Published: 01 August 2024

An effective simulation scheme for the prediction of aerodynamic environment under hypersonic conditions characterized by NACA0012

Fangli Ding1
Lu Yang2
1College of Electrical Engineering, Tongling University, Tongling, China
2Department of Public Security Science and Technology, Anhui Public Security College, Hefei, China
Corresponding Author:
Lu Yang
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Abstract

Currently, aerodynamic environment prediction research into scramjet-propelled vehicles characterized by NACA0012 under hypersonic conditions is relatively sparse. Two-dimensional 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, Spalart-Allmaras, 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.

An effective simulation scheme for the prediction of aerodynamic environment under hypersonic conditions characterized by NACA0012

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 ground-to-orbit reentry, the low-orbit reentry and the scramjet-propelled [1] and the reusable reentry vehicle is the leading research object [2]-[7], while the prediction research for the scramjet-propelled vehicle is relatively sparse. Hence, we focus our research on the scramjet-propelled 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 scramjet-propelled 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 sub-combustion, and the super-combustion 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 scramjet-propelled 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 scramjet-propelled vehicle during flight, enabling enhanced vehicle optimization and thermal protection design.

Fig. 1The typical flight trajectory of the scramjet-propelled hypersonic vehicle

The typical flight trajectory of the scramjet-propelled 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 k-omega and Spalart-Allmaras (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 k-omega 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 k-epsilon turbulence model and Mach number 0.7 to evaluate the NACA0012 anti-icing 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 scramjet-propelled 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 Pt is the static pressure of the free stream. T is the local static temperature, and Tt is the static temperature of free stream. U is the local velocity, and Ut is the velocity of free stream. We adopt the wind tunnel data of P/Pt, T/Tt and U/Ut from Ref. [13] as the reference data, where the location range of P/Pt and T/Tt is x/L [–0.007 0] and the location range of U/Ut is y/L[0.01 0.1] at x/L = 0.95. The initial values are as follows: Reynolds number (Re) is 10e6, the Mach number (Ma) is 10, Pt is 576 Pa, Tt is 81.2 K, and the wall temperature of airfoil (TW) 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 16-core 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 Y-axis 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:

1
y=±0.59470.2983x1/2-0.1271x-0.3579x2+0.2920x3-0.1052x4.

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 12-20 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 12L, 16L, and 20L 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 12L/16L/20L 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 no-slip, 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 wall-bounded flows is heavily influenced by the near-wall 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

Six NACA0012 models

Fig. 3The two types of computational domains

The two types of computational domains

a) Computational domain of the sharp trailing edge and the grid division

The two types of computational domains

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 (yH) being directly correlated to y+ and the calculation process is displayed in Fig. 4. The calculations of Re, Ut, air speed (Cair), friction coefficient (Cf), shear stress (τW), friction velocity (μt), and the distance from the wall to the centroid of the wall adjacent cells (yp) are shown from Eqs. (2-8):

2
Re=ρUtLμ,
3
Ut=Cair×Ma,
4
Cair=20.05Tt,
5
Cf=[2log10(Re)-0.65]-2.3,
6
τW=0.5×ρU2Cf,
7
uτ=τWρ1/2,
8
yp=y+μuτρ.

where Tt is 81.2 K and Pt is 576 Pa, the flow density (ρ) is about 0.0247 kg/m3. Cair is 180.6 m/s and Ma is 10. Hence, Ut is 1806 m/s. Re is 10e06, ρ, L and Ut are substituted into Eq. (2) and flow viscosity (μ) is about 4.46082 Pa·s. Then we could solve yp according to Eqs. (5-8). At last, yH is calculated according to Eq. (9):

9
yH=2yp.

Fig. 4The calculation process of yH

The calculation process of yH

Since the empirical formula (Cf) is used in the above calculation process [34], yH 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 yH is 4.6e-5 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 yH is 1.4e-5 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 yH could potentially lead to a large aspect ratio value. This scenario may result in floating-point 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 12L/16L/20L 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)

Mean 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 k-omega model and ROE flux type as an example. The numerical results of P/Pt, T/Tt, U/Ut at location ranges and related error ratios with wind tunnel data are shown in Fig. 5. At 12L far field distance, the average error ratios of P/Pt, T/Tt, and U/Ut 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/PtT/TtU/Ut) 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 20L 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. 8-9.

Fig. 6Mean error ratios of P/Pt, T/Tt, U/Ut at three far field distances under three grid levels (Airfoil tools)

Mean 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)

Mean 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

The mesh views of the sharp trailing edge

Fig. 9The mesh views of the blunt trailing edge

The 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 time-consuming, necessitating the averaging of the NS equation to reduce turbulence components. The Reynolds Averaged Navier-Stokes (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 k-epsilon, SST k-omega, and Spalart-Allmaras (SA) turbulence models are widely accepted and relatively accurate for most numerical simulation applications [36]. However, the k-epsilon 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 k-omega model demonstrates superior performance over the k-epsilon model in predicting adverse pressure gradients and boundary layer flow, showcasing its enhanced capabilities. Furthermore, the SST k-omega model effectively addresses the sensitivity issue of the original k-omega model to freestream conditions, thereby enhancing its applicability [37]. Moreover, the Spalart-Allmaras (SA) turbulence model is specifically tailored for aerospace applications involving wall-bounded flows and exhibits exceptional predictive capabilities for adverse pressure gradient boundary layers [38]. In conclusion, we employ SA and SST k-omega models.

Regarding the INLET boundary, when utilizing the SST k-omega, 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/Pt, T/Tt, and U/Ut at location ranges and related error ratios with wind tunnel data are shown in Tables 2-4. For the first order, under three far field distances, the mean error ratios of (P/Pt, T/Tt and U/Ut) 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/PtT/TtU/Ut) 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 first-order upwind and second–order upwind of the modified turbulent viscosity

Type of upwind order
x/L locations (m) and P/Pt wind tunnel data (P/Pt 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
First-order
12L
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 %
16L
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 %
20L
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 %
Second-order
12L
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 %
16L
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 %
20L
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 first-order upwind and second-order upwind of the Modified turbulent viscosity

Type of upwind order
x/L locations (m) and T/Tt wind tunnel data (T/Ttnumerical 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
First-order
12L
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 %
16L
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 %
20L
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 %
Second-order
12L
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 %
16L
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 %
20L
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 first-order upwind and second-order upwind of the modified turbulent viscosity

Type of
upwind order
y/L locations (m) at x/L= 0.95 m (U/Utnumerical 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
12L
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 %
16L
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 %
20L
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
12L
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 %
16L
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 %
20L
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 (node-based/cell-based/least cell-based). Out of these three, the least cell-based scheme is advantageous because it provides comparable accuracy to the node-based scheme, has fewer computing resources, and avoids spurious oscillations. Hence, the least cell-based with the standard gradient limiter is applied. In addition, when the Mach number is bigger than 5, the density-based 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 (Pc) is 3.77 MPa, and if the ratio of P and Pc 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 Pc is about 0.019, which satisfies the condition mentioned above. Hence, flow density (ρ) 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 seventy-two sets of numerical calculations are carried out. Tables 5-22 demonstrate the numerical results of P/Pt, T/Tt, U/Ut at sampling locations and Figs. 10-15 demonstrate the corresponding numerical error ratio distributions of P/Pt, T/Tt, U/Ut 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/Pt, T/Tt, U/Ut for six NACA0012 models under different simulation configurations. Through the error ratio comparison, the optimal mean error ratio of (P/PtT/TtU/Ut) of 2.05 % could be achieved based on the configuration of sharp trailing edge (definition formula) + 16L 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/Pt wind tunnel data (P/Pt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Tt wind tunnel data (T/Tt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Ut wind tunnel data (U/Ut 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Pt, T/Tt and U/Ut are shown in Tables 5-7 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+12L 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

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of blunt trailing edge

a) Numerical result distribution of P/Pt of the blunt trailing edge

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of blunt trailing edge

b) Error ratio distribution of P/Pt of the blunt trailing edge

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of blunt trailing edge

c) Numerical result distribution of T/Tt of the blunt trailing edge

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of blunt trailing edge

d) Error ratio distribution of T/Tt of the blunt trailing edge

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of blunt trailing edge

e) Numerical result distribution of U/Ut of the blunt trailing edge

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut of blunt trailing edge

f) Error ratio distribution of U/Ut of the blunt trailing edge

3.1.2. NACA4

The corresponding numerical results and distributions of P/Pt, T/Tt and U/Ut are shown in Tables 8-10 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/Pt wind tunnel data (P/Pt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Tt wind tunnel data (T/Tt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Ut wind tunnel data (U/Ut 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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+12L 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

The 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/Pt of the sharp trailing edge based on NACA4

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on NACA4

b) Error ratio distribution of P/Pt of the sharp trailing edge based on NACA4

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on NACA4

c) Numerical result distribution of T/Tt of the sharp trailing edge based on NACA4

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on NACA4

d) Error ratio distribution of T/Tt of the sharp trailing edge based on NACA4

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on NACA4

e) Numerical result distribution of U/Ut of the sharp trailing edge based on NACA4

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on NACA4

f) Error ratio distribution of U/Utof the sharp trailing edge based on NACA4

3.1.3. Definition formula adopting 132 points

The corresponding numerical results and distributions of P/Pt, T/Tt and U/Ut are shown in Tables 11-13 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+16L 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/Pt wind tunnel data (P/Pt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Tt wind tunnel data (T/Tt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Ut wind tunnel data (U/Ut 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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)

The 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/Pt of the sharp trailing edge based on definition formula (132 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (132 points)

b) Error ratio distribution of P/Pt of the sharp trailing edge based on definition formula (132 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (132 points)

c) Numerical result distribution of T/Tt of the sharp trailing edge based on definition formula (132 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (132 points)

d) Error ratio distribution of T/Tt of the sharp trailing edge based on definition formula (132 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (132 points)

e) Numerical result distribution of U/Ut of the sharp trailing edge based on definition formula (132 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (132 points)

f) Error ratio distribution of U/Ut 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/Pt, T/Tt and U/Ut are shown in Tables 14-16 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+16L 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/Pt wind tunnel data (P/Pt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Tt wind tunnel data (T/Tt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Ut wind tunnel data (U/Ut 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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)

The 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/Pt of the sharp trailing edge based on definition formula (264 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (264 points)

b) Error ratio distribution of P/Pt of the sharp trailing edge based on definition formula (264 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (264 points)

c) Numerical result distribution of T/Tt of the sharp trailing edge based on definition formula (264 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (264 points)

d) Error ratio distribution of T/Tt of the sharp trailing edge based on definition formula (264 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (264 points)

e) Numerical result distribution of U/Ut of the sharp trailing edge based on definition formula (264 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (264 points)

f) Error ratio distribution of U/Ut 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/Pt, T/Tt and U/Ut are shown in Tables 17-19 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/Pt wind tunnel data (P/Pt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Tt wind tunnel data (T/Tt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Ut wind tunnel data (U/Ut 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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)

The 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/Pt of the sharp trailing edge based on definition formula (200 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (200 points)

b) Error ratio distribution of P/Pt of the sharp trailing edge based on definition formula (200 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (200 points)

c) Numerical result distribution of T/Tt of the sharp trailing edge based on definition formula (200 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (200 points)

d) Error ratio distribution of T/Ttof the sharp trailing edge based on definition formula (200 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (200 points)

e) Numerical result distribution of U/Ut of the sharp trailing edge based on definition formula (200 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (200 points)

f) Error ratio distribution of U/Ut 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/Pt, T/Tt and U/Ut are shown in Tables 20-22 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+20L 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/Pt wind tunnel data (P/Pt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Tt wind tunnel data (T/Tt 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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/Ut wind tunnel data (U/Ut 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+12L
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+12L
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+16L
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+16L
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+20L
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+20L
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)

The 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/Pt of the sharp trailing edge based on definition formula (400 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (400 points)

b) Error ratio distribution of P/Pt of the sharp trailing edge based on definition formula (400 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (400 points)

c) Numerical result distribution of T/Tt of the sharp trailing edge based on definition formula (400 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (400 points)

d) Error ratio distribution of T/Tt of the sharp trailing edge based on definition formula (400 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (400 points)

e) Numerical result distribution of U/Ut of the sharp trailing edge based on definition formula (400 points)

The numerical result and error ratio distributions of P/Pt, T/Tt and U/Ut  of sharp trailing edge based on definition formula (400 points)

f) Error ratio distribution of U/Ut 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/Pt
T/Tt
U/Ut
Total mean error ratio of (P/Pt, T/Tt, U/Ut)
P/Pt
T/Tt
U/Ut
Total mean error ratio of (P/Pt, T/Tt, U/Ut)
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 (Rcell) of near-wall 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 Rcell 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 Rcell and aspect ratio of the near shock wave wall grid cells with NACA0012 as the characteristic object, the Rcell 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 Rcell values of 16, 8, and 4 and the related y+ and yHvalues are (0.3 1.4e-5 m), (0.15 7e-6 m), and (0.08 3.5e-6 m). Three numerical simulations are performed and the corresponding mean error ratios of (P/Pt, T/Tt, U/Ut) 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 Rcell 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 near-wall mesh close to the shock at the optimal Rcell 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 Rcell 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/Pt
T/Tt
U/Ut
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/Pt
T/Tt
U/Ut
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 [28-29], 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 12L to 16L. But with the far field distance reaching 20L, the simulation precision declines by 19.88 %. When considering the turbulence model, the SST k-omega model proves to be more effective than the SA model at far field distances of 12L and 20L.

Fig. 16The optimal numerical error ratio comparison among six NACA0012 models

The 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 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 16L 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 16L 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) + 16L + SA turbulence model + ROE, with the Rcell and aspect ratio values of near-wall 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 sub-phases in seconds. Maintain a constant speed within each sub-phase and complete the acceleration process instantly when entering the next sub-phase. The hypersonic conditions can be divided into 11 sub-phases, which first undergo the accelerated flight for 10 s (hypersonic 1-hypersonic 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 Pt = 3467 Pa Tt = 219.65 K μ = 1.438e-5 Pa·s ρ = 0.055 kg/m3
hypersonic2 M = 5.250 Pt = 3218 Pa Tt = 220.15 K μ = 1.441e-5 Pa·s ρ = 0.051 kg/m3
hypersonic3 M = 5.400 Pt = 2972 Pa Tt = 220.65 K μ = 1.444e-5 Pa·s ρ = 0.047 kg/m3
hypersonic4 M = 5.550 Pt = 2753 Pa Tt = 221.15 K μ = 1.446e-5 Pa·s ρ = 0.043 kg/m3
hypersonic5 M = 5.700 Pt = 2549 Pa Tt = 221.65 K μ = 1.449e-5 Pa·s ρ = 0.040 kg/m3
hypersonic6 M = 5.850 Pt = 2361 Pa Tt = 222.15 K μ = 1.452e-5 Pa·s ρ = 0.037 kg/m3
hypersonic7 M = 6.000 Pt = 2188 Pa Tt = 222.65 K μ = 1.454e-5 Pa·s ρ = 0.034 kg/m3
hypersonic8 M = 6.125 Pt = 1880 Pa Tt = 223.54 K μ = 1.459e-5 Pa·s ρ = 0.029 kg/m3
hypersonic9 M = 6.250 Pt = 1610 Pa Tt = 224.53 K μ = 1.465e-5 Pa·s ρ = 0.025 kg/m3
hypersonic10 M = 6.375 Pt = 1390 Pa Tt = 225.52 K μ = 1.470e-5 Pa·s ρ = 0.021 kg/m3
hypersonic11 M = 6.500 Pt = 1197 Pa Tt = 226.51 K μ = 1.475e-5 Pa·s ρ = 0.018 kg/m3
Grid strategy
hypersonic1 Cair = 297 m/s Ut = 1516 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic2 Cair = 298 m/s Ut = 1562 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic3 Cair = 298 m/s Ut = 1608 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic4 Cair = 298 m/s Ut = 1655 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic5 Cair = 299 m/s Ut = 1702 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic6 Cair = 299 m/s Ut = 1748 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic7 Cair = 299 m/s Ut = 1795 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic8 Cair = 300 m/s Ut = 1836 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic9 Cair = 300 m/s Ut = 1878 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic10 Cair = 301m/s Ut = 1920 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
hypersonic11 Cair = 302m/s Ut = 1961 m/s y+= 0.3 (Rcell= 16) yH= 2e-6 m as ratio = 380
Numerical method
Turbulence model
Spalart-allmaras: turbulent viscosity ratio 1
Materials
Density: ideal-gas; viscosity: sutherland law; Cp (j/kg-k): 1006.43
Solver
Density-based solver adopting ROE flux type
Gradient: least-squares cell-based; Flow: second-order upwind
Modified turbulent viscosity: second-order upwind

Fig. 18The aerodynamic environment prediction under hypersonic condition characterized by NACA0012

The aerodynamic environment prediction under hypersonic condition characterized by NACA0012
The aerodynamic environment prediction under hypersonic condition characterized by NACA0012
The aerodynamic environment prediction under hypersonic condition characterized by NACA0012
The aerodynamic environment prediction under hypersonic condition characterized by NACA0012
The 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 low-frequency 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 scramjet-propelled 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 scramjet-propelled 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

Distribution of acoustic signals of different frequencies  under hypersonic conditions characterized by NACA0012
Distribution 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 scramjet-propelled 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 Spalart-Allmaras 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 low-frequency 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

Received
13 March 2024
Accepted
12 July 2024
Published
01 August 2024
SUBJECTS
Flow induced structural vibrations
Keywords
simulation
hypersonic
prediction
model
Acknowledgements

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).

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Author Contributions

Fangli Ding: Conceptualization, methodology, investigation, resources, validation, visualization, writing-original draft preparation. Lu Yang: Conceptualization, methodology, investigation, formal analysis, resources, validation, visualization, writing-review and editing.

Conflict of interest

The authors declare that they have no conflict of interest.