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points into an element, and the number of nodes depends on the type of
element. The number of nodes in each element depends not only on the
number of angles in the element, but also on the type of element
interpolation function. After grid formation, the next step is to choose
interpolation functions that describe the variation of the fi eld variables
over the element. These functions are usually polynomial, because they
can be easily integrated or differentiated. The element equations can be
assembled into a system of equations, with the solution being the
unknown variables at grid points. The most commonly used method for
discretization of differential equations is Galerkin's method of weighted
residuals (Sayma, 2009).
The
fi nite volume method
was developed in the 1970s. This method of
discretization uses the integral forms of the Navier-Stokes and Euler's
equations. The solution domain is divided into control volumes, and the
integral forms of the equations are applied for each volume separately.
The center of control volume, in which fl ow variables are sought, can be
placed in the center of the grid cell when the control volume corresponds
to grid cell, or control volume can be centered on the grid node
(Figure 7.3). The values of variables at control volume boundaries are
determined by interpolation from the values at the centers. The main
advantage of this method is fl exibility. It can be applied both in the case
of structured and unstructured networks, making it suitable for fl ow
analysis in cases of complex geometry (Blazek, 2005; Sayma, 2009).
Illustration of: (a) cell-centered; and (b) node-centered
control volume
Figure 7.3
7.2.3 Numerical grids
Grid generation involves division of physical space into a large number
of geometrical elements, such as grid cells, that are formed by connecting
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