A quasi-conservative discontinuous Galerkin method for multi-component flows using the non-oscillatory kinetic flux

In this paper, a high order quasi-conservative discontinuous Galerkin (DG) method using the non-oscillatory kinetic flux is proposed for the 5-equation model of compressible multi-component flows with Mie-Gr\"uneisen equation of state. The method mainly consists of three steps: firstly, the DG method with the non-oscillatory kinetic flux is used to solve the conservative equations of the model; secondly, inspired by Abgrall's idea, we derive a DG scheme for the volume fraction equation which can avoid the unphysical oscillations near the material interfaces; finally, a multi-resolution WENO limiter and a maximum-principle-satisfying limiter are employed to ensure oscillation-free near the discontinuities, and preserve the physical bounds for the volume fraction, respectively. Numerical tests show that the method can achieve high order for smooth solutions and keep non-oscillatory at discontinuities. Moreover, the velocity and pressure are oscillation-free at the interface and the volume fraction can stay in the interval [0,1].Comment: 41 pages, 70 figure

Dynamic p-enrichment schemes for multicomponent reactive flows

We present a family of p-enrichment schemes. These schemes may be separated into two basic classes: the first, called \emph{fixed tolerance schemes}, rely on setting global scalar tolerances on the local regularity of the solution, and the second, called \emph{dioristic schemes}, rely on time-evolving bounds on the local variation in the solution. Each class of $p$-enrichment scheme is further divided into two basic types. The first type (the Type I schemes) enrich along lines of maximal variation, striving to enhance stable solutions in "areas of highest interest." The second type (the Type II schemes) enrich along lines of maximal regularity in order to maximize the stability of the enrichment process. Each of these schemes are tested over a pair of model problems arising in coastal hydrology. The first is a contaminant transport model, which addresses a declinature problem for a contaminant plume with respect to a bay inlet setting. The second is a multicomponent chemically reactive flow model of estuary eutrophication arising in the Gulf of Mexico.Comment: 29 pages, 7 figures, 3 table

Contributions to the Development of Entropy-Stable Schemes for Compressible Flows

Entropy-Stable (ES) schemes have gathered considerable attention over the last decade, especially in the context of under-resolved simulations of compressible turbulent flows, where achieving both high-order accuracy and robustness is difficult. ES schemes provide stability in a nonlinear and integral sense: the total entropy of the discrete solution can be made non-decreasing, in agreement with the second principle of thermodynamics. Additionally, the amount of entropy produced by the scheme is known and can be modified, making room for analysis and improvements. This thesis delves into some of the challenges currently limiting their use in practice. The current state of the art solves the compressible Navier-Stokes equations for a single-component perfect gas in chemical and thermal equilibrium. This model is inappropriate in aerospace engineering applications such as hypersonics and combustion, which typically involve chemically reacting gas mixtures far from equilibrium. As a first step towards enabling their use for these applications, we formulated ES schemes for the multicomponent compressible Euler equations. Special care had to be taken as we found out that the theoretical foundations of ES schemes begin to crumble in the limit of vanishing partial densities. The realization that ES schemes can only go as far as their theory led us to review some of it. A fundamental result supporting the development of limiting strategies for high-order methods is the minimum entropy principle for the compressible Euler equations. It states that the specific entropy of the physically relevant weak solution does not decrease. We proved that the same result holds for the specific entropy of the gas mixture in the multicomponent case. While entropy-stability is a valuable property, it does not imply a well-behaved solution. One must recall that the second principle is a prescription on the correct behavior of a system at the global level only. To better understand how ES schemes may or may not improve the quality of the numerical solution, we revisited two classical problems encountered in the development of shock-capturing techniques. First, we studied the receding flow problem, which is a simple setup used to study the anomalous temperature rise, termed "overheating", typically observed in shock reflection and shock interaction calculations. Previous studies showed that the anomaly can be cured if conservation of entropy is enforced, but at the considerable price of total energy conservation. Entropy-Conservative (EC) schemes, a particular instance of ES schemes, can achieve both simultaneously and therefore appeared as a potential solution. We showed that while the overheating is correlated to entropy production, entropy conservation does not necessarily prevent it. Second, we studied the behavior of ES schemes in the low Mach number regime, where shock-capturing schemes are known to suffer from severe accuracy degradation issues. A classic remedy to this problem is the flux-preconditioning technique, which consists in modifying artificial dissipation terms to enforce consistent low Mach behavior. We showed that ES schemes suffer from the same issues and that the flux-preconditioning technique can improve their behavior without interfering with entropy-stability. Furthermore, we demonstrated analytically that these issues stem from an acoustic entropy production field which scales improperly with the Mach number, generating spatial fluctuations that are inconsistent with the equations. An important outgrowth of this effort is the discovery that skew-symmetric dissipation operators can alter the way entropy is produced or conserved locally.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155304/1/gouasmia_1.pd

Discrete-continuum hybrid modelling of flowing and static regimes

Bulk handling, transport and processing of granular materials and powders are fundamental operations in a wide range of industrial processes and geophysical phenomena. Particulate materials, which can be found in nature, are usually characterized by grain size which can range across several scales: from nanometre to the order of metre. Depending on the volume fraction and shear strain conditions, granular materials can have different behaviours and often can be expressed as a new state of matter with properties of solids, liquids and gases. For the above reasons both the experimental and the numerical analysis of granular media is still a difficult task and the prediction of their dynamic behaviour still represents nowadays an important challenge. The main goal of the current thesis is the development of a numerical strategy with the objective of studying the macroscopic behaviour of dry granular flows in quasi-static and dense flow regime. The problem is defined in a continuum mechanics framework and the balance laws, which govern the behaviour of a solid body, are solved by using a Lagrangian formalism. The Material Point Method (MPM), a particle-based method, is chosen due to its features which make it very suitable for the solution of large deformation problems involving complex history-dependent constitutive laws. An irreducible formulation using a Mohr-Coulomb constitutive law, which takes into account geometric non-linearities, is implemented within the MPM framework. The numerical strategy is verified and validated against several benchmark tests and experimental results, available in the literature. Further, a mixed formulation is implemented for the solution of granular flows that undergo undrained conditions. Finally, the developed MPM strategy is used and tested against the experimental study performed for the characterization of the flowability of several types of sucrose. The capabilities and limitations of this numerical strategy are observed and discussed and the bases for future research are outlined.El manejo, el transporte y el procesamiento de materiales granulares y polvo son operaciones fundamentales en una amplia gama de procesos industriales y de fenómenos geofísicos. Los materiales particulados, que pueden ser encontrados en la naturaleza, generalmente están caracterizados por el tamaño del grano, que puede variar entre varios órdenes de magnitud: desde el nanómetro hasta el orden de los metros. En función de las condiciones de fracción volumétrica y de deformación de cortante, los materiales granulares pueden tener un comportamiento diferente y a menudo pueden expresarse como un nuevo estado de materia con propiedades de sólidos, de líquidos y de gases. A causa de las observaciones antes mencionadas, tanto el análisis experimental como la simulación numérica de medios granulares es aún una tarea compleja y la predicción de su comportamiento dinámico representa aun hoy día un desafío muy importante. El principal objetivo de esta tesis es el desarrollo de una estrategia numérica con la finalidad de estudiar el comportamiento macroscópico de los flujos de medios granulares secos en régimen cuasiestático y en régimen dinámico. El problema está definido en el contexto de la mecánica de medios continuos y las leyes de equilibrio, que gobiernan el comportamiento del cuerpo sólido, y están resueltas mediante un formalismo Lagrangiano. El Metodo de los Puntos Materiales (MPM), método basado en el concepto de discretización del cuerpo sólido en partículas, está elegido por sus características que lo convierten en una técnica apropiada para resolver problemas de grandes deformaciones donde se tienen que utilizar complejas leyes constitutivas. En el marco del MPM está implementada una formulación irreducible que usa una ley constitutiva de Mohr-Coulomb y que tiene en cuenta no-linealidades geométricas. La estrategia numérica está verificada y validada con respecto a tests de referencia y resultados experimentales disponibles en la literatura. Además, se ha implementado una formulación mixta para resolver casos de flujo granular en condiciones no drenadas. Por último, la estrategia MPM desarrollada está utilizada y evaluada con respecto a un estudio experimental realizado para la caracterización de la fluidez de diferentes tipologías de azúcar. También se presentan unas observaciones y discusión sobre las capacidades y las limitaciones de esta herramienta numérica y se describen las bases de una investigación futura.Postprint (published version

Discrete-continuum hybrid modelling of flowing and static regimes

Bulk handling, transport and processing of granular materials and powders are fundamental operations in a wide range of industrial processes and geophysical phenomena. Particulate materials, which can be found in nature, are usually characterized by grain size which can range across several scales: from nanometre to the order of metre. Depending on the volume fraction and shear strain conditions, granular materials can have different behaviours and often can be expressed as a new state of matter with properties of solids, liquids and gases. For the above reasons both the experimental and the numerical analysis of granular media is still a difficult task and the prediction of their dynamic behaviour still represents nowadays an important challenge. The main goal of the current thesis is the development of a numerical strategy with the objective of studying the macroscopic behaviour of dry granular flows in quasi-static and dense flow regime. The problem is defined in a continuum mechanics framework and the balance laws, which govern the behaviour of a solid body, are solved by using a Lagrangian formalism. The Material Point Method (MPM), a particle-based method, is chosen due to its features which make it very suitable for the solution of large deformation problems involving complex history-dependent constitutive laws. An irreducible formulation using a Mohr-Coulomb constitutive law, which takes into account geometric non-linearities, is implemented within the MPM framework. The numerical strategy is verified and validated against several benchmark tests and experimental results, available in the literature. Further, a mixed formulation is implemented for the solution of granular flows that undergo undrained conditions. Finally, the developed MPM strategy is used and tested against the experimental study performed for the characterization of the flowability of several types of sucrose. The capabilities and limitations of this numerical strategy are observed and discussed and the bases for future research are outlined.El manejo, el transporte y el procesamiento de materiales granulares y polvo son operaciones fundamentales en una amplia gama de procesos industriales y de fenómenos geofísicos. Los materiales particulados, que pueden ser encontrados en la naturaleza, generalmente están caracterizados por el tamaño del grano, que puede variar entre varios órdenes de magnitud: desde el nanómetro hasta el orden de los metros. En función de las condiciones de fracción volumétrica y de deformación de cortante, los materiales granulares pueden tener un comportamiento diferente y a menudo pueden expresarse como un nuevo estado de materia con propiedades de sólidos, de líquidos y de gases. A causa de las observaciones antes mencionadas, tanto el análisis experimental como la simulación numérica de medios granulares es aún una tarea compleja y la predicción de su comportamiento dinámico representa aun hoy día un desafío muy importante. El principal objetivo de esta tesis es el desarrollo de una estrategia numérica con la finalidad de estudiar el comportamiento macroscópico de los flujos de medios granulares secos en régimen cuasiestático y en régimen dinámico. El problema está definido en el contexto de la mecánica de medios continuos y las leyes de equilibrio, que gobiernan el comportamiento del cuerpo sólido, y están resueltas mediante un formalismo Lagrangiano. El Metodo de los Puntos Materiales (MPM), método basado en el concepto de discretización del cuerpo sólido en partículas, está elegido por sus características que lo convierten en una técnica apropiada para resolver problemas de grandes deformaciones donde se tienen que utilizar complejas leyes constitutivas. En el marco del MPM está implementada una formulación irreducible que usa una ley constitutiva de Mohr-Coulomb y que tiene en cuenta no-linealidades geométricas. La estrategia numérica está verificada y validada con respecto a tests de referencia y resultados experimentales disponibles en la literatura. Además, se ha implementado una formulación mixta para resolver casos de flujo granular en condiciones no drenadas. Por último, la estrategia MPM desarrollada está utilizada y evaluada con respecto a un estudio experimental realizado para la caracterización de la fluidez de diferentes tipologías de azúcar. También se presentan unas observaciones y discusión sobre las capacidades y las limitaciones de esta herramienta numérica y se describen las bases de una investigación futura

Combining Discrete Equations Method and Upwind Downwind-Controlled Splitting for Non-Reacting and Reacting Two-Fluid Computations

Lors que nous examinons numériquement des phénomènes multiphasiques suite à un accidentgrave dans le réacteur nucléaire, la dimension caractéristique des zones multi-fluides(non-réactifs et réactifs) s avère beaucoup plus petite que celle du bâtiment réacteur, cequi fait la Simulation Numérique Directe de la configuration à peine réalisable. Autrement,nous proposons de considérer la zone de mélange multiphasique comme une interface infinimentfine. Puis, le solveur de Riemann réactif est inséré dans la Méthode des ÉquationsDiscrètes Réactives (RDEM) pour calculer le front de combustion à grande vitesse représentépar une interface discontinue. Une approche anti-diffusive est ensuite couplée avec laRDEM afin de précisément simuler des interfaces réactives. La robustesse et l efficacité decette approche en calculant tant des interfaces multiphasiques que des écoulements réactifssont à la fois améliorées grâce à la méthode ici proposée : upwind downwind-controlled splitting(UDCS). UDCS est capable de résoudre précisément des interfaces avec les maillagesnon-structurés multidimensionnels, y compris des fronts réactifs de détonation et de déflagration.When numerically investigating multiphase phenomena during severe accidents in a reactorsystem, characteristic lengths of the multi-fluid zone (non-reactive and reactive) are foundto be much smaller than the volume of the reactor containment, which makes the directmodeling of the configuration hardly achievable. Alternatively, we propose to consider thephysical multiphase mixture zone as an infinitely thin interface. Then, the reactive Riemannsolver is inserted into the Reactive Discrete Equations Method (RDEM) to compute highspeed combustion waves represented by discontinuous interfaces. An anti-diffusive approachis also coupled with RDEM to accurately simulate reactive interfaces. Increased robustnessand efficiency when computing both multiphase interfaces and reacting flows are achievedthanks to an original upwind downwind-controlled splitting method (UDCS). UDCS is capableof accurately solving interfaces on multi-dimensional unstructured meshes, includingreacting fronts for both deflagration and detonation configurations.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Maximum-principle-satisfying space-time conservation element and solution element scheme applied to compressible multifluids

2016-2017 > Academic research: refereed > Publication in refereed journal201804_a bcmaAccepted ManuscriptOthersState Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology; Army Research Office (ARO)Publishe

Numerical methods for all-speed flows in fluid-dynamics and non-linear elasticity

In this thesis we are concerned with the numerical simulation of compressible materials flows, including gases, liquids and elastic solids. These materials are described by a monolithic Eulerian model of conservation laws, closed by an hyperelastic state law that includes the different behaviours of the considered materials. A novel implicit relaxation scheme to solve compressible flows at all speeds is proposed, with Mach numbers ranging from very small to the order of unity. The scheme is general and has the same formulation for all the considered materials, since a direct dependence on the state law is avoided via the relaxation. It is based on a fully implicit time discretization, easily implemented thanks to the linearity of the transport operator in the relaxation system. The spatial discretization is obtained by a combination of upwind and centered schemes in order to recover the correct numerical viscosity in different Mach regimes. The scheme is validated with one and two dimensional simulations of fluid flows and of deformations of compressible solids. We exploit the domain discretization through Cartesian grids, allowing for massively parallel computations (HPC) that drastically reduce the computational times on 2D test cases. Moreover, the scheme is adapted to the resolution on adaptive grids based on quadtrees, implementing adaptive mesh refinement techinques. The last part of the thesis is devoted to the numerical simulation of heterogeneous multi-material flows. A novel sharp interface method is proposed, with the derivation of implicit equilibrium conditions. The aim of the implicit framework is the solution of weakly compressible and low Mach flows, thus the proposed multi-material conditions are coupled with the implicit relaxation scheme that is solved in the bulk of the flow. Dans cette thèse on s’intéresse à la simulation numérique d’écoulements des matériaux compressibles, voir fluides et solides élastiques. Les matériaux considérés sont décrits avec un modèle monolithique eulérian, fermé avec une loi d’état hyperélastique qui considère les différents comportéments des matériaux. On propose un nouveau schéma de relaxation qui résout les écoulements compressibles dans des différents régimes, avec des nombres de Mach très petits jusqu’à l’ordre 1. Le schéma a une formulation générale qui est la même pour tous le matériaux considérés, parce que il ne dépend pas directement de la loi d’état. Il se base sur une discrétization complétement implicite, facile à implémenter grâce à la linearité de l’opérateur de transport du système de relaxation. La discrétization en éspace est donnée par la combinaison de flux upwind et centrés, pour retrouver la correcte viscosité numérique dans les différents régimes. L’utilisation de mailles cartésiennes pour les cas 2D s’adapte bien à une parallélisation massive, qui permet de réduire drastiquement le temps de calcul. De plus, le schéma a été adapté pour la résolution sur des mailles quadtree, pour implémenter l’adaptivité de la maille avec des critères entropiques. La dernière partie de la thèse concerne la simulation numérique d’écoulements multi-matériaux. On a proposé une nouvelle méthode d’interface “sharp”, en dérivant les conditions d’équilibre en implicite. L’objectif est la résolution d’interfaces physiques dans des régimes faiblement compressibles et avec un nombre de Mach faible, donc les conditions multi-matériaux sont couplées au schéma implicite de relaxation