Other assumptions include:
Cross sections are uniform and prismatic in shape, ensuring that streamline
curvature remains small. For this analysis, a rectangular channel shape is as-
sumed.
All flow is subcritical. Though there may be considerable changes in depth
and ice thickness between two cross sections (notably at the locations of shov-
ing or thickening fronts), it is assumed that through methods (i.e., with no
explicit representation of fronts) suitably describe these fronts.
Ice-piece properties remain constant (i.e., no heat transfer, phase change, or
freeze-bonding between ice pieces).
Jams are particulate continua, such that forces and stresses are describable
using Mohr-Coulomb stress theory and an average value across the cross sec-
tion.
Jams float with a constant bulk specific gravity, do not ground on the channel
bed, and are not subject to significant motion or accelerations in the vertical
direction.
Development of equations
The integral form of the equations for water and ice flow are developed using a
control volume approach. The Cartesian coordinate system used is depicted in Fig-
ure 24, in which x denotes horizontal distance along the longitudinal river axis, y
denotes vertical distance, and z denotes transverse distance normal to the longitu-
dinal axis.
Conservation of water mass
Conservation of water mass requires the net inflow of water entering a control
volume (bounded by x1, x2, the bed, and the bottom of the jam in Fig. 24) during a
given period be equal to the change in water storage within the control volume for
the period, that is
[
]
]
t2
x2
∫ [(ρuA)x2 - (ρuA)x1
dt + ∫ (ρA)t - (ρA)t dx = 0 .
(34)
2
1
t1
x1
For practical purposes, water is incompressible, such that ρ is constant and eq 34
reduces to
[
]
[(A)
]
t2
x2
∫ (uA)x2 - (uA)x1 dt + ∫
- (A)t dx = 0.
(35)
t2
1
t1
x1
Further simplifications are made subsequently, such as expressing area A in terms
of flow depth d(x,t).
Conservation of ice mass
The net inflow of ice and pore water (between the ice pieces) into the control
volume, bounded by x1, x2, and the bottom and top of the jam in Figure 24, is the
time integral of the difference between the mass flow rates entering the control
volume at x1 and leaving the control volume at x2, i.e.
t2
)x
)x
- (ρυAi si p)x dt
∫ (ρi υAi [1 - p]
+ (ρυAi si p)x - (ρi υAi [1 - p]
2
(36)
t1
1
1
2
30
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