Study of Navigable Flow Conditions in the Intermediate Channel of Decentralized Cascade Locks

Abstract

In this study, the effects of the different conveyance modes of the intermediate channel in decentralized cascade locks on navigation flow conditions were investigated. A new hybrid numerical simulation method was established to evaluate navigable flow conditions in intermediate channels at different water conveyance modes. This hybrid numerical simulation method was reliably compared by physical modeling tests. We used the 33.73 m class high-head intermediate channel filled with water as a study case. The study used the maximum water surface slope and maximum flow velocity as evaluation indexes for navigable flow conditions. The results showed that the navigable flow conditions of the centralized water conveyance mode were worse compared to the decentralized water conveyance mode in the intermediate channel. Especially in the upstream region of the intermediate channel with a centralized outflow, the navigable flow conditions were exceptionally harsh. We recommend the decentralized outflow mode in the high-head intermediate channel. This study provides an effective numerical simulation method for optimizing the water conveyance mode of the high-head intermediate channel of decentralized cascade locks and saving project investment.

1. Introduction

Inland waterway freight systems play a key role in the trade and commerce of many countries. It has low cost and low carbon emission compared to road and rail transportation [1,2,3]. However, the first problem of inland waterway freight systems is how to overcome the huge dams with complex terrains that have large differences between upstream and downstream heights. Some dams are over 100 m tall, and others are over 200 m [4]. For example, four world-class high dams, namely Wudongde, Baihetang, Xiluodu, and Xiangjiaba, have been built in the lower reaches of the Jinsha River in southwest China [5]. Ship locks can help ships successfully overcome high dams (see Figure 1). When the ship sailed downstream in decentralized cascade ship locks, it entered the upstream chamber from the upstream approach channel and closed the upstream gate. The chamber emptied with water into the intermediate channel to lower the water level of the chamber. Once the water level in the upstream chamber matched that of the intermediate channel, the downstream gate was opened, allowing the ship to enter the intermediate channel. The process was repeated in a similar manner when passing through the downstream lock. Sailing upstream was analogous to sailing downstream for ships.

The decentralized cascade lock was a navigation building connecting locks at both ends through the restricted intermediate channel. It effectively reduced the working head of single-stage locks and made full use of the complex terrain [6]. When the upstream and downstream locks filled and emptied water into the intermediate channel, unsteady flow fluctuations occurred in the intermediate channel. Specifically, during water discharge from the upstream lock into the intermediate channel, a positive surge was generated in the intermediate channel [7]. During water discharge from the intermediate channel to the downstream lock, a negative surge formed within the intermediate channel [8]. Unsteady flow fluctuations were reflected and superimposed in propagation to form a complex flow state, thus causing the vessel’s transverse rocking, longitudinal tilt, and droop effect, affecting vessel navigation safety [9]. It was crucial to limit the fluctuations caused by the filling and emptying of locks to ensure the vessel’s safety, where the water surface slope and flow velocity are the important parameters to measure whether the vessel can navigate safely [10].

Traditional intermediate channels used a centralized outflow mode where the filling and emptying of water did not share a common water transmission culvert [11]. It exhibited excellent adaptability to intermediate channels with a small head. However, the concentrated outflow mode seriously deteriorated the intermediate channel’s flow state with a high head. It posed a considerable risk to the safety of vessel mooring and navigation. To optimize the flow condition, we proposed a novel water conveyance mode for water transmission in the intermediate channel—the dispersed outflow. Intermediate channel filling and emptying water shared a common culvert. Water was no longer concentrated in an outflow from one end, but distributed from different locations.

In previous studies, the three-dimensional model was established to calculate the hydraulic characteristic value for the filling and emptying of navigation structures with water [12,13,14,15]. However, the scale of the intermediate channel was significantly larger than that of the ship lock, making it difficult to create a three-dimensional mathematical model with both high quality and efficiency.

A hybrid numerical simulation method of the intermediate channel of dispersed outflow was established in this paper, which was applied to evaluate the effect of different water conveyance modes on the navigable flow condition of the intermediate channel. This hybrid numerical simulation method first established the corresponding unsteady flow energy equation based on the culvert’s flow state. Subsequently, combined with the results of the three-dimensional numerical simulation of the outflow zone, the flow rate at different times of each outflow zone was quantitatively calculated when the upstream locks filled with water to the intermediate channel. Finally, the flow rate of each outflow zone has been used as the boundary for the two-dimensional numerical simulation. This approach allowed us to determine the navigable flow conditions of the intermediate channel during the filling with water. In addition, we verified the reliability of the hybrid numerical simulation method by physical modeling tests. Additionally, the reliability of the hybrid numerical simulation method was verified. According to the research results, the suggestion of the water conveyance mode of the intermediate channel was put forward.

2. Materials and Methods

2.1. Study Area

A 1:30 physical model of the dispersed outflow intermediate channel, which met the gravity similarity criterion, was constructed, as shown in Figure 2. The upstream and downstream chambers were 180 m long and 23 m wide, with the total area being about 5300 m². The intermediate channel was about 930 m long and 55 m wide, with the total area being about 46,400 m². There was a 15 m wide, 4 m high water conveyance culvert at the bottom of the intermediate channel. The distances between the upper boundary of the intermediate channel and the first outflow zone, between the first outflow zone and the second outflow zone, and between the second outflow zone and the third outflow zone were approximately 100 m, 375 m, and 375 m, respectively. This paper described filling the intermediate channel from the upstream lock at a constant 8 min opening rate using a valve. Its working head was 33.73 m, and the initial water depth of the intermediate channel was 11.27 m.

2.2. Hybrid Numerical Simulation Method

In this paper, only the intermediate channel-filling was considered. The physical model test result for filling the intermediate channel with a working head of 33.73 m is shown in Figure 3. Q₁, Q₂, and Q₃ were the culvert flow rate in the previous sections of the first, second, and third outflow zones. Additionally, q₁ and q₂ were the flow rate in the first and second outflow zones. Notably, the flow rate in the third outflow zone matched the culvert flow in its preceding section. Four flow states were shown in Figure 4: from time 0 to t₁, ${\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3,{\mathrm{q}}_1,{\mathrm{q}}_2>0$; from time t₁ to t₂, ${\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3,{\mathrm{q}}_2>0$ and ${\mathrm{q}}_1<0$; from time t₂ to t₃, ${\mathrm{Q}}_2,{\mathrm{Q}}_3,{\mathrm{q}}_2>0$ and ${\mathrm{Q}}_1,{\mathrm{q}}_1<0$; and from time t₃ to t₄, ${\mathrm{Q}}_2,{\mathrm{Q}}_3>0$ and ${\mathrm{Q}}_1,{\mathrm{q}}_1,{\mathrm{q}}_2<0$. This study focused on hybrid numerical simulations within the time interval from 0 to t₄ to simplify the calculation.

2.2.1. Governing Equations

Different split ratios directly affected the magnitude of the resistance loss coefficient [16], and Figure 5 shows six different flow patterns in the outflow zone.

Define the resistance loss coefficient under different flow conditions:

$${\mathrm{k}}_{\mathrm{i}.\mathrm{j}-\mathrm{k}\mathrm{l}}=\frac{{\mathrm{p}}_{\mathrm{i}.\mathrm{k}}-{\mathrm{p}}_{\mathrm{i}.\mathrm{l}}}{\mathrm{g}\rho }/\frac{{\mathrm{v}}^2}{2\mathrm{g}}$$

(1)

where ${\mathrm{k}}_{\mathrm{i}.\mathrm{j}-\mathrm{k}\mathrm{l}}$ is the resistance loss coefficient of water from position k (k = 1, 2, 3) to position l (l = 1, 2, 3) in the flow state of the i outflow zone and the j flow state; ${\mathrm{p}}_{\mathrm{i}.\mathrm{k}}$ is the total pressure on position k; v is the corresponding speed of the maximum flow rate; g is the acceleration of gravity; and $\rho$ is the density of water.

Prior to deriving the governing equation for intermediate channel filling, some assumptions were made: (1) the water surface fluctuation is neglected in the chamber and intermediate channel during water filling, (2) the sum of all frictional and minor head losses in outflow zone is characterized by loss coefficient, and (3) internal inertia term of each outflow zone is ignored.

Taking the case ${\mathrm{Q}}_1>{\mathrm{Q}}_2>{\mathrm{Q}}_3>0$ as an example, as shown in Figure 4a, the other three flow regimes gave the control equations in the Appendix A.

Combining the continuity equation for a culvert,

$${\mathrm{Q}}_1=-\frac{{\mathrm{A}}_1{\mathrm{A}}_2}{{\mathrm{A}}_1+{\mathrm{A}}_2}\frac{\mathrm{d}\mathrm{H}}{\mathrm{d}\mathrm{t}}$$

(2)

where A₁ is the area of the upstream chamber, A₂ is the area of the intermediate channel, and H is the difference between the upstream chamber and the intermediate channel.

The unsteady flow equations between the upstream lock and the first outflow zone are

$$\mathrm{H}=\frac{\xi +{\xi }_{\mathrm{t}}+{\mathrm{D}}_1+{\mathrm{k}}_{1.\mathrm{a}-12}}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_1\right|{\mathrm{Q}}_1+\frac{{\mathrm{L}}_1}{\mathrm{g}\mathrm{S}}\frac{\mathrm{d}{\mathrm{Q}}_1}{\mathrm{d}\mathrm{t}}.$$

(3)

where $\xi$ and ${\xi }_{\mathrm{t}}$ are the resistance coefficient from the upstream chamber to the first outflow zone and the valve resistance coefficient, respectively; D₁ is the resistance coefficient along the first section of the culvert, which is calculated by Darcy’s formula; S is the area of the culvert; and L₁ is converted inertia length from the upstream chamber to the first outflow zone.

Combining the unsteady flow energy equations between the culvert section in front of the 1st outflow zone and the culvert section in front of the 2nd outflow zone

$${\mathrm{H}}_1+\frac{{\mathrm{p}}_1}{\rho \mathrm{g}}+\frac{{\mathrm{v}}_1{}^2}{2\mathrm{g}}={\mathrm{H}}_2+\frac{{\mathrm{p}}_2}{\rho \mathrm{g}}+\frac{{\mathrm{v}}_2{}^2}{2\mathrm{g}}+{\mathrm{k}}_{1.\mathrm{a}-13}\frac{\left|{\mathrm{v}}_1\right|{\mathrm{v}}_1}{2\mathrm{g}}+\frac{{\mathrm{L}}_2}{\mathrm{g}}\frac{\mathrm{d}{\mathrm{v}}_2}{\mathrm{d}\mathrm{t}}+{\mathrm{D}}_2\frac{\left|{\mathrm{v}}_2\right|{\mathrm{v}}_2}{2\mathrm{g}}$$

(4)

where ${\mathrm{H}}_1+\frac{{\mathrm{p}}_1}{\rho \mathrm{g}}+\frac{{\mathrm{v}}_1{}^2}{2\mathrm{g}}$ is the total energy of the culvert section in front of the i (i = 1, 2, 3) outflow zone and ${\mathrm{v}}_{\mathrm{i}}$ is the speed of the culvert section in front of the i outflow zone.

Combining the unsteady flow energy equations between the culvert section in front of the 1st outflow zone and the water surface of the intermediate channel

$${\mathrm{H}}_1+\frac{{\mathrm{p}}_1}{\rho \mathrm{g}}+\frac{{\mathrm{v}}_1{}^2}{2\mathrm{g}}={\mathrm{k}}_{1.\mathrm{a}-12}\frac{\left|{\mathrm{v}}_1\right|{\mathrm{v}}_1}{2\mathrm{g}}+{\mathrm{h}}_{\mathrm{c}}$$

(5)

Combining the unsteady flow energy equations between the culvert section in front of the 2nd outflow zone and the water surface of the intermediate channel

$${\mathrm{H}}_2+\frac{{\mathrm{p}}_2}{\rho \mathrm{g}}+\frac{{\mathrm{v}}_2{}^2}{2\mathrm{g}}={\mathrm{k}}_{2.\mathrm{a}-12}\frac{\left|{\mathrm{v}}_2\right|{\mathrm{v}}_2}{2\mathrm{g}}+{\mathrm{h}}_{\mathrm{c}}$$

(6)

Combining the Equations (4)–(6).

$$\frac{\left({\mathrm{k}}_{1.\mathrm{a}-13}-{\mathrm{k}}_{1.\mathrm{a}-12}\right)}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_1\right|{\mathrm{Q}}_1+\frac{{\mathrm{L}}_2}{\mathrm{g}\mathrm{S}}\frac{\mathrm{d}{\mathrm{Q}}_2}{\mathrm{d}\mathrm{t}}+\frac{{\mathrm{D}}_2+{\mathrm{k}}_{2.\mathrm{a}-12}}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_2\right|{\mathrm{Q}}_2=0$$

(7)

Likewise, ${\mathrm{Q}}_3$ can be derived.

$$\frac{\left({\mathrm{k}}_{2.\mathrm{a}-13}-{\mathrm{k}}_{2.\mathrm{a}-12}\right)}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_2\right|{\mathrm{Q}}_2+\frac{{\mathrm{L}}_3}{\mathrm{g}\mathrm{S}}\frac{\mathrm{d}{\mathrm{Q}}_3}{\mathrm{d}\mathrm{t}}+\frac{{\mathrm{D}}_3+{\mathrm{k}}_{3.\mathrm{a}-12}}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_3\right|{\mathrm{Q}}_3=0$$

(8)

2.2.2. Solution of Governing Equations

Firstly, the governing equation was integrated and then solved by computer. The valve resistance coefficient ${\xi }_{\mathrm{t}}$ and the outflow zone resistance coefficient ${\mathrm{k}}_{\mathrm{i}.\mathrm{j}-\mathrm{k}\mathrm{l}}$ underwent minor changes over shorter $\Delta \mathrm{t}$. In order to facilitate the solution, we assumed that the valve resistance coefficient and the outflow zone resistance coefficient were constant over time $\Delta \mathrm{t}$. When the unsteady flow energy equation was integrated, these two coefficients were regarded as constants.

The first-order Taylor expansion of $\mathrm{H}$ and $\mathrm{Q}$ at the time ${\mathrm{t}}_0$ was carried out.

$${\displaystyle \underset{\mathrm{t}}{\overset{{\mathrm{t}}_0+\Delta \mathrm{t}}{\int }}\mathrm{H}\mathrm{dt}}={\displaystyle \underset{{\mathrm{t}}_0}{\overset{{\mathrm{t}}_0+\Delta \mathrm{t}}{\int }}\left[{\mathrm{H}}_{{\mathrm{t}}_0}+{\left(\frac{\mathrm{d}\mathrm{H}}{\mathrm{d}\mathrm{t}}\right)}_{{\mathrm{t}}_0}(\mathrm{t}-{\mathrm{t}}_0)\right]\mathrm{d}\mathrm{t}}={\mathrm{H}}_{{\mathrm{t}}_0}\Delta \mathrm{t}+{\left(\frac{\mathrm{d}\mathrm{H}}{\mathrm{d}\mathrm{t}}\right)}_{{\mathrm{t}}_0}\frac{\Delta {\mathrm{t}}^2}{2}$$

(9)

$${\displaystyle \underset{\mathrm{t}}{\overset{{\mathrm{t}}_0+\Delta \mathrm{t}}{\int }}\mathrm{Q}\mathrm{dt}}={\displaystyle \underset{{\mathrm{t}}_0}{\overset{{\mathrm{t}}_0+\Delta \mathrm{t}}{\int }}\left[{\mathrm{Q}}_{{\mathrm{t}}_0}+{\left(\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{t}}\right)}_{{\mathrm{t}}_0}(\mathrm{t}-{\mathrm{t}}_0)\right]\mathrm{d}\mathrm{t}}={\mathrm{Q}}_{{\mathrm{t}}_0}\Delta \mathrm{t}+{\left(\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{t}}\right)}_{{\mathrm{t}}_0}\frac{\Delta {\mathrm{t}}^2}{2}$$

(10)

Wylie highlighted the following relationship in the study of unsteady pipe flow [17].

$${\displaystyle \underset{\mathrm{t}}{\overset{\mathrm{t}+\Delta \mathrm{t}}{\int }}\left|\mathrm{Q}\right|\mathrm{Q}}\mathrm{d}\mathrm{t}=\left|{\mathrm{Q}}_{\mathrm{t}}\right|{\mathrm{Q}}_{\mathrm{t}+\Delta \mathrm{t}}\Delta \mathrm{t}$$

(11)

Equations (2), (3), (7), and (8) were integrated.

$${\mathrm{H}}_{\mathrm{t}+\Delta \mathrm{t}}={\mathrm{H}}_{{\mathrm{t}}_0}-\frac{{\mathrm{Q}}_{1\left({\mathrm{t}}_0\right)}\Delta \mathrm{t}}{\mathrm{C}}+{\left(\frac{\mathrm{d}{\mathrm{Q}}_1}{\mathrm{d}\mathrm{t}}\right)}_{{\mathrm{t}}_0}\frac{\Delta {\mathrm{t}}^2}{2\mathrm{C}}$$

(12)

$${\mathrm{Q}}_{1({\mathrm{t}}_0+\Delta \mathrm{t})}=\frac{{\mathrm{H}}_{{\mathrm{t}}_0}\Delta \mathrm{t}+{\left(\frac{\mathrm{d}\mathrm{H}}{\mathrm{d}\mathrm{t}}\right)}_{{\mathrm{t}}_0}\frac{\Delta {\mathrm{t}}^2}{2}+\frac{{\mathrm{L}}_1}{\mathrm{g}\mathrm{S}}{\mathrm{Q}}_{1({\mathrm{t}}_0)}}{\frac{\delta +\xi +{\mathrm{k}}_{1.12}}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_{1({\mathrm{t}}_0)}\right|\Delta \mathrm{t}+\frac{\mathrm{L}}{\mathrm{S}\mathrm{g}}}$$

(13)

$${\mathrm{Q}}_{2({\mathrm{t}}_0+\Delta \mathrm{t})}=\frac{\frac{{\mathrm{k}}_{1.12}-{\mathrm{k}}_{1.13}}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_{1({\mathrm{t}}_0)}\right|{\mathrm{Q}}_{1({\mathrm{t}}_0+\Delta \mathrm{t})}\Delta \mathrm{t}+\frac{{\mathrm{L}}_2}{\mathrm{g}\mathrm{S}}{\mathrm{Q}}_{2({\mathrm{t}}_0)}}{\frac{{\mathrm{L}}_2}{\mathrm{g}\mathrm{S}}+\frac{{\mathrm{D}}_2+{\mathrm{k}}_{2.12}}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_{2({\mathrm{t}}_0)}\right|\Delta \mathrm{t}}$$

(14)

$${\mathrm{Q}}_{3({\mathrm{t}}_0+\Delta \mathrm{t})}=\frac{\frac{{\mathrm{k}}_{2.12}-{\mathrm{k}}_{2.13}}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_{2({\mathrm{t}}_0)}\right|{\mathrm{Q}}_{2({\mathrm{t}}_0+\Delta \mathrm{t})}\Delta \mathrm{t}+\frac{{\mathrm{L}}_3}{\mathrm{g}\mathrm{S}}{\mathrm{Q}}_{3({\mathrm{t}}_0)}}{\frac{{\mathrm{L}}_3}{\mathrm{g}\mathrm{S}}+\frac{{\mathrm{D}}_3+{\mathrm{k}}_{3.12}}{2\mathrm{g}{\mathrm{S}}^2}\left|{\mathrm{Q}}_{3({\mathrm{t}}_0)}\right|\Delta \mathrm{t}}$$

(15)

If we knew the hydraulic parameters at any given moment $\mathrm{t}$, we could subsequently determine the hydraulic parameters for the moment $\mathrm{t}+\Delta \mathrm{t}$. It was essential to highlight that the magnitude $\Delta \mathrm{t}$ directly impacts the precision of the computational outcomes. If the $\Delta \mathrm{t}$ is too large, the previous assumption that the valve resistance coefficient and the outflow zone resistance coefficient were unchanged within time $\Delta \mathrm{t}$ will become invalid.

2.3. The Calculation of the Outflow Zone Resistance Coefficient

2.3.1. Governing Equations

With the development of computer technology, computational fluid dynamics (CFD) has been considered a suitable alternative or auxiliary method [18,19,20]. In this paper, the value ${\mathrm{k}}_{\mathrm{i}.\mathrm{j}-\mathrm{k}\mathrm{l}}$ was determined through the three-dimensional numerical simulation. The controlling continuity equations were the mass conservation equation for incompressible flows and the momentum conservation equation in the Cartesian coordinate system:

$$\nabla \cdot \mathrm{u}=0$$

(16)

$$\frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\left(\mathrm{u}\cdot \nabla \right)\mathrm{u}=\frac{-\nabla \mathrm{P}}{\rho }+\mathrm{v}{\nabla }^2\mathrm{u}+{\mathrm{f}}_{\mathrm{b}}+\nabla \left(\overline{{\mathrm{u}}^{,}}{\mathrm{u}}^{,}\right)$$

(17)

where $\mathrm{u}$ is velocity, $\mathrm{P}$ is pressure, ${\mathrm{f}}_{\mathrm{b}}$ is gravitational body force and external body force, ${\mathrm{u}}^{,}$ is turbulent velocity fluctuation, and $\mathrm{v}$ is dynamic viscosity.

The $\mathrm{k}-\omega$ SST turbulence model was a hybrid model that combines the characteristics of the original $\mathrm{k}-\omega$ model near the wall and the characteristics of the $\mathrm{k}-\epsilon$ model away from the wall. In addition, the $\mathrm{k}-\omega$ STT model could be likened to the convergence of the $\mathrm{k}-\omega$ model [21]. The $\mathrm{k}-\omega$ SST turbulent model more accurately described the motion of the fluid in numerical simulations of large curvature [22,23,24]. In this study, the outflow zone features a sharp curvature of 90 degrees. Consequently, we employed the $\mathrm{k}-\omega$ SST turbulence model for this simulation.

2.3.2. Computational Domains and Mesh

As shown in Figure 6, the outflow zone was axisymmetric. Only half of the outflow zone was considered for numerical simulation to simplify the model. Each outflow zone was 89.6 m in length and 18 m in width. Two branch holes, spaced 4 m apart, were incorporated every 5.6 m along the zone. These branch holes were 4 m in length, 0.3 m in width, and 1.5 m in height. A total of 32 branch holes, each with an energy dissipation cover plate, were present in every outflow zone.

Three structured grid sizes were used to check the grid irrelevance of the computational domain model dx = 0.15 m, 0.20 m, and 0.25 m; the total grid sizes were about 1.07 million, 1.94 million, and 4.81 million, respectively. The grid independence was tested under the condition that the velocity of inlet 1 was 2.4 m/s and the velocity of inlet 3 was −3 m/s. As shown in Figure 7, the velocity along the culvert no longer varied significantly when $\mathrm{dx}\le 0.2$. Therefore, the dx = 0.2 m was selected as the final grid size for model discretization for the precision and time of response.

2.3.3. Boundary Condition

Two velocity inlet boundaries and a pressure outlet boundary controlled different flow regimes in the outflow zone with different splitting ratios (see Figure 6). The wall surface was a no-slip boundary with a viscous bottom standard wall function. The SIMPLEC algorithm and the second-order upwind discrete format were used to solve the momentum equations. The commercial software ANSYS fluent 17.0 was used to help discretize the solution domain and the set of equations to find a more reliable approximate solution.

2.3.4. Simulation Validation

The empirical equation of the single branch hole physical model was chosen to validate the three-dimensional numerical model. The single branch hole outflow model was similar to Figure 5a.

As shown in Figure 8, the results presented through the numerical simulation were in good agreement with the results of the physical experiments in the single branch hole, which indicated that the branch hole resistance coefficient was reflected by the three-dimensional model. Therefore, we were confident in accurately calculating the resistance coefficient for the outflow zone through the three-dimensional model.

2.4. Numerical Simulation of Intermediate Channel

2.4.1. Governing Equations

For the intermediate channel, the one-dimensional hydrodynamic model was difficult to reflect fully the required navigable flow conditions. The two-dimensional hydrodynamic model was well-developed and has been widely used in the numerical simulation of rivers [26,27]. In this study, the horizontal scale was larger than the vertical depth scale. Therefore, we ignored the impact of the Coriolis force and wind. The two-dimensional shallow water equation was selected as the governing equation, which was shown in Equations (19)–(21).

$$\frac{\partial \mathrm{U}}{\partial \mathrm{t}}+\nabla \cdot \mathrm{F}\left(\mathrm{U}\right)=\mathrm{S}$$

(18)

$$\mathrm{U}=\left[\begin{array}{c}\mathrm{h}\\ \mathrm{h}\mathrm{u}\\ \mathrm{h}\mathrm{v}\end{array}\right],\mathrm{F}=\left[{\mathrm{F}}_{\mathrm{x}},{\mathrm{F}}_{\mathrm{y}}\right],{\mathrm{F}}_{\mathrm{x}}=\left[\begin{array}{c}\mathrm{h}\mathrm{u}\\ \left(\mathrm{h}{\mathrm{u}}^2+\mathrm{g}{\mathrm{h}}^2\right)/2\\ \mathrm{h}\mathrm{u}\mathrm{v}\end{array}\right],{\mathrm{F}}_{\mathrm{y}}=\left[\begin{array}{c}\mathrm{h}\mathrm{u}\\ \mathrm{h}\mathrm{u}\mathrm{v}\\ \left(\mathrm{h}{\mathrm{u}}^2+\mathrm{g}{\mathrm{h}}^2\right)/2\end{array}\right]$$

(19)

$$\mathrm{S}=\left[\begin{array}{c}0\\ \mathrm{g}\mathrm{h}\left({\mathrm{S}}_{0\mathrm{x}}-{\mathrm{S}}_{\mathrm{f}\mathrm{x}}\right)\\ \mathrm{g}\mathrm{h}\left({\mathrm{S}}_{0\mathrm{y}}-{\mathrm{S}}_{\mathrm{f}\mathrm{y}}\right)\end{array}\right]$$

(20)

where $\mathrm{u}$ and $\mathrm{v}$ are the velocities in x and y directions, respectively; ${\mathrm{S}}_{0\mathrm{x}}=-\frac{\partial Z}{\partial \mathrm{x}}$ and ${\mathrm{S}}_{0\mathrm{y}}=-\frac{\partial Z}{\partial \mathrm{y}}$ are the bottom slope terms of the x and y directions, respectively; Z is the bottom elevation; ${\mathrm{S}}_{\mathrm{f}\mathrm{x}}={\mathrm{n}}^2\mathrm{u}\sqrt{{\mathrm{u}}^2+{\mathrm{v}}^2}{\mathrm{h}}^{-4/3}$ and ${\mathrm{S}}_{\mathrm{f}\mathrm{y}}={\mathrm{n}}^2\mathrm{v}\sqrt{{\mathrm{u}}^2+{\mathrm{v}}^2}{\mathrm{h}}^{-4/3}$ are the friction slopes of the x and y directions, respectively; and n is the Manning coefficient.

2.4.2. Computational Domains and Mesh

The two-dimensional numerical simulation area was the entire intermediate channel. The unstructured mesh was used to divide the simulation domain. This mesh comprises a total of 5931 elements and 3192 nodes. Enhanced refinement was applied to the mesh at the two ends of the intermediate channel, and the grid and the computational domain, as shown in Figure 9.

The result derived from the numerical simulation in Section 2.2 was used as the flow rate of the two-dimensional numerical simulation outlet zone. The total flow change curve of centralized outflow was consistent with the decentralized outflow, but all the flow had the first outflow partition zone, as shown in Figure 10. Feature points were set at 35 m intervals along the middle channel and 3 m from the convex bank, with a total of 26.

2.5. Simulation Method Validation

To validate the reliability of the simulation method, we extracted the experiment results of the water level of the upstream chamber, the average water level of the intermediate channel, each culver flow rate, and the water level in different positions of the intermediate channel (see Figure 11 and Figure 12). The statistical indicator of Kling Gupta efficiency (KGE) was used to evaluate the preference for the hybrid numerical simulation method [28,29]. The equation for KGE was as follows:

$$\begin{array}{l}\mathrm{K}\mathrm{G}\mathrm{E}=1-\sqrt{{\left(\mathrm{R}-1\right)}^2+{\left(\beta -1\right)}^2+{\left(\gamma -1\right)}^2}\\ \beta =\frac{\overline{{\mathrm{C}}_{\mathrm{s}\mathrm{i}\mathrm{m}}}}{\overline{{\mathrm{C}}_{\exp }}},\gamma =\frac{{\sigma }_{\mathrm{s}\mathrm{i}\mathrm{m}}/\overline{{\mathrm{C}}_{\mathrm{d}.\mathrm{s}\mathrm{i}\mathrm{m}}}}{{\sigma }_{\exp }/\overline{{\mathrm{C}}_{\mathrm{d}.\exp }}}\\ \mathrm{R}=\frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{C}}_{\exp .\mathrm{i}}-\overline{{\mathrm{C}}_{\exp }}\right)\times \left({\mathrm{C}}_{\mathrm{s}\mathrm{i}\mathrm{m}.\mathrm{i}}-\overline{{\mathrm{C}}_{\mathrm{s}\mathrm{i}\mathrm{m}}}\right)}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{C}}_{\exp .\mathrm{i}}-\overline{{\mathrm{C}}_{\exp }}\right)}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{C}}_{\mathrm{s}\mathrm{i}\mathrm{m}.\mathrm{i}}-\overline{{\mathrm{C}}_{\mathrm{s}\mathrm{i}\mathrm{m}}}\right)}}\end{array}$$

(21)

where n is the total number of data, ${\mathrm{C}}_{\exp }$ is the experimented value, $\mathrm{C}\mathrm{s}\mathrm{i}\mathrm{m}$ is the simulated value, $\overline{{\mathrm{C}}_{\exp }}$ is the mean value of the experimented value, $\overline{{\mathrm{C}}_{\mathrm{s}\mathrm{i}\mathrm{m}}}$ is the mean value of the simulated value, ${\sigma }_{\exp }$ is the standard deviation of the experimented value, ${\sigma }_{\mathrm{s}\mathrm{i}\mathrm{m}}$ is the standard deviation of the simulated value, and R is the correlation coefficient.

The results presented through the hybrid numerical simulation were in good agreement with the results of the physical experiments (the KGE of all parameters exceeded 0.95), which indicated this numerical method was reliable.

3. Results

As was seen in Figure 13, the maximum water surface slope in the intermediate channel with a centralized outflow was significantly worse than that in the intermediate channel with a decentralized outflow. The maximum water surface slope of the intermediate channel with a concentrated outflow was 2.65‰ at 300 s, which seriously threatened the ship’s berthing and navigation safety. The maximum water surface slope of the intermediate channel with a dispersed outflow changed more slowly, and the maximum of only 0.62‰ appeared in 300 s.

Additionally, we paid attention to the water surface slope at different locations of the intermediate channel. Consequently, the water surface slope data were extracted for the upstream, middle, and downstream regions in the intermediate channel (see Figure 14). It was quite obvious that the intermediate channel with a centralized outflow was larger than the water surface slope with a decentralized outflow. However, there was little difference in the water surface slope between the middle and downstream regions. A large amount of water for a short period flowed from the upstream chamber to the intermediate channel when the upstream lock emptied. All water flowed out in the first outflow zone for the concentrated outflow mode, which made the water level near the outflow zone higher than other areas. A propulsive wave was formed above the water surface and propagated downstream, which made the water surface slope drop in the upstream region by a much larger amount than that in the middle and downstream regions. However, water flowed out of each outflow zone in a dispersed outflow mode, thereby alleviating the excessive local water surface slope in the upstream region.

The change in the maximum flow velocity in the intermediate channel during water filling is shown in Figure 15, with only the outflow mode changed. The maximum flow velocity of the intermediate channel with a concentrated outflow was larger, and the flow velocity change was more intense. The maximum flow velocity of the intermediate channel with a concentrated outflow was 1.45 m/s, which appeared in 420 s. The maximum flow velocity of the intermediate channel with a dispersed outflow was 0.56 m/s, which appeared in 300 s.

We were also interested in the flow state in the intermediate channel with different outflow modes at 6 min (see Figure 16). Water flowed from the only outflow zone into the intermediate channel and diffused downstream in the concentrated outflow water conveyance mode. Water could flow into the channel from multiple outflow zones in the intermediate channel with a dispersed outflow, meaning it no longer only flowed downstream. Changing the direction of water flow diffusion helps to decrease the discharge for unit width in the intermediate channel. The water level fluctuation was ignored relative to the water depth. The flow velocity was directly related to the discharge for unit width. From the perspective of flow velocity, the choice of a dispersed outflow was better than that of the concentrated outflow, especially in the upper area of the intermediate channel with a concentrated outflow, and the navigable flow conditions were particularly harsh.

4. Discussion

The most common method was to treat all the branch holes as a single entity, and the decentralized filling–emptying system was simplified into a centralized filling–emptying system for calculation [30]. However, this method could not calculate the flow rate in each branch hole, much less understand the flow conditions inside the lock intermediate channel. The hybrid numerical simulation method is the most important innovation in this paper, whose core is to calculate the flow rate of each outflow zone based on the flow patterns in the culvert.

Poor water flow conditions in the upstream region are unavoidable in the intermediate channel with a high head and concentrated outflow mode. The method generally adopted is to increase the resistance coefficient of the filling–emptying system and thus reduce the water flow rate, but this will undoubtedly increase the filling time and reduce the navigable efficiency. Hybrid numerical simulation results show that the intermediate channel with decentralized water conveyance has better navigable flow conditions than the centralized outflow, and so it can consider the safety of the ship’s navigation and berthing and navigable efficiency. These results are similar to the superiority of decentralized filling–emptying systems over centralized filling–emptying in high-head ship locks [31]. However, the water conveyance mode of the dispersed outflow is more complicated in its design. It is necessary to consider the problem of the uneven outflow in each outflow zone due to the inertia of the long water conveyance culvert.

Similar to many studies, this study encountered limitations. This paper discusses two different discharge modes of the intermediate channel in the upstream locks operating alone. The layout of the outflow zone within the intermediate channel, operation modes (the filling and emptying can be carried out separately, simultaneously, or at different times), and others have not been thoroughly discussed. In the future, we are looking for a more efficient and safe operation and arrangement of decentralized locks to provide a reference for constructing high-head, large-scale intermediate channels.

5. Conclusions

This paper established a hybrid numerical simulation method to comprehensively evaluate the influence of the outflow mode of the intermediate channel in decentralized cascade locks on its navigable flow conditions. The maximum water surface slope and the maximum flow velocity were evaluation indices of the navigable flow condition. The hybrid numerical simulations were conducted for the intermediate channel filling with a working head of 33.73 m. The main conclusions are as follows.

1. The flow in each culvert of the chamber’s water level, the average water level of the intermediate channel, and the water level in different positions of the intermediate channel were used as standards. The hybrid numerical simulation method in this paper, to quantify the individual hydraulic parameters of the intermediate channel, was reliable by the 1:30 physical model test.

2. When centralized water conveyance is arranged, the intermediate channel with a high head is more worrying. Both indicators of the centralized outflow are much worse than the decentralized outflow for the intermediate channel. The maximum water surface slope and the maximum flow velocity increased by 327.4% and 158.9% in the numerical simulation, respectively. The navigable flow conditions are extremely poor, especially in the upstream region of the centralized outflow.

3. We recommend that the intermediate channel of the high-head decentralized cascade locks adopt a decentralized outflow mode to safeguard the navigable flow conditions in the channel.