Instability and breakup of cavitation bubbles within diesel drops☆

Ming Lü,Zhi Ning*,Kai Yan,Juan Fu,Chunhua Sun

College of Mechanical and Electrical Engineering,Beijing Jiaotong University,Beijing 100044,China

Keywords:Stability Diesel droplet Cavitation bubble Secondary breakup

ABSTRACT A modified mathematical model is used to study the effects of various forces on the stability of cavitation bubbles within a diesel droplet.The principal finding of the work is that viscous forces of fluids stabilize the cavitation bubble,while inertial force destabilizes the cavitation bubble.The droplet viscosity plays a dominant role on the stability of cavitation bubbles compared with that of air and bubble. Bubble-droplet radius ratio is a key factor to control the bubble stability,especially in the high radius ratio range.Internal hydrodynamic and surface tension forces are found to stabilize the cavitation bubble,while bubble stability has little relationship with the external hydrodynamic force.Inertia makes bubble breakup easily,however,the breakup time is only slightly changed when bubble growth speed reaches a certain value(50 m·s?1).In contrast,viscous force makes bubble hard to break.With the increasing initial bubble-droplet radius ratio,the bubble growth rate increases,the bubble breakup radius decreases,and the bubble breakup time becomes shorter.

1.Introduction

Cavitation phenomenon in injection nozzles has a strong impact on spray formation and atomization,which are directly correlated to the efficiency of the combustion process and extents of pollutant emission[1-4].Study on the stability of cavitation bubbles has important significance and value.Cavitation phenomenon in injection nozzles is divided into two types,onset of cavitation and supercavitation[5-7].Supercavitation in the process of fuel injection refers to a kind of phenomenon that cavitation bubbles exist in the fuel jet leaving the nozzle.Some experiments have proved that more efficient atomization occurs in the liquid jet with the stronger supercavitation[8-16].However,supercavitation turns the fuel jet into gas-liquid two-phase mixture flow,which increases the instability of fuel jet and has an important impact on the breakup and atomization of fuel injection process so that the spray breakup and atomization mechanism under the condition of supercavitation is still unclear.Therefore,the study on how the ligament breaks up from the spray and when the secondary breakup of diesel droplet happens under the condition of supercavitation is meaningful.

Cavitation bubbles,which always exist in diesel droplets generated from breaking jet as a result of cavitation of the diesel,increase the instability of jet and droplets in part due to the two-phase mixture[17],while the mechanism and degree of this effect is uncertain.Due to the small scale of diesel droplets and cavitation bubbles,and the high jet speed of several hundred meters per second,experimental studies on the breakup of cavitation bubble within the droplet are very limited.Likewise,numerical modeling results are not exact in the case of the limited quantitative data,especially for the breakup process of secondary droplets.Therefore,it is obvious that theoretical analysis is promising to explain some observations of practical atomizer performance and improve the understanding of the breakup problem of cavitation bubbles within diesel drops.

Zeng and Lee[18]applied linear stability analysis method to develop a simple mathematical model to solve the secondary droplet character of dimethyl ether,with the atomization being considered as the result of two competing mechanisms:the hydrodynamic force and the bubble growth.In the regimes with intermediate super heating degrees their study considered that sprays are atomized by bubble growth,which produces smaller Sauter mean diameter than hydrodynamic forces alone.However,they neglected the viscous forces of fluids and the surface tension force.In this paper,based on their studies,a modified mathematical model considering the effect of fluids viscosity is used to study the effects of various forces(viscous force,inertial force,internal and external hydrodynamic forces,and surface tension force)on the stability and breakup of cavitation bubbles within the diesel droplet,which is helpful in studying the breakup mechanism of the secondary diesel droplet under the condition of supercavitation.

2.Mathematical Model

Fig.1.Schematic of cavitation bubble within the diesel droplet.

Fig.1 shows that the schematic of cavitation bubble within a diesel droplet.Three modeling objects are present for this bubble/droplet system:stationary diesel droplet,stationary air and expanding cavitation bubble with the velocity of 2 m·s?1in radial direction.Three assumed conditions for building the mathematical model of instability of radial expansion of a droplet and an inside bubble are:only one cavitation bubble within a diesel droplet,no evaporation on the surface of diesel droplet,and the disturbances on the bubble surface are assumed to be spherically symmetric.It is further assumed that the ambient air and the diesel droplet are viscous and incompressible while the cavitation bubble is viscous and compressible.

2.1.Mathematical model

Spherical coordinate system is used in this work.The effects of viscosity including ambient air,diesel droplet and cavitation bubble are considered in this study.For the limit of paper length,the detailed derivation process of mathematical model[19]is omitted,and here is the final form of mathematical model derived by the authors.

For ambient gas outside the droplet,the diesel droplet,and the cavitation bubble,we can get the linearized disturbance governing equations.To establish the mathematical model describing the temporal stability of cavitation bubbles within the diesel droplet,we should give the initial conditions and the corresponding boundary conditions at the interface and at the infinity.Substituting disturbance governing equations into the boundary conditions,which leads to

Eq.(1)is the mathematical model,which can be used to describe the disturbance growth rate of the diesel bubble growth instability.For the purpose of generality and convenience to research,Eq.(1)can be normalized as

where δ=Ro/Ri,is the liquid-bubble radius ratio,ψo= ρ1/ρ2,is the air-liquid density ratio,,is the bubble-liquid density ratio,Mai=Vi/c,is the bubble Mach number,,is the non-dimensional disturbance growth rate,Re2=ρ2VoRo/μ2,is the liquid Reynolds number,,is the bubble Weber number,,is the liquid Weber number,μv,is the bubble Reynolds number,and,is the air Reynolds number.

Eq.(2)is a cubic equation about plural number ω*and characterized by 10 non-dimensional variables and can be solved analytically or numerically.Three roots exist for the mathematical model.The root with the large streal part represents the disturbance growth rate,and the corresponding imaginary part represents the frequency of oscillation.The effects of viscosity including ambient air,diesel droplet and cavitation bubble are considered in this study.So Eq.(2)is more complete than that derived in[18].

Assuming all viscosity coefficients are equal to zero,all terms related to Reynolds numbers disappear,then Eq.(2)degenerates to

which is the mathematical model used by Zeng and Lee[18].

2.2.Bubble breakup criterion

Experimental studies display that the jet spray is atomized primarily by bubble breakup under cavitation conditions,but the exact atomization mechanism concerning when and how the bubble breakups is not very clearly identified up to now.

The breakup model proposed by Zeng and Lee[18]for a bubble droplet system is used in this study.The breakup is assumed to occur when the disturbance grows larger than a certain characteristic length.The film thickness(i.e.,the difference between droplet radius Roand bubble radius Ri)is chosen.So,the breakup variable K is defined as

where η0is the initial disturbance,for the time being,linear stability analysis method cannot be used to solve the value of η0.The initial disturbanceη0is assumed to be proportional to the initial dropletradius Ro0in this paper,η0=fRo0,where f is a constant,which can be determined from experimental data.In this study,f is taken as 0.05.The breakup criterion is chosen to be

namely when the disturbance is up to the fuel thickness,her tbrepresents the breakup time.

3.Results and Discussions

Physical parameters of diesel and air are set at the temperature of 300 K according to the literature[20-23],and they are shown in Table 1.The corresponding non-dimensional parameter values are thus determined as δ =10,ψo=0.00141,ψi=0.0000128,Mai=0.008,Re1=0.06466,Re2=0.28884,Rev=0.06,Wei=0.631,Weo=0.0000631.

Table 1 Summary of physical parameters and standard conditions

Since there are not any available experimental data of diesel cavitation bubble growth,the mathematical model was only compared with Zeng's model.For the purpose of researching on the effects of different fluids viscosity,the reduced mathematical model were listed,namely(1)reduced model A(only ambient air viscosity),(2)reduced model B(only cavitation bubble viscosity),and(3)reduced model C(only diesel droplet viscosity).The compared results are shown in Fig.2.

Note that fluid viscosity reflects the viscous force.As shown in Fig.2,the full model makes the disturbance growth rate smaller compared with Zeng's model,which suggests that viscous forces of fluids stabilize the cavitation bubble.In addition,reduced model C is almost equal to the full model,which indicates that the droplet's viscous force plays a dominant role on the stability of cavitation bubble compared with the viscous forces of air and bubble.As the bubble-droplet radius ratio increases,the normalized disturbance growth rate also increases,which suggests that radius ratio is the key factor to control the bubble stability,especially in high radius ratio.Thereafter,the full model is used throughout in the subsequent work.

3.1.Temporal stability of cavitation bubbles

Fig.2.Model comparison.(ψo=0.00141,ψi=0.0000128,Ma i=0.008,Re1=0.06466,Re2=0.28884).

Bubble Reynolds number Revrepresents the ratio of inertial force and viscous force.If we keep the viscous force unchanged,bubble Reynolds number reflects the inertial force.Fig.3 shows the variation of disturbance growth rate versus bubble Reynolds number at different bubble-droplet radius ratios,which indicates that inertial force destabilizes the cavitation bubble.

Fig.3.Variation of disturbance growth rate with bubble Reynolds number at different bubble-droplet radius ratios(ψo=0.00141,ψi=0.0000128).

The air-liquid density ratio ψoreflects the external hydrodynamic force,while the bubble-liquid density ratio ψireflects the internal hydrodynamic force.Figs.4 and 5 give the variation of disturbance growth rate versus bubble-liquid density ratio at different air-liquid density ratios and different bubble-droplet radius ratios.

As shown in Fig.4,disturbance growth rate decreases linearly with the increasing bubble-liquid density ratio,which indicates that internal hydrodynamic force stabilizes the cavitation bubble.In addition,it is also found in Fig.4 that disturbance growth rate is almost unchanged even at different air-droplet density ratios,which signifies that bubble stability has little relationship with the external hydrodynamic force.

As Fig.5 shows,the variation of disturbance growth rate versus bubble-droplet density ratio at different bubble-droplet radius ratios seems consistent,which suggests that the effects of the internal hydrodynamic force on bubble stability have little relationship with bubbledroplet radius ratio.

Fig.4.Variation of disturbance growth rate with bubble-droplet density ratio at different air-droplet density ratios(δ=10,Ma i=0.008,Re2=0.28884,We i=0.631,We o=0.0000631).

Fig.5.Variation of disturbance growth rate with bubble-droplet density ratio at different bubble-droplet radius ratios(ψo=0.00141,Ma i=0.008,Re1=0.06466,Re2=0.28884).

Bubble Weber number Weirepresents the ratio of inertial force and surface tension force.If we keep the inertial force unchanged,bubble Weber number reflects the surface tension force.Fig.6 shows that the disturbance growth rate increases with the increasing of bubble Weber number,indicating that surface tension force is a factor to stabilize the cavitation bubble.

Fig.6.Variation of disturbance growth rate with bubble Weber number at different bubble-droplet radius ratios(ψo=0.00141,ψi=0.0000128,Ma i=0.008,Re1=0.06466,Re2=0.28884).

3.2.Breakup of cavitation bubbles

The growth of disturbances results in the breakup of cavitation bubbles within diesel drops.

Fig.7 shows the variation of breakup variable K versus bubble growth time at different bubble-droplet radius ratios.Because cavitation bubbles have broken up,the curves are meaningless above the critical breakup line.As shown in Fig.7,the bubble-droplet radius ratio is a key factor to affect bubble breakup.At the beginning of bubble growth,bubble breakup variable is bigger for bubble with large bubble droplet radius ratio than bubble with small radius ratio,which indicates the bubbles of initial larger ratio of bubble-droplet radius are prone to be broken.Referring to Fig.7,the breakup time tbof a bubble with bubble-droplet radius ratio δ?1=0.2 is 3.59 ms,while the breakup time of a bubble with bubble-droplet radius ratio δ?1=0.8 reduces to 0.52 ms sharply.

Bubble breakup time tbdetermines when bubble breakup occurs,which is affected by many parameters.Referring back to forces on the bubble stability,we will study the effects of those forces involved parameters on bubble breakup time.

Fig.7.Variation of breakup variable versus bubble growth time at different bubble droplet radius ratios.(ψo=0.00141,ψi=0.0000128,Ma i=0.008,Re1=0.06466,Re2=0.28884).

Note that bubble growth speed reflects the inertial force.Fig.8 gives the variation of breakup time versus bubble growth speed at different bubble-droplet radius ratios.As shown in Fig.8,the bubble breakup time decreases rapidly with the increasing of the cavitation bubble growth speed,which suggests that inertial force destabilizes the cavitation bubble and makes bubble breakup easily.However,the breakup time almost is slightly changed when bubble growth speed reaches a certain value(50 m·s?1).In addition,the effect of bubble-droplet radius ratio on bubble breakup time is of significance only if the bubble growth speed is relatively low,which indicates that the bubble-droplet radius ratio has a great impact on bubble breakup time when inertial force is low relatively.

Fig.8.Variation of breakup time versus bubble growth speed at different bubble-droplet radius ratios.(ψo=0.00141,ψi=0.0000128).

Note that the diesel droplet viscosity reflects the viscous force.Fig.9 shows the variation of bubble breakup time versus the droplet viscosity at different bubble-droplet radius ratios,suggesting that viscous force stabilizes the cavitation bubble and makes bubble hard to break.In addition,we can also find that the bubble-droplet radius ratio has a great impact on effects of the viscous force on bubble breakup time.

Fig.10 shows the bubble growth process and the final breakup radius of cavitation bubbles which have the same initial radius of 5 μm at different initial bubble-droplet radius ratios.As shown in Fig.10,with the increasing of the initial bubble-droplet radius ratio,the bubble growth rate increases,the bubble breakup radius decreases,and the bubble breakup time becomes shorter.It is also indicated that the bubbles of initial larger ratio of bubble-droplet radius are prone to be broken.

Fig.9.Variation of bubble breakup time versus the droplet viscosity at different bubble droplet radius ratios.(ψo=0.00141,ψi=0.0000128,Ma i=0.008,Re1=0.06466).

4.Conclusions

A modified mathematical model considering the effect of fluids viscosity is used to study on the stability of a cavitation bubble within the diesel droplet.In addition,based on the modified bubble breakup criterion,the breakup time and the breakup radius of cavitation bubbles are evaluated.The principal findings of the work may be summarized as follows.

(1)Viscous forces of fluids stabilize the cavitation bubble.The droplet's viscous force plays a dominant role on the stability of cavitation bubbles compared with the viscous forces of air and bubble.Bubble-droplet radius ratio is a key factor to control the bubble stability,especially in the high radius ratio range.

(2)Inertial force destabilizes the cavitation bubble.Internal hydrodynamic force and surface tension force are found to stabilize the cavitation bubble,while bubble stability has little relationship with the external aerodynamic force.The effects of the internal hydrodynamic force on bubble stability have little relationship with the bubble-droplet radius ratio.

Fig.10.Bubble growth process.(ψo=0.00141,ψi=0.0000128,Ma i=0.008,Re1=0.06466,Re2=0.28884).

(3)Inertial force makes bubble breakup easily.However,the breakup time is almost slightly changed when bubble growth speed reaches a certain value(50 m·s?1).Bubble-droplet radius ratio has great impact on bubble breakup time when the inertial force is relatively low.In contrast,viscous force makes bubble hard to break.

(4)With the increase of the initial bubble-droplet radius ratio,the bubble growth rate increases,the bubble breakup radius decreases,and the bubble breakup time becomes shorter.

Nomenclature

c sound speed,m·s?1

f proportional factor(=0.05)

K breakup variable

Maibubble Mach number(=)

Ribubble radius,m

Rodiesel droplet radius,m

Re1air Reynolds number(=)

Re2liquid Reynolds number(=)

Revbubble Reynolds number(=)

t time,ms

tbbubble breakup time,ms

Weibubble Weber number

Weoliquid Weber number

δ liquid-bubble radius ratio

η0initial disturbance

μ1air viscosity,kg·m?1·s?1

μ2diesel viscosity,kg·m?1·s?1

μvcavitation gas viscosity,kg·m?1·s?1

ρ1air density,kg·m?3

ρ2diesel density,kg·m?3

ρvcavitation gas density,kg·m?3

σ surface tension,N·m?1

ψibubble-liquid density ratio

ψoair-liquid density ratio

ω* non-dimensional disturbance growth rate