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December 26, 2020, 12:07 
Particles following the flow trajectory

#1 
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Shabi
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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 nonuniform 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. 

December 26, 2020, 12:54 

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

January 4, 2021, 08:08 

#3 
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Shabi
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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?


January 4, 2021, 11:26 

#4 
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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?


January 4, 2021, 19:31 

#5  
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q
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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. 

January 5, 2021, 05:29 

#6  
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Filippo Maria Denaro
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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. 

January 5, 2021, 05:45 

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

January 5, 2021, 05:47 

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January 5, 2021, 05:50 

#9 
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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? 

January 5, 2021, 06:15 

#10  
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Filippo Maria Denaro
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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. 

January 5, 2021, 06:26 

#11 
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January 21, 2021, 18:57 

#12  
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Shabi
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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... 

January 22, 2021, 05:53 

#13  
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January 22, 2021, 07:41 

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But if we solve the advection equation exactly with the same configuration, we will get the same value of "C" in the whole domain. 

January 22, 2021, 12:24 

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Filippo Maria Denaro
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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. 

January 23, 2021, 18:21 

#16  
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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. Last edited by Shabi; January 24, 2021 at 07:18. 

January 24, 2021, 08:32 

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I can't make sense of what you're asking. Would you like to clarify this please?
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January 24, 2021, 09:19 

#18  
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Shabi
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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'. 

January 24, 2021, 09:28 

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January 24, 2021, 09:34 

#20 
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Shabi
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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?


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advection equation, particle concentration, transport equation, volume fraction. 
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