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: [email protected]�ce Hours: by appointmentTeaching Assistant: Mr. Kukjin Kim, [email protected]
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;
2
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)
3
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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