# Experimental and Numerical Analysis of the Hydrodynamics around a Vertical Cylinder in Waves

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

_{m}T/D (U

_{m}is the maximum velocity at the bottom just outside of the boundary layer and D is the diameter of the pile), is KC ≈ 6 and the Reynolds number, defined as Re = U

_{m}A/ν (A is the semi-amplitude of the horizontal motion of the water particles just outside of the bottom boundary layer, ν is the kinematic viscosity), is Re ≈ 3 × 10

^{4}. Due to the significant computational cost, we focused on one specific forcing condition characterized by vorticity generation and scour formation, as found in the first experimental campaign [10] for waves with similar values of a Keulegan–Carpenter parameter. Being the vortex flow regimes governed primarily by KC for a cylinder subjected to an oscillatory flow [5], the wave here analyzed was representative of waves within the vortex shedding flow regime. The scour pattern map obtained in the previous experiments [10] was here used in order to relate it with the vortical structures.

## 3. Numerical Setup

^{4}). Note that here the Re number was defined according to the maximum velocity U

_{m}(as usually done for flow induced by waves) and not according to the mean velocity value, which was almost null in an oscillatory flow. From the practical point of view, we found that the computational resources required by DNS are large but significantly lower than those usually adopted for solving a different flow problem (e.g., current) characterized by a similar Re value which was obtained by using the mean velocity. The employed discretization scheme adopted a finite volume method in space and an implicit algorithm in time. More in detail, it used a second order backward time integration scheme, while a second order reconstruction of the numerical fluxes at faces yields a second order accuracy also in space. The volume of fluid (VOF) method was used for locating and tracking the free surface. It consisted of the assumption that the two-phase flow was a mixed fluid whose density and viscosity were weighted functions of the volume fraction which evaluated the fraction of the volume occupied by the two phases and was computed by an advection equation.

#### 3.1. Sensitivity Analysis

^{−3}around the pile and Δh/D = 1.2× 10

^{−2}above the bottom wall, where Δh was the normal to the wall grid size.

#### 3.2. Spectral Analysis

_{p}= 1/T = 0.56 Hz. The frequency spectrum showed a decrease of the spectral energy content typical of turbulent fluctuations up to a frequency (f

_{u}), where viscosity canceled out all the fluctuating motions. The developed range of scales was very tiny, being characterized by a few decades (the ratio f

_{u}/f

_{p}was about equal to 30), suggesting that the turbulent flow emerging from the interaction of the oscillating flow with the vertical cylinder was very weak. Indeed, no evident inertial range, typical of fully developed turbulence where the spectrum was supposed to follow the ${f}^{-5/3}$ power law, was observed.

_{u}/f

_{p}, we argue that the DNS approach is strongly recommended for the solution of the wave action on a pile with no current.

## 4. Results

#### 4.1. Water Surface Elevation

#### 4.2. Particle Velocity

#### 4.3. Pressure Distribution

_{d}evaluated in front of the pile (φ =0°) and behind it (φ = 180°) for the wave phases ωt = 45° and ωt = 270°. The dynamic pressure P

_{d}is given by the difference between the pressure P and the hydrostatic pressure ${P}_{h}=\rho gh$ ($\rho $ is the water density and g the gravity acceleration). The analysis of the dynamic pressure distribution around the pile was very important because its asymmetry with respect to the vertical plane (yz-plane) produced a horizontal force on the cylinder acting along the x-axis. During the growth of the positive water surface elevation (e.g., ωt = 45°), a larger dynamic pressure P

_{d}was observed in front of the pile with respect to the area behind it; hence, a positive horizontal force was expected. During the negative water surface elevation (e.g., ωt = 270°) the pressure distributions in front of and behind the pile were quite similar; hence, the horizontal force along the x-axis was expected to be very small (in absolute value). Moreover, the dynamic pressure data at φ = 90° and φ = 270°, here not shown, were similar, confirming that the horizontal force along the transversal direction y was negligible, as found in Section 4.4.

#### 4.4. Total Force

_{D}), and an inertial component (F

_{I}).

_{D}and C

_{M}are, respectively, the drag and inertia coefficients.

_{W}). Knowing the distance between the load cell and the hinge (b

_{C}= 1.04 m) and estimating for each time the distance b

_{W}, the force on the pile was obtained. These assumptions were verified by means of a calibration process of the load cell, applying different known horizontal forces at different locations.

_{D}= 0.9 and C

_{M}= 2.0). The application of the linear theory as for the evaluation of particle velocities and accelerations could lead to a significant underestimation of the maximum value of the force (here about –30%). The use of higher order theories in Equation (2) (e.g., Stokes V order theory) improved significantly the estimation of the maximum value of the wave force, as shown in Figure 12, and allowed us to better reproduce its trend over the whole wave period (the velocities and the accelerations of water particles were computed according to the methodology proposed by Fenton [18]). Therefore, the use of the Morison equation with higher order wave theories is recommended for the prediction of the wave force on the pile.

#### 4.5. Flow Pattern, Shear Stress, and Scour

## 5. Discussion and Conclusions

^{4}). The use of the Morison’s equation with higher order wave theories improved the prediction of the maximum force and, hence, is recommended. Field conditions were associated with higher values of Re (Re ≈ 10

^{6}÷ 10

^{7}) and, thus, to correctly design a structure, a safety factor for the total force of 1.5 ÷ 2.0 was needed, due to the uncertainties on the determination of the coefficients in such conditions. Full scale experiments at higher Reynolds numbers were useful to optimize the design process.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Sketch of the rigid-bed model setup with the position of the elevation gauges S3, S4, S5, and of the acoustic Doppler velocimetry (ADV) at φ = 0° (

**a**) and of the planar coordinate system, with the position of pressure sensors (

**b**).

**Figure 4.**Force (

**a**) and water level (

**b**) comparisons in correspondence to the pile to evaluate the effectiveness of the numerical beach. Simulations 0 and 0a (continuous lines), 1 and 1a (dashed lines).

**Figure 5.**Comparison of the force over the pile (

**a**) and the wet surface around it (

**b**) among different mesh (1a–3a) with numerical beach.

**Figure 6.**Frequency spectra evaluated in the near-bed (z = 0.003m) at φ = 0° (

**a**) and φ = 90° (

**b**). The inertial subrange ${f}^{-5/3}$ (black dotted line). The red and blue lines report the behavior for mesh 2a and 3a, respectively.

**Figure 7.**Comparison between numerical and experimental data of water surface elevation in correspondence of water gauges S3 (

**a**), S4 (

**b**), and S5 (

**c**).

**Figure 8.**Comparison between numerical and experimental data of water particle velocity in position φ = 0° at z = 0.01 m (

**a**) and φ = 0° at z = 0.24 m (

**b**).

**Figure 9.**Vertical distributions of the pressure P and of the dynamic pressure P

_{d}for the wave phase ωt = 45° (

**a**,

**b**) and ωt = 270° (

**c,d**) in front of the pile at φ = 0° (

**a**,

**c**) and behind it at φ = 180° (

**b**,

**d**). Experimental data: Filled blue circle. Numerical data: Red empty triangle. The dashed line is the hydrostatic pressure.

**Figure 10.**Plan view (xy-plane) of the nondimensional dynamic pressure Pd, for the wave phase ωt = 36° (

**a**), ωt = 54° (

**b**), ωt = 72° (

**c**), and ωt = 90° (

**d**). Pressure gradients dP/dx’ values (positive, null, or negative) in the area φ = 90°–180°.

**Figure 11.**Comparison between the force obtained from the load cell measurements, and as integral of the pressure measurements.

**Figure 12.**Comparison between experimental and numerical data of water level (

**a**) and force over the pile (

**b**).

**Figure 13.**Comparison between the experimental force and the force evaluated by applying Morison’s formula [1] with the linear wave theory (original formulation) and with Stokes V order wave theory (modified formulation).

**Figure 15.**Contour map of the measured scour pattern evaluated in the mobile-bed experiments (

**a**) and of the bed-shear stress magnitude computed from simulation (

**b**). The streamlines in (

**b**) report the behavior of the bed-shear stress vector field (μ∂u/∂z, μ∂v/∂z). Wave direction from left to right.

Name | Num. of Volumes | Length (m) | Numerical Beach |
---|---|---|---|

0 | 432’800 | 6.0 | No |

0a | 492’300 | 7.5 | Yes |

1 | 750’660 | 6.0 | No |

1a | 784’500 | 7.5 | Yes |

2a | 880’900 | 7.5 | Yes |

3a | 1’965’600 | 7.5 | Yes |

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**MDPI and ACS Style**

Corvaro, S.; Crivellini, A.; Marini, F.; Cimarelli, A.; Capitanelli, L.; Mancinelli, A.
Experimental and Numerical Analysis of the Hydrodynamics around a Vertical Cylinder in Waves. *J. Mar. Sci. Eng.* **2019**, *7*, 453.
https://doi.org/10.3390/jmse7120453

**AMA Style**

Corvaro S, Crivellini A, Marini F, Cimarelli A, Capitanelli L, Mancinelli A.
Experimental and Numerical Analysis of the Hydrodynamics around a Vertical Cylinder in Waves. *Journal of Marine Science and Engineering*. 2019; 7(12):453.
https://doi.org/10.3390/jmse7120453

**Chicago/Turabian Style**

Corvaro, Sara, Andrea Crivellini, Francesco Marini, Andrea Cimarelli, Loris Capitanelli, and Alessandro Mancinelli.
2019. "Experimental and Numerical Analysis of the Hydrodynamics around a Vertical Cylinder in Waves" *Journal of Marine Science and Engineering* 7, no. 12: 453.
https://doi.org/10.3390/jmse7120453