# Studying the Wake of an Island in a Macro-Tidal Estuary

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. In Situ Data Collection

#### 2.2. Modelling System and the Turbulence Models

**u**is the vector of velocity; ${v}_{t}$ is the turbulent diffusion coefficient; t is time; ${S}_{h}$ is the source or sink term of fluid mass, ${S}_{x}$, ${S}_{y}$ are the source or sink terms of fluid momentum, representing wind shear, the Coriolis force, and bottom friction within the domain.

^{−4}m

^{2}/s following the same value of other researchers [33,34].

_{l}, and the transver diffusion, i.e., K

_{t}. Those two viscosity values are expressed as

_{t}in the range from 0.6 to 1.0; Steffler and Blackburn [38] set α

_{t}to 0.5 with recommended values from 0.2 to 1.0. Different values could be used for ${\alpha}_{l}$ and ${\alpha}_{t}$ due to the anisotropic features of turbulence structure in the horizontal and vertical directions. For this study, considering the finding of Elder and latter researchers, ${\alpha}_{l}$ and ${\alpha}_{t}$ are assigned values of 6 and 0.6, respectively, following the advised value of Telemac-2D manual.

_{s}is the free surface elevation (m), Z

_{f}is the bottom elevation (m), ${u}_{i}^{\prime}$ is the temporal fluctuation of velocity and the $\overline{{u}_{i}^{\prime}}$ corresponds to the average value of ${u}_{i}^{\prime}$ over time.

#### 2.3. Model Setup

#### 2.4. Analysis Tools

^{2}) and the Root Mean Squared Error (RMSE) were used to assess model performance as follows:

^{2}to assess model performance, namely excellent (R

^{2}> 0.85), very good (0.65 < R

^{2}< 0.85), good (0.5< R

^{2}< 0.65), and poor (0.2 < R

^{2}< 0.5). However, the RMSE value is mainly relevant to scalar quantities, not vector quantities. In this case, the Mean Absolute Error (MAE) was used. The MAE includes errors of both magnitude and direction in a single statistic. For a 2D vector $\overrightarrow{X}=\left({X}_{1},{X}_{2}\right)$,

## 3. Results and Discussion

#### 3.1. Model Calibration and Validation

^{2}) shows a strong correlation between the modelled and observed data, and RMSE value is also encouraging bearing in mind the high tidal range and currents. However, validation data of model against the Newport gauges show relatively poor agreement. This was believed to be due to the inconsistencies in the bathymetric data in this region. The hydrodynamic performance of the model was further validated against data collected using five bed-mounted ADCPs deployed in Swansea Bay between September 2012 and December 2012. The current speeds and velocities were measured throughout the water depth at these sites using seabed mounted ADCPs (Figure 1). The corresponding field data were then integrated over depth to acquire the depth average values. Typical comparisons between the model predictions and measured data for water levels and current speeds and directions and a summary of the statistical analysis are given (Figure 5, Table 2). The statistic values all show good correlation. For water level, all R

^{2}are higher than 0.99 and RMSE is relatively small. For velocity validation, three RMAE locate in ‘excellent’ and two are ‘very good’. In summary, validation of the model shows very good correlation between the model predictions and measured field data and, therefore, the model verification is considered acceptable and appropriate for examining the flow around Flat Holm Island.

#### 3.2. Comparison of Turbulence Schemes

#### 3.3. Comparison of Different k-ε Solvers

#### 3.4. Model Comparison with ADCP Data

#### 3.5. The Evolution of Wake in the Lee of Island

_{0}is the free stream velocity, D is the water depth, L is the diameter of island, and K

_{z}is the vertical eddy diffusion coefficient. When P << 1, friction is dominant and quasi-potential flow results. A relatively stable wake is present when P ≈ 1. For P >> 1, then bottom friction effects are weak, and an unsteady wake is formed, similar to the flow around obstacles at a large Re value in laboratory experiments. For Flat Holm island, the island diameter (L) is about 700 m and kept constant during the rise and fall of tide due to its steep cliff. While the vertical eddy viscosity (K

_{z}) in the Bristol Channel is defined as 0.20 m

^{2}s

^{−1}[50,51]. The free stream velocity U

_{0}and water depth are taken at 400 m upstream away from Flat Holm Island. The island wake parameter (P) corresponding to different tide condition are calculated (Table 6).

## 4. Summary and Conclusions

^{2}. The general performance of the model was first validated against water levels and velocities measured across the domain. Further field surveys were undertaken with vessel-mounted ADCP data being acquired specifically around the island and for different tidal conditions to validate and improve the models predictions. To acquire better model predictions, four different turbulence model were tested and compared, including: a constant eddy viscosity model, an Elder model, a k-ε model, and a Smagorinski LES-based model. The k-ε model showed the best performance when compared with the field measurements and was chosen for this study. Furthermore, six different methods to solve the k-ε model equation were considered and compared. All models showed good predictions compared to the field measurements around the island, while the best results were acquired by using the conjugate residual. The conjugate residual solver was selected and then used in this study.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Model domain, showing calibration points and acoustic Doppler current profilers (ADCPs) measurement lines. Bathymetric data relative to Ordnance Datum Newlyn (ODN).

**Figure 3.**A example where an embedded Large Eddy Simulation (LES) model to calculate the flow containing a hydraulic structure that induces the formation of a region in which large-scale unsteady eddies are present [40].

**Figure 4.**Water level comparison of model predictions and British Oceanographic Data Centre (BODC) measured data.

**Figure 8.**Streamlines in the vicinity of Flat Holm island on 11 July 2011: (

**a**) High water (slack tide); (

**b**) HW + 1.7 h (

**c**) HW + 3.25 h (peak ebb); (

**d**) Low water (slack tide); (

**e**) LW + 1.7 h; (

**f**) LW + 3.25 h (peak flood).

**Figure 9.**Streamlines in the vicinity of Flat Holm island on date 05/07/2011: (

**a**) High water (slack tide); (

**b**) HW + 0.5 h (

**c**) HW + 1.0 h; (

**d**) HW + 1.5 h (

**e**) HW + 2.0 h; (

**f**) HW + 3.0 h (peak ebb).

**Figure 10.**Streamlines in the vicinity of Flat Holm Island at date 05/07/2011: (

**a**) low water (slack tide); (

**b**) LW + 0.5 h; (

**c**) LW + 1.0 h; (

**d**) LW + 1.5 h; (

**e**) LW + 2.0 h; (

**f**) LW + 3.0 h (peak flood).

Date | Time (GMT) | Measure Route | Date | Time (GMT) | Measure Route |
---|---|---|---|---|---|

5 July 2011 | 09:05 | A1 | 1 August 2011 | 09:27 | A4 |

09:49 | A1 | 09:55 | A3 | ||

11:00 | A1 | 10:08 | A2 | ||

11:34 | A1 | 10:40 | A1 | ||

12:29 | A1 | 11:04 | A4 | ||

13:00 | A1 | 11:20 | A3 | ||

13:39 | A1 | 11:40 | A2 | ||

14:22 | A1 | 12:02 | A1 | ||

15:00 | A1 | 30 September 2011 | 09:37 | A2 | |

16:38 | A1 | 10:56 | A1 | ||

7 July 2011 | 15:10 | A2 | 11:49 | A2 | |

15:47 | A2 | 12:33 | A1 | ||

16:55 | B1 | ||||

15:37 | B1 | ||||

18:50 | D1 | ||||

19:31 | D1 |

Water Level Statistical Analysis | ||

Site | Coefficient of Determination(R^{2}) | Root Mean Squared ErrorRMSE (m) |

Avonmouth | 0.992 | 0.359 |

Hinkley | 0.988 | 0.351 |

Mumbles | 0.964 | 0.420 |

Newport | 0.932 | 0.767 |

ADCP L1 | 0.99 | 0.260 |

ADCP L2 | 0.993 | 0.213 |

ADCP L3 | 0.992 | 0.232 |

ADCP L4 | 0.992 | 0.231 |

ADCP L5 | 0.993 | 0.214 |

Swansea Bay ADCPs Measured Velocity Magnitude | ||

Site | Mean Absolute Error(MAE(m/s)) | Relative Mean Absolute Error(RMAE) |

ADCP L1 | 0.122 | 0.222 |

ADCP L2 | 0.083 | 0.145 |

ADCP L3 | 0.057 | 0.142 |

ADCP L4 | 0.045 | 0.191 |

ADCP L5 | 0.076 | 0.230 |

Tidal Gauges | Data Classification | M2 Amplitude (m) | M2 Phase (deg) | S2 Amplitude (m) | S2 Phase (deg) | N2 Amplitude (m) | N2 Phase (deg) |
---|---|---|---|---|---|---|---|

Mumbles | Observation | 3.16 | 59.98 | 1.25 | 227.95 | 0.37 | 282.18 |

Prediction | 3.17 | 57.49 | 1.22 | 225.58 | 0.49 | 271.09 | |

Difference | 0.56% | −4.15% | −2.82% | −1.04% | 33.26% | −3.93% | |

Hinkley | Observation | 3.97 | 66.84 | 1.58 | 243.97 | 0.63 | 285.55 |

Prediction | 4.03 | 65.56 | 1.55 | 238.59 | 0.61 | 283.40 | |

Difference | 1.49% | −1.94% | −2.15% | −2.26% | −2.18% | −0.76% | |

Newport | Observation | 4.18 | 86.51 | 1.65 | 267.42 | 0.63 | 307.12 |

Prediction | 4.26 | 78.43 | 1.62 | 254.62 | 0.64 | 298.68 | |

Difference | 1.96% | −9.35% | −1.63% | −4.79% | 1.57% | −2.75% | |

Avonmouth | Observation | 4.30 | 91.48 | 1.69 | 274.06 | 0.67 | 313.47 |

Prediction | 4.29 | 86.04 | 1.60 | 264.46 | 0.64 | 308.15 | |

Difference | −0.38% | −5.96% | −5.45% | −3.50% | −4.83% | −1.70% |

Scenario | Turbulence Model | MAE (m/s) | RMAE |
---|---|---|---|

1 | Constant viscosity model | 0.3744 | 0.3672 |

2 | Elder model | 0.3950 | 0.3705 |

3 | k-ε model | 0.3597 | 0.3266 |

4 | Smagorinski model | 0.3735 | 0.3708 |

Scenario | Solver in Telemac-2D Model with k-ε Turbulence Model | MAE | RMAE |
---|---|---|---|

1 | Conjugate Gradient | 0.3597 | 0.3266 |

2 | Conjugate Residual | 0.3420 | 0.3129 |

3 | Conjugate Gradient on Normal Equation | 0.3556 | 0.3254 |

4 | Minimum Error | 0.3625 | 0.3298 |

5 | Squared Conjugate Gradient | 0.3607 | 0.3274 |

6 | BICGSTAB (Biconjugate Stabilized Gradient) | 0.3535 | 0.3231 |

7 | GMRES (Generalised Minimum Residual) | 0.3544 | 0.3251 |

Figure | Moment | U_{0}(m/s) | D (m) | K_{z}(m ^{2} s^{−1}) | L (m) | P |
---|---|---|---|---|---|---|

Figure 9a | HW | 0.42 | 16.2 | 0.2 | 700 | 0.79 |

Figure 9b | HW + 0.5 h | 0.51 | 15.4 | 0.2 | 700 | 0.86 |

Figure 9c | HW + 1.0 h | 0.67 | 14.6 | 0.2 | 700 | 1.02 |

Figure 9d | HW + 1.5 h | 0.82 | 14.1 | 0.2 | 700 | 1.16 |

Figure 9e | HW + 2.0 h | 1.05 | 13.5 | 0.2 | 700 | 1.37 |

Figure 9f | HW + 3.0 h | 1.09 | 13.1 | 0.2 | 700 | 1.34 |

Figure 10a | LW | 0.62 | 8.5 | 0.2 | 700 | 0.32 |

Figure 10b | LW + 0.5 h | 0.68 | 9.3 | 0.2 | 700 | 0.42 |

Figure 10c | LW + 1.0 h | 0.79 | 9.9 | 0.2 | 700 | 0.55 |

Figure 10d | LW + 1.5 h | 0.95 | 10.6 | 0.2 | 700 | 0.76 |

Figure 10e | LW + 2.0 h | 0.89 | 11.9 | 0.2 | 700 | 0.90 |

Figure 10f | LW + 3.0 h | 1.1 | 12.4 | 0.2 | 700 | 1.21 |

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## Share and Cite

**MDPI and ACS Style**

Guo, B.; Ahmadian, R.; Evans, P.; Falconer, R.A.
Studying the Wake of an Island in a Macro-Tidal Estuary. *Water* **2020**, *12*, 1225.
https://doi.org/10.3390/w12051225

**AMA Style**

Guo B, Ahmadian R, Evans P, Falconer RA.
Studying the Wake of an Island in a Macro-Tidal Estuary. *Water*. 2020; 12(5):1225.
https://doi.org/10.3390/w12051225

**Chicago/Turabian Style**

Guo, Bin, Reza Ahmadian, Paul Evans, and Roger A. Falconer.
2020. "Studying the Wake of an Island in a Macro-Tidal Estuary" *Water* 12, no. 5: 1225.
https://doi.org/10.3390/w12051225