Surface tension implementation for Gensmac 2D

In the present work we describe a method which allows the incorporation of surface tension into the GENSMAC2D code. This is achieved on two scales. First on the scale of a cell, the surface tension effects are incorporated into the free surface boundary conditions through the computation of the capillary pressure. The required curvature is estimated by fitting a least square circle to the free surface using the tracking particles in the cell and in its close neighbors. On a sub-cell scale, short wavelength perturbations are filtered out using a local 4-point stencil which is mass conservative. An efficient implementation is obtained through a dual representation of the cell data, using both a matrix representation, for ease at identifying neighbouring cells, and also a tree data structure, which permits the representation of specific groups of cells with additional information pertaining to that group. The resulting code is shown to be robust, and to produce accurate results when compared with exact solutions of selected fluid dynamic problems involving surface tension

Nonlocal diffusion, a Mittag-Leffler function and a two-dimensional Volterra integral equation

In this paper we consider a particular class of two-dimensional singular Volterra integral equations. Firstly we show that these integral equations can indeed arise in practice by considering a diffusion problem with an output flux which is nonlocal in time; this problem is shown to admit an analytic solution in the form of an integral. More crucially, the problem can be re-characterized as an integral equation of this particular class. This example then provides motivation for a more general study: an analytic solution is obtained for the case when the kernel and the forcing function are both unity. This analytic solution, in the form of a series solution, is a variant of the Mittag-Leffler function. As a consequence it is an entire function. A Gronwall lemma is obtained. This then permits a general existence and uniqueness theorem to be proved

Numerical study of the stress singularity in stick-slip flow of the Phan-Thien Tanner and Giesekus fluids

Stick-slip flow is a challenging viscoelastic benchmark problem due to the presence of a separation or transition point at the die exit where a sudden change in flow boundary conditions occurs. We present numerical simulations of transient planar stick-slip flow of the Phan-Thien-Tanner (PTT) and Giesekus fluids, investigating the polymer stress behavior around the stress singularity at the stick-slip point, confirming the asymptotic results presented by Evans et al. ["Stresses of the Oldroyd-B, PTT and Giesekus fluids in a Newtonian velocity field near the stick-slip singularity," Phys. Fluids 29, 1-33 (2017)]. In order to improve the numerical knowledge about this viscoelastic benchmark problem, two distinct mathematical methodologies are used for comparison in the computational simulations: the Cartesian and natural stress formulations. The former is widely applied in computational rheology, while the latter is used for the first time in the context of this problem. The natural stress formulation gives improved convergence results both temporally and spatially near to the singularity while maintaining the same global flow characteristics as the Cartesian.</p

O impacto da utilização de propostas didáticas de história da matemática nas aprendizagens da grandeza comprimento por alunos do 1º ciclo do ensino básico

Resumo Este relatório de investigação é realizado no âmbito do mestrado profissionalizante em Educação Pré-escolar e 1.º Ciclo do Ensino Básico, na Escola Superior de Educação, que está integrada no Instituto Politécnico de Viseu. Este trabalho é realizado tendo por base dois propósitos. Por um lado, pretende-se analisar e refletir acerca das práticas adotadas no estágio das áreas curriculares de Prática de Ensino Supervisionada II e III, salientando o que foi feito em ambos os níveis de ensino e as competências pedagógicas que foram sendo adquiridas durante todo este processo. Por outro lado, procura perceber-se, através de uma investigação, o impacto da inserção de propostas didáticas relativas à História da Matemática na aprendizagem de conteúdos relativos à grandeza comprimento por uma turma de alunos do terceiro ano do 1.º ciclo do ensino básico, numa escola da rede pública, situada na região de Viseu. O estudo mostra que a integração de propostas didáticas relativas à grandeza comprimento influencia de forma positiva a aquisição de aprendizagens dos alunos. Assim, é feita uma análise e reflexão acerca das competências por mim adquiridas durante este período de tempo. Este relatório de investigação permitiu melhorar a minha capacidade crítica e reflexiva, fundamentais para a profissão docente, bem como competências relativas à investigação, que muito influenciam na prática de um docente. A investigação permite ainda refletir acerca das práticas adotadas, permitindo alterar práticas menos adequadas, substituindo-as por escolhas mais favoráveis à aprendizagem dos alunos.Abstract This research report is conducted under the professional master's degree in Preschool Education and first Cycle of basic education in Escola Superior de Educação de Viseu, which is integrated into Instituto Politécnico de Viseu. This work is based on two purposes. First, we intend to analyse and reflect the practices adopted in the internship of the curricular areas Prática de Ensino Supervisionada II and Prática de Ensino Supervisionada III, highlighting what was done and the teaching skills that have been acquired throughout this process. On the other hand, we will try to understand, through an investigation, the effect of the inclusion of didactic proposals using the History of Mathematics in learning of measurement contents, namely of the attribute of length, by a group of third year students of the first cycle of basic education of a public school, located near Viseu. This study shows that the use of didactic proposals for the History of Mathematics on measurement of length has a positive influence on the students’ learning. matics related to measure contents has a positive influence on the students’ learnings. This research report has improved the critical and reflective capacity, essential for the teaching profession, as well as skills related with research, which greatly influence the practice of teaching. The investigation allowed us to reflect about the practices, and so to change some practices that aren't so appropriate and replacing them by others that facilitate learnin

A note on the eigenvalues of a special class of matrices

In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1, 1] for all values of m (the order of the matrix) and all values of a positive parameter a, the stability parameter sigma. As the order of the matrix is general, and the parameter sigma lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices. (C) 2010 Elsevier B.V. All rights reserved

Simulation of free surface flows in a distributed memory environment

A parallel technique, for a distributed memory machine, based on domain decomposition for solving the Navier-Stokes equations in cartesian and cylindrical coordinates in two dimensions with free surfaces is described. It is based on the code by Tome and McKee (J. Comp. Phys. 110 (1994) 171-186) and Tome (Ph.D. Thesis, University of Strathclyde, Glasgow, 1993) which in turn is based on the SMAC method by Amsden and Harlow (Report LA-4370, Los Alamos Scientific Laboratory, 1971), which solves the Navier-Stokes equations in three steps: the momentum and Poisson equations and particle movement, These equations are discretized by explicit and 5-point finite differences. The parallelization is performed by splitting the computation domain into vertical panels and assigning each of these panels to a processor. All the computation can then be performed using nearest neighbour communication. Test runs comparing the performance of the parallel with the serial code, and a discussion of the load balancing question are presented. PVM is used for communication between processes. (C) 1999 Elsevier B.V. B.V. All rights reserved

Three-dimensional transient complex free surface flows: Numerical simulation of XPP fluid

In this paper we present a finite difference MAC-type approach for solving three-dimensional viscoelastic incompressible free surface flows governed by the eXtended Pom-Pom (XPP) model, considering a wide range of parameters. The numerical formulation presented in this work is an extension to three-dimensions of our implicit technique [Journal of Non-Newtonian Fluid Mechanics 166 (2011) 165-179] for solving two-dimensional viscoelastic free surface flows. To enhance the stability of the numerical method, we employ a combination of the projection method with an implicit technique for treating the pressure on the free surfaces. The differential constitutive equation of the fluid is solved using a second-order Runge-Kutta scheme. The numerical technique is validated by performing a mesh refinement study on a pipe flow, and the numerical results presented include the simulation of two complex viscoelastic free surface flows: extrudate-swell problem and jet buckling phenomenon. © 2013 Elsevier B.V

Incompressible Turbulent Flow Simulation Using the κ-ɛ Model and Upwind Schemes

In the computation of turbulent flows via turbulence modeling, the treatment of the convective terms is a key issue. In the present work, we present a numerical technique for simulating two-dimensional incompressible turbulent flows. In particular, the performance of the high Reynolds κ-ɛ model and a new high-order upwind scheme (adaptative QUICKEST by Kaibara et al. (2005)) is assessed for 2D confined and free-surface incompressible turbulent flows. The model equations are solved with the fractional-step projection method in primitive variables. Solutions are obtained by using an adaptation of the front tracking GENSMAC (Tomé and McKee (1994)) methodology for calculating fluid flows at high Reynolds numbers. The calculations are performed by using the 2D version of the Freeflow simulation system (Castello et al. (2000)). A specific way of implementing wall functions is also tested and assessed. The numerical procedure is tested by solving three fluid flow problems, namely, turbulent flow over a backward-facing step, turbulent boundary layer over a flat plate under zero-pressure gradients, and a turbulent free jet impinging onto a flat surface. The numerical method is then applied to solve the flow of a horizontal jet penetrating a quiescent fluid from an entry port beneath the free surface