Numerical, Experimental and Theoretical Studies on Mechanism of K-H Instability and Ring Generation bhids behind Supersonic MVG

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1 Nmecal Eemenal and heoecal Sdes on Mechansm of K-H Insably and Rng Geneaon bhds behnd Sesonc MVG Yongha Yan Qn L Chaoqn L Fank L echncal Reo - h://

2 Nmecal Eemenal and heoecal Sdes on Mechansm of K-H Insably and Rng Geneaon behnd Sesonc MVG Yongha Yan Qn L Chaoqn L 3 Fank L 4 Unvesy of eas a AlngonP.O. Bo 948 Alngon X 769 USA hs ae llsaes he domnan mechansm of he ng-lke voe fomaon whch was fond geneaed by mco voe geneaed MVG fo shock-bonday laye neacon conol by o mlc lage eddy smlaon ILES wh hgh ode accacy and o eemen. An aveaged velocy ofle afe MVG s aken as he base flow. he dsbed vscos ncomessble and nvsc comessble sably eqaons have been deved n a 3D coodnaes aally symmec n acla. Afe we nodced he effec of smlfed sanwse cone-oang voces we heoecally conclde ha he sce of voe ngs behnd he MVG s a conseqence of he momenm defc accodng o he asymmec nsably heoem. I I.Inodcon n he sesonc am jes shock bonday laye neacon SBLI can sgnfcanly edce he qaly of he flow feld by ggeng lage-scale seaaon casng oal esse loss makng he flow nseady and dsong. MICRO voe geneao VG s a knd of low-ofle assve conol devces desgned fo he bonday laye conol. Inensve comaonal and eemenal sdes have been made on ecenly. In o sdy a hgh ode LES and a wnd nnel es have been condced fo he nvesgaon on he flow aond mco voe geneao MVG a ch nmbe.5 and Reθ98 wh he bonday laye momenm hckness as he efeence lengh. An aoach named monoone negaed LES MILES was adoed n whch he nmecal dssaon s sed as he sb-gd sess model. he well-known 5h ode WENO s aled as he basc nmecal scheme n he MILES aoach. he flow feld aond he MVG and sondng aea has been sded n deals. Fhe moe 3-D sce of he shocks s also obaned by o nmecal smlaon. he ng-lke voces have been fond fomed and avel downseam. Accodng o he analyses a dynamc voe model was develoed and can be skeched n Fg. n he half doman. he new model can be descbed as follows. he domnan voe nea he MVG s he may voe; ndeneah hee ae fs wo seconday cone-oang voces whch wll lef he body sface lae o become flly 3D seaaons by he way of sal ons n body sface and hese voces seem o be waed by he may voe when oagang downseam; afe ha a new seconday voe wll be geneaed nde he may voe. Ph. D Canddae h Deamen Unvesy of eas a Alngon Alngon X 769 Vsng Pos Doc. h Deamen Unvesy of eas a Alngon Alngon X Pofesso h Deamen Unvesy of eas a Alngon Alngon X 769 AIAA assocae fellow. 4 Pofesso ME/AE Deamen Unvesy of eas a Alngon Alngon X 769 AIAA assocae fellow. Amecan Inse of Aeonacs and Asonacs

Fge. he dynamc voe model A song momenm defc has been fond behnd MVG whch cases a song ccla shea laye as shown n Fg.. he esl s n conssency wh he efeenced comaons and eemens 345. Fge.

3 Fge. he dynamc voe model A song momenm defc has been fond behnd MVG whch cases a song ccla shea laye as shown n Fg.. he esl s n conssency wh he efeenced comaons and eemens 345. Fge. he momenm defc Fo clay he ycal sce of he defc s shown agan n Fg. 3 wh sanwse seamlnes. Insde he defc aea hee ae wo cone-oang may voces whch ae llsaed n Fg.. In nea MVG egon he shae of defc aeas o be a ccle and sally has a oo conneced o he bonday laye. A he ndeneah sdes of he ccle hee ae wo hgh seamwse velocy egons. Amecan Inse of Aeonacs and Asonacs

Fge 3. Sce of he defc and he seamlnes Fg. 4 shows he so-sface of esse. Besdes he eanson wave and he sface wang he be nea he alng-edge he ng-lke voe sces ae fond n he sbseqen downseam egon.

he so-sface of he nsananeos esse he dsbons of aveaged seamwse-velocy ae gven n Fg. 5 along he nomal gd lnes a he cene lane. he seamwse osons of he lnes ae L/h 3.3 6.

4 Fge 3. Sce of he defc and he seamlnes Fg. 4 shows he so-sface of esse. Besdes he eanson wave and he sface wang he be nea he alng-edge he ng-lke voe sces ae fond n he sbseqen downseam egon. Sch ngs aea nally no fa away fom he alng-edge and become lage and egla when movng downseam; meanwhle he seamwse voe bes become weake and dsaea a a cean locaon. Fge 4. he so-sface of he nsananeos esse he dsbons of aveaged seamwse-velocy ae gven n Fg. 5 along he nomal gd lnes a he cene lane. he seamwse osons of he lnes ae L/h and whee L s dsances fom he ae of MVG. he d of he lnes coesonds o he momenm defc whch s smla o ha gven by Babnsky 3.I can be seen clealy ha hee ae a leas wo hgh shea layes n he cenal lane one s locaed a he e edge of he d and he ohe s locaed a he lowe edge. he second ode devave w/y s calclaed o demonsae he esence of he nflecon ons and he esl of he lne a L_fom_ae/h 6.7 s loed n Fg. 6. he esence and coesondence of he nflecon ons a he e and lowe shea laye s llsaed by wo dashed lnes nesecng he dsbon of he seamwse velocy and s second ode devave Fg Amecan Inse of Aeonacs and Asonacs

5 Fge 5. he seamwse velocy dsbon a dffeen locaons Fge 6. he dsbon of he seamwse velocy lef and s second ode devave gh o y coodnae a L fom ae/h 6.7 I s obvos ha he esence of he nflecon ons n shea layes wll case he flow nsably and geneaes voe olles accodng o he Kelvn-Helmholz nsably K-H heoem n D. So he mechansm fo he voe ng geneaon may be cased K-H ye nsably and he los of he sably of he shea laye wll esl n he fomaon of he voe ngs. Snce he analyss of K-H nsably s based on D b o esls show a comlcaed 3D case so o fhe eloe he mechansm of he voe ng we develoed an asymmec nsably analyss whch cold be consdeed as a 3D nsably. II. Insably Analyss he nsably analyss s comosed by wo saons: ncomessble flow and comessble b nvscd flow. he ose of he aangemen of he analyss s de o ha we have manly wo majo dmensonless aamees Reynolds nmbe and ch nmbe wll be ease f we dscss only one of hem seaaely. On he ohe hand alhogh nvscd K-H ye nsably s consdeed as he domnan mechansm of he nsably n o case we sll need o dscss he effec of vscosy o some een. A. Comessble K-H ye nsably Fsly accodng o he dsbon of he seamwse velocy a he e bonday of he defc n Fg 6 we assme hee s an asymmec flow U U U U U anh δ Wh wo consan veloces U > U as shown n Fg. 7 and defnes as he n veco n he seamwse decon. 4 Amecan Inse of Aeonacs and Asonacs

6 5 Amecan Inse of Aeonacs and Asonacs he aamee δ hee s sed o adjs he cve of o aoach he eal dsbon of he seamwse velocy and s se as.6 o f o LES case. If he dsbed flow s comessble b nvscd n boh nsde and osde he momenm defc egon he govenng eqaon can be descbed by nvscd Ele eqaons q D De D D whee s he densy s he velocy veco s he esse q denoes he hea ansfe. A efec gas obeys he efec gas eqaon of sae R 3 We condc he comessble b nvscd nsably analyss whch eends he sandad D K-H nsably analyss o he asymmec cases. he devaon sas fom he nvscd dmensonless Ele eqaons n cylndcal coodnae sysem whch shold also be asymmec 4 whee s he emeae s he hea ao and s he mach nmbe. he sbsc eesens he vale of fee seam flow. Consde ha q q q n whch q can be secfed as and P U q hen he govenng eqaon fo small ebaons wll be Fge 7. Confgaon of he asymmec shea laye U

7 6 Amecan Inse of Aeonacs and Asonacs 5 Fo o oblem he saal dsbance s easy o mease. Acally elaes o he dsance among wo neghbong voces n he cenal seamwse lane as shown by scala feld of he gaden of esse behnd he MVG n Fg 8. So he emoal mode s sed fo he sably analyss. Assme he nomal mode s e q q 6 whee q and he aamee s gven whch s eal and se o be 6. shold be a comle nmbe. Eqaon 5 can be ewen as D D D D D 7 whee and d d D By elmnang and we can oban he nsably eqaon elaed on esse 3 D D D D 8 he base velocy ofle s gven n eqaon and he elaon beween emeae and velocy comes fom Besmann-Coccos eqaon 6 du d 9 hen he elaon beween emeae ofle and velocy can be easly obaned accodng o he condons gven by Fg 7 Fge 8. Scala feld of he gaden of esse behnd he MVG

8 * * U U U * U U * * U U U * whee / - he qoen of he emeae osde and nsde he momenm defc egon * and U U /U. Fge 9. he emeae ofle he second ode cenal dffeence scheme s sed o deve he fne dffeen eqaon fom eqaon 8 D j j j j D j j hen a so called egenvale mehod s aled o ge he vale fo whch shold be a comle nmbe. Accodng o he nomal mode n Eq. 6 shows ha f he magnay a of s s negave. Wh he global mehod bascally he amon of he solons of fom he dffeence eqaon osve he flow s nsable b wll be sable f of 8 s deemned by how many nmecal gds ha have been sed. Fg shows he dsbon of hese solons on he comle lane when and gds ae sed j a n b n Fge. Dsbon of solons of wh dffeen gds Alhogh all of he solons above ae he solons of eqaon 8 mos of hem ae nonhyscal. o dsngsh he hyscal and nonhyscal solons hee ae wo ceons n geneally. Fs he nonhyscal solons wll sally sck o each ohe. In Fg many solons fom a bow and all of 7 Amecan Inse of Aeonacs and Asonacs

9 hese ons shold no be he hyscal ones. Second he hyscal solons wll no change he vales whle he qany of gds s changed. By he wo ceons descbed above we can ge he hyscal solon of he feqency as llsaed n he Fg whch s locally enlaged by Fg whose magnay a s abo.44 fo o case. he osve vale means hs knd of flow s nsable. Fg shows he coesondng shae fncon of whch s smla o hose fond n bonday laye flows -S waves Physcal Solon Fge. Dsbon of solons of nea he ogn.6.4 Fge. Shae fncon of B. he effec of vscosy Ne he effec of vscosy s also consdeed. o smlfy he analyss we assme he flow as ncomessble. If he dsbed flow s ncomessble n boh nsde and osde he momenm defc aea hen he govenng eqaon can be descbed as follows. 8 Amecan Inse of Aeonacs and Asonacs

10 9 Amecan Inse of Aeonacs and Asonacs Re whee Re s he Reynolds nmbe whch s abo 98 f he hegh of he MVG s consdeed as he efeence lengh scale. Fo lnea sably consdeng he nsananeos flow q as he sm of base flow q and he dsbance q we can eess 3 whee ae he small ebaons. If all he second ode ems ae omed he eqaons fo he small ebaons wll be Re 4 hs s a fndamenal eqaon n he lnea sably heoy. A cylndcal coodnae sysem θ s sed fo o oblem θ. Snce s asymmec we have / θ θ and he sysem 4 can be eessed as Re Re 5 n he cylndcal coodnae sysem. By he mehod of nomal modes we assme ha an abay dsbance can be esolved no modes of he fom e φ φ 6 hee φ s. By lggng n 6 he sysem 5 can be ewen as Re Re Re D D L D L 7 he oeaos ae defned as } Re { D D L and d d D By elmnang and û we can ge he sably eqaon fo o oblem

DL D L Re D D D 8 Second ode cenal dffeence scheme s agan sed o deve he fne dffeen eqaon fom eqaon 8 D 4 j j 4 j 6 j 4 4 j j 9 D j j j j D j j j Usng he same mehod dscssed above he vale of can be

11 DL D L Re D D D 8 Second ode cenal dffeence scheme s agan sed o deve he fne dffeen eqaon fom eqaon 8 D 4 j j 4 j 6 j 4 4 j j 9 D j j j j D j j j Usng he same mehod dscssed above he vale of can be obaned whch s abo.6 Alhogh s mch smalle he flow s sll nsable. he dffeence beween nvsc and vscos flow nsably s elavely lage. C. Nmecal and Eemenal Resls of Voe Rngs Snce he vale of s osve n boh saons dscssed above he flow aond he bonday of momenm defc shold be nsable de o he shea laye llsaed n Fg 6. by o LES smlaon. Becase nvcd nsably s essenal o shea laye also he Reynolds nmbe s geae han 9 and mach nmbe s.5 he vscosy effec s only aled o make he flow be less nsable so he domnan mechansm fo he voe ng geneaon shold be he K-H ye nsably and he los of he sably of he shea laye wll esl n he fomaon of he voe ngs aal symmecally snce he shea laye aond he momenm defc aea s nealy a ccle. o eveal he coheen sce of he flow he so-sface of λ scala feld s gven n Fgs. 3 and 4 by LES and Eemen. I s vey clea ha hee s a chan of voe ngs whch sa fom he alng-edge of MVG. he ngs ae laced eecly and n a good ond shae n he nal sage hen hey begn o defom when oagang downseam. Fge 3. Voe ngs shown by so-sface of λ Amecan Inse of Aeonacs and Asonacs

a LES b Usng PIV c Usng he aceone vao Fge 4 he lase-shee flash mage a he cene lane Fom fhe analyss we fnd ha he fs ng s fomed when he han voe avels.

Conclsons Based on o DNS and heoecal sdy he followng conclsons can be made.

he esls show ha he shea layes aond he momenm defc s nsable.

12 a LES b Usng PIV c Usng he aceone vao Fge 4 he lase-shee flash mage a he cene lane Fom fhe analyss we fnd ha he fs ng s fomed when he han voe avels. Conseqenly mlle ngs ae geneaed a almos same me. hese ngs cold domnan he mechansm of MVG fo conol of shock bonday laye neacon. III. Conclsons Based on o DNS and heoecal sdy he followng conclsons can be made.. Insably analyss s made fo he shea layes aond he bonday of he momenm defc geneaed by mco voe geneao fo boh ncomessble vscos flow and comessble nvscd flow. he esls show ha he shea layes aond he momenm defc s nsable.. he domnan mechansm fo he voe ng geneaon whch has been obseved by o LES and eemen shold be K-H ye nsably and he los of he sably of he shea laye wll esl n he fomaon of he voe ngs aal symmecally whch s vefed by he nmecal esl shown by so-sface of λ and eemen. Acknowledgemens hs wok s soed by AFOSR gan FA sevsed by D. John Schmsse. he ahos ae gaefl o eas Advanage Comng Cene ACC fo ovdng comaon hos. Amecan Inse of Aeonacs and Asonacs

13 Refeence C. L and L. Chen Sdy of Mechansm of Rng-Lke Voe Fomaon n Lae Flow anson AIAA Pae Q. L and C. L Nmecal Invesgaons on he Effecs of he Declnng angle of he alng-edge of MVG AIAA ae H. Babnsky Y. L and C. W. P Fod Mcoam Conol of Sesonc Oblqe Shock-Wave/Bonday-Laye Ineacons AIAA J. Vol. 47 No S. Ghosh J. Cho and J. R. Edwads RANS and Hybd LES/RANS Smlaons of he Effecs of Mco Voe Geneaos Usng Immesed Bonday Mehods AIAA ae S. Lee E. Loh and C. Wang LES of Sesonc blen Bonday Layes wh µvg s AIAA Pae Sandham ND and Renolds WC Comessble mng laye: lnea heoy and dec smlaon AIAA J Vol G. Jang and C. W. Sh Effcen Imlemenaon of Weghed ENO Schemes J. Com. Phys. Vol Chales W. P Fod and Holge Babnsky Mco-Ram Conol fo Oblqe Shock Wave/Bonday Laye Ineacon AIAA ae H. Holden and H. Babnsky Effec of Mcovoe Geneaos on Seaaed Nomal Shock/Bonday Laye Ineacons AIAA J S. Lee and E. Loh Sesonc Bonday Laye Ineacons wh Vaos Mco-Voe Geneao Geomees AIAA Pae D. S. Dollng and M. Mhy Unseadness of he Seaaon Shock Wave Sce n a Sesonc Comesson Ram Flowfeld AIAA J. Vol D. S. Dollng Hgh-Seed blen Seaaed Flows: Conssency of hemac Models and Flow Physcs AIAA J. Vol Amecan Inse of Aeonacs and Asonacs

Chapter Finite Difference Method for Ordinary Differential Equations

Chapter Finite Difference Method for Ordinary Differential Equations Chape 8.7 Fne Dffeence Mehod fo Odnay Dffeenal Eqaons Afe eadng hs chape, yo shold be able o. Undesand wha he fne dffeence mehod s and how o se o solve poblems. Wha s he fne dffeence mehod? The fne dffeence