Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T13:15:01.138Z Has data issue: false hasContentIssue false
  • English
  • Français

Shear-induced incipient motion of a single sphere on uniform substrates at low particle Reynolds numbers

Published online by Cambridge University Press:  20 July 2017

J. R. Agudo Open the ORCID record for J. R. Agudo [Opens in a new window] ,
C. Illigmann ,
G. Luzi ,
A. Laukart ,
A. Delgado  and
A. Wierschem
J. R. Agudo
Affiliation:
Institute of Fluid Mechanics, FAU Busan Campus, University of Erlangen-Nuremberg, 618-230 Busan, Republic of Korea
C. Illigmann
Affiliation:
Institute of Fluid Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), D-91058, Erlangen, Germany
G. Luzi
Affiliation:
Institute of Fluid Mechanics, FAU Busan Campus, University of Erlangen-Nuremberg, 618-230 Busan, Republic of Korea
A. Laukart
Affiliation:
Institute of Fluid Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), D-91058, Erlangen, Germany
A. Delgado
Affiliation:
Institute of Fluid Mechanics, FAU Busan Campus, University of Erlangen-Nuremberg, 618-230 Busan, Republic of Korea Institute of Fluid Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), D-91058, Erlangen, Germany
A. Wierschem*
Affiliation:
Institute of Fluid Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), D-91058, Erlangen, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the incipient motion of single spheres in steady shear flow on regular substrates at low particle Reynolds numbers. The substrate consists of a monolayer of regularly arranged fixed beads, in which the spacing between beads is varied resulting in different angles of repose and exposures of the particle to the main flow. The flow-induced forces and the level of flow penetration into the substrate are determined numerically. Since experiments in this range had shown that the critical Shields number is independent of inertia but strongly dependent on the substrate geometry, the particle Reynolds number was fixed to 0.01 in the numerical study. Numerics indicates that rolling motion is always preferred to sliding and that the flow penetration is linearly dependent on the spacing between the substrate particles. Besides, we propose an analytical model for the incipient motion. The model is an extension of Goldman’s classical result for a single sphere near a plain surface taking into account the angle of repose, flow orientation with respect to substrate topography and shielding of the sphere to the linear shear flow. The effective level of flow penetration is the only external parameter. The model, applied to triangular and quadratic arrangements with different spacings, is able to predict the dependence of the critical Shields number on the geometry and on the orientation of the substrate. The model is in very good agreement with numerical results. For well-exposed particles, we observed that the minimum critical Shields number for a certain angle of repose does not depend sensitively on the considered arrangement. At large angles of repose, as expected in fully armoured beds, the model is consistent with experimental results for erodible beds at saturated conditions.

JFM classification

Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Multiphase and Particle-laden Flows: Particle/fluid flow Geophysical and Geological Flows: Sediment transport
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 284 - 314
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agudo, J. R., Dasilva, S. & Wierschem, A. 2014 How do neighbors affect incipient particle motion in shear flows? Phys. Fluids 26, 053303.CrossRefGoogle Scholar
Agudo, J. R., Luzi, G., Han, J., Hwang, M., Lee, J. & Wierschem, A. 2017 Detection of particle motion using image processing with particular emphasis on rolling motion. Rev. Sci. Instrum. 88, 051805.CrossRefGoogle ScholarPubMed
Agudo, J. R. & Wierschem, A. 2012 Incipient motion of a single particle on regular substrates in laminar shear flow. Phys. Fluids 24, 093302.CrossRefGoogle Scholar
Ali, S. D. & Dey, S. 2016 Hydrodynamics of sediment threshold. Phys. Fluids 28, 075103.CrossRefGoogle Scholar
Amini, H., Sollier, E., Weaver, W. M. & Dicarlo, D. 2012 Intrinsic particle-induced lateral transport in microchannels. Proc. Natl Acad. Sci. USA 109, 11593–11598.CrossRefGoogle ScholarPubMed
Armanini, A. & Gregoretti, C. 2005 Incipient sediment motion at high slopes in uniform flow condition. Water Resour. Res. 41, 212431.CrossRefGoogle Scholar
Bleil, S., Marr, D. W. M. & Bechinger, C. 2006 Field-mediated sefl-asembly and actuation of highly parallel microfluidic devices. Appl. Phys. Lett. 88, 263515.CrossRefGoogle Scholar
Bravo, R., Ortiz, P. & Perez-Aparicio, J. L. 2014 Incipient sediment transport for non-cohesive landforms by the discrete element method (dem). Appl. Math. Model. 38, 1326–1337.CrossRefGoogle Scholar
Brayshaw, A. C., Lynne, E. F. & Reid, I. 1983 The hydrodynamics of particle clusters and sediment entrainment in coarse alluvial channels. Sedimentology 30, 137–143.CrossRefGoogle Scholar
Brooks, S. B. & Tozeren, A. 1996 Flow past an array of cells that are adherent to the bottom plate of a flow channel. Comput. Fluids 25, 741–757.CrossRefGoogle Scholar
Buffington, M. & Montgomery, D. R. 1997 A systematic analysis of eight decades of incipient motion studies, with special reference to gravel-bedded rivers. Water Resour. Res. 33, 1993–2029.CrossRefGoogle Scholar
Burdick, G. M., Berman, N. S. & Beaudoin, S. P. 2005 Hydrodynamic particle removal from surfaces, thin solid films. Thin Solid Films 488, 116–123.CrossRefGoogle Scholar
Chang, Y. 1939 Laboratory investigation of flume traction and transportation. Trans. Am. Soc. Civ. Engrs 104, 1246–1284.CrossRefGoogle Scholar
Chaoui, M. & Feuillebois, F. 2003 Creeping flow around a sphere in a shear flow close to a wall. Q. J. Mech. Appl. Maths 56, 381–410.CrossRefGoogle Scholar
Charru, F., Larrieu, E., Dupont, J. B. & Zenit, R. 2007 Motion of a particle near a rough wall in a viscous shear flow. J. Fluid Mech. 570, 431–453.CrossRefGoogle Scholar
Charru, F., Mouilleron, H. & Eiffel, O. 2004 Erosion and deposition of particles on a bed sheared by a viscous flow. J. Fluid Mech. 519, 55–80.CrossRefGoogle Scholar
Cheng, N. S. & Chiev, Y. M. 1998 Pick-up probability for sediment eintrainment. J. Hydraul. Engng 124, 232–235.CrossRefGoogle Scholar
Chepil, W. S. 1961 The use of spheres to measure lift and drag on wind-eroded soil grains. Proc. Soil Sci. Soc. Am. 25, 343–345.CrossRefGoogle Scholar
Chin, C. O. & Chiew, Y. M. 1993 Effect of bed surface-structure on spherical particle stability. J. Waterways Port Coast. Ocean Engng 27, 937–942.Google Scholar
Coleman, N. L. 1967 A theoretical and experimental study of drag and lift forces acting on a sphere resting on a hypothetical streambed. In Proceedings of 12th IAHR Congress (ed. Collins, F.), vol. 3, p. 185. Int. Assoc. of Hydraul. Eng. And Res..Google Scholar
Derksen, J. J. 2011 Simulations of granular bed erosion due to laminar shear flow near the critical shields number. Phys. Fluids 23, 113303.CrossRefGoogle Scholar
Derksen, J. J. & Larsen, R. A. 2011 Drag and lift forces on random assemblies of wall-attached spheres in low-Reynolds number shear flow. J. Fluid Mech. 673, 548–573.CrossRefGoogle Scholar
Dey, S. 1999 Sediment threshold. Appl. Math. Model. 23, 399–417.CrossRefGoogle Scholar
Dey, S., Das, R., Gaudio, R. & Bose, S. K. 2012 Turbulence in mobile-bed streams. Acta Geophys. 60, 1547–1588.CrossRefGoogle Scholar
Dey, S. & Debnath, K. 2000 Influence of streamwise bed slope on sediment threshold under stream flow. J. Irrigat. Drain Eng. 126, 255–263.CrossRefGoogle Scholar
Drake, G. D. & Calantoni, J. 2001 Discrete particle model for sheet flow transport in the nearshore. J. Geophys. Res. 106, 19859–19868.CrossRefGoogle Scholar
Durst, F. 2008 Fluid Mechanics: An Introduction to the Theory of Fluid Flows. Springer.CrossRefGoogle Scholar
El-Kareh, A. W. & Secomb, T. W. 1996 Stokes flow inpinging on a spherical cap on a plane wall. Q. J. Mech. Appl. Maths 49, 179–183.CrossRefGoogle Scholar
Fan, F. G., Soltani, M., Ahmadi, G. & Hart, S. C. 1997 Flow-induced resuspension of rigid-link fibers from surfaces. Aerosol Sci. Technol. 27, 97–115.CrossRefGoogle Scholar
Fenton, J. D. & Abbott, J. E. 1977 Initial movement of grains on a stream bed: the effect of relative protrusion. Proc. R. Soc. Lond. A. 352, 523–537.Google Scholar
Goharzadeh, A., Khalili, A. & Jorgensen, B. B. 2005 Transition layer thickness at a fluid-porous interface. Phys. Fluids 17, 057102.CrossRefGoogle Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall-ii. Chem. Engng Sci. 22, 653–660.CrossRefGoogle Scholar
Groh, C., Wierschem, A., Rehberg, A. & Aksel, N. 2008 Barchan dunes in two dimensions: Experimental tests of minimal models. Phys. Rev. E 78, 021304.Google ScholarPubMed
Guazzelli, E. & Morris, J. F. 2012 Prologue. In A Physical Indroduction to Suspension Dynamics. Cambridge University Press.Google Scholar
Happel, J. & Brenner, H. 1983 Introduction. In Low Reynolds Number Hydrodynamics. Martinus Nijhoff Publisher.CrossRefGoogle Scholar
Hermann, H. J. 2007 Dune formation. In Traffic and Granular Flow. Springer.Google Scholar
Hong, A., Tao, M. & Kudrolli, A. 2015 Onset of erosion of a granular bed in a channel driven by a fluid flow. Phys. Fluids 27, 013301.CrossRefGoogle Scholar
Ippen, A. T. & Eagleson, P. S.1955 A study of sediment sorting by waves shoaling on a plane beach. Tech. Rep. 63. Beach Erosion Board. Corps of Engineers.Google Scholar
James, C. S. 1990 Prediction of entrainment conditions for nonuniform, noncohesive sediments. J. Hydraul. Res. 28, 25–41.CrossRefGoogle Scholar
James, D. F. & Davis, A. M. J. 2001 Flow at the interface of a model fibrous porous medium. J. Fluid Mech. 26, 47–72.CrossRefGoogle Scholar
Julien, P. Y. 2010 Incipient motion. In Erosion and Sedimentation. Cambrdige University Press.CrossRefGoogle Scholar
King, M. R. & Leighton, D. T. 1997 Measurements of the inertial lift on a moving sphere in contact with a plane in a shear flow. Phys. Fluids 9, 1248–1255.CrossRefGoogle Scholar
Kirchner, J. W., Diertrich, W. E., Iseya, F. & Ikeda, H. 1990 The variability of critical shear stress, friction angle, and grain protrusion in water worked sediments. Sedimentology 37, 647–672.CrossRefGoogle Scholar
Krishnan, G. P. & Leighton, D. T. 1990 Inertial lift on a moving sphere in contact with a plane wall in a shear flow. Phys. Fluids 7, 2538–2545.CrossRefGoogle Scholar
Kudrolli, A., Scheff, D. & Allen, B. 2016 Critical shear rate and torque stability condition for a particle resting on a surface in a fluid flow. J. Fluid Mech. 808, 397–409.CrossRefGoogle Scholar
Kuhnle, R. 1993 Incipient motion of sand-gravel sediment mixtures. J. Hydraul. Engng 119, 1400–1415.CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1985 The lift on a small sphere touching a plane in the presence of a simple shear flow. Z. Angew. Math. Phys. 36, 174–178.CrossRefGoogle Scholar
Li, G., McKinley, G. H. & Ardekani, M. 2015 Dynamics of particle migration in channel flow of viscoelastic fluids. J. Fluid Mech. 785, 486–505.CrossRefGoogle Scholar
Ling, C. H. 1995 Criteria for incipient motion of spherical sediment particles. J. Hydraul. Engng 121, 472–478.CrossRefGoogle Scholar
Lobkovsky, A. E., Orpe, A. V., Molloy, R., Kudrolli, A. & Rothman, D. H. 2008 Erosion of a granular bed driven by laminar fluid flow. J. Fluid Mech. 605, 47–58.CrossRefGoogle Scholar
Loiseleux, T., Gondret, P., Rabaud, M. & Doppler, D. 2005 Onset of erosion and avalanche for an inclined granular bed sheared by a continuous laminar flow. Phys. Fluids 17, 103304.CrossRefGoogle Scholar
Mando, M. & Rosendahl, L. 2010 On the motion of non-spherical particle at high Reynolds number. Powder Technol. 202, 1–13.CrossRefGoogle Scholar
Mantz, P. A. 1977 Incipient transport of fine grains and flanks by fluids-extended shields diagram. J. Hydr. Engng Div. ASCE 103, 601–615.Google Scholar
Marsh, A. N. 2004 Comparison of methods for predicting incipient motion for sand beds. J. Hydr. Engng Div. ASCE 130, 616–621.CrossRefGoogle Scholar
Martino, R., Paterson, A. & Piva, M. 2009 Onset of motion of a partly hidden cylinder in a laminar shear flow. Phys. Rev. E 79, 036315.Google Scholar
McEwan, I. & Heald, J. 2001 Discrete particle modelling of entrainment from flat uniformly sized sediment beds. J. Hydraul. Engng 127, 588–597.CrossRefGoogle Scholar
Miller, R. L. & Byrne, R. J. 1966 The angle of repose for a single grain on a fixed rough bed. Sedimentology 6, 303–314.CrossRefGoogle Scholar
Mo, J., Zhengming, G., Zhipeng, L., Yuyun, B. & Derksen, J. J. 2015 Suspending a solid sphere in laminar inertial liquid flow-experiments and simulations. AIChE J. 61, 1455–1469.CrossRefGoogle Scholar
Morsi, S. A. & Alexander, A. J. 1972 An investigation of particle trajectories in two-phase flow systems. J. Fluid Mech. 55, 193–208.CrossRefGoogle Scholar
Mouilleron, H, Charru, F. & Eiff, O. 2009 Inside the moving layer of a sheared granular bed. J. Fluid Mech. 628, 229–239.CrossRefGoogle Scholar
Nezu, I. & Nakagawa, H. 1994 Turbulence in open-channel flows. J. Hydraul. Engng 10, 1235–1237.CrossRefGoogle Scholar
Nguyen, N. K. & Yoon, K. B. 2009 Facile organization of colloidal particles into large, perfect, one- and two- dimensional arrays by dry manual assembly on patterned substrates. J. Am. Chem. Soc. 131, 14228–14230.Google Scholar
Nikuradse, J. 1933 Laws of flow in rough pipes. VDI Forschungsheft 361.Google Scholar
Olayiwola, B. & Walzel, P. 2009 Effects of in-phase oscillation of retentate and filtrate crossflow filtration at low Reynolds number. J. Membr. Sci. 345, 36–46.CrossRefGoogle Scholar
O’Neill, M. E. 1968 A sphere in contact with a plane wall in a slow linear shear flow. Chem. Engng Sci. 23, 1293–1298.CrossRefGoogle Scholar
Ouriemi, M., Aussillous, P. & Guazzelli, E. 2009 Sediment dynamics. Part 1. Bed-load transport by laminar shearing flows. J. Fluid Mech. 636, 295–319.CrossRefGoogle Scholar
Ouriemi, M., Aussillous, P., Medale, M., Peysoon, Y. & Guazzelli, E. 2007 Determination of the critical shields number for particle erosion in laminar flow. Phys. Fluids 19, 061706.CrossRefGoogle Scholar
Paintal, A. S. 1971 A stochastic model of bed load transport. J. Hydraul. Res. 9, 527–554.CrossRefGoogle Scholar
Papanicolaou, A. N., Diplas, P., Evaggelopoulos, N. & Fotopoulos, S. 2002 Stochastic incipient motion criterion for spheres under various bed packing conditions. J. Hydraul. Engng 128, 369–380.CrossRefGoogle Scholar
Pokrajac, D., Manes, C. & McEwan, I. 2007 Peculiar mean velocity profiles within a porous bed of an open channel. Phys. Fluids 19, 098109.CrossRefGoogle Scholar
Price, T. C. 1985 Slow linear shear flow past a hemispherical bum in a plane wall. Q. J. Mech. Appl. Maths 38, 93–104.CrossRefGoogle Scholar
van Rijn, L. C. 1984 Sediment transport, part I: Bed load transport. J. Hydraul. Engng 100, 1431–1456.CrossRefGoogle Scholar
Roache, P. J. 1994 Perspecitve: A method for uniform reporting of grid refinement studies. J. Fluids Engng 116, 405–413.CrossRefGoogle Scholar
Roache, P. J. 1997 Quantification of uncertainty in computational fluid dynamics. Annu. Rev. Fluid Mech. 29, 123–160.CrossRefGoogle Scholar
Roache, P. J. 1998 Verification and Validation in Computational Science and Engineering. Hermoas Publishers.Google Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385–400.CrossRefGoogle Scholar
Sawetzki, T., Rahmouni, S., Bechinger, C. & Marr, D. W. M. 2008 In situ assembly of linked geometrically coupled microdevices. Proc. Natl Acad. Sci. USA 105, 20141.CrossRefGoogle ScholarPubMed
Seizilles, G., Devauchelle, O., Lajeunesse, E. & Metivier, F. 2013 Width of laminar laboratory rivers. Phys. Rev. E 87, 052204.Google ScholarPubMed
Seizilles, G., Lajeunesse, E., Devauchelle, O. & Bak, M. 2014 Cross-stream diffusion in bedload transport. Phys. Fluids 26, 013302.CrossRefGoogle Scholar
Shields, A. 1936 Anwendungen der aehnlichkeitsmechanik und der turbulenzforschung auf die geschiebebewegung. Mitteilungen der Preussischen Versuchsanstalt fuer Wasserbau und Schiffbau 26.Google Scholar
Soepyan, F. B., Cremaschi, S., McLaury, B. S., Sarica, C., Subramani, H. J., Kouba, G. E. & Gao, H. 2016 Threshold velocity to initiate motion in horizontal and near-horizontal conduits. Powder Technol. 292, 272–289.CrossRefGoogle Scholar
Stevanovic, V. D., Stanojevic, M. M., Jovovic, A., Radic, D. B., Petrovic, M. M. & Karlicic, N. V. 2014 Analysis of transient ash pneumatic conveying over long distance and prediction of transport capacity. Powder Technol. 254, 281–290.CrossRefGoogle Scholar
Sugiyama, K. & Sbragaglia, M. 2008 Linear shear flow past a hemispherical droplet adhering to a solid surface. J. Engng Maths 62, 35–50.CrossRefGoogle Scholar
Thompson, J. A. & Bau, H. H. 2010 Microfluidic, bead-based assay: theory and experiments. J. Chromatogr. B 878, 228–236.CrossRefGoogle ScholarPubMed
Valyrakis, M., Diplas, P., Dancey, C. L., Greer, K. & Celik, A. O. 2010 Role of instantaneous force magnitude and duration on particle entrainment. J. Geophys. Res. 115, F02006.CrossRefGoogle Scholar
Vollmer, S. & Kleinhans, M. G. 2007 Predicting incipient motion, including the effect of turbulent pressure fluctuations in the bed. Water Resour. Res. 43, W05410.CrossRefGoogle Scholar
White, C. M. 1940 The equilibrium of grains on the bed of a stream. Proc. R. Soc. Lond. A 174, 322–338.Google Scholar
Whitehouse, R. J. S. & Hardisty, J. 1988 Experimental assesment of two theories for the effect of bedslope on the threshold of bedload transport. Mar. Geol. 79, 135–139.CrossRefGoogle Scholar
Wiberg, P. L. & Smith, J. D. 1987 Calculations of the critical shear-stress for motion of uniform and heterogeneous sediments. Water Resour. Res. 23, 1471–1480.CrossRefGoogle Scholar
Wierschem, A., Groh, C., Rehberg, I., Aksel, N. & Kruelle, C. A. 2008 Ripple formation in weakly turbulent flow. Eur. Phys. J. E 25, 213–221.Google ScholarPubMed
Wu, F. C. & Chou, Y. J. 2003 Rolling and lifting probabilities for sediment entrainment. J. Hydraul. Engng 129, 110–119.CrossRefGoogle Scholar
Yalin, M. S. & Karahan, E. 1979 Inception of sediment transport. J. Hydraul. Engng Div. ASCE 105, 1433–1443.Google Scholar
Yergey, B., Beninati, M. L. & Marshall, J. S. 2010 Sensitivity of incipient motion to fluid flow penetration depth within a packed bed. Sedimentology 57, 418–428.CrossRefGoogle Scholar
Yin, Y., Lu, Y., Gates, B. & Xia, Y. 2001 Template-assisted self-assembly: a practical route to complex aggregates of monodispersed colloids with well-defined sizes, shapes, and structures. J. Am. Chem. Soc. 123, 8718–8729.CrossRefGoogle ScholarPubMed