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# Analyzing the issue of oscillating modes at low Mach numbers

Ibtissem Lannabi, Jonathan Jung, Vincent Perrier

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Ibtissem Lannabi, Jonathan Jung, Vincent Perrier. Analyzing the issue of oscillating modes at low Mach numbers. CEA-SMAI/GAMNI 2024 - la mécanique des fluides numériques, Jan 2024, Paris, France. hal-04885344

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https://inria.hal.science/hal-04885344v1

Submitted on 14 Jan 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Analyzing the issue of oscillating modes at low Mach numbers

Jonathan Jung, Ibtissem Lannabi, Vincent Perrier

CNRS/Univ Pau and Pays Adour/E2S UPPA, Laboratoire de Mathématiques et de leurs Applications de Pau & Cagire team, Inria Bordeaux Sud-Ouest

## Interest

* Low Mach number flows $M \ll 1$.

* Roe solver.

## The low Mach number accuracy problem

$$ 
\left\{ 
\begin{array}{l}
\partial_t \tilde{\rho} + \nabla_{\tilde{x}} (\tilde{\rho} \tilde{u}) = 0, \\
\partial_t (\tilde{\rho} \tilde{u}) + \nabla_{\tilde{x}} (\tilde{\rho} \tilde{u} \otimes \tilde{u}) + \frac{1}{\gamma M^2} \nabla_{\tilde{x}} \tilde{p} = 0, \\
\tilde{p} = p(\rho_0 \tilde{\rho})/p_0.
\end{array}
\right. 
$$

<figure><img src="image_3.png" /><figcaption>Figure 1: Numerical solution computed using the Roe scheme (Euler system).</figcaption></figure>

## The link with the long time limit of a linear wave system

* Two-time scales: $t_0 = x_0/u_0$ & $t_{ac} = x_0/a_0$.

* Acoustic time: $\tau = \tilde{t}/M$.

* Asymptotic expansion of $\tilde{\varphi} \in \{\tilde{\rho}, \tilde{\rho}\tilde{u}\}$:
  $\tilde{\varphi}(\tilde{x}, \tilde{t}, M) = \sum_{n=0}^{N} M^n \tilde{\varphi}^{(n)}(\tilde{x}, \tilde{t}, \tau) + O(M^{n+1})$.

### Continuous level

$$ 
\left\{ 
\begin{array}{l}
\partial_{\tau} \tilde{p}^{(1)} + \gamma \tilde{\rho}^{(0)} (\tilde{a}^{(0)})^2 \nabla_{\tilde{x}} \cdot \tilde{u}^{(0)} = 0, \\
\partial_{\tau} \tilde{u}^{(0)} + \frac{1}{\gamma \tilde{\rho}^{(0)}} \nabla_{\tilde{x}} \tilde{p}^{(1)} = 0.
\end{array}
\right. 
$$

### Discrete level

* Roe scheme $M \to 0$ → Godunov scheme.

* The family of Roe-pressure-centered-type schemes $M \to 0$ → pressure-centered scheme.

## The problem of spurious modes at low Mach numbers

<figure><img src="image_4.png" /><figcaption>Figure 2: Numerical solution computed using a Roe-pressure-centered scheme (Euler system).</figcaption></figure>

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

## Aim & Strategy

* Study of the origin of the appearance of spurious modes.

* Focus on the two dimensional linear wave system.

## Key ingredient: A Hodge Helmholtz Decomposition

Continuous level

Discrete level (Tri)

$$ 
\left\{ 
\begin{array}{ll}
\bar{u} = \bar{u}^{\psi} + \nabla\varphi, & \Omega, \\
\nabla\varphi \cdot \mathbf{n} = \bar{u} \cdot \mathbf{n}, & \partial\Omega.
\end{array}
\right. 
\quad 
\left\{ 
\begin{array}{ll}
\bar{u}_h = \bar{u}_h^{\psi} + \nabla\varphi_h, & \Omega, \\
\nabla\varphi_h \cdot \mathbf{n} = \bar{u}_h \cdot \mathbf{n}, & \partial\Omega.
\end{array}
\right. 
$$

* $\bar{u} \in (L^2(\Omega))^2$

* $\varphi \in H^1(\Omega)$

* $\bar{u}^{\psi} \in (L^2(\Omega))^2$

* $\bar{u}_h$ piecewise constant

* $\varphi_h \in CR$

* $\bar{u}_h^{\psi} \in Z_h^{\text{div}}$

## Filtering of the long time limit

* Godunov [1]: $\bar{u}_h^{\infty} = \bar{u}_h^{0,\psi}$, with $\bar{u}_h^0 = \bar{u}_h^{0,\varphi} + \bar{u}_h^{0,\psi}$.

* Pressure-centered [2]: $\bar{u}_h^{\infty, \text{Spurious}} = \bar{u}_h^{\infty} - \bar{u}_h^{0,\psi}$.

<figure><img src="image_6.png" /><figcaption>Figure 3: Numerical solution computed using Godunov/pressure-centered scheme (linear wave system).</figcaption></figure>

<figure><img src="image_7.png" /></figure>

## Structure of the kernel of the space-discretization matrix

$$ 
\left\{ 
\begin{array}{l}
\frac{d\bar{U}_h}{d\tau} + A_h^{\theta} \bar{U}_h - B_h = 0, \quad \bar{U}_h = (p_h, \bar{u}_h), \\
\bar{U}_h(\tau = 0) = \bar{U}_h^0.
\end{array}
\right. 
$$

$$ \ker A_h^{\theta} = \{0\} \times Z_h^{\theta}, \quad \theta \in \{\text{God, PC}\} $$

### Godunov

$$ Z_h^{\text{God}} = \left\{ \bar{u}_h \in \mathbb{R}^{2\#C} \mid [\bar{u}_h \cdot \mathbf{n}]_{S_{ij}} = 0, \forall S_{ij} \in S_{\text{int}}, \bar{u}_h \cdot \mathbf{n}_{|_{S_{ij}}} = 0, \forall S_{ij} \in S_{\text{b}} \right\} = Z_h^{\text{div}}. $$

### Pressure-centered

$$ Z_h^{\text{PC}} = \left\{ \bar{u}_h \in \mathbb{R}^{2\#C} \mid \sum_{S_{ij} \in \mathcal{V}_{\text{int}}(C_i)} |S_{ij}| [\bar{u}_h \cdot \mathbf{n}]_{S_{ij}} = 0, \forall C_i \in \mathcal{C} \right\} = Z_h^{\text{div}}. $$

## References

* J. Jung and V. Perrier, "Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systems," IMA Journal of Numerical Analysis, 2021.

* J. Jung, I. Lannabi, and V. Perrier, "On the convergence of the godunov scheme with a centered discretization of the pressure gradient," in International Conference on Finite Volumes for Complex Applications, pp. 201–208, Springer, 2023.

## Main results

* The kernel space $A_h^{\text{PC}}$ is larger than it should be

$$ \ker A_h^{\text{God}} \subsetneq \ker A_h^{\text{PC}}. $$

* Existence of a spurious mode in the numerical velocity field

$$ Z_h^{\text{PC}} = Z_h^{\text{div}} \oplus Z_h^{\text{Spurious}}. $$

* The space of spurious modes is defined as

$$ Z_h^{\text{Spurious}} := Z_h^{\text{PC}} \cap \nabla \mathbf{C} \mathbf{R}. $$

* The number of the oscillatory modes is huge
  $\dim Z_h^{\text{Spurious}} \geq \#\mathcal{N}_{\text{int}} + \#\text{holes} - 1$.

## Identification of the spurious modes

<figure><img src="image_8.png" /></figure>

## Perspectives

* Identify the remaining spurious modes: case of odd number of neighboring cells.

* Extend the study of analyzing the kernel space from triangular grids to quads.

## Acknowledgement

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

PAU BÉARN PYRÉNÉES
Communauté d'Agglomération

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skodowska-Curie grant agreement No 945416.