72 Computational Modelling in Hydraulic and Coastal Engineering
4.3.4.1 Initial conditions
The initial conditions refer to the pre-existing steady flow, characterized by
a constant discharge within the conduit and a linear decrease of the pressure head from a maximum value upstream (in the reservoir feeding the
conduit) to a minimum value downstream (zero relative pressure in the case
of a free outflow to the atmosphere). By using the Darcy-Weisbach equation, the initial steady state velocity (u o ) is
u
gDH
LK
o
o
f
= 4
(4.29)
H o is the pressure head at the feeding reservoir, and L is the length of the
conduit.
4.3.4.2 Boundary conditions
The boundary conditions refer to the pressure and velocity values at both
ends of the pipe. Those variables are interrelated in each cross-section via
the local specifics (so two conditions are necessary). As explained in the following, the boundary conditions applied are (1) on the downstream section,
a known relation between time and discharge (the valve closure schedule);
and (2) on the upstream condition (first reach of the conduit inside the reservoir), where pressure head is known (i.e. the water depth of the reservoir).
The valve closure schedule (u nx (t)) is expressed as
u
u
n t
T
nx
o
=
−
1
∆
(4.30)
where n is the number of time steps passed, Δt is the time step and T is the
valve closure time.
4.3.4.3 Numerical algorithm
To avoid the collocated calculation of u and h, we use a staggered computational grid. According to this discretization procedure, the spatial domain
is divided into a number of cells. However, the computational points for u
and h are not the same, since the velocities are calculated at each side of
the cell, while the pressure heads are calculated at the mid-point of the cell
(and remain constant throughout the cell). This type of spatial discretization, known as the Arakawa C-grid, ensures the conservation of mass in
the numerical solution (Figure 4.8).
Indeed, the scheme equates the difference of incoming and outgoing discharges from the two sides of the cell to the change of the volume of water
4.3.4.1 Initial conditions
The initial conditions refer to the pre-existing steady flow, characterized by
a constant discharge within the conduit and a linear decrease of the pressure head from a maximum value upstream (in the reservoir feeding the
conduit) to a minimum value downstream (zero relative pressure in the case
of a free outflow to the atmosphere). By using the Darcy-Weisbach equation, the initial steady state velocity (u o ) is
u
gDH
LK
o
o
f
= 4
(4.29)
H o is the pressure head at the feeding reservoir, and L is the length of the
conduit.
4.3.4.2 Boundary conditions
The boundary conditions refer to the pressure and velocity values at both
ends of the pipe. Those variables are interrelated in each cross-section via
the local specifics (so two conditions are necessary). As explained in the following, the boundary conditions applied are (1) on the downstream section,
a known relation between time and discharge (the valve closure schedule);
and (2) on the upstream condition (first reach of the conduit inside the reservoir), where pressure head is known (i.e. the water depth of the reservoir).
The valve closure schedule (u nx (t)) is expressed as
u
u
n t
T
nx
o
=
−
1
∆
(4.30)
where n is the number of time steps passed, Δt is the time step and T is the
valve closure time.
4.3.4.3 Numerical algorithm
To avoid the collocated calculation of u and h, we use a staggered computational grid. According to this discretization procedure, the spatial domain
is divided into a number of cells. However, the computational points for u
and h are not the same, since the velocities are calculated at each side of
the cell, while the pressure heads are calculated at the mid-point of the cell
(and remain constant throughout the cell). This type of spatial discretization, known as the Arakawa C-grid, ensures the conservation of mass in
the numerical solution (Figure 4.8).
Indeed, the scheme equates the difference of incoming and outgoing discharges from the two sides of the cell to the change of the volume of water