Course Project
Overview
A proper introduction to the art of Computational Fluid Dynamics (CFD) would require several semesters on this topic alone. Clearly this cannot be accomplished in a first graduate course in fluid dynamics, where the semester is already filled with the necessary topics to understand the underlying principles of the subject. On the other hand, the two primary approaches for describing fluid flows that cannot be obtained from the analytical solutions of the governing equations are: the use of experiments and the use of the computer to find numerical solutions. We will speak of experimental techniques throughout the course in each module. The use of computers in fluid dynamics is so pervasive that to ignore this topic completely until a later course is not prudent. The compromise that has been adopted is to provide an introduction to CFD in Module 2 sufficient to allow the solution of two typical engineering problems contained in this course project. Students will develop grids, write computer codes, produce converged solutions, assess the error, plot flow fields and compute derived engineering quantities from the results. From this experience students will gain an appreciation for many of the primary features of CFD. This project will be graded on the basis of 100 points, and the grade will represent 25% of the total course grade.
A good introduction to CFD is gained by focusing attention on a subclass of fluid flow problems governed by elliptic partial differential equations that are steady in time and vary only in space. Examples of such equations are the Poisson and Laplace equations, which are the topic of the first part of the project. The point of the first project problem is to allow the student to develop the needed code to complete a CFD solution of a simple problem. After gaining experience with the first problem, portions of this code are reused, and the rest is expanded upon to tackle an engineering-oriented pipe flow problem which is the subject of the second part of the project. Students are required to complete both parts of the project. This document will concentrate on the assignment of the problems, submission details, required deliverables and a grading rubric. Substantial background information and advice on the approach to these problems may be found in the Module 2 Lecture Notes. Also, several of the discussion activities are designed to assist progress towards the solution.
Objectives
After completion of this project, you will be able to:
• Discretize the governing equation for a fluid flow problem by using central difference estimates for the derivatives
• Develop both rectangular and polar grids for the problems and assign boundary and symmetry conditions appropriately
• Compute solutions using two iterative methods: Jacobi and Gauss-Seidel
• Plot the convergence history as well as contour plots of the solution
• Produce a series of solutions, where preceding computations are required to define the next
• Compute derived quantities of engineering significance that permit appropriate business decisions to be made
Project Assignment – Part 1 – Solution of Elliptic Partial Differential Equations
Introduction
The diffusion of a scalar quantity in a two-dimensional physical domain can be mathematically described by Poisson's equation f (x, y),(1)
where is the gradient operator and (x, y) is a distribution of sources (or sinks) within the domain?
A Poisson equation is an elliptic partial differential equation; by this it is meant that the behavior of the scalar quantity , at any given point in the physical domain, is related to all other values within the domain, as well as to the boundary conditions and to the term, this equation is zero, then the equation is called a Laplace equation.
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