Potential flow

The image represents the stream function of an incompressible inviscid flow past a cylinder. The stream function has a closed form solution in polar coordinates:

$$\psi\left(r,\theta\right) = V_{\infty} \left(r - \frac{a^2}{r}\right) \sin{\theta}$$

where a is the cylinder radius.

Cylinder radius0.1

Streamlines30

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Particles1000

Particle radius3

The pressure field is obtained from Bernoulli's equation. The velocity components in polar coordinates are:

$$\begin{aligned} v_r &= V_{\infty}\left(1 - \frac{a^2}{r^2}\right)\cos\theta, \qquad\\ v_\theta &= -V_{\infty}\left(1 + \frac{a^2}{r^2}\right)\sin\theta \end{aligned}$$

giving the pressure coefficient:

$$\begin{aligned} C_p &= 1 - \frac{v_r^2 + v_\theta^2}{V_{\infty}^2} \\ &= 1 - \left(1 - \frac{a^2}{r^2}\right)^2\cos^2\theta - \left(1 + \frac{a^2}{r^2}\right)^2\sin^2\theta \end{aligned}$$

On the cylinder surface (r = a) this reduces to:

$$C_p\big|_{r=a} = 1 - 4\sin^2\theta$$