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Finite volume method
Main article: Finite volume method
The finite volume method (FVM) is a common approach used in CFD codes.[citation needed] The governing equations are solved over discrete control volumes. Finite volume methods recast the governing partial differential equations (typically the Navier-Stokes equations) in a conservative form, and then discretize the new equation. This guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the form,
where is the vector of conserved variables, is the vector of fluxes (see Euler equations or Navier¨CStokes equations), is the volume of the control volume element, and is the surface area of the control volume element.
[edit]Finite element method
Main article: Finite element method
The finite element method (FEM) is used in structural analysis of solids, but is also applicable to fluids. However, the FEM formulation requires special care to ensure a conservative solution. The FEM formulation has been adapted for use with fluid dynamics governing equations.[citation needed] Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach[4] However, FEM can require more memory than FVM.[5]
In this method, a weighted residual equation is formed:
where is the equation residual at an element vertex , is the conservation equation expressed on an element basis, is the weight factor, and is the volume of the element.
[edit]Finite difference method
Main article: Finite difference method
The finite difference method (FDM) has historical importance[citation needed] and is simple to program. It is currently only used in few specialized codes.[citation needed] Modern finite difference codes make use of an embedded boundary for handling complex geometries, making these codes highly efficient and accurate.[citation needed] Other ways to handle geometries include use of overlapping grids, where the solution is interpolated across each grid.
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