A Numerical Lifting-Line Method Using Horseshoe Vortex Sheets. Douglas F. Hunsaker Utah State University

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1 A Nme Lftg-Le Method Ug Hoehoe otex Sheet Dog F. He Uth Stte Uety A me method ed o the og ftge theoy of Pdt deeoped whh de the fee of hoehoe otex heet. The method ttempt t deeopg hghe-ode method th peo det-fto method of the me type. The defto of hoehoe otex heet gty toded d the eoty ded t ty pot pe y the gty deeoped. No oed-fom oto fo th ded eoty w fod fo pot ot oe wth the od poto of the gty. Addtoy, the eoty ded og the od poto of hoehoe otex heet wth weep detemte. The gty w ed to deeop me method pe of pedtg the eodym foe d momet o ytem of ftg fe. The method ge et wth the y of othe m method, t eqe hghe gd efemet d moe omptto th peo method. P I. Itodto RANDTL S ftg-e theoy, h hd pofod effet o eodym p to th pot tme. I the deeopmet of h theoy, Pdt eted the two-dmeo fo ft t eh eto of the wg to the o to thogh the twodmeo otex ftg w 3,4. Th method h ee fod to e ey te fo wg wth pet to gete th ot 4. Howee, t mted t ppto to wg wth o weep d o dhed. Addtoy, Pdt deeopmet doe ot ot fo the teto of mtpe ftg fe. See me method ooted th og theoy e fod the tete 5,6,7. Thee method he ee deeoped to exted the og theoet of Pdt theoy to wg wth weep d dhed d to mode the teto of mtpe ftg fe. Howee, ome of the mot ommo method ot e odeed det me ogy to the og theoy of Pdt. Pehp the mot we-ow me method tht ft to th tegoy tht peeted y Ktz d Pot 7. Athogh the method peeted y Ktz d Pot ofte efeed to the ftg-e method, t od moe ppoptey e efeed to otex tte method pped g oy ge eemet the hodwe deto. Th tte my e moe ppopte ee the th og the ytem of eqto y etg the to t eh eto to the two-dmeo eto ft, og ody odto mpemeted eqg the om eoty t the thee-qte hod to e zeo. Php d Syde 8 peeted me eqet of Pdt og ftg-e method whh e ed to ete the fow ot wg wth weep d dhed. Th deeopmet moe goo eqet to Pdt og theoy ee the etohp etwee the two-dmeo fo ft t eh eto of the wg d the thee-dmeo otex ftg w 9 ed to ot the o to t eh wg eto. Th method how pomg potet fo oth the ft dty d the wd d wte te dte whee ftg fe oetoy he pet to gete th 4. Addtoy, the method ey e exteded to ow fo tg otex moemet d op ehd the ftg fe d th ot fo oth oottg d ottg fow. Aodg to Php d Syde, th method tey pedt the ft oeffet fo wg wth weep d dhed d eqe oy fto of the ot of oeto omptto fd dym CFD method to ot gd-oeged et. I tdto me ftg-e method, ytem of ftg fe ythezed y detzg the wg to ee of hoehoe ote, eh hg tegth eqet to the od otty t the oto of the eto. Th oeet ee odg to the Hemhoz otex ftg w, otty ot e poded potet fowfed, d otex ot eg o ed the fowfed. Theefoe, hoehoe ote e ofte ed whh exted fom fte dte dowtem, o od poto of the wg, d the fte dte dowtem. Bee eh hoehoe otex h dete e d the otty t eey othe pot the fow ot oped y egmet of the otex zeo, th ppoh ete otty fed ythezed y ee of det fto. Othe det fto type method de the otex g method eqet to the doet pe method fo 3D fow. Thee ote e ged g of ott tegth whh tfe the Hemhoz otex w. A me method ed o det fto gte ofte efeed to zeo th -ode method.

2 A ft-ode method e deeoped y epg the dete hoehoe ote wth hoehoe otex heet of ott tegth. Sh ppoh ow otty to otoy e hed fom the wg, the th detey the e wth tdto method. Th ppe peet h method. II. The Hoehoe otex Sheet A hoehoe otex heet ompoed of two emfte tg otex egmet oeted y oe od otex egmet of fte egth d oe emfte otex heet how Fg.. The emfte otex egmet he tegth d tht gee w dffe fom eh othe t e ott og the epete egth. They e oeted y the od otex egmet whh e ey tegth og t egth egg t d edg t. Thee thee otex egmet fom the ote of tg otex heet of ott tegth. The eoty ded t ty pot P pe y the ete hoehoe otex heet the m of the eoty ded y the two tg otex egmet, the od otex egmet, d the otex heet t Hee t the eoty ded y the two tg otex egmet, the eoty ded y the od otex egmet, d the eoty ded y the tg otex heet. P t B. otex Le Segmet wth Ley yg Stegth The od poto of the hoehoe otex heet otex e egmet wth ey yg tegth egg t pot P wth tegth d edg t pot P wth otex tegth. The geomety fo th otex e egmet how Fg.. The dffeet eoty ded y dffeet deto otex egmet of egth d d d 3 3 Itodg dmeoe dte og the e egmet ge the hge of e d d 4 The e of the geomety ge the eto ψ P d P, P P, P, P, Fge. Poto eto howg the geomety fo otex e egmet wth yg tegth. P γ 5 d d d 6 Fge. Poto eto d geomety fo hoehoe otex heet. A. Sem-Ifte otex Segmet of Cott Stegth The hoehoe otex heet de two emfte tg ote tht exted fom pot P d P to fty the deto of the feetem. The eoty ded y the tg otex egmet deeoped y Php d Syde 8 d e wtte 3 3/ 7 d the o otex fmet tegth e fto whh e expeed 8 Ug Eq. 4 8 Eq. 3 ge d [ ] d d 3 3 / The tot eoty ded y the ete e egmet 9

3 3 4 π Th e tegted to yed [ 3 / d [ ] [ ] / ] [ ] By defto of the geomety, whh ge the eto 3 4 Ug Eq. 4 Eq. yed 5 Appyg tgoomet dette, the etohp e deeoped o ψ ψ 6 Ug Eq. 6 Eq. 5 d mpfyg yed ef expeo fo the eoty ded y otex e egmet wth ey yg tegth 7 4 π Note tht th expeo oy detemte whe the ge etwee d ± π. It ot detemte whe the ge etwee d zeo. C. Sem-Ifte otex Sheet wth Cott Stegth The od d tg egmet of the hoehoe otex heet fom the ote of the tg otex heet. The geomety fo em-fte otex heet wth ott tegth how Fg. 3. ψ P P d Fge 3. Poto eto howg the geomety fo em-fte otex heet wth ott tegth. P The eoty ded t pot y ge emfte otex ge y Php d Syde 8 γ 8 whee the eto fom the egg pot of the em-fte otex to the pot of teet. A emfte otex heet ompoed of oto heet of em-fte ote egg og the eto d extedg ftey og the t eto d hg tegth pe t egth og of γ. A dffeet egmet of the otex heet hg egth of d the deto of de dffeet eoty of γ d d 9 Ag, defg dmeoe dte og the e egmet ge the hge of e d d Ug the geomety defto ge whh ge the eto / Ug Eq. Eq. 9 ge γ d d 3 [ ] The teg of Eq. 3 ge the tot ded eoty fom em-fte otex heet of ott tegth

4 4 [ γ d 4 ] The tegth of the otex heet eted to the hge otty o the od poto of the hoehoe otex heet odg to Ug Eq. 5 Eq. 4 ge whee d γ 5 d 6 4 π [ d 7 ] the deto eto fo eoty ded y the otex heet t the pot of teet d ge Eq.. I gee, Eq. 7 mt e mey tegted ode to ete the fee of the otex heet o ty pot of teet. Howee, f the pot of teet e og the me e, th teg e eted dffeety how the foowg two e.. Ce : A Coe Pot Not Codet wth the otex Sheet Fo the pe e of ξ, Eq. 7 e yty tegted to yed [ ] [ [ d ξ ξ, ] ξ ξ, ] ξ ξ < ξ > 8 Note tht th oto oy good fo ξ < o ξ >. I othe wod, th oto oy d fo pot ot dety o the otex heet. We ee fom th et tht f the pot of teet e og t ot dety o the e egmet tef, the teg yed mpe expeo. Howee, f the pot of teet e dety o the e egmet, th yt expeo ot e ed. Ao ote tht th expeo d depedet of the oetto of the otex heet ete to the feetem. Theefoe, th expeo e ed fo wg wth weep d dhed.. Ce : A Coe Pot Codet wth the otex Sheet See ttempt wee mde to deeop expeo fo the e of ξ whee ξ. Th the e fo the eoty ded t pot yg dety o the od poto of otex heet y the otex heet tef. The mot pomg ttempt, howee, me fom me et. Ug foth-ode Rge-Ktt tegto, Eq. 7 w eted o the ge of ξ fo ee otex heet weep ge eh wth t egth,. The oy modfto to the tdd tegto ote w tht the de of the tegd w goed whe ξ. Th tfed o the god tht otex egmet doe ot de y eoty t y pot og t egth. The teg w eted g oe, medm, fe, d ey fe tegto tep ze of /, /, /4, d /8 epetey. Fge 4 how the tegto et fo the mgtde of Eq. 7 fto of ξ fo weep ge of -.,.,., d 5. degee. The ge ymo epeet the et of the oe tep ze, the ext get epeet the et of the medm tep ze, d o dow to the m ymo whh epeet the et of the ey fe tep ze. Note tht the et oege oy fo the e of o weep. Th how tht f ot peped to, the ded eoty doe ot oege. Th me ee ee fo the e of o weep, t ge poto, ξ, the fee fom the otex egmet t d d d w pefety e. Howee, f ot peped to, the fee fom the otex egmet t d d d w ot pefety e. A the me tegto efed, the eoty ded og the od poto of otex heet wth weep ppohe fty. Ao ote fom Fg. 4 tht the eoty ded t the oe pot of the otex heet e detemte egde of the weep ge deg 6. deg deg 5 deg ξ Fge 4. Mgtde of ded eoty og the od poto of otex heet.

5 5 We ee fom the et peeted hee, tht the fee of otex heet oy e eted og the od poto of the otex heet f the heet peped to the feetem eoty. Theefoe, the et of the ftg-e gothm ded th wo oy e expeted to e te fo wg wth o weep d dhed d t zeo dep ge. A yt ppoxmto tht e ed pe of Eq. 7 fo the fee of otex heet wth o weep o dhed og the od poto of the heet.787 [ h.4o ] π ξ < < ξ 9 Fge 5 how the mgtde of fto of ξ fo otex heet of egth. eted y mey tegtg Eq. 7 g gd-eoed tep ze of /8. The yt et of Eq. 8 d the ppoxmto ge Eq. 9 e o ded fo ompo. The et how tht Eq. 8 d fo ξ < o ξ > fo y weep ge, d Eq. 9 ppopte fo ξ whe the weep ge zeo. whee e eted fom ethe Eq. 7, 8, o 9 depedg o the e. III. Agothm Fomto A ytem of ftg fe ythezed y detzg the ftg fe to ee of hoehoe otex heet how Fg. 6. A ee of hoehoe otex heet ped og the wg, the ght oe of oe heet odet wth the eft oe of the ext heet. The od egmet of eh hoehoe otex heet ped og the o qte-hod, d theefoe ged wth the o weep d dhed. The tg ote d heet e ged wth the feetem eoty eto, the eft-oe tg otex extedg fom fty to the qte-hod, d the ght-hd tg otex extedg fom the qte-hod to fty. Bee the tg ote odet t oetg hoehoe otex heet e oppote deto, f the tegth t the ght oe of oe hoehoe otex heet det to the tegth t the eft oe of the ext hoehoe otex heet, they e exty d the et hed otty t tht pot zeo exept fo the otty hed thogh the tg otex heet...5 Eq. 8 Eq. 9 Eq deg deg.5 deg deg 45 deg ξ Fge 5. Mgtde of ded eoty og pot oe wth the od poto of the otex heet. D. The Compete Hoehoe otex Sheet The eoty ded t ty pot y ompete hoehoe otex heet the m of Eq., 7, d 6 3 Fge 6. Hoehoe otex heet dtto og qte hod of ftg fe. Bee the otex heet h ott tegth, the otty og the od poto of the hoehoe otex heet mt y ey. I ode to pefy the e hge od otty, two oto pot e pefed og the od otex. Thee e deoted P d P d how Fg.. Hee we defe the ete poto of the oto pot to the ed pot of the hoehoe otex heet odg to P P P P 3

6 6 Coe teg 8 ed fo the detzto of the wg to ee of hoehoe otex heet. Addtoy, the oto pot e oe teed h tht eh hoehoe otex heet ot two oto pot, d eh oto pot eey ped the oe dom ete to the othe oto pot. The tegth of the od otex t the oe P d P e eted to the tegth of the od otex t oto pot P d P odg to 3 Ug Eq. 3 Eq. 3 d egg ge 33 whee 4 4 π π 34 Fo eh hoehoe otex heet, thee e two ow, d, whh epeet the tegth of the otex t the oto pot of the hoehoe otex heet. Thee otex tegth e ot ow po d mt e fod y ppyg ome otg odto tht w et ytem of eqto tht e oed fo the otex tegth. If the ytem of ftg fe detzed to hoehoe otex heet, th ytem w he ow. Theefoe, ytem of eqto mt e deeoped to oe fo the ow. Th omphed thogh the ppto of Pdt ftg-e hypothe t the two oto pot ped og the od poto of eh hoehoe otex heet. Pdt ftg-e hypothe ed o the mpto tht eh pwe egmet of the ftg fe h o ft oeffet eq to twodmeo egmet of fte wg t the me ge of tt. The otex ftg w pped to dffeet egmet of ftg fe ge d df 35 Th eqto e pped twe to eh hoehoe otex heet, oe t eh oto pot of the heet. Thee two pot o the hoehoe otex heet w e efeed to the odd d ee oto pot. Bee thee e hoehoe otex heet, thee e oto pot. The o eoty t y oto pot,, the m of the feetem eoty d the eoty ded y eh otex tegth, 36 whee ee,, odd,, 37 the odmeo fee eto of the otex tegth, whh pt of the hoehoe otex heet, o oto pot, / ] [ 4 4 π π d 38 the eto fom oe of the hoehoe otex heet to oto pot, the eto fom oe of the hoehoe otex heet to oto pot, d the hod egth t oto pot. Comg Eq. 36 d 35, the dffeet foe t oto pot,, e wtte

7 7 df d 39 Ug Pdt ftg e theoy, the mgtde of th dffeet foe mt e eq to the mgtde of the two-dmeo dffeet egmet t the me o ge of tt df CL, δ d 4 whee the o ge of tt t 4 d the o eoty t oto pot 4 Eqtg the mgtde of Eq. 39 d 4 ge d CL, δ d 43 The dffeet egth og the od otex d eted to the dffeet egmet p d thogh the o weep ge Λ odg to the eto d o Λ 44 d Defg the dmeoe pmete d, oλ d, 45 the fomto e wtte odmeo fom, CL δ 46 d the o ge of tt e wtte t 47 whee the odmeo o eoty t oto pot 48 Appyg Eq. 46 t eh of the oto pot pode oe depedet eqto. We wh to oe th ytem of eqto g Newto method. The ytem of eqto e wtte ompt fom of whee F R 49 F CL, δ 5 d R the eto of ed whh epeet the eo of the oto. The hge the eto of otex tegth,, whh w deee the eto of ed e ompted fom the ytem [ J] R 5 whee [J] the Jo mtx of pt dete. The Jo mtx e yty oted d wtte J F w δ w w C L whee δ the Koee det d w 5 53 Th oto method geey oege ey pdy to the oto. Howee, t eqe t ge fo the otex tegth. A t ge e oted y ppyg the m ge ppoxmto to the ytem of eqto d ezg the ytem. Fo m ge of tt, the eto ft oeffet e expeed CL CL L εδ 54 The ge of tt t y oto pot the m of the feetem ge of tt d the ded ge of tt

8 8 d, t t 55 Ug the m ge ppoxmto fo the ded ge of tt ge t ge fo the o ge of tt t 56 Ug Eq. 56 Eq. 54 ge L L L C C δ ε t 57 Ug th expeo fo the ft oeffet Eq. 46 d doppg eod-ode tem ge e ppoxmto fo the oto L L L C C δ ε t 58 whh ey e oed fo e t ge to the oe ytem of eqto. I. Aeodym Foe d Momet Oe the otex tegth e ow, the tot eodym foe d momet o the ft e fod y mmg the eodym foe d momet otted y eh hoehoe otex heet. The tot eodym foe the mmto of the eodym foe fom eh hoehoe otex heet S f F 59 d the tot eodym momet the mmto of the eodym momet fom eh hoehoe otex heet S L m M 6 I the deeopmet of th method, we he med tht the otty e ey og the od egmet of the hoehoe otex heet. If we o me tht the ded eoty og the heet e ey, the o otty d o eoty og the od egmet of the hoehoe otex heet e wtte 6 6 Ug thee expeo Eq. 35 ge expeo fo the dffeet foe eto og dffeet egmet of the od otex egmet d df ] ][ [ 63 Appyg the hge of e ge Eq. 4 to Eq. 63 d tegtg og the egth of the od otex egmet of hoehoe otex ge d } ] [ { ] ][ [ 3 F 64 The tegth of the od otex t the oe P d P eted to the tegth of the od otex t oto pot P d P how Eq. 9. Lewe, the eote t thee pot e eted odg to 65 Ug Eq. 3 d 65 Eq. 64 ge the eodym foe fom oe hoehoe otex heet. I odmeo fom, t e wtte ] [ o 3 S S F f Λ 66

9 9 whee Eqto 66 e ed Eq. 59 to te the et eodym foe o the ft. The dffeet eodym momet ed y the dffeet foe fom dffeet egth og the od otex of hoehoe otex heet ot the ete of gty e wtte the foe oed wth the dte of the foe fom the ete of gty, C dm C d 69 Ag mg tht the otty d eoty y ey og the od otex doe the dte to the ete of gty d ppyg the hge of e ge Eq. 4 dm [ C { [ C C ] ][ 7 ] d } Th e tegted d the et ed Eq. 6. If th wo ped fthe, th expeo w e oted d epoted fte pto. Howee, t doe ot ffet the et ded th wo d w ot e ded hee.. Ret The wo peeted hee pode two method fo etg the fee of otex heet o oto pot oted og the od poto of the hoehoe otex heet. Bee the fee detemte whe the heet h ome ompoet of weep, ethe the heet fee o oto pot oted og the od otex mt e goed, o the heet fee e med to e the fee of m heet hg o weep o dhed. I the et tht foow, thoe et tted Wth Sheet de the effet of heet wth o weep d dhed ge Eq. 9. Thoe et tted Wthot Sheet e eqet to ettg. whe the fee of heet mt e eted t pot oted og the od poto of the heet. Ug Pdt ftg-e theoy, oed-fom oto fo ft o ept wg e oted. Th pode good ddte fo eg the y of me ftg-e method. A ept wg how Fg. 7 of pet to 3/π wth ft ope of π w ed fo ompo of the ftge method. Fg. 8 how the ft oeffet fo the ept wg t 5 degee ge of tt pedted y the two method oted fom th wo we the wo y Php d Syde fto of the me of eemet ed to detzed the wg. Note tht thee method ppe to oege to the oet oto. Howee, the method of Php d Syde gee h e eo th the method ed th wo. C L Fge 7. Ept wg of pet to 3/π Wth Sheet.45 Wthot Sheet Php d Syde CoedFom Soto.448 Fge 8. Lft Coeffet fto of me of detzto eemet fo ept wg wth pet to 3/π. Fge 9 how the me dt tht Fge 8, t potted to how the peet eo etwee the ext oto d tht pedted y eh of the method. Note tht the method of Php d Syde doe ft he gfty owe eo th the othe method fo ow gd eoto. Howee, the method of Php d Syde doe ot fy oege to the oet oto. Fom the dt peeted hee, t ppe tht oe o oth of the method deeoped th wo my fy oege to the oet oto, thogh the method wod eed to e t hghe gd eoto to ofm th.

10 C L % Eo 3 Wth Sheet Wthot Sheet Php d Syde Fge 9. Peet eo of the ft oeffet fto of gd eoto. A f ommet o Fg. 9, ote tht the ode of oegee of the hoehoe otex heet method doe ot eem to offe y mpoemet to the ode of oegee of the method of Php d Syde. If the ode of oegee wee mpoemet, the ope of the dt Fg. 9 fom the otex heet method wod e gete th the ope of the dt fom the Php d Syde method. The eho of the method fo wg wth weep o of teet ee Pdt og ftg-e method ot e exteded to wg wth weep o dhed. The method wee ed to pedt the ft o etg wg of pet to 8 hg 45 degee of weep how Fge. to hge wht ppe to e e me o og-e pot. Ogy t w med tht th w de to the ft-ode dotty the ftg e t oe the pe of ymmety of the wg. Howee, the modfto ggeted y Khem emte th poty ee the ft dete of the ftg e fo th modfto oto o the wg pe of ymmety. Fom the et Fg., t mt e med tht thee omethg the deeopmet of thee method tht poht gd efemet fo wg wth weep. Th poe top of fte eeh. C L Wth Sheet Wthot Sheet Php d Syde Khem Fge. Lft oeffet fo wg of pet to of 8 wth 45 degee of weep fto of gd eoto. Fge. Retg wg wth pet to of 8 d 45 degee of weep. Fge how the et of the ft oeffet pedted fo th wg t 5 degee ge of tt. Ao ded e et pedted y the method of Php d Syde, t wth to the pemet of the ftg e ed o modfto ggeted y Khem. A epeette ftge dtto ed o the Khem modfto ded Fg.. Note fom Fg. tht oe of thee method oege to oto f the wg h weep. A the gd efed, the oto ote Fge. Lftg-e dtto ed o the wo of Khem howg ft-ode otty t wg pe of ymmety. A f emet of the y of the method exmed hee, ytem of tetg ftg fe w odeed. Th ytem of ftg fe ded wg wth weep d dhed d oto pot pe em-p wee ed fo the to. Fge 3 how the to dtto og the p of the ftg fe fo the Wth Sheet e. Sm dtto wee oted fo the Wthot Sheet e we the Php d Syde method.

11 The getet dffeee the tot ft oeffet pedted fom the thee me mode w.6%. Bee eodym oeffet geey ot e meed to tht y, the me method peeted th ppe pode deg too tht e expeted to ge et to wth expemet y of thoe pedted y the method of Php d Syde. Fge 3. Cto dtto fo ytem of tetg ftg fe. I. Coo The defto of hoehoe otex heet gty deeoped, d me ftg-e method fo etg the eodym foe d momet o ytem of ftg fe ed o th gty peeted. The eoty ded t ty pot pe y the gty e eted fom Eq. 3 oto wth ethe Eq. 7, 8, o 9 depedg o the poto of the pot of teet. It w fod tht the eoty ded t pot og the od poto of the hoehoe otex heet wth weep g. Two method of metg th dffty wee ggeted whh ded etg the ded eoty og the od poto thogh the heet wee wept, o ettg the ded eoty og the od poto to zeo. I gee, the me method poded et tht gee ey oey wth et ed o the method of Php d Syde 8. Oe dtge of the method e ee the eto of the eodym popete of wg wth o weep d o dhed. Fo wg wth o weep o dhed, the method poded et tht gd eoe to te oto whh my e moe te th thoe of Php d Syde. Fo wg wth weep d/o dhed, the method doe ot pode gd-eoed et whh w o fod to e the e wth the method of Php d Syde. The et how tht the otex heet method peeted hee h ppet dw. Ft, the method ppeed to eqe hghe gd eoto to mth the y of peo method. Seod, t dd ot eem to mpoe the ode of oegee oe zeo th -ode method. Fy, etg the fee of otex heet o ty pot pe geey moe ompttoy expee th othe method ed o ftg-e theoy ee th method eqe me tegto fo pot ot oe wth the od egmet of the hoehoe otex heet. Fte wo my de g the method to ete the eodym ete of wg wth weep d dhed. Commo me ftg-e method f to tey pedt the oto of the eodym ete of wg, d f the method peeted hee od mpoe h etmto, the eodym ete pedted y th method wod e of teet. The gte ed th wo do ot wee wth gd efemet. Th ey the oot poem tht gd eoed oto od ot e oted fo wg wth weep d/o dhed. Fdmety, fte eeh hod e foed o me ftg-e method tht wee gte wth gd efemet. Refeee Pdt, L., Tgfüge Theoe, Nhhte o de eehft de Wehefte z öttge, ehäefthe Mttege, Ke, 98, pp Pdt, L., Appto of Mode Hydodym to Aeot, NACA 6, Je 9. 3 Ktt, M. W., Afteäfte Stömede Fügete, Itete Aeothe Mttege, o. 6, 9, p Joow, N. E., S e Too Adot, Tx de Seto Phyqe de Soete Impee de Am de See Nte, o. 3, No., MCom, B. W., The Lftg Le Mode, Aeodym, Aeot d Fght Meh, d ed., Wey, New Yo, 995, pp Adeo, J. D., J., Cod, S., d We, D. M., Nme Lftg-Le Theoy Apped to Dooped Ledg-Edge Wg Beow d Aoe St, Jo of Aft, o. 7, No., 98, pp Ktz, J., d Pot, A., Lftg-Le Soto y Hoehoe Eemet, Low-Speed Aeodym, fom Wg Theoy to Pe Method, Mw-H, New Yo, 99, pp Php, W. F., d Syde, D. O., Mode Adpto of Pdt C Lftg-Le Theoy, Jo of Aft, o. 37, No. 4,, pp Sffm, P.., otex Foe d Bod otty, otex Dym, Cmdge U. Pe, Cmdge, Egd, U.K., 99, pp Khem, D., A Smpe Method fo Ctg the Sp d Chodwe Lodg o Stght d Swept Wg of y e Apet Rto t So Speed, Aeot Reeh Co, RM-935, Lodo, My, 956.