Uniform flow

$$\phi = u_{\infty}x + v_{\infty}y $$ $$\psi = u_{\infty}x - v_{\infty}y $$

Source

$$\phi = \frac{q}{2\pi} ln \space r$$ $$\psi = \frac{q}{2\pi} \theta $$

Sink

$$\phi = \frac{-q}{2\pi} ln \space r$$ $$\psi = \frac{-q}{2\pi} \theta $$

Rankine half-body

A Rankine half-body is a source added to a uniform flow.

$$ \phi = u_{\infty}x + \frac{-q}{2\pi} ln \space r$$ $$\psi = u_{\infty}y\frac{-q}{2\pi} \theta $$

The stagnation point $s$ of a Rankine half-body is given by $$ s = \frac{q}{2\pi u_{\infty}}$$ $$ q = 2\pi s u_{\infty}$$ The streamline which passes thorugh he stagnation point, passes along $y = 0$ when $\theta = \pi$, as given by $$ \psi = u_{\infty}(y + s\theta \\ \Rightarrow \psi_s = u_{\infty}s\pi \\ = \frac{q}{2}$$

Pressure distribution is given by $$ C_p = - \left(\frac{2s \space cos \theta}{r} + \frac{s^2}{r^2} \right)$$

Doublet

A doublet is a source added on top of a sink, both equal in strength. Streamlines will then circulate from the center of the sink/source outwards, make a loop, and then end in the center.

$$ \phi = \frac{q}{2\pi}\frac{cos \theta}{r}$$

The strength of a doublet can be calculated by $$ B = \frac{q \space a}{2\pi}$$

Note that this flow elements has directionality, which means it is sensitive to the values provided.

Line Vortex

A line vortex causes streamlines to move in a circular fashion around a certain point in the field. The flow of a line vortex has no radial movement. $$ \phi = \frac{\Gamma}{2\pi r}$$ $$ \psi =\frac{\Gamma}{2\pi r} \space ln \space r$$ where $\Gamma$ is a constant representing the circulation.

Flow around cylinder

A flow around a cylinder is built up of doublet placed in a uniform flow. $$ \phi = u_{\infty}r \space sin\theta - \frac{B}{r}sin\theta$$

The pressure distrubution is given by $$C_p = 1-4 \space sin^2 \theta$$

This gives a symmetric pressure distrubution, resulting in no net forces (which is unphysical).

In reality, drag and lift of a cylinder is given by $$ C_D = \frac{F_x}{\frac{1}{2} \rho u^2_{\infty} D^2} \\ C_L = \frac{F_y}{\frac{1}{2} \rho u^2_{\infty} D^2}$$

The drag is caused by skin friction (viscous effects) and pressure drag. A real cylinder in a flow will have a messy wake.

Flow around cylinder with circulation

A flow around a cyliner with circulation is modeled by adding a line vortex on top of the flow around of a cylinder.

Source above wall

Advection of vortices