AERODYNAMIC DESIGN METHOD FOR SUPERSONIC SLENDER BODY USING AN INVERSE PROBLEM

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1 AERODYNAMIC DESIGN METHOD FOR SUPERSONIC SLENDER BODY Kia Matuhima*, Ikki Yamamichi**, Naoko Tokugawa*** * Dept. Engineeing, Univeity of Toyama, Gofuku 3190, Toyama , JAPAN. Phone: , Fa: , kiam@eng.u-toyama.ac.jp ** Gaduate Student, Univeity of Toyama. *** Supeonic Aicaft Section, Japan Aeopace Eploation Agency. Keywod: Supeonic lende bodie, Aeodynamic deign, Invee poblem Abtact Aiming to deign the hape of SST fuelage of good aeodynamic pefomance, an efficient deign ytem uing peonal level compute ha been developed. Thi conit of CFD olve and an invee poblem fo hape deign. The deign ytem wok well on SST noe of aiymmetic fuelage deign a well a nonaiymmetic cae. 1 Intoduction In upeonic flight, high fuel-efficiency and the eduction of onic boom at cuiing ae cucial iue. Both phenomena ae pimaily connected with the peue ditibution on aiplane. In thee two decade, a lot of wok ha been done fo upeonic wing deign to attain good aeodynamic pefomance [1, ]. On the othe hand, the deign of fuelage i not well tudied. In thi aticle, conideing impotant effect of peue ditibution on upeonic fuelage pefomance, an invee poblem i fomulated and utilizing the invee poblem, a imple and efficient aeodynamic deign ytem fo upeonic fuelage hape i developed. The deign ytem wok well on SST noe deign poblem. The tun-aound time to complete a deign i hot. In Section, the fomulation of an invee poblem i dicued. In Section 3, a deign ytem i contucted to utilize the fomulated invee poblem. To eamine the ability of the deign ytem, the deign poblem ae olved in Section 4 and 5. Section 4. Fomulation of an Invee Poblem fo Slende Bodie of evolution in Supeonic Flow.1 The Petubation Potential Equation in Cylindical Coodinate The fomulation tat with a body of evolution located on a upeonic flow field which might follow the aially ymmetic potential equation of Eq. (1) in cylindical coodinate (,, ). 1 (1 M ) 0 (1) In Eq.(1), i the petubation potential fo petubation velocity u, v, w, whee M denote the unifom flow Mach numbe. It velocity peed i U. A fo petubed velocitie, u, v, w coepond to,, coodinate, epectively. Hee, the aial velocity w i 0, becaue we ae thinking aially ymmetic flow field. Fo the upeonic flow, intoducing the notation, M 1 > 0, Eq.(1) could be olved to be the integal equation fom[3]; f ( ) (, ) d () 0 ( ) whee, i a integal vaiable along the -ai. In Eq.(), f ( ) i unknown and hould be detemined by uing a bounday condition, Eq. 1

2 Kia Matuhima, Ikki Yamamichi and Naoko Tokugawa (3). When the aiymmetic body uface contou i given by = R(), the lip condition thee ae decibed a dr v (3) d U u whee ubcipt indicate phyical quantity at the body uface. A eade know v = Function of f ( ), Eq.(3) elate f ( ) to velocity component. It alo elate the fuelage geomety and velocity.. Peue a a Function of Velocity Let u define the elation between peue ditibution and velocity. Fit, we conide ientopic elation between peue and velocity. Then, nomalizing peue by the fee team dynamic peue and applying the fitode mall petubation theoy, it yield the peue coefficient. u v Cp (4) U U.3 Slende Body Appoimation Fo the upeonic flow about a lende body, Eq.() i to be tanfomed into the following fom [1]. (, ) f( )log f( )log( ) d (5) Uing Eq.(5), the adial velocity component i f( ) v (6) On the body uface, Eq.(6) i epeed f( ) v f( ) vr( ) (7) R Multiplying R() and Eq.(3) and uing Eq. (7), we obtain the concete fom of f. dr v R ( ) f ( ) R = d U u U u o d f( ) R d U u { ( )} (8) 0 Futhemoe, the polynomial epanion of u i applied to Eq(8) and take the fit ode appoimation. Then, the aea ditibution function S() of co-flow ectional plane along the ai i intoduced. Finally, the equation of peue coefficient yield the function of geometical paamete that might be pimitive equation of an invee poblem. 0 dr 1 Cp S( )log d R 1 d S( )log( ) d (9) d.4 Solution fo Slende Cone Eq. (8) and (9) i applied to the flow field about a lende cone whoe noe angle i In thi cae, 3 R ( ) tan (10) 3 S ( ) R (11) f ( ) U (1) Theefoe, an equation to elate the peue coefficient (Cp) and the noe angle paamete i fomulated. It i one of eential equation in thi aticle. Cp log (13).5 Invee Poblem fo Slende Cone Thi ection dicue the diffeential fom of Eq.(13). The diffeential fom i needed fo the iteative eidual coection deign method that i eplained late in Chapte 3. The idea i baed on Taylo eie epanion of Eq.(13) in tem of, diffeence in. Neglecting econd and highe ode tem, we obtain imple mathematical model of Eq.(14).

3 AERODYNAMIC DESIGN METHOD FOR SUPERSONIC SLENDER BODY Cp g( ) ( 14) whee g( ) log and g(δ ) 4 (log 1) Accodingly, to compenate the Cp diffeence between a taget and a cuent tate, the noe angle hould be changed by the following. Cp / g( ) (15).5 Invee Poblem fo Geneal Slende Bodie of Revolution The Invee poblem in thi aticle detemine geomety coection to update a baeline hape o that the updated hape ealize a taget Cp ditibution on it uface. Theefoe, a deign uing the invee poblem hee tat with a baeline hape and it uface Cp ditibution. The idea i a follow to deign the hape of geneal lende bodie. The meidian plane hape of a geneal lende body i divided into N inteval and it contou cuve i epeed by the connection of N+1 gid point. The noe leading edge i located at (0, 0) and it gid coodinate i identified a ( 0, R 0 ). The coodinate of the i th gid point i ( i, R i ) a hown in Fig. 4. Then, we egad the meidian plane hape a the integation of tiangle each of which i contucted by the gid point of ( i-1, R i-1 ), ( i, R i ) and ( i, R i-1 ) whee i = 1, N+1. It i called tiangle i and the angle at ( i-1, R i-1 ) i i. We aume that Eq.(15) hould hold fo the Cp i elation to on the piecewie tiangle i becaue the tiangle could be a meidian plane of a cone. Cp i i the diffeence in uface Cp value between the taget and cuent one, calculated a Cp i = Cp i (taget)- Cp i (cuent) (16) Cp i i the uface Cp at the midpoint of the edge line, ( i-1, R i-1 ) and ( i, R i ), i.e. Cp( i-1/ ). Once Cp i i obtained, the hape R i new to ealize the taget Cp i i deigned by uing following equation fo i = 1, N+1. Gaph in Fig. 4 how the updating poce. i Cpi / g( i) (17) new i i i (18) new new new Ri Ri 1 i ( i i 1) (19) new R R R (0) i i i 3 Deign Sytem uing the Invee Poblem In thi ection, a deign ytem i built uing the invee poblem fomulated in The ytem i fo geneal fuelage geometie including aiymmetic and quai-aiymmetic bodie. Figue 5 illutate the iteative poce of the method. We tat with an abitay baeline lende body hape and taget Cp ditibution. The hape i dicetized in the adial () diection into j ma inteval. Fo each tation, one meidian plane eit, then totally thee ae j ma +1 plane. A one can ee, when a aiymmetic body i to be deigned, j ma i 0. Afte CFD flow analyi about the baeline hape, Cp i calculated. At evey meidian plane, the invee poblem of Eq. (1) i olved to obtain the geomety change (f) to compenate the Cp diffeence. Afte updating all the geomety of meidian plane, we obtain a modified hape of the lende body. Uing thi new hape, the CFD analyi and the invee poblem poce i equentially iteated until the uface Cp ditibution of the new hape agee with the taget one. Fo the flow analyi of the method, a linealized potential flow code [4] i ued at the peent. Thu the total time fo one deign poblem i about two hou with an odinay peonal compute. When one pefe high fidelity in analyi and imulation intead of quick tun-aound of deign, one can conduct Navie-Stoke (N-S) flow imulation. Even if we do N-S imulation, the cot and time i not o heavy compaing with common optimization. Actually, the numbe of flow analyi imulation equied to convege the peent deign poce i meely up to thity fo uual deign cae. Theefoe, the imple and low-cot deign fo fuelage can be ealized. 3

4 Kia Matuhima, Ikki Yamamichi and Naoko Tokugawa 4 Deign fo the Noe of an Aiymmetic Fuelage To eamine the ability of the deign ytem, an aiymmetic deign ha been fitly conducted. The taget Cp ditibution ae thoe of a known hape which i the noe of the Sea-Haak body. A baeline (initial) geomety i one of tial poduct of a SST noe eamined in JAXA. It i called FC which indicate a flaed cone [5,6]. It i thee-dimenionally diplayed in Fig. 6. The Mach numbe i 1.6 and the peed of unifom flow (U) i nomalized to 1. The fuelage ha no angle of attack. The caled noe length i 0.5. Figue 7 how the meidian plane contou of the baeline (initial) and taget noe hape a well a thei Cp ditibution along the -ai coodinate, the plane contou cuve ae identical to adiu ditibution along the -ai, R(). The ma diffeence in Cp i in the vicinity of the leading edge while that in R() i at the ea. Fo deign poce, -ai of the body i divided into 800 non-unifom inteval. The leading edge ha fine gid ditibution than the othe pat. Afte the fit iteation of the popoed deign ytem, we obtain eult hown a cuent R and Cuent Cp in Fig. 8. Though the fit iteation of deign give almot conveged olution, we continue the deign iteation until the fifth. Finally we obtain the hape which ealize the identical Cp ditibution to the taget a hown in Fig. 9. The deigned hape alo agee to the taget. The eo between Cp and R( i ) at each iteation tep ae lited in Table 1 and. It take only twenty minute fo a tudent to finih deigning thi eample uing a coei7 PC. Table 1 # Ave ERR Cp Ma ERR Cp E E E E E E E E E E E E-04 Table # Ave ERR in R Ma ERR in R 0.33.E E E E E E E E E E E E-04 It hould be concluded that the deign ytem uing the invee poblem fomulated hee i pomiing a a quick and pecie deign tool fo SST noe and fuelage. 5 Deign fo the Lamina Noe of a SST Fuelage The lamina SST noe ha been deigned uing the invee deign ytem. Thi deign handle a non-aiymmetic body a well a a body which ha the angle of attack of degee. The body length i caled to 0.33 by the length of a whole fuelage. It i athe a ealitic deign poblem. The fee team Mach numbe wa.0. The baeline noe hape wa a cone. The hape wa dicetized into 36 inteval in the adial diection ( 0 ) and 800 inteval in the - ai diection. We have 37 meidian plane to deign, thu we pecify 37 taget Cp fo each plane. Figue 10 how the thee of pecified taget Cp, which ae fo the plane of =0, / and. The Cp ditibution wee devied by the SST R&D team of JAXA to attain lage lamina flow egion on a SST noe [7]. The initial Cp ditibution ae alo thee. They ae the one on the top line which mean the contou line of the meidian plane of =0. Becaue the baeline hape i a cone, the initial Cp ae contant on a taight line. Afte 14 time of invee deign iteation, the deign look to almot each a convegence tate. The deign eult afte the 10th iteation ae epoed in Fig Figue 11 how the eulting Cp and geomety a well a the taget one fo the meidian plane at =0. So do Fig. 1 and 13 fo the plane at = / and epectively. The eult indicate the invee deign ytem popoed hee i pomiing fo the deign of quai aiymmetic body, too. Howeve, mall amount of dicepancy emain 4

5 AERODYNAMIC DESIGN METHOD FOR SUPERSONIC SLENDER BODY in the vicinity of the leading edge on evey plane. The maimum eo i on the top line. We think that the dicepancy could be ettled by uing Eule o Navie-Stoke calculation intead of the linealized potential code in the deign ytem. The longitudinal ection hape of the deigned noe i hown in Fig. 14 a well a the co ection hape at = 0.18 i in Fig. 15. In both figue aially ymmetic Sea-Haack body hape i plotted to check the aymmety of the deigned hape. 6 Concluion Aiming to computationally deign the hape of SST fuelage of good aeodynamic pefomance, an efficient deign ytem ha been developed. Fo the ytem the invee poblem baed on the aiymmetic lende body theoy ha been newly fomulated. The concept of the deign ytem i iteative eidual coection methodology which utilize the invee poblem to let it poible to deign nonaiymmetic fuelage. The deign ytem wok well on SST noe deign poblem. The tun-aound time to complete a deign i hot. It ha been hown the ytem can handle aiymmetic noe a well a non-aiymmetic one. AIAA Jounal, Vol. 5, No. 6, pp , June 014. [3] H. W. Liepmann, A. Rohko. Element of g adynamic. John willey & on, Inc., (Chapte 9) [4] Pivate Communication with D. Kenji Yohida, The head of SST R&D goup in JAXA. [5] T, Kawai, N. Tokugawa et al. Lamina-tubulent tanition of noe bounday laye on upeonic tanpotation. Poc. Annual meeting of Japan ociety of fluid mechanic, p , July 009. (in Japanee) [6] N. Tokugawa, J. Akatuka et. al. Suface peue weauement on aiymmetic bodie in Supeonic Flow. JAXA-RM , JAXA, ISSN , Mach. 01. [7] N. Tokugawa, T. Kawai et. al. Deign of natual lamina flow noe fo upeonic tanpot by wavy defomation. JAXA-RR , JAXA, ISSN , Dec Copyight Statement The autho confim that they, and/o thei company o oganization, hold copyight on all of the oiginal mateial included in thi pape. The autho alo confim that they have obtained pemiion, fom the copyight holde of any thid paty mateial included in thi pape, to publih it a pat of thei pape. The autho confim that they give pemiion, o have obtained pemiion fom the copyight holde of thi pape, fo the publication and ditibution of thi pape a pat of the ICAS 014 poceeding o a individual off-pint fom the poceeding. Acknowledgement The autho thank D. K. Yohida of JAXA fo hi honet advice. They alo appeciate M. A. Tomita, a fome tainee of JAXA and gaduate of Gakuhuin Univeity, fo hi uppot in conducting invee deign on compute. The Contact Autho Adde i kiam@eng.u-toyama.ac.jp Refeence [1] H. Ihikawa, K. Ohia, Y. Inoue, et al. Development upeonic natual lamina flow wing deign uing CFD-baed invee method. Poc. 8 th ICAS conge, Bibane Autalia, ICAS , Sep. 01. [] Y. Ueda, K. Yohida, K. Matuhima, H. Ihikawa. Supeonic natual-lamina-flow wing-deign concept at high-reynold-numbe condition. 5

6 Kia Matuhima, Ikki Yamamichi and Naoko Tokugawa Stat Baeline Fuelage Shape Z u v w θ v Fig. 1 Body of evolution and cylindical coodinate. Flow analyi by CFD Reult Cp Taget Cp Cp= Taget Cp Reult Cp no Cp < At Evey Meidian Plane Solve Invee Poblem Cp f & Update Plane Geomety f = f + f ye Conveged z θ θ=30 θ=60 y Get updated geomety togethe and Fom New Fuelage Shape Fig. 5 Deign ytem of iteative eidual coection method uing the invee poblem. Fig. Co-ectional plane. U Z Z=R() Minf Fig.3 Longitudinal (meidian) ection X Fig. 6 The FC noe hape in the 3D pace. Initial R Taget R Initial Cp Taget Cp ΔR (0 1) Fig. 4 Piecewie linea appoimation of the cuent (thin line) and updated (thick line) hape of a meidian geomety R(). Fig. 7 Initial and taget adiu R and Cp ditibution. 6

7 AERODYNAMIC DESIGN METHOD FOR SUPERSONIC SLENDER BODY Cuent R Taget R R(1 ) Cuent Cp Taget Cp Fig. 8 Cuent and taget adiu R and Cp ditibution afte 1 t iteation. Fig.1 Invee deign eult fo quai aiymmetic hape at Cuent R Taget R Cuent Cp Taget Cp Fig. 9 Conveged adiu R and Cp ditibution afte 5 th iteation. Fig. 13 Invee deign eult fo quai aiymmetic hape at. Initial CP Fig. 10 Taget Cp fo quai ai-ymmetic hape deign with initial Cp of a cone. Fig. 14 Longitudinal plane of a deigned noe compaed with Sea-Haak and cone hape. Fig. 11 Invee deign eult fo quai aiymmetic hape at. Fig. 15 Co ection hape of a deigned noe compaed with Sea-Haak at =0.18 of