Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Forced Hydraulic Jumps Described by Classic Hydraulic Equations Reproducing Cusp Catastrophe Features

  • 55 Accesses

  • 3 Citations


The information content of the classic equations describing the problem of forced hydraulic jumps in open channels is the subject of this paper. The forcing refers to designed structural composites to transform incoming supercritical flows into outgoing subcritical flows through hydraulic jumps. The complete flow history through such systems is described by two stable flow profiles, which are interlocked and one can jump to the other. The interlocked profiles underpin a hydraulic effect known as hysteresis with the prime feature of the dependence of flows on their past history. The paper develops an algorithm by using classic hydraulic equations alone to identify hysteresis by building on tacit knowledge already published. The algorithm reproduces the effects of catastrophe theory and shows that hysteresis has predictable configurations. It is validated for geometries with step-rises by using published experimental data; as well as a flume with abrupt contractions by authors’ experimental data.

This is a preview of subscription content, log in to check access.


  1. 1.

    Abecasis, F.M.; Quintela, A.C.: Hysteresis in steady free-surface flow. Water Power 16, 147–151 (1964)

  2. 2.

    Austria, P.M.: Catastrophe model for the forced hydraulic jump. J. Hydraul. Res. 25, 269–280 (1987)

  3. 3.

    Muskatrovic, D., Batinic, B.: The influence of abrupt change of channel geometry on hydraulic regime characteristics. Paper presented at the meeting of 17th General Meeting International Association for Hydraulic Research, Baden-Baden, Germany, vol. 2, pp. 397–404 (1977)

  4. 4.

    Thom, R.: Stabilité Structurelle et Morphogénèse. Benjamin, New York (1972)

  5. 5.

    Zeeman, E.C.: Euler Buckling, Symposium on Structural Stability, The Theory of Catastrophes and Applications in the Sciences. Lectures Notes in Mathematics 525. Springer, Berlin (1976)

  6. 6.

    Zeeman, E.C.: A catastrophe model for the stability of ships. In: Palis, J., do Carmo, M. (eds.) Geometry and Topology. Lecture Notes in Mathematics, vol. 597, pp. 775–827. Springer, Berlin (1977)

  7. 7.

    Baines, P.G.; Davies, P.A.: Laboratory studies of topographic effects in rotating and /or stratified fluids. In: WMO Orographic Effects in Planetary Flows, vol. 1, pp. 233–299 (1980)

  8. 8.

    Lawrence, G.A.: Steady flow over an obstacle. J. Hydraul. Eng. 113, 981–991 (1987)

  9. 9.

    Baines, P.G.; Whitehead, J.A.: On multiple state of single layer flow. Phys. Fluids 15, 298–307 (2003)

  10. 10.

    Defina, A.; Susin, F.M.: Hysteretic behavior of the flow under a vertical sluice gate. Phys. Fluids 15, 2541–2548 (2003)

  11. 11.

    Defina, A.; Susin, S.M.: Multiple states in open channel flow, invorticity and turbulence effects in fluid structures interactions. In: Brocchini, M., Trivellato, F. (eds.) Advances in Fluid Mechanics, pp. 105–130. Wessex Institute of Technology Press, Southampton (2006)

  12. 12.

    Babaali, H.; Shamsai, A.; Vosoughifar, H.: Computational modeling of the hydraulic jump in the stilling basin with convergence walls using CFD codes. Arab. J. Sci. Eng. 40, 381–395 (2015)

  13. 13.

    Das, R.; Pal, D.; Das, S.; Mazumdar, A.: Study of energy dissipation on inclined rectangular contracted chute. Arab. J. Sci. Eng. 39, 6995–7002 (2014)

  14. 14.

    Cobb, L.: Stochastic catastrophe models and multimodal distributions. Behav. Sci. 23, 360–374 (1978)

  15. 15.

    Rajaratnam, N.; Humphries, J.A.: Free flow upstream of vertical sluice gates. J. Hydraul. Res. 20, 427–437 (1982)

  16. 16.

    Belaud, G.; Cassan, L.; Baume, J.P.: Calculation of contraction coefficient under sluice gates and application to discharge measurement. J. Hydraul. Eng. 135, 1086–1091 (2009)

Download references

Author information

Correspondence to Rahman Khatibi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sadeghfam, S., Khatibi, R., Hassanzadeh, Y. et al. Forced Hydraulic Jumps Described by Classic Hydraulic Equations Reproducing Cusp Catastrophe Features. Arab J Sci Eng 42, 4169–4179 (2017).

Download citation


  • Cusp catastrophe flags
  • Hysteresis
  • Flow profiles
  • Flow regimes (subcritical, critical, supercritical)
  • Forced hydraulic jumps