Experimental Analysis of the Influence of the Cross-Section Coefficient on the Ship Squat While the Ship Exits the Shiplift Chamber

Abstract

Due to the small cross-section coefficient of a shiplift chamber, the hydraulics problem becomes a complex water flow problem in ship operation. To prevent the ship from touching the bottom and ensure the safety of ship navigation and the docking of the chamber, a generalized physical model was used to study the ship squat of a 1000 t ship leaving the chamber. Throughout the experiment, the law of variation law in the ship squat under different chamber widths, water depths, chamber lengths, ship speeds and ship starting acceleration was obtained. In this study, the flow pattern of the ship leaving the chamber was divided into two parts: backflow and bottom flow. According to sensitivity analysis, when the shipping speed was less than 0.9 m/s, the influence of the width of the chamber on the squat was greater than that of the depth of the chamber; when the speed is greater than 0.9 m/s, the influence of the depth of the chamber on the squat was slightly greater than that of the width of the chamber. Based on our theoretical analysis, the basic expression of a ship squat was established.

1. Introduction

A shiplift, as a navigation building with less water consumption and faster running speed, plays an indispensable role in realizing unobstructed inland waterway navigation. Compared with a ship lock, a shiplift is more suitable for navigating high dams and canals [1]. Since 1986, shiplifts have experienced rapid development: at present, more than 60 shiplifts have been built at home and abroad. These include Strépy-Thieu in Belgium, New Niederfenno in Germany, and the Three Gorges and Jinghong shiplifts in China, which have broad application prospects [2]. See Figure 1.

The operation process of a shiplift is mainly divided into two parts: the lifting part and upstream/downstream docking. In the process, the lifting of the shiplift chamber is simple, smooth and steady. Meanwhile, when docking with the upstream/downstream of the shiplift, a series of navigation safety problems are likely to occur when the ship enters and leaves the chamber; for example, the ship may hit the chamber bottom due to the excessive squat, and the fluctuation of the water surface in the ship chamber leads to a weight change and capsizing of the chamber.

This kind of ship navigation safety problem occurs because, in the design process of the ship chamber, in order to reduce the dragging power of the shiplift as well as the engineering cost, the size of the chamber and the water depth in the chamber are usually reduced to a minimum according to the size of the design ship and the freight volume. The cross-section coefficient determined by the design of the ship chamber is generally 0.5~0.6 [3]. According to its definition, the cross-section coefficient n is the area ratio between the cross-section of the ship and the chamber crossing the water. The cross-section area of the ship crossing the water $A_s=B\times d$ is shown in Figure 2, where B is the width of the ship and d is the draft of the ship; the cross-section area of the chamber crossing the water is $A_c=b\times h$, where b is the width of the chamber and h is the depth of the chamber.

In addition, since the chamber is semi-closed, the disturbance and propulsion effect of the ship on the water in the chamber while entering the chamber forms a propulsion wave inside the chamber. When the wave is propelled to the front of the chamber, it has a solid-boundary reflection phenomenon, and forms a superimposed reflection wave in the restricted water area where the chamber is connected to the approach channel. The wave further complicates the ship’s movement (sinking, trim and yaw).

Unlike the non-restricted channel, the shallow water and bank effects have a significant influence on the ship’s hydrodynamics in the process of entering and leaving the shiplift chamber due to its size limitations. Therefore, the process of a ship entering and leaving the chamber is one of the core links of the safe operation of the shiplift. In order to ensure the safe operation of the shiplift and improve the traffic capacity, it is necessary to accurately estimate the squat of a ship entering and leaving the chamber [4].

The Barrass [5,6], Eryuzlu [7] and Huuska/Guliev [8] formulas are recommended in the “Guidelines for Design of Approach Channel” by the PIANC to calculate the ship squat. In addition to the three formulas recommended above, many empirical formulas are used for calculating squat, including those by Tuck [9], Romisch [10], Millward [11,12], Ankudinov [13], ICORELS [14], as well as Yoshimura and Norrbin [15]. All these formulas have limited specific application scopes and conditions, as shown in

Table 1. Conditions for the use of empirical formulas.

Empirical Formulas	Application Scope				Application Channel
	Cb	h/d	L/h	Fd	U	R	C
Barrass (1981)	0.5~0.9	1.1~1.5			✓	✓	✓
Eryuzlu and Hausser (1978)	≥0.8	1.08~2.75			✓	✓
Husska/Guliev (1976)	0.6~0.8	1.1~2.0		≤0.7	✓	✓	✓
ICORELS (1980)	0.6~0.8	1.1~2.0		≤0.7	✓
Millward (1990)	0.44~0.83		6~12		✓
Norrbin (1986)	0.6~0.8			<0.4	✓
Romisch (1981)		1.19~2.25			✓	✓	✓

Note: C_b is the block coefficient; F_d is the Froude number (water-depth-dependent); U denotes infinite water; R is the restricted channel (artificial canal); and C is the vertical wall channel.

.

Scholars have compared the results of the ship squat obtained by different researchers [16]. The section coefficient of the Barrass formula is applicable in the range of 4–10; simultaneously, the Barrass formula does not consider the change in trim, the calculated result is rather large, and it is unsuitable for calculating the ship squat in the shiplift chamber. Similar to the Barrass formula, Eryuzlu and Huuska both provide formulas for calculating the squat of a ship’s bow, and do not consider the influence of the blocking effect on the squat.

In addition to the above studies, Elsherbinya [17] conducted experiments on the New Suez Canal to provide guidance for shipping. With the rapid development of computational fluid dynamics, some results have been obtained in studying the ship behavior using numerical methods. Terziev [18,19] quantified the effect of a step in the channel topography on ship sinkage, trim, and resistance, and elucidated scale effects in shallow water performance predictions using Star-CCM+ RANS solver. Campbell [20] supplemented the trim and draught increase effects on a ship’s resistance while advancing through a restricted waterway. Hadi [21] conducted CFD-based simulations to investigate the effects of different canal types and cross-sections on resistance and wave-generated characteristics of a 750 DWT Perintis Ship.

Other scholars have also conducted relevant studies focusing on the ship squat of a shiplift. Considering the difficulty of prototype observation and complex influencing factors, the three following ship squat calculation formulas were formed based on model test research.

The formula of China’s “Design Code for Shiplift” [22] is proposed by Bao [23] in 1991 based on the fitting of model test data of the Three Gorges, Lushui, Dahua and other shiplifts:

$$S=7.07\times F_d^{1.5}\times n^{2.3}\times d$$

(1)

where S is the maximum squat of the ship’s stern; and n is the cross-section coefficient. Due to the limits of materials, the influence of ship types on the squat is not considered in the formula.

The ship squat empirical formula of NHRI was proposed by Hu [24] in 2011. In order to analyze the main influencing factors of ship squat, he assumed that the ship was sailing in a narrow and shallow area of infinite length, fixed the coordinate system on the ship according to Figure 2 and established the following equations:

$$V{A}_c=(V+u)(A_c-A_s-bS)$$

(2)

$$\frac{V^2}{2g}=\frac{{(V+u)}^2}{2g}-S$$

(3)

where V is the ship speed; and u is the backflow velocity around the ship.

We can draw the P~K relationship curve via the dimensionless processing of the ship squat S, ship speed V, chamber depth h and the reciprocal of section coefficient m of many shiplifts in China, as shown in Figure 3, where $P=S/h$, $K=F_d^2\times \left[{\left(\frac{m}{m-1}\right)}^2-1\right]$. It can be seen that there is a quadratic function relationship between ship squat and ship speed. Therefore, NHRI proposed a new empirical formula that has clearer physical significance.

$$\frac{S}{h}=2.11\times F_d^2\times \left[{\left(\frac{m}{m-1}\right)}^2-1\right]+0.04$$

(4)

In 2005, the German Federal Waterway Design and Research Institute researched the problem of extra-large ships entering and exiting the Lüneburg shiplift [25]. During the field investigation, it was found that the speed of the ship exiting the shiplift was limited by the block coefficient and the expression of maximum squat when the ship exiting the chamber is put forward:

$$\frac{S}{h}=2.03{(m-1)}^{-1.15}{C_b}^{-0.31}{F}_d^{1.63}$$

(5)

This formula’s adaptive range is: $1.17\le m\le 3.26$, $0.018\le F_d\le 3.26$, $0.83\le C_b\le 0.96$.

Although the above three formulas all consider the influence of the blocking effect, they do not analyze the influence of the chamber width on the ship squat. Unlike sailing in unrestricted waterways, water flowing alongside the hull is spatially restricted, and its flow pattern changes from three-dimensional space flow in infinite waters to two-dimensional plane flow on both sides of the hull. As a result, the distribution of water pressure around the hull changes, the ship’s resistance increases, and the magnitude and distribution of hydrodynamic force on the hull surface change.

Unlike foreign ships, which are built according to strict standards, hundreds of different types of ships sail on China’s rivers, with C_b ranging from 0.6 to 1.0. There is no standard for complex and diverse ship types. For navigation buildings in China, further consideration should be given to the influence of the section coefficient of ship shape on the squat. Therefore, it is necessary to conduct specific research on the squat of ships that enter and leave the chamber.

2. A simplified Ship Squat Formulation While Passing through a Shiplift Chamber

Previous studies have found that, at the same speed, the squat of the ship exiting the chamber is larger than that of the ship entering the chamber [26,27]. Therefore, the ship’s squat exiting the chamber is the safety control condition.

It is assumed that the distance between the stern and the closed-end gate of the ship chamber is l₂ when the ship starts. When the ship sails Δl distance in Δt time, the water flow on both sides of the ship and the bottom of the ship moves around the hull. This backflow movement with the loss of velocity produces the water level difference, which forms the squat of the ship. Figure 4 establishes the following equation, where S is the ship squat, h is the depth of the chamber, l₀ is the length of the chamber, Q is the flow rate of the section, μ is the flow coefficient of the flow section and A is the area of the flow section.

$$b{l}_{2}h-b(l_2+\Delta l)(h-S)=Q\Delta t$$

(6)

$$Q=\mu A\sqrt{2gS}$$

(7)

The flow section generated around the ship comprises two parts: the area on both sides of the ship and the chamber and the bottom of the ship, as shown in Figure 5. Therefore, it can be regarded as backflow on both sides of the ship and bottom flow, as shown in Figure 6. In the theoretical analysis, for the convenience of calculation, the influence of the block coefficient of the ship is ignored and the bottom of the ship is considered to be square, thus:

$$b\frac{\Delta l}{\Delta t}(h-S)=Q_1+Q_2$$

(8)

By discussing the backflow on both sides of the ship Q₁ and the bottom flow Q₂ of the ship, it can be concluded that

$$S=\frac{1}{{{\mu }_1}^2}\frac{V^2}{2g}{\left(\frac{b}{b-B}\right)}^2+\frac{1}{{{\mu }_2}^2}\frac{V^2}{2g}{\left(\frac{b}{B}\right)}^2{\left(\frac{h-S}{h-S-d}\right)}^2$$

(9)

3. Experimental Program

Considering the research content, test requirements and the size of the test site, the model scale was selected as 1:16. The physical model test was designed using the similarity criterion of gravity, and must be satisfied with geometric, kinematic, dynamic and Froude number similarities. In this study, the Reynolds number for the model R_em for the 1000 t ship at 0.275 m/s calculated to be 1,157,178.2, and water kinematic viscosity which is equal to 1.010 × 10⁻⁶ for fresh water at 20 °C which was the temperature of water during the test.

We mainly studied the squat in the process of exiting the shiplift chamber. Therefore, the model design does not consider the lifting equipment of the shiplift chamber, the range of model simulation includes the shiplift chamber and the downstream navigation approach channel. The layout of the model test is shown in Figure 7.

The test ship is a typical 1000 t class ship in China; the size and draft of the ship are shown in

Table 2. Parameters of the test ship.

	1000 t Ship	1000 t Ship Model
Length (m)	68	4.25
Length at water line (m)	66.4	4.15
Longitudinal centre of Buoyancy (%)	0.338	0.338
Breadth (m)	11	0.68
Draft (m)	2.4	0.15
Wetted surface area (m2)	682.2	2.66
Displacement (m3)	1580	0.386
Block coefficient	0.88
Froude number	0.05–0.22

. The design of the ship model follows the similarity criterion of gravity, with the same scale as the hydraulic model, and the velocity and displacement similar to those of the real ship. In addition, the ship model must satisfy the similarity principle, conversion method and basic requirements of the model resistance experiment.

Due to the small cross-section coefficient, water flow and the ship itself can interact with each other when ships sail into or out of the shiplift chamber. The variation in water flow around the ship causes the disparity of ship speed, and the unstable speed will cause the difficulty in data analysis. Thus, the towing system is normally adopted to study the hydraulic conditions of ships navigating in restricted channels. It sets the guide device to tow the ship model sailing at the given speed in the designed route. The device can ensure that the speed is stable and the ship model does not deviate from the route, as shown in Figure 8. The movement of the model ship in the horizontal plane can be controlled by small cars, but the model is free in the vertical degrees of freedom, allowing trim and sinkage.

Two ultrasonic displacement meters are installed in small cars. When the towing system drives the ship model in navigation, the ultrasonic displacement meters can measure the distance between the level track and the bow or the stern. Then, the squat at each location, the bow or stern, can be obtained. The frequency of data acquisition is 10 hz. The accuracy of the sensor used for sinkage measurement is 0.6 mm. All the data are gathered by a portable data acquisition system, Wavebook/516E, on the ship model, and transferred to the computer for automatic measurement by wireless transmission technology. The route and speed of the ship model can be determined and measured by the towing system. The test device is shown in Figure 8.

Since it is difficult to change the ship’s size in the physical model test, the chamber size is chosen for this test to change the cross-section. This paper studies the squat variation with various navigation speeds, water depths, chamber widths and chamber lengths. The test parameters are shown in

Table 3. Test parameters.

	Full Scale	Model Scale
Navigation speed (m/s)	0.3, 0.5, 0.7, 0.9, 1.1	0.075, 0.125, 0.175, 0.225, 0.275
Water depths (m)	2.6, 3.0, 3.5, 4.0	0.163, 0.188, 0.219, 0.25
Chamber widths (m)	12, 14, 16, 18	0.75, 0.875, 1, 1.125
Chamber lengths (m)	72, 92, 144	4.5, 5.75, 9

. The acceleration rate of the ship from the static state to the constantly moving state or from the constantly moving to the static is 0.015 m/s², 0.025 m/s² and 0.035 m/s², and the maximum squat was found during the state of constant movement. Specific working conditions are shown in

Table 4. Working conditions of cross-section coefficient (l₀/L = 1.05, l₂ = 8 m, a = 0.015 m/s²).

Number of Cases	b/B	h/d	m	V (m/s)
N12	1.09	1.25	1.36	0.3; 0.4; 0.5
N13		1.46	1.59	0.3; 0.5; 0.7; 0.9; 1.1
N14		1.67	1.82	0.3; 0.5; 0.7; 0.9; 1.1
N21	1.27	1.10	1.40	0.3; 0.4
N22		1.25	1.59	0.3; 0.5; 0.7; 0.9; 1.1
N23		1.46	1.86	0.3; 0.5; 0.7; 0.9; 1.1
N24		1.67	2.12	0.3; 0.5; 0.7; 0.9; 1.1
N31	1.45	1.10	1.60	0.3; 0.4; 0.5; 0.6; 0.7
N32		1.25	1.82	0.3; 0.5; 0.7; 0.9; 1.1
N33		1.46	2.12	0.3; 0.5; 0.7; 0.9; 1.1
N34		1.67	2.42	0.3; 0.5; 0.7; 0.9; 1.1
N41	1.64	1.10	1.81	0.3; 0.4; 0.5; 0.6; 0.7
N42		1.25	2.05	0.3; 0.5; 0.7; 0.9; 1.1
N43		1.46	2.39	0.3; 0.5; 0.7; 0.9; 1.1
N44		1.67	2.73	0.3; 0.5; 0.7; 0.9; 1.1

,

Table 5. Working conditions of chamber lengths (l₂ = 8 m, a = 0.015 m/s²).

Number of Cases	l0/L	b/B	h/d	m	V (m/s)
L1N13	1.05	1.09	1.46	1.60	0.3; 0.5; 0.7; 0.9; 1,1
L2N13	1.35
L3N13	2.12
L1N14	1.05	1.09	1.67	1.82
L2N14	1.35
L3N14	2.12

and

Table 6. Working conditions of acceleration rate (l₂ = 8 m).

Number of Cases	a (m/s2)	l0/L	b/B	h/d	m	V (m/s)
A1L1N13	0.015	1.05	1.09	1.46	1.60	0.3; 0.5; 0.7; 0.9; 1,1
A2L1N13	0.025
A3L1N13	0.035
A1L3N13	0.015	2.12	1.09	1.46	1.60	0.3; 0.5; 0.7; 0.9; 1,1
A2L3N13	0.025
A3L3N13	0.035

.

4. Results and Discussion

4.1. Impacts of Cross-Section on the Squat

An example of the time history of a ship squat is shown in Figure 9. The data (ship squat, sailing distance, ship speed, etc.) are presented and transposed at the ship’s full scale. In order to study the impact of the cross-section coefficient on ship squat, this paper obtained the influence of changes in the ship chamber’s water depth and width on ship squat by changing the water depth and width of the chamber. The specific test results are shown in Figure 10.

A ship with the same speed, with a smaller reciprocal of section coefficient m of the chamber, has a greater maximum squat. For example, when the reciprocal of the section coefficient m = 2.42, the maximum ship squat is 0.05 m, and when the reciprocal of section coefficient m = 1.59, the maximum ship squat is 0.22 m. This is because the ship brings out a large amount of water when exiting the chamber, resulting in a drop in the water surface. The water returning at the front of the bow needs to pass through the narrow cross-section, so it cannot quickly replenish the water at the rear of the ship. Therefore, the smaller the reciprocal of section coefficient m is, the more obvious the blocking effect will be, and the maximum ship squat when the ship exits the chamber will also increase, which is consistent with previous research results.

It can also be seen in Figure 11 that, at the same speed, even if the reciprocal of the section coefficient m is the same, the maximum ship squat is different with a different width–depth ratio. Under the working conditions N34 and N43, that is, when the corresponding reciprocals of the section coefficient m are both 2.42, the ship exits the chamber at the same speed. Although the width–depth ratio of the section of the ship is different, the maximum subsidence of the ship exits is basically the same, and there is a slight difference only when the ship exits at the speed of 1.1 m/s. However, this differs under the working conditions of N31, N22, and N13, that is, when the corresponding reciprocal of section coefficient m are all 1.60. At this time, the maximum ship squat when the ship exits the chamber is significantly different with different combinations of chamber width and depth.

For example, when the speed of the ship exits the chamber V = 0.9 m/s, b/B = 1.09, h/d = 1.46, and the maximum ship squat is 0.37 m. When b/B = 1.27 and h/d = 1.25, the maximum ship squat is 0.46 m. This indicates that, even if the cross-section area of the ship is fixed, the overflow on both sides and bottom of the ship will be affected when the depth and draft ratio and the width ratio of the ship are changed, and the maximum ship squat is also significantly different. By comparing these two combinations, it can be seen that the width of the chamber increases by 16.5%, the water depth of the chamber decreases by 14.4%, and the maximum ship squat increases by 24.3%, indicating that, when the ship speed V = 0.9 m/s, the water depth of the chamber significantly influences the ship squat. With the decrease in the reciprocal of section coefficient m, the influence of the chamber depth is more significant.

4.2. Sensitivity Analysis

In order to further analyze the influence of the width and depth of the cross-section coefficient on the ship squat when exiting the chamber, the sensitivity coefficient method is used. The sensitivity coefficient method assumes that a single factor changes in a certain range, while other factors are constant. The sensitivity degree of the parameter is determined according to the change range of the target physical quantity under the unit change range of the parameter. Therefore, a set of reference parameters is needed to determine the sensitivity of the ship based on the change of the maximum ship squat.

The sensitivity coefficient is defined as follows:

$$M=\frac{{\eta }_1}{{\eta }_2}$$

(10)

$${\eta }_1=\frac{\Delta y}{y_0}$$

(11)

$${\eta }_2=\frac{\Delta x}{x_{\max }-x_{\min }}$$

(12)

where ${\eta }_1$ is the change rate of the squat; Δy is the change value of ship squat caused by the change of a certain factor; y₀ is the base value of squat obtained based on the base value of each influencing factor; ${\eta }_2$ is the change rate of the influencing factor; Δx is the change in factor x; and x_max and x_min are the maximum and minimum values of factor x, respectively. The larger M is, the more sensitive the squat is to the influencing factor.

According to the definition of the sensitivity coefficient, the ship speed V = 0.3 m/s, b/B = 1.64 and h/d = 1.67 are selected as the reference parameters. With the ship squat of the cross-section coefficient as the reference value, the sensitivity coefficients of width and depth at different speeds when the ship exits the chamber were compared and analyzed. The results are shown in

Table 7. Sensitivity coefficient of ship chamber width and depth when the ship is exiting.

V (m/s)	Mb	Mh
0.3	1.00	0.52
0.5	1.76	1.16
0.7	1.93	1.63
0.9	2.22	2.26
1.1	2.35	2.43

. The sensitivity coefficient of chamber width is greater than that of chamber water depth when the ship speed V is less than 0.9 m/s. The sensitivity coefficient of water depth is greater than the width of the chamber when the velocity of the ship is greater than 0.9 m/s. This indicates that the width of the chamber has a great influence on the ship squat when exiting the chamber at low speed. When exiting the shiplift chamber at a higher speed, the influence of the depth of the chamber on the ship squat is greater than that of the width of the chamber, which is consistent with the previous test results.

4.3. Impacts of Chamber Length on the Squat

In view of the influence of the chamber length on the ship squat, the relationship between the maximum ship squat and the length of the chamber under different cross-section coefficients is shown in Figure 12.

Taking the case when m = 1.82 as an example, the comparison of the maximum squat of a ship at different chamber lengths shows that when the ship’s exit speed is lower than 0.7 m/s, the maximum squat of the ship in the three lengths is basically the same. When the exit speed of a ship is greater than 0.7 m/s, the influence of the chamber length on the squat gradually increases. When m decreases, the influence of the length of the chamber on the ship squat intensifies. When the exit speed V = 1.1 m/s and m = 1.82, the ship squat of l₀/L = 1.05 is 70% higher than that of l₀/L = 2.12. When the exit speed V = 1.1 m/s and m = 1.60, the ship squat of l₀/L = 1.05 is 165% higher than that of l₀/L = 2.12.

4.4. Impacts of Acceleration Rate on the Squat

The results obtained by the test with different starting accelerations are shown in Figure 13. It is not difficult to see that, when the length of the chamber is short, the starting acceleration of the ship has a greater impact on the ship squat while, when the length of the chamber is long, the starting acceleration of the ship has little impact on the ship’s sinking. When l₀/L = 1.05, the ship exits at the same speed, and the maximum ship squat gradually increases with the increase in acceleration rate. With the increase in speed, the effect of acceleration on ship squat becomes more obvious.

4.5. The Basic Expression of the Ship Squat

Because a shiplift usually has a long approach channel in practical engineering, the influence of a ship starting acceleration on the squat decreases with the increase in the length of the narrow water area in front of the ship, so the influence of starting acceleration is ignored in this regression analysis. After dimensionless processing and simplifying Equation (9), the relation of ship squat S can be obtained:

$$\frac{S}{h}\sim \frac{1}{{\mu }^2}\frac{V^2}{2gh}\left[{\left(\frac{\frac{b}{B}}{\frac{b}{B}-1}\right)}^2+{\left(\frac{b}{B}\right)}^2{\left(\frac{\frac{h}{d}}{\frac{h}{d}-1}\right)}^2\right]$$

(13)

In the equation, the squat is mainly related to the area of the backflow on both sides of the ship and the underflow area. All the test results of the influence of the cross-section coefficient on the ship squat when exiting the chamber can be obtained by dot plotting, as shown in Figure 14, where $P=\frac{S}{h}$, $X=\frac{V^2}{gh}\times \left[{\left(\frac{\frac{b}{B}}{\frac{b}{B}-1}\right)}^2+{\left(\frac{b}{B}\right)}^2{\left(\frac{\frac{h}{d}}{\frac{h}{d}-1}\right)}^2\right]$. There is a certain clustering of data at all points, which indicates that Equation (13) is basically correct.

The linear regression analysis of the data show that, the flow coefficient is related to the depth and draft ratio h/d, the chamber width and ship width ratio b/B, and the chamber length l₀. When the flow area on both sides of the ship is smaller than the underflow area, the flow coefficient μ is mainly related to h/d; when the flow area on both sides of the ship is larger than the underflow area, the flow coefficient μ is mainly related to b/B. And the empirical calculation formula of the maximum ship squat in the chamber can be obtained as follows, as shown in Figure 15, where $P=\frac{S}{h}$, $K=\frac{1}{{\mu }^2}\frac{V^2}{gh}\times \left[{\left(\frac{\frac{b}{B}}{\frac{b}{B}-1}\right)}^2+{\left(\frac{b}{B}\right)}^2{\left(\frac{\frac{h}{d}}{\frac{h}{d}-1}\right)}^2\right]$.

$$\frac{S}{h}=0.3007\frac{1}{{\mu }^2}\frac{V^2}{gh}\left[{\left(\frac{\frac{b}{B}}{\frac{b}{B}-1}\right)}^2+{\left(\frac{b}{B}\right)}^2{\left(\frac{\frac{h}{d}}{\frac{h}{d}-1}\right)}^2\right]+0.0224$$

(14)

The adaptive range of this formula is 1.09 ≤ b/B ≤ 1.64, 1.25 ≤ h/d ≤ 1.67, 1.05 ≤ l₀/L ≤ 2.12, 0.3 ≤ V (m/s) ≤ 1.1.

We put the test results of this study into the prediction formula proposed by the Nanjing Institute of Hydraulic Science, as shown in Figure 16. The test results are consistent with the relationship of the squat with the correlation coefficient R² = 0.8266, and the ship squat prediction formula proposed in this paper is more accurate.

5. Conclusions

This study used the shiplift chamber as the research object, using theoretical analysis and a generalized physical model test, the ship squat when the ship exits the chamber was studied comprehensively. The relationship between the maximum ship squat and the width, the depth and length of chamber and the ship speed was also analyzed.

(1) Through theoretical analysis, the flow pattern around the ship in the process of the ship exiting the shiplift chamber is divided into two parts: backflow on both sides and bottom flow on the underside. The basic expression of the ship squat was established. As matter of fact, the problem studied in this paper is analyzed, finding that the main factors affecting the squat are the width and depth of the chamber and the ship speed. Based on the design standard of the shiplift at home and abroad, and the ship type selected in the test, a generalized physical model was established, and a test scheme was designed according to the factors affecting the squat.

(2) By analyzing the experimental data, it is found that the cross-section coefficient has a significant impact on ship squat. Under the same ship speed, the smaller the cross-section coefficient, the greater the maximum squat of the ship. When m = 2.42, the maximum ship squat was only 0.05 m, and when m = 1.59, the maximum ship squat was 0.22 m. With the same cross-section coefficient, when m was less than 2.0, the shape of the section through the water had a certain influence on the ship squat.

(3) Sensitivity analysis showed that, when the ship exits the chamber slower than 0.9 m/s, the width of the chamber influenced the squat more than the water depth; when the ship exited the chamber faster than 0.9 m/s, the water depth of the chamber influenced the squat slightly more than the width of the ship chamber.

(4) Based on previous studies, we conducted further research on the influencing factors and variation rules of the ship squat, but there are still many problems to be discussed. The use of CFD numerical simulation for ship squat may be a more flexible approach to solve the problems in practical engineering, as there are various ship types. It is necessary to further study the influence of different ship types so that, for practical engineering, the regression functions from this experimental study can be further enhanced.