FLOW AROUND A CYLINDER - istiarto

– 4.47 –
� reconstruct the mesh, maintain the number of cells in the vertical and their relative
positions to the local depth
� use linear interpolation to get variables at the new cell centers (see Fig. 4.11b) in
order to maintain the continuity:
�z
uP �� � uP �z ��
, v �z
P �� � vP �z ��
, wP �� � w P,
�z
kP �� � kP �z ��
, � �z
P �� � �P �z ��
, pP �� � pP � � gz �h
� continue the computation to the next time level
4.6 Solution procedures
4.6.1 Spatial discretisation
Applying the discretized governing equations, Eqs. 4.52 and 4.68 requires spatial
discretisation of the computational domain. The domain is divided into Ni–2, Nj–2, and
Nk–2 cells in the x-, y-, z-directions, respectively, from which there are Ni, Nj, and Nk
nodes (interior, boundary, and dummy nodes) in the corresponding directions. A typical
spatial discretisation is shown in Fig. 4.12. The cell and node are denoted by a single
index; the cells are indexed by ijk = 2,3,…,Nijkm, going along the y-direction (j =
2,3,…,Nj–1), the x-direction (i = 2,3,…,Ni–1), and the z-direction (k = 2,3,…,Nk–1),
while the nodes are indexed by ijk = 1,2,…,Nijk. With this indexing, the single-index ijk
for a cell and for its six-neighbors can be easily obtained from their position in the (x,y,z)
space (see Table 4.5).
Writing Eq. 4.52 or 4.68 for all cells, ijk = 2,3,…,Nijkm, produces a series of algebraic
linear equations which can be presented in a matrix form as follows:
A � � B (4.97)
The matrix A contains the coefficients of the equations, a nb and a P , � is a column matrix
of the dependent variable, and B is a column matrix of the source terms. The matrix A
has diagonal blocks which are themselves tridiagonal, and sub- and super-diagonal blocks
in which each block has two diagonals, thus it has only 7 non-zero diagonals while the
other elements are zero. It is then not necessary to store all elements of the matrix A; only
those seven diagonals need to be stored. To facilitate the storage, each non-zero diagonal
is stored in a separate column matrix, i.e. A nb , nb = EWNSTB, and A P . Fig. 4.13 shows
the form of the matrix A.