Page 194 - Basic Structured Grid Generation

Moving grids and time-dependent co-ordinate systems 183

as in eqn (1.97), is valid, with k summed over the values 1 to 3. Thus it is not necessary
k
to extend the set of spatial contravariant base vectors g , k = 1, 2, 3.
With suffixes restricted to the values 1, 2, 3, we now have
∂g i k ∂g i k
=[ij, k]g , =[i0,k]g ,
∂x j ∂t
∂g i k ∂g i k
= g k , = g k , (7.16)
ij
i0
∂x j ∂t
where Christoffel symbols of the second kind are
k kl k kl
= g [ij, k], = g [i0,k]. (7.17)
ij
i0
i
Note that substituting f = x in eqn (7.5) gives
∂x ∂x ∂x
i i i
i i i
0 =∇x · W + = g · W + = W + ,
∂t ∂t ∂t
y j y j y j
using eqns (1.11) and (1.48), so that we have the result
i
∂x i
=−W , i = 1, 2, 3, (7.18)
∂t
y j
which states that the time-derivatives of the curvilinear co-ordinates at a fixed spatial
point of the physical domain as the co-ordinate system moves are equal to the negative
of the contravariant components of the grid point velocity. √
i
Another useful formula gives the time-derivative at fixed x of g, where by
eqn (1.31)
√
g = g 1 · (g 2 × g 3 ).
We have, after re-arrangement of scalar and vector products, and with use of eqns (7.8),
(1.8), and (1.134),
1 ∂ √ 1 ∂g 1 ∂g 2 ∂g 3
√ ( g) = √ · (g 2 × g 3 ) + · (g 3 × g 1 ) + · (g 1 × g 2 )
g ∂t g ∂t ∂t ∂t
1 ∂W ∂W ∂W
= √ 1 · (g 2 × g 3 ) + 2 · (g 3 × g 1 ) + 3 · (g 1 × g 2 )
g ∂x ∂x ∂x
∂W 1 ∂W 2 ∂W 3
= · g + · g + · g =∇ · W. (7.19)
∂x 1 ∂x 2 ∂x 3
This is a fundamental identity in time-dependent co-ordinate theory; it is called
the Geometric Conservation Law. In Computational Fluid Dynamics it provides an
additional equation which has to be solved alongside the usual transport equations
when we have moving grids.
From eqn (7.12) we also have

k ∂g 0 k j k j j
∇· W =∇ · g 0 = g · = g · g j = δ 0k = ,
j
0k
0j
∂x k
using (7.16). Hence
j 1 ∂ √
= √ ( g). (7.20)
0j g ∂t