#navier-stokes

[amimmy]

look at my finite elements, boy

[fuckyeahfluiddynamics]

Although turbulent flow is chaotic, it’s not completely disordered. In fact, order can emerge from turbulence, though exactly how this happens has been a long-enduring mystery. Take the animations above. They show the flow that develops between two plates moving in opposite direction that are separated by a small gap. (The formal name for this is planar Couette flow.) The visualization is taken in a plane at a fixed height between the plates.

Initially (top), the flow shows narrow bands of turbulence, shown in green, separated by calmer, laminar zones in black. As time passes, these areas of laminar and turbulent flow self-organize, eventually forming diagonal stripes that are much longer than the gap between plates (bottom), the natural length-scale we would expect to see in the flow. Researchers have wondered for years why these distinctive stripes form. What sets their spacing, and why are they along diagonals?

To answer those questions, researchers explored the full Navier-Stokes equations, searching for equilibrium solutions that resemble the striped patterns seen in experiments and simulation. And for the first time, they’ve found a mathematical solution that matches. What the work shows is that the pattern emerges naturally from the equations; in fact, given the characteristics of the solution, the researchers found that many disturbances should lead to this result, which explains why the pattern appears so frequently. (Image and research credit: F. Reetz et al., source; via phys.org; submitted by Kam-Yung Soh)

[deschosesdanslespace]

Stöwerplätzel

Il clamait au-delà des rues parfois les hasards sont beaux, on est perdu, dans un personnage, on sent la fraicheur du soir, l'eau qui coule doucement dans la fontaine, les ivresses qui déambulent et notre corps, qui n'est plus, s'emballe mécaniquement, les pas arrivent, s'enchainent, serpentent, déposent nos âmes devant le restaurant, la grande silhouette d'un autre, un autre personnage, on discute près de la fontaine, ça parle de marche, de feuilles de papier, du bruit des vagues. Et là sur le fond se pose doucement l'équation -Navier-Stokes - , la turbulence de l'eau, les tourbillons au loin du barrage, les bouchons de pêches qui cahotent et se noient.

[fuckyeahfluiddynamics]

With some help from Physics Girl and her friends, Grant Sanderson at 3Blue1Brown has a nice video introduction to turbulence, complete with neat homemade laser-sheet illuminations of turbulent flows. Grant explains some of the basics of what turbulence is (and isn’t) and gives viewers a look at the equations that govern flow -- as befits a mathematics channel! 

There’s also an introduction to Kolmogorov’s theorem, which, to date, has been one of the most successful theoretical approaches to understanding turbulence. It describes how energy is passed from large eddies in the flow to smaller ones, and it’s been tested extensively in the nearly 80 years since its first appearance. Just how well the theory holds, and what situations it breaks down in, are still topics of active research and debate. (Video and image credit: G. Sanderson/3Blue1Brown; submitted by Maria-Isabel C.)

[fuckyeahfluiddynamics]

Turbulence -- that pestersome, unpredictable, and chaotic state of flow -- has been a thorn in the sides of mathematicians, physicists, and engineers for centuries. It is certainly one of -- if not the -- oldest unsolved problem in physics. Over at Ars Technica, Lee Phillips has a nice overview of the situation, including what makes the problem so difficult:

The Navier-Stokes equation is difficult to solve because it is nonlinear. This word is thrown around quite a bit, but here it means something specific. You can build up a complicated solution to a linear equation by adding up many simple solutions. An example you may be aware of is sound: the equation for sound waves is linear, so you can build up a complex sound by adding together many simple sounds of different frequencies (“harmonics”). Elementary quantum mechanics is also linear; the Schrödinger equation allows you to add together solutions to find a new solution.

But fluid dynamics doesn’t work this way: the nonlinearity of the Navier-Stokes equation means that you can’t build solutions by adding together simpler solutions. This is part of the reason that Heisenberg’s mathematical genius, which served him so well in helping to invent quantum mechanics, was put to such a severe test when it came to turbulence. 

Phillips goes on to describe some of the many methods researchers use to unravel the mysteries of turbulence computationally, experimentally, and theoretically. This is a great introduction for those curious to get a sense of how turbulence, stability theory, and computational fluid dynamics all fit together. (Image credits: L. Da Vinci; NASA; see also: Ars Technica; submitted by Kam Yung-Soh)

[stvff-talks]

Fluid Sim Failing but In Progress

[mathgobbobarb]

Lukaszewicz & Kalita Navier-Stokes C2S1

Let's not give time a special status and instead define (x1,x2,x3,x4) and the transformations x(X). Then we can consider a set of paths pi for allowing any individual component of x to vary while the others remain fixed.

Just because our perception of time seems superficially special doesn't mean I gotta treat it so.

This is the source of my derangement.

Here is my first headache. I wanna just go X = X(x). Does that work?

for i=/= 4 it is the same. So I'm basically just requiring that I can do for time what I do for the others. Which is EXACTLY what I'm wanting to assume from the start so DUH BARBIE.

The determinant form remains unchanged too we just have an extra layer to the matrix

the determinant considered in the text is the determinant of space, the coefficient of the dt/dT portion. the mixed determinants have the dx/dT, dy/dT, etc. if Those are 0 then it collapses to the same that the authors consider.

here's the first annoying deviation. By giving special consideration to time, we have F(X_space, t) instead of F(X_space, T) = F(X).

need to be considerate of the material Time as well as the material Space.

I need a little bit extra to account for t vs T. I generally suspect that in most cases L&K cosider here dT/dt = dt/dT = 1, but in the case of generality I want to keep it in my head that they're using this.

It's probably not *super* important until we consider the material derivative

because we're not giving special consideration to time, pulling df/dt out is meaningless. especially since we have a u_t = 1 sitting in front. so

our material derivative becomes a scaled gradient. Ain't that nice and comfy?

This is probably something else to hold on to

so we can bring in our Material Time derivative of f by scaling our scaling factor and our spacial coordinates.

This probably needs derivation, but it seems like a good idea to make an equivalent material derivative for material space and time. I suspect V and v are the same, but I don't wanna assume that without doing a derivation that I don't have time for right now since I wanna get this chapter re-read done before work.

I'm gonna go out on a limb and say that our revision is more elegant.

Now there is a bit of ??? about if our new divergence works, but if u_t = 1 then of course it does so it's fine.

There are Exercises to do after work. In addition I want to poke at my new divergence and at the v vs V question.

[fatfreddyscat-blog]

Calculus

This is a public confession.  I am afraid of calculus.  The Navier-Stokes equations “for instance, the Navier-Stokes equations” says the man on the video giving a brief overview of how climate change models are made.  “The Navier-Stokes Equations?”.  Used to calculate the motion of viscous fluids.  Sea water and air aren’t viscous I ponder, but the delta signs have already terrified me.  Probably why I dropped out of my chemical engineering course decades ago.  You don’t translate an ease with Mathematics at O level to understanding Pure Maths at A level.  Felicity wasn’t having a lie in on Saturday mornings, her private tutor - no disclosure needed for the Sutton Trust - was cramming her for Physics and Maths A levels: justification? Mummy’s degree was in Film Studies and Daddy has dyslexia. You don’t need to go to a state school to afford maths tutors but you have to be twice as quiet if you’re at a private school.  I’ve digressed.  My state school had very few children studying maths and I could have asked for help.  I want to get over this fear.