It can be seen that since momentum equations are vector equations, the convective and diffusive terms in the equations appear more complicated than the generic transport equations discussed in the previous section. The convective terms are non-linear and the viscous terms contain more than one term. However, all of these terms can be discretized using the methods discussed in the previous section. All the extra non-zero terms not conforming to the generic equations are usually combined in the form of a source term. It must also be noted that all the three momentum equations are strongly coupled because each velocity component appears in all three momentum equations. This coupling can also be handled by the techniques of iterative solution discussed earlier. The unique feature of momentum equations, which distinguish them from the generic transport equation discussed earlier, is the role played by the pressure. The pressure gradients appear in the source terms of the momentum equations but there is no obvious equation to obtain the pressure. The pressure field is indirectly specified via the continuity equation. It is, therefore, necessary to calculate the pressure field in such a way that the resulting velocity field satisfies the continuity equation. Special treatments are needed to convert the indirect information in the continuity equation into a direct algorithm to calculate pressure (algorithms to treat pressure-velocity coupling). Some widely used algorithms are discussed in this subsection. Since the principal variable in momentum equations is a vector, it allows more freedom in the choice of variable arrangements on the grid.

6.3.1. Co-located and Staggered Grid Arrangement

Basic features of grids used for numerical solution are discussed in Section 6.2. When all the variables are stored at the same set of grid nodes, the arrangement is termed as 'colocated'. It is, however, not necessary that all the variables share the same grid. It is possible, and sometimes advantageous, to use different locations for storing values of different velocity components and pressure (staggered grid). The two types of grid arrangement are shown in Fig. 6.9. 'Colocated' seems to be an obvious choice, which has significant advantages in complicated solution domains. However, straightforward application of the finite volume method discussed earlier for momentum equations using the colocated grid fails to recognize the difference between a checkerboard pattern and uniform pressure fields. The staggered grid arrangement is proposed to suit the natural coupling of pressure and velocity. In this arrangement, the velocity field is stored at the faces of CV around a pressure node. In such an arrangement, the pressure and diffusion terms are very naturally approximated by a central difference approximation without interpolation. Also the evaluation of mass fluxes in the continuity equation (on the faces of a pressure CV) is straightforward. With a staggered grid arrangement, the natural coupling between pressure and velocity fields helps to avoid some types of convergence problems and oscillations in the pressure field. Because of these advantages, the staggered grid arrangement has been used extensively to solve momentum equations. In recent years, more and more problems with complex geometry have been tackled using non-orthogonal grids. The staggered grid arrangement for equations in generalized coordinates is complicated because it introduces additional curvature terms, which are difficult to treat numerically. Thus, improved pressure-velocity coupling algorithms were developed which enable the use of colocated grids to solve momentum equations. Most commercial CFD codes now use colocated arrangements.

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