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Investigation of Mach number effects on the Richtmyer-
Meshkov instability using simultaneous PIV and PLIF
imaging diagnostics
Gregory C. Orlicz*, Sridhar Balasubramanian, Katherine P. Prestridge, B.J. Balakumar
Los Alamos National Laboratory, Los Alamos, NM 87545
Experiments are carried out at the Los Alamos Gas Shock Tube facility where a
varicose-perturbed, thin, heavy-gas curtain is impulsively accelerated by planar shock waves
of varying strength within the weak shock regime (M ≤ 1.5). The resulting Richtmyer-
Meshkov instability and subsequent fluid mixing is interrogated using both PIV (particle
image velocimetry) and PLIF (planar laser induced fluorescence) simultaneously. Presented
are the time evolution of density maps and vorticity maps for each Mach number. These 2-D
maps help to elucidate the differences in mixing at both large and small scales in the flow
when Mach number is varied. Several parameters derived from these maps are plotted with
time. It is found that if the time axis is scaled with the convection velocity, the rate of change
of these parameters with distance traveled can be effectively collapsed.
Nomenclature
M = Mach number
χ = instantaneous mixing rate
D = molecular diffusivity
ci,j = 2-D concentration map
A = Atwood number
ρ1 = density of light fluid
ρ2 = density of heavy fluid
x = streamwise position
y = spanwise position
z = vertical position
t = time
u = streamwise velocity
v = spanwise velocity
w = total mixing width
ω = vorticity
u’ = streamwise velocity fluctuation
v’ = spanwise velocity fluctuation
TKE = turbulent kinetic energy
I. Introduction
HE instability generated by the impulsive acceleration of an interface between between two fluids of different
densities where the pressure and density gradients are misaligned is known as the Richtmyer-Meshkov (R-M)
instability.1,2
Any perturbations that exist on the interface will grow with time, eventually leading to mixing of the
two fluids. Initial growth may be linear if the amplitude of the perturbations is small compared to the wavelength.
However, as the instability grows the flow transitions to nonlinear growth for even the simplest initial conditions
(e.g. a plane sinusoid). The R-M instability is important in both engineering applications3 and astrophysical
phenomena,4 and can be realized on a wide range of scales. In inertial confinement fusion, cryogenic capsules
* Graduate Research Assistant, Extreme Fluids Team, Physics Division, Los Alamos National Laboratory, PO Box
1663, MS H803, Los Alamos, NM 87544
T
41st AIAA Fluid Dynamics Conference and Exhibit27 - 30 June 2011, Honolulu, Hawaii
AIAA 2011-3709
Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
American Institute of Aeronautics and Astronautics
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containing deuterium-tritium (DT) are impulsively accelerated inward. The R-M instability arises due to
perturbations that exist between the layer of DT ice on the inner capsule surface, and the interior DT gas, decreasing
the yield of the reaction. In a supernova, an expanding shock wave travels through the stratified layers of the
exploding star, leading to enhanced mixing of the layers, which helps explain observations of supernova remnants.
The current study aims to understand the differences in instability growth and mixing behavior when the strength
of the impulsive force (shock wave) is varied. Previous experimental work has shown that varying Mach number
leads to small and large scale differences in the evolution of the R-M instability.5-9
The observed Mach number
effects have been attributed to differences in (1) the amount of initial condition compression during shock passage,
(2) the amount of vorticity deposited, and (3) the refraction of the incident shock wave as it passes through the
curtain, and the resulting internal reflections of shock waves and expansion waves off each interface.5 Current
evidence has shown that measurements of the total mixing width can be scaled using the velocity jump imparted on
the initial conditions.5,6,9
However, the total mixing width is only a measure of the largest scale in the flow, and fails
to capture the amount of true mixing of the test gases. This necessitates the use of other quantitative measurements
to achieve a more complete view of the instability growth, the ensuing mixing, and the mechanisms that drive it.
The present work represents an extension of previously reported experiments5 carried out at the Los Alamos Gas
Shock Tube facility where a varicose-perturbed, thin, heavy-gas curtain is impulsively accelerated by planar shock
waves of varying strength within the weak shock regime (M ≤ 2), and studied using planar laser induced
fluorescence (PLIF) to acquire 2-D concentration fields. In the previous work, it was found that total mixing width
growth rates scaled with the mean velocity of the curtain between Mach 1.2 and Mach 1.5 experiments, while
measurements of the instantaneous mixing rate (χ(x,y) = D(cc), where D is the molecular diffusivity between
gases and c is the 2-D concentration field) did not collapse with the same scaling. This demonstrated that there is a
disparity in the time scale for small vs. large scale mixing when Mach number is varied. The current work aims to
extend our understanding of the physics governing this disparity through the addition of particle image velocimetry
(PIV) measurements, which provide instantaneous velocity fields. Moreover, the new sets of experiments conducted
compare 3 different Mach numbers (Mach 1.2, 1.35, and 1.5), with instability growth observed for a longer duration
made possible by a new test section with extended optical access.
II. Experimental Setup
The experiments presented here were performed using a horizontal shock tube with a 3 in square cross section
and a total length of approximately 5.4 m. A schematic can be found in Fig. 1. The driven, test, and end sections are
open to atmosphere (11.5 psi). The driver section is initially separated from the rest of the tube by a polypropylene
film and is pressurized with nitrogen or helium gas to the appropriate level to generate the desired Mach number
shock wave. Experimentally, this was determined to be approximately 22 psi (N2), 30 psi (He), and 50 psi (He) for
Mach 1.2, Mach 1.35, and Mach 1.5, respectively. Once the desired pressure is reached, a trigger is sent to a
solenoid driven set of razor blades, which puncture the diaphragm. The rapid depressurization of the driver section
generates a shock wave that becomes planar as it travels down the length of the tube, eventually interacting with the
initial conditions. Four pressure transducers, embedded in the shock tube walls, are located along the path of the
shock wave, and are used to measure shock speed, time of shock interaction with initial conditions, and to
coordinate the timing of imaging
diagnostics. More experimental details can
be found in the available literature.5,10
The initial conditions consist of a thin
fluid layer of SF6 with varicose
perturbations surrounded by air. To seed
the initial conditions, pure SF6 gas is first
bubbled through liquid acetone before
flowing into a settling chamber located
above the shock tube. There, glycol fog
particles are added to the mixture. The
initial conditions are then formed by a
gravity induced flow of the heavy gas
mixture from the settling chamber to a
specially designed nozzle whose exit is
aligned with the top wall of the test section.
The nozzle consists of a row of closely
Figure 1. Experimental schematic.
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spaced holes of 3 mm diameter and 3.6 mm spacing. The initial conditions flow through the test section, where
diffusion between the individual jets creates a heavy gas curtain, and exit at the bottom where there is a mild suction
set just strong enough to prevent overflow. At the measurement plane, the maximum vertical flow velocity of the
initial conditions was measured to be 1 m/s, and the SF6 concentration is estimated at about 60% of pure SF6, with
20% acetone vapor and 20% air by volume. For this composition, the Atwood number is, A=(ρ2-ρ1)/(ρ2+ρ1)=0.60,
where ρ1 is the density of air and ρ2 is the density of the heavy gas mixture.
All images of the initial conditions and the resulting instability were acquired in a plane located 2 cm below the
nozzle exit. Two imaging techniques were used: PLIF and PIV, simultaneously. These measurements were made
using two dual headed Nd-YAG pulsed lasers with outputs of 266 nm for PLIF and 532 nm for PIV. The lasers are
coaligned through a combination of optics and formed into a horizontal laser sheet that enters the shock tube through
a UV-transparent window in the end wall of the end section at a vertical location of 2 cm below the top wall. The
light sheet from the 266 nm laser causes the acetone vapor tracing gas to fluoresce within the visible range (350 to
550 nm), whereas the 532 nm light sheet scatters off of the fog tracer particles. PLIF images are acquired using two
separate Apogee CCD cameras to gain optical access to early time and later time flow structures. A 2184 x 1470
CCD array camera with 3 x 3 binning was used to acquire early times, including the IC’s. Later times were acquired
by a second camera with a 1024 x 1024 CCD array. Both PLIF cameras provide high resolution images with about
50 µm/pixel. PIV images were acquired using a Kodak Megaplus cross-correlation camera with a 2048 x 2048
CCD array, yielding an image resolution of 16.1 µm/pixel. The PIV images were processed using Insight 3G
software with a processing window size of 32 x 32 pixels with 50% overlap. This provided a vector spacing of 258
µm/vector. A mild Gaussian vector smoothing filter was applied during processing with σ = 0.8 over a 3 x 3
neighborhood.
For each run of the experiment, only 2 pulses for each laser were available, yielding two 2-D concentration
maps at different times (PLIF), and a 2-D velocity field at a single time from a pair of PIV images. In order to
acquire simultaneous PIV/PLIF data, the PIV laser pulses were timed to sandwich one of the PLIF pulses.
III. Results
The current data was acquired through hundreds of runs of the experiment. By changing laser pulse timings from
one run to the next, it is possible to construct an extensive time sequence of the evolving instability for each Mach
number. In general, the experiments were performed to capture several images at each time step to get a measure of
the variability introduced by the initial conditions and small changes in shock speed from shot to shot. The images
are then analyzed and classified based on both qualitative and quantitative criteria (i.e. structure symmetry, structure
shape, seeding density, shock speed) to determine whether to include in the data set. The data presented here are a
sample of the data that have met these requirements.
A. PLIF Time Series
A complete time series of about 40 PLIF images has been assembled for each Mach number from the available
data. A subset of these images is presented in Fig. 2. Repeatability of the initial conditions allows for tracking of
specific flow features over time for each Mach number despite only having two PLIF pulses per experimental run.
Images show the initial conditions at t = 0 µs, followed by visualization of the shock wave passage, maximum
compression after shock interaction, and the subsequent evolution of the instability. Perturbations on the upstream
interface begin to grow immediately after shock interaction, while a phase inversion first takes place on the
downstream interface. As the perturbations on either side grow, they begin to interact, leading to a complex flow
pattern that is characterized by the classic mushroom shape that is common in the R-M instability.
It can be seen that at early times the evolution for each Mach number is qualitatively very similar in terms of the
shape of the structures as the main vortex pairs form. The main difference at these times is that the higher the Mach
number the smaller the overall width at a given stage of vortex development. This can be attributed to the higher
degree of compression with higher Mach number, and therefore, the smaller the initial width when the instability
first begins to grow.
As time progresses the structures begin to grow differently. From a qualitative perspective, it appears that the
higher the Mach number, the stronger the main vortices, and the greater the amount of SF6 that is entrained by them.
As the structures continue to roll up, this leads to differences in the structure shape. This is especially evident when
comparing late time structures in Mach 1.2 experiments (t > 900 µs) with those of Mach 1.5 (t > 350 µs). In Mach
1.2 experiments, the main vortices are not strong enough to continue entraining all the SF6 and tend to lag behind
the center of the mixing layer. Alternatively the main vortices in the Mach 1.5 experiments dominate most of the
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Figure 2. Density map evolution of 5 wavelengths with time. White indicates SF6 and black indicates air.
Flow direction is from top to bottom, and spanwise length scale of each structure is 18 mm. Large scale
growth patterns are similar at early times. At later times, dissimilarities develop.
American Institute of Aeronautics and Astronautics
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mixing layer and remain located near
its center. It can also be seen that as
Mach number is increased, small scale
mixing is achieved sooner relative to
the stage of large scale development.
Again, this is evident when comparing
Mach 1.2 and Mach 1.5 experiments,
for example at t = 1050 µs and t = 425
µs, respectively. Similar trends in
structure evolution and small scale
mixing were observed in a previous
study.5
B. Mixing Layer Width
The total width of the mixing layer
is a common quantitative metric used
to compare the mixing between
different experiments, models, and
simulations. In Fig. 3 the integral width
(w) of the mixing layer is plotted
against time for all time series images.
As expected, the growth rate is higher as the Mach number is increased. At late times, secondary flow features
cause a secondary jump in the growth rate for each Mach number. For Mach 1.2, it is mainly due to the lag of the
main vortices, whereas for Mach 1.35 and 1.5 it is due to the protrusion of SF6 spikes ahead of the downstream
interface. Plotting width against streamwise position (equivalent to scaling the time axis with the convection
velocity), as in Fig. 4, achieves an effective collapse of the data in the sense that the growth rate with distance
travelled is equivalent for each Mach number case until the latest times where the Mach 1.2 structures grow faster
after 23.5 cm. The measured convection velocity for the structures is 104, 158, and 222 m/s for M = 1.2, 1.35, and
1.5, respectively. At earlier positions there does exist an offset in width between each data set, as the higher the
Mach number the greater the compression, and the smaller the initial width of the mixing layer after shock passage.
As discussed above, previous studies have also shown that the mixing width tends to scale with the velocity jump
for a variety of initial conditions, both mixing layers5 and single interface experiments,
6,9 so this remains a consistent
trend in the current work.
While integral width is a useful metric to compare between different experiments, simulations, and models, and
is a simple measurement to make, it does not tell the whole story about the mixing and the mechanisms that drive it,
especially at smaller scales in the flow. Previous experiments have shown that quantitative measurements of smaller
scale features, such as the instantaneous
mixing rate, do not scale similarly with
velocity. This highlights the need for
additional metrics to compare between
experiments, simulations, and models,
and to gain an understanding for the
physics involved in the mixing process.
The rest of this study focuses on metrics
based on the PIV data.
C. Vorticity Map Time Series
The time evolution of the 2-D
vorticity field (curl of the velocity field,
yuxvz // ) for each Mach
number is shown in Figure 5. The
vorticity maps that are presented
correspond to velocity fields that were
acquired simultaneously to PLIF
images that belong to the complete
0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 30000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Wid
th (
mm
)
Time (s)
Mach 1.2
Mach 1.35
Mach 1.5
Figure 3. Integral width vs. time shows higher growth rate with
increasing Mach number. Inset image shows an example of the
total mixing width measurement.
0 25 50 75 100 125 150 175 200 225 250 275 3000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Streamwise Position (mm)
Wid
th (
mm
)
Mach 1.2
Mach 1.35
Mach 1.5
Figure 4. Integral width vs. position shows equal growth rate
with distance travelled for each Mach number.
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PLIF time series (images in Fig. 2 only represent a subset). The numbers in parentheses represent the streamwise
position of the structures in centimeters. As expected, values for vorticity are higher in the higher Mach number
case, where the main vortices also dominate a greater region of the flow. It is believed that these differences
account for disparity in large scale flow morphology seen in PLIF images at later times, as discussed above. In each
case, as time progresses, the main vortex pairs eventually lose their symmetry, which then precipitates a quick
transition of the vorticity field to a more disordered state, indicating a transfer of energy to smaller scales.
D. Sum of Vorticity
A sum of all positive and negative vorticity in each vorticity map was calculated over 5 wavelengths, and is
presented in Fig. 6. Here, and in all plots below, solid symbols indicate PIV data acquired simultaneously with PLIF
images from the complete PLIF time series. As can be seen, the amount of vorticity decreases with time for each
Mach number, and at a faster rate for higher Mach numbers. The decrease in absolute vorticity indicates either a
transfer of energy to under-resolved length scales in the mixing layer, or that the flow in the interrogation plane is
losing energy, perhaps due to viscous or 3-D effects. Figure 7 shows the total positive and negative vorticity plotted
against streamwise position. Similar to the mixing width data, the rate at which the vorticity changes with distance
traveled is the same for all Mach numbers.
Figure 5. Vorticity map time evolution. Values in parentheses indicate the position of the structure at the
given time. Red indicates strongest positive vorticity, and blue indicates negative.
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E. Circulation
Circulation, Γ, was calculated from the velocity field via a
line integral of the tangential velocity component dlu .
The calculation was made using a rectangular path enclosing a
selected positive valued vortex. For each velocity field, up to 5
vortices were chosen, and the circulation of each was used to get
an average value for circulation in a given field. At later times,
when the velocity field becomes less ordered, only the strongest,
most defined vortices were selected. Figure 8 shows an example
of the rectangular regions used to calculate average circulation
for a single velocity/vorticity map. Figure 9 shows the average
circulation of the main vortices over time for each Mach number.
Similar to vorticity, the average circulation decreases with time
for all cases, and with a faster rate for higher Mach numbers.
The decrease in circulation over time provides further evidence
that energy is being transferred out of the main vortex pairs and
into smaller scales. Also similar to vorticity and mixing width
data, when plotted against position the rate at which the circulation changes with distance travelled is similar in all
cases, as seen in Fig. 10.
0 250 500 750 1000 1250 1500 1750 2000 2250 2500-3
-2
-1
0
1
2
3x 10
7
Time (s)
Tota
l V
ort
icity (
s- 1
)
Mach 1.2
Mach 1.35
Mach 1.5
Figure 6. Total positive and negative vorticity over
five wavelengths. The absolute value decreases with
time for each mach number. Solid symbols indicate
PIV data acquired simultaneously to PLIF images
that belong to the complete PLIF time series.
2.5 5 7.5 10 12.5 15 17.5 20 22.5 25-3
-2
-1
0
1
2
3x 10
7
Streamwise Position (cm)
Tota
l V
ort
icity (
s- 1
)
Mach 1.2
Mach 1.35
Mach 1.5
Figure 7. The rate of change of total vorticity
with distance travelled is the same for each Mach
number.
0 250 500 750 1000 1250 1500 1750 2000 2250 25000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Time (s)
Circula
tion (
m2/s
)
Mach 1.2
Mach 1.35
Mach 1.5
Figure 9. The average circulation of the strongest
positive valued vortices in the flow decreases with
time
2.5 5 7.5 10 12.5 15 17.5 20 22.5 250
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Streamwise Position (cm)
Circu
latio
n (
m2/s
)
Mach 1.2
Mach 1.35
Mach 1.5
Figure 10. The rate of change of measured
circulation with distance travelled is similar for
each Mach number.
Figure 8. Up to five wavelengths were
used to calculate an average circulation
for each time. The top image shows the
raw PIV data. The bottom image shows
the velocity field with arrows, and
vorticity with the color map.
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F. Turbulent Kinetic Energy
The average turbulent kinetic energy (TKE) is plotted in Fig. 11. Without ensemble averages, the velocity
fluctuations had to be calculated from instantaneous realizations. For each velocity field, the whole field mean of
each velocity component (u and v) was calculated and subtracted from each velocity vector. The TKE at each point
in the field was then calculated as )''(5.0 22 vuTKE , where u’ and v’ are the fluctuations from the mean. As can
be seen in Fig. 11, the average TKE decreases with time for each case, and at a faster rate for higher Mach numbers,
again indicating that energy is being transferred to smaller scales as the flow becomes more mixed. If TKE is
normalized by the earliest measured value (TKE0) for each Mach number case, and then plotted against scaled time,
the data collapse fairly well as can be seen in Fig. 12.
IV. Conclusions
Experiments were performed to investigate the effect of varying Mach number on the R-M instability in a
varicose perturbed, heavy-gas curtain. Qualitative PLIF concentration maps and PIV velocity/vorticity fields are
used to interrogate the evolution of the instability after the initial conditions are impulsively accelerated by Mach
1.2, 1.35, and 1.5 shock waves. The initial conditions were carefully controlled so that a time series for each Mach
number case could be compiled from many different runs of the experiment, and so that isolation of Mach number
effects could be made possible. A series of 18 PLIF images for each Mach number shows that structure growth is
very similar at early times, but quite different at late times, apparently due to differences in vortex strength. Vorticity
maps confirm that as time progresses the main vortex pairs begin to deteriorate, as the motion within the mixing
layer is transferred to smaller length scales.
From the concentration and velocity fields, several quantitative parameters were measured and plotted against
time, including w, ω, Γ, and TKE. As a whole, these plots fit within the anticipated framework that as the instability
grows and induces fluid mixing, energy is transferred to smaller scales, and at a faster rate for higher Mach number.
The decrease in ω, Γ, and TKE with time is likely a combination of energy being transferred to under-resolved
length scales, 3-D effects not measured by planar imaging diagnostics, and viscous effects. The degree to which
each contributes is currently a work in progress.
If plotted against streamwise position, the rate of change of each parameter with distance traveled was found to
be approximately the same for each Mach number. However, images from the spatial maps indicate that there are
differences between how the structures evolve at both large and small scales, even when comparing images from the
same streamwise position. More work is required to find metrics that can capture these differences. Work is
currently underway to process the PLIF images quantitatively, and may provide additional insight.
References 1Richtmyer, R. D., “Taylor instability in shock acceleration of compressible fluids,” Commun. Pure Appl. Math, Vol. 13,
1960, p. 297. 2Meshkov, Y. Y., “Instability of a shock wave accelerated interface between two gases,” NASA TT F-13074, 1970. 3Lindl, J., “Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition
and gain,” Physics of Plasmas, Vol. 2, 1995, p. 3933.
0 250 500 750 1000 1250 1500 1750 2000 2250 25000
20
40
60
80
100
Time (s)
Ave
rage
TK
E (
m2/s
2)
Mach 1.2
Mach 1.35
Mach 1.5
Figure 11. Average turbulent kinetic energy as a
function of time.
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
1.2
Position (cm)
(Avera
ge T
KE
)/(A
vera
ge T
KE
) 0
Mach 1.2
Mach 1.35
Mach 1.5
Figure 12. Average turbulent kinetic energy
plotted as a function of streamwise position.
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4Arnett, W. D., Bahcall, J. N., Kirshner, R. P., and Woosley, S. E., “Supernova 1987A,” Annu. Rev. Astron. Astrophys., Vol.
27, 1989, p. 629. 5Orlicz, G. C., Balakumar, B. J., Tomkins, C. D., Prestridge, K. P., “A Mach number study of the Richtmyer-Meshkov
instability in a varicose heavy gas curtain,” Physics of Fluids, Vol. 21, 2009. 6Jacobs, J. W., and Krivets, V. V., “Experiments on the late-time development of single-mode Richtmyer-Meshkov
instability,” Physics of Fluids, Vol. 17, 2005. 7Ranjan, D., Niederhaus, J., Motl, B., Anderson, M., Oakley, J., and Bonazza, R., “Experimental investigation of primary and
secondary features in high-Mach-number shock-bubble interaction,” Phys. Rev. Lett., Vol. 98, 2007. 8Holmes, R. L., et al., “Richtmyer-Meshkov instability growth: Experiment, simulation and theory,” J. Fluid Mech. Vol. 389,
1999, p. 55. 9Motl, B., Oakley, J., Ranjan, D., Weber, C., Anderson, M., Bonazza, R., “Experimental validation of a Richtmyer-Meshkov
scaling law over large density ratio and shock strength ranges,” Physics of Fluids, Vol. 21, 2009. 10 Balakumar, B. J., Orlicz, G. C., Tomkins, C. D., and Prestridge, K. P., “Simultaneous particle-image velocimetry-planar
laser-induced fluorescence measurements of Richtmyer-Meshkov instability growth in a gas curtain with and without reshock,”
Physics of Fluids, Vol. 20, 2008.