Page 34 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 23

k
We have seen in eqn (1.125) that, for a second-order tensor T, ∂T/∂x can be
3
3
regarded as a linear operator, acting on vectors in E to give vectors in E .When it
k
acts on the contravariant base vector g , the resulting vector is called the divergence
of T, and we write

∂T k
∇· T = g (1.153)
∂x k
with summation over k.Inother words,
ij k ij k ij k ij
∇· T = (T g i ⊗ g j )g = T g i (g j · g ) = T g i δ = T g i , (1.154)
,k ,k ,k j ,j
expressed in terms of the covariant derivatives of the contravariant components of T.

Exercise 13. Verify the formulas
1 ∂ √ ij 1 ∂ √ ij kj i
∇· T = √ ( gT g i ) = √ ( gT ) + T kj g i . (1.155)
g ∂x j g ∂x j

Exercise 14. Show that if p is a scalar field and I is the unit second-order tensor
(defined in eqn (1.87)), then
∇· (pI) =∇p. (1.156)

1.8 Summary of formulas in two dimensions

For two-dimensional situations in which field variables depend only on the rectangular
cartesian co-ordinates x and y but not z, it is straightforward to establish the reduced
form of the above results. We give a summary here of some of the main results for
convenience.
1
2
With y 1 = x, y 2 = y, and curvilinear co-ordinates with x = ξ, x = η (and
3
occasionally finding it useful to put y 3 = x = z = ς), wehavebasevectors
g 1 = ix ξ + jy ξ , g 2 = ix η + jy η , g 3 = k, (1.157)
where suffixes denote partial differentiation, e.g. x ξ = ∂x/∂ξ. The components of the
covariant metric tensor are given by

   2 2 
g 11 g 12 g 13 x + y ξ x ξ x η + y ξ y η 0
ξ
2
 g 12 g 22 g 23  =  x ξ x η + y ξ y η x + y 2 η 0  , (1.158)
η
g 13 g 23 g 33 0 0 1
with determinant
2
g = g 11 g 22 − (g 12 ) . (1.159)
Moreover

x ξ x η
0
√
g = g 1 × g 2 · g 3 = y ξ y η 0 = (x ξ y η − x η y ξ ). (1.160)

0 0 1