For a point vortex at origin,streamlines are concentric circles and velocity potential lines are radial lines.

gam = 1; % Vortex strength

% GRID:

x = -4:.02:4;

y = -4:.02:4;

for m = 1:length(x)

for n = 1:length(y)

xx(m,n) = x(m); yy(m,n) = y(n);

% Velocity potential function:

phi_Vortex(m,n) = (gam/2/pi)*atan2(y(n)+.01,x(m));

% Stream function:

psi_Vortex(m,n) = -(gam/4/pi)*log(x(m)^2+(y(n)+.01)^2);

end

end

% Plots

% Vortex at origin of coordinate system:

contour(xx,yy,psi_Vortex,10,'k'),hold on

contour(xx,yy,phi_Vortex,[-.5:.05:.5],'r')

legend('streamlines' ,'potential')

title(' Point vortex')

axis image

Point vortex at origin