Instability of a heavy gas layer induced by a cylindrical convergent shock

Abstract

The instability of a heavy gas layer (SF 6 sandwiched by air) induced by a cylindrical convergent shock is studied experimentally and numerically. The heavy gas layer is perturbed sinusoidally on its both interfaces, such that the shocked outer interface belongs to the standard Richtmyer–Meshkov instability (RMI) initiated by the interaction of a uniform shock with a perturbed interface, and the inner one belongs to the nonstandard RMI induced by a rippled shock impacting a perturbed interface. Results show that the development of the outer interface is evidently affected by the outgoing rarefaction wave generated at the inner interface, and such an influence relies on the layer thickness and the phase difference of the two interfaces. The development of the inner interface is insensitive (sensitive) to the layer thickness for in-phase (anti-phase) layers. Particularly, the inner interface of the anti-phase layers presents distinctly different morphologies from the in-phase counterparts at late stages. A theoretical model for the convergent nonstandard RMI is constructed by considering all the significant effects, including baroclinic vorticity, geometric convergence, nonuniform impact of a rippled shock, and the startup process, which reasonably predicts the present experimental and numerical results. The new model is demonstrated to be applicable to RMI induced by a uniform or rippled cylindrical shock.

Full text

Phys. Fluids 34, 042123 (2022); https://doi.org/10.1063/5.0089845 34, 042123
© 2022 Author(s).
Instability of a heavy gas layer induced by a
cylindrical convergent shock
Cite as: Phys. Fluids 34, 042123 (2022); https://doi.org/10.1063/5.0089845
Submitted: 01 March 2022 • Accepted: 30 March 2022 • Published Online: 20 April 2022
Jianming Li (李坚明), Juchun Ding (丁举春), Xisheng Luo (罗喜胜), et al.

Instability of a heavy gas layer induced
by a cylindrical convergent shock
Cite as: Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845
Submitted: 1 March 2022 .Accepted: 30 March 2022 .
Published Online: 20 April 2022
Jianming Li (李坚明),
1
Juchun Ding (丁举春),
1,a)
Xisheng Luo (罗喜胜),
1
and Liyong Zou (邹立勇)
2
AFFILIATIONS
1
Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, China
2
Laboratory for Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics,
Mianyang, Sichuan 621900, China
a)
Author to whom correspondence should be addressed: djc@mail.ustc.edu.cn
ABSTRACT
The instability of a heavy gas layer (SF
6
sandwiched by air) induced by a cylindrical convergent shock is studied experimentally and
numerically. The heavy gas layer is perturbed sinusoidally on its both interfaces, such that the shocked outer interface belongs to the
standard Richtmyer–Meshkov instability (RMI) initiated by the interaction of a uniform shock with a perturbed interface, and the inner one
belongs to the nonstandard RMI induced by a rippled shock impacting a perturbed interface. Results show that the development of the outer
interface is evidently affected by the outgoing rarefaction wave generated at the inner interface, and such an influence relies on the layer
thickness and the phase difference of the two interfaces. The development of the inner interface is insensitive (sensitive) to the layer
thickness for in-phase (anti-phase) layers. Particularly, the inner interface of the anti-phase layers presents distinctly different morphologies
from the in-phase counterparts at late stages. A theoretical model for the convergent nonstandard RMI is constructed by considering all the
significant effects, including baroclinic vorticity, geometric convergence, nonuniform impact of a rippled shock, and the startup process,
which reasonably predicts the present experimental and numerical results. The new model is demonstrated to be applicable to RMI induced
by a uniform or rippled cylindrical shock.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0089845
I. INTRODUCTION
When a perturbed interface separating two fluids of different
property is impulsively accelerated by a shock wave, the interface
deforms persistently, giving rise to a continuous increase in perturba-
tion amplitude followed by a flow transition to turbulent mixing.
This type of hydrodynamic instability is often referred to as
Richtmyer–Meshkov instability (RMI) since it was first theoretically
analyzed by Richtmyer
1
and later experimentally confirmed by
Meshkov.
2
The RMI can be considered as an impulsive variant of
Rayleigh–Taylor instability (RTI) that occurs at a perturbed interface
under a finite but sustained acceleration.
3,4
Over the past few decades,
the RMI has received increasing attention due to its wide existence in
natural and industrial situations such as astrophysical problems
5
and
inertial confinement fusion (ICF)
6
as well as the fundamental signifi-
cance in academic research such as compressible turbulence
7
and vor-
tex dynamics.
8
To explore the physical mechanisms of RMI, a large amount of
efforts and attempts have been made from theoretical,
9
experimental,
10
and numerical aspects.
11,12
Most of the previous studies were focused
on the RMI initiated by a planar shock wave (i.e., planar RMI),
13–15
and only a few exceptions were devoted to the convergent counterpart
(i.e., RMI induced by a convergent shock).
16,17
Based on the shock
dynamics theory, a special wall profile that gradually transforms an
incident planar shock into a cylindrical convergent one was designed
by Zhai et al.,
18
and subsequently, numerous experiments on the con-
vergent RMI were successfully conducted by Si et al.
19
and Luo et al.
20
A gas-lens technique for generating cylindrical shocks through shock
refraction was originally proposed by Dimotakis and Samtaney
21
and
then realized by Biamino et al.
22
in a conventional shock tube. These
experimental results exhibited great potential of this technique for
studying the convergent RMI.
23
Recently, a novel semi-annular shock
tube, whose structure greatly facilitates both the formation of cylindri-
cal shocks and the generation of well-characterized gas interfaces, was
designed by Luo et al.,
24
with which accurate measurements of pertur-
bation growths of the convergent RMI at a single-mode interface were
achieved.
25
Previous experiments on convergent RMI were mainly limited to
the situation of a single interface. This is not the situation that
Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-1
Published under an exclusive license by AIP Publishing
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happened in reality such as the instability in ICF, where there at least
exist two interfaces separating three material shells, namely, the outer
ablator, the middle solid deuterium–tritium (DT) ice, and the inner
gaseous DT fuel. As the capsule is irradiated by high-power x-rays or
laser beams, a spherical convergent shock is immediately produced,
which later moves inward and passes across the outer interface (OI)
and inner interfaces (II) successively, triggering the RMI at each inter-
face. As compared to the RMI at an isolated interface, the presence of
two interfaces involves more physical regimes such as interface cou-
pling and the influence of complex waves reverberating between the
two interfaces. Moreover, these new regimes could couple with the
effects of geometric convergence and Rayleigh–Taylor (RT) instability/
stability and further complicate the instability evolution. Therefore,
elaborate experiments on the convergent RMI at two (or more) interfa-
ces are highly desirable for understanding the instability in ICF.
Recently, a novel soap-film technique has been extended to generate
controllable gas layers with well-defined shapes (a gas layer has two
interfaces: one inner and one outer), such that the convergent RMI at
dual interfaces can be realized in experiment.
26,27
The cylindrical gas layers can be categorized into three types
depending on the interface perturbation: outer-perturbed layer (only
the outer interface is perturbed and the inner interface is uniform),
inner-perturbed layer (only the inner interface is perturbed and the
outer interface is uniform), and double-perturbed layer (both the inner
and outer interfaces are perturbed). Recently, the outer-perturbed layer
and the inner-perturbed layer subjected to a cylindrical convergent
shock have been examined in a semi-annular shock tube,
26,28
and rich
phenomena that are absent in the single interface case were observed.
It was found that the initial shape of the gas layer significantly influen-
ces the instability growth at each interface. As the most complex case,
the development of a gas layer with both inner and outer interfaces
being perturbed subjected to a cylindrical shock has never been exam-
ined, which motivates the current study. For this case, the perturbation
at the outer interface could be imparted to the transmitted shock (TS)
generated inside the layer, which later moves inward and impacts the
inner interface. Thus, the instability at the inner interface is of the
problem of a rippled shock accelerating a perturbed interface, which
has never been analyzed in literature due to its high complexity. The
interaction initiated by the interaction of the rippled shock with the
perturbed inner interface is of a type of complex nonstandard RMI
(i.e., a distorted shock impacting a perturbed interface), which contains
much more regimes than the simple nonstandard RMI (i.e., a distorted
shock impacting an unperturbed interface)
29,30
and the conventional
standard RMI (i.e., a uniform shock impacting a perturbed inter-
face).
31,32
This complex nonstandard RMI, particularly in a convergent
geometry, further motivates the present study. Also, the influence of
the initial phase difference of the two interfaces on theinstability devel-
opment remains an open question.
In this work, we examine the convergent RMI at a heavy gas
layer with sinusoidal outer and inner interfaces in a semi-annular
convergent shock tube. High-accuracy numerical simulations are
also performed to provide rich flow field information. Influences
of the layer thickness and the phase difference of the two interfaces
on the instability growth are emphasized. The regimes of the com-
plex nonstandard RMI at the inner interface of the gas layer are
carefully analyzed, and subsequently, a universal theoretical model
will be established.
II. EXPERIMENTAL AND NUMERICAL METHODS
Experiments are carried out in a semi-annular convergent shock
tube originally designed by Luo et al.,
24
which has exhibited good fea-
sibility and reliability in producing cylindrical convergent shock
waves.
26,27
The overall structure of the facility and the formation prin-
ciple of the cylindrical shock have been detailed in previous work,
25
and, thus, are not presented here. Benefiting from the open test section
[Fig. 1(a)], a removable interface formation device can be easily
designed in which single and dual interfaces of well-characterized
shapes can be generated using the existing soap-film technique. As
illustrated in Fig. 1(b), the interface formation device is composed of
two semi-circular Plexiglass plates connected to a rectangular block.
Two pairs of grooves of a sinusoidal shape are carved on the internal
surfaces of the two plates, respectively, to produce desired constraints.
These grooves are 0.30 mm wide and 0.50 mm deep and have a negli-
gible influence on the instability development. In experiment opera-
tion, a thin pipe with its leading end dipped with some soap solution
(made of 78% distilled water, 2% sodium oleate, and 20% glycerin by
mass) is first inserted into the device (located at the region between
the inner and outer grooves). As the test gas (SF
6
)issuppliedthrough
the pipe from its trailing end, a gas bubble is immediately formed.
Later, the bubble expands continuously due to the injection of SF
6
and
then contacts the grooves. Afterward, the soap film (bubble surface)
moves along the grooves, and finally, a gas layer of a desired shape is
FIG. 1. (a) Drawing of the open test section, (b) photograph of the interface formation device, and (c) the experimental configuration corresponding to a cylindrical shock
impacting a SF
6
layer. The single-mode outer interface is described as ro¼Ro
0þao
0cos ðnhÞand the inner interface as ri¼Ri
0þai
0cos ðnhÞ, where R
0
is the radius of the
initial interface, a
0
is the initial amplitude, and nis the azimuthal mode number. Superscript o(i) denotes the outer (inner) interface.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-2
Published under an exclusive license by AIP Publishing

generated. After this, the whole device is immediately inserted into the
test section and equipped tightly with the optical windows, as shown
in Fig. 1(a). The experimental configuration is sketched in Fig. 1(c),
where a cylindrical convergent shock moves inward and later impacts
the perturbed SF
6
layer. In a polar coordinate system, the outer surface
of the layer can be parameterized as ro¼Ro
0þao
0cos ðnhÞand the
inner surface as ri¼Ri
0þai
0cos ðnhÞ. Here, R
0
refers to the mean
radius of the initial interface, a
0
to the initial perturbation amplitude
of the interface, and nto the azimuthal mode number. Superscript o
(i) denotes the outer (inner) interface.
Due to the wide-open inner core of the semi-annular shock tube,
aZ-fold schlieren system combined with the high-speed imaging tech-
nique can be readily adapted to the facility, which greatly facilitates the
flow diagnostic. In this work, the frame rate of the high-speed video
camera is set as 50000 fps. with a shutter time of 1 ls. The pixel reso-
lution of the schlieren images obtained is approximately 0.28 mm/
pixel. The incident cylindrical shock (ICS) has a Mach number of
M¼1.24 60.01 when it arrives at the mean position of the outer sur-
face (i.e., Ro
0¼55:0 mm). The ambient pressure and temperature are
101.33kPa and 293.15 K, respectively.
Numerical simulation is performed to obtain the detailed flow
field required for an in-depth analysis on flow regimes. In this work,
two-dimensional (2D) Euler equations supplemented by the mass con-
servation of either species are adopted as the governing system. The
reasons why 2D simulation is used are as follows. First, the flow in
experiment is nearly 2D during the present timeof interest, and, there-
fore, 2D numerical simulation is acceptable for the instability growth
at early to intermediate stages.
33,34
Second, Euler simulations that
ignore viscosity can provide results that have good agreements with
experimental results for the growth of RMI at early to intermediate
stages.
35,36
In addition, agreements between experimental results and
predictions of inviscid theoretical models for the early-stage perturba-
tion growth rate were achieved in previous studies,
37,38
which demon-
strates again the reasonability of inviscid flow assumption at early to
intermediate stages. At late stages, small-scale structures are generated
due to secondary instabilities and nonlinearity, and the flow exhibits
evident three dimensionality (sometimes the flow even transits to tur-
bulence). As a result, 2D simulation fails to provide sufficiently accu-
rate results, and the numerical results become sensitive to viscosity. If
one aims to examine small-scale structure at late stages or subsequent
turbulent mixing, not only three-dimensionality and physical viscosity
but also thermodynamic nonequilibrium behaviors
39–41
must be con-
sidered. Since the present study is mainly focused on the interface evo-
lution at early to intermediate stages, the employment of 2D Euler
equations is acceptable.
The governing equations (2D Euler equations) in quasi-
conservative form are
@
@tðqÞþ @
@xjðqujÞ¼0;
@
@tðquiÞþ @
@xjðquiujþpdi;jÞ¼0;
@
@tðqeÞþ @
@xjðqeþpÞuj

¼0;
@
@tðqYkÞþ @
@xjðqujYkÞ¼0;
(1)
where q,u,p,andeare the density, velocity, pressure, and total energy
per unit mass of the gas mixture, respectively. qe¼p=ðc1Þþ
1
2quiuiwith cbeing the effective specific heat ratio of the gas mixture,
which is calculated by
c¼XYkck
Mkðck1ÞXYk
Mkðck1Þ:(2)
Here, Y
k
,c
k
,andM
k
are the mass fraction, specific heat ratio, and molar
mass of species k, respectively. Since the mass conservation equations
for gas mixture (q)andspecies1(q
1
) are solved at every time step,
the density of species 2 (q
2
) can be obtained via q2¼qq1.Thus,the
mass fraction of either species can be calculated by Yk¼qk=q.Inthe
present simulations, c1¼1:4andM1¼28:97 g/mol are adopted for
air, and c2¼1:094 and M2¼146:05 g/mol for SF
6
.Atgridpoints
where a gas mixture exists, the effective specific heat ratio is calculated
according to Eq. (2) with the calculated mass fractions of air (Y
1
)and
SF
6
(Y
2
) there. To accurately capture material interfaces and wave pat-
terns, fifth-order weighted essentially nonoscillatory (WENO) construc-
tion
42
combined with the double-flux algorithm of Abgrall and Karni
43
is implemented in the present solver. The third-order total variation
diminishing (TVD) Runge–Kutta scheme is used for time advancing.
The present algorithm has exhibited great capability for the simulation
of shock-interface interaction problems.
34,44
For more details about the
algorithm, readers can refer to Ding et al.
34
The computational setup is sketched in Fig. 2(a), where the initial
and boundary conditions are specified according to the experimental
configuration. To save the computational cost, only half of the physical
domain with an area of xy¼100 100 mm
2
is considered in sim-
ulation. The pre-shock gases are stationary at the beginning, and their
physical properties are the same as those in experiment (Table I). The
post-shock flow is given according to the Rankine–Hugoniot relation.
The left and bottom edges of the computational domain adopt sym-
metric boundary condition, and the right and top ones adopt outflow
boundary condition, which applies a zeroth-order extrapolation of
physical quantities to ghost points. The effect of mesh size on the com-
putational result is first checked. Four uniform meshes with grid spac-
ings of 0.4, 0.2, 0.1, and 0.05 mm are employed. The density profiles
extracted from the y¼xline for case 2 (130 lsaftertheshockimpact)
under different meshes are plotted in Fig. 2(b), where the density of
pre-shock air (q
0
) is employed as the characteristic value. As we can
see, the density distribution curves are convergent as the mesh size
decreases gradually from 0.4 to 0.05 mm. To ensure the computational
accuracy and, meanwhile, to minimize the computational cost, the
mesh with a 0.1 mm grid spacing is adopted in the present simulations
(i.e., the total number of mesh points is 10011001). Since the
Cartesian grid produces initial small steps along a curved interface,
which would influence the instability development, the initial sharp
interface is artificially diffused within four grid cells.
III. RESULTS AND DISCUSSION
A. Evolution of the SF
6
layer with in-phase
perturbations
Three SF
6
layers of different thicknesses subjected to a cylindrical
convergent shock are first examined. All the three layers have in-phase
sinusoidal perturbations on their inner and outer interfaces. Detailed
parameters corresponding to the initial conditions for each case
(cases 1–3) are listed in Table I. The Atwood number is defined as
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Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-3
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Ao¼ðq2q1Þ=ðq2þq1Þfor the outer interface and Ai¼ðq3q2Þ=
ðq3þq2Þfor the inner interface, where q
1
refers to the density of
ambient air, q
2
to the gas inside the layer (SF
6
contaminated by air),
and q
3
tothegasontheinteriorsideoftheinnerinterface(aircon-
taminated by SF
6
).Forthethreelayers,theouterinterfaceislocatedat
a fixed position (Ro
0¼55.0 mm) with a sinusoidal perturbation (ao
0
¼2.0 mm and n¼6) on it. The radius of the inner interface is varied
(Ri
0¼40.0, 30.0, and 20.0 mm) to produce different-thickness gas
layers. To maintain a constant amplitude-over-wavelength ratio
(0.035) for the inner interface, the amplitude of the inner interface is
set to be 1.45, 1.09, and 0.73 mm for cases 1–3, respectively.
Developments of the wave patterns and interface morphologies
for cases 1–3 are displayed by sequences of schlieren images in Fig. 3,
where the left half of each image refers to the experimental result and
the right half to the numerical one. Generally, the numerical simula-
tion reasonably reproduces the experimental results for both the wave
propagation and the interface deformation. It demonstrates good reli-
ability of the numerical algorithm and high fidelity of the numerical
flow field obtained. Despite the overall agreement, there is a visible dis-
crepancy between simulation and experiment at t>300 ls. The dis-
crepancy may lie in uncertainties in both experiment and simulation.
Specifically, in experiment, soap droplets are produced immediately
after the shock impact, which are then entrained in the evolving inter-
faces and contaminate the instability development. Three dimension-
ality and viscosity also influence the instability development to some
extent, particularly at late stages, which are not considered in the pre-
sent 2D simulations. Here, we take case 2 as an example to detail the
instability evolution process. Time origin (0 ls) in this work is defined
as the moment when the incident cylindrical shock arrives at the
mean position of the outer interface (i.e., Ro
0¼55.0 mm). At the begin-
ning, an incident cylindrical shock (ICS) as well as the sinusoidal outer
(OI) and inner interfaces (II) can be clearly observed (−20 ls). Both
interfaces are initially covered by the grooves in the Plexiglass plates
and, thus, seem quite thick prior to the shock impact. After the ICS
impact, the outer interface leaves the grooves and presents a distinct
sinusoidal shape (140 ls), which verifies the feasibility of the experi-
mental method for generating controllable gas layers. After the ICS
passes across the outer interface, it immediately bifurcates into an
outward-moving reflected shock (RS) and an inward-moving trans-
mitted shock (TS
1
). Noticeable perturbations are imparted to the gen-
erated waves (both TS
1
and RS present a sine-like shape similar to the
outer interface) due to the acoustic impedance mismatch across the
interface. It is seen that the perturbation amplitude of the outer inter-
face experiences a sudden drop due to shock compression. Later, it
increases continuously caused by the induction of baroclinic vorticity
and the effect of geometric convergence, as has been reported by Ding
et al.
25
As time proceeds, the distorted transmitted shock (TS
1
)moves
inwards and collides with the sinusoidal inner interface, triggering the
RMI there. The interaction between the distorted TS
1
and the sinusoi-
dal inner interface is of a type of complex nonstandard RMI (i.e., a dis-
torted shock impacting a perturbed interface), which contains much
more regimes than the simplenonstandard RMI (i.e., a distorted shock
impacting an unperturbed interface)
29,30
and the conventional stan-
dard RMI (i.e., a uniform shock impacting a perturbed interface).
31,32
To the authors’knowledge, this complex nonstandard RMI, particu-
larly in a convergent geometry, has never been reported in literature,
and the present result provides a rare opportunity to analyze the
underlying mechanisms.
FIG. 2. (a) Schematic of the numerical setup and (b) the density profiles along the y¼xline for case 2 (130 ls) at different mesh sizes.
TABLE I. Parameters corresponding to the initial conditions for each case. a
0
and k
refer to the initial perturbation amplitude and wavelength, respectively. vfra1 (SF
6
)
and vfra2 (SF
6
) are the SF
6
volume fractions inside the layer and on the interior side
of the inner interface, respectively. Aþ
o(Aþ
i) denotes the post-shock Atwood number
of the outer (inner) interface.
Case
Ri
0
(mm)
ai
0
(mm)
k
i
(mm) ai
0=kivfra1
(SF
6
)
vfra2
(SF
6
)Aþ
oAþ
i
1 40.0 1.45 41.9 0.035 0.54 0.02 0.56 −0.52
2 30.0 1.09 31.4 0.035 0.53 0.06 0.55 −0.46
3 20.0 0.73 20.9 0.035 0.58 0.12 0.57 −0.41
4 40.0 −1.45 41.9 0.035 0.59 0.03 0.58 −0.53
5 30.0 −1.09 31.4 0.035 0.54 0.08 0.55 −0.44
6 20.0 −0.73 20.9 0.035 0.52 0.15 0.55 −0.34
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The driving mechanisms of nonstandard RMI are the velocity
perturbation and pressure disturbance.
45
As shown in Fig. 4(a),the
crest of the transmitted shock TS
1
first passes across the inner interface
and attains a higher speed than its trough part. As a result, the
generated transmitted shock (TS
2
) has a perturbation amplitude much
lower than that of TS
1
and seems even like a uniform cylindrical shock
(120 ls). This indicates a weak influence of pressure disturbance
behind TS
2
on the instability growth. Also, an outward-moving
FIG. 3. Typical schlieren images from experiment (left half of each image) and simulation (right half), illustrating the developments of interfaces and waves for cases 1–3. ICS:
incident cylindrical shock; OI: outer interface; II: inner interface; GR: grooves that exist only in experiment; RS: reflected shock; RW
i
:ith rarefaction wave; TS
i
:ith transmitted
shock; RTS: reflected transmitted shock.
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Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-5
Published under an exclusive license by AIP Publishing

rarefaction wave (RW
1
) is generated, which later interacts with the
evolving outer interface and affects the instability growth there. The
RW
1
is relatively weak and, thus, not clear in the experimental images.
Later, TS
2
focuses at the geometric center (170 ls), and a reflected
transmitted shock (RTS) is immediately generated. Then, the explod-
ing RTS interacts with the deforming inner interface (called reshock),
producing a third transmitted shock (TS
3
). The occurrence of reshock
significantly complicates the instability development at the inner inter-
face, which will be quantitatively discussed hereinafter. With time
going on, TS
3
moves outward and strikes the deforming outer inter-
face, producing an outgoing transmitted shock (TS
4
) and an ingoing
rarefaction wave (RW
2
). At t300 ls, the inward-moving RW
2
encounters the inner interface, affecting the development of the inner
interface again. Impacted by RTS (TS
3
), the inner (outer) interface
slows down quickly and then evolves at a nearly fixed radial location.
Temporal variations of the perturbation amplitude of the outer
interface from experiment and simulation for cases 1–3 are shown in
Fig. 5(a). The uncertainty of the measured interface amplitude
is 60.28 mm before reshock. After reshock, the interface becomes
thicker and the uncertainty is 60.56 mm, which is r epresented by the
error bars in Fig. 5(a). As expected, the instability growth from numer-
ical simulation matches quite well with the experimental one for all
FIG. 4. Detailed observations of the wave patterns and interface morphologies for (a) case 2 and (b) case 5. The symbols are the same as those in Fig. 3.
FIG. 5. Temporal variations of perturbation amplitudes of the (a) outer and (b) inner interfaces for in-phase layers of different thicknesses. Symbols refer to the experimental
data and lines to the numerical results. The time origin (0 ls) is defined as the instant when the ICS (TS
1
) arrives at the mean location of the outer (inner) interface.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-6
Published under an exclusive license by AIP Publishing

the cases at early stages (t<350 ls).Generally,theamplitudegrowth
can be divided into five stages. At stage I, the perturbation amplitude
experiences a sudden drop due to shock compression. Then, it
increases continuously due to the induction of baroclinic vorticity on
theouterinterface(stageII).Atthisstage,theinterfacemovesinward
at a nearly constant velocity, and geometric convergence gives rise to a
continuous increase in the perturbation growth rate, which is in accor-
dance with the analyses of Bell
46
and Plesset.
47
Later, the rarefaction
wave RW
1
interacts with the outer interface, promoting the instability
growth there (stage III). Here, the RW
1
affects the instability growth in
two ways. First, it stretches the perturbed interface, quickly amplifying
the interface amplitude. Second, it deposits additional baroclinic vor-
ticity on the interface due to the misalignment of pressure gradient
across RW
1
and the density gradient at the interface, which further
changes the instability growth. Later, the interface decelerates when it
approaches the geometric center, causing RT stability that evidently
suppresses the instability growth. This explains the continuous drop of
the perturbation growth rate at stage IV. After the transmitted shock
(TS
3
) impacts the evolving outer interface, the interface amplitude
reduces gradually to zero (phase inversion) and then increases in the
opposite direction (stage V). After the reshock, the perturbation ampli-
tude presents a linear growth with time, which is similar to the RMI
under reshock in a planar geometry.
48
After the impact of TS
1
, the perturbation amplitude of the inner
interface reduces gradually to zero and then increases in the opposite
direction, as shown in Fig. 5(b). Later, the RTS hits the evolving inner
interface, causing a considerable rise in the perturbation growth rate.
At 190 ls, the rarefaction wave (RW
2
) encounters the inner inter-
face, further increasing the growth rate. As mentioned above, the RMI
at the inner interface here belongs to the complex nonstandard RMI,
which has never been specifically studied. The detailed wave patterns
and the growth of perturbation amplitude obtained in this work pro-
vide a rare opportunity to carefully analyze the flow regimes, based on
which a theoretical modeling of the perturbation growth can then be
performed (Sec. III C). It can be generally concluded that, for the con-
vergent RMI at a perturbed gas layer, there exist complex wave pat-
ternsthatinteractwiththeinnerandouterinterfacesofthegaslayer
successively, making the instability growth on either interface far more
complicated than that on an isolated interface.
For case 1 with a thinner layer, the rarefaction wave (RW
1
)
arrives at the outer interface earlier than that in case 2, and accord-
ingly, the RTI occurs earlier. This explains the deviation of the instabil-
ity growth of case 1 from cases 2 and 3 at t100 ls. Also, at later
stages, RT stability occurs due to interface deceleration, causing a grad-
ual drop in the perturbation growth rate before the RTS arrival. This is
analogous to the perturbation growth in case 2. For case 3 with a
thicker layer, RW
1
has to travel a longer distance to reach the outer
interface and, thus, promotes the instability growth at a relatively later
time at which the interface deceleration has started (i.e., RT stability
exists). The quasi-linear growth of perturbation amplitude with time
(from 200 to 300 ls) for case 3 in both experiment and simulation is
ascribed to the counteraction between the instability promotion
(caused by both the rarefaction wave and geometric convergence) and
the instability suppression caused by RT stability. It is found that the
growth rates after reshock for all the three cases are nearly identical,
which is consistent with the previous finding that the post-reshock
growth rate is almost independent of pre-reshock amplitude and
wavenumber.
28,49
Although the growth rates of perturbation ampli-
tude of the inner interface after the impact of RTS for cases 1–3are
nearly identical, there exist evident discrepancies at late stages due to
the different effects of RW
2
. Specifically, when RW
2
implodes, its
trough part diverges and becomes gradually weaker with time. For
case 3 with a thicker layer, RW
2
travels a longer distance and, thus,
becomes very weak when it arrives at the inner interface, indicating a
negligible influence of RW
2
on the development of inner interface.
Differently, for cases 1 and 2, RW
2
is stronger when it arrives at the
inner interface and, thus, evidently promotes the perturbation growth
there. The present result suggests that the gas layer thickness deter-
mines both the strength of rarefaction wave and the moment of the
rarefaction wave–interface interaction and subsequently affects the
instability growth.
B. Evolution of the SF
6
layer with anti-phase
perturbations
To examine the influence of perturbation phase on the instability
development, three SF
6
layers with anti-phase perturbations on the
innerandouterinterfaces(cases4–6) are then considered. For cases
4–6, the outer interface is the same as that of cases 1–3, but the inner
interface has a perturbation phase opposite to that of the outer inter-
face. The initial conditions for each case are given in Table I.
Detailed processes of the wave propagation and interface defor-
mation for cases 4–6 are illustrated in Fig. 6.Althoughthewholeevo-
lution process for each case is qualitatively similar to that of the
corresponding in-phase case, there still exist visible differences
between the two configurations. First, the RW
1
has a perturbation
phase opposite to that of the outer interface and, thus, takes a longer
time to pass across the outer interface. Second, the trough of the trans-
mitted shock (TS
1
) first passes across the inner interface, attaining a
higher speed than the crest part. As a consequence, the TS
2
generated
has a perturbation amplitude much higher than that of the TS
1
,indi-
cating a stronger pressure disturbance behind the TS
2
.Foranin-phase
layer, as shown in Fig. 4, the distorted rarefaction wave (RW
1
) has the
same perturbation phase as that of the outer interface, such that RW
1
transverses the outer interface quickly, and the TS
2
is almost uniform.
The schlieren images on the last column of Fig. 6 show that the inner
interface presents distinctly different morphologies (red lines) at late
stages for different-thickness gas layers. Specifically, for a thin layer
(case 4), a pair of vortices is formed at the spike neck on the inner
interface, whereas for a thicker layer (case 5), the inner interface devel-
ops far more slowly, and thus, no roll-up structures are formed during
the present time of interest. For the thickest layer (case 6), the pertur-
bation phase of the inner interface is opposite to that of cases 4–5. A
physical explanation for such different interfacial structures will be
given hereinafter.
Temporal variations of the circulation at the inner interface for
the three anti-phase cases are plotted in Fig. 7(a). The circulation here
is obtained by summing the product of vorticity with area at the region
just enclosing the inner interface from numerical simulation, which is
mathematically expressed as
CðtÞ¼X
IIX
0X
IIY
0
tðiþ1;jÞtði1;jÞ
2Dxuði;jþ1Þuði;j1Þ
2Dy

ðDxDyÞ;
(3)
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where iand jsatisfy RiðtÞaiðtÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðiDxÞ2þðjDyÞ2
qRiðtÞ
þaiðtÞ(the superscript idenotestheinnerinterface).IIX and IIY are
the maximum grid numbers in the xand ydirections, respectively,
Dx¼Dyis the grid spacing, uand tare the velocity components in
the xand ydirections, respectively. After the TS
1
impact, the circula-
tion rises quickly to a peak due to baroclinic torque. Then, it presents
an evident oscillation before the RTS arrival. As shown in Fig. 8,
noticeable transverse waves are formed behind TS
2
, and later, they
FIG. 6. Typical experimental (left half) and numerical (right half) schlieren images showing the evolution of interfaces and waves for anti-phase layers of different thicknesses
(cases 4–6). Symbols are the same as those in Fig. 3.
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interact with the inner interface continuously, causing a gradual reduc-
tion in circulation. At t¼69 ls, the transverse waves encounter each
other at the trough of the interface and then are reflected. Afterward,
the reflected transverse waves sweep the interior interface in the oppo-
site direction, causing a quick rise in circulation. The reverberation of
transverse waves is the major reason for the circulation oscillation.
This also explains the finding that the oscillation frequency of circula-
tion is nearly equivalent to the reverberation frequency of transverse
waves. It can be generally concluded that the thinner the gas layer, the
stronger the interaction between the transverse waves and the inner
interface, and consequently, the greater the oscillation amplitude of
circulation. For case 6, the transverse waves are very weak (cannot be
clearly identified in density contour images), leading to a small oscilla-
tion of circulation.
According to the aforementioned findings, we realize that the
complex nonstandard RMI at the inner interface is dominated by three
mechanisms: velocity perturbation caused by the impact of nonuni-
form distorted shock (Mach number is nonuniform along the shock
front), baroclinic vorticity deposited on the interface, and pressure dis-
turbance behind the transmitted shock. According to Ishizaki et al.,
45
the growth rate ( _
aimp) caused by velocity perturbation is closely related
to the growth rate (_
as) of perturbation amplitude of the distorted
shock, i.e., _
aimp ¼_
asuc=us,whereu
c
refers to the post-shock interface
velocity and u
s
to the shock velocity. In the present experiments, the
complex nonstandard RMI at the inner interface occurs in a conver-
gent geometry, for which there exist additional flow regimes such as
the RT effect and geometry convergence. Considering that the geomet-
ric convergence and the RT effect are weak at early stages,
25
the solu-
tion of Ishizaki et al.
45
can be adopted to approximately estimate the
growth rate caused by velocity perturbation. Temporal variation of
perturbation amplitude of TS
1
for case 6 is given in Fig. 7(b),which
shows a continuous decay of perturbation amplitude of the rippled
shock. It indicates that the nonuniform impact of TS
1
(i.e., velocity
perturbation) yields a negative growth rate for the perturbation ampli-
tude of the inner interface. We can take case 6 as a typical case to illus-
trate the effect of the velocity perturbation because the transmitted
shock in this case traverses the longest distance before it encounters
the inner interface. The development of TS
1
inside the gas layer for
case 6 in experiment is in reasonable agreement with the numerical
one, and also the gas concentration inside the layer exhibits a subtle
difference from case to case in experiment; therefore, Fig. 7(b) that is
plotted based on the numerical and experimental results of case 6 can
be used to characterize the development of TS
1
for all the cases in this
work. Time instant at which TS
1
encounters the inner interface for
each case has been marked in Fig. 7(b) to facilitate the understanding.
As shown in Fig. 7(b), for a thinner layer where the inner interface is
closer to the outer interface, the growth rate of the perturbation ampli-
tude of TS
1
is smaller in magnitude when it arrives at the inner inter-
face, corresponding to a slower instability growth caused by velocity
perturbation.
The inner interface of the present gas layers is of a heavy/light
configuration, and thus, baroclinic vorticity deposited on it would
make the interface amplitude decrease continuously until the occur-
rence of phase reversal. In this work, the amplitude of the inner inter-
face for the anti-phase layers is defined as negative, and the growth
rate of perturbation amplitude caused by baroclinic vorticity is posi-
tive. As indicated by Fig. 7(b), for a thinner layer, the perturbation
amplitude of TS
1
is higher when it meets the inner interface. This
means that the angle between the shock front (pressure gradient) and
interface morphology (density gradient) is larger, and, therefore, more
baroclinic vorticity is produced during the shock-interface interaction.
Hence, for case 4 with a thin layer, baroclinic vorticity is a major
regime dominating the instability growth, and it causes a quick reduc-
tion in interface amplitude until the phase inversion (34 ls). As a
result, the interface amplitude is quite high just before the RTS impact
(129 ls), and thus, a large amount of baroclinic vorticity is produced
by the RTS impact. This explains the formation of roll-up structures at
the spike neck of the inner interface (426 ls). For the layer of an inter-
mediate thickness (case 5), the negative growth rate imparted by veloc-
ity perturbation counteracts the positive one caused by baroclinic
vorticity, and thus, the perturbation amplitude presents a very slow
growth before the reshock. Hence, the amplitude of inner interface is
quite low just before the RTS arrival, and thus, little baroclinic vorticity
will be produced by the RTS impact. This explains the slow growth of
perturbation amplitude and the absence of roll-up structure at late
stages for this case. For the thickest layer (case 6), velocity perturbation
FIG. 7. (a) Variations of circulation with time for three anti-phase cases, and (b) temporal variation of the perturbation amplitude of TS
1
for case 6. t
4
,t
5
, and t
6
are the instant
when the transmitted shock TS
1
arrives at the mean position of the inner interface in cases 4, 5, and 6, respectively.
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is significant (i.e., produces a large negative growth rate), and the phase
inversion has not occurred at the RTS arrival. This results in an oppo-
site phase of interface perturbation for case 6 as compared to that of
cases 4 and 5. It is also found that for the anti-phase cases, the bubble
structure of the inner interface is quite flat as compared to the in-
phase cases, which is attributed to the rarefaction wave RW
2
that
reverberates between the inner and outer interfaces and continuously
influences the development of inner interface. Here, we take case 4 as
an example to illustrate this influence mechanism. As shown in Fig. 8,
the rarefaction wave RW
2
reflected from the outer interface interacts
continuously with the inner interface, depositing additional baroclinic
vorticity on it. It is seen that the baroclinic vorticity near the bubble tip
changes the sign at t¼224 ls. For a thicker layer (case 5), the vorticity
deposited by the RTS is weaker, and thus, the negative vorticity pro-
duced by RW
2
becomes more dominant, producing a pit at the bubble
center (228 ls). For case 6, the RTS impacts the inner interface earlier
than the RW
2
, producing an evident pit on the bubble.
Figure 9 provides temporal variations of the perturbation ampli-
tudesoftheinnerandouterinterfacesforcases4–6. The instability
growth at the outer interface here is similar to that of in-phase cases.
The growth rates of perturbation amplitude for cases 4–6arenearly
identical before the RW
1
arrival. Once the RW
1
encounters the outer
interface, the growth rate experiences an evident rise. Later, the insta-
bility growth is inhibited by the RT stability caused by interface decel-
eration. Similar to the in-phase cases, the promotion of instability
growth by RW
1
is noticeable for thin layers (cases 4 and 5) and is
FIG. 8. The density contour (left half of each image) showing the developments of waves and interfaces and the distribution of baroclinic vorticity (right half of each image) for
the anti-phase cases. Symbols are the same as those in Fig. 3. To clearly show the wave patterns and interfacial structures, the density contour level is varied for different
images.
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relatively weak for the thick one (case 6). Also, the quasi-linear growth
before the reshock is observed for the thick layer (case 6). These results
demonstrate that the perturbation phase at the inner interface produ-
ces a negligible influence on the development of the outer interface.
Since the interaction duration between the RW
1
and the outer inter-
face is longer for the anti-phase layers, the promotion of instability
growth caused by the RW
1
becomes more noticeable. As shown in
Fig. 4(b),therarefactionwave(RW
1
) has a shape similar to that of the
inner interface, i.e., it has an anti-phase perturbation relative to the
outer interface. As a result, the interaction of the RW
1
with the outer
interface takes a longer period of time. Comparison between the in-
and anti-phase cases for gas layers of different thicknesses is shown in
Fig. 10. As we can see, the perturbation phase produces a prominent
effect on the instability growth for thin layers (e.g., cases 1 and 4 where
the interaction duration for the anti-phase case is much longer than
that in the in-phase case). As the layer thickness increases, the RW
1
becomes weaker, and the discrepancy in the interaction duration
between the in- and anti-phase cases diminishes. Particularly, for the
thickest layer (cases 3 and 6), the phase difference can almost be
ignored.
C. Theoretical model for nonstandard convergent RMI
The RMI development involves two phases: the vorticity produc-
tion phase that occurs as the shock wave passes through the interface
and the interface evolution phase that occurs after the shock wave
leaves the interface. Accurate estimation of baroclinic vorticity gener-
ated at the first phase is very important since it dominates the ensuing
instability growth. Taking a curl of the momentum equation for com-
pressible flows, vorticity transport equation can be obtained:
D~
x
Dt ¼~
xr
~
u~
xr~
uþr2~
xþ1
q2ðrqrpÞ;(4)
where ~
urefers to the velocity, ~
xto the vorticity, pto the pressure, qto
the density, and refers to the kinematic viscosity coefficient. The first
term on the right-hand side of Eq. (4),~
xr
~
u, stands for the vortex
stretching, the second term, ~
xr~
u, for the vortex dilatation, and the
third term, r2~
x, for the viscous dissipation. For a 2D flow consid-
ered in this work, the stretching term can be ignored. Also, compress-
ibility and viscosity produce a negligible influence on the vorticity
transport at the pre-turbulence stage. Thus, the vorticity transport
equation reduces to
D~
x
Dt ¼ðrqrpÞ=q2:(5)
For the canonical standard RMI (i.e., a uniform planar shock
impacting a perturbed interface), both the shock strength and shock
shape keep invariant during the shock-interface interaction, and thus
baroclinic vorticity deposited on the interface can be readily calculated.
Baroclinic vorticity deposited at an inclined interface subjected to a
uniform planar shock has been carefully analyzed by Samtaney
et al.,
50,51
and a scaling law that clearly reveals the circulation on per
unit length of the interface was derived based on shock dynamics the-
ory. The circulation in its physical unit is formulated as
dC
ds ¼2c0
cþ111
ffiffiffi
g
p

ð1þM1þ2M2ÞðM1Þsin a;(6)
for a light/heavy interface, and
dC
ds ¼4c0
cþ111
ffiffiffi
g
p

M21
1þ2c
cþ1ðM21Þ
sin a;(7)
for a heavy/light interface. Here, Mis the incident shock Mach num-
ber, gis the density ratio across the interface, c
0
is the sound speed of
pre-shock fluid, and ais the angle between the shock and interface.
For the convergent RMI considered in this work, the shock
strength and the shock angle relative to the interface vary continuously
during the shock-interface interaction, significantly complicating the
baroclinic vorticity calculation. Moreover, for the inner interface
impacted by a rippled shock, the shock strength is nonuniform along
the shock front, and also there exists significant pressure disturbance
behind the rippled shock, which makes the calculation of baroclinic
vorticity extremely difficult. In this work, to facilitate the modeling of
FIG. 9. Temporal variations of perturbation amplitudes of the (a) outer and (b) inner interfaces for anti-phase layers of different thicknesses. Symbols refer to the experimental
data and lines to the numerical results.
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baroclinic vorticity on the inner interface, the following assumptions
are adopted. First, considering the shock-interface interaction takes
place within a very short period of time, the shock Mach number is
assumed to be constant during this stage. Also, according to Ishizaki
et al.,
45
after the shock leaves the interface, pressure disturbance pro-
duces a negligible influence on the instability growth and, thus, can be
ignored. With these assumptions, the deposition of circulation on an
infinitely small element surrounding the point (r,h) is sketched in
Fig. 11(a), where the straight segments L
1
,L
2
,andL
3
are, respectively,
tangent to the radial line, to the shock front, and to the contact surface
and /s(/c) denotes the inclination angle of L
1
relative to L
2
(L
3
). The
shock shape and shock velocity just before the shock-interface inter-
action can be approximately described as
RsðhÞ¼Rs0þascos ðnhÞ;
_
RsðhÞ¼
_
Rs0þ_
ascos ðnhÞ;
((8)
where Rs0and a
s
denote the mean radius of the initial rippled shock
and the amplitude of perturbation on the rippled shock, respectively.
The angle between the shock and interface satisfies a¼/s/c,and
thus, sin a¼sin /scos /csin /ccos /s. For a sinusoidal interface
rðhÞ¼R0þa0cos ðnhÞ,thereis
L3ds ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2ðhÞþðna0sin ðnhÞÞ2
qdh;(9)
and, thus, we have
sin/c¼rðhþdhÞrðhÞ
L3¼na0sinðnhÞdh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2ðhÞþðna0sinðnhÞÞ2
qdh
;
cos/c¼L1
L3¼rðhÞdh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2ðhÞþðna0sin ðnhÞÞ2
qdh
:
(10)
Also, sin /sand cos /scan be calculated by
sin/s¼RsðhþdhÞRsðhÞ
L2¼nassinðnhÞdh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
sðhÞþðnassin ðnhÞÞ2
qdh
;
cos/s¼L1
L2¼RsðhÞdh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
sðhÞþðnassinðnhÞÞ2
qdh
:
(11)
Thus, we get
sin a¼ða0RsðhÞasrðhÞÞnsin ðnhÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2ðhÞþðna0sin ðnhÞÞ2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
sðhÞþðnassin ðnhÞÞ2
q:(12)
FIG. 10. Comparisons between in- and anti-phase gas layers with thicknesses of 15 mm (a), 25 mm (b), and 35 mm (c) for time-varying amplitude of the outer interface.
Symbols refer to the experimental data and lines to the numerical results.
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Substituting Eqs. (9) and (12) into Eqs. (6) and (7),acirculationmodel
for the RMI at a perturbed interface impacted by a rippled convergent
shock is obtained as
dC
dhðhÞ¼ 2c0
cþ111
ffiffiffi
g
p

ð1þM1ðhÞ
þ2M2ðhÞÞðMðhÞ1Þða0RsðhÞasrðhÞÞnsin ðnhÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
sðhÞþðnassin ðnhÞÞ2
q;
(13)
for a light/heavy interface, and
dC
dhðhÞ¼ 4c0
cþ111
ffiffiffi
g
p

M2ðhÞ1
1þ2c
cþ1ðM2ðhÞ1Þ
ða0RsðhÞasrðhÞÞnsin ðnhÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
sðhÞþðnassin ðnhÞÞ2
q;(14)
for a heavy/light interface. Here, MðhÞ¼
_
RsðhÞ=c0denotes the aver-
age Mach number of the rippled shock during the shock-interface
interaction, and Rs0¼R0þja0asj.IfR0ja0asj,Eqs.(13)
and (14) reduce, respectively, to
dC
dhðhÞ¼ 2c0
cþ111
ffiffiffi
g
p

ð1þM1ðhÞ
þ2M2ðhÞÞðMðhÞ1Þnða0asÞsin ðnhÞ;(15)
for the fast/slow interface, and
dC
dhðhÞ¼ 4c0
cþ111
ffiffiffi
g
p

M2ðhÞ1
1þ2c
cþ1ðM2ðhÞ1Þ
nða0asÞsin ðnhÞ;
(16)
for the slow/fast interface.
With Eqs. (13) and (14), the interface velocity induced by the
vortex sheet along the interface can be estimated based on the
Birkhoff–Rott equation:
~
vCðh;tÞ¼ð2p
0
dC
dhð/Þd/
2pð~
rðh;tÞ~
rð/;tÞÞ:(17)
Discretizing the interface into Npoints and also assuming RðtÞ
aðtÞ(i.e., the interface is regarded as a circle), we have
j~
rðhi;tÞ~
rðhj;tÞj ¼ 2RðtÞ
sin hjhi
2
:(18)
Thus, the radial velocity of the interface induced by the vortex sheet is
vCrðhi;tÞ¼ 1
2NRðtÞX
N
j6¼i
dC
dhðhjÞ

sin hjhi
2
cos hjhi
2þhjhijhjhij
2ðhjhiÞp:(19)
In addition to the vorticity-induced velocity, the shock impact
also imparts a velocity to the interface. As mentioned above, for the
case of a rippled shock, the shock Mach number varies along the shock
front (i.e., the rippled shock is relatively strong at the crest and rela-
tively weak at the trough), and thus, the interface velocity imparted by
the shock impact is nonuniform. This is different from the unper-
turbed shock case where the interface attains a uniform velocity imme-
diately after the shock passage. For a sinusoidal interface
parameterized as rðh;tÞ¼RðtÞþaðtÞcos ðnhÞin a cylindrical geom-
etry, its velocity jump caused by the shock impact is
vimpðhÞ¼
_
RimpðtÞþ_
aimp cos ðnhÞ;(20)
where
_
RimpðtÞis the velocity of the mean position of the interface and
_
aimp is the growth rate of interface amplitude caused by the nonuni-
form impact of a rippled shock. As sketched in Fig. 11(a), for a sinusoi-
dal interface that is in phase with the rippled shock, the interface crest
(trough) is impacted by the strongest (weakest) part of the rippled
shock, and thus, the shock-induced velocity at the crest (trough) can
be approximately solved based on one-dimensional shock dynamics
theory, provided the rippled shock strength distribution. Then, _
aimp
can be readily calculated by _
aimp ¼ðvcrest vtroughÞ=2withv
crest
ðvtroughÞbeing the crest (trough) velocity.
To simplify the theoretical analysis, we assume that the interface
moves at a constant velocity (i.e.,
_
RimpðtÞ¼vc) and also ignore the
effect of compressibility. In this work, a virtual sink is set at the geomet-
ric center, which produces a mean velocity for the interface
_
RimpðtÞand,
meanwhile, ensures the mass conservation of fluid enclosed by the inter-
face. The strength of the sink satisfies Q¼2pRðtÞvc,whereR(t)isthe
time-dependent radius of the interface.
The total velocity of the interface is composed of the above three
parts (i.e., the vorticity-induced velocity, the velocity of interface
FIG. 11. (a) The vorticity deposition for a
perturbed cylindrical shock colliding with a
sinusoidal interface and (b) the distribution
of baroclinic vorticity along the interface.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-13
Published under an exclusive license by AIP Publishing

induced by the sink at the geometric center, and the velocity caused by
the nonuniform impact of a rippled shock) and is expressed as
vrðhi;tÞ¼Rðt0Þ
RðtÞvCrðhi;t0ÞþRðtÞvc
rðhi;tÞþ_
aimp cos ðnhiÞ:(21)
For the convergent RMI, the minimum (maximum) interface velocity
is located at nh¼0(nh¼p), and thus, the growth rate of perturba-
tion amplitude is
_
aðtÞ¼vrð0;tÞvrp
n;t

2¼Rðt0Þ
RðtÞ
_
aCðt0ÞaðtÞvc
RðtÞþ_
aimp:(22)
Here, _
aCðt0Þis the initial growth rate induced by baroclinic vorticity
and can be calculated by
_
aCðt0Þ¼vCrð0;t0ÞvCrp
n;t0

2:(23)
The first term on the right-hand side of Eq. (22) represents the ampli-
tude growth induced by baroclinic vorticity, which becomes gradually
greater (smaller) as the interface moves inward (outward). The second
term corresponds to the effect of geometric convergence/divergence,
which is usually called the BP effect.
46,47
The last term denotes the per-
turbation growth rate caused by nonuniform impact of a rippled shock
(i.e., the velocity perturbation mechanism called by Ishizaki et al.
45
).
Different from the previous theoretical models that are developed
based on potential functions, the present model is established based on
clear physical pictures, and each term of Eq. (22) hasaclearphysical
meaning. This facilitates the understanding of flow regimes of RMI
and also offers a reliable path to the modeling of complex nonstandard
RMI.
The present model reduces to the linear theory of Bell
46
for the standard cylindrical RMI (i.e., the RMI induced by a
uniform cylindrical shock) provided the same initial values. The
demonstration is given in the appendix. A major difference
between the present theory and the Bell model
46
lies in the initial
growth rate. For the Bell model, the initial growth rate is calcu-
lated by
_
aBellðt0Þ¼ðnAþþ1Þvcaðt0Þ
Rðt0Þ;(24)
whereas for the present model, the initial growth rate is calculated
by Eq. (A2). A comparison among the predictions of the two mod-
els and the experimental result for case 3 is given in Fig. 12.Aswe
can see, both models overestimate the experimental initial growth
rate, which is similar to the previous finding on the convergent
RMI at an isolated interface.
25
The overestimation is mainly
ascribed to the ignorance of the startup process of the convergent
RMI (the startup process leads to a lower initial growth rate) dur-
ing the theoretical derivation.
As we know, a point vortex can impart a velocity to its surround-
ing fluid only when the information of the point vortex arrives there
(i.e., the startup process). In a polar coordinate system, a slightly per-
turbed interface can be approximately considered as a circle, and thus,
the information of each point vortex (has a radial velocity v
c
)propa-
gates along the arc. The vortex on point jat t0can produce a radial
velocity v
ij
for point iat t(t>t0),
vij ¼Cj
2pRðt0ÞDhij
cos hjhi
2þhjhijhjhij
2ðhjhiÞp

cos hjhi
2þhjhijhjhij
2ðhjhiÞp
;(25)
where C
j
is the circulation at point jand Dhij is the azimuthal angle
between points jand i.Rðt0Þsatisfies
RðtÞRðt0Þ
vc¼Rðt0ÞDhij
cþ
0
;(26)
where cþ
0is the sound speed behind the transmitted shock (TS
1
for the
outer interface and TS
2
for the inner interface in our experiments).
According to Eq. (26),thereis
Rðt0Þ¼ RðtÞcþ
0
cþ
0þvcDhij
:(27)
Since vc<0and0<RðtÞ<Rðt0ÞRðt0Þ,wehave0<Dhij
<cþ
0
vcðRðtÞ
Rðt0Þ1Þ. Assuming the interface moves uniformly, i.e., RðtÞ
¼Rðt0Þþvct, and also let t
0
¼0forsimplicity,weget0<Dhij
<cþ
0t
Rðt0Þ.AccordingtoEqs.(25) and (27), the induction velocity at point
ican be obtained as
v0
Crðhi;tÞ¼ 1
NRðtÞX
N
0<Dhij<cþ
0t
Rðt0Þ
dCðhjÞ
dhðcþ
0þvcDhijÞ
cþ
0Dhij

cos hjhi
2þhjhijhjhij
2ðhjhiÞp

cos hjhi
2þhjhijhjhij
2ðhjhiÞp
:(28)
Hence, the growth rate of the perturbation amplitude considering the
startup process is
FIG. 12. Comparison among the predictions of the two models and the experimen-
tal result for case 3.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-14
Published under an exclusive license by AIP Publishing

_
a0ðtÞ¼v0
Crð0;tÞv0
Crp
n;t

2aðtÞvc
RðtÞþ_
aimp:(29)
As shown in Fig. 12,Eq.(29) considering the startup process
gives much better prediction of the present experimental and numeri-
cal results than Eq. (22) and the model of Bell.
46
To the authors’
knowledge, this is the first time to model the startup process for the
convergent RMI. Comparison among the theoretical, experimental,
and numerical results for the growth of perturbation amplitude at the
inner interface is shown in Fig. 13. As we can see, the present model
reasonably predicts the instability growth for the in- and anti-phase
cases. It is also found that for anti-phase cases, agreement is relatively
worse than that in the in-phase cases. This is mainly ascribed to the
neglecting of pressure disturbance behind a rippled convergent shock
in our theoretical analysis (pressure disturbance in convergent RMI is
complex and very difficult to model). Specifically, for the in-phase
cases, the second transmitted shock TS
2
is quite smooth [Fig. 4(a)],
and thus, pressure disturbance behind TS
2
produces a negligible influ-
ence on the instability growth. Nevertheless, for the anti-phase cases,
TS
2
presents a considerably large perturbation, and thus, the influence
of pressure disturbance becomes pronounced.
Note that RMI in ICF is triggered by a very strong convergent
shock (Ma >20). In laboratory conditions, it is very difficult to gener-
ate such a strong shock. As a fundamental study, the present work is
dedicated to weak shock-induced RMI in a convergent geometry,
focusing on the underlying physical mechanisms of the instability
growth. For strong shock-induced RMI, the compressibility effect is
very strong, and new phenomena would emerge.
52
In fact, the design
of a new shock tube for generating Ma >3 strong shocks is ongoing in
our group, and strong shock-induced RMI will be reported in near
future. Also, RMI-induced turbulent mixing is beyond the scope of the
present study, and it will be investigated experimentally with planar
laser induced fluorescence diagnostic in future work.
IV. CONCLUSIONS
In this work, a series of elaborate experiments of a convergent
RMI has been performed on a SF
6
layer (with sinusoidal outer and
inner interfaces) surrounded by air in a semi-annular shock tube.
An extended soap-film technique is adopted to generate gas layers of
different shapes and thicknesses. The propagation of wave patterns
and the deformation of interfacial structures are clearly captured by a
high-speed schlieren system. High-accuracy numerical simulations are
also carried out with a compressible multi-component Euler solver,
which well reproduces the experimental results. The detailed flow field
obtained from numerical simulation greatly facilitates the flow analy-
sis, and subsequently, a theoretical model for complex nonstandard
RMI is established.
Complex wave patterns are produced during the shock-gas layer
interaction, which later interact with the outer and inner interfaces of
the layer successively, making the instability growth at either interface
far more complicated than that at an isolated interface. Specifically,
the rarefaction wave RW
1
(RW
2
) generated at the inner (outer) inter-
face collides with the outer (inner) interface, evidently promoting the
instability growth there. The gas layer thickness determines the
strength of the rarefaction wave and the moment of the rarefaction
wave–interface interaction and, thus, further affects the instability
growth at either interface. Also, the phase difference between the inner
and outer interfaces affects both the shape of rarefaction wave and the
perturbation amplitude of the second transmitted shock TS
2
. For an
anti-phase (in-phase) layer, the TS
2
has a perturbation amplitude
much higher (lower) than that of TS
1
, indicating a strong (weak) pres-
sure disturbance on the instability growth.
The instability at the inner interface is of a type of complex non-
standard RMI (i.e., initiated by a rippled convergent shock interacting
with a perturbed interface), which contains much more underlying
regimes than the simple nonstandard (i.e., initiated by a rippled planar
shock interacting with a perturbed interface) and standard (i.e., initi-
ated by uniform planar shock interacting with a perturbed interface)
RMIs. It is found that the inner interface presents distinctly different
(similar) morphologies at late stages for anti-phase (in-phase) layers of
different thicknesses. For a thin layer (case 4), a pair of vortices is
formed at the spike neck on the inner interface, whereas for a thicker
layer (case 5), the inner interface develops far more slowly, and thus,
no roll-up structures are formed during the present time of interest.
For the thickest layer (case 6), the perturbation phase of the inner
interface is opposite to that of cases 4 and 5. The circulation deposited
FIG. 13. Comparison among the experimental (symbol), numerical (sold line), and theoretical (dash line) results for the instability growth at the inner interface of (a) in-phase
and (b) anti-phase layers.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-15
Published under an exclusive license by AIP Publishing

on the inner interface presents evident oscillation before the arrival of
RTS, which is mainly ascribed to the reverberation of transverse waves
inside the layer. The complex nonstandard RMI at the inner interface
is dominated by three mechanisms: the baroclinic vorticity deposited
on the interface, the velocity perturbation caused by the nonuniform
impact of a rippled shock, and the geometric convergence effect. A lin-
ear model is developed by taking these effects and the startup process
into account, which reasonably predicts the present experimental and
numerical results. This model is demonstrated to be applicable to both
standard and nonstandard RMIs. The finding in this work is helpful to
understand and model the RMI in ICF.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China (Nos. 12122213, 12072341, and 91952205).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
APPENDIX: THEORETICAL DERIVATION
For the convergent RMI induced by a uniform cylindrical
shock (_
aimp ¼0), Eq. (22) reduces to
_
aðtÞ¼Rðt0Þ
RðtÞ
_
aCðt0ÞaðtÞvc
RðtÞ;(A1)
where the initial growth rate at t¼t
0
is
_
aðt0Þ¼_
aCðt0Þaðt0Þvc
Rðt0Þ:(A2)
Here, aðt0Þ¼a01vc
vs

refers to the perturbation amplitude
shortly after the shock impact with a
0
being the pre-shock perturba-
tion amplitude, vs being the shock velocity and v
c
being the jump
velocity of the interface caused by the shock impact.
Also, assuming the shocked interface moves at a constant
speed in a cylindrical geometry (
€
R¼0), the Bell model
46
can be
expressed as
€
aBellðtÞþ2vc
RðtÞ
_
aBellðtÞ¼0;(A3)
where
RðtÞ¼Rðt0Þþvct:(A4)
Integrating equation (A3), the perturbation growth rate can be
obtained as
_
aBellðtÞ¼R2ðt0Þ
R2ðtÞ
_
aBellðt0Þ:(A5)
Integrating equation (A5), we get
aBellðtÞ¼aBell ðt0Þþ_
aBellðt0ÞRðt0Þ
RðtÞt:(A6)
Let aBellðt0Þ¼aðt0Þand _
aBellðt0Þ¼_
aðt0Þ, Eqs. (A5) and (A6) can
be expressed as
_
aBellðtÞ¼R2ðt0Þ
R2ðtÞ
_
aCðt0Þaðt0Þvc
Rðt0Þ

;(A7)
aBellðtÞ¼aðt0Þþ _
aCðt0Þaðt0Þvc
Rðt0Þ

Rðt0Þ
RðtÞt:(A8)
Substituting (A4) to (A8), we get
aðt0Þ¼RðtÞ
Rðt0ÞaBellðtÞ_
aCðt0ÞRðtÞRðt0Þ
vc
:(A9)
Then, substituting (A9) to (A7), there is
_
aBellðtÞ¼Rðt0Þ
RðtÞ
_
aCðt0ÞaBellðtÞvc
RðtÞ:(A10)
Equation (A10) equals exactly to Eq. (A1), which demonstrates that
the present model reduces to the Bell linear theory for the conver-
gent RMI induced by a uniform cylindrical shock provided the
same initial values.
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Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 042123 (2022); doi: 10.1063/5.0089845 34, 042123-17
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