Numerical Simulation of Hydraulic Jumps. Part 1: Experimental Data for Modelling Performance Assessment
Abstract
:1. Introduction
2. The Hydraulic Jump Case Study
2.1. General Remarks
2.2. Inflow Froude Number and Hydraulic Jump Typology
2.3. Hydraulic Jump Flow Structure and Related Experimental Studies
- Wall jet velocity decay: the high-speed inlet flow impacts the slower moving water body and the shearing reduces the maximum velocity through the hydraulic jump length, likewise a turbulent wall jet decay. The shear layer also expands into the roller region. The analogy between hydraulic jumps and turbulent wall jets was first conducted by Rajaratnam [14]. This parameter is highly relevant as it holds the biggest part of the kinetic energy which, during the decay, is transformed into pressure and potential energy (depth). Experimental data can be found in Wang and Chanson [51], Chanson [56], Liu et al. [57], Chanson and Brattberg [16], Wu and Rajaratnam [58], Hager [2] and Ohtsu et al. [59].
- Roller length: marked in Figure 4 as , it is one of the most distinctive features of a hydraulic jump. Some of the inflow uplifts to the free surface, reaching a stagnation point and reversing, hence falling back again to the toe, where it impacts with the inflow jet. As the stagnation point moves continuously, its visual estimation is affected from a certain uncertainty in a physical model. Experimental data can be found in Murzyn et al. [60]. Wang and Chanson [52], Carollo et al. [36] and Hager [2] presented a nearly parabolic experimental relationship.
- Hydraulic jump length: marked in Figure 4 as . Different methods have been discussed concerning its identification as, for instance: a horizontal free surface can be observed, a hydrostatic pressure distribution is established or where the hydraulic jump is fully deaerated. However, it is also difficult to estimate as the free surface is wavy downstream and tends smoothly to horizontal, which implies a large degree of uncertainty. Pressure distribution is not commonly measured and deaeration may strongly depend on the modelling scale. Bayon et al. [61] argued that the turbulence produced in the toe of the hydraulic jump tended at the end of the hydraulic jump to the values commonly observed in open channel flows, but, for an experimental study, this is even more rare to measure. Given the difficulty of all these methods, visual determination is oftentimes the experimentalist preferred choice.
- Mean free surface profile: it necessarily matches the supercritical flow depth at the toe while it asymptotically tends to the downstream water depth at the end of the hydraulic jump. With an abrupt increase at the toe, it presents a concave curve during the hydraulic jump extension for Froude numbers over the undular jump limit (Table 1). In addition to the aforementioned studies, relations for this parameter were also suggested by Wang [50], Bakhmeteff and Matzke [12] and, analytically, by Valiani [55]. For undular jumps, experimental data can be found in Lennon and Hill [62] and analytical considerations were proposed by Bose et al. [63].
- Mean velocities: the high-speed jet entering the hydraulic jump splits, partly reversing and partly reducing its velocity, thus matching the downstream open channel flow velocity profile. This yields a complex velocity distribution zero-value-crossing at the roller region with an important negative velocity reaching a magnitude of up to 0.4 to 0.6 of the inlet velocity [3,56,64]. Recently, Wang and Chanson [51] suggested that this magnitude has an inverse relation with the Froude number. Empirical velocity distributions were proposed by Chanson and Carvalho [3] and Hager [2]. Literature is rich in experimental data too; see Wang and Chanson [51], Wang et al. [65], among others. Lin et al. [66] studied, separately, the velocity of the water and air phases by using Particle Image Velocimetry (PIV) and Bubble Image Velocimetry (BIV), alternatively. For undular jumps, experimental data can be found in Lennon and Hill [62].
- Aeration (impingement and interfacial): large quantities of air are entrained inside the hydraulic jump via the toe impingement. These large air volumes are subject to break-up and coalescence depending on the surrounding turbulence quantities. A second air entrainment mechanism is related to the interfacial fluctuations occurring in the upper region of the roller at higher Froude numbers. Large quantities of spray and splashing can be visually observed and have a clear footprint in the air concentration profiles. For both mechanisms, different air concentration profiles can be fitted, hence revealing different air entrainment mechanisms [67,68]. Far downstream, the flow deaerates as the velocity decreases and the transport capacity is reduced. The air concentration is, probably, one of the most case sensitive mean flow variables as it is a result of the combination of Froude, Reynolds and Weber—or equivalently, Morton—numbers [69]. Another reason for the case-sensitivity of the air entrained quantities should be related to the level of development of the inflow boundary layer, as reported by Takahashi and Ohtsu [70]. Takahashi and Ohtsu [70] noted that the advective diffusion region showed larger aeration, with greater development of the inlet boundary layer, while the (upper) breaking region was nearly insensitive to this variable. Ervine [71] also discussed on the relevance of the inlet flow free surface perturbations on the total air entrained. Moreover, the interaction between air bubbles diffusion and momentum transfer is not completely understood [68]. Air transport in the streamwise direction is mainly due to advection, whereas transport in the spanwise and normalwise direction is necessarily driven by turbulent diffusion. In the vertical direction, buoyant forces can also play an important role, especially at the end of the hydraulic jump where lower velocities occur. Experimental data on air concentrations and different semi-empirical relations can be found in Wang and Chanson [51], Takahashi and Ohtsu [70], Gualtieri and Chanson [68], Murzyn et al. [67], Chanson and Brattberg [16], among others.
- Inlet boundary layers (water and air): the supercritical flow impacting the hydraulic jump develops from farther upstream as long as its extension allows. Two boundary layers take place, the water boundary layer that can affect the hydraulic jump characteristics [44,70]—and thus it is common to distinguish between partially developed and fully developed inlet flows—and the interfacial air layer flow [72], which could affect the hydraulic jump air entrained quantities according to Ervine [71]. For the supercritical water boundary layer development, several methods exist, such as Castro-Orgaz and Hager [73] and Castro-Orgaz [74] based on spillway prototype scale velocity profiles. For the interfacial air flow, the only data-set was collected by Valero and Bung [72], which also discussed on the occurring free surface instabilities in supercritical flows that were suggested by Ervine [71] to affect the entrained quantities. A general framework for the computation of the free surface perturbations can be found in Valero and Bung [75]. Recently, Bertola et al. [76] showed that inlet disturbances considerably affect the air entrainment rates for planar plunging jets.
- Velocity fluctuations: given the turbulent nature of hydraulic jumps, instantaneous velocities oscillate around the mean value. Velocity fluctuations were first reported by Rouse et al. [13] using an air flow channel. Long et al. [77] studied velocity fluctuations in submerged hydraulic jumps for Froude numbers up to 8.0, highlighting the three-dimensional nature of the hydraulic jump; and Liu et al. [57] presented turbulence measurements for low Froude numbers, including turbulence spectra. Turbulence intensity has also been approximated using the width of the cross-correlation function between the signals of the two tips of phase detection probes; see Chanson and Toombes [78] for further description and limitations and study of Murzyn and Chanson [49] for an application to hydraulic jumps. For undular jumps, experimental data were collected by Lennon and Hill [62]. Velocity fluctuations are of utmost interest to ensure the stability of downstream environment.
- Interfacial oscillations: the hydraulic jump free surface shows a wide range of turbulent motions. These oscillations play an important role in the study of the total air entrained in the hydraulic jump, as discussed above. Recently, some experimental studies have found that the maximum oscillation occurs close to the toe location and that the intensity of these oscillations increases with the Froude number. Experimental data can be found in Wang and Chanson [51], Chachereau and Chanson [79], Murzyn and Chanson [80].
- Toe oscillations: the roller flow backwards periodically as a consequence of all the hydrodynamic processes that occur inside the hydraulic jump. The Strouhal number (dimensionless frequency based on the toe frequency, inflow depth and mean velocity) is oftentimes used to describe this phenomenon. Toe oscillations have been reported by Zhang et al. [64], Chachereau and Chanson [79], Chanson and Gualtieri [69], Gualtieri and Chanson [68], Mossa and Tolve [81], and Long et al. [82].
- Large vortices advection: large vortices are created in the shear region between the inlet high-speed jet and the roller (vortex shedding). These are advected downstream with a velocity around 0.4 times the inlet velocity. It has not been noticed a clear effect of the Reynolds number. Experimental data can be found in Chanson [56] and Wang [50].
- Pressure fluctuations: the high-speed inlet jet has an impact against the slower water body abruptly slowing down while transforming some of the kinetic energy into pressure, which is later converted into potential energy (in the form of increasing depth). Considerable turbulent pressure fluctuations occur inside the hydraulic jump, thereby compromising the structural stability. Fiorotto and Rinaldo [83] discussed on the spatial structure and magnitude of these pressure oscillations at the channel bed level. Abdul Khader and Elango [84], additionally, also provided insight on the pressures spectra. Hydraulic jump related high pressure fluctuations also have a complex impact on the hyporheic flows occurring in river flows [4,5]. Special care must be taken to properly understand some recent experimental data reporting results on total pressure, which accounts both for the pressure and the velocity head and needs to be corrected with simultaneous velocity estimations and air concentrations to extract the real pressure head. Further description on this technique applied to hydraulic jumps can be found in Wang et al. [65,85] and Wang [50].
- Hydraulic jump vorticity: it is produced at the toe of the hydraulic jump. The inlet flow is abruptly subject to large shearing quantities after impinging the roller flow. The shearing reduces as the flow advances streamwise and the turbulence produced in the toe is slowly dissipated. Hornung et al. [86] analytically studied the mean vorticity downstream of a hydraulic jump and its relation with the Froude number.
- Inner turbulent structures: turbulent length and timescales have been measured using arrays of conductivity probes. Several synchronized sensors can be placed inside the flow and correlation of two signals allow a certain physical insight. Integration of the correlation functions (across time or spatial lag) leads to the estimation of the integral turbulent scales (temporary and spatial scales, respectively). These turbulent scales can be understood as an average measure of the turbulent structures taking place inside the hydraulic jump. A complete data bench of experimental data can be found in Wang and Murzyn [17].
- Bubble characteristics: bubble frequencies, chord lengths and times and bubble clustering are the result of an interaction among turbulent processes. Its determination may be one of the most challenging—if not the most—for a numerical model. These variables have been, however, reported in some experimental studies (e.g., [67,68,87,88]).
3. Adequacy of the Hydraulic Jump as a Benchmark for Numerical Modelling
- Deep understanding of the phenomenon, as presented in Section 2.3.
- Large hydraulic tradition, which makes most of the community familiar with this type of flow.
- It represents an extreme case for turbulence, aeration and flow recirculation, which ensures that numerical models representing properly a CHJ are likely to be useful for other environmental applications.
4. Discussion on Experimental Uncertainties
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Bradley and Peterka [46], Hager [2] | Montes [47] | Chow [48], Chanson [42] | |||
---|---|---|---|---|---|
range | Classification | range | Classification | range | Classification |
1–1.7 | Without roller | 1.2–2 | Undular | 1–1.7 | Undular |
1.7–2.5 | Pre-jump | 2–4 | Oscillating | 1.7–2.5 | Weak |
2.5–4.5 | Transition | 4–9 | Stable | 2.5–4.5 | Oscillating |
4.5–9 | Stabilized | 4.5–9 | Steady | ||
>9 | Choppy | >9 | High dissipation | >9 | Strong |
Flow Variables | Hager [2] | Murzyn and Chanson [49] | Wang [50] | Other Relevant Studies |
---|---|---|---|---|
Sequent depths | ER | ED (*) | ED | ER: Palermo and Pagliara [38], |
Chanson [34], | ||||
Pagliara et al. [20] | ||||
Maximum velocity decay | ER | ED | ER, ED | ER: Wang and Chanson [51], |
Chanson and Brattberg [16], | ||||
Rajaratnam [14] | ||||
Roller length | ER, ED | ED | ER, ED | ER: Wang and Chanson [52], |
Carollo et al. [53] | ||||
Jump length | ER | ED | ER: Schultz et al. [54], | |
Peterka [22] | ||||
Mean free surface profile | ER, ED | ED | ER, ED | ER: Wang and Chanson [51], |
Valiani [55], | ||||
Bakhmeteff and Matzke [12] | ||||
Mean velocity profiles | ER | ER, ED | ER, ED | ER: Wang and Chanson [51] |
Air concentrations | ED (**) | ER, ED | ER, ED | ER: Wang and Chanson [51], |
Chanson and Brattberg [16] |
Techniques | Basic Variables Measured | Description | Relevant References |
---|---|---|---|
Point gauge | Static flow depth | Static measurements, low accuracy for turbulent free surfaces | – |
Acoustic Displacement Meters (ADM) or Ultrasonic Sensors (USS) | Instantaneous flow depth | Low-band frequency capped, preference for foreground obstacles | Zhang et al. [94] |
Light Detection and Ranging (LiDAR) | Instantaneous free surface profiles | Both spatial and temporal free surface detection, large surface detection, amounts of outliers | Montano et al. [95] |
Phase detection probes: conductivity or optical fibre probes | Air concentration, Mean velocity | Problems in regions with recirculation, only measures in the probe direction, intrusive | Wang [50] Chanson [96] Felder and Pfister [97] Kramer et al. [98] |
Acoustic Doppler Velocimeters (ADV) | Instantaneous flow velocities | Can only be applied in regions with low presence of bubbles | Liu et al. [57] |
Particle Image Velocimetry (PIV) | Instantaneous flow velocities | Can only be applied in regions with low presence of bubbles | Lin et al. [66] |
Bubble Image Velocimetry (BIV) | Instantaneous flow velocities | Underprediction of flow velocities, when compared against other intrusive techniques | Bung [99] Leandro et al. [100] Bung and Valero [101] |
Optical Flow (OF) | Instantaneous flow velocities | Better match with intrusive techniques, more robust to strong noises present in the bubbly images | Bung and Valero [102,103,104] Zhang and Chanson [105] |
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Valero, D.; Viti, N.; Gualtieri, C. Numerical Simulation of Hydraulic Jumps. Part 1: Experimental Data for Modelling Performance Assessment. Water 2019, 11, 36. https://doi.org/10.3390/w11010036
Valero D, Viti N, Gualtieri C. Numerical Simulation of Hydraulic Jumps. Part 1: Experimental Data for Modelling Performance Assessment. Water. 2019; 11(1):36. https://doi.org/10.3390/w11010036
Chicago/Turabian StyleValero, Daniel, Nicolò Viti, and Carlo Gualtieri. 2019. "Numerical Simulation of Hydraulic Jumps. Part 1: Experimental Data for Modelling Performance Assessment" Water 11, no. 1: 36. https://doi.org/10.3390/w11010036