2.
MATHEMATICAL MODEL
The evolution of the bed or bottom surface elevation due to the transport of sediment as
Henderson, 1966):
∂z
+ ∇ ⋅ qb = 0
(1)
∂t
where z is the elevation of the bed relative to a datum located below the bed (z is positive
upwards) and qb is the bed load sediment transport function vector.
In order to close Eq. (1), a functional form of qb must be specified. It is assumed that the
sediment transport is always in the direction of the flow velocity, U = (u, v) where u and v are the
velocity components of the flow in the x and y directions, respectively. Thus the vector qb is
computed as:
^
qb = U qb
(2)
^
where qb is the magnitude of the sediment transport in the direction of the flow and U is the
^
normalized flow velocity vector (i.e. U = U U ). There are a number of empirical bed load
sediment transport functions available (e.g. Bagnold, Einstein, Meyer-Peter and Mueller, see for
example Sleath, 1984 for a thorough list), most of which can be transcribed in the following form:
qb = A ( U, H ,
)U
n
(3)
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