Waverider Configuration Design With Variable Shock Angle

Waverider Configuration Design With Variable Shock Angle

LILI CHEN*, ZHENG GUO, XIAOLONG DENG, ZHONGXI HOU, AND WENKAI WANG*
College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, China
Corresponding author: Zheng Guo (guozheng@nudt.edu.cn)

Abstract

This work was supported in part by the Hunan Provincial Innovative Foundation for Postgraduate of China under Grant CX2016B004, in part by the National Natural Science Foundation of China under Grant 61703414, and in part by the Hunan Natural Science Foundation under Grant 2018JJ3587.

Abstract

A novel waverider design methodology is proposed with a variable shock angle based on the osculating cones’ theory. The expected shape with special requirements can be generated by arranging the distribution of shock angle using the new method. Contrast investigation is conducted on three waveriders, which are designed with constant, decreasing and increasing shock angle distribution starting from symmetry plane at the average shock angle of $12^{\circ}$. The feasibility of the proposed approach is validated by the computational fluid dynamics (CFD) simulation. The results show that the distribution of shock angle has a significant influence on the lift-to-drag ratio and volumetric efficiency. Waverider with decreasing shock angle owns the greater volume and volumetric efficiency but with weak aerodynamic performance. The waverider with increasing shock angle has a higher lift-to-drag ratio and enhanced the static stability but with lower volumetric efficiency. The performance of waveriders with variable shock angle under off-design and blunted conditions is also investigated, which reveals a great overall aerodynamic performance and good robustness.

INDEX TERMS Hypersonic waverider, variable shock angle, lift-to-drag ratio, off-design condition, bluntness.

I. INTRODUCTION

Hypersonic vehicles [1], [2] with high lift-to-drag ratio (L/D) has been pursued for decades. A waverider is a type of hypersonic lift body which characterizes with high L/D [3]-[5]. The first " $\Lambda$ " waverider was designed by Nonweiler [6] based on two-dimensional flowfield. Initially, the volumetric efficiency of waverider was low and had no practical engineering value, so it was not widely recognized by designers. Until the 1980s, Rasmussen [7] and Rasmussen et al. [8] generated a “cone-derived waverider” from given conical flowfield, which improved the alternative shapes and properties of previous waverider. After that, Sobieczky et al. [9] proposed the osculating cones waverider method with a general shock wave shape of the lower surface, which accelerated the development of waverider design all over the world.

In the past few decades, the waverider configuration has become one of the main options for hypersonic vehicles.

^[1]Meanwhile, some researchers proposed several new design concepts to widen the design method of hypersonic waverider. For instance, He et al. [10] proposed a waverider using the general curved cone as the osculating flow field, named osculating general curved cone (OCC) waverider, the results proved that the OCC waverider had good volumetric characteristics and good flow compression abilities to satisfy the inlet requirements for hypersonic air-breathing vehicles. Ding et al. [11], [12] designed a waverider generated from axisymmetric supersonic flow past a pointed von Karman ogive, named “von Karman waveriders”, he also compared the aerodynamic performance between the new waverider and conventional cone-derived waverider using CFD method, the results showed that the “von Karman waveriders” possessed higher lift-to-drag ratio but lower volumetric efficiency than conventional cone-derived waverider. Liu et al. [13] investigated a wide-ranged multistage morphing waverider design method based on conical flowfield. To improve the performance of hypersonic vehicles with large volume, Cui et al. [14] proposed a family of aircraft with high-pressure capturing wing (HCW) configurations,

the numerical results demonstrated that HCW could gain considerable lift-to-drag ratio and higher volumetric efficiency. In addition, Rodi [15] exploited vortex lift to generate a waverider by the osculating-cone waverider method and geometrical relationships. These waveriders with specific leading edge sweep angles could produce strong leading edge vortices, these vortices could produce low pressure on the leeward surface. Based on dual-cone-derived waveriders, Liu et al. [16] designed a hypersonic gliding-cruising vehicle to achieve vehicle fly at two regimes. Li et al. [17], [18] proposed a parallel vehicle design method and variable Mach number method for a wide-ranged Mach number to improve the overall performance of the aircraft. Zhao et al. [19] put forwards a variable Mach number design approach based on the osculating cone theory and found that the waverider had superior low-speed aerodynamic performance and high-speed aerodynamic performance. Wang et al. [20] expanded waverider design method with a controllable planar shape to inspire the design of wide-speed-range vehicles. Chen et al. [21] proposed the idea of volume-improved osculating-cone waveriders (VOCW), the method was different from conventional design method of osculating-cone waverider, which could release more geometrical constraints by introducing a new design curve named curve of center (COC). The method could design more shapes with expected features by changing the center and radius of shock. The results showed good aerodynamic performance and higher volumetric efficiency.
Although numerous design methods for waverider configuration have been studied, the novel method with variable shock angle has rarely been reported in the open literature. As reported, the shock intensity has significant effect on aerodynamic performance and is directly influenced by shock angle [22], [23]. In addition, the volumetric efficiency is also closely linked with shock angle [24]. For hypersonic waveriders, volume distribution directly create the strong shock waves which should be incorporated into the design process [25]. Therefore, it is very important and interesting to investigate the shock angle distribution on waverider performance. In this study, the design method with variable shock angle is introduced. Then, the waveriders generated from three different shock angle distribution are numerically analyzed and compared on both inviscid and viscous conditions to examine the features of L/D and volumetric efficiency. Furthermore, the aerodynamic performance and volumetric efficiency of off-design condition and blunted prototypes are analyzed. Finally, a summary of the obtained results is provided.

II. DESIGN METHOD

A. BASIC FLOWFIELD

For the hypersonic inviscid conical flow, Taylor and Maccoll [26] described the hypersonic airflow around circular cone as Eq.(1). The Taylor-Maccoll equation is an ordinary differential equation with only one dependent variable,

FIGURE 1. Generating diagram of cone-derived waverider.
the dimensionless expression by the critical sound velocity is given as Eq.(2). The equation must be solved numerically to determine the flow field. In this paper, the fourth-order Runge-Kutta numerical integration method is used to solve the flowfield.

$$\begin{aligned} & \frac{d v_{r}}{d \theta}\left(v_{r} \frac{d v_{r}}{d \theta}+\frac{d v_{r}}{d \theta} \cdot \frac{d^{2} v_{r}}{d \theta^{2}}\right)-\frac{k-1}{2}\left(v_{\max }^{2}-v_{r}^{2}-\left(\frac{d v_{r}}{d \theta}\right)^{2}\right) \\ & \cdot\left(\frac{d^{2} v_{r}}{d \theta^{2}}+\cot \theta \frac{d v_{r}}{d \theta}+2 v_{r}\right)=0 \\ & \left\{\begin{array}{l} \frac{d v_{r}^{*}}{d \theta}=v_{\theta}^{*} \\ \frac{d v_{\theta}^{*}}{d \theta}=\frac{\left(c / c^{*}\right)^{2}\left(v_{r}^{*}+v_{\theta}^{*} \cot \theta\right)}{v_{\theta}^{* 2}-\left(c / c^{*}\right)^{2}}-v_{r}^{*}=f\left(v_{r}^{*}, v_{\theta}^{*}, \theta\right) \end{array}\right. \end{aligned}$$

where $c^{*}$ is the critical sound velocity of freestream, $T_{t}$ is the total temperature, $\gamma$ is specific heat ratio.

$$\begin{aligned} c^{*} & =\sqrt{\frac{2 \gamma R}{\gamma+1} T_{t}} \\ T_{t} & =\left(1+\frac{\gamma-1}{2} M a^{2}\right) \cdot T \\ \left(c / c^{*}\right)^{2} & =\frac{\gamma-1}{2} \cdot\left(\frac{\gamma+1}{\gamma-1}-v_{r}^{* 2}-v_{\theta}^{* 2}\right) \end{aligned}$$

Fig. 1 shows the schematic diagram of generating conederived waverider from given conical flowfield, $\delta$ is half angle of the basic cone, $\beta$ is the angle of shock wave, $\theta$ is the angle measured from the axis of the generating cone to area of interest in the flowfield. The upper surface is typically constructed by tracing the trailing edge inversely to intersect the conical shock surface, the compression surface is obtained by tracing the streamlines rearward to intersect the baseplane. In this paper, the Z -axis is direction of the inflow, the X -axis is perpendicular to Z -axis and points downward in the symmetry. The Y -axis is pointed in the spanwise direction.

B. WAVERIDER DESIGN WITH VARIABLE SHOCK ANGLE

In this paper, the basic waverider is generated based on the design method [21], for the design methodology, the waverider is determined by three curves, which are the Flow Capture Curve (FCC), Inlet Capture Curve (ICC) and the corresponding Curve of center (COC) of exit shock.

FIGURE 2. Design method of waverider with variable shock angle.

FIGURE 3. Sketch for the design of variable shock angle waverider.
The basic philosophy of the method is to relieve the partial geometric constraints of the conventional osculating cone waverider, and make the design parameters become more flexible. Because the method can greatly expand the design space of waverider, especially for the waverider with variable shock angle in this paper. Hence, all the waveriders are generated by the method. The design process of waverider with variable shock angle is illustrated as follows.

As shown in Fig.2, firstly, the coordinates of the trailing edge curve of the upper surface and the exit shock curve
are determined, here that is FCC and ICC, respectively. Then the COC is defined to ensure that each trailing curve point of the upper surface corresponds a sole point on the trailing curve of lower surface, for instance $A_{i} B_{i}$ and $A_{j} B_{j}$ in Fig.3. Herein, a new parameter is defined, named as expansion angle $\varphi$, based on the center of the coordinate (prescribed to be positive to the right and negative to the left). Once the three required curves are given, the basic conical flowfield and shock center for each expansion angle $\varphi_{i}$ can be determined. It is evident that the line equation through the shock radius is $x=y \cdot \cot \left(\varphi_{i}\right)$, then yield three intersection points, $O_{i}^{\prime}, A_{i}, B_{i}$ corresponding to $\varphi_{i}$ in Fig.3, of which $O_{i}^{\prime}$ indicates the center of the local exit shock, $A_{i}$ and $B_{i}$ indicating the point on the FCC and ICC, respectively. By now we can get the shock radius $O_{i}^{\prime} B_{i}$, the local basic shock can be decided by shock angle $\beta_{i}, O_{i}^{\prime}$ represents the origin of the shock. The shock angle distribution $\beta_{i}$ can be designed according to requirement for special $\varphi_{i}$, then we can obtain different waveriders with variable shock angle. The leading edge point can be produced by inverse tracing $A_{i}$, the streamline tracing technology is utilized to generated the lower surface from the leading edge, then we can obtain the trailing edge of the lower surface $C_{0}-C_{i}-C_{j}$. As a result, the configuration of waverider with variable shock angle is generated. In this paper, the FCC and ICC are given as $x=3.0+0.1885 y^{2}-0.0138 y^{4}$ and $x=-0.1583 y^{2}+5$, respectively.

For comparison, three waveriders with different shock angle distribution are attained using the new method. Fig. 4 displays the configurations of three designed waveriders, which share the same FCC, ICC and COC. Case 1 is the waverider with constant shock angle $12^{\circ}$ at each osculation plane. Case 2 is generated by gradually decreasing from $14^{\circ}$ in symmetry plane to $10^{\circ}$ in the plane of endpoint linearly. Case 3 is designed by gradually increasing from $10^{\circ}$ in symmetry plane to $14^{\circ}$ in the plane of endpoint linearly. The average shock angle $12^{\circ}$ is remained for each waverider. In Fig.2, the trailing edge curves of the three waveriders are illustrated and marked with special color and linestyle. The concrete property parameters are listed in Table.1.

FIGURE 4. Designed geometry configurations of different waveriders. (a) Case1. (b) Case2. © Case3.

TABLE 1. Geometric parameters of different waveriders.

From Table.1, we can conclude that the distribution of shock angle has a great effect on the volumetric efficiency.

III. NUMERICAL METHODS AND CODE VALIDATION

A. NUMERICAL APPROACH

The commercial software ANSYS Fluent [27] is selected as the simulation platform to numerically complete the simulation. Three-dimensional implicitly coupled Reynoldaveraged Navier-Stokes (RANS) equations are solved to obtain the results using the density-based solver. The second-order spatially Advection Upstream Splitting Method (AUSM) to the flux-vector is utilized, because it is less susceptible to Carbuncle phenomena and provides exact resolution of contact and shock discontinuities. The least-squares cell-based method is selected to compute the gradients. The SST $k-\omega$ turbulent model [28] is selected to gain the viscous results. The Courant-Friedrichs-Levy (CFL) number is set at 0.5 with proper under-relaxation factors to ensure stability. The boundary conditions imposed on the computational domain are given here. The surface of waverider is set as no slip and adiabatic wall boundary conditions. The pressure far-field boundary condition is selected in the inflow, while the pressure outlet boundary condition is set for outlet. The air is assumed to be a thermally and calorically perfect gas because air is compressible in hypersonic flow. The specific heat ratio $\gamma$ is assumed to be a constant value as 1.4. The viscosity coefficient $\mu$ is calculated by Sutherland law in Eq.(6), the thermal conductivity $k$ is determined by Prandtl number in Eq.(7).

$$\mu=1.458 \times 10^{-6} \frac{T^{1.5}}{T+110.4}$$

TABLE 2. Grid convergence analysis.

$$k=\frac{\mu C_{p}}{\operatorname{Pr}}$$

B. CODE VALIDATION

In order to valid the credibility of the numerical approach in this paper, the numerical approach is verified using the provided waverider configuration by [29], which was an optimized cone-derived waverider using the MAXWARP (Maryland Axisymmetric Waverider Program) code. Moreover, the configuration was also been analyzed using CFL3D code, which can also be used to verify the results from Fluent. For the validation, the flight Mach number is 6 and flight altitude is 30 km , the Reynolds number based on the length of the waverider is $1.294 \times 10^{8}$. The length of the waverider is 60 m . In order to investigate the effect of grid scale, the grid convergence is analyzed, the first layer grid height is set as 0.00001 m , the grid size of different scales are listed in the Table.2. The simulation of convergence analysis is performed by viscous condition. In the validation, the structured and hybrid grid are both adopted to complete and compare the results. A half ellipsoid domain is adopted to surround the waverider, the outlet extent passes through the base plane of waverider because the base drag is not numerically considered in this paper. The computational domain and generated grid are shown in Fig.5.

The grid refinement study shows little effect on the calculated results. Through the grid convergence analysis from Table.2, we can find that when the total grid numbers keeps

FIGURE 5. Computational domain and grid of waverider for verification at $\mathrm{Ma}=6$ (a) computational domain; (b) structured grid; © hybrid grid.

TABLE 3. Comparison between different methods.

FIGURE 6. Pressure contour contrast between results in this paper and results from Ref. [29].
among 1.8 to 5.3 million, the difference caused by grid scale is very little. Therefore, although a refined grid can better capture shock and gain elaborate flowfield, to enhance the computational efficiency, a moderate grid is used for the numerical analysis in the study.

From Table.3, the results show that the numerical results of this paper can be well matched with the literature whatever structured or hybrid grid. For inviscid calculation, the error between Fluent and MAXWARP is only $2.56 \%$, while for viscous calculation, the error between Fluent and CFL3D is $2.3 \%$. The predicted results of Fluent show good agreement with MAXWARP and CFL3D, which verifies the feasibility and accuracy of the numerical approach in this paper. Fig. 6 shows the comparison of the outlet pressure contours obtained by the three methods. Hence, the numerical approach used in this paper is reliable.

IV. RESULTS AND DISCUSSION

A. GRID GENERATION

The commercial software ICEM is employed to generate the grids. For the inviscid calculation, the unstructured grid is adopted in this paper, for the viscous analysis, the unstructured hybrid grid is used. The half model is adopted to save consumption time. The surface mesh size is approximately $1 / 200$ of the length of the special waverider. The grid for inviscid simulation is between 1.5 million and 2 million in number. The distance away from the body to the first cell is 0.00001 m , ensuring that $y+$ based on the Reynolds number per unit is less than 5 . To solve the boundary layer flow in

FIGURE 7. Schematic diagram of grid of the waverider (unstructured grid for inviscid analysis and hybrid grid for viscous analysis).
detail, a 20 -layers boundary layer grid is generated near the wall with a total grid number about 2.5 million. The dynamic adaptive mesh technology is used in the calculation process to achieve better shock resolution. The computational domain is basically same as that in code validation and the generated grid is as shown in Fig. 7.

B. PERFORMANCE ANALYSIS AT ON-DESIGN CONDITION

In this section, the aerodynamic performance has been analyzed by inviscid and viscous calculation. The design point is set at the freestream condition of Mach 6 at an altitude

FIGURE 8. Comparison of aerodynamic parameters of different designed waveriders. (a) Lift coefficient. (b) Drag coefficient. © Lift-to-drag ratio. (d) Relative position of the center of pressure.
of 30 km , the inflow pressure is 1200 Pa , and temperature is 226.5 K . The moment reference point is selected at the leading edge of the symmetry plane of the waverider. The projected area is selected as the reference area, and the length is selected as the reference length. The freestream flow is set as a pressure far-field boundary condition, and the outlet is set as a pressure outlet boundary condition. The wall is set as no slip and adiabatic wall boundary condition. The base pressure of waverider is considered in the analysis and the hypersonic base pressure model [30] of the base pressure coefficient, $C p$, shown in Eq.(8), is used to obtained the base pressure.

$$C p=-1 / M^{2}$$

TABLE 4. Increment percentage of aerodynamic performance of case 2 and case 3 relative to case 1 .

By comparison, it can be seen from Fig.8(a) and (b) that the lift coefficient and drag coefficient of case 2 are larger than case 1 and case3. Fig.8© reveals that the inviscid lift-to-drag ratio of case 2 is less than case 1 , the lift-to-drag ratio of case 3 is greater than case 1 at 0 degrees angle of attack. That is, case 3 owns the highest $L / D$, and case 2 is the

lowest whatever inviscid or viscous condition in an overall view. Through the analysis, we can know that the volumetric efficiency of case 3 and case 2 are the least and the largest, respectively, among the three configurations. It is obvious that there exists a contradiction between $L / D$ and volumetric feature, the tradeoff should be considered in the design process. Consequently, it can be concluded that the distribution sequence of shock angle has a great influence on aerodynamic performance of the waveriders.

By analyzing the position of center of the pressure Fig.8(d), it can be obtained that the positions of the center change remarkably due to variable shock angle. Because the three waveriders are designed with variable shock angle on the different osculation planes. For case1, the shock angle is unchanged, pressure keeps basically the same at different osculation planes, which can be noticed in Fig.11(a). However, for case2, the high pressure is distributed near the central position and close to front position along the waverider, the low pressure distribute on both sides, and thereby the center of pressure move forwards relative to case1. Similarly, the surface pressure of case 3 shows inverted distribution, the high pressure locates on the both sides and close to the outlet of the waverider, therefore its center of pressure moves backwards. When the angle of attack is 0 degrees, the position of the center of case 2 and case 3 is 0.59 and 0.725 , respectively, while case 1 is 0.645 . As the angle of attack increases, the position of center of the pressure for case 1 and case 3 will move forward, however that of case 2 is basically unchanged. Compared to previous investigations [13], conventional waverider is generally generated by constant shock angle, its center of pressure usually locates at the $65 \%$ along the length of waverider and changes at a little range. From the calculated results, the lift coefficient of the viscous calculation is basically consistent with the inviscid result, but the drag coefficient has a significant increment because of viscosity. The angle of attack for the maximum $L / D$ of inviscid and viscous analysis both appears at the angle of attack of 2 degrees. The maximum viscous $L / D$ of case 3 is about 5.8 , while that of case 1 and case 2 is 5.4 and 5.05 , respectively. At the same time, it can also be seen that the viscosity has little effect on the position of the center of the pressure.

In order to investigate the pressure distribution regulation by variable shock angle, non-dimensional pressure $P / P_{\infty}$ of three sections in the spanwise direction are extracted, shown in Fig.9. Fig. 10 depicts the obtained inviscid pressure coefficient distribution of the lower surface under different cross-sections for different waveriders. The horizontal axis is rescaled by the local length at each station. As can be seen, for case 1 , the $P / P_{\infty}$ presents slight variation at different sections along the $z$-axis direction. However, we can find that for case 2 , the pressure at $\mathrm{y}=0 \mathrm{~m}$ is evidently the highest and that at $\mathrm{y}=2.5 \mathrm{~m}$ is the lowest among the three sections. And for case3, the pressure distribution exhibits adverse trend compared with case2. It is obvious that the pressure variation is basically identical to the distribution of the shock angle,

FIGURE 9. Chosen sections of the lower surfaces for different waveriders.

FIGURE 10. Dimensionless pressure distribution of different cross-sections.

TABLE 5. The freestream condition at different Mach.

because larger shock angle leads to stronger shock intensity, the relative small shock angle produces weak shock. Overall, the pressure of case 2 gradually decreases from the symmetry plane to the two sides, and that of case 3 gradually increases from the symmetry to the sides. Consequently, we can design the shock angle to realize expected pressure distribution by the new method.

Additionally, Fig. 11 gives a comparison of the inviscid and viscous pressure contours for different longitudinal crosssections of the waveriders. According to the inviscid pressure distribution of case 1, case 2 and case3, the three waveriders analyzed can almost completely block the high-pressure region on the lower surface and satisfy good waverider characteristics. It can be seen from the viscous cross-section

©

FIGURE 11. Comparison of pressure contours at different slices. (a) Pressure contour slices of case1 (left:invisicd;right:viscous). (b) Pressure contour slices of case2 (left:invisicd;right:viscous). © Pressure contour slices of case3 (left:invisicd;right:viscous).

TABLE 6. Geometry parameters of blunted waveriders.

FIGURE 12. Comparison of pressure contour of different waveriders between inviscid and viscous results. (a) Case1. (b) Case2. © Case3.
pressure contours in Fig. 11 that there exists obvious highpressure spillage on both sides of the waveriders. Because viscosity changes the shock intensity and induces a detached bow shock, the shock wave is no longer attached to the leading edge of the body.
In order to further analyze the flow field characteristics, the pressure contours at the outlet including the inviscid and viscous results, are given in Fig.12. It can be seen that under the condition of inviscid calculation, all the configurations can limit the high pressure to the lower surface, and the waverider characteristics are well maintained. The exit shock curves of case 1, case2, and case 3 are all in good agreement with the design curve. The high pressure zone of case 2 is mainly concentrated in the median region of the lower surface, while the high pressure region of case 3 is dispersed on both sides of the lower surface. And the outlet pressure distribution of case 1 is relatively uniform, which can be a promising design candidate for an engine integrated airframe.
From Table.4, it can be seen that compared with case1, the inviscid and viscous lift-to-drag ratio of case 2 reduces

TABLE 7. The increment change of design parameters of blunted waveriders relative to the original waveriders at zero incidence.

by $10.83 \%$ and $5.15 \%$, respectively, but the volumetric efficiency increases by $6.69 \%$. The inviscid lift-to-drag ratio of case 3 increases by $7.56 \%$ with a little loss of $1.85 \%$ for viscous lift-to-drag ratio, however, the volumetric efficiency decreases by $16.22 \%$. As a result, the inviscid lift-to-drag ratio of case 2 is lower than that of case 1 and case3. The reason could be explained that the inviscid lift-to-drag ratio is related to the volumetric efficiency, the increase of the volumetric efficiency leads to the decreased lift-to-drag ratio at the ondesign condition. Therefore, the distribution of the shock angle can make a significant impact on the aerodynamic performance of the waverider, which must be considered for optimization in the design process.

C. PERFORMANCE ANALYSIS AT OFF-DESIGN CONDITION

The aerodynamic performance of off-design condition for different waveriders is conducted on the key flight Mach numbers and angles of attack. Except for Ma 6, Ma 4 and Ma 8 are also selected to complete the investigation. In order to seek for distinction caused by Mach number, the inflow dynamic pressure, $q=\frac{1}{2} \rho v^{2}=\frac{1}{2} p \gamma M a^{2}$, is fixed as same as that of Mach 6. Here, $q, \rho$ and $v$ are the dynamic pressure, density and velocity of inflow, while $p$ and $M a$ are the static pressure and Mach number of inflow, respectively. Because the Mach number is varied in this study, the static pressure magnitude shows difference. For practical flight, the flight altitude would be different according to the definition of dynamic pressure. Table. 5 lists the specific flight conditions for each Mach number case analyzed.

In Fig.13, the aerodynamic performance of different waveriders under off-design conditions is presented. The effect by different Mach number yields no deviations from expected results. As shown in Fig.13(a) and (b), the inviscid results show that as the Mach number increases, the lift coefficient and drag coefficient decrease. The lift-to-drag ratio monotomically increases with increased Mach number, shown in Fig.13©. It can be seen from Fig.13(d) that the increase of Mach number causes the center of pressure to move slightly backward, that is, the increase of the flight Mach number helps to improve the stability of the waverider. The viscous calculated results of the lift coefficient for different waveriders are the same as the inviscid. The drag coefficient decreases with the increase of Mach number. Compared with inviscid result, the viscous drag coefficient increases greatly because of the existence of viscosity. For example, for case 1, the inviscid lift-to-drag ratio of Mach number 4 is 5.26 , the viscous $L / D$ is 3.93 , the viscosity makes a loss by $25.3 \%$. When the Mach number is 8 , the inviscid

FIGURE 13. Comparison of aerodynamic parameters of different designed waveriders at different Mach. (a) Lift coefficient. (b) Drag coefficient. © Lift-to-drag ratio. (d) Relative position of the center of pressure.

FIGURE 14. Perspective view of dimensionless pressure contour at different slices of case 2 under blunted condition.

FIGURE 15. Dimensionless pressure contours of fringe surface of case2.
lift-to-drag ratio is 7.58 , the viscous lift-to-drag ratio is 5.03 , the lift-to-drag ratio decreases by $33.64 \%$ due to viscosity. However, the inviscid and viscous lift-to-drag ratio increases by $44.1 \%$ and $28.0 \%$, respectively, when the Mach number increases from 4 to 8 . In addition, from Fig.13(d), with the

FIGURE 16. Aerodynamic performance of blunted waveriders and increments relative to original configurations. (a) Lift coefficient. (b) Drag coefficient. © Lift-to-drag ratio. (d) Relative position of the center of pressure.
increase of Mach number, the position of the center of the pressure does not change significantly. Accordingly, it can be concluded that the viscosity has little influence on the position of the center of pressure.

D. BLUNTNESS IMPACT

Restricted by aerodynamic heating and manufacture techniques, the realistic waverider can not be processed with sharp leading edge [31], [32]. Bluntness makes ideal waveriders shapes changed and leads to the perturbation of flowfield. The leading radius is enlarged after bluntness, which can significantly relieve the severe aero-heating problem. Therefore, bluntness is very important and only blunted waverider could be applied to engineering for practical hypersonic configurations. Accordingly, the blunted waveriders should also be
analyzed to predict aerodynamic perfomance close to practical engineering application. Herein, the viscous analysis is executed with the bluntness radius of 10 mm in the leading edge. The ICEM CFD commercial software is employed to generate the structured grid, the first layer grid height is set to 0.00001 m and the total grid number by half computed domain is about 2.0 million. The aerodynamic reference length and area of the blunted waveriders keep consistent with original configurations.

To obtain the flow characteristics, the non-dimensional pressure distribution $P / P_{\infty}$ of case 2 at different slices is illustrated in Fig.14, where these slices are normal to the inflow direction along the waverider. It is obvious that bluntness leads to more high pressure leakage on both sides. The leading edge shock has detached from the leading edge and a

more noticeable bow shock is developed in the leading edge. In addition, Fig. 15 gives the surface pressure distribution of the fringe of the blunted waverider case 2 , it is interesting that the maximum $P / P_{\infty}$ is not located in the symmetry but away from symmetry plane and the maximum $P / P_{\infty}$ reaches 42 . Therefore, it can be concluded that the distribution of shock angle can change the high pressure distribution on the fringe, which is tightly linked with thermal protection.

Fig. 16 shows the aerodynamic force variations and increments relative to original configurations. We can see from Fig.16(a) and (b) that the lift coefficient and drag coefficient of different waveriders increase with the increasing the angle of attack. It can also be observed that bluntness makes the lift coefficient decrease and the increment percentage appears clear difference. However, bluntness leads to the increase of drag coefficient when the angle of attack is below at $2^{\circ}$, the negative increment can be noted once the angle of attack is larger than $2^{\circ}$. The increment percentage decreases with the increase of angle of attack. For case2, the drag coefficient of original and blunted configuration is 0.00773 and 0.00823 at zero incidence, respectively. After bluntness, the drag coefficient increases by $6.5 \%$. Additionally, as shown in Fig.16©, bluntness also leads to the loss of $L / D$, for instance, the $L / D$ of case 2 reduces by $8.2 \%$ primarily due to the detached shock wave. However, the trend slows down with increasing angle of attack, which can be obtained from Fig.16©. Besides, the center of pressure shifts slightly for case 1 and case 2 under blunted condition so the change can be neglected. But the change of case 3 is close to $4 \%$ and should be considered in the engineering application. From above results, the difference by the distribution of shock angle can lead to different intrinsic properties of waveriders. Therefore, the waveriders with variable shock angle should be treated differently.

v. CONCLUSION

In this paper, a novel waverider with variable shock angle based on osculating cones theory is proposed. Three different waveriders are designed and discussed. According to the study, we can obtain the following conclusions:

1. The new waverider can limit the high-pressure to the lower compression surface and maintain good waverider properties. Simultaneously, the new method has the ability to adjust volume and volumetric efficiency at a wide range.
2. The waverider with increasing shock angle can improve $L / D$ and static stability, but with low volumetric efficiency. The waverider with decreasing shock angle appears high volumetric efficiency but with low $L / D$. The waverider with decreasing shock angle shows greater lift and drag coefficient relative to waveriders with increasing or constant shock angle.
3. Under off-design conditions, $L / D$ increases with increasing Mach number, the relative positions of center of pressure show slight changes. Under same blunted conditions, the volumetric efficiency of case 2 shows greater increment than case 1 and case3,

while the negative increment of $L / D$ for case 2 is greater than case 1 and case3.

APPENDIX

CFD Computational Fluid Dynamics
HCW High-pressure Capturing Wing
FCC Flow Capture Curve
ICC Inlet Capture Curve
COC Curve of Center for the exit shock
$X c p \quad$ the position of center of pressure
$C_{l} \quad$ lift coefficient
$C_{d} \quad$ drag coefficient
$L / D \quad$ lift-to-drag ratio
$y+$ dimensionless distance from the wall
$L$ length of vehicle, m
$b$ width of vehicle, m
$\mu \quad$ viscosity coefficient, $\mathrm{kg} /(\mathrm{m} \cdot \mathrm{s})$
$\eta \quad$ volumetric efficiency
$k \quad$ thermal conductivity, $\mathrm{W} /(\mathrm{m} \cdot \mathrm{K})$
Pr Prandtl number
$\beta \quad$ shock angle, ${ }^{\circ}$
$\eta \quad$ volumetric efficiency
$S_{p} \quad$ projected area of vehicle, $\mathrm{m}^{2}$
$S_{\text {wet }} \quad$ wetted area of vehicle, $\mathrm{m}^{2}$
$S_{b} \quad$ base area, $\mathrm{m}^{2}$
$\gamma \quad$ specific heat ratio
$\alpha \quad$ angle of attack, ${ }^{\circ}$

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LILI CHEN received the B.S. degree in aircraft system and engineering and the M.S. degree in aeronautical and astronautical science and technology from the National University of Defense Technology (NUDT), China, in 2013 and 2015, respectively, where he is currently pursuing the Ph.D. degree in aeronautical and astronautical science and technology with the College of Aerospace Science and Engineering under the supervision of Prof. Z. Guo. He has authored or coauthored several articles in the field of aerodynamic configuration design and CFD. His current research interests include aerodynamic design for hypersonic cruise vehicles.

ZHENG GUO received the Ph.D. degree in computational fluid dynamics from the National University of Defense Technology (NUDT), in 2002. He was a Visiting Academic with the University of Cambridge, in 2009. He is currently a Professor and the Head of the UAV Technology Research Group, College of Aerospace Science and Engineering, National University of Defense Technology. His current research interests include the conceptual design and aerodynamic configuration design of near-space vehicles and UAVs, computational aerodynamics, fluid and structure interaction, and hypersonic vehicle design. He has published more than 70 academic papers and three text books. He holds 30 authorized patents.

XIAOLONG DENG received the Ph.D. degree in engineering physics from Ghent University, Belgium, in 2015. He is currently a Lecturer with the National University of Defense Technology of China. His current research interests include the conceptual design and dynamics of near-space vehicles.

ZHONGXI HOU received the Ph.D. degree in computational fluid dynamics from the National University of Defense Technology (NUDT), China, in 2000, where he is currently a Professor in aeronautical and astronautical science and technology and also an Associate Dean of the College of Aerospace Science and Engineering. He has authored a monograph and over 40 international journals. His current research interests include the aircraft conceptual design of high-altitude longendurance (HALE) solar-powered unmanned aircraft systems (UAS) and hypersonic vehicle design.

WENKAI WANG received the B.S. degree in space engineering and the M.S. degree in aeronautical and astronautical science and technology from the National University of Defense Technology (NUDT), China, in 2012 and 2014, respectively, where he is currently pursuing the Ph.D. degree in aeronautical and astronautical science and technology under the supervision of Prof. Z. Hou. He has authored or coauthored several articles in the field of aircraft design and control. His current research interests include preliminary design, flight dynamics, and optimal control for hypersonic cruise vehicles.