numerical modelling of the rise of taylor bubbles through a change in pipe diameter

Accepted Manuscript Numerical modelling of the rise of Taylor bubbles through a change in pipe diameter Stephen Ambrose, Ian S Lowndes, David M Hargreaves, Barry Azzopardi PII: DOI: Reference: S0045-7930(17)30036-1 10.1016/j.compfluid.2017.01.023 CAF 3389 To appear in: Computers and Fluids Received date: Revised date: Accepted date: August 2016 23 January 2017 28 January 2017 Please cite this article as: Stephen Ambrose, Ian S Lowndes, David M Hargreaves, Barry Azzopardi, Numerical modelling of the rise of Taylor bubbles through a change in pipe diameter, Computers and Fluids (2017), doi: 10.1016/j.compfluid.2017.01.023 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain ACCEPTED MANUSCRIPT Highlights • We simulate the rise of Taylor bubbles through expansions in vertical CR IP T pipes • The angle of expansion influence whether the bubble breaks up • The diameter of the upper pipe influences whether the bubble breaks up AC CE PT ED M AN US • Simulations are compared against experimental work ACCEPTED MANUSCRIPT Numerical modelling of the rise of Taylor bubbles through a change in pipe diameter a CR IP T Stephen Ambrosea , Ian S Lowndesa , David M Hargreavesa,∗, Barry Azzopardia Faculty of Engineering, University of Nottingham, Nottingham, NG7 2RD, UK AN US Abstract The rise of Taylor bubbles through expansions in vertical pipes is modelled using Computational Fluid Dynamics The predictions from the models are compared against existing experimental work and show good agreement, both quantitatively and qualitatively Many workers, including the present work, M find that, as the bubble passes through the expansion, it will either remain intact or split into one or more daughter bubbles We find that the critical ED length of bubble, defined as the maximum length that will pass through intact, is proportional to the cosecant of the angle of the expansion Further, PT we show that for an abrupt expansion, the critical bubble length became unaffected by the walls of the upper pipe as the diameter was increased CE Keywords: Numerical Simulation, Taylor Bubble, change in geometry, AC oscillations, CFD ∗ Corresponding author (Tel: +44 (0)115 846 8079) Email address: david.hargreaves@nottingham.ac.uk (David M Hargreaves ) Preprint submitted to Computers and Fluids February 10, 2017 ACCEPTED MANUSCRIPT 1 Introduction The rise of Taylor bubbles is a well–documented and well–studied phe- nomenon in many fields, from chemical reactions in micro-scale systems to the eruption of volcanoes Taylor bubbles are elongated, bullet-shaped gas bubbles that move through stagnant or co-flowing liquid in horizontal, in- clined or, in the present context, vertical pipes (Figure 1) Research in this field has focussed in a variety of topics, in particular the characterisation of the rise rate of the bubbles (Taylor and Davies, 1950; Dumitrescu, 1943), the determination of the flow fields ahead of (Nogueira et al., 2006a), in the 10 liquid film around (Brown, 1965) and in the wake region behind the bubble 11 (Nogueira et al., 2006b) Despite these and numerous other studies, there 12 is a paucity of published experimental or numerical work on Taylor bubbles 13 that encounter a change in pipe diameter as they rise M AN US CR IP T James et al (2006) reported the results of an experimental investigation 15 into the rise single Taylor bubbles through a variety of pipe expansions and 16 contractions (using 0.038, 0.05 and 0.08 m diameter pipe sections) Sugar 17 syrup solutions of different concentrations, with viscosities of 0.001, 0.1 and 18 30 Pa s, were used to compare the rise behaviour across a range of Froude 19 numbers These experiments were monitored quantitatively by means of 20 pressure sensors and force meters and also qualitatively by video recording 21 They observed that when a Taylor bubble encountered an expansion in pipe AC CE PT ED 14 22 diameter, it rapidly expanded both vertically and laterally from the nose It 23 was hypothesised that this resulted in an increase in the flow in the liquid 24 film surrounding the bubble which caused the observed necking or pinching 25 of the bubble For bubbles above a certain length, this necking process splits M AN US CR IP T ACCEPTED MANUSCRIPT (a) (b) ED Figure 1: Examples of an air Taylor bubble rising through (a) water and (b) silicone oil (Clanet et al., 2004) the bubble into two or more daughter bubbles as shown in Figure 2, which 27 is a schematic of the process taken from James et al (2006) The splitting 28 will also generate oscillations in the pressure, which they measured both 29 above and below the expansion The objective of their work was to compare 30 the experimental pressure signals measured against the acoustic seismic data 31 recorded at active volcanic sites; their hypothesis being that the source of 32 pressure oscillations observed in seismic data are caused by a large bubble 33 of gas rising through a sudden expansion in the cross-sectional area of the 34 conduit They were able to show that the pressure changes measured during AC CE PT 26 CR IP T ACCEPTED MANUSCRIPT Figure 2: Sketches of the breakup of a long parent bubble into several daughter bubbles AN US (James et al., 2006) 35 in their experiments exhibited similar behaviour to those recorded in the 36 field, hence adding weight to their hypothesis Kondo et al (2002), whose primary focus was on co-current bubbly liquid 38 gas flow, conducted a number of experiments using single Taylor bubbles in 39 a quiescent liquid In these, a Taylor bubble rises through a pipe of diameter 40 0.02 m which undergoes a sudden expansion to one with a diameter of 0.05 m 41 Figure shows a still video image taken from Kondo et al (2002) showing the 42 bubble during the necking process – the poor quality is due to the standard 43 of photocopy available After the neck of the bubble closes, the rear of the 44 leading bubble bursts through the nose of that part of the bubble This 45 process can be observed in the still video images shown in Figure These 46 images have been taken after the sudden expansion but are cropped to the 47 central 0.02 m of the pipe AC CE PT ED M 37 48 Danabalan (2012) investigated the rise of Taylor bubbles as they move 49 from a straight, vertical pipe into either a rounded glass bowl or else a cubic 50 box – the rationale being that this is an analogue of a conduit in a volcano 51 expanding into a lava lake One novel aspect of the work was that she looked AN US CR IP T ACCEPTED MANUSCRIPT Figure 3: A still video image extracted and cleaned-up from Kondo et al (2002) which shows a Taylor bubble during the necking process while passing through a sudden expan- AC CE PT ED M sion from a pipe of diameter 0.02 m to 0.05 m in water Figure 4: A series of still video images extracted from Kondo et al (2002) which show a Taylor bubble which has passed through a sudden expansion from a pipe of diameter 0.02 m to 0.05 m in water ACCEPTED MANUSCRIPT for the maximum volume of bubble that could pass through the expansion 53 without breaking into two or more daughter bubbles It was found that 54 the critical volume at which this splitting occurred was dependent on the 55 viscosity of the liquid and the geometry of the expansion, with the rounded 56 glass bowl being able to support a larger bubble passing through intact 57 Notice that in Figure there is no evidence of the bursting of the bubble 58 from behind as was reported by Kondo et al (2002) This is due to the much 59 higher viscosity of the liquid in Danabalan’s work AN US CR IP T 52 Another experimental study recently conducted by Soldati (2013) em- 61 ployed a Hele-Shaw cell to investigate the effect of the angle of expansion, 62 fluid viscosity and volume of bubble may have on the rise characteristics A 63 Hele-Shaw cell is made up of two parallel plates some distance apart which 64 are sealed at the sides By varying the volume of air injected into the base 65 of the apparatus, different lengths of Taylor bubbles were generated in the 66 cell and it was possible to find the critical volume of bubble which can pass 67 through the expansion without splitting by the necking of the bubble Sim- 68 ilar to approach of Danabalan (2012), an exact value for the critical length 69 could not be found, but only upper and lower bounds for it Thus the critical 70 bubble volume was deemed to lie between a lower volume, which could pass 71 through the expansion unbroken, and an upper volume, when the bubble did 72 break up CE PT ED M 60 Soldati (2013) concludes that the critical length of a bubble decreases as 74 the angle of expansion, θ, (for a definition of θ, see Figure 7) increases from 75 30◦ to 90◦ , as shown in Figure This is consistent with the findings of Dan- 76 abalan (2012), in which a 90◦ expansion gave a smaller critical length than AC 73 CE PT ED M AN US CR IP T ACCEPTED MANUSCRIPT AC Figure 5: A photographic sequence of a 60 cm3 Taylor bubble injected into a liquid with viscosity of 68 Pa s moving into a rounded bowl (Danabalan, 2012) The upper bowl is filled with clear glucose syrup and the lower pipe is filled with glucose syrup mixed with red dye Images (a) to (f) show the passage of the first daughter bubble while (g) to (l) shows a second daughter bubble rising after a brief hiatus ACCEPTED MANUSCRIPT a more gradual increase in pipe diameter In Figure the filled circles indi- 78 cate the bubbles which did not break up as they rose through the expansion, 79 whilst the unfilled circles indicate those that did CR IP T 77 90 80 AN US Slope 70 60 M 50 PT 30 ED 40 10 12 Bubble volume (ml) 14 16 18 CE Figure 6: A diagram showing Taylor bubble breakup or otherwise for different angles of expansion (adapted from Soldati, 2013) Solid markers indicate bubbles that did not break AC up, hollow markers those that did 80 Figure 7, taken from Soldati (2013), shows a series of diagrams based on 81 still photographs that clearly illustrate the different stages of the breakup 82 mechanism As the nose of the bubble enters the expansion section of the 83 pipe, the nose of the bubble expands to fill the widening diameter as it is CR IP T ACCEPTED MANUSCRIPT Upper bound Lower bound AN US M ED Non-dimensional bubble length, Lb /D 20 CE 10 PT 30 40 50 60 70 Expansion angle, θ 80 90 100 AC Figure 18: A plot of the upper and lower bounds of the non-dimensional critical length, L , of bubble against the angle of the expansion 36 CR IP T ACCEPTED MANUSCRIPT Upper bound Linear fit, R2 = 0.997 Lower bound Linear fit, R2 = 0.998 AN US M ED Non-dimensional bubble length, Lb /D CE 0.5 PT 1.5 cosec θ 2.5 3.5 AC Figure 19: A plot of the upper and lower bounds of the non-dimensional critical length, L , of bubble against cosec θ 37 ED M AN US CR IP T ACCEPTED MANUSCRIPT the expansion The angle is CE 472 PT Figure 20: Schematic illustrating the definition of the angle φ φ = tan−1 ur uz (15) where ur and uz are the radial and axial components of the velocity, as 474 indicated in Figure 20 Figure 21 demonstrates a linear relationship between 475 φ and θ Clearly, the relationship will change as the bubble passes and the 476 extent of the necking increases, but this is a interesting insight nonetheless AC 473 477 Some insight into the splitting process can be garnered from arguments 38 ACCEPTED MANUSCRIPT 30 20 10 10 20 30 40 50 60 Angle of expansion, θ ( o ) 70 80 90 AN US 0 CR IP T o Angle of velocity, φ ( ) 40 Figure 21: Plot showing the linear relationship between φ and θ based purely on continuity either side of the expansion Consider, firstly, the 479 time taken for the bubble to rise through the expansion Assuming a bubble 480 of length, Lb is rising at a speed, wb , then the time, t1 , taken for the bubble 481 to pass through the expansion is ED t1 = Lb /wb (16) PT The rise speed can be recovered from the Froude number, wb Fr = √ , gD (17) CE 482 M 478 which for Taylor bubbles for an air-water system takes a value of 0.351 (Du- 484 mitrescu, 1943) For a bubble in a uniform cylindrical tube, continuity su- AC 483 485 ugests that the vertical velocity component of the falling water film, wfl , well 486 below the nose region, is wfl = wb rb2 r12 − rb2 39 , (18) ACCEPTED MANUSCRIPT l where r1 is the radius of the lower pipe (Figure 13) and the superscript 488 refers to the lower pipe Let us assume that as the bubble moves into the 489 upper pipe it does not immediately expand appreciably laterally and so, by 490 continuity, the vertical component of the flow around the bubble is, rb2 r22 − rb2 wfu = wb , u CR IP T 487 (19) where r2 is the radius of the upper pipe and refers to the upper pipe 492 This assumes local continuity as the flow around the nose of the bubble is 493 redirected down the side of the bubble that remains in the lower pipe If we 494 assume that this flow is guided by the wall of the expansion into the body of 495 the bubble, then the radial component of flow is AN US 491 (20) M ur = wfu sin θ While it passes through the expansion, the bubble is being squeezed by 497 this radial component of liquid flow in the proximity of the base of the ex- 498 pansion Assuming this component of the flow works its way into the bubble 499 at a constant rate, this will take a time, t2 , to pinch off the bubble PT ED 496 rb , wf sin θ (21) CE t2 = where rb is the radius of the bubble Equating t1 and t2 for the bubble of 501 critical length, Lc and rearranging gives AC 500 502 503 Lc = r22 − r12 r1 cosec θ, (22) where we have assumed r1 ≈ rb This result is clearly very simplified and includes no reference to the fluid properties However, if we take the 45◦ 40 ACCEPTED MANUSCRIPT case in Figure 18, which suggests a critical bubble length of approximately 505 3D or 0.114 m Equation 22 predicts a value of 0.092 m, which is surpris- 506 ingly good agreement for such a simplified model The agreement, however, 507 becomes worse for the 90◦ expansion critical lengths of 0.076 and 0.13 m for 508 the simulations and simple model respectively This clearly casts doubt on 509 the simplified model and suggests more work is required CR IP T 504 The experimental results of (Soldati, 2013) were also analysed in the same 511 manner, and are shown in Figure 22 From this it can be observed that a 512 linear relationship may exist between cosec θ and the critical bobble volume 513 (remember Soldati (2013) used a Hele-Shaw cell) However, the fit is not as 514 good as that for the CFD models of the present study In this case, a linear 515 regression analysis leads to a coefficient of determination, R2 of 0.97 This 516 reduced level of agreement may be due to the large increments between the 517 different volumes of gas injected during the experiments and the consequent 518 lack of accuracy 519 3.3 Variation of upper pipe diameter ED M AN US 510 A set of simulations was conducted in which the diameter of the upper 521 pipe was varied The angle of expansion was maintained at 90◦ and the di- 522 ameter of the lower pipe was maintained at 0.038 m during these simulations 523 The purpose of these simulations was to determine the effect of varying the 524 ratio between the diameters of the upper and lower pipes on the critical 525 length of the bubble It was hypothesised that there would be a critical ra- 526 tio at which the effect of the walls of the upper pipe played no role in the 527 splitting of the bubble AC CE PT 520 528 In these simulations, the diameter, D2 (= 2r2 ) of the upper pipe was 41 CR IP T ACCEPTED MANUSCRIPT 18 Upper bound Linear fit, R2 = 0.97 Lower bound Linear fit, R2 = 0.97 17 16 AN US 14 13 12 11 M Bubble volume (ml) 15 10 ED 1.5 cosec θ CE 0.5 PT 2.5 Figure 22: The upper and lower bounds of the critical volume of bubbles which can fully pass through the expansion before the neck closes against cosec θ for the experiments AC performed by Soldati (2013) 42 ACCEPTED MANUSCRIPT varied from 0.06 m to 0.14 m in increments of 0.02 m, which corresponds to 530 a variation in upper to lower pipe diameter ratios, D2 /D1 , of approximately 531 1.58 to 3.68 CR IP T 529 For the narrowest upper pipe, with D2 /D1 = 1.58, Figure 23 clearly shows 533 that the critical length of the bubble has increased when compared with the 534 90◦ expansion used in Section 3.2 and shown in Figure 18 where D2 /D1 = 2.1 535 This suggests that the narrowing of the upper pipe has an effect, allowing 536 longer bubbles through without breakup It is thought that this is due to 537 a reduced expansion in the head of the bubble Also, in this case there is 538 a smaller volume of liquid to cause the split in the bubble As the head of 539 the bubble expands (e.g Figure 16(a)), the flow is guided both around the 540 bulging head but also by the wall of the expansion Both are ultimately key 541 to determining the strength of the radial component that pinches the bubble 542 It should be clarified that the bubble bulges because the liquid flow governs 543 this process since the air is a relatively passive tracer in the process ED M AN US 532 As the ratio D2 /D1 increases, it can be seen that it is not until the ratio 545 gets above approximately 2.6 that the proximity of the walls of the upper 546 pipe cease to have an effect on the critical length of the bubble Further, the 547 experiments of Section 3.2 were conducted at ratio below this asymptotic 548 value It is also noted that at a ratio of 2.6, Equation 22 performs even more 549 poorly, suggesting that the model works only for smaller D2 /D1 ratios and CE PT 544 lower values of the angle of expansion θ AC 550 43 CR IP T ACCEPTED MANUSCRIPT Upper bound Lower Bound 2.8 2.6 AN US 2.4 2.2 M 1.8 ED 1.6 1.4 1.2 CE 1 PT Non-dimensional bubble length, Lb /D 1.5 2.5 3.5 Ratio of upper to lower pipe diameters, D /D AC Figure 23: A plot of the upper and lower bounds of the critical length of bubble against the ratio of the diameter of the upper pipe to the diameter of the lower pipe, D2 /D1 44 ACCEPTED MANUSCRIPT 551 Conclusions and Further Work Firstly, a comprehensive review of the existing literature on Taylor bubble 553 passing through an expansion was presented It became apparent that there 554 are a number of interesting features of this process Some bubbles break 555 into smaller parts, others remain largely intact as they pass through the 556 expansion The bubbles exhibit oscillatory behaviour as they traverse the 557 expansion, resulting in pressure variations in the liquid phase both above 558 and below the expansion AN US CR IP T 552 The qualitative and quantitative behaviour of Taylor bubbles rising thr- 560 ough expansions in pipe diameter observed during the laboratory experi- 561 ments reported by James et al (2006) was modelled using CFD A frequency 562 analysis of the results of the CFD simulations showed comparable dominant 563 frequencies to the experimental results The use of a CFD model also con- 564 firmed the qualitative mechanism proposed by James et al (2006) for the 565 breaking of a Taylor bubble as it passes through an expansion section ED M 559 A variation in the angle of the expansion revealed that much longer bub- 567 bles could pass intact through a more gradually expanding section than could 568 through a sudden expansion All bubbles were seen to “neck” or narrow as 569 they passed through the expansion The extent of this necking determined 570 whether the bubble would split into two daughter bubbles A linear variation 571 was found between the critical length of bubble which could pass through the AC CE PT 566 572 expansion section before the neck closed and the cosecant of the angle of ex- 573 pansion When analysed in the same fashion, the results of Soldati (2013) 574 also exhibited this trend 575 A simple model of the necking process, based purely on continuity argu45 ACCEPTED MANUSCRIPT ments, was shown to work over a very limited range of angle of expansion, but 577 did at least explain the dependency on the cosecant of the angle of expansion 578 For the 90◦ expansion, the simulations showed that varying the ratio of 579 diameter of the upper pipe to that of the lower pipe did result in a variation 580 in the critical length of bubble that could pass through the expansion intact 581 At lower ratios, longer bubbles would remain intact However, as the ratio of 582 upper to lower pipe diameters was increased beyond approximately 2.6, the 583 effect of the walls of the upper pipe ceased AN US CR IP T 576 The majority of the simulations were conducted using an air-water system 585 and there is considerable scope for analysing liquids of different viscosities to 586 examine the range of applicability of these findings 587 Acknowledgements M 584 We would like to thank the NERC for funding this research under grant 589 NE/G016593/1 In addition, the authors would like to acknowledge the use 590 of the High Performance Computer at the University of Nottingham 591 References 592 S Ambrose, D.M Hargreaves, and I.S Lowndes Numerical modeling of os- PT cillating taylor bubbles Engineering Applications of Computational Fluid Mechanics, 10(1):578–598, 2016 AC 594 CE 593 ED 588 595 J.D.P Araujo, J.M Miranda, A.M.F.R Pinto, and J.B.L.M Campos Wide- 596 ranging survey on the laminar flow of individual Taylor bubbles rising 597 through stagnant Newtonian liquids International Journal of Multiphase 598 Flow, 43:131 – 148, 2012 46 ACCEPTED MANUSCRIPT J.U Brackbill, D.B Kothe, and C Zemach A continuum method for mod- 600 eling surface tension Journal of Computational Physics, 100:335 – 354, 601 1992 602 603 CR IP T 599 R Brown The 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Taylor bubbles through a change in pipe diameter a CR IP T Stephen Ambrosea , Ian S Lowndesa , David M Hargreavesa,∗, Barry Azzopardia Faculty of Engineering, University of Nottingham, Nottingham,... pipe diameter ED M AN US 510 A set of simulations was conducted in which the diameter of the upper 521 pipe was varied The angle of expansion was maintained at 90◦ and the di- 522 ameter of the. .. undertaken to examine at which ratio of the 155 diameter of the lower pipe to that in the expansion that the walls of the 156 upper pipe no longer affect the behaviour of the bubble passing through