World Academy of Science, Engineering and Technology 60 2011

Investigation of Undular Hydraulic Jump over Smooth Beds

F. Rostami, M. Shahrokhi, M. A. Md Said and S.R. Sabbagh-Yazdi
surface undulations might be taken place at the downstream of plunging jets, submerged out let or flip bucket [4]. Some researches [4-8] show that an undular jump is fundamentally two-dimensional but close to the sidewalls. The free-surface undulations are quasi-periodic, but the longitudinal profile is neither sinusoidal nor conoidal. Sidewall cross-waves were seen upstream of the first wave crest and sometimes immediately upstream of the downstream wave crests. Fig. 1 shows a typical free surface profile of undular hydraulic jump and illustrates the important parameters in this phenomenon. AbstractUndular hydraulic jumps are illustrated by a smooth
rise of the free surface followed by a train of stationary waves. They are sometimes experienced in natural waterways and rivers. The characteristics of undular hydraulic jumps are studied here. The height, amplitude and the main characteristics of undular jump is depended on the upstream Froude number and aspect ratio. The experiments were done on the smooth bed flume. These results compared with other researches and the main characteristics of the undular hydraulic jump were studied in this article.

KeywordsUndular Hydraulic Jump, low Froude Number, wave

characteristics

I. INTRODUCTION
APID transition from super-critical to sub-critical flow is called Hydraulic jump. This phenomenon is commonly happening in rivers, natural waterway or canals. Hydraulic jump is one of the interesting happenings in the open channel hydraulic. The undular hydraulic jump is a special case of the hydraulic jump. For super critical flow with Froude number near to unity, an undular hydraulic jump is formed by a smooth rise of the surface and undulations that stretch out over a long length. In this type of hydraulic jump, there is not any significant energy dissipation and air entrainment. As the undular hydraulic jump does not have any considerable roller, the energy-loss is retarded forward into a train of stationary waves [1-2]. The knowledge of the feasible flow circumstances of undular jumps is important for designing and managing channels, and hydraulic structures. In addition, information about the formation of undular jumps can be useful in planning water-sport facilities such as rafting chutes, which can also be an attractive feature of flowing water in a landscape. The waves caused by undular hydraulic jump may also affect the downstream discharge measurement structures such as sharp-crested and broad-crested weir. Gibson [3] studied the effect of free surface waves on discharge measurement over weirs, and he found that for the wave heights of about h/dc ~ 0.375, the discharge increase about 2 to 3%. The freeF. Rostami is with the School of Civil Engineering, Universiti Sains Malaysia, Nibong Tebal, Malaysia (e-mail: fa.rostami@gmail.com). M. Shahrokhi is with the School of Civil Engineering, Universiti Sains Malaysia, Nibong Tebal, Malaysia (e-mail: mshotm2000@yahoo.com). M.A. Md Said is with the School of Civil Engineering, Universiti Sains Malaysia, Nibong Tebal, Malaysia (e-mail: azlin@eng.usm.my). S.R. Sabbagh-Yazdi is with the Civil Engineering Department, K.N. Toosi University of Technology, Tehran, Iran (e-mail: syazdi@kntu.ac.ir).

Fig. 1 Free-surface profile of an undular hydraulic jump

Some researchers [9-11] studied hydraulic jump included some undular jump cases, but the respects of undular hydraulic jumps was not considered in their researches. Some other authors have done some experiments on undular jumps [12-16]. Their analyses were based on similarity with undular surge in tranquil flow. Montes [2] and Ryabenko [17] disagree with the hypothesis of similarity between undular hydraulic jump and undular surge. Their results show that the undular hydraulic jump become weak jump at the upstream Froude numbers in the range 1.0 to 3.6 where this conversion is a function of upstream flow conditions (i.e. Froude number, aspect ratio and roughness). On contrary, an undular surge vanishes for surge Froude number higher than 1.5 to 1.8 [1821]. Some of the recent studies about undular hydraulic jump are listed in Table I. The modern instrument like LDV, PIV, and ADV systems record just the velocity field. However, in the undular hydraulic jumps, the pressure field is not hydrostatic and the pressure data must be recorded together with velocity field. The only instrument that can measure velocity, pressure and total energy, and the bed shear stress after appropriate calibration is the PrandtlPitot tube [22, 24, 29].

627

World Academy of Science, Engineering and Technology 60 2011

TABLE I SUMMARY OF THE STUDIES ON UNDULAR HYDRAULIC JUMPS [22] Author Chanson and Montes[6] Montes and Chanson [23] Chanson[24] Ohtsu et al[25] Chanson [26] Lennon and Hill [27] Meftah et al. [28] Chanson and Montes[6] Montes and Chanson [23] Chanson[24] Ohtsu et al[25] Chanson [26] Lennon and Hill [27] Meftah et al. [28] Chanson and Montes[6] Channel L = 20 m, B = 0.25 m, F/D inflow L = 12 to 20 m, B = 0.2 to 0.3 m, F/D inflow L = 20 m, B = 0.25 m, F/D inflow L = 5 to 20 m, B = 0.1 to 0.8 m, P/D & F/D inflow L = 3.2 m, B = 0.5 m, P/D inflow L = 4.9 m, B = 0.3 m, F/D inflow L = 15 m, B = 4 m, F/D inflow L = 20 m, B = 0.25 m, F/D inflow L = 12 to 20 m, B = 0.2 to 0.3 m, F/D inflow L = 20 m, B = 0.25 m, F/D inflow L = 5 to 20 m, B = 0.1 to 0.8 m, P/D & F/D inflow L = 3.2 m, B = 0.5 m, P/D inflow L = 4.9 m, B = 0.3 m, F/D inflow L = 15 m, B = 4 m, F/D inflow L = 20 m, B = 0.25 m, F/D inflow Instrumentation PrandtlPitot tube (3.3 mm) PrandtlPitot tubes PitotPreston tube ( 3.3 mm) Micro-propeller ( 3 mm), PrandtlPitot tube, 1D- LDV Type B PrandtlPitot tube ( 3.3 mm) Particle Image Velocimetry (PIV) Acoustic Doppler Velocimetry (ADV) PrandtlPitot tube (3.3 mm) Type C PrandtlPitot tubes PitotPreston tube ( 3.3 mm) Micro-propeller ( 3 mm), PrandtlPitot tube, 1D- LDV PrandtlPitot tube ( 3.3 mm) Particle Image Velocimetry (PIV) Acoustic Doppler Velocimetry (ADV) PrandtlPitot tube (3.3 mm) Type D Type A Type

TABLE II CLASSIFICATION OF THE UNDULAR HYDRAULIC JUMPS Range of Upstream Froude number Description Undulations of free surface have small amplitude and proportionately long wave length. There is a twodimensional flow without any roller or shock wave in the surface of flow. Lateral shock waves generate at the upstream of the first wave crest. These shock waves cross slightly downstream of the first crest of wave. Lateral shock waves appear from both sidewalls next to the toe of the jump when the upstream Froude number is higher than 1.2 The lateral shock waves intersect at the first crest, and a small roller appears at the first crest, directly after crossing of the shock waves. A cockscomb shape roller is places on the centerline. The roller has small size and does not occur at the later waves. For larger Froude numbers, the undular jump has a similar appearance with shock wave and air bubble enter the water at the top of the first wave. The bubbles trap at the intersection of the roller and lateral shock waves at the short distance (less than a single wave length). A small roller may appear at the second crest but no air bubble entrainment is observed here. In this type of undular jumps, the size and width of the roller (at the first crest of wave) raise and roller are limited by the lateral shock waves. Moreover, the air bubbles appear at the second crest.

1Fr1FrA

FrA Fr1FrB

FrB Fr1FrC

FrC Fr1FrD

Notes. B: channel width; F/D: Fully developed; L: channel length, P/D: partially developed.
Type E FrD Fr1FrE

II. CLASSIFICATION OF UNDULAR JUMPS The undular hydraulic jump characteristics is related to inflow conditions: the upstream Froude number, the aspect ratio, the Reynolds number, the channel slope, the side wall and bed roughness and the boundary layer development at the toe of the jump, e.g., the turbulence level [8]. Chanson and Montes [6] classified the undular jumps base on the upstream Froude number (Table II). For upstream Froude numbers Higher than FrE , the undulation of free surface will be disappeared and roller will be fully developed over the channel width and the jump become a weak jump. Ohtsu et al. [8] divided the undular hydraulic jump into two cases. Case I and II include non-breaking and breaking undular jumps. A non-breaking undular jump has a continuous water surface without any breaking, although a breaking undular jump is referred to as an undular jump with a surface roller at the center part of the first wave. III. CHARACTERISTICS OF UNDULAR HYDRAULIC JUMP A. Experimental Apparatus Experiments were performed in a 10-m long channel of uniform rectangular section made of glass (sidewalls) and

steel(bottom), located in the Hydraulic Laboratory of the Universiti Sains Malaysia (Fig. 2). The channel width is 0.30 m and the sidewall height is approximately 0.6 m. The channel slope can be adjusted using a geared lifting mechanism. Tailwater levels were controlled by a radial gate fitted at the downstream channel end. The upstream flow was controlled by a sluice gate. In this study, the water discharge ranges is 6.94 to 12.5 l/s and the upstream Froude number is more than 1.35.

Fig. 2 A photograph of an experiment on smooth bed

628

World Academy of Science, Engineering and Technology 60 2011

2
Fr1=1.35

Fr* and is:

Fr1=1.67 Fr1=1.87

1.5 h/yc 1 0.5 0.50

Fr * = 1.0591Fr1 0.0033, R = 0.89

(3) (4)

= 35.57 Fr1
3.0

0.0681

, R = 0.909

= 1.0591 1 0.0033 R2 = 0.890

1.00

x/x1

1.50

2.00

2.5
Fr*

Fig. 3 Water surface profile in undular hydraulic jump (for yc/B=0.126)

2.0

0.119

B. Flow Pattern In Type A of undular jumps, the flow is two-dimensional downstream of the top of the first wave. In type B, flow is symmetrical around the centerline of channel. For upstream Froude number higher than FrB, a various shape flow pattern is considered. After the first crest, the free surface is steady and symmetrical pattern and the crest of each wave is placed on the centerline (with similar phase shift and wavelength) but with smaller amplitudes [6, 30-32]. Fig. 1 shows the water surface profile in the undular hydraulic jump for Q=6.94 l/s until 4th undulation. C. Lateral Shock Waves The main properties of undular hydraulic jumps are lateral shock waves. Shock waves were generated in the positive pressure gradient domain. Configuration of vertical velocity and pressure distributions combine with lateral boundary layers, and the sidewall boundary layers caused to an immediate converse pressure gradient which cause a sudden decrease of velocity near the wall and maybe appear separation in this area [6]. The adverse pressure gradient takes place from being of subcritical flow depth in the jump, and the dimension of a sidewall-boundary-layer development is expanded from the toe of the jump. Then, the surface-flow velocity near the sidewalls is inclined to become critical, such that the lateral shock waves emerge from both sides of the toe section. Chanson[22] proposed a relation between Fr*, and Fr1 based on the experimental data:

1.5
Experimental data Chanson (1995)

1.0 1.0

1.5

2.0
Fr1

2.5

3.0

Fig. 4 Flow Froude number Fr* at the inception of the lateral shock waves with the sidewall
50.0

40.0

30.0

20.0
Experimental data Chanson (1995)

10.0 1.0

1.5

2.0
Fr1

2.5

3.0

Fig. 5 Angle of the lateral shock waves with the sidewall

Fr * = Fr1 0.119

(1) (2)

= 28.1 Fr1

0.38

Where, Fr* is the Froude number at the start of the shock waves and is the angle between the sidewall and shockwaves. Properties of shock waves in the undular hydraulic jump were presented in Fig. 5. In Fig. 5, the experimental data was compared with studies of undular hydraulic jump by Chanson [4]. From Fig. 5(a), there are close comparisons between presented result and Chanson [4] and these differences is related to applied aspect ratios. The trend of rising the Fr* with Fr1is near to Chanson [4] experiments but the differences between present and Chanson [4] experiments for is further more (Fig. 5(b)). Based on recent experiments the modified formulations for

D. Wave Properties Wave length (l w): For a fixed Froude number, decrease of discharge reaches rising of dimensionless wave length and relative roughness and friction. Wave length decrease exponentially along the canal and the rate of this reduction is free from the upstream Froude number, the aspect ratio and the type of undular hydraulic jump [6]. Wave amplitude (aw): For Froude number near the critical value, the wave amplitude close to the theoretical solution of Boussinesq equation [6, 12]. With increasing Froude number, the wave amplitude data separating from the solution of the motion equation and arrive in maximum value [1, 6, 10, 13]. For higher Froude numbers, with increasing Froude number, the wave amplitude decreases. Before the free-surface undulations disappear, the wave amplitude does not decrease with increasing Fr1. When the undular jump become a weak jump, the amplitude of the first wave will aproximate to half

629

World Academy of Science, Engineering and Technology 60 2011

of the roller height, and the amplitude of the second wave will be close to zero [6]. Fig. 6 and Fig. 7 show the dimensionless wave amplitude and wave length. The data show the maximum wave amplitude that takes place for Froude number 1.5 to 1.7.

affected by the aspect ratio while Ohtsu et al. [8] did not consider the effect of the aspect ratio in their formulas.
4

3.5

hwc/h1

2.5

1.5

Experimental data Ohtsu et al. (2003)

1 1 1.5 2 2.5

Fr1

Fig. 8 Depth of first crest of undular hydraulic jump Fig. 6 Dimensionless first wave length
3

2.5

hwt/h1

1.5

Experimental data Ohtsu et al. (2003)

1 1 1.5 2 2.5

Fr1

Fig. 7 Dimensionless first wave amplitude

Fig. 9 Depth of first trough of undular hydraulic jump

E. Characteristics of the First Wave in Undular Jumps For case I (non-breaking and breaking undular jumps), the flow parameters are only affected by Froude number [8]: hwc 2 = 0.76 ( Fr1 1) + 2.3 ( Fr1 1) + 1, h1 (5)

1.0 < Fr1 Fr1u hwt h1

= 0.90 ( Fr1 1) + 0.2 ( Fr1 1) + 1,
2.5

(6)

F. Velocity and Pressure Distributions in Undular Jumps In the undular hydraulic jump due to the existence of wavy surface, the velocity and pressure distributions are different in the crests and troughs. Velocity and pressure distributions are presented in Fig. 10 and Fig. 11. The velocities in Fig. 10 were normalized by the mean centerline velocity Ucl and shown as a function of y/h, where y is distance from bed measured normal to channel bottom and h is flow depth in centerline. Fig. 10 shows the main differences of velocity profile happened between upstream flow and first wave crest. In addition, the strong velocity decrease was observed in first and second crest, near the free surface for all cases of smooth bed but in the lowest Froude number, this decrease is small.

1.0 < Fr1 Fr1u

Where, hwc and hwt are depth of flow in first crest and trough, respectively. Fig. 8 and Fig. 9 show the experimental data recognize the trend of hwc /h1 and hwt /h1 but some differences. The reason of this divergence is related to experimental condition. The undular hydraulic jumps are also

630

World Academy of Science, Engineering and Technology 60 2011

Fig. 10 Dimensionless velocity distributions in undular jumps

Fig. 11 Dimensionless pressure distributions in undular jumps

The flow pattern at the first crest of the undular hydraulic jump diverges considerably from those at the consequent wave crest. For later sections, the agreement between measured data and calculated data is improved. The results of velocity distribution show that the flow acceleration next to the bed (i.e. y/dc<0.3) between crest and trough is higher than between crest and trough. The dimensionless pressure in Fig. 11 is shown as a function of y/h. In the undular hydraulic jumps, due to the undulation of free surface, the pressure distribution is not hydrostatic. Where the free surface has concaved shape, the real pressure is larger than hydrostatic and in the convex surface the pressure is less than hydrostatics. This case can be defined by the irrotational flow motion theory [21, 33]. This concept is perceptible in Fig. 11.

IV. CONCLUSION For super critical flow with Froude number near to unity, an undular hydraulic jump is formed by a smooth rise of the surface and undulations. An undular hydraulic jump may happen in irrigation and water supply channels, in narrow or shallow straights and downstream of short drop structures or in the transitional region from steep to mildly sloping channels. The knowledge of the feasible flow circumstances of undular jumps is important for designing and managing channels, and hydraulic structures. When undular jump forms in a channel, large amplitude waves were produced and extended downstream. These phenomena may damage the channel banks and the rising of water level due to these undulations must be considered in design of channels. Furthermore, the generation of free-surface waves might enforce additional impact loads and vibrations on downstream structures. Hence finding some ways to reduce the length of distribution of waves, wavelength and amplitude is very important. The experimental measurements indicate some metrical and kinematical properties of undular hydraulic jumps over the smooth beds. The water surface profile shows a traveling train

631

World Academy of Science, Engineering and Technology 60 2011

of conical waves. These waves propagate along the length of channel. The velocity profile of crests and troughs are different. From hydraulic point of view, due to the curve shape of free surface, the pressure distribution is non-hydrostatic and this concept is obvious on the pressure measurement. REFERENCES Montes, J.S., Undular Hydraulic Jump - Discussion. Journal of the Hydraulics Division, ASCE, 1979. 105(HY9): p. 1208-1211. Montes, J.S., A study of the undular jump profile, in 9th Australasian Fluid Mechanics Conference, AFMC. 1986: Auckland, New Zealand. p. 148-151. Gibson, A.H., The Effect of Surface waves on the Discharge over Weirs, in Proc. Instn Civ. Engrs. 1930: UK. p. 3-18. Chanson, H., Flow characteristics of undular hydraulic jumps. Comparison with near-critical flows. 1995, Dept. of Civil Engineering, University of Queensland: Australia. Chanson, H., Characteristics of Undular Hydraulic Jumps. 1993, Dept. of Civil Engineering, University of Queensland: Australia. p. 109. Chanson, H. and J.S. Montes, Characteristics of undular hydraulic jumps. Experimental apparatus and flow patterns. Journal of Hydraulic Engineering - ASCE, 1995. 121(2): p. 129-144. Ohtsu, I., Y. Yasuda, and H. Gotoh, Characteristics of undular jumps in rectangular channels, in XXVI Int. Association for Hydraulic Research Congress. 1995: London. p. 450-455. Ohtsu, I., Y. Yasuda, and H. Gotoh, Flow Conditions of Undular Hydraulic Jumps in Horizontal Rectangular Channels. Journal of Hydraulic Engineering, 2003. 129(12): p. 948955. Bakhmeteff, B.A. and A.E. Matzke, The Hydraulic jump in terms of dynamic similarity. Trans. ASCE 1936. 101: p. 630-647. Binnie, A.M. and J.C. Orkney, Experiments on the flow of water from a reservoir through an open channel. II. The formation of hydraulic jump. Proc. Roy. Soc. London Ser., 1955. A 230: p. 237-245. Sandover, J.A. and P. Holmes, The Hydraulic Jump in Trapezoidal Channels. Water Power, 1962. Nov: p. 445-449. Andersen, V.M., Undular hydraulic jump. Journal of the Hydraulics Division, ASCE, 1978. 104 (HY8): p. 11851188. Iwasa, Y., Undular jump and its limiting conditions for existence, in Proc. 5th Japan National Congress for Applied Mechanics. 1955: Japan. p. 315-319. Jones, L.E., Some Observations on the Undular Jump. Journall of Hydraulic Division., ASCE, 1964. 90(No. HY3): p. 69-82. Lemoine, R., Sur les ondes positives de translation dans les canaux et sur le ressaut ondul de faible amplitude. J. La Houille Blanche 1948: p. 183-185. Serre, F., Contribution letude des ecoulements permanents et variables dansles canaux. J. La Houille Blanche, 1953: p. 830872 Ryabenko, A.A., Conditions favorable to the existence of an undulating jump. Gidrotekhnicheskoe Stroitelstvo, 1990. 12: p. 29-34. Koch, C. and H. Chanson, Turbulent mixing beneath an undular bore front. J. Coastal Res., 2008. 24(4): p. 999-1007. Henderson, F.M., Open Channel Flow. 1966, New York, USA: MacMillan Company. Koch, C. and H. Chanson, An experimental study of tidal bores and positive surges: Hydrodynamics and turbulence of the bore front. 2005, Dept. of Civil Engineering, The University of Queensland: Brisbane, Australia. p. 170. Rouse, H., Fluid Mechanics for Hydraulic Engineers. 1938, New York, USA: McGraw-Hill Publ. Chanson, H., Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results. European Journal of Mechanics B/Fluids, 2009. 28: p. 191-210. Montes, J.S. and H. Chanson, Characteristics of undular hydraulic jumps: Experiments and Analysis. Journal of Hydraulic Engineering, ASCE, 1998. 124(2): p. 192-205. Chanson, H., Boundary shear stress measurements in undular flows: Application to standing wave bed forms. Water Resource Research. , 2000. 36(10): p. 3036-3076. Ohtsu, I., Y. Yasuda, and H. Gotoh, Hydraulic condition for undularjump formations. Journal of Hydraulic Research, IAHR., 2001. 39(2): p. 203-209. Chanson, H., Physical modelling of the flow field in an undular tidal bore. journal of Hydraulic Research, 2005. 43 (3): p. 234-244.

[1] [2]

[3] [4]

[27] Lennon, J.M. and D.F. Hill, Particle image velocimetry measurements of undular and hydraulic jumps. Journal of Hydraulic Engineering, ASCE., 2006. 132(12): p. 12831294. [28] Meftah, M.B., et al., Analysis of the velocity field in a large rectangular channel with lateral shockwave. Environmental Fluid Mechanics, 2007. 7(6): p. 519-536. [29] Troskolanski, A.T., Hydrometry: Theory and Practice of Hydraulic Measurements. 1960, Oxford, UK: Pergamon Press. [30] Darcy, H.P.G. and H.E. Bazin, Recherches hydrauliques Vol. Parties 1re et 2me. 1865, Paris, France: Imprimerie Impriales. [31] Fawer, C., Etude de Quelques Ecoulements Permanents Filets Courbes ('Study of some Steady Flows with Curved Streamlines.). 1937, Thesis, Lausanne, Switzerland,: Switzerland, Imprimerie La Concorde. [32] Hager, W.H. and K. Hutter, On Pseudo-Uniform Flow in Open Channel Hydraulics. ActaMechanica, 1984. 53(183-200). [33] Liggett, J.A., Fluid Mechanics. 1994, New York, USA: McGraw-Hill.

[5] [6]

[7]

[8]

[9] [10]

[11] [12] [13]

[14] [15]

[16] [17] [18] [19] [20]

[21] [22]

[23]

[24]

[25]

[26]

632