# Linearized Euler Equations for the Determination of Scattering Matrices for Orifice and Perforated Plate Configurations in the High Mach Number Regime

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Test Case Description

## 3. Numerical Procedure

#### 3.1. Mean Flow Computation

#### 3.2. Stabilized Linearized Euler Equations in Frequency Space

#### 3.3. Numerical Setup for Orifice and Perforated Plate Configurations

#### 3.4. Wave Extraction and Evaluation of Scattering Matrix

## 4. Scattering Matrices for Orifice and Perforated Plates

#### 4.1. Acoustic Fields for Perforated Plate and Orifice

#### 4.2. Scattering Matrices and Comparison to Experimental Data

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

LEE | Linearized Euler Equations |

CFD | Computational Fluid Dynamic |

CAA | Computational Aeroacoustic |

CFL | Courant–Friedrichs–Lewy number |

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**Figure 1.**Experimental setup of [33] for the orifice (

**top**) and perforated plate (

**bottom**) and relevant geometrical dimensions in ($\mathrm{mm}$).

**Figure 2.**Spatial Mach number distribution of the perforated plate (

**top**) and the orifice (

**bottom**) for five static pressure ratios ${p}_{r}$.

**Figure 3.**Linearized Euler equations (LEE) setup for the orifice case (perforated plate analogous): boundary conditions, wave extraction zones upstream and downstream and phase reference ${x}_{0}$.

**Figure 4.**Acoustic pressure fields for the real part Re$\left(\widehat{p}\right)$ (

**top**), amplitude |$\widehat{p}$| (

**middle**) and phase $\varphi \left(\widehat{p}\right)$ (

**bottom**) for the orifice case with a pressure ratio of ${p}_{r}=0.97$ and isolines of the axial mean velocity. The values of Re$\left(\widehat{p}\right)$ and |$\widehat{p}$| are normalized by their maximum values. The excitation of $1500\mathrm{Hz}$ is located downstream.

**Figure 7.**Scattering matrices of orifice (

**top**) and perforated plate (

**bottom**) plotted over the pressure ratio ${p}_{r}$.

**Table 1.**Investigated operation conditions. Ma corresponds to the Mach number within the perforated plate/orifice.

Static Pressure Ratio ${\mathit{p}}_{\mathit{r}}$ | Static Pressure Downstream ${\mathit{p}}_{\mathit{d}}$ (Bar) | Mass Flow $\dot{\mathit{m}}$ (g/s) | Ma |
---|---|---|---|

1.0 | 0.937 | 0.0 | 0 |

0.97 | 0.937 | 83.8 | 0.2 |

0.89 | 0.937 | 160.8 | 0.4 |

0.78 | 0.937 | 234.7 | 0.6 |

0.64 | 0.938 | 330.5 | 0.8 |

0.51 | 0.944 | 455.4 | 1.0 |

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**MDPI and ACS Style**

Schulze, M.; Wagner, M.; Sattelmayer, T.
Linearized Euler Equations for the Determination of Scattering Matrices for Orifice and Perforated Plate Configurations in the High Mach Number Regime. *Aerospace* **2016**, *3*, 33.
https://doi.org/10.3390/aerospace3040033

**AMA Style**

Schulze M, Wagner M, Sattelmayer T.
Linearized Euler Equations for the Determination of Scattering Matrices for Orifice and Perforated Plate Configurations in the High Mach Number Regime. *Aerospace*. 2016; 3(4):33.
https://doi.org/10.3390/aerospace3040033

**Chicago/Turabian Style**

Schulze, Moritz, Michael Wagner, and Thomas Sattelmayer.
2016. "Linearized Euler Equations for the Determination of Scattering Matrices for Orifice and Perforated Plate Configurations in the High Mach Number Regime" *Aerospace* 3, no. 4: 33.
https://doi.org/10.3390/aerospace3040033