ANSYS FLUENT 12.0 Theory Guide - 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 $\phi$ . This is demonstrated by the following equation written in integral form for an arbitrary control volume

as follows:

$\int_V \frac{\partial\rho\phi}{\partial t}\,dV + \oint \rho ... ..._{\phi} \, \nabla \phi \cdot d{\vec A} + \int_V S_{\phi} \, dV$

(18.2-1)

where
	$\rho$	=	density
	${\vec v}$	=	velocity vector (= $u \,\hat{\imath} + v \,\hat{\jmath}$ in 2D)
	${\vec A}$	=	surface area vector
	${\Gamma}_{\phi}$	=	diffusion coefficient for $\phi$
	$\nabla \phi$	=	gradient of $\phi$ (= $\partial\phi/\partial x) \,\hat{\imath} + (\partial\phi/\partial y) \,\hat{\jmath}$ in 2D)
	$S_{\phi}$	=	source of $\phi$ 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

$\frac{\partial\rho\phi}{\partial t}\,V + \sum_f^{N_{\rm face... ...amma}_{\phi} \, \nabla \phi_f \cdot {\vec A}_f + S_{\phi} \, V$

(18.2-2)

where
	$N_{\rm faces}$	=	number of faces enclosing cell
	$\phi_f$	=	value of $\phi$ convected through face
	$\rho_f {\vec v}_f \cdot {\vec A}_f$	=	mass flux through the face
	${\vec A}_f$	=	area of face , $\left\vert A\right\vert$ (= $\left\vert A_x \hat{\imath} + A_y \hat{\jmath} \right\vert$ in 2D)
	$\nabla \phi_f$	=	gradient of $\phi$ at face
		=	cell volume

Where $\frac{\partial\rho\phi}{\partial t}\,V$ 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.

**Figure 18.2.1:** Control Volume Used to Illustrate Discretization of a Scalar Transport Equation