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25.2 General Scalar Transport Equation: Discretization and Solution

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 $V$ 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 (25.2-1)

  $\rho$ = 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  25.2-1 is applied to each control volume, or cell, in the computational domain. The two-dimensional, triangular cell shown in Figure  25.2.1 is an example of such a control volume. Discretization of Equation  25.2-1 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 (25.2-2)

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

Where $\frac{\partial\rho\phi}{\partial t}\,V$ is defined in Section  25.3.2. The equations solved by FLUENT take the same general form as the one given above and apply readily to multi-dimensional, unstructured meshes composed of arbitrary polyhedra.

Figure 25.2.1: Control Volume Used to Illustrate Discretization of a Scalar Transport Equation

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