Computational Fluid Dynamics (CFD)

24-718, Fall 2020

Satbir Singh

Simulation of laminar flow around a cylinder

Lecture:

Time: Tuesday and Thursday, 11:40 AM - 1:00 PM
Location: TBA

Instructor Office Hours:

Time: Sunday, 5:00 - 6:00 PM
Location: TBA

TA Office Hours:

Time: Mondays and Wednesdays, 6:30 - 7:30 PM
Location: TBA

Course Description:

Many real-world engineering problems are governed by mathematical equations that cannot be solved using analytical methods. For example, Navier-Stokes (NS) equations that govern the transport of mass, momentum, and energy can only be solved using computers and numerical methods. This course emphasizes numerical methods for solving fluid dynamics equations, hence the name, computational fluid dynamics (CFD). Students will be introduced to many commonly used techniques to find numerical solutions to fluid dynamics equations. Negative and positive aspects of each technique will be compared using mathematical and computational analyses. Each student will write their own computer program to implement numerical algorithms derived in the class. Students can use either MATLAB or C/C++ as programming languages. Once the computer codes are developed, they will be utilized to investigate the physical behavior of the underlying system and the governing differential equations.

Prerequisites:

Undergraduate level knolwege of Numerical Methods and Fluid Mechanics

Reference Books:

Computational Fluid Mechanics and Heat Transfer; by Anderson, Tannehill and Pletcher

Computational Methods for Fluid Dynamics; by Ferziger and Peric

Numerical Heat Transfer and Fluid Flow; by S.V. Patankar

Grading:

Homework (65%)

Project (25%)

Midterm exam (10%)

Tentative Syllabus Outline:

---------------------------------------------------------------------

Introduction to Computational Fluid Dynamics (CFD) ---------------------------------------------------------------------
Week 1	The need for CFD, applications, historic perspective, different methods for CFD, state-of-the-art, challenges, future directions Introduction to Navier-Stokes (NS) equations, physical and mathematical classification of PDEs, system of equations, some key PDEs of interest in CFD
--------------------------------------------------------------------- Basics of Finite Difference Method (FDM)) ---------------------------------------------------------------------
Week 2	Numerical approximation of derivatives using Taylor series, order-of-accuracy, modified wavenumber analysis of numerical derivatives, finite difference representation of a PDE, truncation error, consistency, stability
--------------------------------------------------------------------- FDM for Parabolic PDEs ---------------------------------------------------------------------
Week 3	Numerical solution of 1-D transient diffusion equation, explicit methods, modified wavenumber analysis, von Neumann stability analysis, modified equation method for accuracy and consistency, DuFort Frankel method, implicit methods, types of boundary conditions and their implementation
Week 4	Crank-Nicolson method, 2-D and 3-D transient diffusion equations, approximate factorization and alternate direction implicit (ADI) methods for computational efficiency
--------------------------------------------------------------------- FDM for Hyperbolic PDEs ---------------------------------------------------------------------
Week 5	Numerical solution of 1-D advection equation, upwind explicit and FTCS implicit methods, Courant condition, von Neumann stability analysis, amplitude and phase errors
Week 6	Lax-Friedrichs method, Lax-Wendroff method, trapezoidal method, boundary conditions, linear Burger's equation, advection-diffusion equation, matrix structure for implicit method in 2-D
Week 7	Review for Midterm exam Midterm exam
--------------------------------------------------------------------- FDM for Elliptic Partial Differential Equations ---------------------------------------------------------------------
Week 8	Some common elliptic PDEs, solution using direct methods such as Gauss elimination, iterative methods: point Jacobi, Gauss Seidel, SOR, boundary conditions Line-SOR, method of steepest descent, multigrid acceleration
--------------------------------------------------------------------- FDM for Navier-Stokes (NS) Equations ---------------------------------------------------------------------
Week 9	Derivation of mass, momentum and energy equations, macroscopic and microscopic views Conservative vs non-conservative forms of NS equations, some other simple fluids equations, FDM for incompressible NS equations
Week 10	FDM for incompressible NS equations, vorticity-stream function formulation for 2-D incompressible flows
Week 11	FDM for incompressible NS equations, premitive variable formulation
Week 12	Introduction to Direct Numerical Simulations (DNS), Reynolds-Averaged Navier Stokes (RANS) and large-eddy simulation (LES) techniques for modeling of incompressible turbulent flows, introduction to numerical methods for compressible flows
--------------------------------------------------------------------- Unstructured Grids and Finite Volume Method (FVM) ---------------------------------------------------------------------
Week 13	Introduction to complex geometry and grids, need for unstructured grids, grid generation and storage of grid connectivity, implimentation of boundary conditions, conversion of PDE into numerical equations using FVM
--------------------------------------------------------------------- Project Office Hours ---------------------------------------------------------------------
Week 14	In-class help with projects
--------------------------------------------------------------------- Project Presentations and Submission of Project Reports ---------------------------------------------------------------------
Week 15	15-20 minutes presentation to the class