The gas turbine blade/vane internal cooling is achieved by circulating the compressed air through the cooling passages inside the turbine blade. Leading edge and trailing edge of the turbine blade are two critical regions which need to be properly cooled. Leading edge region receives extremely hot mainstream flow and high heat transfer enhancement is required. Trailing edge region usually has narrow shaped geometry and applicable cooling techniques are restricted. Heat transfer will be investigated in the leading edge and trailing edge cooling channels at high rotation numbers close to the engine condition. Heat transfer and pressure drop has been investigated in an equilateral triangular channel (Dh=1.83cm) to simulate the cooling channel near the leading edge of the gas turbine blade. Three different rib configurations (45°, inverted 45°, and 90°) were tested at four different Reynolds numbers (10000-40000), each with five different rotational speeds (0-400 rpm). By varying the Reynolds numbers (10000-40000) and the rotational speeds (0-400 rpm), the rotation number and buoyancy parameter reached in this study were 0-0.58 and 0-2.3, respectively. 45° angled ribs show the highest thermal performance at stationary condition. 90° ribs have the highest thermal performance at the highest rotation number of 0.58. Heat transfer coefficients are also experimentally measured in a wedge-shaped cooling channel (Dh =2.22cm, Ac=7.62cm2) to model an internal cooling passage near the trailing edge of a gas turbine blade where the coolant discharges through the slot to the mainstream flow. Tapered ribs are put on the leading and trailing surfaces with an angle of attack of 45°. The ribs are parallel with staggered arrangement on opposite walls. The inlet Reynolds number of the coolant varies from 10,000 to 40,000 and the rotational speeds varies from 0 to 500 rpm. The inlet rotation number is from 0 - 1.0. The local rotation number and buoyancy parameter are determined by the rotational speeds and the local Reynolds number at each region. Results show that heat transfer is high near the regions where strong slot ejection exists. Both the rotation number and buoyancy parameter have been correlated to predict the rotational heat transfer enhancement.

This work investigates some aspects of the dynamics of non-invertible quasi-periodic circle maps, from the point of view of rotation numbers and their structure in parameter space.Circle maps and quasi-periodically forced circle maps have been widely used asa model for a broad range of physical phenomena. From the mathematical point of view they have also received considerable attention because of the many interesting features they exhibit.The system used is given by the maps: x_n = [ x_n-1 + a + b/(2pi) sin( 2pi x_n-1) + c sin( 2pi theta_n-1) ] mod 1, and, theta_n = theta_n-1 + omega.Where a, b and c are real constants. In addition, b and omega are restricted, respectively, to values larger than one and irrational.A fundamental part of this thesis consists of numerical approximations of rotation intervals using and adapting of the work of Boyland (1986) to the quasi-periodic case.Particular emphasis was given to the case of large coupling strength in quasi-periodicforcing.Examination of the computed rotation numbers for the large coupling case, together with previous claims suggesting that for large coupling strength the b-term could be neglected (see Ding (1989)), led to the formulation of an ergodic argument which is statistically supported. This argument indicates that, for this case, the qualitative behavior of rotation number depends linearly on a. It is also shown that the length of the rotation interval, when the transition from a trivial rotation interval (invertible case) to a non-trivial rotation interval occurs, it develops locally as a universal unfolding.A different map, piecewise monotone, and structurally similar to the maps defined to calculate the edges of rotation intervals in Boyland (1986), is studied to illustrate how the rotation number grows. The edges of rotation intervals are analytically calculated and matched with numerical observations.

Renormalization is a powerful tool showing up in different contexts of mathematics and physics. In the context of circle diffeomorphisms, the renormalization operator acts like a microscope and allows to study the dynamics of a circle diffeomorphism on a small scale. The convergence of renormalization leads to a proof of the so-called rigidity theorem, which classifies the dynamics of circle diffeomorphisms geometrically: the conjugacy between $C^3$ circle diffeomorphism with Diophantine rotation number and the corresponding rotation is $C^1$. In this thesis, we define the renormalization of circle diffeomorphisms and study its dynamics. In particular, we prove that the renormalization of orientation preserving $C^3$ circle diffeomorphisms with irrational rotation number of bounded type converges to rotations at exponential speed. We also introduce the necessary relevant concepts such as rotation number, distortion and non-linearity and discuss some of their properties. This thesis is a summary and supplement to the book One-Dimensional Dynamics: from Poincaré to Renormalization. / Omnormalisering är en kraftfull teknik som dyker upp i olika sammanhang inom matematik och fysik. I samband med cirkeldiffeomorfier är omnormaliseringsoperatorn ett dynamiskt system, som fungerar som ett mikroskop och gör att vi kan studera dynamiken hos en cirkeldiffeomorfi på en liten skala. Omnormaliseringens konvergens leder till ett bevis för det så kallade rigiditetssatsen, som klassificerar dynamiken hos cirkeldiffeomorfier geometriskt: konjugatet mellan $C^3$ cirkeldiffeomorfi med diofantiska rotationstal och den motsvarande rotationen är $C^1$. I denna avhandling definierar vi omnormaliseringen av cirkeldiffeomorfier och studerar dess dynamik. I synnerhet bevisar vi att omnormaliseringen av orienteringsbevarande $C^3$ cirkeldiffeomorfier med irrationellt rotationstal av begränsad typ konvergerar till rotationer med exponentiell hastighet. Vi introducerar också nödvändiga och relevanta begrepp så som rotationstal, distorsion och icke-linjäritet och diskuterar några av deras egenskaper. Denna avhandling är en sammanfattning och ett komplement till boken One- Dimensional Dynamics: from Poincaré to Renormalization.

O principal objetivo desta tese é estudar o número de rotação de fins primos de homeomorfismos planares que pertencem a uma classe de homeomorfismos H. Tal número de rotação é devido à Carathéordory e semelhante à teoria de Poincaré para homeomorfismos do crculo. Para todo irracional (0, 1), denotando por (h, U ) o número de rotação de fins primos de h H em U , com U a bacia de repulsão do infinito, construiremos um homeomorfismo h H satisfazendo (h, U ) = e que possui uma sela periódica com intersecção homoclnica transversal em U . Além disso, quando h é de classe C 2 e det(Dh| x ) < 1 em todo ponto, mostraremos que existe ponto periódico acessvel em U se, e somente se, (h, U ) é racional. Também será provado que, quando h é uma ferradura de Smale, o número de rotação (h, U ) é racional. Finalizando, provaremos que se for possvel a existência de um difeomorfismo C r , r 1, em um conjunto genérico a ser definido, com U = W u (p) para p uma sela homoclnica com intersecção transversal e tal que o número de rotação (h, U ) é irracional, necessariamente, h deve satisfazer uma propriedade que não é válida para ferraduras de Smale. / The main objective of this thesis is to study the prime ends rotation number of planar homeomorphisms belonging to a class of homeomorphisms H. Such rotation number is due to Carathéordory and similar to the Poincarés theory of homeomorphisms of the circle. For all irrational (0, 1), denoting by (h, U ) the prime end rotation number of h H in U , with U the infinity repulsion basin, we will construct a homeomorphism h H satisfying (h, U ) = and having a homoclinic saddle with transverse intersection in U . Also, when h is class C 2 and det (Dh| x ) < 1 at every point, we will show that there is accessible periodic point in U if, and only if, (h, U ) is rational. It will also be proved that when h is a Smales horseshoe, the rotation number (h, U ) is rational. To conclude, we will prove that if there exists a C r -diffeomorphism, in a generic set to be defined, with U = W u (p) for a saddle point p with transverse homoclinal intersection and such that the rotation number (h, U ) is irrational, then h must satisfy a property that is not valid for Smales horseshoes.

Gas turbine blade/vane cooling is obtained by circulating the high pressure air from compressor to the internal cooling passage of the blade/vane. Heat transfer and cooling effect in the rotating blade is highly affected by rotation. The typical rotation number for the aircraft engine is in the range of 0~0.25 and for the land based power generation turbine in the range of 0~05. Currently, the heat transfer data at high rotation numbers are limited. Besides, the investigation of heat transfer phenomena in the turn region, especially near hub portion is rare. This dissertation is to study the heat transfer in rectangular channels with turns in the tip or the hub portion respectively at high rotation numbers close to the engine condition. The dissertation experimentally investigates the heat transfer phenomena in a two-pass rectangular channel (AR=W/H=2:1) with a 180 degree sharp turn in the tip portion. The flow in the first passage is radial outward and after the turn in the second passage, the flow direction is radial inward. The hydraulic diameter (Dh) of the channel is 16.9 mm. Parallel square ribs with an attack angle (alpha) of 45 degrees are used on leading and trailing surfaces to enhance the heat transfer. The rib height-to-hydraulic diameter ratio (e/Dh) is 0.094. For the baseline smooth case and the case with rib pitch-to-height ratio (P/e) 10, channel orientation angles (beta) of 90 degrees and 135 degrees were tried to model the cooling passage in the mid and rear portion of the blade respectively. Two other P/e ratios of 5 and 7.5 were studied at beta=135 degrees to investigate their effect on heat transfer. The data are presented under high rotation numbers and buoyancy parameters by varying the Reynolds number (Re=10,000~40,000) and rotation speed (rpm=0~400). Corresponding rotation number and buoyancy parameter are ranged as 0~0.45 and 0~0.8 respectively. The dissertation also studies the heat transfer in a two-pass channel (AR=2:1) connected by a 180 degree U bend in the hub portion. The flow in the first passage is radial inward and after the U bend, the flow in the second passage is radial outward. The cross-section dimension of this channel is the same as the previous one. To increase heat transfer, staggered square ribs (e/Dh=0.094) are pasted on leading and trailing walls with an attack angle (alpha) of 45 degrees and pitch-to-height ratio (P/e) of 8. A turning vane in the shape of half circle (R=18.5 mm, t=1.6 mm) is used in the turn region to guide the flow for both smooth and ribbed cases. Channel orientation angles (beta) of 90 degrees and 135 degrees were taken for both smooth and ribbed cases. The heat transfer data were taken at high rotation numbers close to previous test section.

Nós estudaremos o comportamento de difeomorfismos do círculo, tanto do ponto de vista combinatório quanto do ponto de vista topológico e da teoria da medida, seguindo os trabalhos de Michael Herman. A cada homeomorfismo do círculo podemos associar um número real positivo, denominado número de rotação. Mostraremos que existe um conjunto de números irracionais de medida de Lebesgue total na reta tal que, se f é um difeomorfismo do círculo de classe \'C POT. r \' que preserva a orientação, com r maior ou igual a 3 e com número de rotação nesse conjunto, então f é pelo menos \'C POT. r - 2\' -conjugada a uma translação irracional. Além disso, mostraremos que dado um caminho \'f IND. t\' de classe \'C POT. 1\' definido em um intervalo [a;b] no conjunto dos difeomorfismos do círculo de classe \'C POT. r\' que preservam a orientação, com r maior ou igual a 3, o conjunto dos parâmetros em que \'f IND. t\' é \'C POT. r - 2\' -conjugada a uma translação irracional tem medida de Lebesgue positiva, desde que os números de rotação em \'f IND. a\' e \'f IND. b\' sejam distintos / We will study the generic behavior of circle diffeomorphisms, in the combinatorial, topological and measure-theoretical sense, following the work of Michael Herman. To each order preserving homeomorphism of the circle we can associate a positive real number, called rotation number, which is invariant under conjugacy. We will show that there is a set of irrational numbers with full Lebesgue measure on R such that, if f is a circle diffeomorphism of class \'C POT. r\', with r greater or equal 3 and with rotation number in that set, then f is at least \'C POT. r - 2\' -conjugated to an irrational translation. Moreover, we will show that if ft is a \'C POT. 1\' -path defined on a interval [a;b] over the set of the circle diffeomorphisms orientation preserving, with r \'> or =\' 3, then the set of parameters where \'f IND. t\' is \'C POT. r - 2\' -conjugated to a irrational translation has positive Lebesgue measure, since the rotation numbers of \'f IND. a\' and \'f IND. b\' are distinct

Nós estudaremos o comportamento de difeomorfismos do círculo, tanto do ponto de vista combinatório quanto do ponto de vista topológico e da teoria da medida, seguindo os trabalhos de Michael Herman. A cada homeomorfismo do círculo podemos associar um número real positivo, denominado número de rotação. Mostraremos que existe um conjunto de números irracionais de medida de Lebesgue total na reta tal que, se f é um difeomorfismo do círculo de classe \'C POT. r \' que preserva a orientação, com r maior ou igual a 3 e com número de rotação nesse conjunto, então f é pelo menos \'C POT. r - 2\' -conjugada a uma translação irracional. Além disso, mostraremos que dado um caminho \'f IND. t\' de classe \'C POT. 1\' definido em um intervalo [a;b] no conjunto dos difeomorfismos do círculo de classe \'C POT. r\' que preservam a orientação, com r maior ou igual a 3, o conjunto dos parâmetros em que \'f IND. t\' é \'C POT. r - 2\' -conjugada a uma translação irracional tem medida de Lebesgue positiva, desde que os números de rotação em \'f IND. a\' e \'f IND. b\' sejam distintos / We will study the generic behavior of circle diffeomorphisms, in the combinatorial, topological and measure-theoretical sense, following the work of Michael Herman. To each order preserving homeomorphism of the circle we can associate a positive real number, called rotation number, which is invariant under conjugacy. We will show that there is a set of irrational numbers with full Lebesgue measure on R such that, if f is a circle diffeomorphism of class \'C POT. r\', with r greater or equal 3 and with rotation number in that set, then f is at least \'C POT. r - 2\' -conjugated to an irrational translation. Moreover, we will show that if ft is a \'C POT. 1\' -path defined on a interval [a;b] over the set of the circle diffeomorphisms orientation preserving, with r \'> or =\' 3, then the set of parameters where \'f IND. t\' is \'C POT. r - 2\' -conjugated to a irrational translation has positive Lebesgue measure, since the rotation numbers of \'f IND. a\' and \'f IND. b\' are distinct

O conceito de número de rotação originalmente definido para homeomorfismos do círculo S1 que preservam orientação pode ser generalizado para todo homeomorfismo h do anel fechado S1×[0; 1] isotópico à identidade, onde obtemos o chamado conjunto de rotação. Neste trabalho estudamos o caso em que o conjunto de rotação de h se reduz somente a um número irracional ? (neste caso dizemos que h é uma pseudo-rotação irracional), obtendo que para qualquer inteiro positivo n, existe um arco simples ? que une uma componente do bordo do anel à outra, de tal modo que ? é disjunto de seus n primeiros iterados por h: Este resultado é um análogo do Teorema de Kwapisz concernente a difeomorfismos do toro bidimensional [14]. Posteriormente e utilizando o primeiro resultado, provamos que a rotação rígida de ângulo pode ser aproximada por um homeomorfismo conjugado a h. Finalmente, mostramos que ser uma pseudo-rotação irracional é uma propriedade necessária para que um homeomorfismo tenha a propriedade de interseção de curvas e não tenha pontos periódicos. / The concept of rotation number originally defined for orientation preserving homeomorphisms of the circle S1 can be generalized for any homeomorphism h of closed annulus S1×[0; 1] which is isotopic to the identity. In this setting we obtain the so called rotation set. In this work we study the case when the rotation set of h is reduced to a single irrational number ? (we say that h is an irrational pseudo-rotation), and we prove that for any positive integer n, there exists a simple arc ? joining one of the boundary components of annulus to the other, such that ? is disjoint from its n first iterates under h: This result is an analogue of a theorem of Kwapisz dealing with diffeomorphisms of the two-torus [14]. Subsequently and applying the first result, we prove that a rigid rotation of angle can be approximated by a homeomorphism that is conjugate to h: Finally, we prove that to be an irrational pseudo-rotation is a necessary property in order that a homeomorphism has the curves intersection property and no periodic points.

O conceito de número de rotação originalmente definido para homeomorfismos do círculo S1 que preservam orientação pode ser generalizado para todo homeomorfismo h do anel fechado S1×[0; 1] isotópico à identidade, onde obtemos o chamado conjunto de rotação. Neste trabalho estudamos o caso em que o conjunto de rotação de h se reduz somente a um número irracional ? (neste caso dizemos que h é uma pseudo-rotação irracional), obtendo que para qualquer inteiro positivo n, existe um arco simples ? que une uma componente do bordo do anel à outra, de tal modo que ? é disjunto de seus n primeiros iterados por h: Este resultado é um análogo do Teorema de Kwapisz concernente a difeomorfismos do toro bidimensional [14]. Posteriormente e utilizando o primeiro resultado, provamos que a rotação rígida de ângulo pode ser aproximada por um homeomorfismo conjugado a h. Finalmente, mostramos que ser uma pseudo-rotação irracional é uma propriedade necessária para que um homeomorfismo tenha a propriedade de interseção de curvas e não tenha pontos periódicos. / The concept of rotation number originally defined for orientation preserving homeomorphisms of the circle S1 can be generalized for any homeomorphism h of closed annulus S1×[0; 1] which is isotopic to the identity. In this setting we obtain the so called rotation set. In this work we study the case when the rotation set of h is reduced to a single irrational number ? (we say that h is an irrational pseudo-rotation), and we prove that for any positive integer n, there exists a simple arc ? joining one of the boundary components of annulus to the other, such that ? is disjoint from its n first iterates under h: This result is an analogue of a theorem of Kwapisz dealing with diffeomorphisms of the two-torus [14]. Subsequently and applying the first result, we prove that a rigid rotation of angle can be approximated by a homeomorphism that is conjugate to h: Finally, we prove that to be an irrational pseudo-rotation is a necessary property in order that a homeomorphism has the curves intersection property and no periodic points.

On s’intéresse dans cette thèse à des systèmes couplés de type champ moyen en étudiant l’existence de l’état de synchronisation qui se caractérise par une distance uniformément bornée dans le temps entre chaque paire de composantes d’une solution. L’étude se base sur une méthode perturbative. Néanmoins les résultats obtenus ne sont pas évidents dans le cas non-perturbé. En outre dans le cas où le système couplé est périodique et grâce au Théorème du point fixe on montre l’existence d’une solution périodique sur le tore. L’étude de stabilité et de stabilité exponentielle est établie dans le cas linéaire et appliquée à ce type de systèmes couplés / We study in this thesis a class of a perturbed interconnected mean-field system, also known as a coupled systems. Under some assumptions we prove the existence of an invariant open set by the flow of the perturbed system ; in other word, we prove that the distance between the components of an orbit is uniformly bounded, this property is also called synchronization. We use the perturbation method to obtain the result. However the result is not trivial for the not perturbed system. We use the fixed point theorem to prove the existence of a periodic orbit in the torus. We study in addition the stability and the exponential stability of such systems by studying the stability of a linear systems.