Wake and Rumer 1983, Shen et al. 1993). The present
along the boundary are set to be zero. Since the
study uses the Mohr-Coulomb yield criterion (Gutfraind
hydrodynamic model is a depth-averaged model, it is
and Savage 1997), i.e.
assumed that the floating ice boom does not affect the
flow condition. However, if the boom configuration can
(
)
(35)
σ l σ2 = σ l + σ2 sin φ
significantly affect the flow, an internal boundary
condition may be added.
where σ1 and σ2 are the principal stresses and φ is the
internal friction angle. The internal stress is expressed
Ice dynamics
in terms of a nonlinear viscosity , as follows:
Initial conditions for ice dynamic simulations are
ice concentration, ice velocity, and ice thickness in the
(36)
σ ij = 2 ε ij - ε kk δ ij - Pδ ij
channel. Ice thickness, ice concentration, and ice
where εij is the strain rate tensor. Assuming that the
velocity are required upstream boundary conditions.
principal axes of stress and strain rate coincide, we can
External and internal forces acting on the ice govern
express the viscosity as
the boundary ice velocity. However, when the ice
concentration is not very high, the internal ice resistance
is not significant, which is usually the case at the
P sin φ
(37)
= min
, max
upstream boundary, and the ice velocity at the upstream
ε1 - ε2
boundary may be approximated by the water velocity.
where ε1 and ε2 are the principal components of the
Limiting conditions for ice accumulation
strain rate tensor and max, the maximum value of the
behind the boom
viscosity, determines whether the stress state is inside
When surface ice arrives at the boom from upstream,
or on the yield envelope. When = max, the ice will
it may stop behind the boom to accumulate into an ice
flow as a viscous fluid, whereas it will flow in a plastic
cover. However, if the current velocity exceeds a critical
manner when is smaller than max. The following
entrainment velocity, the surface ice will submerge and
expression obtained by extending the constitutive
be transported downstream. If the flow condition
relationship for static ice jams (Shen et al. 1990) is used
permits the accumulation of the ice rubble behind the
to determine the pressure P:
boom, the upstream progression is limited by the
possible entrainment of surface ice at the upstream edge
πφ
1
P = -(σ1 + σ2 ) / 2 = [1 + tan 2 ( )]
of the ice accumulation. In addition, an increase in the
2
42
current velocity beneath the ice accumulation, caused
(38)
ρ ρ gt
N j
either by the thickening of the cover or an increase in
(1 - i ) i i (
)
ρ
water discharge, can erode ice particles from the
2 N max
underside of the cover and limit its progression. Neither
in which j = 15, an empirical constant, and the + and
surface ice entrainment nor undercover erosion was
signs are for convergent and divergent states,
considered in the lake ice boom simulation model of
respectively.
Shen et al. (1997), owing to the low current velocity in
lakes. The critical condition for ice accumulation behind
Initial and boundary conditions
a lake ice boom is the spillover of ice rubble when the
Hydrodynamics
boom load exceeds a critical value. This condition
Initial values of unit discharges qx,qy, and water level
should also be considered for river ice booms. However,
η at every finite-element node have to be specified for
owing to the increased effect of bank resistance, the ice
the hydrodynamic simulation. A steady-state condition
load on river ice booms rapidly approaches a limiting
that corresponds to the specified boundary conditions
value as the ice cover progresses a few channel widths
at t = 0 is generated with the model and used as the
upstream (Latyshenkov 1946, Tuthill and Gooch 1998).
initial conditions for qx,qy, and η. Boundary conditions
In this section, the limiting conditions for ice
used are the water level at the downstream boundary
accumulation behind boom will be discussed.
and discharge at the upstream boundary. The unit normal
Ice entrainment condition at the boom and
discharge distribution across the upstream boundary is
the leading edge of the ice cover
estimated by the stream-tube method. The calculated
unit-discharge distribution is adjusted to ensure that the
When water velocity is high, surface ice arriving at
water level across the width of the boundary is constant.
the boom will submerge and pass under it. Past field
Along land boundaries, such as shorelines and dikes,
experience suggests that ice retention is possible at river
the unit normal discharges at the finite-element nodes
locations where the surface water velocity is at or below
8
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