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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.

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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.

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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.