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J. Fluid Mech. (2021), vol.928, A37, doi:10.1017/jfm.2021.849
Richtmyer–Meshkov instability on
two-dimensional multi-mode interfaces
Yu Liang1,2, Lili Liu1, Zhigang Zhai1,†,JuchunDing
1,TingSi
1and
Xisheng Luo1,†
1Advanced Propulsion Laboratory, Department of Modern Mechanics, University of Science and
Technology of China, Hefei 230026, PR China
2NYUAD Research Institute, New York University Abu Dhabi, Abu Dhabi 129188, UAE
(Received 14 February 2021; revised 1 August 2021; accepted 23 September 2021)
Shock-tube experiments on eight kinds of two-dimensional multi-mode air–SF6interface
with controllable initial conditions are performed to examine the dependence of
perturbation growth on initial spectra. We deduce and demonstrate experimentally that
the amplitude development of each mode is influenced by the mode-competition effect
from quasi-linear stages. It is confirmed that the mode-competition effect is closely
related to initial spectra, including the wavenumber, the phase and the initial amplitude of
constituent modes. By considering both the mode-competition effect and the high-order
harmonics effect, a nonlinear model is established based on initial spectra to predict
the amplitude growth of each individual mode. The nonlinear model is validated by the
present experiments and data in the literature by considering diverse initial spectra, shock
intensities and density ratios. Moreover, the nonlinear model is successfully extended
based on the superposition principle to predict the growths of the total perturbation width
and the bubble/spike width from quasi-linear to nonlinear stages.
Key words: shock waves, turbulent mixing
1. Introduction
Richtmyer–Meshkov (RM) instability is initiated when a shock wave interacts with an
interface between two fluids of different densities (Richtmyer 1960; Meshkov 1969), and
further induces mushroom-shaped flow structures such as bubbles (light fluids penetrating
into heavy ones) and spikes (heavy fluids penetrating into light ones), which finally may
cause a flow transition to turbulent mixing (Zhou, Robey & Buckingham 2003; Zhou 2007;
Zhou et al. 2019). Over the past few decades, the RM instability has become a subject of
intensive research due to its crucial role in various industrial and scientific fields such
†Email addresses for correspondence: sanjing@ustc.edu.cn,xluo@ustc.edu.cn
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
as inertial confinement fusion (ICF) (Lindl et al. 2014) and supernova explosion (Kuranz
et al. 2018). For example, the RM instability determines the seeds of Rayleigh–Taylor
(RT) instability (Rayleigh 1883; Taylor 1950) that develops during the implosion in ICF
(Goncharov 1999). The mixing of hot fuel inside with cooler shell material outside,
induced by RM and RT instabilities in the target of ICF, significantly reduces and
even eliminates the thermonuclear yield (Miles et al. 2004). The RM instability on a
single-mode interface has been extensively studied due to its fundamental significance
(Brouillette 2002; Zhou 2017a,b;Zhaiet al. 2018). However, the initial perturbation in
reality is essentially multi-mode with wavenumbers spanning many orders of magnitude,
and whether the perturbation growth of a multi-mode RM instability depends on the initial
spectrum or not is crucial to ICF (Miles et al. 2004) but remains unclear.
Theoretically, there are mainly six kinds of models describing the perturbation growth of
a multi-mode interface: the linear model, the modal model, the potential model, the vortex
model, the perturbation expansion model and the group theory approach. Based on the
principle that each individual mode develops independently in linear stages, Mikaelian
(2005) proposed the linear model to describe the multi-mode interface evolution by
summing the time-varying amplitude growth of each mode. Haan (1989) found that
the constituent modes with similar wavelengths of a multi-mode interface add up to
create an effective local large amplitude, and, therefore, the onset of the nonlinear stage
of a multi-mode perturbation is earlier than that of the classical single-mode case.
Subsequently, Haan (1991) proposed the modal model with second-order accuracy to
quantify the mode-competition effect on the perturbation growth of each mode in the
early nonlinear stage. The modal model and its extended types have achieved a wide range
of validation in RT instability issues (Remington et al. 1995;Oferet al. 1996; Elbaz &
Shvarts 2018), but their application to the RM instability is still lacking. Assuming that
mode competition is absent before a bubble reaches its asymptotic growth, the potential
model was proposed by Alon et al. (1994) and Layzer (1955) to predict the eventual
average bubble distribution and the growth rate. However, the potential model is invalid
when the Atwood number (defined as A=(ρ2−ρ1)/(ρ2+ρ1),withρ1and ρ2being the
densities of light fluid and heavy fluid, respectively) is low. When the Atwood number
approaches zero, the vortex model (Jacobs & Sheeley 1996) was adopted by Rikanati,
Alon & Shvarts (1998) to make up the bubble asymptotic growth rate. Note that both the
potential model (Alon et al. 1994,1995;Oronet al. 2001) and the vortex model (Rikanati
et al. 1998)involve a self-similar growth of the bubble front which is independent of the
initial spectrum, and both models obtain a 1/tdecay for the late-time bubble growth rate
in a multi-mode RM instability. The perturbation expansion model developed by Zhang &
Sohn (1997) was extended by Vandenboomgaerde, Gauthier & Mügler (2002) to predict
the early nonlinear amplitude growth of the constituent modes of a multi-mode interface
by retaining only the terms with the highest power in time. The group theory approach
(Abarzhi 2008,2010; Pandian, Stellingwerf & Abarzhi 2017) identifies the connection
between the symmetry properties of the interface morphology and the relative phases of
waves constituting the interface perturbation.
Experimentally, shock-tube experiments were performed to investigate two-bubble
competition (Sadot et al. 1998), and the results showed that the growth of the larger (or
smaller) bubble is promoted (or suppressed). Dimonte & Schneider (2000) conducted a
series of three-dimensional linear electric motor experiments to investigate multi-mode
RT and RM instabilities, and found that the density ratio has a limited effect on the
self-similar growth factor for the bubble. When the density ratio is large, the self-similar
growth factor for the spike is clearly larger than that for the bubble counterpart.
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RM instability on 2-D multi-mode interfaces
The multi-mode RM instability of two liquids was investigated by Niederhaus & Jacobs
(2003), and the development of the multi-mode perturbation was found to be strongly
dependent on the relative amplitudes of initial modes. The growth of the multi-mode
interface perturbation created by the gas curtain technique shows a weak dependence
on the initial conditions (Balasubramanian, Orlicz & Prestridge 2013). Experiments of
a dual-mode interface RM instability under high-Mach-number conditions have been
performed (Di Stefano et al. 2015a,b), and the results indicated that new modes are
generated from the mode-competition effect, and the perturbations of these modes grow
and saturate over time. The dual-mode RM instability under weak shock conditions
was also considered, from which the mode-competition effect on the RM instability
development cannot be ignored when the wavenumber of one constituent mode is twice
that of the other constituent mode (Luo et al. 2020). The mixing of a multi-mode interface
was investigated by Mohaghar et al. (2017) using density and velocity statistics, and the
flow shows a distinct memory of initial conditions, the long-wavelength perturbation
having a strong influence on the interface development. Recently, developments of
quasi-single-mode interfaces created by the soap film technique in the early nonlinear
stage have been studied, and the effect of high-order modes on the perturbation growth
was highlighted to distinguish from single-mode perturbation (Liang et al. 2019). A
near-sinusoidal interface dominated by one mode was generated by a novel membraneless
technique where cross-flowing air was separated from SF6by an oscillating splitter plate
(Mansoor et al. 2020), and the effects of the initial amplitude on the perturbation width
growth and mixing transition have been discussed, and earlier mixing transitions for higher
amplitude-to-wavelength ratio cases are noted from experiments.
Numerically, it is commonly realized that the phases of the constituent modes influence
multi-mode perturbation growth (Vandenboomgaerde et al. 2002; Miles et al. 2004;
Pandian et al. 2017). Besides, the self-similar growth factor of the late-time RM
instability has a dependence on the scale of the initial spectrum. Specifically, a broadband
perturbation leads to a larger bubble growth factor than a narrowband counterpart
(Thornber et al. 2010; Liu & Xiao 2016; Thornber 2016; Groom & Thornber 2020).
Although significant progress on the multi-mode RM instability has been made, the
quantitative relation between initial conditions and perturbation growth is still unclear
mainly because a general nonlinear theory for predicting the multi-mode perturbation
width growth is absent, and elaborate experiments on the multi-mode RM instability
with controllable initial conditions are very limited. In our previous work, the extended
soap-film technique was utilized to create a classical two-dimensional (2-D) single-mode
perturbation (Liu et al. 2018), a 2-D multi-mode interface dominated by only one mode
(Liang et al. 2019)anda2-D multi-mode interface dominated by two modes (Luo
et al. 2020). The initial perturbations of these interfaces were precisely designed and
the initial conditions were well controlled. In this work, a 2-D complex multi-mode
interface constituted of various modes is first formed, and shock-tube experiments on
the developments of eight kinds of air–SF6multi-mode interface are performed. Then,
new nonlinear theories based on the initial spectrum, shock intensity and density ratio are
proposed to predict each mode amplitude growth and the total perturbation width growth
of a 2-D multi-mode interface.
2. Experimental method
The extended soap-film technique, which has been widely used in our previous work
(Ding et al. 2017;Liuet al. 2018; Liang et al. 2019), is adopted to generate a periodic
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
x
y
o
Shock
140 mm
Air SF6
100 mm 10 mm
7 mm
Soap film
Rectangular
frame
140 mm
Shock
Filaments
Interface
w0
x
y
z
z= –3.5 mm
z=0 mm
z= 3.5 mm
(a)(b)
Figure 1. Schematics of soap-film interface generation (a) and the initial configuration (b).
multi-mode interface with a controllable initial shape to separate SF6from air. Such a
technique can largely eliminate the short-wavelength perturbations, diffusion layer and
three-dimensionality of the formed interface (Liu et al. 2018; Liang et al. 2019). As shown
in figure 1(a), two transparent devices with an inner height of 7.0 mm and a width of
140.0 mm are first made using acrylic plates (3.0 mm in thickness). A groove (0.7 mm
in thickness and 0.5 mm in width) with a multi-mode shape is then manufactured on the
internal side of each plate by a high-precision engraving machine. Then, two thin filaments
(1.0mm in thickness and 0.5 mm in width) with the same multi-mode shape are mounted
into the grooves of the upper and lower plates, respectively, to produce desired constraints.
Therefore, the bulging of the filament into the flow is less than 0.3 mm, and has a negligible
effect on the flow field. A small rectangular frame wetted by soap solution (78 % distilled
water, 2 % sodium oleate and 20 % glycerine by mass) is pulled along the filaments,
and a quasi-2-D soap-film interface is immediately generated, as shown in figure 1(a).
Subsequently, the auxiliary framework is gently inserted until it is completely connected
to the corresponding device. After that, the framework with a soap film on its surface is
slowly inserted into the test section of the shock tube. To form an air–SF6interface, the
air on the right-hand side of the interface is replaced by SF6. To minimize the effect of
the shock-tube walls on interface evolution, a short flat part with 10mm on each side of
the perturbed interface is adopted, as sketched in figure 1(b), and its effect on the interface
evolution is limited (Luo et al. 2019).
In the Cartesian coordinate system, as sketched in figure 1(b), the multi-mode interface
investigated can be described as a sum of three cosine modes:
y=
3
1
a0
kncos(knx+φkn), x∈[−60,60] mm,(2.1)
where a0
kn,knand φknrespectively denote the initial amplitude, wavenumber and phase
of the nth constituent mode with n=1, 2 and 3. To illustrate the influences of the initial
amplitude, wavenumber and relative phase on the 2-D multi-mode RM instability, eight
different kinds of multi-mode interface are designed in this work. The initial spectrum and
the initial perturbation width (w0, sketched in figure 1b) of the multi-mode interface in
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RM instability on 2-D multi-mode interfaces
Case a0
k1a0
k2a0
k3k1k2k3φk1φk2φk3a0
k1k1a0
k2k2a0
k3k3w0w0bw0s
IP-s 1.0 1.0 1.0 105 209 314 0 0 0 0.1 0.2 0.3 4.3 3.0 1.3
k3AP-s 1.0 1.0 1.0 105 209 314 0 0 π0.1 0.2 0.3 3.5 1.5 2.0
k2AP-s 1.0 1.0 1.0 105 209 314 0 π0 0.1 0.2 0.3 4.3 1.3 3.0
AP-s 1.0 1.0 1.0 105 209 314 0 ππ 0.1 0.2 0.3 3.5 2.0 1.5
IP-h 3.0 2.0 1.0 105 209 314 0 0 0 0.3 0.4 0.3 8.0 6.0 2.0
k3AP-h 3.0 2.0 1.0 105 209 314 0 0 π0.3 0.4 0.3 7.5 4.0 3.5
k2AP-h 3.0 2.0 1.0 105 209 314 0 π0 0.3 0.4 0.3 8.0 2.0 6.0
AP-h 3.0 2.0 1.0 105 209 314 0 ππ 0.3 0.4 0.3 7.5 3.5 4.0
Table 1. Initial spectrum and initial perturbation width of multi-mode interfaces formed in the present work.
Here a0
kn,knand φkndenote the initial amplitude, wavenumber and phase of the nth constituent mode,
respectively; w0,w0band w0sdenote the initial total perturbation width, bubble width and spike width of
the multi-mode interface, respectively. The unit for the amplitude and width is mm and for the wavenumber is
m−1.
different cases are listed in table 1. In this work, φk1is kept as 0 and φk2and φk3are varied.
For convenience, the following notation is adopted: φk2=φk3=0 (in-phase (IP) case);
φk2=φk3=π(anti-phase (AP) case); φk2=0, φk3=π(k3AP case); φk2=π,φk3=0
(k2AP case). The influence of the initial amplitude of mode knon the RM instability can
be studied by varying a0
knkn(Mansoor et al. 2020;Sewellet al. 2021). In this work, the
cases of IP-s, k3AP-s, k2AP-s and AP-s are classified as small-w0cases, whereas the cases
of IP-h, k3AP-h, k2AP-h and AP-h are classified as large-w0cases.
The experiments are performed in a horizontal shock tube with a 140 mm ×13 mm
cross-sectional area. This type of tube has been widely used in shock–interface interaction
studies (Luo, Wang & Si 2013;Zhaiet al. 2014;Luoet al. 2015; Ding et al. 2017).
The ambient pressure and temperature are 101.3kPa and 299.5±1.0 K, respectively. In
experiments, the incident shock wave with velocity (vs)of409±1ms
−1(the incident
shock Mach number (M)is1.18)movesfromairtoSF
6. The ambient air is considered
as pure and the test gas is a mixture of air and SF6,the mass fraction of SF6being
0.97 ±0.01 calculated according to one-dimensional gas dynamics theory. Meanwhile, the
transmitted shock velocity (vt) and the speed jump of the interface (Δv) can be calculated
as 182 ±1ms
−1and 65.5±0.5ms
−1, respectively. The post-shock Atwood number A+
(defined as A+=(ρ+
2−ρ+
1)/(ρ+
2+ρ+
1),withρ+
2and ρ+
1being the densities of shocked
test gas and air, respectively) is 0.66 ±0.01. The flow field is monitored using high-speed
schlieren photography. The frame rate of the high-speed video camera (FASTCAM SA5,
Photron Limited) is 62 500 f.p.s. with a shutter time of 1 μs. The spatial resolution of the
schlieren images is 0.4mm pixel−1. The visualization window of the flow field is within
the range x∈[−50,50] mm, as shown with the grey zone in figure 1(b). For each case, at
least three experimental runs are performed, and the experiments have a good repeatability.
The relative differences of the data among diverse experimental runs are within 3 %.
The three-dimensional feature of the initial soap-film interface is discussed. Because the
gases on both sides of the interface are at ambient pressure, the soap-film interface formed
has a zero mean curvature, and the geometry can be characterized as minimum surface
(Luo et al. 2013; Liang et al. 2021). For instance, half of the perturbation width of the
soap-film interface in case IP-h or k3AP-h is 4 mm (see table 1), indicating that the total
amplitude of the soap-film interface on the boundary slice (i.e. z=±3.5 mm) is 4 mm.
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
The maximum wavelength of the soap-film interface is 60 mm, and the interface height is
7 mm. Based on our previous work (Luo et al. 2013; Liang et al. 2021), the amplitude of
the soap film on the symmetry slice (i.e. z=0 mm) is 3.75 mm. Therefore, the amplitude
ratio of the symmetry slice over the boundary slice is 93.75 %. The absolute difference
between the amplitudes on the symmetry slice and the boundary slice is about 0.25 mm,
and is smaller than the size of a schlieren image’s pixel. As a result, it is believed that the
interface height of 7 mm can reduce the three-dimensional effect in this work.
The boundary layer effect on the interface evolution is also considered. After the
incident shock with Mof 1.18 impacts the interface, the flow behind the transmitted shock
can be regarded as laminar and incompressible. As a result, the displacement thickness of
the boundary layer (δ∗) can be approximately calculated using the following expression:
δ∗=1.72μymax
ρΔv,(2.2)
where ymax (≈100mm measured from experiment) is the maximum distance that the
interface moves when image recording ends. In this study, μ=1.83 ×10−5Pa s (=
1.60 ×10−5Pa s) is the viscosity coefficient of the ambient (test) gas, ρ=1.2kgm
−3
(=5.3kgm
−3) is the density of the ambient (test) gas and Δv≈65.5ms
−1. According
to (2.2), the displacement thickness of the boundary layer is calculated to be about 0.26 mm
for ambient gas and 0.12mm for test gas, which is much smaller than the inner height of
the acrylic plates (7.0 mm). Therefore, the effect of the boundary layer on the interface
evolution is negligible.
After a shock wave impacts the soap film, the soap solution is atomized into tiny droplets
(Cohen 1991; Hosseini & Takayama 2005; Ranjan et al. 2005). Our previous work (Luo
et al. 2013;Siet al. 2015;Leiet al. 2017) revealed that the dimension of the atomized
droplets is within 1–10 μm, and a portion of tiny soap droplets follows the evolving
interface nicely and can be utilized for light scattering illuminated by a laser. Besides,
it is recommended to mix atomized olive oil droplets with a diameter of around 1 μmwith
SF6when injecting the test gas into the test section of a shock tube. Using the atomized
soap droplets and oil droplets as tracer particles, it is worth looking forward to adopting
aparticle image velocimetry system to capture the velocity and vorticity contours of an
evolving interface initially generated with soap-film technology.
3. Results and discussion
3.1. Experimental observation and quantitative results
The schlieren images of the shocked multi-mode interface for small-w0cases are shown
in figure 2. It is evident that the phases of the constituent modes greatly affect the initial
interface shape and the later interface evolution. Taking the IP-s case as an example, after
the transmitted shock just leaves the interface, the shocked interface retains its initial
shape (69 μs). Then, the perturbation on the interface grows gradually but the interface
remains single-valued, which indicates that the interface evolves in early nonlinear
stages. Subsequently, vortices appear on the spikes, and the interface morphology acts
as multi-valued (261 μs). Due to the bubble-merging process (Sadot et al. 1998), the large
spike between the large bubble and small bubble skews towards the large bubble, whereas
the small spike between two large spikes remains symmetric (581 μs), which qualitatively
agrees with the group theory analysis (Abarzhi 2008,2010; Pandian et al. 2017). Finally,
the scales of vortices are comparable to the total perturbation width, and the interfacial
morphology shows a strong nonlinearity (1061μs).
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RM instability on 2-D multi-mode interfaces
69
68
66
261
260
258
581
580
578
574
IP-s
k
3
AP-s
k
2
AP-s
AP-s
Interface
TS
Vortices LS
SS
LB
SB
w
UP DP
x
y
1061
1060
1058
105462 254
(a)
(b)
(c)
(d)
Figure 2. Schlieren images of multi-mode interface evolution for small w0cases. TS, transmitted shock; w,
interface perturbation width; LS, large spike; LB, large bubble;SS, small spike; SB, small bubble; UP, upstream
point of interface; DP, downstream point of interface. Numbers denote time in μs, and similarly hereinafter.
The Schlieren images of the shocked multi-mode interface for large-w0cases are shown
in figure 3. The interface morphologies in large-w0cases are qualitatively similar to
the corresponding small-w0ones. However, for the large-w0cases, there is a greater
misalignment between the pressure gradient of the shock wave and the density gradient of
the interface, resulting in more baroclinic vorticity production and the earlier appearance
of vortices. Besides, at late time in the k2AP case (1062 μs), the spike structures
on the multi-valued interface break, and the whole interface becomes chaotic, which
indicates that the transition may occur earlier when the initial interface amplitude is larger
(Mohaghar et al. 2019; Mansoor et al. 2020).
The captured interface morphology is distinct such that the interface contours in all
cases can be extracted by an image processing program, as indicated by the insets in
figures 4 and 5. The mean ycoordinate in each image is taken as the average position of
the local interface. Spectrum analysis is then performed on the averaged interface contour
before the interface becomes multi-valued, and the amplitudes of three constituent modes
are acquired, as shown in figures 4 and 5. Time is normalized as τn=k1|vR
kn|t,wherevR
kn
is the Richtmyer growth rate of mode kncalculated by the impulsive theory (Richtmyer
1960):
vR
kn=ZckA+Δva0
kncos(φkn), (3.1)
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
261
259
262
IP-h
k
3
AP-h
k
2
AP-h
AP-h
581
579
582
580
1061
1059
1062
1060
69
67
70
69 260
x
y
(a)
(b)
(c)
(d)
Figure 3. Schlieren images of the multi-mode interface evolution for large-w0cases.
in which Zc(=1−Δv/vs) is the shock compression factor and is equalto0.84inall
cases. In the present coordinate system, vR
knis positive if φkn= 0, but becomes negative
if φkn=π. The amplitude is scaled as ηn=k1|akn(t)−Zca0
kncos(φkn)|,withakn(t)the
time-varying amplitude of mode kn.
In small-w0cases, as shown in figure 4, it is clear that the dimensionless amplitudes
of modes k1and k2are larger than those of mode k3in IP-s and AP-s cases, but smaller
than those of mode k3in k3AP and k2AP cases. In other words, the low-order modes
(modes k1and k2) in IP-s and AP-s cases dominate the flow, whereas the high-order
mode (mode k3)ink3AP and k2AP cases dominates the flow, which agree with the
observations in figure 2. For example, in the IP-s case in figure 2(a), the late-time interface
is dominated by long-wavelength structures (a large bubble and two groups of large and
small spikes), but in the k3AP case in figure 2(b), the late-time interface is dominated
by four short-wavelength structures (four pairs of spikes and bubbles). Therefore, the
mode-competition effect plays a role in the amplitude development of the constituent
modes in early stages. Since the small perturbation hypothesis is satisfied for each
constituent mode, the impulsive theory should be valid to predict the amplitude growth
of each constituent mode if the mode-competition effect is ignored. However, compared
with the predictions of the impulsive theory, in the IP-s case, mode k1development is
promoted but mode k3development is suppressed, while mode k2development is not
obviously influenced by the mode-competition effect. Differently, in the k3AP-s case,
the developments of both modes k1and k2are suppressed, while the development of
mode k3is not obviously influenced by the mode-competition effect. Therefore, the
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RM instability on 2-D multi-mode interfaces
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
69 261213165117
k1
Run2
mode k3
Run1
k3
k2
mode k1mode k2
Present
Haan-RM
Ofer-RM Impulsive
theory
IP-s
0.1 0.2 0.3 0.4
0
0.1
0.2
0.3
0.4
68 260212164116
k1
k3
k2
k3AP-s
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
66 258210162114
k1
k3
k2
k2AP-s
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
62 254206158110
k1
k3
k2
AP-s
ηn
ηn
τnτn
(a)(b)
(c)(d)
Figure 4. The dimensionless amplitudes of three constituent modes obtained from small-w0cases. The insets
show the interface contours for the spectrum analysis in which numbers indicate time in μs. Run1 and run2
represent typical experimental runs. The black dashed line represents the impulsive theory (Richtmyer 1960).
The coloured solid lines, dashed lines and dash-dotted lines represent the amplitudes of three constituent modes
calculated by the Haan-RM model (3.4), the Ofer-RM model (3.7)and the present model (3.10), respectively,
and similarly hereinafter.
mode-competition effect is greatly influenced by the initial wavenumber and the initial
phase of the constituent mode.
In large-w0cases, as shown in figure 5, similar to the corresponding small-w0cases,
the amplitudes of both modes k1and k2are larger than those of mode k3in IP-h and
AP-h cases, but smaller than those of mode k3in k3AP-h and k2AP-h cases. However,
compared with the impulsive theory, mode k2development in the IP-h case is suppressed
whereas mode k3development in the k3AP-h case is promoted by the mode-competition
effect, which is different from the results of corresponding small-w0cases. Therefore,
the initial amplitude of the constituent mode also affects the mode-competition effect. In
summary, the effect of mode competition is closely related to the initial spectra, including
the wavenumber, the phase and the initial amplitude of the constituent modes.
Generally, the linear stage is defined by the fact that Fourier modes evolve separately
(Drazin & Reid 2004; Chandrasekhar 2013). Because it is difficult to obtain the starting
point of the linear stage in reality, usually used within the framework of RM instability
is that the linear stage occurs as long as mode ksatisfies ak(t)k<α. This constant α
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
0.1 0.2 0.3 0.4
0
0.1
0.2
ηn
ηn
τnτn
0.3
0.4
52 18014811684
k1
k3
k2
Run2
mode k3
Run1
mode k1mode k2
Present
Haan-RM
Ofer-RM Impulsive
theory
IP-h
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
51 17914711583
k1
k3
k2
k3AP-h
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
54 18215011886
k1
k3
k2
k2AP-h
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
36 22818013284
k1
k3
k2
AP-h
(a)(b)
(c)(d)
Figure 5. The dimensionless amplitudes of three modes obtained from large-w0cases.
depends essentially on the accuracy required. As a result, it is generally accepted that the
necessary condition of the linear RM instability is a0
kk1 (Mügler & Gauthier 1998;
Collins & Jacobs 2002; Mikaelian 2003; Niederhaus & Jacobs 2003; Mariani et al. 2008;
Vandenboomgaerde et al. 2014; Mansoor et al. 2020). However, the linear stage still exists
for a very large initial a0
kkat the expense of a large reduction in the duration of the linear
stage (Rikanati et al. 2003; Dell, Stellingwerf & Abarzhi 2015;Zhaiet al. 2016;Dellet al.
2017). It should be noted that there are harmonics growing with time in these large initial
αcases but they are (almost) negligible. Therefore, this linear stage within large initial α
is actually a quasi-linear stage. As listed in table 2,thecriterion of dimensionless time,
i.e. τ∗
n(=kn|vR
kn|t), of mode knbetween the quasi-linear stage and the nonlinear stage is
evaluated from experiments. It is found that τ∗
nof the three initial modes are less than
the generally accepted criterion of dimensionless time of 0.7 for the single-mode RM
instability (Niederhaus & Jacobs 2003), indicating that the large initial a0
knknresults in a
very short quasi-linear stage and the mode competition advances the nonlinearity of the
multi-mode RM instability.
3.2. Linear and nonlinear theories
To quantitatively describe the 2-D multi-mode RM instability development, linear and
nonlinear theories based on the impulsive theory, the modal model and the interpolation
model have been established.
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RM instability on 2-D multi-mode interfaces
Case IP-s k3AP-s k2AP-s AP-s IP-h k3AP-h k2AP-h AP-h
τ∗
10.06 0.07 0.06 0.08 0.13 0.18 0.17 0.22
τ∗
20.64 0.32 0.32 0.34 0.36 0.40 0.36 0.52
τ∗
30.66 0.72 0.54 0.72 0.30 0.48 0.42 0.66
Table 2. The criterion of dimensionless time (τ∗
n) of mode knbetween the quasi-linear stage and the
nonlinear stage.
I. Impulsive theory. If each mode of a multi-mode interface satisfies |ak(t)k|1, the
whole interface evolves linearly. Previous studies (Sadot et al. 1998; Mikaelian 2005;Di
Stefano et al. 2015a,b; Liang et al. 2019)considered that the mode-competition effect is
negligible in quasi-linear stages. Therefore, for an initial light–heavy interface, the linear
amplitude growth rate (vl
k) of mode kcan be described by the impulsive theory (3.1), i.e.
vl
k=vR
k. For an initial heavy–light perturbed interface, the Richtmyer growth rate should
be modified as vR
k=(Zc+1)kA+Δva0
kcos(φk)/2 (Meyer & Blewett 1972).
When a0
kis comparable to its wavelength or/and the shock intensity is large, the
high-amplitude effect or/and the high-Mach-number effect will inhibit vR
k(Rikanati et al.
2003;Dellet al. 2015,2017;Guoet al. 2020). Here, the high-amplitude effect and the
high-Mach-number effect are considered independently.Then the modified Richtmyer
growth rate (vMR
k)is
vMR
k=βΔvtanh(RkvR
k/βΔv), (3.2)
where Rk(=1/[1 +(ka0
k/3)(4/3)]) is the reduction factor proposed by Dimonte &
Ramaprabhu (2010) to quantify the high-amplitude effect on mode k.Forsmall-w0cases,
Rk1=0.97, Rk2=0.93 and Rk3=0.91; for large-w0cases, Rk1=0.91, Rk2=0.88 and
Rk3=0.91. Parameter β(=1−Δv/vt) is the reduction factor proposed by Hurricane
et al. (2000) to quantify the high-Mach-number effect on all modes, and β=0.64 in all
cases.
II. Modal model. When the mode competition starts to play a role, the interface evolution
enters the early nonlinear stage. Haan (1991) first proposed a modal model applicable to
the 2-D RT instability, and then deduced the modal model for the 2-D RM instability with
zero acceleration (g=0) and assuming |ak(t)||a0
k|, i.e. ignoring the initial amplitude
terms, as
ak(t)=al
k(t)+1
2Ak
k
al
k(t)al
k (t)×1
2−ˆ
k·
k−1
2
k·
k,(3.3)
where k =k−k;k,kand k ∈R2;and ˆ
kis the unit vector k/k.Amplitude al
k(t)is the
linear amplitude of mode k. Taking the first derivative of (3.3) and using the post-shock
physical parameters, the ‘Haan-RM’ model can be obtained to calculate the early nonlinear
amplitude growth rate ven
k(t)of mode kas
ven
k(t)=vMR
k+A+k2
k
vMR
kvMR
k+k−
k<k
vMR
kvMR
k−kt,(3.4)
where k,kand k ∈R+. The first term on the right-hand of the Haan-RM model indicates
that the development of each mode of a multi-mode RM unstable interface is still strongly
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
influenced by its independent perturbation growth. At t≈0, the Haan-RM model reduces
to ven
k(t)≈vl
k=vMR
k, which indicates that the mode-competition effect is very weak and
can be ignored. The second term on the right-hand side is the mode-competition term, and
it is evident that as tincreases, the mode-competition effect plays a more important role in
influencing the RM instability. The first sum term in the mode-competition term represents
the generation of mode kfrom high-order modes and indicates the bubble-merging
process. The second sum term in the mode-competition term represents the generation
of mode kfrom the interaction of low-order modes, which is related to the bubble-spike
asymmetry and the total mixing rate decrement.
However, the initial amplitude terms cannot be ignored in deducing the modal model for
the whole process of the RM instability, especially when the initial amplitude of mode kis
large. Now, the modal model for the RM instability including the initial amplitude terms
is re-derived. Based on the modal model solved by Ofer et al. (1996)to second-order
accuracy with g=constant for the RT instability,
ak(t)=al
k(t)+1
2Ak
k
al
k(t)al
k+k(t)−1
2
k<k
al
k(t)al
k−k(t),(3.5)
we take the second derivative of (3.5) with time, and get
d2ak(t)
dt2=Agka0
k+1
2A2gk
k
[ka0
kal
k+k(t)+2k(k+k)a0
ka0
k+k
+(k+k)a0
k+kal
k(t)]−1
2
k<k
[ka0
kal
k−k(t)
+2k(k−k)a0
ka0
k+k+(k−k)a0
k−kal
k(t)].(3.6)
Similar to Richtmyer (1960), the constant gis replaced by an impulsive acceleration δtΔv
(δt=0whent=0andδt=1whent>0) and the post-shock physical parameters are
adopted. Through integrating (3.6) with time, ven
kfor the RM instability can be expressed
as a superposition of the linear amplitude growth rate vl
kand the weakly nonlinear
modification vwn
k(t):
ven
k(t)=vl
k+vwn
k(t), (3.7)
with
vl
k=vMR
k+1
2A+k
kvMR
kZca0
k+k+vMR
k+kZca0
k1+2k
k+1
−1
2
k<kvMR
kZca0
k−k+vMR
k−kZca0
k1+2k
k−1 (3.8)
and
vwn
k(t)=A+k
k
vMR
kvMR
k+k−1
2
k<k
vMR
kvMR
k−kt.(3.9)
Here, (3.7)–(3.9)arecalled the ‘Ofer-RM’ model. Different from vl
k=vMR
kindicated
by the Haan-RM model, vl
kin the Ofer-RM model is a superposition of vMR
kwith the
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RM instability on 2-D multi-mode interfaces
Case vMR
k1vl
k1vMR
k2vl
k2vMR
k3vl
k3vMR
k4vl
k4vMR
k5vl
k5vMR
k6vl
k6
IP-s 3.75.27.17.910.18.80−3.10−2.40−3.4
k3AP-s 3.73.37.25.1−10.3−11.30−0.10 2.60−3.7
k2AP-s 3.72.1−7.2−6.110.311.40−3.30 2.60−3.7
AP-s 3.74.1−7.1−9.2−10.2−9.50−0.10−2.60−3.6
IP-h 10.115.812.914.210.15.20−10.70−4.90−3.6
k3AP-h 10.412.213.25.2−10.4−15.90−1.80 5.10−3.8
k2AP-h 10.14.9−12.9−12.510.115.50−10.70 4.90−3.6
AP-h 10.38.8−13.4−21.5−10.3−5.20−1.70−4.90−3.6
Table 3. Comparison of the modified Richtmyer growth rate (vMR
kn) calculated by (3.2) with the linear
amplitude growth rate (vl
kn) calculated by (3.7). The unit for the growth rate is m s−1.
mode-competition term that is related to the initial amplitudes of constituent modes. In
other words, the mode-competition effect influences each mode perturbation growth in the
quasi-linear stage of the multi-mode RM instability, which is different from the previous
view that the mode-competition effect can be ignored in the quasi-linear stage (Sadot
et al. 1998; Mikaelian 2005; Di Stefano et al. 2015a,b; Liang et al. 2019). Besides, when
ven
k(t)=0, mode kis fully saturated. Here, additional rules introduced by Ofer et al. (1996)
for the enforced post-saturation treatment in calculating ven
k(t)are adopted: (i) no weakly
nonlinear modifications of a saturated mode to low-order modes and (ii) the phases of the
harmonics generated by the saturated modes are opposite.
The predictions from the Haan-RM model and the Ofer-RM model are calculated as
shown in figures 4 and 5.Forsmall-w0cases, because the initial amplitudes of three
modes are small, both models give reasonable predictions of the experimental results.
Differently, for large-w0cases, the Ofer-RM model provides a better prediction of the
amplitude growth than the Haan-RM model for some constituent modes, such as modes
k1and k3in the IP-h case, mode k1in the k2AP-h case and modes k1and k2in the AP-h
case. Note that the amplitude developments of mode k1in the IP-h case and mode k3in the
AP-h case deviate from predictions of the impulsive theory from the very beginning, which
verifies that the mode-competition effect plays an important role in the early evolution of
a multi-mode interface. In summary, the Ofer-RM model is more applicable to describing
the multi-mode RM instability behaviour in the early nonlinear stage, especially when the
initial amplitudes of constituent modes are large. Meanwhile, the present experimental
results prove that the linear growth of each mode amplitude is also influenced by the
mode-competition effect.
Basedon(3.8), vl
kconsidering the mode-competition effect is calculated and compared
with vMR
kwithout the mode-competition effect, as listed in table 3 for all cases. The
difference between vl
kand vMR
kis larger in large-w0cases than in small-w0cases.
According to the modal analysis (Haan 1991;Oferet al. 1996; Miles et al. 2004), the
coupling of constituent modes will generate new harmonics. In our experiments, k2and
k3are integral multiples of k1, and thus no modes with wavenumber lower than k1will
be generated (Miles et al. 2004). However, three new harmonics with higher order, i.e.
harmonics k4(=k1+k3), k5(=k2+k3)andk6(=k3+k3), are generated if only the first
generation of new harmonics is considered (Ofer et al. 1996). The values of vl
kfor the three
generated harmonics are listed in table 3. It is evident that the new generated harmonics
cannot be ignored in the multi-mode RM instability development in early stages.
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
III. Interpolation model. Although the Ofer-RM model generally gives a better
prediction than the Haan-RM model in large-w0cases, it still underestimates mode k2
development in the k3AP-s case and overestimates mode k2development in the AP-h case
when tis large. According to the Ofer-RM model, when t→∞,vl
kis neglected and ven
kis
proportional to tas long as vwn
kis non-zero. Actually, in the consideration of classical
single-mode RM instability, the late nonlinear amplitude growth rate (vln
k) of mode k
should be t−1decay (Hecht, Alon & Shvarts 1994; Alon et al. 1995; Mikaelian 1998)
because of the suppression of high-order (three orders or greater) harmonics generated by
mode kitself (Velikovich & Dimonte 1996; Zhang & Sohn 1997; Nishihara et al. 2010;
Velikovich, Herrmann & Abarzhi 2014). In the multi-mode RM instability counterparts,
the perturbation width growth in the fully turbulent stage is proportional to tθas a result
of bubble merging (Alon et al. 1994,1995). Although the value of θhas not been
unified, it should be much lower than 1.0, as reviewed by Zhou (2017a,b). Therefore,
the perturbation width growth rate of a multi-mode interface should be proportional to
tθ−1with −1<(θ−1)<0 in the fully turbulent stage. As a result, the Ofer-RM model
with second-order accuracy is not applicable to describing the late-time 2-D multi-mode
RM instability, and its scope should be extended by considering the suppression from
high-order harmonics.
An interpolation model proposed by Dimonte & Ramaprabhu (2010) (DR model) for
predicting 2-D single-mode amplitude growth covers the entire time domain from the
early to late nonlinear stages of the RM instability, and it has been well verified by several
independent experiments (Dimonte et al. 1996;Sadotet al. 1998; Niederhaus & Jacobs
2003; Jacobs & Krivets 2005) through considering diverse amplitude-to-wavelength ratios,
shock intensities and density ratios. In this work, the scope of the Ofer-RM model is
extended in combination with the DR model, and then vln
kcan be expressed as an average
of the bubble amplitude growth rate vln
kb(t)with the spike amplitude growth rate vln
ks(t)of
mode k:
vln
k(t)=1
2[vln
kb(t)+vln
ks(t)],(3.10)
with
vln
kb/ks(t)=ven
k(t)1+(1∓|A+|)|kven
k(t)t|
1+Cb/s|kven
k(t)t|+(1∓|A+|)Fb/s|kven
k(t)t|2,
Cb/s=4.5±|A+|+(2∓|A+|)(kak
0)
4,Fb/s=1±|A+|.
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(3.11)
Finally, (3.2), (3.7)and(3.10) constitute the ‘present’ model for describing the 2-D
multi-mode RM instability in this work. Notably, the self-similar law shows that the
perturbation width growth rate of a multi-mode interface in the fully turbulent stage has
atθ−1dependence and therefore approaches zero when t→∞. According to (3.10)and
(3.11), when t→∞,vln
k(t)also approaches zero but shows a t−1dependence, i.e. θ=0,
which violates the self-similar law. However, the t−1asymptotic dependence agrees with
the potential model (Alon et al. 1994,1995;Oronet al. 2001), the vortex model (Rikanati
et al. 1998)and recent experimental findings (Mansoor et al. 2020).
The predictions of the present model for our experiments are shown in figures 4
and 5, and a good agreement between them is achieved. Besides, data from literature
are extracted to validate the present model. First, the numerical results from figures 9
and 10 of Vandenboomgaerde et al. (2002)withbothM(=1.0962) and A+(=0.764)
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RM instability on 2-D multi-mode interfaces
t (μs)
an (mm)
an (μm)
0 100 200 300 400
–6
–4
–2
0
2
4
6
k1
k3
k3
k2
k1
k2
Figure 9
Vandenboomgaerde et al. (2002)
Figure 10
t (ns)
20 22 24 26 28 30
0
5
10
15
20
k3
k2
k1
Presentexp num Di Stefano et al. (2015)
Figure 5
(a)(b)
Figure 6. The amplitudes of modes obtained from (a) the numerical results in Vandenboomgaerde et al.
(2002)and(b) the experimental and numerical results in Di Stefano et al. (2015b). Coloured lines represent the
predictions calculated with the present model.
similar to those in our experiments are extracted, as shown in figure 6(a). The initial
interface consists of three modes: a0
k1=0.35 ×10−3,a0
k2=1.9055 ×10−3and a0
k3=
1.072 ×10−3m; k1=274.855 m−1,k2=3k1/7andk3=4k1/7. For the data in figure 9
of Vandenboomgaerde et al. (2002), φ1=φ2=φ3=0, while for the data in figure 10,
φ1=φ2=φ3=π. For these two cases, Rk1=0.97, Rk2=0.93 and Rk3=0.95; β=
0.80. It is found that the predictions of the present model agree well with all the numerical
results (Vandenboomgaerde et al. 2002). Further, the experimental and numerical results
(Di Stefano et al. 2015b) with a much larger M(=8) and a smaller A+(=1/3) compared
with our experimental conditions are predicted by the present model. The initial interface
in the literature (Di Stefano et al. 2015b) consists of two modes: a0
k1=5×10−6m,
a0
k2=0.5a0
k1;k1=6.28 ×104m−1,k2=2k1;φ1=0, φ2=π;Rk1=0.91, Rk2=0.91;
β=0.35. The amplitude growths of the two constituent modes and the harmonic k3
generated by the coupling of modes k1and k2are extracted from figure 5 of Di Stefano
et al. (2015b), as shown in figure 6(b). It is found that not only the constituent mode
amplitudes, but also the generated new harmonic amplitude are well predicted by the
present model. Overall, the present model established in this work by considering both the
mode-competition effect and the high-order harmonics effect is applicable to the nonlinear
RM instability before the transition.
3.3. The perturbation width growth
The memory of the perturbation width growth of a multi-mode interface on its initial
spectrum is crucial to RM instability research. The perturbation width w(t)of a
multi-mode interface is defined as the streamwise distance from the upstream point
(UP) to the downstream point (DP) of an interface, as shown in figure 2(b). The
variations of the perturbation width measured from the schlieren images for small-and
large-w0cases are shown in figures 7(a)and8(a), respectively. The time is normalized
as τw=0.5k1vMR
wt,wherevMR
w=3
1vMR
kn[cos(knxDP)−cos(knxUP )],withxUP and xDP
being the xcoordinates of UP and DP, respectively. The half of the perturbation width
is scaled as ηw=0.5k1[w(t)−Zcw0]. To compare the perturbation width between the
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
0.5 1.0 1.5 2.00
0.5
1.0
1.5
Impulsive
theory
DR model
0.5 1.0 1.5 2.0
0
0.5
1.0
1.5
k3AP-s
k2AP-s
IP-s
exp Sum
Bubble Bubble
Spike
Spike
AP-s
0.5 1.0 1.5 2.0
0
1.5
2.0
2.5
3.0
0.5
1.0
0.5
0
1.5
1.0
0.5 1.0 1.5 2.0
0
1.5
2.0
0.5
1.0
0.5
0
1.5
1.0
ηwζw
ηb/sζb/s
τwτw
(a)(b)
(c)(d)
Figure 7. Comparison of the perturbation width (a), growth rate of the perturbation width (b), widths of
bubble and spike (c)and width growth rates of bubble and spike (d)insmall-w0cases. Black dashed lines
and dash-dotted lines represent the single-mode linear and nonlinear amplitude growth rates calculated by the
impulsive theory (Richtmyer 1960) and DR model (Dimonte & Ramaprabhu 2010), respectively. Coloured solid
lines represent the predictions from the sum model calculated with (3.12), and similarly hereinafter.
multi-mode interface and the classical single-mode interface, the predictions from the
impulsive theory and the DR model are calculated, as shown in figures 7(a)and8(a),
respectively. It is evident that the perturbation width growth of a multi-mode interface in
the early stage (τw≤0.5) is close to or even larger than the single-mode linear growth,
which indicates that the mode-competition effect may enhance the initial growth of the
multi-mode perturbation. Later, the perturbation width growth in all cases is smaller than
the single-mode linear growth when τw>0.7 and smaller than the single-mode nonlinear
growth when τw>1.2. Therefore, the mode-competition effect suppresses the multi-mode
interface development at late stages because it enhances local mixing and reduces the
global mixing. Besides, the w(t)growth curves in all cases deviate from each other, which
indicates that the initial spectrum influences the 2-D multi-mode RM instability until the
late nonlinear stage.
To more clearly distinguish the differences of the perturbation growth with diverse initial
conditions, through the direct differentiation of the experimental data, the perturbation
widthgrowthrate ˙w(t)of a multi-mode interface is acquired, and compared with the
single-mode linear and nonlinear amplitude growth rates, as shown in figures 7(b)and8(b)
for all cases. The perturbation width growth rate is scaled as ζw=˙ws(t)/vMR
w. Therefore,
the dimensionless single-mode linear growth rate is ζw=1. In small-w0cases, ˙w(t)
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RM instability on 2-D multi-mode interfaces
in IP-s and AP-s cases is smaller than that of the single-mode growth rate during the
whole process. The value of ˙w(t)in k3AP-s and k2AP-s cases is larger than that of the
single-mode counterpart in early times (τw≤0.5), but smaller than that of the single-mode
counterpart in late stages. In large-w0cases, differently, ˙w(t)in IP-h and k2AP-h cases is
smaller than that of the single-mode growth rate during the whole process. The value
of ˙w(t)in k3AP-h and AP-h cases is larger than that of the single-mode counterpart
in early times (τw≤0.5), but smaller than that of the single-mode counterpart in late
stages. Therefore, both the phase and amplitude of the constituent modes influence the
mode-competition effect on the perturbation width growth of a multi-mode interface.
Besides, the growth rate of a multi-mode interface with all modes in-phase is smaller than
the counterpart in the classical single-mode case and other multi-mode cases. In addition,
under specific conditions, such as in k3AP-s and k3AP-h cases, the mode-competition
effect promotes the perturbation width growth in early stages.
Compared with the reference moving with the post-shock flow velocity, the width
growths of the bubble (wb(t)) and the spike (ws(t)) are separated, as shown in figures 7(c)
and 8(c)forsmall-and large-w0cases, respectively. The bubble/spike width is scaled
as ηb/s=k1[wb/s(t)−Zcw0
0b/0s], in which w0b/0sis the initial bubble/spike width, as
listed in table 1. The bubble width of the multi-mode interface in all cases is lower
than that of the single-mode counterpart in late stages (τw≤0.5), which indicates that
the mode-competition effect advances the saturation of the bubble evolution in the 2-D
multi-mode RM instability. The nonlinear behaviour of the spike of a multi-mode interface
is greatly influenced by the initial spectra. In small-w0cases, ws(t)in IP-s and AP-s
cases are lower while ws(t)in k2AP-s and k3AP-s cases are higher than those of the
single-mode counterpart, which means that the phase of the constituent mode influences
the spike behaviour in the 2-D multi-mode RM instability. In large-w0cases, ws(t)in
k3AP-h, k2AP-h and AP-h cases are close to those of the single-mode counterpart, which
indicates that the initial amplitude of the constituent mode influences the spike behaviour.
The width growth rates of the bubble ˙wb(t)and the spike ˙ws(t)are calculated, as shown in
figures 7(d)and8(d)forsmall-and large-w0cases, respectively. The bubble/spike width
growth rate is scaled as ζb/s=2˙wb/s(t)/vMR
w. It is shown that the bubble width growth
rate of a multi-mode interface is generally lower than that of the single-mode counterpart.
The spike width growth rate reflects whether the mode-competition effect promotes or
suppresses the multi-mode RM instability in early stages.
Since the angle between the incident shock and UP (DP) is zero, no vorticity is deposited
at UP (DP). Therefore, UP and DP on the interface remain single-valued until the late
nonlinear stage before the transition. As a result, the perturbation width growth of a
multi-mode interface can be predicted by superimposing the perturbation growths of
initially constituent modes and the generated harmonics at xUP and xDP. In the present
coordinate system, the time-varying w(t)growth is superimposed by wb(t)and ws(t):
w(t)=wb(t)+ws(t), (3.12)
with
wb(t)=Zcw0b+
nt
0
vln
kncos(knxDP)dt,(3.13)
ws(t)=Zcw0s−
nt
0
vln
kncos(knxUP)dt.(3.14)
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
Case IP-s k3AP-s k2AP-s AP-s IP-h k3AP-h k2AP-h AP-h
˙w0
exp 30 ±232±239±234±237±236±249±239±2
˙w0
theo 28 29 38 32 34 35 52 41
Table 4. Comparison of the experimental initial perturbation width growth rate (˙wexp ) of the multi-mode
interface wit h the theoretical counter part ( ˙wtheo ) calculated with (3.15).
In this work, (3.12)–(3.14) constitute the ‘sum’ model which is used to calculate the
perturbation width by superimposing the amplitudes of three constituent modes and three
generated harmonics. By linear fitting of the perturbation width of a multi-mode interface
before 120 μs in all cases, the experimental initial perturbation width growth rate ( ˙w0
exp)
can be obtained, as listed in table 4. The theoretical initial perturbation width growth rate
of the interface ( ˙w0
theo) can be evaluated by superimposing the amplitude growth rates of
three constituent modes and three generated harmonics:
˙w0
theo =
6
n=1
vln
kn[cos(knxDP)−cos(knxUP)].(3.15)
The values of ˙w0
theo in all cases are listed in table 4 and agree well with the experimental
results. Then, the predictions of the sum model for w(t),wb(t)and ws(t)are shown
in figures 7 and 8, and agree well with the experimental counterparts. Meanwhile, the
differentiation of the sum model is calculated and compared with the experimental ˙w(t),
˙wb(t)and ˙ws(t), and a good agreement is also achieved between them. Note that there are
several break points, for example in k3AP-h and AP-h cases, when the sum model is used
to predict the growth rates, which is ascribed to the enforced post-saturation treatment
adopted when ven
kis calculated. After the incident shock wave impacts the interface,
the transverse waves between the transmitted shock and reflected shock interact with the
interface, especially when the initial interface perturbation is prominent. Therefore, the
non-uniform flow influences the linear interface growth (Guo et al. 2020), resulting in the
perturbation width growth varying around the sum model prediction at an early regime.
To further validate the sum model, the experimental results with M(=1.3) and A+
(=0.67) similar to our experiments extracted from figure 4(a)inSadotet al. (1998)
are adopted as shown in figure 9(a). The initial interface is a two-bubble interface,
which is dominated by the first five order modes: a0
k1=1.49 ×10−3,a0
k2=1.06 ×
10−3,a0
k3=0.40 ×10−3,a0
k4=0.21 ×10−3,a0
k5=0.10 ×10−3m; k1=180 m−1,ki=
ik1with i=2–5; φ1=π,φ2=φ3=φ4=φ5=0; Rk1=0.92, Rk2=0.89, Rk3=0.93,
Rk4=0.95 and Rk5=0.97; β=0.49. The amplitudes of the five constituent modes and
five generated harmonics are calculated by the present model, and then superimposed
according to the sum model. One can see that the predictions of the sum model agree well
with both the large bubble width and the small bubble width. Besides, the experimental
results with M(=1.2) and A+(=0.6) similar to our experiments extracted from
figure 9 in Luo et al. (2019) are shown in figure 9(b). For the CS-1 case (ICS-2 case),
the initial interface is a spike-dominated (bubble-dominated) chevron-shaped interface,
which is dominated by the first five order modes: a0
k1=1.08 ×10−3,a0
k2=0.81 ×10−3,
a0
k3=0.48 ×10−3,a0
k4=0.20 ×10−3,a0
k5=0.04 ×10−3m; k1=52 m−1,ki=ik1with
i=2–5; for the CS-1 case, φ1=φ3=φ5=0, φ2=φ4=π; for the ICS-2 case, φ1=
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RM instability on 2-D multi-mode interfaces
0.5 1.0 1.5 2.52.00
0.5
1.0
1.5
Impulsive
theory
DR model
0.5 1.0 1.5 2.52.0
0
0.5
1.0
1.5
k3AP-h
k2AP-h
IP-s
exp Sum
Bubble Bubble
Spike
Spike
AP-h
0.5 1.0 1.5 2.52.0
0
1.5
2.0
2.5
3.0
0.5
1.0
0.5
0
1.5
1.0
0.5 1.0 1.5 2.52.0
0
1.5
2.0
0.5
1.0
0.5
0
1.5
1.0
ηwζw
ηb/sζb/s
τwτw
(a)(b)
(c)(d)
Figure 8. Comparison of the perturbation width (a), growth rate of the perturbation width (b), widths of
bubble and spike (c)and width growth rates of bubble and spike (d)inlarge-w0cases.
φ3=φ5=π,φ2=φ4=0. For these two cases, Rk1=0.98, Rk2=0.97, Rk3=0.98,
Rk4=0.99 and Rk5=1.0; β=0.67. The amplitudes of the five constituent modes and
five generated harmonics are calculated and the sum model well predicts the experimental
results. Moreover, the numerical results with M(=5) and A+(=0.95) much larger
than in our experiments extracted from figure 4 in Pandian et al. (2017) are shown
in figure 9(c). The initial interface consists of two modes: a0
k1=a0
k2=1.1×10−3m;
k1=1.885 ×103m−1,k2=2k1;φ1=π,φ2=0; Rk1=0.59, Rk2=0.42; β=0.26.
The amplitudes of the two constituent modes and two generated harmonics are calculated
and the sum model also well predicts the numerical results. All these agreements achieved
demonstrate the generality of the sum model.
4. Conclusions
In this work, a 2-D complex multi-mode interface constituted of various modes is first
formed by the soap-film technique, and then elaborate shock-tube experiments on the
developments of eight kinds of air–SF6multi-mode interface are performed. Based
on these well-controlled experiments and several theories in the literature, a general
nonlinear theory is established for predicting multi-mode evolution and mixing, and
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Y. Liang, L. Liu, Z. Zhai, J. Ding, T. Si and X. Luo
0 0.2 0.4 0.8
t (ms) k3vl
wt
k3(w(t)-Zcw0)
0.6 1.0 1.2 1.4
–4
0
4
12
8
0.5 1.0 1.5 2.52.0
0
1
3
2
4
exp Sum
Bubble height (mm)
Sadot et al. (1998) Luo et al. (2019)
CS-1
ICS-2
CS-1 (exp)
ICS-2 (exp)
CS-1 (sum)
ICS-2 (sum)
Figure 4(a)
Pandian et al. (2017)
Figure 4
Small bubble
Large bubble
num
Bubble
num
Sum
0.20.1 0.3 0.4 0.5 0.6
0.1
0
0.2
0.3
2πk1w
2πk1Δvt
(a)(b)
(c)
Figure 9. The perturbation width of the multi-mode interface obtained from (a) the experiments and
simulations in Sadot et al. (1998), (b) the experiments in Luo et al. (2019)and (c) the simulations in Pandian
et al. (2017).
finally a quantitative relation between the initial conditions and the perturbation growth
is constructed.
From the schlieren images of the shocked multi-mode interface, it is found that the
phases of the constituent modes greatly affect the initial interface shape and the later
interface evolution. It is also found that the transition from linear to nonlinear may
occur earlier when the initial interface amplitude is larger. By considering different
wavenumbers, initial amplitudes and phases of constituent modes, the dependence of the
perturbation growth on initial spectra is highlighted.
The captured interface morphology is distinct such that the interface contours in all
cases can be easily extracted by an image processing program. Subsequently, spectrum
analysis is performed on the interface contour before the interface becomes multi-valued,
and amplitudes of constituent modes are then acquired. It is first proved that the
mode-competition effect influences the amplitude growth of each mode from the very
beginning (quasi-linear stage), especially when the initial amplitudes of constituent modes
are large. It is interesting that the mode competition starts to play a role at this quasi-linear
stage although the growing of harmonics is so small as to be negligible. Therefore our
findings differ from previous views. The mode-competition effect is closely associated
with the initial spectra.
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RM instability on 2-D multi-mode interfaces
A nonlinear theory is constructed by considering both the mode-competition effect
and the high-order harmonics effect to predict the amplitude growth of the modes. The
new theory has been validated by our experiments and data in the literature with the
consideration of diverse constituent modes, and a wide range of Mach number and Atwood
number. Further, the nonlinear theory is extended based on the superposition principle to
predict the growths of the total perturbation width and spike/bubble width, and there is
a satisfactory agreement between the predictions and the experimental results. It can be
concluded that the evolution of the shocked multi-mode interface has an evident memory
of the initial conditions from quasi-linear to late nonlinear stages.
The RM instability at the fully turbulent state has been a focus of attention recently.
We will combine the soap-film technique with a time-resolved particle image velocimetry
system to investigate the multi-mode RM instability induced by one shock or two shocks
in the near future. Besides, we look forward to examining the models established in this
work with experiments involving very high Atwood numbers.
Acknowledgements. The authors appreciate the valuable suggestions of the reviewers.
Funding. This work was supported by the Natural Science Foundation of China (nos. 91952205, 12022201,
11772329, 11625211 and 11621202) and Tamkeen under NYU Abu Dhabi Research Institute grant CG002.
Declaration of interests. The authors report no conflict of interest.
Author ORCIDs.
Yu Li a ng https://orcid.org/0000-0002-3254-7073;
Zhigang Zhai https://orcid.org/0000-0002-0094-5210;
Juchun Ding https://orcid.org/0000-0001-6578-1694;
Ting Si https://orcid.org/0000-0001-9071-8646;
Xisheng Luo https://orcid.org/0000-0002-4303- 8290.
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