Introductory Turbulence Modeling

Transcription

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

2 TABLE OF CONTENT age TABLE OF CONTENT... NOMENCLATRE...v.0 INTRODCTION....0 REYNOLD TIME AVERAGING AVERAGED TRANORT EQATION DERIVATION OF THE REYNOLD TRE EQATION THE EDDY VICOITY/DIFFIVITY CONCET 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) ONE-EQATION TRBLENCE MODEL...7 EXACT TRBLENT KINETIC ENERGY EQATION...7 MODELED TRBLENT KINETIC ENERGY EQATION 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 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 EFFECT OF BOYANCY ADVANCED MODEL...54

3 ALGEBRAIC TRE MODEL...54 ECOND ORDER CLORE MODEL: RM...56 LARGE EDDY IMLATION 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

4 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

5 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

6 .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"

7 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.

8 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

9 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

10 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

11 .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

12 . 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

13 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

14 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.

15 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).

16 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)

17 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

18 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)

19 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.)

20 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

21 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

22 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

23 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

24 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

25 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

26 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

27 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 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.

28 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

29 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 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

30 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

31 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 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

32 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

33 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

34 (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

35 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

36 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

37 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

38 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