Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations

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1 ounal of Computational Physics 97 (004) Fully consevative finite diffeence scheme in cylindical coodinates fo incompessible flow simulations Youhei Moinishi a, *, Oleg V. Vasilyev b, Takeshi Ogi a a Gaduate School of Engineeing, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi , apan b Depatment of Mechanical Engineeing, Univesity of Coloado, 47 UCB, Boulde, CO 80309, USA Received 8 Septembe 003; accepted 8 Decembe 003 Available online 4 anuay 004 Abstact A new finite diffeence scheme on a non-unifom staggeed gid in cylindical coodinates is poposed fo incompessible flow. The scheme conseves both momentum and kinetic enegy fo inviscid flow with the exception of the time maching eo, povided that the discete continuity equation is satisfied. A novel pole teatment is also intoduced, whee a discete adial momentum equation with the fully consevative convection scheme is intoduced at the pole. The pole singulaity is emoved popely using analytical and numeical techniques. The kinetic enegy consevation popety is tested fo the inviscid concentic annula flow fo the poposed and existing staggeed finite diffeence schemes in cylindical coodinates. The pole teatment is veified fo inviscid pipe flow. Mixed second and high ode finite diffeence scheme is also poposed and the effect of the ode of accuacy is demonstated fo the lage eddy simulation of tubulent pipe flow. Ó 004 Elsevie Inc. All ights eseved. Keywods: DNS; LES; Consevation popeties; Finite diffeence scheme; Cylindical coodinate; Staggeed gid; Non-unifom gid; Enegy consevation; Pole teatment; Concentic annuli flow; Pipe flow. Intoduction It is well known that enegy-consevative finite diffeence schemes offe eliable and stable flow simulations and, in geneal, ae pefeed ove non-consevative schemes when used fo diect and lage eddy simulations of tubulent flows. Howeve, until ecently the standad second ode accuate staggeed gid finite diffeence scheme of Halow and Welch [] was the only scheme that simultaneously conseved mass, momentum, and kinetic enegy on a unifom mesh. A fully consevative high ode accuate finite diffeence scheme fo unifom Catesian staggeed gids was ecently developed by Moinishi et al. []. The scheme * Coesponding autho. Tel.: ; fax: addesses: youhei.moinishi@nitech.ac.jp (Y. Moinishi), Oleg.Vasilyev@Coloado.EDU (O.V. Vasilyev) /$ - see font matte Ó 004 Elsevie Inc. All ights eseved. doi:0.06/j.jcp

2 Y. Moinishi et al. / ounal of Computational Physics 97 (004) conseves momentum and kinetic enegy simultaneously povided that the flow is inviscid and the discete continuity equation is satisfied. Attempts to genealize the high ode fully consevative scheme of Moinishi et al. [] to non-unifom meshes wee not fully successful. The symmety peseving extension of the scheme poposed by Vasilyev [3] could not simultaneously conseve mass, momentum, and kinetic enegy but, depending on the fom of the convective tem, the consevation of eithe momentum o enegy in addition to mass was achieved. It was shown in [3] that the pesence of non-zeo commutation eo between aveaging and diffeencing opeatos in the non-unifom diections of the mesh esults in non-consevation of eithe enegy o momentum fo, espectively, advective o skew-symmetic foms of the convective tem. Realizing this limitation, Kajishima [4] and Ham et al. [5] wee able to extend the second ode fully consevative scheme of Halow and Welch [] to non-unifom gids by using weighted aveages. Howeve, the use of weighted aveages esulted in the eduction of accuacy on non-unifom meshes. The ultimate objective of ou study is to extend the fully consevative scheme of Moinishi et al. [] to nonunifom meshes such that both the consevation popeties and high ode accuacy ae peseved. The geneal extension of the method to ectangula cuvilinea coodinate system is cuently undeway and is based on the ecognition that in ode to achieve full consevation, the equations of motion should be ewitten in cuvilinea coodinates and the finite diffeence discetization should be pefomed in computational space (mapped cuvilinea coodinates) instead of physical space (non-unifom cuvilinea gid). This allows the constuction of enegy-consevative high ode finite diffeence schemes on non-unifom meshes. The common featue of othogonal cuvilinea coodinates is the pesence of the pole such as the axis of symmety in cylindical coodinates, whee, in geneal, the equations of motion ae singula. The pesence of a singulaity can destoy the consevation popeties of the scheme and, thus, equies special consideation. As a fist step to achieve the ultimate goal of constucting fully consevative high ode schemes on non-unifom meshes, we will conside cylindical coodinates. The cylindical coodinate system is chosen fo two easons. Fist, the Navie Stokes equations witten in cylindical coodinates ðx; ; hþ have a singulaity at the pole, ¼ 0. Second, many impotant flows of physical and engineeing inteest ae descibed in cylindical coodinates, e.g. [6 8]. Eggels et al. [6] and Akselvoll and Moin [7] applied the standad staggeed scheme in cylindical coodinates. Vezicco and Olandi [8] intoduced a special technique to emove the pole singulaity. Howeve, these schemes ae not enegy consevative. Only the ecent scheme by Fukagata and Kasagi [9] conseves enegy on a unifom gid fo inviscid flow. They intoduced the volume-weighted intepolation poposed by Kajishima [4] and Ham et al. [5] to cylindical coodinates. In ode to constuct fully consevative schemes, the special teatment is equied to emove the singulaity at the pole. In paticula, the adial velocity component at the pole is equied fo the flow simulations using the standad staggeed gid configuation. The existing pole teatments ae typically based on cental intepolations. The pole teatments by Fukagata and Kasagi [9] and Giffin et al. [0] ae singlevalued and have bette physical and numeical popeties, while the teatment by Eggels et al. [6] is multivalued. Howeve, thei adial velocity at the pole is not govened by the discete adial momentum equation and, consequently, the kinetic enegy is not conseved in the inviscid limit. In addition, the numeical teatment of pola coodinates by Mohseni and Colonius [], which avoids placing a gid point at the pole, is not possible fo the standad staggeed gid configuation. In this wok we popose to intoduce a discete adial momentum equation at the pole, which esults in enegy consevation. The single-valued popety is satisfied though the econstuction pocess. The objectives of this wok ae manifold. The fist objective concens the genealization of the high ode schemes of Moinishi et al. [] to non-unifom gids in cylindical coodinates. The second objective is to popose a novel pole teatment that in combination with the poposed scheme esults in a fully consevative high ode scheme. The pape is oganized as follows. The govening equations fo incompessible flow in cylindical coodinates and the coesponding tansfomed equations in computational space ae pesented in Section.

3 688 Y. Moinishi et al. / ounal of Computational Physics 97 (004) The consevation popeties of the new fomulation ae discussed thee as well. The adial momentum equation at the pole is intoduced in Section 3. Fully consevative high ode finite diffeence schemes in cylindical coodinates ae poposed in Section 4. In Section 5, the existing pole teatments with cental intepolation ae eviewed and a new pole teatment based on the adial momentum equation is poposed. Finally, in Section 6 numeical tests fo enegy consevation ae pefomed fo an inviscid concentic annula flow and the effect of pole teatments ae studied fo an inviscid pipe flow. Lage eddy simulations of tubulent pipe flow demonstate the meit of the poposed high ode fully consevative scheme.. Govening equations and consevation popeties The govening equations fo incompessible flow ae the continuity and momentum equations. The govening equations fo incompessible flow witten in cylindical coodinates ae given by: ou x ox þ ou o þ ou h oh þ u ¼ 0; ou x ot þ oðu xu x Þ ox ou ot þ oðu xu Þ ox þ oðu u x Þ þ o þ oðu u Þ þ o ¼ os x ox þ os o þ os h oðu h u x Þ oh oðu h u Þ oh oh þ ðs s hh Þ þ f ; þ ðu u x Þ þ op q ox ¼ os xx ox þ os x o þ os hx oh þ s x þ f x; þ ðu u u h u h Þ þ op q o ðþ ðþ ð3þ ou h ot þ oðu xu h Þ ox þ oðu u h Þ þ o oðu h u h Þ þ u u h þ op oh q oh ¼ os xh ox þ os h o þ os hh oh þ s h þ f h; whee u x, u and u h ae velocity components of axial (x), adial () and azimuthal (h) diections in cylindical coodinates, f x, f, and f h ae body foce components, q is the density, and p is the pessue. The components of the viscous tenso, s ij ði; j ¼ x; ; hþ fo the Newtonian fluid ae given by ou x s xx ¼ m ; s ¼ m ox ou h s xh ¼ s hx ¼ m þ ou x ox oh ou ; s hh ¼ m o ; s h ¼ s h ¼ m ou h o ou h oh þ u u h þ ; s x ¼ s x ¼ m : ou oh ou ox þ ou x ; o Many finite diffeence schemes in cylindical coodinates have been constucted in physical space, e.g. [6 8]. On the othe hand, the mapping of independent vaiables is a useful tool fo constucting finite diffeence schemes on non-unifom gids. In this study, ðx; ; hþ coodinates in physical space ae, espectively, mapped into ðf x ; f ; f h Þ in computational space as and x ¼ xðf x Þ; ¼ ðf Þ; h ¼ hðf h Þ ð5þ f x ¼ f x ðxþ; f ¼ f ðþ; f h ¼ f h ðhþ: ð6þ Fo instance, the mapping of the h plane including the pole onto the f f h plane is shown in Fig. o the case of N ¼ 4andN h ¼ 8. Scaling factos and the acobian ae defined as ð4þ

4 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Fig.. Mapping of (a) h plane onto (b) f f h plane fo N ¼ 4 and N h ¼ 8. h x ¼ dx df x ; h ¼ d df ; h h ¼ dh df ; ¼ h xh h h h ¼ dx d dh df x df df : h The deivatives in physical space ae tansfomed into computational space as o ox o h x of x ; o o o h of ; o oh h h o of h : In addition to the deivative mapping, we can use the following elations fo equation tansfomation: o o of x ¼ 0; h x of h ; o ¼ 0: ð9þ of h h h Finally, the tansfomed continuity and momentum equations can be witten as ðcont:þ o u of j j ¼ 0; ð0þ ou x ot þ ou ot þ ou h ot þ o of j o of j o of j u j u x u j u u j u h þ q h x u hu h þ u u h op of x os jx of j þ s x þ f x; þ q þ q h h h op of os j of þ ðs s hh Þ j op of h os jh of j þ s h þ f h: The epeated indices, fo instance, j, imply the summation ove j ¼ x; ; h. In addition, the index fo the scaling facto moves with the accompanying deivative index. The fomulation using the acobian was fist intoduced by Vinoku []. In this study, we will constuct a fully consevative finite diffeence scheme in þ f ; ð7þ ð8þ ðþ ðþ ð3þ

5 690 Y. Moinishi et al. / ounal of Computational Physics 97 (004) cylindical coodinates fo incompessible flow based on Eqs. (0) (3), which implies that all spatial discete opeations ae done in the computational space. The pessue tem in the momentum equation can be epesented symbolically as ðpes:þ i op q h i of : ð4þ i The convective tem in the momentum equation can be witten in the divegence, advective, o skewsymmetic foms. The divegence fom, ðdiv:þ i, is defined as ðdiv:þ x o u of j j u x ðdiv:þ o of j u j u u j u h ðdiv:þ h o of j The advective fom, ðadv:þ i, is given by ; ð5þ u hu h ; ð6þ þ u u h : ð7þ ðadv:þ x ¼ u j ou x of j ; ð8þ ðadv:þ ¼ u j ou of u hu h j ; ð9þ ðadv:þ h ¼ u j ou h of þ u u h : ð0þ j The skew-symmetic fom, ðskew:þ i, is defined as the aveage of the divegence and advective foms: ðskew:þ i ðdiv:þ i þ ðadv:þ i : ðþ The thee foms, ðdiv:þ i, ðadv:þ i, and ðskew:þ i, ae commutable with the following identities, povided that the continuity constaint is satisfied: ðdiv:þ i ¼ðAdv:Þ i þ u i ðcont:þ; ðþ ðskew:þ i ¼ðDiv:Þ i u iðcont:þ ¼ðAdv:Þ i þ u iðcont:þ: ð3þ Next, consevation popeties fo the momentum and the kinetic enegy in cylindical coodinates ae biefly eviewed. The tem witten in a fom, ð=þðo/=of j Þ (heeafte efeed as divegence fom), is consevative in the computational space, since the Gauss theoem is accomplished in the following way: Z V o/ of j dv ¼ Z V o/ of j dfx df df h ¼ Z S j / ds j ; whee dv ¼ df x df df h and ds j ¼ df x df df h =df j. Consequently, any tem witten in the divegence fom is conseved a pioi. It should be noted, that the momentum equations () (4) o the tansfomed mo- ð4þ

6 Y. Moinishi et al. / ounal of Computational Physics 97 (004) mentum equations () (3), in addition to the tems witten in divegence fom, have souce tems such as u h u h = o u u h =. These souce tems, analogously to the body foce components, contibute to the oveall momentum balance and enegy exchange. Howeve, these tems ae physical and eflect the cuvatue of the cuvilinea coodinate system. Consequently, consevation popeties of the finite diffeent schemes should be studied in light of thei analytical countepats, i.e. the tems witten in divegence fom should be conseved, while the souce tems in the discete equations should contibute to the oveall momentum and enegy balance the same way as in the continuous case. The kinetic enegy is defined as K ¼ð=Þu i u i ¼ð=Þðu x þ u þ u hþ. The tanspot equation fo K fo inviscid flow without the body foce is ok ot þ u iðdiv:þ i þ u i ðpes:þ i ¼ 0: ð5þ The pessue tem in the enegy equation is efomed as follows: u i ðpes:þ i o u of i i p p ðcont:þ: ð6þ h i The convective tem with the divegence fom in the enegy equation can be ewitten as follows: u i ðdiv:þ i o u of j j K þ K ðcont:þ: ð7þ In the same manne, the advective and skew-symmetic foms of the convective tem have, espectively, the following foms in the enegy equation: u i ðadv:þ i o u of j j K K ðcont:þ; ð8þ u i ðskew:þ i o u of j j K : ð9þ Theefoe, the convective tem witten in the skew-symmetic fom is consevative a pioi, and the othe convection foms and the pessue tems ae consevative, povided that the continuity constaint is satisfied. Thus, the kinetic enegy is conseved in the inviscid flow limit in the absence of the body foce f i. This enegy consevation should be peseved fo the enegy conseving scheme. 3. Radial momentum equation at the pole In this section, the momentum equation fo the adial velocity component is consideed at the pole, since the component is defined at the pole in the standad staggeed gid configuation. Hee we select the cylindical coodinate system as x ¼ x, y ¼ cos h and z ¼ sin h. Coesponding tansfomation fo the vecto components between h and y z planes is given by u ¼ u y cos h þ u z sin h; u h ¼ u y sin h þ u z cos h: ð30þ ð3þ In the same manne the tansfomation fo the tenso components ae s ¼ s yy cos h þ s zz sin h þ s yz sin h; ð3þ

7 69 Y. Moinishi et al. / ounal of Computational Physics 97 (004) s hh ¼ s yy sin h þ s zz cos h s yz sin h; ð33þ s h ¼ s h ¼ s yz cos h ðs yy s zz Þ sin h: ð34þ Using the single-valued popety at the pole fo the velocity and tenso components in Catesian coodinates, we get ou oh ¼ u h; ou h oh ¼ u at ¼ 0; ð35þ os h oh þ s s hh ¼ 0 at ¼ 0: ð36þ The elation of Eq. (35) was pointed out by Constantinescu and Lele [3]. These elations ae effectively used fo singulaity emoval with the aid of LÕHopitalÕs theoem: lim s h ¼ m o ¼0 oh ou o ; ð37þ os h lim þ s s hh ¼ o os h þ s s hh : ð38þ ¼0 oh o oh Eq. (37) will be adopted fo s h at the pole in the discete equations. The adial momentum equation, Eq. (3), is ewitten at the pole with the aid of Eq. (38) as ou ot þ oðu xu Þ þ oðu u Þ þ o ou h u þ u u u h u h þ op ox o o oh q o ¼ os x ox þ os o þ o os h þ s s hh þ f at ¼ 0; ð39þ o oh whee the convection tem is also ewitten in the same manne as Eq. (38). The above adial momentum equation econfims that the singulaity at the pole is not physical but coodinate oiginated, and it should be used in futue numeical and analytical studies. Howeve in this study, the singulaity emoval is adopted only fo the viscous tem, since the singulaity on the convection tem will be emoved to satisfy the enegy consevation in the coesponding discete equation. Thus, the adial momentum equation at the pole, which we use in this study, is given by ou ot þ o u of j j u u hu h þ q h op of h x os x of x þ h os of þ h o of os h oh þ s s hh þ f at ¼ 0: ð40þ 4. Fully consevative finite diffeence schemes in cylindical coodinates 4.. Second ode finite diffeence scheme In this study we popose the following scheme as the second ode accuate finite diffeence scheme fo the continuity and momentum equations in cylindical coodinates. Definition points fo the velocity components and pessue ae specified in Fig.. The pessue is defined at the cente of each cell,

8 Y. Moinishi et al. / ounal of Computational Physics 97 (004) ðx iþ= ; jþ= ; h kþ= Þ, while the velocity components in the axial, adial and azimuthal diections ae defined at ðx i ; jþ= ; h kþ= Þ, ðx iþ= ; j ; h kþ= Þ and ðx iþ= ; jþ= ; h k Þ, espectively, as the standad staggeed configuation. Fig. 3 shows the staggeed gid aangement in the f f h plane coesponding to Fig. (b). In paticula, the adial velocity is also defined at the pole, ðx iþ= ; 0; h kþ= Þ. The continuity equation is discetized at the pessue point, while the components of the momentum equation ae discetized at the coesponding velocity points: ðcont:-þ ¼0; Fig.. Definition points fo the velocity components and pessue. ð4þ ou x ot þðconv:-þ x þðpes:-þ x d s jx d j þ fx ou ot þðconv:-þ þðpes:-þ d s j d j þ f x s x þ f x ; ð4þ ðs s hh Þ þ f ; ð43þ ou h ot þðconv:-þ h þðpes:-þ h d s jh d j þ fh h s h þ f h ; ð44þ whee - denotes a second ode accuate appoximation on a staggeed gid in cylindical coodinates. Hee, we suppose that the components of the viscous tenso ae given at the equied discete points. In the poposed finite diffeence scheme, all the discete opeations ae done in the computational space. Spatial discete opeatos in the computational space ae defined in Appendix A. The definitions of local and global discete consevation is given thee as well. The finite diffeence appoximations fo the continuity and pessue tems can be witten as follows: ðcont:-þ d d j u j ð¼ 0Þ; ðpes:-þ i ¼ d p fi q h i d : ð46þ i ðconv:-þ i is a geneic fom of the convection tem and thee foms ae possible as in Moinishi et al. []. ð45þ

9 694 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Fig. 3. Staggeed gid configuation in f f h plane. Divegence fom: ðdiv:-þ x fx d d j ðdiv:-þ f d d j ðdiv:-þ h d fh d j Advective fom: " # f x f u j u j x " # u j u j 4 u j h u h j ; ð47þ f 3 5 þ fh u h f f hu h h ; ð48þ h u f f u h h : ð49þ ðadv:-þ x fx " f x #f j d u x u j d ; ð50þ j ðadv:-þ f " f #f j d u u j d j f u h f f hu h h ; ð5þ ðadv:-þ h fh 3 f h f j 4 d u h u j 5 þ d j fh h u f f u h h : ð5þ

10 Skew-symmetic fom: Y. Moinishi et al. / ounal of Computational Physics 97 (004) ðskew:-þ i ðdiv:-þ i þ ðadv:-þ i : ð53þ Note that the divegence fom of the convective tem is conseved a pioi in the momentum equation in the light of momentum consevation discussed in Section. The thee discete convection foms ae commutable with the following identities povided that the discete continuity equation is satisfied: ðdiv:-þ i ¼ðAdv:-Þ i þ u i fi ðskew:-þ i ¼ðDiv:-Þ i u i fi ðcont:-þ f i ; ðcont:-þ f i ¼ðAdv:-Þ i þ u i fi ðcont:-þ f i : Now we shall show the enegy consevation popety of the poposed finite diffeence scheme. To estimate the enegy consevation, a discete kinetic enegy nom defined at the pessue point is intoduced fo the second ode accuate discetization. K nd f f i i ui u i " x f fx u x u x # h f f þ f u u þ fh u h u h The coesponding discete kinetic enegy equation fo inviscid flow without the body foce can be witten as ok nd ot þ i f fi u i ðconv:-þ i þ f i fi u i ðpes:-þ i ¼ 0: The pessue tem in the enegy equation is consevative, povided that the discete continuity is satisfied: ifi fi ui ðpes:-þ d u q d i i p i p h i q ðcont:-þ: ð58þ The consevation popety of the skew-symmetic fom is poved as follows: fi ui ðskew:-þ ifi d f i 4 u d j j ðgu i u i Þ fj 3f i 5 u u h f f f f hu h h ð54þ ð55þ ð56þ ð57þ þ f u h fh h u f f u h h : ð59þ The fist tem on the ight-hand side of Eq. (59) is locally consevative. The sum of the second and thid tems ove a peiodic o zeo-bounded domain is ewitten as (see Appendix A) X f 6 4 u f u h f f hu h h þ u h u f u h f h fh3 f h 7 75 Df x Df Df h ¼ X h f u f u h f h u h f h þ u h f h u f u hi h Df x Df Df h ¼ 0: ð60þ

11 696 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Theefoe, the sum of the second and thid tems in the skew-symmetic fom is globally consevative in the enegy equation. Consevation popeties fo the divegence and advective foms ae the same as that fo the skew-symmetic fom, which follows fom the commutability equation (55). In summay, the poposed finite diffeence scheme satisfies the consevation popety as discussed in Section in a discete sense even fo non-unifom meshes, i.e. the scheme is fully consevative in cylindical coodinates on a non-unifom gid. This study assumes that a tempoal discetization eo is negligible fo the enegy consevation popety. We will intoduce a thid ode Runge Kutta time maching method fo the inviscid pat. Enegy-consevative tempoal discetization by Ham et al. [5] can be combined with the pesent spatial discetization method with the enegy nom of Eq. (56), although a non-linea implicit discete system should be solved at each time step. The acobian and scaling factos ae appoximated as h x ¼ Dx; h ¼ D; h h ¼ Dh; ¼ DxDDh; ð6þ with Df x ¼ Df ¼ Df h, whee Dx, D and Dh ae gid spacings at the defined point in the physical space, espectively. The aveaged acobian, fi, which appeaed in the discete equations, can be eplaced by fo simulations with computational egion, which does not include ¼ Mixed high and second ode finite diffeence scheme The azimuthal gid width is popotional to the adius fo standad gid configuation in cylindical coodinates. Theefoe, high ode finite diffeence schemes in the azimuthal diection may be useful fo lage eddy simulations in a pipe. In this section, a mixed high (x h) and second () ode enegy consevative finite diffeence scheme is pesented. The discete fom fo continuity and pessue tems ae ðcont:-n nþ X n= a ¼ d ð Þ d ð Þ f x u x h x þ d d u þ h X n= a ¼ d ð Þ d ð Þ f h u h h h ¼ 0; ð6þ ðpes:-n nþ x ¼ fx q ðpes:-n nþ ¼ f q ðpes:-n nþ h ¼ fh q X n= h x h ¼ d p d ; X n= h h ¼ a d ð Þ p d ð Þ f x ; a d ð Þ p d ð Þ f h ; ð63þ ð64þ ð65þ whee the a ae the intepolation weights and given as the solution of the following linea system: X n= ¼ ð Þ ði Þ a ¼ d i ; i ; ;...; n=: ð66þ The weights up to n ae summaized in Table. The components of the convection scheme witten in divegence fom ae:

12 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Table Weights fo nth ode intepolations n a a a 3 a 4 a 5 a þ þ 50 5 þ þ 5 45 þ þ 39; þ þ 35 3;768 3;768 3;768 3;768 3;768 0 þ 30;66 76;30 þ ; þ ;44 6;44 6;44 6;44 6;44 6;44 ðdiv:-n nþ x ðdiv:-n nþ ðdiv:-n nþ h Xn= a fx ¼ þ Xn= fx ¼ Xn= a f ¼ þ Xn= f ¼ X n= a fh ¼ þ X n= fh ¼ " # " nth f d ð Þ x d ð Þ f x ð Þf u x u x x þ # nth f d x h x fx d f u u x h " # nth f d ð Þ x ð Þf a u d ð Þ f h h u h x ; ð67þ h h " # " d ð Þ d ð Þ f x ð Þf u x u x þ # d h x f d u u h " # d ð Þ ð Þf a u d ð Þ f h h u h h h f u h nth f h nth f u h h ; ð68þ d ð Þ 4 d ð Þ f x u x h x d ð Þ a 4 u d ð Þ f h h h h nth f h u h ð Þ f x nth f h u h ð Þ f h þ nth f d h 4 fh d u u h 5 h 3 5 þ nth f h u f nth f u h h ; ð69þ whee / nth fx and / nth fh ae the nth ode intepolations that ae defined, espectively, as fh / nth fx ¼ Xn= a / ð Þ fx ; / nth fh ¼ Xn= a / ð Þ fh : ¼ ¼ ð70þ The divegence fom of the convection tem is locally consevative and the enegy is conseved, povided that the discete continuity equation, Eq. (6), is satisfied. Coesponding convection schemes with advective and skew-symmetic foms can also be defined. Analogously to the second ode case, the advective and skew-symmetic foms conseve momentum and enegy, povided that the discete continuity equation is satisfied. The eplacement of fx and fh by nthfx and nthfh in the denominatos impoves fomal spatial accuacy slightly fo non-unifom gids, but is not essential. The intoduction of the mixed high and second ode method is aimed at impoving the modified wave numbes in the steam and azimuthal diections.

13 698 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Pole teatment 5.. Existing pole teatments with cental intepolation In diect numeical simulations of tubulent pipe flow, Eggels et al. [6] and Akselvoll and Moin [7] intoduced the following pole teatment: u ðx; 0; h kþ= Þ¼bu ðx; 0; h kþ= Þ; ð7þ whee bu ðx; 0; h kþ= Þ¼ u ðx; ; h kþ= Þ u ðx; ; h kþ= þ pþ : ð7þ The velocity aangement aound the pole is shown in Fig. 4. The above teatment satisfies the antisymmetic elation u ðx; 0; h kþ= Þ¼ u ðx; 0; h kþ= þ pþ: Howeve, the coesponding velocity components in Catesian coodinates ae multi-valued a ¼ 0. Giffin et al. [0] intoduced a single-valued epesentation of the velocity component at ¼ 0. The tansfomation elations fo the vecto components between h and y z planes ae Eqs. (30), (3), and ð73þ u y ¼ u cos h u h sin h; u z ¼ u sin h þ u h cos h: ð74þ ð75þ Giffin et al. used Eq. (30) to detemine u at ¼ 0as u ðx; 0; h kþ= Þ¼u y ðxþ cos h kþ= þ u z ðxþ sin h kþ= : The coefficients, u y ðxþ and u z ðxþ, ae the aveaged values of u y plane). ð76þ and u z at ¼ 0 fo a given x (on a h Fig. 4. Velocity aangement aound the pole.

14 Y. Moinishi et al. / ounal of Computational Physics 97 (004) u y ðxþ N Xh bu ðx; 0; h kþ= Þ cos h kþ= bu h ðx; 0; h k Þ sin h k ; N h u z ðxþ N Xh bu ðx; 0; h kþ= Þ sin h kþ= þ bu h ðx; 0; h k Þ cos h k ; N h ð77þ whee bu ðx; 0; h kþ= Þ is given by Eq. (7) and bu h ðx; 0; h k Þ is given by bu h ðx; 0; h k Þ¼ u hðx; = ; h k Þ u h ðx; = ; h k þ pþ : ð78þ Giffin et al. intoduced one-sided intepolated values instead of bu and bu h in thei oiginal pape [0]. Recently, Fukagata and Kasagi [9] intoduced a single-valued epesentation of u at ¼ 0 based on the seies expansion of Constantinescu and Lele [3]. Thei pole teatment can be intepeted as Eq. (76) with u y ðxþ ¼ N Xh bu h ðx; 0; h k Þ sin h k ; N h u z ðxþ ¼ N Xh bu h ðx; 0; h k Þ cos h k : N h ð79þ The combinations of coefficients, u y ðxþ and u z ðxþ, ae also deived by using least squae minimization of the eo of Eqs. (30) and (3). The coefficients of Giffin et al. [0] ae obtained by least squae minimization of Eqs. (30) and (3), i.e. by minimizing the following L -eo: Q h ðxþ N Xh bu ðx; 0; h kþ= Þ u y ðxþ cos h kþ= u z ðxþ sin h kþ= N h þ N Xh bu h ðx; 0; h k Þ þ u y ðxþ cos h k u z ðxþ cos h k : N h ð80þ Othogonality elations of the tigonometic functions ae used in the deivation pocess. The coefficients of Fukagata and Kasagi [9] ae deived by least squae minimization of Eq. (3), i.e. by minimizing the following L -eo: Q h ðxþ N Xh bu h ðx; 0; h k Þ þ u y ðxþ cos h k u z ðxþ cos h k : N h ð8þ Thee exists anothe coefficient epesentation, which minimizes the squae eo of Eq. (30), Q : Q ðxþ N Xh bu ðx; 0; h kþ= Þ u y ðxþ cos h kþ= u z ðxþ sin h kþ= : N h ð8þ Coesponding coefficients ae: u y ðxþ ¼ N Xh bu ðx; 0; h kþ= Þ cos h kþ= ; N h u z ðxþ ¼ N Xh bu ðx; 0; h kþ= Þ sin h kþ= : N h ð83þ

15 700 Y. Moinishi et al. / ounal of Computational Physics 97 (004) In this study, the multi-valued pole teatment of Eq. (7) is called Mðu Þ. The single-valued pole teatment of Eq. (76) with Eqs. (77), (79) and (83) ae efeed to as Sðu ; u h Þ, Sðu h Þ and Sðu Þ, espectively. The singlevalued popety is a good indicato fo specifying the adial velocity at the pole. Howeve the intepolated values, bu and bu h, ae not govened by the momentum equation at the pole. 5.. A new pole teatment We believe that the best way to obtain u at the pole is to solve a discete adial momentum equation, which is discetized in the same manne as those fo the neighboing u points. Eq. () (and its oiginal fom, Eq. (3)) is mathematically singula at ¼ 0. Howeve, the oigin of the singulaity is not physical but geometical (coodinate system dependent), as was explained in Section 3. In this study, we popose to intoduce the following discete adial momentum equation that coesponds to Eq. (40) at ¼ 0: o u o t þðconv:-þ þ q d p f h d f h x d s x d x þ h d s d þ h d d f Df h Dh d s h d h þ d h d ðs s hh Þþf at ¼ 0; ð84þ whee ðconv:-þ is a geneic fom and specific foms ae defined in Section 4.. The enegy consevative convection schemes can also be applied fo the adial momentum equation at the pole. The enegy consevation popety of the convection tem in the egion including the pole is confimed in the same way as that in Section 4. with the enegy nom of Eq. (56). The intoduction of the aveaged acobian appeaing in the denominato of the discete equation is necessay fo emoving the singulaity. In the case of a unifom gid, the diffeence between and XN x i¼0 XN x i¼0 NX h = NX h = f xiþ= ;0;h kþ= þ XNx j xiþ= ;0;h kþ= þ XNx i¼0 i¼0 XN XN h j¼ XN XN h j¼ f xiþ= ; j;h kþ= ¼ p N = L x ð85þ j xiþ= ; j ;h kþ= ¼ pð N = = ÞL x ð86þ justifies the poposed acobian teatment, so the contol volume fo u with f fills a whole domain including the pole, while the standad teatment intoduces a hole at the pole because j x;0;h ¼ 0. This also eveals that the scheme with denominato woks fo simulations in the egion excluding ¼ 0 as mentioned at the end of Section 4.. Note that the acobian appeaing as a numeato in the convection tem of Eq. (84) is standad, i.e. zeo at ¼ 0. This condition is equied fo the enegy consevation when the discete continuity equation is solved with ðð=h Þu Þ¼0at ¼ 0. In addition, the application of ðpes:-þ at the pole yields op=o ¼ 0, which is not acceptable, while it offes the complete enegy consevation fo flows including the pole. The eason fo selection of the discete pessue tem (the thid tem on the left hand side of Eq. (84)) at the pole will be explained in Section 6. Some discete vaiables at 6 0 ae equied fo Eq. (84). Fist of all, we suppose u x ðx; ; hþ ¼u x ðx; ; h þ pþ; u ðx; ; hþ ¼ u ðx; ; h þ pþ; ð87þ ð88þ

16 ðx; ; hþ ¼ðx; ; h þ pþ; u ¼ 0: ð90þ h x;0;h Othe discete vaiables at 6 0 ae decided by imposing the condition that Eq. (84) at ðx iþ= ; 0; h kþ= Þ is equal to )Eq. (84) at ðx iþ= ; 0; h kþ= þ pþ, which guaantees the condition of Eq. (88) at ¼ 0. Fo instance, fom the elation, d d we get u h " # u u j h x; ;h x;0;h ¼ d d " # u u j h x;0;hþp ¼ u : ð9þ h x;;hþp In the same manne, the following elations ae specified fo the inviscid tems: u x ¼ u x ; ð9þ h x x; ;h h x x;;hþp u h ¼ h h x; ;h u h f hu h f h u h h h x; ;h Y. Moinishi et al. / ounal of Computational Physics 97 (004) ð89þ ; ð93þ x;;hþp ¼ u h fh f u h h : ð94þ x;;hþp The discete pessue tem satisfies the asymmetic condition. Requied elations fo the viscous tems ae: s x j x;0;h ¼ s x j x;0;hþp ; s j x; ;h ¼ s j x;;hþp ; ð95þ ð96þ Df h Dh d s h d h x; ;h ¼ Dfh Dh d s h d h ; ð97þ x;;hþp ðs s hh Þj x; ;h ¼ðs s hh Þj x;;hþp : ð98þ Supplementay explanations may be equied fo the teatment of viscous tems. The adial deivative of u x in s x at the pole is estimated by ðh Þ ¼0 ¼ = with Df and Eq. (87), so that Eq. (95) is satisfied. Eq. (96) is satisfied with ðh Þ ¼= ¼ðh Þ ¼ = ¼, Df, and Eq. (88). Eqs. (97), (98) and the coesponding discete tem in Eq. (84) ae consistent with Eq. (36) in the sense of second ode cental finite diffeence method. In the staggeed gid configuation, we should estimate os h =o in the discete u h equation at ¼ =, and theefoe we need s h at ¼ 0 fo the standad cental discetization. We intoduce a discete fom of Eq. (37) fo s h at ¼ 0

17 70 Y. Moinishi et al. / ounal of Computational Physics 97 (004) s h ¼ m Dfh d d u Dh d h h d f at ¼ 0: ð99þ This discete fom with Eq. (88) satisfies s x;0;h ¼ s x;0;hþp, which follows fom Eq. (34). Now we can close and solve the discete system fo the flow egion including the pole in cylindical coodinates. Howeve, the velocity components in Catesian coodinates at ¼ 0, coesponding to u obtained fom Eq. (84), ae still multi-valued. Theefoe, the single-valued econstuction by Eq. (76) with u y ðxþ ¼ N Xh u N ðx; 0; h kþ=þ cos h kþ= ; h u z ðxþ ¼ N Xh u N ðx; 0; h kþ=þ sin h kþ= h ð00þ is intoduced. Hee, u ðx; 0; h kþ=þ is the adial velocity at ¼ 0 obtained fom Eq. (84). In the pesent pole teatment, the adial velocity at the pole is obtained based on the adial momentum equation with the enegy-consevative convection scheme. The single-valued popety is satisfied though the single-valued econstuction at the pole. 6. Numeical tests 6.. Numeical method The coupling algoithm of the discete momentum and continuity equations fo the viscous flow is based on the second ode splitting method by Dukowicz and Dvinsky [4]. The esulting discete Poisson equation fo the pessue is solved diectly using FFT in the peiodic diections and ti-diagonal matix algoithm (TDMA) in the adial diection. Theefoe, the discete continuity equation is satisfied completely except fo the ound-off eo of the compute. The tempoal integation scheme is a combined RK3/CN scheme. Fo the tempoal integation scheme the device poposed by Akselvoll and Moin [5] is intoduced, whee the computational domain is divided into two egions in which tems with deivatives in only one coodinate diection ae teated implicitly with the Cank Nicolson scheme. In this study, second ode deivatives in the azimuthal diection ae teated implicitly in 0 6 < N=, while second ode deivatives in the adial diection ae teated implicitly in N= 6 6 N. Fo the inviscid flow simulations the algoithm is adopted with m ¼ Inviscid flow in a concentic annula pipe The objective of the fist numeical test is to study the enegy consevation popety of the poposed finite diffeence scheme. The test flow field is a concentic annula pipe with R =R ¼ 0: and R R ¼ D :0, whee R and R ae inne and oute adii as shown in Fig. 5. The computational domain is 0 6 x 6 L x ð¼ 4pDÞ, R 6 6 R and 0 6 h 6 p, whee the steamwise and azimuthal diections ae peiodic. The gid esolution, N x N N h,is The adial gid distibution is non-unifom with a hypebolic-tangent type stetching function, while the gid spacings in the peiodic diections ae unifom. The initial condition of the simulation is a flow field computed by a viscous simulation with m =360 and f x ¼. The inviscid simulations ae pefomed with m ¼ f x ¼ f ¼ f h ¼ 0. The time incement, Dt, is Fig. 6 shows the evolution of total kinetic enegy fo the inviscid flow simulation. The total kinetic enegy is defined as

18 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Fig. 5. Concentic annula flow. TK XN x XN XN h i¼0 j¼0 " x f fx u x u x # h f f þ f u u þ fh u h u h x iþ= ; jþ= ;h kþ= ; whee ðx iþ= ; jþ= ; h kþ= Þ coesponds to the pessue node, and 0 ¼ R and N ¼ R. The initial value is TK :75 at t ¼ t 0. Numeical simulations wee pefomed fo a numbe of existing finite diffeence schemes in cylindical coodinates. The details of these schemes ae given in Appendix B. The inviscid simulations with the standad, ðdiv:-stþ i, and Vezicco and Olandi, ðdiv:-voþ i, type convection schemes divege soon, while the total kinetic enegy with a Fukagata and Kasagi type convection scheme, ðdiv:-fkþ i, gadually inceases with time. This coincides with the esults epoted in [9]. As it is clealy seen in the figue, the total kinetic enegy of the poposed scheme, ðdiv:-þ i, slightly deceases with time. Fig. 7 shows the time incement dependence of the eo with the pesent scheme at t t The eo slope of ðdiv:-þ i is Dt 3, which is the eo due to the time integation scheme. Theefoe, the pesent finite diffeence scheme itself conseves the kinetic enegy completely with the exception of the tempoal integation eo Inviscid flow in a staight pipe Fig. 6. Total kinetic enegy evolution fo inviscid flow in a concentic annula pipe. The second numeical test is designed to check the enegy consevative popety of the pesent pole teatment. The test flow field is a staight pipe flow with adius Rð:0Þ as shown in Fig. 8. The

19 704 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Fig. 7. Dependence of the total kinetic enegy eo on time incement fo ðdiv:-þ and ðdiv:-fkþ. Fig. 8. Staight pipe flow. computational domain is 0 6 x 6 L x ð¼ 4pRÞ, 06 6 R and 0 6 h 6 p, whee the steamwise and azimuthal diections ae peiodic. The gid esolution, N x N N h,is66 3. The adial gid distibution is also non-unifom with a hypebolic-tangent type stetching function, while the gid spacings in the peiodic diections ae unifom. The initial condition of the simulation is a flow field computed by a viscous simulation with m =80, f x ¼ and f ¼ f h ¼ 0. The inviscid simulations ae done with m ¼ f x ¼ f ¼ f h ¼ 0. The time incement, Dt, is The diffeence of the total kinetic enegy evolution by diffeent pole teatments is illustated in Fig. 9. The total kinetic enegy is defined as in the pevious test case except that 0 ¼ 0and N ¼ R. The initial value is TK 0 ¼ 4804:65355 at t ¼ t 0. The finite diffeence scheme except at ¼ 0 is the pesent one with ðdiv:-þ i. The enegy with the multi-valued pole teatment (Mðu Þ) diveges quickly. The enegy with the single-valued pole teatments (Sðu ; u h Þ, Sðu h Þ, Sðu Þ) gadually incease and finally divege, while they ae bette than the multi-valued teatment. Theefoe, peviously existing pole teatments inject unphysical kinetic enegy poduction at ¼ 0. One of the main objectives of the pesent study is to intoduce the adial momentum equation to define u at ¼ 0. Diect application of the scheme poposed in Section 4. at ¼ 0 offes complete kinetic enegy consevation (zeo in Fig. 9). A

20 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Fig. 9. Kinetic enegy evolution fo inviscid flow in a staight pipe. slight decease comes fom the time integation eo. Howeve, it yields op=o ¼ 0at ¼ 0, which is not acceptable fom the physical point of view. Theefoe, we have examined an altenative teatment of the pessue tem at the pole. Two possible candidates wee the discete pessue tems with stencil o : op o d p h d ; op o d p f h d f : ð0þ The kinetic enegy of the case with Eq. (0), stencil in Fig. 9, gadually inceases with time while it is much bette than the cases with the existing pole teatments. On the othe hand, the case with the discete pessue tem given by Eq. (0), stencil in Fig. 9, conseves the enegy. This is the eason fo selecting the discete pessue tem poposed in Section Lage eddy simulation of tubulent pipe flow The thid numeical test is the lage eddy simulation (LES) of tubulent pipe flow. The computational domain is 0 6 x 6 L x ð¼ 4pRÞ, 06 6 R and 0 6 h 6 p, whee the steamwise and azimuthal diections ae peiodic. The gid esolution, N x N N h,is and The adial gid distibution is also non-unifom with a hypebolic-tangent type stetching function, while the gid spacings in the peiodic diections ae unifom. The numeical paametes ae m =80, R, f x ¼, and f ¼ f h ¼ 0. The coesponding Reynolds numbe is 80 based on the fiction velocity, u s, and the pipe adius, R. The subgid scale model is the dynamic Smagoinsky model of Gemano et al. [6] with the least squae modification of Lilly [7]. Fo detailed implementation of the model please efe to Moinishi and Vasilyev [8,9], whee the model is compaed with othe subgid scale models in plane channel flow. Hee we just use it fo cylindical coodinates. Figs. 0 and show the computational esults fo and gids, espectively. The esults with the espective mixed high and second ode schemes ae labelled as 4th, 8th and th FDM. The symbols (s) indicate the diect numeical simulation (DNS) data of pipe flow by Fukagata and Kasagi [9] with gids fo L x 0R at Re s 80. The LES with the 8th and highe ode simulations maintain the tubulent flow even fo gids, while the flow ð0þ

21 706 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Fig. 0. LES of tubulent pipe flow at Re s 80 with gid: (a) mean velocity pofiles; (b) tubulence intensities. Fig.. LES of tubulent pipe flow at Re s 80 with gid: (a) mean velocity pofiles; (b) tubulence intensities. with the second and fouth ode schemes ae laminaized. This implies a possibility that an LES of pipe flow with a moe sophisticated subgid scale model offes eliable esults with the high ode scheme even fo the coase gid. In addition, the isotopy in the h plane (u 0 ¼ u0 h ) is ecoveed close to the pole. The LES with the second ode simulation maintains the tubulent flow with a gid. Howeve, the eliability of the mean velocity and tubulence intensities pofiles ae not enough. Consideing the gid configuation fo the pipe, inceasing the ode of accuacy only in the peiodic diections is a ecommended device fo LES. The high ode schemes offe bette esults than the second ode scheme. The factos of computational cost of the simulation pe time step with gid fo the 4th, 8th and th ode schemes to the second one ae.6,.78 and.9, espectively. On the othe hand, the factos of the second ode scheme with , and gids to gid ae.08, 3.5 and 4.43, espectively. In addition, the use of fine mesh esults in memoy incease. Theefoe, the mixed high ode scheme is ecommended fo the LES of tubulent pipe flow. The mixed high ode schemes should be beneficial fo the DNS of pipe flow, although the meit of the schemes was tested only fo the LES. The epesentation of high ode statistics with the second ode scheme equies highe gid esolution than one fo mean velocity and tubulence intensity epesentation. Fo instance, the skewness facto of u as shown in Fig. is stongly affected by the ode of the method.

22 Y. Moinishi et al. / ounal of Computational Physics 97 (004) Fig.. Skewness facto of u fo the LES at Re s 80 with gid. 7. Conclusions The main objective of the pesent study was to impove the numeical simulation of incompessible flow in cylindical coodinates. A fully consevative finite diffeence scheme fo staggeed and non-unifom gids is poposed. The complete consevation is achieved by pefoming all discete opeations in computational space. This is an appopiate extension of the fully consevative finite diffeence scheme by Moinishi et al. [] to non-unifom gids in cylindical coodinates. A novel pole teatment is also poposed, whee the adial momentum equation is solved to detemine the velocity at the pole. The singulaity is popely emoved by analytical and numeical techniques. The single-valued popety of the velocity at the pole is satisfied though the poposed econstuction pocess. Reliability and consevation popeties of the poposed scheme ae numeically veified in inviscid flow simulations. The benefits of the poposed mixed high and second ode fully consevative scheme fo lage eddy simulations of tubulent flow in cylindical coodinates ae demonstated fo tubulent pipe flow. Acknowledgements The fist autho (Y. Moinishi) was patially suppoted by the Cente fo Pomotion of Computational Science and Engineeing, apan Atomic Enegy Reseach Institute, whose suppot is gatefully acknowledged. Patial suppot fo the second autho (O.V. Vasilyev) was povided by the National Science Foundation unde Gant Nos. EAR-0459, EAR-03769, and ACI and National Aeonautics and Space Administation unde Gant No. NAG--06. Appendix A. Discete opeatos The fist appendix pesents the discete opeatos used in this pape. The following discete opeatos ae basically the same as those poposed in [] except that the pesent opeations ae done in a computational space in cylindical coodinates. The intepolation opeato with stencil n acting on / in the f x diection is descibed as / n fx ðf x ; f ; f h Þ¼ /ðfx þ ndf x =; f ; f h Þþ/ðf x ndf x =; f ; f h Þ ; ða:þ

23 708 Y. Moinishi et al. / ounal of Computational Physics 97 (004) whee / n f and / n fh ae defined in the same manne as / n fx. The finite diffeence opeato with stencil n acting on / in the f x diection is given by d n / d n f x ðf x ; f ; f h Þ¼ /ðfx þ ndf x =; f ; f h Þ /ðf x ndf x =; f ; f h Þ ndf x ; ða:þ whee d n /=d n f and d n /=d n f h ae defined in the same manne as d n /=d n f x. An intepolation opeato fo the poduct of / and w is descibed as f/w n fx ðf x ; f ; f h Þ /ðfx þ ndf x =; f ; f h Þwðf x ndf x =; f ; f h Þ þ wðfx þ ndf x =; f ; f h Þ/ðf x ndf x =; f ; f h Þ; ða:3þ whee f /w n f and f /w n fh ae defined in the same manne as f /w n fx. Note that nf j, appeaing as a supescipt, does not follow the summation convention. Following ae useful identities associated with the discete opeatos: d n / m fi d n f j ¼ d n / d n f j m f i ; d n / mt d n f j ¼ d n / d n f j mt ; ða:4þ d n w / n fj ¼ w d n f n / j þ / d n w d n f j d n f j d n f j / d n w / n j f d n f j d n w f // n fj d n f j þ // d n w d n f j ða:5þ ða:6þ In this study, we set Df x ¼ Df ¼ Df h in the computational space. We now define two concepts of discete consevation. We say that a discetization of a tem, Q ¼ð=Þðo/=of j Þ,islocally consevative if it can be witten in the following fom: Q ¼ X n d n ðu n Þ : ða:7þ d n f j We say that a discetization of a tem, Q,isglobally consevative if the following elation holds in a peiodic field: X X X Q DV ¼ 0; ða:8þ f x f f h whee DV ¼ Q 3 j¼ Dfj. Note that in the peiodic case the local consevation (A.7) implies global consevation. Also note that the definition (A.8) is a discete analogue of Eq. (4). Finally, the following summation popety is used in the poof of the global enegy consevation in Section 4., Eqs. (59) and (60), fo the non-consevative tems in the peiodic o zeo-bounded diections. NX j = i¼= f j Nj = w / fj ¼ X i¼= w fj /: ða:9þ

24 Appendix B. Existing convection schemes fo cylindical coodinates To indicate the diffeence between the poposed and existing schemes, typical staggeed finite diffeence schemes in a cylindical coodinate system ae intepeted and epesented in the fom of Eqs. (47) (49). Theefoe, the following schemes have the essence of the oiginal fomulation, while they ae not identical to the oiginal ones fo the case of a non-unifom gid. In addition, the following schemes have bette consevation popeties than the oiginal fomulations. Standad type [6,7]: ðdiv:-stþ x fx d d j ðdiv:-stþ f d d j ðdiv:-stþ h fh d d j f f u j xu j x u j u j u j hu h j ; ðb:þ u h fh f u h f h f þ u h u f h Vezicco and Olandi type [8]: To emove the pole singulaity, Vezicco and Olandi [8] intoduced the quantity u. Hee we intepet it as q j ¼ u j. Rewiting the adial convective tem of Eq. (48) in tems of q j we obtain f d d j " # q j q j f " f q h f # h q h f h : The fist tem is still singula when the discete equation is adopted at the u node adjacent to the singula point. Theefoe, they eplaced it as " # " d q j f d j q j f q h f # h q h f h f : This is equivalent to the following fom: ðdiv:-voþ f d d j " # u j ðu Þ j f ðb:þ ðb:3þ u h f f hu h h ðb:4þ We call this scheme as Vezicco and Olandi type. It is appaent that the enegy consevation popety is destoyed at the eplacement stage, since this scheme does not equie u at the pole. Fukagata and Kasagi type [9]: The convective scheme by Fukagata and Kasagi [9] is basically close to ou scheme. Appaent diffeences appea in the adial and azimuthal components. ðdiv:-fkþ v Y. Moinishi et al. / ounal of Computational Physics 97 (004) f d d x þ f d d h " # " u x u x þ # d h x f d u u h " # u h u h h ðu h u h Þ fh ; ðb:5þ h h h

25 70 Y. Moinishi et al. / ounal of Computational Physics 97 (004) ðdiv:-fkþ h d fh d j 4 u j h u h j 3 5 þ u f h u h ; ðb:6þ whee the ad hoc coefficient v appeaed in the fist tem of Eq. (B.5) is defined as v ¼ ðh Þ =ðh Þ. The coefficient v is unity when the adial gid is unifom, and thei finite diffeence convective scheme is equivalent to ous when the gid is unifom. Howeve, thei scheme does not conseve enegy fo nonunifom meshes, although it is bette than the othe existing ones and seems not to cause poblem fo viscous flow simulations. Also the last tems in Eqs. (B.5) and (B.6) conseve enegy, while they ae diffeent fom ou fomulation. Note that these tems conseve enegy globally, even though the authos claimed it as locally consevative. Refeences [] F.H. Halow,.E. Welch, Numeical calculation of time-dependent viscous incompessible flow of fluid with fee suface, Phys. Fluids 8 (965) [] Y. Moinishi, T. Lund, O.V. Vasilyev, P. Moin, Fully consevative highe ode finite diffeence schemes fo incompessible flow,. Comput. Phys. 43 (998) [3] O.V. Vasilyev, High ode finite diffeence schemes on non-unifom meshes with good consevation popeties,. Comput. Phys. 57 (000) [4] T. Kajishima, Finite-diffeence method fo convective tems using non-unifom gid, Tans. SME B (999) (in apanese). [5] F. Ham, F.S. Lien, A.B. Stong, A fully consevative second-ode finite diffeence scheme fo incompessible flow on nonunifom gids,. Comput. Phys. 77 (00) [6].G. Eggels, F. Unge, M.H. Weiss,. Westeweel, R.. Adian, R. Fiedich, F.T.M. Nieuwstadt, Fully developed tubulent pipe flow: a compaison between diect numeical simulation and expeiment,. Fluid Mech. 68 (994) [7] K. Akselvoll, P. Moin, Lage eddy simulation of tubulent confined coannula jets and tubulent flow ove a backwad facing step, Stanfod Univesity Repot, TF-63, 995. [8] R. Vezicco, P. Olandi, A finite-diffeence scheme fo thee-dimensional incompessible flows in cylindical coodinates,. Comput. Phys. 3 (996) [9] K. Fukagata, N. Kasagi, Highly enegy-consevative finite diffeence method fo the cylindical coodinate system,. Comput. Phys. 8 (00) [0] M.D. Giffin, E. ones,.d. Andeson, A computational fluid dynamics technique valid at the centeline fo non-axisymmetic poblems in cylindical coodinates,. Comput. Phys. 30 (979) [] K. Mohseni, T. Colonius, Numeical teatment of pola coodinate singulaities,. Comput. Phys. 5 (000) [] M. Vinoku, Consevation equations of gasdynamics in cuvilinea coodinate systems,. Comput. Phys. 4 (974) [3] G.S. Constantinescu, S. Lele, A new method fo accuate teatment of flow equations in cylindical coodinates using seies expansions, CTR Annual Reseach Biefs 000, Cente fo Tubulence Reseach, NASA Ames and Stanfod Univesity Pess, Stanfod, CA, 00, pp [4].K. Dukowicz, A.S. Dvinsky, Appoximation as a highe ode splitting fo the implicit incompessible flow equations,. Comput. Phys. 0 (99) [5] K. Akselvoll, P. Moin, An efficient method fo tempoal integation of the Navie Stokes equations in confined axisymmetic geometies,. Comput. Phys. 5 (996) [6] M. Gemano, U. Piomelli, P. Moin, W.H. Cabot, A dynamic subid-scale eddy viscosity model, Phys. Fluids A 3 (99) [7] D.K. Lilly, A poposed modification of the Gemano subgid scale closue method, Phys. Fluids A 4 (99) [8] Y. Moinishi, O.V. Vasilyev, A ecommended modification to the dynamic two-paamete mixed subgid scale model fo lage Eddy simulation of wall bounded tubulent flow, Phys. Fluids 3 (00) [9] Y. Moinishi, O.V. Vasilyev, Vecto level identity fo dynamic subgid scale modeling in lage eddy simulation, Phys. Fluids 4 (00)