# Numerical Study of Unstable Shock-Induced Combustion with Different Chemical Kinetics and Investigation of the Instability Using Modal Decomposition Technique

^{*}

^{†}

## Abstract

**:**

_{2}/O

_{2}reaction mechanisms is made for the numerical simulation of SIC with higher-order numerical schemes intended for the use of the code for the hypersonic propulsion and supersonic combustion applications. The simulations show that specific reaction mechanisms are grid-sensitive and produce spurious reactions in the high-temperature region, which trigger artificial instability in the oscillating flow field. The simulations also show that specific reaction mechanisms develop such spurious oscillations only at very fine grid resolutions. The instability mechanism is investigated using the dynamic mode decomposition (DMD) technique and the spatial structure of the decomposed modes are further analyzed. It is found that the instability triggered by the high-temperature reactions strengthens the reflecting compression wave and pushes the shock wave further and disrupts the regularly oscillating mechanism. The spatial coherent structure from the DMD analysis shows the effect of this instability in different regions in the regularly oscillating flow field.

## 1. Introduction

## 2. Numerical Modeling

#### 2.1. Governing Equations

#### 2.2. Numerical Methods

_{2}, O, O

_{2}, H

_{2}O, OH, HO

_{2}, and H

_{2}O

_{2}) along with an inert species N

_{2}, as the nitrogen oxidation is of less significance at such high temperatures. A structured grid system is used for this analysis in the curvilinear flow-field area around the projectile surface, as shown in Figure 2. For efficient computation in the multi-core share memory processors (SMP) machines, the code is parallelized by the OpenMP method.

#### 2.3. Numerical Setup

## 3. Numerical Simulation of Shock-Induced Combustion Using Different Chemical Kinetic Mechanisms

#### 3.1. Hydrogen–Air Combustion Mechanisms

#### 3.2. Comparison of Ignition Delay Time and Laminar Flame Speed

#### 3.3. Comparison of the Mechanisms for SIC Flow Field

## 4. DMD Analysis of the Shock-Induced Combustion Instability

#### 4.1. Description of Modal Decomposition Analysis

#### 4.2. Time-Sequencing of the Snapshots

#### 4.3. Ranking of the Modes

#### 4.4. Modal Decomposition of the Flow Field with Regular Oscillation

#### 4.5. Modal Decomposition of the Flow Field with Instability Phenomena

#### 4.6. Coherent Structure of the Experimental Modes

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Experimental shadowgraph of periodical oscillation observed taken at 12° angle from perpendicular direction for M = 4.48 in Lehr’s experiment [7].

**Figure 2.**Computational domain for the analysis of shock-induced combustion with N

_{x}× N

_{y}grid system.

**Figure 3.**Ignition delays calculated at 1 bar (symbol represents the experimental results from ReSpecTh repository).

**Figure 4.**Ignition delays calculated at 10 bar (symbol represents the experimental results from ReSpecTh repository).

**Figure 5.**Laminar burning velocity calculated for initial temperature 298 K and 1 bar (symbols represent the experimental results from ReSpecTh repository).

**Figure 7.**Temperature profile for various reaction mechanisms calculated from the CV explosion model.

**Figure 9.**Overview of the numerical results for regularly oscillating shock-induced combustion case flying at Mach 4.48. Color scale of x-t diagram is the same as the contour plot.

**Figure 10.**Pressure probed along the stagnation point for various grid resolutions (red—150 × 200; green—200 × 300; blue—300 × 450; and black—400 × 600).

**Figure 11.**Mach contour of the shock-induced combustion at Mach number 4.48 with various reaction mechanisms (300 × 450 grid system) compared with experimental shadowgraph image [7] (min 0, max 4.48, increment 0.04, colored with Mach number).

**Figure 12.**The x-t graph of SIC probed along the stagnation streamline for various reaction mechanism at 150 × 200 grid resolution. Color scale is the same as Figure 9.

**Figure 13.**The x-t graph of SIC probed along the stagnation streamline for various reaction mechanism at 400 × 600 grid resolution. Color scale is the same as Figure 9.

**Figure 14.**Streamline extraction of the temperature contour for the Jachimowski-92 mechanism at various grid levels. Color scale is the same as Figure 9.

**Figure 15.**Streamline extraction of the temperature contour for the Dryer mechanism at various grid levels. Color scale is the same as Figure 9.

**Figure 16.**Attenuation of the instability phenomena observed with the Jachimowski-88 mechanism at 400 × 600 grid system. Color scale is the same as Figure 9.

**Figure 17.**Temporal characteristics of decomposed modes using DMD with UCSD reaction mechanism for 200 × 300 grid resolution (red dot—growth factor; blue line—normalized coherence).

**Figure 18.**Energy distribution among the decomposed modes using POD with UCSD reaction mechanism for 200 × 300 grid resolution.

**Figure 20.**Temporal characteristics and spatial coherent structure of the flow field with DMD analysis for with UCSD reaction mechanism for 400 × 600 grid resolution (red dot—growth factor; blue line—normalized coherence). DMD color scale is the same as Figure 19.

**Figure 21.**Spatial coherent structure of the flow field with POD analysis for with UCSD reaction mechanism for 400 × 600 grid resolution (E represents normalized eigen value.). POD color scale is the same as Figure 19.

**Figure 22.**Temporal characteristics and spatial coherent structure of the flow field with DMD analysis for with the Dryer reaction mechanism for 400 × 600 grid resolution (red dot—growth factor; blue line—normalized coherence). DMD color scale is the same as Figure 19.

**Figure 23.**Spatial coherent structure of the flow field with POD analysis for with the Dryer reaction mechanism for 400 × 600 grid resolution (E represents normalized eigen value). POD color scale is the same as Figure 19.

**Figure 24.**Experimental DMD modes: (

**a**) Dryer mechanism at 408.3 kHz; and (

**b**) UCSD mechanism at 419.2 kHz.

Reaction Mechanism | Ignition Delay (μs) |
---|---|

Conaire | 2.537 |

Kéromnès | 2.983 |

Jachimowski 88 | 2.194 |

Jachimowski 92 | 2.466 |

GRI Mech 3.0 | 7.444 |

Dryer | 3.005 |

UCSD | 2.811 |

USC | 2.676 |

Reaction Mechanism | 150 × 200 | 200 × 300 | 300 × 450 | 400 × 600 |
---|---|---|---|---|

Jachimowski (1988) | 430.2 | 444.6 | 428.1 | 75.0/330.0 |

Jachimowski (1992) | 408.4/41.0 | 416.7 | 76.2/513 | 72.2/360.0 |

Dryer | 397.2 | 80.0/413.8 | 79.0/351 | 74.9/335.0 |

GRI Mech 3.0 | 226.0 | 230.35 | 220.5 | 208.5 |

UCSD | 416.5 | 431.3 | 415.4 | 409.4 |

USC | 416.7 | 427.0 | 411.1 | 398.0 |

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**MDPI and ACS Style**

Pavalavanni, P.K.; Jo, M.-S.; Kim, J.-E.; Choi, J.-Y.
Numerical Study of Unstable Shock-Induced Combustion with Different Chemical Kinetics and Investigation of the Instability Using Modal Decomposition Technique. *Aerospace* **2023**, *10*, 292.
https://doi.org/10.3390/aerospace10030292

**AMA Style**

Pavalavanni PK, Jo M-S, Kim J-E, Choi J-Y.
Numerical Study of Unstable Shock-Induced Combustion with Different Chemical Kinetics and Investigation of the Instability Using Modal Decomposition Technique. *Aerospace*. 2023; 10(3):292.
https://doi.org/10.3390/aerospace10030292

**Chicago/Turabian Style**

Pavalavanni, Pradeep Kumar, Min-Seon Jo, Jae-Eun Kim, and Jeong-Yeol Choi.
2023. "Numerical Study of Unstable Shock-Induced Combustion with Different Chemical Kinetics and Investigation of the Instability Using Modal Decomposition Technique" *Aerospace* 10, no. 3: 292.
https://doi.org/10.3390/aerospace10030292