# Nonlinear Lifting Line method to estimate wing lift distribution

1. Nov 24, 2009

### aero1

Hi all,
I have implemented a nonlinear lifting line method to estimate lift/induced drag of a typical wing. This method utilizes 2D lift coefficient data, and uses a numerical technique to solve the nonlinear equations.

The program I have written in MATLAB is working, though I would like to confirm one key aspect of the results.

When I run the program for a swept back wings, I see that the induced angle of attack goes to negative towards the wing tip (aeff = a(geometry) - a(induced)).
From what I understand, towards the tip a wing, the local angle of attack there is greater than the freestream angle of attack, due to the induced flow, therefore a(induced)should be negative.

So my question is: Is getting a negative value for the induced angle of attack actually correct? I have a hard time trying to grasp this physically.

sample data induced angle of attack at the wing tip
0.024249058497046 (Close to wing tip)
-0.051359819152125
-0.088120053792167
-0.019725856790100
0.457096103863267 (wing tip)

Any help would be much appreciated.
Thank you.

2. Nov 28, 2009

### Phrak

For a swept and highly tappered wing (Is your wing tapered?) the induced vortex is not centered at the tip of the wing, but somewhere inboard. So I don't see how your program model is coming up with effective aoa that is less than the inboard value...

3. Dec 1, 2009

### aero1

When I run cases for taper wings and/or with sweep, it had negative values. But when I ran the model for straight, no sweep, no taper, it was giving somewhat correct results.

I think I have fixed the problem. I had my control points at the 1/4 chord , in the middle of the bound vortex. This would work for straight wings. But for swept and tapered wings, I had to use the 3/4 chord theorem, and located the control points there in the middle of the trailing vortices.

I am not sure of the physical meaning to this, but I will have to go and read up on the very basics of the 3/4 chord theorem to better understand why this is the case.

Thanks again.

4. Dec 1, 2009

### Phrak

Let's try again. I wasn't paying good attention.

You said,
There's a trailing vortex following a wing. Upward on the the wing tips; down toward the root. This induces some amount of vortex flow to the leading edge. The differential is downward inboard and upward outboard. So the outboard wing will experience greater angle of attack. You've simply inverted your thinking as far as I can tell.

5. Dec 22, 2009

### cobalt 60

Hello aero1,

I am in the process of writing the numerical lifting-line method in Matlab for part of my research. Most of the info I am getting is from Mechanics of Flight by Phillips. I would like to know where you did your search fOR the lifting-line therORy. Also I never heard of the 3/4 chord theorem. Can you elaborate more on this. I will plan to do a search regardless. It's a monumental process writing this code, but I think satisfaction of writing your own code far outways the cons.

ty
cobalt 60

6. Dec 22, 2009

### aero1

Hey cobalt,

Look up "Introduction to Aircraft Flight Dynamics or was it flight mechanics..." Google book
has it but has some pages taken off for more info on how to program a simple modified
wessinger lifting line method.
Also, if you can get access, look into AIAA or SAE papers and search word non-linear lifting line or wessinger. I am away now and wont be able to respond to you with the exact references I used till 6th Jan 2009. But those searchs on AIAA and SAE should help and the book too. Oh, and look up AIAA " Full drag Configuration (Or was it estimation..)" by M.H. MASON as co-author. It would prove useful in how order your code.

3/4 chord theorem or pistolesi theorem (not sure if i spelled it right) will elaborate more when I get back.

Happy holidays!
Aero1