Geoscience Reference
In-Depth Information
where
z
s
,1
d
,
z
s
,2
d
, and
z
s
,3
d
are the water levels calculated in the 1-D, 2-D, and 3-D
subdomains, respectively;
B
is the width of the interface; and
y
is the coordinate along
the interface.
Eq. (8.9) contains the conditions for connections between 1-D and depth-averaged
2-D models, between depth-averaged 2-D and 3-D models, and between 1-D and
3-D models. The connections with the width-averaged model are left to the interested
reader.
In the 3-D model that solves the full Navier-Stokes equations, the dynamic pres-
sure needs to be provided at the interfaces, whereas the 1-D and depth-averaged 2-D
models assume a hydrostatic pressure distribution there. To overcome this problem,
the interfaces should be located in the regions where the flow varies gradually and the
hydrostatic pressure assumption is valid.
Flow discharge
The flow discharges at the interfaces should satisfy the continuity condition:
B
Q
1
d
=
U
2
d
h
2
d
dy
=
u
3
d
dydz
(8.10)
0
where
Q
1
d
is the flow discharge calculated in the 1-D subdomain,
h
2
d
and
U
2
d
are
the flow depth and depth-averaged velocity in the 2-D subdomain,
u
3
d
is the local
velocity in the 3-D subdomain,
denotes the interface in the 3-D subdomain, and
z
is
the vertical coordinate.
Flow resistance
The bed shear stresses at the interfaces satisfy
χ
0
τ
b
,2
d
d
χ
0
τ
b
,3
d
d
χτ
b
,1
d
=
χ
=
χ
(8.11)
where
τ
b
,3
d
are the bed shear stresses calculated in the 1-D, 2-D, and
3-D subdomains, respectively; and
τ
b
,1
d
,
τ
b
,2
d
, and
is the wetted perimeter at the interfaces.
In 1-D and 2-D models, the bed shear stress is determined using either the Manning
equation or the log-law approach with the equivalent roughness height. Inserting the
Manning equations for 1-D and 2-D uniform flows into condition (8.11) yields
χ
χ
R
4
/
3
1
d
A
1
d
U
1
d
n
2
d
u
2
d
h
1
/
3
2
d
n
1
d
=
d
χ
(8.12)
0
where
R
1
d
and
U
1
d
are the hydraulic radius and the section-averaged flow velocity
calculated in the 1-D reach, respectively; and
n
1
d
and
n
2
d
are the Manning coefficients
in the 1-D and 2-D models, respectively.
It can be seen from Eq. (8.12) that in order to satisfy condition (8.11) at the interface
between 1-D and 2-D reaches,
n
1
d
and
n
2
d
may have to be given different values.
