Air non-premixed rotating detonation engine under different equivalence ratios

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Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios Hongtao Zheng, Qingyang Meng, Ningbo Zhao*, Zhiming Li, Fuquan Deng College of Power and Energy Engineering, Harbin Engineering University, Harbin, 150001, China

highlights
Detailed formation process under various equivalence ratios is investigated.
Insights into physical mechanism of new detonation front generation.
Effects of equivalence ratio on RDW propagating stability is explored.
Propulsion performance of RDE under various equivalence ratios is compared.

article info

abstract

Article history:

In this paper, three-dimensional numerical simulations are performed to investigate the

Received 4 June 2019

formation and propagation characteristics of rotating detonation wave in a non-premixed

Received in revised form

engine. By changing the mass flow rate of H2 and fixing air mass flow rate, the effects of

25 October 2019

equivalence ratio involving fuel lean and rich operating conditions are mainly discussed.

Accepted 2 November 2019

Numerical results show that equivalence ratio plays a very critical role in the formation

Available online xxx

process and propagation mode, which further affects the propulsion performance of rotating detonation engine significantly. For current numerical geometry and operating

Keywords:

conditions, the lean limit of equivalence ratio for formatting a stable RDW is about 0.4,

Numerical simulation

dual-wave mode (at equivalence ratio of 0.6, 0.8, 1.0 and 1.4) and single-wave mode (at

Rotating detonation wave

equivalence ratio of 1.2) are obtained, respectively. When equivalence ratio is 1.0, rotating

Combustion

detonation engine can exhibit excellent operating performance with the shortest forma-

Equivalence ratio

tion time, best propagation stability, middling class thrust and specific impulse. Besides,

Propagation stability

the pressure contour analysis indicates that the effects of equivalence ratio and mass flow rate of H2 on the collision strength and times during the re-initiation process are the main mechanisms for determining the formation possibility and propagation mode of rotating detonation wave. Besides, the intensity of accumulated pressure wave and distributions of equivalence ratio are two important factors for the generation of new detonation wave front. Furthermore, it is also detected from the comparisons of the propulsion performance that the effects of equivalence ratio on thrust and specific impulse under fuel lean conditions are more significant than those under fuel-rich conditions. © 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

* Corresponding author. E-mail address: [email protected] (N. Zhao). https://doi.org/10.1016/j.ijhydene.2019.11.014 0360-3199/© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article as: Zheng H et al., Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2019.11.014

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Introduction Combustion process plays an important role in the energy conversion and utilization. Compared to the conventional deflagration in most of modern propulsion systems, detonation, consisting of the leading shock wave coupled with chemical reaction zone, is a very complex supersonic combustion phenomenon. In recent years, more and more investigations indicate that detonation-based engine is expected to be one potential alternative technology in the propulsion systems due to its higher thermal cycle efficiency, wider operating range, and more compact engine design [1]. Nowadays, pulsed detonation engine (PDE) [2] and rotating detonation engine (RDE) [3e6] are two common types of detonation-based propulsion systems. For PDE, thrust is generated by the intermittent detonation wave in a tubeshaped combustor, and the corresponding propulsion performance is directly depended on the reactants injection, ignition, deflagration-to-detonation transition (DDT) [7e9], expansion, and purge [10e12]. Such complex operating characteristics result in many practical challenges and impedes of the effective integration of PDE with the existing cycles. As an alternative of PDE, RDE is based on the rotating detonation wave (RDW), which continuously propagates in the circumferential direction within annular combustor [13e15]. Compared to PDE, RDE has many obvious advantages, such as once detonation initiation [16,17], higher frequency (more than several kHz) [18], more uniform and steady thrust [19], wider flight Mach numbers [20], less energy loss associated with DDT [21,22], easier integration with turbine [23], etc. Owing to the above attractions, many different RDE concepts have been developed by researchers from America [24], Russia [25], China [26], Japan [27], Poland [21], Singapore [28] and French [29], respectively. In order to effectively realize the initiation, self-sustaining stable propagation and mode control of RDW, a proper equivalence ratio (ER) of fuel-oxidizer mixture is on demand because it can directly determine the chemical reactivity and further affect the propagation characteristics of RDW. Besides, influenced by the injection strategy of fuel and oxidizer, the local equivalence ratio of non-premixed RDE is generally uneven and obviously different from that in premixed combustor, which leads that the operating characteristics of RDW are usually more complex. Therefore, the effect of equivalence ratio on RDW is always a very challenging topic in the study of RDE [30], so focusing on this, a lot of experimental investigations were carried out by many researchers. Schauer’s research group mainly focused on the experimental investigations of RDE. They were the first to explore and design the premixed RDE which prevented the flashback [31e33]. According to the experimental results, they pointed that fuel selection was crucial for the successful operation of premixed RDE and ethylene-air mixture was the preferred propellant. Subsequently, by employing an optically accessible RDE, they experimentally investigated the characteristics of RDW when the mass flow rate, equivalence ratio and inlet structure were changed [34,35]. They pointed that as the air mass flow rate was over high, single-wave mode would switch to dual-wave mode. Selecting H2 and air as reactant, Li et al. [36] carried

out an experiment to investigate the initiation and propagation process of RDW under different equivalence ratios. Their results showed that the variation of equivalence ratio could significantly affect the propagation features (mode, frequency, velocity and stability) of RDW, and the formation time of stable RDW decreased rapidly and rise up slightly with the increase of equivalence ratio from fuel-lean to fuel-rich condition. Based on the experiment and premixed numerical simulation, Wang et al. [37] studied the effect of equivalence ratio on the propagating speed of H2/O2 RDW. They found that with the increase of equivalence ratio, the detonation speed increased at the beginning and then decreased, which was consistent with the conclusions obtained by Li et al. [36]. Furthermore, Wang et al. [38] experimentally investigated the feasibility of C2H4/air RDW in a non-premixed hollow channel. It was indicated that single RDW could be initiated and selfsustained when the air mass flow rate was 465e689 g/s and the equivalence ratio was 0.47e1.06. Afterwards, considering the effect of air-inlet slot width, Zhou et al. [39] experimentally examined the propagating stability of RDW under three equivalence ratios (0.88, 1.08 and 1.27) and found that the stability of RDW could be improved via increasing equivalence ratio to some extent within the scope of their study. Anand et al. [40] and Xie et al. [41] respectively explored the propagating diagram of H2/air RDW on various operating conditions. Their experimental results consistently showed that due to the comprehensive effects of mass flow rate and equivalence ratio, different RDW modes were observed. They thought one important reason for this phenomenon was the change of the reactant injection height and the detonation cell size. Besides, the results of Anand et al. [40] also indicated that the static pressure gain of RDE was dependent on the equivalence ratio and the air flow rate. In addition, Deng et al. [42] experimentally investigated the effects of equivalence ratio and mass flow rate on the propagation process of RDW and pointed out that changing equivalence ratio was a feasible way to control the propagation modes. Numerical investigations were also employed by many researchers to explore the propagating characteristics of RDW. Schwer and Kailasanath numerically investigated the rotating detonation wave in various perspectives, such as the boundary conditions (stagnation pressure, back pressure) [43], geometric structures of RDE (area ratio of injectors to head-end wall, annular diameter, chamber length, design of inlet structure reducing pressure feedback, converging-diverging nozzle attached to the outlet) [44e46] and the effects of mixing rate on RDW [47]. Especially, they numerically explored the non-premixed RDE by extending a stoichiometric hydrogen and air Induction-time Parameter Model to cover a range of equivalence ratios. The results showed that this approach could reproduce correct trend of cell sizes when the equivalence ratio changed and this approach could also accurately recover the CJ properties, which was treated as a reasonable method for the future work. Frolov et al. [48] were the first to numerically conduct three-dimensional investigation of RDE with non-premixed injection scheme. According to the results, the height of RDW was close to that measured in experiment. However, it was observed two or three RDWs in experiment, while there was only one wave in their numerical results. Furthermore,

Please cite this article as: Zheng H et al., Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2019.11.014

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they carried out further investigations and finally obtained the consistent results with experiments. Using Large Eddy Simulations (LES) coupled to Reynolds-Averaged-Navier Stokes (RANS) method, Cocks et al. [49] conducted the numerical investigations of non-premixed RDE under different mass flow rates. They pointed that it was a challenge for non-premixed RDE to achieve high combustion efficiency due to the compact space of practical device. Yao et al. [50] conducted numerical investigation to study the spontaneous formation of multi-wave mode. They thought the occurrence of new detonation wave depended on the collision of shock waves and whether there was sufficient reactant. High pressure produced by collision of shock waves could re-initiate the reactant and a new detonation wave was established. Later, Yao and Wang [51] numerically explored the stabilization process of RDW by using multiple ignitions method to gain multiple waves. The results revealed that one and two ignition spots would obtain the corresponding number of RDW. However, four ignition spots obtained eight-wave mode finally. They attributed this to the shock collision and the re-initiation of timely feeding reactant. From the above literature review, it is easily found that the previous studies involving the effect of equivalence ratio on non-premixed RDW mainly focus on the experimental analysis, while few on the combustion field measurement and numerical simulation to explain the initiation, propagation and mode switch mechanism of RDW, especially for non-premixed RDW. Considering the complexity and difficulties to experimentally capture the detailed behavior of rotating detonation wave in RDE, this paper performs a three-dimensional numerical simulation to study the operating characteristics and mechanisms of H2/air nonpremixed RDE under a wide equivalence ratio ranging from 0.4 to 1.4. First, combustion field information is discussed in detail to understand the initiation mechanism of RDW with different propagation modes. Second, the effect of equivalence ratio on the propagating mode of RDW is determined by analyzing the typical feature parameters. Finally, the performance of RDE under different equivalence ratios is compared.

Numerical modeling and method Governing equations The governing equations used in current study are listed as following: vr þ V,ðrUÞ ¼ 0 vt

(1)

vr þ V,ðrUUÞ ¼ Vp þ V,ðts Þ vt

(2)


  2 ts ¼ m VU þ VUT  V , UI 3

(3)

3

vðrYi Þ þ V , ðrUYi Þ ¼  V , Ji þ u_ i vt

(4)

! X vðrEÞ þ V , ðUðrE þ pÞÞ ¼ V , keff VT  hi Ji þ ðts , UÞ þ u_ c vt i

(5)

E¼e þ u_ c ¼

 2 U 2

X hi u_ i

(6)

(7)

i

where r is the density, t is time, U is the velocity, p is pressure, ts is the stress tensor and it is given by Equation (3), I is the unit tensor. Yi is the mass fraction of each species, Ji is the diffusion flux of species i, u_ i is the net rate of production of species i by chemical reaction, e is the internal energy. keff is the effective conductivity, T is temperature, hi is the enthalpy of species i.

Computational domain Fig. 1 presents the schematic of the three-dimensional nonpremixed RDE and its detailed cross-section configuration. As displayed in Fig. 1, air is injected into combustor via a converging-diverging slot with an inlet width (D) of 5 mm and a throat width (Wn) of 0.6 mm H2 is injected into plenum firstly and then enters into RDE through 90 cylindrical channel with a diameter (Dh) of 0.8 mm. The other key parameters of RDE including inner radius (Rin), outer radius (Rout), air inlet slot length (Ln) and combustor body length (Lc) are 35 mm, 40 mm, 15 mm and 60 mm, respectively.

Numerical methods The computational fluid dynamics software ANSYS FLUENT is used to perform the numerical calculation. Based on the ideal gas assumption and considering the compactly coupled relationship between temperature and pressure of RDW, density based Navier-Stokes solver is employed to solve the equations [52]. Due to the mixing process occurs in the non-premixed inlet structures, the viscosity of reactant is considered. Consequently, standard k-ε turbulence model and standard wall function are selected [53]. By checking the yþ values of our numerical results, it is found that the yþ values range from 30 to 60, which is reasonable for standard wall function. Besides, the convective fluxes are calculated by a flux-vector splitting scheme called Advection Upstream Splitting Method (AUSM) that is good at providing exact resolution of contact and shock discontinuities. Four-order Runge-Kutta scheme is employed for time advancement. Referring the published numerical investigations on RDE, laminar finiterate model [54] with one-step reaction [55e57] is applied. The reaction rate constant (kf) is calculated by Arrhenius formula as shown in Equation (8), where A denotes the preexponential factor, 1.03  109 s1. T is the temperature. b is the temperature exponent, 0. Ea is the activation energy, 1.26  105 J. R is the specific gas constant, 8.314 J/(mol$K). More

Please cite this article as: Zheng H et al., Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2019.11.014

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Fig. 1 e (a) Three-dimensional geometric model and (b) two-dimensional cross section information of non-premixed RDE. information about numerical method can be found in our previous study [58] and Ref. [59]. kf ¼ ATb expð  Ea = RTÞ

(8)

1.55 MPa, 2942 K and 1964 m/s. Additionally, the location of ignition center is at the point (x ¼ 17.5 mm, y ¼ 37.5 mm, z ¼ 0 mm) and the direction of initial velocity in the ignition zone is towards z axial (see Fig. 2).

Boundary condition As shown in Fig. 1, mass flow inlet boundaries are used for the inlets of air and H2. The outlet of RDE selects pressure-outlet boundary. The walls are treated to be adiabatic and no-slip condition in this study. Table 1 lists the detailed boundary parameters under different equivalence ratios ranging from _ air and m _ H2, 0.4 to 1.4. The mass flow rate of air and H2 are m _ total denotes the total mass flow rate in respectively. m combustor. T means the temperature of reactant, and pb is the outlet back pressure. Moreover, the non-reacting inflation process is simulated before ignition. And one ignition zone is simulated as a Chapman-Jouquet (CJ) detonation sphere kernel (sphere of 5 mm diameter) from the stoichiometric H2/ air pre-detonation tube. The pressure, temperature and tangential velocity in ignition zone are respectively set as

Independence test and model validation Before performing the numerical analysis, independence validations of grid and time step sizes are conducted firstly. As shown in Fig. 3, the hexahedral meshes are generated by ANSYS ICEM to discretize the computational domain. Then,

Table 1 e Boundary parameters of H2 and air under various equivalence ratios. _ air (kg/s) m _ H2 (kg/s) m _ total (kg/s) ER T (K) pb (MPa) Case m #1 #2 #3 #4 #5 #6

0.1 0.1 0.1 0.1 0.1 0.1

0.0012 0.0018 0.0024 0.0030 0.0036 0.0042

0.1012 0.1018 0.1024 0.1030 0.1036 0.1042

0.4 0.6 0.8 1.0 1.2 1.4

300 300 300 300 300 300

0.1 0.1 0.1 0.1 0.1 0.1

Fig. 2 e Position of ignition region.

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Fig. 3 e Grid of computational domain. taking case #4 (ER ¼ 1.0) as the benchmark, Fig. 4 compares the variations of pressure at monitoring point (x ¼ 17 mm, y ¼ 39.8 mm, z ¼ 0 mm) marked in Fig. 1(a) under different maximum grid sizes (Dx) and time step sizes (t). The total cells numbers are 5.2  106 (Dx ¼ 0.1 mm), 3.8  106 (Dx ¼ 0.2 mm) and 2.8  106 (Dx ¼ 0.5 mm) respectively. Obviously, the results can be independent when Dx is 0.2 mm and t is 0.02 ms To ensure the numerical accuracy of other cases, the necessary

Fig. 4 e Effects of (a) grid size and (b) time solved step on pressure in case #4.

independence validations of grid and time step sizes are also performed. Two-dimensional physical model, of which outer and inner diameters are equal to the three-dimensional physical model used in current work, is selected to calculate the detonation wave speed with one-step reaction mechanism under various equivalence ratios. The ignition zone center is at the point (0 mm, 37.5 mm). For the two-dimensional physical model, Fig. 5 shows the contours of pressure when stable detonation wave is formed under various equivalence ratios. Additionally, Fig. 6 shows the variations of pressure with time (Ignition time is 0 ms) at the monitoring point P1 (37.5 mm, 0 mm) and P2 (0 mm, 37.5 mm). Dt is the time interval of RDW propagates through two monitoring points and the detonation speed can be calculated by the circumferential distance between the monitoring points and Dt. Table 2 shows the comparison between two-dimensional numerical speed and three-dimensional numerical speed (non-premixed physical model in current study) with CJ speeds. Note that the detonation wave speed is neglected for ER ¼ 0.4 in three-dimensional non-premixed physical model because the detonation wave is not able to be established in this case. It can be seen that the numerical detonation speed for two-dimensional model is slightly over-predicted than CJ speed but the deviations from CJ values are less than 2%. For three-dimensional non-premixed model, deficit speed is observed in each case because of the non-uniform distributions of reactant ahead of RDW and its deviations from CJ speed are less than 8%. Additionally, Fig. 7 shows the comparisons of ignitiondelay time and laminar flame speed when initial pressure (p0) is 1 atm among one-step mechanism (used in current study), GRI 3.0 mechanism and Experimental data. According to Fig. 7(a), it can be found that the ignition-delay time in onestep mechanism is slightly higher than those in GRI 3.0 mechanism and experimental data but the largest deviation from GRI 3.0 mechanism lowers than 4.4%, from experimental data lowers than 6%. Besides, according to Fig. 7(b), it can been found that the laminar flame speeds in one-step mechanism agree well with GRI 3.0 mechanism and experimental data. The largest deviations of one-step mechanism from experimental data and GRI 3.0 are both around 9%. On this basis, in order to further evaluate the reliability of present numerical approach, Table 3 compares the numerical detonation speed and experimental data published by Anand

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Fig. 5 e Contours of pressure under different equivalence ratios.

et al. [40] and Deng et al. [42] under the equivalence ratios of 1.0 and 1.2. It is found from Table 3 that both the numerical speed and frequencies agree well with the experimental results. Based on above reacting mechanism and numerical method validations, it can be found that present numerical approach is reasonable to investigate the propagation characteristics of RDW.

Results and discussion RDW formation process analysis To clearly understand the formation and propagation mechanism of RDW with different propagating modes, the pressure contours at different moments in case #5 (ER ¼ 1.2, RDW with single-wave), case #6 (ER ¼ 1.4, RDW with dual-wave) and case #1 (ER ¼ 0.4, detonation formation failure) are analyzed respectively in the following. Fig. 8 displays the pressure contours in combustor at different moments when the equivalence ratio is 1.2. When a clockwise CJ detonation kernel is initiated, a single-way detonation wave is established. Later, since the high reacting activity of hydrogen, two obvious detonation fronts can be generated rapidly and propagates in the opposite directions,

as shown in Fig. 8(b). Then after a short time, the above two detonation fronts collide at 66 ms Due to the high energy loss triggered by collision, the leading shock wave and chemical reaction zone decouple, which leads that the detonation fronts degrade to be weak transmit shock waves as seen in Fig. 8(d). These two counter-rotating waves collide at t ¼ 154 ms As a result, the propagating strength of transmit shock wave further decreases at t ¼ 174 ms Similarly, the transmit shock waves collide again at t ¼ 264 ms It is clearly observed from Fig. 8(g) that the pressure in collision region is very weak and lowers than 1 MPa. However, the generated transmit shock waves can still propagate continuously. Subsequently, with the following continuous collisions, one stronger transmit shock wave appears and its pressure front reaches more than 2 MPa as shown in Fig. 8(h). However, the other transmit shock wave is usually so weak that can be disappeared by the stronger one (see t ¼ 356 ms). Afterwards, as shown in Fig. 8(j)e(p), the stronger transmit shock wave eventually evolves into a stable RDW which can self-sustaining propagate in combustor. This kind of RDW formation process is called re-initiation phenomenon by Yao et al. [62], Smirnov et al. [63] and Lau-Chapdelaine [64]. This could be attributed to shock wave compression and pressure wave accumulation near outer wall. Fig. 9 describes the pressure contours at various moments when the equivalence ratio is 1.4. Comparing the results with

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Fig. 6 e Pressure history at monitoring point (a) ER ¼ 0.4, (b) ER ¼ 0.6, (c) ER ¼ 0.8, (d) ER ¼ 1.0, (e) ER ¼ 1.2, (f) ER ¼ 1.4.

those in Fig. 8, it is found that there are some similarities but also obvious differences between the formation processes of RDW with single-wave and dual-wave modes. As shown in Fig. 9(a)e(q), once a clockwise CJ detonation kernel is initiated in combustor, a series of complex phenomena including formation of two detonation fronts, collisions and strength decrease of detonation fronts or transmit shock waves

happen in sequence. But the differences are that the times of collisions in Fig. 9 are obviously increased. Moreover, compared to case #5 (ER ¼ 1.2), a local strong shock wave is generated and then developed to be a new high pressure spot at t ¼ 670 ms After a period of adaptive propagating process, the activated pressure spot evolves into a stable self-sustaining RDW, which forms the dual-wave propagating mode, as

Table 2 e Comparisons of numerical detonation speed with CJ speed. ER

CJ (m/s)

Numerical (2D) (m/s)

Deviation (%)

Numerical (3D) (m/s)

Deviation (%)

0.4 0.6 0.8 1.0 1.2 1.4

1489 1705 1895 1964 2026 2065

1494 1710 1896 2004 2058 2088

0.33 0.29 0.05 1.99 1.55 1.10

e 1576.7 1716.1 1820.5 1956.6 1903.9

e 7.5 7.7 7.3 3.4 7.8

Please cite this article as: Zheng H et al., Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2019.11.014

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According to Fig. 10, the intensity of accumulated pressure wave is about 0.4 MPa in case of ER ¼ 1.2. Besides, fuel-rich conditions are observed near the accumulated pressure wave, which causes the reactant is not able to be ignited. As a result, the reacting rate in this region is very low. However, the intensity of accumulated pressure wave in case of ER ¼ 1.4 is higher than that in case of ER ¼ 1.2 as shown in Fig. 11. Additionally, according to the distributions of equivalence ratio near the accumulated pressure wave, favorable condition (equivalence ratio is close to 1) is observed in case of ER ¼ 1.4. Consequently, fast reacting rate is observed in this region, which is about 3000 kmol/(m3$s). The ignited reactant strengthens the pressure wave and a new detonation wave is established soon in case of ER ¼ 1.4. It can be inferred that the intensity of accumulated pressure wave and distributions of equivalence ratio are two important factors to affect the occurrence of new detonation wave. Fig. 12 shows the pressure contours at various moments in combustor when the equivalence ratio is 0.4. Note that the gradient color scale in Fig. 12 is only half of that in Figs. 8 and 9 in order to show clear contours. From the results of Fig. 12, it is indicated that compared to above two cases, since both equivalence ratio and mass flow rate are lower, the initial clockwise CJ detonation kernel can rapidly degrade to be the low pressure shock waves after a few times of collisions. However, unlike the results in above two cases, the propagating direction of shock waves is changed constantly in present operating condition. This similar phenomenon was also observed in the fuel-lean cases of Ref. [30,40] and thought to be a main mechanism for causing energy loss and propagating instability. Finally, as the reaction proceeds, the energy decreases to a low value, which leads that the shock waves fails and RDW cannot be successfully formed. Fig. 13 shows the distributions of hydrogen ahead of rotating detonation wave under various equivalence ratios. Obviously, as the decrease of equivalence ratio, separate hydrogen streams are observed especially when the equivalence ratio is 0.4. The separate distributions of hydrogen stream indicate the poor mixing condition, which might cause the failure of RDW establishment in case of ER ¼ 0.4. When the equivalence ratio is higher than 1.0, the separate hydrogen stream interacts with each other significantly and the distributions of hydrogen become uniform. However, for case of ER ¼ 1.4, a portion of hydrogen is observed behind the detonation wave as shown in Fig. 13(f). This unburned hydrogen survives and flows towards the outlet presenting a long stripe which connects the detonation front and the outlet. Table 4 shows the incomplete combustion percentage of hydrogen. According to Table 4, it is seen that as the increase of

Fig. 7 e Parameter comparisons between different chemical mechanisms.

shown in Fig. 9(s)e(t). The occurrence of activated pressure spot might be related to the accumulation of pressure wave assembled by the outer wall. In order to explore the physical mechanism of new detonation wave front, Fig. 10 and Fig. 11 show the variations of pressure, reacting rate (u) and equivalence ratio with circumferential degree when the accumulated pressure wave shows up (pointed by the arrow).

Table 3 e Velocity and frequency comparisons of RDW obtained by simulation and experiment. No.

ER

1 2 3 4

1.0 1.2 1.0 1.0

Speed (m/s)

Deviation (%)

Simulation

Experiment [40]

1720 1956 e e

1680 1785 e e

0.2 8.7 e e

Frequency (Hz) Simulation

Experiment [42]

e e 7022 13760

e e 6980 13500

Deviation (%) e e 0.6 1.9

Please cite this article as: Zheng H et al., Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2019.11.014

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Fig. 8 e Pressure contours of combustor at ER ¼ 1.2. equivalence ratio, the percentage of unburned hydrogen increases due to lack of oxidizer.

RDW propagating mode and parameters analysis Fig. 14 presents the pressure and temperature histories at monitoring point (x ¼ 17 mm, y ¼ 39.8 mm, z ¼ 0 mm) in different cases. According to the time-varying characteristics of parameters, it can be found that under each equivalence ratio discussed in this study, RDW is not able to form directly when an initial CJ detonation ignition kernel is triggered in RDE. Instead, detonation front decouples and quenches quickly after collisions due to diffraction effect, which leads that both pressure and temperature decrease dramatically.

Then after a transitory idle time period, failure or initiation of RDW may happen. This kind of detonation re-initiation phenomenon has been observed in many experimental and numerical investigations [60,65]. On this basis, analyzing the pressure and temperature peaks shown in Fig. 14, we can found that except for case #1 (ER ¼ 0.4), all of the other five cases can successfully form the self-sustaining RDW. This means that for the present RDE geometric structure and operating conditions, the lean equivalence ratio limit of forming RDW is about 0.4. In addition, further parameter comparisons in Fig. 14(b)e(f) show that affected by equivalence ratio, RDW exhibits different initiation and formation processes, especially for the initiation time, propagating mode and stability.

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Fig. 9 e Pressure contours of combustor at ER ¼ 1.4.

Defining the time interval from ignition to the formation of stable RDW is Dtf. The formation of stable RDW is the moment when the relative deviation between RDW speed and CJ speed lowers than 9%, which distinguishes the formation process from stable process as shown in Fig. 14. Fig. 15 displays the variations of Dtf with equivalence ratio. Due to the failure of

establishing RDW when the equivalence ratio is 0.4, it is omitted in Fig. 15. The result of Fig. 15 shows that the equivalence ratio has a great effect on the formation time of RDW. With the increase of equivalence ratio, Dtf decreases rapidly and reaches minimum when equivalence ratio is 1.0. Then, as equivalence ratio increases further, Dtf rises up accordingly

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Fig. 10 e Variations of pressure, reaction rate and equivalence ratio with circumferential direction when ER ¼ 1.2.

Fig. 11 e Variations of pressure, reaction rate and equivalence ratio with circumferential direction when ER ¼ 1.4.

while there is little difference when the equivalence ratio is 1.2 and 1.4. According to the trend of Dtf in Fig. 15, as expected, we can find that RDW is easier to initiate when the equivalent ratio is close to be stoichiometric. Besides, a careful inspection of Fig. 15 reveals that due to the common effects of reactivity (equivalence ratio) and mass flow rate, the changes of Dtf under fuel-rich condition are slight in comparison with those under fuel-lean condition. The effective weights of these two factors (equivalence ratio and mass flow rate) on the Dtf might different on lean or rich fuel condition, causing the different trends of Dtf. This effectively verifies the experimental phenomenon of Li et al. [30]. Fig. 16 shows the effect of equivalence ratio on parameter fi ¼ 1/Dti that represents the propagating frequency of rotating detonation waves in combustor. Dti is the time interval between two pressure peaks during stable stage (as shown in Fig. 14). According to Fig. 16, it is clearly seen that when equivalence ratio is 0.6, 0.8, 1.0 and 1.4, the propagation

frequency of RDW ranges from 10 kHz to 15 kHz because of the dual-wave mode, while it is only about 8 kHz when equivalence ratio is 1.2 due to the single-wave mode. From the changes of frequency in Fig. 16, it is known that the propagating stability of RDW can be affected significantly by equivalence ratio. In this study, the following relative standard deviation (sf) of frequency is used to quantitatively analyze the average speed stability of RDW. The smaller sf, the propagating speed of RDW is more stable. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
n 2 i P ðn  1Þ fi  fave sf ¼

1

fave

(9)

Fig. 17 shows the variations of sf with equivalence ratio. It is observed from Fig. 17 that due to the effects of reaction activity, sf under equivalence ratio of 0.6 and 1.0 are respectively maximum and minimum. This indicates that for over fuellean condition, RDW shows very poor speed stability while it

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Fig. 12 e Pressure contours of combustor at ER ¼ 0.4. exhibits the highest speed stability when stoichiometric reactants are filled. Moreover, with the increase of equivalence ratio from 0.6 to 1.2, sf decreases firstly and then increases. Subsequently, when equivalence ratio is further increased, sf decreases as expected since the propagating mode of RDW is switched from single-wave to dual-wave and dual-wave mode shows higher stability. Similar phenomenon also has been found in several other studies [42,66,67]. Besides, Anand et al. [67] explained the above changing characteristic in view of detonation height. They thought that as the increase number of RDWs, both RDW height and velocity decreased, which might enhance the stability. In order to further explore the intensity stability of RDW, Fig. 18 shows the instantaneous pressure peak variations of RDW at various moments during approximately two

propagating periods. It can be seen that when the equivalence ratio is 1.2, relatively large variations of pressure show up, this attributes to the poor stability of single-wave mode compared to dual-wave mode. The rest cases show relatively slight fluctuations. Similarly, in order to quantitatively measure the intensity stability, relative standard deviation (sp) is defined as following: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n 2

P pi  pave ðn  1Þ sp ¼

1

pave

(10)

where pi is the pressure peak of every moment, pave is the average pressure. The high sp means severe fluctuations. Fig. 19 plots the variations of sp with different equivalence

Please cite this article as: Zheng H et al., Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2019.11.014

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Fig. 13 e Distributions of hydrogen ahead of RDW under various equivalence ratios. ratios. It can be found that relatively high value of sp is observed when the equivalence ratio is 1.2. Note that different from the distributions of sf, very low sp is obtained when the equivalence ratios are 0.6 and 0.8. This indicates the intensity Table 4 e Incomplete combustion percentage of hydrogen under different equivalence ratios. ER

0.4

0.6

0.8

1.0

1.2

1.4

Unburned H2 percentage (%)

0.9

1.8

2.1

2.5

7.8

9.2

of RDW is more stable on fuel-lean condition compared to fuel-rich condition. Generally the instability of pressure is affected by the detonation wave propagation instability and the chemical reaction activity of the reactant. Due to the single-wave mode characteristics for case of ER ¼ 1.2, its height of RDW will be the highest because only one detonation wave consumes the injected reactant. Therefore, the pressure intensity is the highest as shown in Fig. 18 and the chemical reaction activity is also higher than other cases, which enhances the inherent instability in the detonation wave front.

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Fig. 14 e Variations of pressure and temperature with times under different equivalence ratios (a) ER ¼ 0.4, (b) ER ¼ 0.6, (c) ER ¼ 0.8, (d) ER ¼ 1.0, (e) ER ¼ 1.2, (f) ER ¼ 1.4.

Fig. 15 e Effects of equivalence ratios on Dtf.

So the highest pressure instability is obtained in case of ER ¼ 1.2. Fig. 20 shows the axial variations of detonation wave pressure, due to the RDW cannot be established when equivalence ratio is 0.4, its axial pressure variations are not plotted in Fig. 20. It can be seen that all the curves present the similar trend, as the axial length increases, the pressure increases firstly and then drops down. The maximal pressure locates at the axial length is about 15 mm for each case. Note that the pressure always shows relatively high level when the equivalence ratio is 1.2 compared to other cases. This is because the reactant has longer time to jet into combustor for single-wave mode compared to dual-wave mode. Therefore, more injecting reactant in front of RDW causes the increase of RDW height, so the axial pressure keeps higher level than other cases. On this basis, Fig. 21 presents the relationship of two normalized perimeters of RDW (htotal/l and wch/l), and

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Fig. 16 e Propagation frequency under various equivalence ratios.

compares it with the single-dual RDW transition boundary proposed by George et al. [66] as displayed in the following formulas:

htotal ¼

VD ¼

m_ total rfill ,VD ,wch

2,p,rP ,fave nD

Fig. 18 e Time-varying of pressure peaks under various equivalence ratios.

Fig. 19 e Variations of sp with equivalence ratios.

Fig. 17 e Variations of sf with equivalence ratios.

htotal wch ¼ 3:7  l l

15

(11)

(12)

(13)

where htotal is the total height of RDW. For dual-wave mode, it will be the mathematical sum of two RDW heights. l is the cell size of the reactant. In this study, l of H2/air with different equivalence ratios is obtained from the experiments of Ciccarelli et al. [66]. wch is the width of RDE and equal to 5 mm

shown in Fig. 1(b). rfill is the density of reactant. VD is the velocity of RDW. rP is the distance from monitoring point to x axis. fave ¼ fi/n is the average value of n statistical frequencies from Fig. 16. nD is the wave front number of RDW. As shown in Fig. 21, the RDW propagating modes obtained in this study agrees with the prediction principle of George et al. [66]. That is, RDW propagates with single-wave mode when htotal/l < 3.7-wch/l. Otherwise, it is dual-wave mode. Besides, unlike to the locations of the cases (ER ¼ 0.8, 1.0, 1.4) in Fig. 21, those of other two cases (ER ¼ 0.6 and 1.2) are very close to the transition boundary. This is because as the equivalence ratio is changed from stoichiometric to lean condition, on the one hand, the reaction activity decreases and the corresponding cell size increases. On the other hand, the total mass flow rate decreases, causing the height of RDW drops. Above two factors both promote the transition of dualwave to single-wave mode, so the cell size or height of RDW in the case #2 (ER ¼ 0.6) might have reached the critical condition

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of mode switch. Unlikely, for case #5 (ER ¼ 1.2), the adjacent cases (ER ¼ 1.0 and 1.4) are both dual-wave mode, which means increasing the mass flow rate of hydrogen (height of RDW increases) or reducing the mass flow rate of hydrogen (move to stoichiometric condition) both promotes the single to dual-wave mode transition. Based on above discussion, these two cases (ER ¼ 0.6 and 1.2) locate at the unstable region of mode switch.

Propulsion performance of RDE Net thrust (F) and fuel-based specific impulse (Isp, fuel) are two important indexes for quantitatively evaluating the propulsion performance of RDW based combustor. In this section, the following formulas will be used. I



F¼ out

Fig. 20 e Axial variations of pressure. Isp;fuel ¼

Fig. 21 e Relationship between numerical results and transition boundary.

 rout V2x þ pout  p∞ dA  m_ air ,Vair;in

(14)

F m_ H2 g

(15)

where rout are density and pout are pressure of reactant on the outlet. Vx is the axial velocity. p∞ is the ambient pressure. _ H2 is the mass flow rate of Vair,in is the air injecting velocity, m H2, equal to 0.1 kg/s g is the gravity acceleration. Fig. 22 shows the effects of equivalence ratio on the time variation of F and Isp, fuel. It can be seen from Fig. 22 that after a short time of ignition, both F and Isp, fuel in different cases are increased quickly and reach a significant high level. This may attribute to the axial propagation of CJ detonation wave at initial stage, as discussed in Fig. 8(b). When detonation wave spreads close to the outlet of combustor, the huge F and Isp, fuel are obtained, respectively. Then, due to the complex collisions and evolutions of various shock waves, F and Isp, fuel decrease and become irregular. Afterwards, with the formation and stable propagation of RDW, F and Isp, fuel gradually trend to the constant values except for case of ER ¼ 0.4. The above parameters still fluctuate during the entire time period in case of ER ¼ 0.4. The main reason for this phenomenon is the long duration of collision and failure of shock waves. Besides, the parameters comparisons in Fig. 22 also indicate that affected by equivalence ratio, the time interval from ignition to the establishment of stable F and Isp, fuel (Dts) are different. The definition of stable F and Isp, fuel is the moment

Fig. 22 e Time-varying of (a) thrust and (b) specific impulse under different equivalence ratios. Please cite this article as: Zheng H et al., Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2019.11.014

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with equivalence ratio and inlet mass flow rate, but specific impulse is mainly dependent on equivalence ratio. This may be meaningful to design and control RDE in practice.

Conclusions To clear the formation and propagation characteristics of H2/ air non-premixed RDW and their effects on propulsion performance of RDE under different lean and rich operating conditions, six cases with equivalence ratio ranging from 0.4 to 1.4 are numerically analyzed in this paper. The main conclusions are drawn as following:

Fig. 23 e Effects of equivalence ratios on Dts.

Fig. 24 e Variations of performance parameters with different equivalence ratios.

when the relative deviations between adjacent local maximum and minimum of thrust and specific impulse lower than 10%. In view of this, Fig. 23 presents the variations of Dts with equivalence ratios. According to Fig. 23, with the increase of equivalence ratio, Dts drops firstly and rises up subsequently. And as expected, the lowest Dts occurs at the equivalence ratio of 1.0. This means that case #4 (ER ¼ 1.0) takes the shortest time to obtain the stable propulsion performance. Furthermore, Fig. 24 shows the variations of average F and Isp, fuel of RDW based combustors on stable stage (from 1200 ms to 2400 ms). According to Fig. 24, it can be seen the maximum thrust and specific impulse is observed in cases of ER ¼ 1.2 and 0.6 respectively. However, it is unexpected that although thrust at equivalence ratio of 0.4 is lower than other cases, the corresponding specific impulse can reach relatively moderate level. In order to better understand this phenomenon, more numerical investigations and detailed analysis of parameters need to be carried out in the future. Combined the present results with those in our previous study [58], it can be found that thrust of RDW based combustors is a complex function

(1) Equivalence ratio plays a crucial role in the formation and propagating process of RDW. The feature parameters of RDW or RDE including formation time, propagating mode, stability and propulsion performance are all obviously related to equivalence ratio. For the geometrical structure of combustor and operating parameters used in this paper, three kinds of RDW phenomena including single-wave (ER ¼ 1.2), dual-wave (ER ¼ 0.6, 0.8, 1.0 and 1.4) and failure (ER ¼ 0.4) are obtained. Besides, our numerical results show favorable agreement with the single-double boundary line experimentally proposed by George et al. Moreover, when equivalence ratio is close to 1.0, RDE can exhibit the shortest formation time, superior stability, middling class thrust and specific impulse. (2) Reinitiation phenomenon is effectively captured. Affected by the comprehensive effects of increasing mass flow rate and decreasing equivalence ratio, the collision strength and times during re-initiation process directly determine the formation possibility and propagating mode of RDW. Moreover, the occurrence of new detonation wave is dependent on the intensity of pressure wave and distributions of equivalence ratio. (3) With the increase of equivalence ratio, thrust and specific impulse of RDE can behave different trends. In this study, the highest thrust and specific impulse are obtained at equivalence ratio of 1.2 and 0.6, respectively. But the lowest ones are obtained at equivalence ratio of 0.4 and 1.4. Due to the parameters limitation of this study, more detailed investigations need to be further discussed to clear this complex phenomena and possible physical mechanism. (4) Auto-ignition phenomenon is a very important factor affecting the propagation of rotating detonation wave and its fundamental investigations have been deeply conducted [69,70]. In the future, it is necessary to numerically explore the auto-ignition characteristics of rotating detonation wave in non-premixed rotating detonation combustor.

Acknowledgments The authors would like to acknowledge the National Nature Science Foundation of China (Grant No. 51709059) and the

Please cite this article as: Zheng H et al., Numerical investigation on H2/Air non-premixed rotating detonation engine under different equivalence ratios, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2019.11.014

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Fundamental Research Funds for the Central Universities (Grant No. 3072019CF0306, Grant No. HEUCFP201719, Grant No. 3072019CFJ0308, Grant No. 3072019CFJ0301) for supporting this work.

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