Page 195 - Basic Structured Grid Generation

184 Basic Structured Grid Generation

7.3 Transformation of generic convective terms

Here we take a typical convective expression in the form

∂ϕ
C = +∇ · (vϕ) (7.21)
∂t
y i
for some field quantity ϕ = ϕ(y 1 ,y 2 ,y 3 ,t),where v is the local fluid (or continuum)
i
velocity, and we transform it to a time-dependent curvilinear co-ordinate system x .
From eqn (7.7) we have immediately

∂ϕ ∂ϕ
C = − W ·∇ϕ +∇ · (vϕ) = + ϕ∇· W +∇ ·[(v − W)ϕ]. (7.22)
∂t x i ∂t x i
Alternatively, using eqns (7.19) and (1.138), we can write
√

∂ϕ ϕ ∂ g 1 ∂ √ i
C = + √ + √ ( gg · (v − W)ϕ)
∂t x i g ∂t g ∂x i
1 ∂ √ ∂ √ i
= √ ( gϕ) + i ( gg · (v − W)ϕ) (7.23)
g ∂t ∂x
x i
in conservative form. It may be convenient to put

U = v − W, (7.24)

which clearly represents the continuum velocity relative to the moving grid points.
Exercise 1. Show that

√ ∂ √ ∂ √ i
gC = ( gϕ) + ( gϕU ). (7.25)
∂t ∂x i
It follows that if the hosted equations are given by
∂ϕ
+∇ · (vϕ) +∇ · (ν∇ϕ) + S = 0, (7.26)
∂t
rather than the time-independent eqns (5.119), the transformed equations (on a fixed
grid in computational space) may be written in conservative form as

∂ √ ∂ √ i ij ∂ϕ √
( gϕ) + g U ϕ + νg + gS = 0, (7.27)
∂t ∂x i ∂x j
with appropriate summation over i and j.

In place of eqn (5.121) we also have the non-conservative form
∂ϕ i 2 i ∂ϕ ij ∂ ∂ϕ i ∂v
+[U + ν(∇ x )] + g ν + ϕg · + S = 0. (7.28)
∂t ∂x i ∂x i ∂x j ∂x i