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Aerospace Science and Technology 42 (2015) 297–308

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Aerospace Science and Technology www.elsevier.com/locate/aescte

Novel approach for design of a waverider vehicle generated from axisymmetric supersonic flows past a pointed von Karman ogive Feng Ding a,b , Jun Liu a,b,∗ , Chi-bing Shen a,b , Wei Huang a,b a b

College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 31 July 2014 Received in revised form 26 January 2015 Accepted 28 January 2015 Available online 2 February 2015 Keywords: Hypersonic vehicle Waverider von Karman ogive Method of characteristics Streamline tracing

a b s t r a c t This paper proposes a novel waverider generated from axisymmetric supersonic flows past a pointed von Karman ogive. First, a pointed von Karman ogive is obtained by pointing the blunt nose of the original von Karman ogive. Then, the axisymmetric supersonic flow field past the pointed von Karman ogive is calculated using the method of characteristics, and this calculated flow field is employed as the new basic flow field of a waverider. A novel waverider is generated from this new basic flow field by a streamline tracing technique that is based on left-running Mach lines. Second, numerical methods are employed to validate the applicability of this new design concept to an aerodynamic configuration. Finally, a comparison model is designed using the conventional design concept and its performance is numerically predicted to analyze the differences between the new and conventional design concepts. The comparison results show that the novel waverider possesses higher lift-to-drag ratios, smaller trim drag, and larger internal volume than the conventional waverider; the results also demonstrate that the proposed design concept can provide greater flexibility in the design and optimization of hypersonic waverider vehicles.  2015 Elsevier Masson SAS. All rights reserved.

1. Introduction Hypersonic vehicles have become one of the key research areas in the field of aerospace engineering in the 21st century in view of the importance of long-range precision strike operations [12–14,16]. A waverider is any supersonic or hypersonic lifting body that is characterized by an attached, or nearly attached, oblique shock wave along its leading edge [21]. Previous studies have shown that a waverider can be designed to have features that may be advantageous as the basis for a hypersonic configuration [15]. Optimized waveriders have been confirmed to generate higher lift-to-drag ratios than comparable lifting bodies and can provide reasonable volumetric efficiency with minimal degradation of performance [3,30]. Computational fluid dynamics (CFD) studies have shown that optimized waveriders also have acceptable off-design performance [20,27]. In the past 50 years, researchers have widened the design space of a waverider by attempting to establish choices of basic flow

*

Corresponding author at: Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, Hunan 410073, People’s Republic of China. Tel.: +86 731 84576452; fax: +86 731 84576447. E-mail address: [email protected] (J. Liu). http://dx.doi.org/10.1016/j.ast.2015.01.025 1270-9638/ 2015 Elsevier Masson SAS. All rights reserved.

fields from which the waverider is generated, and the aim of such studies has been to improve the aerodynamic performance of hypersonic vehicles, particularly their lift-to-drag ratio and volumetric efficiency. Basic flow fields can be classified into two types— axisymmetric and non-axisymmetric. In most of these studies, the basic flow fields were considered as axisymmetric flows for ease of calculation. Axisymmetric flows allow for rapid design of waveriders. Nonweiler [24], Jones et al. [18], Rasmussen [25], Corda and Anderson [5], and He et al. [11] first employed flows past a wedge, circular cone, elliptic cone, power-law body, and curved cone, respectively, as the basic flow fields to generate waveriders at a zero angle of attack. Further, Goonko et al. [10] and Mazhul [22] respectively used the convergent flow inside constricting ducts and an isentropic compression flow as the basic flow fields to generate waveriders. All the above-listed flows can be classified as axisymmetric flows. Takashima and Lewis [28] derived a nonaxisymmetric flow field from a flow past a wedge-cone body and first used it as the basic flow field to generate waveriders. The von Karman ogive [23] is one of the minimum-drag bodies of revolution at a zero angle of attack with supersonic speeds. Since the von Karman ogive offers minimum pressure drag for a given length and diameter, it is often used as the nose cone in the aerodynamic design of a rocket or missile. Then, one might intuitively speculate that a novel waverider with lower drag and higher

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Nomenclature MOC L0 R

δ δA θ x r M0 P0 T0 Ma P P /P0 RS

α L′

Method of characteristics Length of von Karman ogive, m Base radius of von Karman ogive, m Angle of slope, ◦ Angle of slope at point A, ◦ Flow angle, ◦ Axial coordinate, m Radial coordinate, m Free-stream Mach number Free-stream static pressure Free-stream static temperature Flight Mach number Local static pressure, Pa Ratio of local static pressure to free-stream static pressure Radius of shock wave at the base plane, m Angle of attack, ◦ Lift, N

lift-to-drag ratio may be generated by using the flow past a von Karman ogive as the basic flow field. However, the von Karman ogive is a blunt body, and when it travels at a supersonic or hypersonic speed, the blunt nose will induce a detached shock wave and a subsonic flow field will form in the region surrounding the tip of the body. Therefore, it is important to modify the original von Karman ogive in order to obtain an acceptable aerodynamic configuration for hypersonic vehicles. The method of characteristics (MOC) is the most accurate numerical technique for solving hyperbolic partial differential equations [31], and it can save computational cost. Therefore, in the design process of waveriders, the MOC is an effective approach for calculating the basic flow field when the flow is completely supersonic behind the shock wave [11,19]. In the present study, a novel waverider generated from axisymmetric supersonic flows past a pointed von Karman ogive is examined. This pointed von Karman ogive is obtained by pointing the blunt nose of the original von Karman ogive. Next, the axisymmetric supersonic flow field past the pointed von Karman ogive is computed using the MOC, and this flow field is employed as the new basic flow field of waveriders. Specifically, this new basic flow field is used to generate a novel waverider by using the streamline tracing technique. Finally, the novel waverider is validated using numerical approaches.

D′ M′ Sr lr q∞ CL CD C mz L/ D LW HW

Δ Vol S wet Sp Sb

η

Drag, N Pitching moment, N·m Reference area, m2 Reference length, m Freestream dynamic pressure, Pa Lift coefficient Drag coefficient Pitching moment coefficient Lift-to-drag ratio Length of waverider, m Height of waverider, m Increment percentages of drag coefficient and lift-todrag ratio due to the viscous effects, % Internal volume of waverider, m3 Wetted surface area of waverider, m2 Planform surface area of waverider, m2 Base area of waverider, m2 Volumetric efficiency

Fig. 1. Schematic illustration of von Karman ogive.

2. Novel waverider 2.1. Pointed von Karman ogive Von Karman used the slender-body theory to obtain the minimum-drag ogive of a given length L 0 and base radius R at supersonic speeds; this is referred to as the von Karman ogive [23]. The contour of the von Karman ogive is computed from Eqs. (1) and (2) [8], and its schematic illustration is shown in Fig. 1. The distribution of the angle of the slope of the von Karman ogive along the x axis, denoted as δ , is shown in Fig. 2. The initial δ value at x = 0 is 90◦ . Thus, the head of the von Karman ogive is blunt. In other words, the von Karman ogive has a blunt nose. In order to ensure the creation of a completely supersonic flow with an attached shock wave when the ogive travels at a supersonic or hypersonic speed, a von Karman ogive with a pointed nose (hereafter simply called ‘pointed von Karman ogive’) is required.

Fig. 2. Distribution of angle of slope of von Karman ogive along the x axis.

Fig. 3. Schematic illustration of pointed von Karman ogive.

A pointed von Karman ogive is obtained by pointing the blunt nose of the von Karman ogive. As shown in Fig. 3, the blunt nose contour OA of the von Karman ogive is replaced with a straight

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Table 1 Specific parameters of pointed von Karman ogive and properties of free-stream flow. M0

P 0 (Pa)

T 0 (K)

L 0 (m)

6.0

2511.18

221.649

10.0

R (m) 1.6

δ A (◦ ) 19.0

Fig. 5. Grid around pointed von Karman ogive for MOC. Table 2 Comparison of number of grid nodes and speeds between the Euler code and MOC.

Fig. 4. Structured grid around pointed von Karman ogive for the Euler code.

Method

Number of grid nodes

Computation time (min)

MOC Euler code

86 957 50 000

1.5 22.3

cone contour O 1 A at point A, and the angle of the slope at point A is δ A . The contour O 1 AH is the contour of the pointed von Karman ogive.

  2x Θ = arccos 1 − L0  R sin(2Θ) r= √ Θ−

π

2

(1) (2)

2.2. Axisymmetric supersonic flow past the pointed von Karman ogive and validation of MOC Since the flow past the pointed von Karman ogive is completely supersonic at the design Mach number, the MOC can be used to calculate the flow fields. Details of the MOC have been reported elsewhere [31]. The pointed von Karman ogive can be determined when the parameters, L 0 , R, and δ A are given. The specific parameters of the pointed von Karman ogive as well as the properties of the freestream flow in this case are listed in Table 1. In order to validate the accuracy and speed of the MOC, the same flow is also calculated using an Euler code [9] for comparison purposes. This code calculates the inviscid axisymmetric flows by solving two-dimensional (2D) axisymmetric Euler equations [9]. Specifically, these equations are solved using the densitybased (coupled) implicit solver. The second-order spatially accurate upwind scheme, which applies the advection upstream splitting method (AUSM) to the flux vector, is utilized [17], and the leastsquares cell-based method is used to compute the gradients. The Courant–Friedrichs–Lewy number (CFL) is maintained at 5. Fig. 4 shows the generated 2D structured grid around the body for the Euler code, and Fig. 5 shows the characteristic grid based on the Mach lines for the MOC. Table 2 presents a comparison of the number of grid nodes and speeds between the Euler code and the MOC. From Table 2, it is clear that the MOC is almost 15 times faster than the Euler code for the same flow field calculation. A comparison of the Mach number contour lines of the axisymmetric supersonic flow past the pointed von Karman ogive as obtained from the Euler code and MOC is shown in Fig. 6, and a comparison of the non-dimensional wall pressure distributions along the x axis as obtained from the Euler code and MOC is

Fig. 6. Comparison of Mach number contour lines of axisymmetric supersonic flow past pointed von Karman ogive between the Euler code and MOC.

shown in Fig. 7. From Fig. 6, it is obvious that there is no significant difference between the flow field structures obtained from the Euler code and the MOC, and from Fig. 7, it is clear that the wall pressure distributions are nearly identical between the Euler code and the MOC. However, the shock wave locations predicted by these two approaches are different: the MOC fits an infinitely thin shock wave, whereas the Euler-code-based approach creates a shock wave that smears over several grid points. In order to further validate the accuracy of the MOC, the nondimensional wall pressures of a cone as obtained by the MOC are compared with those obtained from the Euler code and Taylor– Maccoll equations, as shown in Fig. 8; the semi-vertex angle of the cone in this case is 9◦ . The Taylor–Maccoll solutions that are solved by the fourth-order Runge–Kutta method can be classified as analytical solutions, unlike the solutions obtained using the Euler code and MOC, both of which have second-order accuracy. Fig. 8 reveals that the solutions obtained using the Euler code and MOC are nearly identical to those obtained using the Taylor–Maccoll equations.

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Fig. 9. Schematic illustration of streamline tracing technique based on left-running Mach lines. Fig. 7. Comparison of non-dimensional wall pressure distributions of pointed von Karman ogive along the x axis between the Euler code and MOC.

using the MOC. The solid nodes represent the points in a streamline, termed streamline points. All the streamline points except for the leading point at the shock wave and the trailing point at the base plane are located on the left-running Mach lines. Once the position coordinates of a streamline point are known, the flow properties can be obtained by the linear interpolation of the values of points j and j + 1 on the left-running Mach lines. Thus, the streamline tracing technique adopted in this study is based on left-running Mach lines. In this study, the numerical integration of Eq. (3) is performed using the modified Euler predictor–corrector method, which is a second-order method for integrating ordinary differential equations [31]. The predictor algorithm can be expressed by Eq. (4), and the corrector algorithm with iterations can be expressed by Eq. (5),

ri0+1 = ri + tan(θi )x Fig. 8. Comparison of non-dimensional wall pressure distributions of a cone along the x axis among the Euler code, MOC, and Taylor–Maccoll equations.

The above analysis indicates that the MOC possesses sufficiently high accuracy and speed for calculating the axisymmetric supersonic flow past a pointed body. These results also show that the calculation precision of the MOC is the same as that of the Euler code but the former is more than 10 times faster than the latter. 2.3. Streamline tracing technique based on left-running Mach lines and its validation By definition, a streamline is a curve whose tangent at any point is in the direction of the velocity vector at that point [2]. The ordinary differential equation for a streamline in two dimensions in the cylindrical coordinate system is

dr dx

= tan(θ )

(3)

where x is the axial coordinate, r is the radial coordinate, and θ is the flow angle. In the streamline tracing technique, Eq. (3) is numerically integrated from a known starting point i, designated as (xi , r i ), where r i = r (xi ). In Fig. 9, the thin lines represent the left-running characteristics, the dashed lines represent the right-running characteristics, and the hollow nodes represent the intersection points of the left-running and right-running Mach lines. The position coordinates and flow properties at these hollow nodes can be obtained

(4)

Here, x is the step size of the finite-difference algorithm, θi is the flow angle at the known starting point i, and r 0i +1 is the predicted value of r at xi +1 = xi + x.

rin+1 = ri + tan



θi + θin+−11  x 2

(5)

Here, rin+1 is the value of r at xi +1 after n applications of the corrector algorithm and θ in+−11 is the value of θ after n − 1 appli-

cations of the corrector algorithm. θ in+−11 is obtained by the linear interpolation of the values of points j and j + 1 on the left-running Mach lines. In order to validate the streamline tracing technique based on the left-running Mach lines, a streamline named streamline B is solved. The r coordinate of the leading point of streamline B is 0.5R S , where R S is the radius of the shock wave at the base plane. Then, a streamline body C is obtained by rotating streamline B around the x axis, and it is a body of revolution. Fig. 10 shows a comparison of Mach number contour lines of the flows past the pointed von Karman ogive with those of flows past streamline body C as obtained by the Euler code. Further, Fig. 11 shows a comparison of non-dimensional wall pressure distributions of streamline body C as obtained by the Euler code with those of streamline B as obtained by the streamline tracing technique. As shown in Fig. 10, the flow structure above streamline B agrees well with the structure of the flow past streamline body C. As shown in Fig. 11, the wall pressure distributions of streamline body C as obtained by the Euler code also agree well with those of streamline B. The streamline tracing technique can well predict the position coordinates and flow properties of the streamline points. These

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Fig. 13. Principle of von Karman waverider.

Fig. 10. Comparison of Mach number contour lines of flows past pointed von Karman ogive with those past streamline body C obtained by the Euler code.

Fig. 14. Geometric model of test case.

Fig. 11. Comparison of non-dimensional wall pressure distributions of streamline body C obtained by the Euler code with those of streamline B obtained by streamline tracing technique.

stream or compression-stream surface is called the trailing-edge curve. In this study, the novel waverider generated from axisymmetric supersonic flows past a pointed von Karman ogive is termed a von Karman waverider. As shown in Fig. 13, the von Karman waverider can be constructed by the following procedure. First, given the parameters M 0 , P 0 , T 0 , L 0 , R, and δ A , the pointed von Karman ogive and the flow past it can be determined. Second, the axisymmetric supersonic flow past the pointed von Karman ogive is calculated using the MOC and is used as the basic flow field to generate the waverider. Third, given the curve equation for the trailing edge of the free-stream surface in the local coordinate system x′ o′ y′ , all points on this curve are pushed in the direction parallel to the free-stream and they intersect with the shock wave; then, the leading edge and free-stream surface can be obtained. Next, the streamline tracing technique based on leftrunning Mach lines is employed for tracing streamlines in the basic flow field from the leading edge downstream up to the base plane. Finally, the compression-stream surface is obtained by lofting these streamlines, and simultaneously, the trailing edge of this surface is also obtained. As a result, the shape of the von Karman waverider is generated. 3. Case verification

Fig. 12. Principle of a waverider.

results validate the effectiveness of using the streamline tracing technique based on left-running Mach lines for our study. 2.4. Design procedure of von Karman waverider As illustrated in Fig. 12, a waverider is composed of three surfaces—the free-stream surface, compression-stream surface, and base surface. The compression-stream surface is obtained by lofting a group of streamlines. The curve where the free-stream surface intersects the shock wave is called ...