Skip to main content Accessibility help
×
Home
Hostname: page-component-79b67bcb76-c2bf7 Total loading time: 0.197 Render date: 2021-05-14T05:50:02.526Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Distribution of gyrotactic micro-organisms in complex three-dimensional flows. Part 1. Horizontal shear flow past a vertical circular cylinder

Published online by Cambridge University Press:  06 August 2018

L. Zeng
Affiliation:
State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
T. J. Pedley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Corresponding

Abstract

As a first step towards understanding the distribution of swimming micro-organisms in flowing shallow water containing vegetation, we formulate a continuum model for dilute suspensions in horizontal shear flow, with a maximum Reynolds number of 100, past a single, rigid, vertical, circular cylinder that extends from a flat horizontal bed and penetrates the free water surface. A numerical platform was developed to solve this problem, in four stages: first, a scheme for computation of the flow field; second, a solver for the Fokker–Planck equation governing the probability distribution of the swimming direction of gyrotactic cells under the combined action of gravity, ambient vorticity and rotational diffusion; third, the construction of a database for the mean swimming velocity and the translational diffusivity tensor as functions of the three vorticity components, using parameters appropriate for the swimming alga, Chlamydomonas nivalis; fourth, a solver for the three-dimensional concentration distribution of the gyrotactic micro-organisms. Upstream of the cylinder, the cells are confined to a vertical strip of width equal to the cylinder diameter, which enables us to visualise mixing in the wake. The flow downstream of the cylinder is divided into three zones: parallel vortex shedding in the top zone near the water surface, oblique vortex shedding in the middle zone and quasi-steady flow in the bottom zone. Secondary (vertical) flow occurs just upstream and downstream of the cylinder. Frequency spectra of the velocity components in the wake of the cylinder show two dominant frequencies of vortex shedding, in the parallel- and oblique-shedding zones respectively, together with a low frequency, equal to the difference between those two frequencies, that corresponds to a beating modulation. The concentration distribution is calculated for both active particles and passive, non-swimming, particles for comparison. The concentration distribution is very similar for both active and passive particles, except near the top surface, where upswimming causes the concentration of active particles to reach values greater than in the upstream strip, and in a thin boundary layer on the downstream surface of the cylinder, where a high concentration of active particles occurs as a result of radial swimming.

Type
JFM Papers
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below.

References

Augustin, L. N., Irish, J. L. & Lynett, P. J. 2009 Laboratory and numerical studies of wave damping by emergent and near-emergent wetland vegetation. Coast. Engng 56, 332340.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bearon, R. N., Bees, M. A. & Croze, O. A. 2012 Biased swimming cells do not disperse in pipes as tracers: a population model based on microscale behaviour. Phys. Fluids 24, 121902.CrossRefGoogle Scholar
Bearon, R. N., Hazel, A. L. & Thorn, G. J. 2011 The spatial distribution of gyrotactic swimming micro-organisms in laminar flow fields. J. Fluid Mech. 680, 602635.CrossRefGoogle Scholar
Bees, M. A. & Hill, N. A. 1998 Linear bioconvection in a suspension of randomly swimming, gyrotactic micro-organisms. Phys. Fluids 10, 18641881.CrossRefGoogle Scholar
Bees, M. A., Hill, N. A. & Pedley, T. J. 1998 Analytical approximations for the orientation distribution of small dipolar particles in steady shear flows. J. Math. Biol. 36, 269298.CrossRefGoogle Scholar
Bourguet, R., Karniadakis, G. E. & Triantafyllou, M. S. 2011 Vortex-induced vibrations of a long flexible cylinder in shear flow. J. Fluid Mech. 677, 342382.CrossRefGoogle Scholar
Bratbak, G., Egge, J. K. & Heldal, M. 1993 Viral mortality of the marine alga Emiliania huxleyi (haptophyceae) and termination of algal blooms. Marine Ecol. Progr. Ser. 93, 3948.CrossRefGoogle Scholar
Canuto, D. & Taira, K. 2015 Two-dimensional compressible viscous flow around a circular cylinder. J. Fluid Mech. 785, 349371.CrossRefGoogle Scholar
Chen, G. Q., Zeng, L. & Wu, Z. 2010 An ecological risk assessment model for a pulsed contaminant emission into a wetland channel flow. Ecol. Model. 221, 29272937.CrossRefGoogle Scholar
Costanza, R., d’Arge, R., de Groot, R., Farber, S., Grasso, M., Hannon, B., Limburg, K., Naeem, S., O’Neill, R. V., Paruelo, J., Raskin, R. G., Sutton, P. & van den Belt, M. 1997 The value of the world’s ecosystem services and natural captial. Nature 387, 253260.CrossRefGoogle Scholar
Cronk, J. K. & Fennessy, M. S. 2001 Wetland Plants Biology and Ecology. CRC Press LLC.CrossRefGoogle Scholar
Croze, O. A., Bearon, R. N. & Bees, M. A. 2017 Gyrotactic swimmer dispersion in pipe flow: testing the theory. J. Fluid Mech. 816, 481506.CrossRefGoogle Scholar
Croze, O. A., Sardina, G., Ahmed, M., Bees, M. A. & Brandt, L. 2013 Dispersion of swimming algae in laminar and turbulent channel flows: consequences for photobioreactors. J. R. Soc. Interface 10, 20121041.CrossRefGoogle ScholarPubMed
De Lillo, F., Cencini, M., Durham, W. M., Barry, M., Stocker, R., Climent, E. & Boffetta, G. 2014 Turbulent fluid acceleration generates clusters of gyrotactic microorganisms. Phys. Rev. Lett. 112, 044502.CrossRefGoogle ScholarPubMed
Dean, R. G. & Bender, C. J. 2006 Static wave setup with emphasis on damping effects by vegetation and bottom friction. Coast. Engng 53, 149156.CrossRefGoogle Scholar
Durbin, P. A. & Medic, G. 2007 Fluid Dynamics with a Computational Perspective. Cambridge University Press.CrossRefGoogle Scholar
Durham, W. M., Climent, E., Barry, M., Lillo, F. D., Cencini, M., Boffetta, G. & Stocker, R. 2013 Turbulence drives microscale patches of motile phytoplankton. Nat. Commun. 4, 3148.CrossRefGoogle ScholarPubMed
Durham, W. M., Kessler, J. O. & Stocker, R. 2009 Disruption of vertical motility by shear triggers formation of thin phytoplankton layers. Science 323, 10671070.CrossRefGoogle ScholarPubMed
Durham, W. M. & Stocker, R. 2012 Thin phytoplankton layers: characteristics, mechanisms, and consequences. Annu. Rev. Mar. Sci. 4, 177207.CrossRefGoogle ScholarPubMed
Ezhilan, B., Shelley, M. J. & Saintillan, D. 2013 Instabilities and nonlinear dynamics of concentrated active suspensions. Phys. Fluids 25, 070607.CrossRefGoogle Scholar
Feagin, R. A., Irish, J. L., Möller, I., Williams, A. M., Colón-Rivera, R. J. & Mousavi, M. E. 2011 Short communication: engineering properties of wetland plants with application to wave attenuation. Coast. Engng 58, 251255.CrossRefGoogle Scholar
Fornberg, B. 1985 Steady viscous flow past a circular cylinder up to Reynolds number 600. J. Comput. Phys. 61, 297320.CrossRefGoogle Scholar
Ghorai, S. & Hill, N. A. 2007 Gyrotactic bioconvection in three dimensions. Phys. Fluids 19, 054107.CrossRefGoogle Scholar
Griffin, O. M. 1985 Vortex shedding from bluff bodies in a shear flow: a review. Trans. ASME J. Fluids Engng 107, 298306.CrossRefGoogle Scholar
Hamann, E. & Puijalon, S. 2013 Biomechanical responses of aquatic plants to aerial conditions. Ann. Bot. 112, 18691878.CrossRefGoogle ScholarPubMed
Hill, N. A. & Bees, M. A. 2002 Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow. Phys. Fluids 14, 25982605.CrossRefGoogle Scholar
Hill, N. A. & Pedley, T. J. 2005 Bioconvection. Fluid Dyn. Res. 37, 120.CrossRefGoogle Scholar
Hopkins, M. M. & Fauci, L. J. 2002 A computational model of the collective fluid dynamics of motile micro-organisms. J. Fluid Mech. 455, 149174.CrossRefGoogle Scholar
Hwang, Y. & Pedley, T. J. 2014a Bioconvection under uniform shear: linear stability analysis. J. Fluid Mech. 738, 522562.CrossRefGoogle Scholar
Hwang, Y. & Pedley, T. J. 2014b Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel. J. Fluid Mech. 749, 750757.CrossRefGoogle Scholar
Ishikawa, T., Locsei, J. T. & Pedley, T. J. 2008 Development of coherent structures in concentrated suspensions of swimming model micro-organisms. J. Fluid Mech. 615, 401431.CrossRefGoogle Scholar
Ishikawa, T., Pedley, T. J. & Yamaguchi, T. 2007 Orientational relaxation time of bottom-heavy squirmers in a semi-dilute suspension. J. Theor. Biol. 249, 296306.CrossRefGoogle Scholar
Ishikawa, T., Simmonds, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119160.CrossRefGoogle Scholar
Jiang, H., Cheng, L., Draper, S., An, H. & Tong, F. 2016 Three-dimensional direct numerical simulation of wake transitions of a circular cylinder. J. Fluid Mech. 801, 353391.CrossRefGoogle Scholar
Kappler, M., Rodi, W., Szepessy, S. & Badran, O. 2005 Experiments on the flow past long circular cylinders in a shear flow. Exp. Fluids 38, 269284.CrossRefGoogle Scholar
Kessler, J. O. 1985 Hydrodynamic focusing of motile algal cells. Nature 313, 218220.CrossRefGoogle Scholar
Kessler, J. O. 1986 Individual and collective fluid dynamics of swimming cells. J. Fluid Mech. 173, 191205.CrossRefGoogle Scholar
Lega, J. & Passot, T. 2003 Hydrodynamics of bacterial colonies: a model. Phys. Rev. E 67, 031906.Google ScholarPubMed
Leonard, L. A. & Luther, M. E. 1995 Flow hydrodynamics in tidal marsh canopies. Limnol. Oceanogr. 40, 14741484.CrossRefGoogle Scholar
Lightbody, A. F. & Nepf, H. M. 2006 Prediction of velocity profiles and longitudinal dispersion in emergent salt marsh vegetation. Limnol. Oceanogr. 51, 218228.CrossRefGoogle Scholar
Luhar, M., Coutu, S., Infantes, E., Fox, S. & Nepf, H. 2010 Wave-induced velocities inside a model seagrass bed. J. Geophys. Res. 115, C12005.CrossRefGoogle Scholar
Manela, A. & Frankel, I. 2003 Generalized Taylor dispersion in suspensions of gyrotactic swimming micro-organisms. J. Fluid Mech. 490, 99127.CrossRefGoogle Scholar
Maull, D. J. & Young, R. A. 1973 Vortex shedding from bluff bodies in a shear flow. J. Fluid Mech. 60, 401409.CrossRefGoogle Scholar
Millie, D. F., Schofield, O. M., Kirkpatrick, G. J., Johnsen, G., Tester, P. A. & Vinyard, B. T. 1997 Detection of harmful algal blooms using photopigments and absorption signatures: a case study of the florida red tide dinoflagellate Gymnodinium breve . Limnol. Oceanogr. 42, 12401251.CrossRefGoogle Scholar
Möller, I. 2006 Quantifying saltmarsh vegetation and its effect on wave height dissipation: results from a UK East coast saltmarsh. Estuar. Coast. Shelf Sci. 69, 337351.CrossRefGoogle Scholar
Mukhopadhyay, A., Venugopal, P. & Vanka, S. P. 1999 Numerical study of vortex shedding from a circular cylinder in linear shear flow. Trans. ASME J. Fluids Engng 121, 460468.CrossRefGoogle Scholar
Mukhopadhyay, A., Venugopal, P. & Vanka, S. P. 2002 Oblique vortex shedding from a circular cylinder in linear shear flow. Comput. Fluids 31, 124.CrossRefGoogle Scholar
Nepf, H. M. 2012 Flow and transport in regions with aquatic vegetation. Annu. Rev. Fluid Mech. 44, 123142.CrossRefGoogle Scholar
Park, J., Kwon, K. & Choi, H. 1998 Numerical solution of flow past a circular cylinder at Reynolds number up to 160. KSME Intl J. 12, 12001205.CrossRefGoogle Scholar
Pedley, T. J. 2010 Instability of uniform microorganism suspensions revisited. J. Fluid Mech. 647, 335359.CrossRefGoogle Scholar
Pedley, T. J. & Kessler, J. O. 1987 The orientation of spheroidal microorganisms swimming in a flow field. Proc. R. Soc. Lond. B 231, 4770.CrossRefGoogle Scholar
Pedley, T. J. & Kessler, J. O. 1990 A new continuum model for suspensions of gyrotactic micro-organisms. J. Fluid Mech. 212, 155182.CrossRefGoogle ScholarPubMed
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming micro-organisms. Annu. Rev. Fluid Mech. 24, 313358.CrossRefGoogle Scholar
Smayda, T. J. 1997 Harmful algal blooms: their ecophysiology and general relevance to phytoplankton blooms in the sea. Limnol. Oceanogr. 42, 11371153.CrossRefGoogle Scholar
Stansby, P. K. 1976 The locking-on of vortex shedding due to the cross-stream vibration of circular cylinders in uniform and shear flows. J. Fluid Mech. 74, 641665.CrossRefGoogle Scholar
Tanino, Y. & Nepf, H. M. 2008 Lateral dispersion in random cylinder arrays at high Reynolds number. J. Fluid Mech. 600, 339371.CrossRefGoogle Scholar
Thorn, G. J. & Bearon, R. N. 2010 Transport of gyrotactic organisms in general three-dimensional flow fields. J. Fluid Mech. 22, 041902.Google Scholar
Williams, C. R. & Bees, M. A. 2011 Photo-gyrotactic bioconvection. J. Fluid Mech. 678, 4186.CrossRefGoogle Scholar
Williamson, C. H. K. 1989 Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 206, 579627.CrossRefGoogle Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477539.CrossRefGoogle Scholar
Williamson, C. H. K. & Govardhan, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413455.CrossRefGoogle Scholar
Wolgemuth, C. W. 2008 Collective swimming and the dynamics of bacterial turbulence. Biophys. J. 95, 15641574.CrossRefGoogle ScholarPubMed
Woo, H. G. C., Cermak, J. E. & Peterka, J. A. 1989 Secondary flows and vortex formation around a circular cylinder in constant-shear flow. J. Fluid Mech. 204, 523542.CrossRefGoogle Scholar
Zeng, L., Zhao, Y. J., Chen, B., Ji, P., Wu, Y. H. & Feng, L. 2014 Longitudinal spread of bicomponent contaminant in wetland flow dominated by bank-wall effect. J. Hydrol. 509, 179187.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Distribution of gyrotactic micro-organisms in complex three-dimensional flows. Part 1. Horizontal shear flow past a vertical circular cylinder
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Distribution of gyrotactic micro-organisms in complex three-dimensional flows. Part 1. Horizontal shear flow past a vertical circular cylinder
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Distribution of gyrotactic micro-organisms in complex three-dimensional flows. Part 1. Horizontal shear flow past a vertical circular cylinder
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *