Page 31 - Basic Structured Grid Generation

20 Basic Structured Grid Generation

referred to background cartesian co-ordinates {y i }, is the scalar defined by div u =
∂U i /∂y i , otherwise denoted by ∇· u, with summation over i. In general curvilinear
co-ordinates this transforms to the sum (summation over i) of covariant derivatives
i
∇· u = u . (1.133)
,i
We may also deduce from eqn (1.119) that
∂u
i
∇· u = g · . (1.134)
∂x i
By eqns (1.120) and (1.118), we have
∂u i i k ∂u i i k
∇· u = + u = + u
ki
ik
∂x i ∂x i
∂u i 1 ∂ √ k ∂u i 1 ∂ √ i
= + √ ( g)u = + √ ( g)u ,
∂x i g ∂x k ∂x i g ∂x i
which gives the useful formula
1 ∂ √ i
∇· u = √ i ( gu ), (1.135)
g ∂x
with summation over i. This is an expression for the divergence in conservative form.
In general, conservative form is preferred for operator expressions when numerically
solving partial differential equations (in particular, transport equations in fluid flow
problems) because numerical accuracy is enhanced. More examples are given below.
A vector identity which recurs frequently in the following is:

∂ ∂ ∂
−
(g 2 × g 3 ) + (g 3 × g 1 ) + (g 1 × g 2 ) = 0. (1.136)
∂x 1 ∂x 2 ∂x 3
i
Exercise 12. By writing each g i as the appropriate ∂r/∂x , performing the differen-
tiations using product rules, and finally exploiting the skew-symmetry of the vector
product, verify eqn (1.136).
We write eqn (1.136) as
3
∂
(g j × g k ) = 0, (1.137)
∂x i
i=1
where for each i it is assumed that j and k are such that i, j, k are in cyclic order 1, 2, 3.
Now from eqns (1.135), (1.48) and (1.32) we have the two conservative forms
3
1 ∂ √ i 1 ∂
∇· u = √ ( gg · u) = √ {(g j × g k ) · u} (1.138)
g ∂x i g ∂x i
i=1
and the non-conservative form
3
1 ∂u
∇· u = √ (g j × g k ) · , (1.139)
g ∂x i
i=1
using eqn (1.137).