An Accurate and Robust Numerical Scheme for Transport Equations

En esta tesis se presenta una nueva técnica de discretización para ecuaciones de transporte en problemas de convección-difusión para el rango completo de números de Péclet. La discretización emplea el flujo exacto de una ecuación de transporte unidimensional en estado estacionario para deducir una ecuación discreta de tres puntos en problemas unidimensionales y cinco puntos en problemas bidimensionales. Con "flujo exacto" se entiende que se puede obtener la solución exacta en función de integrales de algunos parámetros del fluido y flujo, incluso si estos parámetros son vari- ables en un volumen de control. Las cuadraturas de alto orden se utilizan para lograr resultados numéricos cercanos a la precisión de la máquina, incluso con mallas bastas.Como la discretización es esencialmente unidimensional, no está garantizada una solución con precisión de máquina para problemas multidimensionales, incluso en los casos en que las integrales a lo largo de cada coordenada cartesiana tienen una primitiva. En este sentido, la contribución principal de esta tesis consiste en una forma simple y elegante de obtener soluciones en problemas multidimensionales sin dejar de utilizar la formulación unidimensional. Además, si el problema es tal que la solución tiene precisión de máquina en el problema unidimensional a lo largo de las líneas coordenadas, también la tendrá para el dominio multidimensional.In this thesis, we present a novel discretization technique for transport equations in convection-diffusion problems across the whole range of Péclet numbers. The discretization employs the exact flux of a steady-state one-dimensional transport equation to derive a discrete equation with a three-point stencil in one-dimensional problems and a five-point stencil in two-dimensional ones. With "exact flux" it is meant that the exact solution can be obtained as a function of integrals of some fluid and flow parameters, even if these parameters are variable across a control volume. High-order quadratures are used to achieve numerical results close to machine- accuracy even with coarse grids. As the discretization is essentially one-dimensional, getting the machine- accurate solution of multidimensional problems is not guaranteed even in cases where the integrals along each Cartesian coordinate have a primitive. In this regard, the main contribution of this thesis consists in a simple and elegant way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain.<br /

Streamline-averaged mass transfer in a circulating drop

Solute mass transfer is considered from the outside to the inside of a circulating drop in the context of liquid-liquid extraction. Specifically an internal problem is treated with resistance to mass transfer dominated by the liquid inside the drop. The Peclet number of the circulation is large, on the order of tens of thousands. A model is proposed by which the mass transfer into the drop begins in a boundary layer regime, but subsequently switches into a so called streamline-averaged regime. Solutions are developed for each regime, and also for the switch between them. These solutions are far easier to obtain than those of the full advection-diffusion equations governing this high Peclet number system, which are very stiff. During the boundary layer regime, the rate at which solute mass within the drop grows with time depends on Peclet number, with in creases in Peclet number implying faster growth. However larger Peclet numbers also imply that the switch to the streamline-averaged regime happens sooner in time, and with less solute mass having been transferred to date. In the streamline-averaged regime, solute concentration varies across streamlines but not along them. In spite of the very large Peclet number, the rate of mass transfer is controlled diffusively, specifically by the rate of diffusion from streamline-to-streamline: sensitivity to the Peclet number is there by lost. The model predictions capture, at least qualitatively, findings reported in liter ature for the evolution of the solute concentration in the drop obtained via full numerical simulation

Perfectly Matched Layer Absorbing Boundary Conditions for the Discrete Velocity Boltzmann-BGK Equation

Perfectly Matched Layer (PML) absorbing boundary conditions were first proposed by Berenger in 1994 for the Maxwell\u27s equations of electromagnetics. Since Hu first applied the method to Euler\u27s equations in 1996, progress made in the application of PML to Computational Aeroacoustics (CAA) includes linearized Euler equations with non-uniform mean flow, non-linear Euler equations, flows with an arbitrary mean flow direction, and non-linear clavier-Stokes equations. Although Boltzmann-BGK methods have appeared in the literature and have been shown capable of simulating aeroacoustics phenomena, very little has been done to develop absorbing boundary conditions for these methods. The purpose of this work was to extend the PML methodology to the discrete velocity Boltzmann-BGK equation (DVBE) for the case of a horizontal mean flow in two and three dimensions. The proposed extension of the PML has been accomplished in this dissertation. Both split and unsplit PML absorbing boundary conditions are presented in two and three dimensions. A finite difference and a lattice model are considered for the solution of the PML equations. The linear stability of the PML equations is investigated for both models. The small relaxation time needed for the discrete velocity Boltzmann-BC4K model to solve the Euler equations renders the explicit Runge-Kutta schemes impractical. Alternatively, implicit-explicit Runge-Kutta (IMEX) schemes are used in the finite difference model and are implemented explicitly by exploiting the special structure of the Boltzmann-BGK equation. This yields a numerically stable solution by the finite difference schemes. As the lattice model proves to be unstable, a coupled model consisting of a lattice Boltzmann (LB) method for the Ulterior domain and an IMEX finite difference method for the PML domains is proposed and investigated. Numerical examples of acoustic and vorticity waves are included to support the validity of the PML equations. In each example, accurate solutions are obtained, supporting the conclusion that PML is an effective absorbing boundary condition

Simulations of Nanoparticle Synthesis in Laminar Stagnation Flames

A new implementation of a multidimensional solver for studying nanoparticle synthesis in laminar flames is presented. The governing equations are convective-diffusive-reactive partial differential equations that are discretised using the finite volume method. Detailed chemical source terms and transport coefficients are used to close the equations. The implementation of these governing equations is discussed and the numerical algorithm used to solve them is presented. The new solver is verified against analytic solutions and numerical solutions from 1D models for counterflow diffusion flames. The new solver was used to calculate the flame location, shape and temperature of laminar premixed ethylene jet-wall stagnation flames when the equivalence ratio, exit gas velocity and burner-plate separation distance are varied. The simulation results were compared to new experimental 2D measurements of CH* chemiluminescence and temperature. The 2D simulations showed excellent agreement, and correctly predicted the flame shape, location and temperature as the experimental conditions were varied. The new solver was used to study growth of inorganic nanoparticles in premixed, jet-wall stagnation flames. Titanium dioxide, also known as titania and TiO2, is a white powder than has many uses as a pigment, including in paper and cosmetics, and was selected as the system to apply the new solver. TiO2 nanoparticles formed from titanium tetraisopropoxide (TTIP) were simulated using a two step methodology, which enabled insight into the variations of particle properties as a function of the deposition radius. Two different TTIP loadings (280 and 560~ppm) were studied in two flames, a lean flame (equivalence ratio 0.35) and a stoichiometric flame (equivalence ratio 1.0). First, the growth of particles was described with a spherical particle model fully coupled to the conservation equations of chemically reacting flow. Second, particle trajectories were extracted from the 2D simulations and post-processed using a detailed particle model solved with a stochastic numerical method. The simulation produced gas phase predictions of flame location that are in good agreement with available literature. The particle morphologies and size distributions were examined and found to be dependent on the deposition radius. Particles began to have different size distributions at a deposition radius of approximately one and a half times the nozzle radius (1.0 cm), which should be kept in mind when synthesising and modelling nanoparticles for novel applications. This coincided with the growth of total residence time along particle trajectories. It is suggested that experiments critically examine the radially uniformity of deposited particles do not affect the performance for their intended application.Gates Cambridge Foundation, Gates Cambridge Scholarship (OPP1144

Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require $\mathcal{O}(p^{2d})$ storage and $\mathcal{O}(p^{3d})$ computational work, where $p$ is the degree of basis polynomials used, and $d$ is the spatial dimension. Our SVD-based tensor-product preconditioner requires $\mathcal{O}(p^{d+1})$ storage, $\mathcal{O}(p^{d+1})$ work in two spatial dimensions, and $\mathcal{O}(p^{d+2})$ work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in $p$ per degree of freedom in 2D, and reduce the computational complexity from $\mathcal{O}(p^9)$ to $\mathcal{O}(p^5)$ in 3D. Numerical results are shown in 2D and 3D for the advection and Euler equations, using polynomials of degree up to $p=15$. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees $p$.Comment: 40 pages, 15 figure

Application of GPU Accelerated Paired Explicit Runge-Kutta Methods to Turbulent Flow Over a Sphere

The design of next generation aircraft will rely on the use of high-fidelity computational fluid dynamics (CFD). For this purpose, the Paired Explicit Runge-Kutta (P-ERK) time stepping method was developed. It is a variation of explicit Runge-Kutta methods that allows the pairing of multiple methods within a given simulation. This results in higher stability methods being used in stiff regions, and lower cost methods being used in non-stiff regions, resulting in significant performance improvements. This work explores the utility of Graphical Processing Unit (GPU) acceleration combined with P-ERK schemes Subcritical flow over a smooth sphere at a Reynolds number (Re) of 3700 was used as the validation case. The flux reconstruction (FR) spatial discretization scheme was used with the implicit Large Eddy Simulation (ILES) turbulence modelling approach. Instantaneous quantities such as velocity fluctuations, Strouhal numbers, and time-averaged quantities such as drag coefficient, pressure coefficient, back pressure ratio, and Reynolds stresses were obtained. In addition, Q-criterion contours were generated and used to obtain separation angle and recirculation bubble length. This study shows that the P-ERK method can achieve good agreement with both reference simulation, and experimental data. Moreover, when compared to the traditional fourth order Runge-Kutta (RK) method the P-ERK scheme shows an average speedup factor of 4.73 using GPUs and of 5.82 using CPUs with regards to solution polynomial scaling, and with regards to resource scaling it requires approximately 8 times more resources using CPUs, or each CPU has an approximately 45 times greater runtime compared to one GPU. This is a significant reduction in computational times, while maintaining accuracy

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg