4.i. 4 Numerical Model Development 4.1

Transcription

1 4i Chapr 4 4 Numrica Mod Dvopmn 41 Asrac 41 Résumé Govrning quaions 4 4 Souion sragy: h iraiv mhod Numrica mhod: h fini-voum approximaion Grid arrangmn Compuaion of h surfac ara and of h c voum C-fac inrpoaion and gradin compuaion Discrisaion of h im drivaiv rms Discrisaion of h convciv rms Discrisaion of h diffusiv rms Convciv-diffusiv rms: hyrid and powr-aw schms Sourc rms Assmy of h cofficins 4 44 rssur-vociy couping SIMLE agorihm rssur corrcion procdur Undr-raxaion facor and im sp Boundary condiions Boundary pacmn Infow oundary Oufow oundary Wa oundary Symmry oundary Surfac oundary Souion procdurs Spaia discrisaion Marix sovrs Summary 453 Rfrncs 453 Noaions 454

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3 41 4 Numrica Mod Dvopmn Asrac rsnd in his chapr is h dvopmn of a 3D numrica mod inndd o simua fow around a cyindr Th mod is asd on h Rynods-avragd Navir- Soks and coninuiy quaions for incomprssi fow cosd wih h k-ε urunc mod Th working quaion of h mod is oaind y discrizing h govrning quaions wrin in a gnra convciv-diffusiv ranspor quaion using finivoum chniqus on a srucurd coocad oundary-fid hxahdra conrovoum grid Th hyrid Spading 197) or powr-aw aankar 1980) upwind-cnra diffrnc schm comind wih h dfrrd corrcion mhod Frzigr and ric 1997) is mpoyd in h discrisaion of h govrning quaions Th souion of h working quaion is achivd y an iraiv mhod according o SIMLE agorihm aankar and Spading 197) Aong soid oundaris us is mad of h wa funcion mhod whi aong surfac oundaris h prssur dfc is usd o dfin h surfac posiion On ohr oundaris namy in ou and symmry oundaris cassica mhods ar usd such as zro gradins zro srsss or known funcions Résumé C chapir présn un dévoppmn d un modè numériqu pour simur écoumn ridimnsionn auour d un cyindr L modè s asé sur a rprésnaion n voums finis ds équaions d Rynods d coninuié d k-ε Ls équaions sous form d un équaion d ranspor son n sui formués pour un maiag srucuré don s varias primiivs son définis au cnr ds voums d conrô Ls fux convcif diffusif son cacués par s méhods hyrid Spading 197) ou oi d puissanc aankar 1980) avc ds corrcions ds rms non orhogonaux Frzigr and ric 1997) L modè uiis a méhod iéraiv d SIMLE pour résoudr s équaions d ravai ainsi onus Ls condiions aux ords ong d un parois son imposés par a oi ogarihmiqu La surfac d au s dérminé à parir ds prssions résidus dans s cus d surfac En aurs ips ds ords par xmp à nré à a sori aux pans d syméri ds méhods sandards son appiqués soi ds gradins nus sans cisaimn ou ds vaurs connus

4 4 41 Govrning quaions Th fow mod ha is dvopd in his work is asd on h approxima souion of h im-avragd quaions of moion and coninuiy for incomprssi fows y using fini-voum mhod In h Carsian coordina sysm hs quaions rad: u + uu x + vu y + wu z = 1 p ρ x + 1 τ xx ρ x + 1 τ yx ρ y + 1 τ zx ρ z + g x 41) v + uv x + vv y + wv z = 1 p ρ y + 1 τ xy ρ x + 1 τ yy ρ y + 1 τ zy ρ z + g y 4) w + uw x + vw y + ww z = 1 ρ p z + 1 τ xz ρ x + 1 τ yz ρ y + 1 τ zz ρ z + g z 43) u x + v y + w z = 0 44) in which x y and z ar Carsian co-ordinas in h horizona ransvrsa and vrica rspcivy; u v and w ar h corrsponding im-avragd) vociy componns p is h im-avragd) prssur ρ is h mass dnsiy of war g x g y g z ar h x y z componns of h graviaiona accraion and τ ij s ar h j dircion componns of h shar srss acing on h surfac norma o h i dircion Ths srsss ar du o h mocuar viscosiy and urun fucuaion For fows having sufficiny high Rynods numr h viscous srsss ar much smar in comparison wih hos of h urunc and hus can ngcd Using Boussinsq s ddy viscosiy concp hs srsss ar proporiona o h vociy gradins according o h foowing xprssions s Rodi 1984 p 10): τ xx ρ = ν u x 3 k τ yy ρ = ν v y 3 k τ zz ρ = ν w z 3 k τ xy ρ = τ yx ρ = ν v x + u ) + y * τ xz ρ = τ zx ρ = ν w x + u ) z * τ yz ρ = τ zy ρ = ν w y + v ) z* + 45) in which ν is h urun or ddy viscosiy and k is h urun kinic nrgy dfind as k = 1 u! u! + v! v! + w! w! ) whr suprscrips man h fucuaing componns Insring h dfiniions in Eq 45 ino h momnum quaions Eqs 41 o 43 on oains:

5 43 u + uu x + vu y + wu z = 1 p ρ x k 3 x v + uv x + vv y + wv z = 1 p ρ y k 3 y w + uw x + vw y + ww = 1 z ρ z k 3 z + x ν u x + y ν v x + u ) * + y z ν w x + u z + g x + x ν v x + u * + y ) y ν v * + y ) z ν w y + v * + g z ) y + x ν w x + u z + y ν w y + v ) * + z z ν w z + g z Sparaing h norma and cross scond-drivaivs and puing h formr on h fhand sids on gs: u + uu x + vu y + wu z x ν u x y ν u ) y z ν u z = 1 p ρ x + x ν u x + y ν v x + z ν w x + g x 46) u + uv x + vv y + wv z x ν v x y ν v ) y z + x ν u ) + y y ν v ) + y z ν v z = 1 p ρ y 47) w ) + g y y ν w + uw x + vw y + ww z x ν w x y ν w y ) z ν w z + x ν u z + y ν v z + z ν w z + g z = 1 ρ p z 48) Th scond o fourh rms on h f-hand sid of Eqs 46 o 48 rprsn a convciv ranspor and h nx hr rms rprsn a diffusiv ranspor Th rms on h righhand sid ar considrd as sourcs and ar rad as known quaniis whn soving h quaions for h vociy componns u v and w Th urun kinic nrgy gradin ing sma compard o h prssur gradin is ngcd From h k-ε urunc mod Laundr and Spading 1974; Rodi 1984 p 7) h urun viscosiy ν is givn y:

6 44 ν = c µ k ε 49) whr ε is h dissipaion of h urun kinic nrgy Th fid disriuions of h urun kinic nrgy and is dissipaion ar oaind from h foowing ranspor quaions Laundr and Spading 1974; Rodi 1984 p 8): k + uk x + vk y + wk z x ε + uε x + vε y + wε z x ν k σ k x * ν k * ν k * = G ε 410) ) y σ k y ) z σ k z ) ν ε) + ν ε) + ν ε ) + = ε σ ε x* y σ ε y* z σ ε z* k c 1 G c ε in which G is h producion of kinic-nrgy givn y: ) 411) ) + G = ν u * x + + u y + v u x y + u z + w x u z + v x + u v y x + v y + v z + w v y z + w x + u w z x + w y + v w z y + w + z / ) Th mod cofficins c µ c 1 c σ k and σ ε conaind in h aov ranspor quaions ar assumd o consan and ak h vaus givn in Ta 41 Laundr and Spading 1974; Rodi 1984 p 9) Ta 41 Vaus of cofficins in k-ε mod c µ c 1 c σ k σ ε I is mor convnin o cas h coninuiy quaion Eq 44 h momnum quaions Eqs 46 o 48 and h ranspor quaions of k and ε Eqs 410 and 411 ino a gnra ranspor quaion Vrsg and Maaaskra 1995 p 5): φ + φ V) Γ φ ) = R 413) In h aov quaions φ is any dpndn scaar varia V is h vociy vcor Γ is h diffusion cofficin and R is a coumn marix of scaar sourcs s is dfiniion in Ta 4) Ingraing his quaion ovr a hr-dimnsiona discr conro voum yids Vrsg and Maaaskra 1995 p 5): φ dv + V φ V)dV ) dv Γ V φ V = R dv V 414)

7 45 Th voum ingras of h convciv and diffusiv rms h scond and hird rms on h f-hand sid can xprssd as ingra ovr h cosd surfac ounding h conro voum y appying Gauss divrgnc horm Vrsg and Maaaskra 1995 p 6; Hirsch 1988 p 41): φ dv + V φ V ds Γ S φ ds S = R dv 415) V whr S is h surfac vcor norma ouward o h conro voum dv Ta 4 Trms in h gnra ranspor quaion Eq 413 φ + φ V ) Γ φ ) = R 413) φ Γ R 1 u ν 1 ρ v ν 1 ρ 3 w ν 1 ρ p x + x ν p y + x ν p z + x ν k ν σ k G-ε u x + y ν v x + z ν u y ) * + y ν v y) * + z ν u z + y ν v z + z ν ε 6 ε ν σ ε c 1 k G c ε k ε w x w y w z + g x ) * + g y + g z 4 Souion sragy: h iraiv mhod Th ingra form of h gnra ranspor quaion Eq 415 is usd o oain h souion for u v w k and ε y susiuing hs varias o h scaar varia φ Th souion of h quaion is sough a discr im sps; cacuaions ar prformd a vry discr im sps and rpad uni a sady-sa souion is oaind Th im drivaiv rm in Eq 415 faciias h appicaion of h mod o ransin fow proms; in his cas h souion a ach discr im sp mus convrg Whn h proms concrn sady cas ons as is h cas in h prsn work h im drivaiv srvs as an iraion oop In his cas h souion a ach im sp is considrd as an inrmdia souion and h nd-souion h sady-sa on) is oaind whn a varias φ s convrg Sinc h nd-souion ha is sough i is no ncssary o forc h inrmdia souion o convrg a h sam dgr of convrgnc as ha of h nd-souion Th comp compuaiona procdur is dpicd in h fowchar shown in Fig 41 Th im oop from h iniia uni h sady-sa souion is dpicd as h n-iraion Evry varia in h govrning

8 46 quaions Eq 415 is inkd o ach ohr sinc hy appar in vry quaion To oain h souion of vry varia ha saisfis a quaions a a im sp an iraiv procdur is mpoyd; his is cad h m-iraion in Fig 41 Th asic ida of h procdur is o conscuivy sov h quaions for ach varia in h ordr of h momnum h coninuiy and h k-ε quaions Th momnum quaion is sovd succssivy for h u v and w componns Th ordr of h compuaion is no imporan Th souion of ach varia is sough indpndny for xamp whn soving h x-momnum quaion for u h ohr varias apparing in ha quaion h v w k and ε ar assumd as known Whn a vociy componns ar oaind h prssur is compud hrough h coninuiy quaion which in urn wi modify h vociy Nw souions of h momnum quaion ar hn ncssary Whn h vociy and prssur convrg h k-ε quaions ar sovd asd on h as vaus of h vociy componns Th k is sovd firs and h ε foows Whn soving for ε h as vau of k is usd Th procdur is rpad uni vry varia saisfis a govrning quaions Eq 415 howvr canno usd o oain dircy h prssur Th prssur appars in h momnum quaions u dos no hav any quaion of is own Th fourh quaion h coninuiy dos no xpiciy ink h prssur o h vociy Th souion of h prssur is hus no sraighforward; som kind of a ria-and-corrcion procdur is mpoyd This is indicad as h -iraion Firsy h prssur is simad and suppid o h momnum quaion o g h u v and w vociy componns Scondy h coninuiy quaion is imposd upon hos vociis If h vociis do no saisfy h coninuiy quaion h vociis and h prssur ar hn corrcd Th corrcd prssur is usd as h nw sima and h procdur is rpad Upon h compion of h -iraion h compuaion coninus o h k-ε mod Givn h vociy oaind from h -iraion h k-ε quaions ar sovd conscuivy and h ddy-viscosiy is susquny oaind A chck is carrid ou o a nw varias φs If ach φ saisfis a h govrning quaions hos varias ar rgardd as h vaus a h nw im sp ohrwis h compuaion gos ack o h souion of h momnum quaion h m-iraion) Th surfac oundary which drmins h compuaiona domain u is posiion is par of h souion is handd a h n-iraion and kp consan during h m-iraions Thus h posiioning of h fr surfac is carrid ou xpiciy A h nd of h m- iraion h war surfac is movd according o h prssur dfc a h surfac This in urn wi chang h compuaiona domain for h nw im of h n-iraion Th ovra procdur hus invovs hr ocks of iraion Th firs iraion ock is o g souion of h vociy and prssur i soving h momnum quaions and imposing h coninuiy Th scond ock sovs h momnum coninuiy and k-ε quaions wihin a im sp Th as ock is h im marching iraion o g h sady-sa souion

9 47 START Gnra h grid Iniia condiions: uvwpk n = 1 Esima a prssur fid: p = p m = 1 = 1 = +1 Sov h momnum quaions: u v w Impos h coninuiy: prssur corrcion Upda corrc) h vociy and prsssur fids u v w uc vc wc pc uvwp n = n+1 m = m+1 N uvwp convrgd Y? Y Sov h urunc k-quaion k Sov h urunc -quaion Compu h ddy-viscosiy N uvwpk convrgd? Y Sady-sa souion? Y STO N Upda h surfac oundary and h grid Fig 41 Ovra iraiv procdur of h souion of Eq 415

10 48 43 Numrica mhod: h fini-voum approximaion 431 Grid arrangmn Th approximaion o h souion of Eq 415 is sough y fini-voum approach Th compuaiona domain is discrizd in a 3D grid having a fini numr of conro voums cs); h ingraion is hn carrid ou in ach c A non-orhogona hxahdron c is scd in h prsn mod A ypica on is shown in Fig 4 A c is idnifid y is cnr which maks up h nod whr h dpndn varia is o dfind A c has six nighors namd according o hir rspciv compass dircions ing h Eas Ws Norh Eas Top and Boom Th c facs ar idnifid a h fac cnr and namd wih owr-cas rs namy h w n s and Th Carsian coordina sysm is scd for dscriing oh h gomrica and fow propris ing h z axis dfins h oom-o-op dircion I is o nod howvr ha h grid in his mod is scd such ha h c facs w n and s ar para o h z axis This choic is akn o faciia h handing of h surfac oundary Th discrisaion of h govrning quaion nvrhss is carrid ou for gnra non-orhogona cs T 4 W 3 S z zw y yv S Sw s w 8 n 7 Sn N x xu 1 Ss 5 S B S E 6 Fig 4 Typica hxahdron conro voum Th dpndn varias φ uvwpkε) ar dfind a h c cnr hus consiuing a c-cnrd non-saggrd grid Non-saggrd grid varia arrangmns may yid a prom of prssur-vociy dcouping ha cras a spurious osciaion in h souion This prom dos no xis wih h us of saggrd grid Howvr saggrd grids rquir spara conro voums for h vociy and ohr dpndn varias ha

11 49 incras h compur sorag rquirmn For fows wih hr-dimnsiona gomry h sorag spac rquird for addiiona conro voums is normous In addiion h non-orhogonaiy of h cs givs anohr compxiy sinc h vociy componns ar no rad o h aignmn of h c fac This maks h non-saggrd grid is mor suia for 3D proms To avoid h prom of prssur-vociy dcouping us is mad of h inrpoaion mhod according o Rhi and Chow Rhi and Chow 1983) This mhod consiss of drmining h convciv vociis on a non-saggrd grid hrough h us of h discrizd momnum quaion hus couping h prssur fid wih h vociy fid Th sandard mhod for saggrd grids h SIMLE aankar and Spading 197) is hn usd o corrc h prssur Th SIMLE an acronym for Smi-Impici Mhod for rssur-linkd has n succssfuy mpoyd for fow compuaions in wo-dimnsiona proms Koayashi and rira 1991; Oi a 1989; Frzigr and ric 1997) as w as hr-dimnsiona cass Osn and Kjsvig 1998; Wu a 000) Th prsn mod adops a simiar mhod 43 Compuaion of h surfac ara and of h c voum C-fac ara Th surfac vcor of h c facs can vauad from h vcor producs of h diagonas As can sn in Fig 43 h ara of h as fac quadriara 5678 is haf of ha of paraogram ABCD ui on h diagonas 57 and 68 no h us of h cockwis convnion sn from h c cnr o indx h cornrs) Hnc h surfac vcor is Hirsch 1988 p 47):!!! " x y z S 5678 = 1 SABCD = 1 Δx 57 Δy 57 Δz 57 Δx 68 Δy 68 Δz 68 = 1!!! " x y z x 7 x 5 ) y 7 y 5 x 8 x 6 ) y 8 y 6 ) z 7 z 5 ) ) z 8 z 6 ) 416) Whn h c fac is no copanar h aov xprssion givs h projcion ara of wo riangs sharing a common sid 57 or 68 Th uni vcor norma o a c fac is compud as foows:! n = S S 417) Th norma disanc from poin o h as fac can hn dfind as δ n = L whr L is h vcor originaing from o fac cnr no ha L = L )! n )

12 410 C voum Th c voum is oaind y dividing h hxahdron ino six rahdrons sharing on common diagona 17 and on crs 1 Hnc wih L17 = L7 L1 whr L1 and L7 ar h posiion vcors of 1 and 7 h c voum is hus: V = V V V V V V 1487 = 1 6 { L17 [ L18 L15) + L15 L16) + L16 L1 L 1 L 13 ) + L 13 L 14 ) + L 14 L 18 ) + )]} 418)!!! in which Lmn = Δx mn x + Δy mn y + Δz mn z = x n x m )! x + y n y m )! y + z n z m )! z 4 s w 8 3 n 7 8 D 7 C A B Fig 43 Evauaion of h surfac vcor and c voum

13 C-fac inrpoaion and gradin compuaion C-fac inrpoaion Non-saggrd grids dfin a compud varias a h c cnrs Whn vaus a h c fac ar rquird inar inrpoaion appis Wriing for h as fac h inar inrpoaion aks h foowing form: φ) = 1 β )φ + β φ E 419) wih h inrpoaion facor β dfind as: β = L L E This xprssion is xnsivy usd in h mod xcp in wo cass namy a) whn vauaing convciv rms y using upwind diffrncs Sc 435) and ) whn compuing inrpoad cofficins and sarrd vociis in h prssur-corrcion quaion Sc 44) Gradins Th Gauss horm provids h gradin a h c cnr Th gradins aong h x y and z dircions rad Hirsch 1988 p 53): φ 1 x V φ y - 1 V φ 1 z V V φ! x ) dv = 1 V φ! y ) dv V φ! z ) dv V = 1 V = 1 V x ds 1 φ S x S V cf=wns φ! φ! ) cf ) cf S y ds 1 φ S V y cf= wns S z ds 1 φ S V z cf= wns φ! ) cf 40) Th summaion xnds ovr h six c facs cf: h as ws norh souh op and oom Th dpndn varia a h c fac φ cf is oaind y inar inrpoaion of h varias a h wo inrmdia nighoring c cnrs s Eq 419) Th sam inar inrpoaion is appid whn gradins ar ndd a h c fac Using an ovrar symo o dno inar inrpoad vaus h gradins a h as fac rad: φ = 1 β x ) φ +β x φ x E φ ) y * ) = 1 β φ = 1 β z ) φ ) y * φ +β ) y * φ φ +β z z E E 41)

14 Discrisaion of h im drivaiv rms Th psudo-) im drivaiv rm srvs as a goa iraion ha modis h iraiv souion procdur as dscrid in Sc 4 Th im iraion can considrd as an iraion v marking h progrss of h surfac compuaion sinc h surfac oundary is updad a h nd of a im sp Th souion of Eq415 a a givn im sp n-iraion) dsignas an inrmdia souion Th fina souion wi achivd whn h iraion convrgs owards h sady-sa souion For an inrmdia souion a simp firs-ordr fini-diffrnc schm can appropriay usd o vaua h im drivaiv of Eq 415: φ dv φn +1 φ n V n 4) V Δ Th c voum V is xpiciy dfind a im v n V n ) sinc h nw gomry of h compuaiona domain is no y known a priori This appis aso o a gomrica paramrs such as surfac ara S n ) and spaia coordina and disanc x n y n z n L n ) In soving Eq 415 for φ n +1 iraions hav o carrid ou o hand h non-inar rms As shown in Fig 41 hr ar wo iraion oops h - and m-iraions in arriving o φ n +1 from known vaus φ n Whn hs iraions convrg ha is = and m = w hav φ n +1 = φ n m Eq 4 hus can approximad as: φ dv φn +1 φ n V n = V n =1 V Δ Δ φ n+1 n =1 φ ) 43) Th varia indx is usd o rfr ihr - or m-iraion Wih his approach h ranspor quaion Eq 415 can rwrin as: n =1 V ) [ ] n n +1 [ S ] n n +1 [ V ] n n+1 φ n+ 1 n =1 φ + φ V ds Γ φ ds " Δ S " " = " R dv im drivaion convcion diffusion sourc 44) Th psudo-im indx n n +1 is inroducd o indica h progrss of h iraions n m and usd o vaua h rms in rack Sinc h gomrica paramrs ar a vauad a im v n h convcion-diffusion and h sourc rms conain xpici rms Th schm is hus xpici Th psudo-im sp Δ is rad o h undr-raxaion facor usd in h iraiv procdur of h prssur compuaion; his wi discussd ar in Sc 443

15 Discrisaion of h convciv rms Th discr form of h convciv rms in Eq 44 for c rads: F C n n+1 ) = φ +1 V S ) n d S " ) n ) n )* " φ +1 V S ) n 45) ) cf=wns * cf Th usua convnion of h summaion indx appis ha is h summaion runs ovr h six c facs: h as ws norh souh op and oom Th vauaion of h convciv ranspor hrough h as fac is aorad in h foowing paragraphs and a simiar approach appis o h ohr facs F C n n+1 ) = φ V S n n+1 n n +1 ) = V S n n +1 ) φ n n n + w n n S z ) φ +1 = q + 1 φ = u n S x + v n S y 46) In h aov quaion q is h discharg h mass fux pr uni mass) norma o h as fac For simpiciy h im indx n is omid and h noaion q sands for h discharg oaind from u n and S n A inarisaion has n appid o h convciv rm in Eq 46 y sing φ as h ony unknown whi aking h discharg q xpiciy from h prvious iraion Th unknown varia a h as fac φ is simad y using upwind schm ha is y aking is vau a h upsram conro voum which dpnds on h fow dircion Vrsg and Maaaskra 1995 p 115): φ +1 = φ +1 if q 0 φ +1 = φ E +1 if q < 0 47) Th convciv fux across h as fac Eq 46 is hn: F C n n+1 ) = max q 0 [ ] φ +1 max q +1 [ 0] φ E 48) No ha h discharg q is a scaar produc of h vociy and h surfac vcor and i has a posiiv sign whn aving h c Thus h discharg across h ws c fac of c is qua o h opposi vau of ha across h as fac of c W Th sam is ru for h ohr c facs Th discharg across h norh or op facs of c is qua o h opposi vau of ha across h souh or oom fac of cs N or T rspcivy This propry has o kp in h cacuaion of h convciv fux in ordr o mainain h fux consisncy Th convciv fux aving h c across h as fac is qua o ha nring c W; ohrwis a discrpancy occurs wn nighoring cs Th convciv fux across h ws fac hus rads: F C n n+1 ) = max qw 0 w [ ]φ +1 max q +1 [ w 0]φ W 48a)

16 Discrisaion of h diffusiv rms Th discr form of h diffusiv rms rads in Eq 44 for c : n n+1 S[ ] F D ) n n +1 = Γ φ ds ) cf n n+1 Γ φ S 49) cf=wns in which h summaion xnds ovr h six c facs and h non-inar rms ar inarizd as in h vauaion of h convcion rm Th diffusion across h as fac is aorad and a simiar approach appis for h ohr facs In vauaing h diffusiv rm across h as fac i is convnin o us a oca coordina sysm aachd o h as fac as shown in Fig 44 Across h as fac Eq 49 rads: F D n n +1 ) = Γ φ S n n +1 ) = n φ Γ ) n * n+1 " n n S) 430) ξ n E ξ n a) ) E Fig 44 Evauaion of h diffusion rms across h as fac Th vauaion of h norma gradin prsns som difficuis for is y and z componns Bsids h varia a h nighor c E addiiona ons a NE SE S and N migh hav o akn ino considraion This woud incras h numr of unknowns To ovrcom his prom h so cad dfrrd-corrcion approach Frzigr and ric 1997) is scd in h prsn mod whr ony h immdia nighor c nds o considrd In his approach h norma gradin rm is vauad impiciy y a simp approximaion and a corrcion is addd Th corrcion is akn as h diffrnc wn h corrc and approxima gradins; oh ar xpiciy oaind from h prvious iraion This corrcion is pu in h sourc rms a h righ-hand sid Th diffusion rm vauad wih his approach rads Frzigr and ric 1997 pp 18-):

17 415 ) ξ* + F D n n +1 ) = φ Γ S Γ φ) S φ ) 0 / n* ξ* + / " 1 corrcion xpici 431) In h aov xprssion h im indx n is omid for simpiciy and a rm wihou any indx rfrs o h iniia souion of h im sp n n +1 for xamp S is consan during a im sp S = S n =1 ) In h oca coordina ξ is h dircion of a sraigh in joining and E s Fig 44) An approximaion is usd o vaua h gradin in h impici rm of Eq 431 whr a cnra diffrnc is usd Frzigr and ric 1997; p 4) φ ξ ) +1 = φ E φ wih L E = LE 43) L E Susiuing his raion o h impici gradin h diffusiv fux rads: F D n n +1 Γ ) = S L E φ +1 E Γ S - φ +1 L E + Γ φ S φ 0 / n ξ / " 1 corrcion xpici 433) and for h ws fac i rads: F D n n +1 Γ ) w = w S w L W φ +1 W Γ w S w - φ +1 L W + Γ φ w S w φ 0 / n w ξ / w " 1 corrcion xpici 433a) Th xpici pars can asiy oaind sinc h Carsian componns of h gradin ar known from h prvious compuaion φ n φ ξ = φ x = φ x Sx + φ S y ΔxE Sy + φ L E y + φ S z ΔyE S z + φ L E z S ΔzE L E 434) Appying Eq 431 o h six c facs givs 7 unknowns o h diffusion ranspor rm for ach compuaiona c Errors du o h us of a simp cnra diffrnc o oain diffusion across a c fac Eq 43 ar minimizd y h corrcion givn in h xpici par of Eq 431 I sha nvrhss nod ha h rror wi magnifid whn h ξ dircion of h c fac is far from is n-dircion or whn h as fac cnr dos no coincid wih h ξ in s Fig 44)

18 Convciv-diffusiv rms: hyrid and powr-aw schms Th convciv upwind schm Eq 47 is simp and asy o impmn; i accouns for h fow dircion Th schm howvr is firs ordr accura and producs considra rror whn diffusiv ranspor is imporan To avoid ha prom h socad hyrid schm Spading 197) and powr-aw schm aankar 1980) giv formua which comin h convciv and diffusiv ranspors in a spcia way Dpnding on h grid c numr ing h raio of h convciv and diffusiv conducanc = q L ΓS whr L is h noda disanc ihr h upwind-schm convcion cnra-diffrnc diffusion or cominaion of h wo is considrd o ranspor any scaar quaniy φ across a c fac Th hyrid schm Spading 197) uss h upwind schm for arg c numrs ) and cnra diffrnc for sma c numrs < ) According o his schm h oa fux across h as fac F = F C F D is dfind as foows: for = q L E Γ S < ony h convciv ranspor is akn ino accoun: F n n+1 = q φ E ) for = q L E Γ S < 0 a par of h diffusiv ranspor is aso akn ino considraion: ) Γ S F n n +1 = q φ E ) + Γ φ S / L E φ E +1 φ +1 - / n ) φ 07 + ξ) ) for 0 = q L E Γ S < a par of h diffusiv ranspor is aso akn ino considraion: ) Γ S F n n+1 = q φ ) + Γ φ S / L E φ E +1 φ +1 - / n ) φ 07 + ξ) ) for = q L E Γ S > ony h convciv ranspor is akn ino accoun: F n n+1 = q φ ) Th powr-aw schm aankar 1980 p 90-91) ss h imiing vau of whr h diffusion no ongr affcs h ranspor a = 10 insad of = usd in h hyrid schm for = q L E Γ S < 10 : F n n+1 = q φ E )

19 417 for 10 = q L E Γ S < 0: ) 5 Γ S F n n +1 = q φ E ) + Γ φ S / L E φ E +1 φ +1 - / n ) φ 07 + ξ) ) for 0 = q L E Γ S < 10: ) 5 Γ S F n n +1 = q φ for = q L E Γ S >10 : n F n+1 = q +1 φ ) + Γ φ S / L E φ E +1 φ +1 - / n ) φ 07 + ξ) ) 44) Equaions 435 o 44 can comind ino a compac form as foows aankar 1980 pp 94-95): n F n+1 = max[ q 0] φ +1 max q +1 [ 0] φ E f D Γ S L E φ E +1 φ +1 - / n ) + Γ φ S / ) φ ξ) ) whr f D is a facor ha dpnds on h asou vau of grid c numr; i has a diffrn form for h hyrid and powr-aw schms as shown in Ta 43 Ta 43 Hyrid and powr-aw schm diffusion facors Schm Hyrid owr-aw f D = f ) = f ql ΓS) max[ 1 05 ) 0] max[ 1 01 ) 5 0]

20 418 Arranging h rms in Eq 443 on has: n F n+1 = max q D Γ [ 0] + f S φ +1 L + max q D Γ 0 E [ ] f S +1 φ L E " E 3 f D Γ φ S φ n/ ξ/ " xpici impici 443a) and for h ws fac h convciv-diffusiv fux rads: n F n+1 w = max q D Γ [ w 0] + f w S w w φ +1 L + max q D Γ w 0 W [ ] f w S w +1 w φ L W " W 3 f D w Γ φ w S w φ n/ w - ξ / 8 4 w " xpici impici 443) Shorr noaions is usd o wri h xprssion of h convciv-diffusiv fux for n xamp Eq 443a may rwrin as F n+1 = a C D + a )φ +1 + a C D E + a E )φ +1 E + D E in such a way ha y summing up h convciv-diffusiv fuxs across h six facs of c on oains: F n n +1 = a C D E + a E )φ +1 E + a C D W + a W )φ +1 W + a C D N + a N )φ +1 N + a C D S + a S )φ +1 S + 444) a C D T + a T )φ +1 T + a C D B + a B )φ +1 B + a C D + a )φ +1 + D whr h cofficins indica conriuion of convciv-diffusiv rms from nighoring cs as prsnd in Ta 44 Th diffusiv-corrcion rms ing vauad xpiciy ar known from h prvious iraion and ar incudd in h indpndn cofficin D

21 419 Ta 44 Cofficins of h discrizd convciv-diffusiv quaions Convciv rms Diffusiv rms a C = a E C = max q 0 a W C = max q w 0 a N C = max q n 0 a S C = max q s 0 a T C = max q 0 a B C = max q 0 a C n n=ewnstb [ ] a D E = f D Γ S L E [ ] a D W = f D Γ w S w L W [ ] a D N = f D Γ n S n L N [ ] a D S = f D Γ s S s L S [ ] a D T = f D Γ S L T [ ] a D B = f D Γ S L B + q cf a D D = a n cf= wns n =EWNSTB D = f D Γ S cf= wns n - ) φ cf cf φ * ξ + cf / Sourc rms Th sourc rms may consis of scaar quaniis firs drivaivs or scond drivaivs of a scaar quaniy Th sourc in addiion may aso incud h im ingraion rm h diffusion corrcion and known varias from h oundary condiions Somims a rm iniiay considrd as a sourc aks advanag o xprssd as a funcion of h unknown varia a h c cnr such as cs nx o a oundary In ha cas h rm is inarizd which givs Vrsg and Maaaskra 1995 p 87): +1 R dv R V V = + φ 445) in which incuds a known quaniis ihr consans prscrid or known from prvious iraion) and is h cofficin of h unknown varia a Scaar sourc rms Th sourc rm conaining scaar quaniis S coms from h graviy accraions g x g y g z or h urun nrgy producion and dissipaion G and ε Th scaar sourc a rprsns h avrag vau of hos quaniis in h c ing considrd I is known and hus is considrd as a consan Thrfor h sourc rms conaining scaar quaniis can asiy vauad according o h foowing xprssion:

22 40 S = φ dv = φ V V 446) In appying h aov raion o h sourc rms of h k quaion coming from h urun nrgy producion G som approximaions ar ndd Wriing Eq 446 for G w hav: G dv = G V V Sinc h G rm conains non-inar gradin rms s Eq 41) h aov raion impis ha hs rms ar vauad individuay ha is y using Eq 40) This mans ha h ingra of hs rms ar compud in h foowing fashion an xamp is givn hr for h u x ) rm): " u " dv = u V x x " 1 V = V u V x dv V This approximaion of cours wi inaccura whn h vociy gradin is imporan Nvrhss his mhod is scd for is asinss o impmn In h ε quaion h sourc rm is inarizd for h rm conaining ε in h foowing form: " εg " ε = c 1 V k c V " k " S S ε ) h as rm of which wi join h cofficin a Firs drivaiv sourc rms Th sourc rm conaining firs drivaivs 1D is found in h prssur gradin of h momnum quaion and in h vociy gradin of h nrgy producion for h k-ε quaions Foowing h mhod dscrid in Sc 433 h sourc rms conaining firs drivaivs in h x- y- and z-dircions ar vauad as foows h rms wihin racks ar gnray prdominan): 1D φ ) x = dv = φ " V x x ) dv φ " V x ds = φ S S x ) cf cf=wns 448a) = φ S x + φ w S wx ) + φ n S wx + φ s S s x + φ S x + φ S x

23 41 1D φ ) = y dv = φ " V y y ) dv φ " V y ds = φ S S y ) cf cf=wns 448) = φ n S n y + φ s S s y ) + φ S y + φ S y + φ S y + φ w S wy 1D φ ) z = dv = φ " V z z ) dv φ " V z ds = φ S S z ) cf cf= wns 448c) = φ S z + φ S z ) + φ S z + φ w S wz + φ n S wz + φ s S s z Scond drivaiv sourc rms Th sourc rm conaining scond drivaivs D is found in h momnum quaion Ths ar du o h non-orhogona rms of h srsss s Eqs 46 o 48) and h xpici pars of h diffusion rms s Eq 443) An xamp is givn ow for h vauaion of h sourc rms conaining scond drivaivs in h u-momnum quaion D ) x = D ) y = x ν u dv x ) * + u = ν V V + = u ν " ) x x ds u ν + S x cf * + u ) = ν ν S x + ν x u ) s S s x + ν x s u ) w x u ) w x " x S x - cf S wx + ν S x + ν x y ν v ) v dv = ν V x V + x * + = v ν " ) x y ds v ν + S x cf * + v ) n = ν v ) ν S n y + ν x n S x y + ν v ) s s - dv u ) n u ) " y S y - cf S s y + ν x v ) S x y + ν S nx + x x - dv v ) n S x S y + x v ) s x S wy w 449a) 449)

24 4 D ) z = ν w w dv = ν V V + z x ) * + w " ) w = ν x z ds ν S + x cf * + w ) = ν ν S z + ν x w ) w x w w ) S wz + ν x " z S z - cf S z + ν x w ) n n - dv w ) S n z + ν x S z + x w ) s x S s z s 449c) Th as four rms on h righ-hand sid ar du o h grid non-orhogonaiy; hy vanish for orhogona cs Th c fac vaus of h vociy gradins ar oaind y inar inrpoaion s Sc 433 and Eq 41) 439 Assmy of h cofficins Afr vauaing a rms of h convcion diffusion and sourcs ovr h nir compuaiona domain and rarranging h cofficins h discrizd ranspor quaion producs a sris of agraic quaions For h unknown varia φ a h c cnr φ xyz) and a h nighoring c cnrs n φ n xyz) h quaion rads: a φ +1 + a n φ +1 n = 450) n Th cofficins a n a and in h aov quaion ar isd ow: cofficins a n consis of h convciv and diffusiv rms s Ta 44): a E = a E C + a E D = max q 0 D ΓS [ ] f a W = a W C + a W D = max q w 0 a N = a N C + a N D = max q n 0 a S = a S C + a S D = max q s 0 a T = a T C + a T D = max q 0 L E D ΓS [ ] f w L W D ΓS [ ] f n L N D ΓS [ ] f s L S D ΓS [ ] f L T s n w and

25 43 a B = a B C + a B D = max q 0 D ΓS [ ] f L B cofficin a is formd from various rms namy h psudo-mpora ingraion convciv-diffusiv rms Ta 44) and rms coming from h sourc inarisaion s Eq 447): a = a T + a C + a D = V Δ + a n C + q cf ) + a D n) n sourc rms : cf n = S + 1D ) + 1D x ) + 1D y ) + D z ) + D x ) + D y ) + T D z S scaar sourcs: Eq 446 or 447 1D firs drivaiv sourcs: Eqs 448 D scond drivaiv sourcs: Eqs 449 T = V Δ φ n psudo-im drivaiv sourc: Eq 4 and D = f D Γ φ S) cf φ / n cf * ξ diffusiv corrcion rms: Eq 443 cf - cf0 No ha for h oundary cs h cofficins may chang from h aov dfiniions This wi dscrid in Sc 45 Undr-raxaion facor Th souion of Eq 450 for any dpndn varia φ hrough ou h compuaiona domain is achivd y iraiv procdur marching from known vaus a h iraion v o nw vaus a h iraion +1 During h procss osciaion may occur In ordr o avoid such a prom an undr-raxaion facor is appid o updaing h souion from iraion o +1 Suppos ha h souion a a paricuar iraion v is φ hus: a φ + a n φ +1 n = 450a) n Now insad of aking ha souion for h vau of φ +1 on may ak aso ino h considraion is vau a h prvious iraion v φ arguing ha φ +1 shoud no oo much diffrn from φ Th undr-raxaion facor ϖ is hn appid according o h foowing form:

26 44 φ +1 = ϖ φ + 1 ϖ)φ or φ = 1 ϖ φ +1 1 ϖ ϖ φ 451) Susiuing his raion o h rm φ in Eq 450a on finds: a 1 ϖ φ +1 1 ϖ ϖ φ + n a +1 nφ n = which afr som arrangmns of h rms yids: a φ +1 + n +1 a n φ n whr h cofficins ar now: = 45) a = a ϖ and = + 1 ϖ) a ϖ φ = + 1 ϖ) a φ 44 rssur-vociy couping 441 SIMLE agorihm Whn soving h momnum quaion for vociy h prssur is unknown and an simad vau p is firsy usd insad In gnra h vociy ha is oaind dos no saisfy h coninuiy quaion A corrcion o h simad prssur is addd and a nw souion is sough for h nw vociy This procdur is rpad uni i givs prssur and vociy fids saisfying no ony h momnum quaion u aso h coninuiy quaion An iraiv souion procdur known as SIMLE Smi-Impici Mhod for rssur-linkd Equaion) mhod aankar and Spading 197) is widy usd for his vociy-prssur compuaion Th mhod rquirs vociy and discharg a c facs which ar no immdiay avaia wih h us of non-saggrd grids in h prsn mod Th inrpoaion chniqu of Rhi-and-Chow Rhi and Chow 1983) sovs his prom Th chniqu givs inrpoad vociy a c facs from h noda vaus Th sandard SIMLE agorihm is hn usd o prform h prssur corrcion This scion givs som dais of h procdur which foows h drivaion givn y aankar aankar and Spading 197; Vrsg and Maaaskra 1995; Frzigr and ric 1997) In h iraion +1 h discrizd momnum quaion Eq 45 wih φ = u v w can rwrin as: + a u i + a n u in n = 1 ρ V p + 1 x i * ) 453)

27 45 whr h symos u i and x i ar usd o dno h Carsian componns of h vociy and dircion u i = u v w and x i = x y z rspcivy No ha h prssur gradin in h aov xprssion has innionay n xracd from h sourc rm for a rason ha wi vidncd ar in Eq 453 is hus no xacy h sam as ha in Eq 45) Th cofficins a a n and h sourc rms ar funcions of h known varias ihr a h prcdn iraion or im sp n For pracica souions of Eq 453 sinc hr ar ony 3 quaions for 4 unknowns h prssur p is mporariy fixd a is iniia vau Th foowing sysm of quaions is sovd in h firs sag: a u i p = p + a n u in n = V ρ p ) + x i * 454a) 454) Th simad prssur p and h vociis oaind from his prssur u v w ar of cours o corrcd: u i +1 = u i + u i c p +1 = p + p c 455a) 455) and p c wi rsu from h momnum quaions comind wih h coninuiy quaion Th corrcions ar such ha h vociy wi saisfy h coninuiy and momnum quaions whn h iraion convrgs I can hrfor said ha Eqs 454a com Eq 453 whn In ha cas having a sufficiny arg numr of -iraions w hav u i ) = u i +1 ; and his is so for h cofficins and sourc rms W can hrfor oain h raion wn h prssur and vociy corrcions y suracion of Eqs 454a from Eq 453: whr h corrcions u i c c a u i + a n u in = n c V ρ p c * x i ) 456) Eq 456 is a raion in which h corrcions nd owards zro A simpifying approximaion can hn inroducd y ngcing à priori h corrcion rms of h nighoring cs aankar and Spading 197) Th vociy corrcion hus rducs o: c u i = 1 ρ V a p c x i ) 457) Sinc h cofficin a is drivd in such a way ha i is h sam for a vociy componns i a u = a v = a w = a h aov raion is vaid for any vociy

28 46 componn a any poin and hus aso for h norma vociy componn a a c fac Wriing for h as fac on has: c u n = 1 ρ V a p c n ) 458) Th cofficin a h c fac V a ) is dfind as h avrag vau of hos of h nighoring c cnrs and E:! V " a = 1! ) ) " V a! + V " a E * + 459) ) E rprsns h cofficin No ha a a of Eq 45 wrin for h c E ha is no h cofficin a E of Eq 45 wrin for h c No aso ha h inrpoaion in Eq 459 dos no mach h inar inrpoaion in Eq 419 sinc his ar is no rvan for h voums Indd h voum rad o a c for h as fac is composd of h haf voum of c and h haf voum of c E Using h dfrrd-corrcion mhod as in h discrisaion of h diffusiv rms s Sc 436 Eq 431) o compu h norma prssur gradin yids: c u n = 1 V p c c E p * ρ a ) 1 V p c L! " E + ρ a / pc * ) / ) n ξ +! " u imp n : impici no : xpici u n od 460) in which h rms in h scond squar rack ar du o non-orhogonaiy of h c and ar vauad xpiciy From h sarrd-vociy and h vociy corrcion h norma vociy componn can compud: u n = u n c + u n = u n + u imp no n + u n 461) Hr h vociy corrcion du o h c non-orhogonaiy is wrin sparay Whis h vociy corrcion is oaina from Eq 460 h sarrd-vociy unforunay is no dircy avaia a h c fac Inrpoaing h sarrd vociy a h nighoring c cnrs o g h c fac vau woud rsu in h dcouping of h vociy from h prssur ha causs an osciaion of h souion A rmdy o his prom is o us h so-cad Rhi-and-Chow inrpoaion chniqu Rhi and Chow 1983) which rpacs h inrpoad prssur gradin a a c fac wih h on compud from h prssur a h immdia nighoring c cnrs As can sn in h raion ow wriing h quivan of Eq 454a a h as fac y simpy inrpoaing hs sarrd vociis woud rsu in vociis ha hav no dirc raion wih h prssur diffrnc wn and E:

29 47 u n ) = a a n u n n ) 1 ρ V a p ) n 46) In ordr o ra h vociy a h as fac ack o h prssur diffrnc wn and E a corrcion is givn o his inrpoad vociy: u n = - a a n u nn ) 1 V p / ρ a ) 1 n V p ρ a ) 0 - n p E p L E / ) This xprssion can sn as h vociis a h c cnrs and E inrpoad o h as fac and corrcd y a facor du o h inrpoaion: u n RC = u n ) + u n 464) whr h ovrar rm is oaind from inar inrpoaion of h vociis in c cnrs and E Using his xprssion o susiu h sarrd vociy in Eq 461 givs: u n = u n ) + u RC n + u imp no n + u n 465) Nx w nd o dfin h quaion of h prssur corrcion which is carrid ou y using h coninuiy quaion Th discrizd coninuiy quaion can oaind y wriing h govrning quaion Eq 415 wih φ = 1 Γ = 0 and R = 0 ha givs: ) cf V d S! V d S! S = q cf = u ncf S cf ) = 0 466) cf=wns cf= wns cf=wns Using Eq 465 wih h dvopmn of h impici rm in Eq 460 and insring h rsu o h aov coninuiy quaion on gs h prssur corrcion quaion of h form: 1 V ρ a 1 V ρ a S L E S s s L S p c c E p ) 1 ρ V a p c c S p ) 1 V ρ a ) + q cf cf cf= wns[ ] + q RC + q no cf = 0 S w w L W S L T c c p W p ) 1 ρ p c c T p ) 1 ρ V a V a n S L B S n p c c L N p ) N p c c B p ) 467) whr: q ) = u cf cf ) S cfx + v cf ) S cfy + w cf ) S cfz inrpoad discharg

30 48 RC = 1 " V + " p - ρ a ) cf - n q cf cf p n p 0 / 0 S cf corrcion du o c - fac inrpoaion L n no = 1 ρ q cf V a cf + p c - ) - n cf p c ) ξ cf 0 / 0 od S cf corrcion du o non - orhogona rms Afr arranging h rms on has: ) a p p c + a p n p c n = p 468) whr: n a p E = 1 ρ V a a p S = 1 V ρ a s S a p W = 1 L E ρ S s L S a p T = 1 ρ V a V a w S w a p N = 1 V L W ρ a S L T a p = a p n p = q RC + q no cf n cf [ ) cf + q cf ] n a p B = 1 V ρ a w S n L N S L B Th prssur gradins ncounrd in h sourc rms ar compud according o h foowing raions: p n p c n p c ξ = p x = pc x = pc x S x S S x S Δx E L E + p y + pc y + pc y S y S S y S + p z + pc z Δy E L E + pc z S z S S z S Δz E L E 469) in which h inrpoaion of h prssur gradins aong h Carsian coordinas is don using Eq rssur corrcion procdur Th sourc rm p has prssur corrcion rms conaind in h discharg du o nonorhogonaiy of h cs q no Ths rms ar vauad xpiciy y a dou-sp prssur corrcion procdur as foows: Sov Eq 468 for p c y ngcing h non-orhogona rms q cf no = 0 and corrc h vociis and prssur according o Eqs 455a

31 49 Sov again Eq44 wih h non-orhogona rms now avaia from h firs sp and corrc onc again h vociis and prssur 443 Undr-raxaion facor and im sp To avoid insaiiy of h compuaion i is a common pracic o pu an undrraxaion facor o h prssur corrcion 0 ϖ p 1 in updaing h prssur: p = p + ϖ p p c 470) As mniond in Sc 4 h im sp pays aso as an undr-raxaion facor for sady fow cass This yp of appicaion ha is using h ransin quaions o sov sady fows is gnray known as a psudo-ransin compuaion In ordr o achiv h ffcs of undr-raxd iraiv sady-sa compuaions from a givn iniia fid y mans of a psudo-ransin compuaion saring from h sam iniia fid h imsp siz is akn such ha Fchr 1997 p 365): ϖ p 1 = wih E Δ = a Δ 471) 1 + E Δ V 45 Boundary condiions 451 Boundary pacmn Th oundary condiions ha can considrd in h mod ar infow oufow wa war) surfac and symmry oundaris Th spaia discrisaion of h compuaiona domain is don in such a way ha h oundaris coincid wih h c fac s Fig 45) Th c nighoring h oundary has spcia characrisics ha modify h dfiniion of c s forh in Sc 431; i has mor han on nod and ss han six nighors Thr yps of c and nod ar inroducd s Fig 45): Inrior c h whi c) is a compuaiona c whr h dpndn varia φ is unknown and is o compud a h inrior nod h soid circ); an inrior c has ony on nod h inrior nod Boundary c h gray c) is a oundary nighoring c whos on or mor of is facs coincid wih a oundary A oundary c has on inrior nod h soid circ) a h cnr of h c and on oundary nod h gray circ) a h cnr of ach fac ha coincids wih h oundary Th known oundary vaus of a varias φ ar o dfind a h oundary nod ihr givn or xrapoad from h inrior nods Dummy c hachd c) and dummy nod whi nod) ar usd o dno h domain which is xcudd from h compuaion for xamp ockd-rgions cyindrs and cornrs Ths dummy cs and nods ar ncssary in ordr o mainain a coninuous ordring of h c and nod indxs

32 430 Excp of som spcia cass for h k and ε quaions h ffc of h oundaris o h compuaion for h inrior nod of a oundary c is addiiv In h discrizd quaion of u v w and p c h conriuion of ach oundary nod is addd o h sourc rm of h inrior nod of h oundary c and h cofficin rad o his oundary nod is vnuay s o zro Th k and ε for h inrior nods of oundary cs having wa or fr-surfac oundaris howvr ar dfind y a givn xprssion In h foowing scions ar prsnd h mhod of compuaion for h oundary cs wa infow oufow ockd-ara wa wa a) xy-pan ockd-ara wa symmry war surfac infow oufow wa ) xz-pan inrior c oundary c dummy c inrior nod oundary nod dummy nod Fig 45 Boundary condiions impmnd in h mod 45 Infow oundary Suppos ha h infow oundary is a h ws fac w of h oundary c s Fig 46) Th infow oundary vaus across h fac w ar imposd as h oundary

33 431 condiion whos vaus ar dfind a h oundary nod W ocad a h sam pac as w s Sc 451): φ W = φ in 47) Th prssur is assumd o vary inary wn W and E: p W p L W = p E p p L W = 1 + β )p β p E 473) E Th aov raion hods as w for p and p c vaus Th oundary nod dnod y W insad of w) upon which h infow oundary vaus ar dfind aows h discrizd quaions formry saishd for inrior nods o appid o h oundary nod W wihou h nd o chang h noaion T infow φw = φin givn) qw = qin givn) c c c L c c pw = pw = p pe p) LE xrior W w inrior E B Fig 46 Infow oundary Momnum and k-ε quaions In forming h cofficins in Eq 45 h foowing sps appy: a varias a h in ar givn: φ W = φ in vaua h convciv-diffusiv rms as for norma inrior cs: a C W a D C W a D ) W a ) W D ) W 1D ) W D ) W ring h conriuion of nod W o h sourc rm: D s h cofficin a nod W o zro: a W = 0 ) W a W C + a W D )φ W

34 43 rssur and vociy corrcions Th conriuion of h discharg across h ws fac q w o h coninuiy quaion Eq 466 is rpacd y h imposd discharg q in In forming h cofficins in Eq 468 h foowing sps appy: s h cofficin a nod W o zro: a W p s h conriuion of h infowing discharg o h sourc rm: p = 0 ) w = q in For h vociy corrcion Eq 457 h prssur corrcion gradin p c x i is compud y h fini-voum chniqu Eq 40 which rquirs h vau of p c ) w = p c ) W This ar is oaind y Eq Oufow oundary Suppos ha h oufow is a h as fac of h oundary c s Fig 47) Across h oufow fac h convciv fux is compud according o h upwinding princip of Eq 48: F C ) = q φ = q ou φ whi h diffusion fux is s o zro: F D ) = 0 ading o a simpifid form of Eq 443: F = q ou φ +1 whr q ou is ihr imposd or compud from h upwinding of vau a h formr iraion sp: q ou = V ou S = V S Th sam upwinding procss finds h ohr varias a h oundary nod E φ E = φ = φ

35 433 T oufow W w E φe = φ c c p = pe = p c inrior xrior B Fig 47 Oufow oundary Momnum and k-ε quaions s a cofficins rad o nod E in Eq 45 o zro: a C E = a D C E = a D ) = a E ) = D E ) = 1D E ) = D E ) = 0 E xrapoa a varias a o E: φ E = φ h oufowing discharg rsus: q ou = V S = V S rssur and vociy corrcions Th conriuion of h discharg across h as fac q o h coninuiy quaion Eq 466 is rpacd y h oufowing discharg q ou In forming h cofficins in Eq 468 h foowing sps ar don: s h cofficin a nod E o zro: a E p = 0 s h conriuion of h infowing discharg o h sourc rm: p ) = q ou For h vociy corrcion Eq 457 h prssur corrcion gradin p c x i is compud y h fini-voum chniqu Eq 40 which rquirs h vau of p c ) = p c ) E ; his ar is oaind y h upwinding: p c ) = p c E ) 454 Wa oundary Wa funcion approach Th wa funcion approach Laundr and Spading 1974) is appid o h c whos fac is a rigid wa Major assumpions usd in his approach mri o pu forward for prsning h drivaion of h wa funcion; hy ar: 1) h no-sip fow condiion prvais a h wa wih h univrsa ogarihmic vociy disriuion norma

36 434 o h wa ) h producion of h urun kinic nrgy is mry du o h urun) shar srss hus ngcing h ffc of h norma srss and 3) a oca nrgy aanc xiss i h dissipaion of h urun kinic nrgy is qua o h producion Givn in h foowing paragraphs ar h drivaions of h wa funcion in which hos assumpions ar furhr highighd T V n V W wa w n n V n B nn S V V E inrior xrior n n V V V + = n E n ) V k = V = 0 B = k = B B c c B c p = p = p p p p g n = = ) B z z z wa nn n a) ) S Fig 48 Wa oundary 1) In wa oundaris h cnr of h cs is ocad sufficiny cos o h wa u ousid h viscous su-ayr s Fig 48a); h univrsa ogarihmic vociy disriuion hn prvais in h rgion s Fig 48): V = V κ n E δ n + ) or V = κ V + n E δ n ) wih V = V and δ + n = V δ n ν 474) in which V is h shar vociy V is h vociy componn para o h wa κ is h Karman univrsa consan δ n is h norma disanc from h wa ν is h mocuar viscosiy of war and E is h wa roughnss cofficin Th cofficin E in h aov raion accouns for a fow rgims ihr hydrauicay smooh rough or ransiion No h dircions of srsss on h wa fac ; h shar srss τ! n is o h opposi dircion of! whras h norma srss τ! nn is according o! n Ths dircions ar consisn wih h convnion ha shar forcs ar in h dircion of posiiv incrass of vociy posiiv vociy gradins) In h dircion of! n of h

37 435 oca) wa coordina!! n ) h vociy gradin V V n n is posiiv n is ngaiv whras ) Th quaion of h urun kinic-nrgy producion Eq 41 wrin in h wa coordina sysm!! n ) shown in Fig 48 is: G = ν V ) ) + V n + V n + V n * n + 41a) Sinc h vociy is zro vrywhr aong h wa V = 0 no-sip condiion) and V n = 0 no-fow across h wa) a vociy gradins aong h wa h angnia componns) disappar Th aov xprssion hus rducs ino: G = ν V n + ν V n n 475a) Th firs and h scond rms dpic h urun kinic-nrgy producion du o h shar and norma srsss rspcivy In h wa funcion h scond rm is ngcd which impis ha h norma vociy componn canno dvop in h wa rgion This yids h foowing: G = ν V n 475) Th vaidiy of h aov quaion is sricy imid a h wa u is gnray xndd o h c cnr whr i consius an approximaion Th omission of h norma vociy gradin in Eq 475a and h xnsion of Eq 475 o h c cnr ar of cours a rahr rud approximaion noay in h cas of fow around a cyindr Masurmn daa in fron of h cyindr s Chaprs and 3) show ha h radia vociy h norma componn) and h downward vociy h angnia componn) hav h sam ordr of magniud Th norma vociy gradin hrfor shoud accound for in h urun kinic-nrgy producion Howvr h prsn mod adops Eq 475 sinc i ads o a numrica simpificaion 3) Th hird assumpion in h wa funcion is h xisnc of a oca aanc wn h urun kinic-nrgy dissipaion and is producion ε = G s Laundr and Spading 1974; Vrsg and Maaaskra 1995 p 73) This yids: ε = G = * ν )* V n ) - Th ddy viscosiy can found from h Boussinsq concp Eq 45 and h dfiniion of h fricion vociy τ n = ρ V Considring ha h variaion of h shar srss is

38 436 ngigi in h wa rgion τ n = τ n and ha h vociy gradin aong h wa is ngigi V n ) 0 on may wri: V = τ n ρ = τ n ρ = * ν + V n + V n - )/ V = ν 477) n ) Th vociy gradin is oaina from h ogarihmic vociy disriuion Eq 474: " V = V n κ δ n 478) Insring his raion ino Eq 477 on oains: ν = V κ δ n 479) Susiuing Eqs 478 and 479 ino h righ-hand-sid of Eq 476 yids: ε = G = V κ δ n V * κ δ n ) = V 3 480) κ δ n Comining Eqs 479 and 480 o h dfiniion of h ddy viscosiy in h k-ε mod Eq 49 on gs: ν = c µ k ε V κ δ n = c µ k κ δ n V 3 ) * + V = c µ 1 4 k 1 481) Thr ar now wo xprssions of h fricion vociy i Eqs 474 and 481 Boh raions ar usd o vaua h shar srss a h wa:! τ n = ρ V V = ρc µ 1 4 k 1 κv )! + n E δ n 48) Th ngaiv sign is rquird sinc τ! n is acing o h opposi dircion of! s Fig 48) This raion is h on ncssary o vaua h conriuion of h wa oundary o h fow momnum quaion; is impmnaion wi furhr prsnd ar

39 437 Wa funcion: h fina quaions Th ink wn h wa funcion and h k quaion is achivd hrough h urun kinic-nrgy producion Eq 476 h vociy gradin Eq 478 and h shar vociy Eq 481 Comining hs quaions on gs: G = ) ν ) V n * = τ n + ρ V = τ n n ρ V κ δ 3 n = τ n ρ c µ 1 4 k 1 κ δ n 483) wih τ n =! τ n This quaion is usd o dfin h nrgy producion in h sourc rm of h k quaion s Ta 4) for cs nighoring h wa For h ε quaion h nrgy dissipaion is oaind from Eqs 480 and 481: ε = c µ 3 4 k 3 κ δ n 484) Bfor daiing h impmnaion of h wa funcion o h discrizd momnum and k-ε ranspor quaions wo varias nd o dfind namy h wa roughnss cofficin E and h para vociy componns V Wa roughnss cofficin Th wa roughnss cofficin E in h ogarihmic vociy profi is adjusd according o h quivan sandard) roughnss k s whhr i is hydrauicay smooh rough or ransiion wn smooh and rough Th foowing raion is usd o dfin h roughnss cofficin Wu a 000): E = xp[ κ B ΔB) ] 485) whr B is an addiiv consan and ΔB is a roughnss funcion drmind according o h sandard roughnss k s as foows Cci and Bradshaw 1977): 0 for k + s < 5 ΔB = B κ n k + - s ) sin 0458 n k + s 0811) B κ n k s + for k + s 90 / [ ] for 5 k s + < ) wih B = 5 κ = 04 and k s + = V k s ν ing h roughnss Rynods numr Tangnia vociy componn Th vociy a h c cnr nds o dcomposd ino is norma and angnia componns wih rspc o h wa s Fig

40 438 48) Th uni vcor norma o h wa! n has an ouward dircion s Eq 417) whi h uni vcor angnia o h wa! has h dircion of h projcion of V on h wa Boh uni vcors ar prpndicuar in such a way ha any vcor for xamp V ) can dcomposd aong hm in h pan!! n ) which aso conains V For drmining! i is ncssary o find h projcion V which can oaind from: V = V Vn = V V! n )! n 487) Knowing V h uni vcor! can asiy compud:! = V V ) 488) Impmnaion of h wa funcion Wa funcion for h momnum quaion Sinc hr is no discharg across h wa h convciv fux dos no xis across h wa fac; h ony fux is du o h diffusion In h discrizd momnum quaion Eq 45 h diffusion rm is no vauad y Eq 433 u y vauaing his rm as a norma forc pr uni mass) acing on h wa Simiary h wa shar-srss Eq 48 is aso ransformd as a shar forc Boh forcs ar considrd as a sourc rm and ar vauad a iraion which hn inarizd such ha h vociis a h c cnr com unknown varias Th forc du o h norma srss acing on h wa s Fig 48) can compud as: " Fn ρ " " τ = nn ρ " S = ν V n " n n " S ν V n " n n " S = ν V n " δ n n S 489) Th ngaiv sign is ncssary sinc h norma srss! τ nn is in h ngaiv dircion of V n s Fig 48) A rms ar vauad xpiciy ha is h vociy is from h h iraion h kinic nrgy is from h m h iraion and h gomry is from h n h im iraion Th forc du o h shar srss acing on h wa is oaind from Eq 48 u is form is modifid o aow asy compuaion of h urun viscosiy ar on " F ρ " " τ = n ρ ) 1 κ δ n S = c µ 1 4 k m / 1 " V " + n E δ n ) 1 δ - 0 n S 1) )

41 439 Th rms groupd in h firs rack on h righ-hand sid of h aov xprssion hav oghr h dimnsion of a viscosiy hus can considrd as h wa urunviscosiy ν wa : " F ρ " V = ν wa δ S wih ν wa = c µ 1 4 k m n - ) 1 κ δ n n E δ n + ) / ) Boh forcs ar addd o h sourc rm of h oundary c as h conriuion from h wa oundary nod B Th ohr cofficins rad o h conriuion from h oundary nod B ar hn assignd o zro Th foowing sps ar usd in vauaing h cofficins in Eq 45: s a cofficins rad o h conriuion of h oundary nod B o zro: a C B = a D C B = 0 a D ) B = a ) B = D ) B = 0 and D ) B = 0 compu! using Eqs 487 and 488 compu h forcs du o h norma and shar srsss as sourc rms and inaris h sourc: x-momnum: + 1 " + u ) B = F nx ρ + F x ρ [ ] S = ν u nx + v ny + w n z ) nx + ν wa v y + w z ) x δ " n ) B S [ ) + ν wa x x )] ν nx n x δ " n ) B u a) y-momnum: + 1 " + u ) = F ny B ρ + F y ρ [ ] S = ν u n x + v ny + w n z ) ny + ν wa u x + w z ) y δ " n ) B S [ ) + ν wa y y )] ν ny ny δ " n ) B v )