masashi nakagawa 1, masamitsu koyasu 1, tomohiko tanaka2,...

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008

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Transport Mechanisms of Turbulent Kinetic Energy in Upward Bubbly Pipe

Flow

Masashi Nakagawa 1, Masamitsu Koyasu 1, Tomohiko Tanaka2, Yohei Sato 1, Koichi Hishida 1

1: Department of System Design Engineering, Keio University, Yokohama, Japan, [email protected]

2: Central Research Laboratory, Hitachi Ltd., Kokubunji, Japan, [email protected] Abstract Particle tracking velocimetry (PTV) with Sub-Kolmogorov time and spatial resolution was used to investigate the mechanisms of turbulence modification by dispersed bubbles in an upward bubbly pipe flow. Sub-Kolmogorov resolution is high enough to obtain reasonably accurate turbulence kinetic energy dissipation rate, which is the essential quantity to clarify the turbulence modulation. Gaseous-phase shapes and behaviors were also taken by shape projection imaging (SPI). Experiments were conducted for two different bubble diameters of 1.7 mm and 2.6 mm. The void fraction, α, is set to be 0.5%, 1.0%, or 1.5%. The flow in a 44 mm diameter pipe is driven upward by a pump at the bulk velocity, Ub, of 200 mm/s, corresponding to a pipe Reynolds number, Re2R, of 9900. The mean flow profiles were flattened in the presence of bubbles because bubbles, which accumulated near the wall region, accelerated the liquid-phase fluid. The flattened mean flow profiles suppressed the shear-induced turbulence intensities in the pipe middle region, while the bubble wakes tends to augment turbulence intensity at high void fraction. TKE budget was calculated using the accurate form of dissipation rate introduced by Tanaka & Eaton (2007). The dissipation rate augmented as the void fraction increased. However, the augmentation trends are larger for larger bubble case, indicating that the TKE dissipation is strongly depending on the bubble diameter in bubbly flows. By expanding the spatial filtering technique as used in previous studies by Liu et al. (1994, 1999), we characterized the length scales that govern the energy transfer between bubbles and turbulence. Strong energy transport from bubbles can be clearly observed at the scale of from two to three times the bubble diameter, indicating the interactions between bubbles and the fluid are predominant at this scale. The findings above give guidance on the physics and modeling of multiphase flows.

1. Introduction Turbulent bubbly flows are present in many engineering applications such as chemical reactors and power plants. To deal with these applications, it is essential to understand details of the turbulence structures interacting with bubbles. In turbulent bubbly flow studies, there are two fundamental research interests: bubble behaviors and turbulence modification by bubbles. Previous studies for bubble motion have provided important findings. Bubbles tend to accumulate in low-pressure area and form high-density cluster for low void fraction case (Cadot et al. 1995; La Porta et al. 2000; Sridhar & Katz, 1999). Also, a direct numerical simulation (DNS) for uniform-isotropic turbulence in the presence of micro bubbles showed that lift force played important roles in the bubble accumulation (Mazzitelli et al. 2003). The bubble accumulation due to the lift force effect was experimentally observed in the study of bubbly pipe flow by Fujiwara et al.(2004). On the other hand, the experimental study of turbulence modification have shown that dispersed bubbles can either augment or attenuate the liquid-phase turbulent kinetic energy (TKE) (Serizawa et al. 1975; Wang et al. 1987; Lance & Bataille, 1991;Fujiwara et al. 2004). Details of turbulence modulation can be understood by TKE energy spectrum. Lance & Bataille (1991) reported that the energy spectrum exponent -5/3 was substituted by -8/3 at high bubble Reynolds number. In contrast, Mudde et al. (1997) observed the -5/3 power law in a bubble column even at high bubble Reynolds

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numbers, which was controversial. With regards to TKE modification by bubbles, the underlying physical energy transport mechanisms are still not well understood, because of the difficulties capturing an entire bubbly flow system including large scale fluid motions and small scale interactions between bubbles and turbulence in both experiments and simulations. The main objective of the present study is to experimentally investigate the TKE transport mechanism among bubbles, large and small eddies, which affect the overall turbulence modification by using a particle tracking velocimetry (PTV) with Sub-Kolmogorov time and spatial resolution. The premise is that a detailed understanding of the microscopic interactions between bubbles and turbulence will lead to improve understanding and prediction of the macroscopic turbulence modification. Sub-Kolmogorov resolution enables us to obtain reasonably accurate TKE dissipation rate, which is the essential quantity to clarify the turbulence modulation. Bubble shapes and behaviors were obtained by shape projection imaging (SPI). By expanding the spatial filtering as used in previous studies by Liu et al. (1994, 1999), we characterize the length scales that govern the energy transfer between bubbles and turbulence.

2. Measurement system and experimental facility The schematic of the experimental facility used in the present study is described in Figure 1. All experiments were conducted in the same upward fully-developed bubbly pipe flow used for the study of Fujiwara et al. (2004). While a thorough description is described in their article, the important aspects of the experimental facility are recounted below. 2.1 Pipe description The flow in a 44 mm diameter pipe is driven upward by a pump at the bulk velocity, Ub, of 200 mm/s, corresponding to a pipe Reynolds number, Re2R, of 9900. More detailed flow properties are listed in Table 1. The fully-developed pipe flow was assured by extending the pipe development

t = 0[s] t = 2[ms] t = 4[ms]

Pulse generator

CMOS camerafor SPI

Blue LEDsλ = 473 nm

CMOS camera for PTV

Laser sheetNd:YLF Laser

λ = 527 nm

Fluorescent particles

Flow direction

Bubble

z

r

t = 0[s] t = 2[ms] t = 4[ms] t = 0[s] t = 2[ms] t = 4[ms]

Pulse generator

CMOS camerafor SPI

Blue LEDsλ = 473 nm

CMOS camera for PTV

Laser sheetNd:YLF Laser

λ = 527 nm

Fluorescent particles

Flow direction

Bubble

z

r

t = 0[s] t = 2[ms] t = 4[ms]

DensityKinematic viscosityPipe diameterMean velocityBulk velocityPipe Reynolds numberKolmogorov scale

LengthTime: values at the centerline of the pipe

ρf

ν2R<Uc>Ub

Re2R

ητk

[kg/m3][m2/s][mm][mm/s][mm/s][-]

[μm][μs]

997.00.89 × 10-6

442472009900

330120

Water (25ºC)

Table 1 Flow properties

Void Fraction

Bubble diameter

α

Deq

0.51.01.51.72.6

Air

Table 2 Bubble propertis

Fig.1 Schematic of the experimental facility. The pipe is drivenupward by a pump at the bulk velocity at a Reynolds number, Re2R, of 9900. Bubbles are injected by a bubble generator attached at the

Fig.1 Schematic of the experimental facility. The pipe is drivenupward by a pump at the bulk velocity at a Reynolds number, Re2R, of 9900. Bubbles are injected by a bubble generator attached at the entrance of the pipe.


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section for an entry length, xentry, of 34 pipe diameters (xentry /2R > 0.623 Re2R1/4, Kays & Crawford,

1993). The upward flow direction is defined as the positive z axis and the radial direction is defined as the r axis. Optical distortion through the pipe is eliminated using a fluorinated ethylene propylene (FEP) pipe at the test section. FEP has almost the same refraction index as water. The test section is covered with a rectangular acrylic container filled with water to have straight optical access into the pipe. 2.2 Bubble properties The bubble properties in the present study are shown in Table 2. Bubbles are injected by a bubble generator attached at the entrance of the pipe. The bubble generator has 34 stainless steel pipes with an inner diameter of 0.07 mm. The void fraction, α, is set to be 0.5 %, 1.0 % or 1.5 % by adjusting the air pressure of the bubble generator to examine the effects of void fraction on turbulence modification. Two different bubble diameters were examined by adding a surfactant of 3-pentanol (C5H11OH), since the presence of the surfactant avoids bubble coalescence resulting in the smaller bubbles, compared with the bubbles without the surfactant. In the present study, the bubble diameters were about 1.7 mm for case with the surfactant and 2.6 mm for case without it, where the bubble diameter was estimated as an area equivalent diameter, Deq. 2.3 Measurement system The measurement system in the present study consists of a high temporal/spatial resolution PTV and a bubble shape projection imaging (SPI) to capture both liquid and gaseous phases simultaneously as described in Figure 1. Two 500 Hz CMOS cameras(X-Stream, Integrated Design Tools) were located facing each other around the test section and synchronized by a pulse generator. Velocities of liquid phase were measured by PTV, and bubble shapes were obtained by SPI. For PTV, an Nd-YLF laser (Quantronix, Falcon 527DP) was used as the light source. The laser wavelength, λ, was 527 nm and the energy of each pulse was 30 mJ. Fluorescent tracers (Lefranc & Bourgeois, Flashe, Light orange) with an fluorescence wavelength of about 600 nm, were used so that we could capture only fluorescent tracer images by cutting the laser scattering at the bubble surfaces using a color filter. A tracer image is shown at the lower left of Figure 1. The tracer diameter was about 4 μm. A consecutive PTV algorithm was used using the high speed CMOS camera. Since the time duration of the camera frames is small enough to be the time separation of the PTV image pairs any of two consecutive images can be processed. Thus, the PTV time resolution is the same as the camera frame rate of 500 Hz, or 2ms, smaller than the Kolmogorov time scale of 120 ms, which enable us to resolve the dissipative process of turbulence. Each velocity vector measured by PTV was used to reconstruct grid velocity fields by spatially averaging velocity vectors. The grid size of 200 μm is smaller than the Kolmogorov scale of 330 μm at the centerline, which is reasonably high to resolve the TKE dissipative scale and to obtain the TKE dissipation. The PTV experimental error is estimated to be 1.2 % at 95 % confidence limits. For SPI, blue (λ = 473 nm) LEDs illuminated bubbles from behind. The same type of high speed CMOS camera as PTV was used to capture the projected images of bubbles. Three consecutive images taken by SPI are shown at the lower right of Figure 1, where the scattering light by the Nd-YLF laser was filtered out through a color filter. The focal plane was set to be identical to the PTV laser sheet plane. Bubbles on the focal plane were detected by recognizing the edge of the bubbles, which have high contrast.

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3. Results and discussions 3.1 Local void fraction Bubble diameters and local void fractions were obtained by SPI. The local void fraction, αl, which was the ratio of the liquid and gaseous phases, was measured using the area of bubbles, Sb, and the area of the imaged section, S, as:

r

brl S

S⎟⎠

⎞⎜⎝

⎛=)(α (1)

where the subscript, r, denotes an arbitrary position at r. Figure 2 shows the local void fraction in the presence of (a) the larger (<Deq> = 2.6 mm) or (b) smaller (<Deq> = 1.7 mm) bubbles. The horizontal axis is normalized by the pipe radius, R. It is clearly observed that bubbles are not uniformly distributed due to the shear induced lift force acting on bubbles (Ervin E.A et al.); the local void fractions near the pipe wall region are noticeably high. Since the shear induced force is depending on the bubble diameter, the local void fraction distributions differ. For <Deq> = 1.7mm, the local void fraction peak appeared at r/R ∼ 0.95, while it is at r/R ∼ 0.90 for <Deq> = 2.6mm. The non-uniformity trend is greater for smaller bubble case. 3.2 Liquid-phase mean and fluctuating velocities The liquid-phase streamwise mean velocity profiles are shown in Figure 3 for (a) <Deq> = 2.6 mm and (b) <Deq> = 1.7 mm. The vertical axis is normalized by the mean velocity at the centerline, <Uc>. Since bubbles accelerate the fluid velocity near the wall region, the entire mean velocity becomes flat. This trend is greater for <Deq> = 1.7mm case. The flattened mean flow profiles agree to the previous results by Fujiwara et al. (2004). The position where the steep mean velocity gradient appeared was almost the same as the local void fraction peak. This indicates that bubbles block the effect of the shear due to the wall and change the shear-induced turbulence structure. Note that the non-uniformity trends are greater for lower void fraction. Most bubbles are concentrated near the wall for a dilute (α ∼ 0.5 %) bubbly flow to create flattened mean velocity profiles. For

r / R0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Voi

d fr

actio

n [%

]

0.0

2.0

4.0

6.0

8.0

α = 1.5 % α = 1.0 % α = 0.5 %

r / R0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Voi

d fr

actio

n [%

]

0.0

2.0

4.0

6.0

8.0

α = 1.5 %α = 1.0 % α = 0.5 %

r / R

0.0 0.2 0.4 0.6 0.8 1.0

 /



0.0

0.2

0.4

0.6

0.8

1.0

α = 0.0 %α = 0.5 %α = 1.0 %α = 1.5 %

r / R0.0 0.2 0.4 0.6 0.8 1.0

 /



0.0

0.2

0.4

0.6

0.8

1.0

α = 0.0 %α = 0.5 %α = 1.0 %α = 1.5 %

(a) <Deq> = 2.6 mm (a) <Deq> = 2.6 mm

(b) <Deq> =1.7 mm (b) <Deq> = 1.7 mm

Fig.2 Local void fraction, for cases of (a): <Deq> = 2.6 mm and (b): <Deq> = 1.7 mm.

Fig.3 Liquid-phase mean velocity profiles , for cases of (a): <Deq> = 2.6 mm and (b): <Deq> = 1.7 mm.

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higher α, the local void fraction increases more uniformly, and they do not affect the mean velocity profile much. In the present measurements, all the mean velocity profiles in the presence of the bubbles with the same diameter are almost identical regardless of the void fraction, α. The streamwise turbulence intensity profiles of the liquid-phase are shown for <Deq> = 2.6 mm (Figure 4 (a)) and <Deq> = 1.7 mm cases (Figure 4 (b)). The vertical axis is normalized by <Uc>. In the presence of the 2.6 mm bubbles, the streamwise turbulence intensity at α = 0.5 % decreased in the region of 0.4 < r/R < 0.95 due to the flattened mean velocity profiles by the bubbles, while it was increased around the pipe center region due to the presence of bubble wakes (r/R < 0.4). At the higher void fraction than 0.5%, turbulence augmentation at the entire pipe is observed due to the bubble wakes. This is because the local void fraction increases more uniformly for higher α than 0.5 % in the middle of the pipe (r/R < 0.8). Therefore, it can be observed that turbulence intensities are uniformly augmented by bubbles in the pipe middle region. In the presence of the 1.7 mm bubbles, the similar trends to the 2.6 mm bubbles are observed. However, the turbulence attenuation trends are greater than 2.6 mm cases for the larger bubbles due to the greater non-uniformity of the local void fraction. Also, the turbulence augmentation due to the bubble wakes are smaller for <Deq> = 1.7mm case than that for <Deq> = 2.6 mm case. At α up to 1.0 %, the turbulence attenuation was predominant. Though Fujiwara et al. (2004) observed only turbulence attenuation in the pipe middle region, the trends in the present study can be though to be consistent to the their results by considering that they examined smaller bubble diameter of 1 mm and 2mm in the diluter bubbly flow regimes (α = 0.5 % or 1.0 %). 3.3 TKE energy budget To investigate the turbulence modulation by bubbles, the transport equation for the turbulent kinetic energy (TKE), K, is also derived based on the Reynolds decomposition as:

bj

TKE

xDtD

FPTK

−−=∂

∂+ ε (2)

The second term in the left-hand side of Eq. (2) represents the transport term including pressure-diffusion and viscous diffusion. The first term in the right-hand side of Eq. (2) is the TKE production, P, which represents the energy production induced by the mean flow shear stress. The second term, ε, is TKE dissipation by viscosity. The last term, Fb, is the correlation of the fluctuating velocity and the force acting on bubbles. In the present study, these terms were estimated as:

r / R0.0 0.2 0.4 0.6 0.8 1.0



1/2 /



0.0

0.1

0.2α = 0.0 %α = 0.5 %α = 1.0 %α = 1.5 %

r / R0.0 0.2 0.4 0.6 0.8 1.0



1/2 /



0.0

0.1

0.2

α = 0.0 %α = 0.5 %α = 1.0 %α = 1.5 %

(a) <Deq> = 2.6mm case (b) <Deq> = 1.7mm case

Fig.4 Liquid-phase RMS velocity profiles in the (a): <Deq> = 2.6 mm and (b): <Deq> = 1.7 mm.

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rU z

∂

∂≈ rzuu-P (3)

zu

zu

ru

ru

zU

zu

ru

ru zzzzrrrr

∂∂

∂∂

+∂

∂∂

∂+

∂∂

∂∂

+∂

∂∂

∂≈ νε (4)

Note that the TKE dissipation was obtained by using a correction method introduced by Tanaka & Eaton（2007) in order to eliminate the measurement noise, which is critical for high resolution measurements as demonstrated in Saarenrinne & Piirto (2000). The correction method is described as:

32

4ΔxmΔxm εε

ε−

≅ (5)

where the subscript, m, denotes a quantity obtained from measured data, and xm Δε is the TKE

dissipation using 2nd order central difference approximation applied to the PIV data. Similarly, xm Δ2ε is the measured dissipation rate with double grid spacing, (2Δx). The grid resolution should

be higher than the half of Kolmogorov scale to accurately obtain the dissipation rate. In the present study, the grid scale of 0.2 mm was two thirds of the Kolmogorov scale. Thus, the calculated dissipation rates are probably underestimated by about 10 %, based on the results by Tanaka and Eaton (2007). Figure 5 shows the TKE production and dissipation rate profiles for (a) <Deq> = 2.6 mm and (b) <Deq> = 1.7 mm cases. In the bubbly flows, turbulence kinetic energy production prominently reduces due to the flattened mean flow profile. For (a) <Deq> = 2.6 mm case at α = 0.5 %, it can be observed that the TKE production clearly reduced to almost zero in the pipe middle region (r/R <0.8) for all of the three bubble-laden flow cases. On the other hand, the dissipation rate slightly increased for α = 0.5 % case, while it significantly augmented for α = 1.0 and 1.5 % in the middle of the pipe region. This is because that the local void fraction for α = 1.0 or 1.5 % is considerably higher than that for α = 0.5 %. The presence of bubbles in the uniform flow regime (r/R <0.8) significantly augments the TKE dissipation. For <Deq> = 1.7mm case, greater attenuation of the TKE production is observed because of the greater trend of the flattened mean velocity profile due

(a) <Deq> = 2.6 mm (b) <Deq> = 1.7 mm

Fig. 5 TKE budget profiles for cases of (a) <Deq> = 2.6 mm and (b) <Deq> = 1.7 mm. Circles represent the TKE production and squares denotes the TKE dissipation rate. Solid line and dashed lines are the production and dissipation for unladen case, respectively. Black symbols are for the case of α =

r/R0.0 0.2 0.4 0.6 0.8 1.0

P, ε

[mm

2 /s3 ]

-4000

-2000

0

2000

4000

6000 α = 0.0% (production)α = 0.0% (dissipation)α = 0.5% (production)α = 1.0% (production)α = 1.5% (production)α = 0.5% (dissipation)α = 1.0% (dissipation)α = 1.5% (dissipation)

r/R0.0 0.2 0.4 0.6 0.8 1.0

P, ε

[mm

2 /s3 ]

-4000

-2000

0

2000

4000

6000 α = 0.0% (production)α = 0.0% (dissipation)α = 0.5% (production)α = 1.0% (production)α = 1.5% (production)α = 0.5% (dissipation)α = 1.0% (dissipation)α = 1.5% (dissipation)

0.5%, gray ones are at α = 1.0 %, and open ones are for the case at α = 1.5%.

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to the smaller bubbles. The dissipation rate augmented as the void fraction increased. However, the augmentation trends are smaller compared with the case for <Deq> = 2.6 mm, indicating that the TKE dissipation is strongly depending on the bubble diameter in bubbly flows. 3.4 TKE energy spectrum In bubbly flows, there are multiple important scales such as Kolmogorov scale, energy containing eddy scale, bubble diameter and inter-bubble spacing. In each scale, turbulence is modulated by different mechanisms. In order to investigate scale independent turbulence modification, turbulent energy spectra at the pipe center were calculated as shown in Figure 6 (a) for <Deq> = 2.6 mm and (b) <Deq> = 1.7mm cases. By assuming the Taylor's hypothesis, energy spectra were defined as:

∫= −iUT iiii deu)(E 0

2ςςκ κς iUt≡ς (6)

Since the Fourier transform of the intermittent liquid-phase due to the gaseous phase is not strictly defined, we applied a linear interpolation for the intermittence of the liquid-phase. The vertical axis is normalized by the kinematic viscosity of water, ν, the energy dissipation rate at the centerline for unladen case, εc. The horizontal axis is normalized by the Kolmogorov micro length scale, η, where κ is the wavenumber defined as 2π/z. Turbulence kinetic energy spectra at r/R = 0.0 in the streamwise direction are shown in Figure 6 for (a) <Deq> = 2.6 mm and (b) <Deq> = 1.7mm. The horizontal axis is normalized by the Kolmogorov length scale, η, at the centerline for unladen case. For <Deq> = 2.6 mm case, the energy spectra increased at almost entire scales (3 × 10-1< κη) monotonically as the void fraction increased. Since the mean flow profile is flat at the pipe center, the TKE production due to the mean shear is negligible small. Thus, it can be concluded that the turbulence augmentation in this region is mainly caused by the bubble wakes. The augmentation trends are the greatest around κη ∼ 6 ×10-1, which is about two times the bubble diameter. For <Deq> = 1.7mm case, it is noticeable that the turbulence level at the large scale (κη > 4 × 10-1) at α = 0.5 due to the flattened mean flow profile. This attenuation at the small scale means that the energy cascade from the large scale turbulence also reduced. Since the turbulence augmentation at the small scale due to bubble wakes barely increased because of the low local void fraction at the center for α = 0.5 case, the total turbulence intensity attenuated as shown in Figure 4 (b). As the void fraction increases, the TKE increases by the distortion around the bubbles not only at the scale of bubble size but also at the larger scales. This implies that the bubbles induce various size of eddies by clustering. 3.5 Scaling analysis of turbulence modification by bubbles using filtering

κη

10-2 10-1 100

E 11/(

ν5 <ε>)

1/4

10-3

10-2

10-1

100

101

α = 0.0%α = 0.5%α = 1.0%α = 1.5%

κη

10-1 100

E 11/(

ν5 <ε>)

1/4

10-3

10-2

10-1

100

101

α = 0.0%α = 0.5%α = 1.0%α = 1.5%

(a) <Deq> = 2.6 mm (b) <Deq> = 1.7 mmFig.6 Turbulent energy spectra at the pipe center for the case of (a): <Deq> = 2.6 mm and (b): <Deq> = 1.7 mm.


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In the realistic numerical methods to simulate bubbly flows such as Reynolds Averaged Navier-Stokes (RANS) model and LES, the interactions between small scale turbulence behavior and large scale flow structure such as mean flows or large eddies need to be well described to accurately capture entire bubbly flows. In the previous sections, we phenomenologically observed that the bubbles modified the large eddies related to the mean flow and small scale eddies due to the wakes. Although the energy spectrum provides much fruitful information, it is not enough to understand the interactions among the large scale flows, small scale turbulence structures and bubbles. In this section, we investigate the modification of the TKE cascade from large to small eddies in order to clarify the mechanisms of turbulence modification by bubbles and to obtain physical guidance on the modeling for the numerical simulations. We introduce an analytical method to specify the physical mechanisms of turbulence modification at each scale. To conduct the scaling analysis, the filtering techniques derived from LES, which can separate TKE into two scales. The filtering technique used in LES is experimentally applied as used in previous studies by Liu et al. (1994,1999) for jet flows for the purpose to improve LES models by evaluate turbulent kinetic energy flux from large to small eddies, which can be also considered as the energy cascade. In the present study, we examine detailed properties of energy cascade such as TKE dissipation spectrum and an energy transport effects due to bubbles at each scale. Before we described the detailed procedures for the scaling analysis, a general description of LES are recounted. The fluid equations for LES are obtained by filtering the velocity field using a low-pass filtering function, GΔ, which satisfies the normalization condition. In the present study, we used the sharp cutoff filter, Δ

SG , whose property in one dimension is:

)Δπξsin()(G Δ

S πξξ 2

=

where Δ is the specified filter width. The filtered value of an arbitrary turbulent quantity, B is defined as, ,),)(),(~ ξξξ dtGtx -B(xB ∫ Δ= (7)

where ~)(⋅ represents the filtering. The filtered TKE transport equation for the unresolved or

subgrid scale (SGS) can be derived by defining SGS kinetic energy, QSGS, as:

bSGSSGS

j

SGSSGS ΠxtD

DF

TQ−+−

∂∂

= ε~~

(8)

2

~~iii

SGSUUUU −

=Q (9)

where T, ε, and Fb denote the energy transport, the energy dissipation, and the energy transfer from bubbles at each scale. Each term is defined as: SΠ SGS

ij~τ−= (10)

δκ κ

E

Π (κ) Π (κ+δκ)

D (κ)

Fb(κ)

δκ κ

E

Π (κ) Π (κ+δκ)

D (κ)

Fb(κ)

Fig. 7 Concept of energy transport by bubbles in wavenumber space.

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j

i

j

i

j

i

j

iSGS x

UxU

xU

xU

∂∂

∂∂

−∂∂

∂∂

=~~

ννε (11)

where the subgrid scale (SGS) stress, SGSijτ

jijiSGSij UUUU ~~−=τ (12)

The energy flux, Π, represents the energy exchange from resolved eddies to the SGS eddies. Thus, Π is the energy source term for SGS kinetic energy equation similar to the TKE production in RANS. The subscript, SGS, represents subgrid scale. The method for the scaling analysis we introduce is given as follows based on the filtering techniques above. The scaling analysis is conducted in the wavenumber space as described in Figure 7. Considering the TKE energy budget an arbitrary wavenumber, κ, an energy conservation of a tiny control wavenumber domain in between κ and κ + δκ can be written as:

δκκδκκκδκκ )()()()( DΠΠ b −+=+ F (13) where TKE dissipation spectrum, D(κ), and the energy term due to bubbles, Fb, are described as:

κ

κεκ

∂∂

−=)(

)( SGSD (14)

)()()( κκκκ DΠ

b +∂

∂=F (15)

Using these equations introduced above, we can specify the behaviors of the TKE energy cascade process. This method can be applied for a wide range of complicated turbulent flows such as multi-phase flows, even if the flows are not ideal cases; i.e. homogeneous and isotropic turbulence. Also, we emphasize that the current approach can calculated the energy transport by bubbles, which is very complicated term in the TKE equation (Eq. (8)).

κη0.5 1.0 1.5 2.0 2.5

Π[m

m2 /s

3 ]

0

50

100

150

200

250

300α = 0.0%α = 0.5%α = 1.0%α = 1.5%

κη0.5 1.0 1.5 2.0 2.5

Π[m

m2 /s

3 ]

0

10

20

30

40

50α = 0.0%α = 0.5%α = 1.0%α = 1.5%

κη0.5 1.0 1.5

D[m

m3 /s

3 ]

0

100

200

300

400

500

600

700

800α = 0.0%α = 0.5%α = 1.0%α = 1.5%

κη0.5 1.0 1.5

D[m

m3 /s

3 ]

0

100

200

300

400α = 0.0%α = 0.5%α = 1.0%α = 1.5%

(a) <Deq> = 2.6 mm (a) <Deq> = 2.6 mm

(b) <Deq> = 1.7 mm (b) <Deq> = 1.7 mmFig.8 TKE flux from large to small scales in the pipe middle region (0 < r /R < 0.66) for (a): Deq = 2.6 mm and

Fig.9 TKE dissipation spectrum in the pipe middle region (0 < r /R < 0.66) for (a): Deq = 2.6 mm and (b):Deq = 1.7 mm(b): Deq = 1.7 mm

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Each term in Eq. (13) is calculated using the measured velocity fields at 0 < r/R < 0.66. The energy flux, Π, dissipation spectrum, D, and the energy due to bubbles, Fb, are shown in Figures 8, 9, and 10, respectively. Open circles represent the unladen case. Black squares are for the case at α = 0.5 %, gray squares are at α = 1.0 %, and open squares are for the case at α = 1.5 %. The horizontal axis is normalized by the Kolmogorov length scale, η, at the centerline for unladen case. In Figure 8, it is observed that the energy flux, Π, for the unladen case decreased as the wavenumber increases. This indicates that the TKE transported from the large eddies from small scales while it gradually dissipated at each scale. In the presence of bubbles for <Deq> = 2.6 mm case, the energy flux profiles significantly changed. The profiles had peaks around κη ∼ 0.41 - 0.54, while the energy flux for the unladen case monotonically decreased. This implies that the energy from bubbles transports to the fluid flows around this scale, resulting in the augmentation of the energy flux. The energy flux, Π, increased as the void fraction increased because the energy transport between the liquid-phase and bubbles was promoted. In the presence of <Deq> = 1.7mm bubbles, similar trends to the <Deq> = 2.6 mm case are observed; the energy flux, Π, augmented as the void fraction increased and the profiles had peaks around κη ∼ 0.41 - 0.54 as shown in Figure 8 (b). Also, it is interesting to note that the energy flux attenuated for both α = 0.5 % and 1.0 % cases, because of the suppression of the large scale turbulence by the flattened mean flow. Compared with <Deq> = 2.6 mm case, the attenuation effect was larger. This is because the local void fraction is smaller and the augmentation due to the bubble wakes is also small for <Deq> = 1.7mm.

In Figure 9, the energy dissipation spectrum for (a): <Deq> = 2.6 mm and (b): <Deq> = 1.7 mm are shown. The dissipation spectrum, D, had a peak at κη ≈ 0.41. Though the dissipation spectra augmented at the entire scales as α increased, the peak positions barely changed. Also, the scales responsible for the bulk of the dissipation are at 0.32< κη <1.5, which are larger than the Kolmogorov scales. In the presence of 1.7 mm bubbles, D for α = 0.5 % augmented at κη < 0.41 by the bubble wakes, while D augmented at κη < 0.41 due to the turbulence attenuation. For higher void fraction, the dissipation spectrum augmented at the entire scales. In Figure 10, the energy transport term due to bubbles calculated for (a): <Deq> = 2.6 mm and (b): <Deq> = 1.7 mm. Strong energy transport from bubbles can be clearly observed for both <Deq> = 2.6 mm and <Deq> = 1.7mm cases. The bubble term augmented as the void fraction increased. The peaks appeared at κη ≈ 0.32-0.37 for <Deq> = 2.6 mm and at κη ≈ 0.37-0.41 for <Deq> = 1.7 mm. We can confirm that the smaller bubbles induced TKE at smaller scales. By defining the peak wavenumber, κb, we attempt to nondimensionalize the peak wavenumber by bubble diameter. The normalized peak wavenumbers, 2π/(κb <Deq>}, in the present study were:

κη0.5 1.0 1.5

F b[m

m3 /s

3 ]

0

100

200

300

400

500

α = 0.5%α = 1.0%α = 1.5%

κη0.5 1.0 1.5

F b[m

m3 /s

3 ]

0

500

1000

1500α = 0.5%α = 1.0%α = 1.5%

(a) <Deq> = 2.6 mm (b) <Deq> = 1.7 mm

Fig.10 Energy transport effects due to bubbles in the pipe middle region (0 < r /R < 0.66) for (a): Deq = 2.6 mm and (b): Deq = 1.7 mm.

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6.221.2 <<eqb Dκ

π for <Deq> = 2.6 mm (16)

2.329.2 <<eqb Dκ

π for <Deq> = 1.7 mm (17)

Thus, it indicates that the TKE transported due to bubbles are at the scale of from two to three times the bubble diameter for similar flow regimes to the present experiment. In other words, the interactions between bubbles and the fluid are predominant at this scale. We believe that the findings above give guidance on building models of bubbly flows such as LES. Some LES studies for two-phase flows including particle-laden flows were conducted by Armenio et al. (1999), Boivin et al. (2000) and Yamamoto et al. (2001). The SGS fluctuations can have critical effects on dispersed-phase motion, since particles are much smaller than the filter width (Armenio et al. (1999), Boivin et al. (2000) and Yamamoto et al. (2001)). Yamamoto et al. (2001) suggested that the required grid resolution be smaller than the Taylor microscale and similar to the inter-particle spacing. In the present study, we found that the length scale where the bubble-fluid interactions were predominant of about two to three times the bubble diameter. This scale must be the important parameter to determine the filter width, Δ. 4. Conclusions Turbulence modification in an upward bubbly pipe flow at Re2R =9900 was investigated using a sub- Kolmogorov PTV/SPI measurement system. Experiments were conducted at two different bubble diameters of 1.7 mm and 2.6 mm and the void fraction, α, up to 1.5. The findings in the present study are as follows. The mean flow profiles were flatten for all tested cases because bubbles accumulated near the wall region accelerated the liquid-phase fluid. The flattened mean flow profiles suppressed the shear-induced turbulence intensities in the pipe middle region. The turbulence attenuation trend due to the non-uniformity of the local void fraction was greater for smaller bubble case (<Deq> = 1.7 mm). On the other hand, at the higher void fraction than 0.5%, turbulence augmentation at the entire pipe is observed due to the bubble wakes. The trend of turbulence augmentation due to the bubble wakes were larger for larger bubbles. TKE energy budget in bubbly flows was calculated using the accurate form of dissipation rate introduced by Tanaka & Eaton (2007). While the TKE production prominently reduces due to the flattened mean flow profile in the presence of bubbles. The dissipation rate augmented as the void fraction increased. However, the augmentation trends are larger for larger bubble case (<Deq> = 2.6 mm), indicating that the TKE dissipation is strongly depending on the bubble diameter in bubbly flows. The TKE spectra also captured the attenuation trend at the low wavenumber due to the flattened mean flow profiles and the augmentation trend due to the bubble wakes. Scaling analysis using the filtering technique was applied to the fluid-phase velocity fields to characterize the length scales that govern the energy transfer between bubbles and turbulence. The energy flux from large to small eddies augmented as the void fraction increased and the profiles had peaks around κη ∼ 0.41 - 0.54, implying that the energy from bubbles transports to the fluid flows around this scale, resulting in the augmentation of the energy flux. Also, it is interesting to note that the energy flux attenuated for both α = 0.5 % and 1.0 % cases, for <Deq> = 1.7mm because of the suppression of the large scale turbulence by the flattened mean flow. Strong energy transport from bubbles can be clearly observed in the presence of the bubbles at the scale of from two to three times the bubble diameter, indicating the interactions between bubbles and the fluid are predominant at this scale.


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coupling using large eddy simulation. Physics of Fluids 12(8) (2000) 2080-2090. 3. Cadot, O., Douady, S., Couder, Y., Characterization of the low-pressure filaments in a 3-

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masashi nakagawa 1, masamitsu koyasu 1, tomohiko tanaka2,...

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