Committee Feed
https://feeds.library.caltech.edu/people/Meiron-D-I/committee.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:39:27 +0000Diffusion with Varying Drag; the Runaway Problem
https://resolver.caltech.edu/CaltechETD:etd-03192008-095816
Authors: {'items': [{'id': 'Rollins-David-Kenneth', 'name': {'family': 'Rollins', 'given': 'David Kenneth'}, 'show_email': 'NO'}]}
Year: 1986
DOI: 10.7907/mwvx-6b36
<p>We study the motion of electrons in an ionized plasma of electrons and ions in an external electric field. A probability distribution function describes the electron motion and is a solution of a Fokker-Planck equation. In zero field, the solution approaches an equilibrium Maxwellian. For arbitrarily small field, electrons overcome the diffusive effects and are freely accelerated by the field. This is the electron runaway phenomenon.</p>
<p>We treat the electric field as a small perturbation. We consider various diffusion coefficients for the one-dimensional problem and determine the runaway current as a function of the field strength. Diffusion coefficients, non-zero on a finite interval are examined. Some non-trivial cases of these can be solved exactly in terms of known special functions. The more realistic case where the diffusion coefficient decays with velocity is then considered. To determine the runaway current, the equivalent Schrödinger eigenvalue problem is analyzed. The smallest eigenvalue is shown to be equal to the runaway current. Using asymptotic matching a solution can be constructed which is then used to evaluate the runaway current. The runaway current is exponentially small as a function of field strength. This method is used to extract results from the three-dimensional problem.</p>https://thesis.library.caltech.edu/id/eprint/1016Intermediate Reynolds Number Free-Surface Flows
https://resolver.caltech.edu/CaltechETD:etd-03042008-083806
Authors: {'items': [{'id': 'Dandy-David-Stewart', 'name': {'family': 'Dandy', 'given': 'David Stewart'}, 'show_email': 'NO'}]}
Year: 1987
DOI: 10.7907/frtp-v339
<p>The buoyancy driven motion of a deformable viscous drop at intermediate Reynolds numbers has been studied using numerical techniques. The motion was assumed to be steady and rectilinear, and a pseudo-implicit method was used to solve the Navier-Stokes equations. Cases for a variety of values of the Reynolds number, Weber number, viscosity ratio and density ratio have been considered. The calculations reveal that the shape of the drop is most heavily dependent on the Weber number, attaining spheroidal capped shapes at 0(1) Reynolds numbers, and flattened ellipsoidal shapes at higher Reynolds numbers. Two mechanisms are observed for vorticity production at the interface of the drop—curvature and the no-slip condition—and the no-slip mechanism is a more efficient source of vorticity. When there is sufficient vorticity produced, a detached closed streamline wake forms at the back of the drop, in contrast to the attached wakes seen on inviscid bubbles and solid particles.</p>
<p>To further explore the role of vorticity production in wake formation, numerical computations were done on flow past inviscid bubbles of fixed shape. It was found that attached recirculating wakes existed at intermediate Reynolds numbers and these wakes could not be predicted by either low or high Reynolds number asymptotic theories. The numerical results indicate that the mechanism responsible for flow separation at modest Reynolds numbers may be different than that present at high Reynolds numbers.</p>
<p>Because of the inherent difficulties in solving the Navier-Stokes equations using successive approximation schemes, and to investigate the behavior of solutions of these equations on the dimensionless parameters, a Newton's method scheme has been developed and tested successfully on the steady buoyancy driven motion of an inviscid bubble; an arc length continuation method has also been implemented. Calculations indicate that the scheme achieves quadratic convergence.</p>
<p>Last, a numerical technique has been developed for the study of the creeping motion of drops and particles in the presence of a rigid plane boundary. This method is based upon the distribution of point forces on the surface of the body and the use of a Green's function to obtain the unknown velocity and stress on the body surface, without having to consider the rigid boundary.</p>
https://thesis.library.caltech.edu/id/eprint/868Part I: Symmetry Breaking of Water Waves. Part II: On the Superharmonic Instability of Surface Water Waves
https://resolver.caltech.edu/CaltechETD:etd-03052008-090255
Authors: {'items': [{'email': 'jzufiria@yahoo.com', 'id': 'Zufiria-Juan-Antonio', 'name': {'family': 'Zufiria', 'given': 'Juan Antonio'}, 'show_email': 'YES'}]}
Year: 1987
DOI: 10.7907/4JDZ-9Y53
<p>PART I:</p>
<p>A weakly nonlinear Hamiltonian model for two dimensional irrotational waves on water of finite depth is developed. The truncated model is used to study families of periodic travelling waves of permanent form. It is shown that nonsymmetric periodic waves exist, which appear via spontaneous symmetry breaking bifurcations from symmetric waves.</p>
<p>In order to check these results with the full water wave equations, two different methods are used to calculate nonsymmetric gravity waves on deep water. It is found that they exist and the structure of the bifurcation tree is the same as the one found for waves on water of finite depth using the weakly nonlinear Hamiltonian model. One of the methods is based on the quadratic relations between the Stokes coefficients discovered by Longuet-Higgins (1978a). The other method is a new one based on the Hamiltonian structure of the water wave problem.</p>
<p>Another weakly nonlinear model is developed from the Hamiltonian formulation of water waves to study the bifurcation structure of gravity-capillary waves on water of finite depth. It is found that, besides a very rich structure of symmetric solutions, nonsymmetric Wilton ripples exist. They appear via spontaneous symmetry breaking bifurcation from symmetric solutions. The bifurcation tree is similar to that for gravity waves. The solitary wave with surface tension is studied with the same model close to a critical depth. It is found that the solution is not unique, and further nonsymmetric solitary waves are possible. The bifurcation tree has the same structure as for the case of periodic waves. The possibility of checking these results in low gravity experiments is discussed.</p>
<p>PART II:</p>
<p>Saffman's (1985) theory of the superharmonic stability of two-dimensional irrotational waves on fluid of infinite depth has been generalized to solitary and periodic waves of permanent form on fluid of finite uniform depth. The frame of reference for the calculation of the Hamiltonian for periodic waves of finite depth is found to be the frame in which the mean horizontal velocity is zero.</p>
<p>Also, a simple analytical model has been constructed to demonstrate Saffman's (1985) theory. The model shows the change of geometrical and algebraic multiplicity of the eigenvalues and eigenvectors of the stability equation at the critical height. It confirms the existence of Hamiltonian systems with limit points at which there is no change of stability.</p>
https://thesis.library.caltech.edu/id/eprint/880Finite Amplitude Waves in Plane Poiseuille Flow
https://resolver.caltech.edu/CaltechETD:etd-11072007-104253
Authors: {'items': [{'email': 'jpugh@alumni.caltech.edu', 'id': 'Pugh-Jeffrey-David', 'name': {'family': 'Pugh', 'given': 'Jeffrey David'}, 'show_email': 'NO'}]}
Year: 1988
DOI: 10.7907/XHFQ-MJ23
<p>Nonlinear behavior in plane Poiseuille flow has attracted theoretical interest over the last decade, both because of its tractability and because it is believed that some of the results may be applicable to phenomena occurring in the boundary layer. We have investigated the existence of three-dimensional finite amplitude waves in plane Poiseuille flow, in the hope of finding candidates for a class of simple flows which might provide insight into the nature of turbulence. These so-called <i>vortical states</i> would exist as attractors for the turbulent flow and mimic many of its properties.</p>
<p>One of the requisite properties of these simple flows is existence at the low Reynolds numbers observed in experimental studies of transition to turbulence in plane Poiseuille flow. Although no such three-dimensional solutions were found in our study, a number of new insights have been made into the structure and stability of two- and three-dimensional steady wave solutions in plane Poiseuille flow. These in turn suggest new areas of investigation for finding vortical states.</p>https://thesis.library.caltech.edu/id/eprint/4446A Model for Stress-Driven Diffusion in Polymers
https://resolver.caltech.edu/CaltechTHESIS:01232013-124105465
Authors: {'items': [{'email': 'rwcox123@gmail.com', 'id': 'Cox-Robert-William', 'name': {'family': 'Cox', 'given': 'Robert William'}, 'orcid': '0000-0002-0031-1568', 'show_email': 'NO'}]}
Year: 1988
DOI: 10.7907/06tx-qh63
<p>Penetration of solvents into polymers is sometimes characterized by steep concentration gradients that move into the polymer and last for long times. The behavior of these fronts cannot be explained by standard diffusion equations, even with concentration dependent diffusion coefficients. The addition of stress terms to the diffusive flux can produce such progressive fronts. Model equations are proposed that include solvent flux due to stress gradients in addition to the Fickian flux. The stress in turn obeys an concentration dependent evolution equation.</p>
<p>The model equations are analyzed in the limit of small diffusivity for the problem of penetration into a semi-infinite medium. Provided that the coefficient functions obey certain monotonicity conditions, the solvent concentration profile is shown to have a steep front that progresses into the medium. A formula governing the progression of the front is developed. After the front decays away, the long time behavior of the solution is shown to be a similarity solution. Two techniques for approximating the solvent concentration and the front position are presented. The first approximation method is a series expansion; formulas are given for the initial speed and deceleration of the front. The second approximation method uses a portion of the long time similarity solution to represent the short time solution behind the front.</p>
<p>The addition of a convective term to the solvent flux is shown to raise the possibility of a traveling wave solution. The existence of the traveling wave solution is shown for certain types of coefficient functions. The way the initial front speed evolves onto the traveling wave speed is sketched out.</p>
https://thesis.library.caltech.edu/id/eprint/7429Asymptotic Methods in Semiconductor Device Modeling
https://resolver.caltech.edu/CaltechETD:etd-02012007-131948
Authors: {'items': [{'email': 'ward@math.ubc.ca', 'id': 'Ward-Michael-Jeffrey', 'name': {'family': 'Ward', 'given': 'Michael Jeffrey'}, 'show_email': 'NO'}]}
Year: 1988
DOI: 10.7907/08TK-NM84
<p>The behavior of metal oxide semiconductor field effect transistors (MOSFETs) with small aspect ratio and large doping levels is analyzed using formal perturbation techniques. Formally, we will show that in the limit of small aspect ratio there is a region in the middle of the channel under the control of the gate where the potential is one-dimensional. The influence of interface and internal layers in the one-dimensional potential on the averaged channel conductivity is closely examined in the large doping limit. The interface and internal layers that occur in the one-dimensional potential are resolved in the limit of large doping using the method of matched asymptotic expansions. The asymptotic potential in the middle of the channel is constructed for various classes of variable doping models including a simple doping model for the built-in channel device. Using the asymptotic one-dimensional potential, the asymptotic mobile charge, needed for the derivation of the long-channel I-V curves, is found by using standard techniques in the asymptotic evaluation of integrals. The formal asymptotic approach adopted not only provides a pointwise description of the state variables, but by using the asymptotic mobile charge, the lumped long-channel current-voltage relations, which vary uniformly across the various bias regimes, can be found for various classes of variable doping models.</p>
<p>Using the explicit solutions of some free boundary problems solved by Howison and King (1988), the two-dimensional equilibrium potential near the source and drain is constructed asymptotically in strong inversion in the limit of large doping. From the asymptotic potential constructed near the source and drain, a uniform analytical expression for the mobile charge valid throughout the channel is obtained. From this uniform expression for the mobile charge, we will show how it is possible to find the I-V curve in a particular bias regime taking into account the edge effects of the source and drain. In addition, the asymptotic potential for a two-dimensional n⁺-p junction is constructed.</p>
https://thesis.library.caltech.edu/id/eprint/440On the Interaction of Shock Waves with Contact Surfaces Between Gases of Different Densities
https://resolver.caltech.edu/CaltechETD:etd-10302003-102505
Authors: {'items': [{'email': 'martin.brouillette@usherbrooke.ca', 'id': 'Brouillette-Martin', 'name': {'family': 'Brouillette', 'given': 'Martin'}, 'show_email': 'YES'}]}
Year: 1989
DOI: 10.7907/9JGS-ZX78
<p>The interaction of shock waves with a contact surface between gases of different densities has been studied experimentally and theoretically. The basic mechanism for the instability of perturbations at the interface is baroclinic vorticity generation resulting from the misalignment of the pressure gradient of the shock and the density gradient of the interface. In the present study, the effects of interface density contrast and initial thickness, and incident wave strength on the development of the instability at the interface are investigated. The experiments were performed in a new vertical shock tube facility where the interaction of a shock wave with either a discontinuous interface, formed by a thin (0.5 µm) plastic membrane, or a continuous interface, created by retracting a metal plate initially separating the two gases, was studied. Air was used on one side of the interface and either helium, carbon dioxide, refrigerant-22 or sulphur hexafluoride was used on the other side as the test gas.</p>
<p>Experiments to study the time evolution of quasi-sinusoidal perturbations on a continuous interface have shown that the growth rates are reduced as the interface thickness is increased. It has been observed that growth rates of perturbations of wavelength λ ~ 25 mm on interfaces of thickness δ ~ 10 mm are about three times smaller than those predicted by the linear theory for the impulsive acceleration of discontinuous interfaces. A new model that accounts for the growth rate reduction caused by the presence of a finite density gradient on the interface has been proposed, and good agreement was obtained with the present experimental results.</p>
<p>Experiments were also performed to observe the schlieren visual thickness of plane discontinuous or continuous interfaces with random small-scale perturbations after interaction with the incident shock wave and its reverberations. The interface was initially located near the end wall of the shock tube to permit the observation of the development of the interface phenomena after the arrival of the incident shock and its reverberations. It is found that the interaction of a shock wave with a discontinuous interface causes the appearance of a turbulent mixing zone between the two gases, whose growth rate slows down as time increases, owing to a decrease in turbulence intensity and the action of viscosity. Because of the large uncertainty associated with the measurements a short time after the interaction with the incident shock, the accurate determination of a possible universal power law governing the thickening of the interface is not feasible. Results for the interaction of the first reverberation of the primary wave with the already turbulent interface have demonstrated that this growth is sensitive to the initial pre-growth state of the interface. It also appears that the thickening of the turbulent mixing zone is accomplished by the merging of large structures within the interface. However, since the energy available for the turbulent motions at the impulsively accelerated interface remains constant after the interaction with the shock and also depends on the wavelength of the initial perturbation, it is not certain whether the development of mixing at the interface achieves an asymptotic stage of self-similar turbulence independent of initial conditions, as has been observed for the gravity-driven interfaces. Also, it has been found that the growth rates measured in the present experiments with discontinuous interfaces are nearly an order of magnitude lower than those reported by previous investigators. The continuous interfaces formed by the retracting plate are smoothed by molecular diffusion, and thus the combination of low density gradient and small initial perturbations is such that they exhibit growth only after being perturbed by acoustic noise introduced by the reverberation of waves between the interface, the side walls and the end of the shock tube.</p>
<p>The development of viscous boundary layers on the side walls of the test section can cause the bifurcation of waves reflected from the end wall of the shock tube, and, thereafter, the formation of wall bubbles and interface contaminating jets. Moreover, the generation of vortical structures by the baroclinic instability excited by the interaction of reflected waves with the distorted interface within the boundary layer has been demonstrated. Significant contamination of the test gas can by achieved by these structures, even if reflected-wave bifurcation is absent. Moreover, the strain induced by the vorticity in these wall structures tends to thin the interface; the magnitude of this effect on the growth rates in the present plane interface experiments is estimated to be of order 10% for discontinuous interfaces and 50% for continuous interfaces.</p>
https://thesis.library.caltech.edu/id/eprint/4315A Study of Finite Amplitude Bifurcations in Plane Poiseuille Flow
https://resolver.caltech.edu/CaltechETD:etd-02142007-080941
Authors: {'items': [{'email': 'isoibelman@ll.mit.edu', 'id': 'Soibelman-Israel', 'name': {'family': 'Soibelman', 'given': 'Israel'}, 'show_email': 'NO'}]}
Year: 1989
DOI: 10.7907/2xg0-5q97
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>https://thesis.library.caltech.edu/id/eprint/634Effect 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/640Topics in Vortex Methods for the Computation of Three- and Two-Dimensional Incompressible Unsteady Flows
https://resolver.caltech.edu/CaltechETD:etd-11032003-112216
Authors: {'items': [{'id': 'Winckelmans-Grégoire-Stéphane', 'name': {'family': 'Winckelmans', 'given': 'Grégoire Stéphane'}, 'show_email': 'NO'}]}
Year: 1989
DOI: 10.7907/19HD-DF80
<p>Contributions to vortex methods for the computation of incompressible unsteady flows are presented. Three methods are investigated, both theoretically and numerically.</p>
<p>The first method to be considered is the inviscid method of vortex filaments in three dimensions, and the following topics are presented: (a) review of the method of regularized vortex filaments and of convergence results for multiple-filament computations, (b) modeling of a vortex tube by a single filament convected with the regularized Biot-Savart velocity applied on the centerline: velocity of the thin filament vortex ring and dispersion relation of the rectilinear filament, and (c) development of a new regularization of the Biot-Savart law that reproduces the lowest mode dispersion relation of the rectilinear vortex tube in the range of large to medium wavelengths.</p>
<p>Next the method of vortex particles in three dimensions is investigated, and the following contributions are discussed: (a) review of the method of singular vortex particles: investigation of different evolution equations for the particle strength vector and weak solutions of the vorticity equation, (b) review of the method of regularized vortex particles and of convergence results, and introduction of a new algebraic smoothing with convergence properties as good as those of Gaussian smoothing, (c) development of a new viscous method in which viscous diffusion is taken into account by a scheme that redistributes the particle strength vectors, and application of the method to the computation of the fusion of two vortex rings at <i>Re</i> = 400, and (d) investigation of the particle method with respect to the conservation laws and derivation of new expressions for the evaluation of the quadratic diagnostics: energy, helicity and enstrophy.</p>
<p>The third method considered is the method of contour dynamics in two dimensions. The particular efforts presented are (a) review of the classical inviscid method and development of a new regularized version of the method, (b) development of a new vector particle version of the method, both singular and regularized: the method of <i>particles of vorticity gradient</i>, (c) development of a viscous version of the method of regularized particles and application of the method to computation of the reconnection of two vortex patches of same sign vorticity, and (d) investigation of the particle method with respect to the conservation laws and derivation of new expressions for the evaluation of linear and quadratic diagnostics.</p>https://thesis.library.caltech.edu/id/eprint/4385Forced Emissions of Nonlinear Water Waves in Channels of Arbitrary Shape
https://resolver.caltech.edu/CaltechETD:etd-06132005-111317
Authors: {'items': [{'email': 'teng@wiliki.eng.hawaii.edu', 'id': 'Teng-Michelle-Hsiao-Tsing', 'name': {'family': 'Teng', 'given': 'Michelle Hsiao Tsing'}, 'show_email': 'NO'}]}
Year: 1990
DOI: 10.7907/DA4H-XZ13
This thesis is a joint theoretical, numerical and experimental study concentrated on investigating the phenomenon of weakly nonlinear, weakly dispersive long water waves being generated and propagating in a channel of arbitrary cross section. The water depth and channel width are assumed comparable in size and they may vary both in time and space. Two types of theoretical models, i.e., the generalized channel Boussinesq (gcB) two-equation model and the forced channel Korteweg-de Vries (cKdV) model, are derived by using perturbation expansions for quasi-one-dimensional long waves in shallow water. In the special case for channels of variable shape and dimension but fixed in time, the motion of free traveling solitons may be calculated by our models to predict their propagation with modulated amplitude, velocity and phase. In the precence of external forcings, such as a surface pressure distribution or a submerged obstacle moving with a near critical speed, solitary waves can be produced periodically to advance upstream. Analytical solutions for three specific cross-sectional shapes, namely, the rectangular, triangular and semi-circular sections, are obtained in closed form and with the main features of the solutions examined. The specific geometry of the cross section is found to affect only the magnitude of the dispersive terms in the equations. For a submerged moving object taken as an external forcing, its effective strength of forcing is directly related to the blockage-ratio of the cross-sectional area. Our long-wave models have their useful applications to the areas of river dynamics, near-coastal engineering, and other related fields.https://thesis.library.caltech.edu/id/eprint/2569Vortex Simulation of Separated Flows in Two and Three Dimensions
https://resolver.caltech.edu/CaltechETD:etd-08092005-142847
Authors: {'items': [{'email': 'bsgh@tm.net.my', 'id': 'Chua-Kiat', 'name': {'family': 'Chua', 'given': 'Kiat'}, 'show_email': 'YES'}]}
Year: 1990
DOI: 10.7907/9ENS-EP36
<p>This thesis is concerned with the applications of vortex methods to the problem of unsteady, separated flows in two and three dimensions, and can be divided into three parts. In the first part, an improved method for satisfying the boundary conditions on a flat plate is developed and applied to the two-dimensional separated flow problem. In this method, boundary layers on both side of the plate are represented by stacks of multiple vortex panels, the strength of which are determined by enforcing both the no-through flow and no-slip boundary conditions at the plate. Vortex shedding at the sharp edge of the plate is represented as the separation of the boundary vortex elements. Both forced and unforced flows are studied and comparisons to experiments are carried out. For the case without forcing, large discrepancy between calculations and experiments, which is also reported by other workers using a different vortex method or Navier-Stokes calculations, is observed. In the case with forcing, the discrepancy is reduced with lateral forcing at low amplitude; and eliminated, regardless of amplitude, with streamwise forcing (acceleration). In the second part, an improved three-dimensional vortex particle method is developed. In this method, vortex elements of vorticity that move with the local velocity and are stretched and rotated according to the local strain field, are used. To mimic the effects of vorticity cancellations, close pairs of opposite sign vortex elements are replaced by high order dipoles. The method is designed to handle complex high Reynolds number vortical flows and a non-linear viscosity model is included to treat small-scale effects in such flows. Applications to two problems involving strong interactions of vortex tubes are carried out and core deformation with complex internal strucures and induced axial flow within vortex tubes are observed. Qualitative comparison to experiments are encouraging. In the third part, the two-dimensional method developed in the first part is modified and extended to three dimensions. Here, solenoidal condition for vorticity is considered and closed vortex loops are used to represent the boundary layer vorticity and the vorticity at shedding. For the evolution of the vortex wake, the vortex particle method developed in the second part is used. Applications to the flow past a normal square plate is carried out and the early stages of the flow are studied.</p>https://thesis.library.caltech.edu/id/eprint/3064Structure in the Near Field of the Transverse Jet
https://resolver.caltech.edu/CaltechETD:etd-02232007-075829
Authors: {'items': [{'id': 'Fric-Thomas-Frank', 'name': {'family': 'Fric', 'given': 'Thomas Frank'}, 'show_email': 'NO'}]}
Year: 1990
DOI: 10.7907/JVHG-E582
<p>Photographs of an axisymmetric turbulent jet issuing from a wall into a crossflow display the four types of vortical structures which exist in the near field: the jet shear layer vortices, the nascent far field vortex pair, the near wall horseshoe vortices, and a system of vortices in the wake of the jet.</p>
<p>Additionally, results of hot-wire measurements in the wake of the transverse jet are presented. Among these results are characteristic wake Strouhal frequencies, which vary with the jet to crossflow velocity ratio, and wake velocity profiles.</p>
<p>It is found that the wake vorticity is not "shed" from the jet but is formed from vorticity which originated in the wall boundary layer. Therefore, analogies between the wakes of transverse jets and the wakes of solid cylinders are incorrect. Since the jet is not a solid obstacle to the crossflow, as a cylinder is, new vorticity is not generated at the interface between the jet and the crossflow. Instead, the boundary layer on the wall from which the jet issues separates near the downstream side of the jet because it cannot negotiate the adverse pressure gradient imposed on it by the flow around the jet, which is not "separated" as it is for a cylinder. The wake vortices subsequently formed are found to be most coherent near a jet to crossflow velocity ratio of four.</p>
<p>The near field development of the counterrotating vortex pair, which is the dominant structure of the far field jet, is also addressed. It is argued that the source of vorticity for the vortex pair is the vorticity from the boundary layer within the jet nozzle. Estimates for the strength of these vortices are obtained by considering the flux of vorticity emanating from the nozzle.</p>
<p>Possible implications for mixing are briefly discussed.</p>https://thesis.library.caltech.edu/id/eprint/719Study of spherical couette flow via 3-D spectral simulations: large and narrow-gap flows and their transitions
https://resolver.caltech.edu/CaltechETD:etd-04162004-103555
Authors: {'items': [{'email': 'gdumas@gmc.ulaval.ca', 'id': 'Dumas-G', 'name': {'family': 'Dumas', 'given': 'Guy'}, 'show_email': 'YES'}]}
Year: 1991
DOI: 10.7907/Z6DW-4T51
Incompressible, viscous flows in the spherical gap between a rotating inner-sphere and a stationary outer-shell, Spherical Couette Flows (SCF), are studied via direct numerical simulations. The investigation covers both "small-gap" and "large-gap" geometries, and is concerned primarily with the first occurrence of transition in those flows. Strong emphasis is put on the physical understanding of the basic flows and their transition mechanisms.
An alias-free spectral method, based on divergence-free vector expansions for the 3-D velocity field in spherical coordinates, is developed. The vector expansions are constructed with Chebyshev polynomials in the radial direction and Vector Spherical Harmonics for the two angular directions. Accuracy and spectral convergence of the resulting initial-value code are thoroughly tested. Three-dimensional transitional flows in both narrow-gaps and large-gaps as well as axisymmetric transitions in moderate-gaps are simulated.
For small-gap SCF's, this study shows that the formation of Taylor-vortices at transition is a deterministic process and not the result of the instability of initial perturbations. The formation process involves the sub-critical appearance of a saddle-stagnation point within the meridional circulation cell in each hemisphere. A minimum length-scale ratio is shown necessary, and for a given inner-sphere radius, this leads to a theoretical prediction of the largest gap-width in which Taylor-vortices may form.
This investigation confirms that the first transition in large-gap SCF's is caused by a 3-D instability of a linear nature. It is found that the process is characterized by very small growth-rates of the disturbance and by the absence of a "jump" in the friction torque. The supercritical flow is a complex-structured, laminar, time-periodic flow that exhibits traveling azimuthal-waves. The physical mechanism responsible for the large-gap transition is shown to be related to a shear instability of the "radial-azimuthal jet" that develops at the equator of the basic flow. A physical model is proposed in which that jet is viewed as a sequence of adjacent "fan-spreading quasi-2-D plane jets". Predictions from the model are presented and verified from the computed unstable disturbance field. Extension of the model to the transition toward waviness in the Taylor-Couette flow, the Gortler-vortex flow and the Dean-vortex flow is proposed.https://thesis.library.caltech.edu/id/eprint/1397Numerical studies of incompressible Richtmyer-Meshkov instability in a stratified fluid
https://resolver.caltech.edu/CaltechETD:etd-07122007-143228
Authors: {'items': [{'id': 'Pham-T', 'name': {'family': 'Pham', 'given': 'Thu'}, 'show_email': 'NO'}]}
Year: 1991
DOI: 10.7907/ryrb-cg53
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.https://thesis.library.caltech.edu/id/eprint/2864Numerical studies of nonlinear axisymmetric waves on vortex filaments
https://resolver.caltech.edu/CaltechETD:etd-07092007-105408
Authors: {'items': [{'email': 'vm@ctfd.cmmacs.ernet.in', 'id': 'Mudkavi-V-Y', 'name': {'family': 'Mudkavi', 'given': 'Vidyadhar Yogeshwar'}, 'show_email': 'YES'}]}
Year: 1991
DOI: 10.7907/TMK2-ME89
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.
https://thesis.library.caltech.edu/id/eprint/2839New 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/4158Protein structure/function classification using hidden Markov models
https://resolver.caltech.edu/CaltechETD:etd-01142008-095446
Authors: {'items': [{'id': 'Regelson-M-E', 'name': {'family': 'Regelson', 'given': 'Moira Ellen'}, 'show_email': 'NO'}]}
Year: 1997
DOI: 10.7907/rgd9-wz06
<p>Three-dimensional protein structures can be divided into classes in which proteins demonstrate high similarity of structure. While full and accurate determination of a protein's three-dimensional structure from its amino acid sequence is riot feasible at this time, methods seeking to determine this full structure would be aided by a priori information about the sequence's overall structural class. We have utilized Hidden Markov Models on sequences from the SWISS-PROT database in an attempt to determine the structural class of a protein given only its primary amino acid sequence.</p>
<p>Varying representations of the amino acid sequences and the accuracy with which the models using these representations differentiate between classes give some insight into the chemical and physical properties which are significant in the protein folding process. In addition, some representations of the protein sequence can illustrate the redundancy in the protein alphabet and others can capture structural class information with reduced computational requirements. Real vector representations provide an analogy to the problem of speech recognition.</p>
https://thesis.library.caltech.edu/id/eprint/178Unsplit Numerical Schemes for Hyperbolic Systems of Conservation Laws with Source Terms
https://resolver.caltech.edu/CaltechETD:etd-06032005-161139
Authors: {'items': [{'email': 'miltos@term.ucl.ac.be', 'id': 'Papalexandris-Miltiadis-Vassilios', 'name': {'family': 'Papalexandris', 'given': 'Miltiadis Vassilios'}, 'show_email': 'YES'}]}
Year: 1997
DOI: 10.7907/HW7S-AR36
In this thesis, a new method for the design of unsplit numerical schemes for hyperbolic systems of conservation laws with source terms is developed. Appropriate curves in space-time are introduced, along which the conservation equations decouple to the characteristic equations of the corresponding one-dimensional homogeneous system. The local geometry of these curves depends on the source terms and the spatial derivatives of the solution vector. Numerical integration of the characteristic equations is performed on these curves.
In the first chapter, a scalar conservation law with a stiff, nonlinear source term is studied using the proposed unsplit scheme. Various tests are made, and the results are compared with the ones obtained by conventional schemes. The effect of the stiffness of the source term is also examined.
In the second chapter, the scheme is extended to the one-dimensional, unsteady Euler equations for compressible, chemically-reacting flows. A numerical study of unstable detonations is performed. Detonations in the regime of low overdrive factors are also studied. The numerical simulations verify that the dynamics of the flow-field exhibit chaotic behavior in this regime.
The third chapter deals with the development and implementation of the unsplit scheme, for the two-dimensional, reactive Euler equations. In systems with more than two independent variables there are one-parameter families of curves, forming manifolds in space-time, along which the one-dimensional characteristic equations hold. The local geometry of these manifolds and their position relative to the classical characteristic rays are studied. These manifolds might be space-like or time-like, depending on the local flow gradients and the source terms.
In the fourth chapter a numerical study of two-dimensional detonations in performed. These flows are intrinsically unstable and produce very complicated patterns, such as cellular structures and vortex sheets. The proposed scheme appears to be capable of capturing many of the the important details of the flow-fields. Unlike traditional schemes, no explicit artificial-viscosity mechanisms need to be used with the proposed scheme.https://thesis.library.caltech.edu/id/eprint/2427A structured approach to parallel programming
https://resolver.caltech.edu/CaltechETD:etd-01242008-074143
Authors: {'items': [{'id': 'Massingill-B-L', 'name': {'family': 'Massingill', 'given': 'Berna Linda'}, 'show_email': 'NO'}]}
Year: 1998
DOI: 10.7907/5ma9-h225
Parallel programs are more difficult to develop and reason about than sequential programs. There are two broad classes of parallel programs: (1) programs whose specifications describe ongoing behavior and interaction with an environment, and (2) programs whose specifications describe the relation between initial and final states. This thesis presents a simple, structured approach to developing parallel programs of the latter class that allows much of the work of development and reasoning to be done using the same techniques and tools used for sequential programs. In this approach, programs are initially developed in a primary programming model that combines the standard sequential model with a restricted form of parallel composition that is semantically equivalent to sequential composition. Such programs can be reasoned about using sequential techniques and executed sequentially for testing. They are then transformed for execution on typical parallel architectures via a sequence of semantics-preserving transformations, making use of two secondary programming models, both based on parallel composition with barrier synchronization and one incorporating data partitioning. The transformation process for a particular program is typically guided and assisted by a parallel programming archetype, an abstraction that captures the commonality of a class of programs with similar computational features and provides a class-specific strategy for producing efficient parallel programs. Transformations may be applied manually or via a parallelizing compiler. Correctness of transformations within the primary programming model is proved using standard sequential techniques. Correctness of transformations between the programming models and between the models and practical programming languages is proved using a state-transition-based operational model.
This thesis presents: (1) the primary and secondary programming models, (2) an operational model that provides a common framework for reasoning about programs in all three models, (3) a collection of example program transformations with arguments for their correctness, and (4) two groups of experiments in which our overall approach was used to develop example applications. The specific contribution of this work is to present a unified theory/practice framework for this approach to parallel program development, tying together the underlying theory, the program transformations, and the program-development methodology.
https://thesis.library.caltech.edu/id/eprint/321Models of Richtmyer-Meshkov instability in continuously stratified fluids
https://resolver.caltech.edu/CaltechETD:etd-01242008-153024
Authors: {'items': [{'id': 'Meloon-M-R', 'name': {'family': 'Meloon', 'given': 'Mark Robert'}, 'show_email': 'NO'}]}
Year: 1998
DOI: 10.7907/S4XY-H779
The Richtmyer-Meshkov instability occurring when a planar shock wave passes through a sinusoidal region of continuous density gradient is studied numerically. Models are used to calculate the propagation of the shock through the inhomogeneity and to determine the late time behavior of the shocked fluid layer. The results from the models are compared with computations of the nonlinear Euler equations to determine ranges of validity. The models enjoy some success for problems involving weak incident shocks. As the Mach number is increased, however, the complex interactions between the transmitted and reflected fronts and the shocked density layer play an increasingly important role in the development of the flow and cause the models to fail.
The popular impulse approximation is applied to the continuously stratified fluid configuration through the use of a model due to Saffman and Meiron. The predictions of the late time growth rate of the interface and interfacial circulation from the model are compared with calculations from the nonlinear Euler equations. It is shown that for weak incident shocks the model is a very accurate prediction of the asymptotic behavior of the interface for a wide range of problems including those with interfaces of finite amplitude and thickness. For stronger shocks, post-shock values for Atwood ratio, amplitude and layer thickness are used in the model to obtain accurate predictions of late time growth rate for high Atwood ratio configurations. Poor agreement is seen for low Atwood ratios. Comparisons between circulation calculations and pointwise values of vorticity between the model and Euler simulations reveal that the impulse model does not predict the correct vorticity distribution for high or low Atwood ratios. A numerical implementation of the Biot-Savart law is used to calculate the growth rate strictly from the vorticity field in the compressible Euler simulations. The good agreement between the compressible and incompressible growth rates, as well as direct measurement of the discrete divergence in the flow, indicates that compressible effects are only important in the initialization of the instability and that the subsequent evolution is determined from the vorticity distribution. The vorticity generated by subsequent oscillations of the transmitted and reflected shocks is shown to have a non-negligible effect on the interfacial growth rate. It is conjectured that the success of the impulse approximation in predicting the asymptotic growth rate for problems involving moderate to strong shocks and high Atwood ratios is simply the result of fortuitous cancellation between regions of vorticity not computed accurately by the model.
The theory of Geometrical Shock Dynamics is used to propagate the shock through the region of density inhomogeneity and beyond. An important feature of the model is the neglect of interactions between the shock and the flow behind. A procedure for computing circulation in the entire flow from Geometrical Shock Dynamics is developed and implemented. For low to moderate Mach numbers, the initial circulation deposited on the layer is calculated with reasonable accuracy. At late times, however, the shape of the shock front does not agree with Euler calculations, resulting in incorrect calculation of the time evolution of total circulation of the flow. The agreement between the model and Euler simulations becomes poorer with increasing incident shock strength. By performing comparisons of local shock strength between the method and Euler simulation, it is shown that the method of Geometrical Shock Dynamics does not perform as well for problems involving nonuniform sound speed as had previously been believed. This suggests that the nonuniform flow conditions behind the shock, characterized by the vorticity baroclinically deposited on the interface during the shock refraction phase, plays a significant role in determining the evolution of the transmitted shock front.
https://thesis.library.caltech.edu/id/eprint/329Simulation of Controlled Bluff Body Flow with a Viscous Vortex Method
https://resolver.caltech.edu/CaltechETD:etd-03162004-133652
Authors: {'items': [{'id': 'Shiels-Dpouglas-G', 'name': {'family': 'Shiels', 'given': 'Douglas G.'}, 'show_email': 'NO'}]}
Year: 1998
DOI: 10.7907/49A0-VA12
Bluff body flows controlled in various manners are simulated with a high-resolution, gridless vortex method. Two-dimensional, unsteady, viscous simulations are utilized to illuminate the physical phenomenon underpinning certain flows of this class. Flows past a rotationally oscillating circular cylinder and flows past an elastically mounted circular cylinder are studied, providing a variety of new insights about these systems. A computational method facilitating longtime, high-resolution vortex simulations is developed whose grid-free nature enables future extension to complex geometries.
The significant fluid forces experienced by bluff bodies are of much practical concern and are induced by flowfields that are often complex. The studies in this thesis aim to contribute to the understanding of the relation between wake development and forces and how to exploit this relationship to achieve flow control. A circular cylinder undergoing rotational oscillation is known to experience a significant deviation in forces from unforced flow. Computations from Re=150-15000 verify past experimental observation of significant drag reduction for certain forcing parameters. These simulations also illuminate the mechanism which renders this control effective - a forced boundary layer instability triggering premature shedding of multipole vortex structures.
New insights were also provided by studies of flow over a model of an elastically mounted cylinder. A two-dimensional cylinder modeled as a damped oscillator can serve as an approximation to three-dimensional situations such as a cable under tension. Simulations clarified the behavior of such a two-dimensional system and, contrary to a line of classical thinking, revealed an unexpected adaptivity in wake evolution. New scaling is also suggested which better classifies these systems under certain conditions.
Vortex methods are well-suited for incompressible bluff body flow in many ways. However, the handling of viscous diffusion causes complications for such simulations. A relatively unexplored approach, the core expansion method, is studied, extended, and implemented in this work in order to balance accuracy with preservation of the gridless foundation of vortex methods. This viscous technique is found to enable long-time calculations that are prohibitive with other techniques while preserving a high level of accuracy.https://thesis.library.caltech.edu/id/eprint/965Part I. Vortex dynamics in wake models. Part II. Wave generation
https://resolver.caltech.edu/CaltechETD:etd-04052007-141032
Authors: {'items': [{'email': 'djh@cacr.caltech.edu', 'id': 'Hill-D-J', 'name': {'family': 'Hill', 'given': 'David J.'}, 'show_email': 'NO'}]}
Year: 1998
DOI: 10.7907/81N1-6X49
In Part I, steady wakes in inviscid fluid are constructed and investigated using the techniques of vortex dynamics. As a generalization of Foppl's flow past a circular cylinder [5], a steady solution is given for flow past an elliptical cylinder of arbitrary aspect ratio (perpendicular or parallel to the flow at infinity) with a bound wake of two symmetric recirculating eddies in the form of a point vortex pair. Linear stability analysis predicts an asymmetric instability and the symmetric nonlinear evolution is discussed in terms of a Kirchhoff-Routh path function. The wake behind a sphere is represented by a thin cored vortex ring of arbitrary internal structure. Steady configurations are obtained and long-wavelength perturbations to the ring centerline identify a tilting instability. A generalization of the Kirchhoff Routh function to an axisymmetric flow consisting of vortex rings and a body is presented. Using conformal maps and point vortices, translating symmetric two-dimensional bubbles with a vortex pair wake are constructed. An instability in which the bubble and vortex pair tilt away from each other is found as well as a symmetric oscillatory instability. The cross-section of a trailing vortex pair immersed in a cross stream shear is represented by two counter-rotating vortex patches. Numerical and analytical analyses are provided. The method of Schwarz functions as introduced by Meiron, Saffman and Schatzman [13] is used in the computation and stability analysis of steady patch shapes. Excellent agreement is obtained using an elliptical patch model. An instability essentially isolated to a single patch is identified, the nonlinear evolution of the elliptical patch model suggests that the patch whose fluid elements rotate against the shear will be destroyed.
Part II examines a possible mechanism for the generation of water waves which arises from the instability of an initially planar free surface in the presence of a parallel, sheared, inviscid flow. A two-dimensional steady flow comprised of exponential profiles representing both wind and a drift layer in the water is infinitesimally perturbed. The resulting Rayleigh equation is analytically solved by mean of Hyper-geometric functions and the dispersion relation is implicitly defined as solutions of a transcendental equation; stability boundaries are determined and growth rates are calculated. Comparisons are made with the simpler model of Caponi et al. [2] which uses piecewise linear profiles.https://thesis.library.caltech.edu/id/eprint/1274Saffman-Taylor fingers in deformed Hele-Shaw cells
https://resolver.caltech.edu/CaltechETD:etd-02062008-103933
Authors: {'items': [{'id': 'Gallagher-D-A', 'name': {'family': 'Gallagher', 'given': 'Donal A.'}, 'show_email': 'NO'}]}
Year: 1999
DOI: 10.7907/32e1-fp41
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Viscous fingering occurs when an essentially inviscid fluid is used to displace a viscous fluid in a porous medium or a Hele-Shaw cell. Long finger-shaped protrusions of the non-viscous fluid are found to advance into the viscous fluid. The importance of viscous fingering was first realised when attempts were made to retrieve oil from underground reservoirs by water injection.
When one displaces oil in a Hele-Shaw cell of width a with air, an initially plane interface between the two fluids becomes unstable. This interface eventually develops into a finger shaped protrusion of width [...] moving at constant speed. Experimentally, in the presence of small surface tension at the air-oil interface the constant [...] is found to be near 1/2. An exact solution of the model equations was found by Saffman and Taylor in the absence of surface tension.
Until now the fingering problem has only been studied in Hele-Shaw of constant gap corresponding to a porous medium with constant permeability. In order to understand the effect of non-homogenous permeability we look at the problem of fingering in a Hele-Shaw cell of varying gap. We consider variations in the direction perpendicular to the motion of the finger allowing steady fingers to exist.
We extend previous work on the constant gap case to show how the presence of surface tension interacts with variations in the gap to change the solution structure in a non-trivial way. For example values of [...] less than a half are possible. It is even found that solutions cease to exist for small surface tension in some cases. We solve the problem using both numerical methods based on conformal mapping and analytical methods involving asymptotics beyond all orders.https://thesis.library.caltech.edu/id/eprint/527Shock detachment process on cones in hypervelocity flows
https://resolver.caltech.edu/CaltechETD:etd-02082008-162753
Authors: {'items': [{'email': 'ialr2003@gmail.com', 'id': 'Leyva-I-A', 'name': {'family': 'Leyva', 'given': 'Ivett A.'}, 'show_email': 'YES'}]}
Year: 1999
DOI: 10.7907/62TN-GA26
<p>The shock detachment process on cones in hypervelocity flows is one of the most sensitive flows to relaxation effects. The critical angle for shock detachment under frozen conditions can be very different from the critical angle under chemical and thermal equilibrium. The rate of increase of the detachment distance with cone angle is also affected by the relaxation rate.</p>
<p>The purpose of this study is to explain the effects of nonequilibrium on the shock detachment distance and its growth rate on cones in hypervelocity flows. The study consists of an experimental and a computational program. The experimental part has been carried out at Caltech's hypervelocity reflected shock tunnel (T5). Six different free-stream conditions have been chosen, four using N<sub>2</sub> as the test gas and two using CO<sub>2</sub>. About 170 shots were performed on 24 cones. The cones range in diameter from 2 cm to 16 cm with half-angles varying from 55° to 75°. The experimental data obtained are holographic interferograms of every shot, and surface temperature and pressure measurements for the bigger cones. Extensive numerical simulations were made for the N<sub>2</sub> flows and some were also made for the CO<sub>2</sub> flows. The code employed is a Navier-Stokes solver that can account for thermal and chemical nonequilibrium in axisymmetric flows.</p>
<p>The experimental and computational data obtained for the shock detachment distance confirms a previous theoretical model that predicts the detachment distance will grow more slowly for relaxing flows than for frozen or equilibrium flows. This difference is explained in terms of the behavior of the sonic line inside the shock layer. Different growth rates result when the detachment distance is controlled by the diameter of the cone (frozen and equilibrium cases) than when it is controlled by the extent of the relaxation zone inside the shock layer (nonequilibrium flows). The experimental data are also complemented with computational data to observe the behavior of the detachment distance from the frozen to equilibrium limits for a given cone half-angle and free-stream condition. As deduced by a previous simple scaling argument, the ratio of the detachment distance to the diameter of the cone is constant in the two extremes and rapidly switches from one value to the other for cone diameters of about 2 cm to 16 cm. The experimental interferograms are also compared with numerical ones in terms of the detachment distance, the number of fringes in the shock layer, and the shape of the fringes.</p>
<p>The heat flux traces obtained from the temperature measurements show different behaviors for the attached and detached cases, but these effects can be related to the conditions at the edge of and inside the boundary layer and to the Reynolds number of the flow rather than to nonequilibrium effects. The pressure measurements were insensitive to the degree of nonequilibrium.</p>
https://thesis.library.caltech.edu/id/eprint/564A study of viscous flow past axisymmetric and two-dimensional bodies
https://resolver.caltech.edu/CaltechETD:etd-02242008-093110
Authors: {'items': [{'id': 'Kang-S-P', 'name': {'family': 'Kang', 'given': 'Sung Phill'}, 'show_email': 'NO'}]}
Year: 1999
DOI: 10.7907/ky36-f623
In this thesis, we study the behavior of viscous flow past bodies of different shapes. In Chapter 2, we construct a boundary-fitted, numerical grid around a rigid spheroid of various aspect ratios and solve numerically the Navier-Stokes equations in steady, axisymmetric form at various Reynolds numbers. In addition, we use these steady solutions as a base flow and perform a linear stability analysis to determine the critical Reynolds numbers above which the base flow becomes unstable. We are able to confirm the results of Natarajan and Acrivos [26] and extend them to more generalized body shapes.
In Chapter 3, we solve the Navier-Stokes equations to investigate flows past an oblate ellipsoidal bubble of fixed shape, which is characterized by a free-slip boundary condition. We then compare our results with previous results by Dandy and Leal [6] and Blanco and Magnaudet [4] and use the computed steady solutions as the base flow to perform a linear stability analysis. We show that even with a free-slip boundary condition, if the body is sufficiently oblate, the flow can become unstable in a manner similar to that of flows past rigid bodies.
In Chapter 4, we develop an alternative numerical method to compute steady flows past a deforming, axisymmetric bubble. A newly developed conformal grid generation method is applied. We show that our results are in good agreement with those of Ryskin and Leal [34], [35] and then extend some of their results to higher Reynolds number.
In Chapter 5, we modify the method developed in Chapter 4 to compute steady flows past a symmetric, two-dimensional bubble. We show that the bubble deforms to an elliptical shape and that a wake can develop if the deformation of the bubble is sufficiently large.
https://thesis.library.caltech.edu/id/eprint/743A fixed-grid numerical method for dendritic solidification with natural convection
https://resolver.caltech.edu/CaltechETD:etd-11092007-083533
Authors: {'items': [{'id': 'Lahey-P-M', 'name': {'family': 'Lahey', 'given': 'Patrick M.'}, 'show_email': 'NO'}]}
Year: 1999
DOI: 10.7907/6yzn-g986
The solidification of a material into an undercooled melt occurs quite frequently in material processing applications. The interface between the solid and liquid phases in such cases is inherently unstable. This instability can lead to the formation of dendritic growth patterns which may significantly impact the microstructure of the resulting solid. Because the microstructure of materials notably influences their macroscopic properties, there is significant interest in understanding and controlling the formation and evolution of dendrites.
For many years, material scientists have sought to develop a predictive theory that could relate the observed dimensions and characteristics of dendrites to the thermal and fluid dynamics conditions that prevailed during their formation. To date, no such general theory exists. The problem is a difficult one, both from an experimental and mathematical standpoint.
In this work, we develop an accurate numerical method capable of simulating dendritic solidification both with and without natural convection effects. The scheme explicitly tracks and parametrizes the interface between the liquid and solid phases using a series of independent marker particles. Due to the release of latent heat, the derivatives of the temperature of a growing dendrite are discontinuous across the interface. As a consequence, great care is required when discretizing the derivatives at nodes adjacent to the interface. We use a generalized version of LeVeque and Li's immersed interface method to accurately compute the spatial derivatives. We also develop an accurate one-step time marching scheme for problems with derivatives that jump discontinuously across a moving interface. The method is notable because it does not require that the same time discretization scheme be applied to every term in the governing equation.https://thesis.library.caltech.edu/id/eprint/4477The Multiscale Finite Element Method (MsFEM) and Its Applications
https://resolver.caltech.edu/CaltechETD:etd-11102005-090314
Authors: {'items': [{'email': 'efendiev@math.tamu.edu', 'id': 'Efendiev-Yalchin-R', 'name': {'family': 'Efendiev', 'given': 'Yalchin R.'}, 'orcid': '0000-0001-9626-303X', 'show_email': 'NO'}]}
Year: 1999
DOI: 10.7907/2QJN-2S06
<p>Multiscale problems occur in many scientific and engineering disciplines, in petroleum engineering, material science, etc. These problems are characterized by the great deal of spatial and time scales which make it difficult to analyze theoretically or solve numerically. On the other hand, the large scale features of the solutions are often of main interest. Thus, it is desirable to have a numerical method that can capture the effect of small scales on large scales without resolving the small scale details.</p>
<p>In the first part of this work we analyze the multiscale finite element method (MsFEM) introduced in [28] for elliptic problems with oscillatory coefficients. The idea behind MsFEM is to capture the small scale information through the base functions constructed in elements that are larger than the small scale of the problem. This is achieved by solving for the finite element base functions from the leading order of homogeneous elliptic equation. We analyze MsFEM for different situations both analytically and numerically. We also investigate the origin of the resonance errors associated with the method and discuss the ways to improve them.</p>
<p>In the second part we discuss flow based upscaling of absolute permeability which is an important step in the practical simulations of flow through heterogeneous formations. The central idea is to compute the upscaled, grid-block permeability from fine scale solutions of the flow equation. It is well known that the grid block permeability may be strongly influenced by the boundary conditions imposed on the flow equations and the size of grid blocks. We analyze the effects of the boundary conditions and grid block sizes on the computed grid block absolute permeabilities. Moreover, we employ the ideas developed in the analysis of MsFEM to improve the computed values of absolute permeability.</p>
<p>The last part of the work is the application of MsFEM as well as upscaling of absolute permeability on upscaling of two-phase flow. In this part we consider coarse models using MsFEM. We demonstrate the efficiency of these models for practical problems. Moreover, we show that these models improve the existing approaches.</p>https://thesis.library.caltech.edu/id/eprint/4487Magnetohydrodynamic modeling of solar magnetic arcades using exponential propagation methods
https://resolver.caltech.edu/CaltechETD:etd-02062006-154529
Authors: {'items': [{'email': 'mtokman@ucmerced.edu', 'id': 'Tokman-M', 'name': {'family': 'Tokman', 'given': 'Mayya'}, 'show_email': 'YES'}]}
Year: 2001
DOI: 10.7907/PDCS-GX15
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.https://thesis.library.caltech.edu/id/eprint/519Diffusion-Mediated Regulation Endocrine Networks
https://resolver.caltech.edu/CaltechETD:etd-12052003-095049
Authors: {'items': [{'email': 'petrasek@caltech.edu', 'id': 'Petrasek-Danny', 'name': {'family': 'Petrasek', 'given': 'Danny'}, 'orcid': '0000-0003-4178-4844', 'show_email': 'NO'}]}
Year: 2002
DOI: 10.7907/776t-vs55
In endocrine glands, vigorous and coordinated responses are often elicited by modest changes in the concentration of the organist molecule. The mammalian parathyroid gland is a representative case. Small (5%) changes in serum calcium result in tenfold (1000%) changes in glandular parathyroid hormone (PTH) release. In vitro, single isolated cells are observed to secrete fewer hormones than cells residing within a connected group, suggesting that a network has emergent regulatory properties. In PTH secreting tumors however, the ability to quickly respond to changes in calcium is strongly damped. A unifying hypothesis that accounts for these phenomena is realized by extra-cellular modulation of calcium diffusivity. A theoretical model and computational experiments demonstrate qualitative agreement with published experimental results. Our results suggest that in addition to the cellular mechanisms, endocrine glandular networks may have regulatory prowess at the level of interstitial transport. The extra-cellular diffusional mechanism proposed provides a consistent argument for 1) higher secretion of single cells in a connected network compared to isolated cells, 2) the rapid nonlinear response seen in healthy glands as well as 3) the pathological responses seen in hyperplasia and adenoma. Since the proposed diffusional regulation strongly depends on the existence of a connected cell network (gland), it also suggests a rationale for the advantages of cell networks as organs versus a dispersed system of isolated cells (in the case of the parathyroid gland).https://thesis.library.caltech.edu/id/eprint/4788Phase Boundary Propagation in Heterogeneous Media
https://resolver.caltech.edu/CaltechTHESIS:10082010-142653040
Authors: {'items': [{'id': 'Craciun-Bogdan', 'name': {'family': 'Craciun', 'given': 'Bogdan'}, 'show_email': 'NO'}]}
Year: 2002
DOI: 10.7907/JXG6-W865
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>https://thesis.library.caltech.edu/id/eprint/6122Fast, High-Order Methods for Scattering by Inhomogeneous Media
https://resolver.caltech.edu/CaltechETD:etd-08142002-182101
Authors: {'items': [{'id': 'Hyde-Edward-McKay', 'name': {'family': 'Hyde', 'given': 'Edward McKay'}, 'show_email': 'NO'}]}
Year: 2003
DOI: 10.7907/Z0V0-KM52
<p>In this thesis, we introduce a new, fast, high-order method for scattering by inhomogeneous media in three dimensions. As in previously existing methods, the low (O(N log N)) complexity of our integral equation method is obtained through extensive use of the fast Fourier transform (FFT) in evaluating the required convolutions. Unlike previous FFT-based methods, however, this method yields high-order accuracy, even for scatterers containing geometric singularities such as discontinuities, corners, and cusps.</p>
<p>We begin our discussion with a thorough theoretical analysis of an efficient, high-order method recently introduced by Bruno and Sei (IEEE Trans. in Antenn. Propag., 2000), which motivated the present work. This two-dimensional method is based on a Fourier approximation of the integral equation in polar coordinates and a related, generally low-order, Fourier smoothing of the scatterer. The claim that use of this low-order approximation of the scatterer leads to a high-order accurate numerical method generated considerable controversy. Our proofs establish that this method indeed yields high-order accurate solutions. We also introduce several substantial improvements to the numerical implementation of this two-dimensional algorithm, which lead to increased numerical stability with decreased computational cost.</p>
<p>We then present our new, fast, high-order method in three dimensions. An immediate generalization of the polar coordinate approach in two dimensions to a spherical coordinate approach in three dimensions appears less advantageous than our chosen approach: Fourier approximation and integration in Cartesian coordinates. To obtain smooth and periodic functons (which are approximated to high-order via Fourier series), we 1) decompose the Green's function into a smooth part with infinite support and a singular part with compact support; and 2) replace, as in the two-dimensional approach, the (possibly discontinuous) scatterer with its truncated Cartesian Fourier series.</p>
<p>The accuracy of our three-dimensional method is approximately equal to that of the two-dimensional method mentioned above and, interestingly, is actually much simpler than the two-dimensional approach. In addition to our theoretical discussion of these high-order methods, we present a parallel implementation of our three-dimensional Cartesian approach. The efficiency, high-order accuracy, and overall performance of both the polar and Cartesian methods are demonstrated through several computational examples.</p>https://thesis.library.caltech.edu/id/eprint/3116A Phase-Field Model of Dislocations in Ductile Single Crystals
https://resolver.caltech.edu/CaltechETD:etd-05302003-094155
Authors: {'items': [{'email': 'marisol@purdue.edu', 'id': 'Koslowski-Marisol', 'name': {'family': 'Koslowski', 'given': 'Marisol'}, 'orcid': '0000-0001-9650-2168', 'show_email': 'YES'}]}
Year: 2003
DOI: 10.7907/SFMJ-1B50
<p>A phase-field theory of dislocations, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles, resulting in hardening, path dependency and hysteresis.</p>
<p>We introduce a variational formulation for the statistical mechanics of dissipative systems. The influence of finite temperature as well as the mechanics in the phase-field theory are modeled with a Metropolis Monte Carlo algorithm and a mean field approximation.</p>
<p>A chief advantage of the present theory is that at zero temperature it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. The theory predicts a range of behaviors which are in qualitative agreement with observation, including hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect; the evolution of the dislocation density under cycling loading; temperature softening; strain rate dependence; and others.</p>
<p>The model also reproduces the formation of dislocation networks observed in grain boundaries for different crystal structures and orientations. Simultaneously with the stable configurations the theory naturally predicts the equilibrium dislocation density independently of initial values or sources.</p>https://thesis.library.caltech.edu/id/eprint/2287Efficient Algorithms for Solving Static Hamilton-Jacobi Equations
https://resolver.caltech.edu/CaltechETD:etd-05202003-170423
Authors: {'items': [{'id': 'Mauch-Sean-Patrick', 'name': {'family': 'Mauch', 'given': 'Sean Patrick'}, 'show_email': 'NO'}]}
Year: 2003
DOI: 10.7907/5R5P-Y603
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>https://thesis.library.caltech.edu/id/eprint/1888A Numerical and Analytical Study of Detonation Diffraction
https://resolver.caltech.edu/CaltechETD:etd-02122003-152525
Authors: {'items': [{'id': 'Arienti-Marco', 'name': {'family': 'Arienti', 'given': 'Marco'}, 'orcid': '0000-0001-8166-0016', 'show_email': 'NO'}]}
Year: 2003
DOI: 10.7907/MAGN-R628
An investigation of detonation diffraction through an abrupt area change has been carried out via two-dimensional, parallel simulations. The existence of critical conditions for successful diffraction is closely related to the occurrence of localized re-initiation mechanisms, and is relevant to propulsion and safety concepts concerning detonation transmission. Our analysis is specialized to a reactive mixture with perfect gas equation of state and a single-step reaction in the Arrhenius form. The concept of shock decoupling from the reaction zone is the simplest idea used to explain the behavior of a diffracting detonation front. Lagrangian particles are injected into the flow in order to identify the dominant terms in the equation that describes the temperature rate of change of a fluid element, expressed in a shock-based reference system. Conveniently simplified, this equation provides an insight into the competition between the energy release rate and the expansion rate behind the diffracting front. We also examine the mechanism of spontaneous generation of transverse waves along the front. This mechanism is related to the sensitivity of the reaction rate to temperature, and it is investigated in the form of a parametric study for the activation energy. We study in detail three highly resolved cases of detonation diffraction that illustrate different types of behavior, super-, sub-, and near-critical diffraction. We review the applicability of existing shock dynamics models to the corner-turning problem. Numerical results from the parametric study are compared with predictions from these theories in the attempt to find a formula for shock decay in a quenching detonation. This estimate is then used in the simplified temperature rate of change equation to provide a relation between critical channel width and activation energy. We conclude this study by examining the spontaneous formation of transverse waves along the wavefront of a successfully transmitted detonation. The problem is simplified to a planar CJ detonation moving in a channel over a small obstacle to investigate how acoustic waves propagate within the reaction zone. Depending on the reaction kinetics, we show that such waves may be amplified due to feedback between the chemical reaction and fluid motion. The amplification can lead to shock steepening and formation of transverse detonation waves.
https://thesis.library.caltech.edu/id/eprint/610Modeling a Hox Gene Network: Stochastic Simulation with Experimental Perturbation
https://resolver.caltech.edu/CaltechETD:etd-10042002-200303
Authors: {'items': [{'email': 'kastner@alumni.caltech.edu', 'id': 'Kastner-Jason-Christopher', 'name': {'family': 'Kastner', 'given': 'Jason Christopher'}, 'show_email': 'YES'}]}
Year: 2003
DOI: 10.7907/2TWD-EK72
<p>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.</p>
<p>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.</p>https://thesis.library.caltech.edu/id/eprint/3907Chebyshev Spectral Method for Singular Moving Boundary Problems with Application to Finance
https://resolver.caltech.edu/CaltechETD:etd-09042002-131120
Authors: {'items': [{'id': 'Greenberg-Andrei-Yakovlevich', 'name': {'family': 'Greenberg', 'given': 'Andrei Yakovlevich'}, 'show_email': 'NO'}]}
Year: 2003
DOI: 10.7907/J383-YX41
<p>Accurate results for the inherently nonlinear models involving moving boundaries can only be produced by sophisticated, high-quality numerical algorithms. However the most general approaches are usually not very accurate, while those producing accurate results for certain cases are hard to generalize. In an attempt to bridge this gap, we propose a general method, based on a unified framework for arbitrary parabolic operators, which, in particular, can accurately treat singular problems.</p>
<p>Our method consists of front-fixing and a Chebyshev series solution of the resulting nonlinear partial differential equation. An appropriate set of convergent smooth approximations is used in singular cases. For smooth problems, our method is very competitive in both speed and accuracy. At the same time, our method is able to produce accurate solutions in the most general setting, whenever existence theorems for moving boundary problems hold. We establish convergence of numerical solutions to the true solution for a large class of possibly singular initial conditions.</p>
<p>In addition to the general method, we introduce computational techniques which enhance its performance for singular problems. These include derivative evaluations with Padé approximations; prior integration in time; and domain decomposition.</p>
<p>We demonstrate the performance of our method with several regular and singular problems. A comparison with other methods shows that our algorithms produce more accurate results. The additional techniques, which do not use smoothing approximations, significantly shorten computing times while retaining reasonable accuracy.</p>
<p>We present a systematic study of the mathematical finance problem of pricing American options on a dividend-paying asset from the point of view of partial differential equations. A symmetry result, obtained via a simple change of variables, allows to reduce any American option problem to one of the two canonical cases, depending on the relation between the interest rate and the dividend yield. Each of these cases is equivalent to a singular Stefan problem, which can be solved by our method. We present calculations for the classical problems of options written on a single stock and the more complicated examples, such as index options and foreign currency options, thus demonstrating the remarkable practical scope of the proposed approach.</p>https://thesis.library.caltech.edu/id/eprint/3321A Front Tracking Method for Modelling Thermal Growth
https://resolver.caltech.edu/CaltechETD:etd-03042003-115138
Authors: {'items': [{'email': 'beth@jetcafe.org', 'id': 'Howard-Elizabeth-Anne', 'name': {'family': 'Howard', 'given': 'Elizabeth Anne'}, 'show_email': 'NO'}]}
Year: 2003
DOI: 10.7907/HRNV-DA03
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. https://thesis.library.caltech.edu/id/eprint/860Low-Coherence Interferometric Imaging: Solution of the One-Dimensional Inverse Scattering Problem
https://resolver.caltech.edu/CaltechETD:etd-09092003-212358
Authors: {'items': [{'email': 'mario.j.chaubell@jpl.nasa.gov', 'id': 'Chaubell-Mario-Julián', 'name': {'family': 'Chaubell', 'given': 'Mario Julián'}, 'orcid': '0000-0002-8067-6988', 'show_email': 'YES'}]}
Year: 2004
DOI: 10.7907/ED7Y-Q436
Optical coherence tomography (OCT) is a non-invasive imaging technique based on the use of light sources exhibiting a low degree of coherence. Low coherence interferometric microscopes have been successful in producing internal images of thin pieces of biological tissue; typically samples of the order of 1 milimeter in depth have been imaged, with a resolution of the order of 10 to 20 microns in some portions of the sample. In this thesis, I deal with the imaging problem of determining the internal structure of a body from backscattered laser light and low-coherence interferometry. In detail, I formulate and solve an inverse problem which, using the interference fringes that result as the back-scattering of low-coherence light is made to interfere with a reference beam, produces maps detailing the values of the refractive index within the imaged sample. Unlike previous approaches to this imaging problem, the solver I introduce does not require processing at data collection time, and it can therefore produce solutions for inverse problems of multi-layered structures containing thousands of layers from back-scattering interference fringes only. We expect that the approach presented in this work, which accounts fully for the statistical nature of the coherence phenomenon, should prove of interest in the fields of medicine, biology and materials science.https://thesis.library.caltech.edu/id/eprint/3398Compressible Vortices and Shock-Vortex Interactions
https://resolver.caltech.edu/CaltechETD:etd-05262004-145030
Authors: {'items': [{'id': "O'Reilly-Gerard-Kieran", 'name': {'family': "O'Reilly", 'given': 'Gerard Kieran'}, 'show_email': 'NO'}]}
Year: 2004
DOI: 10.7907/FGJD-0Z31
Secondary instabilities on the organized, spanwise, vortical structures in incompressible shear layers, play an important role in generating the onset of three-dimensional turbulence in such flows. The effect of increasing compressibility on these instabilities is examined by using the compressible Stuart vortex as a model for a compressible shear layer. It is found that both two- and three-dimensional subharmonic instabilities cease to promote pairing events even at moderate free stream Mach numbers. The fundamental mode becomes dominant as the free stream Mach number is increased, and a new instability corresponding to an instability on a parallel shear layer is observed. The interaction of a shock with a compressible vortex may be viewed as a simplified model of the general interaction of a shock with the coherent structures in a turbulent flow field. An approximate theory for computing shock-compressible-vortex interactions is developed, based on Ribner (1954). The problem of convection of a frozen patern of vorticity, dilatation, temperature and entropy through a planar shock wave is considered. The refraction and modification of the upstream disturbances into the three basis modes permitted by the linear Euler equations is derived, as well as the perturbation to the shock wave. This theory is used to compute approximate post-shock states corresponding to shock-CSV interactions, a model for shock shear layer interactions. The method is verified by comparing its approximate post-shock fields with those computed explicitly using AMROC, a finite difference, AMR-WENO code. Finally, numerical solutions corresponding to a compressible analogue of the Mallier and Maslowe vortex (a periodic array of counter-rotating vortices) are presented. These solutions admit the existence of large regions of smooth supersonic flow, and could potentially be used to model the counter-rotating vortices arising from the single- and multi-mode Richtmyer-Meshkov instability.https://thesis.library.caltech.edu/id/eprint/2082Spatiotemporal Chaos in Rayleigh-Bénard Convection
https://resolver.caltech.edu/CaltechETD:etd-08062003-162208
Authors: {'items': [{'email': 'ChiamKH@MailAPS.ORG', 'id': 'Chiam-Keng-Hwee', 'name': {'family': 'Chiam', 'given': 'Keng-Hwee'}, 'orcid': '0000-0002-8987-8463', 'show_email': 'YES'}]}
Year: 2004
DOI: 10.7907/2NJV-BB91
<p>Spatiotemporal chaos, or disorder in both the space and time coordinates, is studied in direct numerical simulations of Rayleigh-Bénard convection. In particular, the following investigations pertaining to spiral defect chaos are discussed.</p>
<p>First, in the absence of the mean flow, spiral defect chaos is found to collapse to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wave numbers that approach those uniquely selected by focus-type singularities, which, in the absence of the mean flow, lie at the zig zag instability boundary. In addition, mean flow is shown to contribute to the phenomenon of rolls terminating perpendicularly into lateral walls. In the absence of the mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with the Rayleigh number.</p>
<p>Second, the transport of passive tracers in the presence of advection by spiral defect chaos is found to be characterized by normal diffusion. The enhancement in the tracer diffusivity follows two regimes. When the molecular diffusivity of the tracer concentration is small, the enhancement is proportional to the Péclet number. When the molecular diffusivity is large, the enhancement is proportional to the square root of the Péclet number. This difference is explained in terms of the dependence of the transport on the local wave numbers. It is found that tracer concentrations with small molecular diffusivity experience enhanced longitudinal diffusion and suppressed lateral diffusion at regions of the flow occupied by defects.</p>
<p>Third, perturbations in spiral defect chaos are found to propagate in a localized manner. In particular, they nucleate around the defect structures in the flow. In addition, an oscillatory instability at the spiral core is discovered. Finally, the propagation in pre-chaotic stripe textures is explained in terms of the diffusion of the phase variable of the stripe state.</p>https://thesis.library.caltech.edu/id/eprint/3020Mathematical Modeling and Simulation of Aquatic and Aerial Animal Locomotion
https://resolver.caltech.edu/CaltechETD:etd-05272005-004852
Authors: {'items': [{'email': 'svgaby@gmail.com', 'id': 'Stredie-Valentin-Gabriel', 'name': {'family': 'Stredie', 'given': 'Valentin Gabriel'}, 'show_email': 'NO'}]}
Year: 2005
DOI: 10.7907/8PWE-RK28
<p>In this thesis we investigate the locomotion of fish and birds by applying both new and well known mathematical techniques.</p>
<p>The two-dimensional model is first studied using Krasny's vortex blob method, and then a new numerical method based on Wu's theory is developed. To begin with, we will implement Krasny's ideas for a couple of examples and then switch to the numerical implementation of the nonlinear analytical mathematical model presented by Wu. We will demonstrate the superiority of this latter method both by applying it to some specific cases and by comparing with the experiments. The nonlinear effects are very well observed and this will be shown by analyzing Wagner's result for a wing abruptly undergoing an increase in incidence angle, and also by analyzing the vorticity generated by a wing in heaving, pitching and bending motion. The ultimate goal of the thesis is to accurately represent the vortex structure behind a flying wing and its influence on the bound vortex sheet.</p>
<p>In the second part of this work we will introduce a three-dimensional method for a flat plate advancing perpendicular to the flow. The accuracy of the method will be shown both by comparing its results with the two-dimensional ones and by validating them versus the experimental results obtained by Ringuette in the towing tank of the Aeronautics Department at Caltech.</p>https://thesis.library.caltech.edu/id/eprint/2142Experiments and Modeling of Impinging Jets and Premixed Hydrocarbon Stagnation Flames
https://resolver.caltech.edu/CaltechETD:etd-05242005-165713
Authors: {'items': [{'email': 'jeffrey.bergthorson@mcgill.ca', 'id': 'Bergthorson-Jeffrey-Myles', 'name': {'family': 'Bergthorson', 'given': 'Jeffrey Myles'}, 'orcid': '0000-0003-2924-7317', 'show_email': 'NO'}]}
Year: 2005
DOI: 10.7907/7FQZ-EY88
<p>To model the combustion of long-chain hydrocarbon fuels, an accurate kinetics mechanism must first be developed for the oxidation of small hydrocarbons, such as methane, ethane, and ethylene. Even for methane, a generally accepted mechanism is still elusive due to a lack of kinetically independent experimental data. In this work, a combined experimental and modeling technique is developed to validate and further optimize these mechanisms. This technique relies on detailed measurements of strained flames in a jet-wall stagnation flow using simultaneous Particle Streak Velocimetry (PSV) and CH Planar Laser Induced Fluorescence (PLIF). Stagnation flames are simulated using an axisymmetric, one-dimensional model with accurate specification of the requisite boundary conditions. Direct comparisons between experiment and simulation allow for an assessment of the various models employed, with an emphasis on the chemistry model performance.</p>
<p>The flow field for a cold impinging laminar jet is found to be independent of the nozzle-to-plate separation distance if velocities are scaled by the Bernoulli velocity. The one-dimensional formulation is found to accurately model the stagnation flow if the velocity boundary conditions are appropriately specified. The boundary-layer-displacement-thickness corrected diameter is found to be an appropriate scale for axial distances and allows the identification of an empirical, analytical expression for the flow field of the impinging laminar jet.</p>
<p>Strained methane-air flame experiments confirm that the reacting flow is also independent of the nozzle-to-plate separation distance. Methane, ethane, and ethylene flames are studied as functions of the applied strain rate, mixture dilution, and mixture fraction. Mechanism performance is found to be relatively insensitive to both the mixture dilution and the imposed strain rate, while exhibiting a stronger dependence on the fuel type and flame stoichiometry. The approach and diagnostics presented here permit an assessment of the predictions of strained-hydrocarbon flames for several combustion chemistry mechanisms. The data presented in this thesis are made available to kineticists looking for optimization targets, with the goal of developing a predictive kinetics model for hydrocarbon fuels. The methodology described here can allow new optimization targets to be rapidly measured, reducing the experimental burden required to fully constrain the chemistry models.</p>https://thesis.library.caltech.edu/id/eprint/2004Localized Non-bBlowup Conditions for 3D Incompressible Euler Flows and Related Equations
https://resolver.caltech.edu/CaltechETD:etd-05302005-161405
Authors: {'items': [{'email': 'xinwei.yu@gmail.com', 'id': 'Yu-Xinwei', 'name': {'family': 'Yu', 'given': 'Xinwei'}, 'show_email': 'YES'}]}
Year: 2005
DOI: 10.7907/K5E1-W938
<p>In this thesis, new results excluding finite time singularities with localized assumptions/conditions are obtained for the 3D incompressible Euler equations.</p>
<p>The 3D incompressible Euler equations are some of the most important nonlinear equations in mathematics. They govern the motion of ideal fluids. After hundreds of years of study, they are still far from being well-understood. In particular, a long-outstanding open problem asks whether finite time singularities would develop for smooth initial values. Much theoretical and numerical study on this problem has been carried out, but no conclusion can be drawn so far.</p>
<p>In recent years, several numerical experiments have been carried out by various authors, with results indicating possible breakdowns of smooth solutions in finite time. In these numerical experiments, certain properties of the velocity and vorticity field are observed in near-singular flows. These properties violate the assumptions of existing theoretical theorems which exclude finite time singularities. Thus there is a gap between current theoretical and numerical results. To narrow this gap is the main purpose of the work presented in this thesis.</p>
<p>In this thesis, a new framework of investigating flows carried by divergence-free velocity fields is developed. Using this new framework, new, localized sufficient conditions for the flow to remain smooth are obtained rigorously. These new results can deal with fast shrinking large vorticity regions and are applicable to recent numerical experiments. The application of the theorems in this thesis reveals new subtleties, and yields new understandings of the 3D incompressible Euler flow.</p>
<p>This new framework is then further applied to a two-dimensional model equation, the 2D quasi-geostrophic equation, for which global existence is still unproved. Under certain assumptions, we obtain new non-blowup results for the 2D quasi-geostrophic equation.</p>
<p>Finally, future plans of applying this new framework to some other PDEs as well as other possibilities of attacking the 3D Euler and 2D quasi-geostrophic singularity problems are discussed.</p>https://thesis.library.caltech.edu/id/eprint/2302Mathematical Models of the Developing C. elegans Hermaphrodite Gonad
https://resolver.caltech.edu/CaltechETD:etd-05092006-160328
Authors: {'items': [{'id': 'Goulet-David-Michael', 'name': {'family': 'Goulet', 'given': 'David Michael'}, 'show_email': 'NO'}]}
Year: 2006
DOI: 10.7907/D7C9-PA08
The study of growing and developing organisms is a fascinating branch of experimental biology. Once created, cells must exchange chemical and physical cues with neighboring cells in order to grow, divide, and differentiate properly. In this thesis we study portions of development of the C. elegans hermaphrodite gonad, building mathematical models of the development process. Using our models, we show that vulval precursor cells make fate decisions under a flexible program that takes advantage of inherent chemical oscillations. This flexibility allows the cells to react more sensitively to weak signaling gradients and to the actions of neighboring cells. With our mathematical models, we also show that the development of the anchor cell cannot proceed properly using the currently known decision mechanisms. We draw upon knowledge of homologous proteins in D. melanogaster to propose a modification to the current theory on anchor cell development. Our models suggest that this modified mechanism, though not yet identified in C. elegans, is sufficient to specify anchor cell fates in accordance with experimental observations. In studying our mathematical models, novel analytical techniques were developed to understand the asymptotic behavior of systems of delay differential equations.https://thesis.library.caltech.edu/id/eprint/5184Simulations and Analysis of Two- and Three-Dimensional Single-Mode Richtmyer-Meshkov Instability using Weighted Essentially Non-Oscillatory and Vortex Methods
https://resolver.caltech.edu/CaltechETD:etd-12082006-124547
Authors: {'items': [{'email': 'mlatini@acm.caltech.edu', 'id': 'Latini-Marco', 'name': {'family': 'Latini', 'given': 'Marco'}, 'show_email': 'YES'}]}
Year: 2007
DOI: 10.7907/1397-GZ04
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>https://thesis.library.caltech.edu/id/eprint/4868Modeling and Simulation of Axisymmetric Stagnation Flames
https://resolver.caltech.edu/CaltechETD:etd-04252007-170838
Authors: {'items': [{'id': 'Sone-Kazuo', 'name': {'family': 'Sone', 'given': 'Kazuo'}, 'show_email': 'NO'}]}
Year: 2007
DOI: 10.7907/DYGA-YJ20
<p>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.</p>
<p>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.</p>
<p>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.</p>https://thesis.library.caltech.edu/id/eprint/1500Rotating Rayleigh-Bénard Convection
https://resolver.caltech.edu/CaltechETD:etd-08252006-154116
Authors: {'items': [{'email': 'jscheel@oxy.edu', 'id': 'Scheel-Janet-D', 'name': {'family': 'Scheel', 'given': 'Janet D.'}, 'orcid': '0000-0002-1669-4188', 'show_email': 'YES'}]}
Year: 2007
DOI: 10.7907/961N-6776
<p>Rotating Rayleigh-Benard convection (rRBC) is studied as a paradigmatic example of pattern formation and spatiotemporal chaos. For large enough rotation rates, this system undergoes a supercritical bifurcation from the uniform state to a state known as domain chaos.</p>
<p>In domain chaos, domains of straight parallel rolls change their orientation and size discretely. This roll switching causes an overall counterclockwise precession of the pattern. An additional mechanism of precession, glide-induced precession, is introduced here, by deriving the rRBC amplitude equation to higher order. New terms due to the rotation cause rolls to precess whenever there is an amplitude gradient in the direction parallel to the rolls. Hence, dislocations which are stationary in a nonrotating system will glide in a rotating frame, causing the overall precession.</p>
<p>Theory that includes the Coriolis force but ignores the centrifugal force predicted scaling laws near the transition to domain chaos. However, experimenters found different scaling laws. The scaling laws are studied here by direct numerical simulations (DNS) for the exact parameters as experiments. When only the Coriolis force is included, the DNS scaling laws agree with theory. When the centrifugal force is also included, the DNS scaling laws agree better with experiment; hence the centrifugal force cannot be neglected from theory.</p>
<p>The coefficients of the amplitude equation for the Complex Ginzburg-Landau equation (CGLE) are found for DNS of traveling waves. They agree well with experimental results. The CGLE is chaotic for certain values of the coefficients. However, for the parameters in the DNS, those chaotic regimes were not realized.</p>
<p>Leading order Lyapunov exponents (LLE) and eigenvectors are computed for both rotating and nonrotating convection. For certain parameters, these systems are found to have positive LLEs; hence they are truly chaotic. For time-dependent systems, the leading eigenvector is characterized by localized bursts of activity which are associated with dynamical events. The short-time dynamics of the LLE is correlated with these dynamical events. However, contributions to the LLE are due to non-periodic events only.</p>
<p>Lagrangian particle tracking methods are employed for rRBC. These systems exhibit chaotic advection in that initially localized particle trajectories explore the available phase space.</p>https://thesis.library.caltech.edu/id/eprint/3217On the Non-Local Geometry of Turbulence
https://resolver.caltech.edu/CaltechETD:etd-05092008-173614
Authors: {'items': [{'email': 'ivan.bermejo.moreno@gmail.com', 'id': 'Bermejo-Moreno-Ivan', 'name': {'family': 'Bermejo-Moreno', 'given': 'Ivan'}, 'show_email': 'YES'}]}
Year: 2008
DOI: 10.7907/DH9W-Y894
<p>A multi-scale methodology for the study of the non-local geometry of eddy structures in turbulence is developed. Starting from a given three-dimensional field, this consists of three main steps: extraction, characterization and classification of structures. The extraction step is done in two stages. First, a multi-scale decomposition based on the curvelet transform is applied to the full three-dimensional field, resulting in a finite set of component fields, one per scale. Second, by iso-contouring each component field at one or more iso-contour levels, a set of closed iso-surfaces is obtained that represents the structures at that scale. For periodic domains, those structures intersecting boundaries are reconnected with their continuation in the opposite boundaries. The characterization stage is based on the joint probability density function (jpdf), in terms of area coverage on each individual iso-surface, of two differential-geometry properties, the shape index and curvedness, plus the stretching parameter, a dimensionless global invariant of the surface. Taken together, this defines the geometrical signature of the iso-surface. The classification step is based on the construction of a finite set of parameters, obtained from algebraic functions of moments of the jpdf of each structure, that specify its location as a point in a multi-dimensional 'feature space'. At each scale the set of points in feature space represents all structures at that scale, for the specified iso-contour value. This allows the application, to the set, of clustering techniques that search for groups of structures with a common geometry.</p>
<p>Results are presented of a first application of this technique to a passive scalar field obtained from 512³ direct numerical simulation of scalar mixing by forced, isotropic turbulence (Re<sub>λ</sub>=265). These show transition, with decreasing scale, from blob-like structures in the larger scales to blob- and tube-like structures with small or moderate stretching in the inertial range of scales, and then toward tube and, predominantly, sheet-like structures with high level of stretching in the dissipation range of scales. Implications of these results for the dynamical behavior of passive scalar stirring and mixing by turbulence are discussed.</p>
<p>We apply the same methodology to the enstrophy and kinetic energy dissipation rate instantaneous fields of a second numerical database of incompressible homogeneous isotropic turbulence decaying in time obtained by DNS in a periodic box. Three different resolutions are considered: 256³, 512³ and 1024³ grid points, with k<sub>max</sub>η̅ approximately 1, 2, and 4, respectively, the same initial conditions and Re<sub>λ</sub> ≈ 77. This allows a comparison of the geometry of the structures obtained for different resolutions. For the highest resolution, structures of enstrophy and dissipation evolve in a continuous distribution from blob-like and moderately stretched tube-like shapes at the large scales to highly stretched sheet-like structures at the small scales. The intermediate scales show a predominance of tube-like structures for both fields, much more pronounced for the enstrophy field. The dissipation field shows a tendency toward structures with lower curvedness than those of the enstrophy, for intermediate and small scales. The 256³ grid resolution case (k<sub>max</sub>η̅ ≈ 1) was unable to detect the predominance of highly stretched sheet-like structures at the smaller scales.</p>
<p>The same methodology, but without the multi-scale decomposition, is then applied to two scalar fields used by existing local criteria for the eduction of tube- and sheet-like structures in turbulence, Q and [A<sub>ij</sub>]<sub>+</sub> respectively, obtained from invariants of the velocity gradient tensor and alike in the 1024³ case. This adds the non-local geometrical characterization and classification to those local criteria, assessing their validity in educing particular geometries.</p>
<p>Finally we introduce a new methodology for the study of proximity issues among different sets of structures, based also on geometrical and non-local analyses. We apply it to four of the fields previously studied. Tube-like structures of Q are mainly surrounded by sheets of [A<sub>ij</sub>]<sub>+</sub>, which appear at close distances. For the enstrophy, tube-like structures at an intermediate scale are primarily surrounded by sheets of smaller scales of the enstrophy and structures of dissipation at the same and smaller scales. A secondary contribution results from tubes of enstrophy at smaller scales appearing at farther distances. Different configurations of composite structures are presented.</p>
https://thesis.library.caltech.edu/id/eprint/1713Effective Behavior of Dielectric Elastomer Composites
https://resolver.caltech.edu/CaltechETD:etd-08272007-145455
Authors: {'items': [{'email': 'lixiutian@gmail.com', 'id': 'Tian-Lixiu', 'name': {'family': 'Tian', 'given': 'Lixiu'}, 'show_email': 'YES'}]}
Year: 2008
DOI: 10.7907/CZNF-JB47
<p>The class of electroactive polymers has been developed to a point where real life applications as ``artificial muscles" are conceivable. These actuator materials provide attractive advantages: they are soft, lightweight, can undergo large deformation, possess fast response time and are resilient. However, widespread application has been hindered by their limitations: the need for a large electric field, relatively small forces and energy density. However, recent experimental work shows great promise that this limitation can be overcome by making composites of two materials with high contrast in their dielectric modulus. In this thesis, a theoretical framework is derived to describe the electrostatic effect of the dielectric elastomers. Numerical experiments are conducted to explain the reason for the promising experimental results and to explore better microstructures of the composites to enhance the favorable properties.</p>
<p>The starting point of this thesis is a general variational principle, which characterizes the behavior of solids under combined mechanical and electrical loads. Based on this variational principle, we assume the electric field is small as of order ε½, assume further the deformation is caused by the electrostatic effects; the deformation field is then of order ε. Using the tool of Γ-convergence, we derive a small-strain model in which the electric field and the deformation field are decoupled which results in a huge simplification of the problem.</p>
<p>Based on this small-strain model, employing the powerful tool of two-scale convergence, we derive the effective properties for dielectric composites conducting small strains. A formula of the effective electromechanical coupling coefficients is given in terms of the unit cell solutions.</p>
<p>Armed with these theoretical results, we carry out numerical experiments about the effective properties of different kind of composites. A very careful analysis of the numerical results provides a deep understanding of the mechanism of the enhancement in strain by making composites of different microstructures.</p>https://thesis.library.caltech.edu/id/eprint/3248Nonreflecting Boundary Conditions Obtained from Equivalent Sources for Time-Dependent Scattering Problems
https://resolver.caltech.edu/CaltechETD:etd-05202008-111349
Authors: {'items': [{'email': 'dhoch@acm.caltech.edu', 'id': 'Hoch-David', 'name': {'family': 'Hoch', 'given': 'David'}, 'show_email': 'NO'}]}
Year: 2008
DOI: 10.7907/5M0P-NR33
<p>In many engineering applications, scattering of acoustic or electromagnetic waves from a body of arbitrary shape is considered in an infinite medium. Solving the underlying partial differential equations with a standard numerical method such as finite elements or finite differences requires truncating the unbounded domain of definition into a finite computational region. As a consequence, an appropriate boundary condition must be prescribed at the artificial boundary. Many approaches have been proposed for this fundamental problem in the field of wave scattering. All of them fall into one of three main categories.</p>
<p>The first class of methods is based on mathematical approximations or physical heuristics. These boundary conditions are easy to implement and run in short computing times. However, these approaches give rise to spurious reflections at the artificial boundary, which travel back into the computational domain and corrupt the solution.</p>
<p>A second group consists of accurate and convergent methods. However, these formulations are usually harder to implement and often more expensive than the computation of the interior scheme itself.</p>
<p>Finally, there are methods which are accurate and fast. The drawback of these approaches lies in the fact that the outer boundary must be taken to be either a sphere, a plane, or a cylinder. For many applications of interest, this may require use of a computational domain much larger than actually needed, which leads to an expensive overall numerical scheme.</p>
<p>This work introduces a new methodology in order to compute the fields at the artificial boundary. Like the second class of methods described above, the proposed algorithm is accurate and numerically convergent, yet its computational cost is less than the underlying portion of the volumetric calculation. And, unlike the third category, this new approach allows us to choose the artificial boundary to be arbitrarily close to the scatterer. This method is based on a novel concept of "equivalent source' representations which allows a highly accurate and fast evaluation of the boundary condition.</p>https://thesis.library.caltech.edu/id/eprint/1899A Super-Algebraically Convergent, Windowing-Based Approach to the Evaluation of Scattering from Periodic Rough Surfaces
https://resolver.caltech.edu/CaltechETD:etd-01032008-222910
Authors: {'items': [{'email': 'monro@acm.caltech.edu', 'id': 'Monro-John-Anderson', 'name': {'family': 'Monro', 'given': 'John Anderson'}, 'show_email': 'NO'}]}
Year: 2008
DOI: 10.7907/F9VM-JP39
<p>We introduce a new second-kind integral equation method to solve direct rough surface scattering problems in two dimensions. This approach is based, in part, upon the bounded obstacle scattering method that was originally presented in Bruno et al. [2004] and is discussed in an appendix of this thesis. We restrict our attention to problems in which time-harmonic acoustic or electromagnetic plane waves scatter from rough surfaces that are perfectly reflecting, periodic and at least twice continuously differentiable; both sound-soft and sound-hard type acoustic scattering cases---correspondingly, transverse-electric and transverse-magnetic electromagnetic scattering cases---are treated. Key elements of our algorithm include the use of infinitely continuously differentiable windowing functions that comprise partitions of unity, analytical representations of the integral equation’s solution (taking into account either the absence or presence of multiple scattering) and spectral quadrature formulas. Together, they provide an efficient alternative to the use of the periodic Green’s function found in the kernel of most solvers’ integral operators, and they strongly mitigate the rapidly increasing computational complexity that is typically borne as the frequency of the incident field increases.</p>
<p>After providing a complete description of our solver and illustrating its usefulness through some preliminary examples, we rigorously prove its convergence. In particular, the super-algebraic convergence of the method is established for problems with infinitely continuously differentiable scattering surfaces. We additionally show that accuracies within prescribed tolerances are achieved with fixed computational cost as the frequency increases without bound for cases in which no multiple reflections occur.</p>
<p>We present extensive numerical data demonstrating the convergence, accuracy and efficiency of our computational approach for a wide range of scattering configurations (sinusoidal, multi-scale and simulated ocean surfaces are considered). These results include favorable comparisons with other leading integral equation methods as well as the non-convergent Kirchhoff approximation. They also contain analyses of sets of cases in which the major physical parameters associated with these problems (i.e., surface height, wavenumber and incidence angle) are systematically varied. As a result of these tests, we conclude that the proposed algorithm is highly competitive and robust: it significantly outperforms other leading numerical methods in many cases of scientific and practical relevance, and it facilitates rapid analyses of a wide variety of scattering configurations.</p>https://thesis.library.caltech.edu/id/eprint/19Premixed Hydrocarbon Stagnation Flames: Experiments and Simulations to Validate Combustion Chemical-Kinetic Models
https://resolver.caltech.edu/CaltechETD:etd-05302008-113043
Authors: {'items': [{'email': 'benezech.laurent@gmail.com', 'id': 'Benezech-Laurent-Jean-Michel', 'name': {'family': 'Benezech', 'given': 'Laurent Jean-Michel'}, 'show_email': 'YES'}]}
Year: 2008
DOI: 10.7907/TVB9-4266
<p>A methodology based on the comparison of flame simulations relying on reacting flow models with experiment is applied to C<sub>1</sub>–C<sub>3</sub> stagnation flames. The work reported targets the assessment and validation of the modeled reactions and reaction rates relevant to (C<sub>1</sub>–C<sub>3</sub>)-flame propagation in several detailed combustion kinetic models. A concensus does not, as yet, exist on the modeling of the reasonably well-understood oxidation of C<sub>1</sub>–C<sub>2</sub> flames, and a better knowledge of C<sub>3</sub> hydrocarbon combustion chemistry is required before attempting to bridge the gap between the oxidation of C<sub>1</sub>–C<sub>2</sub> hydrocarbons and the more complex chemistry of heavier hydrocarbons in a single kinetic model.</p>
<p>Simultaneous measurements of velocity and CH-radical profiles were performed in atmospheric propane(C<sub>3</sub>H<sub>8</sub>)- and propylene(C<sub>3</sub>H<sub>6</sub>)-air laminar premixed stagnation flames stabilized in a jet-wall configuration. These nearly-flat flames can be modeled by one-dimensional simulations, providing a means to validate kinetic models. Experimental data for these C<sub>3</sub> flames and similar experimental data for atmospheric methane(CH<sub>4</sub>)-, ethane(C<sub>2</sub>H<sub>6</sub>)-, and ethylene(C<sub>2</sub>H<sub>4</sub>)-air flames are compared to numerical simulations performed with a one-dimensional hydrodynamic model, a multi-component transport formulation including thermal diffusion, and different detailed-chemistry models, in order to assess the adequacy of the models employed. A novel continuation technique between kinetic models was developed and applied successfully to obtain solutions with the less-robust models. The 2005/12 and 2005/10 releases of the San Diego mechanism are found to have the best overall performance in C<sub>3</sub>H<sub>8</sub> and C<sub>3</sub>H<sub>6</sub> flames, and in CH<sub>4</sub>, C<sub>2</sub>H<sub>6</sub>, and C<sub>2</sub>H<sub>4</sub> flames, respectively.</p>
<p>Flame position provides a good surrogate for flame speed in stagnation-flow stabilized flames. The logarithmic sensitivities of the simulated flame locations to variations in the kinetic rates are calculated via the "brute-force" method for fifteen representative flames covering the five fuels under study and the very lean, stoichiometric, and very rich burning regimes, in order to identify the most-important reactions for each flame investigated. The rates of reactions identified in this manner are compared between the different kinetic models. Several reaction-rate differences are thus identified that are likely responsible for the variance in flame-position (or flame-speed) predictions in C<sub>1</sub>–C<sub>2</sub> flames.</p>https://thesis.library.caltech.edu/id/eprint/2316Asymptotic Weight Analysis of Low-Density Parity Check (LDPC) Code Ensembles
https://resolver.caltech.edu/CaltechETD:etd-05202008-094714
Authors: {'items': [{'email': 'sarah@acm.caltech.edu', 'id': 'Sweatlock-Sarah-Lynne', 'name': {'family': 'Sweatlock', 'given': 'Sarah Lynne'}, 'show_email': 'NO'}]}
Year: 2008
DOI: 10.7907/86BY-MA30
<p>With the invention of turbo codes in 1993 came increased interest in codes and iterative decoding schemes. Gallager's Regular codes were rediscovered, and irregular codes were introduced. Protograph codes were introduced and analyzed by NASA's Jet Propulsion Laboratory in the early years of this century. Part of this thesis continues that work, investigating the decoding of specific protograph codes and extending existing tools for analyzing codes to protograph codes.</p>
<p>The rest of this work focuses on a previously unknown relationship between the binary entropy function and the asymptotic ensemble average weight enumerator, which we call the spectral shape of the ensemble. This result can be seen as an extension of the Pless power-moment identities based on the discovery that the convex hull of the spectral shape is the Legendre transform of a function closely related to the moment-generating function of a codeword's weight. </p>
<p>In order to fully investigate this new relationship, tools needed to be designed to calculate the derivatives of the spectral shape as the equation describing an ensemble's spectral shape is rarely straightforward. For Gallager's regular ensembles, a formula for calculating derivatives of functions defined parametrically was required. For repeat-accumulate (RA) codes, a formula was needed for functions defined implicitly through a second function. Both formulas are similar to Faa di Bruno's formula for derivatives of compositions of functions.With the invention of turbo codes in 1993 came increased interest in codes and iterative decoding schemes. Gallager's Regular codes were rediscovered, and irregular codes were introduced. Protograph codes were introduced and analyzed by NASA's Jet Propulsion Laboratory in the early years of this century. Part of this thesis continues that work, investigating the decoding of specific protograph codes and extending existing tools for analyzing codes to protograph codes.</p>
<p>The rest of this work focuses on a previously unknown relationship between the binary entropy function and the asymptotic ensemble average weight enumerator, which we call the spectral shape of the ensemble. This result can be seen as an extension of the Pless power-moment identities based on the discovery that the convex hull of the spectral shape is the Legendre transform of a function closely related to the moment-generating function of a codeword's weight.</p>
<p>In order to fully investigate this new relationship, tools needed to be designed to calculate the derivatives of the spectral shape as the equation describing an ensemble's spectral shape is rarely straightforward. For Gallager's regular ensembles, a formula for calculating derivatives of functions defined parametrically was required. For repeat-accumulate (RA) codes, a formula was needed for functions defined implicitly through a second function. Both formulas are similar to Faa di Bruno's formula for derivatives of compositions of functions.</p>https://thesis.library.caltech.edu/id/eprint/1898A Subdivision Approach to the Construction of Smooth Differential Forms
https://resolver.caltech.edu/CaltechETD:etd-02282008-112022
Authors: {'items': [{'email': 'wang@acm.caltech.edu', 'id': 'Wang-Ke', 'name': {'family': 'Wang', 'given': 'Ke'}, 'show_email': 'NO'}]}
Year: 2008
DOI: 10.7907/K0DE-9P44
<p>Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this thesis we interpret these schemes as defining bases for discrete differential 0- resp. 2-forms, and present a novel subdivision-based method of constructing smooth differential forms on simplicial surfaces. It completes the picture of classic primal/dual subdivision by introducing a new concept named r-cochain subdivision. Such subdivision schemes map scalar coefficients on r-simplexes from the coarse to the refined mesh and converge to r-forms on the mesh. We perform convergence and smoothness analysis in an arbitrary topology setting by utilizing the techniques of matrix subdivision and the subdivision differential structure.</p>
<p>The other significance of our method is its preserving exactness of differential forms. We prove that exactness preserving is equivalent to the commutative relations between the subdivision schemes and the topological exterior derivative. Our construction is based on treating r- and (r+1)-cochain subdivision schemes as a pair and enforcing the commutative relations. As a result, our low-order construction recovers classic Whitney forms, while the high-order construction yields a new class of high order Whitney forms. The 1-form bases are C^1, except at irregular vertices where they are C^0. We also demonstrate extensions to three-dimensional subdivision schemes and non-simplicial meshes as well, such as quadrilaterals and octahedra.</p>
<p>Our construction is seamlessly integrated with surface subdivision. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector filed on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate. The subdivision r-forms can also be used as finite element bases for physical simulations on curved surfaces. We demonstrate the optimal rate of convergence in solving the Laplace and bi-Laplace equations of 1-forms.</p>
https://thesis.library.caltech.edu/id/eprint/812Detonation Stability with Reversible Kinetics
https://resolver.caltech.edu/CaltechETD:etd-06022008-170629
Authors: {'items': [{'id': 'Kao-Shannon-Theresa', 'name': {'family': 'Kao', 'given': 'Shannon Theresa'}, 'show_email': 'NO'}]}
Year: 2008
DOI: 10.7907/H8JN-VS03
<p>Detonation propagation is unsteady due to the innate instability of the reaction zone structure. Up until the present, investigations of detonation stability have been exclusively concerned with model systems using the perfect gas equation of state and primarily single-step irreversible reaction mechanisms.</p>
<p>This study investigates detonation stability characteristics with reversible chemical kinetics models. To allow for more general kinetics models, we generalize the perfect gas, one-step irreversible kinetics, linear stability equations to a set of equations using the ideal gas equation of state and a general reaction scheme. We linearly perturb the reactive Euler equations following the method of Lee and Stewart (1990) and Short and Stewart (1998). Our implementation uses Cantera (Goodwin, 2005) to evaluate all thermodynamic quantities and evaluate generalized analytic derivatives of quantities dependent on the kinetics model.</p>
<p>The computational domain is the reaction zone in the shock-fixed frame such that the left boundary conditions are the perturbed shock jump conditions which we have derived for a general equation of state and implemented for an ideal gas equation of state. At the right boundary, the system must satisfy a radiation condition requiring that all waves travel out of the domain. Unlike the case of a single reversible reaction, in a truly multistep kinetics model, the radiation boundary condition cannot be solved analytically. In this work, we provide a general methodology for satisfying the appropriate boundary condition.</p>
<p>We then investigate the effects of reversibility on the characteristics of the instability in one and two dimensions. These characteristics are quantified by the unstable eigenvalues as well as the shape of the base flow and eigenfunctions. We show that there is an exchange of stability as a function of reversibility. To confirm the results our work, we have performed unsteady calculations. We show that we can match the frequencies predicted by our linear stability calculations near the stability threshold.</p>https://thesis.library.caltech.edu/id/eprint/2409Large-Eddy Simulations of Molecular Mixing in a Recirculating Shear Flow
https://resolver.caltech.edu/CaltechETD:etd-05262008-152803
Authors: {'items': [{'email': 'georgios.matheou@uconn.edu', 'id': 'Matheou-Georgios', 'name': {'family': 'Matheou', 'given': 'Georgios'}, 'orcid': '0000-0003-4024-4571', 'show_email': 'YES'}]}
Year: 2008
DOI: 10.7907/VFKF-SC30
<p>The flow field and mixing in an expansion-ramp geometry is studied using large-eddy simulation (LES) with subgrid scale (SGS) modeling based on the stretched-vortex model. The expansionramp geometry was developed to provide enhanced mixing and flameholding characteristics while maintaining low total-pressure losses, elements that are important in the design and performance of combustors for hypersonic air-breathing propulsion applications. The mixing was studied by tracking a passive scalar without taking into account the effects of chemical reactions and heat release.</p>
<p>In order to verify the solver and the boundary closure implementation, a method utilizing results from linear stability analysis (LSA) theory is developed. LSA can be used to compute unstable perturbations to a flow, subject to certain approximations. The perturbations computed from LSA are used as an inflow condition to the flow computed by the solver been assessed. A projection based metric is constructed that only assumes the shape of the solution and not the growth rate of the perturbations, thus also allowing the latter to be determined as part of the verification. The growth rate of the perturbations for an unbounded (effectively) incompressible shear layer and a confined compressible shear layer is found to be in agreement with the prediction of the LSA.</p>
<p>The flow and mixing predictions of the LES are in good agreement with experimental measurements. Total (resolved and subgrid) probability density functions (PDFs) of the passive scalar are estimated using an assumed beta-distribution model for the subgrid scalar field. The improved mixing characteristics of the expansion-ramp geometry compared to free shear layers are illustrated by the shapes of the PDFs. Moreover, the temperature rise and the probability of mixed fluid profiles are in good agreement with the experimental measurements, indicating that the mixing on a molecular scale is correctly predicted by the LES–SGS model. Finally, the predictions of the LES are shown to be resolution-independent. The mean fields and passive scalar PDFs have essentially converged at the two finer grid-resolutions used.</p>https://thesis.library.caltech.edu/id/eprint/2117Richtmyer-Meshkov Instability in Converging Geometries
https://resolver.caltech.edu/CaltechETD:etd-05302008-140331
Authors: {'items': [{'email': 'manuel.lombardini@polytechnique.org', 'id': 'Lombardini-Manuel', 'name': {'family': 'Lombardini', 'given': 'Manuel'}, 'show_email': 'YES'}]}
Year: 2008
DOI: 10.7907/5SNE-4003
<p>We investigate the Richtmyer-Meshkov instability (RMI) in converging geometries analytically and computationally. The linear, or small amplitude, regime is first covered as it is the onset to subsequent non-linear stages of the perturbation growth. While the plane interaction of a shock with a slightly perturbed density interface is classically viewed as a single interface evolving as baroclinic vorticity have been initially deposited on it, we propose a simple but more complete model characterizing the early interaction between the interface and the receding waves produced by the shock-interface interaction, in the case of a reflected shock. A universal time scale representing the time needed by the RMI to reach its asymptotic growth rate is found analytically and confirmed by ideal gas computations for various incident shock Mach numbers MI and Atwood ratios A, and could be useful especially for experimentalists in non-dimensionalizing their data.</p>
<p>Considering again linear perturbations, we then obtain a general analytical model for the asymptotic growth rate reached by the instability during the concentric interaction of an imploding/exploding cylindrical shock with a cylindrical interface containing three-dimensional orthogonal perturbations, in the azimuthal and axial directions. Stable perturbations, typical of the converging geometry, are discovered. Comparisons are made with simulations where the effects of compressibility, wave reverberations, and flow convergence are isolated. Azimuthal and axial perturbation evolution are compared with results obtained for the plane RMI at comparable initial wavelengths.</p>
<p>A second interaction occurs when the transmitted shock, produced by the incident converging shock impacting the interface, converges to the axis and reflects to reshock the initially accelerated interface. This leads to highly non-linear perturbation growth. To isolate the complex wave interaction process, the interface is considered initially unperturbed so that the flow is radially symmetric. An accurate visualization procedure is performed to characterize the underlying physics behind the reshock event. We study extensively the cylindrical and spherical geometry, for various MI and for the air → SF6 (A=0.67) and SF6 → (A=-0.67) interactions, and draw important differences with the equivalent plane configuration.</p>
<p>A hybrid, low-numerical dissipation/shock-capturing method, embedded into an adaptive mesh refinement framework is optimized in order to achieve large-eddy simulations of the self-similar cylindrical converging shock-driven RMI and the turbulent mixing generated by the reshock. Computations are produced for MI=1.3 and 2.0, and for air -> SF6 SF6 -> air interfaces. We develop statistics tools to study extensively the growth of the turbulent mixing zone using cylindrical averages as well as various measures such as probability density functions of the mixing and turbulent power spectra, with the objectives of understanding the turbulent mixing in this particular geometry.</p>https://thesis.library.caltech.edu/id/eprint/2319Algorithms for Mapping Nucleic Acid Free Energy Landscapes
https://resolver.caltech.edu/CaltechETD:etd-12312008-153810
Authors: {'items': [{'email': 'othmer@acm.caltech.edu', 'id': 'Othmer-Jonathan-Andrew', 'name': {'family': 'Othmer', 'given': 'Jonathan Andrew'}, 'show_email': 'NO'}]}
Year: 2009
DOI: 10.7907/VJX1-6376
To complement the utility of thermodynamic calculations in the design and analysis of nucleic acid secondary structures, we seek to develop efficient and scalable algorithms for the analysis of secondary structure kinetics. Secondary structure kinetics are modeled by a first-order master equation, but the number of secondary structures for a sequence grows exponentially with the length of the sequence, meaning that for systems of interest, we cannot write down the rate matrix, much less solve the master equation. To address these difficulties, we develop a method to construct macrostate maps of nucleic acid free energy landscapes based on simulating the continuous-time Markov chain associated with the microstate master equation. The method relies on the careful combination of several elements: a novel procedure to explicitly identify transitions between macrostates in the simulation, a goodness-of-clustering test specific to secondary structures, an algorithm to find the centroid secondary structure for each macrostate, a method to compute macrostate partition functions from short simulations, and a framework for computing transition rates with confidence intervals. We use this method to study several experimental systems from our laboratory with system sizes in the hundreds of nucleotides, and develop a model problem, the d-cube, for which we can control all of the relevant parameters and analyze our method's error behavior. Our results and analysis suggest that this method will be useful not only in the analysis and design of nucleic acid mechanical devices, but also in wider applications of molecular simulation and simulation-based model reduction.https://thesis.library.caltech.edu/id/eprint/5172Stable High-Order Finite-Difference Interface Schemes with Application to the Richtmyer-Meshkov Instability
https://resolver.caltech.edu/CaltechETD:etd-03132009-095507
Authors: {'items': [{'email': 'rmjkramer@gmail.com', 'id': 'Kramer-Richard-Michael-Jack', 'name': {'family': 'Kramer', 'given': 'Richard Michael Jack'}, 'show_email': 'NO'}]}
Year: 2009
DOI: 10.7907/HXGM-DC92
<p>High-order adaptive mesh refinement offers the potential for accurate and efficient resolution of problems in fluid dynamics and other fields where a wide range of length scales is present. A critical requirement for the interface closures used with these methods is stability in the context of hyperbolic systems of partial differential equations.</p>
<p>In this study, a class of energy-stable high-order finite-difference interface closures is presented for grids with step resolution changes in one dimension. Asymptotic stability in time for these schemes is achieved by imposing a summation-by-parts condition on the interface closure, which is thus also nondissipative. Interface closures compatible with interior fourth- and sixth-order explicit, and fourth-order implicit centered schemes are presented. Validation tests include linear and nonlinear problems in one and in two dimensions with tensor-product grid refinement.</p>
<p>A second class of stable high-order interface closures is presented for two-dimensional cell-centered grids with patch-refinement and step-changes in resolution. For these grids, coarse and fine nodes are not aligned at the mesh interfaces, resulting in hanging nodes. Stability is achieved by again imposing a summation-by-parts condition, resulting in nondissipative closures, at the cost of accuracy at corner interfaces. Interface stencils for an explicit fourth-order finite-difference scheme are presented for each geometry. Validation tests confirm the stability and accuracy of these closures for linear and nonlinear problems.</p>
<p>The Richtmyer-Meshkov instability is investigated using a novel first-order perturbation of the two-dimensional Navier-Stokes equations about a shock-resolved base flow. The computational domain is efficiently resolved using the one-dimensional fourth-order interface scheme. Results are compared to analytic models of the instability, showing agreement with predicted asymptotic growth rates in the inviscid range, while significant discrepancies are noted in the transient growth phase. Viscous effects are found to be poorly predicted by existing models.</p>
https://thesis.library.caltech.edu/id/eprint/947Catalytic Modification of Flammable Atmosphere in Aircraft Fuel Tanks
https://resolver.caltech.edu/CaltechTHESIS:06092010-102349721
Authors: {'items': [{'email': 'inkichoi@caltech.edu', 'id': 'Choi-Inki', 'name': {'family': 'Choi', 'given': 'Inki'}, 'show_email': 'NO'}]}
Year: 2010
DOI: 10.7907/69YQ-XZ33
A facility for investigating catalytic combustion and measurement of fuel molecule concentration was built to examine catalyst candidates for inerting systems in aircraft. The facility consists of fuel and oxygen supplies, a catalytic-bed reactor, heating system, and laser-based diagnostics. Two supplementary systems consisting of a calibration test cell and a nitrogen-purged glove box were also constructed. The catalyst under investigation was platinum, and it was mixed with silica particles to increase the surface area available to react. The catalyst/silica mixture was placed in a narrow channel section of the reactor and supported from both sides by glass wool. The fuels investigated were n-octane and n-nonane because their vapor pressure is sufficiently high to create ammable gaseous mixtures with atmospheric air at room temperature. Calibration experiments were performed to determine the absorption cross-section of the two fuels as a function of temperature. The cross-section values were then used to determine the fuel concentration before the flow entered the reactor and after exposure to the heated catalyst. An initial set of experiments was performed with the catalytic-bed reactor at two temperatures, 255 and 500°C, to investigate pyrolysis and oxidation of the fuel. The presence of the catalyst increased the degree of pyrolysis and oxidation at both temperatures. The results show that catalytic modification of ammable atmospheres may yield a viable alternative for inerting aircraft fuel tanks. However, further tests are required to produce oxidation at sufficiently low temperature to comply with aircraft safety regulations.
https://thesis.library.caltech.edu/id/eprint/5942Detonation Induced Strain in Tubes
https://resolver.caltech.edu/CaltechTHESIS:05142010-174001426
Authors: {'items': [{'email': 'jim.karnesky@gmail.com', 'id': 'Karnesky-James-Alan', 'name': {'family': 'Karnesky', 'given': 'James Alan'}, 'show_email': 'NO'}]}
Year: 2010
DOI: 10.7907/GTKC-FY91
<p>When a detonation wave propagates through a piping system, it acts as a traveling pressure load to the pipe wall. The detonation wave must be followed by an expansion wave in order to bring the combustion products to zero velocity at the ignition end. When it reaches a closed end-wall, a reflected shock is formed which propagates back into the tube with a decaying pressure. The present study aims to develop predictive models for the stresses and strains produced in such a situation. To this end, two series of experiments are discussed. The first series used strain gauges and a laser vibrometer to measure the elastic response of the tube to the incident detonation in thin aluminum tubes. The second series used strain gauges and high speed video to measure the plastic response of steel tubes to incident detonations and reflected shocks. In these experiments a novel mode of plastic deformation was discovered in which the residual plastic deformation in the tube wall had a periodic sinusoidal pattern.</p>
<p>A semi-empirical model of the pressure history was developed for use as a boundary condition in models of the mechanical response of the tube. This model was tested against experiment, and it was found that the pressure and arrival time could not be simultaneously predicted from the simple model. This and the general form of the pressure traces in the experiment seem to suggest an interaction between the reflected shock and the boundary layer behind the detonation resulting in a possible bifurcation in the reflected shock wave.</p>
<p>With these considerations in mind, the model was applied to single degree of freedom and finite element models of the tube wall. The ripples observed in the experiment were present in the 1-D single degree of freedom models, indicating that they are a result of the interaction of the reflected shock wave with the elastic oscillations set in motion by the detonation wave. Strain-rate hardening was found to be an important consideration under detonation loading conditions. With proper consideration of rate hardening, a single material model may be used to arrive at reasonable predictions the plastic strains resulting from detonations and reflections at initial pressures of 2 and 3 bar initial pressures.</p>https://thesis.library.caltech.edu/id/eprint/5809The Simulation of Shock- and Impact-Driven Flows with Mie-Grüneisen Equations of State
https://resolver.caltech.edu/CaltechTHESIS:12162010-115725941
Authors: {'items': [{'email': 'geoff@caltech.edu', 'id': 'Ward-Geoffrey-M', 'name': {'family': 'Ward', 'given': 'Geoffrey M.'}, 'show_email': 'NO'}]}
Year: 2011
DOI: 10.7907/8Q2Q-GT29
<p>An investigation of shock- and impact-driven flows with Mie-Grüneisen equation of state derived from a linear shock-particle speed Hugoniot relationship is presented. Cartesian mesh methods using structured adaptive refinement are applied to simulate several flows of interest in an Eulerian frame of reference. The flows central to the investigation include planar Richtmyer-Meshkov instability, the impact of a sphere with a plate, and an impact-driven Mach stem.</p>
<p>First, for multicomponent shock-driven flows, a dimensionally unsplit, spatially high-order, hybrid, center-difference, limiter methodology is developed. Effective switching between center-difference and upwinding schemes is achieved by a set of robust tolerance and Lax-entropy-based criteria [49]. Oscillations that result from such a mixed stencil scheme are minimized by requiring that the upwinding method approaches the center-difference method in smooth regions. To attain this property a blending limiter is introduced based on the norm of the deviation of WENO reconstruction weights from ideal. The scheme is first demonstrated successfully for the linear advection equation in spatially fourth- and sixth-order forms. A spatially fourth-order version of the method that combines a skew-symmetric kinetic-energy preserving center-difference scheme with a Roe-Riemann solver is then developed and implemented in Caltech's Adaptive Mesh Refinement, Object-oriented C++ (AMROC) [16,17] framework for Euler flows.</p>
<p>The solver is then applied to investigate planar Richtmyer-Meshkov instability in the context of an equation of state comparison. Comparisons of simulations with materials modeled by isotropic stress Mie-Grüneisen equations of state derived from a linear shock-particle speed Hugoniot relationship [36,52] to those of perfect gases are made with the intention of exposing the role of the equation of state. First, results for single- and triple-mode planar Richtmyer-Meshkov instability between mid-ocean ridge basalt (MORB) and molybdenum modeled by Mie-Grüneisen equations of state are presented for the case of a reflected shock. The single-mode case is explored for incident shock Mach numbers of 1.5 and 2.5. For the planar triple-mode case a single incident Mach number of 2.5 is examined with the initial corrugation wave numbers related by k₁=k₂+k₃. A comparison is drawn to Richtmyer-Meshkov instability in fluids with perfect gas equations of state utilizing matching of a nondimensional pressure jump across the incident shock, the post-shock Atwood ratio, post-shock amplitude-to-wavelength ratio, and time nondimensionalized by the Rcithmyer linear-growth rate time constant prediction. Result comparison demonstrates difference in start-up time and growth rate oscillations. Growth rate oscillation frequency is seen to correlate directly to the expected oscillation frequency of the transmitted and reflected shocks. For the single-mode cases, further comparison is given for vorticity distribution and corrugation centerline shortly after shock interaction that demonstrates only minor differences.</p>
<p>Additionally, examined is single-mode Richtmyer-Meshkov instability when a reflected expansion wave is present for incident Mach numbers of 1.5 and 2.5. Comparison to perfect gas solutions in such cases yields a higher degree of similarity in start-up time and growth rate oscillations. Vorticity distribution and corrugation centerline shortly after shock interaction is also examined. The formation of incipient weak shock waves in the heavy fluid driven by waves emanating from the perturbed transmitted shock is observed when an expansion wave is reflected.</p>
<p>Next, the ghost fluid method [83] is explored for application to impact-driven flows with Mie-Grüneisen equations of state in a vacuum. Free surfaces are defined utilizing a level-set approach. The level-set is reinitialized to the signed distance function periodically by solution to a Hamilton-Jacobi differential equation in artificial time. Flux reconstruction along each Cartesian direction of the domain is performed by subdividing in a way that allows for robust treatment of grid-scale sized voids. Ghost cells in voided regions near the material-vacuum interface are determined from surface-normal Riemann problem solution. The method is then applied to several impact problems of interest. First, a one-dimensional impact problem is examined in Mie-Grüneisen aluminum with simple point erosion used to model separation by spallation under high tension. A similar three-dimensional axisymmetric simulation of two rods impacting is then performed without a model for spallation. Further results for three-dimensional axisymmetric simulation of a sphere hitting a plate are then presented.</p>
<p>Finally, a brief investigation of the assumptions utilized in modeling solids as isotropic fluids is undertaken. An Eulerian solver approach to handling elastic and elastic-plastic solids is utilized for comparison to the simple fluid model assumption. First, in one dimension an impact problem is examined for elastic, elastic-plastic, and fluid equations of state for aluminum. The results demonstrate that in one dimension the fluid models the plastic shock structure of the flow well. Further investigation is made using a three-dimensional axisymmetric simulation of an impact problem involving a copper cylinder surrounded by aluminum. An aluminum slab impact drives a faster shock in the outer aluminum region yielding a Mach reflection in the copper. The results demonstrate similar plastic shock structures. Several differences are also notable that include a lack of roll-up instability at the material interface and slip-line emanating from the Mach stem's triple point.</p>https://thesis.library.caltech.edu/id/eprint/6211SPICE²: A Spatial, Parallel Architecture for Accelerating the Spice Circuit Simulator
https://resolver.caltech.edu/CaltechTHESIS:10262010-082537998
Authors: {'items': [{'email': 'nachiket@uwaterloo.ca', 'id': 'Kapre-Nachiket-Ganesh', 'name': {'family': 'Kapre', 'given': 'Nachiket Ganesh'}, 'orcid': '0000-0002-2187-0406', 'show_email': 'YES'}]}
Year: 2011
DOI: 10.7907/QVZR-VB52
<p>Spatial processing of sparse, irregular floating-point computation using a single FPGA enables up to an order of magnitude speedup (mean 2.8X speedup) over a conventional microprocessor for the SPICE circuit simulator. We deliver this speedup using a hybrid parallel architecture that spatially implements the heterogeneous forms of parallelism available in SPICE. We decompose SPICE into its three constituent phases: Model-Evaluation, Sparse Matrix-Solve, and Iteration Control and parallelize each phase independently. We exploit data-parallel device evaluations in the Model-Evaluation phase, sparse dataflow parallelism in the Sparse Matrix-Solve phase and compose the complete design in streaming fashion. We name our parallel architecture SPICE²: Spatial Processors Interconnected for Concurrent Execution for accelerating the SPICE circuit simulator. We program the parallel architecture with a high-level, domain-specific framework that identifies, exposes and exploits parallelism available in the SPICE circuit simulator. This design is optimized with an auto-tuner that can scale the design to use larger FPGA capacities without expert intervention and can even target other parallel architectures with the assistance of automated code-generation. This FPGA architecture is able to outperform conventional processors due to a combination of factors including high utilization of statically-scheduled resources, low-overhead dataflow scheduling of fine-grained tasks, and overlapped processing of the control algorithms.</p>
<p>We demonstrate that we can independently accelerate Model-Evaluation by a mean factor of 6.5X(1.4--23X) across a range of non-linear device models and Matrix-Solve by 2.4X(0.6--13X) across various benchmark matrices while delivering a mean combined speedup of 2.8X(0.2--11X) for the two together when comparing a Xilinx Virtex-6 LX760 (40nm) with an Intel Core i7 965 (45nm). With our high-level framework, we can also accelerate Single-Precision Model-Evaluation on NVIDIA GPUs, ATI GPUs, IBM Cell, and Sun Niagara 2 architectures.</p>
<p>We expect approaches based on exploiting spatial parallelism to become important as frequency scaling slows down and modern processing architectures turn to parallelism (\eg multi-core, GPUs) due to constraints of power consumption. This thesis shows how to express, exploit and optimize spatial parallelism for an important class of problems that are challenging to parallelize.</p>
https://thesis.library.caltech.edu/id/eprint/6159Lagrangian and Vortex-Surface Fields in Turbulence
https://resolver.caltech.edu/CaltechTHESIS:02212011-233246689
Authors: {'items': [{'email': 'yangyue00@gmail.com', 'id': 'Yang-Yue', 'name': {'family': 'Yang', 'given': 'Yue'}, 'show_email': 'NO'}]}
Year: 2011
DOI: 10.7907/DF3E-G629
<p>In this thesis, we focus on Lagrangian investigations of isotropic turbulence, wall-bounded turbulence and vortex dynamics. In particular, the evolutionary multi-scale geometry of Lagrangian structures is quantified and analyzed. Additionally, we also study the dynamics of vortex-surface fields for some simple viscous flows with both Taylor--Green and Kida--Pelz initial conditions.</p>
<p>First, we study the non-local geometry of finite-sized Lagrangian structures in both stationary, evolving homogenous isotropic turbulence and also with a frozen turbulent velocity field. The multi-scale geometric analysis is applied on the evolution of Lagrangian fields, obtained by a particle-backward-tracking method, to extract Lagrangian structures at different length scales and to characterize their non-local geometry in a space of reduced geometrical parameters. Next, we report a geometric study of both evolving Lagrangian, and also instantaneous Eulerian structures in turbulent channel flow at low and moderate Reynolds numbers. A multi-scale and multi-directional analysis, based on the mirror-extended curvelet transform, is developed to quantify flow structure geometry including the averaged inclination and sweep angles of both classes of turbulent structures at multiple scales ranging from the half-height of the channel to several viscous length scales. Results for turbulent channel flow include the geometry of candidate quasi-streamwise vortices in the near-wall region, the structural evolution of near-wall vortices, and evidence for the existence and geometry of structure packets based on statistical inter-scale correlations.</p>
<p>In order to explore the connection and corresponding representations between Lagrangian kinematics and vortex dynamics, we develop a theoretical formulation and numerical methods for computation of the evolution of a vortex-surface field. Iso-surfaces of the vortex-surface field define vortex surfaces. A systematic methodology is developed for constructing smooth vortex-surface fields for initial Taylor--Green and Kida--Pelz velocity fields by using an optimization approach. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Numerical results on the evolution of vortex-surface fields clarify the continuous vortex dynamics in viscous Taylor--Green and Kida--Pelz flows including the vortex reconnection, rolling-up of vortex tubes, vorticity intensification between anti-parallel vortex tubes, and vortex stretching and twisting. This suggests a possible scenario for explaining the transition from a smooth laminar flow to turbulent flow in terms of topology and geometry of vortex surfaces.</p>
https://thesis.library.caltech.edu/id/eprint/6251High Pressure Hugoniot Measurements in Solids Using Mach Reflections
https://resolver.caltech.edu/CaltechTHESIS:05242011-143955754
Authors: {'items': [{'email': 'jlbrown@caltech.edu', 'id': 'Brown-Justin-Lee', 'name': {'family': 'Brown', 'given': 'Justin Lee'}}]}
Year: 2011
DOI: 10.7907/SC1V-PK42
Shock compression experiments provide access to high pressures in a laboratory setting. Matter at extreme pressures is often studied by utilizing a well controlled planar impact between two flat plates to generate a one dimensional shock wave. While these experiments are a powerful tool in equation of state (EOS) development, they are inherently limited by the velocity of the impacting plate. In an effort to dramatically increase the range of pressures which can be studied with available impact velocities, a new experimental technique is examined. The target plate is replaced by a composite assembly consisting of two concentric cylinders and is designed such that the initial shock velocity in a well characterized outer cylinder is higher than in the inner cylinder material of interest. After impact, conically converging shocks are generated at the interface due to the impedance mismatch between the two materials and the axisymmetric geometry. Upon convergence, an irregular reflection occurs and the conical analog of a Mach reflection develops. This Mach reflection grows until it reaches a steady state, for which an extremely high pressure state is concentrated behind the Mach stem. The reflection is studied using a combination of analytical, numerical, and experimental techniques. Ideas from gas dynamics, such as shock polars, are connected to the classic treatment of one-dimensional shocks in solids to form a simple method for treating the oblique reflections in the Mach lens configuration. Numerical simulations provide detailed full-field solutions and illustrate a methodology for extracting EOS information. The technique is validated experimentally by studying the shock response of copper and iron. Two different confining materials, 6061-T6 aluminum and molybdenum, are used to drive the converging shock waves for which the high pressure state is measured through a combination of velocity interferometry and impedance matching techniques.https://thesis.library.caltech.edu/id/eprint/6427Multiscale Modeling and Computation of 3D Incompressible Turbulent Flows
https://resolver.caltech.edu/CaltechTHESIS:05302012-081356007
Authors: {'items': [{'email': 'lanxin0106@gmail.com', 'id': 'Hu-Xin', 'name': {'family': 'Hu', 'given': 'Xin'}, 'show_email': 'YES'}]}
Year: 2012
DOI: 10.7907/K1RZ-1H07
<p>In the first part, we present a mathematical derivation of a closure relating the Reynolds stress to the mean strain rate for incompressible turbulent flows. This derivation is based on a systematic multiscale analysis that expresses the Reynolds stress in terms of the solutions of local periodic cell problems. We reveal an asymptotic structure of the Reynolds stress by invoking the frame invariant property of the cell problems and an iterative dynamic homogenization of large- and small-scale solutions. The Smagorinsky model for homogeneous turbulence is recovered as an example to illustrate our mathematical derivation. Another example is turbulent channel flow, where we derive a simplified turbulence model based on the asymptotic flow structure near the wall. Additionally, we obtain a nonlinear model by using a second order approximation of the inverse flow map function. This nonlinear model captures the effects of the backscatter of kinetic energy and dispersion and is consistent with other models, such as a mixed model that combines the Smagorinsky and gradient models, and the generic nonlinear model of Lund and Novikov.</p>
<p>Numerical simulation results at two Reynolds numbers using our simplified turbulence model are in good agreement with both experiments and direct numerical simulations in turbulent channel flow. However, due to experimental and modeling errors, we do observe some noticeable differences, e.g. , root mean square velocity fluctuations at Re<sub>τ</sub> = 180.</p>
<p>In the second part, we present a new perspective on calculating fully developed turbulent flows using a data-driven stochastic method. General polynomial chaos (gPC) bases are obtained based on the mean velocity profile of turbulent channel flow in the offline part. The velocity fields are projected onto the subspace spanned by these gPC bases and a coupled system of equations is solved to compute the velocity components in the Karhunen-Loeve expansion in the online part. Our numerical results have shown that the data-driven stochastic method for fully developed turbulence offers decent approximations of statistical quantities with a coarse grid and a relatively small number of gPC base elements.</p>https://thesis.library.caltech.edu/id/eprint/7093High-Order Integral Equation Methods for Diffraction Problems Involving Screens and Apertures
https://resolver.caltech.edu/CaltechTHESIS:06072012-004925615
Authors: {'items': [{'email': 'lintner@caltech.edu', 'id': 'Lintner-Stéphane-Karl', 'name': {'family': 'Lintner', 'given': 'Stéphane Karl'}, 'show_email': 'NO'}]}
Year: 2012
DOI: 10.7907/VP8P-DP74
This thesis presents a novel approach for the numerical solution of problems of diffraction by infinitely thin screens and apertures. The new methodology relies on combination of weighted versions of the classical operators associated with the Dirichlet and Neumann open-surface problems. In the two-dimensional case, a rigorous proof is presented, establishing that the new weighted formulations give rise to second-kind Fredholm integral equations, thus providing a generalization to open surfaces of the classical closed-surface Calderon formulae. High-order quadrature rules are introduced for the new weighted operators, both in the two-dimensional case as well as the scalar three-dimensional case. Used in conjunction with Krylov subspace iterative methods, these rules give rise to efficient and accurate numerical solvers which produce highly accurate solutions in small numbers of iterations, and whose performance is comparable to that arising from efficient high-order integral solvers recently introduced for closed-surface problems. Numerical results are presented for a wide range of frequencies and a variety of geometries in two- and three-dimensional space, including complex resonating structures as well as, for the first time, accurate numerical solutions of classical diffraction problems considered by the 19th-century pioneers: diffraction of high-frequency waves by the infinitely thin disc, the circular aperture, and the two-hole geometry inherent in Young's experiment.
https://thesis.library.caltech.edu/id/eprint/7143Simulations of Compressible, Diffusive, Reactive Flows with Detailed Chemistry Using a High-Order Hybrid WENO-CD Scheme
https://resolver.caltech.edu/CaltechTHESIS:12302011-185742249
Authors: {'items': [{'email': 'jackalak@gmail.com', 'id': 'Ziegler-John-Lewis-Jack', 'name': {'family': 'Ziegler', 'given': 'John Lewis (Jack)'}, 'orcid': '0000-0001-6127-5567', 'show_email': 'YES'}]}
Year: 2012
DOI: 10.7907/ZKW8-ES97
<p>A hybrid weighted essentially non-oscillatory (WENO)/centered-difference (CD) numerical method, with low numerical dissipation, high-order shock-capturing, and structured adaptive mesh refinement (SAMR), has been developed for the direct numerical simulation (DNS) of the multicomponent, compressive, reactive Navier-Stokes equations. The method enables accurate resolution of diffusive processes within reaction zones. This numerical method is verified with a series of one- and two-dimensional test problems, including a convergence test of a two-dimensional unsteady reactive double Mach reflection problem. Validation of the method is conducted with experimental comparisons of three applications all of which model multi-dimensional, unsteady reactive flow: an irregular propane detonation, shock and detonation bifurcations, and spark ignition deflagrations.</p>
<p>The numerical approach combines time-split reactive source terms with a high-order, shock-capturing scheme specifically designed for diffusive flows. A description of the order-optimized, symmetric, finite difference, flux-based, hybrid WENO / centered-difference scheme is given, along with its implementation in a high-order SAMR framework. The implementation of new techniques for discontinuity flagging, scheme-switching, and high-order prolongation and restriction is described. In particular, the refined methodology does not require upwinded WENO at grid refinement interfaces for stability, allowing high-order prolongation and thereby eliminating a significant source of numerical diffusion within the overall code performance.</p>
<p>A minimally reduced irregular detonation mixture mechanism (22 species and 53 reversible reactions) is developed and combined with the WENO-CD numerical method to accurately model two-dimensional hydrocarbon (propane) detonations with detailed chemistry and transport. First of its kind, resolved double Mach reflection (DMR) detonation simulations with a large hyrdocarbon mixture are presented. Detailed discussions and comparisons of the influence of grid resolution, lower-order numerical methods, and inviscid approximations are made in addition to the detailed presentation of fluid dynamics found in an unsteady, highly unstable, reactive DMR simulation. Also conducted are direct experimental comparisons to soot foils and schlieren images with an unresolved large-scale propane detonation channel simulation.</p>
<p>The numerical method is also applied to the DNS of two other problems, detonation/shock bifurcations and spark ignited deflagrations. Through the resolution of viscous/diffusive scales, new insights into how a bifurcated foot develops after a detonation end wall reflection, and how geometry can influence the development of a flame kernel after spark ignition are found.</p>
https://thesis.library.caltech.edu/id/eprint/6759Planar Reflection of Gaseous Detonation
https://resolver.caltech.edu/CaltechTHESIS:06112013-153305610
Authors: {'items': [{'email': 'jason.damazo@gmail.com', 'id': 'Damazo-Jason-Scott', 'name': {'family': 'Damazo', 'given': 'Jason Scott'}, 'orcid': '0000-0002-4155-7177', 'show_email': 'NO'}]}
Year: 2013
DOI: 10.7907/4QW7-TK55
<p>Pipes containing flammable gaseous mixtures may be subjected to internal detonation. When the detonation normally impinges on a closed end, a reflected shock wave is created to bring the flow back to rest. This study built on the work of Karnesky (2010) and examined deformation of thin-walled stainless steel tubes subjected to internal reflected gaseous detonations. A ripple pattern was observed in the tube wall for certain fill pressures, and a criterion was developed that predicted when the ripple pattern would form. A two-dimensional finite element analysis was performed using Johnson-Cook material properties; the pressure loading created by reflected gaseous detonations was accounted for with a previously developed pressure model. The residual plastic strain between experiments and computations was in good agreement.</p>
<p>During the examination of detonation-driven deformation, discrepancies were discovered in our understanding of reflected gaseous detonation behavior. Previous models did not accurately describe the nature of the reflected shock wave, which motivated further experiments in a detonation tube with optical access. Pressure sensors and schlieren images were used to examine reflected shock behavior, and it was determined that the discrepancies were related to the reaction zone thickness extant behind the detonation front. During these experiments reflected shock bifurcation did not appear to occur, but the unfocused visualization system made certainty impossible. This prompted construction of a focused schlieren system that investigated possible shock wave-boundary layer interaction, and heat-flux gauges analyzed the boundary layer behind the detonation front. Using these data with an analytical boundary layer solution, it was determined that the strong thermal boundary layer present behind the detonation front inhibits the development of reflected shock wave bifurcation.</p>https://thesis.library.caltech.edu/id/eprint/7890Models of Turbulent Pipe Flow
https://resolver.caltech.edu/CaltechTHESIS:11272012-130849053
Authors: {'items': [{'email': 'jeanloup.bourguignon@gmail.com', 'id': 'Bourguignon-Jean-Loup', 'name': {'family': 'Bourguignon', 'given': 'Jean-Loup'}, 'show_email': 'NO'}]}
Year: 2013
DOI: 10.7907/BPEZ-VM28
<p>The physics of turbulent pipe flow was investigated via the use of two models based on simplified versions of the Navier-Stokes equations. The first model was a streamwise-constant projection of these equations, and was used to study the change in mean flow that occurs during transition to turbulence. The second model was based on the analysis of the turbulent pipe flow resolvent, and provided a radial basis for the modal decomposition of turbulent pipe flow. The two models were tested numerically and validated against experimental and numerical data.</p>
<p>Analysis of the streamwise-constant model showed that both non-normal and nonlinear effects are required to capture the blunting of the velocity profile, which occurs during pipe flow transition. The model generated flow fields characterized by the presence of high- and low-speed streaks, whose distribution over the cross-section of the pipe was remarkably similar to the one observed in the velocity field near the trailing edge of the puff structures present in pipe flow transition.</p>
<p>A modal decomposition of turbulent pipe flow, in the three spatial directions and in time, was performed, and made possible by the significant reduction in data requirements achieved via the use of compressive sampling and model-based radial basis functions. The application and efficiency of compressive sampling in wall-bounded turbulence was demonstrated.</p>
<p>Approximately sparse representations of turbulent pipe flow by propagating waves with model-based radial basis functions, were derived. The basis functions, obtained by singular value decomposition of the resolvent, captured the wall-normal coherence of the flow; and provided a link between the propagating waves and the governing equations, allowing for the identification of the dominant mechanims sustaining the waves, as a function of their streamwise wavenumber.</p>
<p>Analysis of the resolvent showed that the long streamwise waves are amplified mainly via non-normality effects, and are also constrained to be tall in the wall-normal direction, which decreases the influence of viscous dissipation. The short streamwise waves were shown to be localized near the critical-layer (defined as the wall-normal location where the convection velocity of the wave equals the local mean velocity), and thus exhibit amplification with a large contribution from criticality. The work in this thesis allows the reconciliation of the well-known results concerning optimal disturbance amplification due to non-normal effects with recent resolvent analyses, which highlighted the importance of criticality effects.</p>
https://thesis.library.caltech.edu/id/eprint/7287Current Transport and Onset-Related Phenomena in an MPD Thruster Modified by Applied Magnetic Fields
https://resolver.caltech.edu/CaltechTHESIS:01252013-171305685
Authors: {'items': [{'email': 'robertcmoeller@gmail.com', 'id': 'Moeller-Robert-Carlos', 'name': {'family': 'Moeller', 'given': 'Robert Carlos'}, 'show_email': 'NO'}]}
Year: 2013
DOI: 10.7907/39VT-GB93
<p>This work investigated the effects of tailored, externally-applied magnetic fields on current transport and near-anode processes in the plasma discharge of a magnetoplasmadynamic thruster (MPDT). Electrical and plasma diagnostics were used to determine whether applied magnetic fields could mitigate the effects of the "onset" phenomena, including large-amplitude terminal voltage fluctuations and high anode fall voltages associated with unstable operation and anode erosion. A new MPDT was developed and operated with quasi-steady 1 ms pulses from 36 kW to 3.3 MW with argon propellant. Three magnetic configurations studied included self-field operation (without external electromagnets) and two applied poloidal magnetic fields. One configuration used magnetic field lines tangential to the anode lip (and intersecting the anode further upstream) and the other created a magnetic cusp intersecting the anode downstream.</p>
<p>The influence of the applied fields on the discharge current streamlines, current densities, and key plasma properties (electron temperature, number density, and plasma potential) was studied. Key findings included that the current pattern and current densities redistributed to follow the applied magnetic field lines. Also, the anode fall voltage was substantially reduced with both applied fields over a large range of currents (and eliminated at 8 kA). These results occurred because applied magnetic field lines intersecting the anode provided a high conductivity path and reduced the local electric field required to sustain the radial current densities. The applied fields reduced the amplitude and frequency of the terminal voltage fluctuations (up to 49%) over a broad range of currents and also decreased transients in the ion saturation current, which suggest reduction of current filamentation and surface-eroding anode spots. Additionally, the cusp field reduced mean terminal voltages over the entire range of discharge currents (up to 31%), and the tangential field lowered terminal voltages below 10.7 kA. These significant reductions in onset-related behaviors should lead to improved thruster lifetime and increased efficiency. These results suggest a distinctive and more effective approach to influencing the near-anode phenomena and mitigating the effects of onset with appropriately designed applied magnetic fields that differ from those used in the vast majority of conventional, so-called "applied-field MPD thrusters."</p>https://thesis.library.caltech.edu/id/eprint/7442Adaptive Methods Exploring Intrinsic Sparse Structures of Stochastic Partial Differential Equations
https://resolver.caltech.edu/CaltechTHESIS:09182012-175436855
Authors: {'items': [{'email': 'mulin.cheng@gmail.com', 'id': 'Cheng-Mulin', 'name': {'family': 'Cheng', 'given': 'Mulin'}, 'show_email': 'YES'}]}
Year: 2013
DOI: 10.7907/V638-V403
Many physical and engineering problems involving uncertainty enjoy certain low-dimensional structures, e.g., in the sense of Karhunen-Loeve expansions (KLEs), which in turn indicate the existence of reduced-order models and better formulations for efficient numerical simulations. In this thesis, we target a class of time-dependent stochastic partial differential equations whose solutions enjoy such structures at any time and propose a new methodology (DyBO) to derive equivalent systems whose solutions closely follow KL expansions of the original stochastic solutions. KL expansions are known to be the most compact representations of stochastic processes in an L<sup>2</sup> sense. Our methods explore such sparsity and offer great computational benefits compared to other popular generic methods, such as traditional Monte Carlo (MC), generalized Polynomial Chaos (gPC) method, and generalized Stochastic Collocation (gSC) method. Such benefits are demonstrated through various numerical examples ranging from spatially one-dimensional examples, such as stochastic Burgers' equations and stochastic transport equations to spatially two-dimensional examples, such as stochastic flows in 2D unit square. Parallelization is also discussed, aiming toward future industrial-scale applications. In addition to numerical examples, theoretical aspects of DyBO are also carefully analyzed, such as preservation of bi-orthogonality, error propagation, and computational complexity. Based on theoretical analysis, strategies are proposed to overcome difficulties in numerical implementations, such as eigenvalue crossing and adaptively adding or removing mode pairs. The effectiveness of the proposed strategies is numerically verified. Generalization to a system of SPDEs is considered as well in the thesis, and its success is demonstrated by applications to stochastic Boussinesq convection problems. Other generalizations, such as generalized stochastic collocation formulation of DyBO method, are also discussed. https://thesis.library.caltech.edu/id/eprint/7207Simulation of Richtmyer-Meshkov Flows for Elastic-Plastic Solids in Planar and Converging Geometries Using an Eulerian Framework
https://resolver.caltech.edu/CaltechTHESIS:02202013-185004693
Authors: {'items': [{'email': 'alejandro_lopez_ortega@hotmail.com', 'id': 'Lopez-Ortega-Alejandro', 'name': {'family': 'Lopez Ortega', 'given': 'Alejandro'}, 'show_email': 'YES'}]}
Year: 2013
DOI: 10.7907/4WJ6-D795
This thesis presents a numerical and analytical study of two problems of interest involving shock waves propagating through elastic-plastic media: the motion of converging (imploding) shocks and the Richtmyer-Meshkov (RM) instability. Since the stress conditions encountered in these cases normally produce large deformations in the materials, an Eulerian description, in which the spatial coordinates are fixed, is employed. This formulation enables a direct comparison of similarities and differences between the present study of phenomena driven by shock-loading in elastic-plastic solids, and in fluids, where they have been studied extensively. In the first application, Whitham's shock dynamics (WSD) theory is employed for obtaining an approximate description of the motion of an elastic-plastic material processed by a cylindrically/spherically converging shock. Comparison with numerical simulations of the full set of equations of motion reveal that WSD is an accurate tool for characterizing the evolution of converging shocks at all stages. The study of the Richtmyer-Meshkov flow (i.e., interaction between the interface separating two materials of different density and a shock wave incoming at an angle) in solids is performed by means of analytical models for purely elastic solids and numerical simulations when plasticity is included in the material model. To this effect, an updated version of a previously developed multi-material, level-set-based, Eulerian framework for solid mechanics is employed. The revised code includes the use of a multi-material HLLD Riemann problem for imposing material boundary conditions, and a new formulation of the equations of motion that makes use of the stretch tensor while avoiding the degeneracy of the stress tensor under rotation. Results reveal that the interface separating two elastic solids always behaves in a stable oscillatory or decaying oscillatory manner due to the existence of shear waves, which are able to transport the initial vorticity away from the interface. In the case of elastic-plastic materials, the interface behaves at first in an unstable manner similar to a fluid. Ejecta formation is appreciated under certain initial conditions while in other conditions, after an initial period of growth, the interface displays a quasi-stationary long-term behavior due to stress relaxation. The effect of secondary shock-interface interactions (re-shocks) in converging geometries is also studied. A turbulent mixing zone, similar to what is observed in gas--gas interfaces, is created, especially when materials with low strength driven by moderate to strong shocks are considered.https://thesis.library.caltech.edu/id/eprint/7488GRAph Parallel Actor Language: A Programming Language for Parallel Graph Algorithms
https://resolver.caltech.edu/CaltechTHESIS:08192012-145253489
Authors: {'items': [{'email': 'michael@delorimier.org', 'id': 'DeLorimier-Michael-John', 'name': {'family': 'DeLorimier', 'given': 'Michael John'}, 'show_email': 'YES'}]}
Year: 2013
DOI: 10.7907/M3TW-7Y53
We introduce a domain-specific language, GRAph Parallel Actor Language, that enables parallel graph algorithms to be written in a natural, high-level form. GRAPAL is based on our GraphStep compute model, which enables a wide range of parallel graph algorithms that are high-level, deterministic, free from race conditions, and free from deadlock. Programs written in GRAPAL are easy for a compiler and runtime to map to efficient parallel field programmable gate array (FPGA) implementations. We show that the GRAPAL compiler can verify that the structure of operations conforms to the GraphStep model. We allocate many small processing elements in each FPGA that take advantage of the high on-chip memory bandwidth (5x the sequential processor) and process one graph edge per clock cycle per processing element. We show how to automatically choose parameters for the logic architecture so the high-level GRAPAL programming model is independent of the target FPGA architecture. We compare our GRAPAL applications mapped to a platform with four 65 nm Virtex-5 SX95T FPGAs to sequential programs run on a single 65 nm Xeon 5160. Our implementation achieves a total mean speedup of 8x with a maximum speedup of 28x. The speedup per chip is 2x with a maximum of 7x. The ratio of energy used by our GRAPAL implementation over the sequential implementation has a mean of 1/10 with a minimum of 1/80.https://thesis.library.caltech.edu/id/eprint/7188Progress in Numerical Modeling of Non-Premixed Combustion
https://resolver.caltech.edu/CaltechTHESIS:05292014-112456783
Authors: {'items': [{'email': 'yxuan.caltech@gmail.com', 'id': 'Xuan-Yuan', 'name': {'family': 'Xuan', 'given': 'Yuan'}, 'orcid': '0000-0001-9326-2197', 'show_email': 'NO'}]}
Year: 2014
DOI: 10.7907/ZED4-A872
<p>Progress is made on the numerical modeling of both laminar and turbulent non-premixed flames. Instead of solving the transport equations for the numerous species involved in the combustion process, the present study proposes reduced-order combustion models based on local flame structures.</p>
<p>For laminar non-premixed flames, curvature and multi-dimensional diffusion effects are found critical for the accurate prediction of sooting tendencies. A new numerical model based on modified flamelet equations is proposed. Sooting tendencies are calculated numerically using the proposed model for a wide range of species. These first numerically-computed sooting tendencies are in good agreement with experimental data. To further quantify curvature and multi-dimensional effects, a general flamelet formulation is derived mathematically. A budget analysis of the general flamelet equations is performed on an axisymmetric laminar diffusion flame. A new chemistry tabulation method based on the general flamelet formulation is proposed. This new tabulation method is applied to the same flame and demonstrates significant improvement compared to previous techniques.</p>
<p>For turbulent non-premixed flames, a new model to account for chemistry-turbulence interactions is proposed. %It is found that these interactions are not important for radicals and small species, but substantial for aromatic species. The validity of various existing flamelet-based chemistry tabulation methods is examined, and a new linear relaxation model is proposed for aromatic species. The proposed relaxation model is validated against full chemistry calculations. To further quantify the importance of aromatic chemistry-turbulence interactions, Large-Eddy Simulations (LES) have been performed on a turbulent sooting jet flame. %The aforementioned relaxation model is used to provide closure for the chemical source terms of transported aromatic species. The effects of turbulent unsteadiness on soot are highlighted by comparing the LES results with a separate LES using fully-tabulated chemistry. It is shown that turbulent unsteady effects are of critical importance for the accurate prediction of not only the inception locations, but also the magnitude and fluctuations of soot.</p>https://thesis.library.caltech.edu/id/eprint/8421Techniques for Strength Measurement at High Pressures and Strain-Rates using Transverse Waves
https://resolver.caltech.edu/CaltechTHESIS:01152014-115401299
Authors: {'items': [{'email': 'vstolyarfirst@gmail.com', 'id': 'Richmond-Victoria-Stolyar', 'name': {'family': 'Richmond', 'given': 'Victoria Stolyar'}, 'show_email': 'NO'}]}
Year: 2014
DOI: 10.7907/SH60-5659
<p>The study of the strength of a material is relevant to a variety of applications including automobile collisions, armor penetration and inertial confinement fusion. Although dynamic behavior of materials at high pressures and strain-rates has been studied extensively using plate impact experiments, the results provide measurements in one direction only. Material behavior that is dependent on strength is unaccounted for. The research in this study proposes two novel configurations to mitigate this problem.</p>
<p>The first configuration introduced is the oblique wedge experiment, which is comprised of a driver material, an angled target of interest and a backing material used to measure in-situ velocities. Upon impact, a shock wave is generated in the driver material. As the shock encounters the angled target, it is reflected back into the driver and transmitted into the target. Due to the angle of obliquity of the incident wave, a transverse wave is generated that allows the target to be subjected to shear while being compressed by the initial longitudinal shock such that the material does not slip. Using numerical simulations, this study shows that a variety of oblique wedge configurations can be used to study the shear response of materials and this can be extended to strength measurement as well. Experiments were performed on an oblique wedge setup with a copper impactor, polymethylmethacrylate driver, aluminum 6061-t6 target, and a lithium fluoride window. Particle velocities were measured using laser interferometry and results agree well with the simulations.</p>
<p>The second novel configuration is the y-cut quartz sandwich design, which uses the anisotropic properties of y-cut quartz to generate a shear wave that is transmitted into a thin sample. By using an anvil material to back the thin sample, particle velocities measured at the rear surface of the backing plate can be implemented to calculate the shear stress in the material and subsequently the strength. Numerical simulations were conducted to show that this configuration has the ability to measure the strength for a variety of materials.</p>
https://thesis.library.caltech.edu/id/eprint/8051Large-Eddy Simulations of Fully Developed Turbulent Channel and Pipe Flows with Smooth and Rough Walls
https://resolver.caltech.edu/CaltechTHESIS:02142014-112419793
Authors: {'items': [{'email': 'namiko12@icloud.com', 'id': 'Saito-Namiko', 'name': {'family': 'Saito', 'given': 'Namiko'}, 'show_email': 'YES'}]}
Year: 2014
DOI: 10.7907/WKNJ-ET18
Studies in turbulence often focus on two flow conditions, both of which occur frequently in real-world flows and are sought-after for their value in advancing turbulence theory. These are the high Reynolds number regime and the effect of wall surface roughness. In this dissertation, a Large-Eddy Simulation (LES) recreates both conditions over a wide range of Reynolds numbers Re<sub>τ</sub> = O(10<sup>2</sup>)-O(10<sup>8</sup>) and accounts for roughness by locally modeling the statistical effects of near-wall anisotropic fine scales in a thin layer immediately above the rough surface. A subgrid, roughness-corrected wall model is introduced to dynamically transmit this modeled information from the wall to the outer LES, which uses a stretched-vortex subgrid-scale model operating in the bulk of the flow. Of primary interest is the Reynolds number and roughness dependence of these flows in terms of first and second order statistics. The LES is first applied to a fully turbulent uniformly-smooth/rough channel flow to capture the flow dynamics over smooth, transitionally rough and fully rough regimes. Results include a Moody-like diagram for the wall averaged friction factor, believed to be the first of its kind obtained from LES. Confirmation is found for experimentally observed logarithmic behavior in the normalized stream-wise turbulent intensities. Tight logarithmic collapse, scaled on the wall friction velocity, is found for smooth-wall flows when Re<sub>τ</sub> ≥ O(10<sup>6</sup>) and in fully rough cases. Since the wall model operates locally and dynamically, the framework is used to investigate non-uniform roughness distribution cases in a channel, where the flow adjustments to sudden surface changes are investigated. Recovery of mean quantities and turbulent statistics after transitions are discussed qualitatively and quantitatively at various roughness and Reynolds number levels. The internal boundary layer, which is defined as the border between the flow affected by the new surface condition and the unaffected part, is computed, and a collapse of the profiles on a length scale containing the logarithm of friction Reynolds number is presented. Finally, we turn to the possibility of expanding the present framework to accommodate more general geometries. As a first step, the whole LES framework is modified for use in the curvilinear geometry of a fully-developed turbulent pipe flow, with implementation carried out in a spectral element solver capable of handling complex wall profiles. The friction factors have shown favorable agreement with the superpipe data, and the LES estimates of the Karman constant and additive constant of the log-law closely match values obtained from experiment.https://thesis.library.caltech.edu/id/eprint/8077A New High-Order Fourier Continuation-Based Elasticity Solver for Complex Three-Dimensional Geometries
https://resolver.caltech.edu/CaltechTHESIS:10082013-093825165
Authors: {'items': [{'email': 'fpamlani@outlook.com', 'id': 'Amlani-Faisal', 'name': {'family': 'Amlani', 'given': 'Faisal'}, 'show_email': 'YES'}]}
Year: 2014
DOI: 10.7907/V9DQ-P103
This thesis presents a new approach for the numerical solution of three-dimensional problems in elastodynamics. The new methodology, which is based on a recently introduced Fourier continuation (FC) algorithm for the solution of Partial Differential Equations on the basis of accurate Fourier expansions of possibly non-periodic functions, enables fast, high-order solutions of the time-dependent elastic wave equation in a nearly dispersionless manner, and it requires use of CFL constraints that scale only linearly with spatial discretizations. A new FC operator is introduced to treat Neumann and traction boundary conditions, and a block-decomposed (sub-patch) overset strategy is presented for implementation of general, complex geometries in distributed-memory parallel computing environments. Our treatment of the elastic wave equation, which is formulated as a complex system of variable-coefficient PDEs that includes possibly heterogeneous and spatially varying material constants, represents the first fully-realized three-dimensional extension of FC-based solvers to date. Challenges for three-dimensional elastodynamics simulations such as treatment of corners and edges in three-dimensional geometries, the existence of variable coefficients arising from physical configurations and/or use of curvilinear coordinate systems and treatment of boundary conditions, are all addressed. The broad applicability of our new FC elasticity solver is demonstrated through application to realistic problems concerning seismic wave motion on three-dimensional topographies as well as applications to non-destructive evaluation where, for the first time, we present three-dimensional simulations for comparison to experimental studies of guided-wave scattering by through-thickness holes in thin plates.https://thesis.library.caltech.edu/id/eprint/7974Storm Track Response to Perturbations in Climate
https://resolver.caltech.edu/CaltechTHESIS:05112015-075223217
Authors: {'items': [{'email': 'C.Mbengue@wolfson.oxon.org', 'id': 'Mbengue-Cheikh-Oumar', 'name': {'family': 'Mbengue', 'given': 'Cheikh Oumar'}, 'show_email': 'YES'}]}
Year: 2015
DOI: 10.7907/Z9FT8J05
<p>This thesis advances our understanding of midlatitude storm tracks and how they respond to perturbations in the climate system. The midlatitude storm tracks are regions of maximal turbulent kinetic energy in the atmosphere. Through them, the bulk of the atmospheric transport of energy, water vapor, and angular momentum occurs in midlatitudes. Therefore, they are important regulators of climate, controlling basic features such as the distribution of surface temperatures, precipitation, and winds in midlatitudes. Storm tracks are robustly projected to shift poleward in global-warming simulations with current climate models. Yet the reasons for this shift have remained unclear. Here we show that this shift occurs even in extremely idealized (but still three-dimensional) simulations of dry atmospheres. We use these simulations to develop an understanding of the processes responsible for the shift and develop a conceptual model that accounts for it.</p>
<p>We demonstrate that changes in the convective static stability in the deep tropics alone can drive remote shifts in the midlatitude storm tracks. Through simulations with a dry idealized general circulation model (GCM), midlatitude storm tracks are shown to be located where the mean available potential energy (MAPE, a measure of the potential energy available to be converted into kinetic energy) is maximal. As the climate varies, even if only driven by tropical static stability changes, the MAPE maximum shifts primarily because of shifts of the maximum of near-surface meridional temperature gradients. The temperature gradients shift in response to changes in the width of the tropical Hadley circulation, whose width is affected by the tropical static stability. Storm tracks generally shift in tandem with shifts of the subtropical terminus of the Hadley circulation.</p>
<p>We develop a one-dimensional diffusive energy-balance model that links changes in the Hadley circulation to midlatitude temperature gradients and so to the storm tracks. It is the first conceptual model to incorporate a dynamical coupling between the tropical Hadley circulation and midlatitude turbulent energy transport. Numerical and analytical solutions of the model elucidate the circumstances of when and how the storm tracks shift in tandem with the terminus of the Hadley circulation. They illustrate how an increase of only the convective static stability in the deep tropics can lead to an expansion of the Hadley circulation and a poleward shift of storm tracks.</p>
<p>The simulations with the idealized GCM and the conceptual energy-balance model demonstrate a clear link between Hadley circulation dynamics and midlatitude storm track position. With the help of the hierarchy of models presented in this thesis, we obtain a closed theory of storm track shifts in dry climates. The relevance of this theory for more realistic moist climates is discussed.</p>https://thesis.library.caltech.edu/id/eprint/8854Stability of Hypervelocity Boundary Layers
https://resolver.caltech.edu/CaltechTHESIS:06052015-111128842
Authors: {'items': [{'email': 'bitternp@gmail.com', 'id': 'Bitter-Neal-Phillip', 'name': {'family': 'Bitter', 'given': 'Neal Phillip'}, 'show_email': 'NO'}]}
Year: 2015
DOI: 10.7907/Z9Q23X5Z
<p>The early stage of laminar-turbulent transition in a hypervelocity boundary layer is studied using a combination of modal linear stability analysis, transient growth analysis, and direct numerical simulation. Modal stability analysis is used to clarify the behavior of first and second mode instabilities on flat plates and sharp cones for a wide range of high enthalpy flow conditions relevant to experiments in impulse facilities. Vibrational nonequilibrium is included in this analysis, its influence on the stability properties is investigated, and simple models for predicting when it is important are described.</p>
<p>Transient growth analysis is used to determine the optimal initial conditions that lead to the largest possible energy amplification within the flow. Such analysis is performed for both spatially and temporally evolving disturbances. The analysis again targets flows that have large stagnation enthalpy, such as those found in shock tunnels, expansion tubes, and atmospheric flight at high Mach numbers, and clarifies the effects of Mach number and wall temperature on the amplification achieved. Direct comparisons between modal and non-modal growth are made to determine the relative importance of these mechanisms under different flow regimes. </p>
<p>Conventional stability analysis employs the assumption that disturbances evolve with either a fixed frequency (spatial analysis) or a fixed wavenumber (temporal analysis). Direct numerical simulations are employed to relax these assumptions and investigate the downstream propagation of wave packets that are localized in space and time, and hence contain a distribution of frequencies and wavenumbers. Such wave packets are commonly observed in experiments and hence their amplification is highly relevant to boundary layer transition prediction. It is demonstrated that such localized wave packets experience much less growth than is predicted by spatial stability analysis, and therefore it is essential that the bandwidth of localized noise sources that excite the instability be taken into account in making transition estimates. A simple model based on linear stability theory is also developed which yields comparable results with an enormous reduction in computational expense. This enables the amplification of finite-width wave packets to be taken into account in transition prediction. </p>https://thesis.library.caltech.edu/id/eprint/8995Fast Lattice Green's Function Methods for Viscous Incompressible Flows on Unbounded Domains
https://resolver.caltech.edu/CaltechTHESIS:04062016-223108239
Authors: {'items': [{'email': 'sebastian.liska@gmail.com', 'id': 'Liska-Sebastian', 'name': {'family': 'Liska', 'given': 'Sebastian'}, 'orcid': '0000-0003-4139-9364', 'show_email': 'NO'}]}
Year: 2016
DOI: 10.7907/Z9ZC80TG
In this thesis, a collection of novel numerical techniques culminating in a fast, parallel method for the direct numerical simulation of incompressible viscous flows around surfaces immersed in unbounded fluid domains is presented. At the core of all these techniques is the use of the fundamental solutions, or lattice Green’s functions, of discrete operators to solve inhomogeneous elliptic difference equations arising in the discretization of the three-dimensional incompressible Navier-Stokes equations on unbounded regular grids. In addition to automatically enforcing the natural free-space boundary conditions, these new lattice Green’s function techniques facilitate the implementation of robust staggered-Cartesian-grid flow solvers with efficient nodal distributions and fast multipole methods. The provable conservation and stability properties of the appropriately combined discretization and solution techniques ensure robust numerical solutions. Numerical experiments on thin vortex rings, low-aspect-ratio flat plates, and spheres are used verify the accuracy, physical fidelity, and computational efficiency of the present formulations.https://thesis.library.caltech.edu/id/eprint/9658Windowed Integral Equation Methods for Problems of Scattering by Defects and Obstacles in Layered Media
https://resolver.caltech.edu/CaltechTHESIS:08182016-124629380
Authors: {'items': [{'email': 'caperezar@gmail.com', 'id': 'Perez-Arancibia-Carlos-Andrés', 'name': {'family': 'Pérez Arancibia', 'given': 'Carlos Andrés'}, 'orcid': '0000-0003-1647-4019', 'show_email': 'YES'}]}
Year: 2017
DOI: 10.7907/Z9GQ6VQT
<p>This thesis concerns development of efficient high-order boundary integral equation methods for the numerical solution of problems of acoustic and electromagnetic scattering in the presence of planar layered media in two and three spatial dimensions. The interest in such problems arises from application areas that benefit from accurate numerical modeling of the layered media scattering phenomena, such as electronics, near-field optics, plasmonics and photonics as well as communications, radar and remote sensing.</p>
<p>A number of efficient algorithms applicable to various problems in these areas are pre- sented in this thesis, including (i) A Sommerfeld integral based high-order integral equation method for problems of scattering by defects in presence of infinite ground and other layered media, (ii) Studies of resonances and near resonances and their impact on the absorptive properties of rough surfaces, and (iii) A novel <i>Window Green Function Method</i> (WGF) for problems of scattering by obstacles and defects in the presence of layered media. The WGF approach makes it possible to completely avoid use of expensive Sommerfeld integrals that are typically utilized in layer-media simulations. In fact, the methods and studies referred in points (i) and (ii) above motivated the development of the markedly more efficient WGF alternative.</p>https://thesis.library.caltech.edu/id/eprint/9902Spatial Profiles in the Singular Solutions of the 3D Euler Equations and Simplified Models
https://resolver.caltech.edu/CaltechTHESIS:09092016-000915850
Authors: {'items': [{'email': 'pengfeiliuc@gmail.com', 'id': 'Liu-Pengfei', 'name': {'family': 'Liu', 'given': 'Pengfei'}, 'orcid': '0000-0002-6714-7387', 'show_email': 'YES'}]}
Year: 2017
DOI: 10.7907/Z9V9862G
<p>The partial differential equations (PDE) governing the motions of incompressible ideal fluid in three dimensional (3D) space are among the most fundamental nonlinear PDEs in nature and have found a lot of important applications. Due to the presence of super-critical non-linearity, the fundamental question of global well-posedness still remains open and is generally viewed as one of the most outstanding open questions in mathematics. In this thesis, we investigate the potential finite-time singularity formation of the 3D Euler equations and simplified models by studying the self-similar spatial profiles in the potentially singular solutions.</p>
<p>In the first part, we study the self-similar singularity of two 1D models, the CKY model and the HL model, which approximate the dynamics of the 3D axisymmtric Euler equations on the solid boundary of a cylindrical domain. The two models are both numerically observed to develop self-similar singularity. We prove the existence of a discrete family of self-similar profiles for the CKY model, using a combination of analysis and computer-aided verification. Then we employ a dynamic rescaling formulation to numerically study the evolution of the spatial profiles for the two 1D models, and demonstrate the stability of the self-similar singularity. We also study a singularity scenario for the HL model with multi-scale feature.</p>
<p>In the second part, we study the self-similar singularity for the 3D axisymmetric Euler equations. We first prove the local existence of a family of analytic self-similar profiles using a modified Cauchy-Kowalevski majorization argument. Then we use the dynamic rescaling formulation to investigate two types of initial data with different leading order properties. The first initial data correspond to the singularity scenario reported by Luo and Hou. We demonstrate that the self-similar profiles enjoy certain stability, which confirms the finite-time singularity reported by Luo and Hou. For the second initial data, we show that the solutions develop singularity in a different manner from the first case, which is unknown previously. The spatial profiles in the solutions become singular themselves, which means that the solutions to the Euler equations develop singularity at multiple spatial scales.</p>
<p>In the third part, we propose a family of 3D models for the 3D axisymmetric Euler and Navier-Stokes equations by modifying the amplitude of the convection terms. The family of models share several regularity results with the original Euler and Navier-Stokes equations, and we study the potential finite-time singularity of the models numerically. We show that for small convection, the solutions of the inviscid model develop self-similar singularity and the profiles behave like travelling waves. As we increase the amplitude of the velocity field, we find a critical value, after which the travelling wave self-similar singularity scenario disappears. Our numerical results reveal the potential stabilizing effect the convection terms.</p>https://thesis.library.caltech.edu/id/eprint/9920Determining Strength of Materials Under Dynamic Loading Conditions Using Hydrodynamic Instabilities
https://resolver.caltech.edu/CaltechTHESIS:05182017-095600418
Authors: {'items': [{'email': 'zsternberger@gmail.com', 'id': 'Sternberger-Zachary-Martin-Murphy', 'name': {'family': 'Sternberger', 'given': 'Zachary Martin Murphy'}, 'orcid': '0000-0002-7612-673X', 'show_email': 'NO'}]}
Year: 2017
DOI: 10.7907/Z9N877T5
<p>Hydrodynamic instability experiments allow access to material properties at extreme conditions where the pressure exceeds 100 GPa and the strain rate exceeds 10<sup>6</sup> 1/s. Laser ablation dynamically loads a sample, causing a manufactured initial perturbation to grow due to hydrodynamic instability. The instability growth rate depends on the strength of the sample. Material strength can then be inferred from a measurement of the instability growth. Past experiments relied on in-flight diagnostics to measure the amplitude growth, which are not available at all facilities.</p>
<p>Recovery instability experiments, where the initial and final amplitude of the instability are measured before and after the sample is dynamically loaded, obviate the need for in-flight diagnostics. Recovery targets containing copper and tantalum samples coined with 2D (hill and valley) and 3D (eggcrate) initial perturbations were dynamically loaded using the Janus laser at the Jupiter Laser Facility, Lawrence Livermore National Laboratory. The energy of the laser pulse was varied to cover a range of conditions in the dynamically compressed sample with pressures in the range 10 GPa to 150 GPa and strain rates in the range 10<sup>5</sup> 1/s to 10<sup>8</sup> 1/s.</p>
<p>The coupling of laser energy into a loading wave was studied with a combination of laser-matter interaction simulations (Hyades) and velocity interferometry data (VISAR). Laser ablation of the recovery targets generated a blast wave, loading the coined initial perturbations with a shock wave followed by a release wave. Different ablator materials and variations in the amount of laser energy deposited in the ablator lead to variations in the loading wave and consequently variations in instability growth.</p>
<p>The growth of the initial perturbation amplitude from initial to final conditions was studied with hydrocode simulations (CTH). During dynamic loading of the sample, the shock wave caused amplitude growth due to hydrodynamic instability. The release wave accelerated the perturbed interface and slowed amplitude growth, in some cases reversing growth.</p>
<p>The sensitivity of the instability growth to coarse changes in the strength model was demonstrated. However, uncertainty in modeling the laser ablation loading prevented a definitive comparison between simulation and experiment.</p>https://thesis.library.caltech.edu/id/eprint/10182Dynamics and Stability of Spinning Membranes
https://resolver.caltech.edu/CaltechTHESIS:05222017-093718571
Authors: {'items': [{'email': 'melanie.delapierre@gmail.com', 'id': 'Delapierre-Mélanie', 'name': {'family': 'Delapierre', 'given': 'Mélanie'}, 'show_email': 'YES'}]}
Year: 2017
DOI: 10.7907/Z9HT2MC8
<p>Many future space missions require large structures subject to stringent shape accuracy requirements. Spinning membrane-like structures are a cost effective solution for these applications. However, any small deflection of a spinning structure, due to maneuvers or solar radiation pressure, leads to geometrically nonlinear effects on its stability and dynamics. Accurate experiments, simulation tools, and models are required to ensure that buckling and vibrations will not affect mission objectives.</p>
<p>We first focus on the influence of transverse uniform loads on the dynamics and stability of spinning isotropic uniform membranes. A transverse uniform load models the effect of a transverse light beam on flat membranes with small deflections. We present experimental measurements of the angular velocities at which various membranes become wrinkled, and of the wrinkling mode transitions that occur upon spin down. A theoretical formulation to predict the critical angular velocities and critical transverse loads is also presented. The transition between bending dominated and in-plane dominated behavior is identified, and the wrinkling modes are obtained. Next, deflected, non-buckled membranes are further analyzed. Axisymmetric nonlinear oscillations are studied analytically, and a reduced-order model is presented. This model predicts that the deflection of the membrane introduces a hardening behavior at low angular velocities and a softening behavior at high angular velocities. This model is validated through experiments and FEM simulations.</p>
<p>Then, we relax the assumption of uniform membranes loaded by transverse light beams. We present an Abaqus model of foldable membranes and show that for particular types of hinges and at high angular velocities, these structures behave like uniform membranes. Finally, we derive an FEM model for solar radiation pressure for quadrilateral surface elements and 3D problems and present its implementation in Abaqus. We show that this follower load introduces an unsymmetric stiffness matrix and that instabilities known as solarelastic flutter can develop. This new FEM capability enables equilibrium and frequency-based stability analyses for a wide range of spacecraft.</p> https://thesis.library.caltech.edu/id/eprint/10190Analysis of Packaging and Deployment of Ultralight Space Structures
https://resolver.caltech.edu/CaltechTHESIS:05242017-230338904
Authors: {'items': [{'email': '64squared@gmail.com', 'id': 'Wilson-Lee-L', 'name': {'family': 'Wilson', 'given': 'Lee L.'}, 'orcid': '0000-0002-5865-9903', 'show_email': 'NO'}]}
Year: 2017
DOI: 10.7907/Z9B27S96
<p>This thesis presents a new approach to modeling in finite element analysis (FEA) creased thin-film sheets such as those used for drag sails, as well as modeling the packaging behavior of coilable deployable booms. This is highly advantageous because these deployable space structures are challenging to test on the ground due to their lightweight nature and the effects of gravity and air resistance. Such structures are utilized in the space industry due to their low mass and ability to be packaged into a small volume during their launch into space.</p>
<p>It is shown that removing the crease bending stiffness in creased sheets still allows the deployment behavior of a benchmark problem to be captured, including deployment forces and equilibrium configurations. In addition, folding creased sheets from a flat state into a packaged configuration can result in crease crumpling and excessive wrinkling. To avoid this the Momentless Crease Force Folding (MCFF) technique is developed.</p>
<p>Further presented is the behavior of tape springs and Tubular Rollable and Coilable (TRAC) booms when coiled to radii greater than their natural bend radius. Under these conditions the booms can form multiple localized folds which may jam during boom deployment. Understanding this behavior is important for extending the use of these booms to large scale space structures such as orbital solar power stations.</p>
<p>A useful analytical model is developed determining when the localized folds in a tape spring will bifurcate and is verified against simulation results. Additionally, a numerical model of the wrapping an isotropic tape spring around a hub with a radius greater than the localized fold radii is validated against physical experiments. This model is used to predict trends in the force required to fully wrap a tape spring around a given hub radii.</p>
<p>Finally, when examining the coiling and uncoiling behavior of TRAC booms it was found that the tension force required to keep a TRAC boom tightly coiled is significantly less than the force required to initially coil the boom.</p>https://thesis.library.caltech.edu/id/eprint/10206Maximum Entropy Reconstruction for Gas Dynamics
https://resolver.caltech.edu/CaltechTHESIS:05262017-215132894
Authors: {'items': [{'email': 'dustinsummy@gmail.com', 'id': 'Summy-Dustin-Phillip', 'name': {'family': 'Summy', 'given': 'Dustin Phillip'}, 'orcid': '0000-0002-6383-0621', 'show_email': 'YES'}]}
Year: 2017
DOI: 10.7907/Z9GT5K7W
<p>We present a method for selecting a unique and natural probability distribution function (PDF) which satisfies a given number of known moments and apply it for use in the closure of moment-based schemes for approximately solving the Boltzmann equation in gas dynamics.</p>
<p>The method used for determining the PDF is the Maximum Entropy Reconstruction (MER) procedure, which determines the PDF with maximum entropy which satisfies a given set of constraining moments. For the five-moment truncated Hamburger moment problem in one dimension, the MER takes the form of the exponential of a quartic polynomial. This implies a bimodal structure which gives rise to a small-amplitude packet of PDF-density sitting quite far from the mean. This is referred to as the Itinerant Moment Packet (IMP). It is shown by asymptotic analysis that the IMP gives rise to a solution that, in the space of constraining moments, is singular along a line emanating from, but not including, the point representing thermodynamic equilibrium. We use this analysis of the IMP to develop a numerical regularization of the MER, creating a procedure we call the Hybrid MER (HMER). Compared with the MER, the HMER is a significant improvement in terms of robustness and efficiency while preserving accuracy in its prediction of other important distribution features, such as higher order moments.</p>
<p>We apply the one-dimensional HMER to close a fourth order moment system derived from the Boltzmann equation by using a specific set of moment constraints which allow the full, three-dimensional velocity PDF to be treated as a product of three independent, one-dimensional PDFs. From this system, we extract solutions to the problem of spatially homogeneous relaxation and find excellent agreement with a standard method of solution. We further apply this method to the problem of computing the profile within a normal shock wave, and find that solutions exist only within a finite shock Mach number interval. We examine the structure of this solution and find that it has interesting behavior connected to the singularity of the MER and the IMP. Comparison is made to standard solution methods. It is determined that the use of the MER in gas dynamics remains uncertain and possible avenues for further progress are discussed.</p>https://thesis.library.caltech.edu/id/eprint/10214An LES and RANS Study of the Canonical Shock-Turbulence Interaction
https://resolver.caltech.edu/CaltechTHESIS:05212018-165403115
Authors: {'items': [{'email': 'nbraun1027@hotmail.com', 'id': 'Braun-Noah-Oakley', 'name': {'family': 'Braun', 'given': 'Noah Oakley'}, 'orcid': '0000-0002-9710-0686', 'show_email': 'NO'}]}
Year: 2018
DOI: 10.7907/BGX1-C128
<p>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 <i>k</i><sup>-5/3</sup>, 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.</p>
<p>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 <i>M<sub>t</sub></i> = 0.03 - 0.18, shock Mach numbers <i>M<sub>s</sub></i> = 1.2 - 3.0, and Taylor-based Reynolds numbers <i>Re<sub>λ</sub></i> = 20 - 2500. The LES displays minimal Reynolds number effects once an inertial range has developed for <i>Re<sub>λ</sub></i> > 100. The inertial range scales of the turbulence are shown to quickly return to isotropy, and downstream of sufficiently strong shocks this process generates a net transfer of energy from transverse into streamwise velocity fluctuations. The streamwise shock displacements are shown to approximately follow a <i>k</i><sup>-11/3</sup> decay with wavenumber as predicted by linear analysis. In conjunction with other statistics this suggests that the instantaneous interaction of the shock with the upstream turbulence proceeds in an approximately linear manner, but nonlinear effects immediately downstream of the shock significantly modify the flow even at the lowest considered turbulent Mach numbers.</p>
<p>LES allows consideration of high <i>Re<sub>λ</sub></i> 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.</p>https://thesis.library.caltech.edu/id/eprint/10919Resolvent-Based Modeling of Flows in a Channel
https://resolver.caltech.edu/CaltechTHESIS:06012018-114927289
Authors: {'items': [{'email': 'kevin.t.rosenberg@gmail.com', 'id': 'Rosenberg-Kevin-Thomas', 'name': {'family': 'Rosenberg', 'given': 'Kevin Thomas'}, 'orcid': '0000-0001-6101-3823', 'show_email': 'YES'}]}
Year: 2018
DOI: 10.7907/PHDW-Z389
<p>This thesis concerns the continued development of the resolvent framework (McKeon and Sharma, 2010) to model wall-bounded turbulent flows. Herein, we introduce novel modifications and extensions of the framework to improve the compact representation of flows in a channel. In particular, inspired by ideas rooted in classical linear stability theory, we introduce a decomposition of the velocity field into Orr-Sommerfeld (OS) and Squire (SQ) modes in a nonlinear context via the resolvent operator. We demonstrate through the analysis of a number of exact coherent states (ECS) of the Navier-Stokes equations (NSE) in Couette and Poiseuille flow that this decomposition offers a significant improvement in the low-dimensional representation of these flows. With this efficient basis, we are able to develop through the notion of interaction coefficients a method to compute accurate, self-consistent solutions of the NSE with knowledge of only the mean velocity profile. We also highlight the role of the solenoidal component of the nonlinear forcing in the solution process. In addition, the resolvent framework is extended to the analysis of 2D/3C flows. This approach, again applied to ECS, sheds light on the underlying scale interactions which sustain these solutions. Notably, it reveals that lower branch ECS can be effectively described in their entirety with a single resolvent response mode. This discovery is leveraged to construct a method to compute accurate approximations of ECS starting from a laminar profile using a single parameter model. This thesis also utilizes a constant time-step DNS of a turbulent channel to perform a direct characterization of the nonlinear forcing terms. We compute power spectra and confirm that the nonlinear forcing has a non-trivial signature in the wavenumber-frequency domain. We also compute and analyze spectra for the OS/SQ vorticity and discuss the potential benefit of this decomposition technique to the study of fully turbulent flows as well.</p>https://thesis.library.caltech.edu/id/eprint/10998Investigations of Incompressible Variable-Density Turbulence in an External Acceleration Field
https://resolver.caltech.edu/CaltechTHESIS:12052017-154614667
Authors: {'items': [{'email': 'ilana.gat@gmail.com', 'id': 'Gat-Ilana-Batya', 'name': {'family': 'Gat', 'given': 'Ilana Batya'}, 'orcid': '0000-0003-0223-0507', 'show_email': 'NO'}]}
Year: 2018
DOI: 10.7907/Z9JM27V7
<p>Of interest to turbulence modeling is the behavior of variable-density flow at high Reynolds numbers - a flow difficult to model. This thesis provides insight into variable-density flow behavior by examining the dynamics and mixing of variable-density turbulence subject to an externally imposed acceleration field. The flow is studied in the zero-Mach-number limit with a series of direct numerical simulations. The flow configuration consists of alternating slabs of high- and low-density fluid in a triply periodic domain. Density ratios in the range of 1.005 to 10 are investigated. The flow produces temporally evolving shear layers.</p>
<p>A perpendicular mean density–pressure gradient is maintained as the flow evolves, with multi-scale baroclinic torques generated in the turbulent flow that ensues. For all density ratios studied, the simulations attain Reynolds numbers at the beginning of the fully developed turbulence regime.</p>
<p>An empirical relation for the convection velocity predicts the observed entrainment-ratio and dominant mixed-fluid composition statistics. Two mixing-layer temporal evolution regimes are identified: an initial diffusion-dominated regime with a growth rate with the square-root of time followed by a turbulence-dominated regime with a cubic growth rate in time. In the turbulent regime, composition probability density functions within the shear layers exhibit a slightly tilted ('non-marching') hump, corresponding to the most probable mole fraction. The shear layers preferentially entrain low-density fluid by volume at all density ratios, which is reflected in the mixed-fluid composition.</p>
<p>The mixed-fluid orientations of vorticity, baroclinic torques, density gradients, and pressure gradients are presented. Baroclinic torques, the cross product of the density and pressure gradients, tend to be aligned with positive or negative vorticity direction, with vorticity preferentially aligning with the intermediate eigenvector of the local strain-rate tensor, with some variance.</p>https://thesis.library.caltech.edu/id/eprint/10586Numerical Investigations of Transport and Chemistry Modeling for Lean Premixed Hydrogen Combustion
https://resolver.caltech.edu/CaltechTHESIS:05312018-170312588
Authors: {'items': [{'email': 'jason.schlup@gmail.com', 'id': 'Schlup-Jason-Robert', 'name': {'family': 'Schlup', 'given': 'Jason Robert'}, 'orcid': '0000-0002-3121-3477', 'show_email': 'NO'}]}
Year: 2018
DOI: 10.7907/KFN6-7K54
<p>The use of hydrogen as a fuel for power generation applications has been suggested as an additive to, or replacement of, hydrocarbon fuels. The safety of hydrogen combustion has also received recent attention due to nuclear power plant disasters and the rise of hydrogen refuelling stations. In these uses and scenarios, lean hydrogen--air flames are prone to thermo-diffusive instabilities which can be dangerous to equipment and personnel. These instabilities are heavily influenced by two mechanisms: transport properties (e.g., diffusion) and chemical species production rates. This thesis investigates lean premixed hydrogen combustion using direct numerical simulations. A wide range of flame configurations are considered, spanning one-dimensional steady configurations to three-dimensional unsteady laminar and turbulent flames with high curvature. In particular, the two controlling mechanisms of thermo-diffusive instabilities are carefully investigated. </p>
<p>The effects of transport properties, in particular the importance of thermal diffusion in these mixtures, are quantified through global and local evaluations. Thermal diffusion is found to change flame speeds in one-dimensional flat flames, and also modify species profiles due to the increased diffusivity of light reactants. The impact of thermal diffusion is greatly enhanced in the presence of flame curvature, resulting in higher flame speeds (20% to 30% for two- and three-dimensional laminar and turbulent flames), fuel consumption, and flame surface area relative to simulations neglecting thermal diffusion. The mixture-averaged thermal diffusion model proposed by Chapman and Cowling (1970) is found to accurately reproduce global and local flame statistics (including enhanced burning and local extinction) computed using multicomponent transport at significantly reduced costs. Further cost reductions of the mixture-averaged thermal diffusion method are undertaken, and a new model is developed with constant computational requirements for large (~100 species) chemical models. The resulting reduced thermal diffusion model additionally improves upon the accuracy of the mixture-averaged thermal diffusion technique.</p>
<p>The effects of fluctuating chemical source terms on flame instabilities are then investigated using tabulated chemistry. One-dimensional unstretched flames including non-equal diffusion and thermal diffusion are incorporated into a chemistry table. This table successfully captures the interaction of differential diffusion and flame curvature. The chemistry tabulation approach is applied to a similar set of flame configurations, and accurate predictions of global and local statistics are found. The tabulated chemistry method reproduces flame curvature, local enhanced burning, and local extinction of unstable flames using one-dimensional, flat, burning flames in its construction. The proposed reduced-order thermal diffusion and chemistry tabulation models significantly reduce computational costs while simultaneously including physical properties necessary to predict lean premixed hydrogen--air flame instabilities.</p> https://thesis.library.caltech.edu/id/eprint/10991Reconstruction and Estimation of Flows Using Resolvent Analysis and Data-Assimilation
https://resolver.caltech.edu/CaltechTHESIS:05302018-181049042
Authors: {'items': [{'email': 'seansymon@gmail.com', 'id': 'Symon-Sean-Pearson', 'name': {'family': 'Symon', 'given': 'Sean Pearson'}, 'orcid': '0000-0001-9085-0778', 'show_email': 'NO'}]}
Year: 2018
DOI: 10.7907/B4K7-K876
<p>A flow reconstruction methodology is presented for incompressible, statistically stationary flows using resolvent analysis and data-assimilation. The only inputs necessary for the procedure are a rough approximation of the mean profile and a single time-resolved measurement. The objective is to estimate both the mean and fluctuating states of experimental flows with limited measurements which do not include pressure. The input data may be incomplete, in the sense that measurements near a body are difficult to obtain with techniques such as particle image velocimetry (PIV), or contaminated by noise. The tools developed in this thesis are capable of filling in missing data and reducing the amount of measurement noise by leveraging the governing equations. The reconstructed flow is capable of estimating fluctuations where time-resolved data are not available and solving the flow on larger domains where the mean profile is not known.</p>
<p>The first part of the thesis focuses on how resolvent analysis of the mean flow selects amplification mechanisms. Eigenspectra and pseudospectra of the mean linear Navier-Stokes (LNS) operator are used to characterize amplification mechanisms in flows where linear mechanisms are important. The real parts of the eigenvalues are responsible for resonant amplification and the resolvent operator is low-rank when the eigenvalues are sufficiently separated in the spectrum. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The latter is studied by considering well-known turbulent structures while the former contains a marginally stable eigenvalue which drowns out the effect of other eigenvalues over a large range of temporal frequencies. There is a geometric manifestation of this dominant mode in the mean profile, suggesting that it leaves a significant footprint on the time-averaged flow that the resolvent can identify. The resolvent does not provide an efficient basis at temporal frequencies where there is no separation of singular values. It can still be leveraged, nevertheless, to identify coherent structures in the flow by approximating the nonlinear forcing from the interaction of highly amplified coherent structures.</p>
<p>The second part of the thesis extends the framework of Foures et al. (2014), who data-assimilated the mean cylinder wake at very low Reynolds numbers. The contributions presented here are to assess the minimum domain for successfully reconstructing Reynolds stress gradients, modifying the algorithm to assimilate mean pressure, determining whether weighting input measurements contributes to improved performance, and adapting the method to experimental data at higher Reynolds numbers. The results from data-assimilating the mean cylinder wake at low Reynolds numbers suggest that the measurement domain needs to coincide with the spatial support of the Reynolds stress gradients while point weighting has a minimal impact on the performance. Finally, a smoothing procedure adapted from Foures et al. (2014) is proposed to cope with data-assimilating an experimental mean profile obtained from PIV data. The data-assimilated mean profiles for an idealized airfoil and NACA 0018 airfoil are solved on a large domain making the mean profile suitable for global resolvent analysis. Data-assimilation is also able to fill in missing or unreliable vectors near the airfoil surface.</p>
<p>The final piece of the thesis is to synthesize the knowledge and techniques developed in the first two parts to reconstruct the experimental flow around a NACA 0018 airfoil. Preliminary results are presented for the case where <i>α</i> = 0° and <i>Re</i> = 10250. The mean profile is data-assimilated and used as an input to resolvent analysis to educe coherent structures in the flow. The resolvent operator for non- amplified temporal frequencies is forced by an approximated nonlinear forcing. The amplitude and phase of the modes are obtained from the discrete Fourier-transform of a time-resolved probe point measurement. The final reconstruction contains less measurement noise compared to the PIV snapshots and obeys the incompressible Navier-Stokes equations (NSE). The thesis concludes with a discussion of how elements of this methodology can be incorporated into the development of estimators for turbulent flows at high Reynolds numbers.</p>https://thesis.library.caltech.edu/id/eprint/10976On the Kinetics of Materials of Geophysical Interest
https://resolver.caltech.edu/CaltechTHESIS:08282017-105838669
Authors: {'items': [{'email': 'matthew.g.newman@gmail.com', 'id': 'Newman-Matthew-Gregory', 'name': {'family': 'Newman', 'given': 'Matthew Gregory'}, 'orcid': '0000-0003-2752-0121', 'show_email': 'YES'}]}
Year: 2018
DOI: 10.7907/Z9319T35
Knowledge of the equation of state and phase diagram of magnesium silicates and light iron alloys is important for understanding the thermal evolution and interior structure of terrestrial planets. Dynamic compression techniques are the primary viable methods to create the temperature and pressure conditions that are relevant to Earth and super-Earth (1-10 Earth mass) sized planets. However, due to the kinetic constraints imposed by the timescale of dynamic compression experiments, the nature of the state within the dynamically compressed sample (whether equilibrium or metastable) is uncertain. Here, we present the results of a series of dynamic compression experiments performed on both laser driven compression and plate impact facilities to study the nanosecond to microsecond response of forsterite and iron silicide. In situ x-ray diffraction measurements are used to probe the crystal structure of solid phases and test for the presence of melt, from which we investigate the decomposition of forsterite and iron silicide into compositionally distinct phases at high pressure. For forsterite, we do not observe chemical segregation in the solid phase, however the presence of melt speeds up the kinetics and allows chemical segregation to occur on nanosecond timescales. For iron silicide, our results show a textured solid phase upon shock compression to pressures ranging from 166(14) to 282(24) GPa consistent with cubic and hcp structures in coexistence. Above 313(29) GPa, the intense and textured solid diffraction peaks give way to a diffuse scattering feature and loss of texture, consistent with melting along the Hugoniot.https://thesis.library.caltech.edu/id/eprint/10393Towards a priori Models for Differential Diffusion in Turbulent Non-Premixed Flames
https://resolver.caltech.edu/CaltechTHESIS:06062018-163232775
Authors: {'items': [{'email': 'nicholas@burali.it', 'id': 'Burali-Nicholas', 'name': {'family': 'Burali', 'given': 'Nicholas'}, 'orcid': '0000-0002-0733-0577', 'show_email': 'YES'}]}
Year: 2018
DOI: 10.7907/N3VJ-BE39
<p>In this work, progress is made towards the correct modeling of differential diffusion, both for resolved simulations, and for reduced-order combustion models. For resolved simulations, the validity and the limitations of the constant non-unity Lewis number approach in the description of molecular mixing in laminar and turbulent flames is studied. Three test cases are selected, including a lean, highly unstable, premixed hydrogen/air flame, a lean turbulent premixed n-heptane/air flame, and a laminar ethylene/air coflow diffusion flame. For the hydrogen flame, both a laminar and a turbulent configuration are considered. The three flames are characterized by Lewis numbers which are less than unity, greater than unity, and close to unity, respectively. For each flame, mixture-averaged transport simulations are carried out and used as reference data. The analysis suggests that, for numerous combustion configurations, the constant non-unity Lewis number approximation leads to small errors when the set of Lewis numbers is chosen properly. For the selected test cases and our numerical framework, the reduction of computational cost is found to be minimal. Two different methods of evaluating the Lewis numbers are tested, with both performing well, and neither consistently better than the other.</p>
<p>The flamelet-based chemistry tabulation technique is a popular reduced-order chemical model for non-premixed turbulent flames. In this approach, the correct choice of the species Lewis numbers in the flamelet equations plays an important role. Experimental results have highlighted that, in turbulent non-premixed jet flames, turbulent transport becomes gradually dominant over molecular mixing with (i) increasing axial distance from the burner exit plane, and (ii) increasing jet Reynolds number. In the current work, this transition is characterized and a priori models for the effective species Lewis numbers in turbulent non-premixed flames are assessed.</p>
<p>First, a flamelet-based methodology is proposed to extract these effective Lewis numbers from data sets of turbulent non-premixed flames. This methodology is then applied to the Sandia non-premixed methane/air jet flames B, C, D, and E (R. Barlow, Int. Work. Meas. Comput. Turb. Non-Prem. Flames, 2003). The effective Lewis numbers are found to transition from their laminar values, close to the burner exit plane, to unity further downstream. Previously-suggested scalings for the effective Lewis numbers are then assessed.</p>
<p>To overcome the limitations associated with the experimental data, a campaign of Direct Numerical Simulations (DNS) of Sandia flame B is carried out. A baseline grid is carefully designed, and grid independence is assessed through simulations using refined grids in the axial, radial and azimuthal directions. Radiation and differential diffusion effects are systematically isolated by considering radiating and unity Lewis number cases, respectively. The DNS database is then validated using available measured statistics for flame B, and comparisons to the higher Reynolds number flames are carried out. Effective Lewis numbers extracted from the DNS data are found to transition to unity with increasing downstream distance. Finally, the scalings for the effective Lewis numbers are re-computed from the DNS data base, and compared to the higher Reynolds number flames.</p>https://thesis.library.caltech.edu/id/eprint/11030Using the Force: Applications and Implications of Turbulence Forcing Terms in Direct Numerical Simulations
https://resolver.caltech.edu/CaltechTHESIS:06102019-185605511
Authors: {'items': [{'email': 'chandrudiitm@gmail.com', 'id': 'Dhandapani-Chandru', 'name': {'family': 'Dhandapani', 'given': 'Chandru'}, 'orcid': '0000-0002-7319-557X', 'show_email': 'YES'}]}
Year: 2019
DOI: 10.7907/FH31-4468
<p>Most energy requirements of modern life can be fulfilled by renewable energy sources, but it is impossible in the near future to provide an alternative energy source to combustion for airplanes. That being said, combustion in aviation can be made more sustainable by using alternative jet fuels, which are made from renewable sources like agricultural wastes, solid wastes, oils, and sugars. These alternative fuels can be used in commercial flights only after a long certification process by the Federal Aviation Agency (FAA) and ASTM International. Unfortunately, in over 50 years of fuel research, only five fuels have been certified.
This research project aims to speed up the certification process with quicker testing of alternative fuels. Engine testing and even laboratory testing require large amounts of time and fuel. Simulations can make the process much more efficient, but accurately simulating highly turbulent flames in such complex geometries would need large amounts of computational resources. The goal of this thesis is to create an efficient computational framework, that can replicate different engine-like turbulent flow conditions in simple geometries with numerical tractability.</p>
<p>The central idea is to decompose the flow field into ensemble mean and fluctuating quantities. The simulations then resolve only the fluctuations using simple computational domains, while emulating the effect of the mean flow using "forcing" terms. These forcing terms are calculated first for incompressible turbulence, and this method is later extended to turbulent reacting flows. In incompressible turbulence, Direct Numerical Simulations (DNS) performed on simple triply periodic cubic domains reasonably capture the statistically stationary shear turbulence, that is observed in free shear flows. The simulations are also performed in cuboidal domains, that are longer in one direction and with an inflow/outflow along it. Both changes are observed to not have a significant impact on the turbulence statistics. Finally, shear convection is applied to the turbulence simulations with inflow/outflow, which has a significant impact on the turbulence. These simulations accurately capture the turbulence anisotropy in free-shear flows.</p>
<p>The study is extended to DNS of highly turbulent <i>n</i>-heptane-air flames performed under different flow conditions. Turbulent flames involve two-way coupling between fluid mechanics and combustion. The effects of the flame on the turbulence and the impact of the turbulent flow conditions on the flame behavior are analyzed. The focus is placed on the effects of turbulence production, shear convection, and pressure gradients. The anisotropy produced in the turbulence due to the different flow conditions and the flame are also compared and contrasted. While the global behavior and flow anisotropy were affected by these conditions, the local chemistry effects were unaffected, and depend only on the laminar flame properties and turbulence intensity. These findings can help predict turbulent flame behavior, and can expedite the search and testing of sustainable alternatives to conventional jet fuels.</p>https://thesis.library.caltech.edu/id/eprint/11737Plasma Surface Interactions in LaB₆ Hollow Cathodes with Internal Xe Gas Discharge
https://resolver.caltech.edu/CaltechTHESIS:06032019-100503451
Authors: {'items': [{'email': 'p.guerrero.eng@gmail.com', 'id': 'Guerrero-Vela-Pedro-Pablo', 'name': {'family': 'Guerrero Vela', 'given': 'Pedro Pablo'}, 'orcid': '0000-0001-5766-2038', 'show_email': 'NO'}]}
Year: 2019
DOI: 10.7907/4CW7-2K35
<p>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.</p>
<p>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<sub>6</sub> inserts are attractive candidates for long duration EP based space missions. However, the physics behind LaB<sub>6</sub> 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<sub>6</sub> hollow cathode thermal behaviour and the interplay between LaB<sub>6</sub> insert surface chemistry and xenon plasma.</p>
<p>Our investigation of the thermal behaviour of LaB<sub>6</sub> 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<sub>6</sub> 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<sub>6</sub> 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<sub>6</sub> 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.</p>
<p>Our results revealed the unexpected phenomenon of work function reduction upon plasma exposure of LaB<sub>6</sub>. These findings suggest that LaB<sub>6</sub> hollow cathodes may outperform current technologies and become the component of choice in EP thrusters for future space missions.</p>https://thesis.library.caltech.edu/id/eprint/11673Constraining the Mantle's Rheology Using Methods in Uncertainty Quantification
https://resolver.caltech.edu/CaltechTHESIS:05132019-143045769
Authors: {'items': [{'email': 'vratnas@sandia.gov', 'id': 'Ratnaswamy-Vishagan', 'name': {'family': 'Ratnaswamy', 'given': 'Vishagan'}, 'orcid': '0000-0002-2371-807X', 'show_email': 'NO'}]}
Year: 2019
DOI: 10.7907/F6FW-T648
An accurate estimation of the large-scale forces in the mantle has been difficult to obtain as numerical models either do not use an accurate rheology nor reproduce surface observations. While much work has been done in developing high-fidelity forward models that capture the salient physics of shear-thinning and dynamic weakening, they fail to reproduce observations such as plate motions and topography. In this thesis, we develop an optimization methodology that minimizes the misfit in surface observations such as plate motions and average effective viscosity for certain regions of the mantle. We utilize adjoints to calculate the gradient, while using second-order adjoints to construct the Hessian so as to infer the rheological parameters of the mantle's rheology. Furthermore, we build on this optimization scheme by constructing the Gaussian approximation of the posterior distribution for the inferred rheological parameters using the Hessian and establish the trade-offs between each parameter through their conditional distributions. We further extend this Gaussian approximation to infer extrinsic quantities such as the stresses in the fault zones and the average effective viscosity in the hinge zones to not only quantify the uncertainty, but also to see partitioning of the coupling of each subduction zone. https://thesis.library.caltech.edu/id/eprint/11514Linear and Non-linear Interactions in a Rough-Wall Turbulent Boundary Layer
https://resolver.caltech.edu/CaltechTHESIS:04112019-234812867
Authors: {'items': [{'email': 'jpmorgan89@gmail.com', 'id': 'Morgan-Jonathan-Philip', 'name': {'family': 'Morgan', 'given': 'Jonathan Philip'}, 'orcid': '0000-0003-2898-4868', 'show_email': 'NO'}]}
Year: 2019
DOI: 10.7907/7RSR-3277
<p>This thesis explores the linear and non-linear interactions which take place in a rough-wall turbulent boundary through experiments and modeling. In order to derive physics-based models for the relation between roughness geometry and flow physics, two very simple periodic roughnesses were 3D printed and placed in a boundary layer wind tunnel for separate experiments. Hot-wire measurements were taken at a grid of points within a single period of the roughness in order to map the spatial variation of important flow statistics in way that allows correlation back to the roughness geometry. Time averaged streamwise velocity and the power spectrum of instantaneous streamwise velocity were both found to vary coherently with the roughness. The spatial variation of the time averaged velocity was identified as the linear result of the roughness, as it has identical wavenumber and frequency to the static roughness geometry. Modeling the time-averaged velocity field as a response mode of the linear resolvent operator was found to be reasonable for certain wavenumbers. The spatial distribution of the power spectrum was shown to be a non-linear effect of the roughness; the power spectrum only measures the energy of convecting modes, which necessarily have non-zero frequency and cannot correlate linearly to the static roughness. The spatial modulation of the power spectrum was found to be indicative of non-linear triadic interactions between the static velocity Fourier modes and pairs of convecting modes, as allowed by the Navier-Stokes equations. A low-order model for these interactions, and their effect on the power spectrum, was constructed using resolvent response modes to represent all velocity Fourier modes. The model was found to qualitatively predict the modulation of the power spectrum for several sets of wavenumbers. The success of such a simple model suggests that it presents a useful low-order understanding of non-linear forcing between scales in rough-wall boundary layers.</p>https://thesis.library.caltech.edu/id/eprint/11453Optimization of CCD Charge Transfer for Ground and Space-Based Astronomy
https://resolver.caltech.edu/CaltechTHESIS:05302019-172558320
Authors: {'items': [{'email': 'pavan.bilgi@gmail.com', 'id': 'Bilgi-Pavaman', 'name': {'family': 'Bilgi', 'given': 'Pavaman'}, 'orcid': '0000-0002-2642-8553', 'show_email': 'YES'}]}
Year: 2019
DOI: 10.7907/CNKG-8Y84
<p>This thesis will be of particular interest to anyone integrating Charge-Coupled Devices (CCDs) into any precision scientific imaging instrument, especially so in space. The first part of the thesis concerns optimization of a CCD camera as a whole. CCDs for the WaSP imager at the Hale telescope are characterized using a minimal amount of data using just a flat-field illumination source. By measuring performance over the entire parameter space of (clock and bias) inputs and analyzing the multidimensional output (linearity, dynamic range, read noise etc), optimal operating conditions can be selected quickly (and possibly automatically). With ever growing sizes of detector arrays such as the recently launched Gaia mission, the upcoming Euclid mission and ground-based cameras such as the LSST (189 CCDs), the task of streamlining detector optimization will be increasingly important. In the second (larger) part, the optimization of Charge Transfer Efficiency (CTE) is explored in particular. In modern CCDs, CTE is caused by lattice defects in the bulk silicon and is significantly worsened by radiation exposure, which is unavoidable in space. As shown in the literature, just a year of exposure to high energy solar proton radiation at low earth orbit can result in CTE reducing to 0.9999 for a signal level of 10,000e<sup>-</sup> — problematic for most precision astronomical measurements. Here, CTE degrading traps are fully explored in an undamaged CCD to new levels of accuracy. Several unique species are identified, and their population statistics are analyzed by both wafer and sub-pixel location. Subsequently, easily applied CTE measurement techniques are presented, yielding results with new levels of accuracy, concluding in the presentation of a new trap mitigating readout clocking scheme. This scheme can be readily applied to any CCD employing a parallel transfer gate without readout speed penalty. It is proposed that the results herein may be used to construct a simple model to predict CTE given a temperature, readout timing and signal level. This model could then be used to automatically optimize CTE for any CCD, given only its trap parameter statistics.</p>https://thesis.library.caltech.edu/id/eprint/11574Shock Compression of Molybdenum Single Crystals to High Stresses
https://resolver.caltech.edu/CaltechTHESIS:02192020-135417079
Authors: {'items': [{'email': 'o.tomo.oni@gmail.com', 'id': 'Oniyama-Tomoyuki', 'name': {'family': 'Oniyama', 'given': 'Tomoyuki'}, 'orcid': '0000-0001-6097-9917', 'show_email': 'NO'}]}
Year: 2020
DOI: 10.7907/YWPJ-5379
<p>To investigate the role of crystal anisotropy and the impact stress on the shock induced elastic-plastic deformation of BCC single crystals at high stresses, molybdenum single crystals were shock compressed along [100], [111], and [110] orientations. A series of plate impact experiments were conducted with various impact stresses (23 - 190 GPa) along each orientation. Along the [100] and [111] orientations, two-wave structure - an elastic shock wave trailed by a plastic shock wave - was observed to 110 GPa. Along the [110] orientation, the two-wave structure was observed only up to 90 GPa.</p>
<p>Based on the measured quantities, in-material quantities at the elastic limit and at the peak state were calculated. The elastic wave amplitudes were analyzed to determine the crystal anisotropy effects, the impact stress dependence, and the activated slip systems on the elastic limit. The elastic wave amplitude increased linearly with
increasing impact stress, and that was significantly larger along the [111] orientation compared to the other orientations. The difference between calculated maximum resolved shear stresses at the elastic limit and corresponding Peierls stress suggested the activation of {110}<111> slip systems.</p>
<p>At the peak state, the Hugoniot relations were calculated along each orientation and compared with polycrystalline molybdenum Hugoniot relations. The Hugoniot relations along three orientations were in agreement within experimental uncertainties, even though the elastic limit showed considerable anisotropy. Also, they agreed reasonably well with the polycrystalline molybdenum data. This implied that the in-material quantities at the peak state do not depend on crystal orientation or the presence of grain boundaries.</p>
<p>In addition to the plate impact experiments, finite element simulations of shock compressed molybdenum single crystals were conducted using Abaqus Explicit in order to gain insight into deformation mechanisms activated during the elasticplastic
deformation. Shear strains on slip systems were explicitly considered by the crystal plasticity model implemented using Abaqus VUMAT subroutine. The results of FEM simulations indicated that {110}<111> systems were likely to be operating at the elastic limit. This observation was consistent with the experimental results from the present study.</p>https://thesis.library.caltech.edu/id/eprint/13641Surface Reconstruction from Distributed Angle Measurements
https://resolver.caltech.edu/CaltechTHESIS:02282020-192725947
Authors: {'items': [{'email': 'thibaud.talon@gmail.com', 'id': 'Talon-Thibaud', 'name': {'family': 'Talon', 'given': 'Thibaud'}, 'orcid': '0000-0002-8240-1101', 'show_email': 'YES'}]}
Year: 2020
DOI: 10.7907/ZG2D-2K77
<p>This thesis presents an innovative solution to the shape measurement of large structures for space applications. The current state-of-the-art heavily relies on optical solutions such as cameras or lasers to recover the shape of a surface. Because of the impracticality of placing a system in front of a large structure flying in space, new solutions need to be developed. The proposed solution is to embed angular sensors (such as sun sensors) directly on the surface. The sensors provide a collection of distributed measurements that form a discrete map of the angular orientation of the structure. An integration scheme can then estimate the 3D shape of the surface.</p>
<p>A mathematical model to perform the integration from angle measurements to the shape of a 3D surface is presented first. This model is purely geometric and serves as a basis for similar concepts. The surface is known in a reference configuration and is assumed to have deformed inextensibly to its current shape. Inextensibility conditions are enforced through a discretization of the metric tensor generating a finite number of constraints. This model parameterizes the shape of the surface using a small number of unknowns, and thus requires a small number of sensors. We study the singularities of the equations and derive necessary conditions for the problem to be well-posed. The limitations of the algorithm are highlighted. Simulations are performed on developable surfaces to analyze the performance of the method and to show the influence of the parameters used in the algorithm. Optimal schemes which lower the RMS error between the reconstructed shape and the actual one are presented.</p>
<p>An experimental validation of the proposed solution and algorithm is performed on a 1.3 x 0.25 m structure with 14 embedded sun sensors. The sensors measure the two local angles of the surface from a light source placed in front of the surface. A small, lightweight and expandable design of the sensors is shown in this thesis. A calibration procedure accurately correlates the output of the sensor with a 0.5° precision. The procedure also highlights the limitations of the design. The structure was deformed in bending and torsion with amplitudes of a few centimeters, and its shape was reconstructed to an accuracy on the order of a millimeter.</p>
<p>The accuracy of the initial algorithm is found to be limited by local shape deformations caused by the mechanical response of the structure. A new algorithm, replacing the discrete inextensibility conditions with the equilibrium equations derived from a finite-element model, is shown. This new algorithm is tested on the experimental structure and the accuracy of the reconstruction is increased by a factor of 2. The RMS error is under a millimeter on average over the different applied shapes and goes as low as 0.3 mm.</p>
<p>To understand how this solution can apply to large space structures, simulations are performed on a model of a large planar spacecraft. A 25 x 25 m structure representing the current concept for the Caltech Space Solar Power Project satellite is used as an example. Sensors with similar noise properties as the ones built for the experiment are placed on the spacecraft. A finite-element model combining the vibration of the spacecraft with large rigid body rotations is presented. This model is used in a Kalman filter that estimates the shape of the structure by iterative prediction from the dynamic finite-element model and correction from the angle measurements. Simulations are performed around the thruster actuation applied at the corner of the structure to follow a specific guidance scheme that is optimal for space solar power satellites. The actuation creates both vibrations of the structure with amplitudes of few centimeters and large rotations of the spacecraft. The designed Kalman filter can accurately estimate both effects and it is shown that millimeter accuracy is achievable. The relationship between the number of sensors, the reconstructed shape error, as well as potential stiffness deviations in the FE model is studied. The results provide first order estimates of the performance of this measurement system, in order to enable the design of future space missions.</p>https://thesis.library.caltech.edu/id/eprint/13650Aspects of Reduced-Order Modeling of Turbulent Channel Flows: From Linear Mechanisms to Data-Driven Approaches
https://resolver.caltech.edu/CaltechTHESIS:05282020-161209039
Authors: {'items': [{'email': 'rmcmullen54@gmail.com', 'id': 'McMullen-Ryan-Michael', 'name': {'family': 'McMullen', 'given': 'Ryan Michael'}, 'orcid': '0000-0003-1371-7150', 'show_email': 'NO'}]}
Year: 2020
DOI: 10.7907/wayx-eh75
<p>This thesis concerns three key aspects of reduced-order modeling for turbulent shear flows. They are linear mechanisms, nonlinear interactions, and data-driven techniques. Each aspect is explored by way of example through analysis of three different problems relevant to the broad area of turbulent channel flow.</p>
<p>First, linear analyses are used to both describe and better understand the dominant flow structures in elastoinertial turbulence of dilute polymer solutions. It is demonstrated that the most-amplified mode predicted by resolvent analysis (McKeon and Sharma, 2010) strongly resembles these features. Then, the origin of these
structures is investigated, and it is shown that they are likely linked to the classical Tollmien-Schichting waves.</p>
<p>Second, resolvent analysis is again utilized to investigate nonlinear interactions in Newtonian turbulence. An alternative decomposition of the resolvent operator into Orr-Sommerfeld and Squire families (Rosenberg and McKeon, 2019b) enables a highly accurate low-order representation of the second-order turbulence statistics. The reason for its excellent performance is argued to result from the fact that the decomposition enables a competition mechanism between the Orr-Sommerfeld and Squire vorticity responses. This insight is then leveraged to make predictions about how resolvent mode weights belonging to several special classes scale with increasing Reynolds number.</p>
<p>The final application concerns special solutions of the Navier-Stokes equations known as exact coherent states. Specifically, we detail a proof of concept for a data-driven method centered around a neural network to generate good initial guesses for upper-branch equilibria in Couette flow. It is demonstrated that the neural network is capable of producing upper-branch solution predictions that successfully converge to numerical solutions of the governing equations over a limited range of Reynolds numbers. These converged solutions are then analyzed, with a particular emphasis on symmetries. Interestingly, they do not share any symmetries with the known equilibria used to train the network. The implications of this finding, as well as broader outlook for the scope of the proposed method, are discussed.</p>https://thesis.library.caltech.edu/id/eprint/13730Application of Path-Independent Integrals to Soil-Structure Interaction
https://resolver.caltech.edu/CaltechTHESIS:11212019-100323260
Authors: {'items': [{'email': 'ajgarciasuarez@gmail.com', 'id': 'García-Suárez-Antonio-Joaquín', 'name': {'family': 'García Suárez', 'given': 'Antonio Joaquín'}, 'orcid': '0000-0001-8830-4348', 'show_email': 'NO'}]}
Year: 2020
DOI: 10.7907/MMWW-B046
<p>Assessing seismic pressure increment on buried structures is a critical step in the design of infrastructure in earthquake-prone areas. Due to intrinsic complexities derived from the need to match the solution in the far-field to the localized solution around the structure, the near-field, researchers have aimed at finding simplified models focused on engineering variables as the seismic earth thrust. One such model is the so-called Younan-Veletsos model, which pivots on a stringent assumption on the stress tensor.</p>
<p>At the same time, the might of the path-independent integrals of solid mechanics to deal with problems in Geotechnical Engineering at large, and Soil-Structure Interaction in particular, has remained unexplored, despite of a rich landscape of potential applications. The unbridled success of these path-independent integrals in Fracture Mechanics, a discipline which cannot be understood without them currently, may be mirrored in problems in Geotechnical Engineering, since the two fields, despite appearing very detached from each other at first glance, share deep traits: in both cases, the system under consideration can be conceptualized as a domain with simple, easy-to-assess regions (the areas where remote loading is applied and the far-field, respectively) and also with other complex, hard-to-understand regions (the crack tip, the near-field).</p>
<p>We present the first derivation of the exact solution of the Younan-Veletsos problem, which is later analyzed to reveal phenomena not captured by previous approximate solutions. Then, we introduce a novel model which relies on the path-independent Rice’s J-integral, a customary tool in Fracture Mechanics, which is applied here in the Soil-structure Interaction context for the first time. This novel model captures those features of the exact solution that were missed by prior approximations. The capabilities of the J-integral to, first, find an upper bound of the force induced by earthquakes over the walls of underground structures, under some conditions, and, second, to understand the soil-structure kinematic interaction phenomenon are also assessed.</p>
<p>Additionally, the intermediate step of analyzing of the far-field yielded some results concerning Site Response Analysis which are also included in the text.</p>https://thesis.library.caltech.edu/id/eprint/13587Multi-Resolution Lattice Green's Function Method for High Reynolds Number External Flows
https://resolver.caltech.edu/CaltechTHESIS:06072021-162542722
Authors: {'items': [{'email': 'skeleton.yk@gmail.com', 'id': 'Yu-Ke', 'name': {'family': 'Yu', 'given': 'Ke'}, 'orcid': '0000-0003-0157-4471', 'show_email': 'NO'}]}
Year: 2021
DOI: 10.7907/wkc8-se35
<p>This work expands the state-of-the-art computational fluid dynamics (CFD) methods for simulating three-dimensional, turbulent, external flows by further developing the immersed boundary (IB) Lattice Green's function (LGF) method.
The original IB-LGF method applies an exact far-field boundary condition using fundamental solutions on regular Cartesian grids and allows active computational cells to be restricted to vortical flow regions in an adaptive fashion as the flow evolves. The combination of spatial adaptivity and regular Cartesian structure leads to superior efficiency, scalability, and robustness, but necessitates uniform grid spacing. However, the scale separation associated with thin boundary layers and turbulence at higher Reynolds numbers favors a more flexible distribution of elements/cells, which is achieved in this thesis by developing a multi-resolution LGF approach that permits block-wise grid refinement while maintaining the important properties of the original scheme. We further show that the multi-resolution LGF method can be fruitfully combined with the IB method to simulate external flows around complex geometries at high Reynolds numbers. This novel multi-resolution IB-LGF scheme retains good efficiency, parallel scaling as well as robustness (conservation and stability properties). DNS of bluff and streamlined bodies at Reynolds numbers <i>O</i>(10<sup>4</sup>) are conducted and the new multi-resolution scheme is shown to reduce the total number of computational cells up to 99.87%.</p>
<p>We also extended this method to large-eddy simulation (LES) with the stretched-vortex sub-grid-scale model. In validating the LES implementation, we considered an isolated spherical region of turbulence in free space. The initial condition is spherically windowed, isotropic homogeneous incompressible turbulence. We study the spectrum and statistics of the decaying turbulence and compare the results with decaying isotropic turbulence, including cases representing different low wavenumber behavior of the energy spectrum (i.e. <i>k</i><sup>2</sup> versus <i>k</i><sup>4</sup>). At late times the turbulent sphere expands with both mean radius and integral scale showing similar time-wise growth exponents. The low wavenumber behavior has little effect on the inertial scales, and we find that decay rates follow Saffman's predictions in both cases, at least until about 400 initial eddy turnover times. The boundary of the spherical region develops intermittency and features ejections of vortex rings. These are shown to occur at the integral scale of the initial turbulence field and are hypothesized to occur due to a local imbalance of impulse on this scale.</p>https://thesis.library.caltech.edu/id/eprint/14253A Shock Compression Investigation of Failure Waves and Phase Transition in Soda-Lime Glass
https://resolver.caltech.edu/CaltechTHESIS:05282021-233441075
Authors: {'items': [{'email': 'akshay.joshikc@gmail.com', 'id': 'Joshi-Akshay', 'name': {'family': 'Joshi', 'given': 'Akshay'}, 'orcid': '0000-0001-8347-8357', 'show_email': 'NO'}]}
Year: 2021
DOI: 10.7907/b8xs-8r91
<p>Soda-lime glass (SLG) and other silica glasses find use in many technological applications involving high pressures and strain rates, such as systems with laser-matter interactions, transparent armor, etc. An experimentally validated constitutive model for these glasses is required for modeling their mechanical behavior at high pressures and strain rates. Also, due to the abundance of silica in the earth's crust, understanding the behavior of these glasses at high pressures can provide significant insights into many geophysical processes. To this end, shock compression experiments are carried out on SLG to study the material's behavior under impact stresses of 5-10 GPa. These experiments are accompanied by numerical simulations and constitutive modeling of SLG to gain further insights into the reported failure-wave phenomenon and phase transitions associated with the material.</p>
<p>The significant findings of this study in relation to the failure-wave phenomenon were the sudden densification/compaction of SLG associated with the failure-wave and the disappearance of the failure-wave phenomenon for impact stresses above 10 GPa. When viewed in the context of the findings from past experiments, these results seem to suggest that localized densification/compaction of SLG causes nucleation of cracks and subsequent comminution in the material under shock compression. These results and observations offer a potential explanation of the mechanism underlying the failure-wave phenomenon.</p>
<p>Further, the shock compression and release experiments performed in this work provided significant insights into the onset of possible phase-transition in SLG under shock compression. A loading-unloading hysteresis is observed in the material’s stress-strain curve for impact stresses higher than 5.8 GPa, with the permanent/residual strain increasing with impact stress. Further analysis of these results strongly indicates that the hysteresis is more likely due to a gradual, irreversible phase transition of SLG than due to regular inelastic behavior. Thus, the results suggest that the SLG undergoes a gradual phase transition to a stiffer phase, although other properties of this phase remain unclear. It can also be noted that this phase transition is postulated to start occurring under shock compression of SLG to stresses above 5 GPa, which is also the threshold stress for the onset of the failure-wave phenomenon. It is, therefore, possible that the two phenomena are interrelated. The experimental results from this study are further used to construct a constitutive model to capture the unloading behavior of SLG.</p>https://thesis.library.caltech.edu/id/eprint/14198Part I: The Equations of Plasma Physics and the Richtmyer-Meshkov Instability in Magnetohydrodynamics. Part II: Evolution of Perturbed Planar Shockwaves.
https://resolver.caltech.edu/CaltechTHESIS:10042020-174941725
Authors: {'items': [{'email': 'shennaijian@gmail.com', 'id': 'Shen-Naijian', 'name': {'family': 'Shen', 'given': 'Naijian'}, 'orcid': '0000-0002-0533-8081', 'show_email': 'NO'}]}
Year: 2021
DOI: 10.7907/e9b8-y318
<p>Part I: Mitigating the Richtmyer-Meshkov instability (RMI) is critical for energy production in inertial confinement fusion. Suitable plasma models are required to study the hydrodynamic and electromagnetic interactions associated with the RMI in a conducting medium. First, a sequence of asymptotic expansions in several small parameters, as formal limits of the non-dissipative and non-resistive two-fluid plasma equations, leads to five simplified plasma/magnetohydrodynamics (MHD) systems. Each system is characterized by its own physical range of validity and dispersion relations, and includes the widely used magnetohydrodynamic (MHD) and Hall-MHD equations. Next we focus on the RMI in MHD. Using ideal MHD, it has been shown that the RMI is suppressed by the presence of an external magnetic field. We utilize the incompressible, Hall-MHD model to investigate the stabilization mechanism when the plasma ion skin depth and Larmor radius are nonzero. The evolution of an impulsively accelerated, sinusoidally perturbed density interface between two conducting fluids is solved as a linearized initial-value problem. An initially uniform background magnetic field of arbitrary orientation is applied. The incipient RMI is found suppressed through oscillatory motions of the interface due to the ion cyclotron effect. This suppression is most effective for near tangential magnetic fields but becomes less effective with increasing plasma length scales. The vorticity dynamics that facilitates the stabilization is discussed.</p>
<p>Part II: We consider the evolution of a planar gas-dynamic shock wave subject to smooth initial perturbations in both Mach number and shock shape profile. A complex variable formulation for the general shock motion is developed based on an expansion of the Euler equations proposed by Best [<i>Shock Waves</i>, {1}: 251-273, (1991)]. The zeroth-order truncation of Best's system is related to the well-known geometrical shock dynamics (GSD) equations while higher-order corrections provide a hierarchy of closed systems, as detailed initial flow conditions immediately behind the shock are prescribed. Solutions to Best's generalized GSD system for the evolution of two-dimensional perturbations are explored numerically up to second order in the weak and strong shock limits. Two specific problems are investigated: a shock generated by an impulsively accelerated piston with a corrugated surface, and a shock traversing a density gradient. For the piston-driven flow, it is shown that this approach allows full determination of derivative jump conditions across the shock required to specify initial conditions for the retained, higher-order correction equations. In both cases, spontaneous development of curvature singularity in the shock shape is detected. The critical time at which a singularity occurs follows a scaling inversely proportional to the initial perturbation size. This result agrees with the weakly nonlinear GSD analysis of Mostert <i>et al.</i> [<i>J. Fluid Mech.</i>, {846}: 536-562, (2018)].</p>https://thesis.library.caltech.edu/id/eprint/13974Probing the Buckling of Thin-Shell Space Structures
https://resolver.caltech.edu/CaltechTHESIS:05312021-185024653
Authors: {'items': [{'email': 'ffabienroyer@gmail.com', 'id': 'Royer-Fabien-A', 'name': {'family': 'Royer', 'given': 'Fabien A.'}, 'orcid': '0000-0003-2452-2893', 'show_email': 'NO'}]}
Year: 2021
DOI: 10.7907/ksn2-t598
<p>The overarching goal of the research presented in this dissertation is to apply and extend a newly developed methodology to understand the buckling of complex thin shell structures. This methodology enables the determination of tighter buckling criteria and paves the way to the development of more efficient structures, used closer than ever to their buckling load and even beyond. It would result in dramatically lighter structures to be built and has the potential to enable new applications, such as extremely large aperture satellites.</p>
<p>We first analyze the stability of open section thin shell structures under a pure bending moment, through simulations. These structures are composed of longitudinal thin-shell elements connected transversely by thin rods, and inspired by real spacecraft structures. The present study applies and extends recent work on the stability of cylindrical and spherical shells. The role of localization in the buckling of these structures is investigated and early transitions into the post-buckling regime are unveiled using a probe that locally displaces the structure. The probing method enables the computation of the energy input needed to transition early into a post-buckling state, which is central to determining the critical buckling mechanism for the structure. We show that the structure follows stability landscapes also found in cylindrical and spherical shell buckling problems. This initial computational study is the basis for the first ever probing experiment on a complex structure.</p>
<p>In order to test these new structures under bending, a new bending apparatus is designed and implemented. The boundary conditions are chosen such that the apparatus is statically determinate (isostatic), and no state of self stress can develop in the sample during its mounting and testing. This feature is especially desirable in the study of thin shell structures and their elastic instabilities, for which imperfection sensitivity plays a crucial role in the buckling transition and the post-buckling regime. The accuracy of the isostatic bending machine is first assessed through the testing of rods, and its imperfection insensitive behavior is then highlighted in experiments on tape springs, and through numerical studies of the same structures.</p>
<p>The new bending machine is complemented by a probing apparatus, and the stability of the open section thin-shell structures subjected to a pure bending moment is studied experimentally. The experiment confirms that localization of deformations plays a paramount role in the structure's nonlinear post-buckling regime and is extremely sensitive to imperfections. This characteristic is investigated through probing experiments. The range of moments for which the early buckling of the structure can be triggered using this probe perturbation is determined, as well as the energy barrier separating the pre-buckling and post-buckling states. The stability of the local buckling mode is then illustrated by an experimental stability landscape of shell buckling, and probing is then extended to the entire structure to reveal alternate buckling modes disconnected from the structure's fundamental path. These results can be used to elaborate efficient buckling criteria for this type of structures, through the use of transition diagrams determined experimentally.</p>
<p>Finally, the buckling and post-buckling behavior of ultralight ladder-type coilable structures is investigated. These specific structures are used in the Space Solar Power Project at Caltech and are referred to as strips. Similarly to the previous studies, the stability of strip structures loaded by normal pressure is computationally studied by applying controlled perturbations through localized probing. The probing technique is generalized to higher-order bifurcations along the post-buckling path, and low-energy escape paths into buckling that cannot be predicted by a classical eigenvalue formulation are identified. It is shown that the stability landscape for a pressure-loaded strip is similar to the landscape for classical shells, and the open section thin shell structure studied initially in this thesis. While classical shell structures buckle catastrophically, strip structures feature a large stable post-buckling range. Probing enables the full characterization of the structure's unstable behavior, which paves the way to extend its operation closer than ever to the buckling load, and even in the post-buckling regime. It would enable the design of more efficient structures by dramatically reducing their mass, therefore enabling new large spacecraft to be built.</p>https://thesis.library.caltech.edu/id/eprint/14209Control of Wall-Bounded Turbulence Through Closed-Loop Wall Transpiration
https://resolver.caltech.edu/CaltechTHESIS:05272021-055610816
Authors: {'items': [{'email': 's.todtli@posteo.de', 'id': 'Toedtli-Simon-Silvio', 'name': {'family': 'Toedtli', 'given': 'Simon Silvio'}, 'orcid': '0000-0001-9371-9572', 'show_email': 'NO'}]}
Year: 2021
DOI: 10.7907/me3y-te05
<p>Many wall-bounded flows of practical relevance are turbulent, including the flows past airplanes and ships. The turbulent motions enhance momentum mixing and, as a result, the drag force on the engineering surface increases, for transportation vessels typically by at least a factor of two compared to laminar flow. Turbulent flow control aimed at drag reduction therefore has the potential to deliver enormous energetic and economic savings, but many challenges remain despite active research for well over a century. The present thesis aims to contribute towards two open questions of the field: first, what are suitable controller design tools for high Reynolds number flows? And second, how does actuation through closed-loop wall transpiration change the flow physics? We investigate aspects of these questions through direct numerical simulation (DNS) and modal analyses of an example control scheme, which is applied to a low Reynolds number turbulent channel flow. The controller is a generalization of the opposition control scheme, and introduces a phase shift between the Fourier transformed sensor measurement and actuator response.</p>
<p>The first part of the thesis demonstrates that a low-order model based on the resolvent framework is able to approximate the drag reduction results of DNS over the entire parameter space considered. The model is about two orders of magnitude cheaper to evaluate than DNS at low Reynolds numbers, and we present a strategy based on subsampling of the wave number space and analytical scaling laws that enables model-based flow control design at technologically relevant Reynolds numbers. The second part of the thesis shows that the physics of the controlled flow can be understood from two distinct families of spatial scales, termed streamwise-elongated and spanwise-elongated scales, respectively. Wall transpiration with streamwise-elongated scales attenuates or amplifies the near-wall cycle and therefore leads to drag reduction or increase, depending on the phase shift. In contrast, wall transpiration with spanwise-elongated scales only leads to drag increase, which occurs at positive phase shifts and is due to the appearance of spanwise rollers which largely enhance momentum mixing. Both patterns are robust features of flows with closed-loop wall transpiration, and the present study offers a simple explanation of their origin in terms of phase relations at distinct spatial scales. The findings of this study may set the stage for a unifying framework for various forms of wall transpiration, and implications for future flow control design are discussed.</p>https://thesis.library.caltech.edu/id/eprint/14179Dynamics of Ultralight Flexible Spacecraft During Slew Maneuvers
https://resolver.caltech.edu/CaltechTHESIS:05262022-221946560
Authors: {'items': [{'email': 'michael.a.marshall@protonmail.com', 'id': 'Marshall-Michael-Aaron', 'name': {'family': 'Marshall', 'given': 'Michael Aaron'}, 'orcid': '0000-0002-4259-2484', 'show_email': 'NO'}]}
Year: 2022
DOI: 10.7907/w6na-w476
<p>Traditional spacecraft design paradigms rely on stiff structures with comparatively flexible appendages. More recent trends, however, trade deployed stiffness for packaging efficiency to stow increasingly large-area apertures inside existing launch vehicles. By leveraging recent advances in materials and structures, these ultralight, packageable, and deployable spacecraft, hereafter referred to as ultralight flexible spacecraft, are up to several orders of magnitude lighter and more flexible than the current state-of-the-art. They promise to deliver higher performance for a wide range of applications, but this comes at a cost, in this case, due to their very low-frequency structural dynamics. Structural dynamics can negatively interact with spacecraft attitude control systems and degrade pointing performance.</p>
<p>These developments motivate the main objective of this thesis: to demonstrate the feasibility and limitations of maneuvering next-generation ultralight flexible spacecraft. To that end, the thesis proposes a quantitative method for determining structure-based performance limits for flexible spacecraft slew maneuvers using reduced-order modal models. It then develops a geometrically nonlinear flexible multibody dynamics finite element model of a representative ultralight flexible spacecraft based on the Caltech Space Solar Power Project architecture to validate this method. The results demonstrate that contrary to common assumptions, other constraints impose more restrictive limits on slew maneuver performance than the dynamics of the structure. In particular, they show that the available attitude control system momentum and torque are often significantly more limiting than the structure. Consequently, these results suggest that spacecraft structures can either be (i) maneuvered significantly faster, assuming suitable actuators are available, or (ii) built using lighter-weight, less-stiff, and lower-cost construction that moves the structure-based performance limits closer to those of the rest of the system. Thus, there is a significant opportunity to design less-conservative, higher-performance space systems.</p>https://thesis.library.caltech.edu/id/eprint/14629Characterization and Optimization of a Fully Passive Flapping Foil in an Unsteady Environment for Power Production and Propulsion
https://resolver.caltech.edu/CaltechTHESIS:05312022-024822211
Authors: {'items': [{'email': 'morglhooper@gmail.com', 'id': 'Hooper-Morgan-Louise', 'name': {'family': 'Hooper', 'given': 'Morgan Louise'}, 'show_email': 'NO'}]}
Year: 2022
DOI: 10.7907/tymc-1985
<p>This thesis provides an experimental window into the duality between thrust production and energy harvesting by a flapping foil subject to unsteadiness in an oncoming flow. In particular, an airfoil is placed downstream of a circular cylinder, and allowed to interact with the vorticity shed in its wake to produce motions in both the transverse and streamwise directions. It is confirmed that under the right conditions, passive fluid-structure interactions arising from such a configuration give rise to simultaneous extraction of energy from the flow, coupled with net thrust larger than net drag experienced by the airfoil.</p>
<p>Measurements of forces acting on the airfoil and the motion that arises are presented, for cases where the flapping motion is both active (the foil is driven through a pre-planned trajectory) and fully passive (the foil is allowed to react to the fluid forcing it experiences). These are coupled with simultaneous Particle Image Velocimetry (PIV) measurements of the flow field in the region of the airfoil. These measurements allow for direct observation of fluid-structure interactions which give rise to both thrust production and power extraction potential, illuminating the mechanisms driving each. The dynamics of a fully passive flapping foil are largely determined by the mounting system used to facilitate its motion. It is shown that by leveraging Cyber-Physical Fluid Dynamics (CPFD) capabilities to tune these mounting parameters, the behaviour of a fully passive flapping foil can be made similar to that of a representative driven system. A framework based on a simplified linear model for mounting system dynamics is presented, to allow for the optimization of such a system for power extraction potential subject to relevant engineering constraints. The effects of nonlinearity on airfoil behaviour, particularly those due to friction in the mechanism(s) permitting passive flapping, are also explored. Finally, two-dimensional motion of a fully passive flapping foil is demonstrated, allowing for the foil to travel upstream against the oncoming flow solely due to forces induced by interactions with oncoming unsteadiness.</p>https://thesis.library.caltech.edu/id/eprint/14653Folding and Dynamic Deployment of Ultralight Thin-Shell Space Structures
https://resolver.caltech.edu/CaltechTHESIS:05292023-160132013
Authors: {'items': [{'email': 'harshaastronaut@gmail.com', 'id': 'Reddy-Narravula-Harshavardhan', 'name': {'family': 'Reddy', 'given': 'Narravula Harshavardhan'}, 'orcid': '0000-0003-3897-8162', 'show_email': 'NO'}]}
Year: 2023
DOI: 10.7907/m7rd-6s86
<p>Thin-shell structures are becoming increasingly popular for space missions due to their high stiffness-to-mass ratio, easy folding and coiling, and self-deployment using stored strain energy. Broadly, two deployment strategies exist: 1) controlled or deterministic, and 2) unconstrained. Controlled deployment involves carefully orchestrated events using control or guidance systems, while in unconstrained deployment, the structure is simply allowed to self-deploy with minimal guidance. Unconstrained deployment offers lighter deployment mechanisms and better packaging efficiency but the unpredictability of this process has been a significant obstacle to its adoption.</p>
<p>This study focuses on demonstrating the predictability of unconstrained dynamic deployment of thin-shell structures, using the Caltech Space Solar Power Project (SSPP) structures as a case study. The Caltech SSPP uses composite triangular rollable and coilable longerons as the primary building blocks to create large bending-stiff structures. The specific objective is to improve the predictability and robustness of the unconstrained dynamic deployment of the Caltech SSPP structures. Deployment is influenced by the initial conditions and the interaction between the structure and the mechanism during the deployment. To understand these effects, high-fidelity numerical simulations are developed and validated against experiments. The study also examines the sensitivity of deployment characteristics to various design parameters and external influences to ensure the robustness of deployment.</p>
<p>This research demonstrates that the interaction between the structure and the deployment mechanism must be minimal to ensure the predictability of deployment, as thin-shell structures can self-deploy using stored strain energy. This study's sensitivity analysis will inform the design of future SSPP deployment mechanisms and structures. Additionally, the numerical simulation techniques developed have broader applicability beyond this specific case study to any deployable thin-shell structure.</p>
<p>Due to the large aspect ratios of thin-shell structures, a very fine finite element mesh is required to model them accurately. A dense finite element mesh is also required to model the contact interactions between the structure and the rigid components of the deployment mechanism. As large spacecraft structures become increasingly complex, full-scale numerical modeling becomes impractical, necessitating the search for more computationally efficient finite element methods. In this study, NURBS-based isogeometric analysis is explored, and it is shown that it is not yet worth switching to NURBS-based elements for the analysis of thin-shell deployable structures. In addition, h-adaptive meshing for quadrilateral shell elements is investigated, and more efficient refinement indicators and solution mapping techniques for nonlinear analyses are proposed and their superior performance is demonstrated using a test case of quasi-static folding of a tape spring.</p>
<p>This thesis fills a gap in the literature on unconstrained dynamic deployment of space structures, providing crucial insights and numerical modeling tools for further research. It establishes a knowledge and resource foundation to advance space structure design and promote more frequent use of unconstrained deployment, marking a pivotal contribution to the field and enabling safe and efficient space structure deployment. Furthermore, the study provides insights into more computationally efficient finite element methods, such as h-adaptive meshing. These insights are broadly applicable and can inform the design of future deployable structures beyond the tested cases.</p>https://thesis.library.caltech.edu/id/eprint/15223The "Interpolated Factored Green Function" Method
https://resolver.caltech.edu/CaltechTHESIS:07072022-003500251
Authors: {'items': [{'email': 'christoph.bauinger@gmail.com', 'id': 'Bauinger-Christoph', 'name': {'family': 'Bauinger', 'given': 'Christoph'}, 'show_email': 'NO'}]}
Year: 2023
DOI: 10.7907/1cnc-s558
<p>This thesis presents a novel <i>Interpolated Factored Green Function</i> (IFGF) method for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of <i>O</i>(<i>N</i> log <i>N</i>) operations for an <i>N</i>-point surface mesh. The IFGF strategy capitalizes on slow variations inherent in a certain Green function <i>analytic factor</i>, which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. In fact, a (hybrid MPI-OpenMP) parallel implementation of the IFGF algorithm is proposed in this thesis which results in highly efficient data communication, and which exhibits in practice excellent parallel scaling up to large numbers of cores -- without any hard limitations on the number of cores concurrently employed with high efficiency. Moreover, on any given number of cores, the proposed parallel approach preserves the linearithmic (<i>O</i>(<i>N</i> log <i>N</i>)) computing cost inherent in the sequential version of the IFGF algorithm. This thesis additionally introduces a complete acoustic scattering solver that incorporates the IFGF method in conjunction with a suitable singular integration scheme. A variety of numerical results presented in this thesis illustrate the character of the proposed parallel IFGF-accelerated acoustic solver. These results include applications to several highly relevant engineering problems, e.g., problems concerning acoustic scattering by structures such as a submarine and an aircraft-nacelle geometry, thus establishing the suitability of the IFGF method in the context of real-world engineering problems. The theoretical properties of the IFGF method, finally, are demonstrated by means of a variety of numerical experiments which display the method's serial and parallel linearithmic scaling as well as its excellent weak and strong parallel scaling -- for problems of up to 4,096 wavelengths in acoustic size, and scaling tests spanning from 1 compute core to all 1,680 cores available in the High Performance Computing cluster used.</p>https://thesis.library.caltech.edu/id/eprint/14968Numerical Analysis of Folding and Deployment Dynamics of Thin Shell Structures with Localized Folds
https://resolver.caltech.edu/CaltechTHESIS:06012023-233806562
Authors: {'items': [{'email': 'giaf.can@gmail.com', 'id': 'Canales-Escobedo-Fabricio-Gianfranco', 'name': {'family': 'Canales Escobedo', 'given': 'Fabricio Gianfranco'}, 'orcid': '0000-0001-9071-3263', 'show_email': 'YES'}]}
Year: 2023
DOI: 10.7907/gt81-0s18
This thesis focuses on the analysis of tape springs folded in the opposite sense and their dynamic deployment, and aims to use methods to reduce the computational cost of the analysis. The tape spring is a thin shell deployable structure that has features in common with other deployable structures. The deployment process of such structures can be difficult to predict, and the use of numerical models can be a more cost-effective alternative to experimental testing. Approaches to reduce the computational cost of the analysis of tape springs are investigated such as adaptive meshing and reduced order models. The thesis also presents an accurate analysis of tape spring deployment and a detailed study of the energies and the physics of the deployment. This is used to investigate the energy leak observed in previous tape spring deployment work. Overall, this thesis contributes to improving the efficiency and accuracy of the analysis of deployable structures, particularly tape springs, which can have significant applications in spacecraft technology.https://thesis.library.caltech.edu/id/eprint/15271Singularity Formation in the High-Dimensional Euler Equations and Sampling of High-Dimensional Distributions by Deep Generative Networks
https://resolver.caltech.edu/CaltechTHESIS:09202022-034157716
Authors: {'items': [{'email': 'zhangsm1995@gmail.com', 'id': 'Zhang-Shumao', 'name': {'family': 'Zhang', 'given': 'Shumao'}, 'orcid': '0000-0003-3071-3362', 'show_email': 'NO'}]}
Year: 2023
DOI: 10.7907/8had-3a90
<p>High dimensionality brings both opportunities and challenges to the study of applied mathematics. This thesis consists of two parts. The first part explores the singularity formation of the axisymmetric incompressible Euler equations with no swirl in ℝⁿ, which is closely related to the Millennium Prize Problem on the global singularity of the Navier-Stokes equations. In this part, the high dimensionality contributes to the singularity formation in finite time by enhancing the strength of the vortex stretching term. The second part focuses on sampling from a high-dimensional distribution using deep generative networks, which has wide applications in the Bayesian inverse problem and the image synthesis task. The high dimensionality in this part becomes a significant challenge to the numerical algorithms, known as the curse of dimensionality.</p>
<p>In the first part of this thesis, we consider the singularity formation in two scenarios. In the first scenario, for the axisymmetric Euler equations with no swirl, we consider the case when the initial condition for the angular vorticity is C<sup>α</sup> Hölder continuous. We provide convincing numerical examples where the solutions develop potential self-similar blow-up in finite time when the Hölder exponent α < α*, and this upper bound α* can asymptotically approach 1 - 2/n. This result supports a conjecture from Drivas and Elgindi [37], and generalizes it to the high-dimensional case. This potential blow-up is insensitive to the perturbation of initial data. Based on assumptions summarized from numerical experiments, we study a limiting case of the Euler equations, and obtain α* = 1 - 2/n which agrees with the numerical result. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the Euler equations. This one-dimensional model might suggest a possible way to show this finite-time blow-up scenario analytically. Compared to the first proved blow-up result of the 3D axisymmetric Euler equations with no swirl and Hölder continuous initial data by Elgindi in [40], our potential blow-up scenario has completely different scaling behavior and regularity of the initial condition. In the second scenario, we consider using smooth initial data, but modify the Euler equations by adding a factor ε as the coefficient of the convection terms to weaken the convection effect. The new model is called the weak convection model. We provide convincing numerical examples of the weak convection model where the solutions develop potential self-similar blow-up in finite time when the convection strength ε < ε*, and this upper bound ε* should be close to 1 - 2/n. This result is closely related to the infinite-dimensional case of an open question [37] stated by Drivas and Elgindi. Our numerical observations also inspire us to approximate the weak convection model with a one-dimensional model. We give a rigorous proof that the one-dimensional model will develop finite-time blow-up if ε < 1 - 2/n, and study the approximation quality of the one-dimensional model to the weak convection model numerically, which could be beneficial to a rigorous proof of the potential finite-time blow-up.</p>
<p>In the second part of the thesis, we propose the Multiscale Invertible Generative Network (MsIGN) to sample from high-dimensional distributions by exploring the low-dimensional structure in the target distribution. The MsIGN models a transport map from a known reference distribution to the target distribution, and thus is very efficient in generating uncorrelated samples compared to MCMC-type methods. The MsIGN captures multiple modes in the target distribution by generating new samples hierarchically from a coarse scale to a fine scale with the help of a novel prior conditioning layer. The hierarchical structure of the MsIGN also allows training in a coarse-to-fine scale manner. The Jeffreys divergence is used as the objective function in training to avoid mode collapse. Importance sampling based on the prior conditioning layer is leveraged to estimate the Jeffreys divergence, which is intractable in previous deep generative networks. Numerically, when applied to two Bayesian inverse problems, the MsIGN clearly captures multiple modes in the high-dimensional posterior and approximates the posterior accurately, demonstrating its superior performance compared with previous methods. We also provide an ablation study to show the necessity of our proposed network architecture and training algorithm for the good numerical performance. Moreover, we also apply the MsIGN to the image synthesis task, where it achieves superior performance in terms of bits-per-dimension value over other flow-based generative models and yields very good interpretability of its neurons in intermediate layers.</p>https://thesis.library.caltech.edu/id/eprint/15033General Domain FC-Based Shock Dynamics Solver
https://resolver.caltech.edu/CaltechTHESIS:03152024-221312028
Authors: {'items': [{'email': 'dleibovi@gmail.com', 'id': 'Leibovici-Daniel-Victor', 'name': {'family': 'Leibovici', 'given': 'Daniel Victor'}, 'orcid': '0009-0007-8267-4250', 'show_email': 'YES'}]}
Year: 2024
DOI: 10.7907/bd5r-4q30
This thesis presents a novel FC-SDNN (Fourier Continuation Shock-detecting Neural Network) spectral scheme for the numerical solution of nonlinear conservation laws in general domains and under arbitrary boundary conditions, without the limiting CFL constraints inherent in other spectral schemes for general domains. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth artificial viscosity assignments localized in regions detected by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions—as evidenced by an essential absence of spurious oscillations in contour levels. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method, especially in the vicinity of contact discontinuities. The approach, which does not require the use of otherwise ubiquitous positivity-preserving limiters, enjoys a great geometrical flexibility on the basis of an overlapping-patch discretization. This allows its application for the simulation of supersonic and hypersonic flows and shocks, including Euler simulations at significantly higher speeds than previously achieved, such as e.g. Mach 25 re-entry flow speeds, impinging upon complex physical obstacles. This multi-domain approach is suitable for efficient parallelization on large computer clusters, and the MPI implementation proposed in this thesis enjoys high parallel scalability and in particular perfect weak scaling, as demonstrated by simulations on general complex domains. The character of the proposed algorithm is demonstrated through a variety of numerical tests for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains, with results in accordance with physical theory and prior experimental and computational results up to and including both supersonic and hypersonic regimes.https://thesis.library.caltech.edu/id/eprint/16329Linear Amplification in Nonequilibrium Turbulent Boundary Layers
https://resolver.caltech.edu/CaltechTHESIS:08312023-005517217
Authors: {'items': [{'email': 'rey.gomez@berkeley.edu', 'id': 'Gomez-De-La-Cruz-Salvador-Rey', 'name': {'family': 'Gomez De La Cruz', 'given': 'Salvador Rey'}, 'orcid': '0000-0002-7568-721X', 'show_email': 'YES'}]}
Year: 2024
DOI: 10.7907/hn98-c285
<p>Resolvent analysis is applied to nonequilibrium incompressible adverse pressure gradient (APG) turbulent boundary layers (TBL) and hypersonic boundary layers with high temperature real gas effects, including chemical nonequilibrium. Resolvent analysis is an equation-based, scale-dependent decomposition of the Navier Stokes equations, linearized about a known mean flow field. The decomposition identifies the optimal response and forcing modes, ranked by their linear amplification. To treat the nonequilibrium APG TBL, a biglobal resolvent analysis approach is used to account for the streamwise and wall-normal inhomogeneities in the streamwise developing flow. For the hypersonic boundary layer in chemical nonequilibrium, the resolvent analysis is constructed using a parallel flow assumption, incorporating N₂, O₂, NO, N, and O as a mixture of chemically reacting gases.</p>
<p>Biglobal resolvent analysis is first applied to the zero pressure gradient (ZPG) TBL. Scaling relationships are determined for the spanwise wavenumber and temporal frequency that admit self-similar resolvent modes in the inner layer, mesolayer, and outer layer regions of the ZPG TBL. The APG effects on the inner scaling of the biglobal modes are shown to diminish as their self-similarity improves with increased Reynolds number. An increase in APG strength is shown to increase the linear amplification of the large-scale biglobal modes in the outer region, similar to the energization of large scale modes observed in simulation. The linear amplification of these modes grows linearly with the APG history, measured as the streamwise averaged APG strength, and relates to a novel pressure-based velocity scale.</p>
<p>Resolvent analysis is then used to identify the length scales most affected by the high-temperature gas effects in hypersonic TBLs. It is shown that the high-temperature gas effects primarily affect modes localized near the peak mean temperature. Due to the chemical nonequilibrium effects, the modes can be linearly amplified through changes in chemical concentration, which have non-negligible effects on the higher order modes. Correlations in the components of the small-scale resolvent modes agree qualitatively with similar correlations in simulation data.</p>
<p>Finally, efficient strategies for resolvent analysis are presented. These include an algorithm to autonomously sample the large amplification regions using a Bayesian Optimization-like approach and a projection-based method to approximate resolvent analysis through a reduced eigenvalue problem, derived from calculus of variations.</p>https://thesis.library.caltech.edu/id/eprint/16170Numerical Simulations of Cavitating Bubbles in Elastic and Viscoelastic Materials for Biomedical Applications
https://resolver.caltech.edu/CaltechTHESIS:10162023-141935060
Authors: {'items': [{'email': 'jeansebastien.spratt@gmail.com', 'id': 'Spratt-Jean-Sébastien-Alexandre', 'name': {'family': 'Spratt', 'given': 'Jean-Sébastien Alexandre'}, 'orcid': '0000-0002-1962-4214', 'show_email': 'NO'}]}
Year: 2024
DOI: 10.7907/g34e-6p65
<p>The interactions of cavitating bubbles with elastic and viscoelastic materials play a central role in many biomedical applications. This thesis makes use of numerical modeling and data-driven approaches to characterize soft biomaterials at high strain rates via observation of bubble dynamics, and to model burst-wave lithotripsy, a focused ultrasound therapy to break kidney stones.</p>
<p>In the first part of the thesis, a data assimilation framework is developed for cavitation rheometry, a technique that uses bubble dynamics to characterize soft, viscoelastic materials at high strain-rates. This framework aims to determine material properties that best fit observed cavitating bubble dynamics. We propose ensemble-based data assimilation methods to solve this inverse problem. This approach is validated with surrogate data generated by adding random noise to simulated bubble radius time histories, and we show that we can confidently and efficiently estimate parameters of interest within 5% given an iterative Kalman smoother approach and an ensemble- based 4D-Var hybrid technique. The developed framework is applied to experimental data in three distinct settings, with varying bubble nucleation methods, cavitation media, and using different material constitutive models. We demonstrate that the mechanical properties of gels used in each experiment can be estimated quickly and accurately despite experimental inconsistencies, model error, and noisy data. The framework is used to further our understanding of the underlying physics and identify limitations of our bubble dynamics model for violent bubble collapse.</p>
<p>In the second part of the thesis, we simulate burst-wave lithotripsy (BWL), a non- invasive treatment for kidney stones that relies on repeated short bursts of focused ultrasound. Numerical approaches to study BWL require simulation of acoustic waves interacting with solid stones as well as bubble clouds which can nucleate ahead of the stone. We implement and validate a hypoelastic material model, which, with the addition of a continuum damage model and calibration of a spherically- focused transducer array, enables us to determine how effective various treatment strategies are with arbitrary stones. We present a preliminary investigation of the bubble dynamics occurring during treatment, and their impact on damage to the stone. Finally, we propose a strategy to reduce shielding by collapsing bubbles ahead of the stone via introduction of a secondary, low-frequency ultrasound pulse during treatment.</p>https://thesis.library.caltech.edu/id/eprint/16208