Geometric Interpretation of Turbulence

[George444fg]

TL;DR Summary

Geometric Interpretation of Turbulence

I would like to give a geometric interpretation to turbulence. Let's take into consideration for example a Poiseuille flow. The velocity profile resembles a parabolic bullet. As the particles are pushed by other layers of particles, then it must be that in addition to their translation, they would rotate sideways due to the shape of the profile. While as far away the particles are from the centre of the tube the greater the slope of the paraboloid would be and therefore the greater the turbulence. Is my intuition valid?

 

[Chestermiller]

Laminar flow has no turbulence. So your interpretations must be incorrect. Turbulence is characterized by rapid and chaotic temporal fluctuations of the velocity components; such rapid temporal fluctuations are not present in laminar flow.

 

[boneh3ad]

Summary:: Geometric Interpretation of Turbulence

I would like to give a geometric interpretation to turbulence. Let's take into consideration for example a Poiseuille flow. The velocity profile resembles a parabolic bullet. As the particles are pushed by other layers of particles, then it must be that in addition to their translation, they would rotate sideways due to the shape of the profile. While as far away the particles are from the centre of the tube the greater the slope of the paraboloid would be and therefore the greater the turbulence. Is my intuition valid?

Why would particles need to be pushed in the way you describe? The velocity vector does not point normal to the profile you seem to describe. The fluid particles simply move horizontally with a velocity that, when plotted, is parabolic (in the case of Poiseuille flow that you cite).

If you want a "geometric" interpretation, the best I can do for you is to cite a 1922 limerick by Lewis Fry Richardson:

Big whorls have little whorls
Which feed on their velocity,
And little whorls have lesser whorls
And so on to viscosity

 

[A.T.]

Let's take into consideration for example a Poiseuille flow. The velocity profile resembles a parabolic bullet. As the particles are pushed by other layers of particles, then it must be that in addition to their translation, they would rotate sideways due to the shape of the profile. While as far away the particles are from the centre of the tube the greater the slope of the paraboloid would be and therefore the greater the turbulence. Is my intuition valid?

Not sure what you mean by "rotate sideways", but if a test body floats in the Poiseuille flow, it will be rotated by the gradient. The vorticity (curl of the velocity field) can be non-zero, even if the flow velocity vectors are all parallel. But vorticity is different from turbulence.

https://en.wikipedia.org/wiki/Vorticity#Examples

 

[boneh3ad]

Not sure what you mean by "rotate sideways", but if a test body floats in the Poiseuille flow, it will be rotated by the gradient. The vorticity (curl of the velocity field) can be non-zero, even if the flow velocity vectors are all parallel. But vorticity is different from turbulence.

https://en.wikipedia.org/wiki/Vorticity#Examples

True, though it should be pointed out that while turbulence is inherently vortical, the converse is not true.

 

[Arjan82]

Summary:: Geometric Interpretation of Turbulence

I would like to give a geometric interpretation to turbulence. Let's take into consideration for example a Poiseuille flow. The velocity profile resembles a parabolic bullet. As the particles are pushed by other layers of particles, then it must be that in addition to their translation, they would rotate sideways due to the shape of the profile. While as far away the particles are from the centre of the tube the greater the slope of the paraboloid would be and therefore the greater the turbulence. Is my intuition valid?

Allthough, Poiseuille flow is by definition laminar indeed. Also, there is not really rotation in the flow. But as @A.T. already mentioned, there is vorticity in the flow, i.e. draw am off-center rectangle in the flow and compute the integrated tangential velocity, this is not zero. This is a 'kind of' rotation, but only if you subtract the mean flow.

All that said, there is some validity to your intuition. Many turbulence models, used in viscous flow computations (CFD) use the gradient in the flow as a source of turbulence (together with some more complex parameters and modelling). So, the higher the gradient (as is true when you get closer to the wall) the higher the turbulence generation. This is not the entire story, because flow velocity, among others, also plays its part, but it is true to some extent.

 

[A.T.]

All that said, there is some validity to your intuition. Many turbulence models, used in viscous flow computations (CFD) use the gradient in the flow as a source of turbulence (together with some more complex parameters and modelling). So, the higher the gradient (as is true when you get closer to the wall) the higher the turbulence generation. This is not the entire story, because flow velocity, among others, also plays its part, but it is true to some extent.

One example of gradient in the flow as a source of turbulence:

https://en.wikipedia.org/wiki/Kelvin–Helmholtz_instability

 

[boneh3ad]

You have to be very careful suggesting that velocity gradients lead to turbulence, because it is not universally true. The Blasius boundary layer, Poiseuille flow, Couette flow, and many others (including the superposed fluids subject to Kelvin-Helmholtz posted above) have velocity gradients but are only unstable under certain conditions. In other words, the existence of a velocity gradient is not a sufficient condition for the generation of turbulence.

 

[George444fg]

Allthough, Poiseuille flow is by definition laminar indeed. Also, there is not really rotation in the flow. But as @A.T. already mentioned, there is vorticity in the flow, i.e. draw am off-center rectangle in the flow and compute the integrated tangential velocity, this is not zero. This is a 'kind of' rotation, but only if you subtract the mean flow.

All that said, there is some validity to your intuition. Many turbulence models, used in viscous flow computations (CFD) use the gradient in the flow as a source of turbulence (together with some more complex parameters and modelling). So, the higher the gradient (as is true when you get closer to the wall) the higher the turbulence generation. This is not the entire story, because flow velocity, among others, also plays its part, but it is true to some extent.

Excuse me I used the wrong term. We have rotational flow.

 

[George444fg]

You have to be very careful suggesting that velocity gradients lead to turbulence, because it is not universally true. The Blasius boundary layer, Poiseuille flow, Couette flow, and many others (including the superposed fluids subject to Kelvin-Helmholtz posted above) have velocity gradients but are only unstable under certain conditions. In other words, the existence of a velocity gradient is not a sufficient condition for the generation of turbulence.

Excuse me I used the wrong term. I meant rotational flow, not turbulent

 

[Arjan82]

Do you mean rotational flow (large scale structures rotating around some center) or vorticity (local flow parameter, or point value, like velocity and pressure)?

 

[George444fg]

Do you mean rotational flow (large scale structures rotating around some center) or vorticity (local flow parameter, or point value, like velocity and pressure)?

Yep exactly that, I made a mistake

 

[Arjan82]

Yep exactly that, I made a mistake

exactly what? rotation or vorticity?

 

[George444fg]

exactly what? rotation or vorticity?

rotation