# Why is the numerically determined location of the center of pressure inaccurate near Cl = 0?

I am working on an assignment and I need to calculate the center of pressure of an airfoil, through a numerical method (the panel method). I am getting a weird result because it gives its position on a point outside the chord for a determined angle of attack.

When I asked my professor he said that under an angle of attack near the angle of attack that makes Cl=0, it may come out that way, because the center of pressure is calculated through an expression that involves a quotient with the Cl=0 on its denominator.

I am confused because I don't know if it is just that the method fails in that case, or if it has some physical explanation.

• Sounds like a good opportunity for further discussion with the professor. Since you're already paying him (indirectly through tuition), might as well get your money's worth. Commented Apr 18, 2016 at 14:16
• sounds numerical, if you divide by 0 usually bad things(TM) happen. Commented Apr 18, 2016 at 14:26
• Just a very, very general point about modelling and engineering: sometimes there is no "physical explanation", the model in not in the real physical world. You now have a great opportunity to learn something about the limitations of the model, any assumptions implied, when it might be expected to fail etc.
– Andy
Commented Apr 18, 2016 at 14:43
• Thanks for replying, I will try to clarify it with my professor, but I found this on wikipedia: "It is common for the center of pressure to be located on the body, but in fluid flows it is possible for the pressure field to exert a moment on the body of such magnitude that the center of pressure is located outside the body." So isn't it a nonsense to think it to be out of the chord ?
– abcd
Commented Apr 18, 2016 at 22:06

If some parts of the airfoil create lift and others a downforce, the center of pressure can be outside of the airfoil's chord at low lift coefficients. This condition is fulfilled for cambered airfoils, airfoils with a deflected flap and especially for rear loading airfoils, which have low camber in the front and high camber in the rear part. Supercritical airfoils meet this last condition.

In airfoil theory, an airfoil creates lift and a moment, because the lift is assumed to attack at the quarter chord point. This point is special because here the moment coefficient does not change with the angle of attack (at least in the inviscid, linear potential theory, which is sufficiently close to reality at large Reynolds numbers to be useful). In reality, positive camber will cause the resulting lift force to act aft of the quarter chord point, and the pitching moment is negative. When lift becomes small and the pitching moment stays constant, the lever arm of that small lift needs to become large to achieve the same moment, and this is when the center of pressure can slip out of the airfoil's chord.

Below you see the result from XFRL5 V6.0.5. I plotted the local center of pressure as a green line. It's elevation over the plane of the wing shows the amount of lift locally created, and the location in streamwise direction shows that it leaves the local chord when lift becomes low. Note that when moving out in spanwise direction the location jumps from far aft of the airfoil to far ahead when the local lift turns negative. At the point of no local lift you have a division by zero error, patched over here by a straight connection between the results from the single panels.

Location of the local center of pressure on a swept wing (own work).

When angle of attack is increased, all additional lift has the Birnbaum distribution, so the local center of pressure moves towards the quarter-chord location.

• Could you elaborate a bit on the Birnbaum distribution? I tried to Google it, but I could only find references to unsteady flapping aerodynamics such as this one or should I ask this in a new question? Commented Apr 21, 2016 at 7:49
• @ROIMaison: I trust you understand German, so I recommend this page. Commented Apr 21, 2016 at 12:12
• Thank you for such a complete explanation. It still sounds me a bit counterintuitive. Mathematically I see it, when you weight the position "x" with the local lift coefficient (to calculate where the resultant is applied), since both "x" and "cl" can be negative, there may be some contribution that tends to set it out of the chord. Am I missing any important idea or simply there is no more on it?
– abcd
Commented Apr 23, 2016 at 10:46
• @abcd The airfoil is creating both a moment (measured around the quarter chord point) and lift, and the moment is determined by camber, so it does not change with angle of attack (linear theory). When lift is very small, the lever arm to express the moment must become huge. Commented Apr 23, 2016 at 18:24
• So, would I be right in saying that the main problem here is that "centre of pressure" is an aggregate function, rather than a simple force? The wing itself is only providing lift "within itself" but that CoP also includes other components? Commented Mar 11, 2021 at 9:30