9

=0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

  • Upload
    lamhanh

  • View
    215

  • Download
    1

Embed Size (px)

Citation preview

Page 1: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

���������������� ����������� ������� � �!"�#!"$%�'&�!)(*$ !"$ +-,�. � /�+0(*$�!1� 23�4&�+

5-687�6:9<;<6�=?>"@A>)7BDCFEHGIBDJAKLNMPO1QSRUTWVXMPYZT�[]\I^MF_a`1QSYcbd_PQ]efQSY1g0h�MPRW[jikO1Qj_lMnmoYcpjbqYcMFMPRWbqY1p

rsYcbutSMPRaikbvTxwy[S\oz{QSeqbq\|[SRWYcbdQc}�~�RWtZbuYcM~�Rkt�buYcMS})z{h �Z�j���H�<���S���]�

���-�����A���{���`cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl��O"MPRWbqVXMPYZTaQ]e�RkM<iU�ceqTWiN[SY�Q�VXMlTW`c[�gTk`)QHT�MFeqbuVXbqY1Q]TkM<io^�QS_a`3��Q�tSMFi�\|RW[SV�Tk`cM�MP��`1Q]�1iUTo[]\fik�cO"MPRk�ik[SYcbd_��xMlTaiP}j`cMPY1_PM�RkMFVX[HtSMFiIQ�iUTkRW[SYcp�_P[SVXO)[jYcMPYZTo[S\Dik�cO"MPRk�ik[SYcbd_��xMlT�Y1[SbdiUMj �m�eqbuVXbqY1Q]Tkbu[SYXbdioQS_a`cbuMPtjMFg¢¡�w3ik�cRWRk[j�cY1g�buYcpTk`1Mn�xMlT£��bqTk`¤Q]Y¤Q]Y1YZ�1euQSRniUTkRWMFQSV�QHT�OcRWMFiW_lRWbu¡)M<g¥tjMPeu[�_PbvTxwQ]Y)gXTkMFV3O"MPRaQHTW�cRWMNik[�Tk`1Q]T{QSeqe)Tk�cRW¡c�ceuMPYZT�VX[]Tkbu[SY)io¡"MF_l[jVXMbuYjTWRkbuY1ikbu_FQ]eueqw�iU�c¡)iU[jYcbu_S y¦�[VXMF_a`)Q]Ycbd_PQSeoiU�cO1OcRkM<ikik[SRainQ]RWM�1ikMFg§ %~�VXOceuMPVXMFYjTaQHTWbq[jY�[S\£Tk`cM�TkM<_a`cYcbd¨Z�cM�buY©QªTxw�Ocbd_PQSeTk�1Rk¡"[]\«QSY3MFYcpSbuYcM�bdiIMFiUTkbuV¬QHTkM<g¢Tk[�buY1_lRWMFQjiUM{TWQS­SMP�8[S®¬Tk`cRW�1iUT��bqTk`-VXbqYcbuV¬Q]efbuVXO1QS_lT�[SY0[HtSMFRWQSeqe)\|�cMPe�_l[jY1iU�1V3OcTkbu[SYf 

¯o°�¯f± ���A²y³�´-�{�Aµl² ±

z�[SVXV¢�cYcbqTxw?Y1[SbdiUM�bui¶[SYcM�[]\�TW`cM�V¬QH�x[jRfTWMF_a`cYc[jeq[jpSbd_PQSeS`��cRk�g�euMFiX\«Qj_lbuYcp·\|��TW�cRkM�ik�cO"MPRaiU[jYcbd_0TkRaQ]Y1ikO)[jRUTaiP}{iU�)_a`¸QjiXTk`1M¹�bupS`c��º�O"MPM<g»z�but�bqe���RaQ]Y)iUO"[SRkT�¼«¹?º1z{��½¾ o��`1M�`c[STXik�cO"MPRk�ik[SYcbd_�xMPTWi·Ml��`1QS�1iUTkbuYcp�\|RW[SV¿TW`cMMFYcpSbuYcM<i�[S\�ik�1_a`�Q]buRk�_lRaQH\ÀTXQ]RWM�O"[H��MFRU\|�1e�Yc[jbuikM�pjMPYcMFRWQ]Tk[jRWiF}¶M<iUO"MF_PbuQSeqeuwªg��cRWbuYcpTWQS­SMP�8[S®� Ár�YZTkRWMFQ]TkM<g§}ÂTW`cMPwÃ_PRkM<QHTWMªYc[SbdikM�euMPtSMFeui���`cbd_a`Q]RWM�MPY�t�buRk[jYcVXMPYZTWQSeqeuwy�cY1Qj_P_PMPO�TaQ]¡ceuMS {¦��cVXMPRW[S�)i�Tk`cMF[SRWMlTk�bd_PQ]e8}�_l[jV3O1��TWQ]Tkbu[SY1QSe:}�Q]Y1g�Ml��O"MPRWbqVXMPYZTaQ]eDixTW�1g�buMFi{`1Q�tSM?QHTk�TkRWbu¡c��TkM<g�TW`cM¬MPeuMPtHQ]TkMFg�Yc[jbuikMÂTW[^�QS_a`��{Q�tjMFiF}"��`1bu_a`·Q]RWMOcRWMFiWiU�1RkM���Q�tSM<i�pSMFYcMPRaQHTWMFg-¡�w�TW`cMXiU�cO"MPRaik[SYcbd_�VX[]TWbq[jY[]\Tk�1Rk¡c�1eqMFYjT3MFgcgcbqM<i£��bqTk`ªRWMFikO)M<_¾T¢Tk[TW`cM�ik�cRWRk[j�cY1g�buYcp�Q]buR< ¦�[STkM<g`1MPRWM0QSRkMik[SVXM-­SMPw¤�{[SRW­�i3bqY»Tk`1M-Ä1MPedg§ ÅmA��O"MPRk�buV3MFYZTWi�[S\�^¥_PÆ�QS�cpS`1eqbuY�ÇlÈ�ÉHÊqË·Ìq�lÍÎ}¶\|[Seueq[H�{MFg�¡�w�Tk`1[jikM¬[]\�¶Rk[j��TUTIÏÅ^�_PƶQ]�cpj`ceqbuYyÌ ��ÍÎ}HMFiUTWQ]¡1eqbdiU`1MFg?Tk`)QHTAiU[j�cY1g�bdi¶QSV3�OceubvÄ)MFg3bqYXQng�buRkM<_¾Tkbu[SYX_l[jY1ikbuiUTkMFYjTA��bqTk`¢Tk`cM�O1Rk[jO1Q]pZQHTkbu[SYÂ[]\^�Qj_a`���Q�tSM<i3¼PÐkg�buRWMF_¾TWbqt�bqTxw�Ñ�[S\�ik[S�1Y1g1½¾ ��¶QSVÒÏ�Ó��1RUTW[SYÌ �HÍ��1ikMFg£eubuYcMFQSR�iUTWQS¡cbqeubqTxwnTk`cMF[SRWwNTW[?Qj_P_P�cRWQ]TkMFeqwnOcRWMFg�bd_¾T�Tk`1Mik[S�cY1g·O1Rk[�g��1_PMFgÔ¡�wÔ^¥QS_a`·�{Q�tjMXRWQjg�buQ]Tkbu[SYÔ\|RW[SVÒQ_P[SedgRW[S�cY1g¬�xMlTF n��`cMFbqR?Q]Y1QSeqw�ikbuis�{Qji�Ml��TWMPY1g�M<gTW[�`c[ST��xMlTais¡�w��Q]V�ÇlÈ�ÉHÊuËÂÌ Õ]ÍIQ]Y1g¡�wº�MPbuYcMFR3ÇPÈ�É]ÊqË¢Ì �<ÍΠs^[ZixTNRkM<_lMPYZTWeqwj}^bqTW_a`cMFeqeXÇlÈXÉHÊqËÖÌ ×HÍNO)MFRU\|[jRkVXMFg¸g�buRWMF_¾TyY��cVXMFRkbd_PQSesiUbuVÂ�c�edQHTkbu[SY�[]\�RW[S�cY1g�xMlTWiXQSY1g·\|[S�1Y1g¤TW`1QHT¢Tk`cM0YcMFQSR¢QSY1g¤\«QSRÄ1MFeugci�[S\�ik�cO"MPRaiU[jYcbu_��xMPTWi�Q]RWMnVÂ�1_a`0Yc[jbuikbqMFR{Tk`1QSY�TW`c[jikMn[]\ik�c¡1ik[SYcbd_n�xMlTaiP}�Q]pjQSbqY¤g��1M�TW[�bqYZTWMPY1ikM�^�Qj_a`��Î�{Q�tjM¬RWQjg�buQ]�ØWÙ�Ú8Ú:ÛFÜxÝßÞWà:áAâ"ã:ÛaäuáxÚ8Ú:Û¾ãkå<æ�áUæ�ç]áUã"Ù�èÀÙ�Ùé

êDÛ¾ëPìFã8Ýîí¾ïlà Üð�ñUò¾òló çlì�ô�õsâ1ÞaëHÞaæ�Û¾Ú:ÜÎï�Û¾öZõ÷â"ö�ç<øîÝîÚ8ï�áxù�çPì»à8ï�áÙ�æ�áxã:ÝîÜkÞaúÂèÀú�Ú«à8Ý à:ö<à:áNÛaäfÙ¶áUã:Û¾úHÞaö<à:ÝîÜxÚ�Þaú�ùXÙ¶Ú:à:ã8Û¾úHÞaöFà8ÝîÜxÚkåcèÀú<Ülõ�û�Ý à:ïë]áxã8æ�ÝîÚ8Ú:ÝîÛ¾ú

Tkbu[SY� �~�Y�QSgcg�bqTkbu[SYXTW[¢^�Qj_a`��Î�{Q�tjM{Y1[SbdiUMj}ZiW_lRWMPM<_a`3¡"MF_P[SVXMFig�[jV3buY1QSYjTbuY��xMPTWi��bqTk` iUTkRW[SYcp¸ik`c[�_a­�iF}?b:  MS u}nbqY©�cY1gcMPRk�[SR¢[HtSMFRU�ÎMl��O1QSY1g�MFg��xMPTWiF Ô��`cM-_l�1RkRWMPYZT¢iUTk�1gcw�\|[�_P�1iUM<iÂ[jYO"MPRk\|MF_¾TWeqwZ�ÎMl��O1QSY1g�MFgÂ�xMlTaiP}1��`cbu_a`-g�[XYc[ST�O1Rk[�g��1_PM�ik_PRkMFMF_a`Yc[jbuikMS �¶[RWMFg��)_lM·^�QS_a`��{Q�tjM�MPVXbdikikbq[jYf}�Tx�{[¸Q]OcOcRW[jQj_a`cMFi�Q]RWM_l�1RkRWMPYZTkeuw¥O1RkM<g�[SVXbuY1Q]YZTFü3VXbv��buYcp¥MFYc`1QSY1_lMFV3MFYZTF}I�1ik�1Q]eueqwt�buQNeu[S¡"M�VXbq��MFRWiF}]Q]Y)g£Mx�xM<_¾TW[SRAiU`1Rk[j�1gci{Ì �FÍΠoýs\ÀTkMPY�}HTk`cM{Tx�{[VXMlTW`c[�gci£Q]RWM¬_l[jV¡cbuYcMFgf �^bq��buYcpMFYc`1Q]Y)_lMPVXMFYjT£RWMFgc�1_lM<iTk`1M�eqMFYcp]TW`X[]\"Tk`cMs^¥QS_a`��Î�{Q�tjMl�ÎMPVXbqTUTkbuYcp?RkMFpSbu[SY3[]\"Tk`cMI�xMPTF ~ÎT�g�[�MFi?iU[���bvTW`�Tk`1M¢RWbuik­0[]\�QSV3O1eqbq\|w�bqYcp�YcMFQSRU�8Ä1MFeug�^¥QS_a`��Q�tSMFinQSY1g·��bvTW`ªQ]OcO1RkM<_lbdQ]¡ceuM3Tk`cRW�1iUT£O)MFY1Q]eqTkbuMFiF 0þ1[SR£MP���Q]VXOceuMS}AbuYªTW`cM0��[jRk­Ô[]\N¦NQ]pZQ]V¬QHTaiU�4ÇPÈÂÉHÊqË�Ì ÿ]Í8}AM<QS_a`����[]\fY1[SbdiUMNRkM<g��1_¾TWbq[jY���QSi�QS_P_P[SVXO1QSYcbqM<g¬¡�w-���ÃTk`cRW�1iUT{eu[jiWiF þc[jR��������-Yc[jbuikM¥RWMFg��)_¾Tkbu[SY�}�Tk`1M¥TW`cRk�)ixT-eq[Ziki���[j�cedg�¡"M������}1��`cbd_a`buis�cY1Qj_P_lMFO�TWQS¡ceuw�`cbupS`f �mf�xMF_¾TW[SRsik`cRW[S�1g1i�MPY��_PQjiUM�Tk`1M¶�xMPTo�cYZTkbue�bvT�gcMF_lMFeqMFRWQ]TkM<i�TW[�iU�c¡)iU[jYcbu_{ikO)MFMFgciF}HTk`��1iTk`1My^�QS_a`Ô�{Q�tjM¢RaQSgcbuQ]Tkbu[SY¥bdi�_l[jY�Ä1YcM<gÔbqYZTkMFRkY)Q]eueqwj Xmf�xMF_l�Tk[jRWi�OcRW[HtZbdg�M�ik[SVXMTW`cRk�)ixT-Q]�cpjV3MFYZTWQHTWbq[jYÅQHT�eu[H�1bqpj`ZT^�Qj_a`�Y��cV¡"MPRaiP}D¡c�cTnO"MPY)Q]eub��FMÂTW`cRk�)ixT�QHTn^�QS_a`�YZ�1V¡"MPRai`cbupS`cMFRÂTW`1Q]YQ]¡"[S��T �c  ×ªÌ �HÍΠԹsMPY1_PMS}oMU�xMF_¾TW[SRai£V¬Q�w·YcMPM<gTk[¡)M�RWMlTWRWQj_¾TaQ]¡ceuMS}ZQSgcg�buYcpnTW[�Tk`cMN_l[SVXOceuMl��bqTxw3QSY1gX�{MPbupS`ZT[]\ATW`cMXiUw�iUTkMPV- ���`cMFRkM¢`1QSiN¡)MFMPYÔiUbupSYcbqÄ)_FQ]YZTsOcRW[SpjRkM<iki�[jY_l[jV¡cbuY1Q]Tkbu[SYÅ[]\�eu[S¡"M¥VXbq��MFRWiy��bvTW`�Mx�xM<_¾Tk[jRWi�Ìv����ÍΠ z��cRk�RWMPYZT?ik�cOcOcRWMFiWikbq[jY�_l[SY)_lMPOcTWi?RkMFtS[jeqtjM£Q]RW[S�cY1g¥iU�1_a`�_l[SV¢¡cbq�Y1Q]Tkbu[SY1iF}H¡c��T�Tk`cRW�1iUTIQSY1g��{MPbupS`ZT¶O"MPY)Q]eqTkbuMFi�RkMFV¬Q]buYÂiUMFRkbu[S�)i_l[jY1_lMFRkY)iP AýsTk`cMFRoik_a`cMFVXMFiITW[?RWMFgc�1_lM�Yc[jbuikM�`1Q�tSM�buY1_leu�1g�M<gTk`1M{~�Y�tjMPRkTkMFg��AMPeu[�_PbvTxw��oRW[]Ä1euM�QSY1g�TW`cM���`cMPRWV¬Q]e�hs_l[j�1iUTkbd_º�`cbuMPedg§}�[HtjMPRWtZbuMP�{MFgX¡�w�º�MPbuYcMPR�Ï��nRWMFi��FQ�Ì �FÍ���Tk`1MPbuR�g�MPtjMPeq�[SO1V3MFYZT�iUTWQ]Tk�1i�bui�Yc[ST�­�Yc[H��Y�TW[3Tk`cM£QS��Tk`c[jRF ~�Y�Qjgcg�bqTkbu[SY TW[4RWMlÄ1YcbuYcp�Ml��buiUTkbuYcp4TkMF_a`1Ycbu¨Z�cM<iP}ÂTk`cMFRkMbuiYcMFMFgTW[-eq[�[S­Q]TnQSevTWMPRWY1QHTWbqtjMÂ��Q�w�iN[]\{iU�cO1OcRkM<ikikbuYcp�Y1[SbdiUMj ��`cM�V3MPTk`c[�g�g�MFiW_lRWbq¡"MFg�bqYÅTW`cbdi�O)Q]O"MPR��cTkbueqb��PM<i�O1Rk[jOc�ceq�ikbqtjMV3M<Q]Y1iyTk[ªO1RkMFtSMPYZTypjMPYcMFRWQ]Tkbu[SY¸[]\£^�Qj_a`Å�{Q�tjMFiF}�buY_l[jYZTkRaQSiUT£Tk[�Ml��buiUTkbuYcp¥TkMF_a`1Ycbu¨Z�cM<i¢Tk`1Q]T¢�1ikM�buYZTkRW�1ikbqtjM�g�Ml�t�bu_PMFi�Tk[Xik�cOcOcRWMFiWi{[SR�_l[jY�Ä1YcMn^�QS_a`��{Q�tjMFi�QH\ÀTkMFR{Tk`cMFw�Q]RWM\|[SRWVXMFg§ ?��[�Tk`cM¢­�Yc[H��eqM<g�pSMÂ[]\ATk`1M¢Q]�cTk`c[jRF})Tk`cM3r?z�~�MP���O"MPRWbqVXMPYZTaiAg�MFiW_lRWbq¡"MFg¢`cMFRkMFbqYXQ]RWM�TW`cM�Ä1RaixTITW[ngcMPVX[SY1iUTkRaQHTWMTk`1M£_PQ]O)Q]¡cbueqbqTkbuMFi�[S\¶TW`cbdi�Q]OcOcRW[jQj_a`f 

Page 2: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

¯I¯o°�� ������� ���� "!$#Pµ�% µ ± �?�Iµl² ±^�Qj_a`¢��Q�tSM<iP}]Qng�[jV3buY1QSYjToik[S�cRa_lM{[]\§ik�cO)MFRWik[SY1bu_��xMlT�Y1[SbdiUMj}Q]RWM�pSMFYcMPRaQHTWMFg¢¡)M<_PQS�1iUM�TW�cRW¡c�ceuMPYZT�MFg1g�bqM<iobuYXTk`cMo�xMlT�OcRW[SOc�Q]pZQHTWM���bqTk` Q»_l[SY�tjMF_¾TWbqtjM�tjMPeu[�_lbqTxw'&)(0��`cbd_a`©buiik�cO"MPRk�ik[SYcbd_0��bvTW`»RWMFikO"MF_¾TXTW[·Tk`cM�ik�cRWRk[j�cY1g�buYcp·Q]buRXiUTkRWMFQSV-}{Qjiik­SMlTa_a`cMFg0buY¥þ�bqp) ��j �^�MFQSik�cRWMPVXMPYZTai�[]\o_l[jY�tSMF_lTkbutSMntSMFeq[�_l�bqTxwbqYÔiU�cO"MPRaik[SYcbd_¾��iU�1¡1iU[jYcbd_£iU`1MFQ]RNeuQ�wjMPRaiP}"eubq­jM£Tk`1[jikM3ik�cRk�RW[S�cY1gcbqYcpnTW`cM�O"[]TWMPYZTkbdQ]e"_l[SRWM�[]\DTW`cMA�xMlT<}j`)Q�tSM�ik`c[H��Y3Tk`1Q]T&)(�QSOcOcRW[jQS_a`1MFi�ÿ*�H���+����[S\cTk`1M�tjMPeu[�_PbvTxw�[S\cTk`1M�\«QjixTAiUTkRWMFQ]V¼nÌv�j�¾ÍÎ}�Ìq�<�<ÍÎ}�Ìq�F��Í|½¾ ¬��`cbdinbui�_P[SY1ikbdixTWMPYZT?��bqTk`ÔTW`cM¬iUeu[SO"MX[]\^�Qj_a`���Q�tSMFi�MPVXbqTUTkM<gy\|RW[SV ik�cO"MPRaiU[jYcbd_o�xMPTWi�[SR�ik`cMFQSR�edQ�wZ�MPRaiF}�[j¡1ikMPRWtSMFg·¡�w·V¬Q]Y�wÔbqY�tSM<ixTWbqpZQHTW[SRaiy¼«ikMPM�\|[SR3Ml�cQ]VXOceuMÌq�F��ÍÎ}�Ìq�PÕHÍ|½¾ 

Mach waves

a

Uj

jUc

8

U 8 > ac jU -

jM >1

SupersonicEddies

U 8

8

þ¶bupS�1RkMX�SüA^�Qj_a`���Q�tSM?RaQSg�bdQHTWbq[jY�buY-ik�cO"MPRaiU[jYcbd_��xMPTWiF 

a

a 8

8

f

Uj

Uf

U fc

jUc

< a

< a

U 8

Uf

jcU -

fcU -

f

fM

jM >1

U 8

fM

þ�bqpj�cRWM��cü,��RkbuY1_lbuOceuM�[]\"^¥QS_a`3�{Q�tjM�MFeqbuVXbqY1Q]Tkbu[SYfü�_lRWMFQ]Tkbu[SY[]\�_P[-1[H�¸QSgH�UQS_lMFYZToTk[¢VXQSbqY£�xMlT{ik[�TW`1QHT�Q]eue)M<gcg�wXVX[]Tkbu[SY)i¡"MF_l[jVXM?buYZTkRWbqY)iUbd_PQSeqeuw�ik�c¡1ik[SYcbd_] ��`cMOcRWbuY1_lbuOceuM0[S\?^�Qj_a`��Q�tSM0MPeubuV3buY1Q]Tkbu[SYf}�g�MPO1bu_lTkMFg»buYþ�bqp) ��c}�bdi�Tk[�iU�cRWRW[S�cY1g¥Tk`1M��xMlTÂMP��`1Q]�1iUTÂ��bqTk`ªQ-edQ�wSMFR�[]\_l[+1[H��buYcp-pjQSin��`1[jikM¬OcRk[jO)MFRUTWbqM<i£Q]RWM3TWQSbqeu[SRWMFg·ik�1_a`ÔTk`1Q]T¼«QZ½�Tk`cM��xMPT�MFgcg�buMFi�¡"MF_P[SVXM�iU�1¡1iU[jYcbd_s��bqTk`-RWMFikO)M<_¾T�Tk[3Tk`1M_l[+1[H�©Q]Y1g·¼|¡"½�Tk`1M¢_l[+1[H�4M<gcg�buMFiNQ]RWM£iU�c¡)iU[jYcbu_n��bqTk`¥RkMP�ikO)M<_¾T3Tk[�TW`cM-Q]V¡1bqMFYjTXQ]buR3iUTkRWMFQSV- Ôº�buY1_lM-Q]eue{MFgcg�w¤VX[S�Tkbu[SY)i�Q]RWM�ik�c¡1ik[SYcbd_��bqTk`�RWMFikO)M<_¾TyTW[¤Tk`1MPbuR0ik�cRWRk[j�cY1g�buYcpiUTkRWMFQ]V¬iF}SYc[3^�QS_a`¬��Q�tSM<iAQ]RWM�pSMFYcMPRaQHTWMFg§ �~�YXTk`1MNQ]Y)Q]euw�ikbdiTk`)QHTn\|[Seueq[H��iF}�ik�c¡1iW_lRWbqOcT�.gcMPYc[STkMFi?TW`cM?�xMlT£OcRW[SO"MPRkTkbuMFiF}0/Tk`1M�_l[+1[H�ÖOcRW[SO"MPRkTkbuMFiF}IQSY1g21 Tk`cM0Q]V¢¡cbqMFYZT3Q]buRk�ÎiUTkRWMFQSVOcRW[SO"MPRkTkbuMFiF A��`cM?ikw�V¡"[Se0&©RWMPO1RkM<iUMFYjTaioTW`cM31[H��tSMFeq[�_PbvTxwj}4 TW`cM£iUO"MPM<g0[]\�ik[S�1Y1g§}6587:9<; 4 TW`cM�^�QS_a`0Y��cV¡"MPR�QSY1g&)(�Tk`cM£_P[SY�tSM<_¾TkbutSM?tSMFeq[�_PbvTxw¬[S\¶TW`cM�MFgcgcbqM<iP 

��`��1iF}fTk[O1RkMFtSMPYZT�pSMFYcMPRaQHTWbq[jY�[S\�^¥QS_a`·�{Q�tjMFiF}§��M¬V¢�1iUTiWQHTkbdiU\|w &�(�=�>?&�@BA 4 @ ¼x��½Q]Y)g &)( @ >C&)DEA 4 D ¼:�j½��`cM<iUMyRWMPedQHTWbq[jY1i£_FQ]YªQ]edik[¡)M0_PQSiUT£buY¤TkMFRkV¬i£[S\�Tk`cM0_l[SYc�tSM<_¾TWbqtjM¢^�Qj_a`¥Y��cV¡"MPRn[S\oTk`cM3MFg1g�w��bvTW`�RWMFikO"MF_¾T?Tk[�Tk`1Meu[H���ÎikO)MFMFg3iUbdg�Mj}S_l[jV3VX[jYceqwÂRkMP\|MPRWRkM<g£TW[nQji)5CF�G] I¹�MFRkMj}]�{MRWMF¨Z�cbuRkM�5CF�G�HIA'��Q]Y1gJ5CF�GLK�A'�S �¶[�TWRWQSY1ikeuQ]TkM�Tk`cM<iUM-RWMF¨Z�cbuRkMFV3MFYZTWi3bqYZTW[ÔO1RWQj_¾Tkbd_PQSe�pS�cbdg�MP�eubqYcM<i3\|[SRXTW`cM_l[-)[H� OcRW[SO"MPRkTkbuMFiF}��{M��1ikM-RkM<_lMFYjT¬MP��O"MPRWbq�VXMPYZTWQSeNgcQ]TWQ¤[jY»Tk`cMÔ_l[SY�tjMF_¾TWbqtjM-tSMPeu[�_lbqTxwM&�(¬buYÅOcedQ]Y1QSRik`cMFQSR�edQ�wSMFRWi�¡)MPTx��MFMPY'Qik�cO"MPRaiU[jYcbu_�ixTWRkM<Q]V ¼U�<½�Q]Y1g�Qik�c¡1ik[SYcbd_¥iUTkRWMFQSV ¼8�S½¾ Ö~�YÅTW`cM·g�[S�c¡1eqMP�8MP��O"[jik�cRWMS}�_PRk[ZikiU�ikMF_¾TWbq[jY1Q]eSbqV¬Q]pjbqY1p�Ml��O"MPRWbqVXMPYZTaif[]\fÌq�<�<ÍÎ}<Tk`cM�\|[Seueq[H��buYcp�MFV¢�OcbuRkbd_PQSe§VX[�gcMPe§\|[jRI5 F�G �{Qji{_P[SY1iUTkRW�1_lTkMFgfü5CF�GN7 5CF,O �QPSRT �UO4¼ 4 G ; 4WV ½ G ¼«�Z½

��`cMFRkM ��P3R'7YX �*Z � 5 F >[�WZ Õ \ 5 FN] �WZî�j�� \ 5 FN^ �WZî�j� ¼|Õ�½

Q]Y)g 5 F 7 & V >?&)G4_V O 4 G ¼:�j½ikbqpjYcbvÄ)MFiÔTk`1M Ðkikw�V3VXMPTkRWbu_FÑ�_P[SY�tSM<_¾TkbutSMÅ^�Qj_a` Y��cV¢¡)MFRF ��`cM{RWQ]Tkbu[SY1QSeqM�[]\cTW`cbuiIVX[�g�MFej\|[jR`5 F�G bdi¶¡1RkbuMa1w£Qjif\|[jeqeu[H��iFüMl��O"MPRWbqVXMFYjTai0`1Q�tSMÔg�MlTWMPRWV3buYcM<g�TW`1QHT<}?Qji 5 F bqY1_PRkM<QSikMFiF}Tk`1M¥_l[jYZtjMF_lTkbutSM-tSMFeq[�_lbqTxwgcMPO1QSRUTaiX\|Rk[jV TW`cM¥Q�tSMFRWQSpSM0tSMP�eu[�_PbvTxw»Q]Y1gf}�buY»Tk`cM¥_FQSikM-[]\�iU�cO"MPRaik[SYcbd_¾��iU�1¡1iU[jYcbd_�edQ�wSMFRWiF}TkMFY1gcioTk[H��Q]Ragci�& V ¼À\«QSiUT�VX[�g�M<½l �ýNYXTk`cM35 F V �ÎtSMPRaik�1ix�b5 F�GOceu[]T<}§Tk`1Q]T£g�MPO1QSRUTW�cRWM3_FQ]Y¥¡"MXMl��OcRWMFiWiUM<g¥QjisTW`cM¬g�bdixTaQ]Y1_PM��P3R�[]\NTk`cMMP��O)MFRkbuVXMPYZTWQSe�V3M<QSik�cRWMPVXMPYZTX\|Rk[jV bqTWiyikw�V3�VXMlTWRkbd_ÂtHQSeq�cMj Â��`cM¬g�bdixTaQ]Y1_PMc��PSR�Ml��`cbu¡cbqTWi�Q0_l[jY1iUbdiUTkMPYZT<}VX[SYc[STk[jYcbu_FQ]eueqw�bqY1_PRkM<QSikbqY1p¢TWRkMFY1g-tSMFRWik�1i 5 F ¼«m�¨) )ÕZ½�QSY1g_PQSYTk`1MPRWMl\|[SRWM¡"MÂ�)iUM<gTk[�OcRWMFg�bd_¾Td5 F�G  n��`cbdi?QjiUw�VXVXMlTk�RWbu_�¡)MF`1Q�t�bq[jRÂ[]\�TW`cM0euQSRkpjMyMFgcgcbqM<i¢bdi¢Ml��TkRWMPVXMPeuw·RkMFeqMFtHQ]YZTTk[Ô^�QS_a`��{Q�tjMyRWQjg�bdQHTkbu[SY�}¶TW`c[S�1pS`ªTW`cM�O1`Zw�ikbu_FQ]e{VXMF_a`1Q]�YcbdiUV¬iÂ\|[SR¢Tk`cM�\«QSiUT¢VX[�g�MFi0¼«QSY1g¤\|[jR£TW`cM0ikeu[H� VX[�g�M<iÂbuYik�cO)MFRWik[SY1bu_l�Îik�cO"MPRaiU[jYcbu_NiU`1MFQ]R�edQ�wSMFRWia½�RkMFVXQSbqY0MPeu�1ikbqtjMS rsikbuYcp�Tk`1MsVX[�g�MPe"[]\�m�¨ZiF S�]���n�{Ms_FQ]Y¬RWMFQSgcbqeuw3gcMPRWbqtjM�RkMFeuQ]�Tkbu[SY)i§Tk`)QHT�TW`cM�_P[-1[H�¤iUTWQHTWbu_ATkMPVXO"MPRaQHTW�cRkMj}�e<@1}HQ]Y1g�_P[-1[H�^�Qj_a`¬YZ�1V¡"MPR<}Q5C@1}jV¢�1iUT�ikQ]Tkbdix\|w¢Tk[ÂOcRkMFtSMFYjT�^¥QS_a`¬��Q�tSMRaQSg�bdQHTWbq[jYf ���`cMFwªQ]RWM�Oceu[]TkTkM<g�buY»þ�bup1 A�¥\|[SRf5 = 7��*Z �c}e = ;�e<Dg7��WZ ÿ¤QSY1gh5CD TWQS­�bqYcp·TW`cM¥tHQSeq�cM¥[]\d��¼«iUTWQ]Tkbd__l[jY1g�bqTkbu[SY1ia½oQSY1gi�1  ��¼ÀTxw�Ocbd_PQ]e)TWQS­SMP�8[S®0ikO)MFMFg1½l I��`1MNeu[H��MFR¡"[S�cY1g£RkMFOcRkM<iUMFYZTWi�_P[SY1gcbvTWbq[jY1i�Q]T¶��`cbd_a`�5CF�G�H�7 ��QSY1gnTk`1M�cOcO"MPR{¡"[S�cY1g�_l[jY1g�bqTkbu[SY1i{QHT{��`cbu_a`j5CF�GkKl7 �S �ýNYcMsV¢�1iUT­SMFMPObuYVXbqY1gTk`1Q]T�Tk`cMÂRkMFeuQ]Tkbu[SY1i�[S\oþ¶bup1 "�yQ]RWM�¡1QjiUM<g�[jYOcedQ]Y1QSRAiU`cM<Q]RIedQ�wSMPRaiIQ]Y1gÂTk`)QHTITk`cM�_P[]��Q]YcY��cedQ]R)�xMlToV¬Q�w£MP���`cbu¡cbqT�¨Z�cbqTkM�g�bq®DMPRWMPYZT�¡)MF`1Q�t�bq[jRF}�MFikO"MF_lbdQ]eueuw¬bv\¶Tk`cM�Q]Y1YZ�1eq�1i

Page 3: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

bdi�TW`cbuYf o��`cMPRWMl\|[jRkMj}STW`cMPw�Q]RWMs�1ikMFg�[SYceuwyQSi{QÂO1RkMFeqbuVXbqY1QSRkwpS�1bug�MQSY1g»YcMPM<gªTW[¤¡)MRWMlÄ1Y1MFg¡Zwª\|��Tk�cRWMMl��O"MPRWbqVXMFYjTaiP ¦�MFtSMFRUTW`cMPeuMFiWiP}"iU[jVXM£bqVXO"[SRkTWQSYjTsTkRWMPY1gci?Q]RWM�[S¡�t�bu[S�1iF �þc[jRik�c¡1ik[SYcbd_�5 @ }"Tk`cMX_l[+1[H�©pSMFYcMPRaQ]eueqw0VÂ�)ixTn¡)M3`cM<QHTkM<g\|[jR^�Qj_a`��Q�tSM�MPeubqVXbuY1QHTWbq[jY�TW[Ô[�_F_l�cR< »h{T¬YcM<Q]Rk�Îik[SY1bu_j5 @ }`cM<QHTkbuYcp¤RWMF¨Z�cbuRkMFVXMPYZTWiXQ]RWM-VXbqYcbuV¬Q]e?Q]Y1g»O"[jiWiUbu¡ceuwC�PMFRk[) hsim5 @ buY1_lRWMFQjiUM<iP}SMPeubqVXbuY1QHTWbq[jY¬¡)M<_l[SVXM<iAVX[SRWM�Many_lbuMPYZT�buYTkMFRkV¬i�[S\�`cMFQ]TÂQSgcg�bqTkbu[SY� �h{T£TW`cM�iWQ]VXMXTkbuVXMS}¶Tk`cMyMFeqbuVXbv�Y1Q]Tkbu[SYo�P[jYcMS}DQ]Y)g-`cMPY1_PM�Tk`1MÂ[SO"MPRaQHTWbqY1p¬VXQSRkpjbqY�}1¡"MF_l[jVXMY1QSRkRW[H��MFRF qpÅbqTk`�\|[SRW�{QSRWg:)bqpj`jTikO)MFMFg§}NTk`cM·[jO)MFRWQ]TkbuYcpV¬Q]RWpSbuY¸bui¬��bdg�MFR�¡c��T�Tk`cM�eubuYcM�e @ ;re D 7 ��OcRWMFikMPYZTai¬Qeu[H��MFRsOcRaQS_lTkbd_PQ]e�eubqVXbqTX¼«bvT�bdiN�1Y�\|MFQjiUbu¡ceuM£Tk[-_l[�[je�Q0_l[+1[H���buYcp¬ixTWRkM<Q]V buYQ]Y0MPYcpjbqYcM£QSOcOceubu_FQHTWbq[jYf  ½

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

Tf /

T oo

MACH WAVEELIMINATION

(b)(a) Moo

=0

Mj=1.5

Tj/T

oo=2.8

Mf

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

Mf

Tf /

T oo

MACH WAVEELIMINATION

Moo

=0.3

Mj=1.5

Tj/T

oo=2.8

þ�bqpj�cRWMn�1üA^[�g�MPeD\|[jR�^�QS_a`��Î��Q�tSMsMFeqbuVXbqY)QHTkbu[SYj�P[jYcMFiFü�¼«QZ½iUTWQHTWbu_n_P[SY1g�bqTkbu[SY)is��¼«¡)½�Q]V¢¡cbuMPYZT�^�QS_a`0Y��cV¡"MPR�[S\t�1  �1 h�Y0QSgcgcbvTWbq[jY1Q]eDOcRaQS_lTkbd_PQ]e)RkM<¨Z�cbqRWMPVXMFYjT�\|[SR{Tk`1MNVXMlTW`c[�g¬TW[�{[SRW­ÂbdiATk`)QHToTW`cMN_P[-1[H�»¡)MNbqVXVXMFgcbuQ]TkMPeuwXQSg��UQj_lMPYZToTW[�Tk`1M�xMlT<}]b8  Mj q}�TW`cMPRWM{ik`c[j�ceug£¡)M{Yc[?VXQ]�x[SR¶pjQ]O)i�¡"MlTx�{MPMFY£Tk`1M�Tx�{[1[H��iF JunQSO1i��{[S�cedgÔMPY1QS¡ceuMypSMFYcMPRaQHTWbq[jY�[S\�^¥QS_a`·�{Q�tjMFiF}Tk`��1isg�Ml\|M<QHTkbuYcp3Tk`1M�Oc�cRWO)[ZiUMn[]\�Tk`1M£_l[-)[H�� ¯I¯I¯o° !wvjxy ��oµs%z ± �mA��O"MPRWbqVXMFYjTai£�{MPRWM�_P[SY1g��)_¾TkM<g¤buY�Q�YcMF� _l[S�ÎQSYcY��ceuQSR��xMlT\«QS_PbqeubqTxw0g�MFOcbd_¾TkM<g-bqY�þ�bup1 )Õ1 �^bv��TW�cRkM<i�[S\I`1MPeubq�cV Q]Y)gQ]buR�{MPRWMyiU�cO1OceqbuMFgÔTW[�Q�_l[jY1_lMFYZTkRWbu_3Yc[*�s�PeuMyQ]RWRaQ]YcpjMPVXMPYZT£QS_l�_lMFO�TkbuYcp0Q¬tHQ]RWbqMPTxw�[]\obuYcYcMFRNQSY�[S��TWMPRNYc[+���PeuMFiF ���`cM£buYcYcMPRYc[*�s�PeuMFiF}o[S\¢�<�c î���|{j{ MP��bvTyg�bdQ]VXMlTWMPR<}��{MPRWM�g�M<iUbupSY1MFgª¡�wTk`1M0VXMPTk`c[�g»[]\N_a`)Q]RaQS_¾TWMPRWbuiUTkbd_PiÂ\|[SR¬^�Qj_a`Y��cV¢¡)MFRWi0�j  �c}�S ß�]�c}ZQSY1g��c  �1 oýNY1eqw3Tk`1M?^�Qj_a`��S î��RWMFik�ceqTWi{Q]RWMsOcRWMFikMPYZTWMFg`cMFRkMj ª��`cM-[S��TWMPR3Yc[+���PeuMFi¢\|[SRWVXMFg�ikVX[�[]Tk`»_l[jYjTWRWQj_¾TWbq[jY1iTkMFRkVXbuY1QHTWbqY1p3buY-Ml��bqTsg�buQSVXMlTkMFRWi�[S\��H��  ÿ3QSY1g-���S  ×�{}{Ô h�buR{�{Qji�ik�cOcOceubqM<g¬¡�w3_P[SVXOcRWMFiWiU[jRWi�QHT�QÂOcRkM<ikik�cRWM�[]\o�j} �*�+�~_� 4 ¼:Q]eue§OcRWMFiWiU�cRWMFi�QSRkM�Q]¡)iU[jeq��TWM<½l I¹sMPeubq�1V ��QSi�ik�cOcOceubqM<g¡�w£TWMPY¬`cbupS`��ÎOcRWMFiWiU�1RkM�_lw�eubqY)g�MPRaioQ]Y)g¢��QSioRkMFpS�cedQHTWMFgXg�[H��YTk[ª�jÿ+� ~_� 4  4��`cM�ik�cO"MPRaiU[jYcbu_¢�xMPT�_l[jY1iUbdiUTkMFg»[S\?`cMFeqbu�cVVXbv��M<g���bqTk`ÃQ¸ikV¬Q]eueÂQ]VX[S�1YjT�[S\XQSbqRbuYH�xM<_¾TkM<g'Tk`cRW[S�1pS`QÔ�c  Õ]�ÎV3V ik[SY1bu_�[SRWbvÄ"_lMS ��`cM-Oc�cRWO)[ZiUM0[]\NTk`cbdi¬g�bueq��TWbq[jY��QSi�Tk[3¡cRkbuYcpÂTk`cM{�xMlT�tSMFeq[�_PbvTxwyQSY1g¬Ml®DMF_lTkbutSMNTkMFV3O"MPRaQHTW�cRWM

g�[H��Y¤TW[�euMPtjMPedi£Txw�Ocbd_PQSe�[S\Nik�cO)MFRWik[SY1bu_�MPYcpjbqYcM0Ml��`1QS�1iUTWiF �AQ]RkTN[S\�TW`cM?�xMlT£pjQjis��QSing�buRWMF_¾TWMFg¥Tk[�Tk`cMn�xMPT�Yc[+���PeuM3QSY1gTk`1M¬RkM<ixT£��QSinVXbq��M<gÔ��bvTW`ÔO1�cRkMyQSbqR£QSY1g·g�buRkM<_¾TkM<g�Tk[-Tk`1M_l[+1[H�� Å^bq��buYcp·¡"MlTx�{MPMPYTk`1MX�xMlT�pZQSiXQ]Y1g�Tk`1M�Q]buR¬�{Qji_l[jYZTkRW[SeueqM<g¸¡Zw�_FQ]eubq¡1RWQ]TkMFg¸VXMPTkMPRWbuYcp�tHQ]eutSM<iP Ö��`��1iF}�Tk`1M_l[+1[H��_P[SY1ikbuiUTkM<g£[S\"Q?��bugcM{tHQSRkbuMlTxw�[S\"Q]buRa��`cMFeqbu�cVÃ_l[jVXO)[S�ikbvTWbq[jY1iNQ]T?Q�euQSRkpjM£RWQSYcpSMÂ[]\ATk[STWQSe¶OcRWMFiWik�cRkM<iP N��`cM¢_P[-1[H�_l[jY1g�bqTkbu[SY1i¢�{MPRWM�OcRWMFg�bd_¾TWMFg�¡ZwªQ�RWMFQ]eq�8TkbuV3M�pjQji¢g�w�Y1QSV3�bd_Pi3_PQ]ed_l�1euQ]Tkbu[SYf}IO"MPRk\|[SRWV3M<g¤[SYªQ�iUO1RkM<QSgcik`cMPMPTF}I��`cMFRkMF¡ZwTk`1MN_l[+1[H�Åg�bdQ]VXMlTWMPR<}jVXMlTWMPRWbqY1pÂtHQ]eutSM�O"[jikbqTkbu[SY1i{Q]Y1g¬ik�cO��Oceuw·OcRkM<ikik�cRWMFi��{MPRWMypjbqtjMPYªQSiÂbqY1Oc��TWiF ·��`cM�ik�cRWRk[j�cY1g�buYcpQ]buR3�{QjiÂQHTXQ]V¢¡cbqMFYZTF}oiUTkbueqe�_l[jY1g�bqTkbu[SY1i3Q]Y)g·Tk`cM0V¬Q]buY�xMlT��QSiÂQSeq��Q�w�inO)MFRU\|M<_¾TWeqw·Ml��O1QSY1g�M<g§ ¥��`cM�_P[-1[H�Ö��QSi�Y)QHTU��cRaQ]eueqw-OcRWMFiWik�cRkM<g��ÎVXQ]TW_a`cM<gQHTniU�c¡)iU[jYcbu_¢iUO"MPM<gciF}§¡c��Tn�{QjiikeqbupS`ZTkeuwn�cY1g�MFRkMP��O1Q]Y1gcMFgÂQ]TIik�cO"MPRaiU[jYcbd_�ikO)MFMFgciIg��cM�Tk[NTk`1M_l[jY�tSMPRWpSbuYcp¢pSMP[jVXMlTkRWw¬[]\�Tk`1M£_l[-)[H��Yc[+���PeuMS 

JET

AIR

SOLENOIDVALVE

PRESSUREREGULATOR

COMPRESSED AIR (130 PSIG)

AIR

HELIUM FROMCYLINDERS500-2500 PSIG

SHUTOFFVALVE

SONICORIFICE

AIR BYPASSLINE

REMOVABLE M.O.C.SUPERSONIC NOZZLEMACH 1.5, 1.75, AND 2.0

SOLENOIDVALVE

REMOVABLE COFLOW NOZZLE.0.7" and 0.85" EXIT DIAMETERS

0.5" DIA.

METERING VALVES

COFLOW

HELIUM

þ�bqpj�cRWM?Õ)ü�z�[]��Q]YcY��cedQ]R�ik�cO)MFRWik[SY1bu_��xMlT�\«Qj_lbueqbqTxwS ��`cM¶�xMlTA[SRA_P[-1[H�ªg�MFY1ikbvTxw���QSi�TkRaQ]Y1ikedQHTkM<gnTk[NTk`1M{MP®"M<_¾TWbqtjMTkMFVXO)MFRWQ]Tk�cRWM�[]\oQ]Y-M<¨j�1bqtHQ]euMPYZTstHQSRkbdQ]¡ceuMl�8TkMFVXO)MFRWQ]Tk�cRWM�Q]buR1[H�¸t�buQ£Tk`cMNRkMFeuQ]Tkbu[SY1i�e = ;�e<D�7���Di;r� = QSY1gBe<@Q;re<D�7�_Di;��_@c �hsiIiUMFMPY£buY3º�MF_¾TWbq[jYÂ~k~¾}FTW`cM�_lRWbvTWbu_FQ]eZO1Q]RaQ]VXMlTWMPRaifbuY^�Qj_a`0��Q�tSMnMFeqbuVXbqY1Q]Tkbu[SY�QSRkM?Tk`1MÂtSMFeq[�_PbvTxw�QSY1g0Tk`1MÂiUO"MPM<g[]\�ik[S�1Y1g 4 7�� ���3en S��Q]bdiUbuYcp ��� ¡Zw0�1ikbqY1p�Qyeubqpj`ZTspjQji`1Qji{Tk`cM£iWQ]VXMnMl®DMF_lTsQSi�RaQ]bdiUbuYcpce£ h�OcbqTk[]TXOcRW[S¡"M�RWMF_P[SRag�MFg¤Tk`cM-_lMFYjTWMPRWeqbuYcM0OcbqTk[]TXOcRWMFiWiU�1RkMj}��`cbd_a`4��QSi�TkRaQ]Y)iUedQHTWMFg¸TW[¸^¥QS_a`4Y��cV¢¡)MFR-¡Zw�Qjikik�cVXbuYcp��7 �+Zî��  ��`1MÔQS��Tk[jVXQ]TkM<gÅ\«QS_PbqeubqTxw��{Qji�buY1iUTkRW�cVXMPYZTkM<g��bqTk`XOcRWMFiWiU�cRWM{TkRaQ]Y1iWg��1_PMPRaiIRkM<_l[jRWg�buYcpNTk`cM�TW[]TaQ]ecOcRWMFiWik�cRkM<ibuY�TW`cM?�xMlTÂQSY1g�_P[-1[H� Q]Y1g�Tk`1M3O1bvTW[]T£OcRWMFiWiU�1RkMj ¢��Q]¡ceuM-�OcRWMFikMPYZTWiyTk`cMÔRaQ]YcpjM[]\d1[H� _P[SY1g�bqTkbu[SY)i�_l[HtjMPRWMFg»buY�Tk`1M^�Qj_a`��j  �s�xMlT�MP��O)MFRkbuVXMPYZTWiF �U�_�`���C���d�������������t�,���������t��jMlT z�[+1[H�^�QS_a`0Y��cV¡"MPR �S î� �]�W�j  �Q �AMPeu[�_PbvTxw�¼|Vf�Hia½ �j�+� �]���+�+�mA®DMF_¾TWbqtjM?TkMPVXO"MPRaQHTW�cRkM¬¼¡�£½ ÿ]ÕQ� �Sÿ+�H�U�r�*�LsbdQ]VXMlTWMPR£¼«V3Vy½ �<�c î� �H��  ÿc}1���j  ×�S¢�£tH ×+�*�c} �+�*�¤ r�Y)g�MPRWMl��O1QSY1g�MFg�\|[SRI5 ] �j 

Page 4: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

º�_a`ceubuMPRWMPYÅOc`c[STk[jpSRaQ]Oc`�w���QSi��1ikMFg¸TW[g�MPTkMF_lT�TW`cMÔ^¥QS_a`��Q�tSMFiF  ��`cM iW_a`ceubqMFRkMFY ikw�ixTWMPV MFVXOceq[HwjMFg�Q �-�]�Y1QSYc[jikMF_P[SY1giUO1QSRk­iU[j�cRW_PMÔ¼�¥sMFYc[SYŦsQSYc[SedQ]VXO)½Â\|[jRX\«QSiUTOc`c[STk[jpSRaQ]Oc`�wS ���`1M�­�Ycbq\|Ml�ÎMFg�pjM�[SRWbuMPYZTWQ]Tkbu[SY¤��QSiÂQSeqbupSY1MFg��bqTk`�Tk`cMnMl��O"MF_lTkMFg0ikeq[jO)M?[]\¶Tk`cM�^�Qj_a`���Q�tSMFi�bqY0[SRag�MFR�TW[QS_F_lMFYjTW�1QHTWM0Tk`cMFV [SYTk`1M0buV¬Q]pjMFiF ¸h�OcbuYc`c[jeqM��QSiXQSeuik[�1ikMFg�[�_P_PQjiUbu[SY)Q]eueqwj ¶��`cMnbuVXQSpSM<i��{MPRWM?RkM<_l[SRag�M<g¬[SY0Q3gcbqpS�bqTWQ]e�z�z{L _PQSVXMPRaQÔ¼¡�o`c[STk[jV3[�MPTkRWbu_Fi£ºZTWQSR£~U½n��bqTk`Å�<�H�8¡1bvT<}�SÿSÕ0¦��Z�H×£QSRkRaQ�wS ¯¨§ °¥¯ % �d©� «ª��A²��� ��¶�Iµ ± ©þ�bqpj�cRWM¥�·iU`c[H��i¬Q¤_P[SVXO1QSRkbdiU[jY�¡"MlTx�{MPMFY¸Tk`cM��1YjTWRkM<QHTWMFg^�Qj_a`Ô�j  �N�xMPT¢¼|Y1[�_l[+1[H�s½�Q]Y1g0TW`cMÂiWQ]VXM��xMlT?iU�cRWRW[S�cY1gcMFg¡�wÅ_l[-)[H� []\Âg�bdQ]VXMlTWMPR-RaQHTkbu[�¬ @ ;r¬ = 7 �*Zî�ªQ]Y1g4��bvTW`5 @ 7 �*Zq��Q]Y1g?e @ ;�e D 7 �*Z �1}A_l[SY)g�bvTWbq[jY1i£Y1MFQ]RÂTk`cM�TkbuOQ]Y)gnbqY1ikbdg�MATk`1M�OcRWMFg�bd_¾TWMFg�^�Qj_a`��Î�{Q�tjM�MFeqbuVXbqY)QHTkbu[SYd�F[SYcM�[]\þ�bqp) ��1¼«QZ½¾ I~ÎT�bui�ikMPMPYyTk`1Q]T�TW`cM�_l[-)[H�TWRkM<QHTWV3MFYZT{RWMPVX[HtSM<iTk`1M{^�Qj_a`£��Q�tSMFiF <pÔM�[j¡�TWQSbqY¢Q�VX[SRWM�g�MPTWQSbqeuMFgÂtZbuMP�¤[S\cTk`1M��Q�tSMFinQSY1gÔTk`cMFbqRÂMPeubqVXbuY1QHTWbq[jYÔ¡�w�MPYc`1QSY1_lbuYcp-TW`cMybuVXQSpSM<iQS_F_l[jRWg�buYcp¢Tk`cM£iW_a`cMFV3M?TW`1QHT�\|[jeqeu[H��iP ��`cM£O1�cRkO"[jikM£[S\�TW`cM¢gcbqpjbvTaQ]e�buV¬Q]pjM�MPYc`1QSY1_lMFVXMPYZTs��QSi�TW[QS_F_lMFYjTW�1QHTWMXTk`cM��{Q�tjMFinbuY¤Tk`cM�Q]V¡1bqMFYjTÂQSbqR<}�RWQ]Tk`cMFR�Tk`1QSYTk`1MXTk�cRW¡c�ceuMPYZT£ixTWRk�)_¾Tk�1RkMX[]\�Tk`cM?�xMPTF}¶QSY1g¥Tk[�g�[-iU[-buYÔQSYQ]�cTk[SV¬Q]TkMFgf}o_l[SY)iUbdixTWMPYZTÂ\«QjiU`1bq[jY¤\|[SR3Q]eue�TW`cM�buV¬Q]pjMFiF ��¶[Ml��OcedQ]buYyTW`cM�OcRW[�_lMFiWikbqYcp3ik_a`1MPVXMS}��{M?gcMPYc[STkM�Q]Y�bqV¬Q]pjMN¡�wTk`1MÂV¬QHTWRkbq�J­¢¼¯®,\±°c½¾}D��`cMFRkM�®ÔQ]Y1g�°0QSRkM£buYZTkMFpSMPRaiF �þ�buRWiUTF}Tk`1M¢buV¬Q]pSM¢�cY1g�MFRkpj[�MFi�VXM<g�buQSY�Ä1eqTkMFRkbuYcp���bvTW`�QyedQ]RWpSMj}"�1�¦��c�¢V¬QHTkRWbq�D ?r�YceubqTnRWMPpjbq[jY1i�[S\ATk`cM3buVXQSpSM3Q]RWM£Ml�c_leu�1g�M<g\|RW[SV�Tk`cbdiNQSY1gQ]eue¶ik�c¡1ikMF¨Z�cMFYZT�Ä1eqTkMPRWbuYcpy[SO"MPRaQHTWbq[jY1iF ���`1MRWMFik�cevTWbqY1p¥buVXQSpSMj}C²­Â¼¯®,\±°c½¾}�buiÂTk`cMFYik�c¡�TWRWQj_¾TkM<g¤\|RW[SV*Tk`1M[SRWbupSbuY1Q]e"Tk[¬pjbqtjMsTW`cMÐb)�1_¾TW�1QHTWbq[jY1Ñ3bqV¬Q]pjM­I³8¼¯®,\±°c½M7´­Â¼�®0\±°c½`>�²­3¼¯®0\�°c½��`cMJ1�)_¾Tk�)QHTkbu[SYÅbuV¬Q]pjM0`)QSiy_P[SVXOceuMlTkMFeqw�cYcbq\|[SRWV ¡1Qj_a­Z�pSRW[S�1Y1g§}{�cYceubq­jMTk`cM¥[SRWbqpjbqY1QSe�bqV¬Q]pjM��`c[jikM¡1Qj_a­ZpjRk[j�cY1gtHQ]RWbuQ]Tkbu[SY1i¢Q]RWMybuYcMPt�bqTWQ]¡1eqM·¼|M<iUO"MF_PbuQSeqeuw¤QHT¢`1bqpj`�_l[jYjTWRWQjixT¾½g��cM?TW[ÂTW`cMs\«Qj_¾T�Tk`)QHT�Tk`cMnikO1Q]RW­Xbui�Yc[ST�Q¢O"MPRk\|MF_¾T�O"[SbuYjT�[jReubqYcM?iU[j�cRa_lMS ���`cMspjeq[j¡1Q]e1VXQ]��bqV¢�cV QSY1gXVXbqY1bqV¢�cVÖ[]\§Tk`1M1�1_lTk�1Q]Tkbu[SY¢bqV¬Q]pjMS}r­ ³æ�Þ¶µ Q]Y1g�­ ³æ�Ýîú }]Q]RWMoTk`1MPY¢_P[SVXOc��TWMFg§ ��`cM<iUMXMl��TWRkMFVXQ-g�MFO)MFY1g�Tk[edQ]RWpSM3Ml��TkMFYZT£[SY�Tk`cMyg�MFY1ikbvTxwtHQ]RWbuQ]Tkbu[SY1i�buY3Tk`cM��xMlT< AÓ�wÂ�1iUbuYcp?Tk`1MPVÃTW[�ik_FQ]euM�TW`cM�buVXQSpSMj}QSiÂiU`c[H��Y·YcMP��TF}IQg�MPpjRkMFM¬[]\�_l[SY)iUbdixTWMPY1_Pw¥bdi£QS_a`1bqMFtSMFgÔbuYbuYjTWMPRWOcRWMlTkbuYcp¬Q]eueDTk`1M�OcRk[�_PMFiWiUM<g�buV¬Q]pSM<iP �¶[ÔQj_P_lMFYZTk�1Q]TkMyTW`cM�\|MFQHTW�cRWMFi3��bvTW`cbuYQÔ_lMPRkTWQSbqY�bqYZTkMFY1ikbvTxwRaQ]YcpjM{[S\¨­ ³ ¼¯®0\�°c½¾}]TW`cM�eubqY1MFQ]RARaQ]VXO¬Q]eupS[jRkbqTk`1VÃ[]\§þ�bqp) S×nbui�1ikMFg-TW[0_P[SY1iUTkRW�1_lTsTk`1MÂ[S�cTkOc��TnbuVXQSpSMj ?��`cM3iUeu[SO"MÂ[S\oTk`1MRaQ]VXOf}§g�MlÄ)YcMFg¥¡Zw-bqTWiniUTWQ]RkTNQSY1g�MFY1g�O"[SbuYZTWiS­ ³V Q]Y1g�­ ³G }g�MPTkMPRWVXbqY1MFi{Tk`1M£iUMFY1iUbqTkbut�bvTxw

· 7��w> ­ ³G >[­ ³ V­ ³æ�Þ¶µ >[­ ³æ�Ýîú ¼«×Z½

¼«QZ½

¼|¡"½

þ�bqpj�cRWMn�cüIrsYcOcRW[�_lMFiWikMFgyiW_a`ceubqMFRkMFY�buVXQSpSM<iPü�¼«QZ½��1YjTWRkM<QHTWMFg�xMlT���¼|¡"½{TkRWMFQ]TkMFgÂ�xMlT���bqTk`�5 @ 7©�+Zu�<�c}_e @ ;�e D 7¢�j  � 

��`cbd_a`�eubqM<i�buY�TW`cM¥RaQ]Y1pSM¸� ^ · AÁ�j}I�PMFRk[�¡)MFbqYcpTk`1MeuMFQSiUT�ikMPY1ikbqTkbutSMS  ��`cM¥eq[�_PQ]Tkbu[SYÅ[S\?TW`cM¥VXbdg�O"[SbuYjT�[]\�Tk`1MRaQ]VXOf}�­ ³æ�Ýîù }"g�MlTWMPRWV3buYcM<i���`cbd_a`RWQSYcpSMn[S\IbuYZTkMFY1iUbqTkbuMFispSMPTQS_F_lMFYjTW�1QHTWMFg§ ���`cM£MFVXOc`1QSikb��PMnTk`1MÂ^�QS_a`-��Q�tSM<iP}W­ ³æ�Ýîù buiikMlTIMF¨Z�1Q]eZTW[sTW`cM�V3M<Q]Y¢t�QSeq�1M�[S\�­ ³ buY¢QSYÂbuVXQSpSM{iU�c¡1RkMFpSbu[SY[S�cTWikbug�M?TW`cM{�xMlTF ���`cM�ÿH�ΡcbvT�[j��TkOc�cT�buV¬Q]pjM?bdi�_l[jY1iUTkRW�1_¾TWMFgQS_F_l[jRWg�buYcp¢Tk[

­I³Û¾ö<à ¼�®0\±°c½:7 �:O �S�S� ­ ³ ¼¯®0\�°c½`>y­ ³V­ ³G >[­ ³ V \ ¼8�S½

tHQ]eubug-buY-Tk`1MÂRaQ]YcpjMl­ ³V ^ ­ ³ ^ ­ ³G  sýN��TaiUbdg�M�TW`1QHTNRWQSYcpSMj}­ ³Û¾ö<à 7��¬\|[jR�­ ³ A¹­ ³V Q]Y1g?­ ³Û¾öFà 7��j�]×�\|[SRc­ ³ ] ­ ³G  ��`cM�buV¬Q]pjMOcRk[�_PMFiWiUbuYcpªQ]eupS[jRkbqTk`1V ��QSi¬��RWbvTkTkMFYÅbqY�þ1[SRk�TkRaQ]Y� IhseqeDTW`cMnbqV¬QSpSMFi�[]\¶Tk`cbdi�ixTW�1g�wy�{MPRWMsO1Rk[�_lM<ikikMFg���bvTW`ikMPY1ikbvTWbqt�bqTxw · 7º�6Z ÿZ�_��Yc[ªQjgcg�bqTkbu[SY1QSe��1ikMPR¬buYcOc��Taiy�{MPRWMV¬QSg�Mn[jR�RkM<¨j�1bqRWMFg§ 

Õ

Page 5: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

A’A’min A’ A’maxA’A’1 2mid

A’out

256

128

1

þ�bqpj�cRWM£×cü�~�V¬QSpSMnOcRW[�_PMFiWiUbuYcpyiW_a`cMPVXMn\|[jR�Tk`cM�1�1_¾TW�1QHTWbq[jYbuVXQSpSMd­ ³ ¼¯®,\±°c½¾ § °¼» ��I´o#a���þ�bqpj�cRWM��¢ik`c[H��i�Tk`cMnMPY1`1Q]Y1_PMFg�tSMFRWikbq[jY1i�[]\�TW`cMnbqV¬QSpSMFi{buYþ�bqp) ��� X��`cMX¡�wZO)QSiWiNRaQHTWbq[J½7Y¾{o@�;c¾{ = Q]Y)g�TW`cM3Tk`cRW�1iUTRaQHTkbu[�e���¼ÀTW`cRW�1ixT{[]\§TWRkM<QHTkM<g?�xMlT{[HtSMPRoTk`cRW�1iUT�[]\f�1YjTWRkM<QHTWMFg�xMlT¾½�QSRkMÔiU`1[H��YÅbqY4Tk`cM·_PQSO�Tkbu[SY1iF  m�Yc`1QSY1_lMFV3MFYZT�`cMFeqO1i¡cRWbqY1pÅ[S��T�buYZTkMPRWMFiUTkbuYcp»\|MFQ]Tk�cRWMFi�[]\3Tk`cMªO1RkM<ikik�cRWMÔ��Q�tSM<iTk`)QHTÂVXbupS`ZTÂ[]TW`cMPRW��bdiUM¬pj[�cYcYc[STkbd_lMFgf 0~�Y·TW`cM��1YjTWRkM<QHTWMFg_PQjiUMj}<Tk`cM��{Q�tjMFi§MFV3bqTUTWMFg¡ZwNTk`1M{ik�cO"MPRaiU[jYcbu_oMFgcg�buMFi�¼«^�Qj_a`��Q�tSMFia½�QSRkMXikMPMPY·QSiniUTkRaQ]bupS`ZT¬¼«_l[jYcbd_PQ]eÀ½s��bvTW`ÔY1MFQ]RWeuw�_l[SYc�iUTWQ]YZT�iUeu[SO"MS}�buY1g�bd_PQ]TkbuYcp£TW`1QHT{Tk`cMFRkM?bdi{Q¢iUbuYcpjeqMn_l[jYZtjMF_lTkbutSMtSMFeq[�_PbvTxw�\|[SRyTW`cM¥edQ]RWpSMM<gcg�buMFi�bqYÅTW`cM¥ik`cMFQSR�edQ�wSMFRWi¬ik�cRk�RW[S�cY1gcbqYcp�Tk`1MO)[STkMPYZTWbuQSe�_l[SRWMS ¸��`cMikeq[jO)M-[]\NTk`cM�^¥QS_a`��Q�tSMFi{bdi�Q]O1OcRk[���buV¬QHTkMFeqwy�*�Xg�MPpjRkMFMFiF}�\|Rk[jV ��`cbd_a`0�{M�[S¡c�TWQSbqY?5CF�G�Hc¿Ö�_Z �Ô¼|&)(d¿Ö×j�+�}{o;+À�½¾ X��`cMXV3[�g�MFeo[]\�m�¨�iF �H�x�ÂOcRkM<g�bd_¾TWi35CF�G±H�7 �*Z �1 z{Q]RWMl\|�1e�MP�cQ]VXbqY)QHTkbu[SY�[]\oTW`cMXbqV¬QSpSMXQ]ediU[�RkMFtSMFQSeuisTk`1M3MP���bdixTWMPY1_PM3[S\��{MFQ]­jMPR?�{Q�tjMFiN[]\�Q�g�bq®DMPRWMPYZTn\«Q]VXbqeuwDü?_PbqRa_l�1euQSR¼«ikOc`cMFRkbd_PQSed½l}�[jRkbupSbuY1QHTWbqY1p�\|RW[SVÒYcMFQSRnTk`cMyeubuOÔ[S\�TW`cM¬buYcYcMPRYc[*�s�PeuMS Ö��`cMFw¸QSRkM�MPVXbqTUTkM<gÅ\|RW[SV Q�iUTWQ]Tkbu[SY1QSRkw¸ik[S�cRa_lMj}OcRW[S¡1QS¡ceuwtS[SRkTkMP��iU`cM<gcg�buYcp�\|Rk[jV Tk`cMXYc[*�s�FeqM3eqbuOf}fbuY·_l[SYc�TkRaQSiUT�Tk[XTW`cM£^�QS_a`0��Q�tSMFi{��`cbd_a`-Q]RWM?MFVXbvTkTkMFg-¡�wyVX[Ht�buYcpik[S�cRa_lM<iP}<Tk`1M{M<gcg�buMFiF ���`1M{_P[ZMP��buiUTkMFY1_lM{[]\)Tk`cM�Tx��[N\«Q]VXbueqbuMFi[]\I��Q�tSMFi�bdi�bqeueu�1ixTWRWQ]TkM<g�¡�w¬TW`cM£iU­jMlTa_a`�[S\Iþ�bup1 cÿc h�O1Oceqbd_PQ]Tkbu[SYª[S\�Tk`cM0_l[-)[H� buYªþ�bqp) A�-RkMFV3[HtjMFi�¡"[]TW`¤Tk`1M^�Qj_a`���Q�tSMFi�Q]Y1gÃTk`cM»�{MFQ]­jMPRÔiUOc`1MPRWbu_FQ]e3�{Q�tjMFiF Á��`1MQ]buR£ik�cRkRW[S�1Y1g�buYcp�TW`cMn�xMlT3Q]OcO"MFQSRWi?VÂ�)_a`'ÐW¨Z�cbqMPTkMFRWÑ1}fMPtjMPYTk`1[S�cpj`ÅTk`cMÔ_l[jV¡cbuY1QHTWbq[jYÅ[]\��xMPT0Oceu�1i0_l[+1[H� pSMPY1MPRaQHTkM<i�+��� VX[jRkMXTW`cRk�)ixT¢Tk`1QSY¤Tk`cM0iU[jeqMn�xMPTF ÔmoeubqVXbuY1QHTWbq[jY¤`cMPRWMbdiÂQS_a`1bqMFtSMFg���bqTk`�QSY'Ðk�cYc`cM<QHTWMFgcÑ_P[-1[H�ÖQHTÂikeqbupS`ZTWeqwÔiU���O"MPRaiU[jYcbu_¢_l[jY1g�bqTkbu[SY1iF Nþ�bup1 §�yik`c[H��iNQSYc[]TW`cMPRNMPeubqVXbuY1QHTWbq[jYikMF¨Z�cMPY)_lMSü�buV3O1Rk[jO)MFRkeuw3TkRWMFQHTWMFgyQSY1g¬OcRW[SO"MPRWeuwÂTWRkM<QHTkM<g§ I~�Yþ�bqp) f�1¼:Qj½¾}DTW`cM¬_l[+1[H�Ãbdi?ik�cOcO1eqbuMFgÔQHT£_l[jY1g�bqTkbu[SY1i?[j��TWikbugcMTk`1M�OcRWMFg�bd_¾TWMFg©MFeqbuVXbqY1Q]Tkbu[SY RWMPpjbq[jY<�yQS_F_l[SRag�buYcpSeuwS}£^¥QS_a`��Q�tSMFi0g�MFtSMPeu[SO�  ��`1M·_P[-1[H��[]\¬þ¶bup1 ��)¼|¡)½�bdi�ik�cOcOceubqM<gQHT£_P[SY1g�bqTkbu[SY)inbqY1ikbdg�MXTk`cM¬MFeqbuVXbqY)QHTkbu[SY·RWMPpjbq[jYf}§��bvTW`·`cbqpj`Ml®DMF_lTkbutSM�TkMPVXO"MPRaQHTW�cRkM-Q]Y)gªiU�1¡1iU[jYcbd_�^�Qj_a`ªY��cV¡"MPR���Qji

¼«QZ½

¼|¡"½

þ�bqpj�cRWM?��ü`�oRW[�_lMFiWikMFgXik_a`1eqbuMPRWMPYXbuVXQSpSM<io[]\��xMPT���bqTk`�_P[-1[H�g�bdQ]VXMlTWMPRÁ¬ @ ;-¬ = 7©�*Zî��ü�¼«QZ½f�1YjTWRkM<QHTWMFg��1¼«¡)½�TkRWMFQ]TkMFgÂ��bvTW`5 @ 7 �+Zu�<�c}_e @ ;�e D 7¢�S  �c}_½Â7��_Z �c}_e��'7©�*Z �1 OcRWMFg�bd_¾TWMFg§}I^�Qj_a`Ô��Q�tSM<ing�buiWQ]O1O)M<Q]R< �þ�bqpj�cRWM���-iU`1[H��i�QikbqVXbueuQSR�TkRWMFQ]TkVXMPYZTNikMF¨Z�cMFY1_lM£\|[SRsQn�xMPTN��bqTk`¥ikV¬Q]eue¶_P[-1[H�g�bdQ]VXMlTWMPRi¬ @ ;r¬ = 7 �*Z Õ) Å��`cbuiytSMPRWw·Tk`1bqYÅ_l[+1[H� edQ�wSMFROcRW[Ht�bug�M<iyik�c¡1iUTWQ]YZTWbuQSesMPeubuV3buY1Q]Tkbu[SYf}�¡1��T�Txw�Ocbd_PQSeqeuw»Q¤\|MP��{MFQ]­��Q�tSM<i�RWMPV¬Q]buYf ¢~ÎTnbdinMl��O"MF_lTkMFg¥Tk`)QHT£QyTW`cbuYÔ_P[-1[H�pSMPTWi?_l[jY1iU�1V3M<g0\«QSiUTkMFRN¡�w0MPYZTkRaQ]buYcVXMPYZTF}D`cMFY1_lM¢buiNeuMFiWi�Ml\À�\|MF_lTkbutSM0bqY»MPeubuV3buY1Q]TkbuYcpÔ^¥QS_a`���Q�tSM<iP}�TW`1Q]Y»Q�TW`cbu_a­¤[jYcMS ºZTWbqeue:}�TW`cM?Ml��TkMFYjT{Tk[3��`cbd_a`yTW`cbui�TW`cbqY-_l[+1[H�Å�{[SRW­�i�bui�ik�cRk�OcRWbuikbuYcp¬Q]Y1g�buisg�buiW_l�)ikikMFg�VX[SRWM?buY�º�MF_lTkbu[SYj�s~¾ ��`cMnik_a`1eqbuMPRWMPY�RkM<iU�ceqTWi�QSRkM?iU�cVXV¬Q]RWb��PMFg�[SY�Tk`cMSe<@ÂtjMPRaiU�)i5C@¢g�bdQ]pSRaQ]V¬i�[]\�þ�bqp) D�S�S I��`cM?bqV¬QSpSMFi���MFRkM?g�butZbdg�M<g¬bqYZTW[Tk`1RkMFMX_PeuQjikikMFiFü�ÐWmoeubqVXbuY1QHTWbq[jY1Ñ0g�MPYc[STkM<i�Q�tjMPRWw_leuMFQ]YÔbuV¢�Q]pjM�g�MFtS[Sbdg·[]\sQSY�wÔg�bdiW_lMPRWYcbu¡ceuM�^�QS_a`¤��Q�tSM<iP}IQSiÂbqYþ¶bup1 ��¼«¡)½L�sбpÔMFQ]­��Q�tSM<ikѬikbupSYcbqÄ1MFiNTW`cM¢O1RkM<iUMFY1_lM3[]\�tjMPRWw0\|MP��{MFQ]­»^�QS_a`Å��Q�tSM<iP}�QSiybqY4þ�bup1 N���1¼«¡)½L�0ÐWº�TkRW[SYcp·��Q�tSMFiWÑVXMFQSY1iN^¥QS_a`���Q�tSM<i�[]\�iUTkRWMPY1p]Tk`�ikbuV3buedQ]RNTk[yTW`c[jikMÂ[S\oTk`1M�cYZTkRWMFQ]TkM<gÂ�xMPTF}DQSisbqY�þ�bupjiF )�1¼:Qj½{[jR£���)¼«Qj½l {þ1[SRsTk`cM£edQ]RWpSMFR_l[+1[H� g�buQSVXMlTkMFRF}�þ�bup1 s�j�j¼:Qj½¾}{TW`cM¥MP��O"MPRWbuV3MFYZTWQ]esRkMFpSbu[SY[]\?MPeubqVXbuY1QHTWbq[jY�bdiXbqY»pS[�[�g�QSpSRWMPMFV3MFYZT¢��bqTk`»Tk`cM-OcRWMFgcbu_l�

Page 6: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

þ�bqpj�cRWM?ÿ1ü,p·Q�tSM<io\|RW[SV%VX[Ht�bqY1p¢ik[S�cRa_lM<iN¼:iU[jeqbdgyeubqY1MFia½�QSY1g\|RW[SV iUTWQ]Tkbu[SY1QSRkwyik[S�cRa_lM<in¼«gcQjiU`cM<g0eqbuYcMFia½¾ 

Tkbu[SY�[]\Nþ�bup1 I�1¼«QZ½¾ Ô~�Y1ikbug�M�Tk`cbdi3RkMFpSbu[SYf}�TW`cM�tHQSiUTXVXQ]�x[SRk�bqTxw�[S\�buV¬Q]pSM<i�buY1g�bd_PQHTWMÂ_l[jVXOceqMPTkM£MFeqbuVXbqY)QHTkbu[SY�}1��bqTk`tSMFRkw\|MP�ÖbuV¬Q]pSM<i�iU`1[H��bqYcp�{MFQS­���Q�tSMFiF 0ýN��TaiUbdg�MXTW`cMyRWMPpSbu[SY�}iUTkRW[SYcp���Q�tSM<iI��MFRkMsQSOcO1QSRkMFYjT< ¶þ1[SRoTW`cMNikV¬Q]eueqMFR�_l[+1[H�¸g�bv�Q]VXMPTkMPR<}�þ�bup1 I�S�Z¼|¡)½l}�ixTWRk[jYcp0�{Q�tjMFi?iU[jV3MPTkbuVXMFi�Q]O1O)M<Q]RWMFgbuY1iUbdg�MÂTk`cM3OcRWMFg�bd_¾TWMFgMFeqbuVXbqY1Q]Tkbu[SY��F[SYcM3QHTniU�c¡)iU[jYcbu_�5 @  ��`cbdiobuYZTkMPRWVXbvTkTkMFYjT{��M<Q]­jMPYcbuYcp�[]\DTW`cMN_P[-1[H�»Ml®DMF_lTkbutSMPY1MFiWibdi�OcRW[S¡)Q]¡ceuw¬g��cM�TW[¢bqTWi�MPYZTWRWQSbqYcVXMFYjT{¡�wXTk`cMNVXQSbqYÂ�xMlT< oh{T`cbupS`cMFR�5 @ }�QSpSRWMPMPVXMFYjT���bqTk`ÔTk`1MyOcRWMFg�bd_¾TWMFgÔMPeubqVXbuY1QHTWbq[jY�P[jYcMÂbdiN\«Q]buRF}"TW`c[S�cpj`¥Q�\|MP�Ã�{MFQS­���Q�tSM<i�RWMPV¬Q]buYcMFg�[�_F_PQ]�ikbq[jY1Q]eueqwj ~�Y�Qjgcg�bqTkbu[SYªTW[�ik�cOcOcRWMFiWikbqYcp�^�QS_a`ª��Q�tSM¬RaQSgcbuQ]Tkbu[SYf}I[SY1Mbdi�QSeuik[£buYjTWMPRWMFiUTkM<g¬bqY�RWQSOcbdg�eqw¬g�M<_lMFeqMFRWQ]TkbuYcp�Tk`1M��xMPTU1[H�»TW[ik�c¡1ik[SYcbd_�ikO)MFMFgci�ik[�Tk`)QHTI[]Tk`1MPRIiU[j�cRW_PMFi�[]\�Tk�1Rk¡c�1eqMFYjTIYc[jbuikMQ]RWM�VXbqY1bqVXb��PMFgf �~ÎT�buioTk`��1iobuVXO)[jRUTaQ]YZTATk[ÂMFiUTWQ]¡1eqbdiU`3TW`cM�Ml\À�\|MF_lT�[]\�TW`cM�_l[+1[H�Ô[SY?Tk`cM�_lMPYZTWMPRWeqbuYcMAtSMFeq[�_PbvTxw?g�MF_FQ�wS �h���bdQ]eOcbqTk[ST�ik�cRWtSMFw�i?�{MPRWM¢O"MPRk\|[SRWVXMFg§}f\|Rk[jV ��`cbd_a`¥TW`cM¬_lMPYZTWMPRk�eubqYcM¬^�Qj_a`¥Y��cV¢¡)MFR£g�buiUTkRWbu¡c��Tkbu[SYÔ��QSin_PQSeu_P�ceuQ]TkM<g¥QSiWik�cV3�buYcp0QSY¥Q�tSMFRWQSpSMd�?7%�*Z �c n��`1M3_PMPYZTkMFRkeubqY1M3^�Qj_a`�Y��cV¡"MPRg�bdixTWRkbu¡c��TWbq[jY1i�\|[SR0�cYZTkRWMFQ]TkMFg�Q]Y1g�TkRWMFQ]TkMFg4_PQSikMFi�bdi�O1RkMP�ikMPYZTkM<gbqY�þ¶bup1 ��<�c})��`1MPRWM£bvTnbdisikMPMPY�Tk`1Q]TsTk`1M3_P[-1[H�©`1QjiVXbqY1bqV¬Q]esbqVXO1Qj_¾Ty[SY»TW`cM�^�QS_a`»Y��cV¡"MPR�g�MF_FQ�wS �º�_PQSeqM<g\|[SRsTk`cM¢`cbupS`cMFR�Tk`1Rk�1iUTN[S\ATk`cM3_l[jV¡cbuYcMFgj)[H��}1TW`cM¢_P[-1[H�TkRWMFQ]TkVXMPYZTXMl®DMF_lTkbutSMPeuw¤MPY1`1Q]Y1_PMFi¢Tk`1M0_PMPYZTkMFRkeubqY1M�tjMPeu[�_PbvTxwg�M<_PQ�wS �þ1[SR¢Ml�cQ]VXOceuMS}ITW`cMyTWRkM<QHTkM<g0�xMlT3��bvTW`25C@[7 �*Zq���OcRW[�g��1_lM<i��S  ��TWbqVXM<iITk`cM�TW`cRW�1ixTA[S\)TW`cM��cYZTkRWMFQHTWMFgs�xMlT< I��`1MiWQ]VXMATk`1Rk�1iUT¶�{[S�1eugn`1Q�tjMA¡)MFMPY¢OcRk[�g��)_lMFgn¡�wnQ]Y£�1YjTWRkM<QHTWMFg�xMlT¢��bvTW`¤Ml®DMF_¾TWbqtjMyg�bdQ]VXMlTWMPR�¬ Ãb@�@¸7 � �+Z �-¬ = 7 �*Z ���r¬ =  ��`cM^�Qj_a`�Y��cV¡"MPR¬g�M<_PQ�w¤[S\sTk`1bui3edQ]RWpSMPR3�cYZTkRWMFQ]TkM<g�xMlTbdiN[j¡�TWQSbqY1MFg�¡�w�iUTkRWMlTa_a`cbqY1p�QH��bdQ]eueqw0TW`cMX�cYZTkRWMFQHTWMFg¥eqbuYcM3[]\þ�bqp) ��<��¡�w�QÔ\«QS_lTk[jR3[S\3�j  ���� »~�Y»Tk`1Q]T�_PQSikMS}�Tk`cM�gcMF_PQ�w

¼«QZ½

¼|¡"½

þ�bqpj�cRWM��cü hNgcg�bqTkbu[SY1QSeyOcRW[�_PMFiWiUM<g�buV¬Q]pjMFi·[]\X�xMPT���bvTW`_l[+1[H� gcbuQSV3MPTkMFR�¬B@�;r¬ = 7 �+Zß��ü¬¼«QZ½�buVXOcRk[jO)MFRkeuwTWRkM<QHTkM<g��bqTk`Ä5 @ 7 �WZß�]�c}¼e @ ;re D 7¢�S  �c}¼½ 7 �+Zî��}¸e�� 7�+Z Õ6�d¼|¡"½�OcRW[SO"MPRWeqw�TkRWMFQHTWMFg���bvTW`C5 @ 7¹�WZß�]�c}�e @ ;re D 7n�c  �1}½Â7Ã�+Zu�S}_e��Å7©�+Zî�� 

RaQHTkM�[S\ITk`cMÂTkRWMFQHTWMFg3�xMPTsbui�\«QSiUTkMFR�Tk`)Q]Y-Tk`)QHTs[S\�TW`cMÂedQ]RWpSMFR�cYZTkRWMFQ]TkM<g£�xMPTF ��`cM¤OcbqTk[ST¥VXMFQjiU�cRWMPVXMFYjTai-bqY)g�bu_FQHTWM·Tk`1Q]T�Tk`cM¤O"[]TWMPYZTkbdQ]e_l[jRkM�MPY1g1i���bqTk`cbuYc®¨;r¬ = 7Å×1}H��`cbu_a`¢bui�Q]O1OcRk[���buV¬QHTkMFeqw?[SY1M`1QSev\{[]\oTW`cM3Ä1MPedg¥[S\�t�buMP�Ã[S\�Tk`1M3buV¬Q]pjMFiNOcRkM<iUMFYZTkMFg¥`cMPRWMS ��`cM¬_l[jRkRWMFikO"[SY1g�buYcp0iU`cM<Q]RnedQ�wSMPR?pSRW[H��Tk`�RWQ]TkM3buid�1 q�H��}§[jR���?g�MFpSRWMPMFiF}]buY3QSpSRWMPMFV3MFYZT���bqTk`Xt�buik�1QSe�[S¡1ikMPRWtHQHTkbu[SY)iP ���`1Mik[S�cRa_lMn[]\o^�Qj_a`���Q�tSMnRaQSg�bdQHTWbq[jY0ikO1QSY1i�\|RW[SV TW`cM��xMPTsMl��bqTTk[3�l�x�Ng�buQSVXMlTkMFRWi�O1QjixT�Tk`1M{MFY1g¢[]\cTW`cM�O)[STkMPYZTWbuQSe�_P[SRWMS ¶~�V¢�Q]pjMFiA[S\f�cYZTkRWMFQ]TkMFgN�xMPTWi�bqY¬TW`cMsRWMPpjbq[jY-�S�lA�®�;r¬ = AÅ�S�ngc[Yc[ST�iU`c[H�·^�Qj_a`?��Q�tSMIRaQSg�bdQHTWbq[jY<��_l[SY)iUM<¨j�1MPYZTkeuwS}<Yc[�ikbqpjYcbvÄ1�_PQSYZT¶g�bq®"MFRkMFY1_lM<i�Q]RWM�ikMPMFY�¡)MPTx��MFMPYnTkRWMFQ]TkM<g�Q]Y1g��1YjTWRkM<QHTWMFg�xMlTai�buY�TW`1QHT�RWMPpjbq[jYf 

×

Page 7: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

¼«QZ½

¼|¡"½

þ�bqpj�cRWM����1ü'�oRW[�_lM<ikikMFg'buVXQSpSM<i[]\n�xMlTÔ��bvTW`�_l[+1[H� g�bv�Q]VXMPTkMPR¸¬B@�;r¬ = 7��+Z Õ1ü�¼:Qj½-buVXOcRk[jO)MFRkeuw�TWRkM<QHTWMFg'��bvTW`5C@c7w�6Zî�S��}We<@Q;�e<Di7¢�S  �c}W½Æ7w�6Z ÿ1}We��$7 �+Z �W��¼«¡)½�OcRW[SOc�MPRWeuw·TkRWMFQHTWMFg��bvTW`:5�@�7��*Zq����})e<@�;�e<D 7¢�S u�S}�½Ç7��*Zq�j}e��Å7 �*Z ×1 §Ò¯o° ª��A���{�Aµl����#»�Äxoxo#PµP���?�IµP² ± ���`cM·TW�cRk¡"[]\«QSY MPYcpjbqY1M¤bui�Q¸Y)QHTk�1RWQSe£_PQ]Y)g�bug1QHTkM·\|[jR�buV¢�OceuMPVXMPYZTaQHTkbu[SY�[]\ATW`cM3^¥QS_a`���Q�tSM£MFeqbuVXbqY)QHTkbu[SY¥TkM<_a`cYcbd¨j�1MikbqY1_PM3Q0_l[+1[H��buYcp�iUTkRWMFQ]V QSeqRWMFQjg�w0Ml��buiUTWi?buY¥TW`cM¢\|[SRWV []\Tk`1Mn\«Q]YixTWRkM<Q]V- A��`cMPRWMnbuisQ¢V¢�ceqTkbqTk�1g�M�[S\���Q�w�i{bqY-��`cbd_a`Tk`1M£TkM<_a`cYcbd¨Z�cM3_P[S�cedg-¡"M3QSOcOceubqM<g§}"\|Rk[jV�VX[�g�bq\|wZbuYcp0Ml��bdixTk�buYcp0gcMFikbqpjY1i�TW[0_PRkM<QHTWbqYcp�YcMP��[SYcM<iP d�oRWMF_PbuikM¢_P[jiUTU�Ρ)MFYcMlÄ1TÄ1pj�cRkM<iIg�MPO"MPY1g3iUTkRW[SYcpjeqw�[jYÂTW`cM�g�MlTaQ]bueuiI[S\cTk`cM�buVXOceuMPVXMPY��TWQ]Tkbu[SY-Q]Y1g0Q]RWM?¡)MFwS[SY)gyTW`cMnV3M<Q]Y1i�[S\�TW`cbui�bqY�tSM<ixTWbqpZQHTWbq[jYf ��`cM?Oc�cRWO)[ZiUMN[]\§TW`cbui�ikMF_lTkbu[SYybdioTW[¢O1Rk[Ht�bdg�MsQ£O1RkMFeqbuVXbqY1QSRkwO"MPRk\|[SRWVXQSY1_lM�MFiUTkbuV¬QHTkM�Tk`)QHTIVXQ�w?\|[jRkV©TW`cM�¡)QSikbui�\|[SRIVX[SRWMRWMlÄ1YcM<g�_FQ]ed_l�cedQHTWbq[jY1i�buY�TW`cMn\|��TW�cRkMj ��`cMXRWMFik�cevTain[]\�Tk`cM¬OcRWMFikMPYZT�ixTW�1g�w»¼«þ�bqpj�cRWM��j�<½¾}¶_l[S�1OceqM<g��bqTk`3OcRWQj_¾TWbu_FQ]e�g�MFikbqpjY¢_P[SY1ikbugcMPRaQHTkbu[SY)iP}Hik�cpSpjMFiUT¶Tx�{[NeubuV3bqTWibuY£Tk`1M�RaQ]YcpjMo[]\1[jO�Tkbu[SY)i�Q�tHQ]bueuQS¡ceuM�buYÂQ]OcO1eqw�buYcpN^�Qj_a`£��Q�tSM

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.5

1.0

1.5

2.0

2.5

3.0

(a) Df/D

j=1.7

Elimination Weak waves Strong waves Model

Tf /

T oo

Mf

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.5

1.0

1.5

2.0

2.5

3.0

(b) D

f/D

j=1.4

Elimination Weak waves Strong waves Model

Tf /

T oo

Mfþ�bqpj�cRWM0�j�Sü�mA��O"MPRWbqVXMFYjTaQ]eogcQ]TWQ0[]\�^�Qj_a`���Q�tSM3MPeubqVXbuY1QH�

Tkbu[SY�ü�¼«Qj½È¬ @Q;r¬ = 7 �+Zß���f¼«¡)½�¬ @*;-¬ = 7©�*Z Õ) MPeubuV3buY1Q]Tkbu[SY'Tk[�QTk�1Rk¡"[]\«QSY'MFYcpSbuYcMjüüx�<½0�1Yc`cMFQ]TkM<g4\«QSYiUTkRWMFQ]V ��bqTk`3eu[H���Îik�cO"MPRaiU[jYcbu_�Ml��`1QS�1ixTA^�Qj_a`¢Y��cV¢¡)MFR�O1Rk[S�g��1_PMFg¬¡�wXQn\«Q]Y�OcRkM<ikik�cRWM�RWQ]Tkbu[£[]\fQSRk[j�cY1gy�� î�_�"¼:�j½A`cM<QHTkM<g\«Q]Y�iUTkRWMFQSV���bqTk`¥V3[�g�MFRWQ]TkM�MP��`)Q]�1iUTN^�Qj_a`YZ�1V¡"MPRsO1Rk[S�g��1_PMFgÔ¡�w�Q�\«Q]Y·OcRWMFiWiU�cRWM3RWQ]Tkbu[�[S\{Q]RW[S�1Y1gª�j  �c 3¹sMFQHTWbqY1pV¬Q�w¬¡)M�Qj_a`cbqMFtSM<g¬¡�w¬¡c�cRWYcbqY1p¢\|�1MPe§buYyTk`1Mngc�1_¾T<}cbqY]�xMF_¾TWbqY1p`c[STsQ]buR�\|RW[SV Tk`cM£_P[SRWM?_P[SV¢¡c�1iUTk[SR<}�[SR�[STk`cMFR�V3M<Q]Y1iF ýNO"MPRaQHTkbu[SY¸YcM<Q]R0ÆfbuV3bqT���Q]OcO"MFQSRWi¬VX[jRkM¥Q]TUTWRWQj_¾TkbutSM¡"Ml�_PQS�1ikM�bvT�g�[�MFioY1[]T�RWMF¨Z�cbuRkM�`1MFQHTWbqY1p£[]\DTk`1M�\«Q]YyiUTkRWMFQSV QSY1gV¬Q�wMPtjMPYÔRkM<iU�ceqT�bqYÔ¡"MlTUTWMPR�OcRW[SO1�ceuikbutSM3Many_lbuMPY1_Pw¥g��cM3TW[Tk`1MÔ`1bqpj`cMPR0¡�wZO)QSiWi�RaQHTWbq[) �ýNO)MFRWQ]Tkbu[SY�Y1MFQ]R-ÆfbuVXbvT¥��buieuMFiWiÂMsn¬_PbqMFYZT3¡c�cT3V¬Q�wÔ¡)M0VX[SRWM�MFQjiUbueuwÔbuVXOceuMPVXMPYZTkM<gªbuYQ]Y»Ml��bdixTWbqY1p·gcMFikbqpjY¡"MF_FQ]�1ikMbvTyRWMF¨Z�cbuRWMFiXQ�\«Q]Y»OcRkM<ikik�cRWMRaQHTkbu[·QS_a`cbuMPtHQ]¡1eqM-��bqTk`ÅQ·iUbuYcpSeuM-ixTaQ]pSM-[S\N¡cedQSg�M<iP �º�buY1_PMbqTnbuY�tS[jeqtjMFi�[jYceuw`cMFQ]TnQjgcg�bqTkbu[SY�TW[�TW`cM3\«Q]YÔixTWRkM<Q]V-}§��bvTW`Q]eue�[]TW`cMPR3O1Q]RaQ]VXMPTkMPRaiÂiUTWQ�w�buYcp�Ä1��M<g§}�bqTXbuiXQ]edik[�Tk`1M�M<QSiU�buMFiUTX_P[SY1g�bqTkbu[SY�Tk[ÔQ]Y1QSeqw��PMj ¤¦�Ml��TX��M�[S¡�TaQ]buYQ¥OcRkMFeqbuVXbv�

Page 8: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

0 2 4 6 8 10 12 14 16 180.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Potential core

Field of view

Df/D

j=1.7

Untreated Treated, M

f=0.75

Treated, Mf=1.15

M(x

,0)

x/Dj

þ�bupS�cRWMX�<��ü�z�MFYjTWMPRWeqbuYcM�^�Qj_a`�Y��cV¢¡)MFR�g�buiUTkRWbu¡c��Tkbu[SY)iP 

Y1QSRkwNO)MFRU\|[jRkV¬Q]Y)_lMAMFiUTkbuV¬QHTWMA\|[SR¶MPY�MPY1pSbuYcMo[SO"MPRaQHTWbqY1p�YcMFQSRÆfbuVXbvTN�c pÔM0_P[SY1ikbugcMPR¢QSY�MFYcpSbuYcM���bqTk`�\«Q]YOcRkM<ikik�cRWMyRaQHTkbu[�[S\Â�S î�¼|�1Yc`cMFQ]TkM<g[5�@Q7S�1  ÿ1}t&)@�7%�Sÿ+�j{�;-À�½nQ]Y1g�¡�w�O1QSiWinRWQ]Tkbu[[]\��S  �XMl��`1QS�1iUTkbuYcp¬bqYZTW[yQ]V¡1bqMFYjT<}1iUTkbueqe�_l[SY)g�bvTWbq[jY1i£¼�e�DÉ7e¨ÊLËl7?�*�+�+�£½l ¾~�YnTk`cM�Yc[]TaQHTWbq[jYNTW`1QHT�\|[Seueq[H��iF}FO1RkbuVXMFg�¨Z�1Q]Yc�TkbqTkbuMFi3Q]RWMXTk`1[jikM�QH®DMF_lTkMFg·¡�w�TkRWMFQ]TkVXMPYZTF ¦s[]TkMyTk`1Q]TÂTk`1MOcRWMFik�cVXMFg�buikMPYZTkRW[SO1bu_s\«Q]Y-_l[jV3O1RkM<ikikbq[jYyRaQ]bdiUM<i�TW`cM?\«Q]Y�Tk[S�TWQSe"TkMPVXO"MPRaQHTW�cRkMsTk[ce¨Êk@�7Ã�+Zu�<�le¨ÊaË¢ A��`cM?_P[SRWMo�xMPT{OcRW[SOc�MPRkTkbuMFisQ]RWM�pSbutSMPY-buY¥��Q]¡ceuM¬�S �r�YZTWRkM<QHTkM<g§}cTW`cMd1[H�©�{[S�cedgRWMFikMPV¡1eqM¢Tk`1Q]T?[S\{þ�bup1 §�1¼:Qj½¾}§b8  MS u}�^�QS_a`��{Q�tjMFiN��[j�cedg�¡"MMPVXbqTUTWMFg§ �º��cO1O)[ZiUbuYcpTW`1QHTÂ�{M�QSOcOceuw�TkRWMFQ]TkVXMPYZT¢Qj_P_P[SRag��buYcp3Tk[¬þ�bqp) c�1¼|¡"½¾}�Tk`1M£_l[-)[H�ÅTkMFV3O"MPRaQHTW�cRWMNRaQHTWbq[3VÂ�1iUT�¡"Me ³@ ;re<DB7n�c  �1 ���`cM£_P[SRWRkM<iUO"[SY1gcbqYcp3TW[]TWQSev�8TkMFVXO)MFRWQ]Tk�cRWMnRWQ]�Tkbu[3buime ³ÊLK ;�e¨ÊLK+7 ��  �c �~Î\¶`cM<QHT�bui�Qjgcg�MFgyTk[¢Tk`cM?\«QSY�iUTkRWMFQ]VQHT�eq[H� ^�Qj_a`»Y��cV¢¡)MFRF}�TW[]TaQ]eq�8OcRWMFiWik�cRkM-eu[jiWiUM<iyQ]RWM�ikVXQSeqeQ]Y)g·bvTX_PQ]Y�¡)M0iU`c[H��Y·TW`1QHTÂTk`cM�\«Q]Y�ikeqbuO1iUTkRWMFQSVÒtjMPeu[�_PbvTxwbuY1_lRWMFQjiUM<i�QS_P_P[SRag�buYcpXTk[j& ³@ ;*&)@*7cÌ e ³Êk@ ;�e¨ÊL@�})b:  MS u}1\«QSYtSMP�eu[�_PbvTxw�bqY1_PRkM<QSikMFi{`cMFRkMn¡�w�Õ)���� ��`cMXQSgcgcbvTWbq[jY1Q]eIV¬QSiWiN)[H� RWQ]TkM3[]\�­jMPRW[jikMPYcM£RWMF¨Z�cbuRWMFg-TW[RaQ]bdiUMNTk`cMn\«QSY0Tk[]TaQ]eDTWMPVXO)MFRWQ]Tk�cRWMn\|RW[SVÍe¨ÊL@¢Tk[ e ³Êk@ bdi

Î ¾{oÏÂ7 ¾{ @+ÐkÑ ¼�e ³ÊaK >ye ÊaK ½Ò Ï ¼«ÿZ½

��`cMFRkM$¾{ @ bdiITk`cM�\«Q]YXV¬QSiWi`1[H��RaQHTWMS} Ð¶Ñ Tk`1MsiUO"MF_PbvÄ"_�`cMFQ]T\|[SRyQSbqR<}�QSY1g Ò ÏTW`cM-`cMFQ]TkbuYcp·t�QSeq�1M�\|[jRX­SMPRW[jikMPY1MS ��`1MVX[�g�bvÄ)MFg¢Tk`cRW�1iUToiUO"MF_PbvÄ)_{\|�1MPe1_P[SY1ik�cVXO�Tkbu[SY¢\|[SRITW`cM�MPYZTWbqRWMMPY1pSbuYcMnbui

Ó)Ô*Õ�Ö ³�7 ¾{ Ï O Î ¾{ ÏÓ)×ÈÓ¨ØmÙ}Ó`ÚmÛÁÜÈÔ*Ó 7 Ó)Ô*Õ�Ö O Î ¾{ Ͼ{ = & = Oݾ{ @ & ³@¼«�Z½

º��c¡)ixTWbvTW��TkbuYcpÂm�¨) jÿc}jQSY1g3Yc[SRWV¬Q]eub��FbqY1p?¡�wÂTk`cM��cY1V3[�g�bqÄ1M<g

Ó)Ô*Õ�Ö }��{M�[S¡cTWQ]buYÓ)Ô*Õ�Ö ³Ó)Ô*Õ¨Ö 7©��O�Þ Ð¶Ñ e ÊaKÓ)Ô*Õ�Ö Ò Ï & =_ß[à ½�UO�½c& ³@ ;*& =�á:à e ³ÊLKe¨Ê K >»� á

¼U���Z½��`cMFRkMf½�7 ¾{ @ ;�¾{ = bui£Tk`cM�¡�wZO)QSiWi£RaQHTkbu[1 ¥º��c¡1iUTkbqTk��TWbqYcpTk`1M�t�QSeq�1MFiÐ¶Ñ 7 � ~�â ; ~�ã_äLåÒ Ï-7 Õ��-�+�*� ~�â ; ~Qã ¼|7¢�<�+�+�*���kæç9<;rè|�k{½Ó)Ô*Õ�Ö 7 Vé ʶê�Ê¶Ê ~Qã ;-Àr;Lë ¼|7S�c î��è¡�k{�;-ì6í+;rè¡�s/§½& = 7 �Z�-��{o;+À& ³@ 7 ÕQ�+��{o;+Àe ÊLK 7 �SÕ*� äaå½c7 �j  ��{MÔ[j¡�TWQSbqY Ó)Ô*Õ�Ö ³ ; Ó)Ô*Õ¨Ö 7Á�j  Õ��� �¹�MFQ]T¥QSgcgcbvTWbq[jY���bueqe¡"M�Q]O1OceqbuMFgÅ\|[SR0[SYceuw»ik`c[jRUT�buYZTkMFRktHQ]ediF}�TWQS­SMP[S®'QSY1gÅO"[jiU�ikbq¡ceuwXeuQSY1g�buYcp1}�[S\�Q£eu[SYcp�1bqpj`ZTF AhsiWiU�1V3buYcp¢bvT�bui{QS_lTkbutHQHTkM<g\|[SR�Q»Tk[]TaQ]e¢[]\3Tk`cRWMPMªVXbuY���TkM<i�buYÃQ©�Fÿ+�]�8VXbuY���TkMy1bupS`ZTF}Tk`1MPY�TW`cM·Ml®DMF_lTkbutSM Ó)Ô+Õ¨Ö RaQHTkbu[�bdi Ó)Ô+Õ¨Ö ³ ; Ó)Ô*Õ�Ö 7¢�[O¼��1  Õ��S½P¼«�j½¶���<ÿ+�37 �S  �+�Z�� f~�Y3[]Tk`1MPRA�{[SRagciP}�buVXO1QS_lTo[SYÂ[HtjMPRk�Q]eueZ\|�cMPec_P[SY1ik�cVXO�TWbq[jYÂbdiI[SY£TW`cM{[SRag�MPR�[]\¨�c î�*�� �~ÎTAbdi¶Y1[]TkM<gQ]pZQ]buYyTW`1QHT�Tk`1bui�bdi�Q3OcRkMFeqbuVXbqY)Q]RWw¬MFiUTkbuVXQ]TkMn[SY�[SYcMn[S\�Tk`1MeuMFiWix�ÎMany_lbuMPYZT¢TkRWMFQHTWVXMPYZT¢[jO�Tkbu[SY1iF ·��`cM�Õ)����bqY)_lRWMFQSikMybuY\«Q]Y¸MP��`)Q]�1iUT¬tSMPeu[�_lbqTxw�_PQS�1ikMFi¢Tk`cM�ikQSV3M-buY1_lRWMFQjiUM0buY\«QSYTk`1Rk�1iUTF �¹sMPRWMS}1\«Q]YTW`cRk�)ixTNQj_P_P[S�cYZTWi�\|[SRN�1��� []\ITk`1M�Tk[STWQ]eTk`1Rk�1iUTF})iU[3Tk[STWQSeDTk`cRW�1iUT�bqY1_PRkM<QSikMFi{¡�w��<�Q�� §Ò¯I¯o°?î ² ± �N#F´-³�µ ± © » I% ���Áïy���`cbdiARWMFikMFQ]Ra_a`£`1QjiAiU`1[H��YÂTW`1QHT�QSOcOceubu_FQHTkbu[SY3[]\§Q]YXQ]Y1YZ�1euQSR_l[+1[H� Q]RW[S�1Y1g�Q�OcRWMFiWiU�cRWMl�ÎV¬QHTa_a`cMFg¤ik�cO"MPRaiU[jYcbu_£�xMlTy_PQSYikbqpjYcbvÄ"_PQ]YZTWeqwQ]eqTkMFRsTW`cM3OcRkM<ikik�cRWM£Ä1MFeug¥[]\ATW`cM?�xMlTF £h{T?Tk`1MOcRW[SO"MPRA_l[jY1g�bqTkbu[SY1iF}HTW`cM�_l[-)[H�ªMPeubqVXbuY1QHTWMFiA^�Qj_a`Â��Q�tSM�RWQ]�g�bdQHTWbq[jYf}F`1MPY1_PMoRkMFVX[HtSMFi§Q�iUTkRW[SY1p�ik[S�1RW_PMA[]\�Yc[SbdiUMI\|RW[SV�Tk`1M1[H�� {^[jiUT�bqVXO"[SRkTWQSYjTWeqwj}cTW`cM£Ml��O"MPRWbqVXMFYjTai�`1Q�tjMngcMPVX[SY��iUTkRaQHTkM<g�Tk`1Q]T?^�Qj_a`0��Q�tSMnMFeqbuVXbqY1Q]Tkbu[SY�_FQ]Y[�_P_P�cRsQHT?_l[SYc�g�bqTkbu[SY1iÂTk`)QHT¬Q]RWM�OcRaQS_lTkbd_PQ]eueuwÔQSY1g¤MF_P[SYc[jVXbu_FQ]eueqw�\|MFQjiUbu¡ceuM\|RW[SV Q�O1Rk[jOc�cediUbu[SY4iUTWQSY1g�O"[SbuYZTF %h�OcO1eqbuMFg4Tk[»QªTxw�Ocbd_PQSeTk�1Rk¡"[]\«QSY¬MPYcpjbqYcMj}]TW`cbui�Yc[SbdikM�iU�cO1OcRkM<ikikbu[SY¢TkMF_a`1Ycbu¨Z�cMsbui�MFiU�TkbuV¬QHTWMFg¢Tk[�buY1_lRWMFQjiUM�TWQS­SMP�8[S®3TW`cRW�1ixT<}S��bqTk`XtSMFRkw¢iUV¬Q]eue�buV¢�O1Qj_¾T�[SY-[HtSMFRWQSeqe)\|�cMFe�_l[jY1iU�1V3OcTkbu[SYf h�eqTk`1[S�cpj`�^�Qj_a`y�{Q�tjMsMFeqbuVXbqY1Q]Tkbu[SYy��QSi{QS_a`1bqMFtSMFg�QHT�_l[SYc�g�bqTkbu[SY1i¶_leu[jikMITk[�TW`c[jikMoOcRWMFg�bd_¾TWMFg§}<bqT¶bdifixTWbqeuejik�cRWOcRWbuikbqY1p�Tk`1Q]TTk`1MNMP®"M<_¾T�[]\§TW`cM�_l[-)[H��}jM<iUO"MF_PbuQSeqeuwX[]\fTW`cMsTW`cbuY�[jYcMS}ZO)MFRU�ikbuiUTkM<g�ik[Â\«QSR�gc[H��Y1ixTWRkM<Q]V- Áu?bqtjMPYyTW`cM3���H��g�MFpSRWMPMsikOcRWMFQSg��buYcp�RaQHTWM¥[]\ÂTk`cM·iU`1MFQ]R0edQ�wSMPRaiF}�bvT-�{Qji�MP��O)M<_¾TkM<gÅTW`1QHT-Q_l[+1[H��[]\Â�� î���ÎVXV Tk`1bu_a­�YcM<iki¬�{[S�1eug¸¡"M\|�ceueqw¸MFYjTWRWQSbqY1MFg��bqTk`cbuYQ¥eqMFYcp]TW`ª[]\N�S��{j{�}Ab8  Mj q}A��bqTk`1bqY�Tx��[��xMlT3gcbuQSV¢�MlTWMPRaiP  þ1[SR0Tk`1MÔÕ)  ÕS�|{j{��:TW`cbu_a­¸_l[+1[H� Tk`1M·_P[SRWRkM<iUO"[SY)g��buYcp�Ä)pS�cRWM�bdi�\|[S�cRX�xMPT�g�buQSVXMlTkMFRWiF ¹ðoMPTF}?bqY'¡)[STk`©_PQjiUM<i^�Qj_a`?��Q�tSM<iD��MFRkMAMFeqbuVXbqY)QHTkM<gnbqY£Q�RWMPpjbq[jY�iUO1QSYcYcbuYcp�Q]¡"[S�cT

ÿ

Page 9: =0 =0.3 =1.5 /T =2.8 T/T =2 - University of California, Irvinesupersonic.eng.uci.edu/download/aiaa-97-0147.pdf · `cM3O1Q]O"MPR?OcRWMFikMPYZTWisMl O"MPRWbqVXMPYZTaQ]e RkM

MPeuMPtjMPYÂ�xMlT�g�bdQ]VXMlTWMPRaiP A^[jRkMF[HtSMPR<}]TW`cM?pSRW[H��Tk`�RaQHTWMs[S\�Tk`1M�xMlTng�bdg-Yc[]TN_a`)Q]YcpjMnQSOcOcRWMF_PbuQS¡ceqwj UpÔM£V¢�1ixTsTk`cMFRkMP\|[SRWM�bqY��tSM<ixTWbqpZQHTWM¬bv\x}AbuY�QSg1g�bvTWbq[jY·Tk[¥Tk`cM�Ml��O)M<_¾TWMFg¤¡"MPYcMPÄcTWiF}ITk`1M_l[+1[H�ÖOcRW[�gc�1_lM<iÂik�c¡�TkeuM�¡1��T3buVXO)[jRUTaQ]YZT¢_a`1QSYcpSM<i�TW[�Tk`1MTk�1Rk¡c�1eqMFYjT�iUTkRW�1_¾TW�cRkMj AýNY1MsO"[jiWiUbu¡cbueqbqTxwXbui�Tk`)QHT�TW`cMn_l[-)[H��}buYH�xMF_lTkMFg0QHT�`cbupS`J5 @ }�ñkÇkòró ( ÇLôATW`cM�_l[jYZtjMF_lTkbutSMstjMPeu[�_lbqTxw¬[]\Tk`1M£�xMlT¢MFgcg�buMFiF}AiU[jVXMlTk`1bqYcp�_l[S�1YjTWMPRk�8buYZTk�cbqTkbutSM¬¡1��T¢buYªQS_l�_l[jRWg1Q]Y1_PM���bqTk`3Tk`1M�iUeu[H�ªVX[�g�MFiI[S\<&�(IiUMFMPYXbqYXik�cO)MFRWik[SY1bu_l�ik�cO)MFRWik[SY1bu_nik`cMFQSR{edQ�wSMFRWinÌq�S�]}��<�<ÍΠ��`cMnbuV3O"MlTW�1iF}1Q]Y1g��ceqTkbuV¬QHTkMnpj[jQ]e8}�[]\¶Tk`cbdi�RWMFikMFQ]Ra_a`ybdi�iU���O"MPRaiU[jYcbu_£�xMlT¬Yc[jbuikM�RWMFgc�1_¾TWbq[jY���bqTk`c[j��T3VXM<_a`1Q]Ycbd_PQSe{ik�cO��OcRWMFiWiU[jRWiF Ã¹�[H�{MPtjMPR<}{Tk`cMÔ^�QS_a`���Q�tSMMFeqbuVXbqY)QHTkbu[SY4VXQ�wQ]edik[N¡"MPYcMPÄcTATk`1M�_l�cRWRkMFYZT�VXbq��MFRU�ÎMx�xM<_¾TW[SR�TkM<_a`cYc[Seu[SpjwnbqY3Tk`1M\|[SRWV []\�QSeqeuMPt�bdQHTWbqYcp0QS_P[S�1iUTkbd_£eu[jQjgcis[jY�TW`cM¢buYcYcMFRnMx�xMF_lTk[jRik�cRU\«Qj_lMj �h�Y1[]Tk`1MPR3RkMFeuQ]TkMFg�Q]OcOceubd_PQHTWbq[jYª_l[j�ceugª¡)M0RkM<g��1_l�Tkbu[SY�[]\�Qj_l[S�)ixTWbu_seq[ZQSgci�[SY�QSbqRa_lRaQH\ÀT�ixTWRk�1_lTk�cRWMFi�Ml��O"[jikMFgXTW[^�Qj_a`y�{Q�tjMsRaQSg�bdQHTWbq[jYX\|Rk[jV%MPYcpjbqY1MNMP��`)Q]�1iUTF �ýN�cR{Ml��O"MPRk�buV3MFYZTWi¬��bueqe�_P[SYZTkbuY��cM-��bvTW`ÅVXbu_PRk[jOc`c[SY1M0ik�cRWtSMPw�i3[]\?Tk`1MQS_P[S�1iUTkbd_?Ä1MFeug-[S\�TW`cM��xMPTNQ]Y1g-��bqTk`�gcbqRWMF_lTsVXMFQSik�cRWMPVXMPYZTai[]\�Tk`1M£_l[SY�tjMF_¾TWbqtjMstjMPeu[�_PbvTWbqM<i�[]\�Tk`cM�edQ]RWpSM?MFg1g�bqM<iP 

õMöW÷�øNù,ú2ûbü,ýNþ�ÿqü0ø����

��`cM»iU�cO1O)[jRUTÔ¡Zw ¦NhNº�h Æ�QSYcpSeuMPw$��MFikMFQSRW_a` z�MFYZTkMPR�buipSRaQHTWMl\|�ceueuw�Qj_a­ZY1[H��eqM<g�pSM<gÒ¼|u?RWQSYjT'¦shNuN�a�l�a���]�S��VX[jYcbv�Tk[jRkM<g·¡�w·LNR< ¸�j[S`cYª^¤ oº�MFbqY1MPR¾½¾ �^¥Q]Y�w¥TW`1Q]Y1­�i£Tk[Ô^RF ^bd_a`1Q]MFe�Ó? ¨��wjQSY\|[SR?`cbdiNQjikikbdixTaQ]Y1_PM£��bvTW`�TW`cM3_P[SY1iUTkRW�1_l�Tkbu[SY¸[S\NTW`cMX�xMlT¬\«Qj_lbueqbqTxwS 4h O)QHTkMFYZT�Q]OcOceubd_PQHTWbq[jY�\|[jR¬Tk`1MVXMlTW`c[�gÔgcMFiW_lRWbq¡"MFgÔ`cMFRkMFbqY·`1QjinRkM<_lMPbutSM<g�Q-¦s[]Tkbd_lM¬[S\�hsev�eu[H�{QSY1_lMN\|Rk[jV%Tk`1M£r£ îºD 6�AQHTWMPYZT�Q]Y1g-�¶RWQjg�MPV¬QSRk­�ýIny_lMj » ��- s�Á ± �� ��Ìq�lÍ£^�_FÆ�QS�cpS`ceubuYf}{L¢  �X q}�^[SRWRWbuik[SYf}Èu3  L¢ u}�QSY1g¸��RW[S�cTUTF}�n  �� u}3ÐWmA��O"MPRWbqVXMPYZTaiX[SY»Tk`cM¥~�Y1ixTaQ]¡cbueubvTxw pÔQ�tjMFiXbuYQÔº��cO"MPRaiU[jYcbu_}�SMPT3QSY1g·Tk`1MPbuRXhN_l[j�1ixTWbu_}��QSgcbuQ]Tkbu[SYf  Ñ���ró_ñDÉHÊ�� ����Ê ó���ò���Ç (�� É��� ( ôa}��A[je: ]×j�c}S¦s[1 c�j�S}c�<�Z�]�c}OcO� î�]�H���j�� 

Ì ��Í£�¶Rk[j��TUT<}��n  �� u}�Q]Y)g ^¥_PÆ�QS�cpS`1eqbuYf}�L¢  �X q}�� mo��O"MPRWbq�VXMPYZTai�[SYÃTk`cM»þ¶eu[H� Q]Y)g hs_l[j�1iUTkbd_��oRW[SO"MPRkTkbuMFi�[]\Q¤^[�g�MFRWQ]TkMo��MFwZY1[Sedgciy¦��1V¡"MPR0º��cO)MFRWik[SY1bu_o�jMlT<  Ñ1}���ró_ñDÉHÊ�������Ê ó���ò��¥Ç (�� É��� ( ô¾}+�A[je: Z�S�<×c}H^�QSRW_a`¬�<�Sÿj�c}OcO� f�<�S�H�a�<�]×1 

Ì �HÍ£��Q]V-}�zN  �X  p  ¶QSY1gªÓ{�cRkTk[jYf}�L¢  m� u}�ÐWº�[j�cY1gCu?MPY1MPRk�Q]TkMFg4Ó�wÅ~�Y)ixTaQ]¡cbueqbqTxw:pÔQ�tjMFi�[]\3º��cO)MFRWik[SY1bu_�þ¶eu[H��iF �AQSRUT��c Hh���buikw�VXV3MPTkRWbu_m�jMlTWiF  Ñc}����ró_ñDÉHÊ�������Ê ó���ò���Ç �(�� É��� ( ôa}��A[Se8 §�F�jÿc}W�ZQ]Y��1Q]RWw-�F�jÿ]Õ1}�OcO�  �]Õj�H�x�j���S 

Ì Õ]Í£��Q]V-}czN  �X  p  q}cz�`cMFY}Q�I u}ZQSY1gyº�MFbqYcMFRF}��)  ^¤ u}�Ð���MFeuQ]�TWbq[jY1iU`1bqO¤Ó{MlTx�{MPMFYÔ~�Y)ixTaQ]¡cbueqbqTxwypÔQ�tjMFi�Q]Y)g¤¦�[SbdikM¬[]\¹sbqpj`���º�O)MFMFg¼�jMlTaiP  Ñc}"!$#%!$!&���ró_ñDÉHÊß}��A[je: §�*�c}§¦�[) î��}�<�S�j�c}jO1Of f���HÕ����W�H�]�S�c 

Ì ��ͺ�MFbqY1MPR<}c�)  ^ª q}yÓ{`1Q]TF}X�n  �� îºD}£QSY1gÂ�I[jYjTW[SYf}X^¤  �X q}ÐW^�QS_a`�p·Q�tSM3moVXbdikikbq[jY\|RW[SV Q0¹�bupS`�����MFVXO)MFRWQ]Tk�cRWMº��1O)MFRWik[SYcbd_d�jMlTF  Ñc}'!�#�!$!(���-ó_ñ"ÉHÊv}��A[je: "�j��})¦�[1 ¶�<�c}�<�S�]Õ)}jO1Of )�]�SÕZ�H���S�j�-�1 

Ì ×HÍ£^bqTW_a`1MPeue:}�ÓN  m� u}�Æ�MPeuMS}¶ºD  �X u}�Q]Y1g·^[jbqY�}0�I q}sÐkLNbqRWMF_lTz�[jV3O1��TWQ]Tkbu[SY£[S\cTk`cM�º�[j�cY1g�u?MFYcMPRaQHTWMFg¡Zw£º��1¡1iU[jYcbd_QSY1gÔº��cO)MFRWik[SY1bu_3h���bdiUw�VXVXMlTkRWbd_c�jMlTaiP} Ñj��MPO"[SRkT£��þ��×jפ[]\£TW`cM���`1MPRWV3[Zik_PbqMFY1_lM<i�LNbqt�bdiUbu[SY�}NLNMPO1QSRUTWV3MFYZT[S\¢^MF_a`)Q]Ycbd_PQSe?m�YcpSbuYcMPMFRkbuYcp)}NºZTWQSY�\|[SRag4rsYcbutSMPRaikbvTxwj}¦s[Ht" §�<�S�Z�� 

Ìî�<ͺ�MFbqY1MPR<}-�1  ^¤ HQ]Y1g��nRkMU�UikQ1}Hm� u}§ÐWº��cO"MPRaik[SYcbd_��jMlTA¦�[jbuikMQSY1g�Tk`cMy¹sbqpj`¤º�O)MFMFg�z�butZbue��¶RWQSY1ikO)[jRUT<} Ñ0h�~xhNh ÿS�]��S�j�]ÿ1}Q�ZQ]Y� §�F�Sÿj�

Ì ÿHÍ£¦NQ]pjQSV¬QHTWik�f}�¹£  �n u}¢º�`cMFMPR<}l�£  ms q}ÂQSY1g u?bqeue8}Â^¤ îºD q}оz�`1Q]RaQS_lTkMPRWbdixTWbu_FiI[]\�^�ceqTkbqTk�)g�M?^��cevTWbuik`cRW[S�1gyº��cO)MFRU�ik[SYcbd_��jMlT�¦�[jbuikM3º��cOcOcRWMFiWiU[jRF} Ñ)!$#%!$!&���ró_ñDÉHÊß}��A[je: ���c}�¦�[) 1�c}§�<�Z�]�c}�OcOf 1�*�Z�<���c�<�c 

Ì �HÍ��AQSO1Q]VX[jiW_a`c[j�f}?L¢ u}¥ÐkhsY1Q]euw�iUbdi-[]\i�AQSRUTWbuQSeqeuwZ�Î^bq��MFgº��1O)MFRWik[SYcbd_�mf�xMF_¾TW[SR<} Ñ*!$#%!+!,���ró_ñDÉHÊ'���.-mñ%��/�ó�Ê ô�0��É��ò1-���2�Çsñ¾}6�A[Se8 §�<�c}c¦�[) cÕ1}§�<�S�S×1}�OcOf "�]�S×H�U��Õ)�S 

Ìq���HÍ£��bueqeuV¬Q]Yf}"�n  u3 u}��AQ]TkMFRWik[SYf}W�£  p  u}"QSY1go�oRWMFi��S}_p   ^¤ Ðaº��cO"MPRaiU[jYcbu_¬¦s[+���PeuM�^bq��MFR¢mf�xMF_¾TW[SR<} Ñ�!$#%!+!3���ró_ñ�DÉHÊ4� �+-mñ%��/�ó�Ê ô�0��¤É�¨ò5-���2�Çsñ¾}6�A[Se8 1ÿ1}c¦�[1 "��}f�<�S�j�c}OcO� )���<�H�x���F�1 

Ìq�S�lÍ��AQSO1Q]VX[jiW_a`c[j�f}�L �ÐWºZTWRk�1_lTk�cRWM�[]\�Tk`1M3z�[SVXOcRWMFiWiUbu¡ceuM�¶�cRW¡c�ceuMPYZTIº�`1MFQ]R�Æ�Q�wjMPR<} Ñ�h�~xhNh���ÿS�]���1�<�]×1}��ZQ]Yf j�F�jÿS�

Ìq�<��Í£þ1[S�cRWpS�cMPTUTWMS}L ßzN u}�Lsbu¡c¡ceuMS}���  p  u}�Q]Y1g�^�cYcpZQ]e8}^¤  u3 XÐk��buVXMÔm�tS[Seu��TWbq[jY�[]\£TW`cM¤º�`cMFQSR0ƶQ�wSMFR�[]\3Qº��1O)MFRWik[SYcbd_Âh���buikw�V3VXMPTkRWbu_c�SMPT?Q]Tn^�Q]TW_a`cM<gÔz�[jY1g�bq�TWbq[jY1iP} Ñ6!$#%!$!7���ró_ñDÉHÊß})�A[je: A�]�1}o¦�[) A��}��F�S�1�S}IOcOf �j�<�]�]�W�j�F�*�c 

Ìq�F�HÍ£¹NQ]eue:}Â�)  Æo u}�LsbuVX[]TaQ]­�buiF}��I  m� u}�Q]Y1gÝ��[jikMPV¬Q]YcY�}¹Â �}8Ðkmo��O"MPRWbuV3MFYZTWi£buY¦s[SY��ç��M<QS_¾TWbqY1pÔz�[SVXOcRWMFiWiUbu¡ceuMº�`1MFQ]R-Æ�Q�wjMPRaiP} Ñ8!�#�!$!9���ró_ñ"É]Êß}I�A[je: ��1�S}s¦s[1 n�<�c}�<�S�S�1}jO1Of )�S�]Õ������j�S�HÕ) 

Ìq�PÕ]Í£¦NQ]pjQSb:}"^¤ {Ðk^M<_a`1Q]YcbdikV�[S\���ikMP�1g�[S��º�`c[�_a­fp·Q�tSM£buYº��1O)MFRWik[SYcbd_J�SMPTWÑ1}.:Nó�ÊÀÊqÇPÈ;�<=� �-È � Ç>�@?��)A{}��A[Se8 ��]×1}�<�SÿS�1}jO1Of )�-���<�x���j�S 

Ìq�<��Í��AQSO1Q]VX[jiW_a`c[j�f}cL DQ]Y)g�Ó{�cY�wjQH�xbqTkRaQSgc�ceqwZQc}ch �ÐWmotS[S�eu��Tkbu[SYn[S\�edQ]RWpSMIM<gcg�buMFifbuY£_l[jVXOcRkM<ikikbu¡ceqMAik`cMFQSR§euQ�wjMPRaiP} ÑTW[XQSOcO"MFQ]R�buY�- �CB ô� ( ô1���+��Ê ó���òrô¾