Analytical Solutions of One-Dimensional Space-Fractional Advection–Diffusion Equation for Sediment Suspension Using Homotopy Analysis Method
Publication: Journal of Engineering Mechanics
Volume 145, Issue 7
Abstract
In this study, the full analytical solution of one-dimensional space-fractional advection–diffusion equation (fADE) is derived using the homotopy analysis method (HAM). Apart from the several applications of the fADE, it has been applied in sediment-laden turbulent flows to find the distribution of suspended sediment particles when the effect of nonlocal transport is present. In such flows, due to the presence of turbulent-bursting particles ejected from the bed level up to the free surface performing nonlocal vertical jumps, which cannot be captured through traditional integer-order concentration gradient, the fractional derivative of concentration, therefore, comes into promise. The fractional derivative considered here contains a convolution of suspension concentration and concentration gradient which captures the effect over all points throughout the domain. The HAM is a powerful method to solve the nonlinear fractional differential equations and has been applied here to obtain the solutions. The advantage of this proposed HAM solution over previous analytical solutions of fADE, in terms of the Mittag-Leffler function, is that it allows to choose the region of convergence of the series solution with greater freedom, and thus broadens the applicability of the model under many difficult situations. The HAM solutions are obtained for two different choices of sediment diffusivity and they are also compared with analytical solution and numerical solutions. The solutions are also validated with the experimental data. The comparison results are satisfactory and it shows the advantage of using the homotopy analysis method. Finally, the variations of the model with the parameters are also presented and the reasons are justified physically.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The author is very much thankful to the associate editor of this journal and two reviewers for their helpful and constructive comments which helped to improve the quality of this paper. The work is supported by DST (SERB) sponsored Project with File No. ECR/2017/000184.
References
Bechteler, W., and K. Farber 1985. “Stochastic model of suspended solid dispersion.” J. Hydraul. Eng. 111 (1): 64–78. https://doi.org/10.1061/(ASCE)0733-9429(1985)111:1(64).
Bouvard, M., and S. Petkovic. 2010. “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 (F4): F00A09. https://doi.org/10.1029/2009JF001268.
Cao, Z. 1997. “Turbulent bursting-based sediment entrainment function.” J. Hydraul. Eng. 123 (3): 233–236. https://doi.org/10.1061/(ASCE)0733-9429(1997)123:3(233).
Cao, Z., L. Wei, and J. Xie. 1995. “Sediment-laden flow in open channels from two-phase flow viewpoint.” J. Hydraul. Eng. 121 (10): 725–735. https://doi.org/10.1061/(ASCE)0733-9429(1995)121:10(725).
Cellino, M., and U. Lemmin. 2004. “Influence of coherent flow structures on the dynamics of suspended sediment transport in open-channel flow.” J. Hydraul. Eng. 130 (11): 1077–1088. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:11(1077).
Chaves, A. S. 1998. “A fractional diffusion equation to describe Lévy flights.” Phys. Lett. A 239 (1–2): 13–16. https://doi.org/10.1016/S0375-9601(97)00947-X.
Chen, D., H. G. Sun, and Y. Zhang. 2013. “Fractional dispersion equation for sediment suspension.” J. Hydrol. 491: 13–22. https://doi.org/10.1016/j.jhydrol.2013.03.031.
Chiu, C., W. Jin, and Y. Chen. 2000. “Mathematical models for distribution of sediment concentration.” J. Hydraul. Eng. 126 (1): 16–23. https://doi.org/10.1061/(ASCE)0733-9429(2000)126:1(16).
Coleman, N. L. 1986. “Effects of suspended sediment on the open-channel velocity distribution.” Water Resour. Res. 22 (10): 1377–1384. https://doi.org/10.1029/WR022i010p01377.
Cui, H., and V. P. Singh. 2014. “Suspended sediment concentration in open channels using Tsallis entropy.” J. Hydrol. Eng. 19 (5): 966–977. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000865.
Czernuszenko, M. 1998. “Drift velocity for sediment-laden flows.” J. Hydraul. Eng. 124 (10): 1026–1033. https://doi.org/10.1061/(ASCE)0733-9429(1998)124:10(1026).
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. 1950. The bed load function for sediment transportation in open channels. Washington, DC: Soil Conservation Service, USDA.
Einstein, H. A., and N. S. Chien. 1955. Effects of heavy sediment concentration near the bed on velocity and sediment distribution. Washington, DC: Missouri River Division, USACE.
Fischer, H. B., E. J. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks. 1979. Mixing in inland and coastal waters. New York: Academic Press.
Foufoula-Georgiou, E., V. Ganti, and W. Dietrich. 2010. “A nonlocal theory of sediment transport on hillslopes.” J. Geophys. Res. 115 (F4): F00A16. https://doi.org/10.1029/2009JF001280.
Fu, X. D., G. Q. Wang, and X. J. Saho. 2005. “Vertical dispersion of fine and coarse sediments in turbulent open-channel flows.” J. Hydraul. Eng. 131 (10): 877–888. https://doi.org/10.1061/(ASCE)0733-9429(2005)131:10(877).
Grass, A. J. 1971. “Structural factors of turbulent flow over smooth and rough boundaries.” J. Fluid Mech. 50 (2): 233–255. https://doi.org/10.1017/S0022112071002556.
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. 2010. “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, Q., G. Huang, and H. Zhan. 2008. “A finite element solution for the fractional advection dispersion equation.” Adv. Water Resour. 31 (12): 1578–1589. https://doi.org/10.1016/j.advwatres.2008.07.002.
Jackson, R. G. 1976. “Sedimentological and fluid-dynamic implications of the turbulent bursting phenomenon in geophysical flows.” J. Fluid Mech. 77 (3): 531–560. https://doi.org/10.1017/S0022112076002243.
Jaynes, E. 1957. “Information theory and statistical mechanics. Part I.” Phys. Rev. 106 (4): 620–630. https://doi.org/10.1103/PhysRev.106.620.
Johnson, A. R., C. A. Hatfield, and B. T. Bruce. 1995. “Simulated diffusion dynamics in river networks.” Ecol. Modell. 83 (3): 311–325. https://doi.org/10.1016/0304-3800(94)00107-9.
Kovacs, A. E. 1998. “Prandtl’s mixing length concept modified for equilibrium sediment-laden flows.” J. Hydraul. Eng. 124 (8): 803–812. https://doi.org/10.1061/(ASCE)0733-9429(1998)124:8(803).
Kundu, S. 2018a. “Suspension concentration distribution in turbulent flows: An analytical study using fractional advection-diffusion equation.” Phys. A: Stat. Mech. Its Appl. 506: 135–155. https://doi.org/10.1016/j.physa.2018.04.009.
Kundu, S. 2018b. “Two parameter Mittag-Leffler solution of space fractional advection-diffusion equation for sediment suspension in turbulent flow.” J. Environ. Eng. 144 (8): 06018005. https://doi.org/10.1061/(ASCE)EE.1943-7870.0001416.
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., M. Kumbhakar, and K. Ghoshal. 2018. “Reinvestigation on mixing length in an open channel turbulent flow.” Acta Geophys. 66 (1): 93–107. https://doi.org/10.1007/s11600-017-0109-7.
Lane, E. W., and A. A. Kalinske. 1941. “Engineering calculations of suspended sediment.” Trans. Am. Geophys. Union 22 (3): 603–607. https://doi.org/10.1029/TR022i003p00603.
Laursen, E. M. 1982. “A concentration distribution formula from the revised theory of Prandtl mixing length.” In Vol. 1 of Proc., 1st Int. Symp. on River Sedimentation, 237–244. Beijing: Guanghua Press.
Lelouvetel, J., F. Bigillon, D. Doppler, I. Vinkovic, and J. Y. Champagne. 2009. “Experimental investigation of ejections and sweeps involved in particle suspension.” Water Resour. Res. 45 (2): W02416. https://doi.org/10.1029/2007WR006520.
Li, C., Z. Zhao, and Y. Chen. 2011. “Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion.” Comput. Math. Appl. 62 (3): 855–875. https://doi.org/10.1016/j.camwa.2011.02.045.
Liao, K., J. Qu, J. Tang, F. Ding, H. Liu, and S. Zhu. 2010. “Activity of wind-blown sand and the formation of feathered sand ridges in the Kumtagh desert, China.” Boundary Layer Meteorol. 135 (2): 333–350. https://doi.org/10.1007/s10546-010-9469-0.
Liao, S. J. 2004. Beyond perturbation: Introduction to the homotopy analysis method. Boca Raton, FL: Chapman & Hall.
Liao, S. J. 2009. “Notes on the homotopy analysis method: Some definition and theorems.” Commun. Nonlinear Sci. Numer. Simul. 14 (4): 983–997. https://doi.org/10.1016/j.cnsns.2008.04.013.
Liao, S. J., and Y. Tan. 2007. “A general approach to obtain series solutions of nonlinear differential equations.” Stud. Appl. Math. 119 (4): 297–354. https://doi.org/10.1111/j.1467-9590.2007.00387.x.
Lyn, D. A. 1988. “A similarity approach to open-channel sediment laden flows.” J. Fluid Mech. 193 (1): 1–26. https://doi.org/10.1017/S0022112088002034.
Mao, Y. 2003. “The effects of turbulent bursting on the sediment movement in suspension.” Int. J. Sediment. Res. 18 (2): 148–157.
Mazumder, B. S., and K. Ghoshal. 2006. “Velocity and concentration profiles in uniform sediment-laden flow.” Appl. Math. Modell. 30 (2): 164–176. https://doi.org/10.1016/j.apm.2005.03.015.
McTigue, D. F. 1981. “Mixture theory for suspended sediment transport.” J. Hydraul. Div. 107 (6): 659–673.
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. 1991. “Vertical sediment distribution.” J. Hydraul. Eng. 117 (9): 1184–1194. https://doi.org/10.1061/(ASCE)0733-9429(1991)117:9(1184).
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 S. Momani. 2009. “The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics.” Comput. Math. Appl. 58 (11): 2199–2208. https://doi.org/10.1016/j.camwa.2009.03.009.
Odibat, Z. M., and N. T. Shawagfeh. 2007. “Generalized Taylor’s formula.” Appl. Math. Comput. 186 (1): 286–293.
Offen, G. R., and S. J. Kline. 1974. “Combined dye-streak and hydrogen-bubble visual observation of a turbulent boundary layer.” J. Fluid Mech. 62 (2): 223–239. https://doi.org/10.1017/S0022112074000656.
Paradisi, P., R. Cesari, F. Mainardi, and F. Tampieri. 2001. “The fractional Fick’s law for non-local transport process.” Phys. A 293 (1–2): 130–142. https://doi.org/10.1016/S0378-4371(00)00491-X.
Podlubny, I. 1999. Fractional differential equations. San Diego: Academic Press.
Prandtl, L. 1925. “Bericht über Untersuchungen zur ausgebildeten Turbulenz.” Zeitschrift für Angewandte Mathematik und Mechanik 5 (2): 136–139. https://doi.org/10.1002/zamm.19250050212.
Prandtl, L. 1926. “Über die ausgebildete Turbulenz.” In 2e Internationaler Kongress der Technischen Mechanik, Verhandlung. Zürich, Switzerland: Füessli.
Prandtl, L. 1932. Recent results of turbulence research. Washington, DC: National Advisory Committee for Aeronautics.
Roop, J. P. 2008. “Numerical approximation of a one-dimensional space fractional advection dispersion equation with boundary layer.” Comput. Math. Appl. 56 (7): 1808–1819. https://doi.org/10.1016/j.camwa.2008.04.025.
Rouse, H. 1937. “Modern conceptions of the mechanics or fluid turbulence.” Trans. ASCE 102 (1): 463–505.
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., D. A. Benson, M. M. Meerschaert, and B. Baeumer. 2003. “Multiscaling fractional advection-dispersion equations and their solutions.” Water Resour. Res. 39 (1): 10–22. https://doi.org/10.1029/2001WR001229.
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): F00A07. https://doi.org/10.1029/2008JF001246.
Singh, M. K., A. Chatterjee, and V. P. Singh. 2017. “Solution of one-dimensional time fractional advection dispersion equation by homotopy analysis method.” J. Eng. Mech. 143 (9): 1–15. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001318.
Umeyama, M. 1992. “Vertical distribution of suspended sediment in uniform open channel flow.” J. Hydraul. Eng. 118 (6): 936–941. https://doi.org/10.1061/(ASCE)0733-9429(1992)118:6(936).
Vanoni, V. A. 1940. “Experiments on the transportation of suspended sediment by water.” Ph.D. thesis, Engineering and Applied Science Division, Dept. of Civil Engineering, California Institute of Technology.
Velikanov, M. A. 1958. Fluvial process of rivers. Moscow: State Publishing House for Physics and Mathematics Literature.
Villaret, C., and A. Davies. 1995. “Modeling sediment-turbulent flow interactions.” Appl. Mech. Rev. 48 (9): 601–609. https://doi.org/10.1115/1.3023148.
Wang, X., and N. Qian. 1989. “Turbulence characteristics of sediment-laden flows.” J. Hydraul. Eng. 115 (6): 781–800. https://doi.org/10.1061/(ASCE)0733-9429(1989)115:6(781).
Weeks, E. R., J. S. Urbach, and H. L. Swinney. 1996. “Anomalous diffusion in asymmetric random walks with a quasi-geostrophic flow example.” Phys. D 97 (1): 291–310. https://doi.org/10.1016/0167-2789(96)00082-6.
Zhang, R., J. Xie, M. Wan, and J. Huang. 1989. Dynamics of fluvial sediment transport. Beijing: China Water Power Press.
Zhang, X., J. W. Crawford, L. K. Deeks, M. I. Stutter, 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.
Zheng, Y. Y., C. P. Li, and Z. G. Zhoa. 2010. “A note on the finite element method for the space fractional advection dispersion equation.” Comput. Math. Appl. 59 (5): 1718–1726. https://doi.org/10.1016/j.camwa.2009.08.071.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 145 • Issue 7 • July 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Jun 30, 2018
Accepted: Dec 5, 2018
Published online: May 8, 2019
Published in print: Jul 1, 2019
Discussion open until: Oct 8, 2019
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.