Geoscience Reference
In-Depth Information
where
M
i
m
(
i
1, 2, 3) and
M
m
are the cofactors of
i
m
and
=
β
∂
t
/∂ξ
m
in the Jacobian
matrix
B
, respectively.
Under coordinate transformation (4.63), the first and second derivatives of function
f
are given by
∂
f
=
∂
∂τ
+
∂ξ
f
t
∂
f
∂ξ
m
(4.66)
∂
t
∂
m
∂
f
i
∂
f
∂ξ
m
m
x
i
=
α
(4.67)
∂
2
f
∂
i
∂
∂ξ
j
∂
f
∂ξ
m
n
x
j
=
α
α
(4.68)
∂
x
i
∂
m
n
and the substantial derivative is
Df
Dt
=
∂
f
u
i
∂
x
i
=
∂
f
f
∂τ
+ˆ
u
m
∂
f
∂ξ
+
(4.69)
∂
t
∂
m
where
u
i
and
u
m
are the velocities in the (
x
i
,
t
) and (
ˆ
ξ
m
,
τ
) coordinate systems,
respectively. They are related as follows:
=
∂ξ
m
∂
=
∂
x
i
∂τ
+
β
m
i
m
u
m
ˆ
+
α
i
u
i
,
u
i
u
m
ˆ
(4.70)
t
Note that like the Cartesian coordinate index
i
, the curvilinear coordinate index
m
is also subject to Einstein's summation convention.
In the (
) coordinate system, the continuity and Navier-Stokes equations of
incompressible flows are
ξ
m
,
τ
∂τ
+
∂(
∂
J
J
u
m
ˆ
)
m
=
0
(4.71)
∂ξ
j
∂τ
ij
∂
u
i
∂τ
+ˆ
u
m
∂
u
i
∂ξ
m
=
1
ρ
α
i
∂
p
∂ξ
m
+
1
ρ
α
m
m
F
i
−
(4.72)
∂ξ
m
and the scalar transport equation is
α
c
∂τ
+ˆ
∂
u
m
∂
c
∂ξ
j
∂
c
∂
c
∂ξ
m
n
j
m
=
α
ε
+
S
(4.73)
∂ξ
m
n
where
c
is a scalar quantity, such as mass concentration and temperature.
4.2.3.2 Typical coordinate transformations
Boundary-fitted coordinate transformation
A boundary-fitted coordinate transformation was adopted by Thompson
et al
. (1985)
to simulate flows around physical bodies.
In the 2-D case shown in Fig. 4.7,
