Page 197 - Basic Structured Grid Generation

186 Basic Structured Grid Generation

Now eqn (7.32) may be expressed in vector form as
∂v

ρ + ρ(v ·∇)v =∇ · σ =−∇p +∇ · T
∂t
y i
and using eqn (7.7) gives

∂v
ρ − ρ(W ·∇)v + ρ(v·∇)v =−∇p +∇ · T. (7.33)
∂t x i
i
Dropping the subscript x for the moment, we have
k
∂v ∂(v g k ) ∂v k k ∂g k ∂v k j k
= = g k + v = + v j0 g k . (7.34)
∂t ∂t ∂t ∂t ∂t
The second and third terms of eqn (7.33) may be combined to give

∂v j j k
j
j
ρ[(v − W) ·∇]v = ρ(v − W ) = ρ(v − W )v g k , (7.35)
,j
∂x j
where v k is a covariant derivative. The right-hand side of eqn (7.33) may be written,
,j
using eqns (1.12), (1.51), and (1.155), as
∂p 1 ∂ √
ik kj ij k
−g g k + √ ( gT ) + T ij g k . (7.36)
∂x i g ∂x j
Putting eqns (7.34), (7.35), and (7.36) together into eqn (7.33), and observing that
the resulting contravariant coefficients of g k must be identical on both sides, gives
k
∂v j k j j k ik ∂p 1 ∂ √ kj ij k
ρ + v j0 + ρ(v − W )v =−g + √ ( gT ) + T ij .
,j
∂t ∂x i g ∂x j
(7.37)
Exercise 2. Making use of eqns (1.155) and (1.156), show that eqn (7.37) may be
expressed as

k k
∂v j k j j ∂v 1 ∂ √ kj kj
ρ + v j0 + ρ(v − W ) + √ [ g(pg − T )]
∂t ∂x j g ∂x j
k
j
ij
j
i
ij
+[ρv (v − W ) + pg − T ] = 0. (7.38)
ij
In the last two equations there is summation over the i and j suffixes from 1 to 3, and
k can take any value from 1 to 3. We can also express T ij in generalized contravariant
form as

1 ij
ij
T ij = 2µ ε − g ∇· v (7.39)
3
ij
ik jl
ik jl

with ε = g g ε kl = 1/2g g
v k,l + v l,k .
Equations (7.38) are the momentum equations for a general time-dependent curvi-
linear co-ordinate system in non-conservative form.