The ultimate goals of space vehicles are to move faster, further, and more reliably in the space environment. Electric propulsion (EP) has proven to be a necessary technology in the exploration of our solar system ever since its working principle was empirically tested in space in 1964. Thanks to the high exhaust velocities of ionized propellant gases, EP enables efficient utilization of the limited supply of propellant aboard spacecrafts. This technology has opened the possibility of long distance autonomous space missions.

EP devices require electron sources to ionize the propellant gas and to neutralize charges that are leaving the spacecraft. In modern EP thrusters, this is achieved by the use of hollow cathodes – complex devices that employ low work function materials to emit electrons. Hollow cathodes using polycrystalline LaB_{6} inserts are attractive candidates for long duration EP based space missions. However, the physics behind LaB_{6} hollow cathode operation has not been studied in detail, which limits the possibility of their optimization. This work presents an integrated experimental and computational approach to investigate LaB_{6} hollow cathode thermal behaviour and the interplay between LaB_{6} insert surface chemistry and xenon plasma.

Our investigation of the thermal behaviour of LaB_{6} cathodes led to the unexpected discovery of a thermal transient when a new insert is first used. Specifically, we observed that the cathode temperature decreases by approximately 300 degrees over 50 hours before reaching steady state. This finding suggests a beneficial dynamic evolution of the cathode’s chemical state when it interacts with its own plasma. This evolution is intrinsic to cathode operation and can only be precisely understood when the multiphysic nature of the cathode is self-consistently simulated. Thus, we built a numerical platform capable of combining the plasma, thermal and chemical behavior of a discharging hollow cathode. Simulations incorporating different neutralization models, inelastic ion-surface interaction and heterogeneous chemical evolution led to two major conclusions. First, simulations predicted a significant reduction of the LaB_{6} work function (0.42~eV) compared to previously reported baseline values, which is of paramount importance for EP thruster efficiency and operational lifetimes. Second, simulations suggested that the interaction between xenon low energy ions (< 50 eV) and the LaB_{6} surface occurs following a two step neutralization mechanism. The predicted work function reduction was experimentally confirmed by photoemission spectroscopy. Furthermore, using a combination of crystallographic analysis, scanning electron microscopy and profilometry, we demonstrated that work function reduction is caused by the creation of a crystallographic texture at the LaB_{6} surface upon interaction with Xe plasma. In addition, we postulated the existence of a work function enhancing mechanism of secondary importance, which can be explained by forced cationic termination of plasma exposed crystals.

Our results revealed the unexpected phenomenon of work function reduction upon plasma exposure of LaB_{6}. These findings suggest that LaB_{6} hollow cathodes may outperform current technologies and become the component of choice in EP thrusters for future space missions.

The canonical problem of a nearly stationary, nearly planar shockwave passing through isotropic turbulence is investigated within high Reynolds number regimes. The subject flow contains a wide range of turbulent scales and is addressed in Large Eddy Simulation (LES) to relax the otherwise prohibitive computational cost of simulating these flows. Aliasing errors in the LES of the upstream isotropic turbulence are shown to interact with the mean compression of the shock in a problematic matter, and may result in nonphysical behavior such as a reduction in the dissipation rate as the flow crosses the shock. A method for the regularization of LES of shock-turbulence interactions is presented which is constructed to enforce that the energy content in the highest resolved wavenumbers decays as *k*^{-5/3}, and is computed locally in physical space at low computational cost. The application of the regularization to an existing subgrid scale model is shown to remove high wavenumber errors while maintaining agreement with DNS of forced and decaying isotropic turbulence. Comparisons to analytical models suggest that the regularization significantly improves the ability of the LES to predict amplifications in subgrid terms over the modeled shockwave.

The regularization method is then employed in high resolution LES intended to illustrate the physical behavior of the shocked, turbulent flow. Turbulent statistics downstream of the interaction are provided for a range of weakly compressible upstream turbulent Mach numbers *M _{t}* = 0.03 - 0.18, shock Mach numbers

LES allows consideration of high *Re _{λ}* flows, but remains expensive to compute relative to lower cost modeling approaches such as Reynolds-Averaged Navier Stokes (RANS). Conventional RANS models are often not well suited for simulations containing discontinuous features such as shocks and, in an effort to improve the performance of RANS, models for averaged shock corrugation effects and the impact of turbulent entropy or acoustic modes on the energy equation are presented. Unlike previous RANS work that has focused on the modification of turbulent statistics by the shock, the proposed models are introduced to capture the effects of the turbulence on the profiles of primitive variables — mean density, velocity, and pressure. By producing accurate profiles for the primitive variables, it is shown that the proposed models improve numerical convergence behavior with mesh refinement about a shock, and introduce the physical effects of shock asphericity in a converging shock geometry. These effects are achieved by local closures to turbulent statistics in the averaged Navier-Stokes equations, and can be applied in conjunction with existing Reynolds stress closures that have been constructed for broader applications beyond shock-turbulence interactions.

Laminar flame modeling is an important element in turbulent combustion research. The accuracy of a turbulent combustion model is highly dependent upon our understanding of laminar flames and their behavior in many situations. How much we understand combustion can only be measured by how well the model describes and predicts combustion phenomena. One of the most commonly used methane combustion models is GRI-Mech 3.0. However, how well the model describes the reacting flow phenomena is still uncertain even after many attempts to validate the model or quantify uncertainties.

In the present study, the behavior of laminar flames under different aerodynamic and thermodynamic conditions is studied numerically in a stagnation-flow configuration. In order to make such a numerical study possible, the spectral element method is reformulated to accommodate the large density variations in methane reacting flows. In addition, a new axisymmetric basis function set for the spectral element method that satisfies the correct behavior near the axis is developed, and efficient integration techniques are developed to accurately model axisymmetric reacting flow within a reasonable amount of computational time. The numerical method is implemented using an object-oriented programming technique, and the resulting computer program is verified with several different verification methods.

The present study then shows variances with the commonly used GRI-Mech 3.0 chemical kinetics model through a direct simulation of laboratory flames that allows direct comparison to experimental data. It is shown that the methane combustion model based on GRI-Mech 3.0 works well for methane-air mixtures near stoichiometry. However, GRI-Mech 3.0 leads to an overprediction of laminar flame speed for lean mixtures and an underprediction for rich mixtures. This result is slightly different from conclusion drawn in previous work, in which experimental data are compared with a one-dimensional numerical solutions. Detailed analysis reveals that flame speed is sensitive to even slight flame front curvature as well as its finite extension in the radial direction. Neither of these can be incorporated in one-dimensional flow modeling.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Dimotakis, Paul E. and Meiron, Daniel I.}, } @phdthesis{10.7907/1397-GZ04, author = {Latini, Marco}, title = {Simulations and Analysis of Two- and Three-Dimensional Single-Mode Richtmyer-Meshkov Instability using Weighted Essentially Non-Oscillatory and Vortex Methods}, school = {California Institute of Technology}, year = {2007}, doi = {10.7907/1397-GZ04}, url = {https://resolver.caltech.edu/CaltechETD:etd-12082006-124547}, abstract = {An incompressible vorticity-streamfunction (VS) method is developed to investigate the single-mode Richtmyer-Meshkov instability in two and three dimensions. The initial vortex sheet (representing the initial shocked interface) is thickened to regularize the limit of classical Lagrangian vortex methods. In the limit of smaller thickness, the initial velocity converges to the velocity of a vortex sheet. The vorticity on the Cartesian grid follows the vorticity evolution equation augmented by the baroclinic vorticity production term (to capture the effects of the instability on the layer) and a viscous dissipation term. The equations are discretized using a fourth-order in space and third-order in time semi-implicit Adams-Bashforth backward differentiation scheme. The convergence properties of the method with respect to varying the diffuse interface thickness and viscosity are investigated. It is shown that the small-scale structures within the roll-up are more sensitive to the diffuse interface thickness than to the viscosity. By contrast, the large-scale quantities, including the perturbation, bubble, and spike amplitudes are less sensitive. Fourth-order point-wise convergence is achieved, provided that a sufficiently fine grid is used.

In two dimensions, the VS method is applied to investigate late-time nonlinear effects of the single-mode Mach 1.3 air(acetone)/SF_6 shock tube experiment of Jacobs and Krivets. The results are also compared to those from compressible ninth-order weighted essentially non-oscillatory (WENO) simulations. The density fields from the WENO and VS methods agree with the experimental PLIF images in the large-scale structures but differ in the small-scale structures. The WENO method exhibits small-scale disordered structure similar to that in the experiment, while the VS method does not capture such structure, but shows a strong rotating core. The perturbation amplitudes from the two methods are in good agreement and match the experimental data points well. The WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike amplitude. Comparing amplitudes from simulations with varying Mach number shows that as the Mach number increases, the differences in the bubble and spike amplitudes increase due to intensifying pressure perturbations not present in the incompressible VS method. The perturbation amplitude from the WENO and VS methods is also compared to the predictions of nonlinear amplitude growth models in which the growth rate was reduced to account for the diffuse initial interface. In general, the model predictions agree with the simulation amplitudes at early-to-intermediate times and underpredict at later times, corresponding to the late nonlinear regime.

The WENO simulation is used to investigate reshock, which occurs when the transmitted shock reflects from the end wall of the test section and interacts with the evolving layer. The post-reshock mixing layer width agrees well with the predictions of reshock models for short times until the interaction of the reflected rarefaction with the layer.

The VS simulation was also compared to classical Lagrangian and vortex-in-cell simulations as the Atwood number was varied. For low Atwood numbers, all three simulations agree. As the Atwood number increases, the VS simulation shows differences in the bubble and spike amplitudes compared to the Lagrangian and VIC simulations, as the baroclinic vorticity production for a diffuse layer is different from that of a thin layer. The simulation amplitudes agree with the predictions of nonlinear amplitude growth models at early times. The growth models underpredict the amplitudes at later times.

The investigation is extended to three dimensions, where the initial perturbation is a product of sinusoids and the initial vorticity deposition is given by linear instability analysis. The instability evolution and dynamics of vorticity are visualized using the mass fraction and enstrophy isosurface, respectively. For the WENO and VS methods, two roll-ups corresponding to the bubble and spike regions form, and the vorticity shows the formation of a ring-like structure. The perturbation amplitudes from the WENO and VS methods are in excellent agreement. The bubble and spike amplitude are in good agreement at early times. At later times, the WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike. The nonlinear three-dimensional Zhang-Sohn model agrees with the simulation amplitudes at early times, and underpredicts later. In three dimensions, the enstrophy iso-surface after reshock shows significant fragmentation and the formation of small, short, tubular structures. Simulations with different initial amplitudes show that the mixing layer width after reshock does not depend on the pre-shock amplitude. Finally, the effects of Atwood number are investigated using the VS method and the amplitudes are compared to the predictions of the Zhang-Sohn model. The simulation and the models are in agreement at early times, while the models underpredict later.

The VS method constitutes a useful numerical approach to investigate the Richtmyer-Meshkov instability in two and three dimensions. The VS method and, more generally, vortex methods are valid tools for predicting the large-scale instability features, including the perturbation amplitudes, into the late nonlinear regime.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I. and Schilling, Oleg}, } @phdthesis{10.7907/HRNV-DA03, author = {Howard, Elizabeth Anne}, title = {A Front Tracking Method for Modelling Thermal Growth}, school = {California Institute of Technology}, year = {2003}, doi = {10.7907/HRNV-DA03}, url = {https://resolver.caltech.edu/CaltechETD:etd-03042003-115138}, abstract = {Several important thermal growth problems involve a solid growing into an undercooled liquid. The heat that is released at the interface diffuses into both the solid and the liquid phases. This is a free boundary problem where the position of the interface is an unknown which must be found as part of the solution. The problem can conveniently be represented as an integral equation for the unknown interface. However, a history integral must be evaluated at each time step which requires information about the boundary position at all previous times. The time and memory required to perform this calculation quickly becomes unreasonable. We develop an alternative way to deal with the problems that the history integral presents. By taking advantage of properties of the diffusion equation, we can use a method with a constant operation count and amount of memory required for each time step. We show that a numerical algorithm can be implemented for a two-dimensional, symmetric problem with equal physical parameters in both phases. The results agree well with the exact solution for the expanding circle case and microscopic solvability theory. We also extend the method to the nonsymmetric case. Additionally, a stability analysis is done of a simple, parabolic moving front to perturbations on the surface. As the eigenvalues of our problem increase, the interface becomes more increasingly oscillatory.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I.}, } @phdthesis{10.7907/5R5P-Y603, author = {Mauch, Sean Patrick}, title = {Efficient Algorithms for Solving Static Hamilton-Jacobi Equations}, school = {California Institute of Technology}, year = {2003}, doi = {10.7907/5R5P-Y603}, url = {https://resolver.caltech.edu/CaltechETD:etd-05202003-170423}, abstract = {We present an algorithm for computing the closest point transform to an explicitly described manifold on a rectilinear grid in low dimensional spaces. The closest point transform finds the closest point on a manifold and the Euclidean distance to a manifold for the points in a grid. We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm solves the eikonal equation |grad u| = 1 with the method of characteristics. For many problems, the computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold.

Many query problems can be aided by using orthogonal range queries (ORQ). There are several standard data structures for performing ORQ’s in 3-D, including kd-trees, octrees, and cell arrays. We develop additional data structures based on cell arrays. We study the characteristics of each data structure and compare their performance.

We present a new algorithm for solving the single-source, non-negative weight, shortest-paths problem. Dijkstra’s algorithm solves this problem with computational complexity O((E + V) log V) where E is the number of edges and V is the number of vertices. The new algorithm, called Marching with a Correctness Criterion (MCC), has computational complexity O(E + R V), where R is the ratio of the largest to smallest edge weight.

Sethian’s Fast Marching Method (FMM) may be used to solve static Hamilton-Jacobi equations. It has computational complexity O(N log N), where N is the number of grid points. The FMM has been regarded as an optimal algorithm because it is closely related to Dijkstra’s algorithm. The new shortest-paths algorithm discussed above can be used to develop an ordered, upwind, finite difference algorithm for solving static Hamilton-Jacobi equations. This algorithm requires difference schemes that difference not only in coordinate directions, but in diagonal directions as well. It has computational complexity O(R N), where R is the ratio of the largest to smallest propagation speed and N is the number of grid points.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I.}, } @phdthesis{10.7907/2TWD-EK72, author = {Kastner, Jason Christopher}, title = {Modeling a Hox Gene Network: Stochastic Simulation with Experimental Perturbation}, school = {California Institute of Technology}, year = {2003}, doi = {10.7907/2TWD-EK72}, url = {https://resolver.caltech.edu/CaltechETD:etd-10042002-200303}, abstract = {The Hox genes show a striking segment specific pattern of expression in a variety of vertebrate embryos, and have been the topic of many experimental analyses. There are now sufficient data to construct a higher-level model for the interaction and regulation of the Hox genes. This thesis presents the results of an investigation into a regulatory network for the early Hox genes. Instead of using conventional differential equation approaches for analyzing the system, a stochastic simulation algorithm has been employed to model the network. The model can track the behavior of each component of a biochemical pathway and produce computerized movies of the time evolution of the system that is a result of the dynamic interplay of these various components. The simulation is able to reproduce key features of the wild-type pattern of gene expression, and in silico experiments yield results similar to their corresponding in vivo experiments. This work shows the utility of using stochastic methods to model biochemical networks and expands the stochastic simulation algorithm methodology to work in multi-cellular systems. In addition, the model has suggested several predictions that can be tested in vivo.

A tight connection was also created between the modeling and laboratory experiments. To investigate a connection between two components of the network, retinoic acid (RA) and Hoxa1, a novel laboratory experiment was performed to perturb the system. An RA soaked bead was implanted into the neural tube of a developing chick embryo and the effect of the exogenous RA was assayed with an in situ hybridization for the gene Hoxa1. The resulting expression patterns suggested that one aspect of the model design was not accurate, and based on these results the model was modified to encompass the new data, without losing the fit to the original data sets. The thesis work was therefore brought full circle, thus showing the utility of an interconnected effort: the act of constructing and using the model identified interesting biology questions, and the answer to one of those questions was used to enhance the model.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I.}, } @phdthesis{10.7907/JXG6-W865, author = {Craciun, Bogdan}, title = {Phase Boundary Propagation in Heterogeneous Media}, school = {California Institute of Technology}, year = {2002}, doi = {10.7907/JXG6-W865}, url = {https://resolver.caltech.edu/CaltechTHESIS:10082010-142653040}, abstract = {There has been much recent progress in the study of free boundary problems motivated by phase transformations in materials science. Much of this literature considers fronts propagating in homogeneous media. However, usual materials are heterogeneous due to the presence of defects, grains and precipitates. This thesis addresses the propagation of phase boundaries in heterogeneous media.

A particular motivation is a material undergoing martensitic phase transformation. Given a martensitic material with many non-transforming inclusions, there are well established microscopic laws that give the complex evolution of a particular twin or phase boundary as it encounters the many inclusions. The issue of interest is the overall evolution of this interface and the effect of defects and impurities on this evolution. In particular, if the defects are small, it is desirable to find the effective macroscopic law that governs the overall motion, without having to follow all the microscopic details but implicitly taking them into account. Using a theory of phase transformations based on linear elasticity, we show that the normal velocity of the martensitic phase or twin boundary may be written as a sum of several terms: first a homogeneous (but non-local) term that one would obtain for the propagation of the boundary in a homogeneous medium, second a heterogeneous term describing the effects of the inclusions but completely independent of the phase or twin boundary and third an interfacial energy term proportional to the mean curvature of the boundary.

As a guide to understanding this problem, we begin with two simplified settings which are also of independent interest. First, we consider the homogenization for the case when the normal velocity depends only on position (the heterogeneous term only). This is equivalent to the homogenization of a Hamilton-Jacobi equation. We establish several variational principles which give useful formulas to characterize the effective Hamiltonian. We illustrate the usefulness of these results through examples and we also provide a qualitative study of the effective normal velocity.

Second, we address the case when the interfacial energy is not negligible, so we keep the heterogeneous and curvature terms. This leads to a problem of homogenization of a degenerate parabolic initial value problem. We prove a homogenization theorem and obtain a characterization for the effective normal velocity, which however proves not to be too useful a tool for actual calculations. We therefore study some interesting examples and limiting cases and provide explicit formula in these situations. We also provide some numerical examples.

We finally address the problem in full generality in the setting of anti-plane shear. We explicitly evaluate the term induced by the presence of the inclusions and we propose a numerical method that allows us to trace the evolution of the phase boundary. We use this numerical method to evaluate the effect of the inclusions and show that their effect is quite localized. We use it to explain some experimental observations in NiTi.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I. and Bhattacharya, Kaushik}, } @phdthesis{10.7907/PDCS-GX15, author = {Tokman, Mayya}, title = {Magnetohydrodynamic modeling of solar magnetic arcades using exponential propagation methods}, school = {California Institute of Technology}, year = {2001}, doi = {10.7907/PDCS-GX15}, url = {https://resolver.caltech.edu/CaltechETD:etd-02062006-154529}, abstract = {Advanced numerical methods based on exponential propagation have been applied to magnetohydrodynamic (MHD) simulations. This recently developed numerical technique evolves the system of nonlinear equations using exponential propagation of the Jacobian matrix. The exponential of the matrix is approximated by projecting it onto the Krylov subspace using the Arnoldi algorithm. The primary advantage of the exponential propagation method is that it allows time steps exceeding the Courant-Friedrichs-Lewy (CFL) limit. Another important aspect is faster convergence of the iteration computing the Krylov subspace projection compared to solving an implicit formulation of the system with similar iterative methods. Since the time scales in the resistive MHD equations are widely separated, the exponential propagation methods are especially advantageous for computing the long term evolution of a low-beta plasma. We analyze several types of exponential propagation methods and highlight important issues in the development of such techniques. Our analysis also suggests new ways to construct schemes of this type. Implementation issues, including scalability properties of exponential propagation methods, and performance are also discussed. In the second part of this work we present numerical MHD models which are constructed using exponential propagation methods and which describe the evolution of the magnetic arcades in the solar corona. Since these numerical methods have not been used before for large evolutionary systems like resistive MHD, we first validate our approach by demonstrating application of the exponential schemes to two existing magnetohydrodynamic models. We simulate the reconnection process resulting from shearing the footpoints of two-dimensional magnetic arcades and compute the three-dimensional linear force-free states of plasma configurations. Analysis of these calculations leads us to new insights about the topology of the solutions. The final chapter of this work is dedicated to a new three-dimensional numerical model of the dynamics of coronal plasma configurations. The model is motivated by observations and laboratory experiments simulating the evolution of solar arcades. We analyze the results of numerical simulations and demonstrate that our numerical approach provides an accurate and stable way to compute the solution to the zero-resistive MHD system. Based on comparisons of the simulation results and the observational data, we offer an explanation for the observed structure of eruptive events in the corona called coronal mass ejections (CME). We argue that the diversity of the images of CMEs obtained by the observational instruments can be explained as two-dimensional projections of a unique three-dimensional plasma configuration and suggest an eruption mechanism.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I.}, } @phdthesis{10.7907/etyh-8140, author = {Haroldsen, David John}, title = {The numerical calculation of three-dimensional water waves using a boundary integral method}, school = {California Institute of Technology}, year = {1997}, doi = {10.7907/etyh-8140}, url = {https://resolver.caltech.edu/CaltechETD:etd-01102008-153911}, abstract = {In this work, we consider the numerical calculation of water waves in three dimensions. One well accepted method for studying surface waves is the boundary integral method, which defines the fluid velocities at the interface in terms of integrals over the boundary of the domain in which the problem is posed. There exists a considerable body of work on the numerical study of surface waves in two dimensions. However, until recently the numerical study of surface waves was considered intractable because of the high computational cost of approximating the defining integrals.

We discuss the boundary integral formulation for the three-dimensional water wave problem and present the point vortex approximation to the singular integrals which define the particle velocities. We consider three aspects of the point vortex approximation: accuracy of the approximation, efficient means of computing solutions, and numerical stability of the scheme.

Concerning the accuracy of the point vortex method, we analyze the error associated with the approximation and show that it can be expressed as a series in odd powers of the discretization parameter h. We present quadrature rules which are highly accurate.

The efficient computation of the point vortex approximation is achieved through the use of the fast multipole algorithm, which combines long distance particle inter-actions into multipole expansions which can be efficiently evaluated. The underlying periodicity of the problem is reduced to a lattice sum which can be rapidly evaluated. We discuss the implementation of the numerical schemes in both serial and parallel computing environments.

The point vortex method is shown to be highly unstable for straightforward discretizations of the surface. We analyze the stability of the method about equilibrium and discuss methods for stabilizing the numerical schemes for both the linear and nonlinear regimes. We present numerical results which show that the method can be effectively stabilized.

In the final chapter, we present numerical results from several calculations of three-dimensional waves using the methods developed in the previous chapters.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I.}, } @phdthesis{10.7907/s4n5-1v32, author = {Ardalan, Kayvan}, title = {Compressible vortex arrays}, school = {California Institute of Technology}, year = {1996}, doi = {10.7907/s4n5-1v32}, url = {https://resolver.caltech.edu/CaltechETD:etd-02062008-091358}, abstract = {NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

We construct steady, two dimensional, compressible vortex arrays with specified vorticity distributions. We begin by examining the effects of compressiblity on the structure of a single row of hollow-core, constant pressure vortices. The problem is formulated and solved in the hodograph plane. The transformation from the physical plane to the hodograph plane results in a linear problem that is solved numerically. The numerical solution is checked via a Rayleigh-Janzen expansion. It is observed that for an appropriate choice of the parameters […], the Mach number at infinity, and the speed ratio, a, transonic shock-free flow exists. Also, for a given fixed speed ratio, a, the vortices shrink in size and get closer as the Mach number at infinity, […], is increased. In the limit of an evacuated vortex core, we find that all such solutions exhibit cuspidal behaviour corresponding to the onset of limit lines.

The hollow core vortex array corresponds to a vorticity distribution wherein the vorticity is concentrated on the vortex boundary. In the second part of this thesis, we examine vortex arrays with continuous vorticity distributions. In particular, we construct Stuart-type vortices in a channel by requiring the vorticity distribution to be an exponential func-tion of the stream function of the flow. The problem is formulated and solved in the physical plane. The numerical solution is checked via a Rayleigh-Janzen expansion of the unbounded Stuart vortex solution. It is shown that, in the limit of infinite speed of sound (incompressible flow), as the channel walls tend to infinity, […] , the Stuart vortex solutions are recovered. Furthermore, it is shown that unbounded, compressible Stuart vortices exist and that a generalized Stuart vortex solution is the proper incompressible limit. For a given fixed circulation, […], Mach number, […], and […], (a measure of the compactness of the vorticity distribution) it is shown that the limit of a very narrow channel, […]0, is a parallel shear flow. Exact analytical solutions for the compressible parallel shear flow are also found in implicit form.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I. and Pullin, Dale Ian}, } @phdthesis{10.7907/ryrb-cg53, author = {Pham, Thu}, title = {Numerical studies of incompressible Richtmyer-Meshkov instability in a stratified fluid}, school = {California Institute of Technology}, year = {1991}, doi = {10.7907/ryrb-cg53}, url = {https://resolver.caltech.edu/CaltechETD:etd-07122007-143228}, abstract = {Theory and calculations are presented for the evolution of Richtmyer-Meshkov instability in continuously stratified fluid layers. The initial acceleration and subsequent instability of the fluid layer are induced by means of an impulsive pressure distribution. It is shown that such an initial condition is an adequate approximation of the effect of a weak shock impinging on a stratified layer of fluid. We then calculate the subsequent dynamics of the fluid layer numerically using the incompressible equations of motion.

Both initial conditions having single scale perturbations and multiple scale random perturbations are considered. It is found that the growth rates for Richtmyer-Meshkov instability of stratified fluid layers are substantially lower than those predicted by Richtmyer for a sharp fluid interface with an equivalent jump in density. The initial behavior is linear over a time equivalent to the traversal of several layer thicknesses. It is observed that the nonlinear development of the instability results in the formation of plumes of penetrating fluid. Late in the process, the initial momentum deposited by the shock is primarily used in the internal mixing of the layer rather than in the overall growth of the stratified layer.

At intermediate time, the existence of a weak scaling behavior in the width of the mixing layer of the instability is observed for the multiple scale random perturbations, but not for the single scale perturbations. The time variation of the layer thickness differs from the scaling hypothesized by Barenblatt even at low Atwood ratio, presumably because of the inhomogeneity and anisotropy due to the excitation of vortical plumes. The emergence of these plumes at the boundaries of the density layer is characterized by the elongation of the internal spikes which have weak interactions and grow proportionally to their intial perturbed amplitudes. It is conjectured that the formations of the plumes may correspond to weakly interacting single scale modes.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I.}, } @phdthesis{10.7907/TMK2-ME89, author = {Mudkavi, Vidyadhar Yogeshwar}, title = {Numerical studies of nonlinear axisymmetric waves on vortex filaments}, school = {California Institute of Technology}, year = {1991}, doi = {10.7907/TMK2-ME89}, url = {https://resolver.caltech.edu/CaltechETD:etd-07092007-105408}, abstract = {The equations of Moore and Saffman (1971) are examined and are shown to contain the fast time scale equations governing the core waves on a straight vortex filament. The equations so derived are the same as those reported by Lundgren and Ashurst (1989) except for a correction term that allows for variation of the axial velocity structure within the vortex core. Numerical solutions of the Moore and Saffman equations are presented for various initial conditions consisting of wave-like perturbations on a cylindrical vortex, and they all show development of a jump in the core area. This has been advanced to be a mechanism for vortex breakdown by Lundgren and Ashurst. A comparison of the solutions of the Moore and Saffman equations with the solutions of the Navier-Stokes equations at high Reynolds number is presented for three different cases. In the first case a vortex with a very small perturbation is considered. The Moore and Saffman solution shows steepening of the initial wave resulting in the development of jump in the core area (shock). The Navier-Stokes solution shows bulging of the core. But, there is no indication of formation of a shock. In the second case a vortex with moderate perturbation is considered. The Moore and Saffman solution leads to a shock similar to the weak perturbation case. As before, the Navier-Stokes solution does not develop jump in the core area. However, development of a bubble of reversed flow is seen. In the third case, a jump in the core area in the solutions of the Navier-Stokes equations is seen for a strongly perturbed vortex. But the location and the sense of jump disagrees with jump that develops in the Moore-Saffman solution. Thus, the solutions of the Navier-Stokes equations and the Moore-Saffman equations show qualitative disagreement. Next, an extension of steady Kelvin waves for two different types of vorticity profiles is considered. In the first case, steady nonlinear waves are constructed via a perturbation method. In this case, the vorticity is nonzero inside the core and sharply drops to zero across the boundary. The shape of the core boundary is determined as part of the problem. The dependence of the Bernoulli function and the circulation function on the streamfunction are specified. This serves as the additional constraint necessary to determine the solution uniquely. The solutions are free of any vortex sheets. In the second case, nonlinear steady Kelvin waves on smooth vorticity distributions are constructed by means of a direct Newton method and a large order perturbation method. Instead of specifying the dependence of the Bernoulli function and the circulation function on the stream function as in the previous case, the solutions are restricted such that they have the same axial mean as the base flow. In both the approaches, regions of reversed flow are observed. This is the structure of bubble type of vortex breakdown. Next, an analysis of the weakly nonlinear stability of a columnar vortex is presented. It is shown that the amplitude, assumed to vary slowly in time and space, satisfies a cubic-nonlinear Schrodinger equation. Solutions are found to be unstable in the sense that the perturbations grow slowly in time. Solitary wave solutions are possible in this unstable case.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I.}, } @phdthesis{10.7907/2xg0-5q97, author = {Soibelman, Israel}, title = {A Study of Finite Amplitude Bifurcations in Plane Poiseuille Flow}, school = {California Institute of Technology}, year = {1989}, doi = {10.7907/2xg0-5q97}, url = {https://resolver.caltech.edu/CaltechETD:etd-02142007-080941}, abstract = {Plane Poiseuille flow is known to be linearly unstable at a Reynolds number of 5772.22 (Drazin and Reid, 1981). In experiments, however, transition to turbulent flow is seen to occur at a Reynolds number of 1000 (Nishioka and Asai, 1985). In an attempt at resolving this conflict, we search for 2D and 3D nonlinear bifurcations at low Reynolds number.

Because we wish to study secondary bifurcations, we compute the 2D waves which bifurcate from plane Poiseuille flow. These waves were first computed by Zahn, et al., (1975), and the critical Reynolds number, based on constant pressure, was found to be approximately 2900. To find 2D bifurcations, we study the 2D superharmonic stability of the 2D waves. The stability picture is found to change when switching from a constant flux to constant average pressure gradient boundary condition. For both boundary conditions, we find several Hopf bifurcations on the upper branch of the 2D waves.

We calculate the periodic orbits which emanate from these bifurcations and find that no branch extends below the critical 2D wave Reynolds number. We also confirm the results of Jimenez (1988) who detected one of the branches we calculate with a time dependent formulation.

To find 3D bifurcations, we study the 3D stability of the 2D waves. Several branches of 3D waves are calculated. In particular, we study 3D bifurcations at a spanwise wave number of 2. No bifurcations are found to branches which extend to low Reynolds numbers. This result conflicts with those found by Rozhdestvensky and Simakin (1984) with a time dependent formulation.

In addition, we study 3D oblique waves and 3D standing-travelling waves (standing in the streamwise direction) which bifurcate from plane Poiseuille flow. In particular, we study the bifurcation at spanwise wave numbers greater than .365. Contrary to Bridges’ (1988) hypothesis, we find that no branches extend to low Reynolds numbers.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Meiron, Daniel I.}, }