† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No.61431010).
The characteristics of light propagation through turbulent plasma sheath surrounding the hypersonic aircraft have been studied. The turbulent flow fields around a hypersonic aircraft are given by using the Navier–Stokes (NS) equations and k–ε turbulence model. Based on the distribution of flow field, refractive index and density of the plasma sheath for a blunt cone are discussed with different flight velocities and altitudes. The refractive index is mainly influenced by the electrons in the turbulent plasma sheath. The influence of different velocities and altitudes on the features of light propagation in the turbulent plasma sheath is analyzed. The results show that as the flight speed increases or the flight altitude decreases, the refractive index fluctuation becomes larger. It is also found that the refractive index fluctuation varies with the incident wavelength. This study shows how the characteristics of an optical beam propagating through plasma sheath are affected by the incident wavelength, flight velocities, and altitudes.
With the development of hypersonic aircraft in near space, the plasma sheath surrounding the hypersonic aircraft has attracted the attention of many researchers. The plasma sheath is formed by the hypersonic movement of the aircraft.[1,2] When the aircraft moves at the speed of 10 Mach or more, the air around the aircraft begins to be compressed quickly and the temperature rises precipitously, generating a large amount of heat that ionizes the air. As the temperature rises, the ionization becomes intense and eventually forms a layer of plasma[3–5] which contains numerous free electrons, ions, and ablation particles. This layer is the so-called plasma sheath, which is inhomogeneous and variable with time. Meanwhile, recent studies show that the plasma sheath surrounding a hypersonic aircraft exhibits turbulent features.[6] For example, in 2014, Yang et al. studied the propagation characteristics of signals in time-varying plasmas. The results showed that the plasma parameters have time-varying characteristics due to turbulence effects of the plasma sheath or flight altitude adjustment and other factors.[7] In 2016, Shi et al. analyzed the influence of flight conditions (such as attack angle and small scale turbulence) on the transmission performance of the plasma medium. A new adaptive multistate Markov channel modeling method was proposed to present the dynamic effects of re-entry plasma sheaths on wireless channels.[8] In 2016, He et al. studied the dynamic effects of re-entry plasma sheath on electromagnetic wave propagation using the transmission matrix method. The established model included spatial turbulence and changes of electron density in plasma sheath. The amplitude and phase changes caused by the nonlinear characteristics of plasma were numerically simulated. It was found that the amplitude variation obeys logarithmic normal distribution, and the phase change obeys Rayleigh distribution.[9] Liu et al. found that the randomness of turbulent re-entry plasma sheaths can affect the propagation and scattering properties of electromagnetic waves. In terms of wave propagation, the randomness of turbulence caused amplitude and phase fluctuations, and the degree of fluctuation was closely related to altitude.[10] In 2017, Chen et al. used the finite difference time domain method to study the propagation characteristics of the terahertz wave in the plasma sheath of non-uniform space time. The study showed that the penetration of electromagnetic waves in the plasma sheath is closely related to temperature, pressure, and electron density.[11] In 2017, based on the hypersonic turbulence measurement of fractal theory, Li et al. derived the refractive index power spectrum index and the turbulent power spectrum. It was found that the turbulence in plasma sheath can seriously deteriorate the resolution of synthetic aperture radar imaging.[12] The existence of the turbulent plasma sheath has a great influence on the communication between the aircraft and radars on the land, and even results in communication blackout in some circumstances.[13–22]
To reveal the effect of plasma sheath on the communication of hypersonic aircraft, there have been many studies of the propagation of electromagnetic waves through the plasma sheath surrounding a hypersonic aircraft. In the early work, many researchers investigated the plasma sheath based on the aero-optical.[23–29] Within the aero-optical framework, the flow field was assumed to be low speed and the fluid was considered as air that is not ionized. In fact, air has been ionized into plasma in the hypersonic flow field and exhibits turbulent features. Therefore, researchers have attempted to solve hypersonic turbulence in aerospace engineering by employing other approaches. Li et al. studied the refractive index spectrum of plasma sheath and bit error rate of the laser through plasma sheath based on fractal theory.[30,31] Zhang et al. investigated the plasma channelʼs characteristics and performance.[32] In their work, the turbulent plasma sheath was treated as a fast fading wireless channel, and the simulation results showed that error floor took place for phase-shift keying and quadrature amplitude modulation, while frequency-shift keying with non-coherent detection was a promising way to mitigate the blackout problem. It is worth noting that they used low frequency waves. However, plasma sheath is a non-uniform medium with dispersion effect, and its dielectric properties are closely related to the frequency of the incident wave. High-frequency waves are more resistant to blackout effects than low frequency waves. Specifically, studies have shown that the transmission attenuation of terahertz wave is much smaller than that of millimeter wave in the same plasma.[33,34] However, to the best of our knowledge, the influence of electron density on refractive index has not been studied yet. Recently, we used the number density and Gladstone–Dale (G–D) constant of electrons to obtain the relationship between refractive index and total density of fluid instead of the G–D formula.[35] Further study showed that the electrons have a significant effect on the refractive index. Based on this conclusion, in the present work we investigate the propagation characteristics of high-frequency waves through turbulent plasma sheath surrounding the hypersonic aircraft.
The structure of this paper is as follows. In Section
In the Cartesian coordinates, the three-dimensional Navier–Stokes (N–S) equations with the appropriate closure models, including thermochemical non-equilibrium, can be written as[36–40]
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Based on the finite volume method, the N–S equations can be solved by the advection upstream splitting method by pressure-based weight function[42] (AUSMPW+). Moreover, the implicit lower–upper symmetric Gauss–Seidel relaxation[43] (LU-SGS) is regard as time marching algorithm.
The commonly used turbulence two-equation models are k–(ε turbulence model, k–ω turbulence model, and shear stress transport (SST) turbulence model. To select a more suitable turbulence model, we use the electron density along the wall of the radio attenuation measurement (RAM) C-II flight experiment with a speed of 23.9 Mach and a height of 61 km. In the same flight scenario, based on the Navier–Stokes equations, different turbulence models are used to simulate the plasma sheath. The electron density along the wall of numerical simulation is compared with the RAM C-II flight experiment data, as shown in Fig.
![]() | Table 1.
Simulation effects of different turbulence models. . |
From Table
It is worth noting that the k–(ε model, which is a two-equation model which employs partial differential equations to govern the transport of the turbulent energy, is suitable for high Reynolds number circumstances, where k and (ε are turbulence kinetic energy and dissipation rate, respectively.
The square root of k is regarded as velocity scale, and the length scale is represented as
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![](cpb_28_7_074203/cpb_28_7_074203_ieqn12.gif)
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In most cases, the air is partially ionized. So the gas mixture is usually considered as N2, O2, N, O, NO, NO+, and
![]() | Table 2.
Chemical reaction equations and rate coefficients.[44] . |
In Table
Next, we give the parameter of flow field as shown in Table
![]() | Table 3.
Boundary values of different flight parameters. . |
In plasma sheaths, the mass of free electron is extremely small, and its contribution to mass density is negligible. However, it shows a special dominant effect for index of refraction in the ionized gas mixture.[39] The dependence of the index of refractive on electron density is completely different from that on neutral density. According to Alpher and Whiteʼs point of view,[45] the index-of-refractive difference
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According to the Cauchy empirical formula, the refractive index of an atom is[47]
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![](cpb_28_7_074203/cpb_28_7_074203_ieqn37.gif)
The contribution of positive ions to the refractive index is
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According to this analysis, the total refractive index difference of the plasma can be expressed as[47]
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![]() | Table 4.
The As and Bs in Cauchy empirical formula. . |
Based on the numerical results given in Section
Figure
Figure
Figure
![]() | Fig. 5. The refractive index fluctuations along the propagation path at ![]() |
![]() | Table 5.
Flight parameters. . |
![]() | Table 6.
Flight parameters. . |
The biggest difference between turbulent ray tracing and laminar ray tracing is that the refractive index interface must be taken in the corresponding delta neighborhood in turbulent ray tracing. According to the refractive index distribution of the flow field, the turbulent refractive index field of the blunt body is approximately circular refractive index field in the delta neighborhood. Based on the principle of the laminar ray tracing, a schematic diagram of the turbulent head and side ray tracing is established on the side of blunt body, as shown in Fig.
The basic principle of turbulent ray tracing is similar to that of conventional laminar ray tracing. In the two-dimensional Cartesian coordinate system, the schematic of turbulent ray tracing is shown in Fig.
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By using the inverse distance weighting algorithm, the refractive index n1, the densities
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Figure
![]() | Fig. 7. (a) The initial incident angle is 35 degrees. (b) The initial incident angle is 40 degrees. ![]() |
Finally, figure
In this paper, based on the calculation and analysis of the turbulent hypersonic plasma flow field, the relationship between refractive index and density has been discussed. It is has been proven that among all the species, the electrons have a dominant effect on the refractive index fluctuations of the turbulent plasma sheath, and the refractive index fluctuation is also closely related to the wavelength and flight parameters. As flight speed increases or flight altitude decreases, the refractive index fluctuation becomes larger. In addition, the longer the incident wavelength, the larger the refractive index fluctuation. However, the G–D constant of electron is negative and that of the other six species is positive. Moreover, the influence of electrons on refractive index is much larger than that of the other six species. Therefore, the larger the density, the smaller the index of refraction of plasma sheath, which is exactly the opposite of aero-optical effect. Finally, the propagation of light in the turbulent plasma sheath is studied. When the incident wavelength is
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