Comparison of flow regime transitions with interfacial ...mjm/flow_regime.pdfComparison of flow regime transitions with interfacial wave transitions M. J. McCready & M. R. King Chemical

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/


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.


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.


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


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)


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


Wave transition as gas increases

Data ofMinato1996


Wave transition as gas increases (more)

Data ofMinato1996


The transition at increasing liquid flow rate

Data ofKuru,1995


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


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



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


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.


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]


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


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

)


Two-(matched density) liquid, rotatingCouette device

laser

camera

mirror

Couette Cell

mercury


Rotating Couette experiment


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


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.


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



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



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


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


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


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


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


Spectral simulation, Couette flow

Wavenumber

Amplitude


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


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.


Conclusions





Conclusions





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.













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Comparison of flow regime transitions with interfacial ...mjm/flow_regime.pdfComparison of flow regime transitions with interfacial wave transitions M. J. McCready & M. R. King Chemical