Page 196 - Basic Structured Grid Generation

Moving grids and time-dependent co-ordinate systems 185

7.4 Transformation of continuity
and momentum equations

In this section we show the transformation of both the continuity and momentum
equations of continuum mechanics to a general time-dependent curvilinear co-ordinate
system. As indicated in the above sections, an important role is played by the grid-
point velocity vector W, and both the cartesian and the contravariant components of
W will be required at the grid nodes at any instant. With a differential model of
grid generation, a partial differential equation will have to be solved at each time-step
(a Dirichlet boundary-value problem) to obtain the new grid.

7.4.1 Continuity equation

The continuity equation of Continuum Mechanics can be written in terms of the density
function ρ(y i ,t) as
∂ρ

+∇ · (vρ) = 0. (7.29)
∂t
y i
This can be written with respect to a moving grid, using eqn (7.22), as
∂ρ

+ ρ∇· W +∇ ·[(v − W)ρ]= 0, (7.30)
∂t
x i
or, by (7.27), as
i
∂ √ ∂ √ i ∂ √ ∂ √ i ∂x
( gρ) + ( gρU ) = ( gρ) + gρ v + = 0, (7.31)
∂t ∂x i ∂t ∂x i ∂t
y j
using eqn (7.18).

7.4.2 Momentum equations

We start with the standard (non-conservative) cartesian form of the Momentum Equa-
tions for a viscous fluid with viscosity µ:

∂v i ∂v i ∂σ ij
ρ + ρv j = , (7.32)
∂t ∂y j ∂y j
y i
with the stress tensor σ ij given in terms of pressure p and viscous stress T ij by
σ ij =−pδ ij + T ij ,
where T ij is related to the strain-rate ε ij through the equations

1 ∂v k
T ij = 2µ ε ij − δ ij
3 ∂y k
and ε ij = 1/2(∂v i /∂y j + ∂v j /∂y i ).