Metodos Numericos

CFD Inside – Finite Volume Method Applied CFD
Santiago Laín Beatove

Adapted from material of A. Bakker “Applied CFD” & An Introduction to CFD of S. Laín

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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 willfocus 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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Finite difference method (FDM)
• Historically, the oldest of the three. • Techniques published as early as 1910 by L. F. Richardson. • Seminalpaper 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 tosimple grids and does not conserve momentum, energy, and mass on coarse grids.

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Finite difference: basic methodology
• The domain is discretized into a series of grid points. • A “structured” (ijk) mesh is required.

i j i j

• The governing equations (in differential form) are discretized (converted to algebraic form). • First and second derivatives are approximated by truncatedTaylor series expansions. • The resulting set of linear algebraic equations is solved either iteratively or simultaneously.

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Finite element method (FEM)
• 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 flowwas 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.

coextrusion metal insert contours of velocity magnitude
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Finite volume method (FVM)
• First well-documented use was by Evansand 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. • FVM enjoys an advantage in memory use and speed for very large problems, higher speed flows, turbulent flows, and source term dominated flows (like combustion). •Late 70’s, early 80’s saw development of body-fitted grids. By early 90’s, unstructured grid methods had appeared. • Advantages: basic FV control volume balance does not limit cell shape; mass, momentum, energy conserved even on coarse grids; efficient, iterative solvers well developed. • Disadvantages: false diffusion when simple numerics are used.

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Finite volume: basic methodology
•Divide the domain into control volumes.
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• Integrate the differential equation over the control volume and apply the divergence theorem. • To evaluate derivative terms, values at the control volume faces are needed: have to make an assumption about how the value varies. • Result is a set of linear algebraic equations: one for each control volume.• Solve iteratively or simultaneously.

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Cells and nodes
• Using finite volume method, the solution domain is subdivided into a finite number of small control volumes (cells) by a grid. • The grid defines the boundaries of the control volumes while the computational node lies at the center of the control volume. • The advantage of FVM is that the integral conservation is satisfied...