Air bubbles and water droplets entrainment and removal in turbulent water flows

Anthropogenic activities are making considerable influence on the character of natural processes taking place in atmosphere as well as in open water reservoirs. With the continuous extension of manufacturing activities, the anthropogenic factor has an increasing impact on global environment, including both air and water pollution. The challenge we face is to evaluate and determine the maximal anthropogenic impact that the environment, i.e. water and atmosphere, can still be subjected to the level it can resist trough self-purification. Such an objective can not be achieved without investigation of the pure physical processes, which take place in interfaces between open water and atmosphere [1-4] as well as between separate deep water layers. Both interfaces are of significant importance for the water quality [5-7] of open water bodies. Usually those interfaces are considered separately one from another. Some references [8-11] specify that physical processes at the open water-air interfaces are stronger determinants of water quality than those taking place in deep water. Therefore the consideration of the present paper is focused on the open water-air interface. The above mentioned physical phenomena are fully determined by diffusion based processes such as an aeration and reaeration or evaporation and adsorption processes [12-15]. Moreover, those processes can be regarded as separate cases of diffusion itself, where the character of processes rate is determined by ambient properties and their boundary conditions. Investigation and evaluation of those properties and boundary conditions in various specific cases obviously is a pure physical problem [16-18]. The right solution can be achieved only under the investigation of this problem related to diffusion process and its mathematical description. Determined significant dependences between the processes parameters allow to simulate the process mathematically and give practically useful tool for environmental engineering. On the background of above mentioned considerations can be concluded that the investigation of diffusion processes on the air-water interface should be based on the classical theory of diffusion [19-22]. On the other hand, despite of common general diffusion process features, airwater interface has its own specific behaviour [23-25]. The factor determining this behaviour first of all is specific boundary conditions on the air-water interface. These specific boundary conditions are: roughness of water surface, due to it higher friction between air and water, kinetics of the streams rising from deep layers to the surface, etc. One of the most important factors making influence on diffusion processes is mass transfer over the weirs and through the hydro power plants turbines [26-28]. This falling water from weirs or flow through hydro power plants turbines is so called “white water” phenomenon. This water consists of air bubbles and water droplets mixed together in the water surface. The goal of this paper is the mathematical simulation of physical mechanism of air bubbles entrainment and removal of water drops processes evaluating water turbulence generated by wears and hydro power stations.


Introduction
Anthropogenic activities are making considerable influence on the character of natural processes taking place in atmosphere as well as in open water reservoirs.With the continuous extension of manufacturing activities, the anthropogenic factor has an increasing impact on global environment, including both air and water pollution.The challenge we face is to evaluate and determine the maximal anthropogenic impact that the environment, i.e. water and atmosphere, can still be subjected to the level it can resist trough self-purification.Such an objective can not be achieved without investigation of the pure physical processes, which take place in interfaces between open water and atmosphere [1][2][3][4] as well as between separate deep water layers.Both interfaces are of significant importance for the water quality [5][6][7] of open water bodies.Usually those interfaces are considered separately one from another.Some references [8][9][10][11] specify that physical processes at the open water-air interfaces are stronger determinants of water quality than those taking place in deep water.Therefore the consideration of the present paper is focused on the open water-air interface.The above mentioned physical phenomena are fully determined by diffusionbased processes such as an aeration and reaeration or evaporation and adsorption processes [12][13][14][15].Moreover, those processes can be regarded as separate cases of diffusion itself, where the character of processes rate is determined by ambient properties and their boundary conditions.Investigation and evaluation of those properties and boundary conditions in various specific cases obviously is a pure physical problem [16][17][18].The right solution can be achieved only under the investigation of this problem related to diffusion process and its mathematical description.Determined significant dependences between the processes parameters allow to simulate the process mathematically and give practically useful tool for environmental engineering.
On the background of above mentioned considerations can be concluded that the investigation of diffusion processes on the air-water interface should be based on the classical theory of diffusion [19][20][21][22].On the other hand, despite of common general diffusion process features, airwater interface has its own specific behaviour [23][24][25].The factor determining this behaviour first of all is specific boundary conditions on the air-water interface.These specific boundary conditions are: roughness of water surface, due to it higher friction between air and water, kinetics of the streams rising from deep layers to the surface, etc.One of the most important factors making influence on diffu-sion processes is mass transfer over the weirs and through the hydro power plants turbines [26][27][28].This falling water from weirs or flow through hydro power plants turbines is so called "white water" phenomenon.This water consists of air bubbles and water droplets mixed together in the water surface.
The goal of this paper is the mathematical simulation of physical mechanism of air bubbles entrainment and removal of water drops processes evaluating water turbulence generated by wears and hydro power stations.

Hydropower plants and weirs influence on bubbles entrainment
After weirs the water has falling stream or jet shape.As it's known, under these conditions the local aeration is taking place and the air bubbles are entrained into the water due to jet impact to the water lower stream (Fig. 1).

Fig. 1 Air bubbles and water droplets diffusion under falling down jet conditions
Often in the literature such an air bubbles and a water droplets mixing is known as a "self-aerated process".If to be more exact should be noticed that there are two processes: one when the water falls through the wear free and the nextwhen flow impacts in to the water or when the water forces through hydropower stations turbines.In the latter case the jet forms under the water and air bubbles are entrained in to the water as well as small water droplets compose.The air bubbles and water droplets mixing together form a compound layers.Researchers started to investigate this phenomenon only in the middle of the 20 century.The first were Straub and Anderson [29] and Wood [30].Straub and Anderson described a self-aeration process in the open channel water flows.Wood and Chan-son were first who measured concentration and the rising rate of air bubbles.Later Chanson, Toobes, Gonzalez continued the investigation.These authors investigated only air bubbles entrainment into the water as a diffusion phenomenon, though did not investigate and did not describe mathematically dynamical mixing of air bubbles and water droplets.McKeogh first proved that jet's conditions at the moment of air entrainment can be described as a function of jet's turbulence [31].Turbulent air bubbles diffusion appears under falling down jet conditions.This process is also known as advective diffusion [32][33][34][35].This term denotes the movement of air bubbles from domains with higher spatial gas concentration to domains with lower concentration.The mixing of bubbles stimulates this process.

Entrainmentremoval process in the air-water interface
The exchange of gases in the interface between the air and the water may be regarded as limited molecular diffusion.The simplest model of air-water kinetic exchange on liquid side at the boundary between the two air and water phases is shown in Fig. 2.

Fig. 2 Air-water kinetic exchange
With the water streaming over weirs, dams or through hydro turbines, the peculiar entrainment removal processes are taking place in the air-water interface (Fig. 3), generating air and water bubbles within the zone of falling jet.Thus, the layers with mixed air and water bubbles are formed.These layers are shown in Fig. 4.
Under particular conditions the similar mixtures of bubbles can be formed in lakes, large dams and oceans.In these cases breaking waves (wave breaking initiated at certain wind speeds) influence turbulent diffusion of air (oxygen and CO 2 ) and form water microlayers in the interface [36][37][38][39].The following sections deal only with processes influenced by the turbulence of water falling from the weirs or forcing through hydropower plants.
Fig. 3 Air-water interface without mixing Fig. 4 Layers with together mixed air-water bubbles

Equations describing air bubbles and water droplets entrainment and removal in the open channel
The supercritical flows induce a visual impression of "white water".Such flows appear in the channels where the turbulence is sufficient for air bubbles entraining into the flow.
As it is shown in Fig. 4 two fluxes come in to the water interface: water droplets flux and air bubbles flux.We suggest the following expressions of kinetic equations as a suitable for use in this case where 2

C
is concentration of water droplets, air C is con- centration of air bubbles, t is time.12  and 21  are en- trainment rates of water droplets and air bubbles respectively, which are expressed as 2 12 12 , , where α 12 and α 21 are sticking coefficients of water droplets and air bubbles to the surface respectively, which range from 0 to 1 and 2 HO i and air i are the relative water droplets and air bubbles fluxes to the surface respectively.
Solution of differential Eq. ( 1), gives the following dependences During entrainment process the upper layer gets in contact with air and concentration of air bubbles and water droplets changes.Proposed Eq. ( 1) describes character of this change.However, the process of water droplets removal from the surface is not involved in Eq. ( 1).Evaluating removal process the Eq. ( 1) takes the following form where    3) describe removal rate of air bubbles or water droplets from the surface monolayer and next two components describe entrainment rate of air bubbles or water droplets.Constituents with sign minus give the removal rate of air bubbles and water droplets from the surface monolayer and the constituents with sign plus give the entrainment rate of air bubbles and water droplets into the surface monolayer as a result of the entrainment of relocated bubbles also as a result of the arrival of air bubbles and water droplets from the next monolayer.
The monolayer approach of dynamic mixing process was used.Eq. ( 3) expresses this process.The variations of surface concentration in the second layer must be known for solving this equation.The mixing of water and air bubbles between the layers takes place because of continuous entrainment and removal.In case when due to entrainment and removal air and water particles move from K layer to K -1 or K + 1, this concentration rate can be expressed as follows where v e is the total entrainment rate.
It can be proved that Eq. ( 4) has the form of diffusion equation with moving boundary.After simple mathematical rearrangements Eq. ( 4) can be converted into the following form where D * is effective diffusion coefficient, v x is surface movement velocity.
Bubbles exchange between air and water usually takes place in the upper layers.In the case which we investigated the element composition of air bubbles and water droplets was changeable also in lower layers.

Results and discussions
As Fig. 5 illustrates, air bubbles are entrained in to the water at the zone of falling water.The size and number of bubbles pooled in to water depends on weir high and on velocity of jet at the moment when the jets hit at the surface of the water.Jet hit to the water splashes small water droplets, which mix with the air bubbles together and makes mixed air-water layers.We simulated air bubbles and water droplets entrainment and removal processes with Eq. ( 4).Also we calculated Reynolds and Froude numbers according the following formulas where ρ w is density of the water (kg/m 3 ), v is jet impact velocity, d 1 is jet thickness at impact (m), µ w is dynamic viscosity of the water (N s/m 2 ), g is gravity constant (m/s 2 ).
For the simulation we used experimental data published by H.Chanson, T. Brattberg [40], F.Murzyn, H.Chanson [41] and L.Toombes, H.Chanson [42], H.Chanson C.A.Gonzalez [43].These authors obtained experimental data in artificial channel made for special purpose of their experiment.They determined concentration of air and water bubbles by means of conductivity measuring in different zones of the channel with changing height of weir, flow velocity and horizontal distance from weir to bubbles zone (Fig. 5).Determination of bubbles concentration was based on the feature, that conductivity in air burbles zone and in water bubbles zone differs thousands times.

Fig. 5 Schematic meaning of parameters under changing during the experiment
Entrainment and removal processes take place at the moment of interaction.We described this process with Eq. ( 1) and solution of this equation with Eq. ( 2).Fig. 6 illustrates the solution equations calculated for different initial concentrations.The range of initial water droplets relative concentration we assumed from 0.01 to 0.3 and air bubbles initial concentration from 0.7 to 0.99 (Fig. 6).
As are can see from Fig. 6, with water and air sticking to the surface probabilistic coefficients α 12 = 0.8 and α 21 = 0.2 respectively air bubbles and water droplets concentration dependence on time changes according exponential law.Asymptotes values of all curves are equal to probabilistic sticking coefficients.The shape of curves depends on initial concentration of air bubbles and water droplets.As was mentioned above, the process of air bubbles and water droplets entrainment and removal takes place not only in surface layers, but also in deeper layers.We described this process with Eq. ( 4).Fig. 7 illustrates air and water bubbles concentration dependences on weir height, evaluating deeper layers.The extreme points in the curves of Fig. 7 can be explained as follows: with the water flow trough the weir or trough hydro turbine the submerged jet forms (Fig. 8); this jet spreads in the surrounding liquid and at that same time loses its velocity.
Due to jet and surrounding liquid particles velocity transverse pulsation air bubbles are entrained into the jet.Jet velocity distribution is depicted in cross-sections a-a and b-b.Central initial velocity v 1 is constant.Initial sector of the jet is between cross-sections a-a and b-b.Maximum air bubbles concentration is in the centre of the jet while the minimum concentration is at the jet and surrounding liquid interface.Beyond this interface the recirculation zone forms.As can be seen from Fig. 7 at the y = 0.026 air bubbles concentration is maximum, therefore can be concluded that this point coincides the jet centre.Point y = 0.0325 coincides boundary between jet and surrounding water, because at this point air bubbles concentration is minimum.Further air bubbles concentration increases until reaches 1.With y > 0.0325 recirculation zone begins.Naturally that water drops concentration run has opposite character.

Fig. 8 Submerged jet
With the weir height growth the absorption of air bubbles into water grows till reaching maximal air bubbles concentration.Depending on distance x 1 and d 1 (Fig. 5) and on weir height breaking points of the curves change their position (Figs.7 -11).Fig. 9 Air bubbles and water droplets entrainment and removal curves depending on weir height for experimental data of [44] with x 1 = 1.0 m and d 1 = 0.0245 m, Fr = 7.9, Re = 94000, C air -air bubbles concentration and C wwater droplets concentration was simulated with Eq. 4 As can be seen from Figs. 7 and 9 the curves in these figures are different.The reason was that these curves were calculated not for two air and water layers like in Fig. 6, but for many layers.Fig. 9 shows entrainment curve for x 1 = 1.0 m and d 1 = 0.0245 m (experimental data were taken from [44]).Initial air bubbles concentration equals zero.With the weir height growing air bubbles concentration remains uniform and at y = 0.02 m starts to change.The points y = 0.032 and y = 0.04 are the points of breaking of the curves.
Therefore depending on initial conditions and the number of taken layers entrainment and removal curves can be expressed from exponential dependence curves with breaking points and points of maximums and minimums (Figs. 7 -11).Peak maximum on the air bubbles curve correspond water bubbles minimum on the water bubbles curve and vice versa.The curves simulated by us coincide with experimental data [44].As it can be seen from Fig. 10 each air bubbles and water drops curve have maximum and minimum.All these maximums and minimums well coincide with experimentally obtained maximums and minimums [42].As it can be seen from Fig. 11 air and water curves have two extreme points.
Air bubbles and water drops rising rate, size and frequency depend on jet velocity at the moment of jet hit at the water surface.A bubble rising frequency depends on the distance from weir to jet hit point.With the jet hit at the water surface air bubbles are pooled into the water.Some delay in creating air bubbles process appears and it increases with the distance.This delay explains the character of frequency dependence on distance.

Conclusions
1. Water turbulence highly influences processes of air bubbles and water droplets diffusion.These processes can be described by mathematical equations.
2. The bubbles and droplets created in the falling water flow stimulate the processes of air and water entrainment and removal.
3. Initial conditions of air bubbles and water droplets entrainment and removal processes determine run of proposed simulation equations solutions curves.
4. Good coincidence of theoretical and experimental results was achieved with more than two air-water layers on the air-water interface.

AIR BUBBLES AND WATER DROPLETS ENTRAINMENT AND REMOVAL IN TURBULENT WATER FLOWS
S u m m a r y Open reservoir's water pollution is becoming a global problem.Solution of this problem requires investigation of many phenomena that determinate degree and sustainability of the pollution.These phenomena are highly influenced on many factors.Among these factors an important role plays air bubbles and water droplets mixing in turbulent water flows.The aim of this paper was mathematical description of air bubbles and water droplets entrainment and removal process.Produced mathematical simulation allows calculating of air bubbles and water droplets concentration in several layers next to the airwater interface.The obtained results show that mixing process stimulates water aeration which on its own turn reduces water pollution.Also the results show good coincidence with experimental data obtained by independent researchers.

Fig. 7
Fig. 7 Air bubbles and water droplets entrainment and removal dependences on weir height for experimental data [40] (x 1 = 0.5 m, d 1 = 0.014 m, v = 3.47 m/s, Fr = 5.14, Re = 25000).Results -C air air bubbles concentration and C w water droplets concentration is simulated with Eq. 4