Experimental Investigation of Thermal Prediction and Heat Transfer Characteristics of Two-Phase RDE during Long-Duration Operation

: Accurately predicting the thermal characteristics and heat transfer distribution of the rotating detonation engine (RDE) and acquiring a clear understanding of the performance and mechanism of the rotating detonation are of great significance for achieving the safe and reliable long-duration operation of RDEs. Using RP-3 as fuel, a long-duration experimental study is performed on a 220 mm-diameter RDC to investigate the details with respect to the thermal environment. The heat flux at the typical location and the average heat flux of both the inner and outer cylinders are measured, respectively. Meanwhile, the peak pressure of the rotating detonation wave (RDW) and specific thrust are analyzed. When the ER is between 0.5 and 1 (oxidizer 2 kg/s), the stable rotating detonation mode is obtained, and the detonation duration is set as 40 s to accurately calculate the heat released by the detonation combustion. The heat flux in the upstream region of the RDW location ranges from 2.40 × 10 5 W/m 2 to 3.17 × 10 5 W/m 2 , and the heat flux in the downstream area of the RDW location ranges from 1.05 × 10 6 W/m 2 to 1.28 × 10 6 W/m 2 . The results demonstrate the important role of the detonation combustion zone, and the thrust performance of RDC can be improved by making the RDW move forward along the RDC axis, which is the optimal direction of detonation combustion. Through a comparison of average heat flux under different conditions, it is found that the heat released by the RDC is directly related to its thrust. In addition, the average heat flux of the inner cylinder is about three times that of the outer cylinder for the two-phase RDC with a Tesla valve intake structure, indicating that the high-temperature combustion product is closer to the inner wall. Therefore, more thermal protection should be allocated to the inner cylinder, and a more systematic analysis of the two-phase flow field distribution in the annular combustion chamber should be carried out to improve the thrust performance. In this paper, the average heat flux of the inner and outer cylinders of the RDC as well as the typical local heat flux of the outer cylinders is quantitatively measured by means of experiments, which not only deepens the understanding of RDC flow field distribution, but also provides quantitative boundary conditions for the thermal protection design of RDCs.


Introduction
As is well known, the existing propulsion engines in service are organized almost in the form of deflagration combustion.However, it is found that the deflagration combustion cycle has difficulty achieving a significant breakthrough in thermal efficiency after continuous optimization.Actually, Zeldovich points out that there is another kind of combustion in nature, namely detonation [1].Detonation combustion, coupled with a shock wave and an exothermic reaction zone, has a high pressure gain, and propagates at a supersonic speed (2000~3000 m/s).As a result, detonation-based power has the potential to increase thermal performance compared to conventional gas turbine engines characterized by the Brayton cycle [2].At present, the attractive propulsion systems based on detonation mainly consist of pulse detonation engines (PDEs), rotating detonation engines (RDEs), and standing detonation engines (SDEs).The rotating detonation engine (RDE) is generally composed of a double cylinder, and has an annular rotating detonation combustor (RDC), with propellants injecting from the throat section.With a single ignition, one or several detonation waves rotate periodically in the circumferential direction, and the high-temperature and high-pressure combustion products rapidly expand and then exhaust to generate stable thrust.Hence, rotating detonation engines (RDEs), generating continuous thrust by rotating detonation, have been at the forefront of detonation research due to the advantages of a rapid energy release rate, compact structure, and stable thrust.
Since RDEs have attracted extensive attention as potential fuelers of aerospace and weapons, researchers are focusing on further enhancing the performance of RDEs, such as the propellant injection [3][4][5], combustor geometry [6][7][8][9], method of ignition [10][11][12], and formation mechanism of the rotating detonation wave (RDW) [10,13].In addition, the rotating detonation rocket engines [14,15], scramjet engines [16][17][18], and turbine engines [19,20] have been built as experimental prototypes to promote engineering application, but the running time of RDEs is limited to seconds due to the harsh thermal environment.Due to the higher thermal efficiency of RDEs, the heat load in the rotating detonation combustor (RDC) is much higher and more complex than that of detonation engines, and traditional cooling modes are not suitable for the traveling detonation zone.Consequently, the thermal characteristics as well as heat transfer analysis are of great significance for achieving the reliable long-duration operation of RDEs before material limits are exceeded.
To explore the thermal environment of RDCs, researchers have conducted a series of experiments based on the infrared thermal imager, thermocouples, and heat flux sensor, as well as using the calorimetry method.The pioneering reports on thermal analysis were conducted by Bykovskii et al. [21,22] with the thermocouples installed on the combustor outer wall.They pointed out that the average heat flux transferred to the outer wall was almost the same for continuous detonation and deflagration when only the geometry of an RDE was altered.The maximum wall temperature was located in the region between the leading shock front and detonation front, and the peak heat flux could be 2-3 times higher under unsteady operation.Using hydrogen and air, Theuerkauf et al. [23][24][25] measured heat flux by utilizing high-frequency heat flux sensors and revealed periodic heat flux based on both uncooled and water-cooled RDEs.Additionally, the results indicated that the frequencies of the periodic heat flux and pressure wave were basically the same during the stable detonation process.The heat flux was between 9 MW/m 2 and 1 MW/m 2 at a 0.11 kg/s flow rate and an equivalence ratio of 1.0 under steady conditions.They analyzed the influencing factors of heat flux, including the mass flow rate and equivalence ratio (ER), and found that heat load increased with a greater mass flow rate and higher equivalence ratio.Meyer et al. [26,27] carried out experiments to capture the heat transfer coefficient in AFRL'S Detonation Engine Research Facility (DERF) with hydrogen and air, and revealed the parameter's impact on heat transfer from aerospike nozzles, as well as channel width, equivalence ratio, and time from ignition.In the experiments conducted by Stevens et al. [28,29], the bulk heat flux of the outer wall was measured calorimetrically in a laboratory water-cooled RDE with hydrogen and air.The results indicated that the thermal load increased along the RDE axis, and the RDW number was the most significant factor affecting the average heat flux of water-cooled RDEs.Micka et al. [30] concluded that the highest heat flux was up to 25 MW/m 2 (near the propellant injectors) for a rocket RDE using GOX-GH2 and GOX-GCH4 propellants.Lim et al. [31] analyzed the heat flux of an RP-2-GOX RDE in the laboratory, revealing the problems with the thermocouple in terms of contact resistance and response time.Rein et al. [32,33] confirmed that the heat transfer from detonation products to the combustor wall was spatially and temporally non-uniform by laser absorption spectroscopy.A water-cooled jacket was adopted in the experiments by Aliakbari et al. [34], and the results demonstrated that the heat flux was more sensitive to changes in equivalent ratio than mass flow rate.With thermocouples and an infrared thermal imager, Zhou et al. [35] performed experiments using hydrogen as fuel and air as an oxidant to study the characteristics of outer-wall temperature and heat transfer, and verified that the peak heat flux was located in the same region as the propagation region of the detonation wave, and the heat flux of the RDC increased with the increase in ER.In order to guarantee an RDE flight demonstration, Ishihara et al. [36] and Goto et al. [37,38] applied C/C composite as the heat material in an RDC, and succeeded in the long-duration RDC demonstration of 6-10 s with gaseous ethylene and gaseous oxygen as the propellants.On this basis, the cooling scheme of a cylindrical RDE with an injector surface on the side wall of the combustor was tested and demonstrated.Goto et al. [39] measured the wall heat flux using K-type thermocouples and observed an 18-25% increase in heat flux when doubling the mass flow rate.According to the literature, the early studies focused on gaseous fuels.With a growing understanding of detonation, RDEs supplied with liquid kerosene (RP-3) were found to be suitable for practical engineering applications due to the advantages of high energy density and easy storage.Shi et al. [40] carried out experiments on a kerosene two-phase RDC using a mixture of nitrogen and oxygen as the oxidizer.They reported non-uniform temperature distribution on the outer wall of the RDC using an infrared thermal imager.In the lean combustion range, the peak heat flux of the RDC increased with the increase in ER when the mass flow of the oxidizer was the same.
Researchers have also performed numerical calculations and predictions of the RDC thermal environment.Frolov et al. [41] developed a three-dimensional computational fluid dynamic (CFD) model of an annular combustor with a separate supply of hydrogen and air, which simulated the operation process of RDEs, and concluded that the heat flux increased with the inlet pressure and inlet temperature, and the heat flux of the inner wall was larger than that of the outer wall.Through 3-D numerical simulation results, Dubrovskii et al. [42] reported that the maximum heat flux of the inner wall near the bottom of the RDC was about 1.7 MW/m 2 , while the maximum heat flux of the outer wall was about 0.95 MW/m 2 at a distance of 150-200 mm from the bottom of the RDC.In order to quantitatively predict the convective heat flux in an RDE, Braun et al. [43] established a reducedorder model using unsteady Reynolds-averaged Navier-Stokes CFD with premixed hydrogen and air as the propellant.The peak time-averaged heat flux was predicted to be about 6 MW/m 2 at the triple point, followed by a downstream decrease in the oblique shock.Roy et al. [44] developed a 3-D transient conduction model of an RDC to achieve the prediction of transient heat flux and inner-wall surface temperature.The results indicated that the peak heat flux reached about 2-2.5 MW/m 2 , the range of the average heat flux ranged from 0.5 to 0.8 MW/m 2 , and the wall temperature increased to around 600 K after 10 s of operation.Additionally, Roy et al. [45] predicted that the outer-wall temperature could rise as much as 500 K after 3.5 s of operation and reach 1100 K in the detonation region.Randall et al. [46] performed experimental and numerical studies to analyze the heat transfer of an RDE.The wall temperature was measured by inserting thermocouples into sensor ports at different depths of the outer wall, and an axisymmetric model was developed to predict wall temperature evolution.According to the numerical results, they found that the temperature of the inner wall was considerably higher than that of the outer wall.Using a novel open-source conjugate heat transfer solver, Yelken et al. [47] created a numerical model consisting of fluid domain and solid domain based on the OpenFOAM platform with a premixed H2-air mixture, and obtained the temperature variations of the inner and outer walls, as well as the fluid domain.Ladeinde et al. [48] simulated the heat transfer of an RDC by means of an explicit large-eddy simulation (LES), which adopted a dynamic mechanism consisting of eight reaction steps and seven chemical substances.Hou et al. [49] carried out a 3-D unsteady conjugate heat transfer simulation for premixed H2-air with heat conduction in both the inner and outer walls of an RDC.When a single detonation wave propagated, the peak heat flux was 1.98 MW∕m 2 at the inner wall and 2.63 MW∕m 2 at the outer wall under the condition of a mass flow rate of 0.227 kg/s.Wang et al. [50] found that a higher RDW velocity and a thicker deflagration region near the walls were caused by the higher wall temperature based on a two-dimensional numerical model, using hydrogen and air.Jorgensen et al. [51] considered the thermal stress generated by the temperature gradient and optimized the structure of an RDC to minimize thermal mechanical stress.
Although the thermal environment and heat transfer characteristics of RDCs have attracted extensive concerns and research, the critical details are still not fully understood with respect to the actual thermal load in both the inner and outer walls, since most experiments and numerical simulations focus on the heat transfer of the outer wall.In particular, few studies have been conducted on the thermal environment of an RDE supplied by kerosene, which is the most promising fuel for practical applications.The purpose of this paper is to reveal the heat transfer characteristics and thermal environment of a twophase RDC by experimentally studying a 220 mm-diameter RDC with RP-3 kerosene as fuel, which could not only provide support and guidance for the thermal management of RDEs, but also be of great significance to the understanding of RDW characteristics and the mechanism of detonation combustion.

RDC Experimental Facilities
Figure 1 schematically shows the RDC experimental facilities, including the air supply system, kerosene supply system, ignition device, timing control system, thrust measuring system, data acquisition system, and the RDC experimental section.The air supply system provides the oxidizer for the RDC.The air provided by the air compressor and oxygen are added together to achieve an oxygen-rich state.The kerosene supply system adopts the pressurization method, injecting nitrogen into the kerosene storage tank to increase the pressure of the tank.The kerosene is transported to the combustion chamber under the action of the pressure difference, and the pressure in the storage tank is adjusted to control the flow of kerosene.Additionally, the flow rate of kerosene is directly measured by the turbine flowmeter in the pipeline.A non-premixed injection scheme is adopted for the propellants in the experiments.At high a injection pressure, the fuel (RP-3) is injected into the combustion chamber throat through 48 small holes with a diameter of 0.2 mm, and the oxidizer (oxygen-rich air) flows into the annular combustion chamber passage through an axially convergent expansion channel within the throat.A kerosene/oxygen hot jet detonator is installed on the side wall of the annular combustion chamber as an ignition device.A timing control system is used to precisely control the opening and closing of each valve.The thrust measurement system mainly consists of three parts: fixed frame, mobile frame and thrust sensor.The fixed frame is bolted to the ground, and the movable frame is located on top of the fixed frame and connected by four slightly deformable horizontal support reeds.On the movable frame, the RDC and the movable frame are connected as a whole by support ribs.A thrust sensor is installed between the entire moving frame and the fixed frame to measure thrust along the horizontal direction.

Water-Cooling Experiment Facilities
The water-cooling experimental system used to measure the bulk/average heat flux of the RDC is illustrated in Figure 2. The water tank stores a volume of 500 m 3 of softened water.The pump pressurizes the waterway with a back pressure valve to prevent boiling from worsening heat transfer.Two paralleled delivery systems are used to cool the inner and outer cylinders, respectively and cooling water flows are measured by two volumetric.In order to make the flow into the vertical channels of the inner and outer cylinders uniform, the flow is divided into eight paths when entering the water-cooled shell, as shown in Figure 2. The same method is used for outflow.

Thermal Environment Measurement of RDC
The schematic diagram of the kerosene two-phase RDC used in this paper is shown in Figure 3.The inner diameter of the outer cylinder is 220 mm, the outer diameter of inner cylinder is 140 mm, and the length of the combustor is 300 mm.The oxygen-enriched air (oxidizer) is delivered into the combustor through the Tesla valve intake structure, which has the characteristics of less forward flow loss and more reverse flow loss.The kerosene (fuel) is atomized through 48 small holes with the diameter of 0.2 mm, evenly distributed along the circumference of the jet panel.The hot jet for ignition is installed 60 mm from the combustion chamber entrance.The combustion products are discharged into the atmosphere through nozzles.In order to take away heat load generated by combustion, the water-cooled channel structure is designed for both inner and outer cylinders.The inlet and outlet temperature and pressure of the water flowing through the inner and outer cylinders are measured separately.Under a certain cooling water flow condition, when the outlet temperature reaches stability, it indicates that thermal balance has been reached between the heat taken away by the cooling water and the heat generated by combustion.At this moment, the bulk/average heat flux of the inner and outer walls of the RDC is obtained through the temperature rise and flow rate of the waterway.In order to characterize the axial thermal environment of the combustion chamber, thermocouples are arranged along the axial position to measure the fluid and outer-wall temperature.The temperature data are collected by a DEW-Soft Sirius multi-channel integrated data collection cabinet with a sampling frequency of 20 Hz.In addition, to assess the heat flux at a specific location in the combustion chamber, two high-frequency heat flux sensors are installed 85 mm and 190 mm from the inlet of the combustion chamber.The sampling frequency is 200 kHz, the response time is less than 1 ms, and the accuracy is ±1%.The combustion product is directly discharged into the atmosphere through convergent nozzles, and the exhaust temperature is measured by a T-type thermocouple and a laser measuring device.The uncertainty contains several components, including components caused by random factors and components caused by systematic factors.The parameters directly measured in the experiment include fluid temperature, flow rate, and heat flux, and the uncertainty of the above three parameters is simplified by instrument error.The measuring equipment in the experiment system includes a T-type thermocouple and a volume flowmeter.The measuring range, measuring accuracy, and instrument error of the measuring instrument are shown in Table 1.

Pressure Measurement of Detonation Wave
The propagation of a rotating detonation wave (RDW) is the most direct indicator of the rotating detonation performance.The propagation mode, propagation velocity, and corresponding range of equivalent ratio of RDW directly reflect the propagation and stable operation characteristics of a detonation wave.Hence, the mode is mainly analyzed by detecting high-frequency dynamic pressure signals under different working conditions.Four highfrequency piezoelectric pressure sensors (PCB113B24) are adopted to capture the pressure pulsations of the combustion chamber, which are used to measure the dynamic peak pressure of the detonation wave, estimate the propagation velocity of the detonation wave, and identify the propagation mode of the detonation wave.The PCB located 130 mm upstream of the combustion chamber entrance is labeled as PCB1.The three PCBs of the combustion chamber are evenly arranged at 120° intervals and numbered PCB2-PCB4.Moreover, eight static pressure sensors are used to measure the pressure along the air-collecting chamber and combustion chamber, named P1-P8.The sensor layout is shown in Figure 4.The experimental data are collected by a high-frequency data collection system (DEW-Soft Sirius multi-channel integrated data collection cabinet), in which the sampling frequency of the dynamic pressure sensor is set to 200 kHz, and the sampling frequency of the static pressure sensor is set to 50 kHz.

Experimental Procedure
The experimental system is controlled by the operation sequence, indicating where to start the injection and ignition, as shown in Figure 5.The flow rate of air containing about 30% oxygen is set to 2 kg/s in this paper.In order to achieve kerosene pressure stability, the kerosene filling time is about 3 s before ignition.The ignition time is related to the injection of oxygen in the hot jet and is maintained at about 0.1 s.To ensure the safety of the experiment, the running time of the detonation is usually set as 500 ms without the protection of a cooling system.In addition, after detonation combustion, a nitrogen stream is blown through the alcohol and kerosene pipes to prevent coking.

Results and Discussion
On the basis of the above experimental system, the long-term operation of an RDC was carried out under stable rotating detonation combustion conditions.The thermal distribution along the axial direction of RDC was investigated firstly, and the average heat flux of the inner and outer cylinders was studied by carefully analyzing the experimental data.

The Confirmation of Mode
By analyzing the dynamic peak pressure of a detonation wave measured by PCB, there are four possible combustion modes in an experiment, namely the deflagration mode, stable rotating detonation mode (stable-RD), low-frequency instability rotating detonation (LFI-RD), and pop-out, as displayed in Figure 6.It is verified that the stable rotating detonation mode occurs in the equivalent ratio range of 0.5~1 when the oxidizer flow rate is 2 kg/s, which was tested and analyzed carefully in this study.Because the PCB captures the dynamic pressure at the measuring point, the average peak value of the dynamic pressure measured by the PCB is used to describe the intensity of the detonation wave.The change in the average peak value of dynamic pressure with the equivalent ratio is shown in Figure 7.Although the stable detonation wave occurred in different working conditions, the peak pressure increased first and then decreased with the increase in the equivalent ratio.Accordingly, the specific thrust of the RDC reached its maximum in the range of 0.7 to 0.85 for the equivalent ratio, as shown in Figure 8.Therefore, the following study is based on the stable detonation conditions to conduct longterm experiments to reveal the thermal environment.

Long-Duration Operation of RDC
Long-term experiments can be carried out to obtain the thermal load of an RDC after the inner and outer cylinders are cooled by a water system.In order to calculate the heat load accurately, the system needs to reach thermal equilibrium, that is, the outlet temperature of the water remains constant.Figure 9 shows the fluid temperature history of the outer cylinder at the 35 mm position and the outlet position under the condition of a water flow rate of 1.271 m 3 /h.It was found that the fluid outlet temperature reached a stable state after 40 s of continuous detonation, which means that the system was in thermal balance.Using the same method, it was found that the inner cylinder was also in thermal balance after 40 s.Ignoring the heat dissipation of the environment, the heat taken away by the cooling water can represent the heat released by the combustion chamber.Therefore, the detonation duration in the timing sequence of the long-duration experiment was set as 40 s.When the wall temperature and fluid temperature do not change with time, it indicates that the system has reached thermal equilibrium.At this time, the heat transferred by the high-temperature gas in the RDC combustion chamber to the wall is equal to the sum of the heat carried away by the fluid convection and the heat leakage to the environment.

,
, , The heat carried away by the fluid convection conv q can be calculated by the energy gained by the fluid.
The heat leakage to the environment para q can be obtained by the calculation for- mula of natural air convection.
air,cov = para W air q h T T ( -)

Thermal Distribution along Axial Direction of RDC
Since the detonation combustion is coupled with a shock wave and an exothermic reaction zone, the heat distribution along the axial direction of an RDC is of great significance to revealing the flow field as well as the combustion mechanism in the combustion chamber.For the gaseous fuels studied more in the previous stage, such as hydrogen and ethylene, the location of the detonation wave is close to the head of the combustion chamber because of the good mixing effect.However, spray characteristics such as the atomization effect and mixing length in an RDC directly affect the stable self-sustaining process of the RDW. Figure 10 displays the wall and fluid temperature distribution of the outer cylinder along the axial direction with an ER of 0.56 (Figure 10a) and an ER of 0.65 (Figure 10b).It can be seen that both the wall surface and fluid temperature increased with the axial position of the combustion chamber, and the wall temperature increased significantly near the 100 mm position of the combustion chamber.This indicates that the heat flux in the combustion chamber suddenly increased sharply near this position, and the rise in fluid temperature lagged because of the obstruction of heat transfer between the fluid and solid.Figure 11 shows the static pressure of the combustion chamber and the temperature distribution of the outer wall.The location of the RDW created an area of high pressure, causing a rise in static pressure at that location.Hence, the position of the RDW was determined by measuring the static pressure along the combustion chamber and the outer wall temperature in this study.It is seen from Figure 11 that the static pressure was the highest and the wall temperature increased substantially at the 100 mm position, marking the location of the RDW.For a two-phase RDC, the flow field and combustion organization is related to the atomization and mixing of the two-phase flow, so the thermal distribution can directly characterize the atomization performance of a two-phase propellant.
The wall temperature distributions (Figure 12a) as well as fluid temperature distributions (Figure 12b) of the outer cylinder along the axial direction under different ERs are illustrated in Figure 12.The results showed that the wall and fluid temperature at the same position increased with the increase in ER.In order to quantitatively analyze heat flux, Figure 13 displays the comparison of the heat flux at 85 mm/190 mm with the average heat flux of the outer cylinder.Since the detonation wave occurred near the 100 mm position, the heat flux of 2.40 × 10 5 W/m 2 ~3.17 × 10 5 W/m 2 at the 85 mm position was before the RDW, and the heat flux of 1.05 × 10 6 W/m 2 ~1.28 × 10 6 W/m 2 at the 190 mm position was after the RDW.As the equivalent ratio increased, the difference in heat flux between 85 mm and 190 mm gradually increased, and the average heat flux of the outer cylinder increased at first and then stays almost stable over a certain range.It can be seen from Figure 13 that the heat flux in the upstream region of the RDW location (e.g., 85 mm position) was almost unaffected by the ER, the heat flux in the downstream region of the RDW location (e.g., 190 mm position) increased with the increase in ER from 0.56 to 0.85, and the average heat flux of the outer cylinder remained stable when the equivalent ratio increases to a certain degree.This shows that under certain combustion chamber structure, it is possible that increasing fuel injection may increase the local heat flux after a detonation wave, but the average heat flux remains unchanged.This phenomenon can be explained by the general law of combustion.When less kerosene is used, the amount of heat released from combustion increases with the amount of kerosene used.Once the amount of kerosene is increased to a certain extent, the heat released by combustion will not continue to increase because the oxidized air remains unchanged.Hence, the average heat flux is a key thermal parameter of RDCs.

Average Heat Flux of Inner and Outer Cylinders
Almost all current experiments mainly focus on the temperature and heat transfer of the outer cylinder because of the ease of measurement.In fact, the thermal environment of the inner cylinder is equally important for annular RDCs, which is not only the basis for designing the thermal protection structure, but also key to acquiring an in-depth understanding of the detonation combustion flow field.According to the heat balance of the inner and outer cylinders, the average heat flux of the inner as well as outer cylinder under the condition of an equivalent ratio from 0.56 to 0.8 is shown in Figure 14.It is found that the average heat flux of the inner cylinder is higher than that of the outer cylinder for the RDC structure adopted in this paper.When the equivalent ratio is 0.77, the average heat flux of the inner cylinder is 1.02 × 10 6 W/m 2 , and the average heat flux of the outer cylinder is 3.4 × 10 5 W/m 2 .The possible reason for this phenomenon is that the high-temperature combustion products are closer to the inner wall, which is consistent with the numerical results in the literature [41,46].Hence, the thermal protection of the inner wall needs to be paid more attention in the process of RDE operation over long periods.
Figure 15 shows the comparison of the average heat flux of the inner cylinder and the specific thrust of the RDC, and Figure 16 shows the average heat flux of the outer cylinder, specific thrust of the RDC, and the average peak pressure of the RDW under different ERs.The comparison results verified that the heat released by the RDC is directly related to its thrust.Additionally, it is found that the average heat flux of the RDC is a better indicator of thrust performance than the peak pressure of the detonation wave.With the increase in ER, the fuel mass flow rate in the combustion chamber increases, and the activity of combustible reactants increases correspondingly, so the detonation combustion intensity increases, which is manifested as the increase in heat load and thrust.However, when the equivalent ratio increases to a certain extent, the heat load and thrust by combustion remain stable, or even gradually decrease.This is because too much fuel injection cannot generate more heat load and thrust when other conditions are not optimized.According to the above analysis, the detonation combustion zone determines the average heat flux as well as the thrust, and it is of great significance to move the RDW forward along the axis of the RDC to increase the heat release of the fuel.

Conclusions
Long-duration experiments are conducted on a water-cooled RDC with a diameter of 220 mm, revealing the heat transfer characteristics and thermal environment of the twophase RDC.In this paper, the average heat flux of the inner and outer cylinders of an RDC and the typical local heat flux of the outer cylinder are quantitatively measured by means of experiments, which not only deepens the understanding of RDC flow field distribution, but also provides quantitative boundary conditions for the thermal protection design of RDCs.The thermal distribution can directly characterize the atomization performance of a two-phase propellant, and the average heat flux of an RDC is a better indicator of thrust performance than the peak pressure of the detonation wave.After a careful analysis of heat distribution and average heat flux, several conclusions are stated as follows.
(1) For the 40 mm-wide two-phase RDC with a Tesla valve intake structure used in this study, a stable rotating detonation mode is observed with an ER between 0.5 and 1 (oxidizer 2 kg/s).Additionally, the combustion chamber reaches thermal equilibrium after continuous detonation for 40 s when the water flow rate is 1.271 m 3 /h.(2) The heat flux in the upstream region of the RDW location ranges from 2.40 × 10 5 W/m 2 to 3.17 × 10 5 W/m 2 , and the heat flux in the downstream area of the RDW location ranges from 1.05 × 10 6 W/m 2 to 1.28 × 10 6 W/m 2 .Therefore, the thrust performance of an RDC can be improved by making the RDW move forward along the RDC axis with a better fuel injection scheme and air intake mixing.(3) The average heat flux of the inner cylinder (around 1.02 × 10 6 W/m 2 ) is about three times that of the outer cylinder (around 3.42 × 10 5 W/m 2 ) for the RDC with a Tesla valve intake structure, so more thermal protection should be allocated to the inner cylinder.In addition, compared with the local heat flux, improving the overall fuel heat release in the RDC is more efficient for the thrust performance.
Due to the limitation of the number of heat flux sensors in the experiment, only the heat flux of the outer cylinder at 80 mm and 190 mm was measured in this paper to compare the difference between the axial downstream and upstream locations of the detonation wave.In a later stage, the reasonable arrangement of temperature sensors will be used to reveal the heat flux distribution of both the outer cylinder and the inner cylinder.

Figure 1 .
Figure 1.The schematic diagram of RDC test bench system.

Figure 2 .
Figure 2. The schematic diagram of water-cooling experiment system.

Figure 3 .
Figure 3.The layout of thermal measuring sensors in the cross-section of the annular combustor.

Figure 4 .
Figure 4.The layout of pressure-measuring sensors.

Figure 5 .
Figure 5. Experimental time sequence of RDC.

Figure 6 .
Figure 6.Dynamic pressure signal under typical mode of RDC.

Figure 7 .
Figure 7. Variation in average peak pressure with ER.

Figure 8 .
Figure 8. Variation in specific thrust with ER.

Figure 9 .
Figure 9. Fluid temperature history at water flow rate of 1.271 m 3 /h.

Figure 10 .
Figure 10.Wall and fluid temperature distribution of outer cylinder along the axial direction.(a) Temperature distribution with ER of 0.56; (b) temperature distribution with ER of 0.65.

Figure 11 .Figure 12 .
Figure 11.Distribution of static pressure and outer wall temperature distribution.

Figure 13 .
Figure 13.Comparison of heat flux at 85 mm/190 mm and average heat flux of outer cylinder.

Figure 14 .
Figure 14.Average heat flux of inner and outer cylinder with different ERs.

Figure 15 .
Figure 15.Comparison of average heat flux of inner cylinder and specific thrust.

Figure 16 .
Figure 16.Comparison of average heat flux of outer cylinder and specific thrust.