To aid in conceptual understanding and discussion of the unusual aerodynamic characteristics of oblique wings, it is often convenient to refer forces and moments into a coordinate system that is aligned with the principle inertia axes of the vehicle. In this axis system, the y-axis extends along the span, the x-axis points ‘aft’ along the centerline chord, and the z-axis points up, normal to the plane of the wing. In the following discussion of trim, stability, and control, unless otherwise noted, pitching moment will be assumed to be nose-up about the span axis, rolling moment to the right about the chord axis, and yawing moment to the right about the normal axis.
Oblique wings produce rolling moment, yawing moment, and side force that vary as lift coefficient is increased, in addition to more customary lift, drag, and pitching moment variations. Strategies for designing and operating the vehicle for trimmed flight deserve special attention.
To illustrate these trim strategies, an example oblique wing was analyzed with CART3D and the untrimmed forces and moments estimated as shown below. The example wing is shown in figure 3.1. It has an unswept span of 60 ft., a reference area of 360 sq.ft., and a reference chord length of 8 ft.. At supersonic speeds (Mach 1.6), the wing is swept 65 degrees, flies at CL = .18 and incorporates a parabolic wing bend necessary for roll trim at this condition. At Mach 0.8, with 35 degrees of sweep and CL=0.45 the wing is assumed to incorporate the same bending amplitude as in supersonic flight and so is not trimmed in roll without additional control input.
Figure 3.1. Example oblique wing planform for trim strategy evaluation.
In the supersonic case, the forces and moments are summarized below.
CL=0.179
CD=0.01305 (L/D = 13.7 inviscid) + .008 (CDp) = 0.021
CY=-.00174
Wind axes moments:
Crollw=-.0019 (based on span)
Cmw=-.00749 (based on chord)
Cnw=.00064 (based on span)
Principal axes moments:
Croll = Clw cos Λ - Cmw (c/b) sin Λ = 0.0001 (based on span)
Cm = Cmw cos Λ + Clw (b/c) sin Λ = -0.016 (based on chord)
Note that moments must be referred to common reference length.
Trim about the long axis in pitch may be achieved with elevator deflection, which changes Cm0, or by properly locating the center of gravity. For the sections assumed in this example with almost 0 moment at zero lift, the net pitching moment of -0.0161 may be trimmed by moving the c.g. aft of the reference center (here at 25% chord) by:
dx/c = Cm/CL = -0.0161/.1799 = -0.09, which places the c.g. at 34% of the reference chord.
Because the character of the pressure distribution is similar at subsonic and supersonic speeds, very little change in trailing edge deflection or c.g. position is required to trim in the subsonic case.
Swept wings have a nonuniform spanwise distribution of downwash induced by the trailing wake. For an oblique wing, this results in an asymmetrical lift distribution where the rear wing panel produces more lift. At moderate sweep angles and lift coefficients, it is possible to twist the wing to produce a symmetrical lift distribution. However, since the degree of asymmetry in the lift distribution is strongly coupled to the wing sweep, the wing twist will only be correct at one sweep angle. A more useful means of making the lift distribution symmetric is to incorporate a small amount of parabolic dihedral (wing bend) into the 1-g wing shape. As the wing is swept, the dihedral has the effect of increasing the incidence on the forward wing, and reducing the incidence on the rear wing. The incidence angle is approximately proportional to the sine of the sweep angle, so a wing can be designed to have a nearly symmetrical lift distribution over a wide range of sweep angles.
As with a twisted wing, an oblique wing with parabolic bend is only perfectly trimmed at one lift coefficient for a given sweep angle. At other lift coefficients, traditional aileron roll control is required to balance the remaining untrimmed roll moment. In addition to designing the wing with the appropriate parabolic bend to provide roll trim under 1-g loading, 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. A typical value of wing flexibility to maintain passive roll trim is about 1% of span per ‘g’, based on the example in figure 2.9. Of course, the desired stiffness will change with aspect ratio and other design parameters.
Small changes in rolling moment may also be effected by aileron input. If one assumes, for this example, an aileron geometry shown in figure 3.3, then the rolling moment coefficient (about the centerline chord axis) is estimated as follows:
aileron lift = q Δ Cl (5)(5) cos2Λ
So: rolling moment = q (2) ΔCl (5)(5)(25) cos3Λ = q (.000101)(360)(60)
and the rolling moment coefficient is:
ΔCroll = 0.029, which suggests that modest changes in section Cl (and aileron deflection) at transonic conditions are sufficient to produce fine roll trim.
Figure 3.3. Example aileron dimensions.
The aerodynamic forces on an oblique wing include side-force components that are not experienced on symmetrical aircraft. Although the side force is large enough that it cannot be neglected, it is easily trimmed by flying with a small bank angle, on the order of one or two degrees. The sideforce has two origins, one resulting from the wake-induced downwash on the lifting vortex system, the other a result of the supersonic wave drag.
When the bound vortex is swept, the induced downwash produces a side force in addition to an induced drag force. An oblique wing with the right tip forward (a standard that has been followed in most studies over several decades) has a positive side force in subsonic flow. This side force is easily trimmed by flying with a small bank angle. In fact, care must be taken in interpreting computational predictions, since most codes vary the angle of attack by rotation about the normal Y axis that produces a wing bank angle as well (the forward tip is elevated with respect to the aft tip). The resulting side-force component of the lift vector overwhelms the induced side force.
Figure 3.4. Sideforce trim approach using wing bank.
In supersonic flow, there is also a side-force component from the integrated pressure distribution. This is essentially the volume-dependent wave drag of the wing subjected to the supersonic spanwise flow component. An oblique wing with the right tip forward has a negative side force in supersonic flow.
In the example case, at supersonic conditions, the computed bank angle to trim is:
φ = tan-1 CY/CL = tan-1 (-.00174/.179) = 0.55 degrees
At subsonic conditions the result is still less than 1 degree.
The spanwise distribution of induced drag and 2-D section drag results in significant yaw moments. Of the lateral forces and moments generated by the oblique wing, the yaw moment is perhaps the most difficult to deal with. A fin on the aft wing provides the most natural means of yaw trim, but the lift load (and drag) on a fin located above the wing also produces a pitching moment about the span axis. In the example case here, a single vertical fin with 3% of the wing reference area (10.8 sq ft) located at Y=-25 ft requires a lift coefficient of only 0.057 to trim the residual yawing moment. A second fin on the lower surface cancels the pitch coupling effect, but creates ground clearance problems. An upper surface fin may also be canted so that the line of action of the lift load on the fin passes near the span axis, minimizing the pitch coupling. A fin may be undesirable for other reasons, but the alternatives are not very satisfactory.
Split ailerons, similar to those used on the B-2, are effective at low sweep, subsonic conditions, but loose effectiveness rapidly at high sweep angles.
Figure 3.5. Split ailerons have been considered for yaw trim, but are probably inefficient approaches.
In any case, a relatively large drag penalty would be paid for trimmed flight with the significant split flap deflection needed for trim, not just control and/or stability augmentation. Using 2-D results from Horner (subsonic) with:
ΔCd = 0.6 @ δf=60° (ref to flap area) using aileron geometry in figure 3.3 we find that split aileron deflections of about 30 degrees are required, implying that it is difficult to trim at high sweeps with this approach. Even subsonically, the required deflection is similar.
Engine offset may also be used to produce yaw trim, but at high sweep angles, the required engine offset is large enough to cause problems for low-speed trim.
Figure 3.6. Yaw trim with engine thrustline offset -- example calculations.
An offset of 7.4 ft in the unswept long axis direction is required to trim the example at subsonic speeds.
Multiple engines would be required to allow trim over a range of flight conditions by differential thrust. This requires excess installed thrust. Differential thrust vectoring is also rather ineffective because of the short moment-arm of the off-axis thrust component about the c.g. and any thrust-related trim strategy naturally becomes throttle-dependent.
Figure 3.7. Yaw trim with thrust vectoring -- example calculations.
A cascade of very small fins on the wing upper surface may provide a satisfactory compromise with other design objectives. With a total area of 10 sq ft, the fins shown in figure 3.3 would require a local Cl of only 0.076 to trim the example in yaw at supersonic speeds and a moderate loading of 0.33 at the subsonic condition.
Figure 3.8. Yaw trim with small vertical fins may be compatible with some applications.
To aid in rapid trim assessment, an interactive analysis tool has been developed that determines basic trim requirements given a set of untrimmed aerodynamic forces and moments. The inputs can be provided from any source available, such as wind tunnel data, computational methods, or some form of response surface. The wing bank angle for side-force trim, c.g. location for pitch trim, wing bend for roll trim, and fin lift, split flap deflection, or thrustline offset for yaw trim, are all computed and displayed. An example of the graphical user interface for the trim applet is shown in figure 3.9.
Because of the large number of possible oblique flying wing geometries and their complex relationship to the vehicle forces and moments, a simplified parameterization of oblique wings with 6-10 parameters (including aspect ratio, symmetric sweep, asymmetric taper, wing bend, angle of attack, and others) was used to generate a response surface model of forces and moments. Results were generated with multiple runs of the A502 surface panel code and checked with selected runs of CART3D. Geometries were generated automatically for both codes with the Desktop Aeronautics RAGE parametric geometry tool. Example results are shown in figures 3.6 and 3.7 for a 65 deg obliquely swept wing at Mach 1.4 with the moment center at the quarter chord of the root section. (taper 0.5, AR 8, symmetric sweep zero at 1/4 chord, CL approx 0.12).
Figure 3.6. Effect of parabolic bend on rolling moment.
Figure 3.7. Effect of parabolic bend on yawing moment.