Transport a concentration field with convection equation
 

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.

Why should it increase? You are considering a scalar tracer that cannot diffuse and the velocity field is divergence-free. 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.

Thank you. you are right. I confused it with the mass conservation.

Another short question: If we impose the no-slip 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?

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.

Thanks. I get confused when i replace the concentration by massless particles(who travel with streamlines). If we add some numbers of massless particles consistently to the inlet, do we ever get a steady solution, in terms of the number of particles in the domain? Or the number of particles in the domain increases until the infinity?

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.

If you put particles at the inlet they will follow the trajectories (streamlines only for steady flow) until the outlet. They cannot go to an infinite number