Aerodisk effects on drag reduction for hypersonic blunt body with an ellipsoid nose

Aerospace Science and Technology 86 (2019) 599–612

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Aerodisk effects on drag reduction for hypersonic blunt body with an ellipsoid nose Kang Zhong, Chao Yan ∗ , Shu-sheng Chen, Tian-xin Zhang, Shuai Lou National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China

a r t i c l e

i n f o

Article history: Received 3 July 2018 Received in revised form 14 January 2019 Accepted 14 January 2019 Available online 4 February 2019 Keywords: Aerodisk Drag reduction Non-axisymmetric blunt body Ellipsoid nose Hypersonic flow

a b s t r a c t The paper systematically investigates the drag reduction of aerodisk fixed on the non-axisymmetric blunt body with an ellipsoid nose, different from conventional axisymmetric blunt body. The effects of spike length, aerodisk size and shape on drag reduction performance are numerically examined. Based on the analysis, a well-designed aerodisk configuration is determined and a maximum drag reduction of 41.9% is obtained for the whole blunt body. Meanwhile, some meaningful phenomena are discussed: (i) As the spike length increases, the drag of forebody and whole blunt body decreases first, then increases, and then decreases again. (ii) The total drag of the blunt body decreases first and then increases with the increase of aerodisk size. (iii) An elliptic disk with aspect ratio approaching that of the ellipsoid nose is superior to a round disk with the same size. This numerical study is expected to illustrate the significance of aerodisk to drag reduction over the non-axisymmetric blunt body and provide a reference for the design of aerodisk in hypersonic field. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction During the rapid development of aerospace technology, the demand on reduction of drag and aerodynamic heating is increasingly prominent for hypersonic vehicles [1,2]. Imposing bluntness at the nose is extensively applied to reduce the heat load [3], but it gives rise to remarkable increase of the vehicle wave drag and results in more fuel consumptions and higher propulsion system requirements. In order to reduce the aerodynamic drag of hypersonic vehicles, several techniques have been devised, such as energy deposition along the stagnation streamline [4], opposing jet in the stagnation zone [5], aerospike attached to the blunt body [6–8], and the combination of opposing jet and aerospike [9], etc. Among these various techniques, the aerospike first explored in the middle of 20th century is considered as a relatively simple and effective technique for hypersonic vehicles [1,10,11]. An aerospike is usually a thin cylindrical rod attached to the nose of blunt body. The introduction of aerospike yields significant modification to the flowfield upstream of the blunt body. On the one hand, it replaces the strong bow shock with weaker conical oblique shock waves. On the other hand, it induces the separation of boundary layer from the surface and the creation of a shear layer. The separation region with low pressure and low temperature shields the blunt body from the incoming flow, leading to

*

Corresponding author. E-mail address: [email protected] (C. Yan).

https://doi.org/10.1016/j.ast.2019.01.027 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

remarkable drag reduction for the vehicle [3]. To explain the drag reduction mechanism more accurately, the effective body approach was proposed and has been extensively utilized [12–15]. The effective body was first referred to the shape of the shear layer [12], and later, some researchers suggested to identify the effective body through the dividing streamline [13,15]. It is noted that all streamlines above the dividing streamline pass downstream of the forebody shoulder whereas those below the dividing streamline reverse into the recirculation zone [3]. Since proposed by Alexander in 1947 [16], a series of experimental and numerical studies on drag reduction of spiked blunt body have been carried out. Humières et al. [17] experimentally examined a spiked body at Mach number 8.2, and obtained a maximum drag reduction of 77% of the unspiked body. Crawford et al. [18] investigated the aerodynamic drag of a hemispheric cylindrical model with pointed spike at Mach number 6.8. The ratio of spike length to diameter of blunt body (L / D) played a key role in the drag reduction performance [18]. Yamauchi et al. [19] numerically studied the effects of spike length, Mach number and angle of attack on the drag reduction of spiked blunt body, and the computational results were consistent with Crawford’s experimental data [18]. In the examination of spiked cones, Wood [20] found that the shape and size of the recirculation region were primarily controlled by the flowfield in the vicinity of the reattachment point. Similar findings were obtained by Mehta [21], who numerically evaluated the effects of spike length and spike nose configuration on the drag reduction at Mach number 6 and α = 0 deg. His

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study revealed that the reattachment point of the shear layer on the nose should move backwards so that a better drag reduction performance could be obtained. To further enhance the aerospike’s drag reduction efficiency, a disk is mounted at the tip of the spike (called aerodisk). Motoyama et al. [22] confirmed the advantages of aerodisk over aerospike in drag reduction performance. Through the measurements in a hypersonic shock tunnel, Menezes et al. [23] indicated more than 50% reduction in the drag coefficient for a 120-deg apex angle blunt cone with a flat-faced aerodisk at moderate angles of attack. Kharati-Koopaee et al. [24] numerically investigated the influence of aerodisk size on drag reduction of highly blunted body flying at Mach number 5.75. They concluded that the drag coefficient firstly decreases and then increases with the increase of aerodisk size. Moreover, Huang et al. [2] comprehensively analyzed the influences of spike length and aerodisk diameter on drag reduction mechanism. The obtained results suggested that the drag reduction of blunt body was proportional to the extent of recirculation region for all aerodisks. Besides, numerous researches focus on the effects of aerodisk shape [3,25–27]. Qin et al. [3] compared three typical disk shapes, namely the hemispheric-faced, single flat-faced and double flat-faced disks. In their study, the best drag reduction was found for the double flat-faced aerodisk, which gave a pressure drag reduction of 60.5% of the nose part at flight and α = 8 deg. Sudhir et al. [27] simulated the flowfields around single and double aerodisks with different disk shapes (hemispherical, flat triangular and flat disks) at Mach number 6.2. Numerical results clearly exhibited the favorable performances of the multi-disk aerospikes on the reduction of drag and aerodynamic heating. In addition, with the aim of further improving the drag reduction efficiency, optimization has also been used to assist the geometric design of aerodisk and blunt body. Qin et al. [28] optimized a spiked blunt body using the multi-objective NSGA-II algorithm coupled with kriging surrogates. The optimization objectives were minimizing the drag and aerodynamic heating while keeping the flow stable. The Pareto optimal set showed a degree of competition between these two design objectives. Their study also pointed out that the aerodynamic heating of the spiked body was mainly dictated by forebody shape, whereas the spike length mainly controlled the aerodynamic drag response. In recent years, many researches are also conducted to explore the application of aerospike/aerodisk on aircrafts. Wherter et al. [29] performed stability and control analysis to assess the effect of aerospike on stability, controllability, and handling qualities of the F-15B airplane. Khurana et al. [30] investigated a lifting-body configuration with different aerospikes at Mach number 7 in a wind tunnel. The experimental results showed a large increase in the lift-to-drag ratio and a marginal increase in pitching moment as compared to the non-aerospike case. Qin et al. [3] systematically analyzed the effects of spike length, aerodisk shape and installation angle on the drag reduction of the lifting body. They observed that as the spike length increases, the aerodynamic drag decreases first and then increases at α = 8 deg. This is quite different from conventional recognition that the pressure drag coefficients of the vehicle nose always decreases when the spike length increases [3]. In their study, L / D = 2 was the best ratio yielding 49.3% and 4.39% drag reductions for the nose and the whole vehicle respectively. Based on the corresponding literature survey, some summaries can be drawn: (1) the aerospike is an effective way for drag reduction due to the reduced dynamic pressure in the separation region; (2) the aerodisk has better performance in drag reduction as compared to the pointed spike. Therefore, systematical investigation into the drag reduction mechanism of aerodisk and the effects of its parameters is very inspiring. In addition, non-axisymmetric blunt shapes are widely employed in hypersonic vehicles. However, few researches pay attention to the effects of aerospike/aerodisk

fixed on the non-axisymmetric blunt nose, though numerous investigations about the axisymmetric blunt nose (e.g. a hemispherical nose [2,31] and a large-angle apex blunt cone [23,25,32]) have been done. Thus, the motivation of this study is to explore the aerodisk effects on drag reduction for hypersonic non-axisymmetric blunt body. An ellipsoid forebody attached to a truncated elliptic cone afterbody is selected as the basic blunt body. Firstly, hypersonic flowfields over the basic blunt body with and without an aerodisk are compared numerically. The numerical results are based on threedimensional Navier–Stokes flowfield simulations. Secondly, we examine the effects of spike length on drag reduction and make an explanation on the non-monotonous drag variation versus spike length. Thirdly, a detailed analysis on the effects of aerodisk size and shape on drag reduction is performed. An elliptic aerodisk is employed and its aspect ratio effect on drag reduction is discussed. This is quite different from the previous works [2,8,23], since the conventional aerodisk is basically axisymmetric. Some findings are provided at last. This numerical study is expected to illustrate the significance of aerodisk to hypersonic non-axisymmetric blunt body and provide a reference for the design of aerodisk. 2. Physical model and numerical approach 2.1. Physical model In this paper, a non-axisymmetric blunt body with an aerodisk is considered. It contains four parts, namely the aerodisk, the spike, the forebody and the afterbody shown in Fig. 1(a). Dimensions of the geometry are illustrated in Fig. 1(b) and (c). The forebody is an 8:5 elliptic hemisphere with the maximum length of 160 mm in x–z plane and 100 mm (symbol b in Fig. 1) in x– y plane respectively. The afterbody is a truncated elliptic cone with length of 400 mm. At the base of afterbody, the major axis is 288 mm wide, and the minor axis is 120 mm high. The drag reduction effects of spike length, aerodisk size and shape are explored respectively: (1) The length of the cylindrical spike is studied. The values of the length are L /b = 0.4, 0.6, 0.8, 1.0, 1.25, 1.5, 1.875, 2.125 and 2.5, respectively. Meanwhile, the aerodisk is assumed as a flat, round disk with identical diameter of 0.36b. (2) The spike length is fixed and the size of aerodisk is analyzed. The aerodisk remains to be round, and its diameter varies from 0.18b to 0.42b with a step of 0.06b. (3) The attention is focused on the effect of aerodisk shape on drag reduction behaviors. Here, the aerodisk is assumed as an elliptic disk with variable aspect ratio (Lh/ L v). Both of the spike length and the aerodisk size are fixed, and fourteen aspect ratios varying from 0.8 to 4.0 are examined in detail. The freestream conditions of all simulations are the static pressure of 12076 Pa, the static temperature of 216.7 K, and the freestream Mach number of five with no angle of attack. Accordingly, the Reynolds number based on the reference length of 0.12 m (minor axis of afterbody) and freestream conditions is 2.41 × 106 . According to Crawford’s [18] experimental results, the flow can be assumed to be turbulent if the Reynolds number, based on the blunt body diameter and incoming flow parameters, is beyond 1.50 × 106 . This theory has been employed in many simulations on spiked blunt body [2,33,34], and thus, it is utilized in the current study and a fully turbulent flow state is assumed for all calculations [18]. In addition, the drag coefficient is calculated with the reference length of 0.12 m and reference area of 0.0136 m2 respectively. 2.2. Numerical methods In this study, numerical simulations are performed with an inhouse code named MI-CFD [35]. It is a finite volume solver based

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Fig. 1. Schematic diagram of the non-axisymmetric blunt body with an aerodisk (units: mm).

on multi-block structured grids. The code is employed to simulate a variety of complex flows, and its reliability and accuracy have been verified by a series of numerical experiments [35–39]. Main algorithms of the code are presented as follows. 2.2.1. Governing equations and turbulence model To calculate the steady aerodynamics of aerodisked blunt body, three-dimensional compressible Navier–Stokes equations [40] coupled with Menter’s k–ω shear stress transport (SST) turbulence model [41] are adopted. SST turbulence model merges the original k–ω model in the inner region of boundary layer and the standard k–ε model in the outer region and free shear flows, and performs quite well in the predictions of adverse pressure gradient flows and pressure-induced boundary layer separation [41,42]. Therefore, it has been widely employed and shown excellent performance in previous researches on aerospiked blunt body [2,7]. The governing equations are briefly described as follows [37,43,44]: 1) Continuity equation:

∂ ρ ∂(ρ u i ) + =0 ∂t ∂ xi

(1)

2) Momentum equation:

∂ τi j ∂p ∂(ρ u i ) ∂(ρ u i u j ) + =− + ∂t ∂xj ∂ xi ∂xj

(2)

where ρ is the density, u i is the ith velocity component, p is the pressure, τi j is the viscous stress tensor, E is the total energy, H is the total enthalpy, and q˙ j is the heat flux. P k and P ω represent production source terms of the turbulent kinetic energy k and the specific dissipation rate of turbulence ω respectively. μ L and μ T denote the laminar and turbulent viscosity. F 1 is auxiliary function, and σk , σω , β and β ∗ are model constants [44]. 2.2.2. Discretization and boundary conditions The inviscid fluxes are discretized using Roe upwind scheme with second order MUSCL reconstruction and minmod limiter. The viscous fluxes are calculated by second order central difference scheme. The implicit Lower–Upper Symmetric Gauss–Seidel (LUSGS) scheme is employed for the time integration. Besides, the Courant–Friedrichs–Levy (CFL) number is specified as 0.75 to ensure stability. The solid wall condition is assumed as non-slip for velocity, zero normal gradient of pressure and an isothermal wall temperature of 300 K [42]. The velocity normal to the symmetry plane is zero [42]. The remaining outer boundaries are assigned to be farfield. The variables at inflow boundary are specified with freestream values, while variables at outflow boundary are obtained using extrapolation from internal domain. Moreover, the values of k and ω at solid surface for SST turbulence model are determined respectively as [44]

3) Energy equation:

∂ ∂(ρ E ) ∂(ρ Hu j ) + = (u i τi j − q˙ j ) ∂t ∂xj ∂xj

(3)

4) Transport equation of turbulent kinetic energy k:

  ∂ ∂k ∂(ρ k) ∂(ρ u j k) (μ L + σk μT ) + = P k − ρ β ∗ kω + (4) ∂t ∂xj ∂xj ∂xj 5) Transport equation of the specific dissipation rate of turbulence ω :

  ∂ ∂ω ∂(ρω) ∂(ρ u j ω) (μ L + σω μT ) + = P ω − ρ β ω2 + ∂t ∂xj ∂xj ∂xj ρσω2 ∂ k ∂ ω + 2(1 − F 1 ) (5) ω ∂xj ∂xj

kwall = 0,

ωwall =

60μ L

(6)

0.075ρ (d1 )2

where μ L , ρ and d1 are the laminar viscosity, the density and the distance to the solid wall at the first cell centroid from the solid surface, respectively. For inflow boundaries, the freestream levels of k, ω and μ T are specified as [44]

k∞ a2∞

= 9 × 10−9 ,

μT ,∞ = 0.009 μL ,∞

ω∞ = 1 × 10−6 , ρ∞a2∞ /μL ,∞

For outflow boundaries, the values of k and from internal domain.

(7)

ω are extrapolated

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Fig. 2. Computational domain and grid generation for the blunt body with an aerodisk (medium grid).

2.3. Grid generation and sensitivity analysis

Table 1 Grid system information.

Three-dimensional, multiblock-structured meshes are constructed to acquire credible solutions of the intricate flow structures. Only half-geometry meshes are generated in this study due to the symmetric and stable flowfield. The meshes of the solid wall and symmetry plane are displayed in Fig. 2(a), with elaborate refinement arranged near the aerodisk. Besides, the normal grid spacing near the surface is particularly small to capture the details of the boundary layer (see Fig. 2(b)). In order to show the independence of grid resolution, three different grid scales (i.e., coarse, medium and fine) are generated and examined in detail. Here, the spike length is taken as 100 mm, and the disk is assumed to be a flat round disk with diameter of 0.36b. Total number of the coarse grid is up to 2.4 million. Grid independence analysis is performed by increasing the grid numbers in all three directions and decreasing the first cell height on the surface. Detailed grid information is provided in Table 1. It is noted that the defined grid refinement factor (R ∗ ) is based on the coarse grid in each direction. In addition, the first cell y + (non-dimensional) is about one for all three grids and satisfies the viscous solution requirement in turbulent flow RANS simulations [33]. Surface pressure and skin friction coefficient distributions based on the three grids are plotted and compared in Fig. 3 and Fig. 4 respectively. As shown, the grids of medium and fine scales yield remarkably

R∗

(radial) (circumferential) (azimuthal)

y+

Coarse grid

Medium grid

Fine grid

1.0 1.0 1.0

1.4 1.1 1.1

1.6 1.2 1.2

1.36

1.02

0.68

R ∗ : Grid refinement factor, based on the coarse grid in each direction.

similar pressure and skin friction coefficient distributions in both x– y plane and x–z plane. Numerical results indicate that the calculation is grid independent. Therefore, the medium grid scale is employed for all cases to guarantee the numerical precision and save the computational resources. 2.4. Accumulation of error Since the governing equations are discretized and solved on finite grid size, a definite error occurs in integration at each step and the accumulation of error is proportional to the number of integration steps [45,46]. The integration steps in the current study should not exceed the maximal allowable steps that accumulation error exceeds the acceptable value. The maximal allowable number of integration steps is determined by [45,46]

Fig. 3. Surface pressure comparison for the three grid scales.

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Fig. 4. Surface skin friction coefficient comparison for the three grid scales. Table 2 Accumulate error for the aerodisked blunt body. S1

S2

S3

S err

S max

n

nmax

Rs

1.88E−06 1.69E−07 2.19E−06 3.57E−07 3.57E−07 4.41E−06 8.22E−05 8.22E−05 1.45E−05 8.22E−05 9.71E−07 1.45E−05

4.41E−06 2.19E−06 4.41E−06 1.08E−04 1.45E−05 1.69E−07 1.45E−05 1.45E−05 1.04E−04 1.04E−04 4.41E−06 3.64E−06

1.69E−07 4.41E−06 1.88E−06 1.08E−04 4.41E−06 2.33E−05 2.33E−05 2.33E−05 2.33E−05 2.33E−05 1.69E−07 4.41E−06

6.46E−06 6.76E−06 8.48E−06 2.16E−04 1.93E−05 2.79E−05 1.20E−04 1.20E−04 1.41E−04 2.09E−04 5.54E−06 2.26E−05

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000

6.00E+07 5.46E+07 3.48E+07 5.34E+04 6.73E+06 3.21E+06 1.74E+05 1.74E+05 1.25E+05 5.72E+04 8.13E+07 4.91E+06

1999.38 1821.04 1159.46 1.78 224.38 107.07 5.78 5.78 4.17 1.91 2710.40 163.79



nmax = S max / S err

2

(8)

where S max is total error and presumed to be between 1% and 5%. 3 S err ≈ i =1 S i and S i is the relative error of integration in one dimensional case and defined as follows [45,46]

S i ≈ ( L / L i )k+1

(9)

where L is the mean cell size and L i is the domain size in the “i” direction, k is the order of accuracy of numerical scheme. The ratio of maximal allowable number of integration steps and actual number of steps are defined as follows, the ratio tends to unit when the accumulation error tending to the maximal allowable value [45,46].

R s = nmax /n

(10)

In the current study, twelve blocks of structured meshes are generated for the aerodisked blunt body and corresponding aerodynamics is converged by nearly 30,000 steps. The accumulation error of each block for the medium grid is arranged in Table 2 and all the ratios are larger than unit. The results indicate that the accumulation error is not beyond the maximal allowable value. 2.5. Code validation and wall temperature effects analysis Crawford [18] conducted a series of experiments on a spiked blunt body, and Yamauchi [19] numerically reproduced his experiments and obtained a good agreement. To validate the numerical

Fig. 5. Surface pressure comparison along the hemispherical nose.

approaches and CFD code, one of the experiment cases is numerically performed in this study. The selected configuration is a hemispherical cylinder equipped with a pointed cylindrical spike. The spike length is L = 0.5d (d is diameter of the hemispherical nose). The freestream Mach number is 6.8, the angle of attack is zero, and the Reynolds number based on the diameter of hemispherical nose is 1.4 × 105 . Fig. 5 shows the surface pressure comparison for the hemispherical nose with and without the spike. As seen,

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Fig. 6. Surface pressure comparison at different wall temperatures. Table 3 Aerodynamic drag counts (Cd × 104 ) comparison at different wall temperatures. T wall (K)

Total

Forebody

Afterbody

Aerodisk

Spike

300 900

2581 2553 −1.08%

1221 1214 −0.57%

757 737 −2.64%

604 603 −0.17%

−1 −1



–

the predicted pressure is in excellent accordance with Yamauchi’s simulation [19]. Besides, for the case with a spike, the pressure distribution also agrees with the experiment data except one point in the separation region [18]. The numerical results indicate the correctness and credibility of the numerical code used in this study. In hypersonic flow simulations, wall temperature may affect the aerodynamics and aerothermodynamics predictions. In this study, two isothermal wall temperatures of 300 K and 900 K are examined and compared to explore wall temperature effects on drag reduction of aerodisked blunt body. Comparisons of the surface pressure distribution and the aerodynamic drag are given in Fig. 6 and Table 3, respectively. As seen, the effect of wall temperature on surface pressure distribution is almost negligible. The discrepancy of total aerodynamic drag between the two wall temperatures is merely 1.08%, i.e., 28 drag counts. Numerical results indicate the extremely small effects of wall temperature on aerodynamic drag predictions for aerodisked blunt body investigated in this study. Thus, similar to previous calculations [2,21,47], a simple isothermal wall temperature of 300 K is utilized in our simulations. 3. Results and discussion 3.1. Flow structure Before performing the aerodisk study, it is necessary to explore the basic flowfield of the non-axisymmetric blunt body first. In Fig. 7, the overall flow structure around the blunt body without aerodisk is displayed. It is featured by Mach number distribution on the symmetry plane and three streamwise slices. The limiting surface streamlines on the solid wall are also shown. As seen, a strong bow shock is generated near the forebody nose. When the flow travels downstream, crossflow occurs with streamlines gradually deflecting from the leading edge to the centerline. The deflection is due to stronger compression in the leading edge than in the centerline of the elliptic blunt body, and the induced pres-

Fig. 7. Flowfield for the non-axisymmetric blunt body without aerodisk.

sure deviation perpendicular to the mainstream direction leads to the deflection of the streamlines. Typical flowfield around the non-axisymmetric blunt body with an aerodisk is delineated in Fig. 8, with Mach number contour in the upper half and density-gradient contour in the lower half. The streamlines in the symmetry plane and the limiting surface streamlines are presented as well. Here, the spike length is L /b = 1.0, and the aerodisk is assumed as a flat round disk with diameter of 0.36b. As seen in Fig. 8, the hypersonic freestream yields a strong bow shock near the aerodisk. Due to the combined effects of flow viscosity and adverse pressure gradient, large recirculation region is formed in front of the forebody nose. This separation remarkably lowers down the pressure in front of the nose, and therefore, reduces the pressure drag of the forebody greatly. Besides, a shear layer is clearly observed between the recirculation region and the external flows that pass downstream the blunt body. In Fig. 8, we also observe that the flow structure in x–z plane is quite different from that in x– y plane due to the non-axisymmetric blunt body configuration. As seen, flow reattachment occurs and a strong reattachment shock wave generates near the forebody surface in x–z plane. By contrast, no obvious reattachment is observed in x– y plane. The reattachment shock wave leads to high surface pressure near the reattachment point. The forebody surface in x– y

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Fig. 8. Enlarged view of Mach contours and density-gradient contours of the blunt body with an aerodisk.

To assess the aerodisk effect on drag reduction quantitatively, the total and each part’s values of drag coefficient with and without an aerodisk are compared in Table 4. The pressure drag (Cd_pressure), viscous drag (Cd_viscous) and drag reduction coefficient ( ) are presented as well. Note that all drag coefficients mentioned in the current study are presented in drag counts (Cd × 104 ) as it is usually used in aerodynamics. is defined as:

=

Fig. 9. Surface pressure distribution and limiting surface streamlines on the forebody. Left half: without an aerodisk; Right half: with an aerodisk.

plane is completely enveloped with low-speed, low-pressure recirculation flows. These complex flow structures can be better understood with the displayed streamlines (see Fig. 8) and the surface pressure distributions (see Fig. 9). The introduction of aerodisk not only has a great impact on the flowfield, but also on the surface pressure distribution. Fig. 9 shows a comparison of surface pressure distribution at the forebody. The left half represents the case without an aerodisk while the right half represents that with an aerodisk. The limiting surface streamlines are also displayed. For the case without an aerodisk, the non-dimensional pressure is highest on the nose and reaches the maximum of about 32 at the stagnation point. Then, the surface pressure drops due to the gradual flow expansion downstream. Quite different scenario is clearly observed for the case with an aerodisk. The maximum value of non-dimensional pressure is only about 13, that is, reduced by 59% compared with that without an aerodisk. The maximum pressure appears a little downstream the reattachment point, and it is supposed to be caused by the strong reattachment shock in x–z plane (see Fig. 8(b)). Then, the external flows deflect across the reattachment shock wave and expand along the forebody surface, resulting in the gradually decreased surface pressure downstream. Additionally, it is observed that pressure in recirculation region is low and obtains the minimum value of about 1.8.

Cdno_AD − Cdwith _AD Cdno_AD

× 100%

(11)

Here, no_AD represents the case without an aerodisk, with_AD represents the case with an aerodisk. As seen in Table 4, the largest aerodynamic drag comes from forebody part, and the use of aerodisk contributes to the forebody’s and the total drag reduction by up to 65.1% and 39.5%, respectively. In addition, the forebody’s pressure drag is observed to be significantly larger than the forebody’s viscous drag. Therefore, by creating large recirculation region that shades the blunt forebody to reduce its pressure drag, the total drag of the blunt body can be obviously decreased. 3.2. Effects of spike length on drag reduction The investigation on drag reduction effect of spike length is performed on the non-axisymmetric blunt body. A total of nine spike lengths varying from 0.4b to 2.5b are selected. Meanwhile, the aerodisk is set as a round disk with identical diameter of 0.36b. Mach contours for the blunt forebody with different spike lengths are shown in Fig. 10. Note that the upper half represents the flowfield in x– y plane while the lower half represents that in x–z plane. As seen, only one large recirculation region is clearly observed when L /b is less than 1.0. By contrast, two flow separations are predicted when L /b is more than 1.0. The first is relatively small and occurs just behind the aerodisk, and the second is much larger and attached to the nose of the forebody. Moreover, it is seen that the complex flow patterns (e.g. shock/shock interactions) and the size of recirculation region change greatly with the increase of spike length. The variations of total and each part’s drag coefficients versus spike length are shown in Fig. 11. It is noted that only total and forebody’s drags are sensitive to the spike length. As the spike length increases, both drags decrease first, then increase, and then decrease again. This trend is quite interesting because, in previous researches, the drag coefficient of blunt nose generally decreases as

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Table 4 Aerodynamic drag comparison for the blunt body with and without an aerodisk. Cd counts

Total Forebody Afterbody Aerodisk Spike

Cd_pressure counts

Cd_viscous counts

no_AD

with_AD



no_AD

with_AD

no_AD

with_AD

4269 3495 774 – –

2581 1221 757 604 −1

39.5% 65.1% 2.3% – –

4128 3462 666 – –

2450 1199 647 604 0

141 33 108 – –

131 22 110 0 −1

Fig. 10. Comparison of Mach contour for the forebody with different spike lengths.

K. Zhong et al. / Aerospace Science and Technology 86 (2019) 599–612

Fig. 11. Variations of the drag coefficients versus the spike length.

the spike length increases at α = 0 deg. (This indicates aerospike should be designed as long as possible to achieve a better drag reduction performance.) In the current case, however, there is an optimal point when L /b = 1.0. This is an important phenomenon and deserves further investigation. Since the occurrence of non-monotonous drag variation versus spike length, detailed flow analysis is performed to reveal the underlying mechanism. As mentioned above, the forebody’s drag is mainly derived from pressure drag obtained by surface pressure integral. Therefore, pressure distributions along the forebody with different spike lengths are examined and compared. As seen in Fig. 12, the pressure elevates first and then drops along the forebody for all cases. When L /b < 1.0, the pressure decreases with the increase of spike length, and the location of maximum pressure shifts slightly downstream the shoulder of forebody. When L /b = 1.25, the maximum pressure climbs up and moves upstream significantly. Then, the maximum pressure drops and moves downstream as the spike length increases. Distribution trend mentioned above is true in both x– y plane and x–z plane. As recognized, the value of maximum surface pressure along the forebody in x–z plane indicates the strength of reattachment shock, whereas its position shows the size of large recirculation region. Thus, the non-

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monotonous drag variation at L /b = 1.25 is possibly due to the combined effect of the decrease in recirculation region size and the increase in reattachment shock strength in x–z plane. In order to verify the above explanation, flowfield information of L /b = 1.0 and L /b = 1.25 is further extracted and analyzed herein. According to Ahmed and Qin [15], an important parameter influencing the drag reduction of spiked blunt body is the angle of the reattachment point θ , which is measured in the counterclockwise direction from the centerline of the nose to the dividing streamline on the nose surface in x–z plane. As observed in Fig. 13, θ decreases from 33◦ to 27◦ when L /b varies from 1.0 to 1.25, indicating that the effective body becomes much smaller. This leads to the smaller size of low-pressure recirculation region and stronger strength of reattachment shock due to the larger deflection angle along the forebody shoulder. As a result, the pressure of forebody is increased and the aerodynamic drag climbs up remarkably. To show these variations more intuitively, Fig. 14 illustrates the comparisons of surface pressure distribution and limiting surface streamlines. Note that the left half represents the case of L /b = 1.0 whereas the right half represents that of L /b = 1.25. As known, the maximum pressure indicates the strength of reattachment shock occurring just downstream of the reattachment point, whereas the limiting surface streamlines exhibit the size of the recirculation region. As seen in Fig. 14, the reattachment shock strength of L /b = 1.0 is weaker than that of L /b = 1.25, and the recirculation region of L /b = 1.0 is larger than that of L /b = 1.25. Analysis performed herein verifies the rationality of previous explanation for the occurrence of non-monotonous drag variation. In addition, deep understanding of drag reduction performance of aerodisks attached to different blunt bodies is also of great necessity. Thus, the current results are compared with those for an axisymmetric blunt body as well as for an ellipsoid blunt body with higher aspect ratio. The forebody configurations are presented in Fig. 15(b). Here, the area of each cross section of the blunt body is fixed and only the aspect ratio (Lhf /Lvf, see Fig. 1) is changed, that is, Lhf /Lvf = 1:1, 8:5 and 5:2 respectively. The drag variations versus the spike length for different blunt bodies are displayed in Fig. 15(a). As shown, when no spike is utilized (L /b = 0), the drag decreases with the increase of the aspect ratio of the blunt body. Drag reduction caused by the aerodisk is remarkable for all three cases, and its efficiency is reduced as the aspect ratio of the blunt body increases. The axisymmetric blunt body yields the largest drag reduction effect among the three configurations. Besides, the

Fig. 12. Pressure distribution along the forebody with different spike lengths.

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by comparing the effects of aerodisk size to optimize the disk in the next subsection. 3.3. Effects of aerodisk size on drag reduction

Fig. 13. Angle of reattachment point in x–z plane. Upper half: L /b = 1.0; Lower half: L /b = 1.25.

Fig. 14. Surface pressure distribution and limiting surface streamlines. Left half: L /b = 1.0; Right half: L /b = 1.25.

non-monotonous drag variations versus spike length are observed for all three blunt bodies due to the similar reasons mentioned above. Combining above discussion, L /b = 1.0 is deemed as the best point in this subsection, which gives 65.1% and 39.6% drag reduction for the forebody part and the whole blunt body respectively. Based on the results, the case of L /b = 1.0 will be further analyzed

In this subsection, the investigation on drag reduction effect of aerodisk size is performed. The spike length is fixed to be 100 mm, which is L /b = 1.0, according to the result of Sec. 3.2. The aerodisk is set as a round disk with diameter (D) varying from 0.18b to 0.42b with a step of 0.06b. Streamlines colored by Mach number and surface colored by pressure distribution are shown in Fig. 16. It is noted that the upper half represents the flowfield in x– y plane while the lower half represents that in x–z plane. With the increase of aerodisk diameter, the bow shock becomes more detached from the disk, and the separation point (which is located at the aerodisk shoulder) is pushed further away from the forebody axis. At the same time, there is little change in the location of the reattachment point in x–z plane. The combined effect significantly decreases the inclination angle of the dividing streamline (streamline that separates the external flow from the recirculation zone [1]). As a result, the size of expansion fan at the disk shoulder becomes larger, and the downstream pressure is reduced greatly. This causes the reattachment shock wave to be weaker and the pressure near the reattachment point to become lower with the increase of aerodisk size (see Fig. 16). The variations of total and each part’s drag coefficients versus aerodisk size are shown in Fig. 17. As displayed, the forebody’s drag decreases significantly with the increase of aerodisk size. Meanwhile, the drag over the aerodisk itself increases remarkably due to the increased area affected by the strong bow shock. Besides, it is observed that the drag of spike and afterbody is little affected by the aerodisk size. Overall, the total drag decreases first and then increases, and the maximum drag reduction of 39.8% is achieved when the diameter of aerodisk is 30 mm. 3.4. Effects of aerodisk shape on drag reduction For an axisymmetric blunt body, the conventional aerodisk is also axisymmetric. However, based on the ellipsoid forebody investigated in this study, we consider that an elliptic disk may be superior to a round one in terms of drag reduction characteristics.

Fig. 15. Drag coefficient variations versus the spike length for blunt bodies with different aspect ratios.

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Fig. 16. Streamlines colored by Mach number and surface colored by pressure for forebodies with different aerodisk sizes. Upper half: in x– y plane; Lower half: in x–z plane. L v = Lh = D. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

Fig. 17. Variations of the drag coefficients versus the aerodisk size (L v = Lh = D).

Thus, different from previous work, an elliptic aerodisk is chosen and the effect of its aspect ratio on drag reduction is explored in this subsection.

According to numerical results in Sec. 3.2 and 3.3, the spike length is fixed to be 100 mm, and the elliptic aerodisk size is equal to that of a round one with diameter of 30 mm. Fourteen aspect ratios of the disk (Lh/ L v, see Fig. 1) varying from 0.8 to 4.0 are selected and examined in detail. The flowfields around the elliptic aerodisks with different aspect ratios are displayed in Fig. 18. Streamlines are colored by Mach number and surface is colored by pressure. Besides, the upper half represents the flowfield in x– y plane and the lower half represents that in x–z plane. As shown in Fig. 18(a)–(c), a large recirculation region is attached to the forebody. As the aspect ratio Lh/ L v increases, the separation point in x–z plane is pushed further away from the forebody axis, whereas that in x– y plane gets closer to the forebody axis. At the same time, reattachment point in x–z plane shifts a little downstream the forebody surface. When the aspect ratio becomes even higher (e.g. Fig. 18(d)), quite different scenario will be observed. As seen, two flow separations are clearly predicted near the spike. One of them is attached to the forebody nose and it is much smaller than that of Lh/ L v = 2.6 (see Fig. 18(c)), causing the reattachment point in x–z plane to shift upstream the forebody surface. Besides, the reattachment shock in x–z plane is stronger than that of Lh/ L v = 2.6 due to the larger flow deflection angle along the forebody shoulder. Combining the pressure contours (see Fig. 18) and surface pressure distribution along the forebody (see

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Fig. 18. Streamlines colored by Mach number and surface colored by pressure for the forebodies with different aerodisk aspect ratios. Upper half: in x– y plane; Lower half: in x–z plane.

Fig. 19. Pressure distribution along the forebody with different aerodisk aspect ratios.

Fig. 19), it is clear that as the aspect ratio increases, the maximum surface pressure near the reattachment point decreases first and then increases, and its position moves downstream and then

upstream gradually. Moreover, pressure along the forebody in x– y plane increases with the increase of aspect ratio. These phenomena can be roughly understood as the increase of Lh makes better envelop of forebody in x–z plane whereas the decrease of L v leads to less coverage of forebody in x– y plane. The combined effects result in a reduction of aerodynamic drag of the forebody. However, this does not mean the higher the aspect ratio, the better the “coverage” will be. In fact, the drag reduction performance may become worse if the aspect ratio exceeds a certain degree. Fig. 20 presents the variation of aerodynamic drag versus the aspect ratio of aerodisk. As seen, drag reduction effect is closely related to the relation of the elliptic disk shape and the ellipsoid forebody shape. Specifically, when aspect ratio of the elliptic aerodisk is approaching that of the ellipsoid forebody, the total and forebody’s drag decreases and thus the drag reduction performance improves. The aerodynamic drag maintains a relatively low level and varies little until Lh/ L v = 3.2. Above that aspect ratio, the aerodynamic drag climbs up rapidly and the drag reduction effect turns to be worse. Besides, it is observed that the drag over other parts is nearly constant. Within the research scope of this paper, the elliptic disk with aspect ratio of 2.6 is predicted to obtain the minimum total drag counts of 2479, indicating a drag reduction of 8.4% and 3.8% for the forebody and the whole blunt body compared with the round disk case (Lh/ L v = 1.0).

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Fig. 20. Variation of drag coefficient versus the aspect ratio of aerodisk.

4. Conclusion This paper numerically investigates the drag reduction of aerodisk fixed on the non-axisymmetric blunt body with an ellipsoid nose. The flow structure and surface pressure with and without aerodisk are compared in detail. The effects of spike length, aerodisk size and shape on drag reduction characteristics are discussed. Major conclusions are summarized as follows: (1) The ellipsoid blunt forebody results in the flow structure in x–z plane quite different from that in x– y plane. Flow reattachment occurs and a strong reattachment shock wave generates near the forebody surface in x–z plane, while no obvious reattachment is observed in x– y plane. (2) Flowfield analysis reveals that the non-monotonous drag variation versus spike length is due to changes in the size of large recirculation region and the strength of reattachment shock wave. (3) Total drag of the aerodisked blunt body decreases first and then increases with the increase of aerodisk size. Besides, an elliptic disk with aspect ratio approaching that of the ellipsoid forebody is superior to a round disk with the same size. However, if the aspect ratio of the disk becomes even higher, the drag reduction performance may turn to be worse. Conflict of interest statement The authors declare that they have no conflict of interests. Acknowledgements This work was supported by grants from the National Natural Science Foundation of China (No. 11721202). The first author also acknowledges the help provided by Hong-kang Liu from Beihang University, Xi-wan Sun from National University of Defense Technology and Xiao-yong Wang from Chinese Academy of Sciences. References [1] M.Y.M. Ahmed, N. Qin, Recent advances in the aerothermodynamics of spiked hypersonic vehicles, Prog. Aerosp. Sci. 47 (2011) 425–449. [2] W. Huang, L.Q. Li, L. Yan, T.T. Zhang, Drag and heat flux reduction mechanism of blunted cone with aerodisks, Acta Astronaut. 138 (2017) 168–175. [3] F. Deng, Z. Jiao, B. Liang, F. Xie, N. Qin, Spike effects on drag reduction for hypersonic lifting body, J. Spacecr. Rockets 54 (2017) 1–11. [4] K. Satheesh, G. Jagadeesh, Effect of concentrated energy deposition on the aerodynamic drag of a blunt body in hypersonic flow, Phys. Fluids 19 (2007) 031701.

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