Keywords

1 Introduction

With the development of modern shipping industry, infrastructures such as ports, docks, rivers, and lock chambers are increasingly difficult to meet the navigation needs of large ships. The originally open channels such as coastal, inland rivers, ports and locks have become restricted waters. In 2021, the 400-m vessel Ever Given was stuck in the Suez Canal. More than one week blockage resulted in 422 vessels waiting to pass through the canal and caused huge economic losses.

As a result, the issue of restricted waters has been given more attention. In the restricted waters, the ship is in a state of low-speed navigation and is vulnerable to the influence of bottom and bank of the channel, which seriously threatens the navigation safety of the ship. Generally, the differences between the restricted waters and open waters can be concluded into three main aspects: the pressure distribution around the hull, the components of resistance and the recirculation speed of water. These differences may influence the maneuvering performance of the ship, which may lead to the ship collision or grounding in the restricted waters.

Currently, there are many researchers have work on the issue of restricted waters. Toxopeus, S. L investigated the forces and moments acting on a ship as a function of different drift angle, yaw rate and water depth [1]. Liu and Wan analyzed the pure yaw motion of a ship and investigated its forces, moments, and local flow physics in shallow water [2]. Zou et al. focused on the bank effects on a tanker moving straight ahead in a canal and further investigated the “squat” phenomenon [3].

The issue of a ship sailing in a lock is a typical case of restricted waters. Generally, the space of the lock is shallow and narrow, which means the ship in a lock may face both shallow water effect and bank effect. Consequently, the ship-lock interaction is complicated. Most of researchers take the Panama lock as the research target. Based on the third group of locks in Panama, Vantorre et al. conducted a series of experimental studies to analyze the effects of speed, water depth, eccentricity and drift angle on the hydrodynamic characteristics of ships passing through the lock [4, 5]. Meng and Wan used RANS equations in combination with the k-w SST turbulence model to simulate a 12 thousand TEU ship model entering the Third set of Panama locks. His work introduced three important instants to analyze the work, focusing on wave pattern, dynamic pressure [6, 7].

Vortex plays a vital role in the generation and maintenance of turbulence. In marine hydrodynamics, the vortex structure can be generated by large drift angle maneuvering, deep-draft column stabilized floaters and appendages of a fully appended ship [8]. The mechanism around the ship and marine structures can be deeply analyzed through the vortex structure. For a better study on the vortex structures, currently, there are three generations of vortex identification methods. The first-generation method is simply based on the definition of vorticity proposed by Helmholtz in 1858. However, many researchers found that this method cannot accurately reflect the vortex structure of the flow field. Then, by modifying the first generation vortex identification method, the scholars proposed the second generation represented by Q-criterion, λ2, etc. Unfortunately, the second generation vortex identification method is plagued by unclear physical meaning and artificial threshold selection, and cannot accurately reflect the vortex structure in the flow field. Therefore, the latest third generation vortex recognition method represented by ΩR and Liutex raised by Liu’s team [9, 10] initially has been gradually favored by people. Then, Ren, Wang and Wan use four different vortex identification methods to capture vortex structures to analyze the flow mechanism in the viscous large separated flow field, and find the third generation vortex identification methods are more suitable for displaying the vortex structures in a complex viscous flow field [11, 12].

In the present work, a new lock chamber is studied. The complex viscous flow field and the hydrodynamics of the hull are investigated to explore the ship-lock interaction. The RANS equations in combination with the k-w SST turbulence model is applied to simulate the complex viscous flow field on the self-developed CFD solver, naoe-FOAM-SJTU. And the third generation of vortex identification methods are applied to analyze the complex viscous flow field of the restricted waters. The paper is organized as follows: first the governing equations and main numerical methods are introduced, then the settings of numerical domain are established, and the hydrodynamics performance, free surface, dynamic pressure, vortex field are analyzed. Finally, the conclusion of this paper is drawn.

2 Numerical Methods

2.1 Governing Equations

In this paper, the whole work is based on the CFD solver, naoe-FOAM-SJTU, which is developed on the open-source code platform OpenFOAM, mainly composed of a dynamic overset grid module, a 6 degrees of freedom module and a numerical wave tank module. Then, in the numerical computations, the governing equations are the unsteady Reynolds averaged Navier–Stokes equations (URANS). Which are shown as follows:

$${\boldsymbol{\nabla}} \cdot {{\varvec{U}}} = 0$$
(15.1)
$$\begin{aligned} \frac{{{\boldsymbol{\partial}} {{\varvec{\rho}}}{{\varvec{U}}}}}{{{\boldsymbol{\partial}} {{\varvec{t}}}}} + {\boldsymbol{\nabla}} \cdot \left( {{{\varvec{\rho}}}\left( {{{\varvec{U}}} - {{\varvec{U}}}_{{\varvec{g}}} } \right){{\varvec{U}}}} \right) & = - {\boldsymbol{\nabla}} {{\varvec{p}}}_{{\varvec{d}}} - {{\varvec{g}}} \cdot x{\boldsymbol{\nabla}} {{\varvec{\rho}}} + {\boldsymbol{\nabla}} \cdot \left( {{{\varvec{\mu}}}_{{{\varvec{eff}}}} {\boldsymbol{\nabla}} {{\varvec{U}}}} \right)\\ & \quad + \left( {{\boldsymbol{\nabla}} {{\varvec{U}}}} \right) \cdot {\boldsymbol{\nabla}} {{\varvec{\mu}}}_{{{\varvec{eff}}}} + {{\varvec{f}}}_{{\varvec{\sigma}}} + {{\varvec{f}}}_{{\varvec{s}}} \end{aligned}$$
(15.2)

where \(\mathbf{U}\) and \({\mathbf{U}}_{g}\) represent the velocity and the velocity of grid respectively; \({{\varvec{p}}}_{{\varvec{d}}}={\varvec{p}}{-}\rho gx\) represents the dynamic pressure, \(\rho\) represents the density of the fluid, \(g\) is the gravity acceleration, \({\mu }_{eff}\) is the effective dynamic viscosity; \({f}_{\sigma }\) and \({f}_{s}\) are the surface tension term and source term for the wave elimination region.

2.2 Turbulence Model

In order to enclosure the URANS equations, the shear stress transport turbulence model, SST k-w model is selected. The SST k-w model is one of the widely used turbulence models, which can not only deal with the near-wall region efficiently, but also solve the free surface well. The equations of SST k-w model in OpenFOAM are shown as follows:

$$\frac{{{\boldsymbol{\partial}} {{\varvec{k}}}}}{{{\boldsymbol{\partial}} {{\varvec{t}}}}} + {\boldsymbol{\nabla}} \cdot \left( {{{\varvec{U}}}{{\varvec{k}}}} \right) = {\tilde{\user2{G}}} - {{\varvec{\beta}}}^{\varvec{*}} {\varvec{k\omega }} + {\boldsymbol{\nabla}} \cdot \left[ {\left( {{{\varvec{v}}} + {{\varvec{a}}}_{{\varvec{k}}} {{\varvec{v}}}_{{\varvec{t}}} } \right){\boldsymbol{\nabla}} {{k}}} \right]$$
(15.3)
$$\frac{{{\boldsymbol{\partial}} {{\varvec{\omega}}}}}{{{\boldsymbol{\partial}} {{\varvec{t}}}}} +{\boldsymbol{\nabla}} \cdot \left( {{{\varvec{U}}}{{\varvec{\omega}}}} \right) = {\boldsymbol{\gamma S}}^2 - {\varvec{\beta \omega }}^2 + {\boldsymbol{\nabla}} \cdot \left[ {\left( {{{\varvec{v}}} + {{\varvec{a}}}_{{\varvec{\omega}}} {{\varvec{v}}}_{{\varvec{t}}} } \right) {\boldsymbol{\nabla}} {{\varvec{\omega}}}} \right] + \left( {1 - {{\varvec{F}}}_1 } \right){{\varvec{CD}}}_{{\varvec{k\omega }}}$$
(15.4)

where, the k represents the turbulent kinetic energy; \(\omega\) represents the turbulent dissipation rate; \({F}_{1}\) is a mixed function, which can be applied to switch between SST k-w model (near-wall region) and SST k-ε model (far-filed region).

2.3 Free Surface

For the purpose of capturing the change of free surface, the high precision volume of fluid (VOF). The core concept of this method is to calculate the volume fraction of different fluids in the grid element α to determine the interface. The relative proportion of different fluids in the grid cell is shown as follows:

$$\left\{ {\begin{array}{*{20}c} {{\boldsymbol{\alpha}} = 0} \\ {{\boldsymbol{\alpha}} = 1} \\ {0 < {\boldsymbol{\alpha}} < 1 } \\ \end{array} } \right.\begin{array}{*{20}c} {\boldsymbol{air}} \\ {\boldsymbol{water}} \\ {\boldsymbol{interface}} \\ \end{array}$$
(15.5)

where, the α represents the relative proportion. When it equals to 0, it means the grid cell is completely filled with air while when it equals to 1, it means the grid cell is full of water.

2.4 Overset Grid Method

The dynamic overset grid technology is the key point for direct simulating the complex ship motions. Generally, it includes two or more blocks of overlapping structured or unstructured grids. Then, the decomposed parts will be nested into a uniform and orthogonal background grid. Besides, there will be some overlap between the grids of each subpart, so that the information of the flow field can be coupled and matched by interpolation, which is favor of realizing the calculation of the whole filed. Therefore, the overset grid method can not only retain the advantages of structured grids, but also make up for its deficiency on dealing with complex objects.

2.5 Vortex Identification Method

The third generation of vortex identification method generally has two kinds: \({\boldsymbol{\Omega }}_{{\varvec{R}}}\) and Liutex/Rortex. Firstly, the vorticity ω is obtained by Cauchy Stokes decomposition. And it can be further decomposed into rotating part R and non-rotating pure shear S, which is show as:

$${{\varvec{\omega}}} = {{\varvec{R}}} + {{\varvec{S}}}$$
(15.6)

The \({\boldsymbol{\varOmega}}\) is defined as a ratio of the vorticity tensor norm squared over the sum of the vorticity tensor norm squared and deformation tensor norm squared.

$${{\varvec{\varOmega}}} = \frac{{\left\| {{\varvec{B}}} \right\|_{\text{F}}^{2} }}{{\left\| {{\varvec{A}}} \right\|_{\text{F}}^{2} { + }\left\| {{\varvec{B}}} \right\|_{\text{F}}^{2} + \varepsilon }}$$
(15.7)

where, \(\varepsilon\) is a small positive number to prevent division by zero. The value of \(\varepsilon\) can be determined by the Eq. (15.8), which avoids the influence of manual selection on vortex structure.

$$\varepsilon = 0.001 \times \left( {\left\| {{\varvec{B}}} \right\|_{{\varvec{F}}}^2 - \left\| {{\varvec{A}}} \right\|_{{\varvec{F}}}^2 } \right)$$
(15.8)

As a result, the value range of Ω is between 0 and 1. When Ω = 1, it indicates that the fluid is rotating in a rigid body; When Ω > 0.5, the antisymmetric tensor B is dominant. In practical application, generally, Ω = 0.51 or 0.52 can be used as a fixed threshold to identify the vortex structure in the flow field.

For Liutex method, the rotational part of the vorticity is defined as Liutex/Rortex vector. Consequently, the physical quantity Liutex, which is a vector, has its direction and magnitude. It contains both the local rotation axis and the rigid-body angular speed. The Liutex can be defined as:

$${{\varvec{R}}} = \left( \langle \omega ,r \rangle - \sqrt {\langle\omega ,r \rangle^2 - 4\lambda_{ci}^2 } \right)r$$
(15.9)

where \(\omega\) is the vorticity vector; r is the real eigenvector of ▽V; λci is a Galilean invariant; ▽V is the velocity gradient tensor, which can be expressed as:

$$\nabla {{\varvec{V}}} = {{\varvec{Q}}}\nabla {{\varvec{vQ}}}^{{\varvec{T}}} = \left[ {\begin{array}{*{20}c} {\frac{\partial U}{{\partial X}}} & {\frac{\partial U}{{\partial Y}}} & 0 \\ {\frac{\partial V}{{\partial X}}} & {\frac{\partial V}{{\partial Y}}} & 0 \\ {\frac{\partial W}{{\partial X}}} & {\frac{\partial W}{{\partial Y}}} & {\frac{\partial W}{{\partial Z}}} \\ \end{array} } \right]$$
(15.10)

where Q is a rotation matrix while U, V, W represent the velocity components in the XYZ coordinate respectively.

3 Numerical Setup

3.1 Geometry Model

In the present work, an 8000t ship is adopted as the research object. The geometry model of the ship is shown in Fig. 15.1 and the principal particulars are presented in Table 15.1. In the numerical simulations, the reduced model scale is adopted, which is 1:28.97, so that the length between the perpendiculars of the hull is regarded as 4.35 m.

Fig. 15.1
A diagram of the geometry model of a, b and c. It depicts the rectangle shape plane side view of the shape, U-shaped stem figure, and line drawings.

Geometry model of the 8000t ship

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Table 15.1 Principal particulars of the 8000t ship
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The geometry model of the lock is symmetrical and at the entrance of the lock, there is an obvious rise in the bottom. In the top view, the channel changes from wide to narrow. And the water depth in the entrance of the lock has a sharp drop. The effective size of the full lock is 280.0 m × 40.0 m × 8.0 m (length by width by the height of the water in the lock). The geometry model of the ship is shown in Fig. 15.1. The draft of the ship is 5.5 m while the depth of the water in the lock is only 8 m. The ratio of water depth to draft is less than 1.5. According to the 23rd International Towing Tank Conference (ITTC), a ship’s behavior and maneuverability depends on the depth h of the navigation area. When the ratio of water depth to draft is less than 1.5, it means the ship sailing in the shallow water and the effect of depth restrictions can be significant [13] (Fig. 15.2).

Fig. 15.2
2 horizontal rectangles in light shade. It depicts in a thin line to a thick rectangle of top and side view.

Geometry model of the lock

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3.2 Computational Domain

In the numerical simulation, the ship is towed into the lock from open water. The length between the perpendiculars of the ship model is 4.35 m and the speed of ship model is 0.162 m/s, corresponding to Fr = 0.0248. The computational domain and boundary are shown in Fig. 15.3. In the computational domain, the x-axis points stern, the y-axis points starboard, and the z-axis points upward. The dimension of the calculation domain is (taking the length between LPP vertical lines as the reference): −2.46 Lpp \(\leqslant\) x \(\leqslant\)  3.47 Lpp in length; the open water: −0.23 Lpp \(\leqslant\)z\(\leqslant\)0.23 Lpp in height; the lock: −0.063 Lpp\(\leqslant\) z \(\leqslant\) 0.23 Lpp in height; the open water: −0.556 Lpp \(\leqslant\) y \(\leqslant\) 0.556 Lpp in width; the lock: −0.159 Lpp \(\leqslant\) y \(\leqslant\) 0.159 Lpp in width. Plane z = 0 is the free surface, with air above and water below. The origin of the coordinates is located at the intersection of the free surface, the mid-ship section, and the lock entrance section.

Fig. 15.3
An illustration of the computational domain. It depicts as a rectangle with atmosphere covers in the center and with a hull, wall, and overlap with values.

Computational domin

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In the case, the dynamic overset mesh is selected to support the simulation. The grids of lock domain and the hull domain are generated respectively in order to fit the geometry well. The lock grids are made by the commercial software, Pointwise. And the grids of the hull are made by SnappyHexMesh module in OpenFOAM. Due to the special structure at the entrance of the lock, which is an obvious water level rise, the area of the free surface and the shallow water region are refined. The total number of grids is 6.41 M (M means million.), including 5.26 M background grids and 1.15 M hull grids. Figure 15.4. presents the grid distribution and Table 15.2. Shows the details of the mesh generation.

Fig. 15.4
An illustration of 2 grid distribution patterns in rectangle shapes as a global grid with refinement marked. A half-hemisphere shaped with grid distribution.

Grid distribution

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Table 15.2 Details of the grid distribution
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3.3 Typical Instant

In this work, 3 instants are considered as the analysis targets for a better analysis of the flow field and hydrodynamic characteristics of the ship, which represent three conditions of the ship in the lock. As shown in Fig. 15.5. Time 0 is the ship still in the open water; Time 1 represents the condition that the bow of the ship just reaches the entrance of the lock; Time 2 represents the condition that half of the ship is in the lock while Time 3 represents the condition the ship is fully in the lock.

Fig. 15.5
4 light-shaded rectangles with labeled as open water, reach the entrance, half and all in the lock. A single color is depicted as a line in each rectangle.

Typical instants

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4 Hydrodynamic Results

4.1 Hydrodynamic Characteristics

Figure 15.6 shows the time history curve of the hydrodynamic characteristics of the ship. First, it can be seen from the curve of drag force that, when the ship enters the lock from the far-field, the magnitude of the drag force shows a rising trend. Before the ship gets close to the lock, the resistance presents a regularity change. When the ship reaches the gate of the lock at Time 1, the value of the drag force decreases. This is because the backwater phenomenon inside the lock is unobvious in the beginning. After Time 1, with the ship gradually enters in the lock, the water in the lock is accumulated in front of the ship, and due to the narrowing of the channel and the shallowing of the navigation water area, the blockage coefficient increases and the backwater phenomenon gets worse, result in the increase of the pressure resistance and friction resistance. On the contrary, since the ship enters the lock from the middle of the channel, the flow filed is symmetrical. So, the lateral force and yaw moment of the ship is also symmetrical, reflecting on the curve is their values are almost equal to zero. These results are in line with the facts.

Fig. 15.6
3 line graphs of drag, lateral, and yaw moment as a, b, and c. It plots for times 1, 2, and 3. In a drag versus time in fluctuating trends, in b, lateral versus time as vertical line pattern, and in c, yaw moment versus time in vertical.

Time history of the hydrodynamic forces and moment

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4.2 Free Surface and Dynamic Pressure

The free surface distributions of the typical instants are shown in Fig. 15.7. When the ship gradually enters in the lock, there is an increase of the free surface in front of the ship. The flow section decreases rapidly from a wide-open channel to the narrow space between the ship and the lock, especially the space between the ship’s bottom and the lock’s bottom. At Time 2, when the half of the ship is in the lock, the backwater is most severe. In the picture, the lock is almost occupied by the red, which means the high level of the free surface while outside the entrance, the level is lower, colored by blue. At the last station, the ship has entered the lock completely. The area of the high-level free surface reduces due to the accumulated water before evacuated out of the lock. The level of the free surface around the ship is lower than the other area. Besides, the blackwater effect causes a faster velocity of the return flow and a sinkage of the water level around the ship, leading to a general sinkage of the ship.

Fig. 15.7
An illustration of heat scales of a and b for half and all in the lock. It depicts different color range values from negative 5 to positive 2 in a and b negative 7 to 3.

The free surface distributions of the typical instants

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The same results can be obtained in the dynamic pressure distribution on the bottom of the hull. Overall, the distribution of the dynamic pressure is symmetrical because the flow field is symmetrical. When the half of the ship enters in the lock, the part in the lock has a lower pressure owing to the higher velocity according to the Bernoulli principle. At that time, the lower pressure pare will be pulled downward. As a result, the ship moves vertically downward, which is called the ‘squat’ phenomenon. And, with the ship entering the lock little by little, the area of negative pressure zone becomes larger. At Time 3, due to the return water evacuated out of the lock, the pressure on the stern is higher than that on the bow, causing a trim and threating the safety (Fig. 15.8).

Fig. 15.8
An illustration of heat scales of a and b for half and all in the lock. It depicts different color range values from negative 20 to positive 30 in a and b negative 30 to positive 20.

The dynamic pressure distributions of the typical instants

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4.3 Vortex Structures

Vortex structures around the hull are in different shapes and intensities. Four kinds of vortex identification methods from three generation are applied to capture the vortex structures, as is shown in Fig. 15.9. The Fig. 15.9a shows the different instants’ vortex structures captured by the first-generation method: vorticity. From the figure, the hull is covered fully by the vortex structures in all instants. Obviously, these results are incredibility. The results obtained by Q criterion seem to be more convinced, which is shown in the Fig. 15.9b. Then, as a third-generation vortex identification method, the ΩR method captures larger and more vortex structures in the flow field. However, the results of Q criterion might be easily affected by artificial threshold. Figure 15.10. shows the vortex structures captured by Q criterion with different artificial thresholds. In Time 3, when the ship has entered the lock, the vortex structures in the aft of the ship shows a winding shape and diffuses outwardly form the entrance of the lock, due to the sharp narrowing at the entrance. If the Q is larger than 0.2, the winding vortex cannot be capture. On the contrary, if the Q equals to 0.1 or much smaller, there will be more surplus disturbing the correct identification of the vortex.

Fig. 15.9
An illustration of heat scales of a, b, c, and d for open water, mag, Q, and Ohm with values. It depicts different color range values from 0 to positive 1. Values of 0 to 0 dot 25 cover in all.

Vortex structures obtained by different vortex identification methods in the open water

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Fig. 15.10
An illustration of heat scales of a, b, c, and d for all in the lock and Q with values. It depicts different color range values from 0 to positive 1. Values of 0 to 0 dot 25 cover in all.

Vortex structures obtained by Q criterion with different threshold in Time 3

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On the whole, the second generation method Q criterion and the third generation methods ΩR and Liutex can capture the stream-wise vortex structures caused by the upward of the lock bottom at the entrance and the span-wise vortex structures caused by the narrowing channel successfully. Between the two third generations of vortex identification methods, the wake vortex structures are obvious and the Liutex method shows more broken vortex structures (Fig. 15.11).

Fig. 15.11
An illustration of heat scales of a, b, c, and d for half and all in the lock, Mag, Q, and Ohm with values. It depicts different color range values from 0 to positive 1. Values of 0 to 0 dot 5 cover in all.

Vortex structures obtained by different vortex identification methods

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In addition, as the third generation vortex identification method, Liutex vector represents the rotational angular velocity of the rotating part of the local fluid rigid body, which eliminates the shear effect in various previous vortex identification methods. Consequently, the complex viscous flow field of low-speed ships will be further analyzed by using the Liutex vector method. Figure 15.12 shows the vertical Liutex vorticity distribution at different times. When the ship enters the gate, there is a region of sharp change in vortex structures at the entrance. This is because the ship wake is blocked by the narrowing lock wall, and the wake velocity changes dramatically at the entrance of the lock. At the same time, after the ship completely enters the lock, a positive and negative vortex pair like Karmen vortex appears in the wake flow field of the ship.

Fig. 15.12
An illustration of vertical liutex distribution. It depicts half, all in the lock, sailing in the lock, and vortex pair in the wake flow. It illustrates ranges of color spotted.

Vertical Liutex distribution on the plane

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5 Conclusion and Prospect

In this paper, the numerical simulations of a towing naked hull in restricted waters are carried out by coupling with overset grid technology. For a better analysis of the ship-lock interaction when the ship enters the lock, three typical instants are selected. The free surface and dynamic pressure of these instants are obtained. Besides, combined with the four different vortex identification methods, the vortex structures of the hull are gotten. Overall, three conclusions are summarized:

  • As the dynamic pressure of the hull and the free surface shown, the results of the CFD simulations show the most dangerous time is half of the ship entering the lock. At that time, the magnitude of the darg force is the largest and the backwater phenomenon in the lock is the most severe.

  • Through analysis, it is found that the third-generation vortex identification method can more accurately give the vortex structure around the hull. Due to the narrowing of the channel and the shallowing of the water depth, there will be a span-wise vortex structure and a stream-wise vortex structure at the entrance of lock, where the changes is the most significantly.

  • Compared with the other vortex identification methods, Liutex vector taking advantage of its directionality, can be used to analyze the vorticity evolution process.

The future work will focus on the heavy and pitch motion of the hull while in the present work, the hull is fixed expect moving forward. And more towing velocity will be tested to study the impacts of different velocity on the case.