High-amplitude effect on Richtmyer–Meshkov instability at a single-mode heavy–light interface

Abstract

An experimental study is conducted to explore the high-amplitude effect on Richtmyer–Meshkov instability (RMI) at a single-mode heavy–light interface. A wide range of scaled initial amplitude (ka0, where k and a0 are perturbation wavenumber and initial amplitude, respectively) is considered. Qualitatively, nonstandard (standard) indirect phase inversion occurs in experiments with high (low and moderate) ka0. The nonstandard indirect phase inversion exhibits a complex process, and the interface mixing width does not reduce to near zero. Quantitatively, the linear model poorly (accurately) predicts the post-phase-inversion linear amplitude growth rate when ka0 is high (low and moderate). Additionally, a representative theoretical reduction factor fortuitously evaluates the high-amplitude effect on the post-phase-inversion linear amplitude growth rate well. The high-amplitude effect significantly alters the nonlinear evolution law, which differs from the case of RMI at a light–heavy interface. None of the considered nonlinear models can accurately predict the amplitude evolution under all ka0 conditions, regardless of whether their expressions are related to ka0 or not. Based on the current experimental results, an empirical nonlinear model is proposed to describe RMI at a single-mode heavy–light interface across a wide range of ka0 conditions.

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RESEARCH ARTICLE | DE CE MB ER 11 2023
High-amplitude effect on Richtmyer–Meshkov instability at a
single-mode heavy–light interface 
He Wang (王何) ; Hui Wang (王辉); Zhigang Zhai (翟志刚) ; Xisheng Luo (罗喜胜)
Physics of Fluids 35, 126107 (2023)
https://doi.org/10.1063/5.0180581
11 December 2023 12:25:23

High-amplitude effect on Richtmyer–Meshkov
instability at a single-mode heavy–light interface
Cite as: Phys. Fluids 35, 126107 (2023); doi: 10.1063/5.0180581
Submitted: 10 October 2023 .Accepted: 22 November 2023 .
Published Online: 11 December 2023
He Wang (王何),
1
Hui Wang (王辉),
1
Zhigang Zhai (翟志刚),
1,a)
and Xisheng Luo (罗喜胜)
1,2
AFFILIATIONS
1
Advanced Propulsion Laboratory, Department of Modern Mechanics, University of Science and Technology of China,
Hefei 230026, China
2
State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences,
Beijing 100190, China
a)
Author to whom correspondence should be addressed: sanjing@ustc.edu.cn
ABSTRACT
An experimental study is conducted to explore the high-amplitude effect on Richtmyer–Meshkov instability (RMI) at a single-mode
heavy–light interface. A wide range of scaled initial amplitude (ka
0
,wherekand a
0
are perturbation wavenumber and initial amplitude,
respectively) is considered. Qualitatively, nonstandard (standard) indirect phase inversion occurs in experiments with high (low and mod-
erate) ka
0
. The nonstandard indirect phase inversion exhibits a complex process, and the interface mixing width does not reduce to near
zero. Quantitatively, the linear model poorly (accurately) predicts the post-phase-inversion linear amplitude growth rate when ka
0
is high
(low and moderate). Additionally, a representative theoretical reduction factor fortuitously evaluates the high-amplitude effect on the
post-phase-inversion linear amplitude growth rate well. The high-amplitude effect significantly alters the nonlinear evolution law, which
differs from the case of RMI at a light–heavy interface. None of the considered nonlinear models can accurately predict the amplitude evo-
lution under all ka
0
conditions, regardless of whether their expressions are related to ka
0
or not. Based on the current experimental results,
an empirical nonlinear model is proposed to describe RMI at a single-mode heavy–light interface across a wide range of ka
0
conditions.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0180581
I. INTRODUCTION
Richtmyer–Meshkov instability (RMI)
1,2
occurs when a corru-
gated interface separating two fluids of different densities is impulsively
accelerated by a shock wave. Understanding RMI-induced perturba-
tion growth and the resulting material mixing is crucial because
RMI plays a vital role in various applications such as inertial
confinement fusion (ICF),
3–5
supernova explosion,
6,7
and scramjet.
8,9
Comprehensive insights and specific details regarding RMI and its
applications can be found in several reviews.
10–13
In ICF, the interface
separating ablator and deuterium–tritium (DT) ice and the interface
separating DT ice and DT gas are both heavy–light ones. In addition,
the initial perturbations on the interfaces in ICF capsule have various
scaled initial amplitudes (ka
0
,wherekand a
0
represent perturbation
wavenumber and initial amplitude, respectively). Therefore, it is neces-
sary and desirable to study the evolution of RMI at a heavy–light inter-
face (H-L RMI) under diverse ka
0
conditions. Referring to previous
works,
14–16
the initial interface (II) amplitude can reasonably be
regarded as low, moderate, and high when ka01/3, 1/3 <ka0<2/3,
and ka02/3, respectively.
For H-L RMI with low and moderate ka
0
,MeyerandBlewett
17
performed a numerical study and focused on the early-time evolution.
It was found that the impulsive model proposed by Richtmyer
1
pre-
dicts the post-phase-inversion linear amplitude growth rate (_
apo)
poorly. On the basis of the impulsive model and numerical results,
Meyer and Blewett
17
proposed a modified linear model for H-L RMI
(MB model). Subsequently, Yang et al.
18
derived a compressible linear
theory for the low-amplitude H-L RMI (YZS theory) and discussed the
parameter conditions corresponding to the two types of phase inver-
sion (i.e., direct and indirect phase inversions). The direct (indirect)
phase inversion completes its process at (after) the end of the shock-
interface interaction. Experimentally, Meshkov
2
first studied H-L RMI
with low and moderate ka
0
(0.31 and 0.63) using interfaces formed by
nitrocellulose membrane and obtained _
apo. However, no quantitative
theory was available to compare with the experimental results at that
time. Using well-defined interfaces created by membrane deposited on
a stereolithographed grid support, Mariani et al.
19
conducted shock-
tube experiments on H-L RMI with ka
0
of 0.24 and 0.36. It was
observed that the experimental results are in very good agreement with
Phys. Fluids 35, 126107 (2023); doi: 10.1063/5.0180581 35, 126107-1
Published under an exclusive license by AIP Publishing
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theories
1,20
and simulations. Recently, using the soap-film technique to
generate desirable single-mode heavy–light interfaces, Guo et al.
21
studied H-L RMI with low and moderate ka
0
(ranging from 0.16 to
0.63) and provided a direct experimental validation of the MB model.
To understand the high-amplitude effect on H-L RMI, Holmes
et al.
22
numerically studied RMI at beryllium-foam interfaces with ka
0
ranging from 0.25 to 3.14 under high-energy-density (HED) condi-
tions. It was observed that the YZS theory and MB model provide rea-
sonable (poor) predictions for the early-time amplitude evolution
when ka
0
is low and moderate (high). Moreover, the nonlinear model
of Zhang and Sohn,
23
referred to as the ZS model, initiated by the YZS
theory reasonably (poorly) predicts the temporal variation of the
amplitude growth for H-L RMI with ka0<2/3 (ka0>1.5).
Experimentally, Glendinning et al.
24
investigated RMI induced by a
strong laser-driven shock impacting a solid-state heavy–light interface
and considered two different initial perturbations with ka
0
of 0.29 and
0.92, respectively. Similarly, the YZS theory only well predicts the
early-time evolution of the interface with low ka
0
. In addition, for H-L
RMI with low ka
0
, both the ZS model and the nonlinear model pro-
posed by Sadot et al.
25
(SEA model) can reasonably forecast the non-
linear amplitude evolution. For H-L RMI with ka
0
¼0.92, the ZS
model (SEA model) still predicts (fails to predict) the nonlinear ampli-
tude growth well.
In the HED experiment, the perturbation evolution is influenced
not only by hydrodynamic instabilities but also by the phase transition
of the solid materials forming the initial interface,
26
making it difficult
to isolate the contribution of RMI to the perturbation growth. In con-
trast, the shock-tube experiment provides a relatively “simpler”physi-
cal environment for the shock-induced interface evolution and,
therefore, facilitates the investigation of RMI. Jourdan and Houas
27
experimentally studied H-L RMI with ka
0
¼1.5 in a horizontal large-
cross section shock tube using interfaces formed by microfilm mem-
brane. The study found that the impulsive model effectively describes
the bubble inversion of H-L RMI with low post-shock Atwood number
[A¼(q
d
q
u
)/(q
d
þq
u
), with q
d
and q
u
being the densities of fluids
at the downstream and upstream sides of the shocked interface (SI),
respectively]. In addition, it was found that Vandenboomgaerde’sana-
lytic model
20
balanced by a reduction factor (C
r
) proposed by Rikanati
et al.
28
accurately predicts the nonlinear amplitude evolution of bubble
in H-L RMI with high A. However, Jourdan and Houas
27
considered a
quasi-single-mode interface that contains non-periodic portions,
which, as indicated by Luo et al.,
29
cannot be treated as a single-mode
one. Guo et al.
21
investigated high-amplitude H-L RMI (ka
0
¼1.36)
with reshock, and found that the MB model fails to provide a reason-
able prediction for the experimental _
apo. However, only one experi-
ment with high ka
0
was conducted, and the emphasis was on the post-
reshock flow.
In the previous shock-tube experiments, due to the difficulties
encountered in generating a single-mode interface with high ka
0
,the
related studies are scarce. In our previous work,
30
the soap-film tech-
nique was used to generate single-mode air–SF
6
interfaces with diverse
ka
0
, and high-amplitude effect on the RMI at a light–heavy interface
(L-H RMI) was highlighted. For H-L RMI, however, some issues
remain unclear. How does the high amplitude affect the phase inver-
sion, linear amplitude growth, and nonlinear amplitude evolution?
Can existing theories accurately predict the amplitude evolution of H-
L RMI under various ka
0
conditions? Does the high-amplitude effect
in H-L RMI differ from that in L-H RMI? These issues motivate the
present study. In this work, the soap-film technique is also used to cre-
ate well-defined single-mode interfaces with diverse ka
0
. The develop-
ment of H-L RMI under a wide range of ka
0
conditions is obtained
through shock-tube experiments. Subsequently, the interface morphol-
ogy under various ka
0
conditions is discussed. Finally, the pre- and
post-phase-inversion amplitude evolutions are analyzed, and a repre-
sentative theoretical C
r
and several typical nonlinear models are
examined.
II. EXPERIMENTAL METHODS
To highlight the high-amplitude effect, five experimental cases
with the same wavelength (40 mm) but different a
0
(2,4,8,12,and
18 mm) are conducted, with ka
0
of 0.31, 0.63, 1.26, 1.89, and 2.83,
respectively. Note that the experimental case will be referred to as case
“ka
0
,”and cases 0.31 and 0.63 serve as reference cases to demonstrate
the high-amplitude effect.
In this study, a horizontal shock tube and the soap-film tech-
nique, which have been widely verified in our previous works,
30–33
are
adopted to generate the shock wave and the initial interface, respec-
tively. The shock tube consists of a driver section, a driven section, a
transitional section, a stable section, and a test section, as depicted in
Fig. 1. The interface formation devices (devices A and B) shown in
Fig. 2 are manufactured by assembling two transparent acrylic plates
(5 mm in thickness) with pedestals (6 mm in height). Two identical
single-mode-shaped constraint strips are affixed to the pre-carved
grooves on device B to constrain the soap film. The height ratio
between the constraint strips protruding into the flow field and the
entire flow field is 10%, which ensures that the effect of the constraint
FIG. 1. Sketch of the shock tube and high-speed schlieren system.
FIG. 2. Schematic of single-mode SF
6
–air interface formation.
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strips on the post-shock flow is negligible.
33
To create a single-mode-
shaped soap film, a rectangular frame with the soap solution (78%
pure water, 2% sodium oleate, and 20% glycerin by mass) attached is
cautiously drawn along the pre-moistened constraint strips.
Subsequently, devices A and B are carefully connected and inserted
into the test section of the shock tube. For creating an SF
6
–air interface,
air on the upstream side of the soap film needs to be replaced with SF
6
using the following procedure. First, a thin membrane is used to sepa-
rate the transitional and stable sections. Then, SF
6
is charged slowly
into the stable and test sections through the inlet pipe, and air is
released through the outlet pipe. An oxygen concentration detector is
placed at the outlet pipe to monitor the oxygen concentration in the
test section. Once the volume fraction of oxygen drops below 0.5%,
the pipes are removed, and then the holes are sealed. Subsequently, the
experiment can be conducted.
The ambient pressure and temperature are 101.3 60.1 kPa and
301.3 61.0 K, respectively. The flow field evolution is captured by a
high-speed schlieren system, as depicted in Fig. 1.Theframerateof
the high-speed video camera (FASTCAM SA5, Photron Limited) is set
to 50000 fps, with an exposure time of 1 ls. The spatial resolution of
the schlieren images is 0.39 mm pixel
1
.
Some significant experimental parameters are listed in Table I.
We should point out that it is challenging to ensure that the gas on
one side of the interface is pure air or pure SF
6
in experiments. As dis-
cussed in previous studies,
34,35
gas is capable of penetrating through
the soap–film interface. After the interface is formed and before the
experiment is performed, the time interval allows the gas to diffuse
through the soap–film interface. As a result, the gases on both sides of
theinterfaceareamixtureofSF
6
and air, and the volume fractions of
SF
6
on upstream side (VF
1
) and downstream side (VF
2
) of the initial
interface should be first determined. In this work, the volume fractions
of SF
6
are determined by solving the shock–interface interaction using
one-dimensional gas dynamics theory. Through a MATLAB proce-
dure, the experimental incident shock (IS) velocity (ue
i)ischosenasan
input parameter, and VF
1
and VF
2
are altered to match the experimen-
tal transmitted shock (TS) velocity (ue
t) and its theoretical counterpart
while ensuring that the average shock-induced interface velocity (Dve)
and its theoretical counterpart are in reasonable agreement. Once these
two goals are achieved, VF
1
and VF
2
adopted in the procedure are
regarded as the corresponding experimental values. In our experi-
ments, ue
iand ue
tare obtained by linearly fitting the trajectories of the
incident and transmitted shock waves, respectively. Note that the
transmitted shock is disturbed, and its trajectory is obtained by averag-
ing the horizontal coordinates of the most upstream and most
downstream points on it. In addition, since the interface mean position
is regarded as the average of the horizontal coordinates of the spike
(heavier fluids penetrating lighter ones) and bubble (lighter fluids pen-
etrating heavier ones), Dveis calculated as the average of the velocities
of the spike and bubble.
III. RESULTS AND DISCUSSION
A. Flow features and interface morphology
Schlieren images of five experimental cases are provided in Fig. 3.
The temporal origin (t¼0ls) is defined as the moment when the inci-
dent shock (IS) reaches the balanced position of the single-mode
SF
6
–air initial interface (II). It is important to note that II appears thick
due to the presence of the constraint strips. For clarity, the constraint
strips are removed from the images through image processing once
the shocked interface (SI) moves away from them. In addition, it can
be observed that the right side of the interface does not display a uni-
form gray background, especially for case 1.89. Although SF
6
would
leak from the upstream to the downstream sides of the initial interface,
the leakage is limited in our experiments (VF
2
is close to 0.01). The
density gradient caused by the leakage of SF
6
should be limited and
barely visible using schlieren photography. The non-uniform gray
background is possibly caused by the imperfections of the interface
formation devices. In experiments, the interface formation devices are
manufactured by assembling two transparent acrylic plates with pedes-
tals. Due to the limitation of the machining accuracy, the thickness of
the acrylic plate is generally non-uniform, which leads to the non-
uniform background captured by schlieren photograph. In addition,
some other imperfections on the acrylic plate, such as scratches and
attached soap solution, may also lead to the non-uniform background.
Some “shadows”can be seen between the transmitted waves and the
interface (particularly at the top and bottom of the image correspond-
ing to t¼199 ls in case 2.83). Although the non-uniform background
does not affect the flow evolution, it hinders the readers from clearly
obtaining the effective experimental information. Therefore, during
the image post-processing, most of the background non-uniformities
were eliminated on the premise of maintaining effective experimental
information. In Fig. 3(e), due to the complex shock structures, it is dif-
ficult to eliminate all the background non-uniformities while still
maintaining the shock structures. Therefore, some background non-
uniformities near the shock waves and interface were preserved.
For H-L RMI with low to moderate ka
0
, case 0.63 is taken as an
example to illustrate the detailed evolution process. As shown in
Fig. 3(b), IS would directly intersect with the transmitted shock (TS)
during the IS-II interaction (1ls), indicating the occurrence of
TABLE I. Some significant parameters for five experimental cases. VF
1
(VF
2
): volume fraction of SF
6
on the upstream (downstream) side of the initial interface; A: post-shock
Atwood number; ue
iand ue
t: velocities of the incident and transmitted shocks obtained from experiments, respectively; Dveand Dvt: shock-induced interface velocities measured
from experiments and predicted by one-dimensional gas dynamics theory, respectively; ve
tip and v
bt
: experimental spike tip velocity and its fraction induced by baroclinic vorticity,
respectively.
Case VF
1
VF
2
Au
e
i(m/s) ue
t(m/s) Dve(m/s) Dvt(m/s) ve
tip (m/s) v
bt
(m/s) v
bt
/Dve(%)
0.31 0.81 0.01 0.63 192.5 399.7 93.8 93.2 102.8 9.0 9.6
0.63 0.90 0.01 0.66 182.6 396.9 89.5 90.9 112.4 22.9 25.6
1.26 0.83 0.01 0.63 188.3 395.4 89.3 89.3 126.0 36.7 41.2
1.89 0.91 0.02 0.64 182.9 386.9 91.1 90.9 138.6 47.5 52.1
2.83 0.84 0.02 0.62 187.9 384.2 87.4 87.5 148.6 61.2 70.1
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Phys. Fluids 35, 126107 (2023); doi: 10.1063/5.0180581 35, 126107-3
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regular refraction.
36
The baroclinic vorticity generated due to the mis-
alignment of the pressure and density gradients induces downstream-
directed and upstream-directed velocities at the original crest and
trough of SI (OC and OT), respectively. Consequently, the perturba-
tion amplitude (a),definedashalfofthedistancebetweenthetwo
points OC and OT along the streamwise direction, decreases continu-
ously after the IS-II interaction (59–139 ls). In H-L RMI with low to
moderate ka
0
,themixingwidth(w), defined as the distance between
the most upstream and most downstream points on SI along the
streamwise direction, is close to aand also reduces gradually after the
IS-II interaction. A standard indirect phase inversion occurs at approx-
imately t¼139 ls, with both aand wreducing to nearly zero.
Subsequently, OC and OT transform into the post-phase-inversion
trough and crest (PT and PC), respectively (199 ls). After the phase
inversion, the perturbation amplitude is defined as half of the distance
between PC and PT along the streamwise direction. To avoid confu-
sion, both OC and PT (OT and PC) are referred to below as spike
(bubble) tip. After the phase inversion, the asymmetry of SI increases
gradually, followed by the formation of distinct spike and bubble struc-
tures (379 ls), indicating the generation of high-order harmonics.
Subsequently, roll-up structures are formed on SI (519 ls).
The overall flow features in case 0.31 are similar to those in case
0.63. However, there are still some differences. In case 0.31, since the
perturbation of TS imprinted by II is limited, TS maintains a smooth
morphology throughout its evolution process. In contrast, in case 0.63,
the perturbation of TS imprinted by II is relatively large, and TS rap-
idly evolves into a series of Mach reflection configurations (59 ls). The
reflected shocks in the Mach reflection configurations, also known as
transverse shock waves, will continuously affect the interface evolution
and introduce the secondary compression effect. Specifically, the sec-
ondary compression effect includes the direct interaction of the trans-
verse shocks with SI and the effect of the high-pressure zones created
by the intersection of transverse shocks. Furthermore, due to the small
initial perturbation amplitude, SI in case 0.31 evolves slowly, and no
roll-up structures are formed on it within the effective experimental
time.
Case 2.83 is taken as an example to illustrate the detailed evolu-
tion process of high-amplitude H-L RMI. As shown in Fig. 3(e),IS
does not directly intersect with TS during the IS-II interaction (19 ls),
indicating the occurrence of irregular refraction.
36
Compared to H-L
RMI with low to moderate ka
0
, the average angle between IS and II is
larger in high-amplitude H-L RMI, resulting in more baroclinic
FIG. 3. Schlieren photographs of the evolutions of shocked single-mode SF
6
–air interfaces with different ka
0
. IS: incident shock; II: initial interface; TS: transmitted shock; SI:
shocked interface; OT and OC (PT and PC): original (post-phase-inversion) trough and crest, respectively. Yellow and green dotted lines are the extension lines of observed IS
and TS, respectively. Numbers represent time in ls.
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vorticity deposited on SI and, accordingly, higher baroclinic-vorticity-
induced velocity (v
bv
). Due to the high downstream-directed v
bv
near
the spike tip and the prolonged duration of the IS-II interaction, SI
near the spike tip inverts before the IS-II interaction ends (19 ls). As a
result, win high-amplitude H-L RMI would not reduce to nearly zero.
Due to the high upstream-directed v
bv
near the bubble tip, SI near the
bubble tip inverts shortly after the IS-II interaction (299 ls).
Meanwhile, the roll-up structures are formed on the spike tip. We may
refer to this type of phase inversion as nonstandard indirect phase
inversion to differentiate it from the standard one. After the spike and
bubble tips become the most downstream and most upstream points
on SI, respectively, the evolution of SI follows the same tendency as
that in H-L RMI with low to moderate ka
0
, but the nonlinear evolution
features of SI develop faster. In the late stages, the stem of the spike
becomes very slender (499–1019 ls), which differs from the cases of
H-L RMI with low to moderate ka
0
and high-amplitude L-H RMI.
30
The overall flow features in cases 1.26 and 1.89 are similar to
thoseincase2.83.Inthecasewithhigherka
0
, the thinning of the spike
stem and the development of the roll-up structures are more rapid,
which should be attributed to more baroclinic vorticity deposition on
SI. At late stages, the decomposition of the roll-up structures caused by
the secondary instability
37
is apparent in cases 1.89 and 2.83. This
would result in the generation of local mixing zones on both sides of
the spike stem.
37,38
In contrast, in case 1.26, the decomposition of the
roll-up structures and the resulting mixing appear to be still limited at
late stages. Note that in Secs. III B–IIID, the moment of the phase
inversion is defined as the moment when a¼0 to avoid confusion and
facilitate analysis.
B. Spike tip velocity
The movements of the spike tip for different cases are provided
in Fig. 4,inwhichxis the coordinate along the streamwise direction,
and x
i
and t
i
denote the xcoordinate and moment of the first extracted
data, respectively. Note that the error bars describe the uncertainty in
extracting the position of the spike tip from experiments because the
interface has a thickness. The spike tip moves almost linearly in all
cases, with greater velocity in cases with higher ka
0
. The flow field of
RMI can be divided into two parts: a uniform background flow field
and a non-uniform perturbed flow field. The velocity of the perturbed
flow field is induced by baroclinic vorticity deposited by shock waves,
and the velocity of the background flow is Dve. Accordingly, the veloc-
ity of the spike tip measured in the laboratory coordinate system (ve
tip)
is a superposition of Dveand the velocity of the perturbed flow field at
the spike tip (v
bt
): ve
tip ¼Dveþvbt .Sincev
bt
is physically induced by
baroclinic vorticity, it represents exactly the fraction of ve
tip induced by
baroclinic production. ve
tip,v
bt
,andv
bt
/Dvefor different cases are listed
in Table I.v
bt
exhibits a strong positive correlation with ka
0
,and
v
bt
/Dveexceeds 40% when ka01.26.
During the penetration of the spike, the spike tip needs to push
away the lighter fluid in front of it, thus suffering a reacting force that
causes its velocity to decrease. Accordingly, if v
bt
, which also denotes
the velocity of the spike tip with respect to the background flow, is
higher, the spike would be capable of penetrating deeper into the ligh-
ter fluid. In other words, the penetrating capability of the spike is
directly influenced by v
bt
, which is strongly positively related to ka
0
.As
a result, it is essential to minimize high-amplitude perturbations on
the interfaces of ICF target to prevent the ingress of ablative material
into the hot spot.
C. Pre- and post-phase-inversion linear amplitude
evolutions
The temporal variations of perturbation amplitude afor different
cases are shown in Fig. 5 in dimensional form, in which tis the
moment corresponding to the smallest post-phase-inversion ampli-
tude extracted from experiments (a). The overall amplitude evolution
can be divided into three stages, including the pre-phase-inversion lin-
ear stage (stage I), post-phase-inversion linear stage (stage II), and
post-phase-inversion nonlinear stage (stage III). The experimental pre-
FIG. 4. Movements of the spike tip for different cases.
FIG. 5. Temporal variations of perturbation amplitude. Stage I: pre-phase-inversion
linear stage; stage II: post-phase-inversion linear stage; and stage III: post-phase-
inversion nonlinear stage.
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and post-phase-inversion linear amplitude growth rates (_
ae
pr and _
ae
po)
are obtained by linearly fitting the temporal variations of amplitude
within stages I and II, respectively. Relevant results are provided in
Table II. For cases with low to moderate ka
0
,_
ae
pr is lower than _
ae
po.
According to the linear compressible theory of RMI,
1,18
the early-time
evolution of the interface is driven by the pressure perturbations
caused by the distorted transmitted and reflected waves in the vicinity
of the interface. The continuous effect of the pressure perturbations
causes the amplitude growth rate to increase in an oscillatory manner
toward a maximum value. This process is generally referred to as the
startup period.
39
Since the phase-inversion process is short in cases
with low to moderate ka
0
,thesmaller _
ae
pr than _
ae
po is likely because the
amplitude evolution is still within the startup period when phase inver-
sion occurs. For cases with high ka
0
, due to the relatively long duration
of the phase-inversion process, the effect of the startup process on the
post-phase-inversion amplitude evolution should be weak. _
ae
po is
smaller than _
ae
pr , which should be attributed to the nonlinear effect, as
evidenced by the significant nonlinear features of the post-phase-inver-
sion interface shown in Fig. 3. In other words, the interface evolution
in stage II of high-amplitude H-L RMI has a linear imprint on the
amplitude growth even if it corresponds to a nonlinear physical
process.
The MB model,
17
which has been validated numerically
22
and
experimentally
21
in predicting _
apo of low-amplitude H-L RMI, is
employed to provide a theoretical reference for _
ae
po under diverse ka
0
conditions. The MB model can be described as
_
amb ¼kDveAð1þCcÞa0
2;(1)
in which Cc¼1Dve=ue
iis the shock-compression factor. For cases
with low to moderate ka
0
,_
ae
po agrees well with _
amb, with a difference of
less than 7%. However, the discrepancy between _
ae
po and _
amb is greater
in cases with higher ka
0
due to the more significant high-amplitude
and nonlinear effects.
To evaluate the high-amplitude effect on the linear amplitude
growth rate, several different theoretical C
r
have been proposed.
28,40–42
Since the differences between their values are very limited,
30
only the
theoretical C
r
proposed by Dimonte and Ramaprabhu
40
(C
dr
)isadopted
in this work as a representative one. Comparison between the experi-
mental C
r
of H-L RMI (Chl¼_
ae
po/_
amb )andC
dr
is given in Fig. 6.Chl
exhibits reasonable agreement with C
dr
, which, however, could be
fortuitous because the additional nonlinearity introduced by the phase-
inversion process and the secondary compression effect caused by trans-
verse waves
30,43
were not considered when constructing C
dr
.
For exploring the similarities and differences in the high-
amplitude effect between H-L and L-H RMI, Chlis compared with
the empirical C
r
proposed based on the experimental results of L-H
RMI (Clh),
30
as shown in Fig. 6. It is worth mentioning that the most
notable distinction between H-L and L-H RMI lies in that the former
has a phase-inversion process. Chlfor case 0.31 agrees reasonably
with Clh. However, Chlfor the other cases deviate significantly from
Clh, suggesting that the high-amplitude effect in H-L RMI differs
from that in L-H RMI. The separation of TS from SI occurs faster in
H-L RMI compared to that in L-H RMI and, accordingly, the second-
ary compression effect in H-L RMI should be weaker than that in L-H
RMI. Hence, the existence of phase-inversion process in H-L RMI and
the difference in the secondary compression effect are likely the main
reasons for the difference in the high-amplitude effects between H-L
and L-H RMI.
D. Post-phase-inversion nonlinear amplitude growth
As nonlinearity becomes further significant, amplitude growth
enters stage III. The temporal variations of post-phase-inversion ain
dimensionless form for cases with different ka
0
are shown in Fig. 7,in
which tand aare normalized as s¼k_
ae
poðttÞand a¼kðaaÞ,
respectively. The scaling approach fails to collapse the results of differ-
ent cases, indicating that the high-amplitude effect also influences the
nonlinear evolution law. Notably, the same scaling approach collapses
the results of experiments on L-H RMI with various ka
0
,
30
which fur-
ther demonstrates that the high-amplitude effects in H-L and L-H
RMI are different. The additional nonlinearity introduced by the
phase-inversion process to the post-phase-inversion amplitude evolu-
tion, as observed in Subsection III C, is significantly different for H-L
RMI with different ka
0
, which explains the differences in evolution law
between cases. Moreover, this additional nonlinearity is likely the pri-
mary factor causing the distinct dependencies of the nonlinear evolu-
tion law on ka
0
for H-L and L-H RMI.
TABLE II. Experimental and theoretical results of the linear amplitude growth rate.
_
ae
pr and _
ae
po: experimental pre- and post-phase-inversion linear amplitude growth
rates, respectively; _
amb: linear amplitude growth rate predicted by the MB model.
Case _
ae
pr (m/s) _
ae
po (m/s) _
amb (m/s)
0.31 9.23 60.49 13.11 60.97 14.03
0.63 25.02 61.01 28.00 60.13 27.89
1.26 47.68 61.14 45.94 61.82 54.27
1.89 65.46 60.93 56.75 60.72 82.97
2.83 73.32 61.72 63.05 61.48 116.83
FIG. 6. Comparison between experimental C
r
of H-L RMI (Chl), empirical C
r
pro-
posed based on experimental results of L-H RMI
30
(Clh) and theoretical C
r
pro-
posed by Dimonte and Ramaprabhu
40
(C
dr
).
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Phys. Fluids 35, 126107 (2023); doi: 10.1063/5.0180581 35, 126107-6
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Subsequently, a comparative analysis of the experimental results
and predictions by typical nonlinear models is performed. Nonlinear
models including the ZS,
23
MIK,
44
SEA,
25
DR,
40
ZG,
45
and ZG-New
46
models are considered, and their detailed expressions are listed in
Table III. The SEA and ZG models do not consider the effect of the ini-
tial amplitude, and their predictions are compared with the experi-
mental results in Fig. 7(a). Notably, the predictions of the SEA and ZG
models for various cases are slightly different due to the discrepancies
FIG. 7. Comparisons between post-phase-inversion nonlinear amplitude evolutions in dimensionless form obtained from experiments and predicted by nonlinear models. (a)
SEA and ZG models; (b) MIK model; (c) ZS model; (d) DR model; (e) ZG-New model; and (f) mDR model. Symbols and lines represent the experimental results and theoretical
predictions, respectively.
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in parameters other than ka
0
, and only the middle one among the five
theoretical lines is provided for clarity. The SEA model overestimates
the results of all cases except case 2.83, which should be attributed to
its overestimation of the spike acceleration.
40
The relatively good pre-
dictive capability of the SEA model for case 2.83 should be due to the
overestimation of the spike acceleration coincidentally describing the
high-amplitude effect under this specific ka
0
. The ZG model reason-
ably predicts the experimental results of H-L RMI with low to moder-
ate ka
0
but underestimates the results of high-amplitude H-L RMI
from the early times.
The MIK model has different expressions under different ka con-
ditions, thus implicitly describing the effect of the initial amplitude.
For cases with low to moderate initial amplitude, Ccka0is smaller than
1/3 and the prediction of the MIK model is related to the initial ampli-
tude. The predictions of the MIK model for high-amplitude cases are
unrelated to ka
0
, but slightly different due to the discrepancies in
parameters other than ka
0
. Only the middle one among the three theo-
retical lines of high-amplitude cases is provided in Fig. 7(b) for clarity.
The ZS, DR, and ZG-New models are explicitly related to ka
0
,andthe
comparisons between their predictions and the experimental results
are shown in Figs. 7(c)–7(e), respectively. Interestingly, the predictions
of the ZS, DR, and ZG-New models (MIK model) exhibit a negative
correlation with ka
0
(initial amplitude when Ccka0<1/3), whereas the
experimental results show a positive dependence on ka
0
.When
constructing the MIK, ZS, DR, and ZG-New models, only the linear
and nonlinear amplitude growth periods were considered, while the
phase-inversion process was not taken into account. The additional
nonlinearity introduced by the phase inversion may be the reason why
these models fail to describe the dependence of the nonlinear evolution
law of H-L RMI on ka
0
. Among these models, the ZS model exhibits
the poorest predictive performance, and this can be attributed to two
reasons. First, in all cases, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1=2þðCcka0Þ2
qis greater than jAjand,
therefore, the asymptotic _
apredicted by the ZS model satisfies the
1=t2law. However, according to the potential flow model prediction
and previous numerical and experimental works,
40,47
the late-time
behavior of _
ais expected to follow a 1=tlaw. Second, as pointed out by
Dimonte and Ramaprabhu,
40
the ZS model exhibits an excessive
dependency on ka
0
.
The phase-inversion process is complex when ka
0
is high, and its
effect on the post-phase-inversion perturbation evolution is hard to
describe. Consequently, it is difficult to provide a rigorous theoretical
description for the dependence of the nonlinear evolution law of H-L
RMI on ka
0
. In this study, we attempt to propose an empirical model
applicable to H-L RMI across a wide range of ka
0
conditions based on
the current experimental results. Among models explicitly describing
the dependence on ka
0
, the ZS model fails to capture the late-time 1=t
behavior of _
awhen ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1=2þðCcka0Þ2
q>jAj, while the ZG-New
TABLE III. Detailed expressions of considered nonlinear models, in which subscripts “b”and “s”represent bubble and spike, respectively.
Model Expression
ZS _
aZS
b=s¼_
aZS7jAjk_
ae
po2t
1þ2k2Cca0
_
ae
potþ4k2_
ae
po2t2hðCcka0Þ2þ1
3ð1A2Þi,
in which _
aZS ¼
_
ae
po
1þk2Cca0
_
ae
potþmaxh0;ðCcka0Þ2A2þ1
2ik2_
ae
po
2t2
.
MIK _
aMIK
b=s¼_
ae
po when ka<1=3, _
aMIK
b=s¼
_
ae
po
1þ3_
ae
po
16jAj
36jAj

kt
when ka 1=3.
SEA _
aSEA
b=s¼_
ae
po
1þk_
ae
pot
1þð16jAjÞk_
ae
potþ16jAj
1þjAj

k2_
ae
po2t2
2pg
!
, in which g¼1
3pwhen jAj0:5 and g¼1
2pwhen jAj!0.
DR _
aDR
b=s¼_
ae
po
1þð17jAjÞk_
ae
pot
1þdb=sk_
ae
potþð17jAjÞFb=sðk_
ae
potÞ2, in which db=s¼4:56jAjþð27jAjÞCcka0
4and Fb=s¼16jAj.
ZG _
aZG
b=s¼
_
ae
po
1þhb=sk_
ae
pot, in which hb=s¼3
4ð16jAjÞð36jAjÞ
36jAjþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð16jAjÞ
p
4ð36jAjÞ þ ffiffiffi
2
pð96jAjÞð16jAjÞ1=2
ð36jAjÞ2þ2ffiffiffi
2
pð37jAjÞð16jAjÞ1=2.
ZG-New _
aZGN
b=s¼_
ae
poek½ab=sðtÞCca0hb=s
1
3ðhb=s7jAjÞ21þ2Ccka0
hb=s7jAj
!
þkb=s
1
3ðhb=s7jAjÞ21þ2Ccka0
hb=s7jAj
!
þkb=se3k½ab=sðtÞCca0
8
>
>
>
>
>
<
>
>
>
>
>
:
9
>
>
>
>
>
=
>
>
>
>
>
;
1
3kb=s
, in which
kb=s¼h1
hb=s7jAj1
3ðhb=s7jAjÞ2iþh1
ðhb=s7jAjÞ22
3ðhb=s7jAjÞ3iCcka0.
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Phys. Fluids 35, 126107 (2023); doi: 10.1063/5.0180581 35, 126107-8
Published under an exclusive license by AIP Publishing
11 December 2023 12:25:23

model has a rather complex formula. In contrast, the DR model not
only captures the late-time 1=tbehavior of _
abut also has a relatively
simple form. Therefore, we shall construct a new empirical model
(mDR model) by modifying the DR model. The differences between
the DR and mDR models lie in coefficients db=sand Fb=s.InthemDR
model, db=sand Fb=sare modified as ½ð56jAjÞ=43=5lnðCcka0Þ
and 0.8(1 6jAj), respectively. As shown in Fig. 7(f),themDRmodel
predicts well the post-phase-inversion amplitude evolution of H-L
RMI with ka
0
ranging from 0.31 to 2.83.
IV. CONCLUSIONS
The high-amplitude effect on Richtmyer–Meshkov instability
(RMI) at a single-mode heavy–light interface (referred to as H-L RMI)
is finely studied through shock-tube experiments, considering a wide
range of scaled initial amplitude (ka
0
). Qualitatively, nonstandard
(standard) indirect phase inversion, whose process is rather complex
(simple), occurs at the interface with high (low to moderate) ka
0
.In
the late stages, the spike stem in high-amplitude H-L RMI becomes
very slender, which differs from the cases of H-L RMI with low to
moderate ka
0
and RMI at a single-mode light–heavy interface (L-H
RMI) with high ka
0
.
Quantitatively, the velocity at the spike tip is found to exhibit a
strong positive correlation with ka
0
. This proves the importance of
minimizing initial perturbations on the interfaces of ICF target to pre-
venttheformationofintensespikesthatcouldaffecttheignition.In
H-L RMI with low to moderate ka
0
, the pre-phase-inversion linear
amplitude growth rate in experiment is smaller than the post-phase-
inversion one due to the startup process. In H-L RMI with high ka
0
,
however, the post-phase-inversion value is smaller due to the nonline-
arity and high-amplitude effect. The existing theoretical reduction fac-
tor evaluates the high-amplitude effect, which, however, should be
fortuitous because nonlinearity introduced by the phase inversion pro-
cess and the secondary compression effect were not considered when
the reduction factor was proposed.
For the post-phase-inversion nonlinear amplitude growth, a
widely used scaling approach fails to collapse the experimental results
with different ka
0
, indicating that the high amplitude also influences
the nonlinear evolution law. None of the considered nonlinear models
are found to apply to H-L RMI under all ka
0
conditions, regardless of
whether their expressions are related to ka
0
or not. Based on the cur-
rent experimental results, an empirical nonlinear model applicable to
H-L RMI over a wide range of ka
0
conditions is proposed. The present
study demonstrates that the high-amplitude effect on the linear and
nonlinear evolution laws of perturbation amplitude in H-L RMI differs
from that in L-H RMI. This difference is probably attributed to the
presence of phase-inversion process in H-L RMI and to the difference
in the secondary compression effect between H-L and L-H RMI.
Practically, the perturbations on the interfaces in ICF capsule are
generally multi-mode ones instead of single-mode ones. Therefore,
investigating RMI on a high-amplitude multi-mode light–heavy/
heavy–light interface is also necessary and interesting. In the following
studies, relevant shock-tube experiments will be conducted to explore the
coupling of the high-amplitude and mode-coupling/competition effects.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China (Nos. 12372281, 12102425, and 91952205),
Youth Innovation Promotion Association CAS, and the
Fundamental Research Funds for the Central Universities.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
He Wang: Writing –original draft (lead). Hui Wang: Data curation
(lead). Zhigang Zhai: Supervision (equal); Writing –review & editing
(lead). Xisheng Luo: Funding acquisition (equal); Supervision (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from
the corresponding author upon reasonable request.
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Physics of Fluids ARTICLE pubs.aip.org/aip/pof
Phys. Fluids 35, 126107 (2023); doi: 10.1063/5.0180581 35, 126107-10
Published under an exclusive license by AIP Publishing
11 December 2023 12:25:23