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Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities

01 Sep 1987-Journal of Fluid Mechanics (Cambridge University Press)-Vol. 181, Iss: -1, pp 41-76
TL;DR: In this paper, the interaction of a plane weak shock wave with a single discrete gaseous inhomogeneity is studied as a model of the mechanisms by which finite-amplitude waves in random media generate turbulence and intensify mixing.
Abstract: The interaction of a plane weak shock wave with a single discrete gaseous inhomogeneity 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 inhomogeneities 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 inhomogeneities 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.

Summary (6 min read)

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.
  • Clearly, the description of such interactions is complicated and simple analytical models are difficult to formulate.
  • The authors limit their attention to discrete inhomogeneities because the flow visualization is especially graphic.
  • 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

  • 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.
  • I n 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 analogous phenomena of light wave interaction with spherical particles are well documented (e.g. Van de Hulst 1957).

2. Acoustic description of the wave processes

  • The grazing ray at the top of the cylinder defines the boundary of the shadow region behind the cylinder.
  • According to the geometrical theory of diffraction (Keller 1955 (Keller , 1958)) , the wave diffracted into the shadow region springs from a curved diffracted ray on the boundary which sheds straight diffracted rays tangentially into the shadow region.
  • The amplitude of the surface diffracted wave, initially some fraction of the incident amplitude dependent upon the material properties and the geometry of the problem, decreases exponentially as it propagates along the surface.
  • The amplitude of the tangentially shed diffracted waves is, in turn, some fraction of the local surface wave amplitude.

2.1. Divergent case

  • Both the primary and the secondary transmitted wavefronts are tangent to the reflected wave along the reflected ray at the critical angle.
  • Both families of waves (also identified by Brill & Uberall 1970) appear much earlier than the diffracted wave of figure 1 because of the high internal wave speed.

2.2. Convergent case

  • In this case the authors distinguish between two families of backscattered waves.
  • First, the waves resulting from the back reflection of the direct focused wave are very strong and they dominate.
  • Second, the waves that are consequences of the external diffracted wave, though generated first, are much weaker.
  • They are joined at their trailing edges to the strong family.

3.1. Shock tube

  • The initial trigger is obtained from a pressure transducer (transducer 1) mounted on the top wall of the test-section 64mm upstream of the window centre.
  • Thus transducer 2 measures the pressure as the waves downstream of the interaction reflect from the instrument plate.
  • A survey of the wave processes is made by varying the distance between the instrument plate and the volume.

3.2. Cylindrical volumea

  • Spurious effects of the cylindrical cell on the shock wave propagation are due to the various support parts (end windows, connecting beam, etc.) and to the membrane.
  • A measurement of the perturbations reflected upstream was made with the upstream transducer.
  • For a shock wave of Mach number 1.09 incident on a nitrogen-filled cylinder, three short pressure pulses were observed on the pressure plateau behind the incident shock.
  • The first one, corresponding to a wave of Mach number of 1.005, is due to the membrane, and the following two pulses, each from waves of Mach number about 1.01, are due to the waves reflected from the end windows and the connecting beam.

3.3. Spherical volumes

  • Due to reduced optical depth the image of the wave pattern in the axisymmetric configuration appears weaker on the shadowgraphs than in the cylindrical configuration.
  • On the other hand, due to (a) its less intrusive support structure, (b) the smaller volume of gas relative to the size of the test-section, and (c) the fact that after shock passage the film ruptures in a more ideal fashion, the interaction is less affected by spurious effects.
  • It should be noted that the relatively small effect of the structure is locally magnified near the axis in the axisymmetric case by focusing (forward glory).
  • Figure 6 shows some pressure traces recorded just behind bubbles filled with a +-$ mixture of helium and argon which is acoustically equivalent to air but has a different optical index of refraction for flow visualization.
  • In the trace at 1 mm, the initial rise is caused by the transmitted wave, while the first peak is due to the wave diffracted into the shadow region of the soap film.

4.1.1. Cylinder

  • As time passes, the upstream face continues to deform, and after 245 ps the volume has acquired a kidney shape, as shown in figure 7 (9).
  • When the head of the jet impinges on the downstream high-density air interface it spreads out laterally, eventually forming a pair of ill-defined vortical structures.

4.1.2. Sphere

  • Figure 8 ( c ) (145 ps) shows, left to right, the transmitted wave which has been caught up by the glory wave and the diffracted wave just after emerging from the helium volume.
  • The back-scattered wave to the right of the volume is the internal reflected wave which has crossed the flat upstream interface to the exterior.
  • A fine-scale Rayleigh-Taylor instability is seen developing on the surface of the sphere, especially at the downstream interface.
  • Concentrated vorticity also occurs along the conical shear layer a t the boundary of the re-entrant jet in the main body (cf. below).
  • Both the cylindrical and spherical sequences offer some analogies with the buoyancy-.

4.2. Convergent case 4.2.1. Cylinder

  • Though the response to shock excitation of soap bubbles filled with R22 is basically the same as cylinders, many of the details seem to be qualitatively different.
  • Because of the high density of R22, the 'spheres' are in fact oval bubbles hanging down from the support tube.
  • On the other hand, the surrounding vortex ring, which in cylindrical symmetry is a vortex pair, is much more diffuse.
  • Indeed, at late times , the structure at the left expands, becoming sinuous and even more diffuse.
  • In the field between the transmitted wave and the diffracted wave there is a gradual increase of pressure.

5.2. Convergent case

  • The pressure measurements for the case of a relatively strong shock incident on the R22-filled cylinder show that close to the cylinder the first disturbance (the diffracted wave) is smooth.
  • It is followed by the transmitted wave which is very strong, carrying overpressures up to 6.7 bar near its focus.
  • The diffracted wave steepens (27 mm), becoming a shock (43 and 67 mm) and is caught by the transmitted shock 99 mm behind the cylinder.
  • Henceforth the initial disturbance is the combined front.
  • The pulses seen behind the combined front on the pressure traces from 99 mm on are due to various waves reflected from the shock-tube sidewalls, the strongcst one being the reflection of the transmitted wave from the top and bottom walls.

5.3. Wave strengths: cylinders

  • Measured pressure jumps from profiles similar to those shown above have been used to calculate the strength (i.e. local Mach number) of the various waves in the cylindrical configuration.
  • Downstream of merging the Mach number of the combined diffracted and refracted wave was found to be very close to that of the incident shock Mach number.
  • In R22 the transmitted wave strength decreases significantly between the focal area and the merging point.
  • The calculated values in the table are for the interaction of a plane shock wave with a parallel plane gaseous interface, and are not expected to compare well when effects of wave curvature are large.
  • (Conversely, contamination by air of the test gas within the cylinder will lead to a stronger refracted wave in helium and a weaker one in R22, cf $6.).

5.4. Forward glory from the helium spheres

  • The three pressure profiles of figures 17 (a-c) were obtained for a strong incident shock wave ( M , = 1.25) with the transducer located very close behind (2, 11 and 25 mm) the downstream edge of a helium-filled bubble.
  • The first pressure profile indicates a transmitted shock of strength M , = 1.06 followed by a strong N-shaped pulse of about 1.24 bar amplitude, which is the secondary transmitted wave (forward glory).
  • The second and third profiles show how the secondary transmitted wave merges with the transmitted wave.
  • The third profile also shows the pressure rise due to the diffracted wave.

6. Results: velocities

  • The velocities of the shock waves and gas interfaces observed in the shadowgraphs have been estimated by plotting their location against time (x us. t ) and calculating the slopes of straight lines fitted to the data.
  • The positions of the refracted waves, transmitted waves, and interfaces are measured on the axis of the shock tube, while the incident wave is measured a t the top and bottom of the photograph, outside the acoustic shadow of the cylinder.
  • The important features of the waves and volumes are indicated schematically in figure 18 , and conventions for labelling the data on the x-t diagrams are also given.
  • In these experiments obtaining the superior spatial resolution of spark shadowgraphs over that of high-speed motion pictures was of higher priority than the precision in timing afforded by the latter.

6.2. Convergent case

  • Contamination in the R22-filled cylinder was estimated by calculating the speed of sound inside the cylinder from the average velocity of the refracted wave and its measured strength, as with the helium volumes.
  • Similarly, estimates of the contamination of the air surrounding the R22 volumes from the speed of sound indicate only low levels of contamination.

7.1.1. Helium : upstream interface

  • The observed initial rate of distortion is substantially smaller than that predicted for linear instability, even with the correction for compression, while the jet velocity is somewhat larger.
  • While the normalized upstream interface velocities are similar in the spherical and cylindrical cases, the normalized jet velocity is slightly higher in the axisymmetric configuration.
  • Both experiment and theory show that the rates decrease substantially with increasing wave strength.

7.1.2. Helium : downstream interface

  • As demonstrated by the growth of small-scale corrugations on the downstream side of the helium cylinder and sphere the downstream interface is initially destabilized by the shock that has already interacted with the upstream interface.
  • Though the observed distortions on the sphere are again smaller than calculated by linear stability theory, even with the correction for compression, the agreement in the cylindrical case is substantially better, perhaps fortuitously.
  • Again, there is little difference between the results in two dimensions and in three dimensions.

7.1.3. R22 : upstream interface

  • In table 8 the observed normalized initial interface velocity Yui (table 4) is compared with the prediction V of linear stability theory.
  • Implying that the rate of distortion of the interface is again much less than in linear instability.
  • The authors do not treat the behaviour of the downstream interface in the convergent case because the refracted shock focuses just a t the interface, creating a situation very different from the interaction of a plane shock with an interface.
  • The moving, undeformed bubble after the first step plays the role of Taylor's 'dissolved' vortex-generating disk, so Vb is taken as the velocity of the disk.
  • Equating for the first step the impulse per unit volume I experienced by the bubble to that experienced by the surrounding air,.

7.2.1. Helium volume

  • The measured and calculated velocities of the helium volumes are compared in table 9.
  • The Rudingel-Somers (Rs) and the uncorrected Rayleigh-Taylor ( R T ) models predict bubble and vortex velocities in rough agreement with those observed, but both models fail to account for the decrease of the velocities with increasing shock strength.
  • Most important, the flow visualization of the present experiments shows that the helium bubble does not transform into a single vortex ring, as had previously been expected, but, in fact, splits into at least two structures, the largest containing little vorticity, and the smallest being an energetic vortex ring propagating rapidly ahead of the main structure.
  • Thus calculation of the properties of the vortex is best carried out with a piston vortex-generator model (cf. below).

7.2.2. R22 volume

  • The comparison between theory and experiment is made in table 10.
  • And the experimentally observed velocities are generally larger than predicted.
  • As already noted, in the experiments the R22 structure grows so large that it interacts strongly with the shock-tube walls and acquires an abnormally large velocity.
  • The growth of heavy bubbles after shock interaction was already noted for SF, bubbles by Rudinger & Somers who called attention to the explanation by Turner (1957) in terms of the Rayleigh-Taylor instability of rotating flows.

7.3. Helium sphere : vortex-generator model

  • As might be expected, the most difficult quantity to measure is the core radius.; of the photographs presented in this paper only one, figure 8 (g), suggests a well-defined core.
  • Fortunately the result does not depend strongly on this quantity.
  • The numerical simulations discussed below give a similar result.
  • Picone & Boris (1985 , 1986) and Picone et al. (1986) have carried out numerical simulations of their experiments using a finite-difference solution of the conservation equations for inviscid flow (cf. Picone et al. 1984) .
  • Their plots of density and vorticity contours are strikingly similar to the photographs presented in this work, and the magnitude of their calculated flow velocities is generally in good agreement with their observations.

8. Summary and conclusions

  • The microfilm membrane used to separate the two gases did not perturb the geometry of the wave pattern and the motion of the gas interface, but had an effect on the strength of the shock waves measured very close to the cylindrical wall.
  • Better control of the gas composition and use of a high-resolution high-speed motion picture camera would improve the precision of these measurements.
  • The diameter of the volumes studied in this work was a large fraction of the test-section width, so blockage effects were experienced.
  • Waves reflected from the shock-tube sidewalls modified the flow field at intermediate times.

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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

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Journal ArticleDOI
TL;DR: A new numerical method for treating interfaces in Eulerian schemes that maintains a Heaviside profile of the density with no numerical smearing along the lines of earlier work and most Lagrangian schemes is proposed.

1,933 citations

Journal ArticleDOI
TL;DR: A new model and a solution method for two-phase compressible flows is proposed that provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.

906 citations

Journal ArticleDOI
TL;DR: In this article, Zhou et al. presented the initial condition dependence of Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) mixing layers, and introduced parameters that are used to evaluate the level of mixedness and mixed mass within the layers.

606 citations

Journal ArticleDOI
TL;DR: In this paper, the authors summarized a 10-year theoretical and numerical effort to understand the deflagration-to-detonation transition (DDT), which resulted in the development of numerical algorithms for solving coupled partial and ordinary differential equations and a new method for adaptive mesh refinement.

525 citations

Journal ArticleDOI
TL;DR: In this article, a detailed numerical study of the interaction of a weak shock wave with an isolated cylindrical gas inhomogeneity is presented, focusing on the early phases of interaction process which are dominated by repeated refractions and reflections of acoustic fronts at the bubble interface.
Abstract: We present a detailed numerical study of the interaction of a weak shock wave with an isolated cylindrical gas inhomogeneity. Such interactions have been studied experimentally in an attempt to elucidate the mechanims whereby shock waves propagating through random media enhance mixing. Our study concentrates on the early phases of the interaction process which are dominated by repeated refractions and reflections of acoustic fronts at the bubble interface. Specifically, we have reproduced two of the experiments performed by Haas and Sturtevant.

461 citations

References
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Journal ArticleDOI
TL;DR: Light scattering by small particles as mentioned in this paper, Light scattering by Small Particle Scattering (LPS), Light scattering with small particles (LSC), Light Scattering by Small Parts (LSP),
Abstract: Light scattering by small particles , Light scattering by small particles , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

9,737 citations

Book
01 Dec 1981
TL;DR: Light scattering by small particles as mentioned in this paper, Light scattering by Small Particle Scattering (LPS), Light scattering with small particles (LSC), Light Scattering by Small Parts (LSP),
Abstract: Light scattering by small particles , Light scattering by small particles , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

6,623 citations

Journal ArticleDOI
TL;DR: In this article, Spark shadow pictures and measurements of density fluctuations suggest that turbulent mixing and entrainment is a process of entanglement on the scale of the large structures; some statistical properties of the latter are used to obtain an estimate of entrainedment rates, and large changes of the density ratio across the mixing layer were found to have a relatively small effect on the spreading angle.
Abstract: Plane turbulent mixing between two streams of different gases (especially nitrogen and helium) was studied in a novel apparatus Spark shadow pictures showed that, for all ratios of densities in the two streams, the mixing layer is dominated by large coherent structures High-speed movies showed that these convect at nearly constant speed, and increase their size and spacing discontinuously by amalgamation with neighbouring ones The pictures and measurements of density fluctuations suggest that turbulent mixing and entrainment is a process of entanglement on the scale of the large structures; some statistical properties of the latter are used to obtain an estimate of entrainment rates Large changes of the density ratio across the mixing layer were found to have a relatively small effect on the spreading angle; it is concluded that the strong effects, which are observed when one stream is supersonic, are due to compressibility effects, not density effects, as has been generally supposed

3,339 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa.
Abstract: It is shown that, when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa. The relationship between the rate of development of the instability and the length of wave-like disturbances, the acceleration and the densities is found, and similar calculations are made for the case when a sheet of liquid of uniform depth is accelerated.

2,839 citations