Geoscience Reference
In-Depth Information
Figure 4.10
Local coordinate transformation on moving grid.
Because coordinate transformation (4.89) is time-dependent, it has relations (4.85)
and (4.86) as well as the following:
9
9
∂
x
∂τ
=
∂
x
k
∂τ
ϕ
k
,
∂
y
∂τ
=
∂
y
k
∂τ
ϕ
k
,
∂
t
∂ξ
=
t
∂η
=
∂
t
∂τ
=
∂
0,
0,
1
(4.90)
k
=
1
k
=
1
The local coordinate transformation is convenient for complex movable bound-
ary problems. Because at each time (or iteration) step the used grid conforms to
the physical domain, the complex irregular and movable boundaries can be resolved
effectively.
Stretching coordinate transformation
The stretching coordinate transformation, which is also called the
-coordinate
transformation, is an algebraic example of the unsteady coordinate transformation
introduced in Section 4.2.3.1. If the boundaries are simple and vary gradually, the
physical domain can be expanded or contracted along one or more directions by
the stretching coordinate transformation, so that a fixed, regular logical domain is
obtained. For example, for 2-D gradually varied open-channel flows, the stretching
coordinate transformation shown in Fig. 4.11 is often used, which is expressed as
σ
⎧
⎨
⎩
ξ
=
x
z
−
z
b
(4.91)
ζ
=
H
h
τ
=
t
where
h
is the width of the physical domain, either the flow depth or channel width;
H
is the width of the logical domain; and
z
b
is the distance from the lower boundary
to the
x
axis. For the vertical 2-D case,
z
b
and
h
are the bed elevation and flow depth
and vary with
x
and
t
.
