Several example airfoils will be presented in these notes to illustrate the design method. All airfoils in this series will have four segments - the minimum number of segment. Nevertheless, a surprisingly wide variety of airfoils can be designed with just four segments.
The values presented in the tables are suggested values for the design pages. The notes correspond to these values; but, of course, any values can be used.
Segment PHI ALPHA 1 15.5 10 2 32.2 10 3 45.5 4 4 60.0 4 Cm = -0.15 Ks = 0.35After the airfoil is designed, the velocity distributions, airfoil shape, converged input file (with PHI2 adjusted, see details below) and airfoil coordinates are returned for display.
As seen above, the design angle of attack for the second segment is 10 deg. For this angle of attack the corresponding velocity distribution is constant for the second segment. Likewise, for the third segment, the velocity is constant for 4 deg.
(As a reminder, the upper curves in the velocity distribution correspond to the upper surface, while the lower curves are for the lower surface. As the angle of attack increases, the difference between the upper and lower surface velocity increases - the greater the difference between the curves, the higher the lift coefficient. Thus, the top curve and the lower curve correspond to the highest angle of attack, which is 14 deg.)
Segment PHI ALPHA 1 15.5 -> 10.5 10 2 32.2 10 3 45.5 4 4 60.0 4 Cm = -0.15 Ks = 0.35
Segment PHI ALPHA 1 10.5 10 2 32.2 10 3 45.5 4 4 60.0 4 Cm = -0.15 -> -0.25 Ks = 0.35
Segment PHI ALPHA 1 15.5 10 2 32.2 10 3 45.5 4 4 60.0 4 Cm = -0.15 Ks = 0.35 -> 2(Note: A value of 0 for KS for this example produces an error and no figures or output are produced.)
In a direct design method, the approach usually taken to prevent the airfoil from stalling too early is to increase the airfoil camber or thickness at the nose. One advantage of the current inverse method is that the velocity distribution (in particular, the gradient of the velocity distribution) can be specified to avoid an "adverse pressure gradient" (a rapid deceleration in velocity) until the desired angle of attack is reached.
For the baseline airfoil (Example 1), the design angle of attack for the second segment is 10 deg, which corresponds to a lift coefficient near 1. Stall can be expected to occur shortly after this angle of attack (or lift coefficient) is reached. In the current example, the angle of attack on the upper-surface second segment AFLA2 is changed to 12 deg. Now the velocity distributions are favorable up to 12 deg and stall will occur shortly thereafter. In this case, the lift coefficient is near 1.2.
Segment PHI ALPHA 1 15.5 10 2 32.2 10 -> 12 3 45.5 4 4 60.0 4 Cm = -0.15 Ks = 0.35As seen the airfoil is somewhat thicker. This has happened because the lower surface was not changed; it must still operate with a constant velocity on the lower surface down to an angle of attack of 4 deg. Below this angle of attack the velocity distributions become unfavorable, and the airfoil can be expected to separate off the nose of the lower surface.
Segment PHI ALPHA 1 15.5 10 2 32.2 12 3 45.5 4 -> 8 4 60.0 4 Cm = -0.15 Ks = 0.35
To achieve the desired CM and KS values, two of the design parameters are needed for iteration. These two parameters include the leading-edge arc limit PHI2 and the velocity level (not discussed). Thus, PHI2 on the design page will be adjusted by the method in which case making changes to PHI2 on the design page will have no effect.
In the iteration process, the maximum number of iterations is set to 40. If the specified design parameters do not converge within the given number of iterations, the airfoil will not have the desired CM and KS.
For those with some experience with the Eppler code, the pitching moment coefficient CM takes the place of the OMEGAs. The K values have been set equal to 1 in PROFOIL-WWW, but this is not a requirement. The MUs are now determined by the method and cannot be explicitly prescribed unless further iteration performed; that is, in addition to prescribing CM and KS, the MUs can also be specified, provided that additional design variable are allowed to be free for iteration.