Cahn-Hilliard diffuse interface simulations of bubble-wall collisions
aa r X i v : . [ phy s i c s . f l u - dyn ] M a y Cahn-Hilliard diffuse interface simulations of bubble-wallcollisions
Sohrab Towfighi a,b , Hadi Mehrabian a,c, ∗ a Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, V6T1Z3, BC, Canada b Faculty of Medicine, University of Toronto, Toronto, M5S 1A8, ON, Canada c Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, 02139, MA,USA
Abstract
The collision of a rising bubble with a superhydrophilic horizontal surface is studied nu-merically using the Cahn-Hilliard diffuse-interface method. For the studied systems, theBond number, Bo = ρR g/σ , varies between 0.25 to 1.5, and the Galileo number, Ga = ρ ( R g ) / /µ , changes between 8.5 to 50. We assume that the viscosity and density of thebubble are negligible compared to the surrounding medium. We show that our computationsreproduce experimentally observed dynamics for the collision of a bubble with a solid sur-face. Furthermore, for the studied range of parameters, the bubble-wall collision is mainly inthe inertial regime. Our simulations show that even in the absence of substantial viscous dis-sipation, the ratio of rebound-to-collision velocity, the so-called coefficient of restitution, ismuch smaller than one for the studied systems. More importantly, the coefficient of restitu-tion best scales with the Froude number, F r = U t / √ gR , the ratio of inertial to gravitationalforces. Keywords: bubble rise, Cahn-Hilliard diffuse interface method, Froude number,superhydrophilic surface, coefficient of restitution, thin film, added-mass effect
1. Introduction
Bubbles hold great scientific interest because of their occurrence in many physical systemsand their utility in numerous applications. The interaction between a bubble and a solidsurface has received considerable attention experimentally [1–3], computationally [4–6], andanalytically [7, 8]. In this study, we are particularly interested in the impact of bubbles ona non-wetting horizontal surface. Consider a bubble inside a large liquid container, with asuperhydrophilic surface on the ceiling of the tank as is illustrated in Fig. 1. The bubble-wallimpact process for such a system can be best described in three stages: acceleration of the ∗ Corresponding author.
Email addresses: [email protected] (Sohrab Towfighi), [email protected] (Hadi Mehrabian)
September 22, 2018 ubble from the initial static state until its velocity and shape become steady, decelerationon approaching the wall, and rebound. We aim to characterize the somewhat complicatedcollision process with a simple lump parameter called the coefficient of restitution, similarto what has been done for the characterization of the impact of solid particles and dropletswith solid surfaces. Z t* = -11 -7 -4 -1 0 2 4 Figure 1: Snapshots of the bubble rise for a system with Bo = 0 . , Ga = 40 ( Re = 75 , and W e = 1 . ).Time and z-coordinate are in dimensionless form. The initially static bubble accelerates under the action of buoyancy and reaches a steadyshape and velocity after some time, the length of which depends on the balance of viscousand gravitational forces. For a highly viscous surrounding liquid, the Stokes flow regimeoccurs, meaning that the bubble remains spherical, and by equating the drag and buoyancyforces, its terminal velocity can be calculated. For a bubble moving at velocity U t , if theinertial pressure, ρU t / , dominates the Laplace pressure, σ/R , then surface tension willgive way to inertia and the bubble will deform from its spherical shape. The ratio of thesetwo pressures is called the Weber number W e = ρU t R/σ and is an indication of the shapeelongation in the direction normal to the flow, resulting in a non-spherical shape which willbe called a spheroid shape hereafter. In most real cases, and also in our simulations, the
W e number is larger than one, hence the bubble’s shape deviates considerably from perfectsphericity. There are different regimes for the dynamics and shape of a rising bubble whichare nicely delineated by Tripathi et al. [9] in terms of the bubble Bond (Eötvös) number, Bo = ρR g/σ , and Galileo number, Ga = ρ ( R g ) / /µ . At larger Bond number, anotherphenomenon that complicates bubble rise is the onset of path instability in the form of azig-zagging or helical rise trajectory [10]. In this study, we limit ourselves to a regime withspherical or spheroidal bubbles without unsteady motion during the rise, which means thatwe restrict this study to the range of Bond and Galileo numbers which produce spheroidalor spherical bubbles without path instability.The collision of a rigid particle with a solid wall can be characterized by a lumpedparameter called the coefficient of restitution, defined as ε = U r /U t , in which U t and U r are the velocity of the particle before and after the collision, respectively. The coefficientof restitution is indicative of the dissipated energy during the impact. It is shown that thecoefficient of restitution for the impact of solid particles with a rigid wall scales with Stokesnumber, which is the ratio of inertial to viscous effects [11, 12]. Similarly, Legendre et al.[13] found that the impact of the droplets on the solid wall could also be characterized bya modified Stokes number which includes the inertia of the associated surrounding fluid. Itis interesting to examine the universality of this description for the collision of bubbles onto September 22, 2018 igid surfaces. In fact, we will show that the bubble collision follows an entirely differentscaling.In reality, there are complications which bring uncertainty to the experimental resultson bubble-wall collisions; the dynamics are usually affected by an array of subtle effects,including surfactants [14, 15], wall roughness [16] and the presence of micro-bubbles nearthe hydrophobic wall which are unavoidable [17]. This makes a computational study withcomplete control over input parameters necessary. Most intriguingly, Zenit and Legendre [18]suggested that the coefficient of restitution ε can be fitted into two different expressions,either as a function of the modified Stokes number St ∗ = C m ρRU t / µ or as a functionof a modified Ohnesorge number Oh ∗ = µ/ √ ρ ∗ σR , where σ is the surface tension and ρ ∗ = ρ b + C m ρ accounts for the added mass effect with the coefficient C m calculated fromLamb’s classical formula, C m = (( χ − / − cos − χ − ) / ( cos − χ − − ( χ − / χ − ) . Inthis formula, ρ b is the density of bubble and χ is the bubble aspect ratio defined as R h /R v ,in which R h and R v are the horizontal and vertical radius of the bubble spheroid. Howeverdue to uncertainty in their data and limited range of the input parameters they were unableto verify their hypothesis. Therefore, an investigation into the coefficient of restitution iswarranted.We use the Cahn-Hilliard diffuse interface method to simulate the collision of the bub-ble to a super-hydrophilic horizontal surface. It has been shown that this method is verysuccessful in reproducing the experimental interfacial dynamics of a wide range of systems[19–21]. Similar to real interfaces, this model considers the finite thickness of the interfaceand captures its dynamics using an energy-based formalism whose details will be discussedin section 2.2. This paper is organized as follows. We begin by detailing the problem setupand the computational method; then we demonstrate that the model can reproduce experi-mental work well. Subsequent the exposition, we dive into a presentation of the findings bygiving a general description of collision phenomena, the scaling for bubble velocity and thecoefficient of restitution, followed by a discussion.
2. Numerical simulations
The simulations are done in an axisymmetric geometry where a spherical bubble withradius R is released with zero initial velocity at the bottom of a liquid cylindrical column.The bubble then rises along the axis of symmetry, whose height is 11 R . This is illustrated inFig. 1. The bubble is initially placed with its centroid at 1.5 R above the bottom boundaryand 4 R from the side boundaries. Inspired by the experimental setup used by Zenit andLegendre [18], we choose the computational domain size to be sufficiently large so thatthe side and bottom boundaries do not influence the bubble dynamics, and we verify thisthrough numerical tests. This is facilitated by imposing a “relaxed" boundary condition onthe side and bottom boundaries: vanishing tangential velocity and normal stress equal tothe hydrostatic pressure [22]. This allows flow through these boundaries and minimizes theirconfinement effect. The top wall has a no-slip boundary condition, and its contact angle isset to 180 degrees to make it completely non-wetting. September 22, 2018 he physical problem is determined by the following dimensional parameters: the bubbleradius R , surface tension σ , the gravitational acceleration g , and the density and viscosityof the bubble and liquid phases denoted by ρ b , µ b and ρ , µ , respectively. Besides the densityratio ρ b /ρ and viscosity ratio µ b /µ , this system can be described by two dimensionless groups;we use the two groups suggested by Tripathi et al. [9] to categorize the bubble dynamics,namely the Bond number Bo = ρgR /σ and the Galileo number Ga = ρ ( R g ) / /µ . Thebond number shows the ratio of gravitational to surface force, and Galileo number indicatesthe ratio of gravity to viscous force and is an indicator of the bubble terminal velocity. The computational method is based on the diffuse-interface framework [23–25] used todynamically capture the location of the interface. To demarcate the interface between thebubble and the surrounding liquid, a phase-field parameter φ evolves as governed by theCahn-Hilliard equation. This is solved together with the Navier-Stokes equation using afinite element solver on an unstructured triangular grid with the adaptive mesh refinement atthe moving interface. The numerical scheme uses fully implicit time-stepping and Newton’smethod to handle the nonlinear equations. Details of the theoretical model and numericalmethod and parameters can be found in previous publications [23–25]. In particular, thediffuse-interface model introduces two additional parameters to incorporate the nanoscalephysics into macroscale simulations; the Cahn number Cn = ǫ/d which is an indicator of theinterfacial thickness ǫ , and the diffusion length scale S = l d /R which represents the Cahn-Hilliard diffusivity l d . More detailed description of the parameter definitions can be foundin [26]. For the diffuse-interface model to make accurate predictions, the interface must besufficiently thin so that the sharp-interface limit is achieved. This indicates that the resultsare independent of the interfacial thickness and requires the interface to be adequatelyresolved by having locally refined grids [26]. Extensive testing of Cn and S values havebeen carried out before and the results validated against sharp-interface benchmarks [24–26]. Our numerical experiments have shown that for Cn < − , the sharp-interface limit isapproached. In the results to be presented here, we have used Cn = 10 − when the bubbleis far from the wall and we reduce to Cn = 6 . × − when the bubble is close to the wall.This balances the high computational cost with the need for a thinner diffusion interface toaccurately capture the dynamics of the thin layer between the bubble and the wall.The choice of S is a subtle issue [25, 27] and its value should be calibrated by an experi-mental or theoretical data point. We found that S = 5 × − produces excellent agreementwith the experiments as is shown in Fig. 2). Although the liquid in principle never dewetsthe upper wall, Cahn-Hilliard diffusion may exert a minor effect on the interfacial dynamicswhen the liquid film becomes very thin. In this section, we matched the dimensionless parameters used in our model with thosefrom experiments for the rectilinear rise of an ultra-clean bubble and its collision with a glasssurface. In figure 2, we compared the numerically calculated values for the center of massposition, center of mass velocity, and the bubble’s aspect ratio to the experimental data
September 22, 2018 . − − − − u (cid:2) Z t
Figure 2: Comparison of simulation results with experimental data of Ref. [28]. Experimental data is fora non-polar liquid, free from surfactant effects. Experimental values are given as points while simulatedvalues are presented as curves; ( + ), ( (cid:3) ), and ( ◦ ) represent the center of mass position Z , velocity u , anddeformation ζ , respectively. The dimensionless parameters for this case are Ga = 26 , and Bo = 0 . . Theinsets show axisymmetric bubble geometry at t = -1.20, -0.24, 1.50, and 3.80. The arrow points to a pointof discrepancy discussed in the text. of Ref. [28] to validate the computational results. The experimental result is for a bubblewith a clean interface; by using a non-polar oil, Zenit and Legendre were able to avoidcomplications due to surfactants [28]. The Galileo number and Bond number are Ga = 26 ,and Bo = 0 . for this system, respectively. The viscosity ratio matches the experimentalvalue of µ b /µ = 0 . , although numerical experiments show the bubble viscosity to benegligible for µ b /µ < . . An exception to exact parameter matching is the densityratio. The experimental value for air-water density ratio, ρ b /ρ = 0 . , would be too smallfor accurate resolution in a diffuse-interface formalism [24]. In numerical experiments, wefound that the bubble inertia becomes entirely negligible for ρ b /ρ < . , and have used ρ b /ρ = 0 . for the computations. First, we computed the terminal velocity and shape ofthe bubble using different domain sizes and confirmed that the bubble dynamics are freefrom boundary effects. Then, we recorded the center of mass position for the bubble, z c ,as measured from the bottom of the geometry, the velocity of the bubble u = dz c /dt , andthe bubble aspect ratio χ = R h /R v as functions of time. All lengths are scaled by R andvelocities by U t . To determine the aspect ratio, we measured the distance between thebubble’s horizontal extremes (the horizontal radius R h ) and then applied conservation ofvolume to obtain the corresponding spheroid’s vertical radius R v .Figure 2 shows the simulation overlaying experimental data, with the bubble’s center ofmass position, instantaneous velocity, and aspect ratio plotted as functions of time. Theorigin of time is set at the point of zero velocity when the bubble centroid is roughly closest tothe wall and the bubble at maximum deformation. Consistent with the data presentation inRef. [18], the bubble position is indicated by Z = z c − . and its deformation by ζ = χ/χ − , χ being the aspect ratio of the bubble. Figure 2 shows near perfect agreement betweensimulation and experiment. Note that the numerical solution closely tracks the experimentuntil the end of the first cycle. In later cycles, the discrepancy grows, with the numericaloscillation decaying more slowly than the experiment. We suspect that the deviation stems September 22, 2018 rom the extended close interaction between the bubble surface and the wall, which in ourmodel may have introduced extra friction due to interfacial diffusion. Also, we compared thecomputed terminal velocity and shape of the bubble between numerical and experimentalresults [6, 29, 30]. The calculated bubble aspect ratio differs from experiments by . to . and the terminal velocity by 0.3% to 7.4%. These values are well within the scatter ofexperimental data, and the agreement is better than previous numerical results [6, 30].
3. Results and discussion
We can divide the bubble-wall collision process into three stages, namely, the acceleration,approach, and rebound stages [18]. In our simulations, the bubble starts from a static state,shown at t = − in Fig. 1, and accelerates due to the buoyancy force to its terminal velocity, U t , and shape, χ t , at t = − . The acceleration stage is not the focus of this study, so it isnot shown in Fig. 2. The second stage, the approach, occurs as the bubble feels the presenceof the wall which happens between t = − and t = 0 in Fig. 2. During this stage, thebubble velocity decreases while its deformation increases; some part of the incoming kineticand gravitational energy transforms into the surface energy, and part of it gets dissipatedby the viscous forces. As the bubble approaches the wall, its deceleration is accompanied bydrainage of the liquid film between the bubble and the wall. This leads to the formation ofa central dimple and the appearance of the rim at its closest point to the wall. Eventually,bubble velocity vanishes at t = 0 , defined as the point of collision. However, as we willdiscuss later, this does not mean that the flow field around the bubble vanishes at thisstage.The third stage, called rebound, starts when the bubble reaches the zero velocity pointat t = 0 . During the rebound, the bubble’s centroid velocity increases to a maximum valuein the downward direction and then returns to zero at the end of the rebound at t = 1 . Atthe end of rebound, the bubble’s center of mass is at its farthest position from the wall.For an entirely elastic collision, the bubble would be far enough from the wall to have a fullcycle of acceleration, approach, and rebound again. However, in reality, there is energy loss,and another cycle of acceleration, approach and rebound could only occur if the bubble getsfar enough from the wall at the end of its initial rebound. The bubble can undergo severalrounds of oscillation, each time marked by an exchange between surface energy, gravitationalenergy, and kinetic energy while viscous forces dissipate the energy of the system. In thenext section, we will discuss that viscous dissipation is not the only source of the momentumloss for studied bubble-wall collisions. Fig. 2 shows the second round of oscillation. In theend, at t ≈ in figure 2, all the kinetic energy has been used, and the bubble has come torest against the solid top wall. An important parameter that characterizes the collision of solid particles or droplets isthe ratio between their momentum before and after the collision, the so-called coefficientof restitution, ε . This parameter is a measure of the dissipated energy during the collision. September 22, 2018 r = 0.34 Fr - 0.34 Figure 3: Coefficient of restitution as a function of the Froude number.
Analogously, it is interesting to measure this quantity for the bubble-wall collision. Themost natural way to define this ratio for the bubble-wall collision is to use the bubble’sterminal velocity to characterize the incoming momentum and the maximum velocity of thebubble during its rebound ( U r in Fig. 2) to determine the return momentum. ε = − U r /U t , (1)The negative sign accounts for the change in direction during collision. It has beenshown that the coefficient of restitution for the droplet or particle collisions scales with theratio of the inertial to viscous forces, which can be measured either by the Reynolds orStokes number. However, we found that for the bubble-wall collision the scaling is different.We tested all governing dimensionless groups and discovered that the best collapse of ournumerical data towards a single parameter linear expression happens when ε is plottedagainst the Froude number, defined as F r = U t / √ gR . All the data from our simulationsfor the coefficient of restitution is presented in Fig. 3 which show that the ratio of themaximum rebound velocity to terminal velocity increases almost linearly with the ratio ofinertial to gravitational force, i.e., the F r number. The Froude number manifesting itselfas the most dominant control parameter needs further investigation, as this scaling differsfrom that of the collision of droplets and particles. Moreover, if we consider the magnitudeof the coefficient of restitution in Fig. 3, we see that they are less than 0.4, suggesting thatthere is a considerable momentum loss during the collision process. However, both gravityand inertia, whose ratio creates the Froude number, are conservative forms of energy, and itseems contradictory to represent the momentum loss in terms of conservative forces. In otherwords, if inertia and gravity govern the collision, then why is there a considerable reductionin the bubble momentum after the collision? Commonly, when there is an energy loss inthe fluid system, the first culprit is the viscous dissipation. The critical parameter thatrepresents the relative significance of the viscous effects is the Re number of the bubble. Inour simulations, the minimum Reynolds number is 10 which shows the relative dominance of September 22, 2018 − − − − u (cid:0) Z t
Figure 4: Bubble approach is mainly inertial. For the shown systems, Bo = 0 . and the corresponding Ga changes from 20 to 50. the inertial effects. Also, an interesting observation from our simulations is that the timescaleof the approach stage mainly correlates with the inertial timescale. In other words, if we scalethe velocity with the terminal velocity of the bubble and the time with the characteristicinertial time scale defined as t c = RU t then all velocity profiles collapse almost onto a singlecurve. In figure 4, the dimensionless velocity profile, U ( t ) /U t , the position of bubble’s centerof mass, and the aspect ratio are plotted for cases with varying viscosity of surroundingliquid. These results suggest that the approach dynamics is independent of the viscosityof the liquid due to negligible viscous force. This is similar to the inertial drainage of theinterstitial fluid for droplet and solid particle collision with a rigid wall. If viscous dissipationis not considerable, then why is there a significant decrease in the momentum of the bubbleafter the collision?We believe that the answer is rooted in the importance of the added mass effect forbubble dynamics. Due to the negligible density of the bubble, the kinetic energy is storedmainly in the fluid surrounding the bubble. Since the surrounding fluid is not governedby surface tension, not all of the incoming kinetic energy will be transformed into surfaceenergy during the approach. Some part of the kinetic energy will be changed into radialmomentum (the direction parallel to the top wall in Fig. 1) and will not be recovered duringthe rebound to push the bubble the bubble away from the wall. It is important to notethat the bubble rebound starts while the surrounding liquid is still moving toward the wallin the z-direction. This is illustrated in a series of snapshots in Fig. 5. In Fig. 5(a), thebubble reaches its maximum spreading and starts to retract, while the velocity field in thesurrounding fluid is in the upward direction. Such a velocity field will hamper the bubblerebound and creates another dimple on the lower surface of the bubble. Therefore, even inthe absence of the considerable viscous dissipation, the bubble-wall collision could result ina major momentum loss.It is interesting to formulate any problem in terms of the dimensionless input parame-ters. If we fix the surface properties of the wall, the density ratio, and the viscosity ratio,then, of the five-dimensional groups, only two dimensionless groups remain independent.Therefore, any quantities of interest in our problem, such as ε should be determined by two September 22, 2018 igure 5: Flow field around the bubble during the rebound for Bo = 0 . and Ga = 40 . (a) Flow field aroundthe bubble at the end of the approach (beginning of the rebound). (b) and (c) Flow field around the bubbleduring the rebound. dimensionless groups. We showed that F r number is the most important parameter govern-ing the bubble-wall collision for the cases considered in this study. However, we would liketo obtain more accurate scaling by considering other dimensionless groups, which includeviscous effects and accurately account for the inertial forces close to the wall, in order togive a more precise description of the collision. We found it a daunting task, and none of thefamous dimensionless parameters improve the scaling. We believe that the difficulty arisesdue to inaccurate representation of the viscous forces and the complicated contribution ofthe added mass effect during the collision and rebound. In other words, none of the typicaldimensionless parameters can accurately represent the inertial and viscous effects.
The only previous work on the coefficient of restitution for bubble-wall collisions wasdone by Zenit and Legendre [18], in which it is suggested that the coefficient of restitu-tion might scale with the modified Stokes number, St ∗ , or the modified Ohnesorge number, Oh ∗ = p Ca/St ∗ . Therein it is stated that "In particular, it would be important to obtainmeasurements (through experiments or numerical simulations) for systems in which Ca and St ∗ could be varied independently. Moreover, it is necessary to perform experiments consid-ering clean fluids to obtain a precise measurement of the critical value of the Stokes numberbeyond which a rebound can be expected." We tested the suggested scaling by plotting ournumerical data versus these parameters, St ∗ and Oh ∗ , in Fig. 6(a) and (b), respectively,which shows a considerable scatter with respect to either of them. Therefore, our numericalsimulations revealed that such scaling does not hold for the large range of parameters in anentirely clean system. Zenit et al. [18] mentioned that the presence of surfactants in thewater might cause uncertainty in the accurate measurement of bubble dynamics close tothe wall. It has been observed that presence of any contamination can dramatically affectthe dynamics of the bubble during the collision [31]. Accurate modelling in the presence ofsurfactants necessitates inclusion of a history force and modification of boundary conditions September 22, 2018 r = 0.34 Fr - 0.34 Oh * St * Figure 6: Coefficient of restitution is plotted versus the previously suggested scaling parameters, i.e., themodified Ohnesorge number and the modified Stokes number which does not show any clear trend similarto the
F r number scaling. in the lubrication layer between the bubble and wall. Also, the terminal velocity for bubblesin contaminated systems is considerably lower than the clean systems [31]. This makes thecomputational data much more valuable in the sense that we have control over the parame-ters and the measured dimensionless numbers are correct. The computational code and thenumerical method have been successfully applied to the dynamics of the fluid interface innumerous settings. The only probable issue within the framework of the diffuse interfacemethod is that interface and wall interact due to their diffusive nature of interface model,which may produce additional friction during the bubble spreading and retraction. We ver-ified that the numerical simulations could correctly reproduce the experimental results forthe clean system (Fig. 2) but caution is appropriate with respect to the role of diffusion inmodeling the bubble-wall collision within the diffuse interface framework.
4. Conclusions
We used the Cahn-Hilliard diffuse interface method to study the momentum loss ofbubbles during collision with a rigid superhydrophilic horizontal wall. We simulated thebubble starting from the stasis and ensured that it reached a steady state before reachingthe wall. The bubble-wall collision happens in two stages: the approach and rebound stages.For the studied systems (Bond numbers between 0.25 to 1.5, Galileo numbers between 8.5to 50, and Re numbers from 10 to 100) deceleration of the bubble before its collision to thewall is governed by the inertial forces due to relatively weak viscous effects. The interestingobservation is that even in the absence of strong viscous forces, there is a considerablemomentum loss after the impact due to the fact the kinetic energy of the surrounding liquiddoes not get completely converted to the surface energy in the approach stage. Furthermore,the coefficient of restitution for the studied systems best scales with the Froude numbersuggesting that gravitational deceleration is the primary deceleration mechanism during the September 22, 2018 ebound stage.
5. Acknowledgments
This research was partially supported by the NSERC, the Canada Research Chair pro-gram and the Canada Foundation for Innovation. We thank Roberto Zenit for generouslyproviding us experimental data. We are grateful for discussions with James J. Feng, whoplayed a substantial role in forming the work.
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