It is no coincidence that one of the inventors of wing sweep was also the principal proponent of the oblique wing concept. Jones noted that the large wave drag that arose in transonic and supersonic flow, even for a wing with infinite span, could be eliminated if that infinite wing were swept. Finite wings still produce inviscid drag, but the concept that much of the wave drag could be eliminated with sweep was verified in some of the classical NACA wind tunnel tests and remains a key aspect of modern wing design.

Selecting the optimal wing sweep is a starting point for high speed wing design, and for a variable-sweep wing, such as an oblique wing or oblique all-wing design, this may be scheduled with the flight condition. This seems straightforward, based on simple-sweep theory, but as discussed in the following sections, many subtlties arise.

2.1 Optimal Oblique Sweep

If one assumes, as in simple sweep theory, that the wing is sufficiently long that the component of freestream flow parallel to the wing long axis has little effect on the aerodynamics, then the required sweep is set by section performance considerations.

Figure 2.1. For very large aspect ratio wings, simple sweep theory may be used to determine optimal sweep.

In this case the effective Mach number is given by:

M_⊥ = M_∞ cos Λ

So, even at supersonic speeds, the wing may be designed with transonic, supercritical airfoils. Unfortunately, the effect that reduces the section effective Mach number also reduces the effective dynamic pressure, so that the 2-D effective C_l is given by:

C_l⊥ = C_l∞ / cos² Λ

So, although a wing at Mach 1.4 with 60 degrees of sweep at a CL of 0.25 may be created with sections designed for Mach 0.7, the design section Cl is 1.0.

Oblique wings must fly at sufficient sweep angles to keep the normal-component Mach number below the airfoil drag divergence Mach number, yet must fly at low enough sweep angle so that the normal-section lift coefficient does not become excessive.  Recognizing that increasing sweep not only increases the required section lift coefficient, but also reduces the effective span (aspect ratio), it would seem reasonable to expect that it would be best to fly with the lowest sweep angle that allows the airfoil sections to operate below drag divergence.

Table 1 shows the required sweep angle to maintain constant values of normal-component Mach number at various flight Mach numbers. Figure 2.2 shows the effective aspect ratio of an aspect-ratio = 10 wing as a function of sweep.  It can be seen that at a sweep of 60 degrees, the effective aspect ratio is 2.5, still fairly reasonable for an efficient supersonic aircraft, while a further increase in sweep from 65 degrees to 70 degrees reduces the aspect ratio from 2.0 to 1.0.

To a first approximation, these characteristics of the oblique wing effectively prescribe a relationship between sweep and cruise Mach number (within a range bounded by different airfoil thickness and characteristics).

The achievable wing loading is then related to M_⊥² C_l,
since: W/S = L/S = γ/2 P M_⊥² C_L
and: M_⊥ = M cos Λ and C_l = C_L/ cos²Λ.

It would seem to make sense then to design airfoil sections that seek to maximize M_⊥²Cl.  In the absence of any section pitching moment constraint, this would suggest supercritical sections. 

Experience and a survey of existing airfoils indicates that M_⊥²C_l is maximized for Mach numbers between 0.6 and 0.7 with C_l’s of 1.0 or more.  For an all-wing configuration, likely with a severe limitation on section pitching moment, the C_l and/or Mach number are likely to be somewhat lower.  This leaves the question open as to whether it better to design airfoils to operate at high M_⊥, at the expense of high C_l, or the reverse. Figure 2.3 shows a section designed at NASA Ames for an oblique wing-body with a high section lift coefficient but without a constraint on section moment.

Figure 2.3 An oblique wing section (14% thickness) designed to operate at a normal Mach number of 0.7 and a Cl of 1.0. (From Kennelly, reproduced in Ref. 5.)

The achievable normal-component Mach number and lift coefficient are strongly affected by airfoil thickness and allowable section pitching moment.  To illustrate the role of thickness in limiting the normal-component Mach number, consider two representative airfoils that have been used on previous oblique wing designs. (For comparison purposes achievable Cl and Mach combinations are selected with section compressibility drag increments of 0.001.)   Both are supercritical sections designed to produce a high lift coefficient at high normal-component Mach numbers.  The first, the OWRA-70-10-12, is 12% thick, and can tolerate a normal-component Mach number of 0.7 at a lift coefficient of 1.0.  It is closely related to the section shown in figure 2.3. The second is the OAW-60-85-16.5, a much thicker airfoil (16.5%) designed to meet the constraint of fitting passengers in an all-wing transport configuration.  Its design point has a normal-component Mach number of 0.6 at a lift coefficient of 0.85.

The role of section pitching moment will be discussed in more detail, but it is worth noting here that the strong aft loading typical of modern supercritical sections does offer a significant increase in achievable lift coefficient and Mach number.  For example, the OWRA-70-10-12 airfoil has a Cmo = -0.25 and achieves a value of M_⊥²Cl = 0.49.  A typical subsonic transport airfoil, the DSMA 526, is about 11% thick but has a Cmo = -0.07 and produces a Cl = 0.8 at a normal-component Mach number of .65, for a value of M⊥²Cl = 0.34 .

Typical transonic airfoil design points are selected so that the transonic wave drag is not zero, but is not so high as to reduce the aircraft performance or economics. The selection of an acceptable section wave drag is complicated in the case of highly swept wings by the fact that the section wave drag acts perpendicular to the isobars and so the net effect on wing drag is:

C_D = C_d / cos³Λ.

It appears, therefore, that one might push the section to operate much higher into drag rise than is typical for wings at more moderate sweep angles. However, other factors influence the selection of sweep angle to maximize performance at supersonic speeds.  This requires consideration of three-dimensional characteristics, not included in simple sweep theory.

2.2 Beyond Simple Sweep Theory – Three-Dimensional Effects on Oblique Wing Aerodynamics

Three aspects of 3-D wing aerodynamics importnat to oblique wing design are discussed here:

1. The effect of finite span on optimal sweep through 3D lift-dependent drag.
2. The classical effect of sweep on the wing lift distribution. 
3. A significant effect of the wing spanwise thickness distribution (volume) in supersonic flow.

 2.2.1 Finite Span Effects on Oblique Wing Performance

As noted in Chapter 1, the yawed ellipse represents an intriguing solution for minimum lifting drag with an elliptic distribution of area in both the spanwise and streamwise directions. A symmetrically-swept planform with the same span and length may also be created with an elliptical distribution of area in the spanwise and streamwise directions as shown in figure 2.4. Although the symmetrical design with the same span and length can achieve the same lift-dependent drag at low supersonic Mach numbers, it requires higher sweep and smaller chords for a given span, length, and area. This has adverse implications for viscous drag, structural weight, and internal volume, but the higher sweep and lower normal Mach number allow thicker sections, which offset these disadvantages.

Figure 2.4. For minimum drag, lift must be distributed elliptically over the span and length of the planform. The above figure shows two solutions from an interactive "game" for supersonic wing designers (Ref. 4).

The optimal span and length are fundamentally features of the three dimensional character of the wing and are directly related to the selection of the optimal sweep, emphasizing that the optimal sweep may not be chosen solely on the basis of infinite wing simple sweep theory. Increasing the sweep angle above that needed to keep M_^ below drag divergence increases the section Cl, but also increases effective length.  An interesting trade-off arises between the benefit of increased length in reducing the lift-dependent and volume-dependent wave drag and the benefit of decreased sweep to increase span, decreasing the vortex drag.  A further complication arises in that the wave drag depends on the distribution of lift and volume computed from a set of oblique cuts through the geometry at many different roll angles. So, although the simplest theory that applies near Mach 1 suggests that the two geometries in figure 2.4 would have the same lift-dependent wave drag, substantial differences exist above Mach numbers of even 1.4 and it is necessary to specify planform details to provide an accurate account of these more complex 3D effects.

One approach to investigating these trade-offs to determine the best wing sweep as a function of Mach number is to perform a series of optimizations using a realistic wing drag build-up model.  Such a model for oblique wings consists of a sum of skin friction drag, airfoil section compressibility drag, classical finite-wing induced drag, and supersonic lift-dependent and volume-dependent wave drag.  A more detailed discussion of how each of the drag components are modeled in such a drag build-up model is included in Appendix A. 

This drag model was used to determine the sweep angle for best L/D at a series of Mach numbers.  The series of figures that follow illustrates the combination of Mach number, sweep angle, and lift coefficient for best L/D for a wing with aspect ratio = 12, airfoil t/c = 0.12 and airfoil pitching moment (Cmo) equal zero.

Figure 2.5 shows the resulting optimum L/D vs. Mach number.   Figure 2.6 shows the optimal sweep vs. Mach number.  Figures 2.7 and 2.8 show the normal-component Mach number and lift coefficient, respectively, vs. Mach number. 

The optimization process finds best performance with more sweep than would be indicated from consideration of section properties alone. This is illustrated by the solution at Mach 1.4, which yields an optimum sweep of 64.5 degrees resulting in a normal-component Mach number of 0.6 and an average section Cl=0.7.    At this condition, the 2-D section compressible drag component is only 0.0004, the induced drag component is 0.0029, and the L/D is 12.1.  By comparison, if the sweep of the wing was reduced to 62 degrees while fixing CL, the airfoil section would be at drag divergence with a section Cl= 0.6 and a normal-component Mach number of 0.66. The 2-D section compressible drag component increases to 0.001, the induced drag component is 0.0024, and the L/D  is 11.9.  As the sweep is increased from 62 degrees to 64.5 degrees, the decrease in section drag is largely offset by the increase in induced drag, but the volume-dependent wave drag on the wing also drops from 0.0014 to 0.0010 leading to an improvement in L/D.  

Although a 2.5-degree change in wing sweep might seem insignificant, note the substantial change in normal-component Mach number and section Cl.  Based on the results in Figures 2.5-2.8, it appears that the tendency to fly at higher sweep than would be expected based on 2-D section performance becomes even greater at higher Mach numbers.  At Mach 1.8, the optimum normal-component Mach number has dropped to 0.52, with a section Cl=0.82, well below drag divergence for the 2-D section.  The challenge then becomes obtaining such high design lift coefficients with constraints on pitching moment and with 3D viscous effects.

Figure 2.5.  Optimum L/D vs. Mach number for AR=10, t/c=0.12 wing

Figure 2.7.  Normal-component Mach number for best L/D for AR=10, t/c=0.12 wing

Figure 2.8. Normal-section Cl for best L/D for AR=10, t/c=0.12 wing

2.2.2 Effect of Oblique Sweep on Spanwise Lift Distribution

A well-known effect of wing sweep is the variation of induced downwash along the span from the trailing wake that produces an additional lift distribution characterized by increased loading on the aft wing and reduced additional lift on the forward wing.  For a wing with no twist or bend, this results in a significant rolling moment, tending to roll the forward wing downward.  There is also a yawing moment from the asymmetrical distribution of induced drag that tends to unsweep the wing.  These moments scale roughly linearly with CL until the onset of flow separation on the wing. 

Naturally the wing needs to be appropriately twisted (or bent) so that the basic lift distribution combined with the additional lift distribution at the cruise CL results in a more-or-less symmetrical lift distribution (at least, one that has zero net rolling moment).  It turns out that parabolic bend of the wing along its structural axis is especially useful for achieving roll trim.  The combination of bend and sweep has the effect of increasing the angle of attack of the forward wing, and reducing the angle of attack of the rearward wing.  To illustrate the effect of wing bend, sample cases were run with a vortex-lattice model of an aspect ratio 6 oblique wing with 45 degrees sweep.  Wing bend was simulated by adding dihedral to the outboard 50% of each wing panel.  The vertical displacement of the wing tip has been referenced to the wing span.  These results are shown in figure 2.9.

In addition to designing the wing to have an appropriate deflection under 1-g loading to provide roll trim in this way, it is also possible to design the elastic stiffness of the wing so that increased g-loading due to gusts or maneuvers will cause the appropriate additional bend to trim the additional rolling moment.  If, for various reasons, the wing is stiffer than needed to provide this passive trim benefit, it must be recognized that the additional loading on the aft wing will aggravate any tendency toward buffet or separation there. If aileron deflection is used to re-trim the rolling moment of the wing at the higher CL, the localized de-cambering of the section may also aggravate the section to stall.  A well-designed oblique wing with reasonable gust/maneuver margins thus must have somewhat greater buffet (stall) tolerance on the aft wing than might be assumed from simple 2-D arguments.

 Figure 2.9.  Wing Bend Required to Maintain Roll Trim at Various Lift Coefficients for an Aspect Ratio = 6 Wing at 45 Degrees Sweep. 

2.2.3 Additional 3D Effects on Oblique Wing Aerodynamics

The other significant aspect of the finite oblique wing characteristics arises from the supersonic spanwise flow.   Normally when simple sweep theory is applied, an argument is made that the spanwise flow component has an insignificant effect on the wing pressures.  However, for a finite-span oblique wing with a supersonic spanwise flow component, this effect becomes substantial, and strongly influences the section pressure distributions at various locations on the wing.

Consider an equivalent slender body with the same cross-sectional area variation along the span as the oblique wing, operating at a Mach number corresponding to the spanwise Mach component.  Figure 2.10 illustrates a typical pressure distribution over such a body.  This pressure distribution represents a variation in the local pressure field that each 2-D airfoil section is immersed in.

Figure 2.10.  Equivalent slender body pressure distribution in supersonic flow.

 It is evident that the forward wing is immersed in a local flow field that is compressed somewhat above freestream pressure, and somewhat below freestream Mach number.  This results in a more benign flow environment than would be anticipated from simple 2–D analysis.  At the same time, the rear wing is immersed in a local flow field with accelerated flow, higher local Mach number and lower local pressure.  This has the effect of raising the (negative) Cp on both the upper and lower surface, so that although the section lift coefficient may be the same, the shock Mach number is higher and there are more strongly adverse pressure gradients on both the upper and lower surface.   These effects are shown schematically in Figure 2.11 and in computational results in Figure 2.12.  The CFD results show the wing colored based on Cp and the Cp distribution at several chordwise sections.  The acceleration of the spanwise flow is particularly evident in the color change over the slope discontinuity on the wing at mid span.

 2.3 Center of Gravity Location, Supercritical Airfoils, and the Effect of Section Pitching Moment

A challenging aspect of all-wing configurations is the packaging of payload, systems, engines and structure to achieve a center of gravity in the desired location.  Locating the center of gravity forward of the 25% MAC point may be difficult from a packaging standpoint, but leads to a statically stable vehicle with simpler control system requirements.  However, pitch trim with the c.g. so far forward requires airfoils with nearly zero or positive section pitching moments. 

Conversely, exploiting the useful internal volume to contain payload and systems is likely to lead to a rather aft c.g. location.  This allows airfoils with moderately negative section pitching moment that may provide a performance benefit compared to the airfoils required to trim with a forward c.g..  A practical constraint on aft c.g. location may arise, not only from stability requirements, but also from controllability requirements.  The incremental (added) lift resulting from a trailing-edge flap deflection typically has an effective center of pressure at about 40% chord.  As the c.g. is moved aft toward the 40% chord location, the incremental pitching moment produced by the flap deflection diminishes to zero.

Ignoring the packaging, stability, and controllability requirements for the moment, it is worthwhile to explore the performance tradeoffs with respect to c.g. location by enabling more aft-loaded (supercritical) airfoils to be used.  Based on the previous discussion, one may anticipate that supercritical sections would tend to reduce sweep at a fixed lift coefficient.  This should improve performance at subsonic cruise conditions where the increased span is clearly beneficial.  For supersonic cruise, the previous discussion suggests that the benefit would not be realized, since the optimal sweep increases to  reduce the lift- and volume-dependent wave drag. 

To study the tradeoffs affecting performance as c.g. location is changed, the wing drag build-up model previously described was incorporated into a vehicle performance/mission analysis program developed by Desktop Aeronautics (ref. x).  This program includes a structural weight model, engine performance model, available fuel volume constraint, and other operational constraints.  The effect of section pitching moment on the airfoil compressibility drag was modeled by linear interpolation between conventional sections and the supercritical OWRA-70-10-12 airfoils described previously.  This simple model correlates the airfoil section pitching moment to the drag divergence Mach number at a given lift coefficient. 

The center of gravity location was varied parametrically for a family of oblique all-wing vehicles optimized for a particular mission.  As an illustration of the mission analysis and design program's optimization capabiltities, airplanes were optimized for maximum cruise range with fixed take-off gross weight and fixed payload. Wing aspect ratio, t/c, and proportion of wing available for fuel tanks were fixed.  Oblique wing sweep, wing area, thrust, and cruise altitude were allowed design variables, while pitch trim, climb gradient at cruise, and fuel volume constraints were imposed.  The pitch trim constraint determined the required airfoil section Cmo for each specified c.g. location.  The climb gradient constraint effectively determines the required engine size depending on the cruise altitude, while the fuel volume constraint effectively limits the range for a given wing area.   The design optimization was repeated for several c.g. locations, expressed in the form of static stability margin. 

The results of this study are shown in figures 2.13 and 2.14.   Figure 2.13 shows the L/D vs. cruise Mach number, and figure 2.14 shows the optimum wing sweep vs. Mach number.  It is evident that, as expected, for subsonic cruise, there is a small performance advantage for designs with the c.g. located further aft.  At Mach 0.85, a c.g. shift of 10% of the MAC, from s.m. = 0.02 to –0.08, decreases the optimal sweep 1.2 degrees and the L/D improves from 21.2 to 21.6.  However, at supersonic cruise conditions, there is essentially no change in the optimal sweep or L/D associated with the c.g. shift.  It should be noted that the curve for a static margin of –0.18 in these figures represents a c.g. location that violates the controllability constraint described above.  Even with this extreme aft c.g., there is no performance benefit; in fact the L/D is lower for the aft c.g. cases at lower supersonic Mach numbers.  This indicates that characteristics other than the airfoil section moment (and associated drag-divergence Mach number), are determining the optimum cruise configuration.

2.4 Planform Design Features

 The next step in oblique wing design involves more than setting the oblique sweep. Many planform design variables can play an important role in oblique wing performance, stability, and control. A few of the more significant parameters are described in this section.

2.4.1 Symmetric Sweep 

The influence of the spanwise supersonic flow that increases the effective Mach number of the aft wing motivates the idea of introducing symmetric sweep in the wing planform, so that, combined with the oblique sweep, the aft wing is at a higher sweep angle than the forward wing.  A small amount of sweep is introduced by lofting the wing with various chord-fractions located on a straight line. For example, a wing that is lofted with a straight leading edge has forward sweep compared with a wing that is lofted with a straight quarter-chord.  A second potential advantage of symmetric sweep is in adjusting the location of the aerodynamic center relative to a fixed c.g. position.  This may aid in trim and/or allow selection of a different airfoil section Cmo. 

To evaluate the influence of symmetric sweep, four wing planforms (with an unusual wing tip shape, to be discussed later) were lofted with x/c = 0., 0.25, 0.4, and 0.55 fractions as straight lines.  Each case was computed with the Cart3D Euler code at Mach 1.6 and 65 degrees of sweep at a CL = 0.18.  Based on the results shown in figure 2.14, this is a somewhat aggressive flight condition, so the wings with unswept leading edges and quarter chord lines (x/c=0 and x/c=0.25) were also run at Mach 1.4.  Each wing incorporated parabolic bend, adjusted to produce roll trim.  Figure 2.15 shows the Cp distributions on the wing upper surface for each of the wings.  The effect of symmetric sweep on the pressure distribution at 80% semispan on the aft wing is shown in figure 2.16 for the four wings at Mach 1.6. It is evident from both figures that increasing the symmetric sweep reduces the effective Mach number, so the shock occurs farther forward, although for the Mach 1.6 cases, the Cp at the shock is about the same.  Symmetric sweep also reduces the effective sweep of the forward wing, resulting in changes in the normal-section pressure distributions.  In particular, the leading-edge suction peak present on the forward wing with the straight leading edge is suppressed as the symmetric sweep is increased.  Bearing in mind that a complete design process would involve tailoring the airfoil shapes on the forward and aft wing to achieve more desirable pressure distributions,  the purpose here is to show the sensitivity of the pressure distribution to subtle changes in planform shape.

There is a modest effect of this symmetric sweep on L/D, as shown in figure 2.17, but a more interesting effect is the change in yawing moment.  Although there is a modest L/D penalty associated with the reduction in yawing moment, this effect of (negative) symmetric sweep may still provide a means of establishing yaw trim.  Based on the results at Mach 1.6, it was hypothesized that the yawing moment is influenced by the spanwise distribution of normal-section pressure drag.  As the aft-wing sweep is reduced, the airfoils in that region produce more wave drag and a corresponding negative yawing moment, offsetting the natural positive yaw moment (which arrises from the spanwise distribution of induced drag).  To test this hypothesis, the unswept x/c = 0 and x/c = 0.25 wings were run at Mach 1.4, where the normal-section pressure drag should be greatly reduced.  It seemed reasonable then to expect that the trend of reduced yawing moment with reduced symmetric sweep would vanish. 

Figure 2.16.  Normal-section Cp Distributions at Mach 1.6 At 80% Semi-Span on the Aft Wing with Varying Symmetric Sweep.

Figure 2.17 shows the expected increase in L/D as Mach number is reduced from 1.6 to 1.4, due to a substantial reduction in normal-section pressure drag.  However, as shown in figure 2.18, the slope of the yawing moment change with sweep was unchanged.  Figure 2.19 shows the normal-section Cp distributions at Mach 1.4 at 80% semispan for these two wings.  It is evident that there is still a shock present, although the normal-component Mach number at the shock is only M_^ = 1.09, compared with M_^ = 1.32 for the Mach 1.6 cases. The wing with unswept x/c = 0.25 line shows a lower Cp value at the shock as well as a farther forward shock location.  Nevertheless, it seems unlikely that there would be sufficient wave drag at this section to account for the trend in yawing moment.

Figure 2.17 Effect  of Symmetric Sweep on L/D

Figure 2.18. Effect of Symmetric Sweep on Yawing Moment.

 
Figure 2.19. Normal-section Cp Distributions at Mach 1.4 At 80% Semi-Span on the Aft Wing with Varying Symmetric Sweep.

2.4.2 Asymmetric Taper

Another design option to address the issue of the aft wing operating at a higher effective Mach number is to make the planform asymmetrical with a more highly tapered forward wing and less taper on the aft wing.  This increases the local chord length on the aft wing, thus reducing the section lift coefficient for the same lift, and reducing the section t/c (assuming the physical thickness is maintained).  Both these changes should help reduce the shock strength on the aft wing. 

To illustrate the effect of asymmetric taper, the AR=10 trapezoidal wing used to study symmetric sweep was modified by increasing the taper on the forward wing to a taper ratio of 0.4 and decreasing the taper on the aft wing to a taper ratio of 0.6.  The t/c of the defining airfoil sections were modified by the same amount so that the physical thickness would be preserved.   Figure 2.20 shows the upper surface Cp distribution at Mach 1.4 for the baseline wing and the wing with asymmetric taper.   Although it is evident that the asymmetric taper was successful in reducing the shock strength on the aft wing, it also increases the leading edge suction peak on the forward wing.  The L/D of this wing is actually slightly lower.  Once again, a more refined design process would involve tailoring  the airfoil sections to achieve more desirable pressure distributions on both the forward and aft wings.

 
Figure 2.20. Effect of Asymmetric Sweep on Upper Surface Cp Distribution,  Mach 1.4.
Left: symmetric taper ratio = 0.5, L/D = 18.45, Right: asymmetric taper = 0.4/0.6 L/D = 18.39

These studies of symmetric sweep and asymmetric taper demonstrate the complexity of oblique-wing aerodynamic design and the need for a complete three-dimensional aerodynamic shape optimization to achieve  a satisfactory wing design. 

2.4.3 Wing Tip Shape Study

The wing tips of an oblique wing see a variety of flow conditions as the wing sweep is varied. The side edge of the forward tip becomes a leading edge with diminishing sweep as the wing sweep is increased.  At some combination of sweep and Mach number, a normal tip side edge becomes a supersonic leading edge.  Various concepts for modifying the forward wing tip shape based on heuristic arguments have been studied to determine the effect on wing performance, and the results from four different wingtip shapes will be presented here.

The same trapezoidal wing planform described for previous studies, with AR=10, taper ratio=0.5, was used for this tip-shape study. The baseline tip shape was simply the unmodified trapezoidal wing planform.  With 65 degrees of wing sweep, the tip has a supersonic edge swept 25 degrees. Lofting from the 8% thick airfoil at 90% semispan to the sharp side edge results in a maximum included angle of 6.7 degrees.

Assuming that a supersonic leading edge tip would be undesirable, the second tip has the side edge cut at a 40-degree angle to the normal chord axis.  With the wing swept 65 degrees, the tip side edge is now also swept 65 degrees, resulting in the same subsonic normal Mach number as the wing leading edge.  The tip edge is still sharp, and the wing surface was re-lofted by maintaining the maximum physical thickness at the 90% semispan chord.  This increases the included angle of the tip (in the section parallel to the span axis) to about 13 degrees,

A third wingtip shape was formed by maintaining the normal tip edge sweep from the baseline tip, but reducing its length by aggressively tapering the wing trailing edge.  The trailing edge was trimmed on a line parallel to the freestream intersecting the tip edge at 40% of the original tip chord length.  This results in a portion of the wing trailing edge becoming effectively a side edge and maintains the more slender “wedge angle” of the baseline tip.  However, the trimmed trailing edge significantly shortens the chord length at the 90% semispan section, so that airfoil is effectively thickened to a t/c=11%.

The fourth wingtip shape to be studied combines the concepts of the second and third tips, keeping the trailing edge trim line that forms a streamwise edge and trimming the tip edge to have 65 degrees of sweep with respect to the freestream.

 The four wingtip shapes described above are shown superimposed in figure 2.21.  It is difficult to determine which edges belong to which tips unless the description above is followed.  To aid in interpreting the figure, the tips are drawn in different colors.  Tip one is drawn in blue and orange, tip two in green and brown, tip three in red, yellow, and orange, and tip four in red and brown.

 
Figure 2.21. Four Wingtip Shapes Studied For Oblique Wings.  

Each of these tips was analyzed on the same wing at Mach 1.6  and CL=0.18 with the Cart3D Euler code.  Tip 1 and tip 4 were also run at Mach 1.4  In each case, the parabolic bend of the wing was adjusted to achieve roll trim about the center chord axis.  For this wing thickness, sweep, and CL combination, Mach 1.6 is somewhat too fast, resulting in high 2-D section drag.  The Mach 1.4 cases represent a more reasonable flight condition. Table 5 provides the inviscid L/D results for each wing tip.

 Table 5  Results of Wing Tip Shape Study

It appears that all of the alternate wingtip shapes result in lower L/D than the simple baseline tip.  There are two likely explanations for this result.  The second and third tip shapes reduce the effective wingspan, the third more so than the second.  The second tip also has a blunter included angle as a result of the wing lofting.  The fourth tip suffers from both the span reduction of the third tip, and the blunting effect of the second.  The trend of reduced L/D seems to correlate more strongly with the reduction in effective wingspan, with bluntness as a secondary effect.  Because of the highly swept tip edges, the reduction of effective span is likely to be even greater than would be predicted based only on geometry.  At even modest angles of attack, the sharp, highly swept side edge of tips two and four will form a leading-edge vortex sheet that will roll up to initiate the forward tip vortex.  The trajectory of this vortex will thus be displaced somewhat inboard from the extreme lateral extent of the tip. The overly aggressive Mach 1.6 flight condition seems to reduce the relative importance of the tip shape, with only a 4% drop in L/D for tip 4 relative to tip 1.  At Mach 1.4 where the 2-D section drag is much lower, tip 4 reduces the L/D over 8%.

2.5 Simple Drag Estimation for Oblique Flying Wings