5th International Symposium on Hydraulic Structures Hydraulic Structures and Society: Engineering Challenges and Extremes ISBN 9781742721156 - DOI: 10.14264/uql.2014.15
Brisbane, Australia, 25-27 June 2014
Turbulent Fluctuations in Hydraulic Jumps: a Physical Study H. Wang1 and H. Chanson1
1
The University of Queensland, School of Civil Engineering Brisbane QLD 4072 AUSTRALIA E-mail:
[email protected]
Abstract: In an open channel, the transformation from a supercritical flow into a subcritical flow is a rapidly varied flow with large turbulent fluctuations, intense air entrainment and substantial energy dissipation: i.e., a hydraulic jump. New experiments were conducted for a wide range of Froude numbers (3.8 < Fr1 < 8.5) at relatively large Reynolds numbers (2.1×104 < Re < 1.6×105) to quantify the turbulent fluctuations. The time-averaged free-surface profile presented some self-similar profile. The longitudinal movements of the jump were observed and both fast and very slow fluctuations were documented for all Froude numbers. The air-water flow measurements quantified the intense aeration of the roller. Overall the present findings demonstrated the close interactions between the roller turbulence and free-surface fluctuations. Future investigations should be carried out over long durations to account for the very slow fluctuations in jump position. Keywords: Hydraulic jumps, Turbulent fluctuations, Physical modelling, Air-water flows, Jump toe oscillations
1. INTRODUCTION In an open channel, the transformation from a supercritical flow into a subcritical flow is called a hydraulic jump. The transition is a rapidly varied flow with large turbulent fluctuations, intense air entrainment and substantial energy dissipation (Fig. 1 & 2). For a hydraulic jump in a smooth horizontal prismatic channel, the continuity and momentum principles yield a relationship between the upstream and downstream flow depths, d1 and d2 respectively, and the energy principle gives an expression of the total head loss H in the jump:
d2 1 2 1 8 Fr1 1 d1 2 1 8 Fr 2 3 1 H 1 2 d1 16 1 8 Fr1 1
(1) 3
(2)
Figure 1 – Hydraulic jump above the submerged Mt Crosby weir and road bridge on 17 January 2011
Figure 2 – Experimental facility - Flow conditions: Q = 0.0368 m3/s, d1 = 0.0277 m, x1 = 1.083 m, Fr1 = 5.1, Re = 7.4104 - Flow direction from right to left where Fr1 is the upstream Froude number: Fr1 = V1/(gd1)1/2, V1 is the upstream velocity and g is the gravity acceleration (Henderson 1966, Liggett 1994). For inflow Froude numbers greater than 3 to 4, the hydraulic jump is characterised by a breaking roller with surface splashing and recirculation, large scale turbulence development, air entrainment and large rate of energy dissipation (Fig. 2) (Hager 1992, Chanson 2011). A singular point in the free surface profile is the jump toe, also called impingement point. The purpose of this study is to broaden the knowledge of the roller fluctuations and air-water flow properties in turbulent hydraulic jumps. New experiments were conducted for a wide range of Froude numbers (3.8 < Fr1 < 8.5) at relatively large Reynolds numbers (2.1×104 < Re < 1.6×105). The results provide a finer characterisation of the turbulent fluctuations in hydraulic jumps.
2. EXPERIMENTAL APPARATUS AND INSTRUMENTATION New experiments were conducted in a horizontal rectangular flume at the University of Queensland (Brisbane, Australia). The 3.2 m long 0.5 m wide channel was made of smooth PVC bed and glass sidewalls (Fig. 2). The water was supplied by a constant head tank feeding a large intake structure leading to the test section through a vertical rounded gate. The tailwater conditions were controlled by a vertical overshoot gate located at x = 3.2 m where x is the longitudinal distance from the test section's upstream end. The discharge was measured with a Venturi meter calibrated on site. The clear-water flow depths were measured with point gauges with an accuracy of 0.2 mm. The fluctuating free-surface elevations above the hydraulic jump were recorded non-intrusively using acoustic displacement meters MicrosonicTM Mic+25/IU/TC and Mic+35/IU/TC. A total of 15 displacement meters were mounted above the channel (Fig. 2), enabling simultaneous and non-intrusive measurements of instantaneous water elevations. The signals were sampled at 50 Hz for at least 540 s to record both low- and highfrequency free-surface fluctuations. The air-water flow properties were measured with a double-tip conductivity probe (x = 7.46 mm, z = 1.75 mm) equipped with two identical needle sensors with inner diameter 0.25 mm. The probe was excited by an electronic system (Ref. UQ82.518) designed with a response time less than 10 μs and the signal outputs was sampled at 20 kHz per sensor for 45 s. The elevation of the probe was supervised by a MitutoyoTM digimatic scale unit with a vertical accuracy y < 0.05 mm.
The experimental flow conditions are summarised in Table 1 where Q is the water discharge, x1 is the distance between the jump toe and the upstream vertical gate, and h is the upstream undershoot gate opening. A 3×4 array of displacement meters was positioned over the jump (Fig. 2) and three sensors were placed horizontally above the inflow free-surface to record the roller position. The axes of horizontal sensors were roughly 25 mm above the water surface. Figure 2 shows a photograph of the experiment and further details were presented in Wang and Chanson (2013). Table 1 - Experimental investigations of turbulent fluctuations in hydraulic jumps (Present study) Q (m3/s) 0.0160 0.0179 0.0239 0.0356 0.0397 0.0368 0.0463 0.0552 0.0689 0.0815
h (m) 0.012 0.020 0.020 0.020 0.020 0.026 0.030 0.034 0.040 0.045
x1 (m) 0.50 0.83 0.83 0.83 0.83 1.08 1.25 1.42 1.67 1.87
d1 (m) 0.012 0.0206 0.0209 0.0209 0.0208 0.0277 0.0322 0.0363 0.042 0.047
Fr1
Re
5.1 3.8 5.1 7.5 8.5 5.1 5.1 5.1 5.1 5.1
2.1104 3.5104 4.8104 6.8104 8.0104 7.4104 9.2104 1.10105 1.37105 1.63105
Lr (m) -0.28 0.52 0.80 1.0 -0.85 ----
3. FREE-SURFACE CHARACTERISTICS All visual, photographic and video observations showed a sharp rise in water level in the downstream direction above turbulent jump roller (Fig. 2). The time-averaged free-surface profiles were measured with the acoustic displacement meters and the data were supplemented with point gauge measurements. The ratio of conjugate depths d2/d1 presented a good agreement with the momentum principle (Eq. (1)). The roller free-surface profiles were observed to present a self-similar profile. The experimental data are shown in Figure 3 together with the self-similar function:
d1 x x 1 d 2 d1 L r
0.54
(3)
where is the free-surface elevation above the invert, and Lr is the roller length defined as the distance from the jump toe over which the mean free-surface level increased monotically. The experimental results are compared with previous experimental data, Equation (3) and the profiles proposed by Valiani (1997) and Chanson (2011) in Figure 3. For all investigated flow conditions, the hydraulic jump toe was observed to shift about a mean position x = x1 in both fast and slow manner. Both types of fluctuating motion were investigated herein. Some experiments were run for relative long periods between 27 and 160 minutes, and both the longitudinal position of the roller toe and roller surface elevations were recorded. The observations highlighted some temporary changes of jump toe position ranging from -0.28 m to +0.12 m. The roller tended to stay at some remote positions for about 120 to 400 s before returning to its mean position. The horizontal displacement range was larger than and the associated periods were drastically longer than for the rapid jump toe oscillations. A typical data set in terms of the relative jump toe position x-x1 is presented in Figure 4. The data (Fig. 4) illustrated that some major movements were linked to some upstream migration (x-x1 < 0). Over 20% of the instantaneous jump toe positions were recorded at the mean position (x-x1 = 0). The probability distribution function was skewed towards the upstream side (x-x1 < 0), and the cumulative percentages on the two sides were comparable (37.5% upstream and 42.2% downstream). Note that the flume was 3.2 m long and its relatively short length might have restricted the extent of jump movement. Sometimes the jump toe left its mean position for up to 200 to 400 s. In most cases, however, it took 100 to 150 s to return to its mean location. For the data shown in Figure 4, about 36 major shifts in jump position were recorded within the 160 minutes record corresponding to an average frequency about 0.004 Hz, with longitudinal deviation of up to half the
roller length (Fig. 4). Although such slow and large fluctuations in hydraulic jump positions were rarely documented, but for oscillating jumps (Mossa 1999), the present findings provided quantitative evidences of the phenomenon over breaking hydraulic jumps with higher Froude numbers. The fast fluctuations of jump toe position were documented in earlier studies (Long et al. 1991, Chanson and Gualtieri 2008). It is believed that the rapid jump toe oscillation was related to the generation and advection of large turbulent structures in the developing shear layer as well as air entrapment at the impingement point. Herein the longitudinal jump toe oscillations were detected from upstream using horizontally-mounted acoustic displacement meters (Fig. 5). A sketch of the experimental setup is shown in Figure 5A and the results are presented in Figure 5B. The characteristic oscillation frequencies were deduced from a spectral analysis of the displacement meter signals. The data showed both dominant and secondary characteristic frequencies which were denoted as Ftoe,dom and Ftoe,sec respectively. Typical results are shown in Figure 5B as functions of the inflow Froude number and Reynolds number, and compared with previous visual observations of jump toe oscillation frequencies. The present data were obtained with a constant Froude number (Fr1 = 5.1) for different Reynolds numbers. Others were observed with a constant gate opening (h = 0.020 m) for Fr1 = 3.8, 5.1, 7.5 & 8.5. 1
0.22
0.9
0.2
0.8
0.18 0.16 0.14
0.6
PDF
(-d1)/(d2-d1)
0.7
0.5 0.4
0.2 0.1 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (x-x1)/Lr
0.1 0.08
ADM data Point gauge Eq. (3) Chanson (2011) Murzyn & Chanson Chachereau & Chanson Valiani Fr1=8.5
0.3
0.12
0.06 0.04 0.02 1
0 -0.55
-0.4
-0.25 -0.1 (x-x1)/Lr
0.05
0.2
Figure 3 (Left) – Self-similar free-surface profiles within the roller length – Comparison between experimental data (Present study, Murzyn & Chanson 2009, Chachereau & Chanson 2011), Equation (3) and the correlations of Valiani (1997) and Chanson (2011) Figure 4 (Right) - Probability density function distribution of instantaneous jump toe position: slow fluctiuations (Recording time: 160 min.) - Flow conditions: Q = 0.0239 m3/s, d1 = 0.0209 m, x1 = 0.83 m, Fr1 = 5.1, Re = 4.8×104 The experimental data yielded characteristic Strouhal numbers Ftoe,dom×d1/V1 in a range of 0.005 to 0.015. Some dimensionless secondary frequencies were also observed, with Strouhal numbers Ftoe,sec×d1/V1 typically higher than 0.02. Both present and earlier data were close, but for the data of Zhang et al. (2013) and Chachereau and Chanson (2011) at low Froude numbers (Fr1 < 4.4). In these studies, higher frequencies were observed at Froude numbers less than 5 with an exponential decay in dimensionless frequency with increasing Froude number (Fig. 5B, Left). Figure 5B (Right) presents also the results as functions of the Reynolds number between 2.1×104 and 1.63×105. Higher dominant frequencies were seen at larger Reynolds numbers, while no apparent effect of the Reynolds number is shown in terms of the secondary frequencies. The secondary frequencies are thought to link with the vertical free-surface fluctuations above the roller, as they were shown in a same frequency range. Moreover, larger transverse coherent structures were observed in the wavelike impingement perimeter for stronger hydraulic jumps (Zhang et al. 2013). It is supported by the present study showing stronger
correlations between the signals of two transversely separated displacement meters for higher Froude and Reynolds numbers.
(A) Sketch of longitudinal jump toe oscillation measurements: top view (top) and side view (bottom) 0.04
0.04 Ftoe,dom, present study Ftoe,sec, present study Zhang et al. (2013) Chanson (2010) Chachereau & Chanson Murzyn & Chanson Chanson (2007)
0.03
Ftoed1/V1
Ftoed1/V1
0.03
0.02
0.02
0.01
0.01
0
0 2
3
4
5
6
7
8
Fr1
9
0
40000
80000 120000 Re
160000
(B) Dimensionless characteristic frequencies of jump toe oscillation on the channel centreline Comparison with the visual observation results of Chanson (2007), Murzyn and Chanson (2009), Chanson (2010), Chachereau and Chanson (2011) and Zhang et al. (2013) Figure 5 – Measurements of rapid fluctuations of longitudinal jump toe position
4. AIR-WATER FLOW PROPERTIES The experimental observations demonstrated the intense aeration of the jump roller. The air-water flow properties of hydraulic jumps were studied systematically with a phase-detection conductivity probes. Typical vertical distributions of time-averaged void fraction are presented in Figure 6. The void fraction data showed two distinct flow regions, namely the shear layer between the channel bed and an elevation y* of local minimum void fraction, and the upper free-surface region above, in which the void fraction increased monotonically with distance from the invert (Fig. 6). The shear layer was characterised by the advection of highly-aerated large vortical structures generated at the impingement point, and the void fraction distribution showed a local maximum Cmax. The void fraction
profile followed closely an analytical solution of two-dimensional diffusion equation for air bubbles (Crank 1956):
V1
C 2C Dt 2 X y
(4)
where V1 is the inflow velocity, Dt is the air bubble diffusivity, C is the time-averaged void fraction, y is the vertical elevation and X = x-x1+ur/V1y with ur the bubble rise velocity (Chanson 2010). Assuming an uniform velocity field, constant diffusivity and bubble rise velocity, and neglecting the compressibility effects, Equation (4) may be solved analytically to give the following solution:
( y'1) 2 ( y'1) 2 exp exp # 4 D # X' 4 D # X' 4 D X' Q air / Q
C
(5)
where X’ = X/d1, y’ = y/d1, and D# is a dimensionless diffusivity: D# = Dt/(V1d1) (Chanson 2010). Equation (5) is compared with experimental data in Figure 6. 10 9 8 7
y/d1
6 5 4 3 (x-x1)/d1 = 4.15 (x-x1)/d1 = 8.35 (x-x1)/d1 = 12.5 (x-x1)/d1 = 18.75 (x-x1)/d1 = 25
2 1 0 0
0.1
0.2
0.3
0.4
0.5 C
0.6
0.7
0.8
0.9
1
Figure 6 – Vertical distributions of time-averaged void fraction C in the hydraulic jump roller - Flow conditions: Q = 0.0333 m3/s, d1 = 0.020 m, x1 = 0.83 m, Fr1 = 7.5, Re = 6.6×104 - Comparison with Equation (5) (solid lines) For the present data set, the maximum void fraction Cmax in the shear layer was observed to decrease exponentially with increasing distance from the jump toe. Further the location yCmax of the local maximum in void fraction was seen to increase linearly with increasing distance from the impingement. The data showed some dependence upon the Froude number and they were best correlated by:
x x1 C max (0.436 0.014 Fr1 ) exp (0.911 exp( 0.335 Fr1 )) d 1 y C max x x1 1.479 0.084 d1 d1
(6) (7)
The vertical distributions of time-averaged velocity V were recorded using a Pitot tube in the clearwater flow region and the double-tip conductivity probe in the aerated flow region based upon a crosscorrelation analysis. Some typical data are shown in Figure 7. Overall the velocity data exhibited some
vertical profiles similar to those for a wall jet flow (Rajaratnam 1965, Chanson 2010):
y V Vmax y V max
1/ N
for y/yVmax < 1 (8a)
1 V Vrecirc y y V max exp 1.765 Vmax Vrecirc 2 y 0.5
2
for y/yVmax > 1 (8b)
where Vmax is the maximum velocity in the shear layer and yVmax is the corresponding elevation, N is a constant, Vrecirc is the (negative) recirculation velocity in the upper free-surface region and y0.5 is the elevation where V = Vmax/2. The recirculation velocity Vrecirc was found nearly uniform at a given longitudinal position across the recirculation region, while N = 10 typically (Fig. 7). Although the crosscorrelation analysis does not provide meaningful results in the region where the interfacial velocity was about zero, some statistical analysis of instantaneous time lag in the raw probe signals supported the continuous velocity profile prediction by showing small average velocity close to y(V = 0) (Wang and Chanson 2013). The data implied further that the elevation y(V = 0) differed slightly from the characteristic elevation y*. 10
Phase-detection probe Prandtl-Pitot tube Equation (8)
9 8 7
y/d1
6 5 4 3 2 1 0 -0.6
-0.4
-0.2
0
0.2 V/V1
0.4
0.6
0.8
1
Figure 7 – Vertical distributions of time-averaged interfacial velocity V in the hydraulic jump roller Flow conditions: Q = 0.0378 m3/s, d1 = 0.020 m, x1 = 0.83 m, Fr1 = 8.5, Re = 7.5×104 - Comparison with Equation (8)
5. CONCLUSION The turbulent fluctuations in hydraulic jumps were investigated physically in a relatively large size channel. Both non-intrusive acoustic displacement meters and intrusive phase-detection probe were used. The inflow Froude number varied from 3.8 to 8.5 and the Reynolds number ranged from 2.1104 to 1.63105. The time-averaged free-surface profile of the jump roller presented some self-similar profile. The longitudinal movements of the jump were observed and both fast and very slow fluctuations were documented. Long-period shifts in jump position about its mean occurred with periods up to 400 s with movements of up to half the roller length from the mean position. The fast oscillations in jump toe position were found to be around a dominant frequency with some secondary frequency.
The air-water flow measurements quantified the intense aeration of the roller. The void fraction data were closely matched by a theoretical solution of the advective diffusion equation in the shear layer, where a local maximum in void fraction was observed. In the upper free-surface region, the void fraction increased monotonically with distance from the invert towards unity. The interfacial velocity distributions presented a smooth shape close to a wall jet profile, with a negative recirculation motion in the upper flow region. The flow pattern observations highlighted the role of large vortical structures, generated at the impingement point and advected in the developing shear layer. The eddies were highly aerated. Overall the present findings demonstrated the complex nature of turbulent hydraulic jumps and the close interactions between the roller turbulence and free-surface fluctuations. Future investigations of air-water flow properties should be carried out over long durations with advanced signal processing to account for the very slow fluctuations in jump position.
6. ACKNOWLEDGMENTS The authors acknowledge the advice and input of Dr Frédéric Murzyn (ESTACA Laval, France). They thank the technical staff of the School of Civil Engineering at the University of Queensland for their assistance. The financial support of the Australian Research Council (Grant DP120100481) is acknowledged.
7. REFERENCES Chachereau, Y., and Chanson, H. (2011), Free-Surface Fluctuations and Turbulence in Hydraulic Jumps. Experimental Thermal and Fluid Science, Vol. 35, No. 6, pp. 896-909 (DOI: 10.1016/j.expthermflusci.2011.01.009) Chanson, H. (2010), Convective Transport of Air Bubbles in Strong Hydraulic Jumps. International Journal of Multiphase Flow, Vol. 36, No. 10, pp. 798-814 (DOI: 10.1016/j.ijmultiphaseflow.2010.05.006) Chanson, H. (2011), Hydraulic Jumps: Turbulence and Air Bubble Entrainment. Journal La Houille Blanche, No. 1, pp. 5-16 & Front cover (DOI: 10.1051/lhb/2011026) Chanson, H., and Gualtieri, C. (2008), Similitude and Scale Effects of Air Entrainment in Hydraulic Jumps. Journal of Hydraulic Research, IAHR, Vol. 46, No. 1, pp. 35-44 Crank, J. (1956), The Mathematics of Diffusion. Oxford University Press, London, UK Hager, W.H. (1992), Energy Dissipators and Hydraulic Jump. Kluwer Academic Publ., Water Science and Technology Library, Vol. 8, Dordrecht, The Netherlands, 288 pages Henderson, F.M. (1966), Open Channel Flow, MacMillan Company, New York, USA Liggett, J.A. (1994), Fluid Mechanics, McGraw-Hill, New York, USA Long, D., Rajaratnam, N., Steffler, P.M., and Smy, P.R. (1991), Structure of Flow in Hydraulic Jumps. Journal of Hydraulic Research, IAHR, Vol. 29, No. 2, pp. 207-218 Mossa, M. (1999), On the Oscillating Characteristics of Hydraulic Jumps. Journal of Hydraulic Research, Vol. 37, No. 4, pp. 541-558. Murzyn, F., and Chanson, H. (2009), Free-Surface Fluctuations in Hydraulic Jumps: Experimental Observations. Experimental Thermal and Fluid Science, Vol. 33, No. 7, pp. 1055-1064 (DOI: 10.1016/j.expthermflusci.2009.06.003) Rajaratnam, N. (1965), The Hydraulic Jump as a Wall Jet. Journal of Hydraulic Division, ASCE, Vol. 91, No. HY5, pp. 107-132 Valiani, A. (1997), Linear and Angular Momentum Conservation in Hydraulic Jump. Journal of Hydraulic Research, IAHR. vol. 35, No. 3, pp. 323-354 Wang, H., and Chanson, H. (2013), Free-Surface Deformation and Two-Phase Flow Measurements in Hydraulic Jumps. Hydraulic Model Report No. CH91/13, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 108 pages Zhang, G., Wang, H., Chanson, H. (2013), Turbulence and Aeration in Hydraulic Jumps: Free-Surface Fluctuation and Integral Turbulent Scale Measurements. Environmental Fluid Mechanics, Vol. 13, No. 2, pp. 189–204 (DOI: 10.1007/s10652-012-9254-3)
