Abstract
In this study, physical experiments were performed in a horizontal rectangular channel to investigate the evolution of undular surges that propagate in still water and the reflection of these surges on a vertical wall. The generated undular surges were located in the intermediate water depth wave regime. The results demonstrated that the maximum wave steepness attained by undular surges was close to the breaking limit of progressive waves at intermediate water depths. Nevertheless, the dispersion relationship was inconsistent with the wave dispersion trend for intermediate gravity waves and strongly depended on the initial flow conditions and surge type. Furthermore, it was found that the longitudinal current velocity was governed primarily by water surface displacement. A novel and universal formula based on the experimental results was proposed to describe the velocity distributions in different types of undular surge flows. In addition, a non-breaking undular surge behaved as a solitary wave regarding its reflection on a vertical wall. Overall, the findings highlighted the effects of the initial flow conditions on the hydrodynamic characteristics of undular surges and demonstrated the unique features of undular surges that propagate in still water.
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Abbreviations
- a w :
-
Wave amplitude (m)
- C :
-
Dispersion coefficient: 2πg*tanh(2πhw/Lw)/Lw
- Fr 0 :
-
Surge Froude number: Fr0=S/(gh0)0.5
- g :
-
Gravitational acceleration (m s−2)
- H :
-
Drop height (m)
- h :
-
Water depth (m)
- h 0 :
-
Initial water depth (m)
- h 1c :
-
Water depth (m) at the first wave crest
- h 1t :
-
Water depth (m) at the first wave trough
- K conj :
-
Conjugate water depth (m) behind the surge front
- h i :
-
Incident wave height (m)
- h r :
-
Reflected wave height (m)
- H w :
-
Wave height (m)
- h w :
-
Mean water depth (m) behind the surge front
- K :
-
Reflection coefficient: K=hr/hi
- L w :
-
Wave length (m)
- R 2 :
-
Correlation coefficient
- S :
-
Absolute surge velocity (m s−1), positive downstream
- t :
-
Time (s) from gate opening
- t 1c− :
-
Typical moment (s): t1c−=t1c − t1c1t
- t ic :
-
Time (s) corresponding to the passage of the ith wave crest
- t icit :
-
Typical moment (s): ticit=(tic + tit)/2
- t it :
-
Time (s) corresponding to the passage of the ith wave trough
- t it(i+1)c :
-
Typical moment (s): (tit + t(i+1)c)/2
- t v :
-
Time (s) needed to fully open the gate
- T w :
-
Wave period (s)
- Ur :
-
Ursell parameter: Ur=(awLw2)/hw3
- V x :
-
Instantaneous longitudinal velocity component (m s−1), positive downstream
- V y :
-
Instantaneous transverse velocity component (m s−1), positive towards the left sidewall
- V z :
-
Instantaneous vertical velocity component (m s−1), positive upwards
- x :
-
Longitudinal distance (m)
- y :
-
Transverse distance (m)
- z :
-
Vertical distance (m)
- π :
-
3.141592654
References
Benet F, Cunge JA (1971) Analysis of experiments on secondary undulations caused by surge waves in trapezoidal channels. Journal of Hydraulic Research 9(1):11–33, DOI: https://doi.org/10.1080/00221687109500335
Chambarel J, Kharif C, Touboul J (2009) Head-on collision of two solitary waves and residual falling jet formation. Nonlinear Processes in Geophysics 16(1):111–122, DOI: https://doi.org/10.5194/npg-16-111-2009
Chanson H (1995) Flow characteristics of undular hydraulic jumps: Comparison with near critical flows. Report No. CH45/95, The University of Queensland, Brisbane, Australia
Chanson H (2010a) Undular tidal bores: Basic theory and free-surface characteristics. Journal of Hydraulic Engineering 136(11):940–944, DOI: https://doi.org/10.1061/(ASCE)HY.1943-7900.0000264
Chanson H (2010b) Unsteady turbulence in tidal bores: Effects of bed roughness. Journal of Waterway Port Coastal and Ocean Engineering 136(5):247–256, DOI: https://doi.org/10.1061/(ASCE)WW.1943-5460.0000048
Chen YY, Kharif C, Yang JH, Hsu HC, Touboul J, Chambarel J (2015) An experimental study of steep solitary wave reflection at a vertical wall. European Journal of Mechanics B-Fluids 49(A):20–28, DOI: https://doi.org/10.1016/j.euromechflu.2014.07.003
Dean RG, Dalrymple RA (1991) Water wave mechanics for engineers and scientists. In: Advanced series on ocean engineering, volume 2. World Scientific, Singapore
Favre H (1935) Etude théoretique et expérimentale des ondes de translation dans les canaux découverts (Theoretical and experimental study of travelling surges in open channels). Dunod, Paris, France (in French)
Fritz HM, Hager WH, Minor HE (2004) Near field characteristics of landslide generated impulse waves. Journal of Waterway Port Coastal and Ocean Engineering 130(6):287–302, DOI: https://doi.org/10.1061/(ASCE)0733-950X(2004)130:6(287)
Gualtieri C, Chanson H (2011) Experimental study of a positive surge. Part 2: Comparison with literature theories and unsteady flow field analysis. Environmental Fluid Mechanics 11(6):641–651, DOI: https://doi.org/10.1007/s10652-011-9222-3
Gualtieri C, Chanson H (2012) Experimental study of a positive surge. Part 1: Basic flow patterns and wave attenuation. Environmental Fluid Mechanics 12(2):145–159, DOI: https://doi.org/10.1007/s10652-011-9218-z
Hager WH, Castro-Orgaz O (2019) On the undular hydraulic jump and the undular surge. Proceedings of the 38th IAHR world congress, September 1–6, Panama City, Panama
Henderson FM (1966) Open channel flow. MacMillan Company, New York, NY, USA
Hornung HG, Willert C, Turner S (1995) The flow field downstream of a hydraulic jump. Journal of Fluid Mechanics 287:299–316, DOI: https://doi.org/10.1017/S0022112095000966
Khezri N, Chanson H (2012) Undular and breaking bores on fixed and movable gravel beds. Journal of Hydraulic Research 50(4):353–363, DOI: https://doi.org/10.1080/00221686.2012.686200
Koch C, Chanson H (2005) An experimental study of tidal bores and positive surges: Hydrodynamics and turbulence of the bore front. Report No. CH56/05, The University of Queensland, Brisbane, Australia
Koch C, Chanson H (2008) Turbulent mixing beneath an undular bore front. Journal of Coastal Research 24(4):999–1007, DOI: https://doi.org/10.2112/06-0688.1
Koch C, Chanson H (2009) Turbulence measurements in positive surges and bores. Journal of Hydraulic Research 47(1):29–40, DOI: https://doi.org/10.3826/jhr.2009.2954
Leng X, Chanson H (2017a) Upstream propagation of surges and bores: Free-surface observations. Coastal Engineering Journal 59(2): 1750001–1750003, DOI: https://doi.org/10.1142/S0578563417500036
Leng X, Chanson H (2017b) Integral turbulent scales in unsteady rapidly varied open channel flows. Experimental Thermal and Fluid Science 81:382–395, DOI: https://doi.org/10.1016/j.expthermflusci.2016.09.017
Maeck A, Lorke A (2013) Ship-lock-induced surges in an impounded river and their impact on subdaily flow velocity variation. River Research and Applications 30(4):494–507, DOI: https://doi.org/10.1002/rra.2648
Montes JS (1998) Hydraulics of open channel flow. ASCE Press, New York, NY, USA
Ohtsu I, Yasuda Y, Gotoh H (2001) Hydraulic condition for undular-jump formations. Journal of Hydraulic Research 39(2):203–209, DOI: https://doi.org/10.1080/00221680109499821
Peregrine DH (1966) Calculations of the development of an undular bore. Journal of Fluid Mechanics 25(2):321–330, DOI: https://doi.org/10.1017/S0022112066001678
Reinauer R, Hager WH (1995) Non-breaking undular hydraulic jump. Journal of Hydraulic Research 33(5):683–698, DOI: https://doi.org/10.1080/00221689509498564
Soares Frazao S, Zech Y (2002) Undular bores and secondary waves — Experiments and hybrid finite-volume modelling. Journal of Hydraulic Research 40(1):33–43, DOI: https://doi.org/10.1080/00221680209499871
Su CH, Mirie RM (1980) On head-on collisions between two solitary waves. Journal of Fluid Mechanics 98(3):509–525, DOI: https://doi.org/10.1017/S0022112080000262
Treske A (1994) Undular bores (favre-waves) in open channels — Experimental studies. Journal of Hydraulic Research 32(3):355–370, DOI: https://doi.org/10.1080/00221689409498738
Yeh HH, Mok KM (1990) On turbulence in bores. Physics of Fluids 2(5):821–828, DOI: https://doi.org/10.1063/1.857630
Zheng F, Li Y, Xuan G, Li Z, Zhu L (2018) Characteristics of positive surges in a rectangular channel. Water 10(10):1473, DOI: https://doi.org/10.3390/w10101473
Acknowledgments
This work was sponsored by the Natural Science Foundation of Chongqing, China (Grant No. cstc2020jcyj-bshX0043) and the National Key R&D Program of China (Grant No. 2018YFB1600403).
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Zheng, F., Wang, P., An, J. et al. Characteristics of Undular Surges Propagating in Still Water. KSCE J Civ Eng 25, 3359–3368 (2021). https://doi.org/10.1007/s12205-021-0858-3
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DOI: https://doi.org/10.1007/s12205-021-0858-3