1. MECH/BIOE/CEVE 454/554 COMPUTATIONAL FLUID MECHANICS
Fall 2016
Instructor: Tayfun E. Tezduyar Class Hours: 9:25–10:40 TR
236 MEB 713-348-6051 Room: MEB 128
tezduyar@rice.edu
Office hours: 2:00-3:00 TR
TA: Aaron Hartman [Aaron.Hartmann@tafsm.org] (554)
Izzet Sahin [izzet.sahin@tafsm.org] (554)
Reha Avsar [Reha.Avsar@tafsm.org] (454)
Mingyuan Fu [woshifumingyuan@gmail.com] (454)
Ryon Lab 230
TA hours: Only for questions related to already-graded homework and tests. To meet the TAs
for that, 554 students should send email to Aaron.Hartmann@tafsm.org, with copy to
izzet.sahin@tafsm.org, and 454 students to Reha.Avsar@tafsm.org, with copy to
woshifumingyuan@gmail.com.
Grading: 454 554
30% Homework 20% Homework
30% Test 1 40% Test 1
40% Test 2 40% Test 2
Tests: Closed book, Closed notes. No makeup exams (exception: see General
Announcements, paragraph on "Excused Absences").
Homework: Homework is due at the beginning of class on the date due. No late homework
will be accepted (exception: see General Announcements, paragraph on "Excused
Absences"). Homework should show on the first page your name, the course
number and homework set number. Homework should be solely your work.
Attendance: Please be on time. Students are responsible for all information disseminated and
material distributed in class.
Classroom behavior:
Once the instructor is in the classroom, please avoid talking to your classmates until he steps out.
Please avoid any other behavior that might distract the instructor or your classmates. If you have any
questions, ask the instructor; if you have any comments, direct them to the instructor.
Required: Class notes: http://www.tafsm.org/MECH454/
Notes for Lecture Series on Computational Fluid Mechanics
Introduction to the Finite Element Method
You should bring these notes to class for at least two reasons. 1. As the instructor
explains the material in them, you need to add your own notes to those pages.
2. The instructor will quite often refer to notes covered in earlier lectures.
Reference textbook: Y. Bazilevs, K. Takizawa and T.E. Tezduyar, Computational Fluid–Structure
Interaction, Wiley, ISBN 978-0-470-97877-1, 2013

2. MECH/BIOE/CEVE 454/554
COMPUTATIONAL FLUID MECHANICS
Instructor: Tayfun Tezduyar
COURSE OUTLINE:
Introductory and Quick-Use Knowledge in Computational Fluid Mechanics
Lecture Series on Computational Fluid Mechanics Chapter 1
Governing Equations and Boundary Conditions
A brief review of the governing equations of unsteady incompressible flows and the time-dependent
advection-diffusion equation; introductory scale analysis; boundary conditions and computational
boundary conditions
1a. Navier-Stokes Equations of Incompressible Flows and Advection-Diffusion Equation
Understanding the momentum equation and the incompressibility constraint; stress tensor;
Newtonian fluids; advection-diffusion equation
1b. Scale Analysis and Boundary Conditions
Peclet and Reynolds numbers; boundary layer concept; boundary conditions; computational
boundaries and boundary conditions
Lecture Series on Computational Fluid Mechanics Chapter 2
Spatial Discretization
Understanding the concept of spatial discretization for an advection-diffusion equation; how to
discretize the governing equations to obtain algebraic equations or ordinary differential equations
2a. Spatial Discretization with the Finite Difference Method
One-dimensional finite differences; multi-dimensional finite differences
2b. Spatial Discretization with the Finite Element Method
Finite element shape (interpolation) functions; elements; test function; weighted residual
formulation; global equation system; element-level vectors and matrices; assembly of the
element-level vectors and matrices
Lecture Series on Computational Fluid Mechanics Chapter 3
Time-Integration and Related Solution Techniques
A review of time-integration in the context of finite element methods; stability and accuracy;
nonlinear solution techniques; iterative solution of linear equation systems
3a. Time-Integration Techniques
Time-integration of the ordinary differential equations obtained after spatial discretization;
explicit and implicit methods; predictor/multi-corrector method
3b. Nonlinear Solution Techniques and Iterative Solution of Linear Equations
Newton-Raphson method; iterative solution techniques for linear equation systems; how to
compute the residual of the linear equation system; preconditioning; how to update the solution
vector after each iteration

3. Lecture Series on Computational Fluid Mechanics Chapter 4
Spatial Discretization for Incompressible Flows
Finite element discretization of the Navier-Stokes equations of incompressible flows; weighted
residual formulation; element-level vectors and matrices
4a. Finite Element Discretization of the Navier-Stokes Equations of Incompressible Flows
Weak formulation and finite element spatial discretization; obtaining the nonlinear ordinary
differential equation system corresponding to the momentum equation and the algebraic
constraint equation system corresponding to the incompressibility constraint
4b. Element-Level Vectors and Matrices for the Navier-Stokes Equations of Incompressible
Flows
Derivation of the element-level vectors and matrices; element-level matrices for nonlinear
terms; assembly of element-level vectors and matrices
Introduction to the Finite Element Method
Detailed coverage of topics in Finite Elements in Fluids Chapters 1-3
PLUS: Element types; concepts and measures of mesh refinement
Numerical surface and volume integration
Lecture Series on Computational Fluid Mechanics Chapter 5
Stabilized Formulations (for Advection-Diffusion Equation)
Understanding the stabilized formulations in the context of the time-dependent advection diffusion
equation; the streamline-upwind/Petrov-Galerkin (SUPG) stabilization; extension to incompressible
flows; pressure-stabilizing/Petrov-Galerkin (PSPG) formulation
5a. Stabilized Formulation for the Advection-Diffusion Equation
Element Peclet number; element Reynolds number; SUPG stabilization in 1D; stabilization
parameter; SUPG stabilization in 2D and 3D
5b. Stabilized Formulation for the Navier-Stokes Equations
SUPG stabilization; PSPG stabilization; Galerkin/Least-Squares (GLS) stabilization and its
relationship to SUPG and PSPG stabilizations
Lecture Series on Computational Fluid Mechanics Chapter 6
Computational Aspects of the Stabilized Formulations
The nonlinear ordinary differential equation system corresponding to the momentum equation and
the algebraic constraint equation corresponding to the incompressibility constraint; derivation of the
element-level vectors and matrices
6a. Semi-Discrete Forms Corresponding to the Stabilized Formulations
Semi-discrete forms; derivation of the element-level vectors; simplifications
6b. Fully Discrete Forms Corresponding to the Stabilized Formulations
Time-discretization; derivation of the element-level matrices; simplifications
ADDITIONAL TOPICS:
Methods for flows with moving boundaries and interfaces (two-fluid or free-surface flows and
fluid-structure interactions)
Comments on large-scale computing, supercomputers, parallel processing, and visualization.