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2.29

Numerical Fluid Mechanics

4-0-8 units · Graduate

Introduction to numerical methods and MATLAB: errors, condition numbers and roots of equations. Navier-Stokes. Direct and iterative methods for linear systems. Finite differences for elliptic, parabolic and hyperbolic equations. Fourier decomposition, error analysis and stability. High-order and compact finite-differences. Finite volume methods. Time marching methods. Navier-Stokes solvers. Grid generation. Finite volumes on complex geometries. Finite element methods. Spectral methods. Boundary element and panel methods. Turbulent flows. Boundary layers. Lagrangian Coherent Structures. Includes a final research project. Students taking graduate version complete additional assignments.

Spring 2015

Topics

  • Introduction to numerical fluid mechanics: Models to simulations, error types. Approximation and round-off errors. Number representations. Errors of numerical operations.
  • Introduction to MATLAB ®
  • Recursion. Truncation errors, Taylor series and error analysis. Error propagation and estimation. Condition numbers. Roots of non-linear equations. Introduction and bracketing methods.
  • Roots of non-linear equations, bracketing methods: bisection / false position. Open methods: Open-point iteration / Newton-Raphson / secant methods, extension to systems of equations.
  • Review: Navier-Stokes equations and their approximations. Conservation laws, material derivative, Reynolds transport theorem, constitutive equations.
  • Systems of linear equations: Motivations and plans, direct methods, Gauss elimination.
  • Systems of linear equations. Gaussian elimination (special cases, multiple right hand sides). LU decomposition and factorization, pivoting. Error analysis for linear systems. Operations counts.
  • End of Navier-Stokes review (if needed). Compressible and incompressible flows, vorticity, Euler’s equations, potential flows and (boundary) integral equations.
  • Systems of linear equations. Special matrices: LU decompositions, tridiagonal systems, general banded matrices, symmetric, positive-definite matrices. Introduction to iterative methods.
  • Systems of linear equations. Iterative methods: Jacobi method, Gauss-Seidel iteration, convergence, successive over-relaxation methods, gradient methods, stop criteria, examples.
  • Recitation
  • 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.
  • FD schemes: Finite difference based on Taylor series for higher order accuracy differences and examples. Taylor tables or method of undetermined coefficients.
  • FD schemes: Polynomial approximations (Newton, Lagrange, Hermite, and Pade schemes), iterative improvements and extrapolations, boundary conditions, non-uniform grids, grid refinement.
  • Finite-differences: Fourier error analysis, introduction to stability: heuristic, energy and Von Neumann methods, hyperbolic PDEs, characteristics.
  • Stability, hyperbolic equations. Revisited, Courant–Friedrichs–Lewy (CFL) condition and Von Neumann stability, elliptic equations revisited and FD schemes.
  • 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.
  • Finite volume methods
  • Recitation: Finite volume methods (cont.)
  • Methods for unsteady problems. Time-marching methods. Ordinary differential equations (ODEs). Initial value problems (IVPs). Euler’s method. Runge-Kutta methods.
  • Time-marching (cont.): Higher order ODEs, stiffness and multistep methods. Solutions of the Navier-Stokes equation, incompressible and compressible.
  • Solutions of the Navier-Stokes equation: Incompressible and compressible. Pressure-correction, fractional-step.
  • Solutions of the Navier-Stokes equation: Incompressible and compressible. vorticity, artificial compressibility and other methods.
  • Grid generation and complex geometries.
  • Finite volume on complex geometries. Finite element methods: Introduction, fluid applications.
  • Finite element methods (cont.): Continuous and discontinuous Galerkin Methods. Spectral methods.
  • Inviscid flow equations: Boundary element methods. Panel methods.
  • Recitation: Special topics, boundary layer equations, ODEs. Boundary value problems.
  • Turbulent flows: Models and numerical simulations.
  • Final project presentations.

Readings

Spring 2003 · Numerical Marine Hydrodynamics (13.024)

Topics

  • Incompressible Fluid Mechanics Background
  • Some Useful Results from Calculus
  • An Application Using Complex Numbers
  • Root Finding
  • Curve Fitting and Interpolation
  • Numerical Differentiation
  • Numerical Integration
  • Numerical Intergration and Differential Equations
  • Some Examples of Numerical Errors
  • Panel Methods
  • Boundary Layers
  • Oscillating Rigid Objects

Prerequisites & course information

Subject meets with 2.290 Prereq: 18.075 and ( 2.006 , 2.016 , 2.06, 2.20 , or 2.25 ) G (Spring) 4-0-8 units

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