| This research is phase velocity of roll waves on debris flows. Phase velocity of roll waves has
different characteristics with mean velocity. Using the momentum and mass conservation
equations in a coordinate system moving with velocity c for a homogeneous fluid are as
follows
c-U– βU -U-+ c(1 - β )U--A--H = - g sinθ + g cosθ-H + f′U-2
-ξ -ξ A -H - ξ -ξ 2 R
( ) -A--H- -U-
U - c -H -ξ + A -ξ = 0
here, v(x,t) = U(x - ct) = U(ξ), h(x,t) = H(x - ct) = H(ξ), ξ = x - ct, v :
cross-sectional mean velocity, h : depth of flow, A : cross-sectional flow area, g :
acceleration due to gravity, θ : slope angle of the channel, R : hydraulic radius,
β : momentum correction factor, f′ : friction factor, x : coordinate axis of flow
direction, t : time, the occurrence condition of roll waves is obtained. From other
way to obtain the occorrence condition, the ratio U
-0c which U0 is mean velocity
at control section and c phase velocity (velocity of a coordinate system moving)
is obtained, and under the condition that the channel is rectanglar and the width
B of channel is much wider than the depth (B -« H), the ratio of U
c0 is led as
follows,
° ––––––––
( ) S–1-
U0- β––β-β–-1-+-B-Fr2-
c = S 1
β - B-F-2
r
here, Fr = –U––
-gHcosθ : Froude number, S : wetted perimeter.
Experiments have been conducted in an experimental flume of 28m in length, 10cm in
width and 10cm in depth. In a circulation system the tested material is pumped from the
extit of the flume back to the inlet, providing a constant discharge upstream. The
material used consisted of non-cohesive coal particles with a d50 of 0.67mm, a density
Ïă of 1.41g/cm3 and polypropylene particles with a d50 of 2.9mm, a density Ïă of
1.06g/cm3.
The experimental results and theoretical result (equation (3)) are correspondig well on
β = 1.02. |