# The criteria for potential flow theory

I am learning aerodynamics. In this course a potential flow is denoted that a flow in which the rotation is zero everywhere. But the book told me that we can add vortex into a flow field, and we can also use potential theory to analysis it. I get confused about this. If a vortex is added in a flow field, I don't think there exist any potential. The rotation of a vortex is infinite! Could anyone explain to me, why potential flow theory can be used while there are some vortexes in the flow field?

Potential flow usually means, as you correctly said, that your flow is irrotational, so that your velocity field $\vec{u}(\vec{x},t)$ has the property:

$$\vec\nabla\times \vec{u} = 0 \;\;\Longrightarrow\;\; \vec{u}=\vec\nabla\phi$$

However this is not the general case, because you usually have some rotation in your flow which is called vorticity and is defined as:

$$\vec\omega = \vec\nabla\times \vec{u}$$

Now this vector can be whatever it wants, constant and finite for example, there is no reason for it to be infinite. Coming back to potentials, in case a non-null vorticity is present, the velocity field is can still be decomposed into an irrotational part expressed by $\phi$ and a rotational part expressed by the so-called stream function $\vec\psi$ in the following way:

$$\vec{u}=\vec\nabla\phi + \vec\nabla\times\vec\psi$$

In this sense people still talk of potential flow even when the flow is not irrotational.

The reason is that this formalism is very useful in two dimensional flows, where the vorticity and the stream function are vector fields always pointing in the $z$-direction and are linked by the relation:

$$\nabla^2\psi = -\omega$$

I hope this clarifies a bit your doubts, if you have some more just ask away!

• Thanks for your answer. So, do you mean that if there is a flow field which only have one vortex, its $\phi$ must be a constant through out the flow? And the vortex lattice method has nothing to du with potential flow theory? Nov 8, 2013 at 13:16
• @Mattia: I think the OP asked about specific solution: 2D point vortex, where there is $\delta$-like vorticity field. Nov 8, 2013 at 14:54
• @maple: Unfortunately I don't know the vortex lattice method, so I cannot really tell, but looking at the wikipedia page about it I bet guess it's all about the potential flow theory. Nov 9, 2013 at 8:56
• @Mattia: As user23660 pointed out and maple's comment indicated, you simply did not answer the question asked. In fact you confused maple more and he now thinks $\phi$ is constant for a vortex. Nov 13, 2013 at 20:50
• Well, I think I answered the question "why can I use potential flow theory in presence of vortices?". And yes, I didn's read the part of the comment saying that $\phi$ must be constant, sorry. This is not true, of course, and I think that your answer points it out correctly. Nov 14, 2013 at 8:20

As contradictory as it sounds, you can have a vortex that satisfies $$\vec{\omega} = \vec{\nabla} \times \vec{u} = 0$$ and $$\nabla^2\phi=0$$The potential is given in terms of polar coordinates $r$ and $\theta$ by $$\phi = \frac{\Gamma}{2\pi}\theta$$ which gives $u_{\theta}=\frac{\Gamma}{2\pi r}$ and $u_r = 0$. Evaluating the curl everywhere but at the origin gives $$\vec{\omega} = \frac{\partial ru_{\theta}}{\partial r}-\frac{\partial u_r}{\partial \theta} = 0$$

• Sorry, I don't see the contraddiction. You start with $\vec\omega = 0$ and you end with $\vec\omega = 0$, maybe I missed a step in your reasoning. Nov 8, 2013 at 23:38
• No contradiction between $\vec{w}=0$ and having a vortex. That is what I'm conveying in my answer. Having read your answer and comment above, I think your comprehension of the English language is lacking a bit. Nov 12, 2013 at 18:33
• So why should it sound contraddictory? To me your answear is generating confusion because you're mixing vortex and vorticity, which are different objects. In fact, a vortex can be irrotational, as in your case, or not. Intuitively a vortex is irrotational if the fluid element conserves its orientation during the flow, so that the vorticity is zero. In general the vorticity is a local property of the flow, while a vortex is a flow with closed streamlines. Oh but sorry, I shouldn't say this, my comprehension of the English language is lacking a bit... Nov 13, 2013 at 11:24
• I'm glad you understand the difference between vortex and vorticity, but if you read the original question, the author did not understand that distinction: "...the rotation is zero everywhere.... The rotation of a vortex is infinite!" It's not hard to understand why someone new to the subject would think it contradictory to have an irrotational vortex since a vortex by definition involves rotation. While it might be intuitive to you and me, it wasn't to the author of the question. Nov 13, 2013 at 20:42