1. Introduction

With the development of airborne imaging devices on the high-speed vehicle, the aero-optical effect caused by high-speed compressible flow field has attracted more and more attention [1]. As a typical flow structure causing aero-optical effect, turbulent boundary layer has received extensive attention and in-depth research since 1950s [2–7]. At present, the research methods of aero-optical effects in supersonic turbulent boundary layer are primarily focused on numerical simulation and experimental measurement. Tromeur et al. [8,9] used the large eddy simulation (LES) to study the aero-optics of subsonic (Mach 0.9) and supersonic (Mach 2.3) turbulent boundary layers. Based on the Kolmogorov spectral analysis method, Mani et al. [10] constructed a formula to determine the spatial resolution requirement of LES used in aero-optics study. And then, Wang and Wang [11] utilized compressible LES to study the aero-optics of subsonic (Mach 0.5) turbulent boundary layer under different Reynolds numbers. The influences of optical aperture, turbulence scale and beam incidence angle on the aero-optics were confirmed. Gordeyev et al. [5] used Malley probe to measure the 1-D wavefront of subsonic turbulent boundary layers at different Mach numbers. Gordeyev et al. [12] adopted high-speed Shack-Hartmann wavefront sensor to measure the transition behavior of hypersonic (Mach 6) turbulent boundary layer, and expand the application scope of aero-optical wavefront testing method.

Compared with direct acquisition of wavefront information, a more convenient and cost-effective way is wished to be adopted to evaluate the aero-optical effects of the flow field. The aero-optical linking equation was developed, due to the lack of direct testing technique for aero-optical effects in the past. It is hoped that the results of density fluctuations can be used to predict aero-optical effects effectively. At the same time, with the development of aero-optical experiment research, the scaling law currently used to guide aero-optics experiments is not complete. The aero-optical linking equation could better guide the derivation of the scaling law of aero-optics, so it is still developing until now. From the viewpoint of density fluctuation covariance theory, Steinmetz [13] analyzed the light propagation in thin turbulent shear layers, including boundary and shear layer, and put forward the general aero-optical linking equation. Under the assumption of locally homogeneous turbulence, Sutton [14] derived the Sutton’s linking equation. Havener [15] constructed a statistical variance model for wavefront phase of turbulent flow field, and introduced a weighting function to improve the predictive performance of the aero-optical linking equation in complex flow field. Hugo and Jumper [16] validated the application of Sutton’s linking equation in subsonic shear layer with obvious inhomogeneity. Tromeur and Garnier [17] studied the predictive effect of Sutton’s linking equation on the aero-optical effect of subsonic (Mach 0.9) turbulent boundary layer. However, the definition of turbulence integral scale was wrong, which led to the conclusion that the velocity integral scale was better than density integral scale. Wyckham and Smits [18] proposed a prediction model of the root-mean-square value of optical wavefront based on Sutton’s linking equation, and good prediction results were obtained. Wang and Wang [11] adopted the compressible LES to obtain the density data of subsonic (Mach 0.5) turbulent boundary layer, and then studied the predictive effect of Sutton’s linking equation on aero-optical effect in subsonic turbulent boundary layer at different Reynolds numbers. Based on strong Reynolds analogy, Gordeyev et al. [5,6,19] derived statistical scaling laws of aero-optics at the adiabatic and non-adiabatic wall condition respectively. The predicted results were in good agreement with the experimental results.

In recent years, our group has developed the nano-tracer-based planar laser scattering (NPLS) technique, which has been used to measure the density field and the velocity field of the supersonic flow around a body and the supersonic turbulent boundary layer successfully [20–22]. Depending on its high-spatial (μm level) and high-temporal (ns level) resolution, the fine flow field structure could be clearly observed under the frozen-flow assumption. Therefore, this technique offers great convenience for the study of aero-optics [21–23].

The primary coverage of this paper could summarize the following points: firstly, the distribution of density and density pulsation were researched, which had a certain degree of reliability, based on NPLS technique. The amplitude spectrum distribution of supersonic turbulent boundary layer aero-optics was obtained effectively based on the frozen-flow assumption. The Skewness and Kurtosis were used to determine the deviations from Gaussian distribution in the wall normal direction of the supersonic turbulent boundary layer. Secondly, under different temporal and spatial distribution conditions, the prediction results of various aero-optical linking equations were compared with the direct calculation results. Section 1 of Part 3 was mainly focused on the first point. On the one hand, to a certain extent, it verified the feasibility of aero-optical effects obtained based on NPLS technique. One the other hand, it helped us to deepen our understanding of the aero-optical effects of supersonic turbulent boundary layer, and then pave a way for the Section 2 of Part 3 in this paper. In a word, Part 1 of this paper introduces the background, significance and primary coverage of the paper. Part 2 is primarily to explain the experimental methods and related concepts. Part 3 is mainly to analyze the main contents of the paper. Part 4 is primarily a summary of the full paper.

2. Experimental method and procedure

As shown in Fig. 1, the experimental setup used in this study consists mainly of the supersonic (Mach 3.0) wind tunnel and the nano-tracer-based planar laser scattering (NPLS) technique. The wind tunnel is direct-linked in structure, and the nozzle is developed by the method of B-spline function. The size of the test section is 100 mm × 120 mm. The boundary layer transitions into turbulence in the upstream nozzle naturally. In this paper, the lower wall boundary layer at 180 mm downstream of the wind tunnel is chosen as the study object, the detail parameters of which are listed in Table 1. The optical glass at the bottom window of the test section could reduce the influence of light reflection on the measurement results effectively. The flow direction is the forward direction of the x-axis. The direction that is perpendicular to the wind tunnel bottom window is the forward direction of the y-axis. The Cartesian coordinate system is established according to the rule of the right hand, as shown in the local magnify picture of Fig. 1. The x-y plane is defined as the streamwise plane, and flow direction is from left to right.

 

Fig. 1 Schematic of the NPLS / PIV experimental arrangement.

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Table 1. Parameters of the Mach 3.0 turbulent boundary layer [20]

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δ represents the velocity boundary layer thickness; θ represents the momentum boundary layer thickness; u_τ is the wall friction velocity; C_f is the wall friction coefficient; ρ_e is the mainstream density; u_e is the mainstream velocity; μ is the viscosity coefficient, Re_δ = u_eρ_eδ/μ, Re_θ = u_eρ_eθ/μ.

NPLS is a flow visualization technique that uses nano-particles as tracer particles. As shown in Fig. 1, NPLS technique is composed of the nano-particle generator, dual-cavity Nd: YAG laser, interline transfer double-exposure CCD camera, synchronizer, and computer. Particles with nominal size 18 nm are mixed into the air at the entrance of the wind tunnel [23], and a thin pulsed laser sheet with 500 mJ pulsed energy at wavelength λ = 532 nm and 6 ns pulse duration is used to illuminate the flow region of interest, stimulate the nano-particle scattering light, which is received by the CCD camera. 6 ns pulse duration ensures the transient test ability of NPLS technique. The Particle Image Velocimetry (PIV) technique shares the same control system with the NPLS technique as shown in Fig. 1, and a pair of images could be obtained each time. Velocity vector fields could be obtained (the presence of a few larger particles are necessary sometimes) by cross-correlation of the images with a fast Fourier transform-based algorithm.

As shown in Fig. 2, there is a flow visualization result of supersonic turbulent boundary layer obtained by NPLS technique, test range: $x/\delta \approx 0\text{~}2$, $y/\delta \approx 0\text{~}1.5$. The spatial resolution of the test image: 11.4 μm/pixel. Actual measurement exposure time t_exposure = 6 ns.

 

Fig. 2 The flow visualization result of supersonic (Mach 3.0) turbulent boundary layer obtained by NPLS (Streamwise 2δ, spanwise 1.5δ, image spatial resolution 11.4 μm/pixel, t_exposure = 6 ns, flow direction is from left to right).

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The key to obtain the density of supersonic flow based on the NPLS technique is to let the scattering light of tracer particles correctly describe the density. More detail about NPLS technique and its application in density measurement of the supersonic turbulent boundary layer could be found in [20,22]. Based on the density ρ of the supersonic turbulent boundary layer obtained by NPLS technique and Gladstone-Dale equation (Eq. (1)) [24], the distribution of refractive index n could be obtained.

K_GD is the Gladstone-Dale constant. When the wavelength of light is 532 nm, its value is 2.22 × 10⁻⁴ m³/kg approximately.

When a light ray propagates in a field with continuous refractive index, its trajectory satisfies the light equation:

where r is the position vector of the points along the ray trajectory, s referrers to the propagation length, n is the refractive index, and $\nabla n$ is the gradient value of the refractive index. Equation (2) has no analytical result when a light ray propagates in a field composed with the inhomogeneous refractive medium. Based on the ray-tracing method, Eq. (2) is solved by Runge-Kutta method with third-order accurate to obtain the path L of light passing through the refractive index field [25].

According to Eq. (3), optical path length (OPL) is calculated by integrating the refractive index n along the propagation path L,

In aero-optics research, the optical path difference (OPD) is often the quantity of interest rather than the OPL, which is defined as follows,

The angled brackets denote the spatial average over the optical aperture.

3. Experimental result and discussion

3.1 Density distribution characteristics of supersonic turbulent boundary layer

The aero-optical effect of the flow field is induced by the density fluctuation in the direction of light propagation essentially. For the supersonic flow field, this change not only has relatively large amplitude, but also a relatively high frequency (up to MHz). In order to verify the reliability of the density distribution obtained by NPLS technique, the average density $\rho /{\rho }_e$ obtained by NPLS technique was compared with that transformed from the average velocity distribution obtained by PIV technique. In order to achieve the density distribution transformed from the average velocity distribution obtained by PIV technique, under the assumption of zero pressure gradient in the boundary layer, the adiabatic Crocco-Busemann equation (Eq. (5)) and the idea gas equation $\rho \text{=}P/RT$ were used to build the relationship between density and velocity.

_{$T_e$}, $u_e$, and $M_e$ are the mainstream static temperature, the mainstream speed and the mainstream Mach number respectively. T_w represents the wall temperature; the recovery temperature$T_r=T_e+u_e^2/2c_p$; c_p is the specific heat at constant pressure; the recovery coefficient r = Pr^1/3; Pr is the Prandt number; γ is the specific heat ratio; $\overline{\left(\right)}$ represents the time averaged result.

As shown in Fig. 3, density distribution obtained by NPLS technique is similar with that transformed from the velocity obtained by PIV technique. At the same time, for a fully developed turbulent boundary layer, the assumption of zero pressure gradients in the boundary layer is established to a certain extent.

 

Fig. 3 Comparison of averaged density obtained by NPLS and that transformed from the result obtained by PIV.

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According to the definition of OPD, the distribution of density fluctuation is very important for measuring the strength of aero-optical effect. Based on assumptions of zero pressure gradients and the adiabatic wall, a prediction model of the local density fluctuation transformed from velocity fluctuation was proposed by Lutz [26].

C_f is the friction coefficient, = $2{\tau }_w/{\rho }_e{u}_e^2$; ${\tau }_w$ represents wall shear stress. M_e is mainstream Mach number. T_e is mainstream static temperature. The recovery coefficient r = Pr^1/3; Pr represents the Prandt number; γ is the specific heat ratio; $\overline{\left(\right)}$ represents the time average result; ()’ represents the time fluctuation result.

As shown in Fig. 4, the density fluctuation distribution obtained from NPLS technique was compared with that transformed from PIV data based on Eq. (6). The peak position of the density fluctuation calculated by PIV result is near y = 0.4δ, which is similarity with the result measured by Laderman and Demetriades [27]. The peak position of the density fluctuation measured directly by NPLS technique is near y = 0.1δ. Generally speaking, the peak position of the density fluctuation lies usually in the logarithmic law layer. The inner layer thickness of the boundary layer is generally about 0.1δ. In this respect, density fluctuation obtained by NPLS technique has relatively high reliability. According to the velocity distribution obtained by PIV technique [28], He et al. considered that the inner layer of the boundary layer has reached to 0.4δ, which is similarity with the peak position of density fluctuation transformed from PIV technique in this paper. The possible reason is that the size of the particle used in PIV is relatively large. The size is usually up to tens of microns (The size of particles used in NPLS is usually nanometer). The particle used in PIV has poor following performance. It is even more difficult to follow the flow in the boundary layer completely, especially the relatively small velocity fluctuation. This may result in the larger error of velocity fluctuation obtained by PIV, but this method could still better measure the average velocity distribution in the boundary layer.

 

Fig. 4 Comparison of root-mean-square of density fluctuations obtained by NPLS and that transformed from the result obtained by PIV.

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The large-scale structure of a turbulent boundary layer has faster motion speed, which is even reaching to the mainstream speed. The convective velocity u_c, which measures the speed of the large scale structure, can be obtained by Eq. (7) [29],

Based on the average velocity distribution obtained by PIV, we could obtain the convective velocity u_c = 0.87u_e by using Eq. (7). The result is similar to that measured by Gordeyev et al. [29]. For the supersonic turbulent boundary studied in this paper, the characteristic scale of the large-scale structure could reach the velocity boundary thickness $\delta$. The characteristic time could be defined as $T=\delta /u_c$. Based on the Nyquist sampling theorem, the sampling frequency should reach to $F_s=2u_c/\delta \approx 106.2$ kHz at least to record the time-frequency information of aero-optics effectively. Of course, it is for the large-scale structure, in order to achieve the time-frequency information induced by the small-scale structure, a higher sampling frequency is generally required. Considering the vortex structure in the Mach 3.0 turbulent boundary layer, it has the characteristics of fast motion and slow deformation. The large-scale structure takes about 18.85 μs to flow through the streamwise size $\delta$ based on the convective velocity u_c. Taking into consideration the thickness of the boundary layer obtained by the NPLS technique, there are 877 data points. This means that the sampling frequency of aero-optical wavefront based on NPLS technique could reach to 46 MHz. Very high frequency components may only contribute little energy, and which could mainly be induced by signal noise. Here we only intercept the amplitude spectrum analysis results before 1.2 MHz. The result of related NPLS experiment shows that the deformation of the main vortex structure is very minute in 20 μs [30]. The aero-optical wavefront at corresponding time is also mainly shifting along the streamwise direction [31]. Based on the frozen-flow assumption, the amplitude spectrum analysis results of wavefront are shown in Fig. 5.

 

Fig. 5 Wavefront amplitude spectrum of supersonic turbulent boundary layer.

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Strouhal number $S{t}_{\delta }=f\delta /u_e$ was used to make a dimensionless process on frequency f to measure the unsteady characteristics of the aero-optical effect. $\left|O\widehat{P}D\left(f\right)\right|$ is the amplitude spectrum of OPD. As shown in Fig. 5, in the low-frequency region (St_δ ∈ [0.03, 0.4]), $\left|O\widehat{P}D\left(f\right)\right|\text{~}{f}^{\text{-0}\text{.43}}$. Around St_δ = 1 (St_δ ∈ [0.4, 2.5]), $\left|O\widehat{P}D\left(f\right)\right|\text{~}{f}^{\text{-1}\text{.0}}$. In the relative high-frequency region (St_δ ∈ [2.5, 10.2]), $\left|O\widehat{P}D\left(f\right)\right|\text{~}{f}^{\text{-1}\text{.63}}$, and the power exponent −1.63 is approximately close to −5/3 (−1.67). The time-frequency characteristics of the aero-optical wavefront are essentially determined by the time-frequency characteristics of the density in the flow field. At the same time, the density distribution is based on the vortices with different scales in the flow field to a large extent. For the supersonic turbulent boundary layer studied in this paper, because it satisfied the assumption of zero pressure gradients to a certain extent (based on Fig. (3)), according to the ideal gas equation of state, ρ’ ~T’, that is, the density pulsation is directly related to the temperature fluctuation. Tatarski [32] showed that if the optical distortions are due to temperature fluctuations, the spectrum density of two-dimensional wavefront is ${\left|O\widehat{P}D\left(f\right)\right|}^2~f^{-13/3}$ for high-frequency. From the relation between the two-dimensional spectrum density, and one-dimensional spectrum density, it follows that the one-dimensional wavefront amplitude spectrum should behave as $\left|O\widehat{P}D\left(f\right)\right|~f^{-5/3}$ for large frequency. For turbulent flow, the −5/3 slope in the high-frequency region of the wavefront amplitude spectrum may be a reflection of the dominance of Kolmogorov-type turbulence at small-scale. At the same time, in the high-frequency region St_δ ∈ [7.0, 10.2], the power exponent of wavefront amplitude spectrum gradually deviates from −5/3. It is generally considered that the amplitude spectrum of turbulence could characterize the dissipation rate of turbulence. The supersonic turbulent boundary layer studied in this paper is intermittent in time and space, which leads to the unsteady attenuation rate of the amplitude spectrum of the wavefront. This coincides with the conclusion obtained by Marakasov et al. to a certain degree [33,34].

As shown in Fig. 6, St_δ ∈ [0.03, 0.4], $\left|\widehat{\theta }\left(f\right)\right|\text{~}{f}^{\text{0}\text{.45}}$. Around St_δ = 1 (St_δ ∈ [0.4, 2.5]), $\left|\widehat{\theta }\left(f\right)\right|\text{~}{f}^0$, corresponding to the peak position of the deflection-angle amplitude spectrum. St_δ ∈ [2.5, 10.2], $\left|\widehat{\theta }\left(f\right)\right|\text{~}{f}^{\text{-0}\text{.62}}$. The deflection-angle $\theta \left(t\right)$ could be obtained from OPD(t) based on $\theta \left(t\right)=-\left(\partial OPD/\partial t\right)/u_c$. Therefore, the deflection-angle amplitude spectrum $\left|\widehat{\theta }\left(f\right)\right|$ and wavefront amplitude spectrum $\left|O\widehat{P}D\left(f\right)\right|$ could be linked by the following equation.

 

Fig. 6 Deflection-angle amplitude spectrum of supersonic turbulent boundary layer computed from wavefront data.

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The wavefront amplitude spectrum approaches the slope of f ^-v, depending on Eq. (8), we could deduce that the defection-angle amplitude spectrum approaches the slope of f ^–v+1. This rule is satisfied in the calculation result in this paper. The deflection-angle amplitude spectrum has a peak value around St_δ = 1 (St_δ ∈ [0.4, 2.5]), implying that the dominant source of aero-optical distortion are large structures with order of the boundary layer thickness δ. This also explains why the power exponent of the wavefront amplitude spectrum tends to be −1 in this frequency region in Fig. 5. With the increase of St_δ, the deflection-angle amplitude spectrum is more attenuated. This means that contributions of small-scale turbulent structures in the supersonic turbulent boundary layer to aero-optical distortion are relatively weak, and the wavefront distortion is mainly caused by the large-scale turbulent structures. The wavefront amplitude spectrum in the high-frequency region approximately approaches the slope of f ^-5/3, depending on Eq. (8), we could deduce that the deflection-angle amplitude spectrum at high–frequency approximately approaches the slope f ^-2/3. This is also confirmed by calculation results to a certain degree. In the high-frequency region St_δ ∈ [7.0, 10.2], the power exponent of deflection-angle amplitude spectrum also gradually deviates from −2/3.

The application of aero-optical linking equation is based on the density fluctuation obtained by hot wire in the early stage [16]. When the density fluctuation is assumed to accord with Gaussian or exponential distribution, Sutton’s linking equation could be simplified from the general aero-optical linking equation. In order to analyze the temporal distribution characteristics of the density fluctuation of supersonic turbulent boundary layer in this paper, in fact, it is also necessary to convert the spatial distribution of density fluctuation obtained by NPLS into the result of time distribution via using the assumption of flow freezing. Skewness, S and Kurtosis, K can be as the measure index whether the distribution meets the Gaussian (normal) distribution. When the density fluctuation accords with the Gaussian distribution, S = 0, K = 3 [35]. Based on the density fluctuation results obtained in this paper, variation curves of S and K along the direction perpendicular to the wall were shown in Fig. 7. In the outer layer of the boundary layer, the absolute value of S is larger. The increase of absolute value of S indicates that the degree of inhomogeneous energy dissipation is increasing. The inhomogeneous energy dissipation means that the energy dissipation rate ε has a large fluctuation, in fact, which is consistent with the intermittence of the boundary layer. K could be used to measure the degree of probability density function deviating from the Gaussian distribution. $y<0.05\delta$, the test error of NPLS technique is relatively large, which leads to a larger K. $0.05\delta <y<0.5\delta$, the distribution of density fluctuation in the boundary layer agrees well with the Gaussian distribution. As shown in Fig. 2, $y>0.5\delta$, the intermittence of the boundary layer gradually becomes significant. Taking the K_normal = 3 as the standard the Gaussian distribution, $\xi \text{=}{K}_{normal}/K=3/K$, $\xi <1$ means that the intermittency of turbulence began to strengthen. This means that the intermittency of the turbulent boundary layer causes the time distribution of the density fluctuation to deviate from the Gaussian distribution, and then the prediction error of the Sutton's linking equation could be increased.

 

Fig. 7 Skewness and Kurtosis distribution of supersonic turbulent boundary layer along wall-normal direction.

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3.2 Aero-optical linking equation used in supersonic turbulent boundary layer

Based on the covariance of density fluctuation to predict the intensity of aero-optical distortion, the technical basis is the hot wire technique. Measuring points of hot wire are arranged at different locations in the flow field to record the density variation with time at different locations. For NPLS technique, as discussed above, we can transform spatial information to temporal information based on the frozen-flow assumption. To some extent, the reliability of this transformation could be corroborated by the results of spectral analysis. In essence, the aero-optical linking equation can predict the aero-optical effect by measuring the statistical distribution characteristics of the density fluctuation. The general aero-optical linking equation is

OPD_rms is the root–mean-square (rms) of the OPD on the optical aperture. K_GD is the Gladstone-Dale constant. L represents the propagation distance of light in the flow field, which is equal to δ in this paper. y represents the position along the direction of light transmission, and subscript indicates different positions along the direction of light transmission (y₁, y₂ ∈ [0, δ]). It could be found from Eq. (9), the relationship between wavefront variance and density fluctuation covariance along the light transmission direction is constructed by the general aero-optical linking equation. The density fluctuation covariance equation of two points is defined as

E represents the time average. $\rho \left(y\right)$ is the transient density at position y. $\overline{\rho }\left(y\right)$ is the average density at position y.

The general aero-optical linking equation, Eq. (9) does not introduce any hypothesis of flow field in the derivation process. Generally speaking, it is hard to extract the covariance distribution of density fluctuation from experiments directly, so its simplified approximation form is often used. The covariance in Eq. (9) can be approximated by the following two forms under the assumption of homogeneous isotropic turbulence. Assuming that the time distribution of density fluctuation accords with exponential distribution,

or Gaussian distribution,

_{${\rho }_{rms}^2\left(y\right)\text{=}〈\rho {\text{'}}^2〉$} means density fluctuation variance. $\Lambda \left(y\right)$ represents the length scale under coherent condition of density fluctuation. Based on the above approximation method, we could get the Sutton’s linking equation,

_$\alpha$ is a constant number, in the case of the exponential distribution approximation, $\alpha \text{=}2$, in the case of Gaussian distribution approximation, $\alpha \text{=}\sqrt{\pi }$. $\Lambda \left(y\right)$ is the integral scale of density fluctuation. In order to obtain the Sutton’s linking equation Eq. (13), $\Lambda \left(y\right)$ is defined as

The integral scale defined by Eq. (14) is mainly based on the assumption of locally homogeneous turbulence. In fact, it is different from the inhomogeneous turbulent boundary layer studied in this paper. However, the Sutton’s linking equation Eq. (13) could be obtained based on the integral scale defined by Eq. (14).

For the inhomogeneous turbulence, the integral scale is usually defined as

y represents the position along the direction of light transmission, and subscript indicates different positions along the direction of light transmission (y₁, y₂ ∈ [0, δ]). Due to the lack of locally homogeneous assumption, the Sutton’s linking cannot be obtained based on this form of integral scale. Considering the study object in this paper, the integral limit here is no longer from negative infinity to positive infinity as the standard definition, but is modified to the finite thickness of the turbulent boundary layer.

As shown in Fig. 8, with the assumption of locally homogeneous turbulence, the integral scale is basically around 0.1δ. Without the assumption of inhomogeneous turbulence, the integral scale is basically around 0.05δ. The turbulent boundary layer studied in this paper has obvious inhomogeneity in the direction perpendicular to the wall. It is shown that the integral scale of turbulence with inhomogeneity will be larger if the integral scale formula, Eq. (4) is utilized under the assumption of locally homogeneous turbulence. This increase will affect the prediction accuracy of the aero-optical linking equation inevitably.

 

Fig. 8 The integral length $\Lambda \left(y\right)$ computed with/without the assumption of locally homogeneous turbulence.

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The integral scale is brought to Eqs. (11) and (12), with / without the assumption of locally homogeneous turbulence, the covariance results of density fluctuation under exponential distribution approximation and Gaussian distribution approximation are obtained, as shown in Fig. 9. Compared with the exponential distribution, the result of covariance distribution is closer to the direct calculation based on Gaussian distribution approximation. Because it is not good enough to measure quantitatively the similarity of the density fluctuation distribution with the exponential distribution, specific aero-optical distortion prediction results would be compared and analyzed later. At the same time, the turbulent boundary layer studied in this paper has obvious inhomogeneity in the direction perpendicular to the wall, The results of the density fluctuation covariance obtained without the assumption of locally homogeneous turbulence are better than those obtained with the assumption of locally homogeneous turbulence.

 

Fig. 9 Comparison between the covariance of density fluctuation by calculated directly and approximate calculation results in supersonic turbulent boundary layer (a) ${\mathrm{cov}}_{\rho \text{'}}\left(y_1,y_2\right)$, (b) ${\rho }_{rms}^2\left(y_1\right)\exp \left(-\left|\left(y_2-y_1\right)/{\Lambda }_1\left(y_1\right)\right|\right)$, (c) ${\rho }_{rms}^2\left(y_1\right)\exp \left(-\left|\left(y_2-y_1\right)/{\Lambda }_2\left(y_1\right)\right|\right)$, (d) ${\rho }_{rms}^2\left(y_1\right)\exp \left[-{\left(\left(y_2-y_1\right)/{\Lambda }_1\left(y_1\right)\right)}^2\right]$, (e) ${\rho }_{rms}^2\left(y_1\right)\exp \left[-{\left(\left(y_2-y_1\right)/{\Lambda }_2\left(y_1\right)\right)}^2\right]$).

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Wang and Wang [11] studied the subsonic (Mach 0.5) turbulent boundary layer and Hugo and Jumper [16] studied the mixing layer. The results showed that Sutton’s linking equation coud also be employed to predict aero-optics of inhomogeneous turbulence if the integral scale Λ is calculated reasonably. However, the reliability of this application, especially in the supersonic turbulent boundary layer, has yet to be further verified.

Here, based on the aero-optical linking equation, for the supersonic turbulent boundary layer with distinct inhomogeneous characteristics, which distribution approximation and integral scale definition method could be utilized to obtain better prediction results? As shown in Fig. 10, the OPD_rms calculated directly by the OPD obtained by Eq. (4) could be seen as the reference result. The general aero-optical linking equation without any assumptions could achieve better prediction results. Under the approximation of Gaussian distribution, when the integral scale defined by Eq. (14) was used, the aero-optical prediction result of Sutton’s linking equation would be a larger deviation compared with the reference value. When the integral scale defined by Eq. (15) was used, the calculated result agrees well with the reference value. For the supersonic turbulent boundary layer with distinct inhomogeneity, the turbulence integral scale function defined under the inhomogeneous turbulence could also be used to predict aero-optical effects based on Sutton’s linking equation. For the Sutton’s linking equation, the difference between the exponential distribution and the Gaussian distribution is the coefficient α. As shown in Fig. 9, based on the covariance distribution of density fluctuation, more serious deviation compared with the reference value could be found under the approximation of exponential distribution. This deviation is also shown by the change of the coefficients, which makes the prediction result of OPD_rms bigger than that of the Gaussian approximation.

 

Fig. 10 Comparison of OPD_rms calculated by direct integration, general aero-optical linking equation and Sutton’s linking equations.

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The supersonic turbulent boundary layer, the turbulence characteristic scale of the supersonic turbulent boundary layer is far less than the size of the optical aperture. If the variance of density fluctuation ${\rho }_{rms}^2\left(y\right)$ and integral scale $\Lambda \left(y\right)$ are constants, ${\rho }_{rms}^2\left(y\right)\text{=}\overline{{\rho }_{rms}^2}$, $\Lambda \left(y\right)\text{=}\overline{\Lambda }$. The OPD_rms could be roughly estimated based on the following simplified equation (Eq. (16)).

As shown in Table 2, based on the integral scale defined by inhomogeneous turbulence, a better prediction result could be obtained by the simplified Sutton’s linking equation under the Gaussian distribution approximation. Based on the integral scale defined by locally homogeneous turbulence, the simplified Sutton's linking equation has a poor prediction result.

Table 2. Aero-optics of supersonic turbulent boundary layer computed by simplified Sutton’s linking equation

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4. Conclusions

The intermittency of the turbulent boundary layer causes the time distribution of the density fluctuation to deviate from the Gaussian distribution, and then the prediction error of the Sutton's linking equation could be increased. The deflection–angle amplitude spectrum has a peak at $S{t}_{\delta }\approx 1$. With the increase of $S{t}_{\delta }$, the deflection-angle amplitude spectrum is more attenuated. This means that contributions of small-scale turbulent structures in the supersonic turbulent boundary layer to aero-optical distortion are relatively weak, and the wavefront distortion is mainly caused by the large-scale coherent structures. With the assumption of locally homogeneous turbulence, the integral scale is basically around 0.1δ. Without the assumption of inhomogeneous turbulence, the integral scale is basically around 0.05δ. For the supersonic turbulent boundary layer with distinct inhomogeneity, the turbulence integral scale function defined without assumption of locally homogeneous turbulence could also be used to predict aero-optical effects based on Sutton’s linking equation. Based on the covariance distribution of density fluctuation, more serious deviation compared with the reference value could be found under the approximation of exponential distribution. This deviation is also shown by the change of the coefficients, which makes the prediction result of OPD_rms bigger than that of the Gaussian approximation. Based on the integral scale defined without assumption of locally homogeneous turbulence, a better prediction result could be obtained by the simplified Sutton’s linking equation under the Gaussian approximation. Based on the integral scale defined by locally homogeneous turbulence, the simplified Sutton's linking equation has a poor prediction result.

Although the supersonic turbulent boundary layer studied in this paper has obvious inhomogeneity, the overall density distribution law is relatively simple compared with the complex flow field with boundary layer, mixing layer and shock wave structure. It has become the focus of our next step to verify the validity of the existing aero-optical linking equations and their modified forms in this kind of flow field.