Abstract
Currently, aerodynamic environment prediction research into scramjetpropelled vehicles characterized by NACA0012 under hypersonic conditions is relatively sparse. Twodimensional external flow field models are established, and then through validation tests, we perform a systematic investigation between simulation parameters and prediction accuracy, and an effective aerodynamic environment prediction simulation scheme under hypersonic conditions is proposed. Unlike under incompressible conditions, the maximum accuracy decline could be attributed to the inappropriate choice of the sharp trailing edge modeling method, but the definition formula is still preferred. In particular, for the two modeling data point sources, Airfoil tools and NACA4, the numerical performance of the latter is better than the former, and the calculation accuracy negatively correlates with the number of data points offered by both of them. Moreover, for the mesh cells near the shock, the cell Reynolds number and aspect ratio values should be no smaller than 16 and not exceed 380, respectively, and the recommended values for the far field distance, the turbulence model and flux type are 16L, SpalartAllmaras, and ROE flux type. Under hypersonic conditions, the aerodynamic environment characterized by NACA0012 predicts a maximum temperature of approximately 1856.85 °C, with an average temperature change rate of 77 °C/s. Meanwhile, the top sound pressure level and the vibration acceleration could reach up to 145 dB and 182 g, respectively.
Highlights
 The suitable values of cell Reynolds number and aspect ratio for NACA0012 differ from those for a wedge section.
 The proper simulation parameters for NACA0012 at hypersonic speed differs from that at a low speed.
 The sharp trailing edge is preferred.
 The utilization of advanced materials plays a pivotal role in enhancing structural reliability.
1. Introduction
Adapting to the environment becomes crucial as the hypersonic vehicle encounters increasingly harsh flight conditions. NASA has identified environmental failure as the leading cause of previous vehicle launch crashes, emphasizing the criticality of environment testing. Therefore, it is imperative to first predict the aerodynamic environment experienced by the hypersonic vehicle and subsequently design its structure accordingly. By conducting comprehensive environment testing, potential issues can be identified and addressed to enhance flights’ reliability and adaptability in diverse environments. The predicted results of the aerodynamic environment serve as a fundamental basis for both vehicle design and subsequent environment testing. The accurate prediction of the aerodynamic environment is crucial for the design and optimization of the hypersonic vehicle. The main types of hypersonic vehicle include the groundtoorbit reentry, the loworbit reentry and the scramjetpropelled [1] and the reusable reentry vehicle is the leading research object [2][7], while the prediction research for the scramjetpropelled vehicle is relatively sparse. Hence, we focus our research on the scramjetpropelled hypersonic vehicle.
Analyzing the flight trajectory and determining the initial atmospheric conditions is crucial for accurate environmental forecasting. As shown in Fig. 1 [8], the scramjetpropelled vehicle is transported by a large aircraft carrier to the designated location. Subsequently, the solid rocket engine propels the vehicle to achieve Mach 0.8 into the climbing phase, which encompasses the ejection, the subcombustion, and the supercombustion phases, responsible for rapidly increasing the flight altitude and speed and lasts about 34 seconds. Once the vehicle attains the desired altitude and Mach number (30 km and 6.5), it transitions into the cruise phase. Throughout this period, the vehicle will maintain a consistent altitude and velocity, constituting approximately 90 % of its total range. After reaching the designated airspace, it would proceed to initiate the attack phase with precision and accuracy against the intended target. According to Ref. [9], the external flow field could be categorized as either incompressible or compressible based on its velocity. Furthermore, the compressible flow is considered transonic when the Mach number ranges from 0.6 to 1. The flow field is referred to as supersonic when the Mach number falls between 1 and 3, while it is categorized as high supersonic when the Mach number ranges from 3 to 5. Finally, if the Mach number exceeds 5, the flow field is classified as hypersonic. The flight path of the scramjetpropelled vehicle typically encompasses all the aforementioned compressible conditions. Confirmation of specific environmental prediction parameters is required for different flow conditions. Therefore, it is imperative to analyze the simulation parameters' effect on calculation accuracy to ascertain the optimal simulation configuration. Based on the determined optimal simulation configurations, we could more accurately forecast the aerodynamic conditions of the characteristic position of the scramjetpropelled vehicle during flight, enabling enhanced vehicle optimization and thermal protection design.
Fig. 1The typical flight trajectory of the scramjetpropelled hypersonic vehicle
The NACA0012 airfoil is widely employed for investigating the characteristics of the aerodynamic environment [10][14], and the analysis of Refs. [15], [16] reveals a significant correlation between numerical accuracy and the trailing edge shape. NACA0012 has sharp and blunt two trailing edge shapes. For the sharp trailing edge shape, Refs. [17], [18] discusses the external flow field properties of dimpled and square dimpled NACA0012s. The studied Mach numbers are 1.7, 2.2, and 2.7. SST komega and SpalartAllmaras (SA) turbulence models are used. The results show there is a positive correlation between Mach number and aerodynamic condition. Based on different ranges of upper and lower surface temperatures, Ref. [19] extends the previous investigations, and the force coefficients are evaluated. Under the conditions of pitching and plunging NACA0012s, Ref. [20] uses SST komega and SA at very low speed to study the thermal effect on force coefficients around NACA0012. The conclusion indicates that the lift coefficient is increased, and the drag coefficient is decreased due to the temperature variation between extrados and intrados of the airfoil. Similar research is investigated by Ref. [21], and a spectral analysis demonstrates that as the surface temperature increases, the force coefficient amplitudes decrease. Based on the 0.3 Mach number and SA model, Ref. [22] discusses water droplets impact characteristics on NACA0012 type turbine, and the calculation results show the matching degree with the experimental results is good, and the numerical approach is acceptable. With an asymmetric heating surface, Ref. [23] selects the kepsilon turbulence model and Mach number 0.7 to evaluate the NACA0012 antiicing performance. The research conclusions indicate that the aerodynamic performance could be promoted through the extended heating surface of the suction surface. The impacts of Mach number and ambient temperature on the icing shape and icing growth rate are investigated by Ref. [24], which adopts the SA turbulence model. Regarding the blunt trailing edge, existing literature primarily focuses on noise generation and validation of associated prediction methods, while limited attention has been given to investigating the aerodynamic prediction correlation [25], [26].
In summary, the existing research primarily centers on examining the impact of geometrical alterations on the properties of the external flow field based on a trailing edge shape [27]. There is limited investigation into how the shape of the NACA0012 trailing edge affects the precision of aerodynamic predictions, with only a few studies addressing this aspect [28], [29]. But these studies are all focus on incompressible conditions. During the typical trajectory of hypersonic vehicle shown in Fig. 1, over 90 % of the flight path occurs under hypersonic conditions with a typical Mach number exceeding 5, which is the predominant aerodynamic environment encountered by scramjetpropelled vehicle. However, the maximum Mach number achieved in the aforementioned simulations is below 3, indicating potential deviations from established research conclusions regarding simulation parameter selection for hypersonic conditions. To date, we have not found reliable literature analyzing the influence of trailing edge shape under hypersonic conditions. Meanwhile, the existing literature lacks comprehensive details on the various methodologies employed to establish the NACA0012 model, and the associated CFD simulations commonly incorporate a fixed far field distance and turbulence model. Furthermore, the literature reviewed thus far has paid little attention to the appropriate values of crucial grid parameters such as cell Reynolds number [30] and aspect ratio [31] near the shock, which are characterized by NACA0012. Reliable references addressing the impact of these parameters on numerical accuracy are currently lacking. In summary, there is an insufficient investigation on the selection criteria for these key parameters under hypersonic conditions.
In this study, we select the NACA0012 airfoil as the characterized object to generate computational external flow fields. Our objective is to perform simulations to investigate the influence of trailing edge shape, modeling method, far field distance, and turbulence model on prediction accuracy under hypersonic conditions. By comparing numerical results with wind tunnel data, we establish a correlation between prediction accuracy and simulation parameters and identify an optimal simulation configuration for future reference. This research contributes to the field of environmental prediction for hypersonic flight vehicles. Furthermore, based on the identified optimal simulation configuration mentioned above, we are able to predict environmental conditions along the trajectory of hypersonic vehicle during flight. These predictions provide valuable information for vehicle optimization and thermal protection design.
2. Simulation fundamentals
To look for a suitable simulation scheme for the aerodynamic environment prediction, the appropriate parameter values of the grid strategy and numerical method should be identified through validation tests. $P$ is the local static pressure, and ${P}_{t}$ is the static pressure of the free stream. $T$ is the local static temperature, and ${T}_{t}$ is the static temperature of free stream. $U$ is the local velocity, and ${U}_{t}$ is the velocity of free stream. We adopt the wind tunnel data of $P/{P}_{t}$, $T/{T}_{t}$ and $U/{U}_{t}$ from Ref. [13] as the reference data, where the location range of $P/{P}_{t}$ and $T/{T}_{t}$ is $x/L\in $ [–0.007 0] and the location range of $U/{U}_{t}$ is $y/L\in $[0.01 0.1] at $x/L$ = 0.95. The initial values are as follows: Reynolds number (${R}_{e}$) is 10e6, the Mach number (${M}_{a}$) is 10, ${P}_{t}$ is 576 Pa, ${T}_{t}$ is 81.2 K, and the wall temperature of airfoil (${T}_{W}$) is 311 K. Ansys ICEM CFD and Ansys fluent are chosen as the meshing tool and the CFD simulation tool, respectively. The calculation process is 16core parallel, and the precision is double.
2.1. Grid strategy
2.1.1. NACA0012 computational domains
Firstly, we need to choose an appropriate modeling method to design NACA0012 model, which is the basis for establishing computational domain. There exists three NACA0012 modeling methods: NACA4 digital generator, Airfoil tools and definition formula. NACA4 digital generator offers 200 modeling data points and provides the function of the close trailing edge. Therefore, it could be used to design the sharp trailing edge NACA0012 model. Airfoil tools offers 132 modeling data points. The established trailing edge based on this method is not closed and requires a manual connection. So, this method could design the blunt trailing edge. Eq. (1) is the Definition formula, in which the value of $x$ represents the point on the $X$axis and the value of y corresponds to the point on the $Y$axis. The 200 and 132 $X$axis data points offered by NACA4 and Airfoil tools could be substituted into the definition formula to calculate the related Yaxis data points to build the NACA0012 model. In summary, the NACA0012 has two trailing edge shapes, three modeling methods, two data point sources, and different numbers of modeling data points. Therefore, we establish six NACA0012 airfoils, shown in Table 1, to investigate the selection principles of NACA0012 modeling method parameters. Fig. 2 demonstrates the related NACA0012 models, and the airfoil characteristic length ($L$) is 1 m. There exist significant positional differences between different modeling data points:
Table 1The designed six NACA0012s and the corresponding modeling methods
Trailing edge shape  Modeling method  
One blunt trailing edge  Airfoil tools (132 points)  
Five sharp trailing edges  Naca4 (200 points)  
Definition formula  Adopts 132 points from Airfoil tools  
Definition formula  We double 132 points to 264 points, then substitute them into definition formula  
Definition formula  Adopts 200 points from NACA4  
Definition formula  We double 200 points to 400 points, then substitute them into definition formula 
After establishing the NACA0012 models, we need to select the proper far field distance to establish the computational domain of the NACA0012 airfoil to perform numerical simulation. Ansys suggests the far field distance should be 1220 times $L$ [32]. However, Ansys only provides a suggested range without providing specific values or analyzing the impact of different far field distances on numerical calculation accuracy. By selecting far field distances of 12$L$, 16$L$, and 20$L$ and conducting validation tests using wind tunnel data, Ref. [29] examines the correlation between the accuracy of the far field distance in the incompressible external flow field. The findings suggest a discernible relationship between the far field distance and numerical accuracy in the incompressible external flow field. In this study, based on the same research object NACA0012, we still select 12$L$/16$L$/20$L$ far field distances to create computational domains. Our investigation focuses on examining the relationship between these distances and numerical accuracy under hypersonic conditions, while also comparing them to incompressible external flow fields [33]. Fig. 3 demonstrates the established two trailing edge types of computational domains, where the black dot is the coordinate origin. The INLET and OUTLET serve as the input and output boundaries, respectively, employing the Pressure far field condition. The WALL boundary, highlighted in red, is characterized by the noslip, isothermal wall condition. The airfoil surfaces serve as the primary source of the turbulence and the mean vorticity, and the accuracy of numerical predictions for turbulence in wallbounded flows is heavily influenced by the nearwall meshing. Therefore, finer meshing areas are created by further dividing blocks in close proximity to the airfoil surface. As illustrated in Fig. 3, the cyan lines indicate the grid division. For the sharp trailing edge, the block at the end is folded, whereas it remains in place for the blunt trailing edge.
Fig. 2Six NACA0012 models
Fig. 3The two types of computational domains
a) Computational domain of the sharp trailing edge and the grid division
b) Computational domain of the blunt trailing edge and the grid division
2.1.2. Grid parameters
The innermost layer of the near wall is the viscous sublayer, where viscosity has the dominant role in the heat and momentum. The first layer cells of the boundary layer are proposed to exist within the viscous sublayer, with the height value (${y}_{H}$) being directly correlated to ${y}^{+}$ and the calculation process is displayed in Fig. 4. The calculations of ${R}_{e}$, ${U}_{t}$, air speed (${C}_{air}$), friction coefficient (${C}_{f}$), shear stress (${\tau}_{W}$), friction velocity (${\mu}_{t}$), and the distance from the wall to the centroid of the wall adjacent cells (${y}_{p}$) are shown from Eqs. (28):
where ${T}_{t}$ is 81.2 K and ${P}_{t}$ is 576 Pa, the flow density ($\rho $) is about 0.0247 kg/m^{3}. ${C}_{air}$ is 180.6 m/s and ${M}_{a}$ is 10. Hence, ${U}_{t}$ is 1806$$m/s. ${R}_{e}$ is 10e06, $\rho $, $L$ and ${U}_{t}$ are substituted into Eq. (2) and flow viscosity ($\mu $) is about 4.46082 Pa·s. Then we could solve ${y}_{p}$ according to Eqs. (58). At last, ${y}_{H}$ is calculated according to Eq. (9):
Fig. 4The calculation process of yH
Since the empirical formula (${C}_{f}$) is used in the above calculation process [34], ${y}_{H}$ value is an estimate, and it needs to be tested repeatedly through numerical simulation to ensure that the maximum value of ${y}^{+}$ at airfoil surface is less than 1 [35]. The initial value of ${y}^{+}$ is 1, and ${y}_{H}$ is 4.6e5 m could be estimated according to the above Equations. However, tests show that the maximum value of ${y}^{+}$ at the airfoil surface exceeds 1 during simulations, which indicates that the initial value of ${y}^{+}$ is inappropriate. Through repeated testing, ${y}^{+}$ is 0.3 and ${y}_{H}$ is 1.4e5 m could meet the condition.
Secondly, it is crucial to evaluate the quality of the mesh utilized in a simulation, encompassing an examination of diverse metrics such as aspect ratio and determinant. In particular, meticulous attention must be paid to the aspect ratio since an excessively small value of ${y}_{H}$ could potentially lead to a large aspect ratio value. This scenario may result in floatingpoint overflow or calculation divergence, ultimately leading to simulation failure. In this study, we validate the stability of numerical simulations by conducting CFD tests to determine the appropriate aspect ratio and determinant values. At 12$L$/16$L$/20$L$ three far field distances, the maximum aspect ratios and the minimum determinants for the sharp and blunt trailing edge shapes are (2100 0.84), (3310 0.881), (4460 0.873) and (2760 0.894), (3680 0.886), (4770 0.827) respectively. These findings ensure the robustness of our numerical simulations.
Fig. 5Mean error ratios of P/Pt, T/Tt, U/Ut at three far field distances under three grid levels (NACA4)
Thirdly, grid independency should be performed to confirm the appropriate total mesh cells number. Three modeling methods are applied to establish three types NACA0012 models: sharp trailing edge (based on NACA4), sharp trailing edge (based on definition formula), and blunt trailing edge (based on Airfoil tools) and we adopt the sharp trailing edge designed by NACA4, combined with the SST komega model and ROE flux type as an example. The numerical results of $P/{P}_{t}$, $T/{T}_{t}$, $U/{U}_{t}$ at location ranges and related error ratios with wind tunnel data are shown in Fig. 5. At 12$L$ far field distance, the average error ratios of $P/{P}_{t}$, $T/{T}_{t}$, and $U/{U}_{t}$ under three grid levels at all calculating locations are (3.094 % 5.608 % 2.188 %), (2.277 % 3.964 % 1.034 %), and (2.248 % 3.936 % 1.025 %). Hence, the total mean error ratios of ($P/{P}_{t}$$T/{T}_{t}$$U/{U}_{t}$) for the three grid levels are 3.630 %, 2.425 %, and 2.403 %. Similarly, the average error ratios for three grid levels under 16L are (11.570 % 5.638 % 3.038 %), (6.229 % 3.251 % 2.356 %), and (6.212 % 3.218 % 2.338 %) and the corresponding total mean error ratios are 6.749 %, 3.945 %, and 3.923 %. The average error ratios under 20$L$ are (10.198 % 5.949 % 2.835 %), (3.151 % 4.140 % 2.277 %), and (3.141 % 4.106 % 2.249 %) and the corresponding total mean error ratios are 6.327 %, 3.189 %, and 3.165 %. As depicted in Fig. 5, at three far field distance, with the mesh number increases from 608,000 to 850,000, the numerical error ratio hardly changes, and the other two types present similar grid independency performances as shown in Figs. 6 and 7. Therefore, the mesh with 608,000 could meet the requirement of grid independency. The mesh views of the two trailing edge shapes are depicted in Figs. 89.
Fig. 6Mean error ratios of P/Pt, T/Tt, U/Ut at three far field distances under three grid levels (Airfoil tools)
Fig. 7Mean error ratios of P/Pt, T/Tt, U/Ut at three far field distances under three grid levels (Definition formula)
Fig. 8The mesh views of the sharp trailing edge
Fig. 9The mesh views of the blunt trailing edge
2.2. Numerical method
2.2.1. Turbulence model
When working with transonic fluids, it is necessary to manage compression and heat transfer. This necessitates solving control equations such as mass continuity, momentum (the NS equation), and energy. Since turbulent flow exists, additional transport equations must be solved. Turbulence is defined as unsteady random motion in fluids with medium to high Reynolds numbers, as described by the NS equation. However, direct numerical simulation (DNS) calculations can be timeconsuming, necessitating the averaging of the NS equation to reduce turbulence components. The Reynolds Averaged NavierStokes (RANS) model is widely used to average the turbulence fluctuation time term. This method employs turbulent viscosity to calculate Reynolds stress and solve the RANS equations. The kepsilon, SST komega, and SpalartAllmaras (SA) turbulence models are widely accepted and relatively accurate for most numerical simulation applications [36]. However, the kepsilon model exhibits limited sensitivity to adverse pressure gradients and boundary layer separation, resulting in delayed predictions and separation. Consequently, it is unsuitable for investigating the aerodynamic external flow field in this paper. The komega model demonstrates superior performance over the kepsilon model in predicting adverse pressure gradients and boundary layer flow, showcasing its enhanced capabilities. Furthermore, the SST komega model effectively addresses the sensitivity issue of the original komega model to freestream conditions, thereby enhancing its applicability [37]. Moreover, the SpalartAllmaras (SA) turbulence model is specifically tailored for aerospace applications involving wallbounded flows and exhibits exceptional predictive capabilities for adverse pressure gradient boundary layers [38]. In conclusion, we employ SA and SST komega models.
Regarding the INLET boundary, when utilizing the SST komega, we employ the intensity and viscosity ratio turbulence method with values of 1 % and 1. In case of adopting the SA model, the turbulent viscosity ratio method is chosen and set its value to 1. The same actions are done for the OUTLET boundary. Furthermore, careful consideration should be given to selecting an appropriate upwind order for modifying turbulent viscosity in relation to the SA model. According to Ref. [30], there exists a situation where the performance of the first order is better than that of the second order. So, we execute a numerical comparison between these two upwind schemes. We still take the sharp trailing edge designed by NACA4, combined with ROE flux type as an example to carry out the analysis. The numerical results of $P/{P}_{t}$, $T/{T}_{t}$, and $U/{U}_{t}$ at location ranges and related error ratios with wind tunnel data are shown in Tables 24. For the first order, under three far field distances, the mean error ratios of ($P/{P}_{t}$, $T/{T}_{t}$ and $U/{U}_{t}$) at all calculating locations are (9.18 % 4.96 % 3.02 %), (9.38 % 7.71 % 2.06 %), and (7.42 % 7.11 % 2.72 %). The corresponding total mean error ratios of ($P/{P}_{t}$$T/{T}_{t}$$U/{U}_{t}$) are 5.72 %, 6.38 %, and 5.75 %. Similarly, for the second order, the mean error ratios under three far field distances are (4.42 % 2.16 % 1.90 %), (5.33 % 5.24 % 1.61 %), and (4.69 % 4.28 % 1.73 %), and the total mean error ratios are 2.83 %, 4.06 %, and 3.57 %. Hence, the second order upwind is chosen.
Table 2Numerical results comparison of P/Pt between the firstorder upwind and second–order upwind of the modified turbulent viscosity
Type of upwind order  $x$/$L$ locations (m) and $P$/${P}_{t}$ wind tunnel data ($P$/${P}_{t}$ numerical results and error ratios)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
103.63  114.55  117.87  119.40  121.74  122.10  123.39  123.83  
Firstorder  12$L$  81.68  106.49  114.80  110.43  110.71  112.04  112.79  112.42 
21.19 %  7.04 %  2.60 %  7.51 %  9.06 %  8.24 %  8.59 %  9.22 %  
16$L$  63.99  101.22  118.76  117.55  120.54  125.16  134.83  136.33  
38.25 %  11.64 %  0.76 %  1.55 %  0.99 %  2.51 %  9.27 %  10.10 %  
20$L$  107.09  106.96  105.16  105.66  106.74  110.22  118.13  122.88  
3.34 %  6.63 %  10.79 %  11.5 %  12.32 %  9.73 %  4.27 %  0.77 %  
Secondorder  12$L$  93.15  112.16  117.71  119.13  122.20  125.88  134.72  136.43 
10.11 %  2.09 %  0.14 %  0.23 %  0.38 %  3.10 %  9.18 %  10.17 %  
16$L$  106.92  112.50  115.21  115.48  115.45  115.19  113.54  107.39  
3.18 %  1.79 %  2.26 %  3.28 %  5.17 %  5.66 %  7.99 %  13.28 %  
20$L$  102.75  106.52  113.59  117.72  122.04  126.47  136.72  136.17  
0.85 %  7.01 %  3.63 %  1.41 %  0.25 %  3.58 %  10.8 %  9.96 % 
Table 3Numerical results comparison of T/Tt between the firstorder upwind and secondorder upwind of the Modified turbulent viscosity
Type of upwind order  $x$/$L$ locations (m) and $T$/${T}_{t}$ wind tunnel data ($T$/${T}_{t}$_{}numerical results and error ratios)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
18.15  20.19  20.28  20.43  20.52  20.66  21.04  21.33  
Firstorder  12$L$  15.34  18.71  20.36  20.78  20.86  20.79  20.39  19.33 
15.50 %  7.31 %  0.38 %  1.70 %  1.68 %  0.65 %  3.11 %  9.35 %  
16$L$  12.08  17.15  20.02  20.13  20.30  20.53  20.44  20.07  
33.47 %  15.03 %  1.26 %  1.49 %  1.07 %  0.62 %  2.83 %  5.91 %  
20$L$  19.59  19.92  19.86  19.62  19.58  20.02  20.44  14.71  
7.96 %  1.32 %  2.09 %  3.96 %  4.57 %  3.09 %  2.87 %  31.04 %  
Secondorder  12$L$  17.55  19.13  20.06  20.17  20.30  20.55  20.64  20.71 
3.28 %  5.24 %  1.08 %  1.25 %  1.08 %  0.52 %  1.89 %  2.92 %  
16$L$  19.57  20.03  20.26  20.33  20.31  20.47  20.10  15.72  
7.82 %  0.79 %  0.07 %  0.51 %  1.03 %  0.92 %  4.46 %  26.3 %  
20$L$  20.47  20.33  20.20  20.06  20.08  20.11  20.29  19.16  
12.79 %  0.68 %  0.39 %  1.79 %  2.17 %  2.68 %  3.58 %  10.18 % 
Table 4Numerical results comparison of U/Ut between the firstorder upwind and secondorder upwind of the modified turbulent viscosity
Type of upwind order  $y$/$L$ locations (m) at $x$/$L=$0.95 m ($U$/${U}_{t}$_{}numerical results and error ratios)  
0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  
0.6907  0.8199  0.8556  0.8667  0.8705  0.8711  0.8740  0.8746  0.8771  0.8801  
First order  12$L$  0.6315  0.8444  0.8823  0.8837  0.8858  0.8899  0.8920  0.8963  0.9020  0.8997 
8.56 %  2.98 %  3.13 %  1.97 %  1.75 %  2.16 %  2.06 %  2.49 %  2.84 %  2.22 %  
16$L$  0.6392  0.8383  0.8668  0.8606  0.8737  0.8797  0.8883  0.8892  0.8943  0.8998  
7.46 %  2.24 %  1.31 %  0.71 %  0.36 %  0.98 %  1.63 %  1.67 %  1.96 %  2.23 %  
20$L$  0.6487  0.8456  0.8862  0.8895  0.8865  0.8852  0.8904  0.8933  0.8971  0.8983  
6.08 %  3.13 %  3.57 %  2.64 %  1.84 %  1.61 %  1.88 %  2.15 %  2.29 %  2.06 %  
Second order  12$L$  0.6405  0.8374  0.8733  0.8777  0.8795  0.8807  0.8820  0.8843  0.8870  0.8887 
7.26 %  2.13 %  2.07 %  1.27 %  1.02 %  1.10 %  0.92 %  1.12 %  1.13 %  0.97 %  
16$L$  0.6339  0.8357  0.8624  0.8666  0.8707  0.8756  0.8801  0.8851  0.8893  0.8918  
8.23 %  1.93 %  0.79 %  0.01 %  0.01 %  0.51 %  0.69 %  1.20 %  1.39 %  1.32 %  
20$L$  0.6547  0.8416  0.8802  0.8815  0.8795  0.8788  0.8794  0.8816  0.8841  0.8863  
5.21 %  2.64 %  2.87 %  1.71 %  1.03 %  0.88 %  0.62 %  0.81 %  0.81 %  0.70 % 
2.2.2. Flux type and spatial discretization
An appropriate scheme is needed to evaluate the flux component. According to Ref. [39], the cylinder is taken as a research object to investigate the simulation performance of different flux types, and the conclusions demonstrate that the results of the ROE and AUSM are closer to the reference data. Therefore, we adopt these two types to study the capability of the simulation accuracy of aerodynamic prediction characterized by NACA0012 under hypersonic conditions. For the flow discretization, the second order upwind is selected. A suitable gradient calculation scheme is also needed, based on which the cell face scalar values could be constructed. The calculation of related diffusion terms and velocity derivatives can be done. There are three types of schemes (nodebased/cellbased/least cellbased). Out of these three, the least cellbased scheme is advantageous because it provides comparable accuracy to the nodebased scheme, has fewer computing resources, and avoids spurious oscillations. Hence, the least cellbased with the standard gradient limiter is applied. In addition, when the Mach number is bigger than 5, the densitybased solver is employed. Since the Mach number of validation tests is 10, it should be considered whether there exists real gas effect. The air critical pressure (${P}_{c}$) is 3.77 MPa, and if the ratio of $P$ and ${P}_{c}$ is much less than 1, then we could select the ideal gas. During the numerical simulation process, the value of $P$ increases from the initial 576 Pa to the maximum 73728 Pa. The maximum ratio of $P$ and ${P}_{c}$ is about 0.019, which satisfies the condition mentioned above. Hence, flow density ($\rho $) selects the ideal gas.
3. Numerical results and discussion
3.1. Numerical results
Based on the description in Grid strategy and Numerical Method, we apply six NACA0012 models, three far field distances, two turbulence models and two flux types to construct the simulation configurations and a total of seventytwo sets of numerical calculations are carried out. Tables 522 demonstrate the numerical results of $P/{P}_{t}$, $T/{T}_{t}$, $U/{U}_{t}$ at sampling locations and Figs. 1015 demonstrate the corresponding numerical error ratio distributions of $P/{P}_{t}$, $T/{T}_{t}$, $U/{U}_{t}$ compared with the wind tunnel data. The bold black values in the tables below $x$/$L$ sampling positions represent the corresponding wind tunnel test data, and the red dashed diamond shapes in the figures indicate wind tunnel data at those positions. Table 23 demonstrates the mean error ratios of $P/{P}_{t}$, $T/{T}_{t}$, $U/{U}_{t}$ for six NACA0012 models under different simulation configurations. Through the error ratio comparison, the optimal mean error ratio of ($P/{P}_{t}$$T/{T}_{t}$$U/{U}_{t}$) of 2.05 % could be achieved based on the configuration of sharp trailing edge (definition formula) + 16$L$ far field distance + SA turbulence model + ROE flux type.
Table 5Numerical results of P/Pt of blunt trailing edge adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $P$/${P}_{t}$ wind tunnel data ($P$/${P}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
103.6285  114.5508  117.8656  119.3983  121.7447  122.1036  123.3942  123.8330  
SST+12$L$  ROE  103.1758  114.2245  119.3853  122.5377  123.7579  124.7891  127.2207  128.9688 
AUSM  115.2700  114.4670  115.0006  114.7569  113.8172  111.8231  102.0335  97.0301  
SA+12$L$  ROE  6.0925  35.2682  107.9680  118.2251  118.7486  121.5961  129.3444  138.5572 
AUSM  113.6156  114.3060  113.3640  113.4916  114.8708  118.5805  122.1218  124.7841  
SST+16$L$  ROE  101.3087  113.2991  119.0844  122.3194  123.5652  124.6285  127.1412  128.9871 
AUSM  126.2989  125.2276  124.6631  126.3320  127.7393  129.1166  123.0015  109.6105  
SA+16$L$  ROE  112.1930  112.6418  113.9845  114.8477  115.8725  120.3363  125.7635  127.4569 
AUSM  115.5999  116.4176  117.9377  122.8046  126.0157  129.6497  135.4119  138.2230  
SST+20$L$  ROE  108.5238  119.5044  122.4766  125.7646  127.0867  128.2546  131.3245  133.7835 
AUSM  85.9375  103.1402  111.8435  111.6520  111.9114  112.0232  111.3510  110.1554  
SA+20$L$  ROE  88.2687  113.0102  122.2196  123.6521  126.3275  128.9982  134.4160  135.5053 
AUSM  108.2922  118.8877  118.7423  122.1060  125.9117  129.5463  140.4347  152.9953 
Table 6Numerical results of T/Tt of blunt trailing edge shape adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $T$/${T}_{t}$ wind tunnel data ($T$/${T}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
18.152  20.190  20.284  20.427  20.521  20.664  21.043  21.327  
SST+12$L$  ROE  19.918  20.418  20.675  20.792  20.765  20.680  20.151  17.892 
AUSM  20.097  20.334  20.580  20.726  20.290  19.677  18.936  17.701  
SA+12$L$  ROE  1.996  7.765  18.344  18.839  18.886  18.838  18.760  21.042 
AUSM  18.489  18.647  18.768  18.914  19.123  19.572  19.940  20.739  
SST+16$L$  ROE  19.825  20.387  20.678  20.805  20.785  20.708  20.191  17.840 
AUSM  20.592  20.758  20.866  20.959  20.879  20.700  18.750  5.439  
SA+16$L$  ROE  20.483  20.664  20.806  20.898  20.907  20.604  19.867  19.799 
AUSM  21.109  21.046  20.979  21.037  21.168  21.327  21.626  18.600  
SST+20$L$  ROE  18.909  20.412  20.572  20.688  20.685  20.628  20.178  18.384 
AUSM  17.365  19.335  19.951  20.074  20.074  20.443  20.074  15.271  
SA+20$L$  ROE  16.506  19.037  20.212  20.459  20.567  20.657  20.795  20.293 
AUSM  18.681  19.851  20.331  20.683  20.925  21.156  20.720  21.642 
Table 7Numerical results of U/Ut of blunt trailing edge adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $y$/$L$ (m) at $x$/$L$= 0.95 m and $U$/${U}_{t}$ wind tunnel data ($U$/${U}_{t}$ numerical data)  
0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  
0.6907  0.8199  0.8556  0.8667  0.8705  0.8711  0.8740  0.8746  0.8771  0.8801  
SST+12$L$  ROE  0.5939  0.8335  0.8538  0.8620  0.8680  0.8733  0.8774  0.8818  0.8855  0.8876 
AUSM  0.5529  0.7516  0.7981  0.8395  0.8628  0.8714  0.8720  0.8719  0.8731  0.8742  
SA+12$L$  ROE  0.6364  0.8604  0.8746  0.8881  0.8845  0.8826  0.8825  0.8832  0.8849  0.8861 
AUSM  0.6481  0.8350  0.8642  0.8657  0.8711  0.8769  0.8792  0.8796  0.8802  0.8807  
SST+16$L$  ROE  0.6323  0.8421  0.8568  0.8639  0.8695  0.8744  0.8782  0.8823  0.8858  0.8878 
AUSM  0.5359  0.7300  0.7756  0.8271  0.8548  0.8636  0.8694  0.8773  0.8841  0.8879  
SA+16$L$  ROE  0.6659  0.8723  0.8732  0.8730  0.8763  0.8807  0.8840  0.8871  0.8897  0.8912 
AUSM  0.6365  0.7868  0.7936  0.8110  0.8432  0.8701  0.8841  0.8928  0.8965  0.8976  
SST+20$L$  ROE  0.6212  0.8438  0.8581  0.8646  0.8696  0.8740  0.8775  0.8814  0.8849  0.8871 
AUSM  0.5947  0.8718  0.8651  0.8665  0.8782  0.8853  0.8871  0.8863  0.8866  0.8873  
SA+20$L$  ROE  0.6567  0.8747  0.8762  0.8766  0.8772  0.8783  0.8798  0.8823  0.8853  0.8873 
AUSM  0.6249  0.8669  0.8788  0.8777  0.8738  0.8741  0.8762  0.8800  0.8836  0.8858 
3.1.1. Airfoil tools
The corresponding numerical results and distributions of $P/{P}_{t}$, $T/{T}_{t}$ and $U/{U}_{t}$ are shown in Tables 57 and Fig. 10. Table 23 provides the mean error ratios under different simulation configurations. From the aspect of far field distance, the minimum values are 2.696 %, 2.738 %, and 3.436 % in order. From the aspect of turbulence model, the most favorable outcome for SST is 2.696 %, while that of SA is 3.292 %. For the two flux types, the finest value of ROE is 2.696 % and that of AUSM is 3.441 %. In conclusion, the SST+12$L$ configuration, combined with ROE has achieved the minimum total mean error ratio. Compared with other simulation configurations, the corresponding accuracy improvements are 57.249 %, 84.153 %, 21.642 %, 1.532 %, 70.082 %, 18.089 %, 48.237 %, 21.543 %, 55.935 %, 27.146 %, and 27.602 %, respectively.
Fig. 10The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of blunt trailing edge
a) Numerical result distribution of $P$/${P}_{t}$ of the blunt trailing edge
b) Error ratio distribution of $P$/${P}_{t}$ of the blunt trailing edge
c) Numerical result distribution of $T$/${T}_{t}$ of the blunt trailing edge
d) Error ratio distribution of $T$/${T}_{t}$ of the blunt trailing edge
e) Numerical result distribution of $U$/${U}_{t}$ of the blunt trailing edge
f) Error ratio distribution of $U$/${U}_{t}$ of the blunt trailing edge
3.1.2. NACA4
The corresponding numerical results and distributions of $P/{P}_{t}$, $T/{T}_{t}$ and $U/{U}_{t}$ are shown in Tables 810 and Fig. 11. Table 23 provides the mean error ratios under different simulation configurations. From the aspect of far field distance, the minimum values are 2.420 %, 3.797 %, and 3.189 % in order. From the aspect of turbulence model, the most favorable outcome for SST is 2.420 %, while that of SA is 2.829 %. For the two flux types, the finest value of ROE is 2.420 % and that of AUSM is 3.797 %.
Table 8Numerical results of P/Pt of sharp trailing edge based on NACA4 adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $P$/${P}_{t}$ wind tunnel data ($P$/${P}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
103.6285  114.5508  117.8656  119.3983  121.7447  122.1036  123.3942  123.8330  
SST+12$L$  ROE  102.6623  113.7859  119.5540  122.7781  124.0214  125.0571  127.6113  129.5765 
AUSM  125.0601  122.6266  123.4892  130.3869  133.7133  136.7315  138.2960  133.3994  
SA+12$L$  ROE  93.1517  112.1581  117.7100  119.1311  122.1984  125.8803  134.7221  136.4293 
AUSM  120.0194  113.7502  110.6636  109.2987  109.0662  109.8360  111.0621  115.5866  
SST+16$L$  ROE  90.3901  114.5756  124.3798  125.3509  127.3098  128.9722  132.1900  135.2095 
AUSM  123.6560  124.3612  124.9024  124.9201  124.3482  122.1458  119.9241  118.5962  
SA+16$L$  ROE  106.9225  112.5033  115.2072  115.4793  115.4510  115.1886  113.5401  107.3866 
AUSM  118.0516  119.6460  120.2279  120.3639  120.1021  119.3604  115.6705  111.5844  
SST+20$L$  ROE  91.5191  108.2494  117.8706  120.9798  122.3012  123.3865  125.8781  127.7352 
AUSM  98.4080  99.3577  98.2276  97.5850  97.8597  100.8383  103.7027  103.7561  
SA+20$L$  ROE  102.7478  106.5220  113.5859  117.7168  122.0448  126.4707  136.7221  136.1655 
AUSM  37.4234  73.3389  117.5831  118.8793  121.1027  123.9634  132.9692  140.2752 
Table 9Numerical results of T/Tt of sharp trailing edge based on NACA4 adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $T$/${T}_{t}$ wind tunnel data ($T$/${T}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
18.152  20.190  20.284  20.427  20.521  20.664  21.043  21.327  
SST+12$L$  ROE  19.860  20.361  20.640  20.773  20.761  20.693  20.219  18.605 
AUSM  21.769  21.834  21.924  21.832  21.495  20.869  20.137  18.077  
SA+12$L$  ROE  17.555  19.131  20.061  20.174  20.297  20.552  20.642  20.707 
AUSM  20.927  20.926  20.499  20.374  20.430  20.417  20.365  20.634  
SST+16$L$  ROE  17.182  19.264  20.271  20.609  20.704  20.696  20.454  18.930 
AUSM  20.769  21.064  21.367  21.568  21.237  20.772  20.092  15.503  
SA+16$L$  ROE  19.570  20.031  20.265  20.325  20.309  20.469  20.101  15.717 
AUSM  20.084  20.383  20.633  20.941  20.775  20.338  20.090  18.476  
SST+20$L$  ROE  18.834  20.122  20.647  20.833  20.832  20.610  20.262  17.108 
AUSM  17.580  17.965  18.472  18.797  19.136  19.811  19.313  18.382  
SA+20$L$  ROE  20.471  20.327  20.202  20.064  20.075  20.107  20.287  19.159 
AUSM  9.741  15.713  20.595  20.631  20.719  20.863  20.630  20.683 
Table 10Numerical results of U/Ut of sharp trailing edge based on NACA4 adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $y$/$L$ (m) at $x$/$L$= 0.95 m and $U$/${U}_{t}$ wind tunnel data ($U$/${U}_{t}$ numerical data)  
0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  
0.6907  0.8199  0.8556  0.8667  0.8705  0.8711  0.8740  0.8746  0.8771  0.8801  
SST+12$L$  ROE  0.6547  0.8416  0.8802  0.8815  0.8795  0.8788  0.8794  0.8816  0.8841  0.8863 
AUSM  0.6162  0.7966  0.8333  0.8513  0.8659  0.8776  0.8842  0.8870  0.8870  0.8873  
SA+12$L$  ROE  0.5989  0.7913  0.8351  0.8549  0.8648  0.8712  0.8757  0.8803  0.8841  0.8864 
AUSM  0.6515  0.8304  0.8604  0.8849  0.8960  0.8948  0.8907  0.8876  0.8871  0.8872  
SST+16$L$  ROE  0.6405  0.8374  0.8733  0.8777  0.8795  0.8807  0.8820  0.8843  0.8870  0.8887 
AUSM  0.5989  0.8126  0.8572  0.8667  0.8717  0.8770  0.8818  0.8871  0.8913  0.8938  
SA+16$L$  ROE  0.5957  0.7883  0.8337  0.8542  0.8642  0.8708  0.8755  0.8805  0.8845  0.8869 
AUSM  0.6457  0.8428  0.8642  0.8679  0.8722  0.8775  0.8822  0.8875  0.8917  0.8942  
SST+20$L$  ROE  0.6070  0.7921  0.8348  0.8547  0.8650  0.8718  0.8766  0.8813  0.8848  0.8868 
AUSM  0.5712  0.7690  0.7879  0.8170  0.8482  0.8685  0.8784  0.8839  0.8872  0.8889  
SA+20$L$  ROE  0.6339  0.8357  0.8624  0.8666  0.8707  0.8756  0.8801  0.8851  0.8893  0.8918 
AUSM  0.6564  0.8019  0.8492  0.8747  0.8793  0.8809  0.8820  0.8828  0.8836  0.8842 
In conclusion, the SST+12$L$ configuration, combined with ROE has achieved the minimum mean error ratio. Compared with other simulation configurations, the corresponding accuracy improvements are 68.882 %, 14.453 %, 46.120 %, 38.628 %, 55.249 %, 40.390 %, 36.269 %, 24.114 %, 71.737 %, 32.125 %, and 73.366 %, respectively.
Fig. 11The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of sharp trailing edge based on NACA4
a) Numerical result distribution of $P/{P}_{t}$ of the sharp trailing edge based on NACA4
b) Error ratio distribution of $P/{P}_{t}$ of the sharp trailing edge based on NACA4
c) Numerical result distribution of $T/{T}_{t}$ of the sharp trailing edge based on NACA4
d) Error ratio distribution of $T/{T}_{t}$ of the sharp trailing edge based on NACA4
e) Numerical result distribution of $U/{U}_{t}$ of the sharp trailing edge based on NACA4
f) Error ratio distribution of $U/{U}_{t}$of the sharp trailing edge based on NACA4
3.1.3. Definition formula adopting 132 points
The corresponding numerical results and distributions of $P/{P}_{t}$, $T/{T}_{t}$ and $U/{U}_{t}$ are shown in Tables 1113 and Fig.12. Table 23 provides the mean error ratios under different simulation configurations. From the aspect of far field distance, the minimum values are 3.136 %, 2.640 %, and 2.975 % in order. From the aspect of turbulence model, the most favorable outcome for SST is 2.640 %, while that of SA is 2.975 %. For the two flux types, the finest value of ROE is 2.640 % and that of AUSM is 3.136 %. In conclusion, the SST+16$L$ configuration, combined with ROE has achieved the minimum mean error ratio. Compared with other simulation configurations, the corresponding accuracy improvements are72.073 %, 15.821 %, 61.731 %, 58.624 %, 25.987 %, 63.533 %, 45.070 %, 53.338 %, 31.817 %, 11.290 %, and 66.411 %, respectively.
Table 11Numerical results of P/Pt of sharp trailing edge based on definition formula (132 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $P$/${P}_{t}$ wind tunnel data ($P$/${P}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
103.6285  114.5508  117.8656  119.3983  121.7447  122.1036  123.3942  123.8330  
SST+12$L$  ROE  77.9720  97.5837  103.9596  104.2068  105.7531  108.2004  103.7853  93.4325 
AUSM  122.9772  118.5715  117.4966  120.4748  122.3124  123.9481  126.0108  132.5421  
SA+12$L$  ROE  126.8481  127.4266  128.1201  128.9839  128.9838  128.9172  128.3850  127.1818 
AUSM  94.8265  106.4431  109.2256  108.2244  108.7156  108.9666  109.8323  131.9843  
SST+16$L$  ROE  102.8513  114.0935  119.5973  122.7339  123.9256  124.9008  127.3781  129.2507 
AUSM  116.3570  118.2875  117.5792  117.6293  119.3575  122.0409  122.4785  121.0955  
SA+16$L$  ROE  110.5474  142.6264  133.8498  134.7360  135.7823  135.8110  131.7480  133.3460 
AUSM  126.5742  124.6259  124.2431  126.2068  126.8411  127.7232  129.5196  127.2727  
SST+20$L$  ROE  89.3645  103.6540  106.7764  103.4379  104.8373  109.8622  112.3890  112.5213 
AUSM  121.9440  118.9971  117.6949  118.6125  119.4331  121.7822  125.1382  132.7346  
SA+20$L$  ROE  103.6814  114.6149  117.3158  116.9946  118.5445  122.0046  125.1439  132.9575 
AUSM  67.4088  111.4342  132.0675  125.9648  128.9382  132.7707  143.0583  153.6435 
Table 12Numerical results of T/Tt of sharp trailing edge based on definition formula (132 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $T$/${T}_{t}$ wind tunnel data ($T$/${T}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
18.152  20.190  20.284  20.427  20.521  20.664  21.043  21.327  
SST+12$L$  ROE  15.133  16.859  18.502  19.281  20.050  20.660  19.174  18.030 
AUSM  19.944  20.075  20.212  20.495  20.646  20.841  21.900  19.971  
SA+12$L$  ROE  20.816  20.841  20.869  20.909  20.932  20.995  20.806  10.720 
AUSM  17.497  18.624  19.302  19.408  19.484  19.593  19.665  18.654  
SST+16$L$  ROE  19.873  20.384  20.656  20.781  20.761  20.683  20.173  18.277 
AUSM  18.708  19.203  19.657  19.851  20.044  20.498  20.571  14.955  
SA+16$L$  ROE  16.122  20.522  22.666  23.641  23.466  20.439  19.578  20.710 
AUSM  21.319  21.230  21.041  20.932  20.794  20.698  21.178  17.974  
SST+20$L$  ROE  17.271  19.272  20.199  20.584  20.672  20.630  19.591  17.817 
AUSM  21.116  21.097  21.012  20.946  20.862  20.764  20.399  20.262  
SA+20$L$  ROE  18.608  19.239  19.496  19.593  19.692  19.713  19.929  18.019 
AUSM  11.876  16.908  20.229  20.249  20.367  20.476  19.791  21.239 
Table 13Numerical results of U/Ut of sharp trailing edge based on definition formula (132 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $y$/$L$ (m) at $x$/$L$= 0.95 m and $U$/${U}_{t}$ wind tunnel data ($U$/${U}_{t}$ numerical data)  
0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  
0.6907  0.8199  0.8556  0.8667  0.8705  0.8711  0.8740  0.8746  0.8771  0.8801  
SST+12$L$  ROE  0.6204  0.7708  0.8074  0.8356  0.8626  0.8760  0.8801  0.8817  0.8831  0.8839 
AUSM  0.6358  0.7639  0.8122  0.8516  0.8696  0.8724  0.8729  0.8748  0.8781  0.8802  
SA+12$L$  ROE  0.6643  0.7866  0.8004  0.8270  0.8482  0.8617  0.8699  0.8765  0.8810  0.8835 
AUSM  0.6456  0.7430  0.7751  0.8323  0.8619  0.8598  0.8560  0.8571  0.8617  0.8649  
SST+16$L$  ROE  0.6048  0.7918  0.8350  0.8536  0.8629  0.8690  0.8737  0.8786  0.8829  0.8855 
AUSM  0.6191  0.8289  0.8634  0.8757  0.8805  0.8792  0.8783  0.8799  0.8829  0.8849  
SA+16$L$  ROE  0.6488  0.8317  0.8708  0.8756  0.8751  0.8780  0.8819  0.8864  0.8898  0.8918 
AUSM  0.7046  0.8616  0.8639  0.8585  0.8626  0.8700  0.8758  0.8810  0.8846  0.8867  
SST+20$L$  ROE  0.6297  0.8086  0.8422  0.8566  0.8647  0.8707  0.8753  0.8803  0.8846  0.8871 
AUSM  0.6331  0.7768  0.7970  0.8270  0.8533  0.8659  0.8709  0.8750  0.8787  0.8810  
SA+20$L$  ROE  0.6689  0.8531  0.8696  0.8693  0.8744  0.8798  0.8833  0.8867  0.8900  0.8920 
AUSM  0.6855  0.7899  0.7990  0.8160  0.8456  0.8684  0.8788  0.8834  0.8859  0.8872 
Fig. 12The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of sharp trailing edge based on definition formula (132 points)
a) Numerical result distribution of $P/{P}_{t}$ of the sharp trailing edge based on definition formula (132 points)
b) Error ratio distribution of $P/{P}_{t}$ of the sharp trailing edge based on definition formula (132 points)
c) Numerical result distribution of $T/{T}_{t}$ of the sharp trailing edge based on definition formula (132 points)
d) Error ratio distribution of $T/{T}_{t}$ of the sharp trailing edge based on definition formula (132 points)
e) Numerical result distribution of $U/{U}_{t}$ of the sharp trailing edge based on definition formula (132 points)
f) Error ratio distribution of $U/{U}_{t}$ of the sharp trailing edge based on definition formula (132 points)
3.1.4. Definition formula adopting 264 points
The corresponding numerical results and distributions of $P/{P}_{t}$, $T/{T}_{t}$ and $U/{U}_{t}$ are shown in Tables 1416 and Fig. 13. Table 23 provides the mean error ratios under different simulation configurations. From the aspect of far field distance, the minimum values are 2.826 %, 2.758 %, and 3.449 % in order. From the aspect of turbulence model, the most favorable outcome for SST is 2.758 %, while that of SA is 4.511 %. For the two flux types, the finest value of ROE is 2.758 % and that of AUSM is 3.822 %. In conclusion, the SST+16$L$ configuration, combined with ROE has achieved the minimum mean error ratio. Compared with other simulation configurations, the corresponding accuracy improvements are 2.388 %, 27.832 %, 38.853 %, 42.156 %, 78.356 %, 48.882 %, 48.951 %, 20.037 %, 67.078 %, 40.690 %, and 69.247 %, respectively.
Table 14Numerical results of P/Pt of sharp trailing edge based on definition formula (264 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $P$/${P}_{t}$ wind tunnel data ($P$/${P}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
103.6285  114.5508  117.8656  119.3983  121.7447  122.1036  123.3942  123.8330  
SST+12$L$  ROE  119.7998  119.7281  120.2691  122.1656  122.9093  123.4018  123.1883  122.8126 
AUSM  106.5388  105.0963  104.8083  104.9926  105.3805  110.8957  115.5403  123.2807  
SA+12$L$  ROE  111.6394  109.2246  109.1835  109.8389  109.8331  118.7383  132.1181  148.5604 
AUSM  117.0965  117.1610  115.3581  114.1848  114.3075  116.5136  120.0602  131.7145  
SST+16$L$  ROE  104.9188  114.8591  120.1705  123.3651  124.5840  125.6013  128.1834  130.1437 
AUSM  14.2166  57.3405  121.6925  120.0211  121.3962  123.1041  126.7583  131.6546  
SA+16$L$  ROE  113.7103  113.9911  115.5370  115.9321  115.4393  113.6081  102.4842  91.1783 
AUSM  112.5245  111.4754  109.5819  108.2756  109.2258  112.8401  115.5227  120.5992  
SST+20$L$  ROE  96.4637  118.4568  124.6484  125.9054  127.6129  129.0491  132.1726  135.1249 
AUSM  108.7255  108.7527  108.7589  108.7113  108.6527  108.5267  107.3855  104.1970  
SA+20$L$  ROE  120.4941  120.9398  121.3385  121.6577  121.5770  121.3591  116.5592  114.0342 
AUSM  67.4088  111.4342  132.0675  125.9648  128.9382  132.7707  143.0583  153.6435 
Table 15Numerical results of T/Tt of sharp trailing edge based on definition formula (264 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $T$/${T}_{t}$ wind tunnel data ($T$/${T}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
18.152  20.190  20.284  20.427  20.521  20.664  21.043  21.327  
SST+12$L$  ROE  20.265  20.399  20.523  20.751  20.862  20.961  20.889  18.934 
AUSM  19.513  19.793  20.230  20.351  20.493  20.652  19.449  20.337  
SA+12$L$  ROE  20.653  20.674  20.662  20.620  20.616  20.787  19.673  21.209 
AUSM  21.407  21.550  21.067  20.836  20.643  20.455  20.339  20.312  
SST+16$L$  ROE  19.961  20.385  20.629  20.749  20.733  20.661  20.186  18.623 
AUSM  4.826  11.009  19.659  19.813  19.921  19.972  19.857  20.080  
SA+16$L$  ROE  19.189  19.449  20.030  20.322  20.471  20.266  19.039  18.611 
AUSM  20.044  20.421  20.526  20.373  20.137  19.901  20.219  12.186  
SST+20$L$  ROE  17.859  19.600  20.384  20.631  20.707  20.658  20.402  18.762 
AUSM  20.354  20.487  20.628  20.896  20.986  20.840  19.309  13.045  
SA+20$L$  ROE  20.566  20.638  20.702  20.772  20.770  20.691  20.248  13.594 
AUSM  11.876  16.908  20.229  20.249  20.367  20.476  19.791  21.239 
Table 16Numerical results of U/Ut of sharp trailing edge based on definition formula (264 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $y$/$L$ (m) at $x$/$L$= 0.95 m and $U$/${U}_{t}$ wind tunnel data ($U$/${U}_{t}$ numerical data)  
0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  
0.6907  0.8199  0.8556  0.8667  0.8705  0.8711  0.8740  0.8746  0.8771  0.8801  
SST+12$L$  ROE  0.6410  0.8225  0.8565  0.8686  0.8747  0.8778  0.8798  0.8823  0.8851  0.8868 
AUSM  0.6903  0.8497  0.8722  0.8704  0.8689  0.8708  0.8734  0.8766  0.8795  0.8811  
SA+12$L$  ROE  0.7094  0.8025  0.8217  0.8538  0.8764  0.8850  0.8868  0.8865  0.8868  0.8870 
AUSM  0.6704  0.7601  0.7867  0.8084  0.8293  0.8443  0.8526  0.8581  0.8615  0.8633  
SST+16$L$  ROE  0.6214  0.8083  0.8430  0.8569  0.8646  0.8702  0.8743  0.8787  0.8826  0.8849 
AUSM  0.6101  0.8230  0.8696  0.8692  0.8699  0.8727  0.8755  0.8792  0.8831  0.8855  
SA+16$L$  ROE  0.6535  0.7837  0.7866  0.8165  0.8505  0.8676  0.8753  0.8814  0.8857  0.8881 
AUSM  0.6940  0.8642  0.8756  0.8650  0.8656  0.8721  0.8776  0.8826  0.8859  0.8877  
SST+20$L$  ROE  0.6307  0.8099  0.8438  0.8580  0.8660  0.8718  0.8761  0.8809  0.8850  0.8874 
AUSM  0.6090  0.7029  0.7321  0.7638  0.7972  0.8300  0.8515  0.8658  0.8733  0.8771  
SA+20$L$  ROE  0.6877  0.8064  0.8123  0.8399  0.8625  0.8713  0.8750  0.8784  0.8814  0.8833 
AUSM  0.6916  0.7442  0.7411  0.7565  0.7934  0.8282  0.8475  0.8582  0.8653  0.8693 
Fig. 13The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of sharp trailing edge based on definition formula (264 points)
a) Numerical result distribution of $P/{P}_{t}$ of the sharp trailing edge based on definition formula (264 points)
b) Error ratio distribution of $P/{P}_{t}$ of the sharp trailing edge based on definition formula (264 points)
c) Numerical result distribution of $T/{T}_{t}$ of the sharp trailing edge based on definition formula (264 points)
d) Error ratio distribution of $T/{T}_{t}$ of the sharp trailing edge based on definition formula (264 points)
e) Numerical result distribution of $U/{U}_{t}$ of the sharp trailing edge based on definition formula (264 points)
f) Error ratio distribution of $U/{U}_{t}$ of the sharp trailing edge based on definition formula (264 points)
3.1.5. Definition formula adopting 200 points
The corresponding numerical results and distributions of $P/{P}_{t}$, $T/{T}_{t}$ and $U/{U}_{t}$ are shown in Tables 1719 and Fig. 14. Table 23 provides the mean error ratios under different simulation configurations. From the aspect of far field distance, the minimum values are 2.866 %, 2.047 %, and 3.376 % in order. From the aspect of turbulence model, the most favorable outcome for SST is 3.264 %, while that of SA is 2.047 %. For the two flux types, the finest value of ROE is 2.047 % and that of AUSM is 3.264 %. In conclusion, the SA+16L configuration, combined with ROE has achieved the minimum mean error ratio. Compared with other simulation configurations, the corresponding accuracy improvements are 36.670 %, 65.361 %, 28.590 %, 56.800 %, 66.338 %, 37.299 %, 37.549 %, 54.702 %, 64.219 %, 46.723 %, and 39.367 %, respectively.
Table 17Numerical results of P/Pt of sharp trailing edge based on definition formula (200 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $P$/${P}_{t}$ wind tunnel data ($P$/${P}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
103.6285  114.5508  117.8656  119.3983  121.7447  122.1036  123.3942  123.8330  
SST+12$L$  ROE  95.2490  121.0420  122.3321  123.1870  124.9979  126.5275  129.6336  132.2661 
AUSM  129.6510  127.9943  126.6424  127.8131  128.1751  127.3835  125.5274  122.0080  
SA+12$L$  ROE  88.1361  106.7947  112.6416  113.1061  114.1654  117.0814  120.5537  127.4284 
AUSM  92.6078  127.9824  128.0527  127.0169  128.0833  129.2793  131.2632  133.4659  
SST+16$L$  ROE  70.0133  107.5888  126.7207  128.4036  128.3239  127.5474  127.2913  126.4086 
AUSM  123.2248  121.5232  118.3046  119.8646  121.7615  123.6976  125.7666  124.9552  
SA+16$L$  ROE  101.5952  111.8097  117.2720  117.7607  118.0276  118.0976  117.8711  119.7389 
AUSM  119.8220  114.3523  111.4593  112.6008  114.4394  119.3344  121.2641  122.1085  
SST+20$L$  ROE  101.8103  112.8437  115.7255  114.2609  114.3502  113.7401  110.6038  114.4976 
AUSM  129.1016  125.3428  123.7512  125.0469  126.9627  128.7558  133.7061  144.7895  
SA+20$L$  ROE  121.1185  120.9658  120.7203  120.3242  119.6663  118.7319  115.1396  110.0396 
AUSM  119.1405  120.4283  120.6966  122.0053  122.8732  123.2574  123.3283  121.9717 
Table 18Numerical results of T/Tt of sharp trailing edge based on definition formula (200 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $T$/${T}_{t}$ wind tunnel data ($T$/${T}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
18.152  20.190  20.284  20.427  20.521  20.664  21.043  21.327  
SST+12$L$  ROE  16.551  19.384  20.448  20.622  20.666  20.651  20.270  18.759 
AUSM  20.727  21.044  21.272  21.549  21.585  20.283  19.882  16.591  
SA+12$L$  ROE  17.981  19.476  20.274  20.411  20.527  20.688  20.812  20.568 
AUSM  14.238  19.106  20.805  20.829  20.869  20.741  20.767  20.866  
SST+16$L$  ROE  11.797  17.598  20.550  21.057  21.280  21.064  20.717  20.642 
AUSM  19.688  20.028  20.234  20.448  20.627  20.889  21.297  17.054  
SA+16$L$  ROE  19.274  19.891  20.245  20.439  20.516  20.648  20.695  20.217 
AUSM  20.448  20.500  20.518  20.514  20.404  20.293  20.064  19.781  
SST+20$L$  ROE  17.608  18.326  18.700  18.845  19.056  19.702  19.834  19.830 
AUSM  20.882  21.048  21.204  21.310  21.056  20.465  20.164  20.226  
SA+20$L$  ROE  20.707  20.751  20.773  20.766  20.769  20.760  20.145  19.683 
AUSM  18.929  19.318  19.726  19.914  20.181  20.654  20.752  17.893 
Table 19Numerical results of U/Ut of sharp trailing edge based on definition formula (200 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $y$/$L$ (m) at $x$/$L$= 0.95 m and $U$/${U}_{t}$ wind tunnel data ($U$/${U}_{t}$ numerical data)  
0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  
0.6907  0.8199  0.8556  0.8667  0.8705  0.8711  0.8740  0.8746  0.8771  0.8801  
SST+12$L$  ROE  0.6634  0.8368  0.8552  0.8629  0.8684  0.8730  0.8766  0.8808  0.8844  0.8866 
AUSM  0.6104  0.8114  0.8678  0.8737  0.8693  0.8687  0.8720  0.8782  0.8843  0.8877  
SA+12$L$  ROE  0.7054  0.8717  0.8779  0.8770  0.8751  0.8748  0.8762  0.8796  0.8837  0.8866 
AUSM  0.7205  0.8669  0.8665  0.8694  0.8760  0.8803  0.8825  0.8848  0.8872  0.8887  
SST+16$L$  ROE  0.6519  0.8574  0.8755  0.8749  0.8743  0.8756  0.8784  0.8830  0.8877  0.8905 
AUSM  0.6017  0.8144  0.8655  0.8726  0.8749  0.8764  0.8785  0.8823  0.8863  0.8889  
SA+16$L$  ROE  0.7092  0.8739  0.8681  0.8692  0.8746  0.8796  0.8830  0.8863  0.8891  0.8908 
AUSM  0.6745  0.8656  0.8790  0.8745  0.8693  0.8685  0.8698  0.8734  0.8780  0.8816  
SST+20$L$  ROE  0.6450  0.8527  0.8731  0.8746  0.8760  0.8791  0.8821  0.8857  0.8889  0.8909 
AUSM  0.5981  0.7864  0.8374  0.8651  0.8692  0.8700  0.8726  0.8775  0.8826  0.8858  
SA+20$L$  ROE  0.6860  0.8558  0.8776  0.8731  0.8713  0.8733  0.8767  0.8819  0.8868  0.8898 
AUSM  0.7078  0.7781  0.8094  0.8215  0.8393  0.8573  0.8664  0.8720  0.8762  0.8786 
Fig. 14The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of sharp trailing edge based on definition formula (200 points)
a) Numerical result distribution of $P/{P}_{t}$ of the sharp trailing edge based on definition formula (200 points)
b) Error ratio distribution of $P/{P}_{t}$ of the sharp trailing edge based on definition formula (200 points)
c) Numerical result distribution of $T/{T}_{t}$ of the sharp trailing edge based on definition formula (200 points)
d) Error ratio distribution of $T/{T}_{t}$of the sharp trailing edge based on definition formula (200 points)
e) Numerical result distribution of $U/{U}_{t}$ of the sharp trailing edge based on definition formula (200 points)
f) Error ratio distribution of $U/{U}_{t}$ of the sharp trailing edge based on definition formula (200 points)
3.1.6. Definition formula adopting 400 points
The corresponding numerical results and distributions of $P/{P}_{t}$, $T/{T}_{t}$ and $U/{U}_{t}$ are shown in Tables 2022 and Fig. 15. Table 23 provides the calculated mean error ratios under different simulation configurations. From the aspect of far field distance, the minimum values are 4.136 %, 2.460 %, and 2.454 % in order. From the aspect of turbulence model, the most favorable outcome for SST is 2.454 %, while that of SA is 3.482 %. For the two flux types, the finest value of ROE is 2.454 % and that of AUSM is 3.482 %. In conclusion, the SST+20$L$ configuration, combined with ROE has achieved the minimum mean error ratio. Compared with other simulation configurations, the corresponding accuracy improvements are 40.670 %, 80.420 %, 56.911 %, 71.324 %, 0.240 %, 61.911 %, 33.899 %, 29.537 %, 40.134 %, 36.977 %, and 59.327 %, respectively.
Table 20Numerical results of P/Pt of sharp trailing edge based on definition formula (400 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $P$/${P}_{t}$ wind tunnel data ($P$/${P}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
103.6285  114.5508  117.8656  119.3983  121.7447  122.1036  123.3942  123.8330  
SST+12$L$  ROE  121.5352  121.5892  121.8436  122.3692  122.4129  122.2378  116.3225  108.3289 
AUSM  104.5392  104.3224  104.0669  103.6346  103.5846  103.5758  98.7789  94.0892  
SA+12$L$  ROE  120.7982  120.4667  120.3445  119.5408  117.6898  114.7833  100.4419  102.2569 
AUSM  106.9614  106.0633  105.1678  103.4016  102.6120  101.9376  101.4844  103.5008  
SST+16$L$  ROE  103.1487  114.1292  119.7196  122.8894  124.0935  125.0836  127.5916  129.4911 
AUSM  119.4194  119.4679  118.3171  116.7076  114.0483  110.7948  107.6613  109.6274  
SA+16$L$  ROE  84.7344  105.5092  115.0007  116.0551  119.1564  123.3079  135.0119  138.4577 
AUSM  115.7716  116.3642  117.5177  118.0496  118.5284  119.1038  118.8542  116.6833  
SST+20$L$  ROE  101.7360  113.5344  119.1353  122.2907  123.5090  124.5382  126.9285  128.7087 
AUSM  111.4175  120.3351  118.6102  120.8620  125.6442  130.3150  137.6943  143.5232  
SA+20$L$  ROE  85.6777  105.7869  113.6548  114.4745  117.0368  120.6551  134.5757  138.7417 
AUSM  86.0844  105.3288  117.5366  116.7220  118.1160  123.6655  136.1005  146.2670 
Table 21Numerical results of T/Tt of sharp trailing edge based on definition formula (400 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $x$/$L$ (m) and $T$/${T}_{t}$ wind tunnel data ($T$/${T}_{t}$ numerical data)  
–0.007  –0.006  –0.005  –0.004  –0.003  –0.002  –0.001  0  
18.152  20.190  20.284  20.427  20.521  20.664  21.043  21.327  
SST+12$L$  ROE  20.476  20.586  20.682  20.862  20.944  20.889  20.056  18.964 
AUSM  19.928  20.157  20.330  20.269  19.400  16.884  8.214  3.902  
SA+12$L$  ROE  20.513  20.606  20.155  19.781  19.622  19.593  19.235  18.366 
AUSM  18.219  18.129  18.075  18.023  18.011  18.089  18.158  16.021  
SST+16$L$  ROE  19.882  20.374  20.645  20.770  20.751  20.675  20.176  18.464 
AUSM  21.466  21.187  20.172  19.808  19.637  19.458  19.065  14.682  
SA+16$L$  ROE  17.053  18.914  20.053  20.139  20.244  20.500  20.609  20.531 
AUSM  20.506  20.677  20.858  21.126  21.139  20.820  20.126  19.176  
SST+20$L$  ROE  19.831  20.385  20.670  20.797  20.776  20.697  20.190  18.259 
AUSM  18.831  19.597  20.131  20.380  20.680  20.935  21.276  20.721  
SA+20$L$  ROE  17.312  19.017  19.999  20.080  20.182  20.470  20.599  20.551 
AUSM  17.093  18.685  19.730  19.768  19.813  19.906  19.975  12.296 
Table 22Numerical results of U/Ut of sharp trailing edge based on definition formula (400 points) adopting two turbulence models, three far field distances and two flux types
NACA0012 models  $y$/$L$ (m) at $x$/$L$= 0.95 m and $U$/${U}_{t}$ wind tunnel data ($U$/${U}_{t}$ numerical data)  
0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  
0.6907  0.8199  0.8556  0.8667  0.8705  0.8711  0.8740  0.8746  0.8771  0.8801  
SST+12$L$  ROE  0.6754  0.8645  0.8788  0.8759  0.8754  0.8778  0.8805  0.8837  0.8867  0.8885 
AUSM  0.6956  0.8018  0.7984  0.8265  0.8641  0.8782  0.8813  0.8818  0.8821  0.8823  
SA+12$L$  ROE  0.7301  0.8741  0.8743  0.8742  0.8761  0.8785  0.8806  0.8834  0.8863  0.8880 
AUSM  0.7055  0.8441  0.8732  0.8773  0.8712  0.8689  0.8696  0.8732  0.8779  0.8809  
SST+16$L$  ROE  0.6532  0.8335  0.8533  0.8617  0.8677  0.8725  0.8763  0.8805  0.8844  0.8867 
AUSM  0.5980  0.8120  0.8621  0.8703  0.8747  0.8769  0.8778  0.8797  0.8828  0.8850  
SA+16$L$  ROE  0.6972  0.8703  0.8708  0.8670  0.8689  0.8743  0.8789  0.8833  0.8870  0.8892 
AUSM  0.7106  0.8780  0.8699  0.8702  0.8764  0.8799  0.8816  0.8844  0.8878  0.8900  
SST+20$L$  ROE  0.6505  0.8288  0.8516  0.8607  0.8669  0.8721  0.8762  0.8808  0.8850  0.8875 
AUSM  0.6226  0.7503  0.7943  0.8283  0.8415  0.8480  0.8542  0.8626  0.8699  0.8743  
SA+20$L$  ROE  0.6937  0.8678  0.8749  0.8770  0.8783  0.8798  0.8816  0.8844  0.8877  0.8898 
AUSM  0.7392  0.8062  0.8433  0.8700  0.8778  0.8755  0.8724  0.8716  0.8730  0.8741 
Fig. 15The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of sharp trailing edge based on definition formula (400 points)
a) Numerical result distribution of $P/{P}_{t}$ of the sharp trailing edge based on definition formula (400 points)
b) Error ratio distribution of $P/{P}_{t}$ of the sharp trailing edge based on definition formula (400 points)
c) Numerical result distribution of $T/{T}_{t}$ of the sharp trailing edge based on definition formula (400 points)
d) Error ratio distribution of $T/{T}_{t}$ of the sharp trailing edge based on definition formula (400 points)
e) Numerical result distribution of $U/{U}_{t}$ of the sharp trailing edge based on definition formula (400 points)
f) Error ratio distribution of $U/{U}_{t}$ of the sharp trailing edge based on definition formula (400 points)
Table 23The mean error ratios for the six NACA0012 models under different simulation configurations
Simulation configuration  NACA0012 models  ROE (%)  AUSM (%)  
$P/{P}_{t}$  $T/{T}_{t}$  $U/{U}_{t}$  Total mean error ratio of ($P/{P}_{t}$, $T/{T}_{t}$, $U/{U}_{t}$)  $P/{P}_{t}$  $T/{T}_{t}$  $U/{U}_{t}$  Total mean error ratio of ($P/{P}_{t}$, $T/{T}_{t}$, $U/{U}_{t}$)  
SST+12L  Blunt  1.968  4.523  1.598  2.696  8.939  5.909  4.072  6.307 
NACA4  2.276  3.962  1.035  2.424  10.415  8.540  4.376  7.777  
Definition formula (132 points)  16.132  9.272  2.950  9.451  4.316  2.877  2.214  3.136  
Definition formula (264 points)  3.437  3.811  1.230  2.826  7.955  2.814  0.697  3.822  
Definition formula (200 points)  4.860  3.881  0.955  3.232  8.019  7.942  1.766  5.909  
Definition formula (400 points)  6.025  4.722  1.660  4.136  13.597  22.172  1.826  12.532  
SST+16L  Blunt  2.197  4.485  1.532  2.738  8.155  13.631  5.249  9.012 
Naca4  6.228  3.245  2.357  3.943  5.962  8.189  2.072  5.408  
Definition formula (132 points)  2.137  4.212  1.569  2.640  2.779  6.139  1.781  3.566  
Definition formula (264 points)  2.621  4.002  1.652  2.758  18.768  17.841  1.622  12.744  
Definition formula (200 points)  8.584  7.830  1.827  6.080  3.738  4.059  1.996  3.264  
Definition formula (400points)  2.209  4.077  1.093  2.460  7.746  9.685  1.893  6.442  
SST+20L  Blunt  5.272  3.356  1.681  3.436  9.471  6.024  2.860  6.119 
NACA4  3.148  4.142  2.276  3.189  15.305  7.999  2.384  8.562  
Definition formula (132 points)  11.002  4.356  1.612  5.657  4.140  4.644  2.830  3.871  
Definition formula (264 points)  6.034  2.755  1.560  3.449  9.670  8.474  6.990  8.378  
Definition formula (200 points)  5.024  6.543  1.988  4.518  9.843  5.132  2.186  5.720  
Definition formula (400 points)  2.057  4.222  1.082  2.454  6.480  1.713  4.103  4.099  
SA+12L  Blunt  24.038  24.609  2.394  17.014  3.619  5.559  1.145  3.441 
NACA4  4.422  2.164  1.901  2.829  8.525  3.545  1.404  4.491  
Definition formula (132 points)  8.581  9.701  2.410  6.897  8.912  6.314  3.912  6.379  
Definition formula (264 points)  8.416  3.388  1.728  4.511  5.190  5.030  4.085  4.768  
Definition formula (200 points)  5.871  1.176  1.552  2.866  7.828  4.626  1.759  4.738  
Definition formula (400 points)  8.663  6.363  2.057  5.694  12.650  12.065  0.954  8.556  
SA+16L  Blunt  3.519  4.375  1.980  3.292  5.893  6.107  3.625  5.209 
NACA4  5.325  5.244  1.610  4.060  5.116  4.569  1.707  3.797  
Definition formula (132 points)  11.850  8.200  1.664  7.238  7.323  5.833  1.261  4.805  
Definition formula (264 points)  8.819  4.448  2.921  5.396  6.809  8.172  1.229  5.403  
Definition formula (200 points)  2.543  1.859  1.738  2.047  4.789  3.754  1.289  3.277  
Definition formula (400 points)  6.962  2.857  1.317  3.712  3.660  4.980  1.806  3.482  
SA+20L  Blunt  6.399  2.696  2.007  3.701  7.272  1.680  2.220  3.724 
Naca4  4.685  4.283  1.728  3.565  15.453  9.744  2.061  9.086  
Definition formula (132 points)  1.760  5.580  1.586  2.975  13.734  7.500  2.340  7.858  
Definition formula (264 points)  5.111  7.582  1.259  4.651  13.734  7.500  5.673  8.969  
Definition formula (200 points)  5.997  4.323  1.206  3.842  3.514  4.132  2.481  3.376  
Definition formula (400 points)  7.353  2.735  1.592  3.893  7.522  9.223  1.353  6.033 
3.2. Discussion
3.2.1. Cell Reynolds number and aspect ratio
The cell Reynolds number (${R}_{cell}$) of nearwall mesh cells close to the shock are crucial in affecting the numerical error ratio. Taking blunt cylinder as the characteristic object, Ref. [30] points out that the ${R}_{cell}$ value of the near shock wave grid cells should be no less than 8. Moreover, Ref. [31] shows that the aspect ratio of wall cells near the shock significantly impacts simulation performance. In this paper, based on the existing research, to investigate the influences of ${R}_{cell}$ and aspect ratio of the near shock wave wall grid cells with NACA0012 as the characteristic object, the ${R}_{cell}$ of near shock wave wall grid cells is first analyzed while keeping the total mesh number unchanged. Based on the optimal simulation configuration conclude in Section 3.1, we apply three ${R}_{cell}$ values of 16, 8, and 4 and the related ${y}^{+}$ and ${y}_{H}$values are (0.3 1.4e5 m), (0.15 7e6 m), and (0.08 3.5e6 m). Three numerical simulations are performed and the corresponding mean error ratios of ($P/{P}_{t}$, $T/{T}_{t}$, $U/{U}_{t}$) are (2.54 % 1.86 % 1.74 %), (2.62 % 2.09 % 2.01 %), and (6.38 % 4.77 % 2.47 %) respectively, as shown in Table 24. Therefore, the optimal results are obtained by ${R}_{cell}$ value of 16. Next, to study the influence of aspect ratio on numerical accuracy, we made changes to the wall cells' aspect ratio near the shock while keeping the following conditions constant: (1) The total number of mesh cells remained the same. (2) The cell Reynolds number remained the same. (3) The aspect ratio changes were made only in small wall regions near the shock. The aspect ratio value of the nearwall mesh close to the shock at the optimal ${R}_{cell}$ value is 380, through double and halve operations we select four aspect ratio values of 760, 380, 190, and 95. Another four simulations are executed and the corresponding comparison of error ratios are described in Table 25. When the aspect ratio is 380, the minimum simulation result is achieved, which is 2.05 %. Then with the further increase of the aspect ratio, the error ratio is also increased. Compared with the other three aspect ratios, the accuracy improvements are 63.97 %, 46.75 % and 65.37 %. In summarize, unlike the suggestions proposed in the existing research characterized by blunt cylinder, the suitable value for ${R}_{cell}$ characterized by NACA0012 should be no smaller than 16, reducing this value will decrease numerical accuracy. Similar situation applies to the aspect ratio, smaller value would not lead to better numerical calculation and the recommended value is 380.
Table 24Comparison of numerical error ratios under three cell Reynolds numbers
Cell Reynolds number  $P/{P}_{t}$  $T/{T}_{t}$  $U/{U}_{t}$  Mean error ratio 
16 (${y}^{+}=$0.3)  2.54 %  1.86 %  1.74 %  2.05 % 
8 (${y}^{+}=$0.15)  2.62 %  2.09 %  2.01 %  2.24 % 
4 (${y}^{+}=$0.08)  6.38 %  4.77 %  2.47 %  4.54 % 
Table 25Comparison of numerical error ratios under four aspect ratios
Aspect ratio  $P/{P}_{t}$  $T/{T}_{t}$  $U/{U}_{t}$  Mean error ratio  Accuracy improvement among aspect ratios 
95  4.78 %  6.26 %  6.04 %  5.69 %  63.97 % 
190  4.35 %  3.49 %  3.71 %  3.85 %  46.75 % 
380  2.54 %  1.86 %  1.74 %  2.05 %  0.00 % 
760  8.56 %  4.07 %  5.13 %  5.92 %  65.37 % 
3.3. Trailing edge shape and modeling method
Fig. 16 depicts the optimal numerical error ratios comparison among six NACA0012 model, where the left ordinate indicates the numerical error ratio displayed in a column graph, and the right ordinate indicates the accuracy improvement displayed in a line chart. Fig. 17 adopts the same settings. From the aspect of trailing edge shape, based on Airfoil tools, the designed blunt trailing edge’s numerical performance is worse than that of other types of sharp trailing edge (the sharp trailing edge adopting 264 data points definition formula is excluded), with the numerical accuracy decreasing by 11.41 %, 2.14 %, 31.73 % and 9.88 %, respectively. For the three modeling methods, the corresponding finest numerical error ratio is 2.7 %, 2.42 % and 2.05 % in order. It is worth noting that although the smallest error ratio could be obtained using the definition formula of 200 data points, the numerical results of the airfoil designed based on NACA4 are better in other cases. The correlation between the number and source of data points of the definition formula and the calculation precision is further analyzed. Airfoil tools offers 132 data points, with the data points increasing to 264, the numerical accuracy decreases by 4.55 %. NACA4 offers 200 data points, and the increase in data amount also results in a decrease in numerical accuracy of 19.71 %. When using 200 and 400 data points, the optimal numerical error ratios are 2.05 % and 2.454 %, respectively. These ratios are superior to those based on 132 and 264 data points. In summarize, unlike existing research conclusions [2829], firstly, there exists a significant decrease in accuracy may occur due to an incorrect shape of the trailing edge in NACA0012, the maximum value of which is up to 31.73 %. The sharp trailing edge is recommended, and the number of data points adopted for NACA0012 modeling is the key to the selection of definition formula or NACA4. Secondly, the performance of data points provided by NACA4 is superior to that provided by Airfoil tools, and there is no positive correlation between the data points number and the calculation accuracy. Lastly, it is recommended to use the definition formula that utilizes NACA4's 200 points to design the sharp trailing edge shape.
3.3.1. Far field distance, turbulence model and flux type
Fig. 17 depicts the optimal numerical error ratios comparison among far field distances, turbulence models and flux types. As to the far field distance, the numerical precision increases by 15.42 %, with the far field distance increasing from 12$L$ to 16$L$. But with the far field distance reaching 20L, the simulation precision declines by 19.88 %. When considering the turbulence model, the SST komega model proves to be more effective than the SA model at far field distances of 12$L$ and 20$L$.
Fig. 16The optimal numerical error ratio comparison among six NACA0012 models
Fig. 17The optimal numerical error ratio comparison among far field distances, turbulence models, and flux types
The numerical accuracy is also enhanced by 14.3 % and 17.54 %, respectively. However, at 16$L$ far field distance, the SA model obtains the smallest error ratio. From the perspective of flux type, compared with AUSM, the numerical performance of ROE is better. The maximum increase is 37.3 %, and the minimum increase is 22.68 %. In summary, unlike the research conclusions in Ref. [29], firstly, the increase in far field distance is not necessarily positively correlated with the calculation accuracy. Keeping 16$L$ far field distance is recommended. Secondly, the turbulence model selection is associated with the distance of the far field, and according to the ideal value of far field distance, it is recommended to prioritize the SA model. Lastly, the ROE flux type is preferred. Therefore, under hypersonic conditions, the preferred simulation configurations of NACA0012 are the sharp trailing edge (definition formula adopting 200 data points) + 16$L$ + SA turbulence model + ROE, with the ${R}_{cell}$ and aspect ratio values of nearwall mesh near the shock are 16 and 380.
4. Simulation scheme and aerodynamic environment prediction
To simplify the analysis, it is assumed that the flight speed is increased uniformly, and the acceleration process could be completed instantaneously without considering the influence of fuel and engine performance. According to the flight path described in Fig. 1, the flight process is divided into subphases in seconds. Maintain a constant speed within each subphase and complete the acceleration process instantly when entering the next subphase. The hypersonic conditions can be divided into 11 subphases, which first undergo the accelerated flight for 10 s (hypersonic 1hypersonic 10), and then maintain the steady flight at the same height and speed when the flow velocity reaches 6.5 Ma (hypersonic 11). According to the conclusion drawn from the detailed discussion in Section 3.2, an effective simulation scheme for the aerodynamic environment prediction under hypersonic conditions characterized by NACA0012 is shown in Table 26. The applied computational external flow field is shown in Fig. 3(a).
Table 26The simulation scheme under hypersonic conditions characterized by NACA0012
Simulation scheme  Values  
Hypersonic conditions  hypersonic1 $M$ = 5.100 ${P}_{t}$ = 3467 Pa ${T}_{t}$ = 219.65 K $\mu $ = 1.438e5 Pa·s $\rho $ = 0.055 kg/m^{3}  
hypersonic2 $M$ = 5.250 ${P}_{t}$ = 3218 Pa ${T}_{t}$ = 220.15 K $\mu $ = 1.441e5 Pa·s $\rho $ = 0.051 kg/m^{3}  
hypersonic3 $M$ = 5.400 ${P}_{t}$ = 2972 Pa ${T}_{t}$ = 220.65 K $\mu $ = 1.444e5 Pa·s $\rho $ = 0.047 kg/m^{3}  
hypersonic4 $M$ = 5.550 ${P}_{t}$ = 2753 Pa ${T}_{t}$ = 221.15 K $\mu $ = 1.446e5 Pa·s $\rho $ = 0.043 kg/m^{3}  
hypersonic5 $M$ = 5.700 ${P}_{t}$ = 2549 Pa ${T}_{t}$ = 221.65 K $\mu $ = 1.449e5 Pa·s $\rho $ = 0.040 kg/m^{3}  
hypersonic6 $M$ = 5.850 ${P}_{t}$ = 2361 Pa ${T}_{t}$ = 222.15 K $\mu $ = 1.452e5 Pa·s $\rho $ = 0.037 kg/m^{3}  
hypersonic7 $M$ = 6.000 ${P}_{t}$ = 2188 Pa ${T}_{t}$ = 222.65 K $\mu $ = 1.454e5 Pa·s $\rho $ = 0.034 kg/m^{3}  
hypersonic8 $M$ = 6.125 ${P}_{t}$ = 1880 Pa ${T}_{t}$ = 223.54 K $\mu $ = 1.459e5 Pa·s $\rho $ = 0.029 kg/m^{3}  
hypersonic9 $M$ = 6.250 ${P}_{t}$ = 1610 Pa ${T}_{t}$ = 224.53 K $\mu $ = 1.465e5 Pa·s $\rho $ = 0.025 kg/m^{3}  
hypersonic10 $M$ = 6.375 ${P}_{t}$ = 1390 Pa ${T}_{t}$ = 225.52 K $\mu $ = 1.470e5 Pa·s $\rho $ = 0.021 kg/m^{3}  
hypersonic11 $M$ = 6.500 ${P}_{t}$ = 1197 Pa ${T}_{t}$ = 226.51 K $\mu $ = 1.475e5 Pa·s $\rho $ = 0.018 kg/m^{3}  
Grid strategy  hypersonic1 ${C}_{air}$ = 297 m/s ${U}_{t}$ = 1516 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic2 ${C}_{air}$ = 298 m/s ${U}_{t}$ = 1562 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic3 ${C}_{air}$ = 298 m/s ${U}_{t}$ = 1608 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic4 ${C}_{air}$ = 298 m/s ${U}_{t}$ = 1655 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic5 ${C}_{air}$ = 299 m/s ${U}_{t}$ = 1702 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic6 ${C}_{air}$ = 299 m/s ${U}_{t}$ = 1748 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic7 ${C}_{air}$ = 299 m/s ${U}_{t}$ = 1795 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic8 ${C}_{air}$ = 300 m/s ${U}_{t}$ = 1836 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic9 ${C}_{air}$ = 300 m/s ${U}_{t}$ = 1878 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic10 ${C}_{air}$ = 301m/s ${U}_{t}$ = 1920 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
hypersonic11 ${C}_{air}$ = 302m/s ${U}_{t}$ = 1961 m/s ${y}^{+}=$ 0.3 (${R}_{cell}=$ 16) ${y}_{H}=$ 2e6 m as ratio = 380  
Numerical method  Turbulence model  Spalartallmaras: turbulent viscosity ratio 1 
Materials  Density: idealgas; viscosity: sutherland law; Cp (j/kgk): 1006.43  
Solver  Densitybased solver adopting ROE flux type  
Gradient: leastsquares cellbased; Flow: secondorder upwind  
Modified turbulent viscosity: secondorder upwind 
Fig. 18The aerodynamic environment prediction under hypersonic condition characterized by NACA0012
The aerodynamic environment prediction characterized by NACA0012 under hypersonic conditions is shown in Fig. 18. Firstly, the change of aerodynamic heat is analyzed. The maximum aerodynamic heat rises to 2130 K (1856.85 °C), which was consistent with the aerodynamic heat descriptions in Refs. [40], [41]. The temperature increases by 847 °C, with an average temperature change rate of about 77 °C/s. With the flow velocity increases, the aerodynamic thermal variation amplitude and average change rate are higher. The temperature extreme point is in front of the leading edge, and the temperature at the leading edge is higher than that at the trailing edge. Then, the aerodynamic sound pressure is analyzed. The minimum and maximum sound pressure levels are 130.8 dB and 145.3 dB. Fig. 19 demonstrates the distribution of frequencies in acoustic signals. Only in hypersonic 3 the middle and high frequencies above 100 Hz have contribution to the sound signal, and the middle and high frequencies in other substages are basically negligible. In general, it is the lowfrequency signal within 100 Hz dominate under hypersonic conditions. Finally, the vibration analysis is conducted, and we first need to ascertain the material composition of the NACA0012 airfoil. The X43 A and X51 A are representative scramjetpropelled hypersonic vehicles, with the airfoil of the former utilizing haynes230 and that of the latter employing inconel718. In this paper, we adopt these two materials and the standard aluminium to analyze the influence of materials on vibration. Fig. 18 shows that Inconel718 has better compression and temperature resistance and could withstand relatively minimum average vibration acceleration and deformation amplitude. The NACA0012 with Inconel718 alloy experiences a maximum vibration acceleration ranges of 120 g to 182 g. Compared to aluminium alloy, the adoption of Inconel718 results in a vibration/deformation optimization range of 21.99 % to 28.02 % and 18.90 % to 22.99 %, respectively. As the flow velocity and temperature increase, both vibration acceleration and deformation also increase, leading to a decrease in the optimization degree achieved by Inconel718. Compared with aluminium alloy, Inconel718 still could bring a minimum optimization amplitude of 21.99 % for vibration acceleration and 18.90 % for deformation. The utilization of novel materials can yield considerable enhancements in structural reliability. To sum up, the scramjetpropelled vehicle encounters significant aerodynamic challenges under hypersonic conditions during the flight.
Fig. 19Distribution of acoustic signals of different frequencies under hypersonic conditions characterized by NACA0012
5. Conclusions
Aiming at the insufficient research on the aerodynamic environment prediction of scramjetpropelled vehicles characterized by NACA0012 under hypersonic conditions, in this paper, based on the wind tunnel experimental data, a comprehensive analysis is performed to study the influences of key simulation parameters on numerical accuracy and an effective simulation scheme for aerodynamic environment prediction under hypersonic conditions characterized by NACA0012 is proposed. In Section 2, we introduce the adopted grid strategy and numerical method. In Section 3, based on six NACA0012 models, three far field distances, two turbulence models, and two flux types, with additional three cell Reynolds numbers, and four aspect ratios, CFD simulations are conducted and the internal relationship between simulation parameters and numerical accuracy is discussed by comparing the numerical results with the wind tunnel data. Characterized by NACA0012, the optimal simulation configuration under hypersonic conditions is derived and the corresponding aerodynamic environment prediction is carried out in Section 4. Through systematic analysis, the study findings are as follows:
1) Compared with the blunt trailing edge, better numerical results could be obtained with the sharp trailing edge. It's worth noting that an incorrect sharp trailing edge modeling method could result in a higher numerical error ratio than a blunt trailing edge. Therefore, it's essential to select the sharp trailing edge modeling method with great care. Preference is given to the definition formula for designing the sharp trailing edge. The source and number of data points used by the definition formula would directly affect the numerical results. In this paper, the data points used in the definition formula are derived from Airfoil tools and NACA4, and the numerical analysis indicates that NACA4’s data points perform better. These two modeling data point sources have a negative correlation between the data points number and the numerical accuracy.
2) Unlike under incompressible conditions, the recommended values for far field distance and flux type are 16L and ROE flux type, respectively. In particular, the appropriate modified turbulent viscosity for SpalartAllmaras turbulence model is second order upwind. Moreover, unlike taking the blunt cylinder as the characterized object, the proposed values of cell Reynolds number and aspect ratio of airfoil mesh cells close to the shock are no smaller than 16 and no larger than 380, respectively.
3) The extreme aerodynamic environments place high demands on ground environment testing. The temperature of heating device can reach 1900 ℃ and the maximum temperature rise rate can reach 77 ℃/s. The vibration table supports a maximum vibration acceleration of 182 g and the maximum sound pressure level of the sound test reverberation room reaches 145 dB, mainly containing lowfrequency signals within 100 Hz. In addition, the influence of aerodynamic heat on the structural vibration increases with the increase of flow velocity, indicating thermal vibration testing should be conducted jointly during environmental testing to better evaluate the structural stability of the vehicle.
4) The utilization of advanced materials like Haynes230 and Inconel718 plays a pivotal role in enhancing structural reliability. Although the optimization efficacy of these materials diminishes with escalating flow velocity and aerodynamic heat, the growth remains considerable. Inconel718 having better compression and heat resistance performance, which is recommended for airfoil design.
5) The research conclusions proposed in this paper provide the basis for the parameter selection and simulation scheme design for predicting the extreme aerodynamic environment experienced by the hypersonic vehicle during the flight. Researchers could obtain more accurate environmental extremes, conduct more efficient structural design and ground testing, avoid redundant protection design, reduce costs, and improve efficiency.
6) The flight trajectory of hypersonic vehicle spans transonic, supersonic, high supersonic and hypersonic four stages. In this paper, we discuss the hypersonic conditions occupying over 90 % of the flight external flow field, while the other three external flow field are not investigated. To further improve the predictive accuracy of vehicle flight environment and provide more accurate numerical references for vehicle design and optimization, future research directions should be firstly to analyze the internal relationships between simulation parameters and numerical accuracy under the other three external flow fields in order, then perform a complete flight environment prediction based on the optimal simulation configurations concluded under four external flow fields.
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About this article
This research is supported by Anhui Provincial Department of Education Natural Science Key Project (2023AH053014) and Anhui Public Security College Excellent Scientific Research and Innovation Team Project (2023GADT06).
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Fangli Ding: Conceptualization, methodology, investigation, resources, validation, visualization, writingoriginal draft preparation. Lu Yang: Conceptualization, methodology, investigation, formal analysis, resources, validation, visualization, writingreview and editing.
The authors declare that they have no conflict of interest.