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Effects of flow-field structures on the stability of rotating detonation ramjet engine
Acta Astronautica 168 (2020) 174–181
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Research paper
Effects of flow-field structures on the stability of rotating detonation ramjet engine
T
Kevin Wu, Shujie Zhang, Mingyi Luan, Jianping Wang∗ Center for Combustion and Propulsion, CAPT & SKLTCS, Department of Mechanics and Engineering Sciences, College of Engineering, Peking University, Beijing, 100871, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Rotating detonation engine Simulation Instability Propulsion
Due to less attention received by rotating detonation ramjet engine (RDRE), the three-dimensional flow-field structures and their influence on stability are less understood than rocket-mode rotating detonation engine (RDE). This paper presents three-dimensional numerical investigations on the RDRE model with a Laval inlet. The computations are based on detailed H2/air chemistry. The simulation shows that, unlike rocket-mode RDEs, an upstream oblique shock wave is induced by detonation. This oblique shock wave is then prevented from propagating upstream further by the normal shock wave in the inlet. In addition, two instabilities in RDREs, namely, backflow and strip-type fresh fuel layer, are analyzed for the first time. Results indicate that the backflow is related to reflected shock waves and the upstream oblique shock wave that induced by detonation. While the strip-type fresh fuel distribution is formed due to the unstable propagation of detonation. Detonation waves interact with fresh fuel layer, transverse waves, and the normal shock waves, making the irregular fresh fuel layer remains in the combustion chamber.
1. Introduction Detonation combustion has a higher thermodynamic efficiency than traditional deflagration combustion [1,2]. There are some detonationbased propulsion devices, among which the pulse detonation engine (PDE) has been most widely researched. As PDE research is facing several obstacles [3], rotating detonation engine (RDE) has received more and more attention in the propulsion community. This is due to its high combustion efficiency, pressure gain property, the lack of the need for regular ignition, and its compact size. The concept of using rotating detonation waves for propulsion was first proposed by Voilsekhovskii [4] and developed by Wolanski [5,6]. Most RDE studies concentrate on its rocket mode. A typical rocketmode RDE is equipped with a co-axial annulus chamber. Fundamental flow structures of a rocket-mode RDE is shown in Fig. 1 (a) [7]. After initiation, the detonation wave rotates circumferentially along the annulus and sustains against the headwall. In the meantime, fresh mixtures are injected axially into the combustion chamber from fuel plenums. After fuel is consumed by detonation, combustion products expand behind the detonation waves. Thrust is produced by exhausting detonation products axially to the ambient atmosphere. Besides, a oblique shock wave attached with detonation exists in the post-detonation area. Various Injection patterns [8,9], geometries [10,11], reactants ∗
[12–14], nozzles [15,16] and instabilities [17,18] have been investigated for rocket-mode RDEs. Besides the rocket-mode RDE, preliminary studies have been performed for the ramjet-mode RDE in recent years. Schematic of a rotating detonation ramjet engine (RDRE) is shown in Fig. 1(b). Unlike rocket-mode RDEs, RDREs take supersonic oxidants (air) from the atmosphere. Because there is no head wall in RDREs, flow-field structures are different between rocket-mode RDEs. Therefore, conclusions derived for rocket-mode RDE should be re-assessed when applied to RDRE. Zhdan [19] firstly established a two-dimensional unsteady mathematical model for RDRE. Supersonic hydrogen-oxygen mixtures were introduced into the combustion chamber via a divergent inlet. It was shown that the continuous detonation was achievable in supersonic premixed mixtures with Mach = 3. Braun [20] proposed a cycle analysis model for RDREs. Supersonic mixtures were assumed to be taken into a convergent-divergent inlet isentropically. The RDRE model fueled by hydrogen/air operated stably between Mach 1.5–5 and reached a specific impulse of 3800 s in the cycle analysis model. However, flow field structures cannot be captured in the two-dimensional theoretical analysis. In Ref. [21], the feasibility of RDRE was demonstrated experimentally with conditions corresponding to Mach 4 flight. Supersonic hydrogen-air mixtures flowed into the combustor via a divergent inlet. Wind tunnel tests of RDREs were performed by Frolov
Corresponding author. E-mail address: [email protected] (J. Wang).
https://doi.org/10.1016/j.actaastro.2019.12.022 Received 16 October 2019; Received in revised form 27 November 2019; Accepted 17 December 2019 Available online 21 December 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
Acta Astronautica 168 (2020) 174–181
K. Wu, et al.
ζ E hi J nmax p Pi S max t T0 u v w Yi
Nomenclature E F G S U η ωi U ‾ V‾ W ‾ ρ ρi ξ
convective flux vector in generalized coordinates convective flux vector in generalized coordinates convective flux vector in generalized coordinates source vector conservative variable vector one of the three axis of in generalized coordinates mass production of the ith specie contravariant velocity component in ξ direction contravariant velocity component in η direction contravariant velocity component in ζ direction density density of the ith specie one of the three axis of in generalized coordinates
one of the three axis of in generalized coordinates total energy per unit volume specific enthalpy of the ith specie the Jacobian determinant maximal allowable number of time steps static pressure static pressure of ith point maximal allowable total error time static temperature of the inflow velocity component in x direction velocity component in y direction velocity component in z direction mass fraction of the ith specie
Fig. 1. (a) Schematic of a rocket-mode RDE. (b) Fundamental flow field of an air-breathing RDE, with arrow 6 indicating the detonation propagating direction. Arrow 1 the detonation wave, arrow 2 downstream propagating oblique shock wave, arrow 3 upstream propagating oblique shock wave, arrow 4 contact surface, arrow 7 the normal shock wave.
unstably. However, previous numerical and experimental studies focused on the feasibility of RDRE, rather than the three-dimensional flow-field structures and stabilization mechanism. In this paper, a threedimensional numerical simulation is performed to investigate threedimensional flow structures and the instability mechanism of the RDRE. Two types of instability, the backflow and the strip-type fresh fuel distribution, are discussed for the first time. Some criteria and insights for RDRE design are also obtained through our study.
[22] with the airflow of Mach 4–8. Hydrogen was used as fuel. The specific impulse reached 3600 s. Two detonation mode, continuous spin detonation and longitudinal pulse detonation, were identified. However, due to the difficulty in flow field visualization, experimental studies failed to reveal the flow structures of RDRE. Numerical studies on RDREs were performed by Zhdan [23] for the first time. Two-dimensional flow-field structures and dynamics of RDRE with the hydrogen-oxygen mixtures were evaluated. Isentropic and shock-wave compression in a divergent inlet were both assumed. It was found that a detonation-induced “detached” oblique shock wave is formed in the exit of the inlet. However, three-dimensional effect was lost in this study. Numerical studies were also performed by Liu [24]. It was found that the upstream oblique shock wave induced by detonation has an effect on the supersonic flow. However, how does the upstream shock wave influence the behaviour of supersonic mixtures was unclear. In recent work by Smirnov [25], three-dimensional simulations were performed for RDREs. Reactant mixtures were introduced into the annular combustor via orifices. It was shown that for narrow combustor and rich-oxygen mixtures, detonation waves in the RDRE operated stably. However, detailed stabilization process was not analyzed. Compared with the rocket-mode RDEs, RDRE is easier to operate
2. Physical model and numerical method 2.1. Computational model The present RDRE model consists of two parts: the Laval inlet and the co-axial combustion chamber, as shown in Fig. 2. The convergent section of the Laval inlet is for stabilizing, decelerating and pressurizing, while the divergent section is designed to obtain the appropriate conditions of the fresh gases for detonation. Since the flow behind the throat is supersonic, weak disturbances in this section will not propagate upstream into the convergent section, thus guarantee continuous enough air being taken. The co-axial combustion chamber used in this 175
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Table 1 Comparison of the results for three meshes. Mesh
Coarse Medium Fine
Mass flow
Detonation wave height (mm)
(Kg/m2s )
Detonation velocity (m/s)
9.52 8.40 8.33
0.6900 0.6463 0.6443
1910.0 1910.4 1914.3
Fig. 2. Schematic of the physical model.
paper has an outer and inner diameter of 40 mm and 50 mm. The total length of our model is 53.5 mm, with a 38.5 mm long straight section, a 6 mm long convergent section and a 9 mm long divergent section. The width of the throat is 1 mm. 2.2. Governing equations and numerical methodology Oran [26] pointed that Euler equations are sufficient to resolve detonation structures and yield virtually identical results to that of NavierStokes equations, though smallest structures are absent. Therefore, three-dimensional reactive Euler equations in generalized coordinates are used in this paper, like most numerical studies of RDE [10,27,28]. Viscosity, thermal conduction, and mass diffusion are ignored. The Euler equations in generalized coordinates are given by
∂U ∂E ∂F ∂G + + + =S ∂t ∂ξ ∂η ∂ζ
Fig. 4. Fresh fuel layer distribution (Left) from inner to outer wall and (Right) in the inner wall.
(1)
S=
where the conservative variable vector U , the convective flux vectors E , F and G , and the source vector S are defined as:
1 U = [ρ , ρu, ρv , ρw, E , ρi ]T J
(2)
T 1 E = [ρU ‾ , ρUu ‾ + pξ x , ρUv ‾ + pξ y, ρUw ‾ + pξz , U ‾ (p + E ), ρi U ‾] J
(3)
T 1 [ρV‾, ρV‾ u + pηx , ρV‾ v + pηy , ρV‾ w + pηz , V‾ (p + E ), ρi V‾] J
(4)
F=
T 1 G = [ρW ‾ , ρWu ‾ + pζ x , ρWv ‾ + pζ y, ρWw ‾ + pζz , W ‾ (p + E ), ρi W ‾] J
1 [0,0,0,0, ρω˙ i ]T J
(6)
⎛ ξ x ξ y ξz ⎞ J = ⎜ ηx ηy ηz ⎟ ⎜ζ ζ ζ ⎟ ⎝ x y z⎠
(7)
U ‾ = uξ x + vξ y + wξz
(8)
V‾ = uηx + vηy + wηz
(9)
W ‾ = uζ x + vζ y + wζz
(10)
where u, v, w denotes three components of velocity, J is the Jacobian determinant, U ‾ , V‾ and W ‾ are three components of contravariant velocity in generalized coordinates, E and ρi indicate the total energy per
(5)
Fig. 3. Comparison of the detonation flow field for three grid sizes. Pressure contours for (a) coarse mesh, (b)medium size and (c)fine mesh. 176
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Fig. 5. (a) Fresh fuel distribution and (b) contours of the outer (First row) and inner wall (Second row) in 773 μs .
(MPWENO). Three-order total variation diminishing (TVD) RungeKutta method is employed to perform the time integration. A linearly point-implicit method is used for source terms to avoid stiffness problem [27]. Since the objective of the present study is the flow-field structures and operation mechanism of an RDRE, the mixing of fuel and oxidants is not considered. Therefore, premixed hydrogen-air mixtures are used in the present simulations. For an actual RDRE, the detonation waves and flames cannot propagate upstream into the inlet since only air exists in the inlet and no chemical reaction occurs in this area. To model the real-world physics, chemical reactions are closed artificially before the combustion chamber. As a result, chemical reactions are open at 17 mm, with a distance of 2 mm from the exit of the divergent section. As shock waves and transverse waves can still propagate upstream into the inlet, feedback between the inlet and the combustion chamber are modeled. The premixed stoichiometric hydrogen/air mixture flows into the inlet with a speed of Mach = 3. The static pressure p0 and static temperature T0 are set to 0.1 MPa and 300 K, respectively. The boundary conditions in the exit in this article are set carefully. Specifically, when the speed of exhausted gas is subsonic, the exit pressure is set to ambient pressure p∞ = 0.1 MPa. The exit pressure is extrapolated according to the pressure in the combustor if the velocity of exhausted gas is supersonic. Rigid wall conditions are set for the inner and outer wall. The inlet and the combustion chamber are initially filled with the fresh mixtures with a pressure of 0.1 MPa, a static temperature of 300 K, and speed of Mach = 3. Cold flow is taken into the RDRE to obtain the initial pressure and fuel distribution at first. Then at 45 μs , two antisymmetrically distributed 1-D detonation waves are mapped into the head of the combustor. Such an initiation method is widely used in previous numerical studies of RDE [10,24,27,30]. To validate the grid dependency, simulations are performed for coarse, medium and fine meshes in the present study. The computational domain introduced above are meshed with 238 × 906 × 35 grids (coarse mesh), 366 × 1006 × 40 grids (medium mesh) and 438 × 1106 × 50 grids (fine mesh), respectively. Grids near the walls are refined and the total number of grids are 7,546,980, 14, 727, 840 and 24, 221, 400 respectively. Therefore, the average grid size is 0.235 mm × 0.155 mm × 0.143 mm for coarse mesh, 0.153 mm × 0.140 mm ×
Fig. 6. Pressure gradient contour at z = 0.23 mm in 773 μs . A series of reflected shock waves can be observed.
unit volume and the density of the ith specie, respectively, and are defined as:
E=
∑ ρi hi − p +
ρi = ρYi
1 ρ (u2 + v 2 + w 2) 2
(11) (12)
where hi and Yi are the specific enthalpy and the mass fraction of ith specie. Hydrogen-air detailed chemistry [29] including nine species and nineteen elementary reactions are applied to obtain the mass production ωi . The following species are considered: [H, O, OH, H2, O2, HO2, H2O2, N2]. Strang's operator splitting method is used to split flow and chemical reactions. The flux terms are solved by the fifth-order monotonicity-preserving weighting essentially non-oscillatory scheme 177
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Fig. 7. (a) Pressure history for four points in the inlet, with 1 and 2 indicating points in the outer wall, and 3 and 4 representing points in the inner wall. (b) The distribution of Mach number along the middle-radii lines indicated in Fig. 5(b).
Figure 8. Fresh fuel layer distribution of the inner wall in 128, 148, 243 and 393 μs . Zigzags appear in 148 μs and, then, strip-type fresh fuel layer is formed.
allowable total error S max is presumed to be 3%.
0.125 mm for medium mesh and 0.127 mm × 0.127 mm × 0.100 mm for the fine mesh. Fig. 3 depicts the pressure contours of three meshes. It can be seen that two detonation waves exist in the combustion chamber and the flow fields are similar. Table 1 compares the detonation wave height, the mass flow and detonation velocity of the three meshes. It is seen that these values do not change obviously with different grid sizes. Considering the computational cost, the medium mesh is used in the present study. Additionally, the accumulation of errors occurs at each time step. Therefore, it is necessary to estimate the precision to verify the reliability of the simulations. Based on the method proposed by Smirnov et al. [31,32], the maximal allowable number of time steps nmax in the present study is 1.51 × 1016 when the
3. Results and discussion Fig. 1(b) gives the flow field of an RDRE with Mach 3 flight speed. Compared with that of rocket-mode RDEs, the flow field of RDRE is more complicated. Several different behaviors should be pointed. Firstly, a normal shock wave pointed by arrow 7 occurs due to the high pressure in the combustion chamber. It was not demonstrated in Ref. [23] because the coupling between inlet and the chamber was not considered by Zhdan. This normal shock wave fluctuates axially, adjusting its position based on the distance from the detonation waves. 178
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Fig. 9. Flow field evolution in the outer wall from 123 μs to 138 μs . The white line indicates the fresh fuel distribution.
to combustion products flowing back. This phenomenon leads to momentum losses and poses a threat for stable operations of RDRE. Therefore, it is worthwhile to investigate the underlying mechanism of this phenomenon. Fig. 5(b) presents the pressure contours of the inner and outer wall in 773 μs . The detonation waves in the outer wall is stronger than that in the inner wall due to the compressed effect of the concave outer wall. In the inner wall, each detonation wave is followed by a strong shock wave. This shock waves induces a oblique shock waves propagating upstream and downstream respectively, as the detonation wave does. Then, shock wave-induced (SW-induced) oblique shock waves collide with the normal shock wave in the divergent section of the inlet. The next task is to trace the source of the strong shock wave in Fig. 5(b) behind detonation waves. The combustion chamber is sliced in z = 23 mm , shown in Fig. 6. The compressed effect makes the intensity of the detonation wave increases from the inner wall to the outer wall. As a result, a strong reflected shock wave is then produced in the outer wall, as shown in Fig. 6. Continuous reflections happen subsequently between the outer and inner wall. Therefore, the shock wave in the inner wall behind the detonation wave is produced by the reflection of the inner and outer wall. Fig. 7(a) shows the pressure history of four points in the inlet over time. P1 and P2 are points in the outer wall, while P3 and P4 are points in the inner wall. Despite some oscillations, pressure in these four points is nearly constant except for initiation. This is consistent with the results of the experimental studies [21,22]. Frolov [22] and Wang [21] pointed out that flow conditions in the inlet are not affected by disturbances from the combustion chamber except for initiation. To explain why disturbances in the combustor can not further propagate upstream, Fig. 7(b) presents the change of Mach number in the middle radii. Three conditions are shown, as indicated by the red dashed lines in Fig. 5(b). These lines indicate different conditions since they intersect with the detonation-induced upstream propagating shock waves at different conditions. It can be seen in Fig. 7(b) that supersonic and subsonic fresh mixtures are clearly demarcated by the normal shock wave. This means the normal shock wave serves as a “moving wall”, impeding the propagation of disturbances from propagating further. This finding is meaningful in RDRE research for two reasons: (I) Reactions in the combustion chamber of detonation engine are extremely intense. (II) Flow-field structures in RDREs are complicated. Each detonation wave induces an unsteady upstream oblique shock wave that may destroy the inlet of an RDRE. Therefore, Laval-type inlet is a candidate for future RDRE application. Through reasonable parameter design, Laval inlet
Fig. 10. Pressure distribution at z = 0.23 mm in the outer wall for 128, 138, 148 and 158 μs , with only one detonation wave being shown.
Secondly, an upstream oblique shock wave, as indicated by arrow 3 in Fig. 1(b), is induced by the detonation wave. This oblique shock wave is induced by detonation and does not exist in rocket-mode RDEs. It rises in the detonation, propagates upstream into the inlet and finally collides with the normal shock wave before the throat. Two detonation waves exist in the present simulation when stable. Wedge-shaped fresh fuel layer indicates that the detonation blocks the fuel filling temporarily in RDREs, like in a rocket-mode RDE. The fresh fuel distribution of the inner wall in 773 μs is shown in Fig. 4, with the green line indicating the beginning of the combustion chamber. We highlight two interesting instability phenomena. The first one is the concave region, pointed by arrow 1. The second one is the strip-type fresh fuel layer, pointed by arrow 2. These instabilities have not been demonstrated before in rocket-mode RDE studies. 3.1. The concave region in the inner wall Fig. 5(a) shows the fresh fuel distribution of the outer and inner wall, where the white line represents the beginning of combustion chamber. The cylinder surfaces are unfolded based on radian. It is seen that in the inner wall, combustion products exists before the combustion chamber, while this phenomenon does not occur in the fuel distribution in the outer wall. Since chemical reactions occur only in the combustion chamber, the concave region of the inner wall is attributed 179
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Fig. 11. Logarithmic pressure gradient contours of the inner wall in 743 μs , 753 μs , 763 μs , 773 μs . Concentric pressure waves can be observed behind the detonation waves.
3.2. The formation and maintenance mechanism of strip-type fresh fuel layer Fig. 8 presents the evolution process of fresh fuel layer in the inner wall. It can be observed that the fresh fuel layer is nearly regular in 128 μs , after which several zigzags can be observed in 148 μs . Then zigzags grow into irregular strips in 243 μs . As continuous fresh injection is crucial for stable operations of RDE, it is worthwhile to investigate the mechanism of this instability. The pressure contours of the flow field during 123–138 μs is shown in Fig. 9. A four-to-two detonation transition can be observed. At first, four rotating detonation waves exist in the combustion chamber, with two detonation waves are weaker than the other two, as shown in Fig. 9(a). Then, in 128 μs , uncoupling happens due to the lack of fresh fuel. Subsequently, the uncoupled detonation waves propagate upstream and collides with the normal shock wave in the divergent section of the inlet in 133 μs . Disturbances produced by collisions are then prevented from propagating upstream. Fig. 10 presents the pressure distributions in the outer wall. It is seen that the detonation waves attenuate from 44 atm to 36 atm after collisions of uncoupling detonation waves and the normal shock wave during 128 μs to 148 μs . The detonation waves in the outer wall then strengths from 36 atm to nearly 50 atm since the pressure of fresh fuel before detonation waves increases when flowing through collision-caused high-pressure regions. Fluctuations of the intensity of detonation waves in the outer wall lead to fluctuations of the intensity of reflected shock waves. Subsequently, the intensity of the high-pressure region in the inner wall near the normal shock wave fluctuates. Fuel intake is affected during this period. Finally, small zigzags appear and grow into strip-type distribution. To explore the maintenance of strip-type fresh fuel layers, Fig. 11 presents the logarithmic pressure gradient in the inner wall for 743, 753, 763 and 773 μs . It is seen that a serious of periodic concentric weak pressure waves are formed behind the detonation waves and propagate in the opposite direction with detonation waves, as indicated by the red dashed circle. These concentric pressure waves have two potential sources: the reflected shock waves and the detonation.
Fig. 12. Pressure distribution of the inner wall at z = 0.21 mm in 743 μs , 753 μs , 763 μs , 773 μs .
has the potential to prevent pressure feedback in RDREs, while maintains acceptable performance. So far, we can safely come to the conclusion. Strong reflected shock waves and detonation waves in the inner wall induce upstream oblique shock waves which collide with the normal shock wave in the inlet. Since the strength of reflected shock waves and detonation waves are comparable in the inner wall, their collisions are violent. High-pressure regions are thus created. These high-pressure regions are then held by the normal shock wave. Fresh fuel accumulation is suppressed since fresh mixtures decelerate when flowing through these areas. Finally, the concave fresh fuel layer is formed in the inner wall over time. Besides, it is worth noting that the height of fresh fuel along the radii, shown in Fig. 4, reaches its peak in the middle radii for RDRE, due to the suppression effect of the inner wall. While the height of fresh fuel maximizes in the inner wall for rocket-mode RDEs [30].
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To investigate the origin of these concentric pressure waves, Fig. 12 shows the pressure distributions on z = 21 mm in the inner wall from 743 μs to 773 μs . It is seen that the intensity of the detonation wave in the inner wall oscillates over time while the intensity of the reflected shock wave is nearly unchanged. Therefore, weak pressure waves shown in Fig. 11 stem from oscillations of detonation waves in the inner wall. These transverse waves collide with the normal shock wave in the inlet, as shown in Fig. 11. High-pressure points are then produced in the divergent section, followed by a temporarily shutting-off effect on fueling. Subsequently, the strip-type fresh fuel layer is formed over time. The strength of detonation waves oscillates when sweeping from strips, producing transverse pressure waves again. In this way, the striptype fresh fuel layer remains in the flow field. It is worth noting that simulations conducted by Smirnov [33] exhibit that rotating detonation show unsteady behaviour because of the interactions of transverse waves, when using stoichiometric hydrogen/air as fuel. They demonstrate that this is due to a higher chemical sensitivity of stoichiometric mixtures to pressure variations.
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Tsuboi, Study of detailed chemical reaction model of hydrogen-air detonation, 39th Aerospace Sciences Meeting and Exhibit, 2001, p. 478. [30] R. Zhou, J.-P. Wang, Numerical investigation of shock wave reflections near the head ends of rotating detonation engines, Shock Waves 23 (5) (2013) 461–472, https://doi.org/10.1007/s00193-013-0440-0. [31] N. Smirnov, V. Betelin, R. Shagaliev, V. Nikitin, I. Belyakov, Y.N. Deryuguin, S. Aksenov, D. Korchazhkin, Hydrogen fuel rocket engines simulation using logos code, Int. J. Hydrogen Energy 39 (20) (2014) 10748–10756. [32] N. Smirnov, V. Betelin, V. Nikitin, L. Stamov, D. Altoukhov, Accumulation of errors in numerical simulations of chemically reacting gas dynamics, Acta Astronaut. 117 (2015) 338–355. [33] N. Smirnov, V. Nikitin, L. Stamov, E. Mikhalchenko, V. Tyurenkova, Rotating detonation in a ramjet engine three-dimensional modeling, Aero. Sci. Technol. 81 (2018) 213–224, https://doi.org/10.1016/j.ast.2018.08.003.
4. Conclusions A three-dimensional numerical simulation of the Laval-inlet equipped RDRE is performed for premixed hydrogen-air mixtures with detailed H2/air chemical reactions. Our results indicate that Laval inlet is an alternative for RDRE inlet design. It is also found that, even in RDREs, the normal shock wave in the exit of inlet has the capability of anti-pressure feedback. The formation of the backflow in RDRE is related to the reflected shock waves and detonation-induced upstream oblique shock waves. The collisions of these shock waves and the normal shock wave produce a high-pressure area. Fresh fuel accumulation is then suppressed periodically and finally the detonation products are pushed upstream. A four-to-two detonation wave-number switch occurs before stable operations of RDRE. The intensity of detonation waves oscillate due to the interactions of uncoupled detonation waves and the normal shock wave. This results in the oscillations of reflected shock wave-induced oblique shock waves, whose interactions with the normal shock wave makes fueling unstable. Finally, the strip-type fuel distribution is formed. After the RDRE operates stably, the strength of detonation wave oscillates periodically when sweeping across the strip-type fresh fuel layer, producing a series of concentric transverse pressure waves. Instant high-pressure rise are created due to the collisions of transverse waves and the normal shock wave, resulting in a temporarily blockage of fueling. This low-frequency instability remains due to the continuous feedback between the detonation waves, transverse pressure waves, the normal shock wave and the fresh fuel layer. Acknowledgements The present study is sponsored by National Natural Science Foundation of China (Grant No.91741202). References [1] E. Wintenberger, J. Shepherd, Thermodynamic analysis of combustion processes for propulsion systems, AIAA Paper, vol. 1033, 2004, https://doi.org/10.2514/6.20041033 2004. [2] W.H. Heiser, D.T. Pratt, Thermodynamic cycle analysis of pulse detonation engines, J. Propuls. Power 18 (1) (2002) 68–76, https://doi.org/10.2514/2.5899. [3] G. Roy, S. Frolov, A. Borisov, D. Netzer, Pulse detonation propulsion: challenges, current status, and future perspective, Prog. Energy Combust. Sci. 30 (6) (2004) 545–672, https://doi.org/10.1016/j.pecs.2004.05.001. [4] B. Voitsekhovskii, Stationary spin detonation, Sov. J. Appl. Mech. Tech. Phys. 3 (6) (1960) 157–164. [5] P. Wolański, Rotating detonation wave stability, 23rd International Colloquium on the Dynamics of Explosions and Reactive Systems, 2011.
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Research paper
Effects of flow-field structures on the stability of rotating detonation ramjet engine
T
Kevin Wu, Shujie Zhang, Mingyi Luan, Jianping Wang∗ Center for Combustion and Propulsion, CAPT & SKLTCS, Department of Mechanics and Engineering Sciences, College of Engineering, Peking University, Beijing, 100871, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Rotating detonation engine Simulation Instability Propulsion
Due to less attention received by rotating detonation ramjet engine (RDRE), the three-dimensional flow-field structures and their influence on stability are less understood than rocket-mode rotating detonation engine (RDE). This paper presents three-dimensional numerical investigations on the RDRE model with a Laval inlet. The computations are based on detailed H2/air chemistry. The simulation shows that, unlike rocket-mode RDEs, an upstream oblique shock wave is induced by detonation. This oblique shock wave is then prevented from propagating upstream further by the normal shock wave in the inlet. In addition, two instabilities in RDREs, namely, backflow and strip-type fresh fuel layer, are analyzed for the first time. Results indicate that the backflow is related to reflected shock waves and the upstream oblique shock wave that induced by detonation. While the strip-type fresh fuel distribution is formed due to the unstable propagation of detonation. Detonation waves interact with fresh fuel layer, transverse waves, and the normal shock waves, making the irregular fresh fuel layer remains in the combustion chamber.
1. Introduction Detonation combustion has a higher thermodynamic efficiency than traditional deflagration combustion [1,2]. There are some detonationbased propulsion devices, among which the pulse detonation engine (PDE) has been most widely researched. As PDE research is facing several obstacles [3], rotating detonation engine (RDE) has received more and more attention in the propulsion community. This is due to its high combustion efficiency, pressure gain property, the lack of the need for regular ignition, and its compact size. The concept of using rotating detonation waves for propulsion was first proposed by Voilsekhovskii [4] and developed by Wolanski [5,6]. Most RDE studies concentrate on its rocket mode. A typical rocketmode RDE is equipped with a co-axial annulus chamber. Fundamental flow structures of a rocket-mode RDE is shown in Fig. 1 (a) [7]. After initiation, the detonation wave rotates circumferentially along the annulus and sustains against the headwall. In the meantime, fresh mixtures are injected axially into the combustion chamber from fuel plenums. After fuel is consumed by detonation, combustion products expand behind the detonation waves. Thrust is produced by exhausting detonation products axially to the ambient atmosphere. Besides, a oblique shock wave attached with detonation exists in the post-detonation area. Various Injection patterns [8,9], geometries [10,11], reactants ∗
[12–14], nozzles [15,16] and instabilities [17,18] have been investigated for rocket-mode RDEs. Besides the rocket-mode RDE, preliminary studies have been performed for the ramjet-mode RDE in recent years. Schematic of a rotating detonation ramjet engine (RDRE) is shown in Fig. 1(b). Unlike rocket-mode RDEs, RDREs take supersonic oxidants (air) from the atmosphere. Because there is no head wall in RDREs, flow-field structures are different between rocket-mode RDEs. Therefore, conclusions derived for rocket-mode RDE should be re-assessed when applied to RDRE. Zhdan [19] firstly established a two-dimensional unsteady mathematical model for RDRE. Supersonic hydrogen-oxygen mixtures were introduced into the combustion chamber via a divergent inlet. It was shown that the continuous detonation was achievable in supersonic premixed mixtures with Mach = 3. Braun [20] proposed a cycle analysis model for RDREs. Supersonic mixtures were assumed to be taken into a convergent-divergent inlet isentropically. The RDRE model fueled by hydrogen/air operated stably between Mach 1.5–5 and reached a specific impulse of 3800 s in the cycle analysis model. However, flow field structures cannot be captured in the two-dimensional theoretical analysis. In Ref. [21], the feasibility of RDRE was demonstrated experimentally with conditions corresponding to Mach 4 flight. Supersonic hydrogen-air mixtures flowed into the combustor via a divergent inlet. Wind tunnel tests of RDREs were performed by Frolov
Corresponding author. E-mail address: [email protected] (J. Wang).
https://doi.org/10.1016/j.actaastro.2019.12.022 Received 16 October 2019; Received in revised form 27 November 2019; Accepted 17 December 2019 Available online 21 December 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
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ζ E hi J nmax p Pi S max t T0 u v w Yi
Nomenclature E F G S U η ωi U ‾ V‾ W ‾ ρ ρi ξ
convective flux vector in generalized coordinates convective flux vector in generalized coordinates convective flux vector in generalized coordinates source vector conservative variable vector one of the three axis of in generalized coordinates mass production of the ith specie contravariant velocity component in ξ direction contravariant velocity component in η direction contravariant velocity component in ζ direction density density of the ith specie one of the three axis of in generalized coordinates
one of the three axis of in generalized coordinates total energy per unit volume specific enthalpy of the ith specie the Jacobian determinant maximal allowable number of time steps static pressure static pressure of ith point maximal allowable total error time static temperature of the inflow velocity component in x direction velocity component in y direction velocity component in z direction mass fraction of the ith specie
Fig. 1. (a) Schematic of a rocket-mode RDE. (b) Fundamental flow field of an air-breathing RDE, with arrow 6 indicating the detonation propagating direction. Arrow 1 the detonation wave, arrow 2 downstream propagating oblique shock wave, arrow 3 upstream propagating oblique shock wave, arrow 4 contact surface, arrow 7 the normal shock wave.
unstably. However, previous numerical and experimental studies focused on the feasibility of RDRE, rather than the three-dimensional flow-field structures and stabilization mechanism. In this paper, a threedimensional numerical simulation is performed to investigate threedimensional flow structures and the instability mechanism of the RDRE. Two types of instability, the backflow and the strip-type fresh fuel distribution, are discussed for the first time. Some criteria and insights for RDRE design are also obtained through our study.
[22] with the airflow of Mach 4–8. Hydrogen was used as fuel. The specific impulse reached 3600 s. Two detonation mode, continuous spin detonation and longitudinal pulse detonation, were identified. However, due to the difficulty in flow field visualization, experimental studies failed to reveal the flow structures of RDRE. Numerical studies on RDREs were performed by Zhdan [23] for the first time. Two-dimensional flow-field structures and dynamics of RDRE with the hydrogen-oxygen mixtures were evaluated. Isentropic and shock-wave compression in a divergent inlet were both assumed. It was found that a detonation-induced “detached” oblique shock wave is formed in the exit of the inlet. However, three-dimensional effect was lost in this study. Numerical studies were also performed by Liu [24]. It was found that the upstream oblique shock wave induced by detonation has an effect on the supersonic flow. However, how does the upstream shock wave influence the behaviour of supersonic mixtures was unclear. In recent work by Smirnov [25], three-dimensional simulations were performed for RDREs. Reactant mixtures were introduced into the annular combustor via orifices. It was shown that for narrow combustor and rich-oxygen mixtures, detonation waves in the RDRE operated stably. However, detailed stabilization process was not analyzed. Compared with the rocket-mode RDEs, RDRE is easier to operate
2. Physical model and numerical method 2.1. Computational model The present RDRE model consists of two parts: the Laval inlet and the co-axial combustion chamber, as shown in Fig. 2. The convergent section of the Laval inlet is for stabilizing, decelerating and pressurizing, while the divergent section is designed to obtain the appropriate conditions of the fresh gases for detonation. Since the flow behind the throat is supersonic, weak disturbances in this section will not propagate upstream into the convergent section, thus guarantee continuous enough air being taken. The co-axial combustion chamber used in this 175
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Table 1 Comparison of the results for three meshes. Mesh
Coarse Medium Fine
Mass flow
Detonation wave height (mm)
(Kg/m2s )
Detonation velocity (m/s)
9.52 8.40 8.33
0.6900 0.6463 0.6443
1910.0 1910.4 1914.3
Fig. 2. Schematic of the physical model.
paper has an outer and inner diameter of 40 mm and 50 mm. The total length of our model is 53.5 mm, with a 38.5 mm long straight section, a 6 mm long convergent section and a 9 mm long divergent section. The width of the throat is 1 mm. 2.2. Governing equations and numerical methodology Oran [26] pointed that Euler equations are sufficient to resolve detonation structures and yield virtually identical results to that of NavierStokes equations, though smallest structures are absent. Therefore, three-dimensional reactive Euler equations in generalized coordinates are used in this paper, like most numerical studies of RDE [10,27,28]. Viscosity, thermal conduction, and mass diffusion are ignored. The Euler equations in generalized coordinates are given by
∂U ∂E ∂F ∂G + + + =S ∂t ∂ξ ∂η ∂ζ
Fig. 4. Fresh fuel layer distribution (Left) from inner to outer wall and (Right) in the inner wall.
(1)
S=
where the conservative variable vector U , the convective flux vectors E , F and G , and the source vector S are defined as:
1 U = [ρ , ρu, ρv , ρw, E , ρi ]T J
(2)
T 1 E = [ρU ‾ , ρUu ‾ + pξ x , ρUv ‾ + pξ y, ρUw ‾ + pξz , U ‾ (p + E ), ρi U ‾] J
(3)
T 1 [ρV‾, ρV‾ u + pηx , ρV‾ v + pηy , ρV‾ w + pηz , V‾ (p + E ), ρi V‾] J
(4)
F=
T 1 G = [ρW ‾ , ρWu ‾ + pζ x , ρWv ‾ + pζ y, ρWw ‾ + pζz , W ‾ (p + E ), ρi W ‾] J
1 [0,0,0,0, ρω˙ i ]T J
(6)
⎛ ξ x ξ y ξz ⎞ J = ⎜ ηx ηy ηz ⎟ ⎜ζ ζ ζ ⎟ ⎝ x y z⎠
(7)
U ‾ = uξ x + vξ y + wξz
(8)
V‾ = uηx + vηy + wηz
(9)
W ‾ = uζ x + vζ y + wζz
(10)
where u, v, w denotes three components of velocity, J is the Jacobian determinant, U ‾ , V‾ and W ‾ are three components of contravariant velocity in generalized coordinates, E and ρi indicate the total energy per
(5)
Fig. 3. Comparison of the detonation flow field for three grid sizes. Pressure contours for (a) coarse mesh, (b)medium size and (c)fine mesh. 176
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Fig. 5. (a) Fresh fuel distribution and (b) contours of the outer (First row) and inner wall (Second row) in 773 μs .
(MPWENO). Three-order total variation diminishing (TVD) RungeKutta method is employed to perform the time integration. A linearly point-implicit method is used for source terms to avoid stiffness problem [27]. Since the objective of the present study is the flow-field structures and operation mechanism of an RDRE, the mixing of fuel and oxidants is not considered. Therefore, premixed hydrogen-air mixtures are used in the present simulations. For an actual RDRE, the detonation waves and flames cannot propagate upstream into the inlet since only air exists in the inlet and no chemical reaction occurs in this area. To model the real-world physics, chemical reactions are closed artificially before the combustion chamber. As a result, chemical reactions are open at 17 mm, with a distance of 2 mm from the exit of the divergent section. As shock waves and transverse waves can still propagate upstream into the inlet, feedback between the inlet and the combustion chamber are modeled. The premixed stoichiometric hydrogen/air mixture flows into the inlet with a speed of Mach = 3. The static pressure p0 and static temperature T0 are set to 0.1 MPa and 300 K, respectively. The boundary conditions in the exit in this article are set carefully. Specifically, when the speed of exhausted gas is subsonic, the exit pressure is set to ambient pressure p∞ = 0.1 MPa. The exit pressure is extrapolated according to the pressure in the combustor if the velocity of exhausted gas is supersonic. Rigid wall conditions are set for the inner and outer wall. The inlet and the combustion chamber are initially filled with the fresh mixtures with a pressure of 0.1 MPa, a static temperature of 300 K, and speed of Mach = 3. Cold flow is taken into the RDRE to obtain the initial pressure and fuel distribution at first. Then at 45 μs , two antisymmetrically distributed 1-D detonation waves are mapped into the head of the combustor. Such an initiation method is widely used in previous numerical studies of RDE [10,24,27,30]. To validate the grid dependency, simulations are performed for coarse, medium and fine meshes in the present study. The computational domain introduced above are meshed with 238 × 906 × 35 grids (coarse mesh), 366 × 1006 × 40 grids (medium mesh) and 438 × 1106 × 50 grids (fine mesh), respectively. Grids near the walls are refined and the total number of grids are 7,546,980, 14, 727, 840 and 24, 221, 400 respectively. Therefore, the average grid size is 0.235 mm × 0.155 mm × 0.143 mm for coarse mesh, 0.153 mm × 0.140 mm ×
Fig. 6. Pressure gradient contour at z = 0.23 mm in 773 μs . A series of reflected shock waves can be observed.
unit volume and the density of the ith specie, respectively, and are defined as:
E=
∑ ρi hi − p +
ρi = ρYi
1 ρ (u2 + v 2 + w 2) 2
(11) (12)
where hi and Yi are the specific enthalpy and the mass fraction of ith specie. Hydrogen-air detailed chemistry [29] including nine species and nineteen elementary reactions are applied to obtain the mass production ωi . The following species are considered: [H, O, OH, H2, O2, HO2, H2O2, N2]. Strang's operator splitting method is used to split flow and chemical reactions. The flux terms are solved by the fifth-order monotonicity-preserving weighting essentially non-oscillatory scheme 177
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Fig. 7. (a) Pressure history for four points in the inlet, with 1 and 2 indicating points in the outer wall, and 3 and 4 representing points in the inner wall. (b) The distribution of Mach number along the middle-radii lines indicated in Fig. 5(b).
Figure 8. Fresh fuel layer distribution of the inner wall in 128, 148, 243 and 393 μs . Zigzags appear in 148 μs and, then, strip-type fresh fuel layer is formed.
allowable total error S max is presumed to be 3%.
0.125 mm for medium mesh and 0.127 mm × 0.127 mm × 0.100 mm for the fine mesh. Fig. 3 depicts the pressure contours of three meshes. It can be seen that two detonation waves exist in the combustion chamber and the flow fields are similar. Table 1 compares the detonation wave height, the mass flow and detonation velocity of the three meshes. It is seen that these values do not change obviously with different grid sizes. Considering the computational cost, the medium mesh is used in the present study. Additionally, the accumulation of errors occurs at each time step. Therefore, it is necessary to estimate the precision to verify the reliability of the simulations. Based on the method proposed by Smirnov et al. [31,32], the maximal allowable number of time steps nmax in the present study is 1.51 × 1016 when the
3. Results and discussion Fig. 1(b) gives the flow field of an RDRE with Mach 3 flight speed. Compared with that of rocket-mode RDEs, the flow field of RDRE is more complicated. Several different behaviors should be pointed. Firstly, a normal shock wave pointed by arrow 7 occurs due to the high pressure in the combustion chamber. It was not demonstrated in Ref. [23] because the coupling between inlet and the chamber was not considered by Zhdan. This normal shock wave fluctuates axially, adjusting its position based on the distance from the detonation waves. 178
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Fig. 9. Flow field evolution in the outer wall from 123 μs to 138 μs . The white line indicates the fresh fuel distribution.
to combustion products flowing back. This phenomenon leads to momentum losses and poses a threat for stable operations of RDRE. Therefore, it is worthwhile to investigate the underlying mechanism of this phenomenon. Fig. 5(b) presents the pressure contours of the inner and outer wall in 773 μs . The detonation waves in the outer wall is stronger than that in the inner wall due to the compressed effect of the concave outer wall. In the inner wall, each detonation wave is followed by a strong shock wave. This shock waves induces a oblique shock waves propagating upstream and downstream respectively, as the detonation wave does. Then, shock wave-induced (SW-induced) oblique shock waves collide with the normal shock wave in the divergent section of the inlet. The next task is to trace the source of the strong shock wave in Fig. 5(b) behind detonation waves. The combustion chamber is sliced in z = 23 mm , shown in Fig. 6. The compressed effect makes the intensity of the detonation wave increases from the inner wall to the outer wall. As a result, a strong reflected shock wave is then produced in the outer wall, as shown in Fig. 6. Continuous reflections happen subsequently between the outer and inner wall. Therefore, the shock wave in the inner wall behind the detonation wave is produced by the reflection of the inner and outer wall. Fig. 7(a) shows the pressure history of four points in the inlet over time. P1 and P2 are points in the outer wall, while P3 and P4 are points in the inner wall. Despite some oscillations, pressure in these four points is nearly constant except for initiation. This is consistent with the results of the experimental studies [21,22]. Frolov [22] and Wang [21] pointed out that flow conditions in the inlet are not affected by disturbances from the combustion chamber except for initiation. To explain why disturbances in the combustor can not further propagate upstream, Fig. 7(b) presents the change of Mach number in the middle radii. Three conditions are shown, as indicated by the red dashed lines in Fig. 5(b). These lines indicate different conditions since they intersect with the detonation-induced upstream propagating shock waves at different conditions. It can be seen in Fig. 7(b) that supersonic and subsonic fresh mixtures are clearly demarcated by the normal shock wave. This means the normal shock wave serves as a “moving wall”, impeding the propagation of disturbances from propagating further. This finding is meaningful in RDRE research for two reasons: (I) Reactions in the combustion chamber of detonation engine are extremely intense. (II) Flow-field structures in RDREs are complicated. Each detonation wave induces an unsteady upstream oblique shock wave that may destroy the inlet of an RDRE. Therefore, Laval-type inlet is a candidate for future RDRE application. Through reasonable parameter design, Laval inlet
Fig. 10. Pressure distribution at z = 0.23 mm in the outer wall for 128, 138, 148 and 158 μs , with only one detonation wave being shown.
Secondly, an upstream oblique shock wave, as indicated by arrow 3 in Fig. 1(b), is induced by the detonation wave. This oblique shock wave is induced by detonation and does not exist in rocket-mode RDEs. It rises in the detonation, propagates upstream into the inlet and finally collides with the normal shock wave before the throat. Two detonation waves exist in the present simulation when stable. Wedge-shaped fresh fuel layer indicates that the detonation blocks the fuel filling temporarily in RDREs, like in a rocket-mode RDE. The fresh fuel distribution of the inner wall in 773 μs is shown in Fig. 4, with the green line indicating the beginning of the combustion chamber. We highlight two interesting instability phenomena. The first one is the concave region, pointed by arrow 1. The second one is the strip-type fresh fuel layer, pointed by arrow 2. These instabilities have not been demonstrated before in rocket-mode RDE studies. 3.1. The concave region in the inner wall Fig. 5(a) shows the fresh fuel distribution of the outer and inner wall, where the white line represents the beginning of combustion chamber. The cylinder surfaces are unfolded based on radian. It is seen that in the inner wall, combustion products exists before the combustion chamber, while this phenomenon does not occur in the fuel distribution in the outer wall. Since chemical reactions occur only in the combustion chamber, the concave region of the inner wall is attributed 179
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Fig. 11. Logarithmic pressure gradient contours of the inner wall in 743 μs , 753 μs , 763 μs , 773 μs . Concentric pressure waves can be observed behind the detonation waves.
3.2. The formation and maintenance mechanism of strip-type fresh fuel layer Fig. 8 presents the evolution process of fresh fuel layer in the inner wall. It can be observed that the fresh fuel layer is nearly regular in 128 μs , after which several zigzags can be observed in 148 μs . Then zigzags grow into irregular strips in 243 μs . As continuous fresh injection is crucial for stable operations of RDE, it is worthwhile to investigate the mechanism of this instability. The pressure contours of the flow field during 123–138 μs is shown in Fig. 9. A four-to-two detonation transition can be observed. At first, four rotating detonation waves exist in the combustion chamber, with two detonation waves are weaker than the other two, as shown in Fig. 9(a). Then, in 128 μs , uncoupling happens due to the lack of fresh fuel. Subsequently, the uncoupled detonation waves propagate upstream and collides with the normal shock wave in the divergent section of the inlet in 133 μs . Disturbances produced by collisions are then prevented from propagating upstream. Fig. 10 presents the pressure distributions in the outer wall. It is seen that the detonation waves attenuate from 44 atm to 36 atm after collisions of uncoupling detonation waves and the normal shock wave during 128 μs to 148 μs . The detonation waves in the outer wall then strengths from 36 atm to nearly 50 atm since the pressure of fresh fuel before detonation waves increases when flowing through collision-caused high-pressure regions. Fluctuations of the intensity of detonation waves in the outer wall lead to fluctuations of the intensity of reflected shock waves. Subsequently, the intensity of the high-pressure region in the inner wall near the normal shock wave fluctuates. Fuel intake is affected during this period. Finally, small zigzags appear and grow into strip-type distribution. To explore the maintenance of strip-type fresh fuel layers, Fig. 11 presents the logarithmic pressure gradient in the inner wall for 743, 753, 763 and 773 μs . It is seen that a serious of periodic concentric weak pressure waves are formed behind the detonation waves and propagate in the opposite direction with detonation waves, as indicated by the red dashed circle. These concentric pressure waves have two potential sources: the reflected shock waves and the detonation.
Fig. 12. Pressure distribution of the inner wall at z = 0.21 mm in 743 μs , 753 μs , 763 μs , 773 μs .
has the potential to prevent pressure feedback in RDREs, while maintains acceptable performance. So far, we can safely come to the conclusion. Strong reflected shock waves and detonation waves in the inner wall induce upstream oblique shock waves which collide with the normal shock wave in the inlet. Since the strength of reflected shock waves and detonation waves are comparable in the inner wall, their collisions are violent. High-pressure regions are thus created. These high-pressure regions are then held by the normal shock wave. Fresh fuel accumulation is suppressed since fresh mixtures decelerate when flowing through these areas. Finally, the concave fresh fuel layer is formed in the inner wall over time. Besides, it is worth noting that the height of fresh fuel along the radii, shown in Fig. 4, reaches its peak in the middle radii for RDRE, due to the suppression effect of the inner wall. While the height of fresh fuel maximizes in the inner wall for rocket-mode RDEs [30].
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To investigate the origin of these concentric pressure waves, Fig. 12 shows the pressure distributions on z = 21 mm in the inner wall from 743 μs to 773 μs . It is seen that the intensity of the detonation wave in the inner wall oscillates over time while the intensity of the reflected shock wave is nearly unchanged. Therefore, weak pressure waves shown in Fig. 11 stem from oscillations of detonation waves in the inner wall. These transverse waves collide with the normal shock wave in the inlet, as shown in Fig. 11. High-pressure points are then produced in the divergent section, followed by a temporarily shutting-off effect on fueling. Subsequently, the strip-type fresh fuel layer is formed over time. The strength of detonation waves oscillates when sweeping from strips, producing transverse pressure waves again. In this way, the striptype fresh fuel layer remains in the flow field. It is worth noting that simulations conducted by Smirnov [33] exhibit that rotating detonation show unsteady behaviour because of the interactions of transverse waves, when using stoichiometric hydrogen/air as fuel. They demonstrate that this is due to a higher chemical sensitivity of stoichiometric mixtures to pressure variations.
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4. Conclusions A three-dimensional numerical simulation of the Laval-inlet equipped RDRE is performed for premixed hydrogen-air mixtures with detailed H2/air chemical reactions. Our results indicate that Laval inlet is an alternative for RDRE inlet design. It is also found that, even in RDREs, the normal shock wave in the exit of inlet has the capability of anti-pressure feedback. The formation of the backflow in RDRE is related to the reflected shock waves and detonation-induced upstream oblique shock waves. The collisions of these shock waves and the normal shock wave produce a high-pressure area. Fresh fuel accumulation is then suppressed periodically and finally the detonation products are pushed upstream. A four-to-two detonation wave-number switch occurs before stable operations of RDRE. The intensity of detonation waves oscillate due to the interactions of uncoupled detonation waves and the normal shock wave. This results in the oscillations of reflected shock wave-induced oblique shock waves, whose interactions with the normal shock wave makes fueling unstable. Finally, the strip-type fuel distribution is formed. After the RDRE operates stably, the strength of detonation wave oscillates periodically when sweeping across the strip-type fresh fuel layer, producing a series of concentric transverse pressure waves. Instant high-pressure rise are created due to the collisions of transverse waves and the normal shock wave, resulting in a temporarily blockage of fueling. This low-frequency instability remains due to the continuous feedback between the detonation waves, transverse pressure waves, the normal shock wave and the fresh fuel layer. Acknowledgements The present study is sponsored by National Natural Science Foundation of China (Grant No.91741202). References [1] E. Wintenberger, J. Shepherd, Thermodynamic analysis of combustion processes for propulsion systems, AIAA Paper, vol. 1033, 2004, https://doi.org/10.2514/6.20041033 2004. [2] W.H. Heiser, D.T. Pratt, Thermodynamic cycle analysis of pulse detonation engines, J. Propuls. Power 18 (1) (2002) 68–76, https://doi.org/10.2514/2.5899. [3] G. Roy, S. Frolov, A. Borisov, D. Netzer, Pulse detonation propulsion: challenges, current status, and future perspective, Prog. Energy Combust. Sci. 30 (6) (2004) 545–672, https://doi.org/10.1016/j.pecs.2004.05.001. [4] B. Voitsekhovskii, Stationary spin detonation, Sov. J. Appl. Mech. Tech. Phys. 3 (6) (1960) 157–164. [5] P. Wolański, Rotating detonation wave stability, 23rd International Colloquium on the Dynamics of Explosions and Reactive Systems, 2011.
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