Effects of Bubbles on Turbulent Flows in Vertical Channels
ICMF2010-Tampa 2010.6.1.
ICMF2010-Tampa 2010.6.1.
Purpose
Turbulence structure in bubbly flows is very complex due to (1) bubble-induced pseudo turbulence, (2) interaction between bubble-induced and shear–induced turbulences, (3) direct interaction between bubbles and turbulence eddies and (4) modification of shear-induced turbulence due to the modulation of mean velocity distribution due to bubbles.
Effects of Bubbles on Turbulent Flows in Vertical Channels
Bubbly upflows in vertical channnels
dB/lt
1. Index of turbulence modification The eddy viscosity ratio φ ( = dBVR/ltu’ )
Shigeo Hosokawa and Akio Tomiyama
2. Turbulence modification at large dB/lt Experiments & numerical predictions
Kobe University
3. Turbulence modification at small dB/lt and large VR/u’ Turbulence kinetic energy budget
Large φ Turbulence augmentation
Large dB/lt Effect of VR/u’ on turbulence modification VR/u’ Small dB/lt Large VR/u’
Small φ Turbulence attenuation
Turbulence modification due to dispersed phase
αL
Large particle/bubble: Turbulence augmentation
2
Small particle/bubble: Turbulence attenuation Gore and Crowe Critical parameter: d/lt ⇒Ratio of mixing length of particle-induced turbulence to that of shear-induced turbulence
1
ΔTI
•
νTP = f (ν t , ν B , K) φ=
0
νB νt
νt : Shear-induced eddy viscosity νB : Bubble-induced eddy viscosity VR : Relative velocity dB : Diameter of dispersed phase u’ : Turbulence velocity lt : Turbulence length scale
ν B ∝ VR d B
S.I.T.
ν t ∝ u ′lt
-1
-2
ICMF2010-Tampa 2010.6.1.
M α DVL = − L ∇P + ∇α L (ν + νTP )(∇VL + ∇VLT ) + i + α L g ρ ρ Dt
Present data (Gas-Liquid) Present data (Liquid-Solid) Hosokawa et al.(1988)(Gas-Solid) Tsuji et al. (1984) Maeda et al. (1980) Kulick et al. (1994) Lee and Durst (1982)
VR
u’ dB
Turbulence attenuation
Ratio of eddy viscosity of particle-induced turbulence to that of shear-induced turbulence
Eddy Viscosity Ratio φ
ICMF2010-Tampa 2010.6.1.
10-2
Turbulence augmentation
10-1 d/lt
100
Eddy viscosity ratio
φ≡
lt B.I.T.
VR d B VR d B ReB = = u ′lt u ′ lt Ret
ReB =
u ′lt VR d , Ret = ν ν
Indicator of Turbulence Modification
ICMF2010-Tampa 2010.6.1.
ICMF2010-Tampa 2010.6.1.
φ－CTI/Nd 3
Pipe diameter: D = 20, 30 mm Liquid volumetric flux: JL = 0.5 -1.0 m/s Volume fraction of dispersed phase: α = 0.7 - 5.0 % Bubble/particle diameter d = 1 -5 mm
2
2
Solid-Liquid(JL=0.5m/s) d=4.0mm,Js=0.002m/s d=4.0mm,Js=0.004m/s d=2.5mm,Js=0.002m/s d=2.5mm,Js=0.004m/s d=1.0mm,Js=0.002m/s d=1.0mm,Js=0.004m/s
-6
-6 -6 CTI/Nds [X10 ΔTI/N [X10 ]]
3
Gas-Liquid JL=0.51m/s,JG=0.017m/s JL=0.51m/s,JG=0.023m/s JL=0.71m/s,JG=0.017m/s JL=0.71m/s,JG=0.023m/s JL=1.01m/s,JG=0.017m/s JL=1.01m/s,JG=0.023m/s JL=0.50m/s,JG=0.015m/s (D=20 mm) JL=0.90m/s,JG=0.020m/s (D=20 mm)
CTI/Nd [X10 ]
LDV Measurements Gas-Liquid bubbly flow (Air-Water) Solid-Liquid two-phase flow (Ceramic-Water) in vertical circular pipe
1
Gas-Liquid JL=0.5m/s,JG=0.017m/s(D=30 JL=0.5m/s,JG=0.023m/s(D=30 JL=0.7m/s,JG=0.017m/s(D=30 JL=0.7m/s,JG=0.023m/s(D=30 JL=1.0m/s,JG=0.017m/s(D=30 JL=1.0m/s,JG=0.023m/s(D=30 JL=0.5m/s,JG=0.015m/s(D=20 JL=0.9m/s,JG=0.020m/s(D=20
mm) mm) mm) mm) mm) mm) mm) mm)
Liquid-Solid(JL=0.5m/s, D=30mm) d=4.0mm JS=0.002m/s d=4.0mm JS=0.004m/s d=2.5mm JS=0.002m/s d=2.5mm JS=0.004m/s d=1.0mm JS=0.002m/s d=1.0mm JS=0.004m/s
1
Dimensionless change in turbulence intensity per unit number density
′ u′ uTP − SP U TP U SP 2
CTI = Nd
u ′SP U SP
φ=1 0
0
2
u’ : Axial fluctuation velocity U : Axial mean velocity Nd : Number density of dispersed phase Subscript TP : Two-phase flow SP : Single phase flow
Nd
2
Correlation between φ and CTI/Nd
0.5
1 d/lt
Gore & Crowe’s parameter: d/lt
0
2
φ
4
4
6
Eddy viscosity ratio: φ=
νB νt
φ is applicable to gas-liquid and liquid-solid flows. Critical point is close to φ=1, irrespective of a type of two-phase flows.
ICMF2010-Tampa 2010.6.1.
Gas-liquid-solid three phase flow (water+air bubble+aluminum ceramic particle)
Gas-solid two-phase flow
0
ICMF2010-Tampa 2010.6.1.
Turbulence modification depends on φ ( = dBVR/ltu’), that is, it depends not only on dB/lt but also on VR/u’.
CTI/Nd [X10-4]
CTI/Nd [X10-8]
Shakutsui et al. (2001)
2
Effect of VR/u’ on turbulence properties in bubbly pipe flows at large dB/lt ( ~ 1). D=30 mm JL=0.5-1.0 m/s JG=0-0.023 m/s JS=0-0.0038 m/s dB≅2.0-8.0 mm
Air-water bubbly flow dB ~ constant ÆVR ~ constant D ~ constant Æ lt ~ constant JL changes Æ u’ changes Case 2
0
-20
ρd=980 kg/m33 (Polystyrene), m=2.3 ρd=980 kg/m 3(Polystyrene), m=4.6 ρd=2500 kg/m (Glass), m=4.6 ρd=2500 kg/m33 (Glass), m=6.0 ρd=3600 kg/m3 (Ceramic), m=4.6 ρd=3600 kg/m (Ceramic), m=6.9
1
2
φ
3
4
2.5
φ=
5
N PVRP dφP + N BVRB d B (N P + N B )u′lt
7.5
φ is applicable not only to gas-liquid and liquid-solid two-phase flows but also to gas-solid two-phase flows and three phase flows.
Turbulence augmentation dB/lt Large φ Millimeter size bubbles
Case 1 VR/u’
Case 1: turbulence augmentation low JL Æ large VR/u’ Small φ
Case 2: turbulence attenuation high JL Æ small VR/u’ Turbulence attenuation
ICMF2010-Tampa 2010.6.1.
Experimental Apparatus Length of pipe L = 2 m Diameter of pipe D = 25 mm Elevation of measurement point 1.7 m
FEP (Fluorinated Ethylene-Propylene resin) Refractive index: 1.338 cf. water: 1.333 Negligible optical distortion
CCD Camera
Reflection light
D=25mm
LDV
1.7m
2.0m
Camera
Reflection of light from the side reduces contrast of interface image.
Optical filter (Green transmit)
Standard light without optical filter
Optical filter (Red transmit)
B
A
Mixing section
LDV: Liquid velocity Image processing method: Void fraction, Bubble velocity, Bubble diameter, Aspect ratio
Flow
FEP
Water
Contrast increase.
Reflection light
LED (Red) 25mm
LED (Green)
Two color LED with optical filter
FEP pipe
41mm
sor Compres
Original image Binary image Elliptic disk with a pixel height
Acrylic duct
ai
Water
Fluids Liquid: water Gas: air
r Ai Drain
Tank
[Pixel]
ICMF2010-Tampa 2010.6.1.
One pixel
F=0 20
F=1
B 0 0
Curvature takes negative peaks at overlapped points A and B.
ICMF2010-Tampa 2010.6.1.
<JG>= 0.018m/s <JL>= 0.5m/s
A
40
Binarize
Void distribution Bubble diameter etc…
Condition
60
The curvature evaluation method was applied to overlapped bubble image.
∇F ∇F
bi
Binary image 1 Binary image 2
Detection and Separation of Overlapped Bubbles
n = −
ai
Pile up
bi
i
Pump
20
Optical setup 40
60 [Pixel]
0.1 κ[1/m]
κ = ∇ ⋅n
ICMF2010-Tampa 2010.6.1.
Image Processing Method
Camera B Camera A 0
κ: Interface curvature n : Unit normal to interface -0.1 F : Color function representing spatial distribution of interface 0
B
A 50
100
Pixel number
ICMF2010-Tampa 2010.6.1.
Mean Velocity Distributions
0.05
HIgh JLÆ high u’ Low VR/u’, φ
Case 1 JL = 0.5 m/s JG = 0.018 m/s
W,w’
Case 2 JL=1.0 m/s JG=0.036 m/s
1
Single phase flow U/JL V/JL W/JL Bubbly flow U/JL V/JL W/JL αG
0.5
0.2 0.4 0.6 0.8
1 0
0.1
The axial mean velocities were flattened due to the presence of bubbles. The radial and tangential mean velocities were zero.
0.2 0.4 0.6 0.8
r/R
U: Axial mean velocity V: Radial mean velocity W: Tangential mean velocity JL: Liquid volumetric flux JG: Gas volumetric flux αG: Void fraction R: Pipe radius
0 1
Æ fully-developed flow
Case1: Large VR/u’ Turbulence augmentation
Case2: Small VR/u’ Turbulence attenuation
0.5 1.5
0.5 Single phase flow
0.5
Bubbly flow 0.2
0.4
Time (s)
0.2 - 0.3m/s = VR
VR Bubble
Bubbly flow
Velocity data
(JL =0.93m/s,J G=0.018m/s,r/R=0.7)
U(m/s)
(JL =0.63m/s,J G=0.017m/s,r/R=0.7)
Phase distribution function (High: gas, Low: liquid)
Single-phase flow
(JL =0.93m/s,r/R=0.7)
1.5
Velocity data
Bubbly flow
0 0
Case 1 u’u’, v’v’ and w’w' are enhanced over the crosssection.
0
B.I.T. augments T.I. whereas it does not enhance u’v’ so much.
Single phase flow u'v'/JL2 2 u'w'/JL Bubbly flow 2 u'v'/JL 2 u'w'/JL
0.004 0.003
Case 2 Turbulence intensities are attenuated in core region, and it is enhanced in the near wall region.
0.002 0.001 0 0
0.2
0.4 0.6 r/R
0.8
0
0.2
0.4 0.6 r/R
Phase di stribution function (High: gas, Low: liquid)
Bubbly flow
10
4
103
Case 1: Turbulence enhancement
Case 1 -5/3
E(ω) was enhanced
JL=0.63 m/s r/R = 0.7
105
Single phase flow
Case 2: Turbulence suppression High frequency eddy was enhanced Low frequency eddy was attenuated
Bubbly flow
Bubbly flow
0 0
0.2
Time (s)
ICMF2010-Tampa 2010.6.1.
105
1 0.5
0.8
u’v’ is attenuated due to the presence of bubbles except in 1 the near wall region.
Turbulence Energy Spectra
1
Single-phase flow
W,w’ V,v’
u’v’ is enhanced in the near wall region, whereas it in the core region changes not so much.
E(ω)
(JL =0.63m/s,r/R=0.7)
V,v’
VR/u’, φ
0.01
Single phase flow Velocity data
Velocity data
U(m /s)
U(m/s)
Single phase flow
U(m/s)
0.02
1.5
1.5
1
0.03
ICMF2010-Tampa 2010.6.1.
Fluctuating Axial Velocity
1
Single phase2 flow u'u'/JL Low 2 v'v'/JL 2 w'w'/JL Bubbly flow 2 u'u'/JL2 v'v'/JL w'w'/JL2
High VR/u’, φ
0.04
r/R
Void fraction exhibits wallpeaking profile in case 2.
U,u’ W,w’
Case 1 Case 2 (JL=0.5 m/s, JG=0.018 m/s) (JL=1.0 m/s, JG=0.036 m/s)
102
Case 2
-5/3
104
0.4
E(ω)
0 0
V,v’
0.2
αG
U/JL, V/JL, W/JL
1.5
V,v’
u'u'/JL2, v'v'/JL2, w'w'/JL2
Low JL Æ low u’ High VR/u’, φ
Turbulence augmentation Turbulence attenuation
U,u’ W,w’
Case 2
u'v'/JL2, u'w'/JL2
Case 1
ICMF2010-Tampa 2010.6.1.
Reynolds Stress
JL=0.93 m/s r/R=0.7
u’ > VR
Breakup of large scale eddy due to bubbles
103
Turbulence attenuation
Low frequency fluctuation becomes weak
B.I.T 10-1
100
101 ω (Hz)
102
102 103
Numerical Prediction of Bubbly Flows
ICMF2010-Tampa 2010.6.1.
ICMF2010-Tampa 2010.6.1.
Field Equations MultiMulti-fluid model (Tomiyama (Tomiyama et al. 1998, 2005)
• Turbulence modification depends on VR/u’.
Gas phase
N =6 ∂nm + ∇ ⋅ (nmVGm ) = 0 (m = 1, K , N ) ∂t DVGm M im 1 =− ∇P − +g Dt ρGm α Gm ρGm
• This indicates that accurate prediction of VR might be required for accurate prediction of turbulence in bubby flow.
n: V: t: ρ: P: α: g: N: Mi : Mμ:
Liquid phase
∂α L + ∇ ⋅ (α LVL ) = 0 ∂t N
1 DV L = − ∇P + Dt ρL
• Multi-fluid simulation of the bubbly flows To examine effect of drag model on predicted k.
∑M m=1
im
+ Mμ
α Lρ L
number density velocity time density pressure volume fraction gravitational acceleration number of bubble size classes interfacial momentum transfer viscous diffusion
Subscripts G: gas phase L: liquid phase
+g
M μ = ∇ ⋅ α L {μ ( ∇VL + ∇VLT ) + τT }
M im = M Dm + M Lm + M VMm + M TDm
Closure relations: drag, lift, virtual mass & turbulence dispersion models
ICMF2010-Tampa 2010.6.1.
Turbulence Model M μ = ∇ ⋅ α L {μ ( ∇VL + ∇VLT ) + τT } Reynolds stress: τT = τ SI +
∑τ
Drag BIm
m=1
Shear-induced turbulence
(
τ SI = ρ L νT ∇VL + ∇VL
T
)
2 + ρ L ASI k SI 3
Bubble-induced turbulence
τ BIm
k BIm =
Dk SI ν αL = ∇ ⋅ α L T ∇k SI + α L ( pk − ε) Dt σk αL νT
N
m =1 N
k
= k SI + ∑ k BIm = m =1
Cμ
k SI2 ε
k SI
+ +
Shear-induced turbulence
N
∑C m =1 N
μb
Bubble-induced turbulence
~ 1 α GmCVM VGm − VL 2
0 0 ⎤ ⎡4 / 10 ABIm = ⎢⎢ 0 3 / 10 0 ⎥⎥ ⎢⎣ 0 0 3 / 10⎥⎦ 2
σ k = 1.0 σ ε = 1.3 C ε1 = 1.44 C ε 2 = 1.92
6α Gm dm
2l CDSm = CDm α 3− L
(
)
Tomiyama (1995)
ASI = − I
d m α Gm VGm − VL
1 ~ α GmCVM VGm − VL ∑ m =1 2
1 a INTmC DSmρ L VGm − VL (VGm − VL ) 8
2
Effect of bubble aspect ratio 8 C Dm = CD=f(E, Eo) Tomiyama (2002)
Eo F −2 3 E 2 / 3 (1 − E 2 ) −1 Eo + 16 E 4 / 3
Coefficients
⎛ ν εp ε2 ⎞ Dε ⎟ = ∇ ⋅ α L T ∇ε + α L ⎜⎜ Cε1 k − Cε 2 σε Dt k SI k SI ⎟⎠ ⎝
= ν SI + ∑ ν BIm =
M Dm =
aINTm =
Free rising single bubble C = max ⎡min ⎧ 16 1 + 0.15 Re 0.687 , 48 ⎫, 8 Eo ⎤ ⎨ ⎬ Dm ⎢ ⎥ CD=f(Re, Eo) Re ⎭ 3 Eo + 4 ⎦ ⎩ Re ⎣
2 = ρ L ABIm k BIm 3
Standard k-ε model
ICMF2010-Tampa 2010.6.1.
Drag Model M im = M Dm + M Lm + M VMm + M TDm
Lopez de Bertodano (1994)
N
Turbulence model
Cμ = 0.09 Cμb = 0.6 ~ CVM = 2.0
F=
sin −1 1 − E 2 − E 1 − E 2 1− E2
8 3 E 2 / 3 (1 − E 2 ) −1 Eo + 16 E 4 / 3
−2
Case 1: <JL>=0.5 m/s, <JG>=0.018 m/s 0.4
0.4
d=2.50 - 2.75 mm 0.3
0.4
VR [m/s]
0.1
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 d=3.00 E - 3.25 mm
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 d=3.25 E - 3.50 mm
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 3.25 mm d=3.00
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 3.50 mm d=3.25 VR [m/s]
VR [m/s]
VR [m/s]
VR [m/s]
0.1
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 3.75 mm d=3.50
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 4.00 mm d=3.75
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 3.75 mm d=3.50
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 4.00 mm d=3.75
0.3
0.3
0.3
0.3
0.1 0 0.4 0.5
0.2
E
)
,
ΔV L dω Sr = = VR VR
VL
CD/CD0
2
CD/CD0=1+0.55Sr2
1
: Experimental Data
ΔVL
VR[m/s]
F=
⎡ 48 ⎤ 8 Eo ⎤ ⎡ 16 1 + 0.15Re 0.687 , , C D = max ⎢min ⎢ ⎥ Re ⎥⎦ 3 Eo + 4 ⎦ ⎣ Re ⎣
(
Case1
CD =
0.2
F=
0.1
)
sin −1 1 − E 2 − E 1 − E 2 1− E2
0.3
F=
0.1
Legendre & Magnaudet’ Magnaudet’s equation is applicable to turbulent bubbly flows.
JL=1.0m/s JG=0.036m/s
00
0.2 0.4 0.6 0.8 r/R
)
1− E − E 1− E 2 1− E 2 2
C D = C D 0 1 + 0.55S r
VR[m/s]
100
−1
(
Case2
CD =
10-1 Sr
(
sin
0
)
8 Eo F −2 3 E 2 / 3 1 − E 2 Eo + 16 E 4 / 3
JL=0.5m/s JG=0.018m/s
0.2
0 -2 10
ICMF2010-Tampa 2010.6.1.
Relative Velocity
8 Eo F −2 3 E 2 / 3 1 − E 2 Eo + 16 E 4 / 3
(
dω VR
ω: magnitude of liquid velocity gradient d: bubble diameter VR: relative velocity
0.3
CD0 =
sin −1 1 − E 2 − E 1 − E 2 1− E2
Sr =
0.7 0.8 0.9 1 E
3 Case1(JL=0.5m/s, JG=0.018m/s) Case2(JL=0.5m/s, JG=0.025m/s) Case3(JL=1.0m/s, JG=0.020m/s) Case4(JL=1.0m/s, JG=0.036m/s)
F=
Legendre & Magnaudet (2002)
VR
Legendre & Magnaudet (2002)
(
Eo F −2 (1 − E ) Eo + 16 E 4 / 3 2 −1
Single bubble in shear flow Effect of liquid velocity gradient
ICMF2010-Tampa 2010.6.1.
Effect of shear on CD C D = C D 0 1 + 0.55Sr
2/3
0.2
E
2
)
C Dm = C D 0 m (1 + 0.55Sr 2 )
0.1 0.1 VR & CD depend0.1on E. Tomiyama’s C model is applicable to turbulent bubbly pipe 0D 0 0 flows. 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 E
3E
VR [m/s]
VR [m/s]
VR [m/s]
0.1
VR [m/s]
0.1
0.2
Effect of bubble aspect ratio 8 C Dm = CD=f(E, Eo) Tomiyama (2002)
0.1
0.2
2l CDSm = CDm α 3− L
0.2
0.2
6α Gm dm
Tomiyama (1995)
0.3
0.3
1 a INTmC DSmρ L VGm − VL (VGm − VL ) 8
(
0.1
0.2
aINTm =
Free rising single bubble C = max ⎡min ⎧ 16 1 + 0.15 Re 0.687 , 48 ⎫, 8 Eo ⎤ ⎨ ⎬ Dm ⎢ ⎥ CD=f(Re, Eo) Re ⎭ 3 Eo + 4 ⎦ ⎩ Re ⎣
0.2
0.1
0.3
M Dm =
0.3
0.1
0.2
Drag
d=2.75 - 3.00 mm
0.2
0.2
0.3
M im = M Dm + M Lm + M VMm + M TDm
⎦
0.4
d=2.50 - 2.75 mm
0.3
VR [m/s]
VR [m/s]
⎭
Case 2: <JL>=1.0 m/s, <JG>=0.020 m/s
d=2.75 - 3.00 mm 0.3
0.2
⎩
⎣
ICMF2010-Tampa 2010.6.1.
Drag Model
0.687
Dm
VR [m/s]
C Dm =
Correlation between ICMF2010-Tampa 2010.6.1. Relative Velocity and Aspect Ratio ⎡ 48 ⎫ 8 Eo ⎤ Eo ⎧ 16 ), Re ⎬, 3 Eo + 4 ⎥ C = max ⎢min ⎨ (1 + 0.15 Re F Re
1
2
)
Tomiyama et al. (2002)
Legendre & Magnaudet (2002) Sr =
ΔV L dω = VR VR
8 Eo F −2 3 E 2 / 3 1 − E 2 Eo + 16 E 4 / 3
sin
(
−1
)
1− E − E 1− E 2 1− E 2 2
ICMF2010-Tampa 2010.6.1.
Predictions Measured (a) (c)
CD(Re,E,Sr) CD(Re) Measured
Case 2
0.2 0.1
Improvement in drag model
k[m /s ]
0
k = Ψ 2 k SI
0.015 Case 1 (JL=0.5 m/s) k [m2/s2]
VR[m/s]
0.4 Case 1 0.3
ICMF2010-Tampa 2010.6.1.
Turbulence Kinetic Energy
0.01
0.005
Case2 (JL=1.0 m/s)
Measured (Single phase) Predicted (Single phase) Measured (Two-phase) Predicted (Two-phase) Lopez de Bertodano et al. Predicted (Two-phase) Hosokawa & Tomiyama
Two-phase multiplier for k
Ψ = nCTM + 1
2
0.01
0.4
r/R
0.6
0.8
0
0.2
0.4
r/R
0.6
0.8
1
3
Accurate prediction of k
2 -6
0.06 0.04 0.02 0 0
0.2
Turbulence modification per unit number density
Gas-Liquid JL=0.5m/s,JG=0.017m/s JL=0.5m/s,JG=0.023m/s JL=0.7m/s,JG=0.017m/s JL=0.7m/s,JG=0.023m/s JL=1.0m/s,JG=0.017m/s JL=1.0m/s,JG=0.023m/s JL=0.5m/s,JG=0.015m/s(D=20 mm) JL=0.9m/s,JG=0.020m/s(D=20 mm)
The turbulence model proposed by Lopez de Bertodano et al. with the accurate drag C (φ) = 2.54 ×10 −7 (φ − 1.12) model gives good prediction for turbulence modification TM caused by bubbles.
0 0.08
αG
0 0
Accurate prediction of VR
CTM [X10 ]
2
0.02
0.4 r/R
0.8
0.4 r/R
Turbulence modification in bubbly flows with small dB/lt and large VR/u’ in a square duct. dB ~ the Kolmogolv scale lK, VR ~ the bulk velocity JL ~ 10 X u’
φ ~ 0.5
Dk = P−ε+ D Dt ∂U i ∂u ′ ∂u ′ P = −ui′u ′j ε=ν i i ∂x j ∂x j ∂x j Measurement of velocity gradients and velocity components with the spatial resolution higher than the Kolmogorov scale. Molecular tagging velocimetry based on photobleaching reaction (PB-MTV)
N
This model has a potential to predict turbulence 0 modification in turbulent 0 2 4 6 bubbly flows.
0.8
∑ (n m =1
m
ν Gm )
nν SI
=
νT of B.I.T. νT of S.I.T.
φ
ICMF2010-Tampa 2010.6.1.
Measurement of TKE At large dB/lt (1) turbulence augmentation at large VR/u’. (2) turbulence attenuation at small φ. At small φ and large VR/u' ?
Eddy viscosity ratio
φ=
An accurate drag model is necessary for accurate predictions of turbulence in bubbly flows.
0
1
Liquid-Solid(JL=0.5m/s) d=4.0mm JS=0.002m/s d=4.0mm JS=0.004m/s d=2.5mm JS=0.002m/s d=2.5mm JS=0.004m/s d=1.0mm JS=0.002m/s d=1.0mm JS=0.004m/s
dB/lt
Large φ Turbulence augmentation
Turbulence attenuation
Millimeter size bubbles C on st an tφ Small φ Turbulence attenuation
VR/u’
Photobleaching: irriversible change from a fluorescent dye to non-fluorescent dye Fluorescent dye: uranine (Fluoresceine sodium salt) Concentration: 10-6 mol/m3 Mask
?
ICMF2010-Tampa 2010.6.1.
Photobleaching MTV
Laser beam
1.0mm
Open
Camera Close ON
Laser sheet λ=514.5nm
OFF ON
Intense laser λ=488 nm
δt ~5ms Tir ~ 1ms
T=t+δt
OFF
Time 1.0mm
Lens
Low power laser sheet (λ=514.5nm): Visualization of tags
δl
Intense laser beam (λ=488.0nm): Photobleaching reactiion
V= Tag Shutter
CCD camera
δl δt
T=t
ICMF2010-Tampa 2010.6.1.
Measurement of Velocity Gradient
ICMF2010-Tampa 2010.6.1.
Experimental Apparatus
W δR: Difference of displacement vectors of two points v: Velocity v0: Velocity at center of tagged region δr: Position vector from the center δt: Time interval
v0 δt
δr
Original tagged region
∂v ⎛ ⎞ δR = vδt = ⎜ v 0 + ⋅ δr ⎟δt ∂r ⎝ ⎠
Laser Transformed tagged region
t=0
Laser
x,u
Measurement position CCD camera
t=δt
Cross-correlation
∂v / ∂r is constant in tagged region
Pump
Intense laser beam for photobleaching
z, w
z, w y, v
Laser sheet
CCD camera
Laser sheet
Exposure time：1 ～5 ms
Kolmogorov scale 1/ 4
aij+ε
aij
Flowmeter
ICMF2010-Tampa 2010.6.1.
⎛ ν3 ⎞ l k ≈ ⎜⎜ ⎟⎟ ⎝ ε ⎠
Lower tank
Axial mean velocity and Void Fraction 0.1
Tag size: 40 – 100 μm Sample number: 200
y, v CCD camera Intense laser beam for photobleaching
Tag size：40～100 μm
Sample number： 200
Correct value
Experimental Conditions x-z plane
Tank
Air
aij-ε
x-y plane
Reynolds number：5000
Water
50 mm
0.08
W: duct width (50 mm)
0.1 0.06 0.04
JG [m/s] U 0.0x10-5 2.5x10-5 5.0x10-5 7.5x10-5
α
0 0
lt ~ 0.2xW ~10 mm
The void fraction exhibits wall-peaking profile.
JG [x10-5m/s]
d32 [mm]
Case 0
0.06
0.0
-
Case 1
0.06
2.5
0.60
Large VR/u’ ~8,
Case 2
0.06
5.0
0.59
Case 3
0.06
7.5
0.59
Small dB/lt ~ 0.06
φ = VRdB/ltu’ ~ 0.5
0.1
0.2
0.3
0.4
U: axial mean velocity
α: void fraction
0.05
lK ~ 0.2 mm ~ dB
JL [m/s]
ICMF2010-Tampa 2010.6.1.
y: distance from the wall
0.02 VR ~ 0.05 m/s ~ JL ~ 10xu’
≈ 200μm
0.15
α [%]
a12 ⎤ ⎡δx ⎤ ⎞ ⎟δt a22 ⎥⎦ ⎢⎣δy ⎥⎦ ⎟⎠
Experimental condition
Compressor Cross-correlation
⎛ ⎡u ⎤ ⎡ a δR = ⎜⎜ ⎢ o ⎥ + ⎢ 11 ⎝ ⎣ vo ⎦ ⎣a21
50 mm
U [m/s]
2D flow
Flow
z,w y,v
1000 mm
δR
Upper tank
0 0.5
y/W The gradients of the axial velocity in the bubbly flows are larger than that in the single-phase flow in the near wall region. The axial velocity distribution in the core region is flattened as many researches reported.
0.1
0.2
y/W
0.3
0.4
0.4
0.3 y/W
102
y+
εν/uτ4
α [%]
Pν/uτ4
0.2
0
-0.2
101
y
+
102
101
y+
102
0
∂U ∂x j The dimensionless andDε does are strongly The dimensionless not change ⎛ P, Pε and ∂k ⎞⎟ so much. ∂ ⎜ 1 ′ ′ ′ attenuated. ′ ′ − u j ui ui − p u j + ν D= ⎜ ⎟ ∂xenhance ∂xand 2 the frictional The bubbles velocity j ⎝ structure j ⎠ The turbulence is different from
0.1
shear-induced turbulence. shear-induced due to the ∂u ′ ∂turbulence ui′ The bubble-induced pseudo turbulence is by not breakup eddies ε = νof thei shear-induced prominent. ∂x j ∂x j bubbles.
0
Fully-developed Turbulence structure doesflow not change so much Symmetry due to bubbles. with respect to y axis
-0.4
2
α [%]
2 2
-4
α [%]
0.2
0.1
0.2
y/W
0.3
Reynolds shear stress
0.3
0.4
JG [m/s] u'v' α -5 0.0x10 -5 2.5x10 -5 5.0x10 -5 7.5x10
0 0.5
0.3
0.2
0.2
0.1 0.1
φ=10
1 Circular pipe (Attenuation)
Circular pipe (Enhancement)
Neglect p’u’ in D
ICMF2010-Tampa 2010.6.1. Large φ: Turbulence augmentation due to bubble-induced turbulence Small φ: Turbulence attenuation occurs Small VR/u’: Agitated flow due to bubbles is weak Æ Eddy breakup by bubbles ÆTurbulence attenuation
0.5
i In thePcore ′ ′ =the −uregion, Between wall i u j and void peak position,
α [%]
0.2 Dν/uτ4
0.1
0 0 10
0 0.3
JG [m/s]-5 Dν/uτ4 α 0.0x10 2.5x10-5 5.0x10-5 7.5x10-5
0.4
100
0.6
0.2
0.2 101
0.1
0.4
core
0.2
0.2
2
φ=1.0
0.4
0.4
0 0 10
0.3
JG [m/s] w' α 0.0x10-5 in the attenuated 2.5x10-5 -5 5.0x10 7.5x10-5
α
0.4
Conclusions
0.2
0.8
0.1
Spanwise component
0.6
0 0
0 0.5
0.4
Although u’v’ is enhanced in the near 0.4 wall region, the increase is not so 0.1 large. 0.2
JG [m/s] εν/uτ4 α 0.0x10-5 2.5x10-5 5.0x10-5 7.5x10-5
4
0.6
0.3 y/W
ICMF2010-Tampa 2010.6.1.
1
0.2 JG [m/s]-5 Pν/uτ α 0.0x10-5 2.5x10 5.0x10-5 7.5x10-5
0.8
0.2
JG [m/s] v'2 0.0x10-5 2.5x10-5 5.0x10-5 7.5x10-5
at small0 φ, whereas VR/u’ is large. 0 0Turbulence attenuation occurs 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 pipe 0.4flows. 0.5 These trends are y/W y/Wthe same as those in the case 2 in bubbly
Dimensionless P, D and ε 1
0.1
u’v’ is strongly 0.6 region.
0 0.5
0.4
v' [x10 m /s ]
2
The turbulence intensities are attenuated in the core region and enhanced in the near wall region due to bubbles.
α [%]
0.2
2
0.8
0.2
0.1
-4
0 0
0.1
0 0
2
0 0.5
0.3
Wall-normal component
α [%]
0.2 α [%]
w'2 [x10-4 m2/s2]
α
0.1 1
0.3 2
JG [m/s] w' 0.0x10-5 2.5x10-5 -5 5.0x10 7.5x10-5
0.6
0 0
0 0.5
0.4
0.2
u'v' [x10-4 m2/s2]
Spanwise component
0.1 0.2
ICMF2010-Tampa 2010.6.1.
α
α [%]
0.3 y/W
2
2
w'2 [x10-4 m2/s2]
0.2
0.4
JG [m/s]-5 u' 0.0x10 2.5x10-5 5.0x10-5 7.5x10-5
3
0.8
0.3
Streamwise component
dB/lt
0.8
0.1
0.2 α [%]
-4
1
4
α
u' [x10 m /s ]
0.1
0.6
2
2
2
2
0 0
JG [m/s] v'2 0.0x10-5 2.5x10-5 5.0x10-5 7.5x10-5
2
-4
α [%]
0.2
0.3
Wall-normal component
α
2
v' [x10 m /s ]
JG [m/s]-5 u' 0.0x10 2.5x10-5 5.0x10-5 7.5x10-5
3
0.8
0.3
Streamwise component
2
2
u' [x10 m /s ]
4
Turbulence Intensity and Reynolds Shear Stress
ICMF2010-Tampa 2010.6.1.
Turbulence Intensity
φ=0.1
Rich experimental data Small dB/lt: B.I.T. quickly dissipates due to its small scale Æ Eddy breakup by bubbles ÆTurbulence attenuation
0.1 0.05 0.5
Square duct (Attenuation)
1
VR/u'
5
10
Further experimental data is required to understand more detailed turbulence structure in bubbly flows.
ICMF2010-Tampa 2010.6.1.
Purpose
Turbulence structure in bubbly flows is very complex due to (1) bubble-induced pseudo turbulence, (2) interaction between bubble-induced and shear–induced turbulences, (3) direct interaction between bubbles and turbulence eddies and (4) modification of shear-induced turbulence due to the modulation of mean velocity distribution due to bubbles.
Effects of Bubbles on Turbulent Flows in Vertical Channels
Bubbly upflows in vertical channnels
dB/lt
1. Index of turbulence modification The eddy viscosity ratio φ ( = dBVR/ltu’ )
Shigeo Hosokawa and Akio Tomiyama
2. Turbulence modification at large dB/lt Experiments & numerical predictions
Kobe University
3. Turbulence modification at small dB/lt and large VR/u’ Turbulence kinetic energy budget
Large φ Turbulence augmentation
Large dB/lt Effect of VR/u’ on turbulence modification VR/u’ Small dB/lt Large VR/u’
Small φ Turbulence attenuation
Turbulence modification due to dispersed phase
αL
Large particle/bubble: Turbulence augmentation
2
Small particle/bubble: Turbulence attenuation Gore and Crowe Critical parameter: d/lt ⇒Ratio of mixing length of particle-induced turbulence to that of shear-induced turbulence
1
ΔTI
•
νTP = f (ν t , ν B , K) φ=
0
νB νt
νt : Shear-induced eddy viscosity νB : Bubble-induced eddy viscosity VR : Relative velocity dB : Diameter of dispersed phase u’ : Turbulence velocity lt : Turbulence length scale
ν B ∝ VR d B
S.I.T.
ν t ∝ u ′lt
-1
-2
ICMF2010-Tampa 2010.6.1.
M α DVL = − L ∇P + ∇α L (ν + νTP )(∇VL + ∇VLT ) + i + α L g ρ ρ Dt
Present data (Gas-Liquid) Present data (Liquid-Solid) Hosokawa et al.(1988)(Gas-Solid) Tsuji et al. (1984) Maeda et al. (1980) Kulick et al. (1994) Lee and Durst (1982)
VR
u’ dB
Turbulence attenuation
Ratio of eddy viscosity of particle-induced turbulence to that of shear-induced turbulence
Eddy Viscosity Ratio φ
ICMF2010-Tampa 2010.6.1.
10-2
Turbulence augmentation
10-1 d/lt
100
Eddy viscosity ratio
φ≡
lt B.I.T.
VR d B VR d B ReB = = u ′lt u ′ lt Ret
ReB =
u ′lt VR d , Ret = ν ν
Indicator of Turbulence Modification
ICMF2010-Tampa 2010.6.1.
ICMF2010-Tampa 2010.6.1.
φ－CTI/Nd 3
Pipe diameter: D = 20, 30 mm Liquid volumetric flux: JL = 0.5 -1.0 m/s Volume fraction of dispersed phase: α = 0.7 - 5.0 % Bubble/particle diameter d = 1 -5 mm
2
2
Solid-Liquid(JL=0.5m/s) d=4.0mm,Js=0.002m/s d=4.0mm,Js=0.004m/s d=2.5mm,Js=0.002m/s d=2.5mm,Js=0.004m/s d=1.0mm,Js=0.002m/s d=1.0mm,Js=0.004m/s
-6
-6 -6 CTI/Nds [X10 ΔTI/N [X10 ]]
3
Gas-Liquid JL=0.51m/s,JG=0.017m/s JL=0.51m/s,JG=0.023m/s JL=0.71m/s,JG=0.017m/s JL=0.71m/s,JG=0.023m/s JL=1.01m/s,JG=0.017m/s JL=1.01m/s,JG=0.023m/s JL=0.50m/s,JG=0.015m/s (D=20 mm) JL=0.90m/s,JG=0.020m/s (D=20 mm)
CTI/Nd [X10 ]
LDV Measurements Gas-Liquid bubbly flow (Air-Water) Solid-Liquid two-phase flow (Ceramic-Water) in vertical circular pipe
1
Gas-Liquid JL=0.5m/s,JG=0.017m/s(D=30 JL=0.5m/s,JG=0.023m/s(D=30 JL=0.7m/s,JG=0.017m/s(D=30 JL=0.7m/s,JG=0.023m/s(D=30 JL=1.0m/s,JG=0.017m/s(D=30 JL=1.0m/s,JG=0.023m/s(D=30 JL=0.5m/s,JG=0.015m/s(D=20 JL=0.9m/s,JG=0.020m/s(D=20
mm) mm) mm) mm) mm) mm) mm) mm)
Liquid-Solid(JL=0.5m/s, D=30mm) d=4.0mm JS=0.002m/s d=4.0mm JS=0.004m/s d=2.5mm JS=0.002m/s d=2.5mm JS=0.004m/s d=1.0mm JS=0.002m/s d=1.0mm JS=0.004m/s
1
Dimensionless change in turbulence intensity per unit number density
′ u′ uTP − SP U TP U SP 2
CTI = Nd
u ′SP U SP
φ=1 0
0
2
u’ : Axial fluctuation velocity U : Axial mean velocity Nd : Number density of dispersed phase Subscript TP : Two-phase flow SP : Single phase flow
Nd
2
Correlation between φ and CTI/Nd
0.5
1 d/lt
Gore & Crowe’s parameter: d/lt
0
2
φ
4
4
6
Eddy viscosity ratio: φ=
νB νt
φ is applicable to gas-liquid and liquid-solid flows. Critical point is close to φ=1, irrespective of a type of two-phase flows.
ICMF2010-Tampa 2010.6.1.
Gas-liquid-solid three phase flow (water+air bubble+aluminum ceramic particle)
Gas-solid two-phase flow
0
ICMF2010-Tampa 2010.6.1.
Turbulence modification depends on φ ( = dBVR/ltu’), that is, it depends not only on dB/lt but also on VR/u’.
CTI/Nd [X10-4]
CTI/Nd [X10-8]
Shakutsui et al. (2001)
2
Effect of VR/u’ on turbulence properties in bubbly pipe flows at large dB/lt ( ~ 1). D=30 mm JL=0.5-1.0 m/s JG=0-0.023 m/s JS=0-0.0038 m/s dB≅2.0-8.0 mm
Air-water bubbly flow dB ~ constant ÆVR ~ constant D ~ constant Æ lt ~ constant JL changes Æ u’ changes Case 2
0
-20
ρd=980 kg/m33 (Polystyrene), m=2.3 ρd=980 kg/m 3(Polystyrene), m=4.6 ρd=2500 kg/m (Glass), m=4.6 ρd=2500 kg/m33 (Glass), m=6.0 ρd=3600 kg/m3 (Ceramic), m=4.6 ρd=3600 kg/m (Ceramic), m=6.9
1
2
φ
3
4
2.5
φ=
5
N PVRP dφP + N BVRB d B (N P + N B )u′lt
7.5
φ is applicable not only to gas-liquid and liquid-solid two-phase flows but also to gas-solid two-phase flows and three phase flows.
Turbulence augmentation dB/lt Large φ Millimeter size bubbles
Case 1 VR/u’
Case 1: turbulence augmentation low JL Æ large VR/u’ Small φ
Case 2: turbulence attenuation high JL Æ small VR/u’ Turbulence attenuation
ICMF2010-Tampa 2010.6.1.
Experimental Apparatus Length of pipe L = 2 m Diameter of pipe D = 25 mm Elevation of measurement point 1.7 m
FEP (Fluorinated Ethylene-Propylene resin) Refractive index: 1.338 cf. water: 1.333 Negligible optical distortion
CCD Camera
Reflection light
D=25mm
LDV
1.7m
2.0m
Camera
Reflection of light from the side reduces contrast of interface image.
Optical filter (Green transmit)
Standard light without optical filter
Optical filter (Red transmit)
B
A
Mixing section
LDV: Liquid velocity Image processing method: Void fraction, Bubble velocity, Bubble diameter, Aspect ratio
Flow
FEP
Water
Contrast increase.
Reflection light
LED (Red) 25mm
LED (Green)
Two color LED with optical filter
FEP pipe
41mm
sor Compres
Original image Binary image Elliptic disk with a pixel height
Acrylic duct
ai
Water
Fluids Liquid: water Gas: air
r Ai Drain
Tank
[Pixel]
ICMF2010-Tampa 2010.6.1.
One pixel
F=0 20
F=1
B 0 0
Curvature takes negative peaks at overlapped points A and B.
ICMF2010-Tampa 2010.6.1.
<JG>= 0.018m/s <JL>= 0.5m/s
A
40
Binarize
Void distribution Bubble diameter etc…
Condition
60
The curvature evaluation method was applied to overlapped bubble image.
∇F ∇F
bi
Binary image 1 Binary image 2
Detection and Separation of Overlapped Bubbles
n = −
ai
Pile up
bi
i
Pump
20
Optical setup 40
60 [Pixel]
0.1 κ[1/m]
κ = ∇ ⋅n
ICMF2010-Tampa 2010.6.1.
Image Processing Method
Camera B Camera A 0
κ: Interface curvature n : Unit normal to interface -0.1 F : Color function representing spatial distribution of interface 0
B
A 50
100
Pixel number
ICMF2010-Tampa 2010.6.1.
Mean Velocity Distributions
0.05
HIgh JLÆ high u’ Low VR/u’, φ
Case 1 JL = 0.5 m/s JG = 0.018 m/s
W,w’
Case 2 JL=1.0 m/s JG=0.036 m/s
1
Single phase flow U/JL V/JL W/JL Bubbly flow U/JL V/JL W/JL αG
0.5
0.2 0.4 0.6 0.8
1 0
0.1
The axial mean velocities were flattened due to the presence of bubbles. The radial and tangential mean velocities were zero.
0.2 0.4 0.6 0.8
r/R
U: Axial mean velocity V: Radial mean velocity W: Tangential mean velocity JL: Liquid volumetric flux JG: Gas volumetric flux αG: Void fraction R: Pipe radius
0 1
Æ fully-developed flow
Case1: Large VR/u’ Turbulence augmentation
Case2: Small VR/u’ Turbulence attenuation
0.5 1.5
0.5 Single phase flow
0.5
Bubbly flow 0.2
0.4
Time (s)
0.2 - 0.3m/s = VR
VR Bubble
Bubbly flow
Velocity data
(JL =0.93m/s,J G=0.018m/s,r/R=0.7)
U(m/s)
(JL =0.63m/s,J G=0.017m/s,r/R=0.7)
Phase distribution function (High: gas, Low: liquid)
Single-phase flow
(JL =0.93m/s,r/R=0.7)
1.5
Velocity data
Bubbly flow
0 0
Case 1 u’u’, v’v’ and w’w' are enhanced over the crosssection.
0
B.I.T. augments T.I. whereas it does not enhance u’v’ so much.
Single phase flow u'v'/JL2 2 u'w'/JL Bubbly flow 2 u'v'/JL 2 u'w'/JL
0.004 0.003
Case 2 Turbulence intensities are attenuated in core region, and it is enhanced in the near wall region.
0.002 0.001 0 0
0.2
0.4 0.6 r/R
0.8
0
0.2
0.4 0.6 r/R
Phase di stribution function (High: gas, Low: liquid)
Bubbly flow
10
4
103
Case 1: Turbulence enhancement
Case 1 -5/3
E(ω) was enhanced
JL=0.63 m/s r/R = 0.7
105
Single phase flow
Case 2: Turbulence suppression High frequency eddy was enhanced Low frequency eddy was attenuated
Bubbly flow
Bubbly flow
0 0
0.2
Time (s)
ICMF2010-Tampa 2010.6.1.
105
1 0.5
0.8
u’v’ is attenuated due to the presence of bubbles except in 1 the near wall region.
Turbulence Energy Spectra
1
Single-phase flow
W,w’ V,v’
u’v’ is enhanced in the near wall region, whereas it in the core region changes not so much.
E(ω)
(JL =0.63m/s,r/R=0.7)
V,v’
VR/u’, φ
0.01
Single phase flow Velocity data
Velocity data
U(m /s)
U(m/s)
Single phase flow
U(m/s)
0.02
1.5
1.5
1
0.03
ICMF2010-Tampa 2010.6.1.
Fluctuating Axial Velocity
1
Single phase2 flow u'u'/JL Low 2 v'v'/JL 2 w'w'/JL Bubbly flow 2 u'u'/JL2 v'v'/JL w'w'/JL2
High VR/u’, φ
0.04
r/R
Void fraction exhibits wallpeaking profile in case 2.
U,u’ W,w’
Case 1 Case 2 (JL=0.5 m/s, JG=0.018 m/s) (JL=1.0 m/s, JG=0.036 m/s)
102
Case 2
-5/3
104
0.4
E(ω)
0 0
V,v’
0.2
αG
U/JL, V/JL, W/JL
1.5
V,v’
u'u'/JL2, v'v'/JL2, w'w'/JL2
Low JL Æ low u’ High VR/u’, φ
Turbulence augmentation Turbulence attenuation
U,u’ W,w’
Case 2
u'v'/JL2, u'w'/JL2
Case 1
ICMF2010-Tampa 2010.6.1.
Reynolds Stress
JL=0.93 m/s r/R=0.7
u’ > VR
Breakup of large scale eddy due to bubbles
103
Turbulence attenuation
Low frequency fluctuation becomes weak
B.I.T 10-1
100
101 ω (Hz)
102
102 103
Numerical Prediction of Bubbly Flows
ICMF2010-Tampa 2010.6.1.
ICMF2010-Tampa 2010.6.1.
Field Equations MultiMulti-fluid model (Tomiyama (Tomiyama et al. 1998, 2005)
• Turbulence modification depends on VR/u’.
Gas phase
N =6 ∂nm + ∇ ⋅ (nmVGm ) = 0 (m = 1, K , N ) ∂t DVGm M im 1 =− ∇P − +g Dt ρGm α Gm ρGm
• This indicates that accurate prediction of VR might be required for accurate prediction of turbulence in bubby flow.
n: V: t: ρ: P: α: g: N: Mi : Mμ:
Liquid phase
∂α L + ∇ ⋅ (α LVL ) = 0 ∂t N
1 DV L = − ∇P + Dt ρL
• Multi-fluid simulation of the bubbly flows To examine effect of drag model on predicted k.
∑M m=1
im
+ Mμ
α Lρ L
number density velocity time density pressure volume fraction gravitational acceleration number of bubble size classes interfacial momentum transfer viscous diffusion
Subscripts G: gas phase L: liquid phase
+g
M μ = ∇ ⋅ α L {μ ( ∇VL + ∇VLT ) + τT }
M im = M Dm + M Lm + M VMm + M TDm
Closure relations: drag, lift, virtual mass & turbulence dispersion models
ICMF2010-Tampa 2010.6.1.
Turbulence Model M μ = ∇ ⋅ α L {μ ( ∇VL + ∇VLT ) + τT } Reynolds stress: τT = τ SI +
∑τ
Drag BIm
m=1
Shear-induced turbulence
(
τ SI = ρ L νT ∇VL + ∇VL
T
)
2 + ρ L ASI k SI 3
Bubble-induced turbulence
τ BIm
k BIm =
Dk SI ν αL = ∇ ⋅ α L T ∇k SI + α L ( pk − ε) Dt σk αL νT
N
m =1 N
k
= k SI + ∑ k BIm = m =1
Cμ
k SI2 ε
k SI
+ +
Shear-induced turbulence
N
∑C m =1 N
μb
Bubble-induced turbulence
~ 1 α GmCVM VGm − VL 2
0 0 ⎤ ⎡4 / 10 ABIm = ⎢⎢ 0 3 / 10 0 ⎥⎥ ⎢⎣ 0 0 3 / 10⎥⎦ 2
σ k = 1.0 σ ε = 1.3 C ε1 = 1.44 C ε 2 = 1.92
6α Gm dm
2l CDSm = CDm α 3− L
(
)
Tomiyama (1995)
ASI = − I
d m α Gm VGm − VL
1 ~ α GmCVM VGm − VL ∑ m =1 2
1 a INTmC DSmρ L VGm − VL (VGm − VL ) 8
2
Effect of bubble aspect ratio 8 C Dm = CD=f(E, Eo) Tomiyama (2002)
Eo F −2 3 E 2 / 3 (1 − E 2 ) −1 Eo + 16 E 4 / 3
Coefficients
⎛ ν εp ε2 ⎞ Dε ⎟ = ∇ ⋅ α L T ∇ε + α L ⎜⎜ Cε1 k − Cε 2 σε Dt k SI k SI ⎟⎠ ⎝
= ν SI + ∑ ν BIm =
M Dm =
aINTm =
Free rising single bubble C = max ⎡min ⎧ 16 1 + 0.15 Re 0.687 , 48 ⎫, 8 Eo ⎤ ⎨ ⎬ Dm ⎢ ⎥ CD=f(Re, Eo) Re ⎭ 3 Eo + 4 ⎦ ⎩ Re ⎣
2 = ρ L ABIm k BIm 3
Standard k-ε model
ICMF2010-Tampa 2010.6.1.
Drag Model M im = M Dm + M Lm + M VMm + M TDm
Lopez de Bertodano (1994)
N
Turbulence model
Cμ = 0.09 Cμb = 0.6 ~ CVM = 2.0
F=
sin −1 1 − E 2 − E 1 − E 2 1− E2
8 3 E 2 / 3 (1 − E 2 ) −1 Eo + 16 E 4 / 3
−2
Case 1: <JL>=0.5 m/s, <JG>=0.018 m/s 0.4
0.4
d=2.50 - 2.75 mm 0.3
0.4
VR [m/s]
0.1
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 d=3.00 E - 3.25 mm
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 d=3.25 E - 3.50 mm
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 3.25 mm d=3.00
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 3.50 mm d=3.25 VR [m/s]
VR [m/s]
VR [m/s]
VR [m/s]
0.1
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 3.75 mm d=3.50
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 4.00 mm d=3.75
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 3.75 mm d=3.50
0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 E - 4.00 mm d=3.75
0.3
0.3
0.3
0.3
0.1 0 0.4 0.5
0.2
E
)
,
ΔV L dω Sr = = VR VR
VL
CD/CD0
2
CD/CD0=1+0.55Sr2
1
: Experimental Data
ΔVL
VR[m/s]
F=
⎡ 48 ⎤ 8 Eo ⎤ ⎡ 16 1 + 0.15Re 0.687 , , C D = max ⎢min ⎢ ⎥ Re ⎥⎦ 3 Eo + 4 ⎦ ⎣ Re ⎣
(
Case1
CD =
0.2
F=
0.1
)
sin −1 1 − E 2 − E 1 − E 2 1− E2
0.3
F=
0.1
Legendre & Magnaudet’ Magnaudet’s equation is applicable to turbulent bubbly flows.
JL=1.0m/s JG=0.036m/s
00
0.2 0.4 0.6 0.8 r/R
)
1− E − E 1− E 2 1− E 2 2
C D = C D 0 1 + 0.55S r
VR[m/s]
100
−1
(
Case2
CD =
10-1 Sr
(
sin
0
)
8 Eo F −2 3 E 2 / 3 1 − E 2 Eo + 16 E 4 / 3
JL=0.5m/s JG=0.018m/s
0.2
0 -2 10
ICMF2010-Tampa 2010.6.1.
Relative Velocity
8 Eo F −2 3 E 2 / 3 1 − E 2 Eo + 16 E 4 / 3
(
dω VR
ω: magnitude of liquid velocity gradient d: bubble diameter VR: relative velocity
0.3
CD0 =
sin −1 1 − E 2 − E 1 − E 2 1− E2
Sr =
0.7 0.8 0.9 1 E
3 Case1(JL=0.5m/s, JG=0.018m/s) Case2(JL=0.5m/s, JG=0.025m/s) Case3(JL=1.0m/s, JG=0.020m/s) Case4(JL=1.0m/s, JG=0.036m/s)
F=
Legendre & Magnaudet (2002)
VR
Legendre & Magnaudet (2002)
(
Eo F −2 (1 − E ) Eo + 16 E 4 / 3 2 −1
Single bubble in shear flow Effect of liquid velocity gradient
ICMF2010-Tampa 2010.6.1.
Effect of shear on CD C D = C D 0 1 + 0.55Sr
2/3
0.2
E
2
)
C Dm = C D 0 m (1 + 0.55Sr 2 )
0.1 0.1 VR & CD depend0.1on E. Tomiyama’s C model is applicable to turbulent bubbly pipe 0D 0 0 flows. 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 E
3E
VR [m/s]
VR [m/s]
VR [m/s]
0.1
VR [m/s]
0.1
0.2
Effect of bubble aspect ratio 8 C Dm = CD=f(E, Eo) Tomiyama (2002)
0.1
0.2
2l CDSm = CDm α 3− L
0.2
0.2
6α Gm dm
Tomiyama (1995)
0.3
0.3
1 a INTmC DSmρ L VGm − VL (VGm − VL ) 8
(
0.1
0.2
aINTm =
Free rising single bubble C = max ⎡min ⎧ 16 1 + 0.15 Re 0.687 , 48 ⎫, 8 Eo ⎤ ⎨ ⎬ Dm ⎢ ⎥ CD=f(Re, Eo) Re ⎭ 3 Eo + 4 ⎦ ⎩ Re ⎣
0.2
0.1
0.3
M Dm =
0.3
0.1
0.2
Drag
d=2.75 - 3.00 mm
0.2
0.2
0.3
M im = M Dm + M Lm + M VMm + M TDm
⎦
0.4
d=2.50 - 2.75 mm
0.3
VR [m/s]
VR [m/s]
⎭
Case 2: <JL>=1.0 m/s, <JG>=0.020 m/s
d=2.75 - 3.00 mm 0.3
0.2
⎩
⎣
ICMF2010-Tampa 2010.6.1.
Drag Model
0.687
Dm
VR [m/s]
C Dm =
Correlation between ICMF2010-Tampa 2010.6.1. Relative Velocity and Aspect Ratio ⎡ 48 ⎫ 8 Eo ⎤ Eo ⎧ 16 ), Re ⎬, 3 Eo + 4 ⎥ C = max ⎢min ⎨ (1 + 0.15 Re F Re
1
2
)
Tomiyama et al. (2002)
Legendre & Magnaudet (2002) Sr =
ΔV L dω = VR VR
8 Eo F −2 3 E 2 / 3 1 − E 2 Eo + 16 E 4 / 3
sin
(
−1
)
1− E − E 1− E 2 1− E 2 2
ICMF2010-Tampa 2010.6.1.
Predictions Measured (a) (c)
CD(Re,E,Sr) CD(Re) Measured
Case 2
0.2 0.1
Improvement in drag model
k[m /s ]
0
k = Ψ 2 k SI
0.015 Case 1 (JL=0.5 m/s) k [m2/s2]
VR[m/s]
0.4 Case 1 0.3
ICMF2010-Tampa 2010.6.1.
Turbulence Kinetic Energy
0.01
0.005
Case2 (JL=1.0 m/s)
Measured (Single phase) Predicted (Single phase) Measured (Two-phase) Predicted (Two-phase) Lopez de Bertodano et al. Predicted (Two-phase) Hosokawa & Tomiyama
Two-phase multiplier for k
Ψ = nCTM + 1
2
0.01
0.4
r/R
0.6
0.8
0
0.2
0.4
r/R
0.6
0.8
1
3
Accurate prediction of k
2 -6
0.06 0.04 0.02 0 0
0.2
Turbulence modification per unit number density
Gas-Liquid JL=0.5m/s,JG=0.017m/s JL=0.5m/s,JG=0.023m/s JL=0.7m/s,JG=0.017m/s JL=0.7m/s,JG=0.023m/s JL=1.0m/s,JG=0.017m/s JL=1.0m/s,JG=0.023m/s JL=0.5m/s,JG=0.015m/s(D=20 mm) JL=0.9m/s,JG=0.020m/s(D=20 mm)
The turbulence model proposed by Lopez de Bertodano et al. with the accurate drag C (φ) = 2.54 ×10 −7 (φ − 1.12) model gives good prediction for turbulence modification TM caused by bubbles.
0 0.08
αG
0 0
Accurate prediction of VR
CTM [X10 ]
2
0.02
0.4 r/R
0.8
0.4 r/R
Turbulence modification in bubbly flows with small dB/lt and large VR/u’ in a square duct. dB ~ the Kolmogolv scale lK, VR ~ the bulk velocity JL ~ 10 X u’
φ ~ 0.5
Dk = P−ε+ D Dt ∂U i ∂u ′ ∂u ′ P = −ui′u ′j ε=ν i i ∂x j ∂x j ∂x j Measurement of velocity gradients and velocity components with the spatial resolution higher than the Kolmogorov scale. Molecular tagging velocimetry based on photobleaching reaction (PB-MTV)
N
This model has a potential to predict turbulence 0 modification in turbulent 0 2 4 6 bubbly flows.
0.8
∑ (n m =1
m
ν Gm )
nν SI
=
νT of B.I.T. νT of S.I.T.
φ
ICMF2010-Tampa 2010.6.1.
Measurement of TKE At large dB/lt (1) turbulence augmentation at large VR/u’. (2) turbulence attenuation at small φ. At small φ and large VR/u' ?
Eddy viscosity ratio
φ=
An accurate drag model is necessary for accurate predictions of turbulence in bubbly flows.
0
1
Liquid-Solid(JL=0.5m/s) d=4.0mm JS=0.002m/s d=4.0mm JS=0.004m/s d=2.5mm JS=0.002m/s d=2.5mm JS=0.004m/s d=1.0mm JS=0.002m/s d=1.0mm JS=0.004m/s
dB/lt
Large φ Turbulence augmentation
Turbulence attenuation
Millimeter size bubbles C on st an tφ Small φ Turbulence attenuation
VR/u’
Photobleaching: irriversible change from a fluorescent dye to non-fluorescent dye Fluorescent dye: uranine (Fluoresceine sodium salt) Concentration: 10-6 mol/m3 Mask
?
ICMF2010-Tampa 2010.6.1.
Photobleaching MTV
Laser beam
1.0mm
Open
Camera Close ON
Laser sheet λ=514.5nm
OFF ON
Intense laser λ=488 nm
δt ~5ms Tir ~ 1ms
T=t+δt
OFF
Time 1.0mm
Lens
Low power laser sheet (λ=514.5nm): Visualization of tags
δl
Intense laser beam (λ=488.0nm): Photobleaching reactiion
V= Tag Shutter
CCD camera
δl δt
T=t
ICMF2010-Tampa 2010.6.1.
Measurement of Velocity Gradient
ICMF2010-Tampa 2010.6.1.
Experimental Apparatus
W δR: Difference of displacement vectors of two points v: Velocity v0: Velocity at center of tagged region δr: Position vector from the center δt: Time interval
v0 δt
δr
Original tagged region
∂v ⎛ ⎞ δR = vδt = ⎜ v 0 + ⋅ δr ⎟δt ∂r ⎝ ⎠
Laser Transformed tagged region
t=0
Laser
x,u
Measurement position CCD camera
t=δt
Cross-correlation
∂v / ∂r is constant in tagged region
Pump
Intense laser beam for photobleaching
z, w
z, w y, v
Laser sheet
CCD camera
Laser sheet
Exposure time：1 ～5 ms
Kolmogorov scale 1/ 4
aij+ε
aij
Flowmeter
ICMF2010-Tampa 2010.6.1.
⎛ ν3 ⎞ l k ≈ ⎜⎜ ⎟⎟ ⎝ ε ⎠
Lower tank
Axial mean velocity and Void Fraction 0.1
Tag size: 40 – 100 μm Sample number: 200
y, v CCD camera Intense laser beam for photobleaching
Tag size：40～100 μm
Sample number： 200
Correct value
Experimental Conditions x-z plane
Tank
Air
aij-ε
x-y plane
Reynolds number：5000
Water
50 mm
0.08
W: duct width (50 mm)
0.1 0.06 0.04
JG [m/s] U 0.0x10-5 2.5x10-5 5.0x10-5 7.5x10-5
α
0 0
lt ~ 0.2xW ~10 mm
The void fraction exhibits wall-peaking profile.
JG [x10-5m/s]
d32 [mm]
Case 0
0.06
0.0
-
Case 1
0.06
2.5
0.60
Large VR/u’ ~8,
Case 2
0.06
5.0
0.59
Case 3
0.06
7.5
0.59
Small dB/lt ~ 0.06
φ = VRdB/ltu’ ~ 0.5
0.1
0.2
0.3
0.4
U: axial mean velocity
α: void fraction
0.05
lK ~ 0.2 mm ~ dB
JL [m/s]
ICMF2010-Tampa 2010.6.1.
y: distance from the wall
0.02 VR ~ 0.05 m/s ~ JL ~ 10xu’
≈ 200μm
0.15
α [%]
a12 ⎤ ⎡δx ⎤ ⎞ ⎟δt a22 ⎥⎦ ⎢⎣δy ⎥⎦ ⎟⎠
Experimental condition
Compressor Cross-correlation
⎛ ⎡u ⎤ ⎡ a δR = ⎜⎜ ⎢ o ⎥ + ⎢ 11 ⎝ ⎣ vo ⎦ ⎣a21
50 mm
U [m/s]
2D flow
Flow
z,w y,v
1000 mm
δR
Upper tank
0 0.5
y/W The gradients of the axial velocity in the bubbly flows are larger than that in the single-phase flow in the near wall region. The axial velocity distribution in the core region is flattened as many researches reported.
0.1
0.2
y/W
0.3
0.4
0.4
0.3 y/W
102
y+
εν/uτ4
α [%]
Pν/uτ4
0.2
0
-0.2
101
y
+
102
101
y+
102
0
∂U ∂x j The dimensionless andDε does are strongly The dimensionless not change ⎛ P, Pε and ∂k ⎞⎟ so much. ∂ ⎜ 1 ′ ′ ′ attenuated. ′ ′ − u j ui ui − p u j + ν D= ⎜ ⎟ ∂xenhance ∂xand 2 the frictional The bubbles velocity j ⎝ structure j ⎠ The turbulence is different from
0.1
shear-induced turbulence. shear-induced due to the ∂u ′ ∂turbulence ui′ The bubble-induced pseudo turbulence is by not breakup eddies ε = νof thei shear-induced prominent. ∂x j ∂x j bubbles.
0
Fully-developed Turbulence structure doesflow not change so much Symmetry due to bubbles. with respect to y axis
-0.4
2
α [%]
2 2
-4
α [%]
0.2
0.1
0.2
y/W
0.3
Reynolds shear stress
0.3
0.4
JG [m/s] u'v' α -5 0.0x10 -5 2.5x10 -5 5.0x10 -5 7.5x10
0 0.5
0.3
0.2
0.2
0.1 0.1
φ=10
1 Circular pipe (Attenuation)
Circular pipe (Enhancement)
Neglect p’u’ in D
ICMF2010-Tampa 2010.6.1. Large φ: Turbulence augmentation due to bubble-induced turbulence Small φ: Turbulence attenuation occurs Small VR/u’: Agitated flow due to bubbles is weak Æ Eddy breakup by bubbles ÆTurbulence attenuation
0.5
i In thePcore ′ ′ =the −uregion, Between wall i u j and void peak position,
α [%]
0.2 Dν/uτ4
0.1
0 0 10
0 0.3
JG [m/s]-5 Dν/uτ4 α 0.0x10 2.5x10-5 5.0x10-5 7.5x10-5
0.4
100
0.6
0.2
0.2 101
0.1
0.4
core
0.2
0.2
2
φ=1.0
0.4
0.4
0 0 10
0.3
JG [m/s] w' α 0.0x10-5 in the attenuated 2.5x10-5 -5 5.0x10 7.5x10-5
α
0.4
Conclusions
0.2
0.8
0.1
Spanwise component
0.6
0 0
0 0.5
0.4
Although u’v’ is enhanced in the near 0.4 wall region, the increase is not so 0.1 large. 0.2
JG [m/s] εν/uτ4 α 0.0x10-5 2.5x10-5 5.0x10-5 7.5x10-5
4
0.6
0.3 y/W
ICMF2010-Tampa 2010.6.1.
1
0.2 JG [m/s]-5 Pν/uτ α 0.0x10-5 2.5x10 5.0x10-5 7.5x10-5
0.8
0.2
JG [m/s] v'2 0.0x10-5 2.5x10-5 5.0x10-5 7.5x10-5
at small0 φ, whereas VR/u’ is large. 0 0Turbulence attenuation occurs 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 pipe 0.4flows. 0.5 These trends are y/W y/Wthe same as those in the case 2 in bubbly
Dimensionless P, D and ε 1
0.1
u’v’ is strongly 0.6 region.
0 0.5
0.4
v' [x10 m /s ]
2
The turbulence intensities are attenuated in the core region and enhanced in the near wall region due to bubbles.
α [%]
0.2
2
0.8
0.2
0.1
-4
0 0
0.1
0 0
2
0 0.5
0.3
Wall-normal component
α [%]
0.2 α [%]
w'2 [x10-4 m2/s2]
α
0.1 1
0.3 2
JG [m/s] w' 0.0x10-5 2.5x10-5 -5 5.0x10 7.5x10-5
0.6
0 0
0 0.5
0.4
0.2
u'v' [x10-4 m2/s2]
Spanwise component
0.1 0.2
ICMF2010-Tampa 2010.6.1.
α
α [%]
0.3 y/W
2
2
w'2 [x10-4 m2/s2]
0.2
0.4
JG [m/s]-5 u' 0.0x10 2.5x10-5 5.0x10-5 7.5x10-5
3
0.8
0.3
Streamwise component
dB/lt
0.8
0.1
0.2 α [%]
-4
1
4
α
u' [x10 m /s ]
0.1
0.6
2
2
2
2
0 0
JG [m/s] v'2 0.0x10-5 2.5x10-5 5.0x10-5 7.5x10-5
2
-4
α [%]
0.2
0.3
Wall-normal component
α
2
v' [x10 m /s ]
JG [m/s]-5 u' 0.0x10 2.5x10-5 5.0x10-5 7.5x10-5
3
0.8
0.3
Streamwise component
2
2
u' [x10 m /s ]
4
Turbulence Intensity and Reynolds Shear Stress
ICMF2010-Tampa 2010.6.1.
Turbulence Intensity
φ=0.1
Rich experimental data Small dB/lt: B.I.T. quickly dissipates due to its small scale Æ Eddy breakup by bubbles ÆTurbulence attenuation
0.1 0.05 0.5
Square duct (Attenuation)
1
VR/u'
5
10
Further experimental data is required to understand more detailed turbulence structure in bubbly flows.
