Shock Waves (2006) 15(2): 73–80 DOI 10.1007/s00193-006-0011-8
O R I G I N A L A RT I C L E
A. Sayapin · A. Grinenko · S. Efimov · Ya. E. Krasik
Comparison of different methods of measurement of pressure of underwater shock waves generated by electrical discharge
Received: 9 February 2005 / Accepted: 28 November 2005 / Published online: 5 April 2006 C Springer-Verlag 2006
Abstract Different methods for measurement of strong underwater shock waves pressure pulses with peak pressures of up to 200 MPa and rise time of tens to hundreds of nanoseconds are described and compared. The experimental techniques include direct methods of pressure measurement using various electromechanical gauges such as quartz, carbon-based, and commercially available PCB gauges, and nondirect methods based on measurement of the velocity of the shock wave such as time-of-flight and fast-streak photography. Advantages and disadvantages of the used gauges and methods are discussed. The shock waves were produced by underwater electrical discharge (discharge current amplitude ≤100 kA, pulse duration ≤5 µs) initiated by an exploding wire. A good correspondence between the pressure amplitudes measured by the various gauges and methods was observed. The obtained dependence of the shock wave pressure on the distance from the discharge channel was found to be best fitted by a r −0.7 law. It is also shown that none of these methods can be used to determine the time evolution of the pressure behind the front of the shock wave. Keywords Electrical wire explosion · Underwater shock wave measuring techniques PACS 52.35.Tc · 07.35.+k · 52.80.Wg
1 Introduction Numerous experimental investigations have shown that high-current underwater electrical discharge is accompanied by the generation of shock waves (SW) [1–3]. These SWs are intensively used in different important technical applications and scientific research [4, 5]. In order to estimate the Communicated by K. Takayama A. Sayapin · A. Grinenko · S. Efimov · Ya. E. Krasik (B) Physics Department, Technion, Haifa 32000, Israel E-mail:
[email protected] Tel.: +972-4-8293559 Fax: +972-4-8226641
efficiency of the discharge-energy transfer to the generated SWs one has to measure temporal pressure profile of these waves. Different methods and probes are used for this measurement. Among these methods one can mention optical fast streak or frame photography of shock wave propagation [6], laser interferometry of target velocity measurements [7], various probes based on the piezoelectric effect [8], carbonbased probes [9] and the time-of-flight (TOF) method [10]. Each of these methods has its own advantages and disadvantages. For instance, optical streak and frame photography gives information about the SW velocity which can be used for the evaluation of SW pressure, but this information is insufficient for SW energy estimation. The same concern is related to the TOF method. The laser interferometry method provides data about the SW pressure time evolution after careful mathematical treatment of the target velocity time evolution. However, in the case of high-current underwater discharges when the effect of water and the target preheating by powerful radiation [2, 11], is significant, this method becomes doubtful. Furthermore, this method requires a replacement of the target in each electrical discharge. Concerning measurements of SW parameters by various piezoelectric or carbon-based probes one should take into account the non-linear response of these probes. The latter could cause a significant disturbance of the obtained pressure waveforms, which could result in overestimation of the SW pressure and energy [12]. In this paper, a number of mentioned gauges along with the indirect pressure measurement techniques have been used to obtain the peak pressure of the SW generated by the underwater electrical wire explosion. The gauges have been independently calibrated and suited for underwater fast SWs measurement. The reliability of the factory calibration of the used gauges and their applicability to the measurement of the pressure of underwater SWs had to be thoroughly checked. Some of the gauges were not destined for the measurement of a pulsed short duration pressure since these gauges are characterized by long rise time or/and hysteresis. In addition, some of gauges (carbon-based probes) were calibrated under static pressure conditions. Other gauges
74
A. Sayapin et al.
Fig. 2 Typical waveforms of the discharge current a and pressure b obtained by PCB-119A02 gauge placed at a distance of 85 mm from the discharge channel. Cu wire 0.1 mm in diameter and 85 mm in length. Charging voltage 22 kV and generator capacitance 5 µF
Fig. 1 a Experimental setup. b Teflon frame support for exploding wire
(piezoelectric) were calibrated in the air shock tube while acoustic impedance of the air being different from the acoustic impedance of the water. The rise time of these gauges was comparable to the typical rise time of the measured SW which could result in nonflat frequency response in the range of the frequencies of the measured SWs. Such non flat frequency response could distort the shape and the amplitude of the measured signal. The results of the direct gauge-based measurement techniques have been compared to and verified by the indirect methods such as the TOF method and direct measurement of the SW velocity by the fast frame streak cameras.
2 Experimental setup and diagnostics The experimental setup is shown in Fig. 1. It consists of two Maxwell type capacitors (each capacitor of 2.7 µF, 50 nH, 50 kV) connected in parallel and having a common railgap triggered switch [13]. This system allows one to obtain in a short circuit a current with an amplitude of ∼115 kA
(at 35 kV charging voltage, stored energy of 3.3 kJ) with a rise time of ∼3 µs. The total self-inductance of the setup is ∼0.5 µH. A high-voltage, high-current pulse was applied to a thin wire (wire diameter in the range of 0.05–0.8 mm) made of Cu of length in the range of 30–85 mm. This wire was placed between two electrodes immersed in technical water. Voltage and current waveform measurements were carried out using a Tektronix high-voltage, high-impedance probe and a Pierson coil, respectively. Typical waveforms of the discharge current Id and pressure, obtained using a PCB-11902A gauge, are presented in Fig. 2. One can see that the discharge of the capacitors causes fast (∼300 ns) electrical explosion of the wire (Id ≈ 15 kA) (see Fig. 2a) and generation of the first SW (see Fig. 2b). Further, because of the fact that a major part of the energy remains in the capacitors, a discharge plasma channel is formed between the electrodes in the water (Id ≈ 55 kA) with the generation of a second SW. It was found that a time delay τ between the wire explosion (the maximum amplitude of Id which causes electrical explosion of the wire) and the formation of the plasma channel (the maximum amplitude of Id in the discharge channel) depends on the diameter and length of the wire and on the stored energy. Namely, an increase in the amplitude of Id leads to a decrease in τ for the same diameter and length of wire. Also, a decrease in length or in diameter of the wire for a constant Id leads to a decrease in τ . Let us note that, in the case when τ > 10−6 µs, the discharge through the plasma channel has an aperiodical form. At smaller τ the discharge has a periodical form with a fast current decay.
Methods of measurement of underwater SWs generated by electrical discharge
75
The experiments described in this paper were performed with a Cu wire (diameter of 0.1 mm, length of 85 mm) which allows two SWs to be generated by the wire explosion and plasma channel formation. In general, these SWs have different pulse duration and amplitudes; in these experiments τ was ∼7 µs. The latter leads to a delay ≥3 µs between the time of appearance the first and second SWs, at a distance of 85 mm from the discharge channel, for different generator-charging voltage, which were varied in the range of 15–30 kV. At that time (≥3 µs), as shown later, the response of the pressure gauge to the first SW becomes negligibly small. This fact allows one to conclude that the second SW propagates in almost non-disturbed water. Thus, such a setup allows one to carry out a comparison of the data obtained by different gauges and also to compare the response of each gauge to SWs with different parameters. In order to measure the parameters of the generated SWs we used gauges based on carbon solution, quartz crystals, piezoelectric gauges PCB-119A02 and PCB-138A38 produced by PCB Piezotronics Inc., TOF gauges based on miniature piezoelectric CA-1136 produced by Dynasen Inc., and fast framing 4Quik05A and streak EOK-XX cameras for photography of SW propagation in water. Pressure gauges based on carbon solution were made of carbon resistors produced by Bradley Inc. It is known that under increased external pressure P the resistance of the carbon-based resistor r decreases [9, 14]. In order to obtain the dependence of the change of the resistance r on the pressure one has to calibrate a specific resistor for use in the experiment. We tested several types of carbon resistors with power of 0.125 and 0.25 W and with resistance in the range of 5–250 . In the described experiments we used an r = 30 , 0.125 W resistor connected in series with an R0 = 100 resistor. A dc voltage of ϕ = 10 V was applied to these two resistors. It was checked that the dc current power supply does not cause heating of the resistor. We carried out careful static calibration of this type of resistor in a specially made chamber filled with quartz sand using a Carver laboratory press in the range of pressure (0–4)×107 Pa. The results of this calibration showed an almost linear dependence on the pressure (within ± 10% error bar) of r/r = k, where k = 0.67 GPa−1 . The obtained dependence of R/R remained the same for several repeatable calibration tests. The calibration also showed the existence of substantial hysteresis behavior of the resistor during the pressure decrease. The latter renders doubtful the measurements of true pressure waveforms using this type of probe. However, these probes can be used to perform pressure amplitude measurements at the rise of the pressure pulse. The change of the resistance R can be expressed by the change in the potential difference ϕ at the resistor R due to the increase of the external pressure as −1 R R0 ϕ ϕ 1 (1) = 1+ 1+ − R R ϕ ϕ 1 + R/r
inductance capacitor, C = 20 µF. The tested resistor was placed inside the water at distances of 85, 65, 45 and 25 mm from the discharge channel. It was shown that the resistor can sustain several (∼5) electrical discharges if the SW pressure does not exceed 108 Pa. A quartz gauge was prepared similar to the gauge described in [15] and [16]. Namely, we used quartz disks, 25 mm in diameter and 6 mm in thickness covered by a grounded thin gold layer. A collector was made at the rear side of the quartz by a thin circular gold plate with a radius of 5 mm. The collector was separated from the grounded layer by a ring of width 1 mm. A 50 low-inductance resistor, which served as a load, was soldered to the collector. The gauges, placed in a metal tube filled with epoxy, were positioned at distances of 35, 50, 60, and 100 mm from the discharge channel. The temporal pressure profile in the quartz Pq (t) was determined from the measured waveform of the current i(t) in the 50 resistor as [17]: Pq (t)[kPa] = 0.1i(t)l[k SUs ]−1 . Here l is the quartz thickness in meters, S is the area of the collector in square meter, Us = 5.72 × 103 m/s is the wave speed in the quartz in meter per second, and k = 1.86×10−4 C/(m2 kPa) is the piezoelectric coefficient. The pressure in water Pw (t) was determined, taking into account for impedance mismatch between the quartz and water: Pw (t) = Pq (t)[(Z w + Z q )/(2Z q )], where Z w = 1.48 × 106 Rayls and Zq = 15.3 × 106 Rayls are the water and quartz acoustic impedances, respectively. The SW front velocity V was determined by the use of the TOF method. We used four miniature piezoelectric gauges CA-1136 placed at aluminum made rings. The gauges were placed at an angle of 90◦ with respect to one another. We used rings with radii of r = 25, 45, 65 and 85 mm. At each distance five discharges were produced with the same generator charging voltage and same initiating wire between the electrodes. A time delay τav in the appearance of the pressure at the gauges position with respect to the maximum of ID through the wire, averaged over all the gauges, was used to obtain the dependence r = f (τav ). This dependence was used for the calculation of the SW front velocity V = f (r ). Finally, the latter dependence was used for the estimation of the cylindrical SW pressure amplitude P as [18]:
This potential difference ϕ was measured by a digitizing oscilloscope TDS-640A through a decoupling low-
Here V = V −c0 , x is the distance between the probes, t is the time which the SW requires to pass the distance x
P = P0 + 0.48ρV (V − c0 )
(2)
Here P0 , ρ, and c0 are the initial pressure, density, and sound velocity of the water, respectively. Let us note that in order to decrease the error in the estimation of P the error in the measurements of the distance between the probes should be minimized. Indeed, it can be shown that the relative error P/P can be expressed as P V x t = −1 − (3) P V x t
76
between the probes, x and t are the errors in the measurements of the distance x and time t. One can see that in the case of V approaching c0 (the case of weak SWs) even a small error in the measurement of x and t would lead to a significant error in P. For instance, for V = 1.1c0 (P ≈ 108 Pa), P/P increases approximately 10 times with respect to the relative error in x and t. In our case, the relative error in t was ≤0.5% due to the large amplitude of the signal produced by the CA-1136 probe and the broad bandwidth of the digitizing oscilloscope that was used. The relative error in measurements of x was x/x ≤ 2% that gives ≤20% in determination of P (in the range P ≤ 108 Pa). The piezoelectric gauges PCB-119A02 and PCB138A38 were used for measurements of the temporal behavior of the SW pressure amplitude. In addition, these probes were used for the TOF measurements. However, the latter measurements showed a larger relative time error as compared with the time error when we used CA-1136 gauges because of the significantly smaller amplitude of the obtained signal. These probes with flash installation were placed inside the water-filled chamber at the same distances as the other gauges (carbon, quartz, and TOF gauges). The gauges have a linear sensitivity (1 V/108 Pa) in the pressure range up to 7 × 108 Pa. However, these gauges have a narrow bandwidth that limits their linear response beyond 100 and 500 kHz for the PCB-119A02 and PCB138A38, respectively. The latter leads to a significant distortion of the measured waveform relative to the real pressure waveform. This is more relevant to the PCB-109A02 gauge (sensitive element is quartz) which is cheaper than the PCB-138A38 gauge (sensitive element is tourmaline), but suffers from severe resonances which fall within the wavelength of the investigated pressure pulses for the duration of a few microseconds. Therefore, a special signal processing algorithm, based on energy conservation requirements and Fourier analysis for reconstruction of the waveform of the pressure wave was considered [12]. In this paper, we will show that simple rules of thumb can be applied for analyzing gauge response in order to obtain the amplitude of the SW pressure. Let us note that the measurements of the SW parameters were performed at the distance of 25–85 mm from the exploding wire. Therefore, except the described quartz probes, one can neglect curvature of the incident SW relatively to the carbon-resistor and PCB gauges which have dimensions ≤4 mm. Indeed, the deviation of the cylindrical SW from its plane analogue is ≤8 × 10−3 cm at the plane with typical size of 4 mm placed at a distance of 25 mm from the exploding wire. One can estimate roughly that deviation of the SW from its plane analogue in units of time is ≤15 ns (here the SW velocity in the probe material is assumed to be ∼5×105 cm/s). An optical measurement of the SW velocity was performed using the fast framing 4Quik05A and streak EOKXX cameras. The purpose of these measurements was to determine the pressure at the front of the SW at a relatively large distance from the exploding wire. Since this method
A. Sayapin et al.
allows a direct observation of SW it is considered to be the most reliable technique of pressure measurement. In the case of the streak camera the optical setup consists of an argon flash lamp, an imaging lens, and a mirror inclined at 45◦ . The flash lamp acts as a source of parallel light which is used to form a shadow image of the SW (see Fig. 1a). The imaging lens and mirror were used to form an image of the shadow of the SW at the entrance slit of the streak camera. The streak camera and the pulsed current generator are synchronized using a time delay unit. The image which is formed on the phosphor display of the streak camera is stored in the PC using a digital CCD camera. In order to fix a space where the propagating SW is photographed in the case when the distances from the discharge are different, a special wire holder was built. The holder consists of a teflon frame with the rubber stretched in the middle. The exploding wire is stretched at the desired distance from the rubber (Fig. 1b). The streak camera is set such that the image of the rubber appears in the middle of the phosphor screen. Thus, by changing the time delay between the explosion of the wire, which is placed at different distances from the rubber for different generator shots, and the beginning of the streak, one can observe the intersection of the SW and the rubber. An example of such an image is shown in Fig. 3a.
Fig. 3 a Streak photography of two SWs intersecting the rubber. First SW was generated by an exploding Cu wire 0.1 mm in diameter and 85 mm in length at Id = 20 kA. The second SW was generated by the plasma channel (Id = 60 kA) formed at a time delay of 7 µs with respect to the wire explosion. b A sequence of framing photographs (exposure time of 5 ns, time delay of 400 ns between the frames) of a SW generated by explosion of Cu wire (0.1 mm in diameter and 85 mm in length at Id = 20 kA) in one generator shot
Methods of measurement of underwater SWs generated by electrical discharge
77
The instantaneous velocity of SW can be found by extrapolating the slope of the wave from the obtained image. An error in the obtained velocity arises primarily due to the image resolution limitation and its value is ≤1.5%, which results in ≤15% error in the calculated pressure. In the case of fast framing photography, the 4Quik05A framing camera was used in a multiframe mode operation during one generator shot. This allows one to obtain up to 10 exposures with duration of 5 ns and variable time delay with respect to each other, stored at one frame (see Fig. 3b). The framing photographs were used to validate the assumption of SW cylindricity.
3 Experimental results 3.1 Quartz gauges The quartz gauges were the only gauges which were tested in different experimental conditions as compared with the other gauges. In these experiments we used a current generator with C = 10 µF charged up to 30 kV (stored energy of 4.5 kJ) with exploding Cu wire of 0.18 mm in diameter and 85 mm in length. In these generator shots the time delay between the wire explosion and the plasma channel formation in water was zero. The amplitude of the current when the wire explosion occurred was Id = 40 kA and the maximum amplitude of the current in the discharge channel was 90 kA with a rise time of ∼4.5 µs. A typical waveform of the signal produced by the quartz gauge placed at a distance of 60 mm from the exploding wire is presented in Fig. 4a. One can see that during ∼1 µs there is an increase in the obtained signal which corresponds to a pressure up to 4 × 107 Pa. Further, there is a sharp increase and decrease in the signal amplitude. Let us note that in each generator shot the quartz gauge was destroyed. Also, taking the pressure wave velocity in quartz as being approximately equal to a sound wave velocity ∼5.7 × 105 cm/s, one can obtain the wave propagation time in the quartz gauge to be ∼1 µs. Thus, it is reasonable to consider that the sharp increase and decrease in the obtained signal is related to the breakdown of the gauge at the time when the pressure wave reaches the rear side of the quartz sample with significant amplitude. Let us note that the obtained signal is related to the SW generated by the wire explosion. In Fig. 4b we present the dependence of the pressure amplitude obtained by the quartz gauges (till the moment of their “breakdown”) on the distance from the exploding wire. One can see that the amplitude of the SW generated by the wire explosion does not exceed 6 × 107 Pa at a distance of 35 mm. To summarize this section, one can conclude that the use of quartz gauges for measurements of SWs produced in water by high-current electrical discharge is inconvenient because of their fast breakdown and the necessity of replacing them by new ones after each generator shot. In addition, a relatively large size of the quartz gauge does not allow one
Fig. 4 a Typical waveform of the pressure obtained by the quartz gauge placed at a distance of 60 mm from the exploding wire. b Dependence of the pressure amplitude obtained by the quartz gauge (till the moment of the gauge “breakdown”) on the distance from the exploding wire
to use it at smaller distances from the exploding wire where the effect of SW curvature becomes significant.
3.2 Carbon-based gauges Typical waveforms of the signal obtained using a carbon gauge placed at distances of 25 mm and 65 mm from the exploding wire are presented in Fig. 5. The obtained waveforms correspond to the SW produced by electrical discharge through the plasma channel. The first SW was not detected by this gauge because of its insufficient sensitivity. One can see that the obtained signals are characterized by a relatively short (∼1 µs) well-defined first pulse, followed by a pulse of long duration (∼8 µs). The reproducibility of the maximum amplitude of the second pulse and its waveform in identical electrical discharges was poor. At the same time the standard deviation in the amplitude of the first pulse was ±7%. Thus, we considered analyzing only the first pulse for the pressure. At present we do not have a clear explanation of the nature of the second pulse, the appearance of which can be related to several factors (mechanical resonance, design of the gauge support, etc.). In the analysis of the first signal we took into account the acoustic impedance mismatching factor which is equal to ∼0.6 in the case of a carbon resistor (Z c = 7.4 MRayls). The dependence of the pressure on the distance from the discharge channel is shown in Fig. 6a where the data related to the pressure measurement obtained
78
A. Sayapin et al.
surements of the pressure amplitude of SWs produced in water by high-current electrical discharge. However, a careful calibration of each gauge should be performed. Also, these gauges cannot be used for measurements of the temporal dependence of the SW pressure because of the hysteresis effect.
3.3 Time of flight method
Fig. 5 Typical waveforms of the signal obtained by the carbon gauge placed at distances of 25 mm a and 65 mm b from the exploding Cu wire
Fig. 6 a Dependence of the SW pressure on the distance from the discharge channel obtained by different gauges and methods. b Best-fit of pressure–distance dependence averaged over all the gauges and 0.5 radial exponent dependence
using the other gauges and methods are also presented. One can see that the pressure obtained by these calibrated carbon gauges does not exceed 1.2×108 Pa at the distance of 25 mm from a discharge channel. Also, the best fit of the obtained dependence P = f (r ) differs from the theoretical dependence for a cylindrical SW: P ∝ r −0.5 [19]. To summarize this section, one can conclude that simple and cheap carbon gauges can be efficiently used for mea-
Averaged data of the SW pressure dependence on the radial distance from the axis, P = f (r ), obtained by the TOF method are shown in Fig. 6a. One can see satisfactory agreement between these data and those obtained by carbon gauges. Let us note that in our case the calculation of the pressure amplitude was based on averaging the data obtained by azimuthally separated four CA-1136 gauges, placed at different distances from the discharge channel. This calculation does not require knowledge of the discharge channel radius. Also, the result was averaged over five different generator shots. However, in the case when one uses the TOF method based on a single gauge, the uncertainty in the discharge channel radius r0 could lead to a large error in the determination of the SW pressure. Indeed, taking Pd as the pressure which is measured by the pressure gauge at a distance rd from the discharge channel, we assume that the pressure amplitude P(r ) at an arbitrary distance between the gauge and the discharge channel can be expressed as P(r ) = Pd (rd /r )0.5 where the radial exponent 0.5 is taken as for classical ideal zero width source. Thus, taking into account Eq. (2) and neglecting P0 , one obtains the following expression for the time t which the SW requires to pass the distance between the discharge channel and the gauge: rd 1 t = (4) r0 0.5 + 0.25 + 107 P (r /r )0.5 d d The dependence of t = f (Pd ) for different r0 and for rd = 85 mm is presented in Fig. 7. One can see that for r0 ≥ 1 mm the error in the determination of Pd for the measured t could be very large. For instance, for t = 53 µs, one obtains Pd = 2.7 × 107 Pa and Pd = 4.2 × 107 Pa for r0 = 2 mm and r0 = 0.05 mm, respectively. Taking into account the microsecond time scale of the electrical discharge and the relatively fast expansion velocity of the plasma channel which is ≤ 3 × 105 cm/s (in fact, this velocity depends on the amplitude of Id ) [6] the error in the pressure estimation could be ≥50%. Here let us note that the same conclusion can be immediately obtained with the experimentally found radial exponent 0.7. To summarize this section, one can conclude that the TOF method can be efficiently used for the estimation of the SW pressure amplitude. In order to decrease the error in these measurements and to overcome the difficulty of determination of the initial discharge channel radii, one has to use several radially spaced gauges and to make very precise distance measurement.
Methods of measurement of underwater SWs generated by electrical discharge
Fig. 7 Dependence of the time t which the SW requires to pass the distance between the discharge channel having a radius r0 and the gauge placed at the radius rd = 85 mm on the SW pressure for different r0
3.4 Piezoelectric gauges To obtain the temporal behavior of the SW pressure, calibrated PCB-119A02 and PCB-138A38 gauges were used (see Sect. 2). These gauges were placed at the same distances from the discharge channel as the carbon and TOF gauges. The data obtained by these probes were analyzed using known calibration and some rules of thumb as described below. The obtained dependence of the SW on the distance from the discharge channel is shown in Fig. 6a. First, let us note that the obtained pulse duration of the PCB-119A02 and PCB-138A38 gauges responses to the pressure pulses did not depend on the duration of the discharge current pulse. In Fig. 2b we present typical waveforms of the PCB-119A02 gauge response to the two SWs generated by the electrical explosion of the wire and plasma channel. The discharge channel formation was delayed by ∼7 µs with respect to the wire explosion. Similar waveforms were obtained for the PCB-138A38 gauge. One can see that the duration of the gauge responses is almost the same for the first and second current pulses which differed in duration of ∼7 times. We explain the larger duration of the first gauge response by the nature of the first SW generation. Namely, the explosion of the wire, which leads to this SW generation, is accompanied by phase transitions which lead to the formation of low-ionized low-conductive plasma column. The column expansion, which lasts longer than the explosion process, leads to a longer duration of the generated SW. Thus, it is evident that the duration of the current pulse cannot be used for the estimation of SW pressure duration. Second, it was shown that the straightforward interpretation of the signals obtained from these gauges as pressure pulses gives an overestimation in the pressure amplitude by a factor ∼2 and ∼1.5 for the PCB-109A02 and PCB-138A38 gauges, respectively. As described in Sect. 2, we explained this overestimation by the limited bandwidth of the gauges and the amplification of resonant frequencies. Also, careful mathematical analysis of the waveforms obtained by the PCB-109A02 gauge showed that the amplitude of the pressure should be decreased by ∼2 times in order to obtain
79
Fig. 8 Typical waveforms of the SW pressure obtained by the PCB138A38 gauge
the correct amplitude of the pressure [12]. Thus, in order to obtain a good correspondence between the pressure obtained by the PCB-109A02 gauge and that obtained by other gauges, this rule of thumb was used (see Fig. 6a). This simulated coefficient of ∼2 allows one to estimate in situ the SW amplitude without making sophisticated mathematical processing of the obtained signal. The PCB-138A38 gauge possesses a flatter response in the interesting region of mechanical frequencies as compared to the PCB 109A02 gauge. Here another rule of thumb can be applied for the evaluation of SW pressure amplitude from the obtained waveform of this gauge. It was assumed that the peak pressure corresponds to the “step” on the rising edge of the signal (see Fig. 8). This suggestion was made due to the fact that in spite of a large variation in the maximum amplitude of the obtained waveform (standard deviation of ±15%) from one generator shot to another, the amplitude of this “step” remained relatively stable (standard deviation of ±2%). Therefore, it was assumed that the correct gauge response is limited by the pulse duration till this “step”. A comparison with other techniques suggests that this approach is correct to within approximately 20% (see Fig. 6a). 3.5 Fast framing and streak photography In Fig. 3b a set of consequent SWs at different distances from the discharge channel are presented. One can see that these SWs can be considered with a good approximation as cylindrical waves. The data obtained by streak photography concerning the instantaneous SW velocity at different distances from the discharge channel were used for the calculation of the SW pressure amplitude according to Eq. (2) (see Fig. 6a). One can see that the calculated pressure amplitudes coincide with the pressure amplitudes obtained by other probes and TOF method at the same distances.
4 Conclusions The SW pressure amplitudes obtained by different gauges and methods have shown a good correspondence. As can be seen from Fig. 6a the data from all the gauges fits within a
80
A. Sayapin et al.
Table 1 Value of the pressure P0 and the power coefficient α
Streak camera Time of flight PCB-138A38 PCB-109A02 Average
Pin (MPa)
α
1057 ± 124 1504 ± 281 935 ± 161 1190 ± 20 1144 ± 63
0.708 ± 0.032 0.799 ± 0.052 0.637 ± 0.047 0.706 ± 0.005 0.710 ± 0.016
10% error region. Here it should be pointed out that the pressure signals obtained from the piezoelectric gauges required a careful mathematical processing in order to obtain the correct pressure values. An error estimation of 20% on the peak pressure evaluated using the TOF method was obtained from the data scatter. In addition, it is worth mentioning here that although the TOF and optical measurements give the value of the pressure at the SW front, the only technique to determine the shape of the pressure behind the front of the SW is the use of piezoelectric gauges and here rules of thumb are insufficient and mathematical signal analysis is indispensable [12]. The experimentally obtained dependence of the peak pressure on the distance from the origin of the shock wave was fitted by a function of the type P(r ) = Pinr α , where P is the pressure, r is the distance from the exploding wire, and Pin is the initial pressure of the SW at its origin. The values of Pin and the power coefficient α were found by fitting the experimental data. These values, which were obtained for different techniques, are summarized in Table 1. It can be seen that Pin ≈ 109 Pa and α ≈ −0.7. The power α predicted by the classical theory for the decay of cylindrical sound waves is αth ≈ −0.5. This discrepancy may arise due to the nonlinearity of the observed waves since their velocities exceed the sound velocity by a relatively large factor. In addition, the value of Pin and α obtained from the pressure– distance dependence averaged over all the gauges is shown in Fig. 6b. From Table 1 one can see that the best agreement with the averaged dependence is obtained using the streak camera technique and the processed PCB-109A02 gauge. Acknowledgements The authors are indebted to J. Felsteiner and V. Ts. Gurovich for fruitful discussions and to L. Iomin for his assistance in the carbon-based gauges calibration.
References 1. Cole, H.: Underwater Explosions. Princeton, New Jersey (1948) 2. Martin, E.A.: Experimental investigation of a high-energy density high-pressure arc plasma. J. Appl. Phys. 31, 255–267 (1960) 3. Lisitsyn, I.V., Muraki, T., Akiyama, H.: Characterization of a shock wave generated by a wire explosion in water. J. Acoust. Soc. Jpn. E 18, 89–91 (1997) 4. Malushevskii, P.P.: Basis of Pulsed Discharge Technology. Naukova Dumka, Kiev (1983) (in Russian) 5. Krivitskii, E.V.: Dynamics of Electrical Explosion in Liquid. Naukova Dumka, Kiev (1983) (in Russian) 6. Skvorzov, Yu.V., Komelkov, V.S., Kuznezov, N.M.: Expansion of discharge channel in liquid. Sov. J. Tech. Phys. 30, 1165–1177 (1960) 7. Barker, L.M., Hollenbach, R.E.: Laser interferometer for measuring high velocities of any reflecting surface. J. Appl. Phys. 43, 4669–4675 (1972) 8. PCB Piezotronics Inc.: http://www.pcb.com 9. Ginsberg, M.J., Asay, B.W.: Commercial carbon composition resistors as dynamic stress gauges in difficult environments. Rev. Sci. Instrum. 62, 2218–2227 (1991) 10. Lisitsyn, I.V., Muraki, T., Akiyama, H.: Wire induced flashover as a source of shock waves for destruction of solid materials. Jpn. J. Appl. Phys. 36, 1258–1263 (1997) 11. Robinson, J.W., Ham, M., Balaster, A.N.: Ultraviolet radiation from electric discharges in water. J. Appl. Phys. 44, 72–75 (1973) 12. Grinenko, A., Gurovich, V.Ts., Sayapin, A., Efimov, S., Krasik, Ya.E., Felsteiner, J.: Analysis of shock wave measurements by a piezoelectric pressure probe. Rev. Sci. Instrum. 75, 240–244 (2004) 13. Neil, G.R., Post, R.S.: Multichannel, high-energy railgap switch. Rev. Sci. Instrum. 49, 401–403 (1978) 14. Schmidt, R.M., Voss, M.E.: Shock wave measurements in water. In: Proceedings of the Techniques and Theory of Stress Wave Measurements for Shock Wave Applications. American Physics Society of Mechanical Engineers, New York (1987) 15. Rosenberg, G.: Extension of the application of the shorted quartz gauge. Rev. Sci. Instrum. 50, 442–444 (1979) 16. Graham, R.A.: Piesoelectric current from shunted and shorted guard-ring quartz gauges. J. Appl. Phys. 46, 1901–1909 (1976) 17. Rosenberg, G.: Failure of lithium fluoride 111 single crystals subjected to shock wave loading. PhD Thesis, Washington State University (1987) 18. Komissarova, I.I., Ostrovskaya, G.V., Filippov, V.N., Shedova, E.N.: Generation of shock waves in water and in air by CO2 laser radiation focused on the free surface of a liquid. Sov. Phys. Tech. Phys. 42, 247–249 (1997) 19. Landau, L.D., Lifshitz, E.M. Fluid mechanics. Nauka, Moscow (1986) (in Russian)
