# A Three-Dimensional Numerical Study of Wave Induced Currents in the Cetraro Harbour Coastal Area (Italy)

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

^{th}contravariant and covariant component of the generic vector $\overrightarrow{b}$ (or tensor $\underset{\_}{C})\text{}$ are given, respectively, by the scalar product between $\overrightarrow{b}$ (or $\underset{\_}{C}$) and the base vectors ${\overrightarrow{g}}^{\left(l\right)}$ and ${\overrightarrow{g}}_{\left(l\right)}$ [20]. Let us define with $\Delta {V}_{0}=\Delta {\xi}^{1}\Delta {\xi}^{2}\Delta {\xi}^{3}$ a control volume bounded by moving curvilinear coordinate surfaces, which at the generic instant coincides with a material volume of fluid. In a curvilinear coordinates system, in order to obtain an integral form of the governing equations, the rate of change of the momentum of a material volume and the net forces acting on it must be projected onto a direction in space that does not coincide with a coordinate line. In this coordinate system, a direction in space is identified by a vector field $\overrightarrow{\lambda}$ whose covariant components ${\lambda}_{k}$ are not constant in space. In this paper, the vector field $\overrightarrow{\lambda}$ that identifies the abovementioned direction in space (and that is used to obtain a contravariant integral form of the momentum equation) is the one parallel to the l

^{th}contravariant base vector defined at the centre of the volume $\Delta {V}_{0}$. We indicate by ${\overrightarrow{\tilde{g}}}^{\left(l\right)}$ this base vector and by ${\lambda}_{k}={\overrightarrow{\tilde{g}}}^{\left(l\right)}\xb7{\overrightarrow{g}}_{\left(k\right)}$ its k

^{th}covariant component. By following this procedure, the contravariant integral form of the Navier–Stokes equations in a time dependent curvilinear coordinate system can be expressed as:

## 3. Numerical Procedure

#### Surface Boundary Condition

## 4. Results

#### 4.1. Breaking Wave Test Case

#### 4.2. T-head Groin Test Case

#### 4.3. Wave Induced Currents in the Cetraro Harbour (Italy)

#### 4.3.1. Wave Fields and Wave Induced Currents in the Present Configuration

#### 4.3.2. Wave Fields and Wave Induced Currents in the Project Configuration

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Ting and Kirby [21] breaking wave test case. Phase-averaged wave crest elevations. Experimental data (circles) and numerical results obtained by ${y}^{+}$ average fixed value of 40 (solid line).

**Figure 2.**Plan view of the time-averaged velocity field: (

**a**) $z/{z}_{b}=0.894$, near the bottom; (

**b**) $z/{z}_{b}=0.488$, intermediate position along the vertical direction; (

**c**) $z/{z}_{b}=0.081$, near the free surface.b.

**Figure 3.**Long-shore currents: (

**a**) at $\text{}y=22\text{}m$, (

**b**) at $\text{}y=26\text{}m$. Comparison between experimental data by Gravens and Wang [22] (red square) and numerical results of the depth-averaged cross-shore currents (blue line). Numerical results of cross-shore currents near the bottom at a relative distance from the free surface $z/{z}_{b}=0.894$ (light green line) and cross-shore currents, near the free surface at a relative distance from the free surface $z/{z}_{b}=0.081$ (green line). Bottom profile in black.

**Figure 4.**Cross-shore currents: (

**a**) at $\text{}y=22\text{}m$, (

**b**) at $\text{}y=26\text{}m$. Comparison between experimental data by Gravens and Wang [22] (red square) and numerical results of the depth-averaged longshore currents (blue line). Numerical results of longshore currents near the bottom at a relative distance from the free surface $z/{z}_{b}=0.894$ (light green line) and long-shore currents, near the free surface at a relative distance from the free surface $z/{z}_{b}=0.081$ (green line). Bottom profile in black.

**Figure 5.**Significant wave height: (

**a**) section at $\text{}y=22\text{}m$, (

**b**) section at $\text{}y=26\text{}m$. Comparison between experimental data by Gravens and Wang [22] (red square) and numerical results (blue line). Bottom profile in black.

**Figure 6.**(

**a**)

**:**Plan view of Cetraro harbour (Italy). (

**b**): Wave height directional distribution offshore Cetraro harbour.

**Figure 7.**Plan view of the boundary conforming curvilinear grid adopted for the numerical simulation in the present configuration. One curvilinear coordinate line out of 6 is shown.

**Figure 8.**(

**a**) Plan view and (

**b**) three-dimensional view of an instantaneous wave field obtained by the numerical simulation of the significant wave event in the coastal region of Cetraro harbour in the present configuration. $\eta $: free surface elevation.

**Figure 9.**Time-averaged velocity field: (

**a**) near the free surface, (

**b**) at intermediate water depth, (

**c**) near the bottom, obtained by the numerical simulation of the significant wave event in the present configuration of the Cetraro harbour.

**Figure 10.**(

**a**) Plan view and (

**b**) three-dimensional view of an instantaneous wave field obtained by the numerical simulation of the significant wave event in the coastal region of Cetraro harbour in the project configuration.

**Figure 11.**Time-averaged velocity field: (

**a**) near the free surface, (

**b**) at intermediate water depth, (

**c**) near the bottom, obtained by the numerical simulation of the significant wave event in the project configuration of the Cetraro harbour.

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**MDPI and ACS Style**

Cannata, G.; Palleschi, F.; Iele, B.; Cioffi, F.
A Three-Dimensional Numerical Study of Wave Induced Currents in the Cetraro Harbour Coastal Area (Italy). *Water* **2020**, *12*, 935.
https://doi.org/10.3390/w12040935

**AMA Style**

Cannata G, Palleschi F, Iele B, Cioffi F.
A Three-Dimensional Numerical Study of Wave Induced Currents in the Cetraro Harbour Coastal Area (Italy). *Water*. 2020; 12(4):935.
https://doi.org/10.3390/w12040935

**Chicago/Turabian Style**

Cannata, Giovanni, Federica Palleschi, Benedetta Iele, and Francesco Cioffi.
2020. "A Three-Dimensional Numerical Study of Wave Induced Currents in the Cetraro Harbour Coastal Area (Italy)" *Water* 12, no. 4: 935.
https://doi.org/10.3390/w12040935