COMPUTATIONAL MODELING OF PLUME DYNAMICS IN MULTIPLE PULSE LASER ABLATION OF CARBON
A Dissertation Presented to The Graduate Faculty of The University of Akron
In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
K
Kedar Ashok Pathak
August, 2008 COMPUTATIONAL MODELING OF PLUME DYNAMICS IN MULTIPLE PULSE LASER ABLATION OF CARBON
Kedar Ashok Pathak
Dissertation
Approved: Accepted:
Advisor Department Chair Dr. Alex Povitsky Dr. Celal Batur
Committee Member Dean of the College Dr. Erol Sancaktar Dr. George K. Haritos
Committee Member Dean of the Graduate School Dr. Gerald Young Dr. George R. Newkome
Committee Member Date Dr. Minel Braun
Committee Member Dr. Scott Sawyer
ii ABSTRACT
The flow field induced by the ablation plume in the presence of background gas is simulated numerically. The study of plume flow that occurs in laser ablation is impor- tant for it can yield information on ablation process itself and the properties of end product for which the ablation is carried out. Unsteady compressible axisymmetric Navier-Stokes equations govern the plume flow. The major challenge involved, even in this simplified model of plume dynamics, is twofold: (i) the time scale of simu- lation spans six orders of magnitude, from nanosecond to millisecond, and (ii) the high nonlinearity of governing equations because of high pressure, temperature and injection velocity of plume. A computational model is developed that can account for the entire range of time scale and high nonlinearity. This model is a combination of numerical methods and includes multi-time step and multi-size grid technique. The uniqueness of model lies in choosing the combination of numerical methods and han- dling multi-size grid interface in a conservative way. The combination of numerical methods is decided after comparing the results of few numerical methods for a single plume. The plume dynamics for single plume is explained with the help of proposed post-processing model based on vorticity dynamics. The model not only helps in understanding the expansion dynamics of plume but also provides quantitative com- parison amongst numerical methods. The validity of nano-to-micro second range
iii viscous and inviscid models of plume dynamics is discussed by means of evaluation of source terms in the vorticity transport equation. The role of turbulence is evaluated by millisecond-scale modeling of plume expanding in surrounding furnace gas with imposed turbulent gust. The results for multiple plumes typical for real life ablation are presented and discussed. Shielding of laser beam by previously ejected plume in multiple laser hits is important because it changes energy deposition of incident laser pulse at the target surface and in turn influences the ablation dynamics and amount of material removed. To account for this shielding effect, shielding models are developed and implemented. The quantity of ablated mass due to the shielding effect is evaluated. Ionization of carbon plume and its impact on plume dynamics and shielding is studied. An iterative procedure is developed to determine the local equilibrium temperature affected by ionization. It is shown that though shielding due to the presence of ionized particles in carbon plume is small, the effect of ionization on plume dynamics can be considerable. Shielding effect is calculated for laser pulses with different time interval between pulses. The effect of high temperature and low density of plume are controversary and cause shielding behavior to be non-monotonic with pulse number. It is shown that the non-monotonic dependence of the delivered laser energy with the pulse number and the difference in shielding characteristics between planar and axisymmetric formulations increase with the time duration between two consecutive pulses. The developed numerical methodology is employed to study the heat transfer modulation between the Thermal Protection Shield (TPS) and the gas flow occurring because of ejection of under-expanded pyrolysis gases through the cracks in the TPS in hypersonic flight. The simulations are performed for an axisymmetric bluff body flying at Mach 7. The influence of the geometry of the TPS on heat transfer pattern
iv is studied for two representative shapes. The results are presented for three different flight altitudes (low-ground level, moderate-20km and high-30km). At the low alti- tude the plume pressure is lower than the pressure behind the detached front shock wave and the plume propagates slowly along the wall surface. At high and moder- ate altitudes, the plume path and consequently, convective heat transfer between the TPS and the plume depends on the plume interaction with the bow shock wave. The effect of viscosity for the plume injection conditions and free stream Mach number considered is found to be negligible at simulated altitudes. However, the effect of initial pressure of pyrolysis gas on the plume dynamics is significant. The presence of the blast wave associated with under-expanded plume alters the heat transfer and increases mixing. Finally, the enhanced heat transfer caused by the emergence of multiple pyrolysis plumes is investigated.
v ACKNOWLEDGEMENTS
I would like to thank Dr. Alex Povitsky for giving me the opportunity to work with him. His excellent support and expert guidance has made the seemingly wide topic of modeling laser ablation practical for a novice like me. I am also thankful to Dr. Erol Sancaktar, Dr. Gerald Young, Dr. Scott Sawyer and Dr. Minel Braun for their review of the presented material and sugges- tions. I highly value their critical comments and suggestions for this work. The work content of Chapter IV was partially supported by the Air Force Office of Scientific Research (AFOSR) through research grant FA9550-07-1-0457. Dr. Datta Gaitonde’s comments for this chapter helped to improve the material of this chapter which is gratefully acknowledged. Finally, it remains to thank all my colleagues for their valuable direct and indirect help from point to point through out the years of this research.
vi TABLE OF CONTENTS
Page
LIST OF TABLES ...... x
LIST OF FIGURES ...... xi
CHAPTER
I. INTRODUCTION ...... 1
1.1 Research Motivation ...... 2
1.2 Previous Research ...... 5
1.3 Multiple Plume Ejections ...... 21
1.4 Objectives ...... 22
II. MATHEMATICAL MODELING ...... 26
2.1 Set-up of Problem ...... 26
2.2 Governing Equations ...... 28
vii 2.3 Treatment of Boundary Conditions ...... 30
2.4 Modeling of Turbulence ...... 31
2.5 Numerical Methodology and Validation ...... 33
2.6 Ionization and Shielding ...... 44
2.7 Multi-Time Step Modeling ...... 47
III. RESULTS AND DISCUSSION FOR LASER ABLATION MOD- ELING ...... 52
3.1 Single gamma versus Two gamma ...... 53
3.2 Inviscid, Viscous and Turbulence Plume Modeling ...... 54
3.3 Dynamics of Vorticity ...... 62
3.4 Multiple Plumes ...... 67
3.5 Shielding Effect ...... 71
3.6 Multi-time Step Modeling ...... 73
3.7 Axisymmetric vs. Planar Plumes ...... 77
3.8 Ionization Effect ...... 86
IV. APPLICATION OF DEVELOPED MODEL IN MODELING PLUMES GENERATED BY LOCAL INJECTION OF CAR- BON ABLATION PRODUCTS IN HIGH-SPEED FLIGHT . . . . . 91
viii 4.1 Background and Previous Research ...... 91
4.2 Description of Model ...... 94
4.3 The Effect of the Flight Altitude on Plume Dynamics and Heat Transfer ...... 100
4.4 Heat Transfer for the Range of Plume Injection Pressures . . . . . 108
4.5 Heat Transfer for Multiple Plume Injections ...... 110
4.6 Heat Transfer for Different TPS Shapes of Comparable Sizes . . . 112
V. SUMMARY AND FUTURE SCOPE ...... 115
5.1 Future Scope ...... 121
BIBLIOGRAPHY ...... 123
APPENDICES ...... 133
APPENDIX A. MULTI-SIZE MESH FOR MULTI-TIME STEP MODELING ...... 134
APPENDIX B. GRID CONVERGENCE ...... 137
APPENDIX C. IONIZATION PREDICTION ...... 138
APPENDIX D. TWO-TEMPERATURE MODELING ...... 140
ix
ix LIST OF TABLES
Table Page
1.1 A survey of numerical methods for LIP simulation in laser ablation . . 21
3.1 Summary of operating parameters of the process being simulated . . . . 52
4.1 Flight regimes and parameters of surrounding gas ...... 95
4.2 Plume injection parameters ...... 95
4.3 Value of normalization parameter at different altitude ...... 97
B.1 L1-Norm of the errors for Numerical solution of ut + uux = 0, u(x, 0) = sin(πx) ...... 137
x LIST OF FIGURES
Figure Page
1.1 Schematic of experimental set-up for laser ablation ...... 6
1.2 In situ imaging and spectroscopic diagnostic investigations of SWNT’s growth (Taken from Ref. [1] with author’s permission) . . . . 8
2.1 Problem schematic a) process of laser ablation showing the expand- ing plume and propagating shock wave and b) set-up of geometry and boundary conditions; b1 is inflow, b2-b3 is solid wall, b4 is outflow and b5 is target condition ...... 27
2.2 Fluctuating velocity components a) Horizontal b) Vertical ...... 33
2.3 Flow chart of the numerical code ...... 34
2.4 Structure of the solution of the considered Riemann problem ...... 38
2.5 Density contours for the double Mach reflection problem, ∆x = ∆y = 1/120: a) the Godunov method b) the Relaxing TVD method and c) the ENO-Roe method ...... 39
2.6 Density contours for the forward facing step problem, ∆x = ∆y = 1/80: a) the Godunov method b) the Relaxing TVD method and c) the ENO-Roe method ...... 40
xi 2.7 Initial/boundary conditions for shock flowing over backward facing step problem ...... 41
2.8 Density contours for shock flowing over backward facing step prob- lem, ∆x = ∆y = 1/300: a) the Godunov method and b) the Relax- ing TVD method c) the ENO-Roe method ...... 41
2.9 Normalized velocity distributions as obtained for boundary-layer type flow by a) the Godunov method and b) the ENO-Roe Scheme . . 43
2.10 Multi-size mesh for multi-time step modeling: a) schematic showing grid sizes for coarse and fine girds (L and D are furnace length and diameter, respectively) and b) computational domain with bound- ary conditions ...... 48
2.11 Calculation of Flux F (U) a) at time t = 0 b) at time t = dt/2 c) at time t = dt ...... 49
3.1 Plume concentration at time t = 50µs: a) species are assumed to have the same λ and b) two gamma, Ar and C3 with their individual γ and molecular weights ...... 53
3.2 Plume concentrations obtained by the Euler equations at t = 300µs a) the Godunov method, b) the Relaxing TVD method and c) the ENO-Roe method ...... 54
3.3 Vector plot obtained by the Euler equations at t = 300µs a) the Godunov method, b) the Relaxing TVD method and c) the ENO- Roe method ...... 55
3.4 Contours of pressure obtained by Euler equations at t = 300µs a) the Godunov method, b) the Relaxing TVD method and c) the ENO-Roe method ...... 56
xii 3.5 Comparison of plume concentrations at t = 100µs a) experimental, b) numerical NS and c) numerical Euler ...... 57
3.6 Propagation of the leading edge of the ablation plume ...... 58
3.7 Plume concentrations obtained by NS equations at t = 300µs a) the Godunov method, b) the ENO-Roe method and c) the ENO-Roe method with turbulence ...... 59
3.8 Comparison of plume concentration at t = 2 ms for a) viscous and b) turbulent models ...... 60
3.9 Trajectory and temperature of injected particle as a function of time for t ≤ 300µs a) Euler model b) NS model at and c) temperature of particle for NS model ...... 61
3.10 Evaluation of sources in the vorticity transport equation a) baro- clinic vorticity generation and b) viscous generation of vorticity for the ENO-Roe method ...... 64
3.11 Mass flux at the near-boundary point ...... 66
3.12 Plume concentrations at t = 300 µs a) the Godunov method, b) the Roe method, and c) combination of the Godunov and Roe methods . . 67
3.13 Plume concentrations at t = 100 µs a) single plume ejection, b) multiple plume ejections ...... 68
3.14 Vorticity source a) single plume ejection, b) multiple plume ejections . . 68
3.15 Pressure contours at for multiple plume ejections ...... 69
xiii 3.16 Comparison of Euler (left) and NS (right) for multiple plume ejec- tions; concentration (top) and vector field (bottom) ...... 70
3.17 Ablated mass for different time interval between two successive pulses for pressure based and velocity based models ...... 72
3.18 Density contours after ablation of the 4th plume with different time interval between two laser pulses a) τ = 1 µs b) τ = 15 µs ...... 72
3.19 Density contours for configuration 4 (Ref. [2]), 2D Riemann problem . . 74
3.20 Plume concentrations before the start of second pulse and the multi- size grid ...... 75
3.21 Energy reaching the target with increasing number of pulses ...... 76
3.22 Multiple-pulse plume modeling using the multi-size grid: a) plume concentration and b) temperature before the start of the 11th pulse . . 77
3.23 Plume patterns for planar (a,c) and axisymmetric (b,d) cases: (a,b) plume concentrations and (c,d) axial component of velocity along the centerline ...... 79
3.24 Propagation of pressure wave in plume: a) planar case, b) axisym- metric case and c) pressure along the center-line ...... 80
3.25 Plume patterns and temperature after the 7th laser pulse with the time interval τ = 5µs between two plumes: (a,c) planar case; (b,d) axisymmetric case; (a,b) plume material concentrations; (c,d) tem- perature along the center line ...... 81
3.26 Degree of laser beam shielding for different time intervals between two pulses: a) t = 50ns; b) t = 1µs; c) t = 5µs and d) t = 10µs . . . . 83
xiv 3.27 Dynamics of temperature of a representative plume particle for mul- tiple pulse laser hits: a) t = 50ns and b) t = 1µs ...... 84
3.28 Density and temperatures contours after first two pulses: (a,b) den- sity, (c,d) temperature; (a,c) after the second pulse; (b,d) after the first pulse ...... 85
3.29 Degree of ionization ζ and temperature before ionization after the fifth pulse: (a,c) planar; (b,d) axisymmetric; (a,b) degree of ioniza- tion ζ; (c,d) temperature K ...... 86
3.30 Degree of ionization ζ and δ (Eq. 2.29): a) degree of ionization as a function of temperature at 100 atm and b) δ Vs. T ...... 87
−3 3.31 Contours of number density of ions (ζ × Ng) in cm and tempera- ture after ionization shown after the fifth pulse: (a,c) planar; (b,d) axisymmetric; (a,b) ion density; (c,d) temperature ...... 89
4.1 Schematic of the problem for flight at the ground level: a)the shape of the representative TPS and computational domain, b) plume concentration, c) pressure distribution, and d) heat transfer at the TPS wall ...... 98
4.2 Comparison of viscous and inviscid models for flight at the altitude of 20km at t ≈ 45µs: a) plume concentration obtained with Eu- ler equations, b) plume concentration obtained with NS equations and c) heat transfer coefficient obtained by inviscid model, d) heat transfer coefficient obtained by viscous model ...... 99
4.3 Plume concentration at flight altitude of 20 km: a) original velocity vector field at t ≈ 20µs, b) original velocity vector field at t ≈ 25µs, ~ ~ c) modified vector field V − Vref at t ≈ 20µs and d) modified vector ~ ~ field, V − Vref at t ≈ 25µs ...... 101
xv 4.4 Plume density and temperature at t ≈ 20µs (left) and t ≈ 25µs (right) : a-b) temperature, c-d) density ...... 102
4.5 Flowfield near the stagnation point: a) pressure contours after plume injection over the body in still air and b) temperature contours along with the vector field at altitude of flight of 20 km at t ≈ 10µs . . . . . 103
4.6 Plume dynamics at the altitude of flight of 20km: plume concentra- tion at a)t ≈ 10µs, b)t ≈ 22µs, and c)t ≈ 30µs and d) heat transfer at the TPS surface ...... 104
4.7 Plume dynamics at the altitude of flight of 30km at t ≈ 25µs: a) plume concentration and b) temperature contours ...... 105
4.8 Interaction of waves and heat transfer at the altitude of flight of 30 km: a) pressure contours (t ≈ 22µs) and b) heat transfer at wall . . . . 106
4.9 Formation of secondary shock waves: pressure contours at; a)t ≈ 32µs, b)t ≈ 41µs, and c)t ≈ 50µs and d) heat transfer at the TPS wall for all considered instants of time ...... 107
4.10 Heat transfer with inset image of plume concentration at t ≈ 30µs for the range of initial plume injection pressures: a) 8 atm, b) 16 atm and c) 24 atm ...... 109
4.11 Emergence of two consequent plumes at the altitude flight of 20km. The plume material concentration is shown at the following time moments; a)t ≈ 40µs, b)t ≈ 50µs and c)t ≈ 60µs and d) heat transfer at the wall ...... 111
4.12 The effect of TPS shape at the altitude of flight of 20km: a-b) density contours at t ≈ 10µs with inset image showing plume concentration and modified velocity vector field, and c-d) heat transfer at the TPS wall.113
A.1 Multi-size mesh generated for multi-time step modeling ...... 136
xvi C.1 Degree of ionization as a function of temperature ...... 139
xvii CHAPTER I INTRODUCTION
Laser ablation refers to removal of material from a target surface that is irradiated with high-intensity laser pulse. The term ‘ablation’ originates from the latin ‘abla- tum’, which means taken away [3]. Entire process of laser ablation involves many phenomena such as molecular bond breaking, non-equilibrium electronic excitations, multi-photon or saturation processes, gas dynamics, condensation and nucleation etc. In general, the time scales that span in laser ablation range from less than 5 fem- toseconds (10−15s) to several milliseconds. The field of high-intensity laser interaction with matter is two decades old and is still a growing field that is bursting with enough important phenomena, which are still to be understood by researches [4–6]. Out of above mentioned phenomena, it is believed in the laser ablation community that the phenomenon of gas dynamics of ablated plume material is important from the point of view of controlling the end product yield, when ablation is carried out in presence of background gas at near-atmospheric pressure. Ablation in presence of background gas has numerous applications such as nanoparticle manufacturing, thin film fabrication, surface patterning and medical applications. Among the applications the synthesis of carbon nanotubes is the state- of-art application. Typically the pulse duration for such an application is few nanosec- onds with fluence values within the range 1 − 10 J/cm2. In the current study, atten- tion is focused on nanosecond laser ablation. Currently, although the entire process of nanosecond ablation is not fully understood, a general consensus has been reached about its overall features. These features are as follows:
1 1. Absorption of laser light by excitation of electronic or vibrational modes that leads to heating
2. Emission of photoelectrons, ions, atoms, molecules or even clusters into an incipient plume
3. Plume expansion
4. Shielding of next laser beam by prior plumes
The first phase is governed by the optical physics of the irradiated solid. The second and third phase, i.e. the formation of an ablation plume, a weakly or strongly ionized, low or moderate density expanding gas, adds to laser ablation the complication of plasma physics, gas dynamics and laser-induced photochemistry [7]. Again, out of these four phases discussion on second, third and fourth phase is focused in this dissertation. Gas dynamics of ablation plumes in background gas that is typical for production of carbon nanotubes is explored.
1.1 Research Motivation
Laser ablation is proving to be a promising technique for many applications such as pulsed laser deposition [8], micro machining [9], medical applications [10], material analysis [11–13] and production of carbon nanotubes [14]. On the one hand, increased applications encourages advancements in experimental techniques in this field; on the other hand, advancements in the experiments reveal new phenomena occurring in the process. For example, recent spectroscopic studies with fast imaging techniques [15, 16] has advanced a knowledge of plume slowing down in presence of background gas and formation of shock wave. In a more rigorous experiment [17], aimed at laser ablation of carbon for single-wall nanotubes (SWNT) synthesis, formation of plume roll up is observed. It was assumed that this plume forming into a vortex 2 and propagating down is a result of viscous interaction between plume particles and ambient gas. In spite of increased number of experiments, some of the parameters such as velocity and temperature of plume particles and their variation is an open question and plume expansion dynamics is not fully agreed upon in the community. This is due to the fact that experimental techniques have their inherent weaknesses toward measuring the parameters of plume. One of the weaknesses for optical emission spectroscopy (OES) technique and fast imaging technique that are widely used for measuring plume parameters is presented below. OES technique, which is used to measure plume temperature in its early stage of evolution, rely on the intrinsic light emission of the plume. Because the evaluated temperature depends on the model employed in the analysis of this emission spectra, the temperature values are not very accurate. Besides, Doppler effect and the apparatus function of the entire spectroscopic system also contribute to emission spectra [18]. At high background pressures, due to confinement of plume in a smaller volume, strong spatial gradient of the temperature and of the atom, ion and electron concentrations exist in the plume. This leads to the formation of strongly asymmetric self-reversed emission lines. In this case, only by theoretical modeling it is possible to retrieve information from the experimental spectra [19]. In fast imaging, which is best suitable for later stages of plume evolution, the temperature of plume particles is evaluated by measuring their blackbody emission (incandescence) spectra at different times and positions in the furnace. This spectra is not accurate because for each distance a time delay is observed in temporal evolution of line emission. Interpretation of such experimental data needs assumptions. Theoretical studies can prove to be powerful tool in verifying the validity of such assumptions. Experiments provide valuable information. But if this information is not supported by proper theory, then it may render itself in parts and pieces and any
3 complete description of the process remains far from reality. For instance, in one of the experiments [20], it was observed that the ambient pressure and laser intensity have opposite effects on the expansion dynamics of the laser ablated plume. In other experiment [21], it was found that the ablation rate decreases with increasing ambient temperature, however, no physical explanation could be provided in any of the above cases. Further, for every new material as well as new set of input process parameters such as laser parameters, furnace condition, gas flow settings etc. [22] requires state-of-art optimization condition. In such cases, computational and theoretical studies are needed to supplement the experiments and the interpretation of experimental results can be improved by theoretical modeling of plume expansion. Such models can provide a sound basis for the analysis of the structure of the laser- induced plume (LIP) and of the effects of the ambient conditions on laser ablation. Thus, understanding of variation in plume expansion dynamics with above mentioned set of input parameters can give insight into as to how the property of end product yield can be controlled. Some modern applications of laser ablation such as nanotube formation or thin film deposition is carried out with a cross-flow as explained in the experiments [23]. This ‘side-pumped’ geometry was developed after it is discovered that the con- ventional front pumped counterflow geometry was ill suited for high target ablation rate. This experimental feature needs 3-D numerical simulations. The current model of axisymmetric Navier-Stokes equations may serve as a benchmark case for more elaborated set-ups involving 3-D features such as cross-flow. The current model can justly accommodate few of the features of many of the applications of laser ablation. For example, with deep holes or grooves, the ablation rate may strongly decrease with increasing pulse number. In this application, as can be seen from a simple es- timation, the attenuation of incident laser beam can be significant. The velocity of
4 ejected species out of the hole is ∼ 105 −106 cm/s. During a laser pulse of 20 ns, these species can travel ∼ 20−200 µm. Thus, for deep holes/grooves the attenuation can be significant. Similarly, for film deposition and synthesis of nanomaterial that involves multiple laser pulses the attenuation of laser beam by previously ejected plumes is important. The current model of shielding can capture this attenuation. The trans- port of ablated material, the attenuation of laser light and target surface corrugations are sensitive to furnace gas pressure. Influence of an ambient atmosphere on laser ablation can be easily implemented and checked in the current numerical set-up.
1.2 Previous Research
In this section the experimental and theoretical investigations on the expansion of LIP is reviewed. Here, as far as experimental investigations are concerned, the emphasis is put on general trends and not on details. It is intended to give an overall idea about experiments in laser ablation and review the most of theoretical models that are available. Numerical methods that makes the basis for theoretical studies are also discussed.
1.2.1 Experimental Investigations
Experimentally, the dynamics of the shock wave and the expansion of plume have been studied by various techniques. These are different types of space-time-resolved measurements for analyzing the plume evolution. The typical experimental set-up consists of a laser with an optical system in order to steer and focus the laser beam, a furnace, a system for base gas feeding and an optical emission spectroscopic system, as depicted in Fig. 1.1. The essential features of the set-up are depicted in Fig. 1.1. The choice of laser depends on the properties of target to be ablated. The energy density of the laser
5 Figure 1.1: Schematic of experimental set-up for laser ablation can be varied by changing the degree of focusing on the target and the laser energy output. Energy meter is used for laser energy detection and monitoring by directing a portion of the laser beam into it. The simple furnace has a cylindrical geometry. It is usually equipped with a rotational holder for the targets, ports for the laser beam, gas pumping out and feeding, pressure gauging and view ports for the OES. The emission spectrum from the plume is detected by two different methods. Time-integrated OES is utilized to identify chemical species generating in the plume. Spatial-temporal OES resolution is used to investigate the time evolution of a spectral line intensity and broadening. A high resolution monochromator is used to record a spectral distribution of the line intensity in emission spectra, in a specific range. It is equipped with an intensified charge coupled device (ICCD). The detection of the ICCD output signal is accomplished by a programmable pulse generator, connected to a personal computer for data acquisition and processing. A fast photo diode detects a portion
6 of the laser beam to avoid any fluctuations in the laser and spectroscopic systems and to trigger the pulsed generator. An oscilloscope and a photo diode together calibrates and controls the gate width and time delay after the laser irradiation. An optical fiber permits vertical and horizontal micro-movements to collect light emitted from different probe volumes of the plume. The aperture of the optical fiber is aligned with the centerline of the plume to ensure that the emission signal is collected perpendicularly. More about experimental details can be found in references [15, 24– 27]. The experiments carried out to study LIP expansion as observed in synthesis of carbon nanotubes and thin film fabrication, in which cases gas dynamics of plume is important, are reviewed below. The overall expansion and shape of the plume has been investigated by time- integrated photography [28, 29]. The dependence of plume pattern on laser fluence, spot size, and pressure of the ambient gas was studied there. At higher-than-vacuum furnace pressures, these studies found that the expansion of plume resembles a blast wave driven by the high pressure. High pressure is generated to accommodate the large amount of ablated mass. Study [30] has shown shadow graph images of species and shock wave. It is suggested there that through the interaction of plume with furnace gas the dissipation of the kinetic and thermal energies of the plume species, its backward motion, and its collisions with each other occurs that lead to the formation of clusters and particles. In the papers [1, 31–35], the authors have studied the plume dynamics in detail. They study plume evolution for several seconds up till actually no movement of plume particles are found. They show actual images of the laser plume (t < 200 µs) and Rayleigh-scattering images of the plume (t > 200 µs) vs time. This evolution of plume as captured by [1] is shown in Fig. 1.2. In this studies it was found that plume rolls itself in vortex ring and this vortex motion efficiently traps aggregated nanoparticles in a confined volume for long
7 Figure 1.2: In situ imaging and spectroscopic diagnostic investigations of SWNT’s growth (Taken from Ref. [1] with author’s permission)
8 times. This plume roll up can seen at ∼ t = 100 µs in Fig. 1.2. In the study [34], it is observed that the shock wave that travels through the plume results in formation of vortex ring of the ablated material. A typical result of vortex formation is the characteristic shape of mushroom clouds which are formed by ground-based explosions in an atmosphere. The formation of mushroom clouds was even observed in an experiment [36] in which the background pressures were relatively low (.014−.14 atm). Paper [37] also studied gas dynamic aspect of the synthesis of SWNT growth. The time-scale of growth of SWNT and optimal conditions such as, pressure, temperature and flow rate of background gas are explored there. Another experiment that involved time-resolved photography to study the shock wave and expansion of plume is [38]. Here the measured temperature of the shock front is measured in the interval 2500 − 4000 K. It is found there that the shock wave is already present 10 − 15 ns after the end of the incident laser pulse. Gas dynamics of polymer ablation is studied by [39] with nanosecond range laser pulse. Their time-resolved photography revealed that particles forms Knudsen layer (KL) (KL is a layer in which species transform to propagate from non-equilibrium to equilibrium state) and then they pass into unsteady adiabatic expansion (UAE). Their results give explicit support to the gas dynamic description of the problem, and indicate that the KL-UAE model is appropriate. In a novel technique used by [23], a spinning target and a preheated shearing Argon gas jet is used to clear away the nanotubes from the laser beam path. This jet not only helps preventing shielding of laser beam by ablated plume, but it also helps in controlling the quality of nanotubes. In a similar technology [40] that applied a swirling jet, it was shown that the velocity distribution of the swirling flow on the substrate was significantly affected by the swirling strength of the flow and that the plasma plume was removed efficiently and the surface roughness on substrate was
9 significantly reduced by the implementation of swirling flow in laser ablation. The swirl in the flow was generated by directing a tangential air stream on the main flow stream. Thus, these experiments support that the gas dynamic are important for achieving and controlling the quality of end product yield. As the field of laser ablation is flourishing there are numerous experiments being done. Few of them are listed above that gives overall idea and understanding reached about its various features. In the next section the theories that are developed to support and explore the evidenced features are presented.
1.2.2 Theoretical Modeling
The growing interest on LIP as well as the motivation to improve the basic knowledge and determine the best experimental conditions of laser induced plume processes have generated numerous theoretical models in this field. As mentioned earlier, although the main emphasis is put on the expansion dynamics of plume in nanosecond abla- tion, references and comparison to results of plume properties for different range of experimental parameters is also given. One of the review paper concerning plume formation and plume expansion is [41]. According to this paper, the expansion of ablated material, starting when the laser pulse is on and continuing after the end of the laser pulse, can be simulated by means of essentially three different approaches: (i) analytical, (ii) numerical solutions of continum based PDEs and (iii) Molecular methods. These three approaches are discussed below.
• Analytical modeling Analytical modeling mainly aims at finding plume properties and plume shape analytically once the initial plume conditions are given. They are based on one-dimensional (1-D) gas dynamic equations [42]. Full details of the analytical modeling is not given; rather a brief idea is presented here. The expansion
10 dynamics of the plume, R = R(t), is determined by the energy conservation E = const. The total energy of the plume, assumed to be consisting of the thermal energy, Et, and the kinetic energy, Ek, is described by
R a E = E + E = 0 E + ςM R˙ 2 (1.1) t k R p where R0 is the initial radius of the plume. Constant a depends on plume material. The coefficient ς can be obtained by spatial integration over plume ˙ region. Mp is the total mass within plume. R is the expansion velocity at the leading edge. Eq. (1.1) is valid when plume undergoes expansion in vacuum. Propagation of plume in gases with significant density requires simultaneous consideration of both the propagation of the shock wave and the ablated plume. Eq. (1.1) is modified for such an application by adding a third term on right hand side which is the kinetic energy of the shock wave. Once R = R(t) is obtained the calculation of pressure and density follows from the gas dynamic equations.
The expansion of plume in vacuum is treated analytically in [43, 44]. Study [43] accounts for the temperature gradients while [44] considers no spatial variation in plume temperature i.e. ∇T = 0 and no explicit solution of gas dynamic equations is given. Efforts have been made in both of these papers to calculate the profile of a film produced by pulsed-laser deposition from a solid target. In the presence of ambient gas, the expansion dynamics of plume is treated analytically by [42]. Here, the plume is considered to act like a piston that compresses and thereby heats the ambient gas causing external shock wave. Based on energy conservation and shock wave relation the plume propagation is described. However, it does not account for reflected shock wave that can
11 be present in a confined space of ablation furnace. Some authors [45–47] have considered the effect of KL in their analytical approach. They solve the hydro- dynamic equations for an unsteady adiabatic expansion with correct boundary conditions. The origin of these boundary conditions stems in [48, 49].
In reality, the plume injection boundary condition for the ablated plume is not a priori defined; instead, plume injection conditions depend on the amount of laser energy delivered to the target. This amount of laser energy is ultimately responsible for total mass removal from the target. There are theories that predict this mass removal from the target surface. Anisimov [43, 48] suggested a trimodal ansatz molecular distribution function that estimates the thickness of Knudsen layer. Based on this method Ytrehus [48, 49] was able to find the maximum injection velocity beyond the Knudsen layer. The classical Hertz- Knudsen formula can also calculate the mass flow rate at an evaporating target surface [7]. These theories are based on one-dimensional ‘columnal’ plume and near-equilibrium assumption, which is not the case with short high-intensity laser pulses. In the latter case, the plume ejected from the surface may have non- Maxwellian distribution. If analytical models being insufficient to calculate the mass removal for the high intensity ablation, experimental values of velocities and pressures available for single plume are used. For example, the theoretical carbon injection velocity value is adjusted in [50] to match their CFD plume pattern to the experimental plume pattern of [32].
• Numerical solutions of continum-based PDEs Analytical models can not be used for an application that involves longer time interval of plume evolution, such as synthesis of SWNT [34, 37] and film depo- sition [28, 51]—because of their inherent assumptions, for example, adiabatic plume expansion [43] and no mixing between the expanding plume and ambient
12 gas [42]. Numerical models allows calculating spatial and temporal evolution of quantities such as density, temperature, pressure and velocity for a longer pe- riod of time of plume evolution without such assumptions, which can be directly compared with experimental results.
Amongst available models the most rigorous model that describes the plume expansion is three dimensional compressible Navier-Stokes equations for a multi species gas[52]. This model assumed ideal-gas for each species and included two- temperatures, electron temperature and particle temperature. It is shown by these authors that the solution of Navier-Stokes equations is necessary only for expansions in a background gas at non-negligible background pressures or for expansion at longer times. This model, however, considers plume expansion up till about the first 10µs and do not consider the plume expansion for longer run when the plume evolution is affected by the reflected shock waves.
Euler equations can be used to simulate the process in the early stage of the evolution. Euler equations have been extensively used in Refs. [53–59]. For instance, Ref.[55] have described the plume expansion through the compressible and non dissipative conservation equations for mass, momentum and energy. Moreover, study [55] introduces the energy equation for electrons, because the electron component of the plasma plume gains energy from the laser field, so that the electron temperature may be different from the atomic one. This study have calculated the shielding of laser pulse by a plume and have found their results in fair agreement with experiments. They were able to capture the shock wave but their simulation is for less than 10µs. Besides, inviscid Euler equations are solved, so the probable vortex formation of plume may not be feasibly captured by this model. The authors of the study [53] have solved three-dimensional Euler equation including the effect of ionization with
13 the aim to elucidate the spatial and temporal evolution of plume over whole interaction time from expansion start to plume decay. Their simulation covers plume evolution up till ∼ 140µs, but the vortex formation of plume has not been captured. In the study [60] the same authors give an analogy that predicts this vortex formation at the plume periphery. They propose to capture this vortex formation, which is due to the viscous effects, by numerical modeling of a flow field using the 2D Navier-Stokes equations.
In a study [56], the authors have investigated the transient, multidimensional radiation transport in the ejected metal vapor and its effect on the vapor hydro- dynamics. They showed that the hot vapor layer region is under high pressure, generating a compression wave pushing the evaporating particles back to the target surface, and a shock wave is formed that expands into the ambience; but they did not analyze the shock-plume interaction. In the solid-vapor interface modeling by [58] showed the evaporation wave results in a discontinuity similar to a shock wave. They solved the equations of mass, momentum and energy conservation, in the discontinuity layer that is in between solid and vapor phase, to provide boundary conditions for the expansion process of high temperature and high pressure vapor. The expansion process is modeled using Euler equa- tions. They calculate the shielding effect of the plume on incident laser pulse. They show that the expansion of LIP is predominantly one-dimensional. They discussed the effect shock propagation on plume structure. However, their sim- ulations are limited by the time interval of 20 ns.
From the point of view of carbon nanotube synthesis and film growth, it is of crucial importance to know the dynamical behavior of the plume of ablated material. When a powerful laser illuminates the target it explodes the target surface. The resulting explosion produces a plume of rapidly expanding gaseous
14 carbon with embedded catalyst particles. Experiments [17] have shown that the initial gas pressure in the ablated plume can exceed 100 atm that produces shock waves in the ambient gas. The propagation of incident and reflected shock waves in a confined space of laser furnace affect the plume behind the incident shock wave. These effect of furnace geometry on plume evolution has been studied by [61]. The same authors in an another study [62], obtained the temperature of the catalyst particles in plume, which is crucial for modeling of growth of nanotubes. They show the influence of chamber pressure, injection velocity, and periodicity of the plume on the temperature of particles, and discuss physical reasons for non-monotonic temperature behavior of catalyst particles. Their study included multiple plumes, however, they did not consider shielding effect and assumed the same injection velocity, as suggested by [50] for all plumes. Their numerical method [63] was robust to account for high injection velocities. However, their numerical method was dissipative compared to the numerical method used in the current study.
Quite different approaches to simulate the LIP expansion have been proposed [64, 65]. The model in [64] is based on a combination of multiple scattering and hydrodynamic approaches. The plume is allowed to be broken into scattering orders, whose particles can undergo many collisions with the background. These orders of plume are based on the number of collisions with the background gas. The scattered density is estimated based on particle collision cross-section. The total plume density is the sum of density in all orders. Particles can only be transferred from one order to the next higher order by collisions. The densities in the individual orders propagate according to the usual conservation equations to give the overall plume expansion. The authors [65] starts from the kinetic equations for a gas with consideration of various processes of encounters be-
15 tween the two species such as, self-collisions, interspecies collisions, ionization, recombination, and charge exchange; and derives the hydrodynamic equations of motion for partially ionized plume when the charged component and the neu- tral component have different flow velocities and different temperatures. They show that the effect of elastic collisions, can compare to—and even dominate over—the effect of charge exchange reactions. These approaches are useful for modeling the expansion of LIP into low pressure gases where initially the mean free path is long enough for penetration of the plume into the background gas. When the background pressures are near vacuum, molecular methods are used to simulate the expansion of LIP. These methods are reviewed below briefly.
• Molecular methods As said earlier the entire process of laser ablation is a complex multi scale phe- nomenon. The processes that occur in it are at different time and length scales. One of the method proposed to simulate the initial processes in laser ablation is molecular dynamics (MD) [66, 67]. In MD simulations each molecule or an appropriate group of atoms is represented by a single particle and it has specific degrees of freedom that will allow the conversion of energy of molecule excited by the laser to the translational motion of other molecule. MD simulations are feasible only for initial few nanoseconds because of high demand of resources. Simulation of these ablated particles for further few microseconds, in low pres- sure background gas, can be carried by means of Direct Simulation Monte Carlo (DSMC). Because of the high memory requirement of MD calculations the in- put number of molecules is lower than real experimental conditions. For the material with low laser light absorption these can be more severe. For these reasons MD simulations permit little comparison with experimental data [66].
16 DSMC is mainly based on the algorithm given by [68]. In DSMC simulations, molecules are initialized with given initial condition in the domain. They move inside the domain as time evolve. Their collision is based on probabilistic and phenomenological models. The fundamental assumption of the DSMC method is that the molecular movement and collision phases can be decoupled over time periods that are smaller than the mean collision time. Authors [69–71] have studied LIP expansion through DSMC. Three dimensional simulation of desorption from a binary graphite target using Monte Carlo method is done in [71]. Bird’s algorithm is used there. The study was able to capture salient features of the process such as formation of jet and segregation of heavy and light particles in the direction of jet and scattering out of jet respectively. Likewise studies [69, 70] have captured and discussed plume expansion and the effects of background pressure range on plume expansion using Monte Carlo method. But when the DSMC is used to simulate the plume expansion process, most of the computer resources are used to calculate only the initial stage of the expansion. In the presence of a background gas, additional collisions between the plume and the buffer molecules should be considered. As a result, the DSMC becomes computationally expensive in terms of computer memory and cpu time, in the case of a high rate laser ablation and in the presence of a background gas with pressure more than ∼ 10 Pa [72].
Ionization is important in early stage of plume evolution. The majority of the literature on the subject of plasma formation in LIP deals with ablation in vac- uum [73]. For long (ns range) and intense (∼ 1010 − 1014 W/cm2) ablation pulse, the latter part of the pulse is absorbed in the plume by Inverse Bremsstrahlung (IB). Bremsstrahlung stems form the German roots bremsen ”to brake” and strahlung ”ra- diation” meaning breaking radiation or deceleration radiation, Bremsstrahlung refers
17 to emission and inverse Bremsstrahlung refers to absorption. This absorption again dissociate the ablated species resulting in further absorption. This may result in absorption of all laser energy. There are salient factors determined that decide the extent of plasma formation in vacuum. These factors need not necessarily affect the plasma formation in the same way when ablation is carried in presence of background gas. Current modeling, however, is more focused on later stages of plume evolution when the molecules are observed free of ionization. Also, for carbon ablation the de- gree of ionization as compared to ablation of metal target is less due to high ionization potential of carbon. This fact is evident form the Saha equation Eq. C.1. In the next section the numerical methods that can be best suited for mod- eling LIP in presence of background gas are reviewed. The method chosen are those with minimum numerical dissipation ans dispersion. They show shock capturing ability without much oscillations.
1.2.3 Numerical Methods
As seen from the above mentioned studies [52–59] that the solution of full Navier- Stokes equations is legitimate for capturing the basic plume evolution. These equa- tions are conservation laws and numerical algorithms for conservation laws have been extensively researched in the past decades [74]. One of the most successful algorithms for conservation laws is the Godunov method [75], which laid a solid foundation for the development of modern upwind schemes [76] including MUSCL [77], total variation diminishing (TVD) [78], piecewise parabolic method [79], essentially non oscillatory (ENO) [80] and Weighted ENO (WENO) [81, 82] schemes. Another class of schemes for conservation laws are central schemes [83]. Godunov’s method is based on the solution of exact Riemann solver that solves the Riemann problem between two neighboring grid points. If the computa- tional domain involves large number of grid points then this method turns out to be
18 computationally expensive. By reducing the number of grid points the computations can be made feasible while using the Godunov method, but this makes solution dis- sipative because the original Godunov’s method is the first-order accurate. In the current study, an effort is made for finding a method that can be accurate as well as efficient. It is important, especially when dealing with plume dynamics in laser ablation as will be explained shortly. On one hand, to improve the accuracy, an attempt is made to extend the first-order Godunov scheme to second-order [77]; this needed an introduction of a limiter to remove spurious numerical oscillations near steep gradients. On the other hand, to improve efficiency, the exact Riemann solver used in the Godunov scheme was replaced by approximate ‘Riemann solvers’ [84] or flux-splitting procedures [85–87]. Approximate ‘Riemann solvers’ are locally linearized, and because the gov- erning equations for LIP are highly nonlinear, any lineariziation tends to have an error in flux values compared to their physical counterparts. As a result of this the method becomes unaccurate. The other option of extending the scheme to higher order is faced with the problem of oscillations near steep gradient. These oscilla- tions may turn the scheme to be highly unaccurate. To cure this problem, the non oscillatory, high-resolution TVD scheme was developed [78]. These schemes showed excellent shock capturing ability, with no numerical oscillations. However, an unde- sirable property of TVD schemes is that near every extrema, even if it is a smooth extremum, the schemes degrade to the first order accuracy to suppress any spurious numerical oscillations, resulting in clipping of smooth extrema [88]. Because LIP in- volve gradients inside plume and in regions with shock waves, the clipping of extrema is very likely. Thus, neither approximate ‘Riemann solvers’ nor TVD in isolation is suitable for modeling LIP. Therefore, it is important from the point of getting ac-
19 curacy as well as efficiency in modeling LIP to investigate other options of choice of numerical scheme. As a remedy of TVD scheme’s earlier drawback, ENO schemes [80] can be used. ENO schemes are high-order accurate and essentially non oscillatory, i.e. nu- merical oscillations, if any, decay with the order of the truncation error. The basic idea of ENO is adaptive stenciling, i.e. to use the smoothest possible data stencil in the reconstruction. By doing that, stencils containing a discontinuity are usually avoided. ENO schemes does not enforce any artificial condition for preventing max- ima from increasing, or minima from decreasing that makes ENO free from clipping error. Thus, ENO schemes retains high-order accuracy right up to discontinuity. Another feature of TVD schemes is that multidimensional TVD schemes are at most first-order accurate [89]. Therefore, the extension of second-order TVD schemes to multi dimensions is carried out in a dimension-by-dimension manner, i.e. one-dimensional TVD schemes are used in each coordinate direction. The extension of ENO schemes to multi dimensions can be accomplished in two different fashions. One approach is based on cell-averaged state variables, and the other is based on state variables at grid points. It was shown in [90] that the extension using grid- based state variables can be accomplished in a dimension-by-dimension fashion and is more efficient than using cell averaged variables. However, the computational grid must be smooth to maintain the formal order of accuracy. Above mentioned schemes such as, Godunov, TVD, ENO are all essentially upwind schemes. In these schemes upwind point is decided based on the solution of either exact or approximate ‘Riemann solvers’. In either case, solution of ‘Riemann solvers’ is computationally expensive. Central schemes are the class of schemes that are ‘Riemann-solver-free’. The Lax-Friedrichs (LF) scheme is the other canonical first- order scheme, which is the forerunner of all central schemes just as Godunov’s scheme
20 does for all upwind schemes. Like the Godunov scheme, it is based on piecewise con- stant approximate solution. Its Riemann-solver-free recipe, however, is considerably simpler. Unfortunately, the excessive numerical viscosity in the LF scheme yields a relatively poor resolution. There are efforts made [91] to extend central schemes to high-order accuracy, but the governing equations for LIP are strictly hyperbolic and even high-order central schemes proves to be dissipative for such application. The table 1.1 lists numerical methods that have been used to simulate plume dynamics in laser ablation.
Table 1.1: A survey of numerical methods for LIP simulation in laser ablation Authors Numerical Method Governing Equations Le et al. [52] 2nd order Flux Corrected Transport 2D NS Aden et al. [55] 1st order Godunov 2D Euler Bulgakov et al. [53] Lagrange coordinates solver 1D Euler Ho et al. [56] 2nd order Godunov type 2D Euler Zhang et al. [58] 1st order Flux Vector Splitting 2D Euler Povitsky et al. [61] 2nd order Relaxing central TVD 2D Euler Vertes [92] 1st order Godunov 1D Euler
1.3 Multiple Plume Ejections
In the development of new laser technology, such as lasers with high repetition rate and short pulse durations, the dynamics of plume is important from the point of view of how much of laser energy is actually being delivered for the subsequent laser pulse hitting the target. Absorption and scattering of the laser irradiation by previously ablated plumes makes the overall system inefficient. Precise delivery of laser energy is needed for many applications such as cornea ablation for vision correction [93]. The shielding effect impedes this precise delivery. This shielding effect can be evaluated
21 by analysing the plume pattern and concentration of particles in plume. Ablation rate decrease is normally observed with shielding of laser irradiation. This behavior is directly related to the two different causes. The increase of the plume number density in the crater produced by the preceding laser pulses is the main reason for the decrease of ablation rate. With the increase of the number of pulses hitting the same spot on the target surface, the laser energy is not only absorbed by the high number density near the target but the absorption gradually changes to the volume absorption, resulting in the increase in volume absorption in the plume and thus non-monotonic decrease in the ablation rate. The present model of shielding of laser beam can capture ablation rate decrease with increasing number of pulses.
1.4 Objectives
In the present study efforts have been made to accurately model the LIP expansion in presence of near-atmospheric pressure. The objectives of this work can be summarized as follows:
1. In the current study, vapor explosion of carbon target in presence of background gas is studied numerically and the ways to improve numerical accuracy are proposed. As seen in the previous section, the most of the methods used till now for simulating plume dynamics were dissipative. Hence, there is a need for a numerical method, which can be robust and accurate to account for the wide range of plume injection condition and which can still resolve various processes involved such as deceleration of plume, attenuation of laser beam, diffusion of plume components, recombination of charged particles, and formation of shock waves. Therefore, in this study the method is developed that can capture multiple plume roll-up and interaction of plume with shock waves.
22 As observed experimentally, laser ablation into an atmosphere of sufficient pres- sure produces a shock wave. This shock wave is formed by the piston like action of the quickly expanding ablated material pushing out the background gas. As the shock wave expands, more background gas is swept by the shock front and gradually the shock wave decays. Capturing of this shock wave propagation into the domain is important to understand the underlying plume dynamics. Nu- merical methods employed in the current study can correctly capture the shock wave and hence, more complicated processes such as plume-reflected shock wave interaction can be studied through this modeling.
2. By allowing the plume to evolve for long time it is possible to explore various aspects of plume propagation. It is aimed to capture vortex formation in the plume and effect of turbulence on the plume. The phenomena of vortex forma- tion and turbulence are essentially caused by viscous effects in a flow field. The objective is to propose a simple model based on vorticity dynamics that can help estimate different aspects of this plume roll-up. This model can also serve as a base to identify which part of governing equation is dominating and when it is dominating. The result of the numerical simulation along with the proposed model can provide better understanding of the process of plume dynamics.
3. To calculate the shielding effect in a multiple pulse laser ablation. For multi- ple plumes injection, the experimental or theoretical ablation rate needs to be adjusted to account for attenuation of laser beam energy by shielding effect. The coupling of the shielding effect with the ablation rate is proposed in the current study. In nanosecond pulses, the attenuation of the incident laser pulse by the previously ejected plume is of great relevance. Shielding of laser pulse primarily depends on the temperature of species ejected and path length tra- versed by the incident laser pulse through these species. Therefore, capturing 23 the plume shape and plume properties allows to predict the shielding effect for given conditions.
4. Multi-time step modeling is developed that allows to simulate the entire range of time scales present in laser ablation. Laser ablation involves phenomena that range from picoseconds to milliseconds. For example, to resolve the picoseconds- scale processes the time step has to be smaller than picoseconds scale. This is very computationally intensive process. Multi-time step can allow higher time step where small time step is not needed. Thus, it can save much of computational time and at the same time it can provide greater process details for multiple plumes.
5. In the current work, the developed numerical infrastructure is used to simulate ablative pyrolysis plumes typical for re-entry vehicles. This model can be used to simulate other similar processes involving high pressure and/or high speed plumes.
In Chapter II, the physical set-up, the mathematical model of plume dynam- ics and the model of shielding of laser beam are described. The numerical parameters are presented. Validation of numerical methods used for the simulation is discussed there. The results obtained and their discussion is followed in Chapter III. The vorticity-based model of plume dynamics is introduced to interpret the accuracy of numerical methods and their combination. Two proposed shielding models are com- pared that link the attenuation of laser beam and the thermodynamic properties of ablated plume. The intensity of mass removal as a function of the length of time inter- val between plumes is presented there. Planar and axisymmetric formulation as well as effect of ionization on shielding and plume dynamics are also discussed in Chapter III. Dynamics of pyrolysis plumes formed through micro cracks of re-entry vehicles is
24 presented in Chapter IV. Chapter V summarizes the discussions and concludes with recommendations for future work.
25 CHAPTER II MATHEMATICAL MODELING
In this chapter, the problem set-up, governing equations and numerical methods used for modeling laser ablation in presence of a background gas are described. To validate the implementation of these methods few test cases are presented. The cylindrical furnace, as shown in Fig. 1.1, is taken as the problem domain. Multi-time step modeling and the model of shielding are also described. The details of multi-time step modeling are given in this section while the generation of multi-size grid for such modeling is given in Appendix A.
2.1 Set-up of Problem
The problem set-up is shown in Fig. 2.1. Fig. 2.1(a) depicts a laser pulse incident on a carbon target and shows the ejected plume into an Argon gas environment. The base Argon gas is at the pressure of 1 atm and the temperature of 1400 K with the velocity of 1 m/s. The incident laser beam is taken as circular, of 5 mm diameter, and the spatial distribution of the laser beam intensity is assumed to be uniform. In the modling the pulse duration tpulse is taken as 20 ns. Laser pulse carries approximate energy of 300 mJ giving approximate fluence of 1.2 J/cm2. The wavelength of laser pulse is 532 nm. Fig. 2.1(b) depicts boundary conditions as explained shortly.
26 Figure 2.1: Problem schematic a) process of laser ablation showing the expanding plume and propagating shock wave and b) set-up of geometry and boundary condi- tions; b1 is inflow, b2-b3 is solid wall, b4 is outflow and b5 is target condition
27 2.2 Governing Equations
The vaporized material ejected from the target is modeled as a compressible, viscous ideal gas. Therefore, there dynamical state is described by compressible Navier-Stokes (NS) equations as follows:
∂U ∂F ∂G ∂R ∂S + + + αA = + + T + αB (2.1) ∂t ∂x ∂y ∂x ∂y where, 0 for planar flow α = 1 for axisymmetric flow ρ ρu ρv ρu p + ρu2 ρuv 2 ρv ρuv p + ρv U = , F = , G = , V 2 V 2 V 2 ρ e + pu + ρu e + pv + ρv e + 2 2 2 ρc ρc u ρc v 1 1 1 ρc2 ρc2u ρc2v 0 0 0 τ τ 0 xx yx τxy τyy 0 R = , S = , T = , uτxx + vτxy − qx uτyx + vτyy − qy κE0exp (−χ) 0 0 0 0 0 0
28 ρv ρuv ρv2 1 A = y , V 2 pv + ρv e + 2 ρc v 1 ρc2v 0 τ + λy ∂ v yx ∂x y τ − τ + λ v + λy ∂ v 1 yy θθ y ∂y y B = y . v2 ∂ v2 ∂ uτyx + vτyy − qy + λ + λy + λ (uv) y ∂y y ∂x 0 0
The stresses and heat fluxes appeared in above are as follows:
∂u ∂v τxx = (λ + 2µ) ∂x + λ ∂y , ∂v ∂u τyy = (λ + 2µ) ∂y + λ ∂x , ∂u ∂v τxy = τyx = µ ∂y + λ ∂x , ∂u ∂v v ∂T ∂T τθθ = λ ∂x + λ ∂y + (λ + 2µ) y , qy = −k ∂y , qx = −k ∂x . The state equation for an ideal gas in thermodynamic equilibrium can be written as p = (γ − 1)ρe. (2.2)
In the above equations ρ, V , p and e represent density, total velocity, pressure and specific internal energy, respectively. Here χ is transmissivity factor, and κ is the absorption coefficient of plume. Total energy carried by laser pulse is E0. For simplicity the above equation of state considers single species gas with molecular weight, MW = 39.9, that corresponds to Argon, taken as furnace gas. The molecular
29 weight of C3 that is the major gaseous ablation product is equal to 36. The difference in these two molecular weights is comparatively small. The presented computations are performed assuming single species with constant γ. However, for the comparison sample computations are done for the set-up of two species and is discuused in section 3.1.
2.3 Treatment of Boundary Conditions
The boundary conditions that are applied with the above equations (2.1) in the present simulation are depicted in Fig. 2.1(b) and are given below:
• On boundary b1, inflow condition is prescribed—V = Vin, q = qin, q = p, T .
∂q • On boundary b2 and b3, solid wall condition is imposed—V = 0, ∂r = 0, q = p, T .
• On boundary b4, outflow condition is prescribed—V = Vout, q = qout, q = p, T .
• On boundary b5, target conditions are prescribed—t ≤ tpulse: q = qin(t), q =
∂q p, T, V ; p = 100 atm, V = 3200 m/s. t > tpulse: ∂z = 0, q = p, T , V = 0. For
t > tpulse the target area is treated as solid wall and any displacement of target’s outer boundary on account of sublimation of target is neglected because size of crater formed due to sublimation is less than half grid size [94].
The inflow (boundary b1) and outflow (boundary b4) condition for subsonic flows is decided based on Riemann invarients to avoid unphysical wave reflection generated from boundaries [95]. If the system has N eigenvalues with m positive and (N − m) negative eigenvlues then m variables has to be defined at inlet and (N −m) at outlet, remaining variables are extrapolated from the domain. To illustrate specification of a typical set of boundary conditions, consider the following example. Assume that
30 the system has u, u + a and u − a eigenvalues where u is velocity and a is speed of sound. Thus, two veriables have to be defined at inlet. This is done using the Reimann invarients which are
2a R = u + ∞ , (2.3) 1 ∞ γ − 1 constructed from the freestream ∞ state and
2a R = u − 1 , (2.4) 2 1 γ − 1 constructed from the domain. From the above two relation the velocity and sound speed at inlet is determined. Also, with the help of entropy invarient condition,
p∞ p γ = γ , (2.5) ρ∞ ρ other conservative variables at inlet are determined. It should be noted that specifi- cation of boundary condition shown above is general, some approximations may be needed in order to implememnt the specified boundary conditions.
2.4 Modeling of Turbulence
To see the effect of turbulence on plume flow, a simple gust model is employed. This model is based on direct numerical simulation of turbulent eddies. The intital conditions of these eddies are based on K − model [96] of turbulence. The above mentioned K− is suitable for moderate Reynolds number isothermal flow. The effort in the present modeling of turbulence is to see the effect of turbulence in furnace gas flow on plume, and because in most of laser furnaces for synthesis of nanotubes, the furnace Argon gas flows with the speed of order of 1 − 10 m/s the K − model
31 of turbulence is best suited to represent turbulent eddies in this flow. The plume evolution is studied for the time period of the order of a single millisecond, that is comparable to the life time of turbulent eddies. The initial conditons of these eddies are designed based on K − model of turbulence as follows: First, the turbulent kinetic energy and the rate of dissipation are calculated using a given level of turbulence in the laser furnace,
3 3 c K 2 u0 = ηu, K = u02, = µ , (2.6) 2 ` where η is the level of turbulence (the regular level of turbulence is usually 1−3% while the elevated level is up to 10%) ` = 0.1D, and D is the diameter of the laser furnace. Next, using the discrete random model employed in the study [97] for modeling of carbon oxide-based reactor of synthesis of nanotubes, the average life time of turbulent eddies is calculated as, K t = c , (2.7) this provides with the semi period in space of the eddies which is,
L = u0t. (2.8)
The initial turbulent gust can be set as,
πx u0 = u0 sin( ), (2.9) max L
The fluctuating v component of velocity is calculated using the continuity equation. This fluctuating u and v components are set as an initial condition, t = 0, and as a boundary condition at the entrance cross-section adding a proper phase shift. The obtained gust-type turbulent flow field for u = 1 m/s and η = 3% is shown in Fig.
32 2.2. This velocity gust affects the solution of viscous equations (2.1) enhancing the momentum transfer in particular in the y direction. Note that all unsteady features of flow are reflected by direct solution of unsteady equations (2.1), therefore, there is no need for time-averaged models of turbulence.
a) b) Figure 2.2: Fluctuating velocity components a) Horizontal b) Vertical
2.5 Numerical Methodology and Validation
Numerical finite-volume discretization of the coupled system (2.1) of partial differen- tial equations (PDE) is done using the Relaxing TVD, the ENO-Roe and the Godunov methods to choose the right combination of methods. For discretization of the right- hand side of the coupled system (2.1), which include diffusion and viscous stresses terms, second order central difference schemes are used. The left-hand side of system of equations (2.1) is hyperbolic, i.e. it has physical waves as their solution. The flux functions Fz and Gr in equations (2.1) are needed to be evaluated at cell interface for conservative formulation of system (2.1). The major difference between these numer- ical methods used for convective fluxes, i.e. for the left side of system (2.1), is in the way they calculate convective fluxes for the discretization of the left-hand side of the system (2.1). To calculate these fluxes, the ENO-Roe method uses Roe’s approximate Riemann solver, in which Roe-averaged variables are used and the eigenvalues of the 33 Figure 2.3: Flow chart of the numerical code system (2.1) are calculated based on these variables to decide the upwind cell. The general procedure of solving system (2.1) is described in Fig. 2.3. In what follows next is the details of solution of NS equations with a particular method. The Godunov method solves 1D gas dynamic equations semi-analytically. The semi-analytical solution of the Riemann problem at each face of any numerical control volume is very computationally intensive. Therefore, the Roe and Relaxing TVD methods are preferable. However, the Roe discretization is unstable for high injection velocities and the latter is too diffusive. In the current study, the overall
34 accuracy of the ENO-Roe and the Relaxing TVD methods is of the 2nd order and the Godunov method is of the 1st order. The Godunov method is kept of the 1st order because at high values of injection velocities and pressures the nonlinearity of problem is very significant and any numerical interpolation of fluxes deviates considerably from physical fluxes. To explain these methods, it suffices to explain only the operations needed to calculate the x component of fluxs which is a nonlinear function F (U). These methods are explained below.
2.5.1 The ENO-Roe Method
The Jacobian F 0(U) = Aˆ has four real eigenvalues in the 2D case and a complete set of independent eigenvectors such that
R−1(U)ARˆ (U) = Λ(U), (2.10) where Λ(U) is the diagonal matrix with eigenvalues on the main diagonal. R−1(U) and R(U) being left and right eigenvectors, respectively, of Aˆ. With the transformation of variables, w, the PDE (2.1) becomes diagonal
wt + Λwx = R.H.S, (2.11) equations in (2.11) are decoupled and the solution is easily available. After the solution of equations (2.11) is obtained, original set of variables can be regained with the transformation U = Rw. Matrices Aˆ, R, R−1 and Λ are dependent upon U. Thus, to compute the flux at the cell interface x 1 , an approximation to the Jacobian Aˆ at i+ 2 the interface is needed. The Roe averaging is used for such an approximation. The computation of above matrices and their products is the major computational expense
35 for ENO-Roe scheme. The ENO-Roe uses ENO interpolation for achieving higher- order accuracy in space and diminishing oscillations. This interpolation not only discards the stencil giving the oscillations in the flux, but also chooses the smoothest stencil. The choice is made using the Newton divided difference. After choosing the smoothest stencil, reconstruction is done to find the numerical flux function at the cell interface. The numerical flux function is expressed as
k−1 ˆ X F 1 = crjfi−r+j, (2.12) i+ 2 j=0
where r is the number of cells left to the cell i and crj are the constants given in [98]. Second order Runge-Kutta scheme is used for time marching. For the gas with multiple species with different ratios of specific heats, γ, conditions for Roe averaging may not be satisfied.
2.5.2 The Relaxing TVD Method
This method has the form that assumes matrix Aˆ as a constant diagonal matrix and gives the characteristic variables of the form
E + Aˆ1/2U and E − Aˆ1/2U (2.13)
The values U 1 are calculated by applying upwind scheme with respect to signs of i+ 2 E + Aˆ1/2U and E − Aˆ1/2U as follows
E + Aˆ1/2U = E + Aˆ1/2U , E − Aˆ1/2U = E − Aˆ1/2U (2.14) 1 1 i+ 2 i i+ 2 i+1
Using equation (2.13) U 1 can be computed i+ 2
1 1 ˆ1/2 Ui+ 1 = (Ui + Ui+1) − A (Ei+1 + Ei) . (2.15) 2 2 2 36 The advantage of Aˆ being simply a positive constant at all faces of control volumes is that the computational effort is greatly reduced compared to that needed for the ENO-Roe method. For the second order accuracy in space for Relaxing TVD scheme, the MUSCL method is used for the discretization of the characteristic variables (equa- tion (2.13)) and time integration is carried out using implicit-explicit (IMEX) Runge- Kutta scheme. The detailed version of Relaxing TVD scheme can be found in [63].
2.5.3 The Godunov Method
This method solves the Riemann problem formed by PDE (2.1) with equation (2.16) at the interface of two neighboring points.
UL if z < 0, U(z, 0) = (2.16) UR if z > 0
In general, given the conservation system (2.1) for the dynamics, it is left to the statements about the material, i.e. the equation of state, to determine not only the structure of Riemann problem but also the mathematical character of the equations. For ideal gas with equation of state (2.2) the Riemann problem gives three waves as the solution. These waves are shown in Fig. 2.4. The property of convective part of system (2.1) reveals that pressure p∗ and velocity u∗ between the left and right waves are constant, for any combination of waves. Using Rankine-Hugoniot relations for shock and isentropic relations for expansion fan an equation for the unknown p∗ is sought. This equation is [88]
2γ/(γ−1) p∗ ∗ − 1 p pL γ − 1 aR pR = 1 + uL − uR − r . (2.17) pR pR 2aL γ ∗ γ+1 p − 1 + 1 2γ pR
37 Figure 2.4: Structure of the solution of the considered Riemann problem
This equation is nonlinear and does not have a close-form solution. The above equation being nonlinear implicit needs a specific numerical technique to be used at each cell at any time step. In the present implementation of the Godunov method Newton-Raphson iterative procedure to find p∗ is used for this equation. Once p∗ is calculated other parameters to calculate F 1 can be found by using, again, the i+ 2 Rankine-Hugoniot and isentropic relations. The details can be found in [88, 99, 100].
2.5.4 Validation
The test cases described below show the validity of codes for propagation and reflec- tion of shock waves and give, to some extent, a comparison amongst the methods used. The grid convergence study is given in Appendix B. The double Mach reflection problem. This test example (see Fig. 2.5)is that of canonical double Mach reflection [79], for larger Mach number typical for initial stages of plume expansion. The spatial domain is [0, 4] × [0, 1]. The reflecting wall lies at x = 1/6. Initially the shock wave with the Mach number 10 is positioned at x = 1/6, y = 0 and makes an angle of the 600 with the x axis. Boundary conditions are listed as follows: inflow is supersonic inlet, outflow conditions are decided from
38 the solution of Riemann Invariant, at y = 0 solid wall is imposed and at y = 1 the flow values are set to describe the exact motion of the Mach 10 shock. Fig. 2.5 shows the contour plots of density for this test example as obtained by two schemes. It should be noted that there is an increase in accuracy for The ENO-Roe method.
Figure 2.5: Density contours for the double Mach reflection problem, ∆x = ∆y = 1/120: a) the Godunov method b) the Relaxing TVD method and c) the ENO-Roe method
The forward facing step problem. The setting of the problem is the following: A right going Mach 3 uniform flow enters a wind tunnel of 1 unit wide and 3 units long. The step is 0.2 units high and is located 0.6 units from left hand end of the tunnel. The problem is initialized by a uniform, right going Mach 3 flow. Reflective conditions are applied along the walls of the tunnel and inflow and outflow boundary conditions are applied at the entrance and exit of the tunnel, respectively. In Fig. 2.6 contour plots of the density at time t = 4.0 are shown. It can be seen that Relaxing 39 method proves to be too dissipative. ENO-Roe method is able to capture the shock but the location of shock with this method is slightly off set.
Figure 2.6: Density contours for the forward facing step problem, ∆x = ∆y = 1/80: a) the Godunov method b) the Relaxing TVD method and c) the ENO-Roe method
Shock flowing over backward facing step problem. A shock wave diffracts over a 90 degree sharp corner. This test example is designed in [101] to compare various CFD schemes which are useful for simulating shock wave phenomena. The initial/boundary conditions for this problem are shown in Fig. 2.7. The obtained density contours are shown in Fig. 2.8. It can be seen that although the Godunov method is dissipative, it locates the shock correctly and is able to the captures the gradients near the corner. The gradients near the corner as captured by the Relaxing TVD method appears to be somewhat dissipative than the gradients obtained by other methods. 2D Laminar boundary layer problem. Navier-Stokes solution for a laminar
40 Figure 2.7: Initial/boundary conditions for shock flowing over backward facing step problem
Figure 2.8: Density contours for shock flowing over backward facing step problem, ∆x = ∆y = 1/300: a) the Godunov method and b) the Relaxing TVD method c) the ENO-Roe method
41 boundary layer is tested here. The test case is compared with [95]. The computa- tional domain used in this case is [0, 320] × [0, 120]. The flat plate is placed at a lower boundary ranging from x = 80. The undisturbed inflow condition at x = 0 is
9 |ρ, u, v, p| = 1, 3, 0, x=0,y,t γM 2 where M = 0.3 and γ = 1.4. In this test case, the Reynolds number is defined as
ul Re = , (2.18) ν where l is the length of reflecting flat plate. The value of ν is taken as ν = .024 that gives Re = 30, 000. No-slip boundary condition is imposed on the flat plate. Non reflecting boundary condition based on the one-dimensional Riemann invariants is used at the left and upper boundaries. Riemann invariants along with the con- stant entropy condition give the conservative variables at the boundary. The output velocities are defined as
r u v u (x − x ) u = , v = ∞ s (2.19) u∞ u∞ ν
where xs = 80 is the location of the leading edge of the flat plate. The distance y in the y-direction above the flat plate is normalized by
r u η = y ∞ . (2.20) (x − xs)ν
Fig. 2.9 show velocity distributions calculated by the ENO-Roe and Godunov meth- ods.
42 Figure 2.9: Normalized velocity distributions as obtained for boundary-layer type flow by a) the Godunov method and b) the ENO-Roe Scheme
43 For stability the time step, ∆t, in all of the above cases is calculated as
! dxdy C ∆t = min (2.21) pdx2 + dy2 a + V where C is the Courant-Friedrich-Levy (CFL) number taken to be 0.5, a is the local speed of sound and V is the local speed of gas. The minimum is taken over all grid cells to ensure stability.
2.6 Ionization and Shielding
Ionization in the plume is modeled using Saha equation. The details of Saha equation are given in Appendix C. For ionization to occur, the energy has to be supplied to the molecule to remove electrons from the atoms of a molecule. This consumed energy, which is equal to the ionization potential of individual species, will result in lowering the local temperature. The Euler/NS equations (2.1) do not account for the thermal effect of ionization resulting in incorrect elevated temperatures when ionization is present. To account for ionization and its effect on shielding, the ionization could be modeled by straightforward solution of Eq. C.1. However, this would neglect the effect of ionization on temperature. To account for the effect of ionization on energy and temperature field an iterative procedure is adopted. In this procedure the energy is recalculated. As mentioned above, ionization consumes energy hence, recalculated energy is lower than energy defined by Eq. (2.1) by an amount equal to ionization potential multiplied by the degree of ionization, ζ. Based on this updated energy, the new temperature in the domain is evaluated. This new temperature is used to re-calculate ζ by Eq. C.1,
44 which will be lower than the previous value of ζ. This iterative process continue till ζ ≤ 10−6. Shielding effect [7, 13, 102] of incident laser beam by plume is calculated assuming gaseous C3 carbon plume. At the considered temperature generated by a laser of 20 ns, 532 nm pulse of 300 mJ energy, the ionization in gaseous carbon plume as calculated from Saha equation is not negligible, although its effect on shielding of incident laser beam is ngligible. Also, in experiments [103] of carbon ablation with similar laser parameters, ionization in carbon vapor is found negligible and hence, ionization is neglected in the present model. Shielding effect that controls the amount of mass ablated from target is described using two simple models introduced in this Section. In the first model (velocity based), the injection velocity is assumed directly proportional to the energy delivered to the target. That is, the fraction of laser energy consumed by heat conduction in target and radiation is assumed to be the same for single pulse and multiple pulses. The ablated mass is therefore proportional to the amount of energy delivered to the target. In addition, the temperature and pressure of the ablated plume are assumed constant. The mass ablated is given by
m˙ = ρAV. (2.22)
The second model (pressure based) is based on Hertz-Knudsen [7] relation between ablated mass and plume pressure
ps(T ) m˙ = p mjA, (2.23) 2πmjkBT
where kB is the Boltzmann constant, mj is the mass of a single molecule species C3 ablated from area A. Therefore, the change in ablated mass leads to the propor- tional change of pressure. The pressure ps(T ) in the injected plume is related to its
45 temperature by the Clausius-Clapeyron relation
h 1 1 ps(T ) = p1exp − , (2.24) R Tb T
where the reference pressure is p1 = 1 atm, Tb is the boiling point temperature at p1, T is the plume temperature, h is the enthalpy of sublimation of carbon and R is the gas constant for Carbon. Attenuation of incident laser beam is calculated based on Bouger-Lambert- Beer law [7] as follows: dE 0 = −κE , (2.25) dz 0 this gives laser beam attenuation,
! E Z z(r) absorbed = exp − κ (Γ,T ) dz , (2.26) E0 0 where z(r) is an optical path length of plume for incident laser beam as shown in Fig. 2.1(a) and Γ is the wavelength of incident laser beam. The latter is calculated based on the number density of absorbing carbon molecule and its absorption cross section,
κ (Γ,T ) = σn, (2.27) where σ (m2 molecule−1) is the absorption cross section, n (molecules m−3) is the number density. The absorption cross-section σ (Γ,T ) is described by semi-empirical formulae [104] that cross-section can be used for higher temperatures till 6000 K [105].
The semi- empirical coefficients of cross-section were obtained for CO2 in vacuum.
For C3, the coefficients are not found in available literature. However, the major species in ablation of carbon is C3 and hence the obtained absorption cross section is modified, assuming that the cross section is directly proportional to molecular
46 weight. In the study [106], no pressure dependence on absorption cross-section at higher pressures is observed so the calculated absorption cross-section is feasible for the present application. The number density n can be found by
p = nkBT, (2.28) where p is the local partial pressure of gaseous carbon. The partial pressure of gaseous carbon is found from the known carbon concentration. The carbon concentration is known by solving an equation for concentration of carbon, and T is the local temperature. To account for the shielding due to ionized particles, inverse Bremsstrahlung mechanism is employed. The absorption coefficient [3] given by this mechanism is
2 −35 3 z NiNe hυ κIB = 1.37 × 10 λ 1/2 1 − (2.29) T kBT
2.7 Multi-Time Step Modeling
The plume roll-up that appears during the process of laser ablation occurs in the vicinity of the target. To simulate this accurately, fine mesh is needed. Away from the target the flow is relatively smooth. In this region, coarse mesh is good enough to capture the flow. If for the sake of simplicity in computations fine mesh is used in the whole domain then computation time increases, which may cause tremendous computer expenses. Besides, laser ablation involves multiple time scales ranging from picoseconds to milliseconds. To cover the entire range of timescales involved in laser ablation with fine mesh all over the domain is numerically very expensive. One effective way to treat this multiple scales problem is to use multi-size mesh. The computation domain is first divided into sub-domains. One sub-domain
47 contains the fine mesh and the other one contains the coarse mesh. The mesh size changes across the domain interface but is constant inside each domain. Fig. 2.10 show the multiple meshes used in the computations. When mesh sizes are different, it
Figure 2.10: Multi-size mesh for multi-time step modeling: a) schematic showing grid sizes for coarse and fine girds (L and D are furnace length and diameter, respectively) and b) computational domain with boundary conditions has significant impact on time steps used in a computation. Time step ∆t, and mesh sizes dx, dy are linked together by numerical stability requirement. If one wish to use single time step ∆t for the entire domain then this time step, through CFL number, is dictated by smallest grid size in domain and as a result more time steps are needed for a given simulation time. Hence, the tremendous expense of elapsed computation time is inevitable. To optimize the situation, ∆t that meets global numerical stability requirements can be used. For a finer grid the time step is reduced exactly by the ratio between two grid steps. In such situations, to keep the simulation time even over whole domain the solution on coarse mesh, is updated once per several time steps on fine grid and most of the computation is carried out in fine mesh region. In carrying out multi-time step computations, the numerical method in each of the uniform mesh sub domain is the same as if the single mesh is used through out the computational domain. Exception is the sub-domain interface where the mesh changes its size. The calculation of flux at this interface is crucial and is the key 48 point in multi-time step modeling. This is crucial because one side of interface is at a different time level than the other side. This flux calculation is carried out in such a way that the fundamental property of conservation is satisfied. When this global conservation is achieved, then the formulas used to update the solution at an interface between coarse and fine grid becomes consistent with one another. To illustrate this calculation of flux at an interface between two meshes, one dimensional case of two grid levels with a factor of two refinement is explained below. As shown in Fig. 2.11 the coarse grid spacing is 2dx and the corresponding time step
Figure 2.11: Calculation of Flux F (U) a) at time t = 0 b) at time t = dt/2 c) at time t = dt
49 is dt. On the fine grid the grid spacing is dx and time step remains dt/2. The interface corresponds to j = 0. The cycle begins at time t = 0 and ends at time t = dt. First
t=0 at time t = 0, all the fluxes Fj , as indicated in Fig. 2.11, are calculated. The coarse grid values of U are updated as follows,
dt U (t = dt) = U (t = 0) − (F (t = 0) − F (t = 0)) (2.30) j j 2dx j j−1 for j ≥ i + 1. This brings all coarse grid values of U at time t = dt. On the fine grid the values are updated twice as follows,
dt/2 U (t = n) = U (t = n−dt/2)− (F (t = n − dt/2) − F (t = n − dt/2)) (2.31) j j dx j j−1 for j < i and n = dt/2, dt. In doing so, when j = i and t = dt/2 the calculation of
t=dt/2 t=dt/2 flux Fi needs the value of Ui+1 but this value is not available because equation (2.30) gives values of U at time t = dt. For this reason equation (2.30) is solved with the time step t = dt/2 only for j = i + 1 and this value is stored in a dummy variable
Udummy as indicated in Fig. 2.11(b). After both the coarse and fine grid values are updated to time t = dt, these values need to be coordinated. In other words, both these updating should be con- sistent with one another. This can be done by preserving the global mass, which can be expressed by X X dx Uj + 2dx Uj = constant. (2.32) j≤0 j≥i+1
t=dt Ui+1 has to be modified in order to equation (2.32) be applicable as follows,
dt/2 U (t = dt) = U (t = dt) + (F (t = dt/2) − F (t = 0)) (2.33) i+1 i+1 2dx i i
50 This completes the cycle. Equation (2.33) takes care, in coarse grid, of the information at time t = dt/2. This information has been used on fine grids, while coarse gird had no clue of what is happening at time t = dt/2. This is because coarse grid takes values of fluxes at time t = 0 and brings values of U at time t = dt. In this Chapter the details of modeling are given. First, the problem set- up is described; the governing equations for the problem are given along with the details of initial/boundary conditions for solving these equations. Then the details of turbulence modeling are given in Section 2.4. Second, the numerical methods used to solve the governing equations are described followed by the results of standard test cases that validated these methods. Third, the detalis of shielding models are presented. These models are designed such that shielding due to both, the ionized and the neutral molecules is accounted. Finally, the details of multi-time step modeling that is used in the current study is given in Section 2.7. In the next Chapter the results obtained for the problem set-up and the modeling presented in this Chapter are discussed in details.
51 CHAPTER III RESULTS AND DISCUSSION FOR LASER ABLATION MODELING
The modeing of ablated plume is carried out for the following boundary and flow conditions (see Fig. 2.1(b)). The plume starts to emerge immediately after appli- cation of laser and emerges during 20 ns of laser beam. The plume conditions are assumed uniform across the target with the pressure of 100 atm, injection velocity of 3200 m/s and plume temperature of 5000 K. The surrounding co-flowing gas moves with the speed of 1 m/s through the entrance cross-section at the temperature of 1400 K. The table 3.1 summarizes the parameters of the ablation being simulated.
Table 3.1: Summary of operating parameters of the process being simulated Laser operating parameters Wavelength 532 nm Estimated laser energy 300 mJ onto a 5 mm spot Pulse duration 20 ns Furnace conditions Furnace argon flow 1 m/s Furnace temperature 1400 K Furnace pressure 1 atm Furnace length 0.25 m Furnace diameter 0.05 m Plume initial conditions Temperature 5000 K Pressure 100 atm Injection velocity 3200 m/s
52 Figure 3.1: Plume concentration at time t = 50µs: a) species are assumed to have the same λ and b) two gamma, Ar and C3 with their individual γ and molecular weights
3.1 Single gamma versus Two gamma
For comparison of numerical results with single and two species, plume concentrations for single plume ejection obtained with Godunov method are shown in Fig. 3.1. Single species refers to Argon, while two species refers to Argon (furnace gas) and
C3; C3 being the plume material. The value of γ for Argon is taken as γ = 1.67 and that for C3 is γ = 1.25. It can be seen from the figure that the concentration field is practically the same for both the cases. Plume shape is qualitatively the same with a negligible difference in the plume core. Thus, for simplicity of model the equation of state (Eq. 2.2) considers, for all further calculations, single species gas with molecular weight, MW = 39.9 that corresponds to Argon, taken as furnace gas. The molecular weight of C3 that is the major gaseous ablation product is equal to 36. The difference in these two molecular weights is moderate. Therefore the presented computations are performed assuming constant γ and molecular weight corresponding to the background gas Argon.
53 3.2 Inviscid, Viscous and Turbulence Plume Modeling
With the plume initial conditions as given above, the solutions for ablation plume are presented at post-ablation time of t = 300µs (see Fig. 3.2). The plume shape
Figure 3.2: Plume concentrations obtained by the Euler equations at t = 300µs a) the Godunov method, b) the Relaxing TVD method and c) the ENO-Roe method as resulted by solving the Euler equations using the Relaxing TVD method is shown in Fig. 3.2(b). This plume has expanded less in lateral direction than the plume obtained by the ENO-Roe and Godunov methods for Euler equations (compare Fig. 3.2(a) and Fig. 3.2(c)). This is because of more dissipative nature of the Relaxing TVD scheme as compared to the other methods. It should be noted that the dissipa- tion in the Godunov method results from its lower order accuracy compared to the other two methods. This dissipation results in washing out the velocities with smaller magnitudes. As the plume speed in the axial direction slows down with the time, this dissipation becomes critical for plume in its evolution. To see the effect of this 54 Figure 3.3: Vector plot obtained by the Euler equations at t = 300µs a) the Godunov method, b) the Relaxing TVD method and c) the ENO-Roe method dissipation the vector plot of velocity in the domain for the three schemes is shown in Fig. 3.3. From the plots it can be clearly seen that numerical dissipation critically affects the magnitude of velocities inside the plume region. The region with smaller velocities is affected more with this dissipation. It can be seen from this plot that velocity magnitudes around plume region are greater for the ENO-Roe method and hence, the plume length in y direction obtained with this method is greater. Other issue that affects plume evolution is a shock wave. The shock wave drags the flow field behind it affecting base flow structure, which in turn affects the plume evolution. This shock wave appears to be somewhat weaker for the Relaxing TVD method in comparison to the ENO-Roe method, as shown in Fig. 3.4. Owing to the dissipation in the Relaxing TVD method, further comparison and discussion of results for LIP involves only the ENO-Roe and the Godunov methods. The dissipation in the Relax-
55 Figure 3.4: Contours of pressure obtained by Euler equations at t = 300µs a) the Godunov method, b) the Relaxing TVD method and c) the ENO-Roe method
56 Figure 3.5: Comparison of plume concentrations at t = 100µs a) experimental, b) numerical NS and c) numerical Euler ing TVD method is directly related to the values of Aˆ in equation (2.13). In fact, this Aˆ is meant to introduce dissipative nature in otherwise an hyperbolic equation which has pure waves as its solution. It turns out that for multi-dimensional computations the value of Aˆ has to be higher than the square of value of highest eigenvalue of the system. Fig. 3.5 shows the ablation plume as obtained by solving the NS equations. It can be seen from the figure that results obtained by NS equations matches compar- atively well, with experimental observations of plume, than does the plume obtained by solving Euler equations. In the experimental Fig. 3.5a the plume roll-up because of the vortex pair is clearly visible. This plume roll-up process is captured by NS
57 Figure 3.6: Propagation of the leading edge of the ablation plume equations. Some difference between experimental and simulation results can be at- tributed to the fact that the real plume in actual carbon nanotube synthesis involves multiple species of carbon, as opposed to the single-species ideal gas in the present study. The generation of this vortex is discussed in section 3.3. Other experimental result for long run of carbon plume evolution available is the propagation of leading edge of the plume found in study [32]. The leading edge distance was determined in this study by laser-inuduced fluorescence (LIF) of certain species in the carbon plume. Fig. 3.6 indicates that the numerical results show higher speed of plume propagation than experimental data. The difference in this results owes mainly to the two reasons: (i) LIF data are not indicative of actual plume propagation in view of the fact that the region of high concentration of certain species traced by LIF lag the propagation of actual plume front downstream of this region and (ii) since the initial conditions with which the plume emerge are not exactly determined in both the cases the difference in experimental and numerical plumes is inevitable. 58 To see the effect of turbulence, this simulation for NS equations is run further up till time t = 300µs with added turbulence in the flow. Fig. 3.7 compares plume with and plume without turbulence.
Figure 3.7: Plume concentrations obtained by NS equations at t = 300µs a) the Godunov method, b) the ENO-Roe method and c) the ENO-Roe method with tur- bulence
The plume patterns obtained by Godunov method (see Fig. 3.7a) does not capture the plume roll-up. This is due to numerical dissipation introduced by this method. This dissipation smears the gradients in the flow field, as a result of which the the plume roll-up is not seen. Figs. 3.7b,c are obtained by ENO-Roe method that captures the plume roll-up. Fig. 3.7c shows the plume modeled with the effect of turbulent gust considered with the NS equations. It shows that the effect of turbulence is minor for the given conditions of furnace gas flow and the given time interval. For the time span of 300µs the velocity magnitude in the plume region are much higher than that for surrounding gas. And as velocity magnitude of surrounding 59 gas determines the magnitude of gust (see equations (2.6-2.9)), therefore turbulent gusts appears to have no effect on plume evolution for the time span considered. To see the effect of turbulence in later stages of plume evolution the ablated plume is simulated for longer run (t = 1 ms) with the turbulence gust given by equations (2.7-2.9). The results are as shown in Fig. 3.8. It can be seen from Fig.
a) b) Figure 3.8: Comparison of plume concentration at t = 2 ms for a) viscous and b) turbulent models
3.8 in millisecond range there is some effect of turbulence. This effect is seen on the peripheral part of plume. The tail of the plume also appeared to have affected by turbulence. The core of the plume, however, is less affected. The core of the plume travels at comparatively higher speed than surrounding gas. In the core region of the plume most of the velocity vector are still forward directed, whereas the peripheral part is more driven by turbulence. As a result, turbulent gust affect only at the periphery and tail of the plume where the velocities are smaller. The generation of vortex in the plume is due to viscosity and baroclinic vor- ticity. Baroclinic vorticity generation is a major source of vorticity at the considered time span of millisecond (see Section 3.3). Turbulence, being a viscous phenomenon shows up its consequence in the region of viscous gradients. Thus, unless the plume speed becomes comparable with the speed of surrounding gas and the viscous effect become dominant, the plume travels being practically unaffected by turbulence in the surrounding furnace gas.
60 The temperature and trajectories of catalyst particles embedded in plume is crucial for the formation of nanotubes and many other applications. The trajectories is shown in Fig. 3.9.
a) b)
c) Figure 3.9: Trajectory and temperature of injected particle as a function of time for t ≤ 300µs a) Euler model b) NS model at and c) temperature of particle for NS model
The trajectories for particles ejected from centerline and peripheral part (at r/R = 0.1) as obtained by Euler and NS equations are not much different. Par- ticles ejected away from the center, however, experience some trajectory difference. Because of the symmetry of the problem the centerline particles travel straight in both, the Euler and the NS models. The peripheral particles immediately experience the steep gradient of velocity, as velocity in the domain is much lower than particle velocity. This makes peripheral particles trajectory form a loop. These particles have
61 tendencies to flow toward center, because they have favorable pressure gradient in that direction only. As a result of this, particles from center and periphery regions are least affected by viscosity. In Fig. 3.9(c) the maximum temperature in the do- main and the temperature of particles along the path lines as obtained by solving NS equations are shown. The maximum temperature shown in in Fig. 3.9(c) is the maximum spatial temperature at any instant of time in the domain. The maximum temperature is associated with the shock wave. This temperature gradually decreases as the shock wave dissipates. The particle temperature increases first, drops sharply and then again increases at t ∼ 1.8 µs. The first increase in particle temperature indicates the presence of shock wave. When the shock wave leaves the particle, its temperature drops. The shock wave, after overpassing the particle, drags the particle along with it. This again brings particle in the vicinity of shock wave resulting in its increase of temperature. This increase of temperature can seen at t ∼ 1.8 µs in Fig. 3.9(c). In the next section, forming of vortex inside the plume is discussed.
3.3 Dynamics of Vorticity
Since the plume is being eventually split into the pairs of rolling vortex propagating downstream into the domain, it is possible to develop an analytical post processing model that determines the plume shape from the amount of deposited vorticity. Such a model could be viewed as an extension of shock-bubble interaction model by [107]. This model can be used to predict the behavior of plume through vorticity dynamics quantifying baroclinic and viscous vorticity deposition on the plume. It should be noted here that the current problem is significantly more involved than the single shock wave-bubble interaction since the plume interacts with several oblique shock waves and possibly with expansion waves and hence, extensive numerical simulations
62 for various initial conditions of plume ejection are needed to validate such a simplified model. Such an analytical model to predict the behavior of plume is based on vorticity transport equation. The vorticity equation describes the evolution of the vorticity (~ω) of a fluid element as it moves around. The vorticity equation is derived from the conservation of momentum equation. With no body forces in its vector form it may be expressed as follows,
Dω ∇ρ × ∇p = ω · (∇V~ ) − ω(∇ · V~ ) + ν∇2ω + (3.1) Dt ρ2 where, V~ is the velocity vector, ρ is the density, p is the pressure and ν is the kinematic viscosity.
Dω • The term Dt is the material derivative of the vorticity vector ω. It describes the rate of change of vorticity of a fluid particle. This can change due to the
∂ω unsteadiness in the flow captured by ∂t (the unsteady term) or due to the motion of the fluid particle as it moves from one point to another, V~ ·(∇ω) (the convection term).
• The first term on the RHS of the vorticity equation, ω · (∇V~ ), describes the stretching or tilting of vorticity due to the velocity gradients. Note that in 3D ω is a vector (~ω) while in 2D it has just a scalar value. Hence, in 2D the value of ω · (∇V~ ) is zero. Putting it in another way, the phenomenon of stretching or tilting of vorticity is absent in 2D.
• The next term, ω(∇ · V~ ), describes stretching of vorticity due to flow compress- ibility.
• ν∇2ω, accounts for the diffusion of vorticity due to the viscous effects.
63 1 • The term, ρ2 ∇ρ × ∇p is the baroclinic term. It accounts for the changes in the vorticity due to density and temperature changes.
In summary, following the fluid motion in 2D, the increase in vorticity is caused by diffusion by viscosity and density inhomogeneity. These two sources of vorticity in plume are shown in Fig. 3.10. The plume-average magnitude of the vorticity source
a) b) Figure 3.10: Evaluation of sources in the vorticity transport equation a) baroclinic vorticity generation and b) viscous generation of vorticity for the ENO-Roe method term is evaluated by integrating over all cells at any given time moment as follows:
1 1 ZZ 1 ∂ρ ∂p ∂ρ ∂p ∇ρ × ∇p = − dA (3.2) ρ2 A ρ2 ∂x ∂y ∂y ∂x
The area A in above equation includes cells of the upper half-domain with the plume concentration greater than 2 %. The above calculated vorticity source term is shown in Fig. 3.10(a). The plume-average magnitude of the viscosity source term (Fig. 3.10(b)) is evaluated by integrating over all cells at any given time moment as follows:
1 ZZ ∂2ω ∂2ω ν∇2ω = ν + dA (3.3) A ∂x2 ∂y2
In Fig. 3.10(a) the magnitude of vorticity source term is large initially and it decays gradually. Initial level of the source term is caused by the misalignment of 64 the radial pressure gradient behind the Sedov-Taylor blast wave and the non-radial density gradient in the injected plume. Non-uniform pressure field behind the incident shock wave also contributes to baroclinically generated vorticity, but except for the first microseconds after ablation, this contribution is weaker than that caused by the reflected shock waves. This interaction of reflected shock wave with plume causes three peaks as seen from Fig. 3.10(a) at time t ∼= .0006 s, t ∼= .0015 s and t ∼= .0022 s. At these time moments reflected shock waves interacts with the plume. These peak values of vorticity source term decays as shown in figure. This is caused by the lesser strength of secondary reflected shock waves. The Roe method is less dissipative it captures shock better than the Godunov method and as a result it gives higher value of vorticity source. For the first peak in
1 Fig. 3.10(a), the Roe method gives absolute magnitude of vorticity source of ∼ 400 s2 , 1 while the Godunov method being dissipative gives ∼ 100 s2 and their combination 1 (see below) results in ∼ 200 s2 . At higher value of vorticity source obtained by the Roe method, the plume roll-up as evidenced in experiments [1] is obtained (see Fig. 3.7(b)). Unfortunately, the Roe method is unstable for high injection velocities around 5228 m/s of ablated material. This velocity value is obtained by [50] to match the experimental plume of [1]. The Godunov method can handle this high injection velocity. However, the Godunov method is dissipative and could not capture plume-shock interaction at later time moments. This can be seen from vorticity source term shown in Fig. 3.10(a). To avoid this excessive dissipation by the Godunov method and instability by the Roe method, the combination of the Godunov and Roe methods is employed. This combination is decided by carefully analyzing the fluxes given by two methods. In Fig. 3.11 the mass flux is shown at the boundary point at the centerline of domain, at which the difference in mass flux by two methods is the maximum.
65 Figure 3.11: Mass flux at the near-boundary point
It can be seen that the obtained mass fluxes start to merge at around five micro-seconds after beginning of the injection. This difference in mass flux is less for the regions furthest form the boundary. Therefore, either time or space can be kept as switching criterion to switch amongst the two methods. For the single plume injection, the criterion for switching is time period of 5 µs. For multiple plumes, for the first few grid points in the domain the Godunov method is employed while the Roe method is employed for the rest of domain. Plume pattern obtained for single plume by the Godunov method, the Roe method and their combination is shown in Fig. 3.12 It is evident that the experimental plume roll-up, is not captured by the Godunov method, because its vorticity source value (that is responsible for plume roll up) is low compared to the Roe method. Nevertheless, their combination gives comparatively high vorticity source value (see Fig. 3.10(a)) and some plume roll up is observed (see Fig. 3.12(c)). The viscous diffusion of vorticity, Fig. 3.10(b), as opposed to the baroclinic source term, may dominate at later time moments of plume evolution. As shown in Fig. 3.10(b), the value of this term stabilizes.
66 Figure 3.12: Plume concentrations at t = 300 µs a) the Godunov method, b) the Roe method, and c) combination of the Godunov and Roe methods
3.4 Multiple Plumes
The vorticity generation for the multiple ejected plumes can be quite different from that for a single plume as shown in the following numerical experiments with multi- ple plumes. Although in general the baroclinic vorticity generation is dominant for multiple plumes, the relative importance of the incident and reflected shock waves in vorticity generation largely depends upon the time interval between ejections of two plumes. In the following example the interval between two ejections is τ = 50 µs, where the conditions of injection plumes are the same and correspond to those for the single plume (see Table 3.1). The results for multiple plume ejections in comparison to the single plume are shown in Fig. 3.13 at t = 100 µs, right before the third plume is ejected. For the single plume ejection, the plume does not turn into the roll-up struc- ture at the considered time whereas for multiple plumes ejection the plume is split into symmetrical structure. This result of plume roll-up for multiple plume ejections
67 a) b) Figure 3.13: Plume concentrations at t = 100 µs a) single plume ejection, b) multiple plume ejections can be attributed to the fact that for multiple plume ejections the magnitude of vorticity source term is high enough, as compared to single plume ejection, to form much stronger vorticity inside the plume. The vorticity source term, for multiple plume ejections and for the single plume (for comparison), is shown in Fig. 3.14. The vorticity source term, equation (3.2), as plotted in Fig. 3.14 is integrated over where the plume concentration is more than 8 % for first plume.
a) b) Figure 3.14: Vorticity source a) single plume ejection, b) multiple plume ejections
At t = 55 µs the high peak magnitude of vorticity source term can be clearly seen for multiple plume ejections. This peak results form plume-reflected shock wave interaction. This plume interaction with the reflected shock wave is shown in Fig.
68 3.15. Plume interaction with the blast shock wave originated from the second plume gives rise to high baroclinic generation of vorticity.
Figure 3.15: Pressure contours at for multiple plume ejections
The other numerical experiment presented is that of multiple plumes with small time interval between two ejections. In this case the interval between two ejections is taken as τ = 5 µs and the results are shown in Fig. 3.16 as obtained by solving Euler and the NS equations. It can be seen that for both, Euler and the NS equations, the multiple plumes appear as a continuous slightly expanding columnar jet and the effect of viscosity for multiple plume ejections is not noticeable for the considered time interval. In this case, the majority of baroclinic generation is because of the interaction of plumes with the blast waves originated from next injected plumes. Since the propagation of unsteady blast waves is not affected by viscosity, the influence of viscous terms is not noticeable. Despite the high level of generated vorticity by multiple blast waves, the roll-up of individual plumes is not observed in Fig. 3.16 since the plumes are pushed forward by the high velocity of next emerging plume, as shown in Fig. 3.16(c), (d) (at t = 55 µs, after ejection of 11th plume), giving an appearance of single jet to the ablated plumes with short time interval between two ablations. Nevertheless,
69 a) b)
a) b) Figure 3.16: Comparison of Euler (left) and NS (right) for multiple plume ejections; concentration (top) and vector field (bottom)
70 formation of flow vortices is clearly observed in figure as a result of deposition of vorticity.
3.5 Shielding Effect
The laser fluence, spot area, pulse duration and corresponding initial/boundary con- dition for plume of carbon ablation are given in Table 3.1. For these data, the absorption of laser beam energy by the ejected plume material appears to have no significant effect on overall plume dynamics. Nevertheless, the presence of absorbing plume strongly affects the amount of laser energy delivered to the target. In the first model (equation (2.22)), the ablated massm ˙ for subsequent pulses is calculated by lowering the first plume’s ejection velocity. This reduction in velocity is directly proportional to the degree of attenuation given by equation (2.26). In the second model of shielding, based on the Hertz-Knudsen formula, the pressure in equation (2.24) is lowered for subsequent pulses due to attenuation given by equation (2.26).
From equation (2.24), T is calculated for attenuated ps(T ) and both are used to cal- culate ablated mass from equation (2.23). Mass ablated given by both methods is shown in Fig. 3.17 for different time intervals between two successive pulses. The ablated mass is not affected significantly by the choice of method. These methods of accounting for shielding are based on pure molecular absorption data that is taken from the study [104]. These data are valid for temperature range 1500-5000 K. As the aim here is to compare velocity based and pressure based calculations of shielding, the plume absorbs at a constant rate if the temperature is above 5000 K. In the later sections involving discussions on shielding these assumption is rectified to account for shielding by ionized particles. The ablated mass does depend on the time interval between two successive pulses. For the time interval of the order of nanoseconds, the most of laser energy is
71 Figure 3.17: Ablated mass for different time interval between two successive pulses for pressure based and velocity based models
Figure 3.18: Density contours after ablation of the 4th plume with different time interval between two laser pulses a) τ = 1 µs b) τ = 15 µs absorbed near target. In this case, both pressure and temperature of plume are high and, consequently, high values of σ and n imply high absorption. As the time inter- val between laser pulses is increased, the above-mentioned plume parameters reduce giving lower absorption. For longer time interval of 15 µs between two pulses, the P absorption path length, i dxi, becomes an important factor. The density contours for time interval of 1 µs and 15 µs are shown in Fig. 3.18. It can be seen that for a longer time interval between pulses the path length increases. The absorption of laser energy by every fresh plume is a dominating factor because of its high temperature, pressure and concentration. However, the absorption of laser energy by previously ejected plumes can affect the overall mass removal. In
72 Fig. 3.17, the mass removal for time interval 15 µs between two pulses is less than mass removal obtained for time interval 5 µs. Although in both cases the fresh (newest) plume absorbs the same fraction of laser energy, the plume path length, P i dxi, formed by previously ejected plumes absorbs more energy in the former case and ultimately results in low mass removal.
3.6 Multi-time Step Modeling
A following test case validates the multi-time step modeling as discussed in section 2.7. For this test case, i.e. 2D Riemann problem (configuration 4 from Ref. [2]), the initial data consist of a single constant state in each of four quadrants of the x − y plane. The initial conditions for V = (p, ρ, u, v) in the four quadrants are:
Vul = (0.35, 0.5065, 0.8939, 0) Vur = (1.1, 1.1, 0, 0) Vll = (1.1, 1.1, 0.8939, 0.8939) Vlr = (0.35, 0.5065, 0, 0.8939) where the subscripts ll, lr , ul and ur denote lower-left, lower- right, upper-left and upper-right quadrants, respectively. The problem is solved in the x − y region (0, 0.16) × (0, 0.19) and the four quadrants are given by dividing this region by two lines x = 0.08 and y = 0.1. The results computed by the Godunov method are presented in Fig. 3.19. Plume concentration is shown in Fig. 3.20 at time t = 70ns. This concen- tration is shown for the first plume before the ejection of the second plume. The pulse duration is 20 ns while the interval between two pulses is 50 ns ([50]). The smaller grid size is needed to capture the plume parameters (see Fig. 3.20) and thus to evaluate the shielding of subsequent incident laser pulses. The shielding of laser pulse for multi-time step modeling is calculated using
Lambert-Beer type formulation (Eq. 2.26). The ratio Eabsorbed at the target surface is E0 plotted against the number of pulses in Fig. 3.21. It can be seen that the progressive amount of laser energy is absorbed by subsequent pulses. The path length that laser
73 Figure 3.19: Density contours for configuration 4 (Ref. [2]), 2D Riemann problem
74 Figure 3.20: Plume concentrations before the start of second pulse and the multi-size grid
75 Figure 3.21: Energy reaching the target with increasing number of pulses beam has to travel before reaching the target increases with a new plume emergence as can be seen from the comparison of Fig. 3.20 and Fig. 3.22. The absorption cross section, σ, is directly dependent on temperature, and so does κ (Γ,T ) as seen from Eq. 2.27; therefore, the rate of absorption of laser energy decreases for subsequent pulses. There is an additional reason for this decrease as will be explained shortly. Nevertheless, the amount of absorption of laser energy for subsequent pulses increases, as can be seen from Fig. 3.21. The rate of energy decrease with number of pulses is varying because the rate of absorption of laser energy varies with the number of pulses. The rate of absorption depends on the molecular cross-section that depends on temperature. The variation of temperature along the path length P dx is different for every plume ejection. The product κ (Γ,T ) dx in Eq. 2.26 is responsible for shown variation of function in Fig. 3.21. This product changes its strength for every new ejection that makes the function non-smooth. In Fig. 3.22b the temperature before the stat of 11th pulse is shown.
76 Figure 3.22: Multiple-pulse plume modeling using the multi-size grid: a) plume con- centration and b) temperature before the start of the 11th pulse
The present model of shielding, as already mentioned in section 3.5 is based on pure molecular absorption, the data for absorption cross-section in such model are established for temperatures range 1500-5000 K and wavelengths < 532 nm. The gas molecule may undergo thermal decomposition at high temperatures and the experimental absorption cross-section show the deviation from the established curve fit beyond 5000 K as discussed in Ref. [104]. Therefore, the value of cross-section, σ, that enters in Eq. 2.26 is kept constant in the results of shielding to be discussed hitherto for temperature above 5000 K. The absorbing cross-section increases with temperature. Thus, for any new plume ejection if the temperature in the domain results above 5000 K (see Fig. 3.22b) the plume will absorb at a reduced constant rate.
3.7 Axisymmetric vs. Planar Plumes
The difference in plume dynamics and shielding for the axisymmetric and planar plumes is discussed in this section. The axisymmetric and planar plumes are impor-
77 tant because they arise in number of applications such as material removal, material deposition, material transformations, synthesis and structure formation. The results discussed in this section are obtained using multi-time step approach.
3.7.1 Dynamics of Single Plume
Concentration of plume material and the axial component of velocity along the center line for planar and axisymmetric formulations are shown in Fig. 3.23. The planar plume expands to a larger degree in the lateral direction than the axisymmetric plume. The axisymmetric plume travels faster in the axial direction because the plume velocity has higher component of velocity (compare Figs 3.23c and 3.23d). The second peak in velocity profile for planar formulation is due to the presence of the blast shock wave. This peak does not occur for the axisymmetric case on account of fast decay of shock wave. The shock waves for planar and axisymmetric cases can be seen in Fig. 3.24. It can be seen from Fig. 3.24b that the shock strength is much higher for the planar case than that for the axisymmetric case. The reason for faster decay of shock wave in the latter case is a higher ratio of furnace-to-target volume for the axisymmetric case compare to that for the planar case. The higher volume of surrounding gas tends to smooth out the high pressure of injected plume. The axisymmetric plume remains forward-directed jet-like flow resulting in moderate lateral expansion of the plume. On the contrary, the planar plume spreads in radial-direction driven by the Sedov-Taylor type flow. It should be noted that the plume is still forward directed because initially the plume is injected in the x (axial) direction in both cases.
3.7.2 Dynamics of Multiple Plumes
In this section, the plume pattern for multiple laser hits is studied for axisymmetric and planar cases. Plume concentrations with the given time interval, τ = 5µs, be-
78 Figure 3.23: Plume patterns for planar (a,c) and axisymmetric (b,d) cases: (a,b) plume concentrations and (c,d) axial component of velocity along the centerline
79 Figure 3.24: Propagation of pressure wave in plume: a) planar case, b) axisymmetric case and c) pressure along the center-line
80 Figure 3.25: Plume patterns and temperature after the 7th laser pulse with the time interval τ = 5µs between two plumes: (a,c) planar case; (b,d) axisymmetric case; (a,b) plume material concentrations; (c,d) temperature along the center line tween two successive laser hits for axisymmetric and planar formulations are shown in Fig. 3.25. If τ is less than 5µs, the plumes merge together and form a continuous plume pattern. Similar to the single plume case considered above, the planar plume expands more in the lateral direction than the axisymmetric plume. In the axisym- metric case, the plume material has large concentration at the center line, whereas for the planar plume the large concentration of the plume material appears at the near-axial and peripheral areas. Likewise the single-plume case, the shock strength is different in two cases. In the planar case, the high-pressure semi-spherical shock wave drags along with it the previously ejected plumes in the radial direction and,
81 as a result, the plume material is distributed over the domain. The shock wave is weak in the axisymmetric case and does not drag the plume particles; nevertheless, it produces a region of low pressure behind it in the central part of domain (see Fig. 3.24b); and the plume particles gather towards the center of plume as shown in Fig. 3.25b. Low pressure zone is created in the planar case too but the strength of the shock wave is high tending to drag the plume particles in the radial direction to follow the semi-spherical shock wave.
3.7.3 Comparison of Shielding
The effect of plume shielding for planar and axisymmetric cases is shown in Fig. 3.26. With the rectification in accounting for value of σ suggested in section 3.6, the plume shields considerable amount of energy. This can be seen from the comparison of Fig. 3.26 and Fig. 3.17. For high temperatures in the plume, ionization effect on shielding is evaluated and discussed in the later section. The shielding of laser beam by plume is a function of plume temperature and plume concentration. For time interval between pulses τ ≤ .5µs the concentration of plume material and its temperature are nearly the same for planar and axisymmetric cases, as seen in Fig. 3.27a. Therefore, the amount of shielding is nearly the same for two cases. The peaks in the temperature profile in Fig. 3.27a occur due to passing of shock wave from a newly ejected plume. For the larger time interval between laser pulses, τ = 1µs, the planar plume shields more than the axisymmetric one. Although the concentration near the center of plume for τ = 1µs in both cases is the same, the temperatures are different for planar and axisymmetric cases. The temperature of a near-center line particle for both cases with time is shown in Fig. 3.27b. It can be seen that temperature for the planar case is higher than that for axisymmetric case resulting in more pronounced shielding effect. These temperature oscillations are caused by the following phenomena (i) particles are trapped is in the
82 Figure 3.26: Degree of laser beam shielding for different time intervals between two pulses: a) t = 50ns; b) t = 1µs; c) t = 5µs and d) t = 10µs
83 Figure 3.27: Dynamics of temperature of a representative plume particle for multiple pulse laser hits: a) t = 50ns and b) t = 1µs vortex and (ii) particles cross weak reflected shock waves associated with previous pulses. In general, as shown in Fig. 3.25b, the temperature across the plume is higher for planar plume. The degree of shielding is a non-monotonic function of the laser pulse number shown for the time interval τ = 5µs and τ = 10µs between two laser pulses (see Fig. 3.26c,d). To explain this non-monotonic behavior, the density and temperature contours after the first and the second pulses are shown in Fig. 3.28. It can be seen that the plume density is low (Fig. 3.28 (a)) while the second pulse is being hit to the target. Although the temperature at this time moment is high (Fig. 3.28c) as compared to that while the first pulse is being hit, the density and hence the number density Ng that enters in Eq. 2.27 is low. It should be noted that in Eq. 2.27 the effect of temperature on shielding enters through the absorption cross-section σ that is an increasing function of temperature. Thus, the effect of low density is higher than the effect of high temperature at the instant when the second pulse is being hit making the shielding pattern non-monotonic (see Fig. 3.26c).
84 Figure 3.28: Density and temperatures contours after first two pulses: (a,b) density, (c,d) temperature; (a,c) after the second pulse; (b,d) after the first pulse
85 Figure 3.29: Degree of ionization ζ and temperature before ionization after the fifth pulse: (a,c) planar; (b,d) axisymmetric; (a,b) degree of ionization ζ; (c,d) temperature K
3.8 Ionization Effect
As shown in Fig. 3.29, the temperature of plume particles is as high as 10000 K for the second plume ejection, and this temperature can be progressively higher with subsequent plume ejections. For such high temperatures plume particles get ionized. In this section, ionization within the plume and its effect on shielding are described. Fig. 3.29 shows the contours of degree of ionization and temperature for the time interval τ = 5µs between two pulses. It is shown in Fig. 3.29 that the value of ζ is large in the regions with high temperature. Therefore, ζ inside the plume is high whenever the plume interacts
86 Figure 3.30: Degree of ionization ζ and δ (Eq. 2.29): a) degree of ionization as a function of temperature at 100 atm and b) δ Vs. T with shock. This plume-shock wave interaction occurs either from reflected shock wave of previously ejected plumes or from the emerging shock wave of newly ejected plume. From Fig. 3.30a it can be seen that ionization of carbon is important for temperatures above 5000 K at high initial pressure of 100 atm. Let us introduce
1 hν δ = 1/2 1 − exp − , (3.4) T kBT that is the temperature-dependent term of Eq. (2.29). This term is plotted in Fig. 3.30b for the entire possible range of temperatures, and its value for most of the highest temperature range in which the laser beam travels is of the order of 10−2. The typical laser wavelength for carbon ablation is λ = 1.064µm; the single stage of ionization employs Z = 1 and Ne = Ni. Taking these numbers, the absorption coefficient (Eq. 2.29) reduces to the following approximate formulae
−37 2 κIB = 1.37 × 10 · Ni . (3.5)
87 The decisive factor for the absorption coefficient in the considered inverse Bremsstrahlung mechanism at higher temperatures is the number density of ions. This number density is derived from Saha equation (Eq. C.1) and is shown in Fig. 3.31. The maximum number density of ions in both axisymmetric and planar cases is 1017, and by Eq. 3.5 this ion density gives the absorption coefficient value of 10−3. Therefore, the effect of ionization on shielding is negligible. It should be noted that ionization itself is not negligible, but the plume density being low is responsible for negligible amount of shielding by ionized particles. The same gas absorbs considerable laser energy by normal molecular absorption. If for the same temperatures the pressure levels are of one order of magnitude higher in the laser furnace (that is, very dense furnace gas), the number density of ions becomes 1018. In such a case the absorption coefficient due to ionized particles is of the order of 10−1 (see Eq. 3.5) and the shielding by ionized particles becomes important. Fig. 3.31c,d shows that the temperature is reduced due to the presence of ionization. Compared to the temperature field in Fig. 3.29c,d, the temperature is reduced significantly when ionization is accounted for; however, the plume pattern shows only modest changes. In this Chapter the results obtained for LIP are discussed. The results for LIP are compared qualitatively and quantitatively with experimental results for plume evolution up till 100µs. The effect of furnace gas turbulence on plume patterns is investigated in Section 3.2. The relative effects of inviscid and viscous terms on LIP and particle’s trajectories and temperatures of LIP are given in this section. Then in Section 3.3, a post-processing model based on vorticity transport equation is proposed that helps in understanding the plume dynamics in terms of baroclinic generation of vorticity and viscous deposition of vorticity in plume. The plume dynamics for axisymmetric (hole-drilling) and planar (channel-cutting) cases is discussed along
88 −3 Figure 3.31: Contours of number density of ions (ζ × Ng) in cm and temperature after ionization shown after the fifth pulse: (a,c) planar; (b,d) axisymmetric; (a,b) ion density; (c,d) temperature
89 with the results for mulit-time step modeling. In Section 3.5 the amount of incident laser beam shielding due to LIP is calculated for laser pulses with different time interval between pulses. The effect of ionization on laser beam shielding and plume dynamics is also described in Section 3.8.
90 CHAPTER IV APPLICATION OF DEVELOPED MODEL IN MODELING PLUMES GENERATED BY LOCAL INJECTION OF CARBON ABLATION PRODUCTS IN HIGH-SPEED FLIGHT
The front detached bow shock wave causes the heating of external surface of re-entry vehicles and missilles. This subsequently leads to cracking of these surfaces. The cracks formed are of millimeter size, and plumes are formed due to the high pressure pyrolysis gas coming out the surface. The conditions of plume so formed are similar to those for the case of laser ablation. In this chapter the developed computational methodology is used to study the dynamics of these plumes and heat transfer to the vehicle thermal protecting shield (TPS) surface caused by these plumes. This chapter is composed as follows. In Section 4.1, a general introduction to the model problem and related work is given. Section 4.2 deals with further detail description of the model. The effect of flight altitude on plume dynamics and heat transfer is discussed in Section 4.3. The effect of pyrolysis injection pressure on heat transfer is discussed in Section 4.4. The injection of multiple plumes is considered in Section 4.5. The effect of vehicle body shape on heat transfer is given in Section 4.6.
4.1 Background and Previous Research
Extensive research has been conducted in predicting the ablation rates of TPS due to hypersonic flow around aerospace vehicles. When subjected to increasing heat flux or temperature, thermal protection materials may pyrolyze and/or ablate. Pyrolysis
91 is chemical decomposition in the interior of a TPS material, which releases gaseous by-products without consuming atmospheric species. Ablation is a combination of vaporization, sublimation, and reactions (such as oxidation) which convert solid sur- face species into gaseous species (Ref. [108]). The majority of published ablation and pyrolysis studies are limited to the set-up of uniformly distributed ablated mass flux that varies gradually along the TPS surface and does not cause separation of the boundary layer. Local non-uniform ablation can produce local plumes having much higher pressure, injection speed, and density compared to those corresponding to the aver- age ablation speed under the same flight conditions. The severely under-expanded gas escapes through the cracks as a series of high-pressure plumes. Small scale but high-intensity phenomena such as locally non-uniform mass transfer across the TPS surface can greatly affect the overall flow field about the vehicle, and heat exchange between the gas surrounding the vehicle and the TPS surface, and the overall abla- tion and pyrolysis rates. Escape of pyrolysis gases dramatically changes the surface temperature and can trigger transition to turbulence as described in the paper [109]. Prior studies regarding transition to turbulence caused by pyrolysis are based on uni- form diffusion of the produced gas through micro-pores and not through macro-cracks resulting from spallation (that is, formation of cracks through which the pyrolysis gas escapes) [110–112]. The current study is aimed at modeling of the heat transfer mod- ulation caused by the local plume(s) emergence for the range of flight altitude, initial plume pressure and the shape of the TPS. The fluid dynamics of the plume is much more involved compared to the extensively studied ”jet in cross-flow” set-up because of (i) interaction of pyrolysis plume with the detached bow shock wave in hypersonic flight and (ii) enhanced mixing of plume with surrounding gas because of the pressure wave associated with the under-expanded plume.
92 Emerging plumes can alter the pressure gradient and the shape of velocity profile around the TPS causing formation of a separation vortex with enhanced con- vective heat transfer from the hypersonic flow to the TPS. On the one hand, intense cooling from pyrolysis gases may increase thermal stresses and enhance spallation. On the other hand, in principle pyrolysis gases can be used for desired cooling of the TPS if their escape routes are properly organized. Overall, the local heat- and mass- transfer processes will be greatly amplified in comparison to those in the laminar boundary layer of regular ablation. Modeling and scaling of these phenomena is one future step in the current research that will map out the uncharted regimes beyond spatially homogeneous ablation. One reason for non-uniform ablation is impingement of solid objects such as debris, ice, and/or foam into TPS elements[113]. Another reason is the formation of cracks and gas escape routes because of the nature of TPS materials at high temper- ature. The pyrolysis gas of carbon-phenolic TPS can be approximately considered a mixture of H, H2, CO, and CH4 [114]. The temperature of pyrolysis gas ranges from 800K at the inner surface to 1800K at the outer surface of pyrolysis zone, based on flight thermocouple data for Pioneer-Venus[114]. Pyrolysis gas pressure inside the TPS of Pioneer-Venus is between 8 and 30 atm for the four Venus vehicles (see Ref. [114]). Such a high internal pressure in pyrolysis layer induces spallation. The ob- served rough surface with deep cracks in TPS confirms the importance of modeling of non-uniform ablation. Refs. [115, 116] show a view of millimeter-scale cracks in a charring ablation observed with the VKI plasmatron. The typical size of cracks that appear because of intensive heating is of the order of millimeters[115, 116] and this size was adopted in the current study. In a broader sense, the size of initial plume diameter ranges from microns to centimeters depending on whether the ablated gas escapes through the pores in the
93 TPS material or its ablation is caused by mechanical damage. In a composite, the space scales range from individual fiber diameter (7µm) to the apparent diameter of tows and warps (300-500m)[117, 118]. On the limit of small scales of cracks, carbon- carbon composites can generally contain a significant initial amount of cracks, voids and de-bonded fiber/matrix interfaces, which can yield a porosity level as high as 20%. The typical size of cracks/voids is 14 micron and the size of de-bonded interfacial gaps is 0.5 micron. On the other hand, the mechanical damage to the TPS may cause formation of much larger local intense ablation spots in the TPS[119].
4.2 Description of Model
In this section the initial conditions for flight and plume emergence are given, and they discussed for flight at ground level. A comparison between viscous and inviscid models is shown.
4.2.1 Conditions of Flight and Emerging Plume
The axisymmetric ellipsoid bluff TPS shape chosen for this study is shown in Fig. 4.1a. The plume emergence for the TPS shape considered in Ref. [120] is studied in Section 4.6. The Mach number of flight for the current study is 7. For this Mach number a detached bow shock wave exists as seen in Fig. 4.1b. Parameters of the atmosphere for different altitudes that are being investigated are presented in Table 4.1. The initial temperature of the TPS is taken equal to the surrounding air temperature at all altitudes. This temperature is an after-shock temperature for respective altitude as given in Table 4.1. The initial conditions of injected pyrolysis gas are given in Table 4.2 (see Ref. [114]). The pyrolysis gas pressure inside the TPS may vary between 8 and 30 atm. It is assumed that the pyrolysis pressure inside the TPS is independent of the choked
94 Table 4.1: Flight regimes and parameters of surrounding gas Flight Flight Mach number, M = 7 altitude (km) Parameters before and after the shock Flight velocity (m/s) Pressure (atm) Temperature, K Density (kg/m3) 0 1 58 288 3055 1.2 6.61 2381 20 .08 4.64 217 2302 .12 .71 2066 30 .01 .58 233 2472 .01 .08 2141
Table 4.2: Plume injection parameters Summary of plume injection parameters Pressure 30 atm Temperature 1000 K Density 10.45 kg/m3 Velocity 400 m/s Spot size 6 mm Time of Injection 5µs pressure at the TPS surface and the initial plume pressure is taken equal to 30 atm unless specified otherwise. The temperature of pyrolysis gas ranges from 800K at the inner surface to 1800 K at the outer surface of pyrolysis zone. The initial temperature of plume injection is taken equal to 1000 K in the current study. The duration of an individual plume injection is taken equal to 5 microseconds. The heat transfer is evaluated by the difference between initial TPS surface temperature and the convected plume temperature near the wall (see section 4.2.2).
4.2.2 Plume Emergence and Heat Transfer for Flight at Ground Level
The flight at ground level is taken as a baseline case. Figure 4.1a shows the simulated body covered with the TPS, together with the computational domain and boundary conditions. The plume emergence point is taken to be few grid cells above the stag-
95 nation point. The computed plume concentration is shown in Fig. 4.1b, while the distribution of pressure is shown in Fig. 4.1c and heat transfer at the TPS wall for the flight at the ground level is shown in Fig. 4.1d. A detached shock is evident, which is important in the heat transfer mechanism between the plume and the TPS (see Fig. 4.1c). The intensity of heat transfer between the plume and the TPS is estimated by ∆T = TNW − TW , where TW is the TPS temperature for the steady flight before emergence of plume and TNW is the gas temperature at the point located at the distance ∆n (equal to the grid step) in the normal direction to the TPS at the time moment t after the plume emergence. The TPS surface is curvilinear so the TNW is determined by the interpolation of temperature values at neighboring grid points. The value of ∆T is positive if heating of the TPS surface occurs and negative if the
∆T plume cools the TPS surface. The estimates of heat flux Q = ∆n at the TPS surface are listed in Table 4.3 for various cases to be considered. The normalization parameter
Q0 is chosen as the value of ∆T obtained at t ≈ 10µs, which is the estimate of the maximum absolute value at all selected time moments for all altitudes. The values of normalized Q/Q0 are presented in the subsequent discussion. The pressure on ground level behind the shock wave (Fig. 4.1c) is larger than the initial plume pressure. Consequently, the plume does not depart from the wall or traverse into the domain. The plume remains attached to the TPS wall and it is dragged with the flow along the wall (see Fig. 4.1b). It can be seen that the heat transfer attains its peak in the vicinity of plume emergence that is caused by the initial cold temperature of the plume. At ground level, the heat transfer due to the plume convection at the TPS surface is weak (see Fig. 4.1d) because the high pressure does not allow the mushroom-type plume pattern (to be observed later) to be formed. Therefore, the heat transfer observed in figure decreases rapidly as
96 Table 4.3: Value of normalization parameter at different altitude Altitude Estimate of heat flux, Q = ∂T/∂n (km) Location of peak Q(K/m) at dif- along the wall X ferent time mo- (m) at different ments time moments t = 10µs t = 22µs t = 30µs t = 10µs t = 22µs t = 30µs 0 .0213 .0213 .0216 27.09E5 8.88E5 5.21E5 20 .0216 .0251 .0265 51.27E5 13.18E5 15.32E5 30 .0213 .0224 .0227 91.54E5 80.93E5 76.27E5 the plume is convected. The x-coordinate of the maximum heat transfer is shown in Table 4.3. The plume being attached to and dragged along the wall is observed for the altitude of flight up till few kilometers from the ground. Hence, the above described heat transfer pattern remains almost the same for the first few kilometers up the ground. When the altitude approaches 20km, the plume leaves the surface and forms a mushroom-pattern, which is discussed in the following sections. Compared to higher altitudes discussed later, the convection of plume along the TPS is slow because the plume remains in the low-speed near-wall area. At t ≈ 10µs, in Fig. 4.1d some positive heat transfer area is observed upstream and downstream of the negative peak. This is due to the low pressure of plume that draws in the surrounding hot flow near the stagnation point.
4.2.3 Comparison of Viscous and Inviscid Formulations for Plume Dynamics
In order to examine the importance of viscosity in the simulation, the plume patterns and heat transfer at 20 km are obtained by solving the inviscid Euler and viscous Navier-Stokes (NS) equations (see Fig. 4.2). Note that in this and subsequent figures, the plume fluid is distinguished by visualizing γ, the ratio of specific heats. The difference between computational results obtained by two approaches is minor because
97 Figure 4.1: Schematic of the problem for flight at the ground level: a)the shape of the representative TPS and computational domain, b) plume concentration, c) pressure distribution, and d) heat transfer at the TPS wall
98 Figure 4.2: Comparison of viscous and inviscid models for flight at the altitude of 20km at t ≈ 45µs: a) plume concentration obtained with Euler equations, b) plume concentration obtained with NS equations and c) heat transfer coefficient obtained by inviscid model, d) heat transfer coefficient obtained by viscous model the initial size of the ablative plume spot is 6 mm, whereas the boundary layer thickness is of the sub-millimeter scale. Besides, the velocity and pressure of plume injection is one order of magnitude higher than corresponding values of surrounding flow in boundary layer. Thus the plume is developing beyond the near-wall boundary layer. There is no significant difference in the plume pattern for these two formulations and hence, further simulations are performed using the Euler equations because of the lesser amount of required computational time. The peaks in heat transfer at the TPS wall (see Fig. 4.2 (c-d)), are similar for inviscid and viscous formulations.
99 4.3 The Effect of the Flight Altitude on Plume Dynamics and Heat Transfer
Fig. 4.3 shows the plume concentrations as the plume evolves at 20 km. The emer- gence of plume creates the non-uniformity in the flow behind the shock wave. At 20 km the bow shock wave remains strong (see Table 4.1 for pressures before and after the shock wave) and the post-shock flow can be altered only temporarily by the plume emergence. The plume core bends initially because the flow accelerates through it along the streamlined surface of the TPS. At later time moments, the acceleration of flow between the detached front shock wave and the TPS is almost uniform across the plume, as seen from the vector field in Fig. 4.3(a-b). This turns the kidney-shaped plume core formed by two vortices into an oval structure (see Fig.
4.3c). In Fig. 4.3(c-d) the vector field is modified by subtracting a velocity ~vref at the approximate core center of the plumes. The shape is a typical Kelvin oval formed by letting the stream flow normal to the vortex pair. This plume shape is then maintained throughout its course over the body. In terms of plume density, at t ≈ 20µs (Fig. 4.4a) the high density in the flow field is a narrow kidney-shaped region of plume. This region is scattered after few microseconds and turns into oval as shown in Fig. 4.4b. Note that the area of the oval is larger than the area of the kidney-shaped plume and the density of the plume material is smaller (compare Figs. 4.4a and 4.4b). The temperature of oval-shaped plume becomes more close to the temperature of surrounding gas. In the current conditions the temperature of oval plume becomes larger than the initial temperature of the plume (that is maintained for the kidney-shaped plume) (compare Figs. 4.4c and 4.4d). As a result, the cooling effect of the plume is softened. The temporarily disturbed pressure field behind the front shock wave is re- stored between 20µs and 25µs (see Fig. 4.4b). This restoration of pressure field gives the uniform acceleration across the plume turning the core into an oval as shown
100 Figure 4.3: Plume concentration at flight altitude of 20 km: a) original velocity vector field at t ≈ 20µs, b) original velocity vector field at t ≈ 25µs, c) modified vector field ~ ~ ~ ~ V − Vref at t ≈ 20µs and d) modified vector field, V − Vref at t ≈ 25µs
101 Figure 4.4: Plume density and temperature at t ≈ 20µs (left) and t ≈ 25µs (right) : a-b) temperature, c-d) density
102 Figure 4.5: Flowfield near the stagnation point: a) pressure contours after plume injection over the body in still air and b) temperature contours along with the vector field at altitude of flight of 20 km at t ≈ 10µs in Fig. 4.3 (b). In other words, the detached front shock wave smoothes any non uniformity in the flow behind the shock wave at the altitude of 20 km. The interaction of shock wave and injection pressure wave affects the heat transfer to the body by altering the temperature field around the body. The pressure wave associated with the plume is shown in Fig. 4.5a. The strongest interaction of these waves occurs at the stagnation point: thus the high temperature is attained near the stagnation point. The hot gases are entrained by the plume that affects the heat transfer at the TPS surface. Therefore, some positive heat transfer is observed around the stagnation region (see Fig. 4.5b). In the later moments the heat transfer is the maximum near the plume tail when the plume is pushed back to the body by the bow shock wave as explained below. Besides, at all time moments at this altitude the modest positive heat transfer is observed around the stagnation point (x .02m).It can be seen at t 30µs that the area with positive heat transfer around stagnation region is larger as compared to that at t 22µs.
103 Figure 4.6: Plume dynamics at the altitude of flight of 20km: plume concentration at a)t ≈ 10µs, b)t ≈ 22µs, and c)t ≈ 30µs and d) heat transfer at the TPS surface
104 Figure 4.7: Plume dynamics at the altitude of flight of 30km at t ≈ 25µs: a) plume concentration and b) temperature contours
In Fig. 4.6 the plume concentration and the heat transfer are shown at the corresponding time moments at the altitude of flight of 20km. The heat transfer is the maximum (see Fig. 4.6 (d), t ≈ 10µs) at the location of plume emergence in the initial stage of plume evolution. At later time moments the cooling heat transfer reaches its maximum near the plume tail when it touches the TPS wall. It should be noted that the peak in heat transfer is higher at t ≈ 30µs than that at t ≈ 22µs.This occurs because the detached bow shock wave, which was distorted by the plume injection, restores and pushes the plume towards the wall. At the higher altitude of 30km, the evolved plume is shown in Fig. 4.7a. In this case, the strength of shock wave is low in terms of pressure difference across the shock wave (see Table 4.1) and hence the plume is able to penetrate through the shock wave as shown in Fig. 4.7b. The emergence of plume distorts the shock wave to much larger extend at higher altitudes (compare Fig. 4.4b and Fig. 4.7b). The plume remains detached from the TPS and maintains its kidney shape for longer time as can be seen in Fig. 4.7. The heat transfer for the altitude of flight of 30 km is shown in Fig. 4.8b. At 30 km the heat transfer due to the plume convection decreases with time as the plume
105 Figure 4.8: Interaction of waves and heat transfer at the altitude of flight of 30 km: a) pressure contours (t ≈ 22µs) and b) heat transfer at wall stays away from the wall on account of its initial high pressure (see Fig.4.7). The cooling effect is caused by the interaction of the right lobe of kidney-shaped plume with the TPS wall, as shown in Fig. 4.7b. The temperature of the kidney-shaped plume is smaller than that for the oval plume (see discussion in the beginning of this Section); therefore, the cooling effect is more significant compared to that at the altitude of flight of 20 km. The positive maxima seen in the heat transfer graphs (Fig. 4.8b) are caused by the out-going pressure associated with the under-expanded plume (see Fig. 4.5). Unlike at the lower flight altitude of 20 km, these peaks are clearly seen downstream of the plume. In addition there is the interaction of plume- associated pressure wave with the front shock wave, as can be seen in Fig. 4.8a. This interaction of waves significantly affects the heat transfer. For longer time of plume evolution at the altitude of flight of 30 km, sig- nificant heating around the stagnation region of the body is observed. In Fig. 4.9, pressure contours and heat transfer for the corresponding time moments are shown. It can be observed in Fig. 4.9d that at the TPS surface there are noticeable positive peaks (heating) which increase with time. This heating is caused by the shock wave associated with plume emergence. This shock wave interacts with the bow shock
106 Figure 4.9: Formation of secondary shock waves: pressure contours at; a)t ≈ 32µs, b)t ≈ 41µs, and c)t ≈ 50µs and d) heat transfer at the TPS wall for all considered instants of time wave and reflects back on the TPS (see Fig. 4.9 (a-c)). In Fig. 4.9a, the bow shock wave is displaced momentarily due to this interaction and the reflection is smaller. Subsequently, at later time instants, Fig. 4.9 (b,c), this reflection and formation of secondary shocks is more pronounced causing increased heating in the stagnation region. In summary of this Section, at all altitudes, the temperature of emerging plume gives the normalization coefficient for heat transfer, that corresponds to the maximum (cooling) heat transfer rate at that altitude. The intensity of cooling heat transfer along the TPS wall at later time moments is determined by the plume
107 convection. The location of maximum cooling effect is the farthest downstream for the flight at the altitude of 20km and is associated with the tail of the plume. At ground level, the convection is relatively slow whereas at the altitude of 30km the plume propagates farther in the direction perpendicular to the TPS wall and the primary cooling is associated with one lobe of the kidney-shaped structure of plume. Apart from cooling, some heating effect is observed at all altitudes. Except at the ground level, this heating is caused by the out-going pressure wave due to the under-expanded plume injection. At the flight altitude of 20km this effect is modest around stagnation region, whereas at the flight altitude of 30km the significant effect of this pressure wave is seen upstream and downstream of the plume over significantly larger area of the TPS. At later time moments the reflections of out-going pressure wave causes significant heating at stagnation region. This heating becomes more significant than cooling due to plume convection (see Fig. 4.9d).
4.4 Heat Transfer for the Range of Plume Injection Pressures
The heat transfer for three different plume injection pressures (8atm, 16atm, and 24atm) is shown in Fig. 4.10 at the flight altitude of 20km. Initial temperature of the injected plume is the same for all considered cases. The heat transfer between plume and the TPS appears to be different for these injection pressures. In Fig. 4.10, the normalization parameter Q0 for all considered injection pressures is taken equal to the value attained at t ≈ 10µs for the injection pressure of 24 atm. The maximum cooling heat transfer occurs at t ≈ 10µs for all injection pressures except for 16 atm, where the highest cooling location moves downstream, but peak values do not change significantly (in this case the maximum is observed at t ≈ 22µs). The peak heat transfer is largest for 24 atm.
108 Figure 4.10: Heat transfer with inset image of plume concentration at t ≈ 30µs for the range of initial plume injection pressures: a) 8 atm, b) 16 atm and c) 24 atm
109 With increasing injection pressure, the plume propagates farther away from the TPS surface driven by its initial pressure. This is shown in the inset image of plume concentration in Fig. 4.10. For higher injection pressure of 24 atm (see Fig. 4.10c) the plume core is separated from the TPS surface thus reducing the heat convection. For 24 atm the peak in heat transfer increases due to t ≈ 30µs, unlike for the smaller injection pressures where the peak decreases with time (see Fig. 4.10a,b). This increase in heat transfer is caused by the high density plume which is pushed closer to the TPS surface by the recovering detached shock wave. For low injection pressure of 8 atm, the rate of heat transfer decreases fast with time (see Fig. 4.10a). Compared to higher injection pressures, the plume has a lower level of concentration (see inset image in Fig. 4.10a) indicating that the low-pressure plume mixes faster with surroundings. The maximum heat transfer for injection pressure of 16 atm is attained at t ≈ 22µs and the peak in heat transfer shows no significant change in time. Overall, for low injection pressure of plume the dynamics resembles one cor- responding to the flight at ground level (see Section 4.2.2). The plume is dragged along the TPS wall, the convection rate and convective speed of plume are relatively slow, and heat transfer diminishes faster than for larger injection pressures. For higher injection pressures the plume convection can be more important for later time moments. For intermediate plume injection pressure of 16 atm the heat transfer rate is relatively steady in time. The magnitude of heat transfer is the largest for high pressure plume of 30 atm for all time moments.
4.5 Heat Transfer for Multiple Plume Injections
Ejection of multiple high-pressure under-expanded plumes may appear as a sequence of increasingly complex scenarios including multiple plumes originating from the same
110 Figure 4.11: Emergence of two consequent plumes at the altitude flight of 20km. The plume material concentration is shown at the following time moments; a)t ≈ 40µs, b)t ≈ 50µs and c)t ≈ 60µs and d) heat transfer at the wall location and multiple spatially distributed plumes. This section is focused on the interaction of plumes and interaction of pressure waves originating from the ejection of under-expanded plumes for a representative case of a pair of consecutively emerging plumes. Fig. 4.11 shows plume pattern and heat transfer for two plumes when the second plume emerges at the tail of the first plume at t ≈ 35µs after emergence of the first plume. The TPS area in contact with the tail of the first plume may have largest thermal stress because of fast cooling that causes formation of cracks and emergence of the second plume. The normalization parameter Q0 for heat transfer is
111 the value obtained at t ≈ 10µs after the injection of first plume (see Table 4.3). For later time moments the peak in heat transfer is observed at the location of the tail of new injected plume (see Fig. 4.11). These locations are x ≈ .035m for t ≈ 50µs and x ≈ .04m for t ≈ 60µs. For these time moments the heat transfer is dominated by the second plume convection and its pattern is similar to what is observed for single plume. It should be noted, however, that the magnitude of peak cooling heat transfer is greater than unity for multiple plume injections because the second plume emerges into the first cold plume. This shows that the intensity in heat transfer increases with new plume injections. The positive heat transfer pattern upstream of the plume injection is qual- itatively similar to that for the single plume injection. As discussed before, these positive peaks were created by the propagation of plume pressure wave and its inter- action with the detached front shock wave. This pressure wave is able to heat up the flow around the stagnation region (see Fig. 3.20d at x ≈ .02m).
4.6 Heat Transfer for Different TPS Shapes of Comparable Sizes
The heat transfer and density contours for plume injection for two different body shapes are shown in Fig. 4.12. In case A, the body is formed of a segment of an ellipse as in previous cases. In case B the body is formed by a segment of a circle connected to a straight line making the angle of 10 degrees with the horizontal axis corresponding to the first test case in Ref. [120] and is typical of earth re-entry vehicles used for planetary missions.
The normalization parameter Q0 for heat transfer is the value obtained at t ≈ 10µs for case B. It is observed that case B has higher heat transfer compare to case A. Peaks observed in both cases at t ≈ 22µs and t ≈ 30µs appear due to the convective heat transfer of plume with the TPS. In the case B these peaks are
112 Figure 4.12: The effect of TPS shape at the altitude of flight of 20km: a-b) density contours at t ≈ 10µs with inset image showing plume concentration and modified velocity vector field, and c-d) heat transfer at the TPS wall.
113 observed for x ≈ .033m and x ≈ .041m, respectively; whereas in case A, these peaks are observed for x ≈ .029m and x ≈ .033m, respectively. This indicates that the plume travels faster in case B because of higher pressure and velocity gradients along the TPS surface that accelerates the plume. In case A the flow is similar to that about the front portion of the sphere whereas in case B the flow similar to the impinging flow into flat plate prevails at the front TPS surface. The plume patterns are shown in the inset image of Fig. 4.12. The vector ~ ~ field V − Vref is plotted to show the vortices by subtracting the velocity at a given point in the plume core for the respective plumes. It can be seen from the figure that the vortical structure is larger in case B that causes more intense mixing. In both cases positive heat transfer occurs upstream of plume for t ≈ 22µs and t ≈ 30µs. This is due to the pressure wave associated with plume emergence. In case B the strength of front detached shock is lower than that in case A. The initial strength of plume pressure wave is the same in both cases; consequently this wave alters the flow field to a larger degree in case B as compared to the case A. To summarize, in this Chapter the developed numerical infrastructure for LIP is applied to study the pyrolysis plume dynamics typical for re-rentry vehicles. The discussion of pyrolysis plume dynamics for flight at low, medium and high altitude is given. The effect of plume injection pressure on plume dynamics and heat transfer to the TPS is investigated followed by discussion of multiple plume injections. Then, the effect of body shape on heat transfer is given.
114 CHAPTER V SUMMARY AND FUTURE SCOPE
The plume dynamics that occurs in laser ablation in presence of background gas is modeled and numerically simulated. The modeling effort helps in revealing and understanding important features of LIP. The summary of numerical and algorithmic developments and major features noticed in LIP as well as locally ablated pyrolysis plumes are as follows:
1. Numerical algorithmic achievements
1.1 The turbulent, viscous, and inviscid numerical models of plume dynamics for single and multiple laser ablated plumes models have been developed, im- plemented and tested. Comparison of the Godunov, the Relaxing TVD and the ENO-Roe methods for discretization of convective terms in governing equations with strong shock waves is conducted. The comparison is presented in terms of plume patterns obtained numerically. Although the Relaxing TVD method is much simpler for implementation compare to the ENO-Roe and the Godunov methods, there is a noticeable difference in obtained plume patterns. Therefore, the ENO-Roe and the Godunov methods was selected for further investigation of physical phenomena in this study. The ENO-Roe method appeared to be simpler than the Godunov method to implement and the method gave mini- mum dissipation at the same time. However, the former was proved unstable for high injection velocities and pressures typical for LIP. The first-order Godunov method though somewhat dissipative, is stable for such velocities and pres-
115 sures and hence, the combination of two methods is implemented that captured adequetely the millisecond-scale roll-up of LIP.
1.2 The combination of nonlinear Godunov and linearized Roe methods for discretization of plume gas dynamic equations is suitable for modeling plume dynamics in laser ablation of carbon. On the one hand, this combination can handle the high injection velocity and pressure typical for such a process. On the other hand, the interaction of reflected shock waves with plume determines its roll-up and, consequently, the absorption properties of plume can be captured correctly by the combination of two schemes.
1.3 The role of turbulence was evaluated by direct simulation of plume in a given turbulent gust. This model helps in predicting the effects of turbulence. More sophisticated models of turbulence can be implemented only when they are needed which in turn, saves computational efforts and CPU time.
1.4 An analytical model based on vorticity transport equation is proposed. This model helps in understanding the plume roll-up and quantifies the importance of viscosity in plume evolution.
1.5 The multi-time step method is developed and implemented for nanosecond- scale laser ablation. Smaller grid size near the target helps in resolving plume features that can not be captured with a single coarse grid over the entire domain. This method can be useful to model nano-second and sub-nano-second scale phenomena including plume ionization near target and save significant amount of computer time. For the presented research, the fine grid is applied only for the near-target region, whereas the coarse gird is used for the rest of domain.
1.6 The model and numerical infrastructure developed has capability to sim- ulate the multiple plume dynamics that involves high pressures and velocities 116 and for all relatively simple shaped domains/bodies. The model easily accom- modates planar as well as axisymmteric formulations. It incorporates shielding models and has capacity to account for ionization in the plume and its effect on temperature field and on the degree of shielding.
2. Physical observations and quantification of plume dyanmics
2.1 The decay of energy delivered to a target because of the shielding effect for multiple nanosecond-scale laser pulses is evaluated. It is shown that shielding by molecular absorption is more significant than that caused by ionization for plume temperatures below 5000 K. The shielding is evaluated through two different models: (i) velocity based and (ii) pressure based. No significant difference in results of energy delivered to the target by these two models is found.
2.2 The plume interaction with shock waves greatly affects the plume devel- opment by means of baroclinic generation of vorticity. Only in later stages of plume propagation when the interaction of plume with shock waves becomes weak, the viscous generation of vorticity affects the plume development and viscous terms should be included in governing gas dynamics equations. The exact amount of generated vorticity by interaction with reflected shock waves depends on injection velocity and the furnace geometry. Evaluation of relative magnitude of baroclinic versus viscous source terms in vorticity generation is proposed to understand the physical mechanisms of plume roll-up and to choose between viscous and inviscid models of plume.
2.3 Modeling of turbulence revealed that only in the millisecond range when the plume speed is lowered down considerably and become comparable with the flow speed of inert gas in laser furnace, the turbulence generated may be a major factor that affects the plume dynamics. 117 2.4 The modeling of multiple plume injection as a result of laser ablation is important for large-scale production of carbon nanotubes where the laser hits the target many times within a short period. For multiple plumes, the large baroclinic generation of vorticity is caused by the shock wave emerging out of the next laser hit that provides pressure gradient behind it to generate baroclinic vorticity in the previously ablated plumes. For the time interval below 100 microseconds, the viscous mechanisms are again negligible since they practically do not affect the outgoing blast waves that appear to be the major source of vorticity for multiple plumes.
2.5 The gas dynamics of emerged plumes affects the amount of injected mate- rial with next plumes. The laser beam passes through the existing plume and a significant part of laser beam energy is deposited in the plume and does not reach the target, that is, the plume shielding effect significantly affect the laser ablation process. This interaction of a laser beam with the gaseous plume is accounted by two relatively simple shielding models based on the light absorp- tion by a gas molecule. This account for the laser beam absorption shows that the ablation rate can be non-monotonic with the number of pulse and depends on the time interval between the successive laser pulses. This non-monotonic behavior results from the increase of shielding effect with the following two factors: (i) plume pressure and temperature, which are larger for shorter time between pulses, and (ii) the path length that laser beam has to travel through the plume before it reaches the target, which increases with the time interval between pulses.
2.6 The plume dynamics for axisymmetric (typical for laser hole-drilling process) and planar (typical for laser channel-cutting process) cases appears to be quite different. In the axisymmetric case the plume travels faster in the axial direction
118 and the plume does not expand much in the lateral direction. The front shock wave strength is low for this case and the plume particles gather in the low- pressure zone near the centerline thus creating considerable shielding of the central part of the target. On the contrary, in the planar case the plume particles are distributed more uniformly in the laser furnace being dragged by strong semi-spherical blast wave.
The plume temperature and its ionization are high for the planar case, but the high degree of ionization appears at the peripheral part of the plume. Although the plume temperature and plume ionization are smaller for the axisymmetric case, the ionization and shielding appear at the near-axial part of the plume where the laser beam passes and thus may shield the incident laser beam.
Laser beam shielding for both cases are evaluated by pure molecular absorp- tion model and by the inverse Bremsstrahlung mechanism that accounts for ionization in the plume. An iterative procedure is developed and implemented to account for the effect of ionization on the plume dynamics through the en- ergy equation. It is found that although the degree of ionization is significant for multiple laser beams being coupled with the target, its effect on shielding for the considered cases is small. It is also found from Saha equation that if the pressure inside the plume becomes an order of magnitude larger, then the ionized particles become significant for shielding of incident laser beam. It is observed that ionization reduces the temperature of plume significantly.
2.7 Numerical simulations are performed to explore the effect of plumes formed through pyrolysis of carbon TPS in hypersonic flight. The effects of initial plume ejection pressure, flight altitude, body shape and emergence of multiple plumes are discussed. The altitude of the flight significantly affects the heat transfer between the emerging plume and the TPS. The pressure difference across the
119 bow shock wave is higher for lower altitudes. At high altitude of flight the plume distorts the shock wave completely and the flow field behind the shock wave is changed significantly after the plume emergence. At low altitude of flight the pressure behind the shock wave is high enough that the plume is unable to eject away from the TPS surface and propagates along it. At moderate altitude of 20km the plume distorts the shock wave temporarily due to its interaction with the plume. For the given injection conditions of relatively cold pyrolysis gas, the simulation results show no significant difference between plume patterns obtained by the Euler and Navier-Stokes equations.
The plume propagates the farthest distance along the TPS wall at moderate altitudes. Initially a kidney-shaped structure is obtained, which then evolves into an oval shape and approaches the TPS wall after being pushed back by the recovering bow shock wave. Consequently, the maximum cooling by the plume tail occurs farther away from stagnation point compared to the low and high altitudes. At high altitude of 30 km, the plume propagates farther away of the TPS and maintains its double kidney shape. The cooling then occurs by interaction of one of plume lobes with the TPS.
The cooling effect is larger for the high altitude of flight because the temperature of the kidney-shaped plume remains cold. During the transformation of plume from kidney shape to oval its temperature rises and therefore the cooling effect is lesser for the moderate altitude of flight. Apart from cooling, some heating effect is observed at all altitudes. Except at the ground level, this heating is caused by the out-going pressure wave due to the under-expanded plume injection. At the moderate flight altitude of 20km this effect is significant around the stagnation region, whereas at higher flight altitudes the effect of this pressure wave is seen upstream and downstream of the plume over significantly larger area of the
120 TPS. At higher flight altitude of 30km the heating effect due to this wave is more pronounced than the cooling effect at later time moments.
The heat transfer is investigated for the range of initial injection pressure of plume. Despite the fact that the plume is detached from the TPS for higher plume injection pressures, the heat transfer between the plume and the TPS is the most intense in this case. This is caused by the plume convection in later time moments when plume is repelled by the bow shock wave toward the TPS.
For multiple plumes, it is observed that the emergence of the second plume increases the magnitude of heat transfer. The behavior of heat transfer for TPS with different geometry but comparable sizes shows that the heat transfer due to plume injection is lower for body with larger radius of curvature, i.e., with more smooth geometry. The heating effect due to the plume pressure wave is higher for a body that has lower strength of front detached shock.
5.1 Future Scope
In the current model of LIP NS equations characterize the plume expansion, which fairly captures the important features like plume roll-up, shielding of incident laser beam by previously ejected plumes and ionization effects of the plume. However, to shed more light on this complex LIP process future developments can be made. In the very early stages of sublimation of target the assumption of continum fails. This assumption might fail for plumes generating from very small spatial scales such as micro-to-nano cracks. In this case, direct Monte Carlo simulation can be used and it can coupled to the NS equations for later stages of plume evolution. Note that such a hybrid method would need very large amount of computer time. It is worthwhile to include non-equilibrium plasma kinetics in the current model. To begin with simple models of ionized plume dynamics, two-temperature 121 modeling would be the most straight-forward continuation for the presently imple- mented NS equations. The details of this kind of modeling are given in Appendix D. Finally, the development of microscopic models that take into account radia- tion as well as non-equilibrium effects can be a promising tool to model the LIP and other application involving high speed/pressure/temperature plume dynamics.
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132 APPENDICES
133 APPENDIX A MULTI-SIZE MESH FOR MULTI-TIME STEP MODELING
If we assume the computational domain is is divided into small elements, then the most basic task is to represent for a given element, its spatial extents. This is done by providing the coordinate information about discrete points defining the element. To access such information two basic sets of data are crated: (i) point data and (ii) element data. These two need to be related for unique access of the information. This relation is given by the so-called connectivity matrix inpoel(1:nnode,1:nelem), where nnode and nelem denote the number of nodes or points corresponding to one element, and the number of elements of the mesh, respectively. The algorithm for the construction of above matrix is given below. It should be noted that this algorithm changes at boundary elements. Boundary elements refers to the elements at the boundary of domain and the boundary where mesh changes its size. The algorithm presented is meant to provide an overall idea and not the details. Given:
• number of points in y direction, j=1,ny
• number of points in x direction, i=1,nx(j)
• x coordinate, xx(i,j)
• y coordinate, yy(i,j)
Initialize: elem=0 134 do j=1,ny do i=1,nx(j) ielem=elem+i lelem(i,j)=ielem do inode=1,nnode if(inode.ne.3.or.inode.ne.4)then ipoin=inpoel(inode+1,lelem(i-1,j)) elseif(inode.eq.3)then ipoin=inpoel(inode-1,lelem(i,j))+1 else ipoin=inpoel(inode-1,lelem(i-1,j)) endif inpoel(inode,ielem)=ipoin x(ipoin)=xx(i,j) y(ipoin)=yy(i,j) end do end do elem=elem+nx(j) end do npoin=inpoel(3,ielem) nelem=ielem
The generated mesh is shown in the following figure.
135 Figure A.1: Multi-size mesh generated for multi-time step modeling
136 APPENDIX B GRID CONVERGENCE
To show the grid convergence the inviscid Burgers’ equation
1 2 ut + u = 0, (B.1) 2 x is taken with intitial condition
u(x, 0) = sin(πx). (B.2)
Its analytical solution is well-known that develops a shock discontinuity at t ≈ 0.31
[74]. Table B.1 shows the L1 norm of the errors at the pre-shock time t = .15 and post-shock time t = .4. Since the original code is 2D, this test case run in both x and y direction to validate correctness of the code. The results show that the numerical methods are correctly implemented and the computational results converge to the analytical solution when the grid size decreases.
Table B.1: L1-Norm of the errors for Numerical solution of ut + uux = 0, u(x, 0) = sin(πx) Repeated for t = .15 and t = .4 N ENO-Roe Relaxing TVD Godunov ENO-Roe Relaxing TVD Godunov 40 .0008 .0026 .0237 .0008 .0036 .044 80 .0002 .0026 .012 .00027 .0012 .023 160 .000061 .00016 .0062 .000098 .00049 .011 320 .000016 .000043 .00031 .000038 .0002 .0052
137 APPENDIX C IONIZATION PREDICTION
For a gas at a high enough temperature, the thermal collisions of the atoms will ionize some of the atoms. For an electron gas that co-exists with gas, i.e. in dynamic equi- librium with the gas of atomic ions and neutral atoms, the Saha equation describes the degree of ionization of this plasma as a function of the temperature, density, and ionization energies of the atoms.
2 3/2 ζ 2gi 2πmekBT Ei = 2 exp − (C.1) 1 − ζ Nga h kBT
where me is the mass of an electron, T is the temperature of the gas, kB is the
Boltzmann constant, h is Planck’s constant, gi, ga is the degeneracy of states for the ions and molecules and Ei is the energy required to remove electrons from a neutral atom. With ζ = Ne/Ng. Ne and Ng are the number densities of electrons and molecules, respectively. In the case where only one level of ionization is important, we have Ni = Ne where Ni is number of ions and defining Ng = Ne + Na where Na number of neutrals, the Saha equation simplifies to:
5/4 −4 T Ei ζ ≈ 5.8 × 10 1/2 exp − p 2kBT
with p(bar) = NgkBT .
−5 For kB = 8.617343 × 10 eV/K, p = 100 bar and Ei = 11.26 eV following result is obtained.
138 Figure C.1: Degree of ionization as a function of temperature
From the above graph it can be seen that the degree of ionization at T = 5000 K is of the order of 10−6 and the plume is opaque to the incident laser pulse only at degree, ζ ≥ 10−2.
139 APPENDIX D TWO-TEMPERATURE MODELING
It is seen in Chapter III that ionization results in significantly lowering the flow field temperature. More sophisticated approaches can help in estimating the equilibrium temperature accurately. One such approach is to consider a two-temperature model.
In this model the electron temperature Te and molecule temperature T are considered different. This difference in temperature occurs on account of slow rate of energy transfer between electrons and molecules essentially due to their mass disparity. An additional energy equation besides solving the system of equation Eq. 2.1 is solved for electron temperature. The simple form in two dimensional of this equation is as follows: ∂e ∂ (e + p ) u ∂ (e + p ) v e + e e + e e = S . (D.1) ∂T ∂x ∂y e
There are variations in estimating the source term Se in above equation. In the papers [52, 53], this is estimated as given below whose origin stems form the Ref. [122]. 2 3 S = E∗n k − E n k − (k T k ) , (D.2) e 3 2 ba i 1 fa 2 B e fb where E∗ is the energy gained by the electrons during three-body recombination, and is expressed as a function of electron temperature and density. n2 and n1 are the neutral atom and ionized atom densities. kba, kfa, kfb are electron impact ionization rate, three-body recombination rate and photo recombination rate constants. Ei is the ionization potential of the ablated species.
140 Once the temperature T and Te are found, the overall pressure is calculated from equation state,
RT RTe p = pa + pe = ρa + ρe , (D.3) Ma Me where Ma is the molecular weight of ablated molecule species and Me is the molecular weight of electron. The details of two-temperature modeling and estimation of source term Se can be found in Refs. [123–126].
141