# . Reflexed Airfoils

A more elegant way to control the pitching moment of an airfoil is to use reflex camber. Generally, for a given airfoil, the addition of reflex camber to an airfoil with camber in the nose region gives a significant reduction in Cm with only a minor reduction in maximum lift capability and a small drag penalty. Many authors have investigated low pitching moment reflexed airfoils based on cubic caiuberliues — see Glauert (1947) and Houghton & Carpenter

(1993) . A cubic camberline may be expressed as

yc —m a x(x + b)(x — 1), (7.82)

where m is the maximum camber as a fraction of chord, and a and b are general coefficients. While this equation describes a family of camberlines, of specific interest in this case are the values of a and b that will produce a camberline with zero pitching moment. We may define a specific cubic camberline such that dyjdx — 0 at x == p and ydm — 1 at jc — p. This leads to the two simultaneous equations

ap3 + a(b — l)p2 — abp = 1 and Зар2 + 2a(b — l)p — ab = 0, (7.83)

which can be solved for a and b under the assumption that Cmi/4 = 0 (i. e., Aj — A 2 = 0 using the thin-airfoil theory). The results for A and A 2 are given by

Ai = — ^ + ab^ m and A2 = ~m, (7.84)

and by satisfying A — A2 = 0 we find that a = 8.28 and b = 0.875 with p — 0.31. The resulting shape is shown in Fig. 7;28 for m = 0.2. It is apparent that only a small amount of reflex at the trailing edge is required to negate the pitching moment produced by the positive camber near the leading edge. Many modem helicopter rotor designs take advantage of the low pitching moment benefits produced by reflex cambered airfoils.

7.8 Drag

Drag forces on airfoils. operating in attached flow are typically nearly two orders of magnitude less than the lift forces at the same AoA. There are two sources of drag:

2

Figure 7.29 Measurement of drag by wake momentum deficit approach.

1. Pressure drag, and 2. Viscous shear drag. The sum of these is called profile drag. In addition, at higher Mach numbers there is a source of drag known as wave drag, which is important whenever a shock wave forms of the airfoil. Pressure drag may be estimated by the integration of pressure with respect to airfoil thickness as previously described. However, for this to be accurate a good concentration of pressure points is required in the region of high suction pressure (usually the leading edge region). The problem is complicated, however, with the formation of supercritical flow and a shock wave. Alternatively the drag may be measured with a force balance. This has the advantage of measuring both the pressure and viscous shear drag. However, the measurements must be properly corrected for any 3-D effects associated with the airfoil configuration in the test facility.

One common way of estimating the steady drag on a 2-D airfoil section is to estimate the momentum loss in the wake of the airfoil by measuring the velocity profile downstream of its trailing edge, as shown in Fig. 7.29. This is sometimes called the “wake rake” technique. By placing a control volume around the airfoil section and its wake, we can use the momentum equations in integral form to find the time rate of change of momentum into and out of the control volume. This will give a force on the fluid as it passes through the control volume, and by Newton’s third law, this force is equal and opposite to the (drag) force on the airfoil.

By looking at the change of momentum in the streamwise direction, it can be shown that (7.85)

The two integrals can both be expressed in terms of y2 by using conservation of mass through the stream tube pV^dyi = pV^dyi – Therefore, the drag force may be written as      (7.86)

Good accuracy has been demonstrated with this method if the measurement plane is about 15% chord downstream from the trailing edge of the airfoil. (See also Question 7.9.) However, this technique is only valid for angles of attack below stall. When significant flow separation exists above the airfoil, the technique becomes less valid because of other rotational losses that do not appear as a loss of momentum in the downstream wake. In these circumstances, the drag must be measured with a force balance or from pressure integration around the airfoil surface. However, the sectional drag can be measured reliably by means of this wake rake or wake survey technique at low angles of attack (which gives a measure of both the viscous and pressure drag) and by pressure integration to obtain the pressure
drag component alone. The total or profile drag is then the sum of the viscous drag and the pressure drag. The wake rake technique, however, cannot measure the unsteady drag, this only being possible by the integration of unsteady surface pressures.