The First Test: Downwash on the Elevator

The first published application and test of Prandtl’s approach was provided by Otto Foppl in the Zeitschrift fur Flugtechnik of July 19, 1911.57 Foppl had already produced a series of experimental reports using the Gottingen wind channel to test the resistance and lift of flat and curved plates.58 These studies indicated that the laws of resistance depended in a complicated way on the effects at the edges of the plates. It was clear that the move from an infinite to a finite wing would introduce significant new factors into the account of lift and drag. Prandtl had begun to identify these factors in his lectures. Foppl’s aim now was to test a quantitative prediction made by Prandtl on the basis of the new theory he was developing. The prediction kept away from the problematic singularity at the wingtips and concerned the angle at which the air would be moving downward at a specified distance behind the wing. It concerned the induced angle of incidence.

Foppl took for granted the qualitative picture of the two (straight-line) trailing vortices or “vortex plaits” (Wirbelzopfe). The reader of the Zeitschrift was assured that these had been rendered visible in the Gottingen wind chan­nel by introducing ammonia vapor into the flow (184). This “really existing flow” (“tatsachlich vorhandene Stromung”), said Foppl, was the empirical basis on which Prandtl had built his theory (184). The question was how the downwash and the tilt in the local flow could be generated and measured in the wind channel. First it would be necessary to introduce a model wing to generate the vortex system that was under study. It should then be possible to detect the downwash by introducing a flat plate at a distance behind the wing. The angle of the plate could be adjusted until it was aligned with the tilt of the airflow. When the plate was correctly aligned with the flow, there should be no lift force on it. This is the empirical clue giving the angle of the flow. It is important that the plate should be flat, because if it were curved or had the cross section of a normal aerofoil it would still generate a lift even when pointing directly into the local flow. The zero-lift position would not reveal the angle of the flow. With a flat plate, however, the observed angle of zero lift gives the actual tilt of the flow, and this can be compared with the predicted angle.

Having reviewed the logic of Foppl’s experimental design, we can now look at the details of the experiment and the connections that Foppl made with the realities of aircraft construction. Consider the choice of the distance between the wing and the flat plate. The distance used in the prediction and test was selected on the basis of practical considerations about current aircraft design. Increasingly, and unlike the early Wright machines, aircraft were being built with a control surface, called the elevator, located at some distance behind the main wing. The elevator controls the pitch of the aircraft. In the Wright Flyer, the elevator was at the front and the propellers at the rear. By 1911 designers typically put the propeller at the front and the elevator at the tail end of the fuselage. As Prandtl explained, according to the “horseshoe” model, if the el­evator is in a horizontal position behind the wing, it will experience a definite downthrust or negative lift (Abtrieb). This will only disappear if the elevator is rotated by a specific amount which depends on the circulation around the wing. If the elevator is a flat surface (in effect a moving tailplane), then for the reasons just given it will experience a zero-lift force when it is aligned with the downward inclination of the flow behind the wing.59 Foppl therefore built a model airplane with exactly this kind of adjustable elevator. The model is shown in figure 7.10.

The main wing was 60 X 12 cm and had a camber of 1/18, while the eleva­tor was a flat plate of 20 X 8 cm. Both were made of 2.3-mm-thick zinc. The elevator, which was rigidly attached to the wing by two struts, could be piv­oted about its leading edge and fixed at different angles relative to the airflow. There was a distance of 34 cm between the elevator and the main wing, that is, the line running along the span of the wing on which the bound vortex was supposed to be located.

Foppl’s experimental procedure involved four steps, each of them us­ing the Gottingen wind channel. First, Foppl removed the elevator from the model, leaving the main wing still connected to the two struts. He placed the wing and struts in the wind channel at a realistic angle of incidence of 4.6°. The channel was run at a single, fixed speed V, and the lift on the wing was measured. The next step was to reposition the wing (still without an eleva­tor) at a different angle of incidence. This time he chose 7.6°. Again the lift

was measured at the speed V. In both cases Foppl expressed the lift as a coef­ficient Za. (This involved dividing the lift force by the density, the area of the wing, and the square of the speed.) He now had two lift coefficients, one for each of the two angles of incidence. In preparation for the next part of the experiment, Foppl reattached the elevator in order to carry out two sequences of measurements on the whole model. In one sequence the elevator-wing system was suspended so that the angle of incidence of the main wing was
4.6°, while for the other sequence the main wing was at 7.6°. For each of these angles Foppl measured the overall lift of the combined system for a range of different elevator settings. He gave the elevator seven different settings, that is, seven different angles relative to the direction of the free flow. The angles of the elevator to the free airstream ranged from +30° to -10°.

To find the forces on the elevator alone, Foppl subtracted the lift mea­surement for the wing in isolation from that of the wing plus elevator. The remaining lift force (that is, the lift force on the elevator) was then cast into the form of a lift coefficient. This gave Foppl data that could be expressed in terms of two graphs in which the lift on the elevator was plotted against the angle of incidence of the elevator—one curve for each of the angles at which the main wing had been set to the free stream. The most important feature of these graphs was the point at which the curves passed through the x-axis, that is, the angle of the elevator when its lift coefficient was zero. This was the angle at which the elevator should be parallel with the downwash, that is, the local, downward flow of the air induced by the vortex system. The graphs indicated that when the main wing was at an angle of 4.6°, the zero-lift position of the elevator was 2.8°. When the main wing was at 7.6°, the zero – lift position of the elevator, and hence the angle of the downwash, was 4.3°. The question now was whether Prandtl’s theory could predict these angles of downwash from the main wing at the two angles of incidence that Foppl had selected for his test.

Foppl duly announced the predicted value of the angles that had been deduced from the theory—but he did not say on what basis the prediction had been made. He simply informed his readers that in his lectures Prof. Prandtl had derived a formula that gave the tangent of the predicted angle of downwash. The formula was stated, but the deduction that led to it was with­held. The tangent, Foppl said, was given by the ratio w/V, where, according to Prandtl,

w = Ь£а

V nl

As Foppl explained, the coefficients of lift ZA to be entered into the formula were the ones that had been determined experimentally for the isolated wing. All the other dimensions could be taken from the model itself. Thus, b was the chord of the main wing (12 cm), l was the span of the wing (60 cm), and x was the distance along the longitudinal axis of the model from the middle of the main wing to the middle of the elevator (34 cm). With these values for the tangent, the predicted angles themselves came out at 3.3° and 4.2°. Given
that the two measured angles of the downwash (derived from the zero-lift position of the elevator) were 2.8° and 4.3°, Foppl concluded that the result amounted to “a very good confirmation of the theory”—“eine sehr gute Bestatigung der Theorie” (184). The prediction derived from the horseshoe model was correct.

The force of this claim must have been somewhat diminished because the theory used to make the prediction was not revealed. Readers of the Zeitschrift would have known that something was afoot in Gottingen, but Foppl was not going to anticipate Prandtl and expound the theory. He merely said that Prof. Prandtl would soon publish his derivation of the formula in the ZFM. No such derivation was forthcoming, but, with the benefit of hindsight, an examination of the formula makes its origin easy to guess. The formula was simply the result of applying the Biot-Savart law to each of the three straight­line parts of the horseshoe vortex and then doing the trigonometry necessary to relate the formulas to Foppl’s model.60