2. The flow field model of reentry vehicle

In this paper, based on the Navier–Stokes equations, the simulation software USim and the model proposed by Kundrapu et al. (Reference Kundrapu, Loverich, Beckwith, Stoltz, Shashurin and Keidar2015) are used to simulate the flow field around a hypersonic reentry vehicle. The Navier–Stokes equations are shown in (2.1)–(2.3), which are used for conservation of fluid mass, momentum and total energy, respectively:

(2.1)\begin{gather} \frac{\partial \rho }{\partial t}+\boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{\mathsf{u}})=0, \end{gather}

(2.2)\begin{gather}\frac{\partial ( \rho \boldsymbol{\mathsf{u}} ) }{\partial t} +\boldsymbol{\nabla} \boldsymbol{\cdot} ( \rho \boldsymbol{\mathsf{u}} \boldsymbol{\mathsf{u}}+P\boldsymbol{\mathsf{I}} )=\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau} , \end{gather}

(2.3)\begin{gather}\frac{\partial e_{{t}} }{\partial t}+\boldsymbol{\nabla} \boldsymbol{\cdot} ( \boldsymbol{\mathsf{u}} ( e_{{t}}+\rho) )=\boldsymbol{\nabla}\boldsymbol{\cdot}( \boldsymbol{\tau}\boldsymbol{\cdot} \boldsymbol{\mathsf{u}}) +\boldsymbol{\nabla} \boldsymbol{\cdot}( \lambda \boldsymbol{\nabla} T ), \end{gather}

where $\rho$ is the mass density, $\boldsymbol{\nabla}$ is the Hamiltonian, $\boldsymbol{\mathsf{u}}$ is the velocity, $\boldsymbol{\tau}$ is the stress tensor, $p$ is the pressure, $\boldsymbol{\mathsf{I}}$ is the identity matrix, $e_{{t} }$ is the total energy of the fluid which is composed of kinetic energy, chemical energy and thermal energy, $\lambda$ is the thermal conductivity and $T$ is the temperature.

The mass density can be calculated using

(2.4)\begin{equation} \rho =\sum_{q}^{N} n_{q} m_{q} ,\end{equation}

where $N$ is the number of components in the flow field and $n_{q}$ and $m_{q}$ are the particle density and mass of component $q$, respectively.

The pressure is defined as

(2.5)\begin{equation} P=\rho RT,\end{equation}

where $R$ is the gas constant.

The stress tensor is calculated using

(2.6)\begin{equation} \boldsymbol{\tau} ={-}\frac{2}{3} \mu( \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\mathsf{u}} ) \boldsymbol{\mathsf{I}}+\mu ( \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\mathsf{u}}+ ( \boldsymbol{\nabla} \boldsymbol{\mathsf{u}} )^{{T_{r}}} ), \end{equation}

where $\mu$ is the dynamic viscosity and ${T_{r}}$ is the transpose.

The total energy is defined as

(2.7)\begin{equation} e_{{t}} =\frac{P}{\gamma -1} +\frac{1}{2} \rho \boldsymbol{\mathsf{u}} \boldsymbol{\cdot} \boldsymbol{\mathsf{u}}+\sum_{q}^{N}n_{q}H_{q}, \end{equation}

where $\gamma$ is the specific thermal ratio, which is defined as $\gamma ={c_{{p}}}/{c_{{v}}}$ ($c_{{p}}$ is specific heat at constant pressure and $c_{{v}}$ is specific heat at constant volume), and $H_{q}$ is the enthalpy of formation.

The transformation relation of each component in the flow field can be satisfied by the mass conservation expressed by (2.1). Considering (2.4) simultaneously, the density change rate $s_{q}$ of each component is obtained as

(2.8)\begin{equation} \frac{\partial n_{q} }{\partial t} +\boldsymbol{\nabla} \boldsymbol{\cdot} ( n_{q}\boldsymbol{\mathsf{u}} ) =s_{q}.\end{equation}

Besides, other parameters of various components can be expressed as (2.9)–(2.12) according to the gas dynamics:

(2.9)\begin{gather} \lambda _{q} =\frac{5}{2}c_{{v},q}\mu _{q} , \end{gather}

(2.10)\begin{gather}\mu _{q} =\frac{5}{16}\frac{\sqrt{{\rm \pi} m_{q}k_{{b}}T } }{{\rm \pi} \sigma ^{2}\varOmega }, \end{gather}

(2.11)\begin{gather}c_{{p},q} =R_{q} \left( \frac{d_{{f}} }{2} +1 \right) , \end{gather}

(2.12)\begin{gather}c_{{v},q} =c_{{p},q} -R_{q}, \end{gather}

where $\lambda _{q}$ is the fluid thermal conductivity, $\mu _{q}$ is viscosity, $k_{{b}}$ is the Boltzmann constant, $\sigma$ is the collision diameter, $\varOmega$ is the collision integral and $d_{{f}}$ is the number of degrees of freedom.

The RAM-C model as described by Grantham (Reference Grantham1970), which has a length of $L=1.295\ \mathrm {m}$, a blunt head radius of $R=0.1524\ \mathrm {m}$ and a half-cone angle of $\theta =9^{\circ }$, is used in this study. The mesh and flow-field model of the reentry vehicle are shown in figures 1(a) and 1(b), respectively. The incident position of the terahertz wave is represented by the dashed white line in figure 1(b). It is assumed that the angle of attack of free stream is $0^{\circ }$, so the flow field on both sides of the reentry vehicle is symmetrically distributed. Therefore, one side of the flow field is selected for study in this paper. When the altitude is $60$–$90\ \textrm {km}$ and speed is $18$–$27\ \textrm {Ma}$, the air around the aircraft is in a state of thermodynamic equilibrium and chemical non-nequilibrium, the air is ionized and forms a plasma with seven components, which are $\textrm {N}_{2},\textrm {O}_{2},\textrm {NO},\textrm {N},\textrm {O},\textrm {NO}^{+}$ and $\mathrm {e}$ (electrons) (Le Reference Le2005). Seven components and eighteen chemical reactions proposed by Park were used in this study (Park Reference Park1985a, Reference Park1985b).

Figure 1. (a) The grid and (b) the initial field of the flow field around the reentry vehicle.

3. The propagation models of terahertz waves

The SMM (Hu et al. Reference Hu, Wei and Lai1999; Guo et al. Reference Guo, Guo and Li2017) is used to analyse the transmission characteristics of terahertz waves in ‘blackout’. The propagation model of terahertz waves in plasma is shown in figure 2. The boundary of the plasma layer is assumed to be flat, and the plasma sheath is divided into some parallel sublayers equally. The plasma properties among these sublayers are different, but the internal property of each layer is uniform. For better transmission efficiency, it is assumed that the terahertz wave is incident perpendicularly into the plasma sheath. According to figure 2, both incident and reflected waves exist in the incident region (0), and the total electric field can be described as

(3.1)\begin{equation} E_{z,0}=\textit{E}_{{0}}({\rm e}^{{-}j k_{0}z}+ A\, {\rm e}^{j k_{0}z}),\end{equation}

the total electric field in each sublayer plasma is described as

(3.2)\begin{equation} E_{z,i}=\textit{E}_{{0}}( B_{i}\, {\rm e}^{{-}j k_{i} z}+ C_{i} \, {\rm e}^{j k_{i}z}),\end{equation}

there is only transmitted wave but no reflected wave in the transmitted region ($\textit {n}$+1) and the total field is described as

(3.3)\begin{equation} E_{z,n+1}={E}_{0}D\ {\rm e}^{{-}j k_{n+1}z},\end{equation}

where the total reflection coefficient and the total transmission coefficient are represented by $A$ and $D$, respectively. The transmission and reflection coefficients in the $i$th-layer plasma are represented by $B_{i}$ and $C_{i}$, respectively. In addition, $k_{i}=k_{{0}}\sqrt {\varepsilon _{{r},i}}$ is the wavenumber of terahertz wave transmission in the $i$th-layer plasma. Therein, $k_{{0}}=\omega \sqrt {\mu _{0} \varepsilon _{0}}=\omega /c$, $\mu _{0}=4{\rm \pi} \times 10^{-7} \ \mathrm {H}\cdot\mathrm {m}^{-1}$ is the permeability in vacuum and $c=3 \times 10^{8}\ \mathrm {m}\cdot\mathrm {s}^{-1}$ is the speed of light in vacuum.

Figure 2. Schematic diagram of the SMM.

According to the formula deduced by Heald, Wharton & Furth (Reference Heald, Wharton and Furth1965), the relative permittivity of plasma is approximately expressed as

(3.4)\begin{equation} \varepsilon_{{r},i} =\left( 1-\frac{\omega _{{p},i}^{2}}{\omega^{2}+ v_{{e},i}^{2}} \right)-j \frac{v_{{e},i}}{\omega } \frac{\omega_{{p},i}^{2}}{\omega^{2}+v_{{e},i}^{2}},\end{equation}

where the angular frequency of the particles in the plasma is lower than the angular frequency of the electrons, and it can be deduced that $\omega _{{p}} \approx \omega _{{pe}}=\sqrt {N_{{e}}\, e^{2}/\varepsilon _{0}m_{{e}}}$. Here $N_{{e}}$ is the electron density in the plasma sheath, $e=1.602\times 10^{-19}\ \mathrm {C}$ is the electron charge, $\varepsilon _{0}=8.854\times 10^{-12}\ \mathrm {F}\cdot\mathrm {m}^{-1}$ is the dielectric constant in vacuum and $m_{{e}} =9.109\times 10^{-31}\ \mathrm {kg}$ is the electron mass. Parameter $v_{{e}}=5.82\times 10^{12} P/\sqrt {T}$ is the collision frequency of electrons in the plasma (Ouyang et al. Reference Ouyang, Jin, Wu and Deng2021), which is obviously dependent on the temperature $T$ and pressure $P$ of the plasma sheath.

According to the boundary conditions of continuous matching of electric field, the coefficient transformation relationship between adjacent plasma layers can be shown to be

(3.5)\begin{equation} \begin{pmatrix} B_{i}\\ C_{i} \end{pmatrix}=\boldsymbol{\mathsf{S}}_{i} \begin{pmatrix} B_{i-1}\\ C_{i-1} \end{pmatrix}.\end{equation}

The scattering matrix $\boldsymbol{\mathsf{S}}_{i}$ in the $i$th-layer plasma is expressed as

(3.6)\begin{align} \boldsymbol{\mathsf{S}}_{i}& = \begin{pmatrix} {\rm e}^{{-}j k_{i}d_{i}} & {\rm e}^{j k_{i}d_{i}} \\ k_{i}\,{\rm e}^{{-}j k_{i}d_{i}} & -k_{i}\, {\rm e}^{{-}j k_{i}d_{i}} \\ \end{pmatrix}^{{-}1}\nonumber\\ & \quad \times\begin{pmatrix} {\rm e}^{{-}j k_{i-1}d_{i-1}} & {\rm e}^{j k_{i-1}d_{i-1}} \\ k_{i-1}\,{\rm e}^{{-}j k_{i-1}d_{i-1}} & -k_{i-1}\, {\rm e}^{{-}j k_{i-1}d_{i-1}} \\ \end{pmatrix}. \end{align}

The relation between the total incidence field and the total transmission field is expressed as

(3.7)\begin{align} \boldsymbol{\mathsf{S}}_{{g}}\begin{pmatrix} A \\1 \end{pmatrix} = \boldsymbol{\mathsf{V}} D, \end{align}

where

(3.8)\begin{equation} \boldsymbol{\mathsf{V}}=\frac{1}{2k_{{n}}} \begin{pmatrix} k_{{n}}+k_{{n+1}}e^{j ( k_{{n}}-k_{{n+1}})d_{{p}}}\\ k_{{n}}-k_{{n+1}} {\rm e}^{{-}j ( k_{{n}}+k_{{n+1}})d_{{p}}} \end{pmatrix}. \end{equation}

Here $\boldsymbol{\mathsf{S}}_{{g}}$ is the global scattering matrix, which can also be described as

(3.9)\begin{equation} \boldsymbol{\mathsf{S}}_{{g}}=\left( \prod_{i=\mathrm{n}}^{2} \boldsymbol{\mathsf{S}}_{i} \right)\boldsymbol{\mathsf{S}}_{{1}}.\end{equation}

By transforming $\boldsymbol{\mathsf{S}}_{{g}}$ into $( \boldsymbol{\mathsf{S}}_{{g1}},\boldsymbol{\mathsf{S}}_{{g2}})$, (3.7) is derived as

(3.10)\begin{equation} \begin{pmatrix} A\\D \end{pmatrix} ={-}\begin{pmatrix} \boldsymbol{\mathsf{S}}_{{g1}} & -\boldsymbol{\mathsf{V}} \\ \end{pmatrix}^{{-}1} \boldsymbol{\mathsf{S}}_{{g2}}.\end{equation}

The reflection coefficient $A$ and transmission coefficient $D$ are calculated. And further, the transmissivity, reflectivity and absorptivity can be expressed by

(3.11)\begin{equation} \left\{\begin{array}{@{}l} \varGamma_{{t}} =|A |^{2},\\ T_{{t}} =|D |^{2},\\ A_{{t}}=1-\varGamma_{{t}} -T_{{t}}, \end{array}\right. \end{equation}

where $\varGamma _{{t}}$ is the total reflectivity of the plasma sheath to terahertz waves, $T_{{t}}$ is the total transmittivity of terahertz waves through the plasma sheath and $A_{{t}}$ is the total absorptivity of terahertz waves by the plasma sheath.

4. Simulation and analysis

The flow field characteristics of the reentry vehicle surface at different flight altitudes ($H$) and speeds ($V$) are analysed in this simulation. The temperature and density of the atmosphere vary with altitude (NASA, Glenn Research Center 2021). According to the formula of atmospheric properties, the data of atmospheric temperature, pressure and density for different altitudes are shown in table 1.

Table 1. The maximum temperature, pressure and density of the atmosphere at different altitudes.

The simulation of plasma flow field around a blunt reentry vehicle was analysed using Tech-X USim. The simulation results are shown in figures 3 and 4. It is obvious that the free stream covers the blunt head with a sheath much larger than the blunt head due to the hypersonic characteristic of the reentry vehicle. The sheath around the head of the reentry vehicle has higher pressure, temperature and electron density; these three parameters are smaller at the tail of the blunt head. That is why the reentry antenna is installed at the tail of the blunt head. In order to further study the influence of external conditions on the internal characteristics of the flow field, the values of peak temperature and peak pressure for different conditions are shown by the dots in figure 5 based on figures 3 and 4. As shown in figure 5(a), with an increase of altitude, it can be seen that the temperature increases significantly at $51$–$61\ \textrm {km}$ and then decreases slightly at $61$–$71\ \textrm {km}$. As shown in figure 5(b), with an increase of speed, it can be seen that the changes of temperature and pressure are relatively stable for $23$–$25$ and $25$–$27\ \textrm {Ma}$.

Figure 3. Flow-field distribution around the reentry model at a speed of 25 Ma for different flight altitudes: (a) 51 km, (b) 61 km and (c) 71 km.

Figure 4. Flow-field distribution around the reentry model at an altitude of 71 km for different flight speeds: (a) 23 Ma, (b) 25 Ma and (c) 27 Ma.

Figure 5. The maximum temperature and maximum pressure for different flow-field conditions. (a) The flight speed is 25 Ma. (b) The flight altitude is 71 km.

The simulation results and table 1 show that with an increase of flight altitude, the decrease of free-stream density is the main reason for the decrease of electron density. With an increase of speed, the temperature of the flow field increases significantly, which promotes the efficiency of electron generation and indirectly increases electron density.

5. Calculation results and discussion

In this section, the parameters of the numerical calculation are assumed to be that the transmission frequency of the terahertz wave is $0.1$–$1.0\ \textrm {THz}$, the flight altitude is $51$–$71 \textrm {km}$ and the flight speed is $21$–$27\ \textrm {Ma}$. The oscillation characteristics of reflectivity are scaled in decibels.

5.1. Influence of altitude

According to figure 3, the distributions of electron density and electron collision frequency at the tail of the blunt reentry vehicle vary with altitude, as shown in figure 6. It can be seen that the electron density is very low, but the electron collision frequency is very high at the surface of the vehicle. With an increase of altitude, the electron density decreases and the peak position moves gradually away from the aircraft surface from 0.2 to 0.3 m; the electron collision frequency decreases and the peak position moves gradually away from the surface of the aircraft from 0.3 to 0.35 m.

Figure 6. The electron characteristics of plasma for different altitudes. (a) Electron density. (b) Electron collision frequency.

It is shown in figure 7 that the transmission characteristics of terahertz waves in the flow field around the reentry vehicle vary with altitude. It can be seen that the transmissivity decreases with a decrease of altitude, while the reflectivity and absorptivity increase with a decrease of altitude. Physically speaking, a decrease in altitude leads to an increase in atmospheric pressure and temperature, which makes electrons in the plasma collide more frequently. At the same time, the increase of free-stream density in the flow field makes more particles participate in the thermochemical reaction around the reentry vehicle, resulting in an increase of electrons in the reaction products and the electron density in the flow field is increased. A large number of electrons weaken the energy in the terahertz waves, which is manifested as the absorptivity of the plasma sheath to the terahertz waves being increased, and the transmissivity of the terahertz waves in the plasma is decreased.

Figure 7. The transmission characteristics of terahertz waves for different altitudes. (a) Transmissivity. (b) Reflectivity. (c) Absorptivity.

5.2. Influence of flight speed

According to figure 4, the distributions of electron density and electron collision frequency at the tail of the blunt reentry vehicle vary with flight speed, as shown in figure 8. It can be seen that, with an increase of speed, the peak electron density increases and the peak position gradually approaches the aircraft surface from 0.3 to 0.25 m, but the curve of electron collision frequency remains basically unchanged.

Figure 8. The electron characteristics of plasma for different flight speeds. (a) Electron density. (b) Electron collision frequency.

It is shown in figure 9 that the transmission characteristics of terahertz waves in the flow field around the reentry vehicle vary with speed. It can be seen that the transmittance decreases with an increase of speed, while reflectivity and absorptivity increase with an increase of speed. Physically speaking, with an increase of flight speed, the kinetic energy of particles in the flow field increases, which is macroscopically manifested as an increase of flow-field temperature. It transforms the ordered kinetic energy into disordered kinetic energy among particles and reduces the effective collision frequency of electrons. The energy of the terahertz wave is consumed by the electrons with low collision frequency, which leads to an increase of absorptivity of the terahertz wave by the plasma. Meanwhile, the increase of flow-field temperature can promote the thermochemical reaction in the flow field, and improve the electron generation efficiency of the thermochemical reaction. As the electron density in the plasma is increased, the transmittance is decreased and the reflectivity is increased.

Figure 9. The transmission characteristics of terahertz waves for different flight speeds. (a) Transmissivity. (b) Reflectivity. (c) Absorptivity.

6. Summary and conclusion

In this paper, the transmission characteristics of terahertz waves in hypersonic target flow field are studied. Firstly, the control variable method is used to simulate the flow-field distribution around the RAM-C reentry vehicle at different altitudes and speeds. Secondly, the variation of electron density and electron collision frequency in the flow field is analysed according to the simulation results. Finally, the SMM is used to study the propagation characteristics of terahertz waves in the hypersonic reentry flow field under different conditions. The results show that, with a decrease of altitude, the electron density increases with an increase of air density and the electron collision frequency increases with an increase of air pressure. In addition, with an increase of speed, the increase of flow-field temperature increases the electron density and electron collision frequency. In both cases, the absorptivity of the plasma sheath to terahertz waves increases, so that the transmission capacity of terahertz waves in the plasma flow field is weakened. The study provides some reference for mathematical modelling and theoretical analysis of electron density and electron collision frequency in ‘blackout’.