Simulation of the pulsed streamer discharge in water considering the effect of hydrostatic pressure Open Access

Underwater high-voltage pulsed discharge is a process in which a high voltage is applied to an electrode submerged in an aqueous solution within a very short period, resulting in the generation of a high-temperature, high-pressure plasma channel within the gap, and its rapid outward expansion. The ability of underwater discharge to simultaneously generate high-energy electrons, active particles, ultraviolet rays, shock waves, and other physicochemical effects has led to widespread attention in biomedicine,^1,2 wastewater treatment,^3,4 food processing,^5,6 petroleum production enhancement,⁷ and rock fragmentation,⁸ reflecting good application value and prospects. Therefore, the study of the formation process of pulsed discharge plasma in water contributes to a comprehensive understanding of the various physicochemical effects of discharge in water, which is of great significance for practical applications.

Compared with streamer discharge in gas, the behavior of streamer discharge in water is more uncertain. Therefore, a multitude of studies have been conducted on the initiation and propagation of the streamer in water. Wen et al. experimentally measured the spatially resolved emission spectrum of a hydrogen line in a single filament of a pulsed positive streamer discharge in water by combining a high-speed camera with a monochromator, confirming that the electron density in the streamer is 10²³–10²⁴ m⁻³ and tends to decrease along the axial direction of the streamer as the distance from the tip increases.⁹ They also measured the propagation velocities of the primary streamer by combining the high-speed camera with the shadowgraph imaging techniques, studied its relationship with the water conductivity and applied voltage, and found that the initial propagation velocity of the primary streamer was about 4–6 km/s and rapidly decreased to about 1.5 km/s and that the conductivity of the water and the applied voltage did not have a significant effect on the propagation velocity of the primary streamer.¹⁰ The processes of initiation, propagation, and branching of underwater subsonic streamers were studied in detail by Li et al. The results show that the breakdown voltage of positive polarity can be higher than the negative polarity in micro-second underwater discharges, and the amplitude of applied voltage can affect the propagation and morphology of subsonic streamers, which propagate faster and emit more intensively at higher applied voltage.¹¹ A novel framework model was also proposed to describe the dynamic evolution behavior of subsonic streamers.¹² Jones and Kunhardt observed a significant polarity effect of hydrostatic pressure on the breakdown voltage of the liquid, in which the positive breakdown in water is less affected by hydrostatic pressure than negative breakdown, and the breakdown threshold gradually saturates with increasing hydrostatic pressure.¹³ 

Since the development process of discharge in water is very rapid, it is necessary to study the distribution law of energetic electron density, charge density, and electric field strength in the discharge process with the help of numerical simulation methods. Babaeva et al. discussed the discharge process when bubbles of different sizes are immersed in liquids with different dielectric constants with the help of the analytical model of multi-hydrodynamics of the non-PDPSIM and concluded that the transition from electron avalanche to streamer and the formation of streamer in the homogeneous electric field are determined by the applied voltage and the parameters of the size of the bubble, as well as by the position and number of the initial free charges inside the bubble.¹⁴ Aghdam and Farouk simulated the onset process of streamer discharge in water on a 3–5 ns timescale and the change of the density of the aqueous solution under the action of the electric field, charge, and correlation force and found that the electrostrictive force plays a key role in the density change at the initial stage.¹⁵ Wang et al. analyzed the effects of voltage amplitude, liquid conductivity, and the presence of air bubbles on the characteristics of underwater discharges by building a two-dimensional finite element simulation model of a pin-plate gap, and the development speed of the streamer and the density of charged particles increase when the conductivity of the water is high, and the presence of bubbles significantly impacts the development of the discharge morphology, causing the channel to have multiple bifurcations.¹⁶ 

In this paper, a two-dimensional axisymmetric pin-plate fluid model is established, and the formation and development of plasma channels in water under a nanosecond voltage pulse are investigated by simulation based on the field ionization mechanism and the Poisson's equation of the drift-diffusion equation coupled with the Poisson's equation of the electric field of three kinds of charged particles, which is numerically solved by using finite elements. The influence of voltage amplitude, discharge gap distance, rising edge of the voltage pulse, and hydrostatic pressure on the morphology of the discharge plasma channel, electric field distribution, and electron density in the channel is examined. Additionally, the effect of hydrostatic pressure on the generation and propagation of streamer is discussed, and the process of changing electron density in the discharge channel as the hydrostatic pressure increases is provided.

In the process of discharge in water, the formation of the streamer is closely related to the processes of generation, migration, multiplication, diffusion, and compounding of charged particles. In the simulation of the discharge process in water, the following assumptions are made: (1) the water molecules are regarded as the background particles and assumed to remain stationary, that is, the impact of the pulse voltage on the water is not taken into account; (2) the ionization products are considered only hydrogen ions and electrons, and it is considered that the final product of electron adsorption on the water molecule after negative ionization is hydroxide ions; and (3) depending on the hydrogen ions, hydroxide ions and electrons as the fluid mass point, and the three kinds of fluid mass points are equivalent to the flow of continuous medium under the action of the electric field and the difference of the concentration. The motion process of drift-diffusion under the action of the three fluid mass points in the electric field force and concentration difference is equivalent to the flow of a continuous medium.

To simulate the discharge plasma generation in water, a two-dimensional axisymmetric needle-plate discharge model is constructed by COMSOL, and the simulation model is shown in Fig. 1. The needle-plate electrode is a widely used electrode structure in the experimental study of discharge in water, the needle electrode is connected to the high voltage, and the plate electrode is grounded. The electrode material used in this paper is copper, the radius of curvature of the needle electrode is 20 μm, and the distance of the needle-plate electrode is adjustable.

The interaction between charged particles and various dielectric surfaces during the electric process will affect the electric field strength distribution in the discharge gap, so it is necessary to set appropriate boundary conditions to ensure the accuracy of the calculation. The boundary condition settings of the simulation model are shown in Table II.

To observe the initiation and development of the streamer in water, the discharge process in a 3 mm water gap at atmospheric pressure (0.1 MPa), with a pulsed voltage amplitude of 25 kV (the tail time and the rise time of impulses were 600 and 30 ns) was selected for illustration. The evolution of the electric field strength and the charged particles in the water at different moments is shown in Figs. 2 and 3, allowing for observation clearly of the process and morphology of the discharge from the tip of the needle and its progression to the ground electrode.

Figure 2 shows the three-dimensional distribution of electric field strength in water when the voltage is applied to the electrode for only 1 ns. Since the tip of the needle electrode has the smallest radius of curvature, the maximum value of the electric field strength occurs at the tip of the needle electrode. Moreover, the inhomogeneous electric field in the vicinity of the electrodes increases rapidly within a few nanoseconds, leading to the emergence of the ponderomotive force in the water,¹⁸ which acts on the liquid medium and is directed to the region with the stronger electric field (at the electrodes). However, the liquid is unable to move due to inertia, resulting in the local liquid in the instant under a huge negative pressure, so that the local liquid ruptures to produce nano-sized pores or voids, forming low-density areas.²⁷ The pores are deformed along the direction of the electric field line under the action of the strong electric field and provide enough acceleration distance for free electrons to trigger impact ionization, so that the charged particles proliferate rapidly in the low-density region, and the initial ionization process lasts for about 40 ns. As shown in Fig. 3, at t = 40 ns, the electric field strength at the tip of the electrode reaches 1.41 × 10⁸ V/m. In part of the area near the electrode, the density of electrons reaches 3.73 × 10²² m⁻³, the density of hydroxide ions reaches 5.56 × 10²⁰ m⁻³, and the density of hydrogen ions reaches 7.73 × 10²² m⁻³.

When the electric field intensity at the tip of the electrode exceeds 10⁸ V/cm, the streamer begins to develop. In this process, under the effect of the electric field, electrons migrate toward the anode and hydrogen ions migrate toward the cathode. Due to the significantly faster migration rate of electrons compared to that of hydrogen ions, the electrons quickly reach the needle electrode (the anode), causing a weakening of the electric field in that vicinity. Meanwhile, the hydrogen ions essentially maintain their original positions, as depicted in Fig. 4. When the density of hydrogen ions reaches a critical threshold, the electric field generated by these ions superimposes on the electric field vector produced by the anode, thereby deforming the head of the streamer and propelling it toward the cathode. During the propagation of the streamer, the maximum electric field intensity is 1.69 × 10⁸ V/m, the maximum electron density is 3.76 × 10²³ m⁻³, the maximum hydrogen ion density is 1.76 × 10²³ m⁻³, the maximum hydroxide ion density is 1.39 × 10²³ m⁻³, and these maxima do not occur in the same space-time. This is consistent with the conclusion of Wen et al., who obtained by spectral analysis that the electron density of pulsed positive streamer discharge in the plasma channel in water is 10²³–10²⁴ m⁻³.⁹ 

Figure 5 displays that the electron density has five different peaks and fluctuations, whereas the distribution curve of electric field intensity remains relatively smooth throughout the discharge process. The curves from each moment show that the diffusion of electrons is slightly lagging behind the change in field strength, specifically that the electrons do not cluster at the head of the streamer where the field strength is the highest but is in the low field strength region that lags behind the head of the streamer.

In addition, as the remaining gap distance of the streamer propagation decreases and the voltage waveform decays, the growth of the electric field strength at the head of the streamer slows down, and the streamer propagation speed slows down as well. The instantaneous velocity change in the development of the streamer is depicted in Fig. 6. During the development of the streamer, the fastest instantaneous propagation speed is 6.7 km/s, and the average speed is 3 km/s, which is much faster than the supersonic speed.¹⁰ 

To investigate the effect of the applied voltage amplitude on the formation and propagation of the streamer, the simulations were performed under the conditions of setting the voltage pulse tail time to 600 ns, the voltage pulse rise time to 30 ns, and the discharge gap to 1 mm, with the applied voltage amplitudes of 15, 20, and 25 kV, respectively. The impact of voltage amplitude is investigated through the analysis of variations in electric field strength and electron density within the plasma channel, as well as changes in the morphology and propagation velocity of the streamer.

It is evident from Fig. 7 that the larger the voltage amplitude, the longer the development length of the streamer in the same period. This is because the electric field strength at the tip of the needle electrode can reach 10⁸ V/m faster as the voltage amplitude increases (as shown in Fig. 8), but this also makes the streamer more susceptible to the radial expansion and the energy is not fully concentrated at the head of the streamer. This phenomenon was also found experimentally by Xie et al., in which the diameter of the streamer increases with the increase in the applied impulse voltage.²⁸ And because the higher the voltage amplitude makes the electric field strength near the tip of the needle, the stronger the field ionization effect, the greater the initial proliferation of charged particles, and the faster the development of the streamer. Comparing the average speed of streamer development at different voltage amplitudes, the smallest average speed is obtained at 15 kV, which is about 7.09 km/s; the average speed at 20 kV amplitude is between 15 and 25 kV, which is about 8.77 km/s, and the average speed at 25 kV amplitude is the highest among the three, which is about 10.9 km/s. The speed of propagation of a positive streamer is proportional to the applied voltage.¹² However, fast speed means short total ionization time and total energy accumulation time, which makes the peak electron density and the peak electric field strength of the discharge channel not proportional to the voltage amplitude.

To study the effect of different electrode gap distances on the characteristics of streamer discharge in water, simulations are performed for the needle-plate electrode gap distances of 1, 2, and 3 mm, respectively, at 25 kV voltage amplitude (the tail time and the rise time of impulses were 600 and 30 ns). Figure 9 demonstrates the electric field intensity distribution for the streamer development length of 1 mm with different discharge gaps.

As the electric discharge gap widens, there is a corresponding increase in the formation time of the streamer head. With a discharge gap of 1 mm, the electrode tip generates a head of the streamer at 20 ns. When the gap is expanded to 2 or 3 mm, the respective times required for the formation of the streamer head are elongated to 30 and 40 ns. That is because the increase in the discharge gap makes the initial electric field strength in the region near the needle electrode decrease, the initial ionization of water is weakened, the initial proliferation of the number of charged particles decreases, and it takes longer to accumulate to a certain number to meet the generation of the streamer. This is consistent with the conclusions of Hamdan and Cha, who established a relationship between the injected charges and the shape of the plasma and found that increasing the size of the gap reduces the injected charges.²⁹ Wang et al. also determined that the breakdown voltage decreases as the gap distance decreases.³⁰ At the same voltage amplitude, as the distance of the discharge gap increases, the electric field strength in the gap decreases. This makes it more difficult for the water medium to break down and results in a tendency for the average propagation speed of the streamer to decrease. The average propagation velocities under 1, 2, and 3 mm gaps are approximately 10.9, 8.66, and 3.82 km/s, respectively. Furthermore, as the electrode gap increases, the thickness of the region where the field intensity of the streamer head becomes smaller and the radial diffusion distance of the streamer becomes shorter when the streamer develops to the same length. Table III presents the radius of the streamer and the thickness of the region where the field strength of the streamer head is measured when the streamer length is 1 mm.

Further analysis of Fig. 10 shows that as the electrode gap increases, the volume of the water medium between the electrodes increases, allowing more water molecules to be ionized. The total ionization time increases, leading to a higher peak electron density during the streamer development. Compared to the 1 mm gap, the increase in the peak electron density of the 2 mm gap is not very significant. However, the increase in the peak electron density of the 3 mm gap is very noticeable, being about three times that of the 1 mm gap.

To study the effect of the rising edge of the pulse voltage on the characteristics of streamer discharge in water, simulations were performed with impulse rise times varied to 30 and 300 ns, using a voltage amplitude of 25 kV (the tail time of impulses was 600 ns) and a discharge gap of 1 mm. Figure 11 demonstrates the electric field strength and electron density distributions at different rising edges of the pulse voltage for a streamer development length of 0.5 mm.

As shown in Fig. 11, when the streamer reaches the same position, a steeper rising edge of the pulse voltage results in a greater the electric field strength at the head of the streamer, facilitating easier expansion in the radial direction. For the streamer development length of 0.5 mm, the thickness of the region where the field intensity of the streamer head is 0.050 mm and the streamer radius is 0.078 mm at a rising edge of 300 ns, while at a rising edge of 30 ns, the thickness of the region where the field intensity of streamer head is 0.038 mm and the discharge radius is 0.057 mm. The steeper the rising edge of the voltage, the faster the rising rate of the electric field and the faster the ionization rate near the tip of the needle. The increase in the production rate of the charged particles prevents them from migrating out of the ionization region in time, causing them to appear near the tip. As a result, the ionization region expands, and the spatial distribution of the streamer exhibits the characteristics of an increase in the radius of the discharge. Meanwhile, it can be noticed from Fig. 12 that when the discharge develops to the same position, the steeper pulse voltage at the rising edge takes less time but a higher voltage amplitude is required, indicating that the discharge can be realized with lower voltage by increasing the voltage duration.³¹ 

To investigate the impact of hydrostatic pressure on discharge in water, simulations were conducted at hydrostatic pressures of 0.1, 0.2, 0.3, 0.4, and 0.5 MPa, respectively. Figure 13 illustrates the variation of electron density and the field ionization rate with time during the formation of a plasma channel under a voltage amplitude of 25 kV (the tail time and the rise time of impulses were 600 and 30 ns) with a discharge gap of 1 mm.

The hydrostatic pressure of liquids affects the formation and development of the streamer. Beroual and Aka-N'Gnui found that an increase in hydrostatic pressure leads to a higher dielectric strength of liquids and raises the initiation voltage of streamers in liquids.³² Wang et al. also observed that the average breakdown time lag increases consistently and significantly with increasing liquid pressure for positive pulses, while the statistical time lag for negative polarity is greater than that for positive polarity. In other words, an increase in liquid pressure can inhibit pulsed liquid breakdown from initiation and propagation, and hydrostatic pressure has a greater impact on negative polarity.³³ 

At varying hydrostatic pressures, the distance between water molecules shows different values; as the hydrostatic pressure increases, the intermolecular distance decreases, resulting in differences in the rate of field ionization, which ultimately leads to changes in the plasma electron density during the discharge in water, as shown in Fig. 13. On the one hand, the increase in hydrostatic pressure improves the dielectric strength of the liquid, making it difficult to form a low-density region, which was used by Lesaint and Gournay to demonstrate the gaseous nature of filamentary positive streamers;³⁴ on the other hand, the increase in hydrostatic pressure reduces the distance between water molecules, and the probability of collision of free electrons increases but greatly reduces the mean free path of the free electrons, which increases the energy loss for the collisional ionization. It is more difficult to accumulate the energy required for ionization of water molecules.¹⁷ In the case where the discharge gap remains constant, this ultimately leads to a reduction in the peak electron density of the plasma channel produced by the discharge. From Fig. 14, it can be seen that increasing the hydrostatic pressure of the water dielectric decreases the peak electron density in the plasma channel, and its effect on the peak electron density decreases as the hydrostatic pressure increases in steps. That is, as the hydrostatic pressure increases, the inhibition of the liquid discharge increases, and the increasing trend is saturated, which is consistent with the conclusions of Wang et al.³³ 

In this study, we comprehensively analyzed the developmental law of streamer discharge under nanosecond pulses by building a two-dimensional pin-plate underwater discharge model, focusing on the effects of the applied voltage amplitude, the discharge gap distance, the rising edge of pulse voltage, and hydrostatic pressure on the underwater discharge. The results of this study contribute to deepening the understanding of the generation of high-density plasma channels by high-pressure pulsed discharges in water as well as the ionization in liquid dielectrics.

1. As the applied voltage amplitude increases, the plasma channel undergoes radial expansion, the breakdown time becomes shorter, and the propagation of the streamer becomes faster, but the peak electron density and the peak electric field strength are not proportional to the voltage amplitude.

2. Increasing the voltage amplitude, reducing the time of the rising edge of the pulse, and narrowing the discharge gap distance can increase the electric field strength of the plasma channel and affect the electron density, where the discharge gap has the greatest impact on the peak electron density. And under the gap of 3 mm, the peak electron density can reach 3.76 × 10²³ m⁻³; if the discharge gap is narrowed to 1 mm, the peak electron density is reduced to 1.20 × 10²³ m⁻³.

3. Hydrostatic pressure and the distance of water molecules are closely linked. Increasing the hydrostatic pressure can decrease the electric field strength and the peak electron density in the plasma channel, and its effect on the peak electron density saturates with increasing hydrostatic pressure.

The work was supported by the Natural Science Foundation of Fujian Province, China under Grant No. 2021J01638.

The authors have no conflicts to disclose.

Sheng Lan: Data curation (equal); Investigation (equal). Xiaoting Ding: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Jiaxu Wang: Investigation (equal). Longhui Yao: Data curation (equal). Jianan Wang: Data curation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.