ME614: COMPUTATIONAL FLUID DYNAMICS

Fall 2015, MWF 2:30 pm - 3:20 pm, ME2053

Instructor

Dr. Carlo ScaloAssistant Professor of Mechanical EngineeringRoom ME2195, ME BuildingWest Lafayette, IN 47907-2045Work: 765-496-0214, Mobile: 650-739-9506Email: scalo@purdue.eduO�ce Hours: by appointmentTeaching Assistant: Mr. Kukjin Kim, kim1625@purdue.edu

Prerequisites

Prerequisites for the course include basic knowledge of fluid mechanics, linear algebra, partial di↵erential equations andaverage programming skills. The use of Python is strongly recommended but not mandatory. The class content is structuredin 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) while providing exposure to the latestgeneration of high-level dynamic languages and version-control software. The course will cover the following topics:

1. Spatial & Temporal Discretizations2. Linear Advection & Di↵usion Equation3. Poisson and Heat Equations4. Navier-Stokes Solvers

with a focus on incompressible flow and turbulent simulations. Students will be expected to write their own completeNavier-Stokes solver from scratch as a final project.

Sample mesh (left) and flow visualization (right) from a transonic turbulent calculation of the flow around aMcDonnell-Douglas 30P/30N multi-body airfoil. Courtesy of Prof. Julien Bodart (Universite de Toulouse, ISAE, France)

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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• (25%) Homework 1: Spatial Discretization• (25%) Homework 2: Linear Advection & Di↵usion Equation• (25%) Homework 3: First Incompressible Navier-Stokes Solver• (20%) Final Project

Examples of source code will be provided in Python only. The use of Python is strongly recommended but not mandatory.Sharing of ideas on the homework assignments is encouraged but submissions need to be individual. Note that it is trivial tocheck whether parts of source code have been copied.

Textbooks

With the exception of programming tutorials, all of the lecture material will be explained at the blackboard to facilitate adynamic discussion. Some of the course material will be based on selected pages from the following textbooks:

• Ferziger, J., and M. Peric, Computational Methods for Fluid Dynamics, Third Edition, Springer, 2001• 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 Di↵erence and Spectral Methods for Ordinary and Partial Di↵erential Equations, unpublished

text, 1996, available at http://people.maths.ox.ac.uk/trefethen/pdetext.htmlThe first two will be the main reference textbooks for the course. The last two cover more theoretical and advanced topics.

Tentative Schedule

A tentative schedule is included below. The instructor reserves the right to (frequently) update it.

Monday Wednesday Friday

Aug 24th Lecture 1

Introduction• Course Structure Overview• Homework 0:

Python, Linux, Git• Initial Course Participation

26th Lecture 2

Principles of Discretization• Discrete Operators• Matrix Multiplication

Reading: review linear algebra (matrixmultiplications, eigenvalues, ...)

28th Lecture 3

Spatial Discretization• Polynomial Fitting• Taylor Expansion

Reading: review linear algebra;Pletcher, et al. (2011) pp. 43 – 75;Ferziger & Peric (2001) pp. 21 – 52.

31st Lecture 4

Spatial Discretization• Pade Approximants• Modified Wavenumber

Reading:Ferziger & Peric (2001) pp. 45 – 63;

Sep 2nd Lecture 5

Homework 0 Due

Spatial Discretization• Homework 1 overview

4th Lecture 6

Spatial Discretization• Python Session:“Best Practices in Python”

Reading:Python Tutorial, Sections 6, 7 and 8

7th

LABOR DAY

9th Lecture 7

Spatial Discretization• Python Session:Homework 1 Starter

Reading:Python Tutorial, Sections 2, 3, 4, and5

11th Lecture 8

Spatial Discretization• Grid Transformations (1D)• Boundary Conditions:

periodic vs non-periodic

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

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Monday Wednesday Friday

14th

NO CLASS

16th

NO CLASS

18th

NO CLASS

21st Lecture 9

Temporal Discretization• Explicit Euler & Upwind• Modified Equation

Reading:Pletcher et al. (2011) pp. 103 – 124;

23rd Lecture 10

Temporal Discretization• Fourier/Von Neumann Analysis• Implicit Euler, MacCormack,Adams-Bashforth, Leap Frog,Crank-Nicholson

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

25th Lecture 11

Homework 1 Due

Temporal Discretization• Runge-Kutta schemes

Reading:Handouts, Chapter 4Pletcher et al. (2011) pp. 124 – 125

28th Lecture 12

Temporal Discretization• �-roots

Reading:Handouts, Chapter 4

30th Lecture 13

Linear Advection & Diffusion• Homework 2 overview• Catching Up: Periodic vsnon-periodic boundaryconditions

Oct 2nd Lecture 14

Linear Advection & Diffusion• Python Session:Homework 2 Starter

5th Lecture 15

Poisson and Heat Equations• 2D spatial operators (DivGradoperator)

• Direct Methods

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

7th Lecture 16

Linear Systems of Equations• Iterative Methods: Jacobi,Gauss-Seidel, Line Relaxation

Reading:Handouts, Chapter 3Pletcher et al. (2011) pp. 152 – 162

9th Lecture 17

Linear Systems of Equations• Iterative Methods:Over-Relaxation, ADI,Multi-Grid

Reading:Handouts, Chapter 3Pletcher et al. (2011) pp. 152 – 162

12th

OCTOBER BREAK

14th Lecture 18

Linear Systems of Equations• Iterative Methods: Multi-Grid(cont’d), Conjugate Gradient

Reading:Handouts, Chapter 3Pletcher et al. (2011) pp. 166 – 175

16th Lecture 19

Homework 2 Due

Poisson and Heat Equations• Homework 3 overview (Part I)• Python Session: 2Darrays/operators, fast indexing,Homework 3 Starter

19th Lecture 20

Navier-Stokes Solvers• Incompressible Navier-Stokesequations: conservative vsnon-conservative form,Lagrangian derivative

21st Lecture 21

Navier-Stokes Solvers• Finite-Volume Approach,Staggered Variable Collocation,Discretization for continuity andpressure gradient

Reading: Harlow & Welch (1965)

23rd Lecture 22

Navier-Stokes Solvers• Suggested 2nd-orderdiscretization foradvection/di↵usion terms

26th

NO CLASS

28th

NO CLASS

30th Lecture 23

Navier-Stokes Solvers• Projection Method: FractionalStep Method

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

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Monday Wednesday Friday

Nov 2nd Lecture 24

Navier-Stokes Solvers• Algebraic Pressure Segregation

4th Lecture 25

Navier-Stokes Solvers• Vorticity-Streamfunction( � !) formulation (in 2D)

6th Lecture 26

Navier-Stokes Solvers• Boundary conditions in � !:solenoidal condition

9th Lecture 27

Navier-Stokes Solvers• Review Session

11th Lecture 28

Navier-Stokes Solvers• Semi-Implicit TimeAdvancement Methods

13th Lecture 29

Navier-Stokes Solvers• Semi-Implicit TimeAdvancement Methods (cont’d)

16th Lecture 30

Homework 3 Due

Navier-Stokes Solvers• Discussion of Final Project

18th Lecture 31

Navier-Stokes Solvers• Boundary conditions forvelocity-pressure formulation

• Mass conservation in boundarylayers

Reading: : Orlanski (1976),Piomelli & Scalo (2010)

20th Lecture 32

Navier-Stokes Solvers• Pseudo-spectral methods:introduction to DFT

Reading: : Pope (2000), Section 6.4;Ferziger & Peric (2001), Section 3.10

23rd

NO CLASS

25th

NO CLASS

27th

NO CLASS

30th Lecture 33

Navier-Stokes Solvers• Pseudo-spectral methods(cont’d)

• De-aliasing

Dec 2nd Lecture 34

Navier-Stokes Solvers• Python Session: Advectiondi↵usion equation with DFT

4th Lecture 35

Navier-Stokes Solvers• Final Project:o�ce hours (2:00 - 3:30 pm)

7th Lecture 36

Navier-Stokes Solvers• Final Project:o�ce hours (2:00 - 3:30 pm)

9th Lecture 37

Navier-Stokes Solvers• Final Project:o�ce hours (2:00 - 3:30 pm)

11th Lecture 38

14th Lecture 39 16th Lecture 40 18th Lecture 41

Final Project Due

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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.

F. Ham, et al. (2002). ‘A fully conservative second-order finite di↵erence scheme for incompressible flow on nonuniform grids’.J. Comput. Physics 177(1):117–133.

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.

I. Orlanski (1976). Journal of Computational Physics 21:251 – 269.

U. Piomelli & C. Scalo (2010). ‘Subgrid-scale modelling in relaminarizing flows’. Fluid Dynamics Research 42(4):045510.

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

S. Pope (2000). Turbulent flows. Cambridge Univ Pr.

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