Post on 06-Apr-2020
Comparison of flow regime transitionswith
interfacial wave transitions
M. J. McCready & M. R. King
Chemical EngineeringUniversity of Notre Dame
mjm@nd.edu http://www.nd.edu/~mjm/
mjm@nd.edu 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.
mjm@nd.edu 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.
mjm@nd.edu 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
mjm@nd.edu http://www.nd.edu/~mjm/
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)
mjm@nd.edu http://www.nd.edu/~mjm/
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
mjm@nd.edu http://www.nd.edu/~mjm/
Wave transition as gas increases
Data ofMinato1996
mjm@nd.edu http://www.nd.edu/~mjm/
Wave transition as gas increases (more)
Data ofMinato1996
mjm@nd.edu http://www.nd.edu/~mjm/
The transition at increasing liquid flow rate
Data ofKuru,1995
mjm@nd.edu http://www.nd.edu/~mjm/
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
mjm@nd.edu 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
mjm@nd.edu 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
mjm@nd.edu http://www.nd.edu/~mjm/
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.
mjm@nd.edu http://www.nd.edu/~mjm/
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]
mjm@nd.edu http://www.nd.edu/~mjm/
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
mjm@nd.edu http://www.nd.edu/~mjm/
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
)
mjm@nd.edu http://www.nd.edu/~mjm/
Two-(matched density) liquid, rotatingCouette device
laser
camera
mirror
Couette Cell
mercury
mjm@nd.edu http://www.nd.edu/~mjm/
Rotating Couette experiment
mjm@nd.edu http://www.nd.edu/~mjm/
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
mjm@nd.edu http://www.nd.edu/~mjm/
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.
mjm@nd.edu 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
mjm@nd.edu 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
mjm@nd.edu 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
mjm@nd.edu 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
mjm@nd.edu http://www.nd.edu/~mjm/
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
mjm@nd.edu http://www.nd.edu/~mjm/
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
mjm@nd.edu http://www.nd.edu/~mjm/
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
mjm@nd.edu http://www.nd.edu/~mjm/
Spectral simulation, Couette flow
Wavenumber
Amplitude
mjm@nd.edu http://www.nd.edu/~mjm/
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
mjm@nd.edu 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.
mjm@nd.edu 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.
mjm@nd.edu 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.
mjm@nd.edu 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.
mjm@nd.edu 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.
mjm@nd.edu 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.