The Role of Nonlinear Body Force in Electrostatic Propeller Driven by Atmospheric DC Corona Discharges The role of nonlinear body force in electrostatic propeller driven by atmospheric DC corona discharges

In this study, the body force generated by atmospheric positive and negative corona discharges were investigated using a wire-cylinder configuration experimentally and numerically. We provided new insight into the atmospheric electric thruster by introducing a nonlinear term in body force constituent the thrust of the system. It was observed that the direction of both body forces and electric winds is always from the wire to the cylinder irrespective of the applied voltage polarity. It was illustrated that the corresponding thrusts and the electric wind of the positive corona are larger than that of the negative corona discharge. We took into account the nonlinear mechanisms to explain the difference in thrust forces in positive and negative corona discharges. To elucidate the origin of the body force in corona discharges, we performed 2-D simulations via COMSOL Multiphysics and MATLAB software. The results of the numerical simulation showed that in addition to the linear body force (Coulomb force) a strong nonlinear body force is generated around the wire electrode that plays a crucial role in corona thrusters. To verify the direction and magnitude of the thrust, a simple theory was proposed based on variable mass systems and confirmed by published experimental works.


Introduction
Corona discharge occurs when the high electric field is not uniform near the electrodes. In atmospheric pressure, the electric field strength near one or both electrodes should be greater than the rest of the discharge gap. By increasing the applied voltage on the electrodes, corona discharge sets up around the sharp edge until the initiation of the sparks or streamers [1,2]. Corona discharge in high-pressure structures and transmission lines is an undesirable and dangerous phenomenon, but a controlled one can create an electric current and wind by charging particles and transferring their momentum to neutral particles, which has many applications in various fields. Studies on the direct current (DC) corona discharge were typically treated in needle-to-plate or wire-to-cylinder configurations. In the wire-to-cylinder setup, both positive and negative coronas are formed around the wire but with different features [3]. In the literature, the positive and negative coronas are distinguished by the polarity of the applied high voltage to the electrode whose surrounding gas is ionized, partially. In the positive corona, the streamers, and in the negative one the Trichel pulses play the main aspects of the process [4][5][6][7]. Corona discharge has been theoretically and experimentally studied for its many applications such as electrostatic precipitation, water treatment, air purification, surface treatment, and plasma medicine [8][9][10][11][12]. Current researches on corona discharge covered many fields including plasma actuators for flow control [13][14][15] and propulsion by electrohydrodynamic (EHD) thrusters [16][17][18][19][20][21][22][23]. The first observation of the EHD current was made in 1709 by Hauksbee [24]. Hauksbee discovered the production of low-velocity airflow when the voltage was applied between two electrodes, a phenomenon later referred to by Newton as the "electric wind" [25]. In the late 19 century, Chattock first studied electric wind quantitatively [26]. In 1957, Harney investigated the electrical characteristics of corona discharge and the change in flow aerodynamic parameters to measure the generated force [27]. Davis conducted one of the first numerical studies on corona discharges using the finite element method [28]. In 2008, Larson et al. investigated electrohydrodynamic force in a corona discharge and compared their results with that of experimental works [29]. Colas et al. optimized the corona discharge regime in a five-electrode configuration and measured the wind velocity of up to 10 m/s [30]. They also calculated the electrohydrodynamic force using velocity profiles. In an experimental study, Moreau et al. investigated the current-to-thrust conversion with a threeelectrode design. They improved the thruster effectiveness using a proper design. [31]. In another study, Moreau et al. examined the thrust force generated in positive and negative corona discharge by considering the wire-cylinder configuration [32]. An outstanding result was achieved by Xu et al in developing a drone with corona propulsion by applying a DC high voltage of +40 kV to generate a thrust of 3.2N and flew the drone for a distance of 40-45 m with a duration of 8-9 s [33].
Although the accepted theoretical models of body force contribute much to understanding the producing corona propulsions, the lack of true knowledge still restrains the popularization and application of such propellers. Therefore, the understanding of the body force mechanisms could facilitate the development of corona propulsion structures. It is generally accepted that the mechanism for electrohydrodynamic force production is based on the Coulomb force, as a linear term, acting on the plasma space charge in an effective electric field. Our previous theoretical work showed that a new mechanism of electrohydrodynamic force could be a nonlinear body force that arose from the gradient in the effective electric intensity inside the discharge zone. This nonlinear term contributes the electric wind generation independent of electric field polarity and charges sign as follows [34]:

Where
, and ν αn are the angular frequency and effective collision frequency for momentum transfer, respectively. refers to the electron and ion species, and 2 = 4 2 , and are the mass and density of electron and ion, respectively. The effective collision frequency is given by = , where is the momentum transfer cross-section, is the number density of neutral particles and is the average velocity of random motion for the particles of type . It is seen that the nonlinear body force pushes all charged species out of the strong electric field zone independent of the sign and electric field polarity.
In this study, we proposed and examined the contribution of a nonlinear body force mechanism in corona propulsions. The contribution of the nonlinear body force arising from a positive and negative corona discharge was simulated in comparison with the linear Coulombic force. Using the simulation results, we showed that there is a nonlinear body force, acts through a different mechanism, influencing the discharge assembly around the wire electrode due to a strong gradient in the electric field intensity. It was also shown that the direction of the nonlinear body force in both positive and negative corona discharges is always from the wire to the cylinder. To consider the influence of linear (Coulomb force) and nonlinear body force in wire-cylinder configuration, the results of references [34][35][36] were used for both coronas. It was demonstrated that during the running of both coronas the electric current can be decomposed into DC and AC signals that characterize the corona process and distinguish positive and negative coronas. To verify the direction and magnitude of the thrust, we introduced a simple theory based on the variable mass force equation with taking into account the examined wind direction. Finally, it was proved that the corona discharge assembly acts as a system of propellers that produces a thrust via a wind generation like a pusher fan on a boat. This paper is organized as follows: in section 2, materials and methods including simulation procedures, and experimental setup are introduced. In section 3 simulation and experimental results are given, in section 4, a simple theory for corona thrusters are given, and in section 5 and 6, the discussions and results are given taking into account the linear and nonlinear body forces.

Simulation section
Linear and non-linear body forces in positive and negative corona discharge were investigated using COMSOL Multiphysics (plasma module) and MATLAB software at atmospheric pressure.
The linear and nonlinear terms of the body force per unit volume (N/m 3 ) can be written as follows In the above equations, ρ is the net charge density of the species (C.m -3 ), E is the electric field strength in V/m such that E=√E y 2 + E x 2 .
As can be seen in figure 1, in a 2-D simulation, two circles act as electrodes with radii of 0.5 mm and 10 mm were placed at a distance of 30 mm from each other. We set the wire and the cylinder as the powered and ground electrodes, respectively. In the simulations, we set the powered electrode (wire) at a DC voltage of +10 kV and −10 kV for positive and negative corona, respectively. In this simulation, we took into account the electron impact reactions, reaction between heavy species, and surface reactions including the secondary emission from the electrode surface. As can be seen in  Because the COMSOL software cannot solve higher derivatives in the plasma module algorithm, we calculated the value of the nonlinear body force using MATLAB software based on the COMSOL outputs. For this purpose, as depicted in figure 2, we divided the space between the two electrodes into 2601 points and formed 51 × 51 matrices. At each point, we calculated all needed parameters for evaluating the nonlinear body force term including charged species density, electric field distribution, neutral density and averaged velocity of the -type particles. The results of linear and nonlinear terms will be given in the result section. Figure 3 shows the flowchart of the simulation procedures that were used in this work.

DC high voltage power supply and voltage-current measurement
Positive and negative DC high voltage power supplies (0 -±30kV, 3mA, 90W) were made using a two-stage full-wave Cockcroft-Walton voltage multiplier. The input for the Cockcroft-Walton module was a sine wave high voltage with frequency and peak to peak values of 7.2 kHz and 22 kV, respectively. The DC applied voltage with a resolution of 0.1 kV was adjusted manually. The applied voltage to the wire electrode was measured using a high voltage probe (Tektronix P6015A.), and the current signals were monitored by a high-precision analog multimeter (Leybold Demo-Multimeter 53,191, precision±0.05μA). All signals were recorded by a four-channel digital oscilloscope (GW Instek GDS-3354). To run the setup in corona mode, a copper wire electrode having 20 cm length and 0.1mm diameter, together with a cylindrical electrode with 20 cm long and 10 mm diameter were used. According to figure 4, both electrodes were held by an insulating housing (No. 1 in Figure 4(a)) and were placed at a distance of 3 cm from each other. In each experiment, the applied voltage was last for 30 seconds. We obtained the artificial thrust of the corona assembly by this method, then we applied the calibration curve for measuring the actual thrust of the coronas. It was proved that the thrust measurement setup has a linear trend as shown in figure 5. The thrust of the corona was obtained by f b = 0.3004 X + 6 × 10 −5 , is the change in digital scale in Newton. The magnitudes of the consumed electrical power and thrust effectiveness were calculated according to the following equations, respectively [32]: Where T represents the thrust force, P is the consumed electric power, I the electric current, and θ the thrust effectiveness.

Electric wind velocity measurement
The average velocity of the electric wind was measured using a Pitot tube which was connected to a micro manometer (CEM, DT-8920. PONPE, Inc.). The stainless steel Pitot tube might lead to sparking at distances near the wire electrode, then a homemade glass capillary tube with an inner diameter of 0.5mm was used to measure the total pressure. Also, static pressure was calculated by the static pressure tube near the site. As depicted in figures 6(a) and (b), a 1-D y-positioner with 0.1mm accuracy was employed to displace the glass tube along the horizontal axes. With this method, the vertical velocity of the electric wind was measured with an accuracy of ±0.01m/s.

Electric wind direction test (EWD test)
As       The magnitude and direction of the nonlinear body force were calculated using Equation (2).

Experimental Results
In addition to the differences in thrust and electric wind between the positive and negative corona,  current Townsend discharge and high-current glow discharge [6]. The differences in the AC could be attributed to the Trichel pulse in negative corona. Figure 16 shows the voltage-current characteristic curve considering the total current ITotal = IDC + IAC per unit spanwise length ( μA m ⁄ ).
This may be due to the greater mobility of the negative ions than the positive ones in some conditions [41]. Despite a large AC current in the negative corona, the corresponding thrust is less than that of the positive corona. The evolutions of the body force were depicted in figure 17 versus the applied voltage (figure17 (a)), total current ( figure 17 (b)), and the consumed power ( figure 17 (c)). The trend of the thrust versus applied voltage differs little from that versus the power and total current. Up to 11 kV, the negative voltage has a larger thrust than the positive polarity, while beyond 11 kV, the positive corona has a large thrust.    To record the characteristics of the induced flow in both DC coronas, we employed smoke of the incense sticks between the wire and the cylinder. Before running the coronas, the smoke flow was upward, that is towards the wire ( figure 21 (a)). As can be seen from Figures 21 (b) and (

Corona thrusters perform like a pusher fan
In this section, by assuming the induced flow to be incompressible, steady-state, and irrotational, we introduce a simple theory for the thrust evaluations using the variable mass equation. According to figures 23(a) and (b), it has been shown that the induced flow is generated at the wire periphery and extends towards the cylinder with an average speed of ̅ . The maximum velocity was created around the cylindrical electrode. Therefore, it can be assumed that the discharge assembly behaves like a variable mass system that generates the thrust via = ( ̅ )⁄ , where m is the released mass of the system. Using the released mass of = in unit time and the corresponding volume = ̅ = ̅ , where, is the volume of the released flow in unit time, L the length of the wire, D the diameter of the cylinder, is the air density, and S is the cross-section of the cylinder. Then the thrust can be estimated as follows: = ̅ 2 (5) Figure 23. Displays, the mechanism of air mass movement (a) and velocity profiles around the cylinder (b).
Therefore, according to Equation 5, using the average electric wind speed around the cylinder electrode, the thrust force generated in a corona discharge can be calculated. The thrust direction is against the induced wind (upward) just like the thrust in a variable mass system. In another word, the corona thrusters play the role of a pusher fan on a boat irrespective of its polarity. We used the measured velocity by Moreau et al to predict the thrust by equation (5). As can be seen in figure (24), there is a good agreement between our theoretical model and that measured by Moreau et al.  In the case of the negative corona, the induced flow characteristic deviates from laminar flow, therefore, our assumptions do not satisfied with the released flow compared with that of the positive corona. As a result, there are deviations between the predicted and measured thrusts in the negative corona.

Discussion
As can be seen in figure 13(a), the corresponding external electric fields of the positive and negative coronas are screened by the space charge fields. In the positive corona, the screening effect takes place immediately after the wire around point 0.5, then the field strength greatly increased and reaches its maximum at about 0.8 mm from the wire center. In this case, electrons are produced by ionization, then sink into the wire leading to a space charge of positive ions. The positive space charge is configured at two distinct layers separated by a neutral thin layer( figure  13 (b)). The first one attached to the wire possesses a small number of positive species relative to the second layer. The positive layers surrounding the wire create an electric field that is directed against the external field, then reduce the field around the wire, thus screening the external field. As the positive species travel to the ground(cathode) the strong field around the wire is restored. Then, a sudden increase of ionization follows, and an increase in current is observed. This is the reason for the intermittent current(AC) in the positive corona [1]. The electric current between the two adjacent peaks does not drop to zero, therefore we observe a DC component in the positive corona discharge. In contrast to the positive corona, the screening effect in the negative corona is less pronounced and it was happened around the point of 0.8 mm, farther than the case of the positive one. For both coronas, the ionization zone is limited to the close distances to the wire whose electric field strength is considerable for ionization. The simulated space charges were depicted in figure 13(b) for both coronas. In the case of the negative corona, there is an apparent dipole constituted of negative and positive space charges. The induced dipole makes an electric field that is superimposed with the external one, as a result, the electric field is distorted in the gap. It is seen from figure 13(a), in the negative corona, the corresponding fields of the cathode and the induced dipole in the vicinity of the wire add up to give a field stronger than the positive corona.
On the other hand, the resultant field inside the dipole zone is directed in the opposite direction that weakens the external field. The positive space charge locates close to the wire relative to the negative charge. The negative space charge rapidly flows to the anode, then the screening fails and the field strength enhances around the wire. In the atmospheric air, once the electrons travel towards the anode, they experience a weak electric field and attach to the electronegative molecules, forming negative ions like 2 − and 2 − . On the other hand, the positive ions flow to the wire leading to an increased current. In this case, the intermittent current is observed like that of the positive corona. Then the external field is restored and new ionization takes place.
Characteristics of the linear and nonlinear forces were depicted in figures 13(c) and (d) for both coronas. In the case of the positive corona, the direction of the linear force is almost towards the cathode parallel to the induced flow direction. It has a reversed direction with less amplitude in a small extension immediately after the wire, while it keeps the positive direction in the remaining gap. In the case of the negative corona, the direction of the linear force changes as a result of the induced dipole in the vicinity of the wire. The linear force direction is towards the anode after the wire while it reverses at the negative space charge location. The reversed linear force has a relatively large extension, that is, from 0.8 mm to 1.1 mm with a comparable amplitude with that of the former. Therefore, the resultant linear force weakens effectively the speed of the induced flow and makes turbulence on the flow movement. The orders of magnitude of linear force for both coronas are the same although their extensions and directions are different. Characteristics of the nonlinear force were depicted in figure 13(d). In contrast to the linear force, the nonlinear one has a unique direction from the wire towards the cylinder. The nonlinear force can be interpreted as a strong gradient in the intensity of the distorted electric field in the vicinity of the wire. So the magnitude of the nonlinear force strongly depends on the polarity of the corona which the space charges play a crucial role, as can be seen in figure 13(b). For the positive corona, the nonlinear force term is one order of magnitude larger than that of the negative corona. If the linear force term was the only responsible term for the induced flow and thrust, then the difference between the positive and negative corona must be very large, because the linear force term reverses in the corona bulk in the case of negative one. Meanwhile, the experiments revealed that both coronas led to the same flow direction with a little difference in the thrust magnitude. The direction and magnitude of the nonlinear force make the sense that it plays the main role in the wire-cylinder configuration thrusters. It provides unidirectional force independent of the corona polarity acting on both positive and negative species inside the corona medium. In the positive corona state, the nonlinear body force pushes back the electrons into the discharge channel and creates more ionization processes. With increasing ionization collisions, more thrust force and less pulsed electric current with irregular amplitude are created. The nonlinear force at the negative corona also causes a larger negative ion cloud to form around the cylindrical electrode. This creates a larger pulse current.

Conclusions
In this study, we provided new insight into electrostatic propulsion by introducing the nonlinear body force term affecting the induced flow in a wire-cylinder corona discharge configuration. The nonlinear body force arises from the strong gradient in electric field intensity independent of the corona polarity. If the linear force term was the only responsible term for the induced flow and thrust, then the difference between the positive and negative corona thrusts might be very large, because the linear force term reverses in the corona bulk in the case of negative discharge.
In the case of the positive corona, it was observed that the induced flow is almost laminar, but in the negative one, it is fairly chaotic due to the change in linear body force direction between the wire and the cylinder. Our simulations revealed that there takes place a strong screening in the external electric field in the negative corona discharge and an electric dipole originates near the wire electrode. The electric dipole is not observed in the positive corona, instead, a net positive space charge creates near the wire. The results of simulations indicated that the nonlinear body force is about 6 orders of magnitude larger than that of the linear one in positive corona. On the other hand, the nonlinear force term in the positive corona is one order of magnitude larger than that of the negative corona. The experimental results showed that both coronas led to the same flow direction with a little difference in the thrust magnitude. It was also shown that the corona thrusters are compatible with the variable mass systems, which the thrust value could be obtained from the released flow during the corona running. The thrust direction is against the induced flow direction like that of a pusher fan on a boat. The body force is responsible for induction flow in the corona assembly, and in turn, the released flow causes the thrust in the system.