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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:29:37 +0000Compressible flows at small Reynolds numbers
https://resolver.caltech.edu/CaltechTHESIS:03292013-152506285
Authors: {'items': [{'email': 'athyagaraja@gmail.com', 'id': 'Thyagaraja-A', 'name': {'family': 'Thyagaraja', 'given': 'Anantanarayanan'}, 'show_email': 'YES'}]}
Year: 1972
DOI: 10.7907/VVKA-ZS09
<p>The problem of the slow viscous flow of a gas past a sphere is considered. The fluid cannot be treated incompressible in the limit when the Reynolds number Re, and the Mach number M, tend to zero in such a way that Re ~ o(M^2 ). In this case, the lowest order approximation to the steady Navier-Stokes equations of motion leads to a paradox discovered by Lagerstrom and Chester. This paradox is resolved within the framework of continuum mechanics using the classical slip condition and an iteration scheme that takes into account certain terms in the full Navier-Stokes equations that drop out in the approximation used by the above authors. It is found however that the drag predicted by the theory does not agree with R. A. Millikan's classic experiments on sphere drag. </p>
<p>The whole question of the applicability of the Navier-Stokes theory when the Knudsen number M/Re is not small is examined. A new slip condition is proposed. The idea that the Navier-Stokes equations coupled with this condition may adequately describe small Reynolds number flows when the Knudsen number is not too large is looked at in some detail. First, a general discussion of asymptotic solutions of the equations for all such flows is given. The theory is then applied to several concrete problems of fluid motion. The deductions from this theory appear to interpret and summarize the results of Millikan over a much wider range of Knudsen numbers (almost up to the free molecular or kinetic limit) than hitherto Believed possible by a purely continuum theory. Further experimental tests are suggested and certain interesting applications to the theory of dilute suspensions in gases are noted. Some of the questions raised in the main body of the work are explored further in the appendices. </p>
https://thesis.library.caltech.edu/id/eprint/7568A model biochemical reaction
https://resolver.caltech.edu/CaltechETD:etd-08152005-155955
Authors: {'items': [{'id': 'Boa-J-A', 'name': {'family': 'Boa', 'given': 'James Andrew'}, 'show_email': 'NO'}]}
Year: 1974
DOI: 10.7907/QBVP-DG31
Asymptotic solutions are presented to the non-linear parabolic reaction-diffusion equations describing a model biochemical reaction proposed by I. Prigogine. There is a uniform steady state which, for certain values of the adjustable parameters, may be unstable. When the uniform solution is slightly unstable, the two-timing method is used to find the bifurcation of new solutions of small amplitude. These may be either non-uniform steady states or time-periodic solutions, depending on the ratio of the diffusion coefficients. In the limit that one of the diffusion coefficients is infinite, multiple steady states of finite amplitude are found. When one of the parameters is allowed to depend on space and the basic state is unstable, it is found that the non-uniform steady state which is approached may show localized spatial oscillations. The localization arises out of the presence of turning points in the linearized stability equations. When diffusion is absent it is shown how kinematic concentration waves arise. Detailed calculations using singular perturbation techniques are made of the basic oscillation giving rise to these waves, which is a relaxation oscillation. It is found that the equations in its asymptotic approximation are not obtained from the full equations as the result of a limit process.
https://thesis.library.caltech.edu/id/eprint/3125I. Numerical solutions of steady viscous flow past spheres and gas bubbles. II. Numerical solution of singular endpoint boundary value problems
https://resolver.caltech.edu/CaltechETD:etd-08172005-081056
Authors: {'items': [{'email': 'don.brabston@verizon.net', 'id': 'Brabston-D-C', 'name': {'family': 'Brabston', 'given': 'Donald Campbell'}, 'show_email': 'YES'}]}
Year: 1974
DOI: 10.7907/4719-D351
<p>In Part I, numerical solutions of the Navier-Stokes equations are given for steady, viscous, incompressible, axisymmetric flow past a rigid and a spherical gas bubble. The problem is formulated in terms of a stream function and the vorticity which are expanded in finite Legendre series. The coefficients in these series satisfy a finite system of ordinary differential equations. A finite-difference scheme is used to solve the system with Newton's method used to solve the nonlinear difference equations. The results agree very well with low and high Reynolds number theories.</p>
<p>In Part II, systems of ordinary differential equations with singular points of the first kind are considered. The singular point may be at either end, at both ends, or in the interior of a finite interval. A two-point linear system of boundary conditions is imposed at the endpoints. A theory is developed stating the conditions under which a unique solution will exist A numerical method is developed for solving these problems. In this method, a series solution about the singular point is matched to a finite difference solution away from the singular point. Error estimates are developed, and numerical examples are given.</p>https://thesis.library.caltech.edu/id/eprint/3152Electromagnetic wave propagation in almost periodic media
https://resolver.caltech.edu/CaltechTHESIS:03122012-112325331
Authors: {'items': [{'email': 'alan.mickelson@colorado.edu', 'id': 'Mickelson-Alan-Rolf', 'name': {'family': 'Mickelson', 'given': 'Alan Rolf'}, 'orcid': '0000-0003-2529-8301', 'show_email': 'YES'}]}
Year: 1978
DOI: 10.7907/SZMY-A964
The problem of electromagnetic wave propagation in almost periodic
media is investigated and a solution is obtained directly from Maxwell's
equations. Techniques to evaluate this solution are developed. These
techniques involve a generalization to almost periodic media of the
Brillouin diagram of periodic media. The method of invariant imbedding
is applied to the coupled mode equations which determine the Brillouin
diagram for the purpose of transforming them to coupled Riccati equations.
These coupled Riccati equations, when subjected to a single boundary condition,
determine the solutions to both the periodic and almost periodic
boundary value problems. These evaluation techniques are used to place
in evidence similarities and differences of wave propagation in periodic
and almost periodic media. It is shown that although the periodic and
almost periodic theories agree in many cases of interest, there exist
cases in which distinct differences appear. In cases of multi-tone perturbations,
the almost periodic theory yields both simpler and more reasonable
results than the periodic theory.https://thesis.library.caltech.edu/id/eprint/6849Periodic solutions of integro-differential equations which arise in population dynamics
https://resolver.caltech.edu/CaltechTHESIS:03212013-102207354
Authors: {'items': [{'id': 'Simpson-H-C', 'name': {'family': 'Simpson', 'given': 'Henry C.'}, 'show_email': 'NO'}]}
Year: 1979
DOI: 10.7907/cd8g-fq50
<p>The problem of the existence and stability of periodic solutions of infinite-lag integra-differential equations is considered. Specifically, the integrals involved are of the convolution type with the dependent variable being integrated over the range (- ∞,t), as occur in models of population growth. It is shown that Hopf bifurcation of
periodic solutions from a steady state can occur, when a pair of eigenvalues crosses the imaginary axis. Also considered is the existence of traveling wave solutions of a model population equation allowing spatial diffusion in addition to the usual temporal variation. Lastly, the
stability of the periodic solutions resulting from Hopf bifurcation is determined with aid of a Floquet theory.</p>
<p>The first chapter is devoted to linear integro-differential equations with constant coefficients utilizing the method of semi-groups of operators. The second chapter analyzes the Hopf bifurcation providing an existence theorem. Also, the two-timing perturbation procedure is applied to construct the periodic solutions. The third chapter uses two-timing to obtain traveling wave solutions of the diffusive model, as well as providing an existence theorem. The fourth chapter develops a Floquet theory for linear integro-differential equations with periodic coefficients again using the semi-group approach. The fifth chapter gives sufficient conditions for the stability or instability of a periodic solution in terms of the linearization of the equations. These results are then applied to the Hopf bifurcation problem and to a certain population equation modeling periodically fluctuating environments to deduce the stability of the corresponding periodic solutions.</p>
https://thesis.library.caltech.edu/id/eprint/7541Stability Theory of Linear and Nonlinear Stochastic Difference Systems
https://resolver.caltech.edu/CaltechETD:etd-06232005-145221
Authors: {'items': [{'id': 'Ma-Fai', 'name': {'family': 'Ma', 'given': 'Fai'}, 'show_email': 'NO'}]}
Year: 1981
DOI: 10.7907/QZ4T-R653
<p>In this report the stability of linear and nonlinear stochastic difference systems is considered. Explicit criteria for stability are derived. An algorithm is developed for computing the moments of linear stochastic systems when a certain Lie-algebraic condition is satisfied. The relationship between various stability definitions is explored.</p>
https://thesis.library.caltech.edu/id/eprint/2703A Model for the von Kármán Vortex Street
https://resolver.caltech.edu/CaltechETD:etd-05122005-112041
Authors: {'items': [{'email': 'James.Schatzman@futurelabusa.com', 'id': 'Schatzman-James-Carl', 'name': {'family': 'Schatzman', 'given': 'James Carl'}, 'show_email': 'YES'}]}
Year: 1981
DOI: 10.7907/34YN-H995
<p>In the wake of a two-dimensional bluff body placed in a uniform stream, for sufficiently large but not too large flow velocity a distinctive pattern of vorticity is observed. The pattern consists of "vortices" of high vorticity surrounded by nearly irrotational fluid. These vortices are organized in two nearly parallel staggered rows of vortices of opposite direction of rotation. This pattern is called the von Kármán vortex street.</p>
<p>This thesis is a report on the analysis of a model for the von Kármán vortex street. The model is inviscid, incompressible, two-dimensional, and consists of vortices of finite area and uniform vorticity. The first part of this thesis contains a brief survey of the work on this problem, and an explanation of the approach used in the present work; the second part describes calculations of steady solutions of the Euler equations of this kind, and the third part describes an analysis of the stability of these steady solutions to two-dimensional disturbances.</p>
<p>The calculations indicate that the vortex wake can be stabilized by sufficiently large area of the vortices. Data are given which (to some approximation) will permit relating the street to the flow past a body; this is proposed as a suitable study for further work.</p>https://thesis.library.caltech.edu/id/eprint/1751I. Similarity Solutions of the Equations of Three Phase Flow through Porous Media. II. The Fingering Problem in a Hele-Shaw Cell
https://resolver.caltech.edu/CaltechETD:etd-09082006-131345
Authors: {'items': [{'id': 'Romero-Louis-Anthony', 'name': {'family': 'Romero', 'given': 'Louis Anthony'}, 'show_email': 'NO'}]}
Year: 1982
DOI: 10.7907/MR6S-7C08
<p>I</p>
<p>In part I of this thesis similarity solutions to the equations of three phase flow through porous media are examined. The three phases are water, steam, and a noncondensing phase, most likely oil. The main purpose of analyzing such flows is to gain understanding of the steam flooding of oil fields.</p>
<p>Provided steam is being injected at a higher pressure than the initial field pressure, it is shown that there will always be at least two saturation shocks. As one increases the pressure of the injected steam several regimes are encountered; first the flow develops a region where all the steam is completely condensed, then the position of two of the shocks are interchanged, and finally one of the shocks grows weaker and is eventually replaced by an expansion fan.</p>
<p>In sections 12 and 13 the stability of steadily moving condensation fronts in porous media is analyzed. For one special problem it is found that the sign of the jump in pressure gradient at the interface determines whether the interfaces are stable or unstable. This result is applied with some caution to the similarity solutions found in the earlier sections.</p>
<p>II</p>
<p>Recently McLean analyzed the shapes of fingers in a Hele-Shaw cell, including the effects of surface tension. His work resolved the question of the uniqueness of the shapes first brought up by Saffman and Taylor in their analysis that did not include surface tension. It is however felt that there are still unresolved problems.</p>
<p>In determining the pressure jump across an interface there are two principal radii of curvature. McLean only took into account the effect of the larger of these, assuming that the other was constant along the outline of the finger. Unless the smaller radius is very nearly constant, it should in fact give a larger contribution to the jump in pressure. In this thesis the effect of this smaller radius of curvature is modelled by assuming that it is a function of the normal velocity of the mean two dimensional surface of the finger.</p>
<p>It is found that if one only takes into account the smaller radius of curvature, the problem is not uniquely determined, as in the case with no surface tension at all. When both radii of curvature are taken into account, the effect of the smaller radius of curvature is to modify the finger shapes in a way that is qualitatively in agreement with experimental data. Also, McLean's results are checked by an independent numerical scheme, and the results are found to be in excellent agreement. Using both methods of solution a second solution branch other than that described by McLean was also found.</p>https://thesis.library.caltech.edu/id/eprint/3385The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations
https://resolver.caltech.edu/CaltechETD:etd-05042006-103859
Authors: {'items': [{'email': 'rob_scheid@yahoo.com', 'id': 'Scheid-Robert-Elmer-Jr', 'name': {'family': 'Scheid', 'given': 'Robert Elmer, Jr'}, 'show_email': 'NO'}]}
Year: 1982
DOI: 10.7907/4JVY-JB67
<p>We consider systems of ordinary differential equations with rapidly oscillating solutions. Conventional numerical methods require an excessively small time step (Δt = 0(εh), where h is the step size necessary for the resolution of a smooth function of t and 1/ε measures the size of the large eigenvalues of the system's Jacobian).</p>
<p>For the linear problem with well-separated large eigenvalues we introduce smooth transformations which lead to the separation of the time scales and computation with a large time step (Δt = 0(h)). For more general problems, including systems with weak polynomial nonlinearities, we develop an asymptotic theory which leads to expansions whose terms are suitable for numerical approximation. Resonances can be detected and resolved often with a large time step (Δt = 0(h)). Passage through resonance in nonautonomous systems can be resolved by a moderate time step (Δt = 0(√εh)).</p>
https://thesis.library.caltech.edu/id/eprint/1601I. Interactions of Fast and Slow Waves in Problems with Two Time Scales. II. A Numerical Experiment on the Structure of Two-Dimensional Turbulent Flow
https://resolver.caltech.edu/CaltechETD:etd-09182006-090057
Authors: {'items': [{'id': 'Barker-John-Wilson', 'name': {'family': 'Barker', 'given': 'John Wilson'}, 'show_email': 'NO'}]}
Year: 1982
DOI: 10.7907/ynsy-nh46
<p>I. Interaction of Fast and Slow Waves in Problems with Two Time Scales</p>
<p>We consider certain symmetric, hyperbolic systems of nonlinear first-order partial differential equations whose solutions vary on two time scales, a 'slow' scale <i>t</i> and a 'fast' scale <i>t</i>/ε. The large (<i>0</i>(ε<sup>-1</sup>)) part of the spatial operator is assumed to have constant coefficients, but a nonlinear term multiplying the time derivatives (a 'symmetriser') is allowed.</p>
<p>In physical applications, it is often the case that the fast scale motion is of little interest, and it is desired to calculate only the slow scale motion accurately. It is known that solutions with arbitrarily small amounts of fast scale motion can be obtained by careful choice of the initial data, and that an error of amplitude <i>0</i>(ε<sup>p</sup>), where p = 2 for one space dimension or p = 3 for two or three space dimensions, in this choice is allowable, resulting in fast scale waves of amplitude <i>0</i>(ε<sup>p</sup>) in the solution.</p>
<p>We investigate what happens when the initial data are not prepared correctly for the suppression of the fast scale motion, but contain errors of amplitude <i>0</i>(ε). We show that then the perturbation in the solution will also be of amplitude <i>0</i>(ε). Further, we show that if the large part of the spatial operator is nonsingular in the sense that the number of large eigenvalues of the symbol, P(iω), of the spatial operator is independent of ω, then the error introduced in the slow scale motion will be of amplitude <i>0</i>(ε<sup>2</sup>), even though fast scale waves of amplitude <i>0</i>(ε) will be present in the solution. If the symmetriser is a constant, this holds even if the spatial operator is singular, and further if an error <i>0</i>(ε<sup>μ</sup>) is made in the initial conditions, for any µ > 0, the resulting error in the slow scale motion will be <i>0</i>(ε<sup>2μ</sup>).</p>
<p>Our proofs are based on energy estimates which show that spatial derivatives of the solutions are <i>0</i>(1), even if time derivatives are not, and the development of the solutions in asymptotic expansions.</p>
<p>II. A Numerical Experiment on the Structure of Two-Dimensional Turbulent Flow</p>
<p>Some previous theories and numerical calculations pertaining to the problem of two-dimensional turburlence are reviewed, and a new numerical experiment is proposed. The purpose of the experiment is to test the hypothesis that narrow regions of concentrated vorticity are produced in two-dimensional flows by advection of vorticity towards dividing streamlines in regions where the local flow is convergent.</p>
<p>The numerical method to be used is described in detail. It integrates the inviscid Euler equations using a Fourier (pseudo-spectral) method for the space derivatives, and a predictor-corrector method due to Hyman (1979) for time stepping. Dissipation is included, following Fornberg (1977), by a chopping of the amplitudes of the higher Fourier modes every few time-steps. This acts as a high-wavenumber energy sink, allowing very high Reynolds number flows to be simulated with relatively little computational effort.</p>https://thesis.library.caltech.edu/id/eprint/3618Dielectric Waveguides for Millimeter Waves
https://resolver.caltech.edu/CaltechETD:etd-09062006-152644
Authors: {'items': [{'id': 'Schweig-Edgard', 'name': {'family': 'Schweig', 'given': 'Edgard'}, 'show_email': 'NO'}]}
Year: 1982
DOI: 10.7907/05DR-KV58
<p>In this thesis, we analyze high-permittivity dielectric waveguides for use as guiding structures of millimeter waves. Two basic geometries are considered: the circular and rectangular guides.</p>
<p>In Part I, we describe the theory of round fibers surrounded by an infinite cladding. Millimeter wavelengths are comparable to the physical dimensions of the guide. Therefore, a large difference in permittivity between the core and the cladding is required in order to provide for a tight confinement of the fields. We present the results of computations of the propagation characteristics and losses of fibers of very high permittivity. We note that the distribution of the electromagnetic power between the core and the cladding can be deduced from the dispersion curves. Finally, we consider the feasibility of a dielectric fiber made of thallium bromide-iodide (KRS-5) for the long distance transmission of W-band signals (94 GHz). Using our measurements of the dielectric parameters of KRS-5, we find that the losses are several orders of magnitude higher than the losses of conventional metallic waveguides.</p>
<p>In Part II, we analyze rectangular dielectric guides made of high-permittivity materials such as GaAs that would permit the fabrication of active devices directly into the transmission line. We present a new numerical technique base on finite-differences for computing the modes of dielectric guiding structures. This method is simple and efficient in computer storage and computational time. We use it to compute the modes of a rectangular dielectric waveguide and compare the numerical results to those obtained from Marcatili's closed-form solution. We find that this latter one is a good approximation for the dominant mode of a rectangular guide even when the permittivity of the guide is large compared to the outer medium. For higher order modes, Marcatili's solution predicts incorrect propagation curves. We have also observed the presence in our numerical solution of "spurious modes" that are thought to be due to the mathematical indefinitiveness of the problem.</p>
<p>In Part III, we present a waveguide technique for the measurement of complex dielectric constants at millimiter wave frequencies: the shorted-waveguide method. Waveguide methods have been extensively used at lower frequencies but this is the first application at 94 GHZ. We use a novel sample preparation technique that allows for an accurate and gap-free positionment of a ductile dielectric material inside a metallic waveguide. We note that the correct choice of sample lengths is critical to the accuracy of the measurement of the loss tangent. Finally, we summarize the results of our measurement of the dielectric constant and loss tangent of thallium bromide-iodide (KRS-5) and thallium bromide-chloride (KRS-6).</p>https://thesis.library.caltech.edu/id/eprint/3363Solution Adaptive Mesh Procedures for the Numerical Solution of Singular Perturbation Problems
https://resolver.caltech.edu/CaltechETD:etd-09182006-134307
Authors: {'items': [{'email': 'dlb@llnl.gov', 'id': 'Brown-David-Leslie', 'name': {'family': 'Brown', 'given': 'David Leslie'}, 'show_email': 'YES'}]}
Year: 1982
DOI: 10.7907/4DVY-AH34
<p>The accurate numerical solution of singular perturbation problems by finite difference methods is considered. (For efficient computations of this type, refinement of the finite difference mesh is important. The technique of solution-adaptive mesh refinement, in which the mesh is refined iteratively by looking at the properties of a computed solution, can be the simplest method by which to implement a mesh refinement.) The theoretical justification of solution-adaptive mesh refinement for singularly perturbed systems of first order ordinary differential equations (ODEs) is discussed. It is shown that <i>a posteriori</i> error estimates can be found for weighted one-sided difference approximations to systems of ODEs without turning points and to systems of ODEs with turning points that can be transformed to a typical normal form. These error estimates essentially depend only on the local meshwidths and on lower order divided differences of the computed solution, and so can be used in the implementation of solution-adaptive mesh refinement. It is pointed out, however, that not all systems with turning points fall into these categories, and solution-adaptive mesh refinement can sometimes be inadequate for the accurate resolution of solutions of these systems.</p>
<p>Numerical examples are presented in which the solutions of some model equations of fluid dynamics are resolved by transforming the problems to singularly perturbed ODEs and applying weighted one-sided difference approximations with solution-adaptive mesh refinement. In particular, well-resolved steady and moving shock solutions to Burgers' equation and to the equations of one-dimensional isentropic gas dynamics are obtained numerically. The method is further extended to problems in two space dimensions by using the method of dimensional splitting together with careful interpolation. In particular, in this extension the mesh refinement is only used to resolve the one-dimensional problems which are solved within the splitting algorithm. Numerical examples are presented in which two-dimensional oblique shocks are resolved.</p>
https://thesis.library.caltech.edu/id/eprint/3619I. Three Dimensional Ray-Tracing and Ray-Inversion in Layered Media. II. Inverse Scattering and Curved Ray Tomography with Applications to Seismology
https://resolver.caltech.edu/CaltechETD:etd-09082006-092225
Authors: {'items': [{'email': 'John.Fawcett@drdc-rddc.gc.ca', 'id': 'Fawcett-John-Alan', 'name': {'family': 'Fawcett', 'given': 'John Alan'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/QC2Q-6G46
<p>In seismology, the basic problem is that of deducing some knowledge of the geological structure of portions of the Earth from observed seismic signals. This leads to the concepts of seismic interpretation, or more mathematically, the formulation of inverse problems.</p>
<p>Some aspects of seismic wave propagation can be interpreted in terms of asymptotic ray theory. In Chapter 1 of Part I, we describe the numerical ray tracing algorithm we developed for layered media with interfaces that can vary in three dimensions. We describe in Chapter 2, how this ray tracing method is implemented in an inversion procedure. This method is based on the theory of non-linear least-squares inversion.</p>
<p>In Part II of the thesis, we discuss two formulations of seismic inverse problems, which are more analytical in nature. Chapter 1 deals with the use of inverse scattering theory for the Schroedinger operator in the seismological problem. In chapter 2 of Part II, we develop the theory of the tomographical inversion of travel time anomalies to determine velocity anomalies within the Earth. Here, we have extended, in an approximate sense, the Inverse Radon Transform to situations where the "background" velocity field varies with depth.</p>https://thesis.library.caltech.edu/id/eprint/3382Forces on a Whirling Centrifugal Pump-Impeller
https://resolver.caltech.edu/CaltechETD:etd-09152006-083609
Authors: {'items': [{'id': 'Chamieh-Dimitri-Suhayl', 'name': {'family': 'Chamieh', 'given': 'Dimitri Suhayl'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/vnqy-ne26
<p>The present work is an experimental and theoretical investigation of the possible forces of fluid dynamic origin that can act on a turbomachine rotor particularly when it is situated off its normal center position. An experimental facility, the Rotor Force Test Facility, has been designed and constructed in order to measure these kinds of forces acting on a centrifugal pump impeller when the latter is made to whirl in a slightly eccentric circular orbit. The rotor speed, eccentric orbital radii and whirl speed could be varied independently. The scope of the present experimental work consists of measuring quasi-steady forces on the impeller as it whirls slowly about the axis of the pump rotation. These forces are due to interaction between the impeller and volute; they are decomposed into force components relative to the geometric center of the volute and to those proportional to displacement from this center. These latter are interpreted as stiffness matrices. These matrices were measured on two widely differing volute types and both were found to have the property of being skew-symmetric. It can be shown that a stiffness matrix of this type can lead to dynamic instability of the impeller shaft system in certain circumstances. This new experimental finding may explain some operational problems of "high speed" hydraulic machinery.</p>
<p>In the theoretical part of this thesis, a somewhat more physical model of a rotor pump is proposed other than has been used heretofore in most works namely an actuator disk having infinitely many blades. As a simplification it is assumed that the flow field is irrotational. Forces and stiffness matrices are calculated on this basis but the stiffness matrix so found does not reveal the skew-symmetric property of the experiments.</p>https://thesis.library.caltech.edu/id/eprint/3551Optimal Low Thrust, Three Burn Orbit Transfers with Large Plane Changes
https://resolver.caltech.edu/CaltechETD:etd-08152005-091409
Authors: {'items': [{'email': 'Keith.Zondervan@aero.org', 'id': 'Zondervan-Keith-Peter', 'name': {'family': 'Zondervan', 'given': 'Keith Peter'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/NXPK-GE17
<p>During the last twenty-five years, much attention has been devoted to the problem of optimal orbit transfer. The problem has been conveniently divided into two categories - unlimited thrust (or acceleration) orbit transfers and limited thrust (or acceleration) orbit transfers. The unlimited thrust orbit transfers use infinite thrust, zero burn time burns and hence have also come to be known as impulsive burn orbit transfers. In general it has been found that optimal (i.e., minimum fuel, time-free) solutions to these types of transfers require two or possibly three burns. The limited thrust transfers, in contrast, do not use impulsive burns but use burns which have a finite thrust level and a nonzero burn time and, hence, are also known as finite burn orbit transfers.</p>
<p>If our attention is restricted to finite multi-burn transfers which have burn times less than an orbital period, two classes of transfers emerge. These classes of transfers are either Geometrically Similar to the 2-Burn Impulsive (GS2BI) transfers or Geometrically Similar to the 3-Burn Impulsive (GS3BI) transfers. For example, if a 2-burn impulsive solution has a perigee burn followed by an apogee burn, the GS2BI finite burn transfer would use one or more perigee burns followed by one or more apogee burns.</p>
<p>Recent-studies have presented optimal solutions to GS2BI finite burn orbit transfers for various thrust to weight ratios. The current study presents the optimal solutions to GS3BI finite burn orbit transfers between a 28.5° inclined low-earth orbit and a series of 63.4° inclined circular orbits and a series of 63.4° inclined elliptical orbits with twelve hour periods. Also presented are optimal solutions to GS3BI finite burn orbit transfers between 97° inclined high-earth orbits and a 57° inclined low-earth orbit. Optimal solutions are found to be bounded by a lower limit on the initial thrust to weight ratio. It is shown that as the final perigee altitude is increased, the GS3BI finite burn transfer degenerates to a GS2BI finite burn transfer much as it would for the impulsive case.</p>
<p>Analysis of the optimal steering during various burns reveals a natural division of the steering strategies into two categories based on whether a burn results in a predominant change in the orbit size-or-the orbit plane. The similarity of these optimal steering strategies to previously obtained simple "near-optimal" steering strategies is discussed.</p>https://thesis.library.caltech.edu/id/eprint/3123I. Stability of Tchebyshev Collocation. II. Interpolation for Surfaces with 1-D Discontinuities. III. On Composite Meshes
https://resolver.caltech.edu/CaltechETD:etd-06222005-104752
Authors: {'items': [{'email': 'luisgreyna@gmail.com', 'id': 'Reyna-Luis-Guillermo-Maria', 'name': {'family': 'Reyna', 'given': 'Luis Guillermo Maria'}, 'show_email': 'YES'}]}
Year: 1983
DOI: 10.7907/AAG6-MW97
<p>I. Stability of Tchebyshev Collocation</p>
<p>We describe Tchebyshev collocation when applied to hyperbolic equations in one space dimension. We discuss previous stability results valid for scalar equations and study a procedure that when applied to a strictly hyperbolic system of equations leads to a stable numerical approximation in the L<sub>2</sub>-norm. The method consists of using orthogonal projections in the L<sub>2</sub>-norm to apply the boundary conditions and smooth the higher modes.</p>
<p>II. On 2-D Interpolation for Surfaces with 1-D Discontinuities</p>
<p>This problem arises in the context of shock calculations in two space dimensions. Given the set of parabolic equations describing the shock phenomena the method proceeds by discretising in time and then solving the resulting elliptic equation by splitting. The specific problem is to reconstruct a two dimensional function which is fully resolved along a few parallel horizontal lines. The interpolation proceeds by determining the position of any discontinuity and then interpolating parallel to it.</p>
<p>III. On Composite Meshes</p>
<p>We collect several numerical experiments designed to determine possible numerical artifacts produced by the overlapping region of composite meshes. We also study the numerical stability of the method when applied to hyperbolic equations. Finally we apply it to a model of a wind driven ocean circulation model in a circular basin. We use stretching in the angular and radial directions which allow the necessary resolution to be obtained along the boundary.</p>https://thesis.library.caltech.edu/id/eprint/2683An Analytical Study of Electromagnetic Vector Field Propagation in a Nonlinear Electron Plasma
https://resolver.caltech.edu/CaltechETD:etd-10312005-133116
Authors: {'items': [{'id': 'Tatoian-James-Zareh', 'name': {'family': 'Tatoian', 'given': 'James Zareh'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/hzt3-s585
<p>From the equations of hydrodynamics and electrodynamics, a system of a coupled nonlinear equations governing the propagation of plane electromagnetic waves in a collisionless electron plasma is obtained. It is shown that solitary wave solutions exist for both the longitudinal and transverse components of the electromagnetic field. It is found that the velocity of the electromagnetic vector solitary wave depends on the amplitudes of all components of the field linearly. The relations among the longitudinal and transverse components that support the solitary waves are determined for different values of plasma temperature. It is shown that while transverse solitary waves cannot exist, except when they are supported by longitudinal waves, the latter can exist by themselves. The dynamics of the plasma electrons during the passage of a longitudinal wave is analyzed and the interaction of such waves with each other is studied. An upper bound on the amplitudes of these waves is obtained. The uniqueness and stability of the longitudinal waves are demonstrated. A Lagrangian density function and two conservation laws for the longitudinal wave equation are found. Frequency spectra of the solitary waves are calculated and their low frequency content is emphasized.</p>
https://thesis.library.caltech.edu/id/eprint/4338Reduction of Unbounded Domains to Bounded Domains for Partial Differential Equation Problems
https://resolver.caltech.edu/CaltechETD:etd-09062006-104459
Authors: {'items': [{'email': 'hagstrom@math.unm.edu', 'id': 'Hagstrom-Thomas-Michael', 'name': {'family': 'Hagstrom', 'given': 'Thomas Michael'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/5FR1-DS57
<p>Many boundary value problems which arise in applied mathematics are given in unbounded domains. Here we develop a theory for the imposition of boundary conditions at an artificial boundary which lead to finite domain problems that are equivalent to the unbounded domain problems from which they come. By considering the Cauchy problem with initial data in the appropriate space of functions on the artificial boundary, we show that satisfaction of the boundary conditions at infinity is equivalent to satisfaction of a certain projection condition, at the artificial boundary. This leads to an equivalent finite problem. The solvability of the finite problem is discussed and estimates of the solution in terms of the inhomogeneous data are given.</p>
<p>Applications of our reduction to problems whose coefficients are independent of the unbounded coordinate are considered first. For a class of problems we shall term 'separable', solutions in the tail can be developed in an eigenfunction expansion. These expansions are used to write down an explicit representation of the projection, which is useful in computations. Specific problems considered here include elliptic equations in cylindrical domains. Spatially unbounded parabolic and hyperbolic problems are also discussed. Here, the eigenfunction expansions must include continuous transform variables.</p>
<p>We use these 'constant tail' results to develop a perturbation theory for the case when the coefficients depend upon the unbounded coordinate. This theory is based on Duhamel's principle and is seen to be especially useful when the 'limiting' problem possesses an exponential dichotomy. We apply our results to the Helmholtz equation, perturbed hyperbolic systems and nonlinear problems. We present a numerical solution of the Bratu problem in a semi-infinite, two-dimensional, stepped channel to illustrate our method.</p>https://thesis.library.caltech.edu/id/eprint/3356Long Distance Energy Correlations in Random Media
https://resolver.caltech.edu/CaltechETD:etd-11032005-093758
Authors: {'items': [{'email': 'zwillinger@az-tec.com', 'id': 'Zwillinger-Daniel-Ian', 'name': {'family': 'Zwillinger', 'given': 'Daniel Ian'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/G3DN-MZ83
<p>This thesis considers the long distance motion of waves in a random medium. Using the geometrical optics approximation and a stochastic limit theorem, we find evolution equations for rays and for energy correlations, in two and three dimensions.</p>
<p>Our equations are valid on a long distance scale, well after the focusing of rays has become significant. We construct asymptotic expansions of the two point energy correlation function in two and three dimensions.</p>
<p>In two dimensions we numerically solve the partial differential equation that determines the two point energy correlation function. We also perform Monte-Carlo simulations to calculate the same quantity. There is good agreement between the two solutions.</p>
<p>We present the solution for the two point energy correlation function obtained by regular perturbation techniques. This solution agrees with our solution until focusing becomes significant. Then our solution is valid (as shown by the Monte-Carlo simulations), while the regular perturbation solution becomes invalid.</p>
<p>Also presented are the equations that describe energy correlations after a wave has gone through a weakly stochastic plane layered medium.</p>
https://thesis.library.caltech.edu/id/eprint/4388A Numerical Study of Bubble Deformation in Steady Axisymmetric Flows
https://resolver.caltech.edu/CaltechETD:etd-04132004-142806
Authors: {'items': [{'id': 'Ryskin-Gregory', 'name': {'family': 'Ryskin', 'given': 'Gregory'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/rrsq-e746
<p>This work is devoted to the development and application of the numerical technique suitable for solution of the free-boundary problems, i.e. those in which the shape of the boundary should be determined as a part of the solution. The technique is based on a finite-difference solution of the equations of the problem on an orthogonal curvilinear coordinate system, which is also constructed numerically and always adjusted so as to fit the current boundary shape. The same orthogonal mapping approach may also be used to construct orthogonal coordinates fitted to boundaries of known but complicated shapes.</p>
<p>The technique is applied to two classical problems of fluid mechanics -- deformation of a gas bubble rising through a quiescent fluid due to buoyancy, and deformation of a gas bubble in a uniaxial extensional flow. For the rising bubble, the shapes and flow fields are computed for Reynolds numbers 1 ≤ R ≤ 200 and Weber numbers up to 20 at the lower Reynolds numbers and up to 10 at Reynolds numbers 50, 100 and 200. The most interesting results of this part are those demonstrating the phenomenon of flow separation at a smooth free surface. This phenomenon does not appear to have been theoretically predicted before, in spite of its importance for understanding the mechanics of free-surface flows.</p>
<p>In the case of a bubble in a uniaxial extensional flow, the computations show that at Reynolds numbers of order 10 and higher the deformation of a bubble proceeds in a way qualitatively different from the low Reynolds number regime studied previously; the bubble bursts at a relatively early stage of deformation never reaching the highly elongated shapes observed and predicted at low Reynolds numbers. It is shown also that for this problem the solution at Reynolds number of order 100 is already quite close to the potential flow solution which can be easily obtained using the present technique.</p>https://thesis.library.caltech.edu/id/eprint/1369Ordinary and Strong Ellipticity in the Equilibrium Theory of Incompressible Hyperelastic Solids
https://resolver.caltech.edu/CaltechETD:etd-11012005-130640
Authors: {'items': [{'id': 'Zee-Layne', 'name': {'family': 'Zee', 'given': 'Layne'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/jzka-ce91
<p>In this paper explicit necessary and sufficient conditions are established for the ordinary and strong ellipticity of the three-dimensional field equations in the nonlinear equilibrium theory of incompressible, homogeneous and isotropic, hyperelastic solids. The resulting system of inequalities involves the local principal stretches directly and in addition restricts the first and second partial derivatives of the strain-energy density with respect to the deformation invariants or the principal stretches. The conditions of ordinary and strong ellipticity are found to coalesce for materials that obey the Baker-Ericksen inequalities and possess a positive shear modulus at infinitesimal deformations. Various implications of these ellipticity conditions for special classes of materials and deformations are explored.</p>https://thesis.library.caltech.edu/id/eprint/4358Linear Programming Methods for the Numerical Solution of Parabolic Equations Backwards in Time
https://resolver.caltech.edu/CaltechETD:etd-09052006-083506
Authors: {'items': [{'id': 'Prendergast-Michael-David', 'name': {'family': 'Prendergast', 'given': 'Michael David'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/rwq0-z817
<p>This thesis investigates linear programming methods for the numerical solution of parabolic equations backwards in time. These problems are ill-posed. Hence an approximate numerical solution for such problems can only be obtained if additional constraints (called a regularization) are imposed on the solution in order to guarantee its stability under small perturbations. Previous authors have implemented regularizations on the backward heat equation which used (linear or nonlinear) least squares, or linear programming. These regularizations use the exact form of the kernel for the heat equation, however, and so are not generalizable to problems with an unknown kernel or unknown eigenfunction expansion. Furthermore, the least squares methods can not easily handle the nonnegativity constraint that a positive temperature, for example, must have.</p>
<p>In the first part of this thesis, linear regularizations which can be used to solve any linear parabolic equation on a finite domain backwards in time are introduced. It is then shown how a numerical approximation to the solution of the regularized problem can be obtained by using linear programming and any stable and consistent difference method (such as Crank-Nicholson). The convergence of these algorithms is shown to be a direct consequence of the Lax equivalence theorem. The stability, accuracy, and results of actual numerical experiments using this linear programming method are analyzed.</p>
<p>The second part of this thesis shows how these regularizations can be used on weakly nonlinear equations. This is done by introducing a successive approximation method, and solving a linear program at each step in the iteration. The stability, accuracy, and results of numerical experiments for this algorithm are also examined.</p>
https://thesis.library.caltech.edu/id/eprint/3334The Propagation and Arrest of an Edge Crack in an Elastic Half-Space Under Conditions of Anti-Plane Shear: Analytical and Numerical Results
https://resolver.caltech.edu/CaltechETD:etd-09052006-082841
Authors: {'items': [{'id': "O'Sullivan-Timothy-Christopher", 'name': {'family': "O'Sullivan", 'given': 'Timothy Christopher'}, 'show_email': 'NO'}]}
Year: 1983
DOI: 10.7907/jb3j-5460
<p>The motion of an edge crack extending non-uniformly in an elastic half-space under conditions of anti-plane shear is analyzed. An expression for the stress intensity factor at the crack tip is obtained, and an energy balance crack propagation criterion is used to find the equation of motion of the tip. On solving this equation numerically, it is found that crack arrest occurs before the second reflected wave from the boundary reaches the tip.</p>
<p>In the second half of this investigation, a numerical procedure for studying anti-plane shear crack propagation problems using finite differences is developed. To approximate the elastodynamic field as accurately as possible near the moving crack tip, where singular stresses occur, the local asymptotic displacement field near the tip is incorporated into the finite difference scheme. The numerical procedure is applied to the edge crack problem analyzed in the first part of this study, and the numerical and exact results are compared.</p>https://thesis.library.caltech.edu/id/eprint/3333Topics in 2-D Separated Vortex Flows
https://resolver.caltech.edu/CaltechETD:etd-02012007-133412
Authors: {'items': [{'id': 'Tanveer-Saleh-Ahmed', 'name': {'family': 'Tanveer', 'given': 'Saleh Ahmed'}, 'show_email': 'NO'}]}
Year: 1984
DOI: 10.7907/BR0B-QH56
<p>This thesis is concerned with vortices in steady two dimensional inviscid incompressible flow. In the first three chapters, separated vortex flows are considered in the context of inviscid flow past two dimensional airfoils for which the action of the vortex is to induce large lift. In the fourth and last chapter, we consider vortices in uniform flow in the absence of any physical bodies.</p>
<p>In chapter I, we consider two configurations of vortices for flow past a flat plate with a forward facing flap attached to its rear edge. In the first case, case (a), we consider a potential vortex in the vicinity of the airfoil, while for case (b), we consider a vortex sheet coming off the leading edge of the plate and reattaching at the leading edge of the flap such that the region between the vortex sheet and the airfoil is stagnant. For case (a), the Schwarz-Christoffel transformation is used to find exact solutions to the flow problem. It is found that by suitably placing a potential vortex of appropriate strength it is possible to satisfy the Kutta condition of finite velocity at both the leading edges of the plate and the flap in addition to satisfying it at the trailing edge, provided the plate flap combination satisfies a geometric constraint. The action of the potential vortex is to create a large circulatory region bounded by the airfoil and the streamline that separates smoothly at the leading edge of the plate (due to the Kutta condition) and reattaches smoothly at the leading edge of the flap (from the Kutta condition again). The circulation induced at infinity for such a flow and hence the lift on the airfoil is found to be very large. For case (b), where the vortex sheet location is unknown, a hodograph method is used to find exact solutions. It is found that once a geometric constraint is satisfied, flows exist for which the Kutta condition is satisfied at the trailing edge of the plate-flap combination. As in (a), large values of lift are obtained. However, in both cases (a) and (b), the adverse pressure gradient of top of the flap is recognized as a source of potential difficulty in the experimental realization of the calculated flow.</p>
<p>In chapter II, successive modifications are made to the airfoil considered in chapter I. Exact solutions are once again obtained by a variation of the hodograph method of chapter I. The lift for these airfoils is found to be significantly larger than the one in chapter I. Because the trailing edge is no longer a stagnation point, it is felt that these flows may be easier to realize experimentally.</p>
<p>Chapter III is concerned with the so-called Prandtl-Batchelor flow past the plate-flap geometry of chapter I. The flow consists of an inner region which has a constant vorticity. The region outside of the airfoil and the vortex sheet coming off the leading edge of plate and reattaching at the leading edge of the flap (as in chapter I) is once again irrotational. The common boundary between the exterior flow and the inner flow, i.e. the vortex sheet, is unknown a priori and is determined by continuity of pressure, which translates into a nonlinear boundary condition on an unknown boundary. By extending the function theoretic approach of complex variables to this problem, we reduce the entire problem into one of determining one unknown function of one variable on a fixed domain from which everything else can be calculated. This is then solved numerically. Our calculations provide what we believe to be the first such calculation of a Prandtl-Batchelor flow. The calculations also provide a more realistic model for the vortex sheet flow considered in chapter I.</p>
<p>Chapter IV deals with a steadily translating pair of equal but opposite vortices with uniform cores and vortex sheets on their boundaries, moving without the presence of any physical boundary. The solutions were found for such flows using the function theoretic approach introduced earlier in chapter III for flows where the velocity on the vortex sheet is not a constant. The solutions form a continuum between the hollow vortex case of Pocklington (1898) and those of Deem and Zabusky (1978) and Pierrehumbert (1980) who consider uniform core with no vortex sheet. The iterative scheme for numerical calculation, however, turns out to have severe limitations, as it fails to converge for the cases with no vortex sheet or when the vortex sheet strength is small. In the last section of the chapter, a more traditional approach due to Deem and Zabusky is taken to calculate a pair of touching vortices with uniform core and no vortex sheet on the boundary and an error in Pierrehumbert's (1980) calculations is pointed out.</p>
<p>In appendix I, we point out some errors in Pocklington's paper on the motion of a hollow vortex pair. The errors are corrected and the results are found to be then in agreement with results using the method in chapter IV.</p>https://thesis.library.caltech.edu/id/eprint/442Part I. The Numerical Solution of Hyperbolic Systems of Conservation Laws. Part II. Composite Overlapping Grid Techniques
https://resolver.caltech.edu/CaltechETD:etd-03312008-100117
Authors: {'items': [{'email': 'henshaw1@llnl.gov', 'id': 'Henshaw-William-Douglas', 'name': {'family': 'Henshaw', 'given': 'William Douglas'}, 'show_email': 'YES'}]}
Year: 1985
DOI: 10.7907/kz0y-2j77
<p>Part I</p>
<p>A method is described for the numerical solution of hyperbolic systems of conservation laws in one space dimension. The basis of the scheme is to use finite differences where the solution is smooth and the method of characteristics where the solution is not smooth. The method can accurately represent shocks. Results are presented for the solution of the equations of gas dynamics. The examples illustrate the accuracy of the method when discontinuities are present and the code's performance on difficult problems of interacting shocks and shock formation.</p>
<p>Part II</p>
<p>Techniques for the numerical solution of partial differential equations on composite overlapping meshes are discussed. Methods for the solution of time dependent and elliptic problems are illustrated, including a discussion of implicit time stepping and using the multigrid algorithm for the iterative solution of Poisson's equation. Two model problems are analyzed. The first gives insight into the accuracy of the solution to elliptic equations on overlapping meshes. The second deals with the numerical approximation of boundary conditions for vorticity stream function formulations. Computational results are presented.</p>https://thesis.library.caltech.edu/id/eprint/1227Complex Bifurcation
https://resolver.caltech.edu/CaltechETD:etd-03262008-112516
Authors: {'items': [{'id': 'Henderson-Michael-Edwin', 'name': {'family': 'Henderson', 'given': 'Michael Edwin'}, 'show_email': 'NO'}]}
Year: 1985
DOI: 10.7907/JF82-1T64
<p>Real equations of the form g(x,λ) = 0 are shown to have a complex extension G(u,λ) = 0, defined on the complex Banach space 𝔹 ⊕ i𝔹. At a singular point of the real equation this extension has solution branches corresponding to both the real and imaginary roots of the Algebraic Bifurcation Equations (ABE's).</p>
<p>We solve the ABE's at simple quadratic folds, quadratic bifurcation points, and cubic bifurcation points, and show that these are complex bifurcation points. We also show that at a Hopf bifurcation point of the real equation there are two families of complex periodic orbits, parametrized by three real parameters.</p>
<p>By taking sections of solutions of complex equations with two real parameters, we show that complex branches may connect disjoint solution branches of the real equation. These complex branches provide a simple and practical means of locating disjoint branches of real solutions.</p>
<p>Finally, we show how algorithms for computing real solutions may be modified to compute complex solutions. We use such an algorithm to find solutions of several example problems, and locate two sets of disjoint real branches.</p>https://thesis.library.caltech.edu/id/eprint/1159Part I. Fold Continuation and the Flow Between Rotating, Coaxial Disks. Part II. Equilibrium Chaos. Part III. A Mesh Selection Algorithm for Two-Point Boundary Value Problems
https://resolver.caltech.edu/CaltechETD:etd-03262008-150456
Authors: {'items': [{'id': 'Fier-Jeffrey-Michael', 'name': {'family': 'Fier', 'given': 'Jeffrey Michael'}, 'show_email': 'NO'}]}
Year: 1985
DOI: 10.7907/cs9b-ft10
<p>Part I:</p>
<p>We consider folds in the solution surface of nonlinear equations with two free parameters. A system of equations whose solutions are fold paths is formulated and proved to be non-singular in a neighborhood of a fold, thus making continuation possible. Efficient numerical algorithms employing block Gaussian elimination are developed for applying Euler-Newton pseudo-arclength continuation to the system, and these are shown to require fewer operations than other methods.</p>
<p>To demonstrate the use of these methods we calculate the flow between two infinite, rotating disks. For Reynold's number less than 1000, six separate solution sheets are found and completely described. Plots of 47 solutions for three values of the disk speed ratio and for Reynold's number equal to 625 are shown. These are compared with the solutions found by previous investigators.</p>
<p>Part II:</p>
<p>Two ordinary differential equations with parameters whose solution paths exhibit an infinite sequence of folds clustered about a limiting value are studied. Using phase-plane analysis, expressions for the limiting ratios of the parameter values at which these folds occur are derived and the limiting values are shown to be non-universal.</p>
<p>Part III:</p>
<p>A mesh selection algorithm for use in a code to solve first-order nonlinear two-point boundary value problems with separated end conditions is described. The method is based on equidistributing the global error of the box scheme, a numerical estimate of which is obtained from Richardson extrapolation. Details of the algorithm and examples of its performance on non-stiff and stiff problems are presented.</p>
https://thesis.library.caltech.edu/id/eprint/1162Generation of Long Water Waves by Moving Disturbances
https://resolver.caltech.edu/CaltechETD:etd-04012008-151918
Authors: {'items': [{'email': 'sjoonlee@cnu.ac.kr', 'id': 'Lee-Seung-Joon', 'name': {'family': 'Lee', 'given': 'Seung-Joon'}, 'show_email': 'YES'}]}
Year: 1985
DOI: 10.7907/EG1N-VZ69
<p>Several theoretical models are developed to study generation of nonlinear dispersive long waves by moving disturbances. All these models belong to the same class as the original Boussinesq or KdV model. The newly developed models, now with external forcing functions added to the KdV equation and the pair of coupled Boussinesq equations, have been chosen for numerical investigations. A predictor-corrector method is adopted to develop the numerical schemes employed here. In order to make the region of computation reasonably small for the case with moving disturbances, a pseudo-moving frame and the sufficiently transparent open boundary conditions are devised. The numerically obtained surface elevations exhibit a series of positive waves running ahead of the disturbance over a wide range of transcritical speeds of the disturbance. The numerical results show that, for speeds close to the critical value, the generation of such waves appears to continue indefinitely. The numerically obtained wave resistance coefficient is compared to the results given by linear dispersive theory. Numerical solutions have been obtained using the KdV and Boussinesq models with surface pressure and bottom bump as forcing functions. Comparisons are made between these results for various cases. Experiments were conducted for a two-dimensional bottom bump moving steadily in shallow water of a towing tank. Experimental results so attained are compared with the numerical solutions, and the agreement between them is good in terms of both the magnitude and the phase of the waves for the range of parameters used in the current study.</p>https://thesis.library.caltech.edu/id/eprint/1242Numerical Shock Propagation Using Geometrical Shock Dynamics
https://resolver.caltech.edu/CaltechETD:etd-03082008-083041
Authors: {'items': [{'id': 'Schwendeman-Donald-William', 'name': {'family': 'Schwendeman', 'given': 'Donald William'}, 'show_email': 'NO'}]}
Year: 1986
DOI: 10.7907/Q5VX-AF72
<p>Various numerical schemes are developed to calculate the motion of shock waves in gases based on Whitham's theory of geometrical shock dynamics. The basic numerical scheme is used to study the propagation of two-dimensional shock waves along walls and in channels, and the self-focusing of initially curved shock- fronts. This scheme is extended to treat shock wave motion in non-uniform media. The extended scheme is used to examine shock wave refraction at both planar and curved interfaces separating gases with different properties. Precursor-irregular refraction patterns are obtained using geometrical shock dynamics. A general numerical scheme designed to propagate a shock surface in three dimensions is presented. Three-dimensional shock focusing and shock propagation in a curved pipe are considered primarily to demonstrate the use of the three-dimensional numerical scheme. The reflection of planar shock waves from curved walls is studied. The motion of the shock is determined using the combined theories of regular reflection and geometrical shock dynamics. A numerical scheme based on the combined theories is discussed. The numerical scheme is used to calculate the reflection and subsequent focusing of weak planar shock waves. Some of the present results are compared with other solutions to the equations of geometrical shock dynamics obtained using different methods. Recent experimental investigations are discussed and compared with our results calculated using geometrical shock dynamics.</p>
https://thesis.library.caltech.edu/id/eprint/899Relating Thermodynamics to Information Theory: The Equality of Free Energy and Mutual Information
https://resolver.caltech.edu/CaltechTHESIS:03212013-084718067
Authors: {'items': [{'email': 'dif@alumni.caltech.edu', 'id': 'Feinstein-David-I', 'name': {'family': 'Feinstein', 'given': 'David I.'}, 'show_email': 'YES'}]}
Year: 1986
DOI: 10.7907/XVQB-7902
<p>In this thesis we uncover a new relation which links thermodynamics and information theory. We consider time as a channel and the detailed state of a physical system as a message. As the system evolves with time, ever present noise insures that the "message" is corrupted. Thermodynamic free energy measures the approach of the system toward equilibrium. Information theoretical mutual information measures the loss of memory of initial state. We regard the free energy and the mutual information as operators which map probability distributions over state space to real numbers. In the limit of long times, we show how the free energy operator and the mutual information operator asymptotically attain a very simple relationship to one another. This relationship is founded on the common appearance of entropy in the two operators and on an identity between internal energy and conditional entropy. The use of conditional entropy is what distinguishes our approach from previous efforts to relate thermodynamics and information theory.</p>
https://thesis.library.caltech.edu/id/eprint/7537Ray Tracing in Complex Three-Dimensional Earth Models
https://resolver.caltech.edu/CaltechETD:etd-03182008-140018
Authors: {'items': [{'id': 'Girnius-Tomas-Paul', 'name': {'family': 'Girnius', 'given': 'Tomas Paul'}, 'show_email': 'NO'}]}
Year: 1986
DOI: 10.7907/d9ts-vp45
<p>The problem of tracing seismic rays between specified source and receiver is discussed for Earth models consisting of layers, in which velocity varies linearly, that are separated by material interfaces of arbitrary shape. The calculation of travel times, amplitudes, and phase shifts is considered. Fast and efficient numerical algorithms are developed. Computed examples are presented.</p>https://thesis.library.caltech.edu/id/eprint/1000Asymptotic Analysis of Thin Plates Under Normal Load and Horizontal Edge Thrust
https://resolver.caltech.edu/CaltechTHESIS:03212013-094948659
Authors: {'items': [{'email': 'taraathan@gmail.com', 'id': 'Brewster-Mary-Elizabeth', 'name': {'family': 'Brewster', 'given': 'Mary Elizabeth'}, 'show_email': 'NO'}]}
Year: 1987
DOI: 10.7907/DDP9-KW92
<p>We consider the radially symmetric nonlinear von Kármán plate equations for circular or annular plates in the limit of small thickness. The loads on the plate consist of a radially symmetric pressure load and a uniform edge load. The dependence of the steady states on the edge load and thickness is studied using asymptotics as well as numerical calculations. The von Kármán plate equations are a singular perturbation of the Fӧppl membrane equation in the asymptotic limit of small thickness. We study the role of compressive membrane solutions in the small thickness asymptotic behavior of the plate solutions. </p>
<p>We give evidence for the existence of a singular compressive solution for the circular membrane and show by a singular perturbation expansion that the nonsingular compressive solutions approach this singular solution as the radial stress at the center of the plate vanishes. In this limit, an infinite number of folds occur with respect to the edge load. Similar behavior is observed for the annular membrane with zero edge load at the inner radius in the limit as the circumferential stress vanishes. </p>
<p>We develop multiscale expansions, which are asymptotic to members of this family for plates with edges that are elastically supported against rotation. At some thicknesses this approximation breaks down and a boundary layer appears at the center of the plate. In the limit of small normal load, the points of breakdown approach the bifurcation points corresponding to buckling of the nondeflected state. A uniform asymptotic expansion for small thickness combining the boundary layer with a multiscale approximation of the outer solution is developed for this case. These approximations complement the well known boundary layer expansions based on tensile membrane solutions in describing the bending and stretching of thin plates. The approximation becomes inconsistent as the clamped state is approached by increasing the resistance against rotation at the edge. We prove that such an expansion for the clamped circular plate cannot exist unless the pressure load is self-equilibrating.</p>https://thesis.library.caltech.edu/id/eprint/7539Shape and Stability of Two-Dimensional Uniform Vorticity Regions
https://resolver.caltech.edu/CaltechETD:etd-06302004-093810
Authors: {'items': [{'email': 'kammj@lanl.gov', 'id': 'Kamm-James-Russell', 'name': {'family': 'Kamm', 'given': 'James Russell'}, 'show_email': 'YES'}]}
Year: 1987
DOI: 10.7907/NW61-5178
<p>The steady shapes, linear stability, and energetics of regions of uniform, constant vorticity in an incompressible, inviscid fluid are investigated. The method of Schwarz functions as introduced by Meiron, Saffman and Schatzman [1984] is used in the mathematical formulation of these problems.</p>
<p>Numerical and analytical analyses are provided for several configurations. For the single vortex in strained and rotating flow fields, we find new solutions that bifurcate from the branch of steady elliptical solutions. These nonelliptical steady states are determined to be linearly unstable. We examine the corotating vortex pair and numerically confirm the theoretical results of Saffman and Szeto [1980], relating linear stability characteristics to energetics. The stability properties of the infinite single array of vortices are quantified. The pairing instability is found to be the most unstable subharmonic disturbance, and the existence of an area-dependent superharmonic instability (Saffman and Szeto [1981]) is numerically confirmed. These results are exhibited qualitatively by an elliptical vortex model. Lastly, we study the effects of unequal area on the stability of the infinite staggered double array of vortices. We numerically verify the results of the perturbation analysis of Jiménez [1986b] by showing that the characteristic subharmonic stability "cross" persists for vortex streets of finite but unequal areas.</p>
https://thesis.library.caltech.edu/id/eprint/2782Ray Trace Tomographic Velocity Analysis of Surface Seismic Reflection Data
https://resolver.caltech.edu/CaltechTHESIS:08232012-133835865
Authors: {'items': [{'id': 'Stork-Christof', 'name': {'family': 'Stork', 'given': 'Christof'}, 'show_email': 'NO'}]}
Year: 1988
DOI: 10.7907/73RH-5N25
<p>Recent development of two technologies allows application of a generalized formulation of travel time inversion to very large data sets, such as the surface reflection surveys collected for oil exploration. This generalized formulation uses very small cell sizes, effectively eliminating discretization effects. Inversion of an effective continuum that has no built-in <i>a priori</i> constraints is what places this technique in the category of <i>tomography</i>.</p>
<p>In reflection surveys, the generalized formulation investigated here treats the continuous velocity field independently from the reflector locations. The <i>a priori</i> assumption, common with travel time inversions in seismic exploration data, is thus not made: that the velocity field is defined as a series of layers with constant or smoothly varying velocity. This assumption restricts significant velocity variations to occur only at reflector locations. Velocity parameterized as layers is merely one of many geologic constraints that can be added optionally in tomographic inversion.</p>
<p>The technologies that enable this generalized approach to travel time inversion are: 1) a computer program capable of tracing rays through a 2-dimensional grid of points and off reflectors with structure, and 2) iterative schemes that efficiently perform damped, constrained generalized matrix inversions over a user-specified wide eigenvalue range for very large model and data sizes. An argument is presented that a variation of Richardson's iteration is preferred to the Conjugate Gradient Iterative Method for performing the matrix inversion.</p>
<p>With this generalized formulation, Ray Trace Tomography is a first approach to tomographic transmission analysis. Travel times and ray paths are a valid approximation to the wave equation for broad velocity variations. The method efficiently addresses the characteristics of more general but much more expensive transmission techniques. For example, Ray Trace Tomography demonstrates that an iterative application of a transmission velocity analysis technique, tomography, and a scattering reflector location technique, migration, do not necessarily converge to the optimal solution. To resolve the ambiguity between velocity-reflector depth, velocity and reflector locations must be coupled in one inversion technique. Ray Trace Tomography is able to couple the two. Using it to indicate the absolute resolution between velocity and reflector depth, we find that for certain geometries, reflector depths cannot be resolved where most recorded energy travels within 45° of vertical.</p>
<p>Poor resolution of the velocity-reflector depth ambiguity and other problems are inherent to reflection surveys. These problems also exist for other transmission techniques and can be solved only through use of inversion constraints. Ray Trace Tomography can test constraints for possible use in other transmission techniques efficiently.</p>
<p>Ray Trace Tomography has difficulty with non-linearities caused by some types of starting model errors, such as small-scale reflector structure. Improved performance with non-linearities is an objective we should seek in other transmission techniques.</p>
<p>Not only is Ray Trace Tomography a useful intellectual exercise as a preliminary analysis of transmission inversion, but in many cases it is a viable technique for addressing serious problems with surface seismic reflection data. It can determine an accurate two-dimensional velocity field for migration, such as in the case of gas pockets or fault blocks. In addition, it can resolve between certain velocity and reflector ambiguities such as those occuring in the permafrost region of Alaska.</p>
<p>As a comparatively efficient technique, Ray Trace Tomography can serve as a tool for interactive interpretation. The geologist can use the ray tracing to compare various geologic models with the data and then use the inversion to fine-tune the models. The inversion enables the geologist to formulate his geologic knowledge as constraints in the inversion. By analyzing the inversion results, the interpreter will develop an understanding of the validity of the various models and the resolution amoung them.</p>https://thesis.library.caltech.edu/id/eprint/7192Oscillating-Field Current-Drive Schemes for Tokamaks
https://resolver.caltech.edu/CaltechETD:etd-02132007-141701
Authors: {'items': [{'email': 'markschalit@gmail.com', 'id': 'Schalit-Mark-Alan', 'name': {'family': 'Schalit', 'given': 'Mark Alan'}, 'show_email': 'NO'}]}
Year: 1989
DOI: 10.7907/xjrw-pq56
<p>A novel current-drive scheme for steady-state tokamak operation is investigated in which external coils are applied to induce time-periodic fluid-type, fluctuations within the plasma; a nonlinear interaction between these fluctuations results in a time-averaged EMF, which maintains the large-scale magnetic field against Ohmic dissipation. Analytical and numerical modeling of this current-drive scheme is presented for low-frequency schemes (where the nonlinear < u⃗ x b⃗ > EMF is dominant) and for higher-frequency schemes (where the < j⃗ x b⃗ > Hall EMF is dominant). The Hall EMF is dominant at frequencies well above the ion-cyclotron frequency (referred to the strength of the static axial field) - except in the case of the rotamak, where the oscillating electric field is in the same direction as the static axial field.</p>
<p>A figure-of-merit for these current-drive schemes is the ratio of the strength of the static axial current to the strength of the oscillating current. This ratio is always much less than unity in all standard MHD calculations. As the electronion collision frequency vanishes, the ratio approaches infinity for the case of the rotamak. The ratio also approaches infinity for the <i>m</i> = 1 analogue of the rotamak - but only in the restrictive case where the static axial field becomes vanishingly small and where the DC magnetic fields are a small fraction of the AC magnetic fields. For the <i>m</i> = 1 analogue, the currents are confined to a skin layer as the axial field becomes very large, with the ratio of DC current strength to the oscillating current strength approaching unity.</p>
<p>The analysis presented here is compared and contrasted with existing theories and to a number of recent experiments.</p>https://thesis.library.caltech.edu/id/eprint/627An Investigation of the Bursting of Trailing Vortices Using Numerical Simulation
https://resolver.caltech.edu/CaltechETD:etd-02012007-105641
Authors: {'items': [{'id': 'Beran-Philip-Stewart', 'name': {'family': 'Beran', 'given': 'Philip Stewart'}, 'show_email': 'NO'}]}
Year: 1989
DOI: 10.7907/chyb-nk54
<p>Solutions of the Navier-Stokes equations are obtained for the flow of an isolated, trailing vortex, and for the swirling flow through a frictionless pipe. In both cases, the flow is assumed to be steady, incompressible and rotationally symmetric. Solutions are computed using Newton's method and Gaussian elimination for a wide range of values of two parameters: Reynolds number, Re, and vortex strength, V. Pseudo-arclength continuation is employed to facilitate the computation of solution points in the parameter space. The numerical procedure is validated through comparison of solutions with solutions obtained in previous investigations for the case of a trailing vortex. Solutions are also compared with results reported by Brown and Lopez (1988) for the case of flow through a pipe.</p>
<p>Solutions of the quasi-cylindrical equations are obtained for the flow of a trailing vortex. Solutions are computed using an explicit, space-marching scheme, and are compared with solutions of the Navier-Stokes equations.</p>
<p>Provided that Re is about 200, or larger, four vortex states are observed.</p>
<p>1. When V is sufficiently small, the flow is entirely supercritical.</p>
<p>2. As V is increased, the flow at an axial station becomes critical and a transition point forms. At the point, the flow departs from an upstream state that is supercritical to a downstream state that is marked by large-amplitude, spatial oscillations of core radius. When Re is large, the downstream state is nearly periodic. The general features of transition are well described by the conjugate-flow theory of Benjamin 1967). Failure of the quasi-cylindrical equations is found to be a necessary and sufficient condition for the existence of a transition point. As V is further increased, the transition point moves upstream. Reversed flow is not observed.</p>
<p>3. Over a narrow range of vortex strengths, a small bubble of reversed flow is observed downstream of the transition point.</p>
<p>4. When V is large, the entire flow is marked by large-amplitude, spatial oscillations of core radius. A transition point is not evident within the computational domain. Typically, large regions of reversed flow are observed.</p>https://thesis.library.caltech.edu/id/eprint/438Effect of Compliant Boundaries on Weakly Nonlinear Shear Waves in Channel Flow
https://resolver.caltech.edu/CaltechETD:etd-02152007-075746
Authors: {'items': [{'id': 'Rotenberry-James-Michael', 'name': {'family': 'Rotenberry', 'given': 'James Michael'}, 'show_email': 'NO'}]}
Year: 1989
DOI: 10.7907/gva0-r175
<p>There exists a critical Reynolds number (at which a linear instability first appears for an incompressible fluid flowing in a channel with compliant walls (Hains and Price, [1962]). It is proven that, for fixed non-dimensionalized wall parameters, to any unstable disturbance in three dimensions there corresponds an unstable disturbance in two dimensions at a lower Reynolds number. Consequently, the Ginzburg-Landau equation is used to study the weakly nonlinear two-dimensional evolution of a disturbance in a channel with compliant walls for Reynolds number near its critical value. The coefficients of this equation are found by numerically integrating solutions of the Orr-Sommerfeld equation and its adjoint as well as solutions of the perturbation equations.</p>
<p>For rigid walls the finite amplitude two-dimensional plane wave solution that bifurcates from laminar Poiseuille flow at the critical Reynolds number is itself unstable to two-dimensional disturbances. It is found that for compliant walls this solution is stable to disturbances of the same type.</p>
<p>The formalism developed by Landman [1987] is used to study a class of quasisteady solutions to the Ginzburg-Landau equation. This class includes solutions describing a transition from the laminar solution to finite amplitude states and nonperiodic, "chaotic" attracting sets. It is shown that for compliant walls the transition solutions persist while the "chaotic" ones do not.</p>https://thesis.library.caltech.edu/id/eprint/640Runtime systems for fine-grain multicomputers
https://resolver.caltech.edu/CaltechETD:etd-08222007-103344
Authors: {'items': [{'id': 'Boden-Nanette-Jackson', 'name': {'family': 'Boden', 'given': 'Nanette Jackson'}, 'show_email': 'NO'}]}
Year: 1993
DOI: 10.7907/2c3a-k589
During the past decade, our research group has been engaged in experiments in the architecture and programming of multicomputers. This research has progressed steadily toward the ideal of small granularity, both of the computing nodes within a multicomputer, and of the execution units within concurrent programs. The context for the runtime-system and program-behavior experiments reported in this thesis are: (1) the reactive-process, message passing computational model, (2)C+-, a C++ -based, concurrent-programming notation, and (3) the Mosaic C, an experimental, fine-grain multicomputer.
We present first a long-sought solution to the formulation of an unbounded queue of elements within the reactive-process model. This result is applied to allow messages to be received selectively using purely reactive semantics.
The primary contributions of this thesis are distributed algorithms and a design method for runtime systems for fine-grain multicomputers. To evaluate the algorithms and design, a prototype runtime system called MADRE was developed, C+- programs whose behaviors are typical of a variety of applications were written, these programs were executed on the Mosaic C under MADRE, and the program behavior was instrumented.
In addition to conventional operating- and runtime-system functions such as local memory management and quiescence detection, MADRE automatically manages userprocess placement and naming. MADRE can also be configured to include capabilities for distributing resource demands across the nodes of the multicomputer. Buffered messages can be exported from congested nodes so that incoming messages can continue to be received. The code of user programs can be distributed across the ensemble, and accessed automatically. Each of these capabilities depends upon the formulation of selective receive demonstrated in the solution to the unbounded queue.
Our experiments evaluate various automatic process-placement strategies. We show that one algorithm, called k-biased placement, distributes loads nearly as well as random placement, while providing a tunable degree of locality between parent and child processes. Other experiments demonstrate that the message-exportation capability is crucial to finegrain multicomputers; unless messages can be exported, computations fail due to receive queue overflow when only a fraction of the multicomputer's memory resources are being used.
https://thesis.library.caltech.edu/id/eprint/3198New plane shear flows
https://resolver.caltech.edu/CaltechETD:etd-10182005-102648
Authors: {'items': [{'email': 'aconley@ucar.edu', 'id': 'Conley-A', 'name': {'family': 'Conley', 'given': 'Andrew'}, 'show_email': 'NO'}]}
Year: 1994
DOI: 10.7907/T34K-J848
A classical problem in fluid dynamics is the study of the stability of plane Couette flow. This flow experimentally sustains turbulence for Reynolds numbers greater than 1440±40 (see [10],[5]). (The Reynolds number is based on channel width and wall velocity difference). Since plane Couette flow is linearly stable for all Reynolds numbers, obtaining non-trivial mathematical solutions to the plane Couette flow equations is difficult. However, M. Nagata [6] finds a non-trivial numerical solution of the plane Couette flow equations at low Reynolds number. We confirm these solutions. We compute the minimum Reynolds number at which they exist. We study their stability. We also study the effect of a Coriolis force on plane Poiseuille flow.
https://thesis.library.caltech.edu/id/eprint/4158Mathematical modeling of air pollution dynamics by parallel computation
https://resolver.caltech.edu/CaltechETD:etd-12132007-083330
Authors: {'items': [{'id': 'Dabdub-D', 'name': {'family': 'Dabdub', 'given': 'Donald'}, 'show_email': 'NO'}]}
Year: 1996
DOI: 10.7907/k1ap-np35
The use of massively parallel computers provides an avenue to overcome the computational requirements in the study of atmospheric chemical dynamics. General considerations on parallel implementation of air quality models are outlined including domain decomposition strategies, algorithm evaluation and design, portability, modularity, and buffering techniques used in I/O operations. Results are given for the implementation of the CIT urban air pollution model on distributed memory multiple instruction / multiple data (MIMD) machines ranging from a cluster of workstations to a 512 node Intel Paragon.
The central challenge in developing a parallel air pollution model is the implementation of the chemistry and transport operators used in the solution of the atmospheric reaction-diffusion equation. The chemistry operator is generally the most computationally intensive step in atmospheric air quality models. A new method based on Richardson extrapolation to solve the chemical kinetics is presented. The transport operator is the most challenging to solve numerically. Because of its hyperbolic nature non-physical oscillations and/or negative concentrations appear near steep gradient regions of the solution. Six algorithms for solving the advection equation are compared to determine their suitability for use in parallel photochemical air quality models. Four algorithms for filtering the numerical noise produced when solving the advection equation are also compared.
A speed-up factor of 94.9 has been measured when the I/O, transport, and chemistry portions of the model are performed in parallel. This work provides the computational infrastructure required to incorporate new physico-chemical phenomena in the next generation of urban- or regional-scale air quality models.
Finally, the SARMAP model is used to model the San Joaquin Valley of California. SARMAP is the updated version of RADM. It can be considered a state-of-the- art regional air pollution model. Like the CIT model, SARMAP incorporates the following atmospheric phenomena: gas-phase chemistry, advection and diffusion. In addition, SARMAP incorporates aqueous-phase chemistry and transport through cumulus clouds. Sensitivity studies performed show a significant dependence of ozone model predictions on boundary conditions.https://thesis.library.caltech.edu/id/eprint/4987Bifurcations in Kolmogorov and Taylor-vortex flows
https://resolver.caltech.edu/CaltechETD:etd-02122008-090309
Authors: {'items': [{'id': 'Love-P', 'name': {'family': 'Love', 'given': 'Philip'}, 'show_email': 'NO'}]}
Year: 1999
DOI: 10.7907/g2f3-s507
The bifurcation structure of Kolmogorov and Taylor-Vortex flows was computed with the aid of the Recursive Projection Method; see Schroff and Keller [32]. It was shown that RPM significantly improves the convergence of our numerical method while calculating steady state solutions. Moreover we use RPM to detect bifurcation points while continuing along solution branches, and to provide the required augmentation when continuing around a fold, or along a traveling wave branch.
The bifurcations to two and three-dimensional solutions from the shear flow solution of Kolmogorov flow are calculated both numerically, by solving an ordinary differential equation, and analytically, using an approximation method. Our results for the two-dimensional bifurcations agree with the work of Meshalkin and Sinai [26].
We also explain how the branches of Kolmogorov flows observed by Platt and Sirovich [29] are connected together, and observe that our solutions have worm like structures even at relatively low Reynolds numbers. Various statistics of our flows are calculated and compare with those from isotropic turbulence calculations.
Additionally various solution branches of the Taylor Vortex flow were computed, including spiral vortices. Furthermore, it was discovered that the Wavy Taylor Vortex branches arise from sub-critical Hopf bifurcations, and they undergo a fold close to their bifurcation point.
https://thesis.library.caltech.edu/id/eprint/618