International Symposium on Mitigative Measures against Snow ...

<strong>International</strong> <strong>Symposium</strong> on Mitigative Measures against Snow Avalanches
Egilsstaðir, Iceland, March 11–14, 2008
may overflow the dam as the flow front hits the dam. The height needed such that none of the
avalanche front overflows the dam is given by:
Hcr = h1 [1/k+ ½ (kFr sinγ) 2 – ½ (Fr sinγ) 2/3 ] + hs, Equation 4
if effects of pressure impulse are neglected (Peregrine, 2003).
The depth of the bore formed upstream of a catching dam (γ = 90° in Equation 4) is solved
from:
h
Fr =
h
2
1
1 ⎛
⎜ 1
1−
2 ⎜ h
⎝ 2 /h
1
2
3
⎛ 1 ⎞ ⎛ 1 ⎞ ⎞
−
⎟
⎜
⎟ + ⎜
⎟ . Equation 5
⎝ h ⎠ ⎝ ⎠
⎟
2 /h1
h2
/h1
⎠
If the deflecting angle of a deflecting dam is less than a certain maximum deflecting angle a
stationary oblique shock is formed upstream of the dam. The depth of the shock is given by:
where the shock angle β is given by:
tan β
h 2 = h1
, Equation 6
tan
( β −γ
)
2 2
4sin
β cosβ(
1−
Fr sin β)
tanγ
=
, Equation 7
2
2 2
2 2
− 3 + 4cos
β(
1−
Fr sin β)
− 1+
8Fr
sin β
The guidelines (SATSIE, 2006) give exact and approximate solutions to Equations 5 and 6.
The final dam height needs to be larger than the shock depth and the critical dam height
needed for a shock to form such that none of the avalanche overflows the dam:
Dimensionless dam height
25.0
20.0
15.0
10.0
5.0
3 5 7 9
Fr
( H cr ; h ) hs
H max + . Equation 8
= 2
a) Catching dam b) Deflecting dam
k =0.75
λ =1.5
λ =2
Dimensionless dam height
80 The design of avalanche protection dams based on new design criteria:
Three different case studiew
9.0
7.0
5.0
3.0
1.0
k =0.75
λ =1
γ =30°
3 5 7 9
Fr
γ =20°
γ =10°
Figure 1 The dimensionless dam height, H/h1 as a function of the Froude number for the
shallow-layer theory (green lines, Equation 8) and the traditional design (black
lines, Equation 1) for steep a) catching dams and b) deflecting dams with γ = 10°,
20° and 30°.