Andre Weiner,
TU Braunschweig, Institute of Fluid
Mechanics

These slides and most of the linked resources are licensed under a

Creative Commons Attribution 4.0 International License.

- Data-driven modeling overview
- Modeling of reactive mass transfer
- Outlook
- Extracting coherent flow structures
- Closure modeling as a control problem

why and how

CFD

- produces large amounts of complex data
- requires data or representations thereof

ML

- finds patterns in data
- creates useful representations of data

primary data: scalar/vector fields, boundary fields, integral values

`# log.rhoPimpleFoam Courant Number mean: 0.020065182 max: 0.77497916 deltaT = 6.4813615e-07 Time = 1.22219e-06 PIMPLE: iteration 1 diagonal: Solving for rho, Initial residual = 0, Final residual = 0, No Iterations 0 DILUPBiCGStab: Solving for Ux, Initial residual = 0.0034181127, Final residual = 6.0056507e-05, No Iterations 1 DILUPBiCGStab: Solving for Uy, Initial residual = 0.0052004883, Final residual = 0.00012352706, No Iterations 1 DILUPBiCGStab: Solving for e, Initial residual = 0.06200185, Final residual = 0.0014223046, No Iterations 1 limitTemperature limitT Lower limited 0 (0%) of cells limitTemperature limitT Upper limited 0 (0%) of cells limitTemperature limitT Unlimited Tmax 329.54945 Unlimited Tmin 280.90821`

`Checking geometry... ... Mesh has 2 solution (non-empty) directions (1 1 0) All edges aligned with or perpendicular to non-empty directions. Boundary openness (1.4469362e-19 3.3639901e-21 -2.058499e-13) OK. Max cell openness = 2.4668495e-16 OK. Max aspect ratio = 3.0216602 OK. Minimum face area = 7.0705331e-08. Maximum face area = 0.00033983685. Face area magnitudes OK. Min volume = 1.2975842e-10. Max volume = 6.2366859e-07. Total volume = 0.0017254212. Cell volumes OK. Mesh non-orthogonality Max: 60.489216 average: 4.0292071 Non-orthogonality check OK. Face pyramids OK. Max skewness = 1.1453509 OK. Coupled point location match (average 0) OK.`

secondary data: log files, input dictionaries, mesh quality metrics, ...

Example: creating a **surrogate** or **reduced-order** model based on numerical
data.

Example: creating a space and time dependent **boundary** condition based on numerical
or experimental data.

Example: creating **closure models** based on numerical data.

Example: **active flow control** or **shape optimization**.

Creating a mapping from **features** to **labels** based on examples.

Finding patterns in **unlabeled** data.

Create an intelligent agent that learns to map **states** to **actions** such that cumulative rewards are maximized.

$\rightarrow$ mathematical, physical, numerical modeling

A single-phase simulation approach to compute the mass transfer at rising bubbles

https://github.com/AndreWeiner/sgs_model_test_transient

Joint work with D. Bothe, CMY. Claassen, IR. Hierck, JAM. Kuipers, MW. Baltussen.

**micro reactor**

size: millimeter

source: SPP 1740

**prediction** of

- mass transfer
- enhancement
- mixing
- conversion
- selectivity
- yield
- ...

**bubble column reactor**

size: meter

source: R. M. Raimundo, ENI

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

- $Pe = Sc\ Re = \nu_l / D_{O_2} \cdot U_b d_b/\nu_l \approx 10^5 $
- $$ Re\approx 250;\quad \delta_h/d_b \propto Re^{-1/2};\quad\delta_h\approx 45~\mu m $$
- $$ Sc\approx 500;\quad \delta_c/\delta_h \propto Sc^{-1/2};\quad\delta_c\approx 2.5~\mu m $$

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

feasible simulations up to $Pe\approx 10000$ (3D, HPC)

Why we might be interested in a simplified simulation approach:

- perform parameter studies
- validate boundary layer models
- generate data for ML-based models
- ...

1. Two-phase flow simulation (Volume-of-Fluid)

2. Parametrization of shape and interfacial velocity

3. Geometry generation and export (STL format)

4. Single phase mesh

5. Flow solution

6. Species transport

Comparison of two-phase and single-phase flow fields.

Influence of bubble size on local selectivity.

Two-phase velocity field (left half) versus single-phase velocity field (right half); speed-up of **20-40x** with **120x** finer mesh at surface.

Comparison of concentration fields: data-driven vs. free-slip.

Local Sherwood number for selected time instances.

Inlet boundary condition - first model:

$$ \mathbf{u}_{in} = -\tilde{U}_b\mathbf{e}_y $$

$\tilde{U}_b$ - the bubble's terminal velocity

What would be a more suitable Ansatz for $\tilde{U}_b$?

- $\tilde{U}_b = f_\theta(\tilde{t})$
- $\tilde{U}_b = f_\theta(\tilde{t})\tilde{t}$

$\tilde{t}$ - dimensionless time; $f_\theta(\tilde{t})$ - neural network model.

Interface velocity boundary condition - second model:

$$ \mathbf{u}_s = \tilde{u}_t\mathbf{t} $$

$$ \tilde{u}_t = \tilde{u}_t(\vartheta, \tilde{t}) $$

$\tilde{u}_t$ - tangential component of interface velocity vector.

Sine and cosine functions with vertical lines at $0$ and $\pi$.

How can we enforce the correct symmetry in the model?

- $\tilde{u}_t = f_\theta (\vartheta, \tilde{t})\mathrm{sin}(\vartheta)$
- $\tilde{u}_t = f_\theta (\vartheta, \tilde{t})\mathrm{cos}(\vartheta)$
- $\tilde{u}_t = f_\theta (\mathrm{sin}(\vartheta), \tilde{t})\mathrm{sin}(\vartheta)$
- $\tilde{u}_t = f_\theta (\mathrm{cos}(\vartheta), \tilde{t})\mathrm{sin}(\vartheta)$

Comparison of model prediction and reference data.

Interface deformation - third model:

$$ \Delta \mathbf{x} = (\tilde{r} - r_0)\mathbf{e}_r $$

$$ \tilde{r} = \tilde{r}(\vartheta, \tilde{t}) = f_\theta (\mathrm{cos}(\vartheta), \tilde{t}) $$

Comparison of model prediction and reference data for the bubble radius.

Mesh motion and zoom view of concentration boundary layer for $Re=569$ and $Sc=100$.

Global Sherwood number $Sh$ for two different mesh resolutions (3250 and 6500 cells/diameter). ~7h, serial, 2.4 GHz.

What about unsupervised and reinforcement learning?

Modal decomp. = spatial structures + temp. behavior

spatial mode

temporal coefficient

Flow past a surface mounted cube at $Re=40000$.

Forces acting in the cube expressed as force coefficients.

Power spectral density (PSD) of force coefficients.

Spectrum of dynamic mode decomposition (DMD).

Reconstruction of main vortex shedding mode.

Reconstruction of high-frequency vortex shedding mode.

Closed-loop active flow control; variable inlet velocity/Reynolds number $Re(t) = 250 + 150\mathrm{sin}(\pi t)$; video by Fabian Gabriel.

LES with Smagorinsky model:

$$ \nu_t = (C_s \Delta)^2 \sqrt{2\tilde{S}_{ij}\tilde{S}_{ij}} $$

Kurz et al. (2022); $C_s$ - Smagorinsky constant; $\tilde{\mathbf{S}}$ - filtered strain rate tensor.

Energy spectra over wave number; DNS, DRL, SSM; Kurz et al. (2022).

Where to go from here:

- lecture resources: ML in CFD
- OpenFOAM SIG data-driven modeling
- reinforcement learning: drlFoam
- flowTorch documentation