# The Effect of Habitat Structure Boulder Spacing on Near-Bed Shear Stress and Turbulent Events in a Gravel Bed Channel

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

_{0}) of 0.5% located in the Water Resources Engineering Laboratory at Clarkson University. Measurements were taken in a section with a length of 2.40 m located 3.80 m downstream from the flume entrance to ensure that the turbulent flow is hydraulically fully developed. Figure 1 shows a scheme of the flume. Natural semi-spherical shape boulders with an equivalent diameter (D) of 12.50 cm were used as large roughness elements. Experiments were carried out over a gravel bed with d

_{90}= 27.50 mm, d

_{50}= 17.21 mm, and d

_{10}= 9.37 mm, where d

_{p}refers to the grain size for which p% of particle sizes are finer. The critical bed shear stress for the median sediment size (d

_{50}) was calculated according to [46] formula, and it was 12.38 N/m

^{2}, at which incipient sediment motion takes place. As projected, no sediment movement was observed at any set of experiments.

#### 2.2. Data Filtering

#### 2.3. Calculation of Bed Shear Stress

#### 2.4. Quadrant Analysis

- Quadrant 1: outward interactions (${u}^{\prime}>0,{w}^{\prime}0$)
- Quadrant 2: ejection events (${u}^{\prime}<0,{w}^{\prime}0$)
- Quadrant 3: inward interactions (${u}^{\prime}<0,{w}^{\prime}0$)
- Quadrant 4: sweep events (${u}^{\prime}>0,{w}^{\prime}0$).

## 3. Results and Discussion

#### 3.1. Near-Bed Shear Stress

^{2}for the S1-60 and the S1-100 scenarios, respectively. These estimations were higher than the shear stress estimations using the point-methods for no-boulder scenarios. The maximum estimation of the average near-bed shear stress was significantly lower than the calculated critical bed shear stress (12.38 N/m

^{2}). This was expected because no sediment movement was observed in all scenarios. For the no-boulder scenarios (S1-60 and S1-100), the TKE and modified TKE methods resulted in a close estimation for the average near-bed shear stress (a maximum difference of 10% for the S1-60) while estimations using the Reynolds method were slightly higher than the TKE and modified TKE methods. Close estimations using these point-methods were reported by [12]. For scenarios with the largest boulder spacing at 60 L/s (S2-60), the TKE method resulted in a higher bed shear stress while the Reynolds and modified TKE resulted in close estimations. At 100 L/s scenario (S2-100), the Reynolds, TKE, and modified TKE methods showed very similar performance (with a maximum 20% difference between the modified TKE and TKE methods). At 60 L/s, for both medium (S3-60) and small boulder spacing (S4-60) scenarios, both TKE and modified TKE methods showed a very similar performance while the Reynolds method showed a significantly lower near-bed shear stress. The average near-bed shear stress using the Reynolds method was about 200% and 70% less than estimations of other point-methods for the S3-60 and S4-60, respectively. At 100 L/s, for both S3-100 and S4-100, the modified TKE method led to the highest estimates, while the Reynolds and TKE methods showed similar results. Generally, it can be said that at submerged condition (100 L/s), the difference between estimations using the point-methods was not significant for each scenario, although the modified TKE led to slightly higher estimations for boulder-scenarios. At unsubmerged condition (60 L/s), for both no-boulder and large spacing scenarios, the difference between the point-methods was not significant; however, for the medium and small spacing scenarios, the near-bed shear estimations using the Reynolds method was significantly smaller.

#### 3.2. Relative Performance of the Reynolds, TKE, and Modified TKE Methods

^{2}indicating a reasonable similarity between results even after adding boulders and then decreasing the boulder spacing. At 100 L/s (Figure 4b), before adding boulders (S1-100), a low RMSE = 0.18 N/m

^{2}indicated a good agreement between the TKE and modified TKE methods. After adding boulders, the RMSE significantly increased, and this increase was intensified after decreasing the boulder spacing. Generally, the modified TKE method resulted in higher values of the near-bed shear stress in comparison with the TKE method for the boulder-scenarios (S2-100, S3-100, and S4-100). This difference can be related to the values of ${C}_{1}$ and ${C}_{2}$ constants that still are required to be found for the natural streams [11,12].

^{2}for the TKE and modified TKE, respectively. At 100 L/s, RMSE = 0.48 and 0.43 N/m

^{2}for the TKE and modified TKE, respectively). It was reported that the modified TKE and Reynolds method have a relatively good agreement over a Plexiglas and sand bed [12]. However, estimations using the TKE and modified TKE methods were mostly lower than the Reynolds method, but at a lower range of shear stresses, the values were close to 1:1 line. This behavior in a range of high shear stress is in agreement with [12]; however, for the lower shear stresses, similar studies found systematically higher values from the TKE and modified TKE methods [12,40]. For the large boulder spacing at 60 L/s (S2-60), the agreement between the Reynolds and both TKE and modified TKE methods slightly decreased (RMSE = 0.87 and 0.95 N/m

^{2}for the TKE and modified TKE, respectively). By decreasing boulder spacing (S3-60 and S4-60), the RMSE increased to higher values indicating a lower similarity between the methods. A reason for the poor performance can be the presence of negative values in the Reynolds method, while values using two other methods are always positive. After adding boulders, it can be found that both TKE and modified TKE methods led to higher estimations than the Reynolds method (except for the modified TKE method at the large boulder spacing). This difference became more significant for the medium and small boulder spacing scenarios. At 100 L/s for the boulder-scenarios, the RMSE of the Reynolds and TKE methods varied between 1.16 and 1.98 N/m

^{2}, and the RMSE of the Reynolds and modified TKE methods varied between 1.30 to 2.67 N/m

^{2}. The RMSE values for both TKE and modified TKE methods reached their maximum (lower similarity) for the small boulder spacing (S4-100). However, some extreme values resulted from the TKE and modified TKE methods might be the reason behind higher RMSE values. After adding boulders, the TKE method estimates generally were smaller than the Reynolds method, while the modified TKE method usually resulted in higher near-bed shear stress than the Reynolds method.

#### 3.3. Spatial Distribution of Near-Bed Shear Stress

#### 3.4. Near-Bed Turbulent Events

## 4. Conclusions

- For all scenarios, the reach-averaged method led to a significantly higher reach-averaged bed shear stress. For the unsubmerged condition, the Reynolds method resulted in a significantly lower near-bed shear stress between the point-methods, while, at submerged condition, all the point-methods showed very similar results.
- At unsubmerged condition, the effect of the boulder spacing on the variation of near-bed shear stress estimated from the Reynolds method was different from the TKE and modified TKE methods, while, at submerged condition, all of the point-methods showed a similar trend.
- At submerged condition, the densest boulder spacing led to the highest near-bed shear stress for all point-methods. However, for the unsubmerged condition, maximum near-bed shear stress varied for different methods and boulder spacing.
- At unsubmerged condition, the TKE and modified TKE methods can be used interchangeably for estimation of the near-bed shear stress in the presence of boulders; however, applying appropriate ${C}_{1}$ and ${C}_{2}$ coefficients is required to obtain more reliable results. For a comprehensive comparison of the Reynolds method with two other point-methods, more measurements are necessary, especially at unsubmerged condition.
- At unsubmerged condition, denser boulder spacing led to a more uniform contribution of turbulent events to the Reynolds shear stress. At submerged condition, decreased ejection events downstream of boulders in the large and medium boulder spacing may reduce the sediment entrainment and suspension.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**A schematic plan of boulder arrangement in the measurement zone (hatched area) for (

**a**) no boulder, (

**b**) 10D boulder spacing, (

**c**) 5D boulder spacing, and (

**d**) 2D boulder spacing. Cross marks show measurement points for each setup, and large circles show the position of idealized (assumed fully spherical) boulders. Two points X3Y3 and X5Y3 have been marked in (

**a**) as common measurement points in the centerline of boulders in all scenarios.

**Figure 3.**Averaged bed shear stress resulted from different calculation methods for different boulder spacing at (

**a**) 60 L/s flow rate, (

**b**) 100 L/s flow rate.

**Figure 4.**Near-bed shear stress estimated from the turbulent kinetic energy (TKE) method against the modified TKE method for all the measurement points at (

**a**) 60 L/s flow rate, (

**b**) 100 L/s flow rate. The unit of RMSE is N/m

^{2}.

**Figure 5.**Near-bed shear stress estimated from the Reynolds method against the TKE and modified TKE methods for all the measurement points at (

**a**) 60 L/s flow rate, (

**b**) 100 L/s flow rate. The unit of RMSE is N/m

^{2}.

**Figure 6.**Contour plots of the estimated near-bed shear stress at 60 L/s (unsubmerged condition) for (

**a**) the Reynolds, (

**b**) TKE, and (

**c**) modified TKE methods. Gray circles show the idealized boulders, and small black dots represent the measuring points.

**Figure 7.**Contour plots of the estimated near-bed shear stress at 100 L/s (submerged condition) for (

**a**) the Reynolds, (

**b**) TKE, and (

**c**) modified TKE methods. Gray circles show the idealized boulders, and small black dots represent the measuring points.

**Figure 8.**Joint frequency distribution of normalized $u\prime $ and $w\prime $ at point X3Y3 at (

**a**) 60 L/s flow rate, (

**b**) 100 L/s flow rate.

**Figure 9.**Joint frequency distribution of normalized $u\prime $ and $w\prime $ at point X5Y3 at (

**a**) 60 L/s flow rate, (

**b**) 100 L/s flow rate.

Scenario | Flow Rate (L/s) | Boulder Spacing | Reach-Averaged Flow Depth (m) | Submergence Ratio |
---|---|---|---|---|

S1-60 | 60 | Infinity | 0.082 | - |

S2-60 | 10D | 0.092 | 0.73 | |

S3-60 | 5D | 0.098 | 0.78 | |

S4-60 | 2D | 0.093 | 0.74 | |

S1-100 | 100 | Infinity | 0.151 | - |

S2-100 | 10D | 0.157 | 1.25 | |

S3-100 | 5D | 0.165 | 1.32 | |

S4-100 | 2D | 0.161 | 1.29 |

Parameter | ${\mathit{S}}^{\mathit{f}}$ | P | ||||||
---|---|---|---|---|---|---|---|---|

Scenario | Outward | Ejection | Inward | Sweep | Outward | Ejection | Inward | Sweep |

S1-60 | −0.05 | 0.55 | −0.04 | 0.41 | 0.03 | 0.36 | 0.03 | 0.27 |

S2-60 | −0.29 | 0.74 | −0.26 | 0.74 | 0.12 | 0.30 | 0.10 | 0.30 |

S3-60 | −1.50 | 2.17 | −1.50 | 1.83 | 0.21 | 0.30 | 0.21 | 0.25 |

S4-60 | −10.10 | 12.01 | −9.77 | 8.86 | 0.25 | 0.29 | 0.24 | 0.22 |

S1-100 | −0.07 | 0.51 | −0.06 | 0.50 | 0.04 | 0.32 | 0.04 | 0.31 |

S2-100 | −0.74 | 1.25 | −0.64 | 1.11 | 0.18 | 0.31 | 0.16 | 0.27 |

S3-100 | −0.51 | 1.14 | −0.58 | 0.93 | 0.14 | 0.32 | 0.16 | 0.26 |

S4-100 | −12.17 | 12.29 | −11.62 | 12.49 | 0.25 | 0.25 | 0.24 | 0.26 |

Parameter | ${\mathit{S}}^{\mathit{f}}$ | P | ||||||
---|---|---|---|---|---|---|---|---|

Scenario | Outward | Ejection | Inward | Sweep | Outward | Ejection | Inward | Sweep |

S1-60 | −0.02 | 0.45 | −0.03 | 0.38 | 0.01 | 0.33 | 0.02 | 0.28 |

S2-60 | 1.24 | −0.57 | 1.14 | −0.84 | 0.30 | 0.14 | 0.27 | 0.20 |

S3-60 | 4.98 | −2.92 | 3.24 | −4.30 | 0.32 | 0.19 | 0.21 | 0.28 |

S4-60 | 1.41 | −0.90 | 1.48 | −1.00 | 0.28 | 0.18 | 0.29 | 0.20 |

S1-100 | −0.08 | 0.59 | −0.10 | 0.49 | 0.05 | 0.34 | 0.06 | 0.28 |

S2-100 | −0.06 | 0.22 | −0.03 | 0.72 | 0.04 | 0.15 | 0.02 | 0.49 |

S3-100 | −0.02 | 0.23 | −0.03 | 0.62 | 0.02 | 0.17 | 0.02 | 0.45 |

S4-100 | −0.05 | 0.39 | −0.03 | 0.49 | 0.03 | 0.27 | 0.02 | 0.34 |

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**MDPI and ACS Style**

Golpira, A.; Huang, F.; Baki, A.B.M.
The Effect of Habitat Structure Boulder Spacing on Near-Bed Shear Stress and Turbulent Events in a Gravel Bed Channel. *Water* **2020**, *12*, 1423.
https://doi.org/10.3390/w12051423

**AMA Style**

Golpira A, Huang F, Baki ABM.
The Effect of Habitat Structure Boulder Spacing on Near-Bed Shear Stress and Turbulent Events in a Gravel Bed Channel. *Water*. 2020; 12(5):1423.
https://doi.org/10.3390/w12051423

**Chicago/Turabian Style**

Golpira, Amir, Fengbin Huang, and Abul B.M. Baki.
2020. "The Effect of Habitat Structure Boulder Spacing on Near-Bed Shear Stress and Turbulent Events in a Gravel Bed Channel" *Water* 12, no. 5: 1423.
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