Page 193 - Basic Structured Grid Generation

182 Basic Structured Grid Generation

with cartesian components obtained from differentiating eqn (7.1):

∂y i
(g j ) i =
∂x j
and time-derivatives
∂ ∂ ∂y i ∂ ∂y i
(g j ) i = = ,
∂t ∂t ∂x j ∂x j ∂t
giving the vector equations

∂g j ∂W
= , j = 1, 2, 3. (7.8)
∂t ∂x j
This variation of the base vectors with time as well as space complicates the trans-
formation of vector equations (such as the time-dependent Navier-Stokes equations) to
a curvilinear co-ordinate system. The variation of base vectors in space was consid-
ered in Chapter 1 by the introduction of Christoffel symbols, and a similar formalism
is convenient for dealing with time-variation.
0
Here we put t = x and regard this as a fourth generalized co-ordinate. We can also
define an associated covariant base vector

∂r ∂r
g 0 = = (7.9)
∂x 0 x j ∂t x j
and thus we can write
∂r
g i = , i = 0, 1, 2, 3, (7.10)
∂x i
with
2
∂g i ∂ r ∂g j
= = . (7.11)
j
∂x j ∂x ∂x i ∂x i
By the definition of W in Section 7.1, we also have
g 0 = W. (7.12)

The covariant metric tensor can also be extended to include the time variable, so that

g ij = g i · g j
2
with the suffixes i, j ranging over the values 0 to 3, so that g 00 = W · W =|W| and
g 0j = W · g j = W j . (7.13)

Similarly we have the Christoffel symbol [ij, k], which is given, as in eqn (1.108), by
1 ∂g jk ∂g ik ∂g ij
[ij, k]= + − . (7.14)
2 ∂x i ∂x j ∂x k
Here, however, we allow i and j but not k to take the value 0 as well as 1, 2, 3, so
that the equation
∂g i k
=[ij, k]g , i, j = 0, 1, 2, 3, (7.15)
∂x j