# Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries

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## Abstract

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## 1. Introduction

## 2. Characteristic-Based Volume Penalization

#### 2.1. Mathematical Formulation

#### 2.2. Volume Penalization of Euler Equations

#### 2.3. Volume Penalization of Navier–Stokes Equations

#### 2.4. Moving Obstacles

#### 2.5. Adaptive Wavelet Collocation Method

## 3. Numerical Results and Discussion

#### 3.1. Benchmark I: Normal Shock Wave Reflection

#### 3.2. Benchmark II: Oblique Shock Wave

#### 3.3. Benchmark III: Two-Dimensional Supersonic Flows around Blunt Bodies

#### 3.4. Benchmark IV: Galilean Invariance

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AMR | adaptive mesh refinement |

AWC | adaptive wavelet collocation |

BP | Brinkman penalization |

CBVP | characteristic-based volume penalization |

GI-CBVP | Galilean-invariant characteristic-based volume penalization |

IB | immersed boundary |

PDEs | partial differential equations |

VP | volume penalization |

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**Figure 1.**Illustration of different masking functions used for different penalization terms: ${\chi}_{\mathrm{h}}$ is the masking function for the hyperbolic thin layer, ${\chi}_{\mathrm{d}}$ is for the diffusive internal region, and ($1-\chi $) is for the external fluid region.

**Figure 3.**Pressure profiles along the shock tube for (

**a**) coarse ($N={2}^{9}$) and (

**b**) fine ($N={2}^{11}$) grid resolutions. The penalized solution for the reflected shock is compared with that obtained using exact algebraic boundary conditions. The theoretical shock location is also indicated.

**Figure 5.**Supersonic flow reflecting from a wedge with the deflection angle $\theta ={10}^{\circ}$, and the oblique shock angle of $\beta \approx {51}^{\circ}$ indicated by the yellow line: (

**a**) density field, (

**b**) adaptive grid colored by the pressure field.

**Figure 6.**Numerical schlieren image (

**a**) and the adaptive grid colored by the pressure field (

**b**) for supersonic flow past an array of cylinders.

**Figure 7.**Density and pressure field plots for fixed (upper half) and moving (bottom half) solid disks at (

**a**) the beginning and (

**b**) the end of simulations.

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**MDPI and ACS Style**

Kasimov, N.; Dymkoski, E.; De Stefano, G.; Vasilyev, O.V.
Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries. *Fluids* **2021**, *6*, 293.
https://doi.org/10.3390/fluids6080293

**AMA Style**

Kasimov N, Dymkoski E, De Stefano G, Vasilyev OV.
Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries. *Fluids*. 2021; 6(8):293.
https://doi.org/10.3390/fluids6080293

**Chicago/Turabian Style**

Kasimov, Nurlybek, Eric Dymkoski, Giuliano De Stefano, and Oleg V. Vasilyev.
2021. "Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries" *Fluids* 6, no. 8: 293.
https://doi.org/10.3390/fluids6080293