Environmental Engineering Reference
In-Depth Information
B
§
W
U
·
³
''
u
(2
gh
))
'
¨
0
d
t
(4.A14)
¸
h
©
¹
B
c
m
Point
P
' is always in the undisturbed area as it moves from
B
' to a point near
B
, and '(W
0
/U
m
h
) is zero in
the process of integration except at point
B
. Since '(W
0
/U
m
h
) is not infinite at
B
, Eq. (4.A14) gives
or
(4.A15)
'
u
'
(
2
(
gh
)
)
0
'
u
'
(
2
(
gh
)
)
For a low discharge of mudflow, the average velocity is small. If the yield stress of the mud is large,
the second term on the right hand side of Eq. (4.A13) is negligible, and the equation can be rewritten as
§
·
d
g
1
§
W
·
1
W
' '
h
B
B
'
h
(4.A16)
¨
¸
¨
¸
¨
¸
d
t
h
2
U
h
2
U
h
2
©
¹
©
¹
m
m
In the process of deriving Eq. (4.A16) and Eq. (4.A15) and the following formulas
d
'
2( )
gh
2( )
gh
'
h
()
gh
'
h
(4.A17)
d
h
§
W
·
d
d
§
W
·
W
'
B
B
'
h
B
'
h
(4.A18)
¨
¸
¨
¸
U
h
h
U
h
U
h
2
©
¹
©
¹
m
m
m
have been used.
The integration of Eq. (4.A16) yields
W
B
'
h
h
t
ghh
2( )
U
(4.A19)
e
m
'
0
where '
h
0
is the initial perturbation in depth.
Equation (4.A19) indicates that the initial perturbation '
h
0
will grow, and the larger the yield stress W
B
and the smaller the mud depth
h
, the faster the wave will grow. After a perturbation develops into a roll
wave, the continuities of velocity and depth no longer hold, thence Eq. (4.A19) does not hold true. Therefore,
the wave height cannot grow indefinitely.
If the average velocity u and the rigidity coefficient K are large and the yield stress is small, the second
term on the right hand side of Eq. (4.A13) is much larger than the first one, and Eq. (4.A13) may be
rewritten as
§
·
d
1
K
U G
u
()
' '
¨
u
¹
(4.A20)
¸
d
t
2
h
©
m
in which Eq. (4.A18) has been used. Since
§
·
ª
º
ª
º
K
u
w
K
u
w
K
u
K
u
'
u
'
h
K
§
·
'
'
u
'
h
(1
Fr
)
'
u
(4.A21)
¨
¸
«
»
«
»
¨
¸
UG
h
w
u
UG
h
w
h
UG
h
UG
h
u
h
UG
h
©
¹
©
¹
¬
¼
¬
¼
m
m
m
m
m
Equation (4.A20) can be rewritten as
d
K
UG
()
'
u
1
'
Fr
)
u
(4.A22)
d
t
2
h
m
or after integration
K
(1
Fr
)
'
u
u
t
2
U G
h
e
(4.A23)
m
'
0
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