Linear model of water movements for large-scale inverted siphon in water distribution system

A_f

wetted area of full cross-section in m²

a

speed of acoustic wave in m/s

B_sl

width of Pressimann slot in m

h

relative water elevation in m

h_f

frictional head loss of quasi-steady flow in m

h_i

inertia head in m

Imag

resonance frequency in rad/s

q

relative discharge in m³/s

s

Laplace variable in s⁻¹

y

water depth in m

τ

delay time of propagation downstream in s

Under the present changing climate, the conflict between water demand and water supply has become more and more critical. In order to balance the water resources in western and eastern areas of China, some long distance water distribution systems are being built. However, a large amount of water is wasted due to the low management level in these systems. Canal automatic control is an efficient tool to maintain stable flows, improve water-use efficiency and fulfill users' demands in a timely manner. It is achieved by automatic operations of control structures based on periodical observations of canal systems (Malaterre et al. 1998; Clemmens et al. 2005; Clemmens & Strand 2010). Commonly, a water level monitor is fixed at a certain point in an open-channel; this point is referred to as the target point. The water level at the target point is sent to a feedback controller periodically and a control algorithm is adopted to decide the opening of check gates so that the water surface at the target point can keep a designed level. For example, when the water surface at the target point is lower than the designed level, the check gate upstream is raised and more water is let into the open-channel so that the water surface goes up, while the feedback controller determines the movement of check gates according to the water level at the target point. A sketch to illustrate this system is shown in Figure 1.

After the check gate is raised, there is a delay time before the water level at the target point starts to rise due to the wave movement in the open-channel. Therefore, the feedback controller needs to predict the water level variations at the target point corresponding to the movement of the check gate (Clemmens et al. 2015; Wahlin & Zimbelman 2018). Water in open-channels can be simulated using the Saint-Venant (S-V) equations (Cunge et al. 1980; Rogers et al. 1993). However, when it comes to the design of feedback controller, the S-V equations are not easily implementable due to the large number of parameters involved. Based on the understanding of wave movements in open-channels, researchers have proposed a linear model as a simplification of the S-V equations (Schuurmans et al. 1999). Schuurmans's model contains a delay section that represents the delay time of wave movement in the open-channel and an integer section that reflects the effect of water volume in the open-channel on the water level at the target point; therefore, this model is referred to as the ID (integer-delay) model. The parameters of the ID model can be derived using the size of open-channels or field data, the latter of which makes it more robust (Foo et al. 2012, 2014). Furthermore, the ID model can be used in optimization of the feedback controller due to its simplicity (Overloop et al. 2005). Litrico & Fromion (2004) improved the accuracy of the ID model by adding two new parameters, which became the integer-delay-zero (IDZ) model.

In previous studies, canal automatic control is limited to open-channels because only linear models for open-channels have been proposed (Guan et al. 2011; Shahdany et al. 2016). Pressure distribution in an inverted siphon or pipe is influenced by the boundary conditions, the flow rate distribution, and the material of the siphon (or pipe) wall (Sadeghi et al. 2012; Okhravi et al. 2017). The wave movements in an inverted siphon are very different from that in free-surface flows. The speed of acoustic wave can reach 1,000 m/s which is two orders of magnitude higher than the wave speed in open-channels, indicating that the delay time between check gate movement and water level variations at the target point is much smaller (Brito et al. 2017; Li et al. 2018). Moreover, unlike in the open-channel where wave dissipates quickly due to wall friction and gravity, the wave in an inverted siphon may reflect for many times before it is completely damped. In addition, the influence of the wall deformation on pressure head variations must be taken into consideration (Zhu et al. 2018).

In order to apply canal automatic control to complex water distribution systems consisting of a large-scale inverted siphon, a new linear model that is especially tailored to describe the wave movements in an inverted siphon is required for the design of feedback controller. Due to the high speed of acoustic wave, the pressure head variations are very frequent and hard to analyze in the time domain. Frequency analysis has been shown as an efficient tool for evaluating the relation between pressure head variations and flow rate variations. It has been widely used in leakage detection in pressurized pipelines (Duan et al. 2011; Wang & Ghidaoui 2018; Lin et al. 2019). In this paper, frequency analysis is used to evaluate pressure head variations at the downstream end of an inverted siphon using the flow rate at two ends. The Preissmann slot method (PSM) is used in this paper to build a simplified relation between wall deformation and pressure head variations for the first time. The linear model proposed in this paper can be used for other forms of structures, like pipes and tunnels, where a pressurized flow condition is present. It can be used to construct a coupled linear model of open-channels and inverted siphons, which can then optimize controller design and improve the management level of complex water distribution systems.

This paper consists of five sections. The first section introduces the ID model and the IDZ model, as well as the PSM. The following section investigates characteristics of pressurized flows in inverted siphon using frequency analysis. The linear model is proposed and validated in the next section, then a section describes the linear model being used to model an inverted siphon in a virtual water distribution system. Conclusions are drawn in the final section.

The PSM has been widely used in modelling transitions between free-surface flow and pressurized flow (Dazzi et al. 2016; An et al. 2018; Maranzoni & Mignosa 2018). It permits simulation of two flow regimes using one set of governing equations, i.e., the S-V equations, by assuming a narrow slot on the top of pressurized flow and forming a virtual free-surface flow, shown in Figure 2.

As shown by Equation (7), the width of the virtual slot B_sl can be regarded as the ratio of wall deformation to pressure head variations, it is a parameter which reflects the wall material, wall size, and wave speed. Equation (7) is a simplified relation because it ignores the time-dependent nature of stress-strain behavior of the wall material (Bertaglia et al. 2018). In this paper, it is used to derive the linear model of inverted siphon for its simplicity.

The ID model and the IDZ model are not appropriate for inverted siphons due to the different flow characteristics. In this paper, the authors propose a linear model that describes the variations of pressure head in inverted siphon as a combination of a low-frequency part and high-frequency part.

In this paper, two scenarios are investigated. Scenario A is an inverted siphon in the Dianzhong water distribution system. It consists of five parallel closed conduits and each one has a circular cross-section, shown as Figure 5. The parameters of scenario A are listed in Table 1. The speed of acoustic wave is 1,014 m/s according to the Korteweg-Joukowsky formula. The slot width is computed using Equation (1) and is 5.9885 × 10⁻⁴ m.

Scenario B is an experimental horizontal polyvinyl chloride (PVC) pipe, shown as Figure 6. In order to avoid the vibration of PVC pipe, it is fixed to a concrete foundation. Its parameters are listed in Table 2. The speed of acoustic wave which is validated by experiments is 348 m/s, and the slot width computed using Equation (1) is 3.5218 × 10⁻⁶ m.

Bode plot is a common method to study transfer functions in frequency domain. Its abscissa is frequency (Freq rad/s) of input and its ordinate is magnification (Gain dB) and phase difference (Phase degree) of output. Figures 7–10 are bode plots of p₂₁ and p₂₂ which contain 500 frequency points that range from 0.001 rad/s to 100 rad/s.

At low-frequency domain where Freq < 1 rad/s, the magnification line of p₂₁ is straight with a slope of −20 dB while the phase line is horizontal. Thses lines are the same as that of an integrator section. The phase line is below zero in low-frequency domain, which means there is a delay in p₂₁. Thus p₂₁ can be estimated by an integrator section and a delay section. The magnification line of p₂₂ is similar to that of p₂₁ at low-frequency domain. However, the phase line is above zero indicating that there is no delay time in p₂₂. Thus, p₂₂ can be estimated by an integrator section only. For Freq> 1 rad/s, the magnification lines of p₂₁ and p₂₂ change in an oscillatory manner indicating that waves are superimposed in high-frequency domain. Several frequencies of the magnification peaks are studied in the following text.

Imag of the first five poles and frequencies of the first five magnification peaks in two scenarios are listed in Table 3. Where k is the serial number of pole, j² = −1. It is shown that the frequencies are close to the imaginary part of poles with a difference of less than 0.2 rad/s which is introduced due to the selected finite frequencies.

Note that L/α + L/β is the duration time of a wave moving in an inverted siphon back and forth once. It can be derived that for q₁ and q₂, whose frequencies are multiples of wave reflection, frequency causes larger amplitude of h₂ as a resonance. To validate this conclusion, a series frequencies of q₁ around 6.132 rad/s are simulated for scenario A, and a series frequencies of q₁ around 4.048 rad/s are simulated for scenario B. The simulated frequencies are shown in Figures 11 and 12. The amplitudes of q₁ for scenario A and scenario B are 0.01 and 0.001 m³/s, respectively. The results are shown in Figures 13 and 14.

For scenario A, q₁ with a frequency of 6.132 rad/s (solid line) causes the largest amplitude in variations of h₂ among all the tested frequencies. This conclusion can also be applied to scenario B. In order to avoid the large amplitude of pressure head variations in inverted siphons and pipes, flow rate change whose frequencies are close to the frequencies computed using Equation (20) should be avoided.

Although Equations (26) and (27) are similar to the IDZ model in form, the meaning of the parameters involved are completely different. In the IDZ model, the low-frequency variations of water depth are caused by water volume variations in the backwater area, and the high-frequency variations are caused by wave peaks. In the linear model for inverted siphon, the low-frequency variations are caused by wall deformation and the high-frequency variations are caused by reflection of acoustic wave.

Figures 15–18 are comparisons of p₂₁ and p₂₂ with the linear model in frequency domain. Results show that the linear model is accurate in the low-frequency domain but less accurate in the high-frequency domain since it cannot present the oscillations. However, pressure head variations of very high frequencies dampen very quickly and rarely appear. Thus they are less important in the effectiveness of the linear model for practical use.

Figure 20 shows the pressure at ball valve (P_bv), FVM represents P_bv obtained using finite volume method, Exp represents P_bv obtained by experiments, Res represents P_bv obtained by Equation (31). The values of q₁ and q₂ used in Equation (31) are obtained from simulation results of finite volume method.

According to Exp, when the valve is closed a wave is generated at the ball valve. The P_bv changes in a periodical manner due to the reflection of acoustic wave between the inlet and the valve. At the first 5 seconds, some variations which are of higher frequency than wave reflection exist in the experimental data and then quickly disappear. From t = 5 s to t = 20 s, two or three frequencies of variations are present in P_bv while from t = 20 s to 40 s only one frequency of variations remains. In this process, the amplitude of pressure gradually decreases due to the stress-strain behavior.

The abscissa is the wave speed in multiples of the correct wave speed a. It can be seen that the minimum L₂ norm is achieved when a is used; the minimum L₂ norms are 0.177 and 0.044 for scenario A and B, respectively. The linear model is sensitive to the wave speed, when a false speed is used, the L₂ norm can reach to 6 and 7 for scenario A and B, respectively. Thus, a correct value of a is critical in the accuracy of the linear model. The value of a may be affected by the upstream structures in practice (Mitosek & Szymkiewicz 2016), so it needs to be calibrated by experiments.

Finally, the linear model is applied to the model inverted siphon in a conceptualized water distribution system, shown as Figure 23. It consists of two open-channels and one inverted siphon. Open-channels 1 and 2 are prismatic with a trapezoidal cross-section while the inverted siphon has a circular cross-section. The wave speed in the inverted siphon is set at 1,014 m/s, and the slot width is 2.9943 × 10⁻⁵ m according to Equation (1). A gate is placed at the upstream end of open-channel 1 to control the inflow rate of the entire system. A weir is placed at the downstream end of open-channel 2. Model parameters are listed in Table 4. This model is simulated using the finite-difference method with a time step of 0.2 s and a distance step of 10 m for two open-channels and the inverted siphon.

The initial flow rate is 3 m³/s, then at t = 5 min, the flow rate at the gate increases to 8 m³/s and remains constant. Figure 24 shows that, at t = 9 min, h₁ and h₂ of the inverted siphon start to change. The variations of h₁ show obvious oscillations while the variations of h₂ are smaller and mild. This finding indicates that the inverted siphon can filter the high-frequency variations and decrease the amplitude of variations.

In this paper, the second term at the right side of Equation (37) is noted as h_i. The frictional head loss h_f is computed using the flow rate at the upstream end of the inverted siphon under the assumption of a quasi-steady flow condition. The comparisons between h₁–h_f, h₁–h_f–h_i and h₂ are shown in Figure 25. The L₂ norm between h₁–h_f and h₂ is 1.206 while the L₂ norm between h₁–h_f–h_i and h₂ is 0.026. This result shows that an inverted siphon reduces the fluctuation of water level in open canals. The effect can be quantified using Equation (37).

A linear model relating the pressure head variations at the downstream end to the flow rate variations at two ends of an inverted siphon or pipe is proposed in this paper. The linear model divides the pressure head variations into low-frequency part and high-frequency part, which are caused by wall deformation and wave reflection, respectively. In the high-frequency domain, specific frequencies of flow rate variations at the two ends may cause a larger amplitude in pressure head variations as a resonance. An equation is proposed to compute these frequencies, i.e., Equation (20). For the low-frequency variations, the Preissmann slot method is adopted to derive a simplified relation between wall deformation and pressure head variations. While for the high-frequency variations, the water hammer equation is adopted to evaluate the pressure head variations caused by acoustic wave reflection. The parameters of the linear model are computed using L, a, and A_f. Two scenarios are studied in this paper, one is a large-scale inverted siphon and the other one is a PVC pipe. The accuracy of the linear model is verified in frequency domain using Bode plot in two scenarios. Then the pressure head computed using the linear model is compared with the simulation results of FVM in two scenarios. The accuracy of this model is based on an accurate speed of acoustic wave. The minimum L₂ norms are achieved when a correct wave speed is used. The minimum L₂ norms are 0.177 and 0.044 for the inverted siphon and the PVC pipe, respectively. When a false speed is used, the L₂ norms can reach 6 and 7 for the inverted siphon and the PVC pipe, respectively. Thus, a correct speed of acoustic wave is important to guarantee the accuracy of the linear model. The linear model is subsequently applied to describe an inverted siphon in a conceptualized water distribution system. Simulation results show that the inverted siphon can reduce water fluctuations. This effect is quantified by a frictional head loss h_f under the assumption of a quasi-steady flow and an inertia head multipled by L/(A_fg). On the basis of the aforementioned analysis, the following conclusions can be drawn:

1. An equation to compute the frequencies of flow rate variations at the two ends that may cause resonant pressure head variations in an inverted siphon or pipe is proposed. These frequencies are multiples of the wave reflection frequency between the two ends. In order to avoid damage of the siphon or pipe wall caused by pressure variations, flow rate variations around these frequencies should be avoided.

2. The pressure head variations in an inverted siphon or pipe can be divided into a low-frequency part and high-frequency part. The low-frequency part is caused by wall deformation while the high-frequency part is caused by the reflection of acoustic wave.

3. A linear model is proposed to calculate the pressure head variations at the downstream end of the inverted siphon using the flow rate variations at two ends. This model is applicable to other structures, such as pipes and tunnels, where pressurized flows are present. The low-frequency pressure head variations are related to the flow rate at two ends using a virtual slot width B_sl, which represents the ratio of wall deformation to pressure head variations. The high-frequency variations are computed using the water hammer equation. The linear model is sensitive to the speed of acoustic wave, but in practice, it needs to be calibrated by experiments to ensure the accuracy of the linear model.

4. An equation to quantify the reduction effect of an inverted siphon on the water fluctuations is proposed. The reduction effect can be computed using a frictional head loss under the assumption of a quasi-steady flow condition and an inertia head multiplied by L/(A_fg).

The linear model proposed in this study can be used to build a coupled model of open-channel and pressured flows (i.e., syphon and tunnel), which can then optimize the feedback controller. These findings may be a reference for large water delivery systems.

The authors acknowledge the support of the NSFC grant 51979202.

The Supplementary Data for this paper is available online at http://dx.doi.org/10.2166/hydro.2019.053.