Aerodynamic Centers of Arbitrary Airfoils

Transcription

1 Utah State Uverst ll Graduate Theses ad ssertatos Graduate Studes -7 erodac eters of rbtrar rfols Orr ea Pope Utah State Uverst Follow ths ad addtoal works at: Part of the Mechacal Egeerg oos Recoeded tato Pope Orr ea "erodac eters of rbtrar rfols" (7. ll Graduate Theses ad ssertatos Ths Thess s brought to ou for free ad ope access b the Graduate Studes at gtaloos@usu. It has bee accepted for cluso ll Graduate Theses ad ssertatos b a authorzed adstrator of gtaloos@usu. For ore forato please cotact dla.burs@usu.edu.

2 EROYMI ETERS OF RBITRRY IRFOIS b Orr ea Pope thess subtted partal fulfllet of the requreets of the degree of MSTER OF SIEE Mechacal Egeerg pproved: ouglas Husaker Ph.. Major Professor Geord Rchards Ph.. ottee Meber Stepha. Whtore Ph.. ottee Meber Mark R. Mcella Ph.. Vce Presdet for Research ad ea of the School of Graduate Studes UTH STTE UIVERSITY oga Utah 7

3 oprght Orr ea Pope 7 ll Rghts Reserved

4 BSTRT erodac eters of rbtrar rfols b Orr ea Pope Master of Scece Utah State Uverst 7 Major Professor: ouglas Husaker Ph.. epartet: Mechacal ad erospace Egeerg ethod for accuratel predctg the aerodac ceter of a arfol s preseted based o a geeral for for the olear lft ad ptchg-oet of a arfol as a fucto of agle of attack. Ths ethod does ot suffer fro sall-agle sallcaber ad th-arfol approxatos ad s show to atch vscd results to uch hgher accurac tha the tradtoal ethods. It s show that the aerodac ceter of a arfol wth arbtrar aouts of thckess ad caber a vscd flow does ot geeral le at the quarter-chord. Rather t s a sgle deterstc pot depedet of agle of attack whch les at the quarter chord ol the lt as the arfol thckess ad caber approach zero. Furtherore t s show that oce vscous effects are cluded the aerodac ceter s ot geeral a sgle pot as predcted b tradtoal th arfol theor but s a fucto of agle of attack. ffereces betwee olear predctos ad those based o th arfol theor are o the order of -5% whch ca be sgfcat whe predctg arcraft stablt. (35 pages

5 v PUBI BSTRT erodac eters of rbtrar rfols Orr ea Pope The stud of desgg stable arcraft has bee wdespread ad ogog sce the earl das of Orvlle ad Wlbur Wrght ad ther faous Wrght Fler arplae. ll arcraft as the fl through the ar are subject to or chages the forces actg o the. The feld of arcraft stablt seeks to uderstad ad predct how arcraft wll respod to these chages forces ad to desg arcraft such that whe these forces chage the arcraft reas stable. The atheatcal equatos used to predct arcraft stablt rel o kowledge of the locato of the aerodac ceter the pot through whch aerodac forces act o a arcraft. The aerodac ceter of a arcraft s a fucto of the aerodac ceters of each dvdual wg ad the aerodac ceter of each wg s a fucto of the aerodac ceters of the dvdual arfols fro whch the wg s ade. The ablt to ore accuratel predct the locato of the arfol aerodac ceter correspods drectl to a crease the accurac of arcraft stablt calculatos. The erolab at Utah State Uverst has develop ew aaltc atheatcal expressos to descrbe the locato of the arfol aerodac ceter. These ew expressos do ot suffer fro a of the restrctos or approxatos foud tradtoal ethods ad therefore result ore accurate predctos of arfol aerodac ceters ad b exteso ore accurate arcraft stablt predctos.

6 v KOWEGMETS I would lke to specfcall thak r. oug Husaker for hs gudace support ad uedg passo for arcraft ad aeroautcs. I a also thakful to the Utah S Space Grat osortu (award X54H for fudg ths research. I gve specal thaks to colleagues freds ad especall fal who have alwas supported e throughout Master s progra ad through all of egeerg educato. Orr ea Pope

7 v OTETS Page BSTRT... PUBI BSTRT... v KOWEGMETS...v IST OF TBES... v IST OF FIGURES... x OTTIO...x HPTER ITROUTIO.... Research Motvato.... terature Revew..... Tradtoal Th rfol Theor Relatos for the erodac eter..... Geeral Relatos for the erodac eter...5 TERTIVE PPROH TO FIIG THE OTIO OF THE EROYMI ETER...8. The erodac eter as a Fucto of oeffcet of ft...8. The erodac eter as a Fucto of oral-force oeffcet..... Equvalece Proof Saple Results Vortex Pael Method Fte fferece Method... 3 THE EROYMI ETER OF IVISI IRFOIS lasscal Th rfol Theor Geeral rfol Theor Theor evelopet oparso to Ivscd oputatoal Results...38

8 3.3 east Squares Regresso Ft oeffcets-ivscd Flow Ft to Th rfol Theor Equatos Ft to Geeral rfol Theor Equatos The erodac eter of rfols Ivscd Flow Th rfol Theor wth Trgooetrc oleartes Geeral rfol Theor THE EROYMI ETER OF VISOUS IRFOIS The erodac eter of rfols Vscous Flow Thrd Order pproxato Saple Results east Squares Regresso Ft oeffcets-vscous Flow OUSIO...74 REFEREES...77 PPEIES...8 Vscous east Squares Regresso Ft oeffcets...8 B Vortex Pael Method ode...86 altcal Sbolc Solver ode...93 Ivscd erodac eter rfol aber/thckess ode...7 v

9 v IST OF TBES Table Page Vortex pael ethod data for 845 arfol Vortex pael ethod data for 845 arfol east squares regresso coeffcets for 845 arfol Vscous least squares regresso ft coeffcet data for a seres of four dgt arfols RMS error of hgh order soluto ad thrd order approxato usg XFOI data RMS error of hgh order soluto ad thrd order approxato usg XFOI data...66

10 x IST OF FIGURES Fgure Page Forces ad ptchg oet o a arfol...3 The aerodac ceter of a 845 arfol vscd flow as a fucto of agle of attack ad the oral-force... 3 Superposto of a ufor flow ad a curved vortex sheet alog the caber le of a th arfol secto rcular clder the coplexζ - plae RMS error for th arfol theor ad geeral arfol theor lft predctos for 5 arfols vscd flow RMS error for th arfol theor ad geeral arfol theor ptchg oet predctos for 5 arfols vscd flow The aerodac ceter of a 845 arfol vscd flow predcted b th arfol theor geeral arfol theor ad fte dfferecg The ptchg oet about the aerodac ceter of a 845 arfol vscd flow predcted b th arfol theor geeral arfol theor ad fte dfferecg The aerodac ceter of 5 arfols vscd flow as a fucto of arfol thckess ad caber...5 The aerodac ceter locato of a selecto of arfols vscous flow coparg the hgh order soluto ad thrd order approxato usg bbott & Vo oehoff data...6 The aerodac ceter locato of a selecto of arfols vscous flow coparg the hgh order soluto ad thrd order approxato usg XFOI data...65 erodac ceter locatos for a 48 arfol vscous flow data Ptchg oet about the aerodac ceter of a 48 arfol vscous flow...69

11 x OTTIO B coeffcets the expaso gve Eq. (47 secto axal-force coeffcet frst dervatve of secod dervatve of wth respect to wth respect to secto drag coeffcet secto drag coeffcet at zero lft Eq. (67 coeffcet of the parabolc relato for Eq. (67 coeffcet of the parabolc relato for Eq. (67 secto lft coeffcet frst dervatve of wth respect to frst dervatve of wth respect to at secto oet coeffcet about the pot (x ac secto oet coeffcet about the aerodac ceter secto oet coeffcet about the quarter-chord c 4 le O secto oet coeffcet about the leadg-edge secto oet coeffcet about the org O frst dervatve of O wth respect to O secod dervatve of O wth respect to

12 x costat coeffcet used Eqs. (55 ad (68 costat coeffcet used Eqs. (55 ad (68 costat coeffcet used Eqs. (55 ad (68 secto oral-force coeffcet frst dervatve of secod dervatve of wth respect to wth respect to c coplex costats the auret seres expaso secto chord legth F F auret seres expasos used Eqs. (44 ad (45 K K costats defed Eq. (78 O R V w w w secto lft ptchg oet about the org ter the auret seres expaso radus of the crcular clder used for the coforal trasforato freestrea arspeed coplex veloct feld coplex veloct the plae of the crcular clder coplex veloct the plae of the arfol x axal ad upward-oral coordates relatve to the leadg edge x real coordate of the ceter of the crcular clder the coplex plae x ac ac x ad coordates of the aerodac ceter

13 x c coordate of the caber le z z l agar coordate of the ceter of the crcular clder the coplex plae coordate the coplex plae leadg edge of the arfol the z-plae z t tralg edge of the arfol the z-plae z ceter of the crcular clder the coplex plae z x agle of attack zero-lft agle of attack Γ ζ crculato stregth aaltc trasforato fucto ζ l pot the coplex plae that aps to the arfol leadg edge ζ surface coordates of the clder surface ζ t pot the coplex plae that aps to the arfol tralg edge Θ θ θt ρ Φ Φ φ chage of varables for the axal coordate of a arfol agle relatve to the horzotal axs the coplex plae value of θ at the arfol tralg edge flud dest coplex potetal the plae of the crcular clder coplex potetal the plae of the arfol local caber agle

14 HPTER ITROUTIO. Research Motvato orrectl detfg the locato of the aerodac ceter of a lftg surface s extreel portat arcraft desg ad aalss. For exaple the locato of the aerodac ceter of a coplete arcraft relatve to the ceter of gravt s a portat easure of logtudal ptch stablt [-]. Ths locato ofte referred to as the eutral pot s a fucto of the aerodac ceter of each lftg surface or wg. I addto to logtudal ptch stablt accurate kowledge of the locato of the aerodac ceter of a lftg surface or wg has bee show to be a fudaetal paraeter aeroelastc aalss as well as flutter ad dvergece speed calculatos [3]. The portace of correctl detfg the locato of the aerodac ceter of a lftg surface s also apparet supersoc arcraft desg. Efforts to ze tr drag axze load factor capablt ad to provde acceptable hadlg qualtes rel o accurate kowledge of the locato of the aerodac ceter of a supersoc lftg surface [4]. The aerodac ceter of a wg s a fucto of the aerodac ceter of the dvdual arfols fro whch the wg s ade as well as wg sweep dhedral ad plafor. Thus correctl predctg the aerodac ceter or eutral pot of a coplete arfrae durg prelar desg depeds o the accurac to whch we ca predct the aerodac ceters of arfols ad fte wgs.

15 . terature Revew.. Tradtoal Th rfol Theor Relatos for the erodac eter The aerodac ceter s tradtoall defed to be the pot about whch the ptchg oet s varat to sall chages agle of attack.e. ac (. The ptchg oet about a pot the arfol plae ca be foud fro a sple trasforato of forces ad oets about the org to the pot of terest.e. x O (. c c where s the ptchg oet about the org O s the axal force coeffcet ad s the oral force coeffcet. The axal ad oral force coeffcets are related to the lft ad drag coeffcets through a trasforato agle of attack as show Fg. cos s (.3 cos s (.4 Usg Eqs. (.3 ad (.4 Eq. (. the ptchg oet about the aerodac ceter s ac xac ac ( cos s ( cos s (.5 O c c

16 3 For a tpcal arfol the vertcal offset of the aerodac ceter fro the arfol chord le s sall ad the drag s uch less tha the lft. ddtoall the agle of attack s sall for oral flght codtos. Therefore applg the tradtoal approxatos cos >> s s cos gves ac ac ac x ac O (.6 c Takg the dervatve of Eq. (.6 wth respect to agle of attack applg the costrat gve b Eq. (. ad rearragg gves the tradtoal approxato for the aerodac ceter x c ac O ac (.7 c ote that the -coordate s tradtoall assued to be zero due to the approxatos appled the developet of Eq. (.6. Fgure. Forces ad ptchg oet o a arfol.

17 4 Equato (.7 gves the tradtoal approxato for the locato of the aerodac ceter of a arfol. These relatos requre kowledge of the lft ad ptchg oet slopes of a gve arfol. Ths predcto for the locato of the aerodac ceter of a arfol s wdel used toda across the aerospace dustr ad acadea. Furtherore these relatos are tradtoall used to approxate the locato of the eutral pot of a arcraft ad are used to evaluate arcraft statc stablt. The tradtoal th arfol theor approach as developed b Max Muk [5-9] predcts solutos to Eq. (.7 as lg at the arfol quarter chord or 5% aft of the arfol leadg edge ad drectl o the chord le. However solutos to Eq. (.7 suffer fro sallagle sall-caber ad th-arfol approxatos. What s ore the assuptos leadg to the result gve b Eq. (.7 clude a lear lft slope ad a oet slope below stall ad therefore eglect oleartes lft ptchg oet ad drag. Furtherore ths tradtoal approach reduces the olear trgooetrc relatos Eq. (.5 to lear fuctos of agle of attack. These learzg approxatos sgfcatl hder our uderstadg of the effects of oleartes assocated wth ptch stablt of arfols ad arcraft. ttepts have bee ade to develop less restrctve deftos for the locato of the arfol aerodac ceter however do ot accout for all of the trgooetrc ad aerodac oleartes or reove all sall-agle sall-caber ad th-arfol approxatos [-]. I order to provde a ore accurate soluto for the locato of the aerodac ceter we shall ow relax the learzg assuptos a ore geeral developet of the aerodac ceter.

18 5.. Geeral Relatos for the erodac eter Phllps lle ad ewoeher [] preseted geeral relatos for the aerodac ceter whch do ot clude the learzg approxatos used the tradtoal approach. Fro Eq. (. or (.5 the ptchg oet about the aerodac ceter ca be wrtte ac x ac ac O (.8 c c Takg the dervatve of Eq. (.8 wth respect to agle of attack ad applg the tradtoal costrats gve Eq. (. gves x c ac c O (.9 ac ote that applcato of the costrat gve Eq. (. produces a equato for a le ot a pot. The le gve Eq. (.9 s the eutral axs of the arfol []. ll pots alog ths le satsf the costrat gve Eq. (.. Therefore the sgle costrat gve Eq. (. s ot suffcet to specf a sgle pot as the aerodac ceter. Phllps lle ad ewoeher [] suggest a secod costrat to solate the locato of the aerodac ceter ael that the locato of the aerodac ceter ust be varat to sall chages agle of attack.e. x ac ac (. fferetatg Eq. (.9 wth respect to agle of attack ad applg Eq. (. gves

19 6 O ac ac c c x (. The tersecto of the two les specfed b Eqs. (.9 ad (. defes a uque pot where both of the costrats are sultaeousl satsfed ad therefore defes the locato of the aerodac ceter. Solvg Eqs. (.9 ad (. for xac ad ac ad usg the result Eq. (.8 gves the locato of the aerodac ceter ad the ptchg oet coeffcet about the aerodac ceter ac c x O O (. ac c O O (.3 ac ac c c x O ac (.4 Equatos (. ad (.3 offer a ore accurate descrpto of the locato of the aerodac ceter for a lftg surface. The allow both the x ad coordates of the aerodac ceter to be evaluated ulke the tradtoal approxatos gve Eq. (.7 whch alwas predcts a -coordate for the aerodac ceter that les o the chord le. Furtherore Eqs. (. ad (.3 correctl clude the effects of vertcal offsets as well as trgooetrc oleartes ad aerodac oleartes such as drag. ote that Eqs. (. ad (.3 are depedet o frst ad secod aerodac dervatves wth respect to agle of attack whle the tradtoal approxato gve

20 7 Eq. (.7 depeds ol o frst dervatves of aerodac propertes. Therefore ths geeral soluto for the aerodac ceter depeds o accuratel predctg a secodorder aerodac oleartes eve below stall. To estate the aerodac ceter of arfols th arfol theor s ofte appled whch as wll be show hapter 3 eglects these secod-order oleartes. Two uque alteratve fors of Eqs. (. ad (.3 ca be developed whch do ot rel o frst ad secod aerodac dervatves wth respect to agle of attack but rather rel o frst ad secod dervatves wth respect to coeffcet of lft ad the oral-force coeffcet respectvel. These alteratve fors a be useful whe coparg the aerodac ceters of two dfferet lftg surfaces where oe desres to fx the desg for a gve value of lft or oral-force whle allowg varato agle of attack.

21 8 HPTER TERTIVE PPROH TO FIIG THE OTIO OF THE EROYMI ETER. The erodac eter as a Fucto of oeffcet of ft The relatos developed for the locato of the aerodac ceter usg tradtoal th arfol theor ad the ore geeral approach as developed b Phllps [] both are fuctos of agle of attack as ca be see Eq. (.7 ad Eqs. (.-.3 respectvel. The value of these relatos deped largel o wg ad arfol geoetr. osder two wgs wth dfferet geoetr both at the sae agle of attack. Each of these wgs wll have a uque coeffcet of lft ad therefore uque locatos of ther respectve aerodac ceters. Ths s due to the fact that the lft dstrbuto geerated over a rage of agles of attack vares fro wg to wg based o secto ad spa geoetr. It s advatageous therefore to be able to descrbe the locato of the aerodac ceter depedet of wg or arfol geoetr. I order to accoplsh ths we odf the ethod preseted b Phllps [] whereb we redefe the chage ptchg oet coeffcet ad the locato of the aerodac ceter to deped ot o sall chages agle of attack but rather o sall chages coeffcet of lft. We redefe the orgal two costrats gve b Eq. (. ad Eq. (. as follows. The ptchg oet about the aerodac ceter ust be varat to sall chages coeffcet of lft

22 ac 9 (.. The locato of the aerodac ceter ust be varat to sall chages coeffcet of lft xac ac (. Usg these two ew deftos the locato of the aerodac ceter as a fucto of coeffcet of lft wll be developed. osder the defto of the ptchg oet ad force copoets oralzed b spa ad dvded b dac pressure ρv z b/ zb / ac c dz z b / ( cos zb / z b / s c x ( s zb/ ac dz cos c ac dz (.3 pplg the defto for the ea oet coeffcet ad the ea aerodac chord legth ad dvdg b the plafor area S we arrve at the odfed defto for the ptchg oet about the org of a arbtrar wg. O c ref O cos cref x ac ρv S (.4 x s s cos where ac b / / b acc dz ad c ac ac Sc c dz S z z

23 Usg the defto of the ptchg oet about the aerodac ceter ad dvdg t b dac pressure ad the plafor area S gves O c ref c x ( cos s ( s cos (.5 ac ref ac ac obg Eq. (.4 ad Eq. (.5 we obta x ac ( cos s ac ( s ( x cos s cos c ac ref ( x s cos ac c ac (.6 Modfg the defto of the secto chage ptchg oet about the aerodac ceter defed b Eq. (. to be wth respect to coeffcet of lft s gve as ac (.7 Usg Eqs. (. (. ad (.7 the frst dervatves of Eqs. (.4 ad (.6 wth respect to coeffcet of lft we obta x ac ( cos s [ ( x cos ac s c ( s ref ( x O cos s cos ] (.8 s prevousl stated the locato of the aerodac ceter s ot defed b a sgle pot but rather b the tersecto of two les. The frst le s defed b Eq.

24 (.8. Ths equato descrbes a le the plae of setr alog whch ever pot satsfes the frst costrat o the locato of the aerodac ceter (Eq. (.. To uquel defe a pot alog ths le a secod equato s eed that satsfes the secod costrat gve b Eq. (.. To obta ths addtoal equato we frst rewrte Eqs. (.5 ad (.8 ters of axal ad oral coeffcets cos s (.9 cos s (. whch elds O c ref c x (. ac ref ac ac x ac c (. ac O ref Equato. (. s equvalet to Eq. (.8 ad defes a le whch satsfes the frst costrat for gve coeffcets of lft. To obta the secod le whch s ecessar to defe the locato of the aerodac ceter we dfferetate Eq. (. wth respect to coeffcet of lft ad appl the secod costrat. Ths gves x ac c (.3 ac O ref s s the case for the le defed b Eq. (. where ever pot alog the le satsfes the frst costrat ever pot alog the le defed b Eq. (.3 satsfes the secod costrat o the locato of the aerodac ceter. The tersecto of these two les uquel defes a pot where both of the costrats are sultaeousl satsfed

25 ad therefore defes the locato of the aerodac ceter. Solvg Eqs. (. ad (.3 for c x ac ad c ac ad usg the results Eq. (. we obta O O ac c x (.4 O O ac c (.5 ac ac c c x O ac (.6 Thus we see that the locato of the aerodac ceter ca be wrtte as a fucto of coeffcet of lft. Eqs. (.4-.6 are fuctos of coeffcet of lft ad are aalogous to Eqs. (.-.4 whch as prevousl stated defe the locato of the aerodac ceter as a fucto of agle of attack.. The erodac eter as a Fucto of oral-force oeffcet other alteratve approach to fdg the locato of the aerodac ceter volves calculatg ts locato as a fucto of the oral-force coeffcet stead of the tradtoal approach whch depeds o chages agle of attack (Eq. (.7 ad Eqs. (.-.3 respectvel. s stated prevousl Secto. the tradtoal relatos deped largel o wg ad arfol geoetr ad are therefore lted whe atteptg to copare ultple arfols or wgs at a gve agle of attack.

26 3 I order to detere the locato of the aerodac ceter ad the assocated ptchg oet depedet of wg or arfol geoetr we odf the ethod preseted b Phllps []. We redefe the orgal two costrats for the chage ptchg oet coeffcet ad the locato of the aerodac ceter gve b Eqs. (. ad (. to deped ot o sall chages agle of attack but rather o sall chages the oral-force coeffcet as follows.. The ptchg oet about the aerodac ceter ust be varat to sall chages coeffcet of lft ac (.7. The locato of the aerodac ceter ust be varat to sall chages coeffcet of lft xac ac (.8 Usg these two ew deftos the locato of the aerodac ceter as a fucto of oral-force coeffcet s developed. osder the followg equato whch descrbes the ptchg oet coeffcet about the aerodac ceter gve ters of the axal ad oral-force coeffcets ad. ac c (.9 ref cref xac ac

27 4 fferetatg Eq. (.9 wth respect to ad applg the costrats gve b Eqs. (.7 ad (.8 elds O ac ac ref x c (. Equato (. descrbes the eutral axs of the wg alog whch ever pot satsfes the frst costrat as gve b Eq. (.7. To be able to appl the secod costrat requred to descrbe the locato of the aerodac ceter we dfferetate Eq. (. aga wth respect to ad appl the costrats gve b Eq. (.8. Ths gves O ac ref c (. Rearragg to solve for ac we obta O ref ac c (. pplg the result obtaed Eq. (. to Eq. (. ad solvg for ac x we obta O O ref ac c x (.3 Here we have obtaed the locato of the aerodac ceter as a fucto of the oral-force coeffcet as gve b Eqs. (. ad (.3. Usg the results of these two equatos Eq. (.9 we obta the ptchg oet about the aerodac ceter ac O O (.4

28 5 Whle Eqs. (.3 ad (. are aalogous to Eqs.(. ad (.3 the appear to be of a dfferet for. I order to verf the correctess of Eqs. (.3 ad (. a equvalece proof s gve here to show that the locato of the aerodac ceter as a fucto of the oral-force coeffcet s equvalet to the locato of the aerodac ceter as a fucto of agle attack. Ths s portat as saple results coparg these two ethods wll be preseted Secto.3... Equvalece Proof I the alteratve approach preseted Secto. the locato of the aerodac ceter was derved usg costrats whch eforce varace of the ptchg oet about the aerodac ceter ad the locato ( x of the aerodac ceter wth respect to the oral force coeffcet. x c ac ref O O (.5 c ac ref O (.6 These equatos appear to be of a sgfcatl dfferet for copared to the aalogous relatos gve b Eqs. (. ad (.3 whch are fuctos of agle of attack. Here a equvalece proof s gve to show that the locato of the aerodac ceter as a fucto of oral-force coeffcet s deed equvalet to the locato of the aerodac ceter as a fucto of agle attack.

29 6 Frst we defe the uerator of the fracto the frst ter of Eq. (.5 as ad ts deoator as. otce that the uerator ad deoator of ths ter are the sae as the case of Eq. (.6. Startg wth ad expadg ts partal dervatves wth respect to agle of attack gves O O (.7 B expadg the partal dervate o the rght had sde of Eq. (.7 aga wth respect to agle of attack we obta ( ( ( 3 o o o O O O (.8 pplg the sae procedure to ** elds ( ( 3 (.9 vdg b results the followg relato

30 7 O O (.3 otce that the relato gve b Eq. (.3 s equal to Eq. (.3 whch s the vertcal copoet of the aerodac ceter as obtaed b Phllps. Therefore we see that Eq. (.6 whch descrbes the vertcal locato of the aerodac ceter as a fucto of the oral-force coeffcet s deed equvalet to Eq. (.3. We ca use the result obtaed Eq. (.3 Eq. (.5 to obta O O O ref ac c x (.3 Expadg the reag partal dervatves wth respect to agle of attack elds ref ac c x O O O (.3 We apulate ths relato further b perforg all ultplcatve dstrbutos cobg each ter b the lowest coo deoator ad cacellg coo factors to obta ref ac c x O O (.33 otce that we have recovered exactl Eq. (. as derved b Phllps. Therefore we see that Eq. (.5 s equvalet to Eq. (.. We see furtherore that the locato of the aerodac ceter ca deed be descrbed as purel a fucto of the oral-force coeffcet. Ths equvalece ca further be show b calculatg values for ac x ad ac

31 usg Eqs. (. ad (.3 as fuctos of agle of attack ad Eqs. (.5 ad (.6 as fuctos of the oral-force coeffcet. 8.3 Saple Results Usg the relatos descrbg the locato of the aerodac ceter as a fucto of agle of attack ad those whch descrbe t as a fucto of the oral-force coeffcet respectvel values ca be obtaed whch show the equvalece of the two ethods. I both cases ether of these sets of relatos ake restrctos about the tpe of flow whether vscd or vscous for whch the calculate the locato of the aerodac ceter. The saple results preseted here reflect purel vscd flow for whch the effects of drag are gored. Ivscd flow data ca be obtaed b a uber of dfferet ethods both aaltcal ad uercal. Oe such uercal ethod the Vortex Pael Method [ppedx B] provdes accurate ad fast results for vscd flow over arfols..3. Vortex Pael Method The vortex pael ethod uses a sthess of straght-le segets ad cotrol pots alog the top ad botto surface of a arfol. B usg a suffcetl hgh uber of straght le segets ad cotrol pots ths ethod ca accuratel predct the coeffcet of lft ad the ptchg oet of a gve arfol as fuctos of agle of attack. Values for coeffcet of lft ca be related to the axal ad oral force coeffcets va the vscd trasforatos

32 9 s (.34 cos (.35 The followg data was geerated for a 845 arfol usg a vortex pael ethod wth 4 odes cose clusterg ad a o-closed tralg edge. Table Vortex Pael Method data for a 845 arfol gle of ttack (degrees Usg ths data the frst ad secod dervatves Eqs. (. ad (.3 ad Eqs. (.5 ad (.6 ca be approxated uercall order to fd solutos to the locato of the aerodac ceter. Ths uercal approxato ca be acheved b eas of a secod order fte dfferecg ethod.

33 .3. Fte fferece Method I order to approxate the frst ad secod dervatves requred to copute values of the locato of the aerodac ceter Eqs. (. ad (.3 ad Eqs. (.5 ad (.6 dscrete approxatos usg the Talor seres expaso about a pot a be eploed. The Talor seres expaso of a fucto ( φ about a pot for the value ca be wrtte as ( 6 ( ( O φ φ φ φ φ (.36 The secod order approxato of the frst dervatve of ( φ at a pot j s gve b ( ( b a a b a b j a b b a j φ φ φ φ (.37 Where a ad b represet the pots before ad after j respectvel. The secod order approxato of the secod dervatve of ( φ at a pot j s gve b ( ( ( ( ( ( ( ( ( ( ( O c a c c b c b a b b a c b b a c a a c b a a c b j c b a c b a j φ φ φ φ φ (.38

34 Where a represets the pot before j ad b ad c represet the frst ad secod pot after j respectvel. Usg the data fro Table Eq. (.37 ad Eq. (.38 we ca geerate approxatos of the frst ad secod dervatves of ad as fuctos of the tradtoall used agle of attack as well as the oral-force coeffcet as dscussed Secto.. Usg these dervatves ther correspodg equatos for the locato of the aerodac ceter results Fg. Fgure. The ( x locato of the aerodac ceter of a 845 arfol usg the tradtoal ethod as a fucto of agle of attack Eqs. (. ad (.3 ad the odfed ethod as a fucto of the oral-force coeffcet Eqs. (.5 ad (.6. Fro the fgure we see that both ethods gve detcal results to ache precso. otg the scale o the axes for all practcal usages the x ad coordates of

35 the aerodac ceter gve ths fgure descrbe a sgle pot. The scatter the data ca be attrbuted to the uercal ethod used. The results of ths fgure further verf the cocluso gve Secto.. that the locato of the aerodac ceter as a fucto of agle of attack ca be equvaletl descrbed as a fucto of the oralforce coeffcet. However the true sgfcace of ths fgure s ot that the two ethods are equvalet. The true sgfcace of ths fgure s that the aerodac ceter of a arfol a vscd flow s descrbed b a sgle pot. Recall that both the geeral ethod for fdg the locato of the aerodac ceter gve b Eqs. (. ad (.3 ad the odfed ethod gve b Eqs. (.5 ad (.6 allow for evaluato of both the x ad coordates ad clude the effects of vertcal offsets as well as trgooetrc ad aerodac oleartes. Therefore the fact that the case of a arfol a purel vscd flow the locato of the aerodac ceter collapse to a sgle pot s rather rearkable. Ths caot readl be see b exag Eqs. (. ad (.3 or Eqs. (.5 ad (.6 as both cases the relatos appear to be hghl depedet o chages agle of attack. However as has bee show Fg. the locato of the aerodac ceter purel vscd flow s fact a sgle pot depedet of agle of attack. We desre to be able to descrbe the locato of ths pot for a vscd arfol aaltcall wth ew ore sple relatos wthout the eed for uercal approxatos such as the fte dfferecg ethod order to detere values of ukow dervatves. ddtoall we desre that these ew relatos aaltcall deostrate the agle of attack depedece observed the saple results whle stll

36 cludg the effects of vertcal offsets as well as a trgooetrc ad or aerodac oleartes. 3

37 4 HPTER 3 THE EROYMI ETER OF IVISI IRFOIS s show Secto.. Eqs. (. ad (.3 offer a ore accurate descrpto of the locato of the aerodac ceter for a lftg surface. The allow for evaluato of both the x ad coordates of the aerodac ceter ulke the tradtoal approxatos gve Eq. (.7 whch alwas predcts a -coordate for the aerodac ceter that les o the chord le. Furtherore Eqs. (. ad (.3 correctl clude the effects of vertcal offsets as well as trgooetrc ad aerodac oleartes such as drag. These two equatos are depedet o frst ad secod aerodac dervatves wth respect to agle of attack whle the tradtoal approxato gve Eq. (.7 depeds ol o frst dervatves of aerodac propertes. Therefore the geeral soluto for the aerodac ceter depeds o accuratel predctg a secod-order aerodac oleartes eve below stall. To estate the aerodac ceter of arfols th arfol theor s ofte appled whch eglects these secod-order oleartes. 3. lasscal Th rfol Theor Th arfol theor was developed b Max Muk durg the perod betwee 94 ad 9 [5 9]. I ths classcal theor a arfol s stheszed as the superposto of a ufor flow ad a vortex sheet placed alog the caber le of the arfol as show Fg. 3. Sall caber ad sall agle-of-attack approxatos are appled such that

38 5 hgher-order ters ca be eglected. Ths results the classcal th-arfol lft ad ptchg-oet relatos ( (3. O (3. c 4 4 where s the lft slope s the zero-lft agle of attack ad s the ptchg oet about the quarter chord. The coeffcets ad obtaed fro the caber le dstrbuto π ( cosθ π dx θ c 4 c 4 are costats that ca be d c dθ (3.3 π dc c [cos(θ cosθ ] dθ (3.4 4 dx θ where θ represets the chage of varables for the axal coordate gve b x ( θ ( c ( cosθ. The coeffcet s a costat whch fro th-arfol theor s predcted to be

39 6 Fgure 3. Sthess of a th arfol secto fro superposto of a ufor flow ad a curved vortex sheet dstrbuted alog the caber le. π (3.5 The developet of th arfol theor ca be foud ost egeerg text books o aerodacs [3-8]. Usg Eq. (3. Eq. (3. ad applg the result to the tradtoal relato for the aerodac ceter gve Eq. (.7 gves the aerodac ceter as predcted b th arfol theor x ac ac (3.6 c 4 c otce fro Eqs. (3. ad (3. that the lft ad ptchg oet are predcted b ths theor to be lear fuctos of agle of attack. ll hgher-order oleartes agle of attack were eglected the developet of ths theor. Strctl speakg Eqs. (3. (3.5 are ol accurate the lt as the arfol geoetr ad operatg codtos approach those of the approxatos appled the developet of classcal th arfol theor. These assuptos clude a ftel th arfol sall caber ad sall

40 7 agles of attack. However t s cool assued that the for of Eqs. (3. ad (3. are correct for arbtrar arfols at agles of attack below stall. Therefore ad are ofte used as coeffcets to ft Eqs. (3. ad (3. to arfol data obtaed fro c 4 experetal easureets or uercal sulatos. Ths results predctos for lft ad ptchg oet that are lear fuctos of agle of attack ad do ot cota a hgher-order depedece o agle of attack below stall. However as was dscussed above the locato of the aerodac ceter s depedet o secod-order aerodac effects wth respect to agle of attack. Thus order to better uderstad the fluece of olear aerodacs o the locato of the aerodac ceter we ow cosder a ore geeral arfol theor that does ot clude a approxatos for thckess caber or agle of attack. 3. Geeral rfol Theor geeral arfol theor that does ot clude the approxatos of sall caber sall thckess ad sall agles of attack ca be developed fro the ethod of coforal appg [9 ]. The theor preseted here ca be used to ap flow about a crcular clder to flow about a arbtrar two-desoal surface. Pressure dstrbutos ca the be tegrated to evaluate the resultg lft ad ptchg oet as show the followg developet.

41 8 3.. Theor evelopet Flow about a crcular clder of radus R cetered at the pot zcludg the effects of agle of attack ad fte crculato Γ ca be descrbed b dφ w ( z dz V e Γ πv ( z z R e ( z z (3.7 where Φ s the coplex potetal ad w s the coplex veloct the plae of the crcular clder. Usg the ethod of coforal appg we ca appl a arbtrar trasforato of ths flow fro the plae of the clder to the plae of a arfol of the for Φ z Φ [ ζ ( ] (3.8 ( z where ζ (z s a aaltc trasforato fucto. The coplex veloct the arfol plae correspodg to ths coplex potetal s dφ d dζ dζ dz w ( z Φ[ ζ ( z] w[ ζ ( z] w [ ζ ( z] (3.9 dz dζ dz dz dζ Thus the coplex veloct for the trasfored flow feld ca be expressed as dφ Γ dz w ( z V e R e (3. dz πv ( ζ z ( ζ z dζ The potetal-flow soluto about a crcular clder ca be trasfored to obta the potetal-flow soluto about a clder wth a arbtrar cross secto. I geeral a

42 9 arbtrar trasforato requres a fte uber of degrees of freedo ad ca be expressed ters of the auret seres expaso [] z( ζ ζ (3. ζ where the coeffcets are coplex costats. The frst dervatve of ths geeral trasforato s dz dζ ζ (3. The equato for the surface of the crcular clder the ζ -plae ca be wrtte as R e θ surface z ζ (3.3 Usg Eq. (3.3 Eq. (3. the trasfored surface of the clder the z-plae s gve b the relato θ z( ζ surface R e z (3.4 θ ( R e z The dervatve of the coforal trasforato gve Eq. (3. ca be zero at ultple pots depedg o the values of the coplex coeffcets. These values of ζ are ofte referred to as the crtcal pots of the trasforato at whch the trasfored veloct feld gve b Eq. (3. s sgular. I order to ap the flow of a crcular clder to that over a arfol oe of the crtcal pots ust le o the crcular clder the ζ -plae at the pot that aps to the arfol s tralg edge the ζ -plae.

43 ll reag crtcal pots ust le sde the crcular clder the ζ -plae order for the flow exteral to the clder to rea coforal. Here we deote the crtcal pot the ζ -plae that aps to the arfol s tralg edge the ζ -plae as ζ. It s t coveet to choose ζ to be o the postve real axs as show Fg. 4. Ths requres t 3 θt t R x R e ζ z (3.5 where θ s the value of t θ at ζ ζ.e. t θ t s ( R ta R (3.6 Slarl the left-had real-axs tercept of the paret crcular clder s l R ζ x (3.7

44 3 Fgure 4. rcular clder the coplex ζ -plae cetered at ζ z x. ote that for a setrc arfol. The Kutta codto ust be satsfed at the tralg edge of the arfol ad requres that ζ be a stagato pot for the flow the t ζ -plae. The coplex veloct feld s gve Eq. (3. ad wll have a stagato pot at the pot o the paret crcular clder that aps to the arfol tralg edge f e Γ πv ( ζ z t R e ( ζ z t (3.8 Solvg ths relato for Γ gves the crculato that wll satsf the Kutta codto at the tralg edge

45 3 πv Γ R e ( ζ t z e ( ζ t z (3.9 Usg Eq. (3.5 we ca evaluate R x ( x R ζ z (3. t Usg Eq. (3. Eq. (3.9 gves a alterate for for the crculato Γ 4πV R s cos (3. Fro the Kutta-Joukowsk law [] the secto lft ca be coputed fro the secto crculato.e. ρv Γ (3. Usg the crculato fro Eq. (3. Eq. (3. gves 4πρV R s cos (3.3 otce that the predcted secto lft gve Eq. (3.3 s depedet of a partcular trasforato ad s a fucto ol of the radus ad vertcal offset of the crcular clder. O the other had the leadg ad tralg edges of the arfol the z- plae are depedet o the trasforato ad are eeded order to copute the chord legth ad lft coeffcet. For a gve trasforato the secto lft coeffcet ca be obtaed fro Eq. (3.3

46 33 8π R s cos ρ ( (3.4 V ( z z z z t l t l R where c z t z s the arfol chord legth. Thus regardless of the trasforato the lft coeffcet wll be of the for l (s ta cos (3.5 where s the lft slope at zero agle of attack ad s the zero-lft agle of attack. Fro Eq. (3.4 8 R π θ t ta ( z z R (3.6 t l otce that fro Eqs. (3.3 ad (3.6 that the lft ad zero-lft agle of attack do ot deped o ether the trasforato or the real part of the clder offset x. O the other had the lft coeffcet ad lft slope at zero agle of attack deped o the trasforato whch tur depeds o x. I a case Eq. (3.5 s a geeral for for the lft coeffcet of a arbtrar arfol. o assuptos of caber thckess or sall agle of attack were ade the developet of Eq. (3.5. Therefore we would expect ths for of equato to ft the vscd lft propertes of a arfol at arbtrar agles of attack. Fro the Blasus relatos [3-4] the ptchg oet about the org for a arbtrar geoetr s

47 34 [ ] { } O dz z z w ( real ρ (3.7 Usg Eq. (3. Eq. (3. the square of the coplex veloct of the trasfored flow feld s [ ] ( ( ( z e R z V Γ e V z w ζ ζ ζ π (3.8 Usg Eq. (3.8 Eq. (3.7 ad expadg a auret seres of gves [ ] { } O d F F V ζ ζ ζ ρ ( ( real (3.9 where [ ] 4 ( ζ π π ζ π ζ R V Γ V e Γz V Γe e F ( ( ζ ζ ζ ζ ζ ζ ζ ζ F (3.3 ad [ ] ζ π π π ζ ζ ζ 4 ( ( e R V Γ V e Γz V Γe e F F ζ

48 35 (3.3 Therefore the ptchg oet about the org ca be wrtte ters of the seres O real ρv B ζ dζ ` (3.33 where B e Γe Γze Γ B B R e πv πv 4π V (3.34 Usg Eq. (3.34 Eq. (3.33 ad tegratg shows that the ptchg oet s ol a fucto of the frst costat the auret seres ( πρv e ρv Γz e (3.35 O real fter applg the Kutta-Joukowsk law gve Eq. (3. the ptchg oet about the org ca be wrtte as O real ( πρv e z e (3.36 Usg the dett e cos s as well as the defto z x Eq. (3.36 gves O πρv s( (3.37 ( x cos s Because the costat depeds o the trasforato we see that ulke the secto lft the secto ptchg oet does deped o the trasforato. vdg Eq. (3.37 b

49 the dac pressure ad chord legth squared the secto ptchg oet coeffcet relatve to the org used the trasforato ca be expressed as 36 O 4π ( z z t ρv l ( z z t l s( x cos s z z t l (3.38 The oet coeffcet about a arbtrar pot the z-plae ca be foud fro the oet coeffcet relatve to the org ad the lft coeffcet x cos s z O zt l (3.39 Usg Eq. (3.38 Eq. (3.39 gves 4π ( x x cos ( s (3.4 s( ( z z t zl t zl I order to copute the ptchg-oet coeffcet we eed to kow zt ad zl whch ust be foud fro the trasforato. However regardless of the trasforato the ptchg-oet coeffcet for a arfol vscd flow about a pot the doa s a fucto of agle of attack of the for s( cos s (3.4 where ad are costat coeffcets. s ca be see fro Eq. (3.4 the values for the coeffcets ad are a fucto of the x ad locato of the ptchg oet relatve to the org used the trasforato. Sce the org of the

50 trasforato has lttle phscal eag the tradtoal arfol coordate sste we wll defe ad to be the coeffcets wth the ptchg oet evaluated at the arfol leadg edge.e. x ad whch s the org of the tradtoal arfol zl coordate sste. For a gve trasforato the ptchg oet of a arfol a vscd flowfeld about the arfol leadg edge ca be evaluated fro Eq. (3.4 wth the coeffcets 37 4π z x (3.4 z l ( zt zl zt zl zt l The for of Eqs. (3.5 ad (3.4 hold for a arfol trasforato ad therefore for a arbtrar arfol shape. These relatos were developed wthout a approxatos for arfol thckess caber or agle of attack ad are therefore ot costraed uder the sae ltatos that were used the developet of the tradtoal sall-caber ad sall-agle relatos gve Eqs. (3. ad (3.. The coeffcets ad requred Eqs. (3.5 ad (3.4 ca be evaluated aaltcall fro a kow paret clder offset ad trasforato b usg Eqs. (3.6 ad (3.4. For exaple the Joukowsk trasforato s defed as a specal case of Eq. (3. where R x z ( ζ ζ (3.43 ζ Usg Eq. (3.43 as well as a gve paret crcular clder offset of z the ethod descrbed above gves the coeffcets for a Joukowsk arfol

51 38 ( ( ( ( ( 4( ( 4( ( 4 ta ( R x R R x R x x R x R R x R x x R x R z x R z x R l t π π (3.44 The aaltcal soluto for the coeffcets gve Eq. (3.44 for a Joukowsk arfol sheds sgfcat sght o a portat aspect of the coeffcets requred Eqs. (3.5 ad (3.4. ote that for a Joukowsk arfol the etre arfol ad trasforato ca be defed b ol three varables R x ad. ll coeffcets Eq. (3.44 are fuctos of these three varables. Hece the aerodac coeffcets are ot etrel depedet. For exaple after soe algebrac apulato ca alteratvel be expressed as ( (8 / π π. lthough the uber of varables requred to defe a arfol a var depedg o the trasforato the aerodac coeffcets of Eqs (3.5 ad (3.4 wll geeral ot be etrel depedet. 3.. oparso to Ivscd oputatoal Results For arfol geoetres that were ot geerated fro coforal appg techques the coeffcets ad requred for Eqs. (3.5 ad (3.4 ca be evaluated uercall. Ths ca be accoplshed b fttg Eqs. (3.5 ad (3.4 to a set of arfol data obtaed fro experetal or uercal results. vertcal

52 39 least squares regresso ethod for fttg these coeffcets to a set of data s outled Secto 3.3. Ths ethod ca be used to evaluate approprate coeffcets for Eqs. (3. ad (3. or Eqs. (3.5 ad (3.4 for cabered arfols. Because the geeral arfol theor relatos gve Eqs. (3.5 ad (3.4 were developed wthout a assuptos for arfol geoetr other tha that of a sgle tralg edge we should expect the for of these equatos to atch vscd arfol aerodac data ore accuratel tha Eqs. (3. ad (3. whch were obtaed fro th arfol theor. Here we cosder the accurac of each of these equatos for the 4-dgt seres over a rage of caber ad thckess agtudes. Ivscd copressble aerodac lft ad ptchg oet coeffcet data for 5 4-dgt seres arfols were geerated over a wde rage of caber ad thckess usg a uercal vortex pael ethod [ppedx B] eplog lear vortex sheets alog the arfol surface [5]. The arfol surfaces were dscretzed usg 4 paels whch produced grd-resolved solutos. The paels were clustered ear the leadg ad tralg edges of the arfol usg stadard cose clusterg. Results were coputed for each arfol at agles of attack ragg fro - to 5 degrees creets of degree. t each agle of attack the lft ad ptchg oet coeffcet about the arfol leadg edge were recorded. Table shows a saple data set for the 845 arfol.

53 4 Table. Ivscd lft ad ptchg-oet data geerated fro a vortex pael ethod for a 845 arfol. gle of ttack (degrees The least-squares regresso ethod was used to ft the aerodac data for each arfol to the th-arfol equatos gve Eqs. (3. ad (3. ad the RMS value for each case was recorded. Slarl the least-squares regresso ethod was used to ft the aerodac data for each arfol to the geeral arfol theor equatos gve Eqs. (3.5 ad (3.4 ad the RMS value for each case was recorded. Table 3 shows saple resultg coeffcets for the 845 arfol alog wth the assocated RMS error fro both th arfol theor ad geeral arfol theor. Fgures 5 ad 6 show the RMS

54 4 values for all 5 arfols as a fucto of arfol thckess ad varg caber. It should be oted that Eq (3.55 whch ca be used for fttg data fro setrc arfols to Eq. (3.4 produces a sste of equatos that has a fte uber of solutos. The aoal for setrc arfols wll be vestgated future work. Table 3. east-squares regresso coeffcets for a 845 arfol usg relatos fro th ad geeral arfol theor respectvel. Root-ea-square error values are also gve for each theor copared agast results fro the vortex pael ethod. oeffcet Th rfol Theor oeffcet Geeral rfol Theor c / Roo Mea Squared Error Th rfol Theor Geeral rfol Theor.69 ache zero.495 ache zero

55 4 Fgure 5. RMS error for lft predctos fro th arfol theor ad geeral arfol theor for 5 4-dgt arfols. Fgure 6. RMS error for ptchg-oet predctos fro th arfol theor ad geeral arfol theor for 5 4-dgt arfols.

56 43 ote that the RMS error fro the geeral arfol theor s several orders of agtude saller tha that of the tradtoal relatos based o th arfol theor. I fact the RMS error fro the geeral arfol theor s o the order of ache precso. Ths dcates that the geeral arfol theor equatos ca be ft exactl to the vscd solutos ad therefore are of the correct for. The error assocated wth the leastsquares regresso fts to the th arfol theor equatos dcate that the for of the th arfol theor equatos are ot exactl correct. Wth curret easureet techolog for experetal setups the accurac gaed fro the geeral arfol theor equatos for lft ad ptchg oet predctos s clearl uwarrated. Experetal data s geerall ol kow to or 3 sgfcat fgures whch s the sae order of accurac as that obtaed fro th arfol theor. Therefore the sgfcace of the geeral arfol theor s ot that t ca ore accuratel ft experetal data or F sulatos. Ideed the error experetal data or F sulatos ofte falls outsde the rage of accurac to be foud ether the geeral arfol theor or th arfol theor. Thus usg oe theor over the other wll ot gve sgfcatl proved results f we wsh ol to predct lft or ptchg oet over a rage of agles of attack below stall. Rather the sgfcace of the geeral arfol theor becoes apparet whe secod dervatves for lft or ptchg oet as a fucto of agle of attack are eeded. Such s the case the estato of the locato of the aerodac ceter.

57 east-squares Regresso Ft oeffcets Ivscd Flow The ethod of least-squares regresso was used to ft the aerodac data for each arfol to the th-arfol equatos gve Eqs. (3. ad (3. ad the geeral arfol theor equatos gve Eqs. (3.5 ad (3.4. The least-squares regresso ethod s cool used regresso aalss ost portatl data fttg. I geeral the su of the squares S of the vertcal devatos betwee the best-ft equato ad a set of data pots s gve b S [ f ( a ] (3.45 x where x are dscrete data pots a s a vector of the ukow coeffcets to be ( detered ad f ( a s the aaltcal expresso to whch the data s to be ftted. The x vertcal least-squares ethod seeks to ze S for a gve data set ad expresso f ( a. The RMS error for a gve soluto ca be coputed fro x S RMS ( Ft to Th rfol Theor Equatos pplg the tradtoal lft relato gve Eq. (3. to Eq. (3.45 elds S [ ] ( (3.47

58 45 Best-ft values for ad ca be detered b settg the partal dervatves of Eq. (3.47 wth respect to ad equal to zero. Expressos for the partal dervatves are [ ]( { } ( S (3.48 [ ] { } ( S (3.49 Solvg Eqs. (3.48 ad (3.49 sultaeousl for ad eld the best-ft values for these two coeffcets for the tradtoal lft equato gve Eq. (3. ( (3.5 ( ( (3.5 Usg the results fro Eqs. (3.5 ad (3.5 Eq. (3. a estate for the lft coeffcet as a fucto of agle of attack ca be obtaed. Ths vertcal least-squares fttg process ca also be used to evaluate the best-ft quarter-chord ptchg oet for the tradtoal equato gve Eq. (3..e. O c 4 4 / (3.5

59 Ft to Geeral rfol Theor Equatos The best-ft coeffcets requred to ft Eq. (3.5 to a data set ca be solved usg the process descrbed above whch elds the followg expressos for ad cos s ( cos (s cos ( cos (s s ( s cos ( ta (3.53 cos ta (s cos ta (s (3.54 Usg the results fro Eqs. (3.53 ad (3.54 Eq. (3.5 a estate for the lft coeffcet as a fucto of agle of attack ca be obtaed. The least-squares process ca the be repeated to evaluate the best-ft coeffcets for Eq. (3.4 as a fucto of agle of attack ad lft coeffcet. fter soe algebrac apulato ths elds the followg lear sste of equatos whch ca be solved to evaluate ad

60 47 ( ( ( ( ( ( ( ( ( ( ( O O O s cos s( s s cos s( s s cos cos s( cos s( s s( cos ( s (3.55 For setrc arfols Eq. (3.55 becoes sgular. The reaso for ths sgulart s at ths te ot full uderstood but could be related to the fact that ad are ot learl depedet as dscussed Secto 3... Fro geeral arfol theor t ca be show that for setrc arfols. However there potetall exsts a ukow uber of solutos for ad that satsf Eq. (3.55. Therefore whle s kow the case of setrc arfols the relato betwee ad s ot presetl kow ad we coclude ( f (3.56 ethod for uquel evaluatg these two coeffcets for setrc arfols wll be the topc of future research.

61 The erodac eter of rfols Ivscd Flow I geeral the locato of the aerodac ceter ca be correctl predcted usg Eqs. (. ad (.3. Recall that ths defto for the locato of the aerodac ceter s a geeral defto that t does ot clude a learzg or sall-agle approxatos. We shall ow cosder the locato of the aerodac ceter of vscd arfols as predcted b the relatos developed fro classcal th arfol theor gve Eqs. (3. ad (3. copared wth the relatos developed fro geeral arfol theor gve Eqs. (3.5 ad (3.4. Frst we cosder ol the case of vscd flow.e.. Fro Eqs. (.3 ad (.4 ths gves s (3.57 cos ( Th rfol Theor wth Trgooetrc oleartes Predctos for the aerodac ceter fro th arfol theor are tradtoall obtaed b applg aerodac ad trgooetrc learzg approxatos to Eqs. (. ad (.3. Ths ethod was outled prevousl ad results a aerodac ceter locato gve Eq. (.7. fter the dscusso surroudg the geeral defto of the aerodac ceter gve b Eqs. (.-(.4 oe a be cled to appl the lear aerodac equatos fro th arfol theor whle retag the geoetrc oleartes of Eqs. (3.57 ad (3.58. Here we exae the results of such a approach. Usg Eq. (3. Eqs. (3.57 ad (3.58 dfferetatg the result twce alog wth Eq. (3. ad applg the resultg dervatves to Eqs. (. (.4 gves

62 xac c ( s cos 4( 8 49 (3.59 ac c ( 4( cos s 8 (3.6 ac ( 3 c 4 4[( [( ] ] (3.6 ote that the aerodac ceter ad assocated ptchg oet gve b Eqs. (3.59 (3.6 s a olear fucto of agle of attack. We ow copare ths result to that fro geeral arfol theor Geeral rfol Theor The aerodac ceter of a arbtrar vscd arfol ca be ore accuratel foud b usg the lft ad ptchg-oet relatos fro geeral arfol theor. Usg Eqs. (3.5 ad (3.4 Eqs. (3.57 ad (3.58 dfferetatg twce ad applg the resultg dervatves to Eqs. (. (.4 gves x c ac cos (3.6 c ac s( (3.63 s( (3.64 ac

63 5 otce that Eqs. (3.6 (3.64 are depedet of agle of attack. Therefore the locato of the aerodac ceter for a arbtrar arfol vscd flow s a sgle pot depedet of agle of attack. Ths pot does ot geeral le at the arfol quarter chord but s a sgle pot depedet o arfol thckess ad caber. Ths soluto was developed fro geeral arfol theor whch does ot ake a assuptos for sall agles of attack sall caber or sall thckess. It s rather rearkable that whe all geoetrc ad aerodac oleartes are retaed the lft ad ptchg oet equatos alog wth those the defto of the aerodac ceter the relatos for the aerodac ceter reduce to such a sple expresso depedet of agle of attack for vscd flow. Fgure 7 shows the aerodac ceter for the 845 arfol as predcted b the tradtoal th arfol theor gve Eq. (.7 th arfol theor wth trgooetrc oleartes gve Eqs. (3.59 ad (3.6 ad the geeral arfol theor gve Eqs. (3.6 ad (3.63. Fgure 8 shows the ptchg oet about the aerodac ceter predcted b each theor as a fucto of agle of attack. Fgures 7 ad 8 also clude results fro secod-order fte dfferece approxatos obtaed fro vscd uercal solutos for the 845 arfol ad Eqs. (. (.4. Equatos (3.59-(3.6 represet a x of usg lear aerodacs fro th arfol theor whle retag the trgooetrc oleartes gve Eqs. (3.57 ad (3.58. Ths produces o-phscal results ad should ever be used to approxate the aerodac ceter ad the assocated ptchg oet. ote that the aerodac ceter predcted b Eqs. (3.6 ad (3.63 does ot le at the quarter-chord but s a sgle