A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variable u and the auxiliary variable σ with respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis

Semiannual report

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period 1 Oct. 1994 - 31 Mar. 1995

HOLA: a High-Order Lie Advection of Discrete Differential Forms With Applications in Fluid Dynamics

The Lie derivative, and Exterior Calculus in general, is ubiquitous in the elegant geometric interpretation of many dynamical systems. We extend recent trends towards a Discrete Exterior Calculus by introducing a discrete framework for the Lie derivative defined on differential forms, including a WENO based numerical scheme for its implementation. The usefulness of this operator is demonstrated through the advection of scalar and vector valued fields (arbitrary discrete k-forms) in a desirable intrinsic and metric independent fashion. Examples include Lie advection of fluid flow vorticity, and we conclude with a significant discussion on the conservative Lie advection of fluid mass density for robust free surface flows in computer graphics

Detailed Numerical Simulation of Multi-Dimensional Plasma Assisted Combustion

Interaction between flames and plasmas are the guiding thread of this work. Nanosecond Repetitively Pulsed (NRP) discharges are non-thermal plasmas which have shown interesting features for combustion control. They can interact with flames not only through heat, but also chemically by producing active species. In this work, fully-coupled plasma assited combustion simulations are targeted. To achieve this goal, plasma discharge capabilities are built in the low temperature plasma code, AVIP. The corresponding numerical methods, as well as validation cases regarding each set of equations, are first presented. To simulate plasma discharges, the coupled drift-diffusion equations and the Poisson equation are considered. AVIP is coupled to the AVBP code which solves the reactive Navier-Stokes equations to describe combustion phenomena. In a second part, we start by constructing and validating a fully-detailed chemistry for methane-air mixtures in zero-dimensional reactors before reducing it for multi dimensional simulations. The multi-dimensional streamer simulation capabilities of the code are then assessed using simple chemistries. All the validated parts of the code come together in a fully detailed simulation of ignition using NRP discharges. We finish by discussing phenomenological models built upon the knowledge that we gained from fully-detailed simulations. In a last part, finally, attempt to solve the Poisson and generalized Poisson equations using neural networks, which have a potential for speedup compared to classical linear solvers, is carried out

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 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