Page 192 - Basic Structured Grid Generation

Moving grids and time-dependent co-ordinate systems 181

where now subscripts outside brackets indicate (where clarity is needed) which vari-
ables are being held constant when partial differentiation is performed. But the trans-
formation (7.1) also gives

∂y i j ∂y i
dy i = dx + dt,
∂x j ∂t j
t x
so that

∂f ∂y i j ∂f ∂y i ∂f
df = j dx + + dt,
∂y i ∂x ∂y i ∂t j ∂t
t t t x y i
which gives the Chain Rules

∂f ∂f ∂y i
= , (7.3)
∂x j t ∂y i t ∂x j t

∂f ∂f ∂y i ∂f
= + . (7.4)
∂t ∂t ∂t
x j ∂y i t x j y i
j
Note that in eqn (7.4) the left-hand side time derivative is carried out at fixed x ,i.e. at
a fixed point in computational space, which we may imagine corresponding to a given
(moving) grid point in physical space, whereas time derivatives evaluated at fixed y i
on the right-hand side are taken at fixed positions of physical space. This equation may
be written as

∂f ∂r ∂f ∂f
=∇f · + =∇f · W + , (7.5)
∂t x j ∂t x j ∂t y i ∂t y i
where

∂r
W = (7.6)
∂t
x j
is the rate of change of position of a given grid point in the physical domain, and may
be called the grid point velocity. Thus the time-derivative of the quantity f at a fixed
point of the physical domain is related to its time-derivative at a fixed point of the
computational domain by the equation

∂f ∂f
= − W ·∇f, (7.7)
∂t ∂t j
y i x
with ∇f evaluated in the physical domain.

7.2 Time-dependent base vectors

As the curvilinear co-ordinate system moves in physical space, the covariant base
vectors g i must be functions of time t. In fact we have
∂r
g i = , i = 1, 2, 3,
∂x i