Page 156 - Basic Structured Grid Generation

Differential models for grid generation 145

! "
(Delet ∗ ϕ) j,k+1 − (Delet ∗ ϕ) j,k−1
+0.5
2[(g tt ) j,k + (g oo ) j,k ]
 
(g ot ∗ ϕ) j+1,k+1 + (g ot ∗ ϕ) j−1,k−1
 
−(g ot ∗ ϕ) j+1,k−1 − (g ot ∗ ϕ) j−1,k+1
 
+0.25 . (5.116)
2[(g tt ) j,k + (g oo ) j,k ]
 
 
An iterative scheme employing SOR is now given by
n+1 n t n
ϕ = ϕ + ω(ϕ − ϕ ), (5.117)
j,k j,k j,k j,k
where ω is an over-relaxation factor which we have taken in the program to be 1.5.
The program uses this method to solve Laplace’s equation in the region between con-
centric circular arcs shown in Fig. 4.2 of Chapter 4, subject to the Dirichlet boundary
conditions:
1
ϕ = 0 when θ = 0, ϕ = sin α when ϕ = α,
r
1 1
ϕ = sin θ when r = r 1 , ϕ = sin θ when r = r 2 . (5.118)
r 1 r 2
1
For these boundary conditions the exact solution ϕ = sin θ exists, and this enables
r
us to assess the accuracy of the numerical procedure.
5.11.2 More general steady-state equation

Suppose that the basic partial differential equation to be solved in physical space for
the field variable ϕ has the form

∇· (vϕ) +∇ · (ν∇ϕ) + S = 0, (5.119)

where S is a source term, ν could be a diffusion coefficient, and v is a vector field
(fluid velocity). In cartesian co-ordinates this takes the form

∂ ∂ ∂ϕ
(v i ϕ) + ν + S = 0,
∂y i ∂y i ∂y i
which transforms to the consistent generalized tensor form
i ij
(v ϕ) ,i + (νg ϕ ,j ) ,i + S = 0. (5.120)
Using eqns (1.128), (1.122), (1.111), and (1.134), this may be expressed as
i i ij
v ϕ, i +v ϕ + g (νϕ, j ) ,i + S
,i
∂ϕ ij ∂ ∂ϕ k ∂ϕ
i
= v + ϕdivv + g ν − ij ν + S
∂x i ∂x i ∂x j ∂x k
∂ϕ i ∂v ij ∂ ∂ϕ 2 k ∂ϕ
i
= v + ϕg · + g ν + (∇ x )ν + S = 0,
∂x i ∂x i ∂x i ∂x j ∂x k