Course Name : Numerical Methods for CFD

Code(Credit) : CUTM1526(2+1+1)

Course Objectives

To learn fundamentals of computational methods like FDM and FVM for solving linear and non-linear partial differential equations related to fluid dynamics and heat transfer.

Course Outcome

COs	Course outcomes	Mapping Cos with POs (High-3, Medium-2, Low-1)
CO1	· Assess the principles of numerical analysis and concepts of consistency, stability, and convergence.	PO1(3), PO2(2)
CO2	· Evaluate finite difference/volume schemes on model problems of computational fluid dynamics.	PO1(3), PO5(3), PO9(3)
CO5	· Construct program-code using Python to obtain numerical solutions of partial differential equations, relevant to Computational Fluid Dynamics.	PO5(3)

Course Syllabus

MODULE I

Introduction to CFD: Basics of computational fluid dynamics, Definition and overview of CFD- need, advantages, problem areas, Governing equations of fluid dynamics – Continuity, Momentum and Energy equations, Non-Dimensional form of these governing equations, Classifications of PDE: Elliptic, Parabolic and Hyperbolic equations.

MODULE II

Finite Difference Method (FDM): Derivation of Finite difference equations (FDE) of 1^stand 2^nd order derivatives using Taylor series expansion. Explicit method-FTCS Method,Implicit method-BTCS Method, Crank-Nicholson method, Error, Convergence and stability analysis of above numerical Scheme, Keller Box Method.

MODULE III

Solution of Simultaneous Equations: Direct and Iterative methods; Gauss-elimination, Gauss-Jordan, Gauss-Jacobi and Gauss-Seidel methods, Tri Diagonal Matrix Algorithm(TDMA) (Thomas)

Practice 1: Gauss-elimination method using Python

Practice 2: Gauss-Seidel method using Python

Practice 3: Tri Diagonal Matrix Algorithm using Python

Project 1: Solution of Simultaneous Equations using Gauss-Jordan method.

Project 2: Solution of Simultaneous Equations using Gauss-Jacobi method.

MODULE IV

Application of FDM: Solutions of

Elliptic PDE: One-and Two-dimensional steady heat conduction, Laplace’s Equation, Poisson’s equation.

Parabolic PDE: Unsteady heat conduction, Stoke’s 1^st & 2^nd Problems.

Hyperbolic PDE: One-dimensional wave equation.

Practice 4: Solution of One-dimensional steady heat conduction using Python.

Practice 5: Solution of Laplace’s equation using Python.

Practice 6: Solution of Unsteady heat conduction using Python.

Practice 7: Solution of One-dimensional wave equation using Python.

Project 3: Solution of Burger’s equation.

Project 4: Solution of Poisson’s equation.

MODULE V

Finite Volume Method (FVM):

Fundamentals of FVM, Integral Form of 1-D Conservation equation, Finite Volume Method in 2-D

MODULE VI

Application of FVM: Solutions of 1-D steady state Diffusion and Convection equations.

Project 5: Solutions of 1-D steady state Diffusion equation.

MODULE VII

Application of FVM: Solutions of 2-D steady state Diffusion and Convection equations.

Project 6: Solutions of 2-D steady state Convection equation.

Text Books:

Computational Fluid dynamics by John D.Anderson,Jr
Computational Fluid dynamics and Heat Transfer , by John C. Tannehill , Dale A.Anderson ,Richard H. Pletcher
Introduction to finite elements in engineering, by Tirupathi R. Chadraupala, Ashok D.Belegundu.
An introduction to computational fluiddynamics, by HKVersteeg and W Malalasekera.

Session Plan

Session 1

Basics of computational fluid dynamics, Definition and overview of CFD- need, advantages, problem areas,Governing equations of fluid dynamics – Continuity, Momentum and Energy equations.

https://youtu.be/jQHp49OyPn8

https://youtu.be/9x-4tF7ldfA

https://slideplayer.com/slide/3872472/