Steady state confined flow field: Is it cyclic?

[huangdaiyu]

For a fluid that is confined to a finite region with no sources and sinks, are the only options for the flow field a) static, and b) cyclic? The example I have in mind is Rayleigh convection in a shallow dish heated from below, where convection cells are formed beyond a certain temperature gradient. It seems reasonable to me that for a steady flow, the same fluid that leaves a small volume must pass through it again. But I don't know how to prove it. And it may not be true. I'm guessing the most relevant equation is continuity: $\operatorname {div} \rho \mathbf{u} = 0$. But that's all I have so far. Intuitively, I think if fluid did not return to the same volume, then eventually there would be no fluid left that could fill it, and thus a vacuum would result. And that is definitely not steady state. But that's no proof.

 

[pasmith]

Is a chaotic turblulent state (such as is observed at high Rayleigh number) "cyclic"?

 

[Arjan82]

First of all, for there to be happening anything with the fluid in a confined space there must be some kind of forcing (the temperature gradient in your case). That forcing must be either constant or cyclic for your hypothesis to have a chance to be true. So let's assume that first.

But then there is a third option: the flow could be chaotic. This means that the flow does not ever have to return to a given state ever again in the future. This is particular true for turbulent flow.

The way you formulate your question you are obviously thinking about potential flow. The assumptions behind potential flow (no vorticity, no viscosity, conservative forcefield, adiabatic, and a few others) probably make it so that for that case your hypothesis is true. But I don't know how to prove that.

 

[huangdaiyu]

Is a chaotic turblulent state (such as is observed at high Rayleigh number) "cyclic"?

No, I suppose it would not be. So that is a third option I did not consider. What if we restrict to non-chaotic flow regime? Is my assumption true then? Can it be "proved" (like in physics textbooks, not necessarily very rigourous)? All I can give so far is informal arguments. Such as, a streamline must end at a boundary in a stagnation point if at all. Otherwise, it must close back on itself in a finite region, and thus form a cycle. If I knew more about topology, I might could make this more precise. There would be at least one fixed point in the interior of the flow field. I'm not sure if that is useful.

 

[huangdaiyu]

First of all, for there to be happening anything with the fluid in a confined space there must be some kind of forcing (the temperature gradient in your case). That forcing must be either constant or cyclic for your hypothesis to have a chance to be true. So let's assume that first.

But then there is a third option: the flow could be chaotic. This means that the flow does not ever have to return to a given state ever again in the future. This is particular true for turbulent flow.

The way you formulate your question you are obviously thinking about potential flow. The assumptions behind potential flow (no vorticity, no viscosity, conservative forcefield, adiabatic, and a few others) probably make it so that for that case your hypothesis is true. But I don't know how to prove that.

Correct, I did not think about turbulent flow. And yes those are my assumptions. Whatever is necessary to make the velocity, pressure, and density fields independent of time.

 

[Chestermiller]

For a fluid that is confined to a finite region with no sources and sinks, are the only options for the flow field a) static, and b) cyclic? The example I have in mind is Rayleigh convection in a shallow dish heated from below, where convection cells are formed beyond a certain temperature gradient. It seems reasonable to me that for a steady flow, the same fluid that leaves a small volume must pass through it again.

This is not generally correct. And it is not correct to also say that if a streamline does not end at a boundary, it must be a closed streamline. All I need to do is to provide a single example of a steady state flow that refutes all this. That example is a closed rectangular parallelepiped containing very viscous fluid with the upper surface that slides across the top of the cavity. If the upper surface is slid at an angle to the vertical walls of the cavity, the streamlines will never close.

 

[huangdaiyu]

This is not generally correct. And it is not correct to also say that if a streamline does not end at a boundary, it must be a closed streamline. All I need to do is to provide a single example of a steady state flow that refutes all this. That example is a closed rectangular parallelepiped containing very viscous fluid with the upper surface that slides across the top of the cavity. If the upper surface is slid at an angle to the vertical walls of the cavity, the streamlines will never close.

Wow that is an interesting counterexample. Is it from a textbook by chance? I would like to look it up to visualize and understand it better.

 

[Chestermiller]

Wow that is an interesting counterexample. Is it from a textbook by chance? I would like to look it up to visualize and understand it better.

Actually, you would have to slide the top surface in one direction relative to the vertical walls, and the bottom surface in a different direction. You can work out the flow velocities by taking a case where the cavity is shallow so that you can treat it as flow between parallel plates, with a horizontal pressure gradient also present so there is no flow through the vertical walls. Once you have established the velocity profile, you can do fluid particle tracking to map out the tortuous path that a particle takes as a result of interaction at the walls.

 

[huangdaiyu]

Actually, you would have to slide the top surface in one direction relative to the vertical walls, and the bottom surface in a different direction. You can work out the flow velocities by taking a case where the cavity is shallow so that you can treat it as flow between parallel plates, with a horizontal pressure gradient also present so there is no flow through the vertical walls. Once you have established the velocity profile, you can do fluid particle tracking to map out the tortuous path that a particle takes as a result of interaction at the walls.

Thank you. That is very interesting. I will look into that. Going back to the example of Rayleigh convection, is there a way to predict convection cells without having to solve the equations of motion? That was what I was hoping for in guessing that the steady flow solutions for a confined region are cyclic (which apparently is not generally true).

 

[Chestermiller]

Thank you. That is very interesting. I will look into that. Going back to the example of Rayleigh convection, is there a way to predict convection cells without having to solve the equations of motion? That was what I was hoping for in guessing that the steady flow solutions for a confined region are cyclic (which apparently is not generally true).

In my judgment, there is no way of doing it without using the Navier Stokes equations.