Molecular vs Eddy Diffusion Let us assume that we have two boxes that are connected by a thin permeable barrier:


*

*If both boxes are non-convective and all motions are due to the
brownian motion of atoms, then I assume that you would use the
diffusive constant of molecular diffusion (D_mol). This constant has a simple analytic approximation.

*If both boxes are convecting then I believe that you would use the
diffusive constant of eddy diffusion (D_eddy). This constant does not have a simple analytic approximation and requires simulations.
So what diffusive constant would you use if one box has convective motions and the other one only has brownian motion? We will assume that mass is transferred from the convecting box to the non-convecting box. If possible, I would really appreciate an in-depth explanation so I could understand why.
Cheers,
-D
 A: The answer to your question depends on how you intend to design the interface between the two fluids. I would like to clarify a few things so that you will be able to give the answer to your question by yourself.

Molecular diffusion: Fluids at rest and adequately resolved moving fluids
Molecular diffusion occurs due the thermal motion of particles. It is proportional to the difference in concentration and is generally assumed isotropic (any source of anisotropy can be attributed to macroscopic and not microscopic mechanisms) and often constant. If you consider laminar flows (low Reynolds number $Re = \frac{U L}{\nu}$, the precise threshold really depends on the type of flow but generally something like $Re \leq \mathcal{O}(10^3)$ ) then the corresponding molecular diffusion coefficient will be sufficient to describe the mass transfer in a moving fluid. You will have a directed convective macroscopic motion due to the macroscopic motion and a (generally less important) rate of isotropic molecular diffusion. The order of magnitude of the molecular diffusion coefficient will vary greatly but its rate to the molecular viscosity, the Schmidt number $Sc = \frac{\nu}{D}$ will be approximately unity for gases and around $1000$ for liquids.
As flow departs from laminar flow an increasing number of whirls (eddies) will start to distribute the material much faster than one would predict by pure molecular diffusion, turbulent eddy transport. If your numerical models (such as direct numerical simulation DNS) is able to resolve all relevant time and length scales of the flow you still do not have to bother, the only parameter you will have to set is molecular diffusion as well.
Eddy diffusion: Modelling of under-resolved mass-transfer
DNS is though limited to moderate Reynolds numbers around $Re \leq \mathcal{O}(10^4)$ and simple geometries and thus you will introduce models such as Large Eddy Models (LES) and Reynolds Averaged Navier Stokes (RANS) that filter/average the flow and try to introduce terms that account for turbulent motion, that mimic the effects of turbulent motion. I have written a short post on the drawbacks and benefits of the methods if you are interested. One of those terms that emerges in the advection-diffusion equation is the eddy diffusivity, that acts additionally to the molecular diffusion. So to say it is an artificial concept introduced to describe the effects of the fine-scale isotropic chaotic turbulent motion - it simply accounts for unresolved effects of macroscopic isotropic turbulence that distribute material. This again is generally an isotropic constant that has to be set before starting a numerical simulation: Similarly one may introduce an additional turbulent Schmidt number $Sc_t = \frac{\nu_t}{D_t}$ that for any fluid is of order $\mathcal{O}(1)$ and use it for determining the additional turbulent diffusion. Thus, for high Reynolds numbers the effects of turbulent diffusion are likely to clearly dominate over molecular diffusion in areas of convective transport and you might even be able to neglect molecular diffusion. In areas of dead waters where the velocity is close to zero, molecular diffusion is still the main mechanism of mass transport. In case you are interested in how precisely the eddy diffusivity emerges in RANS models you could have a look at "Turbulent diffusion" by P.J.W. Roberts and D.R. Webster.
