![SS instability spread angle as a function of incident shock-wave Mach number. Theoretical predictions in thick lines: θw=45° (solid line), θw=40° (dashed-dotted line), θw=30° (upper dashed line), and θw=20° (lower dashed line). Thin lines around the thick lines mark the error in the model prediction [see Eq. (1)]. Experimental results and error bars are also plotted; (○), (◇), (△), and (▽) mark experiments conducted with θw=45°, 40°, 30°, and 20°, respectively. All of the experiments were conducted with ambient air at P0=10.1 kPa, apart from the M=1.5 experiments conducted at P0=100 kPa.](https://www.researchgate.net/profile/G-Ben-Dor/publication/7070052/figure/fig3/AS:286432661917699@1445302276360/SS-instability-spread-angle-as-a-function-of-incident-shock-wave-Mach-number-Theoretical_Q320.jpg)
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Shock-Wave Mach-Reflection Slip-Stream Instability:
A Secondary Small-Scale Turbulent Mixing Phenomenon
A. Rikanati
Department of Physics, Ben Gurion University, Beer-Sheva 84015, Israel
O. Sadot
Department of Mechanical Engineering, Ben Gurion University, Beer-Sheva 84015, Israel,
and Department of Physics, Nuclear Research Center, Negev 84190, Israel
G. Ben-Dor
Department of Mechanical Engineering, Ben Gurion University, Beer-Sheva 84015, Israel
D. Shvarts
Department of Physics, Ben Gurion University, Beer-Sheva 84015, Israel,
Department of Mechanical Engineering, Ben Gurion University, Beer-Sheva 84015, Israel,
and Department of Physics, Nuclear Research Center, Negev 84190, Israel
T. Kuribayashi and K. Takayama
Interdisciplinary Shock-Wave Research Center, Institute of Fluid Science, Tohoku University, Sendai, Japan
(Received 15 December 2005; published 3 May 2006)
Theoretical and experimental research, on the previously unresolved instability occurring along the slip
stream of a shock-wave Mach reflection, is presented. Growth rates of the large-scale Kelvin-Helmholtz
shear flow instability are used to model the evolution of the slip-stream instability in ideal gas, thus
indicating secondary small-scale growth of the Kelvin-Helmholtz instability as the cause for the slip-
stream thickening. The model is validated through experiments measuring the instability growth rates for
a range of Mach numbers and reflection wedge angles. Good agreement is found for Reynolds numbers of
Re > 2 10
4
. This work demonstrates, for the first time, the use of large-scale models of the Kelvin-
Helmholtz instability in modeling secondary turbulent mixing in hydrodynamic flows, a methodology
which could be further implemented in many important secondary mixing processes.
DOI: 10.1103/PhysRevLett.96.174503 PACS numbers: 47.20.Lz, 47.20.Ft, 47.40.Nm, 47.63.mc
Understanding secondary turbulent mixing in complex
unstable hydrodynamic flows is of great importance in
achieving gain in laser driven inertial confinement fusion
(ICF) as well as in many astrophysical processes [1–3].
Mach reflections (MRs) are a well known shock-wave
related phenomenon occurring when an oblique shock
wave reflects from a rigid wall (or interacts with a second
shock wave) and are of great importance in many hydro-
dynamic flows [4]. The basic feature of a Mach reflection is
that of the three shocks structure, giving rise to the slip-
stream (SS) instability. In this Letter, we show, for the first
time, that the growth of the SS instability is due to sec-
ondary turbulent mixing. Through modeling this instabil-
ity, as described further on in this Letter, we implement a
new technique for understanding secondary turbulent mix-
ing, a technique which could be further implemented for
many other hydrodynamic flows.
The MR three shocks structure is a phenomenon widely
demonstrated in many experimental works (see [4] for
examples), as well as in the example from the current
work illustrated in Fig. 1 (see figure caption for details).
The three shocks structure appears when the inclination
angle, between the incident shock and the bounding wall, is
smaller than a critical angle defined through the detach-
ment criteria [5], depending on the shock-wave Mach
number and the material equation of state (see [4] and
references therein for further details). In the MR structure,
the SS is of a unique hydrodynamic nature, being a dis-
continuity which is not a shock. The SS separates between
two regions (regions 2 and 3 in Fig. 1) of different densities
(
2
3
) and tangential velocities (v
k
2
v
k
3
) but of equal
pressures (p
2
p
3
) and of zero perpendicular velocities
(v
?
2
v
?
3
0), i.e., a shear flow. As can be seen from
Fig. 1, and in contrast with the sharp nature of the three
shock waves, the SS discontinuity thickness increases
downstream from the triple point. To our knowledge, the
full nature of this experimentally observed phenomenon,
known as the SS instability, is still unresolved, with the
leading assumption for the cause for this instability relating
to the viscosity generated boundary layer effect [4].
As is the case in the SS discontinuity, the Kelvin-
Helmholtz (KH) shear flow instability [6] occurs when
two fluids flow with proximity to each other with a tangen-
tial velocity difference, defined as the shear velocity. Under
this instability, small perturbations on the two fluid inter-
face evolve into a formation of vortices causing the two
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fluids to turbulently mix. In this Letter, we demonstrate
that the cause for the SS thickening is the KH instability
evolving on the two fluid interface, generating secondary
small-scale turbulent mixing.
The KH instability large-scale behavior has been thor-
oughly investigated through a wide range of experimental,
numerical, and theoretical work resulting in an understand-
ing of its growth rates and characteristics and of the main
mechanisms dominating its evolution (see, for example,
[6,7]). By implementing previously reported KH large-
scale instability growth rates [6,7], we try to model the
spread angle of the SS instability as a function of the MR
flow parameters. In the following paragraphs, a brief de-
scription of the growth rates of the large-scale KH insta-
bility will be presented, followed by a detailed modeling of
the SS instability evolution. The model predictions are then
verified through comprehensive experimental research.
Finally, a short description of viscous effects and the
relation to the Re number of the flow is presented, again
supported by experimental results.
Apart from revealing the nature of this previously un-
resolved phenomenon (the SS instability), the success of
the process described in this Letter demonstrates, for the
first time, that large-scale models of the KH instability can
be implemented in describing secondary turbulent mixing
in hydrodynamic flows. Similar methodologies could be
further implemented to many other phenomena involving
secondary mixing, such as the ICF relevant Rayleigh-
Taylor and Richtmyer-Meshkov instabilities [2,3]. It
should be mentioned that an attempt to numerically de-
scribe secondary mixing phenomena through the solution
of full 2D or 3D Euler equations is expected to be very
difficult, if not impossible. That is due to the very large
number of computational cells required to describe both
the large-scale and the small-scale mixing processes.
As expected from dimensional considerations, the width
of the KH large-scale turbulent mixing zone (TMZ)
evolves with time according to htcvt, where v
is the shear velocity, t represents the evolution time, and
c 0:19 0:01 is a dimensionless constant derived ex-
perimentally [6], numerically, and recently even theoreti-
cally [7]. It should be mentioned that, in most experiments,
the instability growth is measured spatially, i.e., as a func-
tion of the advection distance from the mixing starting
point. The spatial mixing growth rate is easily related to
the temporal growth by assuming the average flow flows
downstream with the fluid average velocity, resulting in
hx2cv
1
v
2
=v
1
v
2
x, where v
1
and v
2
are
the two fluid velocities and c is the previously mentioned
dimensionless constant. To this equation, two corrections
must be introduced. The first, as reported in Ref. [7], is for
cases of fluids of two different densities, and the second is
for high Mach number flows (
v
a
> 1, where a is the
fluid average sound speed). Following these two correc-
tions, the TMZ width hx becomes
hx0:38 0:02
Sv
1
;v
2
1 2f
d
1
=
2
Sv
1
;v
2
xf
HiMach
v
a
;
f
d
1
;
2
1
2
1
2
=
1
p
1
2
=
1
p
;
(1)
where Sv
1
;v
2
v
1
v
2
=v
1
v
2
. f
HiMach
v
a
0:51 tanh2
v
a
1:2 is the high Mach correction
which is based on a parametric fit of the results shown in
Ref. [8]. Finally, based on Eq. (1), the spread angle of the
spatially growing instability is found according to
spread
arctan
hx
2x
: (2)
When trying to implement Eq. (2) for the growth of the SS
instability, one needs to describe v
1
, v
2
,
1
, and
2
of the
two fluids along the SS. These physical properties are the
velocities (in the frame of reference moving with the triple
point) and the densities of regions 2 and 3 in Fig. 1. All are
analytically calculated using the three-shock theory first
suggested by von Neumann [5]. The theory is based on the
traditional shock-wave conservation equations which are
implemented for a single oblique shock. For an ideal gas
(see [9]), the flow parameters behind an oblique shock can
be analytically derived through a set of analytical trans-
lation functions, as a function of the shock inclination
angle and the preshock conditions. By implementing these
translation functions 3 times, once for each shock in the
FIG. 1. A holographic interferometry image of a M 1:9
Mach reflection in air with a wedge angle of
w
30
(see
text for further experimental details). Marked are the following
features: the incident shock (IS), the Mach stem (MS), the
reflected shock (RS), the triple point (TP), the slip stream
(SS), and the triple-point trajectory (TPT) which is also indicated
by a dashed line. Four flow regions are distinguished: the non-
shocked air (0), the shocked air above the reflection (1), the
shocked air after the reflection (2), and the Mach-stem shocked
air, before the SS (3).
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three-shock complex, and by demanding the closure rela-
tions of pressure equalization between regions 2 and 3,
P
2
P
3
, and zero perpendicular velocity, v
?
2
v
?
3
0,
the flow parameters can be solved in the entire domain. By
additionally assuming that the Mach stem is perpendicular
to the wedge, one can also find the angle of the triple-point
trajectory. Once the flow parameters are known, the four
physical parameters mentioned above are easily obtained.
It can be shown from the translation function mentioned
earlier that the resulting spread angle depends only on the
incident shock-wave Mach number, the reflecting wedge
angle, and the adiabatic index of the gas , while the initial
density and pressure are of no importance. In Fig. 2, one
can see the resulting SS instability spread angles for MR in
air ( 1:4) as a function of the incident shock-wave
Mach number for reflecting wedge angles of
w
20
; 30
; 40
; 45
.
Notice that, initially, the expected spread angle increases
with the increase in Mach number or wedge angle, until a
value of about 8
, after which the spread angle decreases.
This effect is a result of the Mach number reduction factor
of Eq. (1).
In order to verify the model predictions, complementary
experimental research was conducted using a shock-tube
facility at the Interdisciplinary Shock-Wave Research Cen-
ter of the Institute of Fluid Science, Tohoku University.
The tube allows the generation of Mach 1.1–5 shock waves
passing through a rectangular tube with a 10 cm by 18 cm
cross section. Near the end of the tube, a windowed test
section allows the user to implement optical diagnostics. A
steel wedge with a varying angle was placed in the test
section, and holographic interferometry images were taken
of the MR generated from the shock-wedge interaction.
For further details on the shock tube and the interferometry
technique, see [10]. Experiments were done with ambient
air at an initial pressure of P
0
10:1 kPa, wedge angles of
w
20
; 30
; 40
; 45
, and incident shock-wave Mach
numbers of M
i
1:55; 1:9; 2:3; 2:78. One can see two
examples: in Fig. 1 for
w
30
, M
i
1:9 and in Fig. 3
for a constant wedge angle of
w
40
and for all four M
numbers.
It is evident that, as the gas on two sides of the SS flows
away from the triple point, the SS thickness linearly in-
creases. A density profile is evident through the increasing
spacing between the fringes along the SS. Notice that, as
predicted by the model, the spread angle increases with
increasing Mach number. In order to quantitatively analyze
the model predictions, a detailed comparison of the mea-
sured and predicted spread angles was done for all Mach
numbers. Spread angle measurements were taken between
two straight lines bounding the SS thickening region, as
demonstrated in Fig. 3(b). Error estimates were set accord-
ing to half the thickness of a single fringe. The results are
plotted in Fig. 2 as a function of the incident shock-wave
Mach number for reflecting wedge angles of
w
20
; 30
; 40
; 45
. Additional experiments conducted at a
higher initial pressure of 100 kPa and at an incident Mach
number of M 1:5 are also plotted.
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0
1
2
3
4
5
6
7
8
9
10
Mach number
spread angle [deg]
FIG. 2. SS instability spread angle as a function of incident
shock-wave Mach number. Theoretical predictions in thick lines:
w
45
(solid line),
w
40
(dashed-dotted line),
w
30
(upper dashed line), and
w
20
(lower dashed
line). Thin lines around the thick lines mark the error in the
model prediction [see Eq. (1)]. Experimental results and error
bars are also plotted; (), (䉫), (4), and (5) mark experiments
conducted with
w
45
,40
,30
, and 20
, respectively. All
of the experiments were conducted with ambient air at P
0
10:1 kPa, apart from the M 1:5 experiments conducted at
P
0
100 kPa.
FIG. 3. Holographic interferometry images for MRs with a
wedge angle of
w
40
and incident shock-wave Mach num-
bers of M
i
1:55 (a), 1.9 (b), 2.3 (c), and 2.78 (d). Two white
lines in (b) bound the SS, demonstrating the growth angle
measurement technique.
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In the figure, good agreement is demonstrated, apart
from the following experiments at P
0
10:1 kPa: all the
Mach 1.55 experiments, the
w
30
experiments at M<
2, and the
w
20
experiments at M<2:4. In order to
understand the cause for the above disagreements, the
discrepancy between the model predictions and the experi-
mental results is plotted in Fig. 4 against the SS shear
velocity Re number. The Re number is calculated accord-
ing to Re
vl
, where v is the shear velocity, is the
average ideal gas kinematic viscosity calculated at re-
gions 2 and 3 according to
air
181:92 0:536T=
[11], and l stands for a typical length scale of 1 cm (chosen
according to the typical size of the Mach stem).
From the figure, it is clear that all of the experiments
conducted at Re > 2 10
4
show good agreement with
theory. Special attention should be pointed out to experi-
ments conducted at M 1:55 with P
0
10 kPa where no
spread angle could be measured, whereas similar M 1:5
experiments but with P
0
100 kPa show very good
agreement. For the
w
45
experiments, for example,
the resulting physical properties of regions 2 and 3 are
2
0:4kg=cm
3
,
3
0:36 kg=cm
3
, v
2
412 m= sec ,
v
3
257 m= sec , and a 460 m= sec for the P
0
10 kPa experiment and
2
3:8kg=cm
3
,
3
3:5kg=cm
3
, v
2
402 m= sec , v
3
257 m= sec , and
a 454 m= sec for the P
0
100 kPa experiment, both
leading to a very similar spread angle according to
Eq. (1). The important difference between the two classes
of experiments are the higher densities obtained at the
P
0
100 kPa experiments, introducing weaker kinematic
viscosities and larger Re numbers (see caption of Fig. 4).
Together with Fig. 4, it is clear that the disagreements
between the model and the experiments are due to viscosity
effects, which are not accounted for in Eq. (1). Viscosity
effects, serving as a secondary mix cutoff stabilizing
mechanism (through a cutoff Re number), are a well
known phenomenon. The critical Re number, measured
at 2 10
4
, is similar to critical Re numbers demonstrated
for several other mixing phenomena, such as secondary
mixing at the Richtmyer-Meshkov and KH instabilities
[2,3].
Summary.—It is shown that, from the properties of a
large-scale hydrodynamic phenomena, i.e., densities, vis-
cosities, velocities, and average sound speed, one can
predict the existence and width of a secondary small-scale
turbulent mixing zone through a simple analytical proce-
dure based on models of large-scale KH instability growth.
Using this technique, the evolution of the MR SS insta-
bility was analytically characterized. Good agreement is
achieved with an experimental evaluation of the SS spread
angle, conducted at Mach numbers of M
i
1:5–2:78 and
spread angles of
w
20
–45
. A critical Re number of
2 10
4
is found to distinguish between turbulent and
laminar flows, thus indicating secondary turbulent mixing
generated through the evolution of the KH instability as the
cause for the SS discontinuity thickening. The success in
modeling secondary mixing using large-scale models of
the KH instability should provide a guideline for future
research in secondary turbulent mixing phenomena.
The financial support of the Interdisciplinary Shock-
Wave Research Center of the Institute of Fluid Science,
Tohoku University, for conducting the experimental phase
of the presented research and helpful discussions with
D. Oron are acknowledged by the authors.
[1] S. P. Regan et al., Phys. Rev. Lett. 89, 085003 (2002).
[2] Y. Zhou, H. F. Robey, and A. C. Buckingham, Phys. Rev. E
67, 056305 (2003).
[3] P. E. Dimotakis, J. Fluid Mech. 409, 69 (2000).
[4] G. Ben-Dor, Shock Wave Reflection Phenomena (Springer-
Verlag, Berlin, 1992).
[5] J. von Neumann, Navy Department, Bureau of Ordinance,
Washington, DC, Explosives Research Report No. 12,
1943.
[6] G. L. Brown and A. Roshko, J. Fluid Mech. 64, 775
(1974).
[7] A. Rikanati, U. Alon, and D. Shvarts, Phys. Fluids 15,
3776 (2003).
[8] S. A. Ragab and J. L. Wu, Phys. Fluids A 1, 957 (1989).
[9] H. Li and G. Ben-Dor, J. Fluid Mech. 341, 101 (1997).
[10] F. Ohtomo, K. Ohtani, and K. Takayama, Shock Waves 14,
379 (2005).
[11] American Institute of Physics Handbook, edited by
Dwight E. Gray (McGraw-Hill, New York, 1972).
0 2 4 6 8 10 12 14 16
x 10
4
−6
−5
−4
−3
−2
−1
0
1
Reynolds Number
θ
measured
−θ
predicted
[deg]
FIG. 4. Predicted spread angle subtracted from the measured
spread angle, as a function of the SS Re number. (), (䉫), (4),
and (5) mark experiments conducted at
w
45
,40
,30
, and
20
, respectively. Experiments conducted at an initial pressure of
100 kPa resulted in Re numbers of Re 15:3 10
4
, 9:2 10
4
,
and 4:8 10
4
for
w
40
,30
, and 20
, respectively.
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