Abstract
In offshore oil and gas exploration and transportation, it is often encountered that the multifloating structures work side by side. In some sea conditions, there is a strong coupling between the multifloating structures that seriously affects the safety of offshore operations. Therefore, the prediction of the relative motion and force between the multifloating structures and the wave elevation around the multifloatingsystem has become a hot issue. At present, the problem of doublefloatingsystem is mostly based on linear potential flow theory. However, when the gap width between two floating bodies is small, the viscous and nonlinear effects are not negligible, so the potential flow theory has great limitations. Based on the viscous flow theory, using the finite difference solution program of FLOW3D and using volume of fluid technology to capture the free surface, a threedimensional numerical wave basin is established, and the numerical results of the wave are compared with the theoretical solution. On this basis, the hydrodynamic model of sidebyside doublefloatingsystem with a narrow gap is established, and the flow field in the narrow gap of the fixed doublefloatingsystem under the regular wave is analyzed in detail. The law of the gapresonance is studied, which provides valuable reference for the future research on the multifloatingsystem.
1 Introduction
In recent years, with the further development and utilization of marine resources, multifloatingsystem has been widely used in production practice [1,2,3,4]. Common multifloatingsystem includes oil and gas transmission system [5,6,7], double barge system in the floating support installation of a large offshore platform [8], warship replenishment system [9], and offshore super large floating system [10,11,12,13,14]. In such structures, the relative distance between two or more floating structures is small, resulting in obvious hydrodynamic effects. In some cases, because of the interaction between waves and multiplefloatingsystem with a narrow gap, resonance occurs in the narrow gap, the wave elevation increases significantly, and the motion and force of the floating body increase correspondingly. Therefore, there will be a strong coupling problem that seriously affects the stability and safety of the offshore operation. How to effectively predict the hydrodynamic performance of the multiplefloatingsystem is a hot issue in the field of marine engineering in recent years.
At present, there are two main methods for hydrodynamic interference research in the narrow gap of multiplefloatingsystem: (1) numerical simulation based on linear potential flow theory and (2) model test. The above method mainly analyzes the force, motion response of the multifloatingsystem in the wave, and the wave elevation around the floating body. As the potential flow theory ignores the nonlinear and viscous effects, the calculation result of the wave elevation in the narrow gap at the resonance frequency is too large, resulting in the calculation distortion of the wave drift force. In view of the limitations of potential flow theory, many international scholars have improved this. Huijsmans et al. [15], in the study of the lowfrequency slow drift force of the oil and gas transmission system, increased the boundary condition limit in the narrow gap of the two ships, which is equivalent to adding a rigid lid on the free surface of the gap, effectively reducing the peak of the slow drift force at the resonance frequency, but the rise of the free surface does not have a real simulation. The damping lid method is put forward by Chen [16]. By adding damping value at the free surface, the excessive wave elevation in the narrow gap of the doublefloatingsystem is suppressed, and the wave elevation and wave drift force at the resonance frequency are well simulated. Pauw et al. [17] used the damping lid method proposed by Chen to study the hydrodynamic problem of mooring doublefloatingsystem. It is found through experiments that it is difficult to give a pervasive damping value for the case with different narrow gap width. The influence of the damping lid method on the secondorder wave drift force is greater than the firstorder wave elevation and the motion response. Bunnik et al. [18] corrected the damping lid method and studied the mooring sidebyside liquefied natural gas (LNG) carrier. Damping is added to the free surface inside and outside the narrow gap. This method reduces the mesh dependence and provides better results. Xin et al. [19] used a combination of viscous computational fluid dynamics (CFD) and test to study the hydrodynamic interference of the sidebyside barges. On the basis of the threedimensional potential flow theory, Zhou et al. [20] studied the hydrodynamic interference of the doublefloatingsystem with different waves’ direction by the damping lid method. Using different types of incident waves, i.e., white noise waves, transient wave groups, and regular waves, Zhao et al. [21] investigated both the transient and steadystate resonant responses of the fluid in narrow gaps at model scale, based on the potential flow solvers with the addition of calibrated damping [22].
It can be seen that the linear potential flow theory ignores the viscosity of the fluid, although it does not affect the basic law of hydrodynamic performance but causes an overestimation in the numerical simulation results. The researchers found the importance of viscosity and nonlinearity and tried to refine the results of the linear potential flow theory by the damping lid method, but it must depend on the study of the test and difficult to give a pervasive damping value. Therefore, it is necessary to develop a numerical model based on viscous fluid to study the hydrodynamic oscillation in the narrow gap of multifloatingsystem in real marine environment. The details of the research work of this paper are as follows:
The basic theory of CFD method is introduced in detail.
Based on the Flow3D finite difference solution program of viscous flow theory, a threedimensional numerical wave basin is established to simulate the hydrodynamic resonance phenomenon in the narrow gap of the doublefloatingsystem under the action of the wave, and the calculation results are compared with the results of the literature [23,24,25]. The mesh convergence analysis is carried out on the hydrodynamic model of the doublefloatingsystem, and the optimal mesh division scheme is proposed.
On this basis, the wave elevation and fluid motion in the narrow gap of fixed doublefloatingsystem are analyzed.
The influence of narrow gap width and wave period on the wave elevation is analyzed in detail.
2 The basic theory of CFD method
2.1 Fluid flow governing equations
It is assumed that the fluid is incompressible. The governing equations include the continuity equation and the Navier–Stokes equation for incompressible viscous fluid motion. In this paper, the fluid is considered as Newtonian fluid. Because of the unique FAVOR grid processing technology of Flow3D, independent and complex geometry can be defined within the structured grid, so that the simple rectangular grid can be used to represent arbitrary complex geometry, and the shortcomings of the previous finite difference method for building boundary fitting are avoided. The software provides multigrid nesting and local encryption technology, which can reduce the consumption of memory, provide high accuracy, and greatly improve the calculation efficiency. The continuity equation of FAVOR grid processing technology is different from the classical equation. The continuity equation and momentum equation contain volume and area fraction parameters. The specific expressions are as follows.
Continuity equation:
Momentum equation:
In the above formula, V _{F} is the volume fraction of the flowable fluid in the numerical basin. A _{ x }, A _{ y }, and A _{ z } represent the area fraction of the flowable fluid in the three directions x, y, and z, respectively. u, v, and w are the velocity components in the three directions x, y, and z. (Gx, Gy, Gz) is the acceleration of gravity, and (fx, fy, fz) is the viscous force acceleration of the fluid.
2.2 Turbulence model
Flow3D provides five turbulence models: the Prandtl hybrid length model in the zeroequation model, an equation model, the standard k–ε model, the RNG k–ε model, and the large eddy simulation. In this paper, the interaction between wave and threedimensional structure is simulated, and a series of irregular motion shapes such as wavebreaking and deformation will occur. It is suitable to be simulated using the RNG k–ε model. Similar to the continuity equation and momentum equation, the volume fraction V _{F} and area fraction A _{ x }, A _{ y }, A _{ z } are added to the control equation of the RNG k–ε model in Flow3D.
2.3 Freesurface model
The volume of fluid (VOF) method is used to simulate the free surface. The basic idea is to define a fluid volume function F in each unit of the computational domain, and F is a function of time and space. The core of the VOF method is to reconstruct the moving free surface by solving the transport equation of the fluid volume function. The transport equation of the fluid volume function F is as shown in the following equation (5), and the volume and area fraction parameters also need to be considered.
2.4 Discretization of governing equations
Flow3D uses the finite difference method to discretize the governing equations and to discretize the space into threedimensional rectangular staggered grids, as shown in Figure 1. The scalar is defined at the center of the control body, such as the pressure P, the fluid volume function F, the density ρ, the flow volume fraction V _{F}, and so on, while the velocity u, v, w and area fraction A _{ x } , A _{ y } , A _{ z } are defined at the center point of the edge of the grid.
Figure 1
3 Numerical simulation based on Flow3D
3.1 3D numerical wave basin
A threedimensional numerical wave basin is established, and its correctness is verified. When the flow field mesh is divided, the ratio of a cell direction does not exceed 1:3, the ratio of the length to width of the xyz is not more than 1:5, the ratio of the adjacent grid block is best within 1:2, the number of grids in the range of wave height is 10–15, and the 100–120 grids are divided within the wavelength range. The simulation results of the wave are basically the same as the theoretical values. Based on the above experience [26,27], the free outflow boundary of Flow3D is used for preliminary wave elimination in the outlet of numerical wave basin.
3.1.1 Model building and meshing
The threedimensional numerical wave basin has a total length of 5 m in the X direction, a width of 2 m in the Y direction, and a height of 1.3 m in the Z direction. The boundary conditions are set as presented in Table 1.
Table 1
Boundary  Inlet  Outlet  Side  Bottom  Top 

Condition  Velocityinlet  Outflow  Symmetry  Wall  Specified pressure 
Mesh encryption is required at the free surface (z = 1 m), and generally 100 grids are set in the wave height range. The encrypted mesh of the free surface and the overall mesh of the threedimensional numerical wave basin are shown in Figure 2.
Figure 2
3.1.2 Numerical wave generation
A linear regular wave with a wave height of 0.04 m and a period of 0.98 s is selected. The time of simulation is 30 s, and the time step is 1/20 of the wave period (0.049 s). The threedimensional color contour plot of free surface after the wave stability is shown in Figure 3, and the wave height time history curve at x = 0 m is shown in Figure 4. Compared with the theoretical value, it is found that the wave is unstable before 5 s, and the wave is stable between 5 and 19 s. The wave height and phase are basically the same, and the simulation results are good. However, it is affected by the reflected wave after 19 s, which interferes with the normal simulation of the working area. Therefore, using Flow3D to establish a numerical wave basin, only the free outflow boundary condition of the numerical wave basin outlet cannot satisfy the wave elimination, and there is still the influence of the reflection wave. It is necessary to add the additional wave elimination device to meet the requirements of subsequent numerical simulation.
Figure 3
Figure 4
3.1.3 Numerical wave elimination
The additional wave elimination device adopts waveabsorbing block, which has the advantages of good waveeliminating effect and low cost. Waveabsorbing block belongs to the method of imitating physical wave elimination and adds a momentum decay source term in momentum equation.
The first term in the right side of the equation (6) is the viscous loss term, the second term is the inertia loss term, and the S _{ i } is the source term of the momentum equation in the i direction. The v is the velocity size, and the 1/α and the C _{2} are a constant [28,29,30,31,32,33,34]. In fact, only the first term at the right side of the equation usually is taken.
The numerical wave basin is divided into three blocks in the X direction, and the meshes are divided into different sizes, as presented in Table 2. The threedimensional mesh schematic diagram is shown in Figure 5.
Table 2
Mesh block  Block 1 (ordinary basin)  Block 2 (waveeliminating block)  Block 3 (ordinary basin) 

X direction length  0–5 m  5–8 m  8–10 m 
Mesh size  0.04  0.08  0.04 
Figure 5
Block 1 in the table is an ordinary threedimensional numerical wave basin, and there is no structure in the basin. Block 2 is a waveabsorbing block, which only plays the role of wave eliminating and does not need to observe the flow field in the block, so the grid can be appropriately sparse. Block 3 is an ordinary basin for testing the effect of wave eliminating.
The length of the waveeliminating region is about one to two times the wavelength, and the distance from the left side of the waveabsorbing block to the inlet is 5 m. Keep the wave parameters unchanged and also monitor the free surface changes at x = 0 m. Figure 6 shows a threedimensional color contour plot with waveabsorbing device, and the wave height is significantly reduced after the wave passes through the waveabsorbing device. Figure 7 shows the comparison of the wave height before and after wave eliminating and theoretical values at x = 0 m. It can be seen that after the improvement of the waveeliminating method, the wave height of the free surface is hardly affected by the reflected wave. It is in good agreement with the theoretical value. It is feasible and can be used for numerical simulation.
Figure 6
Figure 7
3.2 Numerical simulation validation
The hydrodynamic interference in the narrow gap of the sidebyside doublefloatingsystem is a typical strong nonlinear problem, and its theoretical research remains to be improved, and the reliability of the numerical simulation results needs to be further improved. In order to verify the correctness of the numerical simulation method in this paper, the results simulated in the present model was compared with published numerical and experimental results to examine if they agree well. In 2014, Xu [23] studied the flow field resonance phenomenon in the narrow gap of a doublefloatingsystem under the action of waves by experimental and numerical methods. The case in ref. [23] was numerically simulated using Flow3D software, and the results of the numerical simulation were compared with those in the ref. [23].
3.2.1 Description of numerical simulation case
Referring to the basin size and mesh division in ref. [23], a sidebyside doublefloatingsystem with the gap width of 0.05 m was added at the distance of 3 m from the inlet of the numerical wave basin. The specific parameters of the floating body are presented in Table 3.
Table 3
Parameter  Value  Parameter  Value 

Length L  1.5 m  Horizontal position of center of gravity (from stern)  0.7395 m 
Beam B  0.45 m  Vertical position of center of gravity (form baseline)  0.1272 m 
Depth D  0.15 m  I _{ xx }  0.62352 kg m² 
Draft T  0.0667 m  I _{ yy }  8.9248 kg m² 
Displacement ∆  0.0433 t  I _{ zz }  9.2301 kg m² 
Narrow gap width Φ  0.05 m 
To accurately simulate the motion posture of the floating body in the wave, it is necessary to add a nested encrypted mesh block. The nested mesh block can reduce the grid size as much as possible under the premise of fully including doublefloatingsystem, so as to reduce the total number of grids and reduce the computation time. The mesh in the X–Y plane and the overall mesh are shown in Figure 8.
Figure 8
When the coupled motion of wave and doublefloatingsystem is simulated, the floating body drifts away with the wave or collides with other floating bodies, which will lead to the divergence of numerical simulation. Therefore, the horizontal motions of the two floating bodies (swaying, surging, and yawing) are limited in the validation case, and only the vertical motions (heaving, rolling, and pitching) are retained.
3.2.2 Comparison of numerical simulation results
Five monitoring points in the narrow gap of the doublefloatingsystem are defined. The coordinates of monitoring points A–E are presented in Table 4. The location distribution of the monitoring points is shown in Figure 9.
Table 4
Monitoring points  Point A  Point B  Point C  Point D  Point E 

Coordinate  (−0.5, 0, 0)  (−0.25, 0, 0)  (0, 0, 0)  (0.25, 0, 0)  (0.5, 0, 0) 
Figure 9
The results of the numerical simulation are compared with the results in the ref. [23] in Figure 10. SIM represents the numerical simulation results in this paper, CAL represents the numerical results of the reference, and EXP represents the model test results of the reference.
Figure 10
It can be seen from the figure that the wave height values at four points of A, B, C, and E are basically consistent with the results in the reference. Except for the difference at the individual peaks, the phase is also in good agreement, and the overall error is within 5%. However, the wave height time history curve at point D differs greatly from the experimental results in the reference and is basically consistent with the numerical results of the ref. [35–41]. It may be that the horizontal motion of the two floating bodies in the CFD numerical simulation is completely limited, while the horizontal motion in the model test is not completely fixed, only by the horizontal mooring system.
The motion results of the doublefloatingsystem are compared with the results in the reference. The motion of the doublefloatingsystem under the action of wave is completely symmetrical. Therefore, the comparison of motion results of one floating body is shown in Figure 11.
Figure 11
The figure shows the time history curve of the vertical motion of the floating body, which is almost identical in phase, and only differs in part of the peak, but within the error tolerance. There are differences in the results of numerical simulation, and the possible reason is that the mesh of the validation case is different from that of the reference. In the reference, mesh refinement is performed on the free surface and in the narrow gap, but the validation case in this paper is to add a nested mesh around the doublefloatingsystem. Although the way of mesh encryption is different, the purposes are all to observe the free surface and the flow field in narrow gap.
According to the above two sets of comparisons, the results in the reference are in good agreement with the numerical simulation results in this paper within the range of error tolerance, which verifies the accuracy and reliability of the numerical research method in this paper.
3.3 Mesh convergence analysis
According to the hydrodynamic model of the doublefloatingsystem proposed in this paper, before using the Flow3D software to simulate the hydrodynamic interference in the narrow gap, to select the appropriate number of mesh, it is necessary to meet the high numerical simulation precision and less calculation time. Therefore, it is necessary to carry out the mesh convergence analysis.
3.3.1 Description of numerical simulation case
A numerical wave basin with a hydrodynamic model of a doublefloatingsystem is set up. The original point of the coordinate is taken at the middle point of the two floating bodies. The distance between two floating bodies is 0.05 m, the wave height is 0.035 m, and the period is 0.6455 s. Table 5 presents the division schemes of three different grid sizes. The different mesh schematic diagrams of the threedimensional numerical wave basin are shown in Figure 12. In the simulation of mesh convergence analysis, the horizontal motion (swaying, surging, and yawing) of the two floating bodies is limited, and only the vertical motion (heaving, rolling, and pitching) is retained [42].
Table 5
Mesh  Block 1 (overall mesh size)  Block 2 (nested mesh size)  Total number of mesh 

1  0.04  0.02  782,800 
2  0.02  0.01  1,323,200 
3  0.01  0.005  2,505,336 
Figure 12
3.3.2 Comparison of convergence analysis results
Figure 13 shows the comparison of the motion in the vertical direction of the two floating body and the wave height in the middle of the narrow gap under the three meshing schemes.
Figure 13
It can be seen that the convergence of mesh 2 and mesh 3 is better, the trend of the simulated results is basically the same, and the phase and peak are also basically consistent. However, the mesh 1 and mesh 2 and 3 have great differences in peak value, and some phases are also biased. The simulation effect is not good. To balance the calculation time and the accuracy of the calculation, the mesh 2 scheme is used to study the gapresonance problem when the numerical wave basin size and model size are the same.
4 CFD simulation of sidebyside doublefloatingsystem with a narrow gap
4.1 Analysis of flow field in the narrow gap of fixed doublefloatingsystem
The hydrodynamic interference in the narrow gap of the doublefloatingsystem under the action of regular wave is numerically simulated, and the wave elevation around the fixed doublefloatingsystem and the fluid movement in the narrow gap are mainly analyzed.
4.1.1 Description of numerical simulation case
Two identical homogeneous square box models are selected to simplify the hydrodynamic model of the doublefloatingsystem. The main parameters of the model are presented in Table 6.
Table 6
Parameters  Length  Beam  Depth  Draft  Displacement  Density  Narrow gap width 

Value  1.5 m  0.45 m  0.15 m  0.07 m  47.25 kg  466.67 kg/m³  0.1 m 
The narrow gap width is 0.1 m, and the distance from the inlet of the numerical wave basin to the floating body is generally greater than one wavelength (1.5 m). The length of the numerical wave basin is 4.7 m, the width of the basin is 2 m, the depth of the basin is 1.3 m, the water depth is 1 m, and the origin of the coordinates is 2.2 m from the inlet of the numerical wave basin. A linear regular wave with a wavelength of 1.5 m and a wave height of 0.04 m is selected.
4.1.2 Wave elevation in narrow gap
Figure 14 shows the wave elevation around the doublefloatingsystem in a regular wave period (t = 8.768– 9.748 s). At the beginning of a period (t = 0), the wave elevation appears at the front of the floating body. The value of the wave height (0.033 m) is about 1.65 times that of the incident wave amplitude (0.02 m), and the wave height enlargement is mainly because of the superposition of the reflected wave and the incident wave. The maximum wave elevation appears at t = 1/4 T, and the maximum wave height appears at the front of the narrow gap of the doublefloatingsystem (about the 1/10 of the floating body length), and the wave elevation (0.066 m) is about 3.3 times the incident wave amplitude of 0.02 m. At t = 1/2 T, as the wave continues to propagate, the amplitude of the wave in the narrow gap continues to decrease. At this time, the peak of the incident wave propagates to the middle of the floating body, and the wave height around the doublefloatingsystem is not large. The flow field in the narrow gap tends to be stable. At t = 3/4 T, corresponding to 1/4 T, the front of the narrow gap of the doublefloatingsystem is the valley value of the wave (about 1/10 of the total length of the floating body). When t = T, the fluid goes back to the state of t = 0 at the beginning of the period and begins to repeat the process mentioned above.
Figure 14
The overall effect is that the doublefloatingsystem has a shadowing effect on the flow field in the narrow gap, and the internal flow field of the narrow gap is separated from the external flow field. The flow field at the front of the narrow gap shows strong nonlinearity, and the phenomenon of wave elevation is obvious. The wave elevation in this example mainly occurs at the head of the floating body and the front of the narrow gap of the doublefloatingsystem. While other scholars have not found the wave resonance phenomenon at the head of the floating body, the resonance is mainly concentrated in the rear of the narrow gap. The possible reasons for this phenomenon are as follows: (1) the floating body is simplified into a square box model, and the head of the square box is different from the ship; (2) the selected wavelength is too long, the narrow gap width is too small, and the wave cannot pass through the inside of the narrow gap but directly bypasses the doublefloatingsystem. Therefore, to further explore the resonance phenomenon inside the narrow gap, the next section will study the influence of the narrow gap width and wave period on the flow field around doublefloatingsystem.
When t = 1/4 T and t = 3/4 T, Figure 15 shows the distribution of the wave height values in the narrow gap midline (y = 0) of the doublefloatingsystem along Xaxis direction of the numerical wave basin. Two red dotted lines indicate the position of the head and tail of the floating body. It can be seen that under the coupling of the wave and the doublefloatingsystem, the hydrodynamic resonance phenomenon occurs in the head of the floating body and the front of the narrow gap, and the wave rises obviously (Figure 16). However, the wave height at the tail of the doublefloatingsystem begins to decrease gradually, and after passing through the waveeliminating region, the wave height rapidly decreases and tends to be stable.
Figure 15
Figure 16
The five monitoring points in the narrow gap of the doublefloatingsystem are selected to analyze the wave elevation in the gap. The coordinate values of the A–E points are presented in Table 7, and the wave height value of the A–E monitoring point is plotted as a time history curve, as shown in Figure 17.
Table 7
Monitoring points  Point A  Point B  Point C  Point D  Point E 

Coordinate  (−0.75, 0, 0)  (−0.375, 0, 0)  (0, 0, 0)  (0.375, 0, 0)  (0.75, 0, 0) 
Figure 17
It can be seen from the figure that the numerical simulation results of different monitoring points tend to be stable after 10 s. The wave height value near the A monitoring point is the largest, and the maximum value is about (0.07 m) up to 3.5 times of the incident wave (0.02 m), which is consistent with the results of color contour plot of free surface above. With the continuous propagation of the incident wave along the Xaxis direction in the narrow gap, the peak position of the wave elevation in the narrow gap moves along the direction of the Xaxis as well, but the peak value of the wave elevation is continuously reduced until the free surface tends to be stable. It can be found that the decrease in the peak value of the wave elevation in the Xaxis direction is caused by various factors such as fluid viscous action, wave damping effect, and shadowing effect of the floating body on the flow field in the narrow gap.
4.1.3 Fluid motion in narrow gap
To investigate the fluid motion around the doublefloatingsystem, Figure 18 shows the division of a wave period (t = 8.768–9.748 s) into eight typical moments. At the midship section (x = 0 m) of the doublefloatingsystem, a complete wave period is selected to discuss the velocity and pressure distribution of the fluid around the doublefloatingsystem, as shown in Figure 19.
Figure 18
Figure 19
At the beginning of the period t = 0, it can be seen from Figure 19(a) that the fluid flows from the outside of the two floating bodies through the bottom into the inside of the narrow gap, and the fluid velocity inside the narrow gap is upward, and the free surface is continuously increased. When t = 1/8 T, it can be seen from Figure 19(b) that the upward velocity of the fluid in the narrow gap begins to decrease gradually, and the fluid at the bottom of the floating body begins to flow backward to the outside of the two floating bodies. When t = 1/4 T, it can be seen from Figure 19(c) that the free surface in the narrow gap reaches the highest, at which time the fluid velocity is reduced to zero, and the direction is changed to start generating a downward flow velocity. When t = 3/8 T, it can be seen from Figure 19(d) that the downward velocity of the fluid in the narrow gap is increasing, and flows to the outside of the doublefloatingsystem through the bottom of the floating body. When t = 1/2 T, it can be observed from Figure 19(e) that the downward moving speed of the fluid inside the narrow gap reaches the maximum, and the free surface is basically in the static equilibrium position. When t = 5/8 T, it can be seen from Figure 19(f) that the downward velocity of the fluid in the narrow gap begins to decrease. In addition, when most of the fluid flows downward in the narrow gap, the fluid outside the two floating bodies already has the velocity of backflow, indicating that the fluid has started to move from the outside through the bottom of the floating body to the narrow gap. When t = 3/4 T, it can be observed from Figure 19(g) that the flow in the narrow gap moves to the lowest point, and the downward velocity gradually decreases to zero and then begins to move upward. When t = 7/8 T, it can be seen from Figure 19(h) that the upward velocity of the fluid in the narrow gap starts to increase gradually, and the fluid outside the two floating bodies flows into the narrow gap through the bottom of the floating body, eventually causing the free surface in the narrow gap to rise continuously. When t = T, the fluid returns to the state at the beginning of the cycle and begins to repeat the process mentioned above.
4.2 Parameterized study of flow field in narrow gap
Using the control variable method, the narrow gap width of the doublefloatingsystem and wave parameters is selected as variables to observe the wave elevation of the flow field in the narrow gap. The influence of different gap width and different wave periods on the flow field in the narrow gap is explored.
4.2.1 Description of numerical simulation case
The narrow gap width of the doublefloatingsystem is selected as 0.05 m and 0.1 m, the wave height is 0.04 m, and the water depth 1.3 m remains unchanged. The wavelength is 0.25, 0.5, 1, and 1.5 times the length of floating body, respectively, and the numerical simulation cases are presented in Table 8.
Table 8
Case  Case 1  Case 2  Case 3  Case 4  Case 5  Case 6  Case 7  Case 8 

Narrow gap width (m)  Φ = 0.05  Φ = 0.05  Φ = 0.05  Φ = 0.05  Φ = 0.1  Φ = 0.1  Φ = 0.1  Φ = 0.1 
Wavelength (L)  0.25  0.5  1  1.5  0.25  0.5  1  1.5 
Wave period (s)  0.49  0.699  0.98  1.2  0.49  0.699  0.98  1.2 
4.2.2 Wave elevation of the flow field at four different wave periods when the gap width is 0.05 m
Cases 1–4 are used to compare and analyze the flow field in the narrow gap at typical time. According to the wave parameters and the size of numerical wave basin, the wave propagates to the structure around 3 s and is relatively stable around 8 s. Therefore, a complete wave period after 8 s is selected as a typical time in each case, and the wave period is divided into four typical moments (t = 0, t = 1/4 T, t = 1/2 T, and t = 3/4 T). Figures 20–23 show the color contour plot of free surface obtained by numerical simulation.
Figure 20
Figure 21
Figure 22
Figure 23
The following can be seen from the above color contour plot of free surface:
In case 1, the wave period is 0.49 s. When the incident wave peak propagates to the front of the doublefloatingsystem, the free surface rises significantly because of the superposition of the reflected wave and the incident wave after the wave encounters the floating body. As the wave continues to propagate, the peak of the wave appears at the head of the floating body, and the height reaches a maximum value of 0.056 m, which is about 2.8 times the incident wave amplitude of 0.02 m. The wave continues to propagate in the narrow gap, and the wave amplitude continues to decrease until the free surface is stable. As the amplitude is reduced slowly and the selected wavelength is shorter, there are multiple peak points in the flow field in the narrow gap.
In case 2 with a wave period of 0.699 s, when the incident wave peak propagates to the entrance of the narrow gap, the free surface at the entrance of the narrow gap increases obviously. When the peak of the incident wave propagates to about 1/8 to 1/7 of the length of the floating body inside the narrow gap, the wave height reaches 0.045 m, which is 2.25 times of the incident amplitude. When the incident wave peak propagates to the middle of the floating body, the wave elevation in the narrow gap is about 0.5–1.5 times the wave amplitude. As the wave continues to propagate, the wave height in the narrow gap continues to decrease. When the peak of the incident wave reaches the tail of the floating body, there is a peak value and a valley value in the narrow gap.
In cases 3 and 4, the wave periods are 0.98 and 1.2 s, respectively. The obvious wave elevation is also observed at the entrance of the narrow gap, but the peak value of the wave elevation decreases relative to the cases 1 and 2, which is about 1.6–1.8 times of the incident wave amplitude. Compared with the case with smaller wave period (cases 1 and 2), the decrease in the wave height in the narrow gap is more rapid. When the wave peak is spread to the middle of the floating body, the free surface of the flow field in the narrow gap tends to be stable.
4.2.3 Wave elevation of the flow field at four different wave periods when the gap width is 0.1 m
The flow field in the narrow gap in the typical time of cases 5–8 is compared and analyzed. Figures 24–27 show the color contour plot of free surface obtained by numerical simulation.
Figure 24
Figure 25
Figure 26
Figure 27
These color contour plot of free surface above can be seen as follows:
Case 5 with the wave period of 0.49 s is similar to case 1. When the peak of the incident wave propagates to the front of the doublefloatingsystem, the free surface rises obviously, and the wave height of 0.044 m is about 2.1 times that of the incident wave. When the wave continues to propagate to about 1/10 to 1/9 of the length of the floating body in front of the narrow gap, the free surface rises to a maximum value of 0.061 m, which is about three times the amplitude of 0.02 m. Then, the wave amplitude in the narrow gap begins to decrease until the free surface is stable. Because of the short wavelength, there are several peak points in the flow field in the narrow gap, which is basically consistent with the gap width of 0.05 m.
In case 6, the wave period is 0.699 s. When the peak of the incident wave propagates to the front of the narrow gap at a position of about 1/4 to 1/5 of the floating body length, the middle of the narrow gap is a wave valley, and the front and rear of the gap have a wave peak, respectively, and the wave height (0.04 m) is about twice the incident wave amplitude. As the wave continues to propagate, the peak point moves continuously in the narrow gap, and the height of the peak point continues to decrease at a relatively slow speed. When the incident wave peak propagates to the tail of the narrow gap, the wave height of the free surface in the narrow gap is about 0.5 to 1.5 times that of the incident wave.
In cases 7 and 8, the wave periods are 0.98 s and 1.2 s, respectively. The wave elevation occurs at the head of the doublefloatingsystem and at the entrance of the narrow gap. There is no significant wave elevation in the narrow gap. Moreover, compared with the case with smaller wave period (cases 5 and 6), the decrease in the wave height in the narrow gap is more rapid. When the wave peak is spread to the middle of the floating body, the free surface in the narrow gap has tended to be stable.
5 Concluding remarks
In this paper, the doublefloatingsystem is taken as the research object to study the hydrodynamic interference problem in the narrow gap and try to find the law of the gapresonance phenomenon, to explore the internal causes and mechanism. First, a threedimensional numerical wave basin is established to verify the correctness and feasibility of the viscous CFD method in numerical simulation of gapresonance problem. The mesh convergence analysis is carried out for the hydrodynamic model of the doublefloatingsystem, and the optimal meshing scheme is proposed. Second, the wave elevation and fluid movement in the narrow gap of fixed doublefloatingsystem are analyzed in detail. Finally, the effects of gap width and wave period on the hydrodynamic interference in the narrow gap of the doublefloatingsystem are investigated. The main conclusions are as follows:
The narrow gap width is the same, and the flow field movements between the gaps are different under different wave periods. When the incident wave reaches the resonant period of the doublefloatingsystem, the wave continuously reflects in the narrow gap, and the fluid in the narrow gap produces convection with the fluid at the bottom of the floating body, and the free surface in the narrow gap will rise significantly.
It can be seen from the color contour plot of free surface at different gap width that as the narrow gap width increases, the resonant period of the flow field in the narrow gap of doublefloatingsystem also increases.
The velocity of the fluid in the narrow gap is faster than the velocity of the fluid outside the doublefloatingsystem. At the same time, the flow field in the narrow gap is about 1/4 wavelength ahead of the flow field outside the doublefloatingsystem.
When the wavelength is too large, the incident wave cannot enter the narrow gap of the doublefloatingsystem, and the wave directly bypasses the doublefloatingsystem to continue to propagate. The doublefloatingsystem has a shadowing effect on the flow field in the narrow gap, and there is no obvious wave elevation in the narrow gap, and the peak value of the wave elevation appears at the head of the doublefloatingsystem.
When the incident wavelength decreases to less than the floating body length, the wave enters the narrow gap and causes the wave elevation of the flow field in the narrow gap. There may be one or more peak points in the narrow gap, which are related to the aspect ratio L/B and wavelength ratio λ/L.
With the continuous propagation of the incident wave along the direction of floating body length in the narrow gap, the peak point of the wave elevation in the narrow gap is also moving along the direction of floating body length. However, the peak height of the free surface decreases continuously until the free surface tends to be stable. The gradual decrease in wave height on free surface may be attributed to fluid viscosity, wave damping, and the shadowing effect of doublefloatingsystem on the flow field in the narrow gap and so on.
Acknowledgments
The authors would like to acknowledge the support of National Natural Science Foundation of China (Nos. 52001138; 52071094; 11572094; 5171101175) and National Natural Science Foundation of Jiangsu Province (Nos. BK20201029; 20KJB416005). In addition, the author “Zhiyang Zhang” would like to thank Prof. Zhang for the guidance and assistance in his studies, as well as all the teachers and students of Deepwater Engineering Research Center at Harbin Engineering University.

Conflict of interest: Authors state no conflict of interest.
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