18.2 General Scalar Transport Equation: Discretization and Solution

ANSYS FLUENT uses a control-volume-based technique to convert a general scalar transport equation to an algebraic equation that can be solved numerically. This control volume technique consists of integrating the transport equation about each control volume, yielding a discrete equation that expresses the conservation law on a control-volume basis.

Discretization of the governing equations can be illustrated most easily by considering the unsteady conservation equation for transport of a scalar quantity . This is demonstrated by the following equation written in integral form for an arbitrary control volume as follows:

 (18.2-1)

 where = density = velocity vector (= in 2D) = surface area vector = diffusion coefficient for = gradient of (= in 2D) = source of per unit volume

Equation  18.2-1 is applied to each control volume, or cell, in the computational domain. The two-dimensional, triangular cell shown in Figure  18.2.1 is an example of such a control volume. Discretization of Equation  18.2-1 on a given cell yields

 (18.2-2)

 where = number of faces enclosing cell = value of convected through face = mass flux through the face = area of face , (= in 2D) = gradient of at face = cell volume

Where is defined in Section  18.3.2. The equations solved by ANSYS FLUENT take the same general form as the one given above and apply readily to multi-dimensional, unstructured meshes composed of arbitrary polyhedra.

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