Comparison of flow regime transitions with interfacial wave transitions

Transcription

1 Comparison of flow regime transitions with interfacial wave transitions M. J. McCready & M. R. King Chemical Engineering University of Notre Dame

2 Flow geometry of interest Two-fluid stratified flow gas liquid We will consider the transition from a stratified state in this talk. Examining (linear) stability of other structures (e.g., slugs) is also a viable way to formulate the flow regime problem.

3 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 the transition 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.

4 Some regime transition models 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 1 superficial liquid velocity, m/s channel=2.54 cm liquid gas ρ L =1 g/cm 3 ρ G =.112 g/cm 3 µ L =1 cp µ G =.18 cp superficial gas velocity, m/s

5 Oil-Gas at 2 Atm. channel=2 cm liquid gas ρ L =.9 g/cm 3 ρ G =.232 g/cm 3 µ L =5 cp µ G =.371 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) superficial liquid velocity, m/s superficial gas velocity, m/s

6 The key issue is the growing wave peak at low frequency which can lead to roll waves or slugs. When does it occur? R L = 3 R G = 9975 µ L = 5 cp linear growth k-ε model Data of Bruno and McCready, growth rate (1/s) wave spectrum (cm 2 -s) 1-5 wave 1.2 m 3.8 m 6 m frequency (1/s)

7 Wave transition as gas increases Data of Minato 1996

8 Wave transition as gas increases (more) Data of Minato 1996

9 The transition at increasing liquid flow rate Data of Kuru, 1995

10 Experiments laminar, oil-water channel flow 2.44 m.5-1 cm Oil phase Water phase } Tracings PSD CSD - Wave speeds Bicoherence.2 mm platinum wire probes 3 khz Signal 1 cm

11 Measured wave transitions Linear growth: interfacial mode "shear" mode wave spectrum w (1/s) 2 1 Re oil = 3 Re water = 65 f xx (cm 2 /s) f (Hz)

12 f xx (cm 2 /s) (wave spectrum) Measured wave transitions 5 4 Linear growth: interfacial mode "shear" mode measured spectrum 1-2 w (1/s) (linear growth) Re oil =3 Re water = f (Hz)

13 Long wave stability -- THE criterion? We might be tempted to conclude that long wave stability, if we could get it correct -- would tell us everything. 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.

14 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' [1a] φ = [1b] φ' - u b φ = Φ' - Ub [1c] ub ( ) c u b ( ) c φ'' + k 2 φ = µ (Φ'' + k 2 [1d] 1 i k φ νr (φ''' - 3 k2 φ') + i k (φ u b ' - φ' (u b () - c)) + (F + k2 T) (u b () - c) R 2 = ρ R (Φ''' - 3 i ρ k φ k2 Φ') + ρ i k (Φ U b ' - Φ' σ) + F (u b () - c) R [1e]

15 Theoretical analysis (linear laminar theory) i k (U b -c) (Φ'' - k 2 Φ) - i k U b '' Φ = R -1 (Φ iv - 2 k 2 Φ''+ k 4 Φ), for y 1 [1f] i k (u b -c) (φ'' - k 2 φ) - i k u b '' φ = (ν R) -1 (φ iv - 2 k 2 φ''+ k 4 φ), for -1/d y [1g] φ = φ' y = -d -1. [1h] viscosity ratio ==> µ = µ 2 /µ 1, density ratio ==>ρ = ρ 2 /ρ 1, ratio of kinematic viscosities ==> ν, σ = u b () -c depth ratio ==> d = D 2 /D. 1 liquid average velocity profiles ==> u b wavenumber ==> k gas velocity ==> U b

16 Unstable long waves, stable intermediate region growth rate (1/s) 2 1 R G = 2 µ L = 5 cp R L = 35 R L = 25 R L = 2 R L = 15 R L = 1 growth rate (1/s) wavenumber (m -1 ) wavenumber (m -1 )

17 Two-(matched density) liquid, rotating Couette device laser mirror camera mercury Couette Cell

18 Rotating Couette experiment

19 Wave map for rotating Couette flow experiment Regions of "no waves" exist where long waves are unstable 5 Atomization plate speed (cm/s) stable long waves steady 2-D waves occur in most of this range 1. No waves steady periodic unsteady waves solitary long wave stability boundary.2.4 h Will show spectral simulation later at this condition.6.8

20 Weakly- nonlinear theory Du Dt = -Ñp + 1 R 2 u 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 of modes which have a complex amplitude, A, multiplying a linear eigenfunction, ζ, y = Σ A i ζ i A series of amplitude coupled amplitude equations is integrated. Ai = + * Lk ( i) αjiaa j j i + β jiai j Aj t + γ kji AA i j A k Both the dynamic and steady state behavior are watched.

21 Comparison of oil-water experiments and simulation 1-2 Linear stability Re W =12 Re W =12 Re W =65 Re W = wave amplitude linear growth rate (1/s) Simulated Spectra Re W =65 Re W = wavenumber (1/m) -4

22 Measured wave transitions Linear growth: interfacial mode "shear" mode wave spectrum w (1/s) 2 1 Re oil = 3 Re water = 65 f xx (cm 2 /s) f (Hz)

23 f xx (cm 2 /s) (wave spectrum) Measured wave transitions 5 4 Linear growth: interfacial mode "shear" mode measured spectrum 1-2 w (1/s) (linear growth) Re oil =3 Re water = f (Hz)

24 Nonlinear effect on long wave formation Re w =65, cubic nonlinear coefficients are more balanced Re w =12, large cubic nonlinear coefficients between large and small wavenumbers

25 Gas-liquid simulations rel47 rel24 wave amplitude wavenumber (1/m)

26 Nonlinear coefficients for gas-liquid flow Quadratic interactions with mean flow mode are much Stronger at the higher liquid Reynolds number Re G =98, Re L =24 Re G =98, Re L =47

27 Couette flow linear growth rate growth rate TextEnd Depth ratio=1.5 Viscosity ratio = 55 Density ratio=1 Rotation rate=15 cm/s scaled wavenumber

28 Spectral simulation, Couette flow A m p l i t u d e Wavenumber

29 Couette flow simulations Evolution of wave spectrum with time. No preferred wavenumber exists. Should explain why we see no waves amplitude t=3 t=243 t=183 t=122 t= wavenumber

30 Conclusions 1. All evidence is that long wave stability is a necessary condition for the formation of long wave disturbances. 2. For many flow situations, there is a good correspondence between the long wave stability boundary and the formation of growing long waves. 3. However, there is the nagging problem with Couette flow, where a nonlinear cascade appears to saturate waves at an amplitude too small to see.

31 Conclusions 1. All evidence is that long wave stability is a necessary condition for the formation of long wave disturbances. 2. For many flow situations, there is a good correspondence between the long wave stability boundary and the formation of growing long waves. 3. However, there is the nagging problem with Couette flow, where a nonlinear cascade appears to saturate waves at an amplitude too small to see.

32 Conclusions 1. All evidence is that long wave stability is a necessary condition for the formation of long wave disturbances. 2. For many flow situations, there is a good correspondence between the long wave stability boundary and the formation of growing long waves. 3. However, there is the nagging problem with Couette flow, where a nonlinear cascade appears to saturate waves at an amplitude too small to see.

33 Conclusions (cont.) 4. There is also the complication that the entire long wave region may not be unstable, this probably does prevent significant long wave formation,because 5. The simulations suggest that different kinds of nonlinear interactions (e.g., cubic and or quadratic with various modes) are important in the development of the spectrum Quadratic interactions enhance the formation of the low mode for gas-liquid flows and cubic interactions enhance it for oil-water flows 6. Certainly the situation is complex because large ocean waves are not the most linearly unstable (5 cm waves are), but these do not receive much of their energy from nonlinear processes, wind must directly feed them.

34 Conclusions (cont.) 4. There is also the complication that the entire long wave region may not be unstable, this probably does prevent significant long wave formation,because 5. The simulations suggest that different kinds of nonlinear interactions (e.g., cubic and or quadratic with various modes) are important in the development of the spectrum Quadratic interactions enhance the formation of the low mode for gas-liquid flows and cubic interactions enhance it for oil-water flows 6. Certainly the situation is complex because large ocean waves are not the most linearly unstable (5 cm waves are), but these do not receive much of their energy from nonlinear processes, wind must directly feed them.

35 Conclusions (cont.) 4. There is also the complication that the entire long wave region may not be unstable, this probably does prevent significant long wave formation,because 5. The simulations suggest that different kinds of nonlinear interactions (e.g., cubic and or quadratic with various modes) are important in the development of the spectrum Quadratic interactions enhance the formation of the low mode for gas-liquid flows and cubic interactions enhance it for oil-water flows 6. Certainly the situation is complex because large ocean waves are not the most linearly unstable (5 cm waves are), but these do not receive much of their energy from nonlinear processes, wind must directly feed them.