## 1 Introduction  
Since its introduction by Siemens in 1857 [1], Dielectric Barrier Discharge (DBD) continues to attract extensive research [2]. In recent years, this research has focused on Surface DBD (SDBD), a form of discharge that is generated only within a thin layer on the surface of the dielectric insulator separating the electrodes. In contrast, traditional DBD, sometimes referred to as Volumetric DBD (VDBD), fills the entire volume of the air gap between the electrodes. Electrohydrodynamic (EHD) SDBD plasma actuators can be used to improve the performance of aerodynamic elements such as aircraft wings and turbine blades [3; 4; 5]. By modifying the airflow around these elements, SDBD actuators can influence flow separation, laminar-turbulent transition, lift, or drag [6; 7; 8]. In addition, SDBD plasma reactors have proven useful in other applications, including ozone production [9; 10; 11; 12; 13], fluid pumping [14], surface modification [15; 16; 17], air and water treatment, and others [18; 19; 20]. However, despite its many practical applications, a consistent theory describing the nature of SDBD has not yet been developed, largely due to the complexity and highly dynamic nature of the involved processes, which limits its practical use. Planar high-voltage electrode reactors are the most common type of SDBD reactors in practical applications and are widely reported in the literature. However, as originally proposed by [PERSON] et al. [21; 22], the morphology and parameters of the SDBD plasma can vary significantly depending on the shape of the discharge electrode. This concept was later extended by other researchers. [PERSON] et al. [23] compared the electrical, mechanical, and optical characteristics of the SDBD actuators with different electrode shapes and found that the actuator with a wire-shaped electrode produced a streamer-free discharge and a more homogeneous EHD flow compared to planar and saw-like electrode configurations. Similar results were later reported by [PERSON] et al. [24], who also investigated different electrode geometries for high-performance SDBD actuators. [PERSON] et al. [25] compared the thermal and flow characteristics of SDBD actuators with various electrode shapes and found differences in the EHD flow profiles and temperature distributions. [PERSON] et al. [26] investigated the charge distribution in a wire-shaped SDBD actuator, observing that the number of positive charges accumulating near the high-voltage electrode increases as the wire diameter decreases. Recently, [PERSON] et al. [27] examined the thermal characteristics of different types of SDBD actuators and concluded that the planar electrode reactors are characterized by a slightly higher temperature of the dielectric material compared to the wire electrode for the same discharge power. The properties of the SDBD plasma have also been studied by various research groups in the past. [PERSON] et al. [28] was among the first to study the morphology of the plasma generated in a typical SDBD reactor using a photomultiplier tube. He identified two characteristic discharge modes, which were later confirmed by other research [29; 30; 31]. [PERSON] et al. [32] used a fast-gated and infrared camera simultaneously to study the plasma morphology and correlated the plasma onset points at the electrode edge with temperature hotspots. [PERSON] et al. [33] focused on the extension of streamers in SDBD actuators and reported that in a standard electrode configuration, streamers can easily extend up to 40 mm from the electrode. [PERSON] et al. [34] studied the SDBD plasma morphology generated from the rod-shaped electrode and observed that increasing the applied voltage enhanced streamer filamentation. Recently, [PERSON] et al. [35] used a fast-gated imaging technique to study streamer filamentation in the nanosecond SDBD. He found that the velocity of a single streamer front varies with the applied voltage.