**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 aerooptical 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.

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 *Perspective Chapter: Computational Modeling for Predicting the Optical Distortions… DOI: http://dx.doi.org/10.5772/intechopen.106591*

**Figure 2.** *A sample of the density distributions along the X direction.*

#### **Figure 3.**

*A sample of the density distributions along the Y direction.*

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,

**Figure 4.** *The reference frame.*

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 smallscale 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.

#### **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 *ρ* and the refractive index *n* is modeled by [25]

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

where *KGD* 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 *Perspective Chapter: Computational Modeling for Predicting the Optical Distortions… DOI: http://dx.doi.org/10.5772/intechopen.106591*

refraction. As the constant airflow index is approximately 1, Gladstone-Dale (G-D) relationship can be obtained as

$$n = \mathbf{1} + \mathbf{K}\_{\text{GD}} \rho,\tag{2}$$

where *ρ* 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 *KGD* is just dependent on its wavelength. Its values taken from the IR Handbook are fitted with the formula where *ρ* 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-*KGD* coefficient depends only on the wavelength. Its values taken from the IR Handbook are fitted with the formula as follows

$$K\_{\rm GD} = 2.24 \times 10^{-4} \times \left(1 + \frac{7.52 \times 10^{-3}}{\lambda^2}\right) \left(\text{m}^3/\text{kg}\right) \tag{3}$$

where *λ* is the wavelength in micron. In this chapter, the wavelength of 8 μm is used for simulations.
