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

A note on reducing spurious pressure oscillations in fully conservative discontinuous Galerkin simulations of multicomponent flows

A well-known issue associated with the use of fully conservative schemes in multicomponent-flow simulations is the generation of spurious pressure oscillations at contact interfaces. These oscillations can rapidly lead to solver divergence even in the presence of smooth interfaces that are not fully resolved. In this note, we compare various strategies for reducing such oscillations that do not (a) introduce conservation error, (b) rely on artificial viscosity or limiting, or (c) degrade order of accuracy in smooth regions of the flow. The considered test case is one-dimensional advection of a high-pressure nitrogen/n-dodecane thermal bubble using the thermally perfect gas model. Several results are presented that contradict those corresponding to the more conventional hydrogen/oxygen thermal-bubble case

High-order methods for diffuse-interface models in compressible multi-medium flows: a review

The diffuse interface models, part of the family of the front capturing methods, provide an efficient and robust framework for the simulation of multi-species flows. They allow the integration of additional physical phenomena of increasing complexity while ensuring discrete conservation of mass, momentum, and energy. The main drawback brought by the adoption of these models consists of the interface smearing, increasing with the simulation time, therefore, requiring a counteraction through the introduction of sharpening terms and a careful selection of the discretization level. In recent years, the diffuse interface models have been solved using several numerical frameworks including finite volume, discontinuous Galerkin, and hybrid lattice Boltzmann method, in conjunction with shock and contact wave capturing schemes. The present review aims to present the recent advancements of high-order accuracy schemes with the capability of solving discontinuities without the introduction of numerical instabilities and to put them in perspective for the solution of multi-species flows with the diffuse interface method.Engineering and Physical Sciences Research Council (EPSRC): 2497012. Innovate UK: 263261. Airbus U

Numerical simulation of stochastic two-phase flows with a DEM Method coupled to uncertainty quantification scheme

A new scheme for the numerical approximation of a five-equations model taking into account uncertainty quantification (UQ) is presented. In particular, the Discrete Equation Method (DEM) for the discretization of the five-equations model is modified for including a formulation based on the adaptive Semi-intrusive (aSI) scheme, thus yielding a new intrusive scheme (aSDEM) for simulating stochastic two-phase flows. Some reference test-cases are performed in order to demonstrate the convergence properties and the efficiency of the overall scheme. The propagation of initial uncertainties is evaluated in terms of mean and variance of several thermodynamic properties of the two phases

Solving the system of radiation magnetohydrodynamics for solar physical simulations in 3d

In this study we present a finite-volumen scheme for solving the equations of radiation magnetohydrodynamics in two and three space dimensions. Among other applications this system is used to model the plasma in the solar convection zone and in the solar photosphere. It is a non--linear system of balance laws derived from the Euler equations of gas dynamics and the Maxwell equations; the energy transport through radiation is also included in the model. The starting point of our presentation is a standard explicit first and second order finite-volume scheme on both structured and unstructured grids. We first study the convergence of a finite-volume scheme applied to a scalar model problem for the full system of radiation magnetohydrodynamics. We then present modifications of the base scheme. These make it possible to approximate the system of magnetohydrodynamics with an arbitrary equation of state; they reduce errors due to a violation of the divergence constraint on the magnetic field, and they lead to an improved accuracy in the approximation of solution near an equilibrium state. These modifications significantly increase the robustness of the scheme and are essential for an accurate simulation of processes in the solar atmosphere ...thesi

A compressible Lagrangian framework for the simulation of underwater implosion problems

The development of efficient algorithms to understand implosion dynamics presents a number of challenges. The foremost challenge is to efficiently represent the coupled compressible fluid dynamics of internal air and surrounding water. Secondly, the method must allow one to accurately detect or follow the interface between the phases. Finally, it must be capable of resolving any shock waves which may be created in air or water during the final stage of the collapse. We present a fully Lagrangian compressible numerical framework for the simulation of underwater implosion. Both air and water are considered compressible and the equations for the Lagrangian shock hydrodynamics are stabilized via a variationally consistent multiscale method [109]. A nodally perfect matched definition of the interface is used [57, 25] and then the kinetic variables, pressure and density, are duplicated at the interface level. An adaptive mesh generation procedure, which respects the interface connectivities, is applied to provide enough refinement at the interface level. This framework is then used to simulate the underwater implosion of a large cylindrical bubble, with a size in the order of cm. Rapid collapse and growth of the bubble occurred on very small spatial (0.3mm), and time (0.1ms) scales followed by Rayleigh-Taylor instabilities at the interface, in addition to the shock waves traveling in the fluid domains are among the phenomena that are observed in the simulation. We then extend our framework to model the underwater implosion of a cylindrical aluminum container considering a monolithic fluid-structure interaction (FSI). The aluminum cylinder, which separates the internal atmospheric-pressure air from the external high-pressure water, is modeled by a three node rotation-free shell element. The cylinder undergoes fast transient deformations, large enough to produce self-contact along it. A novel elastic frictionless contact model is used to detect contact and compute the non-penetrating forces in the discretized domain between the mid-planes of the shell. Two schemes are tested, implicit using the predictor/multi-corrector Bossak scheme, and explicit, using the forward Euler scheme. The results of the two simulations are compared with experimental data.El desarrollo de métodos eficientes para modelar la dinámica de implosión presenta varios desafíos. El primero es una representación eficaz de la dinámica del sistema acoplado de aire-agua. El segundo es que el método tiene que permitir una detección exacta o un seguimiento adecuado de la interfase entre ambas fases. Por último el método tiene que ser capaz de resolver cualquier choque que podría generar en el aire o en el agua, sobre todo en la última fase del colapso. Nosotros presentamos un método numérico compresible y totalmente Lagrangiano para simular la implosión bajo el agua. Tanto el aire como el agua se consideran compresibles y las ecuaciones Lagrangianos para la hidrodinámica del choque se estabilizan mediante un método multiescala que es variacionalmente consistente [109]. Se utiliza una definición de interfase que coincide perfectamente con los nodos [57, 25]. Ésta, nos facilita duplicar eficazmente las variables cinéticas como la presión y la densidad en los nodos de la interfase. Con el fin de obtener suficiente resolución alrededor de la interfase, la malla se genera de forma adaptativa y respetando la posición de la interfase. A continuación el método desarrollado se utiliza para simular la implosión bajo el agua de una burbuja cilíndrica del tamaño de un centímetro. Varios fenómenos se han capturado durante el colapso: un ciclo inmediato de colapso-crecimiento de la burbuja que ocurre en un espacio (0.3mm) y tiempo (0.1ms) bastante limitado, aparición de inestabilidades de tipo Rayleigh-Taylor en la interfase y formaron de varias ondas de choque que viajan tanto en el agua como en el aire. Después, seguimos el desarrollo del método para modelar la implosión bajo el agua de un contenedor metálico considerando una interacción monolítica de fluido y estructura. El cilindro de aluminio, que a su vez contiene aire a presión atmosférica y está rodeada de agua en alta presión, se modelando con elementos de lámina de tres nodos y sin grados de libertad de rotación. El cilindro se somete a deformaciones transitorias suficientemente rápidos y enormes hasta llegar a colapsar. Un nuevo modelo elástico de contacto sin considerar la fricción se ha desarrollado para detectar el contacto y calcular las fuerzas en el dominio discretizado entre las superficies medianas de las laminas. Dos esquemas temporales están considerados, uno es implícito utilizando el método de Bossak y otro es explícito utilizando Forward Euler. Al final los resultados de ambos casos se comparan con los resultados experimentales

A compressible Lagrangian framework for the simulation of underwater implosion problems

The development of efficient algorithms to understand implosion dynamics presents a number of challenges. The foremost challenge is to efficiently represent the coupled compressible fluid dynamics of internal air and surrounding water. Secondly, the method must allow one to accurately detect or follow the interface between the phases. Finally, it must be capable of resolving any shock waves which may be created in air or water during the final stage of the collapse. We present a fully Lagrangian compressible numerical framework for the simulation of underwater implosion. Both air and water are considered compressible and the equations for the Lagrangian shock hydrodynamics are stabilized via a variationally consistent multiscale method. A nodally perfect matched definition of the interface is used and then the kinetic variables, pressure and density, are duplicated at the interface level. An adaptive mesh generation procedure, which respects the interface connectivities, is applied to provide enough refinement at the interface level. This framework is then used to simulate the underwater implosion of a large cylindrical bubble, with a size in the order of cm. Rapid collapse and growth of the bubble occurred on very small spatial (0.3mm), and time (0.1ms) scales followed by Rayleigh-Taylor instabilities at the interface, in addition to the shock waves traveling in the fluid domains are among the phenomena that are observed in the simulation. We then extend our framework to model the underwater implosion of a cylindrical aluminum container considering a monolithic fluid-structure interaction (FSI). The aluminum cylinder, which separates the internal atmospheric-pressure air from the external high-pressure water, is modeled by a three node rotation-free shell element. The cylinder undergoes fast transient deformations, large enough to produce self-contact along it. A novel elastic frictionless contact model is used to detect contact and compute the non-penetrating forces in the discretized domain between the mid-planes of the shell. Two schemes are tested, implicit using the predictor/multi-corrector Bossak scheme, and explicit, using the forward Euler scheme. The results of the two simulations are compared with experimental data

Extension de la méthode des Différences Spectrales à la combustion

L'amélioration des outils d'ingénierie utilisés dans le design des dispositifs industriels de combustion est indispensable afin de respecter les demandes de plus en plus restrictives pour réduire les émissions de gaz à effets de serre. Parmi eux, la mécanique des fluides numériques (CFD) est devenue essentielle pour étudier et optimiser les chambres de combustion au cours des dernières décennies. Elle se complète parfaitement aux expériences réelles qui peuvent être très couteuses et avec lesquelles il est impossible d'obtenir des informations sur n'importe quelle quantité d'intérêt en tout point de la chambre de combustion. En utilisant les simulations aux grandes échelles (LES), la CFD décrit directement l'interaction entre les flammes et les structures turbulentes avec une faible modélisation. La qualité des résultats LES est ainsi très dépendante de la discrétisation utilisée incluant à la fois le maillage et également les propriétés de dissipation et de dispersion des méthodes numériques utilisées. Cependant, la plupart des codes LES employés de nos jours dans l'industrie utilisent des schémas de discrétisation spatiale de basordre (LO) à cause de leur faible coût de calcul et leur facilité d'implémentation sur des maillages complexes. Pourtant, les méthodes numériques d'ordres élevés (HO) pour la LES sont développées depuis deux décennies et ont été appliquées sur des écoulements non-réactifs amenant à des résultats plus précis que les méthodes LO avec un plus faible coût de calcul. Bien que les méthodes HO semblent très prometteuses en combustion, en particulier pour mieux décrire le front de flamme, leur utilisation pour des écoulements réactifs restent encore à être démontrée. Au cours de ces travaux, les avantages et les bénéfices des méthodes HO en combustion sont évalués en utilisant la méthode des Différences Spectrales (SD) avec du raffinement $hp$. Premièrement, il est démontré que la formulation originelle des SD est instable pour des écoulements multi-espèces avec des propriétés thermodynamiques variant avec la température et la composition. Il a été constaté que calculer les variables primitives aux points solutions puis de les extrapoler aux points flux, au lieu de faire l'inverse en extrapolant d'abord les variables conservatives, rend stable la méthode SD dans ce cas-ci. De plus, une nouvelle méthodologie, également plus stable pour calculer les flux diffusifs aux interfaces des cellules est détaillée. Enfin, les conditions aux limites caractéristiques et de murs ont été étendues aux écoulements multiespèces dans le formalisme SD. Avec ces développements, des flammes laminaires pré-mélangées 1D et 2D ont été simulées avec des mécanismes réduits à 2 réactions ou des mécanismes réduits analytiquement. Les résultats sont très proches de ceux obtenus avec des solveurs de référence bien établis en combustion. Il est montré que pour un même niveau d'erreur, il est plus efficace d'utiliser des maillages grossiers avec des grandes valeurs de $p$ et non l'inverse. Par conséquent, le raffinement local en $p$, qui applique des grandes valeurs de $p$ dans les régions d'intérêts seulement, permet de garder une bonne précision à un coût de calcul plus faible. Ceci est particulièrement intéressant pour des simulations de combustion où le front de flamme est très localisé et requiert une plus grande précision que le reste de l'écoulement. Il est également observé sur ces cas simples 1D et 2D que la méthode SD est moins sensible à la discrétisation du front de flamme que les solveurs volumes finis comme AVBP. Pour terminer, deux différentes configurations de flammes 3D turbulentes ont été simulées avec l'algorithme des SD étendu aux écoulements réactifs