Time-Space Fractional Governing Equations of Unsteady Open Channel Flow
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Volume 22, Issue 2
Abstract
In this study, the complete governing equations for unsteady open channel flow in fractional time-space are developed from the fractional continuity equation combined with the fractional motion equation, which is based on Newton’s second law of motion, by accounting for the acceleration terms in order to render physically interpretable hydraulic terms. Then the kinematic wave and diffusion wave approximations to unsteady open channel flow with physically interpretable terms in fractional time-space are developed from the fractional continuity and motion equations. Because the powers of all of the derivatives in the conventional governing equations of unsteady open channel flow in integer time-space are one, these conventional equations are derived as special cases of the fractional governing equations when the fractional powers in the fractional equations approach unity.
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References
Baumer, B., Benson, D., and Meerschaert, M. M. (2005). “Advection and dispersion in time and space.” Physica A, 350(2–4), 245–262.
Baumer, B., and Meerschaert, M. M. (2007). “Fractional diffusion with two time scales.” Physica A, 373, 237–251.
Benson, D. A., Wheatcraft, S. W., and Meerschaert, M. M. (2000a). “Application of a fractional advection-dispersion equation.” Water Resour. Res., 36(6), 1403–1412.
Benson, D. A., Wheatcraft, S. W., and Meerschaert, M. M. (2000b). “The fractional-order governing equation of Levy motion.” Water Resour. Res., 36(6), 1413–1423.
Bradley, D., Tucker, G., and Benson, D. A. (2010). “Fractional dispersion in a sand bed river.” J. Geophys. Res., 115(F1), FOOA09.
Chaudry, M. H. (2008). Open-channel flow, 2nd Ed., Springer, New York.
Chow, V.-T. (1959). Open channel hydraulics, McGraw-Hill, New York.
Delleur, J. W. (1973). Advanced hydrology lecture notes, Purdue Univ. School of Civil Engineering, West Lafayette, IN.
Deng, Z.-Q., Bengtsson, L., and Singh, V. P. (2006a). “Parameter estimation for fractional dispersion model for rivers.” Environ. Fluid Mech., 6(5), 451–475.
Deng, Z.-Q., de Lima, J. L. M. P., de Lima, M. I. P., and Singh, V. P. (2006b). “A fractional dispersion model for overland solute transport.” Water Resour. Res., 42(3), W03416.
Deng, Z.-Q., de Lima, J. L. M. P., and Singh, V. P. (2005). “Fractional kinetic model for first flush of stormwater pollutants.” J. Environ. Eng., 232–241.
Deng, Z.-Q., Singh, V. P., and Bengtsson, L., (2004). “Numerical solution of fractional advection-dispersion equation.” J. Hydraul. Eng., 422–431.
Ercan, A., and Kavvas, M. L. (2015). “Fractional governing equations of diffusion wave and kinematic wave open-channel flow in fractional time-space. II: Numerical simulations.” J. Hydrol. Eng., 04014097.
Foufoula-Georgiou, E., Ganti, V., and Dietrich, W. E. (2010). “A non-local theory of sediment transport on hillslopes.” J. Geophys. Res., 115(F1), in press.
Foufoula-Georgiou, E., and Stark, C. (2010). “Introduction to special issue on stochastic transport and emergent scaling on earth’s surface: Rethinking geomorphic transport—Stochastic theories, broad scales of motion and nonlocality.” J. Geophys. Res., 115(F1), F00A01.
Furbish, D. J., Childs, E., Haff, P. K., and Schmeeckle, M. W. (2010). “Rain splash of soil grains as a stochastic advection-dispersion process, with implications for desert plant-soil interactions and land-surface evolution.” J. Geophys. Res., 115(F1), 1–10.
Ganti, V., Meerschaert, M. M., Foufoula-Georgiou, E., Viparelli, E., and Parker, G. (2010). “Normal and anomalous diffusion of gravel tracer particles in rivers.” J. Geophys. Res., 115(F1), F00Al2.
Johnson, H. E. (2001). “Predicting river travel time from hydraulic characteristics.” J. Hydraul. Eng., 911–918.
Kavvas, M. L., and Ercan, A. (2015). “Fractional governing equations of diffusion wave and kinematic wave open-channel flow in fractional time-space. I: Development of the equations.” J. Hydrol. Eng., 04014096.
Kavvas, M. L., Kim, S., and Ercan, A. (2015). “Fractional ensemble average governing equations of transport by time-space nonstationary stochastic fractional advective velocity and fractional dispersion. I: Theory.” J. Hydrol. Eng., 04014039.
Kim, S., and Kavvas, M. L. (2006). “Generalized Fick’s law and fractional ADE for pollution transport in a river: Detailed derivation.” J. Hydrol. Eng., 80–83.
Kumar, S. (2013). “A numerical study for the solution of time fractional nonlinear shallow water equation in oceans,” Z. Naturforsch. 68a, 547–553.
Levy, M., and Berkowitz, B. (2003). “Measurement and analysis of non-Fickian dispersion in heterogeneous porous media.” J. Contam. Hydrol., 64(3–4), 203–226.
Meerschaert, M. M., Benson, D. A., and Baumer, B. (1999). “Multidimensional advection and fractional dispersion.” Phys. Rev. E, 59(5), 5026–5028.
Meerschaert, M. M., Benson, D. A., Scheffler, H. P., and Baumer, B. (2002). “Stochastic solution of space-time fractional diffusion equations.” Phys. Rev. E, 65(4), 041103-1–04113-4.
Meerschaert, M. M., Mortensen, J., and Wheatcraft, S. W. (2006). “Fractional vector calculus for fractional advection-dispersion.” Physica A, 367, 181–190.
Momani, S., and Odibat, Z. (2008). “A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula.” J. Comput. Appl. Math., 220(1–2), 85–95.
Nordin, C. F., and Sabol, G. V. (1974). “Empirical data on longitudinal dispersion in rivers.”, USGS Publications Warehouse.
Nordin, C. F., and Troutman, B. M. (1980). “Longitudinal dispersion in rivers: The persistence of skewness in observed data.” Water Resour. Res., 16(1), 123–128.
Odibat, Z. M., and Shawagfeh, N. T. (2007). “Generalized Taylor’s formula.” Appl. Math. Comput., 186(1), 286–293.
Peaudecerf, G., and Sauty, J.-P. (1978). “Application of a mathematical model to the characterization of dispersion effects on groundwater quality.” Prog. Water Tech., 10(5/6), 443–454.
Podlubny, I. (1999). Fractional differential equations, Academic Press, San Diego, 340.
Schumer, R., Benson, D. A., Meerschaert, M. M., and Wheatcraft, S. W. (2001). “Eulerian derivation for the fractional advection-dispersion equation.” J. Contam. Hydrol., 48(1–2), 69–88.
Schumer, R., Meerschaert, M. M., and Baumer, B. (2009). “Fractional advection-dispersion equations for modeling transport at the earth surface.” J. Geophys. Res., 114(F4), F00A07.
Sidle, C., Nilson, B., Hansen, M., and Fredericia, J. (1998). “Spatially varying hydraulic and solute transport characteristics of a fractured till determined by field tracer tests, Funen, Denmark.” Water Resour. Res., 34(10), 2515–2527.
Silliman, S. E., and Simpson, E. S. (1987). “Laboratory evidence of the scale effect in dispersion of solutes in porous media.” Water Resour. Res., 23(8), 1667–1673.
Sudicky, E. A., Cherry, J. A., and Frind, E. O. (1983). “Migration of contaminants in groundwater at a landfill: A case study. 4: A natural-gradient dispersion test.” J. Hydrol., 63(1–2), 81–108.
Voller, V. R., and Paola, C. (2010). “Can anomalous diffusion describe depositional fluvial profiles?” J. Geophys. Res., 115(F1), F00A13.
Wheatcraft, S. W., and Meerschaert, M. M. (2008). “Fractional conservation of mass.” Adv. Water Resour., 31(10), 1377–1381.
Yen, B. C. (1973). “Open-channel flow equations revisited.” ASCE J. Eng. Mech. Div., 99(5), 979–1009.
Zhang, Y., and Benson, D. A. (2008). “Lagrangian simulation of multidimensional anomalous transport at the MADE site.” Geophys. Res. Lett., 35(7), L07403.
Zhang, Y., Benson, D. A., Meerschaert, M. M., and Labolle, E. M. (2007). “Space-fractional advection-dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the macrodispersion experiment site data.” Water Resour. Res., 43(5), W05439.
Zhang, Y., Benson, D. A., and Reeves, D. M. (2009). “Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications.” Adv. Water Resour., 32(4), 561–581.
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Journal of Hydrologic Engineering
Volume 22 • Issue 2 • February 2017
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Sep 18, 2015
Accepted: Jul 18, 2016
Published online: Aug 30, 2016
Discussion open until: Jan 30, 2017
Published in print: Feb 1, 2017
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