Interfacial waves in steady and oscillatory, two-layer Couette flows

Transcription

1 Interfacial waves in steady and oscillatory, two-layer Couette flows M. J. McCready Department of Chemical Engineering University of Notre Dame Notre Dame, IN Page 1

2 Acknowledgments Students: M. R. King, Massimo Sangalli, C. T. Gallagher, D. D. Uphold, K. A. Gifford, Faculty Colleagues: H. C. Chang D. T. Leighton Funding: NASA, µ-gravity; DOE, Basic Energy Sciences NSF, CTS Page 2

3 Importance of gas-liquid flows (and other fluid-fluid flows) 1. Pipeline and process piping flows, energy exchange devices, oil and gas well risers. 2. Process equipment, reactors, absorbers, etc. 3. Environmental flows (ocean surface, cleanup of contaminated ground water). Page 3

4 Inside closed conduits, different flow regimes exist (Flow regime map of Mandhane et al., 1974) 10 Dispersed bubbles "Dispersed limit" U SL (m/s) Bubbly Flow "Steady" Slug Flow "Large structures" "Separated limit" Annular, Annular Mist "Separated limit" Stratified Wavy 0.01 Mandhane Flow Map "Conceptual" boundaries U SG (m/s) Figure 1a. Flow regime map for horizontal flow (Mandhane et al., 1974) with rough boundaries for different limiting analyses added. Page 4

5 A sensible approach is to think of the flow regime problem in terms of: Limiting Flow Configurations Dispersed gas liquid Separated gas liquid This talk with be confined to separated flows Page 5

6 Why study waves? 1. Application to flow regime transition and overall flowrate pressure drop properties of separated multifluid flows, atomization, heat and mass transfer 2. Interesting academic problems that can be used to advance understanding of analytical and numerical solution techniques, develop new experiments and to educate students. Page 6

7 Flow geometry of interest Fluid 2 H Fluid 1 h Two-layer, horizontal two-fluid flow Process and pipeline flows usually start with an inlet device and then the flow develops with distance presumably reaching a fully developed state. This can take arbitrarily long and may not happen in the length available in laboratory flows. We wanted to do a wave experiment that did not have this limitation and also one that allowed the greatest possible control of the flow. Page 7

8 Measurements of wave spectra in a horizontal channel Growth of roll wave precursor peak with distance is seen. Saturation does not occur R L = 300 R G = 9975 µ L = 5 cp linear growth k-ε model growth rate (1/s) wave spectrum (cm 2 -s) wave 1.2 m 3.8 m 6 m frequency (1/s) Page 8

9 Rotating, Two-(matched density) liquid, Couette Flow, Wave Experiment Outside cylinder is Plexiglas, Inside cylinder is Aluminum painted black Torque transducer Hg Layer Outside cylinder is rotated. Page 9

10 Two-Liquid, Matched density Couette device (another view) laser mirror camera mercury Couette Cell Two-layer Couette experiment showing the high speed video camera set up Page 10

11 Direct view video of waves in Couette device Page 11

12 Theoretical analysis Yih (1967); Renardy (1985, 1987) Hooper (1985) 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] u φ' - b ' φ Ub' Φ = Φ' [1c] ub( 0) c ub( 0) c φ'' + k 2 φ = µ (Φ'' + k 2 [1d] 1 νr (φ''' - 3 i k φ k2 φ') + i k (φ u b ' - φ' (u b (0) - c)) + (F + k2 T) (u b (0) - c) R 2 = ρ R (Φ''' - 3 i ρ k φ k2 Φ') + ρ i k (Φ U b ' - Φ' σ) + F (u b (0) - c) R [1e] 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 0 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 0 [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 (0) -c depth ratio ==> d = D 2 /D 1. liquid average velocity profiles ==> u b wavenumber ==> k gas velocity ==> U b Page 12

13 Measured and predicted wavelengths as a function of rotation rate for Couette flow device U [m/s] increasing speed decreasing speed maximum growing wavelength stability boundary λ [m]x10 3 viscosity ratio = 50 density ratio = depth ratio = M. Sangalli, C. T. Gallagher, D. T. Leighton, H. -C. Chang and M. J. McCready Phys. Rev. Lett. 75, pp (1995). Page 13

14 Weakly- nonlinear theory (Blennerhassett, 1980; Renardy & Renardy, 1993). Du Dt = - p + 1 R 2 u Spectral reduction of Navier-Stokes equations and boundary conditions ψ (u, p, h) If system is such that a single dominant mode exists then: ψ = A ζ + A ζ + ξ A Complex Amplitude Function ζ dominant eigenfunction ξ Linear combination of eigenfunctions of stable modes Page 14

15 Center manifold projection to produce a Stuart - Landau equation. A t = L(λ) A + β A2 A β L(λ) Landau constant Linear mode behavior Page 15

16 Experimental verification Comparison of experiment with weakly-nonlinear theory: Measured and predicted wave amplitude for interfacial waves in rotating Couette device The bifurcation is supercritical 1.0 wave amplitude 1 x10 3 [m] increasing speed decreasing speed theory for maximum growing wavelength theory for observed wavelength U [m/s] 0.30 viscosity ratio = 50 density ratio = depth ratio = M. Sangalli, C. T. Gallagher, D. T. Leighton, H. -C. Chang and M. J. McCready Phys. Rev. Lett. 75, pp (1995). Page 16

17 Computed wave shape height (m)x distance (m)x10 3 Page 17

18 Experiments in channel flows and pipes suggest that a sufficient condition for roll waves or slugs or a regime transition is that the growth curve is unstable all the way down to 0 wavenumber. The situation is different for the rotating Couette flow, as there is a range where long waves are unstable and only small amplitude waves appear. Page 18

19 Wave regime map for rotating Couette flow experiment Linear growth curve No waves Steady periodic Unsteady Solitary Atomization Plate Speed, U p (cm/s) long waves stable 20 Instability boundary Inner depth, l (cm) Linear growth curve We see large regions where long waves are unstable and no large waves occur. It remains to be seen if this should change our thinking about channel flows. Page 19

20 Oscillatory two-layer flows Why study oscillatory flow? While most process flows are not (intended to be) oscillatory, there are many transient flows Oscillatory flows are the easiest to understand transients we can construct. We expect that there are some fundamentally different mechanisms at work in oscillatory flows. Page 20

21 Oscillatory, two-layer Couette flow frequency 0-10 Hz Sin(2 π f t) outside wall silicone fluid µ 2 1 cm water&glycerine µ1 m= µ 1 /µ 2 Note: Because of penetration depth differences with viscosity and frequency, it does matter which fluid is next to the oscillating wall. Previous work: Yih (1968), Coward and Papageorgiou (1994); Coward and Renardy (1997); Coward et al. (1997) Page 21

22 Video image of oscillatory waves Page 22

23 Theory Standard boundary conditions and governing equations for the oscillatory problem are expanded using Chebyshev polynomials. This gives:. Ba = A() t a where B= L, A= Re L i ( UL U yy ), L = α y 2 α 2 and a is the vector of Chebyshev polynomial coefficients. These can be directly integrated in time as an initial value problem. We also do this analysis with Floquet theory. 2π S( t + ) = RS( t) Ω BS = A() t S Page 23

24 Comparison of wave onset with long wave stability boundary 1 m = 1/ Stable D 1 (cm) stable Unstable unstable long wave ω (rad/s) Page 24

25 Comparison of wave onset with long wave stability boundary m = 1/ Stable 0.8 D 1 (cm) Unstable long wave stable single wave unstable ω (rad/s) Page 25

26 Measured vs. predicted (peak) wavelength 10 ω = 24 rad/s 8 6 λ (cm) 4 m=1/63 2 linear theory video data averaged U (cm/s) 0 10 ω = 30 rad/s λ (cm) U (cm/s) 0 m=1/63 Page 26

27 Measured and predicted growth rates growth rate (s -1 ) ω = 24 rad/s theory experiment U (cm/s) ω = 30 rad/s growth rate (s -1 ) U (cm/s) 0 Page 27

28 Mechanism of Instability Internal mode in less viscous phase is excited during certain parts of the cycle m = 1/ y Stable D 1 (cm) U 2 y 0.7 Unstable U inflection pt. long wave ω (rad/s) Fjortoft's theorem is an inviscid theory Page 28

29 Further insight into the mechanism: growth is oscillatory in time h(t) D 1 = 0.5 cm ω = 10 rad/s U 0 = 5 cm/s k = 1 cm-1 max time Page 29

30 Snapshots of wave disturbance M t = I S Internal Mode dominates, strongest growth Page 30

31 Continued snap shots M t = I S Interfacial mode dominates, only weak growth occurs Page 31

32 Appearance of inflection point and internal mode over cycle. D = 0.5 cm, 10 rad/s time D = 0.6 cm, 10 rad/s time D = 0.7 cm, 10 rad/s time D = 0.8 cm, 10 rad/s time Page 32

33 Diagram of mechanism t = 0.89 (1.39) a) t = 1.05 b) t = 1.11 c) Page 33

34 Time of maximum wave growth: Interface is deforming upper fluid disturbance, t = 1.11 S S Page 34

35 Effect of phase on wave growth φ max x,y φ (b) (a) Strength of Disturbances 0.1 (c) φ (c) d h(t) dt (b) Wave growth (a) time Page 35

36 Interesting unresolved issue for interfacial waves Regions of subcritical bifurcation. The interfacial wave instability seems to generally be supercritical. We have three different experiments that we use to study waves that are all usually supercritical 1. gas-liquid channel (µ = , ρ = 1000 ) stabilization is usually cubic self-interaction, β 1 A 1 A * 1 A 1 2. oil-water channel (µ = 1/15, ρ = 1.2) stabilization by mean flow interaction at low oil flow β 0 A 0 A * 0 A 1 stabilization by cubic self interaction at higher oil flow 3. liquid-liquid rotating Couette (µ = 55, ρ = 1.001) stabilization by overtone interaction, P 1 A * 1 A 2 Page 36

37 The natural question that arises is: are there subcritical regions of wave transition somewhere? We cannot check for all cases because there are 7 parameters in the two-layer system and no general theory. We can think about this a bit and get some ideas. Subcritical 1. (obvious and not interesting) If a gas-phase internal mode becomes unstable first for, say, air-liquid, R L = 2, R G = 8000, µ L = 10 cp, this mode is subcritical as expected from single-phase theory. Destabilization is from mean flow interaction as expected. Page 37

38 Subcritical 2. (much more interesting) Oil-water channel flow inclined downward at π/37 in the flow direction. Linear stability shows region where increasing the water flow rate decreases the wave growth!! temporal growth (1/s) R w = 120 R w = 140 R w = 160 R w = 180 R w = 220 R w = 240 R w = 260 R w = 300 R w = 315 R w = 330 R w = 350 R oil = 1.6 θ=π/37 ρ oil /ρ w = wavenumber (1/m) Calculation of Landau coefficient shows that the overtone interaction is source of the destabilization. If this is large enough, it dominates the stabilization from the cubic selfinteraction and the mean flow interaction. Page 38

39 Subcritical 3. Consistent with inviscid theory, close to resonance, that is where a fundamental wave travels at the same speed at its overtone, the overtone interaction dominates. Generally for depths slightly larger than resonance conditions, the overtone interaction is destabilizing and large enough to cause a positive Landau coefficient Re(β) Amplitude Saturation β 0FVF (mean flow:fvf) β 0FPG (mean flow:fpg) β 2 (overtone) β 1 (self) Re L We have tried to do experiments in these regions and not yet seen any significant effects. Page 39

40 Conclusions 1. Rotating Couette experiment can be performed that removes effect of distance evolution while maintaining a simple flow geometry (2-D) 2. Supercritical bifurcation occurs from stratified state in good quantitative agreement with weakly-nonlinear analysis of entire set of governing equations. 3. Oscillatory two-layer wave experiment agrees with Floquet and direct integration* predictions of the wavelength and growth rate. 4. Wave growth mechanism that is unique to transient or oscillatory flows dominates where an internal mode in the more viscous phase feeds the interfacial mode over part of the cycle. 5. Interesting ranges of subcritical behavior exist that would benefit from further experimental scrutiny. *Direct integration provides information not evident from Floquet analysis Page 40