# The Logarithmic Law of the Wall in Flows over Mobile Lattice-Arranged Granular Beds

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## Abstract

**:**

## 1. Introduction

## 2. Laboratory Facilities, Instrumentation and Procedures

- a fixed-bed reach comprising $1.5$ m of large boulders (50 mm average diameter), followed by $3.0$ m of smooth bottom (PVC) and $2.5$ m of one layer of glued spherical glass beads ($5.0$ mm diameter) to ensure the development of a rough-wall boundary layer (Figure 2a,b);
- a mobile reach $4.0$ m long and $2.5$ cm deep filled with $5.0$ mm diameter glass beads, with density ${\rho}_{s}=2490\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}/\mathrm{m}}^{3}$, packed (with some vibration) to a void fraction of $0.325$, expressing the mixed nature of the lattice arrangement (face-centered and body centered), seen in Figure 2c).

^{®}, Skovlunde, Denmark) allowed for processing image pairs with the adaptive correlation algorithm. The initial interrogation area was of 128 × 128 px${}^{2}$, while the final was of 16 × 16 px${}^{2}$, with an overlap of 50%.

## 3. Data Analysis and Results

#### 3.1. PIV Post-Processing

#### 3.2. Calculation of the Parameters of the Log-Law

- the von Kármán parameter is considered flow independent ($\kappa =0.405$), the geometric roughness scale ${k}_{s}$ and the constant B are subjected to a best fit procedure.
- the von Kármán parameter is considered flow independent ($\kappa =0.405$), the constant B is $8.5$ and the roughness scale ${k}_{s}$ is calculated from a roughness function.
- the von Kármán parameter is assumed not universal but a fitting parameter, the geometric roughness scale ${k}_{s}$ and the constant B are subjected to a best fit procedure.
- the von Kármán parameter is assumed not universal but a fitting parameter and, as in scenario 2, the constant B is imposed equal to $8.5$ and the roughness scale ${k}_{s}$ is calculated from a roughness function.

#### 3.3. Discussion of the Values of the Parameters of the Log-Law

#### 3.4. Discussion of Bed Roughness

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Idealized physical system. ${Z}_{s}$ is the elevation of the free-surface, ${Z}_{c}$ and ${Z}_{t}$ are the space-averaged elevations of the planes of the crests and of the bed troughs, respectively. The bed amplitude is $\delta ={Z}_{c}-{Z}_{t}$. All elevations are relative to an arbitrary datum.

**Figure 2.**(

**a**) Scheme of the complete flume setup; (

**b**) general view of the flow over the mobile-bed reach; (

**c**) granular bed, prior to water working, showing the simple lattice arrangement.

**Figure 3.**Double-averaged longitudinal velocity profiles (

**a**) and space–time porosity ${\varphi}_{VT}({x}_{i},t)$ (

**b**). The reference zero in plot (

**a**) corresponds to the elevation of the particle crests (defined in the text). H is the flow depth measured from this zero. The reference zero in plot (

**b**) corresponds to the initial elevation of the particle crests for Test 1 (no bedload transport).

**Figure 4.**Shear rate and two-linear reaches identified respectively for Test 1 (

**a**); Test 2 (

**b**); Test 3 (

**c**); Test 4 (

**d**). The regression lines are represented by dashed and dotted lines (as identified in the legend). The bounds of the regression analysis that maximize the determination coefficient are marked with vertical red dashed lines. Blue lines represent the bounds associated with the minimum admissible coefficient of determination—$98\%$ of the maximum coefficient of determination.

**Figure 5.**Double-averaged longitudinal velocity profiles and regression lines for Scenario 1 ($\kappa $ = $0.405$), where ${z}_{*}$ = $(z-\Delta )/({k}_{s}-\Delta )$. The red vertical line represents the lower bound of the linear reach for all tests, whereas the black vertical lines define the upper bound determined for each test. Test 1 is identified by a solid line, Test 2 by a dash-dot line, Test 3 by a dashed line, while Test 4 by a dotted line.

**Figure 6.**Double-averaged longitudinal velocity profiles and regression lines for the computation of the scale of the roughness elements ${k}_{s}$, for Scenario 2 ($\kappa $ = $0.405$, B = $8.5$). The bounds of the regression lines are marked with the solid line (Test 1), dash-dot line (Test 2), dashed line (Test 3), and dotted line (Test 4).

**Figure 7.**Double-averaged longitudinal velocity profiles and regression lines for Scenario 3, where ${z}_{*}$ = $(z-\Delta )/({k}_{s}-\Delta )$. Vertical line specifications are as in Figure 5.The bounds of the regression lines are marked with solid line (Test 1), dash-dot line (Test 2), dashed line (Test 3), and dotted line (Test 4).

**Figure 8.**Double-averaged longitudinal velocity profiles and regression lines for the computation of the scale of the roughness elements ${k}_{s}$, for Scenario 4 ($\kappa <0.405$, B = $8.5$). The bounds of the regression lines are marked with solid line (Test 1), dash-dot line (Test 2), dashed line (Test 3), and dotted line (Test 4).

**Figure 9.**Variation of the roughness height normalized by the sediment diameter d, as a function of the non-dimensional bedload discharge, for scenario 3. Glass particles data are represented by red open diamonds. Data treated by Ferreira et al. [2] of type E are represented by black filled diamonds, type D by black open diamonds and type T by open circles.

Test | Q (m${}^{3}$/s) | H (m) | ${\mathit{i}}_{\mathit{b}}$ (-) | U (m/s) | ${\mathit{u}}_{*}^{\left(1\right)}$ (m/s) | ${\mathit{\tau}}_{\mathit{b}}^{\left(1\right)}$ (Pa) | ${\mathit{u}}_{*}^{\left(2\right)}$ (m/s) | ${\mathit{\tau}}_{\mathit{b}}^{\left(2\right)}$ (Pa) |
---|---|---|---|---|---|---|---|---|

1 | 0.0150 | 0.071 | 0.00317 | 0.518 | 0.041 | 1.64 | 0.041 | 1.68 |

2 | 0.0167 | 0.068 | 0.00456 | 0.602 | 0.048 | 2.29 | 0.048 | 2.26 |

3 | 0.0208 | 0.074 | 0.00623 | 0.691 | 0.058 | 3.33 | 0.056 | 3.15 |

4 | 0.0214 | 0.070 | 0.00714 | 0.757 | 0.060 | 3.63 | 0.061 | 3.73 |

Test | $\mathit{Fr}$ | $\mathit{Re}$ | ${\mathit{Re}}_{*}$ | $\mathit{\theta}$ | $\mathsf{\Phi}$ |
---|---|---|---|---|---|

1 | 0.62 | 41,405 | 227 | 0.023 | 0.0000 |

2 | 0.73 | 46,057 | 268 | 0.030 | 0.0007 |

3 | 0.81 | 57,571 | 323 | 0.042 | 0.0025 |

4 | 0.92 | 58,999 | 337 | 0.050 | 0.0034 |

Test | $\mathbf{\Delta}$ (m) | ${\mathit{k}}_{\mathit{s}}$ (m) | B (-) | ${\mathit{z}}_{0}$ (m) |
---|---|---|---|---|

1 | 0.0072 | 0.0336 | 14.7 | $8.7\times {10}^{-5}$ |

2 | 0.0020 | 0.0192 | 14.1 | $6.5\times {10}^{-5}$ |

3 | 0.0080 | 0.0420 | 14.5 | $1.2\times {10}^{-4}$ |

4 | 0.0037 | 0.0312 | 14.0 | $1.1\times {10}^{-4}$ |

Test | $\mathbf{\Delta}$ (m) | ${\mathit{k}}_{\mathit{s}}$ (m) | ${\mathit{z}}_{0}$ (m) |
---|---|---|---|

1 | 0.0072 | 0.0021 | $6.7\times {10}^{-5}$ |

2 | 0.0020 | 0.0018 | $5.8\times {10}^{-5}$ |

3 | 0.0080 | 0.0029 | $9.4\times {10}^{-4}$ |

4 | 0.0037 | 0.0030 | $9.6\times {10}^{-5}$ |

**Table 5.**Parameters describing the log-law for Scenario 3, where $\kappa $ is not considered universal.

Test | $\mathbf{\Delta}$ (m) | ${\mathit{k}}_{\mathit{s}}$ (m) | B (-) | $\mathit{\kappa}$ (-) | ${\mathit{z}}_{0}$ (m) |
---|---|---|---|---|---|

1 | $-0.0001$ | 0.0018 | 9.54 | 0.352 | $1.7\times {10}^{-4}$ |

2 | $-0.0002$ | 0.0036 | 10.26 | 0.350 | $1.0\times {10}^{-4}$ |

3 | $-0.0005$ | 0.0048 | 9.78 | 0.355 | $1.5\times {10}^{-4}$ |

4 | $-0.0007$ | 0.0060 | 10.41 | 0.305 | $2.5\times {10}^{-4}$ |

**Table 6.**Parameters describing the log-law for Scenario 4, where $\kappa $ is not considered universal and B = $8.5$.

Test | $\mathbf{\Delta}$ (m) | ${\mathit{k}}_{\mathit{s}}$ (m) | $\mathit{\kappa}$ (-) | ${\mathit{z}}_{0}$ (m) |
---|---|---|---|---|

1 | $-0.0001$ | 0.0041 | 0.352 | $2.1\times {10}^{-4}$ |

2 | $-0.0002$ | 0.0030 | 0.350 | $1.5\times {10}^{-4}$ |

3 | $-0.0005$ | 0.0062 | 0.355 | $3.0\times {10}^{-4}$ |

4 | $-0.0007$ | 0.0074 | 0.305 | $5.5\times {10}^{-4}$ |

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**MDPI and ACS Style**

Antico, F.; Ricardo, A.M.; Ferreira, R.M.L.
The Logarithmic Law of the Wall in Flows over Mobile Lattice-Arranged Granular Beds. *Water* **2019**, *11*, 1166.
https://doi.org/10.3390/w11061166

**AMA Style**

Antico F, Ricardo AM, Ferreira RML.
The Logarithmic Law of the Wall in Flows over Mobile Lattice-Arranged Granular Beds. *Water*. 2019; 11(6):1166.
https://doi.org/10.3390/w11061166

**Chicago/Turabian Style**

Antico, Federica, Ana M. Ricardo, and Rui M. L. Ferreira.
2019. "The Logarithmic Law of the Wall in Flows over Mobile Lattice-Arranged Granular Beds" *Water* 11, no. 6: 1166.
https://doi.org/10.3390/w11061166