ME608

NUMERICAL METHODS IN HEAT, MASS, AND MOMENTUM TRANSFER

Fall 2017, MWF 3:30 pm - 4:20 pm, ME3006

Instructor

Dr. Carlo ScaloAssistant Professor of Mechanical EngineeringRoom ME2195, ME BuildingWest Lafayette, IN 47907-2045Email: scalo@purdue.eduOffice Hours:‘walking from ME to WANG right after class’

Course Website:engineering.purdue.edu/∼scalo/menu/teaching.html

Prerequisites

Prerequisites for the course include basic knowledge of fluid mechanics, linear algebra, partial differential equations andaverage (not beginner!) programming skills. The use of Python is strongly recommended but not mandatory. The classcontent is structured in such a way to allow talented undergraduate students to successfully complete the coursework.

Course Objectives

The course will cover traditional aspects of Computational Fluid Dynamics (CFD) with focus on momentum and mass trans-fer applications, while providing exposure to the latest generation of high-level dynamic languages (such as Python) andversion-control software (such as git).

The course will cover the following topics:1. Mesh Generation, Discretization2. Linear Algebra, Poisson Equation3. Time Advancement Schemes, Unsteady Heat Transfer4. Navier-Stokes Solvers on Unstructured Grids5. Advanced topics: Linear-Stability Theory, Block-Spectral solvers, Finite-Element Methods, etc.

Students will write their own 2D incompressible unstructured Navier-Stokes from scratch as a final project.

Color contour of pressure field (left) and hybrid computational mesh (right) from a two-dimensional unstructuredincompressible Navier-Stokes solver written for ME308 (Danish Patel, ME608-Spring 2016)

1

Grade Distribution

Homework assignments and final reports turned in LATEX and/or with supporting images generated in vector graphics arestrongly encouraged (points will be detracted from messy reports, with unclear figures and text). The grade distribution is:

(5%) Homework 0: Computing Environment Setup– workflow setup via gitlab, “choice” of programming language

(20%) Homework 1: Meshing & Spatial Discretization– create unstructured mesh with ICEM CFD, spatial discretization

(20%) Homework 2: Poisson Equation– solve elliptical problems, compare iterative methods, mesh quality analysis

(20%) Homework 3: Unsteady Heat Transfer– compare different time advancement methods

(20%) Homework 4: Stokes Flow Solver– basic pressure/velocity coupling, first incompressible Navier-Stokes solver at Re = 0

(15%) Final Project– Nonlinear Navier-Stokes Solver: introduce convective flux (Re > 0), coupled heat transfer– Advanced Topic of Your Choice: Linear-Stability Theory, Block-Spectral solvers, Finite-Element Methods, etc.

Examples of source code will be provided in Python only (sorry). The use of Python is strongly recommended but notmandatory. Sharing of ideas on the homework assignments is encouraged but submissions need to be individually prepared.Note that it is trivial to check whether parts of source code have been copied. The grading scale for the course is:

Letter Grade Final GradeA+ 100 =< . . .A 95 <= · · · < 100A- 90 <= · · · < 95B+ 85 <= · · · < 90B 80 <= · · · < 85B- 75 <= · · · < 80C+ 70 <= · · · < 75C 65 <= · · · < 70C- 60 <= · · · < 65

References

With the exception of programming tutorials, all of the lecture material will be explained at the blackboard (or documentcamera) to facilitate a dynamic discussion. Some of the course material will be based on the following references:

• Handouts by Prof. Jayathi Y. Murthy (posted on website)• Ferziger, J., and M. Peric, Computational Methods for Fluid Dynamics, Third Edition, Springer, 2001

Other textbooks that cover more advanced and theoretical topics are:• Pletcher, R. H., Tannehill, J. C., and Anderson, D., Computational Fluid Mechanics and Heat Transfer, Third Edition,

CRC Press, 2011.• R. Leveque, Finite Volume Methods For Hyperbolic Problems, Cambridge, 2004• Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, unpublished

text, 1996, available at http://people.maths.ox.ac.uk/trefethen/pdetext.html

2

Schedule

A tentative schedule is included below. The instructor reserves the right to (frequently) update it in order to adapt to theclass progress.

Monday Wednesday Friday

Aug 21st Lecture 1

Introduction• Course Structure Overview• Homework 0:

Python, Linux, Git

23rd Lecture 2

Introduction• Conserved Variables• Coordinate-Invariant Definition

of Operators• Numerical Flux

Reading: Prof. Murthy’s notes - Sec.1.1 - 1.5

25th Lecture 3

Mesh Generation• Classification of Computational

Grids

Reading: review linear algebra (matrixmultiplications, eigenvalues, ...);Prof. Murthy’s notes - Sec. 2.1 - 2.2

28th Lecture 4

ICEM CFD Session• Getting Started: Geometry,

Blocking, Mesh, Parts

Reading:online tutorials and handouts

30th Lecture 5

Homework 0 Due

Python Session:• Python Basics• Homework 1 starter

Reading:Python Tutorial: sections 2, 3, 4, 5

Sep 1st Lecture 6

Spatial Discretization• Taylor Series• Polynomial Fitting

Reading: review linear algebra;Pletcher, et al. (2011) pp. 43 – 75;Ferziger & Peric (2001) pp. 21 – 52;Prof. Murthy’s notes - Sec. 2.3

4th

LABOR DAY(no class)

6th Lecture 7

Spatial Discretization• Polynomial Fitting (cont’d)• Discrete Operators

Reading: review linear algebra;Ferziger & Peric (2001) pp. 45 – 63;Prof. Murthy’s notes - Sec. 2.3

8th Lecture 8

Python Session:• Multivariate Poly. Fitting• Sparse Matrices

Reading:Python Tutorial: sections 6, 7, 8

11th Lecture 9

Poisson and Heat Equations• Assembly of Discrete Poisson

Equation (Finite Differences)

Reading:Pletcher et al. (2011) pp. 147 –152;

13th Lecture 10

Poisson and Heat Equations• Assembly Discrete Poisson

Equation (Finite Volumes)

Reading:Pletcher et al. (2011) pp. 147 –152;

15th Lecture 11

Linear Systems of Equations• Direct Methods: Gaussian

elimination, TDMA, LUDecomposition

Reading:Prof. Murthy’s notes - Sec. 2.4, Sec.8.1 - 8.4Pletcher et al. (2011) pp. 152 – 162

18th Lecture 12

Homework 1 Due

Linear Systems of Equations• Iterative Methods: Jacobi,

Gauss-Seidel

Reading:Prof. Murthy’s notes - Sec. 8.5 - 8.7Pletcher et al. (2011) pp. 152 – 162

20th Lecture 13

Poisson and Heat Equations• Homework 2 overview

22nd Lecture 14

Linear Systems of Equations• Iterative Methods:

Over-Relaxation, Multi-Grid

Reading:Prof. Murthy’s notes - Sec. 8.8 - 8.9Pletcher et al. (2011) pp. 152 – 162

3

Monday Wednesday Friday25th Lecture 15

Linear Systems of Equations• Iterative Methods: Multi-Grid

(cont’d), Conjugate Gradient

Reading:Prof. Murthy’s notes - Sec. 8.8 - 8.9Pletcher et al. (2011) pp. 166 – 175

27th Lecture 16

Spatial Discretization• Grid Transformations (1D)

Reading:Pletcher et al. (2011) pp. 329 – 337;Ferziger & Peric (2001) pp. 47 – 58;

29th Lecture 17

Temporal Discretization• Explicit and Implicit Euler,

Crank-Nicholson

Reading:Prof. Murthy’s notes - Sec. 3.3 - 3.9Pletcher et al. (2011) pp. 103 – 124;

Oct 2nd Lecture 18

Temporal Discretization• MacCormack, Adams-Bashforth,

Leap Frog,

Reading:Pletcher et al. (2011) pp. 82– 95

4th Lecture 19

Temporal Discretization• Runge-Kutta schemes• the Butcher tableau

Reading: Pletcher et al. (2011) pp.124 – 125

6th Lecture 20

Homework 2 Due

Temporal Discretization• Runge-Kutta schemes (cont’d)

Reading: Pletcher et al. (2011) pp.124 – 125

9th

OCTOBER BREAK

11th

Scalo @ NATO Meeting

13th

Scalo @ NATO Meeting

16th Lecture 21

Navier-Stokes Solvers• Incompressible Navier-Stokes• Boussinesq Approximation• Internal Energy Transport

Reading:Prof. Murthy’s notes - Sec. 4.1 - 4.5(2D diffusion on non-orthogonal grids)

18th Lecture 22

Navier-Stokes Solvers• Conservative Formulation,

Finite-Volume Approach• Tensorial and Vectorial Fluxes

20th Lecture 23

Navier-Stokes Solvers• Staggered Finite-Volume

Approach• Discretization for continuity and

pressure gradient

Reading: Harlow & Welch (1965)Prof. Murthy’s notes - Sec. 6.1 - 6.4(Discretization of Mass andMomentum equations, discussion onstaggered grids)

23rd Lecture 24

Homework 3 Due

Navier-Stokes Solvers• Projection Method: Fractional

Step Method

Reading:Chorin (1969), Kim & Moin (1985)

25th Lecture 25

Navier-Stokes Solvers• Algebraic Pressure Segregation

27th Lecture 26

Navier-Stokes Solvers• Review Session

30th Lecture 27

Navier-Stokes Solvers• Semi-Implicit Time

Advancement Methods(constant viscosity)

Nov 1st Lecture 28

Navier-Stokes Solvers• Semi-Implicit Time

Advancement Methods(constant viscosity)(cont’d)

3rd Lecture 29

Navier-Stokes Solvers• Discretization for Convective

Terms– upwind vs central– modified equation

Reading:Prof. Murthy’s notes - Sec. 5.1 - 5.9(2D convection - diffusion onnon-orthogonal grids)

4

Monday Wednesday Friday6th Lecture 30

Navier-Stokes Solvers• SIMPLE, SIMPLEC, PISO, ...

(cont’d)

8th Lecture 31

Navier-Stokes Solvers• Streamfunction Formulation

10th Lecture 32

Navier-Stokes Solvers• Discussion of Final Projects

13th Lecture 33

Navier-Stokes Solvers• Final Project:

office hours (TBD)

15th Lecture 34

Homework 4 Due

Navier-Stokes Solvers• Final Project:

office hours (TBD)

17th Lecture 35

Navier-Stokes Solvers• Final Project:

office hours (TBD)

20th

Scalo @ APS-DFD

22nd

THANKSGIVING

24th

THANKSGIVING

27th Lecture 36

Navier-Stokes Solvers• Final Project:

office hours (TBD)

29th Lecture 37 Dec 1st Lecture 38

4th Lecture 39 6th

Classes End: Dec 9Grades Due: Dec 19

8th Lecture 40

Final Project Due

11th Lecture 41 13th Lecture 42 15th Lecture 43

References

A. J. Chorin (1969). ‘On the convergence of discrete approximations to the Navier-Stokes equations’. Math. Comp. 23:341– 353.

J. Ferziger & M. Peric (2001). Computational Methods for Fluid Dynamics. Springer.

Harlow & Welch (1965). ‘Numerical calculation of time-dependent viscous incompressible flow of fluid with free surfaces’8(21).

J. Kim & P. Moin (1985). ‘Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations’. J. Comput.Phys. 59:308 – 323.

R. H. Pletcher, et al. (2011). Computational Fluid Mechanics and Heat Transfer. CRC Press.

5