# Numerical Study of the Effects of Injection Conditions on Rotating Detonation Engine Propulsive Performance

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Physical Model

- (a)
- if ${p}_{\mathrm{w}}\ge {p}_{0}$, there is no injection, and the slip wall boundary will be implemented;
- (b)
- if ${p}_{\mathrm{cr}1}\le {p}_{\mathrm{w}}<{p}_{0}$, the inflow velocity is locally subsonic;
- (c)
- if ${p}_{\mathrm{cr}2}\le {p}_{\mathrm{w}}<{p}_{\mathrm{cr}1}$, the throat of the nozzle maintains choking conditions, and the injection is subsonic;
- (d)
- if ${p}_{\mathrm{w}}<{p}_{\mathrm{cr}2}$, the injection is supersonic and not affected by ${p}_{\mathrm{w}}$.

## 3. Extension of 3D CESE Method to Cylindrical Coordinate System

## 4. Results and Discussion

#### 4.1. Grid Sensitivity Analysis

#### 4.2. Effect of Channel Width with Full Injection

#### 4.3. Effect of Inner Radius with Fixed Injection Area

#### 4.4. Effect of Injector Location and Size

#### 4.5. Effect of Stagnation Pressure and Area Ratio of the Injector Nozzle

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. 3D CESE Method

#### Appendix A.1. Upwind CESE Method on 3D Meshes

**Figure A1.**(

**a**) Mesh arrangement in three-dimensional CESE method. A denotes the vertex; B and C denote the centers of line segments and surfaces, respectively; P is the staggered point that needs to be solved in the $1\mathrm{st}$ half-step. (

**b**) Definitions of CE. (

**c**) Definition of sub-CE, with one outer flux ${\mathrm{F}}_{\mathrm{R},2}$ and one inner flux ${\mathrm{F}}_{\mathrm{L},2}$.

#### Appendix A.2. Numerical Outline for 3D Upwind CESE Method

## Appendix B. Validation of the 3D CESE Method in Cylindrical Coordinates

#### Appendix B.1. Sedov Blast Wave Problem

**Figure A2.**The 3D Sedov blast problem in curvilinear coordinates. (

**a**) Density contour, (

**b**) density profile as a function of radius.

#### Appendix B.2. 1D Detonation Velocity

## References

- Peng, H.Y.; Liu, W.D.; Liu, S.J.; Zhang, H.L.; Jiang, L.X. Hydrogen-air, ethylene-air, and methane-air continuous rotating detonation in the hollow chamber. Energy
**2020**, 211, 118598. [Google Scholar] [CrossRef] - Lin, W.; Tong, Y.; Lin, Z.; Nie, W.; Su, L. Propagation mode analysis on H2–air rotating detonation waves in a hollow combustor. AIAA J.
**2020**, 58, 5052–5062. [Google Scholar] [CrossRef] - Zhang, P.; Meagher, P.A.; Zhao, X. Multiplicity for idealized rotational detonation waves. Phys. Fluids
**2021**, 33, 106102. [Google Scholar] [CrossRef] - Yi, T.H.; Lou, J.; Turangan, C.; Choi, J.Y.; Wolanski, P. Propulsive performance of a continuously rotating detonation engine. J. Propuls. Power
**2011**, 27, 171–181. [Google Scholar] [CrossRef] - Hishida, M.; Fujiwara, T.; Wolanski, P. Fundamentals of rotating detonations. Shock Waves
**2009**, 19, 1–10. [Google Scholar] [CrossRef] - Nejaamtheen, M.; Kim, T.; Pavalavanni, P.; Ryu, J.; Choi, J.Y. Effects of the dimensionless radius of an annulus on the detonation propagation characteristics in circular and non-circular rotating detonation engines. Shock Waves
**2021**, 31, 703–715. [Google Scholar] [CrossRef] - Lee, J.H.; Ryu, J.H.; Lee, E.S.; Han, H.S.; Choi, J.Y. Experimental Proof of Concept of a Noncircular Rotating Detonation Engine (RDE) for Propulsion Applications. Aerospace
**2022**, 10, 27. [Google Scholar] [CrossRef] - Schwer, D.; Kailasanath, K. Fluid dynamics of rotating detonation engines with hydrogen and hydrocarbon fuels. Proc. Combust. Inst.
**2013**, 34, 1991–1998. [Google Scholar] [CrossRef] - Tsuboi, N.; Watanabe, Y.; Kojima, T.; Hayashi, A.K. Numerical estimation of the thrust performance on a rotating detonation engine for a hydrogen–oxygen mixture. Proc. Combust. Inst.
**2015**, 35, 2005–2013. [Google Scholar] [CrossRef] - Zhao, M.; Zhang, H. Origin and chaotic propagation of multiple rotating detonation waves in hydrogen/air mixtures. Fuel
**2020**, 275, 117986. [Google Scholar] [CrossRef] - Hayashi, A.K.; Tsuboi, N.; Dzieminska, E. Numerical study on JP-10/air detonation and rotating detonation engine. AIAA J.
**2020**, 58, 5078–5094. [Google Scholar] [CrossRef] - Meng, Q.; Zhao, N.; Zhang, H. On the distributions of fuel droplets and in situ vapor in rotating detonation combustion with prevaporized n-heptane sprays. Phys. Fluids
**2021**, 33, 043307. [Google Scholar] [CrossRef] - Ren, Z.; Zheng, L. Numerical study on rotating detonation stability in two-phase kerosene-air mixture. Combust. Flame
**2021**, 231, 111484. [Google Scholar] [CrossRef] - Yao, S.; Tang, X.; Luan, M.; Wang, J. Numerical study of hollow rotating detonation engine with different fuel injection area ratios. Proc. Combust. Inst.
**2017**, 36, 2649–2655. [Google Scholar] [CrossRef] - Liu, X.Y.; Chen, Y.L.; Xia, Z.J.; Wang, J.P. Numerical study of the reverse-rotating waves in rotating detonation engine with a hollow combustor. Acta Astronaut.
**2020**, 170, 421–430. [Google Scholar] [CrossRef] - Tang, X.M.; Wang, J.P.; Shao, Y.T. Three-dimensional numerical investigations of the rotating detonation engine with a hollow combustor. Combust. Flame
**2015**, 162, 997–1008. [Google Scholar] [CrossRef] - Katta, V.R.; Cho, K.Y.; Hoke, J.L.; Codoni, J.R.; Schauer, F.R.; Roquemore, W.M. Effect of increasing channel width on the structure of rotating detonation wave. Proc. Combust. Inst.
**2019**, 37, 3575–3583. [Google Scholar] [CrossRef] - Kawasaki, A.; Inakawa, T.; Kasahara, J.; Goto, K.; Matsuoka, K.; Matsuo, A.; Funaki, I. Critical condition of inner cylinder radius for sustaining rotating detonation waves in rotating detonation engine thruster. Proc. Combust. Inst.
**2019**, 37, 3461–3469. [Google Scholar] [CrossRef] - Jiang, Y.; Wen, C.Y.; Zhang, D. Space–time conservation element and solution element method and its applications. AIAA J.
**2020**, 58, 5408–5430. [Google Scholar] [CrossRef] - Chang, S.C. The method of space-time conservation element and solution element—A new approach for solving the Navier-Stokes and Euler equations. J. Comput. Phys.
**1995**, 119, 295–324. [Google Scholar] [CrossRef] - Shen, H.; Wen, C.Y.; Zhang, D.L. A characteristic space–time conservation element and solution element method for conservation laws. J. Comput. Phys.
**2015**, 288, 101–118. [Google Scholar] [CrossRef] - Shen, H.; Wen, C.Y. A characteristic space–time conservation element and solution element method for conservation laws II. Multidimensional extension. J. Comput. Phys.
**2016**, 305, 775–792. [Google Scholar] [CrossRef] - Shen, H.; Parsani, M. Positivity-preserving CE/SE schemes for solving the compressible Euler and Navier–Stokes equations on hybrid unstructured meshes. Comput. Phys. Commun.
**2018**, 232, 165–176. [Google Scholar] [CrossRef] - Shi, L.; Uy, K.C.K.; Wen, C.Y. The re-initiation mechanism of detonation diffraction in a weakly unstable gaseous mixture. J. Fluid Mech.
**2020**, 895, A24. [Google Scholar] [CrossRef] - Shi, L.; Shen, H.; Zhang, P.; Zhang, D.; Wen, C. Assessment of vibrational non-equilibrium effect on detonation cell size. Combust. Sci. Technol.
**2017**, 189, 841–853. [Google Scholar] [CrossRef] - Uy, K.C.K.; Shi, L.; Wen, C.Y. Numerical analysis of the vibration-chemistry coupling effect on one-dimensional detonation stability. Aerosp. Sci. Technol.
**2020**, 107, 106327. [Google Scholar] [CrossRef] - Jiang, C.; Feng, X.; Zhang, J.; Zhong, D. AMR simulations of magnetohydrodynamic problems by the CESE method in curvilinear coordinates. Sol. Phys.
**2010**, 267, 463–491. [Google Scholar] [CrossRef] - Fan, E.; Guan, B.; Wen, C.Y.; Shen, H. Numerical study on the jet formation of simple-geometry heavy gas inhomogeneities. Phys. Fluids
**2019**, 31, 026103. [Google Scholar] [CrossRef] - Zhang, Z.; Wen, C.; Liu, Y.; Zhang, D.; Jiang, Z. Application of CE/SE method to gas-particle two-phase detonations under an Eulerian-Lagrangian framework. J. Comput. Phys.
**2019**, 394, 18–40. [Google Scholar] [CrossRef] - Jourdaine, N.; Tsuboi, N.; Ozawa, K.; Kojima, T.; Hayashi, A.K. Three-dimensional numerical thrust performance analysis of hydrogen fuel mixture rotating detonation engine with aerospike nozzle. Proc. Combust. Inst.
**2019**, 37, 3443–3451. [Google Scholar] [CrossRef] - Xia, Z.J.; Luan, M.Y.; Liu, X.Y.; Wang, J.P. Numerical simulation of wave mode transition in rotating detonation engine with OpenFOAM. Int. J. Hydrogen Energy
**2020**, 45, 19989–19995. [Google Scholar] [CrossRef] - Teng, H.; Zhou, L.; Yang, P.; Jiang, Z. Numerical investigation of wavelet features in rotating detonations with a two-step induction-reaction model. Int. J. Hydrogen Energy
**2020**, 45, 4991–5001. [Google Scholar] [CrossRef] - Ma, F.; Choi, J.Y.; Yang, V. Thrust chamber dynamics and propulsive performance of single-tube pulse detonation engines. J. Propuls. Power
**2005**, 21, 512–526. [Google Scholar] [CrossRef] - Harroun, A.J.; Heister, S.D.; Ruf, J.H. Computational and experimental study of nozzle performance for rotating detonation rocket engines. J. Propuls. Power
**2021**, 37, 660–673. [Google Scholar] [CrossRef] - Zhdan, S.A.; Bykovskii, F.A.; Vedernikov, E.F. Mathematical modeling of a rotating detonation wave in a hydrogen-oxygen mixture. Combust. Explos. Shock Waves
**2007**, 43, 449–459. [Google Scholar] [CrossRef] - Wen, C.Y.; Jiang, Y.; Shi, L. Space–Time Conservation Element and Solution Element Method: Advances and Applications in Engineering Sciences; Springer Nature: Singapore, 2023. [Google Scholar]
- Sow, A.; Chinnayya, A.; Hadjadj, A. On the viscous boundary layer of weakly unstable detonations in narrow channels. Comput. Fluids
**2019**, 179, 449–458. [Google Scholar] [CrossRef] - Lin, W.; Shi, Q.; Liu, S.; Lin, Z.; Tong, Y.; Su, L.; Nie, W. Study of thrust vector control for the rotating detonation model engine. Int. J. Hydrogen Energy
**2022**, 47, 1292–1305. [Google Scholar] [CrossRef] - Nakayama, H.; Moriya, T.; Kasahara, J.; Matsuo, A.; Sasamoto, Y.; Funaki, I. Stable detonation wave propagation in rectangular-cross-section curved channels. Combust. Flame
**2012**, 159, 859–869. [Google Scholar] [CrossRef] - Nakayama, H.; Kasahara, J.; Matsuo, A.; Funaki, I. Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channels. Proc. Combust. Inst.
**2013**, 34, 1939–1947. [Google Scholar] [CrossRef] - Zhang, Y.; Sheng, Z.; Rong, G.; Shen, D.; Wu, K.; Wang, J. Experimental research on the performance of hollow and annular rotating detonation engines with nozzles. Appl. Therm. Eng.
**2023**, 218, 119339. [Google Scholar] [CrossRef] - Houim, R.W.; Fievisohn, R.T. The influence of acoustic impedance on gaseous layered detonations bounded by an inert gas. Combust. Flame
**2017**, 179, 185–198. [Google Scholar] [CrossRef] - Wen, H.; Fan, W.; Wang, B. Theoretical analysis on the total pressure gain of rotating detonation systems. Combust. Flame
**2023**, 248, 112582. [Google Scholar] [CrossRef] - Schwer, D.; Kailasanath, K. Feedback into mixture plenums in rotating detonation engines. In Proceedings of the 50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Nashville, TN, USA, 9–12 January 2012; p. 617. [Google Scholar]
- She-Ming Lau-Chapdelaine, S.; Radulescu, M.I.; Hong, Z. Quasi-Two-Dimensional Simulation of a Rotating Detonation Engine Combustor and Injector. J. Propuls. Power
**2023**, 1–8. [Google Scholar] [CrossRef] - Baratta, A.R.; Stout, J. Demonstrated Low Pressure Loss Inlet and Low Equivalence Ratio Operation of a Rotating Detonation Engine (RDE) for Power Generation; Technical Report; Aerojet Rocketdyne: Sacramento, CA, USA, 2020. [Google Scholar]
- Paxson, D.E.; Schwer, D.A. Operational stability limits in rotating detonation engine numerical simulations. In Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 7–11 January 2019; p. 0748. [Google Scholar]
- Li, W.; Ren, Y.X.; Lei, G.; Luo, H. The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids. J. Comput. Phys.
**2011**, 230, 7775–7795. [Google Scholar] [CrossRef] - Zhang, X.; Shu, C.W. On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys.
**2010**, 229, 8918–8934. [Google Scholar] [CrossRef] - Sedov, L.I. Similarity and Dimensional Methods in Mechanics; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]

**Figure 1.**(

**a**) Schematic model of an RDE with an annulus combustion chamber, and injection patterns on the head-end wall, (

**b**) ideal, (

**c**) inner, (

**d**) outer, and (

**e**) middle injector slots.

**Figure 4.**Temperature contours of the RDE with different resolutions. (

**a**) $M1$, (

**b**) $M2$, and (

**c**) M3.

**Figure 5.**Temperature contours with channel width of (

**a**) $1\mathrm{mm}$, (

**b**) $3\mathrm{mm}$, (

**c**) $6\mathrm{mm}$, and (

**d**) 9 $\mathrm{mm}$.

**Figure 6.**(

**a**) Density contour and (

**b**) temperature contour for inner radius of $11\mathrm{mm}$ $(w=9\mathrm{mm})$.

**Figure 7.**(

**a**) Pressure record at the point near the inlet at the outer wall and (

**b**) specific impulse when varying the radius of the inner wall. The injection slot is $3\mathrm{mm}$ near the outer wall.

**Figure 8.**Pressure contour with density lines. (

**a**) Outer injection slot, (

**b**) enlarged contour of (

**a**), (

**c**) inner injection slot, and (

**d**) middle injection slot.

**Figure 9.**The distribution of total pressure along the axial direction. Typical pressure gain relative to the total pressure near the inlet is provided, together with that relative to the stagnation pressure in the reservoir in the brackets.

**Figure 10.**(

**a**) Averaged density profile along the radial–axial plane and (

**b**) averaged heat release rate for channel width of $9\mathrm{mm}$ with different injection patterns. The dashed lines indicate the top of the detonation wave.

**Figure 11.**Total pressure and injection Mach number distributions along the centerline. (

**a**) Full injection, (

**b**) middle injection with $\alpha $ = 0.5.

**Figure 12.**Specific impulses as a function of area ratio of the injector to the head-end wall with different injector locations. ${P}_{0}=10\mathrm{atm},{A}^{*}/A=0.2$. Error bars indicate the standard deviations.

**Figure 13.**Influence of stagnation pressure and area ratio of the injector nozzle on specific impulse. For all the cases, the injector slot is located near the outer wall. Error bars indicate the standard deviations.

w (mm) | p_{0} (atm) | ${\mathit{A}}^{*}/\mathit{A}$ | $\mathbf{\Delta}\mathit{r},{\mathit{R}}_{\mathit{o}}\mathbf{\Delta}\mathit{\theta},\mathbf{\Delta}\mathit{z}$ (mm) |
---|---|---|---|

1 | 10 | 0.2 | 0.5, 0.4, 0.4 |

1 | 10 | 0.2 | 0.2, 0.2, 0.2 |

1 | 10 | 0.2 | 0.1, 0.1, 0.1 |

w (mm) | ${\mathit{w}}_{\mathbf{inj}}$ (mm) | Inj Loc | $\mathit{\alpha}$ | ${\mathit{p}}_{0}$ (atm) | ${\mathit{A}}^{*}/\mathit{A}$ |
---|---|---|---|---|---|

1, 3, 6, 9 | same as w | Full | 1 | 10 | 0.2 |

6 | 3 | Outer | 0.5441 | 10 | 0.2 |

9 | 3 | Outer | 0.3978 | 10 | 0.2 |

w (mm) | $\mathit{\alpha}$ | Inj Loc | ${\mathit{p}}_{0}$ (atm) | ${\mathit{A}}^{*}/\mathit{A}$ |
---|---|---|---|---|

9 | 0.33∼0.75 | Outer | 10 | 0.2 |

9 | 0.33∼0.75 | Middle | 10 | 0.2 |

9 | 0.33∼0.75 | Inner | 10 | 0.2 |

w (mm) | $\mathit{\alpha}$ | Inj Loc | ${\mathit{p}}_{0}$ (atm) | ${\mathit{A}}^{*}/\mathit{A}$ |
---|---|---|---|---|

9 | 0.33∼1 | Outer | 10 | 0.2 |

9 | 0.33∼1 | Outer | 20 | 0.2 |

9 | 0.33∼1 | Outer | 10 | 0.1 |

w (mm) | I_{sp} ± SD (s) | F ± SD (N) | V_{o} ± SD (m/s) | V_{i} ± SD (m/s) |
---|---|---|---|---|

1 | 147.4 ± 0.78 | 59.7 ± 0.18 | 1975.43 ± 0.34 | 1876.66 ± 0.33 |

3 | 147.2 ± 0.38 | 169.1 ± 0.22 | 2066.44 ± 1.08 | 1756.4 ± 0.92 |

6 | 148.9 ± 0.32 | 315.5 ± 0.33 | 2374.99 ± 6.07 | 1662.4 ± 4.25 |

9 | 151 ± 0.9 | 439.7 ± 0.62 | 2960.41 ± 8.23 | 1628.23 ± 4.52 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. 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**

Shi, L.; Fan, E.; Shen, H.; Wen, C.-Y.; Shang, S.; Hu, H.
Numerical Study of the Effects of Injection Conditions on Rotating Detonation Engine Propulsive Performance. *Aerospace* **2023**, *10*, 879.
https://doi.org/10.3390/aerospace10100879

**AMA Style**

Shi L, Fan E, Shen H, Wen C-Y, Shang S, Hu H.
Numerical Study of the Effects of Injection Conditions on Rotating Detonation Engine Propulsive Performance. *Aerospace*. 2023; 10(10):879.
https://doi.org/10.3390/aerospace10100879

**Chicago/Turabian Style**

Shi, Lisong, E Fan, Hua Shen, Chih-Yung Wen, Shuai Shang, and Hongbo Hu.
2023. "Numerical Study of the Effects of Injection Conditions on Rotating Detonation Engine Propulsive Performance" *Aerospace* 10, no. 10: 879.
https://doi.org/10.3390/aerospace10100879