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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: [email protected] 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)
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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– 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
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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
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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)
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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.
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