propertes: the proposed scheme conserves mass and momentum; the nonconservaton of knetc energy s weak. It s a functon of a commutaton error whch s very small for smoothly varyng meshes [21]. The fltered Naver-Stokes equatons, the generalzed SGS turbulent knetc energy balance equaton and the vscous dsspaton balance equaton are ntegrated wth a sxth order fnte dfference scheme on a non-unform staggered grd. A fractonal step method s employed. The computatonal grd n the physcal doman s obtaned by mappng a unform computatonal grd n the computatonal doman to the physcal doman. Let Dand Ω be respectvely the physcal and computatonal domans, x = (x1, x 2, x 3 ) T and ξ = (ξ 1, ξ 2, ξ 3 ) T be coordnates n the physcal and computatonal domans, ξ = f ( x ) be a non lnear map of physcal doman D nto the computatonal doman, and 1, 2, 3 be unform grd spacngs n the respectve drectons n computatonal doman Ω. In ths paper we use only un-drectonal mappngs, ξ = f (x ) ( = 1,..3), and the computatonal grd n the physcal space s constructed as a tensor product of one-drectonal computatonal grds. The dervatve n the physcal space s calculated usng the local Jacoban, whch can be found numercally usng the same stencl and the same order accuracy as fnte dfferencng operator n the computatonal space. The dervatve n computatonal space (n the one-dmensonal case) s approxmate as δφ/δξ = (φ +1 φ 1 ) /2 (49) where s the unform grd spacng. The dervatve n physcal space s found

A new turbulence model for Large Eddy Smulaton 265 as δφ/δx = (1/J) (δφ/δξ) (50) where J s the Jacoban of the transformaton x ξ, whch can be found numercally by substtutng xforφ: J = (δx/δξ) = (x +1 x +1 ) /2 (51) Thus, n ths paper, the fnte dfference n the computatonal doman wth stencl n actng on φwth respect to ξ 1 s (δ n φ/δ n ξ 1 ) φ(ξ 1 + n 1 /2, ξ 2, ξ 3 ) φ(ξ 1 n 1 /2, ξ 2, ξ 3 ) n 1 (52) and the nterpolaton operator wth stencl n actng on φ n the ξ 1 drecton s φ nξ 1 = φ(ξ 1 + n 1 /2, ξ 2, ξ 3 ) + φ(ξ 1 n 1 /2, ξ 2, ξ 3 ) 2 (53) Let NS6 be the dfference between the exact convectve term and ts dscrete approxmaton. The sxth order accurate scheme for the dvergence form of the convectve term s gven by: u j u NS6 150 [( δ 1 150 x j 128 δ 1 x j 128 U 1x j 25 128 U 3x j + 3 ) 128 U 5x j 25 [( δ 3 150 128 δ 3 x j 128 U 1x j 25 128 U 3x j + 3 ) 128 U 5x j + 3 [( δ 5 150 128 δ 5 x j 128 U 1x j 25 128 U 3x j + 3 ) 128 U 5x j ] U 1x j ] U 3x j ] U 5x j (54) where the dscrete fnte dfference operator n the physcal doman s defned as δ n φ/δ n x = (1/J ) (δ n φ/δ n ξ ), where J s the local Jacoban of the transformaton x ξ [21]. Ths numercal scheme has good conservaton propertes and sxth order accuracy. The sxth-order accurate fnte dfference approxmaton of the dvergence form of the convectve terms conserves mass and momentum ed ntroduces a weak producton of knetc energy: ths producton s a functon of a commutaton error whch s very small for smoothly varyng meshes [21]; the dynamc procedures for the calculaton of the closure coeffcents of the producton and dsspaton terms of the SGS vscous dsspaton balance equaton (expressed by equatons (32), (35), (40) and (46)) are able to compensate dynamcally the above mentoned ncrease of resolved knetc energy and then allow the large eddy smulaton of three-dmensonal unsteady flows, also for long tme smulatons.

266 F. Gallerano and G. Cannata 5 Results and dscusson Turbulent channel flows (between two flat parallel plates placed at a dstance of 2L) are smulated wth the proposed Large Eddy Smulaton model at dfferent frcton-velocty-based Reynolds numbers (Re*), rangng from 395 to 2340. In order to valdate the proposed closure relaton for the generalzed SGS turbulent stress tensor, the numercal results obtaned wth the proposed model are compared wth DNS results (Mansour et al. [9]) and wth expermental data (Comte-bellot [2]). Fgure 2: Tme-averaged streamwse veloctes. Comparson between DNS and LES results obtaned wth DMM and the proposed model (TEM). Channel flow, Re* = 395. Fgure 3: Tme-averaged streamwse veloctes. Comparson between expermental measurements and LES results obtaned wth the proposed model (TEM). Channel flow, Re* = 2340. In Fgure 2 s plotted the profle of the tme-averaged streamwse velocty

A new turbulence model for Large Eddy Smulaton 267 component obtaned wth the proposed model compared wth the profle obtaned wth DNS [9] and the Dynamc Mxed Model, DMM [23], for channel flow at Re* = 395. The fgure shows that the profle obtaned wth the proposed model agrees more wth the DNS velocty profle than wth the profle obtaned wth the DMM, both n the boundary layer and n the regon nsde the channel. Fgure 3 shows the profle of the tme-averaged streamwse velocty component for a channel flow at Re*=2340 obtaned wth the proposed model, compared wth the profle of the analogous velocty component measured expermentally [2]. The agreement between the two velocty profles s very good. Fgure 4 compares the profle of the component { u 1u 3} of the Reynolds stress tensor (where ndexes (1) and (3) denote, respectvely, the streamwse and wallnormal drectons), calculated wth the proposed model, wth the profle of the smlar component of the Reynolds stress tensor obtaned from expermental measurements [2], for a channel flow at Re* = 2340. Fgure 3 shows that at Re* = 2340 the proposed model provdes a profle of the component { } u 1u 3 n agreement wth that of the correspondng component of the Reynolds stress tensor obtaned from the expermental measurements. Fgure 4: Reynolds stress { u 1u 3}. Comparson between expermental measurements and LES results obtaned wth the proposed model (TEM). Channel flow, Re* = 2340. Fgure 5 shows nstantaneous profles of the terms of the balance equatons of E averaged over homogeneous planes, for channel flow at Re*=2340. Fgure 5 demonstrates that the balance between producton and dsspaton of the generalzed SGS turbulent knetc energy s confrmed only n a lmted regon between the buffer layer and the log layer (20<z + <40) whlst t s not confrmed n other regons of the doman. The vscous dsspaton of E s balanced n the vscous sublayer (z + <5) by the vscous dffuson term whlst the producton of E s practcally neglgble. Movng away from the wall, n the

268 F. Gallerano and G. Cannata Fgure 5: Instantaneous generalzed SGS turbulent knetc energy balance terms averaged over homogeneous planes. Producton: P E ; Turbulent transport: T E ; Convecton: C E ; Vscous dffuson: D E ; Vscous dsspaton: eps. Channel flow, Re*=2340. frst part of the buffer layer, the producton term of E ncreases untl t reaches ts maxmum value (z + 10) and the terms of turbulent transport and vscous dffuson of E are comparable wth the producton term of E. In the regon between the buffer layer and the log layer (20<z + <40) the convectve and turbulent transport terms and the vscous dffuson term are neglgble compared wth the producton and dsspaton terms. Only n ths lmted regon there s a balance between the producton and the dsspaton of E. Towards the centre of the channel (z + >30) the vscous dsspaton tends towards a mnmum but not neglgble value. In ths regon the producton term of E s balanced not only by the dsspaton but also by the turbulent transport of E. 6 Conclusons The relatons between numercal scheme conservaton property of mass, momentum and knetc energy and the drawbacks of the dynamc Smagornskytype turbulence models are shown. A new turbulence model s proposed. The proposed model: a) s able to take nto account the ansotropy of the turbulence; b) remove any balance assumpton between the producton and dsspaton of subgrd scale turbulent knetc energy; c) s able to elmnate the numercal effects produced by the non conservaton a pror of the resolved knetc energy. New closure relatons for the unknown terms of the subgrd scale vscous dsspaton balance equaton are proposed. The fltered momentum equatons are solved by usng a sxth order fnte dfference scheme. The proposed model s tested for a turbulent channel flow at Reynolds numbers

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