Preliminary Research On The Optimisation of Flying Wings In Forward Flight
October 15, 2019
October 15, 2019
By Read Liston
Longitudinal Stability and Trim
The differentiation of stability and trim is an important factor in the design of an aircraft, specifically a tailless one. In terms of static longitudinal stability, the definition remains the same for a flying wing and a traditional aircraft; a centre of gravity location ahead of the neutral point of the aircraft. For a flying wing, this neutral point is analogous to the aerodynamic centre of the wing/airfoil and is generally located along the 20% to 25% line of the chord along the span of the wing. A statically stable aircraft, however, is insufficient for steady flight, the aircraft must also be trimmed. A trimmed aircraft has all the moments about the centre of gravity balanced. With no horizontal stabiliser, the wing design of a tailless aircraft is significantly more involved, requiring both lifting and trimming capabilities. In the statically stable case, a typical main wing will produce a pitching down moment due to the fore centre of gravity location. From ; there are two ways in which to attain trim:
- Use a combination of wing sweep and wash out.
- Use an airfoil that produces a positive pitching moment.
A traditional cambered airfoil produces a large negative pitching moment, exacerbating the pitching down moment. As such, an airfoil with a positive pitching moment can be used to instead balance the main wing moment. This solution is generally only applicable for a wing with no sweep. A much more feasible method is the combination of wing sweep and washout. Typical washout, or wing twist, is implemented to create a negative angle of incidence on the outboard section of the wing. Its combination with a sweptback wing creates a sufficient pitch-up moment arm to balance that of the main wing, trimming the aircraft. It follows that the larger the wing sweep angle, the less washout required (for a constant CG position). Both excessive sweepback and washout, however, have adverse effects on lift and flight performance. In practice, the most optimal solution is to combine wing sweep and washout with an airfoil that produces a minimally negative to zero pitching moment.
A symmetric airfoil would produce a zero pitching moment, however, the minimum drag for a symmetric airfoil occurs at zero lift, therefore during level flight, an angle of incidence is required and excessive drag will be created. To obtain the desired pitching moment a reflexed airfoil is the most appropriate choice as they have minimal negative pitching moments along with a positive coefficient of lift at zero angle of attack.
The center of gravity (CG) is one of the most important factors in the design of a flying wing. Acting as the pivot of the aircraft, its location is often influenced by many factors, particularly payload. In the case of Wren, to avoid sudden shifts in CG, its optimal location is coincident with the center of the cargo bay. This follows, that upon releasing the UGV, the flight characteristics remain unchanged. Specific internal configurations will be required in order to obtain this.
Additionally, while a forward CG location improves longitudinal stability, it is not without its drawbacks. When the center of gravity of the aircraft is displaced from the center of lift, wing twist is necessary to provide the moment balance, as previously discussed. This twist, resulting in a down lift at the wingtips, adversely affects the maximum lift of the aircraft. The figure below demonstrates the percentage loss in a maximum lift for various CG locations.
More significantly, however, is the impact on the induced drag whose value is quadratically proportional to the distance of the CG to the elliptical center of lift. Figure 3 below, combines the losses of lift as well as the increase in induced drag, to demonstrate the optimal CG locations for a given taper ratio. Early Horten sailplanes are plotted as a reference.
Tests by  showed that the critical cg location ahead of the centre of lift before controls became unflyable was around 6-12% of the mean chord length.
A large aspect ratio is a trade-off between reduced drag and reduced maneuverability, while the opposite is true for a smaller ratio. For a given wing area, as the aspect ratio increases, the wingspan follows suit. The adverse effect of tip vortices then occupies a smaller percentage of the entire wing. To optimise forward flight a larger aspect ratio seems to be the obvious choice however depending on the flight requirements and mission profile of the particular competition, an appropriate aspect ratio can be selected in conjunction with the average operating speed.
Taper Ratio and Stall
Due to the delicate balance of the wing, local stall can disturb the flight significantly. For example, stall at a wing root when ahead of the CG will cause a nose dive while local stall further outboard, behind the CG will cause a pitch up. The ideal location for flow separation can, therefore, be inferred to be in line with the CG A crude approximation to achieve this given an idealised lift distribution, a minimum taper ratio of 0.6 should be selected with an increase to 0.7 perhaps being more desirable, as proved by . Moreover, it was discovered that for a sweptback wing with washout, the optimal taper ratio to minimise parasitic drag lies between 0.7 and 1. Conversely, a strong taper ratio will produce generally all positive results apart from the increased induced drag above and center of lift implications displayed in figure 3.
While it seems that the Horten design neglect the previous fact, by having strong tapers, authors Nickel and Wohlfahrt, stress that a flying wing with a taper ratio less than (greater taper) 0.7 has either:
- No optimal Lift Distribution.
- Unfavourable stall Characteristics.
As mentioned, sweep is necessary to balance the locations of the centre of gravity and aerodynamic centre, below is a brief summary of the effects different amount of sweep has on the aircraft.
For larger sweep:
- Creates a larger moment arm for control surfaces located toward the wingtips.
- Creates a larger moment arm for wingtip washout, increasing its effectiveness.
- Creates less lift when compared to a straight wing by approximately the cos of the sweep angle.
The opposite is true for smaller sweep angles.
The amount of washout required can be either a small amount, needing elevon deflection at low speed, a large amount, needing elevon deflection at high speed or somewhere in between. The selection should depend on the operating lift coefficient. For Wren, this would be cruising conditions given the influence of the quad-motors at low speeds speed, take-off, and landing.
The washout amount can be determined using the moment balance equation:
The value of C_mw can be expressed in the form below, and experimentally found from the graph.
Recent studies from  express the benefits of a bell-shaped lift distribution in its ability to create superior control and proverse yaw. Additionally, it is known that the Horten brothers implemented this lift distribution similarly in their design. The effect of proverse yaw was not attained by the Horten’s however, due to the inboard nature of their elevons explained by . The bell-shaped lift distribution developed by Prandtl and implemented by Horten and NASA , however, produces more drag for a given wing when compared to an elliptical lift distribution as mathematically explained in detail by . Horten and NASA increased the aspect ratio (Span) to reduce this loss in drag to attempt to justify the benefits of proverse yaw. A firm expression from  suggests that, particularly for models (small scale aircraft), other techniques are more effective in reducing adverse yaw and that a bell-shaped lift distribution is inappropriate.
It would worth building models from the theory of both  and  to compare the results first hand.
Horten Design Philosophy 
- Followed the above, used large washout at the wingtips
- Wingtip twist created a bell-shaped lift distribution which was theorised to reduce adverse yaw.
- They used a larger aspect ratio to make up for the loss in lift near the tips.
Northrop Design Philosophy 
- Early designs pre B-2, Chose to avoid excess twist and used pusher propellers relying on vertical stabilisers.
- Pusher props along the trailing edge of the wing created a stabilising effect
- Due to the fly-by-wire relaxed stability excess wingtip twist was not needed and instead, the flight computer handled the numerous control surfaces to stabilise the aircraft.
- The B-2 uses wingtip mounted control surfaces that crack open to create drag and thus a yawing moment. This could potentially be implemented into WREN however the response is known to be very non-linear, resulting in difficulty in control. When the B-2 is cruising the yaw control surfaces are cracked open to the point where they are only just catching the air. The flight computer then can fine-tune the aircraft’s dynamic response.
 K. NIckel and M. W. Wohlfahrt ‘Tailless Aircraft in Theory and Practise’ AIAA educational series
 D. Raymer ‘Aircraft Design A conceptual Approach’ AIAA educational series.
 A.H Bowers ‘On The Wings of the Minimum Induced Drag’ 2016.