What is the physics behind line vortices and their equations of motion?

[etotheipi]

I'm getting really stuck with understanding this example:

(Source: page 98)

What is a line vortex, and how do we derive those equations of motion? All I can tell is that the velocity of a given vortex depends on a contribution from every other vortex in the plane, but I wondered if someone could explain the physics of this problem? Thanks

 

[vanhees71]

Have a look at the classic

H. Lamb, Hydrodynamics, Cambridge University Press

He has a long chapter on vortex motion.

 

[etotheipi]

Thank you, I booked it! I'll post something if I determine the derivation

 

[Ibix]

A line vortex is a line around which the fluid circulates. Roughly speaking, the diagram is an idealisation of three tornadoes - their winds interact, either adding or opposing, and this moves the vortices around. A neat demo you can find on YouTube is blowing smoke (or vape) rings. The rings are line vortices curved round into loops, and two rings interact in visually intriguing ways.

I've long forgotten the maths, I'm afraid.

 

[etotheipi]

Okay, after some reading I think I understand the construction now. Start with the vorticity equation,$$\frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega} \cdot \nabla) \boldsymbol{u}$$where the $D/Dt$ is the material derivative. If the flow is confined to a plane, then $\boldsymbol{\omega} = \nabla \times \boldsymbol{u}$ is orthogonal to the plane, which contains $\nabla \boldsymbol{u}$. That means, the RHS is zero, and $$\frac{D \boldsymbol{\omega}}{Dt} = \boldsymbol{0}$$Vortices with cylindrical symmetry are described by a velocity potential $\phi = k\theta$, i.e. $\boldsymbol{u} = \nabla \phi = (k/r) \boldsymbol{\theta}$. If we have N vortices, then the resulting velocity potential is$$\phi(\boldsymbol{r}) = \sum_i^N k_i\theta_i$$where the $\theta_i$ are the angles of the line segments from the $i$th vortex to the point $\boldsymbol{r}$, w.r.t. the $x$ axis. Helmholtz tells us that any given vortex only experiences the velocity field arising due to all of the other vortices (and not due to itself), so $$\dot{\boldsymbol{r}}_i = \nabla_{\boldsymbol{r}_i} \left( \sum_{j \neq i}^N k_j \theta_j \right) = \sum_{j \neq i}^N k_j \nabla_{\boldsymbol{r}_i} \theta_j = \sum_{j \neq i}^N k_j \left( \frac{\boldsymbol{z} \times (\boldsymbol{r}_i - \boldsymbol{r}_j)}{|\boldsymbol{r}_i - \boldsymbol{r}_j|^2} \right)$$and when we break that into $x$ and $y$ components, I think that agrees with what Prof. Tong wrote (except the $\gamma$ terms are wrapped inside the $k$ terms here).

 

[etotheipi]

A neat demo you can find on YouTube is blowing smoke (or vape) rings. The rings are line vortices curved round into loops, and two rings interact in visually intriguing ways.

These things are indeed pretty neat!

 

[Ibix]

That's a slightly more scientific video than I was thinking of. The one below is a guy doing tricks with an eCig, and he has a bit more control over the rings than the bucket gives you. About 1.50 and 2.35 you can see the rings he makes interacting - the latter is a classic where he blows one ring through another and they begin to orbit each other.