10-04-2017, 09:01 PM
By
Sunil Kumar Rath (10503069
National Institute of Technology Rourkela
Abstract
The multigrid algorithm is an extremely efficient method of approximating the solution to a given problem. The functions involved in the calculations are all discrete, or discontinuous, and are represented by an array of values taken from equally-spaced points along their range - a "grid." The algorithm's efficiency lies in the fact that once an approximate solution to the problem is found its accuracy can be improved using calculations on increasingly sparse grids which require less processing power.
In this project the theory behind the multigrid algorithm was studied and a computer program was written which demonstrates the use of this algorithm in solving the problem of natural convection. Stream function-Vorticity approach and the Bossinesq approximation were used in the programs. Also, the same problem was solved using the Multigrid algorithm using Fluent software. The results obtained were matched with the analytical results.
Introduction to CFD
Computational Fluid Dynamics (CFD) is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the millions of calculations required to simulate the interaction of fluids and gases with the complex surfaces used in engineering. However, even with simplified equations and high speed supercomputers, only approximate solutions can be achieved in many cases. More accurate codes that can accurately and quickly simulate even complex scenarios such as supersonic or turbulent flows are an ongoing area of research. CFD provides a qualitative (and sometimes even quantitative) prediction of fluid flows by means of:
a) mathematical modeling (partial differential equations)
b) numerical methods (discretization and solution techniques)
c) software tools (solvers, pre- and postprocessing utilities)
It also gives an insight into flow patterns that are difficult, expensive or impossible to study using traditional (experimental) techniques.
Structure of CFD Codes
CFD codes are structured around the numerical algorithms that can be tackle fluid problems. In order to provide easy access to their solving power all commercial CFD packages include sophisticated user interfaces input problem parameters and to examine the results. Hence all codes contain three main elements:
1) Pre-processing:
Preprocessor consist of input of a flow problem by means of an operator friendly interface and subsequent transformation of this input into form of suitable for the use by the solver.
2) Solver:
Solver consists of discretization by substitution of the approximation into the governing flow equations and subsequent mathematical manipulation and solution of the algebraic equations
3) Post processing.
These include domain geometry & grid display, vector plots, line & shaded contour plots, 2D and 3D surface plots, Particle tracking, view manipulation (translation, rotation, scaling etc.)
Problem Statement
Natural Convection in a Square Enclosure
A Square box of side L has a hot right wall at 2000K, a cold left wall at 1000K, and adiabatic top and bottom walls. Gravity acts downwards. A buoyant force develops because of thermally induced density gradients. The objective is to compute the flow and temperatu patterns in the box, as well as the wall heat flux. The working fluid has a Prandlt number of approximate 0.71, and the Reyleigh number based L is 5e+05. This means the flow is inherently laminar. The Boussinesq assumption is to used to model buoyancy.
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