Maintenance of the values of Froude, Rey-

are needed when modeling many ice-load situa-

nolds, and Weber numbers for model and proto-

tions, because the full-scale conditions of ice load-

type is the underlying basis of similitude for

ing and material behavior of ice are complex, still

hydraulic modeling. However, explicit simulta-

ill-defined, and subject to scientific discussion.

The important failure modes are flexure, shear,

neous satisfaction of Froude, Reynolds, and

and crushing. All three modes may occur simul-

Weber number similitude criteria is impractical

taneously during icestructure or iceship inter-

when water is the model fluid as well as the pro-

action, though one mode usually dominates. The

totype fluid. For water systems modeled using

waterline shape of a structure or ship and contact

water, the criteria collide: Froude number simili-

tude requires that the velocity scale λ V = λ L ,

conditions, together with the strength and thick-

which leads to Reynolds number similitude giv-

ness properties of an ice sheet, determine which

ing the kinematic viscosity scale λ ν = λ1.5 . Since

mode dominates. The most common dominant

L

inertial and gravitational forces dominate free

mode for hydraulic failure of ice sheets is flexure

surface flow, the Froude number is used as the

caused by change in the water-surface profile of a

principal similitude criterion, and the Reynolds

flow or shoving of ice under or above the sheet.

criterion is relaxed by only requiring that fully

To ensure that model ice deforms in the same

manner as ice at full scale, it is customary (e.g.,

turbulent flow be maintained in the model. For

the references given above) to prescribe that the

free surface flow conditions, the transition be-

ratio of ice strength σ and elastic modulus *E *for a

tween smooth and fully turbulent flow (where

particular loading mode be held constant at mod-

viscous forces become negligible) occurs at a Rey-

el and full scales; i.e.,

nolds number (based on the hydraulic radius) of

500 to 2000. For ice-covered flow, this value is ap-

λ(σ/E) = 1

(14)

proximately halved due to the halving of the hy-

draulic radius with the addition of the ice cover.

and that, at both scales, *E*/σ exceed a minimum

From the Weber number (if Froude similitude

value associated with brittle elastic failure. Many

holds), the scale for surface tension is

modeling guides (e.g., Schwarz 1977, Ashton

λ ψ = λ2 λρ ,

(13)

1986) stipulate a value of about 2000.

L

The Cauchy number, *Ch*, is often used as a si-

which shows that, if λρ is held as unity, surface

militude parameter for prescribing the load and

tension must be greatly reduced. This require-

deformation behavior of level sheets of ice. It is a

ment is very difficult to meet, especially when

convenient ratio of inertial and elastic forces

plastic is the model ice, as plastic typically pro-

whose value ideally should be the same in the

duces more surface tension than natural ice. A

model and prototype:

relaxation of the requirements for the Weber num-

ρ*V * 2

ber can also be made as long as the influence of

(15)

.

surface-tension forces remains small compared

to inertial and gravitational forces. This is almost

always the case in natural systems. In open-water

Dynamic similitude requires

models, surface tension can be assumed to be neg-

λ Ch = 1 = λρλ2 λ E .

ligible when depths of 30 to 50 mm are main-

(16)

V

tained. For ice-covered models, however, surface

tension may become a concern, depending on the

As water normally is used to replicate water in

model ice material used. This concern is dis-

model studies, and as Froude number equiva-

cussed under *Model Distortion *below.

equal to the length scale for undistorted models:

i.e.,

The strength and deformation properties of

λE = λL .

monolithic ice sheets are of primary interest for

(17)

modeling ice-sheet loading. Modeling requires a

model ice that not only satisfies buoyancy and

In accordance with eq 14 and 16, the strength

frictional requirements, but that also deforms and

scale equals the geometric scale; i.e.,

fails in the manner that dominates ice behavior at

λσ = λL .

full scale. Considerable judgment and experience

(18)

4