J.
Fluid
Nech.
(1987),
z,o1.
181,
pp.
41-76
Printed
in
Great
Britain
41
Interaction
of
weak shock waves with cylindrical
and spherical gas inhomogeneities
By
J.-F.
HAASt
AND
B.
STURTEVANT
Graduate Aeronautical Laboratories, California Institute
of
Technology,
Pasadena, CA 91
125,
USA
(Received
26
February 1986 and
in
revised form 10 December
1986)
The interaction of
a
plane weak shock wave with
a
single discrete gaseous inhomo-
geneity is studied
as
a model of the mechanisms by which finite-amplitude waves in
random media generate turbulence and intensify mixing. The experiments are treated
as
an example of the shock-induced Rayleigh-Taylor instability,
or
Richtmyer-
Meshkov instability, with large initial distortions of the gas interfaces. The inhomo-
geneities are made by filling large soap bubbles and cylindrical refraction cells
(5
cm
diameter) whose walls are thin plastic membranes with gases both lighter and heavier
than the ambient air in
a
square
(8.9
cm side shock-tube text section. The wavefront
geometry and the deformation of the gas volume are visualized by shadowgraph
photography. Wave configurations predicted by geometrical acoustics, including the
effects of refraction, reflection and diffraction, are compared to the observations.
Departures from the predictions of acoustic theory are discussed in terms of
gasdynamic nonlinearity. The pressure field on the axis of symmetry downstream of
the inhomogeneity is measured by piezoelectric pressure transducers. In the case of
a cylindrical
or
spherical volume filled with heavy low-sound-speed gas the wave
which passes through the interior focuses just behind the cylinder. On the other hand,
the wave which passes through the light, high-sound-speed volume strongly diverges.
Visualization of the wavefronts reflected from and diffracted around the inhomo-
geneities exhibit many features known in optical and acoustic scattering. Rayleigh-
Taylor instability induced by shock acceleration deforms the initially circular
cross-section
of
the volume. In the case of the high-sound-speed sphere,
a
strong
vortex ring forms and separates from the main volume
of
gas. Measurements of the
wave and gas-interface velocities are compared to values calculated for one-
dimensional interactions and
for
a
simple model of shock-induced Rayleigh-Taylor
instability. The circulation and Reynolds number of the vortical structures are
calculated from the measured velocities by modeling a piston vortex generator. The
results of the flow visualization are also compared with contemporary numerical
simulations.
1.
Introduction
The interaction of shock waves with fluid non-uniformities modifies the geometry
and amplitude of the wave fronts by reflection, refraction, diffraction and scattering,
and modifies the morphology of the inhomogeneities by fluid deformation, vorticity
and entropy production, and transport. The interaction of shocks with non-uniform
t
Present address
:
Commissariat
a
I’Energie Atomique, Centre d’Etudes de Limeil-Valenton,
BP27,94190 Villeneuve Saint Georges, France.
42
J.-F.
Haas and
B.
Sturtevant
media occurs often in science and technology, for example, sonic boom propagation
through the Earth’s turbulent atmosphere (Ffowcs Williams
&
Howe
1973)
shock
boundary-layer interaction on transonic airfoils (Anyiwo
&
Bushnelll982) and shock
wave interactions with unstable interfaces between thermonuclear fuel and outer
shell material of laser fusion pellets (Andronov
et
al.
1979).
Clearly, the description
of such interactions
is
complicated and simple analytical models are difficult to
formulate. As
a
consequence, it
is
necessary to carry out exploratory experiments
to elucidate the important processes involved. The simple gas inhomogeneities
considered in the present study can
be
thought of
as
the building blocks of more
complicated inhomogeneous media. We limit our attention to discrete inhomo-
geneities because the flow visualization
is
especially graphic.
It
is easy to arrange
very large index-of-refraction variations in acoustic media; in the present experi-
ments, with a weak shock wave propagating from atmospheric air into helium or
Refrigerant 22 (R22), the acoustic index of refraction takes on the values
0.35
and
1.88,
while the ratios of acoustic impedances are
0.41
and
1.65,
respectively.
Therefore, the inhomogeneities are very strong acoustic lenses, and focal lengths are
of the order of the diameter of the inhomogeneities.
1.1.
Wave eflects
In the interaction of a shock wave with
a
spherical
or
cylindrical volume of gas of
different density and/or sound speed, wave reflection refraction, diffraction and
focusing
are
important. The refraction of shock waves
at
plane gas interfaces has been
examined by Jahn
(1956),
Abd-el-Fattah, Henderson
&
Lozzi
(1976),
Abd-el-Fattah
&
Henderson
1978a,
b)
and Catherasoo
&
Sturtevant
(1983).
Depending on the angle
of incidence of the shock wave onto the interface and on the strength of the shock,
the refraction can be regular (incident, reflected and refracted waves intersect the
interface at the same point) or irregular (the refracted shock intersects the interface
ahead of the incident shock). Furthermore, in the case of the so-called slow-fast
interface, for which the gas downstream of the interface has a higher sound speed
than the gas upstream, the transmitted wave can run ahead of the first disturbance
in the slow medium, leading to a ‘precursor’ configuration. The refraction of
a
shock
from cylindrical
or
spherical interfaces covers the complete range of angles of
incidence and, therefore, of all types of refraction. In view of the complexity of the
plane refraction problem,
it
is not surprising that little attention has been given to
the interaction of plane shock waves with curved
gas
interfaces. In
a
related problem,
Markstein
(1957a,
b)
and Rudinger
(1958)
studied the interaction of shock waves
with curved flame fronts, and
so
considered the curved slow-fast case. Precursors and
lateral shocks associated with the phenomenon of irregular refraction were identified.
As described in $2, the extensive literature on the refraction, reflection and
diffraction of waves of infinitesimal amplitude (e.g. Pierce
1981
;
Friedlander
1958),
provides a useful basis for considering the distortion of weak shocks by fluid
inhomogeneities that act as lenses. Even relatively rarely observed effects such as
tunnelling
or
glory (Jones
1978;
Marston
&
Kingsbury
1981;
Marston
&
Langley
1983)
have analogues in the nonlinear case. In the field of ultrasonics, considerable
research has been carried out on the properties of cylindrical and spherical sonar
targets. The geometry of the transmitted waves in the case of liquid-filled cylinders
(Brill
&
ffberall
1970)
and in the case of metal cylinders (Neubauer
&
Dragonette
1970)
and
of
the reflected waves (Folds
1971)
has been identified. The configuration
of the internal refracted and reflected waves has been observed in the case
of
Interaction
of
weak shock waves with
gas
inhomogeiieities
43
mechanical impact on liquid-filled cylindrical containers (Bockhoff
&
Rauch
1973).
The analogous phenomena of light wave interaction with spherical particles are well
documented (e.g. Van de Hulst
1957).
1.2.
Distortion
of
the
volume
and
mixing
As
the shock sweeps over the inhomogeneity, the shape of the gas volume changes
due to compression and the differential motions induced. In the simplest case, when
the substance in the volume is not different from that outside, a shape of circular
cross-section with diameter
D
deforms into an ellipse of major axis
D
and minor axis
D(
1
-
V,/
V,)
where
V,
and
V,
are the velocities of the shock and the gas behind the
shock. In the case of different fluids, normal interaction on the axis of symmetry again
leads to compression, while oblique and tangential off-axis interactions lead, in
addition, to shear (Chu
&
Kovasznay
1957).
Vorticity is produced because of the
misalignment of the gradients in pressure and density
(or
entropy), as shown by the
vorticity production equation.
do
VP
x
WP
-
=
(o.V)V--oV*V+
dt
P2
’
where
o
is the vorticity, V the velocity,
p
the pressure, and
p
the density. When
a
shock is incident, say from the right, on a low-density (e.g. helium) sphere, because
the light gas is relatively easier to accelerate, clockwise vorticity is produced at the
top and counterclockwise vorticity
is
generated
at
the bottom of the volume.
Consequently, it might be expected that during the subsequent motion, because of
shear-layer instability and vortex roll-up, the inhomogeneity would transform into
a
vortex-ring-like structure, and by vortex induction the structure would move
downstream (to the left) relative to the surrounding fluid. On the other hand, for a
shock incident on
a
high-density (e.g.
R22)
sphere, the sense of the vorticity is
opposite, and the structure moves upstream against the stream. Thus, the expecta-
tion is that after some time a spherical inhomogeneity becomes
a
vortex ring, while
a cylindrical inhomogeneity becomes
a
pair of vortex lines. This process has been the
object of a previous experimental study (Rudinger
&
Somers
1960)
in which the
velocities of the vortical structures resulting from the shock-induced acceleration of
small-diameter cylindrical inhomogeneities of He
or
SF,
were measured and com-
pared with the predictions of
a
simple theoretical model of vortex generation by
impulsive acceleration of an imaginary plate.
The initial deformation of the inhomogeneity can also be interpreted in terms of
the Rayleigh-Taylor instability of accelerated, curved interfaces separating fluids of
different density (Taylor
1950).
The acceleration is caused by the shock wave and
takes place during the first few reverberation times
7
after shock impact, where, in
the present case,
7
is of the order
Dla,
and
a
is a characteristic wave speed. Thereafter
the acceleration is zero,
so
the violent motion induced by the shock, and the
consequent secondary instabilities and mixing, slowly die out by viscous dissipation.
Shock-induced interfacial instability has been observed in shock waveflame inter-
action experiments by Markstein
(1957a,
b;
cf.
$1.1).
Curved flame fronts, when
accelerated by
a
shock wave, undergo heavy distortions such
as
shape reversal and
spike formation (Markstein
1957b).
The deformation histories are similar to the ones
observed in the case of gas bubbles suddenly released in liquids (Walters
&
Davidson
1962, 1963).
The shock-induced Rayleigh-Taylor instability of a sinusoidally perturbed plane
44
J.-F.
Haas
and
B.
Sturtevant
interface. sometimes referred to
as
the Richtmyer-Meshkov instability, was treated
by Richtmyer
(1960),
and Markstein
(1957~).
In the linear regime, the growth rate
v
of interface perturbations is proportional to the product of
V,
the initial amplitude
rl0,
the wavenumber
k
of the corrugation, and the Atwood number,
where
p,
and
pz
are the gas densities upstream and downstream of the interface,
respectively.
For
highly curved interfaces and very different gases the growth rate
or
perturbation velocity
v
is comparable to
V
and can be very large. Richtmyer’s
(1960)
numerical treatment accounted for the effects of compressibility but otherwise
confirmed this relation.
The deformation under shock-induced acceleration of
a
sinusotdally perturbed
plane interface oriented normal to the direction of shock propagation has been
investigated with experimental conditions rather close to the work presented here
(Meshkov
1970).
The lower-than-expected deformation velocities observed by
Meshkov were attributed to experimental difficulties (imprecision in the measure-
ments, gas contamination) and the neglect of some factors such
as
viscosity in the
theoretical calculations (Meyer
&
Blewett
1972).
Other possible effects such
as
drag
force on the spike, and turbulcncc have also bccn mentioned (Baker
&
Freeman
1981).
While this experimental investigation was in progress,
a
computer simulation of
Markstein’s shock wave-spherical flame interaction experiment was made by
integrating the classical conservation equations, and the vorticity production
equation
(1)
was
analytically integrated (Picone
et
al.
1984).
The same approach has
been used to simulate the experiments described here (Picone
&
Boris
1985, 1986;
Picone
et
al.
1986).
2.
Acoustic description
of
the wave processes
To
a
first
approximation, the wavefronts generated by the interaction of
a
weak
plane shock wave with
a
cylindrical
or
spherical volume can be exhibited and
classified by ray tracing and geometrical acoustics. Then the function of experiments
in which finite-amplitude waves occur is
to
elucidate the effects of nonlinear
propagation and volume deformation. The effect of the perturbing gas is that ofm
acoustic ‘lens’
of
index of refraction.
where
a,
and
a2
are the sound spceds of the
air
outside and of the gas inside,
respectively. The rays
of
the acoustic wavefronts are straight lines in regions of
constant sound speed, and refract
at
the boundaries of such regions according to
Snell’s law,
sin
Oi
=
n
sin
Or,
(4)
where
Oi
and
0,
are the angles of the incident and refracted
rays,
respectively. The
rays reflect
at
the boundaries such that the angle
of
reflection is equal to the angle
of incidence.
For
clarity, only the rays arising from the interaction of the incident wave with
the top half of the volume are shown in the figures of this section. The incident wave
Interaction
of
weak shock waves with gas inhomogeneities
45
FIGURE
1.
External reflected and diffracted rays and wavefronts
:
INC,
incident; RFL,
reflected; DIF, diffracted.
FIGURE 2. Rays and wavefronts characteristic
of
the divergent case: RFR, refracted; IRF,
internally reflected; TR, transmitted
;
TR2, secondary transmitted.
is represented by
a
family of parallel rays incident from the right onto the circular
boundary
at
angles of incidence increasing by steps of
5O,
except in figure
2,
in which
the steps are
lo.
The spacing between the wavefronts
is
chosen to correspond to
a
time interval
of
40
ps for
shock interaction with
a
50
mm diameter cylinder, except
in figure
2,
in which the steps are
20
ps.
Figure
1
illustrates the external reflected and diffracted rays and wavefronts. These