# Kinematics of Particles at Entrainment and Disentrainment

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

## 3. The Reference Velocity

- Approach A: as the spatial average between the two velocity vectors in adjacent interrogation areas located above the top-center of the particle.
- Approach B: as the spatial average of velocity vectors of all interrogation windows directly above the particle (spanning its diameter) located at the same reference height of approach A.

## 4. Observations

#### 4.1. The Big Picture

#### 4.2. Local Observations of Entrainment

- i.
- inspect the performance of other definitions of reference velocity.
- ii.
- place resources on the experimental characterization of drag and lift on sediment particles, taking into account local unsteadiness brought about by turbulence; the study of the inertia of the boundary layer should deserve some attention as this may have a strong impact on lift and drag coefficients.
- iii.
- investigate how representative is entrainment due to fluid-particle momentum transfer, relatively to other forms of imparting momentum to bed particles, e.g., by particle-particle interactions.
- iv.
- investigate in what other ways the flow field can be modified in the vicinity of the entrained bed particle without affecting the reference velocity measured above it.

#### 4.3. Representative Types of Particle Entrainment

- A
- singular events associated with non-locally generated hydrodynamic actions.
- B
- singular events associated with locally generated hydrodynamic actions.
- C
- collective entrainment events due to particle-particle momentum transfer collision.
- D
- collective entrainment associated with strong fluid flow events.

#### 4.3.1. Type A: Non-Local Hydrodynamic Actions

#### 4.3.2. Type B: Locally Influenced Hydrodynamic Actions

#### 4.3.3. Type C: Collective Entrainment Due to Particle Collision

#### 4.3.4. Type D: Collective Entrainment Associated to Strong Fluid Flow Events

- Presence of high-speed gust mobilizing more than one particle at the same time.
- Collisions between travelling particles and sediments at rest promoting the motion of the latter.

#### 4.4. Disentrainment: Data Analysis and Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

FoV | Field of View |

PIV | Particle Image Velocimetry |

VLSM | Very Large-Scale Motion |

## References

- Meyer-Peter, E.; Müller, R. Formulas for bedload transport. In Proceedings of the 2nd Meeting of the International Association for Hydraulic Research, Stockholm, Sweden, 7–9 June 1948. [Google Scholar]
- Wong, M.; Parker, G. Reanalysis and Correction of Bed-Load Relation of Meyer-Peter and Mü ller Using Their Own Database. J. Hydraul. Eng.
**2006**, 132, 1159–1168. [Google Scholar] [CrossRef] [Green Version] - Smart, G.M. Sediment transport formula for steep channels. J. Hydraul. Eng.
**1984**, 110. [Google Scholar] [CrossRef] - Recking, A. An analysis of nonlinearity effects on bed load transport prediction. J. Geophys. Res. Earth Surf.
**2013**, 118, 1264–1281. [Google Scholar] [CrossRef] [Green Version] - Bagnold, R.A. An Approach to the Sediment Transport Problem from General Physics; USGS Numbered Series Professional Paper; U.S. Government Printing Office: Washington, DC, USA, 1966. [CrossRef] [Green Version]
- Engelund, F.; Hansen, E. A Monograph on Sediment Transport in Alluvial Stream; Teknisk Forlag: Copenhagen, Denmark, 1967. [Google Scholar]
- Rijn, L.C.V. Sediment transport part I bed load transport. J. Hydraul. Eng.
**1984**, 110. [Google Scholar] [CrossRef] - Wiberg, P.C.; Smith, J.D. Model for calculating bed load transport of sediment. J. Hydraul. Eng.
**1989**, 115. [Google Scholar] [CrossRef] - Einstein, H.A. Die eichung des im Rhein verwendeten geschiebefangers (The calibration of bed-load traps used in the Rhine). Schweiz. Bauztg.
**1937**, 110, 167–170. [Google Scholar] - Einstein, H.A. The Bed-Load Function for Sediment Transportation in Open Channel Flows; US Department of Agriculture: Washington, DC, USA, 1950; Volume 1026.
- Lakatos, I. Falsification and the methodology of scientific research programmes. In Can Theories Be Refuted? Essays on the Duhem-Quine Thesis; Springer: Berlin/Heidelberg, Germany, 1976. [Google Scholar] [CrossRef] [Green Version]
- Engelund, F.; Fredsoe, J. A sediment transport model for straight alluvial channels. Hydrol. Res.
**1976**. [Google Scholar] [CrossRef] [Green Version] - Cheng, N.S.; Chiew, Y.M. Pickup Probability for Sediment Entrainment. J. Hydraul. Eng.
**1998**, 124. [Google Scholar] [CrossRef] - Yalin, M.S. Mechanics of Sediment Transport; Pergamon Press: Oxford, UK, 1972. [Google Scholar]
- Dancey, C.L.; Balakrishnan, M.; Diplas, P.; Papanicolaou, A.N. The spatial inhomogeneity of turbulence above a fully rough, packed bed in open channel flow. Exp. Fluids
**2000**, 29, 402–410. [Google Scholar] [CrossRef] - Ferreira, R.M.; Hassan, M.A.; Ferrer-Boix, C. Principles of Bedload Transport of Non-cohesive Sediment in Open-Channels. In Rivers-Physical, Fluvial and Environmental Processes; Rowinsky, P., Radecki-Pawlick, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2015; Chapter 13; pp. 323–372. [Google Scholar]
- Paintal, A.S. A stochastic model of bed load transport. J. Hydraul. Res.
**1971**, 9, 527–554. [Google Scholar] [CrossRef] - Schmeeckle, M.W.; Nelson, J.M.; Shreve, R.L. Forces on stationary particles in near-bed turbulent flows. J. Geophys. Res. Earth Surf.
**2007**, 112. [Google Scholar] [CrossRef] - Valyrakis, M.; Diplas, P.; Dancey, C.L. Entrainment of coarse particles in turbulent flows: An energy approach. J. Geophys. Res. Earth Surf.
**2013**, 118, 42–53. [Google Scholar] [CrossRef] [Green Version] - Diplas, P.; Dancey, C.L.; Celik, A.O.; Valyrakis, M.; Greer, K.; Akar, T. The role of impulse on the initiation of particle movement under turbulent flow conditions. Science
**2008**, 322, 717–720. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chepil, W.S. Equilibrum of soil grains at the threshold of movement by wind. Soil Sci. Soc. Am. J.
**1959**, 23, 422–428. [Google Scholar] [CrossRef] - Leonardi, A.; Pokrajac, D.; Roman, F.; Zanello, F.; Armenio, V. Surface and subsurface contributions to the build-up of forces on bed particles. J. Fluid Mech.
**2018**, 851, 558–572. [Google Scholar] [CrossRef] [Green Version] - Rubey, W.W. The Force Required to Move Particles on a Stream Bed; Professional Papers; U.S. Geological Survey: Reston, VA, USA, 1938; pp. 121–141.
- Bridge, J.S.; Dominic, D.F. Bed Load Grain Velocities and Sediment Transport Rates. Water Resour. Res.
**1984**, 20, 476–490. [Google Scholar] [CrossRef] - Papanicolaou, A.; Diplas, P.; Evaggelopoulos, N.; Fotopoulos, S. Stochastic incipient motion criterion for spheres under various bed packing conditions. J. Hydraul. Eng.
**2002**, 128, 369–380. [Google Scholar] [CrossRef] - Recking, A. A comparison between flume and field bed load transport data and consequences for surface-based bed load transport prediction. Water Resour. Res.
**2010**, 46, W03518. [Google Scholar] [CrossRef] [Green Version] - Ancey, C.; Bohorquez, P.; Heyman, J. Stochastic interpretation of the advection diffusion equation and its relevance to bed load transport. J. Geophys. Res. Earth Surf.
**2015**, 120, 2529–2551. [Google Scholar] [CrossRef] [Green Version] - Furbish, D.J.; Fathel, S.L.; Schmeeckle, M.W. Particle Motions and Bedload Theory. In Gravel-Bed Rivers: Processes and Disasters; Wiley: Hoboken, NJ, USA, 2017; pp. 97–120. [Google Scholar]
- Houssais, M.; Ortiz, C.P.; Durian, D.J.; Jerolmack, D.J. Onset of sediment transport is a continuous transition driven by fluid shear and granular creep. Nat. Commun.
**2015**, 6, 1–8. [Google Scholar] [CrossRef] - Ancey, C.; Davison, A.; Böhm, T.; Jodeau, M.; Frey, P. Entrainment and motion of coarse particles in a shallow water stream down a steep slope. J. Fluid Mech.
**2008**, 595, 83–114. [Google Scholar] [CrossRef] - Cecchetto, M.; Tregnaghi, M.; Busolin, A.B.; Tait, S.; Marion, A. Statistical Description on theRole of Turbulence and Grain Interference on Particle Entrainment from Gravel Beds. J. Hydraul. Eng.
**2017**, 143, 06016021. [Google Scholar] [CrossRef] [Green Version] - Coleman, S.; Nikora, V.I. Fluvial dunes: Initiation, characterisation, flow structure. Earth Surf. Process. Landf.
**2011**, 36, 39–57. [Google Scholar] [CrossRef] - Furbish, D.J.; Haff, P.K.; Roseberry, J.C.; Schmeeckle, M.W. A probabilistic description of the bed load sediment flux: 1. Theory. J. Geophys. Res. Earth Surf.
**2012**, 117. [Google Scholar] [CrossRef] [Green Version] - Soares-Frazão, S.; Canelas, R.; Cao, Z.; Cea, L.; Chaudhry, H.M.; Moran, A.D.; el Kadi, K.; Ferreira, R.; Cadórniga, I.F.; Gonzalez-Ramirez, N.; et al. Dam-break flows over mobile beds: Experiments and benchmark tests for numerical models. J. Hydraul. Res.
**2012**, 50, 364–375. [Google Scholar] [CrossRef] - Andreotti, B.; Claudin, P.; Devauchelle, O.; Durán, O.; Fourriere, A. Bedforms in a turbulent stream: Ripples, chevrons and antidunes. J. Fluid Mech.
**2011**, 690, 94–128. [Google Scholar] [CrossRef] [Green Version] - Daubert, A.; Lebreton, J.C. Étude éxperimentale et sur modele mathematique de quelques aspects du calcul des processus d‘erosion des lits alluvionaires en regime permanent et non permanent. In Proceedings of the 12th Congress of IAHR, Fort Collins, CO, USA, 11–14 September 1967; Volume 3, pp. 26–37. [Google Scholar]
- Phillips, B.C.; Sutherland, A. Spatial lag effects in bed load sediment transport. J. Hydraul. Res.
**1989**, 27, 113–115. [Google Scholar] [CrossRef] - Charru, F.; Muilleron-Arnould, H.; Eiff, O. Erosion and deposition of particles on a bed sheared by a viscous flow. J. Fluid Mech.
**2004**, 519, 55–80. [Google Scholar] [CrossRef] - Canelas, R.; Murillo, J.; Ferreira, R.M.L. Two-dimensional depth-averaged modelling of dambreak flows over mobile beds. J. Hydraul. Res.
**2013**, 51, 392–407. [Google Scholar] [CrossRef] - Bohorquez, P.; Ancey, C. Stochastic-deterministic modeling of bed load transport in shallow water flow over erodible slope: Linear stability analysis and numerical simulation. Adv. Water Resour.
**2015**, 83, 36–54. [Google Scholar] [CrossRef] - Cecchetto, M.; Tait, S.; Tregnaghi, M.; Marion, A. The mechanics of bedload particles deposition over gravel beds. In Proceedings of the International Conference On Fluvial Hydraulics (River Flow 2016), St. Louis, MO, USA, 12–15 July 2016. [Google Scholar]
- Ancey, C. Bedload transport: A walk between randomness and determinism. Part 2. Challenges and prospects. J. Hydraul. Res.
**2020**, 58, 18–33. [Google Scholar] [CrossRef] [Green Version] - Kramer, H. Sand mixtures and sand movement in fluvial model. Am. Soc. Civ. Eng.
**1935**, 100, 873–878. [Google Scholar] - 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.L. 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] - Dwivedi, A.; Melville, B.W.; Shamseldin, A.Y.; Guha, T.K. Analysis of hydrodynamic lift on a bed sediment particle. J. Geophys. Res. Earth Surf.
**2011**, 116. [Google Scholar] [CrossRef] [Green Version] - Antico, F. Laboratory Investigations on the Motion of Sediment Particles in Cohesionless Mobile Beds under Turbulent Flows. Ph.D. Thesis, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal, 2019. [Google Scholar]
- Nakagawa, H.; Nezu, I. Prediction of the contributions to the Reynolds stress from bursting events in open-channel flows. J. Fluid Mech.
**1977**, 80, 99–128. [Google Scholar] [CrossRef] - 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] [Green Version] - Kim, K.C.; Adrian, R.J. Very large-scale motion in the outer layer. Phys. Fluids
**1999**, 11, 417–422. [Google Scholar] [CrossRef] - Schoklitsch, A. Handbuch des Wasserbaues; Springer: Berlin/Heidelberg, Germany, 1962. [Google Scholar]
- Gyr, A.; Schmid, A. The different ripple formation mechanisms. J. Hydraul. Res.
**1989**, 27, 61–74. [Google Scholar] [CrossRef] - Séchet, P.; Guennec, B.L. Bursting phenomenon and incipient motion of solid particles in bed-load transport. J. Hydraul. Res.
**1999**. [Google Scholar] [CrossRef] - Nelson, J.M.; Shreve, R.L.; McLean, S.R.; Drake, T.G. Role of near-bed turbulence structure in bed load transport and bed form mechanics. Water Resour. Res.
**1995**, 31, 2071–2086. [Google Scholar] [CrossRef] - Ancey, C.; Böhm, T.; Jodeau, M.; Frey, P. Statistical description of sediment transport experiments. Phys. Rev. E
**2006**, 74, 011302. [Google Scholar] [CrossRef] [PubMed] - Valyrakis, M.; Diplas, P.; Dancey, C.L.; Greer, K.; Celik, A.O. Role of instantaneous force magnitude and duration on particle entrainment. J. Geophys. Res. Earth Surf.
**2010**, 115. [Google Scholar] [CrossRef] - Drake, T.G.; Shreve, R.L.; Dietrich, W.E.; Whiting, P.J.; Leopold, L.B. Bedload transport of fine gravel observed by motion-picture photography. J. Fluid Mech.
**1988**, 192, 193–217. [Google Scholar] [CrossRef] [Green Version] - Böhm, T.; Ancey, C.; Frey, P.; Reboud, J.L.; Ducottet, C. Fluctuations of the solid discharge of gravity-driven particle flows in a turbulent stream. Phys. Rev. E
**2004**, 69, 061307. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**(

**a**) Scheme of the velocity vectors considered in the spatial average in approach A and B; velocity vectors averaged in approach A are reported in blue, while those averaged in approach B in yellow. (

**b**) PIV image with velocity vectors superimposed; velocity vectors averaged in approach B are reported in yellow.

**Figure 3.**Velocity time seriesfor test T2 at $d/2$ above the crests of the initial bed averaged in space over the length of the PIV FoV.

**Figure 4.**Sediment discharge time series measured during test T2 at the channel centerline with the bedload discharge meter described in Mendes et al. [44].

**Figure 5.**Normalized histogram of the lowpass filtered longitudinal velocity obtained at the reference elevation in test T2.

**Figure 6.**Normalized histogram of the bedload discharge, ${Q}_{s}$, in particles per second, registered, in test T2, across the central 10 cm of the flume.

**Figure 7.**Reference velocity fluctuations (red dots) characterizing: (

**a**) test T1-Approach A; (

**b**) test T1-Approach B. Contour lines represent the 2D histogram of the velocity time fluctuations at the reference height for the entire set of PIV images.

**Figure 8.**Reference velocity fluctuations (red dots) characterizing: (

**a**) test T2-Approach A; (

**b**) test T2-Approach B. Contour lines represent the 2D histogram of the velocity time fluctuations at the reference height for the entire set of PIV images.

**Figure 9.**Influence of particle bed topography on particle entrainment. Four cases are reported here, in which the particle at entrainment is identified by red contours and ${e}_{1}$ represents the particle exposure: in case (1) the particle at entrainment is characterized by greater exposure with respect to case (2); in case (3) the exposure is the same as in case (1), but the presence of an obstacle downstream hampers particle entrainment—for the same particle-exposure and fluid pressure a larger impulse is needed to overcome the potential energy wall created by the protruding downstream particle. Finally, at case (4), the particle has a negative exposure and a protruding downstream particle; it will require a strong lift force, sustained in time so that its work is able to overcome the potential energy wall.

**Figure 10.**Ratio between longitudinal velocity fluctuations obtained with approach B and the longitudinal double-averaged velocities (averaged first in time and then in space-along the velocity reference level) as a function of the ratio between particle exposure and particle diameter for: (

**a**) test T1; and (

**b**) test T2. Each entrainment event corresponds to a numbered open circle. Curves corresponding to the theoretical exposure model proposed by Ferreira et al. [16] are preented as red circles model for sliding instability (Equation (6)) and as blue diamonds for rolling instability (Equation (7)).

**Figure 11.**Particle entrained because of hydrodynamic forces (Type A; particle 12, Test T1). Sequence of three PIV images respectively showing (from (

**a**–

**c**)): at time $t={t}_{0}$ (2 time instants, corresponding to $0.1333$ s, before entrainment) the bright particle located in the centre of the image is at rest in the bed; at time $t={t}_{0}+\Delta t$ (1 time instant, corresponding to $0.067$ s, before entrainment) the particle is still at rest in the bed; at time $t={t}_{0}+2\Delta t$ the particle has already left the bed and is rolling. (For clarity only 1 out of 2 vectors are depicted.)

**Figure 12.**Instantaneous Reynolds shear stress maps with vectors superimposed representing velocity fluctuations obtained for Type A; (particle 12, Test T1) for the sequence of three PIV images reported in Figure 11. The sediment bed is masked in white. (For clarity only 1 out of 2 vectors are depicted.) (

**a**) At time $t={t}_{0}$ a sweep event (${u}^{\prime}>0$ and ${w}^{\prime}<0$), is clearly identifiable in blue on the middle/top and left side of the image, is approaching. The remaining of that is observed in the next instant (

**b**) on the upstream side of the particle about to be dislodged. In the next instant (

**c**) the particle has already been dislodged and parcels of fluid characterized by intense Reynolds stresses (in blue) at the location previously occupied by the particle.

**Figure 13.**Particle entrained because of particle passing-by (Type B; Particle 3, Test T2). Sequence of three PIV images respectively showing (from (

**a**–

**c**)): at time $t={t}_{0}$ (2 time instants, corresponding to $0.1333$ s, before entrainment) the bright particle located in the centre of the image is at rest in the bed; at time $t={t}_{0}+\Delta t$ (1 time instant, corresponding to 0.067 s, before entrainment) the particle is still at rest in the bed and a particle perturbating the flow field passes nearby; at time $t={t}_{0}+2\Delta t$ the particle starts its entrainment. (For clarity only 1 out of 2 vectors are depicted.)

**Figure 14.**Instantaneous Reynolds shear stress maps with vectors superimposed representing velocity fluctuations obtained for Type B (Particle 3, Test T2) for the sequence of three PIV images reported in Figure 13. (For clarity only 1 out of 2 vectors are depicted.) (

**a**) Instantaneous Reynolds stresses around particle at rest. (

**b**) Flow disturbed by passing-by particle. (

**c**) Instantaneous Reynolds stresses at the moment particle at rest is entrained.

**Figure 15.**Instantaneous velocity profiles used to compute the spatial mean velocity (solid black line) in the region of interest (Figure 14).

**Figure 16.**Particle entrained because of collective entrainment (Type C, Particle 27, Test T2). Sequence of three PIV images respectively showing (from (

**a**–

**c**)): at time $t={t}_{0}$ (2 time instants, corresponding to $0.1333$ s, before entrainment) the bright particle located in the center of the image is at rest in the bed; at time $t={t}_{0}+\Delta t$ (1 time instant, corresponding to $0.067$ s, before entrainment) the particle is still at rest in the bed and another particle is approaching; at time $t={t}_{0}$ + 2$\Delta $t the travelling particle collides with the particle at rest; the bright particle starts its entrainment. (For clarity only 1 out of 2 vectors are depicted.)

**Figure 17.**Instantaneous Reynolds shear stress maps with vectors superimposed representing velocity fluctuations obtained for Type C (Particle 27, Test T2) for the sequence of three PIV images reported in Figure 16. (For clarity only 1 out of 2 vectors are depicted.) (

**a**) Instantaneous Reynolds stresses and particle at rest. (

**b**) Instantaneous Reynolds stresses with particle at rest while moving particle is approaching. (

**c**) Instantaneous Reynolds stresses during particles’ collision.

**Figure 18.**Particle entrained because of collective entrainment (Type D, particle 4, Test T2). Sequence of three PIV images respectively showing (from (

**a**–

**c**)): at time $t={t}_{0}$ (2 time instants, corresponding to 0.1333 s, before entrainment) the bright particle located in the centre of the image is at rest in the bed (partially hidden by another particle); at time $t={t}_{0}+\Delta t$ (1 time instant, corresponding to $0.067$ s, before entrainment) the particle is still at rest in the bed and travelling particles are about to collide with sediments in the bed; at time $t={t}_{0}+2\Delta t$ several particles, included the bright one, are mobilized because of the impact. (For clarity only 1 out of 2 vectors are depicted.)

**Figure 19.**Instantaneous Reynolds shear stress maps with vectors superimposed representing velocity fluctuations obtained for Type D (particle 4, Test T2) for the sequence of three PIV images reported in Figure 18. (For clarity only 1 out of 2 vectors are depicted.) (

**a**) Initial instantaneous Reynolds stresses. A ejection event is observed close to the particle at rest. (

**b**) Induced instantaneous Reynolds stresses by passing particles. (

**c**) Several particles are mobilized and corresponding instantaneous Reynolds stresses.

**Figure 20.**Instantaneous velocity fluctuations characterizing test T1—Approach B. Red dots represent the turbulent flow field associated with deposited particles. Disentrainment events discussed later are marked with numbered open circles: particle 1 (black) and particle 3 (gray).

**Figure 21.**Particle deposited because of hydrodynamic forces (particle 1, Test T1). Sequence of three PIV images respectively showing (From (

**a**–

**c**)): at time $t={t}_{0}$ (2 time instants, corresponding to $0.1333$ s, before disentrainment) the particle is in motion within the field of view; at time $t={t}_{0}+\Delta t$ (1 time instant, corresponding to $0.067$ s, before disentrainment) the particle is approaching its rest location; at time $t={t}_{0}+2\Delta t$ the particle is at rest in the bed.

**Figure 22.**Instantaneous Reynolds shear stress maps with vectors superimposed representing velocity fluctuations obtained for disentrainment event number 1-Test T1 for the sequence of three PIV images reported in Figure 21. (

**a**) the moving particle starts decelerating and a ejection event is observed downstream of the particle location. (

**b**) The ejection event has passed and particle approaches its rest locatin. (

**c**) The particle is at rest.

**Figure 23.**Particle deposited because of the presence of a cluster (particle 3, Test T1). Sequence of three PIV images respectively showing (From (

**a**–

**c**)): at time $t={t}_{0}$ (2 time instants, corresponding to $0.1333$ s, before disentrainment) the particle is in motion within the field of view; at time $t={t}_{0}+\Delta t$ (1 time instant, corresponding to $0.067$ s, before disentrainment) the particle is approaching its rest location; at time $t={t}_{0}+2\Delta t$ the particle is at rest in the bed.(For clarity only 1 out of 2 vectors are depicted.)

**Figure 24.**Instantaneous Reynolds shear stress maps with vectors superimposed representing velocity fluctuations obtained for disentrainment event number 3-Test T1 for the sequence of three PIV images reported in Figure 23. (For clarity only 1 out of 2 vectors are depicted.) (

**a**) The moving particle enters the FoV. (

**b**) As it aproaches a cluster of particles the instantaneous Reynolds shear stresse are significantly higher, as seen by the blue area close to the cluster. Ejection events are observed. (

**c**) The moving particle stops due to the obstacle.

Test | Q | h | U | ${\mathit{u}}_{\mathbf{\ast}}$ | ${\mathit{\tau}}_{\mathit{b}}$ | ${\mathit{F}}_{\mathit{r}}$ | ${\mathit{R}}_{\mathit{e}}$ | $\mathit{R}{\mathit{e}}_{\mathbf{\ast}}$ | $\mathit{\theta}$ | $\mathbf{\Phi}$ |
---|---|---|---|---|---|---|---|---|---|---|

(m${}^{\mathbf{3}}$ s${}^{\mathbf{-}\mathbf{1}}$) | (m) | (m s${}^{\mathbf{-}\mathbf{1}}$) | (m s${}^{\mathbf{-}\mathbf{1}}$) | (Pa) | (-) | (-) | (-) | (-) | (-) | |

$T1$ | 0.01667 | 0.0684 | 0.6016 | 0.0475 | 2.2557 | 0.7345 | $4.61\times {10}^{4}$ | 267.63 | 0.0301 | 0.0007 |

$T2$ | 0.02135 | 0.0696 | 0.7574 | 0.0610 | 3.7250 | 0.9166 | $5.90\times {10}^{4}$ | 337.06 | 0.0497 | 0.0034 |

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## Share and Cite

**MDPI and ACS Style**

Aleixo, R.; Antico, F.; Ricardo, A.M.; Ferreira, R.M.L.
Kinematics of Particles at Entrainment and Disentrainment. *Water* **2020**, *12*, 2110.
https://doi.org/10.3390/w12082110

**AMA Style**

Aleixo R, Antico F, Ricardo AM, Ferreira RML.
Kinematics of Particles at Entrainment and Disentrainment. *Water*. 2020; 12(8):2110.
https://doi.org/10.3390/w12082110

**Chicago/Turabian Style**

Aleixo, Rui, Federica Antico, Ana M. Ricardo, and Rui M.L. Ferreira.
2020. "Kinematics of Particles at Entrainment and Disentrainment" *Water* 12, no. 8: 2110.
https://doi.org/10.3390/w12082110