Comaioal Flid Dyamics I Nmerical Meods or e Navier-Sokes Eqaios Isrcor: Hog G. Im iversiy o Miciga Fall 00
Comaioal Flid Dyamics I Olie Wa will be covered Smmary o solio meods - Icomressible Navier-Sokes eqaios - Comressible Navier-Sokes eqaios Hig accracy meods - Saial accracy imroveme - Time iegraio meods Wa will o be covered No-iie dierece aroaces sc as - Fiie eleme meods (srcred grid) - Secral meods
Comaioal Flid Dyamics I Icomressible Navier-Sokes Eqaios
Comaioal Flid Dyamics I Icomressible Navier-Sokes Eqaios α 0 v w Te (ydrodyamic) ressre is decoled rom e res o e solio variables. Pysically, i is e ressre a drives e low, b i racice ressre is solved sc a e icomressibiliy codiio is saisied. Te sysem o ordiary diereial eqaios (ODE s) are caged o a sysem o diereial-algebraic eqaios (DAE s), were algebraic eqaios acs like a cosrai.
Comaioal Flid Dyamics I Voriciy-sream cio ormlaio Advaages: - Pressre does o aear exlicily (ca be obaied laer) - Icomressibiliy is aomaically saisied (by deiiio o sream cio) Drawbacks: - Limied o -D alicaios (Revised 3-D aroaces are available)
Comaioal Flid Dyamics I Solio Meods or Icomressible N-S Eqaios i Primiive Formlaio: Ariicial comressibiliy (Cori, 967) mosly seady Pressre correcio aroac ime-accrae - MAC (Harlow ad Welc, 965) - Proecio meod (Cori ad Temam, 968) - Fracioal se meod (Kim ad Moi, 975) - SIMPLE, SIMPLER (Paakar, 98)
Comaioal Flid Dyamics I Ariicial Comressibiliy - Back o a sysem o ODE by Wi roerly-cose, solve il Origially develoed or seady roblems Te erm ariicial comressibiliy is coied rom eqaio o sae Possible merical diiclies or large c α c 0 c : arbirary cosa c 0 ( ( x) ) c
Comaioal Flid Dyamics I Ariicial Comressibiliy - Te coce ca be alied o a ime-accrae meod by sig sedo-ime seig a every sb-ses. β α c 0 A every real ime se, ake sedo-ime seig sig exlici ime iegraio il 0, 0 Sice e sedo ime scale is o ysical, we ca accelerae e iegraio owever we wa.
Comaioal Flid Dyamics I Pressre Correcio Meod - Marker-ad-Cell (MAC) Meod Harlow ad Welc (965) Origially derived or ree srace lows wi saggered grid Exlici iegraio α ( α Takig divergece o momem eqaio, ( ) ) ad α 0 ( Poisso eqaio )
Comaioal Flid Dyamics I WPI Proecio Meod Cori (968), Temam (969) Origially derived o a colocaed grid Ideical o MAC exce or e Poisso eqaio Pressre Correcio Meod - 0 0
Comaioal Flid Dyamics I Pressre Correcio Meod - 3 MAC vs. Proecio. Iegraio wio ressre ( A D ). Poisso eqaio ( ) 3. Proecio io icomressible ield ( A D )
Comaioal Flid Dyamics I Pressre Correcio Meod - 4 SIMPLE Algorim Paakar (98) (Semi-Imlici Meod or Pressre Liked Eqaios) - Ieraive rocedre wi ressre correcio. Gess e ressre ield. Solve e momem eqaio (imlicily) 3. Solve e ressre correcio eqaio 0 0 0 0 0 α 0 ( ) 0 0
Comaioal Flid Dyamics I Pressre Correcio Meod - 5 4. Correc e ressre ad velociy 0 0 5. Go o. Reea e rocess il e solio coverges. Noes: - Origially develoed or e saggered grid sysem. - Te correced velociy ield saisies e coiiy eqaio eve i e ressre correcio is oly aroximae. - Someimes eds o be overesimaed 0 ω ( ω 0.8) derrelaxaio
Comaioal Flid Dyamics I Pressre Correcio Meod - 6 SIMPLER (SIMPLE Revised) - Icororaig e roecio meod (racioal se). Gess e velociy ield 0. Solve momem eqaio (imlicily) wio ressre ˆ 0 ˆ ˆ α ˆ 3. Solve e ressre Poisso eqaio * ( û)
Comaioal Flid Dyamics I Pressre Correcio Meod - 7 5. Solve e momem eqaio wi * 0 6. Pressre correcio eqaio * 7. Correc e velociy, b o e ressre * * α ( * ) 8. Go o. Reea e rocess il solio is coverged. * * *
Comaioal Flid Dyamics I Sabiliy Cosideraio Exlici ime iegraio i -D reqires e sabiliy codiio: 4α < ad < ( ) v 4α Hig-Re low: advecio-corolled Low-Re low: disio-corolled 4 ( v ) /α 4α se imlici scemes or aroriae erms! 0 / α ~ Re a bo limis!
Comaioal Flid Dyamics I PI W Accracy Imroveme Saial Saial Accracy Exlici dierecig - se larger secils ) ( O ) ( 8 8 4 O Tridiagoal - Padé (comac) scemes ( ) ) ( 3 4 4 O Peadiagoal L e d c b a ε δ γ β α
Comaioal Flid Dyamics I Accracy Imroveme Saial 3 Exac E 4E 6E 6T 8E 8T k' 0 0 3 k Re: Keedy, C. A. ad Career, M. H., Alied Nmerical Maemaics, 4,. 397-433 (994).
Comaioal Flid Dyamics I Accracy Imroveme Temoral Temoral Accracy A( ) D( ) Imlici Crak-Nicolso A( ) A( [ ( ) ] / ) α( ) Noliear advecio erm reqires ieraio.
Comaioal Flid Dyamics I PI W Accracy Imroveme Temoral Liearizaio o Advecio Terms For examle, a -D eqaio ca be liearized as 0 y x F E v xy xx v E yy xy v v v F [ ] [ ] 0 y B x A [ ] [ ] F E B A, were Jacobia marix
Comaioal Flid Dyamics I Accracy Imroveme Temoral 3 Fracioal Se Meod Kim & Moi (985) Proecio meod exeded o iger accracy φ 0 Re [ ] 3A( ) A( ) ( ) Adams-Basor (AB) φ Crak-Nicolso Noe a φ is diere rom e origial ressre φ φ Re
Comaioal Flid Dyamics I Accracy Imroveme Temoral 4 Treame o imlici viscos erms [ ] 3 ( ) ( ) A A ( ) Re ( δ xx δ yy δ zz ) Re Re Re 3A( ) A( ) δ xx δ yy δ zz Re Facorizig, ( δ xx δ yy δ zz ) Re Re Re 3A( ) A( ) δ xx δ yy δ Re TDMA i ree direcios [ ] ( ) [ ] ( ) zz
Comaioal Flid Dyamics I Accracy Imroveme Temoral 5 Noes o Fracioal Se Meod Origially imlemeed io a saggered grid sysem Laer imroved wi 3rd-order Rge-Ka meod Re: Le & Moi, J. Com. Pys., 9:369 (99) Te meod ca be alied o a variable-desiy roblem (e.g. sbsoic combsio, wo-ase low) were Poisso eqaio becomes ( ) φ T Eq. o Sae Re: Rlad, P. D. Tesis, Saord iversiy (989) Bell, Collela ad Glaz, JCP, 85:57 (989)
Comaioal Flid Dyamics I Bodary Codiios Bodary Codiios or Icomressible Flows I geeral, bodary codiio reame is easier a or e comressible low ormlaio de o e absece o acosics Tyical bodary codiios: - Periodic:, N N ec. - Ilow codiios: ( x 0) F( y, z, ) - Olow codiios: covecive olow codiio 0 a x L x A da
Comaioal Flid Dyamics I Comressible Navier-Sokes Eqaios
Comaioal Flid Dyamics I WPI 0 z y x G F E E w v x xz xy xx E w v ψ ) ( E y yz yy xy v E vw v v v ψ ) ( F z zz yz xz w E vw vw w w ψ ) ( G T c T c e RT v,, R e T e R c v ) (, ) (, γ γ γ or Cosiive relaios z zz yz xz z y yz yy xy y x xz xy xx x q w v q w v q w v ψ ψ ψ were
Comaioal Flid Dyamics I Solio meods or comressible N-S eqaios E x F y G z ollows e same eciqes sed or yerbolic eqaios For smoo solios wi viscos erms, ceral dierecig sally works. No eed o worry abo wid meod, lx-sliig, TVD, FCT (lx-correced rasor), ec. I geeral, wid-like meods irodces merical dissiaio, ece rovides sabiliy, b accracy becomes a cocer.
Comaioal Flid Dyamics I Exlici Meods - MacCormack meod - Lea rog/dfor-frakel meod - Lax-Wedro meod - Rge-Ka meod Imlici Meods - Beam-Warmig sceme - Rge-Ka meod Mos meods are d order. Te Rge-Ka meod ca be easily ailored o iger order meod (bo exlici ad imlici).
Comaioal Flid Dyamics I WPI Mos o e ime, a imlici iegraio meod ivolves oliear advecio erms wic are liearized as z y x G F E [ ] [ ] [ ] z C y B x A [ ] [ ] [ ] C B A G F E,, ADI, acorizaio, ec.
Comaioal Flid Dyamics I limaely, comressible Navier-Sokes eqaios ca be wrie as a sysem o ODE s d F d ( E F x y (, ( ) ) 0 ) 0 G z Iiial codiio Solio eciqes or a sysem o ODE alies. - Exlici vs. Imlici (Nosi vs. Si) - Mli-sage vs. Mli-se
Comaioal Flid Dyamics I Bodary Codiios Bodary Codiios or Comressible Flows I geeral, bodary codiio or e comressible low is rickier becase all e acosic waves ms be roerly ake care o a e bodaries. Tyical bodary codiios: - Periodic: sill easy o imleme - Bo ilow ad olow codiios reqire reame o caracerisic waves (ard-wall, orelecig, soge, ec).