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Comparison of flow regime transitionswith
interfacial wave transitions
M. J. McCready & M. R. King
Chemical EngineeringUniversity of Notre Dame
[email protected] http://www.nd.edu/~mjm/
[email protected] http://www.nd.edu/~mjm/
Flow geometry of interest
liquid
gas
Two-fluid stratified flow
We will consider the transition from a stratified statein this talk.Examining (linear) stability of other structures (e.g., slugs)is also a viable way to formulate the flow regime problem.
[email protected] http://www.nd.edu/~mjm/
Points of this talk• Examine the use of linear stability models for flow
regime transition– Long-wave stability is most appropriate based on transition data
– Models from different studies don’t agree
• Explore the importance of nonlinear effects on thetransition using a well-defined system– Yes, there are some nonlinear effects that do change the “predicted”
transition.– But -- linear stability might be a good engineering model if done
correctly. It is certainly a useful limiting point.
[email protected] http://www.nd.edu/~mjm/
Some regime transition models
0.0001
0.001
0.01
0.1
1
supe
rfic
ial l
iqui
d ve
loci
ty, m
/s
0.01 0.1 1 10superficial gas velocity, m/s
channel=2.54 cm liquid gas
ρL=1 g/cm3
ρG=.00112 g/cm3
µL=1 cp µG=.018 cp
Slug transition models Kelvin-Helmholtz Wallis & Dobson (1973) Taitel & Dukler (1976) Lin & Hanratty (1986) Barnea (1991) Crowley et.al (1993) Bendiksen & Espedal (1992)
Lin & Hanratty (1987) data slugs
Long wave linear stability laminar-laminar differential
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Oil-Gas at 200 Atm.
0.001
0.01
0.1
1
supe
rficia
l liqu
id v
eloc
ity, m
/s
0.01 0.1 1 10superficial gas velocity, m/s
channel=20 cm liquid gas
ρL=.9 g/cm 3 ρG=.232 g/cm 3
µL=50 cp µG=.0371 cp
Barnea (1991) Kelvin-Helmholtz Taitel-Dulker (1976) Wallis & Dobson (1973) Crowley et al. (1992) Lin & Hanratty (1986) Differential, laminar Bendiksen & Espedal (1992) Ruder & Hanratty (1989)
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The key issue is the growing wave peak at lowfrequency which can lead to roll waves or slugs.• When does it occur?
10-6
10-5
10-4
10-3
10-2
wave spectrum (cm2-s)
0.12 3 4 5 6 7
12 3 4 5 6 7
102 3 4 5 6 7
100frequency (1/s)
40
30
20
10
0
-10
growt
h rat
e (1/
s)
RL = 300RG = 9975
µL = 5 cP
linear growth k-ε model
wave spectrum @ 1.2 m 3.8 m 6 m
Data of Bruno andMcCready, 1988
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The transition at increasing liquid flow rate
Data ofKuru,1995
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Experimentslaminar , oil-water channel flow
Oil phase
Water phase1 cm
2.44 m
30 kHzSignal
TracingsPSD
CSD - Wave speedsBicoherence
}
.2 mm platinum wire probes
.5-1 cm
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Measured wave transitions5
4
3
2
1
0
-1
-2
w
(1/s
)
0.12 3 4 5 6
12 3 4 5 6
102
f (Hz)
10-5
10-4
10-3
10-2
f
xx (cm2/s)
Reoil = 3Rewater = 650
Linear growth: interfacial mode "shear" mode wave spectrum
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Measured wave transitions5
4
3
2
1
0
-1
-2
w
(1/s
) (lin
ear g
row
th)
0.12 4 6 8
12 4 6 8
102 4
f (Hz)
10-5
10-4
10-3
10-2
f
xx (cm2/s) (w
ave spectrum)
Linear growth: interfacial mode "shear" mode measured spectrum
Reoil=3Rewater=1200
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Long wave stability -- THE criterion?
• We might be tempted to conclude that long wavestability, if we could get it correct -- would tell useverything.
• However, there are two problems.– 1. "Long" wave stability does not mean all long waves are
unstable.
– 2. We have strong experimental evidence of a range of existence of unstable long waves where there are no visible waves.
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Theoretical analysis (linear laminar theory)
The complete differential linear problem can be formulated as
U = Φ'(y) Exp[i k (x-c t)] , u = φ'(y) Exp[i k (x-c t)] ,
V = - i k Φ (y) Exp[i k (x-c t)], v = - i k φ(y) Exp[i k (x-c t)]
where Φ(y) and φ(y) are the disturbance stream functions
Φ = Φ' = 0 @y=1, [1a]
φ = Φ, @y=0, [1b]
φ' - ub
bu c
©
( )
φ0 −
= Φ' - U
u cb
b
©
( )
Φ0 −
@y=0, [1c]
φ'' + k2 φ = µ (Φ'' + k2 Φ), @y=0, [1d]
1νR
(φ''' - 3 k2 φ') + i k (φ ub' - φ' (ub(0) - c)) + i k φ
(ub(0) - c) (F + k2 T)
R2 =
ρR (Φ''' - 3 k2 Φ') + ρ i k (Φ Ub' - Φ' σ) +
i ρ k φ(ub(0) - c)
FR2 ,
@y=0, [1e]
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Theoretical analysis (linear laminar theory)
i k (Ub -c) (Φ'' - k2 Φ) - i k Ub'' Φ = R-1(Φiv - 2 k2 Φ''+ k4Φ),
for 0≤y≤1 [1f]
i k (ub -c) (φ'' - k2 φ) - i k ub'' φ = (ν R)-1(φiv - 2 k2 φ''+ k4φ),
for -1/d≤y≤0 [1g]
φ = φ' = 0, @ y = -d-1. [1h]
viscosity ratio ==> µ = µ2/µ1, density ratio ==>ρ = ρ2/ρ1,
ratio of kinematic viscosities ==> ν, σ = ub(0) -c
depth ratio ==> d = D2†/D1
†. wavenumber ==> k liquid average velocity profiles ==> ub gas velocity ==> Ub
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Unstable long waves, stable intermediate region
3
2
1
0
gro
wth
ra
te (
1/s
)
0.1 1 10 100 1000wavenumber (m -1)
RG = 20000µL = 5 cP
RL = 35 RL = 25 RL = 20 RL = 15 RL = 10
-0.15
-0.10
-0.05
0.00
0.05
0.10
gro
wth
ra
te (
1/s
)
5 6 7 80.1
2 3 4 5 6 7 81
2 3 4 5 6 7 810
2 3 4 5
wavenumber (m-1
)
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Two-(matched density) liquid, rotatingCouette device
laser
camera
mirror
Couette Cell
mercury
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Rotating Couette experiment
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Wave map for rotating Couette flow experiment
Regions of "no waves" exist where long waves are unstable
50
40
30
20
10
0
plat
e sp
eed
(cm
/s)
0.80.60.40.20.0
h
No waves steady periodic unsteady waves solitary long wave stability boundary
stablelong waves steady 2-D waves occur in most of this range
Atomization
Will show spectral simulationlater at this condition
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Weakly- nonlinear theoryDuDt = -Ñp + 1
R ∇ 2u
Spectral reduction of Navier-Stokes equations and boundary conditions.
The interface is represented by, y ≡ (u, p, h)
We make the assumption that the waves can be represented by a series ofmodes which have a complex amplitude, A, multiplying a linear eigenfunction, ζ,
y = Σ Ai ζ i
A series of amplitude coupled amplitude equations is integrated.
∂∂
= + + +− −A
tL k A A A Ai
i ji j j i ji i j j( ) *α β γ kji i j kA A A
Both the dynamic and steady state behavior are watched.
[email protected] http://www.nd.edu/~mjm/
Comparison of oil-water experiments and simulation
10-5
10-4
10-3
10-2
wav
e am
plitu
de
12 3 4 5 6 7 8
102 3 4 5 6 7 8
1002 3 4 5 6
wavenumber (1/m)
6
4
2
0
-2
-4
linear growth rate (1/s)
Simulated Spectra ReW=650
ReW=1200
Linear stability ReW=1200 ReW=1200 ReW=650 ReW=650
[email protected] http://www.nd.edu/~mjm/
Measured wave transitions5
4
3
2
1
0
-1
-2
w
(1/s
)
0.12 3 4 5 6
12 3 4 5 6
102
f (Hz)
10-5
10-4
10-3
10-2
f
xx (cm2/s)
Reoil = 3Rewater = 650
Linear growth: interfacial mode "shear" mode wave spectrum
[email protected] http://www.nd.edu/~mjm/
Measured wave transitions5
4
3
2
1
0
-1
-2
w
(1/s
) (lin
ear g
row
th)
0.12 4 6 8
12 4 6 8
102 4
f (Hz)
10-5
10-4
10-3
10-2
f
xx (cm2/s) (w
ave spectrum)
Linear growth: interfacial mode "shear" mode measured spectrum
Reoil=3Rewater=1200
[email protected] http://www.nd.edu/~mjm/
Nonlinear effect on long wave formation
010
2030
4050
1
2
3
4
5
6
70
50
100
150
200
250
010
2030
4050
6070
0
2
4
6
80
100
200
300
400
500
Rew=650, cubic nonlinear coefficients are more balanced
Rew=1200, large cubic nonlinear coefficients between large and smallwavenumbers
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Gas-liquid simulations
0.0001
2
4
68
0.001
2
4
68
0.01
2
4
6
wa
ve
am
plitu
de
2 3 4 5 6 7 8 9100
2 3 4 5 6
wavenumber (1/m)
rel470 rel240
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Nonlinear coefficients for gas-liquid flow
ReG=9800, ReL=240 ReG=9800, ReL=470
Quadratic interactions with mean flow mode are muchStronger at the higher liquid Reynolds number
05
1015
2025
3035
0
10
20
30
400
5
10
15
20
25
05
1015
2025
3035
0
10
20
30
400
5
10
15
20
25
30
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Couette flow linear growth rate
100 101
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
scaled wavenumber
grow
th r
ate
TextEnd Depth ratio=1.5Viscosity ratio = 55Density ratio=1Rotation rate=15 cm/s
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Couette flow simulationsEvolution of wave spectrum with time. No preferredwavenumber exists. Should explain why we see no waves
10-6
10-5
10-4
10-3
10-2
ampl
itude
12 3 4 5 6 7 8 9
102 3 4
wavenumber
t=300 t=243 t=183 t=122 t=24
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Conclusions
• 1. All evidence is that long wave stability is a necessary condition forthe formation of long wave disturbances.
• 2. For many flow situations, there is a good correspondencebetween the long wave stability boundary and the formation of growinglong waves.
• 3. However, there is the nagging problem with Couette flow, where anonlinear cascade appears to saturate waves at an amplitude too smallto see.
[email protected] http://www.nd.edu/~mjm/
Conclusions
• 1. All evidence is that long wave stability is a necessary condition forthe formation of long wave disturbances.
• 2. For many flow situations, there is a good correspondencebetween the long wave stability boundary and the formation of growinglong waves.
• 3. However, there is the nagging problem with Couette flow, where anonlinear cascade appears to saturate waves at an amplitude too smallto see.
[email protected] http://www.nd.edu/~mjm/
Conclusions
• 1. All evidence is that long wave stability is a necessary condition forthe formation of long wave disturbances.
• 2. For many flow situations, there is a good correspondencebetween the long wave stability boundary and the formation of growinglong waves.
• 3. However, there is the nagging problem with Couette flow, where anonlinear cascade appears to saturate waves at an amplitude too smallto see.
[email protected] http://www.nd.edu/~mjm/
Conclusions (cont.)• 4. There is also the complication that the entire long wave region
may not be unstable, this probably does prevent significant long waveformation,because …
• 5. The simulations suggest that different kinds of nonlinearinteractions (e.g., cubic and or quadratic with various modes) areimportant in the development of the spectrum
– Quadratic interactions enhance the formation of the low mode for gas-liquid flows andcubic interactions enhance it for oil-water flows
• 6. Certainly the situation is complex because large ocean waves arenot the most linearly unstable (5 cm waves are), but these do notreceive much of their energy from nonlinear processes, wind mustdirectly feed them.
[email protected] http://www.nd.edu/~mjm/
Conclusions (cont.)• 4. There is also the complication that the entire long wave region
may not be unstable, this probably does prevent significant long waveformation,because …
• 5. The simulations suggest that different kinds of nonlinearinteractions (e.g., cubic and or quadratic with various modes) areimportant in the development of the spectrum
– Quadratic interactions enhance the formation of the low mode for gas-liquid flows andcubic interactions enhance it for oil-water flows
• 6. Certainly the situation is complex because large ocean waves arenot the most linearly unstable (5 cm waves are), but these do notreceive much of their energy from nonlinear processes, wind mustdirectly feed them.
[email protected] http://www.nd.edu/~mjm/
Conclusions (cont.)• 4. There is also the complication that the entire long wave region
may not be unstable, this probably does prevent significant long waveformation,because …
• 5. The simulations suggest that different kinds of nonlinearinteractions (e.g., cubic and or quadratic with various modes) areimportant in the development of the spectrum
– Quadratic interactions enhance the formation of the low mode for gas-liquid flows andcubic interactions enhance it for oil-water flows
• 6. Certainly the situation is complex because large ocean waves arenot the most linearly unstable (5 cm waves are), but these do notreceive much of their energy from nonlinear processes, wind mustdirectly feed them.