## 4 Model Validations  
To validate the present numerical methods, the current on the electrode and the discharge energy in the referred experiment [27] are referred for comparisons. The one-dimensional model is validated by comparing the currents in the experiment and the two-dimensional model. Then the results in the one-dimensional models are validated further. To show the reasonability of the quasi-one-dimensional model, the comparison of the currents on the electrode powered by \(Exp\) waveform in \(R4\) in the one-dimensional, the two-dimensional models and that in the experiment is shown in Figure 4, the voltage waveform is also shown in the figure. Because the \(Exp\) waveform is formed by piecewise linear interpolations and the gradient due to time is discontinues, the current oscillates during the pulse and in the post-discharge. The currents in the one-dimensional and two-dimensional models both agree that the experimental current in the fall phase of the negative pulse and the currents in the two models are nearly the same. However, the current in the one-dimensional model is larger than that in the two-dimensional model, and the current in the one-dimensional model has a closer agreement with the experimental current. The currents in both models agree well with that in the experiment in the ascending and decay phase of the positive pulse, and they are nearly the same. It can be concluded that the one-dimensional model and two dimensional model are both feasible for the prediction of the current in the present plane to plane NS-DBD discharges. Therefore, one-dimensional models are used to study the effects of reactions on the discharge characteristics hereafter. The past numerical simulations [4; 8; 20; 22] of a repetitive NS-DBD focused on the first pulse. The experimental results [27] show that the energy in every pulse is nearly the same in the whole discharge. However, the energy in the first pulse differs from that in the other pulses [27]. In order to know when the discharge energy can become steady, the discharges of the first seven pulses are performed in all simulations. \(Q_{A\_ui}\) in \(R99\) in the 1 st, 2 nd, 3 rd, 5 th, and 7 th pulses of the Gaussian voltage waveform are shown in Figure 5.  
Figure 4: The comparison of the currents by the \(Exp\) waveform in \(R4\) in the quasi-one-dimensional and two-dimensional models, and in the experiment [27]. \(Q_{A\_ui}\) after the 1 st pulse are all translated to 0 ns for clarity. It can be seen that \(Q_{A\_ui}\) are almost the same as each other in each phase of the pulse after the 2 nd pulse. It indicates that \(Q_{A\_ui}\) reaches to periodic state following the discharge pulse, which is similar to the experimental results [27]. Hereafter, the results of the 7 th pulse are adopted in all simulations. To study the effects of voltage waveforms on the current, the currents on the electrode powered by \(Exp\) and \(Gaussian\) waveforms in \(R99\) are compared, as shown in Figure 6. The start time of the experiment curve at 180 ns [27] is translated to 0 ns for clarity. The magnitude of the current using the \(Gaussian\) waveform is smaller than the experimental data near the first negative peak. In the time duration after 75 ns the result of the \(Gaussian\) waveform rapidly reduces to zero without oscillates due to the neglect of voltage oscillations after 75 ns in the \(Gaussian\) waveform. The predicted currents using \(Exp\) and \(Gaussian\) waveforms both agree well with the experimental data during the high voltage pulses. Therefore, the present numerical model predicts good results for the discharge current. Figure 5: \(Q_{A\_ui}\) predicted by \(R99\) in the 1 st, 2 nd, 3 rd, 5 th, and 7 th pulses. Figure 6: The comparison of the currents predicted by the \(R99\) model and the one in the referred experiment [27]. The predicted discharge power \(P_{A\_ui}\) and energy \(Q_{A\_ui}\) are also compared with the ones in the referred experiment [27], which are shown in Figure 7a and Figure 7b, respectively.