1. 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.

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, vers. 6.1) and photos therefor.

The photos of the handbook

2. Calculating 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.

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

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 beats
and the weight of birds
by Heinrich Hertel

But according to 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.

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

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 centre at the wake of the wing.

Tip vortex 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 during upstroke. Therefore, the flapping wing lacking behind the losed vortex strip shows lateral contractions in regular intervals.

Also at birds flying with lift the sideways 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).

5. Formation flight of birds

Drag reduction at the
formation flight of birds

Well known is the hypothesis (please take a look at external link 4) 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 work. 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 5).

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.