2.29 | Spring 2015 | Graduate

Numerical Fluid Mechanics


Session Key

L = Lecture
R = Recitation
Q = Quiz

L1 Introduction to numerical fluid mechanics: Models to simulations, error types. Approximation and round-off errors. Number representations. Errors of numerical operations.  
R1 Introduction to MATLAB®  
L2 Recursion. Truncation errors, Taylor series and error analysis. Error propagation and estimation. Condition numbers. Roots of non-linear equations. Introduction and bracketing methods. Problem Set 1 posted
L3 Roots of non-linear equations, bracketing methods: bisection / false position. Open methods: Open-point iteration / Newton-Raphson / secant methods, extension to systems of equations.  
R2 Review: Navier-Stokes equations and their approximations. Conservation laws, material derivative, Reynolds transport theorem, constitutive equations.  
L4 Systems of linear equations: Motivations and plans, direct methods, Gauss elimination.  
L5 Systems of linear equations. Gaussian elimination (special cases, multiple right hand sides). LU decomposition and factorization, pivoting. Error analysis for linear systems. Operations counts.

Problem Set 1 due

Problem Set 2 posted

R3 End of Navier-Stokes review (if needed). Compressible and incompressible flows, vorticity, Euler’s equations, potential flows and (boundary) integral equations.  
L6 Systems of linear equations. Special matrices: LU decompositions, tridiagonal systems, general banded matrices, symmetric, positive-definite matrices. Introduction to iterative methods.  
L7 Systems of linear equations. Iterative methods: Jacobi method, Gauss-Seidel iteration, convergence, successive over-relaxation methods, gradient methods, stop criteria, examples.  
R4 Recitation  
L8 End of systems of linear equations: Gradient methods, preconditioning. Krylov methods. Finite-differences (FD): Classification of partial differential equations (PDEs) and examples, error types and discretization properties.  
L9 FD schemes: Finite difference based on Taylor series for higher order accuracy differences and examples. Taylor tables or method of undetermined coefficients.

Problem Set 2 due

Problem Set 3 posted

R5 Recitation  
Q1 Quiz 1  
L10 FD schemes: Polynomial approximations (Newton, Lagrange, Hermite, and Pade schemes), iterative improvements and extrapolations, boundary conditions, non-uniform grids, grid refinement.  
R6 Recitation  
L11 Finite-differences: Fourier error analysis, introduction to stability: heuristic, energy and Von Neumann methods, hyperbolic PDEs, characteristics.  
L12 Stability, hyperbolic equations. Revisited, Courant–Friedrichs–Lewy (CFL) condition and Von Neumann stability, elliptic equations revisited and FD schemes.

Problem Set 3 due

Problem Set 4 posted

R7 Recitation  
L13 End of elliptic / hyperbolic equations, special advection schemes (donor cell, flux-corrected transport, weighted essentially non-oscillatory (WENO)), parabolic equations revisited and numerical FD schemes.  
L14 Finite volume methods  
R8 Recitation: Finite volume methods (cont.)  
L15 Methods for unsteady problems. Time-marching methods. Ordinary differential equations (ODEs). Initial value problems (IVPs). Euler’s method. Runge-Kutta methods.  
L16 Time-marching (cont.): Higher order ODEs, stiffness and multistep methods. Solutions of the Navier-Stokes equation, incompressible and compressible.

Problem Set 4 due

Problem Set 5 posted

R9 Recitation  
L17 Solutions of the Navier-Stokes equation: Incompressible and compressible. Pressure-correction, fractional-step.  
Q2 Quiz 2  
R10 Recitation  
L18 Solutions of the Navier-Stokes equation: Incompressible and compressible. vorticity, artificial compressibility and other methods.  
R11 Recitation  
L19 Grid generation and complex geometries.

Problem Set 5 due

Problem Set 6 posted

L20 Finite volume on complex geometries. Finite element methods: Introduction, fluid applications.  
R12 Recitation  
L21 Finite element methods (cont.): Continuous and discontinuous Galerkin Methods. Spectral methods.  
L22 Inviscid flow equations: Boundary element methods. Panel methods. Problem Set 6 due
R13 Recitation: Special topics, boundary layer equations, ODEs. Boundary value problems.  
L23 Turbulent flows: Models and numerical simulations.  
  Final project presentations.  

Course Info

As Taught In
Spring 2015
Learning Resource Types
Lecture Notes
Programming Assignments