Turbulence, aeration and bubble features of air-water flows in macro- and intermediate roughness conditions

Stefano PAGLIARA*, Thendiyath ROSHNI, Iacopo CARNACINA

Department of Civil Engineering, University of Pisa, Via Gabba 22, Pisa 56122, Italy

1 Introduction

Nearly all flows in environment and engineering are hydraulically rough (Stoesser and Nikora 2008). Rough bed elements like pebbles and boulders are the most prevalent microtopographic features of gravel-bed rivers, and these features enhance the river bed stability. Tremendous work has been carried out on the flow characteristics of the two-phase flow over smooth channel beds and a few over hydraulically rough beds (Nikora et al. 2001;Stoesser and Nikora 2008). Studies over rough beds are extensively done nowadays because of their high energy dissipation properties. Intensified roughness creates additional resistance and hence leads to higher kinetic energy dissipation. Flow over rough beds may skim at particular flow depths and slopes (Pagliara et al. 2010b). This condition enhances the presence of stable vortices between the rough bed elements and results in a more complicated three-dimensional flow. Some analogies exist between the skimming flows over stepped spillways (Chanson and Toombes 2001b; Ohtsu et al. 2004; Relvas and Pinheiro 2008)and over rough bed channels(Pagliara et al. 2009, 2010a; Rice et al. 1998). Due to the complexity and importance of turbulent flows, the flow properties and flow structures over stepped spillways have been extensively studied in the past (Chanson and Carosi 2007a; Chanson and Toombes 2001b,2003; Felder and Chanson 2009; Gonzalez and Chanson 2008). Relatively few investigations have been carried out on the turbulent behavior of two-phase flows over different bed arrangements (Aivazian 1996; Djenidi et al. 1999; Nikora et al. 2001, 2004,2007; Stoesser and Nikora 2008). Amongst the few studies under the macro-roughness condition, Pagliara et al. (2009, 2010a, 2010b)studied the aeration characteristics at different relative equivalent depths, but unfortunately the underlying mechanism for the phenomenon has not been well revealed.

This study aimed to analyze the hydrodynamics of the two-phase flow over a naturally occurring rough bed of macro- and intermediate roughness, defined as in Bathurst (1978)and Pagliara and Chiavaccini (2006a), and to investigate the turbulence response to the changes in flow and roughness. A further objective was to identify the normal distribution of the void fraction, bubble frequency, and integral time scale of the two-phase flow over the roughened bed. In addition, this paper provides information on the flow structure in the uniform flow conditions and characterizes the general and individual properties of the two-phase flow.

2 Experimental facility and measuring techniques

To describe the air-water flow characteristics and present new evidence, an experimental setup, with a rough bed channel for high velocity open channel flows, was constructed in the PITLAB center of the University of Pisa. The facility consisted of a rough bed chute of 8 m long and 0.3 m wide, a recirculation pit, which ensured the water supply, and a magnetic flow meter (OPTIFLUX 2000)for the discharge measurement. The base macro-roughness (BMR)configuration was prepared by gluing one layer of rough elements over a stainless sheet with characteristic diameters of D16= 38.17 mm, D50= 43.41 mm, D65= 45.59 mm, and D84=47.17 mm, where Dxxrefers to the particle size for which xx% of the particles by weight are finer. The particles were randomly arranged at the bottom of the channel. A uniformity parameter of σp= (D84D16)12= 1.24 for rough elements was adopted (materials with σp＜ 1.4 are considered uniform (Dey and Raikar 2005)). The elements were placed in a manner that could minimize the gap between them. A diagram sketch of the rough bed chute with notations is illustrated in Fig. 1.

Further details about the experimental setup and the measurement locations are described in Pagliara et al. (2009, 2010b).

Fig. 1 Sketch of aerated flow in skimming flow regime on a rock chute and definition sketch of air and water chord lengths and air-bubble clusters

Experiments were performed at the flow rate per unit width q ranging between 0.03 m2/s and 0.09 m2/s and the slope i ranging between 0.26 and 0.46. Detailed experimental investigations of the present study are shown in Table 1.

Table 1 Experimental parameters

Boulders are often used to reinforce the stability of rock chutes and increase the energy dissipation (Ahmad et al. 2009; Pagliara and Chiavaccini 2006b), owing to their influence on the near-bed turbulence. Hence, the roughness of the bed was further intensified by gluing on it hemispherical boulders with the diameter of Db=55mm, which were positioned over the BMRconfiguration either in row (BC-R)or in staggered (BC-S)arrangement (Pagliara et al. 2010a). The BC-Sand BC-Rarrangements over the BMRconfiguration are shown in Figs. 2(a)and (b).

Fig. 2 Rough bed arrangements

Clear water flow depths over the rough bed were measured with a point gauge and the air-water flow properties were recorded using a void fraction meter, produced by the Bureau of Reclamation of the U.S. Department of the Interior, with the help of an intrusive single tip conductivity probe (tip ?6 mm)(Jacobs 1997; Matos and Frizell 1997). The output voltage signals (0 V when the probe touched water and 5 V when bubbles were detected)received at each time step from PT were analyzed for the estimation of the air-water flow properties. The translation of the conductivity probe and point gauge along and across the channel was controlled by a fine adjustment mechanism. The calibration of the conductivity probe was done before each test and the output signal was scanned for 15 s at a sampling rate of 20 Hz and for 30-40 s at 2 kHz. Due to the three-dimensional pattern of the flow, measurements were performed for each 0.5 m longitudinally from the inlet sections at z=0 m,z =±0 .05 m , and z=± 0.1 m, where z is the transverse coordinate from the center of the channel, and for each 1 mm vertically from PT. For each section, the value of the water depth dewas, therefore,obtained by averaging the water depth toward the transverse direction (Pagliara et al. 2010a,2010b). In the present study, all the flow measurements were performed in the quasi-uniform flow region.

3 Data processing methods

Each sample, digitalized in the form of a square wave, was analyzed for the calculation of the void fraction C, the integral time scale Txx, the bubble frequency F, the air bubble chord length Laand water droplet chord length Lwdistributions, and clustering events, respectively.

The void fraction C was evaluated as the percentage of time in which the signal was above the air-water threshold limit (generally fixed at 50% of the maximum output voltage(Pagliara et al. 2010a)). The bubble frequency F is the number of bubbles detected by the conductivity probe per second. The air and water chord time distributions are given based on the time that the probe tip stays on air bubbles or water droplets (Kucukali and Cokgor 2008),and the chord length can be obtained by multiplying V and the chord time (Fig. 1). The integral time scale Txxcan be computed with Eq. (1)as follows:

which is equal to the integral area of the normalized autocorrelation functionRxxof the voltage signals at each step from t= 0 to t = T ( Rxx= 0), as shown in Fig. 3 (Chanson and Carosi 2007a):

Fig. 3 Auto-correlation integral scale definition

4 Experimental results

4.1 Void fraction and frequency analysis

where D0is a dimensionless diffusivity coefficient and K′ is the dimensionless integration coefficient.

Fig. 4(b)shows that the bubble frequency distribution presents a rising limb, an intermediate region and a recession limb in the direction normal to the flow. It can be seen from Fig. 4 that there are not many discrepancies in the average profile measurements in the MR-SKflow regime over the rock chutes except for a small value of F Fmaxcorresponding to the first peak in the frequency distribution in the inner layers (Fig. 4(b)). This is mainly due to the significant interaction of the free surface flow and stable wakes with the intensely rough channel bed (Pagliara et al. 2010a). The second peak shows that the maximum bubble frequencyFmaxoccurs generally in the range of 0.3 ＜C ＜ 0.6.

The frequency distribution over the void fraction better explains the flow regions in the direction normal to the flow. Fig. 5 shows the dimensionless relation between the bubble count rate and void fraction for i= 0.26, 0.38, and 0.46 and q = 0.07 m2/s. All the present data sets for the three slopes consistently show a characteristic shape with the maximum bubble frequency occurring in the range of 0.3 ＜C ＜ 0.6. The relationship between the bubble frequency and void fraction was approximated as a parabolic shape, shown in Chanson (1997)as

and later Toombes (2002)later extended the parabolic law as follows:

where α and β are correction factors and CFmaxis the void fraction corresponding to the maximum bubble frequency (Toombes 2002). The dimensionless distribution of the bubble count rate F Fmaxwith void fraction clearly illustrates three different flow regions in the direction normal to the flow. A bubbly flow regime appears at C＜0.3, while the spray region occurs at a higher concentration of C＞0.6. An intermediate region exists in the range of 0.3 ＜C ＜ 0.6, in which the maximum bubble frequency occurs. Similar results were observed in flows over smooth chutes and stepped spillway flows (Chanson 1997; Gonzalez and Chanson 2008). All the present data, except for few data at i = 0.26 in the bubbly flow region were correlated reasonably well with the modified parabolic law from Toombes (2002). The higher bubble count rate in the inner regions of the flow over the rough bed is due to the high vortex recirculation between the bed elements, which is well discussed in Pagliara et al. (2009,2010a). In the outer layers, due to the spray formation, a higher dispersion of data was seen and hence resulted in a larger deviation from Eq. (3)and Eq. (4).

Boulder presence, either in the BC-Ror the BC-Sarrangement, yields to large air entrainment in the flow condition similar to that of the BMRconfiguration, leading to different flow features. Fig. 6 and Fig. 7 show the void fraction distribution and frequency analysis at xbDb= 0 and xbDb= 0.9, respectively, in the BC-Rarrangement (Γ = 0.05)for five different transverse sections z=± 0.1 m , z=± 0.05 m, and z=0 m, where xbis the distance from a boulder row, y1is the depth measured from the plane joining the top of the boulder rows (Fig. 6(a)), and y1=at C=0.9.

Fig. 5 Dimensionless relationship between C and F Fm ax for i = 0.46, 0.38, and 0.26, q = 0.07 m2/s, and Γ = 0 and comparison with Eq. (3)and Eq. (4)

Fig. 6 Dimensionless distribution of void fraction C and frequency analysis for different transverse sections in BC-R arrangement (Γ = 0.05)for i=0.46 and q = 0.05 m2/s atxb Db= 0

Fig. 7 Dimensionless distribution of void fraction C and frequency analysis for different transverse sections in BC-R arrangement (Γ = 0.05)for i=0.46 and q = 0.05 m2/s atxb Db= 0.9

The void fraction profiles and the frequency analysis over the boulder top (Fig. 6)were homogeneous, similar to the profile of the MR-SKflow regime over the BMRarrangement (Fig. 4).Unlikely, atxbDb= 0.9, as shown in Fig. 7, a profound disturbance of the flow field immediately after the boulder row was clearly seen, which included strong separation zones below and above the boulder top. Fig. 8 shows the dimensionless distribution of the void fraction with bubble frequency and the Txxvariation normal to the flow direction for i= 0.46 and q = 0.05 m2/s at xbDb= 0.9 in the BC-Rarrangement of boulders (Γ = 0.05).Flow over the BC-Rarrangement is characterized by a wake and jet fall mechanism (Pagliara et al. 2010a).

Fig. 8 Dimensionless distribution of void fraction with bubble frequency and integral time scale normal to flow direction fori=0.46and q = 0.05m2/s at xb Db = 0.9in BC-R arrangement (Γ = 0.05)at differentz B

Fig. 9 shows the vortex shedding in the wake diffusion zone over the BC-Rarrangement for i = 0.46 and Γ = 0.05. Due to the vortex shedding in the wake diffusion zone, immediately after the boulder rows (xbDb= 0.9), a high recirculation zone appears (Fig. 9), where a small jet fall occurs after each boulder row, resulting in a higher F (Lacey and Roy 2008; Pagliara et al. 2010a), as shown in detail in Fig. 8. Air packets are broken up into a number of air bubbles because of the fall effect, resulting in a larger air content and a higher bubble count rate (Fig. 8(a)). The fall effect also causes the occurrence of the high recirculation zone, which further results in a higher Txxin the inner layers, i.e.,y1y9′0＜ 0.25(Fig. 8(b)).

Fig. 9 Vortex shedding over BC-R arrangement for i = 0.46, q = 0.05 m2/s, and Γ = 0.05

4.2 Integral time scale analysis

The characteristics of the flow over stepped chutes depend on the step height h and the chute slope, while the characteristics of the flow over rock chutes depend on the diameter of materials and also the chute slope. In order to compare flow characteristics over the stepped chute and rock chute, the nominal diameter of the bed material was considered equal to the normal step height, i.e.,D84= h cosθ , where θ is the angle between the chute and horizontal plane in degrees. Fig. 10 shows the comparison of rock chute data from several transverse sections in the bubbly and intermediate flow regions with stepped chute data (Felder and Chanson 2008). It can be inferred from Fig. 10 that the flow over the rock chute presents a larger Txxcompared with those observed by Felder and Chanson (2008)in the inner layer in the presence of stepped chutes for similar i and deD84. For either the stepped chute or rock chute,the turbulence decreases with the increase of deD84. When deD84decreases, the interaction between the rough bed elements and the water surface increases, resulting in a higher Txx.Moreover, Txxshows the greater values in the intermediate flow region for all deD84. Chanson and Carosi (2007b)showed that the relation between Txxand C generally displays a parabolic shape in the inner layers (bubbly and intermediate flow regions), and that a large deviation occurs over stepped spillways in the spray region. Since the spray formation appears earlier(C＞0.6)over rock chutes, the data systematically break away from the normal parabolic shape.Hence, the present study data only show data of the inner layers (Fig. 10).

Fig. 10 Comparison of Felder and Chanson’s (2008)data for i = 0.4 with present study for i = 0.38 and Γ = 0 for similar d e D84

In order to visualize the effects of turbulence on aeration, the maximum integral time scale Tx′xobtained in the intermediate flow region is plotted for i=0.26, 0.38, and 0.46 and 0.74 ≤deD84≤1.63 in Fig. 11(a). In addition, Tx′xis compared with Felder and Chanson’s (2008)data. A generally decreasing trend of turbulence with the increase of deD84was found at all the slopes in the test range. Moreover, Felder and Chanson (2008)data show a smaller turbulence scale compared with rock chute data. Fig. 11(b)plots the average concentration Cmas a function offor i = 0.26, 0.38, and 0.46, and Γ= 0, 0.05,and 0.15 in the BC-Sarrangement. Larger values of Tx′xcorrespond to larger Cmover the rough bed, as the shear stress strength overcomes both the buoyant force and the surface tension, leading to a higher volume of air to be entrained and carried by the flow. Indeed, as the slope increases, Cmalso increases, and at a constant slope, Cmincreases with both Tx′xandΓ.

Fig. 11 Dimensionless distribution of Tx′xwithdeD84and CmwithTx′xfor i = 0.26, 0.38, and 0.40,0.74 ≤ de D84≤1.63, andΓ= 0, 0.05, and 0.15in BC-S arrangement

4.3 Chord length distributions and clustering analysis

Fig. 12 Probability distribution functions of air bubble and water droplet chord lengths over BMR (Γ=0)configuration for q = 0.07 m2/s and i=0.46at central transverse sectionz B =0

The streamwise structure of the air-water flow can be further explained by the clustering analysis. Voltage signal outputs provide information on the clustering properties of bubbles in the bubbly flow region. A typical result of the clustering analysis normal to the flow direction in the bubbly flow region (i=0.46, q = 0.07 m2/s , and Γ=0 at z B=0)is presented in Fig. 13.

Fig. 13 Clustering properties of bubbles in bubbly flow region and void fraction and frequency analysis in MR-SK flow regime of BMR configuration (Γ = 0)for i = 0.46 and q = 0.07 m2/s atz B=0

Fig. 14 Probability distribution of air bubble chord length in BC-R arrangement (Γ = 0.05)for i = 0.46 and q = 0.05m2/s in central transverse section (z B=0)

Fig. 15 illustrates clustering analysis in the BC-Rarrangement (Γ = 0.05)for i = 0.46 and q = 0.05 m2/s atxbDb= 0.9. Contrary to previous results in the MR-SKflow regime of the BMRconfiguration (Fig. 13), Fig. 15 shows that Pbcvaries from 25%-75% and Nbcvaries from an average of 2.2 in the outer layer to 4.2 in the vortex recirculation zone of the inner flow region (y1y9′0＜ 0).

Fig. 15 Clustering properties of bubbles in bubbly flow region in BC-R arrangement (Γ = 0.05)for i = 0.46 and q = 0.05 m2/s atxb Db = 0.9for three transversal sections ( z B = - 0.33, 0, and 0.33)

5 Conclusions

The two-phase flow properties over the BMR, BC-S, and BC-Rconfigurations in the uniform flow region were investigated for the selected experimental ranges. The changes of the void fraction, frequency behaviors, and turbulence behaviors of flows over the rough bed arrangement were investigated, and accordingly, the flow structure analysis was performed. A comparison of the turbulence behaviors in the stepped spillway with the rock chute data was also conducted.

The void fraction and frequency analysis over the BMRarrangement in the inner layers reveals that there is a strong interaction between the water surface and rough bed elements,resulting in stable drag vortices and stable shear vortices between the rough bed elements. The turbulence analysis, based on the integral time scale, reveals that the reduction of the relative depth intensifies the interactions between the free surface and bed materials and thus increases the turbulence intensity, resulting in a higher quantity of air entrained by the flow. Moreover,the flow over rough bed chutes shows a higher turbulence as compared with the stepped chute data for similar flow conditions, owing to the presence of complex flows and vortex structures downstream of the rock elements. Chord length and clustering analyses over the BMRand BC-Rarrangements show different behaviors in the inner flow region. The results show that the intensified roughness of the BC-Rarrangement enhances the void fraction by air bubbles of larger chord lengths and higher turbulence levels compared to BMR.

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