| Debris flow is sometimes lots of intermittent surge flows. These many surges are generated in
flow instability and it is a kind of roll wave. It is possible to generate the roll wave of shallow
water on the experimental flume. In case of constant discharge and uniform flow at upstream,
period of generated roll wave is not constant at downstream. A cause for unstable period of
roll wave is discussed in this research. A wave equation for roll wave on inclined channel is
obtained by the perturbation method considered shallow water momentum equation in case
for rectangular cross section, wide width channel B compared with mean depth h0 (
B >> h0 ), channel slope θ tanθ < 1 and Froude number Fr ≥ 1. A obtained
non-dimensional wave equation is
∂η′ ′∂η′ ∂2η′ ∂3η′
∂τ′ + a1η ∂ξ′ + a2∂ξ′2 + a3∂ξ′3 = 0
(1)
where,
a1 = (3∕2)c0′2,
( ′2 ) ′ ′
a2 = − (1∕2{) 1∕c0 − 1∕2 tan θ(c0}∕u0), (2)
a = (1∕2) (2 + c′4)∕(2c ′2)− 3∕2 ,
3 0 0
η′ = η∕h0 : fluctuation from mean depth h0, η′ = η∕h0, ξ′ = ξ∕h0 = ε1
2(x′− t′),
ξ = ε1
2(x−vp0t), x : axis of flow direction, t : time, vp0 : phase velocity, τ′ = (vp0∕h0)t = ε3
2,
τ = ε32t, t′ = (vp0∕h0)t, x′ = x∕h0, c0′ = c0∕vp0, c0 = √ --------
gh0cosθ, u0′ = u0∕c0, u0 :
mean velocity, ε : parameter of perturbative expansion.
Equation (1) is a kind of KdV - Burgers equation. For phase velocity vp0 is long wave
velocity c0, that is vp0 = c0, equation (1) becomes Burgers equation. In this state,
waves with different wave numbers deform to a wave of wave number one and
phase velocity is added some. Then the equation of the phenomenon is shifted
back to KdV - Burgers equation from Burgers equation. c0′ and u0′ are assumed as
constant coefficient in equation (1), however u0′ is fluctuated by depth fluctuation
in flow. Using a2′ by linear approximation as a2′ = a2(1 + b1η′), equation (1)
is
∂η′ ′∂η′ ′ ∂2η′ ∂3η′
∂τ′ + a1η ∂ξ′ + a2(1 + b1η )∂ξ′2 + a3∂ξ′3 = 0.
(3)
Some numerical results of equation (3) using by experimental results and b1 = 0.01 show
shift or drift of peak of surges. Unsteady period of roll wave originates in the fluctuation in
mean velocity influenced by deformed flow surface. |