224 Computational Modelling in Hydraulic and Coastal Engineering
hyperbolic equation may be required to describe the phenomenon. Under
steady-state uniform flow conditions, the water discharge across the canal
per unit canal length, q w , is constant:
q w = uh = constant
(8.30)
Therefore, in order for Equation 8.30 to remain constant, the mean velocity (u) and the average depth (h) would change accordingly. Based on the
Meyer-Peter and Muller formula, the bed load sediment discharge can be
expressed to the flow velocity (u) as
q b = αu 3 + β
(8.31)
where α and β are coefficients depending on the sediment characteristics
(Samaras and Koutitas 2008). Utilizing mass continuity principles for the
movable bed, the bed evolution can be described by any of the following
equations:
∂
∂
= −
∂
∂
ζ b
b
t
q
x
(8.32)
∂
∂
=
∂
∂
h
t
u
x
α
3
(8.33)
∂
∂
=
∂
∂
= −
∂
∂
= −
∂
∂
−
h
t
q
h
x
q
h
h
x
c
h
x
w
w
α
α
3
3
3
4
3
*
(8.34)
Equation 8.34 is a advective transport equation (see Chapter 3, Equation
3.30), where the ‘signal’ h(x,t) propagates with a celerity c* along the
x-direction.
Example 8.5
This application simulates the evolution of a dredged section in a
moveable bed canal by utilizing the bed load sediment transport equations (Equations 8.31 and 8.32). The data used are as follows:
Length of canal section = 400 m
Water depth = 2 m
Water discharge per unit width = 2 m 3 /m/s
Sediment transport coefficient = 0.01 s 2 /m
Location of dredged section = Centre of dredge is 102 m from the
canal entrance
Dredge geometry = Trapezoidal, bottom width 16 m, top width
52 m, depth 2 m below the bed