ρi
Q1
b
Y
Q2
k h
Cc = ϕd1
, , , C1 , C2 , α, θ, , .
, ,
,
(14)
Q2 Db2 gD D 1,3 D 1,2,3 D D
ρ
Equation 14 can be rearranged to relate channel widths in a more meaningful
manner:
Q1
b1 b2 b3 D k h
Q2
Cc = ϕd2
, α, θ, , , , , ,
, C1 , C2 , .
,
(15)
b2 b3 D h D Y 1, 2, 3
Q2 Db2 gD
These parameters are useful for describing how confluence flow conditions il-
lustrated in Figure 8 influence ice movement and jamming in a simple confluence
of prismatic channels. The diagnostic experiments conducted for the present study
investigated the influences on Cc (i.e., jam initiation) of the first five parameters in
eq 15. The experiments showed that jamming would occur for a narrow range of
values for the concentrations C1 and C2; i.e., both values need to be close to the
critical value of ice concentration for each channel. Alternatively, values of ice piece
size relative to channel width, D/b3, need to be sufficiently large such that ice
pieces arch; usually, ice-piece width to channel width must exceed about 1/7 for
arching to occur.
Several additional remarks should be made about the parameters in eq 15. An
alternative combination of the first five parameters would produce the parame-
ters bc/b3 and bbar/b3, which define the maximum widths of the dividing stream-
line bc and the bar bbar relative to the width of the downstream channel b3. These
two terms are useful when discussing jam formation in confluences. The two terms
define the extents of maximum flow contraction in the confluence.
Also, in the simple case of a single channel entering a lake, b1/b3 ≈ 0, b2/b3 =1,
θ = 180, and Q1/(Q1 + Q2) ≈ 1.
Equations 14 and 15 can be made more elaborate by including additional vari-
ables, such as different ice pieces and roughness conditions in the two confluent
channels. For most confluences, the first nine parameters usually will be of far
greater importance than the last parameters in eq 15. Only when the outflow chan-
nel is comparatively shallow or rough will the last parameters be important.
The effects of viscosity and surface tension become important when conducting
hydraulic modeling using small pieces of model ice for simulating ice movement.
Then, eq 15 should be expanded to include values of Reynolds number, QD/(νbY),
and Weber number, ρLQ2/(bY)2σ, for the channels.
Ice layer movement through a confluence
Water flow in a channel with a moving layer of accumulated ice pieces, as illus-
trated in Figure 9, can be defined using its discharge, Q, and the variables Y, b, k
and ki. The volumetric rate of ice layer discharge G (a contiguous layer of accumu-
lated ice pieces extending across the full width of the channel and moving at a
speed less than the surface water speed in a single channel) can be written as a
discharge rate of solid ice in the layer is ηQ, and p is layer porosity. The material
behavior of the layer can be defined using its thickness H, angle of internal resis-
As explained above, the effects of water viscosity and surface tension can be
neglected for the scale of flows in rivers. Water density ρ must be retained for use
with ρi as a variable, and with g in terms of specific weight γ.
20
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