Longitudinal dispersion of multiple Microcystis patches in a turbulent open-channel flow

Abstract

Longitudinal dispersion of multiple Microcystis patches is significant for understanding the mechanisms of Microcystis blooms in ambient waters. Presented in this paper is an analysis of multiple Microcystis patches in a turbulent open channel flow, based on the concentration transport equation. Results show that the total amount of Microcystis on a streamline, centre-of-mass motion, as well as longitudinal dispersion finally stabilise for multiple Microcystis patches as they do for a single Microcystis patch. The centre-of-mass velocity exhibits a weak oscillation with a small amplitude in the vertical direction for multiple Microcystis patches, but it can reach the same migration velocity for long time evolution. The large

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results in an asymptotic increase of centre-of-mass velocity towards a positive value except for very early time for the Microcystis patches in close proximity to the free surface, and an initial reduction to zero, subsequent increase to a maximum, and final decrease to a constant for the bed bottom, where

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represents the relative strength of vertical swimming and the total effective diffusion in the vertical direction. For the case of large

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, the longitudinal dispersion coefficient first increases to a maximum, then decrease to a minimum, and finally increases to a constant for the Microcystis patches near the free surface. The effects of initial distribution on the evolution of the concentration of Microcystis patch mainly appear during the initial stage and can be negligible for the long time evolution. The Microcystis patches tend to accumulate near the free surface due to the vertical migration velocity caused by density differences between the Microcystis and the ambient water.

Keywords

Active particles / Longitudinal dispersion / Open-channel flow / Turbulence

References

[1] Aris R. On the dispersion of a solute in a fluid flowing through a tube. Proc R Soc Lond A, 1956, 235(1200): 67-77

[2] Chatwin PC. The cumulants of the distribution of concentration of a solute dispersing in solvent flowing through a tube. J Fluid Mech, 1972, 51(1): 63-67

[3] Croze OA, Bearon RN, Bees MA. Gyrotactic swimmer dispersion in pipe flow: testing the theory. J Fluid Mech, 2017, 816: 481-506

[4] Elder JW. The dispersion of marked fluid in turbulent shear flow. J Fluid Mech, 1959, 5(1): 63-67

[5] Gill WN. A note on the solution of transient dispersion problems. Proc R Soc Lond A, 1967, 298(1454): 335-339

[6] Gill WN, Sankarasubramanian R. Exact analysis of unsteady convective diffusion. Proc R Soc Lond A, 1970, 316(1526): 341-350

[7] Guan MY, Zeng L, Li CF Transport model of active particles in a tidal wetland flow. J Hydrol, 2021, 593(125): 812

[8] Guo J, Jiang W, Chen G Solute dispersion in an open channel turbulent flow: solution by a generalized model. J Hydrol, 2022, 604(127): 239

[9] Hill NA, Bees MA. Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow. Phys Fluids, 2002, 14(8): 2598-2605

[10] Huai WX, Zhang J, Wang WJ Turbulence structure in open channel flow with partially covered artificial emergent vegetation. J Hydrol, 2019, 573: 180-193

[11] Huisman J, Codd GA, Paerl HW Cyanobacterial blooms. Nat Rev Microbiol, 2018, 16(8): 471-483

[12] Jiang W, Chen G. Dispersion of active particles in confined unidirectional flows. J Fluid Mech, 2019, 877: 1-34

[13] Li M, Xiao M, Zhang P Morphospecies-dependent disaggregation of colonies of the cyanobacterium microcystis under high turbulent mixing. Water Res, 2018, 159: 340-348

[14] Maier RS, Kroll DM, Bernard RS Enhanced dispersion in cylindrical packed beds. Phil Trans R Soc A, 2002, 360: 497-506

[15] Mei CC, Auriault JL, Ng CO. Hutchinson JW, Wu TY. Some applications of the homogenization theory. Advances in applied mechanics, 1996 Elsevier 277-348

[16] Murphy E, Ghisalberti M, Nepf H. Model and laboratory study of dispersion in flows with submerged vegetation. Water Resour Res, 2007, 43(5): W05,438

[17] Nepf H, Ghisalberti M, White B Retention time and dispersion associated with submerged aquatic canopies. Water Resour Res, 2007, 43(4): W04,422

[18] Nepf HM. Drag, turbulence, and diffusion in flow through emergent vegetation. Water Resour Res, 1999, 35(2): 479-489

[19] Nepf HM. Flow and transport in regions with aquatic vegetation. Annu Rev Fluid Mech, 2012, 44(1): 123-142

[20] Ng CO. Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions. Proc R Soc Lond A, 2006, 462: 481-515

[21] Qin B, Yang G, Ma J Spatiotemporal changes of cyanobacterial bloom in large shallow eutrophic Lake Taihu, China. Front Microbiol, 2018, 9: 451

[22] Richard S. China aims to turn tide against toxic lake pollution. Science, 2011, 333(6047): 1210-1211

[23] Taylor GI. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc R Soc Lond A, 1953, 219(1137): 186-203

[24] Wu X, Kong F. Effects of light and wind speed on the vertical distribution of microcystis aeruginosa colonies of different sizes during a summer bloom. Int Rev Hydrobiol, 2009, 94(3): 258-266

[25] Wu X, Noss C, Liu L Effects of small-scale turbulence at the air-water interface on microcystis surface scum formation. Water Res, 2019, 167(115): 091

[26] Wu Z, Chen GQ. Approach to transverse uniformity of concentration distribution of a solute in a solvent flowing along a straight pipe. J Fluid Mech, 2014, 740: 196-213

[27] Wu Z, Foufoula-Georgiou E, Parker G Analytical solution for anomalous diffusion of bedload tracers gradually undergoing burial. J Geophys Res Earth Surf, 2019, 124: 21-37

[28] Wu Z, Singh A, Fu X Transient anomalous diffusion and advective slowdown of bedload tracers by particle burial and exhumation. Water Resour Res, 2019, 55(10): 7964-7982

[29] Wu Z, Furbish D, Foufoula-Georgiou E. Generalization of hop distance-time scaling and particle velocity distributions via a two-regime formalism of bedload particle motions. Water Resour Res, 2020, 56

[30] Wu Z, Singh A, Foufoula-Georgiou E A velocity-variation-based formulation for bedload particle hops in rivers. J Fluid Mech, 2021, 912: A33

[31] Yang Y, Tan S, Zeng L Migration of active particles in a surface flow constructed wetland. J Hydrol, 2020, 582(124): 523

[32] Yang Y, Jiang W, Wu Y Migration of buoyancy-controlled active particles in a laminar open-channel flow. Adv Water Resour, 2021, 156(104): 023

[33] Yasuda H. Longitudinal dispersion of matter due to the shear effect of steady and oscillatory currents. J Fluid Mech, 1984, 148: 383-403

[34] Zeng L, Chen GQ. Ecological degradation and hydraulic dispersion of contaminant in wetland. Ecol Modell, 2011, 222(2): 293-300

[35] Zeng L, Zhao YJ, Chen B Longitudinal spread of bicomponent contaminant in wetland flow dominated by bank-wall effect. J Hydrol, 2014, 509: 179-187

[36] Zeng L, Wu Z, Fu X Performance of the analytical solutions for Taylor dispersion process in open channel flow. J Hydrol, 2015, 528: 301-311

[37] Zeng L, Zhang HW, Wu YH Theoretical and numerical analysis of vertical distribution of active particles in a free-surface wetland flow. J Hydrol, 2019, 573: 449-455

Funding

IWHR Research and Development Support Program(HY0199A112021)

Independent Research Project of State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin(SKL2022TS09)