# Electrohydrodynamic force produced by a wire-to-cylinder dc corona discharge in air at atmospheric pressure

Éric Moreau, Nicolas Bénard, Jean-Daniel Lan Sun Luk, Jean-Pierre Chabriat

► To cite this version:

Éric Moreau, Nicolas Bénard, Jean-Daniel Lan Sun Luk, Jean-Pierre Chabriat. Electrohydrodynamic force produced by a wire-to-cylinder dc corona discharge in air at atmospheric pressure. Journal of Physics D: Applied Physics, 2013, 46 (47), 10.1088/0022-3727/46/47/475204. hal-01691632

HAL Id: hal-01691632

https://univ-reunion.hal.science/hal-01691632v1

Submitted on 21 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Electrohydrodynamic force produced by a wire-to-cylinder dc corona discharge in air at atmospheric pressure

Eric Moreau<fnref n="1" />, Nicolas Benard<fnref n="1" />, Jean-Daniel Lan-Sun-Luk<fnref n="2" />, Jean-Pierre Chabriat<fnref n="2" />

## Abstract

Wire-to-cylinder corona discharges are studied to better understand the electrohydrodynamic (EHD) phenomena that govern the performances of electric propulsion systems. First, theory associated *with* EHD thrusters is presented in order to be compared to experimental results. Secondly, direct thrust measurements are conducted to optimize the electrical and geometrical parameters of such devices. The main results are as follows: (1) the discharge current *I* is proportional to the square root of the grounded electrode diameter and to $1/d^2$ where *d* is the electrode gap; (2) for $d \le 20$ mm, the mobility of negative ions is higher than *that* of positive ions *while the* mobility of both ions is equal for higher gaps;

(3) *therefore, for gap $\le 20$ mm*, positive and negative corona results in the same current-to-thrust conversion; (4) the current-to-thrust conversion is equal to 33 N/A per centimeter of gap, and it is proportional to the gap; (5) the thruster effectiveness $\theta$ increases with $\sqrt{d}$, decreases with the square root of thrust, and reaches about 15 N/kW for $d = 40$ mm; (6) the force computed from experimental velocity profiles is over-estimated compared to the values measured with a balance, showing that this method cannot be used for thrust determination.

<footnote n="1">¹ University of Poitiers, Institut Prime, CNRS, ISAE-ENSMA, Futuroscope, France</footnote>

<footnote n="2">² University of La Réunion, LE2P, Saint Denis, France</footnote>
# 1. Introduction

When a high potential difference is applied between two electrodes in atmospheric air, ionization of the air molecules around the active electrode (with the smallest curvature radius and on which the high voltage is applied) induces a corona discharge due to electron avalanche (Townsend discharge). This ionization zone is composed of electrons and ions. In the case of a positive high voltage, positive ions are produced while negative ions are created predominantly if the applied voltage is negative. Due to the electric field, every ion is subjected to a Coulomb force, resulting in their displacement toward the grounded collecting electrode, in a zone called the “drift region”. The sum of Coulomb forces acting on every ion results in a volumetric electrohydrodynamic (EHD) force (in N/m³) occurring inside the discharge. In the drift region, there are many collisions between charged species in motion and neutral air molecules, resulting in a momentum transfer that produces a gas flow from the active high voltage electrode to the grounded collecting one. This flow is usually called “electric wind” or “ionic wind”. This electrohydrodynamic phenomenon can be used as an electromechanical actuator without moving part, that directly converts electrical energy into mechanical energy.

Electric wind has been theoretically and experimentally studied for its many applications such as electrical blowers without fan [1-10], enhancement of heat transfer [11-13], plasma actuators for flow control [14-16] and propulsion by EHD thrusters [17-24]. Lots of numerical simulations on corona discharges and the resulting electric wind are also available ([25-29] for example).

The theoretical aspect of EHD phenomena occurring inside a corona discharge will be presented in the first part of this paper. The theory of EHD in gas starts with the well-known publication of Robinson in 1964 that investigated the ability of volume corona discharges to perfect blowers in absence of any moving parts, based on electric wind phenomenon [6]. Robinson gave the basis of the EHD theory. However the mean electric wind velocity was rather low, up to 4 m/s. For about 10 years, several others groups have been investigating corona discharges, in order to increase the electric wind velocity, to
enhance the resulting flow rate and the electromechanical efficiency of such devices [6-10]. For instance,
in the case of a simple needle-to-grid corona discharge occurring inside a hollow tube, Moreau and
Touchard [7] measured maximum electric wind velocity up to 10 m/s, flow rate of about 1 liter per
second with an efficiency on the order of 1% (mechanical power of about 100 mW for an electrical power
consumption up to a few mW), in agreement with previous theoretical predictions of Sigmond *et al.* [3, 5]
and Goldman *et al.* [4]. In [10], Colas *et al.* optimized a two-dimension EHD device based on five
electrodes (a positive HV wire, two grounded cylinders and two negative HV plate electrodes). They
measured velocity up to 10 m/s. From the velocity profiles, they computed electrohydrodynamic force up
to 350 mN per meter of electrode span, with a power consumption of 210 W/m, corresponding to an
effectiveness of 1.7 mN/W. With a device composed of six stages in series, where each of them consisted
of an inner ring electrode with multi-pins and a out ring electrode, Kim *et al.* [9] demonstrated that the
electric wind velocity and electromechanical efficiency could be increased with the square root of the
number of stages. With the six stages on, the maximum flow rate was equal to 3.3 liter per second with a
electrical power consumption of 90 W.

**Corona discharges have also been** investigated for about 15 years to be used as plasma actuators for
airflow control applications. Plasma actuation consists in generating a surface discharge at the wall of an
aerodynamic profile in order to manipulate the boundary layer flow with **the** help of the produced EHD
force. The resulting electric wind can then modify the near-wall flow characteristics and interact with
surrounding natural flow structures, in order to reduce drag or enhance lift for instance. More details can
be found in three main review papers [14-16]. Through different optimisations, electric wind velocity up
to 12 m/s and EHD force up to 125 mN/m have been achieved. Moreover, it has been demonstrated that
the **temporal** behaviour of the electric wind follows the high voltage waveform [16], highlighting that the
output electric wind is a “quasi-linear” function of the input applied high voltage, making this type of
electromechanical actuators very useful for in-lab experiments. In **practice**, these actuators showed their
ability to control airflow in realistic industrial applications, for Reynolds numbers higher than $1×10^6$, but still lower than the ones encountered in landing and take-off conditions.

The idea of using a corona discharge as an electrohydrodynamic thruster is born in the 1920s when Townsend Brown found that a X-ray "Coolidge" tube was able to produce a force. He believed that he had discovered a new principle to control gravitation [17]. Several patents followed. In the most well-known one [18], Townsend Brown explained that he had discovered an unknown electrokinetic phenomena able to produce a force and gave some explanations on the Biefeld-Brown effect. In 1967, Christenson and Moller [19] published the first paper on EHD propulsion. They investigated theoretically and experimentally the capabilities of corona discharge as a thruster and they defined a parameter θ, corresponding to the thrust per unit input power (in N/W). They showed that θ was proportional to the electrode gap, and inversely proportional to the applied voltage and ion mobility. Thus ion mobility was found to be the most decisive parameter in the system performance. In 2009, low-speed advanced propulsion systems for long-duration near-space platforms were reviewed [20]. On one hand, EHD thrusters have several advantages: they are lightweight, simple, robust, without moving parts and they can operate over a large range of pressure. On the other hand, their efficiency is low and it is predicted to be lower at higher altitudes because ion mobility increases with altitude. Nevertheless, some researchers claim that efficiency can be improved up to 7.5% by operating with an incoming flow velocity of 50 m/s, as demonstrated in [23]. They explained that EHD propulsion might be used to propel balloons or small aircrafts using solar power and photovoltaic energy conversion systems to produce electricity that supply the corona-based thrusters. In [21] and [24], a model of an ideal EHD thruster is examined theoretically in order to determine the feasibility of such systems. Their conclusion was not favourable and concluded that the feasibility of using an EHD ionic wind pump for thruster applications was rather dim. However, in 2009, NASA published a report on ionic wind propulsion [22]. They concluded that if values of θ = 20 N/kW and a thrust per unit surface area $\phi$ of 20 N/m$^2$ could be attained simultaneously, EHD propulsion might be practically useful. However, their experiments did not allow them to reach both these
values, **indicating that this branch of research required more extensive study.** Indeed, only a few **studies** have been published on this subject and **further** efforts are needed to optimize EHD ionic wind pump devices. For instance, the value of φ can be increased **by the use of** multi-stage devices, as Kim *et al.* [9] used them to enhance electric wind flow rate.

In the present work, experiments are conducted at atmospheric pressure in order to understand and to optimize the electrical parameters and the geometrical configuration of such EHD devices. These works give tendencies that will allow us to perfect a **prototype** for future low-pressure measurements in an altitude chamber.

In this paper, the first part is dedicated to the theory associated **with** EHD thrusters. The developed electrohydrodynamic model is simple and it is averaged in time and in space. **This model is appropriate in reflecting** the time-averaged phenomena occurring in corona discharges. In the second part, the experimental setup and the measurement procedure are presented. The measured mechanical variables are the time-averaged thrust and the time-averaged electric wind velocity. **Here, the design of the EHD device is rather simple** because the objective of this work is to better understand the physical phenomena that govern the EHD thruster effectiveness. The device is composed of a 25-µm diameter wire electrode and a grounded cylindrical one. The experimental results are presented and discussed in the third part. The following parameters and their effect on the EHD thruster **performance** are investigated: the high voltage polarity, the relative humidity, the grounded electrode diameter (from 0.8 to 22 mm) and the electrode gap (from 5 to 40 mm). The use of a multi-needle system instead of a **wire as the high voltage electrode** is also examined.

## 2. Theoretical aspect

### 2.1. Electrical aspect

A corona discharge is a weakly luminous discharge that is produced usually at atmospheric pressure near needles or thin wires, where the electric field is sufficiently strong. In a positive pin-to-plate corona
discharge for instance, the discharge current is composed by two components: an alternative streamer current due to the streamer propagation and a continuous unipolar (or synchronous) current due to the ion drift that are collected by the grounded plate. Ignition of both positive and negative coronas depends on the value of the maximum electric field in the vicinity of the tip where the discharge initiates. The critical value is traditionally given by the empirical Peek's formula. Then ionization of charged particles takes place at the vicinity of the high voltage electrode, in the active corona zone called ionization zone (see [30] for more details). Several authors gave an expression of the resulting electric field distribution from the high voltage electrode. In the case of a pin-to-plate configuration, electric field E versus x (x = 0 corresponding to the active electrode wall) can be approximated by [31]:

$$E(x) = \frac{V}{(x+r) \times \ln \left( \frac{2d+r}{r} \right)} \quad (1)$$

with d the gap between both electrodes and V the applied high voltage. In the case of a wire-to-wire configuration, Lacoste et al. used [32]:

$$E(x) = E_c \frac{r}{x} \quad (2)$$

However, whatever the electric field distribution, the current-voltage characteristic may be simplified by [33]:

$$I = C \times V(V - V_0) \quad (3)$$

where I is the total discharge current, V₀ the onset voltage and C is a constant depending on the electrode geometry and ion mobility μ. Equation (3) highlights that the discharge current increases in a quadratic manner with the applied high voltage. The electrical power consumed by the discharge P is:

$$P = V \times I = C \times V^2 \times (V - V_0) \quad (4)$$

## 2.2. Electrohydrodynamic force
**Electrohydrodynamics** consists in the link between fluid mechanics and electrical phenomena. The EHD force per unit volume (in N/m³) acting on the charged particles inside the discharge volume is given by:

$$f_{EHD} = \rho \times E \qquad (5)$$

where $\rho$ is the space charge density in C/m³. Since the force per unit volume is a force per area per distance, perpendicular to the area in question, it can be considered as a pressure gradient:

$$f_{EHD} = \nabla p = \rho \times E \qquad (6)$$

where $p$ is the pressure in Pa (N/m²).

The ion velocity $v_i$ is the sum of the velocity due to the ion migration and the convective term:

$$v_i = v + \mu E \qquad (7)$$

with $v$ the free gas velocity and $\mu$ the ion mobility. Assuming that the current is due to motion of the space charge between both electrodes, the current density may be expressed as:

$$j = \rho \times v_i = \rho \times (v + \mu E) \qquad (8)$$

If $v \ll \mu E$ (typical values of $\mu E$ equal to at least 100 m/s), one can deduce:

$$E = \frac{j}{\rho \times \mu} \qquad (9)$$

From eqs. (5) and (9), one can give a simplified expression of the electrohydrodynamic force per unit volume:

$$f_{EHD} = \frac{j}{\mu} \qquad (10)$$

By integration of eq. (10) on the discharge volume, the total electrodynamic force $F_{EHD}$ in Newton can be deduced:
$$F_{EHD} = \int_V f_{EHD} \, dV = \frac{j \times A_G \times d}{\mu} = \frac{I \times d}{\mu} \quad (11)$$

where $A_G$ is the cross-section area of the corona discharge. Eq. (11) highlights that $F_{EHD}$ increases with the current $I$ and the gap $d$. Moreover, using equation (3), one can write:

$$F_{EHD} = \frac{C \times V(V - V_0) \times d}{\mu} \quad (12)$$

showing that the EHD force increases in a quadratic manner with the applied high voltage. Therefore, by considering the maximum current density, as given by Pekker and Young [21]:

$$j_{\max} = \frac{9\varepsilon_0 \mu V^2}{8d^3} \quad (13)$$

and equation (11), one can deduce the maximum EHD force:

$$F_{EHDmax} = \frac{9\mu\varepsilon_0 V^2}{8d^3} A_G \frac{d}{\mu} = \frac{9}{8} A_G \varepsilon_0 \times E_{mean}^2 \quad (14)$$

If one considers a maximum mean electric field of 10 kV/cm, a wire-to-cylinder design of one meter in spanwise and a corona discharge width of 1 cm, then $F_{EHDmax} \approx 100$ mN/m.

## 2.3. Electric wind velocity

Inside the corona discharge, the mean velocity of the electric wind mean velocity $v_G$ and pressure $p$ are linked by:

$$p = \frac{1}{2} \rho_G v_G^2 \quad (15)$$

with $\rho_G$ the gas density. From eq. (6), and considering a weak contribution of the transverse pressure gradient:

$$f_{EHD} = \frac{dp}{dx} \quad (16)$$
Then with eqs. (16) and (10), one deduces:

$$p = \int_{0}^{d} \frac{j}{\mu} dx = \frac{j \times d}{\mu} = \frac{I \times d}{\mu \times A_{G}} \quad (17)$$

Finally, from eqs. (15) and (17):

$$v_{G} = \sqrt{\frac{2d \times I}{\mu \rho_{G} \times A_{G}}} \quad (18)$$

This expression shows that the electric wind velocity increases with the square root of both the electrode gap and the discharge current, and when the gas density decreases. If $A_G$ and $\rho_G$ are constant, then:

$$v_{G} = D \sqrt{\frac{d \times I}{\mu}} \quad (19)$$

where D is a constant.

## 2.4. Electromechanical efficiency

Electromechanical efficiency is given by:

$$\eta = \frac{P_{mec}}{P} \quad (20)$$

where $P_{mec}$ is the produced mechanical power. From a theoretical point of view, Sigmond and Lagstad [5] established a model where they considered that:

$$P_{mec} = \frac{1}{2} \rho_{G} \times A_{G} \times v_{G}^{3} = \frac{1}{2} \rho_{G} \times Q_{G} \times v_{G}^{2} \quad (21)$$

And the total EHD force was expressed by:

$$F = \rho_{G} \times Q_{G} \times v_{G} = \rho_{G} \times \frac{Q_{G}^{2}}{A_{G}} \quad (22)$$

with $Q_G$ the flow rate. Electrical power P is given by:
$$P = V \times I = \int_{0}^{d} EI \, dx = EI \times d = \frac{v_i}{\mu} \times Id \quad (23)$$

Taking into account eqs. (11) and (23), one can deduce:

$$I \times d / \mu = \rho_G \times Q_G \times v_G \quad (24)$$

Thus:

$$\eta = \frac{P_{\text{mec}}}{P} = v_G / 2v_i = \frac{v_G}{2\mu \times E} \quad (25)$$

Because $v_i \approx 200$ m/s (depending on the electric field distribution) and $v_G$ a few m/s, then efficiency is equal to a few percent. In practice, to compute more accurate values of efficiency, one must take into account the velocity distribution [7, 14].

## 2.5. Thrust effectiveness

In the case of EHD propulsion, Christenson and Moller [19] defined the thrust effectiveness that is given by:

$$\theta = \frac{F_{\text{EHD}}}{P} \quad (26)$$

In using eqs. (4) and (11):

$$\theta = \frac{I \times d}{\mu} \times \frac{1}{V \times I} = \frac{d}{\mu V} \quad (27)$$

From eq. (12) and considering that $V_0 \ll V$, one can deduce that:

$$V \approx \sqrt{\frac{\mu \times F_{\text{EHD}}}{C \times d}} \quad (28)$$

From the two last equations, one finds:
$$ \theta \approx \frac{d}{\mu} \sqrt{\frac{C \times d}{\mu \times F_{EHD}}} = \sqrt{C} \times \left(\frac{d}{\mu}\right)^{3/2} \times \frac{1}{\sqrt{F_{EHD}}} \quad (29) $$

This expression highlights that effectiveness increases with *d* and decreases according to the square root of $F_{EHD}$. Moreover, eq. (27) shows that effectiveness decreases when ion mobility increases.

## 2.6. Thrust and EHD force

In fact, thrust does not correspond to EHD force, because thrust is equal to EHD force minored by losses caused by viscous friction. In the case of a wire-to-cylinder configuration, the measured thrust *T* can be expressed by:

$$ T = F_{EHD} - F_{D} \quad (30) $$

where $F_D$ is drag force due to the electric wind that flows around the cylinder electrode. In the case of an uniform velocity upstream the cylinder, $F_D$ is given by:

$$ F_D = \frac{1}{2} \rho_G S C_x v_G^2 \quad (31) $$

with *S* the cross-sectional area of the cylinder (diameter multiplied by span wise) and $C_x$ the drag coefficient ($C_x \approx 1$ for a cylinder with a circular cross-section). In the case of a corona established between a wire and a cylinder, the drag force is not so easy to compute because the incoming flow upstream the cylinder is not uniform, but eq. (31) could be used as a first approximation. This point highlights that the drag force should be limited in order to maximize thrust. This point will be discussed later in the paper.

## 3. Experimental setup

The corona discharge is generated between a high voltage 25-µm diameter wire in tungsten and a grounded collecting electrode (cylinder with a circular cross-section, in copper or stainless steel, see
Figure 1). The investigated geometrical parameters are the grounded electrode diameter $φ_G$ (between 0.8 and 22 mm), the electrode gap d (10 mm ≤ d ≤ 40 mm). Both others parameters that are investigated here are the high voltage polarity and the air relative humidity. Finally, in the last experiment, the high voltage wire electrode will be replaced by a set of needles spaced by 20 mm.

As illustrated by Figure 1b (side view), both electrodes are held by two horizontal pieces in Teflon (numbered as (1) in Figure 1). These pieces are screwed to two vertical supports in PVC (2). The gap d can be adjusted in modifying the height of the pieces (1). Both vertical arms are fixed on a PMMA plateau (3). A piece in Teflon is placed between the PMMA plateau and the balance plateau (4) in order to isolate electrically the balance and to avoid any leakage current. Moreover, the balance is placed inside a Faraday cage (5) to cancel electromagnetic perturbations.

The high voltage is applied with the help of both negative and positive dc high voltage power supplies (DEL, ± 40 kV, 3.75 mA, 150 W). The voltage is adjusted manually with a resolution of 0.1 kV. The current is measured between the cylinder electrode and earth with a ammeter (precision ± 1 µA). The electrical connection of both electrodes is realized with a very thin wire (25 µm in diameter and 7 cm in length) in order to avoid any mechanical strength that would disturb the balance measurements. With this system, there is no mechanical strength and leakage current is negligible.

The produced thrust is measured with a laboratory balance (Denver instrument, SI-603, precision of ± 0.02 mN, bandwidth of 60 Hz) controlled by a Labview interface software. For each geometrical configuration, experiments consist of measuring the horizontal thrust for different high voltage values, for negative and positive polarities. It has been verified that the location of the vertical PVC arms compared to the balance plateau has no effect on the measure. The force has not to be produced at the center of the balance plateau, the balance measuring only the vertical component of the force. Furthermore, the experimental procedure is as follows: (i) the high voltage is switched off during 20 seconds and the balance is calibrated to zero, (ii) the discharge is switched on during 20 seconds and the mean thrust is computed by time-averaging the measurements of the last 15 seconds, at a frequency of 100 Hz, (iii) the
discharge is switched off during 20 seconds, the voltage value is increased and the balance is calibrated,
(iii) a new thrust measurement is performed as in (ii). To verify the validity of the measurements, the
same procedure is conducted while the applied high voltage is decreased. This procedure must be
accurately followed to ensure a precise and reproducible measurement. With such a procedure, a precision
of about ± 1 mN/m is reached.

<figure><img src="image_2.png" /><figcaption>Figure 1. Sketch of the experimental setup: (a) front view, (b) side view.</figcaption></figure>

The vertical component of the electric wind velocity is determined by total pressure measurements
with the help of a glass home-made pressure probe (see (6) in Figure 2b, internal diameter of 0.5 mm) that
can be displaced vertically and horizontally. Unfortunately, no measure can be conducted between both
electrodes because the presence of the glass tube disturbs the corona discharge and the velocity measure is not valid. This glass capillary is connected to a pressure sensor (Sensortechnics, HCLA-02X5DU series, full scale 2.5 mbar, precision 0.05% FS) via a 5-cm long flexible tube in Tygon. The velocity is deduced with the Bernoulli equation:

$$v_G = \sqrt{\frac{2 \times p}{\rho_G}} \qquad (32)$$

where p is the dynamic pressure, corresponding to the total pressure minus the static pressure [14].

All the experiments are performed at room temperature (≈ 20 °C) and a relative humidity values ranging from 48 to 62%. The effect of relative humidity will be discussed further.

# 4. Results and discussions

## 4.1. Baseline case

The baseline case is as follows: gap d = 30 mm, grounded electrode diameter φG = 12 mm, relative humidity RH = 62 %. The wire electrode diameter is equal to 25 µm.

Figure 2 shows the discharge current per unit length in span wise (discharge current divided by the electrode span wise, in µA per meter) versus applied voltage in kV, for negative and positive coronas. By abuse of language, this plot will be called the I-V characteristic. Square and circular symbols represent experimental measurements when both solid lines correspond to the data interpolation by using equation (3). First, one can see that the negative corona induces a higher current than the positive one, at the same high voltage value. This may be partly explained by the fact that the mobility of negative ions is sometimes slightly higher than the one of positive ions, depending on the nature of ions that are produced [34, 35]. However this reason cannot be the only one, because the physico-chemical phenomena occurring inside an atmospheric corona discharge are very complex. For instance, these phenomena results in streamers in positive coronas and Trichel pulses in negative ones. Secondly, one can observe that both theoretical curves correctly interpolate the experimental measurements. From the theoretical
expression of these both curves, one can deduce the value of the constant C in equation (3). In the case of the positive corona, one finds $C^+ = 2,27 \ \mu A/kV^2.m$ (corresponding to $0.34 \times 10^{-12} \ A/V^2$). For the negative corona, $C^- = 2,54 \ \mu A/kV^2.m$ (i.e. $0.38 \times 10^{-12} \ A/V^2$). These both values are in agreement with Zouzou *et al.* [34] who found $C \approx 0.2 \times 10^{-12} \ A/V^2$ in the case of a point-to-plate corona discharge with a comparable gap equal to 30 mm.

<figure><img src="image_3.png" /><figcaption>Figure 2. I-V characteristic: discharge current (per unit length) versus applied voltage.</figcaption></figure>

Figure 3 presents the measured thrust $T$ per unit span versus length (in mN/m) as a function of three electrical input electrical parameters: the discharge current $I$ (T-I characteristic, Figure 3a), the applied voltage $V$ (T-V characteristic, Figure 3b) and the electrical power consumption $P$ (T-P characteristic, Figure 4c).

Figure 3a highlights that the measured thrust is proportional to the discharge current, as predicted by equation (11) that indicates that $F_{EHD} \propto I$. Moreover, one can see that the corona discharge induces a similar thrust value whatever the sign of the applied high voltage. By fitting of the measurement points with equation (11), one can deduce the ion mobility. In Figure 3a, the slope of both negative and positive curves is equal to about 0.1 mN/µA i.e. 100 N/A, resulting in $\mu^+ \approx \mu^- \approx 3 \times 10^{-4} \ m^2/Vs$. These values are in agreement with literature where ion mobility usually ranges from about $1.9$ to $3.5 \times 10^{-4} \ m^2/Vs$ [35].
depending on the electric field and the ion considered. For instance, Ryzko [36] found experimentally
μ = 2.9×10⁻⁴ m²/Vs. Moreover, as it is highlighted by equation (30), the measured thrust corresponds to
the electrohydrodynamic force minus the cylinder drag. It will be shown later in the paper that the drag
force can be estimated to about 10% of the total measured thrust. Therefore if the electrohydrodynamic
force F_EHD is greater than the measured thrust T by about 10%, the ion mobility can be more accurately
estimated to 2.8×10⁻⁴ m²/Vs, corresponding to the values used by Bérard for its numerical simulations
[35].

Figure 3(b) presents the thrust versus applied voltage (T-V characteristic), for both negative and positive polarities. As illustrated by both solid curves, they can be fitted by equation (12), showing one more time that the experimental measurements behave as expected by theory. Figure 3(c) shows that the curves of the thrust versus electrical power for both polarities are superposed, highlighting that the input power is the main parameter in the electromechanical conversion.

<figure><img src="image_4.png" /></figure>
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Figure 3. Thrust versus current (T-I characteristic, a), applied voltage T-V characteristic, b) and electrical power consumption (T-P characteristic, c).</figcaption></figure>

Finally, Figure 4 present the thrust effectiveness θ as a function of the applied voltage (θ-V characteristic, Figure 4a) and the produced thrust (θ-T characteristic, Figure 4b). As illustrated by Figure 4a, the effectiveness, ranging from about 5 to 10 mN/W, decreases with the voltage. The interpolation of experimental measurements with equation (27) shows that d/μ⁺ = 107, resulting in value
of µ⁺ equal to about 2.9×10⁻⁴ m²/Vs. This value is in agreement with the one obtained with the interpolation of the T-I characteristic by equation (11).

Figure 4(b) shows that effectiveness θ decreases when the produced thrust increases, as previously found by Wilson *et al.* [22]. On one hand, one can see that equation (29) allows us to interpolate correctly the experimental points (solid line), highlighting that θ can in fact be expressed by $θ = α/\sqrt{F_{EHD}}$ with θ in mN/W, $F_{EHD}$ in N/m and α a constant equal to 37 in this case. On the other hand, considering that $α = \sqrt{C} \times (d/μ)^{3/2}$, one can compute it with the values of C and μ determined previously by interpolation of both I-V and T-I characteristics ($C^+ = 0.34 \times 10^{-12} A/V^2$ and $μ^+ = 3 \times 10^{-4} m^2/Vs$). In this case, $α = 53$, and the resulting plot is represented by the dash curve in Figure 4b. This result demonstrates that equation (29) gives a correct behavior of the θ-T characteristic. However, the θ-T characteristic cannot be numerically solved and predicted with this equation from C and μ values, because the theoretical computation over-estimates the effectiveness. This is certainly due to the fact that equation (29) is obtained with the hypothesis that $V_0 \ll V$, an hypothesis that is not practically verified.

To summarize the results concerning this baseline case, one can assume that the theory developed in the previous section gives a good tendency of the electrical and mechanical behaviors of the EHD device. Now, in the following sections, the influence of a few parameters is investigated.

<figure><img src="image_7.png" /><figcaption>(a)</figcaption></figure>
<figure><img src="image_8.png" /></figure>

Figure 4. Effectiveness versus applied voltage (θ-V characteristic, a) and effectiveness versus thrust (θ-T characteristic, b).

## 4.2. Effect of humidity

All the experiments of the present work have been conducted with a relative humidity (RH) ranging from 48% to 62%. In order to observe the influence of this environmental parameter on the corona discharge behaviour and the performances of the EHD thruster, three characteristics have been plotted in Figure 5 for RH equal to 48, 54 and 62 %: current versus voltage (I-V characteristic), thrust versus current (T-I characteristic) and effectiveness versus thrust (θ-T characteristic). Only the case of a positive corona is presented but similar results have been observed with a negative discharge. For instance, the θ–T characteristics were perfectly superposed for both polarities.

As it can be observed, the value of RH in the range 48-62% does not play a key role in the corona discharge characteristics. In Figure 5a, one can see that the current increases slightly with the relative humidity. This result is surprising because the ion mobility is usually enhanced when RH increases. Moreover, the corona discharge is more stable when RH is higher. When RH decreases below 50 %, the discharge becomes unstable, it induces a whisper and its current varies with time. Figure 5b highlights that the slope of the T-I characteristic is equal to 100 N/A whatever the value of RH, showing that the current is the main parameter in the thrust production. Finally, Figure 5c shows that effectiveness θ is not
significantly affected by the value of the air relative humidity. It seems that it is slightly higher when RH decreases, but the maximum variance of θ is limited to 2 mN/W for thrust smaller than 50 mN/m. Therefore we can consider that the total measure error of θ, including the humidity variations, can be estimated at about ±1 mN/W.

<figure><img src="image_9.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b)</figcaption></figure>
<figure><img src="image_11.png" /></figure>

Figure 5. Current versus voltage (I-V, a), thrust versus current (T-I, b) and effectiveness versus thrust (Θ-T, c).

### 4.3. Effect of electrode gap

The objective is here to investigate the effect of the electrode gap *d* on the different characteristics of the EHD device. For these experiments, gaps ranging from 10 mm to 40 mm have been considered, with relative humidity between 54% and 60%.

Figure 6 presents the I-V curve for five different gaps, in the case of a positive corona discharge. As expected, the discharge current decreases when the electrode gap is increased. This means that the smallest the gap is, the smallest the electrical power consumption *is* in order to produce the same discharge current. In other words, one can say that an increase of the electrode gap results in an increase of the power consumption. Moreover, one can see one more time that equation (3) interpolates perfectly the experimental data.

Table I indicates the values of C and $V_0$ for the different gaps (determined by interpolation of I-V experimental curves by eq. (31)). From interpolation of the plot of C versus *d* (not shown here), we demonstrated that $C = \alpha / d^2$ with $\alpha = 2.9 \times 10^{-16}$, showing that C varies according to $1 / d^2$.
Figure 7 presents the T-I characteristics (thrust versus current). First, one can observe that all the curves can be correctly interpolated by equation (11), the thrust remaining proportional to the discharge current, whatever the gap. However, for a gap equal to 30 and 40 mm, experimental measurements are slightly higher than the computed values for low currents. Secondly, it is evident that the produced force increases with the gap. More especially, as expected by equation (11), the EHD force should increase proportionally with the electrode gap d. To verify this theoretical assumption, the value of the ion mobility $\mu^+$ is computed for each gap from the slope of the fitted curves, the slope being equal to $d/\mu^+$. If the thrust is proportional to the gap, the value of $\mu^+$ should remain constant. As it is indicated in Table II, we can consider that $\mu^+$ remains nearly constant, with the highest value for d = 10 mm because a higher electric field results in a higher ion mobility (see [35, 36] for more details).

Moreover, although both high voltage polarities induced the same thrust when d is equal to 30 mm (see Figure 3), the positive corona discharge is more effective than the negative one for gaps smaller than 30 mm (plots are not shown here). The highest difference between both polarities has been found for d = 10 mm. When the gap increases, the difference between both polarities decreases until it becomes equal to zero for d = 30 mm. The only reason that can explain this phenomenon is that the ion mobility is almost equal for both polarities for high gaps whereas the mobility of negative ions is higher than the mobility of positive ions when the gap decreases above 30 mm. From the T-I characteristics obtained with a negative voltage for the different gaps, the values of $\mu^-$ have been computed (Table II). Our assumption is well confirmed: the mobility of negative ions is higher when the gap decreases.

Figure 8 presents the T-P and θ-T characteristics. Both characteristics show clearly that an increase of the electrode gap results in a better electromechanical conversion and then a better effectiveness. From a theoretical point of view, equation (29) indicates that $\theta = \beta / \sqrt{F_{EHD}}$ with $\beta = \sqrt{C} \times (d / \mu)^{3/2}$. Because $C \propto 1 / d^2$, we can deduce that $\beta \propto d^n$ with $n = 0.5$, highlighting that θ increases with $\sqrt{d}$. From an experimental point of view, if we observe the behaviour of θ versus d at a fixed thrust (T = 20 mN/m for instance), the measured values shows that $\theta = \kappa + \lambda \sqrt{d}$ with $\kappa = -1.9 \times 10^{-3}$ and $\lambda = 57 \times 10^{-3}$. On one hand,
according with theory, we can say that we have experimentally verified that $\theta \propto \sqrt{d}$. This gives the behaviour of $\theta$. On the other hand, the use of equation (22) does not allow us to predict and to compute numerically the value of $\theta$ as a function of thrust $T$.

<figure><img src="image_12.png" /><figcaption>Figure 6. Current versus voltage (I-V characteristic) for different gaps $d$.</figcaption></figure>

<table><caption>Table I. Value of $C$ and $V_0$ for different gaps $d$.</caption><thead><tr><td>$d$ (mm)</td><td>$V_0$ (kV)</td><td>$C$ ($\times 10^{-12}$ A/V<sup>2</sup>)</td></tr></thead><tbody><tr><td>10</td><td>4.8</td><td>2.93</td></tr><tr><td>15</td><td>5.4</td><td>1.14</td></tr><tr><td>20</td><td>6.1</td><td>0.63</td></tr><tr><td>30</td><td>7.5</td><td>0.34</td></tr><tr><td>40</td><td>8.8</td><td>0.18</td></tr></tbody></table>
<figure><img src="image_13.png" /><figcaption>Figure 7. Thrust versus current (T-I characteristic) for different gaps d.</figcaption></figure>

<table><caption>Table II. Value of μ for different gaps d and both polarities.</caption><thead><tr><th>d (mm)</th><th>μ<sup>+</sup> (10<sup>-4</sup> m<sup>2</sup>/Vs)</th><th>μ<sup>-</sup> (10<sup>-4</sup> m<sup>2</sup>/Vs)</th></tr></thead><tbody><tr><td>10</td><td>3.2</td><td>4.6</td></tr><tr><td>15</td><td>2.9</td><td>3.6</td></tr><tr><td>20</td><td>2.7</td><td>3.1</td></tr><tr><td>30</td><td>2.9</td><td>2.9</td></tr><tr><td>40</td><td>2.9</td><td>2.9</td></tr></tbody></table>
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>Figure 8. Thrust versus power (T-P characteristic, a) and effectiveness versus thrust (Θ-T characteristic, b) for different gaps d.</figcaption></figure>

### 4.4. Effect of grounded electrode diameter

The choice of the diameter of the grounded electrode is not easy. On one hand, from an electrical point of view, this diameter $φ_G$ should be as high as possible. In this case, the corona discharge behaves like a wire-to-plate discharge. Moreover, this increases the surface that collects ions, resulting in a wider
discharge. Its cross-section $A_G$ being enlarged, one can assume that the discharge current is enhanced. On the other hand, from an aerodynamic point of view, $\phi_G$ must be small in order to reduce the drag force $F_D$ acting on the cylinder, which is proportional to $\phi_G$, as indicated by equation (31). In fact, we have to find a compromise between these both antagonistic phenomena.

For that, we investigated the performances of EHD devices that use a cylinder with diameters ranging from 0.8 mm to 22 mm as grounded electrode. The others parameters are as follows: $d = 20$ mm, $HR = 61 \%$, positive high voltage.

Figure 9 presents the I-V characteristics for the different electrode diameters. As expected, the current collected by the grounded electrode increases with its diameter. In fact, one assumes that the cross-section $A_G$ of the corona discharge and then the current I increase when $\phi_G$ increases. Analysis of the C values as a function of $\phi_G$ shows that $C = f(\phi_G)$ can be well interpolated by the empirical expression $C = \alpha + \beta\sqrt{\phi_G}$ with $\alpha = 2.9 \times 10^{-13}$ and $\beta = 3.4 \times 10^{-12}$, showing that C increases with $\sqrt{\phi_G}$.

Figure 10, that represents the T-I characteristic, highlights two groups: for $\phi_G = 0.8$, 16 and 22 mm, the slope of the curves is equal to about 57 A/N when it is equal to 60 N/A for $\phi_G = 3$, 8 and 12 mm. On one hand, this shows clearly that the optimum diameter is between 3 and 12 mm. Indeed, with $\phi_G = 0.8$ mm, the dissymmetry between both electrodes is not sufficient to establish a stable discharge and the surface that collects ions is too small. In addition, the electric field at the grounded electrode is certainly high enough to ionize the air, producing a counter-electric wind, from the grounded electrode to the high voltage one. With $\phi_G \ge 16$ mm, the cylinder diameter is high and it induces a drag force $F_D$ which is not negligible compared to the EHD force, resulting in a thrust reduction. On the other hand, one can say that the grounded electrode diameter does not play a key role in the current-to-thrust conversion, compared to the electrode gap for instance.
Finally, Figure 11 gives the most relevant information. It shows clearly that the best diameters are $\phi_G = 8$ and 12 mm. The use of a 3-mm diameter electrode needs a too high voltage that results in a decrease of effectiveness.

<figure><img src="image_16.png" /><figcaption>Figure 9. Current versus voltage (I-V characteristic) for different grounded electrode diameter $\phi_G$.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 10. Thrust versus current (T-I characteristic) for different grounded electrode diameter $\phi_G$.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 11. Effectiveness versus thrust (θ-T characteristic) for different grounded electrode diameter φ<sub>G</sub>.</figcaption></figure>

4.5. *Effect of the active shape*

Corona discharges are usually established between a thin active electrode (tip or wire) and a grounded
plate electrode. The advantage of using a multi-tip electrode instead of a wire is that *one* can better
control the ionization zone, the points from which the corona starts and the electric field can be locally
reinforced. However, this type of electrode induces a three-dimension flow.

In the present work, a set of eight needles, fixed on a vertical plate, replaces the wire used as active
high voltage electrode. The needles are spaced by 20 mm in the span wise direction. Experiments are
conducted for HR = 50 %. Negative and positive high voltages are used, for gaps equal to 20, 30 and
40 mm. In this paper, the results obtained with the eight-needle electrode for both positive and negative
polarities are presented. Because the negative corona discharge is more effective, these results are
compared to the ones obtained with a 25-µm wire supplied by a negative voltage.

Figures 12, 13 and 14 present the I-V, the T-I and the θ-T characteristics, respectively. The electrode gap d is equal to 30 mm. Similar results have been observed for d = 20 and 40 mm. Solid symbols corresponds to the values measured with the eight-needle electrode design (interpolations are represented
by solid lines in Figures 12 and 13) when stars symbols show the quantities obtained with the 25-µm wire
design.

First, let us compare the negative and positive discharges created with the eight-needle electrode. As illustrated by Figure 13, for a same current value, the negative and the positive discharges produced the same thrust, showing one more time that the current-to-thrust conversion is similar whatever the high voltage sign. Moreover, because the negative current is higher than the positive one (Figure 12), the effectiveness is better in the case of a negative corona discharge (Figure 14). This result completely differs compared to the one obtained in the same conditions with a 25-µm wire, that previously shown that both polarities resulted in the same effectiveness (see Figure 4). This may be explained by the fact that the difference between both positive and the negative currents is higher with the eight-needle electrode. Therefore the positive discharge needs a higher voltage, resulting in a power consumption increase.

The comparison between the eight-needle electrode and the 25-µm wire is interesting. First, it highlights that the use of needles results in a smaller discharge current, compared to the one induced with the wire based design (Figure 12). Secondly, the current-to-thrust conversion is slightly higher with the wire. From interpolation of T-I curves of Figure 13, one can deduce the ion mobility values of the discharge occurring with the needles. The slopes of both positive and negative curves are equal to 91.2 N/A and 88.5 N/A, respectively, resulting in $\mu^+ = 3.3 \times 10^{-4} m^2/Vs$ and $\mu^+ = 3.4 \times 10^{-4} m^2/Vs$. These values are a little higher than the ones obtained with the wire electrode. Finally, Figure 14 shows that the $\theta$-T characteristics are rather similar.

In conclusion, one can say that the use of needles instead of a thin wire did not modify significantly the EHD thruster performances. The most relevant parameter concerning the high voltage electrode is certainly its curvature radius that must be as small as possible. In the present case, the radius of the needle tip was in the same order of magnitude than to the one of the wire (25 µm).
<figure><img src="image_19.png" /><figcaption>Figure 12. Current versus voltage (I-V characteristic) for needles-to-cylinder and wire-to-cylinder corona discharges.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 13. Thrust versus current (T-I characteristic) for needles-to-cylinder and wire-to-cylinder corona discharges.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Figure 14. Effectiveness versus thrust (θ-T characteristic) for needles-to-cylinder and wire-to-cylinder corona discharges.</figcaption></figure>

4.6. *Electric wind velocity measurements*

In this last part, the velocity of the flow produced by the corona discharge is measured. It would have
been very interesting and relevant to characterize the flow velocity between both electrodes and
downstream the cylinder grounded electrode. Unfortunately, it was not possible for several reasons. First,
no measurement was able in the electrode gap, where the discharge occurs, because the glass pressure
probe modifies the discharge dynamic and because the discharge itself disturbs the pressure sensor.
Therefore, the measurements will not be able to be compared to equation (19) that gives the electric wind
velocity versus the electrode gap *d* and the discharge current *I*. Secondly, the use of a glass tube as a total
pressure probe does not allow us to measure negative velocity. Then it was not possible to measure the
flow velocity downstream the cylinder and in its wake. In absence of velocity profiles upstream and
downstream the cylinder, it is then not possible to compute accurately the cylinder drag *F*<sub>D</sub> by integral
methods.
Therefore, we measured profiles of the vertical velocity on the side of the grounded electrode, as shown in Figure 15, only in the case of a 25-µm wire active electrode. These measurements keep relevant because they give a tendency on the electric wind occurring in the electrode gap.

First, the effect of the voltage and current values is investigated. The others parameters are as follows: d = 20 mm, φ<sub>G</sub> = 12 mm and HR = 50 %. Figure 16 presents velocity profiles for four different voltage values. It highlights that the maximum produced velocity is typically equal to a few m/s, and that the thickness of the zone where there is a flow is equal to 12-13 mm. Moreover, the electric wind velocity, and then the produced electrohydrodynamic force, increase with the voltage.

In literature, some authors ([37] for instance) used velocity profiles in order to compute the force produced by the flow in using:

$$F = \rho_G \int_0^\infty v_G(y) dy \qquad (32)$$

From the velocity profiles presented in Figure 16, and using equation (32) multiplied by two because the same velocity profile is generated on the other side of the cylinder electrode, the produced force has been computed. All the results are presented in Table III, where are indicated the high voltage and current values, the maximum electric wind velocity, the thrust measured experimentally with the balance and the force computed from velocity profile measurements.

Two main remarks may be done. First, the plot of the maximum velocity v<sub>G</sub> versus current I (not presented here) shows that v<sub>G</sub> ∝ √I as indicated by equation (19) and experimentally verified by many authors ([6, 7, 9] for instance). The second remark concerns the computation of the thrust from velocity profiles. As previously observed in [37], it is clear that this method widely over-estimates the value of the force, by about 70 % in our case. This result was expected because it is not possible to compute as easily the force from a single velocity profile. Indeed, one single velocity profile, which is a local information along the x axis, cannot allow us to determine the force, that acts on the whole volume. One proof is that the computed force depends on the x location of the velocity profile. In conclusion, one can say that
equation (32) does not enable to compute the thrust produced by volume discharge from velocity profiles.

More complex integral methods developed very recently [38-41] are needed.

As indicated previously, it is not possible to compute the grounded electrode drag because velocity profiles upstream and downstream the cylinder are required. However, a rough estimation may be reached in considering a uniform flow in the electrode gap. For instance, in using equation (31), one finds $F_D \approx 0.0072 \times v_G^2$. For velocity between 1 and 2 m/s, then $F_D$ ranges from 7 mN to 29 mN. This value is over-estimated because it considers a uniform velocity profile upstream the cylinder while it is not the case. However these values show that the drag force is not negligible compared to the measured thrust, but the electrohydrodynamic force is higher than the measured thrust.

Figure 17 presents velocity profiles with two different cylinder electrodes, for $\phi_G = 3$ and 12 mm. Surprisingly both velocity profiles are superposed. Indeed, we could expect that the discharge using the smallest electrode diameter results in a thinner profile, with a higher maximum velocity. It is not the case. However, this does not prove that the velocity profiles upstream the cylinder (in the electrode gap) are similar.

Finally, Figure 18 shows velocity profiles for three different gaps ($d = 10, 20$ and $30$ mm). On one hand, one can observe that the profile thickens when the gap increases. This is due to the fact that the smallest the gap is, the thinner the region in which the discharge occurs is. On the other hand, one can see that the electric wind velocity does not increase with the square root of the electrode gap, such as predicted by equation (19). This is not surprising because, to our knowledge, nobody experimentally verified this tendency.
<figure><img src="image_22.png" /><figcaption>Figure 15. Sketch of the velocity profile measurements (not to scale).</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 16. Velocity profiles for different voltage values.</figcaption></figure>
<table><caption>Table III. Thrust measured directly with the help of a balance (T measured) and thrust computed from experimental velocity profiles (T computed) versus high voltage V and current I.</caption><tr><td>Voltage (kV)</td><td>12</td><td>13</td><td>14</td><td>17</td></tr><tr><td>Current (μA/m)</td><td>290</td><td>365</td><td>445</td><td>765</td></tr><tr><td>Wind velocity (m/s)</td><td>1.6</td><td>1.75</td><td>1.85</td><td>2.35</td></tr><tr><td>T measured (mN/m)</td><td>18.3</td><td>22.9</td><td>29</td><td>44.5</td></tr><tr><td>T computed (mN/m)</td><td>32</td><td>39</td><td>49</td><td>78</td></tr></table>

<figure><img src="image_24.png" /><figcaption>Figure 17. Velocity profiles for two grounded electrode diameter $\phi_G$.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Figure 18. Velocity profiles for three gap values.</figcaption></figure>

**5. Conclusion**

Wire-to-cylinder corona discharges have been investigated in order to optimize and better understand the electrohydrodynamic (EHD) phenomena that govern the performances of electrostatic thrusters. After a presentation of the EHD theory, a parametric study consisting mainly in direct thrust measurements with a balance has been conducted. Experimental results have been compared to theory and used to determine several quantities, such as the ion mobility for instance.

First, the expression of the I-V characteristic has been improved. Indeed, the experimental results highlighted that the discharge current changes according to $\sqrt{\phi_G}$ and $1/d^2$ where $\phi_G$ is the cylinder diameter and d the electrode gap.

Secondly, the values of ions mobility have been determined from T-I characteristics and equation (11). On one hand, they showed that mobility of negative ions was higher than the one of positive ions when the electrode gap was small ($d \le 20$ mm). On the other hand, the mobility of both ions is equal for higher gaps. Therefore, positive and negative coronas results in the same effectiveness, excepted for small gaps ($d \le 20$ mm).
The curve that presents the produced thrust versus discharge current (T-I characteristic), that can be perfectly interpolated by the theoretical expression, shows that the current-to-thrust conversion is equal to 33 N/A per centimeter of gap d, and it is proportional to this gap. Therefore, the most relevant parameter to enhance the EHD thrust is the electrode gap. Experiments with high gaps will be conducted very soon.

The θ-T characteristic cannot be numerically solved from eq. (29) because the theoretical computation over-estimates the measured effectiveness. This is certainly due to the fact that the constant $\sqrt{C} \times (d/\mu)^{3/2}$ is over-estimated because eq. (29) is obtained with the hypothesis that $V_0 \ll V$, an hypothesis that is not practically verified. However, it has been demonstrated that θ changed according to $1/\sqrt{F_{EHD}}$, such as in eq. (29). Moreover, the experimental measurements with several electrode gaps allowed us to show that θ was proportional to $\sqrt{d}$, highlighting one more time that an increase of the electrode gap induced an enhancement of the thruster performances. This result does not agree with [19] where it was showed that θ was proportional to the electrode gap.

The maximum measured thrust was equal to about 100 mN/m. However, experiments have been conducted with voltages very smaller than the breakdown voltage where a spark can occur. Therefore, the produced thrust can be easily improved in increasing the gap, as indicated previously, and in increasing the applied high voltage.

The parametric study on the effect of the cylinder grounded electrode diameter showed that we had to find a compromise between both antagonistic phenomena (electric and aerodynamic). The compromise resulted in a diameter ranging between 8 and 12 mm.

The measurements of electric wind velocity indicated that thrust cannot be computed from a single velocity profiles, because this method resulted in a high over-estimation of the force value. More complex integral methods are needed.
Finally, in this paper, we did not reach an effectiveness θ = 20 N/kW, with a thrust per unit surface φ of 20 N/m². However the present paper gives the way to achieve these objectives: the electrode gap must be increased to enhance the effectiveness θ, and a multi-stage design must be used to increase φ.

References

[1] Robinson M 1961 A history of the electric wind 366-372

[2] Robinson M 1964 Movement of air in the electric wind of the corona discharge *AIEE Transactions* **80** 143-150

[3] Sigmond R S 1982 *J. Appl. Phys.* **53** 891

[4] M. Goldman, A. Goldman, and R. S. Sigmond 1985 The corona discharge, its properties and specific uses," *Pure & Appl. Chem.* **57**, 1353–1362

[5] Sigmond RS, Lagstadt IH 1993 Mass and species transport in corona discharges *High Temp. Chem. Processes* **2** 221-229

[6] Rickard M, Dunn-Rankin D, Weinberg F, Carleton F 2006 Maximizing ion-driven gas flows *J. of Electrostatics* **64** 368-376

[7] Moreau E, Touchard G 2008 Enhancing the Mechanical efficiency of electric wind in corona discharges *Journal of Electrostatics* **66** 39-44

[8] Tsubone H, Ueno J, Komeili B, Minami S, Harvel G D, Urashima K, Ching C Y and Chang J S 2008 *J. Electrostat.* **66** 115–21

[9] Kim C, Park D, Noh KC, Hwang J 2010 Velocity and energy conversion efficiency characteristics of ionic wind generator in a multistage configuration *Journal of Electrostatics* **68** 36-41

[10] Colas D F, Ferret A, Pai D Z, Lacoste D A and Laux C O 2010 Ion wind generation by a wire-cylinder-plate corona discharge in air at atmospheric pressure *Journal of Applied Physics* **108** 103306

[11] Go DB, Garimella SV, Fisher TS, Maturana RA 2008 Enhancement of external forced convection by ionic wind *International Journal of Heat and Mass Transfer* **51** 6047-6053.
[12] Huang RT, Sheu WJ, Wang CC 2009 Heat transfer enhancement by needle-arrayed electrodes – An EHD integrated cooling system *Energy Conversion and Management* 50 1789–1796

[13] Lakeh RB, Molki M 2012 Targeted heat transfer augmentation in circular tubes using a corona jet
*Journal of Electrostatics* 70 31-42

[14] Moreau E 2007 *J. Phys. D: Appl. Phys.* **40** 605–36

[15] Corke TC, Enloe CL, Wilkinson SP 2010 Dielectric barrier discharge plasma actuators for flow control *Annu. Rev. Fluid Mech.* 42

[16] Benard N, Moreau E 2012 EHD Force and Electric Wind Produced by Plasma Actuators Used for Airflow Control in *Proc. 6th AIAA Flow Control Conference, AIAA paper 2012-3136, New-Orleans, USA.*

[17] Townsend Brown T 1928 A method of and an apparatus or machine for producing force and motion *GB Patent 300311*

[18] Townsend Brown T 1960 Electrokinetic apparatus *US Patent 2958790*

[19] Christenson E A and Moller P S 1967 Ion-neutral propulsion in atmospheric media *AIAA J.* **5** 1768–73

[20] Young M, Keith S 2009 An overview of advanced concepts for near-space systems in *Proc. 45th AIAA, 2-5 august 2009, Denver, Colorado, paper 2009-4805*

[21] Pekker L, Young M 2009 A model of an ideal electrohydrodynamic thruster in *Proc. 45th AIAA, 2-5 august 2009, Denver, Colorado*

[22] Wilson J, Perkins HD, Thompson WK 2009 An investigation of ionic wind propulsion *NASA Report NASA/TM 2009-215822*

[23] Bondar H, Bastein F 1986 Effect of neutral fluid velocity on direct conversion from electric to fluid kinetic energy in an electro-fluid-dynamic device *J. Phys. D: Appl. Phys.* 19 1657-1663

[24] Pekker L, Young M 2011 Model of ideal electrohydrodynamic thruster *Journal of Propulsion and Power* **27** 786-792
[25] Zhao L and Adamiak K 2005 *J. Electrostat.* **63** 337-50

[26] Zhao L, Adamiak K 2008 Numerical simulation of the electrohydrodynamic flow in a single wire-plate electrostatic precipitator, *IEEE Transactions on Industry Applications* 44 683-691.

[27] Martins A 2012 Simulation of a wire-cylinder-plate positive corona discharge in nitrogen gas at atmospheric pressure *Physics of Plasmas* 19 063502

[28] Callebaut T, Kochetov I, Akishev Yu, Napartovich A, Leys C 2004 Numerical simulation and experimental study of the corona and glow regime of a negative pin-to-plate discharge in flowing ambient air *Plasma source Science and technology* **13** 245-250

[29] Matéo-Vélez JC, Degond P, Rogier F, Séraudie A, Thivet F 2008 Modelling wire-to-wire corona discharge action on aerodynamics and comparison with experiment *J. Phys. D: Appl. Phys.* **41** 035205

[30] Moreau E, Benard N, 2009 Plasmas for Flow Control in "Flow Control : Fundamentals, Advances and Applications", Edited by Von Kerman Institute, Bruxelles

[31] Hartmann G 1977 Spectrographie de la décharge couronne : étude des mécanismes de collisions dans le dard PhD thesis (University of Paris XI)

[32] Lacoste A, Pai D, Laux C 2004 Ion wind effect in a positive DC corona discharge in atmospheric pressure air in Proc. AIAA Meeting (Reno, USA, january 2004) paper #2004-0354

[33] Loeb LB 1965 *Electrical coronas, their basic physical mechanisms* University of California press

[34] Zouzou N, Moreau E, Touchard G 2006 Précipitation électrostatique dans une configuration pointe-plan *Journal of Electrostatics* **64** 537-542

[35] Bérard P 2008 Etude du vent ionique produit par décharge couronne à pression atmosphérique pour le contrôle aérodynamique *PhD Thesis Ecole Centrale Paris*

[36] Ryzko H 1965 Drift velocity of electrons and ions in dry and humid air and in water vapour *Proc. Phys. Soc.* 328-330
[37] Hoskinson AR, Hershkovitz N, Ashpis DE 2009, Comparison of force measurement methods for DBD plasma actuators in quiscent air in *Proc. 47th AIAA 5-8 January 2009, Orlando, USA*

[38] Debien A, Benard N, David L, Moreau E 2012 *Appl. Phys. Lett.* **100**, 013901

[39] Debien A, Benard N, David L, Moreau E 2012 *Appl. Phys. Lett.* **101**, 229903

[40] Neumann C, Friedrich J, Czarske Kriegseis J, Grundmann S 2013 *J. Phys. D: Appl. Phys.* **46**, 042001

[41] Bénard N, Debien A, Moreau E 2013 Time dependent volume force produced by a non-thermal plasma actuator from experimental velocity fields *J. Phys. D: Appl. Phys.* in press