Calculating the cl_max of a wing from airfoil characteristics

[MaAl]

Hi,

is there an appropriate method to get the maximum lift coefficient (cl_max) of a wing from the polars of the section airfoils?

The background is that I cut a arbitrary wing into a certain number of sections. After that I use Xfoil to compute the local cl_max. Since this approach neglects spanwise effects, my 3D wing will not reach the cl_max for my profile cl_max-values.

That's why I wonder if there are some correction methods or other approaches how to deal with this.

Thanks!

 

[berkeman]

Interesting question. I'll page @boneh3ad to see if he's available to comment.

 

[boneh3ad]

Funny story, this. There may well be a rough correction here, as that's a pretty common thing to exist in this field. That said, this sort of thing is often taught in a typical undergraduate aerodynamics course. Since my undergraduate degree was in mechanical engineering, I never actually took such a course and haven't had to (had the privilege to?) teach one, so I actually don't know the answer to the question.

Of course, my PhD is in aerospace engineering, so I am sitting here with that degree but with a relatively sparse knowledge of the "rule of thumb" type tricks in the field, but if you need to know about the actual flow physics, I'm your guy.

Now, all that having been said, I have a whole stack of relevant textbooks here in my office that may offer some clues. I can probably browse a couple of them when I get a chance a little bit later. I'm a bit swamped at the moment, though.

 

[cjl]

So there are rules of thumb about this. A finite wing will have a lift slope that differs from an infinite wing because the wingtip vortices cause an induced downwash on the wing. The shorter the wing, the stronger this downwash is, and therefore the larger this effect. This is because this downwash effectively reduces the angle of attack of the wing by adding a small downward component to the inflow velocity. This lift slope correction is as follows:

a = a_0/(1+a_0/(π*e*AR))

in which a is the lift slope, a_0 is the lift slope of the 2-d airfoil, e is a factor that depends on the wing shape (a reasonable value is likely around 0.8), and AR is the aspect ratio.

If your stall angle remained unchanged, your Cl_max would also change by this same factor, but since the lift reduction is caused by a local reduction in AoA along the wing, your stall will also be delayed. I know that for a wing of aspect ratio ~5 to about 12, the stall delay is about 2 degrees, for ~12 to 20, it's about 1 degree, and above 20, you can neglect this (but at that point you'll be getting pretty close to just 2-d lift behavior anyways). I don't know how one would calculate this easily though, since the AoA change varies along the wing. You can use the estimated stall delay combined with the lift slope change to get a reasonable estimate for Cl_max for the wing though.

Also, wow. For once I know a thing about fluid mechanics that @boneh3ad doesn't. I shall mark this day on my calendar (and have a celebratory beer after work).

 

[boneh3ad]

Also, wow. For once I know a thing about fluid mechanics that @boneh3ad doesn't. I shall mark this day on my calendar (and have a celebratory beer after work).

Heh, I'm not some all-knowing fluids oracle. I'm really just a mechanician (the science-y definition) disguised as an engineer, so sometimes the rule-of-thumb applications are outside of my wheelhouse. One exception is in designing and operating wind tunnels, in which case I am a top-notch guesser with the rest of the wind tunnel engineers out there.