Effect of Multicoaxial Injectors on Nitrogen Film Cooling in a GCH4/GO2 Thrust Chamber for Small-Scale Methane Rocket Engines: A CFD Study

Abstract

Improper film cooling design and positioning of an injector in the face plate cause thermal damage to the thrust chamber wall and lead to rocket engine failures. An experimental combustor with five shear coaxial injectors was damaged owing to inadequate film cooling distribution on the thrust chamber wall. The present study aimed to simulate the experimental test case and investigate the causes of the thermal damage. In the simulation, gaseous methane and oxygen were injected at the inner and outer inlets of the shear coaxial injectors and nitrogen, used as the coolant, was injected near the upstream of the chamber wall. The turbulent chemistry interaction was modeled using a reduced DRM-19 mechanism by incorporating the Eddy Dissipation Concept model. Numerical investigations were conducted to examine the cause of thermal damage. The temperature contours of the thrust chamber wall were compared with the experimental image of the damaged wall. Further, simulations of single-row (SR) and multi-row (MR) injector configurations were conducted to assess the effect on film cooling distribution. The adiabatic film cooling effectiveness and specific impulse were determined for all simulated cases. The results showed that MR simulations with narrow injector angles had poor film cooling performance, while wider angles led to lower specific impulse. The face plate with an angle of 15 degrees between the injector positions showed better performance in terms of considering both the film cooling and specific impulse.

1. Introduction

Liquid rocket engines operate under various conditions of elevated combustion temperatures to meet the thrust requirements necessary for launching space vehicles. Over extended periods of operation, the high heat transfer rates of the scorching gases entering the thrust chamber on its wall result in excessive thermal stress and there is a change in the structure of the material resulting from the interaction with high temperatures, ultimately leading to structural damage. Various thrust chamber cooling methodologies, including regenerative, dump, film, transpiration, ablative, and radiation cooling, have been employed to protect the thrust chamber wall in rocket engine applications [1,2]. Among these techniques, film cooling, whether implemented through the injection of a gaseous or liquid-type coolant, is a significant approach for mitigating the adverse effects of high heat transfer rates. A coolant injection, primarily considering a fuel as the coolant, sometimes an oxidizer, is directed near the injector face plate and sometimes at multiple locations along the chamber and nozzle walls, forming a thin film layer between the inner chamber wall and hot gas flow. This thin-film layer attenuates the heat transfer rates, reducing the resultant thermal stress and heat flux experienced by the chamber and nozzle walls [3,4]. Consequently, this allows maintaining the wall temperature below a critical threshold during extended operation periods, ensuring structural integrity without incurring damage.

Film cooling is an interesting technique that can be employed alone or in combination with regenerative cooling to reduce the hazardous temperature to below the acceptable temperature for walls. Therefore, film cooling can protect walls against high thermal loads to ensure engine safety and a long structural life [5,6]. Typically, liquid rocket engines use a significant amount of fuel as a coolant in a film-cooling environment. This study applies methane and oxygen as fuel and oxidizers, considering reusability [6].

Numerous film cooling studies have been conducted through experiments and simulations [1,2,3,4,5,6,7]. Daimon et al. [7] performed film cooling simulations using slot-type multi-injectors for film cooling [7,8] to investigate the effect of slot dimensions on mixing with the core flow. The simulation results revealed that a large slot dimension led to a low heat flux distribution near the face plate, with high combustion efficiency when the velocity ratio of the gaseous film to the main fuel was 0.5. Daimon et al. [9] tested the performance of discrete hole-type film cooling in a multi-injector combustor using hydrogen and oxygen. In their research, discrete film holes were used between the multi-injectors and not over the injectors near the chamber wall to investigate the wall heat flux and film cooling efficiency. Similarly, Ma et al. performed a cooling analysis using a discrete slot-type multi-injector. Based on experimental and simulation data, they found that the cooling efficiency was proportional to the film flow rate to the power of 0.4. Moreover, they suggested that reducing the slot width compared with the slot thickness led to a high cooling efficiency [10]. Shine et al. [11] conducted experiments to investigate the impact of coolant injector configurations on film cooling effectiveness, cooling length, and uniformity using both gas and liquid coolants. The primary result of the study demonstrated that an optimal coolant injector configuration exists, which maximizes the cooling effect. Daimon et al. [12,13] presented a methodology for simulating combustion and heat transfer using gaseous methane and oxygen. The computed results accurately predicted the pressure profile, heat flux based on the coolant’s enthalpy gain, and wall temperatures. Xu et al. [14] conducted a coupled heat transfer calculation with and without film cooling conditions and found that the heat flow entering the thrust chamber on its wall was significantly reduced with film cooling compared with that without film cooling. Additionally, they illustrated the cooling effect of film cooling combined with regenerative cooling, showing that it had better cooling performance on the thrust chamber wall than regenerative cooling alone. Wei et al. [15] modeled and simulated a rocket combustor using GOX/GCH4 as propellants to gain a more comprehensive understanding of the combustion and heat transfer processes within the combustor. The authors aimed to evaluate the computational techniques by applying the Eddy-Dissipation Concept (EDC) combustion model and a turbulence model to simulate detailed chemical reactions. Recently, Chen et al. [16] numerically investigated supersonic film cooling in a convergent–divergent nozzle. Their study focused on minimizing performance impact while improving cooling in real engine conditions. And found that higher slot heights and pressure ratios improved cooling effectiveness with minimal impact on nozzle thrust performance. Xiang et al. [17] compared slot and circular hole film cooling structures in a GO2/GH2 subscale thrust chamber. Their results showed that the slot structure provided better wall adherence, reducing the wall temperature by approximately 6%. This demonstrates the effectiveness of slot structures in improving insulation against hot gas. However, research on gaseous film cooling performance based on increasing or decreasing the number of multiple injectors still needs to be conducted. Therefore, the present study is focused on this.

There is a lack of research on the impact of multiple injectors on gaseous film cooling in the thrust chamber. This cooling is essential as it prevents damage to the thrust chamber wall. This study aims to investigate the insufficient distribution of nitrogen film cooling on the thrust chamber wall using multi-coaxial injectors for small-scale methane rocket engines. The thrust chamber wall in a GCH4/GOX combustion test with five coaxial injectors was damaged due to the overheating caused by inadequate nitrogen film cooling on the chamber wall, which is considered as the base case for performing CFD simulation to examine the inadequate nitrogen film cooling on the chamber wall. Following this, the numerical simulations are carried out based on various total numbers of injector configurations, with the total flow rate set constant at 8, 10, 12, 16, 20, or 24 in a single-row configuration (denoted as SR8, SR10, etc.) and 24, 30, 36, 48, 60, or 72 in a multiple-row configuration (denoted as MR24, MR30, etc.) to investigate the nitrogen film cooling distribution on the thrust chamber wall using the numerical data of the wall temperature and species mass fraction distribution. Subsequently, the adiabatic film cooling effectiveness and specific impulse for all simulation cases are calculated and compared for performance analysis. Based on the simulation results, design guidelines for multiple rows of multi-injectors are suggested.

2. Computational Setup

2.1. Injector, Combustor, and Nozzle Configuration

A shear coaxial injector consists of two inlets: a center gap for GCH4 and an outer gap for GO2. A schematic of this fuel-center injector configuration is shown in Figure 1a. The combustor has five coaxial injectors in the injector face plate, as shown in Figure 1b. The total area of the injector was calculated using the required flow rate and the differential pressure of the injector, which corresponds to 20% of the combustion pressure. The maximum number of holes that can be manufactured was derived by considering the machining and additive manufacturing tolerance for the calculated area, which corresponds to five. Five injectors were placed at the halfway point of the radius from the center. Near the upstream of the combustion chamber wall, coolant N2 is injected, as shown in Figure 1c. Figure 1d shows the thrust chamber configuration; its dimensions are listed in

Table 1. Dimensions of the thrust chamber (unit: mm).

Name	Symbol	Value
Diameter of GCH4	d1	2.5
Inner diameter of GO2	d2	4.5
Outer diameter of GO2	d3	5.7
Recess length	Lr	2.0
Radius of injector position	R2	15.0
Chamber radius	Rc	42.0
Chamber length	Lc	33.0
Slit length for coolant	Lt	2.0
Radius of slit for coolant	R4	38.8
Outer radius of coolant	Rc	42.0
Inner radius of coolant	Ri	41.8
Throat radius	Rt	10.5
Nozzle exit radius	Re	20.0
Nozzle length	Ln	120.0
Chamber exit to throat		86.0
Throat to nozzle exit		34.0
Angle between injectors at R2	${\theta }_{mr}$	72°
Number of injectors		5

.

Figure 2 shows the thrust chamber configuration in a ground combustion test under the operating conditions presented in

Table 2. Operating conditions for the five-injector thrust chamber configuration.

Injection Pressure, Bar			Combustion Chamber Pressure, Bar
GCH4	GO2	N2
16.38	14.39	12.91	12.04

. During the combustion test, the convergent and throat sections of the nozzle were thermally damaged due to inadequate film cooling. In this present study, this thrust chamber configuration is considered a base case to investigate the thermal problem.

2.2. Injector Configuration

2.2.1. SR5 Base Case Configuration

The thrust chamber configuration delineated in Figure 1 is referred to as the SR5 configuration in this work. The calculated total mass flow rate for the operating conditions presented in Table 2 is listed in

Table 3. Inlet boundary conditions for all simulations.

	Total Mass Flow Rate (g/s)
Configuration	GCH4	GO2	N2
SR5	52.59	114.72	94.79

. The equivalence ratio based on the calculated total mass flow rates of GCH4 and GO2, excluding N2, is 1.833, indicating a fuel-rich condition. A brief flowchart of the present work is described in Figure 3.

Furthermore, the same total mass flow rates of the SR5 are used in the simulation cases of the single-row (SR) and multiple-row (MR) configurations, where the injection area of GCH4 and GO2 is increased as the number of injectors increases in the SR and MR configurations. The injection velocities of GCH4 and GO2 decrease when maintaining the same total mass flow rate of GCH4 and GO2.

2.2.2. Single-Row (SR) Configuration of Multiple Injectors

The SR5 configuration is re-considered as a base design for the SR configurations. To investigate improper film cooling on the thrust chamber wall caused by the propellant jet from the injectors, various cases are considered based on the number of injectors with different positions on the face plate. A set of cases comprising a single row (SR) of multiple injectors based on the number of injectors (8, 10, 12, 16, 20, 24) at the radius of the injector position (R₂ = 15 mm) is considered. Based on the total number of injectors, the new cases are named SR8, SR10, SR12, SR16, SR20, and SR24. In the SR configurations, the number of injectors is increased from 5 to 24 by reducing the angle (${\theta }_{mr}$) from 72° to 15°.

2.2.3. Multiple-Row (MR) Configuration of Multiple Injectors

An additional row of injectors was added to investigate the effect of injector placement on mitigating the high velocity and momentum of propellant injected from the injector. A set of cases comprising multiple rows (MRs) of multi-injectors based on the number of injectors (24, 30, 36, 48, 60, and 72) at three different positions of the injectors on the radii (R₁, R₂, and R₃) are considered to investigate and compare the mixing and film cooling performance of the MR and SR configurations for the same number of injectors. The new cases are considered and named MR24, MR30, MR36, MR48, MR60, and MR72 based on the total number of injectors. Like the SR configurations, many injectors are kept on the injector face plate at three different radii, with various angles between the injectors (${\theta }_{ir}$, ${\theta }_{mr}$, and ${\theta }_{or}$) in the MR configurations. A schematic of the 24-injector MR configuration on the face plate with different angles at three different radii, R₁, R₂, and R₃, is shown in Figure 4a. The remaining MR configurations are shown in Figure 4b–f. The total number of injectors and angles between the injectors at R₁, R₂, and R₃ for all the simulation cases are listed in

Table 4. Dimensions of the multiple injector positions in MR configurations.

Configuration	Total Injectors	Inner Row (R1 = 10 mm)		Middle Row (R2 = 15 mm)		Outer Row (R3 = 25 mm)
		No. of Injectors	${\mathit{\theta }}_{\mathit{i}\mathit{r}}$, °	No. of Injectors	${\mathit{\theta }}_{\mathit{m}\mathit{r}}$, °	No. of Injectors	${\mathit{\theta }}_{\mathit{o}\mathit{r}}$, °
MR24	24	4	90	8	45	12	30
MR30	30	5	72	10	36	15	24
MR36	36	6	60	12	30	18	20
MR48	48	8	45	16	22.5	24	15
MR60	60	10	36	20	18	30	12
MR72	72	12	24	24	15	36	10

.

2.3. Numerical Method

2.3.1. Computational Domain and Setup

In the SR5 configuration, featuring five injectors, the geometry was divided into quintiles based on the injector numbers, resulting in symmetry about an axial x-axis at 72 degrees. The flows within one quintile were presumed to be symmetrical for the remaining quintiles. Hence, only one quintile of the SR5 flow domain was discretized into a computational domain for simplicity and to minimize the computational load. This is shown in Figure 5. Combustion simulations of GCH4/GO2 using the Reynolds-averaged Navier–Stokes equations were performed using a pressure-based solver in the commercial code ANSYS Fluent. The standard k–ε model was used for turbulence modeling with near-wall treatment based on the standard wall function. The compressible flow variables in the computational domain were calculated using the continuity, momentum, and energy equations. The mass flow inlet BC was used for all the inlets using the values listed in Table 3, and the pressure outlet BC was used for the nozzle outlet. The film cooling analysis was conducted to determine the wall temperature without any losses occurring on the thrust chamber walls. Therefore, the adiabatic wall BC with a no-slip condition was imposed on all the walls. To repeat the same flow across the sliced boundary surfaces, the periodic BC was applied on the sliced surfaces. The named boundary conditions are shown in Figure 5. The mass flow, which is the same as the experimental conditions shown in Table 3, was applied as a boundary condition to the inlet, and the pressure out condition was applied to the nozzle outlet to ensure sufficient expansion of the flow through the nozzle. A coupled scheme was employed to solve the momentum and pressure-based continuity equations. The solution accuracy for all the flow variables was improved using second-order spatial discretization.

2.3.2. Species Transport Model

Modeling the turbulence–chemistry interaction is a thought-provoking task to achieve high-fidelity results. In this context, the Eddy Dissipation Concept (EDC) model has the capability to simulate the complex chemistry involved in the combustion process, employing multi-step reaction mechanisms such as GRI 3.0, GRI 1.2, and various reduced mechanisms. Therefore, the EDC model has been widely utilized to model turbulence–chemistry interaction in GCH4/GO2 rocket combustors [17,18,19,20,21]. Wei et al. [17,18] modeled GCH4/GO2 combustion for a combustor with multi-injectors, considering standard and realizable k-ε models with various wall treatment parameters. They compared the calculated wall pressure and wall heat flux from the simulation with the measured experimental data. They concluded that the simulation data obtained by employing the standard k-ε model with enhanced wall treatment agreed well with the experimental data. Following this, Liu et al. [19] applied the same numerical methodology as Wei et al. [18] and investigated the combustion characteristics and heat transfer entering a multi-coaxial combustor on its wall based on injector design parameters such as the injector wall distance, post thickness, and recess length. Similarly, Liu et al. [20] also employed the same numerical methodology and proposed a numerical scheme for calculating wall heat flux for multi-injector combustors with film cooling. Based on a review of the simulation results using the EDC model, it is evident that the EDC model can produce accurate and reliable results. The EDC model [22] developed by Magnusson as an extension of a previous eddy dissipation model [23], was adopted to include chemistry mechanisms in turbulent reactive flows. The chemical reactions were incorporated inside the combustion chamber by selecting the species transport model. The reaction rates for species i based on the different turbulent rates occurring within the fine turbulent structures were calculated by formulating small-scale eddies in the turbulent flow. These were governed by the turbulent kinetic energy ($k$) and turbulent dissipation energy ($\epsilon$). The net rate of production of species ($R_i$) was calculated using Equation (1).

$$R_i=\rho \mathcal{K}{\displaystyle \frac{Y_i^{*}-Y_i}{{\tau }^{*}}}$$

(1)

$$\mathcal{K}={\displaystyle \frac{{\xi }^{*2}}{1-{\xi }^{*3}}}$$

(2)

$${\xi }^{*}={\complement }_{\xi }{\left({\displaystyle \frac{\upsilon \epsilon }{k^2}}\right)}^{1/4}$$

(3)

$${\tau }^{*}={\complement }_{\tau }{\left({\displaystyle \frac{\upsilon }{\epsilon }}\right)}^{1/2}$$

(4)

where $\rho$ is the density of species i, $Y_i^{*}$ is the fine-scale species mass fraction after reacting over the integration time scale ${\tau }^{*}$, $\mathcal{K}$ is the volume fraction of the reactive fine scale, ${\xi }^{*}$ is the length scale with constant ${\complement }_{\xi }$ = 2.1377, $\upsilon$ is the kinematic viscosity, and time scale constant ${\complement }_{\tau }$ = 0.4082.

For methane–oxygen combustion, the fully detailed mechanism, GRI 3.0 [24], has been widely used in numerical simulations. However, it requires additional computational resources because it includes 325 elementary two-way reactions involving 53 species. In contrast, the reduced mechanism, DRM19 [25] based on GRI 1.2 [26], comprises 84 elementary reaction steps involving 21 species, including inert N2 and Ar. The reduced mechanism requires fewer computational resources than the fully detailed mechanism, produces good results, and can be used in numerical simulations [27]. In addition, we assumed N2 as the coolant and injected it into the combustion area near the chamber wall to investigate the distribution of the coolant on the thrust chamber wall. To evaluate the cooling performance, the DRM19 mechanism, in which N2 is not involved in the 84 elementary reactions, was used in the simulations of the present study. The ideal gas law was used to calculate the density of species.

2.3.3. Numerical Validation

To validate the numerical methods used in this study, we considered the experimental setup employing a single coaxial injector from the literature [28,29,30,31] at a chamber pressure of 1.9 MPa. The combustor test case featured a circular section with a diameter of 12 mm, a chamber length of 285 mm, a throat radius of 3.8 mm, and a nozzle exit radius of 6.69 mm. The center hole diameter for GO2 was 4.0 mm, whereas the inner and outer diameters of the annular hole for GCH4 were 5 mm and 6 mm, respectively. The inlet mass flow rates of GCH4 and GO2 were 15.3 g/s and 33.9 g/s, respectively, resulting in an oxidizer-to-fuel ratio of 2.2. The inlet temperatures of GCH4 and GO2 were 268 K and 276 K, respectively. Further details of the combustion chamber can be found in the original study [28]. To validate the simulation results against the experimental data of the wall pressure and wall heat flux [28,29], a two-dimensional-axisymmetric computation domain was employed [31]. Figure 6a shows the temperature contours obtained from the simulations. The pressure and wall heat flux profiles obtained from the simulation were compared with the experimental data from the axial chamber wall locations. This comparison is shown in Figure 6b,c. The comparison reveals that the simulated pressure profile aligns well with the experimental data. However, the simulated wall flux profile at the upstream chamber wall shows lower values compared to the experimental data but agrees well with the experimental data from the middle of the chamber to its end. The overall trend of the wall flux profile concurs with that of the experimental data. Consequently, this analysis indicates that all the simulation data obtained using the numerical methods agree reasonably well with the experimental results, with an uncertainty error of 4% for wall pressure and 10–15% for heat flux [30,31]. The validation test confirms that the numerical methods employed produce good results for further simulations in this study.

2.3.4. Grid Independent Study

For the grid-independent test, the computational domain in Figure 5 was chosen. This domain is a quintile part of the three-dimensional geometry (SR5 configuration in Figure 1) and was considered with four grid numbers: Grid-1, Grid-2, Grid-3, and Grid-4. The simulation domain was meshed with structured hexahedral mesh elements throughout, except at the interface between the injector and combustion chamber, where an unstructured tetrahedron mesh was used to simplify the meshing process and make it easy to apply the periodic boundary condition. The detailed structured and unstructured mesh, along with a zoomed view of a small portion 1 mm from both sides of the injector–chamber interface meshed with the tetrahedral elements, are shown in Figure 7a. The first element distance from the thrust chamber wall was 0.01 mm, with a cell growth rate of 1.2 to generate 5-layer mesh inflation. Thereby, the wall Y⁺ range was 5–65. The detailed summary of element size and total number of mesh elements used in the grid-independent simulation cases are listed in

Table 5. Summary of element sizes and total elements in grid-independent simulations.

	Element Size for Injector, mm	Element Size for Combustor and Nozzle, mm	Total Elements
Grid-1	0.3	0.5	1,016,743
Grid-2	0.2	0.4	1,862,141
Grid-3	0.1	0.3	4,881,788
Grid-4	0.1	0.25	6,063,663

.

For comparing the grid-independent results, an axial location line on the thrust chamber wall at ${\theta }_{mr}$ = 0° was chosen; this line is shown in Figure 7b. The wall temperature profiles based on the four different grids are shown in Figure 8. When comparing Grid-1 and Grid-2, Grid-1 shows a higher wall temperature profile up to 30 mm, beyond which it shows a lower wall temperature profile than Grid-2. Upon increasing the grid elements in Grid-3 and Grid-4, a marginal difference in the wall temperature profiles between Grid-2 and Grid-3 was noted. In contrast, no difference in the wall temperature profiles was observed between Grid-3 and Grid-4. Therefore, the element size in Grid-3 was sufficient for the subsequent simulations in the present study. The flow presumption of the SR5 was extended to the SR and MR configurations to reduce computational load. The size of the computational domain for the SR and MR configurations was considered based on the symmetry angle. The symmetry angle for each simulation case differed because of the increasing injector numbers in the SR and MR configurations, as listed in

Table 6. Summary of symmetry angle and total elements for base, SR, and MR configurations.

Configuration	Symmetry Angle for Periodic BC (°)	No. of Elements in Sliced Angle
SR5	72	4,881,788
SR8	45	4,339,767
SR10	36	3,622,330
SR12	30	3,050,831
SR16	45	4,536,740
SR20	36	3,635,617
SR24	30	3,185,438
MR24	90	11,434,370
MR30	72	9,293,276
MR36	60	7,975,823
MR48	45	6,139,528
MR60	36	5,873,786
MR72	30	4,926,006

. Consequently, the total number of grid elements based on the element size of Grid-3 for each case in both the SR and MR configurations was not the same. The detailed symmetry angles and the total number of grid elements are listed in Table 6.

2.3.5. Performance Analysis Method

By increasing the number of injectors while maintaining the same total mass flow rates, the injection area of GCH4 and GO2 increased, resulting in decreased injection velocities for both propellants. The calculated momentum ($J$) of GCH4 and GO2 using Equation (5) indicated a gradual decrease in the momentum of each propellant. However, the momentum ratio of GCH4 to GO2 remained constant across all cases.

$$J_i={\rho }_i{A}_i{u}_i^2$$

(5)

where $J$, $\rho$, $A$, and $u$ represent the momentum, density, injection area, and velocity, respectively. Subscript $i$ indicates GCH4 or GO2.

For the cooling performance analysis, the adiabatic cooling effectiveness (${\eta }_{eff}$) [32] is defined in Equation (6) and was calculated. For the combustion performance analysis, the specific impulse ($I_{sp}$) was calculated for all the simulation cases using Equation (7).

$${\eta }_{eff}={\displaystyle \frac{T_g-T_{aw}}{T_g-T_c}}$$

(6)

$$I_{sp}={\displaystyle \frac{F}{{\dot{m}}_{e}g}}$$

(7)

$$F={\dot{m}}_e{V}_e+A_e\left(P_e-P_a\right)$$

(8)

In Equation (6), $T_g$ represents the maximum hot-gas temperature of 3450 K in the base case simulation, $T_{aw}$ is the adiabatic wall temperature called the wall temperature in the present study, and $T_c$ denotes the coolant temperature of 300 K. In Equations (7) and (8), $F$, $\dot{m}$, $V$, $A$, $g,$ and $P$ represent the thrust, mass flow rate, velocity, area, gravitational force, and pressure, respectively. Subscript $a$ and $e$ denote the outside atmosphere and nozzle exit, respectively.

3. Results and Discussion

This section discusses the inadequate and improper film cooling design that causes thermal damage to the nozzle wall using the simulation result of the SR5 case. A solution for improving the improper cooling effect on the thrust chamber wall is discussed by considering numerous simulations based on the number of injectors in the face plate.

3.1. Investigation into Nitrogen Film Cooling in the SR5 Configuration

To investigate nitrogen film cooling, the SR5 simulation was conducted using the boundary conditions listed in Table 3. The wall temperature contour of the SR5 simulation is shown in Figure 9a. It can be observed that a high-temperature region occurred in the converging section and throat of the nozzle. The high-temperature region could potentially cause structural damage to the SR5 thrust chamber wall, as visualized in Figure 9b.

The species mass fraction contours of GCH4, GO2, OH, and N2 are shown in Figure 9c–f. It is observed that a high mass fraction of GCH4 is distributed on the nozzle wall where high temperature occurs. The presence of GCH4 at the nozzle wall results in a high temperature by combusting with GO2, as observed in Figure 9c,d. Figure 9e shows the reaction that occurred on the nozzle wall using the mass fraction of the OH contour. The mass fraction of N2 is low on the throat region of the nozzle wall, at the same location where the mass fractions of GCH4 and GO2 and the wall temperature are high, as observed in Figure 9f. The presence of higher mass fractions of GCH4 and GO2 compared to that of N2 indicates that GCH4 and GO2 have higher momentum than N2 and touch the convergent and throat sections of the nozzle wall. This can be inferred from the calculated injection velocities of GCH4, GO2, and N2, which are 279, 157, and 133 m/s, respectively. When the number of injectors is not sufficient or the flow area of the injector is not large enough, the injected fuel and oxidizer have a large flow velocity and momentum to satisfy the required mass flow rate. Due to this strong momentum, the injected fuel and oxidizer collide with the nozzle contraction section and a strong combustion reaction occurs in the collision area. This phenomenon prevents the coolant from flowing along the nozzle wall. It can be concluded that the high velocities of GCH4 and GO2 cause inadequate film cooling in the base case of the SR5 simulation, which can be solved by increasing the number of injectors on the face plate. The effect of increasing the number of injectors was investigated and is discussed in future sections regarding the SR and MR configurations.

3.2. Effect of Injectors in SR Configurations

To investigate the effect of the number of injectors on nitrogen film cooling for the thrust chamber wall, the SR5 case and a set of six new injector face plate cases based on the number of injectors (SR8, SR10, SR12, SR16, SR20, SR24) placed circularly at a fixed radial position of R₂ = 15 mm on the injector face plate were considered. The combustion simulations were conducted using the inlet boundary conditions listed in Table 3. As the number of injectors increased, the injector areas for GCH4 and GO2 increased. Thus, the mass flow rate per injector decreased when maintaining the same total mass flow rates. Therefore, the injection velocities of GCH4 and GO2 decreased and the GCH4 and GO2 momentum values using Equation (5) decreased as the number of injectors increased. These trends can be seen in Figure 10. However, the momentum ratio of GCH4 to GO2 remained the same for all the simulation cases.

The wall temperature and mass fraction of the coolant for the SR configurations are shown in Figure 11 and Figure 12, respectively. Because of the decreasing injection velocity, Figure 11a–d shows the high-temperature hotspot region in the converging section and nozzle throat of the SR5 case decreased significantly with increasing the number of injectors up to the SR12 case. After increasing the number of injectors above the SR12 case, a steady decline in the wall temperature distribution was observed, specifically in the hot region, as shown in Figure 11e–g. Similarly, the mass fraction of the coolant is distributed unevenly on the nozzle wall at the high-temperature region of the nozzle, as shown in Figure 12a. Increasing the number of injectors while decreasing injection velocity gradually reduces the high momentum intensity of GCH4 and GO2 on the nozzle wall. Consequently, the very low intensity of the coolant distribution on the nozzle wall in Figure 12a gradually increases and is eventually evenly distributed after SR12, as shown in Figure 12d–g. The higher wall temperature ranges of the convergent part of the nozzle near the throat section are shown in Figure 13. It was observed that the higher wall temperature ranges decreased as the number of injectors increased.

The effect of the number of injectors on the flow structure of the GCH4 and GO2 was visualized using the streamline contours. The streamline paths from the GCH4 and GO2 inlets for the SR configurations are shown in Figure 14. Figure 14a shows that the GCH4 and GO2 streamlines flow axially and touch the nozzle wall. This causes thermal damage on the nozzle wall in the SR5 case. In Figure 14b–g, it is observed that the streamlines from the GCH4 inlet flow axially from case SR5 up to SR12, after which it starts to deflect up towards the chamber wall in case SR24. The GCH4 and GO2 streamlines near the nozzle wall start to detach from the nozzle wall and form a thin gap between the nozzle wall and streamlines. The size of the gap increases gradually with the number of injectors, and the gap allows the coolant to flow near the nozzle wall and forms a thin film to protect it from thermal damage.

The numerical data of the wall temperature and mass fractions of GCH4, GO2, and coolant were extracted from the axial positions of the chamber and nozzle walls at two different azimuthal angles (θ = 0°, m°). Here, θ = 0° (i.e., ${\theta }_{mr}=0^{°}$) indicates an angle that lies on the y-axis in Figure 7b, and this value is the same for all the simulation cases. In contrast, θ = m° (i.e., ${\theta }_{mr}/2$) indicates an angle from the y-axis to the middle of the injectors, and the value becomes different as ${\theta }_{mr}$ is decreased with an increasing number of injectors for all the simulation cases. Therefore, we refer to this angle as m° for simplification in this paper. The wall temperature profiles at θ = 0° and m° along the axial direction are shown in Figure 15. The axial chamber position starts from −33 to 0 mm, and the nozzle position starts from 0 to 120 mm. It is observed in Figure 15a that the wall temperature profiles for the SR5 case gradually started increasing from x = −33 mm to x = 55 mm, where it reached the maximum value. Beyond x = 55 mm, the wall temperature profile decreased. A similar profile trend for the SR8 case is observed, but the location of the maximum wall temperature is at x = 65 mm. For SR10, SR12, SR16, SR20, and SR24, the locations of the maximum wall temperature are approximately at x = 86 mm (nozzle throat). This reveals that the maximum wall temperature locations for the SR5 and SR8 cases occurred before the nozzle throat, whereas they occurred after the nozzle throat for the remaining SR10, SR12, SR16, SR20, and SR24 cases. The overall wall temperature profile and maximum wall temperature decrease as the number of injectors increases, as shown in Figure 15a. Figure 15b shows the wall temperature profile at θ = m°, where the maximum wall temperature occurs at x = 86 mm (the nozzle throat). The temperature profiles for the SR12, SR16, SR20, and SR24 cases at θ = 0° and m° are similar, indicating a uniform temperature distribution, as shown in Figure 11.

The mass fractions of the GCH4, GO2, and OH distributions along the axial direction of the wall at θ = 0° and m° are shown in Figure 16. In Figure 16a, the mass fraction of GCH4 starts to increase from zero at x = 50 mm for the SR5 case and at x = 60 mm for the SR8 case. In contrast, the increase begins after x = 85 mm for SR10, SR12, SR16, SR20, and SR24. In Figure 16b, the mass fraction of GCH4 starts to increase after x = 85 mm for all the simulation cases. The GCH4 on the wall reacts with GO2, initiating combustion and increasing the wall temperature, as shown in Figure 15. The mass fractions of the GO2 distributions along the axial direction of the wall at θ = 0° and m° are shown in Figure 16c,d. To illustrate the combustion reaction on the wall, the mass fraction of OH distribution along the axial direction of the wall is shown in Figure 16e,f. Notably, the OH peak correlates with the maximum wall temperature in each simulation case. Moreover, the OH peak shifts downstream of the nozzle as the number of injectors increases.

The distribution of the coolant mass fraction is shown in Figure 17. In the SR5 case, the coolant profile is decreased significantly along the axial direction of x = 45 mm at θ = 0°. This decrease indicates inadequate cooling along the nozzle wall, which causes damage to the high-temperature region where the nozzle wall is damaged, as shown in Figure 9. This damage results from the thermal stress induced by hot gases. However, the cooling deficiency is mitigated by reducing the flow momentum, which is achieved by increasing the number of injectors, as shown in Figure 17a. The investigation of the SR configurations shows a reduction in the wall temperature of the nozzle and addresses the damage concerns. However, the wall temperature of the injector face plate remains considerably high, as shown in Figure 11h. This issue can potentially be resolved by employing multiple rows with multiple injectors, which is a solution discussed in the following MR configuration section.

3.3. Effect of Injectors in MR Configurations

To investigate the effect of the number of injectors in multiple rows on nitrogen film cooling for the thrust chamber wall, a set of six new injector plates based on the number of injectors (MR24, MR30, MR36, MR48, MR60, and MR72) placed on the injector face plate circularly at three fixed radial distances of R₁ = 10 mm, R₂ = 15 mm, and R₃ = 25 mm were considered. Combustion simulations were conducted using the inlet boundary conditions listed in Table 3. As the number of injectors increased, the injector areas for GCH4 and GO2 increased. Thus, the mass flow rate per injector decreased when maintaining the same total mass flow rates. Therefore, the injection velocities of GCH4 and GO2 decreased and the respective momentum using Equation (5) also decreased with an increasing number of injectors, as shown in Figure 18.

The wall temperatures and mass fractions of the coolant from the MR configuration simulations are shown in Figure 19 and Figure 20, respectively. As a result of decreasing the injection velocity, a significant low-temperature region occurs in the chamber wall and the wall temperature of the nozzle becomes uniform in the MR48 case, as shown in Figure 19a–d. After the MR48 case, a non-uniform temperature distribution is observed for the MR60 and MR72 cases, as shown in Figure 19e,f. The non-uniform temperature distribution is due to the 30 and 36 injectors located at R₃ = 25 mm for the MR60 and MR72 cases, respectively. A similar trend is observed for the coolant mass fraction distribution, as shown in Figure 20. The injector face plate temperatures for all the cases are shown in Figure 21. The MR24 face plate case has a very high temperature compared to the other cases. Increasing the number of injectors reduces the high temperature of the face plate to a lower value, as in the MR72 case. The high temperature in the MR24 case is due to the larger space between the injectors, whereas the low temperature in the MR72 case is due to the smaller space between the injectors. Visualizing the temperature contours in Figure 19 and Figure 21 shows that the MR48 case can be an optimistic design.

The effect of the number of injectors on the flow structure of the GCH4 and GO2 was visualized using the streamline contours. The streamlines from the GCH4 and GO2 inlets, particularly those located at R₃ = 25 mm and θ = 0°, for the MR configurations are shown in Figure 22. A significant difference in the flow observed is that the GCH4 streamlines start slightly deflecting towards the chamber wall as the number of injectors increased. Consequently, the gap between the streamlines and the chamber wall gradually decreased. Similarly, it is observed that the GO2 streamlines gradually touched the chamber wall as the number of injectors increased. This is due to the decreased space between the injectors at R₃. Consequently, it causes the coolant distribution to become uneven, as shown in Figure 20e,f.

The wall temperatures and mass fractions of GCH4, GO2, OH, and coolant at two different azimuthal angles (θ = 0° and m°) for the MR configurations are shown in Figure 23, Figure 24 and Figure 25. In the MR configurations, θ = m° (i.e., ${\theta }_{or}/2$). In Figure 23, the maximum wall temperatures for all the cases except MR60 and MR72 occurred near the nozzle throat, whereas they occurred before the nozzle throat at the azimuthal angle of θ = m° for the MR60 and MR72 cases. The axial wall temperature profile decreases with the number of injectors from the MR24 case to the MR48 case. However, after increasing the number of injectors in the MR60 and MR72 cases, the axial wall temperature profile from x = 20 mm increases compared to the other cases. The wall temperature increases in the MR60 and MR72 cases are due to the mass fraction of GCH4 reacting with GO2, as shown in Figure 24a–d. The mass fraction of OH in Figure 24e,f shows that the peak OH value is obtained after the nozzle throat for all the cases except MR60 and MR72. The peak OH value obtained in the MR48 case is lower than that in the other cases, indicating that the reaction rate is low, which results in a low wall temperature. The coolant mass fractions are compared in Figure 25, which shows that the coolant wall profiles for all the cases are reasonably good up to an axial position of 90 mm and the coolant mass fraction is above 0.65. After 90 mm, a coolant profile is obtained below a mass fraction of 0.65. Overall, the coolant profile throughout the chamber and nozzle walls (mainly after 90 mm) for the MR48 case is better than that of the other cases, as shown in Figure 25.

3.4. Performance Analysis

For the cooling performance analysis, the adiabatic cooling effectiveness (${\eta }_{eff}$) for all the simulations in the SR and MR configurations is calculated using Equation (6), and the performance comparison is shown in Figure 26 and Figure 27. For the SR configuration in Figure 26, the ${\eta }_{eff}$ profile at the two angles on the nozzle wall is improved as the number of injectors is increased. The SR5 case has a poor ${\eta }_{eff}$ on the nozzle wall, which causes the high wall temperature observed in Figure 11a. The poor ${\eta }_{eff}$ profile is improved as the number of injectors is increased in the SR24 case. Overall, the ${\eta }_{eff}$ profile is enhanced by the reduction in the injection velocities of GCH4 and GO2. For the MR configurations in Figure 27, the ${\eta }_{eff}$ profile on the nozzle wall significantly increases up to the MR48 case. Subsequently, it decreases as the number of injectors is increased to approximately 60 and 72 in the MR60 and MR72 cases, respectively. Additionally, when comparing the cooling effectiveness for the same number of 24 injectors in both the SR24 and MR24 cases in Figure 28, the MR24 case shows superior cooling effectiveness compared with SR24, except on the nozzle wall from 50 to 90 mm, where SR24 performs slightly better than MR24. Furthermore, in comparisons with the MR48 case, overall, the MR48 case shows a better cooling effectiveness profile than the other simulation cases.

To analyze the combustion performance, the specific impulse ($I_{sp}$) for all SR and MR configurations was calculated using Equation (7). The measured chamber pressure and theoretical specific impulse were 12.05 bar and 203.5 s, respectively. In the base case configuration, the specific impulse was 210.48 s for SR5 without nitrogen injection, and it decreased to 184.38 s when nitrogen injection was considered. This resulted in a specific impulse loss of 12.4% when the nitrogen injection was considered. The combustion performance comparison is shown in Figure 29. In the SR configurations, the chamber pressure and specific impulse decrease with the number of injectors. Interestingly, the SR24 case shows a low chamber pressure and specific impulse, whereas the MR24 case shows a high chamber pressure and specific impulse, even though both configurations have the same number of injectors. The low chamber pressure and specific impulse in the SR24 case stem from the narrow spacing between injectors. This limitation impedes proper propellant mixing, ultimately diminishing combustion performance. In contrast, the high chamber pressure and specific impulse observed in the MR24 case are attributed to placing the same number of injectors in multiple rows on the injector face plate, as presented in Figure 4a. This configuration allows for greater spacing between the injectors than in the SR24 case, facilitating thorough propellant mixing and superior combustion performance. In the MR configurations, the chamber pressure is increased up to MR60 and is slightly decreased in MR72. The specific impulse increases with the increasing number of injectors. However, the calculated specific impulse for all the simulations is lower than the theoretical specific impulse. Comparing the MR48 case with the SR5 case, the MR48 case achieves almost the same chamber pressure as the base case SR5 and a higher specific impulse than the base case SR5. Overall, the performance of the MR configuration surpasses that of the SR configuration due to an increased spacing between the injectors. This enhanced spacing facilitates better mixing, leading to more efficient combustion, higher chamber pressure, and an improved specific impulse. In MR configurations, cases with fewer than 48 injectors show a low chamber pressure and specific impulse and a high wall temperature on the face plate, attributable to excessively wide spacing between the injectors. Conversely, cases with more than 48 injectors show a high chamber pressure and specific impulse and an elevated wall temperature on the nozzle wall due to narrower injector spacing. Ultimately, the present work concludes that the MR48 configuration is the optimal choice to mitigate thermal damage based on a comparative analysis of the injector face plate and nozzle wall temperatures and combustion performances.

4. Conclusions

This simulation study investigated the impact of the number of multicoaxial injectors placed in the face plate on the GCH4/GO2 thrust chamber wall using nitrogen film-cooling performance. In an experimental test case of a subcritical combustor with five shear coaxial injectors, thermal damage was observed on the thrust chamber wall. In particular, damaged hotspots were identified in the converging section and throat of the nozzle, indicating inadequate film cooling in the thrust chamber wall. This was further examined and visualized using simulation results. The simulation results revealed high-temperature hot spots on the nozzle wall caused by combustion, owing to the strong momentum of GCH4 and GO2 from the injectors impacting the nozzle wall. This thermal issue was addressed by increasing the number of injectors on the face plate. The simulation results are summarized as follows:

• SR configurations (SR5, SR8, SR10, SR12, SR16, SR20, and SR24).

The simulation results showed a reduction in the high-temperature hotspots on the nozzle wall as the number of injectors increased. This reduction was due to the decreased injection velocities of GCH4 and GO2, with the ratio of the GCH4 injection velocity to the coolant velocity decreasing from 2.1 for the SR5 case to 0.48 for the SR24 case. In addition, the adiabatic film cooling effectiveness of the thrust chamber wall increased. The SR24 case showed superior adiabatic film cooling effectiveness on the nozzle wall compared to the other simulation cases. However, despite the decrease in the thrust chamber wall temperature profile, the face plate temperature remained consistently high across all the simulations. It was unaffected as the number of injectors increased in the SR configuration.

2.

MR configurations (MR24, MR30E, MR36, MR48, MR60, and MR72).

The simulation results showed that high-temperature hotspots were not observed on the nozzle wall as the number of injectors increased up to the MR48 case. However, this was observed in the MR60 and MR72 cases. The ratio of the GCH4 injection velocity to the coolant velocity decreased to 0.15 for the MR72 case from 0.48 for the MR24 case. The performance analysis showed that the adiabatic film cooling effectiveness on the thrust chamber wall also increased up to the MR48 case, after which it decreased. The MR48 case showed superior adiabatic film cooling effectiveness on the nozzle wall and face plate compared with the other cases. In the MR48 case, the outer angle (${\theta }_{or}$) was 15°. Any outer angle that was smaller or larger than ${\theta }_{or}$ = 15° in the other simulations showed poor adiabatic film cooling effectiveness. In the larger ${\theta }_{or}$ angle cases, the face plate had a very high temperature, whereas the smaller ${\theta }_{or}$ angle cases had a lower temperature.

The numerical work in the present study suggests that the MR48 case configuration with ${\theta }_{or}$ = 15° represents an optimistic face plate design for achieving the best adiabatic film cooling effectiveness in terms of the thrust chamber wall temperature. Although replacing nitrogen with gaseous methane could further improve the film cooling performance, which is the focus of ongoing research for reusable methane rocket engines, this study was limited to using nitrogen as the coolant. Future work will investigate the use of gaseous methane, as well as explore the effects of varying chamber pressures and geometric parameters, to further optimize multicoaxial injector performance. These efforts are expected to provide valuable insights for improving cooling efficiency in advanced rocket engine designs.