Theory of aerodynamic heating from molecular collision analysis

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Theory of aerodynamic heating from molecular collision analysis Yi Tao 1 , Zhongwu Li 1 , Quan Han, Chengdong Sun, Yan Zhang, Yunfei Chen ∗ Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing, 211189, China

a r t i c l e

i n f o

Article history: Received 10 July 2019 Received in revised form 23 October 2019 Accepted 24 October 2019 Available online xxxx Communicated by R. Wu Keywords: Aerodynamic heating Temperature rise Hypersonic flight

a b s t r a c t An aerodynamic heating model is derived from molecular collision analysis, in which the rising temperature of a hypersonic flying object as a function of the flying speed in a classical dense monoatomic gas environment is set up. The model predicts that the rising temperature of the hypersonic flying object is independent of the gas density but depends linearly on the gas atomic mass. A nonequilibrium molecular dynamics simulation is carried out to verify the theoretical model. Also, through analyzing the vibrational density of states in the flying objects, it is found that the excited phonon frequency is near the collision frequency, uncovering that the phonons are mainly excited by the face colliding gas atoms. Our study provides a new insight into understanding the intrinsic mechanism of aerodynamic heating and helps to develop the temperature-controllable hypersonic flying vehicle. © 2019 Published by Elsevier B.V.

1. Introduction Hypersonic flight vehicle attracts a lot of research attentions for its potential applications in space transportation [1]. However, the high temperature rise on the nose and thin wings of a vehicle resulted from severe aerodynamic heating has become an obstacle to the developments of hypersonic flight vehicles [2–4]. Two factors are responsible for such high temperature rise. One is attributed to that the shock wave will move closer to the vehicle and the boundary layer will grow thicker rapidly as the speed of a hypersonic flight vehicle increases. The other is attributed to that the speed difference among the outer inviscid flow, the shock wave, and the boundary layer becomes severe, leading to the increasing viscous interactions. Under the combining action of the significant flow compression and viscous dissipation, extreme high temperature appears in both the inviscid flow behind the shock wave and the boundary layer, which will cause immense destruction to a hypersonic flight vehicle [3,5–7]. However, to date the precise computation of the heat transfer between the flow and the vehicle is still difficult to achieve, which limits the design of the vehicle structure and thermal protection system [7–10]. In order to construct proper thermal protection system, the intrinsic mechanism and accurate prediction of aerodynamic heating are required. To obtain the exact solution to the aerodynamic heating problem, unsteady Navier-Stockes (NS) equations are widely used [3,5,

* 1

Corresponding author. E-mail address: [email protected] (Y. Chen). Y.T. and Z.L. contributed equally to the work.

https://doi.org/10.1016/j.physleta.2019.126098 0375-9601/© 2019 Published by Elsevier B.V.

6]. However, there exist many critical numerical issues deteriorating the simulation precision of hypersonic flows with the NS equations. A particular problem is that the simulation of hypersonic blunt vehicle geometries includes a large region of subsonic flow near the stagnation point, which would magnify numerical error generated at the strong shock wave, accumulated in the stagnation region and finally corrupted the solutions [11]. Another problem is the continuum hypothesis of the NS equations, which makes the prediction of aerodynamic heating fail at nanoscale [12–14], rarefied gas flows [15] and high speed [16]. The most well-known nanoscale effects are the velocity slip and temperature jump at the solid-flow interface, which have huge influences on the flow and energy transport, resulting in significant deviation from the prediction of the NS equations [12–14]. Besides, the NS equations are based on the linear relations known as the Newton’s law of friction and Fourier’s law of heat conduction. Once the flying object reaches hypersonic speed, a strong shock wave is generated that induces larger gradients in aerodynamic densities, leading to the breakdown of the linear relations and the failure of the NS equations [16]. In order to overcome the shortages of the NS equations, the direct simulation Monte Carlo method (DSMC) is widely used as an established particle simulation approach [17]. The comparisons of hypersonic flows calculated with the NS equations and DSMC methods have been reported previously in several studies [15,18–20]. Nevertheless, the detail of the interaction between the vehicle and hypersonic flows is still missing due to the assumption of the rigid surface set in the DSMC method [17]. Until 1960, the first nonequilibrium molecular dynamics simulations (NEMD) of a shockwave-like phenomenon was carried out by Gibson et al. [21],

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which heralded a new era in the simulation of hypersonic flows with the NEMD method. Using the same method, it was reported that the heat flux was about 70% larger than expected from the Fourier’s law of heat conduction for high speed with strong shock waves in a hard-sphere gas [16]. Later, Holian et al. [22] proposed that if one used the temperature in the direction of shock propagation calculated by equipartition theorem to substitute for the average temperature in the NS equations, the results would agree more closely with the NEMD model. Based on the previous hypothesis, an equation for the heat-flux vector, which went beyond the Fourier’s law of heat conductance, was proposed. This new treatment improved the NS equations’ agreement with the NEMD model under strong shock waves [23,24]. However, these studies did not give the explicit relationship between the temperature rise of a hypersonic vehicle and its speed. In this work, a new theoretical model is put forward to describe the relation between the rising temperature of a hypersonic vehicle and its speed under nanoscale situations. Compared to the previous work on aerodynamic heating calculation from the NS equations [3,5,6], our model abandons the continuum hypothesis and is derived from the perspective of basic atomic collision, which gives more accurate calculations to the heat transfer between the flow and the vehicle. Based on this model, the rising temperature of a hypersonic flying silicon cube as a function of its flying speed is presented. It is also possible to determine the influential factors of the rising temperature during the process of aerodynamic heating with the help of the model. MD simulations are performed to verify our theoretical model. Based on the MD simulations, vibrational density of states (VDOS) of silicon atoms are analyzed to explore the underlying mechanism of the aerodynamic heating process. This study is expected to further understand the intrinsic mechanism of aerodynamic heating and facilitate the development of hypersonic vehicles in the future.

To derive the relationship between the rising temperature of a flying vehicle and its flying speed, we consider a silicon cube flying with velocity v in a monoatomic gas environment along the z-direction. The temperature of the monoatomic gas environment is set at T G . The velocity probability density for the gas atoms along the z-direction should obey the Maxwell distribution [25]:



1/2

m

 exp −

2π k B T G

mv 2z

 (1)

2k B T G

where v z is the velocity of monatomic gas along the z-direction, m is the mass of gas atom, T G is the temperature of the monatomic gas and k B is the Boltzmann’s constant. The average speed of monatomic gas along the z-direction is calculated as:



∞ v z  =

vz 0

1/2

m 2π k B T G

 exp −

mv 2z





2k B T G

dv z =

kB T G 2π m

(2)

The integral from zero to infinity is due to the hypothesis that the speed of monatomic gas along the negative z-direction is considered zero. Furthermore, considering a moving cube with the speed v along the z-direction, the probability density of monatomic gas relative to the cube in the z-direction can be rewritten as:



f (v z ) =

m 2π k B T G

1/2



exp −

m ( v z − v )2 2k B T G



(3)

The average speed of monatomic gas in the opposite direction of the moving cube is calculated as:



∞  v z+  =

vz 0 

=

m



2π m

exp −



1/2

exp −

2π k B T G

kB T G

mv 2z

 +

2k B T G

m ( v z − v )2

 dv z

2k B T G

1 2

 

v · er f c −



m 2k B T G

v (4)

where erfc(*) is the complementary error function. On the other hand, the average speed of monatomic gas in the same direction of the moving cube is calculated as:



∞  v z−  =

2. Theoretical model

f (v z ) =

Fig. 1. The average relative speed of monatomic gas as a function of the velocity of the cube at 300 K. Argon is used as the monoatomic gas environment, for example. The solid line in black represents the relative speed of gas colliding head-on with the moving cube, while the solid line in red represents that from behind. The dash line represents the velocity of the flying cube. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

vz 0 

=

m 2π k B T G

kB T G 2π m

 exp −



1/2

exp − mv 2 2k B T G

 −

m ( v z + v )2

1 2

2k B T G

 v · er f c

 dv z m 2k B T G

 v (5)

The speed < v z+ > denotes the average relative speed of monatomic gas colliding head-on with the moving cube, while < v z− > denotes the average relative speed of monatomic gas colliding with the moving cube from behind. As shown in Fig. 1, the speed < v z+ > increases with the moving speed of the cube, while < v z− > decreases until the value dips to zero. When the speed of the cube exceeds the acoustic velocity, < v z+ > approaches the speed of the cube and < v z− > descends to zero. Therefore, when the speed of the moving cube is higher than the acoustic velocity, it is convincing to substitute the speed of the cube v with < v z+ >, and make < v z− > equal to zero. Meanwhile, for ideal gas, the number density can be described as:

n=

N V

=

p kB T G

(6)

where V is the volume of ideal gas, N is the total number of gas atoms in the volume V and p is the pressure. Combined with the average speed and number density of monatomic gas in Eq. (2) and Eq. (6), the frequency that the gas atoms collide with the side walls of the moving cube in parallel with its speed can be expressed as:

g = 4n  v z  S

(7)

where S is the single side area of the moving cube. The factor 4 is due to that there are four side walls in a moving cube parallel to

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its speed. Similarly, the frequency that the gas atoms collide headon with the front wall of the moving cube can be expressed as:

g f = knv S

(8)

where k is the gaseous momentum ratio between that near the front wall and that far away from the cube, which can be derived from the NEMD simulation. As shown in Eq. (8), the momentum of gas per volume can be noted as the product of the number density n times the relative velocity of flow v. When the gas flows from far away to the front wall of the cube, due to the finite area of the cube and the significant flow compression at the nose of the cube, the gas will be squeezed out of the front of the moving cube during the flow process, leading to the loss of the momentum. On the other hand, the frequency that the gas atoms collide on the back wall of the moving cube from behind is considered to be zero, because in this situation the relative colliding speed of gas is approximate to zero when the speed is higher than the acoustic velocity. Thus, when the gas atoms collide with the moving cube, the energy transfer from the cube to the gas atoms per time is expressed as:

 P = gf

3 2

kB T −

mv 2 2

 3 − k B T G e + gk B ( T − T G ) e

(9)

2

where e represents the efficiency of energy transfer, and T is the temperature of the flying cube. Positive value of P denotes that the energy transfers from the cube to the gas atoms, while negative value denotes the transfer direction is from the gas atoms to the cube. In addition, considering a static cube, the energy transferring per time can be considered as the heat flux flowing from the cube to the gas atoms. Thus, the heat flux per area can be expressed as:

q = h (T − T G ) =

3 2

n  v z  k B (T − T G ) e

(10)

where h is the thermal conductance between the static cube and the gas, which can be estimated using MD simulations or empirical formulas [26]. Then e can be estimated as follows:



e=

2h 3n  v z  k B

=

2π m

3nk B

kB T G

(11)

3 E S = N S kB T 2

(12)

where N S represents the total atomic number of the cube. By taking the derivative of Eq. (12), the energy lost from the cube per time is linked with Eq. (9) as follows:

dE S dt

3

dT

2

dt

= −N S k B +

3 2

 = gf

3 2

kB T −

mv 2 2

 − kB T G e

gk B ( T − T G ) e

(13)

By solving Eq. (13) with the initial condition that the temperature of the cube equals to T S at the very beginning, the temperature of the cube as a function of time can be expressed as:



T=



g f mv 2 + 2k B T G + 3gk B T G



3g f k B + 3gk B

+ TS −



With time elapse, the final equilibrium temperature of the cube can be expressed as:



T eq =



g f mv 2 + 2k B T G + 3gk B T G 3g f k B + 3gk B

(15)

In the case of the moving cube, it is obvious that the instantaneous temperature depends on the efficiency of energy transfer e in Eq. (14), while the final equilibrium temperature is independent of e. Besides, based on the assumption that the face and back colliding speeds between the monatomic gas and the cube are v and zero, respectively, Eq. (14), (15) are applicable when the speed of the cube is higher than the acoustic velocity of gas. When the speed of the cube is far higher than the acoustic velocity, g can be ignored compared with g f . Then, the final equilibrium temperature of the cube can be simplified as follows:

T eq ∼

2 3

TG +

M v2 3000R

(16)

where M is the relative atomic mass of monatomic gas and R is the gas constant. Eq. (16) shows the parabolic relation between temperature and speed of a hypersonic flying object, which is similar as the parabolic relation between temperature of an adiabatic wall and speed of the flow derived from the NS equations [27,28]:

T aw = T G + R aw

v2 2c p

(17)

where R aw is the recovery factor and c p is the specific heat capacity of ideal gas at constant pressure. It is noted that the theoretical model of Eq. (14)-(16) is derived from the classical kinetic theory of gases, thus the model does not consider thermal radiation and ionization processes. As shown in Eq. (16), it is demonstrated that the final equilibrium temperature is independent of the gas pressure, and linearly dependent on the gas relative atomic mass. Eq. (14) allows one to calculate the temperature of a hypersonic vehicle at both nanoscale and macroscale. 3. Theoretical calculations

2h

In the case of the cube, the total energy can be derived from the equipartition theorem [25]:

−

3



g f mv 2 + 2k B T G + 3gk B T G 3g f k B + 3gk B



 exp −

gf + g NS

 et

(14)

In this section, the temperature of a hypersonic silicon cube flying in a classical dense monoatomic gas environment was calculated using our theoretical model. The silicon cube is used instead of the thermal protection materials since the NEMD simulation is limited by the complex of the thermal protection materials [29]. Besides, the thermal properties of silicon have been widely studied over the last decade, which is convenient to verify the accuracy of our theoretical model and NEMD results by comparing them with the previous literature [30]. The NEMD simulation was carried out to verify this theoretical model using NAMD 2.12 package [31]. The simulation model is schematically illustrated in Fig. 2, which is composed of a silicon cube and classical dense monoatomic gas with tenfold atmospheric pressure. The atomic interactions were modeled by the Lennard-Jones interactions with a cutoff of 8 Å, and the parameters for the Lennard-Jones Potential are listed in Table 1. The lattice constant of silicon is fixed at the optimized value of 5.43 Å. The size of simulation box is 20 × 20 × 110 nm with periodic boundary conditions applied along the three axis directions, and the silicon cube is 4 × 4 × 4 unit cells. The total system includes 512 silicon atoms and 11497 gas atoms. During the MD simulation, the velocity Verlet algorithm with a time step of 2 fs was used for the integration of the motion equations [35]. The system was first equilibrated in a NPT ensemble at 300 K by applying the Nosè-Hoover thermostat on all atoms for 2 ns. Then only gas atoms were coupled with the thermostat at 300 K, and the silicon cube was pulled along the positive

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Table 1 Parameters for the Lennard-Jones Potential V (r ) = 4ε ((σ /r )12 − (σ /r )6 ) to characterize the interatomic interactions among silicon [32] and gas [33] atoms, while the parameters of LJ potential for argon are used to stand for monoatomic gas. The interaction between silicon and gas atoms is calculated using the mixing rule from parameters of silicon and gas [34].

Si-Si Gas-Gas Si-Gas

ε (kcal/mole)

σ (nm)

0.3100 0.2381 0.2717 (= (ε Si × εGas )1/2 )

3.8057 3.4118 3.6088 (= (σ Si + σGas )/2)

Fig. 4. The final equilibrium temperature of a silicon cube as a function of the flying speed when the relative atomic mass of gas is 39.948.

Fig. 2. Schematic diagram of a silicon cube flying in classical dense monoatomic gas environment.

Fig. 3. Temperature rise of a hypersonic flying silicon cube as a function of time under different atmospheric pressure, when the flying speed of the silicon cube is 2000 m/s and the relative atomic mass of gas is 39.948. The solid and dashed curves represent the results calculated from our theoretical model using Eq. (14), while the dots represent the results from NEMD simulations.

z-direction by a driving spring in a NVT ensemble, named Steered Molecular Dynamics method [36]. The normal spring constant was set as 7 N/m. The silicon cube atoms were harmonically restrained in x and y directions in order to avoid the deviation of the flying direction. The simulation was carried out at least for at least 7 ns in order to guarantee the temperature of the silicon rising to a steady state, and then the final temperature is collected to certify the convergence of the NEMD calculations. Fig. 3 demonstrates the temperature rise of a hypersonic silicon cube with a speed at 2000 m/s as a function of time calculated from the derived theoretical model, and the NEMD simulation is used to verify the model. The efficiency of energy transfer e in Eq. (14) are derived from fitting the result of NEMD simulation, which are 0.52, 0.77, 0.80 for the pressure of 20 atm, 10 atm and 5 atm, respectively. As shown in Fig. 3, the NEMD simulation re-

sults show good agreements with the prediction of the theoretical model. Noted that the final equilibrium temperature predicted by our theoretical model is independent of e, which is also consistent with the simulated equilibrium temperature using the NEMD simulation, indicating that our theoretical model can predict the rising temperature of a hypersonic silicon cube well. Besides, it is reported that the interfacial thermal conductance between Pt and gaseous argon under 10 atm is 0.39 MW/m2 K calculated by equilibrium molecular dynamics [26]. According to Eq. (11), the efficiency of energy transfer e can be calculated as 0.7747, which is in line with the fitting value 0.77 of e for the pressure of 10 atm in our work. The processes of temperature rising lasts for about 3 ns and this time decreases with the increasing atmospheric pressure. However, the final equilibrium temperature is independent of the pressure, which is in line with our theoretical model in Eq. (15) and the NS model in Eq. (17). As shown in Fig. 4, the relationship between the final equilibrium temperature and the flying speed of a silicon cube is parabolic, and our theoretical model agrees well with the NEMD results. However, the results calculated from the NS model in Eq. (17) differs largely from the NEMD results, because the NS model assumes an infinite adiabatic wall with no velocity slip at the solid-flow interface, which is different from the real situation. Moreover, for macroscopic scale, where the NS model is usually applied, there is a significant number of collisions in front of the vehicle that reduce the velocity of the gas atoms before they reach the front wall, therefore leading to a reduction in the gaseous momentum ratio k in Eq. (8) and heat transfer. That is the reason why the temperature calculated from the NS model is lower than the NEMD results and the predicted values by the theoretical model as shown in Fig. 4. When the silicon cube speed is below 1500 m/s, the predicted rising temperature from our theoretical model is a little bit smaller than that from the NEMD simulation results. A small gap between our theoretical model and the NEMD results exists. This phenomenon is attributed to the simplification of the gaseous relative velocity in the Eq. (16) of our theoretical model, in which the speed of the flight speed v instead of < v z+ > is used as the collision speed between the gas atoms and the flying vehicle, leading to the underestimate of the actual collision speed in the low speed region. However, as the speed exceeds 1500 m/s, which is in the hypersonic region with a Mach number of ∼5, the final equilibrium temperature of the flying silicon cube calculated by NEMD simulation is well predicted by our theoretical model. Despite the underestimate of the rising temperature of the flying silicon cube, the deviation of the theoretical model from the NEMD results is still smaller than that of NS equation, verifying the valid-

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Fig. 5. The final equilibrium temperature of a hypersonic silicon cube as a function of the gas relative atomic mass when the speed of silicon cube is 2000 m/s. The parameters for the LJ potential of the gas atoms are the same as listed in Table 1. Only the gas relative atomic mass is varied from 5 to 60.

Fig. 6. VDOS of atoms at the front wall and back wall of the silicon cube colliding with the gas atoms when the speed of the silicon cube is 1000 m/s. Each wall consists of monolayer silicon with 32 atoms.

5

the distribution of vibration modes at different frequency, is proportional to kinetic energy and temperature contributed by corresponding vibration modes. As shown in Fig. 6, because the speed of the moving cube is higher than the acoustic velocity, the frequency that the gas atoms collide with the back wall of the silicon cube declines to zero. Thus, the heat of the back wall is transferred only from the vibration of the silicon atoms at the front wall, meaning that the VDOS of the silicon atoms at the back wall is similar as the bulk state without any external excitation. Compared with the VDOS of the silicon atoms at the back wall, there is a dramatic peak of the VDOS at the front wall at 1.82 THz. This value is consistent with the frequency that the gas atoms collide headon with the front wall of the moving cube calculated by Eq. (8) (about 1.78 THz), which means massive phonons with a frequency of 1.82 THz are motivated by colliding with gas atoms. As the heat flux transfers from the front wall to the back wall through the vibration of silicon atoms, the vibration modes at 1.82 THz vanish and modes at other frequencies are enhanced. Thus, the silicon cube is being heated through the inelastic scattering of phonons at the front wall with frequency of 1.82 THz. Moreover, the peak of the VDOS at the front wall of the silicon cube at ∼16 THz is missing compared with that at the back wall in Fig. 6. A similar phenomenon has been reported that considering the heat flux transfers across an interface between germanium and silicon, the VDOS of silicon atoms at the interface is strongly suppressed with the frequency of ∼16 THz [39]. This is because the mass of the germanium atom is much larger than that of the silicon atom, leading to the lower cut-off frequency ∼10 THz of the vibration modes in the germanium side. When the heat flux transfers across the interface from the germanium side to the silicon side, the vibration modes with high frequency at the silicon side of the interface cannot be excited due to the suppression of the vibration modes with low frequency in the germanium side. Similarly, in our work due to the low colliding frequency of the gas atoms, the vibration modes with high frequency at the front wall of the silicon cube cannot be excited, leading to the disappearance of the peak in the VDOS at the frequency of ∼16 THz. It has been reported that the multiscale structure introduces additional phonon bandgaps with corresponding frequency, leading to the reduction in thermal conductance due to phonon filtering effects [40,41]. Thus, according to the mechanism that the hypersonic vehicle will be heated by phonons with a specific frequency due to colliding with gas atoms, we expect that the multiscale structure can be utilized as a new thermal protection material in the future. 4. Conclusions

ity of the theoretical model for describing hypersonic aerodynamic heating process. The rising temperature of a flying silicon cube as a function of monoatomic gas relative atomic mass calculated from NEMD simulation and our theoretical model are presented in Fig. 5. NEMD simulation results show that the temperature of the silicon cube depends linearly on the relative atomic mass of gas, which is highly consistent with our theoretical model using Eq. (15). Moreover, it is worth mentioning that the silicon temperature predicted by the NS model in Eq. (17) is also linearly dependent on the relative atomic mass as c p = 2500R/M considering monatomic gas. However, the temperature predicted by the NS model descends compared with the result of NEMD. Thus, combining Fig. 3–5 it is demonstrated that, our theoretical model can present a more accurate description of the aerodynamic heating than the NS model. In order to explore the underlying mechanism of the aerodynamic heating process, the fast Fourier transform of the normalized velocity autocorrelation functions are used to calculate the VDOS [37,38]. It should be noted that the VDOS, which measures

In summary, a theoretical model to predict the temperature rise of an adiabatic hypersonic flying vehicle due to aerodynamic heating is rigorously derived from the molecular collision analysis. Compared with the NS model, our theoretical model can present a more accurate description about the process of aerodynamic heating, and is well verified by NEMD simulations. The theoretical model demonstrates a parabolic relationship between the temperature rise and the flying speed of a silicon cube, which is more accurate than the NS model. Moreover, considering the silicon cube travels with a constant speed, the rising temperature is independent of the atmospheric pressure, and depends linearly on the relative atomic mass of monoatomic gas predicted from our theoretical model. Further, the VDOS of atoms at front wall and back wall of the silicon cube reveals the heating mechanism indicating that the phonons with the specific frequency are excited by the face colliding gas atoms, which make major contribution to the aerodynamic heating of the hypersonic flying silicon cube. Our study provides an accurate model to understand the intrinsic mechanism of aerodynamic heating and is also helpful to design

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related parameters to control the temperature rise in a hypersonic flying vehicle. Acknowledgements The authors thank the financial supports from the National Natural Science Foundation of China (Grant Nos. 51435003, 51575104, 51665030), the Nature Science Foundation of Jiangsu Province (BK20140627) and the State Key Laboratory Foundation of LSL-1511. This work is also supported by the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBJJ1748) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX17_0058). References [1] P.L. Moses, V.L. Rausch, L.T. Nguyen, J.R. Hill, NASA hypersonic flight demonstrators—overview, status, and future plans, Acta Astronaut. 55 (2004) 619–630. [2] B. Budiansky, J. Mayers, Influence of aerodynamic heating on the effective torsional stiffness of thin wings, J. Aeronaut. Sci. 23 (1956) 1081–1093. [3] J.D. Anderson, Hypersonic and High Temperature Gas Dynamics, AIAA, 2000. [4] F. Chen, H. Liu, S. Zhang, Time-adaptive loosely coupled analysis on fluid– thermal–structural behaviors of hypersonic wing structures under sustained aeroheating, Aerosp. Sci. Technol. 78 (2018) 620–636. [5] J.J. Bertin, Hypersonic Aerothermodynamics, AIAA, 1994. [6] M.L. Rasmussen, D.R. Boyd, Hypersonic Flow, Wiley, 1994. [7] J.J. McNamara, P.P. Friedmann, Aeroelastic and aerothermoelastic analysis in hypersonic flow: past, present, and future, AIAA J. 49 (2011) 1089–1122. [8] B. Shen, W. Liu, L. Yin, Drag and heat reduction efficiency research on opposing jet in supersonic flows, Aerosp. Sci. Technol. 77 (2018) 696–703. [9] J.B.E. Meurisse, J. Lachaud, F. Panerai, C. Tang, N.N. Mansour, Multidimensional material response simulations of a full-scale tiled ablative heatshield, Aerosp. Sci. Technol. 76 (2018) 497–511. [10] M.-C. Trinh, S.-E. Kim, Nonlinear stability of moderately thick functionally graded sandwich shells with double curvature in thermal environment, Aerosp. Sci. Technol. 84 (2019) 672–685. [11] G. Candler, D. Mavriplis, L. Trevino, Current status and future prospects for the numerical simulation of hypersonic flows, in: 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2009, p. 153. [12] S. Ge, Y. Gu, M. Chen, A molecular dynamics simulation on the convective heat transfer in nanochannels, Mol. Phys. 113 (2015) 703–710. [13] Y.-W. Gu, S. Ge, M. Chen, A molecular dynamics simulation of nanoscale convective heat transfer with the effect of axial heat conduction, Mol. Phys. 114 (2016) 1922–1930. [14] C. Bing-Yang, C. Min, G. Zeng-Yuan, Rarefied gas flow in rough microchannels by molecular dynamics simulation, Chin. Phys. Lett. 21 (2004) 1777. [15] H. Xiao, Q.J. He, Aero-heating in hypersonic continuum and rarefied gas flows, Aerosp. Sci. Technol. 82–83 (2018) 566–574. [16] E. Salomons, M. Mareschal, Usefulness of the Burnett description of strong shock waves, Phys. Rev. Lett. 69 (1992) 269–272. [17] G.J. LeBeau, A parallel implementation of the direct simulation Monte Carlo method, Comput. Methods Appl. Mech. Eng. 174 (1999) 319–337. [18] I.D. Boyd, G. Chen, G.V. Candler, Predicting failure of the continuum fluid equations in transitional hypersonic flows, Phys. Fluids 7 (1995) 210–219.

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