Abstract
Turbulent plane flow over a bottom of constant slope is considered for very large Reynolds numbers, very small slopes of the bottom, and Froude numbers close to the critical value 1. In contrast to a previous work (Grillhofer and Schneider, Phys Fluids 15:730–735, 2003), it is not assumed that the flow far upstream is fully developed. The first-order perturbation equations contain unknown functions that are determined from a solvability condition of the second-order equations. Without making use of turbulence modeling or empirical parameters, a third-order ordinary differential equation is obtained for the shape of the free surface. Slow changes of amplitudes and wave lengths, respectively, associated with a small damping parameter are described by a multiple-scales solution, which also reveals the source of peculiarities of numerical solutions. A universal diagram of solutions and a universal map of the initial conditions that lead to undular jumps are given. Both numerical and multiple-scales solutions are compared with experimental data.
Similar content being viewed by others
References
Chow V.T.: Open-Channel Hydraulics. McGraw-Hill, New York (1959)
Henderson F.: Open Channel Flow. Macmillan, New York (1966)
Van Dyke M.: An Album of Fluid Motion. 4th printing. Parabolic Press, Stanford (1988)
Ohtsu I., Yasuda Y., Gotoh H.: Flow conditions of undular hydraulic jumps in horizontal rectangular channels. J. Hydraul. Eng. 129, 948–955 (2003)
Kaufmann K.: Hydromechanik II. Springer, Berlin (1934)
Johnson R.S.: Shallow water waves on a viscous fluid—the undular bore. Phys. Fluids 15, 1693–1699 (1972)
Johnson R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997)
Andersen V.M.: Undular hydraulic jump. J. Hydraul. Div. 104, 1185–1188 (1978)
Hager W.H., Hutter K.: On pseudo-uniform flow in open channel hydraulics. Acta Mech. 53, 183–200 (1984)
Montes J.S., Chanson H.: Characteristics of undular hydraulic jumps: experiments and analysis. ASCE J. Hydraul. Eng. 124, 192–205 (1998)
Caputo J.-G., Stepanyants Y.A.: Bore formation, evolution and disintegration into solitons in shallow inhomogeneous channels. Nonlin. Process. Geophys. 10, 407–424 (2003)
El G.A., Grimshaw R.H.J., Kamchatkov A.M.: Analytic model for a weakly dissipative shallow-water undular bore. Chaos 15, 037102 (2004)
El G.A., Grimshaw R.H.J., Smyth N.F.: Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18, 027104-1–027104-17 (2006)
El G.A., Grimshaw R.H.J., Kamchatkov A.M.: Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction. J. Fluid Mech. 585, 213–244 (2007)
Abd-el-Malek M.B., Helal M.M.: Group method solutions of the generalized forms of Burgers, Burgers–KdV and KdV equations with time-dependent variable coefficients. Acta Mech. 221, 281–296 (2011)
Grimshaw, R.: Korteweg de-Vries Equation. In: Nonlinear Waves in Fluids: Recent Advances and Modern Applications. CISM Courses and Lectures No. 483, pp. 1–28. Springer (2005)
Grillhofer W., Schneider W.: The undular hydraulic jump in turbulent open channel flow at large Reynolds numbers. Phys. Fluids 15, 730–735 (2003)
Grimshaw R.: Internal solitary waves in a variable medium. GAMM-Mitteilungen 30/1, 96–109 (2007)
Steinrück H., Schneider W., Grillhofer W.: A multiple scales analysis of the undular hydraulic jump in turbulent open channel flow. Fluid Dyn. Res. 33, 41–55 (2003)
Steinrück H.: Multiple scales analysis of the steady-state Korteweg–de Vries equation perturbed by a damping term. ZAMM 85, 114–121 (2005). doi:10.1002/zamm.200310162
Steinrück, H.: Multiple scales analysis of the turbulent undular hydraulic jump. In: Steinrück, H. (eds.) Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances. CISM Courses and Lectures No. 523, pp. 197–219 (2010)
Schneider W., Jurisits R., Bae Y.S.: An asymptotic iteration method for the numerical analysis of near-critical free-surface flows. Int. J. Heat Fluid Flow 31, 1119–1124 (2010)
Ohtsu I., Yasuda Y., Gotoh H., Iahr M.: Hydraulic condition for undular-jump formations. J. Hydraul. Res. 39, 203–209 (2001)
Gotoh, H., Yasuda, Y., Ohtsu, I.: Effect of channel slope on flow characteristics of undular hydraulic jumps. In: Brebbia, C.A., do Carmo, J.S.A. (eds.) River Basin Management III, 3rd International Conference on River Basin Management, Bologna 2005, WIT Transactions on Ecology and the Environment, pp. 33–42. WIT Press, Southampton (2005)
Schneider, W.: Near-critical free-surface flows. In: Proceedings of the 2nd Shanghai International Symposium on Nonlinear Science and Application (Shanghai NSA’05), Section 8.6 Free and Moving Boundary Problems (2005)
Jurisits, R., Schneider, W., Bae, Y.S.: The undular hydraulic jump—analytical and numerical investigations of the free boundary value problem. In: Proceedings of the 3rd Shanghai International Symposium on Nonlinear Science and Application (Shanghai NSA’07), pp. 90–91. Fudan University, Shanghai (2007)
Jurisits, R., Schneider, W., Bae, Y.S.: A multiple-scales solution of the undular hydraulic jump problem. Proc. Appl. Math. Mech. (PAMM) 7, 4120007–4120008 (2007). doi:10.1002/pamm.200700755. http://www3.interscience.wiley.com/cgi-bin/fulltext/121560064/PDFSTART
Jurisits, R.: Wellige Wassersprünge bei nicht voll ausgebildeter turbulenter Zuströmung. Dissertation, TU Wien (2012)
Gersten K., Herwig H.: Strömungsmechanik. Grundlagen der Impuls-, Wärme- und Stoffübertragung aus asymptotischer Sicht. Vieweg, Braunschweig (1992)
Kluwick, A. (ed.) Recent Advances in Boundary Layer Theory. CISM Courses and Lectures No. 390, Springer, Wien
Schlichting H., Gersten K.: Boundary Layer Theory. Springer, Berlin (2000)
Nezu I., Rodi W.: Open-channel flow measurements with a laser Doppler anemometer. ASCE J. Hydraul. Eng. 112, 335–355 (1986)
Lennon J.M., Hill D.F.: Particle image velocity measurements of undular and hydraulic jumps. ASCE J. Hydraul. Eng. 132, 1283–1294 (2006)
Gad-el-Hak M., Buschmann M.H.: Turbulent boundary layers: is the wall falling or merely wobbling?. Acta Mech. 218, 309–318 (2011)
Komori S., Nagaosa R., Murakami Y., Chiba S., Ishii K., Kuwahara K.: Direct numerical simulation of three-dimensional open-channel flow with zero-shear gas-liquid interface. Phys. Fluids A 5, 115–125 (1993)
Handler R.A., Swean T.F. Jr, Leighton R.I., Swearingen J.D.: Length scales and the energy balance for turbulence near a free surface. AIAA J. 31, 1998–2007 (1993)
Brocchini, M., Peregrine, D.H.: The modelling of a spilling breaker: strong turbulence at a free surface. In: Proceedings of the International Conference Coastal Engineering, pp. 72–85. ASCE (1998)
Svendsen I.A., Veeramony J., Bakunin J., Kirby J.T.: The flow in weak turbulent hydraulic jumps. J. Fluid Mech. 418, 25–57 (2000)
Grillhofer, W.: Der wellige Wassersprung in einer turbulenten Kanalströmung mit freier Oberfläche. Dissertation, Technische Universität Wien (2002)
Stoker J.J.: Water Waves. Interscience, New York (1957)
White F.: Fluid Mechanics. McGraw-Hill, New York (2009)
Chanson H., Montes J.S.: Characteristics of undular hydraulic jumps: experimental apparatus and flow patterns. J. Hydraul. Eng. 121, 129–144 (1995)
Reinauer R., Hager W.: Non-breaking undular hydraulic jump. J. Hydraul. Res. 33, 683–698 (1995)
Castro-Orgaz, O., Chanson, H.: Near-critical free-surface flows: real fluid flow analysis. Environ. Fluid Mech. doi:10.1007/s10652-010-9192-x (2010)
Dias F., Vanden-Broek J.-M.: Generalised critical free-surface flows. J. Eng. Math. 42, 291–301 (2002)
Dias F., Vanden-Broek J.-M.: Trapped waves between submerged obstacles. J. Fluid Mech. 509, 93–102 (2004)
Binder B.J., Vanden-Broek J.-M., Dias F.: Forced solitary waves and fronts past submerged obstacles. Chaos 15, 037106 (2005)
Chardard F., Dias F.: Stability of some stationary solutions to the forced KdV equation with one or two bumps. J. Eng. Math. 70, 175–189 (2011)
Van Dyke M.: Perturbation Methods in Fluid Mechanics. Annotated Ed. Parabolic Press, Stanford (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Hans Irschik on the occasion of his 60th birthday.
Rights and permissions
About this article
Cite this article
Jurisits, R., Schneider, W. Undular hydraulic jumps arising in non-developed turbulent flows. Acta Mech 223, 1723–1738 (2012). https://doi.org/10.1007/s00707-012-0666-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-012-0666-4