Environmental Engineering Reference
In-Depth Information
Appendix
Mathematical derivation and explanation for the phenomenon of roll waves
Pseudo-one-phase debris flow can be simulated with the Bingham model
KH
B
(4.A1)
in which
W
is the shear stress of the flow, W
B
is the yield shear stress of the fluid, K is the rigidity
coefficient (called Bingham viscosity by some researchers), and H is the shear rate, which is equal to the
velocity gradient in laminar flow.
For unsteady open channel flow, the de Saint Venant equations (one-dimensional continuity equation
and momentum equation) are
W
W
w
h
w
h
w
u
(4.A2)
u
h
0
w
t
w
x
w
x
w
u
w
u
w
h
W
0
(4.A3)
u
g
gJ
w
t
w
x
w
x
U
h
where
u
is the cross sectional average velocity, W
0
is the shear stress of the flow acting on the bed, or the
resistance of the bed to the flow,
h
is the depth of flow,
J
is energy slope, which is equal to bed gradient
s
in steady uniform flows, U is the density of the flowing mixture,
x
is the distance along the flowing
course,
g
is the acceleration of gravity, and
t
is time. Applying the method of characteristics, the two
partial differential equations can change into two groups of normal differential equations in which one
group follows the C1-family of characteristic curves
d
x
t
u
()
gh
(4.A4)
d
d
W
U
0
(
u
2(
gh
))
gJ
(4.A5)
d
t
h
m
and another group follows the C2-family of characteristic curves
d
x
u
()
gh
(4.A6)
d
t
d
W
U
(
u
2(
gh
) )
gJ
0
(4.A7)
d
t
h
m
In steady uniform flow, the velocity
u
and the depth
h
are constants, and the frictional resistance must
then be equal to the tractive force, i.e.
W
U
gJ
0
(4.A8)
h
m
If a perturbation induces increments '
u
, (2 (
'
gh
) )
and '(W
0
/(U
m
h
)), then Eq. (4.A5) becomes
d
W
§
W
·
( 2()
u
gh
' '
u
2( )
gh
gJ
0
'
¨
0
(4.A9)
¸
d
t
U
h
U
h
©
¹
m
m
Subtracting Eq. (4.A5) from Eq. (4.A9) yields
§
·
d
W
U
0
(
''
u
(2
(
gh
)))
'
¨
¹
(4.A10)
¸
d
t
h
©
m
Equation (4.A10) is called a perturbation equation along the C1-family of characteristic curves (Wang,
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