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Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 8 ) 1 e1 0
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Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor Zhijie Xia a, Xinmeng Tang b, Mingyi Luan a, Shujie Zhang a, Zhuang Ma a, Jianping Wang a,* a
Center for Combustion and Propulsion, CAPT & SKLTCS, Department of Mechanics and Engineering Sciences, College of Engineering, Peking University, Beijing 100871, China b Department of Engineering and Applied Sciences, Faculty of Science and Technology, Sophia University, 7-1 Kioiecho Chiyoda-ku, Tokyo 102-8554, Japan
article info
abstract
Article history:
A three-dimensional numerical simulation of rotating detonation engine (RDE) with hollow
Received 2 June 2018
combustor is performed to analyze wave structure evolution systematically. Wave struc-
Received in revised form
ture evolution is classified into four categories, namely two-wave collision (counter-
19 September 2018
rotating waves), abscission of detonation tail, and shock wave to detonation transition.
Accepted 22 September 2018
Two-wave collision consists of symmetric detonation collision, asymmetric detonation
Available online xxx
collision, and detonation/shock collision. Two symmetric detonation waves turn into shock waves after collision. Collision of asymmetric detonation waves creates single
Keywords:
detonation wave. The detonation/shock collision decreases the detonation wave intensity.
Rotating detonation engine
Abscission of detonation tail and shock to detonation transition can both create single
Hollow combustor
detonation wave or two opposite-direction detonation waves, depending on the wave
Wave structure evolution
hitting angle and the amount of fresh gas. All phenomena mentioned above affect the
Two-wave collision
number of detonation waves in the combustion chamber.
Number of detonation waves
© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Detonation is an efficient way of burning fuel-oxidizer mixtures, which has high thermodynamic cycle efficiency and high energy release rate. One way to generate thrust utilizing detonation is the pulse detonation engine (PDE), which was firstly suggested by Nicholls et al. [1] in 1957. In a PDE, the detonation wave is initiated through deflagration-todetonation transition, and is blown out from the detonation tube in each cycle, which is the reason for several major challenges [2]. In recent years, the rotating detonation engine
(RDE, also called continuous detonation engine, CDE) attracts increasing interest as one of the most promising Pressure Gain Combustion (PGC) devices available. In typical RDEs, fuel and oxidizer are continuously fed into the combustor to sustain the detonation wave propagating along the combustor azimuthally. The combustion products are blown out of the combustor to provide a stable source of thrust. Two-wave collision is a very common phenomenon in RDEs. Within the detonation adjustment duration (from ignition to stable state), two-wave collision plays an important role in wave structure evolution. It is also the reason for the generation of multiple detonation waves [3]. In addition,
* Corresponding author. E-mail address: [email protected] (J. Wang). https://doi.org/10.1016/j.ijhydene.2018.09.165 0360-3199/© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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under some circumstances, two-wave collision (or the more commonly used terminology of counter-rotating waves) is a stable working mode for RDEs. Rankin et al. have shown instantaneous images of two-wave collision through chemiluminescence imaging at stoichiometric conditions with an air mass flow rate of 0.15 kg/s. It was shown that the detonation waves continue to propagate and sustain themselves for short distances with no reactants [4]. Bluemner et al. have shown, through pressure measurements and high-speed imaging, two counter-rotating waves with an oxidizer mass flow rate of 0.26 kg/s and an equivalence ratio of 0.62. A very similar operation mode was also found at stoichiometric conditions with an oxidizer mass flow rate of 0.128 kg/s [5]. They also found experimentally four modes, Counter-rotating Waves (2CR), Two Counter-rotating Waves Transitioning (2CRT), Single Wave with Counter-rotating Component(SWCC), and Single Wave (SW) [6]. Zahn et al. examined and discussed stationary and non-stationary counter-rotation modes through the use of pressure signal cross-correlation analysis [7]. As for numerical simulation, few researches focus on counter-rotating waves as a stable working mode. Instead, they are more interested in the wave collision phenomenon itself. Han et al. performed simulations of head-on collision of detonation wave and shock wave [8]. Dubrovskii et al. conducted three-dimensional simulation of RDE with separate feeding of hydrogen and air. They discussed the two-wave collision phenomenon right after the detonation ignition [9]. The researches mentioned above are based on cylindrical combustor, where researchers are more interested in counter rotating waves as a stable working mode. Our previous simulations showed hollow combustor has longer detonation adjustment process and more complex evolution of waves than cylindrical combustor. In this paper we explore these phenomena in adjustment process systematically. In order to do this, a three-dimensional numerical simulation of RDE with a hollow combustor is carried out. The hollow combustor model was first proposed by Shao and Wang [10] and Tang et al. [11]. The feasibility of rotating detonations in hollow combustors has already been proved, e.g., with methaneeoxygen mixtures by Lin et al. [12].
Fig. 1 e Schematic models of RDE combustion chamber. (a) Annular model and (b) hollow model.
[13], ignoring diffusion (viscosity, thermal conduction, and molecular diffusion) is not a problem when considering the detonation front, because there the timescales are so short compared to diffusive scales. Therefore, the governing equations are given as:
Table 1 e Model parameters for detonation of stoichiometric hydrogeneair mixture. Model parameter
Value
Model parameter
Value
g1 g2 R1 ,J/kg K R2 ,J/kg K
1.3961 1.1653 395.75 346.2
q,J/kg A,1/s Ta ,K
5:4704 106 1:0 109 15100
Methodology Hollow model Fig. 1(a) shows the schematic of a typical annular combustor model. Reactants are fed into the combustor from the left headwall, to which one or more detonation waves attach. Fig. 1(b) shows the typical schematic of the hollow combustor. In this model, there is no inner cylinder. Premixed reactants are only fed from the outer region (R > Rinner ) of the headwall. The radius of the solid region on the headwall is 3 cm, represented as Rinner . The radius of the outer circle of the headwall is 6 cm, represented as Router .
Governing equations Currently, most of the numerical simulations on RDEs are based on Euler equations. According to the work of Oran et al.
Fig. 2 e Pressure and temperature contours at t ¼ 14 ms. The lines in (a) and (b) are both the interface between fresh gas and combustion products. (a) Pressure contour and (b) temperature contour. 1-Detonation; 2-deflagration.
Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 8 ) 1 e1 0
rt þ V$ðruÞ ¼ 0
(1)
ðruÞt þ V$ðruuÞ þ Vp ¼ 0
(2)
ðreÞt þ V$ððre þ pÞuÞ ¼ 0
(3)
ðrb1 Þt þ V$ðrub1 Þ ¼ u_
(4)
u (u, v, w) is the velocity vector, r is the density, b1 is the mass fraction of reactants, and u_ is the mass production rate of reactants. The pressure p and temperature T are obtained through the equation of state for perfect gas: 1 p ¼ rðg 1Þ e b1 q u2 þ v2 þ w2 2 T¼
p rR
(5)
(6)
R is the gas constant and g is the heat ratio of the mixture: R¼
X
bi Ri ði ¼ 1; 2Þ
(7)
P bi Ri gi g¼ P
gi 1 bi Ri gi 1
ði ¼ 1; 2Þ
(8)
3
Subscript 1 refers to reactants and 2 refers to products. b2 ¼ 1 b1 is the mass fraction of products. The source term is defined with the Arrhenius one-step chemistry model as: u_ ¼
db1 Ta ¼ Ab1 exp dt T
(9)
The parameters of the above one-step model for hydrogeneair reaction are adopted from the work of Ma et al. [14] and are summarized in Table 1.
Numerical methods To solve the equations above, we use the fifth-order monotonicity-preserving weighted essentially non-oscillatory (MPWENO) scheme to integrate the flux term. In addition, the third-order TVD RungeeKutta scheme is used for time integration. The average grid size in all three dimensions is 0.5 mm. The initial condition is a stable ChapmaneJouguet (CeJ) detonation wave obtained from one-dimensional simulations, mapped onto the head end of the combustor, and filled with stoichiometric hydrogeneair mixture [11]. It is assumed that the fresh gas is injected into the combustor through micro Laval nozzles distributed uniformly on the headwall. The stagnation pressure p0 and temperature T0 of the injection fresh gas are 30 atm and 600 K respectively.
Fig. 3 e Detonation/detonation collision on the r ¼ Router slice. (a) Unwrapped maps of pressure contours of this slice. (b) Pressure distribution on the headwall. (c) Temperature distribution on the headwall. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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Fig. 4 e (a) Bottom view of pressure contour at t ¼ 186 ms (b) Pressure gradient magnitude contour at t ¼ 186 ms.
The area ratio of the nozzle At =Aw is 0.1 where At is the net area of the nozzle throats and Aw is the injection ring area of the combustor. With different local pressure pw on the headwall inside the computation domain, the injection boundary conditions (p, T, u) varies. The side wall boundary conditions are adiabatic, slipping and non-catalytic. The outflow boundary condition is obtained by extrapolation [15]: Yb ¼ Y1 ð1 rÞ þ Y∞ r
(10)
r ¼ 0.05 is the relaxation rate coefficient, Y1 is the current value in the first cell near the boundary, and Y∞ is the ambient fluid
parameters. The environment pressure P∞ is set to be 0.5 atm. Further details and code validation can be found in Ref. [11].
Results Ignition A ChapmaneJouguet (CeJ) detonation wave is used for initiation: one-dimensional simulation results are mapped onto the computational domain near the headwall. In Fig. 2, the initial
Fig. 5 e Unwrapped maps and bottom views of pressure contours at t ¼ 288 ms, 292 ms and 302 ms, respectively. 1-Clockwise detonation wave; 2-counterclockwise detonation wave; 3-remaining detonation wave; 4-abscising part. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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an obvious discontinuity due to the difference in detonation and deflagration velocities, shown by arrow 2. Gradually, the curved detonation wave burns the fresh gas outside of the deflagration wave, accelerated toward the counterclockwise direction and encircle the entire deflagration. At 48 ms, the counterclockwise detonation wave hits the headwall, forming two detonation waves moving toward opposite directions. This way of ignition is ideal and contrived, which makes it impossible to operate in experiments. However, the main purpose of this paper is to make a systematic classification of wave collision phenomena in the course of stabilization. From the calculation results, this way of ignition successfully causes the collision of waves and finally evolves into a stable detonation wave. Therefore, from the perspective of this study, this way of ignition is acceptable. Fig. 6 e 3D fresh gas layer at t ¼ 288 ms, 292 ms and 302 ms, respectively. 1,2,3-Detonation waves; 4,5-fresh gas.
Detonation/detonation collision Collision of symmetric detonation waves
detonation wave propagates clockwise, forming a curved front. Meanwhile, a weak shock wave forms at the tail of the initial detonation wave, propagating counterclockwise, where the fresh gas is burnt by deflagration. Between them, there is
Collisions between detonation waves occur frequently in RDEs. Here we focus on the collision of detonation waves with approximately the same intensity. Fig. 3 shows two detonation waves colliding head-on around 194 ms The local pressure and temperature reach the highest value at the moment of
Fig. 7 e Detonation/shock collision process on the r ¼ Router slice. (a) Unwrapped maps of pressure contours of this slice. (b) Pressure on the headwall. (c) Temperature on the headwall. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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Fig. 8 e Quantities on the r ¼ Router slice and on the headwall. (a) Pressure; (b) axial component of velocity and density; (c) height of fresh gas layer; (d) mass of fresh gas, calculated by integral of density over the height of fresh gas layer.
collision and then decrease rapidly. Two shock waves out of the collision will propagate in opposite directions. During the entire collision period, the resulting pressure is rarely lower than 30 atm (stagnation pressure p0 ), which prohibits the fresh gas from getting into the chamber, causing the subsequent detonation waves extinguished. The collision transforms the detonation waves into shock waves. Rankin et al. [4] and Bluemner et al. [5] found the detonations sustain themselves with no reactants right after the collision. The main reason is
the combustion at the detonation front is insufficient in experiments and the burning of residual gas temporarily sustains the detonation waves, which can be seen in the work of Kawasaki et al. [16]. The detonations only sustain themselves for very short distances before they decouple and turn into shock waves, which is consistent with our conclusion. Fig. 4(a) shows the bottom view of pressure contour at 186 ms and Fig. 4(b) shows the three-dimensional structure of the two detonation waves. The angular velocity of the
Fig. 9 e Bottom views of pressure contours at t ¼ 296 ms, 300 ms, 306 ms and 312 ms, respectively. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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Fig. 10 e Bottom views of velocity contours at t ¼ 306 ms (a) Circumferential component of velocity; (b) radial component of velocity.
detonation wave near r ¼ Router is about 3:4 104 rad/s, while the angular velocity of the detonation wave near r ¼ Rinner is about 5:5 104 rad/s. Therefore, the inner parts collide with each other first. The collision point moves from the inner part to the side wall. Katta et al. [17] found similar results in annular combustors that the detonation front on the inner wall moves ahead of that on the outer wall and causes an inclination to the detonation wave between the walls. They have qualitatively confirmed the predictions by using OH chemiluminescence.
The collision process turns both detonation waves into shock waves, resulting in the number of clockwise and counterclockwise detonation waves both reduced by 1. Their number difference remains unchanged.
Collision of asymmetrical detonation waves Fig. 5 shows the collision of asymmetrical detonation waves happening around 292 ms The clockwise detonation wave marked with 1 exists outside of the dashed circle, while the counterclockwise detonation wave marked with 2 extends to
Fig. 11 e Contours at t ¼ 142 ms, 154 ms and 160 ms, respectively. (a) Unwrapped maps of pressure contours with lines separating fresh gas and combustion products; (b) Bottom views of pressure contours; (c) Unwrapped maps of circumferential velocity contours with lines of pressure. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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the inside. During collision, the two detonation waves cancel each other outside of the dashed circle just like the collision of symmetric detonation waves. However, the inner part maintains a counterclockwise detonation wave marked with 3. An abscising part marked with 4 also appears, which will be discussed later in section Abscission of detonation tail. During this process, the number of clockwise detonation waves is reduced by 1, while that of counterclockwise detonation waves remains unchanged. Their number difference is reduced by 1. Detonation wave 3 remains after collision, which is essential for the change of the number of detonation waves. Fig. 6 shows the three-dimensional fresh gas layer. Before collision, detonation waves 1 and 2 burn fresh gas at positon 4. Detonation wave 1 does not extend to the inner part, therefore the volume of fresh gas at position 5 remains adequate. After collision, high pressure, high temperature and adequate fresh gas lead to the ignition of detonation wave 3.
Detonation/shock collision Fig. 7 shows a clockwise detonation wave and a counterclockwise shock wave colliding around 124 ms Before collision, the pressure of shock front is 21.5 atm, lower than that of the detonation front (53 atm). The temperature of the shock front is 745 K, much lower than that of the detonation
front. The pressure reaches 125 atm at the moment of collision and then decreases dramatically to the level before collision. The temperature of the detonation wave remains almost the same during the entire process while that of the shock wave slightly increases to 1095 K. Bluemner et al. found in experiments that there were counter-rotating waves composed of a primary wave and a much weaker wave. They call this mode Single Wave with CounterRotating Component (SWCC). This mode is in accord with Detonation/shock collision [6]. After collision, the pressure and temperature of the detonation wave change slightly while the height of detonation front decreases dramatically. There are two major causes for the decrease. First, the height of detonation is equal to the height of fresh gas layer at the detonation front, as shown in Fig. 8(c). Fig. 8(b) shows that the axial component of velocity is zero at position 1, which indicates no fresh gas is injected into the chamber at the shock front. Additionally, arrow 2 in Fig. 8(d) shows that the mass of fresh gas decreases when the shock wave passes, which indicates part of the fresh gas is burnt by deflagration. Second, arrow 3 in Fig. 8(b) shows that the shock wave compresses the fresh gas. The decreasing of the amount of fresh gas and the compression by shock waves lead to the decrease in the height of detonation front.
Fig. 12 e Contours at t ¼ 142 ms, 154 ms and 164 ms, respectively. (a) Unwrapped maps of pressure contours; (b) Bottom views of pressure contours; (c) Unwrapped maps of axial velocity contours. Lines in (a) and (c) separates fresh gas and combustion products. 1-Shock wave; 2-surface of pressure discontinuity; 3-position where 1 and 2 intersect. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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Table 2 e Cases of new detonation wave(s) formation under different conditions. Symbol £ means there are new detonation waves, symbol B means no new detonation wave forms. Single Opposite-direction detonation detonation waves Wave Collision of symmetric detonation waves. Collision of asymmetric detonation waves. Detonation/shock collision. Abscission of detonation tail. Shock to detonation transition.
B
B
B
B
B
Abscission of detonation tail Fig. 9 shows the abscission of detonation tail around 300 ms This phenomenon is very common in hollow combustor. At 296 ms, the detonation waves are in arc shape, extending to R ¼ Rinner . No fresh gas is injected inside the region of R < Rinner , leading to the combustion extinguished in this region. The detonation waves are divided into two parts, forming tails 1 and 2. Tail 1 gradually becomes a counterclockwise detonation wave, while tail 2 hits the side wall and becomes two detonation waves propagating in opposite directions. Fig. 10 explains why tails 1 and 2 have different evolution processes. Gas at the position of tail 1 has a big counterclockwise velocity, causing tail 1 to become a counterclockwise detonation wave. However, gas at the position of tail 2 has almost no circumferential component of velocity. Instead, it has a large radial component of velocity, which makes abscised tail 2 hit the side wall vertically and become two detonation waves.
Shock wave to detonation transition Fig. 11 shows how a shock wave becomes a detonation wave. The shock wave 1 created by a detonation/detonation collision propagates in the clockwise direction and burns fresh gas near
9
the headwall, shown by arrow 2. At 154 ms, the flame surface reaches the headwall, creating a region of extremely high pressure and temperature. Indicated by arrow 3 in the circumferential velocity contours, this region has a high clockwise velocity, which is the reason for the production of the clockwise detonation wave. At the same moment, the shock wave colliding with a detonation wave, which we have mentioned above, also turns into detonation waves (shown in Fig. 12). The lines in the figure separates the fresh gas and combustion products. The shock wave is marked with arrow 1. At position 2, there is a surface of pressure discontinuity due to the injection of fresh gas, leading to a region of minus axial velocity. The shock wave marked with arrow 1 moves toward the headwall and reaches the surface of pressure discontinuity at 150 ms, causing the fresh gas to be burnt by deflagration at position 3. The flame surface reaches the headwall vertically at 158 ms, forming two detonation waves propagating in opposite directions.
Changes in the number of detonation waves According to previous discussion, all the cases of new detonation wave(s) formation are listed in Table 2. Collision of symmetric detonation waves decreases the number of detonation waves of both directions. However, it cannot change the number difference. Detonation/shock collision do not change the number of detonation waves. The rest three scenarios can create new detonation waves, which affects the net number of detonation waves in the combustion chamber. In RDEs we are interested in the wave number when stable. Apparently, we should focus on the column of Single Detonation Waves. Collision of asymmetric detonation wave, abscission of detonation tail, and shock to detonation transition can create single detonation wave, which is essential to the change of net wave number. Inspired by this idea, we draw Fig. 13 that shows the number of clockwise and counterclockwise waves as well as the number difference of them. In Fig. 13(c), there exist an increase at 154 ms and 3 decreases at 292 ms, 306 ms and 316 ms, respectively. The increase at 154 ms is caused by shock to detonation transition. The decreases at 292 ms and 316 ms are
Fig. 13 e Number of detonation waves. (a) Clockwise waves (b) counterclockwise waves and (c) clockwise waves minus counterclockwise waves. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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caused by the collision of asymmetric detonation waves. The decrease at 306 ms is caused by the abscission of detonation tail. In the end, the stable condition is one counterclockwise detonation wave.
Conclusion A three-dimensional study on the RDE with a hollow combustor is carried out. We analyze two-wave collision phenomena as well as wave structure evolution systematically. The major points are as follows: (1) Two symmetric detonation waves turns into two opposite-direction shock waves after collision. (2) Collision of two asymmetric detonation waves can only create one detonation wave. (3) Collision of shock wave with detonation wave decreases the detonation height instead of extinguishing the detonation wave. (4) Abscission of detonation tail can create single detonation wave or two opposite-direction detonation waves. (5) A shock wave can transit to one detonation wave or two opposite-direction detonation waves. (6) Collision of asymmetric detonation wave, abscission of detonation tail, and shock to detonation transition are essential to the change of net wave number.
Acknowledgements The present study is sponsored by National Natural Science Foundation of China (Grant No. 91741202). This paper is supported by the opening project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology). The opening project number is KFJJ18-13M.
references
[1] Nicholls JA, Wilkinson JA, Ong RSB. Intermittent detonation as a thrust-producing mechanism. J Jet Propuls 1957;27(5):534e41. [2] Roy GD, Frolov SM, Borisov AA, Netzer DW. Pulse detonation propulsion: challenges, current status, and future perspective. Prog Energy Combust Sci 2004;30(6):545e672.
[3] Wang YH, Wang JP, Li YS, Li Y. Induction for multiple rotating detonation waves in the hydrogeneoxygen mixture with tangential flow. Int J Hydrogen Energy 2014;39(22):11792e7. [4] Rankin BA, Richardson DR, Caswell AW, Naples AG, Hoke JL, Schauer FR. Chemiluminescence imaging of an optically accessible non-premixed rotating detonation engine. Combust Flame 2017;176(1):12e22. [5] Bluemner R, Bohon MD, Paschereit CO, Gutmark EJ. Single and counter-rotating wave modes in an RDC. In: 2018 AIAA aerospace sciences meeting. American Institute of Aeronautics and Astronautics; 2018. [6] Bluemner R, Bohon MD, Paschereit CO, Gutmark EJ. Dynamics of counter-rotating wave modes in an RDC. In: 2018 joint propulsion Conference; 2018. [7] Zahn A, Knight E, Anand V, Jodele J, Gutmark EJ. Examination of counter-rotating detonation waves using crosscorrelation. In: 2018 joint propulsion Conference. American Institute of Aeronautics and Astronautics; 2018. [8] Han G, Jiang Z, Zhang D. Numerical investigation on the collision between detonations and shocks. Chin J Theor Appl Mech 2008;40(2):154e61. [9] Dubrovskii AV, Ivanov VS, Frolov SM. Three-dimensional numerical simulation of the operation process in a continuous detonation combustor with separate feeding of hydrogen and air. Russ J Phys Chem B 2015;9(1):104e19. [10] Shao YT, Wang JP. Three dimensional simulation of rotating detonation engine without inner wall. In: 23rd International Colloquium on the dynamics of explosions and reactive systems; 2011. Irvine. [11] Tang XM, Wang JP, Shao YT. Three-dimensional numerical investigations of the rotating detonation engine with a hollow combustor. Combust Flame 2014;162(4):997e1008. [12] Lin W, Zhou J, Liu SJ, Lin ZY. An experimental study on CH4/ O2 continuously rotating detonation wave in a hollow combustion chamber. Exp Therm Fluid Sci 2015;62:122e30. [13] Oran ES, James Jr WW, Stefaniw EI, Lefebvre MH, John Jr DA. A numerical study of a two-dimensional H 2 -O 2 -Ar detonation using a detailed chemical reaction model. Combust Flame 1998;113(1e2):147e63. [14] Ma F, Choi JY, Yang V. Propulsive performance of airbreathing pulse detonation engines. Jpropulpower 2006;22(6):1188e203. [15] Gamezo VN, Desbordes D, Oran ES. Formation and evolution of two-dimensional cellular detonations. Combust Flame 1999;116(1e2):154e65. [16] Kawasaki A, Inakawa T, Kasahara J, Goto K, Matsuoka K, Matsuo A, et al. Critical condition of inner cylinder radius for sustaining rotating detonation waves in rotating detonation engine thruster. Proceedings of the Combustion Institute; 2018. [17] Katta V, Cho K, John LH, Joshua RC, Frederick RS, William MR. Effect of increasing channel width on the structure of rotating detonation wave. In: 37th International Symposium on combustion; 2018. Dublin, Ireland.
Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor Zhijie Xia a, Xinmeng Tang b, Mingyi Luan a, Shujie Zhang a, Zhuang Ma a, Jianping Wang a,* a
Center for Combustion and Propulsion, CAPT & SKLTCS, Department of Mechanics and Engineering Sciences, College of Engineering, Peking University, Beijing 100871, China b Department of Engineering and Applied Sciences, Faculty of Science and Technology, Sophia University, 7-1 Kioiecho Chiyoda-ku, Tokyo 102-8554, Japan
article info
abstract
Article history:
A three-dimensional numerical simulation of rotating detonation engine (RDE) with hollow
Received 2 June 2018
combustor is performed to analyze wave structure evolution systematically. Wave struc-
Received in revised form
ture evolution is classified into four categories, namely two-wave collision (counter-
19 September 2018
rotating waves), abscission of detonation tail, and shock wave to detonation transition.
Accepted 22 September 2018
Two-wave collision consists of symmetric detonation collision, asymmetric detonation
Available online xxx
collision, and detonation/shock collision. Two symmetric detonation waves turn into shock waves after collision. Collision of asymmetric detonation waves creates single
Keywords:
detonation wave. The detonation/shock collision decreases the detonation wave intensity.
Rotating detonation engine
Abscission of detonation tail and shock to detonation transition can both create single
Hollow combustor
detonation wave or two opposite-direction detonation waves, depending on the wave
Wave structure evolution
hitting angle and the amount of fresh gas. All phenomena mentioned above affect the
Two-wave collision
number of detonation waves in the combustion chamber.
Number of detonation waves
© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Detonation is an efficient way of burning fuel-oxidizer mixtures, which has high thermodynamic cycle efficiency and high energy release rate. One way to generate thrust utilizing detonation is the pulse detonation engine (PDE), which was firstly suggested by Nicholls et al. [1] in 1957. In a PDE, the detonation wave is initiated through deflagration-todetonation transition, and is blown out from the detonation tube in each cycle, which is the reason for several major challenges [2]. In recent years, the rotating detonation engine
(RDE, also called continuous detonation engine, CDE) attracts increasing interest as one of the most promising Pressure Gain Combustion (PGC) devices available. In typical RDEs, fuel and oxidizer are continuously fed into the combustor to sustain the detonation wave propagating along the combustor azimuthally. The combustion products are blown out of the combustor to provide a stable source of thrust. Two-wave collision is a very common phenomenon in RDEs. Within the detonation adjustment duration (from ignition to stable state), two-wave collision plays an important role in wave structure evolution. It is also the reason for the generation of multiple detonation waves [3]. In addition,
* Corresponding author. E-mail address: [email protected] (J. Wang). https://doi.org/10.1016/j.ijhydene.2018.09.165 0360-3199/© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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under some circumstances, two-wave collision (or the more commonly used terminology of counter-rotating waves) is a stable working mode for RDEs. Rankin et al. have shown instantaneous images of two-wave collision through chemiluminescence imaging at stoichiometric conditions with an air mass flow rate of 0.15 kg/s. It was shown that the detonation waves continue to propagate and sustain themselves for short distances with no reactants [4]. Bluemner et al. have shown, through pressure measurements and high-speed imaging, two counter-rotating waves with an oxidizer mass flow rate of 0.26 kg/s and an equivalence ratio of 0.62. A very similar operation mode was also found at stoichiometric conditions with an oxidizer mass flow rate of 0.128 kg/s [5]. They also found experimentally four modes, Counter-rotating Waves (2CR), Two Counter-rotating Waves Transitioning (2CRT), Single Wave with Counter-rotating Component(SWCC), and Single Wave (SW) [6]. Zahn et al. examined and discussed stationary and non-stationary counter-rotation modes through the use of pressure signal cross-correlation analysis [7]. As for numerical simulation, few researches focus on counter-rotating waves as a stable working mode. Instead, they are more interested in the wave collision phenomenon itself. Han et al. performed simulations of head-on collision of detonation wave and shock wave [8]. Dubrovskii et al. conducted three-dimensional simulation of RDE with separate feeding of hydrogen and air. They discussed the two-wave collision phenomenon right after the detonation ignition [9]. The researches mentioned above are based on cylindrical combustor, where researchers are more interested in counter rotating waves as a stable working mode. Our previous simulations showed hollow combustor has longer detonation adjustment process and more complex evolution of waves than cylindrical combustor. In this paper we explore these phenomena in adjustment process systematically. In order to do this, a three-dimensional numerical simulation of RDE with a hollow combustor is carried out. The hollow combustor model was first proposed by Shao and Wang [10] and Tang et al. [11]. The feasibility of rotating detonations in hollow combustors has already been proved, e.g., with methaneeoxygen mixtures by Lin et al. [12].
Fig. 1 e Schematic models of RDE combustion chamber. (a) Annular model and (b) hollow model.
[13], ignoring diffusion (viscosity, thermal conduction, and molecular diffusion) is not a problem when considering the detonation front, because there the timescales are so short compared to diffusive scales. Therefore, the governing equations are given as:
Table 1 e Model parameters for detonation of stoichiometric hydrogeneair mixture. Model parameter
Value
Model parameter
Value
g1 g2 R1 ,J/kg K R2 ,J/kg K
1.3961 1.1653 395.75 346.2
q,J/kg A,1/s Ta ,K
5:4704 106 1:0 109 15100
Methodology Hollow model Fig. 1(a) shows the schematic of a typical annular combustor model. Reactants are fed into the combustor from the left headwall, to which one or more detonation waves attach. Fig. 1(b) shows the typical schematic of the hollow combustor. In this model, there is no inner cylinder. Premixed reactants are only fed from the outer region (R > Rinner ) of the headwall. The radius of the solid region on the headwall is 3 cm, represented as Rinner . The radius of the outer circle of the headwall is 6 cm, represented as Router .
Governing equations Currently, most of the numerical simulations on RDEs are based on Euler equations. According to the work of Oran et al.
Fig. 2 e Pressure and temperature contours at t ¼ 14 ms. The lines in (a) and (b) are both the interface between fresh gas and combustion products. (a) Pressure contour and (b) temperature contour. 1-Detonation; 2-deflagration.
Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 8 ) 1 e1 0
rt þ V$ðruÞ ¼ 0
(1)
ðruÞt þ V$ðruuÞ þ Vp ¼ 0
(2)
ðreÞt þ V$ððre þ pÞuÞ ¼ 0
(3)
ðrb1 Þt þ V$ðrub1 Þ ¼ u_
(4)
u (u, v, w) is the velocity vector, r is the density, b1 is the mass fraction of reactants, and u_ is the mass production rate of reactants. The pressure p and temperature T are obtained through the equation of state for perfect gas: 1 p ¼ rðg 1Þ e b1 q u2 þ v2 þ w2 2 T¼
p rR
(5)
(6)
R is the gas constant and g is the heat ratio of the mixture: R¼
X
bi Ri ði ¼ 1; 2Þ
(7)
P bi Ri gi g¼ P
gi 1 bi Ri gi 1
ði ¼ 1; 2Þ
(8)
3
Subscript 1 refers to reactants and 2 refers to products. b2 ¼ 1 b1 is the mass fraction of products. The source term is defined with the Arrhenius one-step chemistry model as: u_ ¼
db1 Ta ¼ Ab1 exp dt T
(9)
The parameters of the above one-step model for hydrogeneair reaction are adopted from the work of Ma et al. [14] and are summarized in Table 1.
Numerical methods To solve the equations above, we use the fifth-order monotonicity-preserving weighted essentially non-oscillatory (MPWENO) scheme to integrate the flux term. In addition, the third-order TVD RungeeKutta scheme is used for time integration. The average grid size in all three dimensions is 0.5 mm. The initial condition is a stable ChapmaneJouguet (CeJ) detonation wave obtained from one-dimensional simulations, mapped onto the head end of the combustor, and filled with stoichiometric hydrogeneair mixture [11]. It is assumed that the fresh gas is injected into the combustor through micro Laval nozzles distributed uniformly on the headwall. The stagnation pressure p0 and temperature T0 of the injection fresh gas are 30 atm and 600 K respectively.
Fig. 3 e Detonation/detonation collision on the r ¼ Router slice. (a) Unwrapped maps of pressure contours of this slice. (b) Pressure distribution on the headwall. (c) Temperature distribution on the headwall. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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Fig. 4 e (a) Bottom view of pressure contour at t ¼ 186 ms (b) Pressure gradient magnitude contour at t ¼ 186 ms.
The area ratio of the nozzle At =Aw is 0.1 where At is the net area of the nozzle throats and Aw is the injection ring area of the combustor. With different local pressure pw on the headwall inside the computation domain, the injection boundary conditions (p, T, u) varies. The side wall boundary conditions are adiabatic, slipping and non-catalytic. The outflow boundary condition is obtained by extrapolation [15]: Yb ¼ Y1 ð1 rÞ þ Y∞ r
(10)
r ¼ 0.05 is the relaxation rate coefficient, Y1 is the current value in the first cell near the boundary, and Y∞ is the ambient fluid
parameters. The environment pressure P∞ is set to be 0.5 atm. Further details and code validation can be found in Ref. [11].
Results Ignition A ChapmaneJouguet (CeJ) detonation wave is used for initiation: one-dimensional simulation results are mapped onto the computational domain near the headwall. In Fig. 2, the initial
Fig. 5 e Unwrapped maps and bottom views of pressure contours at t ¼ 288 ms, 292 ms and 302 ms, respectively. 1-Clockwise detonation wave; 2-counterclockwise detonation wave; 3-remaining detonation wave; 4-abscising part. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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an obvious discontinuity due to the difference in detonation and deflagration velocities, shown by arrow 2. Gradually, the curved detonation wave burns the fresh gas outside of the deflagration wave, accelerated toward the counterclockwise direction and encircle the entire deflagration. At 48 ms, the counterclockwise detonation wave hits the headwall, forming two detonation waves moving toward opposite directions. This way of ignition is ideal and contrived, which makes it impossible to operate in experiments. However, the main purpose of this paper is to make a systematic classification of wave collision phenomena in the course of stabilization. From the calculation results, this way of ignition successfully causes the collision of waves and finally evolves into a stable detonation wave. Therefore, from the perspective of this study, this way of ignition is acceptable. Fig. 6 e 3D fresh gas layer at t ¼ 288 ms, 292 ms and 302 ms, respectively. 1,2,3-Detonation waves; 4,5-fresh gas.
Detonation/detonation collision Collision of symmetric detonation waves
detonation wave propagates clockwise, forming a curved front. Meanwhile, a weak shock wave forms at the tail of the initial detonation wave, propagating counterclockwise, where the fresh gas is burnt by deflagration. Between them, there is
Collisions between detonation waves occur frequently in RDEs. Here we focus on the collision of detonation waves with approximately the same intensity. Fig. 3 shows two detonation waves colliding head-on around 194 ms The local pressure and temperature reach the highest value at the moment of
Fig. 7 e Detonation/shock collision process on the r ¼ Router slice. (a) Unwrapped maps of pressure contours of this slice. (b) Pressure on the headwall. (c) Temperature on the headwall. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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Fig. 8 e Quantities on the r ¼ Router slice and on the headwall. (a) Pressure; (b) axial component of velocity and density; (c) height of fresh gas layer; (d) mass of fresh gas, calculated by integral of density over the height of fresh gas layer.
collision and then decrease rapidly. Two shock waves out of the collision will propagate in opposite directions. During the entire collision period, the resulting pressure is rarely lower than 30 atm (stagnation pressure p0 ), which prohibits the fresh gas from getting into the chamber, causing the subsequent detonation waves extinguished. The collision transforms the detonation waves into shock waves. Rankin et al. [4] and Bluemner et al. [5] found the detonations sustain themselves with no reactants right after the collision. The main reason is
the combustion at the detonation front is insufficient in experiments and the burning of residual gas temporarily sustains the detonation waves, which can be seen in the work of Kawasaki et al. [16]. The detonations only sustain themselves for very short distances before they decouple and turn into shock waves, which is consistent with our conclusion. Fig. 4(a) shows the bottom view of pressure contour at 186 ms and Fig. 4(b) shows the three-dimensional structure of the two detonation waves. The angular velocity of the
Fig. 9 e Bottom views of pressure contours at t ¼ 296 ms, 300 ms, 306 ms and 312 ms, respectively. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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Fig. 10 e Bottom views of velocity contours at t ¼ 306 ms (a) Circumferential component of velocity; (b) radial component of velocity.
detonation wave near r ¼ Router is about 3:4 104 rad/s, while the angular velocity of the detonation wave near r ¼ Rinner is about 5:5 104 rad/s. Therefore, the inner parts collide with each other first. The collision point moves from the inner part to the side wall. Katta et al. [17] found similar results in annular combustors that the detonation front on the inner wall moves ahead of that on the outer wall and causes an inclination to the detonation wave between the walls. They have qualitatively confirmed the predictions by using OH chemiluminescence.
The collision process turns both detonation waves into shock waves, resulting in the number of clockwise and counterclockwise detonation waves both reduced by 1. Their number difference remains unchanged.
Collision of asymmetrical detonation waves Fig. 5 shows the collision of asymmetrical detonation waves happening around 292 ms The clockwise detonation wave marked with 1 exists outside of the dashed circle, while the counterclockwise detonation wave marked with 2 extends to
Fig. 11 e Contours at t ¼ 142 ms, 154 ms and 160 ms, respectively. (a) Unwrapped maps of pressure contours with lines separating fresh gas and combustion products; (b) Bottom views of pressure contours; (c) Unwrapped maps of circumferential velocity contours with lines of pressure. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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the inside. During collision, the two detonation waves cancel each other outside of the dashed circle just like the collision of symmetric detonation waves. However, the inner part maintains a counterclockwise detonation wave marked with 3. An abscising part marked with 4 also appears, which will be discussed later in section Abscission of detonation tail. During this process, the number of clockwise detonation waves is reduced by 1, while that of counterclockwise detonation waves remains unchanged. Their number difference is reduced by 1. Detonation wave 3 remains after collision, which is essential for the change of the number of detonation waves. Fig. 6 shows the three-dimensional fresh gas layer. Before collision, detonation waves 1 and 2 burn fresh gas at positon 4. Detonation wave 1 does not extend to the inner part, therefore the volume of fresh gas at position 5 remains adequate. After collision, high pressure, high temperature and adequate fresh gas lead to the ignition of detonation wave 3.
Detonation/shock collision Fig. 7 shows a clockwise detonation wave and a counterclockwise shock wave colliding around 124 ms Before collision, the pressure of shock front is 21.5 atm, lower than that of the detonation front (53 atm). The temperature of the shock front is 745 K, much lower than that of the detonation
front. The pressure reaches 125 atm at the moment of collision and then decreases dramatically to the level before collision. The temperature of the detonation wave remains almost the same during the entire process while that of the shock wave slightly increases to 1095 K. Bluemner et al. found in experiments that there were counter-rotating waves composed of a primary wave and a much weaker wave. They call this mode Single Wave with CounterRotating Component (SWCC). This mode is in accord with Detonation/shock collision [6]. After collision, the pressure and temperature of the detonation wave change slightly while the height of detonation front decreases dramatically. There are two major causes for the decrease. First, the height of detonation is equal to the height of fresh gas layer at the detonation front, as shown in Fig. 8(c). Fig. 8(b) shows that the axial component of velocity is zero at position 1, which indicates no fresh gas is injected into the chamber at the shock front. Additionally, arrow 2 in Fig. 8(d) shows that the mass of fresh gas decreases when the shock wave passes, which indicates part of the fresh gas is burnt by deflagration. Second, arrow 3 in Fig. 8(b) shows that the shock wave compresses the fresh gas. The decreasing of the amount of fresh gas and the compression by shock waves lead to the decrease in the height of detonation front.
Fig. 12 e Contours at t ¼ 142 ms, 154 ms and 164 ms, respectively. (a) Unwrapped maps of pressure contours; (b) Bottom views of pressure contours; (c) Unwrapped maps of axial velocity contours. Lines in (a) and (c) separates fresh gas and combustion products. 1-Shock wave; 2-surface of pressure discontinuity; 3-position where 1 and 2 intersect. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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Table 2 e Cases of new detonation wave(s) formation under different conditions. Symbol £ means there are new detonation waves, symbol B means no new detonation wave forms. Single Opposite-direction detonation detonation waves Wave Collision of symmetric detonation waves. Collision of asymmetric detonation waves. Detonation/shock collision. Abscission of detonation tail. Shock to detonation transition.
B
B
B
B
B
Abscission of detonation tail Fig. 9 shows the abscission of detonation tail around 300 ms This phenomenon is very common in hollow combustor. At 296 ms, the detonation waves are in arc shape, extending to R ¼ Rinner . No fresh gas is injected inside the region of R < Rinner , leading to the combustion extinguished in this region. The detonation waves are divided into two parts, forming tails 1 and 2. Tail 1 gradually becomes a counterclockwise detonation wave, while tail 2 hits the side wall and becomes two detonation waves propagating in opposite directions. Fig. 10 explains why tails 1 and 2 have different evolution processes. Gas at the position of tail 1 has a big counterclockwise velocity, causing tail 1 to become a counterclockwise detonation wave. However, gas at the position of tail 2 has almost no circumferential component of velocity. Instead, it has a large radial component of velocity, which makes abscised tail 2 hit the side wall vertically and become two detonation waves.
Shock wave to detonation transition Fig. 11 shows how a shock wave becomes a detonation wave. The shock wave 1 created by a detonation/detonation collision propagates in the clockwise direction and burns fresh gas near
9
the headwall, shown by arrow 2. At 154 ms, the flame surface reaches the headwall, creating a region of extremely high pressure and temperature. Indicated by arrow 3 in the circumferential velocity contours, this region has a high clockwise velocity, which is the reason for the production of the clockwise detonation wave. At the same moment, the shock wave colliding with a detonation wave, which we have mentioned above, also turns into detonation waves (shown in Fig. 12). The lines in the figure separates the fresh gas and combustion products. The shock wave is marked with arrow 1. At position 2, there is a surface of pressure discontinuity due to the injection of fresh gas, leading to a region of minus axial velocity. The shock wave marked with arrow 1 moves toward the headwall and reaches the surface of pressure discontinuity at 150 ms, causing the fresh gas to be burnt by deflagration at position 3. The flame surface reaches the headwall vertically at 158 ms, forming two detonation waves propagating in opposite directions.
Changes in the number of detonation waves According to previous discussion, all the cases of new detonation wave(s) formation are listed in Table 2. Collision of symmetric detonation waves decreases the number of detonation waves of both directions. However, it cannot change the number difference. Detonation/shock collision do not change the number of detonation waves. The rest three scenarios can create new detonation waves, which affects the net number of detonation waves in the combustion chamber. In RDEs we are interested in the wave number when stable. Apparently, we should focus on the column of Single Detonation Waves. Collision of asymmetric detonation wave, abscission of detonation tail, and shock to detonation transition can create single detonation wave, which is essential to the change of net wave number. Inspired by this idea, we draw Fig. 13 that shows the number of clockwise and counterclockwise waves as well as the number difference of them. In Fig. 13(c), there exist an increase at 154 ms and 3 decreases at 292 ms, 306 ms and 316 ms, respectively. The increase at 154 ms is caused by shock to detonation transition. The decreases at 292 ms and 316 ms are
Fig. 13 e Number of detonation waves. (a) Clockwise waves (b) counterclockwise waves and (c) clockwise waves minus counterclockwise waves. Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165
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caused by the collision of asymmetric detonation waves. The decrease at 306 ms is caused by the abscission of detonation tail. In the end, the stable condition is one counterclockwise detonation wave.
Conclusion A three-dimensional study on the RDE with a hollow combustor is carried out. We analyze two-wave collision phenomena as well as wave structure evolution systematically. The major points are as follows: (1) Two symmetric detonation waves turns into two opposite-direction shock waves after collision. (2) Collision of two asymmetric detonation waves can only create one detonation wave. (3) Collision of shock wave with detonation wave decreases the detonation height instead of extinguishing the detonation wave. (4) Abscission of detonation tail can create single detonation wave or two opposite-direction detonation waves. (5) A shock wave can transit to one detonation wave or two opposite-direction detonation waves. (6) Collision of asymmetric detonation wave, abscission of detonation tail, and shock to detonation transition are essential to the change of net wave number.
Acknowledgements The present study is sponsored by National Natural Science Foundation of China (Grant No. 91741202). This paper is supported by the opening project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology). The opening project number is KFJJ18-13M.
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Please cite this article in press as: Xia Z, et al., Numerical investigation of two-wave collision and wave structure evolution of rotating detonation engine with hollow combustor, International Journal of Hydrogen Energy (2018), https://doi.org/10.1016/ j.ijhydene.2018.09.165