1 Introduction

Detonation waves can adiabatically compress the premixed gas and increase its temperature instantly before the start of combustion via shock waves [1]. Thus, a detonation cycle utilizing detonation waves is expected to have a high thermal efficiency because the entropy increase is smaller than that caused by constant-pressure combustion. Therefore, a thermodynamic cycle utilizing detonation has a higher theoretical thermal efficiency than the Brayton cycle utilized in conventional engines [2]. Engines that use detonation waves include a pulse detonation engine (PDE) [3], a rotating detonation engine (RDE) [4], and a reflective shuttling detonation engine (RSDE) [5]. Detonation-based combustors are attracting attention as next-generation propulsion systems for aerospace vehicles and have been demonstrated in space [6, 7].

An RDE is a detonation engine in which thrust is obtained by the continuous rotational propagation of detonation waves through a double cylindrical combustion chamber. Owing to their high theoretical thermal efficiency, RDEs are expected to achieve a pressure gain (PG). PG is the total pressure of the product at the engine exit exceeds the total pressure of the supplied oxidizer. Kaemming and Paxson [8] proposed the equivalent available pressure (EAP), the total exit pressure that can be back-calculated from the experimentally measured thrust, to evaluate PG. Generally, the PG is evaluated utilizing the pressure gain ratio (PGR), which is the ratio of the EAP to the total pressure of the oxidizer plenum. Brophy and Codoni [9] demonstrated that the EAP and static pressure measured at the combustor exit and the total pressure calculated assuming an isentropic flow generally agreed. Bach et al. [10] experimentally measured the total pressure at an RDE outlet utilizing a keel probe and proposed an empirical model in conjunction with other reported EAP results. Walters et al. [11] conducted an experimental study utilizing natural gas and air with increased chamber pressure and air preheating temperature, achieving 90% of the theoretical performance. Brophy et al. [12] increased the combustion chamber width to reduce heat transfer and wall friction losses and confirmed operation at PGR of 96.9%. Stout and Baratta [13] demonstrated the operation of a low-pressure loss RDE, achieving a PGR of 98.6%.

As described above, many institutions have evaluated the PG performance of RDEs with EAP. However, PG has not been achieved experimentally, and there is a large discrepancy between RDE operation under ideal assumptions and experimental results. Many non-ideal phenomena can be considered as possible causes. Chacon and Gamba [14] confirmed parasitic combustion (PC) occurring ahead of the detonation wave and commensal combustion (CC) occurring behind the detonation wave in visualization experiments utilizing a racetrack RDE. The PC was suggested to reduce the propagation velocity and peak pressure of the detonation wave, resulting in a lower thermal efficiency. Burr and Paulson [15] performed a thermodynamic cycle analysis of RDE, including PC and CC, and a sensitivity analysis of the specific impulse and propagation velocity of these non-detonation combustions. Noda et al. [16] experimentally investigated the relationship between the fraction of PC and detonation Mach number utilizing a distributed heat release model [15] and showed that the Mach number decreased with increasing fraction of PC.

Injector geometry strongly affects the stability of detonation waves and mixing of fuel and oxidizer in RDEs. Matsuoka et al. [17] experimentally investigated the backflow process of the burned gas by varying the ratio of the oxidizer injector cross-sectional area to the cross-sectional area of the combustion chamber. The results indicated that as the injector cross-sectional area increased, the backflow and hydrodynamic blockage ratio increased, and the propagation velocity of the detonation wave decreased. Paxson and Schwer [18] demonstrated numerically that there is a tradeoff between increasing the injector cross-sectional area and the operational stability of the RDE. Rankin et al. [19] investigated the fuel–air mixing process utilizing the PLIF method. As a result, it was observed that only fuel injection is stopped by detonation waves, and an oxidizer-rich layer is formed ahead of the fuel–oxidizer mixture.

Recently, Athmanathan et al. [20] employed the PLIF method to visualize the combustion chamber cross section of an RDE and investigated the internal structure of the flow. These results suggest that the recirculation zone near the oxidizer injector outlet mixes the previously burned gas and unburned propellant, thereby changing the structure of the detonation wave. Therefore, it is important to investigate the influence of the recirculation zone on detonation waves. Many studies have been conducted mainly with a channel expansion angle of \(90^{\circ }\) at the injector outlet. However, the flow path from the injector to the combustion chamber should be smooth in terms of total pressure loss of the oxidizer. The study focused on the effect of the sudden expansion on the detonation wave structure and PG performance. In the experiment, the channel expansion angle \(\alpha \) near the oxidizer injector outlet was varied \(90^{\circ }\) and \(30^{\circ }\). As a result, the propagation velocity of the detonation wave at \(\alpha =90^{\circ }\) was approximately 5% higher than that at \(\alpha =30^{\circ }\). A comparison utilizing pressure increase ratios obtained from fluctuating pressure in the combustion chamber suggested that the Mach number of the detonation wave was higher at \(\alpha =90^{\circ }\). The PGR was evaluated utilizing the EAP obtained from the thrust measurement, and it was not confirmed to differ with changes in \(\alpha \).

2 Experimental arrangement

2.1 Experimental apparatus

Fig. 1
figure 1

Schematic cross-sectional view of RDE utilized in combustion test: a overview of RDE, b injector geometry (unit: mm)

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Figure 1a presents a schematic of the RDE utilized in the experiment. The dimensions of the annular-shaped combustion chamber were as follows: outer diameter, 60 mm; inner diameter, 50 mm; width, 5 mm; and length, 30 mm. The polar coordinate system was set at the bottom of the combustor as the origin. As illustrated in Fig. 1b, fuel was injected into the combustor from injectors equally spaced in a 72-hole inner cylinder (diameter \(=0.6\,\hbox {mm}\) at \(z=-1.5\,\hbox {mm}\)). The total area of the fuel injector was \(A_{\textrm{f}}=20.4\,{\hbox {mm}}^{\textrm{2}}\), and the fuel injection direction was the same as that of the r-axis. The oxidizer was injected in the z-axis via a 1.9-mm-wide slit between the outer cylinder and the fuel plenum component on the inner cylinder side. The total area of the oxidizer injector was \(A_{\textrm{ox}}=346.8\, {\hbox {mm}}^{\textrm{2}}\). The ratio of the cross-sectional area of the combustion chamber to the oxidizer inlet was \( A_{\textrm{ox}}/A_{\textrm{ch}}=0.401\). The channel expansion angle can be changed to two patterns: \(\alpha =90^{\circ }\) and \(\alpha =30^{\circ }\).

The static pressure of the combustion chamber was measured at six points (\(z=-22,-10, 0, 5, 10\), and \(20\,\hbox {mm}\)) utilizing pressure sensor ports on the outer cylinder wall. All pressure sensors were 5-kHz pressure transducers (PAA-23, Keller). The fluctuating pressure in the combustion chamber was measured at \(z=0\,\hbox {mm}\) utilizing a high-frequency pressure transducer (113B26, PCB Piezotronics) flush mounted on the outer cylinder wall. The RDE was fixed to a slide-type thrust stand, and the thrust was measured utilizing a load cell (DUD-50K, AIKOH). As illustrated in Fig. 2, the thrust stand with the attached combustor was installed in a 30.1-m\(^3\) vacuum chamber in the same configuration as that utilized in a previous study [21].

Fig. 2
figure 2

Schematic of experimental setup

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2.2 Experimental conditions

In this study, the combustion tests were conducted by maintaining constant \(\textrm{ER}=1.0\pm 0.1\), and the total mass flow rate varied from \(\dot{m}_{\textrm{total}}=53\, \hbox {g}/\hbox {s}\) to \(136\, \hbox {g}/\hbox {s}\). We used gaseous ethylene and oxygen as propellants. The back pressure in all combustion tests was \(p_{\textrm{b}}=10\,\pm 1\, \hbox {kPa}\). The mass flow rates of the fuel and oxidizer were varied by utilizing a choke orifice installed upstream of the supply line. Each orifice was utilized after deriving the calibration coefficient \(C_{\textrm{d}}=\dot{m}_{\textrm{exp}} /{\dot{m}}_{\textrm{theory}}\) via a prior calibration test. For the calibration test, \(\dot{m}_{\textrm{exp}}\) was calculated from the difference in fuel or oxidizer tank mass before and after a cold flow test of a few seconds in the same supply system. The tank was under high pressure and satisfied the choke requirements at the orifice. Therefore, the theoretical mass flow rate \(\dot{m}_{\textrm{theory}}\) can be calculated as follows:

$$\begin{aligned} \dot{m}_{\textrm{theory}}=\frac{p_{\textrm{tank}}A_{\textrm{ori}}}{\sqrt{RT_{\textrm{b}}} }\sqrt{\gamma \left( \frac{2}{\gamma +1} \right) ^{\left( \gamma +1 \right) / \left( \gamma -1 \right) }} \textrm{,} \end{aligned}$$
(1)

where \(p_{\textrm{tank}}\) is the tank pressure and \(A_{\textrm{ori}}\) is the cross-sectional area of the choke orifice. Here, R and \(\gamma \) represent the gas constant and specific heat ratio (\(R_{\textrm{f}}=297\) J K\(^{\mathrm {-1}}\) kg\(^{\mathrm {-1}}\), \(\gamma _{\textrm{f}}=1.24\), \(R_{\textrm{ox}}=260\) J K\(^{\mathrm {-1}}\) kg\(^{\mathrm {-1}}\), \(\gamma _{\textrm{ox}}=1.40)\). The calibration coefficients for the fuel and oxidizer were estimated to be \(C_{\textrm{d,f}}=0.800\) and \(C_{\textrm{d,ox}}=0.798\), respectively.

As illustrated in Fig. 2, self-luminance in the combustion chamber was observed along the z-axis. A high-speed camera (Phantom V 2011, Vision Research) was utilized to observe the wave number and propagation velocity of the detonation with 12-bit gradation. The frame rate was 400 kfps (\(2.5\,\upmu \hbox {s}/\hbox {frame}\)), and the exposure time was 400 ns. The spatial resolution of visualization in the z-axis direction was \(0.6\pm 0.1\,\hbox {mm}/\hbox {pixel}\).

The sampling rate of the pressure sensors for the static pressure measurement and load cell was 1 kHz. The sampling rate of the high-frequency pressure transducers for measuring the fluctuating pressure was either 1 MHz or 10 MHz.

2.3 Estimation of total pressure loss of oxidizer for different channel expansion angle

The total pressure loss in the oxidizer injector of the RDE employed here was estimated utilizing the cold flow tests of the oxidizer. Using the same assumptions as Matsuoka et al. [17], the total pressure \(p_{\textrm{t,}z}\) was transformed from the measured static pressure \( p_{z}\) at each point on the z-axis. In the following, the following assumptions are made: (1) steady flow and (2) total temperature is constant.

The mass flow rate was obtained for the position of the z-axis as follows:

$$\begin{aligned} \dot{m}_{\textrm{ox}}=\left[ \rho uA \right] _{z}\textrm{,} \end{aligned}$$
(2)

where \(\rho \), u, and A are the density, velocity, and cross-sectional area at each point along the z-axis, respectively.

Utilizing the isentropic relationships, the ideal gas equation (\(p=\rho RT)\), the sound speed equation (\(a=\sqrt{\gamma RT} )\), and the Mach number equation (\(M=u/a)\), the following equation was eventually obtained by transforming (2):

$$\begin{aligned} M_{z}^{2}\left( 1+\frac{\gamma _{\textrm{ox}}-1}{2}M_{z}^{2} \right) =\left( \frac{\dot{m}_{\textrm{ox}}}{p_{z}A_{z}} \right) ^{2}\frac{R_{\textrm{ox}}T_{\textrm{t,}z}}{\gamma _{\textrm{ox}}}, \end{aligned}$$
(3)

where \(M_{z}\) is the Mach number at each point along the z-axis, respectively. The total temperature \(T_{\textrm{t,}z}\) was calculated at the ambient temperature, \(T_{\textrm{t,}z} = {T}_{\textrm{b}} = 295\,\pm \, 2\) K. The Mach number near the injector is important because it is closely related to the total pressure loss. Previous experiments [22] that measured the total pressure drop in the abruptly expanding section showed that the total pressure drop increased with increasing injector Mach number.

Fig. 3
figure 3

Mach number, cross-sectional area ratio, and total pressure ratio in cold flow tests: \(\dot{m}_{\mathrm {{ox}}} = 88\pm 1\) g/s, \({p}_{\mathrm {{b}}} =101\)  kPa

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Fig. 4
figure 4

Time variation of frequency, pressure, and thrust: a \(\alpha = 90^{\circ }\), \(\dot{m}_{\mathrm {{total}}} / {A}_{\mathrm {{ch}}} = 97\) kg/(m\(^{\mathrm {{2}}}\cdot \) s), \(\mathrm {{ER}} = 0.95\), b \({\alpha =90^{\circ }}\) , \(\dot{m}_{\mathrm {{total}}} / {A}_{\mathrm {{ch}}} = 148\) kg/(m\(^{\mathrm {{2}}}{\cdot }\) s), \(\mathrm {{ER}} = 0.98\)

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From (3), \(M_{z}\) was obtained and \(p_{\textrm{t,}z}\) was calculated from \(M_{z}\). Figure 3 presents the Mach number \(M_{z}\), cross-sectional area ratio \(A_{z}/A_{\textrm{ch}}\) to the combustion chamber cross-sectional area \(A_{\textrm{ch}}\), and total pressure ratio \(p_{\textrm{t,}z} / p_{\mathrm {t,-22}}\) to the plenum total pressure \(p_{\mathrm {t,-22}}\) at each point along the z-axis under the same mass flow rate. As illustrated in Fig. 3, the total pressure at \(z=5\) and \(10\,\hbox {mm}\) decreased and then increased at \(z=20\) mm. As the flow is not uniform because of middle of the expansion or just downstream of the expansion, the assumption of the model may not be valid. If the flow becomes fully uniform at \(z=20\,\hbox {mm}\), the total pressure loss ratio \(1-p_{\textrm{t,20}} / p_{\mathrm {t,-22}}\) at \(\alpha =90^{\circ }\) and \(30^{\circ }\) was estimated to be \(33.1\pm 0.1\)% and \(31.3\pm 0.1\)%, respectively. Although the difference in total pressure loss due to changes in \(\alpha \) was small, the total pressure loss of approximately 30% is critical for achieving PG.

3 Results and discussion

3.1 Time history of detonation propagation velocity, pressure, and thrust

Figure 4 shows a typical time history of frequency, combustor pressure, and thrust for this experiment. The time histories presented in Fig. 4a, b correspond to different mass flux \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\) conditions; \(t=0\) is when the ignition signal was input into the gunpowder installed in the combustion chamber. The top of Fig. 4 shows the frequency obtained by FFT of the self-luminance image captured by the high-speed camera. This frequency was almost the same as that obtained with a high-frequency pressure transducer. Stable frequency was observed even when the number of detonation waves changed, and the dominant frequency was defined as \(f_{\textrm{w}}\). Under most conditions, detonation waves propagate as a single wave (\(n_{\textrm{w}}=1)\). As the \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\) increased, detonation wave propagated as single or double waves (\(n_{\textrm{w}}=2)\) during combustion. Here, the former mass flux region \((\dot{m}_{\textrm{total}} / A_{\textrm{ch}}=0 \sim 140 \,\hbox {kg}/({\hbox {m}}^{\textrm{2}}\cdot \hbox {s}))\) is defined as the “single wave region,” and the latter mass flux region (\(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}=140 \sim 170\,\hbox {kg}/({\hbox {m}}^{\textrm{2}}\cdot \hbox {s}))\) is defined as the “single/double wave region.” As illustrated in the middle and bottom in Fig. 4, the combustor pressure and thrust waveforms increased sharply at the start of the combustion. A brief vibration component due to the ignition of the gunpowder was detected, but it remained stable at a constant value during combustion. The thrust shown in the bottom of Fig. 4 has a large vibration component, so the time moving average is indicated by red line. The pressure \(p_{\textrm{0,cold}}\) during cold flow was evaluated by time-averaging at \(t=-\, 0.2{\sim }0\,\hbox {s}\). The pressure \(p_{\mathrm {-22,det}}\), \(p_{\textrm{0,det}}\), and thrust \(F_{\textrm{det}}\) during combustion were evaluated by time-averaging over the interval illustrated in Fig. 4. As demonstrated in middle of Fig. 4b, the combustor static pressure increased when changing from \(n_{\textrm{w}}=1\) to \(n_{\textrm{w}}=2\). The increase in plenum static pressure \(p_{\mathrm {-22}}\) and combustion chamber static pressure \(p_{0}\) when \(n_{\textrm{w}}=2\) is considered to be due to the decrease in the effective injection area [17]. The hydrodynamic blockage ratio \(\textrm{BR}_{\textrm{hyd}}\) proposed by Matsuoka et al. [17] was calculated assuming subsonic injection, and \(\textrm{BR}_{\textrm{hyd}}\) was higher at \(n_{\textrm{w}}=2\) than at \(n_{\textrm{w}}=1\). Assuming that the propellant filling velocity does not change with the number of detonation waves \(n_{\textrm{w}}\), \(p_{0}\) is also expected to increase when \(n_{\textrm{w}}=2\). The reason why the thrust does not vary with \(n_{\textrm{w}}\) is that the momentum thrust is smaller when \(n_{\textrm{w}}=2\) compared to \(n_{\textrm{w}}=1\). Figure 5 shows the z-axis distribution of time-averaged static pressure. As shown in Fig. 5, the pressure ratio is lower than the critical pressure ratio and is considered to be choking at the combustion chamber exit. It was not observed that there was any difference in the pressure distribution due to the difference in \(\alpha \).

Fig. 5
figure 5

z-axis distribution of cross-sectional area ratio variation and time-averaged static pressure during combustion: \(\dot{m}_{\mathrm {{total}}} / {A}_{\mathrm {{ch}}} = 91\pm 1\) kg/(m\(^{\mathrm {{2}}}\cdot \) s)

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Fig. 6
figure 6

a Propagation velocity of detonation wave, b time-averaged thrust during combustion time

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Fig. 7
figure 7

Waveform of combustion chamber fluctuating pressure in single wave region: a \({\alpha =90^{\circ }}\), \(\dot{m}_{\mathrm {{total}}}/{A}_{\mathrm {{ch}}} = 91\) kg/(m\(^{\mathrm {{2}}}{\cdot }\) s), \(\mathrm {{ER}} = 0.99\), b \({\alpha =30^{\circ }}\), \(\dot{m}_{\mathrm {{total}}}/{A}_{\mathrm {{ch}}} = 92\) kg/(m\(^{\mathrm {{2}}}{\cdot }\) s), \(\mathrm {{ER}} = 0.99\)

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Figure 6a illustrates the propagation velocity \(D_{\textrm{w}}\) of the detonation wave; \(D_{\textrm{w}}\) was obtained using \(f_{\textrm{w}}\) as follows:

$$\begin{aligned} D_{\textrm{w}}=\frac{\pi d_{\textrm{c}}f_{\textrm{w}}}{n_{\textrm{w}}}\textrm{,} \end{aligned}$$
(4)

where \(d_{\textrm{c}}\) is the diameter in the middle of the outer and inner cylinders (\(d_{\textrm{c}}=55\,\hbox {mm}\)). The quantity \(D_{\textrm{CJ}}\) in Fig. 6a is the variation of Chapman–Jouguet (CJ) detonation velocity with respect to \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\); \(D_{\textrm{CJ}}\) was calculated by inputting the combustion chamber pressure during cold flow \(p_{0,\textrm{cold}}\), the ambient temperature \(T_{\textrm{b}}\), and the equivalence ratio ER, to NASA-CEA [23]. Error bars in the plot are the standard deviations of \(D_{\textrm{w}}\); \(D_{\textrm{w}}\) showed a constant trend regardless of the change in \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\). For the same \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\), the velocity at \(\alpha =90^{\circ }\) was about 5% higher than the velocity at \(\alpha =30^{\circ }\). In the single wave region, \(D_{\textrm{w}}\) at \(\alpha =90^{\circ }\) and \(\alpha =30^{\circ }\) were \(86\pm 2\)% and \(81\pm 3\)% of \(D_{\textrm{CJ}}\), respectively. In the single/double wave region, the velocity at \(n_{\textrm{w}}=2\) was about 20% lower than that at \(n_{\textrm{w}}=1\) [24].

Figure 6b, c shows the variation of thrust \(F_{\textrm{det}}\) and propellant-based specific impulse with respect to \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\): Error bars in the plot are the standard deviations of \(F_{\textrm{det}}\). As illustrated in Fig. 6b, for the same \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\), there was no significant change in the \(F_{\textrm{det}}\) regardless of the expansion angle. For comparison of propulsive performance, the specific impulse \(I_{\textrm{sp}}\) was calculated as follows:

$$\begin{aligned} I_{\textrm{sp}}=\frac{F_{\textrm{det}}}{\dot{m}_{\textrm{total}}g}\textrm{,} \end{aligned}$$
(5)

where g is the gravity acceleration (\(g=9.81\) m/s\(^{\textrm{2}})\). Figure 6c shows the variation of thrust \(I_{\textrm{sp,theory}}\) with respect to \(\dot{m}_{\textrm{total}}/A_{\textrm{ch}}\);    \(I_{\textrm{sp,theory}}\) in Fig. 6c is the variation of the theoretical specific impulse with respect to \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\); \(I_{\textrm{sp,theory}}\) was calculated by inputting \(p_{\textrm{0,det}}\), \(p_{\textrm{0,det}} / p_{\textrm{b}}\) to NASA-CEA [23]. At \(\alpha =90^{\circ }\), \(I_{\textrm{sp}}=161\) to 212 s for a change in \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}=62\) to 157 kg/(m\(^{\textrm{2}}\cdot \hbox {s})\). At \(\alpha =30^{\circ }\), \(I_{\textrm{sp}}=143\) to 218 s for a change in    \(\dot{m}_{\textrm{total}} / {A_{\textrm{ch}}=62}\) to 158 kg/(m\(^{\textrm{2}}\cdot \hbox {s})\). \(I_{\textrm{sp}}\) was almost the same regardless of the change in \(\alpha \), and it was not confirmed to differ in propulsive performance.

3.2 Relationship between channel expansion angle and combustion chamber fluctuating pressure

Figure 7 illustrates pressure waveforms obtained utilizing a high-frequency pressure transducer attached to the bottom of the combustion chamber. Figure 7a, b depicts the pressure waveform at \(\alpha =90^{\circ }\) and \(\alpha =30^{\circ }\), respectively, under same mass flux conditions. The pressure waveforms exhibit a sudden and periodic increase in pressure owing to the detonation wave. The peak pressure \({\Delta }p\) was evaluated by setting the average value to the zero point. Figure 8a illustrates the variation in \({\Delta }p\) with respect to \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\). The plot and error bars are the average and standard deviation of 10 ms in the time-averaged duration shown in Fig. 4, respectively. In the single wave region, \({\Delta }p\) increases with increasing \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\), with larger values for \(\alpha =90^{\circ }\). In the single/double wave region, \({\Delta }p\) was small regardless of \(\alpha \).

Fig. 8
figure 8

a Peak pressure at bottom of combustor, b pressure increase ratio

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Fig. 9
figure 9

Comparison of various published data on pressure gain over the throat to oxidizer injector area ratio

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It is essentially important to evaluate the pressure increase ratio \(\pi _{\textrm{peak}}\) rather than the absolute peak pressure [16]. This is because the following relationship exists between the pressure increase ratio \(\pi _{\textrm{CJ}}\) and the Mach number \(M_{\textrm{CJ}}\) of the CJ detonation [25]:

$$\begin{aligned} \pi _{\textrm{CJ}}=\frac{p_{\textrm{CJ}}}{p_{\textrm{reac}}}=\frac{\gamma _{\textrm{reac}}M_{\textrm{CJ}}^{2}+1}{\gamma _{\textrm{prod}}}\textrm{,} \end{aligned}$$
(6)

where \(\gamma _{\textrm{reac}}\) and \(\gamma _{\textrm{prod}}\) are the specific heat ratios of the reactants and products, respectively. As expressed in (6), \(M_{\textrm{CJ}}\) can be considered to increase with an increase in \(\pi _{\textrm{CJ}}\); \(\gamma _{\textrm{reac}}\), \(\gamma _{\textrm{prod}}\), and sound speed of reactants \(a_{\textrm{reac}}\) were calculated by inputting \(p_{\textrm{0,det}}\), \(T_{\textrm{b}}\), and \(\textrm{ER}\) into NASA-CEA [23]. If \(M_{\textrm{CJ}}\) in (6) is replaced by \(M_{\textrm{w}}=D_{\textrm{w}} / a_{\textrm{reac}}\), utilizing the experimental propagation velocity \(D_{\textrm{w}}\), \(\pi _{\textrm{CJ}}\) changes from 17.6 to 26.7. The experimental pressure increase ratio \(\pi _{\textrm{peak}}\) is defined as follows [16]:

$$\begin{aligned} \pi _{\textrm{peak}}=\frac{\Delta p+p_{\textrm{0,cold}}}{p_{\textrm{0,det}}}, \end{aligned}$$
(7)

where \(p_{\textrm{0,cold}}\) is the pressure at the bottom of the combustion chamber at cold flow before ignition. Figure 8b shows the variation of \(\pi _{\textrm{peak}}\) with \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\). Error bars in the plot were derived from the propagation of errors in \(p_{\textrm{0,cold}}\), \(p_{\textrm{0,det}}\), and \(\Delta p\). Note that the comparison of \(\pi _{\textrm{peak}}\) does not take into account the effects of detonation wave height and mixing degree, although peak pressure is strongly affected by mixing degree and other factors. In the single wave region, \(\pi _{\textrm{peak}}\) at \(\alpha =90^{\circ }\) was larger than that at \(\alpha =30^{\circ }\). Thus, it is suggested that detonation waves with high Mach number \(M_{\textrm{w}}\) propagate at \(\alpha =90^{\circ }\). A possible reason for this is that at \(\alpha =90^{\circ }\), there is locally well-mixed around \(z=0\,\hbox {mm}\) due to the existence of a large recirculation zone. At \(\alpha =90^{\circ }\), \(\pi _{\textrm{peak}}\) increased with \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\), but was almost constant above \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}=80\) kg/(m\(^{\textrm{2}}\cdot \hbox {s})\). The velocity \(D_{\textrm{w}}\) was constant regardless of mass flux, suggesting that \(M_{\textrm{w}}\) reached a maximum around \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}=80\) kg/(m\(^{\textrm{2}}\cdot \hbox {s})\). According to Noda et al. [16], \(\pi _{\textrm{peak}}\) at higher mass flux is constant due to the increase in PC. In contrast, at \(\alpha =30^{\circ }\), \(\pi _{\textrm{peak}}\) increased even at higher \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\). At \(\alpha =30^{\circ }\), the PC at higher \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\) is considered to be suppressed compared to \(\alpha =90^{\circ }\). In the single/double wave region, there was no tendency for \(\pi _{\textrm{peak}}\) to increase with \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\) regardless of \(\alpha \) and \(n_{\textrm{w}}\).

3.3 Estimation of pressure gain performance

Finally, PG performance was evaluated utilizing EAP [8]. The base drag force [9], which is the thrust on the outer and inner cylinder end faces at the combustor exit, is small under low back pressure [26]. If the base drag force can be neglected, EAP can be calculated as follows, assuming that the flow is steady and isentropically choked at the combustor exit:

$$\begin{aligned} \textrm{EAP}={\frac{F_{\textrm{det}}+p_{\textrm{b}}A_{\textrm{exit}}}{\left( \gamma _{\textrm{exit}}+1 \right) A_{\textrm{exit}}}\left( \frac{\gamma _{\textrm{exit}}+1}{2} \right) }^{\frac{\gamma _{\textrm{exit}}}{\gamma _{\textrm{exit}}-1}}\textrm{,} \end{aligned}$$
(8)

where \(A_{\textrm{exit}}\) is the cross-sectional area at the combustion chamber exit and is equal to the cross-sectional area of the combustion chamber (\(A_{\textrm{exit}}=A_{\textrm{ch}})\); \(\gamma _{\textrm{exit}}=1.14\) is the specific heat ratio at the combustion chamber exit [23].

The PGR was calculated as \({\textrm{EAP}} / p_{\mathrm {t,-22,det}}\), where \(p_{\mathrm {t,-22,det}}\) is the total pressure of the oxidizer plenum during the combustion time; \(p_{\mathrm {t,-22,det}}\) was obtained in the same method as in Sect. 2.3, utilizing the measured static pressure \(p_{\mathrm {-22,det}}\). Figure 9 shows the variation of PGR with cross-sectional area ratio \(A_{\textrm{throat}} / A_{\textrm{inj,ox}}\), where \(A_{\textrm{throat}}\) and \(A_{\textrm{inj,ox}}\) are the cross-sectional area of the nozzle throat and the oxidizer injector, respectively. In this study, \(A_{\textrm{throat}}=A_{\textrm{ch}}\) because no nozzle is attached. The error bars in Fig. 9 are based on changes in \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\). Comparisons were made with the results of other studies based on the results summarized by Bach et al. [10], Zhou et al. [27], Brophy and Codoni [9], and Ten Eyek [28]. The plots in this study showed a similar trend to those in other studies. PGR increased gradually as \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\) increased, with maximum values of \(61\,\pm \, 6\)% and \(60\,\pm \, 7\)% at \(\alpha =90^{\circ }\) and \(\alpha =30^{\circ }\), respectively. It was not confirmed to differ in PGR due to changes in \(\alpha \). The approximately 30% total pressure loss shown in Sect. 2.3 is dominant. Thus, it is important to lower the total pressure loss by lowering the injection Mach number at the injector outlet and area ratio \(A_{\textrm{throat}} / A_{\textrm{inj,ox}}\) [8]. However, the fluctuating pressure results suggest that changes in \(\alpha \) affect the structure of the detonation wave and require visualization for a more detailed understanding.

4 Conclusions

This study focused on the channel expansion angle near the oxidizer injector outlet to investigate the effect of the recirculation zone generated in the combustion chamber of an RDE. The channel expansion angle \(\alpha \) was set to two patterns: \(\alpha =90^{\circ }\) and \(\alpha =30^{\circ }\), and the mass flow rate was varied, while a constant equivalence ratio was maintained. The following three conclusions were drawn from the combustion test results.

  1. (1)

    It was not confirmed to differ in thrust due to changes in \(\alpha \). However, the propagation velocity of the detonation wave varied with \(\alpha \). For a single detonation wave number, the propagation velocity at \(\alpha =90^{\circ }\) was about 5% higher than the velocity at \(\alpha =30^{\circ }\). When two detonation waves were propagating, the propagation velocity decreased by about 20% regardless of \(\alpha \).

  2. (2)

    The peak pressure due to detonation from the fluctuating pressure at the bottom of the combustion chamber was compared, and the value was larger at \(\alpha =90^{\circ }\). A comparison of the pressure increase ratio, which is important in considering the propagation Mach number, showed a larger value for \(\alpha =90^{\circ }\). Therefore, it is suggested that the Mach number of the detonation wave was larger at \(\alpha =90^{\circ }\). The pressure increase ratio increased with mass flux \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\) but remained constant from around \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}=80\) kg/(m\(^{\textrm{2}}\cdot \hbox {s})\) at \(\alpha =90^{\circ }\). The propagation velocity was constant, suggesting that the maximum Mach number was reached around \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}=80\) kg/(m\(^{\textrm{2}}\cdot \hbox {s})\). At \(\alpha =30^{\circ }\), the pressure increase ratio showed a simple increasing trend with respect to \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\).

  3. (3)

    The model proposed by Matsuoka et al. [17] was utilized to obtain the axial total pressure distribution and estimate the total pressure loss of the oxidizer. The total pressure loss was estimated to be 33% and 31% for \(\alpha =90^{\circ }\) and \(\alpha =30^{\circ }\), respectively. As a result of evaluation utilizing PGR, the ratio of EAP to total pressure of oxidizer plenum, it was not confirmed to differ in PGR due to changes in \(\alpha \). PGR increased with the \(\dot{m}_{\textrm{total}} / A_{\textrm{ch}}\), and the maximum value recorded in this experiment was 61%. In previous studies, PGR was evaluated by the area ratio of the nozzle to the oxidizer injector, and the PGR in this study could be plotted to a similar point.