Call number: LD2668 .T4 1986 R57 / Master of Science / Mechanical and Nuclear Engineering

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Vita. / Includes bibliographical references. / by Warren Steven Ross. / Ph.D.

This thesis deals with the problem of the asymptotic behavior of solutions to several nonlinear equations from fluid dynamics on both mesoscopic and macroscopic levels, including Boltzmann equation, compressible Navier-Stokes equations and the system of viscous conservation laws with positive definite viscosity matrix. The main purpose is to study the asymptotic behavior of solutions to those equations towards linear and nonlinear waves, such as shock waves, rarefaction waves and contact discontinuities as either the times goes to infinity, or the viscosity and heat conductivity go to zero for the macroscopic equations or the mean free path goes to zero for the mesoscopic equations. Those limit processes are singular. For the system of viscous conservation laws, we show the large time asymptotic nonlinear stability of a superposition of viscous shock waves and viscous contact waves for the system of viscous conservation laws with small initial perturbations, provided that the strengths of these viscous waves are small and of the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a new estimate on the heat equation. For the Boltzmann equation, the main purpose is to study the asymptotic equivalence for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered-rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t = 0) as the mean free path epsilon → 0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. As a by-product, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat-conductivity (depending on the viscosity) is also obtained. / Zeng, Huihui. / Adviser: Zhouping Xin. / Source: Dissertation Abstracts International, Volume: 70-09, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 102-110). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

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Bibliography : leaves 211-216 / 216 leaves : ill ; 20 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1994?

Thesis (Ph.D)--Technische Universiteit Eindhoven, 2005. / Title from document title page. Title from title screen (viewed on Dec. 6, 2007). Includes bibliographical references. Available in PDF format via the World Wide Web.

This paper investigates the atmospheric circulation and transport characteristics of the Martian atmosphere (as modeled by a Mars GCM) for three sets of conditions. The conditions are based on a combination of season and dust loading (as parameterized by the optical depth, τ). The first experiment is for the Northern Spring Equinox with no dust loading (τ=0). Experiment 2 is for Northern Hemisphere Winter Solstice with no dust loading. Experiment 3 is for Northern Hemisphere Winter Solstice under moderately dusty conditions (τ=1.0). These cases allow a comparison between seasons and a look at the effects of dust in the atmosphere on the circulation and transport processes. After presenting some of the theoretical and mathematical background pertinent to atmospheric transport and circulation the results of the study are given. These include analyses of the zonal-mean winds, the time-evolution of the mean tracer field, the mean meridional circulation, and the effective transport circulation [Plumb and Mahlman, 1987]. In addition we estimate the time scales for "stratospheric" overturning and calculate a set of eddy diffusion, coefficients (K[subscript yy] and K[subscript zz]) for each case. These coefficients are a means of parameterizing the strength of eddy mixing. Others [Conrath, 1971; Zurek, 1976; Kong and McElroy, 1977; Toon et al., 1977; Anderson and Leovy, 1987] have estimated, using various methods, values for the vertical diffusion coefficient K[subscript zz] of the order of 10³ m²/s. The results here show that there is no "typical" value of K[subscript zz] (or K[subscript yy]) which can be used to characterize the atmosphere globally, and K[subscript zz] seldom reaches 10³ m²/s except in isolated regions and/or under dusty conditions. Both K[subscript yy] and K[subscript zz] are dependent upon season, dust loading, and location in the atmosphere. In addition to identifying the regions of strong mixing, probable sources of the eddy activity which is responsible for the mixing are discussed. In all three cases the effective transport circulation (which includes both advection and diffusion) is structurally similar to the mean meridional circulation but somewhat more intense. The Martian equinox circulation is structurally similar to Earth's circulation; both are characterized by a dual Hadley cell system with rising branch over the equator, poleward flow aloft, and return flow at low levels. The mean zonal winds are westerly in both hemispheres with easterlies near the ground and at high altitudes over the equator. The jet stream in the northern hemisphere peaks at 45 m/s at equinox. Unlike the Earth, Mars' circulation changes dramatically with the seasons. For solstice conditions the mean meridional circulation is characterized by a large, intense cross-equatorial Hadley cell which dominates the circulation pattern. The mean zonal winds are now predominately westerly in the northern winter hemisphere and easterly in the southern hemisphere. The westerly jet reaches 95 m/s while the easterly jet reaches 30 m/s. There is a band of westerlies (up to 10 m/s) found in low southern latitudes near the ground. Dust in the atmosphere acts to intensify the strength of the circulation (while having little effect on the structure); there is a two- to three-fold increase in the strength of the mean winds between the two winter solstice experiments. / Graduation date: 1995

Benchmark calculations are performed for neutron transport in a two material (binary) stochastic multiplying medium. Spatial, angular, and energy dependence are included. The problem considered is based on a fuel assembly of a common pressurized water nuclear reactor. The mean chord length through the assembly is determined and used as the planar geometry system length. According to assumed or calculated material distributions, this system length is populated with alternating fuel and moderator segments of random size. Neutron flux distributions are numerically computed using a discretized form of the Boltzmann transport equation employing diffusion synthetic acceleration. Average quantities (group fluxes and k-eigenvalue) and variances are calculated from an ensemble of realizations of the mixing statistics. The effects of varying two parameters in the fuel, two different boundary conditions, and three different sets of mixing statistics are assessed. A probability distribution function (PDF) of the k-eigenvalue is generated and compared with previous research. Atomic mix solutions are compared with these benchmark ensemble average flux and k-eigenvalue solutions. Mixing statistics with large standard deviations give the most widely varying ensemble solutions of the flux and k-eigenvalue. The shape of the k-eigenvalue PDF qualitatively agrees with previous work. Its overall shape is independent of variations in fuel cross-sections for the problems considered, but its width is impacted by these variations. Statistical distributions with smaller standard deviations alter the shape of this PDF toward a normal distribution. The atomic mix approximation yields large over-predictions of the ensemble average k-eigenvalue and under-predictions of the flux. Qualitatively correct flux shapes are obtained, however. These benchmark calculations indicate that a model which includes higher statistical moments of the mixing statistics is needed for accurate predictions of binary stochastic media k-eigenvalue problems. This is consistent with previous findings. / Graduation date: 2005

The subject of this thesis is the development of a nodal discretization of the low-order quasi-diffusion (QDLO) equations for global reactor core calculations. The advantage of quasi-diffusion (QD) is that it is able to capture transport effects at the surface between unlike fuel assemblies better than the diffusion approximation. We discretize QDLO equations with the advanced nodal methodology described by Palmtag (Pal 1997) for diffusion. The fast and thermal neutron fluxes are presented as 2-D, non-separable expansions of polynomial and hyperbolic functions. The fast flux expansion consists of polynomial functions, while the thermal flux is expanded in a combination of polynomial and hyperbolic functions. The advantage of using hyperbolic functions in the thermal flux expansion lies in the accuracy with which hyperbolic functions can represent the large gradients at the interface between unlike fuel assemblies. The hyperbolic expansion functions proposed in (Pal 1997) are the analytic solutions of the zero-source diffusion equation for the thermal flux. The specific form of the QDLO equations requires the derivation of new hyperbolic basis functions which are different from those proposed for the diffusion equation. We have developed a discretization of the QDLO equations with node-averaged cross-sections and Eddington tensor components, solving the 2-D equations using the weighted residual method (Ame 1992). These node-averaged data are assumed known from single assembly transport calculations. We wrote a code in "Mathematica" that solves k-eigenvalue problems and calculates neutron fluxes in 2-D Cartesian coordinates. Numerical test problems show that the model proposed here can reproduce the results of both the simple diffusion problems presented in (Pal 1997) and those with analytic solutions. While the QDLO calculations performed on one-node, zero-current, boundary condition diffusion problems and two-node, zero-current boundary condition problems with UO₂-UO₂ assemblies are in excellent agreement with the benchmark and analytic solutions, UO₂-MOX configurations show more important discrepancies that are due to the single-assembly homogenized cross-sections used in the calculations. The results of the multiple-node problems show similar discrepancies in power distribution with the results reported in (Pal 1997). Multiple-node k-eigenvalue problems exhibit larger discrepancies, but these can be diminished by using adjusted diffusion coefficients (Pal 1997). The results of several "transport" problems demonstrate the influence of Eddington functionals on homogenized flux, power distribution, and multiplication factor k. / Graduation date: 2003

The implicit Monte Carlo (IMC) method [16] for radiative transfer, developed in 1971, provides numerical solutions to the tightly-coupled, highly-nonlinear radiative heat transfer equations in many physical situations. Despite its popularity, there are instances of overheating in the solution for particular choices of time steps and spatial grid sizes. To prevent overheating, conditions on teh time step size Δt have been sought to ensure that the implicit Monte Carlo (IMC) equations satisfy a maximum principle. Most recently, a discrete maximum principle (DMP) for teh IMC equations has been developed [32] that predicts the necessary time step size for boundedness given the spatial grid size. Predictions given by this DMP assumed equilibrium thermal initial conditions, was developed using pseudo-analytic and symbolic algebra tools that are computationally expensive, has only been applied to one-dimensional Marshak wave problems, and has not considered the evolution of the DMP predictions over multiple time steps. These limitations restrict the utility of the DMP predictions. We extend the DMP derivation to overcome these limitations and provide an algorithm that can be introduced into IMC codes with minimal impact on simulation CPU time. This extended DMP effectively treats non-equilibrium thermal initial conditions, decreases calculation time by using multigroup approximations in frequency, considers multiple spatial dimensions with an arbitrary number of neighboring sources, and overcomes inherent difficulties for the DMP in time-dependent problems. Disequilibrium in the initial conditions is introduced through a redefinition of existing terms from [32] to different radiation and material temperatures on the first time step. This results in a limiting DMP inequality similar in form to the original. Multifrequency approximations are then applied by assuming separation of variables. Energy deposition from multiple sources is assumed to follow linear superposition and the DMP from [32] is re-derived to incorporate multiple incident sources of energy in multiple dimensions. Lastly, an inherent flaw in the DMP resulting in poor predictions when temperature varies slowly over a region is overcome by developing a threshold temperature difference, above which the DMP operates. We have numerically implemented these improvements and validated the results against IMC solutions, showing the predictive capacity of the more general DMP algorithm. We find the disequlibrium conditions to be properly incorporated into the DMP, and multifrequency approximations to be accurate over a large range of time step and spatial grid sizes. The linear superposition assumption is generally very accurate, but infrequently leads to DMP predictions which are not conservative. We also demonstrate that the temperature difference threshold prevents inaccurate predictions by the DMP while preserving its functionality. / Graduation date: 2013