A numerical model for air concentration distribution in self-aerated open channel flows*

WEI Wang-ru （衛(wèi)望汝）, DENG Jun （鄧軍）, ZHANG Fa-xing （張法星）, TIAN Zhong （田忠）

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 650061, China, E-mail: wangru_wei@hotmail.com

A numerical model for air concentration distribution in self-aerated open channel flows*

WEI Wang-ru （衛(wèi)望汝）, DENG Jun （鄧軍）, ZHANG Fa-xing （張法星）, TIAN Zhong （田忠）

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 650061, China, E-mail: wangru_wei@hotmail.com

(Received January 22, 2014, Revised May 15, 2014)

The self-aeration in open channel flows, called white waters, is a phenomenon seen in spillways and steep chutes. The air distribution in the flow is always an important and fundamental issue. The present study develops a numerical model to predict the air concentration distribution in self-aerated open channel flows, by taking the air-water flow as consisting of a low flow region and an upper flow region. On the interface between the two regions, the air concentration is 0.5. In the low flow region where air concentration is lower than 0.5, air bubbles diffuse in the water flow by turbulent transport fluctuations, and in the upper region where air concentration is higher than 0.5, water droplets and free surface roughness diffuse in the air. The air concentration distributions obtained from the diffusion model are in good agreement with measured data both in the uniform equilibrium region and in the self-aerated developing region. It is demonstrated that the numerical model provides a reasonable description of the self-aeration region in open channel flows.

self-aeration, air concentration, open channel flows, turbulence diffusion, numerical model

Introduction

The free surface aeration is frequently observed in supercritical open channel flows. The air-water structure induces drastic changes of the flow characteristics. First, the bulking caused by the entrained air increases the water flow depth as compared with that of the non-aerated water flow[1]. The aeration can eliminate or minimize the cavitation damage caused by the high velocity flow in spillways and channels[2]. Studies show that the presence of air within the boundary layer can reduce the shear stress between the flow layers[3,4]. As an environmental process, the oxygen concentration is a prime indicator of the quality of the water. The highly aerated flow can be identified for its gas transfer characteristics with the transfer of atmospheric gases into the water and the volatilization of pollutants[5]. Besides, the self-aeration contributes to the air-water mass transfer of atmospheric gases significantly[6-8].

In an open channel flow, the turbulent boundary layer generated by the channel bottom develops to the free surface along the channel, and when the turbulence next to the free surface is great enough to overcome the surface tension and gravity, one sees the self-aeration (Point A in Fig.1). For the development of an air-water two phase flow, there is mainly two parts as shown in Fig.1, that is the self-aerated developing region before the flow settles down in a uniform equilibrium, and the full developed uniform region where the turbulence diffusion normal to the bottom counterbalances exactly the buoyancy effect, and the air concentration and time-mean velocity distributions are independent of the flow direction along the channel. The distribution of the air concentration can be seen as the results of a diffusion process. Some classical models of air concentration were developed from the diffusion theory to predict the air concentration distribution in self-aerated flows[9-12]. Straub and Anderson and Wu took two factors into consideration, which are the air entrainment by the flow turbulenttransport fluctuations (in the inner flow region) and the water droplets ejected from the flowing water (outside the flow region), and the interface between the regions was defined as the position of the maximum gradient of the air concentration normal to the channel bottom[13,14]. And both models contain an assumption that the water droplet movement follows a normal distribution and the use of the two models requires the estimate of several empirical coefficients. Wood and Chanson developed models for predicting the air concentration distribution in the self-aerated open channel flow, and their models were used to compared with the experimental data from the uniform equilibrium flow region and the self-aerated developing region, with the mean cross-section air concentration between 0.1 and 0.75[15,16]. The two diffusion models are both based on the assumption that the airwater structure can be seen as a homogeneous mixture.

Fig.1 Sketch of a self-aerated flow in open channel

Fig.2 Concepts of entrapped and entrained air in self-aerated open channel flow[17,18]

However, in self-aerated open channel flows, there are two physical entities with respect to the air concentration at a certain position in the air-water flow[17,18], as shown in Fig.2: (1) the entrained air, which is transported along with the flow in the form of air bubbles that are pulled into the flowing water through the air entrainment, (2) the entrapped air, which is the air above the water surface that is being transported along with the flow because it is trapped in the surface roughness. The difference must be taken into consideration to predict the air concentration distribution when using the diffusion model.

In this paper, a diffusion model is developed to predict the air concentration distribution at a crosssection, by considering the air-water structure in the self-aerated open channel flow in two different diffusion processes. The diffusion model is compared with experimental data both in the uniform equilibrium region and the self-aerated developing region. On the basis of the natural physical air-water structure in selfaerated open channel flows, the necessity to adopt a diffusion model for the self-aerated flows is analyzed.

1. Diffusion model for self-aerated open channel flows

Within the air-water flows, the mixture of the air and the water can be approximated as a diffusion process caused by the flow turbulence. For a two-dimensional flow, the structure of the self-aeration region in open channel flows is considered as consisting of two parts: the low flow region and the upper flow region. In the low flow region, air bubbles are diffused in the water flow by turbulent transport fluctuations. In the upper region, the water droplets and the free-surface roughness are diffused into the air. The interface between the two regions is defined as y=y50, which is the characteristic depth where the air concentration Ca=0.5. In the low flow region yy50.

To develop the diffusion model, the air-water flow is considered as a two-dimensional flow in the self-aerated uniform equilibrium region, where the air concentration and velocity distributions are independent of the flow direction. The time mean velocity of the water and the air in the flow direction are considered as equivalent and there is no slip velocity between the two phases. Besides, in the present study it is noted that the development of the air bubble diffusion does not concern the air bubble diffusion next to the channel bottom wall.

1.1 Low flow region (y

For a steady uniform, two-dimensional flow, the governing equation is the continuity equation of theair in the air-water flow. It can be written in Cartesian coordinates as

Cais the air concentration.Vxand Vyare the timeaverage velocities in the directions along the flow (xdirection) and normal to the flow (y-direction), respectively.(Dtx)aand (Dty)aare the air diffusivity coefficients in thex -and y -directions, respectively. uris the air bubble rise velocity, positive upwards along the vertical direction,α is the channel bed angle relative to the horizontal plane, andt is the time. In the equation, the compressibility effects of the air in the air water flow are neglected.

Under the steady uniform flow conditions, with?/?x=0and Vy=0, Eq.(1) can be simplified as

According to the Chanson’s study[16], the rise velocity squared for a single air bubble is proportional to the pressure gradient in a quiescent surrounding fluid, and with neglect of the weight of the air bubble, we have

where (ur)His the single air bubble rise velocity under a hydrostatic pressure gradient, andpis the pressure. In the air-water flow, the pressure gradient at any positionyis

where the ρwis the water density, and gis the gravitational acceleration. The air bubble rise velocity in the aerated flow can be assumed as

Inserting Eq.(5) into Eq.(2) and assuming that (ur)His a constant, the following equation is obtained

Introducing the dimensionless distance normal to the flow directiony′and let Dabe a dimensionless air turbulent diffusivity that reflects the ratio of the air bubble diffusion turbulence over the rise velocity normal to the flow direction, by definition:

Considering only the first series of solutions obtained for a zero constant, the integration of Eq.(9) yields

Kais a dimensionless constant, and a relationship betweenDaand Kais deduced under the boundary condition Ca=0.5for y′=1

The air concentration distribution in the low flow region is

With the dimensionless turbulent diffusivity Dadeduced from the mean air concentration in the low flow region and with(Ca)0.5defined in terms of y50, we have

For (Ca)0.5≤0.3, the dimensionless turbulent diffusivity and the mean air concentration in the low flow region are best expressed by

with a normalized coefficient of correlation of 0.9997.

1.2 Upper flow region (y>y50)

For a steady uniform, two-dimensional flow, the continuity equation of the water in the air–water flow can be written in Cartesian coordinates as,

Cwis the water concentration in the air.Vxand Vyare the time-average velocities in x -direction and ydirection.(Dtx)wand (Dty)ware the water diffusivity coefficients inx -and y -directions, respectively.ufis the falling velocity of the water droplets and the free-surface roughness, negative upwards in the vertical direction. The compressibility effect of the water is neglected, and the water density is a constant.

Under the steady uniform flow conditions, with?/?x=0and Vy=0, Eq.(15) can be simplified as

The fall velocity of the water at a certain position is affected by the relative position in y-direction and the air concentration at the position[15]. Then, an assumption of the fall velocity of the water movement in negativey -direction ufis

For the present study,(uf)Cis the fall velocity of a single water droplet in an un-aerated open channel flow. Inserting Eq.(17) into Eq.(16) and assuming that (uf)Cis constant, the following equation is obtained

Fig.3 Comparison of air concentration profile on the crosssection between diffusion model (Eq.(12) and Eq.(25)) and uniform equilibrium region data

In dimensionless terms, we have

where

Fig.4 Comparison of air concentration profiles on the cross-section between diffusion model (Eq.(12) and Eq.(25)) and data in developing region

Dwis the dimensionless water turbulent diffusivity coefficient and it reflects the ratio of the water droplet and globule diffusion turbulence over the water fall velocity normal to the flow direction. Considering only the first series of solutions obtained for a zero constant, the integration of Eq.(19) yields

Kwis a dimensionless constant, and a relationship between Dwand Kwis deduced under the boundary condition Ca=0.5for y′=1

The equation becomes

For the dimensionless turbulent diffusivity Dw, the mean water concentration in the upper flow region (Cw)0.5can be deduced in terms of y50as on the above section in the low flow region

Because Eq.(24) contains a complicated exponential function, the integration is difficult to be obtained analytically. For (Cw)0.5≤0.4, the numerical solution gives an approximate relationship betweenDwand (Cw)0.5as

with a normalized coefficient of correlation of 0.9965. Therefore, the air concentration distribution for the upper flow region is

Fig.5 Experimental versus computed y2/y95from diffusion model in self-aerated region

2. Applications and results

The results from the theoretical diffusion model (Eq.(12) and Eq.(25)) are compared with the physical experimental and prototype data. All data are from measurements on the channel centerline. The compared data cover the uniform equilibrium region[13]shown in Fig.3 and the developing region[16,17,19]in the self-aerated open channel flow shown in Fig.4. In each experimental case, the dimensionless turbulent diffusivities Daand Dware deduced from the mean air concentration in the low flow region (Ca)0.5(Eq.(14)) and the mean water concentration in the upper region (Cw)0.5(Eq.(24)), respectively. In the upper flow regiony>y50for the experimental data, the integral region is from y50to the y95(the position where the air concentration Ca=0.95). The necessity to discriminate the mean air concentrations (Ca)0.5and (Cw)0.5will be illustrated in the next section.

It can be seen that in the self-aerated open channel flow, both data in the uniform equilibrium region and the developing region are in good agreement with the results from the diffusion model (Eq.(12) and Eq.(25)). It must be noted that the diffusion model does not consider the air bubble diffusion next to the channel bottom wall. Figure 5 shows the comparison of y2/y95between the results from the diffusion model and the experimental data, where y2is the position where the air concentrationCa=0.02. The agreement is also good. This indicates that the diffusion model can be used to predict the self-aerated region in open channel flows for the situation without air bubble diffusion next to the channel bottom wall. It is believed that the diffusion model provides a relatively reasonable description of the air bubble diffusion and the water diffusion in both the uniform equilibrium region and the developing region in the self-aerated open channel flow.

3. Discussions

According to Killen[17]and Wilhelms[18], with respect to the self-aerated open channel flow, the total air in the air concentration at certain position contains two parts: the entrained air, which is transported along the flow in the form of air bubbles through the process of air entrainment, and the entrapped air, which is the air trapped in the water free surface roughness and transported along with the flow. The water free surface roughness is the strong deformation of the free surface and is mainly affected by the flow turbulence[20-22], especially in the prototype[23]. In Fig.6, it can be seen clearly that there is still a water free surface remaining intact, but very much contorted for the high-aerated flow, while in the interior part of the water there is mainly the bubble flow. The high-speed stop-action photos of the self-aerated open channel flow are taken in the State Key Laboratory of Hydraulics in Sichuan University with the channel slope α= 30o. There is a transformational relationship between the entrained air and the entrapped air. Once the free surface deformation develops to a specific limit, the free surface will contain closures, which may capture the air into the water flow, or the free surface will break and create water droplets, and the air will entrained into the water flow when the water droplets fall back to the water flow. Both processes can transform the entrapped air into the entrained air bubbles. Figure7 shows the process of the air bubble entrainment by the free-surface deformation.

Fig.6 Water fee surface of self-aerated open channel flow[20](The mean velocity of flow is 5.7 m/s)

Fig.7 The process of air bubble entrainment by free-surface deformation[20](The mean velocity of flow is 4.5 m/s)

Killen[17]did a series of experiments to measure the characteristics of the turbulent water free surface in the self-aerated open channel flow, separating the entrapped air from the total air concentration. The range of the mean cross-section air concentration is from 0.20 to 0.60. Based on the measured data, in the upper flow region (Ca>0.5), the entrapped air occupies a great proportion in the total air concentration, as shown in Fig.8(a). With the increase of the air concentration in this region, the proportion of the entrapped air in the total air concentration increases. WhenCa> 0.7-0.8, the entrapped air counts for more than a half of the total air concentration. This indicates that in the upper flow region, the water turbulent movement is the main part. In the low flow region(Ca<0.5), the entrained air is the main part of the total air concentration. It can be seen in Fig.8(b) that the entrained air proportion counts for more than 60% of the total air concentration. This indicates that in the low flow region, the air bubble is the main part in the air concentration. With the increase of the mean cross-section air concentration, which means that the self-aerated open channel flow develops towards a uniform equilibrium flow, the entrapped air proportion in the whole crosssection reduces and the air bubble proportion increases in the total air concentration (as shown in Fig.9), but it should be noted that the entrapped air proportion still counts for 20%-30% when the mean cross-section air concentration is about 0.5-0.6. Thus, both in the uniform equilibrium region and the developing region, the water free surface deformation movement, caused by the flow turbulence and distinguished from the air entrainment, is always an important part in the selfaerated open channel flow. Based on this observation, the upper flow region in the present diffusion model can be seen as in a diffusion process of water droplets and free surface roughness in the air caused by theflow turbulence.

Fig.8 Entrapped and entrained air percentages for different air concentrations

Fig.9 Entrapped air and entrained air percentages for different cross-section mean air concentrations

Based on the above analyses, the conception of the entrapped air and the entrained air confirms that the air-water structure in the self-aerated open channel flows should be considered as consisting of the low flow region and the upper flow region. Because of the different air concentration structure, the mean air concentration in the two different flow regions should be analyzed separately. And the diffusion model (Eq.(12) and Eq.(25)) provides a reasonable description of the water and air bubble turbulent diffusions in both the uniform equilibrium region and the self-aerated developing region. It is recommended that the future experiments and studies in this area will include the prediction of the mean air concentration in the low flow region and the upper flow region.

4. Conclusion

A diffusion model (Eq.(12) and Eq.(25)) is developed to predict the air concentration distribution in the self-aerated open channel flows. The air-water flow is considered as consisting of the low flow region and the upper flow region. The air concentration on the interface is 0.5. In the low flow region, the air concentration is lower than 0.5, one sees a process that the air bubble diffuses in the water flow by turbulent transport fluctuations, and in the upper region, the air concentration is higher than 0.5, one sees a process that the water droplets and free surface roughness diffuse in the air. The data both in the uniform equilibrium region and the developing region are in good agreement with those obtained from the diffusion model. Based on the analyses of the entrapped air and the entrained air in the total air concentration, the water surface roughness movement in the self-aeration open channel flow should be taken into consideration for the air-water structure, besides the air entrainment process. And the present diffusion model provides a reasonable description of the air-water structure in the self-aerated open channel flows.

[1] MATOS J., FRIZELL K. M. Air concentration and velocity measurements on self-aerated flow down stepped chutes[C]. Conference on Water Resource Engineering and Water Resources Planning and Management. Minneapolis, USA, 2000, 1-10.

[2] PFISTER M., LUCAS J. and HAGER W. H. Chute aerators: Pre aerated approach flow[J]. Journal of Hydraulic Engineering, ASCE, 2011, 137(11): 1452-1461.

[3] CHANSON H. Air-water flow measurements with intrusive phase-detection probes. Can we improve their interpretation?[J]. Journal of Hydraulic Engineering, ASCE, 2002, 128(3): 252-255.

[4] CHANSON H. Compressibility of extra-high-velocity aerated flow: A discussion[J]. Journal of Hydraulic Research, 2004, 42(2): 213-215.

[5] BAYLAR A., BAGATUR T. Experimental studies on air entrainment and oxygen content downstream of sharp-crested weirs[J]. Water and Environment Journal, 2006, 24(4): 210-216.

[6] PFISTER M., HAGER W. H. Self-entrainment of air on the stepped spillways[J]. International Journal of Multiphase Flow, 2011, 37(2): 99-107.

[7] WILHELMS S. C., GULLIVER J. S. Bubbles and waves description of self-aerated spillway flow[J]. Journal of Hydraulic Research, 2005, 43(5): 522-531.

[8] TOOMBES L., CHANSON H. Air-water mass transfer on a stepped waterway[J]. Journal of Environmental Engineering, 2005, 131(10): 1377-1386.

[9] KRAMER K., HAGER W. H. Air transport in chute flows[J]. International Journal of Multiphase Flow, 2005, 31(10): 1181-1197.

[10] ARAS E., BERKUN M. Effects of tail water depth on spillway aeration[J]. Water Statistika of Afrika, 2012, 38(2): 307-312.

[11] CHANSON H., LUBIN P. Verification and validation of computational fluid dynamics (CFD) model for air entrainment at spillway aerators[J]. Canada Journal ofCivil Engineering, 2010, 37(1): 135-138.

[12] CHANSON H. Bubble entrainment, spray and splashing at hydraulic jumps[J]. Journal of Zhejiang University SCIENCE A, 2006, 7(8): 1396-1405.

[13] GIRGIDOV A. D. Self-aeration of open channel flow[J]. Power Technology and Engineering, 2012, 45(5): 351-355.

[14] DENG Jun, XU Wei-lin and QU Jing-xue et al. Measurement and calculation of air concentration distribution of self-aerated flow in spillway tunnel[J]. Journal of Hydraulic Engineering, 2002, (4): 23-36(in Chinese).

[15] SABBAGH-YAZDI S. R., REZAEI-MANIZANI H. and MASTORAKIS N. E. Effects of bottom aerator and self-aeration in steep chute spillway on cell center finite volume solution of depth-averaged flow[J]. International Journal of Mathematical Models and Methods in Applied Sciences, 2008, 2(2): 154-161.

[16] VIDAL L. E. O., RODRIGUEZ O. M. H. and ESTEVAM V. et al. Experimental investigation of gravitational gas separation in an inclined annular channel[J]. Experimental Thermal and Fluid Science, 2012, 39: 17-25.

[17] STEVEN C., GULLIVER J. S. Bubbles and waves description of self-aerated spillway flow[J]. Journal of Hydraulic Research, 2008, 46(3): 420-423.

[18] WILHELMS S. C. Gas transfer, cavitation, and bulking in self-aerated spillway flow[J]. Journal of Hydraulic Research, 2005, 45(4): 532-539.

[19] CHANSON H. Hydraulics of aerated flows: qui pro quo?[J]. Journal of Hydraulic Research, 2013, 51(3): 223-243.

[20] ZHANG Fa-xing, XU Wei-lin and ZHU Ya-qin. Experimental study on formation of air bubbles in self-aerated open channel flows[J]. Journal of Hydraulic Engineering, 2010, 41(3): 343-347(in Chinese).

[21] SIMDES A. L. A., SCHULZ H. E. and PORTO R. M. et al. Free-surface profiles and turbulence characteristics in skimming flows along stepped chutes[J]. Journal of Water Resource and Hydraulic Engineering, 2013, 2(1): 1-12.

[22] SMOLENTSEV S., MIRAGHAIE R. Study of a free surface in open-channel water flows in the regime from‘‘weak’’ to ‘‘strong’’ turbulence[J]. International Journal of Multiphase Flow, 2005, 31(8): 921-939.

[23] PFISTER Michael, CHANSON Hubert. Two-phase airwater flows: Scale effects in physical modeling[J]. Journal of Hydrodynamics, 2014, 26(2): 291-298.

* Project supported by the National Natural Science Foundation of China (Grant No. 51179113), the Doctoral Program of China Education Ministry (Grant No. 20120181110083).

Biography: WEI Wang-ru (1988-), Male, Ph. D.

DENG Jun,

E-mail: djhao2002@scu.edu.cn