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7.2.2 Discretization of the fl ow
governing equations
In order to solve the system of differential equations representing the fl ow,
the fi rst step is to defi ne discrete points in space, called grid points or grid
nodes. These points are connected to form a numerical grid. Numerical
methods further convert the system of continuous differential equations
into a system of algebraic equations that represent the fl ow at the grid
points and interdependency of fl ow at those points and their neighboring
points. The values of the fl ow variables at the grid points are the unknowns
in a system of algebraic equations that have to be solved. The most
commonly used discretization methods are the fi nite difference method,
the fi nite element method, and the fi nite volume method. In unsteady fl ow,
when the solution at a discrete point varies with time, discretization of
time dimensions may also be needed (Blazek, 2005; Sayma, 2009).
The
fi nite difference method
is the simplest and among the fi rst methods
used to discretize the differential equations, and was introduced by Euler
in 1768. This method is applicable only in the case of a uniform,
structured grid, that is, numerical mesh having a high degree of regularity.
This method is based on the application of Taylor series expansions for
discretization of derivatives of the fl ow variables in differential equations.
If we assume that the dependent variable is a function of space coordinate
x, spatial discretization will be performed by dividing the spatial domain
into equal space intervals of Δx (Figure 7.1). The value of the dependent
Figure 7.1
Illustration of fi nite difference grid
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