[Martin Hegedus]

I guess part of the reason Re(Length) needs to be big is that Re(Displacement thickness) needs to get to the point that an instability can occur. For example, assuming a laminar boundary layer, an ReL of 500K is an Re(Displacement thickness) of about 3500. Does that 3500 correlate to the critical Re of Orr-Sommerfeld? If so, then the question is whether there is a qualitative explanation of why Re(Displacement thickness) has to be large for the Orr-Sommerfeld equation to predict an instability. From what I gather, some viscosity effects are required to kick in the instability but too much will cause it to dampen out. So, what is so special about the 3500?

[Ahmed]

Originally Posted by truffaldino

Ok, this is a wrong idea, but why can't we show that critical Re should be big (not to find it, but just demonstrate that it is big) in Kolmogorov's style.

Say, to estimate order of total dissipated energy and find order of Re when it is possible, or some other very rough considerations.

If you non-dimensionalize the NS equation the Reynolds Number (or 1/R) will crop up before each term and then you can estimate the effect of each term

[Martin Hegedus]

I think some care needs to be taken. The Re(L) and and the Re(displacement thickness) are two different things which are linked by the rate of growth of the boundary layer. And I gather transition is more a function of the boundary layer thickness than the distance traveled. But, I could be so very wrong.

[truffaldino]

Originally Posted by Martin Hegedus

I think some care needs to be taken. The Re(L) and and the Re(displacement thickness) are two different things which are linked by the rate of growth of the boundary layer. And I gather transition is more a function of the boundary layer thickness than the distance traveled. But, I could be so very wrong.

Yes it is more dependent on Re_theta and shape factors, but still transitional Re_theta are order of 10^2 -10^3 still much more bigger than 1.

[truffaldino]

Originally Posted by Ahmed

If you non-dimensionalize the NS equation the Reynolds Number (or 1/R) will crop up before each term and then you can estimate the effect of each term

If we apply such a direct approach, assuming that transition happens when contribution from inertial term becomes of the same oredr as from dissipative one, we will get result that Re transition will be of order of 1 which is wrong.

[robo]

But why would you make that assumption? It seems more natural to say that transition will occur when the contribution of the inertial term overwhelms the viscous term, not when they're roughly equal. That would imply transition occurs at some Re >> 1, which is what happens.

[FMDenaro]

first, what do you think about transition concept?
for example, the flow around a sphere becomes transitional at very low Re showing time-periodic (laminar) behaviour but then turbulence occurs.
On the other hand, on an airfoil you can have laminar condition even for Re = O(10^5-10^6)

[Martin Hegedus]

This is an interesting topic, but one that I know very little about.

That being said, from what I understand, transition occurs within the boundary layer. So if one uses the distance from the wall at which it first occurs, it will be less, to some degree, than the 3500. OK, probably not close to Re of 1, but maybe in the 100s. Maybe there is some point at which a laminar shear layer wants to break up into discrete vortices (i.e. a buildup in the oscillation [wiggles] of the normal velocity). Anyway, all arm waving on my part.

[truffaldino]

Originally Posted by robo

But why would you make that assumption? It seems more natural to say that transition will occur when the contribution of the inertial term overwhelms the viscous term, not when they're roughly equal. That would imply transition occurs at some Re >> 1, which is what happens.

This is exactly the main question of this thread, why does it ocuur ar Re>>1. If there a simple explanation? Why does it more natural to think that transition happens when inertial term overhelms (by 3-5 orders) the viscous one?

All the explanations in literature are based on liear stability analysis of NS equation which is in fact solution of this equation. Can we explain it in more simple way?

[FMDenaro]

I try to suggest some ideas...

consider a Fourier component of the velocity (exp(ikx)). You get k^2/Re as coefficient for the diffusive part and 2ik from the convective non linear part.
Supposing that you must have a balance between the two terms Re=O(k) and transition should be onset by high-frequencies so that Re is high, too.

That is a rude analysis...

[truffaldino]

Originally Posted by FMDenaro

first, what do you think about transition concept?
for example, the flow around a sphere becomes transitional at very low Re showing time-periodic (laminar) behaviour but then turbulence occurs.
On the other hand, on an airfoil you can have laminar condition even for Re = O(10^5-10^6)

Maybe I am wrong, but as far as I remember for sphere fully turbulent flow develops at Re=10^4-10^5

http://www.grc.nasa.gov/WWW/k-12/air...ragsphere.html

BTW,for which kind of flow transitional Re is the smallest? Is there any table of examples?

From my experience on airfoils, turbulators (trips) can reduce critical Re up to 10^2-10^3, but it is still >> 1

[truffaldino]

Originally Posted by FMDenaro

I try to suggest some ideas...

consider a Fourier component of the velocity (exp(ikx)). You get k^2/Re as coefficient for the diffusive part and 2ik from the convective non linear part.
Supposing that you must have a balance between the two terms Re=O(k) and transition should be onset by high-frequencies so that Re is high, too.

That is a rude analysis...

It is interesting! Let us think about this

[robo]

Take a shear layer with a smooth interface. If you perturbed the interface with a sine wave shape, the positive phases of the wave would be convected faster then the negative phases due to the strengh of the inertial term, and the sine wave shape would begin to distort. It would also be transported by the convection, so that the wave would appear to move forward while it's distorting. But if the viscous term is just as strong, then the initial sine wave shape would also be smeared at an equal rate as the perturbation was convected and distorted, and the sine wave shape and the distortion would both get smeared out while they're being convected. The result would eventually be just a smeared interface.

In order for a perturbation to grow into turbulence, it must be transported and distorted by the non-linearity in the convective term, and that must happen faster then the perturbation can be dissipated by the viscous term. That all implies that Re must be greater then 1 in order for perturbations to grow into turbulence.

[FMDenaro]

Originally Posted by robo

Take a shear layer with a smooth interface. If you perturbed the interface with a sine wave shape, the positive phases of the wave would be convected faster then the negative phases due to the strengh of the inertial term, and the sine wave shape would begin to distort. It would also be transported by the convection, so that the wave would appear to move forward while it's distorting. But if the viscous term is just as strong, then the initial sine wave shape would also be smeared at an equal rate as the perturbation was convected and distorted, and the sine wave shape and the distortion would both get smeared out while they're being convected. The result would eventually be just a smeared interface.

In order for a perturbation to grow into turbulence, it must be transported and distorted by the non-linearity in the convective term, and that must happen faster then the perturbation can be dissipated by the viscous term. That all implies that Re must be greater then 1 in order for perturbations to grow into turbulence.

I agree, I think that the Re is also a parameter that must be considered in a careful way...
for example, a flow over a semi-infinite flat plate has not a unique characteristic lenght you can estimate a-priori but you must consider the Re_x. It is, therefore, the change in the characteristic lenght to produce incresing in Re up to transition a O(10^5)

[truffaldino]

Originally Posted by FMDenaro

I try to suggest some ideas...

consider a Fourier component of the velocity (exp(ikx)). You get k^2/Re as coefficient for the diffusive part and 2ik from the convective non linear part.
Supposing that you must have a balance between the two terms Re=O(k) and transition should be onset by high-frequencies so that Re is high, too.

That is a rude analysis...

But the question remains, why do transition is set by high frequencies? Looks like just a reformulating the problem in momentum space.

[truffaldino]

Originally Posted by robo

Take a shear layer with a smooth interface. If you perturbed the interface with a sine wave shape, the positive phases of the wave would be convected faster then the negative phases due to the strengh of the inertial term, and the sine wave shape would begin to distort. It would also be transported by the convection, so that the wave would appear to move forward while it's distorting. But if the viscous term is just as strong, then the initial sine wave shape would also be smeared at an equal rate as the perturbation was convected and distorted, and the sine wave shape and the distortion would both get smeared out while they're being convected. The result would eventually be just a smeared interface.

In order for a perturbation to grow into turbulence, it must be transported and distorted by the non-linearity in the convective term, and that must happen faster then the perturbation can be dissipated by the viscous term. That all implies that Re must be greater then 1 in order for perturbations to grow into turbulence.

Yes, it is clear that it must be greater than 1, but why in the case of smooth interfaces (flat plates, airfoils etc) it must be transported by 5 orders of magnitude faster?

[H0T_S0UP]

The honest answer:

We don't know.

We do not know enough about the mathematical characteristics of the N-S equations, we only superficially understand turbulence, and we know even less about
laminar-turbulent transition.

My take on the subject: rederive the equations of fluid motion through statistical mechanics and stochastic calculus and show that it degenerates into the N-S equations. I believe the statistical perturbations that when neglected degenerate into the continuum assumption are what governs turbulent behavior.

Note that turbulence is not just in fluids. Google "microturbulence" to see similar behavior in stellar systems.

[robo]

The actual value of the transitional Reynold's number will depend alot on the nature of the perturbation and the flow conditions, as several posters have already said.

It must be significantly more then one for the same reason it must be more then one in general. The nonlinear growth, distortion, and transport caused by the convection term must have enough time to establish the flow structure that is characteristic of turbulence. This can only happen if it is that much greater then the viscous terms. If it is only mildly greater then one, then the perturbation can transport a bit further and be distorted a bit more, but it will still be dissipated before it can establish the large range of length scales that is characteristic of turbulence. The perturbation needs time to grow, and it will only get that if the nonlinear convection is sufficiently rapid in relation to the viscous dissipation.

[robo]

Hot Soup: we can do DNS of low Re turbulence already and get good results, making it appear that the only barrier to high Re DNS is computing resources. This implies that all of the physics of turbulence is already present in the NS equations as is, without any need for statistical mechanics. We can also see similar behaviour in many other nonlinear dynamic systems without recourse to statistical mechanics, which again implies that all of the required physics can be encompassed by deterministic differential equations. Also, stellar systems are fluid systems, just at larger scales and coupled with electromagnetics.

I guess in general I'm not sure what you mean. I don't believe there is any requirement for statistical mechanics in order to explain turbulence.

[H0T_S0UP]

Originally Posted by robo

Hot Soup: we can do DNS of low Re turbulence already and get good results, making it appear that the only barrier to high Re DNS is computing resources. This implies that all of the physics of turbulence is already present in the NS equations as is, without any need for statistical mechanics. We can also see similar behaviour in many other nonlinear dynamic systems without recourse to statistical mechanics, which again implies that all of the required physics can be encompassed by deterministic differential equations. Also, stellar systems are fluid systems, just at larger scales and coupled with electromagnetics.

I guess in general I'm not sure what you mean. I don't believe there is any requirement for statistical mechanics in order to explain turbulence.

Basically, what I am saying is the continuum assumption takes a time and spacial average over a very small domain, and this leads to an approximation of the momentum of the fluid in this region. You cant have a domain where dxdydz = V =0 or dt = 0, because you would not have a flux through the control volume or not be able to measure velocity. Im sure the errors are minor, but my proposition is the sum of these errors over practical length and time scales leads to small deviations from what continuum mechanics (ie, N-S) predicts. Im working on a stochastic derivative in my free time to see if this approach is fruitful. It is more of a fun project since nobody will pay you to do that kind of work, but the results are kind of neat so far.