Can the Navier-Stokes Equations Prove Long-Term Fluid Stability?

[kurious]

The Clay Institute wants a proof that an initially smooth flowing fluid stays free of turbulence in the long run.
Can the Navier-Stokes equations, which describe a fluid that initially has no turbulence in it, be equivalent to a set of equations describing vortices,
with the vortices cancelling each other out.If so, then could the problem be reduced to that of some vortices which are initially perturbed slightly, so they don't cancel,but which evolve in such a way that at a later time they do cancel out one another's effects?

 

[Clausius2]

An excelent interpretation of the process!. But I really don't know how much true is what you said.

My version is that first of all, turbulence, laminarity, and a vorticity field different of zero are not relationed. You can have a laminar flow with a vorticity field, such the boundary layer flow.

Please describe me again but more explained such proof of what you are talking about. I did not understand the problem. I'll be grateful of helping you.

But in a similar case, sure you know the Reynolds experiment. There is a long pipe, in which initially the flow is laminar. Reynolds checked that the more velocity he provided to the flow, the greater were the unstability of the flow, reaching turbulence regimen at a concrete Reynolds Number. Such unstabilty is nowadays a bit known, mathematically it has to do with eigenvalues problems and perturbation theory. But I have to re-remeber you that no turbulence does not implie zero vorticity field. What you are seen in a boundary layer flow (viscous flow near walls) is a vorticity field which "vorticity eye" is located out of the flow domain. In turbulence regimenes, you can see vortex located and slipped inside the flow.

 

[kurious]

Perhaps Navier Stokes can't be proved because it is impossible for laminar flow to
continue over a long time period.In this case we would have to show that
something equivalent to a Reynolds velocity applies to the system even when fluid moves below the Reynolds velocity threshold.This would have to be a kind of
distance threshold.But for distance to cause instability in the laminar flow,then
there must already be small imperfections in the laminar flow (which will evolve with time into turbulence) so that it isn't truly laminar.
The problem would then reduce to showing that there is in reality no such thing as a perfectly laminar flow.

 

[Clausius2]

Well, I see. It seems I understood you a little bit better.

I think we should analyse two geometries:

i) let's see a boundary layer flow, an initial laminar freestream flowing over a flat plate. You are right when you say it exists a distance threshold. Surely, when the boundary layer thickness is too grown, instabilties are amplified. (Non dimensional) Boundary layer thickness grows as the sqrt of the Reynolds Number based on the x coordinate (parallel to the plat). When you have reached a critical Reynolds (i.e. x=x threshold) the flow separates of plate surface. It will be becoming turbulent as the flow passes a "mixing" region. You might intrepret this as some type of flow inestabilities amplifications.

ii) let's see an internal pipe flow. Below of a critical Reynolds the flow is completely laminar, and aproximately is governed by the Euler Equations in the entrance lenght. As the flow runs into the pipe, viscous terms are becoming progressively important. So that, passed another distance threshold the flow is complete viscous, but you have to know that would remain laminar, because if pipe diameter is not too large, radial boundary layer compresses itself again pipe walls, avoiding its growth and separation.

I mean, laminar flow is possible, or aproximately possible at small Reynolds Numbers. It is possible, and enough accurate calculate it as a laminar flow. I think what you want to say is that laminar flow is something "ideal", but not real. Well, perhaps laminar Navier-Stokes equations do not feel such small perturbations at Re<<<1, in part because such perturbations are not visualizable in experiments, or are all of them worth of being neglected.

Perhaps it is not possible a Newtonian fluid too, a perfect liquid, a perfect conductor...etc. Our real experiencies shows that such perfections does not exists. But they have enough precision to describe physical processes. In this case, you should see Laminar Navier-Stokes equations as an aproximation (very good by the way) of the real world at Re<<<1. A complete generalization of the flow equations have to come in future times. But you should know that we have a lot to do with this equations, in part because there are a lot of problems that are something like an "integration mistery".

 

[Clausius2]

As Enigma says with his "aero-heads", I am very happy to talk with a "fluid-head" over here.

 

[kurious]

You have understood what I was saying perfectly:in reality there are no laminar flows or Newtonian fluids - these are just approximations.
I am not expert in the maths of the Navier-Stokes problem but it seems to me that
for Navier-Stokes to be true then it would have to be shown that each layer of fluid with a given velocity in a pipe keeps its velocity constant over time.Also the wall tension in the pipe must stay constant if turbulence is not forming.Is this correct?

 

[Clausius2]

for Navier-Stokes to be true then it would have to be shown that each layer of fluid with a given velocity in a pipe keeps its velocity constant over time.Also the wall tension in the pipe must stay constant if turbulence is not forming.Is this correct?

I think it is not completely correct. There's something that you have to add in order to be correct. There's several unsteady laminar examples that not yield your statement. Such unsteady flows have changes in time in the velocity field. But they conserves the laminar behaviour.

The correct word is that in laminar flow time-variations are enough continuous and smooth as being described as Navier-Stokes equations tell us. Turbulence causes oscillations in time, in each space position, but such oscillations are random, and very much sharper than in laminar flows. Well, to be honest turbulent flows is not my best (at nowadays), but the little I have read about it, it seems to the sciencist it has something like a "chaotic" behaviour. And I have employed the word "chaotic" as in modern physics sense. Turbulent flows are unsteady flows "per se", but it can be "quasi-steadialized" (I hope this word exist in english) under the hypothesis that mean (in time) values of velocity does not depend on time.

In future, when I learn more about fluids mech., if I have said something stupid I'll report you it.

 

[member 11137]

Can someone tell me what these equations (Navier-Stokes) are exactly looking like or help me with a link where I can find this information ? That was Thanks Blackforest.

 

[Clausius2]

Can someone tell me what these equations (Navier-Stokes) are exactly looking like or help me with a link where I can find this information ? That was Thanks Blackforest.

Could you be a little more explicit?. I don't undertand what do you mean with "looking like".

N-S equations look like an infinite number of things.

 

[da_willem]

Google always helps you find these things...

e.g. http://www.navier-stokes.net/nsfield.htm

 

[dhris]

Here is a link to the official problem statement:

http://www.claymath.org/millennium/Navier-Stokes_Equations/Official_Problem_Description.pdf

You have to show that smooth, bounded-energy solutions exist for smooth initial conditions. I don't think there's any restriction to non-chaotic solutions.

 

[Clausius2]

That text is too mathematic for my sake.

I'm able only to imagine it physically.

 

[member 11137]

Thank you da willem and dhris. I got just a "small" difficulty in reading the link: Description.pdf equation (1) because on the right hand you have a force (applied) and on the left hand you have an acceleration (partial derivation of a velocity). Or do I make a mistake in interpreting the velocity as a speed? Where is the volumetric density of matter of the fluid? Can someone help me to understand the Navier - Stokes equation correctly? Blackforest

 

[Clausius2]

Hi Blackforest! Are you really a dentist?. I've never imagined myself explaining the N_S equations to a german dentist ! What a situation!

Ok... It is very interesting for your part showing curiosity in advanced dynamics. To answer your question, it would be possible if LaTex is enabled yet. Let's go...These are the N_S equations written in the Weak Conservative Form:

Continuity: $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \overline{v})=0$$

Continuity tell us about the mass conservation. The first term on the left is the local time variation of density. This term is active in unsteady and strong compressible processes, like unsteady gas flow at high #Mach and high #Strouhal numbers. The second term on the right is the divergence of the mass flux vector. This term represents the quantity of mass leaving the unit of volume per unit of time.

Momentum: $$\rho \frac{\partial \overline{v}}{\partial t} + \rho \overline{v}\cdot \nabla\overline{v} = -\nabla P + \nabla \cdot \overline{\tau'} + \rho \overline{f_m}$$

Momentum tell us about momentum flux conservation. The first time on the left measures the local acceleration of the flow in unsteady processes. The second term on the left is the dot product of the velocity and gradient tensor of the velocity. It measures the acceleration due to convective transport of momentum. The proper flow accelerates another parts of the same flow. This information is propagated by means of this term. The first term on the right is the gradient of pressure field. It represents the force caused by pressure, modifying the rest of the velocity field. The second term on the right is the divergence of the stress tensor. It represents the viscous forces acting as a stress field inside the proper flow. This term enables the sensivity of the flow about solid boundaries. The third term on the right is the force exerted by the volumetric forces, like gravity (or electromagnetic force?). At high #Reynolds numbers, the convective term is the most important in the equation. At low #Froude numbers, the role of gravity is enough important of not being neglected. At high #Euler numbers, the pressure forces play an important role in the fluid motion.

Internal Energy: $$\rho \frac{\partial cT}{\partial t} + \rho \overline{v}\cdot \nabla(cT) = -P\nabla \cdot \overline{v} + \phi_v - \nabla \overline{q''}+ Q_r$$

This equation shows the Internal Energy conservation. The first term on the left is the local change of temperature. The second term on the left is the temperature variation due to convective transport through the flow field. The first term on the right is the power per volume unit experimented by a fluid particle being expansioned in a hidrostatic surrounding of pressure P. The second term on the right is the Rayleigh function of viscous heat dissipation. The flow has internal irreversibilities, so this term represents the internal heating dissipation due to friction between fluid particles. The third term on the right is the divergence of the heat flux. This term is very important because it modelizes the energy expulsion through the solid boundaries, and enhances heat transport phenomena across the flow field. The last term on the right is the heat released by means of Radiation or Chemical reaction. At high #Reynolds numbers the most important term is the convective transport of energy. The flow does not sense any boundaries to dissipate heat trough. At high #Mach numbers the importance of the pressure term is very important, due to compressive effects.

Any question? I hope it will clear it up to you a little bit more.