Experiments on the Richtmyer-Meshkov instability with...

Experiments On The Richtmyer-Meshkov InstabilityWith An Imposed, Random Initial Perturbation

Item Type text; Electronic Dissertation

Authors Tsiklashvili, Vladimer

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.

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Link to Item http://hdl.handle.net/10150/337310

http://hdl.handle.net/10150/337310

Experiments on the Richtmyer-Meshkov

instability with an imposed, random initial

perturbation

by

Vladimer Tsiklashvili

A Dissertation Submitted to the Faculty of the

Department of Aerospace and mechanical engineering

In Partial Fulfillment of the RequirementsFor the Degree of

Doctor of PhilosophyWith a Major in Mechanical engineering

In the Graduate College

The University of Arizona

2 0 1 4

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THE UNIVERSITY OF ARIZONAGRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dis-sertation prepared by Vladimer Tsiklashvilientitled Experiments on the Richtmyer-Meshkov instability with an imposed, ran-dom initial perturbationand recommend that it be accepted as fulfilling the dissertation requirement for theDegree of Doctor of Philosophy.

Date: November 10, 2014Dr. Jeffrey W. Jacobs

Date: November 10, 2014Dr. Anatoly Tumin

Date: November 10, 2014Dr. Cho Lik Chan

Date: November 10, 2014Dr. Dui-dong Wang

Date: November 10, 2014

Final approval and acceptance of this dissertation is contingent upon the candidate’ssubmission of the final copies of the dissertation to the Graduate College.I hereby certify that I have read this dissertation prepared under my direction andrecommend that it be accepted as fulfilling the dissertation requirement.

Date: November 10, 2014Dissertation Director: Dr. Jeffrey W. Jacobs

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of the requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that an accurate acknowledgement of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: Vladimer Tsiklashvili

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Acknowledgments

I would like to express immensely my gratitude to Dr. Jeffrey Jacobs for giving methe opportunity to be part of his research team in the Experimental Fluids DynamicsLaboratory, for helping to expand my knowledge in the field of fluid dynamics, andalso, for supporting my studies through a Graduate Research Assistantship duringmy master and doctorate degrees. I would like to extend my gratitude to Dr. OlegLikhatchev for working by my side me on this projects and for his constant guidanceand support.

I would like to thank Cole Valancius, Dr. Mike Roberts and Dr. Robert Morganfor giving me suggestions on my experiment, and Matthew Mokler for helping meto machine parts needed for the experimental apparatus. To Dr. Vitaliy Krivets forproviding insightful view regarding instability processes.

I want to thank my family for the strength and constant support they gave meto pass difficult situations. Even though it was a long process, I have been able toreach my goals.

The definition of success by Early Nightingale:Success is the progressive realization of a worthy ideal.

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Table of Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 161.1. Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2. Past research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . 462.1. The shock tube and experimental setup . . . . . . . . . . . . . . . . . 462.2. Test section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.3. Flow visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.4. Pressure transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.5. Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 3. Processing methods and results . . . . . . . . . . . . . . 573.1. Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2. Mixing layer width . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3. Interface velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4. Growth constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5. Re-shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.6. Initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.7. Dominant wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.8. The evolution of the spectral content of the mixing layer . . . . . . . 823.9. Integral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.10. Linear momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.11. Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.12. Time Evolution of the growth exponent θ . . . . . . . . . . . . . . . . 943.13. Tests evaluating the measurement methods . . . . . . . . . . . . . . . 94

Chapter 4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Table of Contents—Continued

6

Appendix A. 1D Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111A.1. Analytical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111A.2. Discretization, The Roe scheme . . . . . . . . . . . . . . . . . . . . . 113A.3. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.4. The computational method with a two-stage Runge-Kutta time inte-

gration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115A.5. Multicomponent flow calculation . . . . . . . . . . . . . . . . . . . . . 115A.6. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Appendix B. Acceleration of the interface . . . . . . . . . . . . . . 128

Appendix C. Code for finding the piston velocity for two gassystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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List of Figures

Figure 1.1. Deposition of the vorticity on the interface between two liquidsdue to baroclinic torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 1.2. a) RMI experiment initiated from two dimensional a single-modeharmonic perturbation, from Collins and Jacobs [23]. b) RMI experimentwith a three-dimensional initial perturbation captured in the present ex-perimental study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 1.3. Drawing of two-liquid system with different densities. . . . . . . 20

Figure 2.1. Rendering of the experimental apparatus with three CMOS cam-eras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 2.2. X-T diagram for an experiment with incident Mach numberM=1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 2.3. Test section with attached speaker. . . . . . . . . . . . . . . . . 51Figure 2.4. Pressure transducer readings. . . . . . . . . . . . . . . . . . . . 54Figure 2.5. Schematic of the experimental systems. . . . . . . . . . . . . . . 56

Figure 3.1. Image sequence taken from a typical experiment using the Miescattering diagnostic. The first frame shows the compression effect dueto the propagating shock wave. The line below the interface, separatingthe bright and dim regions of the same gas, is the propagated shockwave into SF6. The gas behind the shock wave is compressed making itbrighter than the uncompressed gas. . . . . . . . . . . . . . . . . . . . . 58

Figure 3.2. a) Images from an Experiment performed with smoke seededSF6 gas. b) Calibrated image using white and black reference imagesobtained prior to the experiment. The obtained smoke concentratinglevels are multiplied by the bit depth of the original image in order tomake them visible for comparison. . . . . . . . . . . . . . . . . . . . . . 62

Figure 3.3. Row-averaged smoke concentration profiles at different timesrelative to the incident shock wave interface interaction. This sequenceof images shows widening of the mixing layer width as time progresses.a)t=0.3 ms, b) t=2.2 ms, c) t=4.0 ms, d) t=5.7 ms. . . . . . . . . . . . . 63

Figure 3.4. Comparison of the mixing layer width evolution in time usingtwo different calibration methods; both methods provide similar mixinglayer width evolution profiles. . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 3.5. Interface velocity for 10 experiments based on the 50% locationof the averaged concentration measurement. . . . . . . . . . . . . . . . . 65

List of Figures—Continued

8

Figure 3.6. Interface acceleration corresponding to the averaged data of Fig.3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 3.7. The averaged interface velocity, obtained for 10 experimentswith similar incident shock wave strength, compared to the theoreticalprediction modelling presence of boundary layers. The experimental con-ditions for each experiment can be found at Table B.1. . . . . . . . . . 67

Figure 3.8. Mixing layer half width evolution in time for 16 experiments witha power law behavior profile following the incident shock wave interfaceinteraction and linear growth after the de-acceleration by the reflectedshock wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Figure 3.9. From Fig. 3.8 the experiments were divided in two groups basedon their different growth behavior; a) with a more consistent growth rateand b) with very rapid initial growth followed by a slower growth. . . . . 69

Figure 3.10. Logarithmic plot of the mixing layer half width for the two setsof experiments shown in Fig. 3.9. . . . . . . . . . . . . . . . . . . . . . . 69

Figure 3.11. Evolution of the θ(t) of the mixing layer half width for the twosets of experiments shown in Fig. 3.9. . . . . . . . . . . . . . . . . . . . 70

Figure 3.12. Mixing layer half width for the two sets of experiments showingidentical growth rates following re-shock. . . . . . . . . . . . . . . . . . . 71

Figure 3.13. Comparison of measured dimensionless re-shock growth ratesfrom this study with those measured by Leinov et al [62]. . . . . . . . . 72

Figure 3.14. Mie scattering image of the perturbed interface prior to theshock-interface interaction and the corresponding deposited vorticity fortwo experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Figure 3.15. a) Initially deposited vorticity profile. b) Corresponding spectralcontent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Figure 3.16. Initial perturbations prior of the incident shock arrival for severalexperiments illustrating the stochastic nature of the initial conditionsfrom which the RMI evolves. . . . . . . . . . . . . . . . . . . . . . . . . 75

Figure 3.17. Evolution of the initial perturbation in time after initiation ofthe vertical oscillation. a) Evolution of the perturbation amplitude intime. b) Evolution of the dominant wavelength of the perturbation intime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 3.18. Development of mixing layer half width following the shock-interface interaction for a set of 17 experiments. . . . . . . . . . . . . . . 78


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Figure 3.19. a) Image sequence taken from an experiment exhibiting the con-sistent growth behavior. b) Image sequence taken from an experimentexhibiting rapid initial growth followed by a slower growth. . . . . . . . 79

Figure 3.20. a) Mixing layer half width versus time for 5 experiments withconsistent growth rate. b) Mixing layer half width versus time for 5experiments with rapid initial growth followed by a slower growth. . . . 80

Figure 3.21. Dominant wavelength evolution in time. a) Experiments withconsistent growth rate. b) Experiments with rapid initial growth followedby a slower growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Figure 3.22. a) Average deposited vorticity spectrum for experiments withconsistent growth rate. b) Average deposited vorticity spectrum for ex-periments with rapid initial growth followed by a slower growth. . . . . . 81

Figure 3.23. Amplitude evolution for different wavelengths of the mixing layerspectrum. a) Spectral content evolution for experiments exhibiting lineargrowth behavior. b) Spectral content evolution for experiments exhibit-ing rapid initial growth followed by a slower growth. . . . . . . . . . . . 82

Figure 3.24. Total mixing layer width versus time for four experiments cal-culated using Eq. 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 3.25. Integral width evaluated from Eq. 3.5 for (a) spikes and (b)bubbles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Figure 3.26. Log-Log plot of the integral layer mixing width for a) spike andb) bubble, indicating that both exhibit power law growth behavior. . . . 86

Figure 3.27. Evolution of the θ(t) according to (3.2) for the integral layermixing width for a) spike and b) bubble. . . . . . . . . . . . . . . . . . . 86

Figure 3.28. Time evolution of the square of the spike and bubble mixinglayer widths defined by (3.10). . . . . . . . . . . . . . . . . . . . . . . . 89

Figure 3.29. Time evolution of the square of the total mixing layer widthh2LM = h2

b−LM + h2s−LM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Figure 3.30. Time evolution of the averaged square of the total mixing layerwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Figure 3.31. Dimensionless total mixing layer width. The solid line is thesquare root function and the symbols represent the experimental mea-surement of the evolution of the turbulent mixing width. . . . . . . . . . 91

Figure 3.32. Total mixing layer width for experiments performed with a thinlaser sheet and smoke in air. The measurements are based on the locationof 10% and 90% of the horizontally averaged smoke concentration. . . . 92


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Figure 3.33. Dimensionless total mixing layer width for experiments pre-sented in Fig. 3.32. The wavelength is selected based on the depositedvorticity spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Figure 3.34. Dimensionless total mixing layer width for experiments pre-sented in Fig. 3.32. The wavelength is selected based on the lineartheory (1.15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Figure 3.35. a) Average total mixing layer width for 55 experiments fromgroup #5. b)Evolution of the θ(t) of the averaged total mixing layerwidth according to (3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Figure 3.36. a) Average total integral mixing layer width for 55 experimentsfrom group #5. b)Evolution of the θ(t) of the averaged total mixing layerwidth according to (3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Figure 3.37. Image sequences of the RMI evolution for four different visual-ization methods. a) Smoke in air and visualized by a thin laser sheet.b) Smoke in air and visualized with a thick laser sheet. c) Smoke in SF6

and visualized with a thin laser sheet. d) Smoke in SF6 and visualizedwith a thick laser sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Figure 3.38. Time evolution of the mixing layer width for the experimentsperformed with a thick laser sheet (2cm) and light gas (air) seeded withsmoke. Left: Mixing layer width measured by the 10 % and 90% smokeconcentration locations. Right: Integral mixing layer width for the sameset of data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 3.39. Dimensionless mixing layer width in which the characteristicwavenumber is selected based on the deposited vorticity profile. Left:The mixing layer width measured based on 10 % and 90% smoke concen-tration locations. Right: The integral measure of the mixing layer widthfor the same set of data. . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 3.40. Dimensionless mixing layer width in which the characteristicwavenumber is selected based on the initial interface evolution measuredimmediately after the shock-interface interaction. Left: Mixing layerwidth measured based on the 10 % and 90% smoke concentration loca-tions. Right: Integral mixing layer width evaluated for the same data. . 100


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Figure 3.41. Time evolution of the mixing layer width for experiments per-formed with a thin laser sheet (1.5mm) and light gas (air) seeded withsmoke. Left: Mixing layer width measured by the 10 % and 90% smokeconcentration locations. Right: Integral mixing layer width for the sameset of data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Figure 3.42. Dimensionless mixing layer width in which characteristic wavenum-ber is selected based on the deposited vorticity profile. Left: Mixing layerwidth measured by the 10 % and 90% smoke concentration locations.Right: Integral mixing layer width for the same set of data. . . . . . . . 101

Figure 3.43. Dimensionless plot of mixing layer width in which characteris-tic wavenumber is selected based on the initial interface evolution im-mediately after the shock wave propagation. Left: Mixing layer widthmeasured by the 10 % and 90% smoke concentration locations. Right:Integral measure of the mixing layer width. . . . . . . . . . . . . . . . . 101

Figure 3.44. Time evolution of the mixing layer width for experiments per-formed with a thick laser sheet (2cm) and heavy gas (SF6) seeded withsmoke. Left: Mixing layer width measured by the 10 % and 90% smokeconcentration locations. Right: Corresponding integral mixing layerwidth plotted for the same data. . . . . . . . . . . . . . . . . . . . . . . 102

Figure 3.45. Dimensionless plot of the mixing layer width in which scalingwavenumber is selected based on the deposited vorticity profile. Left:Mixing layer width measured by the 10 % and 90% smoke concentrations.Right: Corresponding integral mixing layer width. . . . . . . . . . . . . 102

Figure 3.46. Dimensionless plot of the mixing layer width in which scalingwavenumber is selected based on the initial interface evolution immedi-ately after the shock wave propagation. Left: Mixing layer width mea-sured by the 10 % and 90% smoke concentration locations. Right: Cor-responding integral mixing layer width. . . . . . . . . . . . . . . . . . . 103

Figure 3.47. Time evolution of the mixing layer width for experiments per-formed with a thin laser sheet (1.5mm) and heavy gas (SF6) seededwith smoke. Left: Mixing layer width measured by the 10 % and 90%smoke concentration locations. Right: Corresponding integral mixinglayer width for the same data. . . . . . . . . . . . . . . . . . . . . . . . 103


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Figure 3.48. Dimensionless plot of the mixing layer width in which charac-teristic wavenumber is selected based on the deposited vorticity profile.Left: Mixing layer width measured by the 10 % and 90% smoke concen-tration locations. Right: Corresponding integral mixing layer width. . . 104

Figure 3.49. Dimensionless plot of the mixing layer width in which scalingwavenumber is selected based on the initial interface evolution immedi-ately after the shock wave propagation. Left: Mixing layer width mea-sured by the 10 % and 90% smoke concentration locations. Right: Cor-responding integral mixing layer width for the same data. . . . . . . . . 104

Figure A.1. X-T Diagram of characteristics . . . . . . . . . . . . . . . . . . 118

Figure B.1. Interface Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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List of Tables

Table 3.1. Changes in experimental set-up and number of experiments foreach type of configuration. . . . . . . . . . . . . . . . . . . . . . . . . . 59

Table A.1. Input parameters for the 1D Code . . . . . . . . . . . . . . . . . 117

Table B.1. Experimental conditions and the ideal gas piston velocity. . . . . 130

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Abstract

The Richtmyer-Meshkov instability is studied in vertical shock tube experiment.

The instability is initiated by the passage of an incident shock wave over an interface

between two dissimilar gases. The interface is formed by opposed gas flows in which

air and SF6 enter the shock tube from the top and from the bottom of the shock

tube driven section. The gases exit the test section through a series of small holes in

the test section side walls, leaving behind a flat, diffuse membrane-free interface at

that location. Random three-dimensional perturbations are imposed on the interface

by oscillating the column of gases in the vertical direction, using two loud speakers

mounted in the shock tube wall. The development of the turbulent mixing is observed

as a result of the shock-interface interaction. The flow is visualized using planar Mie

scattering in which the light from a laser sheet is scattered by smoke particles seeded

in one of the experimental gases and image sequences are captured using high-speed

CMOS cameras.

The primary interest of the study is the determination of the growth rate of the

turbulent mixing layer that develops after an impulsive acceleration of the perturbed

interface between the two gases (air/SF6) by a weak M=1.2 incident shock wave.

Measurements of the mixing layer width following the initial shock interaction show

a power law growth h ∼ tθ similar to the those observed in previous experiments

and simulations with θ ≈ 0.40. The experiments reveal that the growth rate of the

mixing width significantly varies from one experiment to another. This is attributed

to the influence of initial perturbations imposed on the interface. However, better

consistency for the mixing layer growth rate is obtained from the mixing generated

by the reflected shock wave. A novel approach that is based on mass and linear

15

momentum conservation laws in the moving reference frame leads to a new definition

of the spike and bubble mixing layer widths, which does not depend on the initial

conditions.

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Chapter 1

Introduction

1.1 Problem description

The Richtmyer-Meshkov instability (RMI) develops when a perturbed interface be-

tween two fluids of different densities is subjected to an impulsive acceleration. The

acceleration causes the initial perturbations to grow and eventually become turbu-

lent. The Ricthmyer-Meshkov instability is related to the Rayleigh-Taylor instability

(RTI), which occurs in the presence of two liquids separated by a well-defined in-

terface that is subjected to a body force directed from a heavy liquid to a light

liquid.

The theory [114] shows that at the initial stage, the instability amplitude, a grows

linearly in time. The growth rate is proportional to the initial amplitude a0, the

dimensionless contrast of the densities of the gases the Atwood number A = ρ1−ρ2ρ1+ρ2

,

the impulsive velocity change 4V and the wavenumber k

da

dt=ρ1 − ρ2

ρ1 + ρ2

k4 V a0. (1.1)

At later time, the initial vorticity deposited on the interface, due to missalignment

of the density gradient and the pressure gradient, governs the development of the

instability. The vorticity production is expressed as follows

D

Dt

(~ω

ρ

)=

(~ω

ρ· ~5)~u+

1

ρ3~5ρ× ~5p. (1.2)

The last term on the right hand side is the vorticity generated due to the baroclinic

torque. The drawing in Fig.1.1 illustrates the baroclinic vorticity deposition due to

17

the misalignment of pressure and density gradient. The baroclinic term relates to

the rate of vorticity generation and its sign depends on the direction of the impulsive

acceleration (shock wave propagating from light gas to heavy or from heavy to light).

It has been shown theoretically that accelerating the interface in the direction from

the heavy gas to the light gas differs from the acceleration of the interface from the

light to the heavy gas by simple interface inversion [103] [51].

Figure 1.1: Deposition of the vorticity on the interface between two liquids due tobaroclinic torque.

The instability initiated with a well defined two-dimensional harmonic initial per-

turbation generates mushroom-like coherent structures (see Fig. 1.2). Bubbles are

formed where light fluid penetrates into the heavy fluid and spikes are formed where

heavy fluid penetrates into the light fluid. The interface rolls up due to deposited

initial vorticity into concentrated vorticity regions, which generates a secondary in-

stability near the vortex cores due to misalignment of the density and the centrifugal

body force due to the fluid rotation. While the instability develops, the coherent

flow structures break down leading to transition to turbulent mixing.

In most practical applications that involve the RMI, the initial state consists of a

18

Figure 1.2: a) RMI experiment initiated from two dimensional a single-mode har-monic perturbation, from Collins and Jacobs [23]. b) RMI experiment with a three-dimensional initial perturbation captured in the present experimental study.

set of randomly distributed modes that result in the fully turbulent mixing layer even

in the early stages of the RMI development. The majority of RMI investigations,

initiated from multimode initial perturbations, have placed emphasis on the study

of the rate of growth of the turbulent mixing region that, based on all experiments

carried out to date, shows power law dependence for the mixing layer width, h ∼

tθ where θ < 1. The experimental and computational studies that exist on the

subject have found widely varying values of θ and have indicated that the growth

exponent is dependent upon the form of the initial conditions. However, the state of

understanding of the dependence of RMI on initial conditions is far from settled.

Both, the RMI and RTI are of great fundamental interest in fluid mechanics and

physics as well as to industry. In nature, the RMI has been observed in supernova

explosions [48], [65] where the shock wave from the collapsing core passes through

layers of different elements initiating their mixing on the interfaces. The RMI is

19

also of importance in supersonic combustion ramjets [87] where interactions between

shock fronts and the fuel-air mixture enhances the mixing, thus altering the burn

rate. The study of this phenomenon is the key to achieving a stable flame location

within the combustion chamber.

The RMI is one of the focuses of study at the National Ignition Facility at the

Lawrence Livermore National Laboratory (LLNL), which houses the largest Inertial

Confinement Fusion project. The NIF utilizes a spherical Deuterium-tritium (DT)

capsule and the goal of this project is to sufficiently compress the DT to initiate

a fusion reaction, generating more energy by fusion than in needed to achieve the

compression. The DT capsule is located at the center of a small gold cylinder, called a

hohlraum, which is illuminated simultaneously by 192 high-powered laser beams. The

laser energy striking the hohlaum results in nearly uniform incident x-ray radiation

on the DT capsule. The radiation oblates the shell of the DT capsule, resulting in an

inward propagating shock. The surface imperfections cause the inward propagating

shock waves to interact with the inner layers of fuel, resulting in the mixing of the fuel

rather than compression. This effect disrupts the compression process, preventing it

from achieving sufficient compression.

1.2 Past research

The first theoretical study of the stability of fluids with different densities in a grav-

itational field was published in 1883 by Rayleigh [100] who was able to derive the

stability conditions and calculate the growth parameters for unstable stratification.

This work was continued by Taylor [114], who derived the velocity potential for the

unstable flow; assuming that the constant gravitational field was acting on the un-

stable interface. Richtmyer [103] extended Taylor’s work to the case of impulsive

20

acceleration. He showed that in initial small harmonic perturbation would grow lin-

early immediately after the impulsive acceleration, and that the growth rate was

proportional to the initial amplitude and the Atwood number.

Taylor considered a system of two overlaying liquids of different densities, the

lighter liquid with the density ρ1 on a top of the heavier liquid ρ2 respectively. The

system is subjected to the constant acceleration of gravity having only one component

~g = (0, g2, 0) that is normal to the interface separating the liquids. In the cartesian

coordinate system (x, y, z) with the origin at the interface the governing equations

of motion have body force terms of the form −g2ρ1y for the upper liquid and −g2ρ2y

for the lower liquid. Considering a small harmonic wave imposed on the interface as

Figure 1.3: Drawing of two-liquid system with different densities.

shown in Fig. 1.3 in the inviscid limit, assuming that the perturbation wavelength in

much less than the depth of the liquid, the velocity potential has the following form

Φ1 = Ae−ky+nt cos kx, (1.3)

Φ2 = −Aeky+nt cos kx, (1.4)

where k is the wavenumber of the small harmonic perturbation and n is the growth

exponent. The surface separating the fluids is expressed as follows

η = Akn−1ent cos kx. (1.5)

21

The assumption is also made that the fluid velocity is small enough that the nonlinear

terms in the Bernoulli equations can be neglected. If the system is accelerated

downward with the acceleration g, the growth constant is given by expression

n2 = −K(g + g2)ρ2 − ρ1

ρ2 + ρ1

. (1.6)

If the downward acceleration is greater than the gravity g2, n2 is positive since

(g + g2) is negative and the perturbation of the interface between the two liquids

grows exponentialy and is thus unstable. For this case the solutions of (1.3) and

(1.4) can be rewritten into the form

Φ1 = (Ae−ky+nt +Be−ky−nt) cos kx, (1.7)

Φ2 = (Aeky+nt +Beky−nt) cos kx. (1.8)

If one assumes that the initial perturbation of the interface has the form

η0 = C cos kx, (1.9)

η0 = 0 (1.10)

The interfacial perturbation and the evolution of the velocity potential is expressed

as

η = C coshnt cos kx, (1.11)

Φ1 = Ae−ky(e−nt − e−nt) cosKx =Cn

ke−ky sinhnt cos kx; (1.12)

Φ2 = −Cnkeky sinhnt cos kx. (1.13)

Finally, it can by shown that the amplitude of the small harmonic interfacial pertur-

bation obeys the equation

d2

dt2a(t) = kg(t)a(t)

ρ2 − ρ1

ρ2 + ρ1

. (1.14)

22

In the case when surface tension and viscosity is negligible Eq. (1.14) holds under

linearity condition ka� 1. One of the properties of linearity is that if several waves

with different wavelengths are imposed on the surface they will grow independently

of each other as long as the condition is satisfied. The linear theory can be extended

to the case of shock wave interaction with the perturbed interface integrating the

equation 1.14 over the impulsive acceleration. In his derivation Richtmyer assumed

that the compression effects by the shock wave are negligible and initial amplitude

remains the same. Since after the shock propagation the interface accelerates to the

value∫g(t)dt = ∆V ,

da

dt= k∆V a0

ρ2 − ρ1

ρ2 + ρ1

. (1.15)

Thus the perturbation growth rate is constant and proportional to the post-shock

Atwood number, wavenumber, post-shock amplitude, and the piston velocity.

The first experimental study of the impulsively accelerated interface between

two gases was done by Meshkov [72]. He performed experiments using a horizontal

shock tube where a shock wave interacted with an interface formed between two

different gases using a thin film to separate the gases. Meshkov studied both the case,

when shock wave traveled from the light gas to the heavy one, and when the shock

traveled from the heavy gas to the light gas, In both cases he found the perturbation

amplitude to grow linearly in time. The experimental apparatus used by Meshkov

consisted of a volume region separated by two 1-micron-thick films. The films had

pre-determined shapes of harmonic function with initial amplitudes of a = 2mm

and a = 4mm, and with the wavelength of λ = 40mm. During the experimental

preparation, the air initially occupying the space between the two membranes was

displaced by the test gas that flowed from the top or the bottom of the volume

depending on the relative density. The purity of the displaced gas was determined

23

with 5% accuracy with respect to mass. This was achieved with help of an air

gap 100 µF condenser. As different gases flowed between the plates, the dielectric

permeability of the medium changed, thus changing the capacitance. Measuring the

capacitance yields the purity level between the condenser’s plates in the test region.

During the study, Meshkov used several gas combinations, resulting in the wide

range of the Atwood numbers. The experiment employed the Schlieren visualization

technique and a high-speed camera which allowed recorded images with 64 µsec

intervals. During Meshkov’s experiments it was observed that the interfaces were

non-sinusoidal after it raveled a distance equal to l = 2λ and would have a more

complicated form; where spikes would form, a filament shape and bubble would take

the form of protuberances which were broadened at the tip. This effect seemed to

occur at earlier stages for the gas combinations with the higher Atwood number. The

experiment showed good qualitative agreement between theory and the experiment,

but the qualitative differences were more than 10%, which could be related to the

experimental conditions differing from that of the idealized theoretical derivations.

The Richtmyer-Meshkov instability has been investigated by Dimonte and Rem-

ington [29] using the NOVA laser. These experiments are characterized by high

compression, for which the pre and post-compression amplitudes were averaged to

compare the instability evolution to the theoretical predictions. The sinusoidal inter-

facial perturbation in a planar two-fluid target was subjected to a high compression

shock wave, and the mixing region was visualized using 2.6 keV Monochromatic X-ray

photography. The Rayleigh-Taylor instability was suppressed by delivering sufficient

power to maintain the constant interface velocity. The experiment was radiographed

from two directions, one to measure the perturbation evolution (face-on) and the sec-

ond 90°offset for shock characteristics (side-on). During the experiment, eight NOVA

laser beams delivered 30 kJ into a cylindrical hohlraum which created uniform X-ray

24

drive to the target mounted in a hole in the hohlraum wall. Experimental results

are described for two different payloads to achieve two different Atwood numbers.

Berillium was used as the ablator which allowed for an Atwood number less than

zero, the ablation region was transparent to the X-rays. The highly unstable target

had pre-shock Atwood number A=-0.9 and used AGAR foam consisting of CH2O

doped with Na2WO4. A more stable target had pre-shock Atwood number A=-0.3

and used CH payload with a Br dopant. The instability growth was measured for

a sinusoidal initial interface with wavelength λ = 100µm and initial amplitude of a

ν0 = 14µm. Since the experiments were performed with an Atwood number less than

zero, the characteristic phase inversion was observed prior of instability development.

For experiments with Be/CH the post shock Atwood number was ∼ 0.05 and for Be

AGAR A∼ 0.68. The experiment was able to show that the growth agreed with the

Mayer-Blewett model [70] for negative Atwood numbers, which is based on averaged

pre- and post-compression amplitudes.

Brouillette and Sturtevant [16] addressed the growth of the interface between two

gases separated by thick interface that is subjected to the impulsive acceleration by

incident shock wave. The experiments were designed to investigate the time evolu-

tion of the single scale perturbation with schlieren photography. These experiments

showed that growth rate decreases as density gradient decreases at the interface and

the given observation was compared to the theoretical derivations.

The effect of the membrane, which separated the experimental gases, on the RMI

evolution was experimentally investigated by Vetter and Sturtevant [122]. In previous

studies, it was observed that boundary layers growing in the test section walls had

a large influence on the measured growth rate; therefore a larger test section was

employed to reduce this effect. Observations of the membrane effect were obtained

by using different orientations of the knife edge of the schlieren system, and by

25

fragmenting the membrane with a wire mesh placed downstream. The experiments

were performed in a squared test section of 27 × 27 cm cross-section. In order to

obtain a high Mach number without over-pressurizing the test section the initial

pressure was reduced below atmospheric. This method required the evacuation of

both, the tube and test section, before it could be filled with the desired gases.

Experimental observations showed that the membrane accelerates more slowly than

the interface, and that the developing boundary layer creates streaky structures that

are visible behind the mixing layer. The effect of the membrane on the visualization

was isolated by rotating the schlieren knife 90◦. It was shown that the mixing rate was

dependent upon on the position of the wire mesh relative to the membrane. Placing

the mesh downstream guaranteed that the membrane will rupture into small pieces.

The observed mixing width was larger than when the mesh was placed upstream

relative to the membrane. During this study, it was observed that the reflected

shock wave interaction followed by the development of the mixing layer width, was

not influenced by the membrane showing a linear growth behavior as it was predicted

by theory.

A study of the development of the mixing layer width in time has been performed

by Houas and Chemouni, [46]. The experiments were carried out using a double di-

aphragm 9 m long shock tube with a test section having a 8.5 × 8.5 cm2 square cross

section. Various test section lengths were used ranging from 0.80m to 1.52m. The

experiments employed several driver gases, depending on the desired Mach number

(hydrogen, helium, nitrogen). Gases used to study the RMI evolution were carbon

dioxide or air upstream and helium or carbon dioxide downstream. A heat trans-

fer gauge was used to determine the incident shock Mach number with up to 3%

accuracy. One set of experiments with high Mach number used the asymmetric vi-

brational modes of CO2 emission to measure the mixing layer width. A second set of

26

experiments with a small Mach number used the schlieren visualization technique. In

past experiments monochrome schlieren had been used to obtain information regard-

ing shock tube flows, but monochrome schlieren was not able to provide information

regarding complex flow structures. The color schlieren technique where the variation

of density corresponds to a color variation in the schlieren image, was used for more

detailed study. This study addressed the effects of the boundary layer and its effect

on the instability development. It was found that the boundary layer stretches the

mixing zone toward the walls and if the length of the test chamber is sufficient, it

induces the thinning of the mixing layer. In addition, Houas and Chemouni were

able to measure the evolution of the interface after interaction with the reflected

shock wave. Previous experimental and theoretical studies showed that the interface

evolves in linear fashion after being re-accelerated by the reflected shock wave; but

the results obtained during this study show a better fit to a power law rather than

linear.

Early experiments in RM instability employed a physical barrier between the

two gases or liquids at the initial stage to prevent them from mixing, creating a well-

defined initial condition from which the instability would grow. But these techniques

introduced additional difficulties during the experiments. Once the membrane is

ruptured as instability is initiated, the pieces of membrane influence flow pattern

and complicates flow visualization. in addition, the growth rates obtained using

this method differs from that found in theoretical predictions. Another method to

initially separate two gases or liquids is to use a thin plate which was removed before

initiating instability. The shortcomings of using this technique are that the wake

produced by removal of the plate leaves the initial interface non-uniform, ill-defined

and significantly diffused. Furthermore, it is hard to reproduce the identical initial

conditions from experiment to experiment. As the initial conditions are responsible

27

for how instability grows, it makes it difficult to compare these experiments with

others.

The Richtmyer-Meshkov instability of a two-liquid system was investigated exper-

imentally by Jacobs and Sheeley. [51] in which they introduced a novel technique that

circumvents many of the previous limitations associated with the study of Richtmyer-

Meshkov instabilities. Their experiment improved upon the existing ones by having

a well-defined, sharp initial interface. The low speeds of the instability allowed the

use of conventional video imaging that captured the non-linear behavior of the flow,

along with formation and development of the mushroom-shaped vortex structures.

In these experiments the initial interface was created by the horizontal oscillation of

the tank, resulting in well defined, two-dimensional sinusoidal standing waves. Even

though that initial state is not motionless. Its affect on the instability growth rate is

insignificant as the initial velocity is much less than the velocity deposited by insta-

bility. Furthermore, the initial velocity can be measured accurately, thus its effects

can be included in comparison with the stability theory. The experimental appara-

tus consisted of a thin rectangular tank with inside dimensions of 2.54 cm 311.75 cm

325.4 cm. The tank was free to move in the vertical direction by mounting it to a

vertical linear rail system with 0.5 m of travel distance. Two sets of liquid combina-

tions were used: the Ca(N03)2 solution/water system provided an Atwood number

of 0.15, while a NaI solution/water system had an Atwood number of 0.03. The

experiment was initiated by releasing the tank above a fixed spring, thus allowing

the tank to fall and bounce off of the spring. The only body force experienced by

the system was a bouncing force with duration of approximately 50 ms. As it was

in free fall before and after interacting with the spring, no other body force acted on

the system. Therefore, the system underwent an impulsive acceleration resulted in

the generation of the RMI. The experiment used a shuttered CCD camera operated

28

at 60 frames per second resulting in 30 frames capturing the evolution of the insta-

bility. The camera was attached to the moving container, thus being fixed relative

to the tank. The obtained instability growth rates were compared with the linear

theory developed by Richtmyer, as well as with a modified theory accounting for

a finite duration of the pulse acceleration. The obtained experiments collapse to a

single curve based on the circulation deposited by the velocity change, indicating the

close correlation between growth rates and initial vorticity values. A model based

on vorticity deposition was also developed to capture the late-time development of

instability. This model predicts the shape of the growth curve, but underestimate its

value. The study obtained an expression for the deposited vorticity for single mode

2D interface by considering an interface between two different fluids with an imposed

periodic perturbation. According to (1.2) if the interface is subjected to an impulsive

acceleration, the vorticity is deposited on the interface due to the misalignment of

the pressure and the density gradients.

D~ω

Dt=

1

ρ2~5ρ× ~5p (1.16)

The linear theory can be used to calculate the deposited vorticity if the perturbation

amplitude is much smaller than the corresponding wavelength. In the limit of a

sharp interface the strength of the vortex sheet γ is equal to the jump of tangential

velocity across the interface

γ =∂Φ2

∂x|y=0 −

∂φ

∂x|y=0 = −2a sin kx. (1.17)

We have already seen that initially the perturbations in RMI will grow linearly in

time. However when the amplitude of the perturbation becomes comparable with

the corresponding wavelength, nonlinear effects will take place. While the instability

develops the deposited vorticity rolls up into concentrated vortices. The vorticity

29

deposited on the half-wavelength of the interface can be estimated from the evaluated

strength of the vortex sheet

Γ =

∫ π/k

0

γdx = −4

ka0+. (1.18)

Experiments [51] show that the vorticity deposited on the interface will grow into

vortices. The nonlinear stage of the instability can be modeled as a row of counter

rotating vortices. Thus the motion of the interface is induced by a counter rotating

array of vortices placed at points midway between the crests and troughs. The stream

function is given by the expression

φ =Γ

4πln

(cosh (ky) + sin (kx)

cosh (ky)− sin (kx)

). (1.19)

The region midway between the vortexes will tend to be pushed perpendicular to

the initial interface plane, resulting in growth of the instability. The regions located

midway between the vortex lines x = 0,±πk,±2πk, ... will have the velocity

v =1

2π

kΓ

cosh (ky), (1.20)

and this expression can be integrated to obtain the mixing width as a function of

time

a(t) =1

ksinh−1

(2

πk ˙a0+t+ sinh ka0

). (1.21)

RMI experiments that studied the evolution of the mixing layer width using an

X-ray absorption technique by were performed Bonazza, and Sturtevant [10]. The

object of the study was to observe the turbulent mixing of an air/xenon gas interface

following interaction from a shock wave. Initially, the two gases were separated

by a metal plate. The disturbance of the gases by removing the plate, due to the

no-slip condition on the plate surface, created quasi-sinusoidal perturbations on the

interface. The action of removing the plate dragged the fluid flow back towards the

30

opposite wall, forming gravity waves. Depending on the time elapsed, between the

plate removal and the incident shock arrival at the interface, resulted in different

initial conditions. The shock tube employed in this study had a 2.04 m long driver

section, with an inside diameter of 16.5 cm, a driven section of 7.12 m followed by

a test section with an 11.4 cm square inner cross-section and 38.7 cm long. For

X-ray visualization, the windows were replaced by a 1 mm thick carbon fiber plates,

supported from the incident shock pressure by a pair of dip-brazed aluminum grid

structures. The generated x-ray burst had a duration of 50 ns and a 6◦ divergence.

The same approach was also used to perform experiments with schlieren imaging.

In this case, the windows used were made of optical grade glass, 38 mm thick, and

the setup consisted of a standard Z-folded layout. The experiments were performed

with two incident shock Mach numbers: M = 1.52 and M = 1.32. Shortly after

the incident shock wave interaction with the interface, the images obtained by this

technique showed rapid transition from the light region to the gray representing

the experimental gases respectively. As the instability developed, the middle region

between light and gray gradually grew. The reflected shock interaction with the

interface showed three or four well-defined crests emerging on the interface which

grew to the order of the wavelength of the perturbations. This study found instability

growth rates that were generally smaller than predicted by the impulsive model, and

this was attributed to the fact that the small amplitude perturbation assumption

is not valid for the observed initial perturbations. After re-shock, measured growth

rates were found to be comparable to those of previously reported studies. The X-ray

technique was also validated and compared to the schlieren measurements.

The Bubble merger model was experimentally tested in an horizontal shock tube

by Sadot et al. [85]. The experiments were performed on an initially perturbed air-

SF6 interface, which was accelerated by an incident shock wave with a Mach number

31

of M=1.3. A double-diaphragm square shock tube with length of 7.5 m and an inner

cross section of 8 cm × 8 cm was used for this study. Experiments were initiated from

two sets of initial conditions; one from a single mode perturbation and the second

from a two bubble shape perturbation. The first set was used to study the single

mode bubble and spike evolution while the second was used to observe the bubble

competition phenomenon. The initial perturbations were created using a 0.1 µm

nitrocelluloid membrane that separated the experimental gases. The desired initial

perturbations were achieved by stretching the membrane over thin copper wires. The

evolution of the interface was observed using the schlieren method and was recorded

at a 10 kHz frame rate. The second order expansion of the flow equations captured

the evolution of the interface in linear, early non-linear and asymptotic behavior of

the bubble and spike regions.

U(t) = U01 +Bt

1 +Dt+ Et2, (1.22)

where B = U0k for both regions, Db/s = (1± A)U0k and Eb/s = [(1± A) / (1 + A)]×

(1/2πC)U20k

2. This formula captures the interface evolution of the A ≤ 0.9 with the

limits of C = 1/3π for A ≥ 0.5 and C = 1/2π for A → 0. The single mode

experiments were performed with an initial perturbation having 2 mm amplitude

and wavelengths of 80, 40, 26, and 16 mm. The observed spike and bubble evolu-

tion was plotted in dimensionless units (h− h0) k vs U0kt, which agreed with the

analytical formula obtained when using C = 1/3π. It was suggested that as the

analytical derivation is based on the incompressible theory then the evolution of the

Richtmyer-Meshkov instability is mainly determined by incompressible effects after

the shock wave passes the mixing region. The bubble-competition experiments were

performed by using an alternating pattern of large (25-27 mm) and small (10-17 mm)

bubbles. The obtained experimental data was shown to agree with the theoretical

32

and computational results. By observing the shape of the small and large bubbles as

they evolve, in addition to the orientation of the spikes that skew toward larger bub-

bles, the bubble-merger process was identified. In the initial linear stage, two bubbles

evolve without interacting with each other but at the late non-linear stage interaction

takes place, which results in faster growth of the larger bubble and reduction of the

smaller bubble. The bubble-competition phenomenon was also observed by measur-

ing the width of the two bubbles as they evolved in time. The larger bubble increased

in width while the smaller bubble reduced its size. The above mentioned experimen-

tal study was extended by including the effect of re-shock on a well-developed RM

instability [86]. This experiment used all the same techniques to create the initial

conditions. The experiments were performed on air-SF6 gas pair with the incident

Mach number of M=1.3. The end wall of the test section was located at 24 cm from

the initial interface. The transmitted shock wave was reflected from the end of the

experimental apparatus and interacted with the traveling interface with a velocity

of 190 m/s. Experimental observations showed that the perturbation developed an

asymptotic bubble-spike shape before it interacted with the reflected shock wave.

An immediate phase inversion was observed following the re-shock interaction turn-

ing spikes into bubbles. This resulted in small bubbles flanked by larger bubbles,

initiating the bubble-competition process. The observed phenomenon was computa-

tionally modeled matching the experimental observations. This work demonstrated

the uniqueness of the neighboring bubble interaction phenomenon for shocked and

re-shocked regions.

The study of the turbulent mixing intensity of a gaseous mixture was conducted

by Poggi et. al. [36]. The experiments were conducted on air-SF6 gas pair with an

incident Mach number of 1.45 to obtain the axial velocity profile of the turbulent

mixing zone. The measurements were performed using a laser Doppler anemome-

33

ter (LDA), which measures velocity and is independent of the viscosity and thermal

conductivity properties of the gases. The experimental study was focused on de-

termining the turbulent intensity after initial shock interaction with the interface,

and its evolution after the turbulent mixing width is de-accelerated by the reflected

shock wave. The LDA probe was placed at 51, 125, 161 169 and 178 mm downstream

from the initial interface location and the end of the test section was located at 30

cm from the initial interface location. The two gases were seeded by two different

particles. Carbon particles, the product of burning incense was used to seed air;

while diffuser generated droplets of olive oil were used to seed SF6. Observations at

51 mm downstream showed 6% of the RMS flactuation. At 125 mm downstream the

axial RMS velocity decreased below the measurable level, indicating that turbulence

has been dissipated at this point. At 161mm downstream a strong increase in axial

velocity variance was observed due to the reflected shock wave interaction with the

interface, and the deposition of additional baroclinic vorticity. These observations

revealed that initial small-scale perturbations had developed into a turbulent mixing

layer after the interface was accelerated by the incident shock wave. The developed

turbulence level decayed before it interacted with the reflected shock wave. Follow-

ing the reflected shock interaction the turbulence amplified significantly due to the

additional vorticity deposition.

Holmes [102] performed a comparison of numerical simulations and analytical

models with the experimental results obtained by NOVA laser experiments. The RMI

was studied for negative Atwood number (Shock wave propagating from the heavy to

the light material) and two dimensional perturbations with strong radiatively driven

shocks with Mach number greater than 10. The mixing layer width evolution was

modeled by three hydrodynamic codes (RAGE, PROMETHEUS and FrontTier).

Also the analytical model proposed by Zhang and Sohn [98] was used for comparison

34

with the computational and experimental results. The experiments were performed

using beryllium and foam. These experiments do not require membranes for initial

separation of the substances and thus are highly useful for numerical validations. This

study also provides data at early and late time scales. On the other hand, other effects

in the experiments, such as radiation, were not included in the numerical simulations,

which modeled fluids as a perfect gases. The results of this study indicated that linear

theory does a good job at predicting the instability evolution at the initial stages

while ka < 1. It was also shown that the Mayer-Blewett model [70] predicted the

evolution of the instability at the initial stage better than the Richtmyer model.

Even though the numerical simulations used the ideal gas assumptions, they were

able to accurately predict the development of the instability.

Sohn [113] developed an analytical solution modeling an interface in a channel of

width L filled with two incompressible and inviscid fluids with different densities. In

this case the bubble pushes the upper fluid and has a rounded shape, while the spike

penetrates the lower fluid and has a mushroom shape due to the vortex generated

by the Kelvin-Helmholtz instability. The interface velocity is defined as average of

the upper and lower interface velocities expressed by vector potential. Additionally

the interface at the bubble tip is assumed to have parabolic shape.

Goncharov [42] considered the RTI of irrotational, incompressible, inviscid fluids

in two dimensions. The fluids are subjected to an external acceleration g directed

from the heavier into the lighter fluid, which generates the unstable stratification.

The solution for the bubble shows that the bubble speed approaches the asymptotic

value

Ub →√

2A

1 + A

g

3k. (1.23)

Applying the given model to the Richtmyer-Meshkov instability, the asymptotic bub-

35

ble velocity in this case becomes

Ub →3 + A

3 + 3A

1

kt. (1.24)

Collines and Jacobs [23] conducted RMI experiments in a vertical shock tube.

An interface was formed by introducing light and heavy gases through plenums lo-

cated at the top and bottom of the driven section. The gases meet and exited the

shock tube via slots located near the top of the test section. The stagnation flow was

created by matching the experimental gases flow rates to 6 l/min resulting in a flat

defuse interface with thickness of∼ 5 mm. Standing waves were created by oscillating

the shock tube horizontally with appropriate frequency. The resulting mixing layer

evolution was visualized by seeding light gas with acetone vapor and illuminating it

with UV laser sheet (266nm). The resulting fluorescence was captured by a thermo-

electrically cooled CCD camera. Due to the low visualization and acquisition rates

only one image was recorded for each expedient. The pressure transducer located

above the gas-gas interface was used to trigger a digital delay generator that subse-

quently triggered the laser and camera. Changing the delay allowed the capture of

the developing interface at different times relative to the shock-interface interaction.

The obtained images were corrected for laser illumination attenuation due to Beer’s

law resulting in gas concentration measurements. RMI was studied for two incident

shock wave Mach numbers M=1.11 and 1.21. Several images were captured at each

time level to observe the repeatability of the experiment. The study used 18 and 20

time levels to study the evolution of the mixing layer width in time. It was observed

that as the amplitude approached in size to the wavelength the interface shape de-

veloped asymmetry due to the large Atwood number. Later in time cusps appeared

that developed into mushroom-shaped structures. The experiments performed with

an incident shock wave having M=1.21 grew at twice the rate as the experiments

36

performed with M=1.11, resulting in much more rollup than with the lower Mach

number. The interface experiences additional accelerations later in time one by the

reflected expansion wave from the top of the driver section and the second by the re-

flected shock wave traveling from the bottom of the test section. The reflected shock

waves deposits additional vorticity and produces much more mixing of the gases.

The evolution of the mixing layer width at early times was in good agreement with

Richtmyer’s incompressible model with the diffuse-interface correction. In addition

very good agreement was achieved for late time growth with the nonlinear models of

Zhang and Sohn [98] and Sadot et al. [85].

Niederhaus and Jacobs, [83] extended the experimental study by Jacobs and Shee-

ley [51] by the use of planar laser-induced fluorescence (PLIF) imaging that provided

the ability to obtain a better understanding of the developing instability by looking

only at a two-dimensional slice of the interface. The experimental technique allowed

one to eliminate the edge effects and other unwanted factors from the recorded im-

ages. The time duration at which the instability is observed was also extended by

retracting the spring system after the tank had bounced of it. These experiments

also improved upon the initial condition formation system by allowing more complex

perturbations from which the instability was initiated. Multimode initial perturba-

tions were generated by oscillating the container horizontally using the superposition

of the oscillations required for two or more individual modes, resulting in well-defined

perturbations. The fluids used in the experiment were a water/isopropanol mixture

as the light liquid and a water/calcium nitrate salt solution as the heavy liquid, re-

sulting in an Atwood number of 0.16. Disodium fluorescein dye was added to the

heavier fluid, which fluoresces when illuminated by the laser sheet. The measured

instability amplitude was found to be in a good agreement with the linear theory, and

also with the solution of Zhang and Sohn [98] when the data are nondimensionalized

37

by making use of the dimensionless time scale ka0t. For the late-time development,

measured amplitudes were compared with the vortex model introduced by Jacobs

and Sheeley [51] and a model developed by Sadot et al. [86] and were found to agree

to within 10% accuracy and 30% accuracy respectively. The study was the first one

to investigate the effect of the Reynolds number on the RM instability, showing that

the Reynolds number has insignificant influence on the instability growth rate. The

main effects are confined to the vortex cores concluding that a decreasing Reynolds

number lowers the vortex turning rate, and a higher Reynolds number creates the

secondary instabilities in the vortex cores. According to their observations, at a high

Reynolds number, the secondary instabilities appear when the core has made three

complete rotations.

The vortex model developed by Jacobs and Sheeley [51] was extended for finite

Atwood numbers by Likhachev and Jacobs [63]. The previous model assumes that

line vortices are evenly spaced in a vortex row with the distance between two vortices

of opposite sign equal to a half of the wavelength of the initial interface perturbation.

This results in the stream function given by Eq. (1.19) where the circulation Γ is

defined by Eq. (1.18). In the modified model the initial vortex configuration is

perturbed by shifting the positive vortices by +ε and the negative vortices by −ε

yielding a new stream function of the form

Ψ =Γ

4πln

(cosh(ky) + cos(k(x+ ε))

cosh(ky)− cos(k(x− ε)).

)(1.25)

Based on this stream function the vertical velocity can be expressed as

v = −∂Ψ

∂x=

(kΓcos(kε)

2π

)cosh(ky)sin(kx)− sin(kε)

[cosh(ky)− cos(k(x− ε))] [cosh(ky) + cos(k(x+ ε))].

(1.26)

The model predicts that the perturbed vortex row will move upward in the direction

38

of the light fluid with the constant velocity

V =kΓ

4πtan(kε). (1.27)

In the system moving upward with the constant velocity V, the bubble and spike

interface locations will evolve in time according to the formula

∂ηs,b∂t

=kΓ

2π

(± cos(kx)

cosh(kηs,b)∓ sin(kε)− 1

2tan(kε)

). (1.28)

Latini et al. [109] modeled the re-shocked single-mode Richtmyer-Meshkov exper-

iments performed by Collins and Jacobs [23]. The instability was simulated in two

spatial dimensions using the fifth- and ninth-order weighted non-oscillatory shock-

capturing method. The initial conditions were matched to the experimental ones.

Single-mode air(acetone)/SF6 interfaces were subjected to a shock with a strength of

Mach 1.21. The numerical results were compared with the planar laser-induced fluo-

rescence experimental data. In addition, the model of Zhang and Sohn [98] showed a

good agreement with the simulated amplitude before re-shock. The qualitative com-

parison of the density evolution showed a good agreement between the experiment

and the simulation, even though a small difference between mushroom “stems” and

width of the “caps” were observed. The experiment had more pronounced structures

inside the roll-ups, which was not present at the fifth-order simulation, but was cap-

tured by the ninth-order simulation. This difference was explained as the effect of

the molecular dissipation in the fifth-order simulation being larger than in the ninth-

order simulation, resulting in suppression of the secondary instabilities. Also several

linear and nonlinear perturbation amplitude growth models were compared to the

simulation results. The impulsive (linear) models captured the development at the

initial stage but over estimated it afterward, as it does not account for non-linear

effects present in the system. Since the linear models predict a constant growth rate,

39

the perturbative approach is used to characterize the growth behavior. It was found

that all non-linear models including those of Zhang-Sohn [94] and Vandenboom-

gaerde [121], [68] provide a good agreement with the experimental and simulation

data both in the linear and nonlinear regimes and thus encourages the use of the high-

order WENO methods for the quantitative investigation of shock-induced complex

hydrodynamic flows and mixing.

A combined experimental and numerical study of the growth of the Richtmyer-

Meshkov instability following re-shock was conducted by Leinov et al. [35]. A 5.5

m long horizontal shock tube, having internal dimensions of 8 cm × 8 cm was used

to produce shock waves of low to moderate Mach numbers in air (M∼ 1-3). The

shock wave impacted the interface formed using a nitrocellulose membrane, which

separated air from SF6 resulting in the production of the RMI and its transition to

the turbulent regime. Experimental images were obtained by the schlieren method

using a high-speed rotating drum camera at an acquisition rate of 50 kHz. The light

source used was a Nd: YAG laser with duration of 240 ns. Since the focus of the

study was to investigate the evolution of the RMI following the de-acceleration by

re-shock, the conditions under which re-shock occurred were varied. Two config-

urations for producing the re-shock conditions were used. The first configuration

consisted of changing the mixing layer width at the time at which re-shock occurred,

while keeping the same re-shock Mach number. This was achieved by placing the

end wall at different distances from the initial interface location, thus resulting in

different evolution time of the instability initiated from the incident shock wave.

The second configuration consisted in changing the strength of the reflected shock

wave, while keeping the incident shock wave strength and evolution time constant.

This was attained by using several elastomeric foams as the end wall. The numerical

comparison was done with LEEOR3D, which is a finite volume ALE hydrodynamics

40

code with interface tracking capacity, solving the inviscid Euler equations. The com-

putation was performed with cross section of 128 × 128 numerical cells representing

10 mm × 10 mm domain. The imposed perturbation had at least eight cells per

wavelength. The aim of the numerical study was to obtain the large-scale evolu-

tion, which dictates the mixing zone behavior. The numerical results included the

instability development prior to and after re-shock. Good agreement was achieved

between the numerical simulations and the experiments at different incident Mach

numbers (M=1.15, 1.4 and 1.5) and at different end wall positions (80 mm, 172

mm and 235 mm). The experiments exhibited good repeatability of the instability

evolution under fixed conditions. During the experiments the values of the incident,

transmitted and reflected shock wave speeds were monitored and were found to be

within 5% of one-dimensional calculations. From experiment to experiment, vari-

ation of the mixing layer width was observed and was attributed to the turbulent

nature of the mixing. All experimental configurations showed that the mixing layer

width appeared to grow linearly with time following the reflected shock interaction

until it was subjected to a reflected rarefaction wave from the end wall. Experiments

clearly indicated that the rate of the mixing layer width evolution was independent of

its width, prior to the re-shock interaction and that it only depended on the strength

of the reflected shock wave.

A numerical study conducted by Dimonte and Ramaprabu [28] was intended to

understand the non-linear evolution of the single mode Richtmyer-Meshkov instabil-

ity for all Atwood numbers and initial amplitudes. The study employed the FLASH

hydrodynamic code in two dimensions, which was based on an adaptive compressible

hydrodynamics code that solves the Euler equations using the piecewise parabolic

method. Numerical results were compared to experiments and nonlinear models.

For experimental comparison, researchers choose 2D RMI studies performed on the

41

NOVA laser by Dimonte et al. [29], vertical shock tube experiments performed by

Jacobs and Krivets [53], and drop tower experiments performed by Niederhous and

Jacobs [83]. The experimental results along with the numerical simulation were com-

pared to the 2D non-linear models proposed by Mikaelian [77], Zhang and Sohn [98]

and Sadot et al. [85]. All three of the models are able to describe the evolution of

the spike and bubble regions from moderate values of initial conditions and Atwood

numbers, but failed to agree at larger initial amplitudes and Atwood numbers A ∼ 1.

For this reason, the authors introduced a new empirical model to describe the RM

instability evolution for large initial amplitude and Atwood numbers. The numerical

results agreed well with the experimental results even though they were performed

at different conditions. The simulation revealed that spikes accelerate for A > 0.5

due to higher harmonics that focused them, and bubbles saturated when they grow

by 2/k in amplitude. The obtained results exhibited excellent agreement with sim-

ulation and the linear theory at early stage. When comparing the simulation and

non-linear models, agreement was demonstrated for moderate Atwood number and

amplitude; however, they diverged for large values A ≥ 0.9 and kh > 1. The authors

outlined the different nature of the RM instability evolution for moderate and high

Atwood numbers. The results showed that for moderate Atwood numbers and small

initial amplitudes both, the spike and bubble, begin growing initially with the value

defined by the linear stability theory and then transition to the asymptotic value

predicted from potential flow models. However, at large initial amplitude the growth

begins at a lower rate than that predicted by the linear theory, even though the

asymptotic value remains the same. At large Atwood numbers, the bubbles evolve

in the same fashion as that for moderate and small Atwood numbers. The spikes

however, differ as they are accelerated. The authors attribute these differences to

the effects of harmonics at non-linear stage due to the focusing at the spikes and

42

broadening at the bubbles, effectively increasing the spike velocity over that of the

bubble.

Bonazza et al. [21] investigated the transitional period and the turbulent state of

the RM instability using PLIF. Two-dimensional images provided information about

the light gas mole fraction, revealing the change in length scales and evolution of

density gradients within the mixing layer. The experimental apparatus consisted of

a vertical shock tube with height of 9.13 m. The driver had a circular cross-section

with 0.41 m diameter and 2.08 m length, while the driven and test sections had a

square internal cross section of 0.25 m × 0.25 m. The experimental gases utilized

in this experiment were a helium acetone mixture, as the light gas and argon as the

heavy gas; resulting in Atwood number of A=0.70. The gases meet and stagnate

next to the slots in the shock tube wall connected to a pair of vacuum pumps, which

removed the excess of gas from the shock tube. The initial perturbation from which

the instability evolves was created by injecting the helium acetone mixture and pure

argon horizontally through separated slots below and above the stagnation level,

respectively. It was determined that the given perturbations maintained repeatable

scale content in time. Recorded experimental images included an image of the initial

interface and two subsequent images of the instability after it was accelerated by the

incident shock wave with Mach number of M=1.57. Two images were taken at 27

cm and 67 cm below the initial interface, which corresponds to 0.88 ms and 2.1 ms

after shock arrival, respectively. The mixing layer width was measured as the distance

between 5% and 95% horizontally averaged mole fraction profile. These experimental

observations confirmed the previous experimental [23] and theoretical findings [77]

regarding the evolution of the re-shocked interface. The growth rate of the re-shocked

interface evolution was within the data reported by previous observations. The

mixing layer width was characterized by an alternate measure, called the entrainment

43

length, he, which is defined as

He =

∫ +∞

−∞Xm(< X >)dz, (1.29)

where

Xm(X) =

{2X for X ≤ 0.5

2(1−X) for X > 0.5, (1.30)

is the mixture composition. This definition does not differentiate between mixed

and unmixed gas, but only provides information regarding interpenetrating gases.

Therefore, it is compared to the ratio of the mixture composition field.

Θ =

∫ +∞−∞ < Xm(X) > dz∫ +∞−∞ Xm(< X >)dz

. (1.31)

The experimental results indicate that Θ asymptotes to value of 0.8, which was

also reported by previous researchers who conducted the same measurements for the

Rayleigh-Taylor instability. The turbulent regime for RMI was also characterized by

observing the one-dimensional energy spectra. The one-dimensional energy spectra

E(k) was computed for regions where turbulent mixing is expected to be occurring.

These regions were selected where the spanwise averaged mole fraction was between

0.1<< X >< 0.5. To reduce the effect of noise, an interlacing technique, where the

Fourier coefficients F (X(x)) are multiplied by the complex conjugate of the adjacent

row was used.

E(k) = F (Xj(x))F ∗(Xj+1(x)). (1.32)

The energy spectra showed a k5/3 inertial range, matching that of homogeneous

isotropic one-dimensional turbulence.

The numerical investigation of the turbulent RMI has been carried out by Thorn-

ber [6], [129]. The study addresses the influence of different three-dimensional multi-

mode initial perturbations on the rate of growth of a mixing layer. The computation

44

was done at grid resolutions up to 3× 109 cells using two different numerical meth-

ods that were also compared to corresponding theoretical predictions. The numerical

methods employed implicit large-eddy simulation. The drawback of the method is

that it provides excessive dissipation, where the transition of an initially small per-

turbation to fully turbulent flow must be resolved. This issue was addressed by

making sure that the large scales evolve independent of numerical viscosity. This

is achieved by a systematic grid refinement such that there is sufficient separation

between the dynamic and dissipative scales so that a physically realistic turbulent

flow field is achieved. The computation was performed with an incident shock wave

having M = 1.84. The initial conditions consisted of narrow-band and wide-band

perturbations with post-shock Atwood number A = 0.5. The numerical simulation

showed that at early time, each mushroom structure consists of a coherent vortex

ring. At late time, Kelvin-Helmholtz instabilities grow exponentially and break the

coherent vortex rings as they interact with neighboring vortices, causing them to lose

coherence. The late time simulations of this turbulent flow field also show that the

vorticity is concentrated in small, worm-like vortex structures. The integral mixing

layer width,

W =

∫ +∞

−∞f1(1− f1)dx (1.33)

is used to study the evolution of the mixing layer in time. Assuming a functional

dependence of the mixing width in time of the form

W = C(t− t0)θ (1.34)

a line of best fit yields the exponent θ = 0.26±0.02 for both computational methods.

The value of the exponent is comparable to some experimental results and is predicted

by Barenblatt et al. [8] with the viscous correction θ = 2/3 − ν, where ν ≈ 0.41

parameter based on viscosity.

45

Richtmyer-Meshkov turbulent mixing was studied by Weber, Cook and Bonazza

[20] for different initial perturbations. The net flux of the mass across the center

plane immediately after shock propagation was employed for determining the rate of

growth and linear and nonlinear theories were used to evaluate the initial growth of

the instability and to calculate the turbulent mixing shortly after the shock-interface

interaction. Length and timescales for the RMI growth rate are introduced and it

is shown that these characteristic parameters can be obtained based on the initial

conditions. The parameters used to nondimensionalize the equations were obtained

from to the large-eddy simulations. The RMI simulations were performed with an

initial perturbation having a Gaussian spectrum with specific peak and bandwidth

wavenumber. It was determined that the peak and bandwidth of the initial pertur-

bation spectrum must be chosen to ensure that the initial perturbations are well-

resolved and that the modes grow without influence from the side boundaries. The

study showed that the mass flux through the equimolar plane calculated using the

model is in a good agreement with the computation and that small discrepancies

were attributed to the assumptions that the shock wave does not deform while pass-

ing over the interface. The simulations performed within the same study with broad

band initial conditions confirmed the results obtained by Thornber [6].

Past research concentrated on investigating mixing layer width growth after it was

impulsively accelerated by incident shock wave and determining the growth constant

θ. Multiple experiments employed membrane to separate experimental gases prior of

incident shock interaction which ruptured membrane and initiated mixing. Several

researchers outlined the effect of the presence of the membrane in the mixing later and

its effect on the evolution behavior. New membrane less experiments and simulations

are not subject to the same shortfall and provide more relabel measurement of RMI

evolution.

46

Chapter 2

Experimental setup

2.1 The shock tube and experimental setup

The shock tube employed in this study is similar to the one used by previous research

performed at the University of Arizona [23] [53] [54] [67] [120] (see Fig. 2.1). It has a

circular 72 in long (182.88 cm) driver section with a 4 in (10.16 cm) inner diameter,

followed by a driven section with a square cross section having an inner width of 3.5

in (8.89 cm) and length of 97.25 in (247.02 cm). At the bottom, of the shock tube is

the test section having a length of 34 in (86.36 cm).

Initially, the driver and driven sections are separated by two polypropylene mem-

branes. To obtain the proper pressure the driver is pressurized with nitrogen and

the membrane is punctured by an arrow head connected to a solenoid, resulting in

a shock wave of the desired strength, which propagates in the driven section. The

initial interface is created by letting the light gas (air) flow into the driven section

from the top below the diaphragm location, and the heavy gas (SF6) to flow in to the

bottom of the test section. To ensure that a uniform stagnation interface is created,

the gases flow with matching flow rates (6 liter/min). The gases exit the shock tube

through a series of small holes in the test section walls, leaving behind a flat and

diffuse interface.

Three Photon Fastcam-APX RS, CMOS cameras are placed 60 in from the test

section in a vertical configuration separated 8 in from each other. Each camera has

a view angle of ∼ 15◦ degrees and thus captures the part of the test section which is

47

Driver section

Driven section

Pressure

transducer

Interface

Test section

CMOS

cameras

Loud speaker

Figure 2.1: Rendering of the experimental apparatus with three CMOS cameras.

48

traveled by the instability. The bottom portion of the shock tube that includes the

test section is enclosed in a dark room to eliminate any ambient light and for safety

purposes, as the laser employed in this experiment is extremely powerful.

The experimental conditions were calculated using a 1-D code, which is based

on the Roe scheme that solves the 1-D Riemann problem for a fluid dynamic sys-

tem (conservation of mass, conservation of momentum, energy conservation and gas

species conservation). It uses a two stage Runge-Kutta time integration scheme to

obtain the values for the next time step. A more detailed description of the code can

be found in Appendix A.

Fig. 2.2 is an X-T diagram depicting the experimental conditions following the

rupture of the membrane. The diagram illustrates the evolution of the J+ and

J− characteristic lines. The point 1 corresponds to the time and location where

the membrane between the driven and driver sections is ruptured. The diaphragm

rupture produces the incident shock wave a, rarifaction wave h and a contact interface

b. The initial interface between the experimental gases (air and SF6), represented by

c, is initially stationary and then is instantaneously accelerated to the piston velocity

when it interacts with the incident shock wave at point 2, denoted by curve d. This

interaction creates a weakly reflected shock wave that travels back into the air and

the transmitted shock wave, curve e. The transmitted shock wave will reflect from

the bottom of the test section resulting in the reflected shock wave f. The latter

will interact with the downward traveling interface d at point 3, re-accelerating it,

g. The focus of the current study is the evolution of the two-gas interface in time,

from the initial incident shock wave interaction with stationary interface at point 2

until point 4 when the interface interacts with the rarifaction wave coming from the

bottom of the test section.

During this time, from point 2 until point 4, the interface undergoes only the

49

Figure 2.2: X-T diagram for an experiment with incident Mach number M=1.2

50

initial acceleration from the incident shock wave and the re-acceleration from the

reflected shock wave, and from the X-T diagram it can be seen that the interface is

not influenced by any other acceleration during this time frame.

2.2 Test section

The test section has the same inner dimensions as that of the driven section and it

is attached to the driven section by flanges. An O-ring is used to ensure that the

connection is sealed, creating a smooth transition from the main driven section to the

test section. At the bottom of the test section, SF6 enters the shock tube through

a plenum. The bottom of the test section is sealed by a metal retainer with an

embedded circular glass window, through which the laser light passes. A rendering

of the test section is shown in the Fig. 2.3. The initially flat defused interface is

created by allowing the light and heavy gasses that stagnate and flow out of the text

section through series of holes located 3 in below the top of the test section. Each

hole has 2mm diameter and they are spaced 0.5 in apart.

Over the course of this study, the test section has been modified, in order to

satisfy specific requirements. Initial experiments were performed by illuminating the

interface from below through the test section end wall, which revealed two issues.

Illumination from below required that Beers law was needed to correct for the light

intensity decay due to scattering. However, Beer’s law correction is made difficult

to implement by the relatively large influence of secondary scattering. The second

issue is that the intensity of the scattered light by Mie scattering is strongly angular

dependent, which results in the recorded light intensity changing along the light

propagation direction, as the interface travels in the test section.

The current setup is intended to minimize the effects of Beer’s law attenuation and

51

angular dependence of scattering light. These issues are addressed by illuminating

the interface from the side by placing a transparent wall on one side of the test

section, allowing the laser light to enter the test section from the side.

Figure 2.3: Test section with attached speaker.

In this configuration, the light enters from the side of the test section, traversing

only 3.5 in of gas, which is not enough to attenuate the light intensity significantly

due to the scattering phenomenon. It is estimated that the majority of the particles

are on the order of 200 nm in size (Particle size distribution was obtained by Scan-

ning Mobility Particle Sizer [120]). The fact that the laser light is polarized is also

advantageous as it can be configured to be polarized perpendicularly to the scattered

52

light, which increases the intensity and angular uniformity of the scattered light.

During the course of this study, it was necessary to characterize the initial condi-

tion from which the interface evolves. In order to achieve this characterization, the

holes in the front window were removed to allow an unobstructed view of the initial

perturbation prior to its interaction with the incident shock wave. As a result the

new configuration only employed small holes on the side walls to generate the flat

interface between the two gases.

The initial perturbation is created using two reinforced TangBang W5-1138SM

5-1/4” Neodynium Subwoofer loudspeakers mounted in the shock tube wall, one near

the bottom of the test section and the second near to the top of the driven section

(Fig. 2.3). The two speakers are oscillated out of phase from each other, producing

vertical oscillation of the gas column within the shock tube. This motion generates

Faraday waves on the interface, which result in a small random three-dimensional

perturbation imposed on the otherwise flat interface. The loudspeakers displace the

necessary volume of gas that is required for the vertical oscillation of the interface.

The system protecting the loudspeakers was designed and built by Matt Mokler and

Justine Schluntz. The speakers are protected from high pressures generated by the

passing shock wave by reinforcing them with a concave steel mesh surface, which is

placed between the speakers and the test section wall, and sealed with a flat thin latex

membrane. For gas displacement, the concave steel mesh allows the deformation of

the membrane while protecting the speakers from excessive pressure spikes produced

during the shock wave propagation. The signal controlling the speakers is generated

by a LabVIEW program, which uses initial data consisting of ambient temperature

and perturbation frequency. The code calculates the proper phase shift and generates

the sinusoidal signals that are output through an audio amplifier and sent to the

appropriate speakers.

53

2.3 Flow visualisation

The flow is visualized utilizing planar laser Mie scattering movies. One of the two

working gases is seeded with incense smoke and illuminated by the second harmonic

of a pulsed Nd:YLF diode laser (Photonics Industry DM50-527). The laser’s output

is passed through spherical and cylindrical lenses and reflected through the clear

side wall. The laser beam is split into two parts, which diverge in the vertical plane

creating two laser sheets that illuminate a thin cross section of the interface in the

same plane. Each plane has a Gaussian intensity profile. The two laser sheets

overlap in the plane to create a wide and more uniform illuminated visualization

zone. The resulting Mie scattering image is captured using three high-speed CMOS

video cameras positioned to cover the entire region of interest. The camera shutter

is fully opened during the entire frame, and the 5 ns laser pulse operating at a rate of

6kHz serves as a shutter. Since the intensity of the Mie scattering light depends on

the angle of observation, the experimental setup is intended to minimize this effect.

The cameras are positioned at a distance of 60” from the light sheet plane, thus the

angle between the incident and the scattered light varies by only approximately 3◦,

which is not enough to produce a significant change in the scattered light intensity.

2.4 Pressure transducers

Four high resolution dynamic ICP pressure transducers (PCB Piezotronics 112A22)

are used to detect the passage of the incident, transmitted and reflected shock waves

in the test section during each experimental run. The two sensors are placed above

the interface to estimate the speed of the primary incident shock wave and the other

two are located below the interface to obtain the speed of the shock wave propagated

through the interface into the heavier gas. The signal from each transducer is am-

54

plified by a PCB Piezotronic model 482A05 ICP signal conditioner, and acquired by

an Agilent U2531 data acquisition board with a sampling rate of 2MHz, and then

recorded by a LabVIEW program. An example of the pressure traces is shown if

Fig.2.4.

Figure 2.4: Pressure transducer readings.

The differences between shock passage times in conjunction with the known dis-

tances between the sensors allows one to calculate the Mach number of the primary

and propagated shock wave. The speed of sound in the gases is calculated based on

temperature measurements made prior to each experimental run. The first pressure

transducer is also used to trigger the cameras by setting the trigger threshold in a

Stanford Research System Inc. model DG 535 delay generator at 0.45 V. The data

is used to monitor the experimental conditions from one experiment to another.

55

2.5 Data acquisition

A LabVIEW code has been written to control and fully automate the experiment.

The program is responsible for measuring the initial ambient temperature from two

thermocouples using NI thermal module (National Instrument High Speed USB Car-

rier model NI USB-9162) and the pressure inside the test section using an MKS

barometer absolute pressure transducer, from which the signal is fed to the MKS

Type 670 signal controller and acquired by an NI USB 618 data acquisition board.

The desired Mach number is entered in the LabVIEW program prior to the experi-

ment. Based on the input data and the sensor measurements, the program calculates

the appropriate driver pressure. Once the air flow is turned on and a uniform smoke

concentration is achieved, the program switches from initial data acquisition to driver

pressure control. While the driver pressure increases, the program signals the user

when to start the SF6 flow and switches to the final stage that includes the initializa-

tion of the perturbation generation and the firing the driver at the desired pressure.

The automation of these processes ensures better accuracy and repeatability of the

experiment. After the shock wave is initiated, the program evaluates and records

the incident and the transmitted shock Mach numbers based on the readings of the

pressure transducers and the corresponding temperature measurements. At the end

of the experiment, the program generates an Excel file that contains all measured

parameters for each experiment.

The cameras and the laser are synchronized by the DaVis software from LaVision.

The necessary parameters are inputted shortly before the experiment is executed.

The imaging is triggered by the first pressure transducer with an appropriate time

delay just prior to the shock wave impacting the interface. After the images have

been acquired MATLAB is used for further image processing.

56

Figure 2.5: Schematic of the experimental systems.

57

Chapter 3

Processing methods and results

3.1 Images

Figure 3.1 shows a sequence of Mie scattering images taken from a typical exper-

iment. In this sequence, the initial perturbation can be observed prior to shock

interaction followed by the development of the instability after interaction with the

incident shock wave. Early in the development the instability retains an imprint

of the initial perturbation. However, as time progresses, the instability develops in

what appears to be a self-similar growth fashion, with length scales increasing with

time. Following re-shock, extremely small scales are generated accompanied by the

accelerated growth of the mixing zone; increasing the turbulent mixing of two gases.

The visualization techniques used in the experiments have changed over time.

Initially mixing layer width was illuminated from below by a laser sheet, sliced di-

agonally through the center of the shock tube. Mixing layer evolution was recorded

by three cameras at 6 kHz. Incident Mach number for those experiments were 1.10

and 1.20 (Experimental group #1 and #2). After modification mixing layer width

was illuminated from below by a laser sheet, sliced through the center of the shock

tube. As in previous set-up holes in clear wall obscured clear view of the initial con-

ditions. The modified set-up recorded the evolution of mixing layer width by three

cameras at 6 kHz. From now on experiments were performed using only incident

Mach number of 1.20 (Experimental group #3). Additional modification of the test

section removed holes from front wall (Experimental group #4).

58

Figure 3.1: Image sequence taken from a typical experiment using the Mie scatteringdiagnostic. The first frame shows the compression effect due to the propagatingshock wave. The line below the interface, separating the bright and dim regions ofthe same gas, is the propagated shock wave into SF6. The gas behind the shock waveis compressed making it brighter than the uncompressed gas.

59

Exp. groups Mach Number of Illumination Recording Number ofin air cameras method frequency experiments

#1 1.10 3 From bellow, 6 kHz 47diagonal,

smoke in air#2 1.20 3 From bellow, 6 kHz 32

diagonal,smoke in air

#3 1.20 3 From bellow, 6 kHz 73through holes,smoke in air

#4 1.20 3 From bellow, 6 kHz 68without holes ,

smoke in air#5 1.20 3 From side, 6 kHz 114


wide laser,smoke in air

#7 1.20 1 From side, 10 kHz 42wide laser,


wide laser,smoke in SF6

#9 1.20 1 From side, 10 kHz 36thin laser,


thin laser,smoke in SF6

#11 1.20 2 From side, 6 kHz 54thin laser,


thin laser,smoke in SF6

Table 3.1. Changes in experimental set-up and number of experiments for eachtype of configuration.

60

Need for Beer’s law correction associated with the laser light attenuation due

to scattering was removed by illuminating mixing layer from a side window (Ex-

perimental group #5). Experimental groups #7 - #10 were used to evaluate the

measurement methods using four different configurations by, varying flow illumina-

tion and which of the two gases is seeded with smoke. These experiments were

recorded by one camera at 10 kHz. Experimental groups #11 and #12 was recorded

by two cameras with maximum resolution at 6 kHz framing rate with different seeded

gases. Table 3.1 shows the chronological changes of the experimental configuration

and the number of experiments performed during each case.

3.2 Mixing layer width

It is assumed that the scattered light is proportional to the smoke concentration that

in turn represents the volume fraction of the seeded gas. To analyze the turbulent

mixing induced by the shock interaction with the perturbed interface, the images

are first corrected for nonuniform illumination. The volume fraction of the smoke

seeded gas in the plane of the laser sheet is evaluated by using white and black

reference images recorded prior to each experiment. The image of the test section,

fully filled with smoke, is used as the white reference image for each location, while

the black reference image is taken with unseeded gas inside of the test section. The

normalized concentration of the smoke and, thus the volume fraction of the seeded

gas is calculated using

C(x, y) =Ie(x, y)− Ib(x, y)

Iw(x, y)− Ib(x, y). (3.1)

Where Iw is the local light intensity of the smoke-filled image, Ib is the light

intensity scattered by the test section walls taken with unseeded gas inside the test

section and Ie is the light intensity recorded during the experiment. An example of

61

the implemented procedure is depicted in Fig. 3.2. The smoke concentration pro-

vides the measure to determine the mixing layer thickness in order to learn how the

turbulent mixing between the gases evolves in time. For this purpose the concentra-

tion is row-averaged to obtain a mean smoke concentration distribution. An example

of the time evolution of the row-averaged smoke concentration is illustrated in Fig.

3.3. The turbulent mixing width is determined as the distance between 10% and 90%

locations of the averaged concentration. Evaluating the row-averaged concentration

for each frame from the image sequence allows the determination of how the mixing

layer width develops in time after the incident shock interacts with the interface.

The row averaged concentration distribution also allows the tracking of the in-

terface between the gasses, as it travels down the test section at the piston velocity.

This is achieved by measuring the location of the 50% point of the averaged profile,

which is equivalent to the evenly mixed region (50% lighter gas and 50% heavier

gas). A second approach for obtaining the appropriate calibration for the heavy gas

volume fraction is to use a different frame from the same recorded sequence, which

contains the pure gas concentration on the same region of interest. It follows from the

experimental set-up that only a small part of each image, the interrogation window,

is used for the mixing analysis. Since the shock wave brings the whole gas system

into motion with a constant velocity, this window moves downward in the laboratory

frame of reference with the same velocity. If air is seeded with smoke then, after the

mixing zone passes a specific location of the test section, this area will be filled with

the smoke-seeded air. Assuming that the smoke is distributed uniformly in the air,

the frame taken later in time can be used to calibrate the mixing zone that passed

the same area a few frames earlier. For calibration, the fifth frame recorded after the

one under consideration is used for air with smoke, where as the fifth frame recorded

before the current one is used for smoke seeded SF6 experiments. Figure 3.4 com-

62

Figure 3.2: a) Images from an Experiment performed with smoke seeded SF6 gas.b) Calibrated image using white and black reference images obtained prior to theexperiment. The obtained smoke concentrating levels are multiplied by the bit depthof the original image in order to make them visible for comparison.

63

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Interface Profile

Inte

rfa

ce

Pro

file

Pixel

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Interface Profile

Inte

rfa

ce

Pro

file

Pixel

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Interface Profile

Inte

rfa

ce

Pro

file

Pixel

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Interface Profile

Inte

rfa

ce

Pro

file

Pixel

Figure 3.3: Row-averaged smoke concentration profiles at different times relativeto the incident shock wave interface interaction. This sequence of images showswidening of the mixing layer width as time progresses.a) t=0.3 ms, b) t=2.2 ms, c)t=4.0 ms, d) t=5.7 ms.

64

5

10

15

20

25

30

35

0 1 2 3 4 5

Calibrated with -5 framesWith external white calibration

h (

mm

)

t (ms)

Figure 3.4: Comparison of the mixing layer width evolution in time using two differ-ent calibration methods; both methods provide similar mixing layer width evolutionprofiles.

pares mixing layer width measurements obtained using this method with those of

the previous method in which calibration is obtained separately through black and

white frames. As can be seen in Fig. 3.4, there is not significant difference between

the obtained mixing layer width evolution for both methods.

3.3 Interface velocity

As stated earlier, the interface between the two gases is impulsively accelerated to the

piston velocity after interaction with the shock wave. When the incident shock wave

impacts the interface from the lighter gas, it generates a weak reflected compression

wave along with a transmitted shock wave. The piston velocity can be theoretically

derived knowing the gases initial conditions and the strength of the incident shock

wave. The theoretical value of the piston velocity for the incident shock Mach number

65

M=1.2 is 78.4 m/s. The interface can be tracked experimentally by observing 50%

location of the horizontally averaged smoke concentration as a function of time.

The piston velocity is measured by taking the first derivative from the displacement

measurement. The measured interface velocity is shown in Fig. 3.5.

0

20

40

60

80

100

0 1 2 3 4 5

V (

m/s

)

t (ms)

Figure 3.5: Interface velocity for 10 experiments based on the 50% location of theaveraged concentration measurement.

The data shows that the interface undergoes further acceleration after being

impulsively accelerated by the incident shock wave. The acceleration behavior corre-

sponding to the interface velocity is represented in Fig. 3.5 and in Fig. 3.6. During

the first millisecond the interface accelerates at a rate of approximately 95 g and con-

tinues to accelerate for the entire duration of the experiment. The initial acceleration

is obtained by fitting the line to the first three data points.

Theoretically, after the shock passage, the material volume surrounding the mix-

ing layer moves with a constant velocity. However, in a real experiment, the column

of gases accelerates due to the boundary layers developing on the test section walls.

66

73

74

75

76

77

78

79

80

0 1 2 3 4 5

y = 72.908 + 3.6238x R= 0.81863

V (

m/s

)

t (ms)

95 g

Figure 3.6: Interface acceleration corresponding to the averaged data of Fig. 3.5.

The growing boundary layers cause the constriction of the effective tube cross-section,

resulting in gas acceleration in the center of the tube. The effect of the material vol-

ume acceleration in the shock tube due to the boundary layers has been investigated

by Roshko [105] and Brocher [11]. The experimental measurements of the average

value of the interface velocity and the theoretical acceleration behavior caused by

the boundary layer growth are shown in Fig. 3.7. It can be observed that the ac-

celeration predicted by the boundary layer theory coincides with the experimental

data. The detailed theoretical derivation of the given phenomenon is presented in

Appendix B.

67

70

75

80

85

90

0 1 2 3 4 5 6

Velocity Exp1Velocity Exp2Velocity Exp3Velocity Exp4Velocity Exp5Velocity Exp6Velocity Exp7Velocity Exp8Velocity Exp9Velocity Exp10Averaged velocity Theoretical velocity profile

V (

m/s

)

t (ms)

Figure 3.7: The averaged interface velocity, obtained for 10 experiments with simi-lar incident shock wave strength, compared to the theoretical prediction modellingpresence of boundary layers. The experimental conditions for each experiment canbe found at Table B.1.

3.4 Growth constant

Figure 3.8 depicts a plot of the mixing layer half width versus time for 16 experiments

showing the instability growth following the initial shock interaction and then re-

acceleration by the reflected shock wave. The mixing layer half width is obtained

by finding the spacing between the 10% and 90% locations of the average smoke

concentration and then halving this distance. As it can be observed in this plot, the

mixing layer width initially shows a power law growth h = Ctθ with growth exponent

θ < 1. After re-shock, however, the growth rate is constant. It is evident that a small

amount of scatter is produced by small inconsistencies in the initial perturbation.

68

0

10

20

30

40

50

0 2 4 6 8 10

h (

mm

)

t (ms)

Figure 3.8: Mixing layer half width evolution in time for 16 experiments with a powerlaw behavior profile following the incident shock wave interface interaction and lineargrowth after the de-acceleration by the reflected shock wave.

It is noted that the amplitude and dominant wavelength of the perturbation

are reasonably reproducible from experiment to experiment. However, there are

statistical variations in the initial perturbation due to the random nature of Faraday

forcing which is difficult to control. Furthermore, in the present study the arrival

of the incident shock wave is not synchronized with the phase of the perturbation

forcing, which results in experiments initiated at different phases of the Faraday wave

oscillation cycle.

In examining individual experimental growth curves, two different growth behav-

iors can be observed, as is illustrated in Fig. 3.9. The plot on the right (b) shows

a very rapid initial growth followed by a much slower growth, while the plot on the

left (a) shows a much more consistent growth rate during the entire experiment.

69

Figure 3.9: From Fig. 3.8 the experiments were divided in two groups based on theirdifferent growth behavior; a) with a more consistent growth rate and b) with veryrapid initial growth followed by a slower growth.

Figure 3.10: Logarithmic plot of the mixing layer half width for the two sets ofexperiments shown in Fig. 3.9.

70

In order to investigate the value of the growth exponent, log-log plots of averaged

values of the data from Fig. 3.9 are shown in Fig. 3.10. The slopes of lines fitted

to the curves of Fig. 3.10 should yield the growth exponent θ, as the growth of the

mixing layer half width is expected to have form h = Ctθ. It can be observed, from

the log-log plots, that both sets of data show two distinct linear regions indicating

two values of the growth exponent. At the initial stage, both perturbations grow

with exponent of θ ≈ 0.5. However, afterward, a more variable growth constant is

observed with θ lying at the range of 0.3 < θ < 0.4.

Assuming the power law growth h = Ctθ the time dependent θ can be obtained

by

θ(t) =d log h

d log t. (3.2)

The time evolution of the time dependent θ(t) growth constant is shown in Fig. 3.11.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6

θ

t (ms)

a)

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6

θ

t (ms)

b)

Figure 3.11: Evolution of the θ(t) of the mixing layer half width for the two sets ofexperiments shown in Fig. 3.9.

71

3.5 Re-shock

After the transmitted shock reflects from the bottom of the test section, it interacts

with the downward traveling mixing layer depositing additional vorticity. The growth

of the re-shocked mixing layer width is linear in time as shown in Fig. 3.12. The linear

development of the mixing layer evolution following its re-acceleration by a reflected

shock wave has been predicted by Mikaelian [77]. The obtained experimental data

agrees with the theoretical prediction and shows a clear linear behavior of the mixing

layer width.

0

10

20

30

40

50

0 2 4 6 8 10

y = -56.015 + 9.2666x R= 0.99502

h (

mm

)

t (ms)

h-h0=9.26(t-t

0) a)

0

10

20

30

40

50

0 2 4 6 8 10

y = -56.218 + 9.2661x R= 0.99642

h (

mm

)

t (ms)

h-h0=9.26(t-t

0) b)

Figure 3.12: Mixing layer half width for the two sets of experiments showing identicalgrowth rates following re-shock.

Contrary to the initial development of the mixing layer by the incident shock

wave interaction, the mixing layer following re-shock is observed to have consistent

growth rate, as seen in Fig. 3.12. In this region, both sets of experiments show

identical values of growth rate of dh/dt = 9.26m/s. In order to compare previous

72

re-shock experimental results, a dimensionless version of the re-shock growth rate is

plotted in Fig. 3.13. Mikaelian [77] and Leinov et al. [62] have suggested that since

dh/dt ∝ A∆U , where ∆U is the re-shock impulse, one can define the dimensionless

re-shock growth rate as C = 2A∆U

dhdt

. Using this definition Mikaelian has indicated

that C = 4α, where α is the Rayleigh-Taylor growth constant.

Figure 3.13: Comparison of measured dimensionless re-shock growth rates from thisstudy with those measured by Leinov et al [62].

Fig. 3.13 is a plot of C versus ∆U for the current experiments along with those

of Leinov et al. showing satisfactory agreement. However, it is noteworthy that

the present experiments have noticeably smaller values of growth rate that could be

attributed to the fact that experiments by Leinov et al used a membrane to initially

separate the experimental gases. The presence of membrane fragments could be

expected to produce greater mixing upon re-shock. Thus, the present experiments

could be seen as exhibiting a better correspondence to a membrane-free behavior.

Also, shown on Fig. 3.13 are two lines indicating the range of C values predicted from

73

Mikaelian’s model using the range of expected α values found in previous Rayleigh-

Taylor instability experiments having the same Atwood number as that used in the

present study.

3.6 Initial condition

The image taken prior the shock interaction with the interface provides informa-

tion about the initial perturbation of the interface from which the turbulent mixing

evolves. Examples of such images taken from different experiments are shown in Fig.

3.16. The assumption that the pressure gradient generated by the plane incident

shock wave is aligned with the test section is valid only for a relatively small am-

plitude of the initial perturbation. However, employing on this assumption, we can

estimate the horizontal density gradient in the x − y plane, which when combined

with the pressure gradient can be used to predict the vorticity deposition on the

interface due to baroclinic torque. Fig. 3.14 shows the results of taking the hori-

zontal derivative of two typical initial perturbations thus giving an estimate of the

initially deposited vorticity. The average circulation per unit width can be obtained

by vertically integrating the baroclinic vorticity as shown in Fig. 3.14 in the x − y

view plane. This averaged vorticity distribution can be used to obtain the dominant

wavelength of the initial perturbations. Fig. 3.15 shows the integral of the deposited

vorticity across the test section and its spectrum.

The imposed initial interface perturbation is stochastic in nature with random

amplitudes and wavelength content, which are function of time. However, it is useful

to investigate the behavior of the dominant wavelength as a function of time. For this

purpose, a separate study was carried out to investigate the evolution of the initial

perturbation in time by continuously exciting Faraday waves on the interface without

74

Figure 3.14: Mie scattering image of the perturbed interface prior to the shock-interface interaction and the corresponding deposited vorticity for two experiments.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 20 40 60 80

ω

x (mm)

a)

0

0.05

0.1

0.15

0.2

0.25

0 500 1000 1500 2000

ω (

k)

k (1/m)

b)

Figure 3.15: a) Initially deposited vorticity profile. b) Corresponding spectral con-tent.

75

Figure 3.16: Initial perturbations prior of the incident shock arrival for several ex-periments illustrating the stochastic nature of the initial conditions from which theRMI evolves.

76

shock wave interaction. Fig. 3.17 shows the evolution of the dominant wavelength

and its amplitude of the initial perturbation in time after initiation of the vertical

oscillation. At an early stage of the excitation, the width of the initially diffuse

interface rapidly increases in time and then stabilizes approaching an asymptotic

value about which the interface width oscillates. Based on this study of the initial

perturbation, a steady state is achieved after approximately 3 seconds after the

oscillation is initiated. While performing experiments, the interface is oscillated

longer than 3 seconds to ensure that the initial conditions are within the asymptotic

region.

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3 3.5 4

h (

mm

)

t (s)

a)

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3 3.5 4

λ (

mm

)

t (s)

b)

Figure 3.17: Evolution of the initial perturbation in time after initiation of thevertical oscillation. a) Evolution of the perturbation amplitude in time. b) Evolutionof the dominant wavelength of the perturbation in time.

77

3.7 Dominant wavelength

The instability develops the interface between two gases can no longer be described

as a single value function due to the emergence of vortices which curl the interface

around them. In order to study the evolution of the interface spectral content in

time, the mixing layer is described as a function of one gas penetrating into another.

This is achieved by considering a window with constant hight 2L centered on the

mixing layer and observing the depth of penetration of one gas into the other defined

by

η(y) =

∫ 2L

0

C(x, y)dx− L (3.3)

where C is the volume fraction of one of the two gases. The result is a single

value function that can be further analyzed by applying a discrete Fourier transform

(DFT), yielding the frequency content of the interface for each experimental frame.

The dominant wavenumber is obtained by taking the centroid of portion of the

frequency spectrum with amplitude values grater than 1/10 of the peak amplitude.

k =

∑a(k)× k∑a(k)

(3.4)

It was observed that wavelengths smaller than 1/20 of the test section width con-

tributed little to the instability frequency content. Early experiments performed

using random initial perturbations showed that the mixing layer width evolution

could be separated into two groups, one with consistent growth rate and the other

with rapid initial growth followed by growth saturation later on. Further modifica-

tion of the test section enabled the recording of the initial perturbation along with

the evolution of the instability following shock interaction. Using this new setup an

experimental study that consisted of 17 experiments was performed and it showed

78

the same two behaviors that was observed in the earlier experiments. The develop-

ment of the mixing layer half width for these experiments is shown in Fig. 3.18. Fig.

3.19 shows image sequences taken from two of these experiments that fall into the

two different categories. Subsets of the experiments of Fig. 3.18 exhibiting these two

behaviors are illustrated in Figure 3.20.

Figure 3.18: Development of mixing layer half width following the shock-interfaceinteraction for a set of 17 experiments.

In addition the evolution of the dominant wavelength has been obtained for these

experiments, using the techniques described above. Plots of time dependence of

average of these dominant wavelengths for each of the two groups are shown in Fig.

3.21, which reveal that the experiments with greater variation in growth rates retain

the initial wavelength, while in experiments with more consistent growth rate the

dominant wavelength increases, as instability progresses.

The spectral content of the deposited vorticity as described in section 3.6 above

79

Figure 3.19: a) Image sequence taken from an experiment exhibiting the consistentgrowth behavior. b) Image sequence taken from an experiment exhibiting rapidinitial growth followed by a slower growth.

80

Figure 3.20: a) Mixing layer half width versus time for 5 experiments with consistentgrowth rate. b) Mixing layer half width versus time for 5 experiments with rapidinitial growth followed by a slower growth.

Figure 3.21: Dominant wavelength evolution in time. a) Experiments with consistentgrowth rate. b) Experiments with rapid initial growth followed by a slower growth.

81

was obtained for this set of experiments. The initial perturbation from which the

instability developed after the shock interaction reveals that experiments with con-

sistent growth behavior are characterized with a broad vorticity spectrum, and that

the experiments with more variable growth rates have a narrower vorticity spectrum.

The averaged vorticity spectral content for the two types of evolution behaviors are

shown in Fig. 3.22.

0

1 104

2 104

3 104

4 104

5 104

0 200 400 600 800 1000

ω(k

)

k (1/m)

a)

0

2 104

4 104

6 104

8 104

1 105

0 200 400 600 800 1000

ω(k

)

k (1/m)

b)

Figure 3.22: a) Average deposited vorticity spectrum for experiments with consistentgrowth rate. b) Average deposited vorticity spectrum for experiments with rapidinitial growth followed by a slower growth.

82

3.8 The evolution of the spectral content of the mixing layer

As described in the previous section, experiments in which the mixing width exhibits

linear behavior are accompanied by a dominant wavelength that increases in size, and

that experiments with saturating growth rate maintain the dominant wavelength in

time. This phenomenon can be studied in more detail by investigating the evolution

of each wavelength separately by considering evolution of amplitudes within a range

of wavelengths in time.

0

2

4

6

8

10

12

0 1 2 3 4 5 6

11.9 cm4.0 cm2.4 cm1.7 cm1.3 cm1.0 cm

λ (

mm

)

t (ms)

a)

0

2

4

6

8

10

12

0 1 2 3 4 5 6

11.9 cm4.0 cm2.4 cm1.7 cm1.3 cm1.0 cm

λ (

mm

)

t (ms)

b)

Figure 3.23: Amplitude evolution for different wavelengths of the mixing layer spec-trum. a) Spectral content evolution for experiments exhibiting linear growth behav-ior. b) Spectral content evolution for experiments exhibiting rapid initial growthfollowed by a slower growth.

Fig. 3.23 shows the evolution of different wavelengths for the two sets of experi-

ments having different growth behaviors. As is shown in the plots the amplitude of

the long wavelength grows linearly in time and the amplitude of the shorter wave-

length saturates for both sets of experiments. However, experiments with consistent

83

growth rate are dominated by long wavelengths at later times and the experiments

with saturation behavior are less dominated by the long wavelengths.

Thus, experiments with broad spectral content of the initial vorticity exhibit lin-

ear growth of the total mixing layer width because a significant amount of energy is

deposited on the long wavelength scales and experiments with narrow initial spec-

tral content with more energetic small perturbations result in a saturated behavior

of the total mixing layer width, because little energy was deposited on the longer

scales. Thus depending on the initial spectral content, the linearly developing long

wavelength perturbations will or will not dominate the mixing process.

The constraining effect of the test section walls was important in experiments that

had strong influence on the long wavelengths perturbations, as they were comparable

to the test section width. Thus these experiments are likely not self-similar. At the

same time, experiments that were dominated by short wavelengths, much shorter

than the width of the test section, likely evolved unaware of the confining effects of

the test section walls.

3.9 Integral method

Another characteristic length scale that is used in the study of RM turbulent mixing

is the integral measure

hint =

∫ −∞∞

f1(1− f1)dx, (3.5)

where f1 is the normalized, horizontally averaged smoke concentration. The integral

length scale has an advantage over the previously used mixing layer width that

depends on the artificially chosen range of concentration values (for example 10%

and 90%).

84

From the data set of 20 experiments analyzed using the integral method four

experiments were selected, two experiments with the largest integral mixing layer

width measure and two with smallest mixing layer width measure out of the entire

set of experiments. In both groups, the mixing layer width shows power law growth,

hint = Ctθ behavior with growth exponent θ < 1. All four experiments provide

almost identical values of θ ∼ 0.4.

0

1

2

3

4

5

0 1 2 3 4 5

hin

t (m

m)

t (ms)

Figure 3.24: Total mixing layer width versus time for four experiments calculatedusing Eq. 3.5.

In addition to the total mixing layer width, the analysis has been extended to

study growth behaviors of bubbles and spikes separately. The width of the spike and

bubble regions are defined as follows: considering a system of two gases after the

shock passage as an incompressible flow, the middle line X0(t) that separates the

spike region from the bubble region can be determined by equating the volume of

air propagated into the SF6 with the volume of SF6 propagated in air,

85

∫ X0

−∞f1dx =

∫ +∞

X0

(1− f1)dx (3.6)

Based on this definition the spike and bubble widths are expressed as follows:

hs−int =

∫ X0

−∞f1dx, hb−int =

∫ +∞

X0

(1− f1)dx. (3.7)

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

h s

-in

t (m

m)

t (ms)

a)

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

h b

-in

t (m

m)

t (ms)

b)

Figure 3.25: Integral width evaluated from Eq. 3.5 for (a) spikes and (b) bubbles.

Fig. 3.25 shows the newly defined width of the spike and bubble regions for

experiments showed in Fig. 3.24. It can be observed that both, the spike and

bubble’s widths, develop in power law fashion. In order to investigate the value of

the growth exponent, log-log plots of the data from Fig. 3.25 are depicted in Fig.

3.26. Note that if growth of the form hb−int;s−int = Ctθb−int;s−int is expected, then the

slopes of the lines fit to the curves of Fig. 3.26 yield the growth exponents θb−int and

θs−int. This analysis shows that θ values of the bubble and spike are θb−int = 0.40

and θs−int = 0.42, respectively. The time evolution of the θs−int and θb−int according

to (3.2) is shown in Fig. 3.27.

86

0.1

1

10

0.1 1 10

h s

-in

t (m

m)

t (ms)

θ s-int

= 0.42 a)

0.1

1

10

0.1 1 10

h b

-in

t (m

m)

t (ms)

θ b-int

= 0.40 b)

Figure 3.26: Log-Log plot of the integral layer mixing width for a) spike and b)bubble, indicating that both exhibit power law growth behavior.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

θs-int

t (ms)

a)

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

θb-int

t (ms)

b)

Figure 3.27: Evolution of the θ(t) according to (3.2) for the integral layer mixingwidth for a) spike and b) bubble.

87

3.10 Linear momentum

A novel definition for the width of spike and bubble was introduced by Dr. Oleg

Likhachev, which considers a finite material volume that consists of a certain mass

of unperturbed gases surrounding the growing mixing layer. After the shock passes

the interface, the turbulent mixing can be studied in the frame moving with the

material volume. Also, the gas system is treated as incompressible after shock-

interface interaction similar to as is done in the previous section. The turbulent

mixing of gases results in the redistribution of masses in the material volume, moving

the center of mass toward the lighter fluid during this process. The evolution in time

of the center of mass relative to the moving frame defined by the integral

~R =1

M

∫ +∞

−∞~rρdv (3.8)

where, ~R(t) is the position vector from the frame origin, which can be set at the center

of the plane separating bubbles and spikes (as described in the previous section). At

the earliest stage of mixing, the center of mass in the moving frame can be expanded

in a Taylor series , yielding∫ X0

−∞xf2(x, t)dx−

∫ +∞

X0

xf1(x, t)dx = Pt+O(t2) (3.9)

The parameters on the right-hand side are proportional to the additional linear

momentum per unit cross sectional area of the shock tube that is deposited on the

mixing layer due to RMI. The second order term is neglected, as it relates to any

external forces imposed on the system. In the present experiments, the force acting on

the gas system is mostly due to viscous shear stresses, and having a small effect in the

duration of the experiments can be neglected at the earlier stages in the development

of the turbulent mixing. A new definition of the mixing width for bubbles and spikes

88

can be introduced based on mass transported thought the volumetric middle line X0.

hb−LM(t) =

(∫ +∞

X0

xf1(x, t)dx

)1/2

, hs−LM(t) =

(∫ X0

−∞x(1− f1(x, t))dx

)1/2

(3.10)

Then, the momentum equation (3.9) in the case of inviscid fluid can be written as

h2b−LM +h2

s−LM = Pt, where P has an unique value in each experiment since it relates

to the initial conditions, and it also defines the linear momentum deposited in the

perturbed interface by the incident shock wave. Using the definitions (3.10) and

the momentum equation (3.9) the growth exponent in the power law for spikes and

bubbles is identical and equal to θLM = 1/2 . An experimental study was performed

to measure the evolution of the as defined by (3.10) mixing width for bubbles and

spikes. Fig. 3.28 depicts the time dependance of these mixing width mesurements

for 8 experiments. Fig. 3.30 represents the total mixing width evolution in time,

where h2LM = h2

b−LM + h2s−LM . The data is split into two distinctive groups that

represent two types of initial conditions.

The deviation from linearity in Fig. 3.28 - 3.30 after 3 ms, is believed to be

due to the measurement technique used to obtain the mixing parameters. Later in

time, as the turbulent mixing progresses, the planar Mie scattering technique cannot

provide a sufficiently good measurement of the highly diluted mixture to properly

account for the mass transfer. The deposited momentum can be used to construct

characteristic time and length scales. The experimental parameter P , a measure of

additional linear momentum deposited to the mixing layer, in conjunction with the

velocity ∆V that the material volume assumes after the shock interaction, may be

used to introduce characteristic time and length scales,

T =P

(∆V )2, δ =

P

∆V. (3.11)

89

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5

-hb

-LM

2 , h

s-L

M

2 (

10

-4 m

2)

t (ms)

Spikes

Bubbles

Figure 3.28: Time evolution of the square of the spike and bubble mixing layer widthsdefined by (3.10).

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

(hs-L

M

2+

hb

-LM

2)

(10

-4 m

2)

t (ms)

Figure 3.29: Time evolution of the square of the total mixing layer width h2LM =

h2b−LM + h2

s−LM .

90

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

<h

s-L

M

2+

hb

-LM

2>

(1

0-4

m2)

t (ms)

Figure 3.30: Time evolution of the averaged square of the total mixing layer width.

The total mixing width reduces to a dimensionless form using the scaling (3.11),

H = τ 1/2, (3.12)

where H = h/δ and τ = t/T . The experimental data plotted using these new

dimensionless variables is shown in Fig. 3.31. One can observe that the given char-

acteristic scales collapse the experimental data into a single line. The divergence of

the dimensionless data from the theoretically predicted value at the late stage of the

experiment’s evolution can be attributed to small randomness emerging due to the

measurement technique.

91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6

H

τ

Figure 3.31: Dimensionless total mixing layer width. The solid line is the square rootfunction and the symbols represent the experimental measurement of the evolutionof the turbulent mixing width.

3.11 Dimensional analysis

As stated above the mixing layer growth exhibits different behavior depending on

the initial conditions. Following the procedure normally used in analyzing single-

mode experiments the mixing layer width evolution can be scaled by introducing

characteristic time and length scales based on the initial dominant wavelength. The

dimensional mixing layer width can be scaled by the dominant wavenumber k, yield-

ing the dimensionless mixing layer width H = k(h(t) − h0) and time can be scaled

using the dominant wavenumber and the initial growth rate h0 yielding of the di-

mensionless time τ = kh0t.

The dominant wavenumber is obtained by analyzing the initial perturbation prior

to the interaction of the shock wave with the perturbed interface. The spectral

92

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5

h (

mm

)

t (ms)

Figure 3.32: Total mixing layer width for experiments performed with a thin lasersheet and smoke in air. The measurements are based on the location of 10% and90% of the horizontally averaged smoke concentration.

approach described in section 3.7 is used to select the dominant wavelength, which

is based on the initial vorticity spectrum that governs the turbulent mixing. It is

important to mention that this method provides the ability to clearly define only the

x-component of the dominant initial wavelength (kx). In this study, it is assumed that

instability is isotropic implying that ky ' kx. Fig. 3.32 shows raw unscaled data for

the mixing width measurement for 10 experiments. The dimensionless mixing width

evolution for the same set of data is shown in Fig. 3.33.

It can be observed that the dimensionless data are better grouped than the di-

mensional plot of Fig 3.32. The existing dispersion can be attributed to limitations

in estimating the dominant wavelength using only a single slice in one direction. This

limitation can be improved upon by defining the dominant wavelength based on the

initial growth rate of the mixing layer. According to the linear theory (1.15). where

93

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10 12

k (

h-h

0)

k dh0/dt t

Figure 3.33: Dimensionless total mixing layer width for experiments presented inFig. 3.32. The wavelength is selected based on the deposited vorticity spectrum.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10 12

k (

h-h

0)

k dh0/dt t

Figure 3.34: Dimensionless total mixing layer width for experiments presented inFig. 3.32. The wavelength is selected based on the linear theory (1.15).

94

∆V is the jump in the interface velocity due to the impulsive acceleration and A is

the post-shock Atwood number A = ρ2−ρ1ρ2+ρ1

. The dimensionless mixing width, based

on this new wavenumber is plotted in Fig. 3.34 where it can be observed that it

provides better collapse of the data than in Fig. 3.33. However it should be stressed

that modeling this instability as if it contained only one dominant mode is probably

overly simplistic as the real initial perturbation contains a range of modes that would

contribute to the variability observed in Fig. 3.34

3.12 Time Evolution of the growth exponent θ

As discussed in previous chapters, small inconsistencies in the initial perturbation

results in a unique evolution of the total mixing layer width growth for each exper-

iment. In order to investigate the time evolution of the growth exponent theta the

average total mixing layer width was obtained for 55 experiments from group #5 as

described in Table 3.1 above , which were done using a thin laser sheet and smoke

in air. Taking an average over this large ensemble eliminates the scatter present in

each experiment, resulting in a smooth instability evolution profile. This average

total mixing layer width was calculated by measurements based on the location of

10% and 90% of the horizontally averaged smoke concentration, as well as by the

total integral mixing layer width measurements based on (3.5). Time evolution of

the growth exponent θ was found by applying equation (3.2) to the averaged total

mixing layer growth profiles. The result is shown in Fig. 3.35 and Fig. 3.35.

3.13 Tests evaluating the measurement methods

During the course of this study, several different measurement techniques for the

determination of the mixing layer width were tested in order to evaluate whether

95

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6

h (

mm

)

t (ms)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

θ

t (ms)

Figure 3.35: a) Average total mixing layer width for 55 experiments from group #5.b)Evolution of the θ(t) of the averaged total mixing layer width according to (3.2).

0

1

2

3

4

5

6

0 1 2 3 4 5 6

hin

t (m

m)

t (ms)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

θ

t (ms)

Figure 3.36: a) Average total integral mixing layer width for 55 experiments fromgroup #5. b)Evolution of the θ(t) of the averaged total mixing layer width accordingto (3.2).

96

the particular technique used had any influence on the obtained data. As a result,

four sets of experiments were performed by using four different configurations by,

varying flow illumination and which of the two gases is seeded with smoke. The first

combination, considered as the baseline experimental set-up, uses incense smoke

mixed with air with the test section illuminated by a thin laser sheet (1.5mm). The

second set-up differs from the first one by employing a thicker laser sheet with a sheet

width of 2 cm. Since the first set-up provides only a thin slice of the mixing process

in the x−y plane, it is thought that the thicker sheet provides additional information

about the smoke distribution, since the laser light is scattered by smoke dispersed

over a significantly larger volume. Another set of experiments were performed with

smoke seeded in the SF6 and these experiments were performed with thin and thick

laser sheets as well.

These four sets of data are processed the same way as in the baseline experiments

(smoke in air, illuminated by a thin laser sheet). Comparison of the mixing layer

widths is obtained by both the 10%-90% method, and the integral method. Fig.

3.37 shows sequences of images obtained for the four experimental set-ups. These

experiments were performed by employing only one camera to capture the interface

evolution. This was necessary to eliminate a bias error in the image light intensity

that occurred during the transition from one camera to another.

Fig. 3.38 -3.49 depict the time evolution of the mixing layer width for these four

sets of experiments. Also plots of dimensionless length scale versus dimensionless

time scale as described in section 3.7 are also shown. It should noted that the wide

laser sheet experiments exhibited less run-to-run variation in their mixing width

measurements when processes using 10% and 90% procedure as they provide an

integrated measure of the total mixing layer width across the entire sheet width

making them less sensitive to effects flow in and out of the laser sheet. Based on

97

the small variation of the 10%- 90% measure for wide laser sheet one would also

expect that the integral measure would exhibit this smaller variation as well. It was

revealed that even though that the thin laser sheet experiments exhibited relatively

larger dispersion in 10% - 90% measure the integral measure inherited non of those

short falls. Thus making this method less sensitive to the small randomness at the

outskirts of the mixing layer. The robustness of the given method is expected as the

mixing profile (f1(1 − f1)) has a bell shape and small randomness at the outskirts

of the mixing layer gets dampened. Plotting the dimensionless length scale versus

dimensionless time scale showed the advantage of using characteristic wavenumber

based on the linear theory assumption over the characteristic wavenumber obtained

by analyzing the deposited vorticity profile, reinforcing the finding described in the

previous section.

98

Figure 3.37: Image sequences of the RMI evolution for four different visualizationmethods. a) Smoke in air and visualized by a thin laser sheet. b) Smoke in air andvisualized with a thick laser sheet. c) Smoke in SF6 and visualized with a thin lasersheet. d) Smoke in SF6 and visualized with a thick laser sheet.

99

Figure 3.38: Time evolution of the mixing layer width for the experiments performedwith a thick laser sheet (2cm) and light gas (air) seeded with smoke. Left: Mixinglayer width measured by the 10 % and 90% smoke concentration locations. Right:Integral mixing layer width for the same set of data.

Figure 3.39: Dimensionless mixing layer width in which the characteristic wavenum-ber is selected based on the deposited vorticity profile. Left: The mixing layer widthmeasured based on 10 % and 90% smoke concentration locations. Right: The integralmeasure of the mixing layer width for the same set of data.

100

Figure 3.40: Dimensionless mixing layer width in which the characteristic wavenum-ber is selected based on the initial interface evolution measured immediately afterthe shock-interface interaction. Left: Mixing layer width measured based on the10 % and 90% smoke concentration locations. Right: Integral mixing layer widthevaluated for the same data.

Figure 3.41: Time evolution of the mixing layer width for experiments performedwith a thin laser sheet (1.5mm) and light gas (air) seeded with smoke. Left: Mixinglayer width measured by the 10 % and 90% smoke concentration locations. Right:Integral mixing layer width for the same set of data.

101

Figure 3.42: Dimensionless mixing layer width in which characteristic wavenumber isselected based on the deposited vorticity profile. Left: Mixing layer width measuredby the 10 % and 90% smoke concentration locations. Right: Integral mixing layerwidth for the same set of data.

Figure 3.43: Dimensionless plot of mixing layer width in which characteristicwavenumber is selected based on the initial interface evolution immediately afterthe shock wave propagation. Left: Mixing layer width measured by the 10 % and90% smoke concentration locations. Right: Integral measure of the mixing layerwidth.

102

Figure 3.44: Time evolution of the mixing layer width for experiments performedwith a thick laser sheet (2cm) and heavy gas (SF6) seeded with smoke. Left: Mixinglayer width measured by the 10 % and 90% smoke concentration locations. Right:Corresponding integral mixing layer width plotted for the same data.

Figure 3.45: Dimensionless plot of the mixing layer width in which scaling wavenum-ber is selected based on the deposited vorticity profile. Left: Mixing layer width mea-sured by the 10 % and 90% smoke concentrations. Right: Corresponding integralmixing layer width.

103

Figure 3.46: Dimensionless plot of the mixing layer width in which scaling wavenum-ber is selected based on the initial interface evolution immediately after the shockwave propagation. Left: Mixing layer width measured by the 10 % and 90% smokeconcentration locations. Right: Corresponding integral mixing layer width.

Figure 3.47: Time evolution of the mixing layer width for experiments performedwith a thin laser sheet (1.5mm) and heavy gas (SF6) seeded with smoke. Left:Mixing layer width measured by the 10 % and 90% smoke concentration locations.Right: Corresponding integral mixing layer width for the same data.

104

Figure 3.48: Dimensionless plot of the mixing layer width in which characteristicwavenumber is selected based on the deposited vorticity profile. Left: Mixing layerwidth measured by the 10 % and 90% smoke concentration locations. Right: Corre-sponding integral mixing layer width.

Figure 3.49: Dimensionless plot of the mixing layer width in which scaling wavenum-ber is selected based on the initial interface evolution immediately after the shockwave propagation. Left: Mixing layer width measured by the 10 % and 90% smokeconcentration locations. Right: Corresponding integral mixing layer width for thesame data.

105

Chapter 4

Conclusion

An experimental study of the temporal evolution of the shock-induced Richtmyer-

Meshkov instability in the turbulent regime initiated with three-dimensional random

perturbations was carried out. The primary interest of the study was the determina-

tion of the growth rate of the turbulent mixing layer that develops after an impulsive

acceleration of the perturbed interface between two gases (air/SF6) by a weak M=1.2

incident shock wave. The initial three-dimensional perturbations on the diffused in-

terface were generated by oscillating the entire gas system vertically. Measurements

of the mixing layer width following the initial shock interaction showed a power law

growth similar to the those observed in previous experiments and simulations.

Planar Mie scattering was used to visualize the flow, and image sequences were

captured using high-speed video cameras. To be able to study the mixing process,

incense smoke was added to one of the two gases. It is assumed that the scattered

laser light captured by the cameras is proportional to the density distribution of the

gas seeded with smoke. To analyze the turbulent mixing induced by shock inter-

action with a perturbed interface, the images were first corrected for non-uniform

illumination, by dividing each image by a smoke-filled image at the same location.

The images were then row-averaged to obtain the mean smoke concentration dis-

tribution that, in turn, was attributed to the distribution of the transverse-plane

averaged volume fraction of the smoke seeded gas. For verification purposes, the

mixing fraction obtained using two different calibration methods were compared and

it was found that both methods yield approximately the same mixing layer width

106

measurements.

During this study, it was observed that the interface continues to accelerate after

the shock-propagation. Theoretically, after the shock passage, the material volume

surrounding the interface moves with a constant velocity. However, in a real exper-

iment, the gas column accelerates due to a turbulent boundary layer developing on

the test section walls. Later, the growing boundary layer causes the constriction of

the effective tube cross-section, resulting in gas acceleration. The boundary layer

thickness depends on the intensity of the turbulent mixing generated by the shock

propagation that, in turn, results in differences acceleration of the material volume.

The averaged interface velocity measured during several experiments clearly matches

the theoretical predictions.

Initially the mixing layer width was determined as the distance between the 10%

and 90% of the observed smoke concentration. The 10% margin on both sides of the

concentration range allowed us to successfully track the mixing layer evolution, while

reducing the influence of concentration fluctuations due to the stochastic nature of

the turbulent mixing.

The measurements revealed that the growth rate of the mixing layer width sig-

nificantly varies from one experiment to another. This is attributed to the influence

of initial perturbations imposed on the interface. The Faraday gravity waves gen-

erated by vertical oscillation of the entire gas system have random phases that, in

turn, results in different initial conditions at the arbitrary moment when the shock

wave impacts the perturbed interface. However, better consistency for the mixing

layer width, and consequently, the mixing layer growth rate was obtained from the

mixing generated by the reflected shock wave. After the interface was subject to

further compression from the reflected shock wave, the growth rate of the mixing

layer matched the Mikaelian model, independently of the mixing generated by the

107

passage of the primary shock wave.

The analysis of the turbulent mixing revealed that the data for the layer width

can be split into two groups. The first group was characterized by a very rapid

initial growth followed by a much slower growth at later time. The second group

showed a much more consistent growth rate for the entire time before the mixing

zone encountered the reflected shock wave. At the beginning of the mixing process,

both groups showed similar behavior with growth exponent θ ∼ 1/2. However, later

the growth exponents were observed within the range of 0.3 ≤ θ ≤ 0.4. Based on the

experimental data, it was concluded that initial conditions played a significant role

in the turbulent mixing generated by the RMI. This conclusion encouraged further

modification of the experimental set-up in order to emphasize the study of initial

conditions.

In the modified set-up, the perturbations imposed on the interface were recorded

by a second video camera with higher resolution. This provided information regard-

ing the pre-shock interface perturbation amplitude, dominant wavelength and the

deposited vorticity spectral content. Further study was performed to determine the

evolution in time of the dominant wavelength. The dominant wavelength was calcu-

lated for each frame, and it was noticed that while the experiments with two growth

rates retained the initial value of the dominant wavelength, the experiments with

a more consistent growth rate demonstrated that the wavelength of the dominant

perturbation increased with time as the instability progressed. The spectral analysis

of the vorticity deposited on the interface immediately after the shock interaction

showed that the spectral content for the experiments with the saturation behavior

of the growth rate was characterized by a narrow initial vorticity spectrum, while

a wider spectrum was characteristic of the experiments with of the more consistent

growth rate. Further analysis demonstrated that independent of the total mixing

108

layer width evolution, the longer wavelength in the perturbation spectrum evolved

linearly in time, while shorter wavelengths saturated. Consequently, observations of

these two different behaviors of the mixing layer width evolution indicated that the

experiments with linear growth rate were characterized by higher energy deposition

into long-wavelength perturbations that governed the total mixing layer evolution.

However, experiments with two distinctive growth rates were subjected to less en-

ergy deposition into long-wavelength perturbations; therefore, the short wavelength

components governed the total mixing layer evolution.

The total mixing width measurements based on the 10% and 90% locations of

the averaged smoke concentration were studied in conjunction with the mixing layer

width evolution by the integral method. This method is less sensitive to small fluc-

tuations in the mixing layer width, which are related to the turbulent nature of the

phenomenon. Since the fluctuations in the mixing layer width were located close to

the 0% and 100% locations of the averaged concentration profile, they contributed

less to the width measurements. The mixing layer width evolution showed a power

growth law h ∼ Ctθ with θ ∼ 0.4.

Considering the system of two gases, after the shock passage, as an incompressible

flow, an imaginary plane separating the bubble region from the spike region was

introduced by equating the volume of air penetrating into SF6 with the volume of

SF6 penetrating into air. This middle line allowed the determination of the spike and

bubble integral mixing widths, and the study of their evolution in time separately.

It revealed that both regions evolved in time according to the power law, and the

growth constants θb−int = 0.40 and θs−int = 0.42 were similar.

A novel approach that considers mass and linear momentum conservation laws in

the moving reference frame, led to a new definition of the spike and bubble mixing

layer widths. For the planar incident shock wave, the newly defined bubble and spike

109

widths increased in time as hb−LM,s−LM ∝ tθLM , with growth exponent θLM = 1/2,

which did not depend on the initial conditions and the physical properties of the gases

composing the interface. The total mixing layer width defined as a combination of

the bubble and spike mixing zones allowed the introduction of a characteristic time

and length scale, thus reducing the functional dependence on time of the total mixing

layer width to a universal dimensionless form. The linear momentum deposited on

the interface by the shock interaction has a unique value for each initial perturbation

and can be used to draw similarity between the initial interfacial structure and the

resulting turbulent mixing.

The dimensionless mixing layer width was studied employing techniques used in

single-mode experiments by using the initial dominant wavelength obtained from

analysis of the initial perturbation as the characteristic length scale. This method

enabled the collapse the experimental data within a small uncertainty. Even better

collapse was achieved by estimating the dominant wavelength based on the linear

stability theory employed immediately after the incident shock wave impacted the

interface.

Appendices

110

111

Appendix A

1D Code

A.1 Analytical aspects

The ideal fluid flow is governed by the Euler equations, consisting of the mass con-

servation equation and the momentum conservation equation, and for an ideal case

the energy equation in which µ = 0 . In one dimension, the Euler equations can be

represented as

Ut + f(U)x = 0 (A.1)

Where no heat addition to the system is considered [125]. The variables in this

conservative form system are

U =

ρmρE

f =

mm2

ρ+ p

mH

(A.2)

where m = ρu, H = h + u2

2, E = e + u2

2and h = e + p

ρ, u is flow velocity. For the

given case, we are assuming a perfect gas, hence we have p = ρRT , cp = dh/dT ,

p = (γ − 1)ρe and γ = cp/cv. These relations will be used later to simplify the

equations.

The system of equations A.1 has real eigenvalues and all eigenvectors are linearly

independent. The system is hyperbolic, and can be rewritten as

Ut + f ′(U)Ux = 0 (A.3)

where f ′(U) is the Jacobian of f(U), and f ′(U) has a full set of eigenvectors. The

112

problem turns into an eigenproblem for f ′(U).

f ′ =

0 1 0γ−3

2u2 (3− γ)u γ − 1

(γ − 1)u3 − γuE γE − 32(γ − 1)u2 γu

(A.4)

From identity γE = H + γ−12u2, f ′ can be rewritten as

f ′ =

0 1 0γ−3

2u2 (3− γ)u γ − 1

(γ−1)2u3 − uH H − (γ − 1)u2 γu

(A.5)

In order to solve the eigenproblem for f ′(U), the equations are transformed into

dependent variables

W =

ρup

(A.6)

by introducing the Jacobian Q = ∂U/∂W , the equation A.1 can be introduced as

QWt + f ′QWx = 0 (A.7)

Multiplying the given equation by Q−1 results in

Wt + f ′Wx = 0 (A.8)

where

f ′ = Q−1f ′Q =

u ρ 00 u 1/ρ0 ρc2 u

(A.9)

c =√γp/ρ is speed of sound. The eigenvalues of f ′ are equal to the eigenvalues of

f ′

λ1 = u− c, λ2 = u, λ3 = u+ c. (A.10)

The eigenvectors of the untransformed Jacobian f ′(U) can be obtained by Rp = QRp,

p = 1, 2, 3.

R1 = ρ2

−1/c1− u/c

u− u2

2c− c

γ−1

, R2 =

1uu2

2

R3 = ρ2

1/c1 + u/c

u+ u2

2c+ c

γ−1

, (A.11)

113

A.2 Discretization, The Roe scheme

The exact Riemann problem can by approximated by

∂U∂t

+ ∂f(U)∂x

= 0 ∂U∂t

+ A∂(U)∂x

= 0 (A.12)

The matrix A depends on only in Ui and Ui+1 and should satisfy the following

conditions:

f(Ui−1)− f(Ui) = A(Ui, Ui+1)(Ui+1 − Ui) (A.13)

Also A has only real eigenvalues and a complete system of eigenvectors.

Roe’s scheme is given by

dUjdt

+1

h

(Fj+1/2 − Fj−1/2

)= 0 (A.14)

Fj+1/2 = F (Uj, Uj+1) = Aj+1/2(Uj, Uj+1)U(0;Uj, Uj+1) (A.15)

Aj+1/2 is the Roe matrix and Fj+1/2 is the Roe flux. U is the exact solution of

the linear Riemann problem A.12 with UL = Ui and UR = Ui+1. The Roe flux is

determined by

Fj+1/2 =1

2(f(Uj) + f(Uj+1))− 1

2

m∑p=1

|λp|αpRp (A.16)

where αp is defined implicitly by UR − UL =∑m

p=1 αpRP , if RqRp = δqp with δqp is

Kroneker delta. Taking the inner product of the previous expression by Rq gives

αq = Rq(Ui+1 − Ui).

The Roe matrix for the given case is

Aj+1/2 =

0 1 0γ−3

2u2 (3− γ)u γ − 1

(γ−1)2u3 − uH H − (γ − 1)u2 γu

(A.17)

114

The averages are defined as follows,

u =√ρjuj+

√ρj+1uj+1√

ρj+√ρj+1

H =√ρjHj+

√ρj+1Hj+1√

ρj+√ρj+1

(A.18)

The eigenvalues and eigenvectors used in the current procedure are

λ1 = u− c, λ2 = u, λ3 = u+ c

R1 =

1u− cH − uc

, R2 =

1uu2

2

, R3 =

1u+ cH + uc

(A.19)

m∑p=1

|λp|αpRp = R−1|Λ|R(Uj+1 − Uj) = |Aj+1/2|(Uj+1/2 − Uj) (A.20)

Therefore, putting the Roe flux formula into a more appropriate form, we obtain

Fj+1/2 =1

2(f(Uj) + f(Uj+1))− 1

2|Aj+1/2|(Uj+1/2 − Uj) (A.21)

Finally,

Un+1j = Un

j +τ

h

(F nj+1/2 − F n

j−1/2

)(A.22)

where τ represents time step and h is the spacial resolution.

A.3 Boundary conditions

For a domain representing a shock tube, boundary conditions at the upper and lower

ends are needed. Numerically, the boundary conditions are imposed by providing

proper Roe fluxes F1/2 and FM+1/2, where M is a number of points on which the

domain is broken down. For this purpose, two ghost points, one on each end, are

required. Also, reflective boundary conditions on both ends are imposed [116]. These

conditions can be described numerically as follows,

ρnM+1 = ρnM , unM+1 = −unM , pnM+1 = pnMρn0 = ρn1 , un0 = −un1 , pn0 = pn1

(A.23)

where the computational domain representing the shock tube is represented by points

i = 1, ...,M ;

115

A.4 The computational method with a two-stage Runge-Kutta time integration

In the given code, a two stage Runge-Kutta time integration with cell averaging

method is used [3, 112, 124]. Where the derivative in the middle of the cell is de-

termined by dWj+1/2 = minmod(dWL, dWR)1, dWL = Wi − Wi− 1 and dWR =

Wi+1 − Wi. The stage extrapolates to i + 1/2 is WL = Wi + 0.5dWi and WR =

Wi+1 − 0.5dWi+1; based on WL and WR allowing the Roe flux Fi+1/2 can be com-

puted.

The two-stage Runge-Kutta time integration method consists in calculating U ′i

with W n and then updating W ’s into W n+1 based on which, U ′′i is computed. Finally,

after averaging the data, Un+1i is obtained, and Un+1

i = 12(U ′i + U ′′i ) resulting in the

final value ofW n+1.

A.5 Multicomponent flow calculation

If the system under consideration consists of several gases, species equations must

be added to describe the conservation of species [56, 90, 117]. A common approach

in the computation of a two or more component flow, is to either determine the

concentration of species or, alternatively, use the quantity γ to identify the species.

It is noteworthy that if the system consists of several species all of which having

same γ no additional equation is required. After evaluating several models, the one

selected for the current study describes the system with two species.

1

minmode(a, b) =

a if (|a| ≤ |b|) and (ab > 0)b if (|a| > |b|) and (ab > 0)0 if (ab < 0)

116

ρρuEργ

t

+

ρu

ρu2 + pρuHρuγ

x

= 0 (A.24)

For this model, the Roe matrix has the following form

A =

0 1 0 0

γ−12u2 − γe (3− γ)u (γ − 1) e

γ−12u3 − uH − uγe H − (γ − 1)u2 γu ue−γ γ 0 u

(A.25)

where e is the internal energy of the system. This results in the subsequent eigen-

vectors and eigenvalues.

r1 =

1

u∗ − a∗H∗ − u∗a∗

γ∗

r2 =

1u∗

u∗2

2

γ

r3 =

00

− e∗

γ∗−1

1

r4 =

1

u∗ + a∗

H∗ + u∗a∗

γ∗

(A.26)

λ1 = u∗ − a∗ α1 = ∆p−ρ∗a∗∆u2a∗2

λ2 = u∗ α2 = a∗2∆ρ−∆pa∗2

λ3 = u∗ α3 = ρ∗∆γ

λ4 = u∗ + a∗ α4 = ∆p+ρ∗a∗∆u2a∗2

(A.27)

117

The average state is obtained by the following expressions

ρ∗ =√ρLρR

u∗ =√ρLuL+

√ρRuR√

ρL+√ρR

H∗ =√ρLHL+

√ρRHR√

ρL+√ρR

e∗ =√ρLeL+

√ρReR√

ρL+√ρR

γ∗ =√ρLγL+

√ρRγR√

ρL+√ρR

a∗2 = (γ∗ − 1)(H∗ − u∗2

2

)

(A.28)

A.6 Results

It is desired to show the calculated results in terms of characteristics of character-

istics, as they relay a more complete picture of the processes inside the shock tube.

Two characteristics J+ = u+ aγ−1

and J− = u− aγ−1

as shown on the X −T diagram

in Fig. A.1. The input parameters for the given code is listed in Table A.1.

Parameters Numerical ValuesDriver section length 1.829 [m]Driven section length 2.470 [m]

Test section 0.863 [m]Distance to Air-SF6 interface 0.102 [m]

Atmospheric pressure 93,835.58 [Pascal]Driver pressure 248,132.66 [Pascal]

Atmospheric Temperature 27.25 [◦C]

Table A.1. Input parameters for the 1D Code

118

Figure A.1: X-T Diagram of characteristics

119

program oneDKarni

implicit none

! Declearing the basic variables and numbers

integer , parameter :: p2 = selected_real_kind(13) !Double Precision

real(p2), parameter :: zero = 0.0_p2

real(p2), parameter :: one = 1.0_p2

real(p2), parameter :: half = 0.5_p2

real(p2), parameter :: gammaAir = 1.4_p2 !Ratio of specific heats for air

real(p2), parameter :: gammaSf6 = 1.09_p2 !Ratio of specific heats for sf6

!Derived data type: these data are stored in each cell.

type cell_data

real(p2) :: xc ! Cell-center coordinate

real(p2) :: u(4) ! Conservative variables = [rho, rho*u, rho*E, rho*gamma]

real(p2) :: u0(4) ! Conservative variables at the previous time step

real(p2) :: w(4) ! Primitive variables = [rho, u, p, gamma]

real(p2) :: dw(4) ! Slope (difference) of primitive variables

real(p2) :: res(4) ! Residual = f_{j+1/2) - f_{j-1/2)

end type cell_data

!Local variables

type(cell_data), allocatable :: cell(:) !Array of cell-data

real(p2) :: xmin, xmax !Left and right ends of the domain

real(p2) :: Driver, Driven, Test

real(p2) :: Patm, Pdriv, rhodriv, rhoair, rhosf6,T0, Tamb, Tdriv,Tsf6,Tair

real(p2) :: R, miuair, miusf6, Rair, Rsf6

real(p2) :: dx !Cell spacing (uniform grid)

real(p2) :: t, tf, tstep !Current time and final time and time step for output

real(p2) :: cfl, dt !CFL number and global time step

integer :: ncells !Total number of cells

integer :: nsteps !Number of time steps

integer :: itime !Index for time stepping

integer :: istage !Index for Runge-Kutta stages

integer :: i, j, os, dumpN

!Local variables used for computing numerical fluxes.

real(p2), dimension(4) :: dwl, dwr !Slopes between j and j-1, j and j+1

real(p2), dimension(4) :: wL, wR !Extrapolated states at a face

real(p2), dimension(4) :: flux !Numerical flux

open(unit=1, file = "visualrho.dat", status="unknown", iostat=os)

open(unit=2, file = "visualu.dat", status="unknown", iostat=os)

open(unit=3, file = "visualp.dat", status="unknown", iostat=os)

open(unit=4, file = "visualgamma.dat", status="unknown", iostat=os)

open(unit=8, file = "timestep.dat", status="unknown", iostat=os)

!Input parameters and initial condition.___________________________________________

!Parameters

ncells = 10000 ! Number of cells

tf = 0.020_p2 ! Finial time

tstep = 0.0001_p2! time step when to dump the files

cfl = 0.5_p2 ! CFL number

120

xmin = 0.0_p2 ! Left boundary coordinate

Driver = 1.829_p2 ! Driver length

Driven = 2.572_p2 ! Driven length

Test = 0.761_p2 ! test section length

xmax = Driver+Driven+Test ! Right boundary coordinate

T0 = 273.15_p2

Tamb = 27.25_p2

Tair = T0+Tamb

Tsf6 = Tair

Tdriv = Tair

R = 8.31_p2

miuair = 0.02895_p2

miusf6 = 0.1456_p2

Rair = R/miuair

Rsf6 = R/miusf6

Patm = 93835.58_p2

Pdriv = 248132.66_p2

rhodriv = Pdriv/(Tdriv*Rair)

rhoair = Patm/(Tair*Rair)

rhosf6 = Patm/(Tsf6*Rsf6)

dumpN = 0

! Allocate the cell array: 2 ghost cells, 0 on the left and ncells+1 on the right.

! E.g., data in cell j is accessed by cell(j)%xc, cell(j)%u, cell(j)%w, etc.

allocate(cell(0:ncells+1))

! Cell spacing (grid is uniform)

dx = (xmax-xmin)/real(ncells)

! The initial condition for Sod’s shock tube problem

do i = 0, ncells+1 !Include ghost cells at i=0 and ncells+1.

if (i <= ncells*Driver/xmax) then

cell(i)%w(1) = rhodriv !Density on the left

cell(i)%w(2) = 0.0_p2 !Velocity on the left

cell(i)%w(3) = Pdriv !Pressure on the left

cell(i)%w(4) = gammaAir !Pressure on the left

else if (i <= ncells*(Driver+Driven)/xmax) then

cell(i)%w(1) = rhoair !Density on the right

cell(i)%w(2) = 0.0_p2 !Velocity on the right

cell(i)%w(3) = Patm !Pressure on the right

cell(i)%w(4) = gammaAir !Pressure on the right

else

cell(i)%w(1) = rhosf6 !Density on the right

cell(i)%w(2) = 0.0_p2 !Velocity on the right

cell(i)%w(3) = Patm !Pressure on the right

cell(i)%w(4) = gammaSf6 !Pressure on the right

endif

cell(i)%u = w2u(cell(i)%w) !Compute the conservative variables

cell(i)%xc=xmin+real(i-1)*dx !Cell center coordinate

121

end do

!--------------------------------------------------------------------------------

! Time stepping loop to reach t = tf

!--------------------------------------------------------------------------------

t = zero !Initialize the current time.

nsteps = 0 !Initialize the number of time steps.

time_step : do itime = 1 , 100000 !100000 is large enough to reach tf=20ms.

if (t==tf) exit !Finish if the final time is reached.

dt = timestep(cfl,dx,ncells) !Compute the global time step.

if (t+dt > tf) dt = tf - t !Adjust dt to finish exactly at t=tf.

t = t + dt !Update the current time.

nsteps = nsteps + 1 !Count the number of time steps.

!---------------------------------------------------

! Runge-Kutta Stages

!

! Two-stage Runge-Kutta scheme:

! 1. u^* = u^n - dt/dx*Res(u^n)

! 2. u^{n+1} = 1/2*u^n + 1/2*[u^*- dt/dx*Res(u^*)]

!---------------------------------------------------

rk_stages : do istage = 1, 2

!(1) Residual computation: compute cell(:)%res(1:4).

! Compute the slopes (as difference) at every cell.

! NB: for uniform meshes, difference (du) can be used in place of gradient (du/dx).

reconstruction : do j = 1, ncells

dwl = cell(j )%w-cell(j-1)%w !Simple central-difference between j and j-1.

dwr = cell(j+1)%w-cell(j )%w !Simple central-difference between j+1 and j.

! Apply a slope limiter.

! (minmod: zero if opposite sign, otherwise the one of smaller magnitude.)

do i = 1, 4

cell(j)%dw(i) = minmod(dwl(i),dwr(i))

end do

end do reconstruction

! Initialize the residuals.

initialize_res : do j = 1, ncells

cell(j)%res = zero

end do initialize_res

! Compute the residuals: residual_j = flux_{j+1/2} - flux_{j-1/2}.

! Here, compute the flux at j+1/2 and add it to the left cell and subtract

! from the right cell. Only the internal faces are considered; the left

! and right most faces are considered later.

flux_comp : do j = 1, ncells-1

wL = cell(j )%w + half*cell(j )%dw !State extrapolated to j+1/2 from j

wR = cell(j+1)%w - half*cell(j+1)%dw !State extrapolated to j+1/2 from j+1

flux = roe_flux(wL,wR) !Numerical flux at j+1/2

122

cell(j )%res = cell(j )%res + flux !Add it to the left cell.

cell(j+1)%res = cell(j+1)%res - flux !Subtract from the right cell.

end do flux_comp

! Add boundary fluxes: left end and right end.

! Left most face: left face of cell i=1.

wR = cell(1)%w - half*cell(1)%dw !State extrapolated to j-1/2 from j=1

wL(1) = wR(1) !The same state

wL(2) =-wR(2) !The reverse velocity



flux = roe_flux(wL,wR) !Use Roe flux to compute the flux.

cell(1)%res = cell(1)%res - flux !Subtract the flux: -flux_{j-1/2}.

! Right most face: right face of cell i=ncells.

wL = cell(ncells)%w + half*cell(ncells)%dw !State extrapolated to ncells+1/2 from j=ncells

wR(1) = wL(1) !The same state

wR(2) =-wL(2) !The reverse velocity



flux = roe_flux(wL,wR) !Use Roe flux to compute the flux.

cell(ncells)%res = cell(ncells)%res + flux !Add the flux: +flux_{j+1/2}.

!(2) Solution update

if (istage==1) then

! 1st Stage of Runge-Kutta: save u^n as u0(:); u^* is stored at u(:).

stage01_update : do j = 1, ncells

cell(j)%u0 = cell(j)%u !Save the solution at n for 2nd stage.

cell(j)%u = cell(j)%u - (dt/dx)*cell(j)%res

cell(j)%w = u2w(cell(j)%u) !Update primitive variables

end do stage01_update

else

! 2nd Stage of Runge-Kutta:

stage02_update : do j = 1, ncells

cell(j)%u = cell(j)%u - (dt/dx)*cell(j)%res

cell(j)%u = half*(cell(j)%u0 + cell(j)%u )

cell(j)%w = u2w(cell(j)%u) !Update primitive variables

end do stage02_update

endif

! Copy the solutions to the ghost cells.

! In this program, the ghost cell values are used only in the reconstruction.

cell(0)%w = cell(1)%w

cell(ncells+1)%w = cell(ncells)%w

end do rk_stages

!---------------------------------------------------

! End of Runge-Kutta Stages

!---------------------------------------------------

write(*,*)

write(*,*) "time t(sec) = ",t," by ",nsteps," time steps"

! call output(ncells)

123

if (t>tstep*dumpN) then

do i = 1, ncells

if (i<=(ncells-1)) then

write(1,’(5es25.15)’,advance=’no’) cell(i)%w(1)




else

write(1,’(5es25.15)’) cell(i)%w(1)

write(2,’(5es25.15)’) cell(i)%w(2)

write(3,’(5es25.15)’) cell(i)%w(3)

write(4,’(5es25.15)’) cell(i)%w(4)

endif

end do

dumpN=dumpN+1

write(8,’(5es25.15)’) t

endif

end do time_step

!--------------------------------------------------------------------------------

! End of time stepping

!--------------------------------------------------------------------------------

!--------------------------------------------------------------------------------

call output(ncells)

close(1)

close(2)

close(3)

close(4)

close(8)

stop

!********************************************************************************

! End of program

!********************************************************************************

contains

!****************************************************************************

function minmod(a,b)

implicit none

real(p2), intent(in) :: a, b !Input

real(p2) :: minmod !Output

!Local parameter


if (a*b <= zero) then

minmod = zero !a>0 and b<0; or a<0 and b>0

elseif (abs(a)<abs(b)) then

124

minmod = a !|a| < |b|

elseif (abs(b)<abs(a)) then

minmod = b !|a| > |b|

else

minmod = a !Here, a=b, so just take a or b.

endif

end function minmod

!--------------------------------------------------------------------------------

!*******************************************************************************

function timestep(cfl,dx,ncells) result(dt)

implicit none

real(p2), intent(in) :: cfl, dx !Input

integer , intent(in) :: ncells !Input

real(p2) :: dt !Output

!Local variables

real(p2) :: u, c, max_speed

integer :: i

max_speed = -one

do i = 1, ncells

u = cell(i)%w(2) !Velocity

c = sqrt(cell(i)%w(4)*cell(i)%w(3)/cell(i)%w(1)) !Speed of sound

max_speed = max( max_speed, abs(u)+c )

end do

dt = cfl*dx/max_speed !CFL condition: dt = CFL*dx/max_wavespeed, CFL <= 1.

end function timestep

!--------------------------------------------------------------------------------

!********************************************************************************

function w2u(w) result(u)

implicit none

real(p2), intent(in) :: w(4) !Input

real(p2) :: u(4) !output

u(1) = w(1)

u(2) = w(1)*w(2)

u(3) = w(3)/(w(4)-one)+half*w(1)*w(2)*w(2)

u(4) = w(1)*w(4)

end function w2u

!--------------------------------------------------------------------------------

!********************************************************************************

function u2w(u) result(w)

implicit none

real(p2), intent(in) :: u(4) !Input

real(p2) :: w(4) !output

w(1) = u(1)

w(2) = u(2)/u(1)

w(3) = (u(4)/u(1)-one)*( u(3) - half*w(1)*w(2)*w(2) )

125

w(4) = u(4)/u(1)

end function u2w

!--------------------------------------------------------------------------------

!********************************************************************************

function roe_flux(wL,wR) result(flux)

implicit none

real(p2), intent(in) :: wL(4), wR(4) ! Input (conservative variables rho*[1, v, E])

real(p2) :: flux(4) ! Output (numerical flux across L and R states)

!Local parameters


real(p2), parameter :: one = 1.0_p2

real(p2), parameter :: four = 4.0_p2


real(p2), parameter :: quarter = 0.25_p2

!Local variables

real(p2) :: uL(4), uR(4)

real(p2) :: rhoL, rhoR, vL, vR, pL, pR ! Primitive variables.

real(p2) :: aL, aR, HL, HR ! Speeds of sound.

real(p2) :: RT,rho,v,H,a ! Roe-averages

real(p2) :: drho,du,dP,dV(4)

real(p2) :: ws(4),Da, R(4,4)

real(p2) :: gammaL, gammaR, gammaS,iS, dgamma

integer :: j, k

uL = w2u(wL)

uR = w2u(wR)

!Primitive and other variables.

! Left state

rhoL = wL(1)

vL = wL(2)

pL = wL(3)

gammaL = wL(4)

aL = sqrt(gammaL*pL/rhoL)

HL = ( uL(3) + pL ) / rhoL

! Right state

rhoR = wR(1)

vR = wR(2)

pR = wR(3)

gammaR = wR(4)

aR = sqrt(gammaR*pR/rhoR)

HR = ( uR(3) + pR ) / rhoR

! NEW Roe Averages

rho = sqrt(rhoL*rhoR)

v = (sqrt(rhoL)*vL+sqrt(rhoR)*vR)/(sqrt(rhoL)+sqrt(rhoR))

H = (sqrt(rhoL)*HL+sqrt(rhoR)*HR)/(sqrt(rhoL)+sqrt(rhoR))

gammaS = (sqrt(rhoL)*gammaL+sqrt(rhoR)*gammaR)/(sqrt(rhoL)+sqrt(rhoR))

a = sqrt( (gammaS-one)*(H-half*v*v) )

iS = (sqrt(rhoL)*uL(3)+sqrt(rhoR)*uR(3))/(sqrt(rhoL)+sqrt(rhoR));

!Differences in primitive variables.

drho = rhoR - rhoL

126

du = vR - vL

dP = pR - pL

dgamma = gammaR-gammaL

!Wave strength (Characteristic Variables).

dV(1) = half*(dP-rho*a*du)/(a*a)

dV(2) = -( dP/(a*a) - drho )

dV(3) = rho*dgamma

dV(4) = half*(dP+rho*a*du)/(a*a)

!Absolute values of the wave speeds (Eigenvalues)

ws(1) = abs(v-a)

ws(2) = abs(v )

ws(3) = abs(v )

ws(4) = abs(v+a)

!Modified wave speeds for nonlinear fields (the so-called entropy fix, which

!is often implemented to remove non-physical expansion shocks).

!There are various ways to implement the entropy fix. This is just one

!example. Try turn this off. The solution may be more accurate.

! Da = max(zero, four*((vR-aR)-(vL-aL)) )

! if (ws(1) < half*Da) ws(1) = ws(1)*ws(1)/Da + quarter*Da

! Da = max(zero, four*((vR+aR)-(vL+aL)) )

! if (ws(3) < half*Da) ws(3) = ws(3)*ws(3)/Da + quarter*Da

!

! Entropy fix comes later

!Right eigenvectors

R(1,1)=one;

R(2,1)=v-a;

R(3,1)=H-v*a;

R(4,1)=gammaS;

R(1,2)=one;

R(2,2)=v;

R(3,2)=half*v*v;

R(4,2)=gammaS;

R(1,3)=zero;

R(2,3)=zero;

R(3,3)=-iS/(gammaS-one);

R(4,3)=one;

R(1,4)=one;

R(2,4)=v+a;

R(3,4)=H+v*a;

R(4,4)=gammaS;

!Compute the average flux.

flux = half*( euler_physical_flux(wL) + euler_physical_flux(wR) )

!Add the matrix dissipation term to complete the Roe flux.

do j = 1, 4

do k = 1, 4

flux(j) = flux(j) - half*ws(k)*dV(k)*R(j,k)

end do

end do

127

end function roe_flux

!--------------------------------------------------------------------------------

!********************************************************************************

function euler_physical_flux(w) result(flux)

implicit none

real(p2), dimension(4), intent(in) :: w !Input

real(p2), dimension(4) :: flux !Output

!Local parameters


!Local variables

real(p2) :: rho, u, p

real(p2) :: a2, gamma

rho = w(1)

u = w(2)

p = w(3)

gamma = w(4)

a2 = gamma*p/rho

flux(1) = rho*u

flux(2) = rho*u*u + p

flux(3) = rho*u*( a2/(gamma-one) + half*u*u ) ! H = a2/(gamma-one) + half*u*u

flux(4) = rho*u*gamma

end function euler_physical_flux

!--------------------------------------------------------------------------------

!********************************************************************************

subroutine output(ncells)

implicit none

integer , intent(in) :: ncells !Input

!Local parameters

! real(p2), parameter :: one = 1.0_p2

!Local variables

integer :: i, os

open(unit=9, file = "xcoordinates.dat", status="unknown", iostat=os)

do i = 1, ncells

write(9,’(5es25.15)’) cell(i)%xc

end do

close(9)

end subroutine output

!--------------------------------------------------------------------------------

end program oneDKarni

128

Appendix B

Acceleration of the interface

The phenomenon related to the air - SF6 interface acceleration after interaction

with the incident shock wave has been investigated. According to ideal gas theory,

after the shock interacts with the interface it will be impulsively accelerated to a

constant “piston” velocity that will remain steady till other shock wave or expansion

wave will interact with it. During the experiments, it was found that the interface

undergoes acceleration well after the shock wave has passed. In order to investigate

this phenomenon, 10 experiments were performed with an initially flat interface and

matching incident Mach number. The experimental conditions are given at Table

B.1. It is believed that the interface acceleration is caused by the boundary layers

development on the test section walls. As the mass between the shock wave and

interface leaks through the boundary layers, and the mass addition from the shock

wave is still constant at a low Mach number, due to conservation of mass, the interface

at the center of the test section must increase in velocity to compensate for mass loss.

This problem has been investigated by Roshko [105] and continued by Brocher [11].

In Fig. B.1 is shown the experimental results with the average value of the interface

velocity, and the theoretical acceleration profile caused by a laminar boundary layer

growth. It can be seen that the acceleration predicted by the theory coincides with

the one observed in the experiments.

The rate of the mass flow passing the contact surface with side d is equal to

mc = 4dρuδ (B.1)

where ρ is the density of the gas between the shock wave and the contact surface,

129

u is the speed of the gas in test section (piston velocity), and σ is the mass flow

boundary layer thickness. In the case of a laminar boundary layer, the form is as

follows:

σ = β

(µl

ρu

) 12

(B.2)

where, µ is the viscosity on the wall, and l is the development distance of the bound-

ary layer from the shock wave to the interface. At the same time the gas has been

added to the shock-interface region by the propagating shock wave at a rate of

ms = (Us − u) ρd2 =ρu

η − 1d2 (B.3)

where the coefficient η is the ratio of the shock wave compressed gas density to the

unperturbed initial gas density, and Us is the shock wave velocity in the laboratory

reference frame.

The rate of mass increase between the shock wave and the contact surface is

m = ms − mc. The latter equation can be expressed as follow:

m = ρd2l (B.4)

Thus the rate of increase of the region between the shock wave and the contact

surface is equal to

dl

dt=

u

η − 1− 4β

u

d

(µ

ρu

) 12

l12 (B.5)

the interface velocity can be expressed as u′ = Us − dldt

. In the present study, the

reduction of the shock wave velocity due to the boundary layer related leakage was

not considered because this effect is negligible for small Mach number.

130

Exp M numb. in Air Atm. press. [PSI] Atm temp. [◦C] Ideal piston vel. [m/s]1 1.2049 13.4844 26.15 74.722 1.2028 13.4864 28.19 75.283 1.2057 13.4701 27.41 74.134 1.2049 13.4684 27.81 74.925 1.2040 13.4669 28.24 75.686 1.2015 13.4642 28.68 74.937 1.2044 13.6056 26.39 74.598 1.2010 13.6097 27.25 75.609 1.2011 13.6097 28.03 75.7210 1.2007 13.6090 28.26 74.62

Table B.1. Experimental conditions and the ideal gas piston velocity.

131

Figure B.1: Interface Velocity

132

Appendix C

Code for finding the piston velocity for two

gas system

In[238]:= R � 8.314472;Μair � 0.02895;Μsf6 � 0.14605;p1 � 93 030;T1 � 273.15 � 26.5;M1 � 1.1;Γair � 1.4;Γsf6 � 1.1;

Rsf6 �R

Μsf6;

Rair �R

Μair;

w1 �M1 Rair T1 Γair

p2 � p12 �M12 � 1� ΓairΓair � 1

� 1

u2 � w1 1 �M12 �Γair � 1� � 2M12 �Γair � 1�

T2 �p2 T1 � p2

p1�Γair�1Γair�1

�p1 � p2 �Γair�1�

p1 �Γair�1� � 1�Solve��p5� p1

2 �M22 � 1� Γsf6Γsf6 � 1

� 1 , u5�2 �M22 � 1� Rsf6 T1 Γsf6

M2 �Γsf6 � 1� , p3� p22 �M32 � 1� ΓairΓair � 1

� 1 ,

u3� u2 �2 �M32 � 1� Rair T1 Γair

M3 �Γair � 1� , u3� u5, p3� p5�, �p5, p3, u3, u5, M2, M3��Out[248]= 381.818

Out[249]= 115 822.

Out[250]= 55.2217

Out[251]= 319.109

Out[252]= ��p5� �1654.41, p3� �1654.41, u3� 751.049, u5� 751.049, M2� �0.168758, M3� 0.361405�,�p5� �312.715, p3� �312.715, u3� �607.915, u5� �607.915, M2� 0.205538, M3� �0.374891�,�p5� 91666.8, p3� 91666.8, u3� �1.83776, u5� �1.83776, M2� 0.992981, M3� �0.906221�,�p5� 92595.2, p3� 92595.2, u3� 0.583381, u5� 0.583381, M2� �0.997767, M3� �0.910004�,�p5� 124694., p3� 124694., u3� 36.8243, u5� 36.8243, M2� 1.15104, M3� 1.03231�,�p5� 219438., p3� 219438., u3� �111.647, u5� �111.647, M2� �1.51559, M3� 1.32921��

133

In[223]:= R � 8.314472;Μair � 0.02895;Μsf6 � 0.14605;p1 � 93 030;T1 � 273.15 � 26.5;M1 � 1.2;Γair � 1.4;Γsf6 � 1.1;

Rsf6 �R

Μsf6;

Rair �R

Μair;

w1 �M1 Rair T1 Γair

p2 � p12 �M12 � 1� ΓairΓair � 1

� 1

u2 � w1 1 �M12 �Γair � 1� � 2M12 �Γair � 1�

T2 �p2 T1 � p2

p1�Γair�1Γair�1

�p1 � p2 �Γair�1�

p1 �Γair�1� � 1�Solve��p5� p1

2 �M22 � 1� Γsf6Γsf6 � 1

� 1 , u5�2 �M22 � 1� Rsf6 T1 Γsf6

M2 �Γsf6 � 1� , p3� p22 �M32 � 1� ΓairΓair � 1

� 1 ,

u3� u2 �2 �M32 � 1� Rair T1 Γair

M3 �Γair � 1� , u3� u5, p3� p5�, �p5, p3, u3, u5, M2, M3��Out[233]= 416.529

Out[234]= 140 785.

Out[235]= 106.061

Out[236]= 338.003

Out[237]= ��p5� �1973.95, p3� �1973.95, u3� 801.108, u5� 801.108, M2� �0.158747, M3� 0.361717�,�p5� 739.358, p3� 739.358, u3� �536.422, u5� �536.422, M2� 0.230306, M3� �0.383873�,�p5� 87893.6, p3� 87893.6, u3� �7.06427, u5� �7.06427, M2� 0.973292, M3� �0.823395�,�p5� 91435.9, p3� 91435.9, u3� 2.15159, u5� 2.15159, M2� �0.991788, M3� �0.836388�,�p5� 162106., p3� 162106., u3� 70.7361, u5� 70.7361, M2� 1.3072, M3� 1.06292�,�p5� 470910., p3� 470910., u3� �229.044, u5� �229.044, M2� �2.20846, M3� 1.7349��

134

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