Ultrasonics 53 (2013) 943–955

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Ultrasonics

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Numerical study of the dielectric liquid around an electrical discharge generatedvapor bubble in ultrasonic assisted EDM

Mohammad T. Shervani-Tabar a,⇑, Nima Mobadersany b

a Department of Mechanical Engineering, University of Tabriz, Tabriz, Iranb Department of Mechanical and Aerospace Engineering, George Washington University, Washington, DC, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 April 2011Received in revised form 30 September 2012Accepted 8 November 2012Available online 29 January 2013

Keywords:Ultrasonic assisted EDMPressure distributionVelocity fieldBoundary integral equation method

0041-624X/$ - see front matter � 2012 Elsevier B.V.http://dx.doi.org/10.1016/j.ultras.2012.11.008

⇑ Corresponding author. Tel.: +98 411 3392456; faxE-mail addresses: msherv@tabrizu.ac.ir (M.T. Sherv

gmail.com (N. Mobadersany).

In electrical discharge machining due to the electrical current, very small bubbles are created in thedielectric fluid between the tool and the workpiece. Increase of the number of bubbles and their growthin size generate a single bubble. The bubble has an important role in electrical discharge machining. Inthis paper the effect of ultrasonic vibration of the tool and the velocity fields and pressure distributionin the dielectric fluid around the bubble in the process of electrical discharge machining are studiednumerically. The boundary integral equation method is applied for the numerical solution of the problem.It is shown that ultrasonic vibration of the tool has great influence on the evolution of the bubble, fluidbehavior and the efficiency of the machining in EDM. At the last stages of the collapse phase of the bub-ble, a liquid jet develops on the bubble which has different shapes. Due to the different cases, and a highpressure region appears just near the jet of the bubble. Also the fluid particles have the highest relativevelocity just near the liquid jet of the bubble.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Electrical discharge machining (EDM) is a powerful techniquefor machining of hard materials. In the process of EDM, due tothe electrical current, very small bubbles are created betweenthe two electrodes; the tool and the workpiece. Electrical dischargecauses local material melting on the opposite ends of the plasmachannel on the surfaces of the two electrodes. During the pulseon-time, the size of gas bubble increases. When the current isstopped, the continuous growth of the gas bubble during the pulseoff-time leads to the sharp drop of pressure inside the bubble andsurfaces of the electrodes which consequently causes superheatedmolten material escape from the crater [1,2].

The gas in the working gap has very different influences on thesequence of discharges. The gas part determines the character ofthe following discharge. Gas bubbles that reach the size of theworking gap move more slowly in the working gap and result ina concentration of contamination, caused by previous discharges,at for the further processing disadvantageous locations. Thereby,following discharge will not take place unconditionally in the re-gion of the minimum working gap. These big gas bubbles lead inthe border area of the working gap to the elimination of the con-tamination. Gas bubbles that are by far smaller than the working

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gap move in the working gap with the flow velocity of the dielec-tric. The gas bubbles transport also contamination out of the work-ing gap. It must be considered, whether the gas bubble thatoccupies the entire working gap result in a pure gaseous dischargeoccurring in a limited spherical gas volume [3].

During micro erosion two alternatives of the gas bubble propa-gation must be considered.

Because of the small size of workpiece and tool electrodesexternal propagation of gas bubbles and extension of the dischargechannel can appear more frequently. In some points, the externalpropagation of gas bubbles corresponds to the behavior of ‘‘free’’gas bubbles in a flow. An internal propagation of gas bubbles takeplace only far very small pulse energies which are used for highprocessing qualities. According to theoretical considerations theinternal propagation of gas bubbles leads to an early collapse dueto the cooling effects of the gas bubble–electrode contact that in-creases with the growing gas bubble. At external propagation thesize and propagation of gas bubbles are determined by the timeelapses to the following discharge which leads to their collapse [3].

The ignition and breakdown behavior in spark erosion pro-cesses are determined by a large number of different effects thatinfluences the machining process. For example the addition ofsome selected groups of additives, which were added to the dielec-tric working fluid can led to modified discharge channels [4].

Many researchers have worked experimentally on the ultra-sonic assisted EDM and showed that the ultrasonic vibration ofthe tool improves the machining process. Kremer et al. reported

Fig. 1. The schematic of the problem.

944 M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955

that ultrasonic vibrations can improve the efficiency, in EDM of al-loy steel with a graphite tool [5]. Some other researchers achievedthe increase of machining rate up to 400% or more through ultra-sonic longitudinal vibrations of tool at micro holes/slots drillingthrough EDM [6]. Lin et al. have showed that ultrasonic vibrationof the tool in EDM of titanium alloy increases the material removalrate [7]. Jia et al. reported that the combination of ultrasonicmachining and EDM improves the material removal rate in ad-vanced ceramics [8]. Ghoreishi and Atkinson have proposed thata combination of rotary and vibratory movements of electrode pro-duces higher material removal rate [9]. Guo et al. showed thatapplying ultrasonic vibration in wire EDM also causes an increasein cutting efficiency [10]. Shervani-Tabar et al. have studied the ef-fect of vibration of the tool in EDM numerically and showed thatwhen the electrical discharge occurs in the closest position of thetool to the workpiece, the rate of material removal rate becomeshigher [11]. They also showed the dynamic of the vapor bubblein ultrasonic assisted EDM after its splitting [12].

It should be said that gas bubble growth is very different for dif-ferent dielectrics, and the amplitude of the eroding current influ-ences mainly on the size of the gas bubbles [13].

Shervani-Tabar et al. have studied the profile of a bubble inEDM by using boundary integral equation method and showed thatthe vapor bubble takes the shape of an hour-glass during the col-lapse phase [14].

Shervani-Tabar et al. studied the velocity and pressure fieldsaround the bubble in the process of EDM by using boundary inte-gral method. According to their studies, a high pressure regiondevelops near the annular jet of the bubble [15]. Abdullah et al.studied the effect of tool electrode ultrasonic vibration on somesurface integrity properties of cemented tungsten carbide in EDMprocess and showed ultrasonic assisted EDM applicability inimproving surface integrity [16]. Shabgard et al. have studied thecombination of EDM with tool vibration to improve the efficiency.From their experimental results it could be seen that ultrasonicvibration of the workpiece can significantly reduce the inactivepulses and improve the stability process and that it is effective inattaining a high material removal rate in finishing regime [17].

Experimental study of Zhang et al. in ED machining of SG4 cera-mic revealed that the sinusoidal vibration of the tool with ultra-sonic frequency improves the machining stability and increasesthe material removal rate [18]. In this paper, the bubble behaviorand the velocity and pressure fields in the dielectric fluid aroundthe bubble in the process of EDM are investigated numericallywhen the tool has ultrasonic vibration. The tool and the workpieceare assumed as two parallel rigid boundaries with dielectric liquidbetween them. The flow of the dielectric liquid is considered as apotential flow. Both the boundary integral equation method and fi-nite difference method is applied for the numerical solution of theproblem.

2. Geometrical definition of the problem

In this study the dynamics of an electrical discharge generatedbubble and the surrounded dielectric fluid is studied in three caseswhen the bubble is located between the tool and the workpiece inits initial minimum volume. In the first case the tool and the work-piece are fixed and the tool has not any vibration. In the secondcase, it is assumed that the tool has an ultrasonic vibration andthe electrical discharge occurs when the tool is far from the work-piece. In this case during the growth and collapse phases of thebubble the tool moves towards the workpiece. In the third casealso the tool has an ultrasonic vibration. In this case it is assumedthat the electrical discharge occurs when the tool is in its closestposition to the workpiece. In this case during the growth and

collapse phases of the bubble the tool moves away from the work-piece. The distance between the tool and the workpiece is gap. Inthe present paper the distance between the tool and the workpiecewhen the bubble is in its initial minimum volume is denoted bygapi. The distance between the centroid of the bubble and theworkpiece is called h. In the present paper the distance betweenthe centroid of the bubble and the workpiece when the bubble isin its initial minimum volume is specified by hi. The centroid ofthe bubble is on the vertical axisymmetric axis and the radial axisis laid on the boundary of the workpiece. Fig. 1 shows the sche-matic of the problem.

3. Ultrasonic vibration of the tool

The vibration of the tool can be expressed as:

uðtÞ ¼ A sinð2pft þuÞ ð1Þ

where u(t) is the displacement of the tool, t is time, f is the fre-quency of the tool vibration, A is the maximum amplitude of thetool vibration and u is phase angle difference. The velocity of thetool vibrating is:

VðtÞ ¼ duðtÞdt¼ 2pfA cosð2pft þuÞ ð2Þ

4. Governing equations of the problem

The fluid around the bubble generated due to a local energy in-put, i.e. an electrical discharge is assumed incompressible, invicidand irrotational. Thus, the boundary integral equation methodbased on potential flow problems is used for the numerical solu-tion of the problem. This method is a powerful technique for sim-ulating the time dependant profiles of the bubble generated due tothe electrical discharge in EDM process. The flow of the dielectricfluid is considered as a potential flow and satisfies the Green’s inte-gral formula which is given in the form of

cðpÞ/i þXN

j¼1

/j

ZSj

@

@n1

jpi � qjj

!ds ¼

XN

j¼1

@

@nð/jÞ

ZSj

1jpi � qjj

!ds

ð3Þ

S is the boundary of the liquid domain which includes the bubbleboundary and the interfaces of the liquid domain with two parallelsolid walls, / is the velocity potential and @/

@n is the normal velocityof the boundary. N is the number of elements on the boundary. Itshould be noted that, the discretization of the upper and lower par-allel rigid boundaries are extended up to physical infinity, wherethe growth and collapse of the vapor bubble has no considerable ef-fects on the behavior of the liquid domain. P is any point in the li-quid domain, or on the boundary and q is any point on the

M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955 945

boundary. c(p) is a coefficient that its value is 2p when p is on theboundary and is 4p when p is inside the liquid domain.

In this paper, surface tension and gravity forces are neglectedand the bubble is assumed to be spherical during its initial expan-sion to a minimum volume, so the equation of the radial motion ofthe bubble is given as:

R Rþ32

_R2 þ p1 � pl

q¼ 0 ð4Þ

As it is explained by Best [19], Pl is the variable pressure insidethe bubble, P1 is the pressure in the far field and R is the radius ofthe bubble.

5. Variation of the pressure inside the bubble

The bubble contains a mixture of non-condensable gases and aconstant pressure vapor. The non-condensable gases inside thebubble are assumed to behave as an ideal gas. Thus the pressure in-side the bubble is given by

Pb ¼ Pc þ Pg ð5Þ

Pb is the pressure inside the bubble, Pc is the constant vapor pres-sure and Pg is partial pressure of the ideal gas inside the bubbleand is given by

Pg ¼ PigVi

V

� �k

ð6Þ

where Pig is the initial pressure of the non-condensable gases, Vi isthe initial volume of the bubble and k is the ratio of specific heats.The unsteady Bernoulli equation in Lagrangian form is used to ob-tain the pressure inside the bubble which is given as:

D/Dt¼ 1

2jr/j2 þ p1 � pb

qð7Þ

that p1 is the pressure in the far field, and pb is the pressure insidethe vapor bubble.

6. Discretization

As it is shown in Fig. 2, the interfaces of the liquid domain withupper and lower parallel rigid boundaries are discretized by linearsegments, while the bubble boundary by cubic spline elements.Also the problem is assumed to be axisymmetric.

The normal velocity on the boundary of the liquid domain isindicated by n and is directed outward. The vertical axis is indi-cated by z, while r is the radial axis. Collocation points are locatedat the midpoint of each element and physical functions, i.e. velocitypotential and normal velocity, are assumed to be constant on eachelement.

Fig. 2. Discretization of the boundary of the liquid domain.

7. Computational implementation

The problem is non-dimensionalised by employing the maxi-mum radius of the bubble, Rm, the dielectric liquid density, q,the pressure in the far field, p1, and the saturated vaporpressure, pc. Eq. (4) is non-dimensionalised in order to obtain arelation between the initial radius of the bubble and the initialhigh pressure inside the bubble. It should be noted that at theinitial small radius, R0, the bubble generated by a high localenergy input contains a mixture of saturated vapor with a con-stant partial pressure and an ideal gas with a very high partialpressure.

The non-dimensional form of Eq. (4) is given as

R Rþ32

_R2 þ 1� eR0

R

� �¼ 0 ð8Þ

e ¼ pi

p1 � pcð9Þ

e is the strength parameter and pi is the initial interior pressureinside the bubble. By integrating the above equation, the relationbetween e and R0 is obtained, so a specified R0 is obtained forany specified strength parameter.

The boundary of the liquid domain is consisted of the bubbleboundary, the workpiece and the tool. Velocity potential is zeroon the bubble surface at the initial time. Normal velocity is alsozero on the workpiece and the tool. Eq. (3) gives a set of linearequations for / on the surfaces of the tool and the workpieceand @/

@n on the bubble boundary. It should be noted that as the prob-lem is axisymmetric, the integral terms of Eq. (3) are simplified in amanner that is given by Taib [20]. By having distribution of thevelocity potential on the bubble boundary and using a finite differ-ence scheme, tangential velocity on the bubble boundary is ob-tained by differentiation of the velocity potential along thesurface of the bubble. Thus, by having distribution of normal veloc-ity and tangential velocity on the bubble boundary and by usingthe second order Runge–Kutta scheme, evolution of the bubbleduring its growth and collapse phases is obtained by using a vari-able time step as

Dt ¼minD/

p1�plp1�pc

þ 12 jr/ij

2 þ gðzi � hÞ

!ð10Þ

8. Discretization of the liquid domain

The liquid domain around the vapor bubble is discretized byfixed nodes. By having the distribution of the velocity potentialon the bubble boundary and distribution of normal velocity onthe surfaces of the workpiece and the tool, the distribution of thevelocity potential in the liquid domain around the vapor bubbleis obtained by employing the Green’s integral formula. Then byhaving distribution of the velocity potential in the liquid domainand by employing a finite difference scheme the pressure andvelocity fields around the vapor bubble and between the work-piece and the tool is obtained. It should be noted that the liquid do-main is discretized by even spaced fixed points. The Green’sformula is used for obtaining the velocity potential on each fixedpoint inside the liquid domain and on its four neighbor points(see Fig. 3).

By having the velocity potential on the four neighbor points ofeach fixed point inside the liquid domain and by employing a finitedifference scheme, the vertical and radial components of the veloc-ity on each fixed point can be obtained by employing the followingequations:

Fig. 3. The schematic of the fixed point specified by (i,j), and its four neighbors witha distance of Dr and Dz far from it.

946 M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955

U ¼ /rði; jÞ � /lði; jÞRrði; jÞ � Rlði; jÞ

; V ¼ /uði; jÞ � /dði; jÞZuði; jÞ � Zdði; jÞ

ð11Þ

where i and j specify the coordinates of each fixed point. U and V arethe horizontal and normal velocities respectively. /r and /l are thepotential velocities in the right-hand side and left-hand side neigh-bors of each fixed point and /u and /d are the potential velocities inthe upper and lower sides of each fixed point respectively. Forobtaining the pressure fields in the dielectric fluid around the vaporbubble, the non-dimensional unsteady Bernoulli equation in Euleri-an form is written as:

@/�

@t�¼ �1

2jr�/�j2 þ p1 � p

p1 � pcð12Þ

where p specifies the pressure of each fixed point.

Fig. 4. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the fixed workpiece and tool in an EDM process whenci ¼

GAPi2 . The corresponding non-dimensional times are shown under each

illustration.

9. Results and discussion

In this part the evolution of the electrical discharge generatedvapor bubble and its surrounded fluid behavior is investigatedfor different distances between the centroid of the initial minimumvolume of the bubble and the workpiece. As it is mentioned above,the problem is investigated in three cases:

Case 1: Bubble between the fixed workpiece and tool.Case 2: Bubble between the workpiece and the tool with ultra-sonic vibration of the tool. In this case the electric dischargeoccurs when the tool is far from the workpiece.Case 3: Bubble between the workpiece and the tool with ultra-sonic vibration of the tool. In this case the electric dischargeoccurs when the tool is in its closest position to the workpiece.

In these cases it is assumed that Rm = 30 lm, P1 = 100,000 Pa,A = 10 lm and e = 10. The distance between the balance positionof the tool and the workpiece is 80 lm and the frequency of thetool vibration is 20 kHz. It should be said that GAPi and ci are thenon-dimensional form of gapi and hi respectively which are ob-tained by employing the maximum radius of the bubble.

Figs. 4–6 show the time dependant profiles of an electrical dis-charge generated bubble, between the tool and the workpiece dur-ing its growth and collapse phases when ci ¼

GAPi2 for the three

cases. This means that in these figures the vapor bubble is locatedin the midpoint of the tool and the workpiece when it is in its ini-tial minimum volume.

As it is shown in Fig. 4 when the bubble is between the fixedtool and the workpiece, the vapor bubble remains almost sphericalduring its growth phase. During the collapse phase, the vapor bub-ble elongates in the direction perpendicular to the parallel bound-aries of the tool and the workpiece. At the last stages of thecollapse phase an annular liquid jet develops around the bubbleand the bubble takes the shape of a twin necked bulbs. This isnecking phenomenon which is followed by splitting of the bubbleinto two parts. The bubble behavior can be ascertained on the dis-tribution of the pressure in the dielectric fluid around the bubble. Itis shown that at the early stages of the growing phase of the bub-ble, the pressure inside the bubble has the highest value and thepressure in the liquid domain is decreasing by going far away fromthe bubble and the bubble expels the fluid to the surrounding.Then as times goes on, the pressure in the vicinity of the bubble

Fig. 5. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the workpiece and tool in an EDM process withultrasonic vibration of the tool when ci ¼

GAPi2 . The electrical discharge occurs when

the tool is far from the workpiece. The corresponding non-dimensional times areshown under each illustration.

Fig. 6. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the workpiece and tool in an EDM process withultrasonic vibration of the tool when ci ¼

GAPi2 . The electrical discharge occurs in the

closest position of the tool to the workpiece. The corresponding non-dimensionaltimes are shown under each illustration.

M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955 947

just near the workpiece and the tool goes higher than the pressureof any point in the liquid domain. The bubble expands till all of itskinetic energy is transferred to the fluid. Thus the bubble reaches a

maximum volume which it is shown in the non-dimensional timet� = 1.55964. As it is shown at the time when the bubble reaches itsmaximum volume, the pressure inside the bubble is less than thepressure of outside and the pressure of the liquid domain increasesby going far away from the bubble. Also as it is evident at the in-stant of the maximum volume of the bubble, the fluid near thewalls in the domain is escaping away from the bubble due to thebubble expansion and is directed toward the bubble at the centerof the gap. After this time, the bubble is going to be collapsed be-cause of the higher pressure outside the bubble. Then a lateral highpressure region develops in the middle of the gap close to the bub-ble which causes the development of an annular liquid jet aroundthe bubble. As times goes on, the high pressure region closes to thebubble and at the last stages of the collapse phase an annular highpressure region appears just near the annular jet of the bubble.Also at the last stages of the collapse phase, it is observed thatthe fluid particles have the highest relative velocity just near theannular liquid jet of the bubble.

Zhang et al. have also studied the behavior of bubble breakupphenomenon between two close walls numerically [17]. Accordingto their studies, during the collapse phase and near the necking re-gion the pressure starts to rise and the fluid particles around thebubble move inwards, which will lead to the breakup of the bubbleinevitably.

948 M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955

Fig. 5 illustrates the dynamic of the fluid around the bubble inthe second case which the tool has an ultrasonic vibration andthe electrical discharge occurs when the tool is far from the work-piece. As it is shown in this case the bubble has chance to reboundbefore splitting into two parts. Thus it is shown that at the non-dimensional time t� = 2.1472, the pressure inside the bubble ishigher than everywhere due to the rebounding of the bubble.

Fig. 6 illustrates the dynamic of the fluid around the bubble inthe third case when the tool has an ultrasonic vibration and theelectrical discharge occurs in the closest position of the tool tothe workpiece.

Fig. 7 illustrates the Variation of the non-dimensional volume ofthe bubble with respect to the non-dimensional time when ci ¼

GAPi2

for the cases of (1) bubble between the fixed workpiece and tool,(2) bubble between the workpiece and the tool with ultrasonicvibration of the tool and the electrical discharge occurs when thetool is far from the workpiece, and (3) bubble between the work-piece and the tool with ultrasonic vibration of the tool and theelectrical discharge occurs at the closest distance between the tooland the workpiece. It is shown that in the third case, the vaporbubble expands to the largest maximum volume and the lifetimeof the bubble is the longest. This, in turn makes the pressure insidethe bubble decrease rapidly to the lowest magnitude. This factcauses the molten material at the sparked point ejects and leavethe crater on the surface of the workpiece. Thus in the third casethe material removal rate is higher than the other cases.

Fig. 8 illustrates the Variation of the pressure inside the bubblewith respect to the non-dimensional time when ci ¼

GAPi2 for the

three cases mentioned above.Figs. 9–11 show the time dependant profiles of an electrical dis-

charge generated bubble, between the tool and the workpiece

Fig. 7. Variation of the non-dimensional volume of the bubble with respect to thenon-dimensional time when ci ¼

GAPi2 for the three cases mentioned above.

Fig. 8. Variation of the pressure inside the bubble with respect to the non-dimensional time when ci ¼

GAPi2 for the three cases mentioned above.

during its growth and collapse phases when ci ¼GAPi2:2 for the three

cases. This means that in these figures the vapor bubble is locatedclose to the workpiece in its initial minimum volume.

Figs. 12 and 13 illustrate the Variation of the non-dimensionalvolume of the bubble and the pressure inside the bubble with re-spect to non-dimensional time when ci ¼

GAPi2:2 for the cases of (1)

bubble between the fixed workpiece and tool, (2) bubble betweenthe workpiece and the tool with ultrasonic vibration of the tool andthe electrical discharge occurs when the tool is far from the work-piece, and (3) bubble between the workpiece and the tool withultrasonic vibration of the tool and the electrical discharge occursat the closest distance between the tool and the workpiece. Likethe previous condition it is shown that in the third case whenthe electrical discharge occurs in the closest position of the tool

Fig. 9. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the fixed workpiece and tool in an EDM process whenci ¼

GAPi2:2 . The corresponding non-dimensional times are shown under each

illustration.

Fig. 10. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the workpiece and tool in an EDM process withultrasonic vibration of the tool when ci ¼

GAPi2:2 . The electrical discharge occurs when

the tool is far from the workpiece. The corresponding non-dimensional times areshown under each illustration.

Fig. 11. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the workpiece and tool in an EDM process withultrasonic vibration of the tool when ci ¼

GAPi2:2 . The electrical discharge occurs in the

closest position of the tool to the workpiece. The corresponding non-dimensionaltimes are shown under each illustration.

Fig. 12. Variation of the non-dimensional volume of the bubble with respect to thenon-dimensional time when ci ¼

GAPi2:2 for the three cases mentioned above.

M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955 949

to the workpiece, the vapor bubble expands to the largest maxi-mum volume and the lifetime of the bubble is the longest.

Figs. 14–16 show the time dependant profiles of an electricaldischarge generated bubble, between the tool and the workpieceduring its growth and collapse phases when ci ¼

GAPi2:4 for the three

cases. This means that in these figures the vapor bubble is locatedvery close to the workpiece in its initial minimum volume. It isshown that in this condition, the vibration of the tool causes achange in the bubble shape for the second case that the electricaldischarge occurs when the tool is far from the workpiece. Fig. 15

Fig. 13. Variation of the pressure inside the bubble with respect to the non-dimensional time when ci ¼

GAPi2:2 for the three cases mentioned above.

Fig. 14. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the fixed workpiece and tool in an EDM process when

ci ¼GAPi2:4 . The corresponding non-dimensional times are shown under each

illustration.

950 M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955

shows that during the collapse phase, a maximum pressure regionis created above the bubble which may cause the bubble compres-sion from the top. Due to the large pressure above the bubble, atthe last stages of the collapse phase a liquid jet is developed onthe far side of the bubble from the workpiece and is directed to-wards it. At the last stage of the collapse phase, at the instant oft� = 2.32867, it is shown that a maximum high pressure region ap-pears above the bubble a little far from the jet, and the relativevelocity of the fluid particles are great just near the jet of thebubble.

Figs. 17 and 18 illustrate the Variation of the non-dimensionalvolume of the bubble and the pressure inside the bubble with re-spect to the non-dimensional time respectively when ci ¼

GAPi2:4 for

the cases of (1) bubble between the fixed workpiece and tool, (2)bubble between the workpiece and the tool with ultrasonic vibra-tion of the tool and the electrical discharge occurs when the tool isfar from the workpiece, and (3) bubble between the workpiece andthe tool with ultrasonic vibration of the tool and the electrical dis-charge occurs at the closest distance between the tool and theworkpiece. Like the previous condition it is shown that in the thirdcase when the electrical discharge occurs in the closest position ofthe tool to the workpiece, the vapor bubble expands to the largestmaximum volume and the lifetime of the bubble is the longest.

Figs. 19–21 show the time dependant profiles of an electricaldischarge generated bubble, between the tool and the workpieceduring its growth and collapse phases when ci ¼

GAPi2:66 for the three

cases. This means that in these figures the vapor bubble is locatedextremely close to the workpiece in its initial minimum volume. Itis shown that in this condition, the vibration of the tool causes thatthe shape of the bubble change for the whole cases.

Figs. 22 and 23 illustrate the Variation of the non-dimensionalvolume of the bubble and the pressure inside the bubble with re-spect to the non-dimensional time respectively when ci ¼

GAPi2:66 for

the cases of (1) bubble between the fixed workpiece and tool, (2)bubble between the workpiece and the tool with ultrasonic vibra-tion of the tool and the electrical discharge occurs when the tool isfar from the workpiece, and (3) bubble between the workpiece andthe tool with ultrasonic vibration of the tool and the electrical dis-charge occurs at the closest distance between the tool and theworkpiece.

Figs. 24–26 show the Variation of the non-dimensional volumeof the bubble for different distances of the initial minimum volumeof the bubble centroid from the workpiece, ci for the three men-tioned cases. It is observed that ci has not considerable affect onthe variation of the bubble volume.

Fig. 27 shows the Variation of the non-dimensional volume ofthe bubble when the initial minimum volume of the bubble is

located at the midpoint of the gap and the electrical discharge oc-curs at the closest distance between the tool and the workpiece. Itis shown that for this case by lowering the frequency and the max-imum amplitude of the ultrasonic vibration of the tool, the maxi-mum volume and the lifetime of the bubble becomes higher.

Fig. 15. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the workpiece and tool in an EDM process withultrasonic vibration of the tool when ci ¼

GAPi2:4 . The electrical discharge occurs when

the tool is far from the workpiece. The corresponding non-dimensional times areshown under each illustration.

Fig. 16. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the workpiece and tool in an EDM process withultrasonic vibration of the tool when ci ¼

GAPi2:4 . The electrical discharge occurs in the

closest position of the tool to the workpiece. The corresponding non-dimensionaltimes are shown under each illustration.

Fig. 17. Variation of the non-dimensional volume of the bubble with respect to thenon-dimensional time when ci ¼

GAPi2:4 for the three cases mentioned above.

M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955 951

Therefore the rate of material removal rate increases when thefrequency and the maximum amplitude of the ultrasonic vibration

Fig. 18. Variation of the pressure inside the bubble with respect to the non-dimensional time when ci ¼

GAPi2:4 for the three cases mentioned above.

Fig. 19. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the fixed workpiece and tool in an EDM process when

ci ¼GAPi2:66. The corresponding non-dimensional times are shown under each

illustration.

952 M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955

of the tool decrease. So in the following figure, the materialremoval rate is the greatest when the frequency and the maximumamplitude of the ultrasonic vibration of the tool are 0.000006 mand 20 kHz respectively.

Fig. 28 shows non-dimensional variation of the bubble centroidwhen the initial minimum volume of the bubble is located at themidpoint of the gap and the electrical discharge occurs at the clos-est distance between the tool and the workpiece. It should be saidthat for the third case when the electrical discharge occurs at theclosest distance between the tool and the workpiece, increasingof the material removal rate cannot be anticipated by loweringor increasing the frequency and the maximum amplitude of theultrasonic vibration of the tool.

In our previous studies, experimental tests were carried out ona die-sinking ED machine with iso-frequency pulse generator. Forultrasonic assisted tests an ultrasonic head with a frequency ofabout 22 kHz was attached to the ED-machine head. The work-piece material used for experiments was tungsten carbide (ISOK15-30) and the material of the tool was forged commercial purecopper. Material removal rate against the pulse on-time with andwithout ultrasonic vibration of the tool were tested for differentcurrent settings. It was shown that the material removal ratewas increased in ultrasonic assisted EDM especially in a short pulseon-time. By considering the nature of the EDM process, the electri-cal discharge occurs between the tool and the workpiece in theclosest two opposite points. Therefore in the ultrasonic assistedEDM the electrical discharge most likely occurs in the closest posi-tion of the tool to the workpiece. It should be noted that if themachining parameters are set up properly and if the frequenciesof the electrical discharge and the tool vibration are synchronized,the electrical discharge between the tool and the workpiece mostlikely occurs in the closest position of the tool to the workpiece[9]. From our numerical results it is predicted that in such a casethe vibration of the tool significantly increases the material re-moval rate in ultrasonic assisted EDM which shows that there isa good agreement between numerical and experimental results.

The validity of our numerical study can also be verified by theresults of Ishida et al. [21]. They studied experimentally the cavita-tion bubble behavior between two solid parallel walls when thedistance between them is small. They showed that the bubble be-tween the walls maintains its spherical shape and reaches themaximum volume. Since then, the bubble laterally shrinks andforms a dumbbell-shaped bubble. They have also shown that whenthe gap between the walls is small, the single bubble is finally di-vided completely into two bubbles owing to the large lateral pres-sure [21].

Zhang et al. have also studied the behavior of bubble breakupphenomenon between two close walls numerically. According to

Fig. 20. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the workpiece and tool in an EDM process withultrasonic vibration of the tool when ci ¼

GAPi2:66. The electrical discharge occurs when

the tool is far from the workpiece. The corresponding non-dimensional times areshown under each illustration.

Fig. 21. Velocity fields and Pressure Contours around an electrical dischargegenerated bubble between the workpiece and tool in an EDM process withultrasonic vibration of the tool when ci ¼

GAPi2:66. The electrical discharge occurs in the

closest position of the tool to the workpiece. The corresponding non-dimensionaltimes are shown under each illustration.

M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955 953

their studies, during the collapse phase and near the neckingregion, the pressure starts to rise and the fluid particles moveinwards, which will lead to the breakup of the bubble inevitably[22].

Fig. 22. Variation of the non-dimensional volume of the bubble with respect to the

non-dimensional time when ci ¼GAPi2:66 for the three cases mentioned above.

Fig. 23. Variation of the pressure inside the bubble with respect to the non-

dimensional time when ci ¼GAPi2:66 for the three cases mentioned above.

Fig. 24. Variation of the non-dimensional volume of the bubble with respect to thenon-dimensional time when the bubble is between the fixed workpiece and tool for

(1) ci ¼GAPi

2 , (2) ci ¼GAPi2:2 , (3) ci ¼

GAPi2:4 and (4) ci ¼

GAPi2:66.

Fig. 25. Variation of the non-dimensional volume of the bubble with respect to thenon-dimensional time when the bubble is located between the workpiece and thetool with ultrasonic vibration of the tool and the electrical discharge occurs when

the tool is far from the workpiece for (1) ci ¼GAPi

2 , (2) ci ¼GAPi2:2 , (3) ci ¼

GAPi2:4 and (4)

ci ¼GAPi2:66.

Fig. 26. Variation of the non-dimensional volume of the bubble with respect to thenon-dimensional time when the bubble is located between the workpiece and thetool with ultrasonic vibration of the tool and the electrical discharge occurs at the

closest distance between the tool and the workpiece for (1) ci ¼GAPi

2 , (2) ci ¼GAPi2:2 , (3)

ci ¼GAPi2:4 and (4) ci ¼

GAPi2:66.

Fig. 27. Variation of the non-dimensional volume of the bubble with respect to thenon-dimensional time when the bubble is located between the workpiece and thetool with ultrasonic vibration of the tool and the electrical discharge occurs when

the tool is far from the workpiece and ci ¼GAPi

2 for different frequencies and differentmaximum amplitude of the tool vibration.

954 M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955

This high jet velocity and the high pressure region around thebubble is also seen for the pressure field at the conclusion ofcollapse near one rigid boundary and between the fixed tool andworkpiece in EDM process [15,23].

This high pressure region is also seen for the pressure field atthe conclusion of collapse near a rigid boundary at the null Kelvinimpulse state that the jet is further accelerated during the latestages of the collapse by a ring-shaped region of high pressuresurrounding the bubble that develops when the bubble wallsare already indented [24].

Fig. 28. non-dimensional variation of the bubble centroid with respect to the non-dimensional time when the bubble is located between the workpiece and the toolwith ultrasonic vibration of the tool and the electrical discharge occurs when thetool is far from the workpiece and ci ¼

GAPi2 for different frequencies and different

maximum amplitude of the tool vibration.

M.T. Shervani-Tabar, N. Mobadersany / Ultrasonics 53 (2013) 943–955 955

10. Concluding remarks

It is shown that ultrasonic vibration of the tool has great influ-ence on the evolution of the bubble and fluid behavior in EDM. Inthis paper the evolution of the electrical discharge generated vaporbubble and its surrounded fluid behavior is investigated in threecases:

Case 1: Bubble between the fixed workpiece and tool.Case 2: Bubble between the workpiece and the tool with ultra-sonic vibration of the tool. In this case the electric dischargeoccurs when the tool is far from the workpiece.Case 3: Bubble between the workpiece and the tool with ultra-sonic vibration of the tool. In this case the electric dischargeoccurs when the tool is in its closest position to the workpiece.

At the last stages of the collapse phase of the bubble, a liquid jetdevelops on the bubble which has different shapes Due to the dif-ferent values of ci and different cases. The bubble behavior can beascertained on the distribution of the pressure in the dielectricfluid around the bubble. It is shown that at the early stages ofthe growing phase of the bubble, the pressure inside the bubblehas the highest value and the pressure in the liquid domain isdecreasing by going far away from the bubble and the bubble ex-pels the fluid to the surrounding. Then at the time when the bubblereaches its maximum volume, the pressure inside the bubble is lessthan the pressure of outside and the pressure of the liquid domainincreases by going far away from the bubble. During the collapsephase a lateral high pressure region develops close to the bubblewhich causes the development of liquid jet on the bubble.

As times goes on, the high pressure region closes to the bubbleand at the last stages of the collapse phase a high pressure regionappears just near the jet of the bubble. Also the fluid particles havethe highest relative velocity just near the liquid jet of the bubble.

It is shown that for the second case when the bubble is betweenthe workpiece and the tool with ultrasonic vibration of the tool andthe electrical discharge occurs when the tool is far from theworkpiece, by lowering the frequency and the maximum

amplitude of the ultrasonic vibration of the tool, the maximum vol-ume and the lifetime of the bubble becomes higher. Therefore therate of material removal rate increases when the frequency and themaximum amplitude of the ultrasonic vibration of the tooldecrease.

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