On time averaging in Organised Eddy Simulation Modelling
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On time averaging in Organised Eddy Simulation modelling Ilya Abalakin and Alain Dervieux. N° 4040 Novembre 2000. ISSN 0249-6399. ISRN INRIA/RR--4040--FR+ENG. THÈME 4. apport de recherche.
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(85) RY[IY¡\Z«V Wf^Mm\^m+LUKV WY[RT\^]\c ¨IKM]Wc] d l2WY[MV¦¨WcVTVW d ogWchM]wYNY[UKmg|KUKVTM]KhMRXbNh\c]Kb+R d MmgM d B
(86) VTMYNUKb d Mb+hm+RT|sM RTYNRT] b+\^LNM d MY[WcRTVXb ÍÍ à SaÉÒ©ÉÒ.
(87) . ( # (*. (*
(88) - '. .
(89) .
(90) "! #
(91) $ . H IKMr
(92) MfE]K\cV d b"Wz^Mm+Wc^M d yqWzERTMmpªQEY[\c¬cMb"bgfEbgY+ML m+MVTfERT]K0Y+\Y+IKM
(93) m+RY+Y[M]RT]WNh\^]KbgMmgzZWZY+RTzcM\^mgL Wcb. xLN\. d MVRTb. %'& B )% (+*,&.- B )% 12*3&4- ) %@ 0% / )% 5 9 %):;*3&4- '% =>*3&4% @: *3&4- B A % @= *,&.- BCB *3&4*687 B B %</ 0% 5 ? <% / 0% 5.
(94) IaMm+M = D &E*F/$GH5'G @ - RTb"W UK]KhY+RX\c]WcV_Wcm+m+Wf
(95) RTY+IzZWcVTUKMbRX]JI :LK ¤¨Y+IKM h\^LN`3\^]KM]wY+b"\c
(96) IKRThtIWcmgM¯Y[IKMe]K\^] d RXLNM]Kb+RT\^]WcVTRTb+M d h\^]KbgMmpzcWY[Rz^MzZWcmgR WZ|KVXMb D (+*,&4- Wc] d 1+*,&4- Wcm+M¯Y+IKMeh\^]wz^MhY+RTzcMUSjUa]KhY+RX\^]ab D :;*3&4- ¤ =>*3&4- Wcm+M
(97) Y[IKMqV WcLNRT]WcmzSRTb+h\^UKb¨KUSjSMb 6>7qRTb¨Y[IaMqVXWcLNRX]WZm¦r
(98) MfE]K\^V d b ]AUKL|3Mm
(99) \^|aY[WcRT]KM d WZY
(100) Y[IKMe]K\c] d RTL(M]Kb+RT\^]WZVXRTb[WZY+RX\c]º D ;@: *3&4- ¤ 8@= *,&4- Wcm+M¯Y+UKm+|KUaVXM]AY_zERXbgh\^Uab_UEjUK]KhY+RX\c]Kb D B *3&4- RTb
(101) Y+IKMeb+\^Uam+hMeY[Mm+L mgMVXWZY[M d Y+\~Y[IKM LN\ d MVC MqRXbgh\^UabY[UKmg|KUKVTM]wY~bpY[m+Mb+b~WZVXbg\RX]wzc\^VTzcMb(W d R Wcc\^]WcV¥Y[Mm+O L NQPSRST VU ' 2WU 'KP Y+IKM R d M]wY[RYfLWY[m+Rj 6¨Y[IWZY
(102) RTb_Wchh\^UK]wY[M d Y+IKm+\^Ua^IWZ]W d IK\EhezZWcmgR WZ|KVXM&htIKWc]K^CM = XYY.
(103) RTY+I. N T B R Z YY[]\0_ ^ )`\ _ C B a$T . ). ^ Z.
(104) IKMmgM. a ) * B R *Fb N *. XY ) *cb - T"d?2+ Y[ \ _ )`T"d?7+ * B * T * BJg - BhT e N fe.
(105) IKMmgM RXb Y[IaM¡`KmgMbgb+UKmgMc¤#_ P Y[IaMY+\cYtWcVwM]KMmgcfe`3Mmz^\^VTUKLNM¨Ua]KRTY¤ST2P Y[IKM d M]ab+RYf^¤ nP Y[IKM"Y+UKm+|aUKVXM]wYN¬ERT]KMkY[RThM]KMm+cfc¤id?(IK\cV d b~\ZY[IKM bg`sMhR£hIKMWY \^mh\^]KbpYtWc]wY z^\^VTUKLNMc\ ¤ +P Y+IKM¡Y+MLN`3Mm[WY[UKmgMc¤ b P Y+IKM¨bg`sMhR£h¦IKMWZY m+WZY[RT\qWZb+b+UaL(M d Wcb2h\^]KbpYtWc]wY ¼ÏͼÐ.
(106)
(107)
(108) ! " #%$&!')(* +-,.. K. ) * "\^miW"`3MmgMhY&^Wcb 6_Wc] d ¤'gWZm+MLNMWc]\{ m+MVXWZY+RX\^]|3MY¡MM] _ WZ] d RXb
(109) hV Wcbgb+RTh = 2b. z^MVX\EhRYfh\^LN`s\^]aM]wY[b¯HIKM. . ^. ) \<^ *Fb * - _ N* T e Bhg e . . . HIaM]h\^]wz^MhY[Rz^MUS\ jS^ Mb_Y[UKmg]Y[\N|sMC= ^ T T 'e B Tg. ). ( * &4-. *_ B T T \.
(110). . . . . \- . . G 1 *,&4 . . . ).
(111). . T#g T g T#g e B. *_ B. . . -g ) T#g9 \ T#g \ . . . . ] d VXWcLNRX]WZm¦zSRTb+h\^UKb
(112) USjSMbiWcmgM¯
(113) m+RY+Y[M] Wcb = WcLNRX]KWcm¦zERXbgh\^UKbUEjaMbqWZm+M¯
(114) m+RY+Y+M] Wcb. . . . . :C*,&.- ). . 6 Bhg C6 B b % %</ %.
(115). . . . %</. G = *,&.- ) . % <% /. . . . . . . . . . 6. "6 6 6FBCg 6 6CB b % . . . . . . . . . . . . . . . . . . ÍÍ à SaÉÒ©ÉÒ. . .
(116). . . . . %)5 % %05. . . % 0% 5. . . . .
(117) . ( # (*. (*
(118) - '. HUKmg|KUKVTM]wY_zSRTb+h\^UKb
(119) USjSMbiWcmgMe
(120) mgRTYgY[M]Wcb =. . . . . . . ). @: *3&4C. . . 6. . b % Bhg 6 B @ %</ B % %</ % %</. . . . . . .
(121). . G. % . . %</. . . . . . . ). . @= *,&4-. . 6 6 6. . b % 6 Bhg 6 6 B @ %05 B % %05 % %05 . . . . .
(122). . . . G. % . %05. . .
(123) IKMmgM. . . . ). ). . 9 % % N % B %</ 0% / ? R %0/ . . 9 %<% / B %0% / ? RN <%% / . ). . . . .
(124) QN. . . ) . . . G ) G ) g e. . 6>7 ). T #% E # EF#% E . #% E. ¼ÏͼÐ.
(125)
(126)
(127) ! " #%$&!')(* +-,.. HIKMb+\cUKm+hMeY+MmgL(biWcmgM =. . .
(128) . . . . .
(129) IKMmgM. 9 % % N % B %0/ <% / ? R %</. ) . ). . . . . 9 %<% / B %<% / ? RN %<% / . ). . . . .
(130) SN. . ). . . . . ). B *3&4-. M. . . . G . ) G e ) g. . 6>7 ). T #%AE #%AE F#%AE #%AE . iMmgM Wc] d WcmgM~m+Mb+`3MhY+RTzcMVfW ]K\^] d RXLNM]ab+RT\^]WcVTRXbgM d VXWcLNRT]WcmqzERTb+h\cb+RYf h\EM hRTM]wY¤¶W"]K\^] d RXLNM]ab+RT\^]WcVTRXbgM d Y[Uam+|KUKVTM]wY¯zERTb+h\^b+RYfxh\EM hRXM]AY¤ eWcm+MY[IKMh\^LN`3\Z ]KM]wY[b¨\cºY[IaM ¦WcUKhtIwf"bgY[mgMbgb¨Y[M]Kb+\^m¤ ¦WZ] d " WZm+MqmgMbg`sMhY[Rz^MVTf"Y[IKMiV WcLNRT]Wcm¨Wc] d Y[UKmg|KUKVTM]wYNl¥m[Wc] d Y[V¡]EUaL|3Mm+bN\^|SYtWcRT]KM d mg\^L ]K\^] d RXLNM]ab+RT\^]WcVTRXb+WZY[RT\^]ºy_\cYtWZY+RX\c]Kb T #% E^¤ #%AEc¤ F#% E(Wc] d #%AE~IK\^V d mgMbg`sMhY[Rz^MVTf \^m¯WmgMMm+M]ahM d M]ab+RYf^¤W m+MMm+M]KhM z^MVT\EhR Yf WNmgMMm+M]ahMVXM]KcY[I¶¤sW~mgMMmgM]KhMzERTb+h\^b+RYf^ 2RX]KWcVXVf^¤aRTY_RTb_bgMY = 9. . . . . . . XYY. YY YY Z YY. ) . YY YY Y[. ÍÍ à SaÉÒ©ÉÒ. ). . . ). T e T B.
(131) .
(132) e T e.
(133) {v. ( # (*. (*
(134) - '. O. Wc] d +Y IKM¦zZWcm+RXWZY[RT\^] \Z6V WcLNRT]Wcm2zERTb+h\^b+RYfNWcb2WqUK]KhY+RX\c]~\c d RTL(M]Kb+RT\^]WZVaY[ML(`3Mm+WZY[Uam+M + RTb d M£]KM d |wf Y+IKMQSUaY[IaMm+VXWc] d V W = XYY * + - ) #%AE + RP + * N + #% E Z 2 OI6. YY[ + N B * + - ) * * N *+ B * * RP + * N N* ** HIKMe`Km+\ d UKhkY[RT\^] Y+MmgL RXb d Mk£]KM d |wf = . . . . . . . . . ). 9 N R T ?. . . . . . . . . 9 %<% / B %0% / RN %%0/. . . % ?? %</ .
(135) IKMmgM E¤
(136) ¤
(137) WcmgM~h\^]abgYtWZ]AY+beMLN`KRTm+RThWcVTVTf d M£]aM d m+\^L MkjS`3Mm+RTLNM]wY[bWc] d ¤ Wc] d (WcmgM d WcLN`KRT]KNUK]KhY+RX\c]Kb e ¨\c]KbgY[Wc]wY[b S¤ ¤ ¤ ¤ Wc] d Y[IKM
(138) UK]KhkY[RT\^]Kb ¤ ¤ WcmgM
(139) Y+\|3Mibg`sMhR£M d e e V WZY+Mm
(140)
(141) L
(142)
(143) . 3. C . . HIKMeIKRXcIr
(144) MfE]K\^V d b_b+RTLUKVXWZY+RX\^] RTb_hWcm+mgRXM d \^USY
(145) RTY+I Y+IKM&\^VXVT\{
(146) RX]aRTb+bgUKMb = XYY . YY Z. YY. ) *. ) * YY Y ) * YY [ ) G Y . e. . . . . ) * G . . #
(147) $ C
(148)
(149) . ) * R G ) * G ) * N . Y¡\ZVXWf^Mm¯\cm+LUaV WZY+RX\c]RX]wY+m+\ d UKhM d |wf ¨IKM]Wc] d lWZY[MV2RT] O M^^RXbihtIK\^b+M]º Y £m+bpY¤KY[IaMe\cVXVT\{
(150) RX]K(]AUKL|3MmgbqWZm+MeRX]wY+m+\ d UKhM d = e G : 6 ) 5 G Wc] d 5! ) E 5 : ) 24}6 ¼ÏͼÐ.
(151)
(152)
(153) ! " #%$&!')(* +-,.. OO.
(154) IKMmgM T xRTbY[IKM d M]Kb+RYf WZY
(155) ¨WcVXV WZ] d xY[IaMVXWcLNRT]Wcm¦zERXbgh\^bgRTYfWZY¨WcVTV4 HIaM¯\^]KM©MAUWY[RT\^]VX\{¦r_MkfS]a\^V d b]EUaL|3Mm¦LN\ d MVs\c 0\^VTbgIwY[MRT]RXb¨UKb+M d RX]"Y+IKM m+McRX\^]ab_]aMWcmY[IaMe¨WcVTV : 6 N ³
(156) ¸¸
(157) "³A´¸ C¹
(158) x ³ : 6 N 8] Y+IKRXb
(159) mgM^RT\^]Y[IKMebpYtWc] d Wcm d LN\ d MVRTb_Uab+M d = T ) % 9 B % B % 9 B % T. B. %05 @ %0/ %0/ ? %)5? T ) % 9 B % B % 9 B % B %05 @ %</ %</ ? %05 ? . . . . . . . . . . . . .
(160) IKMmgM Wc] d. . . . ¤ . . . . T e ). ) *. . . ¤ . . . ) * Ra¤ ) * . ). K¤ . ) * N. T e. ³
(161) ¸¸ ¸S· ³w´¸ C¹
(162) x ³ : 6 N 8] Y[IKRTb~m+M^RX\c]±\c]KVTfY[IaM¬ERT]KMkY[RTh M]KMmgcf MEUKWZY[RT\^] RXbb+\^Vz^M d
(163) IKRTVTM d RXbgb+RT`WZY[RT\^] m[WZY+M RTb d Mm+Rz^M d m+\cL W~LNRjSRT]K"VTM]KZY[I = XYY T % 9 B % B % 9 B % T B. Y @ ) %</ %05 %</ ? %)5 ? Z YY Y[ T ) T
(164) IKMmgM ) * Wc] d Y[IaMeY+UKm+|aUKVXM]wY_zERTb+h\^b+RYfWcm+Me^Rz^M]|wEf = ) T G
(165) IKMm+M ) . . . . . . . . . . . . ÍÍ à SaÉÒ©ÉÒ. . .
(166) §}. ( # (*. (*
(167) - '. O. HIKM&Y¨\(LNRjSRT]K"VTM]KZY[I NWc] d WZm+M d Mk£]KM d |wf = ) 5 Wc] d ) 5 2/"6
(168) IKMmgM NWc] d WZm+M¯Y¡\h\^m+mgMhY+RX\c]UK]KhkY[RT\^]Kb = 9 ) * 7 : 6 ? Wc] d ) * 7 : 6 2O6
(169) RTY+
(170) I ) K¤ )
(171) Wc] d ) N a§
(172) ¸S·~µ ¹
(173) ºµ ³w´ ¸ 4¹
(174) ³ ³
(175) ¸ : 6 6\cm * NQN RTYRTbWcb+bgUKLNM d Y[IWZYY+IKMxM dad fzERXbgh\^bgRTYf;zZWcmgRXMbVXRT]KMWZm+Vf |sMkY¨MM]Y+IKM&zZWcVXUaMb_^RTzcM] |wfY[IKMe\^]KM©MEUKWZY[RT\^]Wc] d bgY[Wc] d Wcm d L(\ d MVXb = . . . . . . .
(176) IKMmgM. . . ). . . ). . G B * * - . . NSN : 6 . . ! ' M~h\^]KhM]wY[m[WY[MN]K\{\c]Y[IKM~bpYtWc] d Wcm d h\c]AY+MkjEY\c¨®¯m+wWc]aRXbgM d°¥dKd fQSRTLUaV WZY+RX\c]º¤ RX]
(177) IKRThtIxY[IKMm+M M©jSRXbpY[bWc]M]KMmgcfbg`sMhY[mgUKL RT]wz^\^VzERX]a \c]KM \^mib+MzcMm+WcV¥`sMWc¬Ab¯h\c]S hM]wY+m[WZY+M d \^]`KWcmgY+RXhUKV WZmqm+MEUaM]KhRXMb¤MWZhtIxh\^mgm+Mbg`s\c] d RT]K Y[\ `am+\^]K\cUK]KhM d AUWcb+R `sMm+RT\ d RThe|sMIWzERX\^mgb HIaM\^mgMY[RThWZV d Mbghm+RT`aY+RX\^]ab\Z3Y+IKMbgM
(178) ¬ERT] d \c3h\c]AY+MkjEYhWc]~|3M¦\^UK] d RX] DPO H<¤ DPO§.} H4 ¤ D F.H4 HIaM® ° Q LNMY[Ia\ d h\c]Kb+RTbgY+bi\cbg`KVXRY+Y+RX]aNY+IKMeM]KMm+cf b+`3MhY+m+UKL Y[\~\^]KM d RXbghm+MkY[M `WcmgY mgM^mg\^UK`KRT]KiWcVTV^Y[IKMn\^mgwWc]KRTb+M d LN\ d MbWc] d h\^IaMm+M]wY2LN\cY+RX\^]\cEY[IKM¥\{ bpfEbgY[ML 2CMc K¨Y+IKM d RXbpY[RT]KhYim+MAUKM]Khf`3MWc¬Ab_\c Y[IaMbg`sMhY[mgUK L 6iWc] d Y[\"W(h\^]wY[RT]AUK\^UKb¯`KWcmgY¤ h\^mgm+Mbg`s\c] d RT]KY+\Y[IaM0RT]Kh\cIKMmgM]wY¤htIWc\cY+RXh`WcmpY\ZY[IaM0LN\cY+RX\^] 2CRT]wz^\^VzERX]K\^m MkjaWcLN`KVTM Y+IKMh\^]wY+RX]AUK\^Uab`WZmgYN\c_Y[IKMbg`sMhY[mgUKL$]KMWcm Y[IKM`3MWc)¬ 6© 6\cm Y[IKMb[WZ¬cM \c_b+RTL(`aVXRThRYf^¤n¨M Wcbgb+UKLNMRT] Y[IKMbgMAUKMV¦Y+IWZY Y+IKMmgMRXb~\c]KVTf \^]KM`3MWc¬ 2CRC Mc \c]KM m+MAUKM]Kh5f 6RX]Y[IaM\cm+wWc]aRXbgM d `WcmgY . . ¼ÏͼÐ.
(179)
(180)
(181) ! " #%$&!')(* +-,.. O /. H IaM £Km+bgY `WZmgY\cY[IKM bg`sMhY[mgUKL RTb`Km+M d RXhkYtWc|aVXM|wfY[IKM UK]abgY[MW d f \^`3Mm+WZY[\cm \c Y[IaM~yqWzERXMmpªQEY[\^¬ZMb&MEUKWZY[RT\^]Kb¤º`Km+\{zER d M d W"`KIwfEb+RThWZVXVfbgUKRTY[Wc|KVTM~Wz^Mm[Wc^RT]K
(182) RY[I m+Mbg`sMhY_Y[\(Y[IKMb+`3MhkY[m+UaL d Mh\^LN`s\^bgRTY+RX\c]ºHIKRTbiRTb_Y+IKM`KIWcbgMkªWzcMm[WZ^RX]aK¤3
(183) IaRXhtIRTb WLNMWZb+UKm+Wc|KVTMiAUWc]wY+RTYfWc] d ]K\cY\^]aVTfNWeL(WZY[IaML(WZY[RThWZV3h\^]KhM`a Y D*O{I} H4 MY |3M
(184) Y+IKM `KIWcbgM
(185) W{zcMm+Wc^M
(186) \c ¤cY[IKM_® ° Q\^m+LUKVXWZY[RT\^]Nm+MVXRTMb¥\^]~Y+IKM¦\^VTVX\{
(187) RT]K d Mh\^LN`s\^bgRTY+RX\c] = EHGUL ) B J$NH "LAJ . H IaM¡h\^]wY[RT]EUa\^UKb2`KWcmgY\cEY[IKMb+`3MhkY[mgUKL IWcbY[\_|3MLN\ d MVTVTM d HIaRXb`WcmpY d \SMb]a\cY m+Mcm+\^UK`N\^]aVTfY[IKMIKRT^I~¨W{zcM]AUKL|sMm+b¦WZb¥RX]~Y[IaMhWcbgM_\Z ° Q |KUaYnM©jSY+M] d bmg\^L Y+IKM VX\{ Y+\(Y+IKMIaRX^I m+MEUaM]Khkfm+Wc]K^MZ¨HIKMm+M\cm+Mc¤6Y[IaMhm+RY[MmgRX\c]\c d RXbpY[RT]KhY+RX\c]\c2Y+IKM bgY[mgUKhY+UKm+Mb_Y+\|3Me`Km+M d RThY[M d mg\^L Y[Ia\^b+M&Y[\N|3MeL(\ d MVXVTM d RX]® ° QRTb_Y+IKMRTm
(188) `KIwfSbgRXhWcV ]WZY[Uam+M&Wc] d ]K\cY¦Y+IKMRTmb+*R ?M 2CWcb¦RT] Y[IKM&hWcb+M&\c ° Q 6© @¦WcbgM d \^] Y[IaMb+M¯Ua] d WcLNM]wY[WcV Wcb+bgUKLN`aY[RT\^]Kb¤3® ° Q"RTb¦]K\cY¨RX]wY+m+RT]Kb+RThWcVTVAf / Wcb¡Y[IKM ° Qs¤S|KUaYhWc]|sM&}
(189) IKM] Y+IKM d RXbghmgMY[M`WcmpY
(190) \cY[IKMeb+`3MhkY[mgUKL h\^mgm+Mb+`3\^] d b
(191) Y[\} h\^IaMm+M]wYqbpY[m+UahY[Uam+Mb 8]Y[IKMY[RTL(M© d \cLWZRX]º¤6Y[IaMUK] d WcLNM]wYtWcV2Wcb+bgUKLN`aY[RT\^]x\cn® ° QVTMW d biY+\"MLN`KVT\{f Y[IKM d Mh\^LN`3\^b+RY[RT\^]\cMWchtI UK]a¬E]a\§
(192) ]0AUWc]wY[RYfxRT] +R,5
(193) +k¤¶Y[IKMN\^]aM |sMRX]aY+IKM `KIWcbgMkªWzcMm[WZ^RX]a 2
(194) IKRThtI RXb&W(Y[RTL(M© d M`sM] d M]AYe`3MmgRX\ d RTh \^`3Mm+WZY[\^m 6_WZ] d Y[IKM\cY[IKMm Y[IKMem[WZ] d \^LUKhY+UWZY+RX\^]¶ HIaMm+Mk\^m+M¯Y+IKM&`KIWZb+MkWz^Mm+Wc^M d yiW{zERTMmªpQEY+\^¬cMb_MAUWZY[RT\^]KbIKW{zcM&Y[IKM¯b+WcLNM¯\^m+L Wcb(Y+IKMMAUWZY[RT\^]Kb\c¯Y+IKMbpYtWZY+RXbpY[RThWcViWz^Mm[Wc^RT]K 2Cr_MkfE]K\^V d bWz^Mm[Wc^RT]K 6©¤¦`KVTUKbY+IKM Y[RTL(M© d M`sM] d M]AY2Y[MmgLL 9i\{¡MzcMm¤^Y+IKM¨`aIAfEbgRXhWcVabgRX^]aRT£hWc]KhM¦\cMWchtIY[Mm+L RTbY[\ZYtWcVTVTf d RT¢sMmgM]wY
(195) m+\^L Y[IKMebgY[WZY[RTbgY+RXhWcV Wz^Mm[Wc^RT]K(MAUWZY+RX\c]Kb @¡UKRXV d RT]K]aM hVT\^bgUKm+Mb¨RT]Y[IaM¯® ° Q(LNMY+IK\ d \^VT\^cfNL(WfNIWz^M¯WZb¨Weh\^]ab+MAUKM]KhM Y[IWZYnY+IKMqmgMb+UaVTY+RX]KL(\ d MV3VT\E\^b+M_b+\^LNMiUK]KRz^Mmgb[WcVTRYf^ 8] d MM d ¤KWZb+b+UaL(M_Y[IWY¡Y+IKM_K\§ RX]wz^\cVTzcMb¡Weh\^IKMm+M]wYAUWcbgR`sMm+RT\ d RThbgY+m+UKhkY[UKmgMY[IWZY m+MbgUKVY[bRX](W¯`3MWc¬ \Z3zcMmpf~b+L(WcVTV WcLN`KVXRY[U d M¦RT]Y[IKM¨M]KMm+cfb+`3MhkY[m+UaL ¤^
(196) IaRXhtI~LNMWZ]Kb2Y[IKWZY2Y+IKM¨M]KMmgcf\ZY[IKM¦`3MmgRX\ d RTh bgY[mgUKhY+UKm+MRTb
(197) zcMmpfb+VTRX^IwY+VTfVXWcmg^MmY[IKWc]Y[IKMeM]aMm+Zf\c2htIKWc\cY[RThe\^]KMb HIaM] |s\cY+IbgYtWY[RTbgY[RThWZVL(\ d MVXVTRX]a"Wc] d ® ° QWc`K`KVf^ 8«Y[IKM® ° Q L(\ d MVRXb¦Y+IKM] ]K\cYR d M]wY[RThWZVY+\ Y+IKM hVXWcb+bgRXhWcVr y¯Q;\^]KMc¤i¨MhWc] h\c]KhVTU d M Y[IWY¨MLNMMYW VXRTLNRTY[WZY[RT\^]\c2Y+IKM® ° Q LN\ d MVC. ÍÍ à SaÉÒ©ÉÒ.
(198) O. ( # (*. (*
(199) - '. O.
(200) % ek'x¦
(201) º
(202)
(203) ! ! 8]\cm d MmºY[\
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(273) IKMmgM =. . . 07# 02GU(1Q )
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(324) RY[I ]KMkhVT\^b+UKmgM"h\SM hRXM]wYN`Km+Mb+M]wY~RT] 2RT^UKmgM }E ®&]aMqhWZ] b+MMqY[IWY¨Y[IaMi\{ IWcb¨]K\hVXMWcm+Vf d M£]KM d `sMm+RT\ d Wc] d Y+IKMmgM¯RXb¨WY[mgM] d Y[\ Y[IKMehtIWc\ZY[RThLN\cY+RX\c]WZb_RY_RXb
(325) RT] Y+IKMehWcbgMe\c2bpY[MW d fY[Uam+|KUKVTM]wY
(326) zERXbgh\^bgRTYfc MkY¦UKb¦`KW{f"WZYgY[M]wY+RX\^] Y+\Y[IaM¯Wc`K`3MWZm[Wc]KhM¯\c ]KMk m+MAUKM]KhRTMbRT] Y[IaM¯b+`3MhkY[m+UaL \c2VTR-Yih\SM hRTM]wY¯h\^]Kb+R d Mm+M d Wcb_WNUK]ahY[RT\^]\^] Y+RXLNMZ¨HIKMb+Mem+MAUKM]KhRTMb¯Wcm+Meh\c]S d RTY+RX\c]KM d |wf Y[IKM^M]KMm[WZY+RX\c]\cW dKd RTY+RX\^]KWcVzc\^mgY+Mkj]KMWZmqY[IKMhfEVTRX] d Mm&b+UamgCWchAM 2CbgMM 2RXcUKm+M& 6k: 8] d \^RT]KNb+\K¤S¨M&\^|SYtWcRT] Wc]RX]wY+M]KbgRTYf\c Y+IKRXb_W dad RTY+RX\c]WcVszc\^mgY+MkjY+IWZYRTb IKRXcIKMm¦RT] Y[IaMqhWZb+M&\c ]KMk hVT\^b+UKmgM&h\EM hRTM]wYY+IWc] Y[IaM¯\^]KMi\^|aYtWZRX]KM d RX]WZ`K`KVfSRT]K Y[IKMehVXWcbgb+RThWcVr y¯Q"Y+UKm+|KUaVXM]AY_zERXbgh\^bgRTYfc b2RY2RTb2bgMM]~RX]Y[IKM 2RXcUKm+MbOZ F
(327) Y+IKM¡M]KMmgcf\caY+IKM¨L(WcRT]z^\cmgY[M©j 2Ch\cm+m+Mb+`3\^] d RT]K Y[\iY+IKM¨AUWZb+RT`sMm+RT\ d RTh¨\{ 6 d Mhm+MWZb+MbnWZ] d Y+IKM¨`Kmg\ShMbgb \cM]KMmgcfmgM d RTbgY+m+RT|KUaY+RX\^]\^m ]KMbgY[mgUKhY+UKm+Mb¨^M]KMm+WZY[RT\^](bgY[WcmgY+bL 8]NY[IKM
(328) hWcb+M
(329) \c¶]KMzcWZVXUKM
(330) \cshVT\^bgUKm+M_h\EM hRTM]wY Y[IKRTb
(331) `Km+\EhMb+b_RXbYtWZ¬ERT]K~Y[IaMIaRX^IaMm
(332) RX]wY+M]KbgRTYfc 8«Y(bgMMLNb(RX]wY+Mm+MbgY+RX]KY[\ h\cL(`KWcm+MY[IaMm+MbgUKVY[bN
(333) RTY+I±Y+IKM\^]KMbN\^|SYtWcRT]KM d |wf W ° Q hWcVThUKVXWZY+RX\^] 2DN/.H<6kHIKM"h\^m+mgMb+`3\^] d RX]Kb+`3MhkY[m+UaL h\^m+mgMVXWZY[RT\^] 2 2RT^UKmgMb K 6 b+IK\{
(334) b¯WN^\E\ d Wccm+MMLNM]wYq|3\cY+IRT]Y+IKMW dKd RY[RT\^]WcVmgMAUKM]ahRTMb¯WZ] d Y+IKMRTm
(335) M]KMm+^RTMb RX]Y[IKMehWcb+Me\cUKb+RT]K Y[Uam+|KUKVTM]ahML(\ d MVº
(336) RTY+I]KMk zZWcVTUKM ~h\SM hRTM]wY HIaMqY[RTLNMq\^m\^|aYtWZRX]KRT]K~Wb+RT^]KR£hWZY+RTzcM&\^UaY+`KUaY¦RTb¦m+WZY[IKMm¨bgLWZVXVC¤KWc|3\^UaY¡Y[M] `sM© m+RT\ d b¤S
(337) IKRThtIRT] d RThWZY+Mb¨Y+IWZY¨Y+IKM¯`Kmg\^`3\^b+M d LNMY[Ia\ d ]KMM d b¦WLN\ d Mm[WZY+Mqh\^LN`KUaY[RT]K Y[RTL(MZHIKM"m[WZY+RX\
(338) RTY+I W ° QhWcVThUKVXWZY+RX\^]0
(339) RTY+I h\^LN`Wcm+Wc|KVTMWchhUKm+Wchf0RXb]KMWZm+Vf Y¨\(\^m d Mm+b
(340) \cL(Wc^]KRY[U d Mc. ÍÍ à SaÉÒ©ÉÒ.
(341) }cv. ( # (*. (*
(342) - '. k-eps: cmu=0.09. 2. lift coefficient. 1. 0. -1. -2 180. 200. 220 240 260 280 Nondimensional time. 300. 320. 2RXcUKm+M O= RT-Y(h\SM hRXM]wYWcbNUK]KhkY[RT\^]\Z¯]K\^] d RXLNM]Kb+RT\^]WcV
(343) Y[RTLNM\c|aYtWcRT]KM d
(344) RY[I MAUWZY+RX\c]Kb
(345) L(\ d MVº
(346) RTY+I ) . . . 4 k-eps: cmu=0.04 3. lift coefficient. 2 1 0 -1 -2 -3 -4 200. 250. 300 350 400 Nondimensional time. 450. 500. 2RXcUKm+M})= RT-Y(h\SM hRXM]wYWcbNUK]KhkY[RT\^]\Z¯]K\^] d RXLNM]Kb+RT\^]WcV
(347) Y[RTLNM\c|aYtWcRT]KM d
(348) RY[I MAUWZY+RX\c]Kb
(349) L(\ d MVº
(350) RTY+I ) . . . ¼ÏͼÐ.
(351) })O.
(352)
(353) ! " #%$&!')(* +-,.. 3 SMOES 2. lift coefficient. 1 0 -1 -2 -3 300. 320. 340. 360 380 400 Nondimensional time. 420. 440. 2RXcUKm+M /5= RT-Y(h\SM hRXM]wYWcbNUK]KhkY[RT\^]\Z¯]K\^] d RXLNM]Kb+RT\^]WcV
(354) Y[RTLNM\c|aYtWcRT]KM d
(355) RY[I Q 4® ° Q . 2 k-eps: cmu=0.09. 1.8. k-eps: cmu=0.04. Spectrum (lift coefficient). 1.6 1.4 1.2 1. 0.8 0.6 0.4 0.2 0 0. 0.05. 0.1. 0.15. 0.2 0.25 0.3 Strouhal number. 0.35. 0.4. 2RXcUKm+M¯O5= ¨\^LN`WcmgRXbg\^]VXR-Yh\EM hRXM]AY_bg`sMhY[mgUKLNb
(356) \^|aYtWZRX]KM d
(357) RTY+I L(\ d MVº
(358) RTY+I Y¡\ d RT¢sMmgM]wYizZWcVTUKMb_\Z 5= (Wc] d . . ÍÍ à SaÉÒ©ÉÒ. . . . 0.45. MAUWY[RT\^]Kb.
(359) }^}. ( # (*. (*
(360) - '. 2 k-eps: cmu=0.09. 1.8. SMOES. Spectrum (lift coefficient). 1.6 1.4 1.2 1. 0.8 0.6 0.4 0.2 0 0. 0.05. 0.1. 0.15. 0.2 0.25 0.3 Strouhal number. 0.35. 0.4. 0.45. 0.4. 0.45. 2RXcUKm+M F)= ¨\^LN`Wcm+RTb+\c]VTRT-Yh\EM hRTM]wY"b+`3MhY+m+UKLNb \c|aYtWcRT]KM d
(361) RY[I;Q54® ° QWc] d MAUWZY+RX\c]LN\ d MV 2 ) . . . 1.2 k-eps: cmu=0.04 LES. Spectrum (lift coefficient). 1 0.8 0.6 0.4 0.2 0 0. 0.05. 0.1. 0.15. 0.2 0.25 0.3 Strouhal number. 0.35. 2RXcUKm+M 5= ¨\^LN`Wcm+RTb+\c]VXR-Yih\EM hRTM]wY¯bg`sMhY+m+UKLNb¯\^|SYtWcRT]KM d
(362) RTY+I ° QWc] d MAUWZY+RX\^] LN\ d MV 2 ) . . . . . ¼ÏͼÐ.
(363) } /.
(364)
(365) ! " #%$&!')(* +-,.. 1.8 SMOES LES. 1.6 Spectrum (lift coefficient). 1.4 1.2 1. 0.8 0.6 0.4 0.2 0. 2RXcUKm+M °Q . 0. K)=. ÍÍ à SaÉÒ©ÉÒ. 0.05. 0.1. ¨\^LN`Wcm+RTb+\c]VTRT-Yh\EM . 0.15. 0.2 0.25 0.3 Strouhal number. 0.35. 0.4. 0.45. hRTM]wY"b+`3MhY+m+UKLNb \c|aYtWcRT]KM d
(366) RY[I;Q54® ° QWc] d.
(367) }ZO. ( # (*. (*
(368) - '. Streamlines SMOES - minimum. Streamlines SMOES - maximum. Streamlines OES (c_mu=0.04) - minimum. Streamlines OES (c_mu=0.04) - maximum. 2RXcUKm+Me5 =¥QEY[mgMWcLNVTRX]KMbqWY¦Y[IKM&Y+RXLNM&h\cm+m+Mb+`3\^] d RT]K(LNRX]KRTL(WcVWc] d L(WjSRTL(WcVºzZWcVXUaMb c\ VTRT-Y¡h\EM hRTM]wY BaY[IKMiUK`K`3Mm¡`WcmgY¡RTb¨hWZVXhUKV WY[RT\^]Kb¨\c] eLN\ d MV6
(369) RY[I ) 5B Y[IKMeVT\§¡Mm
(370) `WZmgY_RXb
(371) hWcVXhUKVXWZY[RT\^]Kb
(372) \^]Q 4® ° Q . . ¼ÏͼÐ.
(373)
(374)
(375) ! " #%$&!')(* +-,.. AVERAGED VISCOSITY. }F. STEADY VISCOSITY. 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 LEVELS. 2RXcUKm+M M5= ¨\cL(`KWcm+RTb+\^]>M dKd fzERXbgh\^bgRTYf \^|aY[WcRT]KM d
(376) RTY+I W zcMm[WZ^M d |Af`sMm+RT\ d \c E UKWcb+RT`sMm+RT\ d RXh~bg\^VXUSY[RT\^] 2 6qWc] d \^|SYtWcRT]KM d
(377) RY[Ixbg\^VXUSY[RT\^] c\ ¥Y[IKM . MAUWY[RT\^]Kb
(378) RTY+ILNMWZ] zcMVT\ShRTY+RXMb 2bgY[MW d fM dKd fzERXbgh\^bgRTYf O3 6. ÍÍ à SaÉÒ©ÉÒ.
(379) }. ( # (*. (*
(380) - '. ' ¦%#k' HIKRXb
(381) bgIK\^mgY_bpY[U d fIKWcb
(382) h\^]KhM]wY[m+WZY[M d \^]Y+IKMeUKb+Me\c + " #¯Y+UKm+|aUKVXM]wYizERTb+h\cb+RYf\^m ® ° QLN\ d MVXVTRT]KK ®&]aM]K\{zcMVYfRTb
(383) Y[IWZY_Y[IaRXb_LN\ d MV RXb_|wfh\^]KbpY[mgUKhY+RX\^]h\^]KbgRXbpY[M]AYq
(384) RTY+IY[IaMhWcVXhUKVXW Y[RT\^]0\Z¨bpY[MW d f LNMWc] \{
(385) b B RT] d MM d ¤ Y[IaMVTMzcMVn\czSRTb+h\^b+RYfRXb d R¢¶Mm+M]AYm+\cL Q 4 L(\ d MVXbNWcb~bg\S\c]WZb~UK]KbgY+MW d RX]KMb+bWc`K`3MWcmgbNWc] d RTb Y[IKMb+WcLNM
(386) IKM]±UK]KbpY[MW d RX]aMb+b d \SMb
(387) ]K\cY_Wc`K`3MWZm 9i\{¨Mkz^Mm¤
(388) IKM] Q 4 `am+\{zER d Mb~W^\S\ d `Km+M d RXhkY[RT\^]º¤2Y+IKM(]KMLN\ d MVn`Kmg\§zER d MbVXMb+b `sMm+RT\ d RThe\^UaY+`KUaY[b¤K
(389) RTY+IWZ]WZhhUam[Wchkf\c2`am+M d RXhkY[RT\^]W~VTRTYgY[VTM d M^m+W d M d 8] Y+IKM¦`Kmg\^`3\^b+M d b+AUWcmgM¦hfEVTRX] d Mm¥MkjaWcLN`KVTMc¤w¡M¦bgY[WcmgY2
(390) RTY+INW¯r y¯QhWcVThUKVXWZY[RT\^] Y[IWZYnRTb¥^\E\ d M]K\^UKcI(\^m2`Km+M d RXhkY[RT]Km+WZY[IaMm ¡MVTVKY[IKM
(391) AUWcbgR`sMm+RT\ d RTh¦\{eL 8](L(Wc]wf \cY[IKMm¯hWZb+Mb¤ºY[IKM~hV WZb+b+RThWZV¥r y¯QhWcVXhUKVXWZY[RT\^] hWc]a]K\cY&`Kmg\{zSR d M~bgUKhtIWc]xUK]KbpY[MW d f \{e> 8«Y_RTb_Y+IKM] ]KMhMbgb[Wcmpf Y+\h\^]KbgR d MmqW~£mgbgY
(392) Y[mgR WcVmgMVfERX]K\^] W(LN\ d R£M d r y¯Q L(\ d MV4¤6RT]
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