•
Consider the transformation:
)
1
ln(
+
=
=
y
x
η
ξ
Inverse transformation
1
−
=
=
η
ξ
e
y
x

Example: Stretched (compressed ) grids
•
The following derivatives are used in the transformation
1,
0
1
0,
1
x
y
x
y
y
ξ
ξ
η
η
∂
∂
=
=
∂
∂
∂
∂
=
=
∂
∂
+
1,
0
0,
x
y
x
y
e
ξ
ξ
η
η
η
=
=
=
=
)
1
ln(
+
=
=
y
x
η
ξ
1
−
=
=
η
ξ
e
y
x

Example: Stretched (compressed ) grids
•
The relation between increments
∆
y and
∆
η
Therefore as
η
increases,
∆
y increases exponentially.
Thus we can choose
∆
η
constant and still have an
exponential stretching of the grid in the y-direction.
η
η
η
η
η
η
∆
=
∆
⇒
=
⇒
=
e
y
d
e
dy
e
d
dy

Example: Stretched (compressed ) grids
•
Transform the continuity equation
Note there is relation between (x,y) plane and (
ξ
,
η
) plane

Example: Stretched (compressed ) grids
Substitute for the derivatives in Eq. (5.54) to get
Eq. (5.57) is the continuity equation in the computational domain.
Thus we have transformed the continuity equation from the
physical space to the computational space.