## Wing downwash calculcation

## Wing downwash calculcation

(OP)

Hello,

I know that this might be something so easy that most of you don't bother, but here's my question:

I have a given wing layout and the circulation/lift distribution over this unswept, untwisted, tapered, iow: "normal" wing. Now I want to calculate the induced drag, which requires the induced angle of attack/induced velocity over the wing.

Looking at the formulas available, e.g. in Theory of Wing Sections, I know that for each location y0 I have to evaluate the integral for alpha_induced (from which I can get w_induced, which I need for the induced drag):

alpha_i (y) = 1/(4 pi V) * INTEGRAL (-b/2, +b/2) [ dGamma/dy' * dy'/(y - y') ]

My calculation is a very easy one, I have only few "check points" along the span and it's all discrete, in fact I use a simple Excel spreadsheet. So I have a row with the spanwise location I'm looking at, and a row with the lift at each of those locations. Of course, I don't want to integrate, but rather I want to do a summation instead of an integration over those spanwise locations.

But when I do that, I seem to have a problem with interpreting the above integration equation.

1) If I use dGamma/dy * dy, it just gives me dGamma... so I'm not sure how to interpret this. I know dy is the integration "width", but it cancels out the dy I have to use for the dGamma calculation. So I'm already confused. Do I just use dGamma (current to next spanwise point) here?

2) I know that for each y0, I have to have y go over the whole span and sum up the inner part of the integral. But when I come to the point where y0 = y, and since y0 - y is in the denominator, I get a division by zero, and that can not be right. Do I instead have to use some really large number here? How do I bypass this problem?

So, on the one hand I'm trying to keep this simple with a spreadsheet and all, but on the other hand I don't seem to have the brain power to get this thing done.

Does this question make sense? Any help? If someone has a code in some recognizable programming language that does this, I would appreciate it if this could be shared.

Thanks.

I know that this might be something so easy that most of you don't bother, but here's my question:

I have a given wing layout and the circulation/lift distribution over this unswept, untwisted, tapered, iow: "normal" wing. Now I want to calculate the induced drag, which requires the induced angle of attack/induced velocity over the wing.

Looking at the formulas available, e.g. in Theory of Wing Sections, I know that for each location y0 I have to evaluate the integral for alpha_induced (from which I can get w_induced, which I need for the induced drag):

alpha_i (y) = 1/(4 pi V) * INTEGRAL (-b/2, +b/2) [ dGamma/dy' * dy'/(y - y') ]

My calculation is a very easy one, I have only few "check points" along the span and it's all discrete, in fact I use a simple Excel spreadsheet. So I have a row with the spanwise location I'm looking at, and a row with the lift at each of those locations. Of course, I don't want to integrate, but rather I want to do a summation instead of an integration over those spanwise locations.

But when I do that, I seem to have a problem with interpreting the above integration equation.

1) If I use dGamma/dy * dy, it just gives me dGamma... so I'm not sure how to interpret this. I know dy is the integration "width", but it cancels out the dy I have to use for the dGamma calculation. So I'm already confused. Do I just use dGamma (current to next spanwise point) here?

2) I know that for each y0, I have to have y go over the whole span and sum up the inner part of the integral. But when I come to the point where y0 = y, and since y0 - y is in the denominator, I get a division by zero, and that can not be right. Do I instead have to use some really large number here? How do I bypass this problem?

So, on the one hand I'm trying to keep this simple with a spreadsheet and all, but on the other hand I don't seem to have the brain power to get this thing done.

Does this question make sense? Any help? If someone has a code in some recognizable programming language that does this, I would appreciate it if this could be shared.

Thanks.

## RE: Wing downwash calculcation

that is a so called "singular integral". See http://en.wikipedia.org/wiki/Singular_integral.

There are specific numerical methods for solving this kind of problem.