Boundary-Fitted Coordinates!
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- Πυθις Ναζωραῖος Ουζουνίδης
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1 Computatoal Flud Damcs I Computatoal Flud Damcs I Computatoal Methods or Domas wth! Comple Boudares-I! Grétar Trggvaso! Sprg 00! For most egeerg problems t s ecessar to deal wth comple geometres cosstg o arbtrarl curved ad oreted boudares! Computatoal Flud Damcs I Outle! How to deal wth rregular domas! Overvew o varous strateges! Boudar-tted coordates!!- Naver-Stokes equatos vortct orm!!- Naver-Stokes equatos prmtve orm! Grd geerato or bod-tted coordates!!- Algebrac methods!!- Deretal methods! - Starcasg! Computatoal Flud Damcs I Overvew! Varous Strateges or Comple Geometres! Prmarl to treat sold boudares! - Boudar-tted coordates! - Immersed boudar method o grd chage! - Ustructured grds: tragular or tetrahedral! - Adaptve mesh reemet AMR! Computatoal Flud Damcs I Computatoal Flud Damcs I Starcasg! Boudar-Ftted Coordates! Appromate a curved boudar b a the earest grd les!
2 Computatoal Flud Damcs I Boudar-Ftted Coordates! Coordate mappg: trasorm the doma to a smpler usuall rectagular doma.! Boudares are alged wth a costat coordate le thus smplg the treatmet o boudar codtos! The mathematcal equatos become more complcated! Computatoal Flud Damcs I 0 0 Boudar-Ftted Coordates! Computatoal Flud Damcs I Chage o varables! Frst-Order Dervatves! / Note: The equatos wll be dscretzed the ew! grd sstem. Thereore t s mportat to ed up! wth terms lke ot.! / Boudar-Ftted Coordates! Computatoal Flud Damcs I Usg the cha rule:! We wat to derve epressos or! the mapped coordate sstem.! / / Boudar-Ftted Coordates! Computatoal Flud Damcs I Solvg or the dervatves! Subtractg! Boudar-Ftted Coordates! Computatoal Flud Damcs I Solvg or the orgal dervatves elds:! where! s the acoba.! Boudar-Ftted Coordates!
3 Computatoal Flud Damcs I A short-had otato:! Boudar-Ftted Coordates! Computatoal Flud Damcs I Rewrtg short-had otato! where! s the acoba.! Boudar-Ftted Coordates! Computatoal Flud Damcs I These relatos ca also be wrtte coservatve orm:! Boudar-Ftted Coordates! Sce:! Ad smlarl or the other equato! Computatoal Flud Damcs I The secod dervatves s oud b repeated applcato o the rules or the rst dervatve! Smlarl! Secod-Order Dervatves! Boudar-Ftted Coordates! Computatoal Flud Damcs I Addg! ad! elds a epresso or the Laplaca:! Boudar-Ftted Coordates! Computatoal Flud Damcs I Boudar-Ftted Coordates!
4 where! Computatoal Flud Damcs I Boudar-Ftted Coordates! q q q q q q q [ q q q q ] Computatoal Flud Damcs I Boudar-Ftted Coordates! Epadg the dervatves elds! q q q where! [ q q ] [ q q ] Dervato o!! Computatoal Flud Damcs I Boudar-Ftted Coordates! hece! smlarl! Computatoal Flud Damcs I Boudar-Ftted Coordates! Puttg them together t ca be show that! prove t!! [ q q ] q [ q q ] q Computatoal Flud Damcs I Boudar-Ftted Coordates! We also have or a ucto ad! g g g g g Computatoal Flud Damcs I Boudar-Ftted Coordates! A comple doma ca be mapped to a rectagular doma where all grd les are straght. The equatos must however be rewrtte the ew doma.! 0 0 Thus:! Ad more comple epressos or the hgher dervatves!
5 Computatoal Flud Damcs I Computatoal Flud Damcs I Computatoal Methods or Domas wth! Comple Boudares-II! Vortct-Stream Fucto Formulato! Grétar Trggvaso! Sprg 00! Computatoal Flud Damcs I Vortct-Stream Fucto Formulato! Computatoal Flud Damcs I Vortct-Stream Fucto Formulato! The Naver-Stokes equatos vortct orm are:! ω ψ ω ψ ω α ω t ψ ω Usg the trasormato relatos obtaed earler! The Naver-Stokes equatos vortct orm become:! ω t ψ ω ψ ω α q ω q ω q ω ω ω q ψ q ψ q ψ ψ ψ ω q q q Boudar Codtos! Ilow! Computatoal Flud Damcs I Vortct-Stream Fucto Formulato! 0 : Ilow N : Outlow 0 M : No slp ψ Q ψ 0 Q ud vd Outlow! dψ ψ ψ Lower wall! 0 Computatoal Flud Damcs I Vortct-Stream Fucto Formulato! Stream ucto:! ψ 0 Vortct:! 0 o-slp! ψ ψ 0 ψ 0 ψ 0 HOT Usg that! ω 0 ψ 0 We have:! ω0 [ ψ0 ψ ]
6 Upper wall! Computatoal Flud Damcs I Vortct-Stream Fucto Formulato! Stream ucto:! ψ Q Vortct:! M L M Q ud vd dψ ψ M ψ 0 0 ωm [ ψm ψm ] 0 Ilet low! Computatoal Flud Damcs I Vortct-Stream Fucto Formulato! 0 Cosderg a ull-developed parabolc prole! u 0 C L L L Q L d C C Q C 0 L Q u u 0 L ω L Q Q L d Q L 0 L L Ilet low! Computatoal Flud Damcs I Vortct-Stream Fucto Formulato! 0 Cosderg a ull-developed parabolc prole! ad assume that! L M Q u 0 L M M ψ 0 Q M M Q ω 0 L M Outlow! Computatoal Flud Damcs I Vortct-Stream Fucto Formulato! N Tpcall assumg straght streamles! ψ 0 I s ormal to the outlow boudar ths elds! ψ 0 ψ I ot the a proper trasormato s eeded or! Computatoal Flud Damcs I Computatoal Flud Damcs I Veloct-Pressure Formulato! Veloct-Pressure Formulato! The Naver-Stokes equatos prmtve orm! u t uu vu p α u u v t uv vv p α v v ad cotut equato! u v 0 Notce that we have absorbed the dest to the pressure ad use α or the vscost!
7 Advecto Terms! Computatoal Flud Damcs I Veloct-Pressure Formulato! uu vu [ uu uu uv uv ] [ uu uu uu uv uv uv u u v uu uv uu uv ] { } u v u { } Cotravarat Veloct! V C v u C U Computatoal Flud Damcs I Veloct-Pressure Formulato! U u v V v u Ut ormal vector! Ut taget vector alog! C U u v u v Thereore U s the drecto.! V s the drecto.! Computatoal Flud Damcs I Veloct-Pressure Formulato! Pressure Term! p p p Duso Term! u u u u u u u u u u u u u Computatoal Flud Damcs I Veloct-Pressure Formulato! u u u u u u u u u u q u q u q u q u q q q Computatoal Flud Damcs I Veloct-Pressure Formulato! u-mometum Equato! U u v V v u u t Uu Vu ρ p p α q u q u q u q u v-mometum Equato! v t Uv Vv ρ p p α q v q v q v q v Computatoal Flud Damcs I Veloct-Pressure Formulato! Cotut Equato! u v 0 Usg! Cotut equato becomes! [ u u v v ] 0 or! u v v u 0 U V 0
8 Computatoal Flud Damcs I Veloct-Pressure Formulato! The mometum equatos ca be rearraged to! u v t t U-Mometum Equato! U Uu Vu Uv Vv t p q q p α q u q u q u q u q v q v q v q v V-Mometum Equato! Computatoal Flud Damcs I Veloct-Pressure Formulato! V t Uu Vu Uv Vv p q q p α where! q v q v q v q v q u q u q u q u u U V v V U Computatoal Flud Damcs I Veloct-Pressure Formulato! Computatoal Flud Damcs I I the plae a staggered grd sstem ca be used.! Vv p Uu Same MAC grd ad! projecto method! ca be used.! Stretched Grds! C C Computatoal Flud Damcs I Computatoal Flud Damcs I Stretched grds! Specal case:! q q 0 q u U v V U U V V U V 0 u v 0 u v 0
9 Computatoal Flud Damcs I Stretched grds! Appromate the coservato equato! j j u v 0 j / / Dee:! j j / j/ / / u / j u / j v j / v j/ j 0 Computatoal Flud Damcs I Stretched grds! u-mometum Equato! u t u vu p α u / j u / j t P j P j / u j α / / u j u / j j u uv uv / j / / j/ u u / j / j j / u / j j u / j u / j u / j u / j j/ u Computatoal Flud Damcs I Stretched grds! v-mometum Equato! v t uv v p α v j / v j / t uv j / uv j / v / v v j j j P P j j α v v j / j / v j / j / / v j / j / v j / j v j / / v j / v j/ j v The pressure equato s:! P P j j / Computatoal Flud Damcs I Stretched grds! P P j j j j / * * u u t / j / j P j P j / Whch ca be solved b terato! P j P j j/ * v j / * v j/ j Computatoal Flud Damcs I Stretched grds! Colocated stretched grds! Re0! Computatoal Flud Damcs I Stretched grds! For oe dmesoal stretched grds we smpl replace the global b the local! u -/j! p j! u /j! /! p j!!
10 Computatoal Flud Damcs I Accurac o Geeralzed Coordates! Computatoal Flud Damcs I Accurac o Geeralzed Coordates! Phscal Doma! r Mapped Doma! Cosderg -D Stretchg! Talor epaso! Computatoal Flud Damcs I << It ma appear d order accurac but! Accurac o Geeralzed Coordates! Computatoal Flud Damcs I [ ] Further algebrac epaso gves! 0.5 O r O j j O r O j j j r r Moral: avod large chages! r Accurac o Geeralzed Coordates! Computatoal Flud Damcs I I s evaluated eactl! The use o eact evaluato o troduces a addtoal! ad domat term the trucato error.! Thereore umercal evaluato s preerred.! Accurac o Geeralzed Coordates! Computatoal Flud Damcs I B mappg a comple doma ca be mapped to a rectagular oe where the techques that we have covered ca be used drectl! The goverg equatos however become more comple! Although the geometr o the doma ca be arl comple t must be possble to map t oto a rectagular doma. For more comple domas t s possble to use block-structured grds but ver comple shapes are more easl doe b ustructured grds!
11 Computatoal Flud Damcs I What about Veldmaʼs results? Should I add?!
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