Streamer evolution in nanosecond pulsed sliding surface dielectric barrier discharge: 2D numerical modelling

Surface dielectric barrier discharge is a promising technique for flow control.^1,2 Classical SDBDs are constructed with one electrode placed on the dielectric layer and the grounded electrode is buried below the dielectric.³ Generally, the discharge initiates as a set of streamers that propagate from the edge of the high-voltage electrode, guided along the dielectric surface by the buried grounded electrode, without a sharp increase in electrical current.

Generally, the physical mechanism of applying AC voltages for flow control is the electrodynamic force induced from the interaction between surface discharges and near-wall flow.^4,5 More recently, nanosecond surface dielectric barrier discharges (nSDBD) are used due to their strong thermal and chemical kinetic effects. Fast gas heating (FGH) from the relaxation of electronically excited states results in pressure waves that propagate above the dielectric surface,^6,7 which can provide effective control of flow separation over a wide Reynolds number range.^8,9 Experimentally, a hemispherical shock wave near the HV electrode caused by FGH and a planar one due to streamer propagation have been observed.^8,10,11 The disturbance region induced by local gas heating is related to the streamer’s propagation length, and controlling this distance is crucial for effective flow control under realistic conditions.

To achieve a large-scale surface discharge, a three-electrode sliding SDBD was proposed.^12–15 Building on the classical SDBD design, a third electrode is added. The third electrode and the high-voltage electrode are flush mounted on opposite sides of the dielectric layer, creating a discharge gap. The third electrode is generally connected to a DC voltage, and this configuration, combined with AC or pulsed voltage applied to the high-voltage electrode, enables plasma extension. In the absence of a DC voltage, this type of sliding discharge can still be generated by applying a sufficient pulse voltage.^16–18 

Bayoda et al. carried out experimental studies on nSDBDs with the third electrode connected to negative DC voltages.^19–21 The DC voltage applied to the third electrode elongates the positive streamer originating from the HV electrode. The discharge gap of 40 mm was overlapped with a 14 kV nanosecond pulse and a $−$16 kV DC voltage, forming a sliding SDBD.¹⁹ The formation of a sliding discharge depends on the surface potential difference, specifically the average electric field, between the two exposed electrodes. Experimental results reported that adding the negative DC voltage increased the input energy up to 300%, directly influencing the gas heating. Shock waves were observed near both electrodes, expanding the effective actuation region.²¹ Compared with negative DC voltages, it is difficult to induce sliding discharges by applying positive DC voltages.²² 

Despite the achievements in experimental research, only few numerical works on sliding SDBDs can be found in the literature. Kourtzanidis et al. developed a self-consistent model of a three-electrode sliding SDBD, with the third electrode powered by a negative DC voltage.²³ Numerical results show that a negative corona is formed in the vicinity of the third electrode, which merges with the positive streamer from the HV electrode, covering the entire discharge gap.

To optimize and characterize this sliding nSDBD, understanding and evaluating the physical mechanisms of streamer formation, propagation, and extension are critical. In this work, a 2D model is employed to provide an in-depth understanding of the characteristics of sliding SDBDs. The model description is provided in Sec. II. Next, the discharge dynamics with DC voltages of different polarities are discussed in Sec. III. Finally, the conclusions are presented in Sec. IV.

In this paper, a two-dimensional model of a three-electrode SDBD in atmospheric pressure is established by COMSOL based on the finite element method (FEM).^24,25 Two typical modules for plasma-based flow control are included: discharge dynamics and gas dynamics.

The kinetics scheme suggested in Ref. 6 for modeling atmospheric nSDBD is used. The kinetics scheme is a combination of kinetics for the description of streamer propagation²⁸ and FGH⁷ in air and was validated in papers.^3,29–31 Following neutral, charged, and excited species are taken into account for air: $e$⁠, $N 2$⁠, $N 2 +$⁠, $N 4 +$⁠, $N 2( A 3 Σ u +)$⁠, $N 2( B 3 Π g)$⁠, $N 2( C 3 Π u)$⁠, $O 2$⁠, $O$⁠, $O ( 1D)$⁠, $O 2 +$⁠, $O 4 +$⁠, $O −$⁠, $O 2 −$⁠, and $O 2 + N 2$⁠. The scheme includes in total 15 species and 38 reactions.

The 2D Cartesian geometry of the three-electrode SDBD, with a computational domain of 20  $×$ 5 mm $2$ is shown in Fig. 1. The exposed high-voltage electrode I and grounded electrode III are both 10 mm in length and 50  $μ$m in thickness, forming a discharge gap of 10 mm. The nanosecond pulse waveform applied on electrode I peaks at $V P$ = 7 kV with a rise and fall time of 3 ns. The grounded electrode II is located at the bottom of the dielectric layer, and a positive/negative DC voltage $V D C$ with the amplitude of 1.5–8 kV is applied on electrode III. The dielectric barrier is made of a PVC (⁠ $ε r=3$⁠) flat plate with the thickness of 0.3 mm.

A structured mesh is assigned. Considering the high density and electric field gradient at the electrodes and dielectric surface, a minimum mesh size of 2  $μ$m is implemented. At atmospheric pressure, the thickness of the streamer channel is usually about 100  $μ$m,³ and the maximum mesh size in this region is set to 10  $μ$m. A coarse mesh is employed in the other domain to reduce the computational resources. Consequently, 45 040 nodes and 52 216 elements are obtained in the configuration. The initial electron density is $n e 0= 10 10 m − 3$ uniformly distributed in the plasma region, and the ion density is determined based on quasi-neutrality. The simulated conditions are 1 atm pressure and 300 K air (80% $N 2$ and 20% $O 2$⁠).

The dielectric surface uses charge accumulation boundary conditions to deal with the deposition of surface charges, which can be expressed as $d ρ s d t= n⋅ J i+ n⋅ J e$⁠. Here, $ρ s$ represents the surface charge density and $n⋅ J i$ and $n⋅ J e$ represent the normal components of the total ion current density and the total electron current density, respectively. The boundaries of the electrodes and dielectric surface are treated as wall. Secondary electron emission induced by the impact of heavy species on the wall is considered.

Charges accumulate on the dielectric surface during surface discharge propagation, and the deposited surface charge significantly affects the development of subsequent discharges.^32–35 Many researchers have investigated surface discharge under AC and DC bias and have found that the discharge behavior is inherently governed by the deposited surface charge.^36,37 A recent numerical work on the influence of surface charge deposition on the characteristics of SDBD shows that the residual surface charge significantly affects the propagation length, electric field, and deposition energy of subsequent pulses.³⁸ 

In the simulation of Kourtzanidis et al., the DC voltage rise time was set to be greater than 1  $μ$s to avoid the formation of streamer near the electrode.²³ However, the simulation results indicate that when the positive streamer started from the HV electrode, there was still a high density of electron clouds near the DC electrode. This occurs because the model may not consider the charging effect of the DC voltage on the dielectric surface over a longer time scale.

With electrode III connected to a DC voltage, the deposited surface charge inevitably shields the electric field generated by electrode III. In the experiments,^19,21 no obvious discharge was observed at electrode III when the pulse voltage started. To explain the effect of the DC voltage applied to electrode III on the positve streamer from electrode I, it is necessary to calculate the charging effect of DC voltage on the dielectric surface. The simulation strategy involves calculating the DC voltage for a sufficiently long time (⁠ $≥$ 1 ms) before the initiation of the nanosecond pulse until the steady state is reached. In the following results, time 0 is defined as the start time of the nanosecond pulse.

The relationship between the polarity of the DC source and the discharge characteristics, including streamer dynamics, the behavior of surface charge and electric field, and generated shock waves, is discussed below.

Before the initiation of the nanosecond pulse, the discharge for DC sources is calculated until steady state is reached. Figure 2 shows the distribution of electron density and reduced electric field in steady state for $V D C=−2$ kV. The electron density is below $10 10 m − 3$⁠. The surface charge deposited on the dielectric weakens the electric field generated by the negative DC voltage. Therefore, the electric field in the plasma region cannot provide sufficient ionization.

Figure 3 shows the surface charge density at steady state for different negative DC voltages without the application of a nanosecond pulse. The polarity of the deposited surface charge is negative, consistent with the polarity of the DC voltage. The deposited surface charge is highest at electrode III, remains constant over a short distance, and then decreases exponentially with increasing distance from electrode III. With the increase in $| V D C|$⁠, the surface charge density becomes higher, which increases from $−$11 nC/ $cm 2$ for $V D C$ =  $−$1.5 kV to $−$70 nC/ $cm 2$ for $V D C=−8.0$ kV near the electrode III. The distance covered by the deposited charge also increases: the length of the region where the surface charge density exceeds 0.1 nC/ $cm 2$ increases from 0.73 mm for $V D C=−1.5$ kV to the entire gap length for $V D C=−8.0$ kV.

After the DC voltage calculation reaches a steady state, the nanosecond pulse voltage is activated on electrode I. Figure 4 illustrates the propagation length and velocity of the streamer head for different $V D C$⁠. The streamer forms at electrode I and propagates toward electrode III. Without a DC bias (⁠ $V D C=0$ kV), the maximum discharge length is approximately 8 mm. The curves for $| V D C|≤2$ kV are only drawn up to 100 ns because the streamer no longer propagates and gradually decays over time. Although the applied voltage drops to 0 after 22 ns, the streamer continues to propagate due to the high potential at the discharge front. When the potential at the streamer head no longer provides sufficient ionization, the streamer stops and gradually decays.³⁹ The propagation length remains constant with the amplitude of the negative DC voltage $| V D C|≤2$ kV. In the sliding transition regime $V D C=−3$ kV, the streamer propagates similarly to the case of $V D C=0$ kV during the first 50 ns, and then the ionized channel extends further. The length curves confirm that the voltage amplitude $V P$ = 7 kV applied to electrode I requires a DC voltage amplitude of 3 kV to establish a fully developed sliding discharge.

The higher $| V D C|$⁠, the earlier the discharge gap is overlapped. The time period required to close the gap decreases from 150 ns for $−$3 kV to 50 ns for $−$8 kV. For $| V D C|≤2$ kV, the streamer velocity decreases over time. For $| V D C|≥3$ kV, no significant changes are observed near electrode I. However, as the streamer propagates close to the edge of the region covered by residual surface charges, its velocity increases, showing an enhancement effect. The larger the amplitude of the $V D C$⁠, the earlier and faster for the streamer acceleration.

Table I presents the acceleration moment and position of the streamer head, along with the deposited negative surface charge density corresponding to that position for the sliding discharge regime, i.e., $V D C=−3$⁠, $−$4, $−$6, and $−$8 kV. It can be seen that the acceleration starts at 54 ns (7.24 mm from electrode I) for $V D C=−3$ kV; the acceleration starts at 2.5 ns (0.25 mm) for $−$8 kV. The position at which acceleration begins corresponds to a deposited negative surface charge density range of 0.325–0.41 nC/ $cm 2$⁠. However, for $| V D C|≤2$ kV, at the position where the streamer stops propagating (8 mm from electrode I), the negative surface charge density is significantly less than 0.1 nC/ $cm 2$⁠.

Figure 5 shows the current on electrode I and electrode III for different negative voltages $V D C$⁠. The current waveform on electrode I exhibits positive and negative peaks, although the negative peak value is significantly lower than that of the positive peak. The positive current corresponds to the rising phase and the initial falling phase of the applied voltage, indicating the formation of the “forward breakdown” surface ionization wave. In the sliding regime, an increase in the amplitude of $| V D C|$ enhances the positive peak current on electrode I. Without a DC component, no significant current can be observed on the electrode III. When negative DC components of $−$4, $−$6, and $−$8 kV are applied, current peak values of 0.7, 1.0, and 2.1 A/cm are obtained, and the current reaches a peak earlier for higher DC voltage. The discharge energy can be calculated by the temporal integration of the discharge power $V P(t) I 1(t)+ V D C(t) I 3(t)$⁠, where $I 1(t)$ and $I 3(t)$ are the current on electrode I and electrode III, respectively. Without a DC bias, the energy consumed is 0.35 mJ/cm. For $V D C=−8$ kV, the total energy increases to 0.42 mJ/cm.

Figure 6 shows the spatiotemporal evolution of the electron density and reduced electric field strength for different $V D C$⁠. The streamers develop from electrode I and propagate above the dielectric surface toward electrode III, with a constant thickness of about 72  $μ$m. The maximum electric field is located at the ionization front, corresponding to the highest electron density of approximately $10 21 m − 3$⁠.

The first image [Figs. 6(a) and 6(d)] corresponding to $V D C=−2$ kV shows the discharge decay with the electron density and electric field at the ionization front decreasing significantly. For a fully established sliding discharge [Figs. 6(b), 6(c), 6(e), and 6(f)], the electric field at the front becomes strong, and the ionized channel can reach electrode III.

Figure 7 shows the spatial and temporal evolution of the surface charge density for different negative DC voltages. The horizontal axis is the dielectric surface along the discharge gap (0–10 mm), and the vertical axis is the discharge time (0–150 ns). The data are presented in logarithmic form, enhancing the visibility of regions with low surface charge density. In the absence of DC bias, as illustrated in Fig. 7(a), the evolution of the surface charge indicates the streamer trajectory. The positive charge is deposited on the surface with the development of the streamer. Then, it decreases and reverses polarity near the electrode I due to the secondary reverse breakdown after 20 ns.

Figures 7(b)–7(d) show the behavior of the surface charges with DC bias. The polarity of the surface charge deposited by the negative DC voltage (shown in Fig. 3) is opposite to that of the streamer head, which enhances the net charge transfer. In the region near the electrode I, the dielectric surface is positively charged during the propagation of the streamer head, as shown in Fig. 7(a). Once the positive streamer enters the region of negative surface charges deposited by the DC voltage, the residual negative surface charges are neutralized. However, with the approach to the electrode III, the dielectric surface still remains a negative polarity due to the combined effects of gradually increasing negative charge density and the decaying positive streamer.

The maximum reduced electric field along the probe line (⁠ $y$ = 25  $μ$m) in the streamer channel for different negative $V D C$ is shown in Fig. 8. In the case of $V D C$ = 0 kV, the electric field at the ionization head gradually decreases with the streamer propagation, and thus, it cannot provide enough ionization. With the application of a negative DC voltage, the positively charged streamer head neutralizes the residual negative surface charge, increasing the electric field at the head. As a result, sufficient ionization is maintained for the streamer to propagate until it reaches electrode III.

Gas heating plays a crucial role in the formation of shock wave structures.²²  Figure 9 shows the gas temperature at 100 ns and the pressure at 1000 ns for different negative $V D C$ values. In the case of $V D C=0$ kV, the highest temperature region is observed near electrode I, reaching approximately 530 K. The temperature gradually decreases with distance from electrode I, and no obvious temperature rise is observed near electrode III. However, once the sliding discharge (⁠ $| V D C|≥3$ kV) is established, the gap overlapping expands the heating region, and a temperature rise occurs near electrode III. For $V D C=−4$ kV, the local temperature near electrode I reaches 530 K, which is almost identical to the result without DC bias. However, a temperature rise is noted near electrode III, though the temperature is only 302 K. For $V D C=−8$ kV, the temperature reaches 720 K near electrode I and 400 K near electrode III. Electrode III connected to a negative DC voltage extends the effective heating region and enhances overall gas heating.

The generation and propagation of compression waves in nSDBD is one of the most typical phenomena observed in experiments and simulations.^6,8 In the hydrodynamic perturbations shown in Figs. 9(a2)–9(c2), strong pressure perturbations are observed. The hemispherical waves at the edge of electrode I are induced by a localized heat source, and the planar waves are attributed to the expansion of the plasma layer. The highest pressure gradient is located at the junction between the hemispherical and planar waves.

With the establishment of the sliding discharge [shown in Figs. 9(b2) and 9(c2)], the shock wave structure consists of two hemispherical waves near electrodes I and III, as observed experimentally in studies.^19,22 The two hemispherical waves propagate at the same speed and are connected by the planar wave. However, due to the temperature difference between electrode I and electrode III, the pressure wave intensity above electrode III is weaker than that above electrode I. With the increase in $| V D C|$⁠, the wave intensity near electrode III can be enhanced. It can be seen from Figs. 9(a2), 9(b2), and 9(c2) that compared with the case of $V D C$ = 0 kV, applying a negative DC bias expands the hydrodynamic perturbation region, which is crucial for large-scale flow control.

Section III A explores the elongating effect of the negative DC voltage on the positive streamer. Next, the effect of applying positive DC voltage (the polarity is the same as that of the pulse) on the discharge is discussed.

Similar to the negative polarity DC voltage, when the dielectric surface is charged until it reaches the steady state, the electron density is below $10 10 m − 3$⁠, and no obvious ionization occurs in the plasma region. Figure 10 shows the surface charge distribution in the steady state for different positive DC voltages. The surface charge density is highest near electrode III and remains constant for a short distance from electrode III. It then decreases exponentially with increasing distance from electrode III, consistent with the pattern observed with negative DC voltage. As the voltage amplitude increases, the surface charge density increases. For $V D C=2$ kV, the length over which the surface charge density exceeds 0.1 nC/ $cm 2$ is about 3.88 mm, whereas for $V D C=6$ kV, the length extends to 9.95 mm, almost covering the entire discharge gap.

The propagation dynamics of the streamer for positive DC voltage with different amplitude are shown in Fig. 11. Applying a positive DC voltage has a suppressive effect on the streamer propagation distance and speed: compared with the case without DC bias, both the maximum propagation distance and speed are reduced. For $V D C=2.0$ kV, the propagation length is nearly the same as without DC voltage; after 60 ns, the propagation speed decreases, and the streamer stops at 7.5 mm. With the increase in $V D C$⁠, the suppression effect becomes stronger. For $V D C=4.0$ kV, the maximum length is 6.0 mm, and the streamer stops propagating at about 37 ns, whereas for $V D C=6.0$ kV, the maximum length is only 3.6 mm, with the streamer stopping at about 26 ns. Unlike the application of negative DC voltage, the positive DC bias suppresses the speed of the streamer and does not facilitate the formation of a sliding discharge, which is consistent with the experimental conclusion in Ref. 22.

In the case of positive DC voltages, due to the failure to form a sliding discharge, no significant current is observed at electrode III. As shown in Fig. 12, the current on electrode I decreases with the increase in the amplitude of DC voltage. The discharge energy decreases from 0.31 mJ/cm for $V D C$ = 2.0 kV to 0.2 mJ/cm for $V D C$ = 6.0 kV.

Figure 13 shows the distribution of electron density and reduced electric field for $V D C$ = 6.0 kV. It can be seen that the streamer decays during propagation, with both the electron density and the electric field at the front decreasing over time. At 40 ns, the electron density drops to $10 19 m − 3$⁠.

The streak images of the time-dependent distribution of the surface charge density and electric field for $V D C$ = 4.0 and 6.0 kV are shown in Fig. 14. The polarity of the surface charge deposited by applying a positive DC voltage is the same as the polarity of the streamer head, which causes the surface charge density to continue rising during streamer propagation. For $V D C$ = 4.0 kV, the surface charge density reaches 37 nC/ $cm 2$ at the point where the streamer head stops propagating; for $V D C$ = 4.0 kV, the surface charge density at the stopping point reaches 50 nC/ $cm 2$⁠. The continuous accumulation of positive charge at the streamer head weakens the electric field, causing the streamer to decay and gradually stop propagating.

The gas temperature at 100 ns and pressure at 1000 ns for different positive $V D C$ are given in Fig. 15. Compared with the case for $V D C$ = 0 kV shown in Fig. 9(a1), the application of positive DC voltage has a significant inhibitory effect on fast gas heating: the region of the gas heating is reduced. Since the positive DC voltage cannot cause sliding discharge, the pressure wave consists of only one circular wave and one plane wave, with no circular wave observed near electrode III.²² 

In the present work, the sliding discharge characteristics of a three-electrode SDBD in air at atmospheric pressure, sustained by nanosecond pulse and DC bias, have been studied by a 2D plasma fluid model. The charging of the dielectric surface by DC voltage is calculated for a long time to reach a steady state. Subsequently, a nanosecond pulse is applied to the high-voltage electrode. The streamer dynamics, surface charge, and electric field evolution, as well as hydrodynamic perturbation for different polarities and amplitude of DC voltage, are discussed in detail.

Without the DC voltage, the streamer propagation stops at 8 mm. With the negative DC voltage applied on electrode III, the surface plasma can extend to electrode III when a minimum of amplitude $V D C=−3$ kV is applied. This DC voltage is the minimum value necessary to ignite a sliding discharge regime for a voltage amplitude of $V P=7$ kV applied on electrode I. During the propagation of the positive streamer, the positively charged front is neutralized by the negatively deposited surface charge, which enhances the electric field at the streamer head. This enhancement can be further reinforced by increasing $| V D C|$⁠, and the propagation speed of the discharge increases with the amplitude of the negative DC component.

With the positive DC voltage applied on electrode III, the positively deposited surface charge weakens the charge transfer from the positive streamer head to the dielectric surface, inhibiting the extension of the discharge. With the increase in the amplitude of DC voltage, the streamer propagation stops earlier and decays more rapidly. Therefore, it is difficult to form a sliding discharge for the plasma actuator excited by a nanosecond pulse and positive DC source.

The current waveforms indicate that by applying a negative DC voltage, not only does the current peak at electrode I increase but a significant current is also observed at electrode III, leading to a considerable increase in total electrical energy. However, applying a positive DC voltage reduces the total electrical energy.

Due to the different effects of positive and negative DC voltages on the positive streamer, the hydrodynamic perturbation resulted from fast gas heating is different. When a negative DC voltage is applied to form a sliding discharge, two circular waves located above the two upper electrodes are observed, connected at their tops by a linear region of high pressure gradient. When a positive DC voltage is applied, no circular waves are observed near electrode III.

This study was supported by Training Program of the Major Research Plan of the National Natural Science Foundation of China (No. 92271116) and Fundamental Research Funds for the Central Universities (No. NE2023001).

The authors have no conflicts to disclose.

Bin Zhang: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Xiaobing Zhang: Validation (supporting); Writing – review & editing (supporting). Shuqun Wu: Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.