I noticed this picture yesterday of the Bird of Prey and wondered about the wing design and what advantages would come with this unique wing design compared to other aircraft?
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The Boeing Bird of Prey is indeed a unique aircraft, wahich was an internal project rather than an X-Project funded by the government. It was primarily used as a technology demonstrator for low observable technologies.
The Bird of Prey is rather unusual for a wing only aircraft (in the sense that it had no stabilizer) that it was statically stable and required no correcting inputs from a computer. This stability was achieved by the lift generated by the chines in front of the fuselage.
Here is closer look at the chines.
The chines produce lift by generating vortices (similar to the delta wing). Basically, the chines produce seperated vortices on either side of the fuselage, which produces lift. According to adg.stanford.edu,
Rather than reducing the lift of the wing, the leading edge vortices, increase the wing lift in a nonlinear manner. The vortex can be viewed as reducing the upper surface pressures by inducing higher velocities on the upper surface
Image from Effects of Vertical Tail and Inlet/Strake Geometry on Stability of a Diamond-Wing Fighter Configuration by Mitchell E. Fossum et al., accessed through http://enu.kz/
To compare how the chine generates lift similar to a delta wing, it is instructive to see the vortex flow patterns on a slender (cropped) delta wing with a chine forebody.
Image from On the Nonlinear Aerodynamic and Stability Characteristics of a Generic Chine-Forebody Slender-Wing Fighter - Configuration by Gary E. Erickson and Jay M. Brandon, NASA Technical Memorandum 89447, labeling mine.
An example of a delta wing with a chine forebody shows the two vortices being generated from the chines and the highly sweptback wing, similar to the Bird of Prey.
Image from Aerodynamic Analysis of a Generic Fighter with a Chine Fuselage/Delta Wing Configuration Using Delayed Detached-Eddy Simulation by Tiger L. Jeans et al, accessed through calpoly.edu
The forces acting on the longitudinal axis of the aircraft look something like this (I'm not sure of the exact values or locations), leading to a statically stable aircraft.
Profile from bj-o23.deviantart.com, others own work.
The chines also help in reducing the radar cross section, according to dictionary.sensagent.com (attributed to The Radar Game by Rebecca Grant):
A standard leading edge extension joins to the fuselage at an angle while a chine is an extension of the curvature of the fuselage. Therefore chines avoid presenting corner reflectors or vertical sides to radars.
This design has since become standard in (Boeing) UAVs like X-45 etc.
Another feature is that the fuselage sides are parallel to each other, helping in reducing the Radar Cross section by reflecting the radar waves away from source.
The wing design is rather unusual, a polyhedral section with dihedral (upwardly angled) inboard sections and anhedral (downwardly angled) outboard sections, which helped in stealth and stability.
The wing has a rather high sweep angle (~65 degrees) for a low speed aircraft, in order to improve stealth.
Since @aeroalias' answer just reproduces what is said in Wikipedia, I will add a few more thoughts (albeit without the pictures, there are already plenty in the other answer).
First of all, the design is meant for transsonic flight, but the development was privately funded by McDonnell-Douglas and had to keep cost down. As a consequence, the demonstrator aircraft used a 3000 lb thrust class engine (Pratt & Whitney Canada JT15D5C) which is used more in military trainers and business jets than in combat aircraft. This kept the aircraft's top speed and the aerodynamic forces low, which helped in keeping the systems simple and the structure small. A production aircraft would be much bigger and use a much more powerful engine.
When McDonnell-Douglas was not selected for the ATF program in the late Eighties, they decided to put themselves on the map by building a demonstrator which should show off their low-observables expertise. They had founded their own design office for black programs, called Phantom Works, mirroring the approach of Lockheed with their Skunk Works branch.
Roughly half of the lift in level flight is produced by the forward fuselage. Wikipedia says it is the chines, but they only provoke the generation of a pair of vortices just like those on a delta wing, and the low pressure caused by the speeds of this vortex creates lift on the forward fuselage. In a way, the Bird of Prey is a canard aircraft.
The other half of lift is contributed by the wings, which again create vortex lift due to the high leading edge sweep angle. The dihedral of the inner wing was chosen to
Regular, upward-pointing winglets have a number of disadvantages, one of them their induction of lift on the wing when their rudders are deflected, which produces an undesired rolling moment and prevents rolling the aircraft into the intended turn. Downward-pointing winglets helped to make the aircraft flyable without computer control, because they help with
The high sweep angle was chosen for the high-subsonic flight regime of any production version and for deflecting the returning radar signal away from the sender. The Bird of Prey is a two-lobe design (see this answer for more details)