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J. Fluid Mech. (2020), vol.902, A3. © The Author(s), 2020.
Published by Cambridge University Press
902 A3-1
doi:10.1017/jfm.2020.584
Convergent Richtmyer–Meshkov instability of
heavy gas layer with perturbed inner surface
Rui Sun1, Juchun Ding1,†,ZhigangZhai1,TingSi1and Xisheng Luo1
1Advanced Propulsion Laboratory, Department of Modern Mechanics, University of Science and
Technology of China, Hefei 230026, PR China
(Received 30 January 2020; revised 12 June 2020; accepted 10 July 2020)
The convergent Richtmyer–Meshkov instability (RMI) of an SF6layer with a uniform
outer surface and a sinusoidal inner surface surrounded by air (generated by a novel
soap film technique) is studied in a semiannular convergent shock tube using high-speed
schlieren photography. The outer interface initially suffers only a slight deformation over
along period of time, but distorts quickly at late stages when the inner interface is close
to it and produces strong coupling effects. The development of the inner interface can
be divided into three stages. At stage I, the interface amplitude first drops suddenly to a
lower value due to shock compression, then decreases gradually to zero (phase reversal)
and later increases sustainedly in the negative direction. After the reshock (stage II), the
perturbation amplitude exhibits long-term quasi-linear growth with time. The quasi-linear
growth rate depends weakly on the pre-reshock amplitude and wavelength, but strongly
on the pre-reshock growth rate. An empirical model for the growth of convergent RMI
under reshock is proposed, which reasonably predicts the present results and those in
the literature. At stage III, the perturbation growth is promoted by the Rayleigh–Taylor
instability caused by a rarefaction wave reflected from the outer interface. It is found that
both the Rayleigh–Taylor effect and the interface coupling depend heavily on the layer
thickness. Therefore, controlling the layer thickness is an effective way to modulate the
late-stage instability growth, which may be useful for the target design.
Key words: shock waves
1. Introduction
Richtmyer–Meshkov instability (RMI) arises when a distorted interface separating two
fluids with different properties is impulsively accelerated by a shock wave (Richtmyer
1960;Meshkov1969). After the shock passage, initial perturbations on the interface
grow first linearly, then nonlinearly accompanied by the formation of finger-like bubbles
(regions where a light fluid rises into a heavy fluid) and spikes (regions where a heavy
fluid penetrates into a light fluid), and finally give rise to a flow transition to turbulent
mixing. In recent years, the RMI has received widespread attention due to its fundamental
significance in scientific research (e.g. in the study of compressible turbulence and
vortex dynamics),aswellasits crucial role in engineering applications such as inertial
† Email address for correspondence: djc@ustc.edu.cn
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902 A3-2 R. Sun, J. Ding, Z. Zhai, T. Si and X. Luo
confinement fusion (ICF) (Lindl et al. 2014) and supersonic combustion (Yang, Kubota &
Zukoski 1993).
Most previous studies on RMI have been devoted to the planar shock-induced case, and
several comprehensive reviews have been conducted (Brouillette 2002; Ranjan, Oakley &
Bonazza 2011; Zhou 2017a,b; Zhai et al. 2018b). Recently, the converging shock-induced
RMI has become increasingly attractive, since its initial setting is more relevant to
ICF. Compared to theoretical and numerical studies, experiments can produce more
realistic flow fields and has been demonstrated to be a promising means for studying
the convergent RMI (Fincke et al. 2004; Biamino et al. 2015;Dinget al. 2017). A
primary concern in performing convergent RMI experiments is to generate an initial stable
converging shock wave. Inspired by the pioneering work of Perry & Kantrowitz (1951),
Hosseini, Ondera & Takayama (2000) designed a vertical diaphragmless coaxial shock
tube, in which a cylindrical shock wave with minimum disturbances can be produced.
Based on shock dynamics theory, a conventional shock tube with a special wall profile
which can smoothly transform an incident planar shock into a cylindrical one was
designed by Zhai et al. (2010). Gas interfaces of various shapes impacted by a cylindrical
shock were then examined in this facility (Si, Zhai & Luo 2014; Zhai et al. 2017;Luo
et al. 2018). Also, a gas-lens technique (Dimotakis & Samtaney 2006), which produces
cylindrically converging shocks by letting a planar shock refract at a specially shaped gas
interface, was realized in an ordinary shock tube, and the first experiments exhibited the
great potential of this technique for studying the convergent RMI (Biamino et al. 2015;
Vandenboomgaerde et al. 2018). Recently, a novel semiannular shock tube was designed
by Luo et al. (2015),and its semistructure was demonstrated to bring great conveniences
for interface formation and flow diagnostics. For instance, an advanced soap film
technique which is able to generate controllable gas interfaces free of three-dimensionality,
short-wavelength perturbations and diffusion layers (Luo, Wang & Si 2013;Liuet al.
2018), developed in a planar geometry, can be readily extended to the convergent test
section of this facility, which has enabled the successful execution of a series of convergent
RMI experiments (Ding et al. 2017;Lianget al. 2017;Dinget al. 2019;Zouet al. 2019).
Experimental results showed that geometric convergence (Bell 1951; Plesset 1954)and
the Rayleigh–Taylor (RT) effect (Rayleigh 1883;Taylor1950) significantly influence the
growth of convergent RMI.
Previous studies on convergent RMI mainly considered a cylindrical shock interacting
with an isolated interface. However, in real-world applications such as ICF, there
usually exist two interfaces separating three concentric shells: an outer ablator, a middle
deuterium–tritium (DT) ice and an inner gaseous DT fuel (Montgomery et al. 2018;
Peterson, Johnson & Haan 2018). Irradiated by X-rays (indirect drive) or laser beams
(direct drive), the capsule presents evident instability growths on both the outer and inner
interfaces. In addition to the common flow regimes (i.e. baroclinic vorticity and pressure
perturbation) presented in the RMI of an isolated interface, the two-interface case involves
new physical mechanisms such as interface coupling (Mikaelian 1995) and the influence
of complex waves reverberating between the two interfaces (Liang et al. 2020), and thus
presents much more complex instability evolution. Moreover, these new mechanisms
could couple with geometric convergence and the RT effect, further complicating the
instability development. It is therefore highly desirable to perform elaborate experiments
on a material layer (two interfaces) interacting with a converging shock; the results would
facilitate the understanding of hydrodynamic instabilities in ICF.
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Convergent Richtmyer–Meshkov instability of heavy gas layer 902 A3-3
Recently, we examined the evolution of an outer perturbation layer (in which the outer
interface is perturbed and the inner interface is uniform), filled with either SF6or helium
and surrounded by air, subjected to a cylindrical shock (Ding et al. 2019;Liet al.
2020), and rich new phenomena not present in the evolution of a single interface were
observed. In particular, the initial perturbation on the outer interface was found to have
either a significant or a negligible influence on the development of the inner interface,
depending on the properties of the gas inside the layer. As a follow-up study, in this
work we consider the inner perturbation case, i.e. the case in which the inner surface
is sinusoidally perturbed while the outer surface is uniform. It should be stressed that the
inner perturbation case is more critical for ICF, because the inner interface of such a layer
may exhibit rapid instability growth, which would directly entrain the inside fuel to the
outside and hence impede the ignition. We show in the present work that the evolution of
an inner perturbation layer is distinctly different from the outer perturbation case (Ding
et al. 2019), and we discuss the reasons for such a difference. The growth behaviours of
perturbations on both the outer and inner interfaces of the inner perturbation layer are
carefully analysed and the underlying flow regimes are discussed. Also, the influences of
the layer thickness and the initial amplitude and wavelength of perturbation at the inner
interface on the instability development are examined.
2. Experimental method
The experiments on a cylindrical shock striking a perturbed SF6layer are performed in
asemiannular convergent shock tube. The overall structure of the shock tube,aswellas
the cylindrical shock formation principle, has been detailed in previous work (Luo et al.
2015) and thus is not repeated here. As shown in figure 1(a), thanks to the semicylindrical
structure of the test section, a removable interface formation device can be designed in
which gas interfaces of various shapes can be created in the same way as in a planar shock
tube (Ding et al. 2017). Figure 1(b) shows a real interface formation device, which consists
of two semicircular Plexiglass plates (100mm in radius) fixed on a rectangular block. Two
pairs of cusps with sinusoidal and circular shapes, respectively, are attached to the internal
surfaces of the Plexiglass plates to produce the desired constraints. The cusps are 0.2mm
in height, which is much lower than the height of the test section (5 mm), and thus they
have negligible influence on the instability development. A soap film technique is adapted
to this device to create gas layers with controllable shapes. Prior to the generation of the gas
layers, the internal surface of each plate enclosed by the inner and outer cusps is first wetted
uniformly with alcohol. Then a thin pipe with its leading end dipped with soap solution
(78 % distilled water, 2 % sodium oleate and 20 % glycerine by mass) and its trailing end
connected to an SF6balloon is inserted into the device and placed in the wetted region. As
SF6is supplied through the pipe, a soap bubble is immediately produced at the leading end.
Afterwards, the bubble expands continuously along the cusps (constraints), and finally an
SF6layer with the desired shape is formed. After this, the whole device with the SF6
layer inside is carefully inserted into the test section and fitted tightly between the optical
windows on both sides of the test section. Note that the present initial conditions, including
the incident shock strength, the layer thickness and shape and the gas concentration, can
be well controlled, which ensures good repeatability and also allows a careful study of the
initial condition dependence.
In a cylindrical coordinate system, the single-mode inner interface can be parameterized
as ri=Ri
0+a0cos(nθ) and the uniform outer interface as ro=Ro
0. Here, R0refers to the
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902 A3-4 R. Sun, J. Ding, Z. Zhai, T. Si and X. Luo
Air
Convergence
centre
Solid wall
C
y
l
i
n
d
r
ic
a
l
s
h
o
c
k
SF6
Outer cusps
Inner cusps
Plexiglass plate
Rectangular block
Observing windows
Insert
Gas shell
r° =R0
°ri=R0
i+a
0cos(nθ)
(a)
(b)(c)
FIGURE 1. Drawing of the open test section (a), a real interface formation device (b)and
experimental configuration for a cylindrical shock impacting a perturbed SF6shell surrounded
by air (c). The single-mode inner interface is parameterized as ri=Ri
0+a0cos(nθ) and the
uniform outer interface as ro=Ro
0,whereR0is the radius of the initial interface, a0is the initial
perturbation amplitude and nis the azimuthal mode number. Superscripts oand idenote the
outer and inner interfaces, respectively.
mean radius of the initial interface, a0to the initial perturbation amplitude and nto the
azimuthal mode number. The superscripts oand idenote the outer and inner surfaces,
respectively. Thanks to the wide-open inner core, a Z-fold high-speed schlieren system can
easily be adapted to the semiannular shock tube, which facilitates the flow diagnostics. The
frame rate of the high-speed video camera (FASTCAM SA5, Photron Limited) is set as
50 000 f.p.s. with a shutter time of 1 µs. The spatial resolution of the schlieren images
is approximately 0.3mmpixel
−1. The ambient pressure and temperature are 101.3 kPa
and 298.0 K, respectively. It should be noted that the present test section is complex in
structure, and thus it is difficult to perform pressure measurements at the test section,
especially synchronized with the schlieren photography.
3. Results and discussion
We shall first consider the evolution of an unperturbed SF6layertolearnthebasicflow
features in the convergent setting, then study three perturbed cases with different layer
thicknessestorevealtheeffects of interface coupling, and finally study three perturbed
cases with different perturbations to discover the effects of the initial amplitude and
wavenumber.
3.1. Unperturbed case
To obtain the non-uniform flow features of the convergent RMI of a gas layer, we first
consider the evolution of an unperturbed SF6layer (i.e. one in which both outer and
inner surfaces are uniform, with ro=55 mm, ri=30 mm), surrounded by air, which
is impacted by a cylindrical shock. As shown in figure 2(a), the outer and inner interfaces
as well as the incident cylindrical shock (ICS) can be clearly discerned at the beginning
(−140 µs), which demonstrates the feasibility and reliability of the experimental method.
Here, the shock wave interacts successively with the outer and inner interfaces, producing
wave patterns much more complex than those of the single interface case (Ding et al.
2017). The incident cylindrical shock initially propagates inwards at a speed of 423 m s−1
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Convergent Richtmyer–Meshkov instability of heavy gas layer 902 A3-5
–150 0 150 300 450
0
20
40
60
80
RW
2
II
Air
r (mm)
t (µs)
ICS
OI
RS
1
TS
1
SF6
Air
TS
2
RW
1
TS
3
RTS
TS
4
ICS OI
II
TS1
TS2
TS4
TS3
RTS
RS1RW1
RW2
–140 µs –60 µs 40 µs
80 µs 140 µs 220 µs
260 µs 380 µs 460 µs
(a)
(b)
FIGURE 2. (a) The evolution of an undisturbed SF6layer surrounded by air accelerated by a
cylindrical shock and (b) the corresponding trajectories of the interfaces and the waves: ICS,
incident cylindrical shock; OI, outer interface; II, inner interface; RSi,theith reflected shock;
TSi,theith transmitted shock; RWi,theith rarefaction wave; RTS, reflected transmitted shock.
in air, then collides with the outer interface (OI), generating an inward-moving transmitted
shock (TS1) and an outward-moving reflected shock (RS1)(−60 µs). After the shock
impact, the outer interface moves inwards at a velocity of approximately 86 m s−1at the
early stage. Note that during our experiments, there is a certain degree of contamination of
the SF6inside the layer by the outside air. Since in the present experiments the velocities
of the incident shock and the shocked interface are nearly constant at the early stage (i.e.
there is a very weak convergence effect on the wave propagation), one-dimensional gas
dynamics theory can be used to approximately estimate the unknown initial conditions.
Specifically, given the measured speeds of the incident shock (423 m s−1) and the shocked
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902 A3-6 R. Sun, J. Ding, Z. Zhai, T. Si and X. Luo
outer interface (86 m s−1), the mass fraction of SF6inside the layer and the speed of
the transmitted shock TS1can be calculated to be 93 % and 208 m s−1, respectively.
It is found that the calculated transmitted shock speed (208 m s−1) compares quite
well with the experimental one (207 m s−1), which demonstrates the reliability of the
theoretical estimate. Next, the transmitted shock TS1collides with the inner interface
(II), and immediately bifurcates into an ingoing transmitted shock (TS2) and an outgoing
rarefaction wave (RW1)(40µs). Here, the rarefaction wave RW1maintains a relatively
sharp front and resembles a shock wave, which is mainly attributable to the non-uniform
flow in the convergent geometry (Lombardini, Pullin & Meiron 2014). Later, the RW1
passes across and also slightly accelerates the outer interface (see figure 1b). The zero
time in this work is defined as the moment when the first transmitted shock TS1arrives at
the mean position of the initial inner interface.
After the impact of the transmitted shock TS1, the inner interface moves inwards at
a velocity of 119 m s−1at early stages, and then enters a deceleration phase similar to
that of the single interface case (Ding et al. 2017). As the second transmitted shock
TS2focuses at the geometric centre, a reflected transmitted shock (RTS) is immediately
formed (80 µs). Later, the RTS re-accelerates the ingoing inner interface (this is known as
reshock),and consequently the interface quickly slows down until it is nearly stationary.
During this process, a transmitted shock (TS3) and a reflected shock (not visible in the
schlieren images due to its very weak intensity) are generated (140 µs). As time proceeds,
the outer interface is also re-accelerated by the third transmitted shock TS3, producing
an outgoing transmitted shock (TS4) and an ingoing rarefaction wave (RW2). Later, the
second rarefaction wave RW2interacts with the inner interface (220 µs) and causes a
slight rise in the interface velocity. It should be noted that the rarefaction wave RW2would
trigger RT stability or instability for a perturbed interface. The trajectories of the waves and
the interfaces measured from the schlieren images are plotted in figure 2(b). The dynamic
range of our digital camera is 0–255 in pixel counts, and the locations of the shock and the
interface are represented by the central position of the layers with pixel counts less than
40. It is seen that at late stages (t>370 µs), both the inner and outer interfaces evolve at
nearly constant radial positions, which indicates that geometric convergence and the RT
effect have negligible influence on the instability development there. The present result
shows the main features of the background flow of convergent RMI at a gas layer, and will
facilitate the analysis of the development of a perturbed layer.
Note that in the present experiments the incident shock is weak, and thus the post-shock
flow can be assumed to be laminar and incompressible. Hence, the thickness of the
boundary layer in the post-shock flow (δ∗) can be estimated by
δ∗=1.72μx
ρv.(3.1)
Here, xis taken to be 20 mm, which corresponds to the maximum distance of interface
travel in the radial direction during the experimental time. The viscosity coefficient and
density of pure air (SF6) under the experimental temperature and pressure are μ=1.83 ×
10−5Pa s (=1.60 ×10−5Pa s) and ρ=1.204 kg m−3(=6.14 3 kg m −3), respectively. The
speeds of the shocked outer and inner interfaces are approximately 86 m s−1and 119 m s−1,
respectively, at the early stage. Using equation (3.1), the maximum thickness of the
boundary layer in the post-shock air (SF6) flow is calculated to be approximately 0.1 mm
(0.04 mm), which is much smaller than the inner height of the test section (5.0 mm).
This indicates that the influence of the boundary layer on the interface development is
negligible.
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Convergent Richtmyer–Meshkov instability of heavy gas layer 902 A3-7
Case d(mm) a0(mm) nλ(mm) a0/λmfra(SF6)A+
1 35 2 6 31.42 0.06 0.94 0.64
2 25 2 6 31.42 0.06 0.94 0.64
3 15 2 6 31.42 0.06 0.89 0.59
4 25 1 6 31.42 0.03 0.91 0.61
5 25 3 6 31.42 0.10 0.91 0.61
6 25 2 10 18.85 0.11 0.92 0.62
TABLE 1. Parameters corresponding to the initial conditions for each case. The parameter λ
refers to the wavelength of initial perturbation, dto the layer thickness, mfra(SF6)tothemass
fraction of SF6inside the layer and A+to the post-shock Atwood number.
3.2. Perturbed SF6layers with different thicknesses
We now examine three perturbed layers (consisting of a uniform outer surface and a
sinusoidal inner surface) of different thicknesses,highlighting the influence of the layer
thickness on the instability development. Detailed parameters corresponding to the initial
conditions for each case (Cases 1–3) are listed in table 1,wheredrefers to the layer
thickness, λto the wavelength of perturbation at the initial inner interface, mfra(SF6)
to the mass fraction of SF6inside the layer and A+to the post-shock Atwood number
(A=(ρ2−ρ1)/(ρ2+ρ1)with ρ1and ρ2being the densities of gases outside and inside
the layer, respectively). As shown in table 1,gascontamination is at an acceptable level
and the discrepancy in Atwood number among different cases is quite small, which ensures
good repeatability of the present experiments.
The dynamic evolution of the interfacial morphologies and the wave patterns for the
three cases are illustrated in figure 3 by representative schlieren images. It is seen that the
layer thickness and the boundary shapes are well controlled, which demonstrates the high
flexibility of the experimental method. The instability developments for the three cases are
qualitatively similar; here we take Case 1 as an example to detail the evolution process.
The incident cylindrical shock moves inwards and first collides with the outer interface,
generating an outgoing reflected shock (which soon exits the observation window) and
an ingoing transmitted shock TS1. During this process, both the shock and the interface
are uniform and thus no instabilities arise. As time proceeds, the transmitted shock TS1
converges continuously with an increasing strength (Guderley 1942;Luoet al. 2015). The
slight acceleration of the TS1at early stages cannot be detected by the present schlieren
photography due to its limited spatial and temporal resolutions. Next, the TS1interacts
with the sinusoidal inner interface (heavy/light), producing an ingoing transmitted shock
TS2and an outgoing rarefaction wave RW1.Unlike in the unperturbed case, the TS2
and RW1here present sine-like shapes due to the transfer of interfacial distortion to the
refracted waves. It is observed that the perturbation on RW1is in phase with that on the
inner interface, while the perturbation on TS2is out of phase. The reason is elucidated
below. As the ingoing transmitted shock TS1first encounters the crest of the perturbation
on the inner interface, a part of TS1transmits into the air on the interior side of the
interface first and attains a higher travelling speed (because air has a smaller acoustic
impedance than SF6), while the other part continues to propagate in SF6at a lower speed.
This velocity difference produces an anti-phase perturbation on the TS2(38 µs). This
is shown more vividly in figure 4. Later, the distorted shock TS2continues to converge
with noticeable disturbance waves produced behind it (58 µs). This is a typical wave
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902 A3-8 R. Sun, J. Ding, Z. Zhai, T. Si and X. Luo
ICS OI
II
TS1
TS2
TS3
TS4
RW
2
RTS
RW
1
–202 µs–144 µs –93 µs
–102 µs–64 µs–33 µs
38 µs 36 µs 27 µs
78 µs 76 µs 67 µs
138 µs136 µs127 µs
258 µs 236 µs 187 µs
518 µs 416 µs 407 µs
Case 1 Case 2 Case 3
FIGURE 3. The evolution of SF6layers with thicknesses of 35 mm (Case 1), 25 mm (Case 2)
and 15 mm (Case 3) subjected to a cylindrical shock. Symbols are the same as those in figure 2.
structure for the implosion of a distorted cylindrical shock, which has also been observed
in previous experiments (Apazidis et al. 2002;Siet al. 2015). As the TS2focuses at the
geometric centre, a reflected transmitted shock RTS with evident disturbance waves in
front of it is immediately generated (78 µs). Detailed observations of the typical wave
patterns and interfacial morphologies are shown in figure 4, where the coloured lines have
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Convergent Richtmyer–Meshkov instability of heavy gas layer 902 A3-9
–182 µs–122 µs38 µs58 µs
98 µs138 µs
118 µs298 µs
ICS
RTS
TS1TS2
TS3TS4
RS RW1
RW2
OI
II
FIGURE 4. Typical wave patterns (coloured lines) and interfacial morphologies (black lines)
for Case 1. Symbols are the same as those in figure 2.
been obtained by depicting the wave fronts and interfacial profiles in schlieren images via
the Bessel curve function in CorelDraw software.
After the shock impingement, the amplitude of perturbation on the inner interface
(heavy/light) decays quickly to nearly zero (38 µs) (phase inversion), and then increases in
the negative direction. Later, re-accelerated by the reflected shock RTS, the inner interface
exhibits rapid instability growth. At late stages, finger-like bubbles and spikes arise due
to the strong nonlinearity. Despite the rapid deformation of the inner interface, the outer
interface presents a nearly perfect cylindrical shape at early stages (t<138 µs). Later,
the distorted shock TS3passes through and also seeds a subtle perturbation on the outer
interface. In Cases 1and 2, this seeded perturbation suffers ver y slow growth for a long
time (t<400 µs), which indicates a negligible coupling effect between the inner and
outer interfaces. We note that this behaviour is completely different from that of the outer
perturbation case, where the initially-undisturbed inner interface deforms quickly from
a very early time (Ding et al. 2019). In Case 3,where the inner and outer interfaces
are much closer together, the outer interface presents rapid instability growth at 407 µs.
This is ascribed to the fact that, for Case 3, at late stages the quickly developing inner
interface is very close to the outer interface and produces strong coupling effects. Note
that for all three cases the ingoing rarefaction wave RW2interacts with the evolving inner
interface, producing RT instability which further promotes the perturbation growth at the
inner interface. Aquantitative discussion of such instability promotion will be given later.
Variations of the overall perturbation amplitude of the inner interface with respect to
time are plotted in figure 5(a). The error bar here refers to the pixel width in schlieren
images, which blurs the interfaces and waves. In terms of the impingement moments of
the typical waves TS1,RTSandRW
2, the instability growth at the inner interface can be
divided into three stages. Impacted by the first transmitted shock TS1, the perturbation
amplitude first drops quickly to a smaller value due to shock compression, then decreases
gradually to zero (≈50 µs) and afterwards increases considerably on account of the
induction of baroclinic vorticity (stage I). Later, re-accelerated by the outgoing shock RTS
(reshock), the inner interface exhibits rapid perturbation growth that is quasi-linear in
time (stage II). Note that the word ‘quasi-linear’ here means simply that the growth curve
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902 A3-10 R. Sun, J. Ding, Z. Zhai, T. Si and X. Luo
0 120 240 360 480
0
4
8
12
16
I II
a (mm)
t (µs)
0 120 240 360 480
t (µs)
Case 1
Case 2
Case 3
TS1
RTS
III
RW2
–14
–7
0
7
14
III
II
Case 1
Case 2
Case 3
TS1
RTS
RW2
I
Spike
Bubble
(a)(b)
FIGURE 5. Temporal variations of (a) the overall perturbation amplitude and (b)theindividual
spike and bubble amplitudes of the inner interface for Cases 1–3. Symbols are the same as those
in figure 2.The labels I, II and III stand for the three stages of the amplitude growth process for
Case 2. For Case 1 or 3, the end of stage II should be shifted to the arrival time of RW2.
is almost linear; it does not refer to a precise mathematical property. Such quasi-linear
growth is similar to previous results on planar RMI with reshock (Mikaelian 1989; Leinov
et al. 2009). This can be explained by the fact that the present inner interface evolves at
a nearly constant radial position after reshock (as indicated in the unperturbed case), and
hence geometric convergence and the RT effect have negligible influence on the instability
growth; i.e. the dominant regimes for the post-reshock growth of convergent RMI are
nearly the same as those in the planar counterpart. More discussion of the post-reshock
growth rate is presented later.
At stage III, the ingoing rarefaction wave RW2collides with the inner interface,
producing RT instability which further promotes the instability growth at the inner
interface. In Case 1,in which the layer thickness is large, just before the interaction with
RW2the inner interface has experienced long-term instability growth and thus presents a
high perturbation amplitude. As a consequence, a strong RT instability is produced, which
causes a noticeable promotion of the instability growth on the inner interface (t>340 µs).
When the layer thickness is reduced to 25 mm (Case 2), the RW2interacts with the inner
interface at an earlier time, which means that the interface experiences a shorter period of
growth than in Case 1 and thus presents a smaller perturbation amplitude just before its
interaction with RW2. As a result, a moderately strong RT instability is produced, which
prolongs the quasi-linear growth stage up to about 360 µs. For the thinnest layer (Case 3),
the inner interface presents a much smaller perturbation amplitude just before the arrival
of RW2, and thus the RT instability produced is the weakest among the three cases. Hence,
despite the presence of RT instability, there is still a continuous reduction in perturbation
growth rate in Case 3 (t>240 µs). This means that the RT effect in this case is too weak to
stop the growth saturation. The present results suggest that one can modulate the late-stage
instability growth at both the inner and outer interfaces by changing the layer thickness.
According to the previous studies of Mikaelian (1995) and Jacobs et al. (1995), there
exists a coupling effect between two adjacent interfaces evolving in a planar geometry. A
recent study (Ding et al. 2019)has demonstrated that such an interface coupling occurs
also in the convergent setting. In the present work, a coupling angle originally proposed
by Mikaelian (1995) is employed to estimate the coupling strength between the inner and
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Convergent Richtmyer–Meshkov instability of heavy gas layer 902 A3-11
outer interfaces of the gas layers; this is written as
sin α=(2W1/W2)/[1 +(W1/W2)2],(3.2)
W1/W2=1+sinh(kd)tanh(kd/2)
+sinh(kd)ρ1
ρ2⎧
⎨
⎩
1+1+ρ2
ρ12
+2ρ2
ρ1cosh(kd)1/2⎫
⎬
⎭
.(3.3)
Here, αis the coupling angle, and kis the wavenumber, which equals n/Rfor the
perturbation in a cylindrical geometry. Based on equations (3.2) and (3.3), the coupling
angles (α) for Cases 1–3 are calculated to be 0.001, 0.011 and 0.077, respectively. For Cases
1 and 2, with relatively larger thicknesses, the coupling angles are smaller, which indicates
weaker coupling effects. For Case 3, even though the value of αis relatively greater, the
coupling effect is still too weak to affect the growth of each interface at early stages, as
shown in figure 3. A good collapse of the perturbation growths at stages I and II for all
three cases is found in figure 5, which demonstrates the negligible influence of interface
coupling at early stages.
The growths of the individual bubble and spike amplitudes of the inner interface is
plotted in figure 5(b). The bubble or spike amplitude is measured by taking the unperturbed
interface trajectory as a reference. It is seen that the post-reshock spike growth (stage II)
is far quicker than the post-reshock bubble growth in each case. A major reason is that
spikes develop inwards (shown in figure 3)andtheir growth is promoted by geometric
contraction, whereas bubbles develop outwards and their growth is inhibited by geometric
expansion. At stage III, bubble growth is apparently accelerated by the rarefaction wave
RW2(particularly for Cases 2 and 3), whereas spike growth is not affected at all. At late
stages, both spikes and bubbles saturate gradually and finally stop growing. This type of
instability freeze-out has also been observed in the outer perturbation case (Ding et al.
2019), and the physical mechanisms are given below. As the circumferentially distributed
spikes (carrying heavy SF6) move to the geometric centre at a high speed (≈61 m s −1), air
in the vicinity of the geometric centre is continuously compressed and produces a drag
force (i.e. an adverse pressure gradient) on the ingoing spikes. As this drag force is strong
enough to balance the driving force of the growing spikes (i.e. induction of baroclinic
vorticity), spikes stop developing. As indicated in the unperturbed case, the inner interface
stays at a nearly fixed radial position after reshock, which means that air enclosed by the
inner interface should obey volume conservation (i.e. have negligible compressibility).
Hence, after spikes stop growing, the outward-moving bubbles have to slow down quickly
to maintain a constant fluid volume. When both spikes and bubbles stagnate, the overall
perturbation amplitude freezes out, as shown in figure 5(a). We claim that this type of
instability freeze-out is a universal phenomenon for convergent RMI at a material interface
close to the convergence centre. The above analysis shows that spike growth at late stages
is dominated by geometric convergence, which reasonably explains the minor influence of
the rarefaction wave on spike development.
3.3. SF6layers with different inner perturbations
To examine the influence of the initial perturbation on the instability growth, additional
three layers with different amplitudes and wavelengths of perturbation on the inner
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902 A3-12 R. Sun, J. Ding, Z. Zhai, T. Si and X. Luo
interface are considered. The parameters corresponding to the initial conditions for these
cases (Cases 4–6) are listed in table 1. Sequences of schlieren images illustrating the
developments of the waves and the interfaces are shown in figure 6.Theoverall instability
evolution processes are similar to those of Cases 1–3 and thus are not detailed here. It is
also observed that the small perturbation on the outer interface seeded by the distorted
shock TS3experiences very slow growth for a long time. This is different from the outer
perturbation case (Ding et al. 2019), where the seeded perturbation on the initially uniform
interface grows considerably from a very early time. Such slow instability growth may be
ascribed to the following factors. Firstly, the interaction between the TS3and the outer
interface is an instance of the non-standard RMI (Ishizaki et al. 1996), in which a distorted
shock interacts with a uniform interface. As reported by Zhai et al. (2018a) and Zou et al.
(2019), the driving mechanisms of such a non-standard RMI are different from those of
its standard counterpart (in which a uniform shock impacts a distorted interface), and the
instability growth is much slower than for the latter. Secondly, due to geometric expansion,
the instability growth induced by a diverging shock TS3is inherently slower than that of the
planar or convergent setting. Thirdly, using equations (3.2) and (3.3), the coupling angles
for Cases 4–6 are calculated to be 0.011, 0.011 and 0.0004, respectively, which indicates
weak coupling effects at early stages; i.e. the deformation of the inner interface produces
a negligible influence on the outer interface.
At late stages (t>400 µs), the inner interface presents a large perturbation and its
crest is very close to the outer interface. As a result, the layer thickness becomes very
small, producing strong coupling effects. Therefore, the outer interface distorts quickly and
persistently from then on. In particular, for Case 5, the quickly developing inner interface
approaches the outer interface, causing rapid deformation of the outer interface (426 µs).
For Case 6, even though the value of coupling angle is small, the outer interface distorts
at late stages due to the increased coupling effect (440 µs). It is also observed that the
perturbation wavenumber at the outer interface is closely related to the initial perturbation
wavenumber of the inner interface. The present results suggest that the coupling effect
between the two interfaces plays an important role in the instability development at late
stages.
Temporal variations of the overall perturbation amplitude of the inner interface for Cases
2, 4, 5 and 6 are plotted in figure 7(a). Following the impact of TS1, the perturbation
amplitude decreases gradually to zero (phase reversal) and then increases continuously.
After the reshock (i.e. the passage of the reflected transmitted shock RTS), the perturbation
amplitude experiences a long period of growth that is quasi-linear in time. Figure 7(b)
shows the growths of the individual bubble and spike amplitudes. It is seen that the bubble
growth shows a weak dependence on the initial perturbation amplitude and wavelength,
whereas the spike growth shows a strong dependence. In particular, the interface with a
larger initial amplitude-to-wavelength ratio presents a higher saturation spike amplitude at
late stages. The reason is given below. The spike dynamics at late stages is dominated by
two mechanisms. One is the driving force produced by baroclinic vorticity, whose strength
is positively correlated with the initial amplitude-to-wavelength ratio. The other is the drag
force caused by geometric convergence, which is independent of the initial perturbation.
For cases with alarger initial amplitude-to-wavelength ratio, stronger baroclinic vorticity
is deposited on the interface, which produces a greater driving force acting on the growing
spike. As a consequence, the spike amplitude can grow to a higher saturation value than
in the case of a small amplitude-to-wavelength ratio. It is also found that the rarefaction
wave RW2promotes bubble growth but has a negligible influence on spike growth.
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Convergent Richtmyer–Meshkov instability of heavy gas layer 902 A3-13
ICS
OI
II
TS1
RS1
TS2
TS
3
TS4
RW2
RTS
RW
1
–153 µs–154 µs–120 µs
–73 µs–94 µs–60 µs
27 µs26 µs40 µs
67 µs 66 µs 80 µs
127 µs126 µs 140 µs
187 µs166 µs200 µs
227 µs226 µs260 µs
407 µs426 µs440 µs
Case 4 Case 5 Case 6
FIGURE 6. The evolution of SF6layers with different initial amplitudes and azimuthal mode
numbers for the inner interface: a0=1mm,n=6(Case4);a0=2mm,n=6(Case5);and
a0=2mm,n=10 (Case 6). Symbols are the same as those in figure 2.
The current work provides very scarce experimental results on the convergent RMI
under reshock at a two-dimensional single-mode interface. The RMI with reshock or
multiple shocks exists widely in reality, and thus has attracted continuous attention in
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902 A3-14 R. Sun, J. Ding, Z. Zhai, T. Si and X. Luo
0 100 200 300 400
0
4
8
12
16
III
a (mm)
t (µs)
0 100 200 300 400
t (µs)
Case 2
Case 4
Case 5
Case 6
Case 2
Case 4
Case 5
Case 6
RTS
TS1
III
RW2
–14
–7
0
7
14
II III
I
TS1
RTS
RW2
Bubble
Spike
(a)(b)
FIGURE 7. Temporal variations of (a) the overall perturbation amplitude and (b)theindividual
spike and bubble amplitudes of the inner interface for Cases 2, 4, 5 and 6. Symbols are the same
as those in figure 2.The labels I, II and III stand for the three stages of the amplitude growth
process.
recent years. Due to the high complexity of this problem, only two types of empirical
models have been developed to estimate the post-reshock perturbation growth (Mikaelian
1989; Charakhch’an 2000; Leinov et al. 2009). One reshock model, applicable to
interfaces with initial three-dimensional multi-mode perturbations, was proposed by
Mikaelian (1989);it is written as
dh2
dt=CV2A+
2.(3.4)
Here, h2is the overall width of the mixing zone; for the present experiments it
represents the distance between the spike and bubble tips. The quantity dh2/dtis the
perturbation growth rate after reshock, V2is the velocity jump caused by reshock, A+
2is
the post-reshock Atwood number and C=0.28 is an empirical constant. In later work, the
empirical constant Cwas found to be dependent on the perturbation randomness and the
domain dimension at the time of reshock (Leinov et al. 2009; Ukai, Balakrishnan & Menon
2011). Equation (3.4) indicates that the pre-reshock growth rate has a negligible influence
on the post-reshock one. Another empirical model, suitable for initial two-dimensional
single-mode interfaces, was proposed by Charakhch’an (2000); it can be written as
dh2
dt=βV2A+
2−dh1
dt,(3.5)
where dh1/dtis the growth rate just before reshock, and the empirical coefficient βis
found to be 1.25 by fitting a large number of numerical results. So far, the value of βhas
never been verified by experiments, since there is a substantial lack of experimental data
on reshocked RMI at a well-defined two-dimensional single-mode interface.
The above models imply that, for planar RMI under reshock, the post-reshock growth
rate is almost independent of the pre-reshock amplitude and wavelength. So far, the
dynamics of convergent RMI under reshock has seldom been studied, and the post-reshock
growth behaviour remains unknown. The present results on the instability growth at the
inner interface provide a rare opportunity to examine the growth behaviour of convergent
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Convergent Richtmyer–Meshkov instability of heavy gas layer 902 A3-15
Case d(mm) a0(mm) nA+
2
V2
(m s−1)
dh1/dt
(m s−1)
dh2/dt
(m s−1)b
Theo.
(m s−1)
1 35 2 6 0.61 91.7 59.4 100.30.73 100.2
2 25 2 6 0.61 93.8 54.3 95.80.72 96.4
3 15 2 6 0.58 86.0 60.3 96.70.73 96.9
4 25 1 6 0.61 86.0 31.3 69.40.73 69.6
5 25 3 6 0.58 93.8 71.4 111.5 0.73 111 .4
6 25 2 10 0.60 93.8 74.6 116.30.74 115.6
Biamino 60.51.5 24 0.66 40.8 14.7 34.00.72 34.3
TABLE 2. The post-reshock growth rates from the present experiments, the experiments of
Biamino et al. (2015) and the theoretical predictions from (3.6) (last column). The quantities
dh1/dtand dh2/dtare respectively the pre-reshock and post-reshock perturbation growth rates,
V2is the velocity jump caused by reshock, A+
2is the post-reshock Atwood number and bis the
empirical coefficient in (3.6).
RMI under reshock at a two-dimensional single-mode interface. The post-reshock growth
rate can be obtained by a linear fitting of the experimental data at stage II for each case,
as listed in table 2. It is found that the post-reshock growth rate is closely related to the
pre-reshock one, which depends on the initial perturbation amplitude and wavelength.
This is in accordance with our observations in figures 3 and 6that the evolving interface
maintains a sharp morphology at the time of reshock. As suggested by Charakhch’an
(2000)andUkaiet al. (2011), for a sharp interface at the reshock, the pre-reshock growth
rate could significantly affect the post-reshock instability growth.
Given these findings, and also assuming that the post-reshock growth rate of the
convergent RMI is independent of the pre-reshock amplitude and wavelength, here we
propose the following empirical model for the convergent RMI with reshock at an initial
two-dimensional single-mode interface (heavy/light):
dh2
dt=bV2A+
2+dh1
dt.(3.6)
Note that no geometric parameters exist in (3.6), since, as stated before, geometric
convergence/divergence has negligible influence on the instability growth after reshock.
It is expected that a constant empirical coefficient bcan be found for cases with different
initial conditions, such that the above assumption and the model can be validated. From
the measured growth rates before and after the reshock, the empirical coefficient bfor each
case is obtained;the results are listed in table 2. It is found that for Cases 1–6, with different
pre-reshock amplitudes and wavelengths, bvaries in a small range, which supports our
assumption. From the calculated coefficients for different cases, an averaged value of
b=0.73 is obtained. It is shown in table 2 that the new empirical model with b=0.73
gives a reasonable prediction of the post-reshock growth rate for all cases considered
in this work. Tab l e 2 also provides a comparison of the model’s predictions against the
experimental results of Biamino et al. (2015), which further demonstrates the model’s
accuracy. The present empirical model is useful for estimating the post-reshock instability
growth of convergent RMI.
It should be stressed that the present empirical model is different from that of
Charakhch’an (2000). The model of Charakhch’an (2000) is applicable to the reshocked
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902 A3-16 R. Sun, J. Ding, Z. Zhai, T. Si and X. Luo
RMI at a light/heavy interface. For this case, the direction of baroclinic vorticity deposited
on the interface during the reshock is opposite to that of baroclinic vorticity produced by
the incident shock. This means that the incident and reflected shocks have opposite effects
on the post-reshock perturbation growth, which is reflected by the different signs of the
two terms on the right-hand side of (3.5). By contrast, in the present experiments, the
inner interface of the gas layer is of a heavy/light configuration. For this case, the incident
and reflected shocks produce baroclinic vorticity in the same direction, and thus the two
termsontheright-handsideof(3.6) possess the same sign. As has been recognized by
the RMI community, the instability evolution processes for the light/heavy and heavy/light
configurations are distinctly different. This is a possible reason for the different values of
coefficient in the two models.
4. Conclusion
In this work, the RMI at a perturbed gas layer with a uniform outer surface and a
sinusoidal inner surface accelerated by a cylindrical shock has been examined using
shock-tube experiments. With a novel soap film technique, the layer thickness and the
boundary shapes can be well controlled. Dynamic evolution of the wave patterns and the
interfacial morphologies is captured by high-speed schlieren photography. The influences
of the layer thickness and the initial perturbation of the inner interface on the instability
growth are highlighted.
After the shock impact, the initially uniform outer interface suffers only a slight
deformation over along period. Nevertheless, it distorts quickly at late stages, when the
quickly developing inner interface is very close to it and produces strong coupling effects,
especially in cases with a larger amplitude-to-wavelength ratio and a smaller thickness.
The instability growth on the inner interface is divided into three stages. At stage I, the
perturbation amplitude first drops suddenly to a smaller value due to shock compression,
then decreases gradually to zero (phase reversal) and afterwards increases sustainedly in
the negative direction. After the reshock, the perturbation undergoes a long period of
growth that is quasi-linear in time (stage II). The quasi-linear growth rate is found to be a
weak function of the pre-reshock amplitude and wavelength, but depends evidently on the
pre-reshock growth rate. A new empirical model for the growth of convergent RMI under
reshock is proposed, which reasonably predicts the present experimental results and those
of Biamino et al. (2015). At stage III, a rarefaction wave reflected from the outer interface
interacts with the deforming inner interface, producing the RT instability, which further
promotes the perturbation growth at the inner interface.
The RT effect and the interface coupling, which dominate the late-stage developments
of the inner and outer interfaces, respectively, depend heavily on the layer thickness.
Therefore, controlling the layer thickness is an effective way to modulate the late-stage
instability growth. Specifically, the thicker the gas layer, the slower the development of the
outer interface, and the quicker the development of the inner interface. The present results
are different from those in the outer perturbation case (Ding et al. 2019), where the layer
thickness affects the instability growth from a very early time. This work is an important
step towards a complete understanding of hydrodynamic instabilities at a perturbed layer,
and the findings may be useful for the target design.
Acknowledgements
This work was supported by the National Natural Science Foundation of China
(nos 11802304, 11625211, 11621202 and 11722222), the Science Challenge Project (no.
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Convergent Richtmyer–Meshkov instability of heavy gas layer 902 A3-17
TZ2016001), the Strategic Priority Research Program of the Chinese Academy of Sciences
(no. XDB22020200) and the CAS Centre for Excellence in Complex System Mechanics.
Declaration of interests
The authors report no conflict of interest.
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