ic repose of the particulate material fully dry or
of qualitative significance and awaits further ex-
immersed in liquid. For a dry, angular cohesion-
amination to define a gradation of values to indi-
less material,
cate the range of ice-particle behavior. Its use is
analogous to the use of a Reynolds number in
π φ
characterizing the drag coefficients of bodies in
k1 = tan 2 + ,
(20)
4 2
moving fluids. In modeling, the aim would be to
ensure that the model-scale value of Bo remains
k0=1sinφ
within a range of values for which the sticky forces
,
(21)
are scaled in correct proportion with the inertial
and gravitational forces. Order-of-magnitude esti-
and
mates can be made for Bo, however. As it ap-
ξ = tanφ.
proaches zero, an ice-piece accumulation behaves
(22)
as if it were a cohesionless assemblage of discrete
When φ increases beyond about 45, such that
pieces; buoyancy dominates. When Bo exceeds
about 106, the accumulation behaves as a fused
extensive interlocking of ice pieces takes place, eq
structure of connected particles. At the extreme,
21 and 22 become less appropriate for estimating
k0 and ξ. Based on the force balance within an ice
the accumulation becomes monolithic ice. The im-
portant message here is that it may not be possible
cover, Zufelt (1992) shows that the value of an
alternate parameter, , the internal resistance coef-
to use ice to model ice, because fine-sized ice piec-
ficient of an ice cover, is related to k1, k0, and ξ by
es do not behave like large ice pieces, just as clay
does not behave like gravel.
= k1ξk0 .
(23)
The geometric factors affecting accumulation
strength are fairly straightforward to identify and
Beltaos (1993) has reported values of for natural
to scale. They include accumulation thickness,
ice jams in the range of 0.8 to 1.6, but these were
accumulation porosity, size and size distribution
back-calculated from estimates of jam thickness
of ice pieces comprising the accumulation, and
assuming equilibrium thickness theory. Little
shape and roughness of constituent ice pieces. It is
work has been accomplished on the interparticle
much less straightforward to scale the material fac-
friction of particulate ice pieces, ξ, or the coeffi-
tors, which include strength and deformation
cient of lateral pressure of wetted particulate
properties of constituent ice pieces and the tem-
perature of ice pieces. All of the aforementioned
masses, k0.
variables affect the angle of internal resistance of
an ice accumulation. Ice-piece size also affects the
MODEL DISTORTION
strength and deformation properties of ice pieces.
The lateral distribution of stress through an ac-
Practical considerations often make it necessary
cumulation, and the friction of accumulated ice
to relax similitude criteria in order to ensure that
pieces against other surfaces, are additional prop-
the model adequately replicates the dominant pro-
erties to be taken into account. Forces attributable
cess under investigation. Limitations in modeling
to lateral stress and ice friction are important for
area or flow capacity commonly constrain the hor-
structures or ships flanked by ice accumulations
izontal space, and thereby the horizontal scale, of a
and for ice-jam formation. The shear force at a slip
model. The necessity for small scale may result in
plane (e.g., along the side of a structure or a river
very small model depths, to the extent that viscous
and surface tension effects become significant. The
bank) depends on the coefficient of lateral pres-
sure, k0 (akin to a Poisson ratio), and the coefficient
remedy is to resort to geometric, or vertical, distor-
of friction of ice rubbing against itself, ξ, or against
tion.
A vertical length scale, βL, is chosen to keep the
some other material forming one side of the plane.
The passive pressure coefficient, k1, relates the
viscous and surface tension forces in the model at
maximum internal resistance to the average verti-
negligible levels. The resulting model distortion,
D = λL/βL, is usually maintained at less than 4 for
cal stress within a material. The parameters k1, k0,
and ξ can be expressed in terms of the angle of in-
open-water hydrodynamic models, although no
ternal friction, φ, of the accumulation. In turn, φ is
strict limit is set. Care must be taken in determin-
related to the shape and size distribution of the ice
ing the appropriate scales for horizontal or vertical
pieces constituting the accumulation. As a lower-
forces acting on horizontal or vertical planes. The
bound estimate, φ can be taken as the angle of stat-
scales are not the same. Consequently, whereas
6
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