Streamlines of A Free Vortex

We think a case of
u_{r} = 0, u_{\theta} = \frac{C}{r}.

Then, when we think about velocity of this flow along x and y axis.
u = -u_{\theta}sin\theta = -\frac{C}{r}\frac{y}{r} = -\frac{Cy}{x^2+y^2}
v = u_{\theta}cos\theta = \frac{C}{r}\frac{x}{r} = -\frac{Cx}{x^2+y^2}

Then we can get Figure1.

However, when we think about inertial frame of reference.
In this particular case, velocity is equivalent to this formula.
[v_{moving} = v_{static} - v_{frame}]
Hence, we just add constant V in u.
u = -u_{\theta}sin\theta = -\frac{C}{r}\frac{y}{r} = -\frac{Cy}{x^2+y^2} - V
v = u_{\theta}cos\theta = \frac{C}{r}\frac{x}{r} = -\frac{Cx}{x^2+y^2}

Thus, we can get Figure 2.

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-2,2,1000);
y = np.linspace(-2,2,1000);
C = 1; V = 1; U = 0;
u = -C*y[:,np.newaxis]/(x**2 + y[:,np.newaxis]**2) - V;
v = C*x/(x**2 + y[:,np.newaxis]**2) - U;
speed = np.sqrt(u*u + v*v)

plt.figure()
plt.streamplot(x, y, u, v, density=(2,2), color='k', linewidth=100*speed/speed.max())
plt.show()

#two free vortices

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-2,2,1000);
y = np.linspace(-2,2,1000);
C = 5; D = -1; V = 0; U = 0; x0 = 1; x1 = -1;
u = -C*y[:,np.newaxis]/(2*np.pi*((x-x0)**2 + y[:,np.newaxis]**2)) - D*y[:,np.newaxis]/(2*np.pi*((x-x1)**2 + y[:,np.newaxis]**2)) - V; 
v = C*(x-x0)/(2*np.pi*((x-x0)**2 + y[:,np.newaxis]**2)) + D*(x-x1)/(2*np.pi*((x-x1)**2 + y[:,np.newaxis]**2)) - U;
speed = np.sqrt(u*u + v*v)

plt.figure()
plt.streamplot(x, y, u, v, density=(2,2), color='k', linewidth=100*speed/speed.max())
plt.show()

Figure1

Figure2