Geoscience Reference
In-Depth Information
The inflow boundary condition is usually a time series of flow discharge. However, a
lateral distribution of velocity at the inlet is required in the depth-averaged 2-D model.
The streamwise (resultant) velocity
U
at each computational point of the inlet located
in a nearly straight reach may be assumed to be proportional to the local flow depth,
i.e.,
U
h
r
. Here,
r
is an empirical exponent;
r
3 for uniform flow. A small
r
value means a fairly uniform distribution of velocity along the inlet cross-section.
Therefore, for a given total inflow discharge
Q
∝
≈
2
/
(
=
0
Uhdy
)
,
U
is determined by
Qh
r
B
0
h
1
+
r
dy
U
=
(6.17)
where
B
is the channel width at the water surface, and
y
is the transverse coordinate.
The inflow velocity direction must also be specified; it essentially determines the
two components of velocity in the
x
- and
y
-directions at each point of the inlet.
The boundary condition at the outlet may be a time series of the measured water
stage, a stage-discharge rating curve measured or generated using the uniform or
critical flow condition, or a non-reflective wave condition, depending on the out-
let configurations. For tidal flow, the tidal level may also be determined using the
major astronomical constituents of tide in the study reach.
If a
k
-
turbulence model is used, boundary conditions should be given for the
turbulent energy and its dissipation rate at the inlet and outlet. At the inlet that is
located in a nearly straight reach and far from hydraulic structures, the turbulence can
be assumed to be at equilibrium; thus, Eqs. (6.10) and (6.11) are simplified to
ε
P
k
ν
−
ε
=
0
(6.18)
in
2
in
k
in
=
c
ε
2
ε
P
εν
−
0
(6.19)
yielding
U
2
∗
U
3
∗
in
c
ε
c
1
/
2
in
k
in
=
,
ε
=
(6.20)
in
c
1
/
4
f
c
1
/
2
f
h
in
µ
where the subscript “
in
” denotes the quantities at the inlet.
At the outflow boundary, located in a reach with simple geometry and far from
hydraulic structures, the gradients of flow velocity, turbulent energy, and dissipation
rate can be given zero.
6.1.3 Numerical solutions
Unlike the Navier-Stokes equations described in Section 4.4, the shallow water
equations (6.1)-(6.3) have a stronger linkage between velocity and pressure (water
level), due to the appearance of flow depth in the depth-integrated continuity equation.
It is apparently easier to solve Eqs. (6.1)-(6.3), as
h
,
U
x
, and
U
y
can be calculated by
these three equations, respectively. However, special care is still needed in handling
the convection and pressure gradient terms. Either a staggered grid approach or Rhie
