Depth-averaged specific energy in open-channel flow and analytical solution for critical irrotational flow over weirs

Free surface flow in open-channel transitions is characterized by distributions of velocity and pressure that deviate from uniform and hydrostatic conditions, respectively. Under such circumstances the widely used expressions in textbooks [e.g.,E=h+U2/(2g) and hc=(q2/g)1/3] are not valid to investigate the changes in velocity and depth. A depth-averaged form of the Bernoulli equation for ideal fluid flows introduces correction coefficients to account for the real velocity and pressure distributions into the specific energy equation. The behavior of these coefficients in curvilinear motion at and in the neighbourhood of control sections was not documented in the literature. Herein detailed two-dimensional ideal fluid flow computations are used to characterize the entire velocity and pressure fields in typical channel controls involving transcritical flow, namely the round-crested weir, the transition from mild to steep slope and the free overfall. The detailed two-dimensional ideal fluid flow solution is used to study the behavior of the depth-averaged coefficients, and a novel generalized specific energy diagram is introduced using universal coordinates. The development is used to pursue a simplified critical flow theory for curved flow, relevant to water discharge measurement with circular weirs

Closure to "Minimum specific energy and transcritical flow in unsteady open-channel flow"

The writers thank the discussers for their interest in the original paper, and the comments offered. During the inspection of the discussers’ assertions, it was found that most of them were unsupported by hydraulic analysis. Detailed replies to each comment are given, with bullet points used to differentiate between specific items to be presented

Ritter’s dry-bed dam-break flows: positive and negative wave dynamics

Dam-break flood waves are associated with major environmental disasters provoked by the sudden release of water stored in reservoirs. Ritter found in 1892 an analytical solution to the wave structure of an ideal fluid released during an instantaneous dam failure, propagating over initially dry horizontal terrain. This solution, though ideal, hence frictionless, is widely used to test numerical solutions of the Shallow Water Equations (SWE), and as educational tool in courses of fluid mechanics, given that it is a peculiar case of the Riemann problem. However, the real wave structure observed experimentally differs in a major portion of the wave profile, including the positive and negative fronts. Given the importance of an accurate prediction of the dam break wave, the positive and negative wave portions originating from the breaking of a dam with initially dry land on the tailwater reach are revisited in this work. First, the propagation features of the dry-front are investigated using an analytical boundary-layer type model (Whitham/Dressler/Chanson model) constructed matching an (outer) inviscid dynamic wave to an (inner) viscous diffusive wave. The analytical solution is evaluated using an accurate numerical solution of the SWE produced using the MUSCL-Hancock finite-volume method, which is tested independently obtaining the solution based on the discontinuous Galerkin finite-element method. The propagation features of the negative wave are poorly reproduced by the SWE during the initial stages of dam break flows, and, thus, are then investigated using the Serre–Green–Naghdi equations for weakly-dispersive fully non-linear water waves, which are solved using a finite volume-finite difference scheme

Undular Hydraulic Jump

[ES] La transición de régimen supercrítico a régimen subcrítico cuando el número de Froude Fo aguas arriba esta próximo a la unidad da lugar a un tren de ondas estacionario llamado resalto hidráulico ondulatorio. La caracterización del resalto ondulatorio es muy compleja, debido a que el tren de ondas invalida la hipótesis de presión hidrostática usada en modelos de flujo gradualmente variado, y a otros fenómenos como las ondas de choque del flujo supercrítico. El objetivo de este trabajo es presentar un modelo para el resalto hidráulico ondulatorio obtenido de las ecuaciones de Reynolds para flujo turbulento, asumiendo que el número de Reynolds R es elevado. Se presentan soluciones analíticas sencillas para mostrar las características físicas de la teoría, así como un modelo numérico para la integración de las ecuaciones completas. El límite de aplicación de la teoría se discute en relación a la rotura de onda y formación de vórtices. La validez del modelo matemático es revisada de forma c[EN] The transition from subcritical to supercritical flow when the inflow Froude number Fo is close to unity appears in the form of steady state waves called undular hydraulic jump. The characterization of the undular hydraulic jump is complex due to the existence of a non-hydrostatic pressure distribution that invalidates the gradually-varied flow theory, and supercritical shock waves. The objective of this work is to present a mathematical model for the undular hydraulic jump obtained from an approximate integration of the Reynolds equations for turbulent flow assuming that the Reynolds number R is high. Simple analytical solutions are presented to reveal the physics of the theory, and a numerical model is used to integrate the complete equations. The limit of application of the theory is discussed using a wave breaking condition for the inception of a surface roller. The validity of the mathematical predictions is critically assessed using physical data, thereby revealing aspects on whiCastro-Orgaz, O.; Roldan Cañas, J.; Dolz Ripolles, J. (2015). Resalto Hidráulico Ondulatorio. Ingeniería del agua. 19(2):63-74. https://doi.org/10.4995/ia.2015.3321OJS6374192Benjamin, T.B., Lighthill, M.J. (1954). On cnoidal waves and bores. Proc. Roy. Soc. London A 224, 448-460.Bose, S.K., Dey, S. (2007). Curvilinear flow profiles based on Reynolds averaging. J. Hydraulic Engng. 133(9), 1074-1079.Boussinesq, J. (1877). Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants à l'Académie des Sciences, Paris 23, 1-680 in French.Bose, S.K., Castro-Orgaz, O., Dey, S. (2012). Free surface profiles of undular hydraulic jumps. J. Hydraulic Engng. 138 (4), 362-366.Castro-Orgaz, O., Chanson, H. (2011). Near-critical free-surface flows: Real fluid flow analysis. Environmental Fluid Mechanics 11 (5), 499-516.Castro-Orgaz, O., Hager, W.H. (2011a). Joseph Boussinesq and his theory of water flow in open channels. J. Hydraulic Res. 49 (5), 569-577.Castro-Orgaz, O., Hager, W.H. (2011b). Observations on undular hydraulic jump in movable bed. J. Hydraulic Res. 49 (5), 689-692.Castro-Orgaz, O., Hager, W.H. (2011c). Turbulent near-critical open channel flow: Serre's similarity theory. J. Hydraulic Engng. 137 (5), 497-503.Castro-Orgaz, O. (2010). Weakly undular hydraulic jump: Effects of friction. J. Hydraulic Res. 48 (4), 453-465.Chanson, H. (1993). Characteristics of undular hydraulic jumps. Res. Rep. CE146. Dept. Civ. Engng., University of Queensland, Brisbane Australia.Chanson, H. (1995). Flow characteristics of undular hydraulic jumps: Comparison with near-critical flows. Res. Rep. CH45/95. Dept. Civ. Engng., University of Queensland, Brisbane Australia.Chanson, H. (1996). Free surface flows with near critical flow conditions. Can. J. Civ. Engng. 23(6), 1272-1284.Chanson, H. (2000). Boundary shear stress measurements in undular flows: Application to standing wave bed forms. Water Resources Res. 36(10), 3063-3076.Chanson, H. (2009). Current knowledge in hydraulic jumps and related phenomena: A survey of experimental results. European J. Mechanics B/Fluids 28, 191-210.Chanson, H., Montes, J.S. (1995). Characteristics of undular hydraulic jumps: Experimental apparatus and flow patterns. J. Hydraulic Engng. 121(2), 129-144. Discussion: 1997, 123(2), 161-164.Chaudhry, M.H. (2008). Open-channel flow, 2nd ed. Springer, New York.Fawer, C. (1937). Etude de quelques écoulements permanents à filets courbes. Thesis. Université de Lausanne. La Concorde, Lausanne, Switzerland in French.Gotoh, H., Yasuda, Y., Ohtsu, I. (2005). Effect of channel slope on flow characteristics of undular hydraulic jumps. Trans. Ecology and Environment 83, 33-42.Grillhofer, W., Schneider, W. (2003). The undular hydraulic jump in turbulent open channel flow at large Reynolds numbers. Physics of Fluids 15(3), 730-735.Hager, W.H., Hutter, K. (1984). On pseudo-uniform flow in open channel hydraulics. Acta Mech. 53(3-4), 183-200. doi:10.1007/BF01177950Iwasa, Y. (1955). Undular jump and its limiting conditions for existence. Proc. 5th Japan Natl. Congress Applied Mech. II-14, 315-319.Mandrup-Andersen, V. (1978). Undular hydraulic jump. J. Hydraulics Div. ASCE 104(HY8), 1185-1188; Discussion: 105(HY9), 1208-1211.Marchi, E. (1963). Contributo allo studio del risalto ondulato. Giornale del Genio Civile 101(9), 466-476.Montes, J.S. (1986). A study of the undular jump profile. 9th Australasian Fluid Mech. Conf. Auckland, 148-151.Montes, J.S. (1998). Hydraulics of open channel flow. ASCE Press, Reston VA.Montes, J.S., Chanson, H. (1998). Characteristics of undular hydraulic jumps: Results and calculations. J. Hydraulic Engng. 124 (2), 192-205.Ohtsu, I., Yasuda, Y., Gotoh, H. (2001). Hydraulic condition for undular jump formations. J. Hydraulic Res. 39(2), 203-209; Discussion 40(3), 379-384.Ohtsu, I., Yasuda, Y., Gotoh, H. (2003). Flow conditions of undular hydraulic jumps in horizontal rectangular channels. J. Hydraulic Engng. 129(12), 948-955.Reinauer, R., Hager, W.H. (1995). Non-breaking undular hydraulic jump. J. Hydraulic Res. 33(5), 1-16; Discussion 34(2), 279-287; 34(4), 567-573.Ryabenko, A.A. (1990). Conditions for the favourable existence of an undulating jump. Hydrotechnical Construction 24(12), 762-770.Serre, F. (1953). Contribution à l'étude des écoulements permanents et variables dans les canaux. La Houille Blanche 8(6-7), 374-388; 8(12), 830-887 [in French].White, F.M. (1991). Viscous fluid flow. McGraw-Hill, New York

Bernoulli theorem, minimum specific energy and water wave celerity in open channel flow

One basic principle of fluid mechanics used to resolve practical problems in hydraulic engineering is the Bernoulli theorem along a streamline, deduced from the work-energy form of the Euler equation along a streamline. Some confusion exists about the applicability of the Bernoulli theorem and its generalization to open-channel hydraulics. In the present work, a detailed analysis of the Bernoulli theorem and its extension to flow in open channels are developed. The generalized depth-averaged Bernoulli theorem is proposed and it has been proved that the depth-averaged specific energy reaches a minimum in converging accelerating free surface flow over weirs and flumes. Further, in general, a channel control with minimum specific energy in curvilinear flow is not isolated from water waves, as customary state in open-channel hydraulics