Help on steady potential flow-Linearly varying vortex method
 

Hi everyone,
I'm working on a project to create an unsteady potential flow code. Essentially I want to test it in steady flow to make sure I'm doing the velocities and influence coefficients correctly.

I'm following the book Low Speed Aerodynamics by Katz.
The issue I guess is I'm not sure how to go about calculating the influence coefficients ( previously in lumped vortex method I dotted the normal with u and v) however following the book for linear vortex method, I convert to local panel coordinates I find "u_p" and "v_p" ( corresponding to velocities in local panel coor.) but each is broken down into u_p1 and u_p2, v_p1 and v_p2 corresponding to velocity due to gamma_1 and gamma_2 respectively.

then the book goes on :
a_ij = (u w)_ij * n_i where n is normal vector.

I'm not sure exactly how to do this, what is u and v ? are they simply u = u_1 + u_2
and v = v_1 + v_2 ? ( where u_1,u_2,v_1 and v_2 are converted back from panel to global coordinates)

I have the code if anyone wants to look at it, its written in MATLAB. Any help would be appreciated. Thanks in advance

You are correct - u=u_1+u_2, v=v_1+v_2
The influence coefficient is the normal velocity induced by the singularities,
(u,v) dot n.

Be sure to start with a simple, known case (circular cylinder).

3D potential flow-panel method
 

Thank you for your reply earlier. I was wondering since you seem to have a handle on the subject if I can ask you for a suggestion. I worked on the steady model using linearly varying vortex method, then extended it to a 2D unsteady model( the shed vortices were modeled using discrete vortices. The main goal behind the project is to finally get it to work in 3D.
I read up on the subject in Katz's book Low Speed Aerodynamics and it seems there's multiple ways to skin a cat. I wasn't sure if I should go with the vortex lattice method, horseshoe or sources and doublets. ( My goal is again to work in a steady model in 3D then work with the unsteady one). I would like it to model thick bodies , from my understanding only the sources and doublets are able to do so correct? It seems that the VLM only models thin wings ?
Your suggestions would kindly be appreciated.

The short answer is that I recommend L. Morino's method (doublet=potential, source=normal velocity). This method works for steady, unsteady, oscillatory flow for wings/bodys/whatever.

One of the thousands of cats I've skinned is attached. It's an airship (with body wake) and fins in a wind tunnel (test section walls removed for picture).
If you used sheet metal to build this model for real, you wouldn't try to buy "lifting" sheet metal for the fins and non-lifting sheet metal for the body. The sheet metal doesn't have to "know" what kind of device it's part of. That's the essence of Morino's method. All 6000 body panels and 1000 wake panels are the same. I'm not interested in how many different ways I can skin a cat, I just want one way that will suffice for all cases. I can't say you can't get there by some other means, but I can say that Morino's method will get you there.

I'm curious as to why you model the wake (discrete vortices) differently from the body (linear vorticity).

That looks beautiful!

There was no specific reason as to why I used discrete vortices for the wake, for some reason I felt like that method (in my head at least) was easier to derive than taking the wake as extra panels.

The reason I asked for the methods was because when reading Katz, it seemed like doublets alone(VLM) are easier to code. And when it came to sources and doublets combined Katz suggested using the codes that are available out there, my intentions are to code it from scratch for practice. I really like your take on "lifting" and "non-lifting" bodies. But I'll definitely look at Morino's method, is there a primer to get me started on his method? I would like to go through the math first and derive the matrices. I would also like to note that I'm using MATLAB so matrix inversions (though limited ofcourse) are easier for me as a beginner.