Page 230 - Introduction to Computational Fluid Dynamics

P1: IWV
CB908/Date
0 521 85326 5
May 25, 2005
0521853265c06
EXERCISES
normal to the cell face are drawn through c and e. Now, imagine a plane through 11:10 209
P parallel to the cell face. This plane will orthogonally intersect the two nor-
mals at P 1 and P 2 . A similar face-parallel plane through E will intersect the
two normals at E 1 and E 2 . Necessary evaluations of face-normal transport can
now be carried out along the line P 2 −c−E 2 . Similarly, the construction of a
control volume at the cell face is shown in Figure 6.31(b) when the cell face is
i
triangular. To evaluate β , while direction ξ 1 is along PE, directions ξ 2 and ξ 3
1
may be chosen along any two sides of the triangle rst with origin at r, s, or, t. The
actual directions are determined by requiring that Jacobian J be positive. Simi-
larly, to affect vector boundary conditions, two tangent vectors t 1 and t 2 must be
defined at the boundary cell face. Out of these, t 1 (say) may be chosen along PP 2
and direction of t 2 can be determined using the direction of the normal to the
boundary cell face so as to form an orthogonal frame t 1 , t 2 , n. The reader may
find these figures useful for developing a 3D unstructured grid procedure [18].
6. Because of its generality, commercial codes are increasingly adopting unstruc-
tured grids. Although generality is welcome, the codes must rely heavily on
polyhedral mesh generators as well as on creation of special routines for pro-
cessing of computed results. Such postprocessors typically create contour, vec-
tor, and/or surface plots. For comparison of computed results with experimental
data, however, one often needs to resort to interpolations. The reader will ap-
preciate this difficulty because whereas most detailed measurements in a flow
are carried out along a single straight line at a time, the grid nodes generated by
packages such as ANSYS may not fall on a single line (in two dimensions) or
even in a single plane (in three dimensions).
7. Despite the above-mentioned difficulty, unstructured grid codes are most versa-
tile and, therefore, suitable for complex domains encountered in industrial and
environmental applications.

EXERCISES
i
1. Derive expressions for β (i = 1, 2, 3 and j = 1, 2, 3) for a 3D curvilinear grid.
j
2. Using Equations 6.24 and 6.27, express dA i and dV for a 3D curvilinear grid.

3. Starting with the p equation in Cartesian coordinates (see Chapter 5), derive
Equation 6.39. Identify the neglected terms in Equation 6.39 and explain how
the effect of these terms can be recovered in a predictor–corrector fashion.

4. Analogous to Equation 6.42, derive an expression for p x 2 ,P .
5. Derive Equations 6.91, 6.92, and 6.93.
6. Derive Equation 6.113.
7. Using Equations 6.121 and 6.122, derive explicit symmetry boundary condi-
tions for u 1,B and u 2,B .