| Viscous debris flow in China is typical intermittent flow with high concentrated
sediment. These intermittent surges flows are observed not only in China of viscous
flow but also in the European Alps and other mountains region. It is important to
obtain wave equation for wave motion of intermittent surges. This research shows
wave equation of shallow water with sediment on inclined channel which include
intermittent debris flow. Using non-dimensional basic equation as follows, Laplace
equation,
-2Ï′ -2Ï′
-x′2 + -y′2 = 0
(1)
bottom boundary condition,
--Ï = 0, (y′ = - 1;bottom ofmean depthh )
-y′ 0
(2)
surface condition ( conservation condition of flow surface ),
-Ï′ -η′ -Ï′-η′
- –′ + –′ +–′–-′ = 0, (y′ = 0;surface ofmeandepthh0)
-y -t -t -x
(3)
and equation of momentum,
-Ï′ 1 (-η′)2 c ′
–′ +- –′ - c0′tanθ -
x′ + c0′2(1+ η′)+ tan θ-0′Ï′ = 0
-t 2 -x u0
(4)
where, Ï = Ï(x,y,t) : potential function, Ï′ = -Ï–-
h0vp0, x : coordinate axis of flow direction,
x′ = x
h0, y : coordinate axis of depth direction, y′ = y
h0, h0 : mean depth, t : time, t′ = tvp0
h0-,
vp0 : velocity parameter in G-M transfer, u0 : mean velocity, u0′ = u
v0p0-, h = h0 + η : depth
of flow, η : deflection from h0, η′ = hη0, g : acceleration due to gravity, θ : slope angle of the
channel, c0 = ––––
- gh0cosθ, c0′ = cv0-
p0.
Using a method of perturbation, Gardner-Morikawa(G-M) transfer, ξ′ = ε12 (x′ - t′),
Ï′ = ε32 and ε =perturbation parameter, the wave equation is obtained as follows,
( )
--η′ 1( ′2) ′-η′ 1 c0′2--2η′ 1 -1– -3η′
- Ï′ + 2 1+ 2c0 η -ξ′ - 2 tanθ u0′ -ξ′2 + 2 c0′2 - 1 -η′3 = 0
(5)
and in case of vp0 = -gh0-cosθ- = c0, the wave equation is as follow,
-η′ 3 ′-η′ 1 -2η′
-Ï′ + 2η -ξ′ - 2 tan θ-ξ′2 = 0
(6)
This is a Burgars equation. These mathematical considerations indicate the monitoring
system of intermittent debris flow which should observe the mean velocity and wave velocity
individually. |