Page 259 - Introduction to Computational Fluid Dynamics

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NUMERICAL GRID GENERATION
with boundary conditions T 1 = 0 (south), T 1 = 1 (north), and ∂T 1 /∂n = 0 (east
and west), where n is normal to the boundary. Similarly, Figure 8.4(b) represents
the probable solution to the second problem T = T 2 (say) governed by
2 2
∂ T 2 ∂ T 2 q 2
+ = , (8.11)
∂x 2 ∂x 2 k
1 2
with boundary conditions T 2 = 0 (west), T 2 = 1 (east), and ∂T 2 /∂n = 0 (north and
south). The solutions to Equations 8.10 and 8.11 therefore can be written as
T 1 = T 1 (x 1 , x 2 ), T 2 = T 2 (x 1 , x 2 ). (8.12)
Each isotherm (T 1 and T 2 ) thus represents sets of values of x 1 and x 2 .In
Figure 8.4(c), the two solutions are superposed. The isotherms now take the appear-
ance of a body-fitted curvilinear grid. Now, as in the previous section, Equation 8.12
can also be written by simply interchanging the roles of T and x. Analogous to Equa-
tion 8.9, therefore, we may state that the appropriate equations for determination
of coordinates x 1 and x 2 are

2 2
2 ∂ ξ 1 ∂ ξ 1
∇ ξ 1 = + = P (ξ 1 ,ξ 2 ), (8.13)
∂x 2 1 ∂x 2 2
2 2
2 ∂ ξ 2 ∂ ξ 2
∇ ξ 2 = + = Q (ξ 1 ,ξ 2 ), (8.14)
∂x 1 2 ∂x 2 2
where ξ 1 and ξ 2 are curvilinear coordinates and P and Q are stretching functions.

8.3.3 Inversion of Determinant Equations
To make Equations 8.9 (in the 1D domain) and 8.13 and 8.14 (in 2D domains)
determinants of Cartesian coordinates, they must be inverted. Thus, for the 1D
domain,wehave
∂ ∂ξ ∂
= . (8.15)
∂x ∂x ∂ξ
Now, if directions x and ξ coincide (∂ξ/∂x = 1) then Equation 8.9 can be written
as
2
∂ x
= C, (8.16)
∂ξ 2
with x = 0at ξ = 0 and x = L at ξ = 1. Grid coordinates x(i) can now be deter-
mined for various choices of C.
For 2D domains, however, the matter is not so simple and requires vec-
tor analysis. Thus, we recall that a covariant base vector (tangent to coordinate