be of far greater importance than the last param-

of water viscosity ν can be neglected if it is as-

eters in eq 9. Only when the outflow channel is

comparatively shallow or rough will the last

sumed that flow in the channels and around ice

parameters be important.

pieces is fully turbulent. For ice pieces in actual

rivers, surface tension σ is negligible. The num-

The effects of viscosity and surface tension

ber of variables finally reduces to 19.

become important when conducting hydraulic

If the inflows and outflow of ice occur as a single

model tests using small pieces of model ice for

layer of ice pieces of a given size conveyed in sub-

simulating ice movement. Then, eq 9 should be

critical flow conditions, the following functional

expanded to include values of Reynolds number

relationship may be written for the areal concen-

tration of ice discharge on the confluence outflow

channels.

channel *C*c, as the dependent variable of interest:

Flow in a single channel with a moving ice cover

, *C*1 , *C*2 , α, β, , ρ, ρi ) .

(7)

(Fig. 4) can be defined using its discharge *Q *and

depth *Y, *width *b, *channel roughness *k, *and ice

Equation 7 assumes that, for a confluence of riv-

roughness *k*i. The volumetric rate of ice-accumu-

ers, *Q*c = *Q*1 + *Q*2, and for a confluence of a river

lation discharge as a contiguous layer of ice ex-

and a reservoir or lake, *Q*2, *G*2 = 0, and eq 5 and 6

tending across the full width of the channel and

pertain. The 19 variables in eq 7 are reducible to

moving at a speed less than the surface water

16 nondimensional parameters, given two basic

speed in a single channel can be written as a volu-

dimensions (length and time) involved with the

volumetric discharge of ice through a confluence.

Q1

Q2

If a dimensional analysis is carried out using *D*,

lowing functional relationship emerges for the lim-

Y1

Y2

iting condition of a single layer of free-drifting ice

b1

b2

discharging through a confluence:

a

h2

h1

* Q*1

*b*

*Y*

,

,

,

H2

H1

*Q*1 + *Q*2 *D * 1,2,c *D * 1,2,c

ρ

, , *C*1 , *C*2 , α, β, , i .

,

(8)

ρ

Moving layer

Equation 8 can be rearranged to relate channel

of ice

widths in a more meaningful manner:

* Q*1

, *C*1 , *C*2 , α, β,

b

ρ

, , , i .

, 1 , 2 , c , ,

(9)

ρ

hc

These parameters are useful for describing vari-

bc

ous confluence conditions. For example, in the

simple case of a single channel entering a lake (Fig.

Yc

1d), *b*1/*b*c ≈ 0, *b*2/*b*c = 1, β = 180, and *Q*1/(*Q*1 +

Qc

elaborate by including additional variables, such

as different ice piece sizes and roughness condi-

tions in the two confluent channels. For most

confluences, the first nine parameters usually will

5