Need help with numerical lifting line method

[Dmalyuta]

Dear all,

I am following Warren F. Phillips' book "Mechanics of Flight" (2nd edition) and am trying to code the numerical lifting line method outlined by him starting on page 96 of Chapter 1.

I will write the code in Matlab, but what is preventing me from doing so is a few variables within the said method which I am confused about how to find.

Variable number one is dl_i, which is defined as the "directed differential vortex length vector"; the boundary of my knowledge of vector calculus is differentiating vectors. Now, the method does not require anything beyond that- but I do not know what the vortex length vector even is, and how one finds it.

Question number two is about Newtons method. What is a relaxation factor?

As I write my code, I might have more questions and I will probably ask them here. For now, I would appreciate your help in explaining to me what variable number one is/how it is found!

For reference, variable number one is cited to come from Saffman 1992- I have the book, but cannot find anything on the variable in it (300+ pages, non-searchable format). The method itself is referenced to come from a more detailed account on it in Phillips and Snyder 2000.

THANK YOU!

 

[RandomGuy88]

I am not certain what the directed differential vortex length vector is and i do not have that book but here is what I think it might mean. When using the lifting line method you model the wing as a series of horseshoe vortices. Each horseshoe vortex is made up of 2 semi infinite vortex filaments and one finite vortex filament. I suspect that this variable represents the length and direction of the finite vortex element. Therefore it can be determined based on the geometry of the problem. If you show me the entire equation I may be able to help more.

Unfortunately I can't help with the relaxation factor.

 

[Dmalyuta]

Hi randomguy!

Thank you for the response; I think you have set me on the right track.

However!

I am only familiar with differentiating vectors which have a variable in them; e.g. the derivative of [t; 2t²; 3t] will be [1; 4t; 3]. Where are the variables in the directed differential vortex length vector? I understand the vortex length vector as the difference of the position vectors of the two nodes of the bound segment running along the quarter-chord of the wing- so then how do you differentiate it? And what does 'directed' mean?

If you have access, the paper on the numerical lifting line method is called 'Modern adaptation of Prantl's classical lifting-line theory' by Phillips and Snyder/2000/AIAA Journal of Aircraft issue #4.

Waiting for a reply! :)

 

[hotvette]

Question number two is about Newtons method. What is a relaxation factor?

I think it refers to adding less than the full correction value. Newton's method is iterative where the new estimate is the old estimate plus a correction value:

x_n+1 = x_n + α*Δx_n

Alpha is the relaxation factor. Nominally it has a value of 1 so often isn't shown. But, it can be less (or greater) than 1.

 

[nbunderson]

The differential length vector is the part of the horseshoe vortex that is bound to the lifting surface, exactly what RandomGuy88 said. This vector will not be defined by a formula that you take the derivative of, it is just a vector connecting two points. To do Phillips method you choose n points which span the wing to be the "nodes", these are usually along the quarter-chord. You also choose n-1 "control points", 1 in between each successive pair of nodes. There will then be n differential length vectors connecting the nodes on either side of each control point. You don't need to take the derivative of anything, you just choose where the nodes go (you should eventually use the cosine distribution Phillips describes but for now just make them equally spaced along the quarter-chord from wing-tip to wing-tip) and then you can easily calculate these "differential" vectors.