HAL Id: hal-02080772
https://hal-amu.archives-ouvertes.fr/hal-02080772
Submitted on 27 Mar 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Nonlinear growth of the converging Richtmyer-Meshkov
instability in a conventional shock tube
Marc Vandenboomgaerde, Pascal Rouzier, Denis Souand, Laurent Biamino,
Georges Jourdan, Lazhar Houas, Christian Mariani
To cite this version:
Marc Vandenboomgaerde, Pascal Rouzier, Denis Souand, Laurent Biamino, Georges Jourdan, et
al.. Nonlinear growth of the converging Richtmyer-Meshkov instability in a conventional shock tube.
Physical Review Fluids, American Physical Society, 2018, 3 (1), �10.1103/PhysRevFluids.3.014001�.
�hal-02080772�
Nonlinear growth of the converging Richtmyer-Meshkov instability in a conventional
shock tube.
Marc Vandenboomgaerde, Pascal Rouzier, and Denis Souffland
CEA, DAM, DIF, F-91297 Arpajon, France
Laurent Biamino, Geor ges Jourdan, Lazhar Houas, and Christian Mariani
IUSTI, Aix Marseille Universit´e, 5 rue Enrico Fermi 13013 Marseille
(Dated: June 22, 2017)
I. INTRODUCTION
Converging shock waves are involved in several fieds of high energy physics such as astro physics and inertial
confinement fusion (ICF). As their strength increases with time, these focusing shock waves can efficiently heat and
compress matter. In the wake of these shock waves, the matter implodes and further compresses. However, the
efficiency of this compression can be tempered if the symmetry of the flow is broken. The Richtmyer-Meshkov (RM)
hydrodynamics instabilities [1, 2] ca n disrupt this symmetry. It is induced by the pass age of a shock wave through
a density gradient. For example, in ICF experiments, such gradient occurs at the interface between the deuterium-
tritium (DT) co re, and the encapsulating spherical plastic shell. A specific sequence of converging shock waves is
launched to compress the DT fuel. If defects are present at the DT/plastic interface, the RM instability will be
triggered. The amplitude of the defect increases, and it could lead to a mixing between the core and the plas tic. A
mixing zone within the DT fuel would damp the nuclear reactions within the core. The control o f the RM instabilities
is crucial to ICF success, and reliable models which predict the growth of the RM instabilities in convergent geometries
are needed.
In the past few decades, the RM insta bility was often studied in planar geo metry. Numerous models have been
derived in the planar geometry to describe the linear and the nonlinear regimes of the instability for compressible
and incompressible flows. Their r e liability have been assessed by comparison with lots of experiments which were
often performed in planar shock tubes . In convergent geometry, the growth of the RM insta bility can b e enhanced.
In order to test the corr esponding theories and numerical simulations, experimental results with c ontrolled initial
conditions are needed. Such experiments with perfect gases are scarce [3]. A few more data are available with
laser experiments [4–7]. These expe riments are extremely elaborated [8, 9] and required heavy facilities [10–12].
Furthermore, high energy physics must be taken into account in order to ex plain the experimental data [13]. Some
models have been derived to study the RM instability in convergent geometry for the canonical problem of a single
mode sinusoidal perturbation between two perfect gases [16, 17, 19, 20]. They do not take into a ccount neither
plasma, radiation, thermal transport or ablation physics. Nonetheless, they are an essential s tep in order to assess
the distinction between the incompressible convergence [14, 15], the acceleration or deceleration of the interface which
leads to the Rayleigh-Taylor (RT) insta bility [21, 22], and the compressibility. Furthermore, nonlinear theories a re
also needed [23], and must be tested in the case of a convergent geometry. In the following, we present the results
of ex periments about the canonica l cylindrical RM instability which starts from the linear regime to the reach the
nonlinear stage.
Since 201 4 [24, 25], we have built a new shock tube facility which generates cylindrical RM instabilities. We study
this instability at the interface between perfect gase s (SF
6
and air) which are seperated by a single-mode sinusoidal
interface. This shock tube is obtained from a conventional planar shock tube by using the ga s lens technique [26].
This technique turns a planar shock wave into a convergent one through the impedance mismatch a t a first shaped
interface. The resulting shock wave is then guided toward a second interface where the RM instability is studied.
The interfaces between gases are materialized by thin nitrocellulosic membranes which are held on shaped stereo -
lithographed grids. In the following, we present and analyze the experimental results which are obtained with such a
facility.
The pap er is orga nize d as fo llows. In Sec. I I , we present the e xperimental set-up. In Sec. III, the 1D flows
are dis c ussed. This leads us to sp ecify the actual composition of the involved gases. In Sec IV, we compare the
exp erimental and numerical shapes of the interface where the RM instability takes place. We explain how the imprint
due to the grid bars on the initial shape of the interface must be taken into account. The different features of the
exp erimental data a re specified. In Sec. V, we focus on the growth of the instability. Experimental and numerical
peak-to-valley amplitudes are co mpared. A good a greement is obtained. A theoretica l analysis of these cylindrical
RM instabilities is also presented. We show that the interface is not purely c oasting after the shock passage, and it
undergoes a slight deceleration before being shocked again (re-shock). As a consequence , the interface is subject to a
2
Rayleigh-Taylor instability which initial c onditions are determined by the Richtmyer-Meshkov instability. Finally, we
will demonstrate that the growth of the instability does not saturate in the nonlinear regime as it would have in the
planar case.
II. EXPERIMENTAL SET-UP
The experimental part of this investigation was carried out using a conventio nal shock tube in horizontal orientation,
which has a total length o f 3 .75 m and a square inner cross section of 80 mm by 80 mm [27]. At the sho ck tube
end-wall, a spe c ific wedge test section was designed, manufactured and installed. It has a half apex angle, θ
0
, of 15
degrees, and ac c ommodates a three-fluid three-zone system: a test cell of a heavy gas (SF
6
) enclosed by light gas
(air) on each side as shown in Fig.1. The shock tube is initially at the atmospheric pressure. The initial expected
densities are ρ
air
= 1.204 kg/m
3
, and ρ
SF 6
= 6.073 kg/m
3
. The corresponding adiabatic exponents are γ
air
= 1.4,
and γ
SF 6
= 1.09. The first air/SF
6
interface forms the gas lens which converts a planar shock wave into a cylindrical
one. The same configuration (air/SF
6
fast-slow case) successfully validated in a previous work [24] has been kept: the
incident planar shock wave propagates with a Mach number o f 1.15 in air, and refracts through an elliptic interface
which polar equation writes as:
r(θ) = r(0)
1 − e
1 − e cos(θ)
(1)
where e = W
t
/W
i
represents the ratio of the transmitted and incident shock wave velocities, and r(0) =0.159 m is the
location of the interface from the apex. The materialization of the gas lens interface was obtained by a 0.5-µm-thick
double layer of nitrocellulose membranes combined with a grid. This grid is obtained by stereo-lithography [REFS].
The second interface (SF
6
/air), located at 100 mm from the apex, presents a s ingle mode perturbation with the
following polar equation:
r(θ) = 0.1 − 0.0015
1 − cos
2πθ
θ
0
(2)
with θ
0
= arctan
0.04
0.15
. This second interface is materialized as the first one. The firs t and second interfaces are hold
by a 8×9 .5 mm
2
mesh elliptic grid, and 9 .5×9.5 mm
2
mesh sinusoidal grid, respectively. For recording the evolving
flow pattern in the convergent test section, a Z-type Schlieren system is coupled with a P hotron Fastcam SA1 high
sp e e d digital c amera. In addition to the optical diagnostic, pressure transducer s recorded pressure histories during
each experiment : four of them are located in the corner of the experimental chamber as shown in Fig.1. In the
following, we will disc us s results o bta ined from the Schlieren pictures (Fig. 2). The shock wave moves from left to
right, and propagates through the air/SF
6
/air three zones. At t = −10 µs, the pla nar shock wave is clearly visible
on the left side in the vic inity of the gas lens. When the incident shock wave c ollides with the air/SF
6
interface,
it bifurcates into a transmitted converging cy lindrical shock wave which moves in the central zone from t = 65 µs
to t = 365 µs. At t = 365 µs, the converging shock wave impacts the sinusoidally perturbed SF
6
/air interface, a nd
the subsequent pictures (from t = 440 µs to t = 815 µs) show the different stages of the converging RM instability.
As the shock wave hits the interfaces, the nitrocellulose membranes are broken into small pieces. These remnants
stay in the vicinity of the interfaces, and a re seen as a foamy opaque zone in the Schlieren pictures. At the second
interface, the converging shock refracts fr om the heavy fluid (SF
6
) to the light one (air). Thus, a phase inversion of
the perturbation at the interface occurs. We can also note the geometrical distortion of the shock during its refraction
at t = 440 µs. It later stabilizes and recovers its cylindrical shape. The horizontal waves behind the moving interface
are the consequence of the support grid. Reaching the apex approximatively at t = 590 µs, the converging shock wave
is reflected back, and expands through the evolving SF
6
/air interface at t = 81 5 µs.
III. MONODIMENSIONAL FLOW
In the following, superscripts
(1)
and
(2)
refer to interfaces 1 (the gas lens) and 2 (location of the RM instability),
respectively. When waves are considered, subscripts
r
and
t
mean “reflected” and “transmitted”. When the geometry
is considered, the symbols r and t stand for the distance from the apex of the shock tube, and the time, resp e c tively.
The origin of times corresponds to the moment when the planar shock wave hits the interface 1.
3
SF
Air Air
FIG. 1: Scheme of the experimental device adapted on the conventional shock tube, and locations of the pressure gauges.
A. Wave velocitie s
In this section, we present the theoretical, and expec ted wave diagram for our exper iments.
Let us recall that the gas lens consists in an air/SF
6
interface. Its purpose is to morph the plana r incident shock wave
into a cylindrical transmitted one. The Mach number of the incident shock wave, M
(1)
i
, is equal to 1.15 in air. The
theoretical initial veloc ity of the trans mitted shock wave in SF
6
is equal to W
(1)
t
= 165.5 m/s (M
(1)
t
= 1.226). The
cylindrical shock wave is plainly formed at r = 0.1552 m. In order to predict the trajectory, and the strengthening of
the imploding shock wave, the Whitham’s appr oximate geometrical theory [28] is used. This theory gives the Eq. (3)
between the Mach number of the converging shock wave, M, and its area, A:
M dM
M
2
− 1
λ(M) +
dA
A
= 0 (3)
where λ(M) =
1 +
2
γ+1
1−µ
2
µ
1 + 2µ +
1
M
2
, with µ
2
=
(γ−1)M
2
+2
2γM
2
−γ+1
. As A is a known function of r, M(r) or M (t)
can be computed from Eq. (3). The second interface is located at r = 0.1 m. The Eq. (3) gives a theo retical velocity
for the shock wave before its impact on the second interface equal to W
(1)
t
(r = 0.1) = 171 m/s (M
(1)
t
= 1.26 8). As the
velocity of the shock wave in SF
6
goes from 165.5 m/s to 171 m/s between interfaces 1 and 2, its acceleration is barely
noticeable. Once this shock wave reaches the second interface, a shock wave is transmitted in air, and a rarefaction
wave is reflected in SF
6
. The velocity of the transmitted shock wave is initially equal to W
(2)
t
(r = 0.1) = 3 96 m/s
(M
(2)
t
= 1.154). The head of the rarefaction wave travels back at a velocity equal to W
(2)
r
(r = 0.1) = −75.86 m/s.
B. Analysis of the experimental (r,t) diagrams
In this section, we explain the discr e pancies between the expected wave diagram, and the a c tua l experimental
results.
The e xperimental data are obtained from two shots which are labeled #961 and #962. The initial conditions of
these shots (Mach number and gaz composition) were planned to be identical. As far as the shock transit in SF
6
is c oncerned, the two shots give redundant data. The velocity is quasi-cons tant, and is equal to 168.8 m/s and
166.7 m/s for shots #961 and #962, respectively (Figs. 3-a and b). These values agree with the theoretical velocity
which starts from 165.5 m/s at the interface 1 to reach 171 m/s at the interface 2.
A clear disagreement between experiments and theory oc curs about the reflected wave at the second interface. The
exp erimental measurements show a reflected shock wave which velocity is e qual to −90.6 m/s and −88 m/s for
shots #961 and #962, respectively. Yet the theory predicts a rarefaction wave. This discrepancy is explained by
the grid bars. They ge nerate the reflected shock wave (see Appendix A). The trajectory of this shock wave ca n be
theoretically estimated. It is in reaso nable agreement with the experimental data: at t = 0.6 ms, its radius is 122 mm
4
FIG. 2: Sequence of schlieren pictures (from run #961) showing the evolution of t he converging Richtmyer-Meshkov instability
at the SF
6
/air interface materialized by 2 layers of 0.5-µm-thick nitrocellulose membrane recovered in sandwich on a grid
support (9.5×9.5 mm
2
).
in the experiment, and the theory predicts 119 mm.
We now focus on the transmitted shock wave in the apex section. The shock wave velocity, W
(2)
t
, and the travel
time of the shock wave b e tween interface 2 and the apex, ∆t
focus
, for the two experimental cases are compared with
the theoretical values in Table I. For the shot #961, the relative discrepancies between experimental and theoretical
data are within the error bar of the experiments. Therefore, we will consider that the gases are pure for shot #961.
For the shot #962, the velocity of the transmitted shock wave is slower than ex pected. The relative discrepancy
with respect to the theoretica l velocity is about 16%. In order to explain this value, two reasons ca n be put forward.
