Jean-Charles Matéo-Vélez, Frédéric Thivet, Pierre Degond * ONERA - Centre de Toulouse * CNRS - Mathématiques pour l'Industrie et la Physique, Toulouse Summer school on Mathematical modelling and computational challenges in plasma physics and applications, Cargese, october 2004 Estimation of the ionic wind created by a wire-to-wire corona discharge

Purpose: modelling the interaction between an electric discharge and aerodynamics Some previous experimental works: –Roth (Univ. Tenessee, 1998): AC discharge with dielectric barrier –Moreau (LEA, 1998): corona discharge 2 thin electrodes on a plate DC or pulsed current atmospheric pressure ionic wind about 3-4 m/s air flow Possible applications: EHD actuators for : drag reduction, flow control, shock waves reduction,...

Wire-to-wire discharge Several regimes - a lot of influencing parameters: –electric potential difference, –shapes of the electrodes, their positions upon or in the dielectric plate, –composition of the dielectric plate, –humidity degree of the air, –air flow, –etc … Present simplified study: –"high spot" regime, the most efficient: many luminescent points on the electrodes, –two corona discharges: one positive corona and one negative corona.

Electric potential (V) –Electric field calculation: no space charge included (i.e. without discharge) –Use of PDEtool library of Matlab®. –This highlights 3 characteristic zones: two chemical active zones and 1 passive zone Wire-to-wire discharge: Electrostatic field Anode : +22 kV,  = 0.7 mm cathode : -10 kV,  = 2.0 mm

–electronic avalanche if E > E d, where E d is the disruptive electric field in air at atmospheric pressure, –the radius r a is defined by E(r a ) = E d, –if r < r a, gas ionisation, –if r > r a, no more ionisation, + e-e- e-e- + E e-e- + u+u+ rara Positive corona Moving positive charges –The calculation of the electric field indicates that r a  1.5 mm, –This order of magnitude is confirmed by two analytical calculations inspired by the works of Raizer(1994) and Li (2004). The anode zone:

–electronic avalanche if E > E d, –if r E d, gas ionisation, the positive ions are absorbed by the cathode (secondary emission due to ionic bombardment), –if r > r c, E < E d, the electrons are evacuated, they rapidly attach to neutrals, –the negative charges (negative ions) are accelerated because of the strong electric field, –The electric field calculation indicates that r c  1 mm. –This order of magnitude is also confirmed by an approximated analytic calculation inspired by the exact solution for one unique wire (Raizer). - E + e-e- - u-u- rcrc Negative corona Moving negative charges (ions) e-e- e-e- e-e- Electronic attachment The cathode zone:

–acceleration of positive and negative ions due to the strong electric field (Lorentz force), –collisions with the neutral molecules of air, –ionic wind effect, –competition between positive and negative ionic wind. + - U res The inter-electrodes space:

1 st wire-to-wire discharge model: Objectives –Verifying the corona discharges hypothesis: both positive and negative ions currents provoke ionic wind, evaluation of positive and negative currents, use of experimental data such as the total current, –Developing the simplest discharge-aerodynamics interaction model, –Analysing the results so as to develop a second model if necessary.

1 st wire-to-wire discharge model: Lorentz force: Electric force model: –force due to positive ions current: –force due to negative ions current: –total electric force: C: ratio of positive ions and negative ions density currents  The electric force is linked to the discharge current I, experimentally obtained,  The section S on which the force is exerted must be determined.

–h = height of the moving charges zone, h must be defined, d = 4 cm f uniform h = 0,1-2 mm +22kV-10kV U e The height of the anode zone which is the more efficient ionisation zone –for C = 2,  + = m 2 V -1 s -1,  - = 2, m 2 V -1 s -1, I / l = 1 mA/m (Moreau, LEA): The inter-electrodes space = force application:

–Use of CEDRE: ONERA code for fluid mechanics, –adding of an external volumetric force in agreement with the model, –Calculation of a laminar flow above a plate, –refined mesh near the wall. Numerical simulation: slip Subsonic inflow U = 15 m/s T = 300 K Subsonic outflow P = 1 atm wall Force application zone: L = 4cm Force application zone: h = mm

Results: flow velocity profiles J. Pons, Gas discharge, 2004 x = 6 cm

for h = 1.5 mm, the flow boundary layer is 30% more thin Boundary layer thicknessDrag reduction for h = 1.5 mm, the drag reduction is about 70% at the end of the plate

Conclusion: –this simple model is in agreement with experimental data, –the effects on the aerodynamics are important, –it encourages us to continue this study, by developing a predictive tool for the calculation of discharge-aerodynamics interaction. Perspectives: –a more precise model of the discharge is being developed, –it takes into account the momentum equation of the fluid, the Poisson equation and conservation equations for the species of the plasma, –an asymptotic analysis enables to simplify the problem, –the most difficult issue is to determine the chemistry of the wire-to-wire corona discharge.

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