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The Handbook

1. Handbook

(handbook How Ornithopters Fly is only in German)

frontispiece of the handbook

How does an ornithopter create thrust and lift - despite of alternating flapping directions? The answer can be found in the handbook based on well-known results of research. Apart from the aerodynamics of up- and downstroke, the dynamics of the flapping wing is also taken into consideration. The correlations are described with equations and diagrams. Your own calculations are made possible, which may be helpful for developing specific ornithopter models. Furthermore, you will find useful tips for ornithopter models in practice.

The relatively simple equations for changing circulation distributions make it possible to vary the lift distribution and to determine the appropriate wing twisting.

vortex system along the flight path
Vortex system behind an ornithopter

The ornithopter subject also extends to the field of bionics. It is an attempt to develop better ornithopters by understanding the biological design principles of birds.

You can download the handbook (in German) and photos.

  • PDF 3.3 MB
  • PDF 5.9 MB
  • PDF 9.2 MB

The handbook was translated in French by Jean-Louis Solignac. With his knowledge as an aerodynamics expert and with his experience he has contributed a lot to the improvement of the handbook.

Jean-Louis Solignac, Maître de Recherche, acted as deputy head of the department Principles of Aerodynamics in the directorate of aerodynamics of the national French research institute O.N.E.R.A. (Office National d'Études et de Recherches Aérospatiales). You can find his translation of the handbook here on the French site.

The photos of the handbook


2. Calculation of flapping wings
under the precondition of quasi-stationary conditions

The equations presented in the handbook are used in several calculating tools. Thereby underlies the following method of calculation.

First, the flapping wing is theoretically devided into stripes with a very small span. Then, for each of these wing sections the aerodynamic forces are calculated under stationary or constant oncoming airflow conditions. Their sum results from a numerical integration over the whole wing span.

Forces at a segment of a flapping wing
Configuration of the forces

This way, you get the total forces of lift and propulsion of the flapping wing at a fixed moment of time of the flapping cycle. The corresponding wing twisting, the profile- and induced drag can be determined in the course of this calculating scheme, too.

locations for calculation
Locations for calculation

This process is repeated in equal time segments of the wing stroke motion. Thereby, the changed factors as for instance the distribution of circulation, conditions of oncoming airflow or the dihedral of the wing form the basis. At the same time, stationary conditions are postulated. It is therefore presumed that the airflow does not change during the time span of calculating. Furthermore, unsteady airflow behaviour is not considered.

That way - thus by stringing together different steady conditions - time force progression under quasi-steady conditions results.

The force of a whole stroke motion can be obtained by numeric integration of the force progression over the considered time span. Thereby, up- and downstroke of the wing are advisably considered separately. Finally, the summary of up- and downstroke forces leads to the total forces of a whole flapping cycle.

frequency of wing beat / weight of birds
Frequency of wing beats
and the weight of birds
by literature Heinrich Hertel

But according to literature Erich von Holst this quasi-steady method> only leads to useful results during a fast forward flight with relatively low flapping frequencies (large birds). Otherwise, the influences of unsteady airstream behavious become too strong. Later publications verify these constraints. As an example also the following analysis by M. Neef.


3. Result of the latest research

Dr.-Ing. Matthias F. Neef has examined in his dissertation Analysis of the flapping flight by numeric flow design engineering the unsteady flow at a moved wing. Thereby, he reached a similar vorticity system as aforesaid. However, his picture with a sinusoidal flapping motion-sequence is more specified and more detailed.

Isolines of the circulation
Isolines of the circulation along the flight path

The dissertation includes a general view about flapping flight and more exciting pictures (Please look at external link 1. and 2.).


4. The tip vortex of the flapping wing

The isolines of circulation of a flapping wing shown above also can be visualized as single vortex filaments.

Vortex filaments runing parallel and with a similar direction of circulation, twist themselves to a single vortex in their shared center at the wake of the wing.

boundary vortices of the flapping wing
Wingtip vortices of the flapping wing

This way, the majority of the vortex filaments combined build up the wing tip vortex. During the flapping cycle its starting point is moving back and forth along the trailing edge of the wing - especially in the upstroke. Therefore, the vortex trail behind the flapping wing in plan view shows lateral contractions in regular intervals.

Also in birds, which are flying in cruise flight (Flying with lift) the lateral movement of the starting point of the vortex along the trailing edge of the wing has already been observed (please look at external link 3, Fig. 1). This continuous-vortex gait is contrary to the vortex-ring gait when birds are Flying with thrust (please look at the discribing of the flight modes).

helical wingtip vortex or slipstream of a bird
Helical wingtip vortices or slipstreams (blast of air) of a bird in continuous-vortex gait during cruise flight

When we imagine the wing tip vortex in the adjacent picture in three-dimensions be aware a surprising view.

The starting point of the vortex of one wing side not only moves back and forth along the trailing edge of the wing. It also follows the flapping motion. Seen in flight direction these both movements together resulted in an approximately circular path line. If now also include the forward motion of the flapping wing one sees the helical shape of the wing tip vortex spreading backwards.

Also the tip vortices of a propeller are arranged in a helical shape (Please look at external link 4). They wrapped the propeller slip stream and are an essential part of it. In opposite to the propeller at the flapping wing simply the windings of the tip vortices are pulled more apart. Hence, in the three-dimensional view of this vortex picture will be visible a slipstream at each side of the flapping wing.

An according vortex structure is desirable also at ornithopters in cruise flight. Therefore also in the upstroke, a large lift must exist - maybe larger than indicated here - and the transition between the lift distributions of up- and downstroke must be smooth. In the movie recording of a flying swan for example, you see that the increase and decrease of the angle of incidence moves like a wave from the wing root to wing tip.

In order to generate large thrust at an ornithopter, the cross-section of the slipstream is to make as large as possible. Shifting of the spanwise distribution of lift is a dominant factor here. At downstroke the lift should be shifted as far as possible towards to the wing tip and at upstroke towards to the wing root. Furthermore the stroke angle of the wing should be chosen relatively large without, however, losing sight of thereby decreasing lift.

In case of very great demand of thrust, the shifting of the spanwise distribution of lift in upstroke can be supported by a strong downward bending and/or backward bending of the hand wing. At the same time birds are using the shortening of the arm wing. For more information, please see the article Arrangements of wing tip vortices on flapping wings (version 4.0, PDF 0.5 MB).


5. Formation flight of birds

Downwash distributions
on the wing of an ornithopter
in cruising flight

V-shaped staggered flight formations result in a measurable energy conservation for each single individual. This is particularly the result of aerodynamic influences. With the aid of the ornithopter theory conclusions can be drawn about the mode of functioning concerning the energy savings.

In connection with its lift the leading bird necessarily generates a wing tip vortex at both wing tips. For it this implies a loss of energy. It is relatively big for birds with high wing loadings and short, tapered wing shapes. But the following bird can try to tap the energy content of one of both wing tip vortices to make its own flight work easier.

formation flight of birds
Drag reduction at the
formation flight of birds

Well known is the hypothesis (Please look at external link 5) that the following bird uses a field of uplift of its leading bird. It is generated by the tip vortex spreading backwards at the outer edge of the flight formation. This up wind enables the following bird to increase its own thrust without performing additional flight energy. But it is more effective to use the angular momentum of the incoming vortex to reduce the wing tip vortex of its own wing (adjacent picture and external link 6).

Flock of birds in flight by Titus Tscharntke
Flock of birds in flight
Photografer Titus Tscharntke

The problem for the following bird is the optimal adjustment of all distances in the three-dimensional space behind the leading bird. It must try to adjust the distances to the flapping wings of its leading bird in a way that the proper part of the leading bird's vortex passes it in a suitable moment and at the optimal position. It can surely feel the best flight position, but thereby it must also make compromises. Anyway, in the theory of formation flight of birds many questions remain open. For more information, please see the handbook, annexe E, (lock above) and the article Arrangements of wing tip vortices on flapping wings (version 4.0, PDF 0.5 MB).