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24.2 Discretization

FLUENT uses a control-volume-based technique to convert the governing equations to algebraic equations that can be solved numerically. This control volume technique consists of integrating the governing equations about each control volume, yielding discrete equations that conserve each quantity on a control-volume basis.

Discretization of the governing equations can be illustrated most easily by considering the steady-state 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:


\begin{displaymath} \oint \rho \phi \, {\vec v} \cdot d{\vec A} = \oint {\Gamma}_{\phi} \, \nabla \phi \cdot d{\vec A} + \int_V S_{\phi} \, dV \end{displaymath} (24.2-1)


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  24.2-1 is applied to each control volume, or cell, in the computational domain. The two-dimensional, triangular cell shown in Figure  24.2.1 is an example of such a control volume. Discretization of Equation  24.2-1 on a given cell yields


\begin{displaymath} \sum_f^{N_{\rm faces}} \rho_f {\vec v}_f \phi_f \cdot {\vec ... ...ma}_{\phi} \, (\nabla \phi)_n \cdot {\vec A}_f + S_{\phi} \, V \end{displaymath} (24.2-2)


where      
  $N_{\rm faces}$ = 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)
  $(\nabla \phi)_n$ = magnitude of $\nabla \phi$ normal to face $f$
  $V$ = cell volume

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 24.2.1: Control Volume Used to Illustrate Discretization of a Scalar Transport Equation
\begin{figure} \psfig{file=figures/fig-1d-controlvol.ps,height=1.75in} \end{figure}

By default, FLUENT stores discrete values of the scalar $\phi$ at the cell centers ( $c0$ and $c1$ in Figure  24.2.1). However, face values $\phi_f$ are required for the convection terms in Equation  24.2-2 and must be interpolated from the cell center values. This is accomplished using an upwind scheme.

Upwinding means that the face value $\phi_f$ is derived from quantities in the cell upstream, or ``upwind,'' relative to the direction of the normal velocity $v_n$ in Equation  24.2-2. FLUENT allows you to choose from several upwind schemes: first-order upwind, second-order upwind, power law, and QUICK. These schemes are described in Sections  24.2.1- 24.2.4.

The diffusion terms in Equation  24.2-2 are central-differenced and are always second-order accurate.




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