An Aero-Optical Effect Analysis Method in Hypersonic Turbulence Based on Photon Monte Carlo Simulation

Abstract

Aero-optical effects caused by hypersonic turbulence will affect the accuracy of optical sensors on aircraft. Traditional analysis methods, which do not consider absorption and scattering effects, cannot easily be used to completely describe the transmission process of light in hypersonic turbulence. In this paper, an aero-optical effect analysis method based on photon Monte Carlo simulation (MC-AOEA) was proposed to explain the distortion characteristics of aero-optical effects from the perspective of photon statistics. The energy distribution of photons in the transmission process was determined by taking a photon packet as a unit, and the microscopic statistics of the photon dissipation energy for all photon packets were calculated. The effectiveness of this method was verified by comparing the photon statistical parameters with the traditional optical distortion physical quantities. MC-AOEA was used to analyze the distortion of aero-optical effects at different altitudes and speeds. Additionally, the simulation results showed that, with the reduction in flight altitude and the enhancement of speed, the distortion of aero-optical effects was aggravated, and the energy loss was more serious, which provides a reference for the evaluation of aero-optical effect errors.

1. Introduction

With growing demand, especially in military applications, high-speed aircraft have become an indispensable and cutting-edge technology [1,2]. Aircrafts rely more and more on the accuracy and autonomy of information acquisition, and the application of optical sensors has become an inevitable trend in the future [3,4,5,6]. However, when high-speed aircraft fly in the atmosphere, complex turbulence formed between the optical window and the incoming flow will cause serious interference to the transmitted light, resulting in distortion phenomena, such as deviation and blurring of the image received by optical sensors [7,8,9,10]. Therefore, research on aero-optical effects is an important challenge for the application of optical sensors in high-speed aircraft.

At present, the research on aero-optical effects mainly focuses on two aspects. One is to study the distortion of received images based on image analysis theory. The premise of this method is that there must be target images, and this method acquires the point spread function (PSF) of the received images to analyze diffusion and blurring, and then this method compensates for the deviation of the optical sensors. The second aspect is to study the light transmission mechanism in hypersonic turbulence based on the flow field theory. In recent years, many scholars have analyzed the distortion characteristics of aero-optical effects through wind tunnel tests and summarized the corresponding distortion regularities. The team of Gordeyev, S and Wyckham, C has obtained the statistical law of optical path difference of turbulent boundary layer through experiments and numerical processing [11,12]. However, this method can only analyze the optical images received. In order to further study the actual process of light transmission in high-speed turbulence, some scholars have used the ray tracing method based on the turbulence density field to analyze aero-optical effects, which is also currently the most commonly used method [13,14]. The density fluctuation field of compressed air flow around the optical window of high-speed aircraft is obtained through experiments or calculations. The refractive index field near the turbulent boundary layer (TBL) is taken for analysis. The phase distortion caused by aero-optical effects was studied using the ray tracing method based on geometric optics, and the optical path difference (OPD) and Strehl ratio (SR) were used to describe distortion of the image [15,16]. In the latest research, Mathews, E and Guo, G also used the macro ray tracing method in the analysis of aero-optical effects [17,18]. However, this method is based on the ideal model of light, without considering the absorption and scattering of gas. Therefore, although the traditional aero-optical effects analysis method can basically describe the distortion law, it cannot fully reflect the optical distortion process in real environments [19,20].

The essence of aero-optical effects was the interaction between photons and gas molecules in hypersonic turbulence, namely, absorption, scattering, and stimulated radiation [21]. Stimulated radiation was not considered in this study because the light source considered was mainly the optical detection signal, which is a weak light source. The photon model instead of the traditional ideal ray model could be utilized to analyze the transmission of light in turbulence in more detail. In recent years, the way to solve the radiation equation of light has been developed [22,23]. However, the three-dimensional radiative transfer equation (RTE) of photons is complicated to solve, especially after adding time-varying turbulence molecules. Therefore, we determined a more convenient and effective method, although it requires intensive calculations. Photon Monte Carlo simulation is an important method to realize photon transmission simulations. Some scholars have proven the correctness of the Monte Carlo simulation method for photon transmission simulations, which has been applied in medical and optical fiber fields [24,25,26,27,28]. Therefore, in order to study the distortion characteristics of aero-optical effects from the perspective of photons, an aero-optical effects analysis method based on photon Monte Carlo simulation (MC-AOEA) is proposed to describe the perturbation of high-speed turbulence on photons, as shown in Figure 1.

The contribution of this paper is to apply the photon Monte Carlo simulation to the analysis of aero-optical effects. The influence of aero-optical effects was studied through the photon transmission in turbulence, and the distortion process of aero-optical effects was determined from the perspective of statistical analysis. The aero-optical effect analysis method proposed in this paper can analyze the distortion rule of aero-optical effects at different flight altitudes and speeds. Although this method increases the amount of calculation, it provides a reference for the more accurate evaluation of aero-optical effect errors.

The remainder of this paper is organized as follows. In Section 2, the transmission process of photons in hypersonic turbulence is presented. The design for MC-AOEA is described in Section 3; in Section 4, the simulation performance of MC-AOEA is verified, and the distortion characteristics of aero-optical effects at different flight altitudes and Mach numbers are analyzed. Finally, the conclusions are drawn in Section 5.

2. Transmission Process of Photons in Turbulence

This section introduces the transmission process of photons in hypersonic turbulence based on Monte Carlo simulation, mainly analyzing the refraction, reflection, absorption, and scattering of photons in the transmission process, as shown in Figure 2. A theoretical basis for the subsequent design of MC-AOEA is presented.

2.1. Refraction and Reflection of Photons

The refraction and reflection of photons can be explained as the synthesis of partial scattering by the Feynman quantum electrodynamics theory. However, in order to facilitate the simulation of photon transmission, the scattering part that can be combined into refraction and reflection is treated as a single part, which conforms to the traditional macro-scale method. Both refraction and scattering are related to the refractive index characteristics of turbulent molecules. In order to facilitate the description of absorption, the refractive index, and $\tilde{n}$ is expressed in the form of Lorenz dispersion theory, which can be written as follows [1]:

$$\tilde{n}=n_R+i{n}_I$$

(1)

where $n_R$ and $n_I$ are the real part and the imaginary part of the refractive index, respectively. According to the Gladstone–Dale law, we can derive the relationship between the density, $\rho$, and the real part of the refractive index, which is as follows [29]:

$$n_R=1+K_{GD}\cdot \rho$$

(2)

where $K_{GD}$ is the Gladstone–Dale constant, and the relationship between $K_{GD}$ and the wavelength, $\lambda$, is as follows:

$$K_{GD}=2.2244\times {10}^{-4}\left[1+{\left(6.7132\times {10}^{-8}/\lambda \right)}^2\right]{\mathrm{m}}^3/\mathrm{kg}$$

(3)

The refraction and reflection models selected in this paper conformed to Snell’s law and Fresnel’s law, respectively, and the relationship between the incidence angle, ${\theta }_i$, and the exit angle, ${\theta }_j$, was as follows:

$$\begin{array}{cc}Reflection:& {\theta }_i={\theta }_j\\ Refraction:& n_{R,i}\sin {\theta }_i=n_{R,j}\sin {\theta }_j\end{array}$$

(4)

where ${\theta }_i$ and ${\theta }_j$ are the boundary incident angle and exit angle, respectively. $n_{R,i}$ and $n_{R,j}$ are the macro refractive index on both sides of the boundary.

2.2. Absorption and Scattering of Photons

Photons passing the hypersonic turbulence have a certain probability of being absorbed or scattered by gas molecules. The quantities describing absorption and the scattering process are defined as the absorption coefficient, ${\mu }_a$, and the scattering coefficient, ${\mu }_s$, which indicate the probability that photons will be absorbed or scattered at a certain transmission distance, $ds$.

The number of molecules involved in hypersonic turbulence is very large; thus, it is difficult to be accurate to the molecular level in order to save computational resources in the actual simulation process. Therefore, we used the complex refractive index and molecular number density to describe the absorption coefficient and scattering coefficient. The relationship between $n_R$ and $n_I$ in Equation (1) satisfies the following:

$$\{\begin{array}{l}{n}_R^2-n_I^2=1+\frac{N{e}^2}{{\epsilon }_0{m}_{\mathrm{e}}}\frac{{\nu }_0^2-{\nu }^2}{{\left({\nu }_0^2-{\nu }^2\right)}^2+{\gamma }^2{\nu }^2}\\ 2n_R{n}_I=\frac{N{e}^2}{{\epsilon }_0{m}_{\mathrm{e}}}\frac{\gamma \nu }{{\left({\nu }_0^2-{\omega }^2\right)}^2+{\gamma }^2{\nu }^2}\end{array}$$

(5)

where $\gamma$ is the damping coefficient, which satisfies $\gamma =e^2{\nu }_0^2/6\pi {\epsilon }_0{m}_{\mathrm{e}}{c}^3$; $e$ is the electronic charge; ${\nu }_0$ is the natural frequency of the polarized oscillator; ${\epsilon }_0$ is the dielectric constant of vacuum; $m_{\mathrm{e}}$ is the electronic mass; $c$ is the speed of light; $N$ is the molecular number density; and $\nu$ is the photon frequency. According to Lorentz dispersion theory, the relationship between the absorption coefficient of photons and the imaginary part of refractive index is as follows:

$${\mu }_a=2\nu n_I/c=\frac{N{e}^2}{4m_{\mathrm{e}}{\epsilon }_{0}c}\frac{\gamma }{{\left({\nu }_0-\nu \right)}^2+{(\gamma /2)}^2}$$

(6)

The scattering of photons in hypersonic turbulence depends on the scale of turbulent molecules. The anisotropic scattering of large-scale turbulent structures can be combined into refractive components using the Feynman quantum electrodynamics theory. Therefore, this study mainly considered the Rayleigh scattering of small-scale turbulent structures, which refer to isotropic scattering. According to the photon scattering theory, the Rayleigh scattering cross-section of a single molecule is as follows:

$${\sigma }_{\mathrm{s}}(\lambda )=\frac{8\pi }{3}\left[\frac{{\pi }^2{\left(n_R^2-1\right)}^2}{N^2{\lambda }^4}\right]$$

(7)

The actual photon transmission process is calculated by grid discretization; therefore, the analysis of the interaction between a single molecule and photons cannot be guaranteed. Thus, the scattering coefficient in turbulence is expressed as:

$${\mu }_s=N{\sigma }_{\mathrm{s}}=\frac{8\pi }{3c^4}\left[\frac{{\pi }^2{\left(n_R{}^2-1\right)}^2{\nu }^4}{N}\right]$$

(8)

3. Design of MC-AOEA

This section presents the design of the aero-optical effects analysis method based on photon Monte Carlo simulation. According to the interaction principle between photons and turbulent molecules, the process of photons passing through hypersonic turbulence is tracked. The design of MC-AOEA is based on the following assumptions.

(a)

Photon transmission is seen as the refraction, reflection, absorption, and scattering of a group of photons in turbulence, excluding the photon-stimulated radiation.

(b)

The absorption and scattering of photons are determined by the absorption coefficient, scattering coefficient, and phase function reflecting the scattering distribution (used to determine the direction of scattering).

(c)

The polarization and interference of photons are ignored, and only the transmission of photon energy in turbulent molecules is tracked.

MC-AOEA can calculate the simulation results of a large number of photons, obtain the photon energy distribution in turbulence, and then analyze the distortion of aero-optical effects.

3.1. Photon Probability Function and Photon Step Size

Photon Monte Carlo simulation depends on the properly defined probability density function and cumulative distribution function. The probability density function, $p(s)$, is defined as the probability of the absorption or scattering of photons through distance, $s$, in this paper. According to the Beer–Lambert law, $p(s)$ is written as follows:

$$p(s)={\mu }_t{e}^{-{\mu }_{t}s},0<x<\infty$$

(9)

$${\displaystyle {\int }_0^{\infty }p}(s)ds=1$$

(10)

where ${\mu }_t={\mu }_s+{\mu }_a$. The cumulative distribution function, $P(s)$, is the probability that the specified event occurs between 0 and $s$ (Because the lowest possible value of $s$ in this article is 0), which is expressed as follows:

$$P(s)={\displaystyle {\int }_0^{s}p}(s)ds=1-e^{-{\mu }_{t}s}, 0\le P(s)\le 1$$

(11)

In this paper, $s$, is defined as the photon step size of Monte Carlo simulation, and a random number, $\xi$, is selected from the uniform distribution, $[0,1]$, to make it equal to the cumulative distribution function, $P(s)$; then, the current photon step size can be calculated as follows:

$$s=\frac{-\ln (1-\xi )}{{\mu }_t}$$

(12)

3.2. Update of Photon Position

For a photon transmitted in rectangular coordinate system, $\mathit{x}-\mathit{y}-\mathit{z}$, its initial position and transmission direction are ${\mathit{r}}_0=(x_0,y_0,z_0)$ and ${\mathbf{\Omega }}_0={[{\eta }_{x0},{\eta }_{y0},{\eta }_{z0}]}^T$, respectively, as shown in Figure 3. The local coordinate system ${\mathit{e}}_x-{\mathit{e}}_y-{\mathit{e}}_z$, is obtained via the translation of the coordinate system $\mathit{x}-\mathit{y}-\mathit{z}$.

The unit direction vector cosines of photons, ${\eta }_{x0}$, ${\eta }_{y0}$ and ${\eta }_{z0}$ are projections of the direction vector on coordinate axes ${\mathit{e}}_x-{\mathit{e}}_y-{\mathit{e}}_z$, which can be expressed as follows:

$$\{\begin{array}{l}{\eta }_{x0}=\sin \theta \cos \phi \\ {\eta }_{y0}=\sin \theta \sin \phi \\ {\eta }_{z0}=\cos \theta \end{array}$$

(13)

Given the transmission direction, ${\mathbf{\Omega }}_0={[{\eta }_{x0},{\eta }_{y0},{\eta }_{z0}]}^T$, and the photon path length, $s$, the photon can be pushed from its previous position ${\mathit{r}}_0=(x_0,y_0,z_0)$ to its next position ${\mathit{r}}_1=(x_1,y_1,z_1)$, which can be expressed as follows:

$$\begin{array}{l}{x}_1=x_0+{\eta }_{x0}s\\ y_1=y_0+{\eta }_{y0}s\\ z_1=z_0+{\eta }_{z0}s\end{array}$$

(14)

3.3. Update of Photon Direction

When photons scatter, the scattering direction is described by the polar scattering angle, ${\theta }_s$, and the azimuthal scattering angle, ${\phi }_s$, as shown in Figure 3. Rayleigh molecular scattering was considered to be isotropic scattering based on the scale of turbulent molecules in this study. The polar scattering angle, ${\theta }_s$, and the azimuthal scattering angle, ${\phi }_s$, are independent; therefore, for any random numbers, ${\xi }_1\in [0,1]$ and ${\xi }_2\in [0,1]$, the polar scattering angle, ${\theta }_s$, and the azimuthal scattering angle, ${\phi }_s$, can be written as follows:

$$\begin{array}{l}\cos {\theta }_s=1-2{\xi }_1\\ {\phi }_s=2\pi {\xi }_2\end{array}$$

(15)

Given the polar scattering angle, ${\theta }_s$, and the azimuthal scattering angle, ${\phi }_s$, the updated direction vector, ${\mathbf{\Omega }}_1={[{\eta }_{x1},{\eta }_{y1},{\eta }_{z1}]}^T$, can be calculated, which can be expressed as follows:

$${\mathbf{\Omega }}_1=\sin {\theta }_s\cos {\phi }_s{\mathbf{\Omega }}_{0,\perp }\times {\mathbf{\Omega }}_0+\sin {\theta }_s\sin {\phi }_s{\mathbf{\Omega }}_{0,\perp }+\cos {\theta }_s{\mathbf{\Omega }}_0$$

(16)

where ${\mathbf{\Omega }}_{0,\perp }$ is the unit vector perpendicular to ${\mathbf{\Omega }}_0$. ${\mathbf{\Omega }}_{0,\perp }$ is not unique; therefore, there is no unique equation for updating the direction cosine. In this study, a unit vector on the plane $\mathit{x}-\mathit{y}$ was chosen as ${\mathbf{\Omega }}_{0,\perp }$, which can be expressed as follows:

$${\mathbf{\Omega }}_{0,\perp }=\pm {[-{\eta }_{y0},{\eta }_{x0},0]}^T/\sqrt{1-{\eta }_{z0}^2}$$

(17)

$${\mathbf{\Omega }}_{0,\perp }\times {\mathbf{\Omega }}_0=\pm {[{\eta }_{x0}{\eta }_{z0},{\eta }_{y0}{\eta }_{z0},-(1-{\eta }_{z0}^2)]}^T/\sqrt{1-{\eta }_{z0}^2}$$

(18)

Then, the updated direction can be written as follows:

$$\{\begin{array}{l}{\eta }_{x1}={\eta }_{x1}\cos {\theta }_s+\sin {\theta }_s\left({\eta }_{x0}{\eta }_{z0}\cos {\phi }_s-{\eta }_{y0}\sin {\phi }_s\right)/\sqrt{1-{\eta }_{z0}^2}\\ {\eta }_{y1}={\eta }_{y0}\cos {\theta }_s+\sin {\theta }_s\left({\eta }_{y0}{\eta }_{z0}\cos {\phi }_s+{\eta }_{x0}\sin {\phi }_s\right)/\sqrt{1-{\eta }_{z0}^2}\\ {\eta }_{z1}={\eta }_{z0}\cos {\theta }_s-\sqrt{1-{\eta }_{z0}^2}\sin {\theta }_s\cos {\phi }_s\end{array}$$

(19)

If the direction of the photon is very close to the $\mathit{z}$ axis (for example, ${\eta }_{z0}>0.9999$), the updated direction can be written as follows:

$$\{\begin{array}{l}{\eta }_{x1}=\sin {\theta }_s\cos {\phi }_s\\ {\eta }_{y1}=\sin {\theta }_s\sin {\phi }_s\\ {\eta }_{z1}=sign({\eta }_{z0})\cos {\theta }_s\end{array}$$

(20)

3.4. Update of Photon Weight

In the process of photon transmission, it is very complex to simulate photons one by one, and the amount of calculation is very large. Therefore, the actual simulation process takes a photon packet containing multiple photons as a unit, and the photons in the photon packet follow the same path. When these photons travel in turbulence molecules, some of them will be absorbed. The method to measure the number of photons absorbed is to assign a weight, $\omega$, to the photon packets. This weight is initially set to $\omega =1$. The probability of photon absorption can also be considered as the percentage of absorption. Therefore, the weight of photon packets can be reduced by the ratio to simulate the absorption of photons by turbulence molecules. The weight is updated as follows:

$${\omega }_1={\omega }_0{e}^{-{\mu }_a{s}_0}$$

(21)

Given a weight threshold, ${\omega }_e$, when $\omega <{\omega }_e$, it is considered that the photon packet has been fully absorbed, and the simulation of the photon packet will be stopped at this time. In addition, this study used a roulette model. When $\omega$ is small enough, if the absorption of the photon packet is directly determined, the results will be inaccurate. In order to avoid premature elimination of these photons, the roulette model based on the roulette parameter, $m$, is used to determine the survival of photon packets. The roulette model conforms to the principle of the Monte Carlo method, using a random number, ${\xi }_m\in [0,1]$. When the random number, ${\xi }_m$, is greater than $1/m$, the photon packet is determined to be discarded; otherwise, the weight, $\omega$, is multiplied by $m$ and the photon packet continues moving. In the simulation process of this paper, $m$ is taken as 10.

3.5. Grid Boundary of Photon Transmission

In the process of photon Monte Carlo simulation, the data of turbulent flow field where photons are located are divided into grids, whose physical quantities, such as the refractive index, absorption, and scattering coefficient, adopt the grid interpolation data of computational fluid dynamics (CFD). Snell’s law and Fresnel’s law are adopted for refraction and reflection of the grid boundary, respectively. Notably, when calculating the incidence angle, if a photon just touches an angle or edge of the grid, the incidence angle is not defined theoretically. Because the definition of photon boundary is vague. If the photons are very close to the corner or edge, the uncertainty will be broken by selecting the first grid through which the photons will travel in a straight line along the original direction. This makes it easy to determine the next grid after the photon changes direction at the boundary.

Compared with the traditional photon simulation, the scattering and absorption coefficient distribution inside the simulation threshold used in this paper is uneven because the transmission medium is a dynamic turbulent flow. Therefore, a grid sometimes cannot contain a photon step size, $s$, obtained by random sampling. At this time, it is necessary to split the photon step size, $s$, as shown in Figure 4. The absorption and scattering coefficients of different grids are different. Therefore, in the transmission process, it is necessary to track the step size, $s_k$, of photons passing through different grids, which satisfies the following condition:

$${\displaystyle \sum _{k=1}^P{\mu }_{tk}{s}_k=-\ln (1-\xi )}$$

(22)

where ${\mu }_{tk}{s}_k$ represents the step size contribution within a grid.

In summary, for the transmission of a photon packet, the flow chart of MC-AOEA is shown in Figure 5.

3.6. Physical Description of Aero-Optical Effects Based on MC-AOEA

In the traditional physical quantities of aero-optical effects, the optical path difference (OPD) and Strehl ratio (SR) obtained by the geometric ray tracing method are generally used to describe the aero-optical distortion. In this section, based on MC-AOEA, a photon evaluation system is established to describe the photon perturbation process generated by aero-optical effects.

(1)

Photon statistical optical path difference (PS-OPD)

The PS-OPD takes a certain transmission plane as an object. Assuming that $M$ photon packets with the vector $\mathit{r}$ arrive at a certain reception plane, $\mathsf{\Lambda }$, the PS-OPD at the point is the mean of multiplying the OPD of each photon packet by its weight, which can be expressed as follows:

$$PS-OPD(\mathit{r},\nu )=\frac{1}{M}{\displaystyle \sum _{i=1}^{M}OP{D}_i(\mathit{r},\nu )\times {\omega }_i(\mathit{r},\nu )},\mathit{r}\in \mathsf{\Lambda }$$

(23)

Compared with the physical description of phase distortion using the optical path difference (OPD) in traditional geometric optics, the description of PS-OPD is statistically significant, and the error of OPD caused by reflection, absorption, and scattering is considered, which is more realistic.

(2)

Energy loss of photons ($E_{loss}$)

The $E_{loss}$ is used to describe the energy loss of photons in turbulent molecules. Assuming that $T$ photon packets in total arrive at a certain reception plane, $\mathsf{\Lambda }$, and that the scattering does not change the frequency of photons during transmission (without considering Raman scattering), the $E_{loss}$ at the certain plane is the ratio of the number of photons on the certain plane to the initial number of photons, which can be expressed as follows:

$$E_{loss}(\mathsf{\Lambda },\nu )=\frac{{\displaystyle \sum _{i=1}^T{N}_p\times {\omega }_i(\mathit{r},\nu )}}{N_0},\mathit{r}\in \mathsf{\Lambda }$$

(24)

where $N_p$ is the initial number of photons for each photon packet and $N_0$ is the initial photons of all photon packets. Compared with the traditional quantity of SR, which only describes the maximum light intensity, the $E_{loss}$ considers the interaction between photons and gas molecules and describes the energy loss in the actual transmission process more comprehensively.

4. Simulation and Analysis

In this section, the MC-AOEA presented in Section 3 is simulated and verified. The hypersonic turbulence was obtained according to the large eddy simulation (LES) [30]. Photon Monte Carlo simulation was used to analyze the aero-optical effects of photon transmission in hypersonic turbulence. Compared with the traditional ray tracing method, the advantages of MC-AOEA are obtained. Finally, by analyzing optical distortion under different flight conditions, the distortion rule of aero-optical effects from the perspective of photons was obtained.

4.1. Hypersonic Turbulence by LES

In this paper, the LES was used to obtain the hypersonic turbulence; the geometry of the aircraft is shown in Figure 6. ${\mathit{o}}_b{\mathit{x}}_b{\mathit{y}}_b{\mathit{z}}_b$ and ${\mathit{o}}_w{\mathit{x}}_w{\mathit{y}}_w{\mathit{z}}_w$ represent the vehicle body coordinate system and window coordinate system, respectively. The head angle of the aircraft was 13.5°, and the structure size of the optical window was 185 × 125 mm².The optical sensor receiving size was 80 × 80 mm².

In order to satisfy the accuracy of photon transmission simulation, the flow field simulation threshold of the aircraft was subdivided into grids, as shown in Figure 7.

The total number of cells was 18.95 million, and the thickness of the first layer of the optical window was 0.1 mm. In order to verify the grid independence, different numbers of grids were used for LES simulations. The weighted average velocity of section ${\mathit{y}}_w-{\mathit{z}}_w$ above the window center was taken as the sampling standard. Most coarse grid ($3.701\times {10}^4 cells$), coarse grid ($1.406\times {10}^5 cells$), medium grid ($2.961\times {10}^6 cells$), slightly fine mesh ($2.368\times {10}^6 cells$), fine grid ($1.895\times {10}^7 cells$), and most fine grid ($1.516\times {10}^8 cells$) were employed to verify the grid independence, as shown in Figure 8. The degree to which the average speed changes with the increase of the number of meshes was less than 0.1% using the fine grid; therefore, to ensure the calculation accuracy and save the calculation amount, the number of grids is determined as 18.95 million.

We used Ansys 2021R2 commercial software to conduct the simulation of the hypersonic turbulence with LES, with the following simulation conditions: flight altitude of 20 km; Mach number of 3.8; angle of attack of 0°; and time step of 10⁻⁷ s. In order to ensure the accuracy of simulation, we used a 256-core high-performance server to calculate the hypersonic turbulence, as shown in Figure 9.

4.2. Simulation Analysis of MC-AOEA

On the premise of obtaining the hypersonic turbulence, the MC-AOEA was simulated and verified to analyze the aero-optical effects. In order to fully simulate the photons that could reach the sensor, the photon simulation threshold was cut as a rectangular flow field of 120 × 120 × 200 mm³, as shown in Figure 10. ${\mathit{x}}_w-{\mathit{y}}_w-{\mathit{z}}_w$ represents the window coordinate system. Additionally, the photon simulation threshold was divided into $480\times 480\times 800$ grid cells. The initial condition of the light source was set as a parallel light source. The initial incidence angle was 90 degrees; the wavelength was 572 nm; the initial number of photons for each photon packet was $1\times {10}^6$; and there were $1\times {10}^{12}$ photons in total.

In order to prove that MC-AOEA can realize the simulation of photons in turbulence, the absorption and scattering coefficients of the simulation threshold were first set to zero, and the absorption and scattering effects of photons and turbulent molecules were not considered. The traditional ray tracing method and the MC-AOEA were used to simulate the aero-optical effects, to obtain the OPD of light and the PS-OPD of photons, respectively, as shown in Figure 11.

Figure 11 shows that the OPD obtained by the ray tracing method was generally consistent with the PS-OPD of photons obtained by MC-AOEA, which indicated that MC-AOEA had the characteristics of traditional macro methods and could also verify the correctness of MC-AOEA for simulation of the optical system. Then, the data of the simulation threshold were adjusted to the true state. When considering absorption and scattering, the PS-OPD of photons was obtained, as shown in Figure 12a. The simulation threshold was decomposed into planes with different receiving heights. The receiving planes ranged from the bottom to the top of the photon simulation threshold, and satisfied $z_w=0:200 mm$, $x_w\in \left(-40 mm,40 mm\right)$ and $y_w\in \left(-40 mm,40 mm\right)$ in the window coordinate system. The $E_{loss}$ of different receiving planes was calculated, as shown in Figure 12b.

It can be determined from Figure 12a that, when absorption and scattering were considered, the PS-OPD of photons had more prominent high-frequency characteristics than the traditional OPD. Figure 12b shows that the $E_{loss}$ of photons on the plane closer to the optical window was larger, which indicates that the absorption and scattering effects of photons were stronger with the increase in the transmission distance. In addition, the $E_{loss}$ changed approximately linearly with the increase in the photon transmission distance. Moreover, the two methods were used to calculate the offset error angle caused by aero-optical effects in different receiving planes, as shown in Figure 13.

The results of the two methods exhibited the same trend, and the offset angle increased with the increase in the transmission distance. However, the offset angle of MC-AOEA was slightly smaller than that of the traditional method because the photon transmission simulation method not only considers the refraction distortion, but also the absorption and scattering effects; the actual energy loss is also considered, which should be more in line with the actual situation than the traditional method. In order to further illustrate the advantages of MC-AOEA, the traditional method and MC-AOEA were used to analyze the aero-optical effects of 500 transient flow fields at different times. Under the condition of simulating the same photon number, the simulation was conducted in a 256-core high-performance server. The simulation results showed that the offset angle of the optical window surface would be overestimated by approximately 5.39% using the traditional method. Although the error is small, the measurement error of the optical sensor will cause errors in the optical navigation and guidance system to gradually add up with time; thus, more accurate error estimations of aero-optical effects in hypersonic turbulence are of great significance.

In order to further verify the simulation method in this paper, the CFD simulation and wind tunnel test results in the literature [31,32] are used for comparison. The root mean square of PS-OPD in this paper is approximated to the root mean square of OPD in the traditional method, expressed as $OP{D}_{rms}$. The incident angle of the light source is $\gamma \in {\left(0,180\right)}^{\circ }$.The simulation results of MC-AOEA are compared with the CFD simulation and wind tunnel test results in literature [31,32], as shown in Figure 14.

It can be seen from Figure 14 that the simulation results of MC-AOEA are basically consistent with the results in the literatures near $\gamma ={90}^{\circ }$. Due to the limitation of the angle of view of the optical sensor, the method in this paper can be considered to meet the needs of the actual environment under the current accuracy conditions.

4.3. Aero-Optical Effects under Different Flight Conditions

After the performance verification of MC-AOEA, the aero-optical distortion under different flight conditions was analyzed. The LES method was used to simulate the turbulence results at the same height and different speeds and at the same speed and different heights, as shown in Figure 15 and Figure 16, respectively.

After obtaining the turbulence results under different flight conditions, MC-AOEA was used to analyze the aero-optical effects and obtain the aero-optical distortion characteristics under different flight conditions, as shown in Figure 17.

It can be seen from Figure 17a that at the same flight altitude, the distortion degree of aero-optical effects increased with the increase in flight speed. This is because the increase in flight speed strengthens the disorder degree of turbulence above the optical window, which increases the error caused by aero-optical effects. Similarly, it can be determined from Figure 17b that, at the same flight speed, the distortion degree of aero-optical effects increases with the decrease in flight altitude. This is because, with the increase in height, the air becomes thinner and thinner, which leads to a decrease in the perturbation effect of gas molecules on photons.

Table 1. Comparison of MC-AOEA and ray tracing methods.

Simulation Conditions	Ray-Tracing		MC-AOEA		Relative Error of Offset Angle
	Offset Angle	Time	Offset Angle	Time
3 Ma, 5 km	101.8 μrad	14.12 s	92.3 μrad	5011.45 s	9.35%
3 Ma, 10 km	52.1 μrad	12.89 s	48.5 μrad	5041.56 s	6.99%
3 Ma, 20 km	25.4 μrad	13.30 s	24.1 μrad	5096.20 s	5.11%
3 Ma, 40 km	4.8 μrad	12.65 s	4.7 μrad	5012.99 s	1.08%
2 Ma, 20 km	10.9 μrad	12.54 s	10.5 μrad	4953.10 s	3.82%
5 Ma, 20 km	47.7 μrad	13.61 s	44.0 μrad	4991.28 s	7.76%
10 Ma, 20 km	60.9 μrad	12.98 s	55.6 μrad	5014.04 s	8.67%

shows the comparison of MC-AOEA and ray tracing methods under different flight conditions. Relative error of offset angle can be expressed as follows:

$$\eta =\left|\frac{offset\_angl{e}_{Ray-tracing}-offset\_angl{e}_{MC-AOEA}}{offset\_angl{e}_{Ray-tracing}}\right|\times 100\%$$

(25)

Table 1 indicates that the error between MC-AOEA and the ray-tracing method varied with different flight conditions. The higher the flight speed, the lower the flight altitude, and the greater the error. Unfortunately, although the MC-AOEA is more in line with the actual flight environment, it consumes far more computing resources than the ray tracing method. Therefore, this method can produce a regular summary of the flow field under different flight conditions offline, and it finally provides a reference for the error estimation of aero-optical effects by listing.

5. Conclusions

In this paper, an aero-optical effects analysis method based on photon Monte Carlo simulation (MC-AOEA) is proposed, which considers the absorption and scattering effects of photons and explains the phase distortion and energy loss caused by aero-optical effects from the perspective of microscopic statistics. The absorption and scattering effects of photons are stronger with the increase in the transmission distance, and the changes occur approximately linearly with the increase in the photon transmission distance. With the reduction in flight altitude and the enhancement in the Mach number, the distortion of aero-optical effects is aggravated, and the energy loss is more serious. The offset angle of the traditional method would be overestimated by approximately 1.08% to 9.35% under different flight conditions. However, although the MC-AOEA is more in line with the actual flight environment, and it consumes far more computing resources than the ray tracing method. Thus, we can present a regular summary of the flow field under different flight conditions offline by MC-AOEA, and we finally provide a reference for the error estimation of aero-optical effects by listing.