Modelling a flow past a symmetric body

[mybrainhurts]

Homework Statement

I need to model the streamlines for an incompressible, irrotational flow about various symmetric bodies, starting with an ellipse

Homework Equations

for a uniform flow in the positive x direction, psi =y
for a source/sink of strength K at (xn,yn)
psi = Karctan((y-yn)/(x-xn))
at any point along the body, there must be no flow across it, and hence psi = 0

The Attempt at a Solution

the stream function for the whole flow = Uy+ SUM K_narctan(y/(x-x_n)) (no ydisplacement as all sources are on axis)
this = 0 at each of the control points y(m) (and everywhere else along the surface of the sphere, but a number of control points = to the number of sources should be appropriate)

hence
K_npsi_n(m) - Uy(m) = 0

and so I should be able to get the strengths from the following MATLAB code (taking U =1)

n = 10;
L = 1;
D = 0.5;
dx = 2*L/(n+1);
xs = -L+dx:dx:L-dx;   % uniformly distributed
xb = -L+dx:dx:L-dx   % uniformly distributed
%
% definition of body geometry
%
   yb = D*sqrt(1-xb.^2./L^2); % ellipse
%
% plot control points
%
figure
plot([-L xb L],[0 yb 0],[-L xb L],[0 -yb 0])
axis equal
%
% compute source strengths
%
% K = strength
%
for i=1:n
    for j=1:n
        a(i,j) = atan(yb(i)./(xb(i)-xs(j))); 
    end
    b(i,1) = -yb(i);
end
K = a\b

However this, results in all negative values for K, meaning a huge total removal of volume from the system instead of the expected no (or approximately no) change, and hence a set of funky streamlines that look nothing at all like a flow around an ellipse.

EDIT: realized I made a mistake, should be b(i,1) = yb(i) (not -yb(i)) however this just results in all strengths being positive, so I get what looks like a funky superposition of flows around multiple semi-infinite bodies.

 

[mighty2000]

After further analysis, I have determined that the issue lies in the assumption that the flow is irrotational. In reality, there will be some degree of vorticity present in the flow around the ellipse, which will affect the streamlines and source strengths. To accurately model the flow, the vorticity must be taken into account and incorporated into the equations. Additionally, the flow around an ellipse is not necessarily symmetric, so the equations used may need to be modified to account for this asymmetry. Further research and experimentation will be necessary to accurately model the flow around an ellipse.