Numerical Modeling of Supercritical Waves in Bends with the Finite Volume Method of Roe-TVD and Appraisal of Analytical Assumptions



In this research, using the finite volume method of Roe-TVD, supercritical flow in the curved channel of Reinauer and Hager was studied and the results were compared with the analytical method of Knapp-Ippen, the numerical method of HLL and the available experimental data of Reinauer and Hager. Then, using the numerical results, the accuracy of the assumptions of the analytical method was evaluated. It was observed that the super-critical oblique standing waves are diffused along the bend way. With an inlet Froude number, Fr0<4.2, the assumptions of constant average cross-sectional velocity along the bend and frictionless flow or constant specific energy is acceptable with an error of around one percent and the velocity variation at the external wall is tolerable with a maximum error of four percent. By increasing the inlet Froude number, flow at the internal wall dries up and the above assumptions are invalidated.


[1]Karman,. V.; "Eine Praktische Anwendung Der Analogie Zwischen Ueberschall-Strömung In Offenen Gerinnen" ZAMM, Vol. 18, pp. 49-56, 1938.
[2]Knapp R.T. and Ippen A.T.; " Curvilinear flow of liquids with free surfaces at velocities above that of wave propagation" Proc. 5th Int. Congr. of Appl.Mech., Cambridge University Press, NewYork, pp.531-536, 1938.
[3]Ghaeini Hessaroeyeh, M. and Tahershamsi, A.; " Analytical model of supercritical flow in rectangular chute bends" J. Hydr. Res., Vol. 47, No. 5, pp. 566-573, 2009.
[4]Reinauer, R. and Hager, H.; "Supercritical Bend Flow" J. Hydr. Engng., Vol. 123(3), pp. 208-218, 1997.
[5]Poggi, B.; "Correnti veloci nei canali in curva" L’Energia electrica, Vol. 33, pp.465–480,1956.(Italian)
[6]Beltrami, G.M., Repetto, R. and Del guzzo, A.; "A Simple Method To Regularize Supercritical Flow Profiles In Bends" J. Hydr. Res., Vol. 45, No. 6,pp.773-786, 2007.
[7]Valiani, A. and Caleffi, V.; "Brief Analysis Of Shallow Water Equations Suitability To Numerically Simulate Supercritical Flow In Sharp Bends" J. Hydr. Engng.,Vol. 131, No. 10, pp. 912-916, 2001.
[8]Chow, V. T.; Open Channel Hydraulics, McGraw-Hill Intern, 680p, 1986.
[9]Knapp, R. T.; "Design Of Channel Curves For Supercritical Flow" 2nd paper in High-velocity flow in open channels: A symposium, Transactions, ASCE,Vol. 116, pp. 296-325, 1951.
[10]Toro, E.; Shock Capturing Methods For Free Surface Shallow Flows, John Wiley, ChiChester, New York,308p, 2001.
[11]Brufau1, P. and Garcia-Navarro, P.; "Two- Dimensional Dam Break Flow Simulation" Int. J.Numer. Meth. Fluids, Vol. 33, pp. 35–57, 2000.
[12]Gomez, L.; "An Unstructured Finite Volume Model For Unsteady Turbulent Shallow Water Flow With Wet-dry Fronts Numerical Solver And Experimental Validation" thesis doctoral, Universidad de a Coruna,248p, 2005.
[13]Leveque, R. J.; Finite Volume Methods For Hyperbolic Systems, Cambridge University press, New York,558p, 2002.