Geoscience Reference
In-Depth Information
Fig. 5.1
Definition sketch for two-dimensional
free surface flow. FS indicates the free
surface of the flowing water.
FS
h
z
θ
x
or the
material
or
particle
derivative; for the mean motion it can be defined as
D
Dt
=
∂
∂
∂
∂
∂
∂
y
+
w
∂
t
+
u
x
+
v
(5.2)
∂
z
where, as in Chapter 2,
u
are the mean (in the turbulence sense) velocity
components of the fluid in the
x
,
y
and
z
directions, respectively, of the velocity vector
v
, v
and
w
v
).
To simplify the argument, consider a two-dimensional motion of water with a free
surface, which is located at a distance, taken normally to the bottom surface,
z
=
(
v
+
t
)
from an arbitrary reference; the water is flowing over a bottom, which is located at a
normal distance
z
=
z
s
(
x
,
t
) from that same reference (see Figure 5.1). Observe that,
contrary to its usage in Chapter 2, here the
z
-axis is not vertical, but has an angle
=
z
b
(
x
,
θ
with it. For the situation shown in the figure the function defining the position of the
water surface is
F
(
x
,
z
,
t
)
=
[
z
s
(
x
,
t
)
−
z
]
=
0; therefore, condition (5.1) becomes for
the water surface
u
∂
z
s
∂
x
−
w
+
∂
z
s
∂
t
=
0at
z
=
z
s
(5.3)
Similarly, the bottom surface can be described by
F
(
x
0, in
which the time dependency allows, in principle at least, for bottom sediment accretion
or erosion; thus Equation (5.1) leads to an analogous condition for the bottom interface
of the fluid, which looks the same as Equation (5.3), but with the subscript s replaced by
a subscript b. Usually, however, the bott
om
can be treated as a solid wall without slip,
so that this bottom condition reduces to
u
,
z
,
t
)
=
[
z
b
(
x
,
t
)
−
z
]
=
0. With the latter bottom condition,
the condition for the free surface (5.3) can also be written in terms of the water depth as
follows
u
∂
=
w
=
h
x
−
w
+
∂
h
t
=
0at
z
=
h
(5.4)
∂
∂
in which the water depth is defined as
h
=
z
s
−
z
b
and the reference level
z
=
0 is placed
at the bottom.
