Investigation on the Influence of Film Cooling Structure on the Flow and Heat Transfer Characteristics of Axisymmetric Plug Nozzle

: Some numerical simulations were used to study the effects of blowing ratio (0.25–0.5), hole diameters(0.65~1 mm), and hole inclination angle (30~60 ◦ ) of the film cooling structure on the cooling and aerodynamic characteristics of the axisymmetric plug nozzle under the condition of transonic. The results showed that the bow shocks appear near the film holes on the plug wall in the supersonic region, which cause the wall cooling effectiveness of the plug to decrease. Compared with the baseline plug, the blowing ratio ranged from 0.25 to 0.5, the wall average temperature of the rear plug decreased by 34.4~48.1%, the thrust coefficient and total pressure recovery coefficient decreased by 0.31~0.61% and 0.52~0.93%, respectively. When the perforated percentage is constant, the wall cooling effectiveness increases with the decrease of the hole diameters. The increase in the inclination angle of the film holes lead to the decrease in cooling effectiveness and aerodynamic performance. This is because the penetration ability of the cooling air to the mainstream is enhanced, and the obstruction to the mainstream boundary layer is increased, resulting in the increase of bow shock intensity near the film holes in the supersonic region.


Introduction
The plug nozzle is equipped with an outwardly extending central plug inside the nozzle, allowing the discharged mainstream to continue expanding into a supersonic jet on the surface of the plug after exiting the nozzle.Compared with conventional convergentdivergent nozzles, the plug nozzle can adjust throat and exit areas by altering the geometric structure and axial position of the plug, ensuring excellent thrust performance for the engine under various conditions.Additionally, the coordinated interaction between the plug and the nozzle surface can achieve vectoring and thrust reversing for the engine.It has attracted considerable attention due to the advantages of the plug nozzle.Many scholars have carried out a lot of research on the flow field characteristics and thrust characteristics of the plug nozzle through the methods of experiment [1][2][3][4] and numerical simulation [5][6][7][8][9][10], and the results are rich.
However, with the continuous improvement of performance requirements for aeroengine, the gas temperature also increases rapidly, especially in afterburning conditions.The plug needs to withstand the significant heat carried by high-temperature gas.Therefore, surface cooling of the plug becomes a crucial issue, and there is limited public literature on plug cooling.Smolak, G. R. et al. [11] studied the cooling effect on the wall of the plug nozzle under the afterburner state of a turbojet engine.When the total air temperature reaches 2840 • F, a total nozzle cooling airflow of 4.5 percent of the afterburner gas flow produced an average plug surface temperature of 1620 • F Clark et al. [12] experimentally studied the heat transfer and flow characteristics of axisymmetric plug nozzle with bleed air Symmetry 2024, 16 cooling from the compressor.The results showed that by bleeding approximately 3.5% of the engine flow for plug cooling, the high-temperature point at the throat surface decreased by approximately 89 K. Jeracki and Chenoweth [13] investigated the influence of three axially positioned cooling slots on the surface of the plug on the aerodynamic performance of axisymmetric plug nozzles.The experimental results indicated that the thrust coefficient during takeoff was approximately 99%, the thrust coefficient was 95% and 97% when the external Mach number was 1.2 and 1.97, respectively.Chenoweth [14] experimentally studied the heat transfer characteristics of axisymmetric plug nozzle with three axially positioned cooling slots on the surface of the plug.The results showed that slots with an adverse pressure gradient upstream had poorer cooling effectiveness.When the same amount of coolant was added, the cooling effect of slots located upstream of the throat was better than that of slots located downstream of the throat.Stepka and Cbambellan [15] conducted a study on the use of fuel-cooling plug nozzle with afterburning turbojet engines for turbojet aircraft.The research results indicated that the overall feasibility of employing a fuel-cooling plug was established, with the temperature of the cooled metal surface controlled below 817 K. Clark and Lieberman [16] investigated the cooling of plug nozzle in supersonic cruise.The results showed that bleeding 2% of the compressor air could lower the plug temperature to below 1222 K at maximum engine afterburning.During supersonic cruise conditions, it was possible to obtain 1.5% of the engine mainstream by bleed-off to cool the plug to around 978 K. Nosek and Straight [17] conducted experimental research on the heat transfer characteristics of the J-85 axisymmetric plug nozzle with an afterburning turbojet engine.The experimental results indicated that, compared with convective cooling, the surface temperature of the plug decreased by approximately 150 K when using an air film cooling structure at the same coolant flow rate.
In the literature mentioned above, the research on the cooling of axisymmetric plug nozzles mainly focuses on convective cooling and slot film cooling.There is limited research on the impact of film cooling structures on the aerodynamic performance and heat transfer characteristics of axisymmetric plug nozzle.Jing Chen et al. [18] conducted a numerical simulation to study the aerodynamic and cooling characteristics of a two-dimensional plug nozzle.Research results showed that the film cooling structure of stagger film holes for plug nozzle cooling has a good effect.In fact, film cooling technology has a wide range of applications.Many scholars have conducted a large number of studies on the effect of film cooling on flat plates [19][20][21][22][23], and the research results show that the blow ratio, hole inclination angle, and the shape of film holes are important factors affecting the effect of film cooling.Film cooling technology has shown excellent cooling effects in the cooling of engine components such as turbine blades [24][25][26], combustion chambers [27,28], and other hot sections.Therefore, exploring and studying the effects of applying film cooling technology to an axisymmetric plug nozzle is worthwhile.Some numerical simulations were conducted to investigate the impact of film cooling structure parameters (blowing ratio, hole diameter, and hole inclination angle) of the film cooling structure on the cooling and aerodynamic characteristics of the axisymmetric plug nozzle under the condition of transonic.It is expected to improve the cooling effectiveness of the plug wall and reduce the loss of aerodynamic performance of the nozzle.

Physical Model
Firstly, the flow characteristics of the baseline axisymmetric plug nozzle without cooling structures were investigated.Figure 1 presents the 1/8 model of the baseline axisymmetric plug nozzle, which has a relatively simple structure.It mainly consists of a mainstream inlet section, a spherical hinged section, an exit section, and an axisymmetric plug, as shown in Figure 1a,b, and provides the geometric parameters of the baseline axisymmetric plug nozzle.This study designs a plug cooling structure on the baseline axisymmetric plug nozzle.The cooling structure primarily consists of a cooling channel, strut, baffler, and film holes, as shown in Figure 2. The plug's interior is partitioned into two chambers by a baffler.Coolant enters into the plug through the cooling channel, passes through the strut, enters the inner chamber, and then exits through narrow slots.Part of the coolant forms impingement cooling on the head of the plug, while the rest flows out through film holes on the downstream surface of the plug, creating film cooling effect to protect the plug surface, as shown in Figure 3.The red arrows represent the direction of mainstream flow, and the blue arrows represent the direction of cooling air flow.This study designs a plug cooling structure on the baseline axisymmetric plug nozzle.The cooling structure primarily consists of a cooling channel, strut, baffler, and film holes, as shown in Figure 2. The plug's interior is partitioned into two chambers by a baffler.Coolant enters into the plug through the cooling channel, passes through the strut, enters the inner chamber, and then exits through narrow slots.Part of the coolant forms impingement cooling on the head of the plug, while the rest flows out through film holes on the downstream surface of the plug, creating film cooling effect to protect the plug surface, as shown in Figure 3.The red arrows represent the direction of mainstream flow, and the blue arrows represent the direction of cooling air flow.This study designs a plug cooling structure on the baseline axisymmetric plug nozzle.The cooling structure primarily consists of a cooling channel, strut, baffler, and film holes, as shown in Figure 2. The plug's interior is partitioned into two chambers by a baffler.Coolant enters into the plug through the cooling channel, passes through the strut, enters the inner chamber, and then exits through narrow slots.Part of the coolant forms impingement cooling on the head of the plug, while the rest flows out through film holes on the downstream surface of the plug, creating film cooling effect to protect the plug surface, as shown in Figure 3.The red arrows represent the direction of mainstream flow, and the blue arrows represent the direction of cooling air flow.This study designs a plug cooling structure on the baseline axisymmetric plug nozzle.The cooling structure primarily consists of a cooling channel, strut, baffler, and film holes, as shown in Figure 2. The plug's interior is partitioned into two chambers by a baffler.Coolant enters into the plug through the cooling channel, passes through the strut, enters the inner chamber, and then exits through narrow slots.Part of the coolant forms impingement cooling on the head of the plug, while the rest flows out through film holes on the downstream surface of the plug, creating film cooling effect to protect the plug surface, as shown in Figure 3.The red arrows represent the direction of mainstream flow, and the blue arrows represent the direction of cooling air flow.Table 1 provides the selected ranges for the study parameters: blowing ratio BR (defined as BR = ρ c u c ρ g u g , where ρ c and u c are the density and flow velocity of the coolant, ρ g and u g are the density and flow velocity of the mainstream), hole diameter d and hole inclination angle α.To ensure the perforated percentage is constant, the longitudinal spacing p and lateral spacing s between film holes vary for different diameters.The specific arrangement of the film holes are illustrated in Figure 4. inclination angle α.To ensure the perforated percentage is constant, the longitudinal spacing p and lateral spacing s between film holes vary for different diameters.The specific arrangement of the film holes are illustrated in Figure 4.There are three assumptions that were made in this study, which are as follows: (1) the mentioned fluid including the mainstream and the coolant is compressible; (2) the flow is in a steady state; (3) the thermodynamic parameters of the solid material such as the heat conductivity coefficient, specific heat capacity, etc., are constants; Table 2 lists the thermodynamic property setting of the solid and fluid materials in FLUENT.

Characteristic Parameters Definition
ηeff is an important parameter for the cooling effectiveness of the film cooling, Nu number is an important dimensionless number reflecting convective heat transfer capacity [29], while Cf and σ are important parameters that reflect the aerodynamic performance of engine exhaust system [18].To assess the cooling effectiveness of the plug surface and the aerodynamic performance of the nozzle, this paper defines the following characteristic parameters.
cooling effectiveness:  eff (1) where g T is the temperature of the mainstream, c T is the temperature of the coolant, and w T is the temperature of the solid surface; There are three assumptions that were made in this study, which are as follows: (1) the mentioned fluid including the mainstream and the coolant is compressible; (2) the flow is in a steady state; (3) the thermodynamic parameters of the solid material such as the heat conductivity coefficient, specific heat capacity, etc., are constants; Table 2 lists the thermodynamic property setting of the solid and fluid materials in FLUENT.

Characteristic Parameters Definition
η eff is an important parameter for the cooling effectiveness of the film cooling, Nu number is an important dimensionless number reflecting convective heat transfer capacity [29], while C f and σ are important parameters that reflect the aerodynamic performance of engine exhaust system [18].To assess the cooling effectiveness of the plug surface and the aerodynamic performance of the nozzle, this paper defines the following characteristic parameters.
where T g is the temperature of the mainstream, T c is the temperature of the coolant, and T w is the temperature of the solid surface; Nusselt number : Symmetry 2024, 16, 689 5 of 18 where D is the diameter of the nozzle outlet, λ is the thermal conductivity of fluid, and h is the convective heat transfer coefficient.
Thrust coefficient : where F ac is the actual thrust obtained through flow field calculations, and F id is the thrust generated by the actual flow rate of the nozzle undergoing isentropic expansion; Total pressure recovery coefficient : σ = P t out P t in (4) where P t out is the total pressure at the nozzle outlet, and P t in is the sum of the total pressure on the inlet the nozzle and the coolant.

Computational Domain and Boundary Conditions
Due to the small size of the film holes relative to the plug nozzle, creating a grid for the entire model would result in an excessively large number of grids, making calculations difficult to complete.Therefore, this paper adopted a 1/8 physical model for numerical simulation.As the nozzle exit flow is not fully expanded, a sufficiently large far field is established downstream of the nozzle exit as the external flow field, the length of the far field is 12D and the radius is 5D, as shown in Figure 5.
where  is the total pressure at the nozzle outlet, and  is the sum of the total pressure on the inlet the nozzle and the coolant.

Computational Domain and Boundary Conditions
Due to the small size of the film holes relative to the plug nozzle, creating a grid for the entire model would result in an excessively large number of grids, making calculations difficult to complete.Therefore, this paper adopted a 1/8 physical model for numerical simulation.As the nozzle exit flow is not fully expanded, a sufficiently large far field is established downstream of the nozzle exit as the external flow field, the length of the far field is 12D and the radius is 5D, as shown in Figure 5.
The computational boundary conditions are as follows: the fluid is treated as an ideal gas; the inlet for the mainstream and coolant in the nozzle are set as a pressure inlet, and the external flow field is set as a pressure outlet; the solid surfaces are assigned no-slip boundary condition and coupled heat transfer condition; the two lateral sides of the computational domain are set as periodic boundary conditions.Specific parameters are provided in Table 3.The computational boundary conditions are as follows: the fluid is treated as an ideal gas; the inlet for the mainstream and coolant in the nozzle are set as a pressure inlet, and the external flow field is set as a pressure outlet; the solid surfaces are assigned no-slip boundary condition and coupled heat transfer condition; the two lateral sides of the computational domain are set as periodic boundary conditions.Specific parameters are provided in Table 3.

Turbulence Model and Grid Independence Verification
In this paper, the CFD method is used to calculate the flow field, in which the SST k-ω turbulence model is applied.One-way simulation is used, which does not consider the thermal stress and deformation of the solid wall, and the heat conduction and heat convection between the fluid and the wall are considered in the CFD calculation.The Symmetry 2024, 16, 689 6 of 18 commercial CFD solver Fluent 19.2 was used for numerical calculations.Reference [18] verified that the two-dimensional (2-D) plug nozzle film cooling results calculated by SST k-ω model were in good agreement with the experimental results.The SST k-ω model can effectively simulate Reynolds stress, accurately model separation phenomena under adverse pressure gradients, and exhibit good simulation performance in both near-surface and far-from-surface regions [30].Therefore, this paper adopted the SST k-ω turbulence model.In addition, the characteristics of the flow field and temperature field are obtained by solving the continuity equation, momentum conservation equation, energy conservation equation, and radiation heat transfer equations.The flow and heat transfer conservation equations are discretized with a second-order upwind scheme.
A grid is generated by using the ICEM software, opting for an unstructured grid due to the complexity of the model.During grid generation, refinement is applied to regions with film holes and areas with dramatic changes in flow parameters.Boundary layers grids are generated on the near-surface, with the first layer having a dimensionless height y + ≈ 1, and a growth rate of 1.1 for a total of 10 layers to ensure coverage of the fluid boundary layer.In the area away from the wall, a tetrahedral grid was generated.Due to the small size of the film holes, the grids between the plug wall and the film holes were refined to a maximum size of 0.4 mm, as shown in Figure 6. Figure 7 illustrates the trend of the average temperature on the plug surface with varying grid quantities.When the grid quantity exceeds 3.1 million, the average temperature on the plug surface is almost unaffected by the grid quantity.To balance computational accuracy and resource effectiveness, this study chose a grid quantity of 3.1 million for the calculation.
In this paper, the CFD method is used to calculate the flow field, in which the SST kω turbulence model is applied.One-way simulation is used, which does not consider the thermal stress and deformation of the solid wall, and the heat conduction and heat convection between the fluid and the wall are considered in the CFD calculation.The commercial CFD solver Fluent 19.2 was used for numerical calculations.Reference [18] verified that the two-dimensional (2-D) plug nozzle film cooling results calculated by SST kω model were in good agreement with the experimental results.The SST k-ω model can effectively simulate Reynolds stress, accurately model separation phenomena under adverse pressure gradients, and exhibit good simulation performance in both near-surface and far-from-surface regions [30].Therefore, this paper adopted the SST k-ω turbulence model.In addition, the characteristics of the flow field and temperature field are obtained by solving the continuity equation, momentum conservation equation, energy conservation equation, and radiation heat transfer equations.The flow and heat transfer conservation equations are discretized with a second-order upwind scheme.
A grid is generated by using the ICEM software, opting for an unstructured grid due to the complexity of the model.During grid generation, refinement is applied to regions with film holes and areas with dramatic changes in flow parameters.Boundary layers grids are generated on the near-surface, with the first layer having a dimensionless height y + ≈ 1, and a growth rate of 1.1 for a total of 10 layers to ensure coverage of the fluid boundary layer.In the area away from the wall, a tetrahedral grid was generated.Due to the small size of the film holes, the grids between the plug wall and the film holes were refined to a maximum size of 0.4 mm, as shown in Figure 6. Figure 7 illustrates the trend of the average temperature on the plug surface with varying grid quantities.When the grid quantity exceeds 3.1 million, the average temperature on the plug surface is almost unaffected by the grid quantity.To balance computational accuracy and resource effectiveness, this study chose a grid quantity of 3.1 million for the calculation.
The convergence criterion for numerical solutions requires that the residual of the energy equation be less than 10 −5 , and the residual of other equations be less than 10 −4 .The convergence criterion was reached after about 65,000 iterations.

Analysis of the Aerodynamic Characteristics of the Baseline Plug Nozzle
The flow characteristics of the baseline axisymmetric plug nozzle without cooling structures are analyzed first.Figure 8 presents the flow field distribution of the baseline plug nozzle.Figure 8a shows the velocity distribution of the flow field in the baseline plug nozzle.As the mainstream approaches the head of the plug, the flow path begins to narrow, leading to an accelerated expansion of the mainstream.The Mach number increases rapidly when passing through the throat.After passing through the expansion fan at the nozzle outlet, the mainstream becomes supersonic.Subsequently, the mainstream transitions back to subsonic after passing through a shock.After leaving the plug surface, it continues to expand and accelerate to supersonic; Figure 8b provides the pressure distribution in the flow field of the baseline plug nozzle.The static pressure of the mainstream changes slightly upstream of the throat.As the mainstream passes through the throat, the static pressure decreases along the flow direction continuously.The pressure of the mainstream is higher than the ambient pressure at the nozzle exit, indicating an under-expanded state.The expansion fan is formed at the nozzle exit, then the pressure of the mainstream rapidly decreases after the expansion fan.A shock is formed after reflection at the free boundary, causing a rapid increase in pressure of the mainstream.The mainstream continues to expand after leaving the plug surface, creating a series of high and low-pressure regions until reaching pressure equilibrium with the external flow field; Figure 8c shows the temperature distribution in the flow field of the baseline plug nozzle.The temperature decreases as the mainstream passes through the expansion fan at the nozzle exit.But the temperature increases again after passing through the shock.This is basically the same as the non-dimensional temperature distribution trend of the 2D plug surface in reference [18].The temperature decreases as the mainstream leaves the plug surface and gradually mixes with the external airflow.
(a) The convergence criterion for numerical solutions requires that the residual of the energy equation be less than 10 −5 , and the residual of other equations be less than 10 −4 .The convergence criterion was reached after about 65,000 iterations.

Analysis of the Aerodynamic Characteristics of the Baseline Plug Nozzle
The flow characteristics of the baseline axisymmetric plug nozzle without cooling structures are analyzed first.Figure 8 presents the flow field distribution of the baseline plug nozzle.Figure 8a shows the velocity distribution of the flow field in the baseline plug nozzle.As the mainstream approaches the head of the plug, the flow path begins to narrow, leading to an accelerated expansion of the mainstream.The Mach number increases rapidly when passing through the throat.After passing through the expansion fan at the nozzle outlet, the mainstream becomes supersonic.Subsequently, the mainstream transitions back to subsonic after passing through a shock.After leaving the plug surface, it continues to expand and accelerate to supersonic; Figure 8b provides the pressure distribution in the flow field of the baseline plug nozzle.The static pressure of the mainstream changes slightly upstream of the throat.As the mainstream passes through the throat, the static pressure decreases along the flow direction continuously.The pressure of the mainstream is higher than the ambient pressure at the nozzle exit, indicating an under-expanded state.The expansion fan is formed at the nozzle exit, then the pressure of the mainstream rapidly decreases after the expansion fan.A shock is formed after reflection at the free boundary, causing a rapid increase in pressure of the mainstream.The mainstream continues to expand after leaving the plug surface, creating a series of high and low-pressure regions until reaching pressure equilibrium with the external flow field; Figure 8c shows the temperature distribution in the flow field of the baseline plug nozzle.The temperature decreases as the mainstream passes through the expansion fan at the nozzle exit.But the temperature increases again after passing through the shock.This is basically the same as the non-dimensional temperature distribution trend of the 2D plug surface in reference [18].The temperature decreases as the mainstream leaves the plug surface and gradually mixes with the external airflow.
Figure 9 presents the pressure and temperature distribution on the plug surface.It can be observed that there is high pressure at the plug head, making it difficult for the coolant to exit through the film holes to protect the surface.Therefore, the cooling scheme for the plug head does not utilize film cooling but rather employs impingement cooling.The pressure decreases on the rear surface of the plug after passing through the expansion fan, resulting in a low-pressure region.Although there is a low-temperature region on the rear surface of the plug after passing through the expansion fan, the overall temperature of the plug surface remains relatively high.surface in reference [18].The temperature decreases as the mainstream leaves the plug surface and gradually mixes with the external airflow.Figure 9 presents the pressure and temperature distribution on the plug surface.It can be observed that there is high pressure at the plug head, making it difficult for the coolant to exit through the film holes to protect the surface.Therefore, the cooling scheme for the plug head does not utilize film cooling but rather employs impingement cooling.The pressure decreases on the rear surface of the plug after passing through the expansion fan, resulting in a low-pressure region.Although there is a low-temperature region on the rear surface of the plug after passing through the expansion fan, the overall temperature of the plug surface remains relatively high.Figure 9 presents the pressure and temperature distribution on the plug surface.It can be observed that there is high pressure at the plug head, making it difficult for the coolant to exit through the film holes to protect the surface.Therefore, the cooling scheme for the plug head does not utilize film cooling but rather employs impingement cooling.The pressure decreases on the rear surface of the plug after passing through the expansion fan, resulting in a low-pressure region.Although there is a low-temperature region on the rear surface of the plug after passing through the expansion fan, the overall temperature of the plug surface remains relatively high.To provide a more intuitive analysis of the aerodynamic parameters on the surface of the plug, pressure, and temperature data were extracted along a generatrix (100 points were extracted from this generatrix) of the plug (as shown by line B in Figure 9), as shown in Figure 10. it can be observed that there is a significant pressure drop near X = 39.1 mm, possibly indicating flow separation in Figure 10.The pressure on the plug surface increases around X = 89.6 mm, which is caused by the presence of shock.The temperature To provide a more intuitive analysis of the aerodynamic parameters on the surface of the plug, pressure, and temperature data were extracted along a generatrix (100 points were extracted from this generatrix) of the plug (as shown by line B in Figure 9), as shown in Figure 10. it can be observed that there is a significant pressure drop near X = 39.1 mm, possibly indicating flow separation in Figure 10.The pressure on the plug surface increases Symmetry 2024, 16, 689 9 of 18 around X = 89.6 mm, which is caused by the presence of shock.The temperature variation pattern in Figure 10 is similar to the pressure pattern, with a higher average temperature on the plug surface.The pressure distribution on the plug surface calculated in this paper is basically the same as the static pressure distribution on the plug surface under different jet total pressure ratios when the outflow Ma = 0 is experimentally studied in reference [31].The accuracy of the calculation method is verified.
Symmetry 2024, 16, x FOR PEER REVIEW 9 of 18 variation pattern in Figure 10 is similar to the pressure pattern, with a higher average temperature on the plug surface.The pressure distribution on the plug surface calculated in this paper is basically the same as the static pressure distribution on the plug surface under different jet total pressure ratios when the outflow Ma = 0 is experimentally studied in reference [31].The accuracy of the calculation method is verified.It can be observed that the plug head is suitable for impingement cooling from the flow condition of the baseline axisymmetric plug nozzle.Therefore, no film holes are arranged on the plug head in this study, only film cooling structures are arranged on the rear surface of the plug.Subsequent analysis focuses on the film cooling effect on the rear surface of the plug and its impact on the aerodynamic performance of the nozzle.

Effect of Blowing Ratio on Aerodynamic and Heat Transfer Characteristics of Axisymmetric Plug Nozzle
In this chapter, the film cooling design parameters d = 0.65 mm and α = 30° are fixed, and the blowing ratio is 0.25, 0.38, and 0.5, respectively.Figure 11 illustrates the pressure distribution on the rear surface of the plug under different blowing ratios.The pressure decreases in the mainstream when passing through the expansion fan at the nozzle exit, forming a low-pressure region on the surface.Then, the pressure of the mainstream increases after the shock, and the surface pressure increases accordingly.From the three Figure 11a-c, it can be seen that the increase in blowing ratio has little influence on the change in wall pressure distribution.It is worth noting that the bow shocks appear near the film holes in the low-pressure region.This phenomenon is attributed to the disturbance caused by the coolant flow from the film holes to the mainstream, which turns supersonic after passing through the expansion fan.The mainstream boundary layer is lifted, generating bow shocks that continuously distribute along the axial direction of the plug.Additionally, a pressure decrease is observed behind the film holes in the supersonic region.This is due to the shear between the mainstream and the coolant, forming vortices and reducing the pressure behind the holes rapidly.It can be observed that the plug head is suitable for impingement cooling from the flow condition of the baseline axisymmetric plug nozzle.Therefore, no film holes are arranged on the plug head in this study, only film cooling structures are arranged on the rear surface of the plug.Subsequent analysis focuses on the film cooling effect on the rear surface of the plug and its impact on the aerodynamic performance of the nozzle.

Effect of Blowing Ratio on Aerodynamic and Heat Transfer Characteristics of Axisymmetric Plug Nozzle
In this chapter, the film cooling design parameters d = 0.65 mm and α = 30 • are fixed, and the blowing ratio is 0.25, 0.38, and 0.5, respectively.Figure 11 illustrates the pressure distribution on the rear surface of the plug under different blowing ratios.The pressure decreases in the mainstream when passing through the expansion fan at the nozzle exit, forming a low-pressure region on the surface.Then, the pressure of the mainstream increases after the shock, and the surface pressure increases accordingly.From the three Figure 11a-c, it can be seen that the increase in blowing ratio has little influence on the change in wall pressure distribution.It is worth noting that the bow shocks appear near the film holes in the low-pressure region.This phenomenon is attributed to the disturbance caused by the coolant flow from the film holes to the mainstream, which turns supersonic after passing through the expansion fan.The mainstream boundary layer is lifted, generating bow shocks that continuously distribute along the axial direction of the plug.Additionally, a pressure decrease is observed behind the film holes in the supersonic region.This is due to the shear between the mainstream and the coolant, forming vortices and reducing the pressure behind the holes rapidly.
Figure 12 shows the distribution of cooling effectiveness on the rear surface of the plug under different blowing ratios.It can be observed that the cooling effectiveness of the supersonic region on the plug surface is significantly higher than that of the subsonic region at a lower blowing ratio, as shown in Figure 12a.The cooling effectiveness of both the supersonic and subsonic regions gradually increases as the blowing ratio increases, and the difference in cooling effectiveness between the two regions decreases, as shown in Figure 12b,c.At blowing ratios of 0.25, 0.38, and 0.5, the average cooling effectiveness on the rear surface of the plug are 0.48, 0.59, and 0.63, respectively.Figure 12 shows the distribution of cooling effectiveness on the rear surface of the plug under different blowing ratios.It can be observed that the cooling effectiveness of the supersonic region on the plug surface is significantly higher than that of the subsonic region at a lower blowing ratio, as shown in Figure 12a.The cooling effectiveness of both the supersonic and subsonic regions gradually increases as the blowing ratio increases, and the difference in cooling effectiveness between the two regions decreases, as shown in Figure 12b,c.At blowing ratios of 0.25, 0.38, and 0.5, the average cooling effectiveness on the rear surface of the plug are 0.48, 0.59, and 0.63, respectively.To further investigate the variation of cooling effectiveness on the rear surface of the plug under different blowing ratios, the data of cooling effectiveness along a perforation generatrix on the plug surface are extracted, as shown in Figure 13.The cooling effectiveness on the plug surface increases gradually with the increase in the blowing ratio.In the supersonic region of the plug, the cooling effectiveness reaches a maximum of 0.69.Bow shocks are generated near the film holes in the supersonic region (region of the 6th to 8th rows of holes) as mentioned above.It can be observed that the cooling effectiveness decreases on the surface after the bow shocks near the film holes, then gradually rises again,  Figure 12 shows the distribution of cooling effectiveness on the rear surface of the plug under different blowing ratios.It can be observed that the cooling effectiveness of the supersonic region on the plug surface is significantly higher than that of the subsonic region at a lower blowing ratio, as shown in Figure 12a.The cooling effectiveness of both the supersonic and subsonic regions gradually increases as the blowing ratio increases, and the difference in cooling effectiveness between the two regions decreases, as shown in Figure 12b,c.At blowing ratios of 0.25, 0.38, and 0.5, the average cooling effectiveness on the rear surface of the plug are 0.48, 0.59, and 0.63, respectively.To further investigate the variation of cooling effectiveness on the rear surface of the plug under different blowing ratios, the data of cooling effectiveness along a perforation generatrix on the plug surface are extracted, as shown in Figure 13.The cooling effectiveness on the plug surface increases gradually with the increase in the blowing ratio.In the supersonic region of the plug, the cooling effectiveness reaches a maximum of 0.69.Bow shocks are generated near the film holes in the supersonic region (region of the 6th to 8th rows of holes) as mentioned above.It can be observed that the cooling effectiveness decreases on the surface after the bow shocks near the film holes, then gradually rises again, To further investigate the variation of cooling effectiveness on the rear surface of the plug under different blowing ratios, the data of cooling effectiveness along a perforation generatrix on the plug surface are extracted, as shown in Figure 13.The cooling effectiveness on the plug surface increases gradually with the increase in the blowing ratio.In the supersonic region of the plug, the cooling effectiveness reaches a maximum of 0.69.Bow shocks are generated near the film holes in the supersonic region (region of the 6th to 8th rows of holes) as mentioned above.It can be observed that the cooling effectiveness decreases on the surface after the bow shocks near the film holes, then gradually rises again, as shown in Figure 13.This is because the reverse pressure gradient behind the shock causes the coolant to separate, weakening the protection of the surface.The subsequent coolant reattaches to the surface, leading to an increase in surface cooling effectiveness.This is similar to the temperature distribution of the "half-covering film cooling scheme" in reference [18].
as shown in Figure 13.This is because the reverse pressure gradient behind the shock causes the coolant to separate, weakening the protection of the surface.The subsequent coolant reattaches to the surface, leading to an increase in surface cooling effectiveness.This is similar to the temperature distribution of the "half-covering film cooling scheme" in reference [18].In this paper, Nusselt number represents the intensity of convective heat transfer between the wall and the mainstream.A smaller Nusselt number means that the heat transfer between the wall and the mainstream is weaker and the temperature of the wall is lower.As shown in Figure 14, when the mainstream expands to supersonic, the Nusselt number downstream of the wall film hole is lower than that in the Subsonic zone, indicating that the convective heat transfer intensity between the wall and the mainstream is weak at supersonic, and the cooling air has a better protective effect on the wall.At blowing ratios of 0.25, 0.38, and 0.5, the average temperature on the rear surface decreased by 34.4%, 43.1%, and 48.1%, respectively.In this paper, Nusselt number represents the intensity of convective heat transfer between the wall and the mainstream.A smaller Nusselt number means that the heat transfer between the wall and the mainstream is weaker and the temperature of the wall is lower.As shown in Figure 14, when the mainstream expands to supersonic, the Nusselt number downstream of the wall film hole is lower than that in the Subsonic zone, indicating that the convective heat transfer intensity between the wall and the mainstream is weak at supersonic, and the cooling air has a better protective effect on the wall.At blowing ratios of 0.25, 0.38, and 0.5, the average temperature on the rear surface decreased by 34.4%, 43.1%, and 48.1%, respectively.
as shown in Figure 13.This is because the reverse pressure gradient behind the shock causes the coolant to separate, weakening the protection of the surface.The subsequent coolant reattaches to the surface, leading to an increase in surface cooling effectiveness.This is similar to the temperature distribution of the "half-covering film cooling scheme" in reference [18].In this paper, Nusselt number represents the intensity of convective heat transfer between the wall and the mainstream.A smaller Nusselt number means that the heat transfer between the wall and the mainstream is weaker and the temperature of the wall is lower.As shown in Figure 14, when the mainstream expands to supersonic, the Nusselt number downstream of the wall film hole is lower than that in the Subsonic zone, indicating that the convective heat transfer intensity between the wall and the mainstream is weak at supersonic, and the cooling air has a better protective effect on the wall.At blowing ratios of 0.25, 0.38, and 0.5, the average temperature on the rear surface decreased by 34.4%, 43.1%, and 48.1%, respectively.Figure 15 shows the changes in the aerodynamic parameters of the nozzle at different blowing ratios.It can be observed that both the total pressure recovery coefficient (σ) and thrust coefficient (C f ) exhibit a decreasing trend as the blowing ratio increases.The cooling flow rate increases with the increase in the blowing ratio, enhancing the mixing between the mainstream and the coolant.This leads to an increased velocity gradient between the mainstream and the coolant, resulting in total pressure losses, thus reducing the aerodynamic performance of the nozzle.Compared with the baseline plug, at blowing ratios of 0.25, 0.38, and 0.5, the thrust coefficients decreased by 0.31%, 0.51%, and 0.61%, and the total pressure recovery coefficients decreased by 0.52%, 0.73%, and 0.93%, respectively.Figure 15 shows the changes in the aerodynamic parameters of the nozzle at different blowing ratios.It can be observed that both the total pressure recovery coefficient (σ) and thrust coefficient (Cf) exhibit a decreasing trend as the blowing ratio increases.The cooling flow rate increases with the increase in the blowing ratio, enhancing the mixing between the mainstream and the coolant.This leads to an increased velocity gradient between the mainstream and the coolant, resulting in total pressure losses, thus reducing the aerodynamic performance of the nozzle.Compared with the baseline plug, at blowing ratios of 0.25, 0.38, and 0.5, the thrust coefficients decreased by 0.31%, 0.51%, and 0.61%, and the total pressure recovery coefficients decreased by 0.52%, 0.73%, and 0.93%, respectively.

Effect of Hole Diameters on Aerodynamic and Heat Transfer Characteristics of Axisymmetric Plug Nozzle
In this chapter, the film cooling design parameters BR = 0.25 and α = 30°are fixed, and the hole diameter is 0.65 mm, 0.79 mm, and 1 mm, respectively.Figure 16 presents the pressure distribution on the rear surface of the plug with different hole diameters.The overall trend of surface pressure distribution is not significantly affected by the hole diameter, there is a larger fluctuation in pressure near the film hole as the hole diameter increases, and the intensity of the bow shock in front of the hole also increases.

Effect of Hole Diameters on Aerodynamic and Heat Transfer Characteristics of Axisymmetric Plug Nozzle
In this chapter, the film cooling design parameters BR = 0.25 and α = 30 • are fixed, and the hole diameter is 0.65 mm, 0.79 mm, and 1 mm, respectively.Figure 16 presents the pressure distribution on the rear surface of the plug with different hole diameters.The overall trend of surface pressure distribution is not significantly affected by the hole diameter, there is a larger fluctuation in pressure near the film hole as the hole diameter increases, and the intensity of the bow shock in front of the hole also increases.Figure 17 shows the distribution of cooling effectiveness on the rear surface of the plug with different hole diameters.The cooling effectiveness in the supersonic region is relatively high for all three diameters in general.However, the region with higher cooling effectiveness decreases and overall cooling effectiveness decreases with increasing hole diameter size.The average cooling effectiveness of the rear surface of the plug is increased by 22.7% when the hole diameter d = 0.65 mm compared with d = 1 mm.In the studied Figure 17 shows the distribution of cooling effectiveness on the rear surface of the plug with different hole diameters.The cooling effectiveness in the supersonic region is relatively high for all three diameters in general.However, the region with higher cooling effectiveness decreases and overall cooling effectiveness decreases with increasing hole diameter size.The average cooling effectiveness of the rear surface of the plug is increased by 22.7% when the hole diameter d = 0.65 mm compared with d = 1 mm.In the studied range of hole diameter parameters, the surface cooling effectiveness increases with the decrease of the hole diameter when the perforated percentage is constant.This is because when the perforated percentage is fixed, the increase in film hole diameter means that the film hole number decreases, the film coverage is reduced, and the cooling effectiveness will be dropped.Figure 17 shows the distribution of cooling effectiveness on the rear surface of the plug with different hole diameters.The cooling effectiveness in the supersonic region is relatively high for all three diameters in general.However, the region with higher cooling effectiveness decreases and overall cooling effectiveness decreases with increasing hole diameter size.The average cooling effectiveness of the rear surface of the plug is increased by 22.7% when the hole diameter d = 0.65 mm compared with d = 1 mm.In the studied range of hole diameter parameters, the surface cooling effectiveness increases with the decrease of the hole diameter when the perforated percentage is constant.This is because when the perforated percentage is fixed, the increase in film hole diameter means that the film hole number decreases, the film coverage is reduced, and the cooling effectiveness will be dropped.Figure 18 shows the distribution of the Nusselt number on the rear surface of the plug for different hole diameters.It can be observed that the cooling film coverage area in the front of the film hole decreases as the hole diameter increases.And the Nusselt number on the surface increases, enhancing convective heat transfer and deteriorating the cooling  Figure 19 shows the variation in aerodynamic parameters for different hole diameters.It can be observed from the figure that the change in hole diameter has a relatively weak impact on the thrust coefficient (Cf) and total pressure recovery coefficient (σ) of the nozzle.Figure 19 shows the variation in aerodynamic parameters for different hole diameters.It can be observed from the figure that the change in hole diameter has a relatively weak impact on the thrust coefficient (Cf) and total pressure recovery coefficient (σ) of the nozzle.

Effect of Hole Inclination Angle on Aerodynamic and Heat Transfer Characteristics of Axisymmetric Plug Nozzle
In this chapter, the film cooling design parameters BR = 0.25 and d = 0.65 mm are fixed, and the hole inclination angle is 30°, 45°, and 60°, respectively.Figure 20 shows the pressure distribution on the surface for different film hole inclination angles.It can be observed that the overall trend of surface pressure distribution remains largely unaffected by the increase in hole inclination angle.However, it can be noticed that the intensity of the bow shock in the film hole region in the supersonic zone as the hole inclination angle increases.This is because the penetration capability of the coolant into the mainstream is enhanced with increased inclination angle, resulting in an increase in the height of the

Effect of Hole Inclination Angle on Aerodynamic and Heat Transfer Characteristics of Axisymmetric Plug Nozzle
In this chapter, the film cooling design parameters BR = 0.25 and d = 0.65 mm are fixed, and the hole inclination angle is 30 • , 45 • , and 60 • , respectively.Figure 20 shows the pressure distribution on the surface for different film hole inclination angles.It can be observed that the overall trend of surface pressure distribution remains largely unaffected by the increase in hole inclination angle.However, it can be noticed that the intensity of the bow shock in the film hole region in the supersonic zone as the hole inclination angle increases.This is because the penetration capability of the coolant into the mainstream is enhanced with increased inclination angle, resulting in an increase in the height of the film cooling layer and stronger hindrance to the mainstream boundary layer, leading to an increase in bow shock intensity.Figure 21 presents the distribution of film cooling effectiveness on the surface for different film hole inclination angle.It can be observed that the average cooling effectiveness is higher for the 30° hole inclination angle.The average cooling effectiveness on the surface decreases as the inclination angle increases.This is mainly due to the increased hole inclination angle allowing coolant to penetrate into the mainstream, enhancing mixing with the mainstream, and adherence of coolant decreases, resulting in a decrease in cooling effectiveness.Compared with the 45° and 60° hole inclination angle models, the Figure 21 presents the distribution of film cooling effectiveness on the surface for different film hole inclination angle.It can be observed that the average cooling effectiveness is higher for the 30 • hole inclination angle.The average cooling effectiveness on the surface decreases as the inclination angle increases.This is mainly due to the increased hole inclination angle allowing coolant to penetrate into the mainstream, enhancing mixing with the mainstream, and adherence of coolant decreases, resulting in a decrease in cooling effectiveness.Compared with the 45 • and 60 • hole inclination angle models, the 30 • hole inclination angle model has 4.8% and 9.6% higher average cooling effectiveness on the surface, respectively.Figure 21 presents the distribution of film cooling effectiveness on the surface for different film hole inclination angle.It can be observed that the average cooling effectiveness is higher for the 30° hole inclination angle.The average cooling effectiveness on the surface decreases as the inclination angle increases.This is mainly due to the increased hole inclination angle allowing coolant to penetrate into the mainstream, enhancing mixing with the mainstream, and adherence of coolant decreases, resulting in a decrease in cooling effectiveness.Compared with the 45° and 60° hole inclination angle models, the 30° hole inclination angle model has 4.8% and 9.6% higher average cooling effectiveness on the surface, respectively.Figure 23 presents the variation of aerodynamic parameters for different film hole inclination angle.The overall trend shows a slight decrease in the thrust coefficient and total pressure recovery coefficient as the hole inclination angle increases.This is attributed to the increased ability of coolant penetration with larger hole inclination angle, leading to higher mixing losses with the mainstream.Additionally, the strengthening of the bow shock intensity due to the film cooling layer before the hole contributes to increased aerodynamic losses.Figure 23 presents the variation of aerodynamic parameters for different film hole inclination angle.The overall trend shows a slight decrease in the thrust coefficient and total pressure recovery coefficient as the hole inclination angle increases.This is attributed to the increased ability of coolant penetration with larger hole inclination angle, leading to higher mixing losses with the mainstream.Additionally, the strengthening of the bow shock intensity due to the film cooling layer before the hole contributes to increased aerodynamic losses.

Conclusions
This study conducted a numerical simulation on an axisymmetric plug nozzle and analyzed the impact of three parameters related to film cooling structure (blowing ratio, hole diameter, and hole inclination angle) on the cooling effectiveness and aerodynamic performance of the nozzle under transonic conditions.The main conclusions drawn from the analysis are as follows: (1) The mainstream continues to expand to supersonic after the outlet of the axisymmetric plug nozzle.It is disturbed by the coolant from the film holes, causing the boundary layer of the mainstream to lift near the film holes in the supersonic region,

Conclusions
This study conducted a numerical simulation on an axisymmetric plug nozzle and analyzed the impact of three parameters related to film cooling structure (blowing ratio, hole diameter, and hole inclination angle) on the cooling effectiveness and aerodynamic performance of the nozzle under transonic conditions.The main conclusions drawn from the analysis are as follows: (1) The mainstream continues to expand to supersonic after the outlet of the axisymmetric plug nozzle.It is disturbed by the coolant from the film holes, causing the boundary layer of the mainstream to lift near the film holes in the supersonic region, resulting in the formation of bow shocks along the axial direction of the plug.These bow shocks cause a local decrease in surface cooling effectiveness.(2) The blowing ratio plays a significant role in the cooling effectiveness of the plug.The coolant flow discharged from the film holes increases as the blowing ratio increases, leading to improved surface cooling effectiveness.Compared with the baseline plug, the average surface temperature decreases by 34.4%, 43.1%, and 48.1% in the blowing ratio range of 0.25~0.5,respectively.However, an increase in the blowing ratio results in a decrease in the aerodynamic performance of the nozzle, with a reduction in thrust coefficient of 0.31%, 0.51%, and 0.61%, and a reduction in total pressure recovery coefficient of 0.52%, 0.73%, and 0.93%, respectively.(3) When the perforated percentage is constant, the surface cooling effectiveness increases with the decrease of the hole diameter, resulting in a better surface cooling performance.The variation in hole diameter has a weak impact on the aerodynamics of the nozzle.(4) With the increase in the inclination angle of the film holes, the ability of the coolant flow to penetrate the mainstream increases, enhancing the mixing with the mainstream.The resulting film cooling layer increases in height, enhancing the obstruction of the mainstream boundary layer, and causing an increase in the intensity of bow shocks, leading to a decrease in the average cooling effectiveness of the surface and an increase in aerodynamic losses.A smaller hole inclination angle results in better surface cooling performance.The research findings in this paper can provide a reference for the design of surface film cooling structures in complex supersonic mainstream conditions in engineering applications.

Figure 1 .
Figure 1.Schematic diagram of the baseline axisymmetric plug nozzle.(a) 1/8 model of the baseline axisymmetric plug nozzle, (b) Geometric parameters of the baseline axisymmetric plug nozzle

Figure 2 .
Figure 2. Schematic diagram of the scale plug with cooling structure.

Figure 3 .
Figure 3. Schematic diagram of the film cooling structure for the axisymmetric plug nozzle.
ρ and c u are the density and flow velocity of the coolant,ρ g and u g are the density and flow velocity of the mainstream), hole diameter d and hole

Figure 1 .
Figure 1.Schematic diagram of the baseline axisymmetric plug nozzle.(a) 1/8 model of the baseline axisymmetric plug nozzle, (b) Geometric parameters of the baseline axisymmetric plug nozzle.

Figure 1 .
Figure 1.Schematic diagram of the baseline axisymmetric plug nozzle.(a) 1/8 model of the baseline axisymmetric plug nozzle, (b) Geometric parameters of the baseline axisymmetric plug nozzle

Figure 2 .
Figure 2. Schematic diagram of the scale plug with cooling structure.

Figure 3 .
Figure 3. Schematic diagram of the film cooling structure for the axisymmetric plug nozzle.
ρ and c u are the density and flow velocity of the coolant,ρ g and u g are the density and flow velocity of the mainstream), hole diameter d and hole

Figure 2 .Figure 1 .
Figure 2. Schematic diagram of the scale plug with cooling structure.

Figure 2 .
Figure 2. Schematic diagram of the scale plug with cooling structure.

Figure 3 .
Figure 3. Schematic diagram of the film cooling structure for the axisymmetric plug nozzle.
ρ and c u are the density and flow velocity of the coolant,ρ g and u g are the density and flow velocity of the mainstream), hole diameter d and hole

Figure 3 .
Figure 3. Schematic diagram of the film cooling structure for the axisymmetric plug nozzle.

Figure 4 .
Figure 4. Schematic diagram of the arrangement of film holes with different diameters.

Figure 4 .
Figure 4. Schematic diagram of the arrangement of film holes with different diameters.

Figure 5 .
Figure 5. Schematic diagram of the computational domain.Figure 5. Schematic diagram of the computational domain.

Figure 5 .
Figure 5. Schematic diagram of the computational domain.Figure 5. Schematic diagram of the computational domain.

Figure 6 .
Figure 6.Schematic of the grids for the axisymmetric plug nozzle.(a) Surface grids of plug nozzle; (b) Local grids of the symmetry plane; (c) Boundary layer grid on the near-surface.

Figure 6 . 18 Figure 7 .
Figure 6.Schematic of the grids for the axisymmetric plug nozzle.(a) Surface grids of plug nozzle; (b) Local grids of the symmetry plane; (c) Boundary layer grid on the near-surface.

Figure 8 .
Figure 8.The flow field distribution of the baseline plug nozzle.(a) velocity distribution; (b) pressure distribution; (c) temperature distribution.

Figure 8 .Figure 8 .
Figure 8.The flow field distribution of the baseline plug nozzle.(a) velocity distribution; (b) pressure distribution; (c) temperature distribution.

Figure 9 .
Figure 9. Pressure and temperature distribution of the baseline plug.(a) pressure distribution.(b) temperature distribution.

Figure 9 .
Figure 9. Pressure and temperature distribution of the baseline plug.(a) pressure distribution.(b) temperature distribution.

Figure 10 .
Figure 10.Pressure and temperature distribution along the generatrix of the baseline plug surface.

Figure 10 .
Figure 10.Pressure and temperature distribution along the generatrix of the baseline plug surface.

Figure 11 .
Figure 11.Pressure distribution on the rear surface of the plug under different blowing ratios.

Figure 12 .
Figure 12.Cooling effectiveness distribution on the rear surface of the plug under different blowing ratios.

Figure 11 .
Figure 11.Pressure distribution on the rear surface of the plug under different blowing ratios.

Figure 11 .
Figure 11.Pressure distribution on the rear surface of the plug under different blowing ratios.

Figure 12 .
Figure 12.Cooling effectiveness distribution on the rear surface of the plug under different blowing ratios.

Figure 12 .
Figure 12.Cooling effectiveness distribution on the rear surface of the plug under different blowing ratios.

Figure 13 .
Figure 13.The curve of perforation cooling effectiveness on the rear surface of the plug under different blowing ratios.

Figure 13 .
Figure 13.The curve of perforation cooling effectiveness on the rear surface of the plug under different blowing ratios.

Figure 13 .
Figure 13.The curve of perforation cooling effectiveness on the rear surface of the plug under different blowing ratios.

Figure 14 .
Figure 14.Nusselt number distribution on the rear surface of the plug under different blowing ratio.

Figure 14 .
Figure 14.Nusselt number distribution on the rear surface of the plug under different blowing ratio.

Figure 15 .
Figure 15.Effect of blowing ratio on thrust coefficient and total pressure recovery coefficient of plug nozzle.

Figure 15 .
Figure 15.Effect of blowing ratio on thrust coefficient and total pressure recovery coefficient of plug nozzle.

Symmetry 2024 ,Figure 16 .
Figure 16.Pressure distribution on the rear surface of the plug under different film hole diameters.

Figure 16 .
Figure 16.Pressure distribution on the rear surface of the plug under different film hole diameters.

Figure 16 .
Figure 16.Pressure distribution on the rear surface of the plug under different film hole diameters.

Figure 17 .
Figure 17.Cooling effectiveness distribution on the rear surface of the plug under different film hole diameters.

Figure 17 .
Figure 17.Cooling effectiveness distribution on the rear surface of the plug under different film hole diameters.

Figure 18 Figure 18 .
Figure18shows the distribution of the Nusselt number on the rear surface of the plug for different hole diameters.It can be observed that the cooling film coverage area in the front of the film hole decreases as the hole diameter increases.And the Nusselt number on the surface increases, enhancing convective heat transfer and deteriorating the cooling effect.The hole diameter increase also leads to an uneven distribution of surface temperature, which is unfavorable for the protection of the surface by the coolant.Symmetry 2024, 16, x FOR PEER REVIEW 14 of 18

Figure 18 .
Figure 18.Nusselt number distribution on the rear surface of the plug under different film hole diameters.

Figure 19
Figure19shows the variation in aerodynamic parameters for different hole diameters.It can be observed from the figure that the change in hole diameter has a relatively weak impact on the thrust coefficient (C f ) and total pressure recovery coefficient (σ) of the nozzle.

Figure 18 .
Figure 18.Nusselt number distribution on the rear surface of the plug under different film hole diameters.

Figure 19 .
Figure 19.Effect of hole diameter on thrust coefficient and total pressure recovery coefficient of plug nozzle.

Figure 19 .
Figure 19.Effect of hole diameter on thrust coefficient and total pressure recovery coefficient of plug nozzle.

Symmetry 2024 ,Figure 20 .
Figure 20.Pressure distribution on the rear surface of the plug under different film hole inclination angle.

Figure 20 .
Figure 20.Pressure distribution on the rear surface of the plug under different film hole inclination angle.

Figure 20 .
Figure 20.Pressure distribution on the rear surface of the plug under different film hole inclination angle.

Figure 21 .
Figure 21.Cooling effectiveness distribution on the rear surface of the plug under different film hole inclination angle.

Figure 22
Figure 22 illustrates the distribution of Nusselt numbers for different film hole inclination angles.It can be observed that the coolant penetrating from the film hole into the mainstream enhances as the hole inclination angle increases, adherence of coolant decreases and a deterioration in the protective effect on the surface.

Figure 21 .
Figure 21.Cooling effectiveness distribution on the rear surface of the plug under different film hole inclination angle.

Figure 22
Figure 22 illustrates the distribution of Nusselt numbers for different film hole inclination angles.It can be observed that the coolant penetrating from the film hole into the mainstream enhances as the hole inclination angle increases, adherence of coolant decreases and a deterioration in the protective effect on the surface.Symmetry 2024, 16, x FOR PEER REVIEW 16 of 18

Figure 22 .
Figure 22.Nusselt number distribution on the rear surface of the plug under different film hole inclination angle.

Figure 22 .
Figure 22.Nusselt number distribution on the rear surface of the plug under different film hole inclination angle.

Figure 23
Figure 23  presents the variation of aerodynamic parameters for different film hole inclination angle.The overall trend shows a slight decrease in the thrust coefficient and total pressure recovery coefficient as the hole inclination angle increases.This is attributed to the increased ability of coolant penetration with larger hole inclination angle, leading to higher mixing losses with the mainstream.Additionally, the strengthening of the bow

Figure 23 .
Figure 23.Effect of hole inclination angle on the thrust coefficient and total pressure recovery coefficient of plug nozzle.

Figure 23 .
Figure 23.Effect of hole inclination angle on the thrust coefficient and total pressure recovery coefficient of plug nozzle.

Table 1 provides
the selected ranges for the study parameters: blowing ratio BR (defined as BR =, where c

Table 1 provides
the selected ranges for the study parameters: blowing ratio BR (defined as BR =, where c

Table 1
provides the selected ranges for the study parameters: blowing ratio BR (defined as BR =, where c

Table 1 .
Parameters of film holes.

Table 2 .
Thermodynamic property setting of materials.

Table 2 .
Thermodynamic property setting of materials.

Table 3 .
Specific parameter table for boundary conditions.

Table 3 .
Specific parameter table for boundary conditions.