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J. Fluid Mech. (2025), vol. 1006, R1, doi:10.1017/jfm.2025.36
Shock-tube experiments on strong-shock-driven
single-mode Richtmyer–Meshkov instability
Ting Si1, Shuaishuai Jiang1
, Wei Cai1
, He Wang1and Xisheng Luo1,2
1Advanced Propulsion Laboratory, Department of Modern Mechanics, University of Science and
Technology of China, Hefei 230026, PR China
2State Key Laboratory of High-Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy
of Sciences, Beijing 100190, PR China
Corresponding author: He Wang, ustchewang@ustc.edu.cn
(Received 20 October 2024; revised 14 December 2024; accepted 3 January 2025)
We report the first shock-tube experiments on Richtmyer–Meshkov instability at a single-
mode light–heavy interface accelerated by a strong shock wave with Mach number
higher than 3.0. Under the proximity effect of the transmitted shock and its induced
secondary compression effect, the interface profile is markedly different from that in
weakly compressible flows. For the first time, the validity of the compressible linear theory
and the failure of the impulsive model in predicting the linear amplitude evolution in
highly compressible flows are verified through experiments. Existing nonlinear and modal
models fail to accurately describe the perturbation evolution, as they do not account for
the shock proximity and secondary compression effects on interface evolution. The shock
proximity effect manifests mainly in the early stages when the transmitted shock remains
close to the interface, while the effect of secondary compression manifests primarily at
the period when interactions of transverse shocks occur at the bubble tips. Based on these
findings, we propose an empirical model capable of predicting the bubble evolution in
highly compressible flows.
Key words: shock waves
1. Introduction
Richtmyer–Meshkov instability (RMI) occurs when a perturbed interface separating two
fluids of different densities is accelerated by a shock wave (Richtmyer 1960; Meshkov
1969). It plays a critical role in various engineering applications and natural phenomena
(Zhou 2017a,b,2024; Zhou, Sadler & Hurricane 2025), such as inertial confinement fusion
© The Author(s), 2025. Published by Cambridge University Press. 1006 R1-1
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T. Si, S. Jiang, W. Cai, H. Wang and X. Luo
(ICF) (Zylstra et al. 2022), supersonic combustion ramjet (Urzay 2018) and supernova
explosion (Abarzhi et al. 2019).
As a foundation for research related to shock-driven interface instability, the RMI
on a single-mode light–heavy interface with a small dimensionless initial amplitude
(ka0, where kand a0are perturbation wavenumber and initial amplitude, respectively)
has garnered significant attention. Theoretically, Richtmyer (1960) first proposed an
incompressible model (impulsive model), which empirically describes the influence of
flow compressibility by using post-shock Atwood number and amplitude instead of their
pre-shock counterparts, to predict the linear RMI evolution. Additionally, Richtmyer
(1960) constructed a compressible theory (Richtmyer theory) to describe the linear
evolution of RMI in flows with arbitrary compressibility. For the nonlinear evolution
period, several empirical models (Zhang & Sohn 1997;Sadotet al. 1998; Mikaelian 2003;
Dimonte & Ramaprabhu 2010; Zhang & Guo 2016) were constructed by matching the
compressible linear and incompressible nonlinear solutions.
Experimentally, previous studies primarily utilized shock-tube facilities to investigate
the RMI induced by a weak shock (weak-shock RMI) with the Mach number (M) typically
less than 1.5. Meshkov (1969) first conducted shock-tube experiments on weak-shock RMI
with M≈1.5 using interfaces formed by nitrocellulose membrane. The experimental
linear amplitude growth was found to be slower than the impulsive model prediction,
which may be attributed to the effects of nonlinearity and wire support. Afterwards,
Jacobs & Krivets (2005) studied the late-time development of RMI on a membraneless
light–heavy interface accelerated by a shock wave with Mof 1.3. It was found that the
nonlinear model proposed by Zhang & Sohn (1997) (ZS model) fails to accurately predict
the experimental results, while that constructed by Sadot et al. (1998) (SEA model) offers
better predictions. Subsequently, Liu et al. (2018) performed shock-tube experiments on
the evolution of a soap-film single-mode light–heavy interface accelerated by a shock with
M≈1.22, and verified the validity of the impulsive model in predicting the linear RMI
evolution in weakly compressible flows. Additionally, it was observed that the models
proposed by Mikaelian (2003) (MIK model) and Zhang & Guo (2016) (ZG model)
reasonably predict the nonlinear evolution.
Currently, weak-shock RMI has been studied extensively through shock-tube
experiments. However, high-intensity shocks are commonly encountered in practical
applications involving RMI. Therefore, investigating RMI driven by a strong shock wave
(strong-shock RMI) is highly desirable. Laser-driven experiments on strong-shock RMI
have been widely conducted (Zhou et al. 2025). Nevertheless, the perturbation evolution
in these experiments would be influenced by processes such as material phase transition
and plasma diffusion, making it challenging to isolate the contribution of RMI to the
perturbation growth. Besides the laser-driven experiments, there are a few shock-tube
experiments on RMI at a light–heavy interface with M≥2.0. Using a solid membrane
with wire support to separate different gases, Sadot et al. (2003) investigated the evolution
of an air–SF6interface impacted by a shock wave with M=2. However, the wire
support may affect the perturbation evolution. Additionally, Motl et al. (2009) studied
the development of membraneless quasi-single-mode light–heavy interfaces accelerated
by shock waves with Mup to 2.86. They observed bubble flattening, slip line generation,
and a reduction in the linear amplitude growth rate compared with the impulsive model
prediction. This reduction was attributed to the effect of the diffusion layer. There are other
experiments with M∼3 that have been conducted by the Wisconsin team (Puranik et al.
2004;Ranjanet al. 2005,2007). For instance, they investigated the evolution of soap-
film gas bubbles (Ranjan et al. 2005,2007) and quasi-single-mode heavy-light interfaces
(Puranik et al. 2004). While these studies provide valuable insights, desirable experimental
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Journal of Fluid Mechanics
Convergence section
Settling section Ⅱ Test section
Tail section
Settling section Ⅰ
Diaphragm
15 mm
Shock
120 mm
Double-sided tape
Device A: a ir
Device B: S F6
SF6
Oxyg en
concentration
detector
Inlet pip e
Outlet pip e
Polyester film
(a)
(b)Planar
shock Convergen t
shock Planar
shock
Polyester film
2 m 1.5 m 4 m 2.16 m 1 m
1.5 m
Initial
interface
262 mm
8 mm
Driver
section
Transitional
section
Driven
section
8 mm
Figure 1. Sketches of the shock-tube facility (a), the shock convergence process and the interface formation
process (b). The inner cross-section of the test section is 120mm×8 mm.
research on strong-shock RMI at a single-mode light–heavy interface remains lacking
until now.
Richtmyer–Meshkov instability at a small-amplitude single-mode light–heavy interface
is the basic RMI scenario: free from high-order initial modes, phase inversion, and high-
initial-amplitude effect. Furthermore, although the evolution of weak-shock RMI has been
widely explored previously, whether and how strong flow compressibility affects the RMI
evolution remains unclear. Therefore, investigating the strong-shock RMI at a small-
amplitude single-mode light–heavy interface is of great significance. The present study
aims to explore the evolution law of this phenomenon and the underlying mechanisms
through fine shock-tube experiments, thereby establishing the foundation for related
research and paving the way toward understanding the underlying physics of strong-shock
RMI in real scenarios such as ICF.
2. Experimental methods
Richtmyer–Meshkov instability is highly sensitive to initial conditions (Zhou 2017a,b,
2024). Therefore, to conduct a desirable RMI experiment, both the shock and interface
generation methods are crucial. The experiments are conducted in a shock-tube facility
with a length of approximately 16 m, as illustrated in figure 1(a). This facility is capable
of generating planar shocks with M>3.0 while maintaining a ‘clean’ experimental flow.
The fundamental mechanism used in the facility to generate a strong shock is that the shock
intensity increases with the reduction of the shock cross-sectional area (Zhai et al. 2010;
Zhan et al. 2018). The contraction of the shock cross-sectional area is realized through the
convergence section, whose wall profile is designed using geometrical shock dynamics.
The convergence section can smoothly transform a planar shock wave into a cylindrical
convergent shock and then back again into a planar shock, as shown in figure 1(b). During
this process, the shock cross-sectional area decreases continuously, thus realizing the
generation of strong shock waves with M=3.20 ±0.05.
In our previous studies on weak-shock RMI (Liu et al. 2018;Chenet al. 2023), the
volume capacity of the driver section is small (1850 cm3) and the gas pressure when the
diaphragm separating the driver and driven sections ruptures is low (304 kPa, for example).
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T. Si, S. Jiang, W. Cai, H. Wang and X. Luo
Case a0(mm) VF A M ve
is (m s−1)ve
ts (m s−1)ue
si (m s−1)ut
si (m s−1)
0.157-30 0.75 0.98 0.795 3.20 1106.4 623.9 573.4 566.8
0.157-40 1.00 0.99 0.797 3.25 1119.9 630.9 561.5 574.1
0.157-60 1.50 0.96 0.792 3.23 1115.4 632.2 569.7 574.3
0.314-40 2.00 0.99 0.797 3.16 1085.7 600.0 557.4 552.2
Table 1. Significant parameters for four experimental cases labelled by ka0-λ:VF, volume fraction of SF6
downstream of the initial interface; A, post-shock Atwood number; M, incident shock Mach number; ve
is
and ve
ts, experimental velocities of the incident and transmitted shocks, respectively; ue
si and ut
si , velocities
of shocked interface obtained from experiments and predicted by one-dimensional gas dynamics theory,
respectively.
Accordingly, the inflation time it takes from the start of gas inflation to the generation of
shock wave is short (less than 1 min). This allows the use of the soap film, which is free
of grid support and diffusion layer but has a short maintainable time (generally less than
5 min), to form well-defined desirable initial interfaces. In contrast, the current shock-tube
facility has a driver section with a large volume capacity (62 800 cm3), and the gas pressure
when the diaphragm ruptures exceeds 6080kPa. Accordingly, the inflation time is long
(more than 10min). Thus, the soap-film technique is not available for present experiments.
In this work, the initial single-mode air–SF6interface is generated through polyester
film, as illustrated in figure 1(b). Note that the solid membrane employed in previous
works (Meshkov 1969;Sadotet al. 2003) is made of nitrocellulose, which is brittle and
susceptible to cracking under strain. As a result, grid support, which can significantly
influence the flow evolution, was required to form the initial interface. In contrast, the
polyester employed in the present study exhibits superior mechanical properties (such as
tensile strength, elasticity and flexibility) compared with nitrocellulose. Moreover, the
low height of the interface (8 mm) allows the profile of the polyester film to be well
maintained through the constraints of the upper and lower boundaries (Jiang et al. 2024),
thus eliminating the need for a grid support. The thickness of the polyester film employed
is 2 µm, a choice made after carefully balancing the success rate of experiments and the
potential impact of polyester film on interface evolution.
The ambient pressure and temperature are 101.3±0.1 kPa and 295.2 ±1.7 K,
respectively. The flow evolution is diagnosed using a high-speed shadowgraph system.
In comparison with the schlieren imaging method widely used in RMI studies, the
shadowgraph imaging method can provide images with better contrast and higher
distinguishability for the present experiments where strong pressure perturbations exist
near the interface. The arrangement of the shadowgraph imaging system is identical to
that in our previous work (Jiang et al. 2024), involving a light source (CEL-PF300-T10), a
slit to shape the light emitted from the source, one optical lens to collimate the beam, two
concave mirrors with a diameter of 400mm to convert a spherical beam to a parallel one
or vice versa, two planar mirrors to direct the light vertically through the test section, and
a high-speed camera (Phantom V2012). The frame rate of the high-speed video camera is
set to 100 000 frames per second, with an exposure time of 1 µs. The spatial resolution of
the shadowgraphs is approximately 0.57mm pixel−1.
Four sets of experiments on single-mode strong-shock RMI with different ka0and
wavelength (λ) combinations, referred to as cases ka0-λfor clarity, are performed. The
significant parameters for these experiments are presented in table 1. For the cases with
ka0=0.157, the small-amplitude criterion (ka 1, where adenotes the perturbation
amplitude) is sufficiently satisfied, ensuring accurate capture of the linear perturbation
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Journal of Fluid Mechanics
evolution and effective examination of linear models. In contrast, for the case 0.314-40, a
relatively large ka0enables a comprehensive capture of the perturbation evolution from the
linear to nonlinear stages and an effective evaluation of the nonlinear models. The volume
fraction of SF6downstream of the initial interface is determined by matching the velocities
of transmitted shock obtained from experiments and predicted by one-dimensional gas
dynamics theory (ve
ts and vt
ts). When vt
ts matches ve
ts, the maximum discrepancy between
the experimental and theoretical velocities of the shocked interface is less than 3% across
all four cases, indicating that the polyester film employed for interface generation has a
limited effect on the experimental flow.
To estimate the thickness of the boundary layer developing along the wall at the interface
region, the following formula proposed by Mirels (1956)isused:
δ∗=2μwx
ρwvw2
πvts
vw
−1,(2.1)
where δ∗represents the boundary-layer displacement thickness at the interface; μwand
ρware the viscosity and density of the shocked gas, respectively; xis the distance
between the interface and the transmitted shock; vts and vware the velocity of the
transmitted shock and its relative velocity with respect to the interface, respectively. Here,
Mand the ambient temperature are taken as 3.20 and 295.0 K, respectively. The gases
upstream and downstream of the initial interface are regarded as pure air and pure SF6,
respectively. Accordingly, μwand ρwfor the shocked air (SF6)are4.4107×10−5Pa s
(2.77 ×10−5Pa s) and 7.39 kg m−3(66.31 kg m−3), respectively. The values of vts and
vwcalculated using one-dimensional gas dynamics are 616.91 m s−1and 56.14 m s−1,
respectively. For x, its maximum value in experiments (30 mm) is used for calculation.
Under such conditions, δ∗in air and SF6are calculated to be approximately 0.64 and
0.17 mm, respectively, which are significantly smaller than the width and height of the
flow cross-section (120 and 8 mm). This indicates that the effect of the boundary layer on
the interface evolution is limited.
3. Flow features under strong-shock conditions
Experimental shadowgraphs of the evolution of single-mode air–SF6interfaces
accelerated by a strong shock wave are shown in figure 2. The temporal origin (t=0µs)
is defined as the moment when the incident shock reaches the mean position of the initial
interface. Note that the observation area (97 mm in width) is narrower than the flow field
(120mm in width), i.e. the boundary of the shadowgraphs is not that of the flow field.
Moreover, since the air upstream of the post-shock polyester-film interface is at a high
temperature (1030K) and oxygen-rich, the pyrolysis and combustion of the polyester occur
instantaneously (Martín-Gullón et al. 2001). Therefore, instead of breaking into ‘flaps’
like the nitrocellulose in weak-shock RMI experiments, the polyester film transforms into
various gases in the present experiments. Due to the small thickness of the polyester film
(2 µm), the effect of the reaction products on the flow evolution is negligible.
To illustrate the general flow evolution process of the cases with ka0=0.157, the case
0.157-40 is detailed as an example. When the incident shock interacts with the initial
interface, a transmitted shock and a reflected shock are generated. In contrast to weak-
shock RMI (Liu et al. 2018), the transmitted shock remains close to the shocked interface
throughout the experiment, and the reflected shock also moves downstream as the flow
Mach number of the region ahead (1.42) exceeds the shock’s Mach number (1.31). After
the shock-interface interaction, the shocked interface starts to evolve, initially maintaining
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T. Si, S. Jiang, W. Cai, H. Wang and X. Luo
0.157-30
–18
172
–23
147
–25
145
150
–20
Incident shock
Initial
interface
Slip
line
Jet
42
252
37
207
35
205
40
Transmitted
shock
Shocked
interface
Reflected
shock
210
Spike
Bubble
332
132
97
257
95
255
100
260
Transverse
shock
0.157-400.157-60
0.314-40
Roll-up
structure
Figure 2. Experimental shadowgraphs of the evolution of single-mode air–SF6interfaces accelerated by a
strong shock wave. Numbers represent time in μs.
a single-mode shape (40 µs) and gradually becoming asymmetric (100 µs). Subsequently,
apparent spike and bubble structures emerge (210 µs), with the bubbles flattening and the
spikes sharpening. At t=260 µs, no roll-up structure forms at the spikes, indicating that
the shocked interface remains in the weakly nonlinear evolution phase.
For the case 0.314-40, the shocked interface evolves fast and deviates from a single-mode
profile rapidly. The disturbed transmitted shock evolves to form Mach reflection structures
containing transverse shocks and slip lines (172 µs). The transverse shocks sweep through
the interface and interact with each other at the heads of bubbles, causing further flattening
of the bubbles. Additionally, downstream-directed jets resulting from the interactions of
slip lines are formed (172 µs), which is a typical phenomenon of strong Mach reflection
(Henderson et al. 2003). In the late stage (332 µs), after multiple impacts from transverse
shocks, the bubble heads become nearly flat and their connections to the other parts of
the shocked interface become sharp. Furthermore, significant roll-up structures form at
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Journal of Fluid Mechanics
Case ˙ae
l(m s−1)˙ai
l(m s−1)Ei˙aR
l(m s−1)ER
0.157-30 21.2 ±0.2 34.5 62.7 % 22.2 4.7 %
0.157-40 21.9 ±0.5 35.0 59.8 % 22.7 3.7 %
0.157-60 21.0 ±0.4 34.7 65.2 % 22.3 6.2 %
0.314-40 41.7 ±1.2 67.9 62.8 % 43.5 4.3 %
Table 2. Comparison of experimental linear amplitude growth rates (˙ae
l) with corresponding impulsive model
predictions (˙ai
l) and Richtmyer theory predictions (˙aR
l). Here, Ei/R=(˙ai/R
l−˙ae
l)/ ˙ae
lis the relative error
between ˙ai/R
land ˙ae
l.
the spike heads, with their overall profile being distinguishable. However, the detailed
vortices are difficult to identify due to intense mixing. At this stage, the interface profile
becomes highly irregular, markedly different from that observed in weak-shock RMI (Liu
et al. 2018) where the profile of the shocked interface remains smooth and the bubble front
maintains a curved shape.
4. Linear, nonlinear and modal evolutions of perturbations
4.1. Linear amplitude growth
In the present experiments, Mreaches 3.20 ±0.05 and the gas on one side of the interface
(SF6) is highly compressible. According to previous studies (Mikaelian 1994; Yang,
Zhang & Sharp 1994), these experiments are suitable for evaluating the validity of the
impulsive model and Richtmyer theory to predict the linear amplitude growth rate (˙al)in
highly compressible flows.
The experimental ˙al(˙ae
l) is obtained by linearly fitting the early-time amplitude variation
extracted from experiments. The impulsive model can be written as
˙ai
l=ka1Aue
si.(4.1)
Here, a1=a0[1−(ue
si/ve
is)]is the post-shock amplitude, with ue
si and ve
is representing
the experimental velocities of the shocked interface and incident shock, respectively, and
Ais the post-shock Atwood number. The Richtmyer theory lacks an analytical solution
and involves a number of nonlinear equations requiring to be solved numerically. Details
on this theory can be found in the pioneering works of Richtmyer (1960) and Yang et al.
(1994). The values of ˙ae
l,˙ai
land ˙alpredicted by the Richtmyer theory (˙aR
l) for different
cases are provided in table 2 for comparison. The results indicate that the impulsive model
significantly overestimates ˙ae
leven when ka0adequately satisfies the small-amplitude
criterion. In contrast, the Richtmyer theory provides reasonable predictions for all cases.
To the best of our knowledge, this is the first direct experimental confirmation of the
validity of the Richtmyer theory and the failure of the impulsive model for predicting
the linear amplitude evolution in highly compressible flows.
4.2. Nonlinear amplitude evolution
As the perturbation evolves into the nonlinear stage, the interface becomes markedly
asymmetric, and the bubble and spike exhibit significantly different evolution behaviours.
The overall interface amplitude ais determined by extracting the horizontal coordinates of
the tips of the spike and bubble. To extract the amplitudes of the bubble and spike (aband
as), the mean positions of the shocked interface in the early stages and ue
si are first obtained
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T. Si, S. Jiang, W. Cai, H. Wang and X. Luo
α
αb/s
2.0 2
Bubble
Spike
1
0
1
2
3
4
(a)(b)
1.5
0.157-30
ZS
SEA
MIK
DR
ZG
ZS
SEA
MIK
DR
ZG
mDR
0.157-40
0.157-60
0.314-40
0.157-30
0.157-40
0.157-60
0.314-40
1.0
0.5
0 0.5 1.0 1.5 2.0 2.5
τ
0 0.5 1.0 1.5 2.0 2.5
τ
Figure 3. Temporal variations of perturbation amplitude in dimensionless form: (a) overall interface,
(b) bubble and spike.
through Fourier analysis. On this basis, the mean positions of the shocked interface in
the latter stages are computed, thus realizing the determination of aband asthroughout
the interface evolution. Since nonlinear models are generally initialized with ˙al,the
experimental data within the start-up period (i.e. amplitude growth rate rises from zero
and gradually reaches ˙alunder the drive of pressure perturbations) (Lombardini & Pullin
2009) needs to be excluded to facilitate comparison with nonlinear model predictions. The
temporal variations of aand ab/sin dimensionless form are illustrated in figures 3(a)and
3(b), respectively. Here, t,aand ab/sare normalized as τ=k˙ae
l(t−t∗),α=k(a−a∗),
and αb/s=k(ab/s−a∗
b/s), respectively, in which t∗is the duration of the start-up period,
with a∗and a∗
b/sbeing the corresponding aand ab/sat t=t∗, respectively. Note that t∗is
obtained via visual observations of the extracted experimental amplitude evolution. This
scaling approach, widely adopted in studies on weak-shock RMI (Zhou 2017a,b), collapses
the results of all four cases, indicating that strong-shock RMI at a small-amplitude
single-mode interface follows a similar evolution law.
Typical nonlinear models, including the ZS, SEA, MIK, DR (Dimonte & Ramaprabhu
2010), and ZG models, are considered for evaluation. These models have been examined
widely in research on weak-shock RMI (Zhou 2017a,b,2024). However, their applicability
to RMI in highly compressible flow remains unclear. Each model considered exhibits
similar predictions across different cases. Accordingly, for each model, only three
theoretical lines, corresponding to the overall interface, bubble and spike, respectively, are
presented in figure 3 for comparison. For the overall interface, the ZS, SEA and DR models
overestimate its amplitude growth, whereas the MIK and ZG models offer reasonable
predictions. For the bubble, its early-time growth is slower than the model predictions due
to the shock proximity effect (Sadot et al. 2003): the transmitted shock remains close to the
shocked interface for a relatively long duration, inhibiting the evolution of the bubbles and
flattening them. Subsequently, under the effect of the pressure perturbations introduced by
transverse shocks, i.e. the secondary compression effect, the bubble growth saturates. All
considered nonlinear models overestimate the bubble evolution since they do not consider
these two effects. For the spike, the amplitude growth rate is relatively high due to the spike
acceleration effect (Dimonte & Ramaprabhu 2010): high-order harmonics concentrate on
the spike under high Aconditions (A≈0.8 in present work), promoting its evolution.
The MIK and ZG models significantly underestimate the spike growth because they do not
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Journal of Fluid Mechanics
account for the spike acceleration effect. Therefore, the seemingly reasonable prediction
of the MIK and ZG models for the overall amplitude evolution results from compensating
errors in their overestimation of bubble growth and underestimation of spike evolution.
The SEA model overestimates the spike evolution due to its significant overestimation
of the spike acceleration effect (Dimonte & Ramaprabhu 2010;Chenet al. 2023). For
the ZS model, it slightly overestimates the spike evolution at late stage because its spike
acceleration term is overly sensitive to ka1(Dimonte & Ramaprabhu 2010). In contrast, the
DR model provides a reasonable prediction as it properly describes the spike acceleration
(Chen et al. 2023).
It is challenging to rigorously describe the shock proximity and secondary compression
effects on bubble evolution. Therefore, we attempt to propose an empirical model for
bubble growth in highly compressible flows based on current experimental findings. The
DR model effectively describes the early-time spike evolution behaviour, the dependence
on ka0, and also the late-time amplitude growth behaviour that has been validated in
previous works (Dimonte & Ramaprabhu 2010; Mansoor et al. 2020). Therefore, we
construct an empirical bubble model (the mDR bubble model) based on the DR model.
The mDR bubble model can be expressed as
˙amdr
b=Nb˙ae
l
1+(1−A)Nbτ
1+CbNbτ+(1−A)Fb(Nbτ)
2Kb,
in which τ=k˙ae
l(t−t∗), Cb=[4.5+A+(2−A)ka1]/4,Fb=1+A,
Nb=1.4e−M+0.45,Kb=1
1+e3[τ−f(M)]and f(M)=0.6M+1
M−1.(4.2)
Here, Cband Fbhave the same expressions to those in the DR model. The ideas leading
to such a model are elaborated as follows. According to the above discussions, a model
capable of describing the bubble evolution under strong-shock conditions should well
capture both the effects of the shock proximity and secondary compression. To describe
the shock proximity effect that mainly affects the early-time amplitude evolution and is
expected to be more significant when Mis higher, ˙ae
ladopted in the DR model is replaced
by Nb˙ae
l, in which Nbis a coefficient that is always smaller than 1 in magnitude and
decreases with increasing M. Then, to capture the secondary compression effect that
manifests mainly at the period when interactions of transverse shocks occur, a time-
dependent function Kbis incorporated into the DR model. Here, τis the dimensionless
time whose origin (τ=0) corresponds to the moment when the start-up evolution phase
terminates and f(M)describes the characteristic time of the interactions of transverse
shocks occurring at the bubble tips. The value of Kbconverges to 1 as τ−f(M)→−∞
andto0asτ−f(M)→+∞, but its value changes significantly only when τ−f(M)
is close to 0. In other words, at approximately τ=f(M),Kbvaries from 1 to 0 rapidly,
thus realizing the description of the secondary compression effect. The determination of
f(M)is based on the observation of previous RMI research (Zhou 2017a,b,2024)that
the secondary compression effect is negligible in the weak-shock limit and the transverse
wave intersection occurs earlier when the shock intensity is higher. This indicates that
f(M)should tend to infinity as M→1, and decrease as Mincreases. Based on these
conditional restrictions on function and also the present experimental results, f(M)
is fitted as (0.6M+1)/(M−1). As illustrated in figure 3(b), the mDR bubble model
predicts the experimental results well from the early to late periods.
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T. Si, S. Jiang, W. Cai, H. Wang and X. Luo
1.2
0.8
Exp-α1
Odd-order
Even-order
0.157-30 0.314-40
Exp-α2
Exp-α3
ZSM-α1
ZSM-α2
ZSM-α3
0.4
0
0.4 0 0.15 0.30 0.45
τ
0.60 0.75 0.90
α
Figure 4. Modal evolutions obtained from experiments and predicted by ZSM model. Here, α1,α2and α3are
the dimensionless amplitudes of the first three harmonics; Exp-α1, Exp-α2and Exp-α3(ZSM-α1,ZSM-α2and
ZSM-α3)areα1,α2and α3obtained from experiments (predicted by ZSM model), respectively.
4.3. Modal evolution
Modal analysis is performed to explore the modal evolution of RMI in highly compressible
flows. Among the cases with ka0=0.157, only the case 0.157-30 is considered for clarity.
The interface contours, before the roll-up structures appear, are first extracted from the
experimental shadowgraphs. Then, the fast Fourier transform is applied to obtain the
modal information. The modal model proposed by Zhang & Sohn (1997) (ZSM model),
which has been validated for describing the modal evolution of weak-shock RMI (Liu
et al. 2018), is considered for evaluation.
The temporal variations of the amplitudes of the first three harmonics (am1,am2and
am3) in dimensionless form, obtained from experiments and predicted by the ZSM model,
are shown in figure 4. Here, am1,am2and am3are normalized as α1=k(am1−a∗
m1),
α2=k(am2−a∗
m2)andα3=k(am3−a∗
m3), respectively, in which a∗
m1,a∗
m2and a∗
m3
are the corresponding am1,am2and am3at t=t∗, respectively. In previous work
on weak-shock RMI (Liu et al. 2018), it was found that the ZSM model reasonably
describes the evolution of α1(α2and α3) until τ=0.6 (1.5). In contrast, the results for
strong-shock RMI, as shown in figure 4, show that the model overestimates α1while
underestimating α2and α3when τ>0.15, as the interface shape is altered by the shock
proximity and secondary compression effects. Specifically, these effects influence the
modal amplitude variations via squeezing the bubbles. The flattening and widening of the
bubbles simultaneously result in variations in the shape of the spikes, making them sharper
and narrower. Note that the shape variation of the spikes does not significantly affect the
positions of their tips, thus the spike amplitude evolution is less affected. The flattening
of bubbles inhibits the growth of the first-order mode and results in the generation of the
second-order mode that has opposite phase to the first-order mode at positions close to the
bubble tips. Additionally, the sharpening of spikes results in the generation of the second-
and third-order modes that have the same phase as the first-order mode at positions close
to the spike tips. As a result, the ZSM model, which is proposed for RMIin incompressible
flows, overestimates the development of the basic mode and underestimates the evolution
of higher-order harmonics.
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Journal of Fluid Mechanics
5. Conclusions
Richtmyer–Meshkov instability at a single-mode light–heavy interface accelerated by a
strong shock wave is studied experimentally in a newly developed shock-tube facility.
The Mach numbers of the shock waves reach up to 3.2. Under the proximity effect
of the transmitted shock and its induced secondary compression effect, the bubble
development is greatly inhibited. In contrast, the spikes sharpen rapidly due to the
spike acceleration occurring under high post-shock Atwood number conditions and the
secondary compression. Due to the distinct evolution behaviours of bubbles and spikes,
the interface becomes highly irregular as nonlinearity grows.
For the linear evolution period, the impulsive model (Richtmyer 1960) fails to
accurately predict the experimental growth rate due to its inadequacy in describing the
effect of compressibility on the interface evolution. In contrast, the Richtmyer theory
(Richtmyer 1960) offers reasonable predictions. For the nonlinear evolution period, the
shock proximity effect inhibits the early-time bubble amplitude development, while the
secondary compression effect results in the late-time saturation of bubble amplitude
growth. None of the nonlinear models considered accurately predict the bubble evolution,
as they do not account for these two effects. Based on current experimental findings
and existing nonlinear models, an empirical model, considering the shock proximity and
secondary compression effects on early-time and later evolution, respectively, is proposed
to predict the bubble evolution in highly compressible flows. For the spike evolution, the
amplitude growth rate is relatively high due to the spike acceleration effect, and the DR
model, which properly describes this effect, provides reasonable predictions. Furthermore,
modal analysis shows that the shock proximity and secondary compression effects inhibit
the fundamental mode growth while resulting in the generation of high-order harmonics,
which leads to the failure of the modal model proposed for incompressible flows (Zhang &
Sohn 1997).
In future studies, the effects of mode coupling/competition, high initial amplitude, and
phase inversion on strong-shock RMI will be investigated based on the findings of the
present work. Furthermore, experiments with different Mwill be conducted by varying
the component and pressure of the driver gas, in order to evaluate and develop the newly
proposed empirical model. Moreover, we are currently developing a novel shock-tube
facility capable of generating shock waves with M>6.0, in which the shock waves are
contracted twice successively in two orthogonal directions. Under these conditions, real
gas effects become significant, thus providing an opportunity to investigate their influence
on the instability evolution.
Funding. This work is supported by the National Natural Science Foundation of China (nos. 12027801,
12102425, 12472228 and 12388101), the Chinese Academy of Sciences Project for Young Scientists in Basic
Research (YSBR-087), the Fundamental Research Funds for the Central Universities, and the Young Elite
Scientists Sponsorship Program by CAST (no. 2023QNRC001).
Declaration of interests. The authors report no conflict of interest.
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