Natural flow wing

The invention is a natural flow wing and a method for constructing the same. The method comprises contouring a three-dimensional upper surface and a three-dimensional lower surface of the natural flow wing independently of one another into a prescribed shape. Experimental data and theoretical analysis show that flow and pressure-loading over an upper surface of a wing tend to be conical about an apex of the wing, producing favorable and unfavorable regions of performance based on drag. The method reduces these unfavorable regions by shaping the upper surface such that a maximum thickness near a tip of the natural flow wing moves aft, thereby contouring the wing to coincide more closely with the conical nature of the flow on the upper surface. Nearly constant compressive loading characterizes the flow field over a lower surface of the conventional wing. Magnitude of these compressive pressures on the lower surface depends on angle of attack and on a streamwise curvature of the lower surface of the wing and not on a cross-sectional spanwise curvature. The method thereby shapes the lower surface to create an area as large as possible with negative slopes. Any type of swept wing may be used to obtain the final, shaped geometry of the upper and lower surfaces of the natural flow wing.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to a wing and, more specifically to a wing 
contoured to match the natural flow of air about the wing, and to a method 
of designing such a wing. 
2. Description of the Related Art 
Future supersonic military and commercial aircraft will be required to have 
high levels of lifting efficiency at subsonic, transonic, and supersonic 
speeds. Present philosophies for the design of wings vary greatly in 
addressing multi-point design conditions. A review of existing 
philosophies of wing design for subsonic, transonic, and supersonic flight 
reveals both contradictions and similarities. 
The contradictions exist mainly between the philosophies for subsonic and 
transonic (low-speed) cruise design and the schemes for supersonic cruise 
design. For low speed designs, the tendency is toward a low wing sweep, 
thick airfoils, and blunt leading edges, with supercritical airfoils being 
most commonly used. On the other hand, supersonic designs typically employ 
high-sweep wings with thin airfoils and sharp leading edges. Methods 
involving linear theory usually provide wing twist and camber. 
At maneuvering conditions, designs for both low-speed and supersonic wings 
utilize variable-camber devices such as leading-edge and trailing-edge 
flaps. At subsonic and transonic speeds, leading-edge flaps have succeeded 
fairly well. At supersonic speeds, however, leading-edge flaps have 
accomplished only minimal benefits in performance. Variable camber devices 
also have the added drawbacks of increased complexity in design, increased 
wing weight, and loss in usable volume. 
An alternate approach for meeting the required maneuvering conditions is 
developing a fixed camber wing. Generally, these wing designs have 
succeeded at their designed lift conditions but have suffered severe 
camber drag penalties at lower lift conditions. 
A conventional uncambered delta wing is conical about the wing tip. 
Experimental data in "Supersonic Aerodynamics of Delta Wings", NASA 
TP-2771, March 1988, and theoretical analysis in "The NCOREL Computer 
Program for 3-D Nonlinear Supersonic Potential Flow Computations", NASA 
CR-3694, August 1983, show, however, that the flow and pressure loading 
over the upper surface of a swept delta wing at subsonic, transonic, and 
supersonic speeds tend to be conical about the apex of the wing. The 
conical nature of the field of flow on the upper surface of the wing 
produces favorable and unfavorable pressure fields based on considerations 
involving drag. 
SUMMARY OF THE INVENTION 
An object of the invention is to address problems in wing design over a 
broad range of Mach numbers and lift conditions. 
Another object of the invention is to capitalize on the naturally-occurring 
flow field and resulting pressure distributions over a wing. 
A further object of the invention is to improve the structural and 
volumetric efficiency of swept wings by eliminating the need for 
variable-camber devices for flow management. 
Yet another object of the invention is to provide a wing with fixed 
geometry that reduces pressure drag from zero-lift through high-lift 
conditions. 
The present invention attains the foregoing and additional objects by 
contouring a three-dimensional upper surface and a three-dimensional lower 
surface of a wing independently of one another into a presecribed shape. 
As discussed earlier, experimental data and theoretical analysis show that 
flow and pressure loading over the upper surface of a swept delta wing at 
subsonic, transonic, and supersonic speeds tend to be conical about the 
apex of the wing. This conical nature of the flow field on the upper 
surface of the wing produces favorable and unfavorable regions of 
performance based on drag. The present invention reduces these unfavorable 
regions of performance by shaping the upper surface such that a maximum 
thickness at a center of the wing moves forward and that a maximum 
thickness near a tip of the wing moves aft, thereby contouring the wing to 
coincide more closely with the conical nature of the flow on the upper 
surface. 
A nearly constant compressive loading characterizes the flow field over the 
lower surface of the wing. Magnitude of these compressive pressures on the 
lower surface depends on angle of attack and a streamwise curvature of the 
lower surface of the wing and not on a cross-sectional spanwise curvature. 
The present invention thereby shapes the lower surface to create an area 
as large as possible with negative slopes. 
The present invention employs a scheme that redistributes an area from the 
lower surface to the upper surface of the wing at a given streamwise 
cross-section. The scheme optimizes a cross-sectional slope of the upper 
surface. 
Any wing defined by a leading-edge sweep, a position of a maximum 
thickness, a distribution of an airfoil thickness and a leading-edge 
radius may be used to obtain the final, shaped geometry of the upper and 
lower surfaces of the present invention. Compared to a conventional 
constant airfoil swept wing, a natural flow wing has an increased 
forward-sloping projected area and more gradual slopes on the upper 
surface forward of the maximum thickness, reduced slopes on the upper 
surface aft of the maximum thickness, and a lower surface with negative 
slopes.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
FIG. 1 illustrates a streamwise cut of a conventional, constant airfoil 
swept wing 10 with an upper surface 11, a lower surface 12, a leading edge 
18, a maximum thickness 13, and a chord length 14. An airfoil shape 15 is 
constant along the span of the wing. The conventional, constant airfoil 
swept wing also has along the span a constant thickness ratio, defined as 
the maximum thickness 13 divided by the chord length 14, or the thickness 
ratio might decrease along the span. 
As shown in FIG. 2, a line of maximum thickness 23 locates the maximum 
thickness 13 of each airfoil shape 15 along the span of the wing. Standard 
application of the thickness 13 to an uncambered swept delta wing 20 
results in a wing which is conical about a wing tip 16 as depicted in FIG. 
2. Area in front of maximum thickness line 23 has a forward-sloping 
surface 21 and area behind line 23 has a rearward-sloping surface 24. 
FIG. 3 shows an upper surface 30 of the swept delta wing 20. Results 
obtained for the delta wing 20 are readily transferable to other planforms 
due to the generic nature of the geometry of the delta wing 20. The upper 
surface 30 extends in a spanwise direction 31 from a root 33 to a wing tip 
35 and in a streamwise direction 32 from an apex 34 to a trailing edge 36. 
A leading edge 37 extends from the apex 34 to the tip 35. 
Experimental data and theoretical analysis show that for flow direction 22 
a pressure loading 25 over the upper surface 30 of the swept delta wing 20 
at subsonic, transonic, and supersonic speeds tends to be conical about 
the apex 34 and not conical about the tip 35. The conical nature of the 
pressure field 25 on the upper surface 30 of wing 20 produces favorable 
and unfavorable pressure fields based on considerations involving drag. 
As demonstrated in FIG. 4, the pressure 25 over upper surface 30 for wing 
20 at lifting conditions is characterized by an outboard region 41 of 
pressure below freestream static pressure and an inboard region 42 of more 
positive pressure. A line of recompression 43, which can be straight as 
shown or curvilinear, emanates from apex 34 and extends toward trailing 
edge 36. The line of recompression 43 is generally independent of wing 
geometry and marks a border between outboard region 41 and inboard region 
42. 
As shown in FIG. 4, maximum thickness line 23 and recompression line 43 
intersect one another and divide upper surface 30 into four zones 46, 47, 
48, and 49 that identify two favorable and two unfavorable zones of 
performance. The two unfavorable zones of performance are inboard forward 
zone 46 and outboard aft zone 48. Inboard forward zone 46 experiences a 
compression of flow ahead of the maximum thickness line 44, resulting in 
more positive pressures acting on the forward-facing surface 21 that 
produce high drag levels. Rearward-facing surface 24 combines with high 
negative pressure coefficients in outboard aft zone 48 to produce high 
drag levels. The two other zones 47 and 49 of upper surface 30 have 
pressure fields 25 which combine favorably with the local surface geometry 
to produce reductions in drag. 
The present invention reduces or eliminates the unfavorable zones with high 
drag 46 and 48 by locating a first maximum thickness line 44 near the 
recompression line 43 as depicted in FIG. 5 for natural flow wing 55. By 
moving the maximum thickness line 44 forward at centerline 51 and sweeping 
line 44 aft to the trailing edge 36, a first forward-sloping surface 52 is 
minimized in inboard region 42 and maximized in outboard region 41. 
Further, a first rearward-sloping surface 54 is maximized in inboard 
region 42 and minimized in outboard region 41. The resulting shape of 
upper surface 30 has a small forward-sloping surface 52 ahead of line 44 
and a large rearward-sloping surface 54 behind line 44 in the inboard 
region 42. Upper surface 30 also has, in outboard region 41, a large 
forward-sloping surface 52 and a small or no rearward-sloping surface 54 
behind line 44. The shape of upper surface 30 thus coincides with the 
conical nature of the pressure field 25. 
FIG. 6 illustrates alternative locations of maximum thickness line 44 on 
upper surface 30 that would provide the favorable results of a natural 
flow wing 55. The alternatives shown in FIG. 6, however, are not 
exclusive. Preferably, the maximum thickness line 44 meets centerline 51 
at a 20 percent chord length to provide a smooth airfoil at the root 33 as 
illustrated in FIG. 5. 
As appears in FIG. 7, a lower surface 70 of the swept delta wing 20 extends 
in the spanwise direction 31 from the root 33 to the wing tip 35 and in 
the streamwise direction 32 from the apex 34 to the trailing edge 36. The 
leading edge 37 extends from the apex 34 to the tip 37. 
Flow 75 over the lower surface 70 of wing 20 behaves quite differently at 
positive angles of attack than the upper surface 30. Lower surface 70 thus 
requires a different type of geometry. FIG. 7 demonstrates that the flow 
75 over lower surface 70 is characterized by a nearly constant compression 
loading 71. The magnitude of compressive pressure 71 primarily depends 
upon wing-surface streamwise curvature and not on spanwise cross-sectional 
curvature. The most beneficial lower surface 70 would thus have as large 
an area as possible with a rearward slope. 
Conventional wing 10 typically has a second line of maximum thickness 77 
for lower surface 70 as illustrated in FIG. 8. Area in front maximum 
thickness line 77 has a second forward-sloping surface 78 and area behind 
line 77 has a second rearward-sloping surface 79. 
The present invention maximizes a second rearward-sloping surface 82 and 
consequently minimizes a second forward-sloping surface 81 by locating a 
second maximum thickness line 84 as demonstrated in FIG. 9 for natural 
flow wing 55. The resulting shape of lower surface 70 has a small 
forward-sloping surface 81 and a large rearward-sloping surface 82 as 
shown in FIG. 9. 
Additional modifications to the conventional wing 10 would further improve 
aerodynamic performance. Referring to FIG. 10, spanwise cuts 100 reveal 
that the thickness 13 of conventional wing 10 typically decreases in the 
spanwise direction 31 and that thickness 13 approaches zero at wing tip 
16. Referring now to FIG. 5, maximum thickness line 44 meets centerline 51 
at a 20 percent chord length to provide a smooth airfoil at the root 33. 
This location of line 44 is not ideal because the more positive pressures 
of inboard region 41 are acting on forward-sloping surface 52 and thereby 
producing high drag levels. To compensate for this drawback while 
retaining a smooth airfoil at the root 33, the present invention decreases 
maximum thickness 13 at root 33 and increases thickness 13 at the tip 35, 
with an example of one possible final shape of natural flow wing 55 
depicted with spanwise cuts 110 in FIG. 11. 
Referring to FIG. 12, streamwise cuts 120 of the conventional wing 10 
illustrate that leading edge 18 has a constant bluntness as the wing 
extends in spanwise direction 31. The present invention preferably 
contours upper surface 30 and lower surface 70 to form leading edge 37, as 
shown in FIG. 13, with increasing bluntness in the inboard-to-outboard 
spanwise direction. The combination of increasing the thickness ratio and 
leading-edge bluntness from the root 33 to the wing tip 35 provides a more 
constant wing thickness 13 from the root 33 to the tip 35. Thus, lower 
drag levels are achieved because the nearly constant low pressure at the 
leading edge 37 has a better surface geometry to act upon. 
Referring to FIG. 14, the forward-sloping surface of upper surface 11 of 
conventional wing 10 has a forward slope 140 and the rearward-sloping 
surface has a rearward slope 142. A steeper forward slope provides greater 
frontal area as discussed above. The present invention preferably changes 
the conventional, constant airfoil wing 10 such that upper surface 30 of 
natural flow wing 55 has a majority of forward-facing slopes 141 steeper 
than forward slope 140 and a majority of rearward-facing slopes 143 less 
steep than rearward slope 142, as shown in FIG. 14. 
FIG. 14 also demonstrates that the forward-sloping surface of lower surface 
12 of the conventional wing 10 has a forward slope 144 and the rearward 
slope of 146. The present invention preferably contours lower surface 70 
of natural flow wing 55 such that a majority of forward-facing slopes 145 
are less steep than forward slope 144 and a majority of rearward-facing 
slopes 147 are steeper than rearward slope 146, as illustrated in FIG. 14. 
Frontal area should be minimized on the lower surface because no region of 
low pressure (expansion) exists. 
Elevation cuts 150 of the conventional wing 10, which are identical for 
both the upper surface and lower surface, appear in FIG. 15. FIGS. 16 and 
17 illustrate preferred elevation cuts 160 and 170 for natural flow wing 
55. Note that elevation cuts 160 for upper surface 30 differ in pattern 
from elevation cuts 170 for lower surface 70. 
A wing geometry generation scheme would allow for a broad range of 
analytical wing surfaces to be developed from a few input parameters. Any 
wing defined by a leading-edge sweep, a position of maximum thickness, a 
distribution of an airfoil thickness, and a leading-edge radius may be 
used to obtain the final, shaped geometry of upper surface and lower 
surface 70 of natural flow wing 55. Distribution of the airfoil thickness 
and maximum thickness position may be defined by the equation: 
##EQU1## 
where a.sub.n and b.sub.n are variables which can be series themselves and 
.eta. is the nondimensionalized spanwise location. For the conventional 
wing 10, a constant airfoil 15 is employed which has a variation in 
airfoil thickness and airfoil thickness position defined by all a.sub.n 
and b.sub.n equal to zero for n&gt;0. Examples of possible wing shapes which 
may be defined by the above equation are demonstrated in FIGS. 18(a), 
18(b), 18(c) and 18(d). Preferably, a wing with a positive leading edge 
sweep greater than 20.degree. and a trailing edge sweep of any value may 
be used to obtain the final, shaped geometry of natural flow wing 55. 
The present invention also employs a scheme that redistributes an area from 
lower surface 70 to upper surface 30 at a given streamwise cross-section 
in order to optimize local cross-sectional slopes of upper surface 30. The 
scheme to redistribute the area is defined by the equation: 
##EQU2## 
where a.sub.n and b.sub.n are variables as discussed above and .xi. is the 
nondimensionalized streamwise location.