# Solitary Wave Generation and Propagation under Hypergravity Fields

^{*}

## Abstract

**:**

## 1. Introduction

## 2. General Scaling Laws

^{1/2}n

^{3/2}, which indicates that the viscosity cannot be scaled unless n is defined as N

^{−1/3}. With this scale, gravity similarity and viscosity similarity can coincide simultaneously. However, when n is considered to be N

^{−1/3}, the model is not a small-scale model; on the contrary, it is a large-scale model. In addition, experiments on large-scale models such as wave-structure interactions cannot be performed in a centrifuge. Therefore, although gravity similarity and viscosity similarity can simultaneously coincide in a hypergravity field, viscosity similarity is ignored in this paper. This is similar to experiments performed under normal gravity fields, because both the generation and the propagation of waves are processes that are dependent upon the gravity field and the model scale. In fact, in a centrifuge model test, the centrifuge acceleration is always labeled as N times the gravity acceleration (hereinafter referred to as N g, where g is the gravity acceleration). 1/N is always defined as the model scaling factor to ensure that the centrifuge acceleration is identical to the prototype acceleration. With these scaling laws, the model scaling factors for the wave celerity and fluid field static pressure are both one, which means that the dynamic and static pressures in a small-scale model are equal to the pressures in a full-scale model. Therefore, the self-weight stresses and gravity-dependent processes can be appropriately reproduced. For this reason, the above-mentioned scaling laws are considered in this study. Additionally, the discrepancies between the simulation results and the theoretical results due to viscosity and surface tension are discussed in this paper.

## 3. Wave Generation

_{wn}is the wave number that is determined as:

_{wn}, the piston stroke S, the wavelength l, and the duration of paddle motion are successively described as:

## 4. Governing Equations and Boundary Conditions

#### 4.1. Governing Equations

_{i}denotes the body forces, and $-\rho \overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ represents the Reynolds stresses, which must be modeled to close the RANS equations. In this study, the RNG k-ε two-equation model is used to solve the Reynolds stresses and close the equations, and it is defined as:

_{k}and α

_{ε}are the inverse effective Prandtl numbers for k and ε, respectively, while μ

_{eff}is the effective viscosity, G

_{k}represents the generation of turbulent kinetic energy due to the mean velocity gradients, and C

_{1ε}= 1.42 and C

_{2ε}= 1.68 are constants. R

_{ε}is defined as:

_{μ}= 0.0845, η

_{0}= 4.38, and β = 0.012 are constant, and η

_{ε}is defined as S

_{ε}k/ε.

#### 4.2. Initial and Boundary Conditions

#### 4.3. Numerical Method

## 5. Results and Discussion

_{1}, the downstream depth is d

_{2}, and a depth ratio (d

_{1}/d

_{2}) of 2.64, which was used by Goring [7], is considered. The front face of the shelf is vertical, and it is located at the far end of the wave tank opposite from the wavemaker at a distance of (2/3) L away from the wavemaker’s initial position, which is located at x = 0, as shown in Figure 1. During the simulation, a solitary wave is generated by moving the wavemaker according to the trajectory defined by Equation (6). The details of the conditions of the experiments that are performed in this paper are shown in Table 2. The solitary wave heights considered under the normal gravity field are chosen based on the wave height presented by Chen et al. [5]; accordingly, the wave heights that are used under other gravity fields are calculated by scaling laws that are based on the wave height used under the normal gravity field. The relative wave heights that are used in this study are 0.2, 0.3, and 0.34, which are within the range of the relative wave height used by Chen et al. [5] and Hsiao et al. [10]. The hypergravity considered in this study are 30 g, 50 g, and 100 g, which represent common centrifuge test cases. The 30 g and 50 g are commonly used by researchers to study water-soil interactions [23,24,25,26]; meanwhile, the scale of the model under the 100 g can match the circular channels of some drum centrifuges [39] and the experimental module of the centrifuge under construction.

_{1}is 25 cm, the downstream depth d

_{2}is 9.46 cm, the stroke is 18.25 cm, and the period is 4.24 s. The vertical front face of the shelf is located at a distance of 13 m from the wavemaker’s initial position, sampling point 1 is placed 5.75 m upstream of the step, sampling point 2 is placed at the step, and sampling points 3, 4, and 5 are placed at intervals of 5.68 m downstream of the step, which are slightly different from the model used in this paper. The numerical results from the present paper are compared with the experiments results of Goring [7] in Figure 3. The numerical results agree well with those of the experiments, both at sampling point 2 where the wave propagates over the step, and at sampling point 3, where the wave propagates on the shelf. The good agreement of the results means that the numerical model that is presented in this paper is accurate.

#### 5.1. Solitary Wave Generation and Propagation in a Constant Water Depth

_{a}that is in consideration for both the model scale and the elapsed propagation time is defined as:

_{i}is the wave height at the sampling point, t

_{i}is the elapsed time since the reference time, and N is the scale of the model. The values of the attenuation coefficient ε

_{a}of the cases are also displayed in Table 3. The reference wave height that was used in Table 3 is the numerical wave height at a dimensionless time of t/τ = 0.8 to avoid the initial instabilities caused by the wave generation. The reference time is 0.8 τ. Cases #011, #301, #501, and #101, which are tabulated in Table 2, are characterized by small wave numbers and long wavelengths. Slight reflection effects are observed at the dimensionless time of t/τ = 2.0, causing the wave height at this time to be slightly larger. Therefore, for these cases, the wave heights at the dimensionless time of t/τ = 2.0 are abandoned. As observed in Table 3, the discrepancies in the wave attenuation among the numerical results for the cases under the same gravity field may be attributed to the highly nonlinear effect caused by a high relative wave height. The discrepancies in the wave attenuation among the numerical results obtained from the simulations under different levels of gravity fields may be attributed to the ignored viscosity similarity, surface tension similarity and other complex reasons. Although the values of the attenuation coefficient displayed in Table 3 with the model scales that were considered do not noticeably vary, the attenuation coefficients for all of the cases performed in this study are less than 1%. The discrepancies in the wave attenuation among the numerical results for the cases with a maximum relative wave height are more appreciable, as shown in Figure 4. It is clear that wave attenuation may be more serious in a small-scale model, and the serious wave attenuation may cause a lower wave height and a more serious time lag. Therefore, it deserves closer attention. In addition, it should be mentioned that the use of Boussinesq’s solution in this paper might exacerbate the wave attenuation phenomenon; in contrast, Grimshaw’s solution and Fenton’s solution may be more exact when the relative wave height is not very high [11].

#### 5.2. Solitary Wave Interacting with Shelf

#### 5.3. Comparison of the Simulated Velocity Fields and Dynamic Pressures under Different Gravity Fields

## 6. Conclusions

- (1)
- The waveform obtained from the simulations performed under different gravity fields exhibit good agreement at the macroscale, indicating that solitary waves can be steadily generated and propagated within hypergravity fields. Consequently, wave breaking, impingement, run-up, and the other phenomena that have been observed during wave-coasting interactions, which have drawn considerable attention, can be studied using a downscaled model.
- (2)
- Although the waveform and static pressure field agree well at the macroscale, some discrepancies are detected within the details. Due to the ignored viscosity and surface tension similarity, some of the features of solitary waves are affected by the scale effect, including wave attenuation and time lag. Wave attenuation may be more serious in a small-scale model experiment performed within a hypergravity field, and can cause the time lag to become more serious. However, wave attenuation and time lag can be offset by a well-considered incident wave condition.
- (3)
- Since both the velocity and dynamic pressure are sensitive to the wave attenuation, time lag and fluid viscosity, some discrepancies can be found in the velocity field and dynamic pressure profiles. However, a well-designed initial incident wave can weaken the discrepancies in the velocity field and dynamic pressure.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Comparison between the numerical and experimental wave profiles at (

**a**) sampling point 2 and (

**b**) sampling point 3.

**Figure 4.**Evolution of the solitary wave propagation profiles at a constant water depth for cases (

**a**) #013 (1 g), (

**b**) #303 (30 g), (

**c**) #503 (50 g), and (

**d**) #103 (100 g).

**Figure 5.**Comparison between the numerical and analytical wave profile at t/τ = 1.1 for cases (

**a**) #013, (

**b**) #303, (

**c**) #503, and (

**d**) #103.

**Figure 10.**The maximum dynamic pressure profiles for cases #013 and #103 at sampling point 2 (

**a**) and sampling point 4 (

**b**).

Parameter | Dimension | Model Scaling (General) | Model Scaling (This Study) |
---|---|---|---|

Acceleration due to gravity | LT^{−2} | N | N |

Wave length | L | n | 1/N |

Wave height | L | n | 1/N |

Water depth | L | n | 1/N |

Piston stroke | L | n | 1/N |

Time | T | N^{−1/2}n^{1/2} | 1/N |

Wave celerity | LT^{−1} | N^{1/2}n^{1/2} | 1 |

Density | ML^{−3} | 1 | 1 |

Static pressure | ML^{−1}T^{−2} | Nn | 1 |

Kinematic viscosity coefficient | L^{2}T^{−1} | N^{1/2}n^{3/2} | 1 |

Case Number | H (m) | d_{1} (m) | d_{2} (m) | H/d_{1} | d_{1}/d_{2} | S (m) | τ (s) | L (m) | D (m) |
---|---|---|---|---|---|---|---|---|---|

Under the normal gravity (1 g) field | |||||||||

011 | 1.2 | 6 | 2.27 | 0.2 | 2.64 | 6.20 | 14.75 | 300 | 10 |

012 | 1.8 | 6 | 2.27 | 0.3 | 2.64 | 7.59 | 11.86 | ||

013 | 2.4 | 7 | 2.65 | 0.34 | 2.64 | 9.47 | 11.91 | 16 | |

Under 30 times the normal gravity (30 g) field | |||||||||

301 | 0.04 | 0.2 | 0.076 | 0.2 | 2.64 | 0.21 | 0.49 | 10 | 0.33 |

302 | 0.06 | 0.2 | 0.076 | 0.3 | 2.64 | 0.25 | 0.40 | ||

303 | 0.08 | 0.23 | 0.088 | 0.34 | 2.64 | 0.32 | 0.40 | 0.53 | |

Under 50 times the normal gravity (50 g) field | |||||||||

501 | 0.024 | 0.12 | 0.045 | 0.2 | 2.64 | 0.12 | 0.29 | 6 | 0.2 |

502 | 0.036 | 0.12 | 0.045 | 0.3 | 2.64 | 0.15 | 0.24 | ||

503 | 0.048 | 0.14 | 0.053 | 0.34 | 2.64 | 0.19 | 0.24 | 0.32 | |

Under 100 times the normal gravity (100 g) field | |||||||||

101 | 0.012 | 0.06 | 0.023 | 0.2 | 2.64 | 0.06 | 0.15 | 3 | 0.1 |

102 | 0.018 | 0.06 | 0.023 | 0.3 | 2.64 | 0.08 | 0.12 | ||

103 | 0.024 | 0.07 | 0.027 | 0.34 | 2.64 | 0.09 | 0.12 | 0.16 |

No. | 0.8 τ | 1.1 τ | 1.4 τ | 1.7 τ | 2.0 τ | ε_{a} (%) | No. | 0.8 τ | 1.1 τ | 1.4 τ | 1.7 τ | 2.0 τ | ε_{a} (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

011 | 121 | 120 | 119 | 118 | − | 0.19 | 501 | 2.39 | 2.34 | 2.31 | 2.28 | − | 0.40 |

012 | 181 | 179 | 177 | 175 | 174 | 0.30 | 502 | 3.53 | 3.43 | 3.37 | 3.32 | 3.28 | 0.62 |

013 | 240 | 237 | 234 | 231 | 230 | 0.34 | 503 | 4.66 | 4.51 | 4.41 | 4.34 | 4.30 | 0.71 |

301 | 3.98 | 3.91 | 3.85 | 3.81 | − | 0.36 | 101 | 1.19 | 1.17 | 1.15 | 1.14 | − | 0.36 |

302 | 5.89 | 5.72 | 5.61 | 5.53 | 5.47 | 0.64 | 102 | 1.77 | 1.72 | 1.68 | 1.66 | 1.64 | 0.65 |

303 | 7.76 | 7.52 | 7.36 | 7.24 | 7.16 | 0.69 | 103 | 2.33 | 2.26 | 2.21 | 2.17 | 2.14 | 0.69 |

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**MDPI and ACS Style**

Wang, Q.-S.; Li, M.-H.; Li, D.-W.
Solitary Wave Generation and Propagation under Hypergravity Fields. *Water* **2018**, *10*, 1381.
https://doi.org/10.3390/w10101381

**AMA Style**

Wang Q-S, Li M-H, Li D-W.
Solitary Wave Generation and Propagation under Hypergravity Fields. *Water*. 2018; 10(10):1381.
https://doi.org/10.3390/w10101381

**Chicago/Turabian Style**

Wang, Qiao-Sha, Ming-Hai Li, and Dai-Wei Li.
2018. "Solitary Wave Generation and Propagation under Hypergravity Fields" *Water* 10, no. 10: 1381.
https://doi.org/10.3390/w10101381