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Uncertainty and sensitivity study on blunt body's drag and heat reduction with combination of spike and opposing jet
Acta Astronautica 167 (2020) 52–62
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Research paper
Uncertainty and sensitivity study on blunt body's drag and heat reduction with combination of spike and opposing jet
T
Sheng Wanga, Wei Zhanga, Fangjie Caia, Qiang Wangb, Chao Yana,∗ a b
National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China China Academy of Aerospace Aerodynamics, Beijing 100074, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Uncertainty quantification Sensitivity analysis Drag and heat reduction Spike Opposing jet
Uncertainty and sensitivity study of three main parameters including length-to-diameter ratio, nozzle radius and total pressure ratio on the drag and heat reduction for the blunt body with spike and opposing jet is conducted. According to the simulation results obtained from the computational fluid dynamics, the relationship between input parameters and aerodynamic force/heat is established via the point-collocation non-intrusive polynomial chaos method. Uncertainties of pressure and heat flux caused by changes of input parameters are quantified. Sensitivity analysis is carried out to determine key parameters affecting pressure and heat flux changes. The results indicate that changes of peak pressure and peak heat flux are resulted from significant change in reattachment shock strength which varies with input parameters. Sensitivity analysis reveals that the length-todiameter ratio has huge influence on the uncertainty in the recirculation zone, while the nozzle radius dominates the uncertainty in the reattachment zone. Moreover, it is found that the nozzle radius has an enormous effect on peak pressure and peak heat flux, and a significant negative correlation between them is identified.
1. Introduction Hypersonic vehicles boast high speed, good maneuverability and strong penetration ability, and have attracted worldwide attention for its potential huge military and economic value [1]. They usually adopt blunt nose to attenuate ablation and provide more space. However, the inevitable strong bow shock wave at high speed will produce huge wave drag and severe aerodynamic heating, which will bring severe challenges to the application of hypersonic vehicles [2,3]. Therefore, it is significant to reduce resistance and heating load of hypersonic vehicles effectively. Various techniques for reducing resistance and heating load have been proposed, such as forward-facing cavity [4,5], spike [6–10], energy deposition [11,12], opposing jet [13–16], and their combinations [17–22]. Among these techniques, installing a spike on the tip of a blunt body first proposed in the 1960s is considered to be the simplest and most promising technique [23–26]. As depicted in Fig. 1, compared with the one without a spike, the flow structure changes obviously under the effect of spike. Strong bow shock waves are reconstructed into several weak oblique shock waves, which greatly reduces pressure and temperature of the main body. Since the concept of using spike to reduce resistance and heat was proposed, the mechanism and key factors have been studied extensively. Kalimuthu and Mehta [27] carried ∗
out experimental studies on the influence of spike shapes and geometric parameters for drag reduction. The authors found that the spike with a suitable geometric parameter has the ability to reduce drag. Yadav and Guven [28] proposed the concept of double-disk, which loaded another aerospike at the tip of the first aerospike. The numerical study of the hemispheric cylinder with Mach 6.2 showed that the double-disk could further decrease peak values of pressure and heating load in the reattachment area compared with a single aerospike. Deng [29] numerically studied the effects of the spike for hypersonic lifting body with angle of attack. Their results indicate that when there is an angle of attack, a certain installation angle is necessary to further reduce the drag and increase the lift-to-drag ratio. Ahmed and Qin [7,30,31] conducted extensive and in-depth numerical studies on the shape, length as well as the oscillating instability modes, and reviewed the recent developments in the aerodynamic force/heat of blunt body with spike. In addition, Xu [25], Huang [32] and Gerdroodbary [33] et al. have also carried out relevant studies and reached meaningful conclusions. Although the spike can reduce peak values of pressure and heating load, it still has shortcomings. The shear layer hits the blunt body and produces reattachment shock wave, which increases local pressure and heat flux. Another problem which cannot be ignored is that the tip of the spike will undergo severe ablation. Strictly speaking, this method
Corresponding author. E-mail address: [email protected] (C. Yan).
https://doi.org/10.1016/j.actaastro.2019.10.045 Received 27 September 2019; Received in revised form 20 October 2019; Accepted 26 October 2019 Available online 05 November 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
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Fig. 1. Flow structures around blunt bodies with and without spikes [8].
can be used to identify key parameters on the basis of contributions to the uncertainty of output. Therefore, the content of this paper is to ascertain key parameters for the combined strategy of spike and opposing jet in terms of resistance and heat reduction by means of uncertainty and sensitivity analysis. According to the preceding analysis, the parameters selected in this paper include L/D, PR and Rj. Firstly, samples are generated by Latin hypercube method [45] and calculated precisely by CFD method. Secondly, uncertainties of the blunt body's pressure and heating load relative to the three parameters is quantified using the NIPC method. Finally, sensitivity analysis is carried out to investigate contributions of each parameter to the total uncertainties.
can only be used as a concept of drag reduction [26]. To reduce drag and heat more effectively, a combination of spike and supersonic jet was proposed. Jiang et al. [34] experimentally studied the application of a combination of aerospike and lateral jet in hypersonic flow. Their research indicates that at 4° angle of attack, the combination method can achieve a 70% reduction in peak pressure at the reattachment point and the lateral jet has the ability to prevent the ablation of the spike. Gerdroodbary et al. [17] numerically calculated the effect of the spike combined with opposing jet on blunt body's heat flux. They found that with the jet total pressure elevating, the peak heat flux declined rapidly, and the effect of using He as the jet medium is better than that using CO2. Huang [35,36] made a detailed numerical study on the combinational strategy and studied the influence of geometric parameters of the spike and parameters of the opposing jet on the resistance and heating load. Eghlima [37,38], Zhu [2] and Qu [21] et al. also carried out further numerical studies on this combined method. According to previous research results, following conclusions can be drawn: 1) The combination of spikes and supersonic jets is an effective method to reduce resistance and heating load of hypersonic blunt body. 2) The length-to-diameter ratio of the spike (L/D), the jet total pressure ratio (PR), and the nozzle radius (Rj) are key parameters for reducing resistance and heating load. 3) The control mechanism for the combined method is very complicated when studying the influence of different parameters, and in-depth analysis with effective methods is indispensable. The control variable method is often adopted in previous studies when studying the parametric influence for spike and opposing jet. That is to say, changing the range of one parameter (such as L/D) and keeping other parameters (PR, Rj, etc.) unchanged to study the effect of this parameter. Since the parameters are independent with each other, there are many combinations among them. And the influence law of one parameter in a particular combination case may be different or even opposite to that in another combination case. Consequently, the parametric study based on control variable method is relatively limited. Moreover, we need to identify the most influential factor among the various factors affecting drag and heat reduction for active control. Therefore, sensitivity analysis for the influence factors is essential. Given that considerable time and cost will be taken to calculate a large number of parameter combinations simply by using CFD method, we can cast this problem into a surrogate model problem. The parameter change can be regarded as the input uncertainty, and is transmitted to output of interest by creating the surrogate model [39]. The point-collocation non-intrusive polynomial chaos (NIPC) technique [40] is an effective method to analyze the uncertainty by construct an surrogate model based on few samples. It has been widely used to quantify the uncertainty of aerodynamic force/heat predictions [41–43]. Furthermore, global sensitivity analysis with Sobol index [44]
2. Numerical method All test cases conducted in this paper are carried out with an inhouse code named MI-CFD, which has been widely used in simulations of hypersonic flows [39,46,47]. The main algorithms used in this study are presented as follow. 2.1. Governing equations The conservative governing equations are written as follow: Continuity equation:
∂ρui ∂ρ =0 + ∂x i ∂t
(1)
Momentum equation:
∂τij ∂ (ρui uj ) ∂p ∂ (ρui ) + =− + ∂x j ∂x i ∂x j ∂t
(2)
Energy equation:
∂ (ρHuj ) μ ∂T ⎞ ∂ (ρE ) ∂ ∂ ⎛ ⎛ μl = + + t⎞ (ui τij ) − ⎜ ⎟ ∂x j ∂x j ∂x j ⎝ ⎝ Pr ∂t Prt ⎠ ∂x j ⎠ ⎜
⎟
(3)
where ρ is the density, ui is the ith velocity component, p is the pressure, T is the temperature, and E is the total energy. μl and μt are laminar and turbulent viscosity coefficient respectively. Pr = 0.72 and Prt = 0.9 are the laminar and turbulent Prandtl number respectively. The viscous stress tensor τij , strain rate tensor Sij and total enthalpy H are given as follows:
1 τij = 2(μl + μt ) ⎛Sij − Skk δij ⎞ 3 ⎝ ⎠ 53
(4)
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2.3. Boundary conditions and discretization
∂uj ⎞ 1 ⎛ ∂ui + ⎜ ⎟ ∂x i ⎠ 2 ⎝ ∂x j
Sij =
(5)
p H=E+ ρ
Boundary conditions of the wall surface are assumed to be isothermal wall conditions with non-slip velocity and zero normal pressure gradient in this study. The definition of heat flux on the wall can be found in the Ref [49]. The opposing jet condition is determined by the jet total temperature T0j, the jet total pressure P0j and the jet Mach number Maj, and P0j is defined as:
(6)
2.2. Turbulence model
P0j = P0∞ ∗ PR
The effect of turbulence fluctuation is considered by using the shear stress transport (SST) turbulence model [48] which is widely used in numerical simulations for the combined strategy of spike and opposing jet configurations [18,21,38]. Transport equations of turbulent kinetic energy k and turbulence specific dissipation rate ω are written as follows:
∂ (ρuj k ) ∂ (ρk ) ∂ ⎡ ∂k ⎤ = + (μl + σk μt ) + Pk − β ∗ρωk ∂x j ∂x j ⎢ ∂t ∂ xj ⎥ ⎣ ⎦
where P0∞ is the total pressure of the freestream. For the discretization, the same strategy is adopted in this paper as in Ref. [50]. Roe's FDS scheme in space with MUSCL interpolation and minmod limiter are utilized for inviscid fluxes. The second order central difference scheme is applied for viscous fluxes. The implicit LowerUpper Symmetric Gauss-Seidel (LU-SGS) scheme is used for the time integration and the Courant-Friedrichs-Levy (CFL) number is set to 0.5 to ensure stability.
(7)
2.4. Numerical validation
∂ (ρuj ω) ∂ (ρω) ∂ ⎡ ∂ω ⎤ = + (μl + σω μt ) + Pω − βρω2 ⎥ ∂x j ∂x j ⎢ ∂t ∂ x j ⎣ ⎦ ρσ ∂k ∂ω + 2(1 − f1 ) ω2 ω ∂x j ∂x j
Test cases are employed to valid the CFD code and the numerical approach used in this paper. Since few experiment data is available from open literatures for the combinational thermal protection systems of aerospike and opposing jet, numerical validations are conducted on the opposing jet and single spike thermal protection systems respectively and compared with experiment data before studying the combined thermal protection system.
(8)
where Pk and Pω are production terms of k and ω respectively, defined as
Pk = μt Ω2 ,Pω = Cω ρΩ2
(9)
where Ω represent the magnitude of vorticity. μt is the eddy viscosity, defined as:
a1 ρk max(a1 ω , f2 ‖Ω‖)
μt =
2.4.1. Validation for the opposing jet model The computational model is the same as that used by Hayashi in the wind tunnel experiment [51]. The model's characteristic diameter is 50 mm and the nozzle radius is 2 mm. The jet species and the freestream are all air. The flow conditions are given as: P0∞ = 1.37Mpa, T0∞ = 397K , Ma∞ = 3.98 and the jet conditions are given as: T0j = 300K , Maj = 1 , PR = 0.4, 0.6, 0.8. In addition, the isothermal wall condition with Twall = 295K is adopted. Intricate flow field requires sufficient numerical resolution to obtain accurate numerical solutions, and the grid cell size in the vicinity of the wall surface has a large impact on the accuracy of heating load calculation. The cell Reynolds number ReΔ is adopted to represent the first cell height (Δh ) in the wall normal direction, ReΔ is defined as:
(10)
σk, σω, β and Cω are constants calculated by the following formula:
φ = f1 φ1 + (1 − f1 ) φ2
(11)
The function f1 and f2 are calculated as follow:
f1 = tanh(Γ14 ), f2 = tanh(Γ24 ) k ⎞ 4ρσω2 k ⎤ ⎛ 500μl Γ1 = min ⎡ ⎢max ⎜ ρωd 2 , 0.09ωd ⎟, CDKω d 2 ⎥, CDKω = max ⎝ ⎠ ⎣ ⎦
ReΔ =
⎛2 ρσω2 ∂k ∂ω , 1 × 10−20⎞ ⎜ ⎟ ⎝ ω ∂x j ∂x j ⎠
(12)
where d is the wall distance, the constant a1 is set to be 0.31, and k is set to be 0.41. The two sets of coefficients φ1 and φ2 are given by
φ2 :
ρ∞ u∞ Δh μ∞
(14)
Preceding research [36,43,52] showed that ReΔ = 5–10 is considered to be a suitable choice in the numerical prediction of aerodynamic heat. Therefore, three different grids are utilized to verify the independence of grids. The cell Reynolds number of these three grids is 6. The total number of the three grids is approximate 1.6 million, 2.9 million and 4.5 million respectively. In addition, the numerical error occurs at each step of the integration and accumulates with the increases of the number of integration steps, so evaluations of the accumulation error in numerical simulations are necessary. According to the method proposed by Smirnov et al.
500μl ⎞ 2 k Γ2 = max ⎛⎜ , ⎟ 0.09 ωd ρωd 2 ⎠ ⎝
φ1:
(13)
σk1 = 0.85 , σω1 = 0.5 , β1 = 0.075 , Cω1 = 0.5332 σk 2 = 1.0 , σω2 = 0.856 , β2 = 0.0828 , Cω2 = 0.4403
Table 1 Assessment of accumulation errors for the opposing jet model. Grids Coarse Medium Fine
block1 block2 block1 block2 block1 block2
S1
S2
S3
Serr
Smax
n
nmax
Rs
1.06E-07 1.06E-07 6.03E-08 6.03E-08 3.52E-08 3.52E-08
5.69E-05 2.79E-06 2.14E-05 1.88E-06 1.45E-05 1.33E-06
2.33E-05 1.17E-06 1.10E-05 5.64E-07 7.54E-06 3.57E-07
8.03E-05 4.07E-06 3.25E-05 2.51E-06 2.21E-05 1.72E-06
0.05 0.05 0.05 0.05 0.05 0.05
50000 50000 50000 50000 50000 50000
3.87E+05 1.51E+08 2.37E+06 3.98E+08 5.13E+06 8.46E+08
7.75 3023.19 47.43 7958.82 102.53 16920.84
54
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in this paper.
2.4.2. Validation for the spike model The computational model is the same as that used by Kalimuthu in the wind tunnel experiment [54]. The geometry and the dimension of the model are shown in Fig. 4. In the figure, D is the diameter of the main sphere and has a value of 40 mm. The flow conditions are given as: P0∞ = 8.345 × 105Pa, T0∞ = 449K , Ma∞ = 6. Three kinds of grids, named coarse grids, medium grids (See Fig. 5) and fine grids, are adopted using the grid strategy introduced in the previous section. The total grid amount of three grids are about 2.6 million, 4.4 million and 6.5 million respectively. The cell Reynolds number of three grids is set to 6. Similarly, accumulation errors of the three grids are evaluated, and the results are shown in Table 2. The ratio Rs is larger than 1 for each grid, indicating that the accumulation error is within the allowable range. Fig. 6 depicts comparisons of the computational density contour with experimental schlieren data when L/D = 1.5. As can be seen, the computational results accurately capture flow structures such as oblique shock, reattachment shock and shear layer, and are consistent with experimental results. Fig. 7 compares the distributions of the computed pressure coefficients (Cp) along the main sphere surface for L/D = 1.5 (left) and L/D = 2 (right) with Kalimuthu's experimental data [54] and Eghlima's CFD results [37]. As conveyed from the figure, the computed results are in good agreement with the experimental data and the reference CFD results. Therefore, validations for the opposing jet model and the spike model suggest that the numerical approach used in this paper can be applied to predictions of pressure and heating load.
Fig. 2. St distributions along the main sphere with different grids when PR = 0.4.
[49,53], accumulation errors of these three grids are calculated and shown in Table 1. From the table, we can see that the ratio between the maximum allowable integration steps and the actual integration steps (Rs) is larger than 1 for each grid, indicating that the accumulation error is within the allowable range. Stanton number (St) is used to describe the distribution of heat flux, which is defined as follows:
St =
qw ρ∞ u∞ cp (Taw − Tw )
Taw = T∞ ⎡1 + ⎣
3
Prw ⎛ ⎝
γ−1 2⎤ ⎞ Ma∞ 2 ⎠ ⎦
(15)
(16) 3. Methodology of uncertainty and sensitivity analysis
where qw is the heat flux of the wall surface, ρ∞ and u∞ denote the density and velocity of the freestream, cp represents the specific heat, Tw is the wall temperature, and Prw denotes the Prandtl number. Fig. 2 presents distributions of St along the main sphere with different grids when PR is equal to 0.4. It can be seen that the distribution trends of St calculated by three grids are consistent. The results of the coarse grid have some differences with those of the other two grids. Hence, the medium grid is utilized in the following validation study. Fig. 3 shows comparisons of the Stanton number calculated in this paper with Hayashi's experimental results and Guo's CFD results [43]. As presented, the values of heat flux obtained from the CFD method is smaller than those obtained from the experimental method at each PR. The variances may be mainly resulted from the surface roughness of the experimental model and the difference of the jet medium used between calculation and experiment [43]. Overall, the results calculated in this paper are between the experimental data and the reference results, which verifies the credibility of the numerical simulation method used
In this work, the jet conditions (PR) and the geometric parameters (Rj, L/D) of the spike are treated as input uncertainties, which are then transmitted to output of interest by creating a surrogate model. The surrogate model is established on the basis of CFD results of the chosen sample points. The response surface relationship between output of interest and input parameters is established by using the stochastic expansions based on the point-collocation NIPC method [55][56]. According to the expression of point-collocation NIPC, the stochastic response value α͠ (such as Cp) can be regarded as a combination of the deterministic part and the stochastic part: ∞
α͠ (x, ξ) =
P
∑ αi (xi) Ψi (ξi) ≈ ∑ αi (xi) Ψi (ξi) i=0
i=0
(17)
The meaning of αi, Ψi , x i and ξi has been described in detail in Ref. [55], it will not be repeated here. The series of Eq. (17) is infinite. In practice, it is truncated to a finite term and a discrete sum is taken over a number of output modes [40]. The total number of samples Ns is given by the following form:
Ns = np·(P + 1) = np·⎡ ⎢ ⎣
(n + p)! ⎤ n ! p! ⎥ ⎦
(18)
Here, n is the number of input parameters, np is the oversampling ratio. It has been indicated that np = 2 is an appropriate choice for comprehensive consideration of computational accuracy and computational cost [57]. Therefore, np is set to 2 in this paper. p is the order of the polynomial chaos expansion and p = 2 is applied in this paper. Then the CFD calculation is carried out at the sample points and the left-hand item of Eq. (17) is replaced by the CFD results to establish the surrogate model. Then the linear system is formulated and the equations can be solved using the least-squares method [45] for the spectral models.
Fig. 3. St distributions along the main sphere under different PR. 55
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Fig. 4. Geometry of the spiked blunt body.
Fig. 6. Comparisons of computed density contour (lower half) with experimental schlieren data (upper half).
The total variance can be decomposed as: i=n
P
D=
i=n−1
∑ αi2 (x ) ⟨Ψi2 (ξ ) ⟩ = ∑ Di + ∑ i=1
i=1
1≤i
i=n−2
∑
Di, j +
Di, j, k + …
1≤i
+D1,2, … , n
(21)
where the partial variances Di can be calculated by: Fig. 5. The computational grids for numerical verification (medium).
∑
Di1, … , is = α∗ (x , ξ )
Ψ (ξ )
⎛ ∗ 0 ⎞ ⎛ 0 0 α (x , ξ1) Ψ (ξ ) ⎜ ⎟=⎜ 0 1 ⎜ α∗ (x ,...ξ ) ⎟ ⎜Ψ (ξ... ) Ns − 1 ⎠ ⎝ 0 Ns−1 ⎝
Ψ1 (ξ0)
...
Ψ1 (ξ1) ... ... Ψ1 (ξ Ns − 1) ...
ΨP (ξ0)
⎞ α0 ΨP (ξ1) ⎛ α1 ⎞ ⎟ ⎜ ... ⎟ ... ⎟ ΨP (ξ Ns − 1) ⎝ α Ns − 1 ⎠ ⎠
1 ≤ i1 < … < is ≤ n (22)
Afterwards, Sobol indices can be utilized in global sensitivity analysis determine the proportion of uncertainty of each parameter in the total uncertainty. Sobol indices can be defined as follow:
(19)
The mean μα and total variance D can be evaluated by
Si1, … , is =
μα = a0 (x ) D=
αβ2 (x ) ⟨Ψβ2 (ξ ) ⟩
β ∈ (i1, … , is )
P ∑i = 1 ai2 (x ) ⟨Ψi2 (ξ ) ⟩
Di1, … , is D Total
1 ≤ i1 < … < is ≤ n
(23)
(20)
Table 2 Assessment of accumulation errors for the spike model. Grids Coarse Medium Fine
block1 block2 block1 block2 block1 block2
S1
S2
S3
Serr
Smax
n
nmax
Rs
6.32E-08 6.32E-08 4.51E-08 4.51E-08 3.32E-08 3.32E-08
3.36E-05 2.28E-06 7.54E-06 1.33E-06 1.88E-06 9.71E-07
1.45E-05 5.64E-07 4.41E-06 3.57E-07 2.79E-06 2.90E-07
4.81E-05 2.91E-06 1.20E-05 1.73E-06 4.71E-06 1.29E-06
0.05 0.05 0.05 0.05 0.05 0.05
40000 40000 40000 40000 40000 40000
1.08E+06 2.96E+08 1.74E+07 8.36E+08 1.13E+08 1.49E+09
26.97 7402.29 434.80 20911.35 2818.64 37309.67
56
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Fig. 7. Distributions of Cp along the main sphere surface for L/D = 1.5 (left) and L/D = 2 (right).
4. Problem description
Table 3 Flow parameters of the baseline conditions.
4.1. Baseline conditions The baseline geometry performed in present study is similar to that used in Section 2.4.2. A slight difference is that the tip of the aerospike becomes flat, and from which the opposing jet is sprayed. The dimensions of the baseline geometry are shown in Fig. 8. The radius of the spike R = 2 mm, and the length L = 60 mm. The freestream conditions are the same as those in Section 2.4.2. The material of the jet is supposed to be air and the total pressure ratio is 0.1. Detailed flow conditions are shown in Table 3. The remaining freestream parameters such as the static pressure and static temperature are determined under the assumptions of the perfect gas model.
Flow parameter
Free stream
Opposing jet
Ma P0(Pa) T0(K)
6 8.345 × 105 449
1 8.345 × 105 × 0.1 449
Table 4 Input parameters and the variation ranges. Input parameter
Baseline value
Varied deviation
Value bound
L/D Rj (mm) PR
1.5 2 0.1
± 10% ± 10% ± 10%
[1.35, 1.65] [1.8, 2.2] [0.09, 0.11]
4.2. Variation sources structure for the baseline case and the no-jet-no-spike case. It can be seen that the flow structure is reshaped under the influence of the spike and the opposing jet. The bow shock wave is reconstructed into several weak oblique shock waves, and the high pressure area is replaced by the recirculation area with low pressure and low speed, which greatly reducing the drag and heat flux. Fig. 10 illustrates the temperature contour for the baseline case and the no-jet-no-spike case. As shown from the figure, with the thermal protection system, the temperature near the stagnation point dropped from 440 K to 367 K, with a decrease of 73 K. Although the surface temperature increases slightly at the shoulder, the effect of heat reduction is obvious overall. Fig. 11 and Fig. 12 illustrate distributions of Cp and St under three conditions, respectively. As shown that Cp and St at the stagnation point are at very high levels in the no-jet-no-spike case. In the other two cases, Cp and St rise to the peak values (at θ ≈ 60∘) and then drop. The peak values are resulted from the reattachment shock wave generated
As aforementioned, LD, Rj and PR are key factors affecting the drag and heat reduction. Changes of these three parameters are regarded as the variation sources. The range of these input parameters is determined by specifying the variation range of baseline values, as shown in Table 4. It is assumed that the input parameters have a deviation of ± 10% from the corresponding baseline reference value in present study. 5. Results and discussions 5.1. Baseline results First, numerical simulations of the baseline case are carried out to illustrate the effect of the combined strategy for reducing resistance and heating load. As a comparison, numerical results for the no-jet case and no-jet-no-spike case are presented. Fig. 9 depicts comparisons of flow
Fig. 8. Dimensions of the spiked blunt body. 57
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Fig. 9. Comparisons of flow structure for the baseline case (Lower half) and the no-jet-no-spike case (Upper half). Fig. 12. St distribution along the hemispheric surface.
5.2. Uncertainty quantification In this section, twenty samples of three design parameters (LD, PR and Rj) are chosen using Latin hypercube method and accurate numerical calculations were performed. The specific values of the three parameters in the twenty samples and the calculated Cpmax and Stmax are shown in Table 5. On the basis of the calculation results, surrogate models are established using the NIPC method for the surface pressure and thermal prediction of the blunt body. According to the chaotic polynomial expansion coefficient of Eq. (19), the mean value μ and standard deviation σ of Cp and St are obtained, and then uncertainties caused by the input parameters are determined. Figs. 13 and 14 present distributions of Cp and St along the hemispheric surface, including the CFD results and results predicted by the NIPC method with a 95% confidence interval μ ± 1.96σ . It can be seen from the figures that the distribution of variation interval width of the Cp and St is consistent. The variation interval width near the recirculation zone is very small, which indicates that parameter changes have little influence on the pressure and heat flux near the recirculation zone. Large variation interval width mainly appears at the reattachment point and its slightly backward position, indicating that peak pressure and peak heat flux are greatly affected by parameter changes. The occurrence of this phenomenon is well explained via Fig. 15, in which
Fig. 10. Temperature contour for the baseline case and the no-jet-no-spike case.
Table 5 Design of experiments and the calculated Cpmax and Stmax.
Fig. 11. Cp distribution along the hemispheric surface.
by the shear layer impinging on the blunt body. Meanwhile, we can see that the peak values of Cp and St in the no-jet case are much higher than that in the with-spike-with-jet case. This is caused by the fact that the reattachment shock in the no-jet case is much stronger, and is closer to the surface which explains the necessity of using a combined thermal protection system.
58
CFD test number
L/D
PR
Rj (mm)
Cpmax
Stmax (×10−3)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1.4763 1.5711 1.5869 1.5237 1.6026 1.4921 1.4448 1.3658 1.3817 1.3974 1.6343 1.6500 1.4289 1.3500 1.6184 1.5552 1.4132 1.5080 1.4606 1.5395
0.10158 0.10263 0.09 0.09947 0.10368 0.11 0.10053 0.09211 0.10474 0.09842 0.09632 0.09737 0.10684 0.10789 0.10579 0.10895 0.09526 0.09316 0.09105 0.09421
2.0948 2.200 1.9894 1.9474 1.8210 2.1158 1.8000 1.8632 2.1790 1.9684 1.8842 2.0736 1.9052 2.0316 2.0526 1.9264 2.1578 1.8422 2.0106 2.1368
0.169 0.161 0.175 0.175 0.174 0.163 0.181 0.179 0.159 0.173 0.173 0.164 0.173 0.163 0.163 0.170 0.167 0.183 0.178 0.169
6.40 6.18 6.72 6.57 6.64 6.23 6.86 6.62 6.17 6.57 6.61 6.33 6.57 6.24 6.27 6.46 6.38 6.87 6.65 6.41
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Fig. 13. Distributions of Cp along the hemispheric surface.
Fig. 15. Pressure contours under two different parameter combinations.
Table 6 The results of uncertainty for peak pressure and peak heat flux. Output quantity
Cpmax Stmax
Baseline value (CFD)
0.173 6.66 × 10−3
Predicted by the NIPC method
μ
μ ± 1.96σ
0.171 6.61 × 10−3
[0.154,0.188] [6.07,7.15] × 10−3
Fig. 14. Distributions of St along the hemispheric surface.
pressure contours under two different parameter combinations are presented. As depicted, the variation of parameters significantly changes the intensity of the reattachment shock, resulting in remarkable changes of Cpmax and Stmax near the reattachment zone, while in both cases, the pressure near the recirculation zone remains basically unchanged. Results of uncertainty quantification for peak pressure and peak heat flux are tabulated in Table 6, including the baseline value obtained from the CFD results, mean value with a 95% confidence interval predicted by the NIPC method. As presented that the baseline value of CFD results are very close to the mean value predicted by the NIPC method and the CFD results of all sample points are within the 95% confidence interval predicted by the NIPC method, indicating that the NIPC method adopted in this paper is feasible.
Fig. 16. Sobol indices for Cp along the hemispheric surface.
rely on the surface location evidently. In the recirculation zone (θ < 40°), Sobol indices for L/D have large values indicating that the spike length has a great influence on the changes of pressure in the recirculation zone. Yamauchi [58] concluded that the flow in the recirculation zone is largely determined by the spike length, and the current sensitivity analysis results are consistent with this conclusion. After passing the location of peak pressure (for baseline condition, θ = 56.6°), contributions from L/D can be almost negligible. On the contrary, contributions from Rj to the variations of pressure in the recirculation zone can be neglected, while in the reattachment zone, the contribution increases rapidly. As for PR, the contribution is small on the whole, especially in the recirculation zone, which is almost
5.3. Sensitivity analysis The sensitivity of the pressure and heat flux for the blunt body involves three input parameters is analyzed to identify key uncertain parameters that affect the output uncertainty. Fig. 16 depicts the distribution of Sobol indices for Cp along the hemispheric surface. As presented from the figure, the Sobol indices 59
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results. The peak position here is obtained from the baseline state for convenience. As can be seen from the table that, for the baseline state selected in this paper, Rj is the key factor affecting the variations of Cpmax and Stmax while L/D and PR have small influences. Additionally, to further verify the results of sensitivity analysis, correlation coefficients were obtained by linear regression method [59,60] in this paper to show the degree of linear relationship between the inputs and outputs. The absolute value of the correlation coefficient is close to 1, indicating that the input and output have a strong correlation, while the positive or negative correlation coefficient represents the positive or negative correlation between the input and output. Fig. 18 illustrates the scatter plots of Cpmax and Stmax versus input parameters. As conveyed from the figure, the correlation coefficients between Rj and Cpmax as well as Stmax are much larger than other parameters, indicating that Rj has significant influence on Cpmax and Stmax. This is consistent with the results of sensitivity analysis using NIPC method, which verifies the effectiveness of the NIPC method.
Fig. 17. Sobol indices for St along the hemispheric surface. Table 7 Sobol indices for Cpmax and Stmax. Quantity of interest
Cpmax Stmax
Surface location
θ = 56.6° θ = 58.2°
6. Conclusions Uncertainty and sensitivity studies of three main parameters (L/D, Rj and PR) on the drag and heat reduction for the blunt body with spike and opposing jet are carried out. Twenty numerical simulations were performed for different parameter combinations using the CFD method. Based on the calculation results, the relationship between input parameters and aerodynamic force/heat is established using the NIPC method. Uncertainties of Cp and St caused by changes of input parameters are quantified. Sensitivity analysis is carried out to determine key parameters affecting the pressure and heat flux changes. Main discoveries are summarized as follows:
Sobol indices L/D
PR
R
0.158 0.136
0.226 0.226
0.616 0.638
ineffective. The variation law of Sobol indices for heat flux is the same with that for pressure coefficient expect that the Sobol index of PR has a slight increase in the recirculation zone (see Fig. 17). In terms of drag and heat reduction, it is usually more concerned with changes in peak pressure and peak heat flux. Therefore, Sobol indices of each input parameter at the peak position is presented in Table 7. The peak position calculated under different input parameters is different, but the discrepancies are very small according to the CFD
(1) Uncertainties for Cp and St are relied on the location of the hemisphere surface. Large variation intervals are distributed at the reattachment point and its slightly backward position, indicating that Cpmax and Stmax are greatly affected by parameter changes. Further
Fig. 18. Scatter plots of Cpmax and Stmax versus input parameters. 60
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analysis demonstrates that changes of peak pressure and peak heat flux are resulted from significant changes in reattachment shock strength which varies with input parameters. (2) Under the influence of the combinational thermal protection system, flow ahead of the blunt body is divided into two different zones, the recirculation zone and the reattachment zone. Sensitivity analysis indicates that L/D dominates the uncertainty of the recirculation zone, while Rj makes major contributions in the uncertainty of the reattachment zone. (3) For Cpmax and Stmax, Rj has an enormous effect. Linear regression analysis shows that a significant negative correlation between Rj and the peak values of pressure and heat flux is identified. Therefore, the value of Rj must be considered reasonably in practical applications, since small changes may have substantial effects.
[17] [18]
[19]
[20]
[21]
[22]
It is worth noting that the uncertainty and sensitivity results obtained in this study rely on the baseline state (geometry and jet parameters) selected in this paper. Substantially altered geometry and jet parameters may result in different aerodynamic characteristics. Meanwhile, various freestream conditions may change the structure of the flow field. Hence, the key factors affecting drag and heat reduction for hypersonic vehicles may change. Further study for aerodynamic uncertainty and sensitivity is necessary.
[23]
[24] [25] [26]
[27]
Declaration of competing interest
[28]
None declared.
[29] [30]
Acknowledgement
[31]
This work is supported by grants from the National Natural Science Foundation of China (No.11721202). The first author also acknowledges the help provided by Ya-Tian Zhao and Kang Zhong from Beihang University.
[32] [33] [34]
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Research paper
Uncertainty and sensitivity study on blunt body's drag and heat reduction with combination of spike and opposing jet
T
Sheng Wanga, Wei Zhanga, Fangjie Caia, Qiang Wangb, Chao Yana,∗ a b
National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China China Academy of Aerospace Aerodynamics, Beijing 100074, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Uncertainty quantification Sensitivity analysis Drag and heat reduction Spike Opposing jet
Uncertainty and sensitivity study of three main parameters including length-to-diameter ratio, nozzle radius and total pressure ratio on the drag and heat reduction for the blunt body with spike and opposing jet is conducted. According to the simulation results obtained from the computational fluid dynamics, the relationship between input parameters and aerodynamic force/heat is established via the point-collocation non-intrusive polynomial chaos method. Uncertainties of pressure and heat flux caused by changes of input parameters are quantified. Sensitivity analysis is carried out to determine key parameters affecting pressure and heat flux changes. The results indicate that changes of peak pressure and peak heat flux are resulted from significant change in reattachment shock strength which varies with input parameters. Sensitivity analysis reveals that the length-todiameter ratio has huge influence on the uncertainty in the recirculation zone, while the nozzle radius dominates the uncertainty in the reattachment zone. Moreover, it is found that the nozzle radius has an enormous effect on peak pressure and peak heat flux, and a significant negative correlation between them is identified.
1. Introduction Hypersonic vehicles boast high speed, good maneuverability and strong penetration ability, and have attracted worldwide attention for its potential huge military and economic value [1]. They usually adopt blunt nose to attenuate ablation and provide more space. However, the inevitable strong bow shock wave at high speed will produce huge wave drag and severe aerodynamic heating, which will bring severe challenges to the application of hypersonic vehicles [2,3]. Therefore, it is significant to reduce resistance and heating load of hypersonic vehicles effectively. Various techniques for reducing resistance and heating load have been proposed, such as forward-facing cavity [4,5], spike [6–10], energy deposition [11,12], opposing jet [13–16], and their combinations [17–22]. Among these techniques, installing a spike on the tip of a blunt body first proposed in the 1960s is considered to be the simplest and most promising technique [23–26]. As depicted in Fig. 1, compared with the one without a spike, the flow structure changes obviously under the effect of spike. Strong bow shock waves are reconstructed into several weak oblique shock waves, which greatly reduces pressure and temperature of the main body. Since the concept of using spike to reduce resistance and heat was proposed, the mechanism and key factors have been studied extensively. Kalimuthu and Mehta [27] carried ∗
out experimental studies on the influence of spike shapes and geometric parameters for drag reduction. The authors found that the spike with a suitable geometric parameter has the ability to reduce drag. Yadav and Guven [28] proposed the concept of double-disk, which loaded another aerospike at the tip of the first aerospike. The numerical study of the hemispheric cylinder with Mach 6.2 showed that the double-disk could further decrease peak values of pressure and heating load in the reattachment area compared with a single aerospike. Deng [29] numerically studied the effects of the spike for hypersonic lifting body with angle of attack. Their results indicate that when there is an angle of attack, a certain installation angle is necessary to further reduce the drag and increase the lift-to-drag ratio. Ahmed and Qin [7,30,31] conducted extensive and in-depth numerical studies on the shape, length as well as the oscillating instability modes, and reviewed the recent developments in the aerodynamic force/heat of blunt body with spike. In addition, Xu [25], Huang [32] and Gerdroodbary [33] et al. have also carried out relevant studies and reached meaningful conclusions. Although the spike can reduce peak values of pressure and heating load, it still has shortcomings. The shear layer hits the blunt body and produces reattachment shock wave, which increases local pressure and heat flux. Another problem which cannot be ignored is that the tip of the spike will undergo severe ablation. Strictly speaking, this method
Corresponding author. E-mail address: [email protected] (C. Yan).
https://doi.org/10.1016/j.actaastro.2019.10.045 Received 27 September 2019; Received in revised form 20 October 2019; Accepted 26 October 2019 Available online 05 November 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
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Fig. 1. Flow structures around blunt bodies with and without spikes [8].
can be used to identify key parameters on the basis of contributions to the uncertainty of output. Therefore, the content of this paper is to ascertain key parameters for the combined strategy of spike and opposing jet in terms of resistance and heat reduction by means of uncertainty and sensitivity analysis. According to the preceding analysis, the parameters selected in this paper include L/D, PR and Rj. Firstly, samples are generated by Latin hypercube method [45] and calculated precisely by CFD method. Secondly, uncertainties of the blunt body's pressure and heating load relative to the three parameters is quantified using the NIPC method. Finally, sensitivity analysis is carried out to investigate contributions of each parameter to the total uncertainties.
can only be used as a concept of drag reduction [26]. To reduce drag and heat more effectively, a combination of spike and supersonic jet was proposed. Jiang et al. [34] experimentally studied the application of a combination of aerospike and lateral jet in hypersonic flow. Their research indicates that at 4° angle of attack, the combination method can achieve a 70% reduction in peak pressure at the reattachment point and the lateral jet has the ability to prevent the ablation of the spike. Gerdroodbary et al. [17] numerically calculated the effect of the spike combined with opposing jet on blunt body's heat flux. They found that with the jet total pressure elevating, the peak heat flux declined rapidly, and the effect of using He as the jet medium is better than that using CO2. Huang [35,36] made a detailed numerical study on the combinational strategy and studied the influence of geometric parameters of the spike and parameters of the opposing jet on the resistance and heating load. Eghlima [37,38], Zhu [2] and Qu [21] et al. also carried out further numerical studies on this combined method. According to previous research results, following conclusions can be drawn: 1) The combination of spikes and supersonic jets is an effective method to reduce resistance and heating load of hypersonic blunt body. 2) The length-to-diameter ratio of the spike (L/D), the jet total pressure ratio (PR), and the nozzle radius (Rj) are key parameters for reducing resistance and heating load. 3) The control mechanism for the combined method is very complicated when studying the influence of different parameters, and in-depth analysis with effective methods is indispensable. The control variable method is often adopted in previous studies when studying the parametric influence for spike and opposing jet. That is to say, changing the range of one parameter (such as L/D) and keeping other parameters (PR, Rj, etc.) unchanged to study the effect of this parameter. Since the parameters are independent with each other, there are many combinations among them. And the influence law of one parameter in a particular combination case may be different or even opposite to that in another combination case. Consequently, the parametric study based on control variable method is relatively limited. Moreover, we need to identify the most influential factor among the various factors affecting drag and heat reduction for active control. Therefore, sensitivity analysis for the influence factors is essential. Given that considerable time and cost will be taken to calculate a large number of parameter combinations simply by using CFD method, we can cast this problem into a surrogate model problem. The parameter change can be regarded as the input uncertainty, and is transmitted to output of interest by creating the surrogate model [39]. The point-collocation non-intrusive polynomial chaos (NIPC) technique [40] is an effective method to analyze the uncertainty by construct an surrogate model based on few samples. It has been widely used to quantify the uncertainty of aerodynamic force/heat predictions [41–43]. Furthermore, global sensitivity analysis with Sobol index [44]
2. Numerical method All test cases conducted in this paper are carried out with an inhouse code named MI-CFD, which has been widely used in simulations of hypersonic flows [39,46,47]. The main algorithms used in this study are presented as follow. 2.1. Governing equations The conservative governing equations are written as follow: Continuity equation:
∂ρui ∂ρ =0 + ∂x i ∂t
(1)
Momentum equation:
∂τij ∂ (ρui uj ) ∂p ∂ (ρui ) + =− + ∂x j ∂x i ∂x j ∂t
(2)
Energy equation:
∂ (ρHuj ) μ ∂T ⎞ ∂ (ρE ) ∂ ∂ ⎛ ⎛ μl = + + t⎞ (ui τij ) − ⎜ ⎟ ∂x j ∂x j ∂x j ⎝ ⎝ Pr ∂t Prt ⎠ ∂x j ⎠ ⎜
⎟
(3)
where ρ is the density, ui is the ith velocity component, p is the pressure, T is the temperature, and E is the total energy. μl and μt are laminar and turbulent viscosity coefficient respectively. Pr = 0.72 and Prt = 0.9 are the laminar and turbulent Prandtl number respectively. The viscous stress tensor τij , strain rate tensor Sij and total enthalpy H are given as follows:
1 τij = 2(μl + μt ) ⎛Sij − Skk δij ⎞ 3 ⎝ ⎠ 53
(4)
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2.3. Boundary conditions and discretization
∂uj ⎞ 1 ⎛ ∂ui + ⎜ ⎟ ∂x i ⎠ 2 ⎝ ∂x j
Sij =
(5)
p H=E+ ρ
Boundary conditions of the wall surface are assumed to be isothermal wall conditions with non-slip velocity and zero normal pressure gradient in this study. The definition of heat flux on the wall can be found in the Ref [49]. The opposing jet condition is determined by the jet total temperature T0j, the jet total pressure P0j and the jet Mach number Maj, and P0j is defined as:
(6)
2.2. Turbulence model
P0j = P0∞ ∗ PR
The effect of turbulence fluctuation is considered by using the shear stress transport (SST) turbulence model [48] which is widely used in numerical simulations for the combined strategy of spike and opposing jet configurations [18,21,38]. Transport equations of turbulent kinetic energy k and turbulence specific dissipation rate ω are written as follows:
∂ (ρuj k ) ∂ (ρk ) ∂ ⎡ ∂k ⎤ = + (μl + σk μt ) + Pk − β ∗ρωk ∂x j ∂x j ⎢ ∂t ∂ xj ⎥ ⎣ ⎦
where P0∞ is the total pressure of the freestream. For the discretization, the same strategy is adopted in this paper as in Ref. [50]. Roe's FDS scheme in space with MUSCL interpolation and minmod limiter are utilized for inviscid fluxes. The second order central difference scheme is applied for viscous fluxes. The implicit LowerUpper Symmetric Gauss-Seidel (LU-SGS) scheme is used for the time integration and the Courant-Friedrichs-Levy (CFL) number is set to 0.5 to ensure stability.
(7)
2.4. Numerical validation
∂ (ρuj ω) ∂ (ρω) ∂ ⎡ ∂ω ⎤ = + (μl + σω μt ) + Pω − βρω2 ⎥ ∂x j ∂x j ⎢ ∂t ∂ x j ⎣ ⎦ ρσ ∂k ∂ω + 2(1 − f1 ) ω2 ω ∂x j ∂x j
Test cases are employed to valid the CFD code and the numerical approach used in this paper. Since few experiment data is available from open literatures for the combinational thermal protection systems of aerospike and opposing jet, numerical validations are conducted on the opposing jet and single spike thermal protection systems respectively and compared with experiment data before studying the combined thermal protection system.
(8)
where Pk and Pω are production terms of k and ω respectively, defined as
Pk = μt Ω2 ,Pω = Cω ρΩ2
(9)
where Ω represent the magnitude of vorticity. μt is the eddy viscosity, defined as:
a1 ρk max(a1 ω , f2 ‖Ω‖)
μt =
2.4.1. Validation for the opposing jet model The computational model is the same as that used by Hayashi in the wind tunnel experiment [51]. The model's characteristic diameter is 50 mm and the nozzle radius is 2 mm. The jet species and the freestream are all air. The flow conditions are given as: P0∞ = 1.37Mpa, T0∞ = 397K , Ma∞ = 3.98 and the jet conditions are given as: T0j = 300K , Maj = 1 , PR = 0.4, 0.6, 0.8. In addition, the isothermal wall condition with Twall = 295K is adopted. Intricate flow field requires sufficient numerical resolution to obtain accurate numerical solutions, and the grid cell size in the vicinity of the wall surface has a large impact on the accuracy of heating load calculation. The cell Reynolds number ReΔ is adopted to represent the first cell height (Δh ) in the wall normal direction, ReΔ is defined as:
(10)
σk, σω, β and Cω are constants calculated by the following formula:
φ = f1 φ1 + (1 − f1 ) φ2
(11)
The function f1 and f2 are calculated as follow:
f1 = tanh(Γ14 ), f2 = tanh(Γ24 ) k ⎞ 4ρσω2 k ⎤ ⎛ 500μl Γ1 = min ⎡ ⎢max ⎜ ρωd 2 , 0.09ωd ⎟, CDKω d 2 ⎥, CDKω = max ⎝ ⎠ ⎣ ⎦
ReΔ =
⎛2 ρσω2 ∂k ∂ω , 1 × 10−20⎞ ⎜ ⎟ ⎝ ω ∂x j ∂x j ⎠
(12)
where d is the wall distance, the constant a1 is set to be 0.31, and k is set to be 0.41. The two sets of coefficients φ1 and φ2 are given by
φ2 :
ρ∞ u∞ Δh μ∞
(14)
Preceding research [36,43,52] showed that ReΔ = 5–10 is considered to be a suitable choice in the numerical prediction of aerodynamic heat. Therefore, three different grids are utilized to verify the independence of grids. The cell Reynolds number of these three grids is 6. The total number of the three grids is approximate 1.6 million, 2.9 million and 4.5 million respectively. In addition, the numerical error occurs at each step of the integration and accumulates with the increases of the number of integration steps, so evaluations of the accumulation error in numerical simulations are necessary. According to the method proposed by Smirnov et al.
500μl ⎞ 2 k Γ2 = max ⎛⎜ , ⎟ 0.09 ωd ρωd 2 ⎠ ⎝
φ1:
(13)
σk1 = 0.85 , σω1 = 0.5 , β1 = 0.075 , Cω1 = 0.5332 σk 2 = 1.0 , σω2 = 0.856 , β2 = 0.0828 , Cω2 = 0.4403
Table 1 Assessment of accumulation errors for the opposing jet model. Grids Coarse Medium Fine
block1 block2 block1 block2 block1 block2
S1
S2
S3
Serr
Smax
n
nmax
Rs
1.06E-07 1.06E-07 6.03E-08 6.03E-08 3.52E-08 3.52E-08
5.69E-05 2.79E-06 2.14E-05 1.88E-06 1.45E-05 1.33E-06
2.33E-05 1.17E-06 1.10E-05 5.64E-07 7.54E-06 3.57E-07
8.03E-05 4.07E-06 3.25E-05 2.51E-06 2.21E-05 1.72E-06
0.05 0.05 0.05 0.05 0.05 0.05
50000 50000 50000 50000 50000 50000
3.87E+05 1.51E+08 2.37E+06 3.98E+08 5.13E+06 8.46E+08
7.75 3023.19 47.43 7958.82 102.53 16920.84
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in this paper.
2.4.2. Validation for the spike model The computational model is the same as that used by Kalimuthu in the wind tunnel experiment [54]. The geometry and the dimension of the model are shown in Fig. 4. In the figure, D is the diameter of the main sphere and has a value of 40 mm. The flow conditions are given as: P0∞ = 8.345 × 105Pa, T0∞ = 449K , Ma∞ = 6. Three kinds of grids, named coarse grids, medium grids (See Fig. 5) and fine grids, are adopted using the grid strategy introduced in the previous section. The total grid amount of three grids are about 2.6 million, 4.4 million and 6.5 million respectively. The cell Reynolds number of three grids is set to 6. Similarly, accumulation errors of the three grids are evaluated, and the results are shown in Table 2. The ratio Rs is larger than 1 for each grid, indicating that the accumulation error is within the allowable range. Fig. 6 depicts comparisons of the computational density contour with experimental schlieren data when L/D = 1.5. As can be seen, the computational results accurately capture flow structures such as oblique shock, reattachment shock and shear layer, and are consistent with experimental results. Fig. 7 compares the distributions of the computed pressure coefficients (Cp) along the main sphere surface for L/D = 1.5 (left) and L/D = 2 (right) with Kalimuthu's experimental data [54] and Eghlima's CFD results [37]. As conveyed from the figure, the computed results are in good agreement with the experimental data and the reference CFD results. Therefore, validations for the opposing jet model and the spike model suggest that the numerical approach used in this paper can be applied to predictions of pressure and heating load.
Fig. 2. St distributions along the main sphere with different grids when PR = 0.4.
[49,53], accumulation errors of these three grids are calculated and shown in Table 1. From the table, we can see that the ratio between the maximum allowable integration steps and the actual integration steps (Rs) is larger than 1 for each grid, indicating that the accumulation error is within the allowable range. Stanton number (St) is used to describe the distribution of heat flux, which is defined as follows:
St =
qw ρ∞ u∞ cp (Taw − Tw )
Taw = T∞ ⎡1 + ⎣
3
Prw ⎛ ⎝
γ−1 2⎤ ⎞ Ma∞ 2 ⎠ ⎦
(15)
(16) 3. Methodology of uncertainty and sensitivity analysis
where qw is the heat flux of the wall surface, ρ∞ and u∞ denote the density and velocity of the freestream, cp represents the specific heat, Tw is the wall temperature, and Prw denotes the Prandtl number. Fig. 2 presents distributions of St along the main sphere with different grids when PR is equal to 0.4. It can be seen that the distribution trends of St calculated by three grids are consistent. The results of the coarse grid have some differences with those of the other two grids. Hence, the medium grid is utilized in the following validation study. Fig. 3 shows comparisons of the Stanton number calculated in this paper with Hayashi's experimental results and Guo's CFD results [43]. As presented, the values of heat flux obtained from the CFD method is smaller than those obtained from the experimental method at each PR. The variances may be mainly resulted from the surface roughness of the experimental model and the difference of the jet medium used between calculation and experiment [43]. Overall, the results calculated in this paper are between the experimental data and the reference results, which verifies the credibility of the numerical simulation method used
In this work, the jet conditions (PR) and the geometric parameters (Rj, L/D) of the spike are treated as input uncertainties, which are then transmitted to output of interest by creating a surrogate model. The surrogate model is established on the basis of CFD results of the chosen sample points. The response surface relationship between output of interest and input parameters is established by using the stochastic expansions based on the point-collocation NIPC method [55][56]. According to the expression of point-collocation NIPC, the stochastic response value α͠ (such as Cp) can be regarded as a combination of the deterministic part and the stochastic part: ∞
α͠ (x, ξ) =
P
∑ αi (xi) Ψi (ξi) ≈ ∑ αi (xi) Ψi (ξi) i=0
i=0
(17)
The meaning of αi, Ψi , x i and ξi has been described in detail in Ref. [55], it will not be repeated here. The series of Eq. (17) is infinite. In practice, it is truncated to a finite term and a discrete sum is taken over a number of output modes [40]. The total number of samples Ns is given by the following form:
Ns = np·(P + 1) = np·⎡ ⎢ ⎣
(n + p)! ⎤ n ! p! ⎥ ⎦
(18)
Here, n is the number of input parameters, np is the oversampling ratio. It has been indicated that np = 2 is an appropriate choice for comprehensive consideration of computational accuracy and computational cost [57]. Therefore, np is set to 2 in this paper. p is the order of the polynomial chaos expansion and p = 2 is applied in this paper. Then the CFD calculation is carried out at the sample points and the left-hand item of Eq. (17) is replaced by the CFD results to establish the surrogate model. Then the linear system is formulated and the equations can be solved using the least-squares method [45] for the spectral models.
Fig. 3. St distributions along the main sphere under different PR. 55
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Fig. 4. Geometry of the spiked blunt body.
Fig. 6. Comparisons of computed density contour (lower half) with experimental schlieren data (upper half).
The total variance can be decomposed as: i=n
P
D=
i=n−1
∑ αi2 (x ) ⟨Ψi2 (ξ ) ⟩ = ∑ Di + ∑ i=1
i=1
1≤i
i=n−2
∑
Di, j +
Di, j, k + …
1≤i
+D1,2, … , n
(21)
where the partial variances Di can be calculated by: Fig. 5. The computational grids for numerical verification (medium).
∑
Di1, … , is = α∗ (x , ξ )
Ψ (ξ )
⎛ ∗ 0 ⎞ ⎛ 0 0 α (x , ξ1) Ψ (ξ ) ⎜ ⎟=⎜ 0 1 ⎜ α∗ (x ,...ξ ) ⎟ ⎜Ψ (ξ... ) Ns − 1 ⎠ ⎝ 0 Ns−1 ⎝
Ψ1 (ξ0)
...
Ψ1 (ξ1) ... ... Ψ1 (ξ Ns − 1) ...
ΨP (ξ0)
⎞ α0 ΨP (ξ1) ⎛ α1 ⎞ ⎟ ⎜ ... ⎟ ... ⎟ ΨP (ξ Ns − 1) ⎝ α Ns − 1 ⎠ ⎠
1 ≤ i1 < … < is ≤ n (22)
Afterwards, Sobol indices can be utilized in global sensitivity analysis determine the proportion of uncertainty of each parameter in the total uncertainty. Sobol indices can be defined as follow:
(19)
The mean μα and total variance D can be evaluated by
Si1, … , is =
μα = a0 (x ) D=
αβ2 (x ) ⟨Ψβ2 (ξ ) ⟩
β ∈ (i1, … , is )
P ∑i = 1 ai2 (x ) ⟨Ψi2 (ξ ) ⟩
Di1, … , is D Total
1 ≤ i1 < … < is ≤ n
(23)
(20)
Table 2 Assessment of accumulation errors for the spike model. Grids Coarse Medium Fine
block1 block2 block1 block2 block1 block2
S1
S2
S3
Serr
Smax
n
nmax
Rs
6.32E-08 6.32E-08 4.51E-08 4.51E-08 3.32E-08 3.32E-08
3.36E-05 2.28E-06 7.54E-06 1.33E-06 1.88E-06 9.71E-07
1.45E-05 5.64E-07 4.41E-06 3.57E-07 2.79E-06 2.90E-07
4.81E-05 2.91E-06 1.20E-05 1.73E-06 4.71E-06 1.29E-06
0.05 0.05 0.05 0.05 0.05 0.05
40000 40000 40000 40000 40000 40000
1.08E+06 2.96E+08 1.74E+07 8.36E+08 1.13E+08 1.49E+09
26.97 7402.29 434.80 20911.35 2818.64 37309.67
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Fig. 7. Distributions of Cp along the main sphere surface for L/D = 1.5 (left) and L/D = 2 (right).
4. Problem description
Table 3 Flow parameters of the baseline conditions.
4.1. Baseline conditions The baseline geometry performed in present study is similar to that used in Section 2.4.2. A slight difference is that the tip of the aerospike becomes flat, and from which the opposing jet is sprayed. The dimensions of the baseline geometry are shown in Fig. 8. The radius of the spike R = 2 mm, and the length L = 60 mm. The freestream conditions are the same as those in Section 2.4.2. The material of the jet is supposed to be air and the total pressure ratio is 0.1. Detailed flow conditions are shown in Table 3. The remaining freestream parameters such as the static pressure and static temperature are determined under the assumptions of the perfect gas model.
Flow parameter
Free stream
Opposing jet
Ma P0(Pa) T0(K)
6 8.345 × 105 449
1 8.345 × 105 × 0.1 449
Table 4 Input parameters and the variation ranges. Input parameter
Baseline value
Varied deviation
Value bound
L/D Rj (mm) PR
1.5 2 0.1
± 10% ± 10% ± 10%
[1.35, 1.65] [1.8, 2.2] [0.09, 0.11]
4.2. Variation sources structure for the baseline case and the no-jet-no-spike case. It can be seen that the flow structure is reshaped under the influence of the spike and the opposing jet. The bow shock wave is reconstructed into several weak oblique shock waves, and the high pressure area is replaced by the recirculation area with low pressure and low speed, which greatly reducing the drag and heat flux. Fig. 10 illustrates the temperature contour for the baseline case and the no-jet-no-spike case. As shown from the figure, with the thermal protection system, the temperature near the stagnation point dropped from 440 K to 367 K, with a decrease of 73 K. Although the surface temperature increases slightly at the shoulder, the effect of heat reduction is obvious overall. Fig. 11 and Fig. 12 illustrate distributions of Cp and St under three conditions, respectively. As shown that Cp and St at the stagnation point are at very high levels in the no-jet-no-spike case. In the other two cases, Cp and St rise to the peak values (at θ ≈ 60∘) and then drop. The peak values are resulted from the reattachment shock wave generated
As aforementioned, LD, Rj and PR are key factors affecting the drag and heat reduction. Changes of these three parameters are regarded as the variation sources. The range of these input parameters is determined by specifying the variation range of baseline values, as shown in Table 4. It is assumed that the input parameters have a deviation of ± 10% from the corresponding baseline reference value in present study. 5. Results and discussions 5.1. Baseline results First, numerical simulations of the baseline case are carried out to illustrate the effect of the combined strategy for reducing resistance and heating load. As a comparison, numerical results for the no-jet case and no-jet-no-spike case are presented. Fig. 9 depicts comparisons of flow
Fig. 8. Dimensions of the spiked blunt body. 57
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Fig. 9. Comparisons of flow structure for the baseline case (Lower half) and the no-jet-no-spike case (Upper half). Fig. 12. St distribution along the hemispheric surface.
5.2. Uncertainty quantification In this section, twenty samples of three design parameters (LD, PR and Rj) are chosen using Latin hypercube method and accurate numerical calculations were performed. The specific values of the three parameters in the twenty samples and the calculated Cpmax and Stmax are shown in Table 5. On the basis of the calculation results, surrogate models are established using the NIPC method for the surface pressure and thermal prediction of the blunt body. According to the chaotic polynomial expansion coefficient of Eq. (19), the mean value μ and standard deviation σ of Cp and St are obtained, and then uncertainties caused by the input parameters are determined. Figs. 13 and 14 present distributions of Cp and St along the hemispheric surface, including the CFD results and results predicted by the NIPC method with a 95% confidence interval μ ± 1.96σ . It can be seen from the figures that the distribution of variation interval width of the Cp and St is consistent. The variation interval width near the recirculation zone is very small, which indicates that parameter changes have little influence on the pressure and heat flux near the recirculation zone. Large variation interval width mainly appears at the reattachment point and its slightly backward position, indicating that peak pressure and peak heat flux are greatly affected by parameter changes. The occurrence of this phenomenon is well explained via Fig. 15, in which
Fig. 10. Temperature contour for the baseline case and the no-jet-no-spike case.
Table 5 Design of experiments and the calculated Cpmax and Stmax.
Fig. 11. Cp distribution along the hemispheric surface.
by the shear layer impinging on the blunt body. Meanwhile, we can see that the peak values of Cp and St in the no-jet case are much higher than that in the with-spike-with-jet case. This is caused by the fact that the reattachment shock in the no-jet case is much stronger, and is closer to the surface which explains the necessity of using a combined thermal protection system.
58
CFD test number
L/D
PR
Rj (mm)
Cpmax
Stmax (×10−3)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1.4763 1.5711 1.5869 1.5237 1.6026 1.4921 1.4448 1.3658 1.3817 1.3974 1.6343 1.6500 1.4289 1.3500 1.6184 1.5552 1.4132 1.5080 1.4606 1.5395
0.10158 0.10263 0.09 0.09947 0.10368 0.11 0.10053 0.09211 0.10474 0.09842 0.09632 0.09737 0.10684 0.10789 0.10579 0.10895 0.09526 0.09316 0.09105 0.09421
2.0948 2.200 1.9894 1.9474 1.8210 2.1158 1.8000 1.8632 2.1790 1.9684 1.8842 2.0736 1.9052 2.0316 2.0526 1.9264 2.1578 1.8422 2.0106 2.1368
0.169 0.161 0.175 0.175 0.174 0.163 0.181 0.179 0.159 0.173 0.173 0.164 0.173 0.163 0.163 0.170 0.167 0.183 0.178 0.169
6.40 6.18 6.72 6.57 6.64 6.23 6.86 6.62 6.17 6.57 6.61 6.33 6.57 6.24 6.27 6.46 6.38 6.87 6.65 6.41
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Fig. 13. Distributions of Cp along the hemispheric surface.
Fig. 15. Pressure contours under two different parameter combinations.
Table 6 The results of uncertainty for peak pressure and peak heat flux. Output quantity
Cpmax Stmax
Baseline value (CFD)
0.173 6.66 × 10−3
Predicted by the NIPC method
μ
μ ± 1.96σ
0.171 6.61 × 10−3
[0.154,0.188] [6.07,7.15] × 10−3
Fig. 14. Distributions of St along the hemispheric surface.
pressure contours under two different parameter combinations are presented. As depicted, the variation of parameters significantly changes the intensity of the reattachment shock, resulting in remarkable changes of Cpmax and Stmax near the reattachment zone, while in both cases, the pressure near the recirculation zone remains basically unchanged. Results of uncertainty quantification for peak pressure and peak heat flux are tabulated in Table 6, including the baseline value obtained from the CFD results, mean value with a 95% confidence interval predicted by the NIPC method. As presented that the baseline value of CFD results are very close to the mean value predicted by the NIPC method and the CFD results of all sample points are within the 95% confidence interval predicted by the NIPC method, indicating that the NIPC method adopted in this paper is feasible.
Fig. 16. Sobol indices for Cp along the hemispheric surface.
rely on the surface location evidently. In the recirculation zone (θ < 40°), Sobol indices for L/D have large values indicating that the spike length has a great influence on the changes of pressure in the recirculation zone. Yamauchi [58] concluded that the flow in the recirculation zone is largely determined by the spike length, and the current sensitivity analysis results are consistent with this conclusion. After passing the location of peak pressure (for baseline condition, θ = 56.6°), contributions from L/D can be almost negligible. On the contrary, contributions from Rj to the variations of pressure in the recirculation zone can be neglected, while in the reattachment zone, the contribution increases rapidly. As for PR, the contribution is small on the whole, especially in the recirculation zone, which is almost
5.3. Sensitivity analysis The sensitivity of the pressure and heat flux for the blunt body involves three input parameters is analyzed to identify key uncertain parameters that affect the output uncertainty. Fig. 16 depicts the distribution of Sobol indices for Cp along the hemispheric surface. As presented from the figure, the Sobol indices 59
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results. The peak position here is obtained from the baseline state for convenience. As can be seen from the table that, for the baseline state selected in this paper, Rj is the key factor affecting the variations of Cpmax and Stmax while L/D and PR have small influences. Additionally, to further verify the results of sensitivity analysis, correlation coefficients were obtained by linear regression method [59,60] in this paper to show the degree of linear relationship between the inputs and outputs. The absolute value of the correlation coefficient is close to 1, indicating that the input and output have a strong correlation, while the positive or negative correlation coefficient represents the positive or negative correlation between the input and output. Fig. 18 illustrates the scatter plots of Cpmax and Stmax versus input parameters. As conveyed from the figure, the correlation coefficients between Rj and Cpmax as well as Stmax are much larger than other parameters, indicating that Rj has significant influence on Cpmax and Stmax. This is consistent with the results of sensitivity analysis using NIPC method, which verifies the effectiveness of the NIPC method.
Fig. 17. Sobol indices for St along the hemispheric surface. Table 7 Sobol indices for Cpmax and Stmax. Quantity of interest
Cpmax Stmax
Surface location
θ = 56.6° θ = 58.2°
6. Conclusions Uncertainty and sensitivity studies of three main parameters (L/D, Rj and PR) on the drag and heat reduction for the blunt body with spike and opposing jet are carried out. Twenty numerical simulations were performed for different parameter combinations using the CFD method. Based on the calculation results, the relationship between input parameters and aerodynamic force/heat is established using the NIPC method. Uncertainties of Cp and St caused by changes of input parameters are quantified. Sensitivity analysis is carried out to determine key parameters affecting the pressure and heat flux changes. Main discoveries are summarized as follows:
Sobol indices L/D
PR
R
0.158 0.136
0.226 0.226
0.616 0.638
ineffective. The variation law of Sobol indices for heat flux is the same with that for pressure coefficient expect that the Sobol index of PR has a slight increase in the recirculation zone (see Fig. 17). In terms of drag and heat reduction, it is usually more concerned with changes in peak pressure and peak heat flux. Therefore, Sobol indices of each input parameter at the peak position is presented in Table 7. The peak position calculated under different input parameters is different, but the discrepancies are very small according to the CFD
(1) Uncertainties for Cp and St are relied on the location of the hemisphere surface. Large variation intervals are distributed at the reattachment point and its slightly backward position, indicating that Cpmax and Stmax are greatly affected by parameter changes. Further
Fig. 18. Scatter plots of Cpmax and Stmax versus input parameters. 60
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analysis demonstrates that changes of peak pressure and peak heat flux are resulted from significant changes in reattachment shock strength which varies with input parameters. (2) Under the influence of the combinational thermal protection system, flow ahead of the blunt body is divided into two different zones, the recirculation zone and the reattachment zone. Sensitivity analysis indicates that L/D dominates the uncertainty of the recirculation zone, while Rj makes major contributions in the uncertainty of the reattachment zone. (3) For Cpmax and Stmax, Rj has an enormous effect. Linear regression analysis shows that a significant negative correlation between Rj and the peak values of pressure and heat flux is identified. Therefore, the value of Rj must be considered reasonably in practical applications, since small changes may have substantial effects.
[17] [18]
[19]
[20]
[21]
[22]
It is worth noting that the uncertainty and sensitivity results obtained in this study rely on the baseline state (geometry and jet parameters) selected in this paper. Substantially altered geometry and jet parameters may result in different aerodynamic characteristics. Meanwhile, various freestream conditions may change the structure of the flow field. Hence, the key factors affecting drag and heat reduction for hypersonic vehicles may change. Further study for aerodynamic uncertainty and sensitivity is necessary.
[23]
[24] [25] [26]
[27]
Declaration of competing interest
[28]
None declared.
[29] [30]
Acknowledgement
[31]
This work is supported by grants from the National Natural Science Foundation of China (No.11721202). The first author also acknowledges the help provided by Ya-Tian Zhao and Kang Zhong from Beihang University.
[32] [33] [34]
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