Analytical and numerical modeling of shallow water flow in a channel: Theoretical approaches and simulations

• Autorzy: Bacharou Taofic , Sabi Takou Daniel , Adetola Jamal

Identyfikatory

DOI

10.59441/ijame/200966

• Języki publikacji: EN

Abstrakty

EN

This study focuses on the modeling and simulation of shallow water flows in a channel through the application of the Saint-Venant equations. Two main approaches were explored: an analytical solution and a numerical method based on finite difference discretization. The analytical solution provides exact expressions for water depth and velocity under simplified conditions, offering a reference point for validating numerical simulations. The numerical method, on the other hand, captures more complex dynamics such as wave propagation and nonlinear interactions between sections of the channel. The simulations reveal an inverse relationship between water depth and flow velocity, confirming the validity of the governing equations. Moreover, the influence of parameters such as channel slope, flow rate, and boundary conditions on the system?s dynamics is clearly illustrated. The comparative analysis of the two approaches shows that the finite difference method is a powerful tool for practical applications in hydraulic engineering, allowing for the accurate modeling of real-world phenomena while offering greater flexibility compared to idealized analytical solutions.

Słowa kluczowe

EN

flow modeling; water depth; rectangular channel; Froude number; numerical simulation

PL

modelowanie przepływu; liczba Froude'a; symulacja numeryczna

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mechanics and Engineering
• Rocznik: 2025
• Tom: Vol. 30, no. 2
• Strony: 31--45
• Opis fizyczny: Bibliogr. 18 poz., wykr.

Twórcy

autor

Bacharou Taofic

• Civil Engineering Department, Ecole Polytechnique d’Abomey-Calavi, Université d’Abomey-Calavi, BENIN

autor

Sabi Takou Daniel

• Fundamental Sciences Department, Ecole Polytechnique d’Abomey-Calavi, Université d’Abomey-Calavi, BENIN

• https://orcid.org/0009-0002-7603-8999

autor

Adetola Jamal

• Mathematical Modelisation Department, Université Nationale des Sciences,Technologie, Ingnierie et Mathéematiques, Abomey, BENIN

Bibliografia

• [1] Chesnokov A.A. (2009): Symmetries and exact solutions of the rotating shallow- water equations.– Euro. J. of App. Math., vol.20, No.5, pp.461-477, https://doi.org/10.1017/S0956792509990064.
• [2] Adanhounme V. and Codo F.P. (2012): Shallow water flow down an inclined open channel: analytical solutions of governing equations.– International Journal Scientific and Engineering Research, vol.3, No.6, p.1-5.
• [3] Ball F.K. (1963): Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid.– J. Fluid Mech, vol.17.pp.240-256, https://doi.org/10.1017/S0022112063001270.
• [4] Ball F.K. (1965): The effect of rotation on the simpler modes of motion of a liquid in an elliptic paraboloid.– J. Fluid Mech., pp.529-545, https://doi.org/10.1017/S0022112065000952.
• [5] Bila N., Mansfield E. and Clarkson P. (2006): Symmetry group analysis of the shallow water and semi-geostrophic equations.– Quart. J. Mech. Appl. Math., pp.95-123, DOI:10.1093/qjmam/hbi033.
• [6] LeVeque R.J. (1998): Gradients in high-resolution Godunov methods.– Journal of Computational Physics, vol.141, pp.541-549, https://doi.org/10.1006/jcph.1998.6058.
• [7] LeVeque R.J. (2007): Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems.– Society for Industrial and Applied Mathematics, vol.33, pp.627-665. DOI:10.1137/1.9780898717839.
• [8] Abbott M.B. and Basco D.R. (1989): Computational fluid dynamics.– J. Fluid Mech. vol.229, pp.689 https://doi.org/10.1017/S0022112091213233.
• [9] García-Navarro P. and Vázquez-Cendón M.E. (2000): On numerical treatment of the source terms in the shallow water equations.– Computers & Fluids, vol.29, pp.951-979, https://doi.org/10.1016/S0045-7930(99)00038-9.
• [10] Marche F., Bonneton P., Fabrie P. and Seguin N. (2007): Evaluation of well-balanced bore capturing schemes for 2D wetting and drying processes.– Int. J. Numer. Methods Fluids, vol.53. pp.867-894, https://doi.org/10.1002/fld.1311.
• [11] Brook B.S., Falle S.A. and Pedley T.J. (1999): Numerical solutions for unsteady gravity driven flows in collapsible tubes: evolution and roll-wave instability of a steady state.– J. Fluid Mech., vol.396. pp.223-256, https://doi.org/10.1017/s0022112099006084.
• [12] Chow V.T. (1959): Open Channel Hydraulics.– McGraw-Hill, New York.
• [13] Saint-Venant A.J.C. (1871): Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et a l’introduction de marées dans leurs lits.– Comptes Rendus des Séances de Académie des Sciences, vol.73, pp.147- 237.
• [14] Madsen P.A. and Sørensen O.R. (1992): A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry.– Coastal Engineering, vol.18, No.3-4, pp.183-204, https://doi.org/10.1016/03783839(92)90019-Q.
• [15] Dingman S.L. (2002): Physical Hydrology.– 2nd Edition, Prentice Hall, Upper Saddle River, p.646.
• [16] Hervouet V. (2007): La mobilité du quotidien dans les espaces périurbains, une grande diversité de modèles de déplacements.– Norois, vol.205, pp.37-52.
• [17] LeVeque R.J. (2002): Finite Volume Methods for Hyperbolic Problems.– Cambridge University Press, Cambridge, pp.64-86, doi.org/10.1017/CBO9780511791253.
• [18] Salingaros N. (1981): Realization, extension, and classification of certain physically important groups and algebras.– Journal of Mathematical Physics, vol.22, pp.226-232, doi.org/10.1063/1.524893.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025)