# Semi-analytical/empirical modelling of wall boundary conditions in advection-diffusion-reaction equation with distributed source

Let's suppose I need to numerically solve a 3D steady-state transport equation of the form $$\nabla \cdot (\mathbf{u} c) = \nabla \cdot (D \nabla c) - \lambda c + S$$ where $$c$$ is the transported quantity, $$\mathbf{u}$$ is a velocity field, $$D$$ is the diffusivity (optionally varying with position), $$\lambda$$ is a uniform linear decay rate and $$S$$ a spatially varying distributed source.
Let's also assume that the equation is to be solved in a pre-existent Finite Volume framework, coupled with the "simultaneous" solution of velocity and other quantities, from which $$S$$ is obtained.

The equation is subject, at specified wall boundaries, to a homogeneous mixed-type BC: $$- D \nabla c \cdot \mathbf{n} + \gamma c = 0$$ where $$\mathbf{n}$$ is the outward pointing wall-normal direction and $$\gamma$$ represent a "deposition velocity". The mixed-type BC is used to stay a little general, but indeed $$\gamma$$ should ideally tend to $$+\infty$$ to approximate a homogeneous Dirichlet BC.

Preliminary analysis has shown that the combination of the values of physical parameters for the problem at hand (e.g. low diffusivity, slow decay) together with large deposition velocity and a non-zero source at walls produces dramatically steep gradients in very thin wall layers. If my interest is in both grasping the 3D distribution of $$c$$ and the deposition fluxes at walls, this would require mesh refinement close to walls up to unaffordable levels.

To give an idea, it can be easily verified that the analytical solution of the corresponding problem in a fully-developed laminar flow between parallel plates, with uniform source $$S$$ and uniform properties, yields a solution (in the limit of infinite $$\gamma$$) which approaches a uniform value of $$S/\lambda$$ everywhere except for close to the walls, where it rapidly falls to zero according to an exponential law with a characteristic length of $$\sqrt{D/\lambda}$$.

My question then is: is there some semi-analytical and/or semi-empirical method to treat the wall boundary conditions in such a situation? Specifically, I'm looking to some way to model the wall gradient instead of resolving it in the wall layer.
I'm thinking of something similar to what is achieved with the Wall Function approach in standard turbulence modelling.
The setup with distributed source term is not very common and I'm struggling a little bit to find something suitable. Maybe I'm just using the wrong keywords.

Anyway, any comment/advice/suggestion/reference would be greatly appreaciated.

• "Boundary layer"? – A rural reader Mar 25 at 19:21
• @Aruralreader can you please elaborate a little more? I may not have mentioned it explicitly, but I know what a boundary layer is. The particular behaviour I described is not strictly related to the boundary layer (and by the way, do you refer to the velocity boundary layer, or what else?). It is a direct consequence of the combination of homogeneous dirichlet BC and distributed source which is not zero at the wall. – adironco Mar 26 at 14:34