The SIMPLE method is the most widely valued member of the family of the pressure-based methods. Its basic idea is to formulate a Poisson equation for pressure correction and then to update the pressure and velocity fields until a divergence-free solution is obtained. The implementation of the SIMPLE method requires a staggered grid arrangement in order to avoid the checkerboard-type pressure oscillations.

The CVP method was pioneered by Hatzikonstantinou [1] and although being quite recent it has been implemented in various flow problems (Sakalis et al [2], Papadopoulos et al [3]). In this method, the corrections of the velocity field are determined via the solution of one Poisson equation for each velocity correction variable. These Poisson equations are formulated within the framework of the CVP method and are based on a continuity and a vorticity variational equations. The velocity corrections that have been obtained are then used for the evaluation of the corresponding pressure corrections. Thus, the new updated fields are calculated for both the pressure and the velocities and this iterative procedure is repeated until convergence is achieved. For the implementation of the CVP method a cell-vertex type grid is used which aids the generation of accurate and robust solutions even for coarse or distorted grids.

The two methods presented above are used to solve the equations that govern the laminar, incompressible, fully developed flow inside curved ducts of circular cross- sections or elliptical cross-sections of aspect ratios 0.8, 0.5 and 0.2. For the solution of the flow problem the governing equations are transformed to a numerically generated, boundary fitted coordinate system.

The produced results are checked for their accuracy by comparison with experimental data found in the literature and by evaluation of the average mass residual that exists at the end of the computational procedure. It is seen that the observed differences are small and it can be stated that both methods reach a satisfactory level of accuracy.

The issue of grid independence is examined by comparison of the predictions of the two methods for different grid sizes. It is found that the CVP method has an undisputed advantage as it can produce accurate and grid independent solutions with coarse meshes for a wide range of Dean numbers. On the other hand, the SIMPLE method requires a fine grid in order to converge for large Dean numbers and in some cases it is impossible to obtain a converged solution regardless of the mesh size.

Besides the grid, another factor affecting the convergence behavior is the use of under-relaxation in the two methods. Concerning the CVP method, a time step is used, constant for all computational cases considered, while for the SIMPLE method an under-relaxation factor ranging between 0.3 and 0.1 is necessary.

The matter of computational efficiency is looked into, in terms of the time required by each method to reach a converged solution. The computational efficiency is found to depend on the grid size, the under-relaxation factor and the geometry of the duct. All these elements are considered, leading to different conclusions for each method.

Finally, in order to compare the predictions of the two methods concerning the flow behavior, contour plots of the axial velocity are presented along with vector plots of the secondary flow and diagrams of the pressure distribution. Thus, the comparison procedure of the two highly competitive methods, applied in this work to a problem of laminar incompressible flow, is completed.

1

P.M. Hatzikonstantinou, "A Computational Procedure for the Incompressible Three-Dimensional Parabolic Flows", in Proceedings of the 4th GRACM Congress on Computational Mechanics, edited by D. T. Tsahalis, University of Patras 68-79, 2002.

2

V.D. Sakalis, P.M. Hatzikonstantinou, "A Numerical-Variational Procedure for Laminar Flow in Curved Square Ducts", Int. J. Numer. Methods in Fluids, Accepted for Publication, 2003.

3

P.K. Papadopoulos, P.M. Hatzikonstantinou, "An improved CVP procedure applied to a curved elliptic duct with internal fins", submitted for publication, 2003.

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