Solitary Wave Diffraction with a Single and Two Vertical Circular Cylinders

Department of Physics, College of Science and Arts at ArRass, Qassim University, Buraydah, Saudi Arabia Department of Mathematics, College of Science and Arts at ArRass, Qassim University, Buraydah, Saudi Arabia Department of Mathematics, College of Science, Qassim University, Buraydah, Saudi Arabia Department of Computer Science, College of Science and Arts at ArRass, Qassim University, Buraydah, Saudi Arabia College of Engineering, Texas A&M University, Higher Education Center at McAllen, 6200 Tres Lagos Blvd, McAllen, TX 78504, USA


Introduction
To ensure the safety of the offshore oil or airport platform, it is necessary to determine the interaction of the nonlinear waves with single or multiple cylinders structures. e main tasks of the wave structure interaction (WSI) problems are the prediction of the wave run-up, forces, breaking, and flow separation. e wave diffraction near multiple cylinders depended on the interference flow fields following the gap distance between the cylinders.
Different approaches are used to solve the WSI including experimental, analytical, or numerical methods. e interaction of a solitary wave (representing a real tsunami wave) with a single circular cylinder was studied experimentally by Yates and Wang [17] for weak and strong nonlinear waves. e development of a numerical wave tank (NWT) and its implementation on a computational fluid dynamics (CFD) code constitutes an alternative to the experiments (Cao and Wan [1], Windt et al. [2], and Ji et al. [3]). e NWT includes those based on full NavierStokes equations or the depthaverage Boussinesq equations the simplified irrotational flow. e diffraction of monochromatic short-crested waves on a single vertical cylinder is studied by Zhu [4]. Based on the irrotational flow assumption, analytical solutions were proposed in this limiting assumption. Lin and Man [5] developed a nonlinear wave interaction around coastal structures by the extended Boussinesq equations of Nowogu. e computed results are compared to the full (3-D) NavierStokes results. Based on the Boussinesq model for wave generation and propagation, Zhao et al. [6] investigate numerically the solitary wave run-up around one, two, and four circular cylinders. e wave diffraction of the cnoidal wave over two side-by-side or tandem arrangement is studied numerically by Wang and Ren [7] using a generalized Boussinesq model. Ning et al. [8] integrate the cut-cell method in a Boussinesq model to study the solitary wave diffraction in Cartesian grid cells.
e Boussinesq model has the advantage that it can be used for nonlinear wave transformation for large-scale problems. However, the nonlinear effects are not well reproduced due to average operation. Hence, the numerical solution of the complete NavierStokes equations in threedimensional (3-D) is necessary. Xie et al. [9] 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. e Smagorinsky subgrid-scale model was used to simulate the turbulence structure. e desired waves are generated by prescribing the analytical solutions at the inlet. e monochromatic wave diffraction through a rectangular cylinder was conducted using a (3-D) free surface model by Li et al. [10]. Zhi and Jie-Min [11] propose to generate a cnoidal wave through the specification of the incident wave characteristics at the inlet (velocity components induced by the desired wave). is three-dimensional model is used to predict the cnoidal wave run-up through a single bottom cylinder. Kriebel [12] presents a theoretical solution of the run-up around a circular cylinder due to regular and nonlinear monochromatic waves. e comparison of experimental and analytical solutions shows that the nonlinear nature of wave propagation has a large effect on wave runup. A Large Eddy Simulation (LES) to predict turbulence structure is used by Mo et al. [12] for monochromatic wave flow interaction with a single circular cylinder. Kamath et al. [14] developed a full three-dimensional model to study the free surface displacements and wave forces on three and four vertical cylinders. Frantzis et al. [15] analyze numerically the reflection and transmission coefficients induced by a breakwater formed by a single row of vertical circular piles closely spaced. e free surface deformation is simulated using the (VOF) method, and the (LES) model was used to reproduce the small scales of turbulence. e present study investigates numerically the interaction of solitary wave with a single and twin circular cylinder. e NWT was first validated for solitary wave propagating on a flat bottom and then used to simulate the wave force exerted on a single and twin circular cylinder in tandem and side-by-side arrangements. e source line used by Hafsia et al. [16] in two-dimensional (2-D) NavierStokes model has been extended to generate a nonlinear wave for a threedimensional (3-D) model. Figure 1 represents the locations of the source line and the different grid regions to control the grid distributions through the computational domain. As shown in this figure, the incident solitary wave propagates along the x-direction and impacts one or two circular cylinders in two arrangements following the angle β: in tandem (β � 0) and side-by-side (β � 90 ∘ ). e ratio of the still water depth to cylinder radius is (h/a) � 3. For a given water depth h � 0.2 m, the cylinder ratio is a � 0.067 m. e two vertical cylinders are closely separated with (G/2a) � 2, where G is the gap distance between two cylinders (center to center). e wave forces are calculated for the tandem and side-by-side arrangements and for the following nondimensional wave heights: (H/h) � 0.1 and 0.2. Figure 2 presents the grid structure for two cylinders arranged in tandem with a refinement around the free surface and the circular cylinders. e two circular cylinders are represented using the cut-cell method in a Cartesian grid as shown by Figure 2(b). For all the cylinder arrangements, the grid number along the x-direction is taken equal to N x � 190 and along the z-direction equal to N z � 94. For a single cylinder and the tandem arrangement, the grid number along the y-direction: N y � 40. However, for the side-by-side cylinders: N y � 60. e same time step is adopted for all simulated cases Δt � 0.01 s.

Governing Transport Equations.
e proposed (NWT) was based on the full three-dimensional (3-D) NavierStokes transport equations coupled to the volume of liquid (VOF) convective transport equation to reproduce the water wave interface. For unsteady flow and incompressible fluid, the mass and momentum conservation equations are written as follows: e mass conservation equation: e momentum transport equation: where x i the Cartesian coordinates, u i is the velocity components, ρ is the density of the mixture, p is the pressure, ] is the kinematic viscosity of the mixture, g is the acceleration due to gravity, and s d,z is a momentum source term added to the momentum equation along zdirection to avoid wave reflection at the open boundaries given by where c(x) is a linear damping function and w is the velocity component along the z vertical direction.

Free Surface
Capture. e air-water interface is modeled using the mixture model flow. If α q denotes the volume fraction of the q th fluid in a cell, then e density of the mixture is given by where ρ q is the density of the water when (q � 1) and air when (q � 2).
And, dynamic viscosity of the mixture is where μ q is the dynamic viscosity of the water if (q � 1) and air when (q � 2). e 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 et al. [18]).

Wave
Generation. e desired solitary wave was generated by an internal source inlet across a source line as proposed by Hafsia et al. [16]. e inlet vertical velocity is prescribed as a time-dependent inlet boundary condition: where L s is the length of the internal source line. e wave celerity is given by Dominguez et al. [19]: e solitary wave surface elevation η(x s , t) is given by the following equation: where H is the incident wave height and t is the time. e distance x s permitting to have a negligible source at t � 0 s is determined by the following equation: e equivalent wave number k is Following this equivalent wave number, the pseudowave length can be determined by Table 1 summarizes the wave parameters and the length of the source line L s for the two considered wave height to water depth ratios. e length L s and the position of the source line are determined by the calibration procedure as indicated by Hafsia et al. [16].

Initial and Boundary Conditions.
e following initial and boundaries conditions are adopted for the governing transport equations. e 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.
To solve this proposed model, we adopt the PHOENICS code (Parabolic Hyperbolic or Elliptic Numerical Integration Code Series). In this code, the SIMPLEST iterative algorithm is used to solve the pressure and velocity coupling in the NavierStokes equations (Artemov et al. [20]). e upwind scheme is used for nonlinear convection terms and an implicit formulation for the transient term.
e 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 is used to detect the solid-fluid interface which is not aligned with the Cartesian grid. e proposed three-dimensional wave generation method is implemented in the PHOENICS code.

Wave Diffraction by a Single Circular Cylinder.
e proposed wave generation method based on an internal source line is validated for two solitary waves having the nondimensional heights H/h � 0.1 and 0.2. e 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 in the center of the computational domain. Figure 3 represents the simulated wave profiles at the center of the computational domain before impacting the cylinder. For these two wave heights, the wave is not affected by the cylinder and the free surface profiles agree very well with the analytical one. e wave crests and the pseudo-wave lengths are in accordance with the analytical one. For (H/h) � 0.1, the wave is generated after t � 3 s. As the wave height increases, the solitary wave is completely generated more rapidly (after t ′ � 2.3 s).
In order to validate the cut-cell method, the maximum run-up is compared to the available literature. e maximum wave run-up is determined from the evolution of the solitary wave profiles along the centerline. In Figure 4, the solitary wave profiles are presented for different times around the maximum run-up which takes place at t3 � 4.3 s. For a nondimensional wave height, H/h � 0.1; the maximum wave run-up is equal to R max � 0.0212 m � 1.060 H which implies that the wave height increased by (R max − H/H) * 100 � 6.0 %. ere is no significant difference between this value and the maximum run-up found by Domínguez et al. [19] that is equal to 1.065 for the same wave flow parameters. e maximum wave run-up increases when the wave height increases and is equal to R max � 0.0466 m � 1.165 H for (H/h) � 0.2 which implies that the wave height increased by 16.5 %. Following Domínguez et al. [19], the computed maximum run-up is 1.147 which is almost the same value given by the present study. e evaluation of the wave forces exerted on the cylinder has great importance on the engineering applications and depends on the wave run-up R at the front of the stagnation point. e wave force F → acting on the cylinder is computed by integrating the water pressure p and the normal component of the viscous stress tensor τ on the wetted surface of the cylinder S: where n → is the normal unit vector pointing into the water.
e time histories of the computed wave force on the xdirection acting on an isolated cylinder are presented in Figure 5 for two different solitary wave heights. e peak of the in-line force becomes sharper when the solitary wave height increases. e computed forces are in good agreement with other numerical studies and the tested cut-cell method is validated.

Wave Diffraction by Two Circular Cylinders.
Two vertical circular cylinders with the same radius are considered with different arrangements: in tandem (β � 0) and side-by-side (β � 90°). e still water depth to the radius ratio is h/a � 3. e cylinder and the nondimensional gap distance is G/2a � 2. In order to explain the wave diffraction between the two cylinders, Figure 6 illustrates the perspective view of the free surface elevation at different instants for nondimensional wave height: H/h � 0.2. e free surface raised at the upstream side of the cylinder and decreases behind it. When the reflected diffracted wave propagates away of the upstream cylinder, the free surface elevation drops significantly. e maximum run-up with partial wave reflection propagating radially occurs at instant t3 � 4.3 s for H/h � 0.1 and at the instant t'3 � 3.72 s H/h � 0.2. is radial wave is also observed by Yates and Wang [17]. At the instant t6 � 5.2 s, the solitary wave almost passes through the cylinder and the portion of the wave nonblocked by the cylinder is diffracted around the cylinder.
In order to compare the wave forces, Figure 7 represents the evolution of the in-line wave forces on the two vertical cylinders and on the isolated cylinder. e maximum run-up on the upstream cylinder is almost the same as the isolated one. However, due to the wave field interference, the maximum run-up on the downstream cylinder is less than that on the upstream cylinder for tandem arrangement and  [17], this interference effect is known as the shielding effect of the upstream cylinder on the downstream one. is mechanism induces the reduction of the in-line force on the downstream cylinder by 15 % compared to the force on the upstream cylinder (Table 2). e shielding effect is not observed for tandem arrangement and H/h � 0.2 where the in-line force on the downstream cylinder is greater than the upstream one (Table 3). For side-by-side arrangement, the wave force on the first cylinder is the same as that on the second cylinder for H/h � 0.1 and H/h � 0.2. e maximum force is observed at the same time for the two cylinders in side-by-side cylinders for the different wave heights. However, due to the gap distance, the maximum force for the downstream cylinder occurs after some time lag equal to 0.22 s.

Conclusions
e proposed numerical wave tank (NWT) was validated for wave generation and wave diffraction with a single and two vertical circular cylinders for two solitary wave heights. All the simulated results are conducted in the Cartesian coordinates using the cut-cell method to represent the cylinder shape. Numerical results show that the solitary wave and the wave forces on the cylinder are well reproduced for a weak nonlinear wave. In the tandem arrangement and for H/h � 0.1, the shielding effect is observed. However, for H/h � 0.2, the in-line force for the downstream cylinder is greater than the upstream one. For the side-by-side arrangement, equally aligned, the in-line force for the two cylinders is the same. e diffracted wave affects this force compared to the single cylinder by some time lag.
e present model can be tested for the extreme solitary wave to extend its application to a more realistic case study as the solitary wave diffraction with an offshore oil platform.

Data Availability
No data were used to support the study.