Rankine Oval – Potential Flow

Plane potential flows allows us to approximate the streamlines of a fluid flow.  It does this  by using Laplace’s equations to determine the basic velocity potentials and stream functions for simple irrotational flows.  In turn, these basic velocity potentials and stream functions are combined to determine the streamlines of more complex flows. readmore

Lecture Note | Lecture Note2
[x,y] = meshgrid(-25:0.1:25); 
m=5; 
%a=2; 
U=1; 
p1 = [3 -8];
p2 = [7 -4];
psi1 = U*(y*cos(pi/4)-x*sin(pi/4))-m*atan2(y-p2(2),x-p2(1))+m*atan2(y-p1(2),x-p1(1)); 
figure(1)
contour(x,y,psi1,100);    % original result 
colorbar
grid on;
z = x+i*y;
z1 = p1(1) + i*p1(2);
z2 = p2(1) + i*p2(2);
u = (z2-z1)/abs(z2-z1);   % direction vector on the unit circle
psi2 = imag(z/u) -m*angle(z-z2) + m*angle(z-z1); 
figure(2)
contour(x,y,psi2,100);    % same result
colorbar
grid on;
psi3 = imag(z/u) -m*angle((z-z2)./(z-z1)); 
figure(3)
contour(x,y,psi3,100);   % new result
colorbar
grid on;

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