Acta Mechanica

, Volume 229, Issue 11, pp 4701–4725 | Cite as

Near-critical turbulent open-channel flows over bumps and ramps

  • Wilhelm SchneiderEmail author
  • Markus Müllner
  • Youichi Yasuda
Open Access
Original Paper


Steady two-dimensional turbulent free-surface flow in a channel with a slightly uneven bottom is considered. The shape of the unevenness of the bottom can be in the form of a bump or a ramp of very small height. The slope of the channel bottom is assumed to be small, and the bottom roughness is assumed to be constant. Asymptotic expansions for very large Reynolds numbers and Froude numbers close to the critical value \({Fr} = 1\), respectively, are performed. The relative order of magnitude of two small parameters, i.e. the bottom slope and \(({Fr}-1)\), is defined such that no turbulence modelling is required. The result is a steady-state version of an extended Korteweg–de Vries equation for the surface elevation. Other flow quantities, such as pressure, flow velocity components, and bottom shear stress, are expressed in terms of the surface elevation. An exact solution describing stationary solitary waves of the classical shape is obtained for a bottom of a particular shape. For more general shapes of ramps and bumps, stationary solitary waves of the classical shape are also obtained as a first approximation in the limit of small, but nonzero, dissipation. With the exception of an eigensolution for a ramp, an outer region has to be introduced. The outer solution describes a ’tail’ that is attached to the stationary solitary wave. In addition to the solutions of the solitary-wave type, solutions of smaller amplitudes are obtained both numerically and analytically. Experiments in a water channel confirm the existence of both types of stationary single waves.



Open access funding provided by TU Wien (TUW). The authors are indebted to Prof. Oscar Castro-Orgaz for his encouragement to include variations in bottom shape into the asymptotic analysis of turbulent free-surface flow. Dr. Richard Jurisits’ yet unpublished numerical solutions of the extended KdV equation helped the authors to cope with numerical problems and find the solutions of the second kind. Dr. Christoph Buchner, Dr. Richard Jurisits, Dr. Bernhard Scheichl and a reviewer provided useful references. The reviewers’ comments led to various improvements of the presentation, including the supply of additional information in three appendices. Mr. Dominik Murschenhofer prepared the Open image in new window file. Finally, financial support by Androsch International Management Consulting GmbH is gratefully acknowledged.

Supplementary material

Supplementary material 1 (mp4 14048 KB)


  1. 1.
    Binder, B.J., Blyth, M.G., Balasuriya, S.: Non-uniqueness of steady free-surface flow at critical Froude number. EPL 105(4), 44003 (2014). CrossRefGoogle Scholar
  2. 2.
    Binder, B.J., Dias, F., Vanden-Broeck, J.M.: Influence of rapid changes in a channel bottom on free-surface flows. IMA J. Appl. Math. 73, 254–273 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bowen, M.K., Smith, R.: Derivative formulae and errors for non-uniformly spaced points. Proc. R. Soc. Lond. Ser. A 461(2059), 1975–1997 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Camassa, R., Wu, Y.T.: Stability of forced steady solitary waves. Philos. Trans. R. Soc. Lond. A 337(1648), 429–466 (1991). MathSciNetCrossRefGoogle Scholar
  5. 5.
    Camassa, R., Wu, Y.T.: Stability of some stationary solutions for the forced KdV equation. Physica D 51(1–3), 295–307 (1991). MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cantero-Chinchilla, F.N., Castro-Orgaz, O., Khan, A.A.: Depth-integrated nonhydrostatic free-surface flow modelling using weighted-averaged equations. Int. J. Numer. Methods Fluids 87, 27–50 (2018). CrossRefGoogle Scholar
  7. 7.
    Castro-Orgaz, O.: Weakly undular hydraulic jump: effects of friction. J. Hydr. Res. 48(4), 453–465 (2010). CrossRefGoogle Scholar
  8. 8.
    Castro-Orgaz, O., Chanson, H.: Near-critical free-surface flows: real fluid flow analysis. Environ. Fluid Mech 11(5), 499–516 (2011)CrossRefGoogle Scholar
  9. 9.
    Castro-Orgaz, O., Hager, W.H.: Observations on undular hydraulic jump in movable bed. J. Hydr. Res. 49(5), 689–692 (2011)CrossRefGoogle Scholar
  10. 10.
    Castro-Orgaz, O., Hager, W.H., Dey, S.: Depth-averaged model for undular hydraulic jump. J. Hydr. Res. 53(3), 351–363 (2015)CrossRefGoogle Scholar
  11. 11.
    Castro-Orgaz, O., Hager, W.H.: Non-hydrostatic Free Surface Flows. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin (2017). CrossRefGoogle Scholar
  12. 12.
    Chanson, H.: The Hydraulics of Open Channel Flow: An Introduction, 2nd edn. Elsevier, Butterworth-Heinemann, Oxford (2004). Google Scholar
  13. 13.
    Chardard, F., Dias, F., Nguyen, H.Y., Vanden-Broeck, J.M.: Stability of some stationary solutions to the forced KdV equation with one or two bumps. J. Eng. Math. 70(1–3), 175–189 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chow, V.T.: Open-Channel Hydraulics. McGraw-Hill, New York (1959)Google Scholar
  15. 15.
    Christov, C.I., Velarde, M.: Dissipative solitons. Physica D 86(1–2), 323–347 (1995). MathSciNetCrossRefGoogle Scholar
  16. 16.
  17. 17.
    Dias, F., Vanden-Broeck, J.M.: Generalised critical free-surface flows. J. Eng. Math. 42(3), 291–301 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dutykh, D.: Visco-potential free-surface flows and long wave modelling. Eur. J. Mech. B Fluids 28(3), 430–443 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dutykh, D., Dias, F.: Viscous potential free-surface flows in a fluid layer of finite depth. C. R. Acad. Sci. Paris Ser. I 345(2), 113–118 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
  21. 21.
    Gersten, K.: Turbulent boundary layers I: fundamentals. In: Kluwick, A. (ed.) Recent Advances in Boundary Layer Theory, CISM Courses and Lectures, vol. 390, pp. 107–144. Springer, Wien (1998). CrossRefGoogle Scholar
  22. 22.
    Gong, L., Shen, S.S.: Multiple supercritical solitary wave solutions of the stationary forced Korteweg–de Vries equation and their stability. SIAM J. Appl. Math. 54(5), 1268–1290 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gotoh, H., Yasuda, Y., Ohtsu, I.: Effect of channel slope on flow characteristics of undular hydraulic jumps. In: Brebbia, C., do Carmo, J.S.A. (eds.) River Basin Management III, vol. 83, pp. 33–42. WIT Press, Southampton (2005)Google Scholar
  24. 24.
    Grillhofer, W.: Der wellige Wassersprung in einer turbulenten Kanalströmung mit freier Oberfläche. Dissertation. Technische Universität Wien, Vienna (2002)Google Scholar
  25. 25.
    Grillhofer, W., Schneider, W.: The undular hydraulic jump in turbulent open channel flow at large Reynolds numbers. Phys. Fluids 15(3), 730–735 (2003)CrossRefGoogle Scholar
  26. 26.
    Grimshaw, R.: Exponential asymptotics and generalized solitary waves. In: Steinrück H. (ed.) Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances, CISM Courses and Lectures, vol. 523, pp. 71–120. Springer, Wien, New York (2010)CrossRefGoogle Scholar
  27. 27.
    Grimshaw, R.: Transcritical flow past an obstacle. ANZIAM J. 52(1), 2–26 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Grimshaw, R., Zhang, D.H., Chow, K.W.: Generation of solitary waves by transcritical flow over a step. J. Fluid Mech. 587, 235–254 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hager, W.H., Castro-Orgaz, O.: Transcritical flow in open channel hydraulics: from Böss to De Marchi. J. Hydr. Eng. 142(1), 02515003 (2016). CrossRefGoogle Scholar
  30. 30.
    Handler, R.A., Swean Jr., T.F., Leighton, R.I., Swearingen, J.D.: Length scales and the energy balance for turbulence near a free surface. AIAA J. 31(11), 1998–2007 (1993)CrossRefGoogle Scholar
  31. 31.
    Hassanzadeh, R., Sahin, B., Ozgoren, M.: Large eddy simulation of free-surface effects on the wake structures downstream of a spherical body. Ocean Eng. 54, 213–222 (2012)CrossRefGoogle Scholar
  32. 32.
    Hös, C., Kullmann, L.: A numerical study on the free-surface channel flow over a bottom obstacle. In: Conference on Modelling Fluid Flow (CMFF’06). The 13th International Conference on Fluid Flow Technologies, pp. 500–506. Budapest, Hungary (2006)Google Scholar
  33. 33.
    Jurisits, R.: Wellige Wassersprünge bei nicht voll ausgebildeter turbulenter Zuströmung. Dissertation, Technische Universität Wien, Vienna (2012)Google Scholar
  34. 34.
    Jurisits, R.: Transient numerical solutions of an extended Korteweg–de Vries equation describing solitary waves in open-channel flow. Period. Polytech. Mech. Eng. 61(1), 55–59 (2017)CrossRefGoogle Scholar
  35. 35.
    Jurisits, R., Schneider, W.: Undular hydraulic jumps arising in non-developed turbulent flows. Acta Mech. 223(8), 1723–1738 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Jurisits, R., Schneider, W., Bae, Y.S.: A multiple-scales solution of the undular hydraulic jump problem. Proc. Appl. Math. Mech. (PAMM) 7(1), 4120,007–4120,008 (2007). CrossRefGoogle Scholar
  37. 37.
    Kichenassamy, S., Olver, P.J.: Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. SIAM J. Math. Anal. 23(5), 1141–1166 (1992)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Kluwick, A.: Interacting laminar and turbulent boundary layers. In: Kluwick A. (ed.) Recent Advances in Boundary Layer Theory, CISM Courses and Lectures, vol. 390, pp. 231–330. Springer, Wien (1998)CrossRefGoogle Scholar
  39. 39.
    Knickerbocker, C.J., Newell, A.C.: Shelves and the Korteweg–de Vries equation. J. Fluid Mech. 98, 803–818 (1980). MathSciNetCrossRefGoogle Scholar
  40. 40.
    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(1), 115–125 (1993)CrossRefGoogle Scholar
  41. 41.
    Lennon, J.M., Hill, D.F.: Particle image velocity measurements of undular and hydraulic jumps. J. Hydr. Eng. 132(12), 1283–1294 (2006). CrossRefGoogle Scholar
  42. 42.
    Lovecchio, S., Zonta, F., Soldati, A.: Upscale energy transfer and flow topology in free-surface turbulence. Phys. Rev. E 91, 033,010 (2015). CrossRefGoogle Scholar
  43. 43.
    Madsen, P.A., Svendsen, I.A.: On the form of the integrated conservation equations for waves in the surf zone. Prog. Rep. 48, 31–39 (1979). (Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark) Google Scholar
  44. 44.
    Marchant, T.R.: Coupled Korteweg–de Vries equations describing, to high-order, resonant flow of a fluid over topography. Phys. Fluids 11(7), 1797–1804 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Miles, J.W.: Solitary wave evolution over a gradual slope with turbulent friction. J. Phys. Oceanogr. 13(3), 551–553 (1983).<0551:SWEOAG>2.0.CO;2 CrossRefGoogle Scholar
  46. 46.
    Miles, J.W.: Wave evolution over a gradual slope with turbulent friction. J. Fluid Mech. 133, 207–216 (1983). CrossRefzbMATHGoogle Scholar
  47. 47.
    Müllner, M.: Solutions of an extended KdV equation describing single stationary waves with strong or weak downstream decay in turbulent open-channel flow. ZAMM Z. Angew. Math. Mech 98(1), 7–30 (2018). MathSciNetCrossRefGoogle Scholar
  48. 48.
    Müllner, M., Schneider, W.: Asymptotic solutions of an extended Korteweg–de Vries equation describing solitary waves with weak or strong downstream decay in turbulent open-channel flow. Proc. Appl. Math. Mech. (PAMM) 15(1), 491–492 (2015). CrossRefGoogle Scholar
  49. 49.
    Müllner, M., Schneider, W.: Stationary single waves in turbulent open-channel flow. Proc. Appl. Math. Mech. (PAMM) 17, 683–684 (2017). CrossRefGoogle Scholar
  50. 50.
    Narayanan, C., Lakehal, D., Botto, L., Soldati, A.: Mechanisms of particle deposition in a fully developed turbulent open channel flow. Phys. Fluids 15(3), 763–775 (2003)CrossRefGoogle Scholar
  51. 51.
    Newell, A.C.: Solitons in Mathematics and Physics. No. 48 in CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1985).
  52. 52.
    Nezu, I., Rodi, W.: Open-channel flow measurements with a laser Doppler anemometer. J. Hydr. Eng. 112(5), 335–355 (1986). CrossRefGoogle Scholar
  53. 53.
    Ohtsu, I., Yasuda, Y., Gotoh, H.: Hydraulic condition for undular-jump formations. J. Hydr. Res. 39(2), 203–209 (2001)CrossRefGoogle Scholar
  54. 54.
    Ohtsu, I., Yasuda, Y., Gotoh, H.: Flow conditions of undular hydraulic jumps in horizontal rectangular channels. J. Hydr. Eng. 129(12), 948–955 (2003)CrossRefGoogle Scholar
  55. 55.
    Pelinovsky, E.N., Stepanyants, Y., Talipova, T.: Nonlinear dispersion model of sea waves in the coastal zone. J. Korean Soc. Coast. Ocean Eng. 5(4), 307–317 (1993)Google Scholar
  56. 56.
    Rednikov, A.Y., Velarde, M.G., Ryazantsev, Y.S., Nepomnyashchy, A.A., Kurdyumov, V.N.: Cnoidal wave trains and solitary waves in a dissipation-modified Korteweg-de Vries equation. Acta Appl. Math. 39(1–3), 457–475 (1995)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Rodi, W.: Turbulence Models and their Application in Hydraulics, 3rd edn. Balkema, Rotterdam (1993)Google Scholar
  58. 58.
    Rostami, F., Yazdi, S.R.S., Said, M.A.M., Shahrokhi, M.: Numerical simulation of undular jumps on graveled bed using volume of fluid method. Water Sci. Technol. 66(5), 909–917 (2012)CrossRefGoogle Scholar
  59. 59.
    Schlichting, H., Gersten, K.: Boundary-Layer Theory, 9th edn. Springer, Berlin (2017)CrossRefGoogle Scholar
  60. 60.
    Schneider, W.: Solitary waves in turbulent open-channel flow. J. Fluid Mech. 726, 137–159 (2013)MathSciNetCrossRefGoogle Scholar
  61. 61.
    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(6), 1119–1124 (2010). CrossRefGoogle Scholar
  62. 62.
    Schneider, W., Yasuda, Y.: Stationary solitary waves in turbulent open-channel flow: analysis and experimental verification. J. Hydr. Eng. 142(1), 04015,035 (2015). CrossRefGoogle Scholar
  63. 63.
    Scott, A.: Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford Texts in Applied and Engineering Mathematics, 2nd edn. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  64. 64.
    Steinrück, H., Schneider, W., Grillhofer, W.: A multiple scales analysis of the undular hydraulic jump in turbulent open channel flow. Fluid Dyn. Res. 33(1–2), 41–55 (2003). In memoriam: Prof. Philip Gerald Drazin 1934-2002MathSciNetCrossRefGoogle Scholar
  65. 65.
    Svendsen, I.A., Veeramony, J., Bakunin, J., Kirby, J.T.: The flow in weak turbulent hydraulic jumps. J. Fluid Mech. 418, 25–57 (2000)CrossRefGoogle Scholar
  66. 66.
    Sykes, R.: An asymptotic theory of incompressible turbulent boundary-layer flow over a small hump. J. Fluid Mech. 101, 647–670 (1980)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Vanden-Broeck, J.M.: Free-surface flow over an obstruction in a channel. Phys. Fluids 30(8), 2315–2317 (1987). CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of Fluid Mechanics and Heat TransferTechnische Universität WienViennaAustria
  2. 2.Institute of AerodynamicsRWTH Aachen UniversityAachenGermany
  3. 3.Department of Civil Engineering, College of Science and TechnologyNihon UniversityTokyoJapan

Personalised recommendations