About Turbulence and Navier-Stokes

[CosmicKitten]

I was discussing this with my friend the other night. I mentioned that proving existence of smooth solutions for the Navier-Stokes equations would win a $1,000,000 prize from the Clay Mathematics Institute, and also that turbulence is considered one of the unsolved problems of physics; as a matter of fact, more than one physicist has been quoted as stating something along the lines of expecting turbulence to be a trickier problem to solve than the mysteries of QED or relativity.

My friend believes this is BS, stating that he can visualize the solution physically but is not mathematically skilled enough to state it in mathematical terms. For the record he claims that he was a child prodigy who had designed state-of-the-art avionics system at a very young age.

I want to know what exactly about turbulence is considered so elusive? I have self-studied ODEs (and to maybe a lesser extent, PDEs) to a point of great proficiency (and I invite readers to test it), and though lately I have mostly been self-studying real analysis, abstract algebra, and quantum mechanics (particularly as relates to chemistry, I've studied the elements a lot as well for the purpose of an ambitious art project of mine), basically just whatever captures my fancy and obsessive drive during that day or week or so (though I have been trying to stick with certain topics at least long enough to gain some knowledge before jumping to another one so I won't be stuck at the start when I get back to it later) and I am about ready to dive into the study of... I believe this is what you call continuum mechanics? I was very much displeased by the introductory classical mechanics course I aced at university and so I am very much wanting for some classical mechanics problems to tackle that are actually at my mathematical level.

From what I matter, fluids in such problems are modeled as a continuum, as though there are no individual particles bouncing around within. But isn't that model flawed in the case of turbulence? I mean, given initial conditions in a continuum sense, one cannot guess, for example, the exact shape that smoke will spread and curl into after a given time such that the flow ceases to be laminar? Would one not have to know the state of the individual particles to be able to determine that?

The motion of the particles, given a body of fluid at a given velocity and pressure and temperature and other relevant conditions, presented with a vacuum or another fluid at different conditions, well the rate at which it diffuses for instance can be modeled stochastically based on the average velocity of the particles, although, if you view the particles closely enough, you will find that there will randomly be great holes in the distribution on the boundary between the two fluids (as an analogy from an unrelated field think van der Waals forces which happen because of moments when the electron distribution is dense enough to attract nonpolar molecules together) and at such a moment a burst of Fluid A will not diffuse into Fluid B but pass as a wisp of Fluid A conditions surrounded by fluid with Fluid B conditions (although diffusion continues to happen through this wisp, and more wisps may trail out of it) So couldn't one model where and when such a wisp is most likely to pass through, how large it is likely to be, and when, taking into account the homogenization of the fluids due to diffusion, more wisps will trail out? The shape of the wisps would be based statistically given the motion of the particles in the wisping fluid, and well, more insight might be gained by using such a model? Please tell me the equations were originally derived taking the many particle model into account...

Pardon if this is all incomprehensible or just plain wrong; I do not know all the proper terminology so I am just describing how it happens in my head, not sure about my friend's head though. Thank you for reading, and I hope to gain some insight from others who might know more than I do.

 

[boneh3ad]

From what I matter, fluids in such problems are modeled as a continuum, as though there are no individual particles bouncing around within. But isn't that model flawed in the case of turbulence? I mean, given initial conditions in a continuum sense, one cannot guess, for example, the exact shape that smoke will spread and curl into after a given time such that the flow ceases to be laminar? Would one not have to know the state of the individual particles to be able to determine that?

It is not flawed at all, and in fact, given a set of initial conditions, the shape of a plume of smoke can be predicted. The Navier-Stokes equations, according to our current understanding, contain within them all the physics required to describe turbulence already. The problem is that they represent a highly-nonlinear, many-degree-of-freedom system whose solutions often exhibit what is often thought to be a form of spatio-temporal chaos. With that in mind, the final shape of a plume of smoke is so initial condition dependent that computers simply don't have the the precision necessary to perfectly predict the exact shape given an exact set of initial conditions, though they can get pretty darn close.

Yes, diffusion occurs in a fluid, but the Navier-Stokes equations and turbulence itself are not particle phenomena, but continuum phenomena. The diffusion of various quantities does play into the equations, however. For example, in the Navier-Stokes equations, the [itex]\mu\nabla^2 \vec{v}[/itex] term represents the diffusion of momentum in the fluid. There is a similar diffusion term in the energy equation. In flows with multiple species, additional governing equations are introduced which contain diffusion terms for each species. So yes, diffusion is important, but it is already covered by the continuum equations.

 

[CosmicKitten]

I assume the consequences of the particle model are all lumped into some constant?

That makes it like the continuum version of the N-body problem doesn't it? On the particulate scale, it IS an N-body problem but with a ridiculously large number of bodies, oh wait except that attractive forces such as gravity, electric etc. can be ignored. However their positions and trajectories are not known so is it not similar in a way to quantum mechanics? Can the fluid be modeled as a probability function of some sort, perhaps an uncertainty principle stuck in somewhere?

I thought that such fluids can be modeled as solids, in a sense; when laminar flow becomes turbulent, is it not like a fracture in an amorphous substance? The curling wisps of smoke being roughly analogous to the different fragments in a conchoidal fracture? Is there any system of equations that describes such properties of solid matter? I mean, I know it's not solid, but the fluid sort of becomes broken...

So the motion of a plume of smoke can be modeled by a computer without knowing the behavior of the particles, just the behavior of the continuum? But chaos means that, despite deterministic conditions, the motion is mathematically intensive to compute because there is no analytic solution and you have to approximate by taking the motion over the course of a small step, then resetting the conditions and taking it over another small step (this is Runge-Kutta and Euler Method right?) like with the N-body problem? Is it in any way like the error function which is trying to take the integral of e^-(x^2) which cannot be done unless it is a definite integral in which case you can only find what the area under the curve between a and b is and not the function for it?

Specify nonlinear? Nonlinear can be like a differential equation that multiplies the function you are solving for itself with its own derivatives, or like nonlinear optics in which properties of materials such as refractivity etc. change depending on conditions, which is maybe expressed by a nonlinear differential equation? So what exactly is nonlinear here? And what are these degrees of freedom? Sorry I get too sleepy at night to study all of this I hate having to sleep. I also hate having to eat, empty belly is a distraction and so is cooking to fill it.

Also when I study I have to know the mechanics of the mathematics involved before I can understand any of what they are discussing and often when I read a harder math book I get lost in thinking about the math details, kind of not seeing the forest for the trees, and so I have to study that and maybe learn how to solve it and then once it's firmly wired into my head I can read the book literately and get the big idea. But most books tend to introduce the purpose, ideas etc. first and I can't understand it at a gut level, that is so that I can visualize and like solving a puzzle fill in the gaps in my attention of what they're talking about unless I have driven the mechanics of the process through my head first.

 

[boneh3ad]

I assume the consequences of the particle model are all lumped into some constant?

You just successfully deduced the nature of viscosity and thermal diffusivity.

I suppose in a sense it is a lot like the N-body problem. Fluids can be and often are modeled stochastically. That is a rather central topic to turbulence modeling and attempts to estimate useful quantities (e.g. shear stress) within a turbulent flow. Since a direct numerical simulation (DNS) is so time consuming and and is time-prohibitive in most full-scale useful flow situations, this is currently one of the most common approaches to dealing with turbulence in a practical sense.

I thought that such fluids can be modeled as solids, in a sense; when laminar flow becomes turbulent, is it not like a fracture in an amorphous substance? The curling wisps of smoke being roughly analogous to the different fragments in a conchoidal fracture? Is there any system of equations that describes such properties of solid matter? I mean, I know it's not solid, but the fluid sort of becomes broken...

I am not nearly as familiar with the solid mechanics side of continuum mechanics, but I am sure there are equations that can model conchoidal fracture at least to some degree. That said, I do not see any reason why turbulence would follow a similar path. The transition to turbulence, in terms of modern thinking, is really more of a wave growth phenomenon. You have a base laminar flow that is perturbed in some way (e.g. tiny free-stream fluctuations) and those perturbations can become unstable waves in the fluid. The Navier-Stokes equations, in one way of looking at them, essentially form one gigantically complicated nonlinear oscillator, and as such, under some conditions they are stable and under other conditions they are unstable. In many, if not most, practical flows, there is at least some region of unstable behavior, so these waves that have become entrained in the flow undergo unstable growth. At some point they grow large enough that nonlinear effects dominate and usually shortly thereafter the transition to turbulence occurs. In terms of the current state of the art, fluid mechanicians are have the initial, linear growth stage of this process down pretty well, and in some cases the nonlinear regime is well-understood. It is really the initial entrainment of perturbations in the flow and the ultimate breakdown to turbulence that are the least well-understood at this point. The kicker is that even if these were narrowed down, if the system really is chaotic as it appears to be, it would still be essentially impossible to simulate on a computer to arbitrary accuracy, so you would only likely get accurate results until the point of transition, where solution trajectories would start to diverge rather rapidly from one another.

So the motion of a plume of smoke can be modeled by a computer without knowing the behavior of the particles, just the behavior of the continuum? But chaos means that, despite deterministic conditions, the motion is mathematically intensive to compute because there is no analytic solution and you have to approximate by taking the motion over the course of a small step, then resetting the conditions and taking it over another small step (this is Runge-Kutta and Euler Method right?) like with the N-body problem? Is it in any way like the error function which is trying to take the integral of e^-(x^2) which cannot be done unless it is a definite integral in which case you can only find what the area under the curve between a and b is and not the function for it?

That is not what chaos means. Chaos, at its most basic, means that the problem is extraordinarily sensitive to the initial conditions. If you pick two arbitrarily close initial points in a chaotic system, their trajectories will (eventually) diverge exponentially. The result of this in simulating such systems is that even if you know your initial conditions exactly, a computer cannot hold them exactly since it has finite precision, so the initial conditions are always going to be an approximation. This is one reason turbulence is commonly believed to be a form of chaos. You can give one set of initial conditions and get the same answer every time that looks remarkably similar to real life, and likely is representative of real life if you could somehow exactly reproduce those initial conditions in an experiment (you can't). Unfortunately, no matter how hard you try, you will never reproduce the same initial conditions so you get a flow field that looks remarkably similar in character to the simulation, but not exactly the same. In the laminar regime where the system is not chaotic, simulations fare much better since two very similar initial conditions produce very similar end results before turbulence begins.

Specify nonlinear? Nonlinear can be like a differential equation that multiplies the function you are solving for itself with its own derivatives, or like nonlinear optics in which properties of materials such as refractivity etc. change depending on conditions, which is maybe expressed by a nonlinear differential equation? So what exactly is nonlinear here? And what are these degrees of freedom? Sorry I get too sleepy at night to study all of this I hate having to sleep. I also hate having to eat, empty belly is a distraction and so is cooking to fill it.

All of the above. If something is nonlinear, almost invariably that means that it is ultimately governed by a nonlinear equation of some sort, in this case, a whole bunch of nonlinear PDEs. In particular, the stress tensor in the Navier-Stokes equations is a highly nonlinear term. Degrees of freedom is essentially referring to the number of variables being solved for in the system.

 

[arildno]

Leaving aside for the moment your idea that the continuum model as such might provide troublesome, I would like to point out a particularly nasty feature of turbulence that makes it understandable why the N-S (whether "true" or not) are so difficult to handle:

VERY simply speaking, in turbulent flows, vortices form, are split up in tinier vortices, which split up again into even tinier, all the way down to the length scale called the Kolmogorov length scale.

And even worse, MOST of the energy dissipation (where transfer of kinetic energy into heat occurs) happens down on the Kolmogorov scale.
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The upshot of this is that the N-S-equations have multiple relevant length (and time) scales upon which "stuff occurs", which meant that previously (that is, prior to sufficiently strong computers), they were largely intractable, because every approximation scheme basically failed very soon.
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[Chestermiller]

Expanding on the responses of the previous two responders, I would like to summarize. Turbulent flow has two key characteristics: it is time dependent and the transient variations in flow velocity occur over multiple length scales, including very fine length scales. So even if you are modeling the fluid as a continuum, the computational demands are beyond current computer capabilities, with regard to both grid resolution and handling non-linearities. I might mention that there are some techniques out there that give pretty good approximations to the solution of many turbulent flow problems, such as k-ε theory. There are CFD packages out there that can be used to solve turbulent flow problems using this theory. If you want to read more about the fundamentals of turbulent flow, see Bird, Stewart, and Lightfoot, Transport Phenomena.

 

[boneh3ad]

the computational demands are beyond current computer capabilities, with regard to both grid resolution and handling non-linearities.

This statement is not true. You can create a grid and a code that can directly solve the Navier-Stokes equations at all relevant length scales using current technology. This is done rather frequently by researchers utilizing DNS. The problem is not being able to create those things, but the ability to solve flows in economically feasible lengths of time. They can currently do it on small scales, such as flow over a flat plate with a pressure gradient, a small slice of a 0.5 m long cone in hypersonic flow and other similarly limited geometries, but over a plane or a car or even just over the vertical stabilizer of a plane, we simply don't have fast enough computers to solve that problem in a reasonable time frame and probably won't any time before quantum computing is a reality.

I might mention that there are some techniques out there that give pretty good approximations to the solution of many turbulent flow problems, such as k-ε theory. There are CFD packages out there that can be used to solve turbulent flow problems using this theory.

k-ε theory can give pretty good engineering approximations for some quantities, so is useful from a design standpoint. From a basic research standpoint, however, it is useless. You won't shed light on the underlying mechanisms in turbulence by averaging out all of the turbulent physics.

If you want to read more about the fundamentals of turbulent flow, see Bird, Stewart, and Lightfoot, Transport Phenomena.

That is a fine text for introduction to fluid mechanics, but if you want to delve into turbulence itself, Turbulent Flows by Pope is the gold standard.

 

[Chestermiller]

This statement is not true. You can create a grid and a code that can directly solve the Navier-Stokes equations at all relevant length scales using current technology. This is done rather frequently by researchers utilizing DNS. The problem is not being able to create those things, but the ability to solve flows in economically feasible lengths of time. They can currently do it on small scales, such as flow over a flat plate with a pressure gradient, a small slice of a 0.5 m long cone in hypersonic flow and other similarly limited geometries, but over a plane or a car or even just over the vertical stabilizer of a plane, we simply don't have fast enough computers to solve that problem in a reasonable time frame and probably won't any time before quantum computing is a reality.



k-ε theory can give pretty good engineering approximations for some quantities, so is useful from a design standpoint. From a basic research standpoint, however, it is useless. You won't shed light on the underlying mechanisms in turbulence by averaging out all of the turbulent physics.



That is a fine text for introduction to fluid mechanics, but if you want to delve into turbulence itself, Turbulent Flows by Pope is the gold standard.

Thanks Boneh3ad.
Chet

 

[arildno]

k-ε theory can give pretty good engineering approximations for some quantities, so is useful from a design standpoint. From a basic research standpoint, however, it is useless. You won't shed light on the underlying mechanisms in turbulence by averaging out all of the turbulent physics.

I remember the expressed frustration of the lecturer when I had a course in this, in which he showed how the various "appoximations" just about only was useful for the extremely narrow engineering problem they were designed to solve, and collapsed utterly when seeking to go beyond those constraints.
He said that it was hardly more than "advanced" curve-fitting.

 

[boneh3ad]

I remember the expressed frustration of the lecturer when I had a course in this, in which he showed how the various "appoximations" just about only was useful for the extremely narrow engineering problem they were designed to solve, and collapsed utterly when seeking to go beyond those constraints.
He said that it was hardly more than "advanced" curve-fitting.

It certainly feels like that. A lot of turbulence models involve seemingly a lot of advanced curve fitting, albeit curve fitting based on a lot of experimental evidence. Models like that are incredibly useful on the engineering side of things for designing products from cars to planes when it is never economical to do a simulation with more fidelity. They just aren't useful for science.

 

[arildno]

It certainly feels like that. A lot of turbulence models involve seemingly a lot of advanced curve fitting, albeit curve fitting based on a lot of experimental evidence. Models like that are incredibly useful on the engineering side of things for designing products from cars to planes when it is never economical to do a simulation with more fidelity. They just aren't useful for science.

Definitely.
On both accounts, the engineering perspective and the scientific one.

Arguably, it is engineering, rather than science, that has produced most of "value" throughout human history, so it is no place to denigrate engineering as such.

 

[Andy Resnick]

I was discussing this with my friend the other night. I mentioned that proving existence of smooth solutions for the Navier-Stokes equations would win a $1,000,000 prize from the Clay Mathematics Institute,<snip>

AFAIK, there are no existence or uniqueness proofs of weak solutions for the full 3-D time-dependent N-S equations. Recent work on 'viscosity solutions' (or super-weak solutions) has resulted in an existence proof if the following simplifications hold:

The domain of interest is infinite (unbounded fluid)
The fluid is homogeneous
The temperature is well-defined (isentropic case)
The fluid is Newtonian

For more reading:

http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0140.0156.ocr.pdf