# Hydraulic Transient Analysis of Sewer Pipe Systems Using a Non-Oscillatory Two-Component Pressure Approach

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## Abstract

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## 1. Introduction

## 2. Theoretical Background: Governing Equations

_{0}is the bottom slope of the channel, S

_{f}is the slope of energy grade line, g is the gravitational acceleration, ρ is the flow density,

**R**is the flow hydraulic radius, and n

_{m}is the Manning coefficient.

## 3. Numerical Solution

_{th}computational cell, respectively; $n$ is the previous time step; and $n+1$ is the current time step.

_{th}cell for which the wave velocity is being calculated. ${Y}_{G}$ is then calculated by multiplying the maximum d by the factor ${K}_{a}$. In this way, numerical viscosity is distributed within a number of cells rather than being injected just in one cell. If all $\mathrm{d}$s are greater than the conduit height, the system is pressurized and a ${K}_{a}$ = 1.001 would provide reasonable results. When there is a pressurization front located within the cells, the numerical viscosity is further intensified by applying ${K}_{a}$ = 1.4. The number of cells (NS) considered depend on the resolution of the computational grid and should be selected such that the numerical viscosity is adequately distributed on either side of the computational cell for which the wave velocity is being calculated. Malekpour and Karney [32] suggested that the number of cells should cover a distance equal to at least three times as large as the conduit height but in any case it should not be less than three cells. If the i

_{th}computational cell is found near a boundary, we should also incorporate the flow depth at the boundary point into Equation (13).

## 4. Numerical Verification

#### 4.1. Test Case 1

#### 4.2. Test Case 2

^{1/6}for the valve head loss coefficient, acoustic pipe speed, and pipe’s Manning head loss coefficient, respectively. It is worth mentioning that both the acoustic speed and the valve head loss coefficient affect the magnitude of the waterhammer pressure spike and were calculated through an iterative procedure in which these parameters were changed until the calculated pressure spike became comparable to that of the experiment. The Manning coefficient of the conduit affects the pipe filling and mass oscillations inside the downstream tank and thus it was independently calculated.

#### 4.3. Test Case 3

_{m}= 0.011. Figure 9 compares the hydraulic grade lines obtained from the proposed HLL solver and from the hybrid flux approach proposed by Vasconcelos at al. [22] at 6 s from the beginning of the filling. As can be seen in this figure, the proposed approach provides oscillation-free solution at acoustic speed of 1000 m/s but the hybrid flux induces spurious numerical oscillations even at the acoustic speed of 100 m/s. It is worth noting that the numerical diffusion produced by the proposed approach in the open channel flow that occurs right after the filling bore is less than that induced by the hybrid flux method, confirming that the proposed approach admits an optimal amount of numerical viscosity to the numerical scheme. Finally, Figure 9 depicts that the location of the filling bore replicated by the proposed approach at the time of 6 s is around 1 m ahead of that calculated by Vasconcelos et al. [22], implying that the filling bore moved faster in the proposed approach than in that of Vasconcelos et al. To identify the root of the difference, we compared the experimented water level time history in the downstream tank with both the water level time histories obtained from the proposed HLL solver and from the hybrid flux of Vasconcelos et al. As can be seen in Figure 10, the hydraulic bore arrival time to the downstream tank is correctly replicated by the proposed model; this means that the hydraulic bore speed is correctly captured by the proposed model.

#### 4.4. Test Case 4

#### 4.5. Test Case 5

_{m}= 0.009. The simulation is continued until the whole system is filled, and as shown in Figure 14, the system reaches a steady-state flow condition with a final flow rate of 0.1712 m

^{3}/s. Figure 13 compares the calculated steady state hydraulic grade line (HGL) with that obtained from a spreadsheet analysis. As it can be seen in Figure 13, the numerical results coincides with the analytical solution, showing that the model correctly converges to the steady state condition. Obviously, the model also correctly captures the negative pressure occurring within the second half of the pipe.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**The numerical results at 10 s after the water level had risen in a test reservoir with an acoustic speed of 50 m/s (conventional HLL solver).

**Figure 4.**The numerical results at 10 s after the water level rises in a test reservoir with an acoustic speed of 1000 m/s (proposed HLL solver).

**Figure 6.**Comparing numerical and experimental velocity time histories at 9.9 m from the upstream tank.

**Figure 7.**Comparing numerical and experimental pressure head time histories at 9.9 m from the upstream tank (acoustic speed = 300 m/s).

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**MDPI and ACS Style**

Khani, D.; Lim, Y.H.; Malekpour, A.
Hydraulic Transient Analysis of Sewer Pipe Systems Using a Non-Oscillatory Two-Component Pressure Approach. *Water* **2020**, *12*, 2896.
https://doi.org/10.3390/w12102896

**AMA Style**

Khani D, Lim YH, Malekpour A.
Hydraulic Transient Analysis of Sewer Pipe Systems Using a Non-Oscillatory Two-Component Pressure Approach. *Water*. 2020; 12(10):2896.
https://doi.org/10.3390/w12102896

**Chicago/Turabian Style**

Khani, David, Yeo Howe Lim, and Ahmad Malekpour.
2020. "Hydraulic Transient Analysis of Sewer Pipe Systems Using a Non-Oscillatory Two-Component Pressure Approach" *Water* 12, no. 10: 2896.
https://doi.org/10.3390/w12102896