# 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

- Townsend, A. The Structure of Turbulent Shear Flow; Cambridge University Press: Cambridge, UK, 1976. [Google Scholar]
- Ferreira, R.M.; Franca, M.J.; Leal, J.G.; Cardoso, A.H. Flow over rough mobile beds: Friction factor and vertical distribution of the longitudinal mean velocity. Water Resour. Res.
**2012**, 48, W05529. [Google Scholar] [CrossRef] - Bear, J. Dynamics of Fluids in Porous Media, 1st ed.; American Elsevier Publishing Company: New York, NY, USA, 1972. [Google Scholar]
- Gaudio, R.; Miglio, A.; Dey, S. Non-universality of von Kármán’s κ in fluvial streams. J. Hydraul. Res.
**2010**, 48, 658–663. [Google Scholar] [CrossRef] - Nikora, V.; Goring, D. Flow turbulence over fixed and weakly mobile gravel beds. J. Hydraul. Eng.
**2000**, 126, 679–690. [Google Scholar] [CrossRef] - Gaudio, R.; Miglio, A.; Calomino, F. Friction factor and von Kármán’s κ in open channels with bed-load. J. Hydraul. Res.
**2011**, 49, 239–247. [Google Scholar] [CrossRef] - Koll, K. Parameterisation of the vertical velocity profile in the wall region over rough surfaces. In Proceedings of the River Flow 2006, Lisbon, Portugal, 6–8 September 2006; pp. 163–172. [Google Scholar]
- Hanmaiahgari, P.R.; Roussinova, V.; Balachandar, R. Turbulence characteristics of flow in an open channel with temporally varying mobile bedforms. J. Hydrol. Hydromech.
**2017**, 65, 35–48. [Google Scholar] [CrossRef] - Ferreira, R.M. The von Kármán constant for flows over rough mobile beds. Lessons learned from dimensional analysis and similarity. Adv. Water Resour.
**2015**, 81, 19–32. [Google Scholar] [CrossRef] - Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Rodi, W. Turbulence modeling and simulation in hydraulics: A historical review. J. Hydraul. Eng.
**2017**, 143, 03117001. [Google Scholar] [CrossRef] - Kalitzin, G.; Medic, G.; Iaccarino, G.; Durbin, P. Near-wall behavior of RANS turbulence models and implications for wall functions. J. Comput. Phys.
**2005**, 204, 265–291. [Google Scholar] [CrossRef] - Fröhlich, J.; Von Terzi, D. Hybrid LES/RANS methods for the simulation of turbulent flows. Prog. Aerosp. Sci.
**2008**, 44, 349–377. [Google Scholar] [CrossRef] - Piomelli, U. Large eddy simulations in 2030 and beyond. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2014**, 372, 20130320. [Google Scholar] [CrossRef][Green Version] - Cannata, G.; Petrelli, C.; Barsi, L.; Camilli, F.; Gallerano, F. 3D free surface flow simulations based on the integral form of the equations of motion. WSEAS Trans. Fluid Mech.
**2017**, 12, 166–175. [Google Scholar] - Cea, L.; Puertas, J.; Vázquez-Cendón, M.E. Depth averaged modelling of turbulent shallow water flow with wet-dry fronts. Arch. Comput. Methods Eng.
**2007**, 14, 303–341. [Google Scholar] [CrossRef] - Williams, H.E.; Briganti, R.; Pullen, T. The role of offshore boundary conditions in the uncertainty of numerical prediction of wave overtopping using non-linear shallow water equations. Coast. Eng.
**2014**, 89, 30–44. [Google Scholar] [CrossRef] - Gallerano, F.; Cannata, G.; De Gaudenzi, O.; Scarpone, S. Modeling bed evolution using weakly coupled phase-resolving wave model and wave-averaged sediment transport model. Coast. Eng. J.
**2016**, 58, 1650011. [Google Scholar] [CrossRef] - Nezu, I.; Nakagawa, H.; Jirka, G.H. Turbulence in open-channel flows. J. Hydraul. Eng.
**1994**, 120, 1235–1237. [Google Scholar] [CrossRef] - Soulsby, R.; Dyer, K. The form of the near-bed velocity profile in a tidally accelerating flow. J. Geophys. Res. Oceans
**1981**, 86, 8067–8074. [Google Scholar] [CrossRef] - Stapleton, K.; Huntley, D. Seabed stress determinations using the inertial dissipation method and the turbulent kinetic energy method. Earth Surf. Processes Landf.
**1995**, 20, 807–815. [Google Scholar] [CrossRef] - Mendes, L.; Antico, F.; Sanches, P.; Alegria, F.; Aleixo, R.; Ferreira, R.M. A particle counting system for calculation of bedload fluxes. Meas. Sci. Technol.
**2016**, 27, 125305. [Google Scholar] [CrossRef] - Ferreira, R.M.; Aleixo, R. Experimental Hydraulics: Methods, Instrumentation, Data Processing and Management: Vol. II: Instrumentation and Measurement Techniques; CRC Press, Taylor and Francis Group: London, UK, 2017; Chapter 3; pp. 35–209. [Google Scholar]
- Antico, F. Laboratory Investigation on the Motion of Sediment Particles in Cohesionless Mobile Beds under Turbulent Flows. Ph.D. Thesis, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal, 2018. [Google Scholar]
- Nikora, V.; Ballio, F.; Coleman, S.; Pokrajac, D. Spatially averaged flows over mobile rough beds: Definitions, averaging theorems, and conservation equations. J. Hydraul. Eng.
**2013**, 139, 803–811. [Google Scholar] [CrossRef] - Ferreira, R.M.; Ferreira, L.M.; Ricardo, A.M.; Franca, M.J. Impacts of sand transport on flow variables and dissolved oxygen in gravel-bed streams suitable for salmonid spawning. River Res. Appl.
**2010**, 26, 414–438. [Google Scholar] [CrossRef] - Ferreira, R.M.; Franca, M.; Leal, J. Flow resistance in open-channel flows with mobile hydraulically rough beds. River Flow
**2008**, 1, 385–394. [Google Scholar] - Schlichting, H. Boundary-Layer Theory, 6th ed.; McGraw Hill: New York, NY, USA, 1968. [Google Scholar]
- Owen, P.R. Saltation of uniform grains in air. J. Fluid Mech.
**1964**, 20, 225–242. [Google Scholar] [CrossRef]

**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