## Data-driven subgrid-scale modeling for convection-dominated concentration boundary layers

Andre Weiner, Dennis Hillenbrand, Holger Marschall, Dieter Bothe
Slides available at: andreweiner.github.io/reveal.js/ofw2019_sgs_modeling.html

## Outline

1. Mass transfer at rising bubbles
2. Subgrid-scale (SGS) modeling
3. Data-driven SGS modeling
4. Validation
5. Outlook
6. Summary

## Mass transfer at rising bubbles

water/air: $d_{eq}=3~mm$
water/air: $d_{eq}=5~mm$

## High fidelity data for closure models

Image source: appliedccm.com/portfolio-item/bubble

## High Péclet number problem

Image source: U. D. Kück et al.: Analyse des Grenzschichtnahen Stofftransports an frei aufsteigenden Gasblasen. CIT (2009), 1599-1606

## Specimen calculation

$d_b=1~mm$ water/oxygen at room temperature

• $Pe = Sc\ Re = U_b d_b/D = 10^4 ... 10^7$
• $$Re\approx 250;\quad \delta_h/d_b \propto Re^{-1/2};\quad\delta_h\approx 45~\mu m$$
• $$Sc\approx 500;\quad \delta_c/\delta_h \propto Sc^{-1/2};\quad\delta_c\approx 2.5~\mu m$$

$\delta_c/\delta_h$ typically 10 ... 100

## Subgrid-scale modeling

What happens if the mesh is not fine enough?

A. Weiner, D. Bothe (2017)

## Solution I

$$c(x,\delta) = c_\Sigma + (c_\infty - c_\Sigma) \mathrm{erf}(x/\delta)$$

## Solution II

$$\langle c \rangle_V \overset{!}{=} \frac{1}{V}\int_V \left[c_\Sigma + (c_\infty - c_\Sigma) \mathrm{erf}(x/\delta)\right] \mathrm{d}x$$

## Complex reactions?

$A+B\rightarrow P\quad A+P\rightarrow S$

## Data-driven SGS modeling

A. Weiner, D. Hillenbrand, H.Marschall, D. Bothe (2019)

## IBV problems

• Single phase incompressible Navier-Stokes, inletOutlet velocity, free slip at $\Sigma$, $\mathbf{u}(t=0)=\mathbf{0}$
• $$\partial_t c + \nabla \cdot (\mathbf{u}c-D\nabla c) = -kc$$ $$c_\Sigma (t) = 1,\quad c_\Omega(t=0) = 0$$

## Parameter variation

132 simultions, $70~GB$ raw data, $16~GB$ reduced

## MLP models

• three models (PyTorch), one model per label
• 353 parameters per model
• 30min training time on a GTX 960

## Model errors

Data compression: $16GB\rightarrow 3\times 353$ parameters

## Inference

• overhead: ~$0.2\%$ per time step
• no iterative inversion
• overhead should be even lower in 3D