Staggered Grid

As it has been seen, the major difficulty encountered during solution of incompressible flow is the non-availability of any obvious equation for the pressure. This difficulty can be resolved in the stream-function-vorticity approach. This approach losses it advantage when three-dimensional flow is computed because of the fact that a single scalar stream-function does not exist in three-dimensional space. A three-dimensional problem demands a primitive-variable approach. Efforts have been made so that two-dimensional as well as three-dimensional problems could be computed following a primitive variable approach without encountering non-physical wiggles in the pressure distribution. As a remedy, it has been suggested to employ a different grid for each of the dependent variables. Such a staggered grid for the dependent variables in a flow field was first used by Harlow and Welch (1965), in their very well known MAC (Maker and Cell) method. Since then, it has been used by many researchers. Specifically, SIMPLE (Semi Implicit method for Pressure Linked equations) procedure of Patankar and Spalding (1972) has become popular. Figure 27.1 shows a two-dimensional staggered grid where dependent variables and with the same indices are staggered to one another. Extension to three-dimensions is straight-forward. The computational domain is divided into a number of cells, which are shown as “main control volume” in Fig. 27.1. The location of the velocity components are at the center of the cell faces to which they are normal. If a uniform grid is used, the locations are exactly at the midway between the grid points. In such cases the pressure difference between the two adjacent cells is the driving force for the velocity component located at the interface of these cells. The finite-difference approximation is now physically meaningful and the pressure field will accept a reasonable pressure distribution for a correct velocity foeld.

Figure 27.1: Staggered grid.

Another important advantage is that transport rates across the faces of the control volumes can be computed without interpolation of velocity components. The detailed outline of the two different solution producers for the full Navier-Stokes equations with primitive variables using staggered grid will be discussed in subsequent sections. First we shall discuss the SIMPLE algorithm and then the MAC method will be described.