# Experimental Investigation on Characteristics of Sand Waves with Fine Sand under Waves and Currents

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Installation and Methods

#### 2.1. Experimental Set-Up

#### 2.2. Water Characteristics Calculation

## 3. Results

#### 3.1. Dynamic Growth of Sand Waves

#### 3.2. Geometry Characteristics of Sand Waves

## 4. Discussion

#### 4.1. Comparison with the Results of Cataño-Lopera et al. [1]

_{ew}) to express measured values. Twenty one data of tests with waves alone and combined currents were used to analyze the relationship between dimensionless geometric configuration of sand waves and R

_{ew}. The wave characteristics were obtained using small amplitude wave theory for all cases though Ursell number was larger than 26 in about half of the experiments. In this paper, Cnoidal wave theory was used to calculate the wave characteristics for the experiments with a large Ursell number. Additionally, two different sediment diameters were used in this paper, and the Reynolds wave number was no longer suitable as the dimensionless parameter because sediment configuration needs to be considered. Therefore, the dimensionless bed shear stress was used as the hydrodynamic parameter in this paper.

#### 4.2. Bedforms with Dimensionless Shear Stress

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notation

$H$ = wave height |

$T$ = wave period |

${u}_{0}$ = depth-averaged current velocity |

$d$ = water depth |

${D}_{50}$ = median diameter of sediment |

$Ur$ = Ursell number |

$L$ = wave length |

${T}_{*}$ = dimensionless bed-shear stress |

${\tau}_{cw}$ = mean bed shear stress due to current and waves |

${\tau}_{cr}$ = critical bed shear stress |

${\rho}_{s}$ = density of sediment |

${\rho}_{w}$ = density of water |

$g$ = gravitational acceleration |

${\theta}_{cr}$ = critical Shields number |

${D}_{*}$ = dimensionless particle diameter |

$\nu $ = viscosity coefficient of water |

${u}_{m}$ = amplitude of the near-bottom wave orbital velocity |

${f}_{w}$ = wave friction factor |

${f}_{c}$ = current friction factor |

${\tau}_{w}$ = wave-averaged bed shear stress due to waves only |

${\tau}_{c}$ = bed shear stress due to current only |

$k$ = wave number |

${a}_{m}$ = near-bed peak orbital excursion |

$\Delta $ = wave related roughness |

${\eta}_{s}$ = ripple height |

$\kappa $ = modulus of elliptic integral |

$K\left(\kappa \right)$ = the first kind complete elliptic integral |

$E\left(\kappa \right)$ = the second kind complete elliptic integral |

$u$ = near-bed instantaneous velocity due to waves |

$\mathrm{sn}\left(\right)$ = Jacobian elliptic sinusoid function |

$\mathrm{cn}\left(\right)$ = Jacobian elliptic cosine function |

$\mathrm{dn}\left(\right)$ = Jacobian elliptic delta function |

${z}_{t}$ = distance from the wave trough to the sea bed |

${z}_{c}$ = distance from the wave crest to the sea bed |

${h}_{sw}$ = sand wave height |

${l}_{sw}$ = sand wave length |

${\sigma}_{sw}$ = sand wave steepness |

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**Figure 3.**Time series of the wave particle horizontal velocity with different wave theories. The red dash line represents the first test and the wave velocity is symmetrical in a wave period. The blue solid line represents the second test and the wave velocity is larger following the wave direction but the duration is shorter.

**Figure 4.**Bed evolution over time during the experiment. (

**a**) initial time; (

**b**) several minutes later; (

**c**) 1 h later; (

**d**) 10 h later.

**Figure 7.**The relationship between the measured dimensionless sand wave height and dimensionless shear stress. The blue circles represent the results of experiments of this paper with a medium diameter of 0.17 mm and the dark ones of 0.18 mm. The red circles represent the results of experiments conducted by Cataño-Lopera et al. [1]. The range of the dimensionless shear stress is about 7–49.

**Figure 8.**The relationship between the measured dimensionless sand wave length and dimensionless shear stress.

**Figure 9.**The relationship between the measured sand wave steepness and dimensionless shear stress. The black line is the fitting curve for all the data.

**Figure 10.**The relationship between the measured dimensionless sand wave length and dimensionless shear stress.

**Figure 11.**The relationship between the measured dimensionless sand wave height and dimensionless wave length.

Scenarios | H (m) | T (s) | u_{0} (m/s) | d (m) | D_{50} (mm) |
---|---|---|---|---|---|

1 | 0.10 | 2 | 0 | 0.6 | 0.17 |

2 | 0.12 | 2 | 0 | 0.6 | 0.17 |

3 | 0.14 | 2 | 0 | 0.6 | 0.17 |

4 | 0.16 | 2 | 0 | 0.6 | 0.17 |

5 | 0.14 | 2 | −0.2 | 0.6 | 0.17 |

6 | 0.14 | 2 | 0.2 | 0.6 | 0.17 |

7 | 0.14 | 2 | 0.25 | 0.6 | 0.17 |

8 | 0.14 | 2 | 0.35 | 0.6 | 0.17 |

9 | 0.20 | 2 | 0 | 0.6 | 0.18 |

10 | 0.22 | 2 | 0 | 0.6 | 0.18 |

_{0}is depth-averaged current velocity; d is the water depth, and D

_{50}is the median diameter of sediment.

Scenarios ^{a} | H (cm) | T (s) | u_{0}(m/s) | d (m) | D_{50}(mm) | h_{sw}(cm) | l_{sw}(m) | Ur | ${\mathit{\tau}}_{\mathit{w}}$ | ${\mathit{\tau}}_{\mathit{c}}$ | ${\mathit{\tau}}_{\mathit{c}\mathit{r}}$ | ${\mathit{T}}_{*}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 10.0 | 2 | 0 | 0.6 | 0.17 | 1.09 | 2.28 | 8.81 | 3.49 | 0.00 | 0.15 | 22.10 |

2 | 12.0 | 2 | 0 | 0.6 | 0.17 | 2.79 | 2.2 | 10.57 | 4.19 | 0.00 | 0.15 | 26.72 |

3 | 14.0 | 2 | 0 | 0.6 | 0.17 | 3.18 | 1.96 | 12.33 | 4.92 | 0.00 | 0.15 | 31.49 |

4 | 16.0 | 2 | 0 | 0.6 | 0.17 | 3.79 | 1.78 | 14.09 | 5.66 | 0.00 | 0.15 | 36.41 |

5 | 14.0 | 2 | −0.2 | 0.6 | 0.17 | 2.65 | 1.84 | 12.33 | 4.92 | −0.27 | 0.15 | 29.71 |

6 | 14.0 | 2 | 0.2 | 0.6 | 0.17 | 3.67 | 2.04 | 12.33 | 4.92 | 0.27 | 0.15 | 33.26 |

7 | 14.0 | 2 | 0.25 | 0.6 | 0.17 | 5.64 | 2.41 | 12.33 | 4.92 | 0.42 | 0.15 | 34.26 |

8 | 14.0 | 2 | 0.35 | 0.6 | 0.17 | 3.81 | 2.41 | 12.33 | 4.92 | 0.82 | 0.15 | 36.93 |

9 | 20.0 | 2 | 0 | 0.6 | 0.18 | 6.2 | 3.9 | 17.62 | 7.22 | 0.00 | 0.16 | 44.05 |

10 | 22.0 | 2 | 0 | 0.6 | 0.18 | 12.5 | 4.53 | 19.38 | 8.03 | 0.00 | 0.16 | 49.10 |

11 | 19.6 | 2 | 0 | 0.56 | 0.25 | 26.9 | 5.8 | 20.12 | 7.47 | 0.00 | 0.22 | 32.57 |

12 ^{b} | 12.8 | 5.2 | 0 | 0.56 | 0.25 | 3.3 | 5.1 | 117.19 | 1.93 | 0.00 | 0.22 | 7.65 |

13 | 19.2 | 6.9 | 0 | 0.56 | 0.25 | 11.9 | 6.8 | 347.56 | 1.80 | 0.00 | 0.22 | 7.07 |

14 | 19.7 | 4.2 | 0 | 0.56 | 0.25 | 9.8 | 3.4 | 123.08 | 3.73 | 0.00 | 0.22 | 15.77 |

15 | 17.7 | 1.6 | 0 | 0.56 | 0.25 | 2 | 1.5 | 10.32 | 7.08 | 0.00 | 0.22 | 30.79 |

16 | 20.9 | 2.9 | 0 | 0.56 | 0.25 | 5.9 | 2.6 | 56.76 | 6.07 | 0.00 | 0.22 | 26.27 |

17 | 10.7 | 3.4 | 0 | 0.56 | 0.25 | 7.7 | 3.4 | 37.88 | 2.63 | 0.00 | 0.22 | 10.81 |

18 | 17.4 | 2.3 | 0 | 0.56 | 0.25 | 39.9 | 8.4 | 24.83 | 6.16 | 0.00 | 0.22 | 26.66 |

19 | 15.1 | 5.9 | 0 | 0.56 | 0.25 | 11.5 | 5.1 | 186.26 | 1.86 | 0.00 | 0.22 | 7.37 |

20 | 10.4 | 4.1 | 0 | 0.56 | 0.25 | 6 | 4.2 | 55.46 | 2.12 | 0.00 | 0.22 | 8.51 |

21 | 14.7 | 4.1 | 0 | 0.56 | 0.25 | 8.6 | 4.1 | 82.12 | 2.95 | 0.00 | 0.22 | 12.24 |

22 | 12.3 | 5.1 | 0 | 0.56 | 0.25 | 7.4 | 5.1 | 107.38 | 1.91 | 0.00 | 0.22 | 7.57 |

23 | 19.6 | 1.9 | 0.17 | 0.56 | 0.25 | 13.5 | 5.1 | 17.75 | 7.60 | 0.20 | 0.22 | 34.04 |

24 | 19.2 | 5.8 | 0.17 | 0.56 | 0.25 | 12.4 | 6.8 | 240.36 | 2.32 | 0.20 | 0.22 | 10.34 |

25 | 19.7 | 3.7 | 0.17 | 0.56 | 0.25 | 8.2 | 4.1 | 92.73 | 4.38 | 0.20 | 0.22 | 19.58 |

26 | 17.7 | 1.5 | 0.17 | 0.56 | 0.25 | 5 | 3.4 | 8.65 | 7.10 | 0.20 | 0.22 | 31.77 |

27 | 11.3 | 3.6 | 0.17 | 0.56 | 0.25 | 3.5 | 4.1 | 45.68 | 2.63 | 0.20 | 0.22 | 11.71 |

28 | 10.7 | 2 | 0.17 | 0.56 | 0.25 | 7.5 | 2.1 | 10.98 | 3.93 | 0.20 | 0.22 | 17.57 |

29 | 12.1 | 5.1 | 0.17 | 0.56 | 0.25 | 14.5 | 6.3 | 105.38 | 1.88 | 0.20 | 0.22 | 8.33 |

30 | 13 | 4.4 | 0.17 | 0.56 | 0.25 | 11.6 | 5.8 | 83.15 | 2.41 | 0.20 | 0.22 | 10.74 |

31 | 23 | 2 | 0.17 | 0.56 | 0.25 | 12.5 | 2.3 | 23.61 | 8.95 | 0.20 | 0.22 | 40.08 |

^{a}The first 10 scenarios were conducted in this paper, the last 21 scenarios were conducted by Cataño-Lopera et al. [1].

^{b}The rows with brown color denote the shallow water, with Cnoidal wave theory.

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**MDPI and ACS Style**

Wang, Z.; Liang, B.; Wu, G.
Experimental Investigation on Characteristics of Sand Waves with Fine Sand under Waves and Currents. *Water* **2019**, *11*, 612.
https://doi.org/10.3390/w11030612

**AMA Style**

Wang Z, Liang B, Wu G.
Experimental Investigation on Characteristics of Sand Waves with Fine Sand under Waves and Currents. *Water*. 2019; 11(3):612.
https://doi.org/10.3390/w11030612

**Chicago/Turabian Style**

Wang, Zhenlu, Bingchen Liang, and Guoxiang Wu.
2019. "Experimental Investigation on Characteristics of Sand Waves with Fine Sand under Waves and Currents" *Water* 11, no. 3: 612.
https://doi.org/10.3390/w11030612