Positive Streamer in the Surface Dielectric Barrier Discharge in Air: Numerical Modelling and Analytical Estimations

According to performed numerical simulation of the surface dielectric barrier discharge driven by positive polarity nanosecond voltage pulse the discharge in this case evolves as a streamer “flying” above the dielectric surface. The distance between the streamer and dielectric surface does not depend on dielectric barrier parameters and applied voltage value. The developed analytical model for surface streamer evolution confirms these results and explains the physics of this phenomenon. The electric field in front of a stationary streamer head is constant and defined only by ionization rate constant of the gas and its density.


Introduction
The study of the surface dielectric barrier discharge (SDBD) in atmospheric air is still of the considerable interest because of its promising applications for flow control aerodynamics [1][2][3] and for plasma assisted ignition and combustion [4,5]. The SDBD driven by nanosecond voltage pulse is a preferable object of investigation for discharge physics understanding, because this discharge develops as a single microdischarge on the leading front of the voltage pulse, in contrast to a set of microdischarges relevant for a case of AC applied voltage. The advantage of studying the one microdischarge generation is in well known initial conditions and, consequently, in the possibility of more reliable interpretation of simulated results and experimental data.
The effectiveness of the nanosecond SDBD application for fuel ignition is strongly affected by the mode of the discharge: whether it is quasi-uniform or filamentary (constricted) mode of discharge development. In quasi-uniform mode the ignition occurs in a spot near the electrode edge [6], whereas in filamentary mode it is spread to longer distance along the discharge filament [7]. The physics of the SDBD transition from one mode to another is not understood yet. In order to get it, more detailed data regarding the discharge spatial structure and temporary evolution is needed.
According to performed numerical simulations [8][9][10][11] a single microdischarge in the SDBD evolves as a sliding transient glow discharge for negative exposed electrode polarity and as a streamer for positive one. Whereas for negative polarity the discharge is adjoined to the dielectric surface, for positive polarity it propagates at some distance above the surface. The gap between the streamer and the dielectric surface is characterized by positive ion charge surplus and extra-high electric field value at the level of 300 kV/cm for atmospheric air [9,12]. The existence of high electric field layers in the SDBD was implicitly confirmed in experiment [12]. This work is addressed to enhance our understanding of one aspect of the SDBD physics -the physics of positive streamer propagation. For this purpose the simplified analytical model for positive streamer maintenance condition is developed and the results of this model are compared with the results of numerical simulations. Both in numerical and analytical study we consider the streamer as a 2D sheet of plasma relevant for quasi-uniform SDBD development.

Numerical simulation of positive streamer in air
The 2D numerical simulation for SDBD streamer evolution in atmospheric air has been done using the numerical model developed in [8]. The results in the form of the extra positive charge density contours above the dielectric surface in units of 10 14 cm -3 are shown in Fig.1 and demonstrate the streamer "flying" above the dielectric surface. Calculations were done for stepwise voltage amplitude V = 5 kV, dielectric thickness d = 0.1 mm, and dielectric relative permittivity  = 2.7.

Analytical model
Numerical simulation has shown that all the quantities inside a streamer body, -the E-field and the charged particle densities, are almost constant in the streamer cross section and strongly vary only in the  -layer near the streamer boundary. Accordingly, for analytical estimation of the quasi-steady streamer maintenance condition we can use 1D approximation for charged particles transport equations and for Poisson equation for electric field. The spatial coordinate s belongs to E-field line at the center of the streamer as shown in schematic picture of the streamer head in Fig.2a, where h is a streamer distance from the dielectric surface. (1) Here n e = electron density, n i = positive ion density, n = n i -n e is a positive ion surplus, v e = K e E is electron drift velocity, K e = electron mobility,  i = gas ionization frequency. We neglect the negative ion formation and the drift of ions, which are not important in the streamer head formation process [13]. The Eqs. (1)  moving with streamer head with velocity v s these equations have the same solution obtained in [14]. According to this solution the electron-ion multiplication mainly occurs inside the  -layer. The electron-ion density reaching the value n eh behind the streamer head (Fig.2b) and the  value read [14] () 4 ( The electric field inside the  -layer falls down from E h value in front of the streamer head to almost zero in the streamer body. The Eqs. (2) and (3) show that electron-ion density n eh and  thickness are the functions of electric field E h in front of the streamer head. Here The commonly used expression for where А and В are the constants defined by gas properties, the book [14] recommends А=15cm -1 Torr -1 , В = 365V/cm/Torr for air, p is a gas pressure. The kV/cm in atmospheric air. Following the approach realized in [15] for volumetric streamer between pin and plain electrodes we formulate the electron balance equation near the streamer head. The region between the dielectric surface and the 2D-streamer bottom boundary near the streamer head may be considered as a flat region with all parameters variation only in the normal to dielectric surface direction s. In this approximation the number of electrons generated due to gas ionization inside a gap between dielectric surface and streamer per unit streamer length is   (11) This value for electric field in front of the streamer head has been confirmed by numerical simulations both for surface [9,12] and volumetric streamers [17,18].

Conclusions
The proposed simplified analytical model for quasi-steady streamer maintenance condition permitted to find out the distance at which the streamer moves above the dielectric surface in the surface dielectric barrier discharge (SDBD) driven by positive polarity voltage pulse. This distance is managed only by the gas ionization rate constant, namely, it is primarily defined by asymptotic for infinite E-field ionization length of the gas. For atmospheric air the streamer remoteness from the dielectric surface is around 0.03 mm. The electric field in front of the quasi-steady streamer head is managed by another parameter of the gas ionization rate constant (Townsend coefficient); it is equal to characteristic electric field value, at which the Townsend coefficient starts to saturate in the growing up electric field. It should be emphasized that this result is valid not only for positive surface streamer in SDBD, but for volumetric streamer between pin and plain electrodes as well. The value of this electric field in atmospheric air is near 300 kV/cm. Both analytical results for streamer remoteness from the dielectric surface and for electric field value in the streamer head are confirmed by numerical simulation.