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shake the mixture since the particle size is quite small. To take into account particle
sedimentation, authors generally consider airborne avalanches as turbulent stratified
flows. Thus, contrary to flowing avalanches, bulk behavior is well identified in the case
of airborne avalanches. The main differences between the various models proposed
result from the different boundary conditions, use of the Boussinesq approximation,
and the closure equations for turbulence. Parker et al. [PAR 86] developed a complete
depth-averaged model for turbidity currents. The motion equation set proposed by
these authors is more complicated than the corresponding set for dense flows presented
above, since it includes additional equations arising from the mass balance for the
dispersed phase, the mean and turbulent kinetic energy balances, and the boundary
conditions related to the entrainment of sediment and surrounding fluid:
∂h
∂t
+ ∂hU
∂x
= E a U,
[2.30]
( Ch )
∂t
+ ( hUC )
∂x
= v s E s − v s c b ,
[2.31]
+ ∂hU 2
∂x
2 Rg ∂Ch 2
∂hU
∂t
1
− u ,
= RCgh sin θ −
[2.32]
∂x
∂hK
∂t
+ ∂hUK
∂x
= 1
1
2 E a URCghRCghv s
2 E a U 3 + u U − ε 0 h −
1
2 Rghv s (2 C + E s − c b ) ,
[2.33]
where U is the mean velocity, h the flow depth, K the mean turbulent kinetic energy,
C the mean volume concentration (ratio of particle volume to total volume), E a a
coefficient of entrainment of surrounding fluid into the current, v s the settlement
velocity, E s a coefficient of entrainment of particles from the bed into the current,
c b the near-bed particle concentration, R the specific submerged gravity of particles
(ratio of buoyant density to ambient fluid density), u
the bed shear velocity, and
ε 0 the depth-averaged mean rate of dissipation of turbulent energy due to viscosity.
The main physical assumption in Parker et al .'s model is that the flow is considered
as one-phase from a momentum point of view but treated as two-phase concerning
the mass balance. Equation [2.30] states that the total volume variation results from
entrainment of surrounding fluid. In equation [2.31], the variation in the mean solid
concentration is due to the difference between the rate of particles entrained from the
bed and the sedimentation rate. Equation [2.32] is the momentum balance equation:
the momentum variation results from the driving action of gravity and the resisting
action of bottom shear stress; depending on the flow depth profile, the pressure
gradient can contribute either to accelerate or decelerate the flow. Equation [2.33]
takes into account the turbulence expenditure for the particles to stay in suspension.
Turbulent energy is supplied by the boundary layers (at the flow interfaces with the
surrounding fluid and the bottom). Turbulent energy is lost by viscous dissipation
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