## Abstract

The optical path length (OPL) of supersonic turbulent boundary layer of Mach number 3.0 is obtained with the nanoparticle-based planar laser scattering technique, and its structure is analyzed within the framework of hierarchical symmetry assumption. Our result offers reasonable evidence for that the OPL obeys this assumption with parameter *β* depending on *q*. The scaling exponent *ζ*(*q*) of structure function is computed and compared with the theoretical prediction of She-Leveque model. The curve *ζ*(*q*) we obtained is convex and smaller than the theoretical value for small *q*, which is attributed to the large scale structure of the OPL.

© 2012 OSA

## 1. Introduction

For the optical systems mounted on high-speed vehicles, the optical aberration due to density variation of air can be a great concern, and this glass of study falls generally under the category of aero-optics [1–3]. In most applications the optical path length (OPL) or optical path difference (OPD) is often used to characterize the aberration [4–6]. The supersonic turbulent boundary layer is one of the most important building block flows for the study of aero-optics, and understanding the structure of its OPL is helpful for the control and correction. A significant early work was due to Stine and Winovich [7]. They performed the photometric measurements of the time-averaged radiation passing through turbulent boundary layer with Mach number ranging from 0.4 to 2.5. This work combined together all that has been known till then on optical propagation through index-variant turbulent flow. Specifically, the dependence of scattering on the integral scale of the boundary layer was emphasized. Recently, the group of Norte Dame University has paid much attention to the aero-optics of turbulent boundary layer [8–10]. Their experimental measurement is mainly based on the Malley probe, with which several forms of scaling laws for optical aberration are suggested. Wyckham *et al*. [11] used the Shack-Hartmann wavefront sensor to study the aero-optical distortion of transonic and hypersonic turbulent boundary layers with and without gas injection. Also, a scaling law for the root-mean-square of phase distortion was proposed, which seems to collapse the data better than previous models. Truman and Lee [12] carried our direct numerical simulation of incompressible boundary layer, and regarded refractive-index as passive scalar. The effects of organized turbulence structures on the phase aberration were analyzed, and its substantial variation with the direction of propagation was found. Tromeur *et al*. [13] and Wand *et al*. [14] did large eddy simulation of aero-optical effects in turbulent boundary layers with transonic and supersonic free streams, and the validity of Sutton’s statistical model [15] relating the optical phase distortion to density fluctuation was discussed.

Because of its importance in aero-optics, the structure of OPL has attracted much attention. The power spectrum analysis is a useful tool for revealing the structure of multi-scale signals. Gordeyev *et al*. [9,10] investigated the spectrum of the jitter signal of subsonic and supersonic turbulent boundary layer obtained with Malley probe, and an evident scaling feature is found, with which many spectra from different experiments can be collapsed into one curve. Freeman and Catrakis [16] measured the OPD of the turbulent shear layer with the laser-induced-fluorescence technique, and demonstrated a *k*^{-5/3} power law scaling. The method of structure function is another powerful tool for the analysis of aberrated wavefront corrupted by turbulent medium, and its application in atmospheric optics is a famous paradigm [17]. Another work of the present authors is devoted to the second order structure function of the OPL of supersonic turbulent boundary layer, and an analytic expression is suggested to fit the experimental data [18]. In the processing of OPL of turbulent boundary layer obtained with the experimental technique described in below, we notice that the multi-scale character of OPL is similar to velocity or energy dissipation signals in turbulence research. For incompressible turbulence, the hierarchical structures of the velocity and energy dissipation are well known [19], and many models, such as multifractal model [20] and log-Poisson model [21], have been suggested to describe the phenomena. In addition, the similar problems for passive scalars in turbulence are also studied by many investigators [22,23]. The primary problem of these studies is investigating the structure function of higher orders, especially the behavior of power exponents. This paper is concerned with the hierarchical structure of OPL of the supersonic boundary layer. We know that OPL is proportional to the integral of density along optical propagation direction, so this work will be helpful for understanding the mechanism of compressible turbulence. Extensive researches on supersonic turbulent boundary layer indicate that its essential dynamics essentially follows the incompressible pattern for moderate Mach numbers (usually less than 5), and some particular forms of Reynolds Analogy can be derived [24]. This may imply that the structure of the velocity or energy dissipation in compressible turbulence is same in nature as the case in incompressible turbulence. However, the structure of density or other scalars in compressible turbulence has not been investigated thoroughly in literature. In addition, the dimension of OPL is deduced by 1 due to the integral of density, which offers great convenience for analysis. More importantly, OPL can be visualized more naturally with optical methods (such as interferometry), and may be an interesting optical subject for turbulence research community.

## 2. Experimental setup

The experimental setup used in this study mainly consists of the supersonic wind tunnel and the nanoparticle-based planar laser scattering (NPLS) system, and the sketch of visualizing the boundary layer is shown in Fig. 1
. The wind tunnel is direct-linked in structure, and the nozzle is designed with the method of B-spline function. The size of the test section is 100 mm × 120 mm, and the Mach number of free stream is 3.0. The four walls of the test section are made up of optical glass, and the measurements are made on the boundary layer developed on the wall. More parameters of the wind tunnel are: total temperature *T*_{0} = 300 K, total pressure *P*_{0} = 101 kPa, free stream velocity *U*_{∞} = 620 ms^{−1}, and unit Reynolds number Re = 7.49 × 10^{6} m^{−1}. Another core component of the setup is the NPLS system. TiO_{2} particles of nominal size 18 nm are mixed into the air at the entrance of the wind tunnel, and a thin pulsed laser sheet (double-cavity laser, wavelength: 532 nm, pulse width: 6 ns, energy per pulse: 400 mJ, thickness of the sheet: 0.5 mm) is used to illuminate the flow region of interest. The light scattered by the nanoparticles is received by an interline transfer CCD. The time interval between two exposures can be controlled by synchronizing the laser and CCD, and the smallest interval can reach 0.2 μs. More details about NPLS and its application in visualizing the boundary layer can be found in [25–27].

In Fig. 2
is shown a NPLS image of the turbulent boundary layer obtained in our experiment, whose size is 2048 × 1350, and the digital resolution is about *h* = 0.011 mm. An estimation of the thickness of the boundary layer gives a value of *δ* = 10.2 mm, and the dissipative scale is about *η* = 0.05 mm. The bright line at the bottom of the image is the crossing line between the laser sheet and the flat plate. Because of the influence of diffuse reflection, the flow structure just above the plate is almost invisible. The part of image above this zone shows the density structure of the supersonic turbulent boundary layer, and the structures of diverse scales can be seen clearly.

Rayleigh scattering is an attractive mechanism for measuring the density of gas media [28,29]. The usual problem of low cross section is overcome by using TiO_{2} particles as tracers and strong laser pulse as light source in our experiment. In fact, the methods of how to mix the tracers into free stream uniformly and how to prevent the nanoparticles from aggregating into clusters of larger size are indeed two primary problems in our experiment. We achieve the first purpose by keeping constant pressure for the tracer generator and offering stabilizing section with enough length for the wind tunnel, and achieve the second one by fluidizing the tracers with supersonic nozzle and using cyclone separating device to filter big particles. Based on the particle dynamics of multiphase fluid, we have measured the diameter of tracer particles in NPLS system with oblique shock wave experiment. The result gives a diameter about 40 nm [25], which indicates that the aggregating effect is surmounted effectively. With these problems solved, we can assume that the pixel value of NPLS image is proportional to the density value. In addition, some image preprocessing techniques are employed to correct the influence of background and non-uniform illumination. Tian *et al*. [30] have suggested a calibration procedure to obtain the dimensional density by placing a wedge with adjustable attack angle, and analyzing the density relationship before and behind the shock wave. For the purpose here, we will not resort to this strategy further, and directly take the pixel value as the relative density without dimension.

The experimental error of NPLS technique mainly results from the tracers. The uniform mixture of tracers into the air is statistical, but one cannot guarantee that there would be an exactly uniform number of particles in any volume, and especially in tiny volumes. This means that the fluctuation of count of tracers exists in a small volume, which can be neglected only when the mean number is large enough (Poisson statistics can be assumed for the count of tracers in a small volume). Rather, because the particles within a pixel volume have random positions, they will behave optically as a diffuse reflector, and a speckle pattern results when a laser illuminates a diffuse reflector. There will be many spackles within a single pixel, and the Chi-squared distribution can be assumed for the pixel value [31]. Thus the overall signal-to-noise ratio (SNR) will be increased by the square-root of the number of speckles within a pixel volume. In the diffraction limit, the Airy disk diameter is given by [32]

where*m*is the magnification of the imaging system, and

*f*

_{#}is the

*f*number of the lens. Substituting our experimental parameters into Eq. (1), we obtain

*d*

_{A}= 6.9 μm. So there will be

*N*= 184 speckles in a pixel volume (11 μm × 11 μm × 500 μm) if a cube

*d*

_{A}long is regarded as a diffuse reflector. Based on the property of Chi-squared distribution, the fluctuation of a pixel value divided by its mean is (2/

*N*)

^{1/2}= 10.4%. If we regard a diffuse reflector as a sphere of diameter

*d*

_{A}, the error will be 7.5%. So the experimental error due to speckle phenomenon is between these two estimates. Other error sources come from the influence of background and that the light intensity on the laser sheet cannot be uniform. The total error, regardless of the source, can be estimated with the part of image far from the boundary layer, and the averaged root-mean-square normalized by the mean is about 10%. This total error is quite close to the error due to speckle, so the latter can be regarded as the dominate factor.

As the last point of this section, we validate the density obtained in our experiment with the Crocoo-Busemann (CB) relationship [24]. The velocity of the boundary layer can be measured with particle imaging velocimetry [26]. With the mean velocity and CB relationship, the mean temperature can be computed. By applying the zero pressure gradient assumption further, the mean density is finally obtained, which is compared with the density obtained from the NPLS images directly. In Fig. 3 is shown such a comparison in vertical direction, and we can see that the agreement between the two is evident. As pointed by an anonymous referee, there is a kink in the measured mean density profile. This phenomenon is mainly due to the diffuse scattering of laser sheet by the boundary. In the post-processing of NPLS images, the background image is subtracted from the original one, which is captured when the tunnel is not running. The diffuse scattering by the boundary contributes dominantly to the background image. So there is a band region above the plate in the processed image which does not connect the region above it so smoothly, and a kink appears when the mean density is computed.

## 3. Hierarchical structure description of OPL

In aero-optics, the OPL of 1-dimension cut can be computed as

where*K*

_{GD}is the Gladstone-Dale constant, and we take the streamwise and vertical direction as

*x*and

*y*direction, respectively. The zero point of

*y*axis is at the plate and

*y*=

*H*is at the upper edge of the image. Because we concerned here is the structure function, the items in the integral is replaced by the pixel value of NPLS images in the computation below. In Fig. 4 is shown such an OPL relevant to the image in Fig. 2. One can find that the fluctuations of large and small amplitude are prevalent. The basic profile of the OPL is due to the large scale structure of the layer, and the fluctuations of small amplitude can be attributed to the fine structure of the flow and the experimental error.

The *q*-th order structure function (SF) of OPL is defined as

*x*

_{0}and many realizations. The number of NPLS images used in this study is 400. With the discussion of the experimental error in Sec.2, we can assume that the error of

*L*(

*ih*) is additive and independent for different

*i*, so its SF is approximately a constant. For the SF concerned here, the effect of noise can be reduced effectively by subtracting

*D*(

_{q}*h*). The SF used in below is obtained by this preprocessing method. In addition, the integral operation in Eq. (2) is also useful for suppressing the experimental error. The SFs of order 2, 8, and 16 are shown in Fig. 5 . We can find that the scaling behavior is quite evident in a wide range of distance for low order SF. The SF of larger order shows some scattered feature, but they approximately locate around a straight line in a noticeable distance range.

Among many phenomenological models, the hierarchical structure model suggested by She and Leveque [21], now known as SL scaling, has received much attention for its successful application in wide ranges [33]. Here we adopt the idea of SL to explore the hierarchical structure of OPL. We define the hierarchy as

With the hierarchical symmetry assumption [21,33],the scaling exponents of SF*ζ*(

*q*),

*D*(

_{q}*x*) ~

*x*

^{ζ}^{(}

^{q}^{)}, obeys a general formulawhere

*γ*describes the singularity index of the most intermittent structure and

*C*is its co-dimension.

One of the most noticeable features of the SL scaling is that the assumption Eq. (5) can be conveniently checked with the relation [33]

With the experimental results of structure function of OPL, the relation of log_{2}(

*H*(

_{q}*x*)/

*H*

_{1}(

*x*))

*vs*. log

_{2}(

*H*

_{q}_{+1}(

*x*)/

*H*

_{2}(

*x*)) for

*q*= 5, 10 and 15 is shown in Fig. 6 , and the distance range is from 10

*h*to 470

*h*. The fitted line with least-squares and its slope are also given in each panel. We can see that the character of the data points clustering around a straight line is remarkable. The parameter

*β*obtained by least square fitting for different

*q*is shown in Fig. 7 . The

*β*for

*q*less than 5 is larger than 1 (not shown in Fig. 7), which is meaningless in SL model. For the

*q*-

*β*curve, there is a valley at about

*q*= 6, and it approaches a constant value about 0.84 for large

*q*. Therefore the hierarchical symmetry assumption is quite reasonable for the OPL of the boundary layer, but with a

*q*-dependent parameter

*β*.

Now we compute the scaling exponents and compare it with the theoretical prediction. We adopt the processing technique similar to extended self-similarity [34] to compute the scaling exponents, and the results are shown in Fig. 8
with red circles. The curve is convex for *q* ≤ 6, and becomes concave for larger *q*. The saddle point locates at approximately *q* = 6. This phenomenon is often explained as “phase transition” in literature [35]. Set *β* = 0.84 and fit the data with Eq. (6), the nonlinear least-squares gives *γ* = 0.26 and *C* = 0.34. The curve of Eq. (6) with these three parameters is displayed in Fig. 8 with blue solid line. We can see that the fitting performance is satisfactory for *q* ≥ 6, but becomes not so good for smaller *q*. Because the straight line feature of the experimental data for large *q* is evident, we also fit it with the *β* model [19]

*D*is the fractal dimension of the set on which the cascade accumulates. The linear fitting gives

*D*= 2.85, and Eq. (8) with this dimension is displayed in Fig. 8 with dashed black line. The fitting performance is similar to the case of SL scaling. The fractal dimension of the most intermittent structure based on SL scaling is about 2.64. This dimension is larger than the value about 2.2 obtained by Ruiz-Chavarria et al [22] or 2.36 obtained by Sreenivasan [36] for passive scalars, also larger than the value about 2.0 for velocity fluctuation in incompressible turbulence. The

*β*model gives a little larger dimension. Let’s note that the way of determining

*γ*and

*C*in above is rather mathematical. Ruiz-Chavarria

*et al*. have shown that these two parameters can be expressed with

*ζ*(1),

*ζ*(2) and

*β*[22]. However, the obvious

*q*-dependence of

*β*here makes this method unavailable.

The convex behavior of *ζ*(*q*) for *q* ≤ 6 is “anomalous”. Based on extensive work on energy dissipation, velocity fluctuation and passive scalars, we know that the scaling exponent is often concave [19–23]. For the boundary layer, the large scale coherent structures dominate the flow. Recently many investigators have offered solid evidence for the existence of large sale structures in the supersonic turbulent boundary layers [37–39]. Our preliminary study based on Rayleigh scattering also implies the existence of large scale density structure, whose size can be about 1.0*δ* in streamwise direction. Now we illustrate the structure of OPL with power spectrum analysis. Given the spectrum *Φ*(*k*) of OPL, we known that the area under the log-linear graph of pre-multiplied spectrum *kΦ*(*k*) versus wavenumber *k* corresponds to the kinetic energy. So we can use the pre-multiplied spectrum to show the contribution of different wavenumber, or equivalently wavelength *λ* = 2*π*/*k*, to the energy. The averaged pre-multiplied spectrum of OPL is shown in Fig. 9
, in which the wavelength has been normalized by the thickness of the layer. The spectrum displays two peaks, locating around 0.2*δ* and 1.2*δ*. The main peak is broad and ranges from 1.0*δ* to 2.3*δ*. Thus the phenomenon that the smaller exponents of low orders than the SL prediction can be attributed to the very large scale structure of OPL.

## 4. Conclusion

In summary, the OPL of supersonic turbulent boundary layer is measured, and its hierarchical structure is analyzed. The NPLS technique is used to visualize the boundary layer, and density distribution without dimension is obtained by proper post-processing. We know that the structure function study of velocity or energy dissipation often relies on hot wire measurement for incompressible turbulence, and the frozen hypothesis is required to obtain the spatial distribution. In contrary, the NPLS technique can give spatial distribution of density directly. Based on the density obtained in this way, the structure function of OPL is computed and analyzed within the framework of SL hierarchical symmetry assumption. Our results indicate that the OPL obeys this assumption, but with parameter *β* depending on *q*. The scaling exponent of SF *ζ*(*q*) is computed and compared with the theoretical prediction of SL model and *β* model. Our experimental results agree with the theoretical model quite well for large *q*. The curve *ζ*(*q*) we obtained is convex for small *q*, which deviates from the theory evidently. We analyzed the structure of OPL with pre-multiplied spectrum, and conjecture that this deviation is due to its large scale structure.

## Acknowledgments

We would like to acknowledge the financial supports of the innovation research foundations for postgraduates of National University of Defense Technology and Hunan Province, and the National Basic Research Program (No. 2009CB724100) of China.

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