Plane Potential Flows

A 2D (plane) potential flow is inviscid, irrotational, incompressible, and steady. Recall that potential flows are solutions to Laplace’s Equation.

Recall that based on the stream function and the velocity potential, we have:

$$\begin{array}{rl}u& =\frac{\partial \varphi }{\partial x}=\frac{\partial \psi }{\partial y}\\ v& =\frac{\partial \varphi }{\partial y}=-\frac{\partial \psi }{\partial x}\end{array}$$

Alternatively, in cylindrical coordinates:

$$\begin{array}{rl}{V}_r& =\frac{\partial \varphi }{\partial r}=\frac{1}{r}\frac{\partial \psi }{\partial \theta }\\ V_{\theta }& =\frac{1}{r}\frac{\partial \varphi }{\partial \theta }=-\frac{\partial \psi }{\partial r}\end{array}$$

Since the row is irrotational, the curl of the velocity must be zero:

$$\frac{\partial v}{\partial x}=\frac{\partial u}{\partial y}$$

Substituting the expressions for $u$ and $v$:

$$\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}=0$$

This is Laplace’s Equation, which is also satisfied by the velocity potential:

$${\nabla }^2\varphi =0\text{and}{\nabla }^2\psi =0$$

• Both the stream function and velocity potential satisfy Laplace’s equation in potential flow, enabling us to solve them independently.

Types of plane potential flows:

• Uniform Flows
• Source and Sink Flows
• Vortex Flow
• [[Doublet Flow