Analysis of aerodynamic and stealth characteristics of aircraft under the action of microwave plasma

This paper investigates the impact of plasma on the aerodynamic and stealth characteristics of hypersonic aircraft. Utilizing the finite volume method, the governing equations of the fluid mechanics were discretized, and the seven-element air chemical reaction model was employed for computations. A technique involving the alteration of the chemical reaction source term rate was employed to replicate the actual plasma concentration distortion phenomena induced by microwaves around hypersonic vehicles, enabling an analysis of the influence of plasma density distortion on the aerodynamic characteristics of these vehicles. In addition, the method of Piecewise Linear JE Recursive Convolution Finite-difference Time-domain (PLJERC-FDTD) is used to calculate the radar scattering cross-section of the target within both the initial and distorted flow fields. This allowed an examination of the variation in the aircraft’s stealth characteristics when exposed to distortion in an ionized environment.


Introduction
The level of research about hypersonic vehicles holds a pivotal position in assessing a country's defense capabilities [1].The high-temperature environment produced by high-speed flight results in the formation of a non-uniform plasma sheath around the aircraft.This plasma sheath has significant implications for the aerodynamic and thermal characteristics of the aircraft [2], and it also has a pronounced impact on the stability and control of high-speed aircraft [3].Simultaneously, plasma exhibits reflection, refraction, and absorption effects on incident electromagnetic waves, significantly altering the electromagnetic scattering characteristics of the aircraft throughout its entire flight.These effects directly impact communication and aircraft detection.Currently, aircraft detection primarily relies on radar technology.The radar scattering cross section (RCS) characterizes the echo intensity generated by the target under radar wave irradiation, which directly determines the stealth performance of the aircraft under radar detection [4].With the change of flight state, the plasma exhibits varying characteristics for the transmission of incident electromagnetic waves.Consequently, the RCS amplitude, radar imaging characteristics, and Doppler characteristics of the hypersonic vehicle enveloped by plasma undergo dynamic fluctuations.These fluctuations present challenges for real-time identification, detection, and localization of the aircraft using radar systems.Therefore, it is imperative to study high-power microwave electromagnetic weapons capable of inducing atmospheric distortion and altering the local ionospheric subenvironment's concentration.Such research has a direct impact on the aerodynamic and stealth characteristics of the aircraft along the electromagnetic wave propagation path Currently, extensive research on radar scattering and aircraft stealth is being conducted internationally.The United States has begun to measure the electromagnetic scattering characteristics of radar targets including hypersonic warheads.Among these, HTV2 stands out as a new generation of hypersonic lifter vehicle jointly developed by the US Department of Defence and the Air Force for experimental verification of hypersonic-related technologies, with a design Mach number of 5-25 and a total flight range of 8, 000 km [5].The European Union and Russia have also launched relevant programs and established comprehensive theoretical simulation test pathways.Persson and Bull [6] also proposed a method to measure and analyze the influence of flight dynamics on the radar cross-section model of an aircraft based on flight data from three different types of aircraft: Piper PA-28 Archer II, Boeing 737, and Saab JAS39 Gripen.Papageorgiou et al. [7] proposed a Multidisciplinary Design Optimization (MDO) framework for the early design phase of Unmanned Aerial Vehicles (UAVs), focusing primarily on trade-offs between mission, stealth, and surveillance performance requirements for calculating radar cross-section (RCS) and sensor performance.
In comparison to foreign countries, China's research on plasma sheath phenomena started relatively late but has witnessed rapid development.Zhao et al. [8] discussed the propagation of electromagnetic waves in the heterogeneous plasma sheath with a double exponential distribution of electron density.Mo et al. [9] used the PLRC-TDTD method to investigate and analyze the absorption and attenuation of electromagnetic waves by colliding plasmas.Liu and Yuan [10] proposed the PLJERC-FDTD method and studied the electromagnetic characteristics of plasma covering targets.Chang et al. [11] studied the electromagnetic scattering characteristics of the flow field around the re-entering blunt cone.Zeng et al. [12] studied the ultra-high-velocity model and its plasma sheath RCS characteristics.Li et al. [13] analyzed the amplitude-frequency characteristics of electromagnetic signals passing through the plasma sheath based on the improved WKB method.To solve the problem of electromagnetic wave propagation in time-varying plasma, Yang et al. [14] introduced the time-varying parameter term into the classical dispersive medium propagation theory to obtain the relationship between the modulation effect, the plasma parameters, and the carrier frequency.Based on PLRC-FDTD and MPI+openMP, Wang et al. [15] studied the electromagnetic scattering characteristics of plasma targets based on parallel computing models.Nie et al. [16] analyzed the electromagnetic scattering characteristics of the flow field around the hypersonic vehicle.Gong et al. [17] discussed the unsolved black barrier problem in near-space vehicles determined by the propagation characteristics of electromagnetic waves within plasma sheaths.Xu et al. [18] studied the enhancement method of radar scattering characteristics of low RCS targets for anti-stealth.Yang [19] proposed a sheath inversion modeling method according to the RAM-C flight test data.Based on the test, he studied the energy transmission interference of different bands.
Currently, both domestic and international research on plasma sheath phenomena has incorporated the PLJERC-FDTD method, and investigations based on this method have been conducted.The integration of electromagnetism and computational fluid dynamics research is of great significance for the design and manufacture of avionics.However, there remains a gap in interdisciplinary research bridging electromagnetism and computational fluid dynamics.This paper addresses this gap by employing a method that modifies the source term of chemical reactions to simulate plasma sheath distortions around aircraft.The research explores the impact of changes in the plasma environment on the aerodynamic characteristics of the aircraft.Additionally, the PLJERC-FDTD method is employed to calculate the relationship between the Radar Cross Section (RCS) and plasma frequency of the aircraft under different incident wave frequencies, and the stealth characteristics of the superb aircraft under the electromagnetic wave propagation path are studied by changing the local concentration of the plasma sub-environment.

Governing equations of flow
The simulation of hypersonic plasma flow requires the construction of momentum conservation equations, energy conservation equations, and mass conservation equations of multi-component gas mixtures [20].In addition, under the assumption of the two-temperature model, it is necessary to establish the conservation relationship between the vibrational and electronic energy of the multi-atomic components to consider the thermodynamic non-equilibrium effect [21][22][23].Consequently, the governing equation employed in this context is the compressible Navier-Stokes equation, which encompasses thermochemical nonequilibrium effects.
 is the density of the gas mixture, i   is the mass generation rate of the component i , , i j D is the diffusion coefficient of component i , vib q is the vibrational heat flux term of the polyatomic molecular component, and i   and vib   are the source term and vibrational energy generation source term respectively.In a finite-rate chemical reaction system, the NR chemical reaction equations containing NS components can be written as:

Physical modeling 2.2.1. Modeling chemical reactions in
Where ji v and " ji v are the stoichiometric coefficients of the positive and reverse chemical reactions, i c is the molar concentration of the component i , and NJ is the sum of the number of components and the catalytic substances (the catalytic substances act as catalysts in the reaction, which can be a linear combination of one or more components)., k represent the positive and reverse chemical reaction rates, which are given by the Arrhenius equations ( k T is control temperature for the reactions).
For a multi-component chemical reaction, determining the rates of component formation involves defining the chemical reaction equations for the system and establishing the corresponding forward and reverse reaction rates.Dunn & Kang [24][25], Park [25][26], Blotter [27], and Gupta [28] have proposed their own reaction and temperature models.Gupta sorted out the Blotter and Dunn & Kang models and gave a table of chemical reaction rates for the eleven components of air, which contained a total of 20 reaction equations.The first 5 reaction equations are typically employed for 5-component chemical reactions, while the first 7 reaction equations are used for 7-component chemical reactions.In this paper, the chemical reaction model of Gupta's 7-component [29] is adopted, and the specific reactions are as follows: Equation ( 3) demonstrates that by altering , f j A and b, j A , we can change the positive and reverse reaction rates accordingly.This adjustment enables us to modify the plasma environment surrounding the aircraft, ultimately achieving the objective of modeling the distortion of the plasma sheath.

Microwave plasma generation and maintenance.
When the field of a high-power electromagnetic wave is stronger than the breakdown threshold of the atmosphere, the strong electric field accelerates the free electrons in the atmosphere to a large enough speed.These high-speed electrons subsequently collide with gas molecules, leading to the ionization of these initially neutral molecules.The avalanche amplification of collision ionization caused by the collision of newly generated electrons and neutral molecules will produce a large amount of plasma in a very short period until the angular frequency of the plasma is close to the incident microwave frequency.The angular frequency is calculated as: When the electromagnetic wave is incident on the plasma, the plasma with uniform density can be treated with a special dielectric, and the Maxwell equation of the electromagnetic wave in the plasma is: IOP Publishing doi:10.1088/1742-6596/2764/1/012005 Where  is the angular frequency of the electromagnetic wave, E is the intensity of the incident electric field, H is the strength of the magnetic field, 0  and 0  are the dielectric constant and permeability in the vacuum, respectively, and   is the relative permittivity of the plasma, which can be solved by the following equation: Under the action of the applied electromagnetic field, the collision frequency of charged particles in the plasma increases, and the electron number density increases.The change in electron number density can be examined through the electron transport equation, which can be expressed as follows: Where e n is the electron density, e  is the electron mobility, e D is the electron diffusivity, and e R is the electron source term, which characterizes the generation and disappearance of electrons caused by the collision reaction within the plasma, can be obtained by the following equation: 1 The applied electromagnetic field not only induces a change in the electron density inside the plasma, but also affects the distribution of other charged particles, including protons, neutrons, and other charged excited ions.These changes can be analysed by using the heavy matter transport equation: Where k  is the molar fraction of the particle k ,  is gas density,  is the average fluid velocity, and k j is the diffusion flux of the particle k , which can be expressed by: , , , , Where , k m D is the average diffusion coefficient of the substances, n M is the average molar mass of the substances, T is the temperature of the gas, T k D is the thermal diffusivity of k particles, k z is the charge number of k particles, and , k m  is the average mobility of k particles.
Due to the directional nature of electromagnetic wave field distribution, the plasma produced by microwaves is stretched in the direction of the electric field to form a filamentous structure.Once the plasma starts exerting a shielding effect, the incident microwave gets reflected, causing the total electric field (comprising the incident and reflected field) to be enhanced in front of the original filamentous structure, pointing towards the microwave source.The enhanced electric field initiates a new ionization amplification process, leading to the formation of new filamentous structures.As a result, we observe an increasing number of filamentous structures aligned in the direction of the microwave source, as illustrated in Figure 1.

Discrete of the governing equations of flow
Integrating the compressible Navier-Stokes equation considering thermochemical non-equilibrium effects in an arbitrary finite control volume yields the following equation: Where v P and P denote the viscous and non-viscous fluxes of the control body units, respectively.
( ) Applying the Gauss equation, it can be written as: Where V is the control volume of the body, S is the control surface area of the body, and n is the vector of external normal units above the control surface.
In the lattice-based Finite Volume Method (FVM), the physical quantities are defined at the center of the grid element, and their average volume within the grid element is expressed as: The governing equation that can be discretized by transformation is:

P P n P P n P P n P P n P P n P P n (17)
In the calculated coordinate system, there are: The finite volume discrete equation of the flow governing equation in the computational coordinate system can be obtained:

Electromagnetic scattering characteristics and calculation of plasma flow field
The main characteristics of the plasma sheath include two parameters: plasma frequency and plasma collision frequency.The plasma frequency p f also known as the Langmuir frequency, which is the sum of the frequency of the electron oscillation pe f and the frequency of the ion oscillation pi f .
In ionized plasma, the electron number density and ion number density are equal.However, due to the significantly smaller mass of electrons compared to ions, and the substantially higher oscillation frequency of electrons (usually three to four orders of magnitude greater than the ion oscillation frequency), the plasma frequency is usually approximated as the electron oscillation frequency.
At a given incident electromagnetic wave frequency, when the actual electron number density in the flow field is below the critical electron number density, the electromagnetic wave can still be attenuated during transmission.In this case, the degree of attenuation of the electromagnetic wave depends primarily on the frequency of collisions between the particles.Collisions between electrons and ions and between electrons and neutral particles are most frequent in the plasma sheath, so the plasma collision frequency is usually the sum of these two collision frequencies.Drawing from the hardball collision model, the collision frequencies between electrons and neutral particles in the non-magnetized plasma can be expressed as follows: m n cm  is the number density of all neutral particles in the flow field and T is the temperature of the gas.

PLJERC-FDTD algorithm
The PLJERC-FDTD method [10,15] used in this paper is an algorithm that combines the piecewise linear recursive convolution time-domain finite-difference algorithm and the JE Convolution Finitedifference Time-domain (JEC-FDTD) [30][31] finite-difference algorithm, which takes the system of Equation ( 15) as the starting point.
Where J is the polarized current density.After obtaining the convolutional expression of the time- domain current density, the processing of the electric field strength is based on the processing method of JEC-FDTD.The calculation program uses Cartesian grids to partition the object's surface and flow field.The definition of incident angle  , double station angle  , and polarization angle  are expressed by a spherical coordinate system.The Radar Cross Section (RCS) is calculated by using the following equation: (dBsm) ( )

Grid irrelevance verification
At present, most scholars analyze the grid Reynolds number Re  as an important parameter when carrying out heat flow calculations [32][33][34], and the grid Reynolds number Re  is defined as shown in Equation ( 24): Re l Where   denotes the viscosity coefficient of incoming motion and l  represents the characteristic scale, usually taking the grid spacing of the first layer in the normal direction of the wall.
Usually, when Re  is between 0.1 and 10, the first layer of grid spacing is more appropriate, but the smaller the grid Reynolds number is, the higher the computational cost is, so it is necessary to take into account both the accuracy and efficiency of the calculation.Taking the 50 km working condition as an example, three sets of grids with different spacing between the first layers are selected for irrelevance verification, and the grid parameters are shown in Table 1.Table 1.The grid reynolds number of different first layer grid spacing.

Incoming stream Mach number
The first layer of grid spacing (m) y  50 79.7818 5e -5 17.8971 1e -5  3.5789 1e -6  0.3579 Table 2 is the calculation result table of grid irrelevance verification, and it is evident from Table 2 that the aerodynamic data remains consistent even with different first-layer grid spacing.The error is less than 1.5%.Given the balance between computational accuracy and efficiency, the subsequent research in this paper utilizes a first-layer grid spacing of 1e -5 .
Table 2. Grid independence verification calculation results.
The first layer of grid spacing (m)

Aerodynamic characteristics of HTV2 plasma sheath after distortion
As shown in Figure 2, the aspect ratio of the HTV-2 model is 4:1. Figure 3 illustrates that the number of full-mode grids is 3.52 million.
Figure 2. HTV2 model.Figure 3. Computational grid.Table 3 shows the distortion calculation conditions.According to the 8 positive and reverse reactions within the Gupta 7-component reaction model, the plasma distribution of the aircraft can be altered by changing the chemical reaction source term, and the plasma sheath distortion can be simulated.This study employs a method of adjusting the frequency factors f , j A and b, j A in the Arrhenius equation, simultaneously modifying the rate of positive and reverse chemical reactions.The research explores the effects of varying these magnitudes.The results of the aerodynamic coefficient are shown in Table 4, where +10/-10/0 indicates that the reaction rate is expanded by 10 times, reduced by 10 times, and does not change, respectively.The aerodynamic calculation results in Table 4 are dimensionless, and the reference area is set to 1 m and the reference length to 1 m.The percentage change in aerodynamic force is shown in Table 5 by calculating the change in aerodynamic coefficient under alternative chemical reaction rates.These results were obtained from the baseline (0/0) calculation.6 presents the aerodynamic calculation results when the positive and reverse reaction rate is +100/-100, while Table 7 displays the percentage change of aerodynamic force at the positive and reverse reaction rates of +100/-100.Upon reviewing Table 4, Table 7, and Figure 6, it becomes evident that after altering the chemical reaction source term, the plasma sheath structure changes greatly, but the changes in aerodynamic force are not obvious.

Eigenfrequencies of HTV2 plasma sheath after distortion
The plasma sheath is primarily determined by the plasma frequency and the plasma collision frequency, meaning that the alterations in the two parameters characterize changes in the properties of the plasma sheath.Consequently, a change in eigenfrequency will directly lead to a change in stealth characteristics.Figure 7, Figure 8, Figure 9, and Figure 10 depict the plasma frequency and plasma collision frequency contours of both the initial flow field and the plasma sheath distortion flow field.Through comparison, it becomes evident that the plasma frequency undergoes significant changes following the distortion of the plasma sheath, while the plasma collision frequency remains relatively stable.Owing to the shock effect, plasma accumulates at the aircraft's leading edge.The shock wave occurs intermittently, and the variations in the flow field are especially pronounced in this area.
The change of eigenfrequency following plasma sheath distortion demonstrates that it is feasible to simulate plasma sheath distortion by using the method of changing the chemical reaction source term used in this study.

Stealth characteristics of HTV2 plasma sheath after distortion
The validation of the PLJERC-FDTD algorithm is extensively outlined in [11,17].Illustrated in Figure 11, the schematic diagram delineates the segmentation of the HTV2 electromagnetic model.The target area comprises 199×47×102 three-dimensional grids, utilizing Gaussian pulse incidence, a time factor of 0.5, a calculation time step of 6000, and a maximum calculation frequency of 2 GHz for computing the target RCS. .

Figure 11.
Electromagnetic model division of HTV2. Figure 12 illustrates the RCS calculated by VV copolarization in the P-band.Compared with the initial flow field, the plasma sheath distortion does not cause a drastic change in RCS when the incident frequency is lower than 1 GHz.However, the RCS of the distorted flow field is significantly reduced compared with the initial flow field when the incident frequency is higher than 1 GHz, with a reduction of 10-20 dBm.
Figure 13 represents the variation of RCS with a double station angle at 2 GHz frequency with 90   o .It can be seen from the figure that the RCS of the distorted flow field decreases significantly compared with the initial flow field at about  = 90°, 180°, and 270°, while it increases at 0°.This behavior may be attributed to the high-pass filtering property of the plasma, and near the aircraft's leading edge with a large plasma concentration, its reflection of electromagnetic waves is enhanced, resulting in larger RCS values than those observed in the initial RCS.

Conclusion
Based on the results and discussions presented above, the conclusions are obtained as below: (1) For the simulation calculation of hypersonic vehicles, altering the chemical reaction rate effectively replicates the plasma sheath distortion around the aircraft.(2) When the chemical reaction rate varies by up to 100 times, the plasma sheath distortion has no substantial effect on the shock wave structure, resulting in less than 3% change in flight state and less than 0.5% change in aerodynamic force-thus, presenting an overall small effect.
(3) Utilizing the PLJERC-FDTD method for calculating the RCS of HTV2 in the P-band reveals that within a certain range, the RCS of the distorted flow field is significantly smaller than that of the initial flow field, with a reduction of 10-20 dBm.Similarly, two-station RCS also experiences significant reduction.

.
a plasma environment.Chemical kinetic models are the focus of plasma flow field simulations.For hypersonic flow, the commonly used air response models at home and abroad mainly include 5-component, 7-component, and 11-component models.The 5-component reaction model is mostly used for the chemical reaction flow fields without gas ionization.The 7component model is mostly used for the high-speed flow fields with a velocity of about 7 km/s, and the components involved The 11-component reaction model is mostly used for the re-entry flight flow field with a velocity of about 11 km/s and usually focuses on the study of radiation phenomena.

Figure 1 .
Figure 1.Simulation results of two-dimensional density distribution in microwave plasma.

Figure 4 10 Figure 4 .
illustrates the contour of the change values of different chemical reaction sources, while depicts the contour of the distortion of the plasma sheath caused by microwave injection.Under the premise of the same hierarchical display of electron number density, the change direction of the chemical reaction source term is determined by comparing the pneumatic calculation result diagram with the microwave injection result diagram.The comparison between Figure 4 and 5 shows that when the positive and reverse reaction rates are +10/-10, there is closer alignment with the actual changes induced by microwave injection in the distribution of plasma sheath sleeves.Therefore, the plasma sheath at the positive and reverse reaction rates of +100/-100 is calculated.The plasma sheath and shock wave structure are shown in Figure 6.(a) positive/reserve reaction=+10/+10 (b) positive/reserve reaction=+10/-10 (c) positive/reserve reaction=+10/+0 (d) positive/reserve reaction=-10/+0 IOP Publishing doi:10.1088/1742-6596/2764/1/01200511 (e) positive/reserve reaction=-10/-Different chemical reaction source term change value simulation calculation.

Figure 6 .
Figure 6.Positive and reverse reaction rate +100/-100 plasma sheath and shock wave structure.Table6presents the aerodynamic calculation results when the positive and reverse reaction rate is +100/-100, while Table7displays the percentage change of aerodynamic force at the positive and reverse reaction rates of +100/-100.Upon reviewing Table4, Table7, and Figure6, it becomes evident that after altering the chemical reaction source term, the plasma sheath structure changes greatly, but the changes in aerodynamic force are not obvious.Table6.Aerodynamic force of positive and reverse reaction rate +100/-100.

Figure 7 .
Figure 7. Plasma frequency of initial flow field.

Figure 8 .
Figure 8. Plasma collision frequency of initial flow field.

Figure 9 .
Figure 9. Plasma frequency of distorted flow field.

Figure 10 .
Figure 10.Plasma collision frequency of distorted flow field.

Table 4 .
Results of aerodynamic calculation.

Table 5 .
Percentage change in aerodynamic force.the chemical reaction source term leads to a substantial alteration in the gas composition at the local end position of the aircraft.This, in turn, affects local gas density, resulting in

Table 7 .
Percentage change in aerodynamic force with positive/negative reaction rate of +100/-100.
124. Analysis of stealth characteristics