Two-Parameter Mittag-Leffler Solution of Space Fractional Advection-Diffusion Equation for Sediment Suspension in Turbulent Flows
This article has a reply.
VIEW THE REPLYPublication: Journal of Environmental Engineering
Volume 144, Issue 8
Abstract
In this paper, a simple derivation of the space fractional advection-diffusion equation (fADE) under the steady condition is proposed from a generalized mixing length concept. The obtained equation is a fractional Fickian-type diffusion equation in the steady-state condition. The space fADE is solved and an analytical solution is obtained in terms of the generalized two-parameter Mittag-Leffler function. The model has been applied to investigate the suspension distribution of sediment particles in open-channel turbulent flows over erodible sediment beds. The obtained results show that the fADE is able to predict the Type II profile (where the maximum concentration appears at some distance above the bottom of suspended-load layer) of the suspension concentration distribution in such flows, unlike the standard advection-diffusion equation. Apart from this, the model also describes the Type I profile (where the maximum concentration appears at the bottom of the suspended-load layer) of the suspension concentration distribution of particles well. The outcome of this study indicates that underlying diffusion of particles can be anomalous and exchange of momentum can happen within nonadjacent layers in the Type II distribution. The model is validated with experimental data available in the literature, and the results are satisfactory. Validation results show that the parameter in the Type II profile depends on specific gravity, diameter, and settling velocity of particles. A nonlinear regression analysis is carried out to express the dependency and practical application of the model.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The author is thankful to the editor and the reviewers for their constructive comments, which helped to improve the quality of the paper. The work is partially supported by DST (SERB) sponsored project with File No. ECR/2017/000184.
References
Bouvard, M., and S. Petkovic. 1985. “Vertical dispersion of spherical, heavy particles in turbulent open channel flow.” J. Hydraul. Res. 23 (1): 5–20. https://doi.org/10.1080/00221688509499373.
Bradley, D., G. Tucker, and D. Benson. 2010. “Fractional dispersion in a sand bed river.” J. Geophys. Res. 115 (F1): F00A09. https://doi.org/10.1029/2009JF001268.
Camenen, B., and M., Larson. 2007. A unified sediment transport formulation for coastal inlet application. Lyon, France: CEMAGREF Lyon.
Chen, D., H. G. Sun, and Y. Zhang. 2013. “Fractional dispersion equation for sediment suspension.” J. Hydrol. 491 (May): 13–22. https://doi.org/10.1016/j.jhydrol.2013.03.031.
Chen, X. Y., and K. W. Chau. 2016. “A hybrid double feedforward neural network for suspended sediment load estimation.” Water Resour. Manage. 30 (7): 2179–2194. https://doi.org/10.1007/s11269-016-1281-2.
Cheng, N. S. 2003. “A diffusive model for evaluating thickness of bedload layer.” Adv. Water Resour. 26 (8): 875–882. https://doi.org/10.1016/S0309-1708(03)00062-9.
Coleman, J. M. 1969. “Brahmaputra river; channel process and sedimentation.” Sediment. Geol. 3 (2–3): 129–239. https://doi.org/10.1016/0037-0738(69)90010-4.
Deng, Z. Q., J. D. Lima, M. D. Lima, and V. P. Singh. 2006. “A fractional dispersion model for overland solute transport.” Water Resour. Res. 42 (3): W03416. https://doi.org/10.1029/2005WR004146.
Einstein, H. A., and N. S. Chien. 1955. Effects of heavy sediment concentration near the bed on velocity and sediment distribution. US Army Corps of Engineers, Missouri River Division.
Fu, X. D., G. Q. Wang, and Z. Dong. 2001. “Theoretical analysis and numerical computation of dilute solid/liquid two-phase pipe flow.” Sci. China Ser. E 44 (3): 298–308. https://doi.org/10.1007/BF02916707.
Graf, W. H. 1971. Hydraulics of sediment transport. New York: McGraw-Hill.
Greimann, B. P., and F. M. Holly. 2001. “Two-phase analysis of concentration profiles.” J. Hydraul. Eng. 127 (9): 753–762. https://doi.org/10.1061/(ASCE)0733-9429(2001)127:9(753).
Greimann, B. P., M. Muste, and F. M. Holly. 1999. “Two-phase formulation of suspended sediment transport.” J. Hydraul. Res. 37 (4): 479–500. https://doi.org/10.1080/00221686.1999.9628264.
Hsu, T. J., J. T. Jenkins, and P. L. F. Liu. 2003. “On two-phase sediment transport: Dilute flow.” J. Geophys. Res. 108 (C3): 1–14. https://doi.org/10.1029/2001JC001276.
Huang, G. H., Q. Z. Huang, and H. B. Zhan. 2006. “Evidence of onedimensional scale-dependent fractional advection-dispersion.” J. Contam. Hydrol. 85 (1–2): 53–71. https://doi.org/10.1016/j.jconhyd.2005.12.007.
Hung, T. C., T. K. Hung, and G. Bugliarello. 1980. “Blood flow in capillary tubes: Curvature and gravity effects.” Biorehology 17: 331–342.
Jiang, J. S., A. W. K. Law, and N. S. Cheng. 2004. “Two-phase modeling of suspended sediment distribution in open channel flows.” J. Hydraul. Res. 42 (3): 273–281. https://doi.org/10.1080/00221686.2004.9728392.
Kundu, S. 2015. “Theoretical study on velocity and suspension concentration in turbulent flow.” Ph.D. thesis, Indian Institute of Technology Kharagpur.
Kundu, S., and K. Ghoshal. 2013. “An explicit model for concentration distribution using biquadratic-log-wake law in an open channel flow.” J. Appl. Fluid Mech. 6 (3): 339–350.
Kundu, S., and K. Ghoshal. 2014. “Effects of secondary current and stratification on suspension concentration in an open channel flow.” Environ. Fluid Mech. 14 (6): 1357–1380. https://doi.org/10.1007/s10652-014-9341-8.
Kundu, S., and K. Ghoshal. 2017. “A mathematical model for type ii profile of concentration distribution in turbulent flows.” Environ. Fluid Mech. 17 (449–472): 1–18. https://doi.org/10.1007/s10652-016-9498-4.
Lane, E., and A. Kalinske. 1941. “Engineering calculations of suspended sediment.” Trans. Am. Geophys. Union 22 (3): 603–607. https://doi.org/10.1029/TR022i003p00603.
Meerschaert, M. M., and H. P. Scheffler. 2001. Limit distributions for sums of independent random vectors: Heavy tails in theory and practice. New York: Wiley.
Meerschaert, M. M., and H. P. Scheffler. 2002. “Semistable levy motion.” Fractional Calculus Appl. Anal. 5: 27–54.
Michalik, A. 1973. “Density patterns of the inhomogeneous liquids in the industrial pipeline measured by means of radiometric scanning.” La Houille Blanche 1 (1): 53–57. https://doi.org/10.1051/lhb/1973003.
Ni, J. R., and G. Q. Wang. 1987. “On the two patterns of vertical distribution of sediment concentration and their formation causes.” J. Hydraul. Eng. 7: 60–68. https://doi.org/10.1016/j.limno.2009.11.003.
Ni, J. R., G. Q. Wang, and A. G. L. Borthwick. 2000. “Kinetic theory for particles in dilute and dense solid-liquid flows.” J. Hydraul. Eng. 126 (12): 893–903. https://doi.org/10.1061/(ASCE)0733-9429(2000)126:12(893).
Nielsen, P., and I. A. L. Teakle. 2004. “Turbulent diffusion of momentum and suspended particles: A finite-mixing-length theory.” Phys. Fluids 16 (7): 2342–2348. https://doi.org/10.1063/1.1738413.
Odibat, Z. M., and N. T. Shawagfeh. 2007. “Generalized Taylor’s formula.” Appl. Math. Comput. 186 (1): 286–293. https://doi.org/10.1016/j.amc.2006.07.102.
Odibat, Z. M., and N. T. Shawagfeh 2007. “Generalized Taylor’s formula.” Appl. Math. Comput. 186 (1): 286–293. https://doi.org/10.1016/j.amc.2006.07.102.
Olyaie, E., H. Banejad, K. W. Chau, and A. M. Melesse. 2015. “A comparison of various artificial intelligence approaches performance for estimating suspended sediment load of river systems: A case study in United States.” Environ. Monit. Assess. 187–189. https://doi.org/10.1007/s10661-015-4381-1.x.
Pachepsky, Y., D. Timlin, and D. A. Benson. 2001. “Transport of water and solutes in soils as in fractal porous media.” Soil Sci. Soc. Am. J. 56: 51–75.
Podlubny, I. 1999. Fractional differential equations. San Diego, CA: Academic Press.
Prandtl, L. 1925. “Bericht ber untersuchungen zur ausgebildeten turbulenz.” Zeitschrift für Angewandte Mathematik und Mechanik 5: 136–139.
Prandtl, L. 1926. “Über die ausgebildete turbulenz.” In Proc., 2e Internationaler Kongress der Technischen Mechanik, Verhandlung. Füessli, Zürich.
Prandtl, L. 1932. Recent results of turbulence research. Washington, DC: National Advisory Committee for Aeronautics.
Rouse, H. 1937. “Modern concepts of the mechanics of turbulence.” Trans. ASCE 102: 463–543.
Schumer, R., D. Benson, M. Meerschaert, and S. Wheatcraft. 2001. “Eulerian derivation of the fractional advection–dispersion equation.” J. Contam. Hydrol. 48 (1): 69–88. https://doi.org/10.1016/S0169-7722(00)00170-4.
Schumer, R., M. M. Meerschaert, and B. Baeumer. 2009. “Fractional advection-dispersion equations for modeling transport at the earth surface.” J. Geophys. Res. 114 (F4), in press. https://doi.org/10.1029/2008JF001246.
Sun, Z. L., Z. F. Sun, and J. Donahue. 2003. “Equilibrium bed-concentration of nonuniform sediment.” J. Zhejiang Univ. Sci. 4 (2): 186–194. https://doi.org/10.1631/jzus.2003.0186.
Vanoni, V. A. 2006. Sedimentation engineering. Reston, VA: ASCE.
Wang, G. Q., X. D. Fu, Y. F. Huang, and G. Huang. 2008. “Analysis of suspended sediment transport in open-channel flows: Kinetic-model-based simulation.” J. Hydr. Eng. 134 (3): 328–339. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:3(328).
Wang, G. Q., and J. R. Ni. 1990. “Kinetic theory for particle concentration distribution in two-phase flows.” J. Eng. Mech. 116 (12): 2738–2748. https://doi.org/10.1061/(ASCE)0733-9399(1990)116:12(2738).
Wang, G. Q., and J. R. Ni. 1991. “The kinetic theory for dilute solid liquid two-phase flow.” Int. J. Multiphase Flow 17 (2): 273–281. https://doi.org/10.1016/0301-9322(91)90020-4.
Wang, X., and N. Qian. 1989. “Turbulence characteristics of sediment-laden flows.” J. Hydraul. Eng. 115 (6): 781–799. https://doi.org/10.1061/(ASCE)0733-9429(1989)115:6(781).
Zagustin, K. 1968. “Sediment distribution in turbulent flow.” J. Hydraul. Res. 6 (2): 163–172. https://doi.org/10.1080/00221686809500227.
Zhang, R., J. Xie, M. Wan, and J. Huang. 1989. Dynamics of fluvial sediment transport. Beijing, China: China Water Power Press.
Zhang, X. X., J. W. Crawford, L. K. Deeks, M. I. Sutter, A. G. Bengough, and I. M. Young. 2005. “A mass balance based numerical method for the fractional advection-dispersion equation: Theory and application.” Water Resour. Res. 41 (7): W07029. https://doi.org/10.1029/2004WR003818.
Zhong, D., G. Wang, and Q. Sun. 2011. “Transport equation for suspended sediment based on two-fluid model of solid/liquid two-phase flows.” J. Hydraul. Eng. 137 (5): 530–542. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000331.
Zhou, L., and H. M. Selim. 2003. “Application of the fractional advectiondispersion equation in porous media.” Soil Sci. Soc. Am. J. 67 (4): 1079–1084. https://doi.org/10.2136/sssaj2003.1079.
Information & Authors
Information
Published In
Copyright
©2018 American Society of Civil Engineers.
History
Received: Jun 14, 2017
Accepted: Mar 7, 2018
Published online: Jun 12, 2018
Published in print: Aug 1, 2018
Discussion open until: Nov 12, 2018
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.