What makes the Navier Stokes equation so difficult?

[jedishrfu]

An article in Quanta Magazine discusses the math behind the Navier Stokes equations, why they are so difficult to solve and whether they truly represent fluid flow:

https://www.quantamagazine.org/what-makes-the-hardest-equations-in-physics-so-difficult-20180116/

 

[Charles Link]

An article in Quanta Magazine discusses the math behind the Navier Stokes equations, why they are so difficult to solve and whether they truly represent fluid flow:

https://www.quantamagazine.org/what-makes-the-hardest-equations-in-physics-so-difficult-20180116/

In the link, I question whether there is a typo, and it should read $-\nabla P$ with a minus sign for the force per unit volume from a pressure gradient. In any case, a good article. $\\$ The Navier-Stokes equation and its solutions are rather complicated. As a graduate student, I saw one professor in a graduate course get stuck on a calculation he was attempting at the chalkboard with the Navier-Stokes equation. Technically, this professor was very astute, but on this calculation he was attempting, he was simply stuck. After 1/2 hour of struggling with it, he was totally exasperated with it. He actually told the class, "If anyone is disgusted they can leave". Everyone stayed, but the fact is, the Navier-Stokes equation is simply not a simple equation to try and solve.

 

[Chestermiller]

The Navier Stokes equation is so hard to solve because it is non-linear. If the inertial terms were not present (either because of the geometry or because the inertial terms are negligible0, it would (and can) be much easier to solve. In turbulent flow, the inertial terms become even more problematic because the flow becomes non-time-steady, with many coupled spatial and time scales of variation. Turbulent flow continues to be a challenging issue (involving simplifying approximations) even today.

 

[Andy Resnick]

An article in Quanta Magazine discusses the math behind the Navier Stokes equations, why they are so difficult to solve and whether they truly represent fluid flow:

https://www.quantamagazine.org/what-makes-the-hardest-equations-in-physics-so-difficult-20180116/

A very readable article about a complicated problem.

The equation that's shown in the article is a very simplified form of the NS momentum equation- the left side has been simplified for the case of incompressible fluids (as stated), and the right side is only for Newtonian fluids in the presence of gravity. There are also no boundary conditions discussed- for example, multiphase fluid systems (surfaces across which material properties may be discontinuous) or the presence of solid boundary surfaces (the no-slip boundary condition is surprisingly problematic). Finally, there are 2 other equations that simultaneously hold- continuity equations for mass and energy. All 3 sets of equations must be satisfied.

Even for the 'simple' case of homogeneous, isothermal, incompressible fluids, the existence of weak solutions to the momentum equation has not yet been proven for ≥ 3 dimensions (AFAIK). Pierre-Louis Lions has done a lot of work in this area, most of it well outside of my expertise.

 

[Lord Crc]

In real systems I assume the turbulence can't be significantly smaller than the interaction radius of the particles, nor can the particles travel faster than the speed of light due to relativity.

Does modifications that address the above make the problem more tractable?

 

[Chestermiller]

In real systems I assume the turbulence can't be significantly smaller than the interaction radius of the particles, nor can the particles travel faster than the speed of light due to relativity.

Does modifications that address the above make the problem more tractable?

Huh? Wr’re Talking about non-relativistic fluid mechanics, right?

 

[Lord Crc]

Huh? Wr’re Talking about non-relativistic fluid mechanics, right?

I mentioned one non-relativistic and one relativistic effect which, from what I can gather, should prevent particles from actually reaching infinite speeds in real systems. The Quanta article highlighted those infinities as one of the core challenges in obtaining solutions to the NS equations, from what I understood. Maybe I read that wrong?

Anyway, I was just curious if either of these effects have been studied in relation to NS and if so what their impact was on the ability to solve the (presumably modified) equations.

Obviously both effects are at play in real fluids, though I would assume the relativistic corrections not to be relevant to most earth-bound fluid problems.

 

[boneh3ad]

I mentioned one non-relativistic and one relativistic effect which, from what I can gather, should prevent particles from actually reaching infinite speeds in real systems. The Quanta article highlighted those infinities as one of the core challenges in obtaining solutions to the NS equations, from what I understood. Maybe I read that wrong?

Anyway, I was just curious if either of these effects have been studied in relation to NS and if so what their impact was on the ability to solve the (presumably modified) equations.

Obviously both effects are at play in real fluids, though I would assume the relativistic corrections not to be relevant to most earth-bound fluid problems.

The Navier-Stokes equations are derived using relatively few assumptions, but one of them is that the fluid behaves as a continuum. In this case, that means that a dimensionless number called the Knudsen number is small. This number is the ratio of mean free path to the relevant scales of motion. In other words, the equations would cease to be valid long before you reach a length scale on the order of a single particle.

In practice, the smallest meaningful length scale in a given flow is what is called the Kolmogorov scale, which usually ranges anywhere from 10^-7 to 10^-3 meters for most Earth-bound problems depending on the situation. Basically, these are the smallest scales at which flow energy is dissipated, and anything much smaller than that doesn't really need to be resolved to capture the relevant physics.

The two major examples where the continuum assumption breaks down is in a rarefied flow (edge of space, for example) or micro-/nanofluids (where the length scales get so small that they approach the mean free path in magnitude).

As for relativistic effects, I believe relativistic fluid mechanics can still use some form of the Navier-Stokes equations provided the flow can be adequately modeled as a continuum and also that some allowance is made for the curvature of spacetime. I don't have any real experience there, though, so I will let someone else comment in depth if they know more.

 

[Lord Crc]

The Navier-Stokes equations are derived using relatively few assumptions, but one of them is that the fluid behaves as a continuum. In this case, that means that a dimensionless number called the Knudsen number is small. This number is the ratio of mean free path to the relevant scales of motion. In other words, the equations would cease to be valid long before you reach a length scale on the order of a single particle.

In practice, the smallest meaningful length scale in a given flow is what is called the Kolmogorov scale, which usually ranges anywhere from 10^-7 to 10^-3 meters for most Earth-bound problems depending on the situation. Basically, these are the smallest scales at which flow energy is dissipated, and anything much smaller than that doesn't really need to be resolved to capture the relevant physics.

So what I'm struggling to see is how these infinities show up, especially if the minimum length scales are so large. Or is it an indication that you're breaking the continuum condition?

In any case, thank you for the informative reply.

 

[Andy Resnick]

So what I'm struggling to see is how these infinities show up, especially if the minimum length scales are so large. Or is it an indication that you're breaking the continuum condition?

In any case, thank you for the informative reply.

Turbulence is definitionally a classical phenomenon with all of the attendant 'simplifications' (continuum, etc). The infinities occur when the flow is perturbed- in laminar flow, any perturbation is spatially and temporally confined and the flow reaches some equilibrium configuration. In turbulent flow, any perturbation 'blows up' in space and time- the flow is always unsteady, there is no equilibrium, and while there may be places where, for example, the velocity is formally infinite (non-zero vorticity), the 'infinities' refer to the lack of an steady-state solution.

The field is wide-open: there's little in the way of turbulent heat transport and I know of no investigations into turbulent plasmas or magnetic fluids (other than Chandrasekhar's book on stability).

Depending on your background, you may find these references interesting:

https://www.amazon.com/dp/0511622708/?tag=pfamazon01-20
https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.56.223
http://rspa.royalsocietypublishing.org/content/164/917/192

 

[boneh3ad]

The field is wide-open: there's little in the way of turbulent heat transport and I know of no investigations into turbulent plasmas or magnetic fluids (other than Chandrasekhar's book on stability).

These are all very active areas of research. For example, Los Alamos and Lawrence Livermore National Laboratories both have very active research communities dedicated to turbulent plasmas, which necessarily also includes the influence of electromagnetic effects. In essence, this is what the National Ignition Facility is studying on a very massive scale.

Turbulent heat transport is a very important and active field within high-speed aeronautics due to its importance in the design of thermal protection systems on spacecraft , missiles, etc. It's also quite important in mechanical engineering where efficient design of heat exchangers or efficient combustion is important.

Chandrasekhar's book is about the stability of fluid flows, i.e. how instabilities grow to eventually break down to turbulence. It says little about the actual behavior of turbulence. Most texts on that subject treat the problem (largely) statistically (e.g. Statistical Fluid Mechanics Vol I&II by Monin and Yaglom, Turbulent Flows by Pope, Turbulence by Davidson, or A First Course in Turbulence by Lumley).

 

[jedishrfu]

In a related article on Navier Stokes, mathematicians find some extreme conditions where the equations appear to fail:

https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/

The Navier-Stokes equations capture in a few succinct terms one of the most ubiquitous features of the physical world: the flow of fluids. The equations, which date to the 1820s, are today used to model everything from ocean currents to turbulence in the wake of an airplane to the flow of blood in the heart.

While physicists consider the equations to be as reliable as a hammer, mathematicians eye them warily. To a mathematician, it means little that the equations appear to work. They want proof that the equations are unfailing: that no matter the fluid, and no matter how far into the future you forecast its flow, the mathematics of the equations will still hold. Such a guarantee has proved elusive. The first person (or team) to prove that the Navier-Stokes equations will always work — or to provide an example where they don’t — stands to win one of seven Millennium Prize Problems endowed by the Clay Mathematics Institute, along with the associated $1 million reward.

 

[Auto-Didact]

In a related article on Navier Stokes, mathematicians find some extreme conditions where the equations appear to fail:

https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/

Saw this article a few days ago, but have not found the time to read the paper yet. Any experts willing to comment on how significant these results exactly are, with regard to existence and smoothness, without giving away too much spoilers?

 

[boneh3ad]

Saw this article a few days ago, but have not found the time to read the paper yet. Any experts willing to comment on how significant these results exactly are, with regard to existence and smoothness, without giving away too much spoilers?

I am not a mathematician, more of a mechanician, but to me, the article sounded like they basically just proved that, if you start out with too coarse of a grid for your solution, then the solution when you refine that grid might be non-unique. That seems almost obvious to me, so perhaps I am missing something.

 

[Somali_Physicist]

Quick question, is this turbulence found In the electromagnetic world?

 

[WWGD]

Sorry if this is OT. Does anyone know if Amador Muriel
https://duckduckgo.com/?q=amador+muriel+navier+stokes&t=hg&atb=v88-7&ia=web
, who claims to have found an exact solution, is taken seriously? I remember he showed
up at my school a few years back and was trying to recruit anyone willing.

 

[boneh3ad]

Now, again, I am not a mathematician working on fluid mechanics but a mechanician/engineer, but to the best of my knowledge his work isn't really taken seriously. He is an electrical engineer who basically had a symbolic math programming language (probably Mathematica or something similar) spit out an answer, it didn't really adhere to the requirements of the Clay problem, and it was published in the first ever volume of an obscure journal and also on arXiv.

 

[Zel'dovich]

... The equation that's shown in the article is a very simplified form of the NS momentum equation- the left side has been simplified for the case of incompressible fluids (as stated)...

Indeed, due to the mass conservation, the left side of the equation is also valid for compressible fluids. It is in the right side where you see the incompressibility, since the stress tensor would have additional terms for compressible fluids .

 

[Tom Kunich]

Indeed, due to the mass conservation, the left side of the equation is also valid for compressible fluids. It is in the right side where you see the incompressibility, since the stress tensor would have additional terms for compressible fluids .

I have a problem with the idea that there is always a solution. I cannot see that turbulent flow can ever be predicted in small scale.

 

[Andy Resnick]

Indeed, due to the mass conservation, the left side of the equation is also valid for compressible fluids. It is in the right side where you see the incompressibility, since the stress tensor would have additional terms for compressible fluids .

I don't think so- density was factored out of the material derivative: D/Dt (ρv) →ρ Dv/Dt. If the density is a constant, you have an incompressible fluid.

 

[Zel'dovich]

Sorry for the constant editing. I am just learning how to use this.

You can write the full L.H.S. as:

\begin{equation}
\frac{\partial (\rho u_j) }{\partial t} + \frac{\partial (\rho u_i u_j)}{\partial x_i} = \rho \frac{\partial u_j}{\partial t} + \rho u_i \frac{\partial u_j}{\partial u_j} + u_j \underbrace{\left(\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u_i)}{\partial x_i}\right)}_{=0}
\end{equation}

But it is not really important for the conversation

 

[Tom Kunich]

Because on the smallest scale velocity is indeterminate within bounds and the transfer of this energy is also indeterminate why would anyone think that you could always have a solution save on a grand and generalized scale? Moreover, most fluids are compressible to some extent making the Navier-Stokes equation only a general solution under controlled conditions.

I suppose we should remind people that the linear flow of a river starts as the turbulent flow of the individual rivulets.

 

[boneh3ad]

Why do you think the smallest velocity scale is indeterminate? Those scales are actually fairly well-defined, and are called the Kolmogorov microscales.

Second, yes, most fluids are compressible. It sure is a good thing that the Navier-Stokes equations make no assumptions about compressibility, then.

 

[Tom Kunich]

On the smallest scale we have the molecules in any liquid often showing nearly as much energy from heat energy as from flow because of turbulence. Doesn't this mean that they are just as likely to transfer more or less energy to another molecule on an indeterminate scale?

For instance: let us assume that a gas is passing a flat plane surface perpendicular to the flow. Given a wide enough surface the flow is only going to go around the sides while the innermost area of the gas is nearly static obtaining very little energy from the passing flow. The turbulent flow would be in the area between the uninterrupted flow and the edges of that flow around the plane. The turbulence at the interface would be strongest and would largely taper off towards the inner surface where the viscosity of the plane interrupts both the flow and the turbulence. I assume that there would be a mix of molecules but almost entirely from heat motion and not the passing energy of flow. However, since turbulence is unpredictable you could have an energy wave travel entirely from one side of the flow to the other completely washing the nearly static particles out. And this could be entirely due to unpredictable molecular motions due to heat energy.

So my question would be - are we trying too hard to reduce everything to a mathematical formula? The world about us isn't an equation but requires more statistical analysis than formulaic reductions to individual atoms. Or perhaps I'm missing the point coming from an engineering point of view.

 

[BWV]

Question - have read that the numerical solutions are computationally intensive and of course exact solutions, if they exist, would eliminate this problem. My question is how computationally expensive are they - are there things we would like to do with NS that are simply impossible, analgous to solving chess by brute force calculation, or just expensive and inconvenient and with another 10 or 20 years of Moore’s law would be feasible with current algorithms?

 

[boneh3ad]

Well, technically nothing is impossible. There is a field in computational fluid dynamics called direct numerical simulation (DNS). In a DND, the NS equations are solved directly on a grid fine enough and large enough to resolve all relevant scales. To do this over, say, an entire car, it would take a modern supercomputer years and years to finish the problem. The computation time scales with the cube of Reynolds number.

Sure, maybe Moore's law will someday catch up, but we aren't there yet.

 

[Tom Kunich]

Well, technically nothing is impossible. There is a field in computational fluid dynamics called direct numerical simulation (DNS). In a DND, the NS equations are solved directly on a grid fine enough and large enough to resolve all relevant scales. To do this over, say, an entire car, it would take a modern supercomputer years and years to finish the problem. The computation time scales with the cube of Reynolds number.

Sure, maybe Moore's law will someday catch up, but we aren't there yet.

Well, that answered my question though I still don't believe that you can actually solve for a grid small enough. But you could probably solve for a grid fine enough as makes no difference.

 

[boneh3ad]

Well, that answered my question though I still don't believe that you can actually solve for a grid small enough. But you could probably solve for a grid fine enough as makes no difference.

In my book, "fine enough as makes no difference" equivalent to "small enough." If it's fine enough to make no difference whether you go finer, why do you think it should go finer?

 

[Tom Kunich]

In my book, "fine enough as makes no difference" equivalent to "small enough." If it's fine enough to make no difference whether you go finer, why do you think it should go finer?

If you are solving a problem you solve it. That means going down to the atomic level if necessary. But in reality you only need to know enough about the flow to reduce drag to an acceptable level.

 

[boneh3ad]

If you are solving a problem you solve it. That means going down to the atomic level if necessary. But in reality you only need to know enough about the flow to reduce drag to an acceptable level.

No, you do not need to go down to the atomic level. Provided there are enough molecules in a given volume, then the atoms behave predictable via statistical averaging and the entire fluid flow can be treated as a continuum. On fact, treating the problem on the atomic level would cause the Navier-Stokes equations to lose validity. Fluid dynamics is largely a classical phenomenon.

I don't follow your comment about drag. What does that even mean?