[Shabi]

Consider a simple 2D channel where the flow enters the domain from one side with the inlet velocity boundary condition with a constant value and it forms a boundary layer.

1) If i add many number of particles consistently to the inlet and neglect all inertial forces and let it to follow the flow trajectories, is it true that the volume fraction of the particle in the domain (C) is the solution of the advection equation, dC/dt + div (vC) = 0? If so,

2) If the inlet boundary condition for C is the constant value of 1, we know that in the steady state whole domain will be filled with the uniform value of C=1. But, it contradicts with what we expect with the particle case. Because if we add particles to the inlet consistently, because of the boundary layer, we expect a non-uniform value for the volume fraction in the steady state, because it is not possible to have a uniform distribution of particles in the domain in the steady state.

Thank you in advance for sharing your ideas.

[LuckyTran]

1) Yes


2) Recall conservation laws. If you add C=1 at the inlet, and there are no sources/sinks, then C=1 everywhere.

[Shabi]

Originally Posted by LuckyTran

1) Yes


2) Recall conservation laws. If you add C=1 at the inlet, and there are no sources/sinks, then C=1 everywhere.

Thanks. But how the particles can go to the area in the domain where the streamline is nearly zero? assume we have a solid circle in the middle of the domain and we have a moderate Reynolds number (no vortex behind the circle). So how the particles can fill the area behind the circle?

[agd]

If you have a boundary layer in the channel, then wouldn't there be gradients of the velocity near the walls? and wouldn't these gradients affect the distribution of C, so that your distribution is not uniform in the steady state?

[q__]

Originally Posted by Shabi

2) If the inlet boundary condition for C is the constant value of 1, we know that in the steady state whole domain will be filled with the uniform value of C=1. But, it contradicts with what we expect with the particle case. Because if we add particles to the inlet consistently, because of the boundary layer, we expect a non-uniform value for the volume fraction in the steady state, because it is not possible to have a uniform distribution of particles in the domain in the steady state.

Thank you in advance for sharing your ideas.

How do you imagine particles while injecting just a single phase defining c=1 at the inlet? There is no distribution of c since there is only one phase.

[FMDenaro]

Originally Posted by q__

How do you imagine particles while injecting just a single phase defining c=1 at the inlet? There is no distribution of c since there is only one phase.

You are wrong, the particle injection obeys to the lagrangian equation Dc/Dt=0 in a single phase, that is each particle follows a trajectori.
In a divergence free velocity field, the eulerian equation counterpart is dc/dt+ div(vc)=0, if you integrate over the domain and apply Gauss, you see at the steady state that the inlet of c is balanced by the outlet of c.
At the steady state (if exists), the trajectory and the streamlines are the same and c is constant along the streamline in the BL.

[q__]

Originally Posted by FMDenaro

You are wrong, the particle injection obeys to the lagrangian equation Dc/Dt=0 in a single phase, that is each particle follows a trajectori.
In a divergence free velocity field, the eulerian equation counterpart is dc/dt+ div(vc)=0, if you integrate over the domain and apply Gauss, you see at the steady state that the inlet of c is balanced by the outlet of c.
At the steady state (if exists), the trajectory and the streamlines are the same and c is constant along the streamline in the BL.

I did not find the claim that he injects particles. Just assumes c=1 on the inlet.

[FMDenaro]

Originally Posted by q__

I did not find the claim that he injects particles. Just assumes c=1 on the inlet.

See the title of the post

[q__]

Originally Posted by FMDenaro

See the title of the post

Yes, I saw it. As can be seen I explicitly referred my reply to that statement:


"2) If the inlet boundary condition for C is the constant value of 1, we know that in the steady state whole domain will be filled with the uniform value of C=1. But it contradicts with what we expect with the particle case... "


Of course, it contradicts. So where am I wrong?

[FMDenaro]

Originally Posted by q__

Yes, I saw it. As can be seen I explicitly referred my reply to that statement:


"2) If the inlet boundary condition for C is the constant value of 1, we know that in the steady state whole domain will be filled with the uniform value of C=1. But it contradicts with what we expect with the particle case... "


Of course, it contradicts. So where am I wrong?

No contradiction... consider for example the analytic solution for the Poiseulle case.
Immagine a uniform distribution of particles at the inlet and let them evolve until the particles reach the outlet. The particles close to a wall will have a slower velocity but they will finally reach the outlet so that all the domain is full of particles. The solution is steady and uniform. Of course, this model does not take into account a finite size of the particles.
Now consider the Eulerian counterpart dc/dt+div (vc)=0. The uniform distribution of particles can be obtained by setting the uniform value c=1 at the inlet. Again if you wait until the transient is ended the final solution will show c=1 everywhere. There is no diffusion and no compressibility effects to change this value.

[FMDenaro]

Many many years ago I worked on this problem


https://www.researchgate.net/publica...mental_results

[Shabi]

Originally Posted by FMDenaro

No contradiction... consider for example the analytic solution for the Poiseulle case.
Immagine a uniform distribution of particles at the inlet and let them evolve until the particles reach the outlet. The particles close to a wall will have a slower velocity but they will finally reach the outlet so that all the domain is full of particles. The solution is steady and uniform. Of course, this model does not take into account a finite size of the particles.
Now consider the Eulerian counterpart dc/dt+div (vc)=0. The uniform distribution of particles can be obtained by setting the uniform value c=1 at the inlet. Again if you wait until the transient is ended the final solution will show c=1 everywhere. There is no diffusion and no compressibility effects to change this value.

I can imagine that the whole domain is filled with particles in the steady state. But is the volume fraction of particles near the wall is same as the volume fraction in the middle of the domain?
Or lets formulate like this. Assume that we don't have a constant inlet velocity (we have e.g. a parabolic one) and then if we inject the particles uniformly, we can clearly see the difference between the volume fraction of the particles near the inlet. However, if we solve the advection equation, we see the uniform constant value of C in the steady state in the whole domain...

[FMDenaro]

Originally Posted by Shabi

I can imagine that the whole domain is filled with particles in the steady state. But is the volume fraction of particles near the wall is same as the volume fraction in the middle of the domain?
Or lets formulate like this. Assume that we don't have a constant inlet velocity (we have e.g. a parabolic one) and then if we inject the particles uniformly, we can clearly see the difference between the volume fraction of the particles near the inlet. However, if we solve the advection equation, we see the uniform constant value of C in the steady state in the whole domain...

What do you mean exactly for volume fraction? Using the passive lagrangian transport you are assuming that the particles are masseless and have no dimension.

[Shabi]

Originally Posted by FMDenaro

What do you mean exactly for volume fraction? Using the passive lagrangian transport you are assuming that the particles are masseless and have no dimension.

I mean the number of particles in an specific volume size. For example, by a finite volume mesh descritization, if all mesh cells have the same volume size, the number of particles in the cells near the inlet located in the height of y=0.1h is different than the number of particles located in the height of y=0.5h (assuming a parabolic inlet velocity profile) .
But if we solve the advection equation exactly with the same configuration, we will get the same value of "C" in the whole domain.

[FMDenaro]

Originally Posted by Shabi

I mean the number of particles in an specific volume size. For example, by a finite volume mesh descritization, if all mesh cells have the same volume size, the number of particles in the cells near the inlet located in the height of y=0.1h is different than the number of particles located in the height of y=0.5h (assuming a parabolic inlet velocity profile) .
But if we solve the advection equation exactly with the same configuration, we will get the same value of "C" in the whole domain.

But you are considering from a side a discrete particle representation and on the other a continuous representation of the convection discretized on a grid. What you see in the latter case is the same phenomenology, only differently represented.

[Shabi]

Originally Posted by FMDenaro

But you are considering from a side a discrete particle representation and on the other a continuous representation of the convection discretized on a grid. What you see in the latter case is the same phenomenology, only differently represented.

Thanks and sorry for lots of questions. I hope it is the last one:
What we know so far:
1) The number of particles in some area per unit volume is more than other area.
2) the scalar field 'C' is constant everywhere in the steady state.

Recalling the start of this topic, i wanted to understand the relation between the volume fraction of the particles and the scalar field 'C'.
I think if 1 and 2 is correct, the only explanation for this conflict is that the number of particles per unit volume does not represent the volume fraction. I hope i could clarify my question. you say that the convection discretized on a grid is the same phenomenology. But, the number of particles per unit volume is not constant, whereas 'C' is constant everywhere.

[Aeronautics El. K.]

Originally Posted by Shabi

Thanks and sorry for lots of questions. I hope it is the last one:
What we know so far:
1) The number of particles in some area per unit volume is more than other area.
2) the scalar field 'C' is constant everywhere in the steady state.

Recalling the start of this topic, i wanted to understand the relation between the volume fraction of the particles and the scalar field 'C'.
I think if 1 and 2 is correct, the only explanation for this conflict is that the number of particles per unit volume does not represent the volume fraction. I hope i could clarify my question. you say that the convection discretized on a grid is the same phenomenology. But, the number of particles per unit volume is not constant, whereas 'C' is constant everywhere.

Having read through the whole post here, I still don't understand if you're referring to single phase flow or multiphase flow.
I can't make sense of what you're asking.
Would you like to clarify this please?

[Shabi]

Originally Posted by Aeronautics El. K.

Having read through the whole post here, I still don't understand if you're referring to single phase flow or multiphase flow.
I can't make sense of what you're asking.
Would you like to clarify this please?

It is a single phase flow. I would like to resemble the continuous scalar field 'C' by 'very small particles'. We can imagine the particle as 'fluid parcels'.
What is clear is how a advection equation is solved. Assuming that the inlet boundary condition of C is 1, we know that in the steady state C is 1 in the whole domain. So, i would like to resemble this process with very small particles, i.e., transport very small particles and make them to follow the flow trajectories. Those particles are not going to have any influence on the flow. They only follow the flow trajectories. Similar to what the continuous scalar field 'C' does. if the number of my particles reaches infinity, i expect to converge to the continuous field 'C'.

[Aeronautics El. K.]

Originally Posted by Shabi

It is a single phase flow. I would like to resemble the continuous scalar field 'C' by 'very small particles'. We can imagine the particle as 'fluid parcels'.
What is clear is how a advection equation is solved. Assuming that the inlet boundary condition of C is 1, we know that in the steady state C is 1 in the whole domain. So, i would like to resemble this process with very small particles, i.e., transport very small particles and make them to follow the flow trajectories. Those particles are not going to have any influence on the flow. They only follow the flow trajectories. Similar to what the continuous scalar field 'C' does. if the number of my particles reaches infinity, i expect to converge to the continuous field 'C'.

Pardon my ignorance but that's essentially a Lattice-Boltzmann method, no?

[Shabi]

Originally Posted by Aeronautics El. K.

Pardon my ignorance but that's essentially a Lattice-Boltzmann method, no?

I am not sure, actually i am not doing any real simulation using any method. It is more related to understanding the basics. I saw the conflict here, and i cannot find the answer for that. We are in an incompressible flow. 'C' fills the domain gradually and in the steady state we get a uniform distribution of 'C'. And it is what we expect from the advection equation and the incompressibility. So, why can't we repeat this process with infinite number of tiny 'massless' particles?