Numerical simulation of multiphase immiscible flow on unstructured meshes

The present thesis aims at developing a basis for the numerical simulation of multiphase flows of immiscible fluids. This approach, although limited by the computational power of the present computers, is potentially very important, since most of the physical phenomena of these flows often happen on space and time scales where experimental techniques are impossible to be utilized in practice. In particular, this research is focused on developing numerical discretizations suitable for three-dimensional (3-D) unstructured meshes. In detail, the first chapter delimits the considered multiphase flows to the case in which the components are immiscible fluids. In particular, the focus is placed on those cases where two or more continuous streams of different fluids are separated by interfaces, and hence, correspondingly named separated flows. Additionally, once the type of flow is determined, the chapter introduces the physical characteristics and the models available to predict its behavior, as well as the mathematical formulation that sustains the numerical techniques developed within this thesis. The second chapter introduces and analyzes a new geometrical Volume-of-Fluid (VOF) method for capturing interfaces on 3-D Cartesian and unstructured meshes. The method reconstructs interfaces as first- and second-order piecewise planar approximations (PLIC), and advects volumes in a single unsplit Lagrangian-Eulerian (LE) geometrical algorithm based on constructing flux polyhedrons by tracing back the Lagrangian trajectories of the cell-vertex velocities. In this way, the situations of overlapping between flux polyhedrons are minimized. Complementing the previous chapter, the third one proposes a parallelization strategy for the VOF method. The main obstacle is that the computing costs are concentrated in the interface between fluids. Consequently, if the interface is not homogeneously distributed, standard domain decomposition (DD) strategies lead to imbalanced workload distributions. Hence, the new strategy is based on a load balancing process complementary to the underlying domain decomposition. Its parallel efficiency has been analyzed using up to 1024 CPU-cores, and the results obtained show a gain with respect to the standard DD strategy up to 12x, depending on the size of the interface and the initial distribution. The fourth chapter describes the discretization of the single-phase Navier-Stokes equations to later extend it to the case of multiphase immiscible flow. One of the most important characteristics of the discretization schemes, aside from accuracy, is their capacity to discretely conserve kinetic energy, specially when solving turbulent flow. Hence, this chapter analyzes the accuracy and conservation properties of two particular collocated and staggered mesh schemes. The extension of the numerical schemes suitable for the single-phase Navier-Stokes equations to the case of multiphase immiscible flow is developed in the fifth chapter. Particularly, while the numerical techniques for the simulation of turbulent flow have evolved to discretely preserve mass, momentum and, specially, kinetic energy, the mesh schemes for the discretization of multiphase immiscible flow have evolved to improve their stability and robustness. Therefore, this chapter presents and analyzes two particular collocated and staggered mesh discretizations, able to simulate multiphase immiscible flow, which favor the discrete conservation of mass, momentum and kinetic energy. Finally, the sixth chapter numerically simulates the Richtmyer-Meshkov (RM) instability of two incompressible immiscible liquids. This chapter is a general assessment of the numerical methods developed along this thesis. In particular, the instability has been simulated by means of a VOF method and a staggered mesh scheme. The corresponding numerical results have shown the capacity of the discrete system to obtain accurate results for the RM instability.Aquesta tesi té com a objectiu desenvolupar una base per a la simulació numèrica de fluids multi-fase immiscibles. Aquesta estratègia, encara que limitada per la potència computacional dels computadors actuals, és potencialment molt important, ja que la majoria de la fenomenologia d'aquests fluids sovint passa en escales temporals i especials on les tècniques experimentals no poden ser utilitzades. En particular, aquest treball es centra en desenvolupar discretitzacions numèriques aptes per a malles no-estructurades en tres dimensions (3-D). En detall, el primer capítol delimita els casos multifásics considerats al cas en que els components són fluids immiscibles. En particular, la tesi es centra en aquells casos en que dos o més fluids diferents són separats per interfases, i per tant, corresponentment anomenats fluxos separats. A més a més, un cop el tipus de flux es determinat, el capítol introdueix les característiques físiques i els models disponibles per predir el seu comportament, així com també la formulació matemàtica i les tècniques numèriques desenvolupades en aquesta tesi. El segon capítol introdueix i analitza un nou mètode "Volume-of-Fluid" (VOF) apte per a capturar interfases en malles Cartesianes i no-estructurades 3-D. El mètode reconstrueix les interfases com aproximacions "piecewise planar approximations" (PLIC) de primer i segon ordre, i advecciona els volums amb un algoritme geomètric "unsplit Lagrangian-Eulerian" (LE) basat en construïr els poliedres a partir de les velocitats dels vèrtexs de les celdes. D'aquesta manera, les situacions de sobre-solapament entre poliedres són minimitzades. Complementant el capítol anterior, el tercer proposa una estratègia de paral·lelització pel mètode VOF. L'obstacle principal és que els costos computacionals estan concentrats en les celdes de l'interfase entre fluids. En conseqüència, si la interfase no està ben distribuïda, les estratègies de "domain decomposition" (DD) resulten en distribucions de càrrega desequilibrades. Per tant, la nova estratègia està basada en un procés de balanceig de càrrega complementària a la DD. La seva eficiència en paral·lel ha sigut analitzada utilitzant fins a 1024 CPU-cores, i els resultats obtinguts mostren uns guanys respecte l'estratègia DD de fins a 12x, depenent del tamany de la interfase i de la distribució inicial. El quart capítol descriu la discretització de les equacions de Navier-Stokes per a una sola fase, per després estendre-ho al cas multi-fase. Una de les característiques més importants dels esquemes de discretització, a part de la precisió, és la seva capacitat per conservar discretament l'energia cinètica, específicament en el cas de fluxos turbulents. Per tant, aquest capítol analitza la precisió i propietats de conservació de dos esquemes de malla diferents: "collocated" i "staggered". L'extensió dels esquemes de malla aptes per els casos de una sola fase als casos multi-fase es desenvolupa en el cinquè capítol. En particular, així com en el cas de la simulació de la turbulència les tècniques numèriques han evolucionat per a preservar discretament massa, moment i energia cinètica, els esquemes de malla per a la discretització de fluxos multi-fase han evolucionat per millorar la seva estabilitat i robustesa. Per lo tant, aquest capítol presenta i analitza dos discretitzacions de malla "collocated" i "staggered" particulars, aptes per simular fluxos multi-fase, que afavoreixen la conservació discreta de massa, moment i energia cinètica. Finalment, el capítol sis simula numèricament la inestabilitat de Richtmyer-Meshkov (RM) de dos fluids immiscibles i incompressibles. Aquest capítol es una prova general dels mètodes numèrics desenvolupats al llarg de la tesi. En particular, la inestabilitat ha sigut simulada mitjançant un mètode VOF i un esquema de malla "staggered". Els resultats numèrics corresponents han demostrat la capacitat del sistema discret en obtenir bons resultats per la inestabilitat RM

Unstructured un-split geometrical Volume-of-Fluid methods -- A review

Geometrical Volume-of-Fluid (VoF) methods mainly support structured meshes, and only a small number of contributions in the scientific literature report results with unstructured meshes and three spatial dimensions. Unstructured meshes are traditionally used for handling geometrically complex solution domains that are prevalent when simulating problems of industrial relevance. However, three-dimensional geometrical operations are significantly more complex than their two-dimensional counterparts, which is confirmed by the ratio of publications with three-dimensional results on unstructured meshes to publications with two-dimensional results or support for structured meshes. Additionally, unstructured meshes present challenges in serial and parallel computational efficiency, accuracy, implementation complexity, and robustness. Ongoing research is still very active, focusing on different issues: interface positioning in general polyhedra, estimation of interface normal vectors, advection accuracy, and parallel and serial computational efficiency. This survey tries to give a complete and critical overview of classical, as well as contemporary geometrical VOF methods with concise explanations of the underlying ideas and sub-algorithms, focusing primarily on unstructured meshes and three dimensional calculations. Reviewed methods are listed in historical order and compared in terms of accuracy and computational efficiency

An unstructured Finite-Volume Level Set / Front Tracking method for capillary flows

In this thesis the unstructured Finite-Volume hybrid Level Set / Front Tracking method (LENT) for immiscible two-phase flows is extended to enable the simulation of capillary flows. The major contributions are a more accurate interface curvature approximation, an accuracy driven pressure velocity coupling algorithm, an approximation technique for consistent mass fluxes for momentum convection and two novel approaches for the computation of volume fractions from triangulated surfaces. All proposed techniques and algorithms are devised for unstructured Finite-Volume meshes. The improved curvature approximation uses a signed distance field as input and utilizes surface-mesh/volume-mesh mappings to reduce curvature variation in interface normal direction. A novel, local correction approach is introduced to further reduce the curvature error in cells intersected by the interface. To ensure a prescribed solution accuracy, an iterative, accuracy driven pressure velocity coupling algorithm is presented that builds on the established segregated solution algorithms. The necessity of consistent mass fluxes for momentum convection in the presence of differing fluid densities is analyzed. For interface advection methods that do not utilize phase-specific volumetric fluxes, a method to obtain approximate, consistent mass fluxes is proposed. The resulting improvements for capillary flows are demonstrated using canonical verification and validation test cases. Two novel algorithms to compute volume fractions on unstructured volume meshes from oriented triangle surfaces meshes are introduced, one based on geometric intersections and one based on approximation and adaptive refinement. Intended for the phase indicator calculation in the context of Level Set / Front Tracking methods, both algorithms are shown to be sufficiently accurate to initialize volume fractions also for the Volume-of-Fluid method. In fact, test cases demonstrate that both approaches’ accuracy is only limited by the resolution of the surface mesh

Ideal GLM-MHD - a new mathematical model for simulating astrophysical plasmas

Magnetic fields are ubiquitous in space. As there is strong evidence that magnetic fields play an important role in a variety of astrophysical processes, they should not be neglected recklessly. However, analytic models in astrophysical either do often not take magnetic fields into account or can do this after limiting simplifications reducing their overall predictive power. Therefore, computational astrophysics has evolved as a modern field of research using sophisticated computer simulations to gain insight into physical processes. The ideal MHD equations, which are the most often used basis for simulating magnetized plasmas, have two critical drawbacks: Firstly, they do not limit the growth of numerically caused magnetic monopoles, and, secondly, most numerical schemes built from the ideal MHD equations are not conformable with thermodynamics. In my work, at the interplay of math and physics, I developed and presented the first thermodynamically consistent model with effective inbuilt divergence cleaning. My new Galilean-invariant model is suitable for simulating magnetized plasmas under extreme conditions as those typically encountered in astrophysical scenarios. The new model is called the "ideal GLM-MHD" equations and supports nine wave solutions. The accuracy and robustness of my numerical implementation are demonstrated with a number of tests, including comparisons to other schemes available within in the multi-physics, multi-scale adaptive mesh refinement (AMR) simulation code FLASH. A possible astrophysical application scenario is discussed in detail