This elliptic pressure equation can be solved by the methods discussed earlier. It is important to note that the numerical approximations of this equation must be consistent with the approximations used in discretizing the original momentum and continuity equations. For example, the outer derivative of pressure in Eq. (6.36) comes from the continuity equation, while the inner derivative arises from momentum equations. These outer and inner derivatives must be discretized using the corresponding schemes used for discretizing the continuity and momentum equations, respectively. Violation of this constraint may lead to incorrect solution of the continuity equation. To maintain consistency, generally the pressure equation is derived directly from the discretized momentum and continuity equations rather than approximating Eq. (6.36). Several methods have been proposed to estimate the pressure field. The most widely used methods for incompressible flows, which are relevant to reactor engineering applications, are implicit or semi-implicit pressure correction methods. In these methods, pressure or pressure correction (or both) equations are derived from the discretized momentum and continuity equations, and used to enforce mass conservation at each iteration (or time step). Some of these methods are briefly discussed below.

One of the popular methods proposed by Patankar and Spalding (1972) is called SIMPLE (semi-implicit method for pressure linked equations). In this method, dis-cretized momentum equations are solved using the guessed pressure field. The discretized form of the momentum equations can be written:

where (5 / Sxi) indicates a discretized version of spatial derivative and * indicates the guess value or the value obtained from the previous iteration. VPi is the volume of CV around the node P. The velocity values obtained by solving these equations will not satisfy the continuity equation since the correct pressure field will not be known beforehand. In order to correct the fields obtained, SIMPLE proposes corrections of the form:

The discretized versions of the momentum equations and Eq. (6.38) lead to discretized equations in terms of velocity and pressure correction:

The corrected velocities are assumed to satisfy continuity equations. If the corrected velocity expressions (Eq. (6.38)) are substituted in the discretized continuity equation, pressure correction equations can be derived. However, velocity corrections as given by Eq. (6.39) involve velocity corrections at neighboring nodes and unless some approximations are made, it is not possible to obtain the desired pressure correction equations. In SIMPLE algorithm, the first term comprising velocity corrections at the neighboring nodes is neglected to yield a simplified expression for velocity corrections:

For a staggered grid arrangement, velocity correction can be related to pressure corrections at the two nodes around it:

Substitution of this velocity correction into the discretized form of the continuity equation then leads to a pressure correction equation of the following form:

where SM is the mass imbalance and * indicates the value obtained from the currently available values of variables. The coefficients of this discretized equation, aP and anb, can be obtained with the help of Eq. (6.40). Equation (6.42) can be solved to obtain the pressure correction field. Once the pressure correction field is known, Eq. (6.40) can be used to obtain velocity corrections. Equation (6.38) can then be used to obtain the corrected pressure and velocity field. The gross assumption of neglecting velocity corrections at the neighboring nodes (first term of Eq. (6.39)), however, has detrimental consequences on the overall performance of the algorithm. The corrected pressure and velocity fields need to be under-relaxed in order to maintain the stability of the algorithm. As mentioned earlier, under-relaxation is a way to control the change in the variable values during the iterative processes. Such under-relaxation for the pressure and velocity field may considerably reduce the rate of convergence. Several methods have been proposed to enhance the rate of convergence.

van Doormal and Raithby (1984) proposed a variation of SIMPLE, called SIM-PLEC (SIMPLE consistent). In this method, instead of neglecting the first term of the right-hand side of Eq. (6.30), it is assumed that the order of magnitude of the velocity corrections at the neighboring nodes will be the same as that of the node under consideration. This assumption is more consistent and leads to the following equation for velocity correction:

The overall algorithm is exactly the same as that of SIMPLE, the only exception being that Eq. (6.43) is used instead of Eq. (6.40) to derive the discretized pressure correction equation. The more consistent approximation proposed in SIMPLEC reduces the need for under-relaxing the pressure.

Patankar (1980) proposed a revised SIMPLE algorithm called SIMPLER (SIMPLE revised). In SIMPLER, the velocity correction part of the SIMPLE algorithm is retained. However, instead of using the pressure corrections to calculate the pressure field, the SIMPLER algorithm uses a separate pressure equation to calculate the pressure field. The discretized momentum equation can be written:

If Eq. (6.44) is substituted into the discretized continuity equation, a discretized equation of the pressure field can be obtained which will be similar to Eq. (6.42)

in form. The coefficients of this pressure equation and pressure correction equation will also be the same. The source terms, however, will be different. In the pressure equation the source terms are calculated in terms of pseudo-velocities, which can be written (drawing an analogy with the earlier derivation):

Since no terms are omitted when deriving the pressure equation used by SIMPLER, the resulting pressure field corresponds with the velocity field. Therefore, unlike SIMPLE, the correct velocity field results in the correct pressure field. Consequently, SIMPLER does not require under-relaxation of pressure and performs significantly better than SIMPLE.

Issa (1986) proposed a two-step corrector algorithm called PISO (pressure implicit with splitting of operators). In this algorithm, the first corrector step is the same as that of the SIMPLE algorithm. The corrected velocity and pressure fields are used to derive the second correction equation. For this second step, the first term in the right-hand side of Eq. (6.39) containing the neighboring velocity corrections is calculated from the first correction step. Application of the discretized continuity equation to the corrected velocities leads to the second pressure correction equation. The coefficients of the second pressure correction equation are the same as those of the first correction equation. The source term, however, contains terms containing neighboring velocity corrections. Details of the derivation of this second correction equation may be found in Issa (1986) and Versteeg and Malalasekara (1995).

Several variants of the SIMPLE family of algorithms have been proposed. Various studies comparing relative performances of these algorithms are available (for example, Jang et al., 1986; Braaten and Shyy, 1987; Wanik and Schnell, 1989; McGuirk and Palma, 1993). It must be noted that there is no single algorithm, which may be identified as the best algorithm for all types of problems. The performance of any algorithm depends on the flow conditions, the degree of coupling between various equations, the amount of under-relaxation used and sometimes, also on details of the numerical technique used to solve the algebraic equations (direction of sweeps and so on). In general, when momentum equations are not coupled with a scalar variable, the PISO algorithm performs better than SIMPLE or SIMPLEC. When such a coupling exists, PISO may show no significant advantage over the other methods. SIMPLER and SIMPLEC have proven to be robust and efficient in strongly coupled problems. Both of these are superior to SIMPLE in many flows. It is, however, difficult to single out a superior algorithm between SIMPLER and SIMPLEC. General experience suggests that SIMPLER is more robust and more suitable for complex applications like multiphase flows.

The solution algorithm for this class of methods can be summarized as follows:

1. Momentum equations are solved using the guessed (or available from the previous iteration) velocity and pressure field.

2. The pressure correction equation is solved and the velocity field is corrected using the derived pressure correction field. For PISO, a second pressure correction equation is solved to correct the pressure and velocities again.

For SIMPLER, the pressure equation is solved based on the updated velocity field.

3. Scalar equations (if any) are then solved using the corrected velocity field (for example, k and e equations when solving the k-e model of turbulence or the enthalpy equation when solving non-isothermal flows).

4. Fluid properties are updated (if not constants).

5. Return to step 1 until a converged solution is obtained.

The algorithms discussed so far can be applied directly when staggered grids are used. In staggered grids, the cell face values required for assembling the discretized continuity equation are available readily and contain the pressure gradient terms naturally. For the colocated grid, however, some modifications to these algorithms are required to avoid oscillations in the pressure field. Although these oscillations can be filtered out (van der Wijngaart, 1990), to devise a compact pressure correction equation similar to those discussed earlier, it is necessary to consider corrections to cell face velocities rather that node velocities (where the values are naturally available in colocated grids). The corrections to cell face velocities can be derived following the methods discussed earlier, the only difference is that the coefficient aP in Eq. (6.40) are not the nodal values, as in the staggered arrangement, but are interpolated cell center values. This procedure may appear unnatural compared to direct application of the staggered arrangement, however, as mentioned in the previous sub-section, a colocated grid arrangement is preferable for flow simulations in complex geometry. Details of the derivation of pressure correction equations and application of SIMPLE-like algorithms to colocated grids may be found in Lilek and Peric (1995) and Ferziger and Peric (1995) among others. In general, the performance of SIMPLElike algorithms (convergence rate, dependence on under-relaxation factors, computing costs etc.) is similar for staggered and colocated grid arrangements. The difference between solutions obtained with different variable arrangements is much smaller than the discretization error.

The overall solution procedure and other finer details of under-relaxation and convergence criteria are briefly discussed in Section 6.5. It is useful to briefly discuss the implementation of commonly encountered boundary conditions when solving flow field equations.

6.3.3. Implementation of Boundary Conditions

Mathematical formulations of various boundary conditions were discussed in Section 2.3. These boundary conditions may be implemented numerically within the finite volume framework by expressing the flux at the boundary as a combination of interior values and boundary data. Usually, boundary conditions enter the discretized equations by suppression of the link to the boundary side and modification of the source terms. The appropriate coefficient of the discretized equation is set to zero and the boundary side flux (exact or approximated) is introduced through the linearized source terms, SC and SP. Since there are no nodes outside the solution domain, the approximations of boundary side flux are based on one-sided differences or extrapolations. Implementation of commonly encountered boundary conditions is discussed below. The technique of modifying the source terms of discretized equation can also be used to set the specific value of a variable at the given node. To set a value at node P, the components of the linearized source terms, SC and SP, are set to very large values, like (0set 1030) and (-1030) respectively. In such a case, the right and left sides of the discretized equation (Eq. (6.12)) are dominated by these large terms yielding approximately:

This fixes the value of variable 4> at node P to 0set. This method can also be used to simulate solid obstacles within a solution domain by fixing 0set to zero. The system of discretized equations can then be solved as usual without considering the obstacles separately.

Convective fluxes are usually specified at the inlet boundaries (and set to zero at impermeable walls and the symmetry axis). Upwind approximations can usually be used. For the staggered grid arrangement, the velocity node is located at the boundary surface. The velocity at such a node on the boundary is then directly specified without the need for solving the discretized equation. For colocated grids, the specified inlet velocity is used to calculate the convective flux from the cell face coinciding with the inlet boundary. For diffusive fluxes, either the flux or the boundary value of the variable is specified. If the boundary value is specified, diffusive fluxes are evaluated using one-sided approximations for normal gradients. If the flux is specified, it is used to calculate the flux, and an approximation to the flux in terms of nodal values can be used to calculate the boundary value of the variable. For the zero gradient boundary condition, no term needs to be added to the source term.

For a staggered grid arrangement, a knowledge of pressure is not required at the boundaries on which the normal velocity is specified. At such boundaries, zero gradient boundary conditions should be used for pressure correction and pressure equations. When boundary pressure is specified, the pressure correction at the boundary needs to be set to zero. When a colocated grid arrangement is used, the boundary pressure needs to be known in order to calculate pressure forces appearing in the momentum equations. This is normally obtained by extrapolation. In most cases, linear extrapolation is sufficient. It must be noted that for incompressible flows, the absolute value of pressure (and, therefore, of pressure correction) is not relevant; only differences in the pressure are meaningful. Usually, the pressure is set to a fixed reference value at a suitable grid point and pressure values at all other nodes are calculated relative to this reference value.

At the outlet, extrapolation of the velocity to the boundary (zero gradient at the outlet boundary) can usually be used. At impermeable walls, the normal velocity is set to zero. The wall shear stress is then included in the source terms. In the case of turbulent flows, wall functions are used near walls instead of resolving gradients near the wall (refer to the discussion in Chapter 3). Careful linearization of source terms arising due to these wall functions is necessary for efficient numerical implementation. Other boundary conditions such as symmetry, periodic or cyclic can be implemented by combining the formulations discussed in Chapter 2 with the ideas of finite volume method discussed here. More details on numerical implementation of boundary conditions may be found in Patankar (1980) and Versteeg and Malalasekara (1995).

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