Experimental investigation of oxygen transfer efficiency in hydraulic jumps, plunging jets and plunging breaking waves

The scope of the paper is to analyse the different similarities of air entrainment among the hydraulic jumps, plunging jets, and plunging breaking waves and to discuss current practices. The measured data are reexamined and scrutinised to investigate the gas exchange phenomena through an air-water interface. In particular, oxygen transfer efficiency and penetration depth by air bubbles are discussed. The calculated results highlight that the oxygen transfer efficiency is decreased with the increase of energy dissipation rate both in plunging jets and breaking waves. In contrast, it is shifted almost parallel in the case of hydraulic jumps. In addition, the aeration lengths in the hydraulic jumps and penetration depths both in plunging jets and plunging breaking waves were dependent on the jet impact velocity.

3% along with the water depth, and this level can rise to 8% during a storm surge. But up to date, very little attention is paid to examining the effects of gas exchange in the surf zone.
Several researchers (Van de Sande & Smith 1976;Biń1993;Chanson 1995;Ohl et al. 2000;Anandraj 2012;Bostan et al. 2013;Harby et al. 2014;Jadhav et al. 2017;Bertola et al. 2018;Estrella et al. 2021) studied the gas exchange phenomena in plunging jets and hydraulic jumps and observed some physical characteristics of air-water flow systems made by the water jet flow. Ohl et al. (2000) investigated the air entrainment at the free surface of a liquid pool and showed that the crests of surface disturbances on a falling jet are a powerful agent for air entrainment. Experimental investigations for oxygen exchange by vertical plunging jets were conducted by Van de Sande & Smith (1976), and they found an empirical correlation between penetration depth and oxygen transfer rate. An experimental study by Kumar et al. (2018) showed a significant effect of hollow jet angle on volumetric oxygen transfer coefficient (K L a) at higher jet velocities. Chanson (1995) and Chanson & Cummings (1994) developed a gas transfer model to predict the dissolved gas contents at the hydraulic jump and plunging breaking waves. Estrella et al. (2021) presented an extensive study of similarity and scale effects in a hydraulic jump for constant Froude number. Bertola et al. (2018) proposed a physical study of vertical supported planar water plunging jets in a relatively large-size facility, and results were compared favourably with the literature. Unsteady air-water flow measurements were conducted in a breaking bore propagating in a large-size channel by Leng & Chanson (2019), and they observed that air entrainment took place in the form of air entrapment at the roller toe.
On the other hand, many researchers attempted to establish relationships between aeration and energy dissipation (Chanson & Cummings 1994;Kucukali & Cokgor 2009;Bostan et al. 2013). Bostan et al. (2013) measured the dissolved oxygen in hydraulic jump using a handy oxygen meter for five different jet flow rates. They obtained the oxygen transfer efficiency, which was increased with relative energy loss. Chanson & Cummings (1994) showed an analogy of air bubble entrainment between vertical circular plunging jets and hydraulic jumps. Wüthrich et al. (2020) investigated the air entrainment mechanisms for wave-breaking surge against Froude numbers. Kucukali & Cokgor (2009) measured the oxygen transfer efficiency for a hydraulic jump of unit width channel, linearly related to the energy dissipation rate. Close similarities in aeration and energy dissipation performances were found between the classical hydraulic and Type B jumps, while Type D jumps showed lower energy dissipation efficiency and aeration (Montano & Felder 2020). Cihat et al. (2014) indicated that high- Uncorrected Proof head gated circular conduits flow system might significantly increase the dissolution of oxygen in the water. Data drivenmodels were employed for estimating hydraulic jump oxygen aeration efficiency (E 20 ) under a sluice gate using experimental data and sensitivity analysis and showed that Reynolds number was a significant parameter in the estimation of E 20 (Tiwari 2019). Witt & Gulliver (2012) estimated the oxygen transfer coefficient from scaled formulations of the liquid film coefficient, mean bubble diameter, air entrainment rate, and turbulent energy dissipation rate. Kamal et al. (2020) showed that the gas transfer efficiency was high when pre-aeration (air entrainment on the spillway face) occurred. Retsinis & Papanicolaou (2020) simulated a classical hydraulic jump in a horizontal open channel, and a comparison of the numerical results with experiments showed the validity of the computations. However, up to date, none of the studies compared the oxygen transfer efficiency among the hydraulic jumps, plunging jets, and plunging wave breakers.
The penetration depth of the entrained bubbles is a significant scale in the plunging jet arrangements, which affects the size of the immersed two-phase area. The performance of the aeration system is greatly affected by the residence time of the entrained air bubbles in the case of plunging jets. Iguchi et al. (1998) investigated higher flow rates with the arrangement of jet-interface variability. They observed that the higher air bubbles are buoyant just below the impact point, escaping rapidly and leading to a shorter penetration. McKeogh & Ervine (1981) proposed an empirical formula of penetration depth for different setup conditions. Similar tests were conducted by Suciu & Smigelschi (1976), and they established a relationship based on measured penetration depth. Miller (1987) pointed out that the penetration depth is greater for plunging breakers than spilling breakers. Chanson (1995) measured the aeration length for the hydraulic jump, which is renamed as penetration depth for plunging jet and plunging breaker. Guyot et al. (2020) derived a simple mathematical expression to predict the bubble cloud penetration depth.
Thus, there is a clear need for some fundamental studies on the characterisation of air entrainment, aeration system, and oxygen transfer efficiency in hydraulic jumps, plunging jets, and plunging wave breakers. In this context, the present study presents a new series of detailed physical experiments of air entrainment in vertical plunging jets. It investigates the data of the above cases taken from several references.
More specifically, this research: • evaluates the existing research in the proposed areas by studying basic ideas and current development towards the understanding of important controlling variables of penetration depth and aeration length; • provides a detailed and comprehensive analysis of oxygen transfer efficiency.

REVIEW OF A GAS EXCHANGE MODEL
The presence of air bubbles inside the water increases the transfer of gases between the atmosphere and oceans (e.g. oxygen, carbon dioxide, nitrogen). For these gases and using Henry's and Fick's law, it is stated that an oxygen balance equation can be written as the rate of change in dissolved oxygen concentration (dC/dt) is proportional to the mass transfer of oxygen from the air to liquid. Mathematically, it is found as where C denotes dissolved oxygen concentration (mg/L), C s is the saturation concentration of dissolved oxygen (mg/L), A represents the area of the air-water interface (m 2 ), K L is the oxygen transfer coefficient (m/s), and V is the aerated volume (m 3 ). Equation (1) is found by integrating within the limits C ¼ C 0 and C ¼ C t ; at t ¼ 0 and t ¼ t as follows which, after integration, Equation (2) becomes where A/V ¼ a represents the surface area per unit volume, C 0 and C t denote the oxygen concentrations at initial and at time t, respectively. Equation (4) displays that the values of K L a can be determined by the measured values of C s , C 0 , C t , and t. An identical base of comparison of K L a is generally known as the volumetric oxygen transfer factor (Deswal & Verma 2007), which is normalised at a 20°C standard temperature.

Hydraulic jumps
Although there are many water flows, a hydraulic jump is a phenomenon in the science of hydraulics that is frequently observed in open channel flow. In a hydraulic jump, the flow is characterised by air entrainment and is the leading oxygen transfer provider. Further, it confirms the air entrainment into the water and dissipates the upstream energy. Air packets are entrapped through the impinge point, broken up into fragile air bubbles. At the hydraulic jump, air bubbles are advected downstream into the free shear layer categorised by concentrated turbulence before reaching the free surface and finally releasing to the air (Figure 1(a)).

Data collection
Detailed information on the experimental setup and method are published by Bostan et al. (2013) for hydraulic jumps. We want to briefly mention experimental conditions to determine the oxygen transfer efficiency (OTE). The tests were done in a channel with the dimensions of 0.4 m Â 12 m Â 0.65 m, where the side walls were made of glass. Dissolved oxygen and water temperatures were taken manually before and after the hydraulic jump by DO200 meter for a half-minute interval. The DO meter was calibrated for each test. The experiment ranges of DO meter and temperature were 0-20 mg/L with the correctness of +2%, and À6 to 46°C with an accuracy of +0.3°C, respectively.

Aeration length (L a )
Air entrainment in the hydraulic jump is generated turbulence and associated energy dissipation along with the surface roller ( Figure 1(a)). The roller is characterised by recirculation, containing the steaming region and the foam layer. The distance of the large-scale turbulence region is usually known as the roller length (L r ), which was derived by Hager et al. (1990). They reanalyzed the primary data of Rajaratnam (1967) and defined the length of aeration (L a ) between impingement point and flow depth as where d 2 and Fr 1 denote the downstream flow depth and upstream Froude number, respectively. The aeration length advances monotonically with the increase of upstream velocity, as shown in Figure 2(a).

Oxygen transfer efficiency (OTE)
The efficiency of oxygen transfer depends on air intake at the hydraulic jump. Gameson (1957) suggested a mathematical formula for finding oxygen transfer efficiency (aeration efficiency) as, An oxygen transfer efficiency (OTE) equal to one means that the hydraulic jump has conveyed oxygen into water, where C 0 is equal to C s . A value of OTE . 1 means that the downstream water is supersaturated which is known, namely C 0 . C s , and OTE ¼ 0 means that no oxygen transfer occurs there. Temperature plays a significant role in the mass transfer mechanism, so oxygen transfer efficiency should be transformed based on the average temperature. Gulliver et al. (1990) developed the mass transfer equation to calculate the oxygen transfer efficiency for the hydraulic jump in terms of dissolved oxygen at 20°C and denoted it as OTE 20 . This can be written as where OTE 20 is the oxygen transfer efficiency at 20°C and T represents the temperature of the water. Holler (1971) estimated a relationship of oxygen transfer efficiency for the hydraulic jump as where ΔU represents the velocity difference before and after the hydraulic jump.   Avery & Novak (1978) indicated that one constraint would not be sufficient to express the oxygen transfer efficiency in the hydraulic jump. They said the formulation of OTE 20 based on the Froude number Fr 1 before the hydraulic jump and Reynolds number R e , which is given as where k ' is a constant and connected with water salinity concentration. Equation (7a) has been applied to calculate the oxygen transfer efficiency in the present study Consider a stationary hydraulic jump as presented in Figure 1(a). The mathematical formulation of the energy dissipation rate for the hydraulic jump is written as where DH ¼ (d 2 À d 1 ) 3 =4d 2 d 1 represents the head loss, and it is positive simply if downstream flow depth d 2 is greater than upstream flow depth d 1 , which is a necessary condition of the second law of thermodynamics. Furthermore, q ¼ V 1 d 1 , V 1 , and ρ w denote the water discharge per unit width, the upstream flow velocity, and the water density, respectively. Using Equations (4), (6) and (8) and data of Bostan et al. (2013) and Anandraj (2012), we have calculated the oxygen transfer efficiency and energy dissipation rate, which is plotted in Figure 2(b). The oxygen transfer efficiency is converted to the standard temperature using Equation (7a). The aeration efficiency is shifted almost parallel to the increased energy loss in Figure 2(b).

Plunging jets
Once a vertical water jet impacts a receiving pool of water at steady phenomena, air bubbles entrainment occurs and carries away beneath the free surface of the water's pool. This phenomenon is known as plunging jet air entrainment. A definition sketch and photo of a vertical circular plunging jet are presented in Figure 1(b). According to the literature, the characteristic of air entrainment into water is related to the jet impact velocity. Moreover, observations on vertical plunging jet show that air starts to be entrained when the jet impact velocity exceeds a critical value; see for examples (Biń1993; Chanson & Cummings 1994;Bertola et al. 2018). That is, when the impact velocity is larger than the onset velocity (V 1 . V e ), air bubbles are entrained at the impingement point between the supported free-falling jet and the pool of still water. The mechanisms of air bubble entrainment differ depending on the jet impact velocities.

Descriptions of measurement
The measurements of dissolved oxygen were performed by oxygen meter (DO) below the free surface within a cubic type of water pool (50 cmÂ50 cmÂ45 cm), and it was constructed with transparent side walls (Figure 3(a)). The plunging water jet was impacted on the free surface of the water in the water pool vertically. The diameter of the circular nozzle was d 0 ¼ 1.3 cm. A high-definition camera was used to take photos of the air entrainment process. The video movies were analysed manually to guarantee the maximum reliability of the data. The camera was positioned beside the plunge pool. Figure 3(a) shows a sketch of the relative positions of the experimental facility and high-speed camera system. The oxygen meter was calibrated before each experiment. The main oxygen electrode was located at the end of the cable. The experiment ranges of DO meter were 0-20 mg/L with the correctness of +0.4 mg/L. A digital thermometer with an accuracy of +0.1°C was used for temperature measurements. Before testing the dissolved oxygen, the meter was calibrated with signals that simulated the ideal solution (based on a 25°C ambient environment). Dissolved oxygen is best measured directly in the water using a calibrated dissolved oxygen meter. This meter can measure the amount of dissolved oxygen directly in the water as mg/L. Three free falling lengths x 1 ¼ 30 cm, x 1 ¼ 20 cm and x 1 ¼ 10 cm was investigated for a constant nozzle flow velocity V N ¼ 1.51 m/s. A visualisation technique was used to determine the penetration depth between the camera and computer. During the experiments, the penetration depths of the entrained air bubbles were constantly more minor than the depth of water in the tank. The water jet impinges onto the water surface vertically, and the jet velocity at an impinging point on the water surface, which the Bernoulli equation can determine as q is the jet velocity at the impinging point, V N represents the nozzle velocity of water, and x 1 denotes the free-falling of water length.

Penetration depth (P d )
The penetration depth is defined as the depth between the free surface and the point where the rising velocity of the bubble equals the vertical water speed. Several researchers measured the penetration depth in the vertical circular plunging jets system (Kumagai & Endoh 1983;Ohkawa et al. 1987;Harby et al. 2014). In the present study, Figure 3(b) shows the three representative bubble swarm images for jets with three different free-falling lengths x 1 ¼ 30 cm, x 1 ¼ 20 cm, and x 1 ¼ 10 cm with the same nozzle flow velocity of V N ¼ 1.51 m/s, respectively. The penetration depths were determined Uncorrected Proof vertically from impinging point to a stable point by a video camera. The lower part of the bubble cloud constantly oscillates with time (the lower part was not fully stable), so the time-averaged penetration depth was determined. It is seen in Figure 3(b) that the jet with the higher free-falling length causes more bubble entrainments and lower penetration depth (P d ). Furthermore, we can see that the penetration depth is decreased with the increase of jet impact velocity in Figure 4(a), which follows the results of Qu et al. (2011) andHoque (2002). Penetration depth, P d, is dependent on x 1 , V N /d N and the nozzle aspect ratio l N /d N (see Kumagai & Endoh 1983;Ohkawa et al. 1987).

Oxygen transfer efficiency (OTE)
The oxygen transfer performance of vertical plunging jets is usually uttered based on standard oxygen transfer rate (O R V), which can be obtained as where O R represents the rate of oxygen transfer (mg/L/h) at 20°C and 1 atm pressure, V represents the aerated volume of water (m 3 ), and E pj is an upstream energy transport rate (J/(m.s)). The mathematical definition of O R is given as below Uncorrected Proof (Deswal & Verma 2007): where C Ã S is the saturation dissolved oxygen concentration in water at standard conditions (mg/L). In the case of a plunging jet, laboratory experiments were conducted to determine the oxygen transfer efficiency (OTE). From a typical sample of the deoxygenated water, C 0 was measured initially. After aeration, a certain period (t ¼ 5 minutes), samples of oxygenated water were taken to calculate oxygen concentration (C t ) after time t. Finally, the values of K L a, O R , and OTE 20 were computed using Equations (4), (11) and (10), respectively.

Upstream energy transport rate (E pj )
Mathematically, the upstream energy transfer rate for a vertical circular plunging jet is defined as where r 1 is the radius of the water jet (m) at the impinging point, and Q w denotes the water discharge or water jet flow rate (m 3 /s). The results regarding aeration efficiency in vertical circular plunging jets are presented in section 4.

Plunging breaking waves
An air entrainment phenomenon occurs at plunging breaking waves which contribute to the oxygen transfer through an airwater interface. At the impact point, Chanson & Lee (1997) showed similarities between plunging jet and plunging breakers (deeper waters). The jet impact velocity may be scaled by 2gH b, where the water jet thickness is approximately 0.05*H b . The aeration characteristics and penetration depth of plunging breaking waves in the sea have similarity with the plunging jets which occurs in the waterfall (Koga 1982;Chanson & Cummings 1992). The relationship of air entrainment between plunging waves and jets is reinvestigated through such evidence.

Data collection
DO is a vital water parameter that was measured by a Senso Direct' DO meter ( Figure 5(a)). The measurement ranges of the oxygen meter were 0-20 mg/L for DO with an accuracy of +0.4 mg/L and 0-50°C for temperature with an accuracy of +0.8°C. Before testing the DO, the meter was calibrated with signals that simulated the 'ideal' solution (based on a 25°C ambient environment). The sampling region of this study is the coastal area of the southeastern part of Bangladesh near the Cox's Bazar, which lies between 21°11 0 6″ N latitude and 92°2 0 52.8″ E longitude. In Figure 5(a), points A to E show the sampling points for the air-water mixture. The sampling implies A, B, C, D and E were fixed with a distance of 1.5 m at the inner surf zone. The data were collected from the mentioning points, analysed by the Gas Analyzer meter. For each sample, measurements of DO were taken by immersing the electrode probe and are plotted in Figure 5(c). Tsoukala & Moutzouris (1996) presented experimental data of oxygen transfer for breaking waves. In their experiment, the beach slope of l:2.3 was positioned at the end of the wave flume, and water depth and wave period were considered 72 cm and 1.55 s, respectively. Wave frequencies with ranges of 0.65-0.90 Hz and wave heights in the ranges of 8.5-20.5 cm were generated. Characteristically, the generated waves were plunging breaker types. The dissolved oxygen in the air-water mixture was measured over time by a handy DO meter brand OXI-96 at 21 examining locations through the flume. The transfer coefficients K L were determined from the wave breaking area at 20°C.

Penetration depth (P d )
In the surf zone, the penetration depth (P d ) of air bubbles for plunging breaking waves is found as the depth between the free surface of the water and the deepest air entrainment point, where the air bubbles are reached during the wave breaking process (Figure 1(c)). Significant fluid turbulence occurs where breaking waves completely disintegrate after their crest impinges against water. At this impinge point, the downward velocity of air bubbles turns to zero, where the buoyancy force works against these. The driven air bubbles are broken into tiny bubbles by buoyancy force, and these air bubbles rise to the free surface. During their experiments, the penetration depths were recorded and determined vertically from the impinging point to a stable point by Chanson & Lee (1997). Their results recommended that the bubbles were entrained downward up to 1.2-2 times the wave amplitude beneath the free surface.

Oxygen transfer efficiency (OTE)
Applying a similar calculation method as the hydraulic jump, the oxygen transfer efficiency is calculated using Equation (6), knowing the gas transfer coefficient from Tsoukala & Moutzouris (1996). They measured the transfer coefficients K L from the breaking area, and after that, it transformed to 20°C using Equation (13), which was derived by Daniil & Gulliver (1988):

Energy transport rate (E pb )
The wave energy transmitted by the water wave motion from the deepwater toward the shore is written by where c g , H 0 , and T represent the group velocity, deepwater wave height, and wave period, respectively. If we assume that the total energy is dissipated by the progressive wave in the surf zone, we can write where V b denotes the water volume per unit width. Water Supply Vol 00 No 0, 11 Uncorrected Proof , the above equation can be rewritten as where i is the beach slope.

RESULTS AND DISCUSSION
A similarity of aeration efficiency, dissolved oxygen, and penetration depth among the hydraulic jumps, plunging jets, and plunging breaking waves are investigated and plotted against the energy loss and jet impact velocity. In Figure 2(a), the aeration length increases with jet impact velocity. The resulting trend in Figure 2(a) is similar to Anandraj (2012) data. Bostan et al. (2013) andHager et al. (1990) measured the aeration length for a hydraulic jump, and they found that it was increased with the jet impact velocity.
On the other hand, Figure 2(b) shows the oxygen transfer efficiency for hydraulic jump, which is presented against the energy dissipation rate. It is seen in Figure 2(b) that with the increase of energy dissipation rate in a hydraulic jump, oxygen transfer efficiency is almost parallel shifted, which is consistent with the result of Anandraj (2012). As Kucukali & Cokgor (2009) mentioned, OTE increased with the increase of energy dissipation along with the jump. This may be because the saturation values of DO and initial DO values of water influence the aeration process. Liu et al. (2004) demonstrated the direct relationship between head loss and turbulence kinetic energy. Ozkan et al. (2014) measured the aeration efficiency related to the air-demand ratio and Froude number.
The penetration depth of air bubbles created by plunging water jet, P d , and the oxygen transfer efficiency results (OTE) is plotted in Figure 4, which are varied with the jet impact velocity V 1 and energy dissipation rate E pj , respectively. Figure 4(a) shows the changes in penetration depths, P d formed by the water jets. It is observed from Figure 4(a) that P d decreases with the increasing V 1 for a particular value of V N . Hoque (2002), Qu et al. (2011) and Harby et al. (2014) also measured the penetration depth against the jet impact velocity, and they found similar results to those that are plotted in Figure 4(a). Cumming (1975) showed theoretically that the penetration depth was linked to the jet velocity. In addition, the oxygen transfer performance of the vertical circular plunging jets was calculated using Equation (10). As shown in Figure 4(b), the measured results of oxygen transfer efficiency are found reasonably consistent with the data of Baylar et al. (2006), although present obtained data are very scarce.
We have also investigated the gas exchange phenomena by plunging breaking waves in the coastal area. As displayed in Figure 5(b), the penetration depth decreases with the increase of plunging breaking jet velocity, which has similarities with the plunging jet. Moreover, it is found that DO is reduced with the increase of a shoreward distance in Figure 5(c). After the breaking point, many bubbles are entrained into the water, producing greater oxygen absorption. Similar results were measured by Hosoi & Murakami (1986). In Figure 5(d), we have also observed that OTE almost keeps constant with the increase of energy dissipation rate for the present study; those are calculated using Equations (13) and (14).
Free surface flows are typically scaled with the Froude similitude keeping identical Fr 1 both in the model and prototype. Usually, the selection of the Froude similitude is derived based upon some primary theoretical considerations and air transport in models are affected by the scale effect. In this study, the comparative analysis of OTE, which is calculated based on the oxygenation for hydraulic jump, has revealed almost identical for Fr 1 ¼ 2.5-4.0 (Figure 2(b)). New air-water flows properties were measured in vertical circular plunging jets. The data set together with the earlier data of Baylar et al. (2006) yielded scale effects in terms of oxygenation (Figure 4(b)) with identical inflow Froude numbers (Fr 1 ¼ 8.0-13.0). The scale effects also affected the oxygenation process by air bubbles in the surf zone and laboratory wave flume ( Figure 5(d)). The entrained air bubbles or dissolved oxygen are not similar to the large-scale physical models compared to laboratory wave flumes.

CONCLUSIONS
The analogies of gas exchange among hydraulic jumps, plunging jets, and plunging waves may be appropriate to work out quantities under modelling efforts. The obtained results from the data analysis can be summarised as below: (1) The length of aeration in hydraulic jumps and penetration depths both in plunging jets and plunging breakers were found to be pretty similar, and those were dependent on the jet impact velocity.
(2) Oxygen transfer efficiency was shifted almost parallel with the increased energy loss in the case of hydraulic jumps. On the other hand, it was found that the oxygen transfer efficiency is attenuated as the increased energy dissipation rate in plunging jets and it kept constant in plunging breaking waves. Further research work is necessary in this regard. (3) Scale effect was observed in terms of the oxygenation for identical inflow conditions of vertical circular plunging jet, whereas it was similar for the hydraulic jump. Furthermore, the scale effect was found distinct between the small-scale laboratory results and field conditions. (4) These investigations and observations might provide valuable evidence for evaluating the oxygen transfer rate and ocean aeration due to spilling and plunging breakers.