Effects of Plasma Jet Parameters, Ionization, Thermal Conduction, and Radiation on Stagnation Conditions of an Imploding Plasma Liner

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EFFECTS OF PLASMA JET PARAMETERS, IONIZATION, THERMAL CONDUCTION, AND RADIATION ON STAGNATION CONDITIONS OF AN IMPLODING PLASMA LINER

MILOŠ STANIĆ

A DISSERTATION

Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy in The Department of Mechanical and Aerospace Engineering to The School of Graduate Studies of The University of Alabama in Huntsville

HUNTSVILLE, ALABAMA

2013

In presenting this thesis in partial fulfillment of the requirements for a doctor of philosophy degree from The University of Alabama in Huntsville, I agree that the Library of this University shall make it freely available for inspection. I further agree that permission for extensive copying for scholarly purposes may be granted by my advisor or, in his/her absence, by the Chair of the Department or the Dean of the School of Graduate Studies. It is also understood that due recognition shall be given to me and to The University of Alabama in Huntsville in any scholarly use which may be made of

~ this thesis. 03/03/.2013

MILOS STANIC (date)

II THESIS APPROVAL FORM

Submitted by Milos Stanic in partial fulfillment of the requirements for the degree of Master of Science in Engineering and accepted on behalf of the Faculty of the School of Graduate Studies by the thesis committee.

We, the undersigned members of the Graduate Faculty of The University of Alabama in Huntsville, certify that we have advised and/or supervised the candidate on the work described in this thesis. We further certify that we have reviewed the thesis manuscript and approve it in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering.

ylr/l3 Committee Chair (Date)

22 Dee 2012 Committee Member (Dr. Snezhana I. Abarzhi) (Date)

I'/.--' 3/8 13 Committee Member (Date)

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Committee Member (Date)

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College Dean

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III

ABSTRACT

School of Graduate Studies The University of Alabama in Huntsville

Degree Doctor of Philosophy College/Dept. Mechanical and Aerospace

Engineering

Name of Candidate Miloš Stanić

Title Effects of Plasma Jet Parameters, Ionization, Thermal Conduction, and Radiation Stagnation Conditions of an Imploding Plasma Liner

The disciplines of High Energy Density Physics (HEDP) and Inertial

Confinement Fusion (ICF) are characterized by hypervelocity implosions and strong shocks. The Plasma Liner Experiment (PLX) is focused on reaching HEDP and/or ICF relevant regimes in excess of 1 Mbar peak pressure by the merging and implosion of discrete plasma jets, as a potentially efficient path towards these extreme conditions in a laboratory. In this work we have presented the first 3D simulations of plasma liner, formation, and implosion by the merging of discrete plasma jets in which ionization, thermal conduction, and radiation are all included in the physics model. The study was conducted by utilizing a smoothed particle hydrodynamics code (SPHC) and was a part of the plasma liner experiment (PLX).

The salient physics processes of liner formation and implosion are studied, namely vacuum propagation of plasma jets, merging of the jets (liner forming), implosion (liner collapsing), stagnation (peak pressure), and expansion (rarefaction wave disassembling the target). Radiative transport was found to significantly reduce the temperature of the liner during implosion, thus reducing the thermal

iv leaving more pronounced gradients in the plasma liner during the implosion compared with ideal hydrodynamic simulations. These pronounced gradients lead to a greater sensitivity of initial jet geometry and symmetry on peak pressures obtained. Accounting for ionization and transport, many cases gave higher peak pressures than the ideal hydrodynamic simulations. Scaling laws were developed accordingly, creating a non- dimensional parameter space in which performance of an imploding plasma jet liner can be estimated. It is shown that HEDP regimes could be reached with ~ 5 MJ of liner energy, which would translate to roughly 10 to 20 MJ of stored (capacitor) energy. This is a potentially significant improvement over the currently available means via ICF of achieving HEDP and nuclear fusion relevant parameters.

Abstract Approval: Committee Chair ( 'S-;z.~-r3 Department Chair (Dr. Keith Hol ings rth) (Date)

Graduate Dean ~ ~~ 'i/ts/13 (Dr. Rhonda Kay Ga lie) (Date)

ACKNOWLEDGMENTS

I would like to acknowledge all of my family, my beloved spouse and my friends for providing me with constant support during my Ph.D. studies, as well as several individuals and institutions:

 Dr. J.T. Cassibry – for the enormous amount of knowledge, tips and opportunities he has provided me with.  Dr. R.F. Stellingwerf – for “on call” support regarding the SPHC code.  Dr. S.I. Abarzhi – for her help and cooperation with RMI, as well as many useful discussions.  R. Hatcher and M. Milosevic – for their help regarding programming.  The whole PLX team – for being great collaborators.  Dr. J.M. Horack and D. Cook – for providing me with lots of professional opportunities and experience.  U.S. Department of Energy – for their funding of the PLX project, which made my Ph.D. a reality.

TABLE OF CONTENTS

LIST OF TABLES ...... xi

LIST OF FIGURES ...... xii

1. INTRODUCTION ...... 1

1.1 High Energy Density Physics ...... 1

1.2 Relevance of HEDP ...... 3

1.2.1 High Energy Density Astrophysics ...... 5

1.2.2 Laser-Plasma, Beam-Plasma Interactions ...... 6

1.2.3 Free Electron Laser Interactions ...... 6

1.2.4 High-Current Discharges ...... 6

1.2.5 Hydrodynamics and Shock Interactions...... 7

1.2.6 Equation of State and Stripped Atom Physics ...... 7

1.2.7 Nuclear Fusion ...... 7

1.3 Nuclear Fusion: Main concepts ...... 8

1.3.1 Fundamental Concepts ...... 8

1.3.2 Nuclear fusion approaches ...... 12

1.4 Primary motivation, objectives and technical approach ...... 20

2. THEORETICAL CONSIDERATIONS ...... 23

2.1 Smoothed Particle Hydrodynamics ...... 23

2.1.1 Smoothed particle hydrodynamics theory ...... 24

2.1.2 Equations of motion ...... 26

2.1.3 Numerical aspects of the SPH approach ...... 29

2.2 Equation of state ...... 31

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2.2.1 Perfect gas law, gas composition and partial thermodynamic

properties ...... 32

2.2.2 Classification of gases ...... 34

2.2.3 The Equilibrium constant and mass action laws ...... 35

2.2.4 Partition functions and fundamental thermodynamic properties ...... 38

2.3 Energy transport models ...... 42

2.3.1 Electron-thermal conduction ...... 42

2.3.2 Radiation basics and black body radiation ...... 45

2.3.3 Radiation diffusion and conduction approximations and LTE ...... 50

2.3.4 Radiation losses (optically thin radiation model)...... 53

2.4 Technical implementation of the equation of state (eos) ...... 56

3. VERIFICATION AND VALIDATION OF THE CODE ...... 63

3.1 The Noh problem ...... 65

3.2 Interferometry ...... 70

3.2.1 Interferometry principles ...... 70

3.2.2 Overview of the PLX interferometer and the synthetic

interferometry tool ...... 72

3.2.3 Jet Model ...... 73

3.2.4 Comparison of Results ...... 74

3.3 Richtmyer-Meshkov Instability (RMI) ...... 79

3.3.1 RMI introduction ...... 79

3.3.2 Self-convergence test ...... 80

3.3.3 Results ...... 83

viii

3.4 Full implosion resolution study ...... 85

4. SIMULATION PARAMETER SPACE AND SET UP ...... 87

4.1 Single jet propagation and two-jet merging ...... 87

4.2 Influence of jet geometry and density gradients within a jet study setup ...... 88

4.3 Liner formation and mixing study ...... 94

4.4 Parameter space ...... 97

4.5 Computational system configuration ...... 100

5. RESULTS AND DISCUSSION ...... 101

5.1 Single jet propagation and two jet merging ...... 101

5.2 Influence of jet geometry and density gradient within a jet upon peak

pressure ...... 108

5.3 Liner formation and mixing ...... 111

5.3.1 Purely hydrodynamic simulation results and hydrodynamic

smearing ...... 112

5.3.2 Energy transport models simulation results ...... 116

5.3.3 Influence on radiative collapse on peak pressures ...... 118

5.3.4 Remarks on mixing and Rayleigh-Taylor instability of the liner ...... 127

5.4 Development of scaling laws for plasma jet liners ...... 130

6. SUMMARY AND CONCLUSIONS ...... 145

7. APPENDIX...... 149

7.1 A.1 - Scale coupling in Richtmyer-Meshkov flows induced by

strong shocks ...... 149

7.1.1 Introduction ...... 150

7.1.2 Theoretical and numerical considerations ...... 152

7.1.3 Results ...... 160

7.1.4 Discussion and conclusion ...... 191

REFERENCES ...... 195

LIST OF TABLES

Table Page 1.1 – Table of physical parameters that correspond to HEDP [1]...... 2 3.1 – Table of Gaussian parameters used for creating a 3D plasma jet ...... 73 3.2: Initial jet parameters for the SPHC interferometry simulation, same as the ones used for Nautilus 3D simulation. Jet 1 is the one closer to the origin and jet 2 is the one trailing it...... 76 4.1: Final form of the linear functions used for implementation of radial density gradients, where R is a variable in the radial direction from the axis (see Figure 4.3). The only hollow jet runs are 51, 61, 52, 62 and 53, 63...... 93 4.2: Full setup of all 18 simulations...... 94 4.3: Table showing all of the relevant values of the chosen parameter space...... 100 5.1: Comparison of experimental results obtained by Hsu et.al. [83], and SPHC results for single jet propagation after approximately 45 cm, (15 µs). The variables are: – number density of particles (electrons for the experiments, ions and neutrals for SPHC), – temperature, – jet velocity, – ionization fraction, – jet length, – jet diameter...... 102 5.2: Average values of peak pressure with oscillations for each of the three subgroup runs...... 109 5.3: Results of all 18 simulations...... 109 5.4: Compact set of initial parameters used for liner formation and implosion study. The simulations are based on the Awe et.al. [92] run number 6 in his Table II: all jets have the same initial temperature of 1 eV, initial jet velocity of 50 km/s, with equivalent total kinetic energies. It is important to stress out that all Helios and Raven simulations were performed in 1D with spherical symmetry, while all SPHC cases were done in 3D...... 112 5.5: Results of the simulations (Awe et.al. [92], run 6, Table II) with energy transport models on...... 117 5.6: Characteristic geometry ratios for the GAr50 and 11Dec03 runs ...... 142

LIST OF FIGURES

Figure Page 1.1 – PJMIF concept drawing, showing elements of the system, [30]...... 17 2.1 - 1D Example of energy smoothing ...... 31 2.2 – Surface defined by , with direction vector of the photons and angle between them, [63]...... 47 2.3: simplified SPHC algorithm, showing the need for two different sets of tables...... 59 2.4: Specific internal energy as a function of number density and temperature...... 59 2.5: Transformed temperature matrix as a function of specific internal energy and number density...... 60 2.6: The solid black lines are lines of constant density, showing the dependency of original total pressure matrix as a function internal energy matrix (where both and ). The blue circles are reinterpolated values adjusted so that the pressure is now . Every 10th data point and every 10th density line is shown for legibility...... 61 3.1 - Graphical description of the planar, cylindrical and spherical geometries...... 66 3.2 - Conditions of the 1D Noh problem case at t = 0 and t = 200 ns...... 67 3.3 - Cut-away of the initial 3D Noh case setup...... 67 3.4 - Qualitative representation of the temperature gradient after 4 microseconds, 3D Noh case...... 68 3.5 - L1 norm vs. resolution plot for the 1D Noh problem ...... 69 3.6 - L1 norm vs. resolution plot for the 3D Noh problem ...... 69 3.7 – Cross-section of the 3D jet in X-Y plane showing Gaussian density distribution. Geometry is not to scale for sake of stressing out the Gaussian density distribution (length: 50 cm, total width (diameter): 5cm). The colorbar is density in kg/m3...... 74 3.8: Isometric view of the two jets (left) and X-Y plane view of the two jet (right). The colorbar scales are densities in kg/m3...... 77 3.9: All 8 chords for experimental results (left) and SPHC synthetic interferometry (right) ...... 77

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3.10: All 8 chords synthetic interferometry results for 1D LSP simulation (left) and 3D Nautilus simulation (right) [69]...... 78

3.11 – Amplitude growth for M = 3, A = 0.6 and a0 = 0.06λ. The only somewhat significant departure is the lowest resolution case with 61623 particles...... 82

3.12 – Amplitude growth rate for M = 3, A = 0.6 and a0 = 0.06λ, measured once the shock refraction phase is over. As obvious, the resolution makes almost no difference to the result...... 82

3.13 – Bubble velocity for M = 3, A = 0.6 and a0 = 0.06λ...... 83 3.14 - Development of Kelvin-Helmholtz instability at the fluid interface for early- time RMI at Atwood number A  0.8, Mach number M  5, initial

perturbation amplitude a0  0.5 ...... 84 3.15 – One of the SPHC results colored for temperature (from 4000K (blue) to 8000K (red)) and showing reversed jets in the base of the spikes between the bubbles. Reverse jets have been confirmed experimentally by Orlicz et.al. [91]...... 85 3.16: L1 norm vs. resolution plot for the full implosion case (L01)...... 86 4.1: General jet geometry...... 90 4.2: Sketch of the density profile for cylindrical jets (below) and an isometric view of the jet for reference (above). The blue arrows are indicating forward direction...... 92 4.3: Radial density gradient in hollow cylindrical jets...... 93 4.4: The buckyball form with marked equilateral lines (pink), [66]...... 96 4.5: Uniform (a) and discrete jet (b) initial setup, [66]...... 96 5.1: Isometric view of the two jets with pronounced x-y plane in which all the slices from Figure 5.2 and Figure 5.4 were made. The marked plane cuts jets in half, passing through both jet axes...... 102 5.2: Time snapshots of 2D slices showing particle number density at a) 0 µs, b) 15 µs, c) 30 µs and d) 45 µs. The colorbar scales are in [cm3] and change from frame to frame to adjust for relatively large expansion, always showing optimal range of values...... 103

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5.3: Mach number a) and temperature fields [eV] b), at 30 µs. Notice the absence of shocks and the uniformity of the temperature field...... 104 5.4: Mach number a), temperature b), particle number density c) and jet velocity d) histories during the jet propagation and merging phase. The data for comparison with the experiments has been extracted at 15 µs...... 106 5.5: Time history of specific internal energy showing smooth behavior...... 107 5.6: Peak pressure values for series 1...... 110

5.7: Comparison of peak pressures for different D0/L ratios (11, 12 and 13) ...... 110 5.8: Comparison of ideal gas eos results (no energy transport) for all three codes (Awe et.al. [92], run 6, Table II), including the SPHC discrete jets run. The black color lines refer to absolute pressure values, while the blue lines show the radius of the liner, [66]...... 112 5.9: Discrete jet case a) and ideal liner case b). Slices have been taken in an x-y plane, with z = 0 m for the ideal liner and z = 0.2 m for the discrete jet case. Time is , with the jets in the discrete jet case being positioned at their merging radius, [66]...... 113 5.10: Discrete jet case a) and ideal liner case b). Both slices are now taken in the x- y plane at z = 0 m, with , [66]...... 114 5.11: Discrete jet case a) and ideal liner case b). Peak pressure time at . Very similar structures in both cases with very close peak pressure values, [66]...... 115 5.12: Discrete jet case a) and ideal liner case b). Post-stagnation time at . The similarity trends between the two cases continue, showing a fairly uniform process of rarefaction after the peak pressure, [66]...... 116 5.13: Set of results from Awe et.al. study showing exploration of influence of different energy transport models on the peak pressure. Major difference in peak pressure occurs when both radiation and electron-thermal conduction are turned on, [92]...... 117 5.14: Contour plots of Mach number microseconds prior to peak compression for Run 1 a) at 4.5 µs, Run 2 b) at 27.2 µs and Run 3 c) at 27.2 µs. The colorbars to the right of each figure indicate the Mach number...... 120

xiv

5.15: Contour plots of particle number density [cm-3] at void collapse, prior to peak compression for Run 1 a) at 4.5 µs, Run 2 b) at 27.2 µs and Run 3 c) at 27.2 µs. The contour and colorbar scale for Run 3 is in logarithmic form so as to allow for better presentation of the density field which has extreme variations. For Run 3, the peak value at the origin and the yellow areas surrounding it differ by more than two orders of magnitude...... 121 5.16: History plots of minimum and mean values of particle velocity for Run 1 a), Run 2 b) and Run 3 c). Notice that Run 3 achieves the lowest mean velocity and seems to remain at that level...... 123 5.17: History plots of minimum, maximum and mean values of particle pressures for Run 1 a), Run 2 b) and Run 3 c). Aside from obvious increase in pressure for Run 3, more important for understanding of the process is the behavior of the minimum pressure value, showing quick expansion after peak mean pressure in Runs 1 and 2 and prolonged compression period in Run 3...... 126 5.18: History plots of minimum, maximum and mean values of particle temperatures for Run 1 a), Run 2 b) and Run 3 c). Note the large differences in temperature between runs, especially between the Run 3 and the first two...... 127 5.19: Front a) and isometric b) views of the jet showing four chords whose particles were monitored to evaluate the mixing, [66]...... 128 5.20: Mixing frequency as a function of dimensionless time. The vertical lines in the plot show the characteristic times of void collapse, maximum pressure and liner stagnation. Maximum mixing frequency occurs approximately 4 µs after the peak pressure, [66]...... 129 5.21: RTI slices at (a) , (b) , (c) and (d) . The planes of the slices correspond to those of Figure 5.9 through Figure 5.12, [66]...... 130 5.22: Plot of the dimensionless pressure versus initial Mach number of the jet, based on Eq. ((5.4). The solid line represents the curve and the dashed lines are 3 times and 1/3 of the value. The dimensionless pressure data points fit in perfectly between the dashed lines...... 133

5.23: Same as Figure 5.22, but with tabular eos and energy transport data points (blue triangles)...... 134 5.24: Same as Figure 5.23 with superimposed results from 1D studies carried out by Awe et.al. [92], Davis et.al. [93] and Kim et.al. [104], all of which have included energy transport models, but Awe et.al. used only ideal gas eos...... 135 5.25: Linear law dependence of absolute pressures as a function of initial kinetic energies for both pure hydrodynamic (black circles) and tabular eos and energy transport (blue triangle) cases. Data points that are off are off due to intentional setup of rail-gun misfire...... 136 5.26: Dimensionless peak pressure as a function of the newly developed function . The marked data points represent special cases whose departure from the common behavior can be explained through distinct features of the particular runs. Solid black line represents the scaling law given by Eq. (5.7) and the upper and lower dashed lines represent 3 times and 1/3 of the scaling law value, respectively...... 138 5.27: Selected case (J01) showing pressure slices at peak compression for the purely hydrodynamic case a) and the case including tabular eos and energy transport b). The colorbar is colored for pressure in kbar. Note the areas of high pressure that are better pronounced in the case of tabular eos and energy transport case...... 141

xvi 1. INTRODUCTION

The purpose of this work is to develop scaling laws for peak pressure and identify important mechanisms of plasma liner implosions. The plasma liners are formed by the merging of a spherical distribution of plasma jets. Such implosions have application to High Energy Density Physics (HEDP) [1] and Magneto-Inertial Fusion

(MIF) [2-4], The primary tool for scaling laws development is a modified Smoothed

Particle Hydrodynamics Code (SPHC) [5-7], in which improved physical models were included, such as a tabular equation of state (eos), optically thin and thick radiation models and electron thermal conduction.

In the following subsections we shall describe the field of HEDP and its significance, provide deeper insight into the fields of Inertial Confinement Fusion (ICF),

Magnetic Confinement Fusion (MCF) and MIF and look into primary motivation behind the development of scaling laws for plasma liner implosions.

1.1 High Energy Density Physics

General consensus among the experts is that in order to call some state of matter

HEDP-relevant, it needs to have energy density on the order of 1011 J/m3 [1]. This energy density corresponds to pressures of about 1 Mbar and can be achieved in multiple ways, involving several different technological approaches. Table 1.1 shows a list of physical parameters, such as pressure, electromagnetic wave intensity, electric field strength, etc. that correspond to HEDP states. The field of HEDP has rapidly expanded, facilitated by technological advancements such as the 5×1014 W laser at the National

Ignition Facility which compresses targets to 100 times solid density [8]. Other devices producing HEDP states include electron beams, like the one at Stanford Linear

Accelerator Center, which can reach beam intensities of 1020 W/cm2 [1], Z-pinch machines, like the one at Sandia National Laboratories that can deliver more than

1.8x1014 W to a few cubic centimeters [1] and similar facilities across the globe. These facilities can perform repetitive, consistent experiments that allow the physicists to analyze and develop the theory about the nature of extreme states of matter.

Table 1.1 – Table of physical parameters that correspond to HEDP [1].

Energy density parameter corresponding to ~1011 J/m3 Value Pressure 1 Mbar Electromagnetic radiation Electromagnetic wave (laser) intensity 3x1015 W/cm2 Blackbody radiation temperature 4x102 eV Electric field strength 1.5x1011 V/m Magnetic field strength 5x102 T Plasma pressure Plasma density for a thermal temperature of 1 keV 6x1026 m-3 Plasma density for an energy per particle of 1 GeV 6x1020 m-3

The parameter space covered by HEDP is generally very broad, with both density and temperature spreading over several orders of magnitude [9]. Such a broad parameter space involves states of matter that are commonly found in the field of astrophysics. The most extreme corners of HEDP parameter space correspond to exotic states of matter that can provide insights to fundamental nature of space and matter (strong field quantum electrodynamics) [10].

There are a few characteristics that are common for all HEDP states of matter and which help their classification. These are: non-linear and collective responses, full or partial degeneracy and the dynamics of the system [1]. HEDP states of matter have strong non-linear responses to the external energy supply. Examples of such behavior involve propagation of electromagnetic waves through plasma which cause different instabilities (Raman, Brilloiun and relativistic instabilities at higher energies).

Full or partial degeneracy of matter is commonly encountered when talking about astrophysics, since such conditions are often met at the cores of brown dwarfs, neutron stars, large planets and similar astronomical bodies. The degeneracy itself refers to the fact that the matter becomes so compressed that its pressure is no longer determined by its temperature, but by Pauli‟s exclusion principle. Degeneracy phenomenon is also important when talking about the ICF, as the target during its compression phase goes through different degenerate states.

Finally, the dynamics of the system is usually defined by the Mach number of the possible propagating shock-waves and the Reynolds number. These parameters play an important role in the studies of hydrodynamic instabilities which are equally important in both astrophysics and engineering.

1.2 Relevance of HEDP

There are four main areas of physics and engineering to which HEDP is highly relevant, and those are: fundamental physics, astrophysics, materials and weapons research and last but not least, the field of nuclear fusion. Aside from these four main areas, it expected that the development and improvement of HEDP-capable machinery will inevitably lead to many spin-off technologies that will find their place in the fields

of modern medicine, optics, material development and production and others. HEDP research requires cutting-edge technology and its theory is found to be extremely complex, thus providing a field of great intellectual challenge. The key questions to which HEDP heavily pertains are summarized in [1] and quoted below.

- “How does matter behave under conditions of extreme temperatures, pressures,

density and electromagnetic field?

- What are the opacities of stellar matter?

- What is the nature of matter at the beginning of our Universe?

- How does matter interact with photons and neutrinos under extreme conditions?

- What is the origin of intermediate-mass and high-mass nuclei in the Universe?

- Can nuclear flames (ignition and propagation) be created in the laboratory?

- Can high-yield ignition in the laboratory be used to study aspects of supernovae

physics, including the generation of high-Z elements?

- Can the transition to turbulence, and the turbulent state, in high energy density

systems be understood experimentally and theoretically?

- What are the dynamics of the interaction of strong shocks with turbulent and

inhomogeneous media?

- Will measurements of the equation of state and opacity of materials at high

temperatures and pressures change models of stellar and planetary structure?

- Can electron-positron plasmas relevant to gamma-ray bursts be created in the

laboratory?

- Can focused lasers “boil the vacuum” to produce electron-positron pairs?

- Can macroscopic amounts of relativistic matter be created in the laboratory and

will it exhibit fundamentally new collective behavior?

- Can we predict the non-linear optics of unstable multiple and interacting

beamlets of intense light or matter as they filament, braid and scatter?

- Can the ultraintense field of plasma wake be used to make an ultrahigh-gradient

accelerator with the luminosity and beam quality needed for applications in

high energy and nuclear physics?

- Can high energy density beam-plasma interactions lead to novel radiation

sources?”

All of these questions are very much fundamental and providing an answer to all of them would certainly represent a paradigm shift in physics and technology which may significantly improve the quality of life on Earth. Subsequent paragraphs briefly summarize the pertinence of HEDP to specific fields, and heavily rely on the contents of

[1].

1.2.1 High Energy Density Astrophysics

Development of HEDP machines, for instance Z-pinch machines, has led to the possibility of recreating important, scaled-down, astrophysics phenomena in laboratory surroundings such as compressible hydrodynamic mixing, strong shocks, magnetically collimated jets, magnetohydrodynamic (MHD) turbulence, opacity studies etc. All of these are relevant to astrophysics since they occur in a variety of astrophysical phenomena such as supernovae and their remnants, accretion discs, planetary and stellar interiors, astrophysical jets etc [9].

1.2.2 Laser-Plasma, Beam-Plasma Interactions

Ultra-powerful, short pulse lasers create enormous charged particle fluxes and consequently create some of the strongest magnetic fields on the planet. Similarly, relativistic electron beams create some of the strongest currents on the record, exceeding the Alfven limit by orders of magnitude. These interactions significantly influence the field of ICF, while further development and study of these phenomena could potentially lead to new ultra-high gradient particle accelerators, new light sources and generally better understanding of fundamental particle behavior.

1.2.3 Free Electron Laser Interactions

There is a large potential for use of relativistic electron beams for generation of directed X-ray sources. Such directed sources of X-rays would essentially represent X- ray lasers which would cause a revolution in the fields of biological, molecular and materials research. In physics, the intensity of such a laser is expected to “boil the vacuum”, creating electron-positron pairs out of fabric of space-time. Needless to say that such a beam would have profound impact on the field of ICF.

1.2.4 High-Current Discharges

Ways of achieving of 20 MA current bursts with duration of 100 ns are nowadays available, which has led to an increasing number of high-current, high-power facilities, most notably Z-pinch machines. Z-pinch machines are good sources of X-rays and are often used for studies of radiation-matter interactions. These facilities are also an excellent base for studies of astrophysical phenomena like accretion discs, provide the

ground for MIF concepts, and represent an excellent platform for studying MHD plasma behavior.

1.2.5 Hydrodynamics and Shock Interactions

The HEDP parameter space is an excellent ground for studying a large variety of flow regimes, ranging from low to extreme Reynolds and Mach numbers. Research of turbulence, its development at extreme flow parameters, understanding of common fluid instabilities, such as Rayleigh-Taylor and Richtmyer-Meshkov instabilities (RTI and

RMI, respectively) is crucial for developing ICF and MIF concepts, high-Mach number atmospheric flight and scramjet engine operation, as well as in understanding of some astrophysical phenomena such as supernovae and astrophysical jet propagation.

1.2.6 Equation of State and Stripped Atom Physics

Under HEDP conditions, matter starts to behave in interesting and often unexpected manner. Understanding of this behavior of course particularly pertains to fundamental physics, but in more practical terms can lead to more efficient fusion energy concepts, development of exotic states of matter, i.e. metallic hydrogen or new complex carbon states and provide essential tools for calculations of stellar interiors.

1.2.7 Nuclear Fusion

Nuclear fusion energy is the “holy grail” of energetic, and controlled demonstration of high-yield nuclear fusion process will surely represent one of the major scientific and engineering triumphs. HEDP is fundamental to the inertial branch of nuclear fusion research. Since the work in this dissertation is partially related to a

particular nuclear fusion approach (Plasma-Jet driven Magneto-Inertial Fusion, PJMIF), basics of nuclear fusion shall be presented in the next section.

1.3 Nuclear Fusion: Main concepts

Nuclear Fusion has been a subject of serious research from 1950‟s and ever since it has been what one would call “holy grail” of science. Countless efforts have been made to achieve fusion, many of which have been successful, but none of which have achieved breakeven (input to output energy ratio equal or greater than 1). Despite the fact that no viable fusion concept (that achieved breakeven) has been demonstrated in practice, the field of nuclear fusion research has been relentlessly pursued across the globe. The fact that the development curve of nuclear fusion surpasses the well know

Moore‟s Law of the microprocessors [11] shows how interesting the field of nuclear fusion is.

1.3.1 Fundamental Concepts

Plasma behavior is in most cases governed by a complex system of non-linear, partial differential equations (PDE), namely the Navier-Stokes and Maxwell‟s equations.

The system of PDE‟s is additionally perplexed by numerous instabilities and quantum effects [12-15]. For the sake of basic understanding primary topics in following subsections, we describe only the elementary principles. After the brief presentation of elementary physics, we discuss the basic differences between two main approaches, ICF and MCF, and we compare them to the third, hybrid approach MIF. Then we proceed to analysis of basic fusion concepts from the previous section, their advantages and disadvantages. It should be mentioned that MCF involves low density plasmas, and the

continuum fluid approximation inherent in Navier-Stokes breaks down for the flow parallel to the magnetic field lines and at the vacuum/plasma boundary, and that

Boltzmann‟s equation and/or particle-in-cell models must be used to properly address those physics.

Thermonuclear fusion is a process of fusing two lighter elements into one, heavier element, while releasing energy, relying on the kinetic (typically thermal) energy of the ions to overcome the interparticle Coulomb repulsion. The repulsion is the main obstacle since scattering collision cross sections are several orders of magnitude larger than fusion cross sections (i.e. for deuterium-deuterium reaction at 20 keV the collision cross section is cm2, while fusion cross section is cm2). The repulsive force requires a large amount of energy so that the particles would accelerate to sufficiently high speeds and that the collisions would result in a fusion reaction.

The following theoretical discussion is included for completeness. Numerous text books provide further details, and we refer the interested reader to the following references [4, 15, 16]. We begin the theoretical presentation by describing the rate of reaction:

(1.1)

3 where n1 and n2 are the number of particles of species per cm 1 and 2, and <v> is the average effective reaction cross section in cm3/s. The average effective reaction cross section essentially represents the average probability of two particles colliding, expressed in units of area. The specific power of a single reaction in W/cm3 is

(1.2)

With being the specific energy of the reaction in J.

Now let us adopt a reaction between deuterium (2H) and tritium (3H) as an example reaction, making and transforming term from Eq.

(1.1) into to make up for that factor 2 in front of . If we assume that the temperature remains constant with time and that plasma is stationary (not moving), when we integrate Eq. (1.1) with respect to time we get the total change in number of particles as a function of time:

(1.3)

with being the total initial number of particles and the subscript DT denoting a cross section value for deuterium-tritium reaction. The total energy produced in a reaction is then energy production rate term times time, multiplied with the total change in the number of particles, as given by Eq. (1.4):

(1.4)

A typical engineering parameter for description of any energy system is the gain, defined by the output over the input energy. In case of nuclear fusion, the expression would be as follows:

(1.5)

with being temperature and defined as: . However, this would work

in case that we had an ideal process, which in reality is not possible. Therefore we must account for a certain energy losses and we define an effective gain as:

(1.6)

So now we can define the specific losses per reaction, which would be:

(1.7)

The parameter is the loss coefficient, saying which fraction of the total power is lost due to different loss phenomena.

One of the crucial fusion parameters is the Lawson‟s criterion (L) [12, 15], which can be used to determine the required parameters necessary for fusion breakeven. This criterion is the product of number density ( ) and confinement time (L). Given specific reaction temperature and reaction species (i.e. DT), if one wants to achieve breakeven

( the Lawson product is defined [4] as

(1.8)

We can also define the minimal Lawson‟s time for a reaction process where we will have a gain equal to one and with a bit of algebra we can express it in terms of loss coefficient :

(1.9)

This tells us that if we would lose 90% percent of our specific power to, for example, radiative heat transfer ( = 0.9), we would have a factor of = 10. This means that in order to achieve fusion we have to hold the system for total of 10 Lawson‟s time scales.

Another important parameter is the specific size of the reaction domain, which is represents a characteristic length (volume) in which the reactions need to take place

when given temperature and density conditions. Specific domain size can be calculated

[4] as:

(1.10)

where is the thermal conduction coefficient, is Bremmstrahlung radiation,  is the temperature-gradient scale coefficient, which is a function of , and is the specific heat ratio. The theory presented above provides sufficient insight into a set of parameters which constitute what is called a „parameter space‟ for nuclear fusion [4]. The parameter space represents an excellent ground for comparison of different fusion concepts as it describes fundamental characteristics that are common for every fusion concept like energy and power needed to ignite the fuel burn and characteristic size of the system, all as function of particle density and temperature. In the next section we provide insight to three primary groups of nuclear fusion and compare their parameter spaces.

1.3.2 Nuclear fusion approaches

Possibly the three most general approaches to nuclear fusion are Magnetic

Confinement Fusion (MCF), Inertial Confinement Fusion (ICF) and Magneto-Inertial

Fusion (MIF). When talking about MIF, in this particular study, we concentrate on a sub-group of MIF known as Plasma Jet Magneto-Inertial Fusion (PJMIF). In order to reduce confusion, MIF and MTF are used as synonyms intermittently.

In the following sections we will provide some basic insights into all three of fusion approaches. An excellent reference on comparison of the three was performed in

2009 by Lindemuth and Siemon [4]. The authors presented information on all three

concepts and defined a parameter space, which represents an abstract plane within which the operating parameters of certain fusion concepts can be plotted and compared.

1.3.2.1 Magnetic confinement fusion

MCF utilizes strong magnetic fields to confine low-density plasma over a large spatial and temporal scale (ion density 1015 cm-3, a volume of hundreds of m3 and a continuous, steady state operation), establishing the conditions for fusion reactions, based on the Lawson Criteria. MCF takes advantage of the fact that magnetic fields keep the ions (which carry a significant portion of the energy) within the “reaction domain” and therefore reduce thermal losses. The primary type of device used to confine the fusion is a tokamak [17-19], although there are numerous spin-offs which differ primarily in geometry of the magnetic field i.e. spheromaks [20], Reversed Field Pinch

(RFP) [21, 22] and stellarators [23, 24]. The Polywell [25] and Field Reversed

Configuration (FRC) [26, 27] fusion approaches can partially be categorized as MCF, although they involve some inertial aspects as well (magneto-inertial hybrids) and their construction and magnetic field structure is quite different from the tokamak type machines. There are currently more than 20 different MCF devices in the World [28], all of which are primarily concerned with performing fundamental plasma research in support of the main “flagship” of MCF – namely ITER (International Thermonuclear

Experimental Reactor), situated in the south of France.

As mentioned previously, MCF relies on the capability of the magnetic field to prevent losses, primarily the ones concerning ion departures from the reaction domain.

MCF has low requirements on power demands, but does require a large amount of energy in order to keep the plasma in steady state. Energy and power needed to maintain 13

fusion conditions along with specific size of the system are convenient factors within the parameter space, with which all of fusion concepts can be well described and evaluated.

Another valuable concept that Lindemuth and Siemon [4] have developed is a simplified economic model for estimating the cost of the fusion system, scaled against current ITER and NIF costs. The estimated cost of ITER, as the MCF flagship, is around

$109 (US) The high costs and long development time of MCF and ICF (discussed below), are part of the primary motivations for studying MIF based concepts.

1.3.2.2 Inertial confinement fusion

ICF uses high-energy laser pulses to compress a solid-state target. It does so by using a large array of lasers, symmetrically distributed across a spherical chamber. The target is placed in the center of the chamber and fired upon with lasers, which compress it to an extremely high density state (1026 cm-3, or roughly a 100 times density of lead) at which fusion occurs. There are numerous ways to accomplish this (direct, indirect, fast ignition etc…), but in essence they all involve high-energy beams and a target. This approach relies on the fact that a large amount of energy is delivered to the target within a very short amount of time and on a small space scale, taking the plasma to a much

“higher than necessary” energy state. Consequently, even if there is no magnetic field to trap the ions within the reaction domain, the conditions are sufficient to achieve fusion due to very short timescales.

Two main drawbacks of this design are the lack of mechanism that prevents heat losses (i.e. magnetic field to trap the ions) and the fact that the driver laser has extremely low “wall-plug” efficiency, delivering only about 0.0025% of the stored driver energy to the target [4]. This in turn requires extremely high gain values and in order to do so, the

solid targets (pellets) need to be cryogenically stored and burn fraction of the fuel has to be high. ICF operates in the particle density region of the order of 1025 cm-3, with pressures on the order of hundreds of billions atmospheres and with temporal scales on the order of nanoseconds. High compression ratios result in occurrences of several instabilities, most notably Rayleigh-Taylor and Richtmyer-Meshkov instabilities, both of which significantly limit the compression process.

On the other hand, the primary positive side of the approach is certainly the high level of maturity of this technology. ICF is well developed both theoretically and practically, with NIF expecting to go on a “credible ignition campaign” [29] within the next few years. 400 MJ of stored energy in the large facility at Lawrence Livermore

National Laboratory, CA, USA has already been put to use by doing numerous experiments within the past few years. NIF for the first hand will be using indirect drive

(pellet inside the hohlraum), but even so, if successful, it shall open a new set of possibilities for other experiments with direct drive or fast-ignition drive. Another important advantage over MCF is the distance of the equipment from the reaction domain, so that the stand-off problem is successfully solved.

Similarly to the MCF, the ICF price tag is on the same order of magnitude, but slightly cheaper at about $3x109 US [4]. Considering that a 400 MJ capacitor bank and

192-beam, state-of-the-art, megajoule laser with remarkable optics to direct the beams is required, as well as a small army of scientists and engineers, it is no wonder that the cost of the facility is so high.

1.3.2.3 Plasma jet magneto-inertial fusion

As mentioned before, PJMIF is a subgroup of MIF, with abbreviation PJMIF becoming popular since it is more specific in explaining the process. PJMIF is an

MCF/ICF hybrid in the following sense: a magnetized plasma target is confined inertially (like in ICF) by imploding plasma jets, while the electron thermal conduction and ion scattering are suppressed by a closed internal magnetic field of the target. The suppression of thermal conduction is important because this way ions and the associated energy are prevented from flowing away from the reaction domain. Another convenience is that there is no material shell close to the plasma, so PJMIF provides an elegant solution for the stand-off problem, similarly to ICF. The time-scale of the whole process is supposed to be on the order of several microseconds, where the confinement time is proportional to the jet velocity over jet length ratio. The higher densities of fuel compared with MCF mean that reacting volumes are smaller by orders of magnitude

(because of the Lawson criteria), and the magnetized target permits lower power drivers for implosion and confinement compared with ICF, resulting in a potentially low cost development path to fusion. Depiction of the process is presented in Figure 1.1.

Plasma jet

Magnetized target

Forming

of the

Plasma gun spherical

Figure 1.1 – PJMIF concept drawing, showing elements of the system, [30].

The target of the PJMIF concept is gaseous plasma, not a solid as in the ICF. An even more important difference is the presence of the magnetic field within the PJMIF target. One of the optional targets is a Field Reversed Configuration (FRC) [31, 32] plasmoid that is formed by merging of two Spheromak plasmoids with opposite toroidal fields that cancel out upon merging. Both spheromaks and FRC‟s plasmoids and their creation are well known, proven processes, used in pulsed plasma thrusters for electric propulsion [33-35].

Another possible solution for target formation is to deploy a small amount of non-magnetized plasma which would be magnetized via laser beam prior to the plasma liner impact. This method relies on resonant acceleration of electrons via lasers and its principle has been experimentally validated, generating plasma currents on the order of

60 A [36]. After the target has been formed, it continues to exist in appropriate state for compression for sufficient periods of time.

Once the main portion of the plasma jets are launched from the plasma rail-guns at the edge of the chamber, they merge as they approach the target, forming a spherical

“piston” which is referred to as the liner (see Figure 1.1). Once the liner reaches the target, the target goes through a compression process, the magnetic flux increases significantly (inversely proportional to the square of the target radius), thereby significantly reducing the thermal losses. Fusion begins to take place at the point of peak compression and, according to [37], lasts as long as the plasma liner keeps incoming

(proportional to the jet velocity over jet length ratio). Once the liner has fully imploded, the confinement time is over and a rarefication wave propagates through the plasma, disassembling the plasma configuration.

Santarius in his 1989 study [38] points out that FRC related fusion is “perhaps the most attractive alternate concept” and that “FRC approaches the ideal fusion system”. Although Santarius was talking about a solid-liner FRC concept, the main principles of the solid-liner FRC and PJMIF are essentially the same. PJMIF can be considered a step forward because it provides the solution to the stand-off problem, by which original solid-liner FRC concept is burdened.

The US Department of Energy (DOE) in 2009 granted the funds for the Plasma

Liner Experiment (PLX), which is a 4-year collaborative project led by Los Alamos

National Laboratory and several other universities and private companies, with

University of Alabama in Huntsville leading the theoretical modeling and simulation efforts. The main goal of PLX is developing fundamental knowledge about plasma behavior in the intermediate fusion regimes and appropriate scaling laws that would provide access to High Energy Density Plasma regimes and the proof of viability of the

PJMIF concept. The first experiments of plasma jet propagation and merging began in second half of 2011.

One of the most important factors in PJMIF is the plasma-gun performance. For

PLX, the plasma guns were developed by Hyper-V Technologies company, with capability of producing fairly high plasma velocities (10 – 200 km/s, depending on the mass of the plasma jet), with a lot of room for improvements [39].

The rail-guns exploit the Lorentz force that is generated by the term when the plasma closes the circuit between two parallel rails or coaxial electrodes. For the

PJMIF application, high performance requirements (densities of 1015 to 1017 m-3, mg of mass and velocities of 10‟s of km/s) exceed the typical states of plasma in other plasma guns utilized in propulsion, railgun weapons, and experimental devices.

In spite of the fact that PJMIF is a relatively new concept, presented in its current form by Thio et.al. in [40, 41], a fair amount of studies on general PJMIF theory have been done involving confinement times [42], scaling laws [43], hydrodynamic efficiencies [44], rail-gun parameters [45], liner formation and mixing [46] etc. So far all theoretical/computational efforts and preliminary experiments of 2D/3D liner formation show promise of the PJMIF concept.

The cost estimates, according to [4] and the current PLX costs are multiple orders of magnitude lower than both ICF and MCF. The total estimated cost of a full fusion-capable facility is between $5x107 and $2x108 US, which is significantly cheaper than estimated ITER and NIF costs. These estimates are largely influenced by three key factors. First the reacting volume is significantly smaller than MCF, so the chamber is much smaller. Next, the presence of the high magnetic field in the target suppresses

cross-field electron thermal transport, which reduces the power required for the drivers to produce the thermonuclear state necessary for fusion. This means that the electrical energy storage and delivery (plasma accelerators) are much smaller compared to ICF devices. Finally, because of the high efficiency of electromagnetic acceleration, and the relatively low energy (~100 MJ) needed for the plasma, the overall scale of the reactor is small compared to the leading approaches (MCF and ICF) by at least an order of magnitude.

1.4 Primary motivation, objectives and technical approach

The previous sections have reviewed the significance of studying fusion and

HEDP conditions. Succinctly, the magnitude of scientific impact of understanding matter behavior at extreme conditions is of great importance for fundamental understanding of our natural surroundings as well as a direct path to potentially solving

Earth‟s energy problems once and for all by developing viable nuclear fusion reactors.

We argue that achieving HEDP conditions and nuclear fusion (MIF) can be assessed efficiently by utilizing plasma liner implosions. Studies of plasma liner based

HEDP concept was to be conducted by the PLX project, led by Los Alamos National

Laboratories, prior to the termination of the project. In order to provide support to the

PLX project we are hereby investigating theoretical plasma liner behavior by employing the 3D SPHC code.

In that sense, primary objectives of this work can be summarized as following:

- Simulate plasma liner formation, implosion, convergence, stagnation and post-

stagnation dynamics.

- Develop scaling laws for achieving HEDP using plasma jets.

This has been carried out by going through the following steps:

- Generation of a realistic equation of state (eos) for Argon (since most of the

PLX experiments are going to use Argon plasma) and the implementation of this

eos into SPHC

- Implementation of energy transport models into SPHC (electron-thermal

conduction, optically thin and optically thick radiation models)

- Code verification and validation by:

- Solution of the Noh problem

- Comparison of synthetic interferometry with real experimental

interferometry and other codes (LSP and Nautilus)

- Simulation and comparison of RMI with existent RMI theory and

experiments

- Full implosion self-convergence test of the code

- Development of the test matrix with sufficient number of simulations (sweeping

the desired parameter space)

- Running the simulations

- Analysis of peak pressure and other parameters of merit

- Identification of important mechanisms during plasma liner evolution

- Recognition of patterns and development of scaling laws

In chapter 2 we shall present the basics of Smoothed Particle Hydrodynamics

(SPH) and show the theory behind the development of the eos, radiation and electron-

thermal modeling. Chapter 3 presents the results of the above listed verification and validation cases, while in chapter 4 we show the simulation setups for all the relevant runs that were carried out. In chapter 5 we present the results of the simulations and facilitate a discussion and comments regarding the data. Finally, in chapter 6 we briefly provide an overview of the work that has been done and make concise remarks about the most important results that were obtained.

2. THEORETICAL CONSIDERATIONS

2.1 Smoothed Particle Hydrodynamics

In order to simulate the liner implosion process, Smoothed Particle

Hydrodynamics (SPH) code SPHC has been employed [5, 47]. SPH is a meshless,

Lagrangian, particle based method specifically designed for overcoming the issues of finite difference and finite element methods, some of which include: free surface flows, deformable boundaries, moving interfaces and large deformations. Also, SPH has been successfully applied to a number of high-velocity problems that include both solid materials and fluids [48-53].

It is important to point out that when talking about SPH particles, one does not refer to physical particles (i.e. individual ions, electrons, atoms, molecules), SPH is not a particle-in-cell method, but rather a meshless Lagrangian fluid approach in which finite

“chunks” of matter, that are advanced in accordance to the fluid mass, momentum, and energy equations. Each particle has its own mass, spherical volume (defined by the parameter called smoothing length), thus density, temperature, pressure and whatever other macroscopic physical parameter we need to assign to it. Like traditional computational fluid dynamic (CFD) methods, SPH particles must involve a sufficient number of ions, atoms etc., to satisfy the continuum assumption. Thus, an SPH particle is analogous to a grid cell or an element in CFD or finite element codes.

2.1.1 Smoothed particle hydrodynamics theory

The two main ideas of SPH that allow relatively easy treatment of the above mentioned problems are the kernel approximation (integral representation method) and particle approximation. In the following discussion, all bolded variables are vectors.

The kernel approximation starts with the fact that a field variable function can be expressed as

∫ (2.1)

where x are the position vectors of the particles, Ω the integral volume that contains x and δ is a unit signal function with properties:

{ (2.2)

If the unit signal function is then replaced by a so-called kernel function W with following properties:

∫ (2.3)

(2.4)

, when (2.5) where is referred to as the smoothing length, and k is a coefficient which determines the support domain boundary (see [54] for details), we obtain

∫ (2.6)

The brackets around f(x) represent the Kernel Approximation Operator (KAO). When the spatial derivative operator  is applied to KAO, it can be easily shown that the resulting expression takes the form

∫ (2.7)

This shows that spatial derivatives of the field function in SPH are actually dependent only on the derivative of the chosen kernel function, which simplifies the coding and saves computational time.

Next, the particle approximation, which provides adaptivity to the SPH method, discretizes the domain. Its essence is simple discretization which can be represented by set of Eqs. (2.8) and (2.9).

∫

(2.8)

∑ ( ) ( )

where ΔVj represents a finite volume of particle j, so that the overall mass of the particle is mj = ΔVj ρj.

Further we have

∑ ( ) ( )

(2.9)

∑ ( ) ( )

It can be seen in Eq. (2.9) that particle mass and density are straightforwardly introduced to the SPH equations, which has a particular benefit for fluid flows since density is one of the key field variables. Following the same logic, it can be shown that the spatial derivative of the field function, when discretized requires only that one takes the gradient of the kernel function in Eq. (2.9),

∑ ( ) ( ) (2.10)

Particle discretization is performed at every time step during the simulation and for all field functions. This results in partial differential equations (PDE) system being broken into a system of ordinary differential equations (ODE) with respect to time only. ODE‟s are then solved using explicit integration.

2.1.2 Equations of motion

In this work we solve time dependent mass, momentum and energy equations, neglecting viscosity, but including radiation and electron-thermal conduction models..

There are several different SPH forms of these equations, all of which have certain advantages, but hereby we will briefly cover our approach in SPHC. For more details see

[54]. The continuity equation is:

(2.11)

where the Greek superscript denotes coordinate directions. The derivative operator with capital D‟s is the so called substantial derivative, defined as:

(2.12)

Where is a scalar variable of choice.

The momentum equation in the case of free external force is:

(2.13)

where σαβ represents the total stress tensor (with and being indices, using Einstein‟s summation convention) and consists of two components

. The first component in the above equation is isotropic pressure stress, with as Kronecker‟s delta function and the other component is viscous shear stress term, the latter one in our case being left out entirely due to no-viscosity assumption.

The energy equation is:

(2.14)

Where and are radiation and electron-thermal terms, respectively. These equation terms are presented in section 2.3.

The most basic form of continuity equation arises when we take Eq. (2.9), then apply it to function of density, which gives:

∑ (2.15)

where Wij = W(xi-xj, h) . This form is called summation density approach and is the simplest form of density approximation. When the substantial derivative is taken from

Eq. (2.15), it transforms into:

∑ (2.16)

which is the continuity equation in SPH form.

When the SPH particle approximation is directly applied to Eq. (2.13), it yields:

∑ (2.17)

and when following identity is added to Eq. (2.17)

∑ (∑ ) (2.18)

we get the conservation of momentum equation as presented in Eq. (2.19).

∑ (2.19)

This form of momentum equation is quite common and has a benefit of reducing errors that arise from the particle inconsistency problem [54]. Another popular form of the momentum is:

∑ ( ) (2.20)

The form of the energy equation is heavily dependent on the way that pressure work is derived and considering that all of the derivations tend to be lengthy, we will directly present just the common form of the energy equation used in SPHC, for more details see

[54].

∑ ( ) (2.21)

αβ where µi is dynamic viscosity and εi represents the viscous strain rate.

If viscous stresses are to be neglected, previously listed equations transform into

Euler equations, which are used in this research. It is easy to show that the form remains the same except that the stress tensor is reduced only to pressure terms.

2.1.3 Numerical aspects of the SPH approach

In order to fully describe the flow, several other important equations and parameters have to be included. First, appropriate equations of state must be used.

Second, if shock related phenomena is simulated, artificial viscosity and energy smoothing must be introduced as well.

Artificial viscosity was originally developed by von Neumann and Richtmyer in

1950 as a way to overcome the issue of shock discontinuities, which numerical methods cannot overcome by using solely Navier-Stokes or Euler equations. Artificial viscosity occurs as a significant term only when the material is under compression. The most widely used form of artificial viscosity was developed by Monaghan et al. [51, 55, 56].

This form of artificial viscosity not only allows kinetic energy to be converted into heat, but also prevents particles from unphysical penetration when approaching each other at very high velocities. The form is:

̅ (2.22)

{ where

(2.23)

| |

̅ ( ) (2.24)

̅ (2.25)

̅ (2.26)

(2.27)

In the equations presented above parameter c is the speed of sound, αΠ and βΠ are constants that are typically set around 1.0 [57]. A factor of has been inserted to prevent numerical divergences when two particles are approaching each other. The αΠ coefficient produces a bulk of viscosity, while the second coefficient βΠ suppresses particle interpenetration at high Mach numbers.

Energy smoothing is a numerical method for stabilizing SPH codes. What energy smoothing essentially does is redistributing energy fluctuations, while simultaneously conserving the total energy of the system. This way energy smoothing is normalizing the energies of the neighboring particles and bringing them closer to the mean value. The principle is presented on a 1-D example shown Figure 2.1. More detailed mathematical derivation can be found in [58].

E Threshold Mean energy

Particle

E Threshold Mean energy

Particle

Figure 2.1 - 1D Example of energy smoothing

Both artificial viscosity and energy smoothing require careful approach when including them in the numerical model. Introducing too large or too small values of key coefficients that define both of these numerical corrections can cause serious errors in the solution. There is no universal rule which prescribes the appropriate coefficient values, however, by using appropriate verification tests the coefficients can be tuned so that they suit the desired case.

2.2 Equation of state

SPHC originally comes with multitude of different eos, most of which are meant for simulations of solids. When talking about gases, the only available eos is calorically perfect gas (ideal gas). This is found to be unsatisfactory for purposes of simulating

HEDP conditions as many important changes occur in the fluid once certain conditions

are met, which are not well approximated by the calorically perfect gas law. In this chapter we shall cover the fundamental thermodynamics behind the eos equations that are used for development of current eos tables generated by Propaceos [59] and present the process of practical implementation of eos tables into the SPHC code.

The total amount of material that governs the development of the eos (high- temperature gas dynamics) is at least a semester worth of intensive course work, so for sake of brevity and efficiency we shall try to select and present only the key laws and equations for the case of local thermodynamic equilibrium (LTE). All of the following discussions are based on Anderson's [60] and Vincenti and Kruger‟s [61] books.

2.2.1 Perfect gas law, gas composition and partial thermodynamic properties

There are a number of ways to represent the perfect gas equation of state, all of which are obtained from simple algebraic manipulations. Most notable are ones presented as Eqs. (2.34), (2.38) and (2.43).

(2.28)

(2.29)

(2.30) where is pressure, is density, is the specific gas constant, is the total number of particles, is the number of particles per volume (number density), is temperature, is volume and is the Boltzmann constant. The above equations stand for total behavior of the gas, including all of the atomic or molecular species within it. When talking about a perfect gas, we generalize molecular behavior by defining the gas constants based upon the average molecular weight of the species involved in the gas. That is the

beginning point of the more serious thermodynamic discussion. Let us now go deeper into the gas and allow ourselves to notice that the gas (i.e. air) is composed of several independent molecular species (nitrogen, oxygen, carbon-dioxide etc.). Each of these species i contributes to the total pressure with what is called a partial pressure .

∑ (2.31)

This, logically leads to the conclusion that partial pressures can be expressed as:

(2.32)

(2.33)

We can then define a set of parameters which describes the composition of the gas:

Concentration: (2.34)

Mole ratio: (2.35)

Mole fraction: (2.36)

Mass fraction: (2.37)

where is the number of moles and is mass. With Eqs. (2.34) to (2.37) and different forms of the perfect gas law, we rearrange some of the parameters into several short and useful expressions:

(2.38)

( ) (2.39)

∑ (2.40)

∑ (2.41)

∑ (2.42) ∑

where is the molecular weight of the species .

These are the basic equations that allow us to determine gas composition with knowing a few simple physical facts about its composition. However, things significantly change after sufficient amounts of energy are added to the system.

2.2.2 Classification of gases

When sufficient energy is added to the system, after a certain point, individual molecules and atoms become highly excited and physical phenomena such as saturation of internal energy modes (i.e. vibrational and electronic), dissociation and ionization begin to take place. We note that the rotational mode is fully excited at a few K, and is thus already accounted for in simpler models. The assumption of constant mixture composition in the perfect gas law model and constant specific heats do not account for these physical changes in the gas encountered at high temperatures, and these assumptions can lead to serious errors. An energetic mixture of gases in which the composition may vary due to chemistry, and vibrational and electronic energy modes may become excited can still be modeled as a perfect gas. In this approach, a chemically reacting mixture of perfect gases is a type of gas where perfect gas law still holds for individual species, however, theory for calculations of gas properties needs to be

significantly broaden as thermodynamics properties of specific heats, internal energies and enthalpy are now functions not only of temperature but of the number of particles of each present species that contribute to the overall state with their partial pressures:

, , and

. In the case of thermodynamic equilibrium, this simplifies to:

, , and .

We acknowledge that HEDP states of matter may involve real gas effects in which intermolecular forces may become important. This is especially important at very high pressures and low temperatures. For such a chemically non-reacting gas the perfect gas equation is no longer valid and needs to be modified. One of the well-known modifications is Van der Waals equation:

( ) (2.43)

Along with Van der Waals equation, the thermodynamic set of properties is generally given by: , , and . Since our application involves very high temperatures, we are neglecting the intermolecular forces and have only included this discussion for completeness.

2.2.3 The Equilibrium constant and mass action laws

Thermodynamic equilibrium is a general term for simultaneous existence of three distinct equilibriums: mechanical, thermal and chemical. Mechanical equilibrium means that the sum of all body forces (of present) is zero. Thermal equilibrium means that the temperatures of the system and the surrounding are equal and chemical equilibrium means that there is no tendency of the system to undergo chemical reactions.

Skipping the detailed derivation of the Gibbs free energy, we only present the important side result of the chemical potential term which contributes to the overall entropy production (in the Gibbs equation). The term is

∑ (2.44)

with being the Gibbs free energy of species i per mole of species i. In thermodynamic equilibrium, the Gibbs energy contribution from the chemical potential term is equal to zero: ∑ . For a general chemical reaction with stoichiometric coefficients and reactants we can write

(2.45) with all the reactant coefficients having a negative sign and all product coefficients having a positive sign. From conservation of species we have: ∑ . It can be concluded that because of the conservation of species and mass, total amount of mass does not change, however, in general non-equilibrium state species number ratios may change, so that i.e. for dissociation of hydrogen-oxide

(2.46)

An infinitesimal change in the mole numbers corresponds to

(2.47)

where is called degree of advancement. In general this can be expressed as

(2.48)

Integrating expression (2.48) for each of the species from some reference state of the mixture we get: . Realizing that ∑ and substituting for based on Eq. (2.48) and pulling out , we get

∑ (2.49)

Now, let us continue the derivation using thermodynamic properties on a per molar base.

By definition

̅ ̅ (2.50)

Substituting the general expression for entropy, whose derivation can be found in [60], we get

̅ * ∫ ̅ ( ) ̅ ̅ + (2.51)

Assume to get

̅ ̅ * ∫ ̅ ( ) ̅ + (2.52)

Subtracting Eq. (2.52) from Eq. (2.51) we get

̅ (2.53)

Substitute Eq. (2.53) into Eq. (2.49), and doing some algebra we finally obtain

∏ ∑ (2.54) ̅

or simplified

∏ (2.55)

Eq. (2.55) represents the so-called mass action laws, where is some unknown function of temperature (which we shall derive later). In combination with particle conservation laws (conservation of atomic nuclei and conservation of charge), they determine the gas mixture composition at a given time. Mentioned conservation laws are given as

Atomic nuclei: ∑ ∑ (2.56)

Charge: ∑ (2.57)

2.2.4 Partition functions and fundamental thermodynamic properties

In the previous section we have shown the general expression for mass action laws, which in combination with conservation laws determine the exact composition of the gas in state of thermodynamic equilibrium. However, if one goes back and looks at the equations from the previous section, he will notice that in order to really carry out the calculation, one needs to evaluate fundamental thermodynamic properties of the mixture components such as enthalpy, entropy and specific heat. Following text attempts to provide some insight to the derivation of mass action laws in their useful shape, involving atomic and molecular energy modes as functions of so called partition functions, which are of crucial importance, but skipping some of the algebra. As previous chapters on eos, this text relies on both [60, 61].

First of all, we need to stress out that we realize that each molecule has several energy modes: translational, rotational, vibrational, electronic and some base energy on

top of that (at ). Atoms can have only translational, electronic and the base energy. The expressions for total molecular and atomic energy are given below, respectively

(2.58)

(2.59)

From the definition of Helmholtz free energy F and its derivative

(2.60)

(2.61) where is total energy of the system, is total entropy of the system and is the chemical potential, we can define

( ) (2.62)

(2.63) ( )

(2.64) ( )

and employing Eq. (2.60) to get

( ) ( ) (2.65)

Manipulating the upper expressions and Boltzman‟s famous , we find that

[ ] (2.66)

( ) (2.67)

(2.68)

(2.69)

(2.70)

Equations (2.66) through (2.70) represent all important macroscopic thermodynamic properties, including total energy of the system, entropy and pressure and all of them are functions of the partition function . We shall now briefly present the way in which is calculated and how it relates to calculating the function from

Eq. (2.55).

Total internal energy of the particle is given by Eq. (2.58). In general form, the partition function is given by

∑

(2.71)

∏

Where is the number of energy states of the atom (see [60] for more details). In order to get the value of the partition function, we need to obtain expressions for all of the energy modes . This is done by solving the famous Schrödinger‟s wave equation for each energy mode. For sake of brevity, we shall skip detailed derivations of each of the energy modes and present only the final expressions for the total internal energy of a molecule

(2.72)

and specific heat (for a constant volume) 40

( )

(2.73)

( )

The in the above equations is the particle frequency. The four terms in both equations come from translational, rotational, vibrational and electronic energy modes, respectively.

Finally, let us derive the mass action laws using an example reaction of

(2.74)

For the macrostates we use the notation: .

Total number of microstates is ∑ , and the conservation laws are

∑ ∑ (2.75)

∑ ∑ (2.76)

Following the procedure laid out by Anderson [60] we get the following expression

(2.77)

which represents the mass action law. In more general terms, Eq. (2.77) can be written as

∏ ∏

(2.78) ∏ ∏

Or when is taken into account

∏ ∏

(2.79) ∏ ∏

Equations (2.28)) through (2.79) lay down the foundations of dealing with high- temperature gasses, their detailed composition and basic thermodynamic properties. We now proceed to the discussion about energy transport models used to modify the SPHC code and after that we shall describe the practical implementation of the eos into the

SPHC code.

2.3 Energy transport models

At extreme conditions, likes the one encountered in HEDP, sharp temperature, pressure and density gradients occur on a regular basis, causing mass and energy flows which are more often than not detrimental to the desired goals. Understanding of these processes is of utmost importance to the field of HEDP, as better understanding promises more successful mitigation of the problems. Mass flows are usually handled by hydrodynamics, whose equations are ground in the SPHC, however, the energy transport mechanisms in SPHC were limited due to the fact that SPHC did not have an appropriate eos for handling HEDP. Parallel with the development of the eos, satisfactory energy transport models had to be implemented into SPHC in order to provide merit to the simulation results. This chapter will present some basic theory behind the equations that have been used for energy transport models.

2.3.1 Electron-thermal conduction

Electron-thermal or electron heat conduction is one of the major mechanisms of heat energy transfer in the field of HEDP. This chapter will present Spitzer‟s solution to the Fokker-Planck treatment of electron-thermal conduction taken from [15].

We start with the Fokker-Planck equation for the electron distribution function

, with x, v and t being position, velocity and time, respectively. The expression is

( ) (2.80)

Where E is the electric field and term is formally describing electron-electron collisions, which shall not be covered here as it is beyond the scope of the discussion.

Coefficient A is

(2.81)

with being the Coulomb logarithm, a parameter with typical values between 10 and 20 (for details about Coulomb logarithm see [15] or [62]). Z is the ionization fraction, n is the number of particles per volume, m is the mass and e is the unit charge value.

If there is a temperature gradient in the x-direction, , we look for a general time and space independent solution of the form

(2.82) where , and is the angle between the electron velocity and the temperature gradient direction. Assuming that the length of the gradient L is much greater than the average mean path of the electron, the second term of Eq. (2.82) will be a small distortion of the isotropic equilibrium distribution in Maxwellian form

(2.83)

where √ . As for , inserting Eq. (2.82) into Eq. (2.80) and collecting the linear terms next to gives

( ) (2.84)

and so for a steady state case we can write

( ) (2.85)

The electric current ∫ , corresponding to has to vanish in order to satisfy charge neutrality and this determines the electric field

(2.86)

Combining Eqs. (2.83), (2.85) and (2.86) we explicitly get the expression for

( ) *( ) + (2.87)

and this allows us to calculate the heat flux

∫ ( ) (2.88)

where

(2.89) ( ) √ is the heat conduction coefficient or simply heat conductivity of fully ionized plasma.

The factor accounts for partially ionized plasmas, when electron-electron collisions become important (aside from electron-ion collisions), and .

In addition to the above, when an extremely sharp temperature gradient occurs, the upper conduction equation does not reflect the effects in reality [15]. Instead of

seeing a surge of electrons drifting downhill the temperature gradient, in truth the electron flow causing electron-thermal conduction saturates in a way, staying below predicted values due to the electric field generated by the hot current (so that the total current is zero). This is why codes usually employ what is called a flux-limiter, which keeps the heat flux beneath the so-called free-streaming limit heat flux, so

. The ratio is usually below 0.1 and is a function gradient length and gradient intensity. In practical applications, including SPHC, this is usually implemented through a condition

(2.90) where is the heat flux calculated by Eq. (2.88).

2.3.2 Radiation basics and black body radiation

At high temperatures in the HEDP regime, radiative heat transfer dominates electron-thermal conduction. Therefore it is important to have functioning radiation models to account for these energy losses in HEDP applications. The following discussion is mainly based on Zel‟dovich and Raizer‟s book [63] due to its thoroughness in covering the topic, but useful and more condensed information is available in both

[60] and [15].

Before proceeding deeper into radiation physics, we shall recall some of the more basic physical notions. First, let us remember that radiation is transported by wave- particle quanta called photons. Photons are characterized by their wavelength and their frequency , which are related through . The absolute value of momentum of a

photon is given by and its direction is given by a vector representing the energy flux in the electromagnetic field.

A radiation field is defined as a distribution function, similar to any other particle distribution function. The distribution function is such that there is a number of photons is in the frequency interval and , at time , contained in a volume (around the point ) and having a direction of motion within an element of solid angle (note that is a scalar) around the unit vector , which defines the direction of the photon movement. Each photon has energy equal to and a velocity , so we can now define what is called a spectral radiation intensity

(2.91)

This spectral radiation intensity corresponds to all the physical notions mentioned in the previous paragraph (time , point , volume etc.). The important bit is that the direction of this radiation intensity is perpendicular to an area which situated at point and perpendicular to the unit vector . We can use the spectral radiation intensity to define spectral radiant energy density

∫ ∫ (2.92)

The net spectral radiant energy flux through an area defined by a unit vector () is defined by

∫ ∫ (2.93)

훀

훝 풏

Figure 2.2 – Surface defined by , with direction vector of the photons and angle between them, [63].

The energy flux is a vector quantity and the integral is taken over the entire solid angle. Total spectral energy flux vector is

∫ (2.94)

Logically, when the radiation is isotropic, the radiant energy density is simply

(2.95)

Integrated values of intensity, energy density and energy flux are given by

∫ (2.96)

∫ (2.97)

∫ (2.98)

Now, let us imagine an infinitesimally small volume of atoms or molecules (but still significantly larger than ). It spontaneously emits photons, and as a matter of fact, quite

uniformly in all directions, across the solid angle, so we can consider the radiation emission isotropic. Parameter describing a number of unit photons emitted of certain frequency, per volume, per unit time is called the emission coefficient . Amount of energy radiated in a unit sold angle direction is .

Opposite to the emission coefficient is the absorption coefficient. When a beam of photons encounters some matter, the intensity of the beam in the original direction of the photons is weakened due to atom-photon interactions (absorption, scattering etc.).

This weakening is exponential in its nature and is a function of traversing distance of the photon beam as follows

∫ (2.99) where is the total absorption or attenuation coefficient, composed of the absorption coefficient and the scattering coefficient . The reciprocal values of these coefficients are the mean free paths of photons, therefore: , and

. The integral term in the exponent of Eq. (2.99) is called optical thickness

( ) of layer with respect to light of frequency , so

∫ (2.100)

or in the case where photon scattering can be neglected

∫ (2.101)

With Eq. (2.101) we finished presenting the basic concepts necessary for further discussion. We proceed with the description of black body radiation.

The concept of a perfect black body is an abstract one. It involves an infinite medium in a complete thermodynamic equilibrium, which in basic principle means a completely isotropic medium in all regards. This also includes the fact that in such a medium, any arbitrary volume is radiated and absorbed photons exactly balance. It also includes the fact that the radiation field is isotropic, independent of direction and specific properties of the medium and depending only on frequency and temperature. The spectral energy density for a specific frequency of such black body was derived by

Planck, in the early days of the quantum revolution in physics. The expression Planck derived is

(2.102)

This equation gives the so called Planck‟s function. The spectral intensity of radiation of a black body is then

(2.103)

When we integrate Eq. (2.102) from zero to infinity we get the more familiar expression for overall black body radiation

∫ (2.104)

where is the Stefan-Boltzmann constant. Because of the isotropy of radiation, total radiation flux is equal to zero, however, if one would like to calculate the one sided flux

(i.e. flux from left to right, meaning integrating only over a half of solid angle, rather than over the whole sphere), the expression is

(2.105)

and when we integrate this one-sided flux across all the frequencies we get

∫ (2.106)

Equation (2.106) is the well-known relation for a perfect black body radiation flux as can be shown by a simple thought experiment. Finally, for sake of completeness, we also introduce three more basic variables: reflectivity ( ), absorptivity ( ) and amount of

radiation emitted by the body per unit time, per unit area ( ). The relations that connect these three parameters are

(2.107)

(2.108)

The latter expression is also known as Kirchhoff‟s law. For a perfect black body, the absorptivity is always equal to one (it absorbs all the radiation that falls upon it).

2.3.3 Radiation diffusion and conduction approximations and LTE

The radiative transfer equation in its full form can be found in numerous text books, such as that of Zeldovich [63]

( ) ( ) (2.109)

The right hand side of the equation are source terms, namely the emission term and the overall absorption term. The left side of the equation is just a total derivative of radiation intensity. It can be shown that the emission term can be expressed simply through

reduction of the absorption terms, because when a certain volume receives a dose of photons it will re-emit a number of those photons back into the medium. The probability of the re-emission is equal to . We therefore introduce the modified absorption coefficient

( ) (2.110) and we can then rewrite Eq. (2.109) as

( ) (2.111)

This is an important result as now we do not require to know the exact value of the emission coefficient. Also, the existence of the modified absorption coefficient implies

the existence of a corresponding average mean free path .

For a steady state case, Eq. (2.111) becomes

(2.112)

When we integrate the above equation over the whole solid angle , we get

( ) (2.113)

Equation (2.113) has the form analogous to the steady state hydrodynamic energy equation in the divergence form. Through some manipulation of Eq. (2.113) and the basic radiation parameter definitions we arrive to

(2.114)

Dividing both sides of Eq. (2.114) by the photon energy , we get the expression for the photon flux as a function of gradient of photon density

(2.115)

(2.116)

The above equation for the photon flux is the diffusion equation.

In cases when assuming LTE, everything that was already pointed out in section

2.2.3 is assumed and in addition we assume a large body of some medium, in which the temperature gradients are large in size in comparison to the mean free path of the photons. The latter assumption can be rephrased as when the radiation at any point is almost the same, yet slight temperature variations exist on a large scale, then an LTE between the radiation and the fluid can be assumed. This assumption allows us to calculate the real flux of radiation based on modified mean free path values and equilibrium radiation density

(2.117)

and when we note that , the total flux is ∫

∫ (2.118)

where we factored out an average value of the photon mean free path (over all frequencies) and we denote it by . We see that the radiation flux is proportional to the gradient of the temperature which analogous to the standard heat conduction equation, and the coefficient of thermal conductivity is a function of temperature. Similarly to the ordinary heat conduction, the losses are determined only by the temperature, average mean free path and their spatial derivatives as shown below

∫ (2.119)

The above expression is the fluid energy loss, in case where the fluid is emitting more radiation that it receives (radiative cooling). Before going further into the topic of radiative losses, let us define what is known as the Rosseland mean free path.

Development of Rosseland mean free path is important as it is the process of development of this mean free path that makes the formal distinction between the optically thin and optically thick radiation models. For the case of optically thick radiation, we derive Rosseland mean free path from the definition of the average photon mean free path

∫ (2.120)

and the Rosseland mean free path is

(2.121)

2.3.4 Radiation losses (optically thin radiation model)

Considering a body that is generally non-uniformly radiating, we can conclude that the overall losses are the integrated losses of Eq. (2.119) over the total volume

∫ ∫ (2.122)

where is the normal component of the flux to the radiating surface and is the

element of the surface. We may also write that , where is the brightness temperature of the surface. For an optically thick body of inner temperature , in which

the average photon mean free path is significantly smaller than the characteristic size of the body , the order of the surface brightness temperature is approximately

( ) (2.123)

This usually means that for optically thick bodies, , while for optically thin bodies it is not so unusual to have . This is because for optically thin bodies the average photon mean free path is on the same scale or larger than the characteristic size of the body, thus most of the photons created in the interior of the body emerge on the surface and only a small portion, on the order of , of the photons is absorbed.

This also implies that the radiation is far from the equilibrium state as

, thus and . Integrating these over ∫ the spectrum and over the angles, introducing average mean free path of the photons for optically thin bodies (to make a distinction from optically thick), we get

. Therefore, the energy lost by an optically thin body (per unit volume per unit time) is equal to the emitted energy (integrated emission coefficient)

∫ ∫ (2.124)

Let us then take out the absorption coefficient in front of the integral to get

(2.125)

where

∫ (2.126)

where the weighting function is

(2.127)

This method of averaging the mean free path is different from the Rosseland method since in Rosseland we are averaging the average mean free path directly, while here we average the absorption coefficient which is inverse from the average mean free path. The average mean free path derived this way is referred to as Planck mean free path.

Respective absorption coefficients are referred to as Rosseland and Planck opacities. The total losses of an optically thin body are therefore

∫ ∫ (2.128)

Comparing Eqs. (2.122) and (2.128), we see formal differences, however, is always valid. The only difference between the optically thin and optically thick body is that in an optically thick body, for all practical purposes, the only losses come from the photons that are “born” on the surface of the body, while in an optically thin body, the photons are emitted volumetrically.

Finally, an important note regarding some additional differences between optically thick and optically thin radiation models. For optically thick models, the mean free path (note not average mean free path) of a photon can significantly vary, depending on its frequency (energy). For example, if we imagine an optically thick environment, the situation is likely to be that for instance low frequency photons will be fully absorbed, while the higher frequency ones will have a much longer mean free path.

These high frequency photons will provide potentially significant flux that will increase the influence of thermal conduction on a local level. In order to avoid “conduction smearing” by integrating the opacity across the whole frequency range, often so-called

multi-group models are used, which calculate the radiation conduction based on several distinct opacity groups (for different frequencies) in order to increase the accuracy of the model. On the other hand, for optically thin models, since most of the photons escape the body (or simply have large mean free paths), it is acceptable to use a single, so-called integrated opacity value.

2.4 Technical implementation of the equation of state (eos)

Sections 2.2 and 2.3 have covered the theoretical grounds on which the tables of thermodynamic properties for the new eos were generated. Because of the fact that tables of thermodynamic properties are used, we refer to the new SPHC eos as the

“tabular eos”. It is clear that generation of a detailed eos is a laborious endeavor, requiring extensive programming, calculations of the partition functions (which in turn require a vast amount of coefficient data) and (multi-group) opacities etc. Thus, we employed Prism Corporation [64] software Propaceos [59], which on the basis of the presented theory generates tables of thermodynamic properties and opacities. In this chapter we shall try to explain the way these tables were practically implemented.

Propaceos uses highly detailed atomic models for the generation of eos tables, i.e. for Ar, the model involves around 14,500 atomic energy levels across the ionization stages, and the model accounts for: translations of ions and electrons, electron degeneracy, configuration effects from Coulomb interactions and ionization and excitation [65]. The input for the Propaceos is the required temperature and number density range over which we desire to simulate. As mentioned before, the PLX experiment is planning to use mostly Argon for the first round of experiments, thus we are using Ar for the simulations as well. After a very simple and straightforward setup,

choosing the chemical element for which the eos is generated, LTE or non-LTE models, opacities or no opacity calculations, Propaceos generates the tables, taking sometimes several hours, depending on the required number of points in the temperature-density parameter space. For the purposes of our eos, we decided to go with a 200x200 grid (200 temperature points, 200 density points) and 10 opacity groups for the optically thick radiation model. This resolution is considered to be relatively high, perhaps even unnecessary, but it would allow for a more precise interpolation in the look up routine.

The temperature range is from 0.01 eV to 1 MeV, while the particle number density ranges from 107 cm-3 to 1023 cm3. Propaceos automatically applies logarithmic scaling on the grid, so sufficient accuracy is achieved at lower parameter regions where changes of thermodynamic properties are more pronounced. Selected parameter range goes sufficiently high to reach fusion conditions, while on the other hand allowing relatively rarified and cool plasma jets to realistically expand and further cool during the jet propagation phase.

Once the tables are created, the raw output from Propaceos comes in several .txt files, which are not really prepared for any practical use. So, the first step requires conversion of these files into a single table. To clarify, first we introduce some terminology that shall be used in the proceeding text. When a vector is mentioned we mean an array with a single row (1, j), and when the term matrix is used, we mean an array with (i, j) elements, the latter one basically always being a 200x200 matrix due to the input vectors of temperature and number density. Once again, the inputs are temperature and number density vectors. The outputs that are used for the eos are matrices of total pressure, total internal energy, ionization fractions and opacity tables

(multi-group Rosseland absorption and integrated Planck‟s absorption). These were found in the original raw output files and exported to separate .txt files. From these basic thermodynamic properties, we needed to develop additional properties, such as specific heats and speed of sound, as speed of sound plays a role in the SPHC code routines. A separate code in Matlab was written to deal with this, which relies on the theory presented in Chapter 2.2. The output of that particular code was a matrix of speed of sound squared.

SPHC has a specific algorithm which the eos tables needed to suit. The algorithm is as follows: in the very first time step, the state of the gas or plasma is specified conveniently in the input file based on the initial density and temperature. The code then determines the value of internal energy and other thermodynamic properties for each of the SPH particles. After this step, SPHC proceeds by advancing the equations of motions. Conservation and mass and energy advanced the mass density and specific internal energy, respectively. Thus, these are the natural variables which fix the state of the particles during the simulation. Consequently, temperature, pressure, and other thermodynamic and transport variables are determined using a table lookup with ρ and e.

The algorithm is shown in Figure 2.3, followed by the two figures (Figure 2.4, Figure

2.5) showing the transformation from to .

0 < t < tfinal

Find: e , p, Equations New: e , Find: Input: T, ρ int int Z, a… (integration) p, ρ, a… tfinal T(eint, ρ)

t = 0 Table set 2

Table set 1 푝 푝 푒푖푛푡 휌 푝 푝 푇 휌 푇 푇 푒 휌 … 푖푛푡 푒푖푛푡 푒푖푛푡 푇 휌 …

Figure 2.3: simplified SPHC algorithm, showing the need for two different sets of tables.

Figure 2.4: Specific internal energy as a function of number density and temperature.

Figure 2.5: Transformed temperature matrix as a function of specific internal energy and number density.

Noting that the original tabular eos is generated as a function of temperature and density, a separate code was written in Matlab to interpolate the table such that the thermodynamic variables become functions of evenly spaced (on a logarithmic scale) increments of internal energy and mass density. To do this, the internal energy matrix was compressed into a vector (with same minimum and maximum values as the matrix), while the temperature vector was “extrapolated” into a matrix corresponding to the specific values of internal energy and density. This is accomplished with the Matlab command interp. The interpolation was carried out in log space, which is more precise when the density and energy vary across several orders of magnitude. Figure 2.6 shows the results of the conversion code, plotting the original dependency of pressure and internal energy (black lines) and new interpolated dependency (blue circles). We see that the interpolation routine was very accurate in interpolating from the original to the new thermodynamic variables.

Figure 2.6: The solid black lines are lines of constant density, showing the dependency of original total pressure matrix as a function internal energy matrix (where both and ). The th blue circles are reinterpolated values adjusted so that the pressure is now . Every 10 data point and every 10th density line is shown for legibility.

Once this important step was taken care of, the new output vectors and matrices were converted into C header files, so they can be readily used in SPHC. The look up routine in SPHC is relatively simple and has been implemented into SPHC instead of the perfect gas eos (literally, the expression is substituted by the look up routine).

In the first time step, we use the original Propaceos tables to find the internal energies and pressures based on the initially desired temperatures and densities. For the remainder of the simulation, we use the new, interpolated tables to find temperatures and pressures based on internal energies and densities. The routines compare the densities and internal energy values of each particle to the values from the previous time step, then proceed into the bilinear interpolation scheme where the new approximate value of the variable is calculated. Routines essentially look the same for all variables, except for the optically thick radiation model, where due to the existence of opacity groups, the look up routine

is repeated for each of the groups. The code then proceeds to the integration step and the loop is repeated.

The look up routines for the opacities are essentially the same as the ones used in the eos. The switch between the optically thin and optically thick radiation models within SPHC is an if statement concerning the optical thickness of the SPH particle

(calculating and comparing the optical thickness of an individual particle to its smoothing length). For a given value of the smoothing length the optical thickness of the

SPH particle would be , so if the optical thickness , the code would use optically thin model and if , the code would use the optically thick model.

3. VERIFICATION AND VALIDATION OF THE CODE

In order to make relevant claims about the final results of this work it is essential that the modified SPHC code is properly verified against known analytic solutions and validated against experimental results wherever possible. This has been done through series of simulations which were then compared to exact, analytic solutions, other code solutions and experimental results. All of the tests that SPHC has been put through test different aspects of the code and combined together provide a solid ground for continuation of this work.

Code convergence tests were carried out for each of the test problems and those results are presented in the appropriate sections. One of the things that was taken into consideration when talking about satisfactory resolution for the final set of simulations was capability to resolve certain scales which are important for the plasma liner implosions. In this sense, due to fact that we primarily worry about peak values of general physical parameters, i.e. pressure, density and temperature, it is not necessary to try and resolve fine details of turbulent flow. However, Cassibry et.al. [66] have shown that even with resolutions as low as 3x104 particles, SPHC is capable of capturing mixing phenomena and large scale turbulence in three dimensions. Hot-spot generated by implosion of the liner at peak pressures is on the order of a few millimeters in diameter, which we define as minimum scale necessary to capture for satisfactory results. The SPH parameter that is relevant to that is the size of the smoothing length (h), in which the volumetric size of the particle is proportional to h3.

The Noh problem [67, 68] has been chosen to test SPHC‟s capability of capturing shocks, since shock phenomena play an important role in HEDP. The shock capturing capability in SPH methods is primarily handled with artificial viscosity.

Furthermore, resolution tests have been performed on the Noh problem, showing clear convergence of the solution and showing that SPHC is capable of capturing shock while providing relatively high accuracy at reasonably small resolutions.

Experimental interferometry data from single plasma jet propagation is supplied directly from the PLX experiment at Los Alamos [69]. A synthetic interferometry routine has been developed for post-processing of SPHC data, which in its essence very much resembles the physical experiment. More details about this will be presented in the appropriate subsection below. Uncertainties in the state of the plasma jets limit the code validation to qualitative comparisons of SPHC synthetic interferometry with the measured data. As shall be seen, SPHC results on integrated-line electron density fall within the bounds set by the experiments, sometimes even showing peculiar details that resemble those captured in real experimental data. Same results were compared to the results of two other codes: LSP [70] and Nautilus [71]. These comparisons were performed to provide a sense of confidence that the hydrodynamic models, eos (which all codes share) and energy transport models are consistent with those of other models, since it is highly unlikely that five different, independently developed codes will suffer from identical code errors.

Finally, detailed 2D study of the Richtmyer-Meshkov instability (RMI) has been performed [72], which had multiple objectives. First, SPHC output is validated for strong shocks and captures complex hydrodynamic phenomena like RMI. Second,

during the course of these series of runs, a previously unexplored regime of highly nonlinear RMI development (high Atwood and Mach numbers) was investigated computationally. Third, we mutually verified some of the newly developed non-liner

RMI theory and SPHC results. All three objectives have been successfully met, providing a large amount of data which gives confidence in SPHC results and new possibilities for further research of RMI.

3.1 The Noh problem

Prior to utilization of a code it is important to test for and eliminate errors in the implementation of an algorithm, and to check the accuracy [73]. This can be done by verifying the numerical output against problems with known analytical solutions, which is what we have done for this paper. The verification tests have been chosen based on their relevance to the matter of PJMIF, which would mean successful demonstration of high Mach number implosions and shocks. Chosen cases were the 1D and 3D Noh problem with spherical symmetry [67, 74].

The Noh problem involves a gas with initially uniform properties and ideal gas behavior. The solution consists of the properties of a reflected shock which depend on the initial conditions, and exists in planar, cylindrical and spherical geometries (Figure

3.1). In the planar case, the gas moves uniformly into a wall. For cylindrical and spherical geometries, the velocity is purely radial and directed inward, and the reflected shock begins at the origin at t > 0. The conditions behind the reflected shock are [68]:

( ) (3.1)

(3.2) where N = 0, 1, 2 refer to planar, cylindrical or spherical symmetry, respectively. Other parameters are s – density after the shock, 0 – initial density, ps – pressure after the shock, p0 – initial pressure,  - specific heat ratio. Temperature after the shock can be easily calculated using the ideal gas law:

(3.3)

,with R as the specific gas constant.

Figure 3.1 - Graphical description of the planar, cylindrical and spherical geometries.

Conditions of the 1D case at t = 0 and t = 200 [ns] are presented in Figure 3.2.

Figure 3.3 represents initial particle positions for the 3D case, showing just the positive x, y, and z octant. Initial conditions in the 1D case were: v0 = 100 km/s, T0 = 0.01 eV, 0

3 = 0.016605 kg/m , Mu = 1 kg/mol. The analytical solution for 1D case with spherical symmetry gives after-shock values of Tan = 35 eV, pan = 8.834 kbar and an = 0.264

3 kg/m . Initial conditions for the fully 3D case were: v0 = 100 km/s, T0 = 10 K, 0 = 1

3 kg/m , Mu = 1 kg/mol. The analytical solution for the 3D case gives after-shock values of

3 Tan = 34.72 eV, pan = 2120.09 kbar and an = 63.29 kg/m . For illustrative purposes, we

show the simulation at in Figure 3.4. In this figure, the volume of the gas can be seen to decrease due to the implosion, and strong temperature gradients are visible.

Figure 3.2 - Conditions of the 1D Noh problem case at t = 0 and t = 200 ns.

Figure 3.3 - Cut-away of the initial 3D Noh case setup.

Figure 3.4 - Qualitative representation of the temperature gradient after 4 microseconds, 3D Noh case.

After running the test in several resolutions - 500, 1000, 2000, 3000, 4000 and

5000 particles for 1D case and 10000, 20000, 40000 and 80000 for the 3D case, the self- convergence and L1 norm analysis have been conducted, where the L1 norm is given by

∑ (3.4)

where is the number of grid points and is a relative error given by

(3.5)

with being the reference solution (analytic solution) and being the numerical solution. An excellent discussion on the procedure can found in [75]. L1 norms for the

1D and 3D case are presented in Figure 3.5 and Figure 3.6.

Figure 3.5 - L1 norm vs. resolution plot for the 1D Noh problem

Figure 3.6 - L1 norm vs. resolution plot for the 3D Noh problem

It is clear that SPHC converges on the analytical solution in the in both 1D Noh problem with spherical symmetry case and the full 3D case. The convergent rates for pressure, temperature and density are N0.64, N0.46 and N0.63, respectively. Since L1 norm is basically an average relative error, it can be said, based on the number of particles used in our simulations, that it is reasonable to make an error estimate about 10% or less at a resolution of 30000 particles.

3.2 Interferometry

3.2.1 Interferometry principles

Interferometry is a diagnostic tool utilizing interference properties of electromagnetic waves for a wide range of purposes such as astronomy [76], quantum mechanics [77], plasma physics [78], biology and medicine [79], etc. There is a multitude of variants of interferometry depending on the field of use, but the main concept involves a coherent light source (laser), one half-silvered mirror, (at least) two fully reflective mirrors and a light detector. The coherent light beam is split in two (or more) beams, of which one travels directly to the detector and is called the reference beam. The rest of the beams travel to the diagnostic area where the medium of interest

(what is being measured) causes a change in the diagnostic beam‟s wavelength, and the diagnostic beams are then reflected back to the detector. When the diagnostic beams return to the detector they shall have a shifted wavelength with respect to the reference beam, causing interference between the waves. We call this shift the phase shift.

Depending on the physics of the problem, this phase shift can be interpreted as a change in the property of whatever is measured. In example, for the case of measuring the density of argon plasma, the phase shift is a function of integrated total particle density and ionization fraction, as shown below [69].

[ ] ∫ (3.6) where is the ionization fraction, a function of number of ions and neutral particles only:

(3.7)

The integral in Eq. (3.6) is across the beam chord length. In order to clarify the interpretation of interferometry results, we shall briefly go over the physics of the interferometry process. The coefficients found in the equation (3.6 result from the independent influences of ions, electrons and neutrons and a number of individual natural constants that are involved in the process. Total phase shift is thus a superposition of electron, ion and neutral contributions. The refraction index of a gas can be found using the Dale-Gladstone equation , where K is the specific refractivity, is the gas density and is the refractive index. Merritt et.al. [69] show that Dale-Gladstone relation is still valid for the ions in the plasma and they calibrate the interferometer signal analysis so that the contribution from any species with will result in and contribution from species with will give . Following that setup, Merritt et.al. [69] define total phase shift as , meaning that electrons will have , while ions and ions and neutrals will have

. This explains the phase shift from positive to negative in the results subsection of this section. Also, based on the expected jet temperature at the position of the cords ( , the expected ionization fraction is close to 1, resulting in approximation, so we can represent and as function of ion density only. This ultimately leads to the fact that , explaining equation (3.7. For more details on the physics behind the process of obtaining equation (3.6 see Ref. [69].

3.2.2 Overview of the PLX interferometer and the synthetic interferometry tool

For the purposes of the PLX experiment, a special 8-chord interferometry tool was developed by Merritt et.al. [78]. An Oxxius 561-300-COL-PP-LAS-01079, 320 mW laser was used for the creation of the initial light beam. The chosen Oxxius laser has a beam coherence of at least 10 m (conservatively estimated by the manufacturer), which helps in minimizing the errors since the total chord length is on the order of several meters. The wavelength of the laser was chosen so that it suits the expected integrated density ranges between 1016 cm-2 and 1018 cm-2 (for two jet merging stage). The use and spatial arrangement of the 8 chords is meant to capture the evolution of the jet parameters, primarily jet density and jet length evolution, which are later used to estimate the expansion rates of the jet. The chords are setup in a linear manner, along the axis of the jet propagation, starting 35 cm away from the rail-gun nozzle and thereafter uniformly spaced on every 6.35 cm. For full description of the interferometry system please see [69] and [78].

As discussed by [69], the interferometry experiment at that stage was unable to predict detailed integrated densities because the ionization fraction remained uncertain.

However, the interferometer successfully measured the total phase shift as the jet intersected all 8 chords, which is the value that we use for comparison.

The synthetic interferometry tool was developed using Matlab software package and it has a significantly different mechanism for obtaining the phase shift. Since SPHC is not capable of simulating laser beams and generally electro-magnetic waves, the phase shift has been obtained indirectly by integrating density across a specified chord through

a plasma jet, measuring the average ionization level and then using Eq. (3.6) to calculate the phase shift. The SPHC model of the plasma jet is discussed in the next section.

3.2.3 Jet Model

Based on available data from PLX, a 3D model of the PLX jet was created [80].

The model is cylindrical in geometry, with a uniform initial temperature of 3 eV and a

3D Gaussian density distribution across the cylinder volume. The 3D Gaussian function is given below in polar coordinates with the origin at the front of the jet.

( )

{ with G given as

( )

{

The z-coordinate represents the axis of the cylinder while r-coordinate is radial distance from the axis.

Table 3.1 shows the appropriate parameter values.

Table 3.1 – Table of Gaussian parameters used for creating a 3D plasma jet

Symbol Value Units Description 3 An 1e17 cm Peak ion density R 1.1 cm Radial scale length for Gaussian profile

rn 2.5 cm Radial cutoff density (gun nozzle inner radius)

zd 4.5 cm Density decay length

zc 50 cm Cutoff distance α 3.8 - Power for initial rise

Figure 3.7 shows a cross-section view of the 3D jet in X-Y plane with Gaussian density distribution being obvious. The front of the jet in the figure is pointed downwards. The ratio of the length to width is not realistic (length: 50 cm, total width

(jet diameter): 5 cm), but this has been done on purpose in order for the Gaussian density distribution to be obvious. This particular jet model has been used only for the interferometry study.

Figure 3.7 – Cross-section of the 3D jet in X-Y plane showing Gaussian density distribution. Geometry is not to scale for sake of stressing out the Gaussian density distribution (length: 50 cm, total width (diameter): 5cm). The colorbar is density in kg/m3.

3.2.4 Comparison of Results

The results of the first single jet propagation experiments were published by

Merritt et.al. [69], and this section presents the comparison of those data with the SPHC numerical output. In the same paper, two other codes, LSP and Nautilus, were used to compare their synthetic interferometry results with the experiment. We find this useful as it simultaneously provides us with additional data for mutual code comparison, thus giving further confidence of our results.

LSP was originally developed by ATK Mission Research Corporation with initial support from the US Department of Energy SBIR program and is currently belonging to the Voss Scientific Company from Albuquerque, New Mexico. Thoma et.al. [65] provide an excellent description of the two-fluid, fully electromagnetic algorithm that was embedded into LSP for purposes of plasma simulations, primarily for support of the

PLX project. LSP is a particle-in-cell code, with highly detailed energy transport physics models, capable of handling large variety of problems. The algorithm developed by

Thoma et.al. [65] solves the set of governing equations implicitly, allowing for larger time steps, while Eulerian particle remap procedure allows fewer particles per cell while maintaining desired accuracy of the simulations. LSP eos for Argon is also generated by the Propaceos software. Optically thin radiation regimes are handled as an electron- cooling term, while the optically thick model is a multi-group diffusion approximation, very similar to the models recently implemented in SPHC.

Nautilus is a state-of-the-art multidimensional code capable of performing magneto-hydrodynamic and multi-species flows with radiation, electron-thermal conduction and tabular equations of state [71]. It is developed by Tech-X Co. from

Boulder, Colorado and utilizes MUSCL [81] and discontinuous Galerkin [82] finite volume methods which allow for simulations of several plasma regimes, including weakly and fully ionized multi-fluid, multi-temperature Hall and resistive MHD plasmas. The code was initially available free of charge (on demand) for testing and debugging purposes, but has recently entered the stage of maturity in which its license will become fully commercial.

It is appropriate to mention here that there is a more recent study of jet evolution being performed by the same group (led by Hsu) [83], but the available data from that work will be presented in the results section of the dissertation since it focuses on the evolution of the jet itself.

During the experiment it has been noticed that the plasma exits the rail gun in two distinct bulks of plasma, rather than one, as initially expected. The first portion of the jet is longer, faster, has a higher temperature and lower density than the second jet. It is important to note at this point that all experimental data, for each chord, have been averaged across 30 rail-gun shots. We based our initial setup on the setup which was done for the 3D Nautilus run, as we are also performing a 3D simulation for these purposes. Initial parameters of the simulation are given in Table 3.2, and the isometric views of the two jets at t = 0 is given in Figure 3.8. The jet in front is denoted as jet number 1 and the following jet is jet number 2. The simulation had all energy transport models turned on, along with using tabular eos.

Table 3.2: Initial jet parameters for the SPHC interferometry simulation, same as the ones used for Nautilus 3D simulation. Jet 1 is the one closer to the origin and jet 2 is the one trailing it. Jet 1 Jet 2

vj0 [km/s] 50 15 -3 15 17 ntot0 [cm ] 7×10 1×10

L0 [cm] 80 50

D0 [cm] 5 5

T0 [eV] 2 0.95

Figure 3.8: Isometric view of the two jets (left) and X-Y plane view of the two jet (right). The colorbar scales are densities in kg/m3.

In order to avoid clutter of the data we present the results in the following manner: first, we show the two all-8-chords figure (Figure 3.9) for experiments and

SPHC data, and second, we present the figure showing synthetic interferometry data from LSP and Nautilus (Figure 3.10).

Figure 3.9: All 8 chords for experimental results (left) and SPHC synthetic interferometry (right)

Figure 3.10: All 8 chords synthetic interferometry results for 1D LSP simulation (left) and 3D Nautilus simulation (right) [69].

As the figures above show, all three codes are in general qualitative and quantitative agreement with the experiments. The slight discrepancies that occur in the results, such as slightly different jet-trail shapes and difference in expansion rates can be attributed uncertainties in the experimental data. Regarding the double peak formation, it is interesting to notice it occurs only in experimental results and SPHC data, but not in the LSP and Nautilus data. The formation shows up at the first, 35 cm, chord when the second jet begins to register on the timeline, and the double peak progressively fades as the second jets passes through the rest of the chords.

To further reinforce claims of SPHC‟s capabilities, in the next section we present a short version of a thorough study on strong shocks Richtmyer-Meshkov instability.

The full study can be found in [72], as well as the appendix of this work.

3.3 Richtmyer-Meshkov Instability (RMI)

3.3.1 RMI introduction

RMI is a fluid instability which occurs when a shock wave refracts across initially perturbed interface between two fluids with different sound impedances. RMI often occurs in HEDP and it tends to dictate, along with RTI, mixing mechanisms and turbulence developing. The most prominent cases of RMI in the field of HEDP can be found in supernovae phenomena and ICF, where a strong shock propagates through the target, refracting across several layers of fluids with different sound impedances

(densities). Occurrence of RMI in the process of ICF target implosion is very detrimental to the goal of achieving fusion conditions since RMI induces mixing and turbulence, providing an efficient mechanism for convecting the heat away from the target hot-spot.

Generally, mitigation of hydrodynamic instabilities is one of the key issues in the field of

ICF. In PJMIF it is expected to see occurrence of both RMI and RTI as well, but due to orders of magnitude lesser compression ratios and the fact that the target density is likely to be lesser than that of the incoming liner (prior to impact), there is a reason to believe that issues of hydrodynamic instabilities will not be so prominent as in ICF. In addition to fusion problems, RMI also plays an important role in the field of scramjet engines, where strong shock interactions improve fuel mixing in the flow, which is of great importance considering the short time fuel has to oxidize in a scramjet engine.

There are several reasons why RMI makes a good test case for this study. First,

RMI is very important in ICF and therefore is expected to be significant in PJMIF since the physics processes are similar. Second, the phenomenon has a complex nature, often pushing the boundaries of powerful Eulerian codes such as RAGE, PROMETHEUS,

Frontier [84] etc. Third, in spite of the complex nature, a tremendous success was achieved in the past two decades on the sides of theoretical and experimental studies of

RMI thus making the problem of linear and non-linear evolution of the RMI applicable for extensive validation and verification studies [85-88].

We studied linear and non-linear evolution of RMI using SPHC, in an extreme parameter regime involving strong shocks (M = 3 - 10), high Atwood numbers (A = 0.6 -

0.95) and amplitude range from a0 = 0.06λ - 0.5λ, where λ stands for the wavelength of the initial perturbation. This parameter regime is of great interest in HEDP and scramjets applications. The paper has been published in published paper on the subject can be found in the appendix of the dissertation, see also [72]. The most important findings and the representative cases for both validation and verification against RMI theory and experiment of the SPHC code are discussed below.

3.3.2 Self-convergence test

According to experiments, simulations and theory, the evolution of RMI in the linear, nonlinear and mixing regimes is described by several power-law dependences and strongly depends on the Mach number, density contrast and initial conditions. There is no a single theory and a single „magic‟ formula for a single exact solution describing

RMI in all flow regimes. There are however several rigorous theoretical approaches that describe the flow evolution in the leading order, linear, weakly nonlinear and highly nonlinear regime. We performed quantitative comparison with these four theories and found excellent agreement with all of them. Our validation study started from a self- convergence resolution test.

In our self-convergence resolution test the reference case was M = 3, A = 0.6 and a0 = 0.06λ. When setting up the SPHC RMI simulations, adaptive meshing technique was used to initially distribute the particles, increasing the particle resolution around the areas of particular interest such as the fluid interface. In the input file, desired resolution is specified, but once the simulation is started, depending on the desired resolution, the code optimizes the number of particles and proceeds with the variable particle distribution. For the purposes of the self-convergence test, desired resolutions were

7.5×104, 2×105, 2.5×105, 3×105 and 5×105 particles, resulting in final resolutions of

61263, 125881, 147758, 168403 and 241767 particles.

As RMI evolution is a multi-parameter problem, there is no single a parameter in our study to which one can refer to (i.e. like pressure or density values for the Noh problem), in order to derive values of the L1 norm. However, as clearly seen from

Figure 3.11, Figure 3.12 and Figure 3.13, there is good convergence of our simulation results over the resolution parameter space and the simulation results are consistent, thus indicating that our numerical method adequately captures the important physical features of RMI, including amplitude growth, growth rate, and bubble velocity.

Figure 3.11 – Amplitude growth for M = 3, A = 0.6 and a0 = 0.06λ. The only somewhat significant departure is the lowest resolution case with 61623 particles.

Figure 3.12 – Amplitude growth rate for M = 3, A = 0.6 and a0 = 0.06λ, measured once the shock refraction phase is over. As obvious, the resolution makes almost no difference to the result.

Figure 3.13 – Bubble velocity for M = 3, A = 0.6 and a0 = 0.06λ.

The resolution appears to almost make no impact on the results, meaning that the desired parameters that have been followed in both convergence tests and in the main portion of the RMI research [72] are quite insensitive to particle resolution.

3.3.3 Results

Two types of results were produced: quantitative and qualitative. Both quantitative and qualitative results were used for verification (against the RMI theory) and validation (against experimental results). As shown in [72], the data obtained by the

SPHC simulations is in good agreement with zero-order, linear, weakly non-linear and highly non-linear RMI theory predictions. Expected discrepancies due to finite amplitude effects and difference in flow regimes are similar to those shown in

experiments of Glendinning et.al. [89]. One separate case with , M = 3 and

A = 0.8 was done for specific purposes of comparing SPHC results to the linear theory of

Wouchuk [90]. SPHC gives the value of non-dimensional growth-rate

, while Wouchuk‟s theory predicts , which is within 4% 83

accuracy and is considered an excellent agreement. It is worth noting that standard CFD methods often experience difficulties to achieve such excellent agreement with the linear theory because of the necessity to balance numerous competing requirements, including shock capturing, interface tracking and accounting for dissipation processes. SPH handles all these issues very well. For more details on quantitative results, please refer to

[72].

As for the qualitative results, basic findings of course involve development of the mushroom-like features, characteristic for RMI, flattening of the bubble front [86] and small scale heterogeneities. SPHC has demonstrated capabilities of capturing fine details of the flow, both at the interface and in the bulk. This includes the Kelvin-Helmholtz instability (KHI) developing up at early stages of the RMI evolution at the fluid interface, Figure 3.14, and occurrence of the so called „reverse‟ cumulative jets in the fluid bulk, Figure 3.15.

Figure 3.14 - Development of Kelvin-Helmholtz instability at the fluid interface for early-time RMI at

Atwood number A  0.8, Mach number M  5, initial perturbation amplitude a0  0.5 .

Figure 3.15 – One of the SPHC results colored for temperature (from 4000K (blue) to 8000K (red)) and showing reversed jets in the base of the spikes between the bubbles. Reverse jets have been confirmed experimentally by Orlicz et.al. [91].

Overall the SPHC results of RMI have shown that SPHC is capable of realistically simulating strong shocks and complex fluid phenomena relevant to the field of HEDP.

3.4 Full implosion resolution study

Even though above cases provide confidence that SPHC is appropriately simulating hydrodynamic phenomena and complex atomic physics, it was necessary to carry out a resolution study of full liner implosions that would include all of the above mentioned phenomena. Run L01 (see Table 4.3) was chosen for the resolution study, as an average case within the explored parameter space. Run L01 is comprised of 30 discrete jets, starting with initial velocity km/s, Mach number 12.7 and initial jet

3 number density of m . Since there is no analytical solution to this sort of complex problems, the test was carried out as a self-convergence study, with the nominal case containing 75000 particles in the domain. The L1 norm was calculated based on the values of peak pressure that is achieved at peak compression, which is also the key

parameter of merit that is explored in this dissertation. The simulations were performed at 15000, 30000, 45000 and 60000 particles. The L1 norm for the given resolution study is given in Figure 3.16.

Figure 3.16: L1 norm vs. resolution plot for the full implosion case (L01).

The four data points in the above figure suggest a convergence rate of N-1.25 and generally show a clear trend of self-convergence.

With this section we end the verification and validation of the SPHC code, with the conclusions that the above presented results indicate that SPHC with its modified eos is capable of simulating plasma jet implosions, reaching HEDP relevant regimes, with satisfactory precision and accuracy.

4. SIMULATION PARAMETER SPACE AND SET UP

In this chapter we shall present the parameter space and rationale for the multitude of preliminary studies: single jet propagation, two-jet merging, influence of jet geometry and density gradients within a jet, liner formation, and PJMIF scaling laws.

The preliminary studies represent an important part of the dissertation for a multitude of reasons, most notably verification against the PLX experiments and understanding of physics in the early stages of liner implosion. Results of all these studies are presented in the results chapter.

4.1 Single jet propagation and two-jet merging

The choice of jet parameters for a single jet propagation study was primarily influenced by a manuscript by Hsu et.al. [83], where the PLX group reports experimental results on the parameters, structure and evolution of the argon plasma jets launched by a plasma rail-gun. The initial jet parameters that were measured at the exit of the rail-gun nozzle are: particle density of ≈ 2×1016 cm-3, electron temperature ≈ 1.4 eV, jet velocity ≈ 30 km/s, Mach number of ≈ 14, ionization fraction of ≈ 0.96, jet diameter ≈ 5 cm and jet length of ≈ 20 cm. These parameters were used in the SPHC simulations for both single jet and two-jet studies, in order to compare the results with the experiment. The created SPHC jets were uniform in density and temperature across the cylinder-shaped jet. The Gaussian density gradient was abandoned due to the relatively high resolutions required in order to models these jets. The study described in section 4.4 of this chapter further justifies the use of uniform density jets (see results chapter) as the stagnation conditions do not appear to be significantly influenced by

gradients in the initial jet properties, so long as the initial length, mass, and velocity are held fixed. It is important to state that no separate single jet simulations were carried out as there was no need for it – the two-jet merging study setup allows for sufficient time to analyze the single jet evolution independently, prior to the merging radius (radius at which jets begin to merge). The experimental setup of the mentioned Hsu et.al. [83] study involved several interferometry setups, spectrometer, CCD camera and a photodiode array. The most important experimental aspect that has been compared to

SPHC results is the 8-chord interferometer stationed about 41 cm away from the rail-gun nozzle exit. Using combined measurements of the mentioned diagnostic system, Hsu et.al. [83] have been capable to derive approximate levels of jet expansion, both axially and radially, which correspond to the SPHC results as will be presented in the appropriate section of the dissertation.

For the two-jet merging simulations, we have the following initial geometry conditions. The jet injection radius is 1.37 m away from the origin and the total angle between the jets is 24°, making the half-angle 12°. We note that in the PLX experiments, the injection radius of the jet is effectively about1.11 m due to the cylindrical nozzle being added to the rail-gun, downstream of the port on the spherical vacuum chamber.

This has been accounted for when analyzing the SPHC data by keeping the distance of the synthetic interferometry chords at the same distance as in they are set up in the experiment (first chord 35 cm away from the nozzle).

4.2 Influence of jet geometry and density gradients within a jet study setup

The scope of this research has been limited to the influence of geometrical shape of the jet and density gradients within the jet on peak pressure. The rationale behind this

is because earlier modeling conducted for planning purposes assumed uniform cylindrical jets. It was decided to investigate effects of variable jet properties on peak pressure to scope out potential benefits or compromise on stagnation pressure and dwell time at stagnation conditions. Observations made during the current investigation must be made with the caveat that further validation is needed before making conclusions.

With this consideration, a total of 18 simulations were ran in smooth particle hydrodynamics software package SPHC [47], under several assumptions:

- Plasma behavior is dominated by hydrodynamic phenomena because of the fact that

ion-electron collision time is on the order of 1 ns while the implosion process takes

place on a s scale.

- No thermal conduction or radiation.

- Perfect gas as the working fluid.

General parameters that were common for all runs are: number of jets – N = 30, number of particles – 30000 (1000 particles per jet), inward velocity – vjet = 53.157 km/s, chamber radius – rC = 137.16 cm, the gas was Xenon for all runs, therefore molecular weight MW = 131.293, and  = 1.2. The chamber radius in this context means the inner most radius of a spherical vacuum chamber surface, from which the jets are launched towards the origin. At the beginning of each simulation, the jets are initialized with the leading face of each jet located at rC.

Two main geometrical forms of the jets that have been explored are cylinder and a hollow cylinder. Coaxial plasma guns are leading candidates for the drivers in a full

scale fusion experiment [36], and these types of guns may produce a hollow jet, because of the coaxial geometry of the electrodes. Therefore, a hollow cylinder was chosen as a plausible jet form. Other aspects of geometrical form that have been explored are proportions of the jet, primarily jet outer diameter (Do) versus jet length (L). Three subgroups were formed, and each had its own Do/L that was kept constant. The outer radius was based on the experimental hardware and the inner radius of hollow cylinders was to be a round number, approximately half of the outer radius, which resulted in ratio

DI/Do = 0.525. General jet geometry with dimensions have been given in Figure 4.1, where for cylindrical cases DI = 0.

Figure 4.1: General jet geometry.

Additionally, we have explored effects of radial and longitudinal density gradients within the initial jets. The following basic cases have been explored: linear radial gradient with peak density along the axis, wave shaped longitudinal gradient with the density peak at the front of the jet and wave shaped longitudinal gradient with the density peak at the back of the jet. The goal of these series of simulations was not

necessarily to try and represent the gradients realistically, but rather to see the influence that density gradients may potentially have on peak pressure. The baseline case for comparison was a simple cylinder with Do11 = 15.24 cm, length L11 = 3.81 cm, density

-5 3 11 = 1.1729×10 kg/m , which, due to the discretization scheme, gave the original mass of the jet m11 = 0.0076 g. Taking into account the number of jets we get that the overall mass of the system is m = 0.2271 g. The jet dimensions were chosen so as to approximately resemble real PLX equipment.

For cylindrical cases, longitudinal density gradients were described by a piecewise density profile. Density values were established in such order that the average value of density is kept constant, therefore the original amount of mass is preserved.

Original density for each longitudinal gradient run is described by:

4.1

where m11 is the initial mass of the jet from the baseline run and Vjet is the volume of the jet of interest. Density fluctuations inside the jets for this group of runs are 0.5 of the original density. A sketch of the longitudinal gradient profiles for the cylindrical cases is presented in Figure 4.2

Figure 4.2: Sketch of the density profile for cylindrical jets (below) and an isometric view of the jet for reference (above). The blue arrows are indicating forward direction.

There were also three cases of a cylindrical jet with radial density gradient and three runs of hollow cylindrical jets with uniform density that have been explored.

Finally, linear functions of radial density gradient as a function of radius, presented as they stand in the input code file, are given in Table 4.1. Small slope coefficients have been used in order to avoid abrupt density changes, which would cause the jet to fade quickly at the outer borders and therefore effectively change its geometrical shape. A sketch of the radial density gradients for hollow cylindrical cases is presented in Figure 4.3.

Figure 4.3: Radial density gradient in hollow cylindrical jets.

The complete overview of dimensions, shapes and density gradient types is given in Table 4.2.

Table 4.1: Final form of the linear functions used for implementation of radial density gradients, where R is a variable in the radial direction from the axis (see Figure 4.3). The only hollow jet runs are 51, 61, 52, 62 and 53, 63.

Run Function

51 -0.00000071 × R + 0.00001235

61 -0.0000007 ×R + 0.00001605

52 -0.000000561 × R + 0.0000135

62 -0.00000045 × R + 0.0000088

53 -0.000000714 × R + 0.00000999

63 -0.000000714 × R + 0.000006536

Table 4.2: Full setup of all 18 simulations. Name of the run: 11 21 31 41 51 61 Hollow Hollow Shape: Cylinder Cylinder Cylinder Cylinder Cylinder Cylinder

DO [cm]: 15.24

DI [cm]: n/a n/a n/a 8 n/a 8 L [cm]: 3.81 bulk in the bulk in Density Gradient uniform front of the back uniform radial radial type: the jet of the jet Name of the run: 12 22 32 42 52 62 Hollow Hollow Shape: Cylinder Cylinder Cylinder Cylinder Cylinder Cylinder

DO [cm]: 10

DI [cm]: n/a n/a n/a 5.25 n/a 5.25 L [cm]: 8.85 bulk in the bulk in Density Gradient uniform front of the back uniform radial radial type: the jet of the jet Name of the run: 13 23 33 43 53 63 Hollow Hollow Shape: Cylinder Cylinder Cylinder Cylinder Cylinder Cylinder

DO [cm]: 7

DI [cm]: n/a n/a n/a 3.675 n/a 3.675 L [cm]: 15.25 bulk in the bulk in Density Gradient uniform front of the back uniform radial radial type: the jet of the jet

4.3 Liner formation and mixing study

This portion of the dissertation is derived from the work done by Cassibry et.al.

[66], showing the evolution of discrete jet liner formation and its tendency evolve towards an equivalent spherically homogenous liner during the course of an implosion.

Here we shall present just the major findings of the study done by Cassibry et.al. [66].

Building upon that work, we carried out a comparison between two discrete jet cases, using same set of initial parameters, with ideal gas eos and tabular eos. These cases were studied in order to attempt to identify important differences in the process of liner

formation due to involvement of atomic processes (accounted by the tabular eos) and heat transport mechanisms such as radiation and electron-thermal conduction.

The study by Cassibry et.al. involved two runs: uniform spherical liner (ideal liner) and 30 discrete jets case. The common attributes of the two runs are total mass of the plasma (m0 = 300 mg), kinetic energy (Ek0 = 376 kJ), temperature (T0 = 1 eV) and initial Mach number (M0 = 25). The uniform liner actually represents a hollow sphere, with the inner diameter being RM = 0.241 m (equivalent to the merging radius of the 30 jet case) and the outer radius of RO = 0.496 m. The density of the uniform liner was

-4 3 calculated to be ρM0 = 6.63×10 kg/m . For the jets, due to their discrete nature, the density needed to be calculated based on the expected experimental radius of the jet,

-3 3 which is rj = 0.05 m, and this results in a jet density of ρj0 = 5×10 kg/m . The insertion radius of the jets is set to be equal to the insertion size of the PLX chamber, which is rC

= 1.37 m. Initial velocities of both liner and the discrete jets were set to v0 = 50 km/s.

The 30 jets distribution is presented in Figure 4.4 and Figure 4.5. Figure 4.4 shows the buckyball (referring to the first discovered Buckminster fullerene, C60, molecule node distribution) with marked equilateral lines. Going from top to bottom, the latitudes are:

20.1°, 43.4°, 59.0°, 80.1°, 99.9°, 121.0°, 136.6° and 159.9°. Figure 4.5 graphically shows the simulation setups for both discreet jets and uniform liner. As can be seen, the jets are distributed in 5-10-10-5 formation, corresponding to latitudes of 20.1°, 59°,

121.0° and 159.9°, respectively.

Figure 4.4: The buckyball form with marked equilateral lines (pink), [66].

(a) (b)

Figure 4.5: Uniform (a) and discrete jet (b) initial setup, [66].

As mentioned in the first paragraph of the section, the same 30-jet setup was used for an additional simulation run using tabular eos, whose results were then compared to the results of the ideal gas eos run to identify the major differences caused by the change of eos and energy transfer mechanisms. The results were then also compared with the two independent 1D liner implosion studies done by Awe et.al. [92] and Davis et.al.

[93]. More information about the comparisons and evolution of the liner implosion are given in the appropriate results section.

4.4 Parameter space

As explained in the introduction chapter of this work, a parameter space represents an abstract plane of operational parameters which, once properly defined, can be used for characterization and comparison of similar technical concepts. For this dissertation, this parameter space is utilized to explore sensitivities to stagnation conditions, especially the peak pressure and the dwell time at these conditions. The salient parameters represent a set of all possible combinations of variable inputs including initial values of jet temperature, jet velocity, jet Mach number, jet density, size

(volume) of the jet, geometrical features of the jet (shape, diameter to length ratio), property gradients within the jet, number of jets and jet distribution on the spherical surface of the initial injection radius. In addition, another derived useful parameter is the kinetic energy. Similarly, a special non-dimensionalizing technique was involved in order to simplify the multi-variable parameter space into something more convenient, and further discussion will be reserved for section 5.

This research builds upon the work of Cassibry et.al. [94], which has explored the scaling laws for PJMIF using an ideal gas equation of state. In the mentioned work by Cassibry et.al. [94] the authors thoroughly cover the reasoning behind the chosen parameter space cases. One of the prime aspects of exploring any parameter space is to study the influence of the limiting cases of important parameters. As predicted by the theory, and shown by simulations and preliminary experiments, the Mach number was found to be one of the key parameters in Cassibry et.al. [94] and thus was varied from

M = 4.3 up to M = 87.9. In connection to the Mach number, initial jet velocity has also been varied appropriately, ranging from 5 to 200 km/s. Number density values (ion density) have been varied almost 4 orders of magnitude, going from 5.43×1021 m-3 to

2.62×1025 m-3, but most runs had densities on the order of 1023 m-3. In the original study,

Cassibry et.al. [94] also explored the influence of the molecular weight of the species used, but due to nature of the code and associated complications of developing multiple tabular equations of state, this study has only concentrated on Argon. The choice of

Argon is logical as the PLX experiment stationed at Los Alamos National Laboratories is using Argon as well, so this would facilitate comparisons. Furthermore, initial jet temperatures in the ideal gas study have varied from 1 eV to 796 eV, in order to provide easier achievement of extreme Mach numbers.

The primary objective of this dissertation is to explore the effects of these parameters on peak pressure, accounting for ionization, radiation, and thermal conduction, and investigating the effects that transport and ionization have on the scaling originally found by Cassibry et al. Including transport and tabular equation of state models poses some difficulties such that not all cases that were performed in Cassibry et al. could be modeled with advanced transport physics. The primary reason for this is due to the very small time steps required when transport physics are included, especially those at high temperatures. Thus, while a number of the initial conditions are identical between Cassibry et al. and the current work, the initial temperature had to be kept relatively low (<5 eV or less) in order to avoid excessively small time steps. The initial temperature of the jets produced in PLX were below this, so the modeling in the current work adequately covers the experimental parameter space.

Further rationale behind the chosen parameters are as follows. Several runs in the original study have been included in order to explore the influence of low and high specific heat ratio, however that is not necessary in this study as the modified SPHC code includes the thermodynamic calculations (see Chapter 2) for dynamic specific heat ratio calculations. Other potentially important parameters that are difficult to explore theoretically are the number of jets and their spatial distribution across the chamber.

Thus, the study involves a number of cases which not only vary the number of jets (from

12 to 60), but also vary their distributions. The most common form of spherical symmetry that can be found in nature is the buckyball, referring to the first discovered

Buckminster fullerene, C60, molecule node distribution. More commonly, the shape is associated with the well-known sewing pattern of the soccer ball (see Figure 4.4). The plasma rail guns that deliver the plasma jet are usually assumed to be stationed at the node points of the buckyball and most of the runs utilize some distribution variation among the given buckyball nodes, regardless whether the number of jets is equal or less than 60. Most of the runs involve a 30 jet distribution as shown in the previous chapter

(Figure 4.4 and Figure 4.5), due to the fact that PLX experiment was budgeted to conduct experiments with up to 30 jets before the project was terminated. Some of the symmetric jet distributions, such as the cases of 12, 18 and 24 jets, have been obtained from [95]. All the cases are summarized in Table 4.3.

Table 4.3: Table showing all of the relevant values of the chosen parameter space.

Jet Jet Number Jet Jet Injection Temp Run name N Velocity Mach Density Diameter Length Radius [eV] [km/s] No. [m-3] [m] [m] [m] Q04a 30 2.5 70 16.3 5.43×1021 0.243 0.243 1.372 Ear12 buck 12 2.5 200 49.7 5.00×1023 0.1524 0.5 1.372 Ear18 buck 18 2.5 200 49.7 5.00×1023 0.1524 0.5 1.372 Ear24 symm 24 2.5 200 49.7 5.00×1023 0.1524 0.5 1.372 Ear24 buck 24 2.5 200 49.7 5.00×1023 0.1524 0.5 1.372 F half 24 2.5 200 50.4 5.00×1023 0.1524 0.25 1.372 F quarter 24 2.5 200 50.4 5.00×1023 0.1524 0.125 1.372 GAr50 24 2.5 50 13.9 2.62×1025 0.05 0.1524 1.372 GAr100 24 2.5 100 26.4 2.00×1024 0.1524 0.5 1.372 GAr150 24 2.5 150 38.5 8.89×1023 0.1524 0.5 1.372 HAr10 24 2.5 200 49.7 5.00×1023 0.1524 0.5 1.372 HAr100 24 2.5 200 49.7 5.00×1023 0.1524 0.5 1.372 HAr1000 24 2.5 200 49.7 5.00×1023 0.1524 0.5 1.372 J01 24 2.5 50 12.7 5.00×1023 0.1524 0.1524 1.372 J02 24 2.5 50 12.7 5.00×1023 0.1524 0.0762 1.372 J03 24 2.5 75 19.0 5.00×1023 0.1524 0.1524 1.372 J04 24 2.5 75 19.0 5.00×1023 0.1524 0.0762 1.372 K02 60 2.5 50 12.7 5.00×1023 0.1524 0.0762 1.372 L01 30 2.5 50 12.7 5.00×1023 0.1524 0.1524 1.372 L02 30 2.5 50 12.7 5.00×1023 0.1524 0.1524 1.372 L03 30 2.5 50 12.7 5.00×1023 0.1524 0.1524 1.372 11Dec03 30 1.0 50 29.6 1.23×1023 0.05 0.05 1.372 11Dec06a 30 0.3 5 4.3 1.23×1023 0.005 0.05 0.137 11Dec07a 30 0.3 10 8.7 1.23×1023 0.005 0.05 0.137 11Dec07 M15 30 0.3 15 13.0 1.23×1023 0.005 0.05 0.137 11Dec08 30 0.2 50 57.2 1.23×1023 0.05 0.05 1.372 11Dec09 30 0.1 50 87.9 1.23×1023 0.05 0.05 1.372 11Dec13 30 1.3 50 21.6 1.23×1023 0.05 0.05 0.300

4.5 Computational system configuration

All of the simulations were carried out on a Dell Alienware platform with the following system components: - Processor: Intel i7 Extreme Edition, 8 cores operating at 2.67 GHz - Working memory: 6 GB DDR2 RAM - Graphics card: nVidia GeForce GTX 480 - Hard drive: 1TB Hitachi, 7200 rpm - OS: Linux Ubuntu 10.04 (Lucid), kernel version 2.6.32-38, Gnome 2.30.2

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5. RESULTS AND DISCUSSION

In this chapter we present the results of several studies that have led to better understanding of the physics behind the plasma liner formation and implosion The sections correspond to the sections in the previous chapter: single jet propagation and two jet merging, influence of geometry and density gradients within a jet, liner formation and mixing, and finally the development of scaling laws.

5.1 Single jet propagation and two jet merging

Since the simulation setup has already been presented we proceed straight to the results. Hsu et.al. in their publication [83] present a table in which they compare jet parameters at the exit of the rail gun nozzle and jet parameters roughly 41 cm down the axis of propagation (jet). We show the contents of that table along with the data extracted from the SPHC simulations, Table 5.1. The data used for comparison with the experiments (Table 5.1) is taken 15 µs after the simulation has started, by which time the jets have travelled around 45 cm. This time was chosen because at that point the jet would have propagated close to the merging radius. The SPHC results for Table 5.1 have been extracted using two means: 2D slices through the x-y plane (plane in which both axes remain throughout the process) and by following the time evolution of minimum, maximum and mean values of each of parameters that are of interest as shown in Figure

5.2 and Figure 5.3. The purpose of the comparison between the experimental and numerical data is intended to demonstrate qualitative agreement between simulation and data. Quantitative agreement is an unrealistic goal due to the shot to shot variability and uncertainties in the radial and axial flow properties and profile.

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Table 5.1: Comparison of experimental results obtained by Hsu et.al. [83], and SPHC results for single jet propagation after approximately 45 cm, (15 µs). The variables are: – number density of particles (electrons for the experiments, ions and neutrals for SPHC), – temperature, – jet velocity, – ionization fraction, – jet length, – jet diameter. Experiment SPHC results Nozzle ( ) Chord ( ) Start ( ) Chord ( ) [ ] 2×1016 2×1015 1.85×1016 3.5×1015 [ ] 1.4 1.4 1.4 1.22 [ ] 30 30 30 30 0.96 0.94 0.96 0.65 [ ] 20 45 20 30 [ ] 5 10 to 20 5 20

Figure 5.1 shows the cross-section plane used for extraction of the jet merging data and analysis.

X [m]

Figure 5.1: Isometric view of the two jets with pronounced x-y plane in which all the slices from Figure 5.2 and Figure 5.4 were made. The marked plane cuts jets in half, passing through both jet axes.

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a) b)

c) d)

Figure 5.2: Time snapshots of 2D slices showing particle number density at a) 0 µs, b) 15 µs, c) 30 µs and d) 45 µs. The colorbar scales are in [cm3] and change from frame to frame to adjust for relatively large expansion, always showing optimal range of values.

One of the main concerns of the liner formation and implosion via discrete plasmas jets is that the jets would undergo severe heating prior to the stagnation of the liner, especially during jet merging, due to expected shock-heating. The shock-heating and general increase in temperature would bring the liner Mach number down, which would have a detrimental effect on the peak pressure of the implosion. However, what we see in SPHC simulations is complete absence of shocks during the two-jet merging. 103

The jets thermally expand as they propagate, but once they begin to merge, steep, yet smooth gradient of density occurs in the interfacial regions of the jets. Looking at Figure

5.2 the density difference between the interfacial jet region and the remainder of the jets vary from 3 to 5 times, since the jets continue to expand in the directions not constrained by the incoming of the other jet, yet continue compress each other from one side.

However, upon a look at the Figure 5.4, we see that at 30 µs, the difference between the mean value of density across both jets and peak values of density is on the order of 10.

The absence of shocks is further confirmed by analyzing the slices of Mach number and temperature fields shown in Figure 5.3.

a) b)

Figure 5.3: Mach number a) and temperature fields [eV] b), at 30 µs. Notice the absence of shocks and the uniformity of the temperature field.

The Mach number shows relative increase toward the origin of the chamber, and respectively a drop in the direction away from the chamber origin. This is due to the thermal expansion of the jet, where the front of the jet accelerates in the direction of

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motion. In the same manner, the back of the jet expands thermally in the direction opposite of the bulk motion. We note that the overall momentum is conserved, and the effect lengthens the jet. The interfacial of the two jets shows no significant signs of changes except for a few minor structures which can be seen in Figure 5.3 (a). Unlike in the case of ideal gas eos [96], where obvious bow shocks occur during the jet merging phase, causing significant increase in temperature, the radiative and conductive energy transfer mechanisms smear the temperature field, causing a fairly uniform speed of sound across the jets, meaning that the Mach number remains influenced by the jet velocity only.

As shown in Table 5.1, one of the discrepancies between the experimental and

SPHC data is the 0.18 eV temperature difference, which also results in a difference in the ionization fraction. The drop in the jet temperature may be attributed, but not limited to, the fact that SPHC does not account for magnetic field effects, SPHC tabular eos inaccuracies, and uncertainties of experimental measurements. Hsu et.al. [96], show that for the propagation distance of less 0.5 m, magnetic fields may have an overall influence on the jet evolution. The decay of the magnetic field potentially preserves the high temperatures during that phase of the jet propagation. The second possible explanation might be the inaccuracy caused by the interpolation scheme, which is indicated by step- like results shown in Figure 5.4. Temperature is one of the variables that has undergone re-interpolation (see Chapter 2) during the table generation phase, and thus inherently carries a certain amount of error. Finally, there is an explanation in the form of experimental uncertainties, where mere 10% - 15% error would be sufficient to explain the discrepancy. Nonetheless, taking into account all the factors and keeping in mind the

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goal of qualitative agreement for these complex phenomena, we find the overall matching between the experimental results and SPHC to be satisfactory.

More results that provides us with useful insights are time histories of relevant parameters, such as Mach number, temperature, density and overall jet velocity. Plots in

Figure 5.4 show time histories of minimum, maximum and mean values for the mentioned parameters.

a) b)

c) d)

Figure 5.4: Mach number a), temperature b), particle number density c) and jet velocity d) histories during the jet propagation and merging phase. The data for comparison with the experiments has been extracted at 15 µs.

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The obvious step-like behavior of temperature is a result of the tabular eos resolution at low temperatures (~1 eV) where the gas is partially ionized and electronic excitation is significant. Thus temperature is very sensitive to small changes in specific internal energy. The step like behavior is not present at higher temperatures and is not observed in the implosion simulations during liner formation, implosion, or stagnation.

This numerical artifact results in an error of up to ~5% at low temperatures, but this is well within experimental uncertainties of temperature measurements. Furthermore, proof that the step-like behavior of temperature (and consequently the Mach number) comes from this process is the time history of internal energy, whose matrix-to-vector transformation, due to nature of the conversion, preserved its accuracy (Figure 5.5).

Figure 5.5: Time history of specific internal energy showing smooth behavior.

What we can see in Figure 5.4 and Figure 5.5 is that jets undergo expansion during their propagation phase, without slowing down. The expansion and cooling of the jets causes a general increase in the Mach number and drop in density and specific internal energies. First signs of jet merging start to occur around 20 µs, primarily indicated by

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the temperature, specific energy and Mach number plots. If we look at the mean particle number density plot, we see that the signs of compression and jet merging are lagging behind temperature drop halting by about 5 – 7 µs. This is likely due to the following mechanism: the jets actually do begin to merge at around 20 µs, as indicated by the internal energy plot. The overall temperature begins to rise in the jet interfacial area, but radiation and electron-thermal conduction quickly smear out the gradients, taking the energy away from the interfacial area. This combination of smearing the temperature gradient and local energy losses due to radiation, allows for more compact storage of the particles, or in other words increase in density. The density increase becomes more obvious after several µs, probably due to low values of speed of sound in which the two jets “communicate”. This mechanism, in which temperatures are kept low due to radiative thermal transport and conduction, allowing for higher densities is called radiative collapse and, as we will see, plays an important role in liner implosions.

5.2 Influence of jet geometry and density gradient within a jet upon peak

pressure

Results for installed density gradients show very little or no influence on the peak pressure, temperature and density under the assumption of ideal hydrodynamic simulations. The same can be concluded for the influence of the initial shape of the jet, since both hollow and regular cylinder-shaped jets achieve the same level of peak values

(Figure 5.6). The dominant geometrical influence appears to be the D0/L ratio. As noted in chapter 4, there have been three subgroups of which each had different Do/L ratio. All of the peak parameters show strong dependency on this ratio, such that peak values significantly decrease as Do/L decreases (Figure 5.7) at fixed jet mass. Average values of

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peak pressure (pav) within each subgroup, along with the statistical spread within that subgroup are presented in Table 5.2, while detailed data are presented in Table 5.3.

Table 5.2: Average values of peak pressure with oscillations for each of the three subgroup runs.

Series number pav [kbar] 1 154.35 23.85 2 77.83 13.60 3 39.19 25.39 Table 5.3: Results of all 18 simulations. Name of the run: 11 21 31 41 51 61 Hollow Hollow Shape: Cylinder Cylinder Cylinder Cylinder Cylinder Cylinder

DO [cm]: 15.24

DI [cm]: n/a n/a n/a 8 n/a 8 L [cm]: 3.81

Pmax [kbar]: 139.2 178.2 171.1 162.9 111.8 162.9

Tmax [eV]: 9297 7729 9516 6296 9334 8243

3 max [kg/m ]: 39.26 32.84 32.06 23.95 19.56 39.91 Name of the run: 12 22 32 42 52 62 Hollow Hollow Shape: Cylinder Cylinder Cylinder Cylinder Cylinder Cylinder

DO [cm]: 10

DI [cm]: n/a n/a n/a 5.25 n/a 5.25 L [cm]: 8.85

Pmax [kbar]: 72.0 91.4 73.7 76.9 86.9 66

Tmax [eV]: 8207 7991 6639 6892 5523 6402

3 max [kg/m ]: 17.50 20.58 16.20 19.5 19.1 16.1 Name of the run: 13 23 33 43 53 63 Hollow Hollow Shape: Cylinder Cylinder Cylinder Cylinder Cylinder Cylinder

DO [cm]: 7

DI [cm]: n/a n/a n/a 3.675 n/a 3.675 L [cm]: 15.25

Pmax [kbar]: 38.2 43.5 64.6 46.1 24.3 18.4

Tmax [eV]: 5131 4907 5530 7246 5112 5397

3 max [kg/m ]: 12.6 12.8 14.7 10.8 7.9 5.9

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Figure 5.6: Peak pressure values for series 1.

Figure 5.7: Comparison of peak pressures for different D0/L ratios (11, 12 and 13)

The results strongly indicate that density gradients within the jet and jet geometry

(except length) do not have a major impact on peak values. These observations are important because in practice it would be impossible to form jets with perfectly uniform density and precisely defined geometry. We found that the jet length had a significant impact on peak pressure, which was inversely proportional to L. A tradeoff will have to be made when sizing the jets, because while high peak pressures are desirable, longer jet lengths increase the dwell time at peak conditions. Further analysis will have to be

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conducted with thermal and radiation transport, and ionization, which will be left to future work.

5.3 Liner formation and mixing

This section will present results which are focusing on understanding of the physical processes and evolution during plasma liner implosions. In order to fully grasp the description of the processes we shall present newly obtained results and compare them results of Cassibry et.al. [66], Awe et.al. [92] and Davis et.al. [93]. All simulations have been based on a single run that was originally created by Awe et.al. [92] and is referred to as run number 6 from his Table II. Raven [92] and Helios [93] codes performed the simulations in 1D with spherical symmetry, while all SPHC simulations were 3D. First, we will present the results of the purely hydrodynamic simulations (ideal gas eos and no energy transport models involved) and describe the process of liner implosions and phenomenon of hydrodynamic smearing. Second, we will present the results of simulations involving radiation and electron thermal conduction. Third, we shall explain the discrepancies between the cases and proceed to elaborate on the physical phenomena behind the liner evolution based on a more detailed qualitative analysis of SPHC results. Finally, we shall make a brief mention of some of the findings from the Cassibry et.al. [66] paper regarding the mixing within the liner and Rayleigh-

Taylor stability of the liner during implosion. The compact set of parameters for the performed simulations is given in Table 5.4.

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Table 5.4: Compact set of initial parameters used for liner formation and implosion study. The simulations are based on the Awe et.al. [92] run number 6 in his Table II: all jets have the same initial temperature of 1 eV, initial jet velocity of 50 km/s, with equivalent total kinetic energies. It is important to stress out that all Helios and Raven simulations were performed in 1D with spherical symmetry, while all SPHC cases were done in 3D.

Electron- Injection Kinetic Equation Number Code Radiation thermal Liner type Radius energy of state Density [m-3] conduction [m] [kJ] SPHC OFF OFF ideal gas ideal liner 1.00E+22 0.241 375 SPHC OFF OFF ideal gas discrete jets (30) 7.45E+22 1.37 374 Raven OFF OFF ideal gas ideal liner 1.00E+22 0.241 375 Helios OFF OFF ideal gas ideal liner 1.00E+22 0.241 375 SPHC ON ON tabular discrete jets (30) 7.45E+22 1.37 374 Raven ON ON ideal gas ideal liner 1.00E+22 0.241 375 Helios ON ON tabular ideal liner 1.00E+22 0.241 375

5.3.1 Purely hydrodynamic simulation results and hydrodynamic smearing

Summary of the ideal gas results was presented in Cassibry et.al. study [66] and we show those results in Figure 5.8. As can be seen, the results of all 3 codes are in very good agreement.

Figure 5.8: Comparison of ideal gas eos results (no energy transport) for all three codes (Awe et.al. [92], run 6, Table II), including the SPHC discrete jets run. The black color lines refer to absolute pressure values, while the blue lines show the radius of the liner, [66].

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The ideal liner SPHC case has almost identical pressure as the 30-jet SPHC discrete case. The reason for this has been thoroughly described in the mentioned

Cassibry et.al. [66] paper and the phenomenon is referred to as hydrodynamic smearing.

The mechanism of hydrodynamic smearing can be attributed to the relatively high speed of sound that occurs within the liner once it starts to implode and jets begin to merge.

Because of the ideal gas eos and lack of energy transport and losses during the implosion process, the liner temperature significantly increases, the speed of sound increases, the

Mach number drops and the jets begin to homogenize due to rapid interactions between the particles. The qualitative description of hydrodynamic smearing can be seen in

Figure 5.9 through Figure 5.12.

a) b)

Figure 5.9: Discrete jet case a) and ideal liner case b). Slices have been taken in an x-y plane, with z = 0 m for the ideal liner and z = 0.2 m for the discrete jet case. Time is , with the jets in the discrete jet case being positioned at their merging radius, [66].

In Figure 5.9 we see both discrete (a) and ideal (b) liners at of the reference time. The time scale in the discrete jet case has been shifted with respect to the 113

original simulation time, such that now represents the time when the jets reach the merging radius. This has been done so that the two cases can be compared in an appropriate manner. We see that by the time the jets reach the merging radius, they have already undergone some merging with the closest neighboring jets and we see well pronounced oblique shock areas in which the pressure jump is obvious. On the other hand, we see a uniform pressure slice of the ideal liner.

a) b)

Figure 5.10: Discrete jet case a) and ideal liner case b). Both slices are now taken in the x-y plane at z = 0 m, with , [66].

In Figure 5.10, at we see both discrete (a) and ideal (b) liners moments before the cavity collapse. The discrete jets have gone through a good portion of the merging process and we see less pronounced areas of high pressures between the jets which is the first sign of the hydrodynamic smearing. The ideal liner goes through a process of radially uniform compression and we see the pressure drop in the center due to the still present cavity at the origin.

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a) b)

Figure 5.11: Discrete jet case a) and ideal liner case b). Peak pressure time at . Very similar structures in both cases with very close peak pressure values, [66].

Figure 5.11 shows peak pressure time of for both liner types. The final structures of the pressure slice at peak compression look remarkably similar, both qualitatively and quantitatively. At this point we are drawn to conclude that the effect of hydrodynamic smearing in case of ideal gas eos without energy transport models involved is definitively present.

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a) b)

Figure 5.12: Discrete jet case a) and ideal liner case b). Post-stagnation time at . The similarity trends between the two cases continue, showing a fairly uniform process of rarefaction after the peak pressure, [66].

Finally, Figure 5.12 shows a post-peak pressure time where the two cases continue to show similarities even in the rarefaction phase of liner disassembly.

5.3.2 Energy transport models simulation results

There are several important differences between SPHC, Raven and Helios to consider when comparing the set of simulations with energy transport turned on. First of all, Helios and Raven are 1D codes, while SPHC simulations were carried out in 3D.

Thus, Helios and Raven both assumed spherically symmetric liners, while SPHC performed a 30 discrete jets case which is a highly 3D problem. Finally, Raven performed a simulation turning the energy transport on, but still kept the ideal gas eos.

SPHC and Helios on the other hand had the energy transport on, but both of them employed a tabular eos created by the Propaceos software. Detailed description of Helios and Raven runs can be found in [92] and [93], respectively.

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The peak pressures achieved in these simulations by the three codes are given in

Table 5.5.

Table 5.5: Results of the simulations (Awe et.al. [92], run 6, Table II) with energy transport models on.

Electron- Equation Code Radiation thermal Liner type P [Pa] of state max conduction SPHC ON ON tabular discrete jets (30) 4.12×109 Raven ON ON ideal gas ideal liner 1.32×1011 Helios ON ON tabular ideal liner 4.63×109

What can be seen when we compare the results shown in Table 5.5 and the purely hydrodynamic simulations in Figure 5.8 is that the codes predict significantly higher pressures when the energy transport models are turned on. We also see that the results of

SPHC and Helios simulations are in excellent agreement, while the peak pressure for the

Raven run seems to be off by a few orders of magnitude. In defense of the Raven code, the full result from Awe et.al. [92] study is shown in Figure 5.13.

Figure 5.13: Set of results from Awe et.al. study showing exploration of influence of different energy transport models on the peak pressure. Major difference in peak pressure occurs when both radiation and electron-thermal conduction are turned on, [92].

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Results from Figure 5.13 show that the combination of radiation and electron- thermal conduction can significantly influence the peak pressure. Furthermore, Awe et.al. [92] do make a distinction between the peak pressure and what they refer to as

“stagnation pressure”. The stagnation pressure term refers to pressure values that are present immediately after the peak pressure values and are perceived as a relevant factor on which they estimate the confinement time of the plasma. The stagnation pressure value in the case when both radiation and electron-thermal conduction are turned on is

Pa, which is considerably closer to the SPHC and Helios results.

The differences in predicted peak pressure were described by Awe et al.. Awe et al. [92] make an important note following the Figure 5.13 in their work that is worth quoting:

“When energy transport is suppressed, the expanded leading edge of the liner, which is composed of a very small fraction of the total liner mass, can have a large, unphysical impact on the achieved stagnation pressure by creating a very low density yet high pressure “hot plasma bubble” of extreme temperature (e.g., hundreds of keV). The trailing mass of the plasma liner stagnates upon the central bubble rather than at the origin, reducing the achieved spherical convergence and, thus, also the stagnation pressure”. This statement is of utmost importance as shall be presented in the next section.

5.3.3 Influence on radiative collapse on peak pressures

Based on the presented results from 3 different codes, switching the energy transport models on causes an order of magnitude increase in peak pressure for the given case. However, what has not been mentioned so far is that the temperature levels at peak compression are about two orders of magnitude lower in the cases with energy transport 118

enabled. This implies that the whole increase in pressure comes from a three orders of magnitude increase in peak density. The phenomenon that is allowing for this to happen has been described in section 5.1, where we show that the early stages of liner formation are already influenced by the radiative collapse effects. We proceed with our liner formation analysis by a comparative study of the three SPHC cases: ideal liner case (no energy transport), discrete jet case (no energy transport) and discrete jet case (with energy transport). For sake of brevity we shall name the three runs “Run 1”, “Run 2” and

“Run 3”, respectively.

In order to understand the conditions as the liner approaches peak compression time, we first created several slices at the time prior to peak compression for all three cases showing the two most relevant parameters: Mach number and density. Mach number is an excellent parameter for analysis as it includes the information such as temperature, speed of sound and velocity, while the particle number density is a strong representative of pressure (due to the fact that temperatures remain low and rather uniform due to radiative losses and thermal conduction). We then continue to analyze the evolution of the liner through parameter history plots, equivalent to those in Figure

5.4. The slices are shown in Figure 5.14 and Figure 5.15.

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a) b)

Figure 5.14: Contour plots of Mach number microseconds prior to peak compression for Run 1 a) at 4.5 µs, Run 2 b) at 27.2 µs and Run 3 c) at 27.2 µs. The colorbars to the right of each figure indicate the Mach number.

What can be seen in Figure 5.14 is that the areas of low Mach number for Runs 1 and 2 are significantly larger than in case for Run 3. The area of very low Mach number

( ) in Run 3 is on the order of 1 cm in diameter, while in Runs 1 and 2, the low

Mach number areas are on the order of 20 cm in diameter. The low Mach number implies high temperatures (speed of sound) and low velocities of the plasma, and we

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clearly see the effects that Awe et.al. have postulated in their study, [92]. Lack of energy transport models creates areas of high temperature around the cavity prior to peak compression resulting in a decrease in maximum density that can be achieved and ultimately lower peak pressures, as will be discussed in section 5.4.

a) b)

1020

1016

Figure 5.15: Contour plots of particle number density [cm-3] at void collapse, prior to peak compression for Run 1 a) at 4.5 µs, Run 2 b) at 27.2 µs and Run 3 c) at 27.2 µs. The contour and colorbar scale for Run 3 is in logarithmic form so as to allow for better presentation of the density field which has extreme variations. For Run 3, the peak value at the origin and the yellow areas surrounding it differ by more than two orders of magnitude.

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When energy transport models are turned on, energy loss due to radiation prior to peak compression is on the order of 10% and by the time simulation is over, plasma radiates away more than 90% of its energy. Furthermore, the effects of radiation conduction and electron thermal conduction are extremely fast and provide efficient mechanisms for smearing the temperature gradients across the plasma. Combination of these mechanisms keeps the overall temperature of the liner very low in comparison to the no-energy-transport-models simulations, which in turn allows for a more efficient compression process in an analogy to compression with intercooling in classic thermodynamic cycle analysis. This is, once again, the radiative collapse phenomenon which seems to be a crucial effect in the evolution of the plasma liner implosion. The consequences of radiative collapse are easily seen when one analyzes the colorbar scales in Figure 5.15. The colorbar scales for Runs 1 and 2 are linear, showing areas of high density ( ) that have a diameter on the order of approximately 10 cm, and relatively mild drop-off rate as we move outward from the origin. The slice for Run

3 has been plotted in logarithmic color scale, due to the fact that the difference between peak compression areas at the origin ( , diameter on the order of 1 cm) and surrounding yellow areas differ by approximately 2 orders of magnitude. For the light blue areas the difference is close to 5 orders of magnitude. Overall, it can be concluded that the jets (or bulk of the mass) penetrate much deeper into the liner in the case when energy transport is on, due to radiative collapse, which in turn results in a significant increase in peak pressure and density, while the temperatures at the origin remains low.

When examining the history plots of the minimum, maximum and mean particle values of velocity, pressure and temperature within the domain, we see several

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interesting details that are consistent with the narrative above. Let us first look at the history of the minimum and mean velocity within the domain (Figure 5.16).

a) b)

Figure 5.16: History plots of minimum and mean values of particle velocity for Run 1 a), Run 2 b) and Run 3 c). Notice that Run 3 achieves the lowest mean velocity and seems to remain at that level.

Figure 5.16 shows histories of mean and minimum velocity values within the domain of the simulation for Runs 1, 2 and 3. We see that for Run 1 the liner collapse is very straightforward as the minimum velocity value plummets down to zero. The mean

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velocity also shows a steady decrease, until about 8 µs, after which we see an equally sharp increase in mean velocity after stagnation, with the liner begins to expand thermally. The minimum mean velocity can be regarded as a measure of the uniformity of the liner at stagnation of the outer liner. For Run 2 we see similar trends, except that we note two things. First, the minimum mean velocity value is significantly higher than in either Run 1 or Run 3 cases. Second, we observe some oscillations in the minimum velocity values during the time of the collapse. The former indicates that upon stagnation of the outer edge of the liner, the inner part of the liner has a higher proportion of the material in motion, compared to Runs 1 and 3. This is indicates an increased amount of mixing, which would be expected by the gradients introduced by the discrete jets. The oscillations, referred to in the 2nd observation, are an indication of the multiple plasma jet merging processes that occur during the course of liner formation, since not all jets will collide at the same point in space. For Run 3 we see a relatively sharp drop in minimum velocity values, indicating less influence of hydrodynamic smearing effect, but what is more interesting is that we clearly see a longer and larger drop in the mean velocity than either Run 1 or Run 2. We see that the minimum value of the mean velocity in Run 3 occurs almost 5 µs after the minimum value of the mean velocity in

Run 2, which indicates that the collapse of the liner lasts longer and that the jets penetrate deeper towards the origin compared to the case where we have the energy transport turned on. The „bounce back‟ velocity in Run 3 is also markedly lower, and this is a consequence of the energy lost to radiation.

Figure 5.17 shows pressure history, and it is important to notice two things, aside from the obvious pressure increase in Run 3. First of all, the minimum pressures in the

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domain in Runs 1 and 2 begin to drop as soon as the mean pressure reaches its peak. For these two cases, the point at which mean pressure reaches its maximum represents the point where the liner has reached full stagnation and after that, the expansion process takes place. However, in case of Run 3, we see that the minimum pressure value maintains its level after the mean pressure reaches its peak. This occurs because the outer liner edge continues to implode after the stagnation shock reaches the outer edge of the liner in Run 3.

Second, note that if we compare Figure 5.16 and Figure 5.17, we see that for

Runs 1 and 2, the peak pressures coincide with the minimum velocity value, at the time of the initial void collapse, while for Run 3 the pressures are offset by a few microseconds after the void collapse. This is again a consequence of the radiative collapse. When the liner comes in onto to the void, what happens without the energy transport models is that we create a strong shock at the void which tremendously increases both density and temperature. Without the efficient energy transport mechanisms, relying solely on convection and compressional work, we see a distinct peak in pressure and temperature values (Figure 5.17, Figure 5.18), that coincides with the initial liner collapse. On the other hand, when we do include the energy transport models, the initial shock and the consequent jump in temperature get smeared by the rapid radiation losses and electron-thermal conduction. This allows for more efficient compression and the liner continues to come in at the origin, slowly but surely increasing the density. This continues for a few microseconds as the liner continuously implodes, while the radiation and conduction are continuously dissipating the energy at the origin.

The final result is several orders of magnitude increase in particle density, while the

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temperatures remain on the order of a few eV. Finally, this trade-off between the density and temperature results in overall increase in peak pressures that can be achieved.

a) b)

Figure 5.17: History plots of minimum, maximum and mean values of particle pressures for Run 1 a), Run 2 b) and Run 3 c). Aside from obvious increase in pressure for Run 3, more important for understanding of the process is the behavior of the minimum pressure value, showing quick expansion after peak mean pressure in Runs 1 and 2 and prolonged compression period in Run 3.

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a) b)

Figure 5.18: History plots of minimum, maximum and mean values of particle temperatures for Run 1 a), Run 2 b) and Run 3 c). Note the large differences in temperature between runs, especially between the Run 3 and the first two.

5.3.4 Remarks on mixing and Rayleigh-Taylor instability of the liner

Mixing within the liner during the implosion process and analysis of Rayleigh-

Taylor instability (RTI) of the liner are not within the primary scope of this work, but deserve to be mentioned as the studies are relevant to the PLX experiment. The study was carried out on a single 30-discrete-jet case without energy transport models, 127

equivalent to the second run in Table 5.4. The details of the mixing and RTI study can be found in [66], but we will here present just the most important conclusions.

The mixing intensity was measured in a specific manner that we shall briefly try to explain now. SPHC constructs cylindrical jet shapes in a certain pattern. It begins by aligning particles along the chord which spreads parallel to the axis of the jet, starting from the most inner chord (the one closest to the axis itself) and then moves outward.

The post-processing code was then programmed to identify the particles along these chords and follow their relative positions. More specifically, radial particle coordinates along four chords and 6 jets were followed in order to make sure that all possible mixing scenarios are taken into account (see Figure 5.19).

a) b)

Figure 5.19: Front a) and isometric b) views of the jet showing four chords whose particles were monitored to evaluate the mixing, [66].

Each time any particle, i.e. particle n with initially larger radial coordinate, overtakes another particle m with initially smaller radial coordinate, we record a “mixing

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event”. The idea is to estimate mixing intensity during liner implosion by counting the mixing events. Ultimately, it was shown that the maximum mixing occurs about the time of liner stagnation, as shown in Figure 5.20.

Figure 5.20: Mixing frequency as a function of dimensionless time. The vertical lines in the plot show the characteristic times of void collapse, maximum pressure and liner stagnation. Maximum mixing frequency occurs approximately 4 µs after the peak pressure, [66].

As for the RTI, it is known that along with RMI, it represents one of the main issues in the fields of ICF. Thus, due to the expected compression rates for the PJMIF

(PLX) it was reasonable to try and evaluate the overall stability of the liner concerning

RTI. The primary criterion for the development of RTI, as shown by Atzeni and Meyer- ter-vehn [15], is

(5.1)

The results (Figure 5.21) show that during the plasma liner implosion, very few areas satisfy the criterion given in Eq. (5.1) and they are very short lived. This is somewhat reasonable to expect for a pure liner implosion without the target in the middle as we

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expect to see consistent directions of both pressure and density gradients during most of the implosion.

Figure 5.21: RTI slices at (a) , (b) , (c) and (d) . The planes of the slices correspond to those of Figure 5.9 through Figure 5.12, [66].

5.4 Development of scaling laws for plasma jet liners

Following our code verification/validation procedure and investigations of plasma liner formation and implosion processes, we proceed to the final set of results – development of scaling laws for plasma jet liners. As mentioned before, this set of results is a natural extension of work currently being carried out by Cassibry et.al. [94], which is dealing with the development of scaling laws for plasma jet liners under ideal hydrodynamic assumptions (calorically perfect ideal gas, no thermal or radiative transport). It is expected that the scaling laws developed on the basis of upgraded SPHC

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code with tabular eos and thermal transport would yield more realistic physics insights, while comparisons with the ideal hydrodynamic simulations would give insights into the effects that ionization, thermal conduction, and radiation have on the results. We will thus first describe the mechanism of obtaining the scaling laws, then show the scaling laws results from the ongoing work of Cassibry et.al. [94] and then finally proceed to the newly obtained results from the simulations created by the upgraded SPHC code.

Dimensional analysis is a very common tool in fluid dynamics and is routinely discussed in a number of textbooks such as that of White [97], [98]. It is useful for characterizing complex flows and describing behavior of otherwise complicated phenomena that usually involve a large number of seemingly unrelated variables. By employing dimensional analysis and related non-dimensionalizing procedures, we find solutions that tend to collapse to one or several groups that express certain types of behavior based on the values of derived dimensionless parameters. As discussed by

Cassibry et.al. [94], the field of nuclear fusion does not have a single standardized approach to dimensional analysis even though there are several studies that discuss different approaches to the matter, but only concerning ICF [99-102]. In both Cassibry et.al. [94] and this work we employ a relatively simple method for dimensional analysis that was developed by Ipsen [103]. The method is straightforward, starting by analyzing the physics of the problem, identifying which parameter we want to evaluate (develop scaling laws for), put the rest of the parameters on the other side of the equation and start taking out the dimensions by manipulating the parameters. The process will become more apparent given the example of our particular problem. We start by realizing that peak pressure is a function of number of parameters:

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(5.2)

where the variables are: – jet radius, – jet length, – chamber radius, – number of jets, – initial jet density, – initial jet velocity and – initial speed of sound of the jet, which incorporates the initial temperature and the species type as well. The original work done by Cassibry et.al. [94] includes specific heat ratio ( as an explicit variable, but when using tabular eos, is not constant but rather dynamically calculated throughout the simulation. The logical next step is to attempt and non-dimensionalize as many parameters as we can, which in our case yields

(5.3)

Through a few trial and error iterations, it was shown by Cassibry et.al. [94] that for purely hydrodynamic cases most of the dimensionless peak pressures collapse (within a factor of 3) to

(5.4)

which is an interesting and elegant result, showing very high influence of the initial

Mach number of the jet on the dimensionless pressure. Graphically, the set of results is presented in Figure 5.22.

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Figure 5.22: Plot of the dimensionless pressure versus initial Mach number of the jet, based on Eq. ((5.4). The solid line represents the curve and the dashed lines are 3 times and 1/3 of the value. The dimensionless pressure data points fit in perfectly between the dashed lines.

As can be seen in Figure 5.22, the developed scaling laws for purely hydrodynamic simulations (for full setup list of the runs please see [94]) almost entirely fit the range between and . The runs that fall out of this range are mostly the ones that had introduced a rail-gun misfire jitter, causing the pressure values to drop.

However, once we superimpose the newly obtained data that uses tabular eos and energy transport, we get Figure 5.23.

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Figure 5.23: Same as Figure 5.22, but with tabular eos and energy transport data points (blue triangles).

What we see in the above figure is that the new data points from the tabular eos and energy transport runs do not agree with the purely hydrodynamic scaling law, which is expected. However, what is not expected is that the new data shows essentially no dependence of the initial Mach number of the jet. Even the low Mach number develop rather high values of dimensionless pressure. Conveniently, Awe et.al. [92],

Davis et.al. [93] and Kim et.al. [104] have also performed independent studies of the scaling laws, but were limited to using 1D simulations with ideal liners only, thus completely neglecting the influence of the geometry, which, as will be seen, is of utter importance. With that being said, we included the results of aforementioned studies and obtained Figure 5.24.

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Figure 5.24: Same as Figure 5.23 with superimposed results from 1D studies carried out by Awe et.al. [92], Davis et.al. [93] and Kim et.al. [104], all of which have included energy transport models, but Awe et.al. used only ideal gas eos.

In above figure we see that SPHC‟s 3D results with tabular eos and energy transport are within the same range as the rest of the 1D studies that included energy transport models, and generally show higher dimensionless peak pressures than the pure hydrodynamic cases. This agreement with the 1D codes (especially Davis et.al. work

[93]) provides us with further confidence that the modified SPHC code is performing well. However, we can see very clear trends in Awe‟s et.al. and Davis‟ et.al. data that are similar to the earlier developed hydrodynamic scaling law of . This is due to the fact that the only thing that was varied in those studies (same set of runs) was the kinetic energy of the liner, and it was varied by the only two ways it could be varied: by

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changing initial velocity of the liner and the initial density of the liner (as the volume was kept constant). The absolute pressure for the purely hydrodynamic cases is a power law function of initial kinetic energy that can be achieved (Figure 5.25) and once the all the geometry parameters are locked, it is expected to see a very clear dependence of pressure on the initial Mach number (jet velocity, kinetic energy).

Figure 5.25: Linear law dependence of absolute pressures as a function of initial kinetic energies for both pure hydrodynamic (black circles) and tabular eos and energy transport (blue triangle) cases. Data points that are off are off due to intentional setup of rail-gun misfire.

Interestingly enough, in case of tabular eos and energy transport cases the linear law relationship between absolute peak pressure and kinetic energy seems to be preserved. This is fortuitous, as it allows easy estimates of absolute peak pressure based solely on kinetic energy. There is another interesting and important result presented in

Figure 5.25 – notice that the values of absolute peak pressure for the pure hydrodynamic

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simulations (black circles) are not always lower than their energy transport counterparts.

In some cases, the purely hydrodynamic cases have higher peak pressures. The effects that might cause this are likely associated to geometry and symmetry of each individual case and are further explained in the discussion below.

Returning back to Figure 5.24, we must realize that for an all-inclusive scaling law for tabular eos and energy transport cases, the original hydrodynamic scaling law will not suffice as it clearly does not match the new data. In the previous sections we have presented results on jet merging and liner formation and evolution and have shown clear discrepancies in both qualitative and quantitative results between the pure hydrodynamic and tabular eos with energy transport cases. It is clear that we need to account for the drastic effects such as (absence of) hydrodynamic smearing and

(introduction of) radiative collapse for the cases that involve tabular eos and energy transport. It has therefore been decided to discard the initial scaling law and attempt to create an entirely new scaling law which will attempt to meaningfully organize the vast parameter space that has been covered by the simulations. The process involved a combination of logical deduction and trial-and-error methods to come up with a meaningful function that would produce clear trends. Final result is shown in Figure

5.26.

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K01 EAr24 symm

GAr50 11Dec03

Figure 5.26: Dimensionless peak pressure as a function of the newly developed function . The marked data points represent special cases whose departure from the common behavior can be explained through distinct features of the particular runs. Solid black line represents the scaling law given by Eq. (5.7) and the upper and lower dashed lines represent 3 times and 1/3 of the scaling law value, respectively.

The new scaling law relationship for the dimensionless peak pressure is

(5.5)

where is given as

(5.6) √

The final form of the scaling law, based on the covered set of 28 simulations is:

(5.7)

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Let us first comment some of the trends that can be derived from the scaling law, then explain the mechanisms that cause such large differences in scaling laws and finally explain the 5 major data point departures from the new scaling law.

In order to achieve high dimensionless peak pressures, it is preferred to have low

ratio, which is consistent with the geometry study that is presented in section 5.2 of this chapter. The second thing that can be seen is that larger ratios are also preferable. We also observe that the peak pressure is proportional to the number of jets,

, which is expected. However, what was not expected is relatively weak influence of initial Mach number. It perhaps could have been somewhat indicated by the large scatter of the new data when plotted against the scaling law, but this result not only differs, but is in sharp contrast to the result obtained for the purely hydrodynamic cases as it shows that high Mach number may actually be detrimental to the peak dimensionless pressure.

This counterintuitive behavior of the dimensionless peak pressure decreasing at very high Mach numbers is due to the fact that kinetic energy and Mach number are coupled through the parameter of initial jet velocity. To show this, let us present the following thought experiment, referencing our dimensionless scaling described above: for a fixed dimensionless pressure Eq. (5.5), the only way to increase the Mach number would be to lower the initial temperature of the jet, as the velocity of the jet needs to remain fixed in order to keep the dimensionless pressure constant, Eq. (5.5). If we lower the temperature, for a given density, we will reduce the amount of internal energy and thus the overall energy of the system, finally resulting in a reduced absolute peak pressure. The tradeoff becomes even more apparent if we realize that the kinetic energy

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of the liner scales with , while the Mach number itself is linearly dependent on jet velocity. If we double the Mach number, keeping the temperature constant, we will quadruple the kinetic energy. At some point, this power increase of kinetic energy will outweigh the increase in Mach number and the dimensionless peak pressure will decrease.

The result begs the question: why is this trend not observed for the hydrodynamic scaling law? The answer as will be shown is straightforward, and it provides insights into the relative complexity of scaling laws for the non-ideal, but more realistic, simulations. Let us start by examining the form of the purely hydrodynamic scaling law, transforming Eq. (5.4) into

(5.8)

where is the geometry factor, given by

(5.9)

Recall that hydrodynamic smearing for the pure hydrodynamic (ideal gas eos, no energy transport) cases is a very strong effect that ultimately provides very high levels of symmetry during the liner implosion. In other words, the effects of jet geometry and jet distribution are not well pronounced and the peak pressure that can be achieved is dominated by the Mach number. Furthermore, the reason why dimensionless peak pressure linearly increases with is because the kinetic energy that is used to scale the

absolute pressure is a function of . Thus, in order for the peak dimensionless pressure to scale monotonically upward, we need to square the Mach number, so we would have

functions of on both sides of the equation. Ultimately, the scaling law for purely

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hydrodynamic cases (Equation (5.4)) is a “repackaged” conclusion that absolute pressure scales as a power law with kinetic energy (Figure 5.25). Results of Awe et.al. [92] and

Davis et.al. [93] are in agreement with this hypothesis since they both lock the geometry to the ideal liner form, keeping the geometry factor constant, and we immediately see that the only thing that dictates the scaling is the kinetic energy (initial jet velocity or

Mach number).

We conclude that for cases involving discrete jets with ionization, thermal conduction, and radiative transport – geometry becomes a dominant factor. Radiative collapse keeps the temperatures of the liner very low during the implosion, which in turn lowers the speed of sound down, and the pressure gradients (as they form the liner and continue on imploding) consequently smear at lower rates. At the point of liner stagnation, larger asymmetries will then be present during the implosion, making the peak conditions at stagnation a much more sensitive function of the initial jet topology.

These effects can be seen in Figure 5.27.

a) b)

Figure 5.27: Selected case (J01) showing pressure slices at peak compression for the purely hydrodynamic case a) and the case including tabular eos and energy transport b). The colorbar is colored for pressure in kbar. Note the areas of high pressure that are better pronounced in the case of tabular eos and energy transport case.

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Since the Mach number is no longer a dominant factor, a new variable that describes the importance of geometry in a proper manner had to be derived. The one that we have found satisfactory is given in Eq. (5.6). However, in Figure 5.26 we do see 4 data points that depart significantly from the curve, those are the runs named:

EAr24_symm, GAr50, 11Dec03 and K02.

The EAr24_symm is a specific case because it employs an ideal spherical jet symmetry (given 24 jets). The jet distribution was not tied to the nodes of the bucky-ball, but simply set up in an optimal manner, spreading the jets equally across the chamber.

The result was an order of magnitude higher peak pressure (both absolute and dimensionless) than its closest related run EAr24_buck, whose jet distribution was restrained by the bucky-ball nodes. This is further supporting the hypothesis that symmetry of the implosion plays a significant role in the stagnation conditions. This is very consistent with results from the inertially confined fusion community, a close cousin of PJMIF [105, 106].

Runs GAr50 and 11Dec03 have one thing in common – the jets are extremely small in comparison to the chamber. The geometry ratios of the runs are given in Table

5.6.

Table 5.6: Characteristic geometry ratios for the GAr50 and 11Dec03 runs

GAr50 2 55 27.4 11Dec03 6.1 55 9

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Aside from the fact that the jets are very small, these runs have relatively low

Mach numbers when compared to the runs of the same “family”, indicating that the

Mach number has a minor influence.

As mentioned above, the only thing that makes K02 special is the fact that involves high number of jets (60), which inherently means better symmetry, resulting in significant increase in dimensionless pressure. It is likely that the developed scaling law does not properly capture the effect of number of jets on the peak dimensionless pressure as only five number of jets were explored (12, 18, 24, 30 and 60), with majority being either 24 or 30 jets (as planned for the PLX experiment).

In summary, the results give insights into hydrodynamic phenomena of the plasma liner implosion, and provide further clarification when we account for atomic processes (using tabular eos) and energy transport models. The effect of radiative transport provides an efficient thermal energy drain, resulting in decreased speed of sound within the liner, which ultimately results in reduced hydrodynamic smearing. The lack of hydrodynamic smearing in cases with tabular eos and energy transport models pronounces the influence of jet geometry and jet distribution. Symmetry of the implosion begins to play a much more significant role and the scaling laws are no longer dominated by the initial Mach number of the jets. This has set up a requirement for the new scaling laws, which need to account for both: much more influential geometries and influence of the Mach number. The parameter space of these geometrical parameters is highly heterogeneous which results in several minor data point departures from the developed scaling law. However, most of the data does fall within a factor of 3 from the predicted

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values of the scaling law and we can thus conclude that the general trends can be derived from it.

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6. SUMMARY AND CONCLUSIONS

In this work we have first presented the first 3D simulations of plasma liner, formation, and implosion by the merging of discrete plasma jets in which ionization, thermal conduction, and radiation are all included in the physics model. We conducted this study utilizing a smoothed particle hydrodynamics code SPHC, and the effort supported and was motivated by the plasma liner experiment (PLX) [36]. PLX was motivated by the potentially efficient means of reaching igh energy density physics

(HEDP) regimes in which pressures reach and may exceed 1 Mbar. HEDP is of large importance in fields of fundamental physics, astrophysics, material and weapons research and inertially confined fusion.

In the second chapter we presented the theory behind the SPHC code, the principles of smoothed particle hydrodynamics (SPH), and show details of advanced physics models that have been used for development of tabular equation of state (eos), optically thin and optically thick radiation models and electron-thermal conduction. The generation and technical implementation of the tabular eos was given, along with a brief description of the look-up routines.

In the third chapter we presented three studies that have been done in order to verify and validate the SPHC code. The first one was the Noh problem, which enables a verification study of shock capturing for spherically imploding gases. SPHC showed excellent convergence for both 1D and 3D cases of the Noh problem. Second, we developed a synthetic interferometry tool for SPHC results and have compared the results with recent experimental results from the PLX experiment at Los Alamos

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National Laboratory. The SPHC results were in good agreement with the experimental interferometry. Furthermore, two other codes (LSP and Nautilus) have performed the same set of simulations and are also in agreement with SPHC. Finally, we have performed a validation/verification test of SPHC by using available theoretical and experimental results for the Richtmyer-Meshkov instability (RMI). RMI was chosen as a relevant phenomenon to the field of ICF and is also expected to be seen to some extent in PJMIF as well. Aside from the relevance of the RMI problem, it is an extremely complex phenomenon, involving heterogeneous flow fields, strong shocks and shock interactions, all of which pose a significant test to any fluid code. The work concerning

RMI validation/verification has resulted in a peer-reviewed publication in Physics of

Plasmas [72], which can also be found in the appendix of the dissertation. Overall,

SPHC gave good agreement with other theoretical models and experiments involving strong shocks, implosions, mixing, and thermal expansion, all of which are relevant features of plasma liner formation and implosion by discrete plasma jets.

In chapter 4 we briefly present the setups of all simulations that were carried out in order to understand the liner formation and evolution, and that has ultimately led to the development of new scaling laws.

In the results and discussion chapter, we have explored the processes of two plasma jets merging, plasma jet liner formation and evolution, identified two crucial effects in the liner evolution: hydrodynamic smearing and radiative collapse and provided new scaling laws for plasma liner peak dimensionless pressures for more realistic scenarios that include tabular eos (which accounts for atomic processes), optically thin and thick radiation models and electron-thermal conduction.

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The 1D jet propagation results are in good agreement with the experimental results carried out by Hsu et al. [83]. The jet merging simulations show a strong influence of radiative and electron thermal energy transport mechanisms. As the jets merge, we see no traces of temperature jump indicative of adiabatic oblique shock, and

Mach number and jet temperature in the interfacial region remain relatively constant.

However, we do see 5 to 10 time increase in density and pressure. The absence of temperature jumps and increase in density in the interfacial region leads to a conclusion that the mechanism of radiative collapse plays an important role in the process of plasma liner implosion.

The new scaling law has been built upon and compared with the original scaling law for purely hydrodynamic simulations that was developed by Cassibry et.al. [94]. We also compare SPHC results to the studies of Awe et.al. [92], Davis et.al. [93] and Kim et.al. [104] and provide detailed explanation for the discrepancies in behavior of the given cases. This scaling law yields clear trends concerning the influence of overall geometry of the jets, their relationship with the chamber radius and symmetry of their distribution. The main mechanisms for achieving high peak pressures have been identified and the discrepancies between the purely hydrodynamic and tabular eos/energy transport models have been thoroughly explained. Thus we find that all of the primary goals that were initially laid out were successfully fulfilled.

Based on the results of this work, we can identify a multitude of possible directions for future studies. Given the newly recognized influence of jet geometry and especially the influence of liner symmetry, we can recommend a more detailed analysis of the parameter space that involves the departure cases (EAr24_symm, GAr50,

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11Dec03 and K02). Perhaps a more detailed mathematical analysis of jet distribution influence could be carried out using Fast-Fourier Transforms in a spherical coordinate system, which would identify potential spherical harmonic modes that could enhance or degrade the peak pressures.

Some other ideas for future work would involve the inclusion of magnetic field effects and full magneto-hydrodynamic simulations of plasma liner implosions, as well as inclusion of a plasma target at the origin. The interaction of a target and the liner, compression process of the target and the preservation of the magnetic field within the target are all of crucial importance for development of a viable full scale PJMIF experiment. Furthermore, tabular equations of state for different elements should also be developed (xenon, hydrogen, deuterium, tritium…) in order to properly simulate a full scale fusion experiment.

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7. APPENDIX

7.1 A.1 - Scale coupling in Richtmyer-Meshkov flows induced by strong shocks

M. Stanic (1), R.F. Stellingwerf (2), J.T. Cassibry (1), S.I. Abarzhi (3) University of Alabama, Huntsville (1); Stellingwerf Consulting, (2); University of Chicago (3).

We perform the first systematic study of the nonlinear evolution and scale coupling in Richtmyer-Meshkov (RM) flows induced by strong shocks. The Smoothed particle hydrodynamics code (SPHC) is employed to ensure accurate shock capturing, interface tracking and accounting for the dissipation processes. We find that in strong- shock-driven RMI the background motion is supersonic. The amplitude of the initial perturbation strongly influences the flow evolution and the interfacial mixing that can be sub-sonic or supersonic. At late times the flow remains laminar rather than turbulent, and

RM bubbles flatten and decelerate. In the fluid bulk, reverse cumulative jets appear and

„hot spots‟ are formed – local heterogeneous microstructures with temperature substantially higher than that in the ambient. Our numerical simulations agree with the zero-order, linear, weakly nonlinear and highly nonlinear theoretical analyses as well as with the experiments, and suggest that the evolution of RMI is a multi-scale and heterogeneous process with a complicated character of scale coupling.

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7.1.1 Introduction

Richtmyer-Meshkov instability (RMI) controls variety of transport processes in high energy density plasmas (HEDP) [1, 2]. Examples include formation of a „hot spot‟ in inertial confinement fusion (ICF), interaction of plasma liner with magnetized target in magneto-inertial fusion (MIF), radial compression of imploding Z-pinches, and core- collapse supernova [3-7]. High energy density plasma flows are often characterized by strong shocks and sharp changes in scalar and vector fields, and by relatively small influences of dissipation and diffusion. This leads to formation of interfaces separating flow heterogeneities at macroscopic (e.g. continuous) scales [8, 9]. The shock-interface interaction results in an extensive interfacial mixing that strongly influences the transport of mass, momentum and energy in the plasma flow and the interplay of its scalar and vector fields [10, 11]. In this work we perform the first systematic study of the evolution and scale coupling in Richtmyer-Meshkov (RM) mixing flow induced by strong shocks, and employ Smoothed Particle Hydrodynamic (SPH) simulations to explore a broad parameter regime [12, 13].

RMI develops when a shock wave refracts a corrugated interface separating fluids with different values of the acoustic impedance [1, 2], whether it propagates from the light to the heavy fluid or oppositely. The shock-interface interaction results in the dynamics that is a superposition of two motions: (1) The interface moves as a whole with velocity v  at which an ideally planar interface would move after the shock passage (the post-shock velocity of the planar interface) [1, 2, 14-17]. (2) The interface perturbations grow because of „impulsive‟ acceleration induced by the shock [14-18]. In the nonlinear regime of RMI the growth-rate decreases and a large-scale coherent

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structure of bubbles and spikes appears [2, 11, 19-22]: The light (heavy) fluid penetrates the heavy (light) fluid in bubbles (spikes). The period of the structure is set by the initial perturbation and its amplitude is defined by the bubble and spike positions. Shear-driven instabilities produce small-scale structures on the sides of evolving spikes [22, 23].

Eventually a mixing zone develops. In the mixing regime the bubbles and spikes both decelerate, and energy injected by the shock gradually dissipates [11, 14, 22]. RM mixing is a heterogeneous and statistically unsteady turbulent process and its properties depart from those of decaying isotropic and homogeneous turbulence [21].

In high energy density plasma phenomena shocks are strong, fluid densities are highly contrasting, and perturbations are large [3-8]. This extreme parameter regime is of great importance in applications with each experiment, simulation and theoretical analysis study being the state-of-the-art [3-8]. However, only limited information is currently available on RMI evolution under these conditions [24]. It is unclear, whether for RM flows induced by strong shocks the nonlinear dynamics is single-scale or multi- scale, and whether such a flow tends to be more laminar or turbulent [20, 24, 25].

Moreover, though it is commonly accepted that asymptotically in time RM mixing is homogenizing, in realistic experimental situations the flow remains in transient state, and its heterogeneities are well pronounced [3-8, 24, 25].

Experimental investigations of RMI induced by strong shocks are relatively rare as they impose demanding requirements on flow implementation, diagnostics and control [11, 14, 22, 26, 27]. Numerical modeling of RMI by means of the Eulerian finite- difference methods is a powerful tool for studies late-time evolution, yet to study the dynamics of strong-shock-driven RMI the numerical models should satisfy numerous

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competing requirements, such as shock capturing, interface tracking, and accurate accounting for the dissipation processes [16, 25, 28, 29]. Furthermore, since at both early and late times the dynamics of RMI are described by power-laws, in the experiments and simulations the flow quantities should be measured with high precision and accuracy and should span substantial dynamic range [20, 30].

To overcome these challenges and to test the theoretical ideas and hypothesis

[15-21, 24, 25], we apply the Lagrangian method - Smoothed Particle Hydrodynamic

(SPH) [12, 13] – to systematically study the nonlinear RMI induced by strong shocks for fluids with contrasting densities and with large amplitude initial perturbation at the fluid interface. The simulation results are compared, wherever possible, with the theory and experiments achieving good agreement. We found that: (i) The amplitude of the initial perturbation significantly influences the flow evolution. (ii) At late times, the flow remains laminar rather than turbulent, and RM bubbles flatten and decelerate. (iii) The dynamics at small-scale is heterogeneous, and is characterized by the appearance of reverse cumulative jets, inhomogeneous velocity field, local microscopic structures, and

„hot spots‟.

7.1.2 Theoretical and numerical considerations

7.1.2.1 Outline of theoretical approaches

Having started with the seminal work of Richtmyer [1], extensive theoretical studies [15-21, 24, 25] significantly advanced our understanding of RM dynamics. For early stages of RMI, when the interface is nearly flat and the amplitude of the initial

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perturbation is negligibly small compared to its spatial period , the a 0 impulsive and heuristic models and the rigorous analysis [1, 15-18] derived the value of the instability growth-rate for a broad range of the Mach number and the Atwood v 0 number , where is the density of the heavy (light) fluid.  The growth-rate is where is the amplitude identified by the positions of the tips of the bubble and spike. For infinitesimally small initial perturbation amplitude,

and , where is the wave-vector, the growth-rate linearly increases with and . For small but finite values of with the dependence of on becomes non-nonlinear, and the values of increases with

slower than linear [15, 39]. For very large amplitude of the initial perturbation the growth-rate may decrease with [15].

For nonlinear stages of RMI with , an adequate theoretical description of the large-scale dynamics was found within group theory consideration assuming that the coherent motion is incompressible [19-21]. The analysis showed that the late-time dynamics of RMI is a multi-scale process, which is governed by two independent length scales -- the spatial period and the amplitude of the front. The qualitative indicator of this multi-scale character is the deceleration and flattening of the bubble front: and as , where is the bubble curvature and is its velocity is the frame of references moving with velocity [11,

20]. For RM mixing, which is the final stage of the flow evolution, many features of the dynamics require better understanding [11, 14, 24, 25, 31]. It is not clear, for instance, what the control parameters are that may trigger the flow transition from the nonlinear to mixing regime, what the characteristic time is for turbulent mixing to occur, or what the

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conditions are for maintaining the power-law growth of the amplitude with time

[11, 14, 24, 25, 31].

Theoretical studies [15-21] emphasized the strong dependence of RMI evolution on the initial conditions, and the sensitivity of the macroscopic coherent dynamics to microscopic processes. Furthermore, since linear, nonlinear and mixing regimes in RMI are described by power-laws and the growth-rate is a function of time, transitions between these regimes are challenging to differentiate and the growth-rate is uneasy to reliably quantify in the experiments and simulations. Thus the analyses [15-21] stressed the necessity to characterize the front dynamics with higher accuracy and precision, to conduct the observations of the robust macroscopic parameters over a substantial dynamic range, and to tighter control the experimental parameters, such as Mach and

Atwood numbers and the perturbation amplitude [20, 21, 31, 28, 29].

7.1.2.2 Smoothed Particle Hydrodynamics numerical approach

To address these challenges and to test the applicability of the theoretical results and hypothesis [15-21, 24, 25] under the extreme conditions, we apply Smoothed

Particle Hydrodynamics. SPH is a Lagrangian method that allows for a relatively easy treatment of the governing equations of the fluid motion – the partial differential equations (PDEs) representing the conservation of mass, momentum and energy [12,

13]. The classical Eulerian finite-difference methods solve these PDEs approximating the flow fields at a finite number of grid points in space. Similarly, SPH applies the particle and kernel approximations to reduce the governing PDEs to a set of ordinary differential equations.

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In the SPH, with being the position vectors of the particles, being the integration volume and being the Dirac delta-function, in particle approximation any field variable can be expressed as

∫

. In kernel approximation, the Dirac delta function is replaced by a kernel function with the following properties:

∫

when . Here is a „smoothing‟ length, and is a coefficient determining the support domain boundary. With representing the kernel approximation operator, one obtains

∫

and

∫

, and, upon discretization,

∑ ( ) ( )

and

∑ ( ) ( )

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Where are particle mass and the overall mass density associated with particle respectively.

The foregoing expressions are substituted into the equations of motions and numerical solutions for these equations are found [12, 13]. In this way, on the one hand, the SPH fully uses the advantages of Lagrangian approach for resolving the flow fields, and, on the other hand, is relatively easy to implement. However, as with any numerical method, SPH has certain limitations. In particular, some effort should be undertaken to find an efficient algorithm for identification of „neighbor‟ particles (e.g. to apply a so- called „oct-tree‟ method [12, 13, 32-34]), and to overcome particle penetration with the use of artificial viscosity. In this work we employ the SPHC code, which is one of the most robust SPH codes and which handles these numerical issues adequately [12, 13, 32-

34].

7.1.2.3 Outline of parameter space, simulations setup and diagnostics

To systematically explore the extreme parameter regime, we consider 27 points in the parameter space of the Mach number, Atwood number and the initial perturbation amplitude. In our simulation the shock is strong with Mach number { }, the

Atwood number is high, { }, and the amplitude of the initial

perturbation is large, { } with {

}. To ensure adequate spatial and temporal resolutions, the overall number of particles in two-dimensional (2D) simulations is up to [12, 13,

32-34]. For the standard shock test problems (Noh problem) the analysis of SPHC code‟s convergence shows that for 3D cases accuracy of 10% can be achieved at

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resolutions as low as particles [33, 34]. In addition, we also apply the variation of the particle spacing across the simulation domain in order to increase the computational efficiency while preserving the high accuracy in the areas of special interest, such as the interface [33, 34].

In our simulations we analyze vector and scalar fields of the fluid flows, including quantitative and qualitative studies of temperature, pressure and velocity.

Aside from that, in order to quantify the evolution of the interface, we obtain data from

31 moving probes, which are designated particles recording their state (flow fields) at certain time steps and exporting the data to the data history file. Initially all probes are evenly distributed along the two-fluid interface. As simulations progress, due to appearance of vortical structures, many probe particles are displaced from their original positions at the tips of the bubble and spike thus causing the mixing of the probes and limiting our opportunity to get useful data. At earlier times, the probes remain at the tips of the bubbles and spikes and allow for the accurate diagnostics of the amplitude and amplitude growth-rate.

In our SPH simulations, fluids are ideal mono-atomic gases with the adiabatic

index . Initially the fluids have the same pressure Pa and

temperature K and distinct gas constants and mass densities .

3 For the light fluid kg/m and J/kmol. For the heavy fluid

and J/kmol.

The fluids are rarefied (and rather „stiff‟) gases with high energy per fluid particle and per proton. Initially the energy ranges from J to J per SPH particle or from eV to MeV per proton. After the shock passage the energy

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ranges from J to J per SPH particle or from keV to

MeV per proton.

The shock propagates from the light to the heavy fluid in the 2D domain with dimensions m. Initially the fluid interface is in the middle of the domain and is normal to the shock. The initial perturbation wavelength (spatial period) is

m. The Mach number is defined relative to the speed of sound of the

light fluid m/s [8, 9, 11, 20, 22, 26, 27, 28, 35]. The duration of our simulations is defined by the time when the reflected shock reaches the outer boundary of the domain and it is about few microseconds.

Some artificial surface tension is present in the simulations in order for the interface between the fluids to behave as a sharp boundary [12, 13, 32, 33, 34]. To ensure that the macroscopic dynamics is not influenced significantly by numerical viscosity and surface tension, we conducted a separate sequence of runs, in which the initial densities, pressure, and temperature were scaled (in few orders of magnitude) to that of the natural gases [27]. Good qualitative and quantitative agreement with the experiments [27] was found thus indicating that the effects of numerical viscosity and surface tension are negligible and the dynamics is indeed scale-invariant.

The characteristic length-scale and time-scale in our problem are set by the wavelength and velocity as [11, 20]. Diagnostic parameters in the simulations include the amplitude and the growth-rate of the amplitude . We also derive from the simulations the position and velocity of the bubble , and spike

, in the frame of reference moving with constant velocity , and evaluate the curvature of the bubble front [19-21]. In order to better illustrate the physical behavior of

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the flow, several other parameters are derived from diagnostic data, such as specific drag force experienced by the bubble.

The results of our SPHC simulations are repeatable and are qualitatively similar for all 27 points in the parameter space of { }. This work reports new qualitative features of RM evolution observed in our SPHC simulations for strong shocks, contrast densities and strongly corrugated interfaces. A detailed quantification of the nonlinear dynamics of RMI in HEDP is the subject of future research.

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7.1.3 Results

7.1.3.1 Growth-rate of the Richtmyer-Meshkov instability

7.1.3.1.1 Background motion

According to observations [20, 26, 36, 37], for weak shocks, , and for small amplitude initial perturbation, , in RM mixing flows the velocity scales are

distributed sparsely: The post-shock velocity of planar interface , the

instability growth-rate is and the bubble/spike velocity in the

nonlinear and mixing regimes is . The ratio decreases whereas the ratio increases with the Atwood number [20].

The post-shock velocity of the planar interface increases with the shock strength , Table 1. We emphasize that velocity remains a fraction of , where is the velocity of the light fluid after the shock passage (e.g. post-shock velocity of the light fluid), Table 1 [35, 38]. The ratio depends on the Atwood number and the

Mach number. We emphasize that for weak shocks with and for very strong shocks with , some simple analytical relations can represent the scaling dependence of on the Mach and Atwood numbers [17]. However for realistic shocks with , the distribution of shock energy among the transmitted and reflected shocks and the background motion has a complicated character, and the dependence of on the Mach and Atwood numbers can be derived directly from the boundary conditions balancing the transports of mass, momentum and energy across the ideally planar interface [17].

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Qualitatively, for strong shocks, ( ), the velocity of the background motion becomes super-sonic, , indicating that significant part of the shock energy goes into the compression of the fluids and their background motion.

For Atwood numbers not very close to 1, the ratio increases with the Mach number so that for , Table 2. For a fixed Mach number, the ratio

decreases two-to-three fold with increase of the Atwood number from 0.6 to 0.95, Table 2. These trends are similar to the case of weak shocks with , Table 2 and [15-

21, 35, 38]. Table 2 also provides the values of derived from the boundary conditions at the planar interface with finding good agreement ( ) between the numerical simulation results and the theory [17].

Table 1: Dependence of and on the Mach number for . M 1.1 1.2 2 2.5 3 5 10

0.0994 0.1748 0.7052 1.0028 1.3536 2.4458 5.0665

0.6941 0.6357 0.6269 0.6368 0.6769 0.6795 0.6824

Table 2: Dependence of on the Mach and Atwood numbers M 3 5 10 A 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

1.3609 1.0391 0.5924 2.4710 1.8998 1.0961 5.1202 3.9512 2.2938

1.3513 1.0321 0.5654 2.4458 1.8873 1.0278 5.0665 3.9016 2.3804 0.6757 0.5161 0.2827 0.6795 0.5243 0.2855 0.6824 0.5255 0.3206

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7.1.3.1.2 Early-time evolution of RMI

As the growth-rate of RMI is a function of time and initial amplitude, its reliable quantification is a challenging task. In the limiting case of infinitesimally small amplitude of the initial perturbation, with , the situation is somewhat simplified because the velocity linearly depends on the initial perturbation amplitude as with where is a function of the Atwood number and the Mach number [15-21, 35, 38].

Table 3 shows comparison of the growth-rate in our simulations and in the linear analysis [17] for small but finite amplitude with

. In the simulations the value of the growth-rate is measured after the shock completely passes the interface. Table 3 indicates that the growth-rate values in the simulations depart on (up to at high Mach and high Atwood number cases) from their corresponding theoretical values. This effect is similar to that observed in the experiments [39], and it is likely due to finite amplitude of the initial perturbation amplitude and secondary shocks.

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Table 3: Dependence of the growth-rate on the Mach and Atwood numbers in the simulations

and in the linear analysis [ ] [17] for with

. M 3 5 10 A 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

( ) 0.3390 0.3859 0.2653 0.5271 0.6743 0.5212 1.2031 1.3774 1.0838

( )

* + 0.4672 0.5372 0.4093 0.7699 0.9182 0.7338 1.5221 1.8498 1.5157

To further validate our numerical approach and perform a comparison with the

linear theory [17], we conducted a separate high-resolution run with and

for and case. The asymptotic value of the growth- rate in the simulations and in the linear analysis [17] are

and [ ] respectively. In our simulations and

in the linear theory [17], the values are obtained by averaging data over the time interval that is after the shock completely passes the interface and is before the nonlinear effects start to play a role. In this time interval the growth-rate saturates and is accompanied by slight oscillations that are induced by reverberations of sound waves

[17,20]. The distinction between the mean values of the growth-rate in the simulations and analysis is . The amplitude of the oscillations is in the analysis [17] and is higher in the simulations. The agreement is good, and the simulations reproduce the results of the linear analysis [17]. The deviations between the numerical and theoretical data are likely due to sensitivity of our diagnostic method, as in the simulations the signal is received from the few probes residing at the tips of the bubble and spike.

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Consideration of an ensemble of particles may allow for the reduction of the slevel of noise and quantification of data with higher accuracy. This issue is addressed to the future.

In the case of small but finite amplitude with , we also qualitatively compare our particle simulations of RMI driven by strong shocks with the results of continuous dynamics simulations of RMI driven by weak shocks employing the level set method for the interface tracking [20]. Similarly to the case of weak shocks with , in the case of strong shocks the growth-rate increases with the Atwood number yet it is relatively insensitive to the Mach number,

Table 4 [20]. Remarkably, for cases of small but finite initial perturbation, the amplitude

growth rate for both weak and strong shocks is on the order of , see

Tables 1-4 and Ref. [20].

Table 4: Dependence of on the Mach and Atwood numbers for . M 3 5 10 A 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

0.0946 0.1410 0.1769 0.0812 0.1347 0.1912 0.0895 0.1331 0.1716

7.1.3.1.3 Effect of initial amplitude on the growth-rate of RMI

According to our simulations, and in agreement with experiments and simulations in Ref. [39], for strong shock driven RMI the interfacial dynamics can be sub-sonic with . Thus in the case of relatively small amplitude, we may compare our simulations with the results of the weakly nonlinear analysis [15] of incompressible RMI. The perturbation theory [15] suggests that for small but finite

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amplitude initial perturbation, after the shock completely passes of the interface, the effect of the initial amplitude on the growth-rate can be accounted for via a power-law expansion in terms of small parameter . In particular,

[ ] [ ] * ( )

( ) +

where [ ] , [ ] are the values of the amplitude growth-rate for finite initial amplitude and for infinitesimally small initial amplitude respectively and [ ]

[15,17]. This expansion captures qualitatively and quantitatively the effect of the initial perturbation amplitude on the post-shock growth-rate of RMI indicating that the growth-rate decreases with the increase of the initial amplitude [15]. It may not capture however secondary shocks [39]. To date not a theory has been developed to capture secondary shocks in RMI in a rigorous and self-consistent manner [15, 17].

Quantitative comparison is presented in Table 5 of the post-shock RMI growth-rate in our SPHC simulations and in the linear and weakly nonlinear analyses [15, 17] in cases of with corresponding values of . The values of the growth-rate in the simulations are smaller than those in the linear and weakly nonlinear analyses [15, 17]. The distinctions are similar qualitatively and quantitatively to those in experiments [39]. They indicate that in case of strong shocks and large initial perturbation amplitudes the secondary shocks may strongly influence the dynamics.

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The amplitude of the initial perturbation is a crucial factor for RMI evolution. In our simulations, in agreement with studies [15, 39], we observe strong dependence of the amplitude growth on the initial perturbation amplitude , Figures 1a-c. For given

{ } the growth-rate (and ) increases several folds (two-to-three and higher) when the initial perturbation amplitude changes from to to . It is remarkable that the growth-rate is less sensitive to the Mach number, Figure 1.

Therefore using velocity as a scaling unit appears an adequate choice for quantification of RMI evolution. This parameter represents the amount of energy deposited by the shock.

For convenience, in Table 6 we present the growth-rate and for all values of { } in the SPHC simulations. In our parameter regime the interfacial mixing remains sub-sonic, , except cases with Mach number and initial amplitudes

, Table 6.

Due to the shock-interface interaction, the interface is first slightly contracted and its amplitude decreases, and then the amplitude start to increase. Figure 1 represent

the amplitude evolution versus for all cases of { }. Here

is the time at which the interface growth starts, and is the value of the amplitude at

time . Values and are identified from the simulations data for each case. The values of the amplitude growth and the amplitude growth-rate are taken from the particle probes that reside at the tips of the bubbles and spikes and move with the interface. The probe particles are initially positioned at the tips of the bubbles and spikes and are located there at earlier times of flow evolution. At later times the particles are displaced

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into the vortical structures developing at the interface and cannot be used for accurate quantification of the amplitude dynamics. In particular, in our simulations in the case of

, , the probe particles are displaced quickly into the vortical structures thus inducing some inaccuracies in the amplitude growth and the growth-rate value in Tables 5 and 6 and in Figure 1. Improvements of data post-processing methods and development of advanced techniques for accurate quantification of RM unstable interface in the SPHC simulations are addressed to the future.

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Table 5: Effect of initial perturbation amplitude on the growth-rate in the simulations and v 0 weakly nonlinear theory [ ] [15, 39] scaled with linear theory growth-rate

[ ] [17]. 

M 3 5 10 A 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

[ ] 0.726 0.718 0.648 0.685 0.734 0.710 0.790 0.745 0.715

[ ] [ ] 0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966

3 5 10 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

[ ] 0.565 0.468 0.379 0.565 0.487 0.372 0.551 0.478 0.396 [ ] [ ] 0.680 0.682 0.684 0.680 0.682 0.684 0.680 0.682 0.684

3 5 10 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

[ ] 0.302 0.253 0.208 0.259 0.275 0.227 0.332 0.103 0.205 [ ] [ ] 0.444 0.455 0.465 0.444 0.455 0.465 0.444 0.455 0.465

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Table 6: The growth-rate of RMI in the SPHC simulations

M 3 5 10 A 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

0.1278 0.1455 0.1000 0.1987 0.2542 0.1965 0.4535 0.5193 0.4086

0.0946 0.1410 0.1769 0.0812 0.1347 0.1912 0.0895 0.1331 0.1716

3 5 10 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

0.4149 0.3952 0.2439 0.6828 0.7017 0.4286 1.3179 1.3886 0.9423 0.3071 0.3829 0.4313 0.2792 0.3718 0.4170 0.2601 0.3559 0.3958

3 5 10 0.6 0.8 0.95 0.6 0.8 0.95 0.6 0.8 0.95

0.4433 0.4271 0.2681 0.6268 0.7938 0.5232 1.5895 0.5988 0.9789 0.3280 0.4138 0.4741 0.2563 0.4206 0.5091 0.3137 0.1535 0.4112

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Figure 1: Dependence of the amplitude on time: vertical axis is ; horizontal axis is , where is time at which the interface starts to grow after the shock completely passes the interface; the Atwood number is (a), (b) and (c); initial perturbation amplitude is (black), (red) and (blue); Mach number is (solid), (dash) and (dash-dot).

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7.1.3.2 Effect of the initial perturbation amplitude on the transmitted shock

In addition to significantly influencing the early time evolution of RM unstable interface, the amplitude of the initial perturbation may also influence the motion of fluids in the bulk. One interesting feature of RMI dynamics induced by strong shocks is the effect of initial perturbation amplitude on the transmitted shock [38]. Figure 2 shows the snapshots of the velocity component in the direction of shock propagation for shock refracting (a) corrugated interface with and (b) planar interface with in the case of , . The characteristic time-scale is µs. For

the growth-rate is with and

, Table 6. Snapshots are taken at the end of the simulation run at time µs.

Figure 2 shows measurements of the fluid velocities in the laboratory frame of references. The velocity values are presented using color with light green color corresponding to fluid at rest and with velocity values changing from m/s

(blue) to m/s (red). The dark red strip on the left marks the reflected shock region, and the light green strip on the right marks the transmitted shock region.

Figure 2 clearly illustrates the necessity to account for the background motion and to „separate‟ the interfacial mixing from the background motion for accurate quantification of RMI.

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a) b)

Figure 2: Snapshot of the velocity component in the direction of shock propagation in the laboratory frame of reference: large amplitude initial perturbation (a) and planar interface (b) for , at µs with µs. Shock propagated left to right. The domain size in each direction is given in cm. The color bar for velocity values is given in cm/s. Light green color corresponds to fluid at rest.

Indeed, as the ratio is relatively small, the interfacial dynamics is a challenge to differentiate from the „global‟ motion, and the flow fields should be quantified with high accuracy and precision. Furthermore, comparison of Figures 2a and 2b shows also that the transmitted shock region (e.g. the width of light green strip on the right) differs significantly in these two cases, and that the transmitted shock propagates faster in the case of large amplitude initial perturbations, , than in the planar case,

This effect was observed in experiments [38].

Figure 3 represents further comparison of the velocity fields in Figures 2, and shows the component of velocity in the direction of shock propagation in the frame of reference moving with velocity accounting for the background motion. The yellow color in Figure 3 corresponds to zero velocity value. Figure 3 clearly illustrates that no

(or little) fluid motion occurs in the bulk of the light fluid far from the interface [rather than background motion of the fluids with constant velocity in the laboratory frame

17 2

of references]. Small-scale heterogeneities are induced by „reverse‟ jets in the bulk of the heavy fluid in Figure 3, see discussion below for details.

Figure 3: Velocity component in the direction of shock propagation for , , at µs in the frame of references moving with velocity of the background motion. Shock propagated from left to right, and time-scale is µs. Color bar is in cm/s.

To quantify the effect of the initial perturbation amplitude on the transmitted shock, we measure the width of „non-overlapping region‟ in the transmitted shock regions in Figures 2 and 3. This „non-overlapping region‟ occurs due to the faster propagation of the transmitted shock in case of a large amplitude initial perturbation compared to the planar case. The measurement is performed such that the position of the transmitted shock for finite amplitude case (Figure 2a) is subtracted from that for planar plane (Figure 2b). The width of the region is measured for all 27 points in the parameter space of { } at . Here is the run time of each simulation and

µs, see Figure 4. As discussed in the foregoing, time is defined by the time when the reflected shock hits the outer boundary of the domain, and it depends on the

Mach and Atwood numbers, Figure 2. The measurement results indicate that the width of the region is greater and the transmitted shock propagation is faster for larger values of

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the initial perturbation amplitude. In particular, according to our simulations results, the width of „non-overlapping region‟, and thus the transmitted shock velocity, grows linearly with the initial perturbation amplitude and is strongly influenced by the Atwood number. It is, however, less sensitive to the Mach number. For instance, for constant

Atwood number and for , the transmitted shock velocity remains the same for all three Mach numbers, . The minor errors in Figure 4 can be attributed to the uncertainty of measurements that occur due to the data post-processing and noise.

Figure 4: Dependence of the width of „no-overlapping region‟ and the transmitted shock velocity on the initial perturbation amplitude for Atwood numbers 0.6 (circle), 0.8 (square) and 0.95 (triangle) and for Mach numbers 3, 5 and 10. The width of the region was measured at .

7.1.3.3 Multi-scale character of the nonlinear dynamics

The other important feature of RMI evolution is the multi-scale character of the nonlinear dynamics that is indicated by deceleration and flattening of the bubble front.

This effect was predicted by analysis [19-21]. The analysis considered nonlinear dynamics of RMI in the limit of under assumption that the flow is incompressible and has no sources of mass, momentum and energy at the outer

174

boundaries of the domain. In the case of strong shocks, when the growth-rate is sub- sonic (Table 6) and the interfacial mixing remains weakly compressible, the analysis results [19-21] can still be applied, at least qualitatively. It is conditional however that the flow dynamics is considered in the frame of reference moving with velocity accounting for the background motion, Figures 2 and 3, similarly to the case of weak shocks [20].

To conduct qualitative comparison of our simulations with highly nonlinear theory [19-21] we select the parameters { } such that, on the one hand, the interfacial dynamics is sub-sonic, Tables 5 and 5, and, on the other hand, the instability develops fast enough to ensure that the spike is formed and the flow transitions into highly nonlinear regime during the simulations run.

7.1.3.3.1 Qualitative comparison of the velocity field

First we compare qualitatively the velocity field in the simulations and in the analysis [19-21]. According to theoretical analysis [19-21], in the highly nonlinear regime of RMI the velocity field has the following characteristics: Background motion is accounted for with proper choice of the frame of reference. The flow has no sources at the outer boundaries of the domain. There is no (or little) motion of the fluids away from the interface. The fluid motion is intense near the interface, and the dynamics is characterized by the amplitude and wavelength of the front. At late times the bubble decelerates and flattens, and for the flattened bubble the normal and tangential components of velocity are continuous at the interface [19-21].

Figure 5 present a sequence of snapshots of the velocity field in the SPHC simulations in the case of , and with sub-sonic growth-rate 175

and with time-scale µs allowing sufficient time for RMI to evolve. The other values are and . Snapshots show the velocity component in the direction of shock (that propagated left to right) with subtracted velocity to account for the background motion. Color marks the velocity value with yellow color corresponding to zero. The red strip on the left marks the reflected shock region, and blue strip on the right marks the transmitted shock region. As is seen from Figure 5, when the reflected shocks moves away from the interface, the velocity field becomes non-uniform: In particular, in Figure 5, snapshot at µs, the fluid motion is intense near the interface in the front region (and it is characterized by the wavelength and amplitude of the front). Effectively no motion occurs far from the interface. Very little motion in the bulk of the light fluid is due to weak compressibility whereas small-scale velocity heterogeneities in the bulk of the heavy fluid are due to reverse jets (see below for details). In the direction normal to shock propagation the velocity component is periodic and is non-zero near the tips of the bubbles and spikes

(see below for details). For flattened bubble the normal and tangential components of velocity are both continuous at the interface.

For other parameter values { } at late times of the evolution of RMI the velocity field is similar to that in Figures 5. This velocity field agrees qualitatively with the velocity field in the analysis [19-21]. Figures 2 and 5 clearly show the necessity to account for the background motion for accurate quantification of RMI dynamics and to separate the background motion from the interfacial mixing.

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Figure 5: Velocity component in the direction of shock propagation for , , with subtracted velocity to account for the background motion. Snapshots are taken at µs (left, top), µs (right, top), µs (left, bottom), and µs (right, bottom). The shock propagated from left to right, and time-scale is µs. Color bar is in cm/s.

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7.1.3.3.2 Flattening of the bubble front

If bubble velocity decays as then it decelerates

as , and the faster the bubble moves, the stronger it decelerates and contra versa. As discussed in Refs. [19-21], and is seen from

Figure 5, flattening and deceleration of the RM bubble front are inter-related processes.

In our simulations we observe flattening of the bubble front at late times of RMI evolution for all Mach and Atwood numbers and for initial amplitudes .

Figure 6 shows the evolution of the interface separating the phases of the fluid flow – the particles of the light gas (red) from the particles of heavy gas (blue) in the case of , , with and time-scale

µs. Flattening of the bubble front starts shortly after the shock passage and is clearly seen from the sequence of images in Figure 6. The decrease of the bubble curvature in the nonlinear RMI is observed in other cases as well. The flattening process is accompanied by two events: (1) slight increase of curvature when spike starts to form;

(2) increase of curvature when secondary shocks on the sides of reverse jets converge at the bubble center (in Ref. [27] they are called „transverse shocks‟, see discussion below).

These events are however brief, and after they pass, bubble curvature steadily vanishes.

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Figure 6: Evolution of the interface separating particles of the light gas (red) from the particles of the heavy gas (blue) for of , , , time-scale is µs. The shock propagated left to right.

For small but finite initial amplitude the time duration of our simulations is insufficient to observe the flattening of the bubble front. For a given

{ } in our parameter regime, the growth-rate of RMI decreases two-to-three folds when the initial perturbation amplitude changes from and to ,

Tables 5 and 6. The duration of our simulations is bounded by the time when reflected shock reaches the outer boundary of the domain. At this time, in all cases the bubble curvature is still finite, , and it continues to evolve, in qualitative and quantitative agreement with continuous dynamics simulations in Ref.[20].

7.1.3.3.3 Deceleration of the bubble front

Figure 7a shows several representative cases of the evolution of the bubble velocity calculated in the frame of reference moving with the velocity : The bubble velocity initially increases due to the impulsive acceleration induced by the shock and then decreases with time. It depends strongly on the initial amplitude and is less sensitive to the Atwood number. It is remarkable that bubble velocity decreases

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even in cases of and when and the interfacial mixing is slightly super-sonic, Table 6.

Figure 7b shows several representative cases of the evolution of specific drag

force experienced by the bubble and estimated as , where , are measured in the frame of references moving with velocity . We see that the effective drag force decreases quickly thus indicating that with time the interfacial motion tends to be inertial. In the nonlinear regime in our parameter regime the flow remains laminar rather than turbulent. Scatter in data points in the nonlinear regime is likely induced by particle diagnostic method. Employment of more advanced diagnostic technique (e.g. consideration of an ensemble of designated particles) may reduce the level of noise.

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Figure 7: Velocity (a) and drag force (b) of the bubble versus time for Atwood , initial perturbation amplitude (black) (red) and Mach number (cross) and (square). Values are calculated in the frame of references moving with velocity to account for the background motion.

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7.1.3.4 Heterogeneous dynamics at small scales

7.1.3.4.1 Kelvin-Helmholtz instability

Influence of small-scale structures on the dynamics of large scales and scale coupling is critical issue for RMI evolution. Traditionally, the small-scale dynamics in

RM flows is considered within the contest of shear-driven Kelvin-Helmholtz instability

(KHI) resulting in appearance of small-scale vortical structures at the fluid interface and leading to homogenization of the flow fields [24]. The SPHC, as Lagrangian method, is capable of resolving the dynamics at very fine spatial and temporal scales. In our SPHC simulations, due to adaptive nature of SPH particle motion and adequate numerical resolution, noticeable development of KHI is observed at the fluid interface at early time, Figure 8. In order to ensure that the observed effect is indeed the shear-driven

Kelvin-Helmholtz instability separate runs were conducted with five times the resolution. To study the influence of KHI on late time evolution of RMI, we consider the interfacial process in more details.

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Figure 8: Development of Kelvin-Helmholtz instability at the fluid interface for early-time RMI at Atwood number , Mach number , initial perturbation amplitude . Snap shot is at µs. Time-scale is µs.

Figure 9 presents the snapshots of particle regions (which are distinguished by thermodynamics properties of the particles at t  0 ) that are taken at times µs for the case of , with initial perturbation amplitude and time- scale is µs.. Slight perturbations are present at the interface, and they trigger the development of KHI after the shock passage. However the effect is not numerical, as the simulations are run at the edge of resolution.

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Figure 9: Early time RMI evolution for , , . Snapshots are takes at times µs (time-scale µs) and show the development of KHI on the sides of the spikes. Color represents post-shock region (red), light fluid (green), heavy fluid (blue).

The curl of velocity field in Figure 10 shows significant shear at the interface on the side of evolving spikes and thus justifies that these patterns are indeed the Kelvin-Helmholtz wave patterns and that they are well resolved.

Figure 10: Evolution of on the sides of the spike in early time RMI leading to development of KHI for , , . Snapshots are takes at µs (time-scale µs). Color bar for curl of velocity is in 1/s and it ranges from to .

On the contrary, on the side of the bubble, where same slight perturbations were present initially, there is little or no shear, and Kelvin-Helmholtz instability does not

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develop, see Figure 11. Therefore, development of shear-driven Kelvin-Helmholtz instability is well captured by our numerical simulations.

Figure 11: Evolution of on the sides of the bubbles in early time RMI leading to development of KHI for , , . Snapshots are takes at µs (time-scale µs). Color bar for curl of velocity is in 1/s and it ranges from to .

For strong-shock-driven flows with , the KH structures appear at earlier stages of RMI evolution and then quickly disappear in a vicinity of the bubble tip while remaining in a vicinity of the spike tip. This is illustrated by Figure 12, which shows curl of the velocity field at late times of RMI. Shear is still present at the interface, Figure 12, but its magnitude is an order of magnitude smaller compared to that at the earlier times,

Figure 10. In Figure 12, the Kelvin-Helmholtz wave pattern is not observed along the stem of the spike, and the mushroom shape of the spike is formed in agreement with experiments [39].

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Figure 12: Late-time RMI for , , with time-scale µs. Snapshot is taken at µs. No KHI is observed along the stem of the spike, and the mushroom-type shape of the spike is formed. Arrows show the flow direction in the laboratory frame of references. Color bar for curl of velocity is in 1/s and it ranges from to .

To conclude the discussion of the development of KHI during the evolution of

RMI, we mention briefly that in some cases in our simulations secondary shocks may cause an increase of shear along the spike and induce the development of secondary KH instabilities.

7.1.3.4.2 Reverse cumulative jets and heterogeneities of scalar fields

Our simulations record the observation of multiple „reverse‟ (cumulative) jets in the flow, Figures 13-14 [41]. These jets occur during the RMI evolution at the base of the spikes (bubbles) and propagate in the direction reverse to the spike (bubble) propagation. Cumulative jets are generated due to collision of converging fluid flows at a small angle of attack and their specific kinetic energy is high, where is the flow velocity and is an angle of attack [41]. Similarly to cumulative jets in Ref.

[41], the reverse jets in our simulations are short and energetic. Their dynamics is

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accompanied by appearance of local „hot spots‟ in the flow and thus produces heterogeneities in the vector and scalar flow fields at small scales. We emphasize that while the reverse jets closely resemble the mushroom shape, they are not interfacial structures and are entirely immersed in the fluid bulk.

Figure 13 represents the temperature field in RM flow for , ,

at with red color marking the highest and blue – the lowest temperature values.

Figure 13: Temperature field ranging from K (blue) to K (red) and showing reversed jets in the base of the spikes between the bubbles (and in the base of the bubbles between the spikes) and the „hot spots‟ (red color) for , , at .

Reverse jets are clearly seen at the base of the spikes between the bubbles (and further in the flow -- at the base of the bubbles between the spikes) and their mushroom shape is induced by shear. The jets are accompanied by local „hot spots‟ that appear in the bulk of the flow with the temperature that is nearly two fold of that in the ambient.

Figure 14 presents density (left image) and pressure (right image) for the same case as in

Figure 13 with red color marking the highest and blue the lowest values of the density and pressure correspondingly. In the vicinity of the cumulative jets, the high temperature

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is accompanied by the low density and nearly uniform pressure [41]. At the base of the spike the temperature is relatively cold and uniform, whereas material is compressed with high pressures and achieves high densities.

Figures 13 and 14 also illustrate flattening of the bubble front and formation of vortical structures on the side of evolving spikes, as discussed in the foregoing.

Evaluation of curvature in a vicinity of the bubble tip shows that the curvature steadily decreases to zero and that the process of flattening is accompanied by two events: (1)

When spike is formed, the curvature slightly increases and shortly after decreases again.

(2) The curvature increases significantly over a short time, when the „side shocks‟ from the reverse jets converge at the bubble center. These shocks were observed in Ref. [27] and also called „transverse shocks‟. They can be seen also in Figures 14. After these side shocks pass the bubble, the curvature value vanishes quickly fast and remains solidly at zero.

Therefore, we find that in the highly nonlinear regime of strong shock driven

RMI the flows fields at small scales are non-uniform, and these heterogeneities are volumetric in nature. The flow dynamics is adjusted to the motion of the interface, and has pronounced heterogeneities at small scales in the fluid bulk.

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Figure 14: Density field (left image) with range from g/cm3 (blue) to g/cm3 (red) and pressure field (right image with range from Pa (blue) to Pa (red) for RMI with , ,

at .

7.1.3.4.3 Structure of the velocity field at the interface and in the bulk

In canonical turbulent flows, shear-driven instabilities induce velocity fluctuations that, in turn, drive density and temperature fluctuations and lead to flow homogenization from large to small scales. Our SPH simulations indicate that in RMI driven by strong shocks the late-time dynamics remain heterogeneous with local microstructures appearing at small scales.

Figure 15 represents the sequence of snapshots of the velocity component in the direction of shock propagation (left) and in the normal direction (right) in the frame of reference that moves with velocity . The color bar is in cm/s and the value ranges are given in the figures. The case is represented of , , with

µs, and snapshots are taken at µs. The shock propagated from the left (light fluid) to right (heavy fluid). Figure 15 on the left clearly shows the transmitted and reflected shock regions. Very little motion in the bulk of the light fluid is due to

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weak compressibility. Small-scale regular patterns in the bulk of the heavy fluid (to the right) are due to reverse jets. The reverse jets are generated at the base of the spikes between the bubbles. Secondary reverse jets are generated further in the flow in the base of bubbles between the spikes. Velocity fields have a checkerboard pattern.

In Figure 15 on the left green large „spots‟ correspond to spikes and large red

„spots‟ correspond to bubbles. Reverse cumulative jets are protruding high-velocity regions between the bubbles. The jets start to develop at early times of RMI, immediately after the shock refracts the interface, and they become well pronounced at late times. The jets are generated by two converging flows, that are induced by the large amplitude perturbation, and that collide between the bubbles and accumulate high momentum mass regions developing later into a jet, similarly to [41]. Secondary reverse jets develop further in the flow.

Figure 15: Snapshots of the velocity component fluid particle velocity component in the direction of shock propagation (left) and in the normal direction (right) in the frame of references moving with velocity . Color bar is in cm/s. The case is , , with time-scale µs. Snapshots are taken at µs. Shock propagated from left (light fluid) to right (heavy fluid). Reverse jets are generated in the bulk of the heavy fluid (to the right) at the base of the spikes between the bubbles. Secondary reverse jets are further generated in the flow in the base of bubbles between the spikes.

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Figure 15 on the right shows the velocity component in the direction normal to shock propagation. The velocity component has a checkerboard pattern that occurs in the regions where the vortical structures form mushroom-type shapes of the sides of the spikes. At later times, when the mushroom structures occur in the reversed jets (see

Figure 8), the smaller checkerboard pattern begins to develop between the bubbles in the region where the reverse jet (and secondary reverse jets) occurs. The reversed cumulative jets were observed experimentally [26, 27]. Our numerical simulations results agree with these observations qualitatively and provide quantitative information on the jet structure and evolution.

Therefore, for strong-shock induced Richtmyer-Meshkov flows the dynamics at small-scale is heterogeneous, and is characterized by appearance of local microstructures, reverse cumulative jets and hot spots. The heterogeneities occur in both vector and scalar fields and lead to a complex character of the scale coupling. Our results indicate that at late times of flow evolution the interfacial mixing keeps a certain degree of order and remains sensitive to the initial conditions.

7.1.4 Discussion and conclusion

We performed the systematic study of the Richtmyer-Meshkov instability induced by strong shocks, and employed SPHC simulations to explore the parameter regime of the high Mach number, high Atwood number and large initial perturbation amplitude. In the broad parameter space the SPHC results were repeatable and qualitatively similar. Good agreement was achieved between the numerical simulations and the existing theories and experiments. These include dependence of RMI growth- rate on the initial perturbation amplitude, development of the KH instabilities, formation

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of vortical structures on the sides of evolving spikes, deceleration and flattening of the bubble front [11, 14, 15-23, 26-29, 36-40].

Our simulation results agree (with over 99% accuracy) with the zero-order theoretical analysis [1, 2, 17, 35], describing the post-shock background motion of the fluids separated by a planar interface. They also agree (within 4%) with the linear theory

[17] providing RMI growth-rate in a broad range of the Mach and Atwood numbers.

Similarly to experiments and simulations [16, 39], the SPHC simulations agree (within

20-30%) with the weakly nonlinear theory [15] accounting for the effect of the initial perturbation amplitude on the instability growth-rate under assumptions that secondary shocks do not influence early-time RMI. They also agree (in a proper parameter regime) with the highly nonlinear theory [19-21] describing the evolution of RM unstable front for incompressible fluids without sources. Theoretical analyses [15-21] do not apply adjustable parameters. Our approach to the problem is thus well justified.

Our results suggest that the velocity of the background motion v is an important  parameter as it quantifies the amount of energy deposited by the shock to the fluids,

Figure 2, Tables 2, 6. Depending on Mach and Atwood  numbers of the flow, the background motion can be sub-sonic or super-sonic [15-17]. An important feature of strong-shock-induced RMI is the supersonic character of the background motion,

, in agreement with experiments [38-40]. The ratio between the growth-rate

and the velocity provides an estimate for the portion of energy available for the interfacial mixing. In our simulations the ratio is in the entire parameter space, Table 6, indicating that significant part of the shock energy goes into the compression and background motion of the fluids, and only a small portion remains for

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the interfacial mixing. This suggests that accurate quantification of RM dynamics in experiments and simulations is a challenging task requiring substantial span of highly resolved scales and a careful separation of the interfacial dynamics from the background dynamics, Figure 2, [16, 27, 38-40].

Depending on the Mach and Atwood numbers and the initial perturbation amplitude, the interface perturbations may grow sub-sonic or super-sonic. In our parameter regime the growth-rate is sub-sonic, , except for the cases of very high v 0 cl Mach number and large initial amplitude, Tables 5, 6, Figure 1. Furthermore, for small

initial perturbation amplitude one has similarly to the case of weak shocks [20]. According to our results, and in agreement with the theory and experiments

[15-21, 22-24, 27, 28, 38-40] the initial perturbation amplitude is one of key factors of

RMI evolution. For small initial perturbation the growth-rate increases with the amplitude as , and for larger amplitudes the increase is slower than linear,

Tables 5, 6, Figure 1. In addition to influencing the interfacial dynamics, large initial amplitude may also influence the dynamics in the bulk and the transmitted shock, in agreement with experiments [38], Figures 3, 4. For an optimal use of experimental and computational resources, there is a strong need in an integrated theoretical description of

RMI evolution from early to late times [15-21] for small and finite initial amplitudes.

When RMI progresses quickly and the interfacial dynamics is weakly compressible, then in agreement with analysis [19-21] and experiments [27, 38-40], our simulations report the flattening and deceleration of the bubble front and a quick decay of its specific drag force, Figures 5-7. At late times, the evolution of RMI is a multi-scale process, the interfacial motion tends to be inertial, and the dynamics remains laminar

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rather than turbulent. Furthermore, special efforts should be undertaken to trigger flow transition to a turbulent state, Figures 8-12. The later can be addressed by means of careful choice of the parameter regime, in particular, initial perturbation [11, 14, 15-23,

26-29, 36-40]. Fairly high particle resolution in our simulations and particle distribution optimization techniques [12, 13, 32-34] both allowed the observation of the reverse cumulative jets accompanying RMI evolution [41]. Similar structures were observed in experiments [26, 27]. According to our simulations, the jets appear because of collision of the convergent fluid flows in the bulk. While closely resembling the

„mushroom‟ shape, the jets are not interfacial structures. Being short and energetic, these jets induce small-scale heterogeneities in the flow fields, such as checkerboard patterns in the velocity field, and are accompanied by local hot spots – microscopic structures with temperature substantially higher than that in the ambient, Figures 13-15.

The SPHC simulation data contain complete information on the structure and evolution of the interface and on the flow fields in the bulk. Development of diagnostic approaches advancing our capabilities of data post-processing and involving particle ensembles would allow for more detailed quantification of the interface dynamics at early and late times and at both small and large scales. We address this issue to the future.

To conclude, the nonlinear evolution of the large-scale coherent dynamics in

Richtmyer-Meshkov mixing is an essentially non-local and multi-scale process. At late stages of flow evolution, the dynamics at small-scale is heterogeneous and is

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characterized by the appearance of local microscopic structures. The coupling between the large and small scales has a complicated character.

Acknowledgments

The authors acknowledge the financial support of the US National Science

Foundation (award 1004330) and the US Department of Energy. The authors express their deep gratitude to Professor Wouchuk for productive discussions and useful comments.

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