# A Novel BEM–FEM Scheme for the Interaction of Water Waves with Multiple Vertical Cylinders in the Presence of Currents

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation of the 3D Problem

**U**denote the steady current field velocity on and near the free surface. Following the standard formulation, the free surface boundary conditions for the wave field are linearized as follows (see, e.g., Belibassakis et al. [3,13]):

_{1}-axis; thus, in the far upstream region, ${\phi}_{I}^{(S)}(x)\approx -{U}_{\infty}{x}_{\hspace{0.17em}1}$, with ${U}_{\infty}$ being the current velocity far from the cylinders. Based on the above, the incident wave potential is defined as:

_{1}-axis; however, generalization of the incident flow directed at an angle is possible. The above considerations allow for the formulation of the problems for the steady ${\phi}_{D}^{(S)}(x)$ and the unsteady ${\phi}_{D}^{(U)}\left(x\right)$ diffraction fields in the domain and the development of suitable 3D Boundary Element Methods for their solution, as described in the following sections.

#### 2.1. Formulation of the 3D Steady Current Flow Problem

#### 2.2. The 3D BEM for the Steady Flow Problem

#### 2.3. Resulting Steady Flow Fields

#### 2.4. Formulation of the 3D Wave Problem

#### 2.4.1. Implementation of the Absorbing Layer Technique

_{1}-axis $(\beta =\pi )$.

#### 2.4.2. An Iterative Scheme for the Additional Scattering Effect Due to Current

## 3. A Simplified 2D Formulation on the Horizontal Plane

#### 3.1. Formulation of the Steady Current Problem

_{1}-axis. The numerical solution to the above problem is obtained using boundary integral equation formulations, based on the single-layer potential; see also Belibassakis et al. [4]. This is achieved by introducing the following integral representation for the steady potential function in D:

#### 3.2. Formulation of the Wave Propagation Problem on the Horizontal Plane

_{1}-axis, which is partially reflected due to the structure; thus, ${\phi \left({x}_{h}\right)|}_{{x}_{1}=-L/2}=\mathrm{exp}\left(i{k}_{0}{x}_{1}\right)+R\mathrm{exp}\left(-i{k}_{0}{x}_{1}\right)$, where $R$ denotes the reflection coefficient. Subsequently, by differentiating with respect to ${x}_{1}$ and eliminating $R$, we obtain the following mixed-type condition, that is used on the entrance boundary $\partial {D}_{\hspace{0.17em}1}$:

#### 3.3. An FEM for the Wave Propagation–Scattering Problem

#### 3.4. Iterative Scheme

## 4. Numerical Results

#### 4.1. Verification of the FEM Scheme without Current

#### 4.2. Verification of the Present FEM in the Case of Waves and Currents

_{2}-error of the calculated solution, defined with respect to the result obtained by the finest grid with N = 171,860 elements for the discretization of the domain, is presented against the number of finite elements used for the domain subdivision. It is clearly observed that the present method exhibits a convergence which is compatible with the expected result for linear approximation of the unknown function based on triangular mesh.

#### 4.3. Resonances of Wave and Current Systems in the Case of a Line Array of Cylinders

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Configuration of the arrangement of vertical cylinders considered as permeable breakwaters.

**Figure 3.**Indicative sparse mesh of the free surface boundary along with a cylindrical scatterer, highlighting the ξ-direction.

**Figure 4.**(

**a**) Color plot and (

**b**) rendered view of the free surface elevation generated by the steady perturbation field for one cylindrical scatterer of diameter $D=0.3\hspace{0.17em}\mathrm{m},\hspace{0.17em}h=1\hspace{0.17em}\mathrm{m},\hspace{0.17em}{U}_{\infty}=\left(-0.7,0,0\right)$.

**Figure 5.**(

**a**) Color plot and (

**b**) rendered view of the free surface elevation generated by the steady perturbation field for four cylindrical scatterers of diameter $D=0.3\hspace{0.17em}\mathrm{m},\hspace{0.17em}h=1\hspace{0.17em}\mathrm{m},\hspace{0.17em}{U}_{\infty}=\left(-0.7,\hspace{0.17em}0,\hspace{0.17em}0\right)$.

**Figure 6.**Incident waves of wavelength and height $\lambda =1.2\hspace{0.17em}\mathrm{m},\hspace{0.17em}H=0.2\hspace{0.17em}\mathrm{m}$ interacting with a vertical cylinder of diameter D = 0.3 m in water depth h = 1 m, in the presence of a parallel flow current ${U}_{\infty}=\left(-0.3\hspace{0.17em}\mathrm{m}/\mathrm{s},0,0\right)$. (

**a**,

**b**) Imaginary and real part of the initial estimation of the disturbance field and corrected patterns after the first iteration (

**c**,

**d**) and after the second iteration (

**e**,

**f**), respectively. The dashed line indicates the absorbing layer region.

**Figure 7.**Incident waves of wavelength and height $\lambda =1.2\hspace{0.17em}\mathrm{m},\hspace{0.17em}H=0.1\hspace{0.17em}\mathrm{m}$ interacting with an arrangement of four cylindrical scatterers of diameter D = 0.3 m in water depth h = 1 m, in the presence of a collinear following current ${U}_{\infty}=\left(-0.25\hspace{0.17em}\mathrm{m}/\mathrm{s}\hspace{0.17em},0,0\right)$, indicated with an arrow. (

**a**,

**b**) Real and imaginary part of free surface elevation of the diffracted and (

**c**,

**d**) the total unsteady field, respectively. The dashed line indicates the absorbing layer region.

**Figure 9.**Equipotential lines and streamlines of the steady field resulting from the interaction of a current propagating along the ${x}_{\hspace{0.17em}1}$ axis at speed ${U}_{\infty}=0.5\hspace{0.17em}\mathrm{m}/\mathrm{s}$ with a configuration of 23 cylinders of radius $a=0.15\hspace{0.17em}\mathrm{m}$ in a 6 m wide tank.

**Figure 10.**Indicative plot of Delaunay triangulation-based mesh of $D$, augmented by a regular triangular mesh in the PML region.

**Figure 11.**Real part and modulus of the symmetric wave field in the waveguide of water depth h = 0.23 m, at frequency $f=1\hspace{0.17em}\mathrm{H}\mathrm{z}$, as calculated by (

**a**,

**c**) the modal BEM developed by Belibassakis, et al. [4] and (

**b**,

**d**) the present FEM scheme.

**Figure 12.**Real part of the symmetric field on the horizontal plane, in water depth h = 1 m, for a wave at frequency $f=1\hspace{0.17em}\mathrm{H}\mathrm{z}$, interacting with a cylinder of radius 0.3 m without current, as calculated by (

**a**) the 3D BEM and (

**b**) the proposed 2D FEM scheme. Calculated field for waves of the same frequency and height in the presence of a current of speed ${U}_{\infty}=0.1\hspace{0.17em}\mathrm{m}/\mathrm{s}$ as calculated by (

**c**) the present 3D BEM and (

**d**) the proposed 2D coupled BEM–FEM scheme.

**Figure 13.**Convergence of the numerical solution obtained by the present FEM scheme in the case of waves with and without current scattered by a cylinder.

**Figure 14.**Convergence of the iterative scheme used to obtain a convergent solution by the present FEM scheme in the case of waves and current scattered by a cylinder. The iteration numbered 0 corresponds to the starting solution obtained without the current.

**Figure 15.**Modulus of the wave field in the 10 m long and 0.3 m wide waveguide of water depth h = 0.305 m, resulting from the interaction of an incident field of frequency $f=1.3\hspace{0.17em}\mathrm{H}\mathrm{z},$ with an array of eight cylinders (

**a**) in the absence of currents and (

**b**) in the presence of a steady following current corresponding to volumetric flow rate $Q=70.5\hspace{0.17em}{\mathrm{cm}}^{\mathrm{3}}/\mathrm{s}$.

**Figure 16.**Real part of the wave field in the 10 m long and 0.3 m wide waveguide of water depth h = 0.305 m, resulting from the interaction of an incident field of frequency $f=1.3\hspace{0.17em}\mathrm{H}\mathrm{z}$, with an array of eight cylinders (

**a**) in the absence of currents and (

**b**) in the presence of a steady following current corresponding to volumetric flow rate $Q=70.5\hspace{0.17em}{\mathrm{cm}}^{3}/\mathrm{s}$.

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**MDPI and ACS Style**

Magkouris, A.; Belibassakis, K.
A Novel BEM–FEM Scheme for the Interaction of Water Waves with Multiple Vertical Cylinders in the Presence of Currents. *Fluids* **2022**, *7*, 378.
https://doi.org/10.3390/fluids7120378

**AMA Style**

Magkouris A, Belibassakis K.
A Novel BEM–FEM Scheme for the Interaction of Water Waves with Multiple Vertical Cylinders in the Presence of Currents. *Fluids*. 2022; 7(12):378.
https://doi.org/10.3390/fluids7120378

**Chicago/Turabian Style**

Magkouris, Alexandros, and Kostas Belibassakis.
2022. "A Novel BEM–FEM Scheme for the Interaction of Water Waves with Multiple Vertical Cylinders in the Presence of Currents" *Fluids* 7, no. 12: 378.
https://doi.org/10.3390/fluids7120378