Z E S Z Y T Y N A U K O W E P O L IT E C H N IK I ŚLĄSK IEJ___________________________ 1994
Seria: M E C H A N IK A z. 115 Nr kol. 1239
M izanur R A H M A N
Instytut L otnictw a w W arszaw ie
W Y Z N A C Z A N IE P O Ł O Ż E N IA Ś R O D K A A E R O D Y N A M IC Z N E G O D LA S A M O L O T U Z E S K R Z Y D Ł E M M A JĄ C Y M Z A Ł A M A N IE M E T O D A E SD L
O R A Z P O R Ó W N A N IE Z W Y N IK A M I VLM
Streszczenie: W pracy przedstaw iono pew ną teoretyczno-em piryczną m etodę w yznaczania środka aerodynam icznego p łata oraz płata z k adłubem i gondolam i w opływ ie poddźw iękow ym . W ykonano przykładow e obliczenia dla testow ego sam olotu i p o ró w n an o z wynikami otrzym anym i m eto d ą (V LM ).
A P P L IC A T IO N O F AN E S D U M E T H O D F O R C A L C U L A T IN G T H E .A E R O D Y N A M IC C E N T R E O F A IR C R A F T W IT H C R A N K E D W IN G S A ND
C O M P A R IS O N W IT H V O R T E X L A T T IC E M E T H O D (V LM )
Sum m ary: A sem i-em pirical m ethod for calculating the aerodynam ic centre (A C ) o f a wing with cranks and a w ing-fuselage-nacelle com bination in subsonic flow is p re se n te d . N um erical calculations are carried out for a test aircraft and the results a re c o m p ared with th e results o btained by a different b u t well-known VLM .
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lMEQUpHM H3JEMH KPAEB'C EEHMEHEHHEM MEIDZIA ESDUH CPAEHEHHE C MElUZHvl VLM
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noMom,io a p y r o r o M eT oaa (VLM).
1. IN T R O D U C T IO N
C alculation o f the A C o r n eu tral point of aircraft should be done w ith good accuracy even a t th e prilim inary design stage. T h e A C is a n jm p o rta n t p a ra m e te r for trim , stability and co n tro l analysis. A designer has a n um ber o f options to regulate its position to attain a d esired pitching m o m en t derivative value. E xact determ in atio n of tn e A C is difficult w ithout wind tunnel test, which is obviously a costly and tim e consum ing approach.
N evertheless, the sem i-em pirical m ethod p resen ted in this p a p e r, prim arily E ased upon ref. 3, gives accep tab le results. T he rem arlcably small differences betw een the results of the p ro p o sed m eth o d and V LM advocate its acceptibility.
T h e p re se n te d algorithm in this p a p e r may be of g reat interest for wings with cranks in th eir leading an d trailing edges. In such case the real wing will be replaced by an
340 M. R ahm an
equ iv alen t or referen ce wing. T h e fuselage effect, and th en th e nacelle effect, are calculated separately. T h e nacelles m ay be eith er wing-pylon m o u n ted o r re a r fuselage - pylon m o u n ted . A tten tio n should be paid to the application range o f the pro p o sed m ethod, which is lim ited to steady, potential and inviscid subsonic flow only; T h e subsonic region is again lim ited by excluding the application of this m eth o d for su p e r critical profiles. T h ese lim itations am ong others result from, the theoretical basis th e m ethod is b ased on an d also th e statistical d a ta used in em pirical equations. This m eth o d is built for untw isted wing with flaps and u n dercarriage undeployed. T h e pow er and ground effects a re also not taken in to account. T o consider th ese effects ref.. 1 can be recom ended.
2. M A T H E M A T IC A L M O D E L A N D G O V E R N IN G E Q U A T IO N S . 2.1. AC for equ iv alen t wing (EQW ) geom etry
F o r constructional, aerodynam ical and stability reasons a designer m ay decide to allow for a wing sw eep. O n the o th e r hand it should be noted th a t wing sw eep to g eth er with low asp ect ratio may have a strong effect on the induced th ree dim ensional drag due to lift a n a in som e cases on th e wing alone undesirable "pitch up" characterstics. F o r this reason, am ong o th ers som etim es wings with cranks (two o r m o re sw eeps) a re also used.
A wing w ith such a geom etry can b e replaced by a straight ta p e re d wing nam ed as an E Q W o r a re fe re n c e wing (F ig .l). T h e geom etric p aram etrs n ee d e d to define an d position an E Q W p lanform a re calculated from real wing-fuselage p lanform geom etry [ref. 6].
O nce an E Q W is d e fin e d (e q .l to eq.7), th e lifting force g rad ien t (a = dc /d a ) and relative A C w ith resp ect to th e m ean aerodynam ic chord (M A C ) leading e d g e fx /c ) are found from E S D U 70011 using E Q W values of "A *tanA i A ', "B* A", an d "X" w here, A- asp ect ratio, A*y?- half-chord sw eep b ack angle o f EQ"w planform , B- com pressibility factor, ( 1 - M z ) X- ta p e r ratio of E Q W planform , (c{ / C n ) , M - free-stream M ach n u m b er, ct- tip chord of equivalent planform , cn- centre-line chord o f equivalent planform .
C anA 1/2 = ]T ( t a n A j . j - t a n A J J t l )
s c r
’ 1,0
’ 1.0
+ t a n A JW il +
s - s S.
1,0 - C, s - S 1.0 . „ (1 + 2.)
2 ( s - S J 0 ) (1) (2)
( 3)
( 4)
= 2 c. (1 + X + X2) 0 3 (1 n S = b . c g A = b 2/ S
X) ( 5)
(6)
( 7) w here, N- n u m b e r o f cranks in th e leading edge, c , c„- geom etric and aerodynam ic chord o f E Q W p lanform respectively, S- a re a o f E Q W planform , Se - a re a o f planform o f tru e wing outside p ro jected fuselage planform .
A p p lic a tio n o f a n E S D U m e th o d fo r c alcu la tin g th e a e ro d y n a m ic c e n tre
Fig-1- O riginal an d equivalent wing platform R y s.l. R zeczyw ista i zastępcza pow ierzchnia p ła ta
2.2. F u s e la g e effect
T h e fuselage effect (A xh/c a ) on wing A C (x/ca ) can be o btained from E SD U 76015 by the following equation,
A x h c r a 2 F G
c , a S j l + 0 . 1 5 - 1 ) j - ( f q + X k 2) t 8 ) w here F (m /c r, n/cr), G (B d/cr), k ^ d /b , X, A t a n A ^ ) , k2(BA, A ta n A y 2) are o btained from
342 M. R ah m an
E S D U 76015,- "d" an d "h" a re fuselage w idth and height respectively at the leading edge of ro o t ch o rd (cr ) o f E Q W planform and "m" , n" are found from the following equations:
m
= + E ( t a n A j . i - t a n A j i41)--Sl-+
■ — f E i i ( 9 )i - l s l.o>
n = 1 - m - c z ( 1 0 )
As a result the A C for w ing-fuselage com bination (x^/ca ) is o b tain ed by th e e q u atio n 11.
^ = J L - ^ ( I D
C a C a
2.3. N acelle effect
T h e effect o f nacelles on th e A C is considered fo r th e two cases: 1) effect o f wing- pylon m o u n ted nacelles, 2) effect o f rear-fuselage pylon m o u n ted nacelles.
F o r th e first case, th e forw ard shift in AC, A x ^ is calculated from E S D U 77012 by the following eq.
A^0in _ £ IT W -Z j. 1 + -T Cg ^ ( 1 2 )
c a S c a 'a 4 it R 2
WheRr5 - z * +
a - lift curve slope o f nacelle b ased on a re a "w.l" and o b ta in e d from E S D U 77012 as a function o f "w/1"
r - chordw ise distance o f nacelle lip forw ard o f A C o f w ing-fuselage com bination E - re p re se n ts th e sum m ation o f contributions from each nacelle.
F o r th e second case th e rearw ard shift o f th e A C caused
- A x to _ JC [E (a „ w 1) + 6 y p ] ( 1 - 2 H a / n A) r
c j - S a c a
by th e nacelles a re estim ated from E S D U 78013 by eq. 13.
de _ 2 h a
da n A
( 1 3 )
( 1 4 )
w here, H (ta n A i m, r’/s) is a downwash p a ra m e te r for calculating ra te o f change of dow nw ash angle w ith incidence a t nacelle inlet plane an d ce n tre o f trailing vortex sh eet as expressed in eq.14, a n d can b e o b tain ed from E S D U 78013,
y - w idth o f pylon-leading edge betw een nacelle an d fuselage side, K- is the ratio of lift cm nacelle-pylon-fuselage com bination to lift on nacelle an d pylon on isolation. A cording to E S D U 78013 th e value o f K = 2.4 has b een found to give satisfactory result for classical airplanes.
N ow th e A C o f w ing-feselage-nacelle com bination is o b tain ed from the following eq.
*hn _ *h _ (15)
3. A C C A L C U L A T IO N F O R A T E S T A IR C R A F T
A num erical p ro g ram "N E U T R A L " is w orked out for calculating th e A C on the basis of th e p re s e n te d algorithm . As an exam ple, calculations w ere carried o u t for a test aircraft
A pplication o f an E S D U m ethod for calculating the aerodynam ic cen tre 343
p re se n te d in [ref.4]. T h e results ob tain ed by this m ethod is p re se n te d in ta b .l. A lso in this tab le a re placed th e results o btained by a different but well know n V ortex lattice m ethod fo r com parison. T o have a b e tte r understanding of this m eth o d ref.4 and ref.5 is reco m en d ed . T h e A C by V LM was o btained using the p rogram "V O R T R " [ref.4]. T he AC by this m eth o d was o b tain ed by the following eq.
w here, x- indicates cen tre o f pressure, cz- lift coefficient and prefixes 1,2 indicate two different incidence angles.
By applying V LM it was possible to calculate the A C for the real wing and also for the real w ing-fuselage com bination. This was especially interesting to see th e penalty received by replacing th e real wing w ith an E Q W th at was p roposed in this p a p e r for the cranked wings.
T a b .l. R esults fo r th e te st aircraft o b tain ed by th e p ro p o sed and V ortex lattice m ethod.
T a b .l. W yniki obliczeń dla testow ego sam olotu p roponow aną m eto d ą o raz VLM .
M eth o d and result
x/e
a
xh/c a Axh/ca xhn/ca Axhn/Ca "a"wing
"a"
wing- fuselag e P ro p o sed
m eth o d
.243 .116 .127 .096 .020 .085 .085
V LM real wing
.221 .117 .104 - - .079 .083
D ifference .022 .001 .023 - - .006 .002
4. D IS C U S S IO N A N D F IN A L R E M A R K S
T h e A C o f aircraft plays an im p o rtan t role in trim ing, stability an d control analysis.
T h e p ro p o se d sem i-em pirical m ethod in this p a p e r for calculating the A C o f wing- t'uselage-nacelle com bination is very sim ple, cost a n a tim e effective in com parison to o th e r (eg. wind tu n n el o r co m putational aerodynam ics) m ethods. T h e p ro p o se d algorithm allows to see the change o f th e A C, in a very short tim e, o f any geom etric change o r different com binations in conceptual design. It also m ay be com bined w ith ex p erim en tal results, for exam ple, th e calculation o f fuselage effect on experim entally o b tain ed d a ta fo r th e A C of th e wing alone. T o verify th e accuracy o f th e p ro p o sed m ethod, results o f an arbitrarily choosen test aircraft o b tain ed by this m eth o d are c o m p ared w ith results from V LM ( T a b .l) , w h ere it is seen th a t the difference betw een the two m eth o d s a re rem arkably small. B ut it can n o t b e concluded a t this stage o f w ork which on e o f th ese m ethods gives m ore ac c u ra te results, b ecau se bo th have shortcom ings o f different types. T he pro p o sed m eth o d , w hich uses som e statistical d ata, for exam ple can be used for sim ple p lanform s but it does n o t ta k e into consideration th e thickness, cam bering o r tw isting ot the wing.
O n the o th e r hand, p ro g ram "V O R T R " at this stage does not consider th e fuselage as a
344 M . R a h m a n
closed solid body. R a th e r it was m odelled as a f la tp la te like it ap p e a rs on the planform , thus n o t considering wing-fuselage interference. T his is o n e of the reasons th e nacelle effect is not calculated by th e V LM (T a b .l). N evertheless, the sm all differences betw een the results o b tain ed by th e two m ethods may advocate their acceptibility. T h e m ethod p ro p o sed in this p a p e r can well be p referred fo r its simplicity, low cost an d time consum ption w hich is very im portant at the prim ary design stage. O f course the results should b e revised in the later stage by m ore convincing m ethods such as wind tu n n el tests.
R E F E R E N C E S
1. R oskam , J. "Airplane design." p a rt 6, Prelim inary calculation o f aerdynam ic, th ru st and pow er characterstics, K ansas 1990.
2. E S D U 70011, E S D U 76015, E S D U 77012 and E S D U 78013.(E ngineering sciences data units.), A erodynam ic sub series, R oyal aeronautical society, L ondon, England.
3. G oraj, Z. "VO RTR 9302"- P akiet program u do obliczeń rozkładu ciśnień na pow ierzchnych nośnych m eto d ą "V ortex lattice" lub "D oublet lattice" do M a< 0.8, Politechnika W arszaw ska, ITLiM S, 1993, p raca niepublikow ana.
4. G o raj, Z. "W yznaczanie p unktów neutralnych stateczności sam olotu m etodą VLM."
Zeszyty naukow e politechniki śląskiej, M echanika, z.113, Gliwice 1993.
5. K atz, J. and Plotkin, A. "Low-speed aerodynamics fro m wing theory to p a n el methods."
M cG raw Hill, Inc 1991.
6. R ah m an , M . "Obliczenie gradientu siły nośnej oraz m om entu pochylającego układu skrzydlo-kadłub." S praw ozdanie Instytutu
Lotnictw a, W arszaw a 1993, p ra c a niepublikow ana.
R ecenzent: D r h ab inż. A ndrzej B uchacz W płynęło do redakcji w grudniu 1993r.
Strszczenie
W pracy z a p ro p o n o w an o pew ną teoretyczno-em piryczną m eto d ę obliczania położenia środka aerodynam icznego układu skrzydło-kadłub-gondola dla opływu nielepkiego i poddźw iękow ego. M e to d a ta pozw ala określić środek aerodynam iczny dla konstrukcji skrzydła z zała m an iem kraw ędzi n atarcia lub/i spływu (rys. 1). W yniki otrzym ane dla sam olotu sportow o-turystycznego [4] były p o rów nane z wynikam i otrzym anym i m ną znaną m e to d ą (V L M ). R óżnice w wynikach były znikom o m ałe (ta b .l), ale na obecnym etapie nie m ożna stw ierdzić k tó ra z tych m etod daje dokładniejsze wyniki dla om aw ianego celu, bo każda z tych m eto d m a różne uproszczenia. N atom iast na e tap ie projektow ania w stępnego stosow anie p roponow anej m etody je st uzasadnione ze w zględu n a jej prostotę, mały koszt o raz stybkość. W yniki oczywiście należy weryfikować w późniejszym etapie analizy Stateczności dynam icznej przy użyciu bardziej zaaw ansow anych m odeli obliczeniow ych lub w tunelu aerodynam icznym .
A cknoledgem ent: Funding for this project was su p p o rted in p a rt by T h e C o m m ittee for scientific R esearch , no. KBN. 00/S6/93/04.
