This thesis is concerned with the new formulation of a finite-volume lattice Boltzmann equation method and its implementation on unstructured meshes. The finite-volume discretization with a cell-centered tessellation is employed. The new formulation effectively adopts a total variation diminishing concept. The formulation is analyzed for the modified partial differential equation and the apparent viscosity of the model. Further, the high-order extension of the present formulation is laid out. Parallel simulations of a variety of two-dimensional benchmark flows are carried out to validate the formulation. In Chapter 1, the important notions of the kinetic theory and the most celebrated equation in the kinetic theory, ‘the Boltzmann equation’ are given. The historical developments and the theory of a discrete form of Boltzmann equation are briefly discussed. Various off-lattice schemes are introduced. Various methodologies adopted in the past for the solution of the lattice Boltzmann equation on finite-volume discretization are reviewed. The basic objectives of this thesis are stated. In Chapter2,the basic formulations of lattice Boltzmann equation method with a rational behind different boundary condition implementations are discussed. The benchmark flows are studied for various flow phenomenon with the parallel code developed in-house. In particular, the new benchmark solution is given for the flow induced inside a rectangular, deep cavity. In Chapter 3, the need for off-lattice schemes and a general introduction to the finite-volume approach and unstructured mesh technology are given. A new mathematical formulation of the off-lattice finite-volume lattice Boltzmann equation procedure on a cell-centered, arbitrary triangular tessellation is laid out. This formulation employs the total variation diminishing procedure to treat the advection terms. The implementation of the boundary condition is given with an outline of the numerical implementation. The Chapman-Enskog (CE) expansion is performed to derive the conservation equations and an expression for the apparent viscosity from the finite-volume lattice Boltzmann equation formulation in Chapter 4. Further, the numerical investigations are performed to analyze the apparent viscosity variation with respect to the grid resolution. In Chapter 5, an extensive validation of the newly formulated finite-volume scheme is presented. The benchmark flows considered are of increasing complexity and are namely (1) Posieuille flow, (2) unsteady Couette flow, (3) lid-driven cavity flow, (4) flow past a backward step and (5) steady flow past a circular cylinder. Further, a sensitivity study to the various limiter functions has also been carried out. The main objective of Chapter6is to enhance the order of accuracy of spatio-temporal calculations in the newly presented finite-volume lattice Boltzmann equation formulation. Further, efficient implementation of the formulation for parallel processing is carried out. An appropriate decomposition of the computational domain is performed using a graph partitioning tool. The order-of-accuracy has been verified by simulating a flow past a curved surface. The extended formulation is employed to study more complex unsteady flows past circular cylinders. In Chapter 7, the main conclusions of this thesis are summarized. Possible issues to be examined for further improvements in the formulation are identified. Further, the potential applications of the present formulation are discussed.

<div>The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well as flows involving further physics such as non-equilibrium internal energy exchange and chemical reactions. Despite its wide applicability, deterministic solutions of the Boltzmann equation present a huge computational challenge, and often the collision operator is simplified for practical reasons, hereby, referred to as linear kinetic models. These models utilize the moment of the underlying probability distribution to mimic some properties of the original collision operator. But, just because we know the moments of a distribution, doesn't mean we know the actual distribution. The approximation of reality can never supersede the reality itself. Because, all the facts (moments) about the world (distribution) cannot explain the world. The premise lies not in the fact that a certain flow behavior can be correctly predicted; but rather that the Boltzmann equation can reveal and explain previously unsuspected aspects of reality.</div><div><br></div><div>Therefore, in this work, we introduce accurate, efficient, and robust numerical schemes for solving the multi-species Boltzmann equation which can model general repulsive interactions. These schemes are high order spatially and temporally accurate, spectrally accurate in molecular velocity space, exhibit nearly linear parallel efficiency on the current generation of processors; and can model a wide-range of rarefied flows including flows involving momentum, heat, and diffusive transport. The single-species variant formed the basis of author's Masters' thesis.</div><div><br></div><div>While the first part of the dissertation is targeted towards multi-species flows that exhibit rich non-equilibrium phenomenon; the second part focuses on single-species flows that do not depart significantly from equilibrium. This is the case, for example, in micro-nozzles, where a portion of flow can be highly rarefied, whereas others can be in near-continuum regime. However, when the flow is in near-continuum regime, the traditional deterministic numerical schemes for kinetic equations encounter two difficulties: a) since the near-continuum is essentially an effect of large number of particles in an infinitesimal volume, the average time between successive collisions decrease, and therefore the discrete simulation timestep has to be made smaller; b) since the number of molecular collisions increase, the flow acquires steady state slowly, and therefore one needs to carry out time integration for large number of time steps. Numerically, the underlying issue stems from stiffness of the discretized ordinary differential equation system. This situation is analogous to low Reynolds number scenario in traditional compressible Navier-Stokes simulations. To circumvent these issues, we introduce a class of high order spatially and temporally accurate implicit-explicit schemes for single-species Boltzmann equation and related kinetic models with the following properties: a) since the Navier-Stokes can be derived from the asymptotics of the Boltzmann equation (using Chapman-Enskog expansion~\cite{cercignani2000rarefied}) in the limit of vanishing rarefaction, these schemes preserve the transition from Boltzmann to Navier-Stokes; b) the timestep is independent of the rarefaction and therefore the scheme can handle both rarefied and near-continuum flows or combinations thereof; c) these schemes do not require iterations and therefore are easy to scale to large problem sizes beyond thousands of processors (because parallel efficiency of Krylov space iterative solvers deteriorate rapidly with increase in processor count); d) with use of high order multi-stage time-splitting, the time integration over sufficiently long number of timesteps can be carried out more accurately. The extension of the present methodology to the multi-species case can be considered in the future. </div><div><br></div><div>A series of numerical tests are performed to illustrate the efficiency and accuracy of the proposed methods. Various benchmarks highlighting different scattering models, different mass ratios, momentum transport, heat transfer, and diffusive transport have been studied. The results are directly compared with the direct simulation Monte Carlo (DSMC) method. As an engineering use-case, the developed methodology is applied for the study of thermal processes in micro-systems, such as heat transfer in electronic-chips; and primarily, the ingeniously Purdue-developed, Microscale In-Plane Knudsen Radiometric Actuator (MIKRA) sensor, which can be used for flow actuation and measurement.</div>

The content of Chapter 1 is an author produced version of a paper published in Physics of Fluids. The final publication is available at AIP via http://dx.doi.org/10.1063/1.3691262. The content of Chapters 2 and 4 is an author produced version of papers published in Journal of Statistical Physics. The final publications are available at Springer via http://dx.doi.org/10.1007/s10955-012-0512-z and http://dx.doi.org/10.1007/s10955-015-1364-0, respectively. / 京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第19682号 / 工博第4137号 / 新制||工||1638(附属図書館) / 32718 / 京都大学大学院工学研究科機械理工学専攻 / (主査)教授 青木 一生, 教授 髙田 滋, 教授 稲室 隆二 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

The experimental pieces of evidence for the existence of Majorana states in topo- logical superconductors have so far been inconclusive despite intense research in the past two decades [Zha+20; Kay+20]. Combined with promising applications in quantum computing [Nay+08; Ali+11] and the resulting technological development of society, the elusiveness of Majorana states keeps motivating theoretical and ex- perimental research to this day. Our analytical findings and numerical explorations in new topological superconducting platforms suggest several tools and solutions for their future realisation in condensed matter systems. The planar Josephson junction (pJJ) introduced in 2017 by F. Pientka et al. [Pie+17] and M. Hell et al. [HLF17] is a versatile platform for topological superconductivity. It harnesses the tunability of the superconducting phase difference across the Josephson junction as an external control parameter that switches the pJJ between the trivial and topological phases of matter. The junction between the (trivial) superconductors is quasi-one-dimensional and hosts one new Majorana zero mode at each of its ends following each topological phase transition. However, the creation of a second Majorana zero mode on one end triggers a return to the trivial regime as both zero modes hybridize into a regular non-topological fermion. It is then crucial to identify the model parameters that lead to topological phases with a single Majorana state per end. Our main result on the pJJ establishes the general constraint on its microscopic parameters—including the phase difference and a magnetic field—to cross the topo- logical phase transitions. The identification of sectors in parameter space leading to a single Majorana mode becomes then straightforward. In some limits the pJJ develops a topological sector at small magnetic field for a phase difference close to the value p while it remains trivial at the same field near zero phase difference. Since the phase is sufficient to turn on and off the topology, we call this feature “switchable topology”. Looking for switchable topology is experimentally relevant as it makes the topology easily tunable while keeping intact the proximitized su- perconductivity otherwise jeopardized by the applied field. Concretely, we found switchable topology in three configurations: in wide junctions with a transparent interface with the superconducting regions, in fine-tuned narrow junctions weakly coupled to the superconducting regions, and in junctions with a strong Zeeman energy when they are ultranarrow and transparent. Thanks to our exact analytical results, setups interpolating between these limits can adjust the desired properties at will. The other important finding about the pJJ concerns the stability of its topological phases, by which we mean the presence of a sizable spectral gap in the topological sector. We observed that the Rashba spin-orbit coupling is responsible for strongly decreasing the gap in the relevant topological sector at low Zeeman field, but sym- metry arguments justify that wide, transparent junctions are generically immune to this effect for large enough Rashba coupling. After 2017, other platforms started to use the Josephson superconducting phase difference as a knob to trigger topological superconductivity [Liu+19; JY21]. We introduce here the stacked Josephson junction (sJJ) as a new platform for topological superconductivity, which is made of two non-centrosymmetric superconductors sandwiching a two-dimensional magnet around which chiral Majorana edge modes propagate. Unlike the Majorana zero modes in the pJJ, chiral Majorana modes can add to each other if they propagate in the same direction, as indicated by the integer Chern number of their topological phase. The bulk-edge correspondence, however, only constrains the net number of topological edge states and allows room for other non-topological states to coexist with the chiral Majorana states without interacting with them. We found that the presence of trivial chiral edge modes in the sJJ restricts access to the Majorana states themselves. The symmetry protection of the trivial modes, fortunately, disappears with an in-plane magnetic field applied through the magnet or with superconducting leads different on the top and at the bottom of the stacked junction. The theoretical investigations of topological platforms have currently outnum- bered the experiments with convincing signatures of Majorana edge states. This imbalance calls for new ways to probe the agreement between topological models and laboratory setups. The critical current of a Josephson junction acts as a link between the microscopic description and macroscopic observables. Thermoelectric measurements, which distinguish between supercurrent and quasiparticle current, modify this model-dependent connection, and would provide an electrical probe to estimate the validity of a model like that of the pJJ. We computed the contribution to the thermoelectric coefficient of the bulk states of a uniform superconductor, that has a similar environment to that of the pJJ (i.e., Rashba coupling and in-plane Zeeman field). The results were not conclusive and motivated us to suggest new analytical and numerical approaches to obtain the thermoelectric response of the pJJ, in particular by including the contribution of the Andreev bound states and non-linear effects.:Foreword — how to read this thesis 1 Preamble A popular short story: pencils and lightbulbs 5 Basics and concepts 1 Introduction to Majorana physics 13 1.1 The electrons & their properties 13 1.1.1 Hamiltonian for the planar Josephson junction 17 1.2 The scattering matrix for bound states 19 1.3 Andreev bound states for topology 24 1.4 Topological superconductivity & Majorana edge states 28 1.5 Induced topological superconductivity 34 1.6 Summary 36 Appendices 37 1.A Microscopic dynamics 37 1.A.1 Origin of spin–orbit coupling 37 1.A.2 Bogoliubov-deGennes symmetrization 37 1.A.3 Andreev reflection below the coherence length 38 1.A.4 Proximity-induced superconductivity 40 1.A.5 From s- to p-wave superconductivity 41 1.B Scattering theory for bound states 44 1.B.1 Bound states as trapped waves 44 1.B.2 Scattering theory for an open region 45 1.B.3 Scattering theory for two open regions 46 1.B.4 Bound states recovered from an open region 47 1.B.5 Numerical scattering theory for bound states 48 2 Perspectives on electronic transport 53 2.1 Electric current in a metal 53 2.2 Quantum-mechanical current 54 2.2.1 Expression for the microscopic current 55 2.3 Thermoelectric current 57 2.3.1 The Boltzmann transport equation 61 2.4 Supercurrents and the superconducting coherence phase 64 2.4.1 Josephson currents 67 Appendices 71 2.A Electric current from a potential difference 71 2.B Scattering and current 71 2.C Hole-based current in metals 73 Introduction Introduction to the Research Projects 77 i Topological properties of Josephson junctions 3 Switchable topology in the planar Josephson junction 85 Motivation & Overview of the Study 85 3.1 The planar Josephson junction and the nanowire setup 87 3.1.1 Comparison with the nanowire setup. 89 3.2 Model 92 3.3 General formula for the phase transitions 94 3.3.1 Spin decoupling for the phase transitions 96 3.3.2 Exact reflection coefficients 97 3.3.3 Exact scattering formula and Andreev reflectivity 98 3.3.4 Andreev approximation 100 3.3.5 Dimensionless formulation 101 3.3.6 Numerical and analytical checks 103 3.4 Three regimes for switchable topology 105 3.4.1 Diamond-shape regime 108 3.4.2 V-shape regime 110 3.4.3 Nanowire regime 111 3.4.4 Summary: extent of the topological transitions 114 3.5 Avoiding regimes with a small topological gap 117 3.5.1 Gapless lines as BDI phase transitions 119 3.5.2 Opening the gap in f = p 120 3.5.3 Role of the Rashba coupling 121 3.6 Conclusion 125 Appendices 129 3.A Limiting cases of the pJJ 129 3.A.1 Andreev approximation 129 3.A.2 Small field limit 131 3.A.3 Delta-barrier junction 131 3.A.4 Semiconductor nanowire 132 3.B Normal reflection via surface impurity and surface refraction 134 3.C Symmetry-constrained gap closings 136 3.D Linear deviation of the gapless line near f = p 138 3.E Calculations for the scattering formula 141 3.E.1 Boundary conditions 141 3.E.2 Combinations of scattering coefficients 142 3.E.3 Andreev coefficients for the phase transitions 143 3.E.4 Formula for B > μ 145 4 Topological and trivial chiral states in the stacked Josephson junction 147 Motivation & Overview of the Study 147 4.1 The basics of the stacked Josephson junction 149 4.2 Continuous and lattice models 151 4.3 Topological index 155 4.3.1 Methodology for the Chern number 155 4.3.2 Interpretation of the results 156 4.4 Topological and trivial edge states 162 4.5 BDI phase transitions 167 4.5.1 Dimensional reduction 168 4.5.2 Link between topological invariants 170 4.5.3 Explaining the low-energy sector 171 4.6 Conclusion 174 Appendices 177 4.A Symmetries of the Hamiltonian 177 4.A.1 Class D 177 4.A.2 Class BDI 177 4.A.3 Gapless line in f = p 178 4.A.4 Symmetry around f = p 179 4.B The parity index in 2D TSC 180 ii Transport properties of the planar Josephson junction 5 An approach to thermoelectric effects in the planar Josephson junction 183 Motivation & Overview of the Study 183 5.1 From the Josephson junction to a homogeneous superconductor 185 5.2 Model and Phenomenology 187 5.2.1 Homogeneous superconductor 187 5.2.2 Analytical spectrum and two-surface approximation 188 5.2.3 Magnetoelectric supercurrent: phenomenology 191 5.3 Electric current in a spin–orbit coupled superconductor 194 5.3.1 Formula for the current 196 5.3.2 Zero-temperature current 198 5.3.3 Small perturbations at finite temperature 200 5.4 Thermoelectric current in a spin–orbit coupled superconductor 206 5.4.1 Distribution imbalance under temperature bias 208 5.4.2 Explicit formula for the thermoelectric current 209 5.5 Discussion and Outlook 213 Appendices 219 5.A The Boltzmann equation in temperature-biased superconductors 219 5.A.1 The linear approximation 220 5.A.2 The low-temperature approximation 220 5.A.3 Integral solution of the Boltzmann equation 223 5.B Diagonalisation of the planar superconductor 225 5.B.1 Eigenstates of spin–orbit coupled superconductor 225 5.B.2 Eigenstates with a small Zeeman field 227 Conclusion Majorana quasiparticles in Josephson junctions 233 Extra Material 6 Mathematical details of Scattering theory 241 6.1 Asymmetric quantum well 241 6.2 Scattering theory for an open region 243 6.2.1 Change in potential over a small region 243 6.2.2 Change in spin-orbit coupling over a small region 245 6.2.3 Change in mass over a small region 245 7 Numerical codes for chapter 4 247 7.1 BDI index 247 7.2 Chern number 255 7.3 Spectral gap 257 7.4 Localized edge states 258 8 Short courses 261 8.1 Two formulations of superconductivity 261 8.1.1 The BCS Hamiltonian 261 8.1.2 The Bogoliubov transformation 263 8.1.3 Bogoliubov-de Gennes symmetrization 264 8.1.4 Building the semiconductor representation 266 8.2 Topological band theory 270 8.3 Majorana physics in 1D 274 8.3.1 The SSH chain 275 8.3.2 The Kitaev chain 277 Bibliography 283

The Lorentz gas model describes a cloud of noninteracting point particles in an infinitely extended array of spherical scatterers with centers in a given locally finite point set of uniform density. In the classical case of identical scatterers, Marklof and Strömbergsson have recently developed a general framework that derives the macroscopic transport equations of this gas for a large class of such point sets. This master's thesis is part of a forthcoming paper by the author that generalizes this framework so as to allow non-identical scatterers. Under certain conditions on the point set and on the distribution of scatterer types, we prove a limit theorem for the free path length and the parameters of the first collision. We also establish bounds on the probability of grazing a scatterer, and sketch how this knowledge gives control of the flight process in the macroscopic limit. As an application, we show that our hypotheses hold for finite unions of affine lattices with one scatterer type per lattice, generalizing a recent result of Palmer and Strömbergsson.

In this thesis we study methods for solving the neutron transport equation (or linear Boltzmann equation). This is an integro-differential equation that describes the behaviour of neutrons during a nuclear fission reaction. Applications of this equation include modelling behaviour within nuclear reactors and the design of shielding around x-ray facilities in hospitals. Improvements in existing modelling techniques are an important way to address environmental and safety concerns of nuclear reactors, and also the safety of people working with or near radiation. The neutron transport equation typically has seven independent variables, however to facilitate rigorous mathematical analysis we consider the monoenergetic, steady-state equation without fission, and with isotropic interactions and isotropic source. Due to its high dimension, the equation is usually solved iteratively and we begin by considering a fundamental iterative method known as source iteration. We prove that the method converges assuming piecewise smooth material data, a result that is not present in the literature. We also improve upon known bounds on the rate of convergence assuming constant material data. We conclude by numerically verifying this new theory. We move on to consider the use of a specific, well-known diffusion equation to approximate the solution to the neutron transport equation. We provide a thorough presentation of its derivation (along with suitable boundary conditions) using an asymptotic expansion and matching procedure, a method originally presented by Habetler and Matkowsky in 1975. Next we state the method of diffusion synthetic acceleration (DSA) for which the diffusion approximation is instrumental. From there we move on to explore a new method of seeing the link between the diffusion and transport equations through the use of a block operator argument. Finally we consider domain decomposition algorithms for solving the neutron transport equation. Such methods have great potential for parallelisation and for the local application of different solution methods. A motivation for this work was to build an algorithm applying DSA only to regions of the domain where it is required. We give two very different domain decomposed source iteration algorithms, and we prove the convergence of both of these algorithms. This work provides a rigorous mathematical foundation for further development and exploration in this area. We conclude with numerical results to illustrate the new convergence theory, but also solve a physically-motivated problem using hybrid source iteration/ DSA algorithms and see significant reductions in the required computation time.

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Three-dimensional large-eddy simulations (LES) of single and multiple jets in cross-flow (JICF) were conducted using the 19-bit Lattice Boltzmann Equation (LBE) method coupled with a conventional Navier-Stokes (NS) finite-volume scheme. In this coupled LBE-NS approach, the LBE-LES was employed to simulate the flow inside jet nozzles, while the NS-LES was used to simulate the cross-flow. The key application area was to study the micro-blowing technique (MBT) for drag control similar to recent experiments at NASA/GRC. A single jet in the cross-flow case was used for validation purposes, and results were compared with experimental data and full LBE-LES simulation. Good agreement with data was obtained. Transient analysis of flow structures was performed to investigate the contribution of flow structures to the counter-rotating vortex pair (CRVP) formation. It was found that both spanwise roller (at the lee side of the jet) and streamwise vortices (at the jet-side) contribute to the generation of the CRVP. Span-wise roller at the corner of the jet experiences high spanwise vortex compression as well as high streamwise vortex stretch. As a result, they get realigned, mix with the jet-side streamwise vortices, and eventually generate the CRVP. Furthermore, acoustic pulses were used to test the proper information exchange from the LBE domain to the NS domain, and vice-versa. Subsequently, MBT over a flat plate with porosity of 25 percent was simulated using nine jets in a compressible cross-flow at a Mach number of 0.4. Three cases with injection ratios of 0.003, 0.02 and 0.07 were conducted to investigate how the blowing rate impacts skin friction. It is shown that MBT suppressed the near-wall vortices and reduced the skin friction by up to 50 percent. This is in good agreement with experimental data.

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No description available.