Geoscience Reference
In-Depth Information
The velocity correction is written as follows:
u
∗
−
t
ρ
∇
u
n
+
1
p
=
(7.15)
where
p
u
∗
is defined in Eq. (6.35). The intermediate velocity
is determined by
−
t
ρ
∇
t
G
u
∗
=
u
n
p
n
+
(7.16)
with
G
representing all the remaining terms in the discretized momentum equations.
Substituting Eq. (7.15) into Eq. (7.1) yields
t
ρ
∇
2
p
=∇·
u
∗
(7.17)
The water level is determined using the kinematic condition (2.71), and the grid is
adjusted in the vertical direction to track the temporal variation of water surface.
7.2 3-D FLOW MODEL WITH HYDROSTATIC PRESSURE
ASSUMPTION
Based on the hydrostatic pressure assumption introduced in Section 2.4, the inertia
and diffusion effects in the vertical momentum equation of gradually varied (shallow
water) flows can be ignored. The resulting 3-D governing equations are written as
∂
u
x
∂
+
∂
u
y
∂
+
∂
u
z
∂
=
0
(7.18)
x
y
z
u
x
)
∂
∂
u
x
∂
+
∂(
+
∂(
u
y
u
x
)
∂
+
∂(
u
z
u
x
)
g
∂
z
s
1
ρ
∂τ
1
ρ
∂τ
xy
∂
1
ρ
∂τ
xx
xz
=−
x
+
+
+
+
f
c
u
y
t
x
y
∂
z
∂
∂
x
y
∂
z
(7.19)
u
y
)
∂
+
∂(
∂
u
y
∂
+
∂(
u
x
u
y
)
+
∂(
u
z
u
y
)
g
∂
z
s
1
ρ
∂τ
1
ρ
∂τ
1
ρ
∂τ
yx
yy
yz
=−
+
+
+
−
f
c
u
x
t
∂
x
y
∂
z
∂
y
∂
x
∂
y
∂
z
(7.20)
where
f
c
is the Coriolis coefficient, determined by Eq. (6.6).
The hydrostatic pressure assumption simplifies the full three-dimensional hydrody-
namic equations (7.1) and (7.2) significantly. However, this assumption is valid only
for gradually varied flows (bottom slope less than about 5%), and a full 3-Dmodel with
dynamic (non-hydrostatic) pressure should be used for rapidly varied flows around in-
stream structures, such as bridge piers, spur-dikes, and bendway weirs. Nevertheless,
gradually varied flows exist widely in rivers, lakes, estuaries, and coastal waters; thus,
the hydrostatic pressure assumption is often adopted. The 3-D models developed by
Sheng (1983), Wang and Adeff (1986), Blumberg and Mellor (1987), Casulli and
