Accurate Model Reduction of Transient Flows
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Accurate Model Reduction of Transient Flows Bernardo Galletti — Alessandro Bottaro — Charles-Henri Bruneau — Angelo Iollo. N° 5676 Septembre 2005. N 0249-6399. ISRN INRIA/RR--5676--FR+ENG. Thème NUM. apport de recherche.
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(41) . lkg[&nwhpWY[ &u@kfhpn|~okFhjf"Dz ©VNWY[}Ynj[&±Pou1lfN}Ynju@Y[&Z ofw[GqFlYo±H|@[GkFhhpu $ kYokY\hjWY[ lYk6 u@k6fihjnj|@okY[3 [&µPhpnj[&Z'lZ,u~hpWY[ lYk6Uhpou@k6|~ 1 J =. Z. Nr TX. 2. (ar (t) − a ˆr (t)) dt +. Z. T. bk [a˙ k (t) − Ak − Clk al (t) + Blsk al (t)as (t)]dt.. ¥W[&nj[ b of¬hjWY[/|@}Y}Ynju@}Ynjor|hp[ %|@@n|~kY1[ Z'lYhpo}Yo[GnG© V%uhpWYorf¬[&k<hpW[/±|@kYofpWYokYu@hjWY[ npm3WY[ h Y[&njo±|hjo±1[GfNu@ ¥ohpW¹np[3f}I[G hhpu¤|@ºu@ ohjf+|~nj@lYZ\[&kFhfNZ'lfhw6['oZ\}Iu1fp[Gª© VNWofN[G|1Yfhpu\hjJWY[ (au@(t), u¥bokY(t),/u@A}ghjo,ZC|@ohi)e¤}Ynju@Y[&Z ( a˙ (t) = A + C a (t) − B a (t)a (t) gonj[G h}Ynpu1Y[&Z 0 r=1. 0. k. r. r. r. r. r. kr k. kr. ksr k. s. ar (0) = (u(x, 0), φr ). (. −b˙ r (t) = [Crk − (Blrk + Brlk )al (t)] bk (t) − 2 [ar (t) − a ˆr (t)] br (T ) = 0. + . |@Hviu1okFh}Ynju@Y[&Z. 1u }ghpoZ|~o¯hie&u@kYo¯hjou1kf D a (t)b (t)dt = 0 ¥W[&nj[`|~Ru~ªhjWY[flY6fp&npo}ghjf@u'npu1Z 1 hju N ©5VNWY[Gfp[`[3qFl|hjou1kf|@np[¬gorfp&np[&hpo5'&[G¤¥o¯hjW| }f[Glgu fp}I[GUhjnj|@F&u@ugG|hpou@k'Z/[&hpWYug |~u@kY+hpWY[ t |~µPorfG©VNWY[lYk hpou@kf a (t), b (t) |~k aˆ (t) |~nj[fj|~Z\}Y[G |~h¤hpW[ N / |@lfpf² ºu@6|hhju·}6u1okFhf t = T /2 (1 − ξ ) ¥ohpW ξ = cos π(i − 1)/(N − 1) |~k ÂGhjW|hºorf |@k © ¸wkok1hj[&nj}6u1|~hpou@k orfº}I[&npu@njZ/[3 i = 1, . . . , N hju\np[&hpnjo[G±@[`hpWY[±|~lY[3fau~=hpWYa[ (tlYk6),Uhpbou@k6=fw|Hb¥(t|He/)nju@Z aˆhpWY[ =kYugaˆY|@(tº)}6u1okFhf t ÂYZ\u1np[¬}Ynj[G orfp[&e (R T 0. RT 0. br (t)dt = 0 k. r. r. r. t. i. t. ir. r. i. ir. i. r. i. ir. r. r. i. r. t. i. i. 2 ψj 1 − t ajr T j=1 N t X 2 br (t) ≈ ψj 1 − t bjr T j=1 N t X 2 ˆjr a ˆr (t) ≈ ψj 1 − t a T j=1 ar (t) ≈. Nt X. ¥W[&nj[ |~k6 |@np[hjWY[ %|~1nj|@kY@or|~kok1hj[&nj}6u1|~hpokY}Iu@ePkYu@Z\or|~rfw|1f[3u@k¡hpWY[ k uPY[Gf ξξ ©5=VN1WY[+−hp2oZ/t/T[¬g[&njo±H|~hpψo±@(ξ) [3f5u~ªhjWY[ $nfihhi¥uokFhp[Gnp}Iu@r|hj[G/lYk hpou@kf|hhpWY[`kYugY|~6±|~lY[Gf|~nj[ hjWY[&k j. i.
(42) Nt Nt X 2X dψj
(43)
(44) a = Dij ajr , jr T j=1 dξ
(45) ξi j=1
(46) Nt Nt X 2X dψj
(47)
(48) b = Dij bjr . b˙ r (ti ) ≈ − jr T j=1 dξ
(49) ξi j=1. a˙ r (ti ) ≈ −. ííÑ >åUã?ã. à .
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(51) : 9
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(53) 2 dψj
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(77) hpWY[ ok&u@Z\}Ynj[GfjfoY[¬]`|H±Fo[&np²icFhpu1´@[Gf[3q1l6|hpou@k6f&Âgok[G|~njo5'&[G|~Iu@lYhhjWY['fhp[3|@ge¤lkfih|~Y[fu1lYhpou@k U Âof l6f[3ª©5V%u hpWYorf[&k[ h L Y[&kYu@hp[`hpWY[ okY[3|~nj(o '3|hjou1k¤u~ºhjWY[ ]+|H±Po[Gn²icFhju@´@[3fgorfj nj[ hpo5'G|~hpou@k¤u1}6[Gnj|~hpu1n ok¡fp}|@&[@ÂYu1k [ hjWY[ok6 u@Z\}Ynj[GfjfpoYoo¯hie u@k6fihjnj|@okFhof`fp|~hpor=f $6[Gª©VNWY[np[3fl¯hjokY¤fp[&Z\orgorfp&np[&hp5o 'G|hjou1k ±|@orokhpWY[±Por okYohieu~ U &|@kI[¥njo¯hphp[&k|1f i3 ψ = Lψ ¥W[&nj[ ψ(t) = U(t) − U of | u@lYZ\k·±@[3Uhju@n ¥WYorWT u1kFhj|~okf hjWY[¤1npor#±H|@l[Gf u@NhpWY[¤±1[&ug ohie &u@Z\}Iu@kY[Gk1hf|~njnj|@kY@[3¥ohpWhpW[¤fp|@Z/[\njlY[Gf|@f¬hjWY[&e#|~nj[\u@nj1|~k(o 'G[G¡ok#hpWY[¤fpk|@}fWu~hjf¬±1[GUhju@nf&© VNW[ goZ\[Gkfou@k6fNu~ |@k |~nj[ |~k × 2M nj[Gfp}6[3Uhjo±1[&e@Âg¥ohpW M hjWY[kPlYZ'I[&nwu@ 1npor}6u1okFhjfG©5VNWY[ fpu@ψlghpou@ku~L9[GqFl|~2Mhpou@k ×1G &|~2M kI[¥npohhj[&k @ ψ(t) = P exp (Σt)P ψ ¥W[&nj[ ψ orf'hpWY[oko¯hjo|@&u@kgohpou@kT&u@njnp[3f}Iu@k6gok¹hpu¡hjWY[fhp[3|@gelYk6fih|~Y[fpu@lghpou@k%Â Σ orf'hpWY[ Yo|@@u@k6|~IZ\|~hpnjo¯µ u@kFh|~okYok'hjWY[ [&o@[&kP±|~lY[3fu@ |@k ofhjWY[Z|hpnjoµ¤¥WYu1fp[ u@lYZ\kfN|~nj[¬hpWY[ &u@njnp[3f}Iu@k6gok'[&o@[GkP±@[G hpu1njffl6WhjW|h LP = P ΣL © P [&kYu@hp[¡Pe hpWY[¡Z|~hpnjo¯µ;¥WYu1fp[¹&u@lYZ\kf¤|@np[¹hjWY[¡s Z/ugg[3fnj[ hj|@ok[G;ok hjWY[¡Z\ugg[&> |@npn|~k@[GTok;±@[3UΦhpu1njf/¥ohpW)hpWY[¡fj|~Z\[npl[3f|@f/hpWu1fp[PeT¥WYorW)hpWY[¡]`|H±Fo[&np²icFhpu1´@[Gf fu1lYhpou@k;of |@npn|~k@[G¹ok ψ © #[/W6|H±@[ Φ Φ = I u@n¬hpWY[\u1nhjWYu@kYu1npZ|@ohieu@hjWY[\s Z\uPY[GfG© º[&h a(t) = |~k ψ(t) Â9hpWY[Gk·[GqFl|hjou1k D }np[GZ'lYhpo}Yo[3·Pe Φ nj[G|1Yf ψe ' e [a (t), . . . , a (t)] = Φ a(t) 3 Fe¡&u@Z\}|@npokYhpWYorf [GqFl|~hpou@k#¥o¯hjW G hjWY[\[GfhpoZ\|~hp[ u@n¬hpW[okY[G|@np5o '&[3 ΦJΦ ψ ]`|H±Po[Gne²icFhpu1´@[3fu@}I[&n|hju@n L orfwu1lYkª© k¡hpWY[\u~hjWY[&n¬W|@kªÂRPe¹&u@kLefhp=njlΦJΦ hpou@kºÂR¥['W|H±1[ hpW6|h J = © Φ LΦ VNWY[¹fpu@lghpou@k)u~whjWY[¹okY[G|@np5o '&[3·u¥N²>u1njg[Gn'Z\ugg[&N[3q1l6|hpou@k
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(85) . %[ h\lf Y[&kYu@hp[¥ohpW |~k hjWY[fplYYZ|~hpnjo&[Gfu1npZ\[3·PehpWY[ $njfh'hi¥uy&u@lYZ\kf u~ @| k)FehpWY[<$6njfh\hi¥unjuP¥wf u@ PP nj[Gfp}I[GUhjo±1[&e@©¢cPoZ\or|~nje1Â[ h¤l6f¤g[&ku~hp[¹¥ohpW Q |@k QP hjWY[fplYYZ|~hpnjo&[Gf u1npZ\[GPeyhpWY[ $nfih hi¥u<&u@lYZ\kf u@ Q |@kPe#hpWY[$6njfh'hi¥unpu¥wf u~ Q  nj[Gfp}I[GUhjo±1[&e@© VNWY[GkºÂNPeT[GqFl|@okyhjWY[}YWPegfor&|@wfu1lYhpou@k i@ hpuohjfnj[G&u@kfhpnjlUhjou1k 3z Â|Æhp[GnW|H±Pok k[&@[G hp[3¤hjWY[ g[G&np[3|@fpok [&µP}Iu@k[&kFhpor|~rf&Âgohu@u¥wfhjW|h ?. ?. −1. −1. P. ". eσ1 t 0. 0 e σ2 t. #. P ? ψ 0 = ΦQ. ". e λ1 t 0. 0 e λ2 t. #. Q? Φ T ψ 0 .. cgok&[ ÂI¥['u1ghj|@ok |@k © )³k<u@ng[&n+hpu1o±1[ |@n|~}YWYor&|@%nj[&}g² jn [Gfp[&kFh|σhpou@k=u~λhpWY[nj[G&u@kfhpnjlUhjP[G=}YWPΦQ ePfpor&|~RlYk6Pfih|~Y=[Q[&o@[GΦkF±1[G hpu@n¥[¬W6|H±@[ gnj|H¥kok $1lYnj[ hjWY[ Z\ugglYlfwu@9hpWY[ $njfh+ u1lZ/ku@ P © )Úh`of+okFhp[Gnp[3fihjok\hpu¤Z|~´1[ |¤ u1Z\}|~njofpu@k¥ohpWhjWY[ nj[G|~ lkfih|~Y['[Go1[&kP±@[3Uhju@nu@Yhj|~okY[3Pe¹fpu@±Pok\hpWY[/[&o@[GkY}Ynju@Y[&Z¦|@fjfug&o|~hp[Ghju¤hpW[ lY9okY[G|@npo5'&[3 ]`|H±Po[Gn²icFhpu1´@[3fwu@}I[&n|hju@n ©'VNW[/kPlYZ\[Gnpor&|@±|~lY[ u@n¬hpW[/r|hphp[Gn¬g[G}YorUhp[3<ok $@lYnj[/ à| ªW|H±1[ I[&[Gk·´PokYey}Ynpu±Porg[GyPe L r|H±Fou /¬or|~kYkY[&hhjo¥WYuW|1f np[G}6[3|hj[Gyu@nhjWY[&lYnpnj[&kFh 1[&u@Z\[&hpnjehpWY[ fj|~Z\[´Fok u~6G|~r lYr|hjou1k }I[&npu@njZ/[3 ok Wof ¥u@nj´¬¥ohpW s5|~u@u ºlWYokYoFu@nhpW[NG|@fp[u@hpW[&on lYr|~n &ePokg[Gn ! 3 " ©VNWY[¤|@@nj[&[&Z\[Gk1horf fj|hporfà|@Uhju@nje@Â%&u@k $6npZ\okY¹hpWY[±H|@orgohie<u@hjWY[¤s ² / |@[Gnp´Pok |@}Y}Ynju1|1W¤okhjWY[ fihjlgeu~%hjWY[ fhj|~oo¯hie1© ¹w* w*
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(90) )³khjWYof+f[3Uhpou@k¥[`ug&lfu@khpW[ W|~kY1[Gfhju/I[Z|@g[okhpWY[s =² / |~[&nj´PokZ\uPY[&ºok¹u@ng[&nhpu u1ghj|@ok|}npu1}6[Gn+foZ'l|~hpou@k¹u~9hpW[ £u¥¨|~nju@lYk6hjWY['fjqFl|~nj[ ePokY[&n+¥W[&kYu¥ok fpl hpou@kof }I[&npu@njZ\[G|@u1kYhpW[ W|@kYkY[& ¥N|~fG©wVNWY['±1[&ug ohie|@u1kYhpW['¥N|~f+of`[&±1[&njeF¥W[&nj[ '&[&nju¤[&µg&[&}gh u1nthi¥u u@}}6uFfohp[+fp[&1Z\[&kFhjfkY[ µPhhjuhpW[`fjq1l6|~nj[¥WY[&nj[wohjfhpn|~kfp±@[Gnjfp[w u1Z/}Iu@k[&kFh±|~njo[3ftok\hpoZ\[@ ¥Wo[+nj[&Z|~okYokY &u@kfhj|@kFh|~u@kY hpWY[ fihjnp[3|~Z ¥ofp[+gonp[3Uhjou1kºÂFhpW|~horf u(x, t) = c(t)j ¥ohpW x ∈ Γ Df[G[ o6©@ ©5VNWY[w|1Uhjl|hju@nf |@np[gnjo±@[&k\ok\u@}Y}Iu1fpo¯hj[}YW|1f[3f&Â~o>© [@©9|h|~kPehjoZ\[hjWY[±@[Gug&o¯hie ±1[G hpu@n u@thpW['hju@}|@Uhjl|hju@n`of¬[GqFl|@9ok#|~Z\}Yo¯hjlg[/hpuhpW|~h¬u@thpW[/Iu~hphpu1Z |@Uhjl|hju@n3ªfuhpW|~h`hjWY[/£u¥ n|hj[¬hpWYnju@lY1W¤hjWY[ W|~kkY[&ºguP[3fkYu~hwW|@kY@[1© %[ h lf g[ $k[ ¥ohpW u1lYn¬|@fp[ fhj|~hp[/¥WY[&k = c = 0.1 ÂIu@Yhj|~okY[3vilfh¬o´@[/ok<hpWY[ lkgu@n [3¤&|1f[w¥W[&k¤hjWY[+ukP(x) lYZ\[Gnpor&|@IfpoZ lYr|hpou@knj[G|1WYc(t) [Gf| Z\okYoZ lYZ±|~lY[+okhjWY[+hjoZ\[¬np[3forgl|@D© )Úh G|~k#6[\u@6f[Gnp±1[G¹hjW|hhjWY[ $¯hj[&nj[G#fu1lYhpou@k u (x) = u (x) − u(x) orfflW<hpW|~h u = 0 u@k |@k c j u@k Γ ©9VNW[+±@[Gug&o¯hi7e $[&ru@ªhpWY[¬fegfhp[&Z ¥WYorW¤W|1ftI[&[&k¤u@n [G\¥ohpW Γ ,Γ −Γ |/1[&kY[Gnpor`&u@kFhpnju@ªur|H¥ = c(t) ÂYorfN[ µg}Ynj[Gfjfp[G|@fok ! G " |@G u@ngokY hpu/hpWY[ u1u¥okY\[ µg}|~k6fou@k 1,2. ?. 1,2. ?. T. c. 0. ?. c. in. w. c. ?. 0. c. u(x, t) = u(x) +. )ÚhNhjlYnpk6fu@lghNhjW|hhjWY[ ±@[&ug ohie¤orf '&[&nju/u@k. Nr X c(t) u (x) + an (t)φn (x). c c? n=1. u@nNhjWY[ $hp[&nj[Gfpk|~}fpWYu@hjf u(x, t ) − u(x) − Γ ,Γ Y  ¥ Y W & [ j n ¬ [ @ u N n G [ 1 | f. [ @ u Y k @ u j h | j h o 1 u k ¥. [ W H | @ ± ¬ [ p n G [ k|~Z\[3 c(t)/c |@f c(t) © efplYfhpohplghjok\hpWY[ c(t )u (x) i. c. ííÑ >åUã?ã. in. w. i. ?.
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(93) ,4(. o@lYnj[¬ 1à| cP}|~hpor|~RgorfihjnpoYlghjou1ku~%hpWY[¬Z\uPYlYl6fu@ºhpW[`lYk6fih|~Y[`[Go1[&kP±@[3Uhju@n DY|h|'&u@lYnphp[3fe u@&r|H±Pou /¬or|~kYk[ hhjo © cP}|hjo|@gorfhpnjoYlYhpou@ku~ hpW[ Z\uPYlYl6fwu~9hjWY[ nj[G u1kfhpnjlUhj[GlYkfhj|@Y[ [Go1[&kP±@[3Uhju@n3© u1npZ\[Gn[ µg}|@kfpou1kokFhpu'hpWY[¬]+|H±Po[&np²³cFhju@´1[Gf5[3q1l6|hpou@k6f|~k6}Ynju~vi[G hpokY u1k1hjuhpW[`s Z\ugg[Gfu@kY[ u1ghj|@ok6f,1 00 00 ak )c ak (t) − Bksr ak (t)as (t) − Er00 c˙ − Fr00 c2 + (G00r − Hkr a˙ r (t) = −(∇p, φr ) + A00r + Ckr ar (0) = (u(x, 0), φr ). ¥W[&nj[ |@RhjWY[ fplYfj njo}ghfNnplknju@Z 1 hju. Nr. |~k. Er00 = (uc , φr ) Fr00 = (uc · ∇uc , φr ). G00r = (∆uc , φr ) /Re − (u · ∇uc , φr ) − (uc · ∇u , φr ). 00 Hkr = (uc · ∇φk , φr ) + (φk · ∇uc , φr ). Äì íºÄÆÅ.
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(97) . o@lYnj[ 15¸Uhjl|hjou1kfjWY[GZ/[ #['WYuPuFf[¬hpuZ/ugg[GªhpW['}Ynju~vi[3Uhpou@khp[GnpZnj[&[&±|~kFhhpu\hjWY[ }Ynj[GfjfplYnp[|@khpWY['lYkYnj[Gfpu@±@[Gfj&|~[Gf Pe|1fpfplYZ\okY/hpW|~h . ¥ W[&nj[\hpWY[}YnpoZ\[GqFl|~kFhjo¯hjo[3f'|~nj[lYkY´PkYu¥k%© ef[&hhpokY A = A − A  C = C − C    |@k H = H − H ÂPhpWY[¬u@njZ\[&nfpePfhp[GZ,&|@k E = E −E F = F −F G = G −G I[nj[&¥njo¯hphp[&k 0 0 −(∇p, φr ) = A0r + Ckr ak (t) − Er0 c˙ − Fr0 c2 + (G0r − Hkr ak )c. 0 r. 00 r. r. 0 r. r. 00 r. 0 r. 00 r. r. 0 r. 0 kr. 00 r. r. kr. 00 kr. 0 kr. 00 kr. kr. a˙ r (t) = Ar + Ckr ak (t) − Bksr ak (t)as (t) − Er c˙ − Fr c2 + (Gr − Hkr ak )c ar (0) = (u(x, 0), φr ). )³ku@hpWY[Gn+¥u@nYfNhjWY[ G|~oYn|hjou1k¹}Ynpug&[GglYnj[ orfw}I[&npu@njZ\[Gu@n`[&±1[&nje}Ynju~vi[3Uhpou@k¹hj[&njZ. [&µY [&}Yhwu@n © B #[· u@k6forg[&nj[G |kPlYZ\[&njor&|~ fpoZ lY|~hpou@k |h|){[GePkYu@rYfkFlZ'I[&n¹u~ Â+¥WYorW orf fpo1W1hje @nj[G|hj[&nhpW|@k hpWY[T&npohpor&|@±H|@l[#u@\FP© )³k hpWYorf¡&|1f[yhpW[·oÆh¡u1Refj or=|hj[G60f¹I[ hi¥[&[Gk ² © Fz·|~k6¢Y© 1z#u1nhjWY[ylYe¢g[G±@[&u@}I[G"£u¥© k [<hpWY[yoZ\o¯h¡ eg&[yW|@f6[G[&k np[3|@WY[3ªÂ¥[ fp[ h t = 0 u1nhpWY[<okfhj|@kFh u1npnj[Gfp}6u1kgokY#hpu 'G[&nju·oÆh|@k"|~}Y}e)hpWY[¡u@u¥okYu@n okY·r|H¥ Â¥WY[Gnp[ Y[&kYu@hp[Gf/hjWY[goZ/[Gkfpou1k|~N}I[&njougTu@ c(t) = −2/π sin(2π t T /T ) arctan(t T ) hjWY[woZ\oh&eP&[+u1fj or|hjou1kf&;© )³k o6©P à| 6u@hpWhjWY[+o¯ÆhN|~Tk/hjWY[wu1nj&ok |~nj[wfWYu¥k¤|1f|¬lYk6Uhpou@k u@%hjWY[ goZ/[Gkfpou1k|~RhjoZ\[@© ¸;kPlYZ\[&njoG|~PY|h|~|1f[W|@fI[&[&k YlYo¯h5Pe fj|~Z\}YokY+hpWY[hjoZ\[Nf}|@k'I[ hi¥[&[Gk +|@k ¥ohpW z1z@z/fpk|~}fpWYu@hjf Df[G[ }YlfZ|~nj´@[GnjfNok o6©R D| |~k¹|\s |@fp[u~ N = 5 Z/ugg[3f19.45 W|1fwI[&[&k [&µPhpn|@Uhj[G'npu1Z«oh¥o¯hjW| np[3 u@k6fihjnpl hpou@k [&njnpu1n u@R|@6u1lghw@© © VNWY[lYkY´PkYu¥k A  C  E  F  G |@k |~nj[ g[&hp[&njZ\ok[G|@G u1njgokY hpuhjWY['fj|~Z\[}Ynjug [GYlYnp[¬u1u¥[3¹|~Iu±@[¬u@nhjWY[ Z\ugg[&okY u@%hjWYH[ hpn|~kfpo[&kFhG©5VNWY[nj[GfplYhpokYgonj[G h g|@Hviu1okFhN}Ynpu1Y[&Z,orfNhpW[&k ( a˙ (t) = A + C a (t) − B a (t)a (t) − E c˙ − F c + (G − H a )c gonp[3Uh}Ynju@Y[&Z ksr. ref. S. ref. S. r. r. kr. r. r. kr k. ar (0) = (u(x, 0), φr ). ííÑ >åUã?ã. ksr k. s. r. r. 2. r. kr k. kr. r. r. r.
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