 # Numerical Methods for CFD Teacher

##### Dr.Ashok Misra Category

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# 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.

## Learning Outcomes

• Assess the principles of numerical analysis and concepts of consistency, stability, and convergence.
• Evaluate finite difference/volume schemes on model problems of computational fluid dynamics.
• Construct program-code using Python to obtain numerical solutions of partial differential equations, relevant to Computational Fluid Dynamics.

## 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 1st  and 2nd   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 1st & 2nd 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:

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

## 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/

## Session 2

Non-Dimensional form of these governing equations, Classifications of PDE: Elliptic, Parabolic and Hyperbolic equations.

https://youtu.be/u68QiAHUs5M

https://youtu.be/cnLH2uIgQRk

https://slideplayer.com/slide/17348538/

## Session 3

Derivation of Finite difference equations (FDE) of 1st and 2nd   order derivatives using Taylor series expansion.

https://youtu.be/oQD-z6QuEp4

https://youtu.be/vWiaOcTnOwo

## Session 4

Explicit method-FTCS Method, Implicit method-BTCS Method.

https://youtu.be/P5wAP5c4YxE

https://youtu.be/d_snLGgfZZM

## Session 5

Crank-Nicholson method, Error, Convergence and  stability analysis of above numerical Scheme.

https://youtu.be/NyTsklFe3YU

https://youtu.be/YnMWAa35Sh4

## Session 6

Keller Box Method.

https://youtu.be/uyrQ_91JBvE

## Session 7

Gauss-elimination method.

https://youtu.be/D6HFnV94cZs

## Sessions 8 & 9

Practice 1:  Gauss-elimination method using Python

## Session 10

Gauss-Jordan method.

https://youtu.be/MJ1B6U5Eaxs

## Session 11 & 12

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

## Session 13

Gauss-Jacobi  method.

https://youtu.be/jPv5pP1kOco

## Session 14 & 15

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

## Session 16

Gauss-Seidel method.

https://youtu.be/gxy6VI1hEfs

## Session 17 & 18

Practice 2:  Gauss-Seidel method using Python.

## Session 19

Tri Diagonal Matrix Algorithm.

https://youtu.be/OkEicUyBnOo

## Session 20 & 21

Practice 3:  Tri Diagonal Matrix Algorithm using Python

## Session 22

Solutions of One-dimensional steady heat conduction.

https://youtu.be/moibMiSXIU4

## Session 23 & 24

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

## Session 25

Solutions of Two-dimensional steady heat conduction.

https://youtu.be/CSAIxrWu5nc

## Session 26

Solution of Laplace’s Equation.

https://youtu.be/2GetEQACjZw

## Session 27 & 28

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

## Session 29

Solution of Poisson’s equation.

https://youtu.be/8cGs-tEVkJY

## Session 30 & 31

Project 3: Solution of Poisson’s equation.

## Session 32

https://youtu.be/DLNnn5ODl3s

## Session 33 & 34

Practice 6:  Solution of Unsteady heat conduction using Python.

## Session 35

Solution of Stoke’s 1st & 2nd Problems.

https://youtu.be/g-5bi7dxHP4

## Session 36 & 37

Project 4: Solution of Burger’s equation.

## Session 38

Solution of One-dimensional wave equation.

https://youtu.be/Jw_aGOrTDPo

## Session 39 & 40

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

## Session 41

Fundamentals of FVM, Integral Form of 1-D Conservation equation.

https://youtu.be/YGuLvNWKk2K

https://www.slideserve.com/tymon/the-finite-volume-method

## Session 42

Finite Volume Method in 2-D.

https://youtu.be/H7X6z5SrQbg

## Session 43

Solutions of 1-D steady state Diffusion  equation.

## Session 44 & 45

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

## Session 46

Solutions of 1-D steady state  Convection equation.

https://youtu.be/Fu1ISSK4gkA

## Session 47

Solutions of 2-D steady state Diffusion equation.

https://youtu.be/ZkxdPzvSUU0

## Session 48

Solutions of 2-D steady state Convection equation.

## Session 49 & 50

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