TOPICAL PROBLEMS OF FLUID MECHANICS 141
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1 TOPIL PROBLEMS OF FLUID MEHNIS 4 DOI: h://dx.do.org/.43/tpfm.6.9 BIPLNE ERODYNMIS REISITED E. Morsh ollege of Engneerng nd Desgn, Shur Insue of Tehnology, 37, Fuksku, Mnum-ku, Sm-sh, , Sm, Jn sr Blne s lssl erodynm rolem. In hs er, we frs nrodue well-known smle nlyl mehod sed on r of vores on eh rfol. Ths mehod gve rey ure resul n se of s smly. We hen revs Mory s onforml mng roh whh s n ex nlyl soluon. Ths mehod s n old one wh some mheml euy nd he resen uhor hnk h s worhwhle o ree here, euse ws wren n hs nve lnguge nd ossly no known o he res of he world. We fnlly use he modern dsree vorex mehod for omrson. The lf ro, lne o wo soled rfols, s oned s funon of h nd sgger. Keywords: erodynms, oenl flow, onforml mng, vorex mehod Inroduon Blne erodynms s lssl erodynm sue. I s one of he exmles of erodynm nerferene. We frs sudy he smles model of lne []. r of vores on eh qurer hord reles lne nd flow ngeny ondons ly o eh hree-qurer hord. The uwsh nd downwsh y he unform flow nd he r of vores nels eh oher he hree-qurer hord. We on he srengh of he wo rulons s funons of erofol h nd sgger. We evlue he momen round he referene on eween he wo rfols. Blne s lso sue of led mhems. Mory nlysed he lne rolem sed on he onforml mng mehod []. Tndem fl les n he -lne re rnsformed o lne n he z- lne y Shwrz-hrsoffel rnsformon. He lso found wy o nlude sgger n he rnsformon. Ths mehod gves n nlyl form of lf nerferene ro of lne relve o wo soled fl les. We here ree Mory s nlyss s referene heory. The hrd mehod s modern numerl one, whh models he rfols s vorex rry. Ths s he exenson of he r of vores nd we on del desron of he flow feld, whh s oherwse very dfful. We omre ove hree nlyl ools nd lrfy lne erodynms n del. r of vores Fgure show he smles model of lne wh h nd sgger ngle n he unform flow U wh n k of ngle []. The vores nd loe on he qurer hord nd he onrol ons nd B on hree-qurer hord, resevely. s h Fgure : r of vores lne B
2 The flow ngeny ondons nd B eome s follow, resevely. sn os ( sn os ( where nd re dsnes, nd nd re ngles shown n Fg., resevely. We on he nerferene ro r from Eqs. ( nd ( s follows. r os os os os (3 where sn U s he rulon of n soled erofol nd os os os sn (4 os os os sn (5 sn (6 sn (7 os (8 os (9 The neron eween wo rulons nd lso rodue erodynms fores s n Fg. whh ndenlly ree no momen ou he md on eween he wo rulon euse he nerve reulsve fores le on he lne. Fgure revels he reson why he lower lf lwys redues[]. The erodynm momen M ou he md on eween nd eomes ( ( ρ ρ sn sn M The momen oeffen m s defned s follows. ( ρ sn sn os os os os M m ( Equon ( s no only funons of h nd sgger, u lso funon of n ngle of k. 4 Prgue, Ferury -, 6
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3 TOPIL PROBLEMS OF FLUID MEHNIS 43 ρ y ρ.9.8 ρ.7 r.6 x 6> [deg.]> >(/>.5 ρ Fgure3: Blne lf ro r Fgure : Blne erodynm fores /( >(/>.5 < [deg.]<6 Fgure 4: Lower erofol lodng Fgure 3 shows he lne lf ro o h of wo soled fl le rfols r n Eq.(3. For gven sgger, he lf ro monoonlly nreses u o one s h nreses. Fgure 4 show he lower erofol rulon relve o he ol lf. I s neresng o ree h he lower erofol lf s lwys smller hn h of he uer erofol [], nd from Eqs.(4 nd (5, kes mnmum vlue rulr h (see doed lne n Fg.4 for gven sgger ngle s follows. ( mn os os os ( [deg.] /->nfny..5 m m [deg.] /4..5.5<(/<4 -.5 < <6[deg.] Fgure:5 Momen oeffen [deg.]
4 Prgue, Ferury -, 6 Fgure 5 shows he momen oeffen. The momen oeffen nreses s h nreses for m onsn sgger nd eomes onsn he nfny. From Eq.(, he momen oeffen s lwys zero when. The momen oeffen m lso eomes lwys zero when here s no sgger,.e.,. From Eq.( wh Eqs.(6~(9, we n show lm m sn sn sn( ( The momen oeffen m even / 4 s lredy very lose o he lmng vlue of Eq.(, see Fg.5. In Fgs.3,4,nd 5, we show he resuls for /. 5. lhough he numerl soluon s ossle for < / <.5, soluons er nrore erodynmlly, nd herefore we exlude he regon. 3 Mory s onforml mng mehod Mory suded lne erodynms dedes go y he onforml mng mehod,.e., Shwrz- hrsoffel rnsformon []. Two fl les n he -lne rnsform o lne n he z-lne n Fg.6. We negre Shwrz-hrsoffel rnsformon of Eq. (3 long he rel xs of he -lne shown n Fg. 6. dz λ ( ( d k where nd re onsns, nd we n derve followng equons []. ( ( k m m (3 (4 k k h K( (5 E( λ K( (6 x k k (7 ζ (8 [ E( ζ, λ K( ζ ] x λ d, ( ( k [ E( ζ, λ K( ] l λ. d ζ x, xl x E( ζ, E( ζ, λ [ K( ζ, K( ζ, ] ( ( s xl x. E( ζ, E( ζ, λ [ K( ζ, K(, ] ( ( -lne - - -λ -k k λ z-lne xl y x s h (9 ( x Fgure 6: onforml mng
5 TOPIL PROBLEMS OF FLUID MEHNIS 45 (, ( 4 ( k λ ± ( k 4 ( λ ( k λ ( where h s lne g, s s sgger, ( < λ nd ( > λ re he soluons of Eq.(3, K ( k K ( z, re he fs omlee nd nomlee ell negrls, E ( nd E ( z (3 nd, re he seond omlee nd nomlee ell negrls, resevely, xl nd x reresen ledng nd rlng edge oordnes of he uer rfol, resevely, nd xk s he x -oordne of he uer rfol when we negre Eq.(3 u o k from. The onsn s sle for deermned from he hord lengh. We frs sefy k, nd, h nd λ n e deermned from Eqs.(5 nd (6. We hen sefy nd ge he res of he vrles n Eqs. (8~(3. onversely when we frs sefy rulr se of g h nd sgger s, erons re neessry o fnd roer vlued for k nd. The omlex onuge veloy n he orgnl -lne s gven y dw λ µ os sn (4 d k k where s unform veloy, s n ngle of k, nd s ol rulon of he wo fl le rfols, nd µ s rel onsn. The seond nd he hrd erms of he rgh hnd sde of Eq.(4 ke forms o ssfy he surfe nd he nfny oundry ondons roerly. The omlex onuge veloy n he hysl z -lne eomes dw λ µ os sn dw d k k (5 dz dz ( ( λ d k When we se, dw os sn e (6 dz K[ ] os K ( os sn (7 sn K ( sn os (8 K (9 sn os n (3 os sn he rlng edges of he lne where nd, he denomnor n Eq.(5 s zero, nd herefore numeror hese ons lso mus e zero. onsderng he veloy dreon n he -lne, hese ondons re os k sn ( λ ( µ (3 os k sn ( λ ( µ (3 From Eqs.(3 nd (3 wh Eqs.(7 nd (8, we ge ( k k K sn (33 µ λ µ (34 ( The rulon of n soled fl le s sn, nd, herefore lf ro r, lne o wo soled rfols, eomes s follows.
6 46 Prgue, Ferury -, 6 r K ( Fgure 7 shows he lf ro r of lne of Eq.(35 s funon of he erofol g h / ( os / wh he rmeer sgger ngle (f. Fg.3. The lf ro r nreses owrd s he g nreses. The lf ro r lso nreses s he sgger ngle nreses. Ths mehod hs dffuly when k rohes zero nd he resuls for smller g ro re no shown. Fgure 8 shows he erofol surfe veloy of Eq.(5. re mus e ken for he sgns of veloy omonen n numerors nd he denomnor o lule veloy euse he seond nd he hrd erms of he rgh hnd sde of Eq.(5 hnges he sngs deends on he r of he surfe. The numerl resuls he sme ondon s lso shown n Fg.8 for omrson. Fgure 8 vldes he Mory s mehod nd/or Mory s mehod ssures he vldy of he numerl vorex mehod. Plo ons re mssng for he onforml mng n he enrl regon of Fg.8 euse even leds o uneven x, see Eq.(3..9 k k [deg] (35 r h/ Fgure 7: Blne lf ro r 3 vorex mehod N6 onforml mng uer erofol lower erofol - x/(/ - o Fgure 8: onforml mng vs numerl vorex mehod o,,.79 ( k., h
7 TOPIL PROBLEMS OF FLUID MEHNIS 47 4 orex nel mehod smle numerl mehod s he vorex nel mehod shown n Fg.9. Eh fl le hs N nels of equl lengh. Srengh of he wo vores on he eh rlng edges re zero o ssfy Ku ondon. The omlex oenl w eomes N w( z ze ln( z z (36 where z ( N for he uer erofol, nd he res of he vores on he lower erofol. nd N orresond he ledng edge vores. We ge veloy omonens s follows. N ( y y u os (37 x x y y N ( ( ( x x ( x x ( y y v sn (38 Flow ngeny ondons he mddle of he nel ( x, y ( N equons s follow. where N, ( (, N led o he N smulneous sn (39 ( x x os ( y y ( x x ( y y sn, reresens he nlnon of nel ( N, whh s zero degree n hs smle seng. y (4 s h x Fgure 9: orex nel γ Fgure : Pnel surfe model γ ( x, y γ Fgure : Surfe veloy dsonnuy
8 48 Prgue, Ferury -, 6 We hen lule nel surfe veloy dsonnuy. orex dsruon γ long he nel of lengh l ssfes he followng equon where γ s he nel verge, see Fg.. l γ l γ dl (4 We my on he suon( nd he ressure (-surfe veloes s follow(fg.. γ ± (4 where s he md nel veloy ( x, y ( N. Fgure show lf ro r y he numerl vorex mehod, lne o wo soled rfols gns g ro h /, ogeher wh he onforml mng resuls for omrson, see Fg.7. The numerl nd he onforml mng mehods oh gve lmos he sme vlues. The vorex mehod mgh e lle very lose o zero g, u he numerl resuls gn nrore very lose o zero g nd herefore no shown n Fg.. Fgure 3 lso show he lne lf ro r y r of vores for omrson. lhough he r vorex mehod s very rude, erodynms of lne s surrsngly well rerodued. Fgure 4 shows he lodng for of he lower erofol y he vorex mehod ogeher wh h of he r vorex n Fg.3 for omrson. The lower erofol lod kes mnmum vlues h ro /. The sme s rue for he r vorex model, Eq.( nd Fg. 4. When he g nd/or h nreses, he erofol lod rohes he even ondons. Fgure 5 shows he momen oeffen round he on n Fg.. For he gven ondons m o o nd 3, he numerl vorex mehod nd he r vorex model oh gve he smlr rend lhough he vorex mehod shows slghly hgher vlues, see lso Fg.4. Fgure : Blne lf ro r (see Fg.7 Fgure 3: Blne lf ro r (see Fg.3
9 TOPIL PROBLEMS OF FLUID MEHNIS 49 Fgure 4: Lodng for m. vorex mehod N6 r of vores deg. 3 deg / Fgure 5: Momen oeffen ( Fgure6: Pressure /, 3o, o ( Fgure 7: Sremlne /, 3o, o
10 5 Prgue, Ferury -, 6 We lule he momen onruon M y vorex ( x, y ( N round he referene on, ( x y ( / 4,,, s follows. M x Fy y Fx 4 F v F x y u where u nd x, y,resevely. Fgures 6 nd 7 show ressure nd sremlnes y he vorex mehod, resevely. I s que ler h he lower erofol lod s smller hn h of he uer erofol. We use sredshee exensvely o ge hese resuls, see [4],[5] nd [6]. v re he x- nd y-veloy omonens ndued ( 5 onludng remrks We revs lne erodynms y lssl oenl mehod. We frs ly r of vores o model lne. Ths mehod gves very ler erodynm hrerss n se of s smly. We on lf, erodynm lod ro nd momen nlylly s funon of h nd sgger. I mgh e nrore o ly he mehod elow / <. 5. We nrodue Mory s mehod nd ree here for referene. We ly he onforml mng y Shwrz-hrsoffel rnsformon o wo ndem fl les o ree lne. From Ku ondon wo rlng edges, we n deermne unknown rulon, nd herey lne lfs. Mory derved he lf ro s n ml funon of g nd sgger lhough he lf equon self hs n exl form. There re some dffules n hs heorel mehod on he nerreon of he sgns n he equons nd he numerl negron of he oordnes. We fnlly ly he numerl vorex mehod o lne nd he greemen wh he onforml mng mehod s que ssfory. We on lf, uer nd lower erofol lod nd momen. The momen oeffen gves smlr rend o h of he r vorex mehod, lhough he numerl vorex mehod shows slghly hgher vlues. lhough we nrodue severl led mheml nd numerl mehods n hs er, we n use hese desgn ehnques o novel wng n ground effe where we refer smller wng lnforms, see [7] nd [8]. Referenes [] Mlne-Thomson L.M.: Theorel erodynms, Dover, New York :( [] Mory, T.: Inroduon o erodynms (n Jnese, Bfu-kn, Tokyo :(97.-. [3] Grrk, I.E.: Poenl flow ou rrry lne wng seons, N Re. No.54 :(936. [4] Morsh, E.: Sredshee flud dynms. J. rrf, vol. 36, no. 4: ( [5] Morsh, E.: Sredshee flud dynms for eronul ourse rolems. In. J. Engng Ed., vol.7, no. 3 :( [6] Morsh, E.: Sredshee flud dynms of lun ody rolem. JSME In. J. B, vol.45, no.4 :( [7] Shuy, T., Ishdo, T. nd Morsh, E.: flyer- novel wng n ground-effe vehle, PIST3, Tkmsu :(3 D [8] Morsh, E. nd Shuy, T.: Tron flgh vehle, Pen lon :(4JP49646 (. (43
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