3B-4
9June 30 - July 3, 2015 Melbourne, Australia

Rapid growth of unsteady finite-amplitude perturbations in a supersonicboundary-layer flow

Daisuke Watanabe
Graduate School of Engineering
Toyama University
3190 Gofuku, Toyama-shi, Toyama 930-8555, Japandwata@eng.u-toyama.ac.jp
Hiroshi Maekawa
Graduate School of Informatics and Engineering
The University of electro-communications
1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japanmaekawa@mce.uec.ac.jp
In this study, a spatial simulation of a supersonic,
isothermal flat plate boundary layer flow at Mach 2.5 isanalyzed. Growth of finite amplitude disturbances ontransition to turbulence in a developing boundary layer
downstream require the application of fully spatial
formation in a numerical simulation without extended
temporal simplifying assumptions. The emphasis of thisstudy is to assess the transitional scenarios for supersonicboundary layer and the late stage of streak breakdown andobserving the transitional structures in a Lagrangian
tracking manner, which leads to finding an inherent
structure generated from finite-amplitude perturbations
which induces transition to turbulence in the supersonicboundary layer.

ABSTRACT
Spatial numerical simulations (SNS) are performed to
study transition to turbulence in a supersonic flat plateboundary layer forced by finite amplitude perturbations,where the freestream Mach numbers is 2.5 with theisothermal wall condition. Three-dimensional isotropic
disturbances are superimposed on the laminar profile for
Reynolds number based on the displacement thickness
*of 1000 at the inlet plane of the computational domain.
The effects of the peak location of the disturbance
spectrum and the magnitude distribution in the wall
normal direction are analyzed. Numerical results indicatethat energy spectra with a peak located at around thewavenumber of kmax=1.0 play an important role to induce
transition to turbulence. The disturbance at lower
wavenumbers with concomitant turbulence in the freestream shows an early appearance of hairpin-like structurein the transition region. The vortical structures on set oftransition to turbulence rapid evolving to a hairpin packet
with fine secondary structures downstream are shown inthis paper, which is an inherent structure inducestransition to turbulence in the supersonic boundary layer.

SMULATION DETAILSIn the SNS of spatially developing boundary layer, thenon-dimensional equations governing the conservation ofmass, momentum, and energy for a compressible
Newtonian fluid are solved. Note that displacement
thickness at the inlet boundary layer (δ*) is chosen to be a
representative length scale.
The computational domain size is Lx×Ly×Lz=300*×30*×33*. The form of the solenoidal disturbanceis given as

INTRODUCTIONA deep understanding of the principal route toturbulence in the wall-bounded shear flows is of great
fundamental and practical interest. Various disturbances
affect transition to turbulence in a laminar boundary layer,
which has been investigated in detail so far. The primarystage of transition in a low-turbulence environment has
been extensively studied by stability theories (Mack 1975)
and flow fluctuation measurements by Kendall (1975) or
Graziousi & Brown (2002). Understanding of the latestage of transition scenario or the nonlinear transition(Graziousi & Brown 2002) due to high level disturbances
for boundary layers is even less pronounced especially forsupersonic one due to experimental difficulties.

E ( k )  k 4 exp  2( k / k max ) 2 ,(1)

where k is wavenumber and the turbulence energy peaklocates at kmax. In order to meet the no-slip boundary
condition at a wall, a window function to impose zero
velocities at the wall is used. High-order compact upwind
biased scheme with boundary schemes (Deng et al. 1996,
Deng & Maekawa 1997) are used for spatial derivatives
and a 4th-order Runge-Kutta scheme for timeadvancement. The Navier-Stokes characteristic boundary
conditions by Poinsot and Lele (1992) are used in thestreamwise and normal directions and periodic boundary
conditions in the spanwise direction. For 3-D spatial
numerical simulations, after a grid-convergence analysis,1

A comparison between Fig.2 (a), (c), (e), and (g)indicates the effect of kmax of the disturbance on thetransitional structure in the boundary layer, when the flow
is forced by the disturbance with the same window widthof 0<y<4, as shown in Fig.1. Figures 2 (b), (d), (f), and (h)also show the effects of kmax of the disturbance on thetransitional structure in the boundary layer, where thesame window of 0<y<10 is adopted.
Figure 2 shows the long streamwise low-speed streaks
shown by blue contours with a wall plane. As shown in
Fig. 2 (a) and (b), low-speed regions are associated quasistreamwise near-wall vortices. Quasi-streamwise vortices
appear on the flank of the streak. Figures 3 (c), (d), (e), (f),(g) and (h) show many arch-vortices and these quasistreamwise vortices as well. Isolated near-wall arch
structures are clearly visualized in Fig.2(c) and (d).
Successive arch structures or hairpin heads on the lowspeed streak are dominant at downstream locations ofx>100 in Figs.2 (e), (f), (g) and (h). These numericalresults indicate that transition is induced effectively by
forcing at the lower wavenumbers of kmax such as
kmax=1.0 and 0.5 with the wide windows of 0<y<10. A
comparison of Fig.2 (g) and (h) indicates that forcing withwide windows is effective to create successive hairpins.
Figure 3 indicates the evolution of the shape-factor of theboundary layer forced by various upstream disturbanceswith three window widths of y<2.5, y<4, and y<10, where
the magnitude of each disturbance is 2.0%.

the computational grid is Nx×Ny×Nz= 901×101×96. The
grid is clustered toward the wall in the wall normal
direction. The finite amplitude of the disturbance is 1, 1.5,
2, 2.5% at the boundary layer inlet and kmax is 1.0 where
the corresponding wavelength is 2*.
Note that a window width affects transition toturbulence in the cases of 1.5, 2, and 2.5%. Figure 1
shows the window widths employed in this simulation,
where the inlet streamwise velocity distribution is alsopresented.
In the preliminary simulations of various disturbances,the effect of kmax was investigated, where the employed
kmax of the disturbances are π, 2.0, 1.0 and 0.5 and themagnitude of each disturbance is 2.0%.
1
0<y<2.50<y<40<y<10uin

, uin

0.8
0.6
0.4
0.2
0
0

5

10

15

y

Fig. 1 The distributions of the window functions  andthe inlet mean streamwise velocity uin presented by red
line.

RESULTSFirst of all, the effect of kmax was investigated. Figure 2
shows a top view of the visualized low-speed streaks (bluecontours) and quasi-streamwise vortices with hairpin
packets (yellow contours).

H

2.5

2

1.5

0

kmax= (2.0%) y<2.5
kmax= (2.0%) y<4
kmax= (2.0%) y<10
kmax=2.0 (2.0%) y<2.5
kmax=2.0 (2.0%) y<4
kmax=2.0 (2.0%) y<10
kmax=1.0 (2.0%) y<2.5
kmax=1.0 (2.0%) y<4
kmax=1.0 (2.0%) y<10
kmax=0.5 (2.0%) y<2.5
kmax=0.5 (2.0%) y<4
kmax=0.5 (2.0%) y<10100x/*in200

300

Fig. 3 Downstream evolution of shape factor for the
disturbance magnitude of 2.0%.
Figures 4 and 5 show the Fourier spectrum of u
velocity fluctuation at the streamwise location of x/*=150(mid-plane in the streamwise direction) for the flow
forced by the disturbance of kmax=2.0 and 1.0, respectively,
where  indicates spanwise wave number. As Fig. 4
shows, the spectrum peak locates at  =1.0 corresponding
to the streaky structures in the instantaneous flow field. As
shown in Fig.2(c), isolated arches appears aroundx/*=150. On the other hand, as shown in Fig.2 (e),successive hairpins appear around x=150. A comparisonof Fig.4 and Fig.5 indicate that streaky structures with theisolated arch on the low-speed streak are dominant at
x=150, however successive hairpins appears in Fig.5,

Fig. 2
Downstream evolution of second invariant
Q(=0.005; yellow) structure and low-speed streak u’(= 0.1; blue) for 2.0% case; a) kmax =, y<4, b) kmax=, y<10,c) kmax =2.0, y<4, d) kmax =2.0, y<10,e) kmax =1.0, y<4, f)
kmax =1.0, y<10, g) kmax=0.5,y<4 and h) kmax=0.5, y<10.

2where Fourier spectrum at 0.5< <2.0 becomes dominant
in this transitional flow. In the turbulent boundary layer
downstream, by the formation of an arch vortex (arch ofhorseshoe) spanning a few streaks, the elongated streak
disappears. The upstream lift-up streak and adjacentstreaks create another hairpin packets (secondary andtertiary hairpins), after a short evolution, these trailinglegs fill the boundary layer inside.

effective to induce transition and no essential difference isobserved in the transitional flow forced by the disturbance
kmax =0.5, as shown in Fig.3, where the decaying shapefactor in the cases of kmax =1.0 and 0.5 are similar.
Figure 6 indicates the evolution of the shape-factor of
the boundary layer forced by various upstream
disturbances, where kmax is 1.0, the disturbancemagnitudes are 1, 1.5, 2, 2.5% of the freestream velocity
and the widths of the window function are 0<y<2.5*,
0<y<4*, 0<y<10*. When the magnitude is very small at
the present Reynolds and Mach numbers, transition toturbulence is not achieved in the simulation, different
paths to transition may be active (see Fedorov (2011)).
However, the magnitude of this disturbance is more than
1.0% seems to leads transition to turbulence.

COHERENT MINIMAL STRUCTURE
In this section, we will focus our attention on theslower transitional structures in the computational domainof 0<x/*<300. For instances, the flow forced by the
disturbance of the magnitude of 2% with all of thewindow width, and the flow forced by the disturbance ofthe magnitude is 1.5% with the window width of 0<y<10
or 0<y<4 are focused in this section.
Figure 4. Fourier spectrum of velocity fluctuation in thecase of kmax =2.0 and the width of window function of
0<y<4*( x/δ*=150) .

2.5

H

2

1.5

1
0

kmax=1.0 (1.0%) 0<y<2.5
kmax=1.0 (1.0%) 0<y<4
kmax=1.0 (1.0%) 0<y<10
kmax=1.0 (1.5%) 0<y<2.5
kmax=1.0 (1.5%) 0<y<4
kmax=1.0 (1.5%) 0<y<10
kmax=1.0 (2.5%) 0<y<2.5
kmax=1.0 (2.5%) 0<y<4
kmax=1.0 (2.5%) 0<y<10

kmax=1.0 (2.0%) 0<y<2.5
kmax=1.0 (2.0%) 0<y<4
kmax=1.0 (2.0%) 0<y<10100x/ in*200

300

Figure 6. Evolution of shape-factor of the boundary
layer forced by various upstream disturbances(Parameters are magnitude of disturbance and width of
the window function)

Figure 5. Fourier spectrum of velocity fluctuation in thecase of kmax = 1.0 and the width of window function of
0<y<4*( x/δ*=150).
As indicated in the evolution of the shape-factor of theboundary layer shown in Fig.3, forcing by the
disturbances at the lower wavenumbers of kmax such as
kmax =1.0 and 0.5 with the wide windows of 0<y<10 iseffective for transition to turbulence. Secondly, in thispaper, effects of the magnitude of the disturbance areinvestigated for the disturbance of kmax =1.0, which is of

Figure 7. Fourier spectrum of velocity fluctuationin the case of the magnitude of 2% and the width ofwindow function of 0<y<10*.
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Figure 7 shows the Fourier spectrum of u velocity
fluctuation, where  indicates spanwise wave number. As
this figure shows, the spectrum peak locates at  =0.8～
1.0 corresponding to the streaky structures appeared in the
instantaneous flow field. In these cases, we can find a few
typical structure and streaky structures as well, wherethese structures grow or develop quickly downstream andthese structures are usually localized or isolated in theflow field.
Duguet et al. (2012) performed direct numericalsimulations of the low-speed boundary layer to track the
dynamics in the region of phase space separatingtransitional from relaminarizing trajectories. They foundthat the structure is dominated by a robust pair of lowspeed streaks, whose instabilities generate hairpin vortices
evolving downstream into transient disturbances.
Cherbini et al. (2013) showed localized flow structuresliving on the edge of chaos to understand transition toturbulence. They defined the structure as the hairpindominated edge state. Recent progress in understandingtransition to turbulence is based on the application ofconcepts from dynamical systems theory. The picture thatemerges builds on invariant solutions of the Navier-Stokes
equations such as fixed points and traveling waves.

transition to turbulence in the boundary layer. Cherubini etal. investigated the linear and nonlinear optimal
perturbations in a low-speed boundary layer. The
nonlinear optimal is characterized by spanwise inclined
vortices, whereas the linear optimal contains streamwise
vortices.
Another evidence of evolution to turbulent
boundary layer is energy spectrum at various stages.
Energy spectrum indicates the statistical feature of thetransitional stages.
Figures 9 a) and b) indicate the energy spectra offluctuating u velocities at x=60* and 120*, where St is
Strouhal number defined by the displacement thickness
and the freestream velocity. The Strouhal number
indicates the corresponding wave number and nondimensional frequency. As these figures show, the peak at
St=0.1 appears at y=0.5* (Black line) in the case at
x=60*, then the rapid growth in the spectrum downstream.
Note that the sampling frequency used for time averaging
is not high enough to capture high frequency phenomenain the simulation. Therefore, the energy spectrum
employed for time averaging to obtain longer time data
series does not follow high frequency phenomena, as
shown in Fig.9 (b). The energy shows no decrease forhigher wavenumbers.

t=800

a)

t =825uh2

t =850t =875

x=0

x=100

x=200

x=300

Figure 8. Snapshots of the vortical structure structure
emerged at the downstream location around x=120 at
nondimensional time t=800. The magnitude of
disturbance is 2% of the freestream velocity and thewindow width is 0<y<2.5*.

10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15 -210

y=0.5*y=1.0*y=2.0*y=3.0*y=4.0*y=5.0*y=6.0*

10-1
St

100

b)

uh2

e
Figure 8 shows the visualized low-speed streaks (color:
blue, isosurface of u=-0.1) and the vortical structures(color: yellow, isosurface of Q=0.005) visualized with isosurfaces of the second invariant Q of the velocity gradienttensor evolving to a hairpin packet at the non-dimensionaltime respectively. The visualized vortical structures on setof transition to turbulence are observed, where spanwiseinclined vortices and hairpin heads appear around x=120*
at t=800 and fine structures evolve along the streak shownin successive snapshots at t=825, 850 and 875. Anotheronset structures are observed around x=220* at t=800.
Hairpin heads observing in the outer boundary layer are
due to the lift-up low-speed streaks (see Cherubini et al. b2011). This fact suggests that an inherent structure induces

10-2
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10-4
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10-6
10-7
10-8
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10-12
10-13
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10-15 -210

y=0.5*y=1.0*y=2.0*y=3.0*y=4.0*y=5.0*y=6.0*

10-1
St

100

Figure 9. Energy spectra of u velocity fluctuations fora) x=60*, b) x=120*. The magnitude of disturbance is
2% and the window width is 0<y<10*.

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a)

10-2
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10-6
10-7
10-8
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y=0.5*y=1.0*y=2.0*y=3.0*y=4.0*y=5.0*y=6.0*

a)

uh2

uh2

streaky structure, which is important for rapid evolving
into structure breakdown.

10-1
St

100

10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
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10-13
10-14
-15
10

10-2

y=0.5*y=1.0*y=2.0*y=3.0*y=4.0*y=5.0*y=6.0*10-1
St

y=0.5*y=1.0*y=2.0*y=3.0*y=4.0*y=5.0*y=6.0*

10-1
St

100

b)

uh2

uh2

b)

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10-5
10-6
10-7
10-8
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10

100

Figure 10. Energy spectra of u velocity fluctuations for a)x=60*, b) x=120*. The magnitude of disturbance is 2%
and the window width is 0<y<2.5*.
The instantaneous flow field shows that the finestructures appear downstream and the oscillatory andbreakdown of the low-speed streaks can be observed in
the cases of high magnitude and the low magnitude withwide width of window function. However, as shown in
Fig.2, in the case of the window function of 0<y<2.5*,the energy spectrum grows gradually. Figure 10 shows theenergy spectra of fluctuating u velocities at x=60* and120* of the flow field forced with the window function of
0<y<2.5*. As Fig.10 (b) shows, the bump peak at St=0.1
at y=0.5* still appears at x=120*, which corresponds tothe slower evolving of isolated spanwise-inclined vortices
with a hairpin. This fact suggests that there is a minimal
characteristic structure generated from the finite amplitude
perturbations that induces transition to turbulence.
Hairpin-shaped vortices found by Duguet et al. (2012)
forms above the region where streaks pinch. Optimalvaricose perturbations grows very rapidly in a boundary
layer. Such optimal disturbances lead to transition for
initial energies. The structure found in this supersonicboundary layer simulation is close to hairpin vortices onpinched streaks in incompressible boundary layers or the
nonlinear optimum stage found by Cherubini et al. (2013).
On the contrary, as shown in Fig. 2 (e), the wider
window of 0<y<4 brings a few successive hairpins on the

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y=0.5*y=1.0*y=2.0*y=3.0*y=4.0*y=5.0*y=6.0*

10-1
St

100

Figure 11. Energy spectra of u velocity fluctuations for a)x=180*, b) x=270*. The magnitude of disturbance is 2%
and the window width is 0<y<10*.
Figure 11 shows the energy spectra at x=180* and270*. The shape-factors of the velocity profile at x=180*
and 270* and are higher and lower values than 1.5,
respectively. As shown in Fig.11, where the straight line
presents the power spectrum of -5/3, the energy is
distributed in a cascade fashion at x= 270*.

SUMMARY
In this paper we present a few coherent minimal
structures rapid evolving to a hairpin packet with finesecondary structures downstream, which is an inherent
structure and induces transition to turbulence in theboundary layer. The structure is characterized spanwizeinclined vortices and successive hairpins on the low-speed
streak. Energy spectrum indicates the statistical feature of
the transitional stages generated by the coherent structure.
The spanwise wave number peak locates at  =1.0, where
the corresponding wavelength is 2*. The Strouhal
number defined by the displacement thickness and thefreestream velocity indicates the corresponding wave

5

number and non-dimensional frequency at St=0.1 inside
the boundary layer.

ACKNOWLEDGEMENTS
This study was partly supported by Grants-in-Aid forscientific research (21560165, 24656518) from Japanese
Ministry of Education, Culture, Sports, Science andTechnology.

REFERENCES
Cherubini S., De Palma P. and Robinet J.-C., 2011a,
The minimal seed of turbulence transition in a boundary
layer flow J. Fluid Mech. Vol.689, pp.221-253.
Cherubini S., De Palma P. and Robinet J.-C. andBottaro A., 2011b, Edge states in a boundary layer,
Physics in Fluids, Vol.23, 051705.
Cherubini S., De Tullio M. D., De Palma P. andPascazio G., 2013, Transient growth of the flow past athree-dimensional smooth roughness element, J. Fluid
Mech. Vol.724, pp.642-670.
Deng, X., Maekawa, H. and Shen, C., 1996, A Classof High Order Dissipative Compact Schemes, AIAA Paper,
No. 96-1972.
Deng X. and Maekawa H., 1997, Compact high-orderaccurate nonlinear schemes. Journal of Computational
Physics, Vol. 130, pp. 77-91.
Duguet Y., Schlatter P., Henningson D. S., Eckhardt
B., 2012, Self-Sustained Localized Structures in a
Boundary Layer Flow, Physical Review Letters, Vol.108,044501.
Fedorov A., 2011, Transition and Stability of HighSpeed Boundary Layers Annu. Rev. Fluid Mech. Vol 43,
pp.79-95.
Graziousi,P.& Brown G., 2002, Experiments onstability and transition at Mach 3, J. Fluid Mech.,Vol.407,
pp.83-124.
Kendall, J.M., 1975, Wind Tunnel Experiments
Relating to Supersonic and Hypersonic Boundary layer
Transition, AIAA J. Vol.13, pp.290-299.
Poinsot T J and Lele S K, 1992, Boundary Conditions
for Direct Simulations of Compressible Viscous Flows,
Journal Computational Physics, Vol.101, pp.104-129.

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