| Phenomenon of intermittent surges is known a debris flow called viscous debris flow in
China, and recently is observed in the European Alps and other mountains region. A purpose
of this research is to obtain a wave equation for wave motion of intermittent surges
with sediment on inclined channel, especially to evaluate influence of momentum
correction factor on flow mechanism. Using non-dimensional basic equations as Laplace
equation, -2Ï′--x′2 + -2Ï′--y′2 = 0 , boundary condition at bottom of flow,
-Ï′--y′ = 0,(y′ = -1; at bottom of mean depth h0 ), surface condition ( conservation
condition of flow surface ),
′ ′ ′ ′
- -Ï-+ -η- + -Ï--η-= 0 (y′ = 0;atsurfaceofmean depth h0 ),
-y′ -t′ -t′-x′
and momentum equation,
′ ( ′)2 ′2
-Ï-+ 1 (2β - 1) -Ï- - c0′2 tanθx ′ +c0′2 (1+ η′)+ tan θ c0-Ï′
-t′ 2 -x′ u0′
-« ( -Ï′)2 -η′ ′ ′ u0 ′ c0
+ (β - 1) -x′ -x′dx = 0, here,u0 = v–, c0 = v–
p0 p0
where, x : coordinate axis of flow direction, x′ = x-h0, y : coordinate axis of depth direction,
y′ = y-h0, h : depth of flow, h0 : mean depth, t : time, t′ = tvp0-h0, u0 : mean velocity, vp0
: velocity parameter in G-M transfer, Ï = Ï(x,y,t) : potential function, Ï′ = Ï-(h0 vp0), g :
acceleration due to gravity, θ : slope angle of the channel, c0 = -––––
gh0cosθ. From these basic
equation, a wave equation is obtained as follow by perturbation method, here neglecting the
term of Ï′ with tanθ -ª 1,
-η′ 1 ′2 ′ -η′ 1 c0′2 -2η′ 1( 1 ) -3η′
-Ï′ + 2 (2β + 1) c0 η -ξ′ - 2 tanθ u-′--ξ′2-+ 2 c-′2– 1--ξ′3 = 0,
0 0
where η : deflection from h0 (h = h0 + η), η′ = η-h0, ξ = ε1-2(x - vp0t), ξ′ = ξ-h0,
Ï = ε3-2t, Ï′ = tvp0-h0, ε: parameter of perturbation method.
In this equation, second term of left side is non-linear term which generates waves of various
periods, third is dissipation term which disappear high frequency wave and forth is dispersion
term which has a characteristic of a soliton on KdV equation. In a case using vp0 = c0, above
equation is expressed as
-η′ 1 ′ -η′ 1tanθ--2η′
-Ï′ + 2 (2β + 1) η -ξ′ - 2 u0′ -ξ′2 = 0.
Usually β varies from 1 to 1.2, then it is expected that the influence of β for wave formation
η′ is small by above equation. For observation on wave characteristic of intermittent surges, it
is indicated to measure phase velocity of wave, mean velocity of the flow, depth fluctuation
and other usual terms. |