[nuclearnice1]

What a wonderfully informative and educational comment. Thank you.

Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity.

The article dances around the term:

> At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.”

> Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.

[prolyxis]

A simple example of a function with a singularity is f(t)=1/t. Note that at t=0, f(t) is undefined due to division by zero. On either side of zero, the absolute value of f(t) approaches infinity.

In this case, we are tracking the flow of an incompressible fluid over time. This flow is represented by a velocity field evolving over time, under the constraint of no net inflow/outflow of material into any region of space. Thus, the singularity corresponds to a portion of fluid speeding up and approaching an infinite speed as you approach some finite time.

Because the fluid cannot be compressed, the only way the singularity can be produced is for a portion of the liquid to swirl, increasingly rapidly, about some point: hence the discussion in the article about vorticity.

As isoprophlex pointed out, this undefined value of the velocity field prevents you from (or at least complicates) computing the further evolution of the fluid.

[nico]

Thank you for the great explanation.

Do these swirls shed energy? Is it considered in these equations that for example friction within the swirls would slow them down (and hence not reach a singularity)?

[pas]

In real fluids yes, absolutely, they basically transform/branch/divide/split into smaller and smaller scale vortices and then those dissipate the energy into the fluid (heating it up a bit).

[wwalker3]

The incompressible Euler equations model a fluid as a two-valued field. This means that at every point in space, the field has two values, density and velocity (1).

To me (2), a singularity in a field like this means that one or more of the field values "blows up", i.e. goes to infinity as you run the time variable forward.

But how could this ever happen? The Euler equations model the "conservation" (i.e. constant-ness) of three real physical quantities: mass, momentum, and energy. If these three quantities are finite and constant when you add them up over the whole field, how can any part of it "blow up" into an infinite value?

The answer is that the blow-up must occupy a volume that shrinks as the blow-up grows, so the conserved quantities are still constant. The singularity would be infinitely small in space, and have an infinite value of density or velocity (or both).

The hard question is, are these blow-ups merely artifacts of a particular numerical simulation technique, or are they essential somehow to the incompressible Euler equations themselves? That's what these papers are trying to figure out.

To me, an "essential" (i.e. inherent-in-the-equations) blow-up seems intuitively reasonable because of the acausal nature of the field. When you simulate the incompressible Euler equations, it superficially looks like it's a physical fluid doing physical-fluid things, swirling and flowing around. But in a real fluid, a change in one part of the fluid propagates to the other parts at finite velocity, creating real cause and effect.

An Euler fluid's time evolution is not a phenomenon that ripples forward through time in a normal way. Instead, every point in the fluid responds to every other point simultaneously. If you poke a cube of incompressible Euler fluid with your finger, there is no pressure wave that ripples through it, where the fluid parcels push each other along and get out of each other's way. Instead, the whole cube of fluid somehow instantly adopts a new flow pattern that conserves mass/momentum/energy in response to that finger-poke.

1) Note that velocity is a vector, since it has a direction. This means that in 2D the velocity is two numbers, and in 3D it's three numbers. So technically the 3D incompressible Euler equations have four values at every point: one density, and three velocity components, one each in the x, y, and z directions.

2) I'm a numerical simulation guy, not a mathematician. Real math experts have rigorous definitions of a singularity, e.g. in https://arxiv.org/pdf/2203.17221.pdf "Singularity formation in the incompressible Euler equation in finite and infinite time," Theodore D. Drivas and Tarek M. Elgindi.

[vba616]

>The incompressible Euler equations model a fluid as a two-valued field. This means that at every point in space, the field has two values, density and velocity

I don't get it. If the fluid is incompressible, how can density have a value at every point in space? Isn't it just a constant?

[wwalker3]

The density can be constant, but it doesn't have to be. If the density field starts out with some variation in it, then those variations move around as the fluid flows. Incompressibility just means that those density variations can't get bigger or smaller, they can only move, shear, and rotate.

[fuzzfactor]

When you work with near-supercritical and supercritical fluids under laboratory conditions, you can turn the pump by hand and you feel when the density hits the ceiling.

So you know something is up.

Systems would be modeled mathematically using a fluid's individual component values, but we were paid for the real-world laboratory data.

[isoprophlex]

I think (not a physicist), simply put, an infinity or NaN value. As these are step-wise methods, having such values show up anywhere will seriously mess up subsequent calculation steps.

[NotYourLawyer]

It’s when some physical quantity of the simulation becomes infinite. Pressure, particle velocity, etc.

[TEP_Kim_Il_Sung]

That usually indicates a phase change, with a separate set of applicable equations.