Aerodynamic Model of Birds Formation
We will do general considerations regarding multiplane lifting systems starting from Munk’s theory. For a given pair lifting lines the stagger angle is defined as the angle between the vector of the lifting lines and its projection onto a plane normal to the direction of flight. If this angle is zero, the system is said to be unstaggered, otherwise the system is staggered. A generic lifting surface is reduced to a single vortex line having a known distribution of circulation. The vortex is closed with the tip vortices and a starting vortex at infinity. This model replaces the wakes with tip vortices, and the circulation with a bound vortex.
Under the above linearizing assumptions, the complex aerodynamics arising from the an arbitrary number of lifting surfaces can be solved without requiring prohibitive computer costs.
The drag mutually induced by a pair of lifting lines flying at a generic speed is a function of the circulation of both lifting lines and of a number of quantities describing the geometric arrangement of the system. The induced drag is independent of the stagger angle (Munk’s stagger theorem). If the system is unstaggered then D_ij = D_ji.
which is Munk’s reciprocity theorem. The induced drag of a multiplane system does not change if the elements are translated in a direction parallel to the direction of flight. Once the elements are unstaggered, the drag mutually induced by the pair has equal contribution from each element. The induced drag and lift vanish as the mutual distance of the elements of a given pair increases.
For elliptic loading and unstaggered system, the problem can be solved in closed form. Indeed, a bird can not be considered a fixed wing, since it does adjust its lifting surfaces in order to produce minimum loading. This is generally obtained by assuming arrangements with spanwise camber, that have a well-known drag reduction effect. For given span non planar lifting surfaces have wave drag considerably lower than the elliptic loading (Munk’s cosine theorem)
The equation describing the lift distribution becomes singular when a pair of lifting lines is co-planar with the free stream, and when two lifting lines are too close. Thus, birds lagging behind may not lie on the same plane but must glide to a different altitude.
According to the stagger theorem for the computation of the total induced drag, the system can be unstaggered. However, unstaggering produces a different distribution of loads.