05-solv

05-solv - Lecture 5 Solution Methods Applied Computational...

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1 Lecture 5 - Solution Methods Applied Computational Fluid Dynamics Instructor: André Bakker © André Bakker (2002-2006) © Fluent Inc. (2002)

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2 Solution methods Focus on finite volume method. Background of finite volume method. Discretization example. General solution method. Convergence. Accuracy and numerical diffusion. Pressure velocity coupling. Segregated versus coupled solver methods. Multigrid solver. Summary.
3 Overview of numerical methods Many CFD techniques exist. The most common in commercially available CFD programs are: The finite volume method has the broadest applicability (~80%). Finite element (~15%). Here we will focus on the finite volume method. There are certainly many other approaches (5%), including: Finite difference. Finite element. Spectral methods. Boundary element. Vorticity based methods. Lattice gas/lattice Boltzmann. And more!

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4 Finite difference method (FDM) Historically, the oldest of the three. Techniques published as early as 1910 by L. F. Richardson. Seminal paper by Courant, Fredrichson and Lewy (1928) derived stability criteria for explicit time stepping. First ever numerical solution: flow over a circular cylinder by Thom (1933). Scientific American article by Harlow and Fromm (1965) clearly and publicly expresses the idea of “computer experiments” for the first time and CFD is born!! Advantage: easy to implement. Disadvantages: restricted to simple grids and does not conserve momentum, energy, and mass on coarse grids.
5 The domain is discretized into a series of grid points. A “structured” (ijk) mesh is required. The governing equations (in differential form) are discretized (converted to algebraic form). First and second derivatives are approximated by truncated Taylor series expansions. The resulting set of linear algebraic equations is solved either iteratively or simultaneously. i j i j Finite difference: basic methodology

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6 coextrusion metal insert contours of velocity magnitude Earliest use was by Courant (1943) for solving a torsion problem. Clough (1960) gave the method its name. Method was refined greatly in the 60’s and 70’s, mostly for analyzing structural mechanics problem. FEM analysis of fluid flow was developed in the mid- to late 70’s. Advantages: highest accuracy on coarse grids. Excellent for diffusion dominated problems (viscous flow) and viscous, free surface problems. Disadvantages: slow for large problems and not well suited for turbulent flow. Finite element method (FEM)
7 First well-documented use was by Evans and Harlow (1957) at Los Alamos and Gentry, Martin and Daley (1966). Was attractive because while variables may not be continuously differentiable across shocks and other discontinuities mass, momentum and energy are always conserved.

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