Mass transfer at rising bubbles

High fidelity data for closure models

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

High Péclet number problem

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?

Solution I

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

Solution II

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

Workflow

Surfactant influence

C. Pesci, A. Weiner, H. Marschall, D.Bothe (2018)

Surfactant + mass transfer

A. Weiner et al. (2019)

Complex reactions?

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

Data-driven SGS modeling

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

Data generation

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

Feature engineering

Sequential backward selection

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

Validation

Inference

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

Local Sherwood number

Global Sherwood number

Outlook

Summary