Perspective Chapter: Computational Modeling for Predicting the Optical Distortions through the Hypersonic Flow Fields

Abstract

Optical aberrations caused by supersonic/hypersonic flow files can lower the guidance accuracy of high-speed flying interceptors with onboard infrared guider. This chapter mainly summarizes the related research achievements on these issues based on the past works. First, the current developments on this important topic are discussed comprehensively. And secondly, the basic theories for predicting the aero-optical distortions used in this chapter, that is the computational flow field dynamics and its characteristics used for establishing the relationships with the flow fields and the optical light, are carefully provided. And then the density field of the flow field acquired from the large-eddy simulation (LES) can be transformed into the refractive index field in terms of the Gladstone-Dale relation. Recursive ray tracing method of optical propagation through the flow of fluids is given out. In the following, the chapter discusses the information optical modeling approach for the solutions to this issue. In the flow fields, every CFD grid is thought of as a uniform and isotropic cell. This chapter utilized the angular spectrum propagation theory for considering the optical waves propagating cell by cell. The suggested method can give out the optical transformation function (OTF), which can be directly used for modeling the aero-optical image. In the end, this chapter concludes the research works and points out the future development and potential applications of these presented research works.

1. Introduction

The interceptor mounted with inferred detector moves at speed of the supersonic, and an aerodynamic window will be formed at the face of the detector. Supersonic flows produce time- and position-dependent density fields, which directly lead to changes in optical properties dominated by the refractive index. When light passes through a field of varying refractive index, the initial optical path changes, causing distortions and phase errors in the light. This causes optical distortions such as blurring, shifting, jitter, loss of brightness, and loss of resolution. These image distortions are often referred to as the aero-optical effect (AOE). The aberration will affect the image quality of the aeronautical optical sensor, seriously affect the guidance accuracy, and even cause the interceptor to fail. The study of the principle of aero-optical effects and the measurement of aero-optical aberrations are of great importance for the endo-atmospheric aircraft. Consequently, it is necessary to study the influence of the supersonic flow fields on optical propagation and imaging in order to acquire higher guidance accuracy. The research on the aero-optical transmission not only has theoretical merit, but also has important merits in the instruction of the optical system design and restoration of the turbulence-degraded images.

Unlike atmospheric optics, aero-optics is near-field optics [1], which includes turbulent boundary layers, wake layers, and shear layers. Sutton [2] carried out detailed studies of the fundamentals and applications of aero-optics. Aero-optics is a phenomenon of fluid-optic interactions. The refractive index of air and many other fluids is linearly related to the fluid’s density through the Gladstone-Dale relationship. In general, supersonic flows are turbulent. Density fluctuations are the root cause of optical aberrations. Liepmann first studied the aero-optical effects on turbulence in 1952 [3]. After this, methods for simulating and measuring aero-optical effects have been widely developed, and research in aero-optics has a history of almost 60 years. However, there are many difficult problems in numerical modeling of aero-optical images based on computational simulation of flow fields and optic transmission, which can be used to adjust imaging sensors’ measurement, predict potential distortion, and improve guidance accuracy. It is worth our attention that this effectively reduces experimental costs and helps guide wavefront sensor design in the field of adaptive optics. There is still a difficult problem in aero-optical research, and a lot of researchers around the world have fared better in such field.

In the 1990s, researchers improved dynamic measurement and analysis of aero-optical interactions to obtain wavefront phase variance, Strehl ratio, and optical transmission function (OTF) to compensate for images degraded by turbulence. Shack-Hartmann wavefront sensors [4] have been used to measure wavefront distortion for many years. The sensor frequency has recently reached 1Mz [5, 6], almost meeting the requirements of dynamic wavefront phase measurement. Jumper [4] provided a brief perspective on traditional approaches to measure and quantify aerial-optical interactions. Meanwhile, the theory of numerical analysis from aero-optics is integrated into the CFD codes. Sutton [3, 7, 8, 9] pointed out that the procedure of aero-optic, and he devoted efforts to aero-optical performance predictions and analyzed the effect of nonuniform turbulence on the point-spread function (PSF) for imaging through turbulent flow fields. Clark and Farris [10] employed CFD codes and wave optics to provide a numerical method for calculating the aero-optical performance of a hypersonic flow field. Lockheed Martin Aeronautics has published a CFD-based aero-optical analysis of unstable aerodynamic flow fields [11, 12] that has been successfully applied to programs such as ARROW, THAAD, and ENDO LEAP. Catrakis et al. [13] studied aero-optical interactions along the propagation path in shear layers of turbulent compressible separation through direct imaging experiments of refractive-index fields, and the amount and RMS values of the differences in the optical path of interaction, that are a function of the distance traveled in the direction of the beam and a function of the laser aperture size. Roberto et al. [14] described an experimental imaging technique in which the refractive index field and the propagation optical wavefront can be measured simultaneously and based on the results of quantitative image analysis of the refractive index field and the calculated optical wavefront. Frumker et al. [15] proposed a general method to calculate the average MTF flux for a supersonic flying spherical dome using Code V and FLUENT. Monteiro and Jarem [16] studied the mutual interference function in the theory of strong fluctuations when light passes through a nonuniform layer of optical turbulence of gas, and deduced the point scattering function, optical transfer function, and related imaging equations. Michael [17] solved the Laplacian and Runge-Kutta integral parabolic beam equations along the beam path in aero-optics using higher-order compressed differentials. Zhang and Fan [18] and Wang et al. [19, 20, 21] used a grid-based model to study aero-thermal optical effect and aerodynamic optical effect near side-mounted optical windows. Juan and David et al. [22] performed a 1:1 scale validation study of a computational fluid dynamics-based aero-optics model in a wind tunnel experiment and found that the overall performance of the CFD-based predictive model was better. These studies facilitated the study of aero-optics. Numerical simulation of aero-optics propagation and imaging is an important topic in the experimental study of aero-optics physics in the wind conditioning process, and the two are considered complementary to each other.

This chapter makes use of the CFD grid model respectively with geometrical optics and information optics in order to describe a computation model of the light propagation through the supersonic flow field. The CFD grid model is thought of as the foundation of the computational simulation. The first method is based on geometrical optical so as to build up a ray tracing model for optic transmission through the supersonic flow fields. By tracking the ray path in the turbulent flow field, the wavefront aberrations can be calculated and the aero-optical performances were predicted. The algorithms in the cases of the normal incidence and the oblique incidence are worked out, and accurate ray tracing is done well. Provided data from CFD numerical simulations on certain conditions, the optical path differences (OPDs), wavefront phase variances, and the Strehl ratios used for measuring the effect of the high-speed flow fields on the optical intensity are calculated. In addition, the maximum offset angles of the line of sight (LOS) are figured out. The influences of the initial incident angle, the altitude, and the Mach number on the optical transmission through the high-speed flow fields are discussed. The results show the coincident with the prior knowledge on the characteristics of aero-optical phenomena. The second method integrates the CFD grid model with angular spectrum propagation model so as to study the aero-optical imaging through the supersonic flow fields directly. In this point of view, the aero-optical propagation is viewed as the optic angular spectrum of plane wave transmitting grid by grid, and the total optical transfer function of such flow fields can be derived and further digital image processing method is utilized to simulate the aero-optical imaging through supersonic flow fields. Finally, theoretical studies of the side-mounted IR window aero-optical imaging are made and figure out a way to model the imaging through the hypersonic flow fields.

Three kinds of computational simulation methods of aero-optics have been developed: One is to use the ray tracing method, which uses the wave delay phenomenon to measure the change in the direction of the light, but it cannot give the light deviation or the blurring of the uncertain image. One is physical optics, which predicts diffraction caused by interference between light waves; the other is wave optics, which calculates the transmission between wavefronts along the optical path and calculates the complex amplitude distribution on each wavefront. Aero-optics itself studies the interaction of light and fluids, and the application of optics theory is associated with numerical simulation methods of fluids. The density and other related data are obtained through the CFD method to simulate the flow field, and the refractive index field is calculated. Combining geometric optics theory and wave optics to quantitatively study the occurrence of light wavefront through the flow field has always been the focus of aero-optics computational simulation research. The wavefront can accurately compensate for the imaging. In adaptive optics applications, such as the Shack-Hartmann wavefront sensor, the wavefront is directly measured and used to reconstruct the wavefront. The geometry of the turbulent degraded light wavefront accurate prediction of the structure is crucial for inferring and controlling the aero-optical phenomena existing in aerospace applications and assisting in the design of optical systems.

The arrangement of this chapter is described as follows. The first section is the introduction to research on the computational study of aero-optical transmission through supersonic flow fields. In Section 2, the computational fluid dynamics model is analyzed and the Gladstone-Dale relationship used for transforming the density fields into the refractive index fields is figured out. Then, the method based on geometrical optics used for modeling aero-optical transmission is illustrated in detail in Section 3. The corresponding computational results are also given out in Section 3. In Section 4, the method using the angular spectrum propagation model for studying the aero-optical imaging is shown and the simulation results are presented. In the end, the conclusions are described.

2. CFD analysis and Gladstone-Dale relationship

2.1 CFD analysis

A simple flat was used to represent the side-mounted infrared window and CFD grids were constructed to evaluate the CFD/aero-optical analysis method. Solving the dominant flow equations in these CFD meshes is a method for numerically simulating flow fields. The grid is more uniform and rectangular without losing generality. Nonuniform grids used for physical planes must be converted to uniform meshes. If the grid resolutions are good, it can be assumed that the gaseous medium within a single grid is homogeneous and isotropic. Otherwise, the CFD data must be interpolated to increase resolution and obtain approximate streaming data. Ali Mani et al. [23] and Haris et al. [24] have respectively discussed resolution requirement of aero-optical simulation from the theoretical and experimental point of view.

In this chapter, the grids generated from CFD are uniform and hexahedral, the size of which is equal to 1 mm. This chapter considers each CFD hexahedral grid as an index cell with a uniform refractive index, respectively. Each grid is considered as a thin plate glass here. Consequently, the flow field model has cell configuration. Figure 1 describes the optical transmission through the flow fields. The supersonic flow field data used in this chapter are calculated through large-eddy simulation (LES). Figures 2 and 3 show samples of the computed density fields. Generally, supersonic flow fields should be completely viewed as turbulent flows. It is known that turbulence shows violent inhomogeneity and anisotropy with time and space changes. And turbulence can be theoretically seen as the flow fields consisting of mean flows and fluctuations. The blur and centroid shift of image degraded by aero-optical effects can be brought about by the mean flows.

Figure 1.

The relationship between the CFD calculation mesh and the optical transmission.

Figure 2.

A sample of the density distributions along the X direction.

Figure 3.

A sample of the density distributions along the Y direction.

The accurate modeling of high temperature, high pressure, and high-speed complex flow fields considering turbulence has always been a scientific problem in fluid mechanics, and until now, there are still some basic problems that have not been solved. This chapter does not involve the mechanism research of fluid mechanics, nor does it pursue the innovation of flow field modeling and solution methods. Instead, it adopts the most mature and reliable calculation model provided by fluid mechanics and widely accepted in the industry to obtain the flow field data that can be used for this chapter to calculate the imaging migration, and then explore the internal relationship between the related physical quantities such as height, line of sight angle, optical propagation path in the flow field, and the negative value of imaging migration. There are scientific problems behind the application, such as the internal reasons why the variation law of imaging migration is disturbed at different heights. Although the Navier-Stokes (N-S) equation can be used to describe turbulence, the nonlinearity of the N-S equation makes it extremely difficult to accurately describe all the details related to three-dimensional time with analytical methods. Even if these details can be really obtained, it is not of great significance for solving practical problems. From the point of view of engineering applications, it is important that the change in the average flow field caused by turbulence is the overall effect. In engineering calculation, geometric optics is used to calculate the imaging offset caused by the aero-optical effect, which is exactly the result of the action of the average flow field.

Large eddy simulation divides turbulence into large-scale turbulence and small-scale turbulence. By solving the three-dimensional modified N-S equation, the motion characteristics of large eddies are obtained, and the above model is also used for small eddies. Large eddy simulation has unparalleled advantages in the following aspects: (1) prediction of transition from laminar flow to turbulence; (2) prediction of unsteady turbulence; and (3) prediction of high-speed turbulence. However, it must be emphasized that the application of LES in industrial fluid simulation is still in its infancy.

The reference frame between the computational meshes and the incident rays is shown in Figure 4. The computational mesh has 64 × 64 × 80 grid points, ranging from 69 to 132 in the X direction, from −31 to 32 in the Y direction, and from 0 to 79 in the Z direction.

Figure 4.

The reference frame.

2.2 Gladstone-Dale relationship

The Lorentz-Lorenz formula provides the bridge of linking Maxwell’s electromagnetic theory with the micro-substances. The relationship between the flow field density $\rho$ and the refractive index $n$ is modeled by [25]

$$\left(\frac{n^2-1}{n^2+2}\right)\frac{1}{\rho }=\frac{2}{3}{K}_{\mathit{GD}}.$$E1

where $K_{\mathit{GD}}$ is the Gladstone-Dale constant. In general, the refractive index of air depends on its density at room temperature. When the air temperature is high, the refractive index mainly depends on the temperature and fluid composition. This chapter ignores the effects of aerodynamic heating and ionization on the index of refraction and only considers the effects of different current densities on the index of refraction. As the constant airflow index is approximately 1, Gladstone-Dale (G-D) relationship can be obtained as

$$n=1+K_{\mathit{GD}}\rho ,$$E2

where $\rho$ is the local density of the flow field. In the ideal air, the G-D relation is a universal description of the connection between the light rays and the air. Particularly for the infrared, the Gladstone-Dale coefficient $K_{\mathit{GD}}$ is just dependent on its wavelength. Its values taken from the IR Handbook are fitted with the formula where $\rho$ is the local density of the flow field. In ideal air, the G-D relation is a general description of the connection between light and air. Particularly for infrared, the Gladstone-Dale$K_{\mathit{GD}}$ coefficient depends only on the wavelength. Its values taken from the IR Handbook are fitted with the formula as follows

$$K_{\mathit{GD}}=2.24\times {10}^{-4}\times \left(1+\frac{7.52\times {10}^{-3}}{{\lambda }^2}\right)\left({\mathrm{m}}^3/\mathrm{kg}\right)$$E3

where $\lambda$ is the wavelength in micron. In this chapter, the wavelength of 8 $\mathrm{\mu m}$ is used for simulations.

3. Geometrical optical method for modeling aero-optical transmission

3.1 Ray tracing model

Geometrical optics is used in this chapter because wavelengths are considered to be negligible. According to the principle of geometrical optics, light exists in the form of straight lines in a uniform medium. When light passes through two homogeneous media with different refractive indices, its behavior can be determined by the laws of refraction and reflection.

According to the Gladstone-Dale relationship above, CFD grids can be converted to indexed grids. The beam axis is oriented parallel to the negative Z direction. As geometric optics show, the refracted/reflected rays are in the same plane as the incident rays. Therefore, the transport of light in a 3D flow field can be seen as consisting of the transport of multiple layers in a 2D cross-section of the flow field. Light incident on CFD grid points now starts at the top of the computer field. Figure 5 shows how light travels in a plane.

Figure 5.

Geometry of the optical transmission in two dimensions.

The index of refraction at grid point 1–1 is denoted by $n_{11}$ (see Figure 3). Likewise, the index of refraction at point i-j is denoted by $n_{\mathit{ij}}$. Points 1–1 to 1–2 are chosen as $n_{11}$ for the refractive index of the region above the interface, and the index in the next grid of the threshold is selected as $n_{21}$. $d$ is the size of the square grid taken from a hexahedral grid plane. Assuming that the initial angle of incidence is ${\theta }_0$, the angle of refraction of the ray as it passes through the interface is${\theta }_1$. Thus, when a ray passes through the grid in the negative Z direction, the angle of refraction at point k is written as${\theta }_k$. The ray coordinate is (x_k, z_k) and its offset is denoted by $\mathrm{\Delta }{x}_k$. The offsets are expressed by$\mathrm{\Delta }{x}_1=\left|X_{1}O\right|=x_1-O$, $\mathrm{\Delta }{x}_2=\left|X_2{X}_1\right|=x_2-x_1$ … $\mathrm{\Delta }{x}_k=\left|X_k{X}_{k-1}\right|=x_k-x_{k-1}$. Here, the total real offset is marked by$\mathrm{\Sigma \Delta }{x}_k$. Through the geometrical relation, the angle of incidence at point 2 is equivalent to the angle of refraction at point 1. According to the Snell’s law, an equation at grid point 1–1 $n_{11}sin{\theta }_0=n{}_{21}\mathrm{s}\text{in}\theta {}_1$ is acquired. At point 2′, an equation $n_{31}cos{\theta }_2=n_{32}sin{\theta }_2^{\prime }$ is obtained. The point where the transmission is similar to point 2′ is denoted by l′. The optical path length (OPL) from point 1 to point 2 is indicated as ${\mathit{OPL}}_1$. Likewise, the length of the path from point k to point k + 1 is denoted by${\mathit{OPL}}_k$. Total reflection can occur when light is transmitted from an optically denser layer to an optically thinner layer due to the introduction of an interface. But interfaces obviously do not apparently exist in real airflow. Total reflection causes light to rotate. This definitely increases the complexity of the algorithm.

Rayleigh pointed out that the wavefront cannot be changed when the wavefront error between the real and reference wavefronts was less than a quarter wavelength. The refractive indices of two adjacent grids are denoted $n_i$ and $n_j$, respectively. According to the Rayleigh criterion, the wavefront error is negligible if the following equation is satisfied, that is, if the light transmittance is not deformed. That is, the initial light path can be considered unchanged. Rayleigh’s criterion is applied to the algorithm. Resort to judging the criterion when total reflection occurs. If yes, the hypothetical interface is considered nonexistent and the light propagates in the same direction. Otherwise, the virtual interface is assumed to exist and full reflection occurs.

$$\left|n_j-n_i\right|s<\lambda /4$$E4

where $s$ is the geometrical path length and $1\le s\le \sqrt{2}$ (mm) The results show that the approach is useful and enough for simplifying the calculations. Modeling the propagation through the CFD grids, we can derive the recursive algorithm for tracing the light rays. First of all, the relationship at point 1 is expressed by Eq. (5).

$$\left\{\begin{array}{l}\mathrm{\Delta }{x}_1=dtan{\theta }_1\\ n_{21}sin{\theta }_1=n_{11}sin{\theta }_0\end{array}\right.$$E5

At point k, the ideal relation of the light transmission was gained as

$$\left\{\begin{array}{l}\mathrm{\Delta }{x}_k=dtan{\theta }_k\\ n_{k+1,l}sin{\theta }_k=n_{k,l}sin{\theta }_{k-1}\\ {\mathit{OPL}}_k=n_{k+1,l}d/cos{\theta }_k\end{array}\right.\left(l,k=1,2,\dots \right)$$E6

If $\mathrm{\Sigma \Delta }{x}_k>l\times d$, where $l\in N$and l is counter, we should modify the offset $\mathrm{\Delta }{X}_k$ and the ${\mathit{OPL}}_k$. Through the triangular relation, Eq. (7) is obtained by

$$M=\left(\mathrm{\Sigma \Delta }{x}_k-l\times d\right)cot{\theta }_k$$E7

The modified offset and OPL would be acquired by

$$\left\{\begin{array}{l}\mathrm{\Delta }{x}_k=Mcot{\theta }_k^{\prime }+\left(d-M\right)tan{\theta }_k\\ {\mathit{OPL}}_k=n_{k+1,l}\left(d-M\right)/cos{\theta }_k+n_{k+1,l+1}M/sin{\theta }_k^{\prime }\end{array}\right.$$E8

If no reflection over the boundary is produced, the relation between ${\theta }_k$ and ${\theta }_k^{\prime }$ should be given by

$$n_{k+1,l}cos{\theta }_k=n_{k+1,l+1}sin{\theta }^{\prime }$$E9

$${\theta }_{k+1}={90}^{\circ }-{\theta }_k^{\prime }$$E10

In the case of total reflection, it is assumed that the transmission direction has no changes if Eq. (4) is satisfied. Then, Eq. (9) should be changed into Eq. (11).

$${\theta }_k={\theta }_k^{\prime }$$E11

Integrated with Eqs (4)–(11), the recursive algorithm for tracing the ray based on the CFD grids is derived. For the fine resolution, a method to interpolate the discrete flow field data is shown in Figure 6.

Figure 6.

Interpolated mesh cell.

The index at point 1 is interpolated by $n_1=\left(n_A+n_D\right)/2$.

The index at point 2 is gained by $n_2=\left(n_A+n_B+n_C+n_D\right)/4$.

The index at point 3 is expressed by $n_3=\left(n_A+n_B+n_C+n_D+n_E+n_F+n_G+n_H\right)/8$.

The rest can be inferred by analogy and higher resolution indexed fields are obtained. Figure 7b shows the interpolated index field. This method helps to make the data more contiguous with the initial index data and to adopt the algorithm described above.

Figure 7.

(a) The initial index data (b) the interpolated index data.

3.2 Aero-optical analysis

The aero-optical quantities, measured and calculated, mainly are the wavefronts’ distortion, Strehl ratio, and the line of sight error. There is a general assumption that the mean flow fields produce time-averaged blurring and the line of sight error, whereas the turbulence produces jitter and blurring. In the sight of geometrical optics, the wavefronts’ aberrations arise from the OPL changing. OPL is expressed by Eq. (12).

$$\mathit{OPL}=\underset{\mathit{ray}}{\int }\mathit{ndl}$$E12

As the rays penetrate the disturbed air, they pick up the absolute OPD as follows¹¹:

$$\mathit{OPD}={\int }_0^L\left(n-1\right)\mathit{dl}=K_{\mathit{GD}}{\int }_0^L\mathit{\rho dl}$$E13

where L is the total geometrical path length, and ρ is the averaged-density value of the local flow. Additionally, if the central ray is considered as the reference ray, whose OPL is marked ${\mathit{OPL}}_{\mathit{ref}}$, relative OPD will be obtained by Eq. (14) [26].

$$\mathit{OPD}=\mathit{OPL}-{\mathit{OPL}}_{\mathit{ref}}$$E14

Then, OPD data acquired can be transformed to wavefront phase distortion by using the following formula:

$$\mathrm{\Delta }\varphi \left(\overrightarrow{r},t\right)=\mathit{\text{kOPD}}$$E15

where k is the wave number, and $k=2\pi /\lambda$. Therefore, optical path differences directly reflecting the variations of the wavefront phase errors. Another quantity of wavefront aberrations is the root-mean-squared (RMS) optical path difference denoted by ${\sigma }^2$, that is, it is wavefront variance gained as follows [27].

$${\sigma }^2=2{K_{\mathit{GD}}}^2{\int }_0^L{\left({\rho }^{\prime }\right)}^2{l}^{\prime }\mathit{dl}$$E16

where ρ′ is the fluctuation density and $l^{\prime }$ is the turbulence length scale calculated by CFD. Wavefront variance is a measure of the dispersion along with the mean optical path length of a wavefront and is important for modeling aero-optical parameters such as Strehl ratios and OTF for the turbulent flows. Only if the RMS optical path difference is not very large, is the time-averaged Strehl ratio for a given wavefront phase shift approximated as [28].

$$\mathit{SR}=exp\left[-{\delta }_{\varphi }^2\right]$$E17

Here, ${\delta }_{\varphi }^2$ is denoted the wavefront phase variance and it is obtained by the following relationship [29]

$${{\delta }_{\varphi }}^2=k^2{\sigma }^2.$$E18

In Figure 6, the line of sight error is described in terms of optical beam path reversibility. The LOS errors result in the coordinate position excursion of the image formed by the light propagation through the supersonic flow fields. Generally, q is of the small scale. So, the arc length s is approximately equal to q. If R is selected as the radius, the LOS deviation angle denoted by $\tau$ will be expressed as

$$\tau =q/R.$$E19

Here, the unit of $\tau$is rad. q is acquired by Eq. (20)

$$q=htan\theta -\mathrm{\Sigma \Delta }{X}_i.$$E20

Here, the maximum displacement angle ${\tau }_{\max }$ representing the maximum displacement of the image position. In Figure 8, when R = |OB|, the value of $\tau$ is smaller than the real angle, and when R = |OA|, the value of $\tau$ is large. Using each CFD grid as a CCD imaging unit, the detection window is abstracted into an image plane consisting of 64 × 64 pixels. The maximum displacement angle directly reflects the maximum displacement of the intensity distribution, that is, the maximum displacement of the image.

Figure 8.

The LOS deviation angle.

3.3 Simulation results

The supersonic flow field CFD data were calculated using the large-eddy simulation (LES) method. All angles of attack in a flow field simulation are the same. The distribution of density fluctuations in the flow field at a height of 35 km and Mach number 7 are shown in Figures 9 and 10. In Figure 9, density fluctuations near the detection window along the flow direction become bigger and bigger. The distribution of density fluctuations in the positive Y direction is symmetrical. The RMS OPD distribution obtained from Eq. (18) shows the change in phase shift of the wavefront, which in turn shows the characteristics of the flow field in Figures 11 and 12. The RMS optical path difference increases along the flow direction, whereas, in general, in the Y direction, the RMS OPD is the greatest at the center of the window and decreases from the center to the other.

Figure 9.

Solution of density fluctuation along flow direction. Solution at H = 35, and Ma = 7.

Figure 10.

Solution of density fluctuation in the Y direction. Solution at H = 35, and Ma = 7.

Figure 11.

The RMS wavefront errors in the flow direction (H = 35 km, Ma = 7).

Figure 12.

The RMS wavefront errors in the Y positive direction (H = 35 km, Ma = 7).

Strehl ratios are shown in Figures 13 and 14. On the whole, light intensity is weakened along the flow direction, while in Y positive direction, the light intensity at the center of the window is reduced to minimum and is decreased from here to other two sides. Apparently, all the calculation results are qualitatively correct.

Figure 13.

Strehl ratio in the flow direction (H = 35 km and Ma = 7).

Figure 14.

Strehl ratio in the Y positive direction (H = 35 km and Ma = 7).

The absolute differences of the optical path in the positive Y direction are shown in Figures 15–17. It can be seen in Figure 15 that the optical path difference increases as the angle of incidence increases. Therefore, it can be concluded that the sensor within the detection window needs better incident light to reduce wavefront distortion. It can be seen in Figure 16 that the optical path difference value decreases as the height increases. It can be seen from the standard atmospheric table that the atmospheric density in the range from 0 to 80 km decreases with increasing altitude. Of course, as the free air flow density becomes thinner, the density near the detection window inevitably becomes thinner at higher altitudes. Although the supersonic flux field near the window is compressible, the change in density is small. Therefore, the aberration of the accumulated optical wavefront is still reduced. Figure 17 shows the optical path differences at different Mach numbers, at the same height and at the same angle of incidence. In aerodynamics, the effect of the compression ratio becomes greater as the velocity of the flow field increases. As the Mach number increases, the density necessarily increases. Therefore, the optical path difference becomes large.

Figure 15.

Comparison of OPD at different angles of incidence (H = 40 km and Ma = 7).

Figure 16.

Comparison of OPD at different altitudes (Ma = 7, and θ₀ = 5^o).

Figure 17.

Comparison of OPD at different Mach Numbers (H = 30 km, and θ₀ = 5^o).

The maximum deviation angles calculated at different flow fields are given in Table 1. It is magnified with the increasing velocity of the flow and reduced with the altitude being higher.

	H = 30 km	H = 35 km	H = 40 km
Ma = 7	2.17884e-3/rad	2.17878e-3/rad	2.17865e-3/rad
Ma = 5	2.17880e-3/rad	2.17871e-3/rad	2.17848e-3/rad

Table 1.

The maximum deviation angles.

4. Information optical method for modeling aero-optical imaging

4.1 Angular spectrum propagation model

Figure 18 depicts the propagation of the angular frequency spectrum of plane wave. As to the optical wave, the complex amplitude of the plane wave is expressed by

Figure 18.

The angular spectrum propagation of plane wave in uniform optical media.

$$\begin{array}{l}U\left(\mathbf{r},t\right)=aexp\left(\mathit{ik}\cdot \mathbf{r}\right)\\ =aexp\left[\mathit{ik}\left(xcos\alpha +ycos\beta +zcos\gamma \right)\right]\end{array}$$E21

where $a$ is the constant amplitude; $\left(cos\alpha ,cos\beta ,cos\gamma \right)$ denotes the direction cosine; $k=2\pi /\lambda$ is the wave vector [30].

The angular frequency spectrum is just the 2-D Fourier transformation of the complex amplitude of optical wave. Given that a monochromatic light injects the X-Y plane along with the direction of Z axis, its angular spectrum can be expressed as

$$A\left(f_x,f_y,z\right)=\iint U\left(x,y,z\right)exp\left[-i2\pi \left({\mathit{xf}}_x+{\mathit{yf}}_y\right)\right]\mathit{\text{dxdy}}$$E22

where $U\left(x,y,z\right)$ is the complex amplitude distribution. On the other hand, the inverse Fourier transform of the angular spectrum is a complex amplitude distribution. Here, the transmission of aerial optics through the flow fields can be seen as the transmission of each spectrum. Each plane wave spectrum in $\left(x,y,0\right)$ is denoted by $A_0\left(f_x,f_y,0\right)$, and each spectrum in $\left(x,y,z\right)$ is denoted by $A_0\left(f_x,f_y,z\right)$, where $f_x=\frac{cos\alpha }{\lambda }$, $f_y=\frac{cos\beta }{\lambda }$. According to the Helmholtz scalar equation, angular spectrum propagation function can be obtained as

$$A_1\left(\frac{cos\alpha }{\lambda },\frac{cos\beta }{\lambda },z\right)=A_0\left(\frac{cos\alpha }{\lambda },\frac{cos\beta }{\lambda },0\right)exp\left(\mathit{jkz}\sqrt{1-{cos}^2\alpha -{cos}^2\beta }\right)$$E23

which describes the propagation between the two parallel planes. As a matter of fact, a transfer function of frequency filtering is obtained by

$$\begin{array}{l}{H}_c\left(f_x,f_y\right)=\frac{A\left(f_x,f_y\right)}{A_0\left(f_x,f_y\right)}\\ =\left\{\begin{array}{c}exp\left[\mathit{ikz}\sqrt{1-{\left(\lambda f_x\right)}^2-{\left(\lambda f_y\right)}^2}\right]\\ 0\end{array}\right.\begin{array}{c}{f_x}^2+{f_y}^2<\frac{1}{{\lambda }^2}\\ \mathit{\text{others}}\end{array}\end{array}$$E24

The model of aero-optical imaging proposed in this chapter is inspired by the above analysis. Hence, the aero-optical transmission is translated into spatial filtering with limited bandwidth of the lights.

4.2 Linear filter model of aero-optical imaging

The light source described in this chapter may be too far away from the built-in detector, causing the output wave to appear as a plane wave. Therefore, the transmission of light at hypersonic speed can be considered as the transmission of plane waves. The term “diffraction” is conveniently described by Sommerfeld as “ any deviation of light rays from rectilinear paths that cannot be interpreted as reflection or refraction.”. Furthermore, the results obtained from the scalar diffraction theory approximate the real effect if the wavelength is smaller than the diffraction aperture and the observation point is far from the diffraction aperture [31]. The supersonic flow field transmission process considering the above discussion, aero-optics transmission can be seen as a scalar diffraction problem, so angular spectrum propagation can be used for aero-optics research.

From the point of view of information optics, each cubic grid can be seen as an optical system that composes any optical filtering system that characterizes the flow fields. After these considerations, the supersonic flow fields can be divided into $63\times 63\times 79$ optical filtration subsystems, which are serially connected in the negative direction along the Z axis. Figure 19 maps a sketch of the plane optical wave through CFD cubic grids.

Figure 19.

The sketch map of CFD cubic grids.

Figure 20 shows the structure of one cubic CFD grid. Each cubic grid has eight points with the determined index of refraction. The following equation gives out the characteristic parameter ${n^i}_{\mathit{oc}}$ of the cubic optical filtering system.

Figure 20.

The distribution of CFD grid points in one cubic optic system.

$${n^i}_{\mathit{oc}}=\frac{\left(n_1+n_2+\dots +n_8\right)}{8}$$E25

where $i$ is the order number of the optical filtering system ($i=63\times 63\times 79$). One of these serial systems is described in Figure 21. $H_i\left(f_x,f_y,z_i\right)$ denotes the transfer function of the ith optical system, which can be gained through Eq. (23) based on angular spectrum propagation model.

Figure 21.

The diagram of serially connected systems representing the transmission of angle spectrum.

In the frequency domain, the output of such serially linear filtering system can be obtained through

$$A_i\left(f_x,f_y,z_i\right)=A_0\left(f_x,f_y,0\right)H_1{H}_2\cdots H_i$$E26

However, Eq. (23) is actually called as coherence transfer function (CTF). As to aero-optics, it should be viewed as a noncoherent imaging system, which is a linear system concerning on the distribution of light intensity. To the noncoherent imaging system, optical transfer function (OTF) is utilized for the study of light propagation. The OTF can be derived from CTF through the following equation

$$H\left(u,v\right)=\frac{H_c\left(u,v\right)\otimes H_c\left(u,v\right)}{\int {\int }_{-\infty }^{\infty }\left|H_c\left(\alpha ,\beta \right)\right|\mathit{\text{dαdβ}}}$$E27

where CTF is $\mathbb{F}\left[h_c\left(x,y\right)\right]=H_c\left(u,v\right)$, $u=f_x,v=f_y$ and OTF is $\mathbb{F}\left[h\left(x,y\right)\right]=H\left(u,v\right)$. Thus, OTF of the flow fields is equal to the 2D Fourier transformation of the point-spread function (PSF, $\mathit{PSF}=h\left(x,y\right)$) of light intensity distributions. Relationship between the input light intensity distribution $I_o\left(x,y\right)$ and output light intensity distribution $I\left(x,y\right)$ is obtained by

$$\mathbb{F}\left[I_o\left(x,y\right)\right]\cdot \mathbb{F}\left[h\left(x,y\right)\right]=\mathbb{F}\left[I\left(x,y\right)\right]$$E28

where $I\left(x,y\right)={\left|U\left(x,y\right)\right|}^2$ is the light intensity distribution.

Through the above theoretical analysis, a discrete OTF matrix $63\times 63$ could be acquired. In other words, PSF can be gained for the spatial filtering of image. The relationship between PSF and the light intensity distributions is satisfied by

$$I\left(x,y\right)=I_o\left(x,y\right)\ast h\left(x,y\right)$$E29

Here, the transmission of right incident light through the supersonic flow fields is considered. And position of the image centroid can be calculated by

$$x_c=\frac{{\sum }_{i}i\cdot I\left(i,j\right)}{{\sum }_i{\sum }_{j}I\left(i,j\right)},y_c=\frac{{\sum }_{j}j\cdot I\left(i,j\right)}{{\sum }_i{\sum }_{j}I\left(i,j\right)}$$E30

where $\left(i,j\right)$ is the coordinate of the corresponding image pixel at the image coordinate, and $I\left(i,j\right)$ is the corresponding pixel value. Thus, the centroid shift of the degraded image can be evaluated through,

$$\mathrm{\Delta }x=x_c^{\prime }-x_c^o,\mathrm{\Delta }y=y_c^{\prime }-y_c^o$$E31

where ${\left(x_c^{\prime },y^{\prime }\right)}_c,\left(x_c^o,y_c^o\right)$ are respectively the centroids of the degraded image and original image. To estimate the total aberrations induced by the flow fields, Euclidean distance (ED) of image shift is used for an implicit assessment through

$${\mathrm{r}}_{\mathit{ED}}=\sqrt{{\left(x^{\prime }-x\right)}^2+{\left(y^{\prime }-y\right)}^2}$$E32

4.3 Simulation results

Digital images of aircraft obtained from the Internet are used to simulate aero-optical images. To study the shift of image centroid without considering the effect of temporal integration on the image, only snapshots were numerically investigated. Figure 22 shows the original image. Figures 23–25 show the degradation results of a supersonic flow fields with the same Mach number (Ma = 7) and height of 30, 35 and 40 km, respectively. Figures 23 and 26 show the results when the height is 30 km and the Mach numbers are 7 and 5, respectively. Table 1 shows the results of calculating the shifts of image centroid. Although the evaluation method cannot calculate the total aero-optical distortion, the validity of the proposed model of the aero-optical imaging method based on optical information can be easily verified.

Figure 22.

The original image.

Figure 23.

The aero-optically degraded image (H = 30 km, Ma = 7).

Figure 24.

The aero-optically degraded image (H = 35 km, Ma = 7).

Figure 25.

The aero-optically degraded image (H = 40 km, Ma = 7).

Figure 26.

The aero-optically degraded image (H = 30 km, Ma = 5).

It must qualitatively satisfy the aero-optical effect of light propagation in a supersonic flow fields. That is, at the same Mach number, the lower the height, the stronger the aero-optical effect. The higher the Mach number at the same height, the stronger the optical effect of air. The image is blurred and the centroid shift is larger in the quantitative analysis.

Furthermore, from the results in Table 2, it can be seen that the Mach number in the aero-optical images has a greater weight than the flight altitude. That is, the compressive effect of the flow fields by the flight speed compared with atmospheric density at different altitudes is a key factor influencing the change in the density field. Through the above analysis, the simulation results are consistent with the basic facts of the aero-optic effect.

	Δx	Δy	Euclidean distance $r_{\mathit{ED}}$
H = 30 km Ma = 7	2.260931	0.181078	2.2682
H = 30 km Ma = 5	−0.327743	0.174146	0.3711
H = 35 km Ma = 7	−1.229878	−0.048415	1.2308
H = 40 km Ma = 7	−0.535493	−0.131637	0.5514

Table 2.

Image centroid shifts related to the original image.

5. Conclusion

This chapter concentrates on the numerical study of modeling aero-optical transmission and imaging through the supersonic flow fields. We have developed two computational models for predicting aero-optical performance of the supersonic flow fields, respectively, using geometrical optical method and information optical method. Firstly, this model combining the CFD model with the geometrical optics is discussed in detail. The calculation results are coincident favorably with prior knowledge from the completed research about aero-optics. The model has been compared with the experimental knowledge about the influences of the supersonic flow fields on optical transmission. Due to test complication and lack of experiment facilities, a complete comparison cannot be allowed. The model has merits not only in predicting optical performance of supersonic flow field but also in understanding the aero-optical characteristic of a particular design.

In addition, this chapter also provides a solution to aero-optical problems in terms of information optics. A computational model for studying aero-optical imaging through the supersonic flow fields is presented. This model integrates the CFD grids with the model of angular spectrum propagation to construct serially connected optical subsystems for representing the supersonic flow fields. The simulation results are qualitatively coincident with prior knowledge about aero-optical effects. The proposed model can be directly helpful in restoring the image degraded by the supersonic flow fields. Compared with the geometrical optical method, the provided approach based on the information optics can overcome the complexity of ray tracing, and give the explicit shifts of the image and describe the blur circle of the degraded image. However, the computational results are discussed qualitatively due to unavailability of the corresponding wind tune experiments. In future, more extensive computational experiments will be done so as to study the imaging under different field of view and do the comparisons under different premises.

The research methods in this chapter are also suitable for the other optical transmitting through the air turbulence and high-speed flow fields. If the issues discussed above can be take consideration into the passive aero-optical system, the other similar problems will be sure to be faced in the active optical systems. As we all know, in the near the future, airborne active laser emitter/laser communication systems will come true. There are optical beams emitting through the high-speed flow fields, and then, airborne lasers are adversely affected by this flow field when operating. In the active system, the flow field causes the projected beam energy to attenuate and deviate from the illumination target, and the laser imaging system will cause image blur and jitter. Similarly, the approaches provided in this chapter will still have the advanced technical merits in the solutions of the aero-optical effects.