A novel higher-order moment-closure method is applied for the Eulerian treatment of gas-particle multiphase flows characterized by a dilute polydisperse and polythermal particle phase. Based upon the polydisperse Gaussian-moment model (PGM) framework, the proposed model is derived by applying an entropy-maximization moment-closure formulation to the transport equation of the particle-number density function, which is equivalent to the Williams-Boltzmann equation for droplet sprays. The resulting set of first-order robustly-hyperbolic balance laws include a direct treatment for local higher-order statistics such as co-variances between particle distinguishable properties (i.e., diameter and temperature) and particle velocity. Leveraging the additional distinguishing variables, classical hydrodynamic droplet evaporation theory is considered to describe unsteady droplet vaporization. Further, studying turbulent multiphase flow theory, a first-order hyperbolicity maintaining approximation to turbulent flow diffusion-inertia effects is proposed. Investigations into the predictive capabilities of the model are evaluated relative to Lagrangian-based solutions for a range of flows, including aerosol dispersion and fuel-sprays. Further, the model is implemented in a massively parallel discontinuous-Galerkin framework. Validation of the proposed turbulence coupling model is subsequently performed against experimental data, and a qualitative analysis of the model is given for a qualitative liquid fuel-spray problem.

Finite element(FE) method has been used to model cutting process since 1970s. However, it requires special techniques to cope with the difficulties in simulating extremely large strain when compare to static or small deformation problems. With the advancement of FE techniques, researchers can now have a deeper insight of the mechanism of material flow and chip formation of metal cutting process. Even the stagnation effect of the workpiece material in front of the cutting edge radius can be captured by using FE techniques such as Remeshing and Arbitrary Lagrangian Eulerian(ALE) formulation. However most of this models are limited to plane strain assumption which means they are 2-dimensional. Although 3D models are existing in the literatures, most of them employ Remeshing technique which is very computationally intensive and has many critics regarding its accuracy due to its frequent remeshing and mapping process. The rest of the 3D models employ Lagrangian formulation. The 3D models by Lagrangian formulation have the same limitations and drawbacks as in 2D models, as it requires failure criteria and in most of the cases predefined partition surfaces are also required. ALE technique on the other hand resolves all the drawbacks of the other formulations, it not only inherits the advantages of the other techniques but also has its own unique advantages such as it can simulate a longer time span up to couple seconds more economically by fixing the number of elements used. Although it's commonly accepted that ALE formulation is superior to other formulations of techniques in modeling metal cutting process, its usage is only limited to 2D models. Limited 3D ALE metal cutting models is available in the literature. Thus the main objective of this research is to explore the possibility of building a 3D metal cutting model with ALE formulation. The reliability and limitations will also be studied. Furthermore, Couple Eulerian-Lagrangian(CEL) formulation is a recent developed formulation that has a lot of potential in modeling metal cutting process in 3D. It will be compared with ALE models to study its potential and limitations in modeling metal cutting process. A new frictional model will also be proposed, which suggests that the frictional phenomenon in metal cutting is a consolidated effect of both friction between material interface and shear yield of the workpiece material. This idea provide a brand new perspective of viewing the friction phenomenon of metal cutting compared to those existed models. / Thesis / Master of Science (MSc)

This paper consists of a discussion of the properties and applications of certain improper integrals, namely the gamma function and the beta function. There are also specific examples of application of these functions in certain fields of applied science.

Le contexte général de cette thèse est la Simulation Numérique Directe des écoulements diphasiques dilués anisothermes. Un accent particulier est mis sur la détermination précise de la dispersion des particules et du transfert de chaleur entre la phase porteuse et dispersée. Cette dernière est décrite à l’aide d’une approche Eulérienne aux moments : le Formalisme Eulérien Mésoscopique (FEM) [41, 123], récemment étendu aux écoulements anisothermes [78]. Le principal objectif de ce travail est de déterminer si ce formalisme est capable de prendre en compte de manière précise l’inertie dynamique et thermique des particules dans un écoulement turbulent, et particulièrement dans une configuration avec un gradient moyen. Le code de calcul utilisé est AVBP. La simulation numérique d’un spray dilué avec une approche Eulerienne soulève des questions supplémentaires sur les méthodes numériques et les modèles employés. Ainsi, les méthodes numériques spécifiques aux écoulements diphasiques implémentées dans AVBP [69, 103, 109] ont été testées et revisitées. L’objectif est de proposer une stratégie numérique précise et robuste qui résiste aux forts gradients de fraction volumique de particule provoqués par la concentration préférentielle [132], tout en limitant la diffusion numérique. Ces stratégies numériques sont comparées sur une série de cas tests de complexité croissante et des diagnostics pertinents sont proposés. Par exemple, les dissipations dues `a la physique et au numérique sont extraites des simulations et quantifiées. Le cas test du tourbillon en deux dimensions chargé en particules est suggéré comme une configuration simple pour mettre en évidence l’impact de l’inertie des particules sur leur champ de concentration et pour discriminer les stratégies numériques. Une solution analytique est aussi proposée pour ce cas dans la limite des faibles nombres de Stokes. Finalement, la stratégie numérique qui couple le schéma centré d’ordre élevé TTGC et une technique de stabilisation, aussi appelée viscosité artificielle, est celle qui fournit les meilleurs résultats en terme de précision et de robustesse. Les paramètres de viscosité artificielle (c'est-à-dire les senseurs) doivent néanmoins être bien choisis. Ensuite, la question des modèles nécessaires pour d´écrire correctement la dispersion des particules dans une configuration avec un gradient moyen est abordée. Pour ce faire, un des modèles RUM (appel´e AXISY-C), proposé par Masi [78] et implémenté dans AVBP par Sierra [120], est validé avec succès dans deux configurations: un jet plan diphasique anisotherme 2D et 3D. Contrairement aux anciens modèles RUM, les principales statistiques de la phase dispersée sont désormais bien prédites au centre et aux bords du jet. Finalement, l’impact de l’inertie thermique des particules sur leur température est étudié. Les résultats montrent un effet important de cette inertie sur les statistiques mettant en évidence la nécessité pour les approches numériques de prendre en compte ce phénomène. Ainsi, l’extension du FEM aux écoulements anisothermes, c’est-à-dire les flux de chaleur RUM (notés RUM HF), est implémentée dans AVBP. L’impact des RUM HF sur les statistiques de température des particules est ensuite évalué sur les configurations des jets 2D et 3D. Les champs Eulériens sont comparés à des solutions Lagrangiennes de référence calculées par B. Leveugle au CORIA et par E. Masi à l’IMFT pour les jets 2D et 3D, respectivement. Les résultats montrent que les RUM HF améliorent la prédiction des fluctuations de température mésoscopique, et dans une moindre mesure la température moyenne des particules en fonction de la configuration. Les statistiques Lagrangiennes sont retrouvées lorsque les RUM HF sont pris en compte alors que les résultats sont dégradés dans le cas contraire. / This work addresses the Direct Numerical Simulation of non-isothermal turbulent flows laden with solid particles in the dilute regime. The focus is set on the accurate prediction of heat transfer between phases and of particles dispersion. The dispersed phase is described by an Eulerian approach : the Mesoscopic Eulerian Formalism [41, 123], recently extended to non-isothermal flows [78]. The main objective of this work is to assess the ability of this formalism to accurately account for both dynamic and thermal inertia of particles in turbulent sheared flows. The CFD code used in this work is AVBP. The numerical simulation of dilute sprays with an Eulerian approach calls for specific modelling and raises additional numerical issues. First, the numerical methods implemented in AVBP for two-phase flows [69, 103, 109] were tested and revisited. The objective was to propose an accurate and robust numerical strategy that withstands the steep gradients of particle volume fraction due to preferential concentration [132] with a limited numerical diffusion. These numerical strategies have been tested on a series of test cases of increasing complexity and relevant diagnostics were proposed. In particular, the two-dimensional vortex laden with solid particles was suggested as a simple configuration to illustrate the effect of particle inertia on their concentration profile and to test numerical strategies. An analytical solution was also derived in the limit of small inertia. Moreover, dissipations due to numerics and to physical effects were explicitly extracted and quantified. Eventually, the numerical strategy coupling the highorder centered scheme TTGC with a stabilization technique –the so called artificial viscosity– proved to be the most accurate and robust alternative in AVBP if an adequate set-up is used (i.e. sensors). Then, the issue of the accurate prediction of particle dispersion in configurations with a mean shear was adressed. One of the RUM model (denoted AXISY-C), proposed by Masi [78] and implemented by Sierra [120], was successfully validated in a two-dimensional and a three-dimensional non-isothermal jet laden with solid particles. Contrary to the former RUM models [63, 103], the main statistics of the dispersed phase were recovered at both the center and the edges of the jet. Finally, the impact of the thermal inertia of particles on their temperature statistics has been investigated. The results showed a strong dependency of these statistics to thermal inertia, pinpointing the necessity of the numerical approaches to account for this phenomenon. Therefore, the extension of the MEF to non isothermal conditions, i.e. the RUM heat fluxes, has been implemented in AVBP. The impact of the RUM HF terms on the temperature statistics was evaluated in both configurations of 2D and 3D jets. Eulerian solutions were compared with Lagrangian reference computations carried out by B. Leveugle at CORIA and by E. Masi at IMFT for the 2D and 3D jets, respectively. Results showed a strong positive impact of the RUM HF on the fluctuations of mesoscopic temperature, and to a lesser extent on the mean mesoscopic temperature depending of the configuration. Neglecting the RUM HF leads to erroneous results whereas the Lagrangian statistics are recovered when they are accounted for.

In 2010 Chung-Graham-Knuth proved an interesting symmetric identity for the Eulerian numbers and asked for a q-analog version. Using the q-Eulerian polynomials introduced by Shareshian-Wachs we find such a q-identity. Moreover, we provide a bijective proof that we further generalize to prove other symmetric qidentities using a combinatorial model due to Foata-Han. Meanwhile, Hyatt has introduced the colored Eulerian quasisymmetric functions to study the joint distribution of the excedance number and major index on colored permutations. Using the Decrease Value Theorem of Foata-Han we give a new proof of his main generating function formula for the colored Eulerian quasisymmetric functions. Furthermore, certain symmetric q-Eulerian identities are generalized and expressed as identities involving the colored Eulerian quasisymmetric functions. Next, generalizing the recent works of Savage-Visontai and Beck-Braun we investigate some q-descent polynomials of general signed multipermutations. The factorial and multivariate generating functions for these q-descent polynomials are obtained and the real rootedness results of some of these polynomials are given. Finally, we study the diagonal generating function of the Jacobi-Stirling numbers of the second kind by generalizing the analogous results for the Stirling and Legendre-Stirling numbers of the second kind. It turns out that the generating function is a rational function, whose numerator is a polynomial with nonnegative integral coefficients. By applying Stanley's theory of P-partitions we find combinatorial interpretations of those coefficients

Une étude précise des écoulements gaz-particules poly-solides et réactifs rencontrés dans les lits fluidisés circulants (LFC) appliqués au procédé de Chemical Looping Combustion (CLC) est indispensable pour prédire un point de fonctionnement stable et comprendre l'influence de la réaction et de la polydispersion sur l'hydrodynamique des LFC. Dans ce but, des simulations avec le code NEPTUNE_CFD ont été confrontées aux expériences menées à l'Université Technologique de Compiègne par ALSTOM. Cette modélisation a été validée sur des LFC non réactifs mono-solides et poly-solides. L'influence des caractéristiques des particules et de la position des injecteurs sur l'entrainement de solide est étudiée. Un modèle de prise en compte de la production locale de gaz au cours de la réaction est présenté. L'étude locale de l'écoulement a permis de comprendre l'influence des collisions interparticulaire et de la production locale de gaz sur l'écoulement. Finalement, un point de fonctionnement a été proposé pour le pilote CLC en construction à Darmstadt. Ce travail a montré que NEPTUNE_CFD pouvait prédire l'hydrodynamique de LFC poly-solides à l'échelle du pilote industriel et participer au dimensionnement de centrales de types CLC. / This work deals with the development, validation and application of a model of Chemical Looping Combustion (CLC) in a circulating fluidized bed system. Chapter 1 is an introduction on Chemical Looping Combustion. It rst presents the most important utilizations of coal in the energy industry. Then, it shows that because of the CO2 capture policy, new technologies have been developed in the frame of post-combustion, pre-combustion and oxy-combustion. Then, the Chemical Looping Combustion technology is presented. It introduces multiple challenges: the choice of the Metal Oxide or the denition of the operating point for the fuel reactor. Finally, it shows that there are two specicities for CFD modeling: the influence of the collisions between particles of different species and the local production of gas in the reactor due to the gasication of coal particles. Chapter 2 outlines the CFD modeling approach: the Eulerian-Eulerian approach extended to flows involving different types of particles and coupled with the chemical reactions. Chapter 3 consists in the validation of the CFD model on mono-solid (monodisperse and poly-disperse) and poly-solid flows with the experimental results coming from an ALSTOM pilot plant based at the Universite Tchnologique de Compiegne (France). The relevance of modeling the polydispersity of a solid phase is shown and the influence of small particles in a CFB of large particles is characterized. This chapter shows that the pilot plant hydrodynamics can be predicted by an Eulerian-Eulerian approach. Chapter 4 consists in the validation of the CFD model on an extreme bi-solid CFB of particles of same density but whith a large particle diameter ratio. Moreover, the terminal settling velocity of the largest particles are twice bigger than the fluidization velocity: the hydrodynamics of the large particles are given by the hydrodynamics of the smallest. An experiment performed by Fabre (1995) showed that large particles can circulate through the bed in those operating conditions. Our simulations predicted a circulation of large particles, but underestimated it. It is shown that it can be due to mesh size eect. Finally, a simulation in a periodic box of this case was dened and allowed us to show the major influence of collisions between species. Chapter 5 presents the simulation of a hot reactive CLC pilot plant under construction in Darmstadt (Germany). The simulations account for the chemical reactions and describe its eect on the hydrodynamics. Different geometries and operating conditions are tested.

Dans certaines simulations numériques exigeantes de mécanique des fluides, ilest nécessaire de simuler des écoulements multiphasiques impliquant de nombreuses contraintes simultanées : nombre de fluides important, évolutions compressibles à la fois isentropes et fortement choquées, équations d’états variables et contrastées, déformations importantes et transport surdes longues distances. Afin de remplir ces objectifs de manière robuste, il est nécessaire que la cohérence thermodynamique du schéma numérique soit vérifiée.Dans le premier chapitre, un schéma de type Lagrange plus projection est proposé pour la simulation d’écoulements diphasiques avec un modèle squelette à six équations et sans termes de dissipation. L’importance de la propriété de préservation des écoulements isentropiques est mise en évidence à l’aide d’une comparaison avec des résultats issus de la littérature pour le test deRansom. Ce chapitre souligne aussi certaines limitations de l’approche Lagrange plus projection pour simuler des modèles multiphasiques.Afin de pallier à ces limitations, une nouvelle procédure de dérivation est proposée afin de construire un schéma mimétique pour la simulation d’écoulements instationnaires compressibles dans un formalisme ALE direct (Arbitrary Lagrangian–Eulerian). La possibilité de choisir a prioriles degrés de liberté permet de s’inscrire dans une continuité avec les schémas historiques décalés, tout en imposant les conservations au niveau discret. L’équation de quantité de mouvement discrèteest obtenue par application d’un principe variationnel, assurant par construction la cohérence thermodynamique des efforts de pression. Cette approche est appliquée au cas d’écoulements monofluides comme preuve de concept au Chapitre 3, puis elle est étendue au cas d’écoulements à Nphasescompressibles au Chapitre 4. Des tests mono et multiphasiques montrent un comportement satisfaisant en terme de conservativité, versatilité aux mouvements de grilles et robustesse. / In some highly demanding fluid dynamics simulations, it appears necessary tosimulate multiphase flows involving numerous constraints at the same time : large numbers of fluids, both isentropic and strongly shocked compressible evolution, highly variable and contrasted equations of state, large deformations, and transport over large distances. Fulfilling such a challengein a robust and tractable way demands that thermodynamic consistency of the numerical scheme be carefully ensured.In the first chapter, a Lagrange plus remap scheme is proposed for the simulation of two-phase flows with a dissipation-free six-equation bakcbone model. The importance of the property of isentropic flow preservation is highlighted with a comparison with Ransom test results fromthe literature. This chapter also also point out certain limitations of the Lagrange plus remap approach for multiphase simulations.In order to overcome these limitations, a novel derivation procedure is proposed to construct a mimetic scheme for the simulation of unsteady and compressible flows in a direct ALE (ArbitraryLagrangian-Eulerian) formalism. The possibility to choose a priori the degrees of freedom allows to obtain a continuity with historical staggered scheme, while imposing conservativity at discretelevel. The discrete momentum evolution equation is obtained by application of a variational principle, thus natively ensuring the thermodynamic consistency of pressure efforts. This approach is applied to single-fluid flows as a proof of concept in Chapter 3, then it is extended to N-phasecompressible flows in Chapter 4. Single- and multi-phase tests show satisfactory behavior in terms on conservation, versatility to grid motions, and robustness.

Ce sujet de thèse s’inscrit dans le cadre des études actuellement menées au CEA/DAM concernant des schémas numériques ALE (Arbitrary-Lagrangian-Eulerian)de type « Lagrange + Projection », dans le contexte des simulations hydrodynamiques mutli-matériaux en grandes déformations. Ces schémas doivent respecter les équations de conservation de la masse, de la quantité de mouvement et de l’énergie totale.Les schémas décalés en temps et en espace sont très utilisés dans les codes industriels. Ils sont robustes et permettent une bonne approximation des comportements complexes, mais sont connus pour ne pas conserver exactement l’énergie totale. Cela pose un problème dans le traitement des chocs, sur maillages raffinés ou dans la simulation des milieux réactifs.En 2016, des travaux originaux on été proposés par A. Llor et. al. pour rendre conservatif ce type de schéma dans un contexte lagrangien (sans projection), notamment en proposant une correction pour retrouver la conservation de l’énergie totale.Le travail de cette thèse a été d’étendre ce schéma lagrangien dans un contexte ALE multi-matériaux (avec interface), en garantissant la conservation de toutes les quantités, le respect du second principe de la thermodynamique et la robustesse. De nombreux cas tests ont été menés (en 2D plan et en 2D axisymétrique) et comparés aux méthodes existantes afin de montrer la pertinence de cette approche. / This PhD subject comes within actual studies managed by CEA/DAM about ALE (Arbitrary-Lagrangian-Eulerian) schemes (with a splitting of Lagrangian and Remapping steps) in the context of hydrodynamic simulations. These numerical schemes have to respect mass, momentum and total energy conservation, which are the fundamental equations of the studied systems.Space- and Time-Staggered are widely used in industrial codes for their simplicity androbustness despite their known lack of exact energy conservation. This is a major drawbackin presence of strong shocks. Among all existing schemes, none of them meet the expectations of robustness, conservation,thermodynamic consistency (both shocks and relaxations capture), accuracy andadaptibility to complex behaviors. Recently, some novel works have been proposed by A.Llor et. al. in order to make conservative this type of scheme in a Lagrangian context (without remapping step). Current remap methods, necessary in large deformations, donot guarantee simultaneously total energy conservation and thermodynamic consistency.This work aims at extending this conservative Lagrangian space- and time-staggeredscheme to a multi-material ALE methodology, keeping its good properties (conservation,accuracy, thermodynamic consistency, robustness) intact. Classical, but demanding, test cases have been performed (both in plane and axisymmetric 2D geometries) and have been compared to existing numerical methods in order to assess the relevance of our approach.

Underwater explosions are often modeled with Arbitrary Lagrangian-Eulerian (ALE) Finite Element Methods. The objective of this thesis is to validate the simulation method, with respect to the propagating shock wave. A two-dimensional axisymmetric model of a spherical TNT charge submerged in water is simulated using LS-DYNA. The explosive is modeled with the Burn Fraction technique and the Jones-Wilkins-Lee equation of state. Water is modeled as a non-viscous fluid, with the Grüneisen equation of state. The convergence for different mesh resolutions, the effect of different advection methods, and varied constants in the artificial viscosity are examined. Generally, the simulations agree well with empirical results, but the maximum pressure diminishes more rapidly with distance compared to experiments. The excessive dampening is most notable in the early stages of the propagation. Also, unexpected oscillations are observed near the discontinuity. The choice of advection scheme and constants in the artificial viscosity do not resolve the issues suggesting that other numerical techniques for treating the discontinuity should be considered. / Undervattensexplosioner simuleras ofta med ALE-baserade finita elementmetoder. Detta examensarbete avser att validera simuleringsmetoden med hänsyn till stötvågens utbredning i vattnet. En tvådimensionell axisymmetrisk modell av en sfärisk TNT-laddning nedsänkt i vatten simuleras i LS-DYNA. Laddningen modelleras med hjälp av brinnfraktioner och Jones-Wilkins-Lee tillståndsekvation. Vattnet modelleras som en inviskös fluid tillsammans med Grüneisens tillståndsekvation. Nätkonvergens, val av advektionsmetod och ändring av konstanter i den artificiella viskositeten studeras. Övergripande resultat stämmer väl överens med empirisk data, men stötvågens topptryck avtar fortare än väntat. Denna dämpning är tydligast i utredningens tidiga skeden. Dessutom observeras oväntade oscillationer kring stötvågens diskontinuerliga tryckprofil. Val av advektionsmetod och konstanter tillhörande artificiella viskositeten verkar ha liten betydelse för resultaten. En alternativ numerisk metod för behandling av stötvågens diskontinuitet bör implementeras.

A general framework is described for the solution of two-phase fluid-object interaction problems on the basis of coupling a distributed-Lagrange-multiplier fictitious domain method and a level-set method, intended for application to the problem of seabed scour by ice ridges. The resulting equations are discretized in space using stabilized finite-element methods and integrated in time using the generalized-α method. This approach is simple to implement and applicable to both structured and unstructured meshes in two and three dimensions. By means of examples, it is shown that despite the simplicity of the approach, good results are obtained in comparison with other more computationally demanding methods. A robust approach is utilized for constructing signed-distance functions on arbitrary meshes by introducing artificial numerical diffusivity to improve the robustness of classical signed-distance construction approaches without resorting to common pseudo-time relaxation. Under this approach, signed-distance functions can be rapidly constructed while preserving the numerical convergence properties and, generally, having minimal interfacial perturbation. The method is then applied with a modified deformation procedure for fast and efficient mesh adaptivity, including a discussion how it may be used in computational fluid dynamics. The two-phase fluid-object interaction approach is then customized for modeling of the seabed scour and soil-pipe interaction. In this approach, complex history-dependent soil constitutive models are replaced with a simple strain-rate dependent model. Utilization of this constitutive model along with the framework developed earlier leads to the treatment of seabed scour as a two-phase fluid-object interaction, and the soil-pipe interaction as a fluid-structure interaction problem without the need for remeshing. Good agreement with past experimental and numerical studies are obtained using our approach. The dissertation is concluded by conducting a parametric study of seabed scour in two- and three-dimensions. / text