# Two-Dimensional Free-Surface Flow Modeling for Wave-Structure Interactions and Induced Motions of Floating Bodies

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Methods

#### 2.1. Two-Phase Flows Using Level Set Formulation

**f**denotes hydrodynamic forces induced by an immersed body, ${\mathrm{A}}_{\mathrm{b}}$ as defined later represents the wave absorbing coefficient,

**g**is the gravitational acceleration vector, t is time, and $\mathsf{\varphi}$ denotes the level set function. The 2D velocity vector, $u,$ describes the velocity components in x and z directions, and p is the pressure. More detailed descriptions of the physical variables given in Equation (2) can be found in Sethian and Smereka [37].

#### 2.2. Numerical Formulations

**RHS**= $1.5{\mathrm{A}}^{\mathrm{n}}-0.5{\mathrm{A}}^{\mathrm{n}-1}$:

**f**, is computed by a continuous forcing IB method with the computed velocity vectors, $\widehat{u}$. By imposing a body force,

**f,**in those cells, which has the property of a signed distance function near the immersed boundary, the intermediate velocity field is updated from Equation (10).

#### 2.3. Immersed Boundary (IB) Method

**f**) at the Eulerian grid points can be calculated by using a spreading scheme involving the Lagrangian marker points as follows:

**f**) from Equation (27) is then substituted into Equation (10) for solution computation.

#### 2.4. Rigid Body Dynamics

## 3. Results and Discussions

#### 3.1. Flow Passing through a Cylinder

_{D}, as given in Table 1, are compared with the published experimental [42] and numerical [43,44,45,46,47] results. Additionally, to ensure the grid independence, included in Table 1 are the present results for the separately used mesh sizes of ∆h = 1/30 and ∆h = 1/40. As can be seen in Table 1 for the cases of Re = 20 and 40, the converged solutions in L

_{w}and C

_{D}are obtained, and they are in good agreements with other published results. By refining the computational grids, it is shown to be able to improve the solutions for the flow conditions at a larger Reynolds number, such as Re = 100 or 200; this confirms the global accuracy of the present method. Comparisons between the present and other numerical results [43,46,47,48,49,50] of the drag and lift coefficients (C

_{D}and C

_{L}), and Strouhal number (St) for the cases of Re = 100 and 200 are summarized in Table 2. The present results were computed by using a refined grid size, i.e., ∆h = 1/50. It can be seen the present results are shown to have very good agreements with others given in the literature. For the cases of Re = 100 and 200, the transient solution procedure was carried out to reach the prescribed time, i.e., t = 200. Time variations of the dimensionless drag and lift coefficients at Re = 200 are presented in Figure 3. Due to the induced vortex motions, the transition processes until the C

_{D}and C

_{L}approach to the stabilized periodic variations are clearly depicted in Figure 3. Figure 4 exhibits the instantaneous distribution patterns of pressure and vorticity at t = 200 for various Reynolds numbers of 40, 80, 100, and 200. The patterns of vortex shedding for the cases of Re = 80, 100, and 200 can be clearly observed. The predicted results of vortex shedding demonstrate the proposed method can predict accurately the transient flow patterns. As can be seen, the wake triggered at the cylinder surface is asymmetric in reference to the direction of flow. The formation of vortices on either side of the flow direction does not simultaneously occur. In fact, the shedding of vortices alternates from side to side, which had been observed in experiments. Thus, the pressure distribution around the obstacle is asymmetric about the flow direction.

#### 3.2. Two Cylinders Moving Against Each Other in Viscous Fluid

_{D}) and lift coefficient (C

_{L}) for the upper cylinder under the conditions of various Reynolds numbers are shown in Figure 7. From Figure 7, it is noted that, for all cases, the minimum values of the drag coefficient occur roughly at $\text{}\tilde{\mathrm{t}}=16$, where $\text{}\tilde{\mathrm{t}}$= Ut/(D/2); meanwhile, for the lift coefficient, the minimum values take place at a delayed time of around $\text{}\tilde{\mathrm{t}}$ = 18. In general, the results show that the drag coefficient and the positive portion of the lift coefficient decrease with an increase in the Reynolds number. The transitions of the drag and lift coefficients become particularly pronounced at the timeframe from $\text{}\tilde{\mathrm{t}}$= 14 to $\text{}\tilde{\mathrm{t}}$ = 20. In addition, to demonstrate that the present model is capable of successfully simulating the flow fields, Figure 8 depicts the instantaneous vorticity contours of two moving cylinders for Re = 100 at four selected instants. The color-coded vorticity contours range from −5 to 5. In this transient simulation study, the results of hydrodynamic effect and vortex shedding phenomena under the condition of two cylinders moving against each other in viscous fluid are found to be completely different when compared to those for a single moving body at the same Reynolds number. Due to the viscous effect within the boundary layers of the solid bodies, the moving rigid bodies are noticed to play an important role in generating the vortex flow in the immersed viscous fluid domain.

#### 3.3. Dam-Break Problems

^{−4}sec were used to simulate the dam-break flows for the determination of the nondimensionalized leading-edge position. Figure 9 shows the comparisons between the present solutions on the time varying leading-edge position and those from measurements [52] and other numerical results [53]. For the Martin and Moyce [52] dam-break flow case, the corresponding Reynolds number was about 42,792. The computed surge front positions versus the nondimensionalized time reveal a good agreement with the data from Martin and Moyce [52] and the numerical results from Nomeritae et al. [53]. Moreover, the tests of the grid size effect indicate that the converged and nicely fit solutions are obtained when the refined grid size of Δx = Δz = 0.005 m is used.

_{1}at (2 m, 0.16 m), P

_{2}at (4 m,0.16 m), and P

_{3}at (5 m,0.26 m) (see Figure 10), are compared with the Flow-3D model results. The results obtained from the Flow-3D was based on the two-phase laminar flow model with the consideration of water and air phases. Figure 11 presents the comparisons between the present model results (velocity in x direction and pressure) and those from Flow-3D for the case of a submerged semicircular cylinder at point P

_{1}. The time variations of both the velocity and pressure, except at the time near the end of the interaction process, are in good agreements with the Flow-3D results. The results obtained from the arrangements of two different grid sizes reveal the consistent and reliable model predications. The comparisons of time varying pressures at points P

_{2}and P

_{3}under the cases of either a semicircular or a rectangular cylinder are presented in Figure 12. It is found the variation trends of the pressure at P

_{2}and P

_{3}under the influence of a submerged semicircular cylinder are similar to those under the case of a submerged rectangular cylinder. However, due to a more severe geometry change that appeared in the rectangular cylinder, the pressures at P

_{2}and P

_{3}under the case of a submerged rectangular cylinder are greater than those with a semicircular cylinder after the surge waves impact on the cylinders.

#### 3.4. Wave Decomposition Process over a Submerged Trapezoid Breakwater

#### 3.5. Free Falling Wedge

_{b}= 466.6 kg/m

^{3}is selected for model simulations. It is positioned with its vertex of the obtuse angle pointing downward. The length of the wedge base is 1.2 m, and the side angle is 25°. The computational results in terms of the position of wedge’s bottom vertex and its velocity are compared with the experimental data collected by Yettou et al. [56]. A 2D computational domain is considered, in which the length of the domain is 20 m and the height is 4 m, including a fluid domain with water depth of 1 m. Following the experimental setup, initially, the distance from the tip of the bottom vertex to the free-surface is 1.3 m. The wedge is suddenly released into the 20 m long tank. Two grid sizes as Δh = 0.04 m and Δh = 0.02 m are used for the numerical simulations. Figure 18 shows the induced velocity fields of the fluid domains at nine selected instants under the action of a free-falling wedge and its impact on the free surface. The variations of the interface also reflect the motions of the free-surface. The induced severe interfacial oscillations and coupled air motions after the wedge entering the water body can be observed. As the present numerical approaches treats more closely the two-phase flow conditions as a process of fully nonlinear fluid-structure interaction, it is believed more accurate and physically fitted results can be generated by the present model when compared to other models considering only the single-phase flow conditions without the inclusion of the effect of air phase. The use of orthogonal mesh system is highly essential to capture the vortex generation at the regions near or around a solid structure in the cases involving the interactions of air/water/solid. The comparisons between the simulated results and measured data from Yettou et al. [56] in terms of the time varying vertical position in z coordinate and vertical velocity of the bottom vertex for a free-falling wedge described above are presented in Figure 19a,b, respectively. The results obtained by using either the grid size of Δh = 0.04 m or Δh = 0.02 m are concluded to have nearly identical values, confirming the converged solutions. In terms of the model performance, the present results are shown to have good agreements with the published numerical solutions [36] and experimental data [56]. The maximum vertical velocity reaches 4.8 m/s at t = 0.56 s. This computational case demonstrates the model’s capability in simulating the coupled motions between the dynamic responses of a free-falling wedge and the induced free-surface waves.

#### 3.6. Wave Packet Interacting with a Floating Body

^{3}, and the mass of the body was 0.986 kg. As shown in Figure 20, the center of the body was situated at a distance of 2.11 m away from the wavemaker. The computational domain is set as a 13 m long and 0.8 m high channel (water depth = 0.4 m). In Hadzic et al.’s [57] experiments, a flap-type wavemaker was controlled to produce a wave packet with a focusing point at the original location of the object. The grid sizes of Δx = 0.02 m and Δz = 0.005 m were used for the 2D numerical computations. Temporal variations of the body motions in three DOFs are computed during the interaction process between the wave packet and the floating body. Figure 21 depicts the time-based changes of the oscillating angle of the flap wavemaker that generate a wave packet. The simulated time varying wave elevations at the locations of x = 1.65 m and x = 2.66 m during the evolution of the wave packet are presented in Figure 22. The experimentally measured data from Hadzic et al. [57] are also included in Figure 22 for comparisons. The simulated wave packet is shown to fit nicely with the measured data. The maximum values of the wave elevation occur roughly at t = 5.8 s and t = 7.7 s at the locations of x = 1.65 m and x = 2.66 m, respectively. In general, the results show the increasing trend of the wave elevation until t = 5.8 s at the location of x = 1.65 but delayed to t = 7.7 s at the location of x = 2.66 m. Afterward, the wave elevation is shown to have a trend of periodic decay with time.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

English Symbols | |

k | curvature of an interface |

f | hydrodynamic forces |

${\mathrm{A}}_{\mathrm{b}}$ | wave absorbing coefficient |

g | gravitational acceleration vector |

t | Time |

$u$ | velocity vectors |

P | Pressure |

${\mathrm{x}}_{\mathrm{xt}}$ | starting position of the absorbing region |

${\mathrm{x}}_{\mathrm{ab}}$ | length of the absorbing region |

${u}^{*}$ | velocity vectors at each of the intermediate (between n and n+1) time level |

$\widehat{u}$ | velocity vectors without considering the effect of immersed boundary |

$F$ | force density vector at the Lagrangian points |

${X}_{l}$ | Lagrangian points with coordinates |

${U}^{\mathsf{\Gamma}}$ | Lagrangian point velocity of body |

${U}_{\mathrm{b}}$ | translational velocity vectors of body |

${\mathrm{V}}_{\mathrm{b}}$ | body volume |

${\mathrm{I}}_{\mathrm{b}}$ | body moment of inertia |

$r$ | position vector relative to the body centroid |

D | cylinder diameter |

U | uniform flow velocity |

C_{D} | drag coefficient |

C_{L} | lift coefficient |

St | Strouhal numbers |

Greek | |

Symbols | |

$\mathsf{\delta}$ | Dirac delta function |

$\mathsf{\Sigma}$ | surface tension coefficient |

$\mathsf{\varphi}$ | level set function |

$\mathrm{H}\left(\mathsf{\varphi}\right)$ | Heaviside function |

$\mathsf{\rho}$ | fluid density |

$\mathsf{\mu}$ | fluid viscosity |

$\mathsf{\epsilon}$ | interfacial thickness |

$\Delta {\mathrm{V}}_{l}$ | control volume defined about the m-th Lagrangian marker |

${\mathsf{\rho}}_{\mathrm{b}}$ | body density |

${\omega}_{\mathrm{b}}$ | body angular velocity vectors |

Re | Reynolds number |

υ | fluid kinematics viscosity |

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**Figure 1.**Computations of flow containing two-phase, including immersed boundary. The governing equations are solved on a fixed grid, but the interface between two fluids is represented by a moving front in which it is consisting of connected marker points, and the immersed object is included in the governing equation by the immersed boundary method.

**Figure 2.**Arrangements of staggered Cartesian grid and embedded interface with marker points in a two-dimensional case.

**Figure 4.**Instantaneous distribution patterns of (

**a**) vorticity and (

**b**) pressure at t = 200 for a uniform flow past a cylinder at Re = 40, 80, 100, and 200.

**Figure 7.**Comparisons of drag and lift coefficients on upper cylinder at various Reynolds numbers: (

**a**) drag coefficient and (

**b**) lift coefficient.

**Figure 8.**Instantaneous vorticity contours of two moving cylinders at Re = 100, at different times: (

**a**)$\text{}\hat{\mathrm{t}}=\text{}12$ (

**b**)$\text{}\hat{\mathrm{t}}=\text{}16$ (

**c**)$\text{}\hat{\mathrm{t}}=20$ (

**d**)$\text{}\hat{\mathrm{t}}=\text{}32$.

**Figure 10.**Schematic diagram of dam-break waves over a submerged structure: (

**a**) semicircular cylinder and (

**b**) rectangular cylinder.

**Figure 11.**Comparisons between results (velocity and pressure) from the present model and Flow 3D simulations for the case of semicircular cylinder at point P1: (

**a**) velocity and (

**b**) pressure.

**Figure 12.**Comparisons of pressure variations for a dam-break flow over a semicircular and a rectangular cylinder at fixed points of (

**a**) P2 at (4 m, 0.16 m) and (

**b**) P3 at (5 m, 0.26 m).

**Figure 13.**Time variations of drag force (N): (

**a**) semicircular cylinder vs. rectangular cylinder and (

**b**) rectangular cylinder.

**Figure 14.**Comparisons of free-surface elevation at t = 0.5 s and t = 2.0 s: (

**a**) semicircular cylinder and (

**b**) rectangular cylinder.

**Figure 15.**Density distributions of a dam-break wave over submerged structures at different times: (

**a**) semicircular cylinder and (

**b**) rectangular cylinder.

**Figure 16.**Velocity distributions of a dam-break wave over submerged structures at different times: (

**a**) semicircular cylinder and (

**b**) rectangular cylinder.

**Figure 17.**Free surface elevations on six wave gauges (

**a**) x = 18.5 m; (

**b**) x = 20.5 m; (

**c**) x = 21.5 m; (

**d**) x = 22.5 m; (

**e**) x = 23.5 m; (

**f**) x = 25.5 m.

**Figure 18.**Velocity vector of free-falling wedge; the contour shows the density distribution of two-phase at different times: (

**a**) t = 0 s; (

**b**) t = 0.3 s; (

**c**) t = 0.5 s; (

**d**) t = 0.55 s; (

**e**) t = 0.65 s; (

**f**) t = 0.8 s; (

**g**) t = 1.2 s; (

**h**) t = 1.4 s; (

**i**) t = 2 s.

**Figure 19.**Motion of a free-falling wedge: (

**a**) vertical position of the bottom vertex; (

**b**) vertical velocity of the bottom vertex.

**Figure 21.**Time-based variations of the flap wavemaker angle used for the generation of a wave packet.

**Figure 22.**Time-based variations of wave elevation during the evolution of the wave packet at (

**a**) x = 1.65 m and (

**b**) x = 2.66 m.

**Figure 23.**Time-based variations of the floating body motions during the interaction with the wave packet (

**a**) sway, (

**b**) heave, and (

**c**) roll.

**Figure 24.**Density distributions and velocity vectors of floating body at t = 7.2 s (

**left**) and t = 7.6 s (

**right**).

**Table 1.**Drag coefficient and reattachment length for flow past a stationary cylinder at Re = 20 and Re = 40.

Authors | Re = 20 | Re = 40 | ||
---|---|---|---|---|

C_{D} | L_{w} | C_{D} | L_{w} | |

Tritton [42] | 2.22 | - | 1.48 | - |

Calhoun [43] | 2.19 | 0.91 | 1.62 | 2.18 |

Russel and Wang [44] | 2.22 | 0.94 | 1.63 | 2.35 |

Silva et al. [45] | 2.04 | 1.04 | 1.54 | 2.55 |

Xu and Wang [46] | 2.23 | 0.92 | 1.66 | 2.21 |

Kolahdouz et al. [47] | 2.10 | 0.93 | 1.58 | 2.31 |

Present ∆h = 1/30 Present ∆h = 1/40 | 2.15 2.15 | 0.94 0.94 | 1.57 1.57 | 2.25 2.25 |

**Table 2.**Drag and lift coefficients and Strouhal number for flow past a stationary cylinder at Re = 100 and Re = 200.

Authors | Re = 100 | Re = 200 | ||||
---|---|---|---|---|---|---|

C_{D} | C_{L} | St | C_{D} | C_{L} | St | |

Calhoun [43] | 1.330$\pm \text{}$0.014 | $\pm $0.298 | - | 1.172$\text{}\pm \text{}$0.058 | $\pm $0.668 | - |

Xu and Wang [46] | 1.423$\pm \text{}$0.013 | $\pm $0.340 | 0.171 | 1.420$\text{}\pm \text{}$0.040 | $\pm $0.66 | 0.202 |

Kolahdouz et al. [47] | 1.370$\pm \text{}$0.015 | $\pm $0.351 | 0.168 | 1.390$\text{}\pm $ 0.060 | $\pm $0.75 | 0.198 |

Liu et al. [48] | 1.350$\pm \text{}$0.012 | $\pm $0.339 | 0.164 | 1.170$\text{}\pm \text{}$0.058 | $\pm $0.67 | 0.202 |

Choi et al. [49] | 1.340$\pm \text{}$0.011 | $\pm $0.315 | 0164 | 1.360$\text{}\pm \text{}$0.048 | $\pm $0.64 | 0.191 |

Griffith and Luo [50] | - | - | - | 1.360$\text{}\pm \text{}$0.046 | $\pm $0.70 | 0.195 |

Present ∆h = 1/50 | 1.400$\pm \text{}$0.014 | $\pm $0.341 | 0.166 | 1.380$\text{}\pm \text{}$0.050 | $\pm $0.67 | 0.199 |

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**MDPI and ACS Style**

Lo, D.-C.; Wang, K.-H.; Hsu, T.-W. Two-Dimensional Free-Surface Flow Modeling for Wave-Structure Interactions and Induced Motions of Floating Bodies. *Water* **2020**, *12*, 543.
https://doi.org/10.3390/w12020543

**AMA Style**

Lo D-C, Wang K-H, Hsu T-W. Two-Dimensional Free-Surface Flow Modeling for Wave-Structure Interactions and Induced Motions of Floating Bodies. *Water*. 2020; 12(2):543.
https://doi.org/10.3390/w12020543

**Chicago/Turabian Style**

Lo, Der-Chang, Keh-Han Wang, and Tai-Wen Hsu. 2020. "Two-Dimensional Free-Surface Flow Modeling for Wave-Structure Interactions and Induced Motions of Floating Bodies" *Water* 12, no. 2: 543.
https://doi.org/10.3390/w12020543