Overview

The course deals with the numerical solution of equations governing fluid flow and would be of interest to engineers and scientists? both aspiring and professional? with chemical/ mechanical/ civil/ aerospace engineering applications. In all these fields, one needs to deal extensively with fluid flow related phenomena and one needs to resolve flow-related features of the processes and equipment. Although the equations governing fluid flow have been formulated more than 150 years ago, it is only in recent years that these are being solved in the practical applications in which the flow occurs. The course deals with the basic techniques that enable the numerical solution of these equations.

Syllabus

Motivation.
Flow in a rectangular duct: Problem formulation.
Flow in a rectangular duct: Discretiztion of flow domain.
Tutorial 1: Converting PDE to algebraic equation using FD approximation.
Tutorial 1 contd.: Solution for algebraic equations using Gauss- Seidel Method.
Flow in a triangular duct: Problem formulation.
Flow in a triangular duct: Discretiztion of flow domain.
Tutorial 2: Converting PDE to algebraic equation using Finite Volume method.
Tutorial 2 contd.: Description of FV method and solution using G-S Method.
Effect of grid spacing & upcoming course outline.
Mass conservation equations.
Momentum conservation equations.
Equations governing fluid flow in incompressible fluid.
Navier-Stokes equation for simple cases of flow.
Forces acting on control volume.
Kinematics of deformation in fluid flow.
Energy conservation equations.
Practical cases of fluid flow with heat transfer in CFD point of view.
Practical cases of fluid flow with mass transfer in CFD point of view.
Equations governing fluid flow with chemical reactions.
Concept of wellposedness of mathematical problems.
Introduction to finite difference methods.
Finite difference approximation on an uniform mesh.
Higher order and mixed derivatives.
Solution of Poisson equation in rectangular duct- Turorial.
Discretization of time domain.
FD approx. on a non-uniform mesh and need of analysis of obtained discretization.
Need for the analysis of discretized equation.
Properties of Numerical Schemes: Accuracy, Conservation property, Boundedness, Consistency.
Properties of Numerical Schemes: Stability analysis.
Tutorial on Stability Analysis.
Analysis of Generic 1-d scalar transport equation.
Introduction to the solution of coupled N-S equations.
N-S equation in compressible flow- Mac Cormack Scheme.
Stability limits of Mac-Cormack Scheme and the intro to Beam-Warming Scheme.
Implicit Beam-Warming Scheme.
Compressible flow to Incompressible flow.
Solution of coupled equations: Incompressible flow.
Artificial compressiblity method, Stream function-vorticity method.
Pressure equation method, Staggered grid system.
Pressure Correction Method.
Tutorial on Pressure Correction Method.
Tutorial on Pressure Correction Method contd..
Introduction to the basic numerical methods.
Direct Methods: solution of the system of algebraic equations.
Tri-diagonal Matrix Algorithm: Derivation.
TDMA and other iterative methods.
Recap of basic iterative methods..
Mod 5_ Week 2_Lec 5.6_Convergence analysis of basic iterative methods.
Successive Over Relaxation (SOR) method.
Alternating Direction Implicit (ADI) method.
Strongly Implicit Procedure (ILU) method.
Multigrid method.
Body Fitted Grid Approach.
Methods For Unstructured Grid Generation.
Formulation Of Finite Volume Method.
The Advancing Front Method continuation.
Triangulation: The Advancing Front Method.
Time and length scale of turbulance.
The turbulent closure problem.
The generic formulation for turbulence.
More generic formulation and summary.