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Transport a concentration field with convection equation 

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December 19, 2020, 14:07 
Transport a concentration field with convection equation

#1 
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Shabi
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Assume that we solve a simple convection equation for an incompressible flow, by which a concentration scalar field (between 0 and 1) is transported with the flow field.
The geometry is a simple 2d channel with the height of H connected to a bigger channel with Y=2H, and flow enters the domain from one side of the bigger channel and exits from the smaller channel. I set the inlet velocity boundary condition with a constant velocity and i set the inlet concentration boundary condition to 1. My question is about the conservation of the concentration field. Intuitively, when concentration field enters the smaller channel, its value should increase because of the incompressibility. But we know that the concentration field cannot exceed 1. Can anybody please clarify a bit about this confusion? Thanks. 

December 19, 2020, 14:28 

#2  
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Filippo Maria Denaro
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Quote:
Why should it increase? You are considering a scalar tracer that cannot diffuse and the velocity field is divergencefree. Thus, the equation is dC/dt + div (vC) = 0 > DC/Dt=0 Hence, the concentration C cannot exceed the initial values. Conservation is guarateed if your numerical method is conservative but you can have undershoot and overshoot using high order method without flux limiters. 

December 19, 2020, 18:39 

#3  
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Shabi
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Quote:
Another short question: If we impose the noslip boundary condition on the walls for the velocity, should we expect a boundary layer for concentration field, similar to the one we have for velocity? or the concentration field 'fills' the whole domain after reaching the steady situation? 

December 20, 2020, 05:30 

#4  
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Filippo Maria Denaro
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The advection equation says the answer, the concentration is transported along the trajectory. If the value of the concentration is uniform (for example 1) and starts from the steady inlet BC you will see a transient where the concentration profile is somehow similar to a BL but not in the values that remains uniform. After the transient is ended, the whole BL region will be filled by the uniform concentration. 

December 20, 2020, 16:37 

#5  
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Shabi
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Because if the concentration is gonna fill the whole domain with the value of 1 (where the velocity field is not uniform), I think we might get the above conclusion that the number of particles increase until infinity. 

December 20, 2020, 16:50 

#6  
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Filippo Maria Denaro
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Quote:
Last edited by FMDenaro; December 21, 2020 at 16:32. 

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concentration, convection, transport equation 
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