(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant 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

3. The attempt at a solution

the stream function for the whole flow = Uy+ SUM K_{n}arctan(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_{n}psi_{n}(m) - Uy(m) = 0

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

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.Code (Text):

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

EDIT: realised 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.

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# Homework Help: Modelling a flow past a symmetric body

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