Transient electrohydrodynamic flow with concentration dependent fluid properties: modelling and energy-stable numerical schemes

Transport of electrolytic solutions under influence of electric fields occurs in phenomena ranging from biology to geophysics. Here, we present a continuum model for single-phase electrohydrodynamic flow, which can be derived from fundamental thermodynamic principles. This results in a generalized Navier-Stokes-Poisson-Nernst-Planck system, where fluid properties such as density and permittivity depend on the ion concentration fields. We propose strategies for constructing numerical schemes for this set of equations, where solving the electrochemical and the hydrodynamic subproblems are decoupled at each time step. We provide time discretizations of the model that suffice to satisfy the same energy dissipation law as the continuous model. In particular, we propose both linear and non-linear discretizations of the electrochemical subproblem, along with a projection scheme for the fluid flow. The efficiency of the approach is demonstrated by numerical simulations using several of the proposed schemes

Parallel Multiphase Navier-Stokes Solver

We study and implement methods to solve the variable density Navier-Stokes equations. More specifically, we study the transport equation with the level set method and the momentum equation using two methods: the projection method and the artificial compressibility method. This is done with the aim of numerically simulating multiphase fluid flow in gravity oil-water-gas separator vessels. The result of the implementation is the parallel Aspen software framework based on the massively parallel deal.II . For the transport equation, we briefly discuss the theory behind it and several techniques to stabilize it, especially the graph laplacian artificial viscosity with higher order elements. Also, we introduce the level set method to model the multiphase flow and study ways to maintain a sharp surface in between phases. For the momentum equation, we give an overview of the two methods and discuss a new projection method with variable time stepping that is second order in time. Then we discuss the new third order in time artificial compressiblity method and present variable density version of it. We also provide a stability proof for the discrete implicit variable density artificial compressibility method. For all the methods we introduce, we conduct numerical experiments for verification, convergence rates, as well as realistic models

Discontinuous Galerkin Method Applied to Navier-Stokes Equations

Discontinuous Galerkin (DG) finite element methods are becoming important techniques for the computational solution of many real-world problems describe by differential equations. They combine many attractive features of the finite element and the finite volume methods. These methods have been successfully applied to many important PDEs arising from a wide range of applications. DG methods are highly accurate numerical methods and have considerable advantages over the classical numerical methods available in the literature. DG methods can easily handle meshes with hanging nodes, elements of various types and shapes, and local spaces of different orders. Furthermore, DG methods provide accurate and efficient simulation of physical and engineering problems, especially in settings where the solutions exhibit poor regularity. For these reasons, they have attracted the attention of many researchers working in diverse areas, from computational fluid dynamics, solid mechanics and optimal control, to finance, biology and geology. In this talk, we give an overview of the main features of DG methods and their extensions. We first introduce the DG method for solving classical differential equations. Then, we extend the methods to other equations such as Navier-Stokes equations. The Navier-Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing

Approximation Techniques for Incompressible Flows with Heterogeneous Properties

We study approximation techniques for incompressible flows with heterogeneous properties. Speci cally, we study two types of phenomena. The first is the flow of a viscous incompressible fluid through a rigid porous medium, where the permeability of the medium depends on the pressure. The second is the ow of a viscous incompressible fluid with variable density. The heterogeneity is the permeability and the density, respectively. For the first problem, we propose a finite element discretization and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence is exponential, we propose a splitting scheme which involves solving only two linear systems. For the second problem, we introduce a fractional time-stepping scheme which, as opposed to other existing techniques, requires only the solution of a Poisson equation for the determination of the pressure. This simpli cation greatly reduces the computational cost. We prove the stability of first and second order schemes, and provide error estimates for first order schemes. For all the introduced discretization schemes we present numerical experiments, which illustrate their performance on model problems, as well as on realistic ones

Ensemble time-stepping algorithms for natural convection

Predictability of fluid flow via natural convection is a fundamental issue with implications for, e.g., weather predictions including global climate change assessment and nuclear reactor cooling. In this work, we study numerical methods for natural convection and utilize them to study predictability. Eight new algorithms are devised which are far more efficient than existing ones for ensemble calculations. They allow for either increased ensemble sizes or denser meshes on current computing systems. The artificial compressibility ensemble (ACE) family produce accurate velocity and temperature approximations and are fastest. The speed of second-order ACE degrades as $\epsilon \rightarrow 0$ or $\Delta t \rightarrow 0$ due to the iterative solver. However, first-order ACE has a uniform solve time since $\gamma = \mathcal{O}(1)$. The ensemble backward differentiation formula (eBDF) family are most accurate and reliable. The penalty ensemble algorithm (PEA) family are strongly affected by the timestep and are least accurate. In particular, $\gamma = \mathcal{O}(1/ \Delta t^2)$ for second-order PEA leads to solver breakdown. We also propose an ACE turbulence (ACE-T) family of methods for turbulence modeling which are both fast and accurate. A complete numerical analysis is performed which establishes full-reliability. The analysis involves techniques that are novel and results that subsume, elucidate, and expand previous results in closely related fields, e.g., iso-thermal fluid flow. Numerical tests show predicted accuracy is consistent with theory. Predictability is a highly complex and problem-dependent phenomenon. Predictability studies are performed utilizing the new second-order ACE algorithm. We perform a numerical test where the flow reaches a steady state. It is found that increasing the size of the domain increases predictability. Also, spatial averages increase predictability with increasing filter radius. We also study a problem with a manufactured solution. Sufficiently large rotations increase the predictability of a flow. Further, spatial averages decrease predictability with increasing filter radius

Écoulement bi-fluide avec interface diffuse : présentation d'une nouvelle méthode de projection pour le modèle Navier-Stokes/Allen-Cahn

Cette thèse porte sur la simulation numérique des écoulements bi-fluides par l'approche à interface diffuse. La description mathématique d'un écoulement bi-fluide par l'approche à interface diffuse consiste en une équation de Navier-Stokes modifiée couplée à un modèle de capture de l'interface mobile entre les deux fluides. Dans cette thèse, pour la capture de l'interface mobile nous avons porté notre attention sur le modèle de Allen-Cahn. En premier lieu nous nous sommes intéressés de prime abord à la résolution numérique du système semi-discrétisé en temps et totalement implicite Navier-Stokes/Allen-Cahn (NSAC). Pour ce faire nous avons développé un algorithme itératif basé sur la méthode du point fixe. Nous avons montré qu'à chaque itération, le système d'équations (contenu dans l'algorithme) est bien défini ; de plus, la solution de l'équation de Allen-Cahn satisfait le principe du maximum. Par la suite nous avons montré que l'algorithme de point fixe converge et que sa limite est la solution du système NSAC semi-discret. Si cette première méthode itérative nous a donné une méthode de résolution, elle n'est pas satisfaisante quant à la performance. En second lieu nous proposons un nouveau schéma de discrétisation en temps à pas fractionnaire inconditionnellement stable. Utilisant une approche de type point fixe (une projection couplée) nous avons montré la convergence à chaque pas de temps et que la limite correspond à la solution du système semi-discret (donnant ainsi l'existence et l'unicité de la solution). Nous concluons enfin avec des applications numériques aux fins d'illustrer la pertinence et les potentielles limitations du modèle d'une part, puis les performances de notre méthode de résolution du système Navier-Stokes/Allen-Cahn d'autre part

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on April 2-7, 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The vibrancy and diversity in this field are amply expressed in these important papers, and the collection clearly shows the continuing rapid growth of the use of multigrid acceleration techniques

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal