Numerical solution for lid driven cavity flow exploring finite difference techniques for partial differential equations

• Grant number: 23/00695-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: February 01, 2023
• End date: December 31, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Gilcilene Sanchez de Paulo
• Grantee: Thiago Burgo Ikeda
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil

Abstract

Applied and Computational Mathematics has been contributing a lot to the robust solution of problems of Engineering and of industrial interest through the evolution of numerical methods applied to the solution of the Navier-Stokes Equations.This project aims to contribute to the initial training of human resources for this line.The student will numerically solve incompressible flows of Newtonian fluids in a lid-driven cavity flow.The Navier-Stokes equations will not be solved directly in their primitive variables, velocity and pressure (v - p), but in the variables related to the flow and the vorticity of the flow, being known as the mathematical formulation ``Current-Vorticity' ' (psi - w).The equations, valid inside the domain, will be approximated by the finite difference equations considering a uniform and Cartesian mesh.The imposition of boundary conditions will also be handled by finite differences, and in this respect Dirichlet and Neumann type conditions will be involved.The problem will be handled and resolved for the planar case.To ensure a solid education and that the student is able to develop the minimum skills necessary for this area, the problem will be initially simplified. Simplification will be carried out in order to generate prototypes of partial differential equations (PDE), sometimes parabolic, sometimes hyperbolic and sometimes elliptic. In this way, general and peculiar aspects of each discretization will be widely discussed, such as concepts of convergence, stability, consistency, use of boundary conditions, interpretation and presentation of results, verification (accuracy and robustness) of the employed methodologies. The computational resource that will be used is the Python programming language, free, simple access and installation, widely disseminated and accepted by the scientific community.