Computational Fluid Dynamics CFD. Solving system of equations, Grid generation

Transcription

1 Compaoal ld Dyamcs CD Solvg sysem of eqaos, Grd geerao

2 Basc seps of CD Problem Dscrezao Resl Gov. Eq. BC I. Cod. Solo OK??,,...

3 Solvg sysem of eqaos he ype of eqaos decdes solo sraegy Marchg problems Eqlbrm problems Parabolc Doma of depedece P Rego of flece Ellpc Hyperbolc P Every po fleces all oher pos Doma of depedece P Rego of flece

4 Solvg sysem of eqaos Parabolc Hyperbolc Ellpc Marchg mehods may be sed sce he solo oly depeds o prevos daa. Eamples: Ivscd spersoc flow Has o be solved for he whole doma smlaeosly, sce all pos deped o each oher. Relaao echqes. Eamples: Seady compressble flow Noe! me depede compressble flow has a med characer: ellpc space ad parabolc me.

5 Marchg mehods,,, -,, -,,, s order Cosder he vscd Brger eqao 0 N, N,, -,,, N, Coserved form 0 ( ) Sar wh a aylor epaso arod (,) (, ) (, ) HO,

6 Marchg mehods s order he dea s o replace he me dervaves he epaso by spacal oes rs dervave:,,, -, N, Apply d order ceral dfferecg:, -,,, N,, -,,, N,

7 Marchg mehods La-Wedroff scheme Cosder he vscd Brger eqao 0 Coserved form ) ( 0 Sar wh a aylor epaso arod (,) ( ) ( ) HO,,,,

8 Marchg mehods La-Wedroff scheme he dea s o replace he me dervaves he epaso by spacal oes, whch gves a scheme ha s d order accrae space ad me. rs dervave: Secod dervave: Sce s a fco of we ca wre A Jacoba A

9 Marchg mehods La-Wedroff scheme Hece, A he aylor epaso ca ow be wre as: ( ) ( ) HO,,,, A

10 Marchg mehods La-Wedroff scheme Apply d order ceral dfferecg: ( ) ( ) [ ] A A / / Sce A he Jacoba s calclaed as / A A sably aalyss gves ( ) θ θ s cos A A G Sable f he CL-codo

11 Marchg mehods MacCormac scheme hs s a wo sep verso of he L-W wh he advaage ha o Jacobas are eeded. Oherwse has decal properes o he L-W ( ) ( ) Predcor Correcor

12 Solvg sysem of eqaos Marchg problems Eqlbrm problems Parabolc Doma of depedece P Rego of flece Ellpc Hyperbolc P Every po fleces all oher pos Doma of depedece P Rego of flece

13 3 Relaao echqes S S O ) (,,, -, N,,,, -, N, S S 3 S S

14 Relaao echqes Basc echqes for solvg a sysem of eqaos A b Sysem of eqaos N N NN N N b b b a a a a a a Drec mehods Cramer Gass elmao Heavy Error accmlao homas algorhm r-dagoal sysems Ierave mehods

15 homas algorhm N N N N N c c c d b a b a d b a d c d b c c a d b d d P o pper raglar form: Uows comped sg bac-sbso: N N N a d c d c,...n,3..., N N

16 Jacob Easy b slow b A N b a a a b I erao sep : S S,,, -, N,,,, -, N,

17 Gass-Sedel b A N b a a a a b > < I erao sep : Always ses he bes vale avalable, gves faser solo S S,,, -, N,,,, -, N,

18 8 Sccessve Over-Relaao (SOR) Accelerae covergece ω > overrelaao ω < derrelaao (for sably) eac eac ω * * ) ( ω S S ω

19 Resdals R Whe shold we sop he eraos? m, m L R, Jacob SOR Gass-Sedel Ierao Nr.

20 Relaao echqes Po relaao y y φ φ 0,,,, y φ φ φ φ φ φ ( ) ( ),,,, β φ φ β φ φ φ Eample: Poeal flow Gass-Sedel, po relaao: y β

21 Relaao echqes Le relaao I le relaao a whole le s solved a oce sg a drec mehod, for eample he homas algorm. Gass-Sedel, le relaao : y φ φ, φ, ( β φ ), φ, ( ) β - -

22 Relaao echqes ADI, alerag dreco mplc rher mproveme of mercal covergece speed. Compaoal me ca be redced wh p o 0-40 % as compared o Gass-Sedel wh SOR Gass-Sedel, ADI y rs alog -dreco / / / φ, φ, he alog y-dreco φ /, φ, ( / β φ ), φ, ( ) β ( β φ ), φ, ( ) β - -

23 Mlgrd mehods Accelerae covergece

24 Mlgrd mehods Accelerae covergece

25 Mlgrd mehods Accelerae covergece

26 Mlgrd mehods Accelerae covergece

27 Mlgrd mehods Mlgrd mehods are sed o crease he compaoal effcecy of a mplc mehod d d Cosder he eqao: f ( ) Perodc bodary codos ( ) ( 0) 0 Creae a grd: Dscrese h 0 h f h Gass-Sedel m m m h f

28 Mlgrd mehods vo Nema sably aalyss Use he mercal error ξ m m m* e Remember: a b e a (cos( b) s( b)) o rewre he eqao ξ m m ξ ξ m orer modes of he error: ξ m α 0 c m α e θ α πα θ α παh lm G θ h 0 ( ) Wha does hs ell s? Amplfcao facor

29 Mlgrd mehods lmg θ h 0 ( ) Shor wavelegh (hgh freqecy) errors damps faser Creae grds wh dffere resolos Low freqecy errors o fe grds are hgh(er) freqecy errors o coarser grds (damps faser whe relaed o coarse grds)

30 Mlgrd mehods Eample of a lear problem, he Laplace eqao 0 y,,,,,,, y L O each grd, m, we solve: m m R L,, Procedre for he Correco Sorage (CS) scheme:. O he fes grd, M, do a few relaaos (eraos) of o redce he shor wave legh error modes., 0 M L. Calclae he resdal ad rasfer o he e coarser grd, resrco: M M M M R I R,, Resdal

31 Mlgrd mehods Eample of a lear problem, he Laplace eqao 3. O he coarser grd solve L R m m,, 0 m m m, ˆ,, correco Prevos solo o grd m 4. Repea seps ad 3 l he coarses grd s reached 5. O he coarses grd, solve he problem eacly. 6. rasfer he correco bac o fer grd, prologao,, ad do a few relaaos o each grd l he fes grd s reached m m m m ˆ I, m,

32 Mlgrd mehods Mlgrd cycles mm V-cycle: resrco prologao relaao m

33 Geomerc mlgrd Several grds eplcly geeraed Sable for srcred grds Several ype of cycles: V, W,... 33

34 Algebrac mlgrd Coarser levels bl o-le Ca be sed for srcred meshes Mosly for ellpc problems oo may/coarse levels o eccessarly help 34

35 Dscresao ad grd Qesos: How comple s he geomery? Wha accracy s reqred? Grd qaly? Wha abo sably? Grd refeme?

36 Grd geerao-classfcao Srcred Usrcred Nmber of blocs Herarchy erahedral Polyhedral Moo Ml Caresa Ocree Shape Orhogoaly H C O Orhog oal Bodyfed 36

37 Cell ypes raglar (r) -dmesoal 3-dmesoal erahedro (e) Qadrlaeral (qad) Heahedro (he) pyramd wedge

38 Grd ypes Srcred grd Usrcred grd

39 Grd ypes Srcred grd Ml-bloc Usrcred grd

40 Grd ypes Hybrd grd

41 Grd ypes Hybrd grd

42 Srcred Easy o geerae Ibl opology Low memory foopr as solo algoryhms Easy o se hghorder schemes Dffcl for comple geomeres Usrcred Dffcl o geerae opology has o be sored Lower mesh qaly Slow algoryhms Dffcl for hghorder Sable for comple geomeres 4

43 Mesh qaly measres dy dy d α Srechg dy/dy Bes: Aspec rao dy/d Bes: Sewess α Bes: eq-agle 43

44 Effce grds grd srechg Large grades e resolo 44

45 Effce grds grd srechg ) ( ) ( r r r ) ( ) ( ) ( O r r No secod order ay more! Ca be correced Era wor

46 Grd srechg

47 Grd srechg

48 Effce grds local refeme 48

49 Effce grds adapve refeme Rgh marer parameer P,,, d/d, ec. No oo ofe Comp. me Errors rodced No oo lae ollow he flow feares 49

50 Refeme echqes Easer o mpleme Mesh qaly decreases Neghbors affeced Beer qaly 50

51 Oher grd relaed sses Grd geerao sraeges Movg bodares