Computational Fluid Dynamics. Computational Methods for Domains with Complex Boundaries

Transcription

1 h:// h:// Comuaonal Flud Dnamcs Lecure 6 March, 7 Comuaonal Mehods or Domans wh Comle Boundares Gréar Trggvason Oulne How o deal wh rregular domans Overvew o varous sraeges Boundar-ed coordnaes revsed Grd generaon or bod-ed coordnaes Caresan Adave Mesh Renemen AMR Unsrucured grds For mos engneerng roblems s necessar o deal wh comle geomeres, conssng o arbrarl curved and orened boundares Grd Generaon Overvew Varous Sraeges or Comle Geomeres and o concenrae grd ons n secc regons Orgnall, recangular sarcasng on recangular Caresan grds was used o reresen comle boundares Ths was ollowed b bod ed grds, he grds are sll srucured bu grd lnes are no sragh. In commercal codes, unsrucured grds have now mosl relaced bod ed grds From: Regular srucured grds are sll o neres, arcularl when couled wh Adave Mesh Renemen AMR and mmersed boundares

2 Sarcasng Unsrucured Grds Unsrucured versus srucured grds Aromae a curved boundar b a he neares grd lnes srucured grds: an ordered laou o grd ons. unsrucured grds: an arbrar laou o grd ons. Inormaon abou he laou mus be rovded Overvew h:// Fne Derence mehods on bod ed grds are generall derved b mang he equaons Fne Volume mehods are generall derved b usng arbrarl shaed conrol volumes. Boundar-Fed Coordnaes or Comle Domans Boundar-Fed Coordnaes In man raccal alcaons s necessar o deal wh comle domans. For relavel smle domans, he mehods ha we jus develoed or recangular domans and grds can be eended o non-recangular domans hrough coordnae mang. Ths was he aroach aken n earl commercal codes and s sll wdel used n aerodnamcs and urbomachner comuaons Boundar-Fed Coordnaes Coordnae mang: ransorm he doman no a smler usuall recangular doman. Boundares are algned wh a consan coordnae lne, hus smlng he reamen o boundar condons The mahemacal equaons become more comlcaed

3 Boundar-Fed Coordnaes,,,,,, Frs consder he D case: d d d d d d d d d d Boundar-Fed Coordnaes For he rs dervave he change o varables s sraghorward usng he chan rule: For he second dervave he dervaon becomes consderabl more comle: Boundar-Fed Coordnaes The second dervave s gven b d d d d d d d d d d d d d d d d d d d d Where we have used he eresson or he s dervave or he nal se. However, snce he equaons wll be dscrezed n he new grd ssem, s moran o end u wh erms lke /, no /. d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d Boundar-Fed Coordnaes To do so, we look a he second dervave n he new ssem d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d Solvng or he orgnal dervave whch s he one we need o ransorm we ge: d d d d d d d d d d d d d d d d d d B he chan rule we have d d d d d d Boundar-Fed Coordnaes Oen we need he dervaves o he ransormaon sel: d d d d For he second dervave we derenae he above: d d d d d d d d d d d d d d d d d d d d d d d d Gvng: Boundar-Fed Coordnaes D: Frs Dervaves Change o varables,,,,, The equaons wll be dscrezed n he new grd ssem,. Thereore, s moran o end u wh erms lke /, no /. d d d d d d d d d d d d d d d d d d

4 Usng he chan rule, as we dd or he D case: We wan o derve eressons or n he maed coordnae ssem. /, / Boundar-Fed Coordnaes Solvng or he dervaves Subracng Boundar-Fed Coordnaes,, Solvng or he orgnal dervaves elds: where s he acoban. Boundar-Fed Coordnaes A shor-hand noaon: Boundar-Fed Coordnaes Rewrng n shor-hand noaon where s he acoban. Boundar-Fed Coordnaes These relaons can also be wren n conservave orm: Boundar-Fed Coordnaes Snce: And smlarl or he oher equaon

5 Smlarl Boundar-Fed Coordnaes D: Second Dervaves The second dervaves s ound b reeaed alcaon o he rules or he rs dervave Boundar-Fed Coordnaes Addng and elds an eresson or he Lalacan: Boundar-Fed Coordnaes where Boundar-Fed Coordnaes q q q q q q q q q q q Boundar-Fed Coordnaes Eandng he dervaves elds q q q where [ q q ] [ q q ] Dervaon o hence smlarl Boundar-Fed Coordnaes

6 Boundar-Fed Coordnaes Pung hem ogeher, can be shown ha rove [ q q ] q [ q q ] q Boundar-Fed Coordnaes We also have, or an uncon and g g g g g Boundar-Fed Coordnaes Summar: A comle doman can be maed no a recangular doman where all grd lnes are sragh. The equaons mus, however, be rewren n he new doman.,,,, Thus: And more comle eressons or he hgher dervaves Vorc-Sream Funcon Formulaon Vorc-Sream Funcon Formulaon Vorc-Sream Funcon Formulaon The Naver-Sokes equaons n vorc orm are: ω ψ ω ψ ω ν ω ψ ω Use he ransormaon relaons obaned earler o wre he equaons n he new varables The Naver-Sokes equaons n vorc orm become: ω ψ ω ψ ω ν q ω q ω q ω ω ω q ψ q ψ q ψ ψ ψ ω q q q

7 Boundar Condons Inlow Vorc-Sream Funcon Formulaon Δ Δ, M : No sl ψ Q ψ Q ud vd Oulow dψ ψ ψ For & N sar b usng erodc boundares Lower wall Sream uncon: Vorc-Sream Funcon Formulaon ψ Vorc: no-sl ψ, ψ, ψ, ψ, HOT Usng ha ω, ψ, We have: ω, [ ψ, ψ, ] Vorc-Sream Funcon Formulaon Uer wall M Sream uncon: ψ Q Vorc: L M Q ud vd dψ ψ M ψ ω,m [ ψ,m ψ,m ] Grd Generaon b Blnear Inerolaon Blnear Inerolaon Smles grd generaon s o break he doman no blocks and use blnear nerolaon whn each block Blnear Inerolaon Consder an arbrar shaed quadrlaeral block. Selec he and drecon. As an eamle, we wll wre a smle code o grd he doman o he rgh 8, 8, 7, 7, 4, 4 5, 5, 6, 6. Dvde he, oose sdes M evenl wh N ons n he, drecon and M n he drecon and draw sraghs lne beween he ons on he oose sdes N,,

8 Blnear Inerolaon Along he edge beween ons and, N N N Along he edge beween ons and,m N N N Then nerolae agan or ons beween he edges, M N M N N N M N N The -coordnae s ound n he same wa Blnear Inerolaon For a sngle block, we hereore have:, M N M N N N M N N, M N M N N N M N N M,,, N, Blnear Inerolaon The grd Blnear Inerolaon Somemes he grd can be mroved b smoohng. The smles smoohng s o relace he coordnae o each grd on b he average o he coordnaes around. Ths rocess can be reeaed several mes o mrove he smoohness., j.5*, j, j, j, j, j.5*, j, j, j, j The eec o wo smoohng eraons Inlow and Oulow Boundar Condons Vorc-Sream Funcon Formulaon Inlow and oulow Boundar Condons Inlow Δ Δ : Inlow N : Oulow, M : No sl ψ Q ψ Q ud vd Oulow dψ ψ ψ

9 Inle low Vorc-Sream Funcon Formulaon Consderng a ull-develoed arabolc role u, C L L L 6Q L d C C Q C 6 6Q u u, L ω L 6Q Q L d Q L L L L Inle low Vorc-Sream Funcon Formulaon Consderng a ull-develoed arabolc role and assume ha L M 6Q u, L M M ψ, Q M M Q ω L M, 6 Oulow Vorc-Sream Funcon Formulaon N Tcall, assumng sragh sreamlnes ψ n I s normal o he oulow boundar, hs elds ψ I no, hen a roer ransormaon s needed or ψ n Resuls % Se u he grd ab.a.7 w. or :N, sds*-end zerosn,nzerosn,n or :N, bs.5*ab*sn**s*. bab*cos4**s*. s.*a*sn**s wa*cos**s*. end or :N, or j:n,jbj-*-b/n-,jbj-*-b/n- end,end Smle grd generaon hold on or j:n, lo:n,j,:n,jend or :N, lo,:n,,:nend lo,,'k','lnewdh',, lob,b, 'k','lnewdh', segca,'bo','on' segca,'fonsze',8, 'LneWdh', segca,'xtcklabel','' as equal, as[ -..], ause The res o he code s smlar o he vorc sreamuncon code, ece longer. The grd mercs are re-comued Usng he sreamuncon vorc ormulaon o he Naver Sokes equaons makes relavel sragh orward o develo a code or comle boundares. I he boundar s known analcall and s ossble o connec wo boundares b sragh lnes, hen he oher grd lnes can be generaed b smle nerolaon. Bod ed grds Vorc Vorc Sreamuncon

10 Veloc-Pressure Formulaon The Naver-Sokes equaons n rmve orm Veloc-Pressure Formulaon u uu vu ρ ν u u v uv vv ρ ν v v and connu equaon u v Advecon Terms Veloc-Pressure Formulaon uu vu [ uu uu uv uv ] [ uu uu uu uv uv uv u u v uu uv uu uv ] { } u v u { } Conravaran Veloc V C v, u, C U Veloc-Pressure Formulaon U u v V v u, n Un normal vecor, Un angen vecor along C U u, v, u v Thereore, U s n he drecon. V s n he drecon. Veloc-Pressure Formulaon Pressure Term Duson Term u u u u u u u u u u u u u Veloc-Pressure Formulaon u u u u u u u u u u q u q u q u q u q q q

11 Veloc-Pressure Formulaon u-momenum Equaon U u v V v u u Uu Vu ρ ν q u q u v-momenum Equaon q u q u v Uv Vv ρ ν q v q v q v q v Veloc-Pressure Formulaon Connu Equaon u v Usng, Connu equaon becomes [ u u v v ] or u v v u Veloc-Pressure Formulaon The soluon algorhm s he same as or regular grds. Frs we nd he redced veloces u * u n & Uu Δ Vu ' * ν - / q u q u. v * v n & Uv Δ Vv ' * ν - / q v q v. Then we correc he veloc b addng he ressure u n u * ' Δ ρ v n v * ' Δ ρ q u q u *, *, u n v n v n u n q v q v Whch s und b subsung he correcons no he ncomressbl condons Veloc-Pressure Formulaon When we work drecl wh he Caresan veloc comonens, a colocaed grd s generall used, snce he veloces are no longer erendcular o he boundar. Veloc-Pressure Formulaon In he lane, a saggered grd ssem can be used he conravaran veloces are used U u v The connu equaon s V v u u v v u Whch can also be wren as U V Usng hs ormulaon, he same MAC grd and rojecon mehod can be used. V C C U Veloc-Pressure Formulaon The momenum equaons can be rearranged o u v q q U-Momenum Equaon q U Uu Vu Uv Vv q q ν q u q u q v q v q u q u q v q v

12 Veloc-Pressure Formulaon q V-Momenum Equaon Uu Vu Uv Vv V ν q q Sl as beore: q * n U /, U /, j j q V,*j/ V,nj/ qv q v q v q v v V U n, j/ wrng ou V.. THE NAVIER-STOKES EQUATIONS PRIMITIVE FORM.. THE NAVIER-STOKES EQUATIONS IN PRIMITIVEINFORM n U/,j U/,j q @ /,j n /,j U n U/,j U/,j @ 8 /,j n V,j/ V,j/ Dnamcs q 9 Comuaonal @,j/ n,j/ V n V,j/ @ 9 qwe need@ or9 wrng ou he ressure o accoun he ac ha we do n we or he,j/ wrng ou he ressure dervaves ac q ha do Incomressbl we Van o,j/ no have heu ressures /, j and, j / so hose need o be nerno have he ressures a /, jand, j / so hose need o be nerolaed. Thus@/@,j,j q,j /4 and,j /,j,j he ac,j ha we do olaed. /4 ou he ressure dervaves weneed orand,j o accoun wrng ou he wrng ressure dervaves we need accoun he ac ha /,j,j o,j or,j /4 we,j,j/ need o be ner@/@,j ha ressures j,jand /4 Δ Dscreze, usng no have he /,, j / so hose,j Δa,j,j/ q no have he ressures a /, j and, j / so hose need o be ner /4 and /,j nthus n n,j olaed. n,j /4 and /,j U n U /, V V,j,j U/,j U,j n /, j,j/ j j/, /4 /,j,j,j U,j/ /,j,j,j /,j,j /4 /,j,j,j /,j,j,j /,j Subsue /,j n U/,j n U/,j U/,j U/,j /,j /,j,j,j,j,j,j /4,j,j /,j /,j,j,j,j,j /4 /,j,j /,j,j,j /4 /,j,j,j /4,j n V,j/ %% q " " /, j subsung he correcon equaon8 no /,j connu n # # q V,j/ V,j/ * he, j/ 9 U n V n dervaves $ we needo$accoun or he ac ha we do q # q,j/ & n U/,j U/,j /,j /,j,j /,j,j,j n V,j/,j,j,j /4 V,j/ Comuaonal Flud,j/,jDnamcs,j,j/,j/,j,j,j,j /4 Resulng n a ressure equaon ha can be solved n sandardgves: was Subsung /,j /,j /,j,j /,j,j,j/,j,j/,j /,j /,j,j,j,j,j /,j,j,j,j,j /,j,j,j,j,j,j,j U /,j Subsung gves:,j /,j,j,j,j,j /4 Subsung gves:,j /,j,j /,j,j,j /4 /,j,j,j,j /,j /,j /,j,j,j /,j,j,j /4,j,j,j,j,j,j /4,j /,j /,j /,j,j,j,j /4,j,j /,j,j,j,j,j,j /4,j,j,j /,j /,j /,j /,j /,j /,j /4 /4,j,j,j,j,j,j,j/,j/,j,j,j,j,j,j /,j /,j Formulaon,j/,j,j,j/,j,j,j,j/4 /4 Oulne,j/,j Veloc-Pressure,j,j,j,j,j /,j /,j /,j,j/ /,j,j/,j,j,j/,j,j,j,j/4 /4 How o deal wh rregular domans,j,j,j,j,j /,j /,j,j,j,j/ /4,j,j/,j,j,j,j,j,j /4 /,j,j,j,j,j,j /,j /,j/,j,j/ / Overvew o varous sraeges,j /,j U,j U V,j/,j V/4,j / The resulng ressure U/,j U /,j/,j V,j/ V,j,j,j,j /,j,j,j,j,j/,j /4,j /,j,j / /,j,j / V Boundar-ed coordnaes revsed,j / equaon s smlar as or U Grd generaon or bod-ed coordnaes regular grds and can be U/,j U /,j V,j/ U/,j U /,j V,j/ V,j / V,j / solved n smlar was. Caresan Adave Mesh Renemen AMR Once he conravaran veloces are ound, he veloces n he orgnal coordnaes can be ound and loed as beore Unsrucured grds C C /4,j /4,j /4,j /4 V,j/,j /4 /,j,j,j/,j,j U/,j,j/ n V,j/ n V,j/ V,j/ V,j/,j/,j,j,j/,j,j,j/4,j,j/,j,j,j/,j,j /4,j/,j,j,j/,j,j,j/ gves: Subsung,j/,j,j,j Subsung gves: Derve ressure /, j # /, no have he ressures a j and, j & / so hose need o be nerδ " ",j %,j %, j/,j,j /4 and olaed. /4,j V,j/,j,j/,j,j/,j/,j Δ.. THE NAVIER-STOKES EQUATIONS IN PRIMITIVE FORM.. THE NAVIER-STOKES EQUATIONS IN PRIMITIVE FORM /,j $ # U A D " /,j %, j/ n * U /, U /, j j where n V,j/ n U/,j Δ u q u u qu u U V $ # A D &.. THE Δ NAVIER-STOKES equaon b " EQUATIONS % IN PRIMITIVE FORM V,j /