**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31107

##### A Hybrid Artificial Intelligence and Two Dimensional Depth Averaged Numerical Model for Solving Shallow Water and Exner Equations Simultaneously

**Authors:**
S. Mehrab Amiri,
Nasser Talebbeydokhti

**Abstract:**

Modeling sediment transport processes by means of numerical approach often poses severe challenges. In this way, a number of techniques have been suggested to solve flow and sediment equations in decoupled, semi-coupled or fully coupled forms. Furthermore, in order to capture flow discontinuities, a number of techniques, like artificial viscosity and shock fitting, have been proposed for solving these equations which are mostly required careful calibration processes. In this research, a numerical scheme for solving shallow water and Exner equations in fully coupled form is presented. First-Order Centered scheme is applied for producing required numerical fluxes and the reconstruction process is carried out toward using Monotonic Upstream Scheme for Conservation Laws to achieve a high order scheme. In order to satisfy C-property of the scheme in presence of bed topography, Surface Gradient Method is proposed. Combining the presented scheme with fourth order Runge-Kutta algorithm for time integration yields a competent numerical scheme. In addition, to handle non-prismatic channels problems, Cartesian Cut Cell Method is employed. A trained Multi-Layer Perceptron Artificial Neural Network which is of Feed Forward Back Propagation (FFBP) type estimates sediment flow discharge in the model rather than usual empirical formulas. Hydrodynamic part of the model is tested for showing its capability in simulation of flow discontinuities, transcritical flows, wetting/drying conditions and non-prismatic channel flows. In this end, dam-break flow onto a locally non-prismatic converging-diverging channel with initially dry bed conditions is modeled. The morphodynamic part of the model is verified simulating dam break on a dry movable bed and bed level variations in an alluvial junction. The results show that the model is capable in capturing the flow discontinuities, solving wetting/drying problems even in non-prismatic channels and presenting proper results for movable bed situations. It can also be deducted that applying Artificial Neural Network, instead of common empirical formulas for estimating sediment flow discharge, leads to more accurate results.

**Keywords:**
Artificial Neural Network,
shallow water equations,
morphodynamic model,
sediment continuity equation

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1317404

**References:**

[1] E.F. Toro, A. Siviglia, “PRICE: primitive centred schemes for hyperbolic system of equations,” International Journal for Numerical Methods in Fluids, 2003, 42, pp.1263–1291.

[2] W. Wu, “Depth-averaged two-dimensional numerical modeling of unsteady flow and nonuniform sediment transport in open channels,” Journal of Hydraulic Engineering, 2004, 130 (10), pp. 1013–1024.

[3] S. S. Li, R. G. Millar, “Simulating bed-load transport in a complex gravel- bed river,” Journal of Hydraulic Engineering, 2007, 133(3), pp.323-328.

[4] A. Canestrelli, A. Siviglia, M. Dumbser, E. F. Toro, “Well-balanced high-order centered schemes for non-conservative hyperbolic systems, Applications to shallow water equations with fixed and mobile bed,” Advances in Water Resources, 2009, 32, pp. 834–844.

[5] J. Murillo, P. Garcia-Navarro, “An Exner-based coupled model for two-dimensional transient flow over erodible bed,” Journal of Computational Physics, 2010, 229, pp. 8704–8732.

[6] M. Postacchini, M. Brocchini, A. Mancinelli, M. Landon, “A multi-purpose, intra-wave, shallow water hydro-morphodynamic solver,” Advances in Water Resources, 2012, 38, pp. 13–26.

[7] S. Cordier, M. H. Le, T. Morales de Luna, “Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help,” Advances in Water Resources, 2011, 34, pp. 980–989.

[8] J. Smagorinsky, “General circulation experiments with the primitive equations, I. The basic experiments,” Monthly Weather Review, 1963, 91(2), pp. 99-164.

[9] E. F. Toro, Shock-capturing methods for free-surface shallow flows. Wiley: West Sussex, England, 2001.

[10] E. F. Toro, Riemann solvers and numerical methods for fluid dynamics. Third Edition, Springer: Germany 2009.

[11] J. G. Zhou, D. M. Causon, C. G. Mingham, D. M. Ingram, “The surface gradient method for the treatment of source terms in the shallow water equations,” Journal of Computational Physics, 2001, 168(1), pp. 1–25.

[12] S. M. Amiri, N. Talebbeydokhti , A. Baghlani, “A two-dimensional well-balanced numerical model for shallow water equations,” Scientia Iranica, 2013, 20 (1), pp. 97–107.

[13] I. K. Nikolos , A. I. Delis, “An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography,” Computer Methods in Applied Mechanics and Engineering, 2009, 198, pp. 3723–3750.

[14] W. R. Brownlie. Compilation of alluvial channel data: laboratory and field. W. M. Keck Laboratory of Hydraulics and Water Resources Division of Engineering and Applied Science California Institute of Technology Pasadena, California 1981; Report No. KH-R-43B.

[15] P. Brufau, P. Garcia-Navarro,“Two-dimensional dam break flow simulation,” International Journal of Numerical Methods in Fluids 2000, 33, pp. 35–57.

[16] N. Goutal, F. Maurel, “A finite volume solver for 1D shallow-water equations applied to an actual river,” International Journal of Numerical Methods in Fluids, 2002, 38, pp. 1–19.

[17] N. A. Alias, Q. Liang, G. Kesserwani, “A Godunov-type scheme for modeling 1D channel flow with varying width and topography,” Computers and Fluids, 2011, 46, pp. 88–93.

[18] A. Mohamadian, D. Y. Le Roux, M. Tajrishi, K. Mazaheri, “A mass conservative scheme for simulating shallow flows over variable topographies using unstructured grid,” Advances in Water Resources, 2005, 28, pp. 523–539.

[19] J. A. Alvarez, Towards the numerical simulation of ship generated waves using a cartesian cut cell based free surface solver, Ph. D. Thesis: Centre of Mathematical Modeling and Flow Analysis Department of Computing and Mathematics Manchester Metropolitan University 2008.

[20] D. M. Causon, D. M. Ingram, C. G. Mingham, G. Yang, R. V. Pearson “Calculation of shallow water flows using a cartesian cut cell approach,” Advances in Water Resources, 2000, 23, pp. 545–562.

[21] A. Mahdavi, N. Talebbeydokhti, “Modeling of non-breaking and breaking solitary wave run-up using FORCE-MUSCL scheme,” Journal of Hydraulic Research, 2009, 47 (4), pp. 476–485.

[22] Q. Liang, “Flood simulation using a well-balanced shallow flow model,” ASCE Journal of Hydraulic Engineering, 2010, 136(9), pp. 669–675.

[23] B. Spinewine, Y. Zech, “Dam-break waves over movable beds: a flat bed test case,” 2nd IMPACT workshop, Statkraft Grøner, Mo-i-Rana 2002.

[24] R. Ghobadian, M, Shafai Bajestan, “Investigation of sediment patterns at river confluence,” Journal of Applied Science 2007, 7(10), pp. 1372-1380.