Cartesian grid

Multiblock, either with Cartesian or boundary-fitted grid in Fig. 11.8, a two-block Cartesian grid is shown. 

When flow in empty rooms is to be computed, a Cartesian grid is sufficient. If the ceiling, floor, or walls of a room are inclined or curved, a boundary-fitted grid should be used. For computing the flow in an apartment or a building with multiple rooms, a multiblock grid should be used. With a grid as... 

Upon the reconstruction of the full 3D velocity field, the deformation of the Cartesian grid is recreated on selected slices (Figure 4.4.6(c)) in order to verify that the algorithm is properly implemented. 

Fig. 1. The level sets of distance function for a smooth interface over a Cartesian grid.

The -coordinates of points in the resulting surface projected on an x — y Cartesian grid satisfy the 2D equation. The average value of V, i.e. z, on a circle of radius R will now be equal to V at the central point x, y... 

FIGURE 6.2 Typical finite volume CV and the notation used for a Cartesian grid. 

For a uniform Cartesian grid, this approximation is of second-order accuracy. Even for a non-uniform grid, the error reduction with respect to grid refinement is similar to that of a second-order approximation. Higher order polynomials can be used to estimate the required gradients. For example, a fourth-order approximation for the gradient at face e on the uniform Cartesian grid can be written ... 

Some commonly used interpolation practices are discussed here. A simple and straightforward approximation to the value at the CV face center is linear interpolation between the two nearest nodes. At location e on a Cartesian grid (Fig. 6.2), general interpolation coefficients (yS) for such a scheme can be written ... 

On a Cartesian grid, n = x at the e face and the usual schemes can be used to estimate gradient at e. For example, a central differencing scheme will give ... 

Li, Z Zhao, H. and Gao, H. (1999) A Numerical Study of Electro-migration Voiding by Evolving Level Set Functions on a Fixed Cartesian Grid. J. Comput. Phys., 152, 281-304. 

In the FVM a general 3D cell containing the central node P has six neighboring nodes identified as west, east, south, north, bottom and top nodes (W, E, S, N, B, T), as sketched in Fig 12.2). The notation, w, e, s, n, b and t are used to refer to the west, east, south, north, bottom and top cell faces, respectively. The algorithms are illustrated using Cartesian grids. 

A Taylor series expansion about P gives (for Cartesian grid and (v n)g > 0) [49] ... 

Another classical approximation for the value at GCV face center is obtained by linear interpolation between the two nearest nodes. The linear interpolation corresponds to the central difference approximation of the first derivative in FDMs. At location e on a non-uniform Cartesian grid we have [49, 202] ... 

The quadratic upstream interpolation for convective kinetics (QUICK) scheme of Leonard [106] uses a three-point upstream-weighted quadratic interpolation for the cell face values. In the third order QUICK scheme the variable profile between P and E is thus approximated by a parabola using three node values. At location e on a uniform Cartesian grids, tpe is approximated as ... 

The three relevant Taylor series expansions about the e face value in a Cartesian grid are given by [202] ... 

Fig. C.l. Staggered Cartesian grid arrangement in the scalar cell notation. The scalar variables are located in the cell centers, while the velocity components are centered around the cell faces. The velocity components are located in the centers of their own grid cell volumes (not shown) which are staggered in one dimension compared to the scalar grid.

Fig. C.2. A sketch of a scalar Cartesian grid cell showing the distribution of the variables in the grid and the configuration of the staggered velocity grids. In cylindrical coordinates equivalent grid cells can be defined.

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