[FluidKo]

Hello.

I've learned Dean vortex.
What I've heard is fluid in center of cross section of pipe(red arrow line in uploaded picture) is moved to outside(concave region) by centripetal force.
Because of this motion, surrounding fluid must be moved also by molecule viscosity.
In other words, Dean vortex is just similar to cavity flow.
Dean vortex is just vertical symmetry of cavity flow.
I can understand until here.

But I've seen some figures that Dean vortex is biased like uploaded picture.
There is no symmetry at there.
There is only right-handed secondary flow or left-handed secondary flow.
I want to know why this secondary flow with only one direction can occurs.

Thank you

[LuckyTran]

The lid driven cavity is a primary flow example. It's not even in the same category as secondary flows. Secondary flows are driven by small variations in the flow (and most of them are perpendicular/orthogonal to the primary flow) and are sensitive to small perturbations.

There isn't any context of where these photos come from to make any specific explanations. But in general, nothing says that the solution should be or must be symmetric. Even when a symmetric solution exists, it may be unstable. Even in statistically stationary cases, mode switching occurs where the flow switches from one mode to another. Formally/mathematically these effects are studied under bifurcation theory. If there is a little dent in the pipe on one side, it can make the entire downstream vortex turn one way. Turbulent flows are an extreme example where there is an extremely broad range of scales of vortices both left-handed and right-handed occurring intermittently all the time.

[FluidKo]

Originally Posted by LuckyTran

The lid driven cavity is a primary flow example. It's not even in the same category as secondary flows. Secondary flows are driven by small variations in the flow (and most of them are perpendicular/orthogonal to the primary flow) and are sensitive to small perturbations.

There isn't any context of where these photos come from to make any specific explanations. But in general, nothing says that the solution should be or must be symmetric. Even when a symmetric solution exists, it may be unstable. Even in statistically stationary cases, mode switching occurs where the flow switches from one mode to another. Formally/mathematically these effects are studied under bifurcation theory. If there is a little dent in the pipe on one side, it can make the entire downstream vortex turn one way. Turbulent flows are an extreme example where there is an extremely broad range of scales of vortices both left-handed and right-handed occurring intermittently all the time.

Hello LuckyTran!
Actually I have No idea relation between centripetal force and secondary flow.
In the case of flow separation in curved pipe, I can understand relation between occurence of flow separation and curvature(centripetal force).
What I know is below.
In uploaded picture, whenever fluid flows in a curved path, there must be a force acting radially inwards on the fluid to provide the inward acceleration, known as centripetal acceleration .
This results in an increase in pressure near the outer wall of the bend, starting at some point A and rising to a maximum at some point B.
There is also reduction of pressure near the inner wall giving a minimum pressure at C and a subsequent rise from C to D.
Therefore between A and B and between C and D the fluid experiences an adverse pressure gradient (the pressure increases in the direction of flow).
So cause of this adverse pressure gradient, flow is separated similarly to airfoil.
I'm not sure that my explanation is enough but anyway I can understand physically.
But I can't understand the physical relation between centripetal force and secondary flow.
Although I've read paper by Dean and get lots of help from you to understand that paper, I can't understand well because of my low mathematics and English.(Sorry...)
But as I understand physical cause and effect relation between centripetal force and flow separation without mathematical equation, I think there can be way that I can understand physical cause and effect relation between centripetal force and secondary flow.
But I couldn't find clue to find physical relation between them...
So may I ask some help to you?
Thank you

[LuckyTran]

Flow separation is a feature of flow in curved pipes and bend but occurs independently of a Dean Vortex.


Dean vortex is rather intuitive. So the curved pipe results in high pressure on the outer side of the bend and lower pressure on the inner side of the bend providing the centripetal acceleration for the flow to follow the curve. Now draw two lines from the high pressure to low pressure region. One line goes straight across directly through the center and the other curves along the upward or downward side of the pipe. The shortest distance is the path through the center. Flow takes the path of least resistance and prefers going through the shorter central path than along the outward trajectory. Therefore, the pressure gradient sets up a secondary flow (i.e. the dean vortex) with the outward-to-inward movement along this central path flowing with a strong pressure gradient. Due to mass conservation, the flow returns along the outward trajectory against a relatively weaker pressure gradient. A small mismatch in the relative strengths of the pressure gradient along different paths is what causes the Dean vortex and secondary flow.

[FluidKo]

Originally Posted by LuckyTran

Flow separation is a feature of flow in curved pipes and bend but occurs independently of a Dean Vortex.


Dean vortex is rather intuitive. So the curved pipe results in high pressure on the outer side of the bend and lower pressure on the inner side of the bend providing the centripetal acceleration for the flow to follow the curve. Now draw two lines from the high pressure to low pressure region. One line goes straight across directly through the center and the other curves along the upward or downward side of the pipe. The shortest distance is the path through the center. Flow takes the path of least resistance and prefers going through the shorter central path than along the outward trajectory. Therefore, the pressure gradient sets up a secondary flow (i.e. the dean vortex) with the outward-to-inward movement along this central path flowing with a strong pressure gradient. Due to mass conservation, the flow returns along the outward trajectory against a relatively weaker pressure gradient. A small mismatch in the relative strengths of the pressure gradient along different paths is what causes the Dean vortex and secondary flow.

I can understand what are you meaning.
But why patter of dean vortex usually looks like that?

In your picture, there is one linear path and because that linear path is shorter than curve path, fluid prefers to move to center of rotation with shorter path(linear line).

By the way, can you see uploaded picture by me?
In this picture, there are 2 linear path lines(Red line).
These 2 linear pathes are also shorter than longer curved path(Green line).

So what I'm wondering is why dean vortex usually shows pattern like your uploaded picture not my uploaded picture.
Is it is hard to explain without mathematical equation?

Thank you

[LuckyTran]

The dean vortex with a symmetric one pair of left-handed and right-handed cells is simply one mode. There are infinite number of other modes: two pairs, three pairs, and so on. The higher order modes have a tendency to generate more vorticity which enhances viscous effects and are more damped by viscous forces. You can do a similar cartoon to show the vorticity is greater. If you draw a closed loop representing the vortex cell and assume that they all have the same circulation, a smaller loop means the vorticity is greater.

These are just the quick physics-based explanations. If you want to know for an arbitrary flow in an arbitrarily shaped duct how many vortex cells will actually appear, then it requires a detailed analysis. You can get from reading that the theory for predicting which ones will occur are rather limited and it is taken somewhat for granted. The explanation for it can be simple but to predict it can be a tedious effort.


You've seen a similar outcome before with laminar flows. Nothing says the fluid can't rotate in either direction but the viscous forces damp pretty much all of them out and a laminar flow ends up flowing in sheet-like layers with very little vorticity; and even the vorticity that does appear occurs in the handed-ness of the shear layer until you get to really really high Reynolds numbers where the inertial forces can finally overcome the viscous forces.

[FluidKo]

Originally Posted by LuckyTran

The dean vortex with a symmetric one pair of left-handed and right-handed cells is simply one mode. There are infinite number of other modes: two pairs, three pairs, and so on. The higher order modes have a tendency to generate more vorticity which enhances viscous effects and are more damped by viscous forces. You can do a similar cartoon to show the vorticity is greater. If you draw a closed loop representing the vortex cell and assume that they all have the same circulation, a smaller loop means the vorticity is greater.

These are just the quick physics-based explanations. If you want to know for an arbitrary flow in an arbitrarily shaped duct how many vortex cells will actually appear, then it requires a detailed analysis. You can get from reading that the theory for predicting which ones will occur are rather limited and it is taken somewhat for granted. The explanation for it can be simple but to predict it can be a tedious effort.


You've seen a similar outcome before with laminar flows. Flows can rotate in either direction but the viscous forces damp pretty much all of them out and a laminar flow ends up flowing in sheet-like layers with very little vorticity until you get to higher Reynolds numbers.

Ok then finally I want to ask only one more thing.
In some paper, they describe the right-handed secondary flow.
I'm not sure that whether there is a very very small portion of left-handed secondary flow or not.
But it looks like there is only right-handed secondary flow.
(It looks like there is secondary flow with only one direction.)

I want to know why this can occur.
I want to know physical cause and effect relation of this one direction secondary flow.
If it is hard or it can be tedious effort to explain it,
then can you let me know some paper that explains this one direction secondary flow physically without handling too hard mathematics?
(Because as you know, I'm not good at mathematics...)

Thank you

[LuckyTran]

I need a physical context for where these pictures are from if you want a physical explanation. I can guess that each picture is at different flow conditions. It would be helpful to what changed in each... I can surmise that it's probably not a simple flow in a bend. If the pipe is rotating, or if the pipe is vertical, then coriolis forces can cause the flow to prefer one direction of rotation over another. If there are two bends in the pipe, that can also lead to a similar effect. Two-bends in the same plane versus two-bends not in the same plane also lead to different secondary flow characteristics. If I install a swirler, I can also produce a rotational flow like shown.

All I can tell you is that these are all allowed flow solutions and you really have to examine what is going on to explain it. I can't just shake a magic 8 ball and tell you.

[LuckyTran]

Actually nevermind (google image search does wonders).

One of my guesses was correct. The pipe is rotating which therefore gives a tendency of the flow to rotate, which is independent and superimposed on the secondary flow.

When the pipe is not rotating, then you get your expected pair of Dean vortices. If the pipe is rotating very very very fast, then the rotation imparted onto the flow by the pipe exceeds the centripetal force driving the dean vortex and you see a single celled rotation pattern. Btw, this flow rotation pattern caused by the rotation of the pipe is considered a primary flow and not a secondary flow.

There is obviously some critical swirl number where these two are in equal balance. This much we can explain immediately with simple thought experiments and doesn't require any mathematics. Now if you ask me at what Swirl number this transition occurs, then we need to do the heavy duty analysis.

[FMDenaro]

In general, simmetry in the flow is broken by the effect of the convective terms. That is, when a certain Re number is reached.
For example, the lid-drive cavity at low Re, i.e. O(1), but also separation behind a bluff body at Re=O(10).
The prevalence of the non linear term is the response.

[FluidKo]

Originally Posted by LuckyTran

Actually nevermind (google image search does wonders).

One of my guesses was correct. The pipe is rotating which therefore gives a tendency of the flow to rotate, which is independent and superimposed on the secondary flow.

When the pipe is not rotating, then you get your expected pair of Dean vortices. If the pipe is rotating very very very fast, then the rotation imparted onto the flow by the pipe exceeds the centripetal force driving the dean vortex and you see a single celled rotation pattern. Btw, this flow rotation pattern caused by the rotation of the pipe is considered a primary flow and not a secondary flow.

There is obviously some critical swirl number where these two are in equal balance. This much we can explain immediately with simple thought experiments and doesn't require any mathematics. Now if you ask me at what Swirl number this transition occurs, then we need to do the heavy duty analysis.

Oh...I'm sorry.
I had to tell you example.
I'm studying TKE in aorta.
As you can see from uploaded pictire, shape of aorta is similar to double pipe bend.
And there are many experiment results that show secondary flow with one direction.
https://www.researchgate.net/publica...t-based_models
https://www.vki.ac.be/simbio2011/papers/18.pdf
https://cds.ismrm.org/protected/18MP...06458_Fig2.png
https://cds.ismrm.org/protected/18MP...iles/2932.html
https://link.springer.com/article/10...1-2353-6#Sec12
https://d-nb.info/116397997X/34
https://www.ahajournals.org/doi/pdf/....CIR.88.5.2235
(Although you don't read whole text in these paper, you can understand situation with looking only some figures)

But although I read the papers that carriy out this kind of experiment, they don't explain secondary flow in terms of physics and fluid dynamics.
So this is why I ask biased secondary flow to you.

I think the reason why fluid prefers one direction in aorta is caused by geometry of aorta not fast rotation of pipe.
I guess because shape of aorta is similar to double pipe bend, secondary flow with only one direction occurs.
So I want to know why there secondary flow with only one direction occurs in double pipe bend and their physical cause and effect relation.

Thank you

[FluidKo]

Originally Posted by FMDenaro

In general, simmetry in the flow is broken by the effect of the convective terms. That is, when a certain Re number is reached.
For example, the lid-drive cavity at low Re, i.e. O(1), but also separation behind a bluff body at Re=O(10).
The prevalence of the non linear term is the response.

Sorry...I can't understand.
What is the physical relation between convective terms, Reynolds number and break of symmetry?

[FMDenaro]

Originally Posted by FluidKo

Sorry...I can't understand.
What is the physical relation between convective terms, Reynolds number and break of symmetry?

The Re number is the measure of the ratio between the convective and diffusive fluxes. At low Re number the equations tend to become linear.


However, from your question I see you are talking about RANS solution. In such a case the appearence of a secondary flow should be considered only in the statistical meaning.

[FluidKo]

Originally Posted by FMDenaro

The Re number is the measure of the ratio between the convective and diffusive fluxes. At low Re number the equations tend to become linear.


However, from your question I see you are talking about RANS solution. In such a case the appearence of a secondary flow should be considered only in the statistical meaning.

Yeah you are right.
I have to do RANS simulation for research.
But I've thought that before understanding secondary flow in RANS, I have to understand secondary flow in laminar flow at first.
When I start to learn secondary flow in RANS, I'm going to write new threads at here.
At that time I wish that I can get help from you.

Thank you

[LuckyTran]

When the diffusion term dominates the flow behaves very Laplacian, i.e. they look like solutions to the heat equation. Fundamentally diffusion is gradient driven. In this limit you have potential flow behavior, flow goes from high potential to low potential and you don't have minima/maxima inside the domain. Hence you can just look at the boundary condition to figure out what the symmetry in the solution will look like. For flow in pipes, you have the same no slip condition on the wall everywhere. So if the pipe is circular, you expect circularly symmetric solutions. If the pipe is a square then you expect solutions that fill four quadrants and so on.

This symmetry is broken when the convective term starts to play a role and the diffusion term is no longer the only actor. The non-linear convective term is what allows the solution to have minima/maxima. In a straight pipe, the diffusion term dominants at low Reynolds number. At high Reynolds numbers the convective term becomes more significant. The convective term has more influence at higher Reynolds numbers because the convective term goes like u whereas the diffusion term goes like grad(u). There are other ways the convective term can be more significant not related to simply increasing the velocity, such as the pipe bend. Changing the geometry can also change the Reynolds number because it changes the length scale in the Reynolds number.

For two bends not in the same plane a coriolis force appears and this coriolis (if it is strong enough) determines the handedness of the resulting helical flow pattern which leads to the single vortex cell. The direction of the coriolis force and the rotation of the helix can be determined quickly using the right hand rule. But the threshold of this onset leads to a single-cell rotation, and not greater numbers like 3, 5, or 7 because those would be viscously damped (and not even numbers of cells because now we have an additional coriolis force in addition to the centripetal force). The coriolis force needs to be of similar strength to the centripetal force otherwise the you get the local solution of a single bend with the double-celled dean vortex. That is, if the coriolis force is not present, the solution after a bend has a tendency to follow whatever was the most recent bend. The coriolis force is significant when the bends occur within a short distance of each other.

Btw so far we have discussed secondary flows of the first kind, i.e. the laminar ones. These can be modeled pretty much exactly using CFD and are straightforward.

Secondary flows of the second kind are their own monster, these are driven by anisotropy in the turbulence, driven by differences in turbulence fluctuations. These cannot be modeled at all using your standard two-equation model and the anisotropic models that are commonly available are not well calibrated. If you need accurate CFD for these, you pretty much have to do LES. You can get qualitative results from an RSM model, but RSM is not well calibrated and the cost is already approaching the cost of doing an LES.

[FluidKo]

Originally Posted by LuckyTran

When the diffusion term dominates the flow behaves very Laplacian, i.e. they look like solutions to the heat equation. Fundamentally diffusion is gradient driven. In this limit you have potential flow behavior, flow goes from high potential to low potential and you don't have minima/maxima inside the domain. Hence you can just look at the boundary condition to figure out what the symmetry in the solution will look like. For flow in pipes, you have the same no slip condition on the wall everywhere. So if the pipe is circular, you expect circularly symmetric solutions. If the pipe is a square then you expect solutions that fill four quadrants and so on.

This symmetry is broken when the convective term starts to play a role and the diffusion term is no longer the only actor. The non-linear convective term is what allows the solution to have minima/maxima. In a straight pipe, the diffusion term dominants at low Reynolds number. At high Reynolds numbers the convective term becomes more significant. The convective term has more influence at higher Reynolds numbers because the convective term goes like u whereas the diffusion term goes like grad(u). There are other ways the convective term can be more significant not related to simply increasing the velocity, such as the pipe bend. Changing the geometry can also change the Reynolds number because it changes the length scale in the Reynolds number.

For two bends not in the same plane a coriolis force appears and this coriolis (if it is strong enough) determines the handedness of the resulting helical flow pattern which leads to the single vortex cell. The direction of the coriolis force and the rotation of the helix can be determined quickly using the right hand rule. But the threshold of this onset leads to a single-cell rotation, and not greater numbers like 3, 5, or 7 because those would be viscously damped (and not even numbers of cells because now we have an additional coriolis force in addition to the centripetal force). The coriolis force needs to be of similar strength to the centripetal force otherwise the you get the local solution of a single bend with the double-celled dean vortex. That is, if the coriolis force is not present, the solution after a bend has a tendency to follow whatever was the most recent bend. The coriolis force is significant when the bends occur within a short distance of each other.

Btw so far we have discussed secondary flows of the first kind, i.e. the laminar ones. These can be modeled pretty much exactly using CFD and are straightforward.

Secondary flows of the second kind are their own monster, these are driven by anisotropy in the turbulence, driven by differences in turbulence fluctuations. These cannot be modeled at all using your standard two-equation model and the anisotropic models that are commonly available are not well calibrated. If you need accurate CFD for these, you pretty much have to do LES. You can get qualitative results from an RSM model, but RSM is not well calibrated and the cost is already approaching the cost of doing an LES.

Oh, I've thought there will be similarity between secondary flow of laminar flow and secondar flow of turbulent flow.
If there is no similarity between them, I have to learn secondary flow turbulent flow not laminar flow.
I think I have to read paper of secondary flow in turbulent flow.

Thank you