1 Introduction

Over the past few decades, detonation combustion-based engines have been intensively studied for the development of a new propulsion or power generation system [1,2,3]. Detonation engines have attracted attention because of their higher theoretical thermal efficiency with the potential to realize pressure gain combustion [4], which brings innovation to gas turbine engines [5, 6]. Rotating detonation engines (RDEs) involve continuous propagation of detonation waves in a combustor, resulting in higher performance, and are technically more feasible as compared to pulse detonation engines. In generic RDEs, which have an annular combustion chamber, a propellant is provided in the axial direction of the combustor and combustion gases are exhausted in the same axial direction.

A disk-type RDE (DRDE) in which a combustible mixture flows radially inward before turning to the axial direction has been recently proposed [7]. It has an advantage of length savings, besides its better thermal efficiency than conventional combustors with deflagrative combustion mode. Huff et al. [8] originally developed a modular-designed DRDE with a constant chamber area and a controlled injection area of fuel and air. They studied the relationship between the mass flow rate of the mixture, the wave number of the rotating detonation waves, and the operating frequency for a hydrogen–air mixture [9]. Boller et al. [10] also studied DRDE operation and reported that the detonation wave is located closer to its inner radius for a single-wave mode, while for a double-wave mode the detonation wave propagates at the outer radius. Muraleetharan et al. [11] constructed a combustion chamber with a constant chamber height, which results in a convergent flow path. Their pressure measurements confirmed that rotating detonation waves propagated along the periphery of the combustion chamber.

Fig. 1
figure 1

Schematic of combustor of the disk-type rotating detonation engine

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We have studied performance of the DRDE experimentally and numerically over the last few years [12,13,14] and have reported effects of the equivalence ratio on operation frequency and the flow structure in the DRDE combustion chamber. In the present work, operation characteristics of the DRDE with a constant cross-sectional channel area were experimentally studied to investigate the performance of the present combustor. We measured the thrust and static pressure and then developed a flow model in the combustion chamber for thrust evaluation to estimate the pressure gain.

2 Experimental apparatus

The DRDE combustor used in the present work is shown in Fig. 1. Hydrogen, which is used as fuel, was introduced through 120 axial orifice holes 0.51 mm in diameter, while air was introduced through a circumferential slit 0.5 mm wide. This configuration of the circumferential slit and orifices results in a vertical interaction of the injected hydrogen and air, leading to enhanced mixing in the combustion chamber. The hydrogen–air mixture flows along the curved chamber wall, changing the gas direction gradually from a radial to an axial direction. It should be noted that the present combustor was designed so that the cross-sectional area of the channel was constant, similar to a generic annular combustion chamber RDE. An initiator tube with an inner diameter of 4.57 mm and a length of 200 mm, in which a stoichiometric oxy-hydrogen mixture was initially charged, was used to detonate the main mixture in the combustion chamber. A conventional spark plug was placed at the one end of the initiator tube to ignite the mixture, resulting in the generation of a fast deflagration or detonation.

To capture the behavior of the rotating detonation waves, the transient-pressure was measured by two pressure transducers (Kistler 603B1), P1 and P2 in Fig. 1, flush-mounted on the top wall of the combustor. In addition, static pressure in the combustion chamber was evaluated with capillary tube average pressure (CTAP) measurement [15]. The CTAP tube, made of a 1/8-inch stainless-steel tube 500 mm long, was installed on the same circumference as P1. The CTAP tube was instrumented with a conventional pressure sensor (Keller PAA-23SYEi).

Fig. 2
figure 2

Mixture supply system

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Fig. 3
figure 3

Propagation of rotating detonation waves. The quantities \(\dot{{m}}\) and \(\phi \) denote the mass flow rate and the equivalence ratio, respectively

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Replacing a portion of the top wall with a plexiglas window makes it possible to visualize the propagation of the rotating detonation waves. The bottom-left photograph in Fig. 1 shows the DRDE equipped with a window for visualization tests. The behavior of rotating detonation waves was recorded with a high-speed camera (nac Image Technology, MEMRECAM fx-K4). The frame rate was set to \(4\times 10^{4}\) fps at a resolution of \(192 \times 144\) pixels.

Figure 2 shows the mixture supply system used in the present study. Hydrogen and air were initially charged in gas storage tanks at desired pressures and were connected to the RDE combustor. Each gas flow passage was equipped with a solenoid valve to control the gas supply timing. While the mass flow rate was varied by changing the pressure in the gas storage tanks, the operation time was set to 226 ms in the present study because of their limited volume. The pressure in the gas storage tanks gradually decreases during operation, which affects the instantaneous mass flow rate. Because it was found to decrease by 8% over the test time, the mass flow rate was evaluated with the time-averaged value calculated two different ways. In the first method, the average mass flow rate was based on the pressure difference between the pre-test and post-test gas storage tanks. The second method involved time integration of the instantaneous mass flow rate, which was calculated from supply pressures measured with PS1 and PS2. The mass flow rates estimated from these two methods agreed with each other within 5% accuracy.

For thrust measurement, the DRDE combustor was placed on a frame so that it can be moved horizontally with a set of linear sliders. A total of 200 N preload ensured constant contact of the DRDE with the load cell.

3 Experimental results

Figure 3 shows the wave mode of rotating detonation waves observed in the present study with nearly the same total mass flow rates, but different mixture equivalence ratios. For a leaner mixture (shown in Fig. 3a), a single detonation wave is found to rotate clockwise, while a double-wave mode is detected for the slightly richer mixture (shown in Fig. 3b). In both modes, the brightest area is located near the outer wall of chamber regardless of the wave mode, in disagreement with Boller et al. [10], who reported that the rotating detonation wave propagates near the inner radius of the combustion chamber for the single-wave mode. This is probably due to the difference in the inlet area ratio. In the present combustor configuration, \(A_{3.1}/A_{3.2}\) was set to 0.1, leading to the stable propagation of rotating detonation waves, where \(A_{3.1}\) is a minimal cross-sectional area of the combustor inlet and \(A_{3.2}\) is a cross-sectional area of the combustion channel. In Boller’s work [10], \(A_{3.1}/A_{3.2}\) was set to 0.2, which may be responsible for the different location of the rotating detonation wave.

Fig. 4
figure 4

Typical pressure time-histories and frequency analysis. The quantity n denotes the wave number

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Typical pressure time-histories and their frequency analysis are shown in Fig. 4. It is found that when the wave number changes from 1 to 2, the operation frequency increases by a factor of about 1.7. As the wave number increases, the peak pressure corresponding to the passage of the detonation wave slightly decreases. Figure 4c indicates that there is a transition mode in which the operation frequency decreases from 6.2 to 3.4 kHz during the test time. Correspondingly, the wave number decreases from 2 to 1, and the peak pressure increases. The appearance of this transition mode seems to be closely related to the gradual decrease in the mass flow rate during the test time.

Figure 5 shows the effect of the equivalence ratio on the operating frequency of the DRDE. In the present configurations, only single- and double-wave modes were observed. Two detonation waves (wave number 2) were observed over a wide range of equivalence ratios around unity, while a stable wave number of 1 appears in the case of leaner mixtures. As the equivalence ratio is increased from 0.6, the wave number increases from 1 to 2 between 0.68 and 0.83, depending on the mass flow rate. For an equivalence ratio greater than 1.45, a transition mode, in which the operation mode changes from a double wave to a single wave, was observed; two operating frequencies corresponding to the two modes are plotted in Fig. 5. For the wave number of 2, the operation frequency tends to increase slightly as the equivalence ratio increases. It is found that the range of equivalence ratios for which the wave number of 2 is stable narrows as the mass flow rate increases.

Figure 6 shows the time-averaged static pressure in the combustion chamber measured with the CTAP technique for different total mass flow rates. Regardless of the mass flow rate, the static pressure for the wave number of 1 is higher than that for the wave number of 2, while the equivalence ratio has a small effect on the chamber pressure for the same wave number.

Fig. 5
figure 5

Operating frequency of rotating detonation waves for different total mass flow rates. The quantity f denotes the operation frequency

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Fig. 6
figure 6

Time-averaged chamber static pressure for different total mass flow rates

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Fig. 7
figure 7

Flow model in combustion chamber

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4 Flow model

For thrust estimation from the CTAP measurement, a flow model, composed of sequentially homentropic flow, Rayleigh flow, and chemically frozen flow with heat loss and wall friction, has been constructed as shown in Fig. 7. Here, the following assumptions have been made:

  1. (1)

    Air in the storage tank (position 0) expands isentropically and is injected at the inlet of the combustion chamber (position 1).

  2. (2)

    The chemical reaction ends before position 2, which corresponds to the location of the static pressure sensor. This assumption is supported by the combustion front visualization that shows that the rotating detonation waves are located outside of the static pressure sensor.

  3. (3)

    The ambient pressure at the chamber outlet (position 3) (\(p_{3}\)) is set to atmospheric pressure of 101.3 kPa (\(p_\textrm{a})\), if the Mach number at the position 3 (\(M_{3}\)) is less than unity. In the case that \(M_{3} >1\) and \(p_{3} > p_\textrm{a}\), recalculation is made so that the choking condition of \(M_{3} = 1\) is satisfied.

Taking the position of the static pressure 2 as the origin and the distance downstream as x, the infinitesimal distance as dx, the hydraulic diameter as D, the specific heat at constant pressure as \(c_{p}\), the heat loss per unit mass as q, and the pipe friction factor as \(f_\textrm{P}\), the flow velocity w, static temperature T, static pressure p, and the Mach number M can be expressed using the influence coefficient [16] for steady one-dimensional flow for one-dimensional flow through a constant area with heat transfer and wall friction as follows:

$$\begin{aligned}{} & {} \frac{\textrm{d}w}{w}=\frac{1}{1-M^{2}}\frac{{\delta }q}{c_{p} T}+\frac{\gamma M^{2}}{2(1-M^{2})}\frac{4f_{\textrm{P}} }{D}\textrm{d}x, \end{aligned}$$
(1)
$$\begin{aligned}{} & {} \frac{\textrm{d}T}{T}=\frac{1-\gamma M^{2}}{1-M^{2}}\frac{{\delta }q}{c_{p} T}-\frac{\gamma (\gamma -1)M^{4}}{2(1-M^{2})}\frac{4f_{\textrm{P}} }{D}\textrm{d}x, \end{aligned}$$
(2)
$$\begin{aligned}{} & {} \frac{\textrm{d}p}{p}=-\frac{\gamma M^{2}}{1-M^{2}}\frac{{\delta }q}{c_{p} T}-\frac{\gamma M^{2}\left[ {1+(\gamma -1)M^{2}} \right] }{2(1-M^{2})}\frac{4f_{\textrm{P}} }{D}\textrm{d}x, \nonumber \\ \end{aligned}$$
(3)
$$\begin{aligned}{} & {} \frac{\textrm{d}M^{2}}{M^{2}}=\frac{1+\gamma M^{2}}{1-M^{2}}\frac{{\delta }q}{c_{p} T}+\frac{\gamma M^{2}\left( {1+\frac{\gamma -1}{2}M^{2}} \right) }{1-M^{2}}\frac{4f_{\textrm{P}} }{D}\textrm{d}x.\nonumber \\ \end{aligned}$$
(4)
Fig. 8
figure 8

Spatial variation of hydraulic diameter, calculated static pressure, and flow Mach number for choking condition (\(\dot{{m}}= 440\,\text {g/s}, \varphi = 0.90, p_{2} = 184\,\text {kPa}\)) and no-choking condition (\(\dot{{m}}= 270\,\text {g/s}, \varphi = 1.00, p_{2}= 129\,\text {kPa}\))

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The hydraulic diameter D, which is the cross-sectional area divided by the wetted perimeter of the cross section, was calculated as a function of x from the flow path geometry as shown in Fig. 8. The pipe friction factor \(f_\textrm{P}\) was set to 0.009 [17], which is slightly larger than that for a smooth pipe with a Reynolds number of the order of \(10^{4}\) in the Moody diagram with the Darcy–Weisbach friction factor \(f_\textrm{D} = 4f_\textrm{P}\). The ratio of specific heats \(\gamma \) was given by the equilibrium composition of the combustion gas when the stoichiometric hydrogen–oxygen mixture is burned at a constant pressure. The heat transfer coefficient h was obtained from the Nusselt number Nu, which was determined by the following Dittus–Boelter equation:

$$\begin{aligned} \textrm{Nu}=0.023\textrm{Re}^{0.8}{\Pr }^{0.5}, \end{aligned}$$
(5)

where Re is the Reynolds number in the combustion chamber and Pr is the Prandtl number and the thermal conductivity is assumed to be 0.10 W/(m K) based on measurements in a hydrogen combustion engine [18]. In the Rayleigh section between the positions 1 and 2 (see Fig. 7), the following equations hold:

$$\begin{aligned}{} & {} \frac{w_{2} }{w_{1} }=\frac{M_{2}^{2}(1+\gamma M_{1}^{2})}{M_{1} ^{2}(1+\gamma M_{2}^{2})}, \end{aligned}$$
(6)
$$\begin{aligned}{} & {} \frac{p_{2} }{p_{1} }=\frac{1+\gamma M_{1}^{2}}{1+\gamma M_{2}^{2}}. \end{aligned}$$
(7)

Integrating equations (1)–(4) and (6)–(7) in the flow direction, the flow velocity at the combustion chamber outlet \(w_{3}\) and the pressure \(p_{3}\) can be calculated. As initial conditions, the static pressure at \(x= 0\) is the value measured at the position 2, and the temperature at \(x = 0\) is given as the calculated adiabatic flame temperature.

Figure 8 shows the hydraulic diameter D and two typical calculated results of the static pressure p and the flow Mach number for choking and no-choking conditions at the position 3. If the static pressure is high enough as compared to \(p_\textrm{a}\), the flow is choked at the chamber outlet. This is the case for the choking condition in Fig. 8. Although, in the case of unchoked condition, the flow Mach number increases with decrease in \(p_{2}\) due to the wall friction, the static pressure is balanced with \(p_\textrm{a}\) at the chamber outlet.

Once the flow velocity at the combustion chamber outlet, \(w_{3}\), is obtained, the thrust, \(F_\textrm{calc}\), can be calculated from the following equation, taking the cross-sectional area at the chamber outlet as A and the mass flow rate as \(\dot{{m}}\):

$$\begin{aligned} F_{\textrm{calc}} =\dot{{m}}w_{3} +(p_{3} -p_\textrm{a} )A. \end{aligned}$$
(8)

Figure 9 shows the relationship between \(F_\textrm{calc}\) and the thrust \(F_\textrm{exp}\) measured with a load cell, indicating that \(F_\textrm{calc}\) estimated using the present flow model is reasonable. That is, if the equivalence ratio, the static pressure, and the mass flow rate are known, thrust can be calculated with good accuracy using the flow model.

Fig. 9
figure 9

Thrust evaluation by flow model in combustion chamber

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Fig. 10
figure 10

Pressure gain for various equivalence ratios and mass flow rates

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The present 1-D model does not consider the detonation structure and the shock dynamics by using Rayleigh flow, which treats detonation as a simple heat addition process. Despite this significant assumption, the model performs well as shown in Fig. 9. This is mainly because the static pressure measured at P1 is used as an initial condition, which essentially serves as a calibration parameter. Although the application of the present model is limed, it is found to be useful in predicting the thrust.

5 Pressure gain

The pressure gain PG was calculated as follows: first, since the stagnation pressure at the chamber outlet is the equivalent available pressure (EAP), the EAP can be calculated from the following equation:

$$\begin{aligned} \textrm{EAP}=p_{3} \left( {1+\frac{\gamma -1}{2}M_{3}^{2}} \right) ^{\left( {\frac{\gamma }{\gamma -1}} \right) }, \end{aligned}$$
(9)

where \(M_{3}\) is the Mach number at the chamber outlet. The pressure gain (PG) is then evaluated as a gain with respect to the pressure in the air storage tank \(p_{0}\), as follows:

$$\begin{aligned} \textrm{PG}=\frac{\textrm{EAP}-p_{0} }{p_{0} }. \end{aligned}$$
(10)

The calculated pressure gain, shown in Fig. 10, shows a negative pressure gain, ranging from \(-81\) to \(-69{\%}\), depending on the equivalence ratio. This result indicates that the single-wave mode is superior to the double-wave mode in terms of pressure gain, which is consistent with the results of the static pressure in the combustion chamber in Fig. 6. From Fig. 10a, b, the pressure gain is found to be governed mainly by the wave number determined by the equivalence ratio, regardless of the mass flow rate.

There are two main reasons why the pressure gain is negative under all test conditions. The first one is that the combustion chamber has a constant channel cross-sectional area, i.e., there is no throat at the chamber outlet. The absence of a channel constriction corresponds to condition \(A_{8}/A_{3.2} = 1\) in [19], in which case the static pressure in the combustion chamber cannot be kept high, where \(A_{8}\) is a nozzle throat area. Another reason is the narrow width of the circumferential slit supplying the air; if the value of \(A_{3.1}/A_{3.2}\) is small, the total pressure loss at the air supply slit is excessive, although stable propagation of the rotating detonation wave is achieved. Therefore, improved pressure gain is expected by attaching a nozzle at the chamber outlet to form a throat condition and enlarging the circumferential slit width.

6 Conclusions

The operation characteristics of a hydrogen–air disk-type rotating detonation engine with a constant flow cross-sectional area were experimentally studied using a hydrogen–air mixture.

  1. 1.

    The rotating detonation wave propagates near the outer wall of the combustion chamber regardless of the wave number.

  2. 2.

    For the present test conditions, the propagation mode with a wave number of 1 was generally obtained for equivalence ratios of 0.68 to 0.83, while the propagation mode with a wave number of 2 was obtained for higher equivalence ratios up to 1.5. The range of equivalence ratios for which the wave number of 2 is stable narrows as the mass flow rate increases.

  3. 3.

    Regardless of the equivalence ratio, the static pressure in the combustion chamber is higher for the wave number of 1 than for the wave number of 2.

  4. 4.

    A negative pressure gain was obtained for all the test conditions. The pressure gain was higher for the wave number of 1 than for the wave number of 2.