Plateau-Rayleigh Instability (PRI) is the well known phenomena of the breakup of a liquid column or cylinder. Such a process is integral to the operation of a range of natural and anthropogenic systems, such as gas-liquid and liquid-liquid separators, fuel cells, the accumulation of dewdrops on spider webs, and many more. Volume Of Fluid (VOF) methods, such as available in OpenFOAM, should be able to accurately resolve PRI in such systems. One such system, in which PRI is integral, is the filtration of oil or water aerosol mists using fibrous filters. In many cases, entrainment (or carryover) of liquid from fibers occurs. The mechanisms behind such entrainment are poorly understood. This work will validate the OpenFOAM VOF against classical PRI theory, both with and without a secondary fluid phase flowing through the system (e.g. air). Furthermore, the work will utilise the validated two-phase VOF solver to examine the phenomena of liquid reentrainment from mist filters

King, Andrew

Mead-Hunter, Ryan

Mullins, Benjamin

English

espace@Curtin

18th Australasian Fluid Mechanics ConferenceLaunceston, Australia3-7 December 2012Simulating Plateau-Rayleigh instability and liquid reentrainment in a flow field using a VOFmethodB. J. Mullins1,2, R. Mead-Hunter1,2 and A. J. C. King1,1Fluid Mechanics Research GroupCurtin University, GPO Box U1987, Perth, Western Australia 6845, Australia2Curtin Health Innovation Research InstituteCurtin University, GPO Box U1987, Perth, Western Australia 6845, AustraliaAbstractPlateau-Rayleigh Instability (PRI) is the well known phenom-ena of the breakup of a liquid column or cylinder. Such a pro-cess is integral to the operation of a range of natural and an-thropogenic systems, such as gas-liquid and liquid-liquid sepa-rators, fuel cells, the accumulation of dewdrops on spider webs,and many more. Volume Of Fluid (VOF) methods, such asavailable in OpenFOAM, should be able to accurately resolvePRI in such systems. One such system, in which PRI is in-tegral, is the filtration of oil or water aerosol mists using fi-brous filters. In many cases, entrainment (or carryover) of liq-uid from fibers occurs. The mechanisms behind such entrain-ment are poorly understood. This work will validate the Open-FOAM VOF against classical PRI theory, both with and withouta secondary fluid phase flowing through the system (e.g. air).Furthermore, the work will utilise the validated two-phase VOFsolver to examine the phenomena of liquid reentrainment frommist filters.IntroductionFine liquid jets and liquid films on cylindrical elements will,under most circumstances, be unstable and break up into a se-ries or array of small droplets via Plateau-Rayleigh instability[7]. There are a number of applications where the simulation ofthis phenomena may be advantageous, examples include, fibermanufacturing, wire coating, electro-spinning, fuel cells, opticfibers, surgical textiles and mist/droplet filtration.The convex surfaces of a cylindrical element (such as a fiber)indicate that positive curvatures will exist on any coating filmsthat develop; leading to a positive Laplace excess pressure act-ing on the film-air surface [6]. The Laplace pressure will gen-erally try to spread a liquid over a surface. In the case of afiber, where the film radius at the solid-liquid interface is dif-ferent from that at the liquid-vapor interface, this pressure willact to force the liquid out of the film and in doing so intro-duce an instability. Here, the liquid surface tension will act tominimize the free surface area, thereby creating the instability[8]. This instability will cause the liquid film to undulate andbreak up into an array of droplets. This was first observed byPlateau, who demonstrated that all axisymmetric wavelengthsgreater than the liquid cylinder (film) circumference will be un-stable. Of all the possible wavelengths, the fastest mode willpredominate [6]. This is the commonly known phenomena, ofPlateau-Rayleigh instability and for a thin film on a fiber thepredominant wavelength (λ) will be 2π√2r f [6], where r f isthe radius of the fiber.Therefore the thickest possible film which could exist, on a thincylindrical fiber, would be when the wavelength was equiva-lent to the circumference of the liquid cylinder coating the fiber.Such that 2π√2r f = 2π(r f +ht ).Mullins et al. [4] showed that this imposes a limit on the filmthickness, ht , of ht = r f (√2−1).The accurate resolution of Plateau-Rayleigh instability is vitalfor realistic 3D simulation of systems such as liquid (mist) fil-ters [1]. In order to develop CFD models for complex droplet-fiber (and fluid flow) systems, such as fibrous filters and ProtonExchange Membrane (PEM) fuel cells, we need to first ensurethat the simulations of the micro-scale physico-chemical pro-cesses are accurate. Given that coalescing filters work by cap-turing liquid aerosol droplets and that these droplets (assumingthey wet the fibre) spread over the fiber to form a thin film, theaccurate simulation of Plateau-Rayleigh instability will be im-portant in any attempt to simulate such droplet-fiber systems asa whole. We seek to demonstrate that such simulations can beaccomplished using an open-source Computational Fluid Dy-namics (CFD) Volume-Of-Fluid (VOF) solver; thereby demon-strating that CFD is an appropriate, accessible and viable tech-nique for the simulation of droplet fibre systems. The simula-tion then, does not rely on the derivation of an appropriate nu-merical expression, but on a more fundamental solution to themomentum and continuity equations.MethodologyOur simulations utilised the open-source OpenFOAM CFDpackage (Silicon Graphics Inc. Sunnyvale, USA). An unstruc-tured hexahedral mesh was generated over a single 10 µm diam-eter cylinder, located centrally and oriented either vertically orhorizontally within a square prism, with the longest side beingequal to (or in the reentrainment studies greater than) the lengthof the fiber. A fluid column (film) was placed around the fibrewith a specified thickness. The flow field within the film wasset to have zero velocity; i.e. it is stationary. A cyclic boundarycondition was set between the two faces of the geometry inter-sected by the fiber. The solver used was a three-dimensionalunsteady solver, that utilised the Multidimensional UniversalLimiter with Explicit Solutions (MULES) algorithm to handlethe transport equation and the Pressure Implicit with Splittingof Operators (PISO) algorithm for the coupled pressure-velocityfields.The transport models implemented in OpenFOAM allow for thecontact angles of the two-phase interface to be defined. This re-quired specification of the parameters θ0, µθ, θA and θR. Thevalue of θ0, the static contact angle, was selected based on pub-lished experimental data [1]. The value of the dynamic contactangle velocity scaleµθ was set to 1. The values of θA and θR (thelimiting advancing and receding contact angles, respectively)were set to 1.047 and 0.01 radians. These values are not the dy-namic contact angles of the system, but limiting ranges for thesimulation and were chosen to give the system a wide permissi-ble range and also to avoid negative numbers and zeros, whichmay have caused the calculation to become unstable.Table 1: Film thicknesses and observed break-up, without flowfield. The maximum theoretical ht is 2.07 µmht (µm) Break up Notes t (s) to Break upu=0 m/s / u=0 m/s / u=0.1 m/su=0.1 m/s1 no / no unstable* - /-1.1 no / no unstable* - /-1.2 no / no unstable* - / -1.3 no / no unstable* - / -1.4 yes / yes stable (2)+ 0.00054 / 0.000411.5 yes / yes stable (2)+ 0.00055 / 0.000411.6 yes / yes stable (2)+ 0.00074 / 0.000621.7 yes / yes stable (2)+ 0.00074 / 0.000621.8 yes / yes stable (2)+ 0.00102 / 0.000861.9 yes / yes stable (2)+ 0.00108 / 0.000862 yes / yes stable (2)+ 0.00108 / 0.000873 yes / yes stable (2)+ 0.0011 / 0.00095 yes / yes stable (2)+ 0.0011 /0.0009- the film does not break up into droplets, however the surfaceprofile changes over time.+ - the film breaks up into 2 axisymmetric dropletsIn order to test the onset of Plateau-Rayleigh instability, thethickness of the film was varied between simulations. The max-imum possible film thickness for a given system can be deter-mined using Equation ht . Therefore by simulating film thick-nesses either side of this maximum value, the effectiveness ofOpenFOAM in simulating the Plateau-Rayleigh instability canbe determined.In addition to the “static” cases examined, a laminar (steady)flow solver was used to impose a second fluid (flow) field (airfrom left-right of geometry) on the original fluid (droplet/film)phase (oil) which allowed droplet motion and displacement in aflow field to be examined, and permitted comparison with pre-vious results [1]. Gravitational forces were not considered insimulations, as it has been proven insignificant for dd < 100µm.Results and DiscussionSimulations were conducted on fiber lengths of 175, 225 and500 µm to ensure that the (cyclic) geometry boundary did notinfluence the film break-up. According to The equation for ht ,the thickest film which should exist on a 10 µm diameter fiberis 2.07 µm. Therefore film thicknesses around this value wereexamined. The results of the simulations run without the pres-ence of a flow field, are given in Table 1. Under-relaxation wasalso utilized, by placing relaxation factors of 0.8 and 0.3 on theinitial estimates of the pressure and velocity respectively.Fig. 1 shows the break-up of a 2 µm thick film on a 10 µmdiameter fiber. Fig. 1a shows the initial film (i.e. at t = 0 s),which simply displays the liquid film on the fiber. Fig. 1b showsthe film at t = 0.00078 s, which has begun to undulate and nolonger appears completely cylindrical. Fig. 1c, d and e show thefairly rapid formation of axisymmetric droplets at t = 0.000845,0.00086 and 0.00087 s, respectively. Fig. 1f shows a distinctcentrally located axisymmetric droplet at t = 0.00092 s, withdroplets immediately above and below extending to the boundsof the geometry.The results in 1 show that the simulated break up occurs at alower value of ht than predicted by ht . This however is per-missible, as the value obtained from ht represents a maximumallowable film thickness. Therefore it is clear that the solverwill not allow a physically unrealistic film to exist on the fiber.The films that do not break-up are considered unstable, as thefluid surface does not attain a consistent conformation, i.e. re-mains clearly as a film, however the surface position exhibitsslight fluctuations over time.(a) (b)‘(c) (d)(e) (f)Figure 1: A series of images showing the simulation ofthe break-up of a liquid film under the influence of Plateau-Rayleigh instability, from t=0 (a) to t=0.00092 (f)(a) (b)Figure 2: The space between droplets formed by the simulationof the break up of a 2 µm film via the Plateau-Rayleigh insta-bility, in absence of flow, (a), and in the presence of flow, (b)The film break-up was also simulated in the presence of flow,using the same film thicknesses. When compared with the re-sults in 1, however, it was found that the maximum film thick-ness that the solver will allow is ultimately no different in thepresence of a flow field; however, the break-up of the film doeshowever occur at an earlier time-step in the simulation. So thepresence of air flow velocity (u) (0.1 m/s in this case) will in-fluence the time to break up of a liquid film but not the filmstability. Additionally, the droplet positioning on the fibre isdifferent in the presence of flow; this can be seen in Fig. 2. Thiseffect was verified over multiple simulations and is likely dueto the system acting to minimize overall drag force.The simulation of the break up of a liquid film into an array ofdroplets via Plateau-Rayleigh instability could be further vali-dated using experimental observations [6], where droplet spac-ing was found theoretically to (usually) correspond to the lengthof the predominating wavelength. This wavelength as deter-mined by Quere [6], is 2π√2r f , and therefore on a 10 µm di-ameter fiber, the average droplet spacing will be 44.43 µm. Thishowever, relies on ht << r f , which is incorrect in our case sincewe have a 2 µm film on a 5 µm radius fiber. As is shown inFig. 2 we can measure droplet spacing from our results. Herethe droplet spacing can be seen to be greater in the presence offlow, which is likely related to the earlier film break up. Meandroplet spacing for multiple simulations was found to be 71.9± 5 µm for u=0 and 81.2 ± 5 µm for u=0.1, with no significantdifference observed for higher values of u.The droplet spacing was found to be consistent, regardless of fi-bre length and is not influenced by the presence (or absence) ofcyclic boundary conditions. However, given the difference be-tween the simulated and theoretical droplet spacing on this fibrediameter, it was decided to explore the simulated droplet spac-ing further. Fig. 3 shows the simulated (VOF) and theoretical(Quere [6] theory) droplet spacing resulting from the break-upof a film of thickness ht on a number of fibre diameters. It canbe seen that good agreement between theory and simulation ex-ists between 15 and 20 µm and near 30 µm. Some deviationoccurs above and below these values. This indicates that thebreak-up of a liquid film into a droplet array can be simulatedreasonably well.Figure 3: Simulated and theoretical [6] droplet spacing, result-ing from the break-up of a liquid film of thickness ht on differentfiber diametersMead-Hunter et al. [1] introduced a theoretical model, whichdescribed the force required to move a droplet along a fiber. Inthis model a term describing the displacement (r) of the masscentre of the droplet away from the center of the fiber was pro-vided. Increasing r was found, both theoretically and experi-mentally, to decrease the force required to move a droplet. Thiseffect should then be present in the simulated system.Droplet displacement (r) in air flow was examined by varyingthe flow velocity (u) acting on the system. r=0, when u=0, andincreases as u increases.A droplet that does not drain at u=0.1 m/s, may drain at u=0.2m/s. This is due to the combined effects of increased air flowforce acting on the droplet and the effect of r, which decreasesthe force needed to drain a droplet of a given size.At u=0.1 m/s the droplet appears slightly displaced, with a smallvalue of r. At u=0.2 m/s, the value of r is slightly larger andthe droplet drains down the fiber much more readily. Henceless force is required in order to initiate droplet motion. Theinfluence of r may be seen in Fig. 5a, where the relationshipbetween u, F and r is shown. Here r is considered in terms ofr/b (where b is the droplet radius) as this allows droplets andfiber of different sizes to be compared [5].The force acting on a droplet in the simulation may be calcu-(a) (b)(c) (d)(e)Figure 4: A series of images showing the movement of a coa-lesced droplet down a fiber under the influence of flow (u=0.2m/s), and the merging of this droplet with the droplet immedi-ately belowlated by extracting the gradients from the surface normals atthe two-phase interface and multiplying by the liquid viscosity(performed using a custom script). The simulated force can bereadily compared to the force predicted by classical drag equa-tions (assuming a spherical droplet). The drag force (Fd) on asphere is given by; Fd = 3πρddu, and the drag on a cylinder(fiber) by; Fc = 12 ρu2dcLCD, where dc is the cylinder diameter,L the cylinder length, and CD the drag coefficient.Both previous drag equations hold for low values of Reynolds’number (Re, <1); Re = ρuddµ , where, ρ is the density of the fluid(air), dd is the droplet diameter and µ is the fluid viscosity. Thedrag force of a “barrel” droplet on a fiber may then be approxi-mated as Ft = Fd −Fc.Fig. 5b shows the simulated forces acting on the droplets com-pared to the drag force on a sphere and cylinder. Re is plotted onthe x-axis as this accounts for both u and b. The difference canbe seen to become more pronounced as the velocity increases,most likely due to the simulated droplet not being truly spheri-cal and also able to be deformed in the flow field, whereas in ??the particle remains perfectly spherical. It is possible to obtainthe relationship between F and Re and that between F and r/bfrom the linear fits shown in Fig. 5, such that; F = 96.31Re,and F = 59.28 rb . Thus, the force can be considered in termsof dimensionless parameters. Additionally, we can empiricallyderive an expression for r, such that; r = 1.625bRe, therefore,the droplet displacement may be found from the droplet sizeand flow parameters. This expression should hold for a range ofdroplet sizes, provided that the Reynolds number is low andthe surface tension of the liquid similar. Previous work byMullins et. al.[5] attempted to measure the force required topull droplets off a fiber, however, these forces were significantlyless than those simulated in the current work. Fig. 6 shows the(a)(b)Figure 5: Forces calculated from results and Ft (a) shows therelationship between u, r/b and F for the simulations. (b) showsF obtained from the simulation and the drag equations againstRe, the values shown are of a series of repeated simulations. Theerror bars on (a) indicated the error associated with measuringr and b from the simulationre-entrainment of a 50 µm droplet after axial motion along thefibre, and a plot of the axial droplet velocity vs (parallel) airflowvelocity. The forces simulated in the current work, however, arecomparable with those predicted by the drag equations and ofthe same order of magnitude as forces measured on similar sys-tems presented in the authors’ previous work [1, 2]. Thereforewe assert that the VOF solver can accurately simulate the forcesacting on a liquid droplet.ConclusionsThis work has shown that it is possible to accurately simulatethe break-up of a liquid film coating a cylinder under the influ-ence of Plateau-Rayleigh instability. The VOF solver in Open-FOAM is able to simulate this break up based on solutions to thetransport equations, without the need to develop detailed numer-ical models. The solver allows the force acting on the systemat any point (e.g. the droplet surface, to be calculated). Realis-tic droplet displacement under the influence of the velocity fieldis also shown. Therefore, CFD can be seen as a valuable toolin the modelling of droplet-fiber systems [3], including morecomplex systems, or even completely disordered systems, suchas mist filters. Furthermore, it has been shown that the Open-FOAM VOF can also resolve re-entrainment of liquid dropletsfrom fibres and mist filters.AcknowledgementsThis work was supported by an Australian Research CouncilLinkage Grant (LP0883877) and MANN+HUMMEL GmbH. *References[1] R. Mead-Hunter, B. J. Mullins, T. Becker, and R. D. Brad-dock. Evaluation of the force required to move a coalesced(a)(b)Figure 6: (a) Re-entrainment of a droplet after axial motion atu=50 m.s−1. (Airflow L-R parallel to fibre) (b) Droplet axialvelocity ud vs airflow velocity u (prior to re-entrainment)liquid droplet along a fiber. Langmuir, 27(1):227–232,2011.[2] R. Mead-Hunter, T. Bergen, T. Becker, R. A. O’Leary,G. Kasper, and B. J. Mullins. Sliding/rolling phobicdroplets along a fiber: Measurement of interfacial forces.Langmuir, 28(7):3483–3488, 2012.[3] R. Mead-Hunter, A. J. C. King, and B. J. Mullins. Plateaurayleigh instability simulation. Langmuir, 28(17):6731–6735, 2012.[4] B. J. Mullins, I. E. Agranovski, and R. D. Braddock. Effectof fibre orientation of fibre wetting processes. J. Coll. Int.Sci., 269(2):449–458, 2003.[5] B. J. Mullins, A. Pfrang, R. D. Braddock, T. Schimmel, andG. Kasper. Detachment of liquid droplets from fibres - ex-perimental and theoretical evaluation of detachment forcedue to interfacial tension effects. Journal of Colloid andInterface Science, 312(2):333–340, 2007.[6] D Quere. Fluid coating on a fiber. Annu. Rev. Fluid Mech.,39:347–384, 1999.[7] Ryong-Joon Roe. Wetting of fine wires and fibers by a liq-uid film. Journal of Colloid and Interface Science, 50(1):70 – 79, 1975.[8] A. L. Yarin, G. G. Chase, W. Liu, S. V. Doiphode, and D. H.Reneker. Liquid drop growth on a fiber. Aiche Journal, 52(1):217–227, 2006.

Simulating Plateau-Rayleigh instability and liquid reentrainment in a flow field using a VOF method

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Simulating Plateau-Rayleigh instability and liquid reentrainment in a flow field using a VOF method

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