# Why does a wing's pitching moment distribution tend towards zero at the tip instead of towards cm0?

I am currently playing around with XFLR5 and am trying to wrap my head around the following:

For airfoils, my understanding that the pitching moment has a component that is independent of lift, which is cm0, or the pitching moment at zero lift.

In XFLR5, I loaded a NACA2412 airfoils and ran the analysis, yielding the following plots: cm is pretty constant between -1<cl<1 (which I think indicates that the aerodynamic center is close to 25%c) and consequently, cm0 is roughly -0.05.

Now, when I apply this airfoil to a simple trapezoidal wing and run the analysis, I get the following lift and pitching moment distributions (at 0 and 10 degrees AOA): The local lift coefficient varies between 0 and a little more than 1, so far so good. From the airfoil 2D data, I would now have expected to see a somewhat constant pitching moment distribution across the span (somewhat like a lookup funtion: local cl = 0.5, therefore local cm = cm.2D(cl=0.5)). However, the analysis shows this: In the center of the wing, we're seeing what I'd expect. At the tip however, cm goes to zero, or rather seems to scale with the cl distribution, instead of being constant.

Why are we seeing this behaviour at the wing tip, which seems to be in contradiction with the 2D data?

• In you last picture, which plot (line, line with circles) is which? And which one exactly would you expect? Apr 6 at 20:43

You are thinking that a section at the tip of a wing will behave like a 2D airfoil. This somewhat happens at the center line due to symmetry (and distance from the tips), but the opposite is true at the wing tips.

Lift is the integral of the local load across the span of a wing.

$$L = 0.5\,\rho\,V^2\,\int_{-b/2}^{b/2} c\,c_l\,dy$$

But that includes the local lift coefficient, which is an integral of pressure across the chord...

$$c_l = \frac{1}{c} \int_0^c \left(C_{p,l}-C_{p,u}\right) dx$$

Where $$C_{p,l}$$ and $$C_{p,u}$$ are the pressure coefficient on the upper and lower surface of a wing.

We can combined these two integrals together to think of Lift as an integral of the difference in pressure coefficient from the top to bottom of the wing over the entire wing area.

$$L = 0.5\,\rho\,V^2\,\int_{-b/2}^{b/2} \int_0^c \left(C_{p,l}-C_{p,u}\right) dx\,dy$$

This makes the most sense if you think of a wing as a thin camber surface (like from thin airfoil theory). This is very much the case when you use VLM or LLT theory (two options in XFLR5), but is a little bit of an abstraction if you're using a thick-surface theory like a panel code or CFD.

In those cases, things are a little more complex, but it still works out pretty much the same. For now, think of a wing like a 3D version of thin airfoil theory -- a thin camber surface plus a symmetrical thickness form plus a flat plate at an angle of attack. Remember the thickness form does not contribute to lift or moment, so we're safe to ignore it here.

Shrink yourself down to that differential element $$dx\,dy$$.

If you were to walk a line at the wing tip, you would find that you are in a place where pressure and a surface doesn't work like normal. One infinitesimal increment outboard (off of the wingtip) and you clearly have no surface to push on, so the air pressure does not contribute to the force/moment of the wing. However, without a surface to push on, the lower and upper pressure coefficients are equal $$C_{p,l}=C_{p,u}$$.

One infinitesimal increment inboard (into the interior of the wing) and you clearly contribute to the integral like normal. The surface of the wing can support the pressure difference that you integrate up to become a lift force.

However, when you are exactly at the wing tip, what happens to $$(C_{p,l}-C_{p,u})$$? Is there a surface to support a pressure difference? If there was a pressure difference on that edge, what would happen (wouldn't the air just escape around the edge? To the point that there would be no pressure difference.

As it works out, aerodynamics requires that the upper and lower surface pressures are equal exactly at the wingtip. Consequently, that exact line can not contribute to the forces and moments.

Although I've constructed this argument thinking about lift (not moment), the equation is the same -- except the integral includes a moment arm as a distance from the reference point (typically c/4) to the point of integration (x,y). This term would look something like $$(x-c/4)$$ or maybe $$(c/4-x)$$ multiplied by the $$(C_{p,l}-C_{p,u})$$ term. This contribution to pitching moment will still go to zero because $$(C_{p,l}-C_{p,u})$$ always goes to zero at the tip.

Check out the distribution of spanwise induced drag coefficient. It too will go to zero at the tip.

So, like I started out -- flow at the centerline of a wing behaves very similar to a 2D wing. However, at the tip, flow is so dominated by 3D effects (the presence of the tip) that it can't even support a pressure difference from top to bottom.

• "This is a VLM code that represents the wing as an infinitely thin lifting surface" the wake is represented as a thin surface, not the wing nor the fuselage Apr 6 at 23:13
• Check this doc xflr5.tech/docs/… slides 22, 23. It appears that XFLR5 has an option for a thick-surface panel formulation in addition to the thin-surface VLM and LLT formulations. We do not know which the OP is using. However, that does not matter for the rest of my answer. Even at a thick wingtip, wings can not support a load and the upper/lower pressures must be equal. Apr 6 at 23:32

According to the user manual, XFLR5 lacks the simulation of viscous boundary layer; this implies that you should "limit the analysis to conditions of limited flow separation:

• High Reynolds numbers
• Low angles of attack
• Low flap deflections"

Furthermore, "the wake is modelled as a straight extension of flat panels behind the wing’s trailing edge" but in reality "the wake takes the shape of the streamlines, and the wake panels should take the shape of the streamlines".

This implies that the primary viscous, 3D effects at the wing tips are completely lost.

Whatever you get outside these conditions is just random numbers.

• Yes, but both at 0° and 10° AOA the sim should still be well within the linear part of the c_L-alpha-curve, so I don't expect any separated flow anywhere. Apr 6 at 20:45

Let's set the computer and the formulas aside for a moment and try to approach the issue with a mechanistic understanding of wingtip airflow, but first ...

what type of airfoil are you using???

The NACA 2412 has zero lift at -3 degrees AoA. Slightly non symmetrical. The Cm increases with AoA, but remains negative through stall, in agreement with your first 2 graphs.

why would Cm increase near the wing tips?

You have 2 plots, 0 and 10 degrees. We see decrease in coefficient of lift near the wing tips and an increase in Cm.

could this be real?

Because the wing tip vortex is expected to curl up and over the wing tip, striking the back of your trapezoidal wing, this is exactly what one might expect.

how do designers help this?

WINGLETS

Notice the wing tips of the seagull, tapering down to a point.

Try this type of wing tip on your XFLR5 and please let us know how your 0, 10 degree Cl/Cm plots turn out.