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Some Definitions

The variables described here are later used on the forms that define the velocity distribution from which the airfoil shape is derived. As depicted in the figure below, the airfoil is divided into a given number of segments, which are numbered in counter-clockwise order. The arc limits PHI1, PHI2, PHI3 and PHI4 on the circle are mapped to the points S1, S2, S3 and S4 on the airfoil. PHI is measured from the point at PHI4. In the method, the minimum number of segments is four - two on the upper surface of the airfoil and two on the lower surface as shown. The circle arc limit PHI begins at PHI = 0 and ends at PHI = 60 (rather than 360 deg); thus, the circle is divided into 6 deg sectors.

Associated with each segment is a corresponding "design angle of attack," which is best described by way of example. Suppose that the design angle of attack for the second segment ALPHA2 is set at 10 deg. When the resulting airfoil is then operated at 10 deg, the velocity distribution along the second segment will be constant. For the third segment, the design angle of attack can be set to 0 deg, which would give a constant velocity along that segment of 0 deg.

It should be noted that the design angle of attack is referenced to the airfoil zero-lift line (rather than the airfoil chord line). Thus, specifying the design angle of attack allows one to prescribed a constant velocity distribution on a segment for a given lift coefficient, since the lift coefficient of the airfoil is approximately 2 pi ALPHA.

For the first and last segments (segments 1 and 4 in the example), termed the recovery regions, the velocity distributions are not constant when the airfoil operates at the corresponding design angles of attack. Morever, the ALPHAs in these regions have little effect on the velocity distributions. In part, the recovery velocity distributions are determined by the method so that the airfoil closes at the trailing edge. This is a requirement of the method. (In fact, any inverse method must allow for some degrees of freedom in the velocity distribution so that (1) the airfoil trailing edge closes and (2) the airfoil velocity distribution is consistent with the freestream.)

The designer, however, does have some control over the velocity distributions (and hence the airfoil shape) in the recovery regions. First, the pitching moment CM of the airfoil can be specified, which in turn determines the amount of aft camber or "aft loading." Second, a trailing edge closure parameter KS is specified to control the relative thickness of the trailing edge. These two design parameters CM and KS (in addition to the PHIs and ALPHAs) will appear on design pages to follow.

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