Question

# How to scale the diffusivity coefficient for different length scales (m/µm)?

Dear all,
i'm concerned with fluid dynamic simulations on a µm-scale. Since the modelling software can't handle calculations on a µm-sized grid i used the idea of dynamic similarity to set up a m-sized model that behaves the same way as the µm-sized example. This works really well so far.
Now i want incorporate a diffusive scalar transport equation. My software therefore requires the input of a diffusivity coefficient D. I chose D = 2.1e-9, which equals to oxygen being dissolved in water. This, again, works fine for the simple µm-sized test model.
To get the same result in my m-sized model, i tried:
- keeping the same Schmidt number on both scales (as the idea of dynamic similarity is that on both scales the dimensionless numbers should be the same)
- having the same diffusion velocity (210 µm/s and 210 m/s)
- having the same diffusion velocity (210 µm/s on both models, as my m-sized model behaves like it is µm-sized)
- using the Stokes-Einstein equation
But the different approaches yield diffusion coefficients between 1.44e-15 to 2.1e3.
The question: If i want to use my m-sized model to simulate the same scalar transport as on the µm-sized scale, how do i scale my diffusivity coefficient?

12th Sep, 2017
Cees Haringa
Delft University of Technology
My advise would be to use dimensionless number similarity - so Schmidt, Peclet, Reynolds and anything else that is of relevance for your system. I am somewhat curious to why you can't directly solve on the micron scale - for dimensional solvers, those scales are not yet that exotic right?
1 Recommendation
12th Sep, 2017
Robert Büssing
MARUM Center for Marine Environmental Sciences
"I am somewhat curious to why you can't directly solve on the micron scale - for dimensional solvers, those scales are not yet that exotic right? "
Most certainly not, i assume. It seems to me that OpenFOAM struggles with the cell size at some point. Using a background mesh size of 0.5L (L being the length scale) and a snappyHexMesh refinement of up to level 5 seems to be to small when L already is 1e-6 m. But to be honest, after finding the convenient solution of using dynamic similarity, i didn't bother with trying the µm-scale again. Might give it a shot for the next batch of experiments, though.
12th Sep, 2017
Cees Haringa
Delft University of Technology
Ok, I have only done exploratory work with 2D microflows in FLUENT myself, and there I never had any issues. My MSc thesis supervisor some years ago did simulations of 2-phase microflow (100 um channel diameter typically) without issues in OpenFOAM. I don't know how he did the meshing there, I haven't worked with openFOAM unfortunately.

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