Wing Sizing: Part 2

Why flaps, and which kind?

The purpose of the flaps, as a high-lift device, is to minimize the wing area (which costs us weight and drag) while also keeping a reasonable stall speed. A variety of flap designs are available to us, but we’re going to make use of a single-slotted flap. Plain flaps and split flaps were considered, since they’re easier to construct, but a single-slotted flap isn’t that much more work to build and offers a good return in terms of additional lift coefficient.

Wing sizing methods

I’ve worked through several variations on several methods for sizing the wing, including flaps, and of course they all return slightly different results. I’ll include two here–one pretty simplified and another that’s more in-depth.

• John Roncz method: In the March 1990 issue of Sport Aviation, Roncz outlined a way to size the wing and include flaps if you want. It’s worth reading just for his style and other insights alone. Ultimately, I won’t be using this method because it’s very generalized and doesn’t take into account differences caused by aspect ratio or planform, but it’s worth seeing.
• Raymer method: We’ll follow the equations provided in Aircraft Design: A Conceptual Approach. It’s a bit more labor-intensive, but I think it’s worth it. As an adjunct, we’ll bring in some empirical flap data from the USAF Data Compendium (DATCOM).

Roncz method

I’ll only summarize Roncz, because you can–and should!–read his article for yourself. It’s available free to EAA members at eaa.org. Roncz designed the wing and flaps for the RV-9, -10, and -14 in addition to lots of Rutan designs, although I suspect his method in those projects was a little more sophisticated than what he presented in Sport Aviation.

• Roncz uses the ratio of . This shows up in the denominator of the equation below. I’m inclined to believe that our higher aspect ratio and elliptical planform might help keep wing area down.
• He states that if the flaps that span 65% – 70% of the wing, the wing makes 93% of the lift that it would if the whole wing were flapped. This too shows up in the denominator.
• A generic single-slotted of 2.6 is provided. The method here could be modified to incorporate whatever value you think is appropriate.

$$\scriptsize S_{ref} = \frac{L}{q \left( \normalsize \frac{\pi}{4} \scriptsize \right) (0.93) c_{lmax}} $$

Substituting in our values of and ,

$$\scriptsize S_{ref} = \frac{2,080 \text{ lb}}{8.60 \text{ lb/ft}^{2} \left( \normalsize \frac{\pi}{4} \scriptsize \right) (0.93) (2.60)} = 127 \text{ ft}^{2}$$

Raymer method

Raymer presents a way to find that includes to increase the lift for the flapped portion of the wing. One criticism of this method is that the lift coefficient doesn’t step suddenly at the end of the flap, but Peery provides a method for fairing this discontinuity, and I’m persuaded it comes out in the wash since the area subtracted very nearly equals the area added.

The equation provided by Raymer is:

$$\scriptsize C_{Lmax} = \overbrace{0.9}^{\small \frac{C_{L}}{\small c_{l}} \text{ratio}} \left[ \scriptsize c_{lmax} \cdot \cos{\Lambda_{c/4}} + \Delta c_{lmax} \cdot \cos{\Lambda_{HL}} \left( \frac{S_{flapped}}{S_{ref} } \right) \right] $$

is the maximum section lift coefficient. We’re going to use Riblett’s GA 37A315 as a placeholder. Riblett notes in fig. 23 of GA Airfoils that the predicted by the Eppler code he used is overstated by about 5%, so we’ll make an adjustment:

$$\scriptsize c_{lmax} = 0.95 \cdot 1.55 = 1.47$$

is the wing sweep measured at the quarter-chord. Because the quarter-chord wing sweep of an elliptical wing varies continuously along the span of a crescent planform, we’ll take the quarter-chord sweep at the mean aerodynamic chord (MAC). (What’s the MAC of an ellipse? We’ll look at that in a coming post.)

$$\scriptsize \Lambda_{c/4} = 3.7^{\circ} $$

is the additional provided by the flaps. We’ll look at how we arrived at this figure from DATCOM in a future post. The assumptions we’ll use when we do that are that the flap is 25% of the chord and has a max deflection of 45°. Deflecting the flap beyond 30° won’t provide much additional lift, but the extra drag can be useful.

$$\scriptsize \Delta c_{lmax} = 1.34 $$

is the sweep at the flap hinge line. For our crescent-shaped wing, this will be zero.

$$\scriptsize \Lambda_{HL} = 0 $$

is the ratio of the flapped area of the wing to the total wing reference area. In our case, we’ll assume the flaps are 60% of the span. Subtracting the fuselage from this flapped span gives us approximately:

$$\scriptsize S_{flapped}/S_{ref} = 0.56 $$

Substituting these values into the first equation,

$$\scriptsize C_{Lmax} = 0.9 \left[ c_{lmax} \cdot \cos{\Lambda_{c/4}} + \Delta c_{lmax} \cdot \cos{\Lambda_{HL}} \left( \frac{S_{flapped}}{S_{ref} } \right) \right] $$

$$\scriptsize C_{Lmax} = 0.9 \left[ 1.47 \cdot \cos{(3.7)} + 1.34 \cdot \cos{(0)} \cdot 0.56 \right] $$

$$\scriptsize C_{Lmax} = 2.00 $$

Finally, substituting this into the lift equation,

$$\scriptsize S_{ref}=\frac{L}{qC_{Lmax}} $$

$$\scriptsize S_{ref}=\frac{2,080 \text{ lb}}{8.60 \text{ lb/ft}^{2} \cdot 2.00 } = 121 \text{ ft}^{2}$$