Дисертації з теми "Smooth numbers"

Cette thèse aborde plusieurs questions liées à la fonction somme des chiffres et aux entiers friables. Le premier chapitre est consacré à une introduction qui rassemble les origines des thèmes principaux abordés dans cette thèse, ainsi que les rappels théoriques et les notations nécessaires pour la suite du travail. Les principaux résultats obtenus au cours de cette recherche y seront également présentés. Le deuxième chapitre est consacré à l'étude des propriétés de l'ensemble ({ n leq x : n ext{ est } k ext{-libre}, , s_q(Q(n)) equiv a pmod{m} }), où ( a in mathbb{Z} ), ( k ), et ( m ) désignent des entiers naturels supérieurs ou égaux à 2. La fonction ( s_q ) représente la somme des chiffres en base ( q ), les entiers ( k )-libres sont ceux qui ne sont pas divisibles par la ( k )-ième puissance d'un nombre premier, et ( Q ) est un polynôme de degré supérieur ou égal à 2. Afin de montrer notre résultat principal, nous évaluons des sommes exponentielles du type (sum_{n leq x atop{ n ext{ est } k ext{-libre}}} e(alpha s_q(Q(n)))), où ( alpha ) est tel que ((q - 1)alpha in mathbb{R} setminus mathbb{Z}). À la fin, nous montrons un résultat d'équirépartition modulo 1. Le troisième chapitre se concentre sur l'équirépartition de Zeckendorf et la somme des chiffres des entiers friables dans des classes de congruence. Un entier est dit ( y )-friable si tous ses facteurs premiers sont inférieurs ou égaux à ( y ). Nous utiliserons systématiquement la notation ( P(n) ) pour désigner le plus grand facteur premier de ( n ), et ( S(x, y) := { n leq x : P(n) leq y } ) pour désigner l'ensemble des entiers ( y )-friables inférieurs ou égaux à ( x ). L'objectif principal de ce chapitre est d'évaluer l'ensemble ( { n in S(x, y) : s_varphi(n) equiv a pmod{m} } ), où ( a in mathbb{Z} ) et ( m ) désigne un entier naturel supérieur ou égal à 2. Ici, ( s_varphi ) est la fonction de la somme des chiffres en base Fibonacci. Comme nous le faisons dans le deuxième chapitre, pour prouver le résultat principal, nous utilisons les sommes exponentielles, ainsi, nous profiterons de la propriété de décomposition des entiers friables dans des intervalles pour nos démonstrations afin d'évaluer la somme exponentielle(sum_{n in S(x, y)} e(vartheta s_varphi(n))), où ( vartheta in mathbb{R} setminus mathbb{Z} ). Le quatrième chapitre porte sur la moyenne des sommes de certaines fonctions multiplicatives sur les entiers friables. Dans ce chapitre notre objectif est de déterminer des estimations pour les expressions suivantes : sigma_s(n) = sum_{d mid n} d^s, varphi(n) = sum_{d mid n} mu(d) n/d, et psi(n) = sum_{d mid n} mu^2(n/d) d, où ( s ) est un nombre réel non nul, lorsque n parcourt l'ensemble S(x,y). Le dernier chapitre présente une application de l'inégalité de Turán-Kubilius. Il est bien connu que cette inégalité traite des fonctions additives et qu'elle a également permis de démontrer le théorème de Hardy-Ramanujan pour la fonction additive (omega(n)), qui compte les diviseurs premiers de l'entier (n). Dans ce chapitre, nous nous déplaçons dans l'espace des entiers friables et nous nous intéressons à la fonction additive ilde{omega}(n) = sum_{p mid n atop{s_q(p) equiv a pmod{b}}} 1,où ( a in mathbb{Z} ) et ( b geq 2 ) sont des entiers. Nous fournissons une estimation de (ilde{omega}(n)), lorsque (n) parcourt l'ensemble (S(x,y)), puis nous utilisons l'inégalité de Turán-Kubilius dans l'espace des entiers friables proposée par Tenenbaum et de la Bretèche, et présentons quelques applications
This thesis addresses some questions related to the sum of digits function and friable integers. The first chapter is dedicated to an introduction that gathers the origins of the main topics covered in this thesis, as well as a background and the necessary notations for the rest of the work. The main results obtained during this research will also be presented. The second chapter focuses on the behaviour of the set ({ n leq x : n ext{ is } k ext{-free}, , s_q(Q(n)) equiv a pmod{m} }), where ( a in mathbb{Z} ), ( k ), and ( m ) are natural numbers greater than or equal to 2. The function ( s_q ) represents the sum of digits in base ( q ), ( k )-free integers are those not divisible by the ( k )-th power of a prime number, and ( Q ) is a polynomial of degree greater than or equal to 2. To show our main result, we evaluate exponential sums of the type(sum_{n leq x atop{ n ext{ is } k ext{-free}}} e(alpha s_q(Q(n)))), where ( alpha ) is a real number such that ((q - 1)alpha in mathbb{R} setminus mathbb{Z}). In the end, we establish an equidistribution result modulo 1. The third chapter, we focus on the distribution of the Zeckendorf sum of digits over friable integers in congruence classes. An integer is called ( y )-friable if all its prime factors are less than or equal to ( y ). We use the notation ( P(n) ) to denote the largest prime factor of ( n ), and ( S(x, y) := { n leq x : P(n) leq y } ) to denote the set of ( y )-friable integers less than or equal to ( x ). The main objective of this chapter is to evaluate the set ( { n in S(x, y) : s_varphi(n) equiv a pmod{m} } ), where ( a in mathbb{Z} ) and ( m ) is a natural number greater than or equal to 2. Here, ( s_varphi ) is the sum of digits function in the Fibonacci base. As in the second chapter, to prove the main result, we use exponential sums, and we utilize the property of decomposition of friable integers into intervals for our demonstration to evaluate the exponential sum(sum_{n in S(x, y)} e(vartheta s_varphi(n))), where ( vartheta in mathbb{R} setminus mathbb{Z} ). The fourth chapter deals with the average of sums of certain multiplicative functions over friable integers. In this chapter, our goal is to determine estimates for the following expressions: sigma_s(n) = sum_{d mid n} d^s, varphi(n) = sum_{d mid n} mu(d) n/d, and psi(n) = sum_{d mid n} mu^2(n/d) d, where ( s ) is a non-zero real number, when (n) runs over the set (S(x,y)). The last chapter presents an application of the Turán-Kubilius inequality. It is well known that this inequality deals with additive functions and has also been used to prove the Hardy-Ramanujan theorem for the additive function (omega(n)), which counts the prime divisors of the integer (n). In this chapter, we move into the space of friable integers and focus on the additive function ilde{omega}(n) = sum_{p mid n atop{s_q(p) equiv a pmod{b}}} 1, where ( a in mathbb{Z} ) and ( b geq 2 ) are integers. Firstly, we provide an estimate of ( ilde{omega}(n)) when (n) runs through the set (S(x,y)), we then use the Turán-Kubilius inequality in the space of friable integers established by Tenenbaum and de la Bretèche to present few applications

Most heat transfer tubes are designed for either fully uniform wall temperature or fully uniform wall heat flux boundary conditions under forced convection. Several applications, including but not limited to the solar collectors of renewable energy systems, do however operate with non-uniform boundary conditions. Limited research has been conducted on non-uniform wall heat flux heat transfer coefficients in circular tubes, especially for mixed convection conditions. Such works are normally numerical in nature and little experimental work is available. In this experimental investigation the effects of the circumferential heat flux distribution and heat flux intensity on the single phase (liquid) internal heat transfer coefficient were considered for a horizontal circular tube. Focus was placed on the laminar flow regime of water within a stainless steel tube with an inner diameter of 27.8 mm and a length to diameter ratio of 72. Different outer wall heat flux conditions, including fully uniform and partially uniform heat fluxes were studied for Reynolds numbers ranging from 650 to 2 600 and a Prandtl number range of 4 to 7. The heat flux conditions included 360˚ (uniform) heating, lower 180˚ heating, upper 180˚ heating, 180˚ left and right hemispherical heating, lower 90˚ heating, upper 90˚ heating and slanted 180˚ heating. Depending on the angle span of the heating, local heat fluxes of 6 631 W/m2 , 4 421 W/m2 , 3 316 W/m2 , 2 210 W/m2 and 1 658 W/m2 were applied. Results indicate that the local and average steady state Nusselt numbers are greatly influenced by the applied heat flux position and intensity. Highest average heat transfer coefficients were achieved for case where the applied heat flux was positioned on the lower half (in terms of gravity) of the tubes circumference, while the lowest heat transfer coefficients were achieved when the heating was applied to the upper half of the tube. Variations in the heat transfer coefficient were found to be due to the secondary buoyancy induced flow effect. The relative thermal performance of the different heating scenarios where characterised and described by means of newly developed heat transfer coefficient correlations for fully uniform heating, lower 180° heating, and upper 180° heating.
Dissertation (MEng)--University of Pretoria, 2018.
Mechanical and Aeronautical Engineering
MEng
Unrestricted

Junction flows originate from the interaction between a fluid moving over a wall with an obstacle mounted on the same surface. Understanding the physics of such flows is of great interest to engineers responsible for the design of systems consisting of wall-body junctions. From aerodynamics to turbomachinery and electronics to bridge hydraulics, a number of phenomena (drag, heat transfer, scouring) are driven by the behavior of the most prominent feature of junction flows: the horseshoe vortex system (HVS). Focusing on turbulent flows, the complex dynamics of the HVS is established through its unsteadiness and non-uniformity. The fundamentals of this dynamically-rich phenomenon have been described within the body of a rapidly-expanding literature. Nevertheless, important aspects remain inadequately understood and call for further scrutiny. This study emphasized three of them, by investigating the effects of: model scale, wall roughness, and bed geometry. High-resolution experiments were carried out using Particle Image Velocimetry (PIV). Statistical analyses, vortex identification schemes, and Proper Orthogonal decomposition were employed to extract additional information from the large PIV datasets. The time-averaged topology of junction flows developing over a smooth and impermeable wall was independent of the flow Reynolds number, Re (parameter that expresses the effects of scale). On the contrary, time-resolved analysis revealed a trend of increasing vorticity, momentum, and eruptions of near-wall fluid with Re. New insights on the modal dynamics of the HVS were also documented in a modified flow mechanism. Wall roughness (modeled with a permeable layer of crushed stones) diffused turbulence and vorticity throughout the domain. This effect manifested with high levels of intermittency and spatial irregularity for the HVS. Energetic flow structures were also identified away from the typical footprint of the HVS. Finally, a novel implementation of PIV allowed for unique velocity measurements over an erodible bed. It was demonstrated that, during the initial stages of scouring, the downflow at the face of the obstacle becomes the dominant flow characteristic in the absence of the HVS. Notwithstanding modeling limitations, the physical insight contributed here could be used to enhance the design of systems with similar flow and geometrical characteristics.
Ph. D.

Complex interactions in porous media play an important role on many industrial and geotechnical applications, such as groundwater treatment, porous catalysts, carbon geosequestration, and oil recovery. Rate-dependent wetting effects are of great significance in understanding the multiphase behaviours of porous media thus further throw light on engineering solutions to the above problems. In this thesis, a modified smoothed particle hydrodynamics (SPH) model is applied to simulate (1) the contact angle dynamics and (2) stretching of liquid bridge at meso-scale. This SPH model adopted an inter-particle force formulation with short-range repulsive force and long-range attractive force to take into account single-phase and multiphase interactions. Particularly, a newly-introduced viscous force is imposed at the liquid-solid interface to capture the rate-dependent behaviours of contact angle without prescribing additional arbitrary condition or force. After identification of model parameters, the rate-dependent contact angle behaviours are studied for both wetting and dewetting phenomena. By analysing the contact angle results of fluid at triple-line region with different moving speeds, the dynamic contact angles and corresponding capillary numbers can be correlated by power law functions. The derived correlation and constants are compared with different forms of empirical power law functions and the results are satisfactory. Moreover, we investigated the properties of stretching liquid bridges, including shape evolution, liquid transfer ratio and flow condition under dynamic loading. Different stretching rates are applied, and the shapes of liquid bridge at same breakup distance is presented. By differentiating the wettability of top and bottom substrates, the liquid transfer ratio regarding wettability difference and substrate moving speed is studied.

Generating random variates as generalisation of a given sample is an important task for stochastic simulations. The three main methods suggested in the literature are: fitting a standard distribution, constructing an empirical distribution that approximates the cumulative distribution function and generating variates from the kernel density estimate of the data. The last method is practically unknown in the simulation literature although it is as simple as the other two methods. The comparison of the theoretical performance of the methods and the results of three small simulation studies show that a variance corrected version of kernel density estimation performs best and should be used for generating variates directly from a sample. (author's abstract)
Series: Preprint Series / Department of Applied Statistics and Data Processing

The topic of this dissertation is the derivation, development, and evaluation of novel hybrid algorithms for process control that use a limited number of measurements and that are suitable to operate in the presence of large amounts of process noise. As an initial step, affine and neural network statistical process models are developed in order to simulate the steady-state system behavior. Such models are vitally important in the evaluation, testing, and improvement of all other process controllers referred to in this work. Afterwards, fuzzy logic controller rules are assimilated into a mathematical characterization of a model that includes the modes and mode transition rules that define a hybrid hierarchical process control. The main processing entity in such framework is a closed-loop control algorithm that performs global and then local optimizations in order to asymptotically reach minimum bias error; this is done while requiring a minimum number of iterations in order to promptly reach a desired operational window. The results of this research are applied to surface mount technology manufacturing-lines yield optimization. This work achieves a practical degree of control over the solder-paste volume deposition in the Stencil Printing Process (SPP). Results show that it is possible to change the operating point of the process by modifying certain machine parameters and even compensate for the difference in height due to change in print direction.

L'algorithme de Levenberg-Marquardt (LM) est parmi les algorithmes les plus populaires pour la résolution des problèmes des moindres carrés non linéaire. Motivés par la structure des problèmes de l'assimilation de données, nous considérons dans cette thèse l'extension de l'algorithme LM aux situations dans lesquelles le sous problème linéarisé, qui a la forme min||Ax - b ||^2, est résolu de façon approximative, et/ou les données sont bruitées et ne sont précises qu'avec une certaine probabilité. Sous des hypothèses appropriées, on montre que le nouvel algorithme converge presque sûrement vers un point stationnaire du premier ordre. Notre approche est appliquée à une instance dans l'assimilation de données variationnelles où les modèles stochastiques du gradient sont calculés par le lisseur de Kalman d'ensemble (EnKS). On montre la convergence dans L^p de l'EnKS vers le lisseur de Kalman, quand la taille de l'ensemble tend vers l'infini. On montre aussi la convergence de l'approche LM-EnKS, qui est une variante de l'algorithme de LM avec l'EnKS utilisé comme solveur linéaire, vers l'algorithme classique de LM ou le sous problème est résolu de façon exacte. La sensibilité de la méthode de décomposition en valeurs singulières tronquée est étudiée. Nous formulons une expression explicite pour le conditionnement de la solution des moindres carrés tronqués. Cette expression est donnée en termes de valeurs singulières de A et les coefficients de Fourier de b
The Levenberg-Marquardt algorithm (LM) is one of the most popular algorithms for the solution of nonlinear least squares problems. Motivated by the problem structure in data assimilation, we consider in this thesis the extension of the LM algorithm to the scenarios where the linearized least squares subproblems, of the form min||Ax - b ||^2, are solved inexactly and/or the gradient model is noisy and accurate only within a certain probability. Under appropriate assumptions, we show that the modified algorithm converges globally and almost surely to a first order stationary point. Our approach is applied to an instance in variational data assimilation where stochastic models of the gradient are computed by the so-called ensemble Kalman smoother (EnKS). A convergence proof in L^p of EnKS in the limit for large ensembles to the Kalman smoother is given. We also show the convergence of LM-EnKS approach, which is a variant of the LM algorithm with EnKS as a linear solver, to the classical LM algorithm where the linearized subproblem is solved exactly. The sensitivity of the trucated sigular value decomposition method to solve the linearized subprobems is studied. We formulate an explicit expression for the condition number of the truncated least squares solution. This expression is given in terms of the singular values of A and the Fourier coefficients of b

We develop and evaluate algorithms for generating random variates for simulation input. One group called automatic, or black-box algorithms can be used to sample from distributions with known density. They are based on the rejection principle. The hat function is generated automatically in a setup step using the idea of transformed density rejection. There the density is transformed into a concave function and the minimum of several tangents is used to construct the hat function. The resulting algorithms are not too complicated and are quite fast. The principle is also applicable to random vectors. A second group of algorithms is presented that generate random variates directly from a given sample by implicitly estimating the unknown distribution. The best of these algorithms are based on the idea of naive resampling plus added noise. These algorithms can be interpreted as sampling from the kernel density estimates. This method can be also applied to random vectors. There it can be interpreted as a mixture of naive resampling and sampling from the multi-normal distribution that has the same covariance matrix as the data. The algorithms described in this paper have been implemented in ANSI C in a library called UNURAN which is available via anonymous ftp. (author's abstract)
Series: Preprint Series / Department of Applied Statistics and Data Processing

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
It is well-known that planes and quadric surfaces in the projective space contain in - nitely many lines. For smooth cubic surface Cayley and Salmon, 1847, (and Clebsch later) proved that it has exactly 27 lines. For degree 4, in 1943 Segre proved that the maximum number of lines contained in a smooth quartic surface is 64. For surfaces of degree greater than 4 this number is unknown. In this work, we are going to explore what is the maximum number of lines that a smooth complex surface of degree d of the family Fd ; may contain. Thus, we obtain a lower bound to the maximum number of lines that non singular surfaces of degree d in P3 may contain. We emphasize that the determination of this numbers is based on the Klein's classi cation theorem of nitte subgroups of Aut(P1) and the study of 􀀀C; the subgroup of Aut(P1) whose elements leaves invariant the nite subset C of P1:
Sabe-se que planos e superf cies qu adricas no espa co projetivo cont em in nitas retas. No caso de uma superf cie c ubica n~ao singular Cayley e Salmon, em 1847, (e Clebsch, mais tarde) provaram que ela cont em exatamente 27 retas. No caso de grau 4, em 1943 Segre provou que o n umero m aximo de retas contidas numa superf cie qu artica n~ao singular e 64. Para superf cies de grau maior que 4 esse n umero e desconhecido. Neste trabalho vamos explorar qual e a quantidade m axima de retas que uma superf cie complexa n~ao singular de grau d na fam lia Fd ; pode conter. Assim obtemos uma cota inferior para o n umero m aximo de retas que as superf cies n~ao singulares de grau d em P3 podem conter. Salientamos que a determina c~ao destes n umeros tem como base o Teorema de Classi ca cao de Klein dos sugbrupos nitos de Aut(P1) e o estudo dos subgrupos 􀀀C de Aut(P1) que deixam invariante um subconjunto nito C de P1:

The context of this thesis is the numerical simulation of turbulent flows at moderate Reynolds numbers and the improvement of the capabilities of an in-house 3D unsteady and incompressible flow solver called SFELES to simulate such flows.

In addition to this abstract, this thesis includes five other chapters.

The second chapter of this thesis presents the numerical methods implemented in the two CFD solvers used as part of this work, namely SFELES and PHASTA.

The third chapter concentrates on the implementation of a new library called FlexMG. This library allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods, which require much less memory to solve linear systems than a direct sparse LU solver available in SFELES. Multigrid is an iterative procedure that relies on a series of increasingly coarser approximations of the original 'fine' problem. The underlying concept is the following: low wavenumber errors on fine grids become high wavenumber errors on coarser levels, which can be effectively removed by applying fixed-point methods on coarser levels.

Two families of algebraic multigrid preconditioners have been implemented in FlexMG, namely smooth aggregation-type and non-nested finite element-type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non-nested finite element-type multigrid so that our approaches can be considered as hybrid. Our aggregation-type multigrid is smoothed with either a constant or a linear least square fitting function, whereas the non-nested finite element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled with a GMRES (Generalized Minimal RESidual) method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics (unsteady flow past a circular cylinder at low Reynolds number), their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in this work.

The fourth chapter is devoted to the study of subgrid scale models for the large eddy simulation (LES) of turbulent flows.

It is well known that turbulence features a cascade process by which kinetic energy is transferred from the large turbulent scales to the smaller ones. Below a certain size, the smallest structures are dissipated into heat because of the effect of the viscous term in the Navier-Stokes equations.

In the classical formulation of LES models, all the resolved scales are used to model the contribution of the unresolved scales. However, most of the energy exchanges between scales are local, which means that the energy of the unresolved scales derives mainly from the energy of the small resolved scales.

In this fourth chapter, constant-coefficient-based Smagorinsky and WALE models are considered under different formulations. This includes a classical version of both the Smagorinsky and WALE models and several scale-separation formulations, where the resolved velocity field is filtered in order to separate the small turbulent scales from the large ones. From this separation of turbulent scales, the strain rate tensor and/or the eddy viscosity of the subgrid scale model is computed from the small resolved scales only. One important advantage of these scale-separation models is that the dissipation they introduce through their subgrid scale stress tensor is better controlled compared to their classical version, where all the scales are taken into account without any filtering. More precisely, the filtering operator (based on a top hat filter in this work) allows the decomposition u' = u - ubar, where u is the resolved velocity field (large and small resolved scales), ubar is the filtered velocity field (large resolved scales) and u' is the small resolved scales field.

At last, two variational multiscale (VMS) methods are also considered.

The philosophy of the variational multiscale methods differs significantly from the philosophy of the scale-separation models. Concretely, the discrete Navier-Stokes equations have to be projected into two disjoint spaces so that a set of equations characterizes the evolution of the large resolved scales of the flow, whereas another set governs the small resolved scales.

Once the Navier-Stokes equations have been projected into these two spaces associated with the large and small scales respectively, the variational multiscale method consists in adding an eddy viscosity model to the small scales equations only, leaving the large scales equations unchanged. This projection is obvious in the case of a full spectral discretization of the Navier-Stokes equations, where the evolution of the large and small scales is governed by the equations associated with the low and high wavenumber modes respectively. This projection is more complex to achieve in the context of a finite element discretization.

For that purpose, two variational multiscale concepts are examined in this work.

The first projector is based on the construction of aggregates, whereas the second projector relies on the implementation of hierarchical linear basis functions.

In order to gain some experience in the field of LES modeling, some of the above-mentioned models were implemented first in another code called PHASTA and presented along with SFELES in the second chapter.

Finally, the relevance of our models is assessed with the large eddy simulation of a fully developed turbulent channel flow at a low Reynolds number under statistical equilibrium. In addition to the analysis of the mean eddy viscosity computed for all our LES models, comparisons in terms of shear stress, root mean square velocity fluctuation and mean velocity are performed with a fully resolved direct numerical simulation as a reference.

The fifth chapter of the thesis focuses on the numerical simulation of the 3D turbulent flow over a re-entry Apollo-type capsule at low speed with SFELES. The Reynolds number based on the heat shield is set to Re=10^4 and the angle of attack is set to 180º, that is the heat shield facing the free stream. Only the final stage of the flight is considered in this work, before the splashdown or the landing, so that the incompressibility hypothesis in SFELES is still valid.

Two LES models are considered in this chapter, namely a classical and a scale-separation version of the WALE model. Although the capsule geometry is axisymmetric, the flow field in its wake is not and induces unsteady forces and moments acting on the capsule. The characterization of the phenomena occurring in the wake of the capsule and the determination of their main frequencies are essential to ensure the static and dynamic stability during the final stage of the flight.

Visualizations by means of 3D isosurfaces and 2D slices of the Q-criterion and the vorticity field confirm the presence of a large meandering recirculation zone characterized by a low Strouhal number, that is St≈0.15.

Due to the detachment of the flow at the shoulder of the capsule, a resulting annular shear layer appears. This shear layer is then affected by some Kelvin-Helmholtz instabilities and ends up rolling up, leading to the formation of vortex rings characterized by a high frequency. This vortex shedding depends on the Reynolds number so that a Strouhal number St≈3 is detected at Re=10^4.

Finally, the analysis of the force and moment coefficients reveals the existence of a lateral force perpendicular to the streamwise direction in the case of the scale-separation WALE model, which suggests that the wake of the capsule may have some

preferential orientations during the vortex shedding. In the case of the classical version of the WALE model, no lateral force has been observed so far so that the mean flow is thought to be still axisymmetric after 100 units of non-dimensional physical time.

Finally, the last chapter of this work recalls the main conclusions drawn from the previous chapters.
Doctorat en Sciences de l'ingénieur
info:eu-repo/semantics/nonPublished

Cooling channels with a developing flow entrance condition and aspect ratios of 1:4 and 2:1 were studied. The range of the rotation number and buoyancy parameter for the selected AR channels was extended. The maximum Ro and Bo for the 1:4 channel was 0.67 and 1.9, respectively. For the 2:1 channel, these values were 0.45 and 0.85, respectively. The effect of rib spacing and rib height on heat transfer in the 1:4 channel is investigated. Three rib spacing configurations were considered: P/e=2.5, 5, 10 with a constant e/Dh ratio of 0.078. To investigate the effect of rib height, a rib configuration with an e/Dh ratio of 0.156 and P/e ratio of 10 was considered. For the 2:1 channel, a smooth channel surface condition was studied. For each channel aspect ratio and surface condition, five Reynolds numbers were studied up to 40K. At each Re, five rotational speeds are considered up to 400 rpm. The results of this research work indicate that rotation can cause a significant increase in heat transfer on the first pass trailing surface of both aspect ratio channels. The leading surface in ribbed channels has shown a dramatic decrease in heat transfer with rotation in the first pass. Reductions in heat transfer by as much as 50% were observed. In the second pass, the leading and trailing surfaces with ribs showed very similar effects of rotation. Also, the effect of rotation seems to vary with the rib spacing. The strength of rotation showed to be greater in the tight rib spacing of P/e=2.5. The rib height in the 1:4 channel had minimal impact due to the large distance between the leading and trailing surfaces. The tip cap heat transfer for both channels showed large increases with rotation. This is very beneficial since tip cooling is an important part of maintaining the life a turbine blade. Finally, the buoyancy parameter proved to be very useful in predicting heat transfer in rotating conditions. The correlations developed showed very acceptable accuracy when compared to the experimental data.

Dans le premier chapitre de cette thèse, nous passons en revue les outils de la théorie analytique des nombres qui seront utiles pour la suite. Nous faisons aussi un survol des entiers y−friables, c’est-à-dire des entiers dont chaque facteur premier est plus petit ou égal à y. Au deuxième chapitre, nous présenterons des problèmes classiques de la théorie des nombres probabiliste et donnerons un bref historique d’une classe de fonctions arithmétiques sur un espace probabilisé. Le problème de Erdos sur la table de multiplication demande quel est le nombre d’entiers distincts apparaissant dans la table de multiplication N × N. L’ordre de grandeur de cette quantité a été déterminé par Kevin Ford (2008). Dans le chapitre 3 de cette thèse, nous étudions le nombre d’ensembles y−friables de la table de multiplication N × N. Plus concrètement, nous nous concentrons sur le changement du comportement de la fonction A(x, y) par rapport au domaine de y, où A(x, y) est une fonction qui compte le nombre d’entiers y− friables distincts et inférieurs à x qui peuvent être représentés comme le produit de deux entiers y− friables inférieurs à p x. Dans le quatrième chapitre, nous prouvons un théorème de Erdos-Kac modifié pour l’ensemble des entiers y− friables. Si !(n) est le nombre de facteurs premiers distincts de n, nous prouvons que la distribution de !(n) est gaussienne pour un certain domaine de y en utilisant la méthode des moments.
The object of the first chapter of this thesis is to review the materials and tools in analytic number theory which are used in following chapters. We also give a survey on the development concerning the number of y−smooth integers, which are integers free of prime factors greater than y. In the second chapter, we shall give a brief history about a class of arithmetical functions on a probability space and we discuss on some well-known problems in probabilistic number theory. We present two results in analytic and probabilistic number theory. The Erdos multiplication table problem asks what is the number of distinct integers appearing in the N × N multiplication table. The order of magnitude of this quantity was determined by Kevin Ford (2008). In chapter 3 of this thesis, we study the number of y−smooth entries of the N × N multiplication. More concretely, we focus on the change of behaviour of the function A(x,y) in different ranges of y, where A(x,y) is a function that counts the number of distinct y−smooth integers less than x which can be represented as the product of two y−smooth integers less than p x. In Chapter 4, we prove an Erdos-Kac type of theorem for the set of y−smooth integers. If !(n) is the number of distinct prime factors of n, we prove that the distribution of !(n) is Gaussian for a certain range of y using method of moments.

碩士
國立臺灣大學
電機工程學系
86
In this thesis, we conduct a simulation to study the application of Global Positioning System (GPS) on aero-navigation. Our simulation considers the total number, the Horizontal Dilution of Precision (HDOP) and the Vertical Dilution of Precision (VDOP) of the satellites in one day over the eight airports in Taiwan. When applying the GPS on navigation, we use the positioning method of Difference GPS (DGPS). It is not suitable on aero-navigation, because the noise of DGPS will result in the violent variation of position. Therefore we use the method of Carrier Smoothed Code (CSC) to improve the violent variation of position On the other hand, we also discuss that when the cycle slip happens, it inevitably will cause the situation of violent variation of positioning due to reset of CSC.To resolve the problem, we use Kalman Filter as Step Filter to prevent the violent variation of positioning.

碩士
中華大學
機械工程學系碩士班
100
Five versions of low Reynolds number turbulent models for the prediction of heat transfer under a fully developed turbulent pipe were analyzed by comparison with the experimental data. None of the low Reynolds number turbulent models are capable of predicting the local Nusselt number in good agreement with experimental results beyond Reynolds number 106. The error in Nusselt number at low and high Prandtl numbers is more pronounced than the Prandtl number at 1. Regardless the poor performance of various low Reynolds number models in fully developed turbulent pipe problems, the Launder and Sharma model was used to study the transient length, constant wall or constant heat flux, and wavy wall in a two-dimensional configuration. Several conclusions were drawn based on the current study.