# Algebraic momentum preserving method with a fully-coupled 3D solver

Mohamed El Ouafa, Stéphane Vincent, Vincent Le Chenadec, Benoît Trouette

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Mohamed El Ouafa, Stéphane Vincent, Vincent Le Chenadec, Benoît Trouette. Algebraic momentum preserving method with a fully-coupled 3D solver. 4th International Conference on Numerical Methods in Multiphase Flows (ICNMMF), Sep 2022, Venise, Italy. hal-04079984

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Submitted on 24 Apr 2023

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<figure><img src="image_1.png" /></figure>

# Algebraic momentum preserving method with a fully-coupled 3D solver

S. El Ouafa, S. Vincent, V. Le Chenadec and B. Trouette

simon.elouafa@univ-eiffel.fr

<figure><img src="image_2.png" /></figure>

## PROBLEM

Investigate the accuracy and robustness of Fully-coupled 3D parallel solver for incompressible two phase flows at large density and viscosity ratios.

## NUMERICAL METHODS

* Finite-Volumes on staggered grids.

* Fully-coupled solver for Pressure / Velocity coupling.

* Penalty methods.

* Conservative VOF method.

* MPI based parallelizations.

<figure><img src="image_3.png" /></figure>

## MATHEMATICAL MODEL

$$ \frac{\partial \rho \mathbf{u}}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = -\nabla p + \nabla \cdot [\mu(\nabla \mathbf{u} + \nabla \mathbf{u}^T)] + \mathbf{f} \Rightarrow \textbf{Conservation of momentum} $$

$$ \nabla \cdot \mathbf{u} = 0 \Rightarrow \textbf{Mass conservation} $$

$$ \frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{u}C) = C\nabla \cdot \mathbf{u} \Rightarrow \textbf{Advection of volume fraction} $$

$$ \mu = \mu_1 C + (1 - C)\mu_2 \Rightarrow \textbf{Mixture viscosity} $$

$$ \frac{\partial \phi}{\partial t} + \nabla \cdot (\phi \mathbf{u}) = 0 \Rightarrow \textbf{Algebraic Momentum Preserving (AMP) } (\phi = \rho, \phi = \rho \mathbf{u})[1] $$

Fully coupled solver [2]

$$ \left( M_u^{(\rho)} + L_u^{(\mu)} \quad B_p^T \right) \left( \mathbf{u}^{n+1} \quad p^{n+1} \right) = \left( \begin{matrix} g \\ 0 \end{matrix} \right) \quad \text{or} \quad \mathbf{Ax} = \mathbf{b} $$

Transforming the original system $\mathbf{Ax} = \mathbf{b}$ into an equivalent one $\mathbf{P}^{-1} \mathbf{Ax} = \mathbf{P}^{-1} \mathbf{b}$.

$$ \mathbf{P} \approx \begin{pmatrix} \hat{\mathbf{F}}_{\mathbf{u}} & B_p^T \\ 0 & \hat{\mathbf{S}}_p \end{pmatrix} $$

$\hat{\mathbf{F}}_{\mathbf{u}}$ by Neumann series expansion and $\hat{\mathbf{S}}_p$ by the PCD operator [3]

<figure><img src="image_4.png" /></figure>

Eulerian Implicit (WY) in 2D

<figure><img src="image_5.png" /></figure>

Matrix Structure

## RESULTS

<figure><img src="image_6.png" /><figcaption>Dam-Break Flow with obstacle</figcaption></figure>

Changing the air/water interface for different times. (Mesh: $322 \times 100 \times 100$, CFL=0.4.)

<figure><img src="image_7.png" /><figcaption>Viscous jet buckling</figcaption></figure>

Propergol/air interface for different times. Mesh $120^3$ refined at the injector. The aspect ratios of H/D>3 $\pi$ and Re<0.52.

<figure><img src="image_8.png" /><figcaption>Phase inversion problem</figcaption></figure>

Oil/water interface over time. Mesh $256^3$. HPC DNS calculations $\approx 12800$ processors of the HPC Occigen cluster for CINES.

<figure><img src="image_9.png" /><figcaption>Drop impact on viscous liquid film</figcaption></figure>

Impact of water droplet on a liquid film. Crown formation and delayed splashing. (Mesh: $1024 \times 1024 \times 512$, CFL=0.4). HPC DNS calculations $\approx 8000$ processors of the HPC Jean Zay cluster of IDRIS.

## ACKNOWLEDGMENT

We are grateful for access to the computational facilities of the French CINES and TGCC granted by GENCI under project numbers A0092B06115. We thank the technical and administrative teams of these supercomputer centers and to S. Fereka (MSME).

## REFERENCES

[1] N. Nangia et al. A robust incompressible navier-stokes solver for high density ratio multi-phase flows. *Journal of Computational Physics* 2019.

[2] M. El Ouafa et al. Navier-stokes solvers for incompressible single- and two-phase flows. *Communications in Computational Physics*, 2021.

[3] N. Bootland et al. Preconditioners for Two-Phase Incompressible Navier-Stokes Flow. *SIAM Journal on Scientific Computing*, 2019.