Proceedłngs of the 2nd EAA InternationalSymposium on Hydroacoustics24-27 May 1999, Gdańsk-JurataPOLAND

Thermal Cavitation Mechanism for Generation of Underwater Sound

S. Buogo, G. B. Cannelli

CNR - Istituto di Acustica "o. M. Corbino"Underwater Acoustics Laboratory

Via deI Fosso del Cavaliere 100,00133 Roma, ltalye-mail: siIvano@idac.rm.cnr.it

The implosion of a vapour bubble generated by spark discharges in seawater can generate high powerpulses in the frequency band up to a few hundred klłz. useful to obtain high-resolutton imaging of the seasubbottom. In this paper, a physical model is illustrated to explain the ortgin of the process generating thebubble in a conducting liquid, together with a brief review of the hypothized ideas on (he growth andimplosion mechanism of the bubble. A description is given of the impIementation oj a numerical scheme forthe solution of a set of equations describing the temperature and electric field distributions in the liquid inthe prebreakdown stage. According to this model, electrical breakdown and subsequent thermal cavitationin seawater is govemed by Joule heating with a temperature dependent electrical conductivity.Measurements of the discharge ofthe paraboloidal sparker-based source eonfirm the validity of the energymodel, showing a good agreement ot tne predicted time required for liquid vaporization witb the observedbreakdown time.

1. Introductlon

Thermal cavitation is meant here as amechanism of bubble generation in a fluid wherethermal effects dominate, rather than inertial.Plesset and Prosperetti employed preferably theterm "vapour bubble", as opposed to "cavitationbubble", when describing the dynamics of vapourcavities in simi1ar cases [1]. Bubble generation byan underwater spark is a typical example ofphenomenon where thermal effects play the leadingrole. Sparkers have been more and more used in thepast decades to generate underwater high-powerpressure pulses as a convenient altemative toexplosives. Recently, at the Istituto di Acustica "O.M. Corbino" (IDAC), significant advances havebeen made in their technology by usingparaboloidal sparker-based sources [2]. Thesesources are more versatile than traditional ones,being adaptable to different applications in a rangeof low and middle frequency band not available intraditional sonar devices. Exploiting the implosionof a vapour bubble, generated by the sparkerdischarge at the paraboloid focus, this type ofsource can generate high power pulse in the bandcentered around some hundred kHz, useful toobtain a high-resolution imaging of the near surface

anomalies and objects of the sea subbottom [3].This latter unique feature stimuIates furtherinvestigation on the bubble phenomenon both froma theoretical and experimental point of view. Thestarting point is to investigate the genesis of thevapour bubble, namely the physical processesoccurring during the pre-breakdown and breakdownphases of the e1ectrical discharge. The process ofelectrlcal discharge in water has found someinterest in the seventies when a number ofinvestigations based mainly on optical techniquesaimed at explaining the nature ot breakdown andprebreakdown phenomena [4,5]. A difference inbehaviour was found between dielectric liquid andconducting liquid. Pulsed discharges in dielectricliquids were shown to initiate with the developmentof a leader, that is a plasma channel starting fromone of the electrodes [6,7]. Breakdown was seen tooccur in plasma and not in the liquid, as theconscquence of the formation of an arc when thechannel bridges the electrode gap. The leadergrowth from electrode to electrode in .theprebreakdown phase was found to be dependent onthe e1ectrical conductance of the liquid, and wasabsent in low resistance electrolytes such as saltwater at large eoncentranons. From these early

207

observations it was argued that the principal rolefor the onset of breakdown in this latter liquid couldbe played by thermal processes [8]. In such cases anelectric arc develops in a bubble filled withelectrolytic gas and water vapour near one of theelectrodes. This bubble is due to vaporization,caused by the large electric current flowing in theliquid.

In this paper, a physical model is illustrated toexplain the origin of the process generating thebubble in a conducting liquid, together with a briefreview of the hypothized ideas on the growth andimplosion mechanism of the bubble. This study iscrucial in view of improving the performance of theinnovative paraboloidal sparker source.

2. Thermal Cavłtatłon In a Conductłng Llquld

Very few investigations have been made on thegenesis of the vapour bubble process. An attempt todevelop a numerical model of the electricaldischarge in a conducting liquid was made to fmdthe temperature and electric field perturbationsduring the preleader stage for the particular case ofconcentric spherical electrodes and infinite sourcecapacity [9]. The duration of the preleader andleader stage of the discharge was observed todepend on the applied voltage and the electricalconductivity of the liquid, the latter being afunction of temperature. As a consequence, thedevelopment of thermal instability in the liquid wasproposed as the initiating mechanism of thedischarge. As a confumation of the validity of athermal model, an experimental study showed thatbreakdown in seawater is strictly correlated withvaporization [10]. Although vaporization could notbe observed directly, a threshold energy wasdetermined based on the minimum energy requiredto observe a bubble collapse pulse in the acousticsignature. Breakdown was shown to take place onlyafter this threshold energy has been dissipated inthe liquid. The proposed model for breakdown inseawater based on vaporization showed to beconsistent up to field strengths of 50 kV/cm. Anopen question remains on how large a vapourbubble must grow to initiate breakdown, andwhether the dielectric strength of the liquid plays anessential role in the process.

High-power sparkers generate a lot of bubbl~,due to the extremely high rate of energy transfer malittle portion of liquid. In these sources, a typicalenergy of several hundred Joule is dissipated in le~sthan 1 ms inside a volume of the order of 1 cm,which gives a mean energy density of the order oflMW/cm'. A large portion of this energy IStransferred to the liquid after breakdown time,causing aviolent expansion of the plasma channeldue to the very large electric current (order of kA)flowing in h. This causes tbe first shock wave

208

travelling in tbe liquid, that is recorded as tbeprimary pressure pulse in tbe acoustic signature oftbe source. It was argued in previous preliminarystudies [11,12] tbat following this violent outwardmotion, an inertial expansion may bring a vapourbubble to a maximum radius, after which tbemotion is reversed and a collapse is started. Thebubble dynamics is slow enough so that no acousticwave is generated. At the end of the collapse phase,when condensation of vapour inside the bubblecannot compensate for the decrease of the bubblevolume, the liquid motion is suddenly stopped andanother shock wave is generated. Vapour inside tbebubble may contain residual non-condensable gas,and one or more rebound may take place after thefirst collapse. More controlled pulses are generatedin the paraboloida! sparker source designed atIDAC [2]. In tbis source tbe electrodes are placedin the focal point of a paraboloidal reflector, toprovide a high-intensity directional response. Arigid surface is only a few cm away from the pointwhere cavitation occurs, and this must have someinfluence on bubble evolution, compared to tbe freefield case. Indeed, tbe acoustic signature of theparaboloidal source shows two distinct pulses: tbefirst one is attributed to tbe shock wave produced atbreakdown time and the second one, occurringseveral ms later, is very likely due to the collapse ofa single large vapour bubble. The time delaybetween the two pulses was seen to be increasingwith tbe electrostatic energy, and was higher forsmaller electrode gap distances. From the analysisof experimental data, a model was proposed toexplain the growth and collapse of a sphericalbubble under the effect of tbe internal vapourpressure [11]. The delay was seen to ~e ~ligh~yincreasing with tbe temperature of tbe Iiquid: thisvariation was in good agreement with thecorresponding variation of the vapour pressure ofwater [13]. To exploit thermal cavitation as a wayto produce controlled bigh-power pulsed sound withtbe maximum acoustic efficiency, an improvementof the electromechanical part of the paraboloidalsource was made tbat allowed the primary pulse tobe minimized concentrating the acoustic energy ina very high-intensity single pulse. One possiblecause for tbe considerably simpler acousticsignature of the paraboloidal source is tbe presenceof the rigid surface tbat makes the bubble collapsewith a highly asymmetric shape [14]. The surfacemight influence the bubble evolution after tbe firstcollapse, suppressing a subsequent rebound andoscillation. This is the objective of a researchproject, still in progress at IDAC, whi~h aims to ~edesign anovel high-power, high-resolutiontransducer for sea subbottom prospecting. In Fig. 1,a comparison is shown between the time signatureof a traditional sparker source and that of theparaboloida! sparker for the two differentprototypes: the old one and the improved one.

a.E-c

I'd ~~ ,/ -----Bubbler-

(4)

o 10 Time(m.) 15

(b)

a.E«

O 5 OTimelmsl 15

~ 1.0.,-------,-----------,~ 00:::;Cl. 0.5

'"'«w::=: 0.0 +__-J-~

o.TlME (uS)

Fig. l. Comparison among the acoustic signaturesof the old prototype of paraboloidal sparker-basedsource (a), of a traditional sparker source (b) andof the improved paraboloidal prototype (c).

8. o.

3. Analytlcal Approach to Breakdown Process

In recent papers, an attempt has been made todevelop an analytical approach to breakdown andcavitation process in seawater [15,16]. A number ofpreliminary tests were done, using the paraboloidalsource, to measure the characteristics of theelectrical discharge and to validate some hypothesison the involved mechanisms. One major featurethat was noted expeńmentally, for spark dischargesin seawater by paraboloidal sources, is theirrelatively long breakdown time (of the order of tento hundred us), in comparison with that found intrue dielectric breakdown occurring at time scalesof less than one us. Another aspect of thermalbreakdown is the considerably lower field sttengthwith respect to that for the true dielectricbreakdown: its value can be of 10-20 kV/cm onIycompared to many hundreds ofkV/cm.

Prom the analysis of the voltage and currentplots recorded during discharges with chargevoltage 2.25 kV, energy 200 J, it was found that aconsiderable portion of the total electtostatic energyis ttansferred into heat in the discharge phase afterbreakdown [15]. This amount of energy is availablefor plasma channel expansion and subsequentvapour bubble growth, which gives an indication ofthe conversion efficiency of electrical energy into

mechanical energy, namely the acoustic efficiencyof the source.

In the following, a description is given of theimplementation of a numerical scheme for thesolution of a set of equations describing thetemperature and electric field distributions in theliquid in the prebreakdown stage. The starting pointis to evaluate how long it takes for an appliedelectric field to raise the liquid temperaturę byJoule effect up to a critic temperature when theliquid starts to vapońze. If we assume thatbreakdown takes place when vapour regions bridgethe electtode gap, the evaluated time to reach thecritic temperature should be comparable with, andno greater than, the breakdown time, that is thetime delay between field application and the onsetof breakdown. In the following, convective effectsare neglected since the time scale is too small toallow signiticant motion of the Iiquid. Under thisassumption, the process of electrical discharge canbe described by Ohm's law with a temperaturedependent electrical conductivity a (T) . Thestatement of conservation of electric chargebecomes

20

120V.[cr(T)VV]=O. (1)

After some passages, this equation takes the form

...,2 l da..., ...,y V=---yT· yV

o dT '(2)

which is a Poisson equation for the potential in theliquid to be solved with boundary conditions±V(t)/2 on the surface of the twa electtodes.

The therma1 power density dissipated in theliquid is given by

(3)

and the temperature evolves according to

dT =_I_(W +kV2T)dt g ,PCp

(4)

where p, cp

and k are the density, specific heat andthermal conductivity of the liquid. The conductiveterm in the right-hand side of eq. (4) can be shownto be much less important than the heat generationterm for field strengths of 10' - lOS V/cm [9].Therefore, boundary conditions for T in equation(4) are not critical and one can assume, forsimplicity, that there is no heat exchange betweenthe liquid and the electtodes. Due to the presence ofthe a(T) term the coupled system of eqs. (2) and (4)is non-linear. The functional form of o is assumedto be

(T) A -Brra =: , (5)

209

where A and B are empirical constants. Accordingto this model, breakdown is the result of aninstability due to the increased electricalconduetance as the temperature is raised by theflow of electric current. For a finite storage ofelectrostatic energy, the potential difference acrossthe electrodes is not a constant, but variesaccording to the electrical parameters of the chargecircuit. Assuming a simple RLC model, theequation grvmg the approximate boundarycondition for Eq. (4) is

1 f dlV =v: -- l dt-L-o C dt'

where Vo is the initial charge voltage of thecapacitor. Here the electrode gap is assumed to bepurely resistive, and the circuit resistance isneglected. This is true for the pre-breakown stage,when the gap resistance dissipates most of theelectrostatic energy.

The electrical current can be evaluated byintegration of the current density over a closedsurface S around one of the electrodes:

1= Ss vve zs.

where o is the unit vector normal to S.

Equations (2), (4), (6) and (7) form a ser'ófcoupled nonlinear equations to be solvedsimultaneously for T and V. Each differentialequation is solved with a numerical scheme, using abispherical coordinate transfonnation to map thespace outside the electrodes in a rectangular space.The first step is to solve the Poisson equation (2)

o 40 80 10020 60t tusec)

2500 r--~----~JTT17t1"'=::-o

2000 .1.6.-\(\1:----

E (J)~~~~ � ---~---~:.~:tĘ~~:~-~~~500 ..J~:~::::.~.:~:;.~~::-..~..~.~:.~...~..~:~:~:~:-

oO 20 8040 60

t (usec)

Fig. 2. Maximum temperature T_ and dissipatedenergy E as a function of time, for four values ofcharge voltage V". Gap distance: 1 cm, electroderadiu s: l cm, capacitance: 12 pF, induetance: 2.5uu. lnitial temperature: 296 K.

210

(6)

with initial uniform temperature and conductivity toobtain an estimate of the potential V, with boundaryconditions obtained by integrating the circuitequation (6). The energy balance equation (4) isthen solved using this estimate, and an iteration isstarted until the solutions converge within aspecified error. Upon convergence, the totaI currentis evaluated by equation (7) and a new potentialdifference for the next time step is obtained fromequation (6), for which a new iteration for V and Tcan be started. This method showed to besufficiently fast and accurate, provided that theelectrode gap/radius ratio is not » 1. Somedrawings are shown to better illustrate the model. InFig. 2, the maximum temperature, T_, anddissipated energy, E, predicted by the aboverelations, are given as a function of time, for fixedvalues of electrodes geometry, spark gap andelectrical parameters. Moreover, two typicaltemperature profiles obtained with differentgap/radiu s ratios are shown in Fig. 3.

To validate the model, electrical and acousticalmeasurements were taken during the discharge ofthe paraboloidal sparker-based source at IDAC. TheexperimentaI apparatus consisted of alaboratorytank with dimensions l m by 0.64 m and a depth of0.66 m. The paraboloidal source was positionedhorizontaIly, and a 180 kHz bandwidth hydrophonewas placed along the paraboloid axis. The liquidwas ftesh water with added sodium chloride, with asalinity of 3.4%. The electrostatic energy wassupplied by a high-voltagę generator (2250 Volt)charging a capacitor bank whose capacitance could

(7)

T (K)400

(a)

T (K)400350

3100

Fig. 3. Temperature profiles in the piane ofcylindricai coordinates {r.z} after t = 3.5 ms jortwo dijferent gap distances: 14 mm (a) and 8 mm(b). Electrode radius: 5 mm, charge voltage: 2250V, capacitance: 360 pF, inductance: 2.5 uu.lnitial temperature is 293 K. The curves in thebase piane indicate the projections oj theisothermallines T = T,.p= 373 K.

be varied in the range 40 - 360 ~. An auxiliary airspark gap was used to trigger the discbarge.Particular care was taken for voltage and currentmeasurements: voltage was measured directiyacross the two bemispberical tips of the electrodesexcluding the inductive load of tbe connectingwires. A properly designed Hall effect detector wasused to measure the current and, at tbe same time,to bave a de-conpling between voltagę and currentwires. In tbis way, it was possible to verify tbatvoltage and current are in phase during the plasmadiscbarge, indicating tbat tbe spark gap bebavesessentially as a purely resistive load. The totalcircuit inductance was kept to a minimum andestimated to be 2.5 ~, being essentially tbat of tbecables from tbe capacitor bank to tbe sparker.Tbeoretical curves computed using equations (2),(4), (6), and (7) were compared witb experimentaldata of voltage and current measured across tbeelectrode gap. Fig. 4 is one of a series of similarplots [15] showing voltage and current as a functionof time up to the observed breakdown time, fordifferent values of hemispherica1 electrode radiusand gap distance (In tbis case R = 5 mm, d = 4mm).Tbe comparison generally shows a good agreementbetween tbeoretical and experimental data. Themajor differences were noted from tbe currentvalues for cases witb tbe smaller gap distance,where the deviation is possibly due to an alterationof tbe liquid cbemical properties after severalsparks, giving an actual bigher conductivity. This isexpected to be more evident in cases wbere asmaller portion of liquid is exposed to tbe electricfield: indeed, tbe deviation is greater for both tbesmaller R and d.

A confrrmation of the validity of the energymodel arises from tbe comparison of tbe predicted

2500-.--------------------------,2000

2:- 1500'"Ol

~ 1000o> 500

o1500 o

~ 1000-c~ 500 -'"u

oo

100 200

-------- f- - - - - - - - - - - -.- '::"...Jvap fi.

1~0 200 300Time (J.ls)

Fig. 4. Voltage and current as a function of timeduring the prebreakdown discharge: theoreticalmodel (dashed line) and experimental data (solidline). Electrode radius: 5 mm, gap distance: 4 mm.Charge voltage: 2250 li; capacitance: 80 jJF.

time required for liquid vaporization witb tbeexperimental breakdown time. Vaporization time isdefined as

(8)

where tbe standard value T = 373 K may beassumed. A simple way to ~ilinate tbe tbeoretica1breakdown time t'. is to continue tbe vaporizationmodel until tbe vaporization temperature is reachedat the midpoint between the electrodes, where thefield strengtb and temperature are tbe lowest alongtbe electrode gap. Wben tbis happens, a vapourcbannel witb substantially lower density and higberelectrica1 conductivity develops, leading to plasmageneration and breakdown. In Fig. 4, two vertica1segments represent tbe predicted t and (: tbese..• .values are in good agreement witb tbe observedbreakdown times in tbe experimenta1 curves.

300

4. DlscusslonTbe above model suggest tbat electrical

breakdown and subsequent tbermal cavitation in aconducting liquid such as seawater can be describedby a simple model of Joule heating with atemperature dependent electrica1 conductivity. It isproposed tbat breakdown mechanism is essentiallytbat of a discbarge in a low-density, highlyconductive vapour, and tbat a sufficient amount ofenergy bas to be delivered to tbe liquid in order tovaporize it and allow the discharge to take place ina gaseous pbase. The electrica1 parameters of tbediscbarge circuit may influence tbe time rate of tbeenergy transfer, yielding low-power or bigh-powerdischarges. As a result, vaporization and breakdownmay be slower or faster even witb tbe same appliedvoltage. Discbarges with a lower voltage and bighercapacitance source may be equivalent to highervoltagę discharges, only with a longer time scale.

From an analysis of plots in Fig. 2, it can beseen tbat tbe energy curves reach asymptotica1lytbeir limiting value CV:/2, and a criticaltemperature Tvap is reacbed when a given energyhas been dissipated, defmed as

Evap = J:"" I(t)V(t)dt. (9)

If tbe energy available in tbe source does notexceed E ,regardless of tbe initial applied voltage,the dischi'rge proceeds witb no vaporization andsubsequent breakdown. If the available energy isgreater tban E , but not sufficient to raise thetemperature at'" tbe gap midpoint above T..,.,vaporization (i.e. bubble growth and collapse) maytake place, This is in agreement witb experimentalresułts [10].

211

The same can happen when the geometry of theelectrodes does not allow high values of fieldstrength over an extended region, that is for largeelectrode radius or gap distance. It can be pointedout from an inspection of the plots of Fig. 3 thatincreasing the gap distance over d = R thetemperature rise is greatly reduced, which suggeststhat there is a limiting value for d, beyond whichthe source ability to deliver energy to the liquid isgreatly reduced. On the other band, if d is sma1lcompared with R, the discharge may take placelargely in vapour with little dissipated beat due tothe high electric conductivity. In this case, thepotential energy left in the source at t = t.., is notused for further heating of the liquid, but oscillatesaccording to the usual RLC law between thecapacitive and the inductive component of the load.Therefore, it seems reasonable to suppose that tooptimize the efficiency of energy conversion intoacoustical pressure, the vaporization should bereached when almost all the available energy basalready been dissipated into the liquid.

For a fixed gap distance d, the model shows thatthe vaporization mechanism is optimal if theelectrode radius R is smaller by approximately afactor of 2/3. If this condition is met, during thefirst stage of the discharge most of the heating isproduced near the electrode surfaces, where thetemperaturegradient is larger, which implies afaster growth rate for the vaporized regions towardsthe midpoint of the gap. If one relates theprobability of a breakdown event to the temperaturereached after a fixed time at the midpoint along thegap distance, an optimal value for R can beobtained once the gap distance d is given, for whicha maximum temperature is attained. For smallervalues of R the heating is highly localized near theelectrodes, which could possibly imply somethermal and electrical insulation effects due to thegas layer surrounding the surfaces. For largervalues of R more liquid is exposed to the electricfield inside the gap, and the local value of fieldstrength is lower, which implies that the heatingproces s is less efficient.

References

[1] M. S. Plesset, A. Prosperetti, Bubbledynamics and cavitation, Ann. Rev. Fluid Mech.(9), pp. 145-185 (1977).

[2] G. B. Cannelli, E. D'Ottavi, Method of high-resolution sea bottom prospecting and tuned arrayof paraboloidal electroacoustic transducers to carryout such method, Japan Patent N. 5-505235 (August5, 1993), USA Patent N. 5,398,217 (March 14,1995), Canadian Patent N. 2,065,457 (July 14,1998).

[3] G. B. Cannelli, E. D'Ottavi, L. Pitolli, G.Pontuale, Advances in the technology ofparaboloidal sparker-based sources: lmaging of

212

underwater objects using bubb1e implosion as theacoustic source, in Proc. 4th European Conferenceon Underwater Acoustics, Rome, pp. 407-412(1998).

[4] A. P. A1khimov, V. V. Vorob'ev, V. F.Klimkin, A. G. Ponomarenko, R. I. Soloukhin, Thedevelopment of electrical discharge in water, Sov.Phys. DokI. 15 (10), pp. 959-961 (1971).

[5] E. V. Yanshin, I. T. Ovchinnikov, Yu. N.Vershinin, Optical study of nanosecondprebreakdown phenomena in water, Sov. Phys.Tech. Phys. 18 (10), pp. 1303-1306 (1974).

[6] E. V. Yanshin, I. T. Ovchinnikov, Yu. N.Vershinin, Mechanism of the pulsed electricalbreakdown of water, Sov. Phys. DokI. 19 (2), pp.95-96 (1974).

[7] LM. Gavrilov, V. R. Kukhta, V. V. Lopatin,P. G. Petrov, V. Ya. Ushakov, Impulsive-dischargeformation in water, Soviet Physics Journal 32, pp.74-78 (1989).

[8] N. P. MeI'nikov, G. A. Ostroumov, M. Yu.Stoyak, Formation of electrical breakdown inaqueous sodium chloride solutions, Sov. Phys.Tech. Pbys. 9 (5), pp. 730-733 (1964).

[9] V. G. Zhekul, G. B. Rakovskii, Theory ofelectrical discharge formation in a conductingIiquid, Sov. Phys. Tech. Phys. 28 (1), pp. 4-8(1983).

[10] A. H. Olson, S. P. Sutton, The physicalmechanism leading to electrical breakdown inunderwater arc sound sources, J. Acoust. Soc. Am.94 (4), pp. 2226-2231 (1993).

[11] G. B. Cannelli, E. D'Ottavi, A. Prosperetti,Bubble activity induced by high-power marinesources, in: Proc. ofOCEANS '90, Washington DC,pp. 533-537 (1990).

[12] A. Prosperetti, G. B. Cannelli, E. D'Ottavi,Dynamics of a spark-generated bubble, Proc. ofWorkshop on thermal acoustic generation, Anstm,Texas, Vol. 1, p. 133 (1991).

[13] Cannelli, G. B., and D'Ottavi, E., Physicalphenomena involved in seawater plasma basedsound source, in: Proc. European Conference onUnderwater Acoustics, Luxembourg, pp. 635-638(1992).

[14] 1. R Blake, B. B. Taib, G. Doherty,Transient cavities near boundaries. Part 1. Rigidboundary, 1. Fluid Mechanics, Vo1. 170, pp. 479-497 (1986).

[15] S. Buogo, G. B. Cannelli, E. D'Ottavi, L.Pitolli, G. Pontuale, Bubb1e Influence on theBehaviour of Sea-Water PIasma-Based SoundSources, ACUSTICA - Acta Acustica, Vol. 84, pp.1025-1030 (1998).

[16] S. Buogo, G. B. Cannelli, E. D'Ottavi, L.Pitolli, G. Pontuale, Seawater Breakdown andBubb1e Generation in Marine Sparker Based SoundSources, in: Proc. 4th European Conference onUnderwater Acoustics, Roma, pp. 119-124 (1998).