Don makes some comments about wing sweep vs. dihedral!

From : Don Stackhouse

Regarding the current discussion about wing sweep being equivalent to dihedral, there is an old rule of thumb that says about 3 deg. sweep is equal to about 1 deg. dihedral. However, this is a GROSS oversimplification.

Sweep actually affects the lift-curve slope. The lift curve slope ("dCl/d_alpha" in engineering jargon) is nothing more than how much the lift coefficient ("Cl") changes for a given change in the angle of attack ("alpha"). For most airfoils, the plot of Cl vs. alpha is nearly a straight line except for some rounding off near the stall, so the dCl/d_alpha is nearly constant for most of the operating range. A typical value is about 0.1 (i.e.: the lift coefficient increases by about 0.1 for each 1 degree increase in angle of attack).

Dihedral causes a direct change in the angle of attack on both wings when the airplane is yawed. For positive dihedral angles, the alpha increases on the forward-yawed wing, and decreases on the aft-yawed wing. It has the same effect as ailerons, except that (unlike ailerons) it does not change the camber of the wings, only the angle of attack. The amount of local alpha change for a given amount of yaw is the same REGARDLESS of what the airplane's initial alpha was before the yaw, even for inverted flight. You get the same control response to a given yaw, regardless of whether the model is right side up, upside down, or halfway in between.

Sweep isn't quite so simple. Sweeping a wing changes its lift curve slope, so the new dCl/d-alpha is equal to the unswept dCl/d-alpha times the cosine of the sweep angle. For example, if the unswept dCl/d-alpha was 0.1, a sweep angle of 30 degrees would result in a dCl/d-alpha of only 0.087 . BTW, this also explains why swept-wing airplanes can get to much higher angles of attack before stalling. The airfoil stalls at pretty much the same lift coefficient regardless of what the sweep angle is (until we get to extreme sweep angles that result in significant amounts of vortex lift, which gets even more complicated), but the decrease in dCl/d-alpha due to the sweep means that we need to have a higher angle of attack to reach the stalling Cl.

Lets assume we have a 30 degree swept wing flying at an initial angle of attack of about 5.77 degrees, so the lift coefficient is about 0.5 . If we yaw this wing 5 degrees to the left, the right wing now has a sweep angle of 25 degrees and the left has a sweep angle of 35 degrees relative to the freestream flow. Because of the change in effective sweep angle and its influence on the dCl/d-alpha's for the two wings, the forward-yawed (25 deg. effective sweep) right wing now has a lift coefficient of 0.52, and the aft-yawed left wing (effective sweep = 35 degrees) now has a Cl of only 0.47 . The difference in Cl between the two wings of 0.05 causes the initiation of a nice roll rate to the left. Once the roll rate is established, the helical flow around the two wings causes enough change in the local alphas to make the Cl's on both wings once again equal.

Now lets try it again, but this time with the airplane flying ballistically, so that the starting Cl is zero. The cosine of any angle times zero is still zero, so no matter what amount of yaw you input, the rolling effects of the sweep will be zero. You can run the rudder all the way to the stop, and the airplane will continue to fly along sideways, with no corresponding roll response. However, give it a little elevator (making Cl no longer equal to zero), and you will get a roll response.

So the bottom line is that there is no clear single relationship between sweep and dihedral. The amount of dihedral effect you get from wing sweep depends on what overall angle of attack you have during the yaw.

Conventional aft-sweep also tends to shift the lift distribution outward, relieving some of the load in the center of the wing and increasing the lift at the tips. This is what makes aft-swept wings more prone to tip stall. The opposite effect occurs on forward-swept wings, which is what makes them more resistant to tip stall. Note also, that if you use some washout in an aft-swept wing to prevent this, you are decreasing the Cl at the tips in positive-G flight, which will also lessen the dihedral effects of the sweep. In negative-G flight, the dihedral effects of the sweep will be increased, but so will the tendency to tip stall. Swept wings can be a real can of worms if you're not careful.

For a more in-depth discussion of this and other sweep effects, I recommend "Tailless Aircraft in Theory and Practice" by Nickel and Wohlfahrt, ISBN no. 1-56347-094-2.

Don Stackhouse
DJ Aerotech