Vortex line, tube, sheet, and filament

Question

In my study of fluid dynamics I have across these terms quite often and I am so far unable to create a physical picture in my head. Suppose we have a 3D velocity field $u$, and its corresponding vorticity field is $\omega = \nabla \times u$. What exactly is a vortex line, a vortex tube, a vortex sheet, and a vortex filament? While we are at it, what is the difference between a point vortex and a vortex line?

My current understanding is that a vortex line is simply an integral curve through the vorticity field (i.e. choose a point and "follow" the vector field to see where the point came from and where it is going). If this physical picture is correct, then a vortex tube is defined as a bundle of vortex lines. What is the precise definition of a bundle here? I have seen a vortex sheet as a surface that is everywhere tangent to $\omega$. Is this now simply an integral surface? As for vorticity filaments, I have no idea what that is currently.

I suppose another way of wording my question is that I understand, in most cases, what these things mean mathematically, but physically I cannot create a picture of what they are.

Answer 1

Before starting with a list of definitions, let me tell you that these concepts are particularly interesting the limit of high-Reynolds flows, with limited regions of the domain where the flow is rotational, i.e. $\omega \ne \mathbf{0}$. A typical example of this kind of flow, is a high-Reynolds flow with no separation around streamlined bodies, where the only vortical regions are the thin boundary layer on the surface of solid bodies and the wakes shed by the bodies.

Vorticity in the limit of inviscid flows

In these conditions, the dynamics of vorticity is governed by:

• Helmholtz's theorems for the vorticity providing some conditions on the continuity/conservation of vorticity. The intensity of a vortex tube is constant in space an time. Vortex tubes are closed, or are created or destroied by solid bodies, since they can't be created or destroied inside the flow: you can see some examples in the videos below, where vortex tubes in waters are closed vortex rings, while vortex tubes originates from solid bodies at the blade tip of an helicopter rotor and evolves in the domain until viscosity dissipates it;
• Reynolds' transport theorem for circulation, providing some conditions on the transport/stretching/tilting of the vorticity through the domain. A full understanding of these two theorems leads to the physical meaning and an easy interpretation of the evolution of vorticity and vortical regions.

Models

If these assumptions hold, vorticity is usually lumped to a infinitely thin regions of the domain, usually either vortex sheets or vortex tubes.

Definitions

A vortex line is a field line of the vorticity field $\omega = \nabla \times \mathbf{u}$.

A vortex tube is ...

(Some visualizations

• dolphins playing with vortex rings https://youtu.be/ks3aQhEohTE
• other air vortex rings in water https://youtu.be/lJfqgmbiqnQ
• vortex tube at the tip of helicopter blades, that can be visualized through condensation of water vapour https://youtu.be/lslarZiRJhg

)

A vortex sheet is ...

(Some visualizations

)

A vortex particle can be interpreted as a small amount of vorticity that moves in the flow, and it's an element often used in mesh-free solvers. Try to have a look at the Vortex Particle Method (VPM)