the separation zone. The streamline delineates an interface profile for the merging
streams of ice. The separation zone delineates a further contraction at the conflu-
ence. It also delineates the approximate extent of the confluence bar, on which ice
may ground.
Confluence flow and bathymetry have received fairly extensive investigation.
Notable studies on confluence flows are Taylor (1944), Modi et al. (1981), Rama-
murthy et al. (1988), Fujita and Komura (1986), Best and Reid (1984, 1987), Hager
(1989), Gurram et al. (1997), and Hsu et al. (1998). Notable studies on confluence
bathymetry include those reported by Komura (1973), Mosely (1976), Ashmore
and Parker (1983), Best (1988), and Biron et al. (1996).
For the purpose of the present study, there is little point in furnishing an exten-
sively detailed summary of these studies, except to briefly summarize what is
known about the location of the dividing streamline, the magnitude of the separa-
tion zone, and the extent of the bar. Those features of confluence flow directly
affect ice movement through a confluence of concordant bed channels.
Location of dividing streamline
No simple analytical relationship exists for predicting the location of the divid-
ing streamline between confluent flows in an alluvial channel. A method for esti-
mating its position in a confluence of flat bottom channels, however, has been
developed by Fujita and Komura (1986).
Fujita and Komura (1986) used a mathematical model developed earlier by Modi
et al. (1981) to analyze flow in confluences of rectangular channels. By applying
potential flow theory and conformal mapping, the location of the dividing and
separating streamline is obtained through numerical integration of complex func-
tions. The equation for the dividing streamline is
ξ2
η2
xd = ∫ FR (ξ + iη2 ) dξ - ∫ FI (1 + iη) dη
(1)
1
0
ξ2
η2
yd = π + ∫ FI (ξ + iη2 ) dξ + ∫ FR (1 + iη) dη
(2)
1
0
for ∞ < ξ < 1 and 0 < η < ∞. FR and FI are the real and imaginary parts of the
complex function
[
]
1 1 - Qr
Qr
2α
bc
F(ζ) =
ln (ζ - 1) + ζ1/2 - αi - +
1/2
+
exp
(3)
π
π
ζ ζ + ck ζ - cl
where bc is the effective width of the channel (the width of the main channel dimin-
ished by the width of the separation zone, bc = b3 bs), Qr is the ratio of the tribu-
tary channel to the downstream discharge, and α is the confluence angle. The
axis of the upper half plane on to which the conformal mapping is represented,
and ε and η are the coordinates of the downstream corner of the confluent chan-
nels in the mapped plane obtained using the Schwarz-Christoffel theorem. The
resultant expression for streamline location agrees well with the experimental re-
sults obtained by Fujita and Komura (1986) for a wide range of discharge and
width ratios as well as confluence angles.
Equations 1 through 3 are cumbersome for most purposes related to ice convey-
ance and ice jam formation in confluences. A simpler, approximate procedure is
12
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