Analysis of the flow field of kerosene-fueled rotating detonation engine with film cooling

The advance of the rotating detonation engine (RDE) toward practical applications demands the integration of effective cooling schemes. In this study, a three-dimensional simulation of the hydrogen-enhanced kerosene-air RDE with inclined cylindrical film cooling holes is conducted to analyze the influence of the cooling flow on the two-phase rotating detonation flow field based an Eulerian–Lagrangian model. The liquid kerosene is injected at the ambient temperature with hydrogen-assisted combustion enhancement. Results suggest that a stable propagation of the kerosene-fueled rotating detonation wave can be maintained after the introduction of cooling air and the three-dimensional structure of the flow field is analyzed. It is found that the periodic sweeping action of the detonation wave leads to temporary blockages of the film cooling holes, causing interruptions in the outflow of cooling air. Additionally, the investigation highlights the intensified heating and evaporation of kerosene droplets near the outer wall of the RDE, whereas the presence of cooling air prevents the accumulation of kerosene vapor near the outer wall. It is revealed that the film cooling efficiency exhibits a lower value in the vicinity of the fuel injection surface, but gradually increases along the length of the combustion chamber.


Introduction
Detonation engines, as a new type of propulsion device, possess advantages such as simple structure, higher thermal cycle efficiency, and rapid heat release [1,2].The rotating detonation engine (RDE) is a detonation-based propulsion system that operates based on one or multiple rotating detonation waves (RDWs) within the combustion chamber, as depicted in figure 1.The RDE exhibits the capability to function across a broad range of conditions, making it suitable for diverse applications, such as rocket or ramjets.In order to expedite the practical application of the RDE, the recent research has gradually shifted from the implementation of explosive gaseous fuels [3][4][5][6] to more practical liquid fuels [7][8][9][10], which necessitates the implementation of combustion enhancement techniques such as hydrogen addition [7,[11][12][13][14] to establish and sustain detonations effectively.However, the practical applications of the RDE are faced with another challenge -the need of thermal protection for long-term operation [15].The research on effective cooling schemes for the chamber walls of the rotating detonation engine (RDE) is currently limited, despite the reported high-frequency heat flux experienced by these walls [16,17].A widely used cooling technique in conventional gas turbines, known as film cooling, has been numerically verified and proven effective [18,19].Nevertheless, the existing studies still focused on RDEs using gaseous fuels instead of the more practical liquid fuels such as kerosene; there are, however, notable distinctions between gaseous and two-phase rotating detonation waves.Consequently, the primary objective of this study is to examine a hydrogenenhanced kerosene-fueled RDE integrated with cylindrical film cooling holes through a threedimensional numerical simulation.

Numerical methods
The configuration of the RDE is characterized by a co-axial annular design with a diameter of 40 mm.On the outer wall of the RDE, a column of 10 film cooling holes with a diameter of 1 mm are uniformly distributed and equally spaced along the axial direction.The film holes are inclined at a 30° angle with respect to the wall surface.Monitoring points are positioned along the axial direction at a distance of 0.1 mm from the center of the film cooling holes for temperature and pressure measurements.figure 2 depicts the configuration of the combustion chamber of the RDE along with the arrangement of film cooling holes and the mesh grid.The numerical simulation is conducted using the Eulerian-Lagrangian approach.The governing equations are the two-dimensional Navier-Stokes equations with the realizable k- model used for turbulence modeling and the kinetics of kerosene droplets are described by the Discrete Phase Model (DPM) approach.The mass flow rate of kerosene is 0.006 kg/s, and the initial droplet diameter and temperature are 10 μm and 300 K, respectively.The mass fraction of hydrogen in the fuel mixture (kerosene/hydrogen) is approximately 40%, falling within the specific range for the experiment [13], and the mass flow rate of air is 0.24 kg/s, resulting in a global fuel-air equivalence ratio of 1.The secondary flow (cooling air) uses a pressure inlet of 0.6 MPa.C12H23 is employed as a surrogate of kerosene and the chemical reactions are described by the global one-step models [20] for both hydrogen and kerosene, as summarized in Table 1.The geometry is meshed with a hexahedral structured grid, with an average size of 0.3 mm for the combustion chamber and a finer mesh of 0.01 mm in the vicinity of the film cooling hole locations.The total number of grids in the entire model reaches1.057million.The simulation employs the density-based solver provided by ANSYS Fluent.The validity of the numerical setup and a grid sensitivity analysis can be found in our previous study [19].

Main characteristics of the flow field
Figure 3 illustrates the establishment of a stable self-sustained RDW and provides a visualization of its instantaneous three-dimensional structure.The plot also highlights the presence of dispersed liquid droplets within the combustion chamber.As the droplets progress through the chamber, their diameter diminishes from 10 μm at the injection surface to less than 2 μm above the RDW.Downstream of the fuel-refill zone, the liquid kerosene undergoes gradual breakup into smaller droplets, followed by evaporation.After being swept by the detonation wave, the droplets are rapidly heated and consumed.However, a fraction of the droplets remains unconsumed and enters the expansion zone, as also noted by ref. [7].Within this region, these droplets undergo evaporation upon contact with the hightemperature products, leading to the formation of kerosene vapor.In contrast to the results obtained from previous two-dimensional simulations [8,14], It is also demonstrated that the strengthening of the detonation wave towards the outer wall results in enhanced heating and evaporation of kerosene droplets in close proximity to the outer wall.Consequently, the kerosene vapor tends to concentrate along the outer wall.On the other hand, near the film cooling holes, due to the presence of the cooling air, the kerosene vapor will not concentrate near the outer wall.Only a small amount of kerosene vapor accumulates.4(a) presents the flow field surrounding the film cooling holes at two consecutive snapshots, which provide insights into the outflow characteristics of the cooling air before and after it is influenced by the sweeping action of the RDW.Notably, the dense presence of injected droplets at normal temperature is observed in the fuel-refill zone.However, shortly after being swept by the RDW, the film cooling holes located within the fuel-refill zone become obstructed, leading to the interruption of the outflow of cooling air.Specifically, the film cooling holes near the injection surface are first affected by the detonation wave, followed by the oblique shock wave sweeping over the remaining downstream film cooling holes.
Figure 4(b) represents the variation of the averaged mass flow rate of cooling air with time.When the film cooling holes are swept by the RDW, they will be temporarily blocked.Moreover, if the pressure of the hot-temperature burned products exceeds the local pressure at the outlet of the film cooling holes, they may invade the film holes.This results in the periodic nature of the mass flow rate of cooling air, including small periods of negative values when the burned products enter the film cooling holes.The fluctuation frequency of the mass flow rate of cooling air is close to that of the propagation of the RDW, which propagates at a speed of 1582.9 m/s.Furthermore, minor secondary peaks in the flow rate profile have been observed prior to reaching the peak value.This can be attributed to the sequential interaction of the detonation wave and the subsequent oblique shock wave with the film cooling holes.The time lag results in the appearance of these secondary peaks in the mass flow rate profile.

Cooling effectiveness
In the preceding section, we undertake an examination to analyze the fundamental attributes and features concerning the flow field of the two-phase rotating detonation wave (RDW).Here we will focus on the cooling effectiveness of the film cooling holes.To this end, the following cooling efficiency  [21] is used: where   ̅̅̅̅ denotes the average wall temperature without film cooling,   ̅̅̅ represents the average wall temperature with film cooling, and   is the inlet temperature of the cooling air. Figure 5(a) presents the average temperature profiles along the length of the combustion chamber, comparing the cases with and without film cooling.The plot includes the height of the rotating detonation wave h to indicate its position, which is also the height of the full-refill zone.The plot clearly demonstrates that within the fuel-refill zone, where two film cooling holes are located, there is a negligible temperature difference between the cases with and without film cooling.This can be attributed to the fact that the fuel-refill zone is filled with cold reactants having the same temperature as the cooling air (300 K).Further downstream, the cooling air discharged from the film holes becomes increasingly effective in terms of cooling effect, resulting in a significant decline in the average temperature along the outer wall.It is noted that the temperature profile exhibits a recurring periodic pattern over time due to the distributed placement of the film cooling holes along the axial direction.Similarly, the film cooling efficiency (Fig. 5(b)) remains at a critically low level near the fuel injection surface, which is less than 0.2. starts to increase slightly away from the injection surface and reaches about 0.4 near the front height of the RDW.Beyond the fuel-refill zone, however, the film cooling efficiency exhibits a noticeable upward trend, with a significant increase observed, indicating an overall effective cooling performance, particularly near the outlet of the combustion chamber.Based on this, it may not be appropriate to position film cooling holes within the region covered by the fuelrefill zone as this could potentially diminish the flammability of the liquid fuel.Besides, the cold reactants in the fuel-refill zone may be able to fulfill a similar role to that of cooling air.

Conclusions
To summarize, this study provides some insights into the characteristics of the flow field and cooling effectiveness in a hydrogen-enhanced kerosene-fueled RDE with film cooling.The three-dimensional analysis reveals an intensified heating and evaporation of kerosene droplets near the outer wall, mitigated by the presence of cooling air which prevents the accumulation of kerosene vapor near the wall.Furthermore, the periodic sweeping action of the RDW may result in temporary blockages of the film cooling holes, causing interruptions in the outflow of cooling air.
The investigation of average temperature profiles along the length of the combustion chamber reveals negligible temperature differences inside the fuel-refill zone with and without film cooling holes.This can be attributed to the presence of cold reactants with temperatures similar to that of the cooling air.Consequently, the placement of film cooling holes within the fuel-refill zone may prove unnecessary due to the observed low cooling effectiveness, potentially compromising the flammability characteristics of the liquid fuel.Further downstream, the cooling air exhibits increasing effectiveness, resulting in a substantial reduction in the average temperature along the outer wall and an improved film cooling efficiency.

Figure 2 .
Figure 2. Configurations of the RDE and cylindrical film cooling holes.

Figure 3 .
Figure 3. Flow field of the RDW and the distribution of kerosene droplets.1-Detonation front, 2-Contact surface, 3-Oblique shock wave, 4 -Film cooling holes.Snapshots at t 1 = 710 , t 2 = 720 , t 3 = 730 .Figure4(a) presents the flow field surrounding the film cooling holes at two consecutive snapshots, which provide insights into the outflow characteristics of the cooling air before and after it is influenced by the sweeping action of the RDW.Notably, the dense presence of injected droplets at normal temperature is observed in the fuel-refill zone.However, shortly after being swept by the RDW, the film cooling holes located within the fuel-refill zone become obstructed, leading to the interruption of the outflow of cooling air.Specifically, the film cooling holes near the injection surface are first affected by the detonation wave, followed by the oblique shock wave sweeping over the remaining downstream film cooling holes.Figure4(b) represents the variation of the averaged mass flow rate of cooling air with time.When the film cooling holes are swept by the RDW, they will be temporarily blocked.Moreover, if the pressure of the hot-temperature burned products exceeds the local pressure at the outlet of the film cooling holes, they may invade the film holes.This results in the periodic nature of the mass flow rate of cooling air,

Figure
Figure 3. Flow field of the RDW and the distribution of kerosene droplets.1-Detonation front, 2-Contact surface, 3-Oblique shock wave, 4 -Film cooling holes.Snapshots at t 1 = 710 , t 2 = 720 , t 3 = 730 .Figure4(a) presents the flow field surrounding the film cooling holes at two consecutive snapshots, which provide insights into the outflow characteristics of the cooling air before and after it is influenced by the sweeping action of the RDW.Notably, the dense presence of injected droplets at normal temperature is observed in the fuel-refill zone.However, shortly after being swept by the RDW, the film cooling holes located within the fuel-refill zone become obstructed, leading to the interruption of the outflow of cooling air.Specifically, the film cooling holes near the injection surface are first affected by the detonation wave, followed by the oblique shock wave sweeping over the remaining downstream film cooling holes.Figure4(b) represents the variation of the averaged mass flow rate of cooling air with time.When the film cooling holes are swept by the RDW, they will be temporarily blocked.Moreover, if the pressure of the hot-temperature burned products exceeds the local pressure at the outlet of the film cooling holes, they may invade the film holes.This results in the periodic nature of the mass flow rate of cooling air,

Figure 4 .
Figure 4. (a) Outflow conditions of film cooling hole before and after the passage of the two-phase RDW.Film cooling protected area is denoted by temperature iso-surfaces within 300 -1500 K. (b) Variation of the average mass flow rate of the cooling air with time. .

Figure 5 .
Figure 5. (a) Temperature (with and without film cooling) and (b) cooling efficiency  along the axial direction.h -Height of the RDW.Similarly, the film cooling efficiency (Fig.5(b)) remains at a critically low level near the fuel injection surface, which is less than 0.2. starts to increase slightly away from the injection surface and reaches about 0.4 near the front height of the RDW.Beyond the fuel-refill zone, however, the film cooling efficiency exhibits a noticeable upward trend, with a significant increase observed, indicating an overall effective cooling performance, particularly near the outlet of the combustion chamber.Based on this, it may not be appropriate to position film cooling holes within the region covered by the fuelrefill zone as this could potentially diminish the flammability of the liquid fuel.Besides, the cold reactants in the fuel-refill zone may be able to fulfill a similar role to that of cooling air.

Table 1 .
Chemical reaction equation