Study of bifurcation in coalscence of bubbles using front-tracking method
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Study of bifurcation in coalscence of bubblesin turbulent flows
Seyedmehdi Abtahi, Mehrdad Izadjoo , † University of Illinois at Chicago, Darab University(Received xx; revised xx; accepted xx)
There are many applications of multiphase flow in important fields such as biological,chemical and power processes. Bubble coalescence is of a significant importance insimulating multiphase fluid flows. Weber number ( W e ), Reynolds number (Re) andcollision parameter play important role in the coalescence of bubbles. In the presentwork, front-tracking method is applied to simulate bubble coalescence. Moreover, theresults are presented for different collision parameters and changes in the coalescence ofbubbles are discussed. Key words:
Keywords: Front-tracking method, bifurcation, coalescence
1. Introduction
Bubbly flows play a prominent role in physical, chemical and biological processes,all of which invlove bubble-bubble and bubble-bubble interactions. Bubble coalescenceis of a significant importance in determining the interfacial area, thereby affecting themass and heat transfer between the two phases. Knowledge of the coalescence of twobubbles is consequential to better comprehend the bubbly flow behavior which results inmore efficient designs in multiphase flows (Esfahanian et al. b ). During coalescence,two drops combines through the liquid bridge between them, then it grows up to thesize of drops. Many researchers have studied the dynamics of coalescence of bubbles(Chi & Leal 1989; Basaran 1992). The coalescence of two bubbles rising in a verticalline is studied by (Ram´ırez-Mu˜noz et al. et al. a ). The colescenceconsists of the following steps: the collision of bubbles, thinning of a liquid film, andrupture of the film at some critical thickness (Chen et al. et al. a ). Researches on two-risingbubbles has been performed for high and low Reynolds number (Rushton & Davies 1978;Yuan & Prosperetti 1994). The steps of coalescence after collision is difficult to studybecause the rapture of liquid is so complex that mechanisms are derived by simplifyingthe coalescence model. Interaction and coalescence between a bubble and a free surfacehave been studied numerically and experimentally (Dehrouyeh-Semnani et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] F e b Mehrdad Izadjoo
Figure 1: Geometry of two bubbles which collide to each other. et al. et al. et al. et al. b ),the level set method (Lakdawala et al. et al. W e ) and collisionparameters. It is shown that by changing Weber number and collision parameter, thecoalescence behavior will change. The rest of this paper is structured as follows: front-tracking method is explained in section 2. Results of simulations are shown in section 3and they will be discussed. Conclusion is provided in section 4.
2. Front-tracking method
The Navier-Stokes equations are numerically solved for both gas and liquid phases: ∂ρ∂t + ∇ . ( ρV V ) = −∇ P + ρg + ∇ .µ eff ( ∇ V + V (cid:105) T ) + (cid:73) ∆ Sσ eff kn ( r − r f ) da where V , ρ and P are normalized velocity, density and pressure, respectfully. Thevariables are normalized with characteristic bubble velocity, V C , liquid density ρ , thedynamic pressure ρ c aV c and t is normalized with L c /V c , where L c is the characteristiclength. µ eff is the reciprocal of Reynolds number, Re = ρV c L c µ l , where µ l is the liquidviscosity. σ eff is the reciprocal of Weber number, W e = ρV c L c σ , where σ is the surface tudy of bifurcation in coalscence of bubbles using front-tracking method Figure 2: Reflexive separation of bubbles with I = 0 in t = 0,t = 2.61,t = 3.86,t = 5.42,t = 9.29,t = 11.07, t = 13.88, t = 21.5, t = 25.37. tension of liquid. k is twice the mean curvature, n is the outwardly directed unit normalvector at the bubble surface, and r is the space vector with the subscript f designatingthe interface. By considering both fluids to be incompressible, continuity can be writtenas below: ∇ .V = 0Moreover, density and viscosity stay constant for incompressible fluid, ∂ρ∂t = 0 and ∂µ∂t = 0.
3. Result
In the present work, we simulate coalescence of two bubbles in three dimension, whichare shown in Figure 3. Regimes of coalescence of bubbles are mainly dependent on W e and collision parameter, which is defined as I = Xds , where X is the vertical distancebetween the velocity of center of mass of bubbles and d is the diameter of bubbles. Thefirst regime happens when the two bubbles collide, but they do not have sufficient energy
Mehrdad Izadjoo
Figure 3: Reflexive separation of bubbles with I = 0.1 in t = 0,t = 3.65,t = 3.86,t = 5.53,t =9.08, t = 11.69, t = 16.49, t = 21.5, t = 25.57. to become separate again. The other important one is reflexive separation, in which thecollided bubble separates in three steps. At first, the collided bubble grows in verticaldirection and shape as a disk. Then, because of surface tension and curvature differencein longitudinal and radial directions, the bubble changes back to be a sphere. The inertiaof bubble makes an longitudinal expansion. Another regime is stretching separation, inwhich the two bubbles rub each other and after being combined, they separate from eachother. Reflexive and stretching separation happen in low and high collision parameters,respectfully. In the simulation, we consider W e = 33 and Re = 163.1. Size of solutiondomain is 1 × ×
2, while there are 96 cells in each unit. Moreover, the height of channelis 8 times larger than the radius of bubble. The result for reflexive separation and I =0 is shown for different time steps in Figure 3. The result for reflexive separation andI = 0.1 is shown for different time steps in Figure (3). As it is shown in Figure 3, thebubble grows in a different angle and separate in a different angle than the case I = 0. tudy of bifurcation in coalscence of bubbles using front-tracking method Figure 4: Permanent coalescence of bubbles with I = 0.4 in t = 0,t = 3.23,t = 4.07,t = 5.64,t =8.98,t = 10.34,t = 11.8,t = 13.26,t = 38.41.
Permanent coalescence is shown in Figure 3 with I = 0.4, which indicates that the bubbledo have the sufficient energy to increase length in vertical direction. Figure 3 indicatesthe results for stretching separation with I = 0.7. It must be noticed that in this case anew bubble appears between the two separated bubbles. As it is shown in Figures 3 to3, by changing I, the coalescence of bubbles will differs (bifurcation in the system).
4. Conclusion
In the present study, front-tracking method is applied to simulate the coalescence ofbubbles. Collision parameter play an important role in coalescence. Different collison pa-rameters result in reflexive separation, permanent coalescence and stretching separation.In each case results was shown for different time steps to show how the bubble changethe shape during coalescence to finally become either separated into two or three bubbles
Mehrdad Izadjoo
Figure 5: Stretching separation of bubbles with I = 0.7 in t = 0,t = 2.92,t = 4.07,t = 6.05,t =8.56,t = 9.5,t = 10.96,t = 13.26,t = 18.37. or stay united as a single bubble. By changing the collision parameter, the results differs,which indicates bifurcation in the system.
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