## Hierarchical structure of the optical path length of the supersonic turbulent boundary layer |

Optics Express, Vol. 20, Issue 15, pp. 16494-16503 (2012)

http://dx.doi.org/10.1364/OE.20.016494

Acrobat PDF (1021 KB)

### 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

3. M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. **44**(1), 299–321 (2012). [CrossRef]

4. M. M. Malley, G. W. Sutton, and N. Kincheloe, “Beam-jitter measurements of turbulent aero-optical path differences,” Appl. Opt. **31**(22), 4440–4443 (1992). [CrossRef] [PubMed]

6. A. Mani, M. Wang, and P. Moin, “Statistical description of the free-space propagation of highly aberrated optical beams,” J. Opt. Soc. Am. A **23**(12), 3027–3035 (2006). [CrossRef] [PubMed]

*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

12. C. R. Truman and M. J. Lee, “Effects of organized turbulence structures on the phase distortion in a coherent beam propagating through a turbulent shear flow,” Phys. Fluids A **2**(5), 851–857 (1990). [CrossRef]

*et al*. [13

13. E. Tromeur, E. Garnier, and P. Sagaut, “Large eddy simulations of aero-optical effects in a spatially developing turbulent boundary layer,” J. Turbul. **7**, N1– N28 (2006). [CrossRef]

*et al*. [14

14. K. Wang and M. Wang, “Aero-optics of subsonic turbulent boundary layers,” J. Fluid Mech. **696**, 122–151 (2012). [CrossRef]

15. G. W. Sutton, “Aero-optical foundations and applications,” AIAA J. **23**(10), 1525–1537 (1985). [CrossRef]

## 2. Experimental setup

*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

25. Y. X. Zhao, S. H. Yi, L. F. Tian, and Z. Y. Cheng, “Supersonic flow imaging via nanoparticles,” Sci. China Ser. E **52**(12), 3640–3648 (2009). [CrossRef]

27. H. Lin, S. H. Yi, Y. X. Zhao, L. F. Tian, and Z. Chen, “Visualization of coherent structures in a supersonic flat-plate boundary layer,” Chin. Sci. Bull. **56**(6), 489–494 (2011). [CrossRef]

*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.

28. M. Smith, A. Smits, and R. Miles, “Compressible boundary-layer density cross sections by UV Rayleigh scattering,” Opt. Lett. **14**(17), 916–918 (1989). [CrossRef] [PubMed]

29. M. Smith and A. Smits, “Visualization of the structure of supersonic turbulent boundary layers,” Exp. Fluids **18**(4), 288–302 (1995). [CrossRef]

_{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

25. Y. X. Zhao, S. H. Yi, L. F. Tian, and Z. Y. Cheng, “Supersonic flow imaging via nanoparticles,” Sci. China Ser. E **52**(12), 3640–3648 (2009). [CrossRef]

*et al*. [30

30. L. F. Tian, S. H. Yi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G **52**(9), 1357–1363 (2009). [CrossRef]

32. R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. **23**(1), 261–304 (1991). [CrossRef]

*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.

26. L. He, S. H. Yi, Y. X. Zhao, L. F. Tian, and Z. Chen, “Experimental study of a supersonic turbulent boundary layer using PIV,” Sci. China Ser. G **54**(9), 1702–1709 (2011). [CrossRef]

## 3. Hierarchical structure description of OPL

*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.

*q*-th order structure function (SF) of OPL is defined aswhere the bracket < > denotes averages over

*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.

21. Z. S. She and E. Leveque, “Universal scaling laws in fully developed turbulence,” Phys. Rev. Lett. **72**(3), 336–339 (1994). [CrossRef] [PubMed]

33. Z. S. She and Z. X. Zhang, “Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems,” Acta Mech. Sin. **25**(3), 279–294 (2009). [CrossRef]

21. Z. S. She and E. Leveque, “Universal scaling laws in fully developed turbulence,” Phys. Rev. Lett. **72**(3), 336–339 (1994). [CrossRef] [PubMed]

33. Z. S. She and Z. X. Zhang, “Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems,” Acta Mech. Sin. **25**(3), 279–294 (2009). [CrossRef]

*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.

33. Z. S. She and Z. X. Zhang, “Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems,” Acta Mech. Sin. **25**(3), 279–294 (2009). [CrossRef]

_{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*.

*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

23. C. Sun, Q. Zhou, and K. Q. Xia, “Cascades of velocity and temperature fluctuations in buoyancy-driven thermal turbulence,” Phys. Rev. Lett. **97**(14), 144504 (2006). [CrossRef] [PubMed]

37. B. Ganapathisubramani, N. T. Clemens, and D. S. Dolling, “Large-scale motions in a supersonic turbulent boundary layer,” J. Fluid Mech. **556**, 271–282 (2006). [CrossRef]

39. G. E. Elsinga, R. J. Adrian, B. W. Van Oudheusden, and F. Scarano, “Three-dimensional vortex organization in a high-Reynolds-number supersonic turbulent boundary layer,” J. Fluid Mech. **644**, 35–60 (2010). [CrossRef]

*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

*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

## References and links

1. | K. G. Gilbert and L. J. Otten, eds., |

2. | E. J. Jumper and E. J. Fitzgerald, “Resent advances in aero-optics,” Prog. Aerosp. Sci. |

3. | M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. |

4. | M. M. Malley, G. W. Sutton, and N. Kincheloe, “Beam-jitter measurements of turbulent aero-optical path differences,” Appl. Opt. |

5. | R. J. Hugo and E. J. Jumper, “Experimental measurement of a time-varying optical path difference by the small-aperture beam technique,” Appl. Opt. |

6. | A. Mani, M. Wang, and P. Moin, “Statistical description of the free-space propagation of highly aberrated optical beams,” J. Opt. Soc. Am. A |

7. | H. A. Stine and W. Winovitch, “Light diffusion through high-speed turbulent boundary layers,” NACA Res. Mem. A56B21 (NACA, 1956). |

8. | S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, “Aero-optical characteristics of compressible, subsonic turbulent boundary layer,” AIAA paper 2003–3606 (American Institute of Aeronautics and Astronautics, 2003). |

9. | S. Gordeyev, J. A. Cress, and E. J. Jumper, “Aero-optical properties of subsonic, turbulent boundary layers,” Preprint. |

10. | S. Gordeyev, E. J. Jumper, and T. E. Hayden, “Aero-optics of supersonic boundary layers,” AIAA paper 2011–1325 (American Institute of Aeronautics and Astronautics, 2011). |

11. | C. M. Wyckham and A. J. Smith, “Comparison of aero-optics distortions in hypersonic and transonic, turbulent boundary layers with gas injection,” AIAA paper 2006–3067 (American Institute of Aeronautics and Astronautics, 2006). |

12. | C. R. Truman and M. J. Lee, “Effects of organized turbulence structures on the phase distortion in a coherent beam propagating through a turbulent shear flow,” Phys. Fluids A |

13. | E. Tromeur, E. Garnier, and P. Sagaut, “Large eddy simulations of aero-optical effects in a spatially developing turbulent boundary layer,” J. Turbul. |

14. | K. Wang and M. Wang, “Aero-optics of subsonic turbulent boundary layers,” J. Fluid Mech. |

15. | G. W. Sutton, “Aero-optical foundations and applications,” AIAA J. |

16. | A. P. Freeman and H. J. Catrakis, “Direct reduction of aero-optical aberrations by large structure suppression control in turbulence,” AIAA J. |

17. | V. I. Tatarski, |

18. | Q. Gao, Z. F. Jiang, S. H. Yi, L. He, and Y. X. Zhao, “Structure function of the refractive index of the supersonic turbulent boundary layer,” Submitted. |

19. | U. Frisch, |

20. | G. Parisi and U. Frisch, “On the singularity structure of fully developed turbulence,” in |

21. | Z. S. She and E. Leveque, “Universal scaling laws in fully developed turbulence,” Phys. Rev. Lett. |

22. | G. Ruiz-Chavarria, C. Baudet, and S. Ciliberto, “Scaling laws and dissipation scales of a passive scalar in fully developed turbulence,” Physica D |

23. | C. Sun, Q. Zhou, and K. Q. Xia, “Cascades of velocity and temperature fluctuations in buoyancy-driven thermal turbulence,” Phys. Rev. Lett. |

24. | A. Smits and J. Dussauge, |

25. | Y. X. Zhao, S. H. Yi, L. F. Tian, and Z. Y. Cheng, “Supersonic flow imaging via nanoparticles,” Sci. China Ser. E |

26. | L. He, S. H. Yi, Y. X. Zhao, L. F. Tian, and Z. Chen, “Experimental study of a supersonic turbulent boundary layer using PIV,” Sci. China Ser. G |

27. | H. Lin, S. H. Yi, Y. X. Zhao, L. F. Tian, and Z. Chen, “Visualization of coherent structures in a supersonic flat-plate boundary layer,” Chin. Sci. Bull. |

28. | M. Smith, A. Smits, and R. Miles, “Compressible boundary-layer density cross sections by UV Rayleigh scattering,” Opt. Lett. |

29. | M. Smith and A. Smits, “Visualization of the structure of supersonic turbulent boundary layers,” Exp. Fluids |

30. | L. F. Tian, S. H. Yi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G |

31. | G. R. Osche, |

32. | R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. |

33. | Z. S. She and Z. X. Zhang, “Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems,” Acta Mech. Sin. |

34. | R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, “Extended self-similarity in turbulent flows,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

35. | B. Davidovitch, M. H. Jensen, A. Levermann, J. Mathiesen, and I. Procaccia, “Thermodynamic formalism of the harmonic measure of diffusion limited aggregates: phase transition,” Phys. Rev. Lett. |

36. | K. R. Sreenivasan, “Fractals and multifractals in fluid turbulence,” Annu. Rev. Fluid Mech. |

37. | B. Ganapathisubramani, N. T. Clemens, and D. S. Dolling, “Large-scale motions in a supersonic turbulent boundary layer,” J. Fluid Mech. |

38. | M. J. Ringuette, M. Wu, and M. P. Martin, “Coherent structures in direct numerical simulation of turbulent boundary layer at Mach 3,” J. Fluid Mech. |

39. | G. E. Elsinga, R. J. Adrian, B. W. Van Oudheusden, and F. Scarano, “Three-dimensional vortex organization in a high-Reynolds-number supersonic turbulent boundary layer,” J. Fluid Mech. |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(030.7060) Coherence and statistical optics : Turbulence

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 20, 2012

Manuscript Accepted: June 22, 2012

Published: July 5, 2012

**Citation**

Qiong Gao, Shihe Yi, Zongfu Jiang, Lin He, and Yuxin Zhao, "Hierarchical structure of the optical path length of the supersonic turbulent boundary layer," Opt. Express **20**, 16494-16503 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16494

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### References

- K. G. Gilbert and L. J. Otten, eds., Aero-Optical Phenomena (American Institute of Aeronautics and Astronautics, 1982).
- E. J. Jumper and E. J. Fitzgerald, “Resent advances in aero-optics,” Prog. Aerosp. Sci.37(3), 299–339 (2001). [CrossRef]
- M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech.44(1), 299–321 (2012). [CrossRef]
- M. M. Malley, G. W. Sutton, and N. Kincheloe, “Beam-jitter measurements of turbulent aero-optical path differences,” Appl. Opt.31(22), 4440–4443 (1992). [CrossRef] [PubMed]
- R. J. Hugo and E. J. Jumper, “Experimental measurement of a time-varying optical path difference by the small-aperture beam technique,” Appl. Opt.35(22), 4436–4447 (1996). [CrossRef] [PubMed]
- A. Mani, M. Wang, and P. Moin, “Statistical description of the free-space propagation of highly aberrated optical beams,” J. Opt. Soc. Am. A23(12), 3027–3035 (2006). [CrossRef] [PubMed]
- H. A. Stine and W. Winovitch, “Light diffusion through high-speed turbulent boundary layers,” NACA Res. Mem. A56B21 (NACA, 1956).
- S. Gordeyev, E. J. Jumper, T. T. Ng, and A. B. Cain, “Aero-optical characteristics of compressible, subsonic turbulent boundary layer,” AIAA paper 2003–3606 (American Institute of Aeronautics and Astronautics, 2003).
- S. Gordeyev, J. A. Cress, and E. J. Jumper, “Aero-optical properties of subsonic, turbulent boundary layers,” Preprint.
- S. Gordeyev, E. J. Jumper, and T. E. Hayden, “Aero-optics of supersonic boundary layers,” AIAA paper 2011–1325 (American Institute of Aeronautics and Astronautics, 2011).
- C. M. Wyckham and A. J. Smith, “Comparison of aero-optics distortions in hypersonic and transonic, turbulent boundary layers with gas injection,” AIAA paper 2006–3067 (American Institute of Aeronautics and Astronautics, 2006).
- C. R. Truman and M. J. Lee, “Effects of organized turbulence structures on the phase distortion in a coherent beam propagating through a turbulent shear flow,” Phys. Fluids A2(5), 851–857 (1990). [CrossRef]
- E. Tromeur, E. Garnier, and P. Sagaut, “Large eddy simulations of aero-optical effects in a spatially developing turbulent boundary layer,” J. Turbul.7, N1– N28 (2006). [CrossRef]
- K. Wang and M. Wang, “Aero-optics of subsonic turbulent boundary layers,” J. Fluid Mech.696, 122–151 (2012). [CrossRef]
- G. W. Sutton, “Aero-optical foundations and applications,” AIAA J.23(10), 1525–1537 (1985). [CrossRef]
- A. P. Freeman and H. J. Catrakis, “Direct reduction of aero-optical aberrations by large structure suppression control in turbulence,” AIAA J.46(10), 2582–2590 (2008). [CrossRef]
- V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
- Q. Gao, Z. F. Jiang, S. H. Yi, L. He, and Y. X. Zhao, “Structure function of the refractive index of the supersonic turbulent boundary layer,” Submitted.
- U. Frisch, Turbulence (Cambridge University Press, 1995).
- G. Parisi and U. Frisch, “On the singularity structure of fully developed turbulence,” in Proc. of Int. School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, M. Ghil, R. Benzi, and G. Parisi eds. (Amsterdam, 1985), pp. 84–87.
- Z. S. She and E. Leveque, “Universal scaling laws in fully developed turbulence,” Phys. Rev. Lett.72(3), 336–339 (1994). [CrossRef] [PubMed]
- G. Ruiz-Chavarria, C. Baudet, and S. Ciliberto, “Scaling laws and dissipation scales of a passive scalar in fully developed turbulence,” Physica D99(2-3), 369–380 (1996). [CrossRef]
- C. Sun, Q. Zhou, and K. Q. Xia, “Cascades of velocity and temperature fluctuations in buoyancy-driven thermal turbulence,” Phys. Rev. Lett.97(14), 144504 (2006). [CrossRef] [PubMed]
- A. Smits and J. Dussauge, Turbulent Shear Layers in Supersonic Flow (Springer, 2006).
- Y. X. Zhao, S. H. Yi, L. F. Tian, and Z. Y. Cheng, “Supersonic flow imaging via nanoparticles,” Sci. China Ser. E52(12), 3640–3648 (2009). [CrossRef]
- L. He, S. H. Yi, Y. X. Zhao, L. F. Tian, and Z. Chen, “Experimental study of a supersonic turbulent boundary layer using PIV,” Sci. China Ser. G54(9), 1702–1709 (2011). [CrossRef]
- H. Lin, S. H. Yi, Y. X. Zhao, L. F. Tian, and Z. Chen, “Visualization of coherent structures in a supersonic flat-plate boundary layer,” Chin. Sci. Bull.56(6), 489–494 (2011). [CrossRef]
- M. Smith, A. Smits, and R. Miles, “Compressible boundary-layer density cross sections by UV Rayleigh scattering,” Opt. Lett.14(17), 916–918 (1989). [CrossRef] [PubMed]
- M. Smith and A. Smits, “Visualization of the structure of supersonic turbulent boundary layers,” Exp. Fluids18(4), 288–302 (1995). [CrossRef]
- L. F. Tian, S. H. Yi, Y. X. Zhao, L. He, and Y. Z. Cheng, “Study of density field measurement based on NPLS technique in supersonic flow,” Sci. China Ser. G52(9), 1357–1363 (2009). [CrossRef]
- G. R. Osche, Optical Detection Theory for Laser Applications (Wiley-Interscience, 2002).
- R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech.23(1), 261–304 (1991). [CrossRef]
- Z. S. She and Z. X. Zhang, “Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems,” Acta Mech. Sin.25(3), 279–294 (2009). [CrossRef]
- R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, “Extended self-similarity in turbulent flows,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics48(1), R29–R32 (1993). [CrossRef] [PubMed]
- B. Davidovitch, M. H. Jensen, A. Levermann, J. Mathiesen, and I. Procaccia, “Thermodynamic formalism of the harmonic measure of diffusion limited aggregates: phase transition,” Phys. Rev. Lett.87(16), 164101 (2001). [CrossRef] [PubMed]
- K. R. Sreenivasan, “Fractals and multifractals in fluid turbulence,” Annu. Rev. Fluid Mech.23(1), 539–604 (1991). [CrossRef]
- B. Ganapathisubramani, N. T. Clemens, and D. S. Dolling, “Large-scale motions in a supersonic turbulent boundary layer,” J. Fluid Mech.556, 271–282 (2006). [CrossRef]
- M. J. Ringuette, M. Wu, and M. P. Martin, “Coherent structures in direct numerical simulation of turbulent boundary layer at Mach 3,” J. Fluid Mech.594, 59–69 (2008). [CrossRef]
- G. E. Elsinga, R. J. Adrian, B. W. Van Oudheusden, and F. Scarano, “Three-dimensional vortex organization in a high-Reynolds-number supersonic turbulent boundary layer,” J. Fluid Mech.644, 35–60 (2010). [CrossRef]

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