Three-Dimensional Simulations of Offshore Oil Platform in Square and Diamond Arrangements

The interaction of the solitary wave with an oil platform composed of four vertical circular cylinders is investigated for two attack angle of the solitary wave β = 0 ° (square arrangement) and β = 45 ° (diamond arrangement). The solitary wave is generated using an internal source line as proposed by Hafsia et al. (2009). This generation method is extended to three-dimensional wave ﬂ ow and is integrated into the PHOENICS code. The volume of ﬂ uid approach is used to capture the free surface evolution. The present model is validated in the case of a solitary wave propagating on a ﬂ at bottom for H / h = 0 : 25 where H is the wave height and h is the water depth. Compared to the analytical solution, the pseudowavelength and the wave crest are well reproduced. For a solitary wave interacting with square and diamond cylinders, the simulated results show that the maximum run-ups are well reproduced. For the diamond arrangements, the di ﬀ raction process seems to not a ﬀ ect the maximum run-ups, which approached the isolated cylinder. For the square arrangement, the shielding e ﬀ ect leads to a maximum wave force more pronounced for the upstream cylinder array.


Introduction
In the last decades, many researchers have focused on searching different wave structures of nonlinear partial differential equations. The interested readers can see [1][2][3][4]. The offshore oil platforms and the coastal bridges are composed of multiple cylinders disposed in different arrangements. When the wave run-up and the following wave forces exceed the expected values, the safety of these structures is compromised. The available analytical solutions in the literature are only valuable under some limiting assumptions. For this reason, experimental and numerical methods are adopted to solve this wave-platform interaction problem.
The interaction of a solitary wave (representing a real tsunami wave) with a single circular cylinder was studied experimentally by Yates and Wang (1994) in [5]. The effect of a single row of circular cylinders on the transmission and reflected coefficients is studied experimentally by Huang (2010) in [6]. The consequent results are that the wave force acting on a coastal structure protected by these cylinders can be reduced to about 60% for S = 1:2 D, where S is the distance between the centers of adjacent cylinders, and D is the cylinder diameter. An experimental study was conducted by Huang (2007) in [7] to measure the reflection and transmission coefficients in the case of single and twin rows of rectangular cylinders, and simplified analytical expressions of these coefficients are proposed. Wang et al. (2021) in [8] investigate experimentally, the back and front run-up and wave forces induced by solitary wave for different truncated vertical cylinders. Secondary peak for both the front run-up and wave forces are observed due to the return flow.
An alternative method to the experimental measurements of the free surface evolution is the use of a twodimensional (2-D) or three-dimensional (3-D) numerical method. The first one is based on the depth averaged Boussinesq equations and can be used for a long-term simulation. Lin and Man (2007) in [9] validate the Boussinesq model for one-dimensional and two-dimensional wave transformations. Mohapatra et al. (2020) in [10] studied the regular wave diffraction by a floating fixed vertical cylinder by two methods: using a CFD code and analytically based on a Boussinesq model. Among the various Boussinesq models existing in the literature, the nonlinear effects cannot well be reproduced (Zhao et al., 2007 in [11], Wang and Ren, 1999 in [12], and Liu et al., 2012 in [13]). Hence, a full threedimensional (3-D) Navier-Stokes model is required. In order to study the regular wave run-ups for a single and a group of vertical cylinders, numerical virtual wave probes were used by Cao and Wan (2017) in [14]. Frantzis et al. (2020) in [15] adopted a (3-D) numerical wave tank (NWT) to study the wave breaking induced by a single row of vertical cylinders for different ranges of cylinder diameter to depth ratios. The large eddy simulation (LES) model was used to reproduce the small scales of turbulence. Numerical results show that the effect of the cylinder diameter is more significant for larger values of solitary wave heights. Wang et al. (2018) in [16] conducted a series of laboratory experiments on the internal solitary wave (ISW) loads upon semisubmersible platforms in a density stratified fluid tank, and investigated the load components induced by different factors. The wave loads on a platform composed of 2 × 2 circular cylinders in side-by-side and tandem arrangements are numerically studied using the Reynolds-Averaged Navier-Stokes equations by Yang et al. (2015) in [17]. The desired monochromatic wave was generated by the prescribed velocity components at the inlet of the computational domain. Using a CFD code, Kamath et al. (2015) in [18] investigate the diffraction of sinusoidal wave by 3 × 3 square array cylinders placed in proximity and show that the wave force is highest when the distance between the cylinder center is less than half of the incident sinusoidal wave. The numerical results show that if S > 4 D, there is no interaction between the platform cylinders. Xie et al. in [19] used a cut-cell method in a fully (3-D) code to simulate solitary wave interaction with a vertical circular cylinder and a thin horizontal plate. Several computational fluid dynamics (CFD) implemented a cut-cell algorithm permitting to identify the contribution of a portion of a rectangular grid to the convective and diffusive fluxes.
The main task of the present study is to investigate the interaction of the solitary wave with one or four circular cylinders in a square or diamond arrangement using a (3-D) numerical wave tank (NWT). The proposed wave generation method is based on an internal mass source. The cut-cell implemented in the PHOENICS code is used to reproduce the circular cylinder shape. In all simulated cases, the depth to cylinder radius ratio is taken: h/a = 1, and the distance from the center of cylinders is S = 3 a. For the square platform, there is one cylinder in each corner of the square as indicated in Figure 1(b). The two upstream cylinders are denoted 3 and 4, and the two downstream cylinders are 1 and 2. The square arrangement corresponding to the attack angle of solitary wave β = 0. This angle is measured between the propagating direction and the symmetric line of the platform (the line parallel to the line joining 1 and 3). The diamond configuration of the platform is shown in Figure 1(c) and corresponds to β = 45°. The effective computational domain has a length of L = 55 a. Two dissipative zones are added in each open boundary to avoid wave reflection having a length of ð25 aÞ. The considered width is ð10 aÞ, ð13 aÞ, and ð12:25 aÞ, respectively, for a single cylinder, square platform, and diamond platform.

Mathematical Formulation
The overall grid and the grid around the oil platform are shown in Figure 2 for the diamond arrangement. Fine grids are adopted around the four cylinders, and coarse grids are imposed in the two dissipation regions. The cut-cell method is used to mesh the circular cylinders in the Cartesian coordinates system. Figure 2(b) shows the details of the cut-cell grids around the four cylinders. The number of grid in x, y, and z directions is, respectively NX × NY × NZ = 220 × 110 × 102 for the diamond arrangement and 220 × 100 × 102 for the square arrangement. For these two arrangements, the time step is Δt = 0:01 s.
(ii) The momentum transport equation: where x i is the Cartesian coordinates, u i is the velocity components, ρ is the density of the mixture, p is the pressure, ν is the kinematic viscosity of the mixture, g is the 2 Advances in Mathematical Physics acceleration due to gravity, and s d,z is a momentum source term added to the momentum equation along z-direction to avoid wave reflection at the open boundaries given by: where γðxÞ is a linear damping function and w is the velocity component along the z-vertical direction.

Free Surface
Capture. The air-water interface is modeled using the mixture model flow. If α q denoted the volume fraction of the q th fluid in a cell, then, (i) The density of the mixture is given by: where ρ q is the density of the water when ðq = 1Þ and air when ðq = 2Þ.
(ii) And dynamic viscosity of the mixture is where μ q is the dynamic viscosity of the water if ðq = 1Þ and air when ðq = 2Þ.
The volume fraction of fluid is determined by the following mass conservation equation for each phase: When α q = 0, the cell is occupied by air, α q = 1, the cell is occupied by water, and 0 < α q < 1, the cell contains the interface (Hirt and Nichols, 1981 in [20]).

Wave
Generation. The desired solitary wave was generated by an internal source inlet across a source line as proposed by Hafsia et al. (2009) in [21]. The inlet vertical velocity is prescribed as a time-dependent inlet boundary condition: where L s is the length of the internal source line. The wave celerity is given by (Dominguez et al., 2019 in [22]): The solitary wave surface elevation ηðx s , tÞ is given by the following equation: where H is the incident wave height and t is the time. The distance x s permitting to have a negligible source at t = 0 s is determined by the following equation: The equivalent wave number k is

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Following this equivalent wave number, the pseudowavelength can be determined by: The where n ! is the normal unit vector pointing into the water.
From the component of this wave force along the x -direction, the force coefficient can be calculated as: 2.6. Initial and Boundary Conditions. The following initial and boundary conditions are adopted for the governing transport equations. The imposed initial condition is still water with a depth h. For the top boundary, the pressure P is set equal to the atmospheric pressure. Two dissipation zones are adopted at the open boundaries (Figure 1). At all the other boundaries of the computational domain, symmetric boundary conditions are imposed.

Numerical Schemes.
To solve this proposed model, we adopt the PHOENICS code (Parabolic Hyperbolic or Elliptic Numerical Integration Code Series). In this code, the SIM-PLEST iterative algorithm is used to solve the pressure and velocity coupling in the Navier-Stokes equations (Artemov et al., 2009 in [23]). The upwind scheme is used for nonlinear convection terms and an implicit formulation for the transient term. The (VOF) method is used to predict the interface between the water wave and air. For all the presented simulation results, the cut-cell within the PARSOL (PARtial SOLid) treatment detects the solid-fluid interface, which is not aligned with the Cartesian grid. The proposed three-dimensional wave generation method is implemented in the PHOENICS code.

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simulated results show that before reaching the vertical cylinder, the wave profile is invariant in the transverse direction and can be represented by two-dimensional profiles. Figure 3 represents the simulated wave profiles at the center of the computational domain before impacting the cylinder. The wave is not affected by the cylinder, and the free surface profiles agree very well with the analytical one. The wave crest and the pseudowavelength are in accordance with the analytical one. When a solitary wave passes around the cylinder, run-up occurs at the front of the cylinder and the maximum run-up depended on the incident wave energy. Then, the water level drops at the front producing the rise of the water level at the rear of the cylinder by wave diffraction. In order to validate the cut-cell method, the maximum run-up is compared to the available literature. The maximum wave run-up is   determined from the evolution of the solitary wave profiles along the centerline and is equal to R max = 0:068 m corresponding to the ratio R max /H = 1:36. Following the numerical study of Zhao et al. (2007) in [11] for the same H/h = 0:25 and h/a = 1, this ratio is found equal to 1.37. There is no significant difference between this nondimensional run-up. The maximum wave force occurs at the same time as the maximum run-up that is equal to t = 2:9 s. Figure 4 shows the time histories of the computed wave force on the x-direction acting on an isolated cylinder. The maximum wave force at this instant is equal to F max = 37:5 N corresponding to the following force coefficient C f ,x = 0:479. The isolated cylinder is taken as a reference case for the computed force coefficient acting on each cylinder of the platform in the two studied configurations.

Solitary
Wave Diffraction by an Oil Platform. The flow field due to the solitary wave diffraction by square and diamond platform is analyzed at the instant of the maximum run-up in terms of the free surface elevation, the maximum run-up R max at each cylinder, and the maximum wave force F max .
The perspective view of the free surface elevation is shown in Figure 5 at the instant of the maximum run-up at cylinder 1. The solitary wave crest has been altered by the diffraction process. Impacting the cylinder obstacle, the wave run-up is observed due to the transformation of the incident wave to potential energy. The R max depends on the incoming    Advances in Mathematical Physics wave and the nature of the obstacle. This interaction leads to a complicated flow field as shown in Figure 5. For square arrangement, the R max for the first array (cylinders 3 and 4) occurs at the instant t = 2:90s and the for second array (cylinders 1 and 2) at t = 3:45s. For diamond arrangement, the first R max is observed for the most upstream cylinder (cylinder 4) at t = 2:88s. The maximum run-ups for cylinders 2 and 3 occur at the same instant (t = 3:16s) and due to nonlinear effect, the R max for cylinder 1 is observed at t = 3:48s. Figure 6 represents the free surface elevation at the instant of the maximum run-up R max for each cylinder in the square arrangement (zero solitary wave incidence). The R max on the downstream cylinders (1 and 2) is smaller than those on the upstream cylinders (3 and 4). The R max on cylinders 1 and 2 is smaller than on the corresponding isolated cylinder. This can be explained by the fact that some of the incident wave energy has been reflected back by other cylinders before the solitary wave impacting the downstream cylinders array. The numerical results of Zhao et al. (2007) in [11] confirm these conclusions, and the calculated R max /H is closely the same as shown by Table 1.
The free surface elevation at the instant of the maximum run-up R max for each cylinder in the diamond arrangement is shown in Figure 7. The R max /H for cylinders 4 and 1 is located at the centerline approach to that on the isolated cylinder. The R max /H for cylinders 2 and 3 is slightly greater than on the isolated cylinder. Table 2 shows good agreement between the present simulations and Zhao et al. in [11] results for the diamond arrangement.
The time evolution of the wave force in the positive xdirection for each cylinder is presented in Figure 8 for the square arrangement. The maximum wave force F max on the most downstream array of the cylinders (1 and 2) is slightly greater than on the isolated cylinder. The increase of the F max relative to the isolated cylinder is more pronounced for the first cylinder array (cylinders 3 and 4). The platform and wave interactions lead to F max for the first cylinders (3 and 4) array greater than on the second array (cylinders 1 and 2). This can be attributed to the shielding effect of the upstream cylinder array. These conclusions are in concordance with the computed run-ups previously discussed.
For the diamond arrangement, Figure 9 presents the time evolution of the wave force for each cylinder. The aligned cylinders 4 and 1 have the same F max . Compared to the isolated cylinder, this F max is significantly smaller. The two symmetric cylinders about the centerline of the computational domain are having the same F max as the isolated cylinder. The approaching solitary wave for these cylinders seems to be not disturbed by the diffraction process.

Conclusions
A full three-dimensional numerical wave tank (NWT) was integrated on the PHOENICS code in order to study the solitary wave diffraction with diamond or square cylinders arrangements. The solitary wave was generated by an internal line source, and the cylinder structures are discretized using the cut-cell method. For the diamond platform arrangement, the maximum wave run-ups approach to that on the isolated cylinder indicating that the diffraction process does not affect the four cylinders. However, for a square arrangement, the shielding effect of the upstream cylinders leads to a maximum wave force for the first cylinders array greater than the most downstream array.
The cut-cell method can be generalized for more complex geometric coastal structures in the Cartesian coordinates system. The proposed model based on an internal line source for wave generation can be used to study the combined effects of wave and current on the forces acting on multiple cylinders. Different incident waves can be tested (monochromatic, Stokes, and cnoidal waves). The present model can be used to test the effect of the turbulence model, the effect of the wave height, and the distance between the cylinders on the wave forces and wave run-ups. Further wave-structure interaction cases can be studied such as the wave diffraction with floating structures.

Data Availability
No data were used to support the study.

Conflicts of Interest
This work does not have any conflicts of interest.

Acknowledgments
The third author would like to thank all the professors of the Mathematics Department at the University of Annaba in Algeria, especially his professors/scientists Pr. Mohamed Haiour, Pr. Ahmed-Salah Chibi, and Pr. Azzedine Benchettah for the important content of masters and PhD courses in pure and applied mathematics that he received during his studies. Moreover, he thanks them for the additional help they provided to him during office hours in their office about the few concepts/difficulties he had encountered, and he appreciates their talent and dedication for their postgraduate