The Finite Wing
Enrique Ortega [email protected] Finite span wings. Background
• The solution of a three-dimensional finite span wing was first tackled by the English engineer F. W. Lanchester (1878-1946) [1]. • Lanchester was a practical engineer (and amateur mathematician) mainly involved in the automotive industry. Together with Kutta and Joukowsky, he was one of the pioneers in the development of the circulation theory. • Lanchester devised that if a wing creates a circulatory motion around itself, it must behave like a vortex. Thus, he replaced the wing by a so-called bound vortex, where the wing is the vortex core. • Due to Helmholtz’s theorem, he realized that the bound vortex cannot end at the tips, and the continuation must be in the fluid, in a form Lanchester’s drawing of the wing vortex system published in his book Aerodynamics (1907). of free tip vortices flowing downstream. Extracted from [1]. It is interesting to note that he supposed that all the free vorticity springing from the wing merged into the tip vortex. 2 – The finite wing • The system of free vortices creates an induced velocity field, whose vertical component is called downwash. • By mechanical principles, the force acting on the wing must have a counterpart in the momentum imparted to the air. Thus, the momentum created by the air moving downward (per unit time) can be assimilated to the wing lift force. • This concept helped Lanchester to answer the long standing question about the work needed to obtain sustentation. He pointed out that the work must be equal to the kinetic energy of the downwash field. He also realized the importance of the wing aspect ratio and its connection with the work needed for sustentation. • Lanchester’s ideas were systematized, expanded and set up in a practical mathematical form by L. Prandtl (1878-1953). • It is not clear how much of the credit in the modern wing theory is due to Lanchester, and how much to Prandtl. However, in a lecture to the Royal Aeronautical society in 1927, the latter said that the basic ideas of his theory had occurred to him before he saw Lanchester’s book (see a transcription in [1]).
3 – The finite wing Prandtl’s lifting line theory
Like Lanchester, Prandtl used a vortex system for the wing composed by a bound vortex plus two semi-infinite free trailing vortices extending downstream (horseshoe vortex). This system of vortices generates vertical downward velocities behind the wing, giving origin to an induced velocity field.
starting vortex
Note that the starting vortex is required for conservation of circulation. However, as its effects vanishes with the distance to the
Extracted from [2]. wing (usually b< wy() 4(y (bb /2)4(/2 y ) b Extracted from [2]. 22 (1) 4(/2)by where the magnitude of the distances is used. This model has some limitations: e.g. Eq. (1) shows that w(y) goes to infinite at the tips and, more importantly, is constant along the span. This would cause that L(y) = const., which is not realistic. To solve these deficiencies, the single horseshoe vortex is subdivided into a large number of horseshoe vortices having smaller span and variable circulation. The heads of these horseshoes vortices are aligned along the original bound vortex, forming the so called lifting-line. 5 – The finite wing Extracted from [2]. The trailing vortices (represented as a single line) are composed by pairs of vortex lines, each of them associated to a horseshoe vortex. By conservation, the circulation along each trailing vortex should be equal to the difference of circulation of the adjacent horseshoe vortices (i.e. equal to the change of circulation along the lifting-line). If the span of the horseshoe vortices becomes infinitesimally small, the circulation is continuous and the change of circulation along a segment dy is given by d ddy (2) dy which is precisely the circulation shed into the trailing vortices at any station along the lifting-line. Then, the total induced velocity at an arbitrary wing station y0 is 1(b/2 ddydy ) wy() (3) 0 4 b/2 yy0 The minus sign makes dw positive if d/dy<0 6 – The finite wing Disregarding for the moment the problem of determining the circulation, it is known from the Kutta-Joukowsky theorem that (y) will generate a distribution of lift along the span given by dL() y Vy() (4) dy which should vanish at the tips because they cannot hold any pressure difference. From this point of view, any wing section y0 is expected to operate with 2( y ) 0 (5) Cyl ()0 Vcy ()0 where, from results of the 2D theory, the section lift coefficient can be written by Cleffla()2yyyy00000 () () 2 ()(6) being eff(y0) the effective angle of attack (which accounts for the effects of the induced velocity field) and l0(y0) the zero-lift angle, both measured at the spanwise station y0. In Eq. (6), a(y0) is the absolute angle of attack. 7 – The finite wing The effective angle of attack at any spanwise station y canbewrittenas eff()yy00 i ()(7) eff where is the geometric angle of attack and V w i(y0) is the induced angle of attack. Assuming the latter to be small (w< using Eq. (3), i results wy() 1(b/2 dd y )dy ()y 0 (8) i 0 VV4 b/2 yy0 Hence, the absolute angle of attack of the section becomes aeffl()yyy0000000 () () il () yy () 1(b/2 ddydy ) ()y (9) l00 4 Vyy b/2 0 8 – The finite wing Introducing Eq. (9) into Eq. (6), and equating the result to Eq. (5), the fundamental equation of the Prandtl’s lifting line theory is obtained ()y 1(b/2 ddydy ) ()yy0 ()(10) 000 l Vcy()400 Vb/2 y y a i which is an integro-differential equation for the unknown (y). Its solution allows computing the lift of each section (see Eq.(4)), which integrated along the span gives the total wing lift and its derived aerodynamic quantities. Eq. (10) is solved next for: 1. Elliptical distribution of circulation (a particular case) 2. Arbitrary distribution of circulation (general case) 9 – The finite wing 1) Elliptical distribution of circulation Extracted from [5]. We assume that the circulation is given by 0 2 1/2 2y ()y 0 1 (11) b Note that = 0 @ y=b/2 where 0 = (y=0). According to the Kutta-Joukowsky theorem, the section lift (per unit span) can be obtained with Eq. (11) as 2 1/2 2y (12) Ly() V () y V 0 1 b and the lift distribution is also elliptical. From Eq. (3), the downwash results 1(())bb/2dydydy /2 ydy wy() 0 (13) 0 2221/2 4(14/)()bb/2yy00 b /2 yb yy where y0 denotes a given spanwise station and –b/2 y b/2. 10 – The finite wing Extracted from [3]. Using the following transformation (with 0) bb y cos ;dy sind (14) 22 Eq. (13) can be integrated(*) to obtain two important results: • The downwash and the induced angle of attack 0 (15) are constant along the span (here is a w()0 0 2b dummy variable, which represents any station along the span). 0 •Ifb, both the downwash and the induced (16) i 2bV angle of attack go to zero (this result is consistent with the 2D airfoil theory). (*) Hints: use the result of the Glauert’s integral and invert the order of integration. The induced angle results from Eq.(8). 11 – The finite wing The sectional lift (Eq.(12)) can be integrated along the span to obtain the total wing lift. This yields, b/2 2 0 b CydyC() (17) LL VSb/2 2 VS From Eq. (17), the value of circulation 0 can be obtained in terms of the wing geometry and flight conditions. Replacing 0 into Eq. (16), the induced angle of attack results 2 0 SCLL C b (18) where i 2 A 2bV b A S The parameter A is the aspect ratio of the wing (it can be also denoted by AR). 12 – The finite wing The induced drag According to the Kutta-Joukowsky theorem, the section lift acts normal to the local relative wind velocity V (which has components V and w(y)). This velocity vector makes an angle i with respect to the aerodynamic vertical axis. Thus, the local lift vector is rotated back at each section, and contributes with a component of force in the direction of V (note that the drag is always defined in the direction of the incident freestream velocity). The total force generated in this way along the span is named induced drag or drag due to lift. The section lift and induced drag (per unit span) are L()yVyy ()cos() i (19) Dii()yVyy ()sin() and assuming that w< 13 – The finite wing Accordingly, from Eq. (20) the induced drag of a section can be obtained by Dii()yLy () (21) where i is constant. Introducing Eq. (14) into Eq. (21), and integrating the latter along the span, it is possible to obtain C 2 C L (22) Di A Eq. (22) shows that: • The induced drag of a wing is proportional to the square of the CL (note that the generation of lift and the trailing vortices causing the rotation of the lift vector are both caused by the pressure difference across the wing). • The induced drag is inversely proportional the wing aspect ratio (CDi0when A (2D case)). 14 – The finite wing It is interesting to note that being i constant along the span, the resultant lift distribution is also constant (if there is no aerodynamic or geometrical twist), i.e. Clleffllill() y C,0,0 ( ) C ( ) C const . (23) Then, the lift per unit span results L(y) = Clqc(y). As the lift distribution is elliptical (see Eq. 14), the last result shows that the chord distribution must be elliptical too. In other words, an elliptical chord distribution is needed to obtain an elliptical lift distribution in an untwisted wing. If the local Cl is constant, from Eq. (23) we obtain CCll ,0 () il C L(24) Introducing Eq. (20) into Eq. (24), and differentiating with respect to , the wing lift slope results C Cl, L (25) 1 CAl, Extracted from [5]. 15 – The finite wing 2) Arbitrary distribution of circulation A general distribution of circulation can be expressed in terms of a trigonometric expansion (Fourier series). This leads to y b cos 2 () 2bV A sin n (26) with (27) n dy b sin d n1 2 and 0 (note that (0)=()=0). The unknown coefficients An must satisfy the lifting-line equation (10), thus, at any wing station y0 2( y ) 1(b/2 ddydy ) ()yy0 ()0(28) 000 l Va00()()4 y cy 0 Vb/2 y 0 y where a0 is the sectional lift slope. Eq. (28)canbewrittenintermsof using Eqs. (26), (27) and ddd d 2cosbV nA n (29) dy d dy n dy 16 – The finite wing Then, after some manipulation, it is possible to obtain (hint: use the Glauert’s result for the integral term) 4sinbA n sin n () n nA 0 ()0(30) 000ac()() nl sin 00 0 0 a i It should be noted that Eq. (30) has been set at a given spanwise station y0 (or 0) and has n unknowns (An). Thus, in order to solve them, it is necessary to evaluate Eq. (30) at n different spanwise stations (up to the desired number of terms). This procedure leads to an algebraic system of equations which can be solved for the unknown An coefficients. Symmetric terms of the expansion (1,3,5,…). Extracted from [5]. 17 – The finite wing From Eq. (30), and by analogy with Eq. (9), it is easy to recognize that sin n () nA (31) in sin where is a dummy variable along the span. Using the distribution of circulation (26), the wing lift coefficient can be obtained by 22b/2 b2 C ()y dy An sin sin d Ln SV b/2 S 0 /2 if n = 1 , zero otherwise 2b2 (32) CAARAL 11 S 2 where AR is the aspect ratio. It is important to note that the coefficient A1 cannot be determined in an isolated manner; the complete system (30) for the n coefficients must be solved. 18 – The finite wing The induced drag can be obtained as follows 2 b/2 Cyydy() () (33) Dii SV b/2 2b2 Ansin ( )sin dand introducing Eq. (31) n ni S 0 2 2b A sinnkAkd sin nknk S 0 2b2 kA Asin k sin n d nk S nk 0 /2 if n = k , zero otherwise Therefore, Eq. (33) results 2 2b 22 CnAARnA(34) Di nn S 2 nn 19 – The finite wing Eq. (34) can be rewritten as 22 CARAnA (35) Di 1 n n2 and introducing Eq. (32) into Eq. (35), the induced drag results 2 2 C A elliptic CnCL 11n (36) DDii 2 1 ARn2 A1 where 1 is a geometric parameter > 0 which depends on the wing planform shape. It is customary to define 1 (37) e 1 1 which is called the span efficiency or Oswald’s efficiency factor. Then, C 2 C L (38) Di ARe where the term AR·e can be seen as an effective aspect ratio. 20 – The finite wing The parameter 1 is computed from the An coefficients. As an example, it is plotted below as a function of the wing taper ratio ( =ct/cr). Extracted from [4]. Note that an aspect ratio 0.35 is optimal from the point of Extracted from [2]. view of the induced drag. The numerical result obtained for the elliptical wing is 0.4% less than the theoretical result (N = 200 is used). 21 – The finite wing The wing lift slope for an arbitrary distribution of circulation can be obtained by modifying slightly Eq. (25). This results C C L l, (39) C 1(1)l, AR where Cl, is the airfoil’s lift-slope and is a parameter which depends on the An coefficients and is constant for a given wing planform ( usually ranges between 0.025 and 0.25). For trapezoidal wings, reference [4] gives the following approximation CL AR Prandtl’s data for rectangular wings. Cl, (40) Extracted from [2]. 24AR2 22 – The finite wing Wing’s total drag estimation The induced drag CDi given by Eq. (38) is only a part of the wing’s total drag CD. The latter can be obtained by CCDDPDi C (41) where CDp is the contribution of the profile drag Cdp (viscous friction+pressure) along the span, i.e. 1 b/2 CCCDP dp(())() l y c y dy (42) S b/2 The Cdp in Eq. (42) can be obtained, for example, from experimental data. Note that this value depends on the local Cl and Re (it varies with the chord). 23 – The finite wing Computing lift distributions. The basic and additional lift The section lift coefficient at a given spanwise station can be written as Cyll() C,0 ( il () y () y ()) y (43) and its value depends on the geometric characteristics of the wing and the flight conditions ( or CL). In order to determine Eq. (43) for any particular condition with efficiency, the following lift distributions are defined: • Basic lift distribution: generates a CL=0 and depends on the wing planform, l0(y) and (y). 1 b/2 Cy() Cycydy ()() 0 (44) llbbS b/2 • Additional lift distribution: generates a CL=1 and only depends on the (untwisted) wing planform. 1 b/2 C ()y C ()()y c y dy 1 (45) llaaS b/2 24 – The finite wing Then, using definitions (44) and (45), the lift distribution at any flight condition can be expressed as Cy() C () y C () yC (46) llba lL For a given wing planform, l0(y) (can include flap or aileron deflection) and (y), the basic and additional lift distributions can be found as follows 1. Two lift distributions are obtained for different angles of attack C ()y C ()y C ()y CC ; ()y C ()y C ()y C (47) l1122 lba lL l l ba lL 2. Subtracting the lift distributions (47) Cy() Cy () Cy() Cy () CyC ()( C ) Cy () ll12 (50) ll12 lLLlaa 12 CC LL12 3. The additional lift distribution can be obtained by replacing Eq. (48) into any of Eqs. (47). The concept of basic and additional lift is useful for calculating loads for structural purposes and analyzing some wing characteristics, e.g. maximum lift and roll due to aileron deflection. When high non-linear effects are present in the flow (e.g. due to wake interaction in low aspect ratio wings, extensive flow separation, etc.), the applicability of Eq. (46) should be carefully evaluated. 25 – The finite wing Example: basic and additional lift distributions for the Piper Cherokee with 0 and 40º of flaps (see [4]) and analysis of CLmax. can be obtained from the two runs using CL =CL0 +CL, · Clmax depends on the section characteristics and Re(c)! increasing Cy() C () y C () yC C llba lLlmax CCy () Cy() llmax b Ls Cy() la CL(ys) is the wing’s CL for which Clmax is achieved at a section ys. Thus, the wing CLmax corresponds to the minimum of the values CL(ys). 26 – The finite wing Wing’s pitching moment The pitching moment about a point A located at cR/4 (see figure) can be obtained by summing the contributions of the wing sections along the quarter chord line, i.e bb/2 /2 M qCycydyqCycyydy()2 () ()()tan (49) Ambb/2ac /2 l where it is considered CzCl. In coefficient form 1 bb/2 /2 C C() yc2 () ydy C ()()tan ycyy dy (50) MmAacbb/2 /2 l Scref Using Eq. (46) (Cl(y)=Clb(y)+Cla(y)·CL), Eq. (50) results 1 bb/2 /2 CCycydyCycyydy ()2 () ()()tan MmAacbb/2 /2 lb Scref b/2 CCycyydy()()tan (51) Llab/2 27 – The finite wing Extracted from [4]. As it can be observed in Eq. (51), the wing’s total pitching moment about the point A can be recast in the form of CM CCA C(52) M A MacC L where L 1 bb/2 /2 C C() yc2 () ydy C ()()tan ycyy dy (53) Mmacbb/2 ac /2 lb Scref and CM 1 b/2 A Cycyy()()tan dy (54) b/2 la CScLref Eq. (53) is the free or pure moment of the wing, composed by the free moment of the sections and a couple caused by the basic lift distribution. Note that Clb contributes only if the wing has sweep (0). The part of the wing’s moment depending on lift is given by Eq. (54) x CL. As seen before, by definition of the aerodynamic center, Eq. (54) gives its position with respect to the point A. 28 – The finite wing Characteristics of finite wings. I. Taper ratio • Rectangular wings have a larger downwash near the tip, which reduces the efficiency of the wing. • Lowering the taper ratio reduces the downwash at the tip and the lift distribution tends to be elliptical. Also, the section load reduces towards the tip, which is beneficial from the structural point of view. • However, the local Cl (divided by the chord) increases with lower taper ratio and Re(y) reduces. Thus, an undesirable wing’s stall behavior can be obtained. Wing twist is frequently used. L()y Cyl () i ()y Extracted from [6]. 29 – The finite wing Extracted from [5]. II. Wing twist • Wing twist makes the angle of attack of the wing sections vary along the span. • The twist can be geometric (variation of the geometric angle of attack due to the torsion of the wing) or aerodynamic (variation on the wing section along the span l0(y)). • The twist is negative when the angle of attack of the tip is less than that at the root. Negative twist is known as wash-out. • The wash-out concentrates the aerodynamic loads inboard the wing. Since the twist does not modify the chord distribution, the effects on L(y) and Cl(y) are similar. • The twist can be used to improve the stall behaviorofthewingortoachievealift Extracted from [7]. distribution with reduced induced drag or other particular characteristics. 30 – The finite wing Effects of wing twist on L0 Suppose a wing with no geometrical twist and constant airfoil, which has all the sections’ ZLL aligned in a plane, such that CL=0 at =0. Then, fixing the root chord, imagine that an upward rotation is applied to the tip in such a way that 2 ()yy t (55) b After that, the CL will increase because most of the wing is now at (y)>0. Hence, at any spanwise section we can write Cyll() C,0 ( il () y ()) y (56) Now, suppose that CL=0 is recovered by rotating the overall wing nose down, such that = L0. For this condition we can assume i CL/A 0, and Eq. (56) gives CyllLl() C,00 ()( y ()) y (57) Then, the total wing lift can be obtained by integrating Eq. (57), i.e. b/2 Lq 2()()0 Cycydy (58) 0 l 31 – The finite wing From Eq. (58) it is possible to obtain S b/2 ( ) ()()y c y dy (59) 2 Ll00 0 which allows computing the variation of the zero-lift angle of attack of the wing L0 in terms of the airfoil l0 and the wing geometrical characteristics. Supposing a trapezoidal wing, we have cc cc c cy() crt y and c rt r (1 ) (60) r b /2 2 2 Then, defining the following change of variables ydy2;y 2dy (61) bb the chord distribution can be written as cy() cr (1(1 )) y (62) 32 – The finite wing Introducing Eqs. (61) and (62) into Eq. (59), it is possible to obtain 1 Sb2 () c ((1))yy dy (63) 22Ll00 tr0 and integrating Eq. (63), the zero-lift angle of the wing results 12 t (64) Ll00 31 Typical values of wash-out in general purpose aircrafts are about -3 or -4 degrees. 33 – The finite wing III. Wing sweep (at low speed) • Back sweep increases the downwash near the root (reducing lift) and the contrary effect is caused by forward sweep. • In swept-back wings, a higher lift near the tip causes a poor stall behavior and higher structural loads for the same CL (also the torsion moment of the wing increases). • Swept-forward wings have an opposite behavior, and the stall always tends to Extracted from [5]. start near the root. • Sweep wings generally need a higher twist to improve the aerodynamic and structural characteristics. • Sweep has also important effects on the stability and aeroelastic behavior of the airplane. 34 – The finite wing The sweep angle is usually defined with respect to the leading edge, or to the line passing through the quarter chord points. To obtain the sweep angle at an arbitrary chord position in a trapezoidal wing we can state b/2 bbtan tanfc fc , 0 f 1 (65) 22fLEtr LE f·cr Hence, 2 f f tan tan cc (66) f LEb r t cc c 2c ccrt r (1) 22r (1) Using the following definitions (67) Sb cAbS;(/) 2 bA it is possible to obtain 4 f 1 tan tan (68) fLEA1 35 – The finite wing IV. Wing mean aerodynamic chord • Wings have different geometric and aerodynamic characteristics, and their chord generally varies along the span. • From the point of view of the analysis (particularly for moments) it is necessary to define a characteristic length. The latter should have a physical meaning and has to be related to the distribution of lift (load) along the span. • This can be done by defining an equivalent rectangular wing, with no twist nor sweep, that has similar aerodynamic characteristics to the original wing (L and M). The chord of this equivalent wing is called the mean aerodynamic chord. 36 – The finite wing The wing aerodynamic chord can be obtained by equating lift and moment between the original and the equivalent rectangular wing. Focusing on the wing lift we state b/2 2()()qCycydyqCS (69) 0 lL Similarly, for the pitching moment we have bb/2 /2 2()()()2()()qCycyxydyqCycydyqSCXqSCc2 (70) 00lmoLM0 where x(y) is the line connecting the aerodynamic center of the sections andc is the mean aerodynamic chord sought. Supossing that CL=0, from Eq. (69) we have that Cl(y)=Clb(y). If the wing has not geometric or aerodynamic twist, then Clb(y) = 0 and Cm0 = constant. Therefore, Eq. (70) reduces to b/2 2 2()CcydyCSc (71) mo 0 M0 37 – The finite wing From Eq. (71), the mean aerodynamic chord results b/2 2 2 ccydy () (72) S 0 Eq. (72) can be integrated for a given chord distribution. Assuming a linearly tapered wing, this procedure leads to 21 2 cc (73) 31r It is important to note that Eq. (72) is obtained for a rectangular wing having the same lift and moment, and supposing CL=0 and no twist (Clb(y) does not contribute to the pitching moment unless the wing has sweep). In other cases, Eq. (72) is not accurate in the sense of the derivation here presented, but it is accepted as the standard definition of the mean aerodynamic chord. 38 – The finite wing V. Aerodynamic center of the wing From the geometry of the figure below, the pitching moment about a point A can be written as bb/2 /2 M qCycydyqCycyydy()2 () ()()tan (74) Ambb/2ac /2 l where CzCl. Supposing that XA is the distance from A to the aerodynamic center, then (75) M A MLXac A Differentiating Eq. (75) with respect to and equating to zero gives M X A (76) A L which can be solved for the position of the aerodynamic center; see for instance Eq. (54). 39 – The finite wing Extracted from [4]. Also, an approximated form of Eq. (76) can be easily obtained by considering an elliptic wing (recall that Cl(y)=const.=CL). This yields b/2 cyydy() 0 (77) XyA tanb/2 tan cydy() 0 wherey is the spanwise position of the centroid of the half-wing area. For a linearly tapered wing, Eq. (77) results [4] b 12 X A tan (78) 61 For the computation of the mean aerodynamic chord and the aerodynamic center location in general planform wings, see for instance NACA Report nº 751 (1942). 40 – The finite wing Extracted from [6]. References 1. von Karman, T. Aerodynamics. New York: McGraw-Hill (1963). 2. Anderson J. D. Jr. Fundamentals of aerodynamics. McGraw-Hill Book Company (1984). 3. Kuethe, A. M., Chow, C. Y. Foundations of aerodynamics. Bases of aerodynamic design. 5th edition. John Wiley and Sons (1998). 4. McCormick, B. W. Aerodynamics, aeronautics, and flight mechanics. 2nd edition. John Wiley and Sons (1995). 5. Katz J., Plotkin A. Low speed aerodynamics: from wing theory to panel methods. McGraw-Hill series in aeronautical and aerospace engineering (1991). 6. Hoerner, S. F., Borst, H. V. Fluid-dynamic lift: practical information on aerodynamic and hydrodynamic lift (1985). 7. Roskam, J., Lan, C. T. Airplane aerodynamics and performance. DAR Corporation (1997). 41 – References and complementary material Enrique Ortega [email protected]