This case is a simulation of the classic flapping-wing propulsion study of Schmidt, and his so-called wave-propeller. Due to the work of Theodorsen and Garrick, it was well known by the mid 1930's that a single flapping wing generated thrust at an efficiency of between 50 and 100 percent; 50 percent at higher freqencies, and a theoretical limit of 100 percent as the frequency is reduced to zero (a rather meaningless limit as both the thrust and the required work go to zero). In practice, to generate significant thrust values with a flapping wing, relatively high frequencies were needed, reducing the efficiency to around 50 percent.

The loss in efficiency at higher frequencies was primarily due to energy lost into the flow in the form of shed vorticity, indicated by the magenta and cyan rotating squares in these animations. In the 1950's, Schmidt noted the experimental results of Katzmayer (1922), where it was shown that a stationary airfoil in an oscillatory flowfield also produced thrust. Schmidt took advantage of this by placing a stationary wing in the oscillatory wake of a flapping-wing, recapturing a portion of the energy lost into the wake. Since the second airfoil was not moving, it required no work, so any thrust it provided was essentiallu free (not quite free, as the second airfoil has additional profile drag).

In the sequence we are moving the flapping airfoil along a circular path, with the geometric angle of attack fixed at zero degrees, as Schmidt did (traveling along a circular path eliminates the mechanical acceleration normally required for a flapping wing). In our simulation the reduced frequency (based on chord length) is 1.0, and the radius of the motion is 1.0 chord lengths. This is an interesting case as the airfoils radial speed is exactly the free-stream speed in the flow. That is, on the lower part of the arc, the airfoil sees a flow speed of twice the free-stream, and on the upper part of the arc it's essentially moving with the flow. In the middle of the up and down stroke, the induced angle of attack reaches 45 degrees, well beyond the expected stall limit of the airfoil. Therefore we can assume the the panel-code simulation is predicting a higher performance than can be expected in a real fluid. The thrust coefficient predicted by the panel code is shown in the figure below.

Figure 1: Thrust coefficient for the Schmidt wave-propeller, k=1.0, r=1.0, x=2.2