ρi

*Q*1

*b*

*Y*

, , , *C*1 , *C*2 , α, θ, , .

, ,

,

(14)

*Q*2 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:

*Q*1

, α, θ, , , , , ,

, *C*1 , *C*2 , .

,

(15)

*Q*2 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 *C*c (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 *C*1 and *C*2; 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*/*b*3, 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 *b*c/*b*3 and *b*bar/*b*3, which define the maximum widths of the dividing stream-

line *b*c and the bar *b*bar relative to the width of the downstream channel *b*3. 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, *b*1/*b*3 ≈ 0, *b*2/*b*3 =1,

θ = 180, and *Q*1/(*Q*1 + *Q*2) ≈ 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*/(ν*bY*),

and Weber number, ρ*LQ*2/(*bY*)2σ, for the channels.

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 *k*i. 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 η*Q*, 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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