No description available.

There has been an increasing focus on using nanoscale devices for various applications ranging from computer components to biomechanical sensors. In order to effectively design devices of this size, it is important to understand the properties of materials at this length scale and their relevant transport equations. At everyday length scales, heat transport is governed by Fourier’s law, but at the nanoscale, it becomes increasingly inaccurate. Phonon kinetic theory can be used to develop more accurate governing equations. We present the moment method, which takes integral moments of the phonon Boltzmann kinetic equation to develop a set of equations based on macroscopic properties such as energy and heat flux. The advantage of using this method is that transport properties in nanodevices can be approximated analytically and efficiently. A number of simplifying assumptions are used in order to linearize the equations. Boundary conditions for the moment method are derived based on a microscopic model of phonons interacting with a surface by scattering, reflection or thermalization. Several simple, one dimensional problems are solved using the moment method equation. The results show the effects of phonon surface interactions and how they affect overal properties of a nanoscale device. Some of these effects were observed in a recent experiment and are replicated by other modeling techniques. Although the moment method has described some effects of nanoscale heat transfer, the model is limited by some of its simplifying assumptions. Several of these simplifying assumptions could be removed for greater accuracy, but it would introduce non-linearity into the moment method. / Graduate

This research is concerned with the development of a mathematical approach for energy and mass transport in solidification modelling involving a control volume (CV) technique and finite element method (FEM) and incorporating non-physical variables in its solution. The former technique is used to determine an equivalent capacitance to describe energy transport whilst the latter technique provides temperatures over the material domain. The numerical solution of the transport equations is achieved by the introduction of two concepts, i.e. weighted transport equations and non-physical variables. The main aim is to establish equivalent transport equations that allow exact temporal integration and describe the behaviour of non-physical variables to replace the original governing transport equations. The variables defined are non-physical in the sense that they are dependent on the velocity of the moving CV. This dependence is a consequence of constructing transport equations that do not include flux integrals. The form of the transport equations facilitate the construction of a FEM formulation that is applicable to heat and mass transport problems and caters for singularities arising from phase-change, which can prove difficult to model. However, applying the non-physical enthalpy method (NEM) any singularity involved in the solidification process is precisely identified and annihilated.

First, I attempt to present a unified approach to the study of transport phenomena in multicoponent anisotropic space plasmas. In the limit of small temperature anisotropies this system of generalized transport equations reduces to Grad's 13-moment system of transport equations. In the collisionless limit, the generalized transport equations account for collisionless heat flow, cillisionless viscosity, and large temperature anisotropies. Also, I show that with the appropriate assumptions, the system of generalized transport equations reduces to all of the other major systems of transport equations for anisotropic plasmas that have been derived to date. Next, for application to aeronomy and space physics problems involving strongly magnetized plasma flows, I derive momentum and energy exchange collision terms for interpenetrating bi-Maxwellian gases. Collision terms are derived for Coulomb, Maxwell molecule, and constant collision cross section interaction potentials. The collision terms are valid for arbitrary flow velocity differences and temperature differences between the interacting gases as well as for arbitrary temperature anisotropies. The collision terms have to be evaluated numerically and the appropriate coefficients are presented in tables However, the collision terms are also fitted with simplified expressions, the accuracy of which depends on both the interaction potential and the temperature anisotropy. In addition, I derive the closed set of transport equations that are associated with the momentum and energy collision terms. Finally, I study the extent to which Maxwellian and bi-Maxwellian series expansions can describe plasma flows characterized by non-Maxwellian velocity distributions, with emphasis given to modeling the anisotropic character of the distribution function. The problem considered is the steady state flow of a weakly-ionized plasma subjected to homogeneous electric and magnetic fields, and different collision models are used. In the case of relaxation collision model, a closed form expression is found for the ion velocity distribution function, while for more regorous models (polarization and hard sphere) I have to use the Monte Carlo simulation. These provided a basis for determining the adequacy of a given series expansion. I find that, in general, the bi-Maxwellian-based expansions for the velocity distribution function is better suited to describing anisotropic plasmas than the Maxwellian-based expansions. (166 pages)

Computational Fluid Dynamics (CFD) has been widely used as a tool to analyze physical phenomena involving fluids. To perform a CFD simulation, the governing equations are discretized to formulate a set of nonlinear algebraic equations. Typical spatial discretization schemes include finite-difference methods, finite-volume methods, and finite-element methods. Error introduced in the discretization process is called discretization error and defined as the difference between the exact solution to the discrete equations and the exact solution to the partial differential or integral equations. For most CFD simulations, discretization error accounts for the largest portion of the numerical error in the simulation. Discretization error has a complicated nonlinear relationship with the computational grid and the discretization scheme, which makes it especially difficult to estimate. Thus, it is important to study the discretization error to characterize numerical errors in a CFD simulation. Discretization error estimation is performed using the Error Transport Equations (ETE) in this work. The original nonlinear form of the ETE can be linearized to formulate the linearized ETE. Results of the two types of the ETE are compared. This work implements the ETE for finite-volume methods and Discontinuous Galerkin (DG) finite-element methods. For finite volume methods, discretization error estimates are obtained for both steady state problems and unsteady problems. The work on steady-state problems focuses on turbulent flow modelled by the Spalart-Allmaras (SA) model and Menter's $k-omega$ SST model. Higher-order discretization error estimates are obtained for both the mean variables and the turbulence working variables. The type of pseudo temporal discretization used for the steady-state problems does not have too much influence on the final converged solution. However, the temporal discretization scheme makes a significant difference for unsteady problems. Different temporal discretizations also impact the ETE implementation. This work discusses the implementation of the ETE for the 2-step Backward Difference Formula (BDF2) and the Singly Diagonally Implicit Runge-Kutta (SDIRK) methods. Most existing work on the ETE focuses on finite-volume methods. This work also extends ETE to work with the DG methods and tests the implementation with steady state inviscid test cases. The discretization error estimates for smooth test cases achieve the expected order of accuracy. When the test case is non-smooth, the truncation error estimation scheme fails to generate an accurate truncation error estimate. This causes a reduction of the discretization error estimate to first-order accuracy. Discussions are made on how accurate truncation error estimates can be found for non-smooth test cases. / Doctor of Philosophy / For a general practical fluid flow problem, the governing equations can not be solved analytically. Computational Fluid Dynamics (CFD) approximates the governing equations by a set of algebraic equations that can be solved directly by the computer. Compared to experiments, CFD has certain advantages. The cost for running a CFD simulation is typically much lower than performing an experiment. Changing the conditions and geometry is usually easier for a CFD simulation than for an experiment. A CFD simulation can obtain information of the entire flow field for all field variables, which is nearly impossible for a single experiment setup. However, numerical errors are inherently persistent in CFD simulations due to the approximations made in CFD and finite precision arithmetic of the computer. Without proper characterization of errors, the accuracy of the CFD simulation can not be guaranteed. Numerical errors can even result in false flow features in the CFD solution. Thus, numerical errors need to be carefully studied so that the CFD simulation can provide useful information for the chosen application. The focus of this work is on numerical error estimation for the finite-volume method and the Discontinuous Galerkin (DG) finite-element method. In general, discretization error makes the most significant contribution to the numerical error of a CFD simulation. This work estimates discretization error by solving a set of auxiliary equations derived for the discretization error of a CFD solution. Accurate discretization error estimates are obtained for different test cases. The work on the finite-volume method focus on discretization error estimation for steady state turbulent test cases and unsteady test cases. To the best of the author's knowledge, the implementation of the current discretization error estimation scheme has only been applied as an intermediate step for the error estimation of functionals for the DG method in the literature. Results for steady-state inviscid test cases for the DG method are presented.

This thesis examines the processes of longshore sediment transport in the swash zone of a mixed sand and gravel shoreline, Lake Coleridge, New Zealand. It focuses on the interactions between waves and currents in the swash zone and the resulting sediment transport. No previous study has attempted to concurrently measure wave and current data and longshore sediment transport rates on a mixed sand and gravel lakeshore beach in New Zealand. Many of these beaches, in both the oceanic and lacustrine environments, are in net long-term erosion. It is recognised that longshore sediment transport is a part of this process, but very little knowledge has existed regarding rates of sediment movement and the relationships between waves, currents and swash activity in the foreshore of these beach types. A field programme was designed to measure a comprehensive range of wind, wave, current and morphological variables concurrently with longshore transport. Four electronic instruments were used to measure both waves and currents simultaneously in the offshore, nearshore and swash zone. In the offshore area, an InterOcean S4ADW wave and current meter was installed to record wave height, period, direction and velocity. A WG-30 capacitance wave gauge measured the total water surface variation. A pair of Marsh-McBirney electromagnetic current meters, measuring current directions and velocities were installed in the nearshore and swash zone. Data were sampled for 18 minutes every hour with a Campbell Scientific CR23x data-logger. The wave gauge data was sampled at a rate of 10 Hz (0.1 s) and the two current meters at a rate of 2 Hz (0.5 s). Longshore sediment transport rates were investigated with the use of two traps placed in the nearshore and swash zone to collect sediment transported under wave and swash action. This occurred concurrently with the wave measurements and together yielded over 500 individual hours of high quality time series data. Important new insights were made into lake wave processes in New Zealand's alpine lakes. Measured wave heights averaged 0.20-0.35 m and ranged up to 0.85 m. Wave height was found to be strongly linked to the wind and grew rapidly to increasing wind strength in an exponential fashion. Wave period responded more slowly and required time and distance for the wave length to develop. Overall, there was a narrow band of wave periods with means ranging from 1.43 to 2.33 s. The wave spectrum was found to be more mixed and complicated than had previously been assumed for lake environments. Spectral band width parameters were large, with 95% of the values between 0.75 and 0.90. The wave regime attained the characteristics of a storm wave spectrum. The waves were characteristically steep and capable of obtaining far greater steepness than oceanic wind-waves. Values ranged from 0.010 to 0.074, with an average of 0.051. Waves were able to progress very close to shore without modification and broke in water less than 0.5 m deep. Wave refraction from deep to shallow water only caused wave angles to be altered in the order of 10%. The two main breaker types were spilling and plunging. However, rapid increases in beach slope near the shoreline often caused the waves to plunge immediately landward of the swash zone, leading to a greater proportion of plunging waves. Wave energy attenuation was found to be severe. Measured velocities were some 10 times less at two thirds the water depth beneath the wave. Mean orbital velocities were 0.30 m s⁻¹ in deep water and 0.15 m s⁻¹ in shallow water. The ratio difference between the measured deep water orbital velocities and the nearshore orbital velocities was just under one half (us/uo = 0.58), almost identical to the predicted phase velocity difference by Linear wave theory. In general Linear wave theory was found to provide good approximations of the wave conditions in a small lake environment. The swash zone is an important area of wave dissipation and it defines the limits of sediment transport. The width of the swash zone was found to be controlled by the wave height, which in turn determined the quantity of sediment transported through the swash zone. It ranged in width from 0.05 m to 6.0 m and widened landward in response to increased wave height and lakeward in response the wave length. Slope was found to be an important secondary control on swash zone width. In low energy conditions, swash zone slopes were typically steep. At the onset of wave activity the swash zone becomes scoured by swash activity and the beach slope grades down. An equation was developed, using the wave height and beach slope that provides close estimates of the swash zone width under a wide range of conditions. Run-up heights were calculated using the swash zone width and slope angle. Run-up elevations ranged from 0.01 m to 0.73 m and were strongly related to the wave height and the beach slope. On average, run-up exceeds the deep water wave height by a factor of 1.16H. The highest run-up elevations were found to occur at intermediate slope angles of between 6-8°. Above 8°, the run-up declined in response to beach porosity and lower wave energy conditions. A generalised run-up equation for lake environments has been developed, that takes into account the negative relationship between beach slope and run-up. Swash velocities averaged 0.30 m s⁻¹ but maximum velocities averaged 0.98 m s⁻¹. After wave breaking, swash velocities quickly reduced through dissipation by approximately one half. Swash velocity was strongly linked to wave height and beach slope. Maximum velocities occurred at beach slopes of 5°, where incident swash dominated. At slopes between 6° and 10°, swash velocities were hindered by turbulence, but the relative differences between the swash and backswash flows were negligible. At slope angles above 10° there was a slight asymmetry to the swash/backswash flow velocities due to beach porosity absorbing water at the limits of the swash zone. Three equations were developed for estimating the mean and maximum swash velocity flows. From an analysis of these interactions, a process-response model was developed that formalises the morphodynamic response of the swash zone to wave activity. Longshore sediment transport occurred exclusively in the swash zone, landward of the breaking wave in bedload. The sediments collected in transit were a heterogeneous mix of coarse sands and fine-large gravels. Hourly trapped rates ranged from 0.02 to 214.88 kg hr⁻¹. Numerical methods were developed to convert trapped mass rates in to volumetric rates that use the density and porosity of the sediment. A sediment transport flux curve was developed from measuring the distribution of longshore sediment transport across the swash zone. Using numerical integration, the area under this curve was calculated and an equation written to accurately estimate the total integrated transport rates in the swash zone. The total transport rates ranged from a minimum of 1.10 x 10-5 m³ hr⁻¹ to a maximum of 1.15 m³ hr⁻¹. The mean rate was 7.36 x 10⁻² m³ hr⁻¹. Sediment transport was found to be most strongly controlled by the wave height, period, wave steepness and mean swash velocity. Transport is initiated when waves break at an oblique angle to the shoreline. No relationships could be found between the grain size and transport rates. Instead, the critical threshold velocities of the sediment sizes were almost always exceed in the turbulent conditions under the breaking wave. The highest transport rates were associated with the lowest beach slopes. It was found that this was linked to swash high velocities and wave heights associated with foreshore scouring. An expression was developed to estimate the longshore sediment transport, termed the LEXSED formula, that divides the cube of the wave height and the wave length and multiplies this by the mean swash velocity and the wave approach angle. The expression performs well across a wide range of conditions and the estimates show very good correlations to the empirical data. LEXSED was used to calculate an accurate annual sediment transport budget for the fieldsite beaches. LEXSED was compared to 16 other longshore sediment transport formulas and performed best overall. The underlying principles of the model make its application to other mixed sand and gravel beaches promising.

Une part importante de l’activité en calcul scientifique et analyse numérique est consacrée aux problèmes de transport d’une quantité par un champ donné (ou lui-même calculé numériquement). Les questions de conservations étant essentielles dans ce domaine, on formule en général le problème de façon eulérienne sous la forme d’un bilan au niveau de chaque cellule élémentaire du maillage, et l’on gère l’évolution en suivant les valeurs moyennes dans ces cellules au cours du temps. Une autre approche consiste à suivre les caractéristiques du champ et à transporter les valeurs ponctuelles le long de ces caractéristiques. Cette approche est délicate à mettre en oeuvre, n’assure pas en général une parfaite conservation de la matière transportée, mais peut permettre dans certaines situations de transporter des quantités non régulières avec une grande précision, et sur des temps très longs (sans conditions restrictives sur le pas de temps comme dans le cas des méthodes eulériennes). Les travaux de thèse présentés ici partent de l’idée suivante : dans le cadre des méthodes utilisant un suivi de caractéristiques, transporter une quantité supplémentaire géométrique apportant plus d’informations sur le problème (on peut penser à un tenseur des contraintes dans le contexte de la mécanique des fluides, une métrique sous-jacente lors de l’adaptation de maillage, etc. ). Un premier pan du travail est la formulation théorique d’une méthode de transport de telles quantités. Elle repose sur le principe suivant : utiliser la différentielle du champ de transport pour calculer la différentielle du flot, nous donnant une information sur la déformation locale du domaine nous permettant de modifier nos quantités géométriques. Cette une approche a été explorée dans dans le contexte des méthodes particulaires plus particulièrement dans le domaine de la physique des plasmas. Ces premiers travaux amènent à travailler sur des densités paramétrées par un couple point/tenseur, comme les gaussiennes par exemple, qui sont un contexte d’applications assez naturelles de la méthode. En effet, on peut par la formulation établie transporter le point et le tenseur. La question qui se pose alors et qui constitue le second axe de notre travail est celle du choix d’une distance sur des espaces de densités, permettant par exemple d’étudier l’erreur commise entre la densité transportée et son approximation en fonction de la « concentration » au voisinage du point. On verra que les distances Lp montrent des limites par rapport au phénomène que nous souhaitons étudier. Cette étude repose principalement sur deux outils, les distances de Wasserstein, tirées de la théorie du transport optimal, et la distance de Fisher, au carrefour des statistiques et de la géométrie différentielle. / In applied mathematics, question of moving quantities by vector is an important question : fluid mechanics, kinetic theory… Using particle methods, we're going to move an additional quantity giving more information on the problem. First part of the work is the theorical formulation for this kind of transport. It's going to use the differential in space of the vector field to compute the differential of the flow. An immediate and natural application is density who are parametrized by and point and a tensor, like gaussians. We're going to move such densities by moving point and tensor. Natural question is now the accuracy of such approximation. It's second part of our work , which discuss of distance to estimate such type of densities.

We present a novel approach for deriving analytical solutions to transport equations expressedin similarity variables. We apply a fixed-point iteration procedure to these transformedequations by formally solving for the highest derivative term and then integrating to obtainan expression for the solution in terms of a previous estimate. We are able to analyticallyobtain the Lipschitz condition for this iteration procedure and, from this (via requirements forconvergence given by the contraction mapping principle), deduce a range of values for the outerlimit of the solution domain, for which the fixed-point iteration is guaranteed to converge.

This thesis examines the processes of longshore sediment transport in the swash zone of a mixed sand and gravel shoreline, Lake Coleridge, New Zealand. It focuses on the interactions between waves and currents in the swash zone and the resulting sediment transport. No previous study has attempted to concurrently measure wave and current data and longshore sediment transport rates on a mixed sand and gravel lakeshore beach in New Zealand. Many of these beaches, in both the oceanic and lacustrine environments, are in net long-term erosion. It is recognised that longshore sediment transport is a part of this process, but very little knowledge has existed regarding rates of sediment movement and the relationships between waves, currents and swash activity in the foreshore of these beach types. A field programme was designed to measure a comprehensive range of wind, wave, current and morphological variables concurrently with longshore transport. Four electronic instruments were used to measure both waves and currents simultaneously in the offshore, nearshore and swash zone. In the offshore area, an InterOcean S4ADW wave and current meter was installed to record wave height, period, direction and velocity. A WG-30 capacitance wave gauge measured the total water surface variation. A pair of Marsh-McBirney electromagnetic current meters, measuring current directions and velocities were installed in the nearshore and swash zone. Data were sampled for 18 minutes every hour with a Campbell Scientific CR23x data-logger. The wave gauge data was sampled at a rate of 10 Hz (0.1 s) and the two current meters at a rate of 2 Hz (0.5 s). Longshore sediment transport rates were investigated with the use of two traps placed in the nearshore and swash zone to collect sediment transported under wave and swash action. This occurred concurrently with the wave measurements and together yielded over 500 individual hours of high quality time series data. Important new insights were made into lake wave processes in New Zealand's alpine lakes. Measured wave heights averaged 0.20-0.35 m and ranged up to 0.85 m. Wave height was found to be strongly linked to the wind and grew rapidly to increasing wind strength in an exponential fashion. Wave period responded more slowly and required time and distance for the wave length to develop. Overall, there was a narrow band of wave periods with means ranging from 1.43 to 2.33 s. The wave spectrum was found to be more mixed and complicated than had previously been assumed for lake environments. Spectral band width parameters were large, with 95% of the values between 0.75 and 0.90. The wave regime attained the characteristics of a storm wave spectrum. The waves were characteristically steep and capable of obtaining far greater steepness than oceanic wind-waves. Values ranged from 0.010 to 0.074, with an average of 0.051. Waves were able to progress very close to shore without modification and broke in water less than 0.5 m deep. Wave refraction from deep to shallow water only caused wave angles to be altered in the order of 10%. The two main breaker types were spilling and plunging. However, rapid increases in beach slope near the shoreline often caused the waves to plunge immediately landward of the swash zone, leading to a greater proportion of plunging waves. Wave energy attenuation was found to be severe. Measured velocities were some 10 times less at two thirds the water depth beneath the wave. Mean orbital velocities were 0.30 m s⁻¹ in deep water and 0.15 m s⁻¹ in shallow water. The ratio difference between the measured deep water orbital velocities and the nearshore orbital velocities was just under one half (us/uo = 0.58), almost identical to the predicted phase velocity difference by Linear wave theory. In general Linear wave theory was found to provide good approximations of the wave conditions in a small lake environment. The swash zone is an important area of wave dissipation and it defines the limits of sediment transport. The width of the swash zone was found to be controlled by the wave height, which in turn determined the quantity of sediment transported through the swash zone. It ranged in width from 0.05 m to 6.0 m and widened landward in response to increased wave height and lakeward in response the wave length. Slope was found to be an important secondary control on swash zone width. In low energy conditions, swash zone slopes were typically steep. At the onset of wave activity the swash zone becomes scoured by swash activity and the beach slope grades down. An equation was developed, using the wave height and beach slope that provides close estimates of the swash zone width under a wide range of conditions. Run-up heights were calculated using the swash zone width and slope angle. Run-up elevations ranged from 0.01 m to 0.73 m and were strongly related to the wave height and the beach slope. On average, run-up exceeds the deep water wave height by a factor of 1.16H. The highest run-up elevations were found to occur at intermediate slope angles of between 6-8°. Above 8°, the run-up declined in response to beach porosity and lower wave energy conditions. A generalised run-up equation for lake environments has been developed, that takes into account the negative relationship between beach slope and run-up. Swash velocities averaged 0.30 m s⁻¹ but maximum velocities averaged 0.98 m s⁻¹. After wave breaking, swash velocities quickly reduced through dissipation by approximately one half. Swash velocity was strongly linked to wave height and beach slope. Maximum velocities occurred at beach slopes of 5°, where incident swash dominated. At slopes between 6° and 10°, swash velocities were hindered by turbulence, but the relative differences between the swash and backswash flows were negligible. At slope angles above 10° there was a slight asymmetry to the swash/backswash flow velocities due to beach porosity absorbing water at the limits of the swash zone. Three equations were developed for estimating the mean and maximum swash velocity flows. From an analysis of these interactions, a process-response model was developed that formalises the morphodynamic response of the swash zone to wave activity. Longshore sediment transport occurred exclusively in the swash zone, landward of the breaking wave in bedload. The sediments collected in transit were a heterogeneous mix of coarse sands and fine-large gravels. Hourly trapped rates ranged from 0.02 to 214.88 kg hr⁻¹. Numerical methods were developed to convert trapped mass rates in to volumetric rates that use the density and porosity of the sediment. A sediment transport flux curve was developed from measuring the distribution of longshore sediment transport across the swash zone. Using numerical integration, the area under this curve was calculated and an equation written to accurately estimate the total integrated transport rates in the swash zone. The total transport rates ranged from a minimum of 1.10 x 10-5 m³ hr⁻¹ to a maximum of 1.15 m³ hr⁻¹. The mean rate was 7.36 x 10⁻² m³ hr⁻¹. Sediment transport was found to be most strongly controlled by the wave height, period, wave steepness and mean swash velocity. Transport is initiated when waves break at an oblique angle to the shoreline. No relationships could be found between the grain size and transport rates. Instead, the critical threshold velocities of the sediment sizes were almost always exceed in the turbulent conditions under the breaking wave. The highest transport rates were associated with the lowest beach slopes. It was found that this was linked to swash high velocities and wave heights associated with foreshore scouring. An expression was developed to estimate the longshore sediment transport, termed the LEXSED formula, that divides the cube of the wave height and the wave length and multiplies this by the mean swash velocity and the wave approach angle. The expression performs well across a wide range of conditions and the estimates show very good correlations to the empirical data. LEXSED was used to calculate an accurate annual sediment transport budget for the fieldsite beaches. LEXSED was compared to 16 other longshore sediment transport formulas and performed best overall. The underlying principles of the model make its application to other mixed sand and gravel beaches promising.

No description available.