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MODEL VERIFICATION AND CALIBRATION
Figure 7. Comparison of simulated jam profiles and the analytical solution
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2
2D
Result
1D
Result
0
1.06
m
Ice
Cover
2
4
River
Bed
6
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Distance
Along
River
(m)
Figure
6.
Comparison
of
water
surface
profiles.
Without
bank
friction
With
bank
friction
An analytical solution
for
the
widthaveraged
ice
The
steadystate
ice
thickness
profile
can
be
obtained
jam
thickness
profile
can
be
derived
by
extending
the
by
simplifying
the
momentum
equation,
eq
22.
When
only
the
water
drag
in
x
direction,
i.e.,
F
wx
= ρ
c
w
V
wx
,
2
solution of
Pariset
and
Hausser
(1961).
is
considered
and
the
bank
friction
is
neglected,
the
ice
2
1
1
momentum
equation
is
simplified
to
R
x
+
F
wx
= 0.
The

x
t
i
=
t
eq
(1 
e
B
)
2
(51)
internal
ice
resistance
reduces
to
where
B
is
the
channel
width
and
the
equilibrium
ice
R
x
=
(σ
xx
Nt
i
) +
(σ
xy
Nt
i
) =
(
PNt
i
)
.
(49)
thickness
is
x
y
x
Using
eq 37
for
the
pressure
term
leads
to a simple
1
2
(52)
BNC
w
V
wx
t
eq
= (
analytical
solution
for
the
static
ice
accumulation
)
2
ρ
i
ρ
i
(1 
g
2
thickness
profile
)
ρ
ρ
2ρ
c
w
V
wx
2
1
where
t
i
= (
t
i2
+
2
x
)
(50)
0
ρ
i
φ
tan
2
(
π
+ )(1 
)ρ
g
2
=
N
tan
φ(1 +
sin
φ)
ρ i
4
2
and
in
which
t
i0
=
single
layer
ice
thickness
and
x
=
distance
1
=
tan
φ(1
sin
φ)
(Beltaos
1995)
from
the
leading
edge
of
the
jam
where
t
i
=
t
i0
.
The
simulated
results
and
analytical solution
are
compared
For
in
Figure
7.
Figure
7a
shows
progressive
thickening
φ = 46
and
compressing
of
the
simulated
ice
cover
with
time.
2
=
1.068
Figure 7b
compares
the
analytical
solution
to
the
1
=
0.29.
simulated
ice
thickness
profile
on
and
after
t
= 4
hours.
Table
1.
Parameters
used
in
the
ice
dynamic
simulation.
Parameter
Description
Value
Maximum
ice
concentration
0.6
N
max
φ
46
Internal
friction
angle
of
ice
tan
φ
Boundary
friction
coefficient
1.04
Empirical
constant
15
j
Water
drag
coefficient
on
ice
0.02
c
w
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