Are closed streamlines necessary to have vortices?

Question

I am studying fluid dynamics, and all the examples I have found so far about vortices show closed streamlines rotating around an axis.

All these examples suggest that having closed streamlines is a necessary condition to have vortices. Is there a rigorous way to show this formally?

EDIT: The answers provided below show that having closed streamlines is a sufficient but not necessary condition to have vortices. I was looking for an intuitive way to distinguish vortical flow from irrotational flow. Is there an intuitive way to visualise zero vorticity on streamlines?

Answer 1

You can have laminar flow with non zero vorticity, in which case streamlines are not closed, but are rather lines. Take for example any: $$ \vec u = u_y(x)\vec e_y $$ with vorticity: $$ \vec \omega = u_y'\vec e_z $$ so most $u_y$ will do the job, like $u_y = x/\tau$ for example.

Hope this helps.

Answer 2

Take a closed stream-line, denote it as $\gamma$. By default the normalized tangent to $\gamma$, $\vec{e}_{\gamma}$, is proportional to the vector field $\vec{V}$ in question, $\vec{e}_\gamma \propto \vec{V}$. Now we can choose the orientation of the curve so that $\vec{e}_{\gamma}\cdot \vec{V} >0$. We realize that this implies that the circulation around the loop is necessarily positive: $$\oint_\gamma \vec{V} \cdot \mathrm{d}\vec{\gamma} \equiv \oint_\gamma \vec{V} \cdot \vec{e}_{\gamma} \mathrm{d}s >0$$ We now apply Stokes' theorem to see that this implies that the curl of $\vec{V}$ must necessarily be non-zero in the region bound by the curve.

As for "rigor", there are a lot of unspoken assumptions here. For example, the vector field $\vec{V}$ should be smooth etc. Also, the curl of $\vec{V}$ can be non-zero in the sense of distributions for isolated vortices. However, once such "usual" assumptions are dealt with, the statement holds rather generally.

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Indeed, the derivation above only shows that non-zero circulation is sufficient for non-zero curl. However, circulation is not necessary in the case of a non-zero curl, as the example of user lpz shows.

In some special cases, on the other hand, it is possible to show that non-zero curl implies circulation. For example, if your field is also divergence-free (such as the magnetic field $\vec{B}$ or the velocity $\vec{v}$ of incompressible fluids), and vanishing smoothly at the boundaries/infinity, there has to be at least a countably-infinite number of closed field lines. This is because in that case we can choose a potential so that the equations of the streamline can be cast into the form of a Hamiltonian dynamical system with bound trajectories, which is guaranteed to have a phase-space densely knitted with periodic orbits ($\sim$ closed field-line loops). More details can be found e.g. in Section 2.4 of Zaslavsky (2005) - Hamiltonian Chaos and Fractional Dynamics. So really the closed loops are eliminated mainly by boundary conditions and field divergences.

Answer 3

Closed streamlines seem necessary for a vortex, but there is an issue with that. Streamlines depend on the frame of reference being used. The streamlines may only be closed in a frame of reference centered on the vortex itself. As an example, consider a translating vortex. In a frame of reference moving with the vortex itself, the streamlines are closed, but in a "laboratory" frame of reference for someone sitting and watching the vortex move along, the streamlines are not closed.

An Album of Fluid Motion depicts this for a moving sphere in plates 8 and 9. This does not depict this for a vortex, but it does demonstrate the frame dependence of the streamlines. Plate 8 is for a frame of reference centered on the sphere and plate 9 depicts an "absolute" frame of reference. Plate 8 is likely what people are used to seeing --- the flow goes around the sphere --- but plate 9 will look weird until you realize that the sphere is not held stationary here.

In any case, even if closed streamlines are necessary for defining a vortex, the streamlines depend on the frame of reference, so they may not in fact be closed in all frames of reference.

Answer 4

For an intuitive picture of vorticity, understand that the vorticity of a flow field $\mathbf{v}$ is its curl, the $\nabla \times \mathbf{v}$ operation from vector calculus. There is a longer description and a set of animated visualizations on this page. Mathematically, the curl is computed by drawing a small loop in the flow field and checking for net flow around the loop rather than merely past it, or between the inside & outside. Imagine immersing a cube in the flow and fixing its center of mass but allowing it to spin. Will it spin or will it pick one orientation and stay with that? If it spins, you've got vorticity.

From Helmholtz's theorem, every flow field can be decomposed into a divergence-free and an irrotational part. The "divergence-free" portion is the vorticity or curl. The "irrotational" part has no vorticity / curl. So, if you're trying to build up intuition about the sign and magnitude of the vorticity, you can try mentally subtracting the irrotational, divergence-only part. That portion of the flow will look like all the fluid's moving together from one place to another. Simple expansion/compression and flow along a straight pipe without interactions between the wall and fluid are both irrotational flows that lack vorticity.

For a third option, you can ask whether the flow exhibits shear. Shear occurs when two nearby parcels of fluid move with different velocity components perpendicular to their separation. It is thus also equivalent to the curl and vorticity. Sheared flow may occur when drag is generated by a fluid moving past a fixed obstruction (like an airplane wing or a ship hull), due to friction / surface interactions between the fluid and the obstruction / wing / vessel. Shear (and vorticity) may also occur when different portions of a fluid experience different forces and thus move past each other.

All three views are equivalent to each other and to vorticity: Shear will cause a suspended object to rotate "downstream" on the faster-moving side and "upstream" on the slower-moving side. This reflects a rotation of that object and the fluid parcels nearest it. Equivalently, it can be seen as a bulk flow at the mean velocity plus a rotation to compensate for the different sides of the sheared flow. Depending on your frame of reference, any of the options may be helpful in building up intuition.

Answer 5

No, streamlines don't need to be closed.

A classic example is the one of an airfoil. One of the simplest ways to model the flow field around an airfoil is in potential theory via lifting line, so by superposition of a vortex at the t/4-line with the far-field (flight) velocity. That way the flow speed differences over the airfoil is described. The streamlines still enter from the inflow boundary on one side of the control volume and leave on the other (outflow) side.

Somewhat closely related to Helmholtz's theorem #2, https://en.wikipedia.org/wiki/Helmholtz%27s_theorems,

"A vortex line cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path."

So the other side of the closed vortex line will also affect the stream lines, but they still leave the control volume through the outflow boundary.

Note, however, that Helmholtz is valid only on inviscid flow. Depending on viscosity & the problem at hand, that can be a good approximation. Also note, that streamlines are only clearly defined in potential theory (that means inviscid & incompressible). As soon as you add viscosity, the flow field will exhibit non-zero curl, hence general integrability is lost, and in a strict definition stream lines cease to exist.

Answer 6

1. The terminology 'vortex-flows' is a bit misleading. The term means flow in circular paths. Obviously, a circular path is a closed path.
2. Vorticity is the curl of the velocity vector field. Physically vorticity represents rotation of a fluid element about its own axis, independent of whether the fluid particle is moving on circular paths. If vorticity is identically zero everywhere in the domain, then its an irrotational flow; else its a rotational flow.
3. A 'vortex-flow' may or may not have vorticity. We can have both rotational and irrotational vortex flows. Both categories are flows in circular streamlines. When you compute the curl, you may end up with zero curl in which it is an irrotational vortex-flow; or you may end up with non-zero curl in which it is a rotational vortex-flow.
4. The term 'vortices' used in you EDIT may be interpreted as sub-domains in the flow field where there is non-zero vorticity (i.e. rotationality). The flow itself may or may not be in circular paths.

I hope this post clears the difference between vortex-flows, vorticity and vortices.

(Reference: Fluid Mechanics Kundu and Cohen, 4th Edition)