Geoscience Reference
In-Depth Information
C
D
A
B
Q
in
A downstream-narrowing channel
with constant discharge (
Q
)
The field situation depicted involves absolutely steady discharge, so that
at each section AB and
CD there is no variation of discharge or velocity
with time … but there is a change in space
In symbols
d
u
/d
t
= 0
C
D
s
Water
surface
A
B
Bed of channel
Discharge in,
Q
in
, equals discharge out,
Q
out
. Area of cross-section AB
>> area of
cross-section CD. By continuity: mean flow velocity
u
AB
<< mean flow velocity
u
CD
.
Therefore a spatial acceleration takes place over distance,
s,
between cross-sections AB and CD.
Magnitude of the spatial acceleration is
u
AB
(
u
CD
-
u
AB
)/s
In differential form,
u
d
u
/d
s
The full expression for acceleration is written below. It looks rather complicated, but is not. The numbered terms
are discussed in the text.
D
u
∂
u
∂
u
∂
u
∂
u
∂
u
≡
+
u
+
w
+
v
≡
+
u
⋅
∇
u
D
t
∂
t
∂
x
∂
z
∂
y
∂
t
term
1
2
3
4
Fig. 3.6
Spatial or advective acceleration and the full expression for total acceleration that includes the unsteady term. The Scale dog is Alaska,
1m long.
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