A new dynamic k-ε subgrid scale model

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1 Proceedngs of the 4th WSEAS Internatonal Conference on Flud Mechancs and Aerodynacs Elounda Greece August (pp ) A new dynac -ε subgrd scale odel F. GALLERANO G. CANNATA L. MELILLA Dpartento d Idraulca Trasport e Strade Unverstà d Roa "La Sapenza" Va Eudossana 8 Roa ITALY francesco.gallerano@unroa.t Abstract: - A new LES odel s proposed. The proposed closure relaton for the generalzed SGS turbulent stress tensor: coples wth the prncple of turbulent frae ndfference; taes nto account both the ansotropy of the turbulence velocty scales and turbulence length scales; reoves any balance assupton between the producton and dsspaton of SGS turbulent netc energy. In the proposed odel the generalzed SGS turbulent stress tensor s related exclusvely to the generalzed SGS turbulent netc energy (whch s calculated by eans of ts balance equaton) and the odfed Leonard tensor. The fltered oentu equatons are solved by usng a staggered fourth order fnte dfference schee. The proposed odel s tested for a turbulent channel flow at Reynolds nubers (based on frcton velocty and channel half-wdth) rangng fro 395 to 340. Key-Words: - LES closure relaton -ε subgrd odel Introducton Aong the ost coon LES odels present n lterature are the Dynac Sagornsy-type SGS Models (e.g. Dynac Sagornsy Mode DSM [] Dynac Mxed Model DMM [] DMM [3] Lagrangan Dynac Model LDM [4] Dynac Two-paraeter Model DTM [5]) n whch the generalzed SGS turbulent stress tensor s related to the resolved stran-rate tensor by eans of a scalar eddy vscosty. It s assued n these odels that the eddy vscosty s a scalar proportonal to the cubc root of the generalzed SGS turbulent netc energy dsspaton and that such dsspaton s locally and nstantaneously balanced by the producton of the generalzed SGS turbulent netc energy (.e. by the rate of netc energy per unt of ass transferred fro the large scales larger than the flter sze to the unresolved ones). Consequently t s evdent that the dynac Sagornsy-type SGS odels are fraught wth three relevant drawbacs. The frst drawbac s represented by the scalar defnton of the eddy vscosty; the second one concerns the local balance assupton of the generalzed SGS turbulent netc energy producton and dsspaton whlst the thrd drawbac s related to the dynac calculaton of the coeffcent used to odel the eddy vscosty (Sagornsy coeffcent). The scalar defnton (frst nconsstency) of the eddy vscosty s equvalent to assung that the prncpal axes of the generalzed SGS turbulent stress tensor or the unresolved part of t (represented by the cross and Reynolds ters) are algned wth the prncpal axes of the resolved stran-rate tensor. Ths assupton has been dsproved by any experental tests and by DNS whch deonstrate that there s no algnent between the generalzed SGS turbulent stress tensor or the unresolved part of t and the resolved stran-rate tensor [6]. Moreover the eddy vscosty s proportonal to the product of two ters of whch the densons are respectvely those of a length and a velocty [7]. These ters whch represent respectvely the turbulence length scales and turbulence velocty scales are ore generally second-order tensors of whch the product s a fourth-order tensor whch represents the eddy vscosty [8]. The scalar defnton of the eddy vscosty used n the above-entoned dynac Sagornsy-type SGS odels presupposes the exstence of a sngle turbulence velocty scale and a sngle turbulence length scale. Ths s equvalent to assung 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 anner the turbulence ansotropy nduced by the contnuous transfer of energy fro the ean flow towards the turbulent fluctuatons whch s generally extreely ansotropc s not consdered. Even though the energy cascade process causes a reducton of the turbulence ansotropy any authors [9] deonstrated that even n the dsspaton range of the sallest turbulent scales where vscous dsspaton occurs there s a hgh ansotropy level even at hgh Reynolds nubers. The second nconsstency of the Sagornsy dynac odels s related to the assupton of a local and nstantaneous balance between producton and dsspaton of the generalzed SGS turbulent netc energy forulated n the above-entoned odels to obtan the turbulent vscosty expresson. Ths assupton s confred statstcally and never nstantaneously and only locally at the scales assocated wth wavenubers wthn the nertal subrange and the latter exsts only for sotropc turbulence and at hgh Reynolds nubers. Moreover snce the dsspaton of the generalzed SGS turbulent netc energy s by defnton postve the assupton of local balance

2 Proceedngs of the 4th WSEAS Internatonal Conference on Flud Mechancs and Aerodynacs Elounda Greece August (pp ) ples that also the producton of generalzed SGS turbulent netc energy s postve. However the assupton that the producton s always postve ples that the energy transfer always occurs fro the largest to the sallest scales and prevents postve transfers of netc energy fro the subgrd scales to the resolved ones (bacscatter). Snce the energy exchange processes between the resolved and unresolved scales generally occur n both drectons (forward scatter and bac scatter) as has been observed by varous authors [0] the assupton that the producton of generalzed SGS turbulent netc energy s always postve does not enable the coplexty of the energy exchange processes whch characterze the turbulence to be adequately taen nto account. The thrd nconsstency of the dynac odels concerns the calculaton of the above entoned Sagornsy coeffcent Cs. It s calculated wth varatonal ethods (e.g. wth a least squares nzaton ethod [] or Lagrangan ethod [4]). These ethods dentfy a sngle value of the scalar coeffcent Cs fro a syste of fve ndependent scalar equatons relatng the coponents of the ansotropc part of the generalzed SGS turbulent stress tensor to the coponents of the resolved stran-rate tensor. Ths procedure does not provde copletely acceptable results. Moreover when sulatng confned flows at hgh Reynolds nuber the results of the dynac procedure are of doubtful relablty n the regon close to the wall ncludng both the vscous sublayer and the buffer layer []. In ths regon the flter wdth used n the dynac procedure s larger than ost eddes that govern the oentu and energy transfer. Consequently the dynac procedure used under these condtons for the calculaton of the coeffcent Cs s not able to fully account for the local subgrd dsspatve processes that affect the entre doan. In ths paper the an drawbacs of the large eddy sulaton odels present n lterature are overcoe and a new LES odel s proposed. The closure relaton for the generalsed SGS turbulent stress tensor: a) coples wth the prncple of turbulent frae ndfference [3]; b) taes nto account both the ansotropy of the turbulence velocty scales and turbulence length scales; c) reoves any balance assupton between the producton and dsspaton of SGS turbulent netc energy. In the proposed odel: a) the closure coeffcent whch appears n the closure relaton for the generalsed SGS turbulent stress tensor s theoretcally and unquely deterned wthout adoptng Gerano s dynac procedure; b) the generalsed SGS turbulent stress tensor s related exclusvely to the generalsed SGS turbulent netc energy (whch s calculated by eans of ts balance equaton) and the odfed Leonard tensor. The calculaton of the vscous dsspaton s carred out by ntegratng ts balance equaton. The closure relatons (whch ntervene n the above entoned vscous dsspaton equaton) are forulated n such a way that the odeled equaton respects the for nvarance and frae dependence of the exact equaton. For the sulaton of the unsteady three-densonal turbulent flow t s very portant to control the dsspaton produced by the nuercal schee. The nuercal dsspaton reoves energy fro the dynacally portant sall-scale eddes; for ths reason unsteady threedensonal turbulent sulatons are uch less tolerant of nuercal dsspaton [4]. On the other hand the nuercal schee ust be accurate. Mornsh et al. [4] proposed a staggered fourth order fnte dfference schee. Vaslyev [5] showed that the extenson of the schee suggested by Mornsh et al. to non unfor eshes produces a fourth order accurate fnte dfference schee that s not fully conservatve. In ths paper the nuercal ntegraton of the fltered equatons s perfored by the staggered fourth order fnte dfference schee proposed by Mornsh et al. The turbulence odel Accordng to Bardna's scale slarty assupton the generalzed SGS turbulent stress tensor can be expressed by = ( r )L () where r s an unnown scalar coeffcent and L s the odfed Leonard tensor. In ths paper t s deonstrated that startng fro the scale slarty assupton n () (by sple atheatcal calculatons) a closure relaton s reached for the generalzed SGS turbulent stress tensor n whch there appears no coeffcent to be calbrate or to be calculate dynacally and whch s gven by the followng relaton: E = L () L It s easy to verfy that as by defnton the generalzed SGS turbulent netc energy equal to half the trace of the generalzed SGS turbulent stress tensor [7] E = (3) fro Equaton () s obtaned E L = ( r )L where r= (4) L Introducng (4) nto () gves: E L E = L = L (5) L L The closure relaton (5) s obtaned wthout any assupton of local balance between the producton and dsspaton of generalzed SGS turbulent netc energy and ay thus be consdered applcable to LES wth the flter wdth fallng nto the range of wave nubers greater than the wave nuber correspondng to the axu turbulent netc energy. The closure relaton (5) for the generalzed SGS turbulent stress tensor: a) coples wth the prncple of turbulent frae ndfference gven that t relates only obectve tensors; b) taes nto account both the ansotropy

3 Proceedngs of the 4th WSEAS Internatonal Conference on Flud Mechancs and Aerodynacs Elounda Greece August (pp ) of the turbulence velocty scales and turbulence length scales; c) assues scale slarty; d) guarantees an adequate energy dran fro the grd scales to the subgrd scales and guarantees bacscatter; e) overcoes the nconsstences lned to the dynac calculaton of the closure coeffcent used n the odellng of the generalzed SGS turbulent stress tensor. The generalzed SGS turbulent netc energy E s calculated by solvng ts balance equaton defned by the followng equaton: DE (u u u ) ( pu) = Dt (6) E ν ( F O u ) ν The st and 3rd ters of the rght-hand sde of Equaton (6) express the turbulent dffuson of the generalzed SGS turbulent netc energy: ( pu ) ( FE) ( u u u ) = The followng equaton s used for the calculaton of ( F E) E ( FE) = D E (8) x Scalar coeffcent D s dynacally calculated by eans of a Gerano dentty appled to the st and 3rd ters on the rght-hand sde of Equaton (6) T ² ( F ) ( ) ² E FE = T( u u u ) ( u u u ) (9) T pu ² pu ( ) ( ) where the frst ter on the left-hand sde of Equaton (9) s the turbulent dffuson of the generalzed SGS turbulent ±. ndcates the netc energy at the test level the sybol ( ) flter operaton at the test level and T(fg) = ² f g % fg (0) T(fgh) = ² f gh f gh % () ft(gh) % gt(fh) % ht(fg) are respectvely the generalzed second and thrd order central oent at the test level [6]. Accordng to (0) and () Equaton (9) reads T ( ) ² ( ) ² F %%% E FE = uuu uuu % ut ² ( ) ( ) % uu u uu ut ( uu ) u uu T pu ² pu () ² ( ) ( ) ( ) Usng Equaton (8) the left-hand sde ter of Equaton (9) taes the for (7) T ² T T E E F FE = D E D E T ² ( E ) ( ) (3) T where E s the generalzed SGS turbulent netc energy at the test level. The coeffcent D s calculated by ntroducng (3) nto (). The last ter on the rght-hand sde of Equaton (6) s defned as vscous dsspaton: u u ε ν = (4) In the proposed LES odel a further balance equaton s ntroduced for the subgrd vscous dsspaton ε. Ths equaton expressed n ters of the generalzed central oents taes the for [7]: ε ε ε u u ν ν u t x x x x x p ν u ν x x x x x u u u ν ν x x x x x ν ν x x x x x ν ν x x (5) F0 ν ν = 0 In ths paper an orgnal expresson s proposed for the odeled for of the balance equaton for the generalzed SGS turbulent netc energy dsspaton n whch the unnown tensors are odeled by adoptng the hypothess of scale slarty and Equaton (5) taes the for: ε ε ε E Ll ε ν CF ε t ε L ν u E ε C F δ q ε ε q u u n x n u xn ε E ( L S ) ν L C P ε x x Lqq L ν l ε ε ν ν x x u ν x x q q q q n n n n

4 Proceedngs of the 4th WSEAS Internatonal Conference on Flud Mechancs and Aerodynacs Elounda Greece August (pp ) ε ν ν x x u E u ν L nn q u x L x x s (6) xs xq ε = 0 CD ε E where δ = ( ) and n whch the closure coeffcents are calculated dynacally by eans of the Gerano denttes. For the sulaton of the unsteady three-densonal turbulent flow t s very portant to control the dsspaton produced by the nuercal schee. The nuercal dsspaton reoves energy fro the dynacally portant sall-scale eddes; for ths reason unsteady threedensonal turbulent sulatons are uch less tolerant of nuercal dsspaton [4]. On the other hand the nuercal schee ust be accurate. In ths paper the nuercal ntegraton of the fltered equatons s perfored by the staggered fourth order fnte dfference schee proposed by Mornsh et al. [4]. As t s shown by Vaslyev [5] the extenson of the schee suggested n [4] to non-unfor eshes produce a fourth order accurate fnte dfference schee that s not fully conservatve. Let y be the non-unfor drecton wth pont dstrbuton y. The followng dfference operator wth stencl n actng on the generc quantty φ wth respect to y s used δ nφ φ ( y n / ) φ ( y n / ) δ y y y n y n / n / and the followng nterpolaton operator s gven by φ ny y ( y y ) φ ( y ) ( y y ) φ ( y ) n / n / n / n / y y n / n / Let NS4 be the dfference between the exact convectve ter and ts dscrete approxaton. The fourth order accurate schee for the dvergence for of the convectve ter s gven by: u 9 δ 3 9 x x x NS4 U U U 8δx 8 8 δ x x x U U U 8 δ 3 x 8 8 (7) Ths nuercal schee has good conservaton propertes and fourth order accuracy and enables the ntegraton of the fltered oentu equaton and of the fltered SGS netc energy and vscous dsspaton balance equatons. 3 Result and dscusson Turbulent channel flows (between two flat parallel plates placed at a dstance of L) are sulated wth the proposed Large Eddy Sulaton odel at dfferent frcton-veloctybased Reynolds nubers (Re*) rangng fro 395 to 340. Fg.. Te-averaged streawse veloctes. Coparson between DNS and LES results obtaned wth DMM and the proposed odel (TEM). Channel flow Re* = 395. In order to valdate the proposed closure relaton for the generalzed SGS turbulent stress tensor the nuercal results obtaned wth the proposed odel are copared wth DNS results [8]and wth experental data [9]. Fg.. Reynolds stress <u 'u 3 '>. (ndexes () and (3) denote respectvely the streawse and wall-noral drectons) Coparson between DNS and LES results obtaned wth the dynac xed odel (DMM) and the proposed odel (TEM). Channel flow Re* = 395. In Fgure s plotted the profle of the te-averaged streawse velocty coponent obtaned wth the proposed odel copared wth the profle obtaned wth DNS [8] and the Dynac Mxed Model DMM [] for channel flow at Re* = 395. The fgure shows that the profle obtaned wth the proposed odel agrees ore 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 shows the profles of the coponent <u 'u 3 '> of the Reynolds stress tensor (where ndexes and 3 denote respectvely the streawse and wall-noral drectons) obtaned fro the sulatons carred out wth the proposed

5 Proceedngs of the 4th WSEAS Internatonal Conference on Flud Mechancs and Aerodynacs Elounda Greece August (pp ) odel copared wth the profles of the analogous coponent obtaned fro DNS and fro sulatons carred out wth the DMM at Re* = 395. As can be seen fro fgure the profle of the coponent <u 'u 3 '> calculated wth the proposed odel yelds a slar profle to that of the correspondng coponent of the Reynolds stress tensor obtaned by the DNS whlst the DMM provdes values whch are greatly underestated. Fgures 5 and 6 show the profles of the varous ters of the balance equaton of the generalzed SGS turbulent netc energy E (producton ter: P E ; turbulent transport ter: T E ; convecton tern: C E ; vscous dffuson ter D E ; vscous dsspaton: eps) calculated wth ths odel and averaged over te and over hoogeneous planes plotted n ters of the dstance fro the wall (expressed n wall unts z) for channel flow at Re* of 395 and 655 respectvely. Fg. 3. Te-averaged streawse veloctes. Coparson between experental easureents and LES results obtaned wth the proposed odel (TEM). Channel flow Re* = 340. Fgure 3 shows the profle of the te-averaged streawse velocty coponent for a channel flow at Re* = 340 obtaned wth the proposed odel copared wth the profle of the analogous velocty coponent easured experentally [9]. The agreeent between the two velocty profles s very good. Fgure 4 copares the profle of the coponent <u 'u 3 '> of the Reynolds stress tensor calculated wth the proposed odel wth the profle of the slar coponent of the Reynolds stress tensor obtaned fro experental easureents [9] for a channel flow at Re* = 340. Fg. 5. Generalzed SGS turbulent netc energy balance ters averaged over te and over hoogeneous planes. Producton: P E ; Turbulent transport: T E ; Convecton: C E ; Vscous dffuson: D E ; Vscous dsspaton : eps. Channel flow Re*=395. Fgure 7 shows nstantaneous profles of the ters of the balance equatons of E averaged over hoogeneous planes for channel flow at Re* = 340. Fgure 5 6 and 7 deonstrate that the balance between producton and dsspaton of the generalzes SGS turbulent netc energy s confred only n a lted regon between the buffer layer and the log layer (0<z<40) whlst t s not confred n other regons of the doan. Fg. 4. Reynolds stress <u 'u 3 '>. Coparson between experental easureents and LES results obtaned wth the proposed odel (TEM). Channel flow Re* = 340. Fgure 4 shows that at Re* = 340 the proposed odel provdes a profle of the coponent <u 'u 3 '> n agreeent wth that of the correspondng coponent of the Reynolds stress tensor obtaned fro the experental easureents. Fg. 6. Generalzed SGS turbulent netc energy balance ters averaged over te and hoogeneous planes. Producton: P E ; Turbulent transport: T E ; Convecton: C E ; Vscous dffuson: D E ; Vscous dsspaton: eps. Channel flow Re*=655. The vscous dsspaton of E s balanced n the vscous sublayer (z<5) by the vscous dffuson ter whlst the producton of E s practcally neglgble. Movng away fro the wall n the frst part of the buffer layer the producton ter of E ncreases untl reachng ts axu value (z

6 Proceedngs of the 4th WSEAS Internatonal Conference on Flud Mechancs and Aerodynacs Elounda Greece August (pp ) 0) and the ters of turbulent transport and vscous dffuson of E are coparable wth the producton ter of E. In the regon between the buffer layer and the log layer (0<z<40) the convectve and turbulent transport ters and the vscous dffuson ter are neglgble copared wth the producton and dsspaton ters. Only n ths lted regon there s a balance between the producton and the dsspaton of E. towards the center of the channel (z>30) the vscous dsspaton tends towards a nu but not neglgble value. In ths regon the producton ter of E s balanced not only by the dsspaton but also by the turbulent transport of E. Fg. 7. Instantaneous generalzed SGS turbulent netc energy balance ters averaged over hoogeneous planes. Producton: P E ; Turbulent transport: T E ; Convecton: C E ; Vscous dffuson: D E ; Vscous dsspaton: eps. Channel flow Re*= Concluson In ths paper a new LES odel s proposed. The proposed closure relaton for the generalzed SGS turbulent stress tensor: coples wth the prncple of turbulent frae ndfference; taes nto account both the ansotropy of the turbulence velocty scales and turbulence length scales; reoves any balance assupton between the producton and dsspaton of SGS turbulent netc energy. In the proposed odel the generalzed SGS turbulent stress tensor s related exclusvely to the generalzed SGS turbulent netc energy (whch s calculated by eans of ts balance equaton) and the odfed Leonard tensor. The fltered oentu equatons are solved by usng a staggered fourth order fnte dfference schee. The proposed odel s tested for a turbulent channel flow at Reynolds nubers (based on frcton velocty and channel half-wdth) rangng fro 395 to 340. References: [] M. Gerano U. Poell P. Mon W.H. Cabot A dynac subgrd scale eddy vscosty odel. Phys. Fluds A3 99 pp [] Y. Zang R.L. Street J.R. Koseff A dynac xed subgrd-scale odel and ts applcaton to turbulent recrcultng flows Physcs of Fluds A pp [3] B. Vrean B. Geurts H. Kuerten On the forulaton of the dynac xed subgrd-scale odel Physcs of Fluds pp [4] C. Meneveau T.S. Lund W.H. Cabot A Lagrangan dynac subgrd-scale odel of turbulence. J. Flud Mech pp [5] V.M. Salvett V.M S. Baneree A pror tests of a new dynac subgrd-scale odel for fnte dfference largeeddy sulatons Phys. Fluds pp [6] B. Tao J. Katz C. Meneveau Geoetry and scale relatonshps n hgh Reynolds nuber turbulence deterned fro three-densonal holographc velocetry Phys. Fluds 000 pp [7] K. Tennees J.L. Luley A frst course n turbulence MIT Press 97 [8] A.S. Monn A.M.Yaglo Statstcal Flud Mechancs: Mechancs of Turbulence MIT Press 97. [9] C.G. Spezale T.B. Gats Analyss and odellng of ansotropes n the dsspaton rate of turbulence J Flud Mech pp [0] U. Poell W.H. Cabot P. Mon S. Lee Subgrdscale bacscatter n turbulent and transtonal flows Phys. Fluds A3 99 pp [] D.K. Llly A proposed odfcaton of the Gerano subgrd scale closure ethod Phys. Fluds A4 99 pp [] F. Sarghn U. Poell E. Balaras Scale-slar odels for large-eddy sulaton Phys. Fluds 999 pp [3] K. Hutter K. Joen Contnuu Methods of Physcal Modelng Sprnger 004. [4] Y. Mornsh T.S. Lund O. Vaslyev P. Mon Fully Conservatve Hgher Order Fnte Dfference Schees for Incopressble Flow Journal of Coputatonal Physcs pp [5] O. Vaslyev Hgh Order Fnte Dfference Schees on Non-unfor Meshes wth Good Conservaton Propertes Journal of Coputatonal Physcs pp [6] M. Gerano Turbulence: the flterng approach Journal of Flud Mechancs pp [7] F. Gallerano E. Pasero G. Cannata A dynac twoequaton Sub Grd Scale odel Contnuu Mechancs and Therodynacs Vol. 7 No 005 pp [8] N.N. Mansour R.D. Moser J. K Fully developed turbulent channel flow sulaton AGARD Advsory Report 343. [9] G. Cote-Bellott Hgh Reynolds nuber channel flow experent AGARD Advsory Report 343.