Question
Asked 12th Sep, 2017

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?

All Answers (3)

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
Thanks for your answer Cees. I'll try that approach.
"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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