ME614: COMPUTATIONAL FLUID DYNAMICS Spring 2017, MWF 2:30 pm - 3:20 pm, WANG2579 & WANG2563 Instructors Dr. Carlo Scalo (left) Assistant Professor of Mech. Engineering Room ME2195, ME Building Email: scalo@purdue.edu Mr. Danish Patel (middle) Email: patel472@purdue.edu Office Hours: Tuesday s, 1 PM 3 PM Venue: Outside WANG 4564 Mr. Kukjin Kim (right) Email: kim1625@purdue.edu Office Hours: Wednesday s, 3.30 PM 5.30 PM Venue: Outside WANG 4564 Prerequisites Prerequisites for the course include basic knowledge of fluid mechanics, linear algebra, partial differential equations and average (not beginner!) programming skills. The use of Python is strongly recommended but not mandatory. The class content 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 transfer applications, while providing exposure to the latest generation of high-level dynamic languages and version-control software. The course will cover the following topics: 1. Spatial & s 2. Linear Advection & Diffusion Equation 3. Poisson and Heat Equations 4. 5. Introduction to Compressible Flow Students will be expected to write their own complete Navier-Stokes solver from scratch as a final project. Density contours of supersonic flow past a cylinder confined in a duct (courtesy of Prof. Guido Lodato). 1
Grade Distribution Homework assignments and final reports turned in L A TEX and/or with supporting images generated in vector graphics are strongly 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 git, ssh and Linux (20%) Homework 1: fundamentals of local/global 1D discretization schemes on uniform/non-uniform grids (10%) HPC Homework: Introduction to MPI getting to grips with Purdue s supercomputing resources, first MPI code (20%) Homework 2: Linear Advection & Diffusion Equation compare different time advancement schemes, numerical stability (20%) Homework 3: Poisson Equation and Navier-Stokes Solver solve elliptical problems, compare iterative methods, first incompressible Stokes solver (25%) Final Project write a complete incompressible or compressible Navier-Stokes solver, pick one of the suggested final projects or propose one yourself Examples of source code will be provided in Python only. The use of Python is strongly recommended but not mandatory. Note that it is trivial to check whether parts of source code have been copied or shared. The grading scale for the course is: 97 score A+ 80 score < 83 B- 63 score < 67 D 93 score < 97 A 77 score < 80 C+ 60 score < 63 D- 90 score < 93 A- 73 score < 77 C score < 60 F 87 score < 90 B+ 70 score < 73 C- 83 score < 87 B 67 score < 70 D+ Policy Regarding Plagiarism Sharing of ideas on the homework assignments is encouraged but submitted reports and source codes need to be individually prepared. If established that two (or more) students have shared source code and/or parts of the write-up, a zero score will be given those assignments. The instructor will send a report to the Office of Dean of Students (ODOS) for every instance of plagiarism. Incidents reported to ODOS stay permanently on record. Textbooks With the exception of programming tutorials, all of the lecture material will be explained at the blackboard to facilitate a dynamic discussion. Some of the course material will be based on selected pages from the following textbooks: Ferziger, J., and M. Perić, 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.html The first two will be the main reference textbooks for the course. The last two cover more theoretical and advanced topics. 2
Tentative Schedule A tentative schedule is included below. The instructor reserves the right to (frequently) update it. Monday Wednesday Friday Jan 9th Lecture 1 Introduction Course Structure Overview Review of Syllabus Homework 0: Python, Linux, Git 16th MARTIN LUTHER KING JR. DAY 23rd Lecture 6 Homework 1 overview 11th Lecture 2 Introduction to Supercomputing HPC Session by Dr. Xiao Zhu (RCAC) 18th Lecture 4 Principles of Discretization Discrete Operators Matrix Multiplication review linear algebra (matrix multiplications, eigenvalues,...) 25th Lecture 7 Introduction to Supercomputing HPC Session by Dr. Xiao Zhu (RCAC) 13th Lecture 3 Python Session: Introduction to Python Python Tutorial, Sections 1 5 20th Lecture 5 Homework 0 Due Polynomial Fitting review linear algebra; Pletcher, et al. (2011) pp. 43 75; Ferziger & Perić (2001) pp. 21 52. 27th Lecture 8 Python Session: Homework 1 Starter Python Tutorial, Sections 6 8 30th Lecture 9 Taylor Expansion Boundary Conditions: periodic vs non-periodic review linear algebra; Pletcher et al. (2011) pp. 43 75; Ferziger & Perić (2001) pp. 21 52. 6th Lecture 12 Homework 1 Due Explicit Euler & Upwind Modified Equation Pletcher et al. (2011) pp. 103 124; 13th Lecture 15 Fourier/Von Neumann Analysis Implicit Euler, MacCormack, Adams-Bashforth, Leap Frog, Crank-Nicholson Pletcher et al. (2011) pp. 82 95 Feb 1st Lecture 10 Padè Approximants Splines Ferziger & Perić (2001) pp. 45 63; 8th Lecture 13 Introduction to Supercomputing HPC Session by Dr. Xiao Zhu (RCAC) 15th 3rd Lecture 11 Modified Wavenumber Spectral Discretization Pletcher et al. (2011) pp. 329 337; Ferziger & Perić (2001) pp. 47 58; 10th Lecture 14 Modified Equation (cont d) Round Off Error 17th Lecture 16 Runge-Kutta schemes Handouts, Chapter 4 Pletcher et al. (2011) pp. 124 125 3
Monday Wednesday Friday 22nd Lecture 18 HPC Homework Due 20th Lecture 17 Linear Advection & Diffusion Python Session: Homework 2 Starter 27th Lecture 20 Poisson and Heat Equations 2D spatial operators (DivGrad operator) Direct Methods Pletcher et al. (2011) pp. 147 152 6th Lecture 21 Linear Systems of Equations Iterative Methods: Jacobi, Gauss-Seidel, Line Relaxation Handouts, Chapter 3 Pletcher et al. (2011) pp. 152 162 13th SPRING BREAK Linear Advection & Diffusion Homework 2 overview Periodic vs non-periodic boundary conditions Mar 1st 8th Lecture 22 Linear Systems of Equations Iterative Methods: Over-Relaxation, ADI, Multi-Grid Handouts, Chapter 3 Pletcher et al. (2011) pp. 152 162 15th SPRING BREAK 24th Lecture 19 Non-uniform grid generation in 1D 3rd 10th Lecture 23 Homework 2 Due Linear Systems of Equations Iterative Methods: Multi-Grid (cont d), Conjugate Gradient Handouts, Chapter 3 Pletcher et al. (2011) pp. 166 175 17th SPRING BREAK 20th 27th Lecture 26 Finite-Volume Approach, Staggered Variable Collocation, Discretization for continuity and pressure gradient Harlow & Welch (1965) Apr 3rd 22nd Lecture 24 Incompressible Navier-Stokes equations: conservative vs non-conservative form, Lagrangian derivative 29th Lecture 27 Suggested 2 nd -order discretization for advection/diffusion terms Projection Method: Fractional Step Method Chorin (1969), Kim & Moin (1985) 5th Lecture 29 Boundary conditions in Ψ ω: solenoidal condition (Mr. Danish Patel) 24th Lecture 25 Poisson and Heat Equations Homework 3 overview (Part I) Python Session: 2D arrays/operators, fast indexing, Homework 3 Starter 31st Lecture 28 Vorticity-Streamfunction (Ψ ω) formulation (in 2D) 7th Lecture 30 Compressible flow solvers 1D Euler equations for compressible flow 4
10th Monday Wednesday Friday 12th Lecture 31 High-Order Block-Spectral Methods Dr. Jean-Baptiste Chapelier 17th Lecture 33 Pseudo-spectral methods (cont d) Python Session: Advection diffusion equation with DFT 24th Lecture 36 Fundamentals of Linear Acoustics (cont d) 19th Lecture 34 High-Order Block-Spectral Methods Dr. Jean-Baptiste Chapelier 26th 14th Lecture 32 Homework 3 Due Pseudo-spectral methods: introduction to DFT : Pope (2000), Section 6.4; Ferziger & Perić (2001), Section 3.10 21st Lecture 35 Fundamentals of Linear Acoustics 28th May 1st Lecture 37 3rd Lecture 38 5th Lecture 39 CLASSES END April 29 th 8th Lecture 40 Final Project Due (May 5, 2017) 10th Lecture 41 Grades Due: May 9 th 12th Lecture 42 5
References A. J. Chorin (1969). On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comp. 23:341 353. J. Ferziger & M. Perić (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. 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. 6