# Selection Rules for Resonant Longitudinal Injector-Coupling in Experimental Rocket Combustors

## Abstract

**:**

## 1. Introduction

## 2. Acoustic Analysis of an Injector-Combustion-Chamber System

## 3. Derivation of Mode Selection Rules for Resonant Coupling

#### 3.1. Resonant Mode Coupling with an Open Injector

#### 3.2. Resonant Mode Coupling with a Closed Injector

## 4. Results

#### 4.1. Open Injector Coupling during Ignition Testing of a Subscale LOx-Methane Rocket Thrust Chamber

#### 4.2. Closed Injector Coupling in Numerical Simulations of a Subscale H${}_{2}$/O${}_{2}$ Rocket Thrust Chamber

## 5. Discussion

## 6. Conclusions

## Funding

_{2}flames” The author also appreciates financial support from the German Aerospace Center (DLR) project AMADEUS (Advanced Methods for Reusable Aerospace Vehicle Design using Artificial Intelligence and Interdisciplinary Numerical Simulation) focusing on the development of numerical methods for LOx-methane-based engine concepts in future space transportation systems.

## Data Availability Statement

## Conflicts of Interest

## References

- Harrje, D.; Reardon, F. Liquid Propellant Rocket Combustion Instability; Report SP-194; National Aeronautics and Space Agency (NASA): Washington, DC, USA, 1972. [Google Scholar]
- Yang, V.; Anderson, W.E. Liquid Rocket Engine Combustion Instability; American Institute for Aeronautics and Aerospace (AIAA): Washington, DC, USA, 1995; Volume 169. [Google Scholar]
- O’Connor, J.; Acharya, V.; Lieuwen, T. Transverse combustion instabilities: Acoustic, fluid mechanic, and flame processes. Prog. Energy Combust. Sci.
**2015**, 49, 1–39. [Google Scholar] [CrossRef] [Green Version] - Conrad, E.; Parish, H.; Wanhainen, J. Effect of Propellant Injection Velocity on Screech in 20,000-Pound Hydrogen-Oxygen Rocket Engine; Report TN-D-3373; National Aeronautics and Space Agency (NASA): Washington, DC, USA, 1966. [Google Scholar]
- Armbruster, W.; Hardi, J.; Oschwald, M. Impact of shear-coaxial injector hydrodynamics on high-frequency combustion instabilities in a representative cryogenic rocket engine. Int. J. Spray Combust. Dyn.
**2022**, 14, 118–130. [Google Scholar] [CrossRef] - Gröning, S.; Hardi, J.S.; Suslov, D.; Oschwald, M. Injector-driven combustion instabilities in a hydrogen/oxygen rocket combustor. J. Propuls. Power
**2016**, 32, 560–573. [Google Scholar] [CrossRef] - Nunome, Y.; Onodera, T.; Sasaki, M.; Tomita, T.; Kobayashi, K.; Daimon, Y. Combustion instability phenomena observed during cryogenic hydrogen injection temperature ramping tests for single coaxial injector elements. In Proceedings of the 47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, San Diego, CA, USA, 31 July–3 August 2011; p. 6027. [Google Scholar]
- Kawashima, H.; Kobayashi, K.; Tomita, T.; Kaneko, T. A Combustion Instability Phenomenon on a LOX/Methane Subscale combustor. In Proceedings of the 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Atlanta, GA, USA, 30 July–1 August 2012. [Google Scholar] [CrossRef]
- Martin, J.; Armbruster, W.; Hardi, J. Flame-acoustic interaction in a high-pressure, single-injector LOX/H2 rocket combustor with optical access. In Proceedings of the Symposium on Thermoacoustics in Combustion: Industry meets Academia (SoTiC 2021), Garching, Germany, 6–10 September 2021. [Google Scholar]
- Gröning, S.; Hardi, J.; Suslov, D.; Oschwald, M. Influence of hydrogen temperature on the stability of a rocket engine combustor operated with hydrogen and oxygen. CEAS Space J.
**2017**, 9, 59–76. [Google Scholar] [CrossRef] - Yu, Y.; Koeglmeier, S.; Sisco, J.; Anderson, W. Combustion instability of gaseous fuels in a continuously variable resonance chamber (CVRC). In Proceedings of the 44th AIAA ASME SAE ASEE Joint Propulsion Conference & Exhibit, Hartford, CT, USA, 21–23 July 2008; p. 4657. [Google Scholar]
- Klein, S.; Börner, M.; Hardi, J.S.; Suslov, D.; Oschwald, M. Injector-coupled thermoacoustic instabilities in an experimental LOX-methane rocket combustor during start-up. CEAS Space J.
**2020**, 12, 267–279. [Google Scholar] [CrossRef] [Green Version] - Schuller, T.; Durox, D.; Palies, P.; Candel, S. Acoustic decoupling of longitudinal modes in generic combustion systems. Combust. Flame
**2012**, 159, 1921–1931. [Google Scholar] [CrossRef] - Candel, S. Combustion dynamics and control: Progress and challenges. Proc. Combust. Inst.
**2002**, 29, 1–28. [Google Scholar] [CrossRef] - Noiray, N.; Durox, D.; Schuller, T.; Candel, S. A unified framework for nonlinear combustion instability analysis based on the flame describing function. J. Fluid Mech.
**2008**, 615, 139–167. [Google Scholar] [CrossRef] - Marble, F.; Candel, S. Acoustic disturbance from gas non-uniformities convected through a nozzle. J. Sound Vib.
**1977**, 55, 225–243. [Google Scholar] [CrossRef] - Horchler, T.; Armbruster, W.; Hardi, J.; Karl, S.; Hanneman, K.; Gernoth, A.; De Rosa, M. Modeling Combustion Chamber acoustic using the DLR-TAU Code. In Proceedings of the Space Propulsion Conference, Cincinnati, OH, USA, 9–11 July 2018. [Google Scholar]
- Martin, J.; Armbruster, W.; Stützer, R.; General, S.; Knapp, B.; Suslov, D.; Hardi, J. Flame characteristics of a high-pressure LOX/H2 rocket combustor with large optical access. Case Stud. Therm. Eng.
**2021**, 28, 101546. [Google Scholar] [CrossRef] - Schwamborn, D.; Gerhold, T.; Heinrich, R. The DLR-TAU-Code: Recent Applications in Reasearch and Industry. In Proceedings of the European Conference on Computational Fluid Dynamics (ECCOMAS), Egmond aan Zee, The Netherlands, 5–8 September 2006. [Google Scholar]
- Jameson, A. Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. In Proceedings of the 10th Computational Fluid Dynamics Conference, Honolulu, HI, USA, 24–26 June 1991; p. 1596. [Google Scholar]
- Thornber, B.; Mosedale, A.; Drikakis, D.; Youngs, D.; Williams, R. An improved reconstruction method for compressible flows with low Mach number features. J. Comput. Phys.
**2008**, 227, 4873–4894. [Google Scholar] [CrossRef] - Rossow, C.C. Extension of a compressible code toward the incompressible limit. AIAA J.
**2003**, 41, 2379–2386. [Google Scholar] [CrossRef] - Rumsey, C. Turbulence Modeling Resource. Available online: https://turbmodels.larc.nasa.gov/ (accessed on 22 August 2022).
- Gaffney, J.R.; White, J.; Girimaji, S.; Drummond, J. Modeling turbulent/chemistry interactions using assumed PDF methods. In Proceedings of the 28th Joint Propulsion Conference and Exhibit, Nashville, TN, USA, 6–8 July 1992; p. 3638. [Google Scholar]
- Kim, S.K.; Choi, H.S.; Kim, Y. Thermodynamic modeling based on a generalized cubic equation of state for kerosene/LOx rocket combustion. Combust. Flame
**2012**, 159, 1351–1365. [Google Scholar] [CrossRef] - Horchler, T.; Fechter, S.; Hardi, J. Numerical Investigation of Flame-Acoustic Interaction at Resonant and Non-Resonant Conditions in a Model Combustion Chamber. In Proceedings of the Symposium on Thermoacoustics in Combustion: Industry meets Academia (SoTiC 2021), Garching, Germany, 6–10 September 2021. [Google Scholar]
- Jovanović, M.R.; Schmid, P.J.; Nichols, J.W. Sparsity-promoting dynamic mode decomposition. Phys. Fluids
**2014**, 26, 024103. [Google Scholar] [CrossRef] [Green Version] - Schmid, P.J. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech.
**2010**, 656, 5–28. [Google Scholar] [CrossRef] - Webster, S.; Hardi, J.; Oschwald, M. One-Dimensional Model Describing Eigenmode Frequency Shift During Transverse Excitation. In Progress in Propulsion Physics; EDP Sciences: Les Ulis Cedex A, France, 2019; pp. 273–294. [Google Scholar]
- Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover Books on Mathematics; Dover Publications Inc.: Mineola, NY, USA, 1965. [Google Scholar]

**Figure 1.**Longitudinal acoustic eigenmodes of a resonator cavity setup. Unburnt gas (labeled by the subscript u) at temperature ${T}_{u}$, speed of sound ${c}_{u}$ and density ${\rho}_{u}$ is injected at the injector inlet location (0) and enters the combustion chamber at location (1). After combustion at the location of the flame, the burnt gas in the combustion chamber (subscript b) is described by the new thermodynamic state ${T}_{b}$, ${c}_{b}$ and ${\rho}_{b}$.

**Figure 2.**Value of the dispersion relation for an open injector configuration. The length and speed of sound of the injector and chamber have been adjusted such that the first longitudinal eigenfrequencies are ${\omega}_{u}=0.87\phantom{\rule{0.166667em}{0ex}}$ s${}^{-1}$ and ${\omega}_{b}=1.0\phantom{\rule{0.166667em}{0ex}}$ s${}^{-1}$.

**Figure 5.**Normalized pressure mode shapes for frequencies approaching the resonance frequency $\omega ={\omega}_{res}+\delta \omega $. Note that the maximum normalized injector pressure remains essentially constant, whereas the maximum amplitude in the combustion chamber approaches zero ${\tilde{p}}_{c}\to 0$ for $\delta \omega \to 0$.

**Figure 7.**Injector and combustion chamber mode shape for the additional closed-closed injector mode coupling.

**Figure 8.**Combustion chamber modes for Case C from Klein et al. [12]. Licensed under CC BY 4.0 https://creativecommons.org/licenses/by/4.0.

**Figure 9.**LOx post modes for Case C from Klein et al. [12]. Licensed under CC BY 4.0 https://creativecommons.org/licenses/by/4.0.

**Figure 10.**3D computational mesh for the injection area of BKN. The central oxygen injector is surrounded by the coaxial hydrogen injector. The flame is anchored at the injector lip in the recessed region of the injector, which is attached to the combustion chamber face plate. The computational model also includes the window cooling manifold, which is connected to the annular window cooling system via a set of small tubes. The window cooling is necessary to protect the combustion chamber side windows from excessive heat loads during the experimental test run.

**Figure 14.**Power spectral density of the combustion chamber pressure signal (red) and the injector signal (blue). The power spectral densities are computed in the limit-cycle oscillation.

**Figure 15.**Normalized spatial pressure mode shapes $p\left(\mathbf{x}\right)/\mathrm{max}\left[p\right(\mathbf{x}\left)\right]$ of the first three cham- ber eigenmodes.

**Figure 16.**Normalized spatial pressure mode shapes $p\left(\mathbf{x}\right)/\mathrm{max}\left[p\right(\mathbf{x}\left)\right]$ of the three strongest injector eigenmodes.

Quantity | Value | Unit |
---|---|---|

Ratio oxidizer-over-fuel ROF | 4 | - |

Chamber pressure | 64.6 | bar |

O${}_{2}$ mass flow rate | 0.327 | kg/s |

O${}_{2}$ injection temperature | 112 | K |

H${}_{2}$ mass flow rate | 0.082 | kg/s |

H${}_{2}$ injection temperature | 168 | K |

H${}_{2}$ window film cooling | 0.199 | kg/s |

H${}_{2}$ film cooling temperature | 168 | K |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Horchler, T.
Selection Rules for Resonant Longitudinal Injector-Coupling in Experimental Rocket Combustors. *Aerospace* **2022**, *9*, 669.
https://doi.org/10.3390/aerospace9110669

**AMA Style**

Horchler T.
Selection Rules for Resonant Longitudinal Injector-Coupling in Experimental Rocket Combustors. *Aerospace*. 2022; 9(11):669.
https://doi.org/10.3390/aerospace9110669

**Chicago/Turabian Style**

Horchler, Tim.
2022. "Selection Rules for Resonant Longitudinal Injector-Coupling in Experimental Rocket Combustors" *Aerospace* 9, no. 11: 669.
https://doi.org/10.3390/aerospace9110669