Acta Polytechnica doi:10.14311/AP.2017.57.0149 Acta Polytechnica 57(2):149–158, 2017 © Czech Technical University in Prague, 2017 available online at http://ojs.cvut.cz/ojs/index.php/ap EXPERIMENTAL DETERMINATION OF TEMPERATURES IN SPARK-GENERATED BUBBLES OSCILLATING IN WATER Karel Vokurka Physics Department, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic correspondence: karel.vokurka@tul.cz Abstract. The surface temperatures of the plasma core in the final stages of the first contraction phase of spark-generated bubbles oscillating under ordinary laboratory conditions in a large expanse of water are determined experimentally. The measurement method is based on an analysis of the optical radiation from the bubbles and on the assumption that the plasma core is radiating as a black-body. It is found that the maximum surface temperatures of the plasma core range 4300–8700 K. Keywords: spark-generated bubbles; temperatures in bubbles; bubble oscillations. 1. Introduction Bubble oscillations remain an important topic in fluid dynamics. While they are traditionally asso- ciated with erosion damage [1, 2], recent efforts have been aimed at medical applications, such as contrast- enhancing in ultrasonic imaging [3] and shock wave lithotripsy [4]. Both spark-generated bubbles [2, 5– 7, 9–11] and laser-generated bubbles [1, 8, 12–18] are very useful tools in experimental studies of free bubble oscillations. The temperature in the interior of a bubble is a very important quantity, and it is necessary to know its variation with time in order to understand the phys- ical processes, such as light emission, taking place in an oscillating bubble. However, it is very difficult to determine the temperatures in spark-generated and laser-generated bubbles experimentally. Until now, only a limited number of estimates based on spec- tral analysis of the light emitted from the bubbles have been reported, in [5, 8, 11, 12, 14, 15]. The dif- ficulties associated with temperature measurements have their origin in the fact that the temperature in a bubble interior varies very rapidly in the final stages of the contraction phase, and the emitted light used for spectral analysis therefore has the form of a very short flash of light. Another difficulty in the measurements originates from the fact that experi- mental spark-generated bubbles have very low repro- ducibility (in terms of bubble size and intensity of oscillation). Although the reproducibility of laser- generated bubbles may be better, it remains rela- tively low. This paper will show that useful estimates of the instantaneous surface temperature of the plasma core in the bubble interior can be determined using the experimental method described in [19]. In this way, au- tonomous behaviour of the plasma can also be proved. The analysis presented here is devoted to free bub- ble oscillation under ordinary laboratory conditions in a large expanse of liquid. The results discussed here have been presented in a shortened form at a conference in Svratka [20]. 2. Experimental setup Freely oscillating bubbles were generated by discharg- ing a capacitor bank via a sparker submerged in a laboratory water tank with dimensions of 6×4×5.5 m (length×width×depth). The experiments were per- formed in tap water at constant hydrostatic pressure p∞ = 125 kPa, at room temperature Θ∞ = 292 K, and far from any boundaries. The capacitance of the capacitor bank could be varied in steps by connecting 1–10 capacitors in parallel. Each of these capaci- tors had a capacitance of 16 µF. The capacitors were charged from a high-voltage source of 4 kV. An air-gap switch was used to trigger the discharge through the sparker. A schematic diagram of the experimental setup is given in Figure 1. Both the spark discharge and the subsequent bub- ble oscillations were accompanied by intensive optical radiation and acoustic radiation. The optical radia- tion was monitored by a detector, which consisted of a fiber optic cable, a photodiode, an amplifier, and an A/D converter. The input surface of the fiber cable was positioned in water at the same level as the sparker, at a distance of 0.2 m aside, pointing perpendicular to the sparker gap and the electrodes. A Hammamatsu photodiode type S2386-18L was po- sitioned at the output surface of the fiber optic cable. The usable spectral range of the photodiode is 320– 1100 nm. An analysis of the optical spectra given in the literature showed that the maximum temperatures in spark-generated and laser-generated bubbles range 5800–8150 K [5, 11, 15, 18]. Then, using the Wien and Planck Law, it can be verified that the spectral max- ima of the optical radiation are within the photodiode bandpass, and that the prevailing part of the radiation is received by the detector. The load resistance of the photodiode was 75 Ω, so the rise time of the mea- sured pulses is about 50 ns. A broadband amplifier (0–10 MHz) was connected to the photodiode output terminals. The output voltage from the amplifier was recorded using a data acquisition board (National In- struments PCI 6115, 12 bit A/D converter) with a sampling frequency of 10 MHz. 149 http://dx.doi.org/10.14311/AP.2017.57.0149 http://ojs.cvut.cz/ojs/index.php/ap Karel Vokurka Acta Polytechnica Figure 1. Experimental setup used to generate oscillating bubbles and to record the optical and acoustic radiation from them (DAQ – data acquisition board, hv – high voltage). The acoustic radiation was monitored using a Re- son broadband hydrophone type TC 4034. The hy- drophone was positioned with the sensitive element at the same depth as the sparker. The distance between the hydrophone acoustic centre and the sparker gap was r = 0.2 m. The output of the hydrophone was connected via a divider 10 : 1 to the second channel of the A/D converter. Prior to the measurements reported here, a limited number of high-speed camera records were taken with framing rates ranging 2800–3000 fps (frames per sec- ond). A more detailed description of the experimental setup is given in an earlier work [19]. In the experiments, a larger number of almost spherical bubbles freely oscillating in a large ex- panse of liquid were successively generated. The size of these bubbles, as described by the first max- imum radius RM 1, ranged 18.5–56.5 mm, and the bubble oscillation intensity, as described by the non- dimensional peak pressure in the first acoustic pulse pzp1 = (pp1/p∞)(r/RM 1), ranged from 24 to 153. Here pp1 is the peak pressure in the first acoustic pulse p1(t), p∞ is the ambient (hydrostatic) pressure at the place of the sparker, and r is the hydrophone distance from the sparker centre. Both RM 1 and pzp1 were determined in each experiment from the re- spective pressure record, using an iterative procedure described in [21]. 3. Results and discussion Examples of several frames from a film record taken with a high-speed camera are given in Figure 2. The first frame was taken in the bubble growth phase. The second to sixth frames correspond to the first contraction phase, and the seventh and eighth frames correspond to the first expansion phase. In the frames taken in the growth and first contraction phases, the glowing plasma core in the bubble interior can be seen. The small bright objects floating in the vicinity of the bubble are plasma packets [19]. The variation of the bubble radius R with time t is shown in Figure 3. The experimental data were determined from the frames, as are the data shown in Figure 2. As the spark-generated bubble is not ideally spherical (it is slightly elongated in the vertical direction), the data points represent an average from two perpendicular directions – one in the horizontal direction and the other in the vertical direction. The individual frames in Figure 2 can be traced to the corresponding points on the plot of bubble radius vs. time, given in Figure 3. The record of the optical radiation (represented by voltage u(t) at the output of the optical detector) consists of a pulse u0(t) that is radiated during the electric discharge and the subsequent explosive bubble growth, and the pulse u1(t) that is radiated during the first bubble contraction and the subsequent bubble expansion [19]. The dynamic range of the optical detector (the photodiode, amplifier, A/D converter) was not sufficiently high to record both u0(t) and u1(t) in a single experiment with good fidelity. Two sets of experiments were therefore performed. The first set of experiments was aimed at recording the pulse u0(t) undisturbed, and the second set of experiments was 150 vol. 57 no. 2/2017 Determination of Temperatures in Spark-Generated Bubbles Figure 2. Selected frames from a film record of a spark-generated bubble. The bubble is RM 1 = 51.5 mm in size, and oscillates with intensity pzp1 = 70.3. The times below each frame refer to the time origin, which is set at the instant of liquid breakdown. The spots of light on the sides of the frames are due to the illuminating lamps. 0 2 4 6 8 10 12 14 16 0 10 20 30 40 50 60 t [ms] R [m m ] R M1 R M2 t 2 t 1 t m1 R m1 Figure 3. Variation of the bubble radius R with time t: ‘◦’ experimental data, ‘–’ fit to the experimental data (in the vicinity of tm1 this fit is only very approximate). The time origin is set at the instant of liquid breakdown. The time at which the bubble attains the first maximum radius RM 1 is denoted as t1; the time at which the bubble is compressed to the first minimum radius RM 1 is denoted as tm1; and the time when the bubble attains the second maximum radius RM 2 is denoted as t2. The growth phase is defined to be within the interval (0, t1), the first contraction phase is within the interval (t1, tm1), and the first expansion phase is within the interval (tm1, t2). 151 Karel Vokurka Acta Polytechnica 0 2 4 6 8 10 12 −0.5 0 0.5 1 1.5 t [ ms ] u (t ) [ m V ] u 0 (t) u 1 (t) u M1 t 2 t u1t 1 Figure 4. Voltage u(t) at the output of the optical detector. The spark-generated bubble is RM 1 = 55.2 mm in size, and oscillates with intensity pzp1 = 153.2. In this figure, the time origin is set at the instant of liquid breakdown, and this instant coincides with the beginning of the steep growth of the initial pulse u0(t). The time at which the bubble attains the first maximum radius RM 1 is denoted as t1, and the time when the bubble attains the second maximum radius RM 2 is denoted as t2. Pulse u0(t) is defined to be within the interval (0, t1), and pulse u1(t) is defined to be within the interval (t1, t2). The blip before t = 0 is a noise due to the air-gap switch. aimed at recording the pulse u1(t) with acceptable noise. A link between the two sets of experiments is achieved by using statistical averages from the first set of records to compute the respective values for the second set of records. An example of the optical pulse u1(t) from the second set of experiments is given in Figure 4. In Figure 4 the pulse u0(t) is clipped due to the limited dynamic range of the optical detector. The maximum value of pulse u1(t) is denoted as uM 1, and the time of its occurrence is denoted as tu1. As can also be seen in Figure 4, the optical radiation from the bubble decreases rapidly to zero after tu1. Another interesting fact that can be observed in Figure 4 is the occurrence of optical radiation from the bubble during the whole first oscillation, i.e., in the interval lasting approximately (0, tu1). As can be observed in the photographs presented in Figure 2, the source of this persisting optical radiation is the plasma core. It can also be seen in these photographs that the bubble interior is filled with two substances. First, there is some transparent matter, which is most probably hot water vapour. Second, there is opaque plasma at the bubble centre. The existence of this hot plasma core during the whole first bubble oscillation, i.e., even long after the electric discharge has termi- nated (in the case of the experimental data shown in Figure 4 the electric discharge lasted approximately only 0.5 ms) is an astonishing phenomenon, observed already by Golubnichii et al. [9, 10]. Golubnichii et al. called this outlasting plasma core “long-living lu- minescence formations”. Similar long-lasting optical radiation has also been observed by Baghdassarian et al. [18]. Baghdassarian et al. explain this radi- ation as the “luminescence from metastable atomic and molecular states injected into the water during or just after the plasma flash, which then recombine very slowly”. However, it can be seen in Figure 2 that the light was emitted from the bubble interior, and not from the surrounding water. A direct comparison of Figure 4 with Figure 1 in [18] can also be used as further proof that in both spark-generated and laser-generated bubbles the glowing plasma core is present in their interior throughout the first bubble oscillation. And since there is no discharge current flowing through the laser-generated bubbles, it follows that the persisting plasma core in spark-generated bubbles is not due to a persisting discharge current. Under the assumption that the hot plasma core in the bubble centre radiates as a black-body (this 152 vol. 57 no. 2/2017 Determination of Temperatures in Spark-Generated Bubbles 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 t [ ms ] θ [ k K ] , R /1 0 [ m m ] R(t) θ(t) θ M1 R M1 R M2 Figure 5. Time variation of the plasma core surface temperature Θ and of the bubble wall radius R. The size of the experimental bubble is RM 1 = 49.0 mm, and the bubble oscillation intensity is pzp1 = 142.1. The time origin is set at the instant when the bubble radius equals RM 1, i.e., it coincides with t1. The maximum surface temperature of the plasma core at the first bubble contraction is denoted as ΘM 1. assumption is justified, e.g., by the results published in [12]), an equation enabling the determination of the plasma surface temperature Θ(t) has been de- rived in [19]. The derivation is based on the Stefan– Boltzman Law, the equation of energy partition during the electric discharge, the time variation of the bubble radius R(t), and the voltage u(t) at the output of the optical detector. Particularly, for the voltage record u1(t) from the second set of experiments the corre- sponding temperature Θ(t) is given by the following equation Θ4(t) = 〈ΘM 0〉4〈RM 0〉2 〈uM 0〉 u1(t) R2p(t) . (1) Here uM 0 is the maximum voltage in pulse u0(t), and this voltage corresponds to the bubble radius RM 0. The surface temperature of the plasma, when the bubble during its growth attains radius RM 0, is ΘM 0. The angle brackets 〈〉 denote the average values on the first set of experiments. For a given bubble size RM 1, these average values can be com- puted using the regression lines and the polynomial derived in [19]: 〈ΘM 0〉 = −0.11RM 1 + 17.4 [kK, mm], 〈RM 0〉 = 0.1836RM 1, and 〈uM 0〉 = 1.25 · 10−4R2M 1 [V, mm]. In (1), Rp is the radius of the light-emitting hot plasma core. An estimate of radius Rp can be ob- tained from knowledge of the bubble wall radius R and of the volume that the plasma core occupies in the bubble interior. Denoting the reduction factor as q (q < 1), then Rp = qR. The variation of the bubble wall radius R with time can be computed us- ing a theoretical bubble model. The exact value of reduction factor q is not known at present. A value of q = 0.2 has been used in [19] for the region in the vicinity of the maximum bubble radius RM 1. This value of q has been estimated from high speed film records of the oscillating bubbles. When photographs of the bubble (e.g., in Figure 2) are inspected, it can be seen that during the contraction phase both the transparent vapour and the opaque plasma are gradu- ally compressed, and remain mutually separated. The proportion of vapour and plasma changes in such a way that the value of reduction factor q is increasing in the final stages of the contraction phase (the volume occupied by plasma decreases more slowly at later times of the contraction phase than the volume of vapour). Unfortunately, no estimate of q can be made from the high speed camera frames for the important time interval covering the closest surroundings of in- stant tu1, when the bubble is compressed to its first minimum radius Rm1. The reason why no estimate of q can be obtained in this interval is that the frames were overexposed by the intensive light emitted from the bubble there (see the sixth frame in Figure 2). However, Ohl [13] has studied the luminescence from 153 Karel Vokurka Acta Polytechnica 0 20 40 60 80 100 120 140 160 180 200 10 −2 10 −1 10 0 p zp1 [ − ] Z m 1 [ − ] Figure 6. The scaling function Zm1 = f (pzp1) computed with Herring’s modified model of an ideal gas bubble oscillating in a large expanse of liquid. a laser-generated bubble and gives (Figure 6 in [13]) the size of the plasma core. The bubble studied by Ohl was RM 1 = 0.8 mm in size, and the radiating plasma core at the maximum contraction of the bubble had dimensions of 31 µm (horizontal diameter) and 44 µm (vertical diameter). This gives the mean value of the first minimum plasma core radius Rpm1 = 14 µm. When expressed in non-dimensional form, one obtains Zpm1 = Rpm1/RM 1 = 0.023. An estimate of the non- dimensional first minimum bubble wall radius given in [19] is Zm1 = Rm1/RM 1 = 0.03. This gives an estimate of the reduction factor q = Zpm1/Zm1 ≈ 0.8, and this is the value of q that will be used in this work for the closest surroundings of Rm1. An example of the variation of the plasma core surface temperature Θ with time t during the first bubble contraction and the subsequent expansion, as computed with (1), is given in Figure 5. Equation (1) has been derived under the assumption that the plasma core is a black-body radiator. This assumption seems to be correct in those instants when the pressure and temperature in the bubble interior are high. And this is fulfilled only in the vicinity of tu1. Hence the computed temperature Θ(t) shown in Figure 5 is correct only in the vicinity of the maximum value ΘM 1. In other instants, the computed temperatures represent just a rough estimate. In (1), only voltages u1(t) and uM 0 are measured directly. Radii R(t) and RM 0 are computed using a theoretical model. In this work, Herring’s modified model of a bubble freely oscillating in a compress- ible liquid far from boundaries is used [21]. In this model, the bubble is assumed to be an ideal gas bub- ble which behaves adiabatically. However, it has been shown earlier, by evaluating experimental pressure records [22], that spark-generated and laser-generated bubbles are vapour bubbles. This means that liquid evaporation and vapour condensation at the bubble wall play an important role during the oscillation of the bubble. It has been further shown by evaluating exper- imental optical records [19], that spark-generated and laser-generated bubbles do not behave adiabatically. Finally, in real spark-generated and laser-generated bubbles there are energy losses, the nature of which has not yet been clarified [19]. They are therefore not encompassed in the theoretical model used to compute R(t). However, it can be shown [23], that during an inter- val lasting for almost all the time of bubble oscillation (except for short intervals at the beginning of the bubble growth and in the final stages of bubble con- traction), the bubble wall motion is governed by the transform of the bubble potential energy into liquid kinetic energy, and vice versa. And neither the bub- ble potential nor the liquid kinetic energy depends on any of the processes mentioned above. That is, these energies and their transforms do not depend on the compressibility of the liquid, the liquid evap- 154 vol. 57 no. 2/2017 Determination of Temperatures in Spark-Generated Bubbles oration and the vapour condensation at the bubble wall, an adiabatic assumption concerning the bubble thermal behaviour, or on any energy losses from the bubble. The computed variation of R(t) (and thus also the computed values of RM 0) may therefore be considered to be relatively correct in this long inter- val. There is only a very short time interval in the vicinity of the minimum radius Rm1, in which all the above-mentioned processes manifest themselves and play an important role. Unfortunately, both the exper- imental description and the theoretical description of the bubble are still insufficient precisely in this short interval. It is a very difficult task to make an experimental determination of Rm1. For example, the framing rate of the high speed camera used here was too low to enable Rm1 to be determined. Even those researchers who have succeeded making an experimental obser- vation of Rm1 encountered extreme difficulties. It is therefore not surprising that few experimentally deter- mined values of Rm1 are given in the literature on the oscillation of spark-generated and laser-generated bub- bles and those values that have been given vary greatly. In addition, unfortunately, the authors usually do not provide further data enabling the computation of pzp1. Because knowledge of the variation of Rm1 with pzp1 is important in this work, a theoretical computation of Rm1 is preferred in the following. The following procedure will be used to determine the value of Rpm1, which is needed in (1) for the computation of temperature ΘM 1. First, using Her- ring’s modified bubble model [21], the scaling function Zm1 = Rm1/RM 1 = f(pzp1) is determined. The com- puted scaling function Zm1 = f(pzp1) is shown in Fig- ure 6. The physical constants used in the computation are: water density ρ = 103 kg m−3, the polytrophic exponent of the gas in a bubble interior γ = 1.25, the velocity of sound in water c = 1480 m s−1, and the hydrostatic pressure p∞ = 125 kPa. The scaling function Zm1 = f(pzp1) has been com- puted for pzp1 ranging from 1 to 200, as this is the range of the bubble oscillation intensities encountered in experiments with spark-generated bubbles [21]. In the experiments that are analysed in this work, pa- rameters RM 1 and pzp1 have been determined for each record u1(t). Then, using the scaling function Zm1(pzp1) given in Figure 6, an estimate of the first minimum radius Rm1 = Zm1RM 1 can be obtained for each experimental record u1(t). An estimate of the cor- responding plasma core radius is then Rpm1 = 0.8Rm1. Thus, using the measured values of uM 1, RM 1, and pzp1 from the second set of experiments and the aver- age values of 〈ΘM 0〉, 〈RM 0〉, and 〈uM 0〉 determined for a given bubble size RM 1 from the regression lines and the polynomial given above, temperature ΘM 1 can be computed. The values of ΘM 1 determined in this way for different bubble sizes RM 1 and for differ- ent bubble oscillation intensities pzp1 are displayed in Figures 7 and 8. Figures 7 and 8 show that the temperatures ΘM 1 determined in this work range c. 4300–8700 K. The regression lines for the mean values of temperature ΘM 1 in dependence on RM 1 and pzp1 are 〈ΘM 1〉 = −0.04RM 1 + 7.57 [kK, mm] and 〈ΘM 1〉 = 0.01pzp1 + 4.61 [kK, –]. It can be seen that the temperatures ΘM 1 vary only moderately with bubble sizes and with the oscillation intensities. Whereas the small varia- tion of ΘM 1 with RM 1 is as expected (the bubbles studied here have sizes for which it can be assumed that the scaling law is valid [24]), the small depen- dence of ΘM 1 on pzp1 is surprising. It shows that the plasma in a bubble behaves rather autonomously, i.e., the plasma temperature varies very little with the pressure in the interior of the bubble. This can be shown easily in the following numerical examples. Let us denote the peak pressure at a bubble wall at the final stages of the first contraction as Pp1. A rough estimate of the value of Pp1 can be obtained from the experimentally determined values of pzp1 and the computed scaling function Zm1 = f(pzp1) displayed in Figure 6. At a bubble wall it holds that r = Rm1 and pp1 = Pp1. After substituting these equalities into the definition formula of pzp1 given in Section ?? (i.e., into the relation pzp1 = (pp1/p∞)(r/RM 1)), one obtains pzp1 = Pp1Zm1/p∞. This equation can be rearranged to give Pp1 = pzp1p∞/Zm1. Then, for pzp1 = 25 and p∞ = 125 kPa it can be read from Figure 6 that Zm1 ≈ 0.1 and thus Pp1 ≈ 31 MPa. For pzp1 = 150 and p∞ = 125 kPa, it can be read from Figure 6 that Zm1 ≈ 0.032 and thus Pp1 ≈ 586 MPa. It follows then that the variation of the peak pressure at the bubble wall Pp1 from 31 MPa to 586 MPa, i.e., by a factor of approximately 19, is accompanied by variation of the mean temperature 〈ΘM 1〉 from 4860 K to 6110 K, i.e. only by a factor of 1.26 (the tempera- tures 〈ΘM 1〉 have been computed using the regression lines given above). The differences in the variation of Pp1 and ΘM 1 provide surprising evidence about the autonomous behaviour of the plasma in the bubble interior. Finally in this Section, the temperatures ΘM 1 given in Figure 7 can be compared with the experimental results of other researchers obtained with bubbles os- cillating in water under ordinary laboratory conditions and far from boundaries. The first rough estimate of ΘM 1 can be derived from data published by Buzukov and Teslenko [8]. These researchers studied bubbles generated by a laser, and found that the optical ra- diation associated with the first optical pulses u1(t) had continuum spectra with spectral maxima occur- ring in the range 375–440 nm. Using Wien’s Law, it can be calculated that the temperatures ΘM 1 ranged 6600–7700 K. Unfortunately, the sizes of the bubbles generated in these experiments are not available. Gol- ubnichii et al. [5] studied bubbles generated by an ex- ploding wire technique, and found that the maximum in the spectrum of the optical radiation associated with the first optical pulse u1(t) lies approximately 155 Karel Vokurka Acta Polytechnica 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 9 10 R M1 [ mm ] θ M 1 [ k K ] BW, Br Ba Go Figure 7. Variation of experimentally determined maximum surface temperatures of the plasma core during the first bubble contraction ΘM 1 with bubble size RM 1: ‘◦’ — the values of ΘM 1 determined in this work, ‘∗’ — the values of ΘM 1 determined in [5] (Go), [12] (Ba), [14] (BW), [15] (Br); these values are discussed at the end of this section. 0 20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 7 8 9 10 p zp1 [ − ] θ M 1 [ k K ] Figure 8. Variation of the experimentally determined maximum surface temperatures of the plasma core during the first bubble contraction ΘM 1 with bubble oscillation intensity pzp1. 156 vol. 57 no. 2/2017 Determination of Temperatures in Spark-Generated Bubbles at 500 nm. Then, using Wien’s Law, temperature ΘM 1 = 5800 K is obtained. In this case, the size of the bubbles was RM 1 = 30 mm. Baghdassarian et al. [12] studied laser-generated bubbles RM 1 = 0.6–0.8 mm in size, and from the gated optical spectra they de- termined that ΘM 1 = 7800 K. Finally, Brujan and Williams [14], and Brujan et al. [15] also studied laser- generated bubbles, and from the gated optical spectra they determined that ΘM 1 = 8150 K. In this case, the bubble sizes RM 1 ranged from 0.65 mm to 0.75 mm. Unfortunately, no data are given in the quoted references to enable the bubble oscillation intensity to be determined. It should be pointed out that the temperatures ob- tained in this work represent instantaneous values, and can be determined for any instant at which the assumption of black-body radiation is valid. In the works of other authors [5, 8, 12, 14, 15], however, no exact instant is given when the temperature was determined. It is only assumed that the measured temperature corresponds to the maximum bubble con- traction. However, when the autonomous plasma behaviour is taken into account, this assumption may not always be correct, and experimental verification is still required. The temperatures measured in this work can also be assigned to particular experiments. This is again in contrast with other works [12, 14, 15], where the spectra have been averaged over 25–50 ex- periments. Although care was taken in those works to average only the spectra corresponding to bubbles of almost the same size, the variation of the temperature with the bubble oscillation intensity (and maybe even with some other as yet unknown factors) has been lost during the averaging procedure. 4. Conclusions The surface temperatures of the plasma core inside spark-generated bubbles in the final stages of the first contraction phase have been determined experimen- tally. It was found that these temperatures range 4300 K to 8700 K. As the statistical averages from another set of experiments were used in the computa- tions, the method employed gives only approximate results. Nevertheless, it has been shown that the val- ues that were obtained are in good agreement with the temperatures published by other researchers. How- ever, unlike the results presented by other researchers, the results presented here were obtained on a large set of bubbles of different sizes that oscillated with different intensities. It has also been shown that the plasma inside bub- bles behaves rather autonomously, i.e., the surface temperature of the plasma varies only very little with the pressure at the bubble wall. The average maxi- mum surface temperature 〈ΘM 1〉 increases moderately with bubble oscillation intensity pzp1, and decreases moderately with bubble size RM 1. In any case, it can be concluded that plasma in bubbles behaves rather differently from the ideal gas that has so often been considered. The experimental method described in this work offers an alternative to spectral methods. As has been discussed here, the method has certain advantages and disadvantages. 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Acustica 60, 269-276, 1986. 158 http://dx.doi.org/10.1063/1.4974452 http://dx.doi.org/10.1103/PhyRevLett.86.4934 http://dx.doi.org/10.1063/1.1489682 http://dx.doi.org/10.1103/PhysRevE.72.016304 http://dx.doi.org/10.1103/PhysRevE.72.066310 http://dx.doi.org/10.1063/1.4818516 http://dx.doi.org/10.1063/1.4793193 http://dx.doi.org/10.1103/PhysRevLett.83.2437 http://dx.doi.org/10.1016/j.expthermflusci.2013.07.004 http://dx.doi.org/10.1016/j.jsv.2010.04.030 http://dx.doi.org/10.1007/BF01596516 http://dx.doi.org/10.1016/S1001-6058(16)60731-X Acta Polytechnica 57(2):149–158, 2017 1 Introduction 2 Experimental setup 3 Results and discussion 4 Conclusions Acknowledgements References