The remaining forces to be determined are the boundary resistance or shear forces.

The total shear stress produced by the water flow (averaged for the control-volume

reach) is

τ = ρ*gRS*f

(52)

associated with the water flow. The Darcy-Weisbach definition of friction slope is

(53)

8*Rg*

where *f*o is the composite (bed and ice cover) Darcy-Weisbach resistance factor.

Equation 53 substituted into eq 52 gives

ρ*gRf*ou2 ρ*f*ou2

τ=

=

.

(54)

8*Rg*

8

This shear stress is a total value, generated by the differences in the velocities of

water flow relative to the velocities of the other boundaries of the control volume.

It can be split into two parts: τb, the shear stress at the bed and bank boundary, and

τi, the shear stress at the ice boundary. Prior formulations (e.g., Beltaos 1983) have

shown that the simple case of a static ice cover can be analyzed approximately

using a "two-layer approach," separating the total flow area into one layer domi-

nated by shear stress on the bed and banks, and another layer dominated by shear

Pi

υ=0

α

i

α

b

Pb

a)

υ=0

τi

τ=0

u

τb

b)

34

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