30 27e 27p

Such higher order approximation for the gradient makes sense, only if the integrals are also approximated using higher order approximations. For most complex flow simulations, second-order approximation of the gradient (given by Eq. (6.8)) is considered satisfactory.

Velocity and other variables at cell faces can be obtained by employing suitable interpolation practices. Numerous alternative interpolation methods have been developed. Generically, a value of general variable <p at the cell face can be expressed in terms of two neighboring nodes and one additional upstream node (the need to include an additional upstream node is discussed later in this chapter). For example, the value of <p at cell face e can be written:

{A0p + &0E + fc&w v Ue> 0 0e + Mp + A0EE v Ue< 0 (6a0)

where j3 are coefficients dependent on the interpolation method (refer to Fig. 6.2 to clarify the notation of node points), which obey the following restriction:

Substitution of gradient terms (Eq. (6.8)), interpolated values and the linearized source term in Eq. (6.6) gives the discretized form of Eq. (6.6) for a uniform grid:

ap 0p = aE 0e + aw 4W + aww 0WW + «N^N + as0s + assess + SC (6.12) where, aE = De- (ftFe) (6.13)

ap = aE + aw + aN + as + aww + ass - Sp + (Fe - Fw + Fn - Fs) (6.19)

Ax Ay

When deriving these expressions, it was assumed that velocity at all the cell faces is positive. In other cases, suitable modifications to include appropriate upstream nodes (in place of pww and pSS) should be made. It can be seen that the continuity equation indicates that the last term inside the bracket of Eq. (6.19) will always be zero for constant density flows. The behavior of numerical methods depends on the source term linearization employed and interpolation practices. Before these practices are discussed, a brief discussion of the desired characteristics of discretization methods will be useful. The most important properties of the discretization method are:

• Conservativeness: To ensure the overall conservation of p, the flux of p leaving a CV across a given face, must be equal to the flux of p entering the adjacent CV through the same face. Therefore, the flux through the common face must be represented by one and the same expression in adjacent CVs.

• Boundedness: Numerical solution methods must respect the physically consistent bounds on variable values (bounded by minimum and maximum boundary values when there is no source). An essential requirement of boundedness is that all the coefficients of the discretized equation should be of the same sign and (usually) positive. If this condition is not satisfied, it is possible to observe unphysical 'wiggles' in the solution. It can be seen from Eqs. (6.13) to (6.18) that some of the coefficients may become negative if values of P are not chosen carefully. For example, aE can become negative if Fe > (De/p2). It must also be noted that source term linearization practices should ensure that Spp is always negative in order to possess the boundedness property (otherwise the value of ap may become negative). Diagonal dominance of the discretized equations is a desirable feature for satisfying the 'boundedness' criterion. Scarborough (1958) gave a sufficient condition for diagonally dominant set of equations as:

Diagonal dominance and all positive coefficients ensure boundedness. Special procedures are invoked to ensure the boundedness of many higher order schemes, which otherwise, may produce wiggles and unbounded solutions. Some of these methods are discussed in the following. Transportiveness: Transportiveness can be illustrated by considering the distribution of p in the vicinity of its source. The contours of constant p are shown in Fig. 6.3 for different values of Peclet number (ratio of strengths of convection and diffusion, Pe = F/D). For a process with zero Peclet number (pure diffusion), contours of constant p are circular and therefore conditions

Pe = 0, only diffusion

Pe = 0, only diffusion

FIGURE 6.3 Distribution of <p around its source. (from Versteeg and Malalasekara, 1995. Printed with permission from the Publishers.)

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