Discretization Methods in Fluid Dynamics

Transcription

1 Corse : Fld Mechacs ad Eergy Coverso Dscretzato Methods Fld Dyamcs Mayak Behl B-tech. 3 rd Year Departmet of Chemcal Egeerg Ida Isttte of Techology Delh Spervsor: Dr. G.Bswas Ida Isttte of Techology Kapr 3/8/007 Ido-Germa Wter Academy

2 OUTLINE. Bascs of PDE. Mathematcal Overvew of fld flow 3. Need For Comptatoal Methods 4. Basc Dscretzato Methods se 5. Fte Dfferece Method 6. Fld Flow Modelg 3/8/007 Ido-Germa Wter Academy

3 Partal dfferetal eqatos (PDEs) are sed to formlate ad solve problems related to ay pheomea that s dstrbted space ad tme : Heat flow φ φ φ φ φ A B C D E Fφ = G(, y Fld flow y y y propagato of sod electrodyamcs, Elastcty. Most of the fld ad related trasport pheomea ca be modeled sg PDEs ) 3/8/007 Ido-Germa Wter Academy 3

4 Classfcato of PDEs φ y A B C φ φ y D φ E φ y Fφ = G(, y) If A,B,C,D all costats or f(,y) If ay of A,B,C,D cotas Φ, Φ etc Lear PDE No-lear PDE B 4AC<0 B 4AC=0 Ellptc eqato B.V.P. Parabolc eqato I.B.V.P; φ φ y Irrotatoal flows steady Heat Trasfer Usteady vscos flows B 4AC>0 Hyperbolc eqato Vbrato problems = φ φ = α t φ = β φ t 0 3/8/007 Ido-Germa Wter Academy 4

5 Bodary codtos Drchlet Bodary codto: Nema Bodary codto: φ = φ ( r) Gve bodary temperatre φ = φ ( r ) Gve bodary heat fl Med Bodary codto: φ aφ b = φ3( r ) Bodary heat fl depedet o bodary temperatre 3/8/007 Ido-Germa Wter Academy 5

6 Mathematcal descrpto of flows; Goverg Eqatos of Fld Dyamcs Eqato of cotty: a dfferetal mass balace ρ ρv y ρw z = ρ t z I vector otato: [ ρ ] = ρ t y (,y,z) For costat desty of the fld: = 0 3/8/007 Ido-Germa Wter Academy 6

7 Goverg Eqatos of Fld Dyamcs () Eqato of moto: t ρ ρ = p τ ρg for compressble, Newtoa Flds, der lamar flow: ρ = p μ t ρg Naver-Stokes Eqato 3/8/007 Ido-Germa Wter Academy 7

8 Ido-Germa Wter Academy 8 3/8/007 The Naver-Stokes eqatos (No lear PDEs) g z y p z w y v t = ρ μ ρ g y z v y v v y p z v w y v v v t v = ρ μ ρ g z z w y w w z p z w w y w v w t w = ρ μ ρ I geeral, Aalytcal Solto Not Possble Solto???

9 Approaches to Fld Dyamcal Problems. Smplfcatos of the goverg eqatos AFD: Aalytcal Fld Dyamc. Epermets o scale models EFD: Epermetal Fld Dyamcs 3. Dscretze goverg eqatos ad solve by compters CFD Comptatoal Fld Dyamcs CFD s the smlato of flds egeerg system sg modelg ad mercal methods 3/8/007 Ido-Germa Wter Academy 9

10 Need for Dscretzato methods I geeral, o aalytcal solto est for o-lear models Oe of the way ot s to appromate the soltos sg Nmercal methods Advet of dgtal compter ad advaced mprovemets compter resorces Bass for CFD, whch ca provde lmted detals of reslts Sbstatally more cost effectve ad more rapd tha EFD ablty to stdy systems der hazardos codtos 3/8/007 Ido-Germa Wter Academy 0

11 Varos Dscretzato Techqes Fte Dfferece Method (focsed ths lectre) Fte Volme Method Fte Elemet Method 3/8/007 Ido-Germa Wter Academy

12 Dscretzato Methods Fte Dfferece Method (focsed ths lectre).goverg eqatos dfferetal form doma wth grd replacg the partal dervatves by appromatos terms of ode vales of the fctos oe algebrac eqato per grd ode lear algebrac eqato system.. Appled to strctred grds Fte Volme Method. Goverg eqatos tegral form solto doma s sbdvded to a fte mber of cotgos cotrol volmes coservato eqato appled to each CV.. Comptatoal ode locates at the cetrod of each CV. 3. Appled to ay type of grds, especally comple geometres Fte Elemet Method. Solto doma s sbdvded to a fte mber OF ELEMENTS, the goverg eqato s solved for each elemet ad the overall solto s obtaed by assembly.. Eqatos are mltpled by a weght fcto before tegrated over the etre doma. 3/8/007 Ido-Germa Wter Academy

13 Bascs aspects of Dscretzato methods Cosstecy A dscretzato method s sad to be cosstet f t ca be show that the dfferece betwee PDEs ad ts fte dfferece (FDE) vashes as the mesh s refed. Trcato error (TE): Dfferece betwee the dscretzed eqato ad the eact oe O( ); O( t) Lm mesh 0 (TE) 0 O( t/ ) Covergece: solto of the dscretzed eqatos teds to the eact solto of the dfferetal eqato as the grd spacg teds to zero. 3/8/007 Ido-Germa Wter Academy 3

14 Bascs aspects of Dscretzato methods Stablty : Dfferecg Method shold ot magfy the errors that appear the corse off mercal solto process ε ε Coservato:. The mercal scheme shold o both local ad global bass respect the coservato laws.. Atomatcally satsfed for cotrol volme method, ether dvdal cotrol volme or the whole doma. 3. Errors de to o-coservato are most cases apprecable oly o relatvely coarse grds, bt hard to estmate qattatvely 3/8/007 Ido-Germa Wter Academy 4

15 Fte Dfferece Method Replacg the dervatves of goverg PDE wth fte, algebrac dffereces qotets. It volves followg steps: Grd geerato defg geometrc doma Dscretzato of goverg eqato sg Taylor seres appromato. Solto of smltaeos algebrac eqatos. φ φ -,j φ -,j φ,j φ,j y Dscrete grd pots 3/8/007 Ido-Germa Wter Academy 5

16 Grds Fte Dfferece Method:grds Strctred grd all odes have the same mber of elemets arod t oly for smple domas Ustrctred grd for all geometres rreglar data strctre 3/8/007 Ido-Germa Wter Academy 6

17 Ido-Germa Wter Academy 7 3/8/007 Fte Dfferece, appromato of the frst dervatve Taylor Seres Epaso: Ay cotos dfferetable fcto, the vcty of, ca be epressed as a Taylor seres: ( ) ( ) ( ) ( ) ( ) ( ) H Φ Φ Φ Φ = Φ Φ!... 3!! ( ) H Φ Φ Φ Φ Φ = Hgher order dervatves are kow ad ca be dropped whe the dstace betwee grd pots s small X -X s small

18 Ido-Germa Wter Academy 8 3/8/007 By wrtg Taylor seres at dfferet odes, -,, or both - ad, we ca have: O( ) Φ Φ Φ Backward Dfferece Scheme-BDS O( ) Φ Φ Φ Forward Dfferece Scheme-FDS order of accracy O( ) Φ Φ Φ Cetral Dfferece Scheme-CDS O( )²

19 Ido-Germa Wter Academy 9 3/8/007 Fte Dfferece, appromato of the secod dervatve Geometrcally, the secod dervatve s the slope of the le taget to the crve represetg the frst dervatve. Φ Φ Φ Estmate the oter dervatve by FDS, ad estmate the er dervatves sg BDS, we get For eqdstat spacg of the pots: ( ) ( ) ( ) ( ) ( ) Φ Φ Φ Φ ( ) Φ Φ Φ Φ Hgher-order appromatos for the secod dervatve ca be derved by cldg more data pots, sch as -, ad, eve -3, ad 3

20 Dscretzato sg Eplct Method Cosder steady vscos flow Φ t 3/8/007 a gve tme terval Φ Φ = α t Φ Φ = α Φ ( ) The Oly kow :Φ Eplct method, vales at tme compted from vales at tme Advatages: solto algorthm s smple.e. - drect comptato wthot solvg system of algebrac eqato - few mber of operatos per tme step Dsadvatage: strog codtos o tme step for stablty Reqres may tme steps to carry ot the calclatos over Φ Ido-Germa Wter Academy 0

21 Ido-Germa Wter Academy 3/8/007 Implct Method Crak-Ncholso Method ( ) t = α t = α vales at tme compted sg the kow vales at tme Secod order accrate tme ad space to be solved smltaeosly at all the grd pots as a system of algebrac eqatos Ucodtoally stable rearragg the above eqato as follows. O( t),o( )

22 3/8/007 Ido-Germa Wter Academy

23 A = C Ths geerates a Trdagoal Matr system whch ca be solved sg Thomas algorthm. However, for hgher dmesoal flows, matr system obtaed are mch more comple ad reqre sbstatally more compter tme 3/8/007 Ido-Germa Wter Academy 3

24 Ido-Germa Wter Academy 4 3/8/007 Alteratg Drecto Implct Method (ADI) for -D Flows ( ) = /, /, /,,,,, /, / y t j j j j j j j j α = y t α Each tme cremet s eected to two steps: Frst step ( ) = /, /, /,,,, /,, / y t j j j j j j j j α Secod step

25 Frst drecto Y Secod drecto X ADI method reslts Trdagoal Eqatos (for -d flow systems) f t s appled alog the dmeso that s mplct. Ths o frst step t s appled alog Y as ad o secod step alog o X as 3/8/007 Ido-Germa Wter Academy 5

26 Advatages: Implct method Stablty ca be mataed over larger tme steps Dsadvatages More comptato tme per tme step, every tme step reqres solvg a system of eqatos. Trcato error s ofte large, sce larger tme steps are employed. Overall, a more volved procedre 3/8/007 Ido-Germa Wter Academy 6

27 Error ad Stablty aalyss Nmercal solto s fleced by followg: Dscretzato errors or Trcato errors: Aalytcal Solto (A) eact Fte Dfferece Solto (D) = A D Comptatoal Rod-Off Error (e) ε=nmercal Solto (N) eact Fte Dfferece Solto (D) N = D ε By defto, N,D follow the Dscretzed Eqato Ths, ε shold also follow the eqato: ε ε t ε = α ε ( ) ε 3/8/007 Ido-Germa Wter Academy 7

28 Errors ad Stablty Aalyss (Cot.) Sbstttg the vale of ε m (,t) the F.D.E., 3/8/007 Ido-Germa Wter Academy 8

29 Stablty: Dfferecg Method shold ot magfy the errors that appear the corse off mercal solto process For stablty: ε ε Vo Nema Stablty Aalyss : assmg error dstrbto follows Forer seres space doma ad epoetal tme ε (, t) = ε at m ε k m Pttg a dvdal term FD eqato ad sg the above stablty crtera α( t) ( ) 3/8/007 Ido-Germa Wter Academy 9

30 Fld Flow Modelg Usg Fte Dfferece Method Nolear Fld Flow eqatos volve: tme depedet mltple varables Covectve terms Dffsve terms Sample fld flow eqato : Brger s Eqato t = ν Ivscd Brger s Eqato t = 0 3/8/007 Ido-Germa Wter Academy 30

31 Propertes of Fld Flow Eqatos Effect of Fte Dfferece method o propertes of fld flow eqatos Coservatve Property: To mata the tegral coservato relato of cotm fte dfferece represetato. It depeds o the form of cotm eqato beg dscretzed. Coservatve form: = t ( ) No-coservatve form: = t 3/8/007 Ido-Germa Wter Academy 3

32 Ido-Germa Wter Academy 3 3/8/007 ) ( = = = = I I I I I I t / / ) ( ) ( = I I t = ) ( ) ( α Itegratg the CONSERVATIVE FORM I I

33 Ido-Germa Wter Academy 33 3/8/007 Week Coservatve Form Fld Flow Coservatve form of advectve part s of prme mportace for fld flow modelg. Naver Stokes week coservatve form: g z y p z w y v t = ρ μ ρ g y z v y v v y p z vw y v v t v = ρ μ ρ g z z w y w w z p z w y vw w t w = ρ μ ρ

34 Ido-Germa Wter Academy 34 3/8/007 Dscretzg Fld Flow eqatos sg Upwd Scheme Upwd Scheme of dscretzato s ecessary for covecto domated flows t = t = t = Backward Dfferecg Scheme for >0 Forward Dfferecg Scheme for <0

35 Why se Upwd scheme Upwd Scheme of dscretzato s ecessary for covecto domated flows for obtag mercally stable reslts Cetral Dfferece Scheme appled to hghly covectve flows gve codtoally Ustable Reslts CDS: FDS: t t = = 0, 0 > 0 Volates Vo Nema Stablty Crtera 3/8/007 Ido-Germa Wter Academy 35

36 Ido-Germa Wter Academy 36 3/8/007 Why se Upwd scheme t = ν CDS: t t ν = 0 t ν 0 ν ν t = ν Re 0 * Re ν C C Re CELL Reyold s No. For steady, covecto-dffso eq.

37 Upwd Scheme ad Trasportve Property = t sg pwd scheme ( > 0) let a pertrbat o δ occrs at m The for at a dowstream locato ( ) t locato (m ) = m m 0 δ δ = = At (m) locato t whch follows the ratoale for the trasport ve proprerty At (m) locato m m δ 0 δ = = t pertrbat o beg trasport ed ot of At (m -) locato m m 0 0 = = 0 t o pertrbato carred pstream th the rego At (m) locato At (m-) locato 3/8/007 Ido-Germa Wter Academy 37

38 Problems of Usg Forward Tme Cetral Space method (FTCS) At (m -) locato t δ 0 δ m m = = 0 Whch dcates that Trasportve Property s volated Hece, oly pwd method matas drectoal flow of formato. 3/8/007 Ido-Germa Wter Academy 38

39 Ido-Germa Wter Academy 39 3/8/007 Upwd Method ad Artfcal Vscosty(/) t = t = = t ν

40 Upwd Method ad Artfcal Vscosty (/) < For algorthm s stablty, artfcal vscosty wll ecessarly be preset Sorce of accracy v e > 0 3/8/007 Ido-Germa Wter Academy 40

41 Upwd Dfferecg Method Advatages ad Lmtatos Upwd effect ecessary to mata trasportve property of flow eqatos Althogh space cetered dffereces (CDS) are more accrate tha pwd scheme (as dcated by Taylor Seres epaso), the whole system s more realstcally modeled sg Upwd Dfferecg Scheme. Leads to Artfcal vscosty or false dffso whch s a case of accracy. v e > 0 Plasble Improvemet: Hgher order Upwd Method 3/8/007 Ido-Germa Wter Academy 4

42 Secod Upwd Dfferecg Scheme Let s cosder where 3/8/007 Ido-Germa Wter Academy 4

43 Cosderg mometm eqato η Here a factor s trodced whch epresses a weghted average of cetral ad pwd dfferecg 3/8/007 Ido-Germa Wter Academy 43

44 Here 0<η< For η=0, above eqato becomes cetered space For η=, above eqato follows fll pwd Hece, η brgs abot a pwd bas the dfferece qotet Accracy of the dfferecg scheme ca be adjsted sg η vale Secod pwd formlato possesses both the coservatve ad trasportve property 3/8/007 Ido-Germa Wter Academy 44

45 Smmary Fte Dfferece methods dscretzes the dfferetal form of goverg eqato sg Taylor Seres epaso. The partal dervatves of goverg PDE are replaced wth fte, algebrac dffereces qotets at the correspodg odes. Leads to lear algebrac eqato system, wth oe algebrac eqato per grd ode. Best Sted for Strctred Grds oly. Flow eqato propertes ca be mataed sg smple schemes lke pwd dfferecg. Dscretzato methods provde more cost effectve ad more rapd approach for solvg Fld Flow problems as compared to epermetal methods 3/8/007 Ido-Germa Wter Academy 45

46 3/8/007 Smmary Fte Dfferece Approach FD Approaches Ca Be Dvded Ito Varos Categores Depedg Upo Type of PDE Beg Solved:.Ellptc PDE: Algebrac System, Gass Sedel.Oe-dmesoal Parabolc PDE A) Eplct B) Implct 3. Two-dmesoal Parabolc PDE ADI Method Ido-Germa Wter Academy 46

47 Ackowledgemet Ths lectre has bee spred by the stdy materal provded by Prof. G.Bswas 3/8/007 Ido-Germa Wter Academy 47

48 Refereces Chapra ad Caale, Nmercal Methods for Egeers,4 th Ed., Tata McGraw-Hll. Som ad Bswas, Itrodcto to Fld Mechacs ad Fld Maches, Tata McGraw-Hll Aderso.J; Comptatoal Fld Dyamcs 3/8/007 Ido-Germa Wter Academy 48

49 Thak yo! 3/8/007 Ido-Germa Wter Academy 49