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.eduOffice Hours: by appointmentTeaching Assistant: Mr. Kukjin Kim, kim1625@purdue.edu

Prerequisites

Prerequisites for the course include basic knowledge of fluid mechanics, linear algebra, partial differential 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 & Diffusion 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 & Diffusion 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.Collaboration on the homework assignment 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 Difference and Spectral Methods for Ordinary and Partial Differential 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

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• Pade Approximants (cont’d)• Homework 1 overview

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

4th Lecture 6

Spatial Discretization• Python Session:

Homework 1 Starter

Reading:Python Tutorial, Sections 2,3,4, and 5

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Monday Wednesday Friday7th

LABOR DAY

9th Lecture 7

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;

11th Lecture 8

Temporal Discretization• Explicit Euler & Upwind• Modified Equation

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

14th Lecture 9

Spatial Discretization• Python Session:

“Best Practices in Python”

Reading:Python Tutorial, Sections 6,7 and 8

16th

NO CLASS

18th

NO CLASS

21st 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

23rd Lecture 11

Homework 1 Due

Temporal Discretization• Runge-Kutta schemes

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

25th Lecture 12

Temporal Discretization• σ-roots

Reading:Handouts, Chapter 4

28th Lecture 13

Linear Advection & Diffusion• Homework 2 overview

30th Lecture 14

Linear Advection & Diffusion• Python Session:

Homework 2 Starter

Oct 2nd Lecture 15

Poisson and Heat Equations• 2D spatial operators (DivGrad

operator)• Direct Methods

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

5th Lecture 16

Linear Systems of Equations• Iterative Methods: Jacobi,

Gauss-Seidel, Line Relaxation

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

7th Lecture 17

Linear Systems of Equations• Iterative Methods:

Over-Relaxation, ADI,Multi-Grid

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

9th Lecture 18

Linear Systems of Equations• Iterative Methods: Multi-Grid

(cont’d), Conjugate Gradient

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

12th

October Break

14th Lecture 19

Poisson and Heat Equations• Homework 3 overview (Part I)• Python Session: 2D

arrays/operators, fast indexing,Homework 3 Starter

16th Lecture 20

Homework 2 Due

Poisson and Heat Equations• Python Session

(cont’d)

19th Lecture 21

Navier-Stokes Solvers• A traumatic introduction to the

incompressible Navier-Stokes(MoureauBP JCP 2007) insteadof Ham JCP 2002

21st Lecture 22

Navier-Stokes Solvers• Incompressible Navier-Stokes

equations: conservative vsnon-conservative form,Lagrangian derivative

23rd Lecture 23

Navier-Stokes Solvers• Finite-Volume Approach,

Staggered Variable Collocation

Reading: Harlow & Welch (1965)

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Monday Wednesday Friday26th

NO CLASS

28th

NO CLASS

30th Lecture 24

Navier-Stokes Solvers• Projection Method: Fractional

Step Method

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

Nov 2nd Lecture 25

Navier-Stokes Solvers• Algebraic Pressure Segregation

4th Lecture 26

Navier-Stokes Solvers• Semi-Implicit Methods• suggested 2nd-order

discretization foradvection/diffusion terms

6th Lecture 27

Navier-Stokes Solvers• Semi-Implicit Time

Advancement, prediction step:Explicit-Euler, Runge-Kutta

9th Lecture 28

Homework 3 Due

Navier-Stokes Solvers• Kinetic energy conservation

properties of the incompressibleNavier-Stokes equations

11th Lecture 29

Navier-Stokes Solvers• Discrete kinetic energy

conservation, review of Ham,et al. (2002)

13th Lecture 30

Navier-Stokes Solvers• Homework 3 tutorial

16th Lecture 31

Navier-Stokes Solvers• Vorticity-Streamfunction

(Ψ − ω) formulation (in 2D)

18th Lecture 32

Navier-Stokes Solvers• Boundary conditions in Ψ − ω:

solenoidal condition

20th Lecture 33

Navier-Stokes Solvers• Boundary conditions for

velocity-pressure formulation• Mass conservation in boundary

layers

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

23rd

NO CLASS

25th Lecture 34

Navier-Stokes Solvers• Discussion of Final Project

27th Lecture 35

Navier-Stokes Solvers• Pseudo-spectral methods:

introduction to DFT

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

30th Lecture 36

Navier-Stokes Solvers• Pseudo-spectral methods

(cont’d)• De-aliasing

Dec 2nd Lecture 37

Navier-Stokes Solvers• Python Session: Advection

diffusion equation and Poissonequation with FFT

4th Lecture 38

Navier-Stokes Solvers• Final Project:

office hours (2:00 - 3:30 pm)

7th Lecture 39

Navier-Stokes Solvers• Final Project:

office hours (2:00 - 3:30 pm)

9th Lecture 40

Navier-Stokes Solvers• Final Project:

office hours (2:00 - 3:30 pm)

11th Lecture 41

Final Project Due

Grades Due:December 22

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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 difference 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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