Numerical Methods for CFD

Teacher

Dr.Ashok Misra

Category

Core Courses

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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.

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 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.

Session 2

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

Session 3

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

Session 4

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

Session 5

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

Session 6

Keller Box Method.

Session 7

Gauss-elimination method.

Sessions 8 & 9

Practice 1:  Gauss-elimination method using Python

Session 10

Gauss-Jordan method.

Session 11 & 12

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

Session 13

Gauss-Jacobi  method.

Session 14 & 15

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

Session 16

Gauss-Seidel method.

Session 17 & 18

Practice 2:  Gauss-Seidel method using Python.

Session 19

Tri Diagonal Matrix Algorithm.

Session 20 & 21

Practice 3:  Tri Diagonal Matrix Algorithm using Python

Session 22

Solutions of One-dimensional steady heat conduction.

Session 23 & 24

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

Session 25

Solutions of Two-dimensional steady heat conduction.

Session 26

Solution of Laplace’s Equation.

Session 27 & 28

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

Session 29

Solution of Poisson’s equation.

Session 30 & 31

Project 3: Solution of Poisson’s equation.


Session 32

Solution of Unsteady heat conduction.

Session 33 & 34

Practice 6:  Solution of Unsteady heat conduction using Python.

Session 35

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

Session 36 & 37

Project 4: Solution of Burger’s equation.

Session 38

Solution of One-dimensional wave equation.

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.

Session 42

Finite Volume Method in 2-D.

.

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.

Session 47

Solutions of 2-D steady state Diffusion equation.

Session 48

Solutions of 2-D steady state Convection equation.

Session 49 & 50

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

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