Base Flow and Drag Characteristics of a Supersonic Vehicle with Cold and Hot Jet Flows of Nozzles

Abstract

Base drag has a significant effect on the overall drag of a projectile in a supersonic flow. Herein, the base drag and flow characteristics of cold and hot gas flow in a supersonic flow are analyzed via numerical simulations. The hot gas flow is simulated using a chemical equilibrium application code based on hydrogen combustion. Two types of nozzle configurations, namely conical and contoured, are chosen for the simulation. The simulation results reveal that the change in base drag is 5–85% according to the injection gases. In the over-expanded and slightly under-expanded conditions, the base drag decreases in the hot gas flow, owing to the weak expansion fan caused by the high-temperature nozzle flow expansion, whereas in the highly under-expanded condition, the base drag decreases, owing to the strong shock wave near the base caused by the deflection of the recirculation region toward the body wall. In addition, the variations in base flow structures are observed differently compared with the cold flow; for example, a weak oblique shock wave at the nozzle exit, an increase in the distance between the shock wave and base, and deflection of the recirculation region based on the body wall are observed.

1. Introduction

Precise aerodynamic data must be acquired to design flight systems such as aircraft and space launch vehicles. In particular, drag considerably affects the achievement of mission requirements; thus, it should be accurately predicted. The drag forces acting on an aircraft are typically composed of pressure drag, friction drag, and base drag. Among these, the base drag is defined as the drag generated due to the difference in pressure between the base part of the aircraft and the freestream flow. However, estimating the base drag is challenging because the base flow forms in a complex manner depending on the nozzle flow conditions. Therefore, the base drag leads to errors in drag prediction and contributes to high effects of the total drag in a supersonic flow. Figure 1 shows the base flow characteristics under the nozzle flow conditions. Depending on the over-expanded or under-expanded conditions, a shock wave or expansion fan is generated near the base, thereby resulting in differences in the base flow structure and base drag. Therefore, several studies have attempted to understand base flow and predict base drag based on the injection conditions of the nozzle flow.

Several experiments and numerical simulations have been performed to predict base drag. Such studies have been conducted for subsonic [1,2,3], transonic [4,5,6,7,8,9], and supersonic [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] flow. Because the current study focuses on supersonic flows, only those on supersonic flows are discussed herein. Previous studies can be divided into wind tunnel studies [10,11,12,13,14,15,16,17,18] and numerical simulations [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. Wind tunnel studies with cold flow were conducted to create base pressure data for the combustion chamber pressure in supersonic flow [10,11]. Some studies measured the base pressure based on the effect of the thrust in supersonic flow [12,13,14,15,16]. Some studies reported that the observed base drag and flow depend on the base configuration [17,18]. Some studies investigated the base pressure according to the base and nozzle configurations in supersonic flow [19,20,21,22].

The base drag of the aircraft body according to the on/off power of the thrust has been reported [23,24,25,26]. Additionally, hot gas flow experiments and numerical simulations have been conducted. A hydrogen combustion study was conducted to observe the characteristics of the hot gas nozzle flow [27]. The axial thrust when a high-temperature jet is injected at different mass flow rates was studied to observe the operation of the thrust vector control system [28,29]. In addition, the base-bleed effects of hot gas on base drag in supersonic flows were studied [30]. Furthermore, research efforts aimed at controlling base pressure and drag using various methods have also been conducted. Studies have been conducted to reduce base drag via the use of base bleed [31] and to control base pressure via passive control methods such as the cavity technique [32,33]. Recently, research has been conducted using machine learning techniques such as genetic algorithms and artificial neural networks to predict base pressure and drag [34,35].

As discussed, numerous wind tunnel experiments and numerical simulation studies predicting the base drag have been conducted. However, most studies on base drag have been performed under no-jet or cold-flow injection conditions. In actual propulsion systems, high-pressure and high-temperature gases are injected through nozzles to obtain thrust. Hot gas from the nozzle can affect the base flow structure and base drag, thus rendering difficulty in predicting the base drag directly by utilizing the results of cold gas conditions. Thus, the effects of hot nozzle flow must be investigated. Nonetheless, only a few studies using flight tests have been performed on the base drag under hot gas flow injections, most of which focused on subsonic and transonic flows, and the data are not publicly available. Therefore, the differences between cold and hot gas flows under the same pressure conditions must be compared and the major factors that lead to flow variations for different nozzle configurations in a supersonic flow must be analyzed.

In this study, the effect of hot gas flow on the base drag was observed, and the difference in the base drag under cold gas conditions was confirmed. The numerical simulation was based on the 2D axisymmetric RANS equation, and the properties of the hot gas were calculated based on the chemical equilibrium. Base pressures under different nozzle flow expansion conditions (over- and under-expansion) were observed when cold and hot gases were injected, and the base flow structure was analyzed. The base drags with cold and hot nozzle flows were compared, and the base drag ratio was analyzed. As a result of the analysis of the base drag ratio, a prediction method for the base drag when the hot gases are injected was proposed.

2. Numerical Method

2.1. Physical Model

The nozzle configuration used in this study is based on a previous wind tunnel study that used cold gas [10]. Figure 2 and

Table 1. Specification of nozzle geometry.

Parameters	Values
Nozzle divergence angle (${\theta }_e$)	0°–20°
Base diameter ($d_b$)	63.50 mm
Nozzle exit diameter ($d_e$)	50.90 mm
Nozzle throat diameter ($d_t$)	28.52 mm
Curvature radius ($R$)	14.27 mm

present the schematics and specifications of the nozzle geometry, respectively. A nozzle divergence angle is defined as an angle of nozzle contour at the nozzle exit. A conical nozzle was used to observe the effects of nozzle divergence angle. In addition, a contoured nozzle was applied to compare the case of zero divergence angle, and the geometry of the contoured nozzle was obtained using the characteristic curve method [36]. The throat and exit diameters of the nozzle were fixed, and the nozzle length was varied according to the nozzle divergence angle. The design exit Mach number of the nozzle was approximately 2.7 based on the cold gas flow.

2.2. Numerical Approach

This study performed base drag analysis via 2D axisymmetric analysis. For a more accurate analysis of real fluid, three-dimensional analysis may be more suitable. In the case of two-dimensional analysis, there is a possibility of differences in boundary layer effects compared to three-dimensional analysis, which could consequently result in variations in the base pressure. However, the differences are not likely to be significant, and it has been confirmed that the results of 2D axisymmetric analysis match the experimental results well in previous research. Therefore, it is considered reasonable to obtain results via 2D axisymmetric simulation. The numerical techniques utilized in this study were validated in a previous study [25]. A two-dimensional axisymmetric steady-state RANS equation and density-based coupled solver were applied as follows:

$$\begin{gathered} \frac{\mathit{\partial }}{\mathit{\partial }t}{\int }_{V}WdV+\oint \left[F-G\right]·da={\int }_{V}HdV \\ W=\left[\begin{array}{c}\rho \\ \rho v\\ \rho E\end{array}\right],F=\left[\begin{array}{c}\rho v\\ \rho vv+pI\\ \rho vH+pv\end{array}\right] \\ G=\left[\begin{array}{c}0\\ T\\ T·v+\dot{q}″\end{array}\right],H=\left[\begin{array}{c}{S}_u\\ f_r+f_g+f_p+f_u+f_{\omega }+f_L\\ S_u\end{array}\right] \end{gathered}$$

(1)

where $\rho$, $\mathbf{v}$, $E$, $p$, $\mathbf{T}$, $\dot{q}″$, $\mathbf{H}$, $\overline{\mathbf{V}}$, $\mu$, and $S$ denote the density, velocity, total energy per unit mass, pressure, viscous stress tensor, heat flux vector, body force vector, mean velocity, dynamic viscosity, and source term, respectively. Recently, more accurate computational fluid dynamics (CFD) simulations have been performed by applying detached eddy simulations (DES) or large eddy simulations (LES). However, the main objective of this study was to compare the difference in the base drag between cold and hot gases. Thus, the RANS method was applied by considering the numerous variables to be observed. The RANS method is widely used in aerospace studies, and sufficient validation was performed to ensure the reliability of the numerical method used in this study. The turbulence model was a realizable k–ε model, which was selected via validation [25]. The k–ε turbulence model solves transport equations for the turbulent kinetic energy $k$ and the turbulent dissipation rate $\epsilon$, and the realizable k–ε model contains a new transport equation for the turbulent dissipation rate. This includes turbulence compressibility correction for dilatation dissipation, as follows:

$$\frac{\mathit{\partial }}{\mathit{\partial }t}\left(\rho k\right)+\nabla ·\left(\rho k\overline{\mathit{V}}\right)=\nabla ·\left[\left(\mu +\frac{{\mu }_k}{{\sigma }_k}\right)\nabla k\right]+P_k-\rho \left(\epsilon -{\epsilon }_0\right)+S_k$$

(2)

$$\frac{\mathit{\partial }}{\mathit{\partial }t}\left(\rho \epsilon \right)+\nabla ·\left(\rho \epsilon \overline{\mathit{V}}\right)=\nabla ·\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\nabla \epsilon \right]+\frac{C_{\epsilon 1}{P}_{\epsilon }}{T_e}-C_{\epsilon 2}{f}_2\rho \left(\frac{\epsilon }{T_e}-\frac{{\epsilon }_0}{T_0}\right)+S_{\epsilon }$$

(3)

where $\overline{\mathbf{V}}$ is the mean velocity; $\mu$ is dynamic viscosity; ${\sigma }_k, {\sigma }_{ϵ},C_{ϵ1},{\mathrm{a}\mathrm{n}\mathrm{d} C}_{ϵ2}$ are model coefficients; $P_k and P_{ϵ}$ are production terms; and $S_k \mathrm{a}\mathrm{n}\mathrm{d} S_{ϵ}$ are source terms. Furthermore, third-order spatial discretization with a monotonic upwind scheme for conservation laws was applied to maintain accuracy as follows:

$${\left(\dot{m}\varphi \right)}_f=\left\{\begin{array}{c}\dot{m}{\varphi }_{FOU}\\ \dot{m}(\sigma {\varphi }_{MUSCL3}+\left(1-\sigma \right){\varphi }_{CD3})\end{array}\right.\begin{array}{c}\mathit{for} \xi <0\mathit{or}\xi >1\\ \mathit{for} 0\le \xi \le 1\end{array}$$

(4)

The advection upstream splitting method was used for the inviscid convection term, as expressed in Equation (9).

$$f_f=f^{\left(c\right)}+P=m_i^{+}{\left(1,u,v,\dots ,H\right)}_0^T+m_i^{-}{\left(1,u,v,\dots ,H\right)}_1^T+P_i$$

(5)

The second-order central difference method was used to obtain the diffusion term, whereas the ideal gas law was used to calculate the gas density. A multi-component material model was established to enable the mixing of different gases. Air was used as the freestream and nozzle gas for cold flow conditions, whereas combustion gas was used as the nozzle gas for hot flow conditions. The wall conditions were set to adiabatic no-slip conditions. To accurately calculate the turbulent flow and boundary layer, the Y+ value at the wall was ensured to be less than 1 by setting the first-layer thickness from the wall to 2 μm. Thus, the convergence of the continuity residual was confirmed to consistently be less than 10⁻³.

The computational domain is illustrated in Figure 3. The domain was designed to be sufficiently large to avoid nozzle flow interaction at the outlet, and the domain length was 65 times in the axial direction and 15 times in the vertical direction compared with the diameter of the body geometry.

The boundary conditions are presented in

Table 2. Boundary conditions.

Boundary	Type	Parameter	Value
Freestream	Freestream	Gas	Air
		Static pressure [bar]	1.0
		Static temperature [K]	300.0
		Mach number	1.5, 2.0, 2.5, 3.0, 4.0
Nozzle flow	Stagnation inlet	Gas	Air (Cold flow), Combustion gas (Hot flow)
		Stagnation pressure [bar]	Determined by NPR
		Stagnation temperature [K]	Obtained from chemical equilibrium analysis
Outlet	Pressure outlet	-	Extrapolated

. Mach numbers of 1.5–4.0 were applied to observe the effects of various freestream Mach numbers in the supersonic region. The stagnation pressure of the nozzle flow was determined using the nozzle pressure ratio (NPR), and the stagnation temperature was derived via a chemical equilibrium analysis at each NPR. The NPR is a ratio of stagnation pressure of the combustion chamber and freestream static pressure and was selected considering the nozzle expansion conditions based on cold gas. The design NPR of the nozzle is 23.3 as an optimum expansion condition. The NPR used in this study are referred from previous experimental data [10] and are listed in

Table 3. NPR values for nozzle flow.

NPR	Pressure Ratio at Nozzle Exit	Nozzle Expansion Condition
4.9	0.21	Over-expanded
9.8	0.42
29.6	1.27	Slightly under-expanded
88.8	3.81
148.0	6.35
246.7	10.59	Highly under-expanded
444.1	19.08
690.8	29.66

.

For the hot gas flow simulation, the properties of the hot gas were derived from the chemical equilibrium analysis (CEA) code [37]. The hot gas was assumed to be generated by the combustion of hydrogen and air. The combustion conditions applied to the CEA maintained an equivalence ratio of 1.06. The properties of the hot gases used are listed in

Table 4. CEA simulation results.

NPR	Temperature (K)	Molecular Weight (1/n)	Specific Heat Ratio	Prandtl Number
4.9	2436.2	24.09	1.200	0.62
9.8	2449.0	24.11	1.208	0.64
29.6	2463.9	24.14	1.220	0.66
88.8	2473.4	24.16	1.228	0.67
148.0	2476.5	24.16	1.231	0.68
246.7	2478.9	24.17	1.233	0.68
444.1	2481.0	24.17	1.236	0.68
690.8	2482.3	24.17	1.237	0.69

. The CEA product, containing various gas components, was assumed to be a single gas mixture injected from the combustion chamber inlet. Since the change in gas properties with temperature has a negligible effect on the base drag, the specific heat ratio was applied as a constant for each NPR.

2.3. Grid Dependency and Validation

Figure 4 shows the mesh around the base applied to analyze the grid dependency. The fine mesh size around the nozzle and base was 0.5, 0.25, 0.2, and 0.1 mm, and the total mesh number was approximately 0.04 M, 0.13 M, 0.49 M, and 0.75 M, respectively. Figure 5 shows the comparison of the mesh dependency. The dimensionless number in Figure 5 is obtained by dividing the base pressure of each grid number based on the base pressure corresponding to the case with 0.75 M grids. The results indicate that the difference in the average base pressure between the 0.49 M and 0.75 M cases was approximately 2%. Therefore, 0.49 M grids were used in the hot gas flow simulation.

The numerical techniques used in this study were validated in a previous study on cold flows [25]. In a previous study, different turbulence models, namely realizable k–ε, SST k–ω, and Spalart–Allmaras models, were compared and the realizable k–ε model was confirmed to be the most accurate with the experimental data. However, the effect of hot gas can be different from that of cold gas; therefore, the hot gas simulations must be validated. However, no studies thus far provide detailed information on base drag in supersonic nozzle flow experiments or simulations with hot gas. Thus, the simulation of the reference was conducted by hot gas injection via the base bleed and compared with the experimental data [38]. Figure 6a presents a comparison of the temperature contour, and Figure 6b shows a comparison of the base pressure results. The injection parameter $I$ is defined by the injection mass flow rate, base area, and free-stream properties, as shown in Equation (6). The simulation of the base bleed of 2200 K hot gas at Mach number 2.0 condition was conducted in a two-dimensional axisymmetric condition.

$$I=\frac{{\dot{m}}_j}{A{\rho }_{\infty }{V}_{\infty }}$$

(6)

Figure 6b presents the results of the comparison between previous [30] and present studies. It was calculated in a manner similar to that of the experimental data and was found to be closer to the simulation results. Therefore, numerical simulation techniques are valid for analyzing nozzle flows with cold and hot gases.

3. Results and Discussion

3.1. No-Jet Condition

Figure 7 and Figure 8 show the base flow and pressure under the no-jet condition, respectively. As shown in Figure 7, a recirculation region is generated around the base of the body and nozzle exit due to the absence of the nozzle jet flow, and the shear layer is generated on the boundary of the recirculation region, as shown in the density gradient contour. Since the recirculation region has low pressure, the freestream flow expands at the corner of the base turning toward the nozzle exit. Therefore, an expansion fan is formed near the base, and the base pressure becomes lower than that of freestream flow. The same results related to the recirculation region and expansion fan were presented in previous studies [19,20]. Figure 8 shows the ratio of the base pressure to the freestream static pressure according to the radial distance from the nozzle exit to the base corner. Evidently, the distribution of the base pressure was constant in the radial direction. The base pressure decreases as the Mach number increases. This phenomenon is attributed to the increasing expansion rate at the base corner with higher Mach numbers. However, at divergence angles between 10 and 20 degrees, the base pressure remains constant regardless of the Mach number. In contrast, at 0 degrees, a significantly higher base pressure is observed. As depicted in Figure 7, when the divergence angle is not 0 degrees, the influence of viscosity causes the flow of the recirculation region to extend to the base and nozzle wall. Conversely, when the divergence angle is 0 degrees, the flow of the recirculation region is unable to reach the base, resulting in an increase in base pressure.

3.2. Over-Expanded Condition (4.9 ≤ NPR ≤ 9.8)

Figure 9 shows the base pressure of the cold and hot gas conditions according to the nozzle divergence angle under the over-expanded condition in the freestream Mach number of 1.5 and NPR of 4.9. Even when the flow was injected through the nozzle, the pressure distribution in the radial direction was constant, similar to that in the no-jet condition [25]. As shown in Figure 9, the base pressure was lower than that under the no-jet condition, regardless of the nozzle geometry and injection conditions. In addition, for all nozzle divergence angles, the base pressure under hot gas conditions was higher than that under cold gas conditions. In both hot and cold gas conditions, the base pressure was maximized when the nozzle divergence angle was 0°. Except for the case of a zero-divergence angle, the base pressure generally increased as the divergence angle increased. However, in the case of a nozzle divergence angle of 10°, the base pressure was significantly higher than that under other nozzle divergence angle conditions.

Figure 10 shows the flow field according to the nozzle divergence angle when the freestream Mach number was 1.5 and NPR was 4.9. The angle shown in the figure is the deflection angle, which is the angle between the body wall extension line and the shear layer. A decrease in the deflection angle indicates that the nozzle flow expands less, and the freestream flow expands more at the base. As shown in Figure 10, under both cold and hot gas conditions, a shear layer was generated, owing to the flow expansion at the base corner regardless of the nozzle divergence angle. In addition, a jet boundary was formed by the gas injected from the nozzle and a recirculation region formed at the base between the jet boundary and the shear layer. However, when the nozzle flow was over-expanded, the nozzle exit pressure was lower than that in the no-jet condition; therefore, a stronger expansion fan was generated at the base to balance the pressure as shown in the density gradient contour [19,20]. Therefore, the base pressure under the over-expanded condition was always lower than that under the no-jet condition, and as the deflection angle decreased, the base pressure decreased, as shown in Figure 9. In Figure 10a, the deflection angle under the hot gas conditions was larger than that under cold gas conditions. This is because the nozzle flow at the exit expands more in the hot gas condition, and the effect of pushing the shear layer outward occurs. Consequently, the strength of the expansion fan at the base corner decreases, and the base pressure becomes higher than that of the cold gas. Evidently, from Figure 10a,b, when the nozzle divergence angle decreased in the hot gas condition, the deflection angle decreased because the expansion of the nozzle flow decreased, even if the nozzle exit Mach number was the same. Thus, as shown in Figure 9, the base pressure decreased as the nozzle divergence angle decreased. However, when the cold gas was injected at a nozzle divergence angle of 10°, as shown in Figure 10b, the nozzle exit pressure increased abnormally because the nozzle flow did not normally expand, owing to the separation inside the nozzle, thus considerably increasing the base pressure, unlike in other conditions. Consequently, as shown in Figure 9, the base pressures at the nozzle divergence angles of 0° and 10° in the cold gas condition were the same.

Figure 11 and Figure 12 show the base pressure ratio and flow field at a freestream Mach number of 3.0 and NPR of 4.9, respectively. Comparing the results in Figure 9 and Figure 11, the base pressure for the same NPR and nozzle divergence angle decreased as the freestream Mach number increased because, as shown in Figure 8, a lower ambient pressure was formed around the base when the freestream Mach number was higher. In addition, even if the freestream Mach number increased, the base pressure under the hot gas conditions was higher than that under cold gas conditions, regardless of the nozzle divergence angle for the same NPR. As shown in Figure 12, when hot gas was injected, the deflection angle increased compared with the cold gas condition because the flow at the nozzle exit expanded more under the hot gas condition. In Figure 11, the base pressure increased as the nozzle divergence angle increased. Unlike the freestream Mach number of 1.5, the base pressure showed a minimum value when the nozzle divergence angle was 0° in the freestream Mach number of 3.0. In the case of a freestream Mach number of 1.5 with cold gas, flow separation occurred inside the nozzle when the nozzle divergence angle was 10° or less, whereas in the case of the freestream Mach number of 3.0, owing to the low basic ambient pressure, the nozzle flow expanded sufficiently, and the separation did not occur inside the nozzle. Consequently, as the nozzle divergence angle decreased, the base pressure also decreased continuously.

Figure 13 shows the average value of the base pressure according to the freestream Mach number for each condition when the nozzle flow was over-expanded. Overall, as the freestream Mach number increased and the NPR decreased, the base pressure tended to decrease, and the base pressure under the hot gas conditions was higher than that under cold gas conditions. As the NPR increased, the nozzle exit pressure increased and the nozzle flow expanded, thus causing the deflection angle to decrease and base pressure to increase. In addition, the base pressure decreased as the nozzle divergence angle decreased; however, in Figure 13c,d, when the nozzle divergence angle was 10° or less, the base pressure increased abnormally at low Mach numbers, owing to the flow separation occurring inside the nozzle, as shown in Figure 10. However, in all cases, the base pressure was lower than that in the no-jet condition, and the base pressure ratio was less than 1.0. Therefore, in conclusion, the nozzle flow in the over-expanded condition always acts as a base drag regardless of the injection condition, and the base drag is higher than that in the no-jet condition.

3.3. Base Drag and Flow in the Slightly under-Expanded Condition (29.8 ≤ NPR ≤ 148.0)

Figure 14 and Figure 15 show the results of the flow field and base pressure when the nozzle flow was injected under slightly under-expanded conditions, respectively. When the nozzle flow was under-expanded, the nozzle exit pressure was higher than the back pressure, which caused the nozzle flow to expand beyond the body diameter. Therefore, a shock occurred immediately before the freestream flow reached the base and the overall base pressure was determined by the strength of the shock [25]. As shown in Figure 14a,b, when the nozzle divergence angles were the same, the hot gas condition expanded more at the nozzle exit than the cold gas condition, and the deflection angle increased accordingly. As the deflection angle increased, the strength of the shock increased, thus leading to a larger base pressure, as shown in Figure 15. In addition, when the nozzle divergence angle decreased, the expansion of the nozzle flow decreased; thus, the deflection angle and shock strength decreased. Consequently, as the nozzle divergence angle decreased, the base pressure also decreased, as shown in Figure 15. In the cold gas condition of Figure 14b, despite the under-expanded condition, the expansion of the nozzle flow was small because of the zero nozzle divergence angle. Therefore, the deflection angle became negative, thus causing the expansion fan at the base corner.

Figure 16 and Figure 17 show the flow field and base pressure results for the slightly under-expanded condition when the freestream Mach number is 3.0. As shown in Figure 16a, when the freestream Mach number increased, the pressure in the post-shock region increased, owing to an increase in shock strength. Because the back pressure increased in terms of the nozzle flow, the size of the jet boundary decreased; essentially, the nozzle flow expanded less, and the deflection angle of the flow also decreased. In the case of the cold gas conditions in Figure 16b, the deflection angle became negative, and an expansion fan was generated, similar to that at an inflow Mach number of 1.5. Eventually, as shown in Figure 17, the trend according to the nozzle divergence angle and gas temperature was the same as the freestream Mach number of 1.5; however, as the freestream Mach number increased, the expansion of the nozzle flow, deflection angle, and base pressure all decreased.

Figure 18 shows the base pressure results according to the freestream Mach number for each analysis condition when the nozzle flow was slightly under-expanded. Similar to the results for the over-expanded condition in Figure 13, the overall base pressure tended to decrease as the freestream Mach number increased, the NPR decreased, and the base pressure in the hot gas condition was higher than that in the cold gas condition. In all cases of the over-expanded condition in Figure 13, the base pressure was small compared with the no-jet condition, and the base pressure ratio was less than 1.0; however, in the slightly under-expanded condition, the base pressure was generally higher than that in the no-jet condition, where the base drag was expected to decrease. In addition, when the base pressure ratio was 1.0 or more, the nozzle flow acted as thrust. As the nozzle divergence angle and NPR increased, the freestream Mach number at which the base pressure ratio exceeds 1.0 increased. This was because the base pressure increased as the deflection angle increased when the nozzle divergence angle and NPR increased, even when the Mach number increased.

3.4. Base Drag and Flow in the Highly Under-Expanded Condition (246.7 ≤ NPR ≤ 690.8)

Figure 19 shows the results of the base pressure according to the freestream Mach number for each analysis condition when the nozzle flow was highly under-expanded. Similar to the results in Section 3.2 and Section 3.3, the base pressure tended to decrease as the NPR decreased, and the base pressure in hot gas conditions was higher than that in cold gas. In addition, most of the base pressure ratios were above 1.0, except for the cold gas with an NPR of 246.7 and a nozzle divergence angle; this implies that the nozzle flow generally acts as a thrust in the case of highly under-expanded conditions. However, unlike the over-expanded and low under-expanded conditions, the base pressure increased as the freestream Mach number increased for a specific nozzle divergence angle and NPR. This tendency was more pronounced as the nozzle divergence angle and NPR increased. When the nozzle divergence angle was 20°, the base pressure increased as the freestream Mach number increased, even under cold gas conditions.

Figure 20 and Figure 21 are the flow field results for the highly under-expanded condition when the freestream Mach numbers are 2.0 and 3.0, respectively. When the nozzle flow was highly under-expanded, the nozzle exit pressure was much greater than the freestream static pressure; therefore, the flow at the nozzle exit expanded significantly.

Consequently, the deflection angle increased compared with the slightly under-expanded condition for the same freestream and nozzle injection conditions. However, in the case of a nozzle divergence angle of 20°, the position of the shock moved upstream, owing to the excessive expansion of the nozzle flow. As the inflow Mach number increased, the shock was pushed downstream; however, it was still located in the body because of the high NPR. Eventually, the base pressure became equal to the post-shock pressure. As the freestream Mach number increased, the base pressure increased because the strength of the shock wave increased, although the deflection angle decreased. However, when the shock was formed at the corner of the base, owing to a decrease in the expansion of the nozzle flow, as in the nozzle divergence angle of 0°, the base pressure decreased as the freestream Mach number increased, similar to the slightly under-expanded condition.

3.5. Simulation Results of Base Drag Coefficients

Based on the analysis results according to the nozzle expansion conditions in Section 3.2, Section 3.3 and Section 3.4, the base drag coefficients in cold and hot gas conditions were compared. The base drag coefficient was calculated based on the base area of the model using Equation (7) [30].

$$C_{D_B}=-\left(\frac{2}{\gamma M_{\infty }^2}\right)\left(\frac{P_B}{P_{\infty }}-1\right)\left[{\left(\frac{d_b}{d_r}\right)}^2-{\left(\frac{d_e}{d_r}\right)}^2\right]$$

(7)

where $d_r$ is the diameter of the reference body, $d_b$ the diameter of the base, and $d_e$ the diameter of the nozzle exit, $P_B$ is the averaged base pressure, $P_{\infty }$ is the freestream static pressure, $\gamma$ is the specific heat ratio, and $M_{\infty }$ is the freestream Mach number.

Figure 22 shows the base drag coefficient according to the freestream Mach number for each NPR and nozzle expansion angle. According to Figure 22, regardless of the freestream Mach number, nozzle divergence angle, and NPR, the base drag coefficient under the hot-gas condition was always smaller than that under the cold-gas condition, which is consistent with the base pressure results in Section 3.2, Section 3.3 and Section 3.4. In addition, for all nozzle divergence angles and freestream Mach numbers, as the NPR increased, the difference in the base drag coefficient between the cold- and hot-gas conditions generally increased. Regardless of the nozzle divergence angle and NPR, as the freestream Mach number increased, the base drag coefficient tended to converge to zero under both cold and hot gas conditions, and the coefficient value approached zero as the nozzle divergence angle decreased. For the same nozzle divergence angle, the base drag coefficient decreased as the NPR increased under both hot-gas and cold-gas conditions. In the highly under-expanded condition, where the nozzle expansion ratio increased to 10 or more, the base drag coefficient became negative and acted as thrust rather than drag. As the nozzle divergence angle increased and the temperature of the nozzle flow increased, the base drag coefficient became negative, even at a low NPR.

Figure 23 presents the results of the base drag coefficient according to the freestream Mach number for each nozzle divergence angle and nozzle expansion condition. As shown in Figure 22, the base drag coefficient under the hot gas condition was lower than that under the cold gas condition, regardless of the freestream Mach number, nozzle divergence angle, and NPR. For the same NPR, the base drag coefficient according to the nozzle divergence angle exhibited an irregular tendency; however, overall, the base drag coefficient increased as the nozzle divergence angle decreased in both the over-expanded and under-expanded conditions because the base pressure decreases as the nozzle divergence angle decreases. In the hot gas condition, the base drag coefficient was clearly distinguished as positive in the over-expanded condition and negative in the highly under-expanded condition. In addition, the coefficient had a negative value even for a small nozzle expansion angle in the slightly under-expanded condition. In summary, when the freestream and nozzle expansion conditions are the same, the base drag coefficient in the hot gas condition is always lower than that in the cold gas condition. Regardless of the gas temperature and nozzle divergence angle, the base drag coefficient exhibits a positive value in the over-expanded condition and a negative value in the highly under-expanded condition. In the slightly under-expanded condition, the condition in which the sign changes depends on the NPR and nozzle divergence angle. However, in the hot gas condition, the NPR, along with the nozzle divergence angle where the base drag coefficient changes to a negative, both decrease.

The ratios of the base drag coefficients were compared to observe whether there was a correlation between the base drag coefficients under hot and cold gas conditions. In Figure 24, the base drag coefficient ratios of the cold and hot gas conditions for each nozzle divergence angle are shown for each nozzle expansion condition. In this study, because two or more NPRs were applied for each nozzle expansion condition, the graph in Figure 24 shows the average value according to the NPR, and the maximum and minimum values under each nozzle expansion condition are displayed as error bars. The base drag coefficient ratio was calculated using Equation (8).

$$C_{Db} ratio=\frac{C_{Db,Hot}}{C_{Db,Cold}}$$

(8)

As shown in Figure 24, the base drag coefficient ratio tended to differ depending on the nozzle expansion conditions. In the over-expanded condition shown in Figure 24a, the overall drag coefficient ratio tended to increase as the Mach number increased. Positive values less than 1.0 were observed in all cases, thus implying that the drag in the cold gas condition is always greater than that in the hot gas condition when the nozzle flow is over-expanded. In the case of the slightly under-expanded condition at a nozzle divergence angle of 15° and the case of the highly under-expanded condition at a nozzle divergence angle of 0 and 10°, as shown in Figure 24b,c, the base drag coefficient ratio increased and decreased rapidly, owing to the change characteristics of the base drag coefficient ratio according to the NPR for each nozzle divergence angle, respectively.

Figure 25 shows the base drag coefficient according to the NPR for the freestream Mach number at different nozzle divergence angles. In the over-expanded condition, the base drag coefficient increased as the NPR increased; however, in the under-expanded condition, the coefficient tended to decrease continuously as the NPR increased. In addition, the base drag coefficient increased with the nozzle divergence angle for the same freestream Mach number when the nozzle flow was under-expanded. Accordingly, the point where the coefficient changed from negative to positive was located in the slightly under-expanded condition region when the nozzle divergence angle was greater than 15°; however, the point moved to the highly under-expanded condition region as the nozzle divergence angle decreased. The base drag coefficient ratio defined in this study is the ratio of the coefficient in the hot gas condition to that in the cold gas condition. As the base drag coefficient under the cold gas condition approached zero, the base drag coefficient ratio increased rapidly. Thus, the base drag coefficient ratio increased or decreased sharply at a divergence angle of 15° in the slightly under-expanded condition and at a divergence angle of 0° in the highly under-expanded condition, as shown in Figure 24.

The purpose of this study was to understand the relationship between the base drag and cold and hot gas conditions. However, deriving a correlation as a specific constant value is challenging because the base drag coefficient ratio between the cold and hot gas conditions tends to vary continuously depending on the nozzle divergence angle and nozzle expansion conditions. Therefore, regression analysis was used for each nozzle expansion condition to derive a formula for calculating the base drag coefficient ratio according to the nozzle divergence angle and freestream Mach number, and the sensitivities of the nozzle divergence angle and inflow Mach number were analyzed. The mean values of the base drag coefficient ratio, presented in Figure 24 were used for the regression analysis.

Table 5. Formulas for ${\mathit{C}}_{\mathit{D}\mathit{b}}$ ratio prediction.

Nozzle Expansion Condition	${\mathbf{C}}_{\mathbf{D}\mathbf{b}}\mathbf{Ratio}$ $\mathbf{(}\mathbf{x}:\mathbf{nozzle}\mathbf{DiverGence}\mathbf{Rangle}\left({\mathsf{\theta }}_{\mathbf{e}}\right)[\mathbf{rad}]$ $, \mathbf{y}:\mathbf{Mach}\mathbf{Number}$)
Over-expanded	$0.851+0.161x-0.015y$
Slightly under-expanded	$5.608+5.620x-1.890y$
Highly under-expanded	$-14.120+29.99x+2.803y$

presents the formula for calculating the base drag coefficient ratio for each nozzle expansion condition derived via regression analysis, and Figure 26 shows the data distribution and regression analysis results. Under the over-expanded and slightly under-expanded conditions, the base drag coefficient ratio increased as the nozzle divergence angle increased, and the freestream Mach number decreased. By contrast, under the highly under-expanded condition, the base drag coefficient ratio increased as the nozzle divergence angle and freestream Mach number increased. In addition, in the over-expanded condition, the sensitivity of the nozzle divergence angle and freestream Mach number was small because the base drag coefficient ratio was small, whereas in the under-expanded condition, the absolute value of the coefficient ratio increased, thus leading to a higher sensitivity of the nozzle divergence angle and freestream Mach number. Thus, to predict the base drag with hot gas injection from the base drag prediction formula derived using cold gas, different correction factors are required according to the nozzle expansion conditions. Additionally, the correction constant is considered to increase as the NPR increases.

4. Conclusions

In this study, the effect on the base drag when hot gas is injected from the nozzle of an aircraft in a supersonic flow field was analyzed. An appropriate NPR was selected to compare the flow according to the nozzle expansion conditions, and the flow field and base pressure according to the injection gas temperature were observed by varying the nozzle divergence angle and freestream Mach number using CFD analysis. The analysis revealed that the base pressure under hot gas conditions was higher than that under cold gas conditions, regardless of the nozzle injection conditions or nozzle geometry. When the nozzle flow was over-expanded, an expansion fan was generated at the base corner and the base pressure depended on the strength and angle of the expansion fan. When the nozzle flow was under-expanded, shock occurred on the body surface as the nozzle flow expanded beyond the diameter of the aircraft, and the base pressure depended on the strength and location of the shock. However, the base pressure was generally dependent on the deflection angle of the flow, and the deflection angle and base pressure tended to increase as the nozzle divergence angle increased, NPR increased, and freestream Mach number decreased, regardless of the nozzle expansion conditions. However, when the cold gas was injected under over-expanded conditions from a nozzle with a divergence angle of 10° or less, and when the cold and hot gases were injected under highly under-expanded conditions from a nozzle with a divergence angle of 15° or more, the base pressure increased, owing to the separation inside the nozzle or the position of the surface shock. The base drag was calculated based on the base pressure according to the nozzle expansion conditions, and the overall tendency of the base drag was identical to that of the base pressure. In the over-expanded condition, the nozzle flow acted as a drag force, whereas in the highly under-expanded condition, it acted as a thrust force. In the slightly under-expanded condition, the condition in which the sign changes was found to depend on the NPR and nozzle divergence angle. However, in the hot gas condition, the NPR and nozzle divergence angle where the base drag coefficient changes to a negative were found to decrease. A comparison of the base drag ratios between the cold and hot gases for each nozzle expansion condition revealed an irregular tendency. In the over-expanded condition, the base drag coefficient ratio was less than 1.0; however, the value increased rapidly in the under-expanded condition, thus implying that the base drag with hot gas was much larger than that with cold gas. The derivation of the formula for predicting the base drag ratio for each nozzle expansion condition via regression analysis confirmed that the effects of the nozzle divergence angle and freestream Mach number are different in the over- and under-expanded conditions, and the sensitivity of the variable is higher in the under-expanded condition. These results can be used to determine the tendency when predicting the base drag under the hot gas conditions using the base-drag prediction method with cold gas.