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

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    [CrossRef]

2012 (2)

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

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

2011 (2)

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]

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]

2010 (1)

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]

2009 (3)

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]

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]

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]

2008 (2)

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]

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]

2006 (4)

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]

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]

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. 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]

2001 (2)

E. J. Jumper and E. J. Fitzgerald, “Resent advances in aero-optics,” Prog. Aerosp. Sci. 37(3), 299–339 (2001).
[CrossRef]

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]

1996 (2)

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]

G. Ruiz-Chavarria, C. Baudet, and S. Ciliberto, “Scaling laws and dissipation scales of a passive scalar in fully developed turbulence,” Physica D 99(2-3), 369–380 (1996).
[CrossRef]

1995 (1)

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

1994 (1)

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

1993 (1)

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 48(1), R29–R32 (1993).
[CrossRef] [PubMed]

1992 (1)

1991 (2)

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

K. R. Sreenivasan, “Fractals and multifractals in fluid turbulence,” Annu. Rev. Fluid Mech. 23(1), 539–604 (1991).
[CrossRef]

1990 (1)

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]

1989 (1)

1985 (1)

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

Adrian, R. J.

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]

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

Baudet, C.

G. Ruiz-Chavarria, C. Baudet, and S. Ciliberto, “Scaling laws and dissipation scales of a passive scalar in fully developed turbulence,” Physica D 99(2-3), 369–380 (1996).
[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. Topics 48(1), R29–R32 (1993).
[CrossRef] [PubMed]

Benzi, R.

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 48(1), R29–R32 (1993).
[CrossRef] [PubMed]

Catrakis, H. J.

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]

Chen, Z.

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]

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]

Cheng, Y. Z.

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]

Cheng, Z. Y.

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]

Ciliberto, S.

G. Ruiz-Chavarria, C. Baudet, and S. Ciliberto, “Scaling laws and dissipation scales of a passive scalar in fully developed turbulence,” Physica D 99(2-3), 369–380 (1996).
[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. Topics 48(1), R29–R32 (1993).
[CrossRef] [PubMed]

Clemens, N. T.

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]

Davidovitch, B.

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]

Dolling, D. S.

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]

Elsinga, G. E.

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]

Fitzgerald, E. J.

E. J. Jumper and E. J. Fitzgerald, “Resent advances in aero-optics,” Prog. Aerosp. Sci. 37(3), 299–339 (2001).
[CrossRef]

Freeman, A. P.

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]

Ganapathisubramani, B.

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]

Gao, Q.

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.

Garnier, E.

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]

Gordeyev, S.

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

He, L.

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]

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]

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.

Hugo, R. J.

Jensen, M. H.

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]

Jiang, Z. F.

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.

Jumper, E. J.

Kincheloe, N.

Lee, M. J.

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]

Leveque, E.

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

Levermann, A.

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]

Lin, H.

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]

Malley, M. M.

Mani, A.

Martin, M. P.

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]

Massaioli, F.

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 48(1), R29–R32 (1993).
[CrossRef] [PubMed]

Mathiesen, J.

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]

Miles, R.

Moin, P.

Procaccia, I.

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]

Ringuette, M. J.

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]

Ruiz-Chavarria, G.

G. Ruiz-Chavarria, C. Baudet, and S. Ciliberto, “Scaling laws and dissipation scales of a passive scalar in fully developed turbulence,” Physica D 99(2-3), 369–380 (1996).
[CrossRef]

Sagaut, P.

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]

Scarano, F.

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]

She, Z. S.

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]

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

Smith, M.

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

Smits, A.

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

Sreenivasan, K. R.

K. R. Sreenivasan, “Fractals and multifractals in fluid turbulence,” Annu. Rev. Fluid Mech. 23(1), 539–604 (1991).
[CrossRef]

Succi, S.

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 48(1), R29–R32 (1993).
[CrossRef] [PubMed]

Sun, C.

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]

Sutton, G. W.

Tian, L. F.

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]

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]

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]

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]

Tripiccione, R.

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 48(1), R29–R32 (1993).
[CrossRef] [PubMed]

Tromeur, E.

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]

Truman, C. R.

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]

Van Oudheusden, B. W.

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[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Sketch of visualizing the supersonic turbulent boundary layer in side view plane with NPLS technique.

Fig. 2
Fig. 2

A NPLS image of the supersonic boundary layer with Mach number 3.0. The flow is from left to right. This image has been enhanced by Gamma transform with γ = 0.7 (The images used in our computation below are the original ones).

Fig. 3
Fig. 3

Comparison of the density between the one obtained with Crocco-Busemann relationship and the one obtained directly from the NPLS image.

Fig. 4
Fig. 4

The OPL relevant to the NPLS image in Fig. 2.

Fig. 5
Fig. 5

SF of the OPL with order 2 (red circles), 8 (black squares), and 16 (blue triangles).

Fig. 6
Fig. 6

Check the hierarchical symmetry assumption with Eq. (6) for q = 5 (top), q = 10 (middle), and q = 15 (bottom).

Fig. 7
Fig. 7

The parameter β in SL scaling obtained by linear fitting for different q.

Fig. 8
Fig. 8

The scaling exponent ζ(q) obtained from experiment (red circles) and its comparison with SL model (black line) and β model (dashed line).

Fig. 9
Fig. 9

The pre-multiplied power spectrum of OPL.

Equations (8)

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d A =2.44(m+1)λ f # ,
L(x)= 0 H (1+ K GD ρ(x,y))dy,
D q (x)=<|L(x+ x 0 )L( x 0 ) | q >, q=0, 1,
H q (x)= D q+1 (x)/ D (q) (x), q=0, 1,
H q+1 (x)= A q H q (x) β H (x) 1β , β1,
ζ(q)=γq+C(1 β q ),
H q+1 (x)/ H 2 (x)=( A q / A 1 ) ( H q (x)/ H 1 (x)) β .
ζ(q)=q/3+(3D)(1q/3),

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