aiaa 2001-0545 development of a decay model for vorticescfdbib/repository/tr_cfd_01_59.pdffor...

10
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344 AIAA 2001-0545 Development of a Decay Model for Vortices Interacting with Turbulence Henri Moet, Denis Darracq, CERFACS, Toulouse, France and Alex Corjon EADS Airbus SA Toulouse, France 39th Aerospace Sciences Meeting & Exhibit January, 8 - 11, 2001 / Reno, NV

Upload: others

Post on 23-Jan-2021

1 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344

AIAA 2001-0545

Development of a Decay Model for VorticesInteracting with Turbulence

Henri Moet, Denis Darracq,CERFACS,Toulouse, Franceand Alex CorjonEADS Airbus SAToulouse, France

39th Aerospace SciencesMeeting & Exhibit

January, 8 - 11, 2001 / Reno, NV

Page 2: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

� �����������������������! "�!��#$ �%'&(��)*���+���,.-/�0,1�324#���5768���9��,: �#;�<2=�?>A@B2��DCEGF*,IH*F������?#��

JLK=MNPO<Q�R�K=S�T UV;K=MOXWLV�YZNPN�Y\[=]^<YZM_a`LbXK=c�deR1NgfhR1MIij�kml<nZo�j�p�qrj�n�sutwvyxZz|{}�~?xZ�=�;j��Z�P�X�8�����3n�q��1�Z�Z�1��tw�Z�Z���Z�Z��v�{n��xZ�:�Pv

������h�4�I�Z��D���+���8�� �¡£¢y��¤8¥����8¦=�Z§w¢©¨�ª9«¬�P­g®� �¡�¯Z¢P­�¤1§���¤8¥�°Z¤Z��¢y­�¦��±��­X§1¢P�±��¯Z¦=¢P­g§�²´³D­£¢��µ¢� ��=¶ ·¡��Z§�¦=��¯Z���¸ ·�y�8¹¬¢y¤¹��º�§���¯��y­X®�»�¡��*¢� ��=¶ �¡£�¼§·¦=�?¹½�±¦=¯8¾¸®�¤w¹�±¡À¿¸�D�·��¤1¶��P�±��°8¯Z¢y­�¤1§·�©®�¯Z¹��¹I ���­X§�²�¢����*¯1§�¯Z¡��P­���¤8¥�¢�������±�� �¡�¢P������¤8³<�8¹Á¢����*¯1»»½�±¯Z��¯1§�¦=��¤8¥$�P�±°Z�8��¯Z¡D»���Z§�¤1®��Z§�¯�¢���¯Z¢�¹½¤1®�­g§�¯Z¢P��¢����*°Z¤Z��¢P�±Âù�8¦=¯8¾Z¿¸Ä�§¯Z¦=¦=¤1®�»�¯1§w¾w­g§�²�»�¯Z��¯1®��±¢P�±�Å�y¢� ·¹¾����±�� �¡�¢P�±¹�­g§�¢����LÆ� �¯1§w¢y­ÇºI¦=¯Z¢y­£¤1§�¤8¥È»½�8��¢P­g§��Z§w¢È»�¯Z��¯1®��8¢y�8������ �¦=��¯Z�¢����r¢� ��=¶ �¡£�¼§�¦��<­g§w¢y�Z§��P­�¢�¾ZÉZ¢��·�D­X§1¢P�±²Z��¯Z¡�¡��Z§�²Z¢����P¦=¯Z¡��r¤8¥�¢����e¢� ·�h¶½ �¡��Z§�¦=�<¯1§·¹?¢����D°Z¤Z��¢P�±Â��P�¼»·¯Z��¯Z¢P­�¤1§�¿�D���8�y��»�¯Z��¯1®��±¢P�±���<¯Z����­g§�¦�¤1¤Z�=»¤Z��¯Z¢P�±¹�­g§´¯�§��8³Ê¢� ��=¶ �¡£�¼§�¦��$¹½�±¦=¯8¾Ë®�¤w¹�8¡�¢���¯Z¢r­£�̶½�8�y¢$®�¯Z¢P¦=���±¹¢y¤�¢�����§� ®��8��­�¦=¯Z¡��y­X®? ·¡�¯Z¢y­£¤1§����±�� �¡�¢P�=¿

͸Î�ÏmÐ8ÑZÒmÓÅÔÖÕ\б×ÀÒIÏØ OXM[=KrY;Ù�K=ÚµÛ1K±Y\NyW<SPÜKrK�Ý�K=[=SPWrRZÙmYZSyÞ*R1WPßÜK=NPOg[e[=RwM�à

_OXSyOXRwMW�R1M¬Y\OX[=N�Y¼Ù�S�SyN�YZOXbXOXMá+â1RwNySPOg[�K=W?Ü�Y8â1K�ã�K=K=MäM�å�àÞ�K�NyOX[±YZbXbXÛ�OXM:âwK=WySPOXá:YZSPK=_mæ£ç�è�é*êrÜK$Y\OgÞ R\Ù�WPå[�ÜËWPSyå_½OgK�WOXW?SPR¸]�åIY\M�SyO£Ù�Û¸Y\OgNP[=N�Y¼Ù�S�ÚëY\ì1K�âwR1NPSyK=cÁã�K=ÜIY8â�OXR1N�íÀO�æ KwæWySPNyK=M½á1SyÜäYZM_ÁSPN�Y=fhK=[=SyRwNyÛî�OgMÁY�bXR�[±YZb3ÚDK±Y\SyÜK�N?WPS�YZSPKíïO�æ K1æ9[=NPR1WPWyÚrOXM_mU½â1K=NPSyOX[±YZb�ÚrOXM_�WPÜK±YZN8UWPSyN�Y\SyO£ðI[±Y\SyOXR1MÅUYZSPÞ�RwWyßÜK�NyOX[�SyåNPãåbXK=M[=KwUmßNPK=[=OXßOXS�Y\SyOXR1MÅægæXæñî�OXM+RwNy_K�NSyR´_K�ðIMK*W�Y¼Ù�K�YZOXNy[�N�Y8Ù�S�à�WyßIY\[=OXMá+NyK�á1åbïYZSPOXR1MW;Ù�RwN�`LOXNê9N�Y¼ò�[$Q+Y\MIYZáwK=Þ�K=M�S�íï`<êLQ´î=æØ åß½ßIRwNySyK�_óã�Û/ô9åNyRwßIK8YZMóNPK=WyK±Y\Ny[�ÜóßNPR1áwN�YZÞ*Wõí�d3àö¬YZìwK1U Ø à�ö¬YZìwK1ULô9÷�ørùrö+`;ú�ô<î�drô<øLûw`Ld Ø OgW*OXM�àâ1RwbXâ1K=_*OgM�SPÜOXWrM:å½Þ�K=NPOg[8YZb�K�Ý�RwNyS8æü�M�ßNyK�â:OXRwåWrßIYZß�K=NPW�ý�þ�ÿPè��ÚDK�ÜIY8âwK�ßNyK=WPK=M�SyK=_��8àyV ÚrYZì1K�âwR1NySPK=c�WPOXÞ�åbïYZSyOXRwMW

ÚrOXSyÜ+áwNyRwåM_ÃK�Ý�K�[=S±U9deNyRZÚ OgM½WyS�Y\ãOXbXOgSPOXK=W±U½[=NyRwWyW�ÚrOXM_WyܽK±YZN�YZM_�YZSyÞ*R1WPßÜK=NPOg[�SPåNyãå½bgK�M[=K1æ

ü�M SyܽOgW ßIYZß�K=N±U M:å½Þ�K=NPOg[8YZb�NyK=WPåbXSyW.R\Ù ÚrYZìwK.â1R1N4àSyOX[=K�WËOXM�SyK�N�YZ[�SyOXMáÁÚrOXSyÜäÙ�åM_IY\Þ�K=M�S�Y\brSyåNPãåbXK=M[�K¸OXWOXM�SyK=MWPOXâ1K=bXÛGOXM�â1K=WPSyOXá:Y\SyK=_óíÀWyK=K¬Q R:K�S¸K=SÃYZb�æ �hîAYZM_K=c�ßbXR1OXSPK=_¬OXM¬RwNy_K�N�SyRÃ_K�ðIMKÃY+Þ*R:_½K=bDR\ÙL_K=[±Y8ÛµRZÙ[=OXNy[�åbïYZSyOXRwMmæ����� ��Ô����ÑZ×�Õ������ ÒmÓ�����Ï<ÓõÎhÏÈ×�Ð8×�������ÒIÏ<ÓÅ×�б×�ÒIÏ�����ïÍ����� ��Ô����ÑZ×�Õ������ ÒmÓ���êrÜK+ßNPK=WyK�M:S !#"%$ &('B[=R�_K*)´WyRwbgâwK=W�SPÜK +�Y8â:OXK=N4à

Ø SPR1ì1K�WÈK�]:å�YZSyOXRwMWmR\Ù�Y��8àyV?U�åMWPSyK±Y\_Û?YZM½_�[=R1Þ*ßNyK�WyW�àOXãbXK�,IRZÚ.ÚrOXSyÜÃÞ�åbXSyO£à�Wyß�K=[=OXK=W8USyN�YZMWPßIRwNyS?Y\M_+[�ÜK=Þ0àOXWySPNyÛ�æëù;ã�â�OXR1åWPbgÛ�U�Ù�R1NrSyܽOgW;WySPå_Û�UIÞ�å½bgSPO£à WPßIK=[�OgK�WLYZM_-/.10�243�576�8�9;:<5;=?>A@ABDC<ED>%FG=H9/IKJMLONQPRIKS*9/5UT�V;9/WYXQJMV;3/Z X[W/=\ 0�SM:�9]P_^�`M`Ka�b�c�c;d�`]e�f*a�=gX[JMhGP1^�`M`ia�bjc�c;d�`Me�eMek F<9;:�LOSMW�BD9;379;J]W�V(0�9/W;=�EDlmEDEnIK9/I�oG9;W/=p>A@1B�C*ED>%FG=A9;IKJMLONQP8�JMW7W�J]V;q�TDV/9;WmX[JMV(3/Z X[W/= \ 0�SM:�9]P1^D`]`ra�b�cKc;dK`]ersge�=<XQJMhpP%^�`M`rab�c�c;d�`Me�e]et EDlYE�E�IK9/I�oG9;W/=*@AED2�F�EDLOW7o�6�3�F<Eu=�`�c;bKBDS]6�5�9�8�9�v%Jxw]ySM:�:�9�C�yY`�c;eMbMeuz{S]6�NOSM6�3�9�>A9;8�9/hK`�=*C�W7JM:�V;9>AS \ w*W7LO|M0<5 V} sgeMejc~o<w�570�9DE�IK9;W7LOV;JM:�lm:�3�57LO5�6�579ASgX[E�9;W7SM:�JM6�57LOV;3JM:�8�ED375�W7SM:�JM6�57LOV;3/=xlm:�VMZ�EDNON�W�LO|]0*5731W79;379;W7�M9/8�Z

[�ÜK�Þ�OXWySPNyÛ�[±Y\ßIYZãOXbXOXSyOXK=WrYZNPK$M½R1SråWPK=_¬æ` ÜOXá1ÜbXÛ+YZ[�[=åN�Y\SyK�ðIMOXSyK�_O£Ý�K=NyK=M½[=K�Þ�K=SPÜR�_+RZâ1K�NÌY[8YZNySPK=WyOïY\M¸áwNyOX_¸OXW;åWyK=_ÁSPR*Ù�åbXbXÛ�ÜIY\M_bXK?Þ*R1WPSrRZÙrSyÜK_½O�Ý�K=NPK=M�SDWP[±YZbXK=WrOXM�SyܽKr�8àyV�,IRZÚ��mWPßIYZSPOïYZb_K�NyOXâ1YZSPOgâwK=WY\NyK+[�R1Þ�ß½åSyK=_äÚrOXSyÜGY��j���ÃR1NP_K=N�[�R1Þ�ß�YZ[=S�WP[�ܽK=Þ�Kí7�9YZ_K�K�WP[�ÜK=Þ�K8î�ç;��ÚrܽOg[�Ü+Ü�YZWLSPÜK�ßNyRwßIK=NPS�Û+RZÙ<ã�K=OXMábXRZÚeà�_OXWyß�K=NyWPOXâ1KDY\M_?M½R1M�à�_OXWyWPOgß�YZSyOXâwK1æ�êrÜK=WPKrWy[�ÜK=Þ*K=WY\bXbgRZÚ Ù�R1N;WyR1Þ*K�]�åIYZWPO£à WPßIK=[�SyN�Y\bDYZ[�[=åN�Y\[=Û�æ�êrOgÞ*K�YZ_�àâ1Y\M[=K=Þ*K=M�S�OXW+YZ[�ÜOXK=âwK=_ÊÚrOXSPÜÊY����x�ÃRwNy_K�N¸øråM½á1K�àú�åSPS�Y Þ�K=SPÜR�_mæ êrܽK¬[=R�_KäOXW¸WPåßß�R1NPSyOXMá _O£Ý�K=N4àK�M:S*S�Û�ßIK=W�R\Ù?ãIRwåM_IY\NyÛä[=RwM_OXSyOXR1M½W�SyÜ�YZMì�W�SyRµSyÜKå½WyK�RZÙDSPÜKH+ Ø d��rd.Þ�K�SyÜR�_mæ ç�ç ü�M+ßIY\NySyOX[=å½b Y\N±U�M½R1M�àNPKg,IK=[�SyOXMá�RwßIK=M�ã�R1åM_�YZNyÛ�[=R1M_½OgSPOXR1MW<YZW9ÚeK=bXbYZW<YZ_O£àY\ãIYZSPOX[LR1NrOXWPR1SyܽK=NyÞ�YZb�WyR1bXOX_�ÚrYZbXbXWrYZNyK�OgÞ*ßbXK=Þ�K=M�SyK�_mæêrܽK��mô Ø YZß½ßNyR�YZ[�Ü.OXW¸ãIY\WyK=_ R1M SyÜK ðIbXSyK=NPOgM½áÊR\ÙSPÜK.[=RwÞ�ßbXK=SPKÊ[=RwÞ�ßNPK=WyWPOXãbXK�+�Y8â:OXK=N4à Ø SyR1ìwK=WGK=]�åIY¼àSPOXR1MW�íï_åK�SyR*SyÜK�ð�MOXSyK�Þ*K=WyÜ�à�WyOQ�=K8î=æ�êrÜK�OXM�,IåK=M½[=KR\ÙIMRwM�à NPK=WyRwbXâ1K=_�Wy[±Y\bXK=W<R1M�SyܽKDNPK=WyRwbgâwK=_�Wy[8YZbXK=W<R\ÙIÞ�R\àSPOXR1M�OXWLS�Y\ì1K=M´OgM�SPR�YZ[=[=RwåM�SLã�Û�Þ�K±Y\MWrRZÙDY�WPåã�à�á1NyOX_WP[±YZbXK�Þ�R�_K�bIãIY\WyK=_�RwMËSPÜK�OXM:SPNyR�_å[�SyOXR1M*RZÙDY�SPåNyã½å�àbXK=M�Srâ�OXWy[=RwWyOXS�Û?� � æ£çyý���[�ÁÎ�Ï9×�б×7������ÒIÏ<ÓÅ×�Ð8×�ÒIÏ��ê9RË_½K�ðIMK�SyܽK�OgM½OgSPOïYZb<[=RwM_OXSyOXRwMW;Ù�R1N�SPÜK�WPOXÞ�åbïY8à

SPOXR1MW8UDÚeK�ÜIY8â1K´åWyK=_ YZMIY\bgÛ�SPOg[�K=c�ßNPK=WyWPOXR1MW*Ù�RwN�SyÜKâwR1NPSyOX[=K=W8æ�êrÜK��mY\Þ�ã�à4ù;WPK=K=MÃâ1R1NPSyK=c´ÚëY\WL[�ÜR1WPK=Mmæ�ü�SOXWDY\M�åMWPSyK±Y\_Û�WyRwbgå½SyOXR1M�RZÙmSPÜKi+�Y8â:OXK=N4à Ø SyR1ìwK=W<K=]�åIY¼àSPOXR1MW�Ù�RwN�SyÜKeK=âwR1bXåSyOXRwMLRZÙIY\M$OXMOXSPO Y\bZbXOgM½KÈâwR1NPSyK=c�[�ÜIYZN4àY\[=SyK�NyOQ�=K=_Ãã�Û Y�[�OgNP[=åbïYZSPOXR1M�� íïWyK�K��ÅYZÞ�ã�ç�éLßA�p�1î�ærü�SÞ*R�_K=bXWÅÙ YZOXNybXÛ�ÚeK=bXb�SyÜK;â1R1NPSyOX[=K=WrWPÜK=_*ã:Û�Y\M�YZOXNy[�N�Y8Ù�S*�]�åIY\WyO£à�WyR1bXOX_�NPR1S�Y\SyOXR1M�OXM�SyÜKr[�R1NyK�YZM_�SyܽKL[=OXNy[�åbïYZSyOXRwMNPK±YZ[�ÜK�WDYr[=RwMWyS�YZM�S9â1YZbXåKDY\S9OgM·ðIMOXS�Û:æ ü�ÙISyÜKe[=Û�bXOXM_NyO£à[8YZb1[�R:RwNy_OXMIY\SyK=Wrí7���x���g�1î�Y\NyK<[�R1MWPOg_½K=NyK=_ÅU1SPÜKDOXM_½å[=K=_âwK=bXR�[=OXS�Û��� OXW�RwMbXÛ+Y��=OXÞ�åSPÜIYZb9YZM_ÃOXSyW�WyßIY\SyOïYZbX_OXWPSyNyO£àã½åSyOXR1MÃ_K=ß�K=M_½W?RwMbXÛ+R1Mn��UmY\M_ÁSyܽK�ßNyK=WPWyåNPK �óOXWáwOXâ1K=M*ã:Û*OXM:SPK=á1N�YZSPOgRwMD�

��   �A¡ í7�Zî�¢¤£u¥ � �;¦�¤§p¨�© K=c�ßÈí ©Kª í �� ¦ î ý �/­�/� ¢¯®D°p±²�¨

`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W

Page 3: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

öGÜK�NyKK���9OXWmSPÜK<[=RwNyKeN�YZ_½Ogå½WÈYZM½_�¥ � OXWmSPÜKDÞ�YZc�OXÞ?å½ÞS�Y\Má1K�M:SPOïYZb9â1K=bXR�[=OXS�Û¸Y\Sr���¼æ�£äYZM½_ ª [±Y\M¸ã�K�WyOXÞ�ß½bgÛÙ�RwåM_ÊY\M_GÜIY8âwK+SyÜK´Ù�RwbgbXRZÚrOXMá+â1Y\bXåK=W*� £ ¢ ¨�� ���YZM½_ ª ¢ ¨�� �p����íïWPK=K?deRwNgf�RwM�K=SLYZb�æ£ç�ÿhî�æ9ü�SrWyÜRwåbX_�ã�KMRwSyK=_�SyÜ�YZSrMR�Y\c:OïY\bIâwK=bXR:[�OgS�Û*OXWDOXMOXSyOïY\bgOQ�=K�_mæêrÜK�bXOXMK±Y\NLWPåßIK�Nyß�R1WyOXSPOgRwM¸R\Ù<S�ÚDR?�ÅYZÞ�ã�àyù;WyK=K=M´â1R1N4àSyOX[=K�WËOXW*MK=OXSyÜK�N Y WyS�YZSyOXRwMIYZNPÛõWPR1bXåSPOgRwMäRZÙ?SPÜK¸ô9å�àbXK=N�K=]�åIYZSPOXR1MW*MR1N+YÁ]�åIY\WyO£à�WySyK8YZ_ÛäWyRwbXåSyOXR1M RZÙ?SPÜK+$Y8â�OXK=N Ø SPR1ìwK=WÁK=]�åIYZSPOXR1MW8æ êrÜK=NyKhÙ�R1NPK1U�ÚeKGYZ_IY\ßSSyܽK¬S�ÚeR��mY\Þ�ã�à4ù;WPK=K=M/âwR1NPSyOX[=K=WÃSyRäÙ�R1NyÞ YäâwR1NPSyK=c_OXß�R1bXKGÚrÜOX[�ÜóOXWGY.Þ�K�Þ�ãIK=NÁR\Ù Y.å½MOX]:å½KGÙ Y\Þ�OXbXÛRZÙ�_OXß�R1bXK=W�SPÜIYZS�Y\NyK+]�åIY\WyO£à WPSyK±Y\_Û¬WPR1bXåSPOgRwMW�RZÙ�SPÜK+$Y8â�OXK=N4à Ø SPR1ìwK=W+K=]�åIY\SyOXR1MW8æ êrܽK¬YZ_IY\ßS�Y\SyOXR1M OXW¸YZMK=M�SyOXNPK=bXÛËS�ÚeRZà�_OXÞ�K�MWyOXR1M�YZbmYZM½_+â�OgWP[=R1å½WLßNPR:[�K=WyW�íÀWyK=KØ OXßßÃK=S?Y\b æ£ç ��î=æ�êrÜK�âwR1NySPK=c+_OXß�R1bXK�OgW�WPåãWyK�]:å½K=M�SybXÛWyå½ßIK=NPOXÞ�ß�R1WyK�_ËRwM+Y�ðIK=bX_�RZÙ<ÜR1Þ*R1áwK=MK=RwåWDOXWPR1SyNPR1ßOX[Syå½NyãåbXK=M½[=KDÚrܽOg[�Ü�OXW�SyÜK3OgM½OgSPOïYZbZ[=RwM_OXSyOXR1M;RZÙR1å½N�WyOXÞ�àåbïYZSPOXR1Mmæ�êrÜK0ðIK=bX_ÁRZÙLSPåNyã½åbXK=M[=K�ÜIY\W?ã�K=K=Mäá1K�MK=N�àYZSPK=_�OXM�SyÜKrWPßIK�[=SyN�YZb�ßbïYZM½KråWyOXMá�SPÜKeÙ�RwNyÞ�åbïYZSPOXR1M�RZÙ�<RwM�ú �Y\NyÞ �YZM?Y\M_?�9YZRIæ Ø K=âwK=N�Y\b�Wyß�K=[=SPN�Y�ÜIY8âwKrbgK�_�SyRÚeK=bXb<_K=âwK=bXR1ß�K=_ÁÜR1Þ*R1áwK=MK=RwåWLOXWyRwSyNPR1ßOX[�SyåNPãåbXK=M[=KðIK=bX_W�íÀWyK=K�ûmOXá�æ ¨ î;[�ÜIY\N�YZ[�SyK=NPO[��K=_¬ã�Û+_K�WyOXNyK=_Áâ1Y\bXåK=WRZÙ9SyÜK�SyåNPãåbXK=M[=K�OgM�SPK=MWyOXS�Û�U1ÚrܽOg[�Ü�OXWr[�ÜIYZN�YZ[=SPK=NyOQ�=K=_ã�Û�SyܽKrNyR�R1S4à Þ*K±YZM�à�Wy]�åIY\NyK<âwK=bXR�[=OXS�Û:U � � ��� U1Y\M_�bgRwMá1O£àSyå½_OXMIYZbwOgM�SPK=á1N�YZb�bXK=Má1SPÜ�Wy[8YZbXK1U���U8ÚrÜOX[�Ü�OgW9YrÞ*K±YZWPåNyKÙ�RwNrSyÜK?Y8âwK=N�Y\á1K�WPOQ�=K�RZÙ<SPÜK�Syå½NyãåbXK=M�S;K=__Û�ßNPK=WyK=M�SOXMÁSyÜK�SPåNyã½åbXK=M[=K#,IRZÚ ðIK=bX_mæ#+LRwSyK*SyÜIY\S�SyÜK*NyR�R1S4àÞ�K8YZM�à�Wy]�åIY\NyKLâwK=bXR�[=OXS�Û�ÜIY\WrãIK=K�M+_K�ðIM½K=_¸YZW<�

� � ��� ¢�� ��� ý� ¢ � � � ý�� � � ý�� � � ý í ¨ î���H¢! #"�%$ í��Zî'&{� í îÚrOXSyÜ $ í��Zî;SPÜK�bXR1MáwOgSPå_OXMIY\b�[=R1NPNyK=bïYZSPOXR1M�Ù�å½M[=SyOXRwM+RZÙSyܽK$SPåNyã½åbXK=M[=K�_K�ðIM½K=_¸YZW<�( $ í7�Zî�¢*)�+,+ - �'. ��. ��. �,/°�0 ± ÚrOXSyÜ1 �2 í�� �43�î�¢ � �� í65 �73�î � �2 í65 � � �43�îV;åNPOXMá�SyÜKe[=R1Þ*ßåS�YZSyOXRwMWmSPÜKrK=MK=NPá1Û�[=R1M�S�YZOXMK=_�ã�ÛSyܽKËSPåNyã½åbXK=M[=K+OXW�MRwS©Y\NySPO�ð�[=OïYZbXbXÛ¸Þ�YZOXM�S�YZOXMK�_¬YZM_OXSyWrOXM�SyK=M½WyOXS�Û�ÚrOgbXb�_K=[±Y8Û�OXM�SyOXÞ�Kwæ9êrÜOXWLOXWrßIK�NyÞ�OXSPSyK=_ÚrÜK=M*[=R1Þ*ßIYZNPOXMá�SyÜK;SyOXÞ�KeWy[±Y\bXKLRZÙmSPåNyã½åbXK=M[=K�ÚrOXSyÜSyܽK<SyOXÞ�KÅWy[±Y\bXK<RZÙSyܽK<â1RwNySyOX[=K�W±æÅ`Á[�Ü�YZN�Y\[=SyK�NyOXWySPOg[eSyOXÞ*KWy[8YZbXK�Ù�RwNDSPåNyãå½bgK�M[=K�OgWrSPÜK�K=_½_Û�SPåNyM�à�RZâ1K�NLSyOXÞ*K�378ÚrÜOX[�Ü�OXW<_K�ðIM½K=_�ã:Û�SyܽKLN�Y\SyOXR�RZÙmSPÜKLOXM�SyK=áwN�Y\bIbXK=MáwSyÜWy[8YZbXK9�!SPRÁSyÜK�N8æ ÞÃæ W±æÃâ1K=bXR�[=OXS�Û � � ��� æµêrÜK�[=R1NPNyK�àWyß�R1M½_OXMá�[�ÜIYZN�YZ[=SPK=NyOXWySPOX[�SyOXÞ�K;Wy[±Y\bXK�Ù�R1NeSyܽK�â1RwNySyOX[=K�W3;:.[�R1åbX_Áã�K�_K�ð�MK=_¬WPOXÞ�OXbïYZNybXÛ*ã:ÛÃSyܽK�N�Y\SyOXR�RZÙLSPÜKOXMOXSyOïYZb3[=R1NPK+N�Y\_OXåW���� <�SPRÁSyÜK�Þ�YZc�OXÞ�YZb<Y��=OXÞ�åSPÜIYZbâ1K�bgR�[=OXS�Û ¥ � YZSrOXMWPS�Y\M:S=3�¢>�½æ

3783;: ¢ �i¥ ���� < � �'�?� í7�1îÚrOXSyÜ � � ��� ¢A@ �B YZM_ B SyÜK*ì:OXMK�SyOX[�K=M½K=NyáwÛ¬[=RwM�àS�Y\OXMK=_¬ã�ÛÃSyÜK*SyåNPãåbXK=M[�K1æ¬û½R1N�RwåN?[�R1Þ�ß½åS�Y\SyOXR1MW

C=DFE�GIH�G & J�K�L7J�MONQP�R�SUT�J>K'PVRXW�DQR�Y[Z\K�N�],DQS\DQ],^`_ acb�_edf�GFghfO_ acbh_ i�j'klK Pm]onhT�p�&�"rq�T�Yts�u�M�J�]�vwN�DQK�Nx]6KyDQz�u�TUS\]6D{z�E]onhT9Z\K�N�],DQSUT�J|G "}n�T[]6MON�~\M�YtT\zhSUT9DQzh],TXzOJ�DQ],^9� ° d�H�f�GQ�X�R�zhs�]tn�TIYoK�z�E�DQ],M�s�DQzhR�Y�D{zh]6T7E�N R�Y�YtT\z�E�]tn!J�SUR�YoT9�?�G���\d��Go���Of�GSPÜK�N�Y\SyOXRÁOXW?R\ÙÌSPÜK�R1NP_K=N�R\Ù��*í����1î�ÚrÜOX[�ÜäOgÞ*ßbXOXK=WSPÜIYZS�SPÜKÃSyåNPãåbXK=M[=KÃ_K=[8Y8ÛõOXW*Wyå�ò�[=OXK=M�SPbgÛµWybXRZÚBSyRRwãS�Y\OXM?â1Y\bgOX_�NPK=Wyå½bgSPW±æ3`LM�YZM�YZbXÛ�WyOXW<RZÙ�SyܽKL[=OXNy[�åbïYZSyOXRwMK�â1R1bXåSPOXR1M�R\ÙëY�â1RwNySPK=c�[=R1M�ð�á1åN�YZSyOXRwM+OXWrÞËYZ_½K$OXM�R1N4à_½K=NmSPRLK=WPSyOXÞËY\SyK9SyÜK3K�Ý�K�[=SyW<R\ÙSyåNPãåbXK=M[=KrYZM½_�â1RwNySyK�cß�YZN�Y\Þ�K=SPK=NyW9WPK=ßIYZN�YZSPK=bXÛ?RwM�â1R1NPSyK=c�_K=[8Y8Û:æÅêrÜKr[=OXNy[�å�àbïY\SyOXR1M�ÚrYZW;Ù�R1å½M_+ã�Û+OXM�SyK=áwN�YZSPOXMá�SPÜK©Y\c�O Y\b9[=R1Þ*ßIR\àM½K=M�S�RZÙ�â1RwNySyOX[=OXS�ÛµRZâ1K=N�YÃ[=OXNP[=åbïYZN*NyK=áwOgRwMäÚrOgSPÜäSyÜK[�K=M�SyNyK�OXM+SPÜK�â1R1NPSyK=c+Y\c:OXW8æ�êrÜK?NPK=Wyå½bgS�OXW$Y0[=OXNy[=å½b Y¼àSPOXR1M�ß½NyRZð�bgK�Y\WLY�Ù�å½M[=SyOXRwMËR\ÙÈSPÜK�N�YZ_½Ogå½WrS�YZìwK=M*Ù�NyRwÞSPÜK�â1RwNySyK�cËYZc�OXW±æ �<í7�Zî�¢ * � a b &�� í{�:îêrܽK�� ý à [=NPOXSyK=NPOgRwM�_K�ð�MK=_�ã�Ûm�1K=RwMárK=S9YZb�æ£ç4��ÚrYZW�åWPK=_SPR�bXR�[±YZSPKLSyܽKLâ1RwNySPK=c�YZc�OXW<ÚrÜOXbXKrSyÜK;â1RwNySyOX[=K�WëY\NyKL_½K�àÙ�R1NyÞ*K=_�YZM½_�Þ�RZâ�OXMá�OgM*SyÜK;[=R1Þ*ßåS�YZSyOXRwMIYZb_½R1ÞËY\OXMmæ

���*Ô����ÑZ× Õj���������Ô���Ð�����ïÍI��×�Ïc�A�7l�$Ò�ÑZÐ<�����Ò�Ï�����Ô9Ñ��б×�Ò�ÏêrÜK�M:å½Þ�K=NPOg[8YZbmÚeR1NPì�ÜIYZW;WyÜRZÚrMÁY�â1Y\NyOXK=S�Û�RZÙDYZW4à

ß�K=[=SPW�[=R1M[�K=NyMOXMáäSPÜK+K=âwR1bXåSyOXRwMõR\Ù?â1RwNySPOg[�K=W�OXM�SyK=N4àY\[=SyOXMá ÚrOgSPÜ/SPåNyãå½bgK�M[=K1æ êrÜOXWÃÚëY\W Y\[�ÜOXK=â1K=_ ã�Û[�R1M_å½[=SyOXMá �¨ �Åô Ø W�RZÙLY�,�RZÚeðIK=bX_¸[�R1MWPOgWPSyOXMá+R\Ù|YâwR1NPSyK=c ,IRZÚ K=Þ�ãIK�__K=_µOgMäY�ð�K=bX_¬RZÙLܽR1Þ�Rwá1K=M½K=R1åWOXWPR1SyNPR1ßOX[ÃSyåNPãåbXK=M[�K1æ êeÛ:ß½Og[8YZb�ßNyR�[=K�WyWyK�W¸WPå[�Ü/YZWâwR1NPSyK=c¸Y\á1OXMá�[�ÜIY\N�YZ[�SyK=NPO[��K=_¬ã�Û+áwN�YZ_½åIYZb9[=OXNy[�åbïYZSyOXRwM_½K=[±Y8Û�U�âwR1NySPK=c/WPSyNPK=Sy[�ÜOXMá�U�[=NyK±Y\SyOXR1M RZÙ Yj�=OXÞ�åSyÜIY\bWPSyNPå[=Syå½NyK=W�RZÙ9â1RwNySyOX[=OXS�Û�bXR1M½áZàÖY\M_�WyÜRwNyS4à ÚrY8â1K�OXMWyS�Y8àã½OgbXOXSyOXK=WeÜIY8âwKLã�K=K=M+K=M½[=R1åM�SPK=NyK=_Åæ`LM K=c�SyK=MWPOXâ1KäßIYZN�YZÞ*K=SyK=NÃWySPå_Û.ÜIY\W¸ã�K=K=M ßIK=N4à

Ù�R1NyÞ*K=_aÙ�R1N Y WPOgM½á1bXKóâ1RwNySPK=c [=R1M�ð�á1åN�YZSyOXRwM íïWPK=Kê<Y\ãmæ ¨ î�æGêrÜK+NPK=Wyå½bgSPW�bXK=_GSPRÁSyÜK´[=R1M[�bgå½WyOXR1MäSyÜ�YZSSPÜK�SPåNyãå½bgK�M[=K�OgM�SPK=MWyOXS�Û�UÚrÜOX[�Ü+OXWr_K�ð�MK=_ÁYZW�� ° ¢� � ��� �_¥ � UZÜIY\WëY�WPOXá1MO£ðI[±Y\M�SÈKhÝ�K=[�SrR1M�SyܽKL[=OXNy[�åbïYZSyOXRwM_½K=[±Y8Û�R\Ù<SyÜK�âwR1NPSyK=c�æ<êrܽK�_K=[±Y8Û�OXWrßNPR1ß�R1NySPOXR1MIY\b�SyR� ° íÀWyK=K?ûÅOXáIæ î=U�SPÜK=NPK�Ù�R1NPK�ßNyRwßIRwNySyOXRwMIYZb�SyR�SyÜK;NyR�R1S4àÞ*K±YZM·à WP]:å�YZNyKrâwK=bXR�[=OXS�Û:U � �'�?� U1R\ÙÈSyܽKLSyå½NyãåbXK=M½[=K�YZM_SPR�SyÜK�NyK=[=OXßNPR�[±YZb9RZÙ<SPÜK�â1RwNySPK=c�WySyNPK=MáwSyÜÃK=c�ßNyK=WPWyK=_

`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W

Page 4: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

SV−1

SV−3

t*=30 t*=60 t*=90 t*=120

t*=30 t*=60 t*=90 t*=120C=DFE�G���G & J�K�L7J�MONQP�R�SUT�JrK'P � ý K'PrR J�DQz�E�YtT9ZXK�N�],T;W R�]xJ�T\ZXT\N R�Y�]6D � T9DQzOJ�],R�zh]{J�����¥ ç�� ¥ � ���dmf�Go� � � ��� ����d�f�GQf��hH ���p���Xdm��Go����f���u¥ é�� ¥ � ���dmf�G�� � � ��� ���dmf�GQf\gO�� �?�G���\d���G�������G 3�Ti¢ 3 :�<�;¦ Gã�Û ¥ � U8ÚrÜOX[�Ü�OXWmSPÜKDÞ�YZc�OXÞ?å½ÞõS�YZMáwK=M�SyOïYZb�Y��=OXÞ�åSPÜIYZbâ1K�bgR�[=OXS�Û�R\ÙÖSPÜK�â1RwNySyK�c�,IRZÚ.YZSeSyÜK�[=R1NPK�N�Y\_OXåW±U����ZæêrÜKr[=OXNP[=åbïYZSPOgRwM�YZbXWyR�WPK=K=Þ�WeSyR�_K�[±Y8Û�Ù YZWPSyK=NeÚrOXSyÜ�OXM�à[=NPK±YZWPOgM½á¬bXR1MáwOXSyå_OXMIY\bDOXM�SyK�á1N�Y\brbXK=Má1SPÜäWy[±Y\bgKwU���UeRZÙSyܽK¸SPåNyãå½bgK�M[=K1æóû½R1N*Þ�K=_OXåÞ�YZM½_GWySPNyR1M½á¬Syå½Nyãå�àbXK=M[=K�ÚDK�NyK�[=R1áwMOQ�=K=_+SPÜK�[=NPK±YZSPOgRwM�RZÙÈâwR1NPSyOX[±YZbÅWySPNyå[�àSyå½NyK=W¸Y\NyRwåM_ SyÜKÃßNyOXÞ�YZNyÛµâ1RwNySyK�cG[±YZå½WyK=_.ã�ÛGSPÜKßNPR:[�K=WyW�RZÙ9â1RwNySyK�c�WySPNyK=SP[�ÜOXMá¸íÀWyK=K?ûÅOXáIæ��wî=æêrÜK�_K=[±Y8Û´RZÙëY*WyOXMáwbXK�â1RwNySyK�c+OXWLÞËY\OXMbXÛ�[�Ü�YZN�Y\[=SyK�N�àOQ�=K=_�ã�Û�SPåNyãå½bgK�M:S3_O£Ý�åWPOXR1MmUZSPÜOXW<bXK±YZ_½W<SyR�SyܽKrSyN�Y\MW�àß�R1NySrR\Ù<â1R1NPSyOX[=OXS�Û�Ù�NPR1ÞBSyÜK�bïYZNPá1K�ßNyOXÞ�YZNyÛ�â1RwNySPK=c�SyRSyܽKLSyܽKLWyÞ�YZbXbXK=N9SyåNPãåbXK=M�SeWySyNPå[=SPåNyK=W8æeöÁKDRwãWyK=NPâ1K=_SyÜ�YZS<SPÜKr_K=[±Y8Û�OXM?SPÜKr[=RwNyKrNyK�á1OXR1M�OXW9bïYZNyáwK=N±UwSyܽOgW9[±Y\Mã�K�K=c�ßbïYZOXMK=_Ãã�Û+SyÜK0Ù YZ[�S$SPÜIYZS�SPÜK�SPåNyãå½bgK�M[=K�ÜIY\WSyܽK$ß½NyR1ß�K=NPS�Û�RZÙ9NyK=_½å[=OXMá�SyܽK$Þ�YZc�OXÞ�åÞ.S�Y\Má1K�M:SPOïYZbâ1K�bgR�[=OXS�Û�RZÙ�SyÜKeâ1RwNySyK�c�,IRZÚ�æ�öGܽK=M?ÚeKeSyN�Y\MWybïY\SyKeSyÜOXWã�K=ÜIY8â�OXR1åN�ã�ÛäOXM:SPK=á1N�YZSPOgM½á¸SPÜK¸âwR1NPSyOX[=OXS�Û:UeYZM_äSPÜ:å½WSyܽK¸âwK=bXR:[�OgS�ÛäßNPRZðIbXK+R\Ù?SyܽK¸âwR1NySPK=c�UrSyRµR1ãS�YZOXMGSPÜK[=OXNy[�åbïYZSyOXRwMmUZSyܽKDbïYZNPá1K=N9[=OXNP[=åbïYZSPOgRwM�_K=[8Y8Û?OXM�SPÜKr[=R1NPKNyK�á1OXR1M�ã�K=[=RwÞ�K=We[=bXK±YZN8æ9`LMRwSyÜK�NrOgÞ*ßIRwNyS�YZM�S�Ù�K8YZSyå½NyK=WÚeKDÜIY8âwKrK=M[=RwåM�SyK=NPK=_�OXW9SyÜ�YZS9SyÜKrKhÝ�K=[�SDR\Ù�SPÜKrbïYZNyáwK=NbXR1MáwOXSyå_OXMIY\bIOXM�SyK�á1N�Y\bmbXK=MáwSyÜ�Wy[8YZbXK=W�R1M�SyܽK�_K=[±Y8Û�OXWÞ�RwWySPbgÛ�Ù�K=bXSeR1åSPWyOX_K�SyܽK$[�R1NyK�NyK=áwOgRwMmæ9êrÜOXWeOgWr_½åK�SyRSyܽKeÙ YZ[=S9SPÜIYZS<SPÜKrbïYZNPá1KeWy[±Y\bgK�W<RZÙ�SyåNPãåbXK=M[�KLWyK�K=Þ SyR_K�Ù�R1NyÞaSyÜK�âwR1NPSyK=c+bXK±Y\_OXMá�SyR�bXR1WPW$R\Ùë[�R1ÜK=NPK=M[=K*RZÙSyܽK?âwR1NySPK=cÃ[±YZåWPOXMá�Ù Y\WySPK=N�_O£Ý�WyOXR1M+R\ÙëâwR1NPSyOX[=OXS�Û+YZM_Wyå½ãWyK=]�åK�M:SPbXÛ�SyR�SyÜKLY\[=[=[=K�bgK�N�YZSPK=_�_½K=[±Y8Û�ærí Ø K=KLQ R:K�SK=SLY\b æ �±î���[� ��ÒIÑZÐ<O���i�½×�Ñ#�$Ò�Ï ���IÔ9Ñ��Ð8×�Ò�ÏêrÜK�ßÜK=MRwÞ�K=M�Y�ÚeK?ÜIY8âwK�WyK�K=MÁÚrÜOXbXK�OXM:âwK=WySPOXá:YZS4à

OXMá¬SPÜK+_K�[±Y8ÛGRZÙ�YÁâ1RwNySPK=cäßIYZOXN¸íÀWyK=KÁê<Y\ãmæ î�YZNPKâ1RwNySPK=c?YZáwOXMárã�Û�SyåNPãåbXK=M�S9_O£Ý�å½WyOXR1MmU8SPÜKD[�NyK±Y\SyOXR1M�RZÙY���OgÞ�å½SyÜIY\bIWPSyNyå½[=SyåNPK=W�R\ÙÈâ1RwNySPOg[�OgS�Û�YZM_�_KhÙ�R1NPÞËYZSPOXR1MßNPR:[�K=WyWPK=W?_½åK?SPR�SPÜK?ß½NyK=WPK=M[=K*RZÙ<WySPNyR1M½á Y\M_+bïYZNPá1KWy[8YZbXK?SPåNyã½åbXK=M�S$WPSyNPå[=Syå½NyK=W8æ�êrÜK=WPK�ßÜK=MRwÞ�K=M�Y�YZNPKWyOXÞ*OgbïY\NLSyR´SyÜRwWyK�K�M[=R1å½M:SPK=NyK�_õOXMÁSPÜK�WPOgM½á1bXK?âwR1NPSyK=c[=RwM�ðIá1å½N�YZSPOXR1MmæäöÁK�ÜIY8âwKËYZbXWPRÁR1ãWPK=NyâwK=_GbXR1Má\à?YZM_WyܽR1NyS4à ÚrY8â1K�OXMWyS�YZãOXbXOXSyOXK=W;ÚrÜOX[�ÜÃYZNyK�OXMܽK=NyK=M�S�SyR�SPÜK

drYZWyK ¥ � ��� � � �� � ° ���p� ���¥ � � � � ��� ���¥ ç � � � � � � �¨ � � � � � �����¥ ý � � � � � � ¨ ��� � �[¨ ��¨��¥ é � � � � ��� � �O� � � �� ���¥ ÿ � � � � � �¨ ¨ � � ��� � � � �����¥ � � � � � ��� ¨ � � � � ��

" R�~\G�H�G�� xZ��xZzLv��Àv=���r���e���X���8��vr�=�Z� �ïv"!��y�Zz$#I���ïx��À�g�Z�w�y�

0.0 25.0 50.0 75.0 100.0 125.0 150.0 175.0t*=tV0/rc

0.00

0.20

0.40

0.60

0.80

1.00

(Γ/Γ

0)r2

r2=4rc

0.0 25.0 50.0 75.0 100.0 125.0 150.0 175.00.00

0.20

0.40

0.60

0.80

1.00

(Γ/Γ

0)rc

SV−5;Iu=1%SV−1;Iu=7%SV−4;Iu=10.5%SV−3;Iu=24%

r=rc

C=DFE�G���G&%�Z\K�YoMO]6DQK�z K P ]tn�T[SXD{N S\M�YoR�],DQK�z9P�K�N9R J|D{z�E�YoTZ\K�N�]6TUW SUK�z�q�E�MON'R�],DQK�z DQz p&�"VP�K�N J�TXZ\TXN'R�Y�� ° �I],KUvN�d �����~UK�],],K � N�d�gp���XGâwR1NPSyK=cÁßIY\OXN?[=RwM�ðIáwåN�Y\SyOXR1MmæµêrÜK�[=NPK±YZSPOgRwMõY\M_äSyÜKß�R1WPOXSyOXR1MäRZÙ�SPÜK Yj�=OXÞ�åSyÜ�YZbrWySPNyå[�SyåNPK=W¸R\Ù�â1R1NPSyOX[=OXS�Û[8YZM+ã�K�K=c�ßbïYZOXMK�_Ëã�Û�bXR�R1ì�OXMá©Y\SrSyÜK�WySPNyK±Y\Þ�bXOXMK=WrR\ÙSPÜK ,IRZÚ/OXMÁY�Ù�N�YZÞ*K=ÚeR1Nyì*Þ�RZâ�OXMá�_RZÚrM�ÚrYZNy_½W$ÚrOXSPÜSPÜK�âwR1NPSyOX[=K=W8æ+êrÚeR+_OXWySPOgM½[=S�Ü�Û�ßIK=NPãIRwbXOg[*WyS�YZá1M�YZSyOXRwMß�R1OXM�SyW9[±Y\M�ãIKrRwãWyK�Nyâ1K�_mU�SyÜK�Û�K=c�K=NyS<ã�R1SPÜ�WySPNyK=SP[�ܽOgM½áY\M_/[�R1Þ�ß½NyK=WPWyOXR1M R1M/SPÜK ,IRZÚ ðIK�bg_BíïWPK=KÊûÅOXá>��î=æêrܽOgW�NPK=WyåbXSPW�OXM¸SPÜK�[=NyK8YZSyOXRwMÁRZÙDâwR1NySPOX[±YZb<WPSyNPå[=Syå½NyK=WY\NyRwåM_óYZM_ OXM ãIK�S�ÚDK=K�MóSyÜK â1R1NPSyOX[=K=WµÚrÜOX[�ÜuY\NyKNPK=Wyß�R1M½WyOXãbXK;Ù�R1NrSPÜK�_K=[±Y8Û*RZÙÖY�â1RwNySPK=c�ßIY\OXNDSPÜNyRwåá1Ü

�`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W

Page 5: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

SyܽK?ßNPR�[=K=WyWPK=W�R\ÙDSyåNPãåbXK=M�SL_½O�Ý�åWPOgRwM¸YZM½_+K=c�[�ÜIYZMáwKRZÙGâwR1NySPOX[=OXS�Û ãIK=S�ÚeK=K=M7â1RwNySPOg[�K=W/YZM_ WPåNyNPR1åM_½OgM½á,IRZÚ�æ

drYZWyK � � ��� � ° �<� ¦ �?�G� �¥r� � � ��� � �¥r� ç � � � ¨ � � � � � � ���¥r� ý � � � � ¨ ��� � � �[¨ ��¨¥r� é � � ��� � ��� � � � �� �¥r� ÿ � � � ¨ ¨ � � ��� � � � � ���¥r� � � � ��� ¨ � � � � � ��¥r� � � � � � ¨ ��� � � �[¨ ��¨¥r� � � � ��� ¨ � � � � � ��¥r� � � � � ¨G¨ � � ��� � � � � ���¥r� ) � � � ¨G¨ � � ��� � � � � ���¥r� ç¬� � � � ¨ ¨ � � ��� � � � � ���¥r� ç�ç � � � ¨ ¨ � � ��� ¨ � � � � ���¥r� ç4ý � � ��� ¨ � � � � � ���¥r� ç é � � � ¨G¨ � � ��� � � � � ���¥r� ç ÿ � � � � ¨ ��� � � � � ���¥r� ç4� � � � ¨ ¨ � � ��� � � � � ���¥r� ç4� � � ��� � ��� � � � � ���

" R�~\G �OG�� xZ��xZzLv��ïv���� ���Z�e�=�Z� �ïv"! #:xZ�X�e�g�����±t��

C=DFE�G�g�G "}nhT SXN'T;R�]6D{K�z K P R��\D � M�]onhR�YmJ|],N�MOS\]6MON T�J[K PZ\K�N�],DQS\DQ]6^ zhTUR�NlR ZXK�N�],T;W v R�DQN�G &'J|K�L SUK�zh],K�MONUJlK PlZ\K�NUL],DQSXD{]6^�R�zhsxJ|],N TUR � YoDQzhT�J P�K�N��=K � K�ZXD{z�E��DQ]on�]onhT�Z\K�NUL],DQSUT�J�GêrÜKÁâwR1NySPK=cGßIY\OXNË[�R1Þ�ß½åS�Y\SyOXR1MW�ß�R1OXM�SyK�_.R1åS�SyÜ�YZSSyܽKGÞËY\OgM.ß�YZN�Y\Þ�K=SPK=N¬Y8Ý�K=[=SPOXMá/SPÜKG_K=[8Y8Û!OXWÁSPÜKâ1RwNySPK=c¸WPßIYZ[�OgM½á � � RwN�SyܽK�N�Y\SyOXR����;��� � íÀWyK=K�ûmOXá�æD�1î�æØ å�ò�[�OgK�M:SPbXÛ WPÞËYZbXbµWyßIY\[=OXMá1WBbXK±YZ_7SyR SPÜK N�YZßOX_á1NPRZÚrSyÜ+R\ÙDbXR1Má\àëYZM½_+WyÜRwNyS4à ÚrY8â1K�OXMWyS�YZãOXbXOXSyOXK=W±æ|êrÜKbXR1Má\à ÚrY8â1K*OXMWyS�YZãOXbXOXS�Û¬[±Y\åWyK=W�bïYZNPá1K�à�Wy[8YZbXK+_K�Ù�R1NyÞ�Y8àSyOXRwMÁRZÙrSyÜK*â1R1NPSyOX[=K=W�YZM_ÁSPÜK�åbXSPOgÞ�YZSPK�_K=WPSyNyå½[=SyOXR1MRZÙ+SPÜKG[=RwÜK=NyK�M:SµßIYZOXNÁR\Ù âwR1NPSyOX[=K=W íïWyK�K ¨X îÁYZM_SyܽK WyÜRwNyS4à ÚrY8â1K OXMWyS�YZãOXbXOXS�Û Þ�Y8Û Y\Þ�ßbXO£Ù�Û å½M:SPOXbW�Y\SyåN�YZSPOgRwMóR�[=[=åNPWG[=RwM[=åNPNyK=M�SybXÛóNPK=Wyå½bgSPOXMá/OXMóSPÜKSyN�YZMWPOXSyOXR1M*RZÙmSyܽKLßNPOgÞ�YZNPÛ?âwR1NySPOX[=K=WeOgM�SPR?SPåNyãå½bgK�M[=K1æü�M¬Y�ßIY\NySPOg[�åbïYZN�[±Y\WyK*SyÜK*WyÜRwNyS�à�ÚrY8â1K=bXK=MáwSyÜÁK=bXbXOXßSyOX[OXMWyS�YZãOXbXOXS�ÛGÚrYZWÃ[=bXK±YZNPbgÛ R1ã½WyK=NPâ1K=_mæ êrÜK¬_K�[±Y8Û.RZÙ

Y â1RwNySyK�c!ß�YZOXN¬OXW¬ß½NyR1ß�R1NPSyOXR1M�YZb�SyR SyÜK SyåNPãåbXK=M�SOXM�SyK�MWyOXS�ÛBOgM YuWyOXÞ�OXbïYZNµÚëY8Û�Y\WÊSPÜK/_½K=[±Y8Û RZÙõYWPOXMá1bXK�â1RwNySPK=c�æ¸êrܽK�[=OXNy[�åbïYZSyOXRwMÁK=â1Rwbgå½SyOXR1MÁWPÜRZÚrW©YáwN�Y\_åIYZb�Y\M_ ]�åIYZWPO�à�bXOXMK±YZNÃ_K=[8Y8Û!å½M:SPOXb�OgM½WyS�Y\ãOXbXOgS�Ûß½NyR�[=K=WPWyK=WÊY\NyKäS�Y\ì�OgM½á.RZâ1K=NÁSPÜKG_K=[8Y8Û!_½K=ßIK�M_OXMáRwM.SyÜKäâwR1NySPK=c.[=RwM�ðIá1å½N�YZSPOXR1Mmæ�û½åNySPÜK=NyÞ*R1NPK1U�SyÜK_½K=[±Y8Û+R\Ù|Y*â1RwNySPK=c+ßIY\OgN�OXW�Y8Ý�K=[=SPK=_¬ã�Û¸Y\M_¸ß½NyR1ß�R1N4àSPOXR1MIY\b�SyR/SPÜK bgRwMá1OXSPå_OXMIYZb�OXM�SyK�á1N�Y\b�bXK=MáwSyÜóWP[±YZbXKR\Ù©SPåNyã½åbXK=M[=K1æ7Q�R1NPK=RZâ1K�N±U�ã�ÛG[�R1M_å½[=SyOXMá.YµSyK=WPSÚrOXSPÜ � �'�?� � ¢ �������|3�����3ËOXS�ÚëY\W�WPÜRZÚrM.SyÜIY\S�SyÜKKhÝ�K=[�S$R\Ù � � ��� R1M�SPÜK�[=OXNy[�åbïYZSyOXRwM�_K=[8Y8ÛËOXWrWPSyNyRwMá1K�N[�R1Þ�ß�YZNyK�_¸SPRËSPÜK�KhÝ�K=[�S?R\Ù��$æ�êrܽK � �'�?� WPK=K=Þ�W�SyRã�KÁSyܽKÁOgÞ*ßIRwNyS�YZM�S�Ù YZ[�SyR1N�OXM SyÜKÁWPOgÞ*ßbXK+_K=[8Y8ÛÊR\Ù[�OgNP[=åbïYZSPOXR1M�ã�Û�SyåNPãåbXK=M�S9_O£Ý�åWPOXR1M�YZM_�ã�Û?K=c�[�ÜIY\Má1KR\ÙÈâwR1NySPOX[=OXS�Û�ÚrOgSPÜ�SyÜK�YZÞ�ã½OgK�M:SeSyå½NyãåbXK=M½[=K1æ�� SyK=M_½WSPR�Y8Ý�K=[=S�SyܽK?_K�[±Y8Û�ã:Û*_K�Ù�RwNyÞ*OgM½á?SPÜK?âwR1NPSyOX[=K=W�YZM_Y\[=[=K=bXK=N�YZSPOgM½á�SyÜK�bXRwMáZà�ÚëY8âwKrOXMWyS�YZãOXbXOXS�Û?ßNPR�[=K=WyW8ædeR1Þ*ßIYZNPOgM½á$SPÜK�NPK=WyåbXSPWrRZÙÈSPÜK�WPOgM½á1bXKLâwR1NySPK=c�ÚrOXSPÜ

SPÜKÃâ1R1NPSyK=c ßIY\OgN*[=R1Þ*ßåS�YZSyOXRwMW�ÚeK¸Ü�Y8â1KÃR1ãWPK=NyâwK=_Y Ù Y\WySPK=N¸_½K=[±Y8Û RZÙ�SyÜKÁâwR1NySPK=c ßIYZOXN+OXM_å½[=K=_.ã�Û YWPSyNPR1MáwK=NÁWySPNyK=SP[�ܽOgM½áIæ�ü�SÃÜIYZWÁYZbXWPRGãIK�K=M R1ãWPK=NyâwK=_SPÜIYZS;WyK=âwK=N�Y\b9OgM½WyS�Y\ãOXbXOgS�Û�ß½NyR�[=K=WPWyK=W�[8YZM+R�[=[=å½N$ÚrܽOg[�Ü[8YZåWPKËY�WPOgáwMO£ðI[±Y\M:S�OXM[=NPK±YZWPK�OgMÃSyܽK�_K=[8Y8ÛÁN�YZSPK1æ�ü�SOXWÃK=WyWPK=M�SyOïYZb*SyR Þ�K=M�SPOgRwM.SyÜIY\SÃÙ�R1NÃSyܽKõ[8YZWyK�W¬SyÜ�YZSÚeK=NPK�ÜIYZM½_bXK=_mU�ÚDK�[=R1MWPOX_K=NyK�_ Y�[=RwM�ðIáwåN�Y\SyOXR1M+ÚrOXSPÜY NPK=bïYZSyOXâwK=bXÛ!WPÞËY\bgb*â1RwNySyK�cóWyßIY\[=OXMá í������� � bïYZNyáwK±î[�R1Þ�ß�YZNyK�_�SyR�NyK8YZbXOXWySyOX[�[=RwM�ðIá1å½N�YZSPOXR1MW8æ9êrÜK=NyKhÙ�R1NPK�OXMNPK±YZbXOXWySPOX[D[=RwM�ðIáwåN�Y\SyOXR1MW9SPÜKrâ1RwNySyOX[=K�WÈÚrOXbXb�K=c�ßIK�NyOXK=M[=KY´_K=[8Y8Û¬WyOXÞ*OgbïY\N$SPR¬Y+WPOXMá1bXK�â1RwNySyK�cõYZM½_¬SyܽKËÞ�YZOXMß�YZN�Y\Þ�K=SPK=NËY¼Ý�K�[=SyOXMáµSyÜKÃ_K=[±Y8Û ÚrOgbXbrã�K¸SPÜKI� ° R1N� � ��� æ"ü�٠ܽRZÚDK=âwK=N¬ÚeK ÞËYZM�YZá1KäSPR/OXM[=NPK±YZWPK.SyÜK� � ��� � K=OXSyÜK=N´ã:Û K=MbïYZNPá1OXMá¬SPÜKÁâ1RwNySPK=cG[=RwNyKÁR1N*ã�Û_½K=[=NyK8YZWyOXMáµSyÜK´â1R1NPSyK=cäWPßIYZ[=OXMáµÚDK�Þ�Y8ÛÃÙ Y8â1RwN�SyÜKR�[=[�åNyNPK=M[=K RZÙ�WPÜR1NPS�à Y\M_/bXR1M½áZà�ÚëY8âwK¸OXMWPS�YZã½OgbXOXSyOXK=WNPK=Wyå½bgSPOXMá�OXM�N�Y\ßOX_�_K=[±Y8Û*RZÙÈSPÜK�ÚrYZìwK1æ

0.0 25.0 50.0 75.0 100.0 125.0t*=tV0/rc

0.00

0.20

0.40

0.60

0.80

1.00

(Γ/Γ

0)b0

/2

starboard−side vortex0.0 25.0 50.0 75.0 100.0 125.0

t*0.00

0.20

0.40

0.60

0.80

1.00

(Γ/Γ

0)b0

/2

port−side vortex

VP−8;b0/rc=8VP−9;b0/rc=4VP−13;b0/rc=6

C=DFE�G���G?"}n�T T���TUSX]=K P�� � �G���yK�z[]onhTlS\DQN'SXM�YtR�]6D{K�z9TXZ\K�LYoMO]6DQK�z � �'�?� ���K¢ � � � ¨G¨ ���p����¢�� � � ����G�n���� �$×�Ñ1ÕZÔ��7�Ð8×�Ò�Ï��#·Õj����� ÒmÓ���

ûRwNDSPÜK�WyOXMáwbgK;â1RwNySyK�c�[=RwÞ�ßåS�YZSPOgRwMW<ÚeKLÜIY8âwK�R1ã�àWPK=NyâwK=_õY á1N�Y\_åIY\bDYZM_Á]�åIY\WyO£à�bgOXMK8YZN�_K�[±Y8Û¸ÚrܽOg[�ÜÁOXW

�`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W

Page 6: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

ÞËY\OXMbXÛ?_K�ßIK=M½_K=M�S?YZM½_Ëß½NyR1ß�R1NPSyOXR1M�YZbmSPR�SPÜK�Syå½Nyãå�àbXK=M[=K´OgM�SPK=MWyOXS�Û�U�� ° UrY´WyÞËY\bXbgK�N?_K�ßIK=M½_K=M[=KÃR1MäSPÜKbXR1MáwOXSyå_OXMIY\b�OXM:SPK=á1N�YZbÅbgK�Má1SPÜ+Wy[±Y\bXK?OXWrÙ�K=bXS±ærêrÜOXW;ãIK�àÜIY8â�OXR1åNeÜIY\WëY\bgWPR?ã�K=K=M´K=M[=RwåM�SyK=NPK=_+Ù�R1NeSyÜK�WyOXÞ�åbïY¼àSyOXRwMWrRZÙ<SyܽK�â1RwNySyK�cËß�YZOXNr[=R1M·ðIá1åN�YZSPOgRwM�ÚrOgSPܸY�NyK�b Y¼àSyOXâwK=bXÛ+bïYZNyáwK�â1RwNySyK�cÁWyßIY\[=OXMáIU � � �p� � ¢ � æ�ü�M¬ûÅOgá�æ �YZM½_õûÅOXáIæ��¸Y�[�R1Þ�ß�YZNyOXWPR1MÁRZÙ�SyÜK*_K=[±Y8ÛµãIK�ÜIY8â�OgRwåNRZÙËYµWPOgM½á1bXKÁâ1RwNySyK�c/YZM_/YäâwR1NPSyK=c ßIYZOXN+[�R1M�ðIáwåN�Y¼àSyOXRwMËOXWrWPÜRZÚrMmU�YZM_�_K�Þ�R1M½WySyN�YZSPK=WLSPÜK�bXOXMK±YZNr_½K=[±Y8Û�æêrÜK=NPKLOXW<MR�WyOXá1MO£ðI[8YZM�S<_O£Ý�K=NyK=M½[=K�OgM�_½K=[±Y8Û�ãIK=S�ÚeK=K=MSyܽK�WyOXMáwbgK�âwR1NySPK=c�RwNLâwR1NySPK=c�ß�YZOXNr[=R1M·ðIá1åN�YZSPOgRwM+YZM_SyܽK=NyK�Ù�R1NyK�OgSeOXWDß½b Y\åWyOXãbXK;SyR�W�Y8Û�SPÜIYZSe[=OXNy[�åbïYZSyOXRwM�_K�à[±Y8Û+R\Ù|Y WyOXMá1bXK�â1RwNySyK�c¸RwNÌY â1R1NPSyK=c´ßIYZOXN�ÚrOXSyÜÃNyK�b Y¼àSyOXâwK=bXÛLbïYZNPá1K<WPßIYZ[=OXMá;OXWmWyOXÞ*OgbïY\NIYZM_�S�Û:ß½Og[8YZbXbXÛrá1N�Y\_åIY\bYZM½_Ë]�åIY\WyO£à�bgOXMK8YZNe_K=[±Y8Û�æ

0.0 50.0 100.0 150.0t*=tV0/rc

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Γ/Γ 0

SV−3VP−3; port sideVP−3; starboard side

C=DFE�G���G���N R�s�MOR�Y�R�zhs��\MOR�J�DoL7YoDQzhTUR�NxSXDQN'SXM�YoR�],DQK�z s�TUSUR�^K P J�DQz�E�YtT�Z\K�N�],TUWmR�zhs=vhK�N�]QL�R�zhsmJ�],R�N�~UK\R�N's�L7J�DQs�T�Z\K�N�]6TUWK P ZXK�N�]6TUW�v R�DQN� � �'�?� ¢!� � ��� � R�zhs ��¢ � � ��� �hG

0.0 50.0 100.0 150.0t*=tV0/rc

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Γ/Γ 0

SV−4VP−4; port sideVP−4; starboard side

C=DFE�G���G���N R�s�MOR�Y�R�zhs��\MOR�J�DoL7YoDQzhTUR�NxSXDQN'SXM�YoR�],DQK�z s�TUSUR�^K P J�DQz�E�YtTmZXK�N�],T;WyR�z�slZXK�N�]6TUWxvhR�DQN � �'�?� ¢ � � � �lR�z�s�n¢ � � � ���hGöGÜK�MõbXR�R1ì�OXMáÁYZS�SPÜK=WyK´ðIáwåNyK=W*ÚDK´ÞËY8ÛÃ[=R1M[�bgå½_KSyÜ�YZSrâ1RwNySPK=c�_K�[±Y8Û�áwRZâ1K=NPMK=_�ã�Û*SyåNPãåbXK=M�Sr_O£Ý�å½WyOXR1MYZM½_?K=c�[�ÜIY\Má1KeRZÙIâwR1NPSyOX[=OXS�Û�ãIK=S�ÚeK=K=M�SPÜKeâ1R1NPSyOX[=K=WeYZM_SyܽK�SyåNPãåbXK=M[�K�OXWrãIK=WPSL_K�Wy[=NPOgã�K=_+ã�Û�Y�bgOXMK8YZNe_K=[±Y8ÛÚrÜOX[�Ü�OXWeßNyRwßIRwNySyOXRwMIYZb�SyR�SyÜK�NPR:RwS�à�Þ�K±Y\M�à�Wy]�åIYZNPKLâwK�àbXR�[=OXS�Û¬YZM_µSyÜK+bXRwMá1OXSyå½_OXMIYZb<OXM�SyK�á1N�Y\brbXK=Má1SPÜäWy[±Y\bgKwædeR1M½WyK=]�åK=M�SPbgÛ�UZSPÜKr[=OXNy[=å½b Y\SyOXR1M�_K=[8Y8Û?ÚrOXbXb�ãIKLY|Ù�åM½[�à

SPOXR1M�R\ÙÈSPÜK=WyK�ßIY\N�YZÞ*K=SyK�NyW*�&��&�3 ¢ $ í � � ��� �'�ëî í7�1î�rYZWPK=_.R1M R1åN �mô Ø NPK=WyåbXSPW+ÚrÜOX[�ÜÊÜIY8âwK¸ã�K=K=M._½K�à

WP[=NyOXã�K=_¬YZã�RZâ1K�ÚeK�ßNyRwßIRwWyKËY*bXOXMK±Y\N$S�Û�ß�K�RZÙrâ1RwNySyK�c_½K=[±Y8Û�æ9êrÜK�[=OXNy[=åbïY\SyOXR1M*OgWLY\ßßNPRZc:OXÞ�YZSyK�_�YZWeÙ�RwbgbXRZÚrW

�3ío3�î�¢ � � ¨�© í�£ � � ���¥ � � ª ���� î,3� í7�1îÚrܽK=NyKi£ÃYZM_ ª YZNPKLK=Þ*ßOXNyOX[±YZbw[=R1M½WyS�Y\M�SyW±æ3êrÜK=WPKrK=Þ�àß½OgNPOX[±YZb�[=RwMWyS�YZM�SyWrÜ�Y8â1K�ãIK�K=M�Ù�RwåM_�ã�Û*_R1OXMá�Y�ã�K=WPSð�S?R\ÙLSyÜK�Þ�R�_K�bDSPR Y\bgbeSyܽK�[=OXNy[�åbïYZSyOXRwM¬_IY\S�Y´[=R1Þ0àOXMáÃÙ�NyR1Þ7SyܽK¸WPOgÞ�å½b Y\SyOXR1M½W±æäêrÜKÃÞ�R�_K=br[�R1MWPS�YZM�SPWWPK=K=Þ�SPR ,Iå½[=SyåIY\SyKÃÚrOXSyܬâ1Y\NyÛ�OXMá � �'�?� Y\M_��óYZM_Þ�YZìwK=W�OgS�_O£ò�[=å½bgS�SyR+[=RwÞ�K�å߬ÚrOXSPܸåM½Og]�åK*â1YZbXåK=WR\Ù £ YZM½_ ª æ�êrÜK�NPK=WyåbXSPOgM½á+â1YZbXåK=W�RZÙLSyܽK=WyK*[=R�K�ò�à[�OgK�M:SPW�ÚrÜOX[�ÜÃNyK�ßNyK=WPK=M�S�ãIK�WyS�SPÜK�â1RwNySyK�c+_K=[±Y8ÛÃYZNyKG�£ ¢ � � ��� ��¨ ������YZM_ ª ¢ �h� � � � ¨ ��� �� ¨ � ��é?`;W�[±YZMã�KLWPK=K=M*OgM�ûÅOXáIæ � SyÜK;Þ�R�_K=b½ßNyK=_½Og[�SyW3Ù YZOXNybXÛ�ÚeK=bXb�SyÜK[�OgNP[=åbïYZSPOXR1MÁÙ�R1N�SyÜK�WyOXMáwbXK�â1RwNySPK=c¬[=RwM�ðIáwåN�Y\SyOXR1MäYZWÚeK=bXbmYZWrSPÜK�â1RwNySyK�c�ßIY\OgNr[�R1M�ðIáwåN�Y\SyOXR1MÅæ

0.0 50.0 100.0 150.0t*=tV0/rc

0.00.20.40.60.81.0

Γ/Γ 0

sv3vp6

0.0 50.0 100.0 150.00.00.20.40.60.81.0

Γ/Γ 0

sv3vp6

0.0 50.0 100.0 150.00.00.20.40.60.81.0

Γ/Γ 0

sv3vp6

0.0 50.0 100.0 150.00.00.20.40.60.81.0

Γ/Γ 0

sv2vp6

C=DFE�G ��Gi$ K\s�T�Y vwN TUs�DQSX],DQK�z[K PmJ�DQz�E�YoT Z\K�N�],TUWrR�zhs9ZXK�NUL],TUWmvhR�DQN���� � �G� � ¢ ��� s�TUS;R�^�Gö ÜK=MËYÌâ1R1NPSyK=c�ß�YZOXN<OXWDß½b Y\[=K=_*OgM�WPå�ò�[=OXK=M�SybXÛ�WPSyNyRwMáSPåNyã½åbXK=M[=K+YZM½_õSPÜK�âwR1NPSyK=cµWyßIY\[=OXMá � � �G� � OXW?WPå�ò�à[�OgK�M:SPbXÛ¸WPÞËYZbXb�U�SPÜK�[=OXNP[=åbïYZSPOgRwMÁ_K=[±Y8ÛÁÚrOXbXbeWyåãWPS�Y\M�àSPOïYZbXbXÛ+[�ÜIYZMáwK1æÁêrÜOXW�[±YZMµãIK�K=c�ßbïY\OgM½K=_¬ã�ÛÁbXR�R1ì�OXMáY\S�ûmOXá�æ � OXM¸ÚrÜOX[�ÜÃSyÜK�S�Û�ßOX[±Y\bÈ_½K=[±Y8Û+ã�K=ÜIY8â�OXR1åN�OXWWPÜRZÚrM+R\ÙÖâwR1NPSyOX[=K=W�K=c�ß�K=NyOXK=M[�OgM½á Y\M+OXMWyS�YZãOXbXOXS�Û�ßNyR\à[�K=WyW8æ<öÁKr[±Y\M�WySyOXbXb�_OXWySyOXMáwåOXWyÜ�Y�áwN�Y\_åIYZb�YZM½_�]:å�YZWyO£àbXOXMK±Y\ND_½K=[±Y8Û�OXM�SyܽK�OXMOXSyOïYZbWPS�Y\á1K=W;RZÙ<SyÜK�WPOgÞ�å½b Y\SyOXR1M_½R1Þ�OXMIY\SyK=_�ã�Û*SyåNPãåbXK=M�SL_O£Ý�åWyOXRwMmæ<`|Ù�SyK=N�Y�[=K=NPS�YZOXMY\Þ�RwåM�SDR\ÙÖSPOXÞ�KLWPR1Þ*K$OXMWPS�Y\ãOXbgOXS�Û*ßNyR�[=K=WPW�OgW;S�Y\ì:OXMáRZâwK=N�YZM_GYZ[�[=K=bXK=N�Y\SyK=W*SyÜK�_K�[±Y8Û:æµêrÜK=M SyÜK�[=OXNy[�å�àbïY\SyOXR1MGYZß½ßIK±Y\NyW+SPRÁÙ�RwbXbgRZÚ YµK=c�ßIRwMK=M�SyOïY\bLÙ�åM[=SPOgRwMmUSPÜOXWeÙ�K±Y\SyåNPK?ÚrOXbXb�ãIK�S�YZìwK=M+OXM�SyR�Y\[=[=R1å½M:S;OXMËSPÜK�_KhÙïàOXMOXSPOgRwM+RZÙrSyܽK�Þ�R�_K�bmÙ�R1N�SPÜK�[=OXNy[=å½b Y\SyOXR1MÃK=âwR1bXåSyOXRwMY¼Ù�SyK�N�3�¢�3�� ����� æ?êrܽK�bgOXMì�OXMá*SyOXÞ�K1U�3�� ����� OXW�_K�ðIM½K=_ܽK=NyKLY\WÈSPÜKrSyOXÞ*KDYZS9ÚrÜOX[�Ü?SPÜKrbXR1Má\à ÚrY8â1K3OgM½WyS�Y\ãOXbXOgS�Û

�`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W

Page 7: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

0.0 100.0 200.0t*=tV0/rc

0.0

0.2

0.4

0.6

0.8

1.0Γ/

Γ 0

tlinkC=DFE�G ��G " ^'vwDQSUR�Y�s�TUSUR�^ ~;TUnhR�ZXD{K�MON K'P Z\K�N�]6TUWmvhR�D{N�T;W�LvhTXN�D{TXzhS\DQz�E�D{zOJ�]6R�~XDoYoDQ],^ � s�T,q�zhD{]6DQK�zVK P9YoDQz���DQz�E�],D � T�3 � ����� GOXW$Y\Þ�ßbXO£ðIK=_+YZM½_+SyÜK�NyK�[=R1MM½K=[=SyOXRwMÁOgW;S�Y\ì:OXMá*ßbïYZ[=KwæêrÜOXW�bXOXMì�OXMá�SPOgÞ*K?[8YZMÁã�K�_K=SyK�NyÞ�OXMK�_¬YZW�_K�ßOX[=SyK=_OXM¬ûÅOgá � æ�êrÜK�âwR1NySPOX[=K=W�K=c�ßIK=NPOXK=M[=K+bïY\Nyá1Khà WP[±YZbXK�_K�àÙ�RwNyÞËY\SyOXR1M½W?_å½K+SyRÁSPÜK¸NPK=bïYZSPOgâwK=bXÛ¬WPÞËYZbXbXK=N�WyßIY\[=OXMáÚrÜOX[�ÜËY\Þ�ßbXO£ðIK=W9SyܽK$OXMWPS�Y\ãOXbgS�Û�ßNyR�[=K=WPW$Y\M_�K=âwK=M�Syå�àYZbXbXÛ�bXK±Y\_W�SyR�SyܽK?NPK=[=R1M½MK=[=SPOgRwMÁRZÙDSPÜK�â1R1NPSyOX[=K=W�YZM_SyܽK�[=NPK±YZSPOgRwM¬R\Ù|Y�âwR1NPSyK=cÁNPOgM½á¬íïWPK=K+ûmOXá�æ ¨X î=æ�êrÜOXWâ1RwNySPK=c�NyOXMáLÚrOXbXb1_½K=[±Y8Û�YZM_�_OXWyOXM�SyK=áwN�YZSPKDOXM�SyRrWPÞËY\bgbXK=NWy[8YZbXK�SPåNyãå½bgK�M[=K�O£ÙLSyÜK*WyåNPNyR1å½M_OXMáÃSyåNPãåbXK=M[=K�OXWWyå·ò�[=OXK=M�SybXÛ�WySPNyRwMáIæÅü�M?NPK±YZb½[=OXNy[=åÞ*WyS�YZM[=K�WDY;SyN�Y\OXM�RZÙâ1RwNySPK=c+NyOXMáwWLÚrOXbXbmã�K�Ù�RwNyÞ�K�_ËÚrܽOg[�ÜÃÚrOXbgb�K�c:ß�K=NPOgK�M[=KWySPN�Y\OgM½OgM½á¬YZM_ä_OXWySPR1NPSyOXR1MäåM_½K=N�Þ�åSPåIYZbeOXM_å[�SyOXR1MYZM½_+K=â1K�M:SPåIYZbXbXÛ�SPÜK=OXNLK�MK=NyáwÛ+ÚrOXbXb�ãIK�SPN�Y\MW�Ù�K�NyNyK�_ÁSyRSyå½NyãåbXK=M½[=K1æ�`rSLSPÜK�ðIM�YZbmWPS�Y\á1K=W�RZÙeSyÜK�WPOXÞ�åbïYZSyOXRwMWSyܽK�[=OXNy[=åbïY\SyOXR1MÃNyK±Y\[�ÜK=W�Y�â1Y\bXåK�RZÙ ¨ ���BRZÙrSyÜK�OXMO£àSyOïY\bÈâ1Y\bXåK©Y\M_ÃSyÜK�âwR1NySPOX[=K=W?Y\NyK�YZbXÞ�RwWySL[�R1Þ�ß½bgK�SyK=bXÛ_K=WPSyNPRZÛ1K=_äYZM_ÃÜIY8â1K�K=c�ß�K=NyOXK=M[�K=_GY�WPOXá1MO£ðI[±Y\M�S$bXRwWyWOXM¸WPSyNPK=Má1SPÜmæ*êrÜK?bXOXMì�OXMá�SPOXÞ�K�OXSyWPK=b£ÙëÜ�YZW�ã�K=K=M¬Y\ß�àßNPRZc:OXÞ�YZSyK�_+ã�Û Y�WPOXÞ�ßbXK�b Y8Ú SyÜIY\S�ÜIY\W?ã�K=K=M ðISySPK=_R1M�SPR+SyÜK*M�åÞ�K=NPOX[±YZb<NPK=Wyå½bgSPW±æ´êrÜOXW?NPK=bïYZSyOXRwM¸[±Y\M¬ã�KK=c�ßNPK=WyWPK=_ OgM.SPK=NyÞ*W+RZÙ � � ��� U ��U ¥ � YZM½_ � � U�ÚrܽOg[�ÜÞ�K8YZMWeSyÜIY\SrOXSDS�YZì1K�WrOgM�SPR©Y\[=[=R1å½M:SrSPÜK�ßNyRwßIK�NySyOXK=W;RZÙSyܽK�SyåNPãåbXK=M[=K�Y\M_ÃSyÜK�âwR1NySPK=c+[=RwM�ðIáwåN�Y\SyOXR1MmUÅYZM_OXWLYZWeÙ�RwbXbgRZÚrW<�

3 � ����� ¢ � § � <� ¦ «°������: <��< í � î

æ�rYZWyK�_ËRwM�SyÜK;Ù�K8YZSyå½NyK=Wr_K�Wy[=NPOgã�K=_+YZã�RZâ1K�í�ûÅOXáIæ � î9Ù�R1N_K=[8Y8Û�RZÙÈY�â1R1NPSyK=c�ß�YZOXN<OXM�SyK=N�YZ[=SPOgM½á?ÚrOXSyÜ*Wyå�ò�[�OgK�M:SPbXÛWySPNyRwMá�SyåNPãåbXK=M[�K?YZM_*ÚrOXSyÜËY�NyK�b Y\SyOXâ1K�bgÛ�WPÞËY\bgb½â1R1N4àSyK�c�WPßIYZ[=OXMá SPÜK.[=OXNy[=å½b Y\SyOXR1MBK=âwR1bXåSyOXRwMóÞ�R�_K=b�OXWÞ�R�_O£ðIK�_¬Ù�R1N 3� 3�� ����� æ êrÜK´Ù�RwbgbXRZÚrOXMáÁÞ*R:_½K=bLOXW

ß½NyR1ß�R1WPK=_D����� ��  �<ío3�îu¢�� � § ¨�© í7£ °������: < � ª ��;¦ î,3 « Ù�R1N 3� 3�� ������<ío3�îu¢�� � § ¨�© í7£ ° �����:�< � ª �� ¦ î,3 «���� 5�� §�� í 3 � ����� © 3�î �;¦� < «� � �[¨ � � � �1������������ Ù�R1N 3�I3 � �����` MK=Ú.K=Þ*ßOXNyOX[±Y\bm[=RwMWyS�YZM�SLÜIY\W�ãIK=K�M¸OXM�SyNPR:_½å[=K=_ÃOXMRwNy_K�N�SyRÃßNyRwßIK=NPbXÛ¬Þ�R�_K�bDSPÜK+K�Ý�K=[=S*RZÙ$SPÜK�â1RwNySyK�cWPßIYZ[�OgM½á?YZM_*ÜIYZWeSyܽKLÙ�R1bXbXRZÚrOgM½áLâ1YZbXåKG� � ¢!� � �æÅêrÜKbïY\WySeSyK=NPÞóOXM�SyܽK$Þ*R�_K=bOXWeOgM�SPNyR�_å[=K�_�Ù�R1NeSyÜK�[=OXNy[�å�àbïY\SyOXR1M�SPR?NPK±YZ[�Ü ¨ ���/R\ÙÈSPÜK�OgM½OgSPOïYZbâ1Y\bgå½K1U{� � UY\WDÜ�YZWã�K=K=M*R1ãWPK=NyâwK=_�Ù�RwNDYZbXb�SyÜK;[±YZWPK=WeÜIYZM_½bgK�_Ëí��¢ ¨�� � ¨ î=æö ÜK=MÃSyÜOXW�ðIM�YZb<â1Y\bgå½K?OXW�NyK±Y\[�ÜK=_¬RwMbXÛ�WyÞËY\bXbÈWP[±YZbXKSPåNyã½åbXK=M[=K;NyK=Þ�YZOXMW9SyÜOXW�,IRZÚäÚrOXbXb:_½K=[±Y8Û�YZSDY;WybXRZÚeK=NN�YZSPK$SPÜIYZM�SPÜK�N�YZSPK�â1RwNySPK=c�_K�[±Y8Û:æ9êrܽK$_½K=[±Y8Û�R\Ù<SyÜK,�RZÚGÚrOXbgb�ã�K�áwRZâ1K=NPMK=_�ã�Û�WyÞ�YZbXb�SyåNPãåbXK=M�SL_O£Ý�åWyOXRwMY\M_+ÚrOXbXbm_K�[±Y8Û¸YZS�Y�N�YZSyK�SyÜ�YZSLOXW;[�Ü�YZN�Y\[=SyK�NyOQ�=K=_µSyÜKK�__Û�SPåNyM·à RZâwK=NrSyOXÞ�KwU�3 8 UIYZM½_ËOXWeá1OXâ1K�MËOXM*K=]�æmí��wî=æü�MÃûmOXá�æ ¨ �*YZM_ÃûmOXá�æ ¨p¨ ÚDK�WyÜRZÚ.SPÜK?[�OgNP[=åbïYZSPOXR1M+R\Ùß½NyK=_OX[=SPOXR1M¬R\Ù$SPÜK�Ù�RwN?SPÜK�â1R1NPSyK=cµßIYZOXN�WyOXÞ�åbïYZSPOXR1MWÚrOXSPܬY+âwR1NySPK=cÁWyß�YZ[=OXMá � � �G� � ¢ �¸YZM_ � � �p� � ¢4�æêrܽK$áwK=MK=N�YZb Ù�K±YZSPåNyK�WLRZÙ9SyÜK�_K�[±Y8Û:U�M�YZÞ�K�bgÛ�SyܽK$OXMO£àSPOïYZbebgOXMK8YZN�_K=[±Y8ÛäYZM_µWyåãWPK=]�åK=M�S�K�c:ß�R1M½K=M�SyOïYZbLY\[�à[�K=bXK=N�Y\SyK=_+_½K=[±Y8Û�UYZNyKÌÙ YZOXNybXÛ�ÚDK�bgb�NPK=ßNPR:_½å[=K=_mæ

�$Ò�Ï<Õ���Ô��=×�ÒIÏêrÜK M�åÞ*K=NyOX[±Y\b OXM�â1K=WPSyOXá:Y\SyOXR1M RZÙ â1RwN�à

SPK=c"!ZSPåNyãå½bgK�M[=K�OXM�SyK=N�YZ[=SPOgRwM ÚrOXSyÜ �Åô Ø ßIK=NPÞ�OXSyWSPÜKr_K�ð�MOXSyOXR1M�R\ÙÈY�Þ�R�_K=b�SPR�[�ÜIYZN�YZ[=SPK=NyOQ�=K�SyܽKL_K=[8Y8ÛR\Ù¸ÚrYZì1K â1RwNySPOg[�K=WGOXM�YZÞ�ã½OgK�M:SµSyåNPãåbXK=M[�K1æ êrÜKÞ*R�_K=b�WyÜRwåbX_ OXM[=RwNyß�R1N�YZSyK SyÜK WySPNyR1M½áóOXM�,IåK=M½[=KR\Ù�SyÜK â1R1NPSyK=c.WPßIYZ[�OgM½áóíïOXM/SPÜKä[±YZWPKõR\Ù Y â1RwNySyK�cß�YZOXN�î�U1SPÜK � � ��� YZM_�SPÜKm�¬RZÙ9SyÜK;SyåNPãåbXK=M[�K$ÚrܽOg[�ÜY\NyKäSyܽKGR1SPÜK=N¬Þ�YZOXM/ß�YZN�Y\Þ�K=SPK=NyWÁSPÜIYZS¬Y¼Ý�K�[=SäSyÜK[�OgNP[=åbïYZSPOXR1M�_K=[±Y8Û�ærêrÜK�Þ�R�_K=b�NPK=ßNyR�_å½[=K=W�ÚDK=bXbÅSyÜKÞ�YZOXM�Ù�K±Y\SyåNPK=W�RZÙ<WyOXMáwbXK$âwR1NPSyK=c�_K=[8Y8Û Y\M_�_K=[±Y8Û�R\ÙY�â1RwNySPK=c�ßIYZOXNeÚrOgSPÜ�NyK=bïYZSPOXâ1K=bXÛ�bïY\Nyá1K;â1RwNySyK�cËWPßIYZ[�OgM½áIUM�YZÞ�K�bgÛ´SyÜK*á1N�Y\_åIY\bëY\M_¬]�åIY\WyO£à bXOXMK±Y\N�_K=[8Y8Û:æÃêrÜKÞ*R�_K=brOXW�YZãbXK+SPR¬Þ*OgÞ�OX[�SPÜK+ã�K=ÜIY8â�OXR1åN*RZÙ�â1RwNySyK�cß�YZOXNyWeÚrOXSyÜ�NyK=bïY\SyOXâ1K=bXÛ�WPÞËYZbXbXK=N<âwR1NPSyK=c*WyßIY\[=OXMá?ÚrܽOg[�ÜK�c:ß�K=NPOgK�M[=KGOXMWPS�YZã½OgbXOXS�ÛGßNPR�[=K=WyWPK=W±U*Wyå½[�Ü ãIK�ÜIY8â�OgRwåNOXW�[�ÜIY\N�Y\[=SyK=NPOQ�=K=_Gã�ÛÁOXMOXSyOïYZb9]�åIY\WyO£à bXOXMK±Y\N�_K=[8Y8Û¬YZM_WPåãWPK=]�åK=M�S K=c�ß�R1MK�M:SPOïYZb/YZ[�[=K=bXK=N�Y\SyK=_ [=OXNy[�åbïYZSyOXRwM_½K=[±Y8Û�æ$#�bXRwãIYZbXbXÛ:U\SyÜK;Þ�R�_K=b�NyK=ß½NyR�_å[=K�WLÙ�YZOXNybXÛ�á1R�R�_NPK=Wyå½bgSPW¬YZbXSyܽR1åáwÜ.OgSÃOXW Y ßNPK=bXOgÞ*OXMIYZNPÛGÞ�R�_K=b�YZM_OXÞ*ßNyRZâwK=Þ�K=M�SyW�[±Y\MÁWySyOXbXb<ã�K�Þ�YZ_K1U9YZW�SyÜK*_K=[±Y8ÛÁR\ÙâwR1NPSyOX[=K=WÁOXWÃ[=R1Þ*ßbXK=c Y\M_/áwRZâ1K=NPMK=_ ã�Û.M�åÞ�K=NPR1åWß�YZN�Y\Þ�K=SPK=NyW�YZM_ÁßÜ�Û�WyOX[±Y\bDß½NyR�[=K=WPWyK=W8æ¬êrÜOXW�ì�OXM_ÁR\ÙÞ*R�_K=b�Þ�Y8ÛGãIKµOXM:SPK=á1N�YZSPK=_/OXM�SyRGÚrYZìwK¬âwR1NySPK=c.ã�K�àÜ�Y8â:OXRwåNLYZM_´K=M[=RwåM�SyK=N�Þ�R�_K�bgWLY\M_+Þ�Y8Û�ãIK�å½WyK�Ù�å½bSPR�NPK�à�_K�ðIMK W�Y8Ù�K!WPK=ßIY\N�YZSPOXR1M _OXWyS�YZM[=K�W/ã�K=S�ÚDK�K=MY\OXNy[=N�Y8Ù�S*OgM S�YZìwK�à�RZÝ.R1N�bïYZM_OXMáÁ[�R1M�ðIáwåN�Y\SyOXR1M YZM_[8YZM�ãIK�åWyK=_´SyR�_K=âwK=bXR1ß+Y�NyK8YZb£à SPOXÞ�KLÞ*K=SyK�R1NyRwbXR1á1OX[±Y\bWPÛ�WySyK�ÞóÙ�RwNrMRZÚe[±YZWPSyOXMá�OgM�SPK=NyÞ*OgM�YZb�YZNyK8Yæ

�`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W

Page 8: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

0.0 50.0 100.0 150.0t*=tV0/rc

0.0

0.2

0.4

0.6

0.8

1.0

Γ/Γ 0

VP−9model

0.0 50.0 100.0 150.0 200.00.0

0.2

0.4

0.6

0.8

1.0

Γ/Γ 0

VP−1model

0.0 50.0 100.0 150.0t*=tV0/rc

0.0

0.2

0.4

0.6

0.8

1.0

VP−10model

0.0 50.0 100.0 150.00.0

0.2

0.4

0.6

0.8

1.0

VP−2model

C=DFE�G�H�f�G $ K\s�T�Y vwN TUs�DQSX],DQK�z K'P�Z\K�N�],T;W�vhR�DQNms�TUSUR�^xP�K�NmSUK�z�q�E�MON'R�],DQK�zOJ��DQ]on J6vhR�S\DQz�E K P�� � �p����¢!� G

0.0 50.0 100.0 150.0t*=tV0/rc

0.0

0.2

0.4

0.6

0.8

1.0

Γ/Γ 0

VP−15model

0.0 50.0 100.0 150.0 200.00.0

0.2

0.4

0.6

0.8

1.0

Γ/Γ 0

VP−13model

0.0 50.0 100.0 150.0t*=tV0/rc

0.0

0.2

0.4

0.6

0.8

1.0

VP−16model

0.0 50.0 100.0 150.00.0

0.2

0.4

0.6

0.8

1.0

VP−14model

C=DFE�G�H�H�G $ K\s�T�Y vwN TUs�DQSX],DQK�z K'P�Z\K�N�],T;W�vhR�DQNms�TUSUR�^xP�K�NmSUK�z�q�E�MON'R�],DQK�zOJ��DQ]on J6vhR�S\DQz�E K P�� � �p����¢ �hG�

`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W

Page 9: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

� Õ���ÏÈÒ��H�7Ó��A���ÏmÐ��êrÜOXW�NyK=WPK±YZNP[�Ü�ÜIYZWÅãIK=K�M?K=c�K=[�åSyK=_�YZWÅßIYZNPSmR\Ù·SPÜK Ø à

ö¬YZìwK9ßNyR±fhK=[=Sríï`;WyWyK�WyWyÞ*K=M�SmR\ÙIö¬YZìwK��<RwNySyK�c Ø Y8Ù�K=S�Ûî�UÚrÜOX[�Ü�ÚrYZWLíÀßIYZNPSybXÛî9Ù�åM_K=_*ã�Û�SyÜK�ô9åNyRwßIK8YZMËdeRwÞ�àÞ�OXWPWyOXR1Mmæ

� ��;�Ñ��Ï<Õ���¨ æK�9NyR�[=SPR1N±U�ûeæZJ?æXU��/+LåÞ*K=NyOX[±Y\b1WPOgÞ�å½b Y\SyOXR1M;RZÙ�ÚëY\ì1Kâ1R1NPSyOX[=K=W Þ�K±Y\WyåNPK=_ _åNPOgM½á ü�_IYZÜR û�YZbXbXW.YZM_Q�K=Þ�ß½ÜOXWrðIK=bX_ÃßNyRwá1N�YZÞ�W8U �OgM�H�gh]on�&� �=v Lv YtD{T;s�xTXN'KXs�^�zhR � DQS�J���K�z�P�TXN'TXz�S;T �!�T ���NUYtTUR�z J���� � U�`Lü�`;`.ßIYZß�K=N � �8à � � �½U ¨ ��� �½æ

ærdeR1NgfhR1MmU�`�ægU�V�YZNPN�YZ[�]�UÃV�ægU �<K=M �±YZ[1U �9æXU+YZM_�DRwåá1K±Y\åbXS±U �9æXU��yêrܽNyK=K�à�_OXÞ�K�MWyOXR1M�YZb �ÅYZNyáwKô9__Û Ø OXÞ�åbïYZSyOXRwM RZÙõö¬Y\ì1K �<RwNySyOX[=K�W±æ¸deRwÞ�àßIYZNPOgWPR1M ÚrOgSPÜ!ûÅOXK=bX_!Q�K8YZWyå½NyK=Þ*K=M�SyW±U OXM H���]on�&���?v�v YoDQTUs�xTXN'KXs�^�zhR � DQS�J���K�z�P�T\N T\zhSUT �m]QLYtR�z�]6R � �� � U�`;ü�`L`.ßIY\ßIK�N ��� à ��� � U ¨ ���� æ�½æ Ø ßIYZbïY\NyS±UG�9ægU���ùLM�SPÜKrQ�R1SPOgRwM?R\Ù�`;OXNy[=N�Y8Ù�S9ö¬YZìwK=WOgM �mYZÞ*OXMIYZN�öGOXMáLö¬Y\ì1K=WÅOXM?Y Ø SPN�Y\SyO£ðIK=_�ûÅbgå½Og_ÅU ��K�MON�zhR�YmK PrC YoM�D{s�$ TUS;n�R�zhDQS�J=U��<Rwb æ�� �� U ¨ ��� �Ußßmæ ¨ � ���%¨ ���½æ��ærV$Y\NyN�YZ[=]�U<V�æXU<deR1NgfhR1MmUÅ`?æXU<V;å[=NPR1W±UeûeægU9ú�K±Y\MK1UQÁæXU1YZM_ �eå[�ì:SPR1MmUwV?æXU��PêrÜNyK�K�à _½OgÞ*K=MWPOgRwMIYZb�+;å�àÞ�K=NPOg[8YZb Ø OgÞ�å½b Y\SyOXR1M½W�RZÙIö¬Y\ì1K=�<R1NPSyK=c?V;K=SPK=[=SPOgRwMÚrOgSPܸSPÜK�Q+û���`$Q´ô Þ*OX[=NyRwM � OX_IYZN8U �OXMI� ��]on�T\N K�J6vhR�SUT! S\DQT\zhSUT�J�$ TUT\]6DQz�EmR�zhs %�W\nhDQ~\DQ] #"�TXz�K��!%$ � U�`Lü�`;`.ßIYZß�K=N ��� à �p�G�p�½U ¨ ���� æ�½ærørOXWyWPRIU±ûeæXUZdeR1NXf�R1MmU�`?æXUZYZM_ Ø SPR�K=WyWPK=b�U1`�ægU���V;OXNyK=[�S+LåÞ*K=NyOX[±Y\b Ø OXÞ�åbïY\SyOXR1M�RZÙ9ö¬YZì1K��3R1NySPOX[=K=WLOXM*ü�M�àSyK=MWPK Ø ÞËYZbXb£à Ø [±YZbXKrJ;R1Þ�Rwá1K=M½K=R1åWrê9åNPãåbXK=M[�K1U �&��&�OK�MON�zhR�YgUh�3R1b�æ{�G�U�� ¨ ���� æ�½ærdeR1NgfhR1MmU `�æ Y\M_ Ø SPR�K=WyWPK=b�U `�æXU �yêrܽNyK=K�àVLOXÞ*K=MWyOXRwMIYZbZü�MWPS�Y\ãOXbgOXS�Û;RZÙIö¬Y\ì1K��<RwNySPOg[�K=W�+LK±Y\NSyÜK #�NPR1åM½_mU +OXM ����]on'�&�� C YtMODQs)(�^�zhR � D{S�J��K�z�P�T\N T\zhSUT � zhK � R�J;J �*�+� � U8`;ü�`L`äßIYZß�K=N ��� ਠ���� U ¨ ���� æ

� ærV$Y\NyN�YZ[=]�U9V?æXUmQ R:K�S±U9J?æXU9YZM_¬deRwNgf�RwMmU�`�ægU���ôÈÙïàÙ�K=[=SPWõR\Ù deNyRwWyWPÚrOgM½_ Ø Ü½K±YZNGYZM½_ó`rSyÞ*R1Wyß½ÜK=NyOX[Ø SyN�YZSyO£ðI[8YZSyOXRwMGR1MG`;OXNy[=N�Y8Ù�S�ê9N�Y\OgbXOXMáI�<RwNySPOg[�K=W±U OgM!� ��]tn,�T\N K�J6vhR�SUT) SXDQT\zhSUT�J $ TUT\]6DQz�E R�zhs %�W�LnhDQ~\DQ] -"�T\zhK �7!%$ � U1`;ü�`L` ßIYZß�K=N �� à � ��� �U ¨ ���� æ

� ærQ�R�K=S±U�J?æXUV�YZNPN�Y\[=]�U�V�æXU��mY\ßIRwNySPK1U�ûDæXUY\M_¸deR1N4àf�R1MmU\`?æXU��yü�M�âwK=WySPOgá�YZSPOgRwM?R\Ù�`;Þ�ãOXK=M�S9êÈå½NyãåbXK=M½[=KôÅÝ�K�[=SyWÃR1M �3R1NySPK=cÊô9âwR1bXåSPOgRwMGåWPOgM½á��Åô Ø U ¬OXM����]on�xT\N K�J,vhR�SUT. S\DQT\zhSUT�J�$ TUT\]6D{z�ErR�zhs % WXn�DQ~XD{]�"�TXz�K �7!�$ � U�`;ü�`L` ßIYZß�K=N �����¼à4� � �p�½U �����æ

� æ Ø SyR�K=WPWyK=b�U�`?æXU/�y`;M+ôÅò�[=OXK=M�SLê9R�R1b�Ù�RwNrSyÜK Ø SPå_ÛRZÙr�1V ê9åNyã½åbXK=M�S©deRwÞ�ãåWySPOXR1M��9ÜK=MRwÞ�K=M�Y�R1MQ ����[=RwÞ�ßå½SyK=NPW±U LOXM�0�N'KXSUTUTUs�DQz�E�J�K'P�]onhT�p�0.�D!����SUK�z�P�T\N T\zhSUT8U?í�Q�OXb Y\M.íïü�S�Y\bXÛ·îPî=Urß½ßmæK���p�

�� ¨p¨ UØ ßNyOXMáwK=N�à7�<K=NPbïYZáIU ¨ ��� �æ

¨ �æ���K=bXK1U Ø æ ú�ægU1��deRwÞ�ßIY\[=SaðIMOXSyK _O£Ý�K=NPK=M[=KWy[�ÜK=Þ*K=W�ÚrOXSyܵWyß�K=[=SPN�YZb£à�bXOgìwK�NyK=WPR1bXåSyOXRwMmU 2��K�MONULzhR�Y�K P���K � v}M�],R�],DQK�zhR�Y�0�nh^\J�DQS�J=UO�<R1b�æ ¨ �p�½U ¨ ��� Ußßmæ ¨ � � � æ¨G¨ æ��mRwOXMWyRwS±U+ê�æ�Y\M_ � K=bXK1U Ø æXU3�;�eR1å½M_IYZNPÛó[=RwM�à_OXSyOXR1MW Ù�R1Nä_OXNyK�[=SÊWPOXÞ�åbïYZSyOXRwMW¬R\Ù [=RwÞ�ßNPK=WyW4àOgã½bgK�â�OXWy[=RwåWK,�RZÚrW±U ��OK�MON�zhR�Y�K'P���K � vwMO],R�],DQK�z�R�Y0�nh^\J�DQS�J=Uh�<Rwb æ ¨ � ¨ UO�wåbXÛ ¨ ��� U1ßßÅæ ¨ ��� �%¨X�� æ¨\ æeVLå½[=NyRwW±UGûeæXUõdeRwÞ�SyK1U��9æXUõY\M_ � K=WyOXK=å½N±UGQÁæXU�x�mY\NyáwK�à4ôÖ__Û Ø OgÞ�å½b Y\SyOXR1M RZÙ Ø ßIYZSPO Y\bXbgÛ #�NyRZÚeàOgM½án�eR1åM_�YZNyÛ �mY8Û1K�N�RZâ1K�NËYZM `L_OïYZã�YZSyOX[¸ûÅbïYZS�9bïYZSyKwU &Uzh]6T\N�zhR�]6D{K�zhR�Y4�OK�MON�z�R�Y K'P>p�TUR�]#R�zhsC�YoMODQs C YtK�<UO�<Rwb æ ¨ �½U ¨ �� �U�ß½ßmæ���� ¨5� ��� � æ¨ �æ��mYZÞ�ãÅU�J?æXU p ^�s�N K\s�^�z�R � DQS�J=æ�+LK=Ú76eR1Nyì%�rV;RZâ1K=NßåãbXOX[±YZSPOXR1MW8U ¨ � � æ¨ �IæedeR1NgfhR1MmU `?æXUGørOXWyWPRIU ûDæXU Ø SPR�K=WyWPK=b�UÊ`�æXUGYZM_�mRwOXMWyRwS±U�ê�æXU��x�wV V;OXNyK=[=S�+;åÞ�K�NyOX[±YZb Ø OXÞ�åbïY8àSyOXR1MWµRZÙ¸ö¬Y\ì1K��<RwNySyOX[=K�W*� `rSPÞ�R1WPßÜK=NPOX[Gê9åN�àãåbXK=M[=KBôÅÝ�K�[=SyWóYZM½_ ø|K=ã�R1åM_ ÚrOXSyÜ deNPR1WyW4àÚrOgM½_mU GOXM �U�O]tn C YoM�D{s4( ^�zhR � DQS�J80�R�zhT�Y9 ^ � Lv K�J|D{M � K�z êrÜKGdeÜIY\N�Y\[=SyK=NPOQ�±YZSPOgRwMóYZM_ Q�R�_�àO�ð�[±YZSPOgRwM RZÙ+ö¬YZìwK=WÃÙ�NyRwÞ � O£Ù�SyOXMá �<K=ÜOX[=bXK=WµOXMûmbXåOX_W8UO�<R1b�æG+L`<êLù|à�` #�`$ø;V|à4dK�Åà;� � �IU�íïê9NyRwM_�àÜK=OXÞ ím+LRwNyÚrY8Û·îPî=Ueßßmæ ���%¨ U ��:� �¨ UÈQ´Y8Û �8à �¨ ��� �æ¨ �æ Ø OXßß V�æXU �<æy��YZ[=]�åOXM YZM_ d�æ�deR1WyWPåmU2� Ø K=b£ÙïàYZ_IY\ßS�Y\SyOXR1M.YZM_.â�OXWy[�R1åWÃWyK=bXK=[�SyOXR1M OgM [=RwM[=K=M�àSyN�Y\SyK=_ VGâwR1NySPK=c�_½Ogß�R1bXK=W8U � MO~ � DQ],]6TUsl],K;0�nh^\J�GC�YoMODQs�J=U ¨ ���� æ¨ �æ �1K=RwMáIU��IæYZM½_ËJ;åWyW�YZOXMmUûeæXU/��ù;M�SPÜK�OX_K=M�SPO�ð�[±Y8àSyOXR1M�R\ÙÈY�â1RwNySyK�c�U ��OK�M�N�zhR�Y}K P�C YtMODQs�$ T;SUnhR�zhDQS�J=U�<R1b�æ �� �U ¨ �� �U�ß½ßmæ�� �:� � ��æ

�`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W

Page 10: AIAA 2001-0545 Development of a Decay Model for Vorticescfdbib/repository/TR_CFD_01_59.pdfFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

���������������

VP−9 VP−10

t*=30

t*=60

t*=90 t*=90

t*=60

t*=30

x

z

x

z

C=DFE�G�H �OG & J�K�L7J�MONQP�R�S;T�JlK P � ý K P RIZXK�N�]6TUW v R�DQN �DQ]onJ,v R�SXD{z�E � � �G� � ¢ � R�]xJ|T\Z\TXN'R�Y�],D � TrDQzOJ�],R�zh]{J���¥� ) �¥ � ��K¢V� � � � ��� ��K¢V� � � ¨G¨ }�?�G���K¢ � � � ��� �¥r� ç;���¥ � ���¢ � � � �'�?� ����¢ � � � �¨ ����p���r¢ � � � ���hG�3�TH¢3 :�<�;¦ G

�`LÞ*K=NyOX[±Y\M�ü�MWPSyOXSyå½SyK�R\ÙÈ`LK�NyR1M�YZåSPOg[�W$YZM½_Ë`;WySPNyR1M�YZåSPOg[�W