As a demonstration of the modeling methodology, we perform Simulink/MatLab simulations for a section of the Bourne channel, from the Mondy gate (a mixed gate and spillway structure) to the branching junction with secondary channel S2 (see Fig. 1.2 and Fig. 1.8). This section
1.4 Fluid simulations | 25
f2_am
f3_av f1_av f2_av f1_am f3_am h u
Reach
f 1_am f 2_am f 1_av f 3_av Qp
f 3_am
f 2_av
Pump
f1_am f2_am f1_av f3_av theta1 theta2
f3_am
f2_av
Mixed-2G1SW
f1_am f2_am f1_av f3_av theta1
f3_am
f2_av
Mixed-1G1SW
f1_am f2_am f1_av f3_av theta
f3_am
f2_av
Gate
Fig. 1.7 A Simulink/MatLab Model Library is created for the modeling and the simulation of an irrigation channel.
consists of three segments interconnected by a pumping station at “Monts du Matin” and a mixed structure (composed of two gates and one spillway) at “Orme”. The flow rate and level meters are available at certain locations and periodically give the flow rates and water heights at different points in the considered section. By coupling different components of the Model Library (shown in Fig. 1.7) such as reach, pump, and mixed components through their interactions, a model of the considered section is created as shown in Fig. 1.9. Based on the model, the simulations require the geometric information of the corresponding section and boundary conditions at upstream and downstream ends to perform the hydraulic calculations for steady or transient flow. The selected geometric and simulation parameters are presented in Tables 1.1, 1.2 and 1.4. In simulations, we use a global flow axis with the beginning at Auberives barrage (see Fig. 1.2). Consequently, the relative location of the first considered section is between 3.25e4mto 3.62e4m. Assuming that the boundary conditions of the first considered section impose the water flow rate at the upstream end of Mondy gate,QM, and the water height at the S2 junction,hSas in Table 1.3 and Fig. 1.10.
Fig. 1.8 A section of Bourne channel is considered for modeling and simulation.
u2
u2
[time' theta2_Orme']
theta 2 Orme
[time' theta1_Orme']
theta 1 Orme
[time' theta1_Mondy']
theta 1 Mondy
h3
h3
h2
h2
f 2_am
f 3_av f1_av f2_av f1_am f3_am h u
Segment 3 f2_am
f3_av f1_av f2_av f1_am f3_am h u
Segment 2 f2_am
f3_av f 1_av f 2_av f1_am f3_am h u
Segment 1
f 1_am f 2_am f 1_av f 3_av theta1 theta2
f3_am
f2_av
Orme Mixed-2G1SW f1_am
f2_am f1_av f3_av Qp
f 3_am
f2_av
Monts du matin Pumping station
-K-
1/vB1
[time' Qp_MdM']
Pumping discharge
[time' Q_M']
Flow rate at Mondy
u3
u3
h1
h1
u1
u1 f 1_am
f 2_am f 1_av f 3_av theta1
f3_am
f2_av
Mondy Mixed-1G1SW
-K-
1/vB3
[time' Q_S']
Flow rate at S2
Fig. 1.9 A Simulink/MatLab model is created for the simulation of the first considered section.
Table 1.1 Geometric parameters of the irrigation channel.
Segment Length (L) Width (B) Slope (I) Manning coef. (n) Space st. (∆x)
1 100 m 5.12 m 2.400e-4 0.033 100 m
2 2600 m 5.20 m 2.441e-4 0.033 100 m
3 1000 m 5.40 m 2.440e-4 0.033 100 m
Table 1.2 Structural parameters of mixed hydraulic structures.
Mixed structure
Gate width (Bg)
Gate coeff. (α)
Spillway width (Bsw)
Spillway height (Hsw)
Spillway coeff. (αsw)
at Mondy 3.97 m 0.66 0.80 m 1.3 m 0.35
at Orme 2.8 m & 0.9 m 0.66 1.91 m 1.7 m 0.35
Table 1.3 Boundary conditions and initial values are used in simulation scenarios.
hS θ1−Orme θ2−Orme QP θ1−Mondy QM 1.64m 0.20m 0.20m 0m3/s 0.60m 2m3/s
Table 1.4 Simulation parameters of the irrigation channel.
Simulation time (Ts) Time step (∆t) Relaxation time (τ)
8000 s 10 s 0.55
1.4 Fluid simulations | 27 For numerical time-interpretation of the resulting equations, the system firstly needs to be initialized at steady state of flow by the following procedure:
• Choose the initial water height of steady flow at S2, hS(lS,0), the steady flow rate is calculated byQS(lS,0) =B3∗hS(lS,0)∗u(in the case of a rectangular section, otherwise QS(lS,0) =S(hS(lS,0))∗u).
• Integrate the steady-flow Eq(s). (1.5) with the boundary conditions at S2 to obtain the water heighthS(l), the steady-state profile forl∈[lO,lS].
• Using the gate relation Eq(s). (1.23), calculate the flow rate through the mixed at Orme, QgO(lO,0), by selecting gate openings,θ1−Ormeandθ2−Orme.
• Integrate the Eq(s). (1.5) with the boundary conditions at Orme to obtain the water height h(l), the steady-state profile forl∈[lMdM,lO].
• The flow rate at “Monts du Matin”,QMdM, is calculated by the pump relation, Eq(s). (1.26) where the pumping flow rate isQP(lP,0).
• Integrate the Eq(s). (1.5) with the boundary conditions at “Monts du Matin” to obtain the water heighth(l), the steady-state profile forl∈[lM,lMdM].
• By selecting the initial flow rate at the Mondy gate, QM(lM,0), and the gate opening, θ1−Mondy, Eq(s). (1.23) can determine the water height at upstream of the Mondy gate.
• The initial water height profile for the considered section is initialized.
The initial values used in simulations are presented in Table 1.3 and an overview of the initialized section is shown in Fig. 1.11. Starting with this equilibrium profile for the considered section,
0 2000 4000 6000 8000
1 1.5 2 2.5 3
Time (s) QM (m3/s)
Boundary conditions
0 2000 4000 6000 8000
0 1 2 3
Time (s) hS (m)
Fig. 1.10 Boundary conditions impose the water flow rate at Mondy gate, QM, and water height at S2 junction,hSfor the first considered section.
3.25 3.3 3.35 3.4 3.45 3.5 3.55 3.6 x 104 0
0.5 1 1.5 2 2.5
Flow axis (m)
Initial water heights (m)
Initialization of water height profile
Fig. 1.11 The first considered section is initialized. The specified flow axis (i.e., 3.25e4mto 3.62e4m) is relative to global flow axis beginning at Auberives barrage.
we propose the following scenario for the simulation of the transient behavior of the LB model.
At timet =80(s), the opening of the Mondy gate,θ1−Mondy, is changed in a way to linearly increase the flow rate at the upstream by 50% from its initial valueQM(lM,0)at time 1000(s).
At timet =1200(s), the gate openings at Orme,θ1−Ormeand θ2−Orme, increase gradually to 0.3(m)att=2000(s). At “Monts du Matin”, the pumping flow rate is increased from 0(m3/s) to 0.2(m3/s)betweent=5000(s)andt=5500(s)and stabilized. The simulation scenario is illustrated in Figs. 1.17 and 1.18.
The simulations are done with Matlab R2016b®on a computer Intel®Core™i5-4310U CPU 2.0GHz. The simulation results presented in Fig. 1.12 (for water height profile along the section) and Fig. 1.13 (for flow rate profile) show a coherent behavior of the considered section. Indeed, we see that (1) there is a wave propagation phenomenon at a constant speed from the upstream to the downstream; (2) the water heights change when the flow passes through the mixed structures at “Mondy” and at “Orme”; and (3) the discontinuity of flow rates corresponds to the withdrawal of the pumping station at “Monts du Matin”. The first coherent results motivate a model of a more complex section or the whole channel, which can potentially be constructed in the same way (see Section 1.4.3).
0 2000 4000 6000 8000 3.3
3.4 3.5
3.6 x 104
1 2 3 4
Time (s) Water height profile along the considered section
Flow axis (m) (h + hp) (m)
Fig. 1.12 Simulation results - Water height profile in the first considered section. The specified flow axis (i.e., 3.25e4 m to 3.62e4m) is relative to global flow axis be- ginning at Auberives barrage.
0 2000 4000 6000 8000 3.3
3.4 3.5
3.6 x 104
1 2 3 4
Time (s) Flow rate profile along the considered section
Flow axis (m) Q (m3/s)
Fig. 1.13 Simulation results - Flow rate pro- file in the first considered section. The spec- ified flow axis (i.e., 3.25e4mto 3.62e4m) is relative to global flow axis beginning at Auberives barrage.
Finally, let us remark that the model still needs to be calibrated and validated during the field campaigns, with surveys and orders from channel operators and precise corresponding experimental measurements.