A set of second-moment closure turbulence models are implemented for the study of wake evolution in an oceanic environment. The effects of density stratification are considered, and the models are validated against laboratory experiments mimicking the stratified ocean environment, and against previous experimental study of wakes subjected to a density stratification. The turbulence models are found to reproduce a number of important behaviors which differentiate stratified wakes from those in a homogeneous environment, including the appropriate decay rates in turbulence quantities, buoyant suppression of turbulence length scales, and canonical stages in wake evolution. The existence of background turbulence is considered both through the introduction of production terms to the turbulence model equations and the replication of scale-resolved simulations of wakes embedded in turbulence. It is found that the freestream turbulence causes accelerated wake growth and faster decay of wake momentum. Wakes are then simulated at a variety of Re and Fr representative of full-scale vehicles operating in an ocean environment, to downstream distances several orders of magnitude greater than existing RANS studies. The models are used to make some general predictions concerning the dependence of late-wake behavior on these parameters, and specific insights into expected behavior are gained. The wake turbulence is classified using "fossil turbulence" and stratification strength criteria from the literature. In keeping with experimentally observed behavior, the stratification is predicted to increase wake persistence. It is also predicted that, regardless of initial Re or F r, the wake turbulence quickly becomes a mixture of overturning eddies and internal waves. It is found that the high Re wakes eventually become strongly affected by the stratification, and enter the strongly-stratified or LAST regime. Additional model improvements are proposed based on the predicted late wake behavior. / Doctor of Philosophy / A set of advanced turbulence models are implemented and used to study ship wakes in an oceanic environment. The flows in the ocean are subject to a density stratification due to changes in temperature and salinity; the associated effects are included in the turbulence models. The models are validated against laboratory experiments mimicking the stratified ocean environment, and against previous experimental study of wakes subjected to a density stratification. The turbulence models are found to reproduce a number of important behaviors expected under such conditions based on experimental study. Additional modifications are made to the models to include the effect of pre-existing freestream turbulence. Wakes are then simulated under conditions representative of full-scale vehicles operating in an ocean environment. The models are used to make some general predictions concerning late-wake behavior. Specific insights into expected behavior are gained. The wake turbulence is classified using ``fossil turbulence'' and stratification strength criteria from the literature. In keeping with experimentally observed behavior, the stratification is predicted to increase wake persistence. Additional model improvements are proposed based on the predicted late wake behavior.

Results from numerical simulation were performed to study flow and heat transfer in two types of rotating multi-pass cooling channels. Second moment closure model was used to solve flow in domain generated from Chimera method. The first type was a four-pass channel with two different inlet settings. The main flowing channel was rectangular channel (AR=2:1) with hydraulic diameter (Dh ) equals to 2/3 inch (16.9 mm). The first and fourth channel were set as different aspect ratio (AR=2:1; AR=1:1). Reynolds number (Re) used in this part was 10,000. The rotating angle was set as 90 degrees. The density ratio was set as 0.115. The rotation number varied from 0.0 to 0.22. It was showed that inlet effect only caused influence to flow and heat transfer in first two passages. The second type was a four-pass channel with/without addition of vane in smooth turn portion. The main flowing channel was rectangular channel (AR=2:1) with hydraulic diameter (Dh) equals to 2/3 inch. The first and fourth passages were set to be square duct (AR=1:1). The Reynolds number (Re) used in this part was 20,000. Three rotation numbers were set here (Ro=0.0; Ro=0.2; Ro=0.4). The density ratio and rotating angle varied from 0.12 to 0.32 and from 45 degrees to 90 degrees respectively. According to numerical results, it was revealed that the addition of vane in smooth turn portion did not cause influence to part before it. However, it caused significant influence to flow and heat transfer in smooth turn portion and part after it.

In this study, the behavior of pressure in the Rapid Distortion Limit, along with its concomitant modeling, are addressed. In the first part of the work, the role of pressure in the initiation, propagation and suppression of flow instabilities for quadratic flows is analyzed. The paradigm of analysis considers the Reynolds stress transport equations to govern the evolution of a dynamical system, in a state space composed of the Reynolds stress tensor components. This dynamical system is scrutinized via the identification of the invariant sets and the bifurcation analysis. The changing role of pressure in quadratic flows, viz. hyperbolic, shear and elliptic, is established mathematically and the underlying physics is explained. Along the maxim of "understanding before prediction", this allows for a deeper insight into the behavior of pressure, thus aiding in its modeling. The second part of this work deals with Rapid Pressure Strain Correlation modeling in earnest. Based on the comprehension developed in the preceding section, the classical pressure strain correlation modeling approaches are revisited. Their shortcomings, along with their successes, are articulated and explained, mathematically and from the viewpoint of the governing physics. Some of the salient issues addressed include, but are not limited to, the requisite nature of the model, viz. a linear or a nonlinear structure, the success of the extant models for hyperbolic flows, their inability to capture elliptic flows and the use of RDT simulations to validate models. Through this analysis, the schism between mathematical and physical guidelines and the engineering approach, at present, is substantiated. Subsequently, a model is developed that adheres to the classical modeling framework and shows excellent agreement with the RDT simulations. The performance of this model is compared to that of other nominations prevalent in engineering simulations. The work concludes with a summary, pertinent observations and recommendations for future research in the germane field.

In this work the predictive capability of a number of Reynolds stress transport(RST) models was first tested in a range of non-equilibrium homogeneous flows, comparisons being drawn with existing direct numerical simulation (DNS) results and physical measurements. The cases considered include both shear and normally strained flows, in some cases with a constant applied strain rate, and in others where this varied with time. Models were generally found to perform well in homogeneous shear at low shear rates, but their performance increasingly deteriorated at higher shear rates. This was attributed mainly to weaknesses in the pressure-strain rate models, leading to over-prediction of the shear stress component of the stress anisotropy tensor at high shear rates. Performance in irrotational homogeneous strains was generally good, and was more consistent over a much wider range of strain rates. In the experimental plane strain and axisymmetric contraction cases, with time-varying strain rates, there was evidence of an accelerated dissipation rate generation. Significant improvement was achieved through the use of an alternative dissipation rate generation term, Pε , in these cases, suggesting a possible route for future modelling investigation. Subsequently, the models were also tested in the inhomogeneous case of pulsating channel flow over a wide range of frequencies, the reference for these cases being the LES of Scotti and Piomelli (2001). A particularly challenging feature in this problem set was the partial laminarisation and re-transition that occurred cyclically at low and, to a lesser extent, intermediate frequencies. None of the models tested were able to reproduce correctly all of the observed flow features, and none returned consistently superior results in all the cases examined. Finally, models were tested in the case of a plane jet interacting with a rectangular dead-end enclosure. Two geometric configurations are examined, corresponding a steady regime, and an intrinsically unsteady regime in which periodic flow oscillations are experimentally observed (Mataoui et al., 2003). In the steady case generally similar flow patterns were returned by the models tested, with some differences arising in the degree of downward deflection of the impinging jet, which in turn affected the level of turbulence energy developing in the lower part of the cavity. In the unsteady case, only two of the models tested, a two-equation k-ε model and an advanced RST model, correctly returned purely periodic solutions. The other two RST models, based on linear pressure-strain rate terms, returned unsteady flow patterns that exhibited complex oscillations with significant cycle-to-cycle variations. Unfortunately, the limited availability of reliable experimental data did not allow a detailed quantitative examination of model performance.

In this work, a three-dimensional turbulent boundary layer experiment was set up with alternating stream-wise and span-wise pressure gradients. The pressure gradients are generated as a result of the test section wavy side wall shape. Each side had six sine waves with a trough to peak magnitude to wavelength ratio of 0.25. Boundary layer control was used so that the flow over the side walls remains attached. The mean flow velocity components, static and total pressures were measured at six plane along the stream-wise direction. The alternating mean span-wise and stream-wise pressure gradients created alternating stream-wise and span-wise vorticity fluxes, respectively, along the test section. As the flow developed downstream the vorticity created at the tunnel floor and ceiling diffused away from the wall. The vorticity components in the stream-wise and span-wise directions are strengthened due to stretching and tilting terms in the vorticity transport equations. The positive-z half of the test section contains large areas that generate positive vorticity flux in the trough region and smaller areas generating negative vorticity around the wave peak. The opposite is true for the negative-z half of the test-section. This results in a large positive stream-wise vorticity in the positive-z half and negative stream-wise vorticity in the negative-z half of the test-section. The smaller regions of opposite sign vorticity in each half tend to mix the flow such that as they diffuse away from the wall, the turbulent stresses are more uniform. Turbulent fluctuating velocity components were measured using Laser Doppler Velocimetery. Mean velocities as well as Reynolds stresses and triple velocity component correlations were measured at thirty stations along the last wave in the test section. Profiles at the center of the test section showed three dimensionality, but exhibited high turbulence intensities in the outer layer. Profiles off the test section centerline are highly three dimensional with multiple peaks in the normal stress profiles. The flow also reaches a state where all the normal stresses have equal magnitudes while the shear stresses are non-zero. Flow angles, flow gradient angles and shear stress angles show very large differences between wall values and outer layer vlaues. The shear stress angle lagged the flow gradient angle indicating non-equilibrium. A turbulent kinetic energy transport budget is performed for all profiles and the turbulence kinetic energy dissipation rate is estimated. Spectral measurements were also made and an independent estimate of the kinetic energy dissipation rate is made. These estimates agree very well with those estimates made by balancing the turbulence kinetic energy transport equation. Multiple turbulent diffusion models are compared to measured quantities. The models varied in agreement with experimental data. However, fair agreement with turbulence kinetic energy turbulent diffusion is observed. A model for the dissipation rate tensor anisotropy is used to extract estimates of the pressure-strain tensor from the Reynolds stress transport equations. The pressure-strain estimates are compared with some of the models in the literature. The comparison showed poor agreement with estimated pressure-strain values extracted from experimental data. A tentative model for the turbulent Reynolds shear stress angle is developed that captures the shear stress angle near wall behavior to a very good extent. The model contains one constant that is related to mean flow variables. However, the developed expression needs modification so that the prediction is improved along the entire boundary layer thickness. / Ph. D.

[pt] O presente trabalho propõe um método estatístico computacionalmente eficiente (chamado Green-FOSM) para consideração de incertezas em problemas geomecânicos, com o objetivo de melhorar o processo de tomada de decisão ao analisar problemas associados com o processo de injeção ou depleção de fluídos. A novidade do método proposto está associada com a utilização do método das funções de Green (GFA), que, com o auxílio do método estatístico first-order second-moment (FOSM), é utilizado para propagar as inerentes incertezas associadas às propriedades mecânicas do material para o campo de deslocamento da formação geológica. Além disso, através dos conceitos de grid estocástico e função de autocorrelação, o método proposto permite a consideração da variabilidade espacial de variáveis aleatórias de entrada que representam essas propriedades mecânicas. O GFA utiliza as soluções fundamentais da mecânica clássica (solução fundamental de Kelvin, solução fundamental de Melan, entre outras) e o teorema da reciprocidade para determinar o campo de deslocamento de uma formação geológica com geometria irregular e diferentes tipos de materiais. A grande vantagem deste método em relação ao clássico método dos elementos finitos (MEF) é que ele não requer a imposição de condições de contorno e a análise do problema pode ser realizada considerando apenas o domínio do reservatório ou outras regiões de interesse. Esta estratégia de modelagem diminui os graus de liberdade do modelo e o tempo de processamento da análise. Desta forma, como o GFA requer menos esforço computacional, este método torna-se ideal para ser utilizado na propagação de incertezas em problemas geomecânicos. Inicialmente, baseado no método das funções de Green original proposto por Peres et al. (2021), foi proposto uma versão iterativa do método Green-FOSM, que apresenta resultados estatísticos semelhantes aos encontrados através da clássica simulação de Monte Carlo (SMC). Nesta versão original, o campo de deslocamento é PUC-Rio - Certificação Digital Nº 1912634/CA calculado usando um esquema numérico iterativo que diminui o desempenho computacional do método e pode gerar problemas de convergência. Tais limitações tem dificultado a aplicação do GFA original e do método Green-FOSM iterativo em problemas reais. Assim, o presente trabalho desenvolveu uma nova versão do GFA que utiliza um esquema numérico não-iterativo. Para os problemas de validação analisados, o método não-iterativo demonstra ser até 17.5 vezes mais rápido do que a versão original. Além disso, esta versão demonstra ser capaz de expandir a aplicabilidade do GFA, pois os problemas de convergência foram eliminados e os resultados obtidos por este método, ao analisar um perfil geológico representativo do pré-sal brasileiro, são semelhantes aos encontrados via MEF. Por fim, a partir do GFA não-iterativo foi proposta uma versão não-iterativa do método Green-FOSM. Esta versão não-iterativa é capaz de analisar probabilisticamente formações geológicas complexas, como é o caso das formações geológicas do présal brasileiro. Utilizando os mesmos recursos computacionais, o método GreenFOSM não-iterativo é no mínimo 200 vezes mais rápido que o método iterativo. De forma geral, os resultados encontrados nas análises realizadas (determinísticas e probabilísticas) são próximos dos resultados obtidos pelo método de referência (MEF e SMC, respectivamente). / [en] The present work proposes a computationally efficient stochastic statistical method (called Green-FOSM) that considers uncertainties in geomechanical problems, with the objective of improving the decision-making process related to problems associated with the process of fluid injection or depletion. The novelty of the method lies in the use of the Green s function approach (GFA), which, together, with the first-order second-moment statistical method (FOSM), is used to propagate uncertainties associated with the mechanical properties of material to the displacement field of the geological formation. Furthermore, using the concepts of stochastic grid and autocorrelation function, the proposed method allows the consideration of the spatial variability of random variables that represent these mechanical properties. The GFA uses the fundamental solutions of classical mechanics (Kelvin fundamental solution, Melan fundamental solution, among others) and the reciprocity theorem to calculate the displacement field of a geological formation with irregular geometry, and different types of materials. The great advantage of this method compared to the classical finite element method (FEM) is that it does not require the imposition of boundary conditions and the analysis of the problem can be performed considering only the reservoir or other regions of interest. This modeling strategy decreases the degrees of freedom of the model and the CPU time of the deterministic analysis. In this way, as the GFA requires less computational effort, this approach becomes ideal for propagating the uncertainties in geomechanical problems. Initially, an iterative version of the Green-FOSM method was proposed, which presents statistical results similar to those found through the classic Monte Carlo simulation (MCS). In this initial version, the displacement field is calculated using an iterative numerical scheme, which decreases the computational performance of the method and can generate convergence problems. Such limitations would restrict the application of the PUC-Rio - Certificação Digital number 1912634/CA original GFA and the iterative Green-FOSM method in real problems. Thus, the present work also developed a new version of the GFA, which uses a non-iterative numerical scheme. For the proposed validation problems, the non-iterative method proved to be up to 17.5 times faster than the original version. This version is able to expand the applicability of the GFA, since the convergence problems were eliminated and the results obtained by this method, when analyzing a representative geological profile of the Brazilian pre-salt, are similar to those found via FEM. Finally, based on the non-iterative GFA, a non-iterative version of the Green-FOSM method was proposed. This non-iterative version is capable of probabilistically analyzing complex geological formations, such as the Brazilian pre-salt geological formations. Using the same computational resources, the non-iterative GreenFOSM method is at least 200 times faster than the iterative Green-FOSM method. In general, the results found in the investigated analyzes (deterministic and probabilistic) are close to the results obtained by the reference method (FEM and MCS, respectively).

[pt] Os projetos geotécnicos são, em geral, avaliados através do cálculo de um coeficiente de segurança, obtido de forma determinística. A presente pesquisa tem como objetivo a comparação de dois métodos probabilísticos (Estimativas Pontuais e Segundo Momento) empregados em análises de estabilidade. Os métodos probabilísticos quantificam as incertezas oriundas da variabilidade dos parâmetros geotécnicos, calculando-se um índice de confiabilidade (beta), que expressa o quanto o fator de segurança é confiável. No entanto, o índice b é um parâmetro relativo, pois não exprime todas as incertezas contidas em um projeto geotécnico. O trabalho apresenta uma comparação das análises probabilísticas de estabilidade da barragem de Curuá-Una, no Pará, e de um muro de arrimo, em São Fidélis, no Estado do Rio de Janeiro. No caso da barragem, estudou-se a influência de um eventual aumento no nível d´água do reservatório no cálculo da probabilidade de ruptura. Para o muro, foram consideradas diversas alturas do nível d´água no retroaterro a montante. As análises enfocaram, ainda, detalhes dos métodos probabilísticos, no que se refere à aplicabilidade para projetos práticos. Verificou-se que o método do Segundo Momento é de mais fácil utilização em geotecnia. Recomenda-se, portanto, a sua aplicação em conjunto com as análises determinísticas, mais usuais na prática. / [en] Geotechnical projects are generaly assessed through the calculus of the safety factor obtained by deterministic methodologies. The purpose of the present research work is to compare two probabilistic methods (Estimate Punctual and Second Moment) employed in stability analyses. Probabilistic methods quantify the uncertainties derived from the variability of geotechnical parameters, calculating a confidence ratio that expresses how reliable the safety factor is. However, the ratio is a relative parameter, since it does not account for all the uncertainties contained in a geotechnical project. In view of this situation, the study compares the stability probabilistic analyses of Curuá-Una dam, in Pará, and those of a retaining wall in São Fidélis, Rio de Janeiro. Regarding the dam, the influence of a possible rise in the resevoir water level in the calculus of failure probability has been investigated. In respect to the wall, several water levels have been considered in the upstream backfill. Moreover, the analyses have focused on the details of probabilistic methods and their application to practical projects. It has been verified that the Second Moment Method is more easily employed in geothecnics. Therefore, it is recommended that it should be applied together with the deterministic analyses currently used in practice.

Le problème de satisfaisabilité booléenne 3-SAT est connu pour présenter un phénomène de seuil en fonction du quotient entre le nombre de clauses et le nombre de variables. Nous donnons des estimations de la valeur de ce seuil au moyen de méthodes combinatoires et probabilistes: la méthode du premier moment et la méthode du second moment. Ces méthodes mettent en jeu des problèmes d'optimisation sous contraintes et nous amènent à employer de façon intensive la méthode des multiplicateurs de Lagrange. Nous mettons en œuvre une forme pondérée de la méthode du premier moment sur les affectations partielles valides de Maneva ainsi que des variantes. Cela nous conduit à élaborer une pondération générale pour les problèmes de satisfaction de contraintes qui soit compatible avec la méthode du premier moment. Cette pondération est constituée d'une graine et d'un répartiteur, et nous permet d'obtenir une pondération des affectations partielles valides meilleure que celle de Maneva. Nous comparons aussi dans certains cas les performances de la pondération et de l'orientation de l'espace des solutions des problèmes de satisfaction de contraintes relativement à la méthode du premier moment. Nous développons la première sélection non uniforme de solutions pour majorer le seuil de 3-SAT et nous montrons sa supériorité sur ses prédécesseurs. Nous construisons un cadre général pour appliquer la méthode du second moment à k-SAT et nous discutons des conditions qui la font fonctionner. Nous faisons notamment fonctionner la méthode du second moment sur les solutions booléennes et sur les impliquants. Nous étendons cela au modèle distributionnel de k-SAT.

This study presents the results of probabilistic modeling of plane and wedge types of slope failures, based on the &rdquo / Advance First Order Second Moment (AFOSM)&rdquo / reliability method. In both of those failure types, two different failure criteria namely, Coulomb linear and Barton Bandis non-linear failure criteria are utilized in the development of the probabilistic models. Due to the iterative nature of the AFOSM method, analyzing spreadsheets have been developed in order to carry out the computations. The developed spreadsheets are called &ldquo / Plane Slope Analyzer (PSA)&rdquo / and &ldquo / Wedge Slope Analyzer (WSA)&rdquo / . The developed probabilistic models and their spreadsheets are verified by investigating the affect of rock and slope parameters such as, ground water level, slope height, cohesion, friction angle, and joint wall compressive strength (JCS) and their distribution types on the reliability index (&amp / #946 / ), and probability of slope failure (PF). In this study, different probability distributions are used and the inverse transformation formulas of their non-normal variates to their equivalent normal ones are developed as well. In addition, the wedge failure case is also modeled by using system reliability approach and then the results of conventional probability of failure and the system reliability approach are compared.

Turbulent dynamics at two sites (C and D) in a hypoxic zone on the Texas- Louisiana continental shelf were studied by investigating turbulence quantities i.e. turbulence kinetic energy (TKE), dissipation rate of TKE (E), Reynolds stress (τ ), dissipation rate of temperature variance (χ), eddy diffusivity of temperature (ν't), and eddy diffusivity of density (ν'p). Numerical models were also applied to test their capability of simulating these turbulence quantities. At site D, TKE, E, and τ were calculated from velocity measurements in the bot- tom boundary layer (BBL), using the Kolmogorov’s -5/3 law in the inertial subrange of energy spectra of vertical velocity fluctuations in each burst measurement. Four second-moment turbulence closure models were applied for turbulence simulations, and modeled turbulence quantities were found to be consistent with those observed. It was found from inter-model comparisons that models with the stability functions of Schumann and Gerz predicted higher values of turbulence quantities than those of Cheng in the mid layer, which might be due to that the former stability functions are not sensitive to buoyancy. At site C, χ, E, v’t, and ν’p were calculated from profile measurements throughout the water column, and showed high turbulence level in the surface boundary layer and BBL, as well as in the mid layer where shear stress was induced by advected non-local water above a hypoxic layer. The relatively high dissolved oxygen in the non-local water resulted in upward and downward turbulent oxygen fluxes, and the bottom hypoxia will deform due to turbulence in 7.11 days. Two of the four models in the study at site D were implemented, and results showed that turbulence energy resulting from the non-local water was not well reproduced. We attribute this to the lack of high-resolution velocity measurements for simulations. Model results agreed with observations only for χ and E simulated from the model with the stability function of Cheng in the BBL. Discrepancies between model and observational results lead to the following conclusions: 1) the stability functions of Schumann and Gerz are too simple to represent the turbulent dynamics in stratified mid layers; 2) detailed velocity profiles measurements are required for models to accurately predict turbulence quantities. Missing such observations would result in underestimation,