Plasma current tomography for HL-2A based on Bayesian inference

Zijie LIU; Tianbo WANG; Muquan WU; Zhengping LUO; Shuo WANG; Tengfei SUN; Bingjia XIAO; Jiangang LI

doi:10.1088/2058-6272/ad1980

Plasma Science and Technology > 2024 > 26(5): 055601. > DOI: 10.1088/2058-6272/ad1980

Zijie LIU, Tianbo WANG, Muquan WU, Zhengping LUO, Shuo WANG, Tengfei SUN, Bingjia XIAO, Jiangang LI. Plasma current tomography for HL-2A based on Bayesian inference[J]. Plasma Science and Technology, 2024, 26(5): 055601. DOI: 10.1088/2058-6272/ad1980

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Plasma current tomography for HL-2A based on Bayesian inference

1.
College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, People’s Republic of China
2.
Southwestern Institute for Physics, Chengdu 610200, People’s Republic of China
3.
Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, People’s Republic of China
4.
University of Science and Technology of China, Hefei 230026, People’s Republic of China

More Information

Author Bio:
Tianbo WANG: wangtianbo@swip.ac.cn

Muquan WU: wumuquan@szu.edu.cn
Corresponding author:
Tianbo WANG, wangtianbo@swip.ac.cn

Muquan WU, wumuquan@szu.edu.cn
Received Date: September 19, 2023
Revised Date: December 27, 2023
Accepted Date: December 27, 2023
Available Online: May 20, 2024
Published Date: April 22, 2024

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Abstract

Abstract

An accurate plasma current profile has irreplaceable value for the steady-state operation of the plasma. In this study, plasma current tomography based on Bayesian inference is applied to an HL-2A device and used to reconstruct the plasma current profile. Two different Bayesian probability priors are tried, namely the Conditional AutoRegressive (CAR) prior and the Advanced Squared Exponential (ASE) kernel prior. Compared to the CAR prior, the ASE kernel prior adopts non-stationary hyperparameters and introduces the current profile of the reference discharge into the hyperparameters, which can make the shape of the current profile more flexible in space. The results indicate that the ASE prior couples more information, reduces the probability of unreasonable solutions, and achieves higher reconstruction accuracy.
- plasma current tomography,
- Bayesian inference,
- machine learning,
- Gaussian distribution

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1. Introduction

As re-entry vehicle speed reaches a level faster than Mach 10 in near space (20–100 km altitude), a hypersonic plasma sheath is formed, covering the surface of the vehicle due to thermal ionization [1, 2]. The sheath is characterized by features of high density (> 10¹⁹ m⁻³), strong collision frequency (~ GHz), and typical inhomogeneity [3, 4]. The EM waves can be transmitted, absorbed, reflected, refracted, and scattered by the sheath, resulting in the loss of communication and/or control between re-entry vehicle and ground station, which is called the RF blackout problem [5].

Improving the effectiveness of EM wave transmission in the plasma sheaths is essential for mitigating the RF blackout problem, and has attracted widespread attention since spacecraft re-entry tasks began in the 1960s. Parameter distributions in the sheaths were generally calculated in numerical simulations by solving Navier-Stokes (N-S) equations, while the real profiles were difficult to measure in flying experiments. Candler and MacCormack [6] numerically investigated flow distributions in a two-dimensional (2D) axisymmetric model for the RAM C-II re-entry vehicle. The results were in good agreements with experiments. On the other hand, Kundrapu et al, from Tech-X Corporation [7], developed a commercial fluid dynamics (CFD) software, USim, for hypersonic flow calculation, and studied formation process and flow distribution characteristics of the plasma sheaths for RAM C-II re-entry. Yuan et al [8–10] and Ouyang et al [11–13] studied the effect of Mach numbers, altitudes, nose cap radius, body flare angle, attack angle, and ionization rate on the distributions in plasma sheaths. Takahashi et al [14–17] applied a CFD-FDTD (finite-difference time-domain) combined model based on the fast aerodynamic routine (FaSTAR) code developed by the Japan Aerospace Exploration Agency (JAXA) and analyzed the attenuation of EM waves in plasma sheaths for the atmospheric re-entry demonstrator (ARD) and RAM C-II missions. Miyashita et al [18, 19] studied the injection of N₂ for reducing electron density and mitigating the RF blackout through numerical simulation and experiment. Ouyang et al [20, 21] analyzed the influence of chemical reaction rate and hypersonic vehicle shapes on the plasma sheath, and their corresponding terahertz transmission characteristics. They also experimentally studied the interaction between terahertz waves and plasma based on the atmospheric-pressure plasma jet (APPJ) source [22, 23]. In addition to the FDTD method [24], the scattering matrix method (SMM) [25–27] and finite element method (FEM) [28–30] are also used in analyzing the propagation characteristics of EM waves in inhomogeneous plasma sheaths.

Considering that the EM transmission is dominated by the profiles of plasma sheaths in a direction perpendicular to the re-entry vehicle surface, a simplified one-dimensional (1D) geometry, where the plasma was distributed only in the direction perpendicular to the re-entry vehicle surface while other two dimensions were homogeneous, was usually assumed in previous studies [31–33]. In such a 1D model, the variation of the 1D (perpendicular) plasma distribution in principle partially corresponds to different attack angles. The re-entry vehicles have various attack angles (e.g., the attack angle of the ARD vehicle is ~ 20° [34], and that for RAM C-II is less than 5° during the re-entry [35]). The preceding studies provide a wide research basis for the transmission characteristics in hypersonic plasma sheaths and the mitigation of RF blackout. According to the work of Ouyang et al [13], the attack angle may greatly affect the hypersonic plasma sheaths around the re-entry vehicle, thereby affecting the transmission characteristics of EM waves in the sheaths. Nevertheless, the specific relations between the EM waves transmission in the plasma sheath and the attack angle are still not directly associated. The EM wave may be reflected or absorbed in the plasma sheaths. The variation of attack angle may lead to a change in the EM wave attenuation mechanism. It is necessary to establish an integrated model, involving the distribution characteristics of the plasma sheath and the corresponding EM transmission characteristics, to explore the influence of plasma sheath, under various attack angles, on the EM wave transmission of different frequencies. The laws and mechanisms of EM waves in plasma sheaths for various attack angles should be revealed. Analyzing the effect of attack angle on transmission characteristics in the hypersonic plasma sheath of re-entry vehicles can also provide an approach for the prediction and mitigation of blackout.

Therefore, in this paper, we propose an integrated model by combining a global 3D plasma fluid model, with varied attack angles and realistic flying conditions, and a local (near the antenna) 1D EM transmission model. Then, the effects of attack angle on the plasma sheath distributions are given by the global 3D model, while the EM wave transmission characteristics and corresponding mechanisms are studied in the local 1D model with the plasma distributions introduced from the global model result at the antenna position. The effect of various attack angles on the distributions of key parameters in plasma sheaths in typical flying conditions is first analyzed, and then the laws and mechanisms of EM wave propagations in the local sheath distribution under various attack angles are revealed. Furthermore, the optimal position of antennas mounted on the re-entry vehicle can be estimated. The novelty of the study is that an integrated model, including 3D plasma fluid model and 1D EM transmission model, is established. By analyzing the reflection, transmission, and absorption coefficients of the plasma sheath at different attack angles, and a direct correlation between the EM wave transmission characteristics and the vehicle attack angle is established. Meanwhile, the qualitative recommendations regarding the choice of communication frequency band are given for various attack angles. Furthermore, the optimal position of antennas mounted on the re-entry vehicle can be estimated. These results can provide theoretical support for mitigating the RF blackout problem.

The layout of the paper is as follows. Modeling and basic settings are introduced in section 2. Section 3 presents the simulation results, and section 4 corresponds to analysis and discussion. The paper is then concluded in section 5 with a brief summary.

2. Modeling and basic settings

2.1 Distribution characteristics of plasma sheath

The distribution characteristics of the plasma sheath under various attack angles are numerically studied by USim, based on the finite volume method (FVM) in a global 3D fluid model. The N-S equation is solved in our plasma sheath model, as shown in equation (1), where q is the conserved quantity to be solved, including the density, velocity, and energy. Also, F, G, and H are the convection flux components along x-, y-, and z-directions, respectively. F_v, G_v, and H_v are corresponding components of the gradient-induced transport flux. The flow field calculation model is based on the following assumptions.

(1) The air environment is assumed to be a mixture of perfect gases with a volume ratio 79:21 of N₂ to O₂.

(2) The thermodynamic states include the translational energy model and vibrational energy model. The thermochemistry is in a non-equilibrium state. The flow state of the mixture air is assumed to be continuous and laminar.

(3) A two-temperature (translation-rotation plus vibration) assumption is used in the calculation. The relaxation processes of the gas rotational and translational energy modes will reach equilibrium rapidly. The vibration temperature of each component is the same. The electron temperature is equal to the vibration temperature.

(4) The effects of electromagnetic fields, polarization, current, and heat radiation on the gas environment are ignored.

(5) The relative velocity between the components in the gas mixture is 0. In other words, the gas mixture is a multi-component single-velocity flow in our calculation model.

Based on these assumptions, the expression of q is shown in equation (2) and the equations of states are shown in equations (3)–(9). ρ is the gas density (kg·m⁻³). u, v, and w are the velocity (m·s⁻¹) in x-, y-, and z-directions, respectively. E and E_v are the total energy and total vibrational energy (J·m⁻³) per unit volume. n_s is the number density (m⁻³) of each component s in gas mixture. P is the gas pressure (Pa). τ_ij (τ_xx, τ_yy, τ_zz, τ_xy, τ_xz and τ_yz) represent the air shear stress tensor. q_i (q_x, q_y, and q_z) and q_vi (q_vx, q_vy, and q_vz) represent the total heat transfer flux and vibration heat transfer flux (J·m⁻³·s) in the x-, y- and z-directions, respectively. d_s is the diffusion coefficient (m²·s⁻¹) of component s. h_s and ns are the enthalpy per unit mass (J·kg⁻¹) and the number of components in gas mixture, respectively. e_vs is the vibrational energy per unit mass (J·kg⁻¹) of component s. N_A = 6.02 × 10²³ is the Avogadro’s constant. M_s is the molar mass of component s (kg·mol⁻¹). c_s = ρ_s/ρ is the mass fraction of component s. ρ_s is the gas density (kg·m⁻³) of component s. ω_v is the source term of vibrational energy. ω_s is the number density generation rate of component s. τ_ij is shown in equations (10)–(15). μ is the viscosity coefficient of the gas mixture (N·s·m²). λ = −2/3 is the second viscosity. The expressions of q_i and q_vi are shown in equations (16)–(19). q_tri and q_ei represent the translational-rotational and electron heat transfer flux (J·m⁻³·s) in the x-, y-, and z-directions, respectively. T and T_v are the translational-rotational and vibrational temperature (K), respectively. K_tr, K_v, and K_e are the translational-rotational, vibrational, and electron heat transfer coefficients (J·m⁻¹·s⁻¹·K⁻¹), respectively. The viscosity coefficient, μ, is calculated through the Wilke’s formula [36], as shown in equations (20)–(23). A_s, B_s, and C_s are the fitting coefficients, which are related to the components. There are seven components (N₂, N, O₂, O, NO, NO⁺, and e) in our calculation model (ns = 7) and the fitting coefficients can be referred to reference [37]. M represents the molecules in the components (N₂, O₂, NO, and NO⁺).

$\frac{{\partial q}}{{\partial t}} + \frac{{\partial F}}{{\partial x}} + \frac{{\partial G}}{{\partial y}} + \frac{{\partial H}}{{\partial z}} = \frac{{\partial {F_{\text{v}}}}}{{\partial x}} + \frac{{\partial {G_{\text{v}}}}}{{\partial y}} + \frac{{\partial {H_{\text{v}}}}}{{\partial z}} + S ,$

(1)

$Q = \left[ {\begin{array}{*{20}{c}} \rho \\ {\rho u} \\ {\rho v} \\ {\rho w} \\ E \\ {{E_{\text{v}}}} \\ {{n_s}} \end{array}} \right] ,$

(2)

$F = \left[ {\begin{array}{*{20}{c}} {\rho u} \\ {P + \rho {u^2}} \\ {\rho uv} \\ {\rho uw} \\ {(P + E)u} \\ {{E_{\text{v}}}u} \\ {{n_s}u} \end{array}} \right] ,$

(3)

$G = \left[ {\begin{array}{*{20}{c}} {\rho v} \\ {\rho vu} \\ {P + \rho {v^2}} \\ {\rho vw} \\ {(P + E)v} \\ {{E_{\text{v}}}v} \\ {{n_s}v} \end{array}} \right] ,$

(4)

$H = \left[ {\begin{array}{*{20}{c}} {\rho w} \\ {\rho uw} \\ {\rho vw} \\ {P + \rho {w^2}} \\ {(P + E)w} \\ {{E_{\text{v}}}w} \\ {{n_s}w} \end{array}} \right] ,$

(5)

${F_{\text{v}}} = \left[ \begin{split} &\qquad\qquad\qquad\quad 0 \\&\qquad\qquad\qquad\quad {{\tau _{xx}}} \\&\qquad\qquad\qquad\quad {{\tau _{xy}}} \\&\qquad\qquad\qquad\quad {{\tau _{xz}}} \\& {u{\tau _{xx}} + v{\tau _{xy}} + w{\tau _{xz}} - {q_x} - \displaystyle\sum\limits_{s = 1}^{ns} {\rho {d_s}{h_s}\dfrac{{\partial {c_s}}}{{\partial x}}} } \\ &\qquad\quad { - {q_{{\text{v}}x}} - \displaystyle\sum\limits_{s = M} {\rho {d_s}{e_{{\text{v}}s}}\dfrac{{\partial {c_s}}}{{\partial x}}} } \\ &\qquad\qquad\quad { - \dfrac{{\rho {N_{\text{A}}}}}{{{M_s}}}{d_s}\dfrac{{\partial {c_s}}}{{\partial x}}} \end{split} \right] ,$

(6)

${G_{\text{v}}} = \left[ \begin{split} & \qquad\qquad\qquad\quad0 \\ &\qquad\qquad\qquad\quad {{\tau _{xy}}} \\ &\qquad\qquad\qquad\quad {{\tau _{yy}}} \\ &\qquad\qquad\qquad\quad {{\tau _{yz}}} \\ & {u{\tau _{xy}} + v{\tau _{yy}} + w{\tau _{yz}} - {q_y} - \displaystyle\sum\limits_{s = 1}^{ns} {\rho {d_s}{h_s}\dfrac{{\partial {c_s}}}{{\partial y}}} } \\ &\qquad\quad { - {q_{{\text{v}}y}} - \displaystyle\sum\limits_{s = M} {{\rho _s}{d_s}{e_{{\text{v}}s}}\dfrac{{\partial {c_s}}}{{\partial y}}} } \\ &\qquad\qquad\quad { - \dfrac{{\rho {N_{\text{A}}}}}{{{M_s}}}{d_s}\dfrac{{\partial {c_s}}}{{\partial y}}} \end{split} \right] ,$

(7)

${H_{\text{v}}} = \left[\begin{split} & \qquad\qquad\qquad\quad0 \\ & \qquad\qquad\qquad\quad {{\tau _{xz}}} \\ & \qquad\qquad\qquad\quad {{\tau _{yz}}} \\ & \qquad\qquad\qquad\quad {{\tau _{zz}}} \\ & {u{\tau _{xz}} + v{\tau _{yz}} + w{\tau _{zz}} - {q_z} - \displaystyle\sum\limits_{s = 1}^{ns} {\rho {d_s}{h_s}\dfrac{{\partial {c_s}}}{{\partial z}}} } \\ & \qquad\quad { - {q_{{\text{v}}z}} - \displaystyle\sum\limits_{s =M} {\rho {d_s}{e_{{\text{v}}s}}\dfrac{{\partial {c_s}}}{{\partial z}}} } \\ & \qquad\qquad\quad { - \dfrac{{\rho {N_{\text{A}}}}}{{{M_s}}}{d_s}\dfrac{{\partial {c_s}}}{{\partial z}}} \end{split} \right] ,$

(8)

$S = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ {{\omega _{\text{v}}}} \\ {{\omega _s}} \end{array}} \right] ,$

(9)

${\tau _{xx}} = 2\mu \frac{{\partial u}}{{\partial x}} + \lambda \mu \left(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}}\right) ,$

(10)

${\tau _{yy}} = 2\mu \frac{{\partial v}}{{\partial y}} + \lambda \mu \left(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}}\right) ,$

(11)

${\tau _{zz}} = 2\mu \frac{{\partial w}}{{\partial z}} + \lambda \mu \Bigg(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}}\Bigg) ,$

(12)

${\tau _{xy}} = {\tau _{yx}} = \mu \Bigg(\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}}\Bigg) ,$

(13)

${\tau _{xz}} = {\tau _{zx}} = \mu \Bigg(\frac{{\partial u}}{{\partial z}} + \frac{{\partial w}}{{\partial x}}\Bigg) ,$

(14)

${\tau _{yz}} = {\tau _{zy}} = \mu \Bigg(\frac{{\partial v}}{{\partial z}} + \frac{{\partial w}}{{\partial y}}\Bigg) ,$

(15)

${q_i} = {q_{{\text{tr}}i}} + {q_{{\text{v}}i}} + {q_{{\text{e}}i}} ,$

(16)

${q_{{\text{tr}}i}} = - {K_{{\text{tr}}}}\frac{{\partial T}}{{\partial i}} ,$

(17)

${q_{{\text{v}}i}} = - {K_{\text{v}}}\frac{{\partial {T_{\text{v}}}}}{{\partial i}} ,$

(18)

${q_{{\text{e}}i}} = - {K_{\text{e}}}\frac{{\partial {T_{\text{v}}}}}{{\partial i}} ,$

(19)

${\mu _s} = 0.1\exp \left[ {{A_s}{{\left( {\ln T} \right)}^2} + {B_s}\left( {\ln T} \right) + {C_s}} \right] ,$

(20)

${\Phi _s} = \sum\limits_{j = 1}^{ns} {\left\{ {{{{X_j}{{\left[ {1 + {{\left( {\frac{{{\mu _s}}}{{{\mu _j}}}} \right)}^{0.5}}{{\left( {\frac{{{M_j}}}{{{M_s}}}} \right)}^{0.25}}} \right]}^2}} \mathord{\left/ {\vphantom {{{X_j}{{\left[ {1 + {{\left( {\frac{{{\mu _s}}}{{{\mu _j}}}} \right)}^{0.5}}{{\left( {\frac{{{M_j}}}{{{M_s}}}} \right)}^{0.25}}} \right]}^2}} {{{\left[ {8\left( {1 + \frac{{{M_s}}}{{{M_j}}}} \right)} \right]}^{0.5}}}}} \right. } {{{\left[ {8\left( {1 + \frac{{{M_s}}}{{{M_j}}}} \right)} \right]}^{0.5}}}}} \right\}} ,$

(21)

${X_j} = \frac{{{n_j}}}{{\displaystyle\sum\limits_{i = 1}^{ns} {{n_i}} }} ,$

(22)

$\mu = \sum\limits_{s = 1}^{ns} {\frac{{{X_s}{\mu _s}}}{{{\Phi _s}}}} .$

(23)

The pressure, P, of the gas mixture around the vehicle is calculated through Dalton’s partial pressure law, as shown in equations (24) and (25), where P_h and P_e are the pressure of heavy particles and electrons, respectively. The energy of the gas mixture, including the internal and free chemical energy, is calculated through equations (26)–(32). e_ts, e_rs, and e_es are the translational, rotational, and electronic excitation energy per unit mass of component s. ∆h_fs is a constant (J·kg⁻¹) representing the standard enthalpy of formation per unit mass of component s at 0 K. e_s is the internal energy per unit mass of component s. R is the universal gas constant (J·k⁻¹·mol⁻¹). R_s is the gas constant (J·K⁻¹·kg⁻¹) of component s. θ_vs is the vibration characteristic temperature (K) of component s. The total heat transfer coefficients (K_tr, K_v, and K_e) and the heat transfer coefficients of each component (K_tr,s, K_v,s, and K_e,s) in the thermochemical non-equilibrium flow field are calculated according to Eucken’s semi-empirical formula [38], as shown in equations (33)–(41). C_v,vib,s and C_v,e are the specific heat at constant volume for vibrational energy and electron energy per unit mass (J·K⁻¹·kg⁻¹) of component s. C_v,ts and C_v,rs are the specific heat at constant volume for translational and rotational energy per unit mass (J·K⁻¹·kg⁻¹) of component s. e₀ is the natural constant. The diffusion coefficient, d_s, of each component is calculated through equation (42), where S_c is the Schmidt number, with S_c = 0.5 for molecules or atoms and S_c = 0.25 for ions. Eighteen chemical reactions among seven components (N₂, N, O₂, O, NO, NO⁺, and e), which is proposed by Park et al [39], are included in the model. The 18 chemical reactions, including dissociation, displacement, and association ionization reactions, are shown in table 1. The source term of reaction R_i is shown in equation (43). k_fi and k_bi represent the rate constants for forward and backward reactions of R_i, respectively. α_ij and β_ij are the reaction coefficients of reactants and products. The expressions of k_fi, k_bi, and K_eq are shown in equations (44)–(46). The kinetic coefficient A_fi, temperature index B_fi, and activation energy E_fi can be referred to references [40, 41], T_f = T, and K_eq is the equilibrium rate constant. Z = 10000/T and A₁–A₅ are the fitting coefficients. They can be referred to reference [42].

Table 1. The 18 chemical reactions proposed by Park et al [39].

Reaction	Chemical reaction	Reaction	Chemical reaction
R₁	N₂+N₂ $\rightleftharpoons$ N+N+N₂	R₁₀	O₂+O $\rightleftharpoons$ O+O+O
R₂	N₂+O₂ $\rightleftharpoons$ N+N+O₂	R₁₁	NO+N₂ $\rightleftharpoons$ N+O+N₂
R₃	N₂+NO $\rightleftharpoons$ N+N+NO	R₁₂	NO+O₂ $\rightleftharpoons$ N+O+O₂
R₄	N₂+N $\rightleftharpoons$ N+N+N	R₁₃	NO+NO $\rightleftharpoons$ N+O+NO
R₅	N₂+O $\rightleftharpoons$ N+N+O	R₁₄	NO+N $\rightleftharpoons$ N+O+N
R₆	O₂+N₂ $\rightleftharpoons$ O+O+N₂	R₁₅	NO+O $\rightleftharpoons$ N+O+O
R₇	O₂+O₂ $\rightleftharpoons$ O+O+O₂	R₁₆	N₂+O $\rightleftharpoons$ NO+N
R₈	O₂+NO $\rightleftharpoons$ O+O+NO	R₁₇	NO+O $\rightleftharpoons$ O₂+N
R₉	O₂+N $\rightleftharpoons$ O+O+N	R₁₈	N+O $\rightleftharpoons$ NO⁺+e⁻

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$P=\sum\limits_{s=1}^{ns}P_s=P_{\text{h}}+P_{\text{e}}=\sum\limits_{s\ne\mathrm{e}}^{ }n_sRT+P_{\text{e}},$

(24)

${P_{\text{e}}} = {n_{\text{e}}}R{T_{\text{v}}} ,$

(25)

$E = \sum\limits_{s = 1}^{ns} {{\rho _s}{e_s} + \frac{1}{2}\rho } {\vec V^2} ,$

(26)

${E_{{\text{ve}}}} = {E_{\text{v}}} + {E_{\text{e}}} = \sum\limits_{s = M} {{\rho _s}{e_{{\text{v}}s}} + {\rho _{\text{e}}}{e_{\text{e}}}} ,$

(27)

${e_s} = {e_{{\text{t}}s}} + {e_{{\text{r}}s}} + {e_{{\text{v}}s}} + {e_{{\text{e}}s}} + \Delta {h_{{\text{f}}s}} = {e_{{\text{tr}}s}} + {e_{{\text{v}}s}} + {e_{{\text{e}}s}} + \Delta {h_{{\text{f}}s}} ,$

(28)

$e_{\text{t}s}=\left\{\begin{split} & \frac{3}{2}R_sT\qquad\left(s\ne\mathrm{electrons}\right) \\ &\frac{3}{2}R_sT_{\text{v}}\qquad\left(s=\mathrm{electrons}\right) \end{split} \right.,$

(29)

$e_{\text{r}s}=\left\{\begin{gathered}R_sT\qquad\left(s=\mathrm{molecules}\right) \\ 0\qquad\left(s=\mathrm{atoms,\ electrons}\right) \\ \end{gathered}\right.,$

(30)

$e_{\text{v}s}=\left\{\begin{gathered}R_s\frac{\theta_{\text{v}s}}{\exp\left(\theta_{\text{v}s}\mathord{\left/\vphantom{\theta_{\text{v}s}T_{\text{v}}}\right.}T_{\text{v}}\right)-1}\begin{array}{*{20}{c}} & \end{array}\left(s=\mathrm{molecules}\right) \\ 0\begin{array}{*{20}{c}} & \end{array}\left(s=\mathrm{atoms,\ electrons}\right) \\ \end{gathered}\right.,$

(31)

${R_s} = \frac{R}{{{M_s}}} ,$

(32)

$K_{\text{tr}}=\sum\limits_{s\ne\mathrm{e}}^{ns}\frac{X_sK_{\text{tr,}s}}{\Phi_s},$

(33)

${K_{\text{v}}} = \sum\limits_{s = M} {\frac{{{X_s}{K_{{\text{v,}}s}}}}{{{\Phi _s}}}} ,$

(34)

${K_{\text{e}}} = \frac{{{X_{\text{e}}}}}{{{\Phi _{\text{e}}}}}{\mu _{\text{e}}}{C_{{\text{v,e}}}} ,$

(35)

${K_{{\text{tr,}}s}} = {\mu _s}\left( {\frac{5}{2}{C_{{\text{v,t}}s}} + {C_{{\text{v,r}}s}}} \right) ,$

(36)

${K_{{\text{v,}}s}} = {\mu _s}{C_{{\text{v,vib,}}s}} ,$

(37)

${C_{{\text{v,t}}s}} = \frac{{\partial {e_{{\text{t}}s}}}}{{\partial T}} ,$

(38)

${C_{{\text{v,r}}s}} = \frac{{\partial {e_{{\text{r}}s}}}}{{\partial T}} ,$

(39)

${C_{{\text{v,vib,}}s}} = \frac{{\partial {e_{{\text{v}}s}}}}{{\partial {T_{\text{v}}}}} = \frac{R}{{{M_s}}}{\left( {\frac{{{\theta _{{\text{v}}s}}}}{{{T_{\text{v}}}}}} \right)^2}\frac{{{{\mathrm{e}}_0^{{{{\theta _{{\text{v}}s}}} \mathord{\left/ {\vphantom {{{\theta _{{\text{v}}s}}} {{T_{\text{v}}}}}} \right. } {{T_{\text{v}}}}}}}}}{{{{\left( {{{\mathrm{e}}_0^{{{{\theta _{{\text{v}}s}}} \mathord{\left/ {\vphantom {{{\theta _{{\text{v}}s}}} {{T_{\text{v}}}}}} \right. } {{T_{\text{v}}}}}}} - 1} \right)}^2}}} ,$

(40)

${C_{{\text{v,e}}}} = \frac{{\partial {e_{\text{e}}}}}{{\partial {T_{\text{v}}}}} = \frac{3}{2}{R_{\text{e}}} ,$

(41)

${d_s} = \frac{{\left( {1 - {c_s}} \right)\mu }}{{\left( {1 - {X_s}} \right)\rho {S_{\text{c}}}}} ,$

(42)

${\omega _i} = {k_{{\text{f}}i}}\prod\limits_{j = 1}^{ns} {{{\left( {\frac{{{\rho _j}}}{{{M_j}}}} \right)}^{{\alpha _{ij}}}} - } {k_{{\text{b}}i}}\prod\limits_{j = 1}^{ns} {{{\left( {\frac{{{\rho _j}}}{{{M_j}}}} \right)}^{{\beta _{ij}}}}} ,$

(43)

${k_{{\text{f}}i}} = {A_{{\text{f}}i}}T_{\text{f}}^{{B_{{\text{f}}i}}}\exp \left( {\frac{{ - {E_{{\text{f}}i}}}}{{R{T_{\text{f}}}}}} \right) ,$

(44)

${k_{{\text{b}}i}} = \frac{{{k_{{\text{f}}i}}}}{{{K_{{\text{eq}}}}}} ,$

(45)

$\ln \left( {{K_{{\text{eq}}}}} \right) = {A_1}Z + {A_2}{Z^2} + {A_3}{Z^3} + {A_4}{Z^4} + {A_5}{Z^5} .$

(46)

The internal energy exchanges in the plasma sheath model include translational-vibrational energy exchange and translational-electron energy exchange. The conversion power between translational and vibrational energy is according to Landau-Teller model, as shown in equations () and (). $_{ }^{ }e_{\mathrm{v}s}^{\ast}\left(T\right)$ and $_{ }e_{\mathrm{v}s}\left(T_{\mathrm{v}}\right)$ represent the equilibrium and non-equilibrium vibrational energy of component s, respectively. Q_tv is the vibrational and translational energy conversion power per unit time (J·m⁻³·s⁻¹) of component s. τ_vs is the relaxation time of translational and vibrational energy (s). τ_si is the vibrational relaxation time of components s and i (s). The conversion power between translational and electron energy is shown in equation (49). Ω_es is the collision frequency between electron and particles (neutral particles or ions). For neutral particles, the expression of collision frequency is shown in equations (50) and (51). k_B is the Boltzmann’s constant. a_s, b_s, and c_s are the coefficients, which can be referred to reference [43]. For ions, the collision frequency is calculated through equation (52).

${Q_{{\text{tv}}}} = \sum\limits_{s = M} {{\rho _s}\frac{{e_{{\text{v}}s}^ * \left( T \right) - {e_{{\text{v}}s}}\left( {{T_{\text{v}}}} \right)}}{{{\tau _{{\text{v}}s}}}}} ,$

(47)

${\tau _{{\text{v}}s}} = \frac{{\displaystyle\sum\limits_{j = M} {\frac{{{\rho _j}}}{{{M_j}}}} }}{{\displaystyle\sum\limits_{i = M} {\frac{{{\rho _i}}}{{{M_i} \cdot {\tau _{si}}}}} }} ,$

(48)

$Q_{\text{TE}}=3\rho_{\text{e}}R\left(T-T_{\text{v}}\right)\sum\limits_{s\ne\mathrm{e}}^{ }\frac{\Omega_{\text{e}s}}{M_s},$

(49)

${\Omega _{{\text{e}}s}} = {n_s}{\sigma _{{\text{e}}s}}{\left( {\frac{{2{k_{\text{B}}}{T_{\text{v}}}}}{{{m_{\text{e}}}}}} \right)^{1/2}},$

(50)

${\sigma _{{\text{e}}s}} = {a_s} + {b_s}{T_{\text{v}}} + {c_s}T_{\text{v}}^2,$

(51)

$\Omega_{\mathrm{e}s}=n_se^4\frac{1}{32\varepsilon_0^2\sqrt{\text{π}}\sqrt{m\mathrm{_e}}\left(k_{\mathrm{B}}T_{\mathrm{v}}\right)^{3/2}}\ln\left(\frac{\left(k_{\mathrm{B}}T\mathrm{_v}\right)^{3/2}}{n_{\mathrm{e}}^{1/2}e^2}\right).$

(52)

The geometry of the re-entry vehicle is shown in figure 1(a). The RAM C-II like vehicle is axisymmetric about the x-axis, where its length, radius of the nose cap, and body flare angle are 1.30 m, 0.15 m, and 9°, respectively. The simulation domain is a cylinder of radius 1.10 m and length 2.35 m around the vehicle symmetry axis x. The inflow is in the (x, y) plane, opposite to the moving direction of the vehicle, with v₀ and φ denoting velocity and attack angle of the re-entry vehicle, respectively. The velocity components (v_0x, v_0y, v_0z) are then easily written as (v₀cosφ, v₀sinφ, 0). Then in figure 1(b), boundary conditions on the left and bottom sides (marked in red) are for the inflow, while on right and top (marked in blue) are for the outflow. The vehicle surface is set as a non-slip boundary and the temperature of the vehicle surface is set as 1200 K. The ions and electrons will recombine immediately at the boundary. The flow velocity and the mass fraction normal gradient of each component at the surface can be referred to references [7, 8]. The initial inflow boundary conditions, including atmosphere density and temperature at different altitudes, refer to the standard atmosphere parameters used by NASA [44], and the boundary inflow velocity is according to the realistic flight velocity of RAM C-II [45]. It is easy to expect that the parameter distributions are asymmetrical in the (x, y) plane if the attack angle φ $\ne$ 0°. The hexahedral girds generated by Gmsh, which is an open access grid-generation software, are used in our global 3D fluid model, as shown in figure 1(c). The model will reach convergence after 0.003 s, consuming 2165455 s of computational time under 40 CPU cores for each case. We then verify the correctness of the 3D model by comparing the calculation results with RAM C-II experimental results at 71 km as an example. Reflectometers with different frequencies and electrostatic probes are mounted on the vehicle to diagnose the electron density during the flight [45]. The comparison of simulation results and reflectometers, which is the peak value of the electron density in the direction perpendicular to the re-entry vehicle, is shown in figure 1(d). The calculated electron density profile is also compared with the results of electrostatic probes, as shown in figure 1(e). It can be observed that the numerical results are in good agreement with experimental data, which demonstrates that our model is a reasonable approximation for the realistic conditions.

Figure 1. (a) Geometry of the re-entry vehicle. (b) Boundary condition settings. (c) Computational grids. The calculation results are compared with the (d) reflectometers and (e) electrostatic probes [45].

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2.2 Transmission characteristics of EM waves in plasma sheath

For the EM transmission property studies, we use the local 1D model similar to the previous work [46], as presented in figure 2. The initial profiles of the plasma sheath distribution (along l₁) at the antenna position, obtained by the global 3D plasma fluid model in section , are imported into the local 1D EM transmission model. We consider the 1D approximation to be reasonable when the EM wave is incident perpendicular to the plasma sheath and the flow field distribution is locally homogeneous in the vicinity of the antenna. The results of the 1D model can provide a qualitatively reasonable description and a fundamentally sound understanding for attack angle effect on re-entry vehicle communication. The EM wave transmission characteristics in the plasma sheath are calculated by COMSOL Multiphysics 5.6 software based on the FEM of the triangular mesh with a mesh size of 1/15 of the EM wavelength to achieve a high spatial resolution. The governing equation is the Helmholtz equation, as shown in equation (). μ_r = 1 is the magnetic permittivity for the non-magnetized plasma in this paper. $\overrightarrow E$ , k₀, and ω are the electric field, wavenumber in vacuum, and angular frequency of the EM wave, respectively. ε₀ is the dielectric constant in vacuum. ε_p and σ are the dielectric permittivity and conductivity of plasma sheath, whose expressions can be approximately given by the cold plasma model, as shown in equations (54)–(56), where ω_p and v are the plasma frequency and collision frequency of the plasma sheath, and n_e, m_e, and e are the electron density, electron mass, and element charge, respectively. v represents the collision frequency of electron and N₂, calculated through equations (50) and (51), considering that the neutral particles in the gas mixture are mainly N₂. The left side of the plasma sheath (figure 2) is set as the receiving port and the right side is the transmitting port. The periodic boundary condition is used to make the parameters of plasma sheath uniform in the y direction. It can be easily observed from equations (53)–(56) that the transmission of EM waves in the plasma sheath is mainly determined by its ω, ε_p, and σ. The n_e and v of the plasma sheath will affect the ε_p and σ parameters. Meanwhile, as a dispersive medium, EM waves with different angular frequencies (ω) will also exhibit different transmission characteristics in the plasma sheath. The EM transmission model is solved in the frequency domain to calculate the electric field at the receiving port, ${\overrightarrow E _{\text{t}}}$ , and the electric field at the receiving port, ${\overrightarrow E _{\text{r}}}$ , through the governing equation (equation (53)). Then we can calculate the transmission (T), reflection (R), and absorption (A) coefficients of EM waves in the plasma sheath by comparing the ${\overrightarrow E _{\text{t}}}$ and ${\overrightarrow E _{\text{r}}}$ with the incident electric field, ${\overrightarrow E _{\text{i}}}$ , as shown in equations (57)–(59). T, R, and A represent the proportion of EM wave energy that is transmitted, reflected, and absorbed, respectively.

Figure 2. Simplified 1D EM transmission model in the plasma sheath.

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$\nabla\times\mu_{\text{r}}^{-1}\left(\nabla\times\overrightarrow{E}\right)-k_0^2\left(\varepsilon_{\text{p}}-\frac{\mathrm{j}\sigma}{\omega\varepsilon_0}\right)\overrightarrow{E}=0,$

(53)

$\varepsilon_{\text{p}}=\varepsilon_0\left(1-\frac{\omega_{\text{p}}^{\text{2}}}{\omega^2+v^2}\right),$

(54)

${\omega _{\text{p}}} = \sqrt {\frac{{n_{\text{e}}^{\text{2}}e}}{{{m_{\text{e}}}{\varepsilon _0}}}} ,$

(55)

$\sigma = \frac{{{n_{\text{e}}}{e^2}v}}{{{m_{\text{e}}}\left( {{\omega ^2} + {v^2}} \right)}} ,$

(56)

$T = {{{{\left| {{{\overrightarrow E }_{\text{t}}}} \right|}^2}} \mathord{\left/ {\vphantom {{{{\left| {{{\overrightarrow E }_{\text{t}}}} \right|}^2}} {{{\left| {{{\overrightarrow E }_{\text{i}}}} \right|}^2}}}} \right. } {{{\left| {{{\overrightarrow E }_{\text{i}}}} \right|}^2}}} ,$

(57)

$R = {{{{\left| {{{\overrightarrow E }_{\text{r}}}} \right|}^2}} \mathord{\left/ {\vphantom {{{{\left| {{{\overrightarrow E }_{\text{r}}}} \right|}^2}} {{{\left| {{{\overrightarrow E }_{\text{i}}}} \right|}^2}}}} \right. } {{{\left| {{{\overrightarrow E }_{\text{i}}}} \right|}^2}}} ,$

(58)

$A = 1 - T - R .$

(59)

3. Results

3.1 Effect of attack angle on the plasma sheath

We take two typical flying conditions as examples. The n_e distribution profiles of the plasma sheath in (x, y) plane at the altitudes of 71 and 31 km, with attack angle φ ranging from 0° to 30° are shown in figure 3. The input freestream parameters, according to realistic flying conditions [44, 45], are shown in table 2. According to the results of our previous study [46], the plasma sheath exhibits high electron density and weak collision frequency characteristics at 71 km when the attack angle is 0°. The collision frequency will become more severe due to the increase of atmosphere density when the altitude falls to 31 km, and the plasma sheath exhibits high electron density and strong collision frequency characteristics. We analyze the effect of angle of attack on the transmission characteristics of EM waves in the plasma sheath at these two typical flying conditions.

Table 2. Flying conditions of RAM C-II re-entry vehicle at altitudes of 71 km and 31 km [44, 45].

Altitude(km)	Atmosphere density (kg·m⁻³)	Atmosphere temperature (K)	Flying velocity (km·s⁻¹)
71	7.64×10⁻⁵	216.00	7.65
31	1.84×10⁻²	226.50	6.65

| Show Table

DownLoad: CSV

Figure 3. The n_e (m⁻³) distributions of the plasma sheath at various altitudes and attack angles. 71 km: φ = 0° (a), 10° (b), 20° (c), 30° (d), and 31 km: φ = 0° (e), 10° (f), 20° (g), 30° (h). Two 3D photographs on the left serve as a schematic.

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It can be observed that the n_e of the plasma sheath is asymmetrically distributed along the y-axis when re-entry flying with a finite attack angle. The n_e on the windward side increases and the sheath thickness decreases compared with φ = 0°. The n_e distribution on the leeward side shows an opposite trend, where the n_e decreases while the sheath thickness increases. The peak values of n_e for attack angles of 0°, 10°, 20°, and 30° are 3.19×10¹⁹, 3.19×10¹⁹, 3.25×10¹⁹, and 3.22×10¹⁹ m⁻³ respectively at 71 km, indicating that the attack angle effect of on the peak n_e is negligible, as shown in figures 3(a)–(d). The rise of the atmosphere density at lower altitudes leads to the increase of n_e accordingly, as shown in figures 3(e)–(h) for the altitude of 31 km. The peak value of n_e can reach 1.02×10²¹ m⁻³ at the flying condition. The thermal ionization mainly occurs around the head of the re-entry vehicle when the attack angle is 0°. Thus, the electron density in the body area tends to be 1–2 orders of magnitude lower than that of the head. Nevertheless, not only does the ionization occur at the head but also at the windward side when the attack angle is finite, resulting in a growth of the electron density on the windward side. Meanwhile, the plasma sheath thickness is decreased due to the compression of the bow shock on the windward side. The windward shift of the flow stationary position also leads to the reduction of the electron density and an increase of plasma sheath thickness on the leeward side.

The collision frequency between electron and neutral particle, namely v, is also an important parameter affecting the transmission of EM waves in the plasma sheath, besides the electron density according to equations (53)–(56). The v-distributions of the plasma sheath at altitudes of 71 and 31 km, with various attack angles, are shown in figure 4. Similar to the distribution of n_e, the peak position of v is shifted to the windward side when the attack angle exists. The v is asymmetrically distributed in (x, y) plane, and obviously higher on the windward than that on the leeward side. The collision of electron to neutral particle is dominated by the collision between electron and N₂, considering that the proportion of N₂ in the freestream is 79%. The increase in electron density will also lead to an increase in the collision frequency of electron to neutral particle. The collision frequency increases due to the increase of atmosphere density and number of particles involved in the collision when the altitude falls to 31 km.

Figure 4. Collision frequency v (Hz) distributions of the plasma sheath at various altitudes and attack angles. 71 km: φ = 0° (a), 10° (b), 20° (c), 30° (d), and 31 km: φ = 0° (e), 10° (f), 20° (g), 30° (h). Two 3D photographs on the left serve as a schematic.

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The communication antenna is commonly mounted on the body area of the vehicle, considering the high thermal load and electron density around the vehicle head [47]. We first take the case in which the axial distance between antenna and vehicle nose is 0.5 m (middle part of body area, see figure 5(a)) as an example. The n_e profile on the (x, y) plane in the direction perpendicular to the vehicle surface is shown in figure 5, and the thickness of the plasma sheath is estimated with the criteria of n_e > 10¹⁶ m⁻³. Clearly, the n_e increases first and then decreases with the distance from the vehicle surface, due to the recombination of electrons and ions near the surface. The peak values of the n_e profile on the windward for attack angles of 0°, 10°, 20°, and 30° are 4.50×10¹⁷, 8.19×10¹⁷, 3.64×10¹⁸, and 8.60×10¹⁸ m⁻³, respectively at 71 km. The peak value of n_e on the windward side can be increased by an order of magnitude when the attack angle reaches 20°. The plasma sheath thickness decreases obviously from 0.15 to 0.07 m when the attack angle varies from 0° to 20° at 71 km, as shown in figure 5(a), but approximately keeps its level as the attack angle increases from 20° to 30°. On the leeward side, the peak values of the n_e profiles for attack angles of 0°, 10°, 20°, and 30° are 4.50×10¹⁷, 2.47×10¹⁷, 1.93×10¹⁷, and 1.71×10¹⁷ m⁻³, respectively, at 71 km and approximately keep the level of ~ 10¹⁷ m⁻³ although they decrease with the attack angle, as shown in figure 5(b). The plasma sheath thickness, on the other hand, increases continuously with the attack angle. The trends of the n_e profiles at various altitudes are similar to each other, as shown in figures 5(c) and (d). The v profiles in the direction perpendicular to the surface of re-entry vehicle at the antenna position are shown in figure 6. The peak collision frequency on the windward side increases with the increase of the electron temperature and also with the increase of attack angle, e.g., by an order of magnitude as the angle reaches 20°, as shown in figure 6(a). Similar to the n_e profiles, the peak collision frequency approximately remains at its order of magnitude on the leeward side. The trend of v profiles at 31 km is similar to that at 71 km, as shown in figures 6(c) and (d). The peak positions of the collision frequency at 71 and 31 km for various attack angles are at the edge of the shock wave. This is due to the high gas density at the edge of the shock wave, where the density of N₂ reaches its peak.

Figure 5. The n_e profiles perpendicular to the vehicle surface at x = 0.5 m and 71 km windward (a) and leeward (b), and 31 km windward (c) and leeward (d), with two photographs on the left as a schematic.

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Figure 6. The v profiles perpendicular to the vehicle surface at x = 0.5 m and 71 km windward (a) and leeward (b), and 31 km windward (c) and leeward (d), with two photographs on the left as a schematic.

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3.2 Effect of attack angle on the transmission characteristics of EM waves in plasma sheaths

The n_e and v profiles in figures 5 and 6 at the antenna position are imported into COMSOL Multiphysics for the calculation of EM wave transmission characteristics. The reflection, transmission, and absorption coefficients of EM waves in plasma sheaths for various attack angles on windward and leeward sides versus communication frequencies ranging from 2 to 40 GHz at the altitude of 71 km are shown in figure 7.

Figure 7. Transmission characteristics of plasma sheaths at 71 km versus various communication frequencies. On the windward side: (a) reflection, (b) transmission, (c) absorption; on the leeward side: (d) reflection, (e) transmission, (f) absorption.

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The transmittance of the EM waves in plasma sheaths increases with communication frequency, as shown in figures 7(b) and (e), and becomes negligible as the frequency exceeds 9 GHz in the zero attack angle case. The reduction of EM waves in plasma sheaths is mostly due to the reflection in f < 9 GHz regime, as shown in the blue lines in figures 7(a) and (c). At the high altitude, e.g., at 71 km, the collision is low. Then the effect of absorption is negligible, as shown in figures 7(c) and (f), where the maximum of absorption is less than 10 dB. The re-entry communication is then further deteriorated as the attack angle increases, as shown in figure 7(b). The electron density on the windward side increases with the attack angle, resulting in more serious reflection of EM waves. The RF blackout is, on the other hand mitigated, on the leeward side where the reflection is reduced obviously, as shown in figures 7(d) and (e).

The reflection, transmission, and absorption coefficients of EM waves in the plasma sheath, with various attack angles on the windward and leeward sides, versus communication frequency band ranging from 2 to 200 GHz at the altitude of 31 km are shown in figure 8. The attenuation is more serious than that at 71 km due to the increase of the electron density and collision frequency.

Figure 8. Transmission characteristics of plasma sheaths at 31 km versus various communication frequencies. On the windward: (a) reflection, (b) transmission, (c) absorption; on the leeward: (d) reflection, (e) transmission, (f) absorption.

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The attenuation in this lower altitude is due to the multiple effects of reflection, absorption, and sub-wavelength effect in certain frequency bands. Besides normalized plasma frequency and collision frequency, the EM wave transmission characteristics in the plasma sheath are also influenced by the plasma sheath thickness. The sub-wavelength effect may be generated when the plasma sheath thickness is equivalent to or even lower than the EM wavelength. The transmission coefficients may be increased in the sub-wavelength condition due to the skin effect, when the EM wave is vertically incident into the plasma sheath. The transmission coefficient is first decreased and then increased with the increase of EM wave frequency, as shown in figures 8(d) and (e). The decrease of transmission coefficient in the low EM wave frequency regime is due to the sub-wavelength effect. Nevertheless, similar to the transmission characteristics at 71 km, the RF blackout is deteriorated on the windward side and mitigated on the leeward side for attack angles ranging from 0° to 30°. When the attack angle increases from 0° to 30°, the peak value of n_e decreases from 3.21×10¹⁹ to 1.40×10¹⁹ m⁻³, and the corresponding plasma frequency also decreases from 50.9 to 33.6 GHz.

4. Discussions

Considering that the peak value of n_e is 4.50×10¹⁷ m⁻³ (plasma frequency ≈ 6 GHz) for the zero attack angle plasma sheath at 71 km in figures 5(a) and (b), we chose a typical communication frequency (f = 4 GHz) lower than the plasma frequency to analyze the effect of attack angle on the EM wave transmission characteristics. The plasma frequency and collision frequency are then normalized by f = 4 GHz, as shown in figure 9. The reflection, transmission, and absorption at 71 km for f = 4 GHz, as well as the normalized plasma frequency and collision frequency on the windward side are shown in the left columns of figures 9(a), (c), and (e). The f_p-peak, v_p-peak, and f_p-wall represent peak values of f_p, v, and f_p on the vehicle surface in figures 5 and 6, where the plasma frequency f_p = ω_p/(2π). One can find that when f_p-peak/f > 1 and v_p-peak/f < 0.1 for the zero attack angle, the reflection is dominating due to the cutoff effect. The electron density and collision frequency are both increasing on the windward side with the increase of attack angle, resulting in the increase of f_p-peak/f and v_p-peak/f. The transmission is getting worse as the attack angle rises. The v_p-peak/f is on the other hand weak (less than 0.1) on both the windward and leeward sides at 71 km for various attack angles, which means that the absorption can be negligible. On the leeward side (the right column of figure 9), the plasma frequency is lower than the communication signal frequency (f_p-peak/f < 1) as the attack angle reaches 20°, as shown in figure 9(d). The EM waves then propagate through the plasma sheath and are received by the antenna. The RF blackout mitigation in this regime is due to the reduction of reflection.

Figure 9. For varied attack angles at the higher altitude of 71 km, (i) the transmission properties for f = 4 GHz on (a) windward and (b) leeward sides, (ii) peak normalized frequencies on (c) windward and (d) leeward sides, and (iii) peak collision frequency on (e) windward and (f) leeward sides.

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The reflection, transmission, and absorption of the plasma sheath at 31 km for f = 40 GHz on the windward and leeward sides are shown in figures 10(a) and (b). The plasma frequency and collision frequency in this altitude are normalized by 40 GHz. The attenuation of EM waves at 31 km is due to both reflection and absorption when the attack angle is 0°, as shown in figure 10(a). However, as the attack angle increases to ~ 20°, the absorption becomes negligible (< −15 dB) on the windward side because the EM waves are mainly reflected. Furthermore, the plasma frequency significantly exceeds the communication frequency (f_p-wall/f > 1), resulting in an almost total reflection (R $\approx$ 1, 10lgR $\approx$ 0) for the attack angle > 10°. On the leeward side, the attenuation is decreased due to the reduction of the electron density and collision frequency with the increase of the attack angle. Therefore, similar to that at 71 km, the RF blackout can be mitigated significantly if the antennas are mounted on the leeward side of the middle part body area.

Figure 10. For varied attack angles at the lower altitude of 31 km, (i) the transmission properties for f = 4 GHz on (a) windward and (b) leeward sides, (ii) peak normalized frequencies on (c) windward and (d) leeward sides, and (iii) peak collision frequency on (e) windward and (f) leeward sides.

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According to the preceding discussions, the communication antennas should be mounted on the leeward side for the re-entry vehicle. We further analyze the influence of the axial installation position of the antennas on the transmission characteristics. We choose the cases in which the re-entry vehicle’s attack angle is 20°, with the distance between the axial position of the antenna and the vehicle nose ranging from x = 0.4–1.2 m. The transmission coefficients of EM waves with various positions of the antennas on the leeward side at the altitudes of 71 and 31 km are shown in figure 11. The direction of EM waves is assumed always to be perpendicular to the plasma sheath and the case with the attack angle φ = 0° is made as a reference for comparison. The transmittance of EM waves is improved obviously with the increase of the distance x. The RF blackout will be mitigated obviously at 71 km if the attack angle is 20°, by comparing figure 11(b) with figure 11(a). This is due to the decrease of electron density on the leeward side. However, the attenuation is shown to be more serious on the rear of the body area (1.0–1.2 m), when the altitude falls to 31 km, by comparing φ = 20° with φ = 0° (figure 11(d) with figure 11(c), Phase I with II). This is due to the increase of plasma sheath thickness while the decrease of electron density and collision frequency is not obvious on the leeward side compared with the cases in which φ = 0°. The increase in the thickness of the plasma sheath will lead to an increase of absorption. The antennas are in general mounted at different locations of the re-entry vehicle []. The traditional view is that the antenna should be mounted on the leeward side, and the communication will be better if the antenna is closer to the backward of the body area. However, our results indicate that at certain altitudes, the communication of φ = 0° is better than the leeward side of φ $\ne$ 0°, for antennas mounted on the backward side. During the re-entry, proper trajectory planning and reasonable antenna placement can mitigate the RF blackout problem to an extent. Taking 31 km as an example, the communication will be improved for the antennas mounted on the middle of the body area when the vehicle re-enters with an attack angle, and the vehicle should try to avoid re-entry with a large attack angle for the antennas mounted on the back.

Figure 11. Transmission coefficient (colorbar) of the plasma sheaths on leeward side versus the axial position of the antenna and communication frequency, with various parameters of (a) φ = 0°, 71 km; (b) φ = 20°, 71 km; (c) φ = 0°, 31 km; and (d) φ = 20°, 31 km.

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5. Conclusion

In this study, we develop an integrated model by combining the global 3D plasma fluid model, considering various attack angles and realistic flying conditions of the re-entry vehicle, and the local simplified EM wave-transmission model at the antenna position by importing the sheath parameter profiles from the global model simulation outputs. With such an integrated approach, we analyze the effect of the attack angle on the distributions of plasma sheath parameters and the transmission characteristics of EM waves. A direct correlation between EM wave transmission and vehicle attack angle is established through the reflection, transmission, and absorption analysis. The qualitative recommendations regarding the choice of communication frequency band for various attack angles are also given. The main results are as follows:

(1) The distribution profiles of the plasma sheaths are asymmetrical when the vehicle flies with a finite attack angle. The electron density and collision frequency are increased on the windward side and decreased on the leeward side. Meanwhile, the plasma sheath thickness is thinned on the windward side and thickened on the leeward side.

(2) The RF blackout to the communication is worsened on the windward side and mitigated on the leeward side as the attack angle increases from 0° to 30°, for antennas on the middle part of body area (e.g., 0.5 m). The weakening of communication signal (e.g., f = 4 GHz) is mainly due to reflection at the higher altitude of 71 km, and gets worse on the windward side as the attack angle rises. At the lower altitude of 31 km, on the other hand, both reflection due to the high electron density and absorption due to the high collision can intensively weaken the communication signal (e.g., f = 40 GHz).

(3) Taking the antenna at the position 0.5 m from the re-entry vehicle head (altitude 31 km) as an example, the communication frequency should reach the terahertz band when installed on the windward side for various attack angles, while exceeding the Ka band on the leeward side. By analyzing the reflection, transmission, and absorption coefficients, a qualitative optimization proposal for antenna position can be given. The communication antennas should be mounted on the leeward side. The communication may be worsened for antennas on the leeward side in certain attack angles compared with the case in which the attack angle is 0° due to the increase of plasma sheath thickness. Reasonable choice of flight trajectory and antenna installation position can substantially mitigate the RF blackout problem.

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