Outline. Review Solution Approaches. Review Basic Equations. Nature of Turbulence. Review Fluent Exercise. Turbulence Models

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1 Trblence Models Larry areo Mechancal Engneerng 69 ompaonal Fld Dynamcs Febrary, Olne Revew las lecre Nare of rblence Reynolds-average Naver-Soes (RNS) Mng lengh heory Models sng one dfferenal eqaon Two-eqaon models, especally Reynolds sress models Large-eddy smlaon (LES) Drec nmercal smlaon (DNS) Revew Basc Eqaons Have general eqaon o se n nmercal analyss approaches φ ϕ ( ϕ ) ϕ ( ϕ ) c S Γ Transen onvecve Dffsve " Sorce" Momenm eqaons separae pressre graden from oher sorce erms S ** for consan, and B P ** S ** S B ( κ ) Δ Revew Solon Approaches Solve momenm and conny for velocy and Pressre a low Mach nmbers where densy does no depend on pressre Densy a hgh Mach nmbers where pressre s fond from eqaon of sae Densy solon approach U E F G H v U y z w ( e V / ) 4 ( K ) W Revew Flen Eercse Noe here are wo fles case and daa ase fle has grd nformaon Daa fle has resls an open and save boh a same me Wll learn abo rblence models hs wee and nmercal algorhms ne wee Use oral fles as eamples of how o se Flen 5 Nare of Trblence haracerzed by flcaons n he flow Imperfec ably o model rblence s maor problem n praccal applcaons of FD Trblen eddes are srcres ha es n he flow Larges scale srcres ge energy from man flow and ransfer energy o smaller Smalles scale srcres ge energy from larger and dsspae energy o vscosy 6 ME 69 ompaonal Fld Dynamcs

2 Nare of Trblence II No smong n SUN classrooms, b he smoe from a cgaree shows how lamnar flows can ranson no rblen flows and he eddy nare of he rblen flow srcres landsa.gsfc.nasa.gov/ Trblen flows n clods from earh saelle shows rblen srcres Trblen ppe flow vdeo Phoo cred: Hmphrey_Bogar_by_Karsh_(Lbrary_and_Archves_anada).pg 7 8 Flcaon Qanes Model flow varables n erms of man flow properes and flcaons Insananeos vale, φ Flcaon vale φ Δ Defnon of mean vale ϕ ϕd Basc resl: Δ ϕ ϕ ϕ Appled o velocy componens somemes se U for mean velocy componen U 9 Trblen Knec Energy The nec energy per n mass de o flcaons n velocy v v w w Trblen nensy rms / V [( ) ( v ) ( w ) ] rms V ( ) ( v) ( w ) Trblen Energy Transfer Trblen flows have a seres of lengh scales whch ransfer energy A larges scales, rblen srcres ge nec energy from man flow Knec energy ransfers from larger o smaller lengh scales Energy s dsspaed by vscos effecs a he smalles lengh scales Large Reynolds nmbers or Grashof nmbers mar ranson o rblence More on Trblen Flcaons Defnon shows ha average flcaon s zero Δ ϕ ϕ ϕ ϕ ϕd Δ Δ Δ Δ ϕ ϕ ϕ ϕ ϕ ϕ ϕ Δ d Δ d Δ d ( ) The mean vale s a consan and he average of a consan s s ha consan ME 69 ompaonal Fld Dynamcs

3 Average of a Prodc The mean of he prodc of wo flow properes φ and ψ, wren as ψϕ, s he sm of wo erms: The prodc of he means of each ndvdal erm ϕ and ψ The mean of he prodc of he wo flcaon qanes ϕψ (correlaon erm) ψϕ ϕ ψ ϕψ Alhogh ϕ and ψ are zero ϕψ s no zero Dervaon on ne slde Average of a Prodc Two flow properes φ and ψ Δ Δ ψϕ ( ϕ ϕ )( ψ ψ ) d ϕψd Δ Δ Δ Δ Δ ψ Δ ϕ ψd Δ Δ ϕ d Δ Δ Δ ϕ ψ d ϕψ Δ Δ ϕ ψ d ϕψ Δ ψϕ ϕ ψ ϕψ Δ d ϕ Δ Δ Δ ϕψ d ψ d ϕ ψ d 4 More on Averages Roo-mean sqare properes based on rblen flcaons ϕ rms Δ ( ϕ ) Δ ( ϕ ) d Average of a space dervave s space dervave of average Δ Δ ϕ ϕ ϕ d d Δ Δ ϕ 5 Reynolds-Average Reynolds average ranspor eqaon (ncldng Naver Soes, called RANS) Sar wh general ranspor eqaon φ ϕ c Γ ϕ ( ϕ ) ( ϕ ) S Loo a seady-sae, zero-sorce, consan properes ( φ ) ( φ ) Γ ϕ ( ϕ ) ϕ γ γ c 6 Wha s γ (φ) Γ (φ) /c? Recall defnon of Γ (φ) as general ranspor coeffcen φ v w e h T Γ (φ) /c v /c p c c v γ (φ) / / / /c v /c p /c v Dmensons for γ (φ) are L /T Knemac vscosy, / Thermal dffsvy α /c p T c p /c p s hermal condcvy 7 Reynolds Average II Tae me average of las eqaon Δ Δ ϕ ( ϕ ) ϕ d γ d Δ Δ Tme average of dervaves are dervaves of me average ϕ ( ϕ ) γ ϕ 8 ME 69 ompaonal Fld Dynamcs

4 Reynolds Average III Use epresson for average of a prodc o compe average of φ ϕ ( ϕ ϕ ) ( ϕ ) ϕ γ ompare o eqaon before averagng Overall vales replaced by averages Add a new erm: he average of he prodc of wo flcaons Have o compe hs prodc erm 9 Reynolds Average IV ombne flcaon prodc erm wh dffsve fl ϕ ( ϕ ) ϕ γ ϕ Bossnesq appromaon: he rblen flcaon s proporonal o he graden of he mean propery wh emprcal rblen ranspor coeffcen, γ (φ) ϕ γ (ϕ) ϕ Reynolds Average IV ombne flcaon prodc erm wh dffsve fl ϕ ( ϕ ) ϕ ( ϕ ) ϕ ( ϕ ) ( ϕ ) ϕ γ l γ γ l γ Trblen ranspor coeffcen, γ (φ) now conans or gnorance abo rblence See ways o model hs erm Usally mch larger han lamnar erm so ha laer s ofen negleced ( ) Momenm Terms Apply general formlaon o momenm eqaon where φ n drecon For momenm eqaons, γ (φ) Also have me averaged pressre graden p/ whch s p / Tme average of momenm eqaon gves p Momenm Terms II Move flcaon prodc o rgh sde p Reslng erms are called he Reynolds sress erms Nne sch erms b only s are nqe Smple models loo a only one erm Advanced models ry o compe all s Trblen Vscosy Argmens based on dmensonal analyss Defne characersc velocy scale,, and lengh scale, l onsder nemac vscosy,, wh dmensons of L /T Dmensons of and l and L/T and L Dmensonally correc choce for n s prodc of and l or T l 4 ME 69 ompaonal Fld Dynamcs 4

5 Trblence Modelng Reynolds averagng erms le φ modeled by a rblen ranspor coeffcen γ (φ), e.g., rblen vscosy To se hs approach we have o fnd ways o compe and he general γ (φ) Varos rblence models proposed Some se smple conceps Ohers reqre nmercal solon of one or more paral dfferenal eqaons (PDE) PDEs have same form as oher FD eqaons 5 Mng Lengh Theory Orgnally proposed by Prandl n 95 Basc dea onsder a rblen fld wh a mean emperare graden A fld parcle ha moves from he cold regon o he ho regon wll ae on he characerscs of he ho regon afer moves hrogh one mng lengh, l The emperare flcaons are relaed o he mean graden as T T l y 6 Mng Lengh By dmensonal argmens / l, where s dmensonless consan For smple flows have only one mporan Reynolds sress, - v If lengh scale s measre of larges eddy sze, one possble dmensonally correc velocy scale s cl U/y c s dmensonless consan whch s dfferen from consan n l 7 Mng Lengh II If / l, and cl U/y, hen cl U/y l Kl U/y Ths gves he Reynolds sress as follows v l y y Oher flow properes se rblen ranspor rao, /Γ (φ) v φ l φ y y 8 Bondary Layers Trblen flow ne o a wall has lamnar sblayer and ranson o flly rblen flow Epermenal daa and smplfed analyss gve emprcal eqaons for velocy profle of rblen bondary layer Smplfed heory sed for rblen wall fncons o gve bondary condons for rblen flows n FD 9 Bondary Layer Profle Flow n drecon (velocy ) wh y as dsance perpendclar o wall y, Wall shear sress τ w τ s called frcon velocy wall Defne dmensonless varables y s dmensonless dsance from wall s dmensonless velocy parallel o wall τw y τ y τ y τ yτ ME 69 ompaonal Fld Dynamcs 5

6 Bondary Layers II Law of he wall: f(y ) No eqaon 5 < y < In oher regons y y < 5 ln( y ) ln( Ey ) B < y < 5 κ κ y ma ln A y > 5 κ δ B 5.5; E 9.8, κ.4 y 5 y. Delo and A.J. Sms, Volmerc vsalzaon of coheren srcre n a low Reynolds nmber rblen bondary layer, Inernaonal Jornal of Fld Dynamcs (997), Vol., Arcle, Fgre 6, downloaded // from hp:// Spalar-Allmaras Model Solves one PDE for ~ ~ ~ whch s sed o fnd from eqaon a rgh ~ v Defal vale: v 7. In mos of flow away from walls ~ Developed for aerospace applcaons o eernal flows wh walls Has recenly been appled o oher flow saons -epslon (-) Model Mos poplar model of rblence Orgnal and modfed versons Wors well n confned flows wh no roang pars Less applcable n eernal flows and oscllang flows Reqres solon of wo PDEs one for rblen nec energy,, and one for rblen dsspaon, 4 Wha s? Dsspaon rae,, s rae of nec energy ransfer from smalles eddes worng agans he vscos forces Defned n erms of deformaon raes, e e τ e ( κ ) Δδ Defnon of s e e Dmensons of are energy dvded by [(mass)(me)], e.g. m /s 5 ompng from and Bac o basc eqaon ha s he prodc of a lengh scale and a velocy scale: / l, Use / as velocy scale Lengh scale, l / / Resl for vscosy s / / Solve PDEs for and an derve form of eqaon for 6 ME 69 ompaonal Fld Dynamcs 6

7 ME 69 ompaonal Fld Dynamcs 7 7 Eqaon an derve followng balance eqaon an denfy ransen, convecon and dffson erms; remander s sorce Terms wh hree flcang componens canno be comped who nrodcng erms wh for flcang componens p 8 Modelng Eqaon Terms Trblen dffson of nec energy Emprcal consan: Prodcon erm, P, ses model for Reynolds sresses Impled smmaon over wo ndces p P 9 The Eqaon Fnal form wh modeled erms Usally gnore lamnar vscosy P P 4 The Eqaon Dervaon smlar o ha for Oban paral dfferenal eqaon wh erms ha have o be modeled Emprcal consans n - model P Bondary ondons Ms specfy vales of and a all bondares (walls, nles, oles, ec.) Sold walls se wall fncons o provde correc resl a frs node Based on bondary layer resls for fla walls s velocy parallel o wall y s drecon normal o wall κ κ τ τ Ey Ey ln ) ln( 4 Wha s τ? Frcon velocy, τ (τ w /) / Need a vale for τ w Assme prodcon and dsspaon of rblence are nearly eqal near wall Derve followng eqaon: τ w ( ) /4 ombne wh eqaon o ge resl for a frs node n from he wall τ τ Ey Ey 4 4 ln ln

8 Inle and Ole ondons Ideally have daa on smlar flows ha relae and o nle properes Falng ha se he followng eqaons nle l.7l, proporonal o sqare of rblence nensy, ypcally abo. o.5 L s a characersc lengh of he nle (e.g., he hydralc dameer) Use zero graden condons for ole for free sream 4 4 l Low Reynolds Nmbers Specal wall fncons are reqred for low Reynolds nmbers flows Defne wall dampng fncons, f Use f o compe f / Use f and f o modfy prodcon and dsspaon erms n eqaon Use lamnar vscosy n addon o rblen vscosy n boh and eqaons Dfferen forms for dampng fncons 44 Reynolds Sress Models Solve a dfferenal eqaon for each nqe Reynolds sress PDEs for, v, w, v v, v w w w Fnd ( v v w w )/ Also solve eqaon Shold wor well for srongly ansoropc flows, b modelng assmpons may lm accracy of model Usally appled o flows wh roang flow or swrl 45 Reynolds Sress Models II Oher ranspor eqaons compe rblen vscosy,, and se Prandl nmber, φ, o ge γ (φ) / φ Algebrac Reynolds sress model s smplfcaon ha solves and eqaons hen ges Reynolds sress by s smlaneos algebrac eqaons Eqaons for, v, w, v v, v w w w Oher rblence qanes: γ (φ) / φ 46 Large Eddy Smlaon (LES) Use small grd scale o compe larges eddes as par of he calclaon Use sb grd scale models o ge resls for he fnes rblence scales Now avalable n prodcon codes Generally no worh he era compaonal cos ecep for comple flows Drec Nmercal Smlaon (DNS) Use a grd fne enogh o resolve he smalles rblence srcres whch are -4 o -5 m A volme of m for flow arond a moorcycle wold reqre nodes DNS no praccal for engneerng calclaons, b an mporan research ool for eamnng rblence properes and esng oher rblence models ME 69 ompaonal Fld Dynamcs 8

9 Model Gdance Use model whch has been sed prevosly for yor problem Prevos wor a yor organzaon or from lerare research onsl ser s manal for FD code regardng ncreased comper me and memory se for more comple models Use defal consans n model nless yo have specfc daa o sfy alernave vales 49 Model Gdance II Revew maeral on rblence models o see f hey can handle nsal feares of he flow yo are modelng Low Reynolds nmber (non-eqlbrm) rblence Hgh sran raes Adverse pressre gradens Roang machnery ompressble flows Oher comple flows 5 Model Gdance III onclsons Whaever model yo se, mae sre ha yo have proper bondary condons Use specal wall fncons for noneqlbrm rblence when lamnar sblayer s no resolved Use correc grd spacng for frs node from wall for choce of wall fncons or resolvng lamnar sblayer hec hs afer calclaons - mos common rblence model for non-aerospace engneerng applcaons wdely regarded as havng many shorcomngs n represenng rblence mos wdely valdaed model probably bes choce for applcaons who srong dreconal effecs or roaonal flows Renormalzable grop and realzable - models can gve beer resls 5 5 onclsons II Spalar-Allmaras model developed especally for aerospace applcaons wh wall-bonded flows One eqaon model Relavely new model now seeng applcaons n areas oher han s nal aerospace applcaons Adverse pressre gradens Trbomachnery 5 onclsons III Reynolds sress model has had sccess for flows wh dreconal effecs and roaonal flows Reqres solon of seven paral dfferenal eqaons o compe rblen vscosy Algebrac verson has been sed Oher models avalable whch may have beer accracy for lmed range of flows 54 ME 69 ompaonal Fld Dynamcs 9

10 onclsons IV LES sed for comple flows parclarly ransen and oscllang flows No sally reqred for common engneerng problems hoce of rblence model shold be based on prevos sccess of model n smlar applcaons No one rgh model o choose 55 ME 69 ompaonal Fld Dynamcs