Minimum drag wing configuration for aircraft operating at transonic speeds

A wing extension or tip fin is disclosed wherein the tip fin is joined to an aircraft wing to form a nonplanar wing configuration which minimizes induced drag during both low speed and high speed operation of an aircraft. The tip fin which is of generally trapezoidal geometry, extends streamwise along the end of the aircraft wing and is canted to project upwardly and outwardly therefrom. Additionally, the tip fin is twisted to toe-out relative to the freestream direction with the angle of twist varying along the lower portion of the tip fin length. Viewed from the side, the tip fin has a sweep angle at least equal to the sweep angle of the aircraft wings with the leading edge of the tip fin intersecting the wing tip chord at a position aft of the wing leading edge. A strake which extends along the upper surface of the wing from the wing leading edge to the tip fin leading edge, forms a smooth transition between the wing and tip fin. To provide maximum aerodynamic efficiency, the length and cant angle of the tip fin are established to reduce the induced drag of the wing-tip fin combination below that exhibited by the wing alone or by a conventional wing of area and span equivalent to that of the combined wing-tip fin. Interference and compressibility drag of the combined wing-tip fin is minimized by controlling the chordwise position of the tip fin and by the strake which not only provides an aerodynamically smooth wing to tip fin transition, but establishes a vortical flow pattern that maintains boundary layer attachment under high speed flight conditions. Further, the area of the tip fin is established for minimum profile drag, the variation in tip fin thickness ratio further minimizing interference drag and the tip fin twist compensates for spanwise loading on the wing to reduce induced drag.

BACKGROUND OF THE INVENTION 
This invention relates to high efficient aerodynamic lifting surfaces. More 
particularly, this invention relates to airfoils such as wings and control 
surfaces which are configured to reduce or minimize induced drag. 
It has long been recognized in the art that the aerodynamic efficiency of a 
lift producing airfoil such as an aircraft wing is affected not only by 
the profile drag of the airfoil, but by a drag component commonly called 
induced drag. This induced drag results from the pressure on the upper and 
lower surfaces of the wing and, from the airflow direction caused by the 
lift producing wing-like structure. With respect to induced drag, this 
airflow is most significant at and near the ends or tips of the wings 
since the pressure differential produces airflow that is transverse to the 
stream of lift-producing air. Since induced drag is most significant at 
the wing tips, it has been recognized in the art that, considering a wing 
having an area S, the aerodynamic efficiency can be increased by 
increasing the length of the wing, i.e., maximizing the wing aspect ratio 
b.sup.2 /S, where b is the wing span. 
Simply increasing the wing span to decrease the effect of induced drag and 
thereby increase aerodynamic efficiency is subject to structural 
constraints which limit the efficiency increases that can be attained. For 
example, to achieve structural integrity of the aircraft, the thickness of 
the wing structure must be increased as the wing is made longer. Such an 
increase in thickness increases the weight of the wing. Not only does the 
increase in profile drag which results from increasing wing aspect ratio 
detract from efficiency gains effected by a higher wing aspect ratio, but 
a point is eventually reached where the increase in profile drag totally 
offsets the benefits attained by a higher aspect ratio. Further, the 
attendant increase in wing weight often means that the wing must be 
operated at a higher coefficient of lift C.sub.L in order to provide the 
desired aircraft performance. Since the induced drag term is not only an 
inverse function of the wing aspect ratio, but is also directly 
proportional to C.sub.L.sup.2, it can again be seen that high aerodynamic 
efficiency cannot be attained simply by increasing wing aspect ratio. 
Because of the limitations associated with planar wings of increased aspect 
ratio, various other means of decreasing induced drag have been proposed. 
For example, in U.S. Pat. No. 1,724,110, which issued to E. G. Reid, on 
Aug. 13, 1929, relatively thin fins or shields that extend streamwise 
along the wings and project outwardly from the wing surfaces to prevent or 
impede transverse flow along the wing surfaces are utilized. In this 
respect, endplates that project orthogonally above and below the wing tips 
are disclosed along with similarly constructed shields or plates that can 
be utilized at various locatiions along the wingspan. 
Additionally, in U.S. Pat. No. 3,270,988, which issued to C. D. Cone, Jr., 
on Sept. 6, 1966, various other non-planar wing designs are described, 
which increase the effective aspect ratio of the wing to thereby attain 
higher aerodynamic efficiency. More specifically, although the disclosure 
of the Cone patent is primarily addressed to appartus and methods for 
analyzing nonplanar wing configurations with respect to ascertaining the 
effective aspect ratio and thereby determine the aerodynamic efficiency of 
the configuration, varius nonplanar wing geometries are suggested within 
this patent. Among these wing configurations are tubular sections of 
circular and elliptical cross-sectional geometry which are mounted at the 
tips of the planar wing with the axis of such tubular sections being 
substantially parallel to the flow direction of the freestream air and, a 
nonplanar wing which is divided at the tip into a number of branches that 
extend arcuately upward such that the wing terminus is effectively a 
number of "winglets" of different curvature which radiate from a planar 
wing. 
Although endplates and other structures that have been previously proposed 
are generally satisfactory with respect to low speed aircraft, various 
disadvantages and drawbacks have prevented such apparatus from being 
incorporated in the design of modern high-speed aircraft. First, since 
modern high-speed aircraft are generally designed to operate at relatively 
high coefficients of lift, the profile drag associated with endplates or 
other nonplanar wing geometries is of even greater importance than it is 
with respect to low speed aircraft. Further, since such endplates are 
located in and influence the flow field of the wing, interference drag is 
created which at least partially cancels the benefits achieved by 
decreasing induced drag. Such interference drag often increases and 
becomes more of a problem in high speed aircraft since flow separation can 
easily occur at the boundary or transition between the wing and endplate 
structure. Additionally, since such high speed aircraft generally cruise 
at transonic flight velocities, shock waves can be induced by such 
boundary or transition regions which result in compressibility drag that 
can completely offset any increase in aerodynamic efficiency that is 
effected under low speed conditions. Even further, and especially with 
respect to nonplanar configurations in which the structure for reducing 
induced drag also provides lift, the spanwise pressure distribution along 
the wing can be affected to greatly increased wing bending moments. For 
example, not only does the weight increase that is often brought about by 
the use of endplates increase the wing bending moment, but if the section 
of the wing which reduces induced drag also produces lift, such as the 
branched winglet arrangement in the patent to Cone, further increases in 
bending moments occur. 
Because of the above-mentioned drawbacks and others, there has been 
relatively little interest in applying endplates (or, as they have come to 
be called, tip fins or winglets) to modern high speed aircraft until the 
potential exhaustion of petroleum resources became apparent and, as a 
consequence, the cost of aircraft fuel increased. Recognizing that an 
increase in aircraft operating efficiency not only conserves fuel, but is 
important in providing aircraft that can be operated economically, those 
skilled in the art have thus begun to consider the design of more 
efficient aircraft engines and aircraft structure. 
In this respect, it should be noted that both the military and commercial 
operators of high speed aircraft presently possess a substantial number of 
such aircraft and a large number are also currently in production. Thus, 
not only are designs necessary and desirable for a new generation of 
highly efficient aircraft, but designs are required for retrofitting to 
existing aircraft and for incorporation in aircraft currently being 
produced without requiring major design changes. 
Accordingly, it is an object of this invention to provide a nonplanar wing 
configuration which exhibits high aerodynamic efficiency on both low speed 
and high speed flight of the aircraft. 
It is another object of this invention to provide a wing configuration 
wherein induced drag is reduced relative to a conventional wing of the 
same aspect ratio and wherein a wing tip fin produces minimal profile drag 
and causes minimal interference and compressibility drag. 
It is yet another object of this invention to provide a wing tip fin for 
reducing induced drag wherein a tip fin is also configured for causing 
minimal increase in wing bending moment normally attendant to adding 
structure to the outboard portions of the wing. 
Even further, it is an object of this invention to provide a wing tip fin 
of the above-described type which is readily incorporated in present 
aircraft designs and readily retrofitting to existing high speed aircraft. 
SUMMARY OF THE INVENTION 
These and other objects are achieved in accordance with this invention by a 
tip fin that is joined to the end portion of the wing of an aircraft to 
form an upwardly and outwardly projecting extension of the wing. Viewed 
from the side, the tip fin is generally trapezoidal in geometry with the 
leading edge of the tip intersecting the wing at a position aft of the 
wing leading edge. A fairing or strake extends rearwardly along the upper 
surface of the wing from the leading edge of the wing to the leading edge 
of the tip fin to form an aerodynamically smooth transition between the 
wing and the tip fin and control airflow within the transition region. 
As the tip fin extends upwardly and outwardly from the wing, the tip fin is 
not of constant airfoil section, but slightly decreases in camber over at 
least the lower portion of the tip fin. Additionally, the angle of 
incidence between the free stream air and the tip fin is not constant 
relative to the tip fin length. Rather, the tip fin includes twist such 
that, when the tip fin is viewed along its chordal plane, the root chord 
of the tip fin is directed outwardly or "toed-out" relative to the 
freestream direction by a predetermined twist angle with the twist angle 
smoothly decreasing over at least the lower portion of the tip fin length. 
In accordance with this invention, the outward projection or cant angle of 
the tip fin and the length of the tip fin relative to the semispan of the 
aircraft are established to provide greatly reduced induced drag relative 
to that of the wing alone. Further, the maximum thickness ratio of the tip 
fin and the variation in camber along with the variation in twist angle 
being established to further reduce induced drag. Additionally the leading 
edge sweep angle of the strake is established to create flow vortices 
which maintain boundary layer attachment througout the interface region 
between the wing and the tip fin to thereby further minimize interference 
and compressibility drag. Even further, the tip fin area, taper ratio, and 
leading edge sweep angle are established to minimize tip fin profile drag 
and interference drag, with the position at which the tip fin leading edge 
intercepts the wing chord also being controlled to further minimize 
induced drag.

DETAILED DESCRIPTION 
Referring now to FIGS. 1, 2 and 3, the tip fin of this invention 
effectively forms an extension of an aircraft wing to provide a nonplanar 
wing geometry having a tip section that projects upwardly and outwardly in 
a manner which minimizes induced drag and also minimizes interference and 
compressibility drag that is normally associated with nonplanar wing 
configurations. More specifically, and as can be seen in FIGS. 1 and 3, 
the tip fin (generally denoted by the numeral 10) extends outwardly and 
upwardly from the end portion of a wing 12 at an angle from the vertical 
which is denoted by .phi. in FIG. 3. 
As can most clearly be seen in FIG. 2, when viewed from the side, the tip 
fin 10 is of a generally trapezoidal geometry with the base of the tip fin 
10 extending from the trailing edge 14 of the wing 12 to a point aft of 
the wing leading edge 16. An aerodynamically smooth strake 18, which 
extends along the upper surface of the wing 12 from the wing leading edge 
16 to the leading edge 20 of the tip fin 10 forms a smooth transition 
between the tip fin 10 and the wing 12. As shall become clear upon 
attaining a fuller understanding of the invention, the point at which the 
tip fin leading edge 20 (if extended as shown in FIG. 2) intercepts a 
chord located at the tip of the wing 12 is determined by the area of the 
tip fin 10 and by considerations relative to minimizing interference drag. 
Further, as shall be described in more detail hereinafter, the position at 
which the strake 18 intercepts the tip fin leading edge 20 (i.e., the 
leading edge sweep angle of the strake 18) is established such that flow 
vortices are created in the wing-tip fin interface region which maintain 
boundary layer flow attachment during high speed flight of the aircraft. 
As can also be seen in FIG. 2, the tip fin leading edge 20 and the tip fin 
trailing edge 22 are inclined or swept relative to a vertical reference 
coordinate which is substantially perpendicular to the tip cord of the 
wing 12. In this respect, the tip fin leading edge angle (.LAMBDA. in FIG. 
2) is usually on the order of 35 degrees and, since the tip fin 10 is 
located in the wing flow field should generally be at least as large as 
the sweep angle of the aircraft wings. It has been found, however, that 
substantial variation in the tip leading edge sweep angle can be effected 
without detrimentally affecting the drag characteristics of the invention, 
such variation being useful in controlling the position of the tip fin 
center of gravity to alleviate any wing flutter problems that may occur in 
a particular wing design. The angle formed by the tip fin trailing edge 22 
is relatively unimportant to drag considerations of the invention and is 
established by other criteria such as the area of the tip fin and the tip 
fin taper ratio, i.e., the ratio of the length of the tip fin chord 34 to 
the length of the root chord 32. 
As can be seen in FIG. 3, the interface region between the tip fin 10 and 
the wing 12 is radiused relative to the fore and aft directions to form a 
smooth boundary region between the upper surface 24 of the wing 12 and the 
inboard boundary surface 26 of the tip fin 10 and a smooth arcuate 
transition between the lower surface 28 of the wing 12 and the outboard 
boundary surface 30 tip fin 10. As can also be seen in FIG. 3, the 
cross-sectional area of the tip fin 10 decreases gradually over the length 
of the tip fin 10 to form a relatively thin smoothly rounded or other 
aerodynamically smooth surface at the upper terminus of the tip fin. 
The variation is streamwise cross-sectional geometry along the length of an 
embodiment of the tip fin 10 of this invention is more specifically 
illustrated by FIGS. 4a, b and c, which depict the airfoil geometry of the 
tip fin at various locations along the tip fin length. In viewing FIGS. 
4a, b and c, it should be recognized that the horizontal coordinate is a 
line that lies in the free stream direction which passes through the tip 
fin trailing edge 22 with this coordinate being graduated in terms of the 
decimal percent of chord length at the particular location depicted, i.e., 
X/C in FIG. 4. Further, the ordinate (Z/C) in FIGS. 4a, b and c represents 
the decimal ratio between the distance from the above noted reference 
coordinate to the inboard and outboard boundary surfaces 26 and 30 of the 
tip fin 10 and the chord length at that particular location. Thus, the tip 
fin airfoil sections depicted in FIG. 4 are normalized relative to chord 
length at each depicted location to clearly depict the relative thickness 
ratio (i.e., thickness divided by chord length) and the cross-sectional 
geometric configuration of the tip fin 10. As can be seen by comparing 
FIGS. 4a, b and c, the maximum thickness ratio of the streamwise airfoil 
section of the depicted embodiment of the tip fin 10 is substantially 
constant at each position along the length of the tip fin 10. To minimize 
the profile drag of the tip fin 10 under high speed flight conditions, it 
has been found advantageous to contour the boundary walls 26 and 30 such 
that the airfoil camber decreases gradually throughout the lower portion 
of the tip fin (i.e., .eta.=0 to .eta.=0.417 in FIG. 4, where .eta. is a 
normalized length parameter that is equal to the ratio between the 
distance from the tip fin root chord 32 to the position at which the 
depicted streamwise airfoil is taken and the total length 1 of the tip fin 
10), and remains substantially constant throughout the upper portion of 
the tip fin 10. Further, by viewing the position of the tip fin leading 
edge 20 (X/C=0 in FIGS. 4a, b and c), it can be noted that the angle of 
incidence between the tip fin 10 and the freestream air varies over the 
lower portion of the tip fin length 1. More specifically, and with 
reference to FIG. 4, the root chord 32 of the tip fin 10 is effectively 
toed-out or twisted such that the tip fin leading edge 20 is outboard of 
the tip fin trailing edge 22 (i.e., is further from the centerline of the 
aircraft fuselage then the tip trailing edge). With reference to FIGS. 4b 
and 4 c, it can be noted that the amount of toe-out or twist of the 
depicted embodiment generally decreases along the length of the tip fin 10 
to the position .eta.=0.47 and is substantially constant throughout the 
remaining length of the tip fin. 
The twist relationship of the depicted embodiments is more clearly 
illustrated in the graph of FIG. 5 which depicts the angle of twist (i.e., 
the angle formed between the freestream direction and the chord of the tip 
fin 10) as a function of position along the tip fin length (i.e., .eta.). 
As can be seen in FIG. 5, the twist angle is slightly in excess of 
-3.degree. (where the minus sign denotes twist away from the aircraft 
fuselage) at the base of the tip fin 10 (.eta.=0) and gradually decreases 
to approximately -2.2.degree. at a position of approximately .eta.0.3. The 
twist angle then smoothly increases relative to position along the length 
of the tip fin to attain a value of approximately -2.5.degree. at the 
position .eta.=0.417. As previously described, the twist angle remains 
substantially constant throughout the remaining length of the tip fin 10. 
Having described the basic structure of a tip fin constructed in accordance 
with this invention, various parametric relationships of the invention 
will now be set forth so that those skilled in the art can recognize 
structural and dimensional relationships that are critical to maximizing 
the aerodynamic efficiency of an aircraft wing utilizing the invention. 
First, it has been found that the length 1 of the tip fin 10 and the cant 
angle .phi. are the most important dimensional parameters relative to 
minimizing induced drag. In particular, it has been determined that to 
achieve minimum induced drag, the ratio 1(b/2), where 1 is the tip fin 
length and b/2 is the semi-span of the aircraft, (i.e., the length of the 
wing 12), is directly related to the section design lift coefficient 
(C.sub.IDES) and is inversely related to the wing cruise lift coefficient 
(C.sub.LCRUISE) of the aircraft which is used to utilize the invention. In 
this respect, FIG. 6 provides an accurate estimate of the proper tip fin 
length for a range of C.sub.IDES between 0.4 and 0.8 and for cruise 
coefficients of lift equal to 0.35, 0.45, 0.55 and 0.65. Viewing FIG. 6, 
and considering the typical lift coefficients of transonic aircraft, it 
can be seen that the length ratio (1/(b/2)) of the tip fin of this 
invention ranges between 0.068 1/(b/2) and 0.25 1/(b/2) except for those 
very few aircraft which have a relatively high C.sub.IDES and which 
operate at relatively low C.sub.LCRUISE. 
In utilizing a tip fin having a length dimensioned in accordance with FIG. 
6, it has been found that the cant angle .phi. should not exceed 25 
degrees. In particular, although cant angles in excess of 25 degrees would 
sometimes appear to provide further reduction in induced drag, such cant 
angles cause undesirably high wing bending moments that would necessitate 
relatively thick wing structure to withstand high pressure loading. 
Further, the aircraft pitching moment also increases as a function of the 
cant angle .phi. and, with respect to cant angles in excess of 25 degrees, 
relatively large trim drag can be encountered to even further offset any 
potential reduction in induced drag. 
As previously mentioned, high aerodynamic efficiency of a wing-tip fin 
combination cannot be effected simply by structuring the tip fin for 
maximum reduction in induced drag, but interference and compressibility 
drag that is caused by the structural interface or juncture between the 
tip fin 10 and the wing 12 must be minimized. In accordance with this 
invention, several dimensional and structural constraints are utilized to 
minimize interference and compressibility drag terms. First, as will be 
recognized by those skilled in the art, the transition between the tip fin 
10 and the wing 12 must be aerodynamically smooth and structured such that 
shock waves are not generated under high speed flight conditions. In 
accordance with this invention, this criteria is met by the substantial 
radius formed between the upper surface 24 of the wing 12 and the inboard 
boundary surface 26 of the tip fin 10 and by the smooth arcuate transition 
between the lower surface 28 of the wing 12 and the outboard boundary 
surface 30 of the tip fin 10. Additionally, and as previously described, 
the strake 18 also provides an aerodynamically smooth transition between 
the leading edge 16 of the wing 12 and the leading edge 20 of the tip fin 
10. 
Beyond these general considerations, it has been found that the angle 
formed by the leading edge 36 of the strake 18 is of prime importance in 
effecting minimum interference and compressibility drag. In particular and 
referring again to FIG. 2, the leading edge 36 of the strake 18 is 
inclined or swept relative to the vertical such that a chord, drawn 
between the juncture of the strake 18 with the upper surface of the wing 
12 and the juncture of the leading edge of the strake 18 with the leading 
edge 20 of the tip fin 10 is inclined at an angle of 55.degree. to 
75.degree. relative to the vertical reference. In this respect, it has 
been discovered that utilizing such a sweep angle generates a vortical 
flow pattern along the base of the tip fin 10 which maintains boundary 
layer attachment during high speed flight to thereby prevent drag 
penalties that could at least offset the gains in aerodynamic efficiency 
effected by the invention. 
As previously noted, maximum aerodynamic efficiency of a nonplanar wing 
configuration which includes the tip fin 10 of this invention requires 
that the profile drag of the tip fin 10 be minimized. In this respect, it 
should be recognized that minimum profile drag can be effected by 
minimizing the area S of the tip fin 10 within the constraints imposed by 
the previously described length requirements and by other considerations. 
More specifically, the tip fin area S is geometrically a function of the 
tip fin length 1, the leading edge sweep .LAMBDA. and the tip fin taper 
ratio. Considering the previously set forth constraints on tip fin length 
and cant angle and the spanwise loading that the tip fin 10 exerts on the 
wing 12, it has been found that the ratio between the area of the tip fin 
10 and the wing 12 should be within the shaded region of the graph of FIG. 
7. More specifically, and with reference to FIG. 7, it can be seen that 
the ratio of the tip fin area to the area of the wing should be greater 
than or equal to 0.092 1/(b/2) and less than or equal to 0.138 1/(b/2). 
The exact tip fin area S that is utilized should fall within the above 
stated range and should be determined for each particular embodiment of 
the invention in view of secondary considerations which affect 
interference drag and/or tip fin profile drag. For example, it has been 
found that a slight improvement in interference drag can be attained when 
the projection of the tip fin leading edge 20 intersects the upper surface 
24 of the wing 12 as far aft as possible. Additionally, the tip fin taper 
ratio should be maintained large enough to prevent flow separation which 
could be experienced if the sectional lift coefficients of the outboard or 
upper portion of the tip fin 10 became excessive. In this respect, it has 
been found that tip fin taper ratios, between 0.25 and 0.35 are generally 
satisfactory. 
As previously described, the twist and the streamwise aerodynamic 
cross-sectional of the tip 10, including the variation in twist angle 
along the length of the tip fin, are primarily configured to minimize the 
loading effect of the tip fin 10 on the wing 12. Thus, it can be 
recognized that the exact thickness ratio and twist distribution are best 
determined by a consideration of the loading effects of the tip fin 10 on 
the wing 12 to which the tip fin is to be installed by experimentally 
varying these parameters to obtain as near to an optimum elliptical load 
distribution as is possible. However, since the tip fin cross-sectional 
geometry also affects interference and compressibility drag, it has been 
found that it is necessary to maintain the maximum thickness ratio of the 
tip fin 10 within a certain range which is graphically depicted in FIG. 8. 
Viewing FIG. 8, it can be observed that for an aircraft cruising at Mach 
0.65 or less, the maximum thickness ratio should lie between 0.02 and 
0.12, whereas for aircraft cruising at Mach 0.65 and higher, the maximum 
thickness ratio should be between 0.02 and approximately 0.12+0.4 
(0.65-M.sub.C) where M.sub.C is the aircraft cruise velocity in Mach 
number. 
In view of the previous description of the invention, it can be recognized 
that this invention is well suited for incorporation in existing aircraft 
designs and for retrofitting to aircraft presently in service to thereby 
improve aircraft operating efficiency and provide fuel savings. In 
particular, it can be recognized that, knowing the cruise coefficient of 
lift C.sub.ICRUISE and the section design lift coefficient L.sub.IDES, the 
length of the tip fin (in terms of the ratio it bears to the semispan of 
the aircraft) can be determined from FIG. 7, the exact cant angle, tip fin 
area and maximum thickness ratio can then be determined along with the 
remaining geometric relationships in accordance with FIGS. 7 and 8 and the 
design considerations previously described herein. 
The increase in aerodynamic efficiency typically attained with this 
invention can be exemplified by an embodiment of the invention which is 
configured for installation on a military transport aircraft which 
normally cruises at Mach 0.77 with a cruise coefficient of lift of 
approximately 0.426. In this embodiment, the twist distribution and the 
tip fin streamwise airfoil geometry corresponds to that depicted in FIGS. 
4 and 5, with the length of the tip fin being 0.135 (b/2) and the tip fin 
being canted at an angle of 20.degree. relative to the vertical. Utilizing 
these more critical dimensional values in the manner hereinbefore 
described, the total area of the tip fin (i.e., including the area of the 
trapezoidal region of FIG. 2 and the area encompassed by the strake 18) 
was established at approximately 1.5% of the aircraft wing area, the 
maximum thickness ratio was established at 0.066, the aspect ratio of the 
tip fin was established at approximately 2.33, the taper ratio was 
established at approximately 0.338, and the tip fin leading edge sweep 
angle was approximately 37.degree.. An analysis of this particular 
embodiment of the invention revealed an estimated reduction in total 
aircraft drag of approximately 7.8 percent, which is estimated to provide 
a decrease in fuel consumption on the order of 8.1 percent. In this 
embodiment of the invention, the twist distribution of FIG. 5 limits the 
increase in wing root bending moment to less than approximately 4.5% to 
thereby permit installation of the tip fin without major modification of 
the aircraft wing.