Pressure-correction Methods

The general pressure-correction method is characterised by a formulation in which the momentum equations are solved sequentially for the velocity components using the best available estimate for the pressure distribution. Such a procedure does not yield a velocity field that satisfies the continuity equation unless the correct pressure distribution is employed. The variety of the methods differ primarily in the algorithms used to solve the component equations and the strategies employed to develop an equation to be solved for an improved pressure, which most often is a

Poisson equation. Some of the most commonly used variations of the pressure-correction methods are the SIMPLE family methods and the fractional-step methods.

In the SIMPLE family of methods the procedure is based on a cyclic series of guess-and-correct operations to solve the governing equations. The velocity components are first calculated from the momentum equations using an initiail pressure field. The pressure and velocities are then corrected, so as to satisfy the continuity equation. This procedure continues until the solution converges. The main distinction between this method and the projection methods is in the way in which the pressure and velocity corrections are achieved.

The SIMPLE algorithm is a semi-implicit scheme based on the general pressure correction idea to satisfy the momentum and continuity conservation equations as well as the equations containing the quantities which influence the flow field (e.g. turbulence quantities and species concentration) at the end of each time step when a converged solution is obtained. A comprehensive discussion on the pressure-correction equation has been presented in Patankar (1980). One major point of this discussion is that the omission of the term representing an implicit influence of the pressure correction on velocity from the equation leading to the velocity-correction equation does not ultimately affect the results. The omission of this term is in favour of avoiding the involvement of the pressure correction at all grid points in the calculation domain in the velocity-correction equation. This involvement would lead to direct solution of the set of the momentum and continuity equations, which does not follow the original idea of the SIMPLE algorithm. The omission of the implicit influence term of the pressure correction on velocity explains the name which has been given to the SIMPLE algorithm. A great many number of studies for two- and three-dimensional problems have been based on SIMPLE family methods.

For instance, Zhou (1995) used a modified SIMPLE-like algorithm to treat the velocity-depth coupling in a depth-averaged model and Sung et al. (1999) used a Chorin-type SIMPLE algorithm. Ouillon and Dartus (1997) and Stansby and Zhou (1998) also used SIMPLE family algorithms.

A revised algorithm to improve the rate of convergence for a faster solution has been proposed by Patankar (1981). The motive behind the revised algorithm lies in overcoming the problems associated with the approximation made in the derivation of the pressure-correction equation, which were introduced by the omission of the term representing the implicit influence of the pressure correction on velocity. This

omission leads to exaggerated pressure corrections, which the latter in turn lead to obtaining an early and correct velocity field, but also result in many iterations before a converged pressure field is established. The revised algorithm, SIMPLER, consists of solving the pressure equation to obtain the pressure field and solving the pressure-correction equation only to correct the velocities. One iteration of the SIMPLER algorithm involves about 30% more computational effort (Patankar, 1981), but it requires fewer iterations for convergence. On the whole the computational time required for convergence by the SIMPLER algorithm is noticeably less than SIMPLE algorithm. Other pressure-correction methods (e.g. SIMPLEC) can be found in Versteeg and Malalasekera (1995).

4.6.1.1 Projection (Fractional-step) Methods

Many variations to this type of splitting the solution procedure have been suggested, among them are the time-splitting approach used for solving the compressible Navier-Stokes equations (Batten et al., 1996), split operator approach (Komatsu et al. ,1997; Yu and Li, 1998; Lin and Li, 2002), semi-implicit time-splitting (Zhou and Stansby, 1999), and the method proposed by Chorin (1968) and Temam (1969) which is known as the projection method, or the method of fractional steps. The method may generally be accomplished in two steps. The pressure gradient terms are omitted from the momentum equations in the first step, and the unsteady equations are advanced in time to obtain a provisional velocity V . In the ^* second step, the provisional velocity is corrected by accounting for the pressure gradient and the continuity equation as follows:

Δ 0

1

*

1   _ 



n n

t p V

V (4.38)

subject to the continuity constraint:

0

divV^n1  (4.39)

By taking the divergence of equation (4.38), subject to the continuity constraint (Eq.

4.39), a Poisson equation is obtained:

pⁿ t

Δ div ^*

1

2  V

 ^ (4.40)

The solution procedure consists of first computing V^* from the momentum equations while neglecting the pressure gradient terms. The pressure Poisson

equation is then solved for the pressure field, after which the velocities are computed from equation (4.38). E and Liu (1995) provided a comprehensive discussion on projection methods.

Fractional-step (projection) methods have been widely studied for solving the incompressible Navier-Stokes equations. Casulli and Stelling (1998) considered the hydrostatic and the dynamic components of the pressure separately in a two-step method. Daubert et al. (1982) and Daubert and Cahouet (1984) used a three-step fractional method. Four-step fractional methods were used in the studies of Choi et al. (1997) and Sung et al. (2000). Blasco et al. (1998) studied a first-order-accurate method in time and Brown et al. (2001) studied accurate projection methods. A variety of projection methods were studied by Vincent and Caltagirone (1999), Minev (2001) and Chang et al. (2002). Armfield and Street (2003) studied the pressure accuracy of fractional-step methods. The method of algebraic splitting was used by Henriksen and Holmen (2002), which can be seen as the matrix equivalent of the fractional-step or projection method.

A fractional-step method has been used in the present study. However the scheme that has been deployed for three-dimensional problems and in conjunction with the provisional velocity fields, obtained from the computation of the advection and the diffusion terms, together with the variable density and eddy viscosity, is a novel approach for solving the three-dimensional stratified flow problems in ALE by the projection (fractional-step) method.