Introductory Turbulence Modeling

Inrodcory Trblence Modelng Lecres Noes by Ismal B. Cel Wes Vrgna nversy Mechancal & Aerospace Engneerng Dep..O. Bo 606 Morganown, WV 6506-606 December 999

TABLE OF CONTENT age TABLE OF CONTENT... NOMENCLATRE...v.0 INTRODCTION....0 REYNOLD TIME AVERAGING...6 3.0 AVERAGED TRANORT EQATION...9 4.0 DERIVATION OF THE REYNOLD TRE EQATION... 5.0 THE EDDY VICOITY/DIFFIVITY CONCET...7 6.0 ALGEBRAIC TRLENCE MODEL: ZERO-EQATION MODEL...0 FREE HEAR FLOW... WALL BONDED FLOW...3 Van-Dres Model...4 Cebec-mh Model (974)...4 Baldwn-Loma Model (978)...5 7.0 ONE-EQATION TRBLENCE MODEL...7 EXACT TRBLENT KINETIC ENERGY EQATION...7 MODELED TRBLENT KINETIC ENERGY EQATION...9 8.0 TWO-EQATION TRBLENCE MODEL...33 GENERAL TWO-EQATION MODEL AMTION...33 THE K-ε MODEL...35 THE K-ω MODEL...38 DETERMINATION OF CLORE COEFFICIENT...39 ALICATION OF TWO-EQATION MODEL...40 9.0 WALL BONDED FLOW...4 REVIEW OF TRBLENT FLOW NEAR A WALL...4 TWO-EQATION MODEL BEHAVIOR NEAR A OLID RFACE...43 EFFECT OF RFACE ROGHNE...45 REOLTION OF THE VICO BLAYER...45 ALICATION OF WALL FNCTION...46 0.0 EFFECT OF BOYANCY...5.0 ADVANCED MODEL...54

ALGEBRAIC TRE MODEL...54 ECOND ORDER CLORE MODEL: RM...56 LARGE EDDY IMLATION...57.0 CONCLION...60 BIBLIOGRAHY... Error! Boomar no defned. AENDICE...68 AENDIX A: TAYLOR ERIE EXANION...68 AENDIX B: BLAYER ANALYI...69 AENDIX C: CONERVATION EQATION...76 C. Conservaon of Mass...76 C. Conservaon of Momenm...76 C.3 Dervaon of he -Eqaon from he Reynolds resses...78 C.4 Dervaon of he Dsspaon Rae Eqaon...80 AENDIX D: GOVERNING EQATION WITH THE EFFECT OF BOYANCY...83 D. Mass...83 D. Boyancy (Mass) Eqaon...84 D.3 Momenm...84 D.4 Mean Flow Energy Eqaon...85 D.5 TKE Eqaon wh he Effecs of Boyancy...86 AENDIX E: REYNOLD-AVERAGED EQATION...89 E. Reynolds-Averaged Momenm Eqaon...89 E. Reynolds-Averaged Thermal Energy Eqaon...90 E.3 Reynolds-Averaged calar Transpor Eqaon...9

NOMENCLATRE Englsh B B ε C f D D ε l mfp l m prodcon of rblen nec energy by boyancy prodcon of rblen dsspaon by boyancy frcon coeffcen desrcon of rblen nec energy desrcon of rblen dsspaon specfc rblen nec energy mean-free pah lengh mng lengh pressre p p* modfed pressre prodcon of rblen nec energy prodcon of rblen dsspaon ε q * y Reynolds fl sorce erm mean sran rae ensor me velocy dmensonless velocy (wall varables) frcon velocy horzonal dsplacemen (dsplacemen n he sreamwse drecon) dmensonless vercal dsance (wall varables) Gree ymbols δ bondary layer hcness δ v * dsplacemen hcness δ Kroenecer dela fncon ε dsspaon rae of rblen nec energy φ generc scalar varable φ flcang componen of me-averaged varable φ Φ mean componen of me-averaged varable φ φ Reynolds me averaged varable φ Γ dffson coeffcen Γ rblen dffsvy κ von-karman consan moleclar vscosy v

ν Π σ σ τ τ w ω Ω eddy vscosy pressre-sran correlaon ensor densy rblen randl-chmd nmber randl-chmd nmber for Reynolds sress ensor wall shear sress dsspaon per n rblen nec energy (specfc dsspaon) mean roaon ensor v

.0 INTRODCTION Theorecal analyss and predcon of rblence has been, and o hs dae sll s, he fndamenal problem of fld dynamcs, parclarly of compaonal fld dynamcs (CFD). The maor dffcly arses from he random or chaoc nare of rblence phenomena. Becase of hs npredcably, has been csomary o wor wh he me averaged forms of he governng eqaons, whch nevably resls n erms nvolvng hgher order correlaons of flcang qanes of flow varables. The sem-emprcal mahemacal models nrodced for calclaon of hese nnown correlaons form he bass for rblence modelng. I s he focs of he presen sdy o nvesgae he man prncples of rblence modelng, ncldng eamnaon of he physcs of rblence, closre models, and applcaon o specfc flow condons. nce rblen flow calclaons sally nvolve CFD, specal emphass s gven o hs opc hrogho hs sdy. There are hree ey elemens nvolved n CFD: () grd generaon () algorhm developmen (3) rblence modelng Whle for he frs wo elemens precse mahemacal heores es, he concep of rblence modelng s far less precse de o he comple nare of rblen flow. Trblence s hreedmensonal and me-dependen, and a grea deal of nformaon s reqred o descrbe all of he mechancs of he flow. sng he wor of prevos nvesgaors (e.g. randl, Taylor, and von Karman), an deal rblence model aemps o capre he essence of he relevan physcs, whle nrodcng as lle compley as possble. The descrpon of a rblen flow may reqre a wde range of nformaon, from smple defnons of he sn frcon or hea ransfer coeffcens, all he way p o more comple energy specra and rblence flcaon magndes and scales, dependng on he parclar applcaon. The compley of he mahemacal models ncreases wh he amon of nformaon reqred abo he flowfeld, and s refleced by he way n whch he rblence s modeled, from smple mng-lengh models o he complee solon of he fll Naver-oes eqaons. The hyscs of Trblence In 937, Taylor and von Karman proposed he followng defnon of rblence: "Trblence s an rreglar moon whch n general maes s appearance n flds, gaseos or lqd, when hey flow pas sold srfaces or even when neghborng sreams of he same fld flow pas or over one anoher"

ome of he ey elemens of rblence are ha occrs over a large range of lengh and me scales, a hgh Reynolds nmber, and s flly hree-dmensonal and me-dependen. Trblen flows are mch more rreglar and nermen n conras wh lamnar flow, and rblence ypcally develops as an nsably of lamnar flow. For a real (.e. vscos) fld, hese nsables resl from he neracons of he non-lnear neral erms and he vscos erms conaned n he Naver-oes eqaons, whch are very comple de o he fac ha rblence s roaonal, hree-dmensonal, and me-dependan. The roaonal and hree-dmensonal nares of rblence are closely lned, as vore srechng s reqred o manan he consanly flcang vorcy. As vore srechng s absen n wo-dmensonal flows, rblence ms be hree-dmensonal. Ths mples ha here are no wo-dmensonal appromaons, hs mang he problem of resolvng rblen flows a dffcl problem. The me-dependan nare of rblence, wh a wde range of me scales (.e. freqences), means ha sascal averagng echnqes are reqred o appromae random flcaons. Tme averagng, however, leads o correlaons n he eqaons of moon ha are nnown a pror. Ths s he classc closre problem of rblence, whch reqres modeled epressons o accon for he addonal nnowns, and s he prmary focs of rblence modelng. Trblence s a connos phenomenon ha ess on a large range of lengh and me scales, whch are sll larger han moleclar scales. In order o vsalze rblen flows, one ofen refers o rblen eddes, whch can be hogh of as a local swrlng moon whose characersc dmenson s on he order of he local rblence lengh scale. Trblen eddes also overlap n space, where larger eddes carry smaller ones. As here ess a large range of dfferen scales (or rblen eddy szes), an energy cascade ess by whch energy s ransferred from he larger scales o he smaller scales, and evenally o he smalles scales where he energy s dsspaed no hea by moleclar vscosy. Trblen flows are hs always dsspave. Trblen flows also ehb a largely enhanced dffsvy. Ths rblen dffson grealy enhances he ransfer of mass, momenm, and energy. The apparen sresses, herefore, may be of several orders of magnde greaer han n he correspondng lamnar case. The fac ha he Naver-oes eqaons are non-lnear for rblen flows leads o neracons beween flcaons of dfferen wavelenghs and drecons. The wavelenghs of he moon may be as large as a characersc scale on he order of he wdh of he flow, all he way o he smalles scales, whch are lmed by he vscos dsspaon of energy. The acon of vore srechng s manly responsble for spreadng he moon over a wde range of wavelenghs. Wavelenghs whch are nearly comparable o he characersc mean-flow scales nerac mos srongly wh he mean flow. Ths mples ha he larger-scale rblen eddes are mos responsble for he energy ransfer and enhanced dffsvy. In rn, hese large eddes case random srechng of he vore elemens of he smaller eddes, and energy s cascades down from he larges o he smalles scales.

Fre chapers wll eamne some of he above aspecs of rblence as hey relae o case specfc sses. A Bref Hsory of Trblence Modelng The orgn of he me-averaged Naver-oes eqaons daes bac o he lae nneeenh cenry when Reynolds (895) pblshed resls from hs research on rblence. The earles aemps a developng a mahemacal descrpon of he rblen sresses, whch s he core of he closre problem, were performed by Bossnesq (877) wh he nrodcon of he eddy vscosy concep. Neher of hese ahors, however, aemped o solve he me-averaged Naver-oes eqaons n any nd of sysemac manner. More nformaon regardng he physcs of vscos flow was sll reqred, nl randls dscovery of he bondary layer n 904. randl (95) laer nrodced he concep of he mng-lengh model, whch prescrbed an algebrac relaon for he rblen sresses. Ths early developmen was he cornersone for nearly all rblence modelng effors for he ne weny years. The mng lengh model s now nown as an algebrac, or zero-eqaon model. To develop a more realsc mahemacal model of he rblen sresses, randl (945) nrodced he frs one-eqaon model by proposng ha he eddy vscosy depends on he rblen nec energy,, solvng a dfferenal eqaon o appromae he eac eqaon for. Ths oneeqaon model mproved he rblence predcons by ang no accon he effecs of flow hsory The problem of specfyng a rblence lengh scale sll remaned. Ths nformaon, whch can be hogh of as a characersc scale of he rblen eddes, changes for dfferen flows, and hs s reqred for a more complee descrpon of he rblence. A more complee model wold be one ha can be appled o a gven rblen flow by prescrbng bondary and/or nal condons. Kolmogorov (94) nrodced he frs complee rblence model, by modelng he rblen nec energy, and nrodcng a second parameer ω ha he referred o as he rae of dsspaon of energy per n volme and me. Ths wo-eqaon model, ermed he -ω model, sed he recprocal of ω as he rblence me scale, whle he qany ω served as a rblence lengh scale, solvng a dfferenal eqaon for ω smlar o he solon mehod for. Becase of he compley of he mahemacs, whch reqred he solon of nonlnear dfferenal eqaons, wen vrally who applcaon for many years, before he avalably of compers. Roa (95) poneered he se of he Bossnesq appromaon n rblence models o solve for he Reynolds sresses. Ths approach s called a second-order or second-momen closre. ch models narally ncorporae non-local and hsory effecs, sch as sreamlne crvare and body forces. The prevos eddy vscosy models faled o accon for sch effecs. For a hreedmensonal flow, hese second-order closre models nrodce seven eqaons, one for a rblence lengh scale, and s for he Reynolds sresses. As wh Kolmogorovs -ω model, he comple nare of hs model awaed adeqae comper resorces. 3

Ths, by he early 950s, for man caegores of rblence models had developed: () Algebrac (Zero-Eqaon) Models () One-Eqaon Models (3) Two-Eqaon Models (4) econd-order Closre Models Wh ncreased comper capables begnnng n he 960s, frher developmen of all for of hese classes of rblence models has occrred. The mos mporan modern developmens are gven below for each class. Algebrac (Zero-Eqaon) Models Van Dres (956) devsed a vscos dampng correcon for he mng-lengh model. Ths correcon s sll n se n mos modern rblence models. Cebec and mh (974) refned he eddy vscosy/mng-lengh concep for beer se wh aached bondary layers. Baldwn and Loma (978) proposed an alernave algebrac model o elmnae some of he dffcly n defnng a rblence lengh scale from he shear-layer hcness. One-Eqaon Models Whle employng a mch smpler approach han wo-eqaon or second-order closre models, one-eqaon models have been somewha npoplar and have no showed a grea deal of sccess. One noable ecepon was he model formlaed by Bradshaw, Ferrs, and Awell (967), whose model was esed agans he bes epermenal daa of he day a he 968 anford Conference on Compaon and Trblen Bondary Layers. There has been some renewed neres n he las several years de o he ease wh whch one-eqaon models can be solved nmercally, relave o more comple wo-eqaon or second-order closre models. Two-Eqaon Models Whle Kolmogorovs -ω model was he frs wo-eqaon model, he mos eensve wor has been done by Daly and Harlow (970) and Lander and paldng (97). Landers -ε model s he mos wdely sed wo-eqaon rblence model; here ε s he dsspaon rae of rblen nec energy. Independenly of Kolmogorov, affman (970) developed a -ω model ha shows advanages o he more well nown -ε model, especally for negrang hrogh he vscos sblayer and n flows wh adverse pressre gradens. 4

econd-order Closre Models De o he ncreased compley of hs class of rblence models, second-order closre models do no share he same wde se as he more poplar wo-eqaon or algebrac models. The mos noeworhy effors n he developmen of hs class of models was performed by Donaldson and Rosenbam (968), Daly and Harlow (970), and Lander, Reece, and Rod (975). The laer has become he baselne second-order closre model, wh more recen conrbons made by Lmley (978), pezale (985, 987a), Reynolds (987), and many oher hereafer, who have added mahemacal rgor o he model formlaon. Whle he presen sdy s no nended o be a complee caaloge of all rblence models, more dealed descrpon s gven for some of he above models n laer chapers. The concep of he closre problem wll also be nvesgaed, along wh a dscsson of case specfc sses as hey relae o dfferen ypes of flows. We shold noe here ha nfornaely here are no many e boos n he lerare whch can be sed for eachng rblence modelng, n conras o he esence of hndreds of hosands of ornal and conference papers n he lerare abo hs sbec. We wold menon he hree boos ha presen ahors consder as he bes for eachng prposes. These are de o Lander and paldng (97), Rod (980), and Wlco (993). 5

.0 REYNOLD TIME AVERAGING A rblence model s defned as a se of eqaons (algebrac or dfferenal) whch deermne he rblen ranspor erms n he mean flow eqaons and hs close he sysem of eqaons. Trblence models are based on hypoheses abo he rblen processes and reqre emprcal np n he form of model consans or fncons; hey do no smlae he deals of he rblen moon, b only he effec of rblence on he mean flow behavor. The concep of Reynolds averagng and he averaged conservaon eqaons are some of he man conceps ha form he bass of rblence modelng. nce all rblen flows are ransen and hree-dmensonal, he engneer s generally forced o develop mehods for averaged qanes o erac any sefl nformaon. The mos poplar mehod for dealng wh rblen flows s Reynolds averagng whch provdes nformaon abo he overall mean flow properes. The man dea behnd Reynolds me-averagng s o epress any varable, φ(,), whch s a fncon of me and space, as he sm of a mean and a flcang componen as gven by φ(, ) Φ ( ) φ(, ) (.) Here we se he noaon ha he ppercase symbols denoe he me average of ha qany. For saonary rblence, hs average s defned by τ Φ( ) φ(, ) lm (, ) d τ φ τ (.) where, by defnon, he average of he flcang componen s zero. For engneerng applcaons s assmed ha τ s mch greaer han he me scale of he rblen flcaons. For some flows he average flow may vary slowly wh me when compared o he me of he rblen flcaons. For hese flows he defnon gven n Eq. (.) may be replaced by where φ(, ) Φ (, ) φ(, ) (.3) τ Φ(, ) φ(, ) φ( d, ) τ (.4) and s assmed ha he me scale of he rblen flcaons s mch less han τ, and ha τ s mch less han he me scale relave o he mean flow (.e. perod of oscllaons for an oscllang flow or wave). Wh he precedng defnon n mnd, he followng rles apply o Reynolds me averagng. 6

. The me average of any consan vale (scalar or vecor) s eqal o he vale of he consan gven by A a a (.5). The me average of a me-averaged qany s he same as he me average self a A (.6) 3. Becase me averagng nvolves a defne negral, s a lnear operaor n ha he average of a sm eqals he sm of he averages as n a b a b A B (.7) 4. The me average of a mean qany mes a flcang qany s zero snce s smlar o a consan mes he average of a flcang qany. φ Ψ φ Ψ 0 (.8) 5. The me average of he prodc of wo varable qanes s gven by whch can be wren as φψ ( Φ φ )( Ψ ψ ) (.9) φψ ΦΨ Φψ Ψφ φ ψ Here we se he fac ha he wo ppercase qanes are averages, and se Eq. (.6) and (.8) o se he second and hrd erms o zero. Also, realzng ha he average of he prodc of wo flcang qanes s no necessarly zero gves φψ ΦΨ φ ψ (.0) 6. The me average of a spaal dervave s gven by φ ( Φ φ ) Φ φ ( Φ) ( φ ) (.) nce he average of any flcang componen s zero, he las erm on he rgh s zero. Ths ndcaes ha he average of he spaal dervave of a varable s eqal o he dervave of he average of he varable, or 7

φ Φ Φ (.) 7. The Reynolds average of a dervave wh respec o me s zero for saonary rblence. φ (, ) For non-saonary rblence, he erm gven by s he average of he me-dervave of a scalar qany φ. Applyng Eq. (.) o he erm gven above yelds φ(, ) φ (, ) (.3) Applyng Reynolds decomposon o he scalar n Eq. (.3) yelds (, ) Φ(, ) φ (, ) Φ(, ) φ (.4) becase he me-average of a mean qany yelds he mean qany, whle he meaverage of a flcang componen s zero. bsng no Eq. (.3) yelds φ(, ) Φ (, ) (.5) Whch shows ha he average of a me dervave s eqal o he me dervave of he average. 8

9 3.0 AVERAGED TRANORT EQATION Wh he concep of Reynolds me-averagng and he rles defnng s applcaon, we rn or aenon o he general conservaon eqaons governng fld flow and ranspor phenomena. Frs, we consder ncompressble flds wh consan properes. The eqaon of conny s gven by 0 (3.) In general, he eqaon of conny s ( ) 0 (3.) Here and afer, he ensor (nde) noaon s sed sch ha repeaed ndces ndcaes smmaon (e.g. 3 n hree dmensons). Tang he Reynolds me-average of Eq. (3.) gves 0 ) ( (3.3) or 0 (3.4) For ncompressble flows, follows from Eq. (3.3) ha he dvergence of he flcang velocy componens s also zero and s gven by 0 (3.5) In addon o he conny eqaon, he oher governng eqaons for ncompressble flow are he momenm eqaon gven by ( ) [ ] g s p ) ( (3.6) where ( ) ( ) s (3.7) and he smplfed hermal energy eqaon whch s gven by ( ) T T T α ) ( (3.8) when c p s consan.

0 A generc scalar ranspor eqaon can also be nclded n hs se of eqaons and s gven by ( ) φ φ φ φ p c Γ ) ( (3.9) Tang he Reynolds me average for Eqs. (3.6), (3.8), and (3.9) gves he Reynolds me averaged momenm eqaon as ( ) ( ) ( ) ( ) (3.0) he Reynolds me averaged hermal energy eqaon as ( ) T T T T ) ( α (3.) and he Reynolds averaged scalar ranspor eqaon as ( ) Φ Φ Γ Φ Φ p c ) ( φ (3.) The fll dervaons for Eqs. (3.0), (3.), and (3.) are gven n Append C. The Reynolds averaged conny eqaon s bascally he same as he naveraged eqaon n ha here are no new erms. However, addonal fl erms arse n he momenm and scalar eqaons. The era erms n he momenm eqaon are gven by τ (3.3) whch are nown as he Reynolds sresses, and he era erms n boh he scalar ranspor and energy eqaon ae he form φ q (3.4) whch are referred o as Reynolds (or rblen) fles. These addonal fles arse from he convecve ranspor de o rblen flcaons. When an eqaon s me-averaged, he nflence of he flcaons over he averagng me perod s nclded va hese addonal fl erms. In he corse of Reynolds averagng of he conservaon eqaons, hese addonal fles have been generaed b no new eqaons were obaned o accon for hese new nnowns. Trblence models provde closre o Eqs. (3.0), (3.), and (3.) by provdng models for he fles gven by Eqs. (3.3) and (3.4).

4.0 DERIVATION OF THE REYNOLD TRE EQATION In he Reynolds averaged momenm eqaon, as was gven n Eq. (3.0), he era erms, whch are commonly called he Reynolds sresses, can be epressed as a ensor τ (4.) where he frs nde ndcaes he plane along whch he sress acs, and he second gves he coordnae drecon. Here he prmes ndcae ha hs average sress s obaned from he rblen flcaon par of he nsananeos velocy, whch s gven by (4.) To fnd an eqaon for he Reynolds sresses, frs consder he Naver-oes eqaons for ncompressble flds, gven by or p p (4.3) (4.4) Noe ha hese eqaons are wren n a non-conservave form sng he eqaon of conny. Wh he evenal goal of fndng he maeral, or sbsanal, me dervave of he Reynolds sress eqaons, realze ha he maeral me dervave of he non-averaged erms can be wren as D( D ) ( ) D ( ) (4.5) D D D sng he chan rle, and also ha he Naver oes eqaons wren n erms of conan he flcang erms. Wh hs n mnd, nvely mlply Eq. (4.3) by and mlply Eq. (4.4) by yeldng and p p (4.6) (4.7)

To oban he dervave as gven n Eq. (4.5), ae he me average of Eqs. (4.6) and (4.7) and add hem ogeher o gve (I) (II) (III) p p (4.8) (IV) where he ransen, convecve, pressre, and vscos sress erms have been groped ogeher. To p Eq. (4.8) n he form of he Reynolds eqaon, each se of erms s consdered separaely. Frs consder he nseady erm gven by (I) (4.9) whch can be re-wren as (I) ( ) ( ) or (I) ) ( ) ( ) ( ) ( Applyng he rle ha he average of a flcang qany mes a mean vale s zero, he frs and hrd erms dsappear. sng he chan rle, and he fac ha he average of a me dervave s eqal o he me dervave of he average, gves he fnal form as (I) ( ) (4.0) econd, consder he convecve erm n Eq. (4.8) gven by (II) (4.) bsng for,, and, he sm of her mean and flcang pars gves

3 (II) ( ) ( ) ( ) ( ) whch can be re-wren as (II) (4.) sng he rle for he mean of an already averaged qany mes a flcang varable (ee Eq. (.8)), he frs and ffh erms n Eq. (4.) are zero. The second and sh erm may be combned sng he chan rle, and he forh and egh erms may also be combned sng he chan rle, reslng n (II) ( ) ( ) (4.3) Now for Eq. (4.3) consder he frs, second and forh erms. In he frs, may be reaed as a consan and may be removed from nder he average sgn; also he rle for a spaal dervave may be sed. In he second and forh erms he dervave of he mean veloces may be reaed as a consan. sng hese, Eq. (4.3) may be wren as (II) ( ) ( ) (4.4) The hrd erm n Eq. (4.4) can be modfed by realzng ha from he conny eqaon, Eq. (3.5), follows 0 (4.5) sng Eq. (4.5) and he chan rle, Eq (4.4) may be wren n s fnal form as (II) ( ) ( ) (4.6)

4 Thrd, consder he pressre erm n Eq. (4.8) gven as (III) p p (4.7) whch can be re-wren as (III) p p Here he frs and hrd erms are zero. The second and forh erms can be re-wren sng he chan rle as p p p ( ) (4.8) and p p p ( ) (4.9) sng he Kronecer dela fncon, δ, he pressre epressons may be wren n her fnal form as (III) [ ] δ δ p p p ( ) ( ) (4.0) Fnally, consder he vscos sress erm n Eq. (4.8) gven by (IV) (4.) Cancelng he average erms, as has been done n prevos dervaons, gves (IV) and applyng he chan rle wce yelds (IV) - (4.)

5 and (IV) - (4.3) whch complees he manplaon of he vscos erm. Collecng he ransen, convecve, pressre, and vscos erms ogeher gves ( ) ( ) ( ) [ ] δ δ p p p ( ) ( ) (4.4) - whch s eqvalen o Eq. (4.8). Mlplyng Eq. (4.4) by - and re-arrangng erms gves ) ( ) ( ) ( (4.5) [ ] p p p δ δ ) ( ) ( where he Reynolds ress erms may be seen n he dervaves on he LH. Applyng he defnon gven by Eq. (4.) he Reynolds sress eqaon may now be wren n s mos recognzable form as τ ) ) ( τ τ τ ε Π C ) ( τ υ (4.6) where ε

and Π p [( ) ( ) δ ( ) δ ] C p p As has been seen from he momenm eqaon, when rblen flows are consdered he averaged fl of momenm de o he rblen flcaons ms be aen no accon. Thogh he nflence of hese flcaons s nown, no drec means of calclang hem ess. Wh he dervaon of he Reynolds sress eqaon, he nflences on he sress erm can be denfed, b wh he dervaon new erms (hgher order correlaons) are generaed whch of hemselves are nnown. Whle he Reynolds eqaons provde nsgh no he nare of he rblen sresses, he engneer ms fnd some way o close he eqaons before hey can be sed. Fndng closre eqaons for calclang hese era erms s he bass of rblence modelng. 6

5.0 THE EDDY VICOITY/DIFFIVITY CONCET The oldes proposal for modelng he rblen or Reynolds sresses rns o o be a sgnfcan par of mos rblence models of praccal se oday. The man dea behnd hs model s Bossnesqs eddy-vscosy concep, whch assmes ha, n analogy o he vscos sresses n lamnar flows, he rblen sresses are proporonal o he mean velocy graden. Ths approach sems from reang rblen eddes n a smlar way ha molecles are reaed and analyzed n nec heory. Here eddes replace molecles as carrers of hermal energy and momenm. The eddy vscosy concep s bes consdered n lgh of moleclar ranspor of momenm. For a prely shearng flow, he average sress de o moleclar moon acng on a plane can be gven by τ y v (5.) where and v denoe moleclar veloces. A ypcal dervaon of hs shear sress can be accomplshed by consderng he shear flow llsraed n Fgre 5. (Wlco, 993). Here he sress eered n he horzonal drecon by he fld parcles, or molecles, a pon B on a plane a pon A, whose normal s n he y-drecon, can be gven by where he area n Eq. (5.) s ha of he vercal plane a A. m B A ) τ y ( (5.) Area A A l mfp y B B Fgre 5. - Typcal shear flow For a perfec gas, he average vercal velocy can be aen o be he hermal velocy, v h. A ypcal parcle wll move wh hs velocy along s mean free pah, l mfp, before colldng wh anoher molecle and ransferrng s momenm. 7

If a molecle s consdered o move along s mean free pah along he vercal dsance from B o A, hen he shear sress on he lower sde of plane A may be wren as τ y d Cvhlmfp (5.3) dy where C s a proporonaly consan. From nec heory, for a perfec gas hs consan can be shown o be 0.5. Ths hen allows he vscosy of a perfec gas o be defned by Now he shear sress gven n Eq. (5.) may be wren as v h l mfp (5.4) d τ y v (5.5) dy Realze ha n Eq. (5.3) he Taylor seres defnng he velocy has been rncaed afer he frs erm. Ths appromaon reqres ha (Wlco, 993) d d l mfp << dy (5.6) dy Ths appromaon also assmes ha he horzonal velocy remans essenally consan a any plane. nce molecles wll be ransferrng horzonal momenm o and from a plane, he velocy a any plane may be consdered effecvely consan only f he molecles eperence many collsons on he me scale relave o he mean flow. Ths splaon reqres d lmfp << vh (5.7) dy Boh of hese splaons are sasfed for vrally all flows of engneerng neres, gven ha v h s on he order of he speed of sond n he fld and l mfp s relavely small. They are menoned here snce anyone performng rblence modelng and ryng o mmc eddy ranspor, by analogy o moleclar ranspor, ms a leas be aware of hese splaons. sng a concep smlar o he moleclar vscosy for moleclar sresses, he concep of he eddy vscosy may be sed o model he Reynolds sresses. For general flow saons he eddy vscosy model may be wren as ν 3 δ (5.8) 8

where ν s he rblen or eddy vscosy, and s he rblen nec energy. In conras o he moleclar vscosy, he rblen vscosy s no a fld propery b depends srongly on he sae of rblence; ν may vary sgnfcanly from one pon n he flow o anoher and also from flow o flow. The man problem n hs concep s o deermne he dsrbon of ν. Inclson of he second par of he eddy vscosy epresson assres ha he sm of he normal sresses s eqal o, whch s reqred by defnon of. The normal sresses ac le pressre forces, and hs he second par conses pressre. Eq. (5.8) s sed o elmnae n he momenm eqaon. The second par can be absorbed no he pressre-graden erm so ha, n effec, he sac pressre s replaced as an nnown qany by he modfed pressre gven by p* p. (5.9) 3 Therefore, he appearance of n Eq. (5.8) does no necessarly reqre he deermnaon of o mae se of he eddy vscosy formlaon; he man obecve s hen o deermne he eddy vscosy. In drec analogy o he rblen momenm ranspor, he rblen hea or mass ranspor s ofen assmed o be relaed o he graden of he ranspored qany, wh eddes agan replacng molecles as he carrers. Wh hs concep, he Reynolds fl erms may be epressed sng φ φ Γ (5.0) Here Γ s he rblen dffsvy of hea or mass and has ns eqvalen o he hermal dffsvy of m /s. Le he eddy vscosy, Γ s no a fld propery b depends on he sae of he rblence. The eddy dffsvy s sally relaed o he rblen eddy vscosy va ν Γ (5.) σ where σ s he rblen randl or chmd nmber, whch s a consan appromaely eqal o one. As wll be shown n laer secons, he prmary goal of many rblence models s o fnd some prescrpon for he eddy vscosy o model he Reynolds sresses. These may range from he relavely smple algebrac models, o he more comple models sch as he -ε model, where wo addonal ranspor eqaons are solved n addon o he mean flow eqaons. 9

6.0 ALGEBRAIC TRLENCE MODEL: ZERO-EQATION MODEL The smples rblence models, also referred o as zero eqaon models, se a Bossnesq eddy vscosy approach o calclae he Reynolds sress. In drec analogy o he moleclar ranspor of momenm, randl s mng lengh model assmes ha rblen eddes clng ogeher and manan her momenm for a dsance, l m, and are propelled by some rblen velocy, v m. Wh hese assmpons, he Reynolds sress erms are modeled by d v vmlm (6.) dy for a wo-dmensonal shear flow as s shown n Fgre 5.. Ths model frher poslaes ha he mng velocy, v m, s of he same order of magnde as he (horzonal) flcang veloces of he eddes, whch can be sppored hrogh epermenal resls for a wde range of rblen flows. Wh hs assmpon v m d v w lm (6.) dy or, n erms of he eddy, or rblen, vscosy for a shear flow d ( l m ) (6.3) dy whch can be mpled from Eq. (6.). Ths defnon for he eddy vscosy can also be mpled on dmensonal gronds. Wh hese defnons n mnd, he obecve of mos algebrac models s o fnd some prescrpon for he rblen mng lengh, n order o provde closre o Eqs. (6.) and (6.3). The dea ha a rblen mng lengh can be sed n a smlar way ha he moleclar mean free pah s sed o calclae he vscosy for a perfec gas provdes a reasonable approach o calclang he eddy vscosy. However, hs approach ms be eamned n lgh of he same assmpons made for he analyss of he moleclar vscosy n econ 5.0. The appropraeness of hs model can be qesoned by consderng he wo reqremens ha were consdered for he case of moleclar mng (Tennees and Lmley, 983), namely and d d l m << dy (6.4) dy d lm << vh (6.5) dy 0

I has been shown epermenally, ha close o a wall l m s proporonal o he normal dsance from he wall. Also, near a sold srface he velocy graden vares nversely wh y, as dedced from he law of he wall for a rblen velocy profle. Consderng hese facs n lgh of Eq. (6.4), he mng lengh model does no gve sold sfcaon for he frs order Taylor seres rncaon ha s sed n he moleclar vscosy eqaon. The fac ha he average me for collsons v m d (6.6) l dy m s large, also garanees ha he momenm of an eddy wll ndergo changes de o oher collsons before ravels he fll dsance of s mng lengh (Tennees and Lmley, 983). Ths fac s no aen no consderaon snce he moleclar lengh of ravel s defned as he ndsrbed dsance ha a molecle ravels before collson. These comparsons show ha he mng lengh model does no have a srong heorecal bacgrond as s sally perceved. Despe he shorcomngs, however, n pracce can acally be calbraed o gve good engneerng and rend predcons. Free shear flows A flow s ermed "free" f can be consdered o be nbonded by any sold srface. nce walls and bondary condons a sold srfaces complcae rblence models, wo-dmensonal, free shear flows form a good se of cases o sdy he applcably of a rblence model. Ths sems from he fac ha only one sgnfcan rblen sress ess n wo-dmensonal flows. Three flows ha can be consdered are he far wae, he mng layer, and he e. A wae forms downsream of any obec placed n he pah of a flowng fld, a mng layer occrs beween wo parallel sreams movng a dfferen speeds, and a e occrs when a fld s neced no a second qescen fld. A far wae s shown n Fgre 6.. Fgre 6. - Far Wae

For each of hese cases he assmpon can be made ha he mng lengh s some consan mes he local layer wdh, δ, or l m α δ() (6.7) Ths consan ms be deermned hrogh some emprcal np, as well as he governng eqaons. For all hree flows he sandard bondary layer (or hn shear layer) eqaons can be sed. Two assmpons governng he evenal solon of he problem are ha he pressre s consan and ha moleclar ranspor of momenm s neglgble compared o he rblen ranspor. From he smlary analyss and nmercal solons ha are obaned for hs class of flows, he vales governng he mng lengh are gven n Table 6.. Table 6. - Mng lengh consans for free shear flows (Wlco, 993) Flow Type: Far Wae lane Je Radal Je lane Mng Layer l m δ 0.80 0.098 0.080 0.07 For each of he free shear cases, he analycal predcon of he velocy gves a sharp rblen/non-rblen nerface. These nerfaces, whle esng n realy, are generally characerzed by me flcaons and have smooh properes when averaged, no sharp nerfaces. Ths nphyscal predcon of he mng lengh model s characersc of many rblence models where a regon has a rblen/non-rblen nerface. The nermen, ransen nare of rblence s herefore no acconed for a he nerface n hese solons.

Wall bonded flows In free shear flows he mng lengh was shown o be consan across a layer and proporonal o he wdh of he layer. For flows near a sold srface, a dfferen prescrpon ms be sed, noably snce he mng lengh can no physcally eend beyond he bondary esablshed by he sold srface. For flow near a fla wall or plae, sng he epermenal fac ha momenm changes are neglgble and he shear sress s appromaely consan, can be shown (Wlco, 993), sng bondary layer heory, ha ν.0 (.0 ν where, y y ν ( * τ w, τ w wall shear sress) are he famlar dmensonless velocy and vercal dsance (also nown as wall varables) as sed n he law of he wall. In he vscos sblayer, where vscos forces domnae rblen flcaons, Eq. (6.8) becomes (see Append B) y (6.9) For wall flows, based on epermenal observaons, s aen ha he mng lengh s proporonal o he dsance from he srface, or n erms of he eddy vscosy as gven by Eq. (6.3) ) d dy (6.8) d ν κ y (6.0) dy Also, n he flly rblen zone, effecs of moleclar vscosy are low compared wh ν. Ths allows Eq. (6.8) o be wren as.0 d κ y (6.) dy whch can be negraed (wh he assmpon ha he rblen shear sress s consan) o gve ln( y ) C (6.) κ and shows ha he mng lengh concep s conssen wh he epermenal law of he wall f he mng lengh s aen o vary n proporon o he dsance normal from he srface. nce Eq. (6.) s a good esmae only n he log layer, and no close o he wall n he vscos sblayer or n he oer layer, several modfcaons o he appromaon for l m have been devsed. Three of he beer nown modfcaons are he Van-Dres, Cebec-mh (974), and Baldwn-Loma (978) models. 3

Van-Dres Model To mae l m approach zero more qcly n he vscos sblayer, hs model specfes [ ep( y A )] l κ / (6.3) m y o where A o 6. Ths model s based prmarly on epermenal evdence, b s also based on he dea ha he Reynolds sress approaches zero near he wall n proporon o y 3. Ths has been shown o be he case n DN smlaons. Cebec-mh Model (974) For wall bondary layers, hs model provdes a complee specfcaon of he mng lengh and eddy vscosy over he enre range of he vscos sblayer, log layer, and defec or oer layer. Here he eddy vscosy s gven n he nner regon by wh / V ν ( l m ) y y y m (6.4) [ ep( y / )] l m κ y A (6.5) / d / d A 6 y (6.6) τ The oer layer vscosy s gven by wh and * ν αeδ vfkleb( y; δ) y > y m (6.7) α 0.068 y FKleb ( y; δ) 55. δ 6 (6.8) Here y m s he smalles vale of y for whch he nner and oer eddy vscoses are eqal, δ s he bondary layer hcness, and δ v s he velocy or dsplacemen hcness for ncompressble 4

flow. F Kleb s Klebanoff s nermency fncon, whch was proposed o accon for he fac ha n approachng he free sream from whn a bondary layer, he flow s somemes lamnar and somemes rblen (.e. nermen). ome general mprovemens of hs model are ha ncldes bondary layers wh pressre gradens by modfyng he vale of A n Van-Dress mng lengh formla. Ths model s also very poplar becase of s ease of mplemenaon n a comper program; here he man problem s he compaon of δ and δ v *. I s also worh nong ha, n general, he pon where y y m and he Reynolds sress s a mamm occrs well nsde he log layer. Baldwn-Loma Model (978) The Baldwn-Loma model (Baldwn and Loma, 978) was formlaed o be sed n applcaons where he bondary layer hcness, δ, and dsplacemen hcness, δ v *, are no easly deermned. As n he Cebec-mh model (Cebec and mh, 974), hs model also ses an nner and an oer layer eddy vscosy. The nner vscosy s gven by ν ( l m ) ω y y m (6.9) where he symbol, ω,s he magnde of he vorcy vecor for hree dmensonal flows. Here he vorcy provdes a more general parameer for deermnng he magnde of he mng velocy han he velocy graden as s gven n Eq. (6.). The mng lengh s calclaed from he Van-Dres eqaon: The oer vscosy s gven by m [ ep( y / A )] l κ y (6.0) o where ν αc F F ( y, yma / C ) y > y m (6.) cp Wae Kleb Kleb [ ma ma ma ma] F mn y F, C y / F Wae w df (6.) F ma ma( lm ω ) κ (6.3) y Ths model avods he need o locae he bondary layer edge by calclang he oer layer lengh scale n erms of he vorcy nsead of he dsplacemen or hcness. By replacng e δ * v n he Cebc-mh model by C cp F Wae 5

f FWae y ma F ma hen δ * v ymzω e (6.4) f F C y / F Wae w ma df ma hen δ df ω (6.5) Here y ma s he vale of y a whch l m of ω acheves s mamm vale, and df s he mamm vale of he velocy for bondary layers. The consans n hs model are gven by κ 0.40 α 0.068 A o 6 C cp.6 C Kleb 0.3 C w Boh he Cebc-mh (974) and he Baldwn-Loma (978) models yeld reasonable resls for sch applcaons as flly developed ppe or channel flow and bondary layer flow. One neresng observaon for ppe and channel flow s ha sble dfferences n a model s predcon for Reynolds sress can lead o mch larger dfferences n velocy profle predcons. Ths s a common accracy dlemma wh many rblence models. Boh models have been fne ned for bondary layer flow and herefore provde good agreemen wh epermenal daa for reasonable pressre gradens and mld adverse pressre gradens. For separaed flows, algebrac models generally perform poorly de o her nably o accon for flow hsory effecs. The effecs of flow hsory accon for he fac ha he rblen eddes n a zone of separaon occr on a me scale ndependen of he mean sran rae. everal modfcaons o he algebrac models have been proposed o mprove her predcon of separaed flows, mos noably a model proposed by Johnson and Kng (985). Ths model solves an era ordnary dfferenal eqaon, n addon o he Reynolds eqaons, o sasfy a noneqlbrm parameer ha deermnes he mamm Reynolds sress. In general, zero-eqaon algebrac models perform reasonably well for free shear flows; however, he mng lengh specfcaon for hese flows s hghly problem dependen. For wall bonded flows and bondary layer flows he Cebec-mh (974) and Baldwn-Loma (978) models gve good engneerng predcons when compared o epermenal vales of he frcon coeffcens and velocy profles. Ths s parally de o he modfcaons hese models have receved o mach epermenal daa, especally for bondary layer flows. Neher model s relable for predcng eraordnarly comple flows or separaed flows, however hey have hsorcally provded sond engneerng solons for problems whn her range of applcably. The Johnson-Kng (985) model menoned above gves beer predcon for separaed flows by solvng an era dfferenal eqaon. Ths model s referred o as a half-eqaon model de o he fac ha he addonal eqaon solved s an ordnary dfferenal eqaon. As wll be shown n he ne secon on one and wo eqaon models, a maor classfcaon of rblence modelng s based on he solons of addonal paral dfferenal eqaons ha deermne characersc rblen velocy and lengh scales. 6

7.0 ONE-EQATION TRBLENCE MODEL As an alernave o he algebrac or mng lengh model, one-eqaon models have been developed n an aemp o mprove rblen flow predcons by solvng one addonal ranspor eqaon. Whle several dfferen rblen scales have been sed as he varable n he era ranspor eqaon, he mos poplar mehod s o solve for he characersc rblen velocy scale proporonal o he sqare roo of he specfc nec energy of he rblen flcaons. Ths qany s sally s referred o as he rblence nec energy and s denoed by. The Reynolds sresses are hen relaed o hs scale n a smlar manner n whch τ was relaed o v m and l m n algebrac models. nce he modeled eqaon s generally he bass for all one and wo eqaon models, he eac dfferenal eqaon for he rblence nec energy, and he physcal nerpreaon of he erms n he rblence nec energy eqaon, wll be consdered frs. Then, wh some ndersandng of he eac eqaon, he mehods sed o model he eqaon wll be consdered. Eac Trblen Knec Energy Eqaon As has been menoned, an obvos choce for he characersc velocy scale n a rblen flow s he sqare roo of he nec energy of he rblen flcaons. sng he prmed qanes o denoe he velocy flcaons, he Reynolds averaged nec energy of he rblen eddes can be wren on a per n mass bass as ( vv ww) (7.) An eqaon for may be obaned by realzng ha Eq. (7.) s s / mes he sm of he normal Reynolds sresses. By seng and n Eq. (4.6), whch s eqvalen o ang he race of he Reynolds sress ensor, he eqaon for an ncompressble fld can be dedced as ( ) τ p I II III IV (7.) The fll dervaon of Eq. (7.) s gven n Append C. As was seen n he dervaon for he Reynolds sress eqaon, he eqaon nvolves several hgher order correlaons of flcang velocy componens, whch canno be deermned. Therefore, before aempng o model Eq. (7.), s benefcal o consder he physcal processes represened by each erm n he eqaon. 7

The frs and second erm on he lef-hand sde of Eq. (7.) represen he rae of change of he rblen nec energy n an Eleran frame of reference (me rae of change pls advecon) and are famlar from any generc scalar ranspor eqaon. The frs erm on he rgh-hand sde of Eq. (7.) s generally nown as he prodcon erm, and represens he specfc nec energy per n volme ha an eddy wll gan per n me de o he mean (flow) sran rae. If he eqaon for he nec energy of he mean flow s consdered, can be shown ha hs erm acally appears as a sn n ha eqaon. Ths shold be epeced and frher verfes ha he prodcon of rblence nec energy s ndeed a resl of he mean flow losng nec energy. The second erm on he lef-hand sde of Eq. (7.) s referred o as dsspaon, and represens he mean rae a whch he nec energy of he smalles rblen eddes s ransferred o hermal energy a he moleclar level. The dsspaon s denoed as ε and s gven by (II) ε ν (7.3) The dsspaon s essenally he mean rae a whch wor s done by he flcang sran rae agans flcang vscos sresses. I can be seen ha becase he flcang sran rae s mlpled by self, he erm gven by Eq. (7.3) s always posve. Hence, he role of he dsspaon wll be o always ac as a sn (or desrcon) erm n he rblen nec energy eqaon. I shold also be noed ha he flcang sran rae s generally mch larger han he mean rae of sran when he Reynolds nmber of a gven flow s large. The hrd erm n Eq (7.) represens he dffson, no desrcon, of rblen energy by he moleclar moon ha s eqally responsbly for dffsng he mean flow momenm. Ths erm s gven by (III) (7.4) and can be seen o have he same general form of any generc scalar dffson erm. The las erm n Eq. (7.), conanng he rple flcang velocy correlaon and he pressre flcaons, s gven by (IV) ( p ) (7.5) The frs par of hs erm represens he rae a whch rblen energy s ranspored hrogh he flow va rblen flcaons. The second par of hs erm s eqvalen o he flow wor done on a dfferenal conrol volme de o he pressre flcaons. Essenally hs amons o a ranspor (or redsrbon) by pressre flcaons. Each of he erms nclded n Eq. (7.4) and 8

(7.5) have he endency o redsrbe he specfc nec energy of he flow. Ths s drecly analogos o he way n whch scalar gradens and rblen flcaons ranspor any generc scalar. I s mporan, hen, o realze ha he ranspor by graden dffson, rblen flcaons, and pressre flcaons can only redsrbe he rblen nec energy n a gven flow. However, he frs wo erms on he RH of Eq. (7.) acally represen a sorce and a sn by whch rblen nec energy may be prodced or desroyed. Modeled Trblen Knec Energy Eqaon If he eqaon s gong o be sed for any prpose oher han lendng some physcal nsgh no he behavor of he energy conaned by he rblen flcaons, some way of obanng s solon ms be sogh. As wh all Reynolds averaged, rblence eqaons, he hgher order correlaons of flcang qanes prodce more nnowns han avalable eqaons, reslng n he famlar closre problem. Therefore, f he eqaon s o be solved, some correlaon for he Reynolds sresses, dsspaon, rblen dffson, and pressre dffson ms be specfed n lgh of physcal reasonng and epermenal evdence. The specfcaon of hese erms n he specfc rblen nec energy eqaon s generally he sarng pon n all one and wo eqaon rblence models. The frs assmpon made n modelng he eqaon s ha he Bossnesq eddy vscosy appromaon s vald and ha he Reynolds sresses can be modeled by τ δ (7.6) 3 for an ncompressble fld. The eddy vscosy n Eq. (7.6) s based on a smlar concep ha was sed for algebrac models and s generally gven by / l (7.7) where l s some rblence lengh scale. Ths lengh scale, l, may be smlar o he mng lengh, b s wren who he sbscrp m o avod confson wh mng lengh models. The characersc velocy scale n Eq. (7.7) s aen o be he sqare roo of, snce he rblen flcaons a a pon n he flowfeld are represenave of he rblen ranspor of momenm. The prescrpon for gven by Eq. (7.7) s very smlar o ha gven by v l (7.8) m m for he algebrac or mng lengh model, where m v was generally aen o be relaed o a sngle velocy graden n he flow. Implc n eher prescrpon for he eddy vscosy s he assmpon ha he rblence behaves on a me scale proporonal o ha of he mean flow. Ths s no a rval assmpon and wll be shown o be responsble for many of he naccraces 9

ha can resl sng an eddy vscosy model for ceran classes of flows. I shold be noed ha Eq. (7.7) s an soropc relaon, whch assmes ha momenm ranspor n all drecons s he same a a gven pon n space. However, f sed appropraely, hs concep yelds predcons wh accepable accracy for a wde range of engneerng applcaons. The physcal reasonng sed n modelng he dsspaon erm s no dsncve; however, he same general conclson of how epslon shold scale can be nferred by several consderaons. A frs appromaon for ε may be obaned by consderng a seady, homogenos, hn shear layer where can be shown ha prodcon and dsspaon of rblen nec energy balance (.e. flow s n local eqlbrm). Mahemacally hs may be wren as ν ε (7.9) from Eq. (7.). nce homogeneos, shear generaed rblence can be characerzed by one lengh scale, l, and one velocy scale,, he lef hand sde of Eq. (7.9) may be aen o scale appromaely as 3 /l. Hence, for rblen flows where prodcon balances dsspaon, f he rblence can be characerzed by one lengh and one velocy scale, he dsspaon may be aen o scale accordng o ε 3 l (7.0) where he lengh scale l, and he velocy scale, are characersc of he rblen flow. Anoher nerpreaon of he dsspaon erm may be o consder ha he rae of energy dsspaon s conrolled by nvscd mechansms, he neracon of he larges eddes. The large eddes cascade energy o he smaller scales, whch ads o accommodae he energy dsspaon. Therefore he dsspaon shold scale n proporon o he scales deermned by he larges eddes (see, e.g. Tennees and Lmley, 983). sng dmensonal analyss, n erms of a sngle rblen velocy scale,, and a sngle rblence lengh scale, l, leads o Eq. (7.0). The esmae obaned n hs fashon ress on one of he prmary assmpons of rblence heory; clams ha he large eddes lose mos of her energy n one rnover me. I does no clam ha he large eddy dsspaes he energy drecly, b ha s dsspave becase creaes smaller eddes. Ths nerpreaon s ndependen of he assmpon ha prodcon balances dsspaon and may be aen as a good appromaon for many ypes of flows. The absence of hs assmpon may be mpled from he fac ha s already acconed for snce a he small scales, rblence prodcon effecvely balances dsspaon. In oher words, a hgh Reynolds nmbers, he local eqlbrm assmpon s sally vald. 30

Whe (99) comes o he same conclson based on a somewha more physcally based argmen. If an eddy of sze l s movng wh speed, s energy dsspaed per n mass shold be appromaed by ( drag)( velocy) ( l ) ε mass 3 (7.) l l Gven hese argmens for he scales of he dsspaon, and ang he sqare roo of he rblen nec energy as he characersc velocy scale, he dsspaon s generally modeled by ε l 3 / 3 (7.) The rblen ranspor of, and he pressre dffson erms, are generally modeled by assmng a graden dffson ranspor mechansm. Ths mehod s smlar o ha sed for generc scalar ranspor n a rblen flow as was gven n econ 5.0. Whle hs mehod s normally appled o he erms nvolvng he rblen flcaons, s common pracce o grop he pressre flcaon erm, whch s generally small for ncompressble flows, wh he graden dffson erm. Wh hese assmpons he rblen ranspor of and he pressre flcaon erm are gven by p (7.3) σ where σ s a closre coeffcen nown as he randl-chmd nmber for. Combnng he modeled erms for he Reynolds sress, dsspaon, rblen dffson, and pressre dffson wh he ransen, convecve, and vscos erms gves he modeled eqaon as ( ) τ ε ( (7.4) σ wh he dsspaon and eddy vscosy gven by 3 / ε C D l (7.5) where C D and C are consans. C / l (7.6) An obvos reqremen s sll he deermnaon of he lengh scale, l, whch for a one-eqaon model ms sll be specfed algebracally n accordance o some mean flow parameers. nce he one-eqaon model s based prmarly on several of he same assmpons as he mng lengh model, for eqlbrm flows can be shown ha he lengh scale l can be made 3

proporonal o l m. For many one-eqaon models, he algebrac prescrpons sed for he mng lengh provde a logcal sarng place. One poplar verson of hs model (Emmons, 954; Glsho, 965) nvolves seng σ.0, wh C D varyng beween 0.07 and 0.09, along wh lengh scale prescrpons smlar o hose for he mng lengh model. Ths model s somewha smple, nvolvng less closre coeffcens han several mng lengh models; however, he necessary prescrpon of l by algebrac eqaons, whch vary from problem o problem, s a defne lmaon. Hence even wh a characersc velocy scale, one-eqaon models are sll ncomplee. Typcal applcaons for one-eqaon models nvolve prmarly he same ypes of flows as dd mng lengh models. One-eqaon models have a somewha beer hsory for predcon of separaed flows; however, hey share mos of he falres of he mng lengh model. The specfcaon of he mng lengh by an algebrac formla s sll almos enrely dependen on emprcal daa, and s sally ncapable of ncldng ranspor effecs on he lengh scale. As wll be seen n econ 8.0, he desre o nclde ranspor effecs on he lengh scale wll be he prmary reason for nrodcng wo-eqaon models sch as he -ε and -ω models. 3

8.0 TWO-EQATION TRBLENCE MODEL Two-eqaon models have been he mos poplar models for a wde range of engneerng analyss and research. These models provde ndependen ranspor eqaons for boh he rblence lengh scale, or some eqvalen parameer, and he rblen nec energy. Wh he specfcaon of hese wo varables, wo-eqaon models are complee; no addonal nformaon abo he rblence s necessary o se he model for a gven flow scenaro. Whle hs s encoragng n ha hese models may appear o apply o a wde range of flows, s nsrcve o ndersand he mplc assmpons made n formlang a wo-eqaon model. Whle complee n ha no new nformaon s needed, he wo-eqaon model s o some degree lmed o flows n whch s fndamenal assmpons are no grossly volaed. pecfcally, mos wo-eqaon models mae he same fndamenal assmpon of local eqlbrm, where rblen prodcon and dsspaon balance. Ths assmpon frher mples ha he scales of he rblence are locally proporonal o he scales of he mean flow; herefore, mos woeqaon models wll be n error when appled o non-eqlbrm flows. Thogh somewha resrced, wo-eqaon models are sll very poplar and can be sed o gve resls well whn engneerng accracy when appled o approprae cases. Ths chaper wll frs focs on he second varable, whch s solved n addon o he rblence nec energy; specfcally he wo addonal varables, ε and ω, wll be consdered. Generally, ε s defned as he dsspaon, or rae of desrcon of rblence nec energy per n me, and ω can be defned eher as he rae a whch rblen nec energy s dsspaed or as he nverse of he me scale of he dsspaon. Boh varables are relaed o each oher and o he lengh scale, l, whch has hs far been assocaed wh zero and one-eqaon models. Here we nrodce he mahemacal epresson for ω for frher reference (Kolmogorov, 94) where c s a consan. c l ω (8.) General Two-Eqaon Model Assmpons The frs maor assmpon of mos wo-eqaon models s ha he rblen flcaons,, v, and w, are locally soropc or eqal. Whle hs s re of he smaller eddes a hgh Reynolds nmbers, he large eddes are n a sae of seady ansoropy de o he sran rae of he mean flow, hogh, v, and w are almos always of he same magnde. Implc n hs assmpon s ha he normal Reynolds sresses are eqal a a pon n he flowfeld. The second maor assmpon of mos wo-eqaon models s ha he prodcon and dsspaon erms, gven n he -eqaon, are appromaely eqal locally. Ths s nown as he local eqlbrm assmpon. Ths assmpon follows from he fac ha he Reynolds sresses ms be esmaed a every pon n he flowfeld. To allow he Reynolds sresses o be calclaed 33

sng local scales, mos wo-eqaon models assme ha prodcon eqals dsspaon n he - eqaon or τ ε or ε (8.) τ where τ s he rblen sress ensor. Ths mples ha he rblen and mean flow qanes are locally proporonal a any pon n he flow. Hence, wh he rao of rblen and mean flow qanes eqal o some local qany, he eddy vscosy s defned o be he proporonaly consan beween he Reynolds sresses and he mean sran rae. Ths s he essence of he Bossnesq appromaon, whch was gven n Chaper 5.0 by ν 3 δ (8.3) where he second qany on he rgh nvolvng has been nclded o ensre he correc normal Reynolds sress n he absence of any sran rae. nce he rblence and mean scales are proporonal, he eddy vscosy can be esmaed based on dmensonal reasonng by sng eher he rblen or mean scales. Ths mples ha for he -ε model for he -ω model (8.4) ε (8.5) ω If prodcon does no balance dsspaon locally hen he rao of he Reynolds sresses o he mean sran rae s no a local consan and wll be a fncon of boh rblen and mean scales. Anoher way o ndersand hs assmpon of local eqlbrm s o realze ha ranspor effecs, whle nclded for he rblen scales, are negleced for he rblen Reynolds sresses. Local scales are hen sed n esmang he Reynolds sresses. Oherwse he Reynolds sresses wold depend on he local condons wh some hsory effecs. In predomnanly parabolc (n-dreconal) flow, f he flow s slowly varyng, hen psream scales are appromaely he same as local scales of rblence, and he eqlbrm assmpon wll wor. Also, f he rblence s evolvng a a sffcenly rapd rae, sch ha he effecs of pas evens do no domnae he dynamcs, he esmaes based on local scales wll wor. Whle here are many cases where he local eqlbrm assmpon does es for a wde range of shear flows, wo-eqaon models shold be careflly appled as he models can be epeced o perform poorly for non-eqlbrm flows. Typcal flows where wo-eqaon models have been shown o 34

fal are flows wh sdden changes n mean sran rae, crved srfaces, secondary moons, roang and srafed flds, flows wh separaon, and hree dmensonal flows. The -ε Model The frs sep n formlang he -ε model s n he consderaon of he second varable, ε. nce ε has already been defned by ε ν (8.6) he sarng pon for calclang he second varable shold be he eac eqaon for epslon. Ths can be done by performng he operaon L( ) ν () (8.7) o boh sdes of he nsananeos, ncompressble, Naver-oes eqaons. The dervaon of he ε-eqaon s gven n Append C. Afer a edos manplaon, he eac eqaon of ε can be wren as [ ] ε ε ν, m, m, m, m, ε νp,,,,,, m, m m, m (8.8) The physcal processes represened n he ε eqaon nvolve several doble and rple correlaons of flcang velocy componens. Whle somewha formdable, he followng physcal nerpreaons es for he erms n he epslon eqaon. The frs wo erms on he rgh-hand sde of Eq. (8.8) represen he prodcon of dsspaon de o neracons beween he mean flow and he prodcs of he rblen flcaons. The ne wo erms on he rghhand sde represen he desrcon rae of dsspaon de o he rblen velocy flcaons. Fnally, he las hree erms n Eq. (8.8) represen he ranspor or spaal re-dsrbon of dsspaon de o vscos dffson, rblen flcaons, and pressre-velocy flcaons. As already menoned, he -ε rblence model s based on Eq. (8.8). I shold be realzed ha wha s acally needed n he model s a lengh or me scale relevan o he large, energy conanng eddes, whch are responsble for he maory of he rblen sresses and fles. However, he mechansm of he smalles eddes, whch physcally accomplshes he dsspaon, 35

are wha s acally represened by Eq. (8.8). Ths leads o some qesons as o how relevan he eac dsspaon eqaon s when he desred qany s a lengh scale, characersc of he large eddes. A coner-argmen n favor of he se of Eq. (8.8) sems from he same scale esmaes for ε sed n Chaper 7.0, whch arge ha ε s deermned by he energy ransfer from he large scales, and ha Eq. (8.8) blds a ln beween he small-scale moon and he large eddes. Acally, snce he closre appromaons sed n modelng Eq. (8.8) are based prmarly on large-eddy scales, s mpled ha he modeled eqaon s acally more of an emprcal eqaon represenng he ransfer of energy from he large eddes o he small eddes. nce he modeled -eqaon was already nvesgaed n Chaper 7.0, he mehod sed o model he ε-eqaon wll be consdered here. The frs erm on he RH sde of Eq. (8.8) s gven by [ ] ε,,,,, (8.9) and s generally aen o gve he prodcon of dsspaon as was prevosly menoned. Ths nerpreaon follows from he presmpon ha he prodc of he rblen flcaons normally have negave correlaons n regons of posve velocy gradens. As hs erm s regarded as a sorce, or prodcon, of ε, he frs sep n modelng he erm s o consder he assmpon of local eqlbrm, namely whch mples τ ε (8.0) ε τ ch ( ) ch (8.) where ch s a characersc me scale for he prodcon of ε. nce, for local eqlbrm, he rae a whch s prodced s eqal o ε, hen he only me scale ha can defne he rae of change of ε ms scale wh. Ths mples ha ch ε (8.) whch gves Cε (8.3) ε ε Hence, sng he assmpon of local soropy and local eqlbrm, Eq. (8.9) s replaced by Eq. (8.3), whch closes he nnown velocy correlaons. In a smlar manner, he qany represened by he hrd and forh erms on he RH of Eq. (8.8), gven by 36

37 D m m m m ε ν,,,,, (8.4) can be aen o represen he me rae a whch dsspaon s desroyed or dsspaed. sng dmensonal analyss, hs can be hogh of as beng gven by D ch ε ε (8.5) whch mples from Eq. (8.) ha D C ε ε ε (8.6) Fnally, he dffson de o moleclar acon and rblen flcaons s modeled sng a sandard graden dffson approach. sng a rblen randl nmber for ε, gven by σ ε, hs mples m m m m p ε σ ν ε ε (,,, (8.7) Collecng he modeled erms for he ε-eqaon along wh he modeled -eqaon, he sandard -ε rblence model s gven by he -eqaon ε τ σ ) ( (8.8) he εeqaon C C ) ( ε τ ε ε σ ε ε ε ε ε (8.9) and he eddy vscosy ε C (8.0) wh he closre coeffcens gven by C ε.44, C ε.9, C 0.09, σ.0, σ ε.3 As eplaned laer, hese coeffcens are deermned n a sysemac manner applyng a sememprcal procedre and opmzaon (see, e.g. Rod, 980 for deals).

The -ω Model Anoher poplar wo-eqaon model s he -ω model, whch wll be presened here n he form gven by Wlco (988a). In conras o he -ε model, whch solves for he dsspaon or rae of desrcon of rblen nec energy, he -ω model solves for only he rae a whch ha dsspaon occrs. Dmensonally, ω can be relaed o ε by ω ε /. Anoher nerpreaon s he nverse of he me scale on whch dsspaon aes place, whch s se by he larges eddes. The eqaon governng ω has radonally been formlaed based on physcal reasonng n lgh of he processes normally governng he ranspor of a scalar n a fld. Consderng he processes of convecon, dffson, prodcon, and desrcon or dsspaon, he model eqaon for ω s gven by ω ω ω σ ) D ( ω ω (8.) The prodcon and dsspaon of ω are modeled by analogy wh he reasonng sed o model he prodcon and dsspaon of ε n he epslon eqaon. In lgh of he dmensons nvolved n, ε, and ω, he prodcon and dsspaon erms for ω are gven by and ω ω α (8.) ω α τ D ω βω (8.3) Combnng hese wh he ω-eqaon and he modeled -eqaon gves he -ω model (Wlco, 988a) as he -eqaon: he ω-eqaon: ( ) τ β * ω σ (8.4) ω ω ω ω ( ) α τ βω σ ω (8.5) and he eddy vscosy: (8.6) ω 38

wh he closre coeffcens gven by α 5/9, β 3/40, β* 9/00, σ ω /, σ / Deermnaon of Closre Coeffcens The deermnaon of he closre coeffcens sed n rblence models s no rgorosly esablshed, especally snce he models nvolve many assmpons and argmens based on physcal reasonng. Therefore, he poplar approach for he deermnaon of closre coeffcens s o se he vales n sch a way ha he model obans reasonable agreemen wh epermenally observed properes of rblence. Ths approach nrnscally adops a hgh degree of presmpon n ha a consan deermned from one applcaon may no necessarly be he bes one for a wde range of rblen flows. The mehod presened by Wlco for he -ω model s gven as an eample. Two rblen flow cases are consdered n esablshng he closre coeffcens for he -ω model. These flows are he decayng (no prodcon) of homogeneos, soropc rblence, and a bondary layer where he coeffcens are chosen o ensre agreemen wh he law of he wall. For decayng, homogeneos, soropc rblence, he asympoc solon for s gven from Eq. (8.9) and (8.0) by β*/ β (8.7) From he bondary layer verson of Eq. (8.9) and (8.0), a relaonshp beween he von Karman consan, κ, from he law of he wall, and he coeffcens n he -ω model, may be dedced as β κ α σ (8.8) β * β * and from he log-layer can be shown ha he Reynolds sress ms sasfy τ y β* (8.9) For decayng, homogeneos, soropc rblence, epermenal daa shows β /β 6/5. Also, n he log-layer he rao gven by τ y / 3/0. Analyss of he defec layer and sblayer ndcae ha σ σ ω / shold be chosen. sng hese vales, along wh he sandard vale of he von Karman coeffcen, κ 0.4, yelds he fve closre coeffcens gven above when combned wh Eq. (8.7), (8.8), and (8.9). 39

Applcaons of Two-Eqaon Models In nvesgang he seflness of wo-eqaon models, resls from he -ε and -ω model wll be dscssed n lgh of her ably o mach epermens. Ths secon wll gve some overvew on how he models perform raher han gve an n deph reamen on how o oban solons o he eqaons, as mos solons ms be generaed nmercally anyway. Namely, free shear flows, bondary layers, ppe flows, and separaed flows wll be eamned (Wlco, 993). For he free shear flows gven by es, waes, and mng layers boh models perform reasonably well. preadng raes for hese flows as predced by he -ε model are generally predced o whn 30% for he far wae and rond e, 5% for he mng layer, and 5% for he plane e. The -ω model predcons are somewha closer for each of he for cases; however, s qe sensve o free sream bondary condons. Ths sensvy does allow a degree of freedom n calbrang he model, and hence he ncreased accracy obaned by sng he -ω model shold be dged wh caon. For ppe flows a Re 40,000, he -ω model has been shown o gve predcons whn 6% of DN predcons for sch qanes as he mean velocy, Reynolds sress, and sn frcon. The model s slghly naccrae a predcng he vale of near he wall, b does accraely predc he vale of whn 4% over 80% of he ppe dameer. One commen on hese wo models s necessary before comparng her predcons for bondary layer flows. The -ω model has been shown o relably predc he law of he wall when he model s sed o resolve he vscos sblayer, hereby elmnang he need o se a wall fncon, ecep for compaonal effcency. The -ε model, n s presen form, has been shown o necessae eher a low Reynolds nmber modfcaon, or he se of a wall fncon, when appled o a wall bonded flow. In lgh of hese characerscs of he models, he followng comparsons were made sng a -ω model ha resolved he vscos sblayer, and a -ε model ha sed a wall fncon. Ths may bas he -ω model somewha n he followng evalaon gven by Wlco; a he anford Conference on Comple Trblen Flows (98). I was conclded ha dfferenal mehods ha negrae o he wall gve beer resls han models sng wall fncons. A more obecve comparson beween he -ω and -ε models may be for versons of he models boh negrang o he wall. Model characerscs for wall bonded flows wll be eamned n more deal n Chaper 9.0 on nmercal consderaons for wall bonded flows. For a consan pressre ncompressble bondary layer flow a a Reynolds nmber close o 0 6, boh models perform very well, predcng vales of he frcon coeffcen and mean velocy whn 5%. However, for ncompressble bondary layers wh adverse pressre gradens, he - ω model s clamed o be somewha speror o he -ε model. For one case smlaed a a Reynolds nmber arond 0 6, he frcon coeffcen for he -ω model gave predcons whn 5%, whle he -ε model predcons showed error as hgh as 0%. 40

An addonal flow wh adverse pressre graden, he bacward facng sep, agan shows he -ω model o gve speror performance o he -ε model. Wlco arbes he poor performance of he -ε model for flows wh adverse pressre gradens o he rblence lengh scale predced by he -ε model n he near-wall regon. sng a perrbaon analyss of he defec layer, vales of he rblence lengh scale were predced o be oo large near he wall for he -ε model. Ths can be recfed by sng he correc lengh scale as prescrbed by van Dres (see Eq. 6.3). In conclson, wo-eqaon models have proven ha hey perform well for a wde range of flows of engneerng neres. Ther applcaon s lmed, however, o flows ha closely follow he mplc assmpons pon whch mos wo-eqaon models are based. The mos mporan assmpons for hese models are ha he flow does no depar far from local eqlbrm, and ha he Reynolds nmber s hgh enogh ha local soropy s appromaely sasfed. The -ω model s spposedly beer han he -ε model for flows wh adverse pressre graden, hogh many varans of he -ε model wh correcon for oher facors sch as sreamlne crvare, boyancy, swrl, ec. for sch cases es. Generally, neher model s capable of gvng qanavely good resls for more complcaed flows sch as flows wh sdden changes n mean sran rae, crved srfaces, secondary moons, and separaon. Whle wo-eqaon models may be able o gve qalave resls for sch flows, generally a frher level of compley s needed n he model o oban close agreemen wh epermens. As wll be shown n Chaper.0, a class of models, nown as algebrac sress models, aemp o end he need o assme local soropy, whle second order closre models aemp o accon for hsory effecs on he sresses and remove he need for assmng local eqlbrm. Anoher mehod called Large Eddy mlaon (LE), where he mos mporan eddes responsble for he rblen flcaons are resolved eplcly, wll also be presened n Chaper.0. 4

9.0 WALL BONDED FLOW Mos flows of engneerng neres nvolve saons where he flow s consraned, a leas o some degree, by a sold bondary. nce wall bonded flows are raher common, approprae mehods for applyng wo-eqaon models o sch flows are mporan and wll be deal wh n some deal n hs chaper. Inally, some fndamenal properes of rblen flows over sold bondares wll be consdered, ncldng properes of he rblence n he vscos sblayer and log layer for flows wh zero or small pressre gradens. The behavor of, ε, and ω wll be consdered n lgh of near wall rblence phenomena, and he ably of he modeled eqaons o capre hs phenomena wll be nvesgaed. The se and mplemenaon of wall fncons, as well as mehods for resolvng he vscos sblayer, wll be dscssed for boh he -ω and -ε models. Ths chaper wll also nclde a dscsson of he nmercal mplemenaon of wall fncons. Revew of Trblen Flow Near a Wall In a flow bonded by a wall, dfferen scales and physcal processes are domnan n he nner poron near he wall, and he oer poron approachng he free sream. These layers are ypcally nown as he nner and oer layers, hogh radonally he flow near a wall has been analyzed n erms of hree layers:. The nner layer, or sblayer, where vscos shear domnaes. The oer layer, or defec layer, where large scale rblen eddy shear domnaes 3. The overlap layer, or log layer, where velocy profles ehb a logarhmc varaon The machng of he nner and oer regons has been sefl for esablshng flow properes n hs layer separang he areas of vscos and neral domnaon. In he nner layer, convecon and pressre gradens are assmed neglgble. Ths s n lne wh he orgnal reasonng of randl and von Karman. These early researchers reasoned ha n he nner layer he mporan varables deermnng he mean velocy are he vscosy, densy, wall shear sress, and normal dsance from he wall, y. The combnaon of hese varables mples a form for he mean velocy, whch can be wren as f ( τ,,, y) (9.) w For he oer layer, vscos effecs are assmed neglgble b he pressre graden and convecon ms be nclded. Then he relevan varables ha wll nflence he mean velocy are he wall shear sress, densy, bondary layer hcness, pressre graden, and he dsance from he wall. Combnaon of hese varables gves f ( τ y w,, δ,, ) (9.) 4

The log layer, where he nner and oer layers merge, can be deermned from fnconal analyss (see e.g. chlchng, 979) by reqrng he same asympoc solon for he frs dervave of he velocy wh respec o he normal dsance from he wall. Ths analyss leads o he logarhmc velocy profle law of he wall, gven by where κ 0. 4, B 5. 0, ln( y ) B (9.3) κ τ w τ / and y y τ ν and τ (9.4a, b, c) I shold be emphaszed ha Eq. (9.3), referred o as he log law, has been epermenally confrmed by many nvesgaors. Whle hs s a somewha vage revew of he physcs nderlyng mos wall bonded flows, serves o se he sage for nvesgang how bondary condons may be appled a sold srfaces for wo-eqaon rblence models. Two-Eqaon Model Behavor Near a old rface For nvesgang he behavor of wo-eqaon models near a sold bondary, Wlco (993) employs perrbaon analyss o deermne lmng forms of he eqaons very near he wall n he vscos sblayer, and n he neral par of he sblayer where he law of he wall s vald. These nvesgaons are nformave n ha hey gve nsgh no how sch qanes as, ε, and ω behave near sold srfaces, and wha nd of nmercal reamen s necessary for avodng sgnfcan errors n sch cases. Esablshng flow properes n he log layer, alhogh s no formally a separae layer, perms ndependen analyss of he sblayer and defec layer. Knowledge of flow properes n he log layer s also mporan n ha forms he bass for bondary condons for many wo-eqaon models. In he log-layer regon, where effecs of he pressre graden, moleclar dffson, and convecon are small compared o oher qanes, he momenm eqaon redces o 0 ν (9.5) y y Redcng he, ε, and ω eqaons, and assmng ha he eddy vscosy vares lnearly from he wall as was done for he mng lengh model, he eqaons for he -ω model n he log layer are gven by τ ln( y ) cons (9.6) κ 43

( ) τ β * and ω τ β* κy (9.7a, b) and he log layer eqaons for he -ε model are gven by τ ln( y ) cons (9.8) κ ( ) τ C and ( ) ε τ κ y 3 (9.9a, b) These relaons defne a ey se of eqaons ha can be sed o mplemen bondary condons for solons where s ndesrable o resolve he enre vscos sblayer. Very close o he wall n he lamnar sblayer, he effecs of rblence de o very qcly. The lmng vale of, ε, and ω approachng he wall from a pon s osde he vscos sblayer can be nferred by consderng he mean and nsananeos momenm and conny eqaons and by epandng he nsananeos flcang velocy componens n a Taylor eres (Wlco, 993). In hs way, can be shown ha, near he wall y (9.0) κν ε (9.) y ν ω (9.) β * y Neher he -ε nor he -ω model reprodces hese heorecal rends, hogh he -ω model does come close enogh o allow he vscos sblayer o be resolved who sgnfcan error. Resolvng he vscos sblayer sng he sandard -ε model canno be done who correcons, whch are sally added o he model as dampng of he lengh scales as he wall s approached. Wh some bacgrond abo how he heorecal vales behave near a sold bondary, he as of nmercally acconng for sold bondares wll now be consdered. The mos poplar choce, as has been prevosly menoned, s o mach he velocy profle o he law of he wall a he frs grd node above he sold bondary. Effecvely hs approach ses he law of he wall as a consve eqaon for he wall shear sress n erms of he velocy a he frs grd pon. sng hs approach, he vale of he velocy may be sed o deermne τ w, whch may n rn be sed o se he vales of and ε or ω sng Eqs. (9.7a, b) and (9.9a, b). I shold be noed ha he applcaon of wall fncons may no yeld good resls when appled nder condons dfferng drascally from hose nder whch he law of he wall s derved. rmarly, effecs of 44

pressre gradens, boyancy fles, and non-eqlbrm ype saons may aler he velocy profle near he wall n a manner ha s no acconed for by he sandard law of he wall. Effecs of rface Roghness When he bondary s smooh, he shear sress a he srface s ransmed o he flow va a vscos sblayer. The velocy n hs sblayer vares lnearly, as n lamnar Coee flow, le τ y/ν, whch mples ha he hcness of he sblayer s on he order of δ lam (cons)ν/ τ. Here he consan s he vale of he velocy where ceases o behave accordng o a lnear relaonshp. As has been menoned already, hs mples a law gven by ln( y ) B (9.3) κ where he consan of negraon B accons for condons a he bondary; he effec of B on he flow s o add a nform velocy o he enre flow wh no change n s nernal srcre. For a rogh srface, he roghness hegh may be larger han δ lam. In ha case, he sress s ransmed by pressre forces n he waes of he roghness elemens, raher han by vscosy. The form of he profle gven n he log layer s hen y ln( ) 85. (9.4) κ where s called Nradses eqvalen roghness hegh, and s relaed o he roghness geomery a he bondary. s Resolon of he Vscos blayer The alernave o sng wall fncons s o solve he governng eqaons flly hrogh he vscos sblayer p o he wall, where no-slp bondary condons can be appled. As prevosly menoned, hs approach reqres low Reynolds nmber modfcaons for he -ε model, whch nclde vscos dampng fncons smlar o he Van Dres formlaon gven n Chaper 6.0. everal sch modfcaons es, he mos poplar beng he Jones-Lander model, he Lander- harma model, he Lam-Bremhors model, and he Chen model. All for of hese models gve he correc asympoc behavor prevosly dscssed. One advanage of he -ω model s ha reqres no modfcaon and can be drecly negraed o he wall wh accepable resls. ome areas of concern when resolvng he vscos sblayer nclde compaonal demands and possble convergence dffcles. Resolon of he vscos sblayer reqres a maor ncrease n he nmber of compaonal cells near he wall o accraely predc he velocy profle. In 45

addon, some of he vscos dampng fncons have been nown o be qe sff when he eqaons are beng solved nmercally; hs can be arbed o he dsspaon s me scale ha s enconered approachng he wall. Eqaon (9.9b) shows ha ε appears o approach nfny near he wall as y approaches zero. Applcaon of Wall Fncons Nmercally, a possble sorce of error may occr n he epslon eqaon near he wall as vares n nverse proporon o y as s ndcaed by Eq. (9.9b) for he case where wall fncons are beng sed. ch a fnconal relaonshp may case sgnfcan errors f he grd s no sffcenly fne near he frs grd node ne o he wall. Esmaes of y can be made before a nmercal smlaon from nown eqaons for flows sch as zero pressre bondary layers, ppe flows, and channel flows s performed. everal sefl eqaons and epermenal correlaons are hence nclded here as a reference for mang sch appromaons. Generalzed Wall Fncons: To avod compaonal cos, we can se wall fncons o brdge he log-layer o he wall,.e. sp he sb-layer and he bffer layer and enforce bondary condons a yy c. andard approach (ee Append B): n y w e s y c wall Enforce bondary condons a y y c ** Coee flow analyss; eqlbrm,.e. Ε and consan shear sress assmpons lead o (ee wor boo for deals) 46

y c >.6 c ln y c κ c T * ln κ (mach pon) ( Ε ); Ε 9.0, Κ 0. 4 C 3, Εc * ω κ, y c ( Ε y ); Ε fnc(r) c h c h c * ( β ) 4κy Defcences of he sandard approach: τ w 0 a separaon pons, hence 0!. and hea fl q h 0 Accordng o epermens, ma, q. 0 a separaon pons. mamm (hea ransfer) a reaachmen pon and separaon pon reaachmen pon y c Remedy: *, f v 0. 3 C * le * c * c (frcon facor) κ c ln Ey ν * c κ ln E Rec The above eqaon can be wren, for compaonal prposes, n varos ways. 47

48 For erave solon: c c c c c C E C E 4 4 Re ln Re ln κ κ For eplc non-erave solon: Κ ν 4 4 ln c c c c y EC C o calclae and *, hence τ w w c τ sa facor for appromang he dervave : * α α c c c y y y v Also, he prodcon and dssapaon erms n he -eqaon need o be calclaed from he followng relaons: () 3 4 3 Ky C ε ln ) ( 3 4 3 3 * * Κ ν α α α Ey y C y y c c c c c c (ee also wo- and hree layer models, Nallasamy (987), as well as ael e al. (985).) TEACH Manal CAT Manal Hea Transfer: ( ) > Κ 0.5 r.3 0.5 r 5.3 ;.ln r.5r ) ( ln * * 3 4 4 B B C C y C Q C H H Q r Q r w w c ν H ac (agnaon) enhalpy Q w y H r Q w Wall Hea Fl

h r H ; ν h r T ν ϑ r y also noe H y For y c <. 6 se sb-layer eqaons T ~y ν, ε~ y * y y0 C 3 * ε 4 C y τ e c y c ; w yc ( τ ) w Tc c r y ; T 0 Alernavely se 0, a he wall, y0 or mooh Bondary Layers: ε w y 0.6 ε 0 or ε ν y y Qw ϕ* ν y Gven (longdnal dsance along he bondary layer), some esmae for e,, and Re e 0. 455 C f ν ln ( 006. Re ) C f τ e / C f τ w / e τ w τ / y y τ ν can be sed o deermne y vales ha gve 30 < y < 0.δ (), whch are sally acceped as vald vales for y n he log layer. The bondary layer hcness, f needed, may be esmaed from 49

δ 037. Re / 5 Flly Rogh Bondary Layers: For bondary layers ha are consdered flly rogh ( τ ν > 5) C f 87. 58. log s 5. may be sed o oban some esmae of he frcon coeffcen for zero pressre bondary layers f comparsons o model predcons are needed. Nmercally, he wall node ms be placed a some dsance above he eqvalen roghness hegh. Ths follows from he physcal argmen ha he flow canno es nsde he wall, whose edges effecvely prorde a dsance s no he flow. Flly developed smooh ppe flow: For favorable pressre gradens, he law of he wall closely holds over he enre ppe. sng hs assmpon: avg Q A avg τ rppeτ ln B κ ν 3 κ and / Λ / ( D Λ ) 0. 8686ln Re 08. Λ 8 τ w avg τ w avg Λ 8 τ w τ / y y τ ν 50

0.0 EFFECT OF BOYANCY Whle mos rblence s generaed as a fncon of shear, n flows wh apprecable densy gradens he rblen eddes may receve or lose energy de o he effecs of boyancy. By dvdng he densy no a flcang qany and a mean qany as was done for he oher varables, can be shown ha he mean momenm eqaon s gven by ( ) ( ) ( ) g (0.) where he densy flcaons are negleced n every erm ecep n he gravy erm, as n he Bossnesq appromaon. The specfc rblen nec energy eqaon needs o be modfed o nclde boyancy effecs as ( ) τ g p (0.) The era erm represened by B g (0.3) represens he prodcon of rblen nec energy by boyancy, and s he rae of wor done agans boyancy forces by rblen moon. Ths s essenally a ransfer of eher poenal energy o rblen nec energy, as wold be he case n an nsably srafed fld, or a ransfer of rblen nec energy o poenal energy, as wold be he case for he mng of a heavy fld wh a lgher fld agans he acon of gravy. A fll dervaon of he mean mass, momenm, and rblen nec energy eqaons are gven n Append D. The boyan prodcon erm s sally modeled as g g (0.4) σ Here σ s a nd of rblen randl nmber. The densy dervave s sed snce a negave densy graden sally corresponds o a sably srafed flow and he erm acs as a sn on rblen nec energy. For posve densy gradens an nsable srafcaon sally ess and hs erm acs as a sorce n he eqaon. 5

A smlar modfcaon s made o he epslon eqaon by addng he sorce erm gven by (Rod, 980) B ε Cεε Cε 3 ma( B,0) (0.5) where C ε3 s a consan. Ths erm ncreases ε for nsable srafcaon, and gves no change n ε for sable srafcaon. Wh hs modfcaon, he epslon eqaon s gven by ( ε ) ( ε ) C τ C ( G, ) ε ε ε ε 3 0 reamlne Crvare and Roaonal Effecs: ε ε ma Cε (0.6) σ The effecs of sreamlne crvare and roaon can be acconed for by analogy o he boyancy effecs (see Bradshaw, 969; Rod, 979, loan e al., 986). Here approprae Rchardson nmbers ms be defned. For a sreamlne wh crvare R, and velocy angen o he sreamlne he approprae Rchardson nmber s R g ( R) n R n where he sbscrp,n denoes dfferenaon w.r.. o he normal drecon o he sreamlne. For sable flow R g >0. For roang (or swrlng) flows Bradshaw (969) sgges a graden Rchardson nmber defned by R g ( W ) W r r r r where W Ωr s he swrl velocy n angenal drecon, Ω s he anglar velocy, s he velocy n aal drecon, and r s he dsance n he radal drecon. In he lerare varos Rchardson nmbers are sed o accon for he nflence of roaon. For eample Rod (979) sggess he se of a so called fl Rchardson nmber defned by 5

53 ( ) f r W r W r r W r r W R I s sally he sorce erms of he ε-eqaon ha s modfed by mang he model consans a fncon of Rchardson nmber o accon for he above menoned effecs. For roang flows he consna Cε s modfed as (C ε ) mod C ε ( - C g R g ) For zero-eqaon models s he mng lengh ha s modfed (see Rod, 980)

.0 ADVANCED MODEL As was menoned n Chaper 8.0, he eddy vscosy appromaon for deermnng he Reynolds sresses s no a good model for flows wh sdden changes n mean sran rae, crved srfaces, secondary moons, roang and srafed flows, separaed, and hree- dmensonal flows. The prmary assmpons of local soropy and local eqlbrm are he man reasons ha wo-eqaon models ypcally fal o gve anyhng more han qalave resls for sch applcaons. Inheren n hese assmpons s ha he normal Reynolds sresses are eqal and ha flow hsory effecs on he Reynolds sresses are neglgble. Two ypes of models amed a mprovng he predcve capables for flows where sch assmpons are no vald are algebrac sress models and second momen closre models. Algebrac ress Models The prmary dsncon of an algebrac sress model (AM) s he assmpon ha he Reynolds sresses are gven by a seres epanson of fnconals, of whch he Bossnesq appromaon only conans he frs erms. Whle here have been nmeros effors n hs area by a nmber of dfferen nvesgaors, and he forms of dfferen algebrac sress models may vary, he sresssran/vorcy consve relaonshp for models whch carry he qadrac or cbc erms n he epanson can be wren n he followng canoncal form for ncompressble flows (Apsley e al, 997): a C ~ ε C C C 3 5 7 ~ ε ~ ε ~ ε C 3 3 3 3 Ω Ω Ω l ~ ε Ω Ω Ω l l l Ω 3 Ω 3 l Ω Ω l l l δ l l δ C C 4 Ω 3 ~ ε l 3 3 ~ ε ( Ω Ω ) ( Ω Ω ) lm Ω m l δ C 6 l ~ ε 3 3 l l l (.) where a s he ansoropy ensor, gven by a δ (.) 3 Here, and Ω are he mean sran-rae and mean vorcy ensors. (see Eqaon.8) Noe ha n Eq. (.) allows for models (e.g. Craf e al, 997) ha mae a dsncon beween he dsspaon rae ε and s homogeneos poron ε ε ν. ~ y 54

There are many dfferen varans of hs model whch adop dfferen vales for he coeffcens, e.g. Len and Leschzner (993); Craf e al (997), and whch may carry only he qadrac erms n he epanson or proceed hrogh he cbc erms. hh e al.s (993) model employs Eq. (.) only hrogh he qadrac erms (p o and ncldng he forh erm on he RH), and lzes he followng vales for he coeffcens: 3.5 0.9Ω C (.3) ( C, C, C3 ) ( 3,5,9 ) (.4) 3 000 where and Ω are he sran and vorcy nvarans, respecvely, gven by ε Ω Ω Ω ε (.5) The cbc eenson of he model, ncldng all of he erms on he RH of Eq. (.), ses he followng vales for he remanng coeffcens: 3 ( C C, C, C ) ( 80, 0, 6, ) C (.6) 4, 5 6 7 6 Gas and pezale (993) derve an eplc algebrac sress model, ncldng he qadrac erms n he epanson, n erms of he rblen nec energy κ. Here, he Reynolds sress ensor s gven by 3 κ ( ) β δ 3 * κ κ τ κδ C β Ω Ω mn mn (.7) 3 ε ε ε 3 4 3 * where β g ( C ) C, β g ( C ) C, and g C, whch has an 4 appromae vale of 0.09. Here, 4 3 3 C 4 3 v v v v Ω (.8) where he overbars denoe ensemble averagng, and g C (.9) ε 55

where τ v represens he prodcon of rblen nec energy. In he above epressons, C, C, C 3, and C 4 are he model consans ha arrve n he pressresran correlaon ensor, gven by Π ( b W b W ) C εb Cκ C3κ b b bmnmnδ C4κ 3 (.9) where s he ansoropy ensor, and b W τ 3 κδ (.0) κ Ω η ω (.) m m s he absole vorcy ensor, where ω m and η m represen he anglar velocy and permaon ensor, respecvely. Whle he model consans gven above have been eamned by many nvesgaors, he vales deermned n wo noeworhy models are gven below, as deermned by Lander, Reece, and Rod (975), and Gbson and Lander (978): Lander, Reece, and Rod Model C 3.0 C 0.8 C 3.75 C 4.3 (.) Gbson and Lander Model C 3.6 C 0.8 C 3. C 4. (.3) everal formlaons whch are smlar o, or varans of, Eq. (.) and Eq. (.7) es. The prmary advanage of sng sch a non-lnear consve eqaon s he ably o predc ansoropy n he normal Reynolds sresses. Ths s very sefl for cases where secondary flows occr, sch as when a ppe or channel s crved. The prmary dsadvanage of sng a formlaon smlar o Eq. (.) or Eq. (.7) s ha he local eqlbrm assmpon s sll presen, leavng flow hsory effecs nacconed for. econd Order Closre Models: RM Wh he prce beng a raher large degree of compley, second order closre models see o model he eac dfferenal eqaon descrbng he Reynolds sress ensor, whch was gven n Chaper.0 by τ ) τ ( ) τ τ ε Π υ τ ( ) C (.4) 56

Inheren n sch a model are he effecs of flow hsory, as he Reynolds sress eqaon does nclde convecon and dffson erms for he sresses. Also, he Reynolds sress eqaons nclde prodcon and body force erms ha can respond o effecs gven by sreamlne crvare, roaon, and boyancy. Anoher advanage s ha here s no reason why he normal Reynolds sresses as predced from he modeled eqaon shold be eqal. The maor compaonal reqremen for hese models s ha for a hree-dmensonal flow, s addonal ranspor eqaons for he Reynolds sresses ms be solved n addon o he ypcal Naver- oes and conny eqaons. In mos cases, he and ε eqaons ms also be solved. A maor dffcly n worng wh sch models s ha he ser ms be slled n ensor calcls o ndersand dfferen models. Large Eddy mlaon A he hghes level of compley n nmercal smlaons of rblence, Large Eddy mlaon (LE) rans second only o drec nmercal smlaon (DN) as beng compaonally nensve. The prmary dea behnd LE s o smlae only he larger scales of rblence ha are se by he geomery or specfc flow condons, and o accon for he nflence of he negleced smaller scales by se of a model. As sch hey need o be hreedmensonal and ransen calclaons wh relavely accrae nmercal schemes. nce rblence s nown o be more soropc a smaller scales, s beleved ha he sal model assmpons nvolved n he eddy vscosy models shold no maer as long as he grd s sffcenly small, snce he negleced scales are roghly proporonal o he mesh sze. The frs sep n LE s o fler he Naver-oes eqaons o deermne whch scales wll be ep and whch scales wll be dscarded. Ths may be nerpreed as wealy analogos o Reynolds averagng where only he me averaged vales are ep; however, LE dffers n he fac ha he flerng eeps he rblence occrrng on scales above a desred mnmm fler wdh. A generalzed fler can be defned by 3 (, ) G( ξ : ) ( ξ, ) d ξ (.5) where he fler fncon s nerpreed o eep vales of occrrng on scales larger han he fler wdh. The fler fncon G, s bascally some fncon whch s effecvely zero for vales of occrrng a he small scales. Here < > ndcaes a flered varable. Consder he Naver-oes eqaon for ncompressble flow n erms of he mean flow qanes ν (.6) 57

58 By Lebnz rle, he flerng operaon commes wh paral dfferenaon wh respec o me,.e., (.7) Noe ha φ φ boh mply spaal flerng. Assmng ha he flerng operaor also commes wh spaal dfferenaon, hen (.8) Wh he assmpon ha he densy does no vary (ncompressble), hen he pressre graden can be flered by (.9) The dffson erms de o moleclar vscosy are flered accordng o ν ν ν ν (.0) Fnally, he flered Naver-oes eqaons are gven by ν (.) Noe ha n Eq. (.), he overbar denoes a flered varable raher han a me-averaged qany. The dffcly n Eq. (.) comes from he nonlnear erm,. To handle hs erm, an analogy o Reynolds decomposon for me averagng s sed, whereby he nsananeos velocy s broen no a large-scale componen and a sbgrd-scale componen by (.) where denoes he large-scale componen of he velocy feld ha s resolved, and denoes he small-scale componen whch s flered o. Wh hs, he nonlnear advecon erm becomes (.3)

bsng Eq. (.3) no Eq. (.) yelds where τ ν (.4) τ (.5) C R The erms represened by C are called "cross-erms" and physcally represen random forcng prodced by sbgrd-scales on he large scales. The erms represened by R are called "sbgrd Reynolds sresses", analogos o he rblen sresses ha resl from Reynolds averagng of he Naver-oes eqaons. I s he prpose of sbgrd scale (G) models o resolve he nflence of hese addonal nnown sbgrd scale sresses ha resl from he flerng process. bgrd scale (G) models are smlar o he rblence model sed n RAN (Reynolds Averaged Naver-oes) calclaons, n ha s prmary prpose s o provde he nflence of he small scales, sally based on some graden dffson hypohess. Here, he raonale follows ha he role of he small eddes s predomnanly o accep energy from he larger scales and dsspae, and ha hs ransfer of energy s consdered o be a one-way process. Ths nflence acs prmarly as a sn for he energy of he large scales o ensre ha he sascs of he large scales are correc n ha hey connally dsspae her energy. ome poplar sbgrd scale models nclde he magornsy (see e.g. Cafalo, 994; omell, 993a), -ε, and dynamc magornsy model (see e.g. Ghosal e al, 995; omell, 993b). 59

.0 CONCLION Wh ncreased comper resorces and speed now avalable a reasonable cos, many rblence models are beng developed and esed. Wh he adven of faser and more powerfl compers, he lmaons of some of he more comple modelng sses are beng elmnaed. The presen sdy eamned many of he nmercal, heorecal, and praccal sses nvolved n compaonal effors a accraely resolvng rblen flows. Mch of he focs of hs wor was devoed o eamnng he closre problem and he more poplar algebrac and wo-eqaon models, as well as nvesgang case specfc sses as hey relae o dfferen ypes of flows. The addonal erms (.e. he Reynolds sresses and rblen fles) ha arse as a resl of meaveragng he eqaons of moon form he hear of he closre problem n rblence modelng. These era erms nvolve correlaons beween flcang velocy componens, and are no nown a pror. The solon of he conservaon eqaons reqres some emprcal np o formlae mahemacal models for hese addonal erms, and hs close he se of eqaons. Ths s he prpose of rblence modelng. The earles proposal for modelng he Reynolds sresses s sll n se n many rblence models oday. I nvolves he Bossnesq eddy vscosy/dffsvy concep, whch assmes ha he rblen sresses are proporonal o he mean velocy graden. The prmary goal of many rblence models, hen, s o provde some prescrpon for he eddy vscosy o model he Reynolds sresses. These may range from relavely smple algebrac models, o he more comple wo-eqaon models, sch as he -ε model, where wo addonal ranspor eqaons are solved n addon o he mean flow eqaons. There are for man caegores of rblence models: () Algebrac (Zero-Eqaon) Models () One-Eqaon Models (3) Two-Eqaon Models (4) econd-order Closre Models Each of hese has receved some aenon from varos researchers n he pas, whle algebrac models and wo-eqaon models have receved he mos aenon and me wh he mos sccess n praccal applcaon. I was no nl recenly ha compaonal resorces were sffcen o effecvely apply second-order closre models, de o he large nmber of eqaons and compley nvolved. Two-eqaon models, especally he -ε model, have enoyed he mos sccess and poplary. Whle hs sdy s no nended o be a complee caaloge of all rblence models, here are several models ha are eamned n hs wor. The reader s referred o oher sorces gven n he bblography for more dealed nformaon regardng specfc models. The nflence of geomerc effecs and flow condons are also eamned (see Chapers 9.0 and 0.0). pecfcally, he presen sdy eamnes wall-bonded flows, ncldng flow near a sold 60

srface, srface roghness, and resolon of he vscos sblayer, as well as he effecs of boyancy, whch may effec he manner n whch rblen eddes receve or dsspae rblen nec energy. More comple sses sch as second order closre models, and large eddy smlaon (LE) are brefly dscssed n Chaper.0. Agan, for more dealed nformaon he reader s referred o oher sorces. Whle no beng a complee eamnaon of all rblence models * n se, hs sdy provdes a fondaon for ndersandng he man conceps nvolved n rblence modelng, as well as case specfc sses for applcaon o ceran ypes of rblen flows. * A long ls of commonly sed rblence models s presened n Append F for frher reference. ACKNOWLEDGEMENT: Ths wor boo has been prepared wh he help of my former gradae sdens, Mr. Mahew R. mbel and Wesley M. Wlson. Ther conrbons are grealy apprecaed. Many hans are de o even Rowan, and my sden Andrew Br for her help n ypng many addonal pars o hs docmen. 6

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[5] hh, T-H., Zh, J., and Lmley, J. L. (993), "A Realsable Reynolds ress Algebrac Eqaon Model," NAA TM05993. [53] hh, T-H. (996), Developmens n Compaonal Modelng of Trblen Flows, NAA Conrac Repor #98458, febrary, Lews Research Cener. [54] palar,.r. and Ahmaras,.R. (994), La Recherche Aérospaale, Vol., pp. 5-. [55] loan, D.G., mh,.j., and moo, L.D. (986) "Modelng of wrl n Trblen Flow ysems," rog, Enenrgy Combs. c., Vol, pp. 63-50 [56] pezale, C. G. (985), "Modelng he ressre-graden-velocy Correlaon for Trblence," hyscs of Flds, Vol. 8, pp. 69-7. [57] pezale, C. G. (987a), "econd-order Closre Models for Roang Trblen Flows," Q. Appl. Mah., Vol. 45, pp. 7-733. [58] pezale, C. G. (987b), "On Nonlnear -l and -ε Models of Trblence," Jornal of Fld Mechancs, Vol. 78, pp. 459-475. [59] pezale, C. G., (998) Trblence Modelng for Tme-Dependen RAN and VLE: A Revew, AIAA Jornal, Vol. 36, No., pp. 73-84. [60] Tennees, H. and Lmley, J. L. (983), A Frs Corse n Trblence, MIT ress, Cambrdge, MA. [6] Van Dres, E. R. (956), "On Trblen Flow Near a Wall," Jornal of he Aeronacal cences, Vol. 3, p. 007. [6] Vandromme, D., (995) Overvew of Trblence Models for CFD n Indsry, von Karman Inse for Fld Dynamcs, Lecre eres 995-03, Aprl 3-7, 995. [63] Van Dye, M., (98) An Albm of Fld Moon, The arabolc ress, anford, CA. [64] Von Karman, T. (930), "Mechansche Ahnlche nd Trblenz," roc. In. Congr. Appl. Mech., 3 rd, ocholm, ar, pp. 85-05. [65] V. Yaho and.a. Orszag, (986) Renormalzaon grop analyss of rblence:. Basc heory. Jornal of cenfc Compng (): 3-5. [66] V. Yaho,.A. Orszag,. Thangam, T.B. Gas, and C.G. pezale, (99) Developmen of rblence models for shear flows by epanson echnqe. hyscs of Flds A, 4(7):50-0. [67] Whe, F. M. (99), Vscos Fld Flow, nd Edon, McGraw-Hll, Inc., New Yor, NY. 66

[68] Wlco, D. C. (988a), "Reassessmen of he cale Deermnng Eqaon for Advanced Trblence Models," AIAA Jornal, Vol. 6, No., pp. 99-30. [69] Wlco, D. C. (993), Trblence Modelng for CFD, DCW Indsres, Inc., La Canada, CA. [70] Woods, W. A., and Clar, D. G., Ed., (988) Vsalzed Flow: Fld Moon n Basc and Engneerng aons Revealed by Flow Vsalzaon, ermagon ress, Oford, K. [7] Zeman, O. (990) Dlaaon Dsspaon: The Concep and Applcaon n Modelng Compressble Mng Layers, hyscs of Flds A, Vl., No., pp. 78-88. 67

AENDICE Append A: Taylor eres Epanson The Taylor eres epansons of he flcang velocy componens s gven below: and hen where so, and v w y ( y ) y ϑ y 0.0 y (A.) v y ( y ) v y ϑ y 0.0 y (A.) y ( y ) w w y ϑ 0.0 y (A.3) y (, z, ) y ϑ( y ) A (A.4) y (, z, ) y ϑ( y ) B (A.5) v y (, z, ) y ϑ( y ) C (A.6) w y 3 4 ( y ) ( B ) ~ y A C y ϑ (A.7) 4 ( B y ) 0 3 ( ) ~ y A C y ϑ (A.8) ε ~ ν A C ϑ ( y) ~ ν y w y (A.9) Assmng ε β w as y 0. 0 ~ y ν ε ~ y w ~ ν β y (A.0) 68

Append B: blayer Analyss () ymmery: w 0.0 z (B.) () In he sblayer, convecon and he pressre graden n he -drecon (domnan flow drecon parallel o he wall) are neglgble. o, from he mean -momenm 0.0 y (B.) and from he nsananeos -momenm 0.0 y (B.3) Eqs. (B.) and (B.3) mply ha 0.0 y (B.4) and applyng he condon ha 0 hen 0 y or A y (, z, ) A, (B.5) (, z ) y (B.6) (3) Insananeos conny eqaon: w v 0 z y (B.7) bsng Eqs. (B.) and (B.6) no Eq. (B.7) yelds or y v B y v y [ A(, z, ) ] 0 (B.8) (, z, ) y (B.9) 69

(4) sng a smlar argmen o ha sed n (): (a) Convecon s neglgble; he pressre graden n he z-drecon s neglgble (b) The Reynolds sresses are neglgble (c) The z-dervave of any qany s neglgble (symmery - Eq. (B.)) (d) and Wh hs hen or Mean Flow n he blayer: w w >> v v y y w v >> y y w y W v y w 0 y C(,z,) (B.0) (B.) (B.) (B.3) Inegrang Eq. (B.) yelds τ C w y ν ν (B.4) Inegrang once more gves y C ν (B.5) B a y0, 0, hence C 0. Then follows ha or y ν y (B.6) whch s Eq. (6.9). 70

Near-Wall Vales of -ε, and wall-fncons Very near he wall we assme frs he eqaons of moon can be smplfed o one-dmensonal Coee flow. I follows ha shear sress s consan n hs regon τ τ w consan and n he log-layer lamnar conrbon o shear sress s neglgble. All hs amons o assmng an eqlbrm wall layer,.e. ε. Hence, follows ha v y ε (B.7) τ v ν τ y w (B.8) Noe also ν C ε From B.7 and B.8 ε τ y bsng for ε from B.9 and sng B.8 we oban or τ τ C w * c τ w C * C (B.9) (B.0) (B.a) (B.b) From log law λ * y κy ln y B * κ bsng (B.b) no B.0 yelds 3 * ε κy, 0.4, B 5.5 (B.a) (B.b) (B.3a) (B.3a) can be wren n erms of by mang se of B., whch gves 7

* C (B.3b) C ε 3 4 κy 3 As a sde noe, B.4 mples a dsspaon lengh scale, gven by κy l e 3 C 4 (B.4) (B.5) In wha follows we presen how wall-fncons, e.g. logarhmc velocy profle (Eq. B.b) can be employed (or mposed) n a ypcal CFD applcaon. There are wo dsnc approaches ha se he same nformaon n dfferen ways. The frs alernave s referred o as sandard wall fncons and he second as generalzed wall fncons () andard wall fncon approach We le s * n Eq. (B.a) and rearrange o oban K s (B.6) ln( Es Re ) y Here E g (from ln E B ), Re, κ ν The sbscrp denoes a vale a he frs grd node nsde he calclaon doman near he wall beng consdered. For crved walls s he velocy componen angen (or parallel) o he wall, and y s he normal dsance from he wall. Afer calclang s from (B.6) eravely, * and hence τ w are calclaed from s (B.7) * τ w * (B.8) From Eq. (B.8) s seen ha τ w * * y ν (B.9) y y nce ( * κ ) ln( Ey ), (B.9) can also be wren as κ * y ν (B.30) ln( Ey ) I follows from (B.9) and (B.30) ha τ w κ * (B.3) ln( Ey ) Once * s calclaed he near wall vales of and ε are prescrbed from Eqs. (B.b) and (B.3) respecvely. 7

() Generalzed wall fncon approach Eperence show ha approach () does no wor well when here s flow separaon and/or reaachmen on he walls. In sch saons * (hence τ w ) s calclaed from Eq.(B.3b) and ε s calclaed from (B.4). Frhermore, he prodcon of,, near he wall (.e. a he frs node) s modfed o nclde τ w drecly where τ w s deermned from (B.3) hence mposng log-law. The rblen eddy vscosy s sll modfed accordng o (B.9) and ε s calclaed from (B.4). In general from Eq.c.3.8 and c.3..9 h X (B.3) ν δ (B.33) For compressble flow ν X X (B.34) X Near he wall we separae he shear sress n Eq. (B.34) and replace wh τ w. A frher modfcaon may be necessary o avod possble snglary n Eq.(B.6) whch s o wre ln( Ey ) ln( Ey ) (B.35) Calclaon of he rodcon Term The prodcon erm n he rblen nec energy eqaon s defned as τ (B.36) where he rblen sresses are modeled by V τ ν δ,.e. τ ν ν 3 y (B.37) and Ω (B.38) where The symmerc sran rae ensor Ω An-symmerc mean vorcy ensor hs, (B.39) 73

and Ω (B.40) Therefore he rblen nec energy prodcon s gven by ν δ ( Ω ) (B.4) 3 Epandng Eq.(6) for we ge II IV I III ν ν Ω Ω (B.4) 3 3 Term III s eqal o zero becase nvolves mlplcaon of a symmerc ensor by an ansymmerc ensor. Term IV s also zero becase Ω 0, hs can be easly seen by nspecng eqaon (5). Ths he rblen nec energy prodcon epresson redces o ν (B.43) 3 where s epanded as follows 3 3 3 3 3 3 33 3 3 33 ( 33 33) 33 (B.44) The dfferen componens of he sran ensor are calclaed as follows V W V W, 3, 3 y z z y (B.45a) V W,, 33 y z (B.45b) ν ( 33) 4ν ( 3 3) 3 (B.46) Normal resses b V W 33 y z sbsng Eq.(B.47) n Eq.(B.46) and sng Eq.(B.37) for τ yelds (B.47) ( 33) ( 33) ( τ τ33 τ 3 3 The shear parallel o y-plane s gven by 3 τ τ ˆ τ ˆ τ ˆ w 3 3 3 τ 3 ˆ ) (B.48) (B.49) 74

τ w can be calclaed from he log-law sng an whch s defned as hen V an V, θ arcsn (B.50) an τ τ τ bw 3 bw 3 bw τ τ τ bw 3 bw 3 bw τ τ bw bw 0; τ cosθ snθ bw y τ bw y 0 The shear sresses a he soh wall are calclaed from (B.5) W an W, θ arcsn (B.5) an sw sw τ τ τ cosθ τ τ τ sw 3 sw 3 τ τ sw 3 sw 3 τ sw sw 0; τ snθ τ sw z τ yz sw z y 0 (B.53) 75

Append C: Conservaon Eqaons C. Conservaon of Mass The conservaon of mass, or conny, eqaon s gven as ( ) 0 (C..) Breang he densy and velocy no mean and flcang componens yelds bsng Eqs. (C..a) and (C..b) no Eq. (C..) yelds (C..a,b) ( ) [( )( )] 0 (C..3) Neglecng, and wh ( ) eqal o zero when Eq. (C..3) s me-averaged, hs yelds he mean conny eqaon. ( ) 0 (C..4) (*) - The dvergence of he flcang velocy s also zero (ncompressble only), by sbracng Eq. (C..4) from Eq. (C..). C. Conservaon of Momenm For he case of ncompressble flow, he conservaon of momenm eqaon s gven by ( ( ) g ( τ ) ) (C..) The sress ensor may be rewren n erms of he sran rae ensor by where τ (C..) 76

77 (C..3) bsng Eq. (C..) no Eq. (C..) yelds ( ) ( ) g ) ( (C..4) By assmng ha s consan (ncompressble), and applyng he chan rle o he second erm n Eq. (C..4) we fnd ( ) ( ) g (C..5) The hrd erm n Eq. (C..5) s zero becase of he ncompressbly condon. Therefore, ( ) ( ) g (C..6) The nsananeos qanes can be broen p no mean and flcang componens by (C..7a,b,c) (C..7d) Then, sbsng Eqs. (C..7a) - (C..7d) no Eq. (C..6) yelds ( ) ( )( ) [ ] ( ) ( ) [ ] g (C..8) Epandng he second erm on he LH of Eq. (C..8) yelds ( ) [ ] ( ) ( ) [ ] g (C..9) Tang he me-average, or Reynolds average, of Eq. (C..9), realzng ha d T (by defnon of he me-average of a consan) 0 d T (he me-average of a consan mlpled by a flcang componen s eqal o zero) 0 d T also 0 d T

78 Wh he above defnons, hen he me-average of Eq. (C..9) becomes ( ) [ ] ( ) [ ] g (C..0) or ( ) [ ] ( ) ( ) [ ] g (C..) The second erm n Eq. (C..) can be rewren as [ ] ( ) ( ) (C..) B he las erm n Eq. (C..) s zero from he ncompressbly condon. Then [ ] ( ) (C..3) bsng Eq. (C..3) no Eq. (C..) yelds ( ) ( ) ( ) [ ] g (C..4) where he las erm,, s he Reynolds sress ensor. C.3 Dervaon of he -Eqaon from he Reynolds resses The specfc rblen nec energy s defned accordng o w v (C.3.a) If he rblence s soropc (.e. w v ) hen 3 (C.3.b) The eqaon of conny for ncompressble flow s gven by

79 0 (C.3.) or eqvalenly by 0. sng conny, he Reynolds sresses may be wren as ( ) ( ) ( ) ( ) ( ) [ ] ( ) δ δ ν (C.3.3) If we now se and, as wold be done o derve he eqaons for τ, τ yy, and τ zz, hen Eq. (C.3.3) becomes ( ) ( ) [ ] ( ) δ ν (C.3.4) The forh erm on he RH of Eq. (C.3.4) s zero from he ncompressbly condon. Also, snce he Kronecer dela fncon s only non-zero when he sbscrps are eqal (.e. ), hen he ffh erm on he RH can be rewren as ( ) [ ] ( ) [ ] δ (C.3.5) bsng for Eq. (C.3.5), and mlplyng boh sdes of Eq. (C.3.4) by -/, hen ( ) ( ) ( ) ( ) ( ) ν (C.3.6) where smmaon s mpled by repeaed ndces. By defnon, he dsspaon of rblen nec energy, ε, s gven by ν ε (C.3.7) sng hs, along wh he defnon of he rblen nec energy gven by Eq. (C.3.), hen Eq. (C.3.6) becomes

80 ( ) ( ) ε (C.3.8) or ( ) ( ) ε τ (C.3.9) C.4 Dervaon of he Dsspaon Rae Eqaon The Naver- oes eqaon can be wren as: (C.4.) bsng he mean and he flcang componens n o (C.4.) yelds ( ) ( ) ( ) p (C.4.) Afer me averagng we ge (C.4.3) bracng Eq. (C.4.3) from Eq. (C.4.) yelds ( ) p ( ) byconny 0 Ths becomes

8 ( ) (C.4.4) Dfferenaon of hs eqaon wh respec o l and mlplcaon by l and averagng of ha eqaon yelds p p 0 Hence, he fnal eqaon becomes p ν (C.4.5a) Noe ha ν ν ν

8 ν ν If we now le ν ε, and mlply by ν, hen he ε-eqaon becomes ( ) () p ν ε ν ε ε (C.4.5b)

83 Append D: Governng Eqaons wh he Effecs of Boyancy D. Mass The eqaon of conny s gven by 0 D D (D..) Ths mples ha 0 (D..) and 0 (D..3) Then he eqaon of conny can be wren as ( ) 0 (D..4) By dvdng he densy and velocy no mean and flcang componens and hen meaveragng, Eq. (D..) becomes ( ) ( )( ) [ ] 0 (D..5) Applyng he rles regardng me-averaged varables, he mean conny eqaon becomes ( ) ( ) 0 (D..6) Also, ( ) 0 (D..7) ( ) ( ) 0 ) ( (D..8) ( ) 0 (D..9)

84 D. Boyancy (Mass) Eqaon Consder Eq. (D..6), whch can be rewren as ( ) ( ) ( ) 0 r r (D..) where r s some (consan) reference densy. We can rewre Eq. (D..) as ( ) [ ] 0 r r r r r g g g (D..) If we now defne a boyancy varable b, gven by r r g b (D..3) hen we can rewre Eq. (D..) as ( ) r g b b (D..4) where he erm r g Trblen Boyancy Fl (D..5) D.3 Momenm The conservaon of momenm eqaon s gven by g (D.3.) If we now se a reasonng smlar o a Bossnesq appromaon, by dvdng he densy no mean and flcang componens, and hen dvdng boh sdes by he mean densy, Eq. (D.3.) can be rewren as g (D.3.)

85 If we hen neglec he conrbon of n he neral erms, and drop he overbars for he mean qanes, hen Eq. (D.3.) becomes ( ) g (D.3.3) If we now dvde he velocy erms no mean and flcang componens by (D.3.4) so ha he mean velocy s gven by (D.3.5) and me-average Eq. (D.3.3), hen hs yelds he mean momenm eqaon: g (D.3.6) D.4 Mean Flow Energy Eqaon The mean flow energy eqaon can be wren as g (D.4.) Wh he defnon of he nec energy, K (D.4.) and nowng ha A A A (D.4.3) hen Eq. (D.4.) can be rewren as g K K (D.4.4)

86 whch becomes g K K K (D.4.5) The forh erm on he RH of Eq. (D.4.5) represens he prodcon of rblen nec energy (D.4.6) and he ffh erm on he RH of Eq. (D.4.6) represens vscos dsspaon, whch s sally small Φ (D.4.7) bsng Eqs. (D.4.6) and (D.4.7) no Eq. (D.4.5) yelds Φ K g K K (D.4.8) D.5 TKE Eqaon wh he Effecs of Boyancy If we dvde he veloces, densy, and pressre no mean and flcang componens, and mlply boh sdes by ( ), he conservaon of momenm eqaon can be wren as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) g p (D.5.) Tme-averagng Eq. (D.5.) yelds ( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) g g p p (D.5.)

87 The second erm on he RH of Eq. (D.5.) s zero from he ncompressbly condon. Ths yelds () ( ) ( ) ( )( ) [ ] ( ) ( ) ( ) ( ) g g p (D.5.3) () Consder he hrd erm on he LH of Eq. (D.5.3), whch can be epanded o gve () [ ] (D.5.4) B he second and forh erms on he RH of Eq. (D.5.4) are zero. Then Eq. (D.5.4) becomes () [ ] (D.5.5) Applyng he defnon for he rblen nec energy, Eq. (D.5.5) becomes () [ ] K (D.5.6) or () [ ] K (D.5.7) Ne, consder he hrd erm on he RH of Eq. (D.5.3), whch can be wren as () ( ) ( ) (D.5.8) bsng for erms () and (), and applyng he defnon of he rblen nec energy o he frs wo erms n Eq. (D.5.3) yelds ( ) ( ) ( ) ( ) ( ) g g K p K K (D.5.9)

88 Addng and sbracng he erm from Eq. (D.5.9) and rearrangng yelds ( ) g p g K K K (D.5.0) bracng he mean flow eqaon, represened by he braceed erms n Eq. (D.5.0), yelds g p (D.5.) whch s he mean TKE eqaon where he Bossnesq appromaon has been employed. *Noe: The erm g s generally modeled as σ ν τ p g g (D.5.) where ε ν τ C (D.5.3) and s sed snce z negave s a sn sally, and z posve s a sorce sally, of TKE. Then z (D.5.4) where posve (*) nsable srafcaon negave (*) able srafcaon

89 Append E: Reynolds-Averaged Eqaons E. Reynolds-Averaged Momenm Eqaon The conservaon of momenm eqaon s gven by ( ) ( ) ( ) g p (E..) From conny (for ncompressble flow), he sran-rae ensor can be wren as (E..) If we dvde he veloces, pressre, and sran-rae ensor no mean and flcang componens, gven by (E..3a,b) p p s (E..3c,d) and frher assme ha densy flcaons are neglgble, so ha (E..3e) hen sbsng no Eq. (E..) and me-averagng yelds ( ) ( )( ) [ ] ( ) ( ) [ ] s g p (E..4) Ne, we wll consder he erms n Eq. (E..4) ndvdally. Transen erm: ( ) (E..5) Convecve erm: ( )( ) [ ] [ ] ( ) ( ) (E..6)

ressre erm: ( p ) (E..7) Vscos erm: [ ( s )] ( ) (E..8) In he above relaons, we have sed he deny ha he me-average of a flcang componen s zero. Collecng all of he above erms ogeher yelds he Reynolds-averaged momenm eqaon, gven by [( ) ] g (E..9) where he erms gven by are ypcally called he Reynolds sresses. E. Reynolds-Averaged Thermal Energy Eqaon The eqaon of hermal energy s gven by h T ( h ) (E..) Assmng agan ha he densy flcaons are neglgble, and dvdng he velocy, enhalpy, and emperare no mean and flcang componens gven by h H h (E..a,b) hen he me-average of Eq. (E..) becomes T T T (E..c) [ ] ( T T ) ( H h ) ( H h )( ) (E..3) Agan, we wll consder each erm ndvdally. Transen erm: 90

9 ( ) H h H (E..4) Convecve erm: ( )( ) [ ] [ ] ( ) ( ) h H h h H H h H (E..5) Dffsve erm: ( ) T T T (E..6) In he above relaons, we have sed he deny ha he me-average of a flcang qany s zero. Collecng all of he above erms and sbsng no Eq. (E..3) yelds ( ) h T H H (E..7) And realzng ha for an deal fld, h c p T, hen Eq. (E..7) becomes ( ) p c T T T T α (E..8) E.3 Reynolds-Averaged calar Transpor Eqaon The generc ranspor eqaon, ncldng a sorce, for some scalar qany φ s gven by ( ) ( ) φ φ φ φ C Γ (E.3.) Assmng agan ha he densy flcaons are neglgble, we can dvde he velocy and scalar no mean and flcang componens. Tme-averagng he reslng eqaon yelds ( ) [ ] ( )( ) [ ] ( ) ( ) φ φ φ φ Φ Φ Γ Φ Φ C (E.3.) Agan, we wll consder each erm separaely.

9 Transen erm: ( ) [ ] ( ) Φ Φ φ (E.3.3) Convecve erm: ( )( ) [ ] ( ) ( ) ( ) ( ) φ φ φ Φ Φ Φ (E.3.4) The second and hrd erms on he RH of Eq. (E.3.4) are zero, becase he average of a flcang componen mes a mean qany s zero. Then Eq. (E.3.4) becomes ( )( ) [ ] ( ) ( ) φ φ Φ Φ (E.3.5) Dffsve erm: ( ) Φ Γ Φ Γ φ (E.3.6) orce: ( ) Φ Φ C C φ (E.3.7) In all of he above relaons, we have sed he deny ha he me-average of a flcang qany s zero. Collecng all of he above erms and sbsng yelds ( ) ( ) ( ) Φ Φ Γ Φ Φ C φ (E.3.8) In a smlar manner, an eqaon can be derved for he varance of he flcaon of a conserved scalar qany,.e. φ η ( ) ( ) [ ] η η η η η D G (E.3.9) where G η and D η represen he generaon and desrcon rae of η, respecvely.

AENDIX F: TRBLENCE MODEL The followng s a ls of models from Haase e al., 997. A shor descrpon of he models can be fond n he same reference. Algebrac Trblence Models. Inrodcon. The Baldwn-Loma [BL] Model.3 The Granvlle Model for se n Baldwn-Loma [GRB].4 The Cebec-mh [C] Model.5 The Granvlle Model for se n Cebec-mh [GRC].6 The Goldberg Bacflow [GB] Model.7 The Algebrac Yaho-Orszag [YA] Model Half-Eqaon Trblence Models. Inrodcon. The Johnson-Kng [JK] Model.3 The Johnson-Coaley [JC] Model.4 The 3D Eenson of he JK Model by Abd-Vasa-Johnson-Wedan [AVJW].5 The Horon [HH] Model.6 The Le Baller -half-eqaon - v [LBA.KV] Model 3 One-Eqaon Models 3. Inrodcon 3. The Wolfshen [W] Model 3.3 The Hassd-oreh [H] Model 3.4 The Baldwn-Barh Model [BB] 4 Two-Eqaon Models 4. Inrodcon 4. The Jones/Lander and Lander/harma [JL] Model 4.3 he Chen [C] Model 4.4 The Lam-Bremhors [LB] Model 4.5 The Len-Leschzner [LL] Model 4.6 The andard Hgh-Reynolds-nmber Jones-Lander [JL]. Model 4.7 The Hang-Coaley [HC] Model 4.8 The Yaho/Orszag [YO] RNG Model 4.9 The Wlco [WX] Model 4.0 The Kalzn-Gold -τ [KG] Model 5 Non-Lnear Eddy Vscosy (-ε) Models 5. Inrodcon 5. The pezale Model [] 5.3 The Rbensen-Baron [RB] Model 5.4 The hh-lmley-zh [LZ] Model 6 Reynolds-ress Models 6. Inrodcon 93

6. The Lnear Gbson-Lander [LR] Reynolds-ress Model 6.3 The F-Craf-Lander [NLR] Reynolds-ress Model 6.4 The Rod AM [R] Model 7 Mean Flow Closre Models 7. Inrodcon 7. The Drela and Gles Model 7.3 The Le Baller Inegral [LBA.INT] Model 94