# Modeling of the Free-Surface-Pressurized Flow of a Hydropower System with a Flat Ceiling Tail Tunnel

^{*}

## Abstract

**:**

## 1. Introduction

- Pressurized flow. If the tail water level is higher than the top elevation of the tail tunnel outlet, the flat ceiling tail tunnel is always in a typical pressurized state.
- Free-surface-pressurized flow. If the tail water level is obviously lower than the top elevation of the tail tunnel outlet, the flat ceiling tail tunnel is always in free channel flow under both steady and transient states, and a unique interface between pressurized flow and free surface flow is located along the connecting tunnel. Particularly as the tail water level is slightly lower than the top elevation of tail tunnel outlet, the interface between the pressurized flow and the free surface flow is located along the flat ceiling tail tunnel under steady states, and possible mixed free-surface-pressurized flow will inevitably happen under transient states, with one or more air masses existing along the crown of the tail tunnel in some cases.

## 2. Material and Methods

#### 2.1. Governing Equations

_{f}is the hydraulic gradient calculated from the Manning formula.

#### 2.2. Characteristic Implicit Format

_{i}, b

_{i}, c

_{i}, d

_{i}(i = 1, 2), and right items u

_{i}(i = 1, 2) are determined by the known parameters and variables of the relevant sections.

_{i}

_{j}, b

_{i}

_{j}c

_{i}

_{j}, d

_{i}

_{j}, and right term e

_{i}

_{j}(i = 1, 2) is the calculated section.

**AX**=

**E**

**A**is the 2 m × 2 m coefficient matrix presented as a band matrix;

**X**is a 2 m column vector,

**X**= (Δh

_{1}, ΔQ

_{1}, Δh

_{2}, ΔQ

_{2}, …, Δh

_{j}, ΔQ

_{j}, …, Δh

_{m}, ΔQ

_{m})

^{T}; and

**E**is also a 2 m column vector,

**E**= (e

_{11}, e

_{21}, e

_{12}, e

_{22}, …, e

_{1j}, e

_{2j}, …, e

_{1m}, e

_{2m})

^{T}.

#### 2.3. Boundary Conditions

#### 2.3.1. Inlet Section

^{+}equation for section k is:

_{P}and B

_{P}are constants calculated from the piezometric head and volumetric discharge of the neighboring section at time t − △t; H

_{Pk}and Q

_{Pk}are the piezometric head and volumetric discharge of section k at time t.

_{1}and Q

_{1}are the piezometric head and volumetric discharge of section 1 at time t, ∇

_{1}is the bottom elevation at the inlet section, and then Equation (11) is reorganized into the following form:

**A**and the corresponding right-hand item can be obtained, which describe the hydraulic characteristics of the inlet section.

#### 2.3.2. Gate Shaft in Mid-Section of the Flat Ceiling Tail Tunnel

_{P}is piezometric head at the bottom tunnel of gate shaft; H

_{j}and H

_{j+}

_{1}are piezometric heads at sections j–j and (j+1)–(j+1); Q

_{j}and Q

_{j+}

_{1}are volumetric discharges at sections j–j and (j+1)–(j+1); Z

_{PS}and Z

_{PS}

_{0}are the instantaneous and initial water levels in the gate shaft, respectively; Q

_{PS}and Q

_{PS}

_{0}are the instantaneous and initial volumetric discharge flowing into gate shaft, respectively; R

_{S}is the head loss coefficient for water flowing into or out of the gate shaft; A

_{S}is the effective area of gate shaft.

**A**and the corresponding right-hand items can also be obtained, which describe the hydraulic characteristics of the gate shaft section.

#### 2.3.3. Outlet of the Flat Ceiling Tail Tunnel

_{w}, and the outlet’s bottom elevation, ${\nabla}_{m}$. This boundary meets the following equation

_{m}is the sectional area of the outlet, and the subscript m represents the outlet section of the flat ceiling tail tunnel.

**A**and the corresponding right-hand items can be modified according to Equation (25).

#### 2.4. Analysis Model of Transient Flow in System

## 3. Results and Discussions

#### 3.1. Experimental Research

#### 3.1.1. Experiment Description

_{L}= 60.0 and two pressure transducers are installed along the flat ceiling tail tunnel, as shown in Figure 5. One transducer is at the bottom of the combined section with the original diversion tunnel, and another is at the bottom of tail gate shaft section. The prototype length between these two transducers is 286.82 m.

#### 3.1.2. Wave Speed Analysis along the Flat Ceiling Tail Tunnel

_{H}= λ

_{L}= 60.0 and time scale λ

_{t}= $\sqrt{{\lambda}_{L}}$ = 7.746. Figure 6 and Figure 7 give the time histories of the piezometric head at monitoring sections A and B under these two cases, together with the specified time as the first peak value at which pressure oscillations appear.

_{f}, can approximately reflect the pressure propagation characteristics from section A to section B, particularly in the process of the free-surface-pressurized flow, so according to the prototype distance between sections A and B (286.82 m), the approximate wave speed a

_{f}for the free-surface-pressurized flow can be calculated and analyzed, which is listed in Table 1.

_{f}in the free-surface-pressurized flow is from 40 to 60 m/s, and its approximate value can be set to a

_{f}= 50 m/s, which can partly describe the propagation characteristics of the free-surface-pressurized flow.

#### 3.2. Numerical Simulation by Using the Characteristic Implicit Method

#### 3.2.1. Effect of Wave Speed on Transient Process

_{f}for the free-surface-pressurized flow in the tail tunnel, which is used to calculate the width of the Preissmann slot, ${B}_{P}=\raisebox{1ex}{$gA$}\!\left/ \!\raisebox{-1ex}{${a}_{f}^{2}$}\right.$. Traditionally, through reference to pressurized flow, empirical data is often used for the wave speed a

_{f}, which is relatively large and may result in an inevitable error in terms of simulation results, particularly for pressure oscillations. Hence, before detailed hydraulic transient computation and analysis, sensitivity analysis of the wave speed a

_{f}or simulation of the free-surface-pressurized flow is carried out. Figure 8 gives the piezometric head at the combining section A for two computing cases, in which three different wave speeds a

_{f}of 25 m/s, 50 m/s, and 100 m/s are introduced into the unified mathematical model, respectively.

_{f}for the free-surface-pressurized flow has an evident effect on the water behavior in the flat ceiling tail tunnel, basically on the pressure oscillation during the free-surface-pressurized flow; with the increase of wave speed a

_{f}, the maximum pressure varies greatly, while the minimum pressure varies less. For computing case C1, because the maximum pressure is controlled by the maximum oscillation pressure in the 1st rising pressure period, the deviation of wave speed a

_{f}will result in inaccurate evaluation of maximum pressure in the given sections, while for computing case C2, with the increase of wave speed a

_{f}, the increasing maximum pressure in the 2nd rising pressure period tends to be greater than that in the 1st pressure rising period, leading to misinterpretation of the maximum pressure and its occurrence time. In summary, to accurately evaluate the hydraulic characteristics of the free-surface-pressurized flow along the flat ceiling tail tunnel, the premise is to take a relatively exact wave speed a

_{f}for detailed numerical computation.

#### 3.2.2. Comparative Analysis with Experimental Results

_{f}of the free-surface-pressurized flow, the wave speed a

_{f}is set to 50 m/s, and further numerical computation and analysis are implemented for two computing cases, C1 and C2. The obtained dynamic curves of the water level in the downstream surge tank and the piezometric head at combining section A are given in Figure 9 and Figure 10. In Figure 9 and Figure 10, the corresponding curves obtained from experimental research are also drawn for comparative analysis. The analysis of maximum and minimum values of water levels in the surge tank and piezometric head at combining section A for two computing cases C1 and C2 is shown in Table 2, in which the data in parentheses is the occurrence time in s for the corresponding maximum or minimum value, and error = numerical data − experimental data.

_{f}for detailed numerical computation, the established unified model can accurately show typical water behavior for water-surface-pressurized flow along a flat ceiling tail tunnel, and can reveal the effects on the hydropower system’s dynamic characteristics.

## 4. Conclusions

- Based on the characteristic implicit method for modeling of the free-surface-pressurized flow in the tail tunnel together with Newton–Raphson linearization, the linear algebraic equations with a band coefficient matrix are constructed, with the introduction of necessary boundary conditions for transient simulation of the free-surface-pressurized flow. Then, a unified mathematical model is established for hydraulic transient analysis of the given hydropower systems. This unified model can accurately reveal typical water behaviors in the water-surface-pressurized flow.
- With the built experimental setup in the lab and further data analysis, considering the dynamic curves of the piezometric head at two typical reference sections along the flat ceiling tail tunnel, the wave speed a
_{f}for the free-surface-pressurized flow is experimentally analyzed, which is used for the correctness in the unified model. It is found that the wave speed a_{f}for the mixed water-surface-pressurized flow in the flat ceiling tail tunnel is close to 50 m/s. - After the sensitivity analysis of wave speed a
_{f}in the free-surface-pressurized flow, the detailed hydraulic characteristics of the free-surface-pressurized flow in the flat ceiling tail tunnel are further investigated and then confirmed by comparative analysis with experimental data. With appropriate correctness of wave speed a_{f}, the numerical results are in good agreement with the experimental results.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 8.**Time histories of the piezometric head at section A with different wave speeds: (

**a**) computing case C1; (

**b**) computing case C2.

**Figure 9.**Time histories of the water level in the surge tank and piezometric head in section A under C1: (

**a**) water level in surge tank; (

**b**) piezometric head at section A.

**Figure 10.**Time histories of the water level in the surge tank and piezometric head in section A under C2: (

**a**) water level in surge tank; (

**b**) piezometric head at section A.

Typical Cases | Reference Period | Peak Time (s) | Propagating Time (s) | Wave Speed, a_{f} (m/s) | |
---|---|---|---|---|---|

Section A | Section B | ||||

C1: Load rejection | 1st pressure increase period | 105.1 | 111.9 | 6.8 | 42.2 |

2nd pressure increase period | 266.9 | 272.8 | 5.9 | 48.6 | |

C2: Load acceptance | 1st pressure increase period | —— | —— | —— | —— |

2nd pressure increase period | 153.6 | 158.7 | 5.1 | 56.2 |

Typical Cases | Transient Variables | Experimental Data | Numerical Data | Error | |
---|---|---|---|---|---|

C1: Load rejection | Water level in surge tank (m) | Max. | 602.32 (134.9) | 603.29 (126.8) | 0.97 |

Min. | 585.33 (43.9) | 585.49 (41.8) | 0.16 | ||

Piezometric head in section A (m) | Max. | 605.10 (105.1) | 605.18 (103.0) | 0.08 | |

Min. | 590.69 (62.0) | 590.72 (57.9) | 0.03 | ||

C2: Load acceptance | Water level in surge tank (m) | Max. | 602.98 (50.3) | 602.25 (46.8) | −0.73 |

Min. | 593.17 (126.0) | 593.42 (120.6) | 0.25 | ||

Piezometric head in section A (m) | Max. | 598.22(153.6) | 598.09(160.3) | −0.13 | |

Min. | 594.52(125.2) | 594.51(128.6) | −0.01 |

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**MDPI and ACS Style**

Zhou, J.; Li, Y.
Modeling of the Free-Surface-Pressurized Flow of a Hydropower System with a Flat Ceiling Tail Tunnel. *Water* **2020**, *12*, 699.
https://doi.org/10.3390/w12030699

**AMA Style**

Zhou J, Li Y.
Modeling of the Free-Surface-Pressurized Flow of a Hydropower System with a Flat Ceiling Tail Tunnel. *Water*. 2020; 12(3):699.
https://doi.org/10.3390/w12030699

**Chicago/Turabian Style**

Zhou, Jianxu, and Yongfa Li.
2020. "Modeling of the Free-Surface-Pressurized Flow of a Hydropower System with a Flat Ceiling Tail Tunnel" *Water* 12, no. 3: 699.
https://doi.org/10.3390/w12030699