L'UNIVERSITE DE LYON

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1 Année 2009 Thèse pour obtenr le ttre de DOCTORAT de L'UNIVERSITE DE LYON délvré par L'ECOLE CENTRALE DE LYON par Le FANG Applyng the Kolmogorov equaton to the problem of subgrd modelng for Large-Eddy Smulaton of turbulence Soutenance le 23 Jullet devant la Commsson d'examen Jury: M. J.-P. Bertoglo LMFA - Drecteur de thèse M. L. Shao LMFA M. P. Sagaut Unversté Pars 6 - Rapporteur M. E. Levêque ENS Lyon - Rapporteur Mme. G. Cu Unversté Tsnghua

2 Abstract The am of the current work s to nvestgate a seres of new subgrd models by employng the Kolmogorov equaton of ltered quanttes (KEF), whch s an exact relaton of turbulence n physcal space. Derent formulatons of KEF are derved, ncludng the forms n velocty eld (homogeneous sotropc turbulence, nhomogeneous ansotropc turbulence, homogeneous shear turbulence, homogeneous rotatng turbulence), n scalar turbulence and n magnetohydrodynamc turbulence. The correspondng subgrd models are then formulated, for example: ˆ The mult-scale mprovement of CZZS model. ˆ A new ansotropc eddy-vscosty model n homogeneous shear turbulence. ˆ The mproved velocty ncrement model (IVI). ˆ The rapd-slow analyss and model applcaton n nhomogeneous ansotropc scalar turbulence. ˆ The attempt n magnetohydrodynamc (MHD) turbulence. Besdes, there are also other mportant conclusons n ths thess: ˆ The ansotropc eect of mean shear n physcal space s analyzed. ˆ Analytcal correctons to the scalng of the second-order structure functon n sotropc turbulence n ntroduced. ˆ It s shown that the two-pont dstance of velocty ncrement must be much larger than the lter sze, n order to satsfy the classcal scalng law. Otherwse, the classcal scalng law can not be drectly appled n subgrd modelng. ˆ A thought-experment s descrbed to analyse the tme-reversblty problem of subgrd models. ˆ A rapd algorthm for Tophat lter operator n dscrete eld s ntroduced.

3 Acknowledgements There are many people wthout whom ths work would not have been possble. I would rst lke to thank my supervsor, Jean-Perre Bertoglo, for takng me on as hs student and allowng me to pursue my research nterests, and for hs support and frendshp durng the course of ths thess. He s, wthout queston, a great master wth genune knowledge on turbulence. I would also lke to thank my co-supervsor Lang Shao, who has qute good deas for helpng me mprove my work. Besdes I wsh to thank hm for teachng me a lot on the atttude of research. I would lke to gve a specal thanks to Wouter. J. T. Bos, who helped me very much n both the work and the language. I'm really qute grateful for hs kndness. I want to thank the two referees of my manuscrpt, professor Perre Sagaut and Dr. Emmanuel Levêque for the tme and energy they have put n readng my manuscrpt and ther nterestng and valuable comments on my work. Also I would lke to thank professor Guxang Cu, professor Zhaoshun Zhang and professor Chunxao Xu. They helped me on both the work and the lfe when I was n Tsnghua Unversty of Chna, and they are also concerned wth my thess. I am also grateful to professor Cu for acceptng to be members of the jury. Madam Benedcte Cuenot accpeted to be a member of the jury, but a serous llness prevents her from vstng Lyon. I really wsh that she wll recover soon. I would lke to thank my collaborators Chunhong Pan, professor Stephane Cotn, Dr. Hongtao Wang, Yng Wang and Yy We. The projects extend my vson and my undertandng of ud mechancs.

4 My former classmates Xaozhou Zhou and Yunke Yang helped me on some problems of my thess. We Ma gave me lots of help wth my artcles. Aurelen Hemon corrected my french language n the abstract. Dr. Jérome Boudet helped me on the nte volumn method of large-eddy smulaton. Thanks also to the frends from Lyon, wth a specal mentonng of Dabng Luo and Chuantao Yn. They are the most close frends of mne, and they made the lfe n Lyon more than pleasant. The most specal thanks to my grl frend (and wll be my wfe soon) Jeyng Hong, for her support n these three years, thank you. And of course to my parents, I dedcate ths thess to them.

5 Contents 1 Introducton Background Mathematcal methods n large-eddy smulaton Krachnan's spectral theory n LES Kolmogorov's theory of physcal space and applcatons n LES Structure of the thess Formulatons of Kolmogorov equaton of ltered velocty (KEF) Flter and ensemble average KEF n homogeneous sotropc turbulence Energy budget and error analyss n physcal space Energy budget analyss n spectral space KEF n nhomogeneous ansotropc turbulence KEF n homogeneous shear turbulence Equaton formulaton Analyss of the shear eect n spectral space KEF n a movng coordnate system Doubt on the assumpton of local sotropy Analyss of two-pont skewness KEF n homogeneous rotatng turbulence The scalng law of ltered velocty Analyss n physcal space Analyss n spectral space

6 3 Applyng KEF on eddy-vscosty models Dscussons on CZZS model One-scale models Mult-scale models n nertal subrange Mult-scale model wth separated scales A new dynamc method to determne the coecent of Smagornsky model SGS models for homogeneous ansotropc turbulence Model applcatons n rotatng turbulence Model applcatons n wall-bounded shear turbulence The results of plane Couette ow The results of plane Poseulle ow Dscusson Improved velocty ncrement model (IVI) Formulatons of the ltered velocty ncrement tensor Improved velocty ncrement model Model analyss n hgh Reynolds number turbulence Model analyss n moderate Reynolds number turbulence Numercal vercatons A Pror analyss n homogeneous sotropc turbulence A Pror analyss on sotropy n homogeneous shear ow A Pror analyss on wall behavor n wall-bounded ow A Posteror tests n homogeneous sotropc turbulence A Posteror tests n Poseulle channel turbulence Energy backscatter n homogeneous sotropc turbulence Energy backscatter n Poseulle channel turbulence Dscusson

7 5 Applyng KEF on ansotropc eddy-dusvty models Basc equatons n nhomogeneous ansotropc scalar turbulence Governng equatons of scalar varance Governng equatons of scalar ux Rapd-and-slow splt of subgrd scalar ux Magntude of the mean rapd subgrd scalar ux A Pror rapd-and-slow decomposton n Couette ow Vector level analyss of rapd-and-slow subgrd scalar ux Mean subgrd ux magntude and ts ansotropy Fluctuatng subgrd scalar ux magntude and ts ansotropy Rapd-and-slow scalar dsspaton n the equatons of scalar varance Subgrd scalar dsspaton of mean scalar varance Subgrd dsspaton of uctuatng scalar varance Rapd-and-slow scalar transport n the equatons of scalar ux Subgrd transport of mean scalar ux Subgrd transport of uctuatng scalar ux A Pror evaluaton of subgrd models Extended formulaton of Cu Model Evaluaton of subgrd scalar dsspaton Applyng KEF on subgrd modelng of magnetohydrodynamc (MHD) turbulence Governng equatons of resolved knetc energy Kolmogorov equaton of ltered quanttes MHD subgrd models n velocty and magnetc elds Dscusson n the Elsässer elds Perspectves Concluson 135 A The dervatons of the Karman-Howarth equaton, Kolmogorov equaton and Yaglom equaton of ltered velocty and scalar, n homogeneous sotropc turbulence and scalar turbulence 144

8 B Numercal method for homogeneous sotropc turbulence 149 C Numercal method for channel turbulence 154 D Correctons for the scalng of the second-order structure functon n sotropc turbulence 159 D.1 Analytcal solutons D.1.1 Correctons for the scalng n the dsspatve range D.1.2 Correctons for the scalng n the nertal range D.2 Comparson of the correctons wth exstng results D.2.1 Comparson wth the approxmaton of a constant skewness D.2.2 Comparson wth Batchelor's formula E Tme-reversblty of Naver-Stokes turbulence and ts mplcaton for subgrd-scale models 170 F A rapd algorthm for Tophat lter operator n dscrete eld 176 Bblography 182 v

9 Lst of Fgures 2.1 Exact two-pont energy transfer budget n homogeneous sotropc turbulence, wth lter sze = ξ Compensate energy spectrum n homogeneous sotropc turbulence Exact two-pont energy transfer budget n homogeneous sotropc turbulence, wth lter sze = 8h Exact two-pont energy transfer wth small lter sze budget n homogeneous sotropc turbulence, wth lter sze = ξ Normalzed two-pont energy budget of forced turbulence for ltered and full velocty eld Tme develop n numercal smulaton, n derent coordnate systems Flter operator n physcal space, n movng coordnate system Sphere coordnate system used for analyzng the ansotropy of the shear term snθcosφ aganst derent angles θ and φ, n a sphere coordnate system Velocty prole n turbulent Couette ow Structure functon D 12 n derent drectons of two-pont dstance Second-order longtudnal structure functon D ll n derent drectons of two-pont dstance Thrd-order longtudnal structure functon D lll n derent drectons of two-pont dstance < two pont 2.14 Two-pont skewness S k n derent drectons of two-pont dstance Von-Karman energy spectrum, wth E(k) = 1.0, k p = Summaton of structure functons of ltered velocty, wth derent cut-o lters of a von-karman energy spectrum v

10 3.1 A Posteror two-pont skewness values, n normal drecton of channel ow A Posteror values of Smagornsky coecent C s, n normal drecton of channel ow A Posteror statstcal results of channel ow Dynamc values of the coecent of Smagornsky model Decay of turbulent knetc energy n sotropc turbulence wth 64 3 LES Tme varaton of large-scale knetc energy for derent rotatng rates, grds: Skewness varaton versus mcro Rossby number at rotatng rate Ω = Skewness versus mcro Rossby number wth ner grds Evoluton of energy spectrum wth grds Mean velocty prole for plane Couette ow wth grds Dstrbuton of turbulent statstcs wth grds A Posteror results of mean velocty prole n channel Poseulle ow A Posteror results of turbulent knetc energy n channel Poseulle ow A Posteror results of Reynolds stress n channel Poseulle ow Comparson of energy spectrum between DNS ( ) and LES Comparson of the statstcal propertes of turbulent plane Couette ow between the new model and no model, Re = 3200, grd Comparson of the statstcal propertes of turbulent plane Poseulle ow between the new model and no model, Re = 7000, grd comparson between exact value and dynamc model value of C f, aganst derent lter szes and two-pont dstances = ξ, n homogeneous sotropc turbulence Dynamc model value of C f, aganst derent two-pont dstance, n homogeneous shear turbulence Flter sze / two-pont dstance = ξ = 2h n derent poston of normal drecton n channel ow A Pror test of the eddy vscosty by usng derent SGS models n channel ow, wth lter sze = 2h v

11 4.5 A Pror eectve ν t n channel ow, by usng IVI model Energy spectrum n decayng sotropc turbulence A Posteror statstcal results usng derent models (Table 4.1) n channel ow Trad nteracton causng energy backscatter A Pror results of energy backscatter, n homogeneous sotropc turbulence Contrbutons of components for subgrd energy dsspaton A Pror coecent values usng derent dynamc methods, n channel ow A Posteror statstcal results of channel ow, the model coecent s determned by derent methods Sketch of computatonal doman, velocty prole and scalar prole Comparson between two DNS results Scalar ux proles Normalzed mean SGS ux components n streamwse drecton Components of the rapd and slow parts of scalar ux Rms of normalzed SGS ux components n streamwse drecton Rms of the components of the rapd and slow parts of scalar ux Contrbuton of subgrd dsspaton n the transport equaton of resolved scalar varance Contrbuton of subgrd dsspaton n subgrd scalar transport equaton Contrbuton of subgrd transport n mean scalar ux equaton The contrbuton of subgrd transport n resolved scalar ux equaton, n the streamwse drecton The contrbuton of the subgrd transport n the resolved scalar ux equaton, n the normal drecton The contrbuton of the subgrd transport n the resolved scalar ux equaton, n the span drecton Comparson between exact values and model values, of slow subgrd scalar dsspaton Mean and uctuatng parts n the ECM v

12 5.16 Comparson between exact and model values of scalar dsspaton Knetc and magnetc energy n decayng turbulence Le budget d'énerge en deux ponts normalsé dans la turbulence forcée avec ltre et sans ltre B.1 Coordnate system of ntal velocty eld n spectral space C.1 Computatonal doman of channel turbulence D.1 Schematc of the velocty ncrement skewness n sotropc turbulence. 161 D.2 Comparsons between the numercal soluton and the analytcal solutons n dsspatve range D.3 Comparsons between the numercal soluton and the perturbatve solutons when r η D.4 Comparsons between Batchelor's formula and: (a) numercal solutons of constant skewness. (b) analytcal solutons of constant skewness D.5 Comparsons of the structure functons, between Batchelor's formula the numercal solutons of constant skewness D.6 Comparsons between Batchelor's formula the analytcal solutons of constant skewness, n nertal range E.1 Schematc representaton of the LES approach E.2 Energy spectrum at the tme of reversng t = E.3 Evoluton of grd-scale energy: (a) n the whole calculaton. (b) around the pont of reversng E.4 Energy spectra of derent cases at t = E.5 Evoluton of subgrd-scale energy v

13 Nomenclature b, b Magnetc nducton C f Coecent n velocty ncrement model C m Coecent n Matas-Leseur model C s Coecent n Smagornsky model D Laplace operator n a movng coordnate system D ll Second-order structure functon of non-ltered velocty D ll < Second-order structure functon of ltered velocty D lll Thrd-order structure functon of non-ltered velocty D lll < Thrd-order structure functon of ltered velocty D θθ < Second-order structure functon of ltered scalar D < Compensatng structure functon of ltered velocty E(k) Energy spectrum F (k) Forcng term n spectral space F n n th order structure functon F (k) Forcng term n physcal space G Mean scalar gradent G( x) Flter kernel h Grd sze J j < Stran rate of resolved scale magnetc eld k, p, q Wave number k c Cut-o wave number k p Characterstc wave number n von-karman energy spectrum L Integrated scale L j Leonard stress tensor n Scalng exponent of the second-order structure functon n, n Unt vector n the normal drcton of the surface p Pressure Pr Prandtl number Pr t Turbulent Prandtl number Q j Fltered velocty ncrement tensor r Two-pont dstance R j Correlaton functon of ltered velocty Re Renolds number Ro Rossby number S Surface of a local sphere S k Skewness S j < Stran rate of resolved scale turbulence t Temporal varable

14 T (k) Energy transfer spectrum T l,ll Subgrd energy transfer term n KEF U Mean velocty u, u Velocty V Volumn of a local sphere x, x Spatal varable y Coordnate n the normal drecton of channel y +, Y + Normalzed coordnate n the normal drecton of channel z, z Elsässer varable γ Mean shear Flter sze (except n chapter 5) Grd sze (only n chapter 5) Two-pont dstance used n velocty ncrement model f Flter sze δ u Velocty ncrement δ j Kronecker Delta ϵ jk Lev-Cvta symbol ε Total dsspaton ε < Resolved dsspaton ε f Subgrd dsspaton ζ, ζ Spatal varable n a movng coordnate system ζ(n) Scalng exponent of n th order structure functon η Kolmogorov scale (except n chapter 6) η Magnetc dusvty (only n chapter 6) θ Scalar (except n Sec ) κ Molecular dusvty κ t Eddy dusvty ν Molecular vscosty ν t Eddy vscosty ξ, ξ Two-pont dstance n KEF τ Temporal varable n a movng coordnate system τ j <, τ j Subgrd stress tensor τj b Subgrd stress tensor n magnetc eld of MHD τj u Subgrd stress tensor n velocty eld of MHD τ θj Subgrd scalar ux Ω Rotatng rate x

15 Chapter 1 Introducton The large-eddy-smulaton (LES) technque of turbulent ows s now recognzed as a powerful tool, and the applcatons n several engneerng elds are becomng more and more frequent. Ths technque has been developed durng the past 40 years, amng at solvng the large-scale structures by employng a subgrd-scale (SGS) model. Large and small scales are solated by a lterng operaton, whch can ether be n physcal or spectral space. The ltered velocty eld represents the moton of large eddes, and the subgrd motons are denoted by the SGS stress tensor n the ltered Naver-Stokes equatons. [1, 2] The sprt of LES s to model the subgrd moton by usng the nformaton contaned n the resolved quanttes, so as to calculate the resolved part by drect numercal smulaton (DNS) and calculate the subgrd part by usng SGS models. In ths thess, we consder each subgrd modelng has two steps: 1. Any subgrd model must be based on a certan assumpton on the subgrd moton,.e. assume a formulaton for subgrd quanttes (especally, subgrd stress). However, there are always undetermned factors n ths assumpton. 2. A complete subgrd model must employ a certan method to determne the unknown factors mentoned n the rst step. Varous assumptons for subgrd tensor were ntroduced, such as the eddy-vscosty assumpton [3], the formulaton of scale-smlarty [4], the gradent duson assumpton [5] and the formulaton of velocty ncrement [6]. The detals of these assumptons wll be shown at the begnnng of chapter 4. What should an assumpton am at? 1

16 As we know, t s mpossble to smulate the correct subgrd moton n each dscrete pont. Usually the subgrd assumptons focus on two physcal behavors: rst, a proper dsspaton, whch could represent the strong dsspatve eect at small scales, s a drect concluson of the knetc energy equaton, and s also qute mportant for numercal stablty [7]; second, some physcal mechansm,.e. the nteracton between resolved and subgrd scales, usually representng some self-smlarty behavors n nertal subrange, whch can accurately descrbe the physcal propertes n hgh Reynolds number turbulence [8, 9]. However, few subgrd assumptons can satsfy both these condtons. For nstance, the scale-smlarty assumpton [4] has very good relatvty, but t does not generate enough subgrd dsspaton. Other pure dsspatve models, such as the mplct model (MILES) [10], could not well satsfy the correct physcal mechansm. In chapter 3 of ths thess, we utlze the eddy-vscosty assumpton snce t usually dsspates well. Besdes, the velocty ncrement assumpton, whch denotes a two-pont self-smlarty between scales, s employed n chapter 4 of ths thess. From many exsted assumptons, we can nd that each assumpton could assume a relaton between subgrd and resolved motons, but the subgrd moton could not be completely xed by ths assumpton. For example, n eddy-vscosty assumpton the subgrd eddy vscosty can not be determned, and n velocty ncrement assumpton the dynamc coecent can not be determned. It ndcates that a certan method must be appled to acheve closure. Thus, a complete pure subgrd modelng should contan both the assumpton and the method. The concepts of method and model are usually confused n many researches. In ths thess, we would lke to pont out that the methods, whch could determne the coecents of an assumpton, can not be consdered as complete models. For nstance, the coecent of eddy-vscosty assumpton could be estmated by usng ether EDQNM theory [8] or Germano procedure [11], and they should only be consdered as derent methods. One method can also be appled n derent assumptons, for nstance both the eddy vscosty and the coecent of velocty ncrement assumpton can be determned by Germano procedure [6, 11]. Many methods are already avalable and wll be ntroduced as a background n the next secton, ncludng mathematcal methods and the physcal methods n spectral and physcal spaces. In Sec we revew varous mathematcal methods 2

17 employed n practcal LES modelng [1214]. Besdes, the physcal theory n spectral space, ntroduced by Krachnan [15, 16] and employed n many spectral SGS models, wll be ntroduced n Sec In addton, we revew the Kolmogorov's theory n physcal space. The orgnal K41 theory s avalable for homogeneous sotropc turbulence for non-ltered eld [9]. Furthermore, Meneveau's work showed that t s possble to apply the Kolmogorov's theory n SGS modelng, snce the Kolmogorov equaton descrbes a physcal law that should be satsed n LES [17]. However, few works are proposed to derve subgrd models n physcal space from Kolmogorov's theory. In secton we wll ntroduce the exsted works. 1.1 Background Mathematcal methods n large-eddy smulaton The Gallean nvarance for the spatal lterng approach was employed to determne the unknown coecent of scale-smlarty model [4]. Although the result s corrected n later works [18, 19], ths method could be consdered as an mportant attempt of mathematcal methods n LES. One of the most famous mathematcal tools n subgrd modelng s the Germano procedure based on Germano dentty [12]. It s based on the decomposton of lter operatons. When appled at two lters, t can be named as the multplcatve Germano dentty [20]. It can also be extended to the case of N lter levels,.e. the multlevel Germano dentty [21]. Germano dentty represents the characterstcs of lters and provdes a dynamc method to determne the undetermned factors. There are also researchers who ntroduced other mathematcal methods to provde subgrd models. For example, a seres of regularzed methods are ntroduced by Holm et al. [14, 22]. The regularzaton s acheved by mposng an energy penalty whch damps the scales smaller than a threshold scale α, whle stll allowng for non lnear sweepng of the small scales by the largest ones. The regularzaton appears as a nonlnearly dspersve modcaton of the convecton term n the Naver-Stokes equatons. Generally speakng, these mathematcal methods try to provde mathematcal approach n subgrd modelng. The lack of physcal background s an obvous dsadvantage, whch may lead to numercal unstablty. For nstance as dscussed n Leseur [8], the Germano procedure can yeld strongly varyng coecent value and 3

18 the correspondng model may be unstable. Many researchers beleve that a wellperformng subgrd model should be derved from physcal propertes, so as to ensure the turbulent behavor. In these studes, the most wdely appled method s Krachnan's theory n spectral space Krachnan's spectral theory n LES The sprt Krachnan's orgnal theory s the trad nteracton between three wave vectors, whch causes the energy transfer between scales. When employ ths theory n LES, derent types of trad nteractons n subgrd transfer should be consdered: local trads and non-local trads. Local trads correspond to nteractons among wave vectors of neghborng modules, and therefore to nteractons among scales of slghtly derent szes. In contrast, non-local trads are all those other nteractons,.e. nteractons among scales of wdely derng szes. Another mportant result of ths Krachnan's theory s the energy backscatter, whch corresponds to non-local trad of the R type accordng to Walee's classcaton [23]. It denotes a backward knetc energy cascade, whch s not mentoned from Kolmogorov 1941 (K41) theory. Walee renes the analyss of ths phenomenon: a very large part of the energy s transferred locally from the ntermedate wave number located just ahead of the cuto toward the larger wave number just after t, and the remanng fracton of energy s transferred to the smaller wave number. The energy backscatter has also been vered n numercal cases [2426]. In the research of largeeddy smulaton, t s also mportant to evaluate the backscatter propertes of subgrd models. Krachnan's theory was further developed by Orszag [27] as an analytcal closure, named as Eddy-Damped Quas Normal Markovan (EDQNM). In LES, the EDQNM theory can also help us n subgrd modelng. In spectral space, Chollet and Leseur proposed an eectve vscosty model usng the results of EDQNM,.e. the spectral eddy-vscosty model [28]. The asymptotc value of the eectve vscosty s extended to the case of spectra of slope m by Metas and Leseur [29]. Bertoglo proposed a spectral stochastc subgrd model based on the EDQNM analyss, whch could represent the backward energy transfer [30]. The spectral result could also be employed 4

19 nto physcal space, for nstance the structure functon model proposed by Metas and Leseur [29]. In general, wth a certan assumpton, for example by assumng eddy-vscosty, we could employ spectral analyss to determne the unknown subgrd statstcal quanttes dynamcally. Especally wth the help of EDQNM theory n spectral space, great success has been obtaned n subgrd modelng. However, the formulatons of these models are n spectral space, and there are few researches n physcal space. Therefore, n the next secton, we ntroduce one of the most mportant turbulent theores n physcal space and ts applcatons n large-eddy smulaton Kolmogorov's theory of physcal space and applcatons n LES Kolmogorov's theory (K41) [9] was ntroduced much earler than Krachnan's spectral theory. The sprt of K41 theory s the four-fth law, whch represents the energy transfer property between two ponts. Ths four-fth law could be wrtten as a formulaton of Kolmogorov equaton. There are already lots of work on extendng ths theory. For nstance, t was extended nto passve scalar by Yaglom [31]. In addton, n K41 theory, the two-pont statstcal quanttes,.e. the structure functons, requre the recognton of scalng law, whch also attracted lots of researchers [3237]. However, most these works focus on non-ltered elds, and few works are done for the ltered elds n LES. The mportance of Kolmogorov equaton n LES s emphaszed by Meneveau [17]. He derved the formulaton of Kolmogorov equaton of ltered velocty, and regarded t as a necessary condton n LES. It can also be consdered as a method for evaluatng two-pont subgrd model behavors. The rst attempt n subgrd modelng by employng Kolmogorov equaton s the CZZS model by Cu et al. [38]. Wth eddy-vscosty assumpton, the eddy vscosty could be determned dynamcally. Furthermore, the smplcaton of CZZS model yelds the skewness-based models [39]. have not been well developed. However, these methods are stll new and In ths thess, we contnue ths method and have publshed more results [4043]. The man structure of the thess s descrbed n the next secton. 5

20 1.2 Structure of the thess As ntroduced n the last secton, the SGS modelng methods n physcal space has not been well developed yet. Therefore, n ths thess, we prmarly am at summarzng and nvestgatng new modelng methods n physcal space, by employng the Kolmogorov equaton for the ltered quanttes (KEF). In chapter 2, varous formulatons of KEF for derent cases of ows are derved. The formulaton n homogeneous sotropc turbulence, ntroduced by Meneveau [17] and Cu et al. [38], s analyzed manly on the energy budget wth derent lter szes and two-pont dstances. The formulaton n nhomogeneous ansotropc turbulence contans rapd and slow terms, whch could be consdered as a complete form of KEF, but too complex to be appled n subgrd modelng. Two more smple cases are the homogeneous shear ow and the rotatng turbulence, the former n whch s analyzed partcularly. Comparng wth the classcal analyss for the correlaton functons n spectral space, we nvestgate the shear eect n physcal space. The ansotropc propertes of structure functon between derent drectons are emphaszed. Besdes, n order to employ the classcal scalng law, we study the behavor of ltered structure functons, n physcal and spectral spaces, respectvely. In chapter 3, we employ KEF on eddy-vscosty models. Wth homogeneous sotropc assumpton, the CZZS model s further dscussed. Varous subgrd models are proposed and vered n numercal cases. Smlar as Germano procedure, another dynamc method s ntroduced to determne the coecent of Smagornsky model, whch s much less expensve n terms of computatonal cost and has clear physcal background. In homogeneous ansotropc turbulence, for nstance the rotatng turbulence and wall-bounded shear turbulence, a new ansotropc model s derved. A Posteror tests are made to evaluate ths model. The most mportant advantage s that the mean velocty s explctly contaned n the formulaton of subgrd eddy vscosty. In chapter 4, another assumpton of subgrd stress s ntroduced,.e. the velocty ncrement assumpton. Comparng wth eddy-vscosty assumpton, ths two-pont assumpton s easer to be combned wth KEF, whch s also a two-pont equaton. The mproved ncrement model (IVI) s then derved, by employng KEF to determne the model coecent dynamcally. In real ow, when scales are not well separated, 6

21 the model coecent has a dynamc form; n deal hgh Reynolds number turbulence, we could obtan a constant coecent. Ths model s extremely smple and low cost. The IVI model s vered n A Pror and A Posteror tests. In chapter 5, KEF s employed n ansotropc eddy-dusvty models of passve scalar. The transfer processes of scalar energy and scalar ux are manly nvestgated. We splt the subgrd scalar ux nto rapd and slow parts, and do A Pror tests on them n a channel Couette ow. The new ansotropc eddy-dusvty model based on KEF s then vered for ts propertes of reproducng subgrd scalar dsspaton. As a sde result, we also analyze the ablty of the scale-smlarty model to properly reproduce the rapd part of the subgrd term, extendng the approach of Shao et al. to the case of a scalar eld. In chapter 6, dynamc subgrd models n magnetohydrodynamc turbulence are derved by employng KEF. The magnetc eld s coupled wth velocty eld, but the smlar Kolmogorov equaton could be obtaned. Thus the subgrd eddy vscosty and magnetc dusvty could be determned by varous methods. The models are vered n A Posteror tests of homogeneous decayng turbulence. In chapter 7, we gve the concluson. 7

22 Chapter 2 Formulatons of Kolmogorov equaton of ltered velocty (KEF) Ths chapter prmarly focus on the basc concepts and equatons of the Kolmogorov equaton of ltered velocty (KEF). In Sec. 2.1, the concepts of lter and ensemble average are ntroduced. From Sec. 2.2 to 2.5 we derve the derent formulatons of KEF: the formulaton n homogeneous sotropc turbulence s shown n Sec. 2.2 and the formulaton n general nhomogeneous ansotropc turbulence s shown n Sec. 2.3; however the general ansotropc formula s too complcated, and two smpled cases are dscussed n Sec. 2.4 and 2.5 respectvely. Besdes, the scalng law of ltered velocty s dscussed, whch could be employed to smplfy the sotropc formulaton n the followng chapters. 2.1 Flter and ensemble average In ths thess, the operators of lter and ensemble average should be clared. They could be appled to any physcal varable ϕ. A lter operator dvdes the varable nto two parts: ϕ = ϕ < + ϕ > (2.1) n whch ϕ < s the grd-scale (GS) part (or resolved part) and ϕ > s the subgrd-scale (SGS) part. In ths thess we only consder the spatal lters, and do not consder tme-dependent temporal lters. A lter n physcal space can be represented by ntroducng a lter kernel G( x x )d x = 1, (2.2) 8

23 and the GS part (ϕ < ) could be denoted as ϕ < ( x) = G( x x )ϕ( x )d x. (2.3) In general, the lter operator has the followng propertes: (ϕ + ψ) < =ϕ < + ψ <, ( ) < ϕ = ϕ< t t. (2.4) Besdes, ϕ << ϕ <, (ϕ < ) > 0 except when lter s performed wth a spectral cuto lter, whch s an exact low-pass lter n spectral space. By contrares, the ensemble average operator s dened n full-developed turbulence. Every physcal varable can be dvded nto the mean part and the uctuatng part: ϕ = ϕ + ϕ (2.5) n whch the symbol s the arthmetc mean from experments. Comparng wth the lter operator, the ensemble average has derent propertes: ϕ + ψ = ϕ + ψ ϕ = ϕ s s, s = x, t ϕ = ϕ (2.6) ϕ =0 The symbol represents a lnear operaton, thus the followng commutatons can be obtaned: ϕ ϕ < = ϕ <, = ϕ. (2.7) x x However, lterng and derentaton do not commute when the lter wdth s nonunform n space [44]. A general class of commutatve lters was ntroduced by Vaslyev [45] and Marsden [46] to decrease the commutaton error n LES equatons, especally for nhomogeneous lter wdth. Leonard et al. [47] studed the commutaton error wth tme-dependent lter wdth. In ths thess, we assume that the meshes are homogeneous, so that the lterng and derentaton could commute: ( ) < ϕ = ϕ< (2.8) x x 9

24 2.2 KEF n homogeneous sotropc turbulence The formulaton of KEF n homogeneous sotropc turbulence was derved by Meneveau [17] and Cu [38]. The process of devaton n detals s shown n appendx A. From the ltered Naver-Stokes (N-S) equaton and wth only the assumptons of sotropy and homogenety, we can obtan the orgnal Karman-Howarth equaton of ltered velocty: (u < 1 ) 2 t 1 2 D < ll t = 1 6ξ 4 ξ (ξ4 D lll < ) ν ( ) ξ 4 D< ll 1 ξ 4 ξ ξ ξ 4 ξ (ξ4 T l,ll ), (2.9) n whch the subscrpt l denotes a component n the drecton of the two-pont dstance ξ. Notce that n homogeneous sotropc turbulence, u = u, thus the prmes are gnored. Dene the velocty ncrement δ u( x, ξ) = u( x + ξ) u( x). In equaton (2.9), D < ll (ξ) = δu< 1 (ξ)δu < 1 (ξ) s the second order longtudnal structure functon, D < lll (ξ) = δu< 1 (ξ)δu < 1 (ξ)δu < 1 (ξ) s the thrd order longtudnal structure functon, and T l,ll (ξ) = u < 1 (x 1 )τ < 11(x 1 + ξ) s the subgrd energy transfer. In order to smplfy the rst term n equaton (2.9), we wrte the ltered N-S equaton: u < t + u < j u < n whch the subgrd stress s dened n a mplct form: = 1 p < + ν 2 u < τ j <, (2.10) ρ x τ < j = (u u j ) < u < u < j. (2.11) The decay of turbulent knetc energy n resolved scale turbulence could be represented as (u < )2 t where S < j + u < j (u < )2 = 2ν u< u < + 2τ j < S j < + (ν (u< )2 2τ j < u < the stran rate of resolved scale turbulence S < j = 1 2 ( u < 2 ρ p< u < j ), (2.12) ) + u< j. (2.13) x After ensemble averagng n homogeneous sotropc turbulence, ths becomes (u < 1 ) 2 t = 2 3 ε< τ < j S < j, (2.14) 10

25 s the resolved dsspaton. τ j < S< j s the subgrd dss- u < where ε < = ν u < paton, denoted as ε f. Substtutng equaton (2.14) nto equaton (2.9) and ntegratng t over ξ, and neglectng the tme-dependent term wth the assumpton of steadness of the small scale turbulence n the classc Kolmogorov equaton, we obtan the formulaton of KEF n homogeneous sotropc turbulence: 4 5 ε fξ 4 5 ε< ξ = D < lll 6T l,ll 6ν D< ll ξ. (2.15) When Reynolds number s hgh, the molecular vscosty could be neglected, therefore the KEF n hgh Reynolds number turbulence reads: 4 5 ε fξ = D < lll 6T l,ll. (2.16) In order to represent ths equaton by usng only 11 component, we propose Meneveau's method [17]. Neglectng the molecular terms n equaton (2.9) leads to (u 1 ) <2 t = 1 6ξ 4 ξ (ξ4 D < lll ) 1 ξ 4 ξ (ξ4 T l,ll ). (2.17) In ths equaton, the followng relatons are satsed: D lll ξ=0 =0, D lll ξ ξ 0 ξ 2 = 0, D lll ξ ξ=0 ξ 2 = 0, T l,ll ξ=0 =0, (2.18) T l,ll ξ ξ 0 = T lll(ξ) T lll (0) ξ 0 = S < ξ 11τ 11, T l,ll ξ ξ=0 = S < 11(x 1 )τ 11 (x 1 + ξ) ξ=0 = S < 11τ < 11. From equaton (2.17) and (2.18), we could obtan (u 1 ) <2 t = 5 S < 11τ < 11. (2.19) 11

26 Therefore, the formulaton of KEF could also be wrtten as 6 S 11τ < 11 < ξ = D lll < 6T l,ll. (2.20) In homogeneous sotropc turbulence, f we select the two-pont dstance ξ n x 1 drecton, the equaton s smpled to denote the behavor n one drecton, that s 6 S 11τ < 11 < ξ = D 111 < 6T 1,11. (2.21) Note that T l,ll s a correlaton term between u 1 (x) and τ 11 (x + ξ), t tends to 0 when ξ s large. From scalng law, we can also obtan T l,ll = 0 n nertal subrange [17]. As wll be dscussed n Sec. 2.6, when ξ, ths term could vansh and the KEF formulaton s 4 5 ε fξ = D lll <. (2.22) Ths qute smple formulaton s appled n the deal modelng analyss n ths thess Energy budget and error analyss n physcal space In order to show the two-pont energy budget n physcal space, two DNS cases of homogeneous sotropc turbulence wth spectral method are used for A Pror test. The computaton doman has grd ponts. The grd sze s denoted as h. The two derent Reynolds numbers Re λ are 50 and 70. The descrpton n detal wll be shown n secton and n the appendx. We apply the Kolmogorov equaton at the lter sze,.e. ξ = = π/k c, to nd where the equaton s well satsed. The energy budget wth derent lter szes s shown n Fg. 2.1(a). The error of energy transfer s dened as T error = D lll < 6T l,ll ε fξ, (2.23) whch s shown n Fg. 2.1(b), normalzed by the subgrd dsspaton term. The mnmum error occurs n the regon where the lter /η s nether too large nor too small, wth η beng the Kolmogorov scale. Ths regon could be consdered as the nertal subrange. It s about 10 < /η < 30. However, because the Reynolds numbers are not large enough, ths regon s not obvous to dstngush, and the energy transfer equaton s not qute well satsed. The compensated energy spectra of DNS cases are shown n Fg. 2.2, where the plateaus represent the nertal subrange 12

27 250 4/5ε f ξ Energy budget D lll < T error 5/4 T error /ε f ξ T l,ll /η, ξ/η /η, ξ/η (a) (b) Fgure 2.1: Exact two-pont energy transfer budget n homogeneous sotropc turbulence. Flter sze = ξ. Sold lne: Re λ = 50; dashed lne: Re λ = 70. (a) Vercaton of each term n Kolmogorov equaton of ltered velocty. (b) Error of subgrd energy transfer, normalzed by the subgrd dsspaton term. n spectral space. Although not obvous, the correspondng wave number s about 0.1 < k c η < 0.3. Accordng to the two cases, the regon s wder when Reynolds number s hgher. In addton, the ntegrated scale L 70η n the two cases, thus we could reasonably consder that = ξ L n the regon of mnmum error. Anyway, the nertal subrange exsts when η = ξ L. If the lter s n the dsspatve range, the resolved dsspaton can not be neglected; f the lter sze s larger than 30η, the error s stll acceptable from Fg. 2.3(b), however, the lter sze s not small enough compared wth the ntegrated scale, and we can not easly consder t the nertal subrange. Notce that ξ and are ndependent. Thus we can x a lter sze, and search for a sutable two-pont dstance ξ to mnmze the error. The lter sze s gven as = 8h, whch s about 15.2η when Re λ = 50, and 26.3η when Re λ = 70. Both lter szes are much greater than Kolmogorov scale and do not need a correcton. The budget of the two-pont energy transfer wth derent two-pont dstances s shown n Fg. 2.3(a). The mnmum error occurs n the regon where the two-pont dstance ξ. Thus wth a sutable lter sze 10 < /η < 30, t seems reasonable to apply ξ =. Ths mnmzed error strongly supports the correspondng subgrd modelng. 13

28 Fgure 2.2: Compensate energy spectrum n homogeneous sotropc turbulence. Sold lne: Re λ = 50; dashed lne: Re λ = /5ε f ξ T error D lll < -6T l,ll ξ/ 5/4 T error /ε f ξ Re λ =50 =15.2η Re λ =70 =26.3η ξ/ (a) (b) Fgure 2.3: Exact two-pont energy transfer budget n homogeneous sotropc turbulence. Flter sze = 8h. Sold lne: Re λ = 50, = 15.2η; dashed lne: Re λ = 70, = 26.3η. (a) Vercaton of each term n Kolmogorov equaton of ltered velocty. (b) Error of subgrd energy transfer, normalzed by the subgrd dsspaton term. 14

29 4/5ε < ξ Energy budget 0 T error T c error 5/4 T c error /ε < ξ ν dd ll /dξ /η, ξ/η /η, ξ/η (a) (b) Fgure 2.4: Exact two-pont energy transfer wth small lter sze budget n homogeneous sotropc turbulence. Flter sze = ξ. Sold lne: Re λ = 50; dashed lne: Re λ = 70. (a) Vercaton of the correcton terms n Eq. (2.15). (b) Corrected error of two-pont energy transfer, normalzed by the resolved dsspaton term. In Fg. 2.1, we nd that the great error n the regon /η < 15 can be corrected by consderng the vscous eects of the resolved part. From equaton (2.15), the corrected error value can then be calculated as T error c = D < lll 6T l,ll ε fξ 6ν D< ll ξ ε< ξ. (2.24) The last two terms and the corrected error value are shown n Fg. 2.4, n the regon /η < 15. The error s obvously reduced, comparng wth the orgnal T error term n Fg. 2.4(a). And n Fg. 2.4(b), the nondmensonal value s approxmately zero. It means that the Kolmogorov equaton of ltered velocty for LES (2.15) can only t for a lter sze much greater than Kolmogorov scale. Otherwse, vscous correctons are necessary. In fact the analyss n ths secton s very rough, whle there are lots of errors: low Reynolds number, forcng, statstcal samples... However the analyss also shows that there s a regon where the KEF s satsed, thus we can employ KEF n subgrd modelng n ths regon. Also n ths regon, even when we neglect the molecular vscosty at low Reynolds number, the two-pont dstance ξ = can be acceptable n subgrd modelng. 15

30 2.2.2 Energy budget analyss n spectral space In order to analyze the Kolmogorov equaton n spectral space, n forced turbulence we can wrte the energy transfer equaton as de(k) dt = 2νk 2 E(k) + T (k) + F (k) = 0, (2.25) n whch E(k) s the energy spectrum, T (k) s the energy transfer spectrum, and we force the system by keepng the energy constant for k < k f,.e. the forcng term s F (k) = { 2νk 2 E(k) T (k), k < k f, 0, k k f. (2.26) Also we can wrte the equaton for the second-order structure functon n forced turbulence: D < lll (ξ) 6T l,ll(ξ) 6ν dd< ll (ξ) dr + F < (ξ) = 0, (2.27) where F (ξ) s the correspondng forcng term, whch can be represented as the dsspatve terms and tme-dependent terms n decayng turbulence. In order to calculate the terms n Eq. 2.27, we have the followng relatons: D lll < (ξ) =12ξ T < (k)g(kξ)dk, 0 dd < ll (ξ) dξ =4 F < (ξ) =12ξ T l,ll (ξ) = ke < (k)f(kξ)dk, F < (k)g(kξ)dk, ( D lll < (ξ) 6ν dd< ll (ξ) ) + F < (ξ), dξ (2.28) n whch f(kξ) = sn(kξ) (kξ) 2 3 cos(kξ) (kξ) sn(kξ) (kξ) 4, g(kξ) = 3 sn(kξ) 3kr cos(kξ) (kξ)2 sn(kξ) (kξ) 5. (2.29) The spectra of E(k) and T (k) are generated usng EDQNM calculaton by W.J.T. Bos. The correspondng terms n very hgh Reynolds number turbulence are shown 16

31 1 normalzed by 4/5εξ F D lll < T l,ll -6ν dd ll < / dξ D lll ξ/ Fgure 2.5: Normalzed two-pont energy budget of forced turbulence for ltered ( < ) and full velocty eld (wthout < ). EDQNM calculaton. The vertcal lne ndcates the locaton of lter sze. n Fg. (2.5). The symbol r s the same as the two-pont dstance ξ. The vertcal lne ndcates the locaton of the lter sze. The forcng term s almost equal to 4/5εξ, or say 4/5ε < ξ because of hgh Reynolds number. The T l,ll term s the same order as D < lll when ξ =, and tends to zero when ξ s large. The vscosty term only takes eect at very small dstance, and can be neglected when ξ s the same magntude as. Another fact s shown that there s only qute short range n whch the scalng law D lll < (ξ) ξ s satsed, snce no obvous plateau of D< lll (ξ)/ξ s shown n ths gure. Ths fact wll also be dscussed n Sec. 2.6 for more detals. 2.3 KEF n nhomogeneous ansotropc turbulence In nhomogeneous ansotropc turbulence, the mean velocty s not zero and should should be focused on. The N-S equaton of velocty uctuaton can be wrtten as u t + u u j + u u j = 1 p + ν 2 u ρ x (u u j u u j ). (2.30) When the lter s appled, the resolved scale equaton of the velocty uctuaton s obtaned: u < t + (u u j ) < + ( u u j) < = 1 p < + ν 2 u < ρ x 17 ( (u u j) < u u j <). (2.31)

32 We could also wrte equaton (2.31) n another pont x, where x = x + ξ: u < t + (u u j ) < x j + ( u u j ) < x j = 1 ρ p < x +ν 2u < ( (u x j x j x u j ) < u u j <). j (2.32) Subtractng equaton (2.32) from (2.31), and denng δu = u u, we obtan: δu < + (u u j ) < t (u u j ) < + ( u u j) < x j ( u u x j j ) < = 1 p < + 1 p < ρ x ρ x + ν 2 u < ν 2 u < x j x j (2.33) ( (u u j) < u u j <) + x j ( (u u j ) < u u j <). Because ξ = x x, the followng relatons are satsed: u x = 0, u = 0, =, x x ξ x = 2, = ξ x x 2 x x = 2 ξ ξ. (2.34) Note that the average operator can not reduce the varable x, and each term of equaton (2.33) s always a functon of two varables: x and ξ. Multplyng equaton (2.33) by δu <, and employng the ensemble average, the rst term becomes δu < δu < 1 (δu < δu < ) = = 1 t 2 t 2 n whch D < ( x) = δu < δu <. D < ( x), (2.35) t The terms nvolvng pressure on the rght hand sde could be smpled as ( δu < 1 p < + 1 ) p < = 1 δu < δp < 1 δu < δp < ρ x ρ x ρ x ρ x = 1 ρ = δ and the vscosty terms read: δu < (ν 2 u < ν 2 u < x j x j ) δu < δp < ( x) + 1 ξ ρ ( 1 ρ δu < δp < ξ =ν D< ( x) ξ j ξ j ) ( x), 2ε < ( x) δu < δp < ( x ) ξ ν ( ) D < 2 δ ( x) + δε < ( x), ξ j ξ j (2.36) (2.37) 18

33 u < n whch ε < ( x) = ν u <. Followng Shao [48], we dene the slow subgrd stress τ slow< j = u < u < j (u u j) <, (2.38) and obtan a two-pont equaton: ( 1 D < (u 2 t + δu < u j ) < (u u j ) < + ( u u j) < ( u u x j x j + = δ + ( (u < δu < u < j ) (u < δu < u < j ) ( 1 ρ δu < δu < δp < ξ ( τ slow< j (u < u < x j j ) (u < u < x j j ) ) j ) < ) ) ( x) + ν D< 2ε < ν ( ) D < ξ j ξ j 2 δ ( x) + δε < ( x) ξ j ξ j ) slow< τj δu < x j Applyng the property called Reynolds rule the terms n equaton (2.39) read (u < δu < u < j ) (u < u < δu < τ slow< j x j τ slow< j slow< τj. x j (2.39) ϕ ψ = ϕ ψ, (2.40) j ) (u < = δu < u < j ) slow< τj τ slow< = δu < j x j The convecton term on the left hand sde s wrtten as ( (u < δu < u < j ) (u < u < ) j ) = 1 2 x j = (u < j δu < δu < ) D j < ( x) 1 ξ j 2 δ (u < u < x j j ) slow< τj = 0. x j = 0 + (u < j δu < δu < ( D < j ξ j x j ) ( x), ) (2.41) (2.42)

34 n whch D j < ( x) = δu < j δu < δu < s the thrd order structure functon. Next, consder the four terms n equaton (2.39). Dene the rapd subgrd stress τ rapd< j = u < u < j ( ) u u < j + uj < u < ( u j u ) <, (2.43) equaton (2.39) reads 1 D < 2 t +1 D j < 1 ( D < 2 ξ j 2 δ j ξ j ( (u + δu < < u j < ) = δ + ( 1 ρ δu < δu < δp < ξ ( τ slow< j ) ( x) (u < u j < ) x j + ( u < u < j ) j ) ( u < u < x j ) ) ( x) + ν D< 2ε < ν ( ) D < ξ j ξ j 2 δ ( x) + δε < ( x) ξ j ξ j ) slow< τj + δu < x j Employng the Reynolds rule (2.40), we obtan ( (u δu < < u j < ) (u < u j < ) ) = 1 (δu 2 u j < < δu < ) x j + 12 u j < (δu < δu < x j ( τ rapd< j ) ) rapd< τj. x j (2.44) Because = 1 2 u j < D< ξ j ( x) 1 2 u j < D< ξ j ( x ) = 1 2 δ u j < D< ( x) 1 ( ) D ξ j 2 u j < < δ ( x) ξ j = 1 2 δ u j < D< ( x) 1 ( ) D ξ j 2 u j < < δ ( x) + 1 ( ) D ξ j 2 δ u j < < δ ( x). ξ j δu < and smlarly ( u < u < j ) δu < = u < j δu < ( u < u < x j j ) δ u < ξ j (2.45) = u < j δu < δ u < ( x), (2.46) ξ j = u j < δu < δ u < ( x ), (2.47) ξ j 20

35 thus from equaton (2.46) - (2.47): u δu < < u < j u δu < < u < j x j = δu < δu < δ u < j + u < j δu < ξ j = δu < δu < δ u < j + u < j δu < ξ j = δ u < D ξ j( x) < + u < j δu < j Equaton (2.44) becomes 1 D < 2 t +1 D j < 1 ( D < 2 ξ j 2 δ j ξ j δ u j < D< ξ j + δ u < D j < + u < j δu < ξ j = δ + ( 1 ρ δu < δ ( δ u < ξ j δ ( δ u < δ ( δ u < ) ( x) ξ j ξ j ) ( x) ) ( x) δ ) ( x) δu < δu < j ( δ u < ξ j δ ( δ u < ξ j ) ( x)d < j( x). 1 ( ) D 2 u j < < δ ( x) + 1 ( ) D ξ j 2 δ u j < < δ ( x) ξ j δu < δp < ξ ( τ slow< j δ ( δ u < ξ j ) ( x) δ ( δ u < ξ j ) ( x)d < j( x) ) ( x) (2.48) ) ( x) + ν D< 2ε < ν ( ) D < ξ j ξ j 2 δ ( x) + δε < ( x) ξ j ξ j ) slow< τj + δu < x j ( τ rapd< j ) rapd< τj. x j (2.49) In fact, we could dene the terms whch are caused by the nequalty φ = φ : H rapd ( x) = 1 ( ) D 2 u j < < δ ( x) 1 ( ) D ξ j 2 δ u j < < δ ( x) ξ j ( δ u < ) +δ ( x)d ξ j( x) < j H slow ( x) = 1 ( D < ) 2 δ j ( x) u < j δu < ξ j δ ( 1 ρ δu < δp < ξ δ ( δ u < ξ j ) ( x) ) ( x) ν ( ) D < 2 δ ( x) + δε < ( x) ξ j ξ j 21 (2.50)

36 Equaton (2.49) could be wrtten as 1 D < 2 t +1 D j < 2 ξ j δ u j < D< + δ u < ξ j ξ j D < j =H rapd ( x) + H slow ( x) + ν D< ξ j ξ j 2ε < + δu < ( τ slow< j ) slow< τj + δu < x j ( τ rapd< j ) rapd< τj x j (2.51) If we consder the local homogeneous assumpton, there wll be φ = φ, thus H rapd ( x) = H slow ( x) = 0, the prevous equaton wll become much more smple. It wll be dscussed n the next secton, and wll be appled to buld new subgrd models. 2.4 KEF n homogeneous shear turbulence Wth constant shear, the formulaton of KEF can be smpled. In ths secton, we rst wrte the formulaton n a general xed coordnate system n Sec , and ntroduce the classcal analyss n spectral space n Sec ; then n Sec we rewrte the formula of KEF n a movng coordnate system, n order to explan the eect of the coordnate-depended term; n Sec the assumpton of local sotropy s studed, and proved to be not satsed n homogeneous shear turbulence; nally n Sec we ntroduce the concept of skewness and nd t approxmately satses local sotropy n shear turbulence Equaton formulaton In homogeneous shear turbulence, we assume u = γx 2 δ 1 terms n equaton (2.51) could be wrtten as = u <, and two 1 2 δ u j < D< = γξ 2 D <, (2.52) ξ j 2 ξ 1 δ u < ξ j D < j = γ 2 D< 12. (2.53) 22

37 (a) (b) Fgure 2.6: Tme develop n numercal smulaton, n derent coordnate systems. (a) Fxed coordnate system. (b) Movng coordnate system. As dscussed n the last secton, n homogeneous turbulence H rapd ( x) = H slow ( x) = 0. Therefore, equaton (2.51) can be smpled: 1 D < 2 t +1 D j < + γξ 2 D < + γ 2 ξ j 2 ξ 1 2 D< 12 ( =ν D< τ < 2ε < + δu < j ξ j ξ j τ j < ), x j (2.54) n whch τ < j = τ rapd< j + τ slow< j corresponds to the total SGS stress n homogeneous shear turbulence. The detaled process can be found n the paper of Cu et al. [40] Analyss of the shear eect n spectral space There are already lots of researches on the two-pont correlaton equaton n spectral space. Snce the correlaton functon could be translated from Fourer transform easly, we follow Hnze's method and nally wrte the governng equaton for two-pont correlaton functons: R < j t +γ ( ) δ 1 R 2j < + δ j1 R 2 < + ξ 2 R j < ξ 1 =transfer + pressure + dsspaton + subgrd, 23 (2.55)

38 n whch R < j ( ξ) = u < ( x)u< j ( x + ξ). In order to clarfy the eect by mean ow, we manly note the terms related wth γ. Contracton of equaton (2.55) vanshes the pressure terms and yelds R < t +γ ( ) 2R 12 < + ξ 2 R < ξ 1 =transfer + dsspaton + subgrd, (2.56) n whch the two shear terms 2γR 12 < and γξ 2 R < are smlar as the terms n equaton ξ The two-pont correlaton functon could be expressed n spectral space: R j( < ξ) = S < je ι ξ k d k, (2.57) where S < j read s the ltered correlaton tensor n spectral space. Thus the shear terms 2γR 12( < ξ) =2γ S < 12e ι ξ k d k, γξ 2 R < = ξ 1 k 1 S < k 2 e ι ξ k d k. (2.58) The term 2γS < 12 s regarded as a producton term. The term k 1 S < k 2 s a energy transfer term whch does not generate or dsspate, snce ntegrated over all wavenumbers t yelds a zero ntegral contrbuton: snce t s equal to k 1 S < k 2 = 0, (2.59) lm ξ R < 2 = 0. (2.60) ξ 2 0 ξ 1 Batchelor [49] suggested dong the same n a ansotropc but homogeneous turbulence by averagng the correlaton and spectrum functons over all drectons of ξ and k n the correspondng spaces, thus the mean eect on a sphere face could be evaluated. However, there s also a turnng eect caused by the mean shear [50]. Any ow of an ncompressble ud, whose velocty dstrbuton s a lnear functon of the space coordnates, can be decomposed nto a pure rotaton and a pure deformaton. For nstance the smple case u = γx 2 δ 1, the ow conssts of a bodly rotaton 24

39 wth angular speed 1 2 γ, and a pure deformaton wth a maxmum stran rate +1 2 γ along the prncpal axs n the drecton π/4, and a mnmum stran rate 1 2 γ (.e. a compresson) n the drecton 3π/4 [51]. Lumley [52, 53] calculates the energy budget of shear ow n spectral space. The shear terms are found to be obvous n small wave numbers,.e. n large scales. Thus ths large-scale regon s consdered to be ansotropc. Also, drect nteracton between the smaller eddes n the equlbrum range and the mean moton, resultng n drect energy transfer from the mean moton to these edded, s neglgbly small. space. These works were done many years ago, whch explaned the shear eect n spectral In the followng parts, we shall analyze the shear eect n physcal space, manly on the evoluton and ansotropy of structure functons KEF n a movng coordnate system We could also study the two-pont behavor n a movng coordnate system. Followng Rogallo [54] and Gualter [55], a coordnate transformaton s ntroduced that ζ 1 = x 1 γx 2 t, ζ 2 = x 2, ζ 3 = x 3, τ = t, (2.61) hence the ( ζ, τ) coordnate system s movng wth the mean ow. The derent coordnate systems are shown n Fg. (2.6). Compared wth the orgn coordnate system ( x, t), the relatons of dervatves are satsed: = γτδ j2, ζ j ζ 1 N-S equatons n the movng coordnate system are t = τ γζ 2. (2.62) ζ 1 u ζ = γτ u 2 ζ 1, (2.63) u τ + u u j = p p u + γτδ j2 + νdu + γτu 2 γu 2 δ 1, (2.64) ζ j ζ j ζ 1 ζ 1 2 where D = 2 2γτ + γ 2 τ 2 2. ζ j ζ j ζ 1 ζ 2 ζ 1 ζ 1 If the varable of tme τ s gven as 0, the equatons become: u ζ =0, u τ + u u j = p + ν 2 u γu 2 δ 1. ζ j ζ j ζ j ζ j (2.65) 25

40 If the lter operator s dened n the movng coordnate system, n order to derve the formulaton of KEF, the lter operator should commute wth the tme dervatve. In physcal space t means that the lter should reman ts shape, for nstance n Fg. 2.7, f at tme τ = 0 the lter s a crcle at A 1, after τ t locates at A 2 but should reman a crcle. Thus the lter operators are the same between the xed and movng coordnate systems, and we could obtan the KEF formulaton by applyng a smlar method. It leads to: 1 D < 2 τ D < j ς j + γ ( 2 D< 12 = ν D< τ < 2ε < + δu < j ς j ς j ζ j < ) τ j, (2.66) ζj n whch ς = ζ ζ s the two-pont dstance n the movng coordnate system. If the tme τ s consdered as 0, the lter operator s the same as n the xed coordnate system n ths moment. From equaton (2.62), we could also wrte = ζ j, = (2.67) ξ j ς j n ths τ = 0 moment. Thus the derences between equaton (2.54) and (2.66) are the tme-dependng term and the shear term γξ 2 D <. In order to analyze the eect 2 ξ 1 of ths shear term, we substract (2.54) from (2.66), and obtan γξ 2 D < ξ 1 = D< (movng) τ D< (fxed) t D < (2.68) Ths equaton means that the shear term γξ 2 denotes the derences of tme 2 ξ 1 development of second order structure functon of ltered velocty D <. As shown n Fg. (2.6), f we apply the movng coordnate system n numercal computng, the dscrete derence has to be employed between two ponts of two moments (τ and τ + τ). It could be treated as a knd of Lagrange descrpton [51, 56, 57], but the equaton s wrtten wth mean ow (not wth a real partcle). D < We could wrte the shear term γξ 2 n a sphercal coordnate system to verfy 2 ξ 1 ts ansotropy. The sphercal coordnate s ntroduced as (see Fg. 2.8): The gradent operaton of D < ξ 1 = rcosθ, ξ 2 = rsnθcosφ, ξ 3 = rsnθsnφ. (2.69) s then wrtten as D < = D< r e r + 1 D < r θ e θ + 1 D < rsnθ φ e φ, (2.70) 26

41 A 1 u (τ = 0) A 2 u (τ = τ) A 3 u (τ = 0) u (τ = τ) Fgure 2.7: Flter operator n physcal space, n movng coordnate system. Fgure 2.8: Sphere coordnate system used for analyzng the ansotropy of the shear term. 27

42 and the component n x 1 drecton reads: 0 D < ξ 1 0 = cos ( 1 D < r θ The ntegraton of the shear term from 0 to radus r s: r 1 2 γξ D < 2 dr = 1 r ( ) 1 ξ 1 2 γ D < D < rsnθcosφcos r θ r dr = 1 2 γsnθcosφ [rcos r 1 0 r D< sn ) D < r. (2.71) ( ) 1 D < r ( ) 1 D < D < D < cos dr r θ 0 r θ ( ) ] 1 D < D < r θ θ dr. If local sotropy s satsed, D < s only a functon of r, thus D< θ value s smpled as r γξ D < 2 dr = 1 ξ 1 2 γsnθcosφ (2.72) = 0, and the ntegral ( r ) rd < D < dr. (2.73) 0 The value veres wth the coordnates θ and φ (see Fg. 2.9). It means ths shear term s ansotropc n the radus r sphere. In fact, the ansotropc property mght also aect the local sotropy assumpton of D <. Although the problem of local sotropy has no drect relaton wth our SGS model desgnng, we want to have a smple dscusson on t. The assumpton of local sotropy s quered on Doubt on the assumpton of local sotropy Local sotropy was ntroduced by Kolmogorov [9] as homogenety plus sotropy of the small scales of turbulent motons, and t remaned an mplct assumpton n hs renements [58]. It s a cornerstone of the theory of unversal self-smlarty, closely connected wth the assumpton of complete ndependence of the small-scale structure of the turbulent eld from ts large-scale structure and mean shears, and also wth the random character of the energy cascade. Obukhov [59] and Corrsn [60] have extended the assumpton to the small scales of scalar elds mxed by turbulence, apparently as a consequence of ther propertes as passve contamnants. Although some scentsts had doubt about the assumpton n shear ow [6163], Mestayer thought that there are at least four reasons that these hypotheses are true [64]: 28

43 6 φ θ Fgure 2.9: snθcosφ aganst derent angles θ and φ, n a sphere coordnate system. 1. The hypotheses seem necessary for the self-smlarty theory to hold - and ths theory seem too coherent and ecent to be wrong. 2. Local sotropy brngs such great smplcatons n equatons, and also n the expermental estmaton of some operators lke the dsspaton rates of knetc energy and of scalar varances that could perhaps not be estmated otherwse. 3. In numerous ud-mechancs problems sotropy has been proved to be a very ecent rst approxmaton. 4. There seems to be lttle expermental evdence aganst the hypotheses. Mestayer' experments show that the local sotropy assumpton of moment spectrum s satsed only at the scales less than 20 tmes of Kolmogorov scale η, but not n the nertal subrange. For the scalar eld (the temperature eld n Mestayer's experments), the local sotropy assumpton s not satsed n almost all scales. Most of the later researches are based on group theory [65, 66]. Numercal and expermental results are analyzed to verfy the local sotropy and ansotropy of hgh-order structure functons [6769]. However, the physcal mechansm leadng to local sotropy and ansotropy has not been clared yet. Ubero thought that the vortcty should 29

44 be hghest and lowest n the drecton of contracton. These drectons are at ±45 to the mean ow for parallel or nearly parallel ows. In most of the studes, turbulence s regarded as 2-D, where only the streamwse drecton and normal drecton are consdered. The spanwse drecton was consdered to be homogeneous, whch s the same as the streamwse drecton. However, the followng analyss shows the ansotropy among the three drectons of 3-D turbulence. In homogeneous shear ow, f the local sotropy assumpton s satsed, there should be: for non-ltered velocty. D (ξ 1 ) = D (ξ 2 ) = D (ξ 3 ) (2.74) If the lter s sotropc (such as n the xed coordnate system, shown n Fg. 2.7), we could also wrte: for the ltered velocty. D < (ξ 1 ) = D < (ξ 2 ) = D < (ξ 3 ) (2.75) We dscuss the case of non-ltered velocty as an example. If the lter sze tends to zero, equaton (2.54) could be rewrtten to represent the structure functons of non-ltered velocty: 1 D + 1 D j + γξ 2 D + γ 2 t 2 ξ j 2 ξ 1 2 D 12 = ν D 2ε. (2.76) ξ j ξ j If the assumpton of local sotropy s satsed, we could wrte ths equaton n the drectons of x 1 and x 3, respectvely: 1 D (ξ 1 ) + 1 D j (ξ 1 ) + γ 2 t 2 ξ j 2 D 12(ξ 1 ) = ν D (ξ 1 ) 2ε, (2.77) ξ j ξ j 1 D (ξ 3 ) + 1 D j (ξ 3 ) + γ 2 t 2 ξ j 2 D 12(ξ 3 ) = ν D (ξ 3 ) 2ε. (2.78) ξ j ξ j The terms of D and D j / ξ j should then be sotropc, thus one can obtan D 12 (ξ 1 ) = D 12 (ξ 3 ). However, notce that D 12 (ξ 1 ) s a correlaton functon between the longtudnal and transverse veloctes, whle D 12 (ξ 3 ) s a correlaton functon between two transverse veloctes. A derence mght exst. The longtudnal-transverse correlaton functon was studed by Kuren [68]. It should be zero n sotropc turbulence, but Kuren's experments show that D 12 (ξ 1 ) ξ n nertal subrange. However, there s no further research on the transverse-transverse correlaton functon D 12 (ξ 3 ). 30

45 30 20 U H Fgure 2.10: Velocty prole n turbulent Couette ow. An A Pror test n turbulent Couette ow s done. The DNS database s calculated n Tsnghua unversty of Chna. A pseudo-spectral method s used. The Reynolds number Re H = 3200, based on the center velocty U m and channel halfwdth H. The grd number s The computaton doman s 4πH, 2H and 2πH n streamwse, normal and spanwse drectons. The mean velocty s shown n Fg In the center of the channel, the mean velocty could be consdered to be lnear, thus we calculate the D 12 ( ξ) when ξ are n three axs drectons respectvely, by makng statstcs n ths lnear regon. The values are normalzed by the second-order moment R 12 = u < 1 u < 2. The results are shown n Fg There s the derence between D 12 (x 3 ) and the others. However, n the 10 < ξ/η < 20 range, the 1.22 scalng s approxmately satsed for all three quanttes. Another ansotropc behavor s the derent scalng law n dsspatve range,.e. ξ/η < 10. The structure functon n normal drecton s not the same as the other two. In ths thess we attempt to explan t, but unfortunately the database s not ne enough, and we could not obtan more nformaton n the dsspatve range. We consder the eect of the shear term mentoned n the last secton. As analyzed, the shear term has an eect of dstorton. If the two-pont dstance s n x 2 drecton, 31

46 ξ 1.22 D ξ 2 D 12 (ξ 1 ) D 12 (ξ 2 ) D 12 (ξ 3 ) ξ/η Fgure 2.11: Structure functon D 12 n derent drectons of two-pont dstance. we can wrte both the equatons n xed and movng coordnate systems: 1 D + 1 D j + γξ 2 D + γ 2 t 2 ξ j 2 ξ 1 2 D 12 = ν D 2ε, ξ j ξ j 1 D 2 τ + 1 D j 2 ς j + γ 2 D 12 = ν D ς j ς j 2ε. (2.79) D There s no reason to assume that the shear term γξ 2 s sotropc. Comparng 2 ξ 1 wth equaton (2.77), t s more probable that the local sotropy between x 1 and x 2 drectons exsts n the movng coordnate system only. Ths eect s obvous when the two-pont dstance s large and the mean velocty gradent s strong. Varous nteractons of mean shear wth turbulent uctuatons cause the ansotropy of structure functon. The second-order longtudnal structure functons n derent drectons are shown n Fg The values are normalzed by the second-order longtudnal moment R ll = u < l u< l. Smlarly, the thrd-order longtudnal structure functons n derent drectons are shown n Fg The values are normalzed by the thrd-order longtudnal moment R lll = u < l u< l u< l. Derence always exsts among the three drectons. Notng the relaton between the second- and thrd-order structure functons, we wll analyze the propertes of skewness n secton The analyses for ltered velocty are also smlar. However, there are more terms of the subgrd stress. In many subgrd models, the subgrd stress s assumed to 32

47 D ll 10-2 D 11 (ξ 1 ) D 22 (ξ 2 ) D 33 (ξ 3 ) ξ/η Fgure 2.12: Second-order longtudnal structure functon D ll n derent drectons of two-pont dstance D lll D 111 (ξ 1 ) -D 222 (ξ 2 ) -D 333 (ξ 3 ) 10-5 ξ/η Fgure 2.13: Thrd-order longtudnal structure functon D lll n derent drectons of two-pont dstance. 33

48 be sotropc, and s represented only wth the velocty uctuatons. For example, the Shear-Improved Smagornsky Model (SISM [70]) s a great mprovement of the orgnal Smagornsky model, by consderng the uctuaton parts. However, more and more works show that the mean velocty also has a contrbuton on subgrd scales, although t s dcult to be modelled [40,48]. We beleve that t s the smlar problem as the queryng of local sotropy. More analyses of the SGS stress n homogeneous shear ow wll be shown n the next chapter. Therefore, the assumpton of local sotropy s quered. It s not completely satsed for the second-order structure functon n homogeneous shear ow Analyss of two-pont skewness Skewness s an mportant physcal quantty of turbulence analyss n physcal space. The one-pont skewness of a non-ltered sotropc velocty eld s dened as ( u1 ) 3 one pont Sk = x 1 ( u1 x 1 ) (2.80) one pont From EDQNM theory, the value of Sk s about [8]. It has been vered n experments and numercal smulatons [71, 72]. Results show that the skewness s not just 0.457, but depends slghtly on the Reynolds numbers. Smlarly, the two-pont longtudnal skewness of a non-ltered sotropc velocty eld s dened as two pont Sk (ξ) = (u1 (x 1 + ξ) u 1 (x 1 )) 3 (u1 (x 1 + ξ) u 1 (x 1 )) 2 3. (2.81) 2 In the dsspatve range for small ξ, Taylor expanson can be employed, that u 1 (x 1 + ξ) u 1 (x 1 ) ξ u 1 x 1, thus the two-pont denton equals the Eq. (2.80). However, the experments of P. Tabelng [73] show that the two-pont longtude skewness s not constant for derent two-pont dstances ξ. When ξ ncreases, ths value decreases, and t s n the nertal subrange. It s also n agreement wth Cerutt's experment [74]. 34

49 In large-eddy smulaton, the skewness s dened as < two pont S k (ξ) = D< lll (ξ) (D ll < (ξ)). (2.82) 3 2 when the lter sze s much less than the two-pont dstance ξ, t could be consdered as the same as n non-ltered velocty eld. Ths wll be proved n secton 2.6. The extended self-smlarty theory (ESS) also strongly supports constant twopont skewness. ESS theory s not a scalng law, but a self-smlarty result between second-order and hgh-order structure functons. It s well satsed n derent regons. From ESS theory, the skewness value (equaton 2.82) s a constant, n both dsspatve range and nertal subrange. skewness should be vered. However, n ansotropc turbulence, the sotropy of In ansotropc turbulence, the two-pont skewness of ltered velocty n an axs drecton x s dened as (wthout summaton conventon): (u < (x + ξ) u < (x )) 3 < two pont S k (ξ ) = (u < (x + ξ) u < (x )) 2. (2.83) 3 2 In the center regon of Couette ow (the database was descrbed n secton 2.4.4), the two-pont skewness values n derent drectons are calculated (Fg. 2.14). Comparng wth the second- and thrd-order structure functons mentoned n secton 2.4.4, the skewness values are almost the same among the three drectons. The value s approxmately between 0.3 and 0.2, and t decreases when ξ ncreases. They are n agreement wth P. Tabelng's experments. < two pont Snce the mean velocty ncrement vanshes, S k < 0 mples that the p.d.f p(δu < l ) s skewed, wth negatve δu< l occurrng less frequently than postve values but reachng larger ampltude. The second-order structure functon denotes energy, and the thrd-order structure functon denotes the transfer, thus skewness could be consdered as a property of energy transfer between resolved and subgrd scales. Batchelor thought that the skewness can represent the non-lnear transfer and dsspaton of the contrbuton of small structures [49]. Therefore, n the modelng applcatons, we attempt to employ the skewness, to represent the local-sotropc behavor n subgrd scale. 35

50 ξ/η S k S k (ξ 1 ) S k (ξ 2 ) S k (ξ 2 ) -0.5 n derent drectons of two-pont ds- < two pont Fgure 2.14: Two-pont skewness S k tance. 2.5 KEF n homogeneous rotatng turbulence Rotatng turbulence s a typcal case of ansotropc turbulence wth practcal nterest n geophyscal and astrophyscal ows, as well as turbulent ows n turbomachnery. Consderable eorts have been made to nvestgate the behavor of rotatng turbulence and ts structure by numercal smulaton [75], expermental measurements [76] and theoretcal analyses [77]. A number of pecular propertes of rotatng turbulence have been revealed. The transfer of knetc energy from large scales to small ones s reduced; consequently energy dsspaton s decreasng wth ncreasng rotatng rates. It has also been found that the energy spectrum shows a steeper slope than the Kolmogorov 5/3 law, approachng a spectral exponent of 3 asymptotcally at nnte Reynolds number and zero Rossby number. It s possble that the feature of rotatng turbulence results from ansotropc nonlnear transfer of turbulent knetc energy among velocty uctuaton components. The governng equaton of LES for homogeneous rotatng turbulence can be wrtten as u < t + u < j u < = 1 p < + ν 2 u < + 2ϵ j3 Ωu < j ρ x τ < j, (2.84) 36

51 n whch Ω s the rotaton rate n x 3 drecton, and ε jk s the Lev-Cvta symbol. In order to derve the formulaton of KEF, we could use the same method as descrbed n secton 2.3. Most of the processes are smlar. Note that ϵ j3 = ϵ j3, (2.85) ths property vansh the rotatng term, whch could mean that the rotaton eect s mplctly nvolved n the energy transfer. The detals can be found n Shao's paper [78]. Fnally the formulaton of KEF s smlar as equaton (2.54), but wthout any forcng term or shear term : 1 D < 2 t D < j ξ j = ν D< ξ j ξ j 2ε < + δu < ( τ slow< j 2.6 The scalng law of ltered velocty ) slow< τj. (2.86) x j From the KEF equatons n derent turbulence, the structure functons of ltered velocty always play mportant roles. In fact, among many other derent analyses whch may eventually be performed [7982], the statstcal propertes of homogeneous sotropc turbulence are usually nvestgated by studyng the velocty structure functons F n (r) = δu(r) n, (2.87) where δu(r) = u 1 (x + r) u 1 (x) s the ncrement of non-ltered velocty. At hgh Reynolds number, one observes that there exsts a range n r, called the nertal subrange, where F n (r) has a power law behavor that s F n (r) r ζ(n). (2.88) The nertal subrange corresponds to length scales where vscosty eects are neglgble,.e. to an nterval η r L, where η = (ν 3 /ε) 1/4 s the Kolmogorov dsspaton scale, ε s the mean energy dsspaton rate and L s the ntegral scale of moton. The Kolmogorov theory [9] predcts that the statstcal propertes of δu(r) depend only on ε and r. It then follows by dmensonal analyss that ζ(n) = n/3. (2.89) 37

52 In fact, many experments and numercal results n hgh Reynolds number turbulence have shown that the n/3 scalng law s not satsed completely. A few models [3237] have been proposed n order to explan the eect of ntermttency on the scalng exponents of the structure functons. Two analytcal solutons of the scalng exponent are also proved n the appendx D. However, these models are usually complex and not easy to be employed n real practce. The correctons for n = 2 and 3 wth respect to K41 are usually very small. These orders are the most mportant n predctng the large scale behavor of turbulent ows. In ths thess, we therefore apply the classcal scalng law theory (2.89) n nertal subrange. The followng problem s that the scalng law of ltered velocty mght not be the same as the non-ltered velocty. In the followng part, we analyze the behavor of structure functon of ltered velocty, n physcal space and spectral space respectvely Analyss n physcal space In order to nvestgate the derence, a lter n physcal space s represented by ntroducng the kernel G( x x )d x = 1, (2.90) and the velocty ncrement δu < ( x) = u< ( x + r) u< 1 ( x) can be wrtten as δu < ( x) = G( x x )δu ( x )d x. (2.91) Followng Germano [83], t s easy to verfy that δu < δu < j = (δu δu j ) < 1 2 G( x x )G( x x ) (u ( x ) u ( x )) (u j ( x ) u j ( x )) d x d x. (2.92) Usng the same coordnate transformaton as Germano, and assumng that the lter s Gaussan wth lter sze, nally we obtan δu < δu < j = (δu δu j ) < 1 2 G( s; ( ) ( ) 2 ) δu ( s) δu ( 0) δu j ( s) δu j ( 0) d r. (2.93) 38

53 Because of the eect of lter kernel G( s; 2 ), the last term of equaton (2.93) s mostly eectve when s. When r, the producton of the velocty ncrements n the last term could be evaluated that ( ) ( ) ( ) ( ) δu ( s) δu ( 0) δu j ( s) δu j ( 0) δu ( r) δu ( 0) δu j ( r) δu j ( 0). (2.94) Notce that the rght hand sde s at most the same order of magntude as the second order structure functon ( ) ( ) ( ) ( ) δu ( r) δu ( 0) δu j ( r) δu j ( 0) u ( r) u ( 0) u j ( r) u j ( 0), thus t could be then estmated approxmately: (2.95) ( ) ( ) δu ( r) δu ( 0) δu j ( r) δu j ( 0) δu δu j. (2.96) In homogeneous turbulence, the rst term n the rght-hand sde of equaton (2.93) s (δu δu j ) < = δu δu j < = δu δu j, (2.97) Therefore when r, there s the approxmate relaton that δu < δu < j δu δu j, (2.98) whch means the propertes of second-order structure functons are approxmately the same between ltered and non-ltered veloctes. In homogeneous sotropc turbulence, we extend ths concluson onto hgher-order structure functons, that (δu < 1 ) n F n (r) (2.99) when r. And generally, n nhomogeneous ansotropc turbulence, there s the concluson that (δu < 1 δu < 2... δu < n ) (δu 1 δu 2... δu n ) (2.100) when r. Therefore, n equaton (2.21) f ξ, the thrd-order structure functon term wll be D 111(ξ) < ξ. (2.101) 39

54 The dsspaton term has the same trend that 6 S < 11τ < 11 ξ ξ. The subgrd transfer term n the rght hand sde 6T 1,11 = 6 u < 1 (x 1 )τ < 11(x 1 + ξ) s a two-pont correlaton functon, whch would tend to zero when ξ s large. Thus the balance of magntude n equaton (2.21) s proved to be satsed. For the two-pont skewness mentoned n secton 2.4.5, f ξ, we could obtan D < lll (ξ) (u 1 (x 1 + ξ) u 1 (x 1 )) 3, D < ll (ξ) (u 1 (x 1 + ξ) u 1 (x 1 )) 2, (2.102) thus the skewness values of ltered and non-ltered velocty eld could be consdered the same Analyss n spectral space A statstcally homogeneous velocty eld can be represented n spectral space: ( ) 3 1 u( x) = u( 2π k)e ι k x d k. (2.103) A cut-o lter n spectral space s dened as u( k) < = { u( k), k kc, 0, k > k c, (2.104) where k c s the cut-o wave number. If the turbulence s sotropc, the summaton of structure functon could thus be denoted as [51, 84]: D < (ξ) = 4 where E(k) s the energy spectrum and E(k) < 0 [ E(k) < 1 snkξ ] dk, (2.105) kξ represents a cut-o lter appled. Approxmately, we could assume a von-karman energy spectrum that E(k) = E(k p )2 17/6 (k/k p ) 4, (2.106) [1 + (k/k p ) 2 17/6 ] n whch we x E(k) = 1.0 and k p = 1.0 (see Fg. 2.15). Suppose that k c0 = 500 corresponds wth a grd sze h of DNS, we have h = 1/k c0 = When the lter sze s 2h, 4h,..., nh, the correspondng cut-o wave numbers are k c0 /2, k c0 /4,..., k c0 /n. Wth the lter szes = h, 2h, 4h, 8h, 16h and 32h, the summaton of structure functons of ltered velocty D < are calculated and shown n Fg. 2.16(a). The ξ 2/3 40

55 k c0 /32 k c0 /16 k c0 /8 k c0 /4 E 10-2 k c0 /2 k c k Fgure 2.15: Von-Karman energy spectrum, wth E(k) = 1.0, k p = 1.0. range s obvous when s small, and t s less obvous when s large (for nstance = 32h). In order to nvestgate the behavor clearly, we dene the compensatng structure functon: D < (ξ) = ξ 2/3 D < (ξ), (2.107) whch are shown n Fg. 2.16(b). The plateau corresponds to the range whch satses the ξ 2/3 scalng law. It becomes more narrow when ncrease. Furthermore, note that wth each lter sze, the start pont of the correspondng plateau s about ξ 10. Ths phenomenon s n agreement wth the analyss n physcal space (secton 2.6.1), that ξ should be satsed to obtan the classcal scalng law. Note that n ths secton only the summaton of structure functon s dscussed. The Fourer transforms of the components of structure functon tensor are much more complex (see P. 206 n Hnze's book [51]). In fact, as long as the sotropc behavor s satsed n each secton of wave number (.e. n each scale), the smlar results for the tensor components are doubtless. In order to employ the classcal scalng law n LES smulaton, ths behavor should be notced and we propose the two-pont dstance s much larger than lter sze,.e. ξ. 41

56 h 32h h 4h 2h 32h 8h 16h 8h 1.5 D < (ξ) 10-1 h 2h 4h ξ -2/3 D < (ξ) ξ ξ (a) (b) Fgure 2.16: Summaton of structure functons of ltered velocty, wth derent cuto lters of a von-karman energy spectrum. In each gure from left to rght: = h, 2h, 4h, 8h, 16h, and 32h. (a) Structure functons. (b) Compensatng structure functons. 42

57 Chapter 3 Applyng KEF on eddy-vscosty models In ths chapter, KEF s appled n eddy-vscosty modelng. From the eddyvscosty assumpton, the SGS stress tensor s algned wth the resolved stran rate tensor,.e.: where ν t s the SGS eddy-vscosty coecent. S < j τ < j 1 3 τ < mmδ j = 2ν t S < j, (3.1) s the stran rate for resolved scale turbulence: S j < = 1 ( ) u < + u< j. (3.2) 2 x When appled n ansotropc turbulence, the stran rates can be dened usng the uctuaton parts of the velocty: S < j = 1 2 ( u < ) + u < j. (3.3) x There s already lots of research on determnng the eddy vscosty. In spectral space, models are all eectve vscosty models drawng upon the analyses of Krachnan. The subgrd vscosty s a functon of wave number k. From Fourer transformaton, ntroducng a cut-o wave number k c, ν t could be represented as ν t (k k c ) = 1 T sgs (k k c ) 2 2k 2 E(k), (3.4) where E(k) s the knetc energy (the superscrpt denotes the conjugaton operaton, and S( k s a sphercal shell of radus k): E(k) = 1 u( 2 k) u ( k)ds( k), (3.5) 43

58 and T sgs (k k c ) s the subgrd energy transfer term [16]. Chollet and Leseur proposed an eectve vscosty model usng the results of the EDQNM closure on the canoncal case of sotropc turbulence [28]. The model formulaton s obtaned by approxmatng the exact soluton wth a law of exponental form: ν t (k k c ) = [ exp( 3.03k c /k)] E(k c ) k c. (3.6) Ths form makes t possble to obtan an eectve vscosty that s nearly ndependent of k for wave numbers that are small compared wth k c, wth a nte ncrease near the cuto. Another smpled form of the eectve vscosty could be derved ndependently of the wave number k [85]. By averagng the eectve vscosty along k and assumng that the subgrd modes are n a state of energy balance, we could obtan: ν t (k k c ) = 2 E(k c ) 3 K 3/2 0. (3.7) k c In physcal space, varous subgrd vscosty models are avalable. The well-known Smagornsky model s very smple to mplement [3]. It s generally used n a local form n physcal space, n order to be more adaptable to the ow beng calculated. Ths model can be expressed as: ν t = (C s ) 2 S <, (3.8) n whch the constant theoretcal value C s should be determned. Usng an sotropc energy spectrum, we could obtan C s = 0.17 [86]. Clark uses C s = for a case of sotropc homogeneous turbulence [87], whle Deardo uses C s = 0.1 for a plane channel ow [88]. Studes of shear ows usng expermental data yeld smlar evaluatons that C s [17, 89]. In order to determne the model coecent, a dynamc method s to employ the Germano Identty [11, 12]. The dynamc method depend on two lters: the ltered part at mesh scale s denoted as, the correspondng subgrd stress s τ j = ũ u j ũ ũ j ; the ltered part as a test scale α (α > 1) s denoted as, the correspondng subgrd stress s τ α j = u u j u u j. A subgrd stress tensor s dened wth the two lters L j = ũ ũ j ũ ũ j. Germano assumes: L j = τ α j τ j. (3.9) 44

59 Consder that the subgrd stress could be formulated by usng Smagornsky model (3.8), we could obtan the followng equaton: where M j = 2 2 [ ( ) C α 2 α 2 s C S S s j S S j]. If the model coecent s ndependent wth grd sze,.e. C α s L j 1 3 L kkδ j = (C s ) 2 M j, (3.10) model coecent could be solved dynamcally: = C s, employng the least-squares procedure, and the (C s ) 2 = L jm j M j M j. (3.11) The dynamc Smagornsky model s wdely appled n numercal smulaton. It could yeld good result, especally on the near-wall behavor. However, ths method s a mathematcal closure, but not a physcal method, there s no physcal relaton mpled. When appled n ansotropc shear turbulence, the classcal Smagornsky model can not well represent the shear eect. Levêque et al. ntroduced a shear-mproved model, whch s based on results concernng mean-shear eects n wall-bounded turbulence [70]. The Smagornsky eddy-vscosty s moded as ν t = (C s ) 2 ( S < S < ), (3.12) where the magntude of the mean shear S < s subtracted from the magntude of the nstantaneous resolved rate-of-stran tensor S <. Thus the shear-mproved Smagornsky model could represent the slow part of energy transfer equaton. The structure functon model s a transposton of Metas and Leseur's constant eectve vscosty model nto physcal space, and can consequently be nterpreted as a model based on the energy at the cuto, expressed n physcal space [29]. second-order structure functon s dened as D ll ( x, r) = [ u( x) u( x + x )] 2 d 3 x, (3.13) x =r and the structure functon model takes the form: The ν t (r) = A(r/δ) D ll (r), (3.14) 45

60 n whch A(x) = 2K 3/2 0 3π 4/3 9/5Γ (1/3) x 4/3 ( 1 x 2/3 H sf (x) ) 1/2. (3.15) In the case where r =, the model takes the smpled form: ν t ( ) = D ll ( ). (3.16) In addton, there are other vscosty models whch are ntroduced n derent ways [90, 91]. In ths chapter, we ntroduce the CZZS model (.e. the thrd-order structure functon model [2]), whch s the rst modelng attempt by employng the homogeneous sotropc KEF. The smplcatons and varatons of CZZS model are dscussed n Sec The KEF can also been appled n determnng the coecent of Smagornsky model n Sec Besdes, ths model s extended nto homogeneous shear turbulence and rotatng turbulence n Sec In shear turbulence, the subgrd vscosty s aected by the mean shear. The ansotropc model s vered n largeeddy smulaton. The correspondng work has been publshed n Journal of Flud Mechancs [40]. 3.1 Dscussons on CZZS model In secton 2.2, we have obtaned the homogeneous sotropc formulaton of KEF: 4 5 ε fξ = D < lll 6T l,ll. (3.17) Employng the eddy-vscosty assumpton (3.1), The sotropc part of subgrd stress can be ncluded n the pressure of knetc equaton and T l,ll n equaton (3.17) could be evaluated as T l,ll = 2ν t u < (x) u< (x + ξ) x = 2ν t ξ u< (x)u x (x + ξ) (3.18) = 2ν t R ll ξ. Snce D < ll = 2 u <2 2R ll, we could wrte T l,ll = ν t D < ll ξ. (3.19) 46

61 The dsspaton rate of resolved scale turbulence ε f equals τ j < S< j, and n the case of constant subgrd eddy vscosty, we have ε f = 2ν t S < js < j. (3.20) Insertng equatons (3.19) and (3.20) nto equaton (3.17) one obtans the formulaton for subgrd eddy vscosty [38] ν t = 5D < lll 8 S < j S< j ξ 30 D< ll ξ. (3.21) Introducng the skewness S k (dscussed n secton 2.4.5): S k = (D < ll D< lll )3/2, (3.22) the subgrd model could be expressed as ν t = 5S k (D ll < )1/2 8 S< j S< j D ll < ξ 30 D ll < D ll < ξ. (3.23) As dscussed n the last chapter, the T l,ll vansh when ξ. Thus the followng dscuss contan two parts: the one-scale models when ξ =, and the mult-scale models when ξ One-scale models The second-order structure functon can be approxmated by the scalng law: D < ll (ξ) ξn, (3.24) where n s the scalng exponent. As dscussed n secton 2.6, the classcal scalng law could not be appled when ξ =. However, from Eq. (3.24) we can obtan D < ll ξ = nd< ll ξ. (3.25) For small ξ, the second nvarant of the stran rate tensor S j < S< j could be evaluated n sotropc turbulence as S < js < j = 15 2 ( u ) < 2 1 x D< ll 2ξ 2. (3.26)

62 Insertng (3.25) and (3.26) nto the subgrd eddy vscosty formulaton (3.23), we have a smlar formulaton to Smagornsky model: where ν t = (C s ) 2 S 2 j < S< j, (3.27) C s = S k (12 6n) 15. (3.28) If S k = and n = 2/3 are accepted then the equvalent Smagornsky coecent C s In fact, the deal values maybe not correct for ths problem. In Fg we have the two-pont skewness 0.3 < S k < 0.2. From secton 2.6 the classcal scalng law s not satsed, thus we have 2/3 < n < 2. Therefore, the coecent C s has dynamc values C s > These values are n magntudal agreement wth the classcal coecent values. The subgrd eddy vscosty can also be wrtten as ν t = S k D ll < ξ. (3.29) 12 6n Snce ξ =, t s n the same formulaton as Metas and Leseur's structure functon model: ν t = C m D ll < ξ. (3.30) If 0.3 < S k < 0.2 and 2/3 < n < 2, the model coecent C m > In ths range, t depends on derent mesh scales, and t's also n magntudal agreement wth Metas and Leseur's result C m = Both the smpled CZZS formulatons (3.28) and (3.29) are expressed by the two-pont skewness S k. An advantage s that the skewness s approxmately sotropc n homogeneous shear ow (as analyzed n secton 2.4.5). It s also a drect result from the ESS theory, whch has been vered to be more unversal than the classcal scalng law. Thus t mght be better n LES applcatons than the structure functon D < ll. A Posteror tests are performed n channel turbulence. Two numercal methods, one s the spectral method and the other s the nte volume method, are employed. The spectral method utlzes the Chebyshev polynomal. The mesh contans grds, n streamwse, normal and spanwse drectons respectvely. The numercal 48

63 (a) Spectral method (b) Fnte volume method Fgure 3.1: A Posteror two-pont skewness values, n normal drecton of channel ow. detals are shown n the appendx. The nte volume code utlzes a precson of spatal resoluton, whch s 4 th order n homogeneous drectons (.e. streamwse and spanwse drectons), and s 3 th order n normal drecton. The temporal proceed s the 3 th order Runge-Kutta method. The mesh contans grds. The Reynolds number Re H = 7000, based on the bulk velocty U m and channel half-wdth H. The subgrd model employed n spectral method s the skewness-based CZZS model of Smagornsky formulaton (3.28). Let n = 2/3, the model could be represented as C s = S k (3.31) The nte volume method utlzes the Shear-Improved Smagornsky Model (SISM): where C s = 0.16 accordng to EDQNM theory. ν t = (C s ) 2 ( S < S < ), (3.32) The skewness values n normal drecton are shown n Fg In the center of channel, the value s about 0.3, whch s n agreement wth Cerutt's experment [74]. In the regon 10 < y + < 20, the skewness s found to be about 0.7 n both cases, but the peak locaton s not exactly the same. The skewness values tend to zero n near-wall regon n the two cases. 49

64 (a) Spectral method (b) Fnte volume method Fgure 3.2: A Posteror values of Smagornsky coecent C s, n normal drecton of channel ow. The correspondng coecent values of the Smagornsky model s calculated by usng equaton (3.31). The values n both the two cases are also n agreement. In the center of channel, the values are about 0.1, whch n n agreement wth Deardo's proposton [88]. The values tend to zero n near-wall regon, whch s the correct wall behavor. A Posteror statstcal results of spectral method are shown n Fg. 3.3, compared wth Moser's DNS results [92], LES results usng classcal Smagornsky model (SM) and the dynamc Smagornsky model (DSM). The behavor of the present subgrd model,.e. Eq. (3.31), s as good as the DSM, and s much better than SM Mult-scale models n nertal subrange Accordng to eddy-vscosty assumpton, we can obtan the mult-scale models, f all scales are n the nertal subrange. From equaton (3.25), we have T l,ll (ξ) ξ n 1, (3.33) where n s the scalng exponent of second-order structure functon. n = 2/3 we have Especally f T l,ll (ξ) ξ 1/3. (3.34) 50

65 (a) (b) (c) (d) (e) Fgure 3.3: A Posteror statstcal results of channel ow. LES cases are usng Eq. (3.31), Smagornsky model (SM), dynamc Smagornsky model (DSM), respectvely. (a) Mean velocty prole. (b) Turbulence ntensty n streamwse drecton. (c) Turbulence ntensty n normal drecton. (d) Turbulence ntensty n spanwse drecton. (e) Reynolds stress prole. 51

66 Therefore, f two space ncrements ξ 1 and ξ 2 both satsfy ths scalng law, we obtan the mult-scale relaton [39]: 0.8ε f ξ 1 + D < lll (ξ 1) 0.8ε f ξ 2 + D < lll (ξ 2) = Thus the subgrd vscosty s denoted as ( ξ1 ξ 2 ) 1/3. (3.35) ν t = ( ) 1/3 D lll < (ξ ξ1 1) D lll < ξ (ξ 2) 2 ( ) ) 4/3. (3.36) ξ1 0.4 S < (1 2 ξ 1 ξ 2 Furthermore, we can derve a smlar formulaton wth three or more scales n the nertal subrange. Suppose three space ncrements ξ 1, ξ 2 and ξ 3 all satsfy the scalng law (3.34), we have T l,ll (ξ 1 ) T l,ll (ξ 2 ) ( ξ1 ξ 2 ) 1/3 = T l,ll(ξ 2 ) T l,ll (ξ 3 ) ( ξ2 The subgrd vscosty s the root of the followng equaton: ξ 3 ) 1/3. (3.37) (Aα 2 2 α 1 α 3 )ν 2 t + (Aα 2 β 2 α 1 β 3 α 3 β 1 )ν t + Aβ 2 2 β 1 β 3 = 0, (3.38) n whch α = 4 ( ) ξ 5 S< ξ, β = D lll < 2 1/3 (ξ ), A = 2. (3.39) ξ 1 ξ 3 These mult-scale models are attempts to obtan a relatve relaton between dfferent scales. However, the same scalng exponents are not exactly satsed when ξ s of the same magntude as. Therefore, another mult-scale model s proposed, consderng two separated scales Mult-scale model wth separated scales In the smplcatons of CZZS model (secton 3.1.1), both the scalng law and the Taylor expanson are employed. However, the scalng law should be appled n nertal subrange, and the Taylor expanson should be used n very small two-pont dstance. Thus n ths secton, we consder that the lter sze and the two-pont dstance ξ n KEF are separated,.e ξ. Assume that ξ s n nertal subrange and the scalng law s satsed: D < ll (ξ) ξ2/3. (3.40) 52

67 Then, the subgrd transfer term could be wrtten by employng scalng law: D < ll ξ = 2D< ll (ξ). (3.41) 3ξ And the dsspaton term could be expressed by employng Taylor expanson at small scale : S < js < j = 15 2 ( u ) < 2 1 x 1 15D< ll ( ) 2 2. (3.42) Therefore, the formulaton of ths mult-scale subgrd model could be: ν t = D < lll (ξ) 12 ξ 2 D< ll ( ) 4D< ll (ξ) ξ. (3.43) Furthermore, when ξ s large, the second term n the denomnator can be negelcted by comparng wth the rst term, and we can obtan a smple expresson: ν t = 2 D lll < (ξ) 12ξD ll <. (3.44) ( ) In LES applcatons, we could consder as the same as grd sze h, and the twopont dstance ξ much larger than h, to calculate the subgrd vscosty dynamcally. 3.2 A new dynamc method to determne the coecent of Smagornsky model The KEF can not only be employed to brng up new subgrd models (such as the CZZS model), but also appled to determne the model coecents. In ths secton we determne the coecent of Smagornsky model by usng the homogeneous sotropc formulaton of KEF. In homogeneous sotropc turbulence, consderng that ξ, the subgrd transfer term T l,ll n equaton (2.16) vanshes. Employng the formulaton of the Smagornsky model (3.8), the subgrd dsspaton can be represented as ε f = τ < j S < j =2 ν t S < js < j (3.45) =2(C s ) 2 (S < kl S< kl )1/2 S < js < j. 53

68 C s ξ/η (a) Re λ = 50 C s ξ/η (b) Re λ = 70 Fgure 3.4: Dynamc values of the coecent of Smagornsky model. From top to bottom: = h, 2h,..., 7h. Thus from equaton (2.16), we solve the dynamc coecent C s as C 2 s = 5D < lll (ξ) 8ξ 2 (S < j S< j )3/2. (3.46) Thus the coecent s determned dynamcally. Note that we should let ξ n order to neglect the T l,ll term. Ths formulaton s smlar wth Canuto's result [93]: C 2 s q2 sgs ε S < 2. (3.47) In order to evaluate the behavor of ths new method, A Pror numercal tests are made. The two DNS cases of homogeneous sotropc turbulence wth spectral method are utlzed. The lter scale and the ncrement dstance ξ are selected ndependently. The lter scales are between 1 and 7 tmes of grd sze h. Results are shown n Fg It shows that the values are of the same magntude as the classcal theory. When s not large (for nstance = 2h) and 10 < ξ < 30 s n the nertal subrange, we could obtan C s 0.1 and 0.06, when Re λ = 50 and 70, respectvely. Both these values could be appled n numercal smulaton. It also shows the trend that the coecent value decreases when lter sze ncreases. In concluson, n ths secton we propose another dynamc method to determne the coecent of Smagornsky model. The results show the correct magntude and scale behavor. As long as we select the approprate lter sze and ncrement 54

69 dstance ξ, ths method mght be appled n large-eddy-smulaton. It s much less costly than the dynamc Smagornsky model based on Germano dentty [11], and t has clear physcal background. 3.3 SGS models for homogeneous ansotropc turbulence The governng equatons for both homogeneous rotatng turbulence and shear turbulence can be wrtten n a uned form for convenent dervaton of subgrd eddy vscosty as follows u t + U u + u j x 1 u = 1 p + ν 2 u + s, ρ x u x =0, (3.48) n whch U = γx 2, s = γu 2δ 1 for homogeneous shear turbulence, and U = 0, s = 2ϵ j3 Ωu j n homogeneous rotatng turbulence. It should be emphaszed that the reference frames for rotatng turbulence and shear turbulence are derent, although ther governng equatons are wrtten n same formulae. Ths ndcates that the subgrd eddy vscosty for rotatng turbulence s reconstructed n a rotatng frame whereas t s n an nertal frame for homogeneous shear turbulence. as From equaton (2.54) and (2.86), the general formulaton of KEF can be wrtten Note that the SGS stress τ < j part. D < t + D< j ξ j + (δud< ) ξ 1 2 δu < δs < ( =2ν D< τ < 4ε < + 2 δu < j ξ j ξ j The last two subgrd terms can be represented as ( τ < < ) δu < j τ j x j = δu < τ j < τ j < ). x j (3.49) contans both the homogeneous rapd part and the slow δu < τ j < x j 55 τ < j δu < τ j < δu <. x j (3.50)

70 Introducng the eddy-vscosty assumpton (3.1), where the stran rate s dened as the uctuaton part,.e. equaton (3.3), the rst two terms on the rght-hand sde of equaton (3.50) can be derved as δu < τ < j δu < τ j < x j = δu < τ < j ξ j + δu < τ j < ξ j =ν t 2 δu < δu< ξ j ξ j + 2ν t 2 δu < j δu< ξ j ξ. (3.51) The last term of the above equaton s equal to zero n homogeneous turbulence as can be proved as follows 2 δu < j ν δu < 2 δu < t =ν t ξ j ξ x δu < =ν k t j δu < δu < x x k = ν t 2 (δu < j δu < x = ν t u < k ) u < x x k = ν t 2 u < k u < x k x. (3.52) Snce the turbulence s homogeneous, the dervatves of one-pont statstcs are equal to zero,.e. 2 δu < j ν δu < 2 u < t = ν k u < t = 0. (3.53) ξ j ξ x k x The rst two terms n (3.50) are then smpled as δu < τ < j δu < τ j < x j = ν t 2 δu < δu < ξ ξ = ν t D < ξ ξ. (3.54) The last two terms of (3.50) are manpulated as follows τ j < δu < τ < δu < j x j = = ( u < ( u < u < = 2ν t u < n whch ε f = ν t + u < j x + u < j x ) ( δu < u < + x j ) ( u < u < + x j + u < j x + u < j x u <, x k x k u < s the subgrd dsspaton. x k x k ) δu < x j ) u < x j (3.55) 56

71 Therefore, the KEF (3.49) could be rewrtten as D < t + D< j ξ j + (δud< ) ξ 1 2 δu < δs < =2(ν + ν t ) D< ξ j ξ j 4ε < 4ε f. (3.56) Followng the classc theory that the tme dervatve of D < ll and the molecular vscosty duson can be neglected n the dynamc equaton of structure functons for hgh- Reynolds-number ows [94], the nal formula of the generalzed Kolmogorov equaton for ansotropc resolved scale turbulence s then wrtten as D < j ξ j + (δud< ) ξ 1 2 δu < δs < = 2ν t D < ξ j ξ j 4ε f. (3.57) In order to obtan the eddy vscosty for homogeneous turbulence, we take the local volume average n dsplacement space as Hll [95] and Cascola [96] dd, as follows ( D < ) j + δud< dv ξ j ξ 1 V ( 2 D < u < = 2ν t 4ν u < t V ξ j ξ j ) + 2 δu < δs < dv. (3.58) The volume ntegraton on the left-hand sde can be transferred to surface ntegraton by the Gauss formula such that v D j < dv = D ξ jn < j da, j S (3.59) δud < dv = δud < n 1 da. v ξ 1 S The rst term on rght-hand sde can also be transferred to surface ntegraton such that 2 D < D < 2ν t dv = 2ν t n k da. (3.60) V ξ j ξ j S ξ k The lnear sze of the ntegraton volume should be wthn the nertal subrange. In practce, t s equal to twce the mesh length n the followng computatonal cases. Dene the local volume average and surface average, respectvely, as Q V = 1 Qdv, V Q S = 1 S 57 V S QdA. (3.61)

72 Equaton (3.58) s then rewrtten as S(D < jn j ) S +S(δUD < n 1 ) S ( ) D < S =Sν t n j 4ν t V ξ j u < u < V + 2V δu < δs < V. (3.62) Fnally, the subgrd eddy vscosty s equal to ν t = (D< j n j) S + (δud < n 1) S + 2V/S δu < δs < ( ) V D < S 2 u < n j 4V/S u < V. (3.63) ξ j The structure functons D <, D< j are the ensemble average of the products of velocty ncrement n physcal space, n practce the ensemble average can be taken n homogeneous drectons; hence, the structure functons are ndependent of x and varyng wth dsplacement ξ. The local volume average s taken n dsplacement space ξ whch s ndependent of physcal space x. After local space averagng, the terms nvolvng structure functons are constants both n physcal and dsplacement spaces and, nally, the subgrd eddy vscosty s a constant n homogeneous turbulence. In practce, the local volume-average method s dependent on the ow geometry so that t can be taken n a sphere for homogeneous rotatng turbulence or n a rectangular box for plane shear ow. In homogeneous rotatng turbulence δu = 0 and δu < δs < = 2ϵ j3 δu < δu < j Ω = 0. The subgrd eddy vscosty for homogeneous rotatng ow s then equal to ν t = (D j < n j) S ( ) D < S 2 u < n j 4V/S u < V. (3.64) ξ j Note that the rotaton rate does not appear explctly n the formula of eddy vscosty. It s not surprsng because the sold rotaton does not contrbute drectly to the total transfer of turbulent knetc energy; nether does t to the transfer of velocty ncrement varance. However, rotaton can redstrbute knetc energy among uctuatng velocty components and the turbulence becomes ansotropc. Equaton (3.64) represents correct transfer of knetc energy n ansotropc resolved scale turbulence through the local volume average of transfer and dsspaton terms. 58

73 In homogeneous shear turbulence, δu = γξ 2 and δu < δs < = γd< 12. The subgrd eddy vscosty becomes ν t = (D< j n j) S + (γξd < n 1) S 2γV/S(D 12) < V ( ) D < S 2 u < n j 4V/S u < V. (3.65) ξ j Model applcatons n rotatng turbulence In ths secton, we are tryng to smulate rotatng turbulence numercally by LES. The tradtonal subgrd models fal to predct rotatng turbulence snce they do not consder the ansotropc eect of turbulence n the model. Yang [97] used truncated NaverStokes (TNS) as a model for large-eddy smulaton of homogeneous rotatng turbulence wth consderable success; for nstance, they obtaned a k 3 energy spectrum at hgh Reynolds numbers and small Rossby numbers. The dea of TNS s to try to model the energy transfer from large (non-truncated) to small-scale (truncated) turbulence by the estmaton method. The new model presented here s derent from thers n that we formulate the eddy-vscosty model wth the correct turbulent energy transfer from the NaverStokes equaton wthout any assumptons on the velocty uctuatons, wth the excepton of homogenety. Moreover, we wll gve more sgncant and crtcal statstcal propertes of rotatng turbulence, such as the skewness of the velocty dervatve, whch was not gven n Yang's paper. The numercal smulaton of homogeneous rotatng turbulence s performed n a rotatng frame by the pseudo-spectrum method wth rotaton n the x 3 drecton. The ntal turbulence eld s generated by the method proposed by Rogallo [54] wth a von-karman spectrum. The computatonal doman s a rectangular box whch s four tmes longer n the rotatng drecton than n the horzontal drecton, snce the turbulence scale s ncreasng greatly n the rotatng drecton. A fourth-order RungeKutta ntegraton s used for tme advancement. The tme step s set to be small enough to resolve the nertal waves. The ow Reynolds number s assumed to be nnte by prescrbng zero molecular vscosty. 59

74 Fgure 3.5: Decay of turbulent knetc energy n sotropc turbulence wth 64 3 LES. In homogeneous rotatng turbulence, the local average s taken n a sphere wth radus r and the subgrd eddy vscosty s derved from (3.64) so that ν t = ( D < 6 r 3(D r < )S r ) Sr u < 4 u < Vr. (3.66) r In practcal computaton, the local volume average s taken n a sphere crcumscrbng a cube wth twce the mesh length. The rst test case s a decayng sotropc turbulence wthout rotaton wth 64 3 grds n order to valdate the model and numercal method. The detals of the calculate condton can be found n appendx B. The smulaton s run for more than 10 6 ntal turn-over tme and the classc decayng law, t n, s found n a short tme wth n 1.25, whch s n agreement wth prevous numercal and experment results. The exponent of the decayng law approaches 2 at the nal stage of decay (Fg. 3.5), and ths result s consstent wth the expermental study by Skrbek [98], and the numercal analyss of Toul et al. [99]. To perform large-eddy smulaton of decayng rotatng turbulence at hgh Reynolds number, a purely decayng turbulence wthout rotaton was computed for sucent tme and a sold-body rotaton was then swtched on when the tme decay of large-scale knetc energy k reached a relable power law n pre-computaton. Fgure 3.6 shows 60

75 Fgure 3.6: Tme varaton of large-scale knetc energy for derent rotatng rates, grds: the tme varaton of the large-scale turbulence knetc energy for derent rotaton rates. The numercal grd s n spectral space. As the rotaton reduces the spectral energy transfer, the decay of the turbulent knetc energy s becomng slower. Ths phenomenon s pronounced when the rotatng rate s ncreasng. The crtcal examnaton of the applcablty of the new model s to check the varaton of the dervatve skewness wth tme. The dervatve skewness,.e. equaton (2.80), represents the balance between transfer of turbulent knetc energy and ts dsspaton. The drect comparson of the dervatve skewness between DNS and LES results s meanngless, but the varaton of S k /S k0 wth tme under the nuence of rotaton ndcates the capablty of the self-adjustment of a subgrd model to the rotaton eect. In decayng turbulence, Cambon [75] proposed a scalng law of S k /S k0 versus Rossby number, as follows: S k (t) S k0 = 1 (1 + 2 ( Ro Ω (t) ) 2 ) 1/2, (3.67) n whch Ro Ω = (ε/kω)/(k 2 /νε) 1/2 s based on the Taylor mcro scale and called the mcro Rossby number. In LES, the turbulent dsspaton ε and molecular vscosty ν are replaced by ε f and ν t, respectvely, and k s the large-scale turbulence knetc 61

76 Fgure 3.7: Skewness varaton versus mcro Rossby number at rotatng rate Ω = 10. energy. Ths replacement s vald so that the governng equaton of LES for homogeneous turbulence wth constant eddy vscosty s n a smlar form to DNS wth replacement of molecular vscosty by eddy vscosty. Followng the same deducton procedure as was performed by Cambon [75], t s reasonable to accept the above parameters n scale law (3.67). Comparson of the predcted results among derent subgrd models s presented n Fg The ntal mcro Rossby number s equal to 1.82 at Ω = 10 and drops to 0.72 at the end of computaton. All models produce a sudden reducton of dervatve skewness to the level predcted by (3.67) at the begnnng when rotaton s appled and the Rossby number s relatvely hgh. However, the later development ders greatly between the new model and the others. The MetasLeseur model [29] and the CZZS model cannot predct the reducton of skewness after a sudden reducton. The MetasLeseur mode predcts nearly constant level of skewness, whereas the CZZS model shows a small reducton of skewness. In contrast, the skewness predcted by the new model, whch nvolves the ansotropc transfer of knetc energy, ts the scalng law (3.67) very well, partcularly at Rossby number around 1. A ner grd wth nodes was used to check the resoluton nuence. The mcro Rossby number s nearly 2.0 n the ntal state and reaches 0.22 at the 62

77 Fgure 3.8: Skewness versus mcro Rossby number wth ner grds Lne: Cambon et al. (1997); symbols: ansotropc model. end of computaton. The result s shown n Fg. 3.8 and s much better than n the lower resoluton run. The energy spectrum E(k) s checked and presented n Fg. 3.9 for rotatng rates of 10 and 100. The ntal mcro Rossby numbers are nearly 2.22 and and approachng to and at the end of computaton. Once the rotaton s swtched on, the spectral slope shfts gradually from 5/3 to 3 as shown n Fgs. 3.9(a) and (b). In Fgs. 3.9(c) and (d), the compensated spectra for both cases are plotted at the end of computaton tme. The plateau s shown clearly n both gures Model applcatons n wall-bounded shear turbulence The proposed new model s based on the assumpton of homogeneous turbulence. The homogenety of turbulence can be accepted approxmately n the major part of wall-bounded turbulence, wth the excepton of the near-wall regon. The proposed new model has correct asymptotc behavor n the near-wall regon where ν t s proportonal to y 3 ; hence, the molecular vscosty s domnant there and the computaton of wall-bounded turbulence can be performed by LES properly wth ne normal resoluton to the wall wth the proposed new model. In fact, the prevous CZZS model s capable of predctng turbulent channel ow n farly good agreement wth DNS 63

78 (a) Ω = 10 (b) Ω = 100 (c) compensated Ω = 10 (d) compensated Ω = 100 Fgure 3.9: Evoluton of energy spectrum wth grds

79 results [38]. Here, we wll show that the proposed new model mproves the predcton precson consderably. In plane wall-bounded turbulent ows, the local volume average s taken n a rectangular element volume. The terms of local volume average, (3.63) can be (D jn < j ) S = 1 D S 1dξ < 2 dξ 3 D 1dξ < 2 dξ 3 + D 2dξ < 1 dξ 3 S + 23 S 23 S + 13 D 2dξ < 1 dξ 3 + D 3dξ < 1 dξ 2 D 3dξ < 1 dξ 2, S 13 ( ) D < S n j = 1 ξ j S S + 23 S 13 S + 12 D < dξ 2 dξ 3 ξ 1 S 23 D < dξ 1 dξ 3 + ξ 2 S + 12 S 12 D < dξ 2 dξ 3 + ξ 1 S + 13 D < dξ 1 dξ 2 ξ 3 S 12 D < ξ 2 dξ 1 dξ 3 D < dξ 1 dξ 2, ξ 3 (3.68) n whch S = 2( x 1 x 2 + x 1 x 3 + x 2 x 3 ). (3.69) Snce the turbulence s homogeneous n the x 1 and x 3 drectons n plane wall-bounded ows (x 2 s assumed to be normal to the wall), two surface ntegrals on the element perpendcular to x 1,.e. on S + 23 and S 23, cancel each other; the same results are obtaned on S 12 + and S12. Therefore, these terms can be smpled as (D jn < j ) S = 1 D S 2dξ < 1 dξ 3 D 2dξ < 1 dξ 3, S + 13 S 13 ( ) D < S n j = 1 ξ j S S + 13 D < dξ 1 dξ 3 ξ 2 S 13 D < dξ 1 dξ 3. ξ 2 (3.70) The ntegrals (D < j n j) S and ( ) D < S n j are then denoted by (D 2 < ξ )A 13 and ( D < / ξ 2) A 13, j 65

80 respectvely. The rato of the local volume to ts surface area s equal to V S = x 1 x 2 x 3 2( x 1 x 2 + x 1 x 3 + x 2 x 3 ) x 2 = 2( x 2 / x 3 + x 2 / x 1 + 1). (3.71) It can be expressed as V/S = c y x 2 wth c y 0.5 near the wall, and c y 1/6 at the central part of the channel when we used the Gauss-Lobatto collocaton n the normal drecton. Wth smlar manpulaton of the other volume average terms, the subgrd eddy vscosty can be wrtten as ν t = (D 2 < )A γ(D 12) < V c y y ( D< ) ξ A 13 u < 4 u < V, (3.72) c y y 2 n whch y = 2 x 2 s twce the local normal mesh length and γ = du/dy s the local mean stran rate. Snce the magntude of the velocty has asymptotc estmatons at the wall n ncompressble ows, as u < 1 y, u < 2 y 2, u < 3 y, (3.73) t s easy to show that (3.72) leads to ν t y 3 n the near-wall regon, and ths s correct asymptotc behavor for eddy vscosty. Note that the thrd and second structure functons are D < 2 and D< 12 n the numerator of (3.72). Ths means that the knetc energy s transferred by normal velocty uctuatons whch s a correct mechansm n the near-wall regon. The numercal method used n smulatng turbulent plane Couette ow and channel ow s a pseudospectral method wth Fourer decomposton n the x 1 and x 3 drectons and Chebyshev polynomal n the normal drecton snce the turbulence s homogeneous n the x 1 and x 3 drectons. The detals of the numercal method s descrbed n appendx The results of plane Couette ow The plane Couette ow s a good test case of the subgrd eddy-vscosty model for the homogeneous shear ow, snce t has almost constant shear n the major part of the ow, apart from the near-wall regon, whch s a thn layer at hgh Reynolds 66

81 (a) Y + scaled by wall unts (b) Y scaled by H Fgure 3.10: Mean velocty prole for plane Couette ow wth grds number. The ow Reynolds number s dened as UH/ν n whch U s the movng speed of the upper plate and Re = 3200 s accepted n the numercal smulaton whch s equvalent to a Reynolds number of n Kawamura's DNS results [100]. The grd ponts are n the streamwse, normal and spanwse drectons, respectvely. The predcted mean velocty prole by the proposed new model s shown n Fg together wth the DNS results by Kawamura, the Smagornsky model, the dynamc Smagornsky model and the CZZS model. In the computaton wth the Smagornsky model, we use model coecent C s = 0.08 and the van Drest dampng functon n the near-wall regon. Fgure 3.11 shows the dstrbuton of turbulent knetc energy and Reynolds stress between two plates. In the plots, both turbulent knetc energy and Reynolds stress nclude subgrd counterparts wth the correctons gven by Pope [1]. The results show that the new model s better than the others The results of plane Poseulle ow Turbulent Poseulle ow s another case for whch to examne the applcablty of the proposed model. The turbulent channel s nhomogeneous n the drecton normal to the wall; however, the mean shear rate s proportonal to 1/y + n the logarthmc layer whch s not so large, and the mean shear rate s much less above the logarthmc layer. Therefore, local homogenety would be a good approxmaton n the major part 67

82 (a) Turbulent knetc energy (b) Reynolds stress Fgure 3.11: Dstrbuton of turbulent statstcs wth grds of the channel ow. In the near-wall regon, the model has the correct asymptotc behavor as proved before, and t s expected that the proposed model s applcable n turbulent channel ows. The ow Reynolds number s dened as U m H/ν, n whch U m s the bulk velocty n the channel and t s unchanged durng the computaton so that we use the constant- ow-rate condton n numercal smulaton [101]. Two test cases are computed at Reynolds numbers of 7000 and 10000, whch are equvalent to Re τ = 395, 590, respectvely, n DNS performed by Moser et al. [92] wth constant pressure gradents. The grd ponts are n the streamwse, normal and spanwse drectons, respectvely, for Re = 7000 and for Re = Although the CZZS model predcts farly good statstcs n turbulent channel ow [38], the new model gves better results. Ths ndcates that the ncluson of ansotropc transfer of turbulent knetc energy s necessary and, ndeed, mproves the predcton. Fgure 3.12 presents the mean velocty proles n whch the predcton by the present subgrd model ts the DNS results well at both Reynolds numbers, and s better than prevous CZZS model. Fgure 3.13 shows the dstrbuton of turbulent knetc energy n whch the correcton of subgrd counterpart s added. The mprovement of the present new model s evdent; n partcular, the locaton of the peak s close to that for the DNS results. Fgure 3.14 shows the dstrbuton of Reynolds 68

83 (a) for Re = 7000 (b) for Re = Fgure 3.12: A Posteror results of mean velocty prole n channel Poseulle ow. stress n whch the correcton s also added. All results show that the proposed model s the best Dscusson The present model s establshed based on the homogenety of turbulence, and s then appled to non-unform shear turbulence. The extenson of the homogeneous model,.e. constant subgrd eddy vscosty, to non-unform shear turbulence s reasonable when the ow can be consdered to be locally homogeneous, and the eddy vscosty s accepted as locally constant. In a large part of the wall shear layer, the mean shear rates are small so that n the logarthm layer the mean shear rate s proportonal to 1/y + and t s much smaller above the logarthm layer. Therefore, the dervatve of the mean shear rate s small and the local homogenety can be acceptable. In the near-wall regon, the proposed new model has correct asymptotc behavor,.e. ν t y 3, and t can be used n the wall shear layer wthout any dampng functons or wall models. The good predcted results from the present model n plane Couette and channel ows ndcate that local constant subgrd eddy vscosty, whch s a functon of the normal dstance to the wall, s acceptable, at least for the attached wall shear turbulence. In rotatng turbulence, the transformaton property of the subgrd stress model should be concerned between the nertal frame and the rotatng frame [102]. The 69

84 (a) for Re = 7000 (b) for Re = Fgure 3.13: A Posteror results of turbulent knetc energy n channel Poseulle ow. (a) for Re = 7000 (b) for Re = Fgure 3.14: A Posteror results of Reynolds stress n channel Poseulle ow. 70

85 Fgure 3.15: Comparson of energy spectrum between DNS ( ) and LES present model s establshed for homogeneous rotatng ow n a rotatng frame by use of an sotropc lter and t can be proved that the generalzed Kolmogorov equatons for resolved scale turbulence of the homogeneous uctuatng moton are the same n both nertal and rotatonal frames. Therefore, the present subgrd-stress model satses the transformaton property. In practce, some errors mght be ntroduced by any slghtly non-sotropc operatons n numercal computaton, but the errors are expected to be small, at least smaller than the modelng errors. To make sure that the nuence of the practcal numercal method on LES results s neglgble, we compare the LES wth DNS results at lower Reynolds number Re λ = 50 and rotatng rate at 10 rad s 1. The comparson of energy spectrum s shown n Fg n whch DNS s performed wth grd number and LES s computed wth coarse grds. The rotaton s swtched on at t = 2k 0 /ε 0 = 2 and the energy spectra shown n Fg are at tme t = 2k 0 /ε 0 = 7 and 14, respectvely. The results of the LES are n good agreement wth DNS. In practcal computaton, the ntal condton of velocty uctuatons s mportant n DNS and LES. For the new model, we need approxmately correct ntal structure functons, n partcular (D j < n j) S. Numercal computaton wth an mproper ntal uctuatng eld s bound to fal. There s no problem for homogeneous turbulence, snce the ntal structure functon s approxmately correct f we use some well-known 71

86 spectrum as the ntal condton, e.g. Comte-Bellot spectrum or von-karman spectrum. Durng the tme advancement, the spectrum wll automatcally evolve nto the correct spectrum. For the wall-bounded turbulence, we have already checked from DNS and LES results that the skewness of the velocty ncrement, or the thrd-order structure functon, s negatve across the channel (see secton 2.4.5). Varous ways can be appled to satsfy ths condton n practcal computaton. One possble way s to use the data bank of lower-reynolds-number cases as the ntal condton for hgher- Reynolds-ows n a channel wth a non-dmensonal correcton. Ths s what we have done n test cases for turbulent plane Couette and channel ows. Another way s to start the computaton by use of an easly accessble subgrd model, for nstance the Smagornsky model, and the new model wll be swtched on when the turbulent ow becomes nearly fully developed. We have tested ths method and obtaned results as good as by the rst method. There may be other ways n practce, snce the ntal condton for LES, also n DNS, s a techncal ssue. In numercal computaton, the length of the local average volume, equvalent to the lter length, s equal to twce the mesh lengths h n the test cases. It has been proved that /h = 2 s approxmately optmum n numercal errors see [1, 103]. For homogeneous rotatng turbulence, the grd resoluton depends on the turbulent Rossby number rather than on the Reynolds number. For nstance, the grds are adequate for Ro ω < 0.5 whereas grds are margnally adequate for Ro ω 0.2 (Fgs. 3.7 and 3.8). As far as the wall turbulent shear ows are concerned, hgher spatal resoluton s requred n the wall-normal drecton than n the horzontal drectons for adequate smulaton of the near-wall behavors wthout wall model. For nstance, 64 non-unform grd ponts are enough for channel ow at Re = 7000, whereas at least 96 non-unform grd ponts are requred at Re = The requrement of resoluton n streamwse and spanwse drectons,.e. n homogeneous drectons, s not as serous as n normal drecton, 32 unform grd ponts are enough n LES, whereas at least 256 grd ponts are requred n DNS for Re = 7000 to To check the eectveness of the SGS stress n the computaton, the comparson of the the predcted results between the new model and no model has been made (Fgs and 3.17). The devaton of no model results s obvous 72

87 (a) mean velocty proles (b) Reynolds stress proles Fgure 3.16: Comparson of the statstcal propertes of turbulent plane Couette ow between the new model and no model, Re = 3200, grd from the DNS results n the mean velocty proles and the devaton s great n the Reynolds stress. 73

88 (a) mean velocty proles (b) Reynolds stress proles Fgure 3.17: Comparson of the statstcal propertes of turbulent plane Poseulle ow between the new model and no model, Re = 7000, grd

89 Chapter 4 Improved velocty ncrement model (IVI) In the last chapter, the eddy-vscosty models assume that the SGS stress tensor s algned wth the stran rate tensor. A less classcal approach conssts n modelng the SGS stress tensor on the bass of a tensor whch possesses a hgher level of correlaton wth τ < j than the stran rate tensor S< j. Such models are developed on the bass of scale nvarance propertes of turbulent ows [7]. The orgnal scale smlarty formulaton s related to the Leonard stress tensor L j = ( ) < u u < u< < j u < j. The only lmtaton of ths approach les n the necessty of an explct lterng procedure. Indeed, the use of a specc lter yelds specc eects of the SGS model. Ths s demonstrated n A Pror tests wth varous local lters [18, 104, 105]. The reconstructon of L j based on Taylor expansons yelds the gradent duson model [5, 106, 107]. Another scale smlarty formulaton s related to the Reynolds type stress tensor R j = u < u < j based on the statstcal velocty uctuaton u = u u. One of the recent attempts s consderng the velocty ncrement n SGS modelng. Metas and Leseur [29] have transposed the spectral eddy-vscosty concept to physcal space on the bass of the second order velocty structure functon whch represents the energy of the ow stored at the mesh sze (.e. the lter scale n spectral space). The dmenson of an eddy-vscosty requres two scales, a lengthscale ξ = and a velocty scale δu < 1 = u 1 (x 1 + ξ) u 1 (x 1 ). However, the sprt of ths model s based on eddy-vscosty assumpton, whch mght be not good n some areas. From spectral analyss, there s weak energy backscatter n homogeneous sotropc turbulence [9]. Ths phenomenon s obvous n some complex turbulence [2426]. Brun proposed an 75

90 ncrement model (VI), n whch the subgrd stress n formulated by usng the ltered velocty ncrement [6]: τ < j = fq j, Q j = δu < δu < j. (4.1) Ths model has good performance of tensor correlaton. Wth ths model, the forward and backward scatters could be both smulated. However, some arbtrarness remans n the model, for example the model coecent was not determned physcally. In ths chapter, we further dscuss the tensor formulatons of the ltered velocty ncrement Q j. Then KEF s appled to determne the coecent values, n sotropc turbulence and shear ow respectvely. The mproved SGS models are tested n homogeneous sotropc turbulence and ansotropc channel ow. The correspondng work has been publshed n Physcs of Fluds [42]. 4.1 Formulatons of the ltered velocty ncrement tensor Brun has dened a formulaton of ltered velocty ncrement, that δu < ( x) = 3 k=1 [ u < Thus the tensor Q j s dened as ( x + 12 ) r k e k u < ( x 12 r k e k )]. (4.2) Q j = δu < δu < j = [ 3 [ u < ( x + 12 ) r k e k u < ( x 1 )] ] [ 3 [ 2 r k e k u < j ( x + 12 ) r k e k u < j ( x 1 2 r k e k )] ], k=1 k=1 (4.3) where r s a gven dstance vector. In fact, ths denton concludes the relatons among four ponts: x, x + 1/2r 1 e 1, x + 1/2r 2 e 2 and x + 1/2r 3 e 3. In numercal applcatons, f the lters and grds are homogeneous,.e. 1 = 2 = 3 = h n the three axs drectons, Brun suggested that the dstance vector s taken as 3 r = 2h e k, (4.4) k=1 76

91 and the ltered ncrement velocty reads as δu < = 3 [u < (x k + h) u < (x k h)]. (4.5) k=1 When h s small, Taylor expanson s employed, and ths model could be wrtten τ < j = fh k h l u < x k u < j x l. (4.6) Comparng wth Leonard's subgrd model [5]: τ < j = C n h 2 u< x k u < j x k, (4.7) Brun's ncrement model contans more terms. It s n agreement wth the results of Taylor expanson of lters. Consderng a three-dmensonal low-pass lter wth the characterstcs [ x k, a (k), b (kl) ], where x k s the lter wdth, a (k) x k and b (kl) x k x l are the rst and second moments of lter, respectvely, we could obtan an expanson of the ltered varable: h < h 2 h ( x) = h( x) + a (k) x k ( x) + b (kl) x k x l ( x) + o( x 2 x k x k x k), (4.8) l where o( x 2 k ) s small compared wth x2 k. In order to further smplfy Brun's formulaton, equaton (4.6) s wrtten on a normal component specally, n sotropc turbulence: ( u τ 11 < = fh 2 < 1 u < 1 u < u < 1 u < u < 1 x 1 x 1 x 2 x 2 x 1 x 2 ) u < u < 1. x 2 x 3 (4.9) In homogeneous sotropc turbulence, both the last two terms equal zero. Besdes, the followng ncompressble condton s satsed: u < 1 u < 1 = 2 x 2 x 2 u < 1 x 1 u < 1 x 1. (4.10) It means that the rght hand sde of equaton (4.9) could be treated by smply consder u < the rst term 1 u < 1. Ths smplcaton also agrees wth the classcal eddyvscosty assumpton, that the 11 component of subgrd stress could be x 1 x 1 formulated by only the S < 11 component of resolved stran rate. Therefore, the smpled tensor of velocty ncrement n 11 drecton reads [ Q 11 (x 1, h) = u < 1 (x ) h u < 1 ( x h )] 2. (4.11) 77

92 In order to further smplfy the calculatons, another formulaton s wrtten at a gven dstance : Q 11 (x 1, ) = 1 2 [δu< 1 (x 1, )δu < 1 (x 1, ) + δu < 1 (x 1, )δu < 1 (x 1, )], (4.12) where δ u( x, ξ) = u( x + ξ) u( x). Ths nal formulaton s used n ths thess, and the tensor s dened as Q j ( x, ) = 1 [ (u < 2 ( x + e ) u < ( x)) ( u < j ( x + e j ) u < j ( x) ) + (u < ( x) u < ( x e )) ( u < j ( x) u < j ( x e j ) )]. (4.13) Although derent from Brun's denton, t also satses the tensor symmetry Q j ( x, ) = Q j ( x, ), Q j = Q j. (4.14) Comparng wth Brun's denton, t s smpled by omttng the transverse components. The man problem of ths smplcaton s that the subgrd stress vanshes when the ow s locally characterzed by a pure shear ow u < 1 (x 2 ) somewhere. However, n ths stuaton, the eect of dsspaton are the same between dentons (4.14) and (4.3), both of them yeld the zero subgrd dsspaton τ j < S< j = 0. Ths behavor can also be explctly acheved n the Clark model [108]. In the KEF formulaton (2.21), all terms can be represented by the longtude components. Thus t could be approprate to characterze the energy transfer usng the longtude components, n homogeneous sotropc turbulence. The formulaton of the velocty ncrement model could be wrtten as τ < j ( x, ) = C f ( )Q j ( x, ), (4.15) where C f s the dynamc model coecent, dependng on another dstance. In Brun's paper [6], he assumed =. However, n ths thess we analyze the relatons, whch could brng good result n nertal subrange. 78

93 4.2 Improved velocty ncrement model The KEF formulaton n homogeneous sotropc turbulence s dscussed n secton 2.2. We only concern the longtudnal components, thus from equaton (2.21), (4.11), (4.15), and sotropy assumpton, the subgrd dsspaton could be denoted as: 4 5 ε fξ = 6C f Q 11 ( )S 11 < ξ = 6C f δu < 1 ( )δu < 1 ( ) u< 1 ξ x 1 = 6C f δu < 1 ( )δu < 1 ( ) δu< 1 ( ) ξ = 2C f ξ D< lll (r) r r=, (4.16) and the the subgrd transfer term reads: 6T l,ll (ξ) = 6C f u < 1 (x 1 )Q 11 (x 1 + ξ, ). (4.17) Therefore, equaton (2.21) could be wrtten: 2C f ξ D< lll (r) r = D lll < (ξ) 6C f u < 1 (x 1 )Q 11 (x 1 + ξ, ), (4.18) r= and the model coecent s: C f = 2ξ D< lll (r) r D lll < (ξ) + 6 u < 1 (x 1 )Q 11 (x 1 + ξ, ) r= (4.19) Three derent scales mpled n ths result should be clared: the physcal quanttes are ltered at a lter sze ; VI model assumes C f and Q j at a dstance ; and the equaton tself s based on a dstance ξ. In the followng parts, the three scales wll be analyzed prmarly n an deal hgh Reynolds number turbulence, and then n a real ow Model analyss n hgh Reynolds number turbulence In secton 2.6, the scalng law of ltered velocty s dscussed. If the scalng law s satsed, the rst term of the denomnator n equaton (4.19) could be smpled 79

94 when and s n nertal subrange: 2ξ D< lll (r) = 2ξ r D< lll ( ). (4.20) r= If ξ s also n nertal subrange, we could obtan: 2ξ D< lll (r) r = 2D lll < (ξ). (4.21) r= The second term of the denomnator n equaton (4.19) corresponds to the transfer term of KEF. Accordng to Meneveau's analyss [17], t tends to zero when ξ s large. In fact, t s a correlaton functon between velocty u < 1 and velocty ncrement Q 11 at dstance, thus t tends to zero when ξ s large. From equaton (4.11) and (4.17), we could also easly evaluate the magntude when ξ: u < 1 (x 1 )Q 11 (x 1 + ξ, ) δu < 1 (ξ)δu < 1 ( ) u < 1 ( ) D lll < (ξ). (4.22) Thus the transfer term n equaton (4.19) could be neglected, and the coecent s C f = D< lll (ξ) 2D < lll (ξ) = 1 2. (4.23) The value does not depend on ξ. It agrees wth the model assumpton (4.15), where C f s only a functon of and. Notce that ths result s satsed only when the nertal subrange s wde enough, and and ξ are both n nertal subrange. Besdes, n order to neglect the molecular terms n orgnal KEF (Cu et al. [38]), the lter sze of LES should be much larger than the dsspaton scale η. Therefore, we could wrte the combned mult-scale relaton: where L s the ntegrated scale. η ξ L, (4.24) Model analyss n moderate Reynolds number turbulence In the numercal smulaton of LES, the Reynolds number mght not be hgh enough, and the nertal subrange s not wde enough. Equaton (4.24) s dcult to be satsed, thus we x the scales as = = ξ, and smplfy the form of VI model n another way. The subgrd energy transfer T l,ll could be represented by usng 80

95 velocty ncrement. In homogeneous turbulence, t s only a functon of ξ, and does not depend on the locaton x 1. It could be represented as: T l,ll (ξ) = u < 1 (x 1 )τ < 11(x 1 + ξ) =C f u < 1 (x 1 )Q 11 (x 1, 2ξ) = 1 2 C f [ u <3 1 2 u < 1 (x 1 )u < 1 (x 1 )u < 1 (x 1 + ξ) + 2 u < 1 (x 1 )u < 1 (x 1 + ξ)u < 1 (x 1 + ξ) + u < 1 (x 1 )u < 1 (x 1 + 2ξ)u < 1 (x 1 + 2ξ) 2 u < 1 (x 1 )u < 1 (x 1 + ξ)u < 1 (x 1 + 2ξ) ]. (4.25) The thrd order moment of velocty u <3 1 = 0. The last term n the rght hand sde of equaton (4.25) could be proved to be zero n sotropc turbulence: u < 1 (x 1 )u < 1 (x 1 + ξ)u < 1 (x 1 + 2ξ) = u < 1 (x 1 ξ)u < 1 (x 1 )u < 1 (x 1 + ξ) = u < 1 (x 1 + ξ)u < 1 (x 1 )u < 1 (x 1 ξ) (4.26) = u < 1 (x 1 )u < 1 (x 1 + ξ)u < 1 (x 1 + 2ξ). Dene the thrd order correlatons R l,ll (2ξ) = u < 1 (x 1 )u < 1 (x 1 + 2ξ)u < 1 (x 1 + 2ξ), R ll,l (2ξ) = u < 1 (x 1 )u < 1 (x 1 )u < 1 (x 1 + 2ξ). (4.27) In sotropc turbulence R l,ll (2ξ) = R ll,l (2ξ). The subgrd energy transfer T l,ll reads: T l,ll (ξ) = 2C f R ll,l (ξ) 1 2 C fr ll,l (2ξ). (4.28) Snce n sotropc turbulence D < lll transfer = 6R ll,l, the term of two-pont subgrd energy T l,ll (ξ) = 1 3 C fd lll < (ξ) 1 12 C fd lll < (2ξ). (4.29) It s nterestng to notce that the energy transfer has a lnear relaton wth the thrd order structure functon. In equaton (4.16) and (4.29) there mght be energy backscatter because of the thrd order structure functon D lll <, whch wll be further analyzed n the next secton. 81

96 However, notce that n CZZS model, the eddy vscosty subgrd dsspaton s always postve. The subgrd energy transfer depends on the gradent of the second order structure functon, that s: From the classcal scalng law D< ll ξ T l,ll > 0 s purely dsspatve. D ll < T l,ll (ξ) = ν (ξ) t. (4.30) ξ > 0 and the eddy vscosty ν t > 0, t mples From equaton (2.21), (4.16) and (4.29), the coecent C f could be obtaned: C f = 4ξ D< lll (r) r 2D lll < (ξ). (4.31) 4D lll < (ξ) D< lll (2ξ) r=ξ In applcatons, f the scalng law D lll < (ξ) ξn s satsed, we could obtan: D lll < (r) r = n D< lll (ξ), (4.32) r=ξ ξ nally the dynamc form of IVI model could be C f = 2D lll < (ξ) 4(n 1)D lll < (ξ) (4.33) D< lll (2ξ). When,, ξ tend to zero, the scalng exponent n 3. Also we have D < lll (2ξ) = 8D lll < (ξ), thus the denomnator tends to zero. It agrees wth the concluson of Meneveau [17] that T l,ll (ξ) = τ < 11S < 11 ξ n small ξ. However, notce that n depends on ξ, thus n s not a constant, and we could not have the relaton D lll < (2ξ) = 2n D lll < (ξ). In real practce, the scalng exponent n could be calculated by any numercal derence methods. However, we suggest to x n = 2.5 n real practces, snce t s smple and numercal stable, and s vered n A Pror cases, whch wll be dscussed n Sec When = = ξ, the deal nertal subrange condton (4.24) s not satsed, thus we should use the dynamc model form (4.33) nstead of the constant model form (4.23). However, the low-cost constant model form s more convenent n calculaton. In order to check whether the constant model form could be appled approxmately n ths case, derent model forms are then vered by A Pror and A Posteror tests n the next secton. 82

97 4.3 Numercal vercatons A Pror analyss n homogeneous sotropc turbulence Two DNS cases of homogeneous sotropc turbulence are used for A Pror test. In these two cases, spectral method s appled, and a determnstc forcng method s employed to smulate a statstcally statonary turbulence. The computaton doman has grd. The grd sze s denoted as h. Reynolds numbers Re λ are 50 and 70 respectvely. The compensate energy spectrums of DNS cases were shown n Fg. 2.2, where the plateaus represent the nertal subrange n spectral space. The correspondng wave number of the plateaus s about 0.1 < k c η < 0.3. The relevant lter sze s 10 < /η < 30. In addton, the ntegrated scale L 70η n the two cases. Thus we could reasonably consder that η = ξ L, whch could mnmze the model error mentoned n secton The plateau n the case Re λ = 70 s wder than Re λ = 50. In homogeneous sotropc turbulence, the exact subgrd stress s calculated wth derent cut-o wave numbers. The coecent of VI model s calculated as C f = τ < 11S < 11 Q 11 S < 11. (4.34) The dynamc value of IVI model (4.33) s compared wth the exact value (4.34) n Fg. 4.1, where the scalng exponent n s calculated by usng the rst-order central derence method. It shows that the exact coecent value ncreases wth and ξ, and t s about 1/2 when 10 < ξ/η 18, whch s a small regon of nertal subrange. regon. Moreover, the dynamc model value s also n good agreement n ths Therefore, n a practcal LES case, f the lter sze and the two-pont dstance ξ are xed n ths regon, the constant form (4.23) of IVI model mght be appled to obtan the low cost n calculaton. When ξ s small, the constant form s n dsagreement. It stems from the eect of molecular vscosty, whch s neglected n KEF. When ξ s too large, the error s also obvous A Pror analyss on sotropy n homogeneous shear ow As analyzed n secton 2.4.4, the sotropy of thrd-order structure functon mght not be satsed n homogeneous shear turbulence. Thus, the ansotropy of C f should 83

98 Fgure 4.1: Comparson between exact value and dynamc model value of C f, aganst derent lter szes and two-pont dstances = ξ, n homogeneous sotropc turbulence. Sold lne wth symbols: Re λ = 50; dashed lne wth symbols: Re λ = 70. The horzontal lne s the theoretcal value. be vered. The DNS database s the center regon of Couette ow, whch was descrbed n secton The dynamc coecent C f s calculated wth derent twopont drecton, and derent two-pont dstance length. Snce t s not easy to change lter sze, we xed t as the grd sze,.e. = h, only to nvestgate the ansotropy among three drectons. Here the scale condton = ξ s not satsed, thus the results could only show a trend. As shown n Fg. 4.2, the values are not the same among derent drectons. Therefore, n applcatons of shear ow, the ansotropc propertes should be notced. In the followng cases usng dynamc model form, we smply set the coecent value as the average of three axs drectons A Pror analyss on wall behavor n wall-bounded ow KEF s based on homogeneous sotropc turbulence, but n wall-bounded shear ow, the local sotropy could also be satsed by consderng the slow parts [38, 48]. In a local regon, the two-pont velocty ncrement s consdered as a slow part for subgrd modelng, whch provdes a good near-wall property [70]. A DNS case of channel ow s used for A Pror test. The pseudo-spectral method s appled. The Reynolds number Re H = 7000, based on the bulk velocty U m and channel half-wdth 84

99 2 1.5 x 1 drecton x 2 drecton x 3 drecton C f / ξ/η Fgure 4.2: Dynamc model value of C f, aganst derent two-pont dstance, n homogeneous shear turbulence. Flter sze s xed as grd sze. The horzontal lne s the theoretcal value. H (Re τ 395). The grd number s The computaton doman s 4πH, 2H and 2πH n streamwse, normal and spanwse drectons. The DNS grd sze s denoted by h. The lter sze of LES and the two-pont dstance n VI model are the same: = ξ = 2h. Ths sze s compared wth the Kolmogorov scale and the ntegrated scale respectvely, whch are shown n Fg Accordng to the pror result n homogeneous sotropc turbulence, ths two-pont dstance, whch satses η ξ L n most of the channel regon, could be appled to mnmze the error of two-ponter energy transfer. In IVI model, the eectve eddy vscosty ν t could be dened as τ < j = 2ν ts < j, n whch the prme means the uctuated part wthout mean ow. To compare wth other models, t could be calculated as: Q j S j < ν t = C f 2, (4.35) S < j S < j where C f s regarded as a constant 1/2. The other SGS models used for comparson are the orgnal Smagornsky model (SM) wth model constant C s = 0.1, the Germano Dynamc Smagornsky model (DSM), the Smagornsky model wth van Drest dampng functon n near-wall regon (SM Dampng) and the CZZS model. A Pror 85

100 Fgure 4.3: Flter sze / two-pont dstance = ξ = 2h n derent poston of normal drecton n channel ow. (a) Compared wth Kolmogorov scale. (b) Compared wth ntegrated scale. test of the magntude of eddy vscosty s done, wth the lter sze = 2h. Results are shown n Fg The orgnal Smagornsky model causes a false vscosty at the wall, whch does not tend to zero. An emprcal method for solvng ths problem s to use van Drest dampng functon. An alternatve mathematcal way s the Germano dynamc model, whch depends on the use of a test lter and a reference model. However, t doesn't t well for the emprcal locaton of peak Y + 25 [70]. In contrast, wth KEF appled n SGS models, the near-wall property s naturally satsed and the locaton of peak s well captured. The relaton between the two-pont dstance and the eectve eddy vscosty of IVI model s then analyzed. Fgure 4.5(a) shows the eectve ν t wth ξ from h to 6h, under the condton = ξ. Only when ξ 2h, the near-wall regon (about 10 < Y + < 50) has a large eddy vscosty, whch satses the emprcal peak locaton. In fact, when ξ > 2h, n ths regon we could not consder ξ L approxmately (see Fg. 4.3(b) ). It means that the two-pont energy transfer could only be satsed n a proper two-pont dstance. Addtonally, the near-wall Y +3 law s satsed well, whch s shown n Fg. 4.5(b) A Posteror tests n homogeneous sotropc turbulence Two A Posteror tests are tred n ths secton. Frstly, we employ IVI model n LES of homogeneous sotropc turbulence. Our LES cases correspond to the decay 86

101 SM DSM SM Dampng CZZS model constant IVI model 8E-05 ν t 6E-05 4E-05 2E Y + Fgure 4.4: A Pror test of the eddy vscosty by usng derent SGS models n channel ow, wth lter sze = 2h. Symbols: Smagornsky model wth or wthout derent correctons; lnes: models based on KEF. (a) (b) Fgure 4.5: A Pror eectve ν t n channel ow, by usng IVI model. Two-pont dstances ξ = h, 2h, 4h, and 6h. Flter sze = ξ. (a) Orgnal coordnate. (b) Log-log coordnate. 87

102 LES runs 64 3 LES runs 96 3 LES runs Comte-Bollot 96M Comte-Bollot 172M E(k) k Fgure 4.6: Energy spectrum n decayng sotropc turbulence. Squares: Comte- Bellot 96 M; damonds: Comte-Bellot 172 M; lnes wth trangles: 48 3 LES runs; dashed lnes: 64 3 LES runs; sold lnes: 96 3 LES runs. turbulence at low Reynolds number, smlar to the results of Comte-Bellot experment [109]. These smulatons are run on 48 3, 64 3 and 96 3 grds respectvely. The ntal eld s generated by usng Rogallo's method [54] wth randomly dstrbuted velocty phases. Fgure 4.6 shows the spectrum n comparson wth experment data (the lled symbols) at 96 and 172 M. All cases are calculated wth the constant coecent,.e., equaton (4.23), and gve satsfactory results. In these calculatons, the energy backscatter s omtted n spectral space. In applcatons, too strong backscatter wll cause the ow nstablty. The study n detal s dscussed n secton A Posteror tests n Poseulle channel turbulence Secondly, another attempt s done n channel ow. The parameters of Reynolds number, computaton doman and numercal method are the same as those n our DNS channel case. The grd numbers and subgrd models are shown n Table 4.1. The DNS case s already ntroduced n the last secton. The no model case apples the DNS method at a coarse grd ( ), whch s compared wth the LES cases to shown the model eect. Two LES cases are also executed at

103 Case Grd Model DNS case No model case Dynamc IVI model case Equaton (4.33) Constant IVI model case Equaton (4.23) Table 4.1: The parameters mplemented n A Posteror tests 22 DNS No Model 20 Dynamc IVI model 18 Constant IVI model 16 fltered DNS 7 No model Dynamc IVI model 6 Constant IVI model fltered DNS No model Dynamc IVI model 1 Constant IVI model U k gs Reynolds stress Y Y Y+ (a) (b) (c) Fgure 4.7: A Posteror statstcal results usng derent models (Table 4.1) n channel ow. (a) Mean velocty. (b) Resolved-scale turbulence knetc energy. (c) Resolved-scale Reynolds stress. grd, wth the dynamc IVI model formulaton (4.33) and the constant IVI model formulaton (4.23), respectvely. The scalng exponent s gven as n = 2.5 n Eq. (4.33). A Posteror results are shown n Fg In Fg. 4.7(a), the two SGS models both yeld good agreement wth DNS velocty prole, they are better than the no model case. In Fgs. 4.7(b,c), the DNS results are ltered to the resoluton, n order to be compared wth others at the same grd scale. Both the two LES cases show qute better performances than the no model case. There are only slght derences between these two LES cases. The dynamc IVI model case s n slghtly better agreement wth the ltered DNS results n near-wall regon. The peak locatons of the knetc energy and the Reynolds stress are a lttle better smulated n the dynamc IVI model case. However, the constant IVI model works qute well. Compared wth other works, the results are as good as other dynamc subgrd models (the same channel ow were calculated usng other models n Refs. [38,40]), and all these models are much better than the orgnal Smagornsky model, especally n near-wall regon. Therefore, the most valuable mprovement n 89

104 Fgure 4.8: Trad nteracton causng energy backscatter (R-type trad accordng to Walee's classcaton). IVI model s not the accurate dynamc formulaton, but s the constant coecent ntroduced. It has as good performance as the other dynamc subgrd models, and the constant formulaton s also greatly low-cost. In addton, n the engneerng projects of complcated geometres, t s dcult to calculate the structure functons usng spatal average, but the constant coecent value wll be approprate Energy backscatter n homogeneous sotropc turbulence Energy backscatter n homogeneous sotropc turbulence could be analyzed n spectral space. Both DIA and EDQNM theores pont out that n dstant trad nteracton, when k p, q k c, k c s the cut-o wave number, there could be energy backscatter [2, 8, 110]. Backward transfer of energy s from wave numbers p and q to k (Fg. 4.8). The backscatter s k 4 magntude at small wave numbers, and could be reasonably well approxmated be a k 4 term [30,111,112]. Analyss shows that energy backscatter only takes eect at very small wave numbers. The subgrd energy could be represented as ε f = τ < j S < j = 3 τ < 11S < 11 6 τ < 12S < 12. (4.36) The normal component could be denoted as ε norm f component s ε dev f = τ < 11S < 11, and the devatonc = τ < 12S < 12. Thus at any moment the nstant energy backscater could be obtaned by calculatng the negatve dsspaton ponts [24]: ε norm f (backscater) = mn( τ < 11S < 11, 0), ε dev f (backscater) = mn( τ < 12S < 12, 0). (4.37) 90

105 The energy backscatter values n homogeneous sotropc turbulence are shown n Fg Two DNS cases (Re λ = 50, 70) are employed n A Pror tests. The lter sze s gven as = 2h, 4h, 8h, 12h, respectvely. At small wave numbers, the k 4 law are all satsed well. However, when the wave number s large, there s also strong eect of energy backscatter. It mght cause the nstablty n LES calculaton. Ths s the reason that we have to omt the energy backscatter n numercal smulaton of homogeneous sotropc turbulence. In order to mprove ths model, the model propertes n spectral space must be further analyzed. However, they are not mentoned n ths thess Energy backscatter n Poseulle channel turbulence Further analyss s on the forward and backward ux of energy transfer n Poseulle channel turbulence. The SGS dsspaton ε f represents the energy transfer between resolved and subgrd scales [24]. The contrbutons of the components of SGS dsspaton were analyzed by Hartel et al. [26], where the nteractons between uctuatng stresses and uctuatng rates of stran are studed. From ther analyss, the SGS energy s dsspated n streamwse and normal drecton, and the backscatter of SGS energy s n spanwse drecton. Whle C f s always postve, the nondmensonalzed subgrd dsspaton tensor components ε + f j caused by uctuatons could be calculated as (wthout summaton conventon) ε + f j = τ j < S < j rms(τ j <)rms(s < j ) = Q j S j < rms(q j )rms(s < j ). (4.38) The components of SGS dsspaton are shown n Fg A Pror results are from our DNS case of channel ow, and A Posteror results are from the ne grd case 1. In these gures, postve value means SGS energy dsspaton,.e. forward scatter; negatve value represents the energy backscatter. Though the Reynolds numbers are derent (Re τ = 395 n our case and Re τ = 180 n the work of Hartel et al.), the results can be qualtatvely compared. Among the normal components, ε f 11 s the most mportant postve component whch leads to the dsspaton, whle the spanwse components ε f 33 causes a strong energy backscatter. Compared wth Hartel's analyss (see Fg. 9 of [26]), the dsagreement only occurs n the Y + < 8 range, where the slght energy backscatter of ε f 11 s not shown by our model. It mght be caused by 91

106 10 0 k k 4 ε f backscatter ε norm f Re λ =50 ε dev f Re λ =50 ε norm f Re λ =70 ε dev f Re λ =70 ε f backscatter ε norm f Re λ =50 ε dev f Re λ =50 ε norm f Re λ =70 ε dev f Re λ =70 k k (a) = 2h (b) = 4h 10 1 ε norm f Re λ =50 ε dev f Re λ =50 ε norm f Re λ =70 ε dev f Re λ =70 ε norm f Re λ =50 ε dev f Re λ =50 ε norm f Re λ =70 ε dev f Re λ = ε f backscatter 10 0 k 4 ε f backscatter k k k (c) = 8h (d) = 12h Fgure 4.9: A Pror results of energy backscatter, n homogeneous sotropc turbulence. Flter szes = 2h, 4h, 8h, 12h. Sold lne wth symbols: Re λ = 50; dashed lne wth symbols: Re λ = 70. Squares: normal component; trangles: devatonc component. 92

107 ε f11 + ε f22 + ε f33 + ε f11 + ε f22 + ε f33 + A Pror A Posteror ε f12 + ε f13 + ε f23 + ε f12 + ε f13 + ε f23 + A Pror A Posteror Y Y + (a) (b) Fgure 4.10: Contrbutons of components for subgrd energy dsspaton, normalzed by usng equaton (4.38). Lnes: A Pror results; symbols: A Posteror results. (a) Normal components: ε + f 11, ε+ f 22, and ε+ f 33. (b) Devatonc components: ε+ f 12, ε+ f 13, and ε + f 23. the strong nhomogenety n ths regon. Among the devatonc components, the most specal component s ε f 12, whch has the backscatter n the regon 10 < Y + < 20 and dsspates n the other ranges. Ths result s n good agreement wth Hartel's analyss, ncludng the peak locatons. Wth each component, both A Pror and A Posteror results assert the smlar SGS dsspaton. The backscatter propertes of IVI model could stem from the thrd order structure functon. From equaton (4.16) and (4.29), the two-pont energy dsspaton and subgrd transfer are represented by the thrd order structure functon, whch could have derent values, ether negatve or postve, n derent drectons and n derent regons. 4.4 Dscusson The most mportant mprovement n IVI model s to determne the model coecent C f by employng the two-ponter energy transfer equaton,.e. KEF. The dynamc procedure proposed by Brun s based on Taylor expanson, t can be wrtten as: C f = 1 α 2 L kk (α) Q, (4.39) n whch L j (α) = û< u< j û< û< j s the Leonard stress, s the test lter at sze α. 93

108 Another method ntroduced by Brun s to employ the Germano dentty L j (α) = C f (Q j (, α ) Q j (, )), by consderng C f be then represented as: C f = as a constant. The coecent can L (α) Q kk (, α ) Q kk (, ). (4.40) Although these methods are ntroduced wth denton (4.3), they can also be appled wth our denton (4.14). A Pror tests usng derent methods are shown n Fg The lter sze s selected as = 2h,.e. the LES mesh has grds. The values usng Taylor expanson are much smaller then other methods, snce Taylor expanson can not been employed n large two-pont dstance. Germano dentty yelds C f 0.5 n the mddle of the channel, whch s n agreement wth the dynamc IVI model when n = 2.5. In the near wall regon, derences exst snce the ansotropc eect s strong. However, the constant value 0.5 s also acceptably satsed. In addton, n = 3 yelds much smaller value than 0.5, and n = 2.5 s n better agreement. That s the reason we propose n = 2.5 n numercal calculatons when utlzng the dynamc formulaton (4.33). A Posteror results n channel ow are shown n Fg. 4.12, n whch the model coecents are calculated by employng derent methods. The parameters n ths case are the same as descrbed n the last secton. From the gures, both Germano dentty and the constant IVI model are better than the method based on Taylor expanson. Both of them are n good agreement wth the DNS results, the constant coecent s as good as the dynamc coecent usng Germano dentty. However, the constant value s much less lost and easy to be mplemented. Notce that ths IVI model s tme-reversble snce t remans when we change all u < nto u <. It s not lke some other models, for nstance the Smagornsky model, whch s tme-nonreversable. Ths ssue s partcularly dscussed n the appendx E. They can represent derent propertes of turbulence, dependng what we want to smulate n LES. 94

109 0.8 α=2 α=4 α=6 α= α=2 α=4 α=6 α=8 0.5 C f 0.4 C f Y Y + (a) (b) n=2.5 n=3 0.5 C f Y+ (c) Fgure 4.11: A Pror coecent values usng derent dynamc methods, n channel ow. Flter sze = 2h. (a) Based on Taylor expanson (4.39), wth derent test lters. (b) Based on Germano dentty (4.40), wth derent test lters. (c) Dynamc formulaton of IVI model (4.31), gvng the scalng exponent n = 2.5 and 3. The dashed lne s the constant value n IVI model. 95

110 22 DNS Taylor expanson 20 Germano dentty 18 Constant IVI model fltered DNS Taylor expanson Germano dentty Constant IVI model U k gs Y Y+ (a) (b) 1.2 fltered DNS Taylor expanson Germano dentty 1 Constant IVI model Reynolds stress Y+ (c) Fgure 4.12: A Posteror statstcal results of channel ow, the model coecent s determned by derent methods. (a) Mean velocty. (b) Resolved-scale turbulence knetc energy. (c) Resolved-scale Reynolds stress. 96

111 Chapter 5 Applyng KEF on ansotropc eddy-dusvty models The understandng of small-scale uctuatons n scalar elds, such as temperature, pollutant densty, chemcal or bologcal speces concentraton, advected by turbulent ow s of great nterest n both theoretcal and practcal domans [113]. Ther dynamcal propertes have been a subject of very accurate expermental nvestgatons carred out n the last few years both n the atmosphere [114,115], n the ocean [116] and for laboratory turbulent ow [117,118]. There are many specal behavors n passve scalar, such as the obvous ansotropy [64, 119, 120]. Many numercal attempts have been performed [121124]. However, each subgrd model has ts lmtaton n calculaton. In chapter 3, we have dscussed the applcatons of KEF on subgrd eddy-vscosty modelng. Smlarly, the KEF formulaton of passve scalar (.e. the Yaglom equaton) n homogeneous sotropc turbulence could be derved (see appendx). Zhou ntroduced a homogeneous sotropc eddy-dusvty model n hs thess [125]. The eddy dusvty assumpton reads θ < τ θj = κ t, (5.1) where the subgrd scalar ux s dened as τ θj = (u j θ) < u < j θ <. (5.2) Note that n ths chapter, we utlze τ θj and τ j nstead of τ θj < and τ j <, but n fact they both represent the same subgrd quanttes. 97

112 From the homogeneous sotropc formula of KEF (see appendx A), the eddydusvty model could be nally represented as κ t = 3D < lθθ θ < θ <. (5.3) 4 ξ 6 D< θθ ξ Ths model has been tested n the LES of channel Couette turbulence, but the results are not qute satsfactory. In ths chapter, we do not menton much about ths homogeneous sotropc model. Instead, we attempt to study the physcal mechansm between mean and uctuaton parts, n nhomogeneous ansotropc scalar turbulence. The scalar varance equaton and the scalar ux equaton are decomposed nto rapd and slow parts. Furthermore, the subgrd transfer of scalar ux contan the nteractons between resolved velocty and subgrd scalar, and between subgrd velocty and resolved scalar. DNS database s employed to show A Pror results. Fnally, KEF s employed n ansotropc model, t s then vered by comparng wth the DNS results. 5.1 Basc equatons n nhomogeneous ansotropc scalar turbulence In order to show the exact nteractons between velocty and scalar n subgrd transfer, t s necessary to derve the basc equatons of nhomogeneous ansotropc scalar turbulence. The lter and the ensemble average operators are dened as the same as n secton 2.1. The governng equatons of scalar energy and scalar ux are descrbed, respectvely Governng equatons of scalar varance For a passve scalar θ, the moton equaton could be wrtten as Takng ensemble average, the equaton becomes θ t + u θ j = κ 2 θ. (5.4) θ t + u j θ = κ 2 θ u x jθ. (5.5) j 98

113 (5.4) (5.5), we could wrte the equaton for uctuated scalar: θ t + u j θ + u θ j = κ 2 θ ( u θ u jθ ). (5.6) j Multply θ to each sde of equaton (5.5), we could obtan the transfer equaton of resolved scalar varance θ 2 t + u j θ2 = 2κ θ 2 θ ( u ) j θ 2. (5.7) Because (θ 2 ) = 2 θ θ + θ 2, the last term could be expressed as ( u ) j θ 2 = u θ 2 j 2 u ( θ θ ) j = 2 u jθ θ 2 u x jθ θ 2 u j θ θ j = 2 u jθ θ 2 θ u jθ (5.8) = 2 u jθ θ u θ θ 2 θ u j θ j From equaton (5.6), the transfer equaton of uctuated scalar varance can be wrtten as θ 2 t + u j θ 2 = 2 u jθ θ + 2κ θ 2 θ u x jθ θ. (5.9) j Multply θ to each sde of (5.5), or substract equaton (5.7) wth (5.9), we could obtan θ 2 + u j θ 2 t We can also wrte the last term as = 2κ θ 2 θ 2 θ u j θ. (5.10) 2 θ u jθ = 2 u jθ θ 2 ( θ u j θ ) (5.11) Now consder the energy transfer n LES. The governng equaton for the resolved scalar could be wrtten as θ < t + u< j θ < = κ 2 θ < τ θj, (5.12) 99

114 where τ θj < s the subgrd-scale scalar ux dened as τ θj = (u j θ) < u < j θ <. (5.13) The energy transfer of resolved scalar uctuaton could then be wrtten as 1 θ < t 2 u< j θ <2 = u < j θ < θ< < θ < + κ θ θ τ < θj θ < ( u < θ <). j (5.14) The term u < j θ < θ< can be regarded as an exchange term between mean and uctuaton scalar varances. And for the last two terms, we have θ < θ < τ θj = ( u < θ <) = j τ θj u < j θ θ < < θ < ( τ θj θ <) ( u θ θ ) j (5.15) Smlarly, we could wrte the governng equaton for resolved mean scalar varance: 1 θ < t 2 u< j θ< 2 =κ θ < θ< + τ θj θ< + u < j θ < θ < ( τ θj θ < ) ( u < θ < θ < ) j Governng equatons of scalar ux (5.16) In order to wrte the governng equaton for scalar ux u θ, rst of all we could wrte the momentum equaton of velocty as u t + u u j = 1 p + ν 2 u. (5.17) ρ x Wth the denton τ j = (u u j ) < u < u< j, the governng equaton for LES could be u < t + u < j u < = 1 p < + ν 2 u < τ j. (5.18) ρ x 100

115 Takng ensemble average, the averaged equaton can be wrtten as u t + u j u = 1 p + ν 2 u u ρ x x ju, (5.19) j and the governng equaton of velocty uctuaton s u t + u j u + u u j = 1 p + ν 2 u ρ x ( u u j u u j ). (5.20) The governng equaton for the uctuaton of scalar ux could be represented as u θ + u j u θ θ p = + κ u 2 θ t ρ x (5.21) + ν θ 2 u u x jθ x u j x u j θ u j x u jθ. j In LES applcatons, the ltered mean velocty reads u < t + u < j u< = 1 p < + ν 2 u < τ j u < u < j. (5.22) ρ x Thus the resolved scalar ux s u < θ< t + u < j u< θ< = 1 ρ θ< p< x + ν θ < 2 u < + κ u < 2 θ < θ < τ j u < τ θj θ < u < u < j u < u < j θ < (5.23) Smlarly, the governng equaton for subgrd scalar ux could be represented as u < θ < + u < j u < θ < = 1 θ p < + κ u < 2 θ < + ν θ < 2 u < t ρ x u < u < j θ< u < j θ < u< u < τ θj θ < τ j u < j (u < θ < ) (5.24) 101

116 5.2 Rapd-and-slow splt of subgrd scalar ux Shao [48] has ntroduced a method of rapd-and-slow splt for subgrd stress. The SGS tensor τ j s splt nto two parts: a rapd part that explctly depends on the mean ow and a remanng slow part. The term rapd s used by analogy to the termnology ntroduced by Rotta [126] and Lumley [127] n the context of Reynolds-averaged modelng, where the component of the pressurestran term that explctly depends on the mean velocty gradent s referred to as the rapd part and the remander as the slow part. Unlke the other researches for the Reynolds-stress transport equaton [128130], the rapd and slow decomposton s appled n LES, and s vered n a turbulent mxng layer by Shao. The classcal Smagornsky model and the scale smlarty model are then evaluated. However, subgrd scalar transfer s not analyzed yet. The subgrd scalar ux τ θj could be splt nto rapd and slow parts as: n whch τ θj = τ rapd θj + τ slow θj, (5.25) τ rapd θj = ( u j θ ) < u j < θ < + ( u j θ ) < u j < θ < + (u j θ ) < u < j θ < τ slow θj = (u jθ ) < u < j θ <. (5.26) In order to descrbe the eect of rapd and slow components of the SGS stress n the scalar transport process, the rapd and slow parts are furtherly decomposed as: τ rapd θj τ slow θj = τ rapd θj + τ rapd θj = τθj slow + τ θj slow It s clear that the mean rapd part s gven by (5.27) whle the uctuatng rapd part s τ rapd θj = ( u j θ ) < u j < θ <, (5.28) τ rapd θj = ( u j θ ) < u j < θ < + (u j θ ) < u < j θ < (5.29) Smlarly, the mean slow part s τ slow θj = (u jθ ) < u < j θ < (5.30) 102

117 and the uctuatng slow part s τ slow θj = τ slow θj τ slow θj (5.31) Magntude of the mean rapd subgrd scalar ux In channel turbulence, the rapd part s produced by the lter n normal drecton. In order to study the propertes of lters, dene the 1-D lter operaton n normal drecton: φ(y) < = 1 (y) b a ( ) y y G (y), y φ(y )dy (5.32) where (y) s the lter wdth and G(η, y) s the locaton dependent lter functon. Let η = y y, (5.32) can be wrtten as (y) φ(y) < = y a (y) y b (y) G(η, y)φ(y (y)η)dη (5.33) Followng the processes of Marsden [46] and Vaslyev [45], takng the Taylor seres expanson of φ(y (x)η) n powers of, we could obtan φ(y) < = φ(y) + ( 1) l l (y)m l (y)d l l! yφ(y) (5.34) l=n where y a M l (y) (y) = η l G(η, y)dη y b (y) { 1, l = 0 n whch M l (y) = 0, l = 1,..., n 1 (5.35) D l y = dl dy l, 103

118 n whch n could represents a property of the lter. Thus the terms of subgrd scalar ux are expressed as θ (y) < u (y) < = θ (y) u (y) ( 1) l + l (y)m l (y) ( θ (y)d l l! y u (y) + u (y)dy θ (y) ) l l=n ( 1) l+r + l+r (y)m l (y)m r (y)d l l!r! y θ (y)dy u r (y) l=n r=n ( θ (y) u (y)) < = θ (y) u (y) + ( 1) l l (y)m l (y) l! l=n l k=0 Cl k Dy l k θ (y)dy k u (y) (5.36) and the magntude of the mean rapd part of subgrd scalar ux could be evaluated as: and θ (y) < u (y) < ( θ (y) u (y)) < = { O( n (y)), n > 1 O( 2 (y)), n = 1 (5.37) In ths chapter, we employ the top-hat lter n physcal space, whch has n = 1, θ (y) < u (y) < ( θ (y) u (y)) < θ u y y 2 (y) (5.38) It means the mean rapd term s about 2 magntude. It wll be further vered n numercal smulaton. Marsden has gven the analyss on the 3-D nhomogeneous lter, and obtaned the same concluson [46]. drectons are both homogeneous. drecton. normal drecton. However, n channel ow, the streamwse and spanwse We should only consder the eects n normal Thus n the followng treatment, all statstcal quanttes are shown n 5.3 A Pror rapd-and-slow decomposton n Couette ow A DNS case of Couette ow s used n evaluatng the rapd and slow parts of scalar transport. The grds number s n streamwse, normal and spanwse drectons respectvely, and the correspondng computaton doman s 4πH 2H 104

119 Fgure 5.1: Sketch of computatonal doman, velocty prole and scalar prole. 2πH. The pseudo-spectral method s employed n calculaton. The numercal detals can be found from Xu et al. [101]. The Reynolds number s Re H = 3200 based on the bulk velocty U m and the half wdth of the channel H, whch s equvalent to a Reynolds number of n Kawamura's DNS results [121]. The scalar value s xed to be 1 n the upper plane and 0 n the lower plane. The molecular Prandtl number s Pr = 0.7. The sketch of computatonal doman, velocty prole and scalar prole s shown n Fg The orgnal mesh s nhomogeneous n the normal drecton. In order to avod the non-commutatvty descrbed n Sec. 5.1, nterpolaton s done n normal drecton, and a new mesh of grds s obtaned for the followng A Pror tests. Tophat lter s employed n physcal space. The grd sze s denoted as and the lter sze s f, n each drecton. Note that the grd szes of DNS are the same between streamwse drecton and spanwse drecton, but not the same n normal drecton, snce normal drecton needs more grds n calculaton. Ths lmtaton may causes the derence of magntude when comparng the results among the three drectons. However, when the lter sze s xed, all results are comparable n normal drecton. Our DNS results are compared wth the DNS results of Kawamura [121]. Fgure 5.2 shows the comparson between them. The scalar proles are n qute good agreement, both n the near wall regon and the channel center. The scalar varance θ 2 are also n agreement, except some derences far from wall. Ths mght stem from the lack of grds by employng Chebyshev sample-pont method n the normal drecton. However, n ths secton we manly pay attenton to the near wall regon, where the nhomogeneous eect could cause the rapd terms. 105

120 present computaton DNS by Kawamura θ Mean θ varance present computaton DNS by Kawamura Y Y + (a) Scalar prole (b) Scalar varance Symbols: DNS wth pseudo- Fgure 5.2: Comparson between two DNS results. spectral method; lnes: DNS by Kawamura. The proles of scalar ux are shown n Fg In our LES case, the scalar ux s postve n the streamwse drecton, and negatve n the normal drecton. In the spanwse drecton t s approxmately zero. Note that the sgn of the value depends on the drecton of the coordnate axs. Ths gure wll be needed n the followng parts when we analyze the subgrd ux transfer Vector level analyss of rapd-and-slow subgrd scalar ux In secton 5.2, the subgrd scalar ux s splt nto rapd and slow parts. Furthermore, the mean and uctuaton parts are dened. A smple analyss of the magntude s shown n secton In the followng part, A Pror tests are made to show the behavor of rapd-and-slow subgrd scalar ux. In the evaluatons, we use the norm of overall slow scalar ux Π θ = uθ to normalze each ux components Mean subgrd ux magntude and ts ansotropy Fgure 5.4(a) shows the components of subgrd ux n streamwse drecton, τ rapd θ1 and τ slow θ1, for derent lter szes f varyng from 2 to 8 tmes of the grd sze. Note that the rapd part exsts mostly n the near-wall regon, especally n the regon Y + < 25, where t may have the same magntude as the slow part. And n the 106

121 7 6 5 <u 1 θ> + <u 2 θ> + <u 3 θ> + 4 scalar flux Y + Fgure 5.3: Scalar ux proles. center part of the Couette ow, the rapd part s neglgble because of the nearly homogeneous velocty and scalar elds here. From equaton (5.38), the mean rapd part can be rescaled by the lter sze ( f / ) 2, and the results are shown n Fg. 5.4(b). The scalng law of Eq. (5.38) s reasonably satsed n the Y 10 regon, where the normalzed curves have the smlar values wth each other. Another mportant fact we should notce s the strong ansotropy of the mean rapd subgrd scalar ux τ rapd θj. Ths rapd parts n derent drectons are shown n Fg. 5.5(a). The lter sze s xed as 4 tmes of grd sze, for nstance. As mentoned n equaton (5.38), the only sgncant component of the rapd mean SGS ux s τ rapd θ1, and other components are neglgbly small. Ths s smply because of the scalar and velocty proles are n the x 1 drecton. It could also be consdered as an example that the large scale structure could gve an ansotropc rapd eect on the subgrd scalar ux. Ths eect s strong n the near-wall regon. The components of the mean slow subgrd scalar ux τθj slow are shown n Fg. 5.5(b). The results show the strong ansotropy among the three drectons. The derences could also stem from the eect of mean ow and scalar. In streamwse drecton, the scalar ux s postve and very strong. In normal drecton, the scalar ux s negatve and not strong n near-wall regon, for nstance Y + < 10. The values n spanwse drecton are almost zero. These behavors are smlar as the total scalar ux (Fg. 5.3). 107

122 <τ θ1 rapd >/2Πθ, <τ θ1 slow >/2Πθ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / =8 <τ θ1 rapd >(2 / f ) 2 /2Π θ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / = Y Y + (a) Rapd and slow parts of τ θ1 (b) Rapd part of τ θ1, rescaled by the lter sze Fgure 5.4: Normalzed mean SGS ux components n streamwse drecton, τ θ1 /2Π θ, wth f / = 2, 4, 6, and 8. Sold lnes: rapd parts. Dashed lnes: slow parts. Thus we could say that the mean ow could eect the mean slow subgrd scalar ux manly n streamwse drecton and n Y + < 10 regon. However, n the channel center, there are also ansotropc contrbutons on the mean slow subgrd scalar ux. The negatve values occur n normal drecton. Note that the negatve subgrd scalar ux do not mean scalar backscatter. The analyss of scalar varance wll be shown n secton Fluctuatng subgrd scalar ux magntude and ts ansotropy The root-mean-square (rms) value of the subgrd scalar ux s closely related to the behavor of the small scales. Fgure 5.6 shows the rms values of normalzed subgrd scalar ux components n streamwse drecton. The lter szes are 2, 4, 6 and 8 tmes of grd sze, respectvely. The magntude of rms values between rapd and slow parts are compared. Unlke the mean values n Fg. 5.4, the slow parts of rms values are also large n near-wall regon, especally when the lter sze s large. However, n the center regon of the channel (Y + > 20), derent lter szes yeld the same rms magntude of slow subgrd scalar ux. The near-wall eect could stem from the strong nhomogenety of velocty and scalar under spatal lters. In addton, the rms values of rapd parts have the smlar behavor as n Fg

123 <τ θ rapd >/2Πθ <τ θ1 rapd > <τ θ2 rapd > <τ θ3 rapd > Y + <τ θ slow >/2Πθ <τ θ1 slow > <τ θ2 slow > <τ θ3 slow > Y + (a) Components of τ rapd θj (b) Components of τ slow θj Fgure 5.5: Components of the rapd and slow parts of scalar ux, wth f / = 4 Usually, n the theory of subgrd scalar modelng, the small scales are assumed to be sotropc whch allows the use of the sotropc relatonshp concernng the scalar transport between scales. In the present case, snce a spatal lter s employed, the uctuatng part of the SGS moton contans explctly the mean ow. Therefore, the scalar transport could devate from sotropy. Fgure 5.7 shows the rms values of all components of the rapd and slow subgrd scalar ux, for the case wth f / = 4. Among the rapd components n Fg. 5.7(a), the most mportant contrbuton s n the streamwse drecton, and the component n normal drecton s almost zero. Comparng wth Fg. 5.5(a), there s also the uctuatng contrbuton n spanwse drecton, although the mean value s almost zero. In Fg. 5.7(b), the all three components of rms values of slow subgrd scalar ux are not zero. They have the close values n channel center. In the near-wall regon (Y + < 20), the contrbutons of both rapd and slow rms values are manly n the streamwse drecton Rapd-and-slow scalar dsspaton n the equatons of scalar varance In secton 5.1.1, the governng equatons of scalar varance are already derved. The subgrd scalar ux has nteracton wth the scalar gradent vector. Ths nner product could be regarded as subgrd scalar dsspaton. In the followng part, the 109

124 rms of τ θ1 rapd /2Π θ, rms of τ θ1 slow /2Π θ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / = Y + Fgure 5.6: Rms of normalzed SGS ux components n streamwse drecton, τ θ1 /2k, wth f / = 2, 4, 6, and 8. Sold lnes: rapd parts. Dashed lnes: slow parts τ θ1 rapd τ θ2 rapd τ θ3 rapd τ θ1 slow τ θ2 slow τ θ3 slow rms of τ θ rapd /2Π θ rms of τ θ slow /2Π θ Y Y + (a) Components of τ rapd θ (b) Components of τ slow θ Fgure 5.7: Rms of the components of the rapd and slow parts of scalar ux, wth f / =

125 rapd and slow parts of subgrd scalar dsspaton are studed. The subgrd terms are θ θ normalzed by usng the overall scalar dsspaton ε θ = κ Subgrd scalar dsspaton of mean scalar varance In the governng equaton of resolved mean scalar varance (5.16), the subgrd term could be decomposed nto rapd and slow parts: τ θj θ< = τ rapd θj θ < + τθj slow θ <, (5.39) whch are shown n Fg The lter szes are 2, 4, 6 and 8 tmes of grd sze, respectvely. In parallel to the observaton at the vector level comparson, the contrbuton of the mean rapd part, τ rapd < θ θj, s neglgbly small compared wth the slow part, τθj slow θ <, because the subgrd ux τ rapd θj only has non-zero value when j = 1, but the scalar gradent θ< only has non-zero value when j = 2. However, the mean slow part could been generated snce τθ2 slow s not zero, whch s shown n Fg. 5.5(b). Although the value of subgrd scalar ux n normal drecton s negatve, n Fg. 5.5(b), the value of scalar dsspaton s postve, whch means that the resolved scalar varance s manly dsspated s subgrd scale. The larger lter szes generate the greater mean slow subgrd dsspaton Subgrd dsspaton of uctuatng scalar varance In the governng equaton of resolved uctuatng scalar varance (5.14), the subgrd term could be smlarly decomposed as τ θj θ < = τ rapd θj θ < + τ slow θj θ < (5.40) whch s shown n Fg The rapd contrbutons are strong n the near-wall regon,.e. Y + < 10, and there s lttle rapd eect n the homogeneous regon of channel center. The slow uctuatng subgrd scalar dsspaton, however, has specal behavor n the buer layer,.e. there are negatve values n the regon 10 < Y + < 20. Ths means the backscatter n ths regon. It s n agreement wth Fredrch's nvestgaton of velocty eld [26], where he found strong eect of backscatter of velocty eld also n ths regon. Ths phenomenon mght stem from the turbulent structures n buer 111

126 -<τ θj rapd > d<θ < >/dxj /2ε θ, -<τ θj slow > d<θ < >/dxj /2ε θ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / = Y + Fgure 5.8: Contrbuton of subgrd dsspaton n the transport equaton of resolved scalar varance, wth f / = 2, 4, 6, and 8. Sold lnes: rapd parts. Dashed lnes: slow parts. layer. Wth derent lter szes, the backscatter remans, but the postve peak value and peak locaton can be derent, n the 30 < Y + < 70 regon Rapd-and-slow scalar transport n the equatons of scalar ux In secton 5.1.2, the governng equatons of scalar ux are derved. In the governng equatons, the subgrd nteractons could exst between the subgrd scalar ux and the velocty gradent tensor, or between the subgrd stress tensor and the scalar gradent vector. The former could represent the nteracton between GS velocty and SGS scalar; whle the latter represents the nteracton between GS scalar and SGS velocty. There are already researches on these contrbutons. Yeung splts the scalar varance transfer term nto four parts n spectral space (the superscrpt denotes the conjugate) [131]: 1. θ u < θ < s the nteracton between GS velocty and GS scalar (GVGS). It causes GS transfer. 2. θ u < θ > s the nteracton between GS velocty and SGS scalar (GVSS). It causes SGS transfer. 112

127 -<τ θj rapd dθ < /dx j >/2ε θ, -<τ θj slow dθ < /dx j >/2ε θ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / = Y + Fgure 5.9: Contrbuton of subgrd dsspaton n subgrd scalar transport equaton, wth f / = 2, 4, 6, and 8. Sold lnes: rapd parts. Dashed lnes: slow parts. 3. θ u > θ < s the nteracton between SGS velocty and GS scalar (SVGS). It causes SGS transfer. 4. θ u > θ > s the nteracton between SGS velocty and SGS scalar (SVSS). It causes SGS transfer. In the three parts causng SGS transfer, the nteracton between GS velocty and SGS scalar (GVSS) s the man part, whch s much more obvous than the other two parts. Especally GVSS has much more contrbuton than SVGS term has. Ths phenomenon was vered n sotropc scalar turbulence by Yeung [131] and n ansotropc scalar turbulence by Fang [132]. In the followng part, we would lke to nvestgate ths behavor n the transport of scalar ux n nhomogeneous ansotropc channel ow, but wth a spatal lter n physcal space. The subgrd terms are normalzed by usng the norm of overall scalar ux ε uθ = (ν + κ) u θ Subgrd transport of mean scalar ux There are two subgrd dsspaton terms n the transport equaton of resolved mean scalar ux (5.23), whch could be denoted as τ θj u< and τ j θ<. They could 113

128 -<τ θj rapd > d<u1 < >/dxj /ε uθ, -<τ θj slow > d<u1 < >/dxj /ε uθ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / = Y + (a) τ rapd θj u < 1 and τθj slow u < 1 -<τ 1j rapd > d<θ < >/dxj /ε uθ, -<τ 1j slow > d<θ < >/dxj /ε uθ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / = Y + (b) τ rapd 1j θ < and τ1j slow θ < Fgure 5.10: Contrbuton of subgrd transport n mean scalar ux equaton, at f / = 2, 4, 6, and 8. Sold lnes: rapd parts. Dashed lnes: slow parts. be decomposed nto rapd and slow parts, respectvely as follows: τ θj u< = τ j θ< = τ rapd θj τ rapd j u < θ < + τθj slow u <, + τj slow θ <, (5.41) where the rapd and slow parts of subgrd stress τ j are dened smlar as the subgrd scalar ux τ θj. The detals could be found n Shao's paper [48]. These four terms are shown n Fg Note that n the mean scalar ux equaton, only the component n streamwse drecton, u < 1 θ <, s not equal to zero. All the rapd terms have zero values. Between gures 5.10(a) and 5.10(b), the magntude of the slow terms are almost the same. It shows that the contrbuton on mean scalar ux of GS velocty and SGS scalar s almost the same as the contrbuton of SGS velocty and GS scalar. It s reasonable snce n Eq. (5.38), the magntude of rapd subgrd s expressed by the mean gradents of velocty and scalar. We could smlarly wrte the expresson for subgrd stress, usng the same method as n Sec , nally the terms are of the same magntude between Fgs. 5.10(a) and 5.10(b). In addton, n both gures the subgrd transfer ncreases when the lter sze ncreases. 114

129 Subgrd transport of uctuatng scalar ux Smlarly, there are two subgrd dsspaton terms n the transport equaton of resolved uctuatng scalar ux (5.24), whch are τ θj u < and τ θ < j. They could be decomposed nto rapd and slow parts, respectvely: τ θj u < = τ rapd u < θj + τ slow u < θj, τ j θ < = τ rapd j θ < + τ slow j θ <. (5.42) These four terms are shown n Fg. 5.11, 5.12 and 5.13, n derent drectons respectvely. The derent nteractons are both shown. Each gure on the left denotes the nteracton between GS velocty and SGS scalar (GVSS), whle on the rght represents the nteracton between SGS velocty and GS scalar (SVGS). Smlar as analyzed before, the rapd parts manly exst n the streamwse drecton, and are almost zero n normal and spanwse drecton. For the slow parts n the streamwse drecton, n the regon 10 < Y + < 20 there s also backscatter, t s smlar as the analyss of energy varance. Both GVSS and SVGS terms show the same behavor. For the slow parts n the normal drecton, GVSS term has negatve value whle SVGS term s postve n most part of the channel. Snce the total scalar ux s negatve (see Fg. 5.3), here GVSS term s the forward transfer and SVGS term s backscatter. Ths behavor of backscatter s qute obvous n the regon Y For the slow parts n the spanwse drecton, all terms are almost zero. Besdes, we can focus at the mostly homogeneous regon (Y + > 100), the prncpal contrbuton among the sx gures s the GVSS term n streamwse drecton,.e. Fg. 5.11(a). It s much stronger than the SVGS term n the same drecton,.e. Fg. 5.11(b). Ths phenomenon s n agreement wth the results of Yeung [131] and Fang [132]. Thus n the homogeneous ansotropc regon, the GVSS term could be consdered as the major contrbuton of subgrd scalar ux. 5.4 A Pror evaluaton of subgrd models From the dscusson of the prevous sectons, t s clear that, through the rapd part, the mean velocty and scalar gradent drectly aects the SGS scalar ux and the 115

130 -<τ θj rapd du 1 < /dx j >/ε uθ, -<τ θj slow du 1 < /dx j >/ε uθ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / =8 -<τ 1j rapd dθ < /dx j >/ε uθ, -<τ 1j slow dθ < /dx j >/ε uθ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / = Y + (a) τ rapd θj u < 1 and τθj slow u < Y + (b) τ rapd 1j θ < and τ slow 1j θ < Fgure 5.11: The contrbuton of subgrd transport n resolved scalar ux equaton, n the streamwse drecton, wth f / = 2, 4, 6, and 8. Sold lnes: rapd parts. Dashed lnes: slow parts. (a) GVSS term. (b) SVGS term. -<τ θj rapd du 2 < /dx j >/ε uθ, -<τ θj slow du 2 < /dx j >/ε uθ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / =8 -<τ 2j rapd dθ < /dx j >/ε uθ, -<τ 2j slow dθ < /dx j >/ε uθ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / = Y + (a) τ rapd θj u < 2 and τθj slow u < Y + (b) τ rapd 2j θ < and τ slow 2j θ < Fgure 5.12: The contrbuton of the subgrd transport n the resolved scalar ux equaton, n the normal drecton, wth f / = 2, 4, 6, and 8. Sold lnes: rapd parts. Dashed lnes: slow parts. (a) GVSS term. (b) SVGS term. 116

131 -<τ θj rapd du 3 < /dx j >/ε uθ, -<τ θj slow du 3 < /dx j >/ε uθ Rapd f / =2 Rapd f / =4 Rapd f / =6 Rapd f / =8 Slow f / =2 Slow f / =4 Slow f / =6 Slow f / =8 -<τ 3j rapd dθ < /dx j >/ε uθ, -<τ 3j slow dθ < /dx j >/ε uθ Rapd f / = Rapd f / = Rapd f / = Rapd f / = Slow f / =2 Slow f / =4 Slow f / =6 Slow f / = Y + (a) τ rapd θj u < 3 and τθj slow u < Y + (b) τ rapd 3j θ < and τ slow 3j θ < Fgure 5.13: The contrbuton of the subgrd transport n the resolved scalar ux equaton, n the span drecton, wth f / = 2, 4, 6, and 8. Sold lnes: rapd parts. Dashed lnes: slow parts. (a) GVSS term. (b) SVGS term. assocated scalar transfer to the small scales. The queston that then arses s whether SGS modelng has to explctly account for the eect of the mean velocty and scalar gradent manfested by the rapd SGS stress. In fact, because the rapd parts strongly depends on the type of lter, and only exsts n nhomogeneous scalar turbulence, t can not be represented by most of the SGS models. In the followng part, we follow Shao's concluson [48], to represent the rapd parts by usng the scale-smlarty model (SSM): [ τ rapd θj = C m ( θ < u j < ) < θ << u j << + ( θ < u j < ) < θ << u << j + (θ < u j < ) < θ << u j <<], (5.43) where C m = 1. The second lterng operaton s done wth a test lter that s the same as the orgnal grd top-hat lter. Besdes, n order to smulate the slow subgrd scalar dsspaton, we apply the Smagornsky model wth constant turbulent Prandtl number, whch can be wrtten as τ slow θj = κ t θ <, κ t = ν t Pr t. (5.44) Two classcal models are appled to determne the eddy vscosty coecent. s the Smagornsky model ν t = (C s ) 2 (2 S < j S < j )1/2 wth C s 117 One = 0.1 and wth van

132 0.1 Slow exact Slow SM dampng Slow DSM -<τ θj slow dθ < /dx j >/2ε θ Y + Fgure 5.14: Comparson between exact values and model values, of slow subgrd scalar dsspaton. f / = 4. Smagornsky model wth constant turbulent Prandtl number s appled. Drest dampng functon n near-wall regon (SM dampng). Another s the Germano Dynamc Smagornsky model (DSM) [11]. In both cases the turbulent Prandtl number s xed as Pr t = Pr. The comparson wth exact slow subgrd scalar dsspaton s shown n Fg It s found that these SGS models are not enough dsspated n the channel center, and they can not represent ether the backscatter around Y + = 15, or the scalar dsspaton at about Y + = 5. In order to propose a better SGS model, we choose a recent ansotropc model by Cu et al. (Cu Model) [40] to represent the slow parts. Comparng wth other SGS models, ths ansotropc model explctly denotes the eddy duson as a functon of mean velocty and scalar, whch can not be analyzed by rapd-and-slow decomposton. We also evaluate the model performance of backscatter performance n near-wall regon Extended formulaton of Cu Model The orgnal form of scalar model n Ref. [40] dd not consder the mean part of scalar. In ths secton we consder that the turbulence s homogeneous wth mean shear γ and mean scalar gradent G, thus the velocty and scalar can be decomposed 118

133 to mean and uctuaton as: u =u + γx 2 δ 1, θ =θ + Gx 2. (5.45) as The equaton of large eddy smulaton for scalar turbulence can then be wrtten θ < t + ( ) ( ) u < θ < j + γx 2 δ j1 + Gδ j2 = κ 2 θ < τ θj. (5.46) Followng the same process as n Ref. [40], denng the structure functons D < θθ = δθ 2, D < jθθ = δu < j δθ 2 and D < 2θ = δu < 2 δθ, neglectng the molecular dusvty, and makng average n a local sphere of radus r, nally the SGS eddy dusvty can be obtaned as (Extended Cu Model, denoted as ECM) κ t = 6 (D< rθθ )A + 6γr (D θθ < n 1n 2 ) A + (D 2θ < )V Gr ( ) dd < A 6 θθ θ < θ < V (5.47) 4r dr n whch the notatons ( ) A and ( ) V are local surface average and local volume average. We can also dvde t nto two parts: κ fs t = κ ms 6 ( dd < θθ dr 6 (D rθθ < ) )A A θ < θ < 4r V, t = 6γr (D< θθ n 1n 2 ) A + (D 2θ < )V Gr ( ) dd < A 6 θθ θ < θ < V, 4r dr (5.48) n whch κ fs t represents only the nteractons of subgrd scale (uctuatng part), and κ ms t contans the nformatons of mean ow and mean scalar (mean part),.e. γ and G, explctly. Comparng wth the orgnal model formulaton n Ref. [40], the mean velocty s addtonally consdered. Because only homogeneous velocty and scalar elds are consdered n Eq. (5.46), we approxmately consder that ECM only generates the slow SGS scalar ux. Ths model formulaton shows that the slow SGS scalar ux s not only aected by SGS uctuatons, but also relatve wth the mean proles. The model s derved by employng the Yaglom equaton of ltered quanttes, and the backscatter can also be denoted. 119

134 5.4.2 Evaluaton of subgrd scalar dsspaton From equaton (5.48), the subgrd eddy-dusvty s splt nto mean and uctuatng parts. Thus from the eddy-dusvty assumpton, the mean and uctuatng subgrd scalar ux are expressed as [26]: τ ms θj τ fs θj = κ ms t = κ fs t θ, θ. The relatve terms of subgrd scalar dsspaton read τ θj ms θ < θ =κ ms < θ <, τ fs θj θ < θ =κ fs < θ <. (5.49) (5.50) Wth an deal subgrd model, there should be τ θj τ slow θj =τ ms θj θ < = τ ms θj + τ fs θj, θ < + τ fs θj θ <. (5.51) The mean and uctuatng parts of subgrd scalar dsspaton are shown n Fg. 5.15, by employng the ECM. The lter szes are 2, 4, 6 and 8 tmes of grd sze, respectvely. The mean parts have postve values, whch means scalar varance could be dsspated by mean velocty and mean scalar. The peak locatons s about Y The uctuatng parts also have postve value n most regon of channel, whch means the subgrd nteractons manly dsspate scalar varance. In the channel center, ow s almost homogeneous, and the subgrd dsspaton ncreases when lter sze ncreases. However, there are also negatve values n Y + 15 regon,.e. the buer layer. It means that the scalar backscatter could also be generated by subgrd nteractons. The total slow subgrd scalar dsspaton could be calculated by employng the total eddy dusvty κ = κ ms + κ fs. The SSM s used to smulate the rapd subgrd scalar dsspaton. Wth derent lter sze, the comparsons between exact values and modelled values are shown n Fg. 5.16, respectvely. 120

135 -<τ θj ms dθ < /dx j >/2ε θ, -<τ θj fs dθ < /dx j >/2ε θ Mean f / =2 Mean f / =4 Mean f / =6 Mean f / =8 Fluctuatng f / =2 Fluctuatng f / =4 Fluctuatng f / =6 Fluctuatng f / = Y + Fgure 5.15: Mean and uctuatng parts n the ECM, wth f / = 2, 4, 6, and 8. Sold lnes: mean parts. Dashed lnes: uctuatng parts. For the slow parts of subgrd scalar dsspaton, the model results have the smlar trend wth the exact values. In the channel center, where the turbulence s almost homogeneous, ther magntudes are n good agreement, especally when f s 4 and 6 tmes of grd sze. In the near-wall regon, all the model values are smaller than the exact values, whch means that the shear model does not dsspate enough n the vscous sublayer. Comparng wth the results of SM and DSM n Fg. 5.14, there s no mprovement n ths range. It could really be a serous problem for large-eddy smulaton [133], whch s stll not enough studed. However, an advantage could be found n the ECM, that n buer layer (Y + 15), the behavor of scalar backscatter s smulated. The peak locatons and values of shear model are n qute good agreement wth the exact results, especally when f s 4 and 6 tmes of grd sze. When the lter sze s too large (for nstance, f / = 8), ths scale s probably out of nertal subrange, and there s no good agreement. In bref, both the agreement n channel center and n the range of backscatter could be consdered as an great advantage than other scalar models. For the rapd parts, the scale-smlarty model smulates qute well the strong dsspaton n near-wall regon. Therefore, we propose the ECM to be employed n wall-bounded scalar turbulence. It could represent the behavor of subgrd scalar backscatter. The mean velocty and scalar proles could aect the subgrd scalar ux explctly. In addton, n the 121

136 -<τ θj rapd dθ < /dx j >/2ε θ, -<τ θj slow dθ < /dx j >/2ε θ Slow exact Slow ECM Rapd exact Rapd SSM -<τ θj rapd dθ < /dx j >/2ε θ, -<τ θj slow dθ < /dx j >/2ε θ Slow exact Slow ECM Rapd exact Rapd SSM Y Y + (a) f = 2 (b) f = 4 -<τ θj rapd dθ < /dx j >/2ε θ, -<τ θj slow dθ < /dx j >/2ε θ Slow exact Slow ECM Rapd exact Rapd SSM -<τ θj rapd dθ < /dx j >/2ε θ, -<τ θj slow dθ < /dx j >/2ε θ Slow exact Slow ECM Rapd exact Rapd SSM Y Y + (c) f = 6 (d) f = 8 Fgure 5.16: Comparson between exact and model values of scalar dsspaton, wth f / = 2, 4, 6, and 8. Sold lne and lled symbols: exact value of slow subgrd scalar dsspaton. Sold lne and hollow symbols: slow subgrd scalar dsspaton by usng the shear model. Dashed lne and lled symbols: exact value of rapd subgrd scalar dsspaton. Dashed lne and hollow symbols: rapd subgrd scalar dsspaton by usng scale-smlarty model on the velocty and scalar proles. 122

137 nhomogeneous regon, the SSM could be employed on the mean velocty and scalar, to smulate the rapd part of subgrd scalar ux and subgrd dsspaton. 123

138 Chapter 6 Applyng KEF on subgrd modelng of magnetohydrodynamc (MHD) turbulence Many plasmas n turbulent moton observed n astrophyscal systems, as well as nuclear fuson devces, and ows of conductng uds, can be descrbed wthn the framework of magnetohydrodynamcs (MHD). Snce MHD turbulence s a vast research area, we wll not try to gve an exhaustng overvew of lterature on the subject. Rather wll we refer to some textbooks on the subject for the reader nterested n more detals. Intally, nterest n MHD focused on the dynamo problem, notably n Batchelor's early paper [134] (see also the book of Moatt [135] on the subject). The book by Davdson gves a more recent survey of the subject, n partcular focusng on the ows of lqud metals [136]. Bskamp's book focuses on turbulence n MHD ows [137]. The nteracton of the velocty and magnetc eld completely changes the dynamcs. For example, the classcal scalng theory by Kolmogorov [9] s changed due to the presence of magnetc (Alfvén) waves. A mlestone n the fundamental scalng theory was the ntroducton of the Alfvén eect proposed ndependently by Iroshnkov [138] and Krachnan [15]. Ths descrbes small-scale turbulent uctuatons as weakly nteractng Alfvén waves propagatng along the large-scale eld. Because of the reducton of the correspondng spectral transfer the energy spectrum was predcted to be somewhat atter, k 3/2 nstead of Kolmogorov spectrum k 5/3. From the observaton sde, however, the energy spectrum of solar-wnd turbulence was found to be clearly closer to a Kolmogorov law. The soluton of the paradox les n the ntrnsc ansotropy of 124

139 MHD turbulence emphaszed by Goldrech [139] and Vorobev [140], but ths remans a subject of actve research and debate. In lqud metal ows most numercal nvestgatons focus on the Hartmann layer [141] n channel turbulence. The magnetc eld s xed as a unform vector B 0, whch s n the normal drecton of the wall [142144]. In astrophyscs and geophyscs the Reynolds number can become very large so that LES becomes a useful tool. There are already attempts to develop SGS models. In spectral space, models based on EDQNM theory are proposed [145, 146]. In physcal space, the most common subgrd model s assumng eddy-vscosty n both velocty and magnetc elds, where the subgrd eddy vscosty and magnetc dusvty are determned by usng ether Germano dentty [147] or dmensonal consderatons [148]. However, no physcal background exsts n these physcal-space models. In ths chapter, we attempt to derve the homogeneous sotropc formula of KEF n Elsässer varables, and employ them to obtan new physcal subgrd models for coupled MHD problems. 6.1 Governng equatons of resolved knetc energy Frst of all, the dsspaton terms n LES should be derved. The governng equatons of ltered quanttes n homogeneous sotropc MHD turbulence could be wrtten as u < t = u< u< j + b< b< j + ν 2 u < p< x τ u j, (6.1a) b < t = b< u< j + u< b< j + η 2 b < τ b j, (6.1b) n whch b s the magnetc nducton, η s the magnetc dusvty. The subgrd stress tensors n velocty and magnetc elds read τj u = ( ) ( ) (u u j ) < u < u < j (b b j ) < b < b < j τj b = ( (6.2) ) ( ) (b u j ) < b < u < j (u b j ) < u < b < j. 125

140 The resolved energes n these elds are then wrtten as 1 2 u <2 t = 1 2 u < u< u< j + u< b< b< j b < b < j u < (6.3a) + ν 2 2 u <2 ν u< u < p< u < x u < τ u j, 1 2 b <2 t = 1 2 b < b< u< j + u< b< b< j u < b < j + η 2 b <2 η b< u < τ b < j b. 2 Takng ensemble average, we obtan the governng equaton of resolved mean knetc energy: n whch ε u< =ν u <2 t b <2 t = 2 b < b < j = 2 u < b < j u < 2ε u< 2ε u x f, j b < 2ε b< 2ε b x f, j b < u < u <, ε u f = τ x js u j, < S j < = 1 ( u < j 2 ) + u< j, x b ε b< < =η b <, ε b f = τ x jj b j <, J j < = 1 ( ) b < + b< j. j 2 x Add (6.4a) to (6.4b), we elmnate the product terms. Thus we can wrte u <2 t + b<2 t (6.3b) (6.4a) (6.4b) (6.5) = 2ε u< 2ε u f 2ε b< 2ε b f. (6.6) In hgh Reynolds number turbulence, the terms of resolved dsspaton can be neglected. It leads to u <2 + b<2 = 2ε u f 2ε b t t f. (6.7) Smlarly, from equaton (6.1), we wrte the followng equatons: b < u < t = b < u < u< j + b < b < b< j + νb < 2 u < p< b < x b < τ u j, (6.8a) u < b < t = u< b < u< j + u < u < b< j + ηu < 2 b < u < τ b j. (6.8b) 126

141 Add (6.8a) to (6.8b), we wrte the resolved cross energy as: u < b< t = u< b< b< j b <2 b < j u <2 b < j p< b < x ( ) b < x τj u + u τj b + νu + ηb + τ u j J j < + τjs b j, < j (6.9) n whch U = b < becomes Snce we obtan 2 u <, B = u < u < b< t 2 u < b< 2 b <. Takng ensemble average, equaton (6.9) = ν U + η B + τ u jj < j + τ b js < j. (6.10) = 2 u< b < + u < ν U = 2ν 2 b < + b < u < b < ν B, u < η B = 2η b < η U. 2 u <, (6.11) (6.12) Therefore, equaton (6.10) can also be wrtten as u < b< u < = 2(ν + η) b < ν B η U + τ t x jj u j < + τjs b j. < (6.13) j Add (6.10) to (6.13), we obtan 2 u< b< t = 2(ν + η) u < b < + (ν η) ( U B ) + 2 τjj u j < + 2 τjs b j. < (6.14) In hgh Reynolds number turbulence, the terms of resolved dsspaton are then neglected. It leads to n whch 2 u< b< t = 2ε c f, (6.15) ε c f = τ u jj < j τ b js < j. (6.16) 127

142 6.2 Kolmogorov equaton of ltered quanttes For the Elsässer elds z ± = u ± b, there s z ± t + z z ± j where P = p + b 2 /2 s the total pressure, ν ± = (ν ± η)/2. = P 2 z ± 2 z + ν + + ν, (6.17) x Wth a gven lter, the governng equaton of resolved scale elds reads z <± t + z < j z <± = P < 2 z <± + ν + x where the subgrd tensor τ z± j = (z ± z j )< z <± 2 z < + ν z < j τ z± j, (6.18). We note the followng relatons: τ z± j = τ u j ± τ b j. (6.19) Wrtng equaton (6.18) n another pont x, and ξ = x x, δ z ± ( x, ξ) = z ± ( x ) z ± ( x), followng the smlar process as n secton 2.3, we obtan ( δz <± + δz < δz <± 2 δz <± 2 δz < τ z± j j = 2ν + + 2ν t ξ j ξ j ξ j ξ j ξ j x j In hgh Reynolds number turbulence, t s smpled as δz <± t + δz < j δz <± ξ j Multply equaton (6.21) by δz <±, t leads to 1 2 δz <±2 t δz <±2 δz < j τ z± = δz <± j ξ j x j ) z± τj. (6.20) z± z± τj τj = +. (6.21) x j τ z± + δz <± j. (6.22) Takng ensemble average, n homogeneous sotropc turbulence the rght hand sde becomes δz <± τ z± j x j δz <± τ z± j = τ z± j δz <± x j τ z± j δz<± + τ z± j ( ) ( ) τ z± j + τ z± j z <± z <± = + ξ j δz <± x j τ z± j τ z± j δz <± z <± + x j τ z± j z <± (6.23) = ( T z±,j T ) z± j, 2ε z± f ξ, j 128

143 n whch T z±,j (ξ) = z<± (x 1 )τ z± j (x 1 + ξ), T z± j, (ξ) = z<± (x 1 + ξ)τ z± j (x 1), ε z± f = τ z± j z <±. (6.24) In the left hand sde of equaton (6.22), the tme-dependent term vanshes for small dstance ξ. Takng ensemble average, the other term s wrtten as 1 δz <± 2 δz < j = 1 D <± j, (6.25) 2 ξ j 2 ξ j n whch D <±,j = δz<± δz <± δz < j. (6.26) In sotropc turbulence, T z± z±,j (ξ) = Tj, (ξ), and the thrd-order tensors can be expressed by usng ther longtudnal components [51, 137, 149, 150],.e. T z± T z±,jk ( l,ll ξ ξ) = T z± l,ll ξ 2ξ 3 ξ ξ j ξ k + z± T 2T z± l,ll l,ll + ξ ξ 4ξ (ξ k δ j + ξ j δ k ) T z± l,ll 2ξ ξ δ jk, D <± T z± n,nl = 1 4ξ ξ,k ( ξ) =D <± ξ k,l ξ, ( ) ξ 2 T z± l,ll, (6.27) D <±,l = 2 ( ξ 4 ( ) ) D <± ξ 3 ll,l 2C <± ll,l, ξ 4 C <± ll,l = z <± l (x 1 )z <± l (x 1 )z < l (x 1 + ξ). From equaton (6.22), (6.23), (6.25) and (6.27), we obtan ( ( 1 2 ξ 4 ( ) )) D <± 2ξ 2 ll,l 2C <± ll,l = 2 ( 1 ( ) ) ξ 4 T z± ξ ξ ξ 4 ξ 2 l,ll 2ε z± f ξ 2ξ ξ. (6.28) Integral both sdes from 1 to ξ twce, nally we obtan the KEF formulaton 8 15 εz± f ξ = D<± ll,l 2C <± ll,l 4T z± l,ll. (6.29) 129

144 Ths scalng law s derent from the result n a homogeneous sotropc velocty eld. Partcularly, n the hydrodynamc lmt z <± = u <, we obtan the 4/5 law as for the velocty eld. When ξ, as analyzed n secton 2.6 and 3.2, the term T z± l,ll n equaton (6.29) vanshes. Thus the smpled KEF reads 8 15 εz± f ξ = D<± ll,l 2C <± ll,l. (6.30) Notng that D <± ll,l 2C <± ll,l = 4 z <± l (x 1 )z < l (x 1 )z <± l (x 1 + ξ). We wrte equaton (6.30) for z <+ and z <, respectvely. Returnng to the basc elds, we obtan the followng equatons: 4 5 εt f r = δu <3 l 6 b <2 l (x 1 )u < l (x 1 + ξ), 4 5 εc f r = δb <3 l + 6 u <2 l (x 1 )b < l (x 1 + ξ), (6.31) n whch 2ε T f =ε z+ f 2ε C f =ε z+ f + ε z f, ε z f. (6.32) Comparng wth the energy analyss n secton 6.1, there are the relatons ε T f =ε u f + ε b f, ε C f =ε c f. (6.33) 6.3 MHD subgrd models n velocty and magnetc elds The eddy vscosty assumpton s appled. tensors are modeled by assumng a lnear relaton: τj u = 2ν t S j, < S j < = 1 ( u < 2 τ b j = 2η t J < j, J < j = 1 2 Followng Agullo [147], the subgrd ) + u< j, x ( ) b < + b< j. x (6.34) 130

145 From equatons (6.5), (6.16) and (6.33), we obtan ε T f = 2ν t S < js < j + 2η t J < j J < j, (6.35) ε C f = 2(ν t + η t ) S < jj < j. (6.36) From equaton (6.31), (6.35) and (6.36), the model coecent can be descrbed as: n whch ν t = 5 A T S j < J j < AC J j < J j < 8ξ S j < J j < ( J j < J j < S< j S< j ), η t = 5 A T S j < J j < AC S j < S< j 8ξ S j < J j < ( S j < S< j J j < J j < ), A T = δu <3 l ( ξ) 6 b <2 l ( x)u < l ( x + ξ), A C = δb <3 l ( ξ) + 6 u <2 l ( x)b < l ( x + ξ). (6.37) (6.38) However, because we can have E C 0, ths dynamc method mght be not stable n numercal calculaton. Therefore, the eddy vscosty s calculated by usng CZZS model, assumng that the magnetc eld doesn't have aect wth the subgrd velocty stress drectly. The cross equaton (6.36) s gnored. Fnally we obtan ν t = 5 8ξ η t = 5 8ξ δu <3 l S j < S< j, 6 b <2 l ( x)u < l ( x + ξ) J j < J j <. (6.39) A Posteror test at 64 3 meshes s done n homogeneous sotropc decayng turbulence, by employng model (6.39). The ntal velocty and magnetc elds are generated by employng Rogallo's method [54], respectvely. It s known snce the work by Tng et al. [151], that derent ntal condtons can lead to completely derent ows dependng on the ntal values of u b, u 2 and b 2. We consder one partcular case n whch the ntal energy spectra are the same,.e. E K = E M, and E C s ntally small. The temporal evoluton and energy spectra at t = 16 are shown n Fg The magnetc energy ncreases a lttle before decayng, correspondng to the deformaton of the magnetc eld lnes before the Lorentz force starts to act. The 5/3 energy spectrum s approxmately obtaned for nter n both velocty and magnetc elds. Ths s only a very smply example to verfy the SGS model (6.39). 131

146 E K E M 1.2 E K, E M E K E M E K, E M t K Fgure 6.1: Knetc and magnetc energy n decayng turbulence. evoluton. Rght: energy spectra. Left: temporal Dscusson n the Elsässer elds To understand the unstablty of (6.37), we dscuss the model now expressed n Elsässer varables. In the eddy vscosty assumptons (6.34), the velocty subgrd stress tensor s also aected by magnetc eld from denton, but t vanshes n the eddy vscosty assumpton. In order to explan t, n ths secton we drectly employ KEF n Elsässer elds. We can also obtan the correspondng subgrd models. Consderng that the subgrd stress tensor take the smlar parts as the molecular dsspaton, from equaton (6.18) we can easly wrte the eddy vscosty assumpton as τ z± j = 2ν + t S <z± j 2ν t S <z j, (6.40) n whch S <z± j = 1 2 ( z <± ) + z<± j = S j < ± J j <. (6.41) x From equaton (6.24) and ths new eddy vscosty assumpton, the subgrd dsspaton n Elsässer elds yelds ε z± f = τ z± j S<z± j = 2ν + t S <z± j Then from equaton (6.30), we obtan ( ν + t S <z± j S <z± j + ν t S <z+ j S <z± j S <z j + 2ν t S <z+ j S <z j. (6.42) ) ξ = D <± ll,l 2C <± ll,l. (6.43) 132

147 The eddy vscostes are solved as ν + t ν t = 15 16ξ = 15 16ξ D <+ ll,l D < ll,l 2C <+ ll,l + 2C < ll,l S <z+ j S <z+ j S <z j S <z, j S <z+ j S <z+ j ( ) ( ) D < ll,l 2C < ll,l S <z j S <z j D <+ ll,l 2C <+ ll,l ( ) S <z j S <z+ j S <z. S <z+ j S <z+ j j S <z j (6.44) Although there s a sngular pont when the magnetc eld vanshes and z + = z, the whole eddy vscosty tends to a lmtng value that n whch D < ll,l ν + t + ν t 15 16ξ = 1 2 (D<+ ll,l + D < ll,l ), C< ll,l D ll,l < 2C< ll,l, (6.45) S <z j S<z j = 1 2 (C<+ ll,l + C < ll,l Alternatvely f u b = 0, we can also obtan S <z+ j S <z+ j result. ), S<z j = 1 2 (S<z+ j = S <z j S <z j + S <z j ). and the same Therefore, f we assume the cross energy s zero,.e. u b = 0, and assume ν t = 0, we can obtan the followng model ν + t = 15 16ξ D ll,l < 2C< ll,l. (6.46) S <z j S<z j Ths model formulaton actually mples the eddy vscosty assumpton and ν t = η t. These results, n fact, are the same as the models expressed n velocty and magnetc elds, whch are dscussed n the prevous secton. Consderng the eddy vscosty assumptons (6.34), equatons (6.32), (6.35) and (6.36) lead to ε z+ f ε z+ f + ε z f =2ε T f = 4ν t S < js < j + 4η t J < j J < j, ε z f =2ε C f = 4(ν t + η t ) S < jj < j. And we can also wrte equaton (6.42) n the followng formulatons: (6.47) ε z+ f + ε z f =2ν + t ( S <z+ j S <z+ j + S <z j S <z j ) + 4νt S <z+ j S <z j ε z+ f ε z f =4(ν + t + ν t ) S < js < j + 4(ν + t ν t ) J < j J < j, =2ν + t ( S <z+ j S <z+ j S <z j S <z j ) (6.48) =8ν + t S < jj < j. 133

148 Comparng (6.48) wth (6.47), the followng relatons are obvous: ν t =ν + t + ν t, η t =ν + t ν t. (6.49) Therefore, the models n ths secton are n fact the same as dscussed before. Although two unknown coecents are solved by two equatons (ε T f of ε C f and εc f can be relatvely small, and the numercal soluton may be unstable. ), the value 6.4 Perspectves In ths chapter we extended the KEF formula to homogeneous sotropc MHD turbulence, and obtaned the correspondng SGS models n physcal space. The models drectly based on the KEF equatons,.e. Eqs. (6.37) and (6.44), are n fact the same as can be seeen by usng the relatons (6.49). Both of them are not numercal stable f E C 0,.e. u b 0. It was shown that, to get a properly workng SGS models for MHD turbulence, more physcal constrants are needed than the KEF equaton only, for nstance, n Eq. (6.39) we assume that the magnetc eld doesn't have aect wth the subgrd velocty stress drectly. Better deas may be employng the scalng law of structure functons, and ths topc wll be further nvestgated n the future. 134

149 Chapter 7 Concluson In subgrd modelng, not only mathematcal approach, but also physcal propertes should be mpled. Comparng wth plenty of researches n spectral space, there are few works based on energy transfer propertes n physcal space, n subgrd modelng. Ths thess manly ams at employng Kolmogorov equaton of ltered quanttes (KEF) n subgrd modelng. It s consdered as a general method for any subgrd model assumpton. The followng results are obtaned: 1. Derent formulatons of KEF are derved, ncludng the forms n velocty eld (homogeneous sotropc turbulence, nhomogeneous ansotropc turbulence, homogeneous shear turbulence, homogeneous rotatng turbulence), n scalar turbulence and n magnetohydrodynamc turbulence. 2. The ansotropc eect of mean shear n physcal space s analyzed. The local sotropc assumpton of structure functon s quered. Instead, the two-pont skewness s proposed to represent the sotropy n small scale. 3. The structure functon of ltered velocty s analyzed n physcal and spectral spaces, respectvely. Results show that n order to satsfy the classcal scalng law, the two-pont dstance of velocty ncrement must be much larger than the lter sze. Otherwse, the classcal scalng law can not be drectly appled n subgrd modelng. 4. The sotropc formulatons of eddy-vscosty models are analyzed. The CZZS model could be smpled n derent ways, whch yelds the one-scale and mult-scale models. Partcularly, the skewness-based sotropc formulaton s 135

150 proposed. Besdes, by employng KEF, we could also determne the coecent of Smagornsky model dynamcally. Ths dynamc procedure s much less cost then Germano procedure and has clear physcal background. 5. The ansotropc formulatons of eddy-vscosty models are derved n homogeneous shear turbulence and homogeneous rotatng turbulence. A Posteror tests are made to evaluate ths model. The most mportant advantage s that the mean moton s explctly contaned n the formulaton of subgrd eddy vscous. 6. Wth the velocty ncrement assumpton, the mproved velocty ncrement model (IVI) s proposed. In real ow, when scales are not well separated, the model coecent has a dynamc formulaton; n deal hgh Reynolds number turbulence, we could obtan a constant coecent, whch s extremely smple and low cost. The IVI model s vered n A Pror and A Posteror tests. Wall behavor s well satsed, and the energy backscatter could be smulated properly. 7. For ansotropc passve scalar eld, the transfer processes of scalar energy and scalar ux are manly nvestgated. We splt the subgrd scalar ux nto rapd and slow parts, and do A Pror tests on them n a channel Couette ow. The new ansotropc eddy-dusvty model based on KEF s then vered for ts propertes of smulatng subgrd scalar dsspaton. The energy scalar backscatter s well represented. The mean velocty and scalar proles could eect the subgrd scalar ux explctly. In addton, durng nhomogeneous regon, the scale-smlarty model could be employed on the mean velocty and scalar, to smulate the rapd part of subgrd scalar ux and subgrd dsspaton. 8. In magnetohydrodynamc turbulence, the formulaton of KEF s derved n the Elsässer elds. Correspondng subgrd models are then founded. However, the present models are not numercal stable when the correlaton between velocty and magntude elds s small. Ths work s expected to be further nvestgated. 136

151 Résumé La smulaton numérque des grandes échelles (LES) est actuellement un outl prometteur pour la prédcton des écoulements turbulents ndustrels. Les grandes et pettes échelles sont solées par une opératon de ltrage, qu peut être applquée dans l'espace sot physque sot spectral. Le champ de vtesse ltrée représente le mouvement des grandes échelles et les équatons de Naver-Stokes ltrées contennent le tenseur de sous-malle (SGS) qu résulte des mouvements des pettes échelles. [1,2] L'esprt de la LES est de modélser les pettes échelles en utlsant les quanttés résolues des grandes échelles, à travers un modèle de sous-malle (SGS). Il y a deux étapes dans chaque modèle de sous-malle : 1. Chaque modèle de sous-malle dot se baser sur une certane hypothèse de mouvement de sous-malle, c'est-à-dre assumer une formulaton pour les quanttés de sous-malle (en partculer, le tenseur de sous-malle). Cependant, l y a toujours des facteurs ndétermnés dans cette hypothèse. 2. Un modèle de sous-malle complet dot fare appel à certanes méthodes an de détermner les facteurs nconnus mentonnés c-dessus. Pluseurs hypothèses de tenseur de sous-malle ont été ntrodutes, telles que l'hypothèse de vscosté turbulente [3], la formulaton de smltude d'échelle [4], l'hypothèse de gradent de duson [5] et la formulaton par ncréments de vtesse [6]. Pour évaluer les performances de ces propostons, l faudrat dénr des crtères d'évaluaton. Il est mpossble de smuler le mouvement correct de sous-malle pour chaque pont dscret, mas au mons deux comportements physques devraent être mplqués

152 dans une hypothèse de sous-malle : d'abord, l'hypothèse devrat produre une dsspaton approprée, qu pourrat représenter le comportement dsspatf fort des pettes échelles; deuxèmement, l'hypothèse devrat représenter un certan mécansme physque, à savor l'nteracton entre les échelles résolues et de sous-malle. Des premères hypothèses de sous-malle n'ont pas prêté attenton à ces condtons. Par exemple, le modèle de smltude d'échelle [4] approche ben certanes caractérstque des tenseur de sous-malle, mas l ne produt pas assez de dsspaton. Pusque la dsspaton de sous-malle est très mportante dans le calcul numérque, la plupart des travaux sur les modélsatons de sous-malle se focalsent sur la dsspaton. Cependant, un modèle dsspatf pur, comme le modèle mplcte (MILES) [10], ne peut pas représenter le mécansme physque correct. Dans ces hypothèses, un type de relaton entre les échelles de sous-malle et les mouvements résolus est supposé, mas le mouvement de sous-malle ne peut pas être complètement xé. Il reste des coecents à détermner. Par exemple, la vscosté turbulente peut être calculée par le modèle de Smagornsky. Dans l'hypothèse d'ncréments de vtesse, un coecent dot auss être calculé. Essentellement cela sgne qu'une fos les hypothèses formulées, une autre certane méthode dot être applquée pour obtenr une fermeture des équatons. Dans cette thèse, nous ne sommes pas ntéressés aux dérentes hypothèses, mas plutôt aux méthodes de fermeture qu permettent de précser les coecents, une fos les hypothèses formulées. De cette façon, l n'y a aucun pur modèle de sous-malle. Pour chaque modèle, on dot détermner les facteurs ndétermnés, tels que la vscosté turbulente. Par exemple, le coecent du modèle de Smagornsky pourrat être évalué en utlsant la théore EDQNM et pourrat auss être détermné dynamquement selon la procédure de Germano. En outre, les dérents détals d'une méthode pourraent donner des résultats dérents, par exemple beaucoup de travaux xent la vscosté turbulente par la théore de EDQNM, mas obtennent des formulatons de modèles dérentes, pusque les détals de ces dérvatons ne sont pas les mêmes. En fat, 138

153 nous préférons consdérer ces "modèles" comme "les méthodes dérentes de détermner les facteurs ndétermnés de l'hypothèse de vscosté turbulente". Dans cette thèse, nous nous concentrons sur "les méthodes", mas pas sur "les suppostons". En général, l y a beaucoup de méthodes mathématques employées dans la modélsaton de sous-malle, qu réalsent la fermeture des équatons mathématques. De plus, l y a la théore dans l'espace spectral présentée par Krachnan [15,16], qu est employée pour donner beaucoup de modèles SGS spectraux. Dans l'espace physque, la seule théore est présentée par Kolmogorov [9], dont les dérentes échelles sont séparées. Il décrt auss une lo physque qu devrat être satsfate dans les smulatons des grandes échelles [17]. Cependant, peu de travaux ont été réalsés dans la modélsaton de sous-malle. Dans cette thèse, nous employons l'équaton de Kolmogorov de quanttés ltrées (KEF) dans la modélsaton de sous-malle. C'est une méthode pour détermner les facteurs nconnus de n'mporte quelle hypothèse de sous-malle. Dans le chaptre 2, les formulatons dverses de KEF dans des champs de vtesse sont présentées. La formulaton en turbulence sotrope homogène, proposés par Meneveau [17] et Cu [38], peut être écrte comme sut: 4 5 ϵ fξ 4 5 ϵ< ξ = D < lll 6T l,ll 6ν D< ll ξ. (7.1) Le budget de l'énerge correspondante dans une turbulence forcée est ndqué dans la Fg. (7.1). Le terme de sous-malle de transfert T l,ll est du même ordre de grandeur que la foncton de structure de la trosème ordre D < lll lorsque la dstance est égale à la talle du ltre, c'est-à-dre ξ =, et tend vers zéro lorsque ξ est grande. Le terme de vscosté est en eet seulement mportant à très pette dstance et peut être néglgé quand ξ est du même ordre de grandeur que. La formulaton en turbulence ansotrope nhomogène content des termes rapdes et lents, que l'on pourrat consdérer comme une forme complète 139

154 1 normalzed by 4/5εξ F D lll < T l,ll -6ν dd ll < / dξ D lll ξ/ Fgure 7.1: Le budget d'énerge en deux ponts normalsé dans la turbulence forcée avec ltre ( <) et sans ltre (sans <). Calcul EDQNM. La lgne vertcale ndque la talle de ltre. de KEF, mas trop complexe pour être applquée dans la modélsaton de sous-malle: 1 D < 2 t +1 D j < 2 ξ j δ u j < D< + δ u < ξ j ξ j D < j =H rapd ( x) + H slow ( x) + ν D< ξ j ξ j 2ε < + δu < ( τ slow< j ) slow< τj + δu < x j ( τ rapd< j ) rapd< τj x j (7.2) La turbulence homogène csallée est partculèrement analysée. En comparant avec l'analyse classque pour la foncton de corrélaton dans l'espace spectral, nous examnons l'eet du csallement dans l'espace physque. Dans la formulaton de KEF, les proprétés ansotropes des fonctons de structures entre des drectons dérentes sont soulgnées. Les tests a pror sont réalsés en turbulence de canal, montrant les dérences au nveau des fonctons de structures et des valeurs de la d-symétre (skewness). Parce que les fonctons de structures ltrées exstent dans KEF, nous devons examner la lo d'échelle correspondant. Nous pouvons obtenr la 140

155 concluson que la lo d'échelle classque pour des fonctons de structures non-ltrées peut seulement être utlsée quand la dstance est beaucoup plus grande que la talle de ltre dans KEF, c'est-à-dre ξ. Dans le chaptre 3, les applcatons de KEF sur des modèles de vscosté sont dscutées. Un problème du modèle CZZS orgnal, tent aux hypothèses dérentes à des échelles dérentes. L'améloraton content les modèles à une échelle, les modèles mult-échelles dans la sous-gamme nertelle et le modèle de mult-échelle avec les échelles séparées. Les modèles à une échelle ont une formulaton tout à fat smple. Nous pouvons auss présenter le skewness en deux ponts pour amélorer le modèle. On propose auss le modèle mult-échelle avec les échelles séparées. Il utlse une pette échelle pour l'expanson de Taylor et une grande échelle pour applquer la lo d'échelle. L'dée de KEF peut auss être applquée dans la détermnaton du coecent du modèle Smagornsky. Dans la sous-gamme nertelle, les valeurs des résultats sont semblables aux valeurs classques. Tant que nous chosssons la talle de l'ncrément approprée, cette méthode pourrat être applquée dans LES. Un autre traval sur le champ de vtesse est le modèle avec vscosté ansotrope, qu s'appue sur KEF ansotrope nhomogène. La méthode de la moyenne sphérque est employée, et la vscosté turbulente peut être représentée par: ν t = (D< j n j) S + (δud < n 1) S + 2V/S δu < δs < ( ) V D < S 2 u < n j 4V/S u < V. (7.3) ξ j L'avantage prncpal de ce modèle est que la vscosté turbulente peut être écrte comme une foncton de la moyenne du csallement, qu devrat être très mportant dans la turbulence csallée. Ce modèle est véré dans la turbulence de rotaton et la turbulence de paro respectvement. 141

156 Dans le chaptre 4, on propose un modèle améloré d'ncréments de vtesse (IVI). Une hypothèse très smple a été présentée par Brun: τ j < = fq j, Q j = δu < δu < j, (7.4) où Q j est l'ncrément de vtesse. C f est le coecent dynamque, on peut applquer KEF pour le détermner. Dans la turbulence à grand nombre de Reynolds et quand les échelles (la talle de ltre, la talle de l'ncrément de vtesse et la dstance de deux ponts dans KEF) sont ben séparées, nous pouvons obtenr la valeur constante : C f = 1/2. (7.5) Dans la turbulence à nombre de Reynolds modéré, l n'est généralement pas possble de séparer les échelles, et l'on peut obtenr une formule dynamque où les échelles sont xés de la même façon: C f = 4ξ D< lll (r) r 2D lll < (ξ). (7.6) 4D lll < (ξ) D< lll (2ξ) r=ξ Ces modèles sont vérés en turbulence sotrope homogène et dans le cas du canal plan. Les résultats montrent un bon accord avec les résultats de DNS et les expérences. En partculer, la formulaton du coecent constant est tout à fat smple et bon marché. Ce modèle peut être utlsé dans les projets d'ngénere. Dans le chaptre 5, nous nous concentrons sur l'étude du mécansme physque entre des partes moyennées et uctuantes, dans la turbulence scalare ansotrope nhomogène. L'équaton d'énerge scalare et l'équaton de ux scalare sont décomposées en partes rapdes et lentes. Le phénomène de backscatter est observé dans la régon 10 < Y + < 20, qu est semblable aux résultats de Hartel et al. [26] dans le champ de vtesse. Le transport de ux scalare est découpé en quatre partes. Dans la régon homogène du canal, la parte de GVSS (la vtesse résolue et le scalare de sous-malle) est beaucoup plus forte que la parte de SVGS (la vtesse de sous-malle et 142

157 le scalare résolu), ce qu est en d'accord avec les résultats d'yeung [131] et Fang [132]. Il est ntéressant de vor s les modèles exstants peuvent représenter l'nuence de la vtesse et du scalare moyens. Nous testons la parte lente en utlsant ECM (la modèlsaton Cu étendue), dans laquelle on montre explctement la vtesse et la scalare moyenne; la parte rapde est dénotée en utlsant SSM (le modèle de smltude d'échelle), à la sute de Shao [48]. Il est trouvé que ECM peut ben représenter la dsspaton scalare lente, de sous-malle, et cec même sur le backscatter dans Y Le SSM représente ben la parte rapde. Ans, on pourrat consdérer un modèle mélangeant des partes lentes et rapdes dans des applcatons de LES. Un problème restant est que la dsspaton présente près de la paro, qu exste dans les deux champs scalares et de la vtesse, ne peut être représentée par aucun modèle de sous-malle. Ce problème devrat être examné à l'avenr. Dans le chaptre 6, la formulaton de KEF est dérvée dans les champs de Elsässer, pour la turbulence MHD en présence de uctuaton de champ magnétque. Des modèles de sous-malle correspondants sont alors formulés. Cependant, les modèles présents ne sont pas numérquement stables quand la corrélaton entre des champs de la vtesse et magnétque est pette. Ce sujet devrat être davantage étudé. 143

158 Appendx A The dervatons of the Karman-Howarth equaton, Kolmogorov equaton and Yaglom equaton of ltered velocty and scalar, n homogeneous sotropc turbulence and scalar turbulence Ths process of dervaton was ntroduced by Cu et al. [38]. We consder the tensors n ncompressble homogeneous sotropc turbulence. The second- and thrdorder correlaton functons of resolved velocty uctuatons could be wrtten as R < j( ξ) = u < ( x)u < j ( x + ξ), R < j,k ( ξ) = u < ( x)u < j ( x)u < k ( x + ξ). (A.1) In the followng content of ths appendx, we use the sgnals R j and R j,k nstead of R j < and R< j,k. They also represent the resolved parts. Another thrd-order tensor s the correlaton between the ltered velocty and subgrd stress: T,jk ( ξ) = u < ( x)τ < jk ( x + ξ). (A.2) As suggested by Lumley [127], n homogeneous sotropc turbulence the secondorder tensor has only 2 ndependent components: R j ( ξ) = f 1 (ξ)ξ ξ j + f 2 (ξ)δ j, (A.3) 144

159 and the thrd-order tensor has 4 ndependent components: R j,k ( ξ) =g 1 (ξ)ξ ξ j ξ k + g 2 (ξ)ξ δ jk + g 3 (ξ)ξ j δ k + g 4 (ξ)ξ k δ j, T,jk ( ξ) =h 1 (ξ)ξ ξ j ξ k + h 2 (ξ)ξ δ jk + h 3 (ξ)ξ j δ k + h 4 (ξ)ξ k δ j. (A.4) Because of symmetry R j,k ( ξ) = R j,k ( ξ), T,jk ( ξ) = T,kj ( ξ), we could obtan that The ncompressble condton s employed, that R j ( ξ) ξ j = 0, g 2 = g 3, h 3 = h 4. (A.5) R j,k ( ξ) ξ k = 0, T,jk ( ξ) ξ k = 0. (A.6) Notce that there s the relaton of transform between Cartesan and Sphercal coordnate systems, that for any sotropc vector ϕ, ϕ = ϕ r x r + 2ϕ r r. Therefore R j ( ξ) ( = 5f 1 + ξ df ) 1 ξ + df 2 = 0, ξ j dξ dξ R j,k ( ξ) ( = 5g 1 + ξ dg 1 ξ k dξ + 2 ξ T,jk ( ξ) ( = 5h 1 + ξ dh 1 ξ dξ + 2 ξ ) ( dg 2 ξ ξ j + 2g 2 + 3g 4 + ξ dg ) 4 δ j = 0, dξ dξ ) ( dh 3 ξ ξ j + 2h 3 + 3h 2 + ξ dh ) 2 δ j = 0. dξ dξ (A.7) Fnally the correlaton functons could be smpled by usng the longtudnal components [149]: R j ( ξ) = ( R ll + ξ 2 ) R ll δ j R ll ξ ξ j ξ ξ 2ξ, R ll,l ξ R ll,l R j,k ( ξ) ξ = ξ 2ξ 3 ξ j ξ k + 2R ll,l + ξ R ll,l ξ 4ξ (ξ δ jk + ξ j δ k ) R ll,l 2ξ ξ kδ j, (A.8) T l,ll ξ T l,ll T,jk ( ξ) ξ = ξ 2ξ 3 ξ j ξ k + 2T l,ll + ξ T l,ll ξ 4ξ (ξ k δ j + ξ j δ k ) T l,ll 2ξ ξ δ jk. The governng equaton of homogeneous sotropc LES could be wrtten as u < t + u < k u < x k = p< x + ν 2 u < x k x k τ < k x k. (A.9) 145

160 In another pont x, t leads to u < j t u < + u < j k x k = p< x j + ν 2 u < j x k x k < τ jk. (A.10) x k (A.9) u < j + (A.10) u <, and takng ensemble average, we have the equaton of correlaton R j as follows: R j t 1 ρ + u< j u < u< k x k u < j u < j p < 1 u < x ρ τ k < u < x k + u< u< j u < k x k = p < + ν x j τ jk <. x k u < j 2 u < + ν x k x k u < j For homogeneous turbulence t could be further smpled as R j t 1 ρ + u< j u < u< k ξ k + u< u< j u < k = ξ k p < u < j 1 p < u < x ρ x j τ < k u < j τ < jk u< ξ k ξ k. + ν 2 u < u< j + ν ξ k ξ k 2 u < x k x k 2 u < j u< ξ k ξ k (A.11) (A.12) The pressure-velocty correlaton term s vanshng n sotropc turbulence. Thus the correlaton equaton becomes R j t R k,j ξ k + R,jk ξ k = 2ν 2 R j ξ k ξ k + T k,j ξ k T,jk ξ k. (A.13) Insertng equaton (A.8) to equaton and applyng the same manpulaton n dervaton of classcal Karman-Howarth equaton [49], we could then obtan the moded Karman-Howarth equaton for resolved-scale turbulence: R ll = 1 t ξ 4 ξ (ξ4 R ll,l ) + 2 ν ( ξ 4 R ) ll 1 ξ 4 ξ ξ ξ 4 ξ (ξ4 T l,ll ). (A.14) In comparson wth the classcal Karman-Howarth equaton of sotropc turbulence, the last term n the rght-hand sde s an addtonal term representng energy transport between resolved- and unresolved-scale turbulence n the moded Karman- Howarth equaton. 146

161 The second- and thrd-order correlaton functons could be expressed n the secondand thrd-order structural functons for sotropc turbulence: D < ll =2 u <2 2R ll, D < lll =6R ll,l, (A.15) n whch u < s the longtudnal resolved-scale velocty, D < ll = (u < (x + ξ) u < (x)) 2 and D < ll = (u < (x + ξ) u < (x)) 2 are the second- and thrd-order longtudnal structural functons respectvely. Replacng correlaton functon n equaton (A.14) by structural functons, we could obtan the followng equaton for structural functons: (u 1 ) <2 1 D ll < = 1 t 2 t 6ξ 4 ξ (ξ4 D lll < ) ν ( ) ξ 4 D< ll 1 ξ 4 ξ ξ ξ 4 ξ (ξ4 T l,ll ). (A.16) Ths process of dervaton of the Yaglom equaton of ltered scalar, n homogeneous sotropc scalar turbulence was ntroduced n Zhou's PhD thess [125]. The passve scalar s denoted as θ. The governng equaton of resolved scalar could be wrtten as where τ < θj θ < t + u< j θ < = κ 2 θ < + τ < θj, s the subgrd scalar ux. The correlaton functons are dened as (A.17) R < θθ ( ξ) = θ < ( x)θ < ( x + ξ), R < θj,θ ( ξ) = θ < ( x)u < j ( x)θ < ( x + ξ). (A.18) The second-order correlaton functon denotes the correlaton between ltered scalars, and the thrd-order correlaton functon s the mxed functon among one velocty and two scalars. In the followng part, we use the sgnals R θθ and R θj,θ nstead, respectvely. Another thrd-order correlaton functon between the resolved scalar and the subgrd scalar ux s denoted as T θ,θj ( ξ) = θ < ( x)τ < θj ( x + ξ). (A.19) Wrte the governng equaton (A.17) n another pont x, t leads to θ < t + u < j θ < x j = κ 2 θ < x j x j + < τ θj. (A.20) x j 147

162 (A.17) (A.20), and takng ensemble average, we have the equaton of correlaton R θθ n homogeneous turbulence: R θθ t = ξ j (R θj,θ R θ,θj ) + 2κ 2 R θθ ξ j ξ j. (A.21) Employng the sotropc condton, the three-order tensors are smpled as R θj,θ = R θl,θ ξ j ξ T θ,θj = T θ,θl ξ ξ j (A.22) where the longtudnal components are R θl,θ (ξ) = θ < (x 1 )u < 1 (x 1 )θ < (x 1 + ξ), T θ,θl (ξ) = θ < (x 1 )τ < θl (x 1 + ξ). (A.23) Therefore, equaton (A.21) could be rewrtten as ( R θθ = 2 t ξ + 2 ) ( R θl,θ + κ R θθ ξ ξ T θ,θl ). (A.24) In sotropc scalar turbulence, the second- and thrd-order correlaton functons could be represented by the second- and thrd-order structure functons: D < θθ =2 θ<2 2R θθ, D < lθθ =4R θl,θ, (A.25) n whch D < θθ = (θ < (x 1 + ξ) θ < (x 1 )) 2, D < lθθ = (u < 1 (x 1 + ξ) u < 1 (x 1 )) (θ < (x 1 + ξ) θ < (x 1 )) 2. (A.26) Thus we could obtan the Yaglom equaton of ltered scalar for homogeneous sotropc scalar turbulence: θ <2 t 1 2 D < θθ t = 2 [ ( ξ 2 Dlθθ ξ 2 ξ 4 κ D θθ 2 ξ T θ,θl )]. (A.27) 148

163 Appendx B Numercal method for homogeneous sotropc turbulence Lots of researchers have attempted to smulate homogeneous sotropc turbulence usng DNS or LES. One of the recent DNS dababase s a case oered by Meneveau's group [152]. In ths thess, we follow Zhou's methods [125] to generate a homogeneous sotropc turbulence. The DNS cases have grds, and the LES cases have 96 3, 64 3 or 48 3 grds. Governng equaton The governng equatons n DNS are u x =0, u t + u u j = 1 p + ν 2 u + f, ρ x (B.1) where f s a random forcng ntroduced to generate stable homogeneous turbulence. The numercal method has been ntroduced by Eswaran [153] and Overholt [154]. If f = 0, the turbulence s naturally decayng. The governng equatons n LES are u < t + u < j u < x =0, u < = 1 p < + ν 2 u < τ j < + f <, ρ x (B.2) 149

164 where τ j < s the subgrd stress. The man smulatng processes are the same between DNS and LES, thus n the followng content, the DNS calculaton s prncpally ntroduced. Dscretzaton method The dscretzaton n physcal space could be wrtten as x = (x 1, x 2, x 3 ) = (x, y, z) = (l 1, l 2, l 3 ), (B.3) n whch l 1, l 2, l 3 are ntegers, 0 l 1, l 2, l 3 computng doman, N s the number of grds. The dscretzaton n spectral space could be wrtten as N 1, = L, L s the length of N k = (k1, k 2, k 3 ) = (m 1 k mn, m 2 k mn, m 3 k mn ), (B.4) n whch m 1, m 2, m 3 are ntegers, N 2 m 1, m 2, m 3 N 2 1, k mn s the mnmal non-zero wave number, k mn = 2π L. The velocty and pressure could be expressed n spectral space: u ( x, t) = k p( x, t) = k û ( k, t ) e ι k x, ( ) p k, t e ι k x, (B.5) n whch denotes the physcal quantty n spectral space. Therefore, the governng equatons n spectral space are k û =0, û t + ιk jû u j = 1 ρ ιk p + νι 2 k 2 û. (B.6) Intal condton The ntal velocty eld s generated n spectral space, by employng Rogallo's method [54]. The ntal velocty eld should satsfy both the contnuous condton and the gven energy spectrum. For any wave number vector k n spectral space, the 150

165 Fgure B.1: Coordnate system of ntal velocty eld n spectral space. velocty vector must be n the vertcal plane, to satsfy the contnuous condton (see Fg. B). For any wave number k, a coordnate system s set as { e 1, e 2, e 3}, n whch e 3 s parallel to k, and { e 1, e 2 are n the vertcal plane of k. The ntal velocty could be wrtten as ( ) ( ) ( ) u k, 0 = α k e 1 + β k e 2, n whch α and β are determned by the spectrum condtons,.e. û û da(k) = E(k), A(k) (B.7) (B.8) where û s conjugate to û. When the followng relatons are gven, the spectrum condton s satsed: E(k) α = 4πk 2 eιθ 1 cosϕ, E(k) β = 4πk 2 eιθ 2 cosϕ, where θ 1, θ 2 and ϕ are random varables satsfyng unform dstrbuton n (0, 2π). The parallel axs e 3 s calculated as (B.9) e 3 = k 1 k e 1 + k 2 k e 2 + k 3 k e 3. (B.10) 151

166 We could smply let e 1 e 3, t leads to e 1 = e 3 e 3, e 2 = e 3 e 1. (B.11) Therefore, the ntal velocty eld n spectral space s ( ) ( ) ( αkk 2 + βk 1 k 3 u(k 1, k 2, k 3 ) = k αk 2 k 3 βkk 1 e 1 + k1 2 + k2 2 k β k1 2 + k2 2 e 2 k1 2 + k2 2 k ) e 3. (B.12) The Von-Karman energy spectrum s employed to generate ntal velocty eld. It could be wrtten as E(k) = E(k p )2 17/6 (k/k p ) 4, (B.13) [1 + (k/k p ) 2 17/6 ] where E(k p ) and k p are the peak value and peak locaton of energy spectrum, respectvely. Derent values could lead to homogeneous sotropc turbulence wth derent characterstc scale and knetc energy. Boundary condton Perodc boundary condton s appled n all drectons. It could be wrtten as u ( x + nl, t) =u ( x, t), p( x + nl, t) =p( x, t), (B.14) where n = n 1 e 1 + n 2 e 2 + n 3 e 3, n 1, n 2, n 3 are ntegers. Therefore, Fourer transform could be appled n all three drectons, and the governng equaton could be solved n spectral space. Temporal ntegral The solvng process could be smply wrtten as du dt = f(u, t), (B.15) where f(u, t) contans the convecton term, pressure term and vscous term n the governng equaton. 152

167 Fourth-order explct Runge-Kutta method n appled n temporal teraton: u n+1 =u n + t 6 (k 1 + 2k 2 + 2k 3 + k 4 ), k 1 =f(t n, u n ), k 2 =f(t n + t 2, u n + t 2 k 1), (B.16) The temporal step length should satsfy the stablty condton,.e., CFL condton: k 3 =f(t n + t 2, u n + t 2 k 2), k 4 =f(t n + t, u n + tk 3 ). velo max t x where velo max = max ( u max, v max, w max ). = 0.1 < 1, (B.17) Parallel algorthm Parallel calculaton s an eectve method for extensve numercal smulatons. The hghest-cost subroutne n the DNS program s Fourer transform. Therefore, Zhou ntroduced a parallel algorthm to reduce the cost [155]. The exsted seral lbrares of 1-D Fourer transform are used to mplement 3-D parallel Fourer transfer, by dvdng regons and exchange messages between computer nodes. Because the Fourer transform of a real functon s conjugate symmetrcal, we could store up only half of the data. Zhou's algorthm avods the message exchange of the conjugate symmetrcal operatons. 153

168 Appendx C Numercal method for channel turbulence The computer program of channel turbulence used n ths thess was developed by Xu [156]. The scalar part are wrtten by Zhou [125]. Governng equaton In channel ow, the Naver-Stokes equatons and the transport equaton of passve scalar could be wrtten as V t = V ω Π + ν 2 V 1 dp ρ dx e x, V =0, (C.1) θ t + V θ =κ 2 θ, where V s the velocty vector, θ s the passve scalar, ω = V s the vortcty, dp p s the mean pressure, s the mean pressure gradent n streamwse drecton, dx Π = p ρ + V 2 s the uctuaton part of total pressure. The characterstc velocty 2 s the mean velocty n the cross-secton U m, and the characterstc length s half channel wdth H. Reynolds number s dened as Re H = U m H/ν, Prandtl number s Pr = ν/κ. The computatonal doman s shown n Fg. C.1. x, y, z are coordnate axes, u, v, w are velocty components of each axs, respectvely. 154

169 Fgure C.1: Computatonal doman of channel turbulence. as The non-dmensonal governng equaton and boundary condton could be wrtten du dt =F x Π x + 1 Re 2 u, dv dt =F y Π y + 1 Re 2 v, dw dt =F z Π z + 1 Re 2 w, (C.2) dθ dt =F 1 θ + Re Pr 2 θ, F x =v ω z w ω y 1 dp ρ dx, F y =w ω x u ω z, F z =u ω y v ω x, F θ = u θ x v θ y w θ z, u x + v y + w z = 0, u(x + L x, y, z, t) =u(x, y, z, t), (C.3) (C.4) v(x + L x, y, z, t) =v(x, y, z, t), w(x + L x, y, z, t) =w(x, y, z, t), (C.5) θ(x + L x, y, z, t) =θ(x, y, z, t), 155