Turbulence Closure Schemes

Transcription

1 /5/5 Trblene n Flds Trblene Closre Shemes Beno Cshman-Rosn Thayer Shool of Engneerng Darmoh College Reall: Eqaons governng he Reynolds sresses and rblen hea fl Problem srfaes! When wrng he eqaons governng and 'b', he eqaons nvolve rple orrelaons of he ype ' ' and ' ' b. Wrng eqaons for hese rple orrelaons does no advane oward a losed framewor of eqaons bease hey nlde qadrple orrelaons. Ths goes on for ever. The problem s alled he Trblene Closre Problem. ' ' ' '

2 /5/5 A frs and qe rdmenary mehod of losre s o epress he rblen fles of momenm and hea as dffsve fles b wh enhaned dffsvy oeffens Pope, sar of Chaper : For momenm: ' ' wh For hea: ' b' b wh These dffsves represen he srrng generaed by he rblen feld. I follows herefore ha hey shold be greaer f he rblen feld s more nense. The nensy of he rblen feld an be measred by he Trblen Kne Energy TKE or n shor. Ths, we ae: and Ths leads s o wre a eqaon for. Reall he bdge for he rblen ne energy: ' ' w' b' ' ' ' ' ' ' p' ' adveon by he average flow shear prodon boyany prodon or qenhng vsos dsspaon always negave spaal redsrbon bease hs s he dvergene of a fl, whh amons o ero afer negraon over he enre doman

3 /5/5 Wh he nrodon of an vsosy, shear prodon beomes: ' ' always onrbng as a sore of rblene Wh he nrodon of an dffsvy, boyany prodon beomes: w ' b ' b T g < qenhng rblene n ase of sable srafaon warmer on op > enhanng rblene n ase of hermal nsably older on op Trblene Closre odels Varos losre shemes abo an lassfed n erms of her nmber of ranspor eqaons solved n addon o he Reynolds Averaged Naver Soes RANS eqaons: Zero eqaon/algebra models: ng lengh, Cebe Smh, Baldwn Loma One eqaon models: Wolfsen, Baldwn Barh, Spalar Allmaras, model Two eqaon models: ε, ω, τ, L, e. 4 Three eqaon models: ε A 5 For eqaon model: v f model

4 /5/5 odel : The mng lengh model Pope, Seon.., pages 66 In hs model, he vsosy and dffsvy are epressed n erms of he veloy graden nder he argmen ha he greaer he shear, he sronger he nsables ha feed rblene and hene he sronger he rblen srrng effe: S S n whh S s he overall shear, defned as S S S S The lengh l, alled he mng lengh, s needed o have he orre dmensons and n m /s. I s eernally mposed, eher as a onsan relaed o he se of he doman or as a fnon of spae, sh as he dsane o he neares sold bondary. odel : Varaon on he mng lengh model Pope, page 67 In hs model, he vsosy and dffsvy are agan epressed n erms of he veloy graden nder he argmen ha he greaer he shear, he sronger he nsables ha feed rblene and hene he sronger he rblen srrng effe, b he overall veloy graden s defned dfferenly: n whh s he overall vory, defned as The mng lengh l s agan eernally mposed, eher as a onsan relaed o he se of he doman or as a fnon of spae, sh as he dsane o he neares sold bondary. odels and are only as good as her defnon of he mng lengh l, and hs omes lose o feedng he desred answer. 4

5 /5/5 odel : The model Pope, Seon., pages 69 In hs model, he vsosy and dffsvy are epressed as T n whh s he rblen veloy defned as Here, oo, he lengh l, alled he mng lengh, s needed o have he orre dmensons and n m /s. I s eernally mposed, eher as a onsan relaed o he se of he doman or as a fnon of spae, sh as he dsane o he neares sold bondary. The dmensonless faors for momenm and T for emperare are nng parameers of he model, ha s, hey have o be adsed emprally nl he bes resls are aheved. Ths model wold no be omplee who sayng somehng abo he redsrbon erms and he energy dsspaon erms. The redsrbon erms are modeled as dsspave erms. The hnng s ha rblen ne energy s re arranged s as momenm s: ' ' ' ' ' ' p' The energy dsspaon erm s modeled as ' E whh s based on mere dmensonal analyss n m /s a he os of nrodng an addonal nng faor, E. Noe: Inrodon of a denomnaor, an addonal nng parameer. 5

6 /5/5 6 Wh hese hoes made o lose he problem, The vsosy and dffsvy are made dependen n he rblen ne energy The eqaon for he rblen ne energy s wren as T / / / / b E T Adveon Shear Boyany Redsrbon Dsspaon prodon prodon or qenhng A onseqene of he model s ha onsan / / E E E Verfaon: Pope, Fgre., page 7

7 /5/5 odel : The l model no n Pope s boo The model had he nheren weaness of needng a mng lengh l spefed orgh. Ths was neessary o epress several epressons, sh as he vsosy and energy dsspaon, n erms of he rblen ne energy n a dmensonally orre way. One dea o remedy hs weaness s o have he mng lengh l beome a varable wh s own evolon eqaon, aally an eqaon for he prod l. Ths has been red ellor & Yamada, 98, and he resl s he l model, whh s very wdely sed n geophysal fld dynams, espeally n he one of med layers. odel 4: The model Pope, Seon.4, pages 7 In he model, he dea s do away wh l alogeher and se nsead, as follows: The dmensonless faors for momenm and T for emperare are nng parameers of he model, wh vales ha are no he same as n he model. T / If a lengh sale needs o be derved aferwards, s done sng. A new eqaon has o be formlaed for he energy dsspaon, b here s none ha old emerge from he Naver-Soes eqaons. Insead, one s manfared by paernng afer he eqaon for : 7

8 /5/5 8 T b b T The eqaon n he model: The eqaon n he model: These erms assme ha s proded and dsspaed n he ea same way as rblen ne energy eep for some oeffens of proporonaly! There s lle physal sppor for hs assmpon. Fne nng of he model has led o he followng vales for he dmensonless onsans n he formalsm: =.9 =.44 =.9 =. =.. Parlar ase: Deayng rblene Pope, pages In he absene of mean veloy gradens, boyany effes and rblene gradens homogeneos rblene, he prodon, boyany and redsrbon erms vansh, and he rblene nensy deays. ahemaally, d d d d The solon o hs par of eqaons s: n n n wh n n n Ths, epermenal observaons of n an provde a vale for he nable parameer.

9 /5/5 Lab daa ohamed & LaRe, 99 ed by Pope on page 76 sgges ha mos of he daa are onssen wh n =. wh spread from.5 o.45. Ths provdes =.77 wh spread from.69 o.87. Ths dffers from =.9 qoed above. Applaon o shear rblene along a wall Pope, pages Consder: Seady sae / = ean flow only n dreon, Only graden n dreon, away from wall / =, / y = No hermal effes The and eqaons rede o: d d d d d d 9

10 /5/5 Observaons sgges ha, alhogh he mean flow s sheared and hs he saon non homogeneos, he rblen ne energy s farly nform aross he flow = onsan. Ths perms s o negle he erm onanng d/dy : d d d d d Ths mples ha shear prodon he sore of rblene s balaned by dsspaon he sn of rblene. The eqaon hen beomes: and s solon s: onsan d d / /.8 d ln d Ne, we an solve for he veloy shear: Defnng he rblen veloy as we dd earler eep for a faor, we an rewre hs as wh ln ln.9 Noe: The rblen veloy s defned here as half he sqare roo of he rblen ne energy o mah he so alled fron veloy ha we wll defne laer from he wall sress.

11 /5/5 Sore: hp://web.sanford.ed/lass/me469b/handos/rblene.pdf Dssson of he rblene losre model Pope, page 8, qoed verbam The model s argably he smples omplee rblene model, and hene has he broades range of applably. IsnorporaednmosommeralCFD odes, and has been appled o a dverse range of problems nldng hea ransfer, ombson, and ml phased flows. A dssson of s aray s deferred o he ne haper Seon., where s performane s ompared wh ha of oher rblene models. Brefly; alhogh s sally aepably arae for smple flows, an be qe narae for omple flows, o he een ha he allaed mean flow paerns an be qalavely norre. The naraes sem from he rblen vsosy hypohess and from he eqaon. Reall ha applably does no mply aray.

12 /5/5 Pros & Cons of ε model + Smple + Affordable + Reasonably arae for wde varey of flows who separaon + Hsory effes Overly dffsve Canno pred dfferen flows wh he same se of onsans la of nversaly Sore erms are sff nmerally No arae n he regon lose o no slp walls where and ε ehb large peas DNS and epermenal observaons Near wall reamen Sore: hp://web.sanford.ed/lass/me469b/handos/rblene.pdf odel 5: The model Pope, Seon.5., pages 8 84 In he model, he dea s agan do away wh he mng lengh l. Insead a freqeny w s sed, as follows: T The dmensonless faors for momenm and T for emperare are nng parameers of he model, wh vales ha are no he same as n he prevos models. An eqaon governng he evolon of w s hen wren: b T In some applaons, hs model behaves beer han he model. If one does no now wha o epe, s mpossble o pred whh model wll wor bes.

13 /5/5 Sore: hp://web.sanford.ed/lass/me469b/handos/rblene.pdf odel 6: The Spalar Allmaras model Pope, Seon.5., pages The dea here s o do away wh he hoe of pls anoher dmensonal varable o epress he vsosy by smply wrng an eqaon for he vsosy! Ths leads o a sngle eqaon for nsead of wo eqaons, one for and one for he oher varable l, or, of he form: S Ths model was developed spefally for aerodynam flows and has proved qe sessfl, b s no a model of wde applably.