| Intermittent surges of debris flow are observed in mountains region in Europe, Aisia and
others. A purpose of this research is to obtain characteristic of wave equation on shallow
water for debris flow surges.
Considering a flow in a rectangular straight channel, where the width is very large
compared to a flow depth, momentum correction factor β = 1 , constant friction factor over
mean depth h0, a channel slope tanθ < 1, Froude number Fr > 1 , and a long wave
condition by results of observations and experiments, a wave equation is obtained
with
/η′ ′/η′ /2η′ /3η′
/Ï′ + a1η /ξ′ - a2/ξ′2 + a3/ξ′3 = 0
(1)
where, a1 = (3/2)c0′2, a2 = (1/2)( )
1/c0′2 - 1/2tanθ (c0′/u0′),
a3 = (1/2){ 4 2 }
(2 + c0′ )/(2c0′)- 3/2 ,
and η : fluctuation of mean flow depth, h0 : mean depth, h = h0 + η : flow depth, η′ = η/h0,
x : coordinate axis of flow direction, x′ = x/h0, ξ = ε1/2(x - vp0), ξ′ = ξ/h0, vp0 :
phase velocity, the velocity parameter of Gardner - Morikawa transformation, y :
coordinate axis of depth direction, y′ = y/h0, t : time, t′ = tvp0/h0, Ï = ε3/2t,
Ï′ = (vp0/h0)Ï, g : acceleration due to gravity, θ : slope angle of the channel,
c0 = /––––
gh0cosθ : wave velocity of a long wave, c0′ = c0/vp0 u0 : mean velocity,
u0′ = u0/c0.
Using for vp0 = c0 under a long wave condition by observations and experiments, above
equation is expressed as
/-η′ 3 ′ /η′ 1tanθ-/2η′
/ Ï′ + 2 η /ξ′ - 4 u0′ /ξ′2 = 0.
(2)
This equation is a kind of Burgers equation. Analytical solutions for different wave number
k = 1/2,Â3/2,Â5/2 and k = 1,2,3 on initial conditions were obtained, and calculated by
numerical analysis. These results show that the wave shape are deformed to a wave of wave
number k = 1 for not multiple wave number. This indicates that a surge is formed with a
wave length from the wave of a lot of wave numbers in initial state on actual surges or
experimental surge flows. |