jap pinheiro martins

8/13/2019 JAP Pinheiro Martins

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Electrical and kinetic model of an atmospheric rf device for plasmaaerodynamics applications

Mario J. Pinheiro1,a and Alexandre A. Martins2,b1Department of Physics, Institute for Plasma and Nuclear Fusion, Instituto Superior Tecnico,

Av. Rovisco Pais, 1049-001 Lisboa, Portugal2Institute for Plasma and Nuclear Fusion, Instituto Superior Tecnico, Av. Rovisco Pais,

1049-001 Lisboa, Portugal

Received 21 December 2009; accepted 11 March 2010; published online xx xx xxxxThe asymmetrically mounted flat plasma actuator is investigated using a self-consistent

two-dimensional fluid model at atmospheric pressure. The computational model assumes the

drift-diffusion approximation and uses a simple plasma kinetic model. It investigated the electrical

and kinetic properties of the plasma, calculated the charged species concentrations, surface charge

density, electrohydrodynamic forces, and gas speed. The present computational model contributes to

understand the main physical mechanisms, and suggests ways to improve its performance. 2010

American Institute of Physics.doi:10.1063/1.3383056

I. INTRODUCTION

There has been a growing interest in the field of plasma

aerodynamics related to its outstanding importance in activeflow control, overriding the use of mechanical flaps.

18

Plasma actuators create a plasma above a blunt body that

modify the laminar-turbulent transition inside the boundary

layer,912

even at a high angle of attack4,13

they induce or

reduce the fluid separation, reducing drag1

and increasing

lift.3,14,15

They also allow sonic boom minimization

schemes,16,17

avoiding unwanted vibrations or noise,18,19

sterilizing or decontaminating surfaces,20,21

jet engine or

wind turbine, through pure electromagnetic control. Its

promising potential extend to flow control at hypersonic

speeds22,23 while still using a jet-reaction aircraft propeller.

The asymmetric dielectric barrier discharge DBDplasma actuator is a normal glow discharge that, like allnor-mal glow discharges, operates at the Stoletow point.

24,25This

guarantees that the generation of the ion-electron pairs at one

atmosphere is done efficiently. In air, the minimum energy

cost is 81 eV/ion-electron pair formed in the plasma. In

plasma torches, this energy cost can be of the order of 1

keV/ion-electron pair; in arcs, it can range from 1050 keV/

ion-electron pair.

Particle and fluid simulations have been performed2628

for a plasma actuator in pure oxygen and pure nitrogen

showing the formation of an asymmetrical force that accel-

erates the ions dragging the neutral fluid in the direction of

the buried electrode. A net force arises because the plasmadensity, and consequently, momentum transfer are greater

during the second half of the bias cycle, due partially to the

ion density greater by a factor of 10 during the second half-

cycle. Thus, in each cycle there is induced a total unidirec-

tional force toward the buried electrode that can create neu-

tral fluid flow velocities on the order of 8 m/s.4,29,30

Two-dimensional fluid models of a DBD plasma actuator

have been envisaged29,30

which calculate the total force on

ions and neutral particles, and showing that the generated

force is of the same nature as the electric wind in a corona

discharge; the difference is that the force in the DBD is lo-

calized in the cathode sheath region of the discharge expand-

ing along the dielectric surface. While the intensity of this

force is much larger than the existing force of a dc corona

discharge, it is active during less than hundred nanoseconds

for each discharge pulse and, consequently, the time aver-

aged forces are of the same magnitude in both cases.2630

The use of voltage pulses in plasma actuators, e.g., by modu-

lation of the high frequency excitation voltage carrier wave

by a square wave, introduces mean and unsteady velocity

components, and the air momentumcomposed of both time-mean and oscillatory components gains efficiency.31

It is becoming clear from different models that actuatorsplaced on the leading edge of an airfoil can control the

boundary layer separation, while if located at the trailing

edge can control lift.28

In particular, Enloe and

co-workers32,33

have found experimentally that thrust Tand

maximum induced speed umax are proportional to the input

powerP , which depends nonlinearly on the voltage drop V

across the dielectric TumaxPV7/2.

At our knowledge the first comprehensive fluidmodel of

a DBD plasma actuator was published by Roy,34,35

since En-

gel et al.36

seminal paper. Particle-in-cell and Monte Carlo

calculations27,28

have shown that negative oxygen ions gen-

erated in the plasma modify the interplay between species

diminishing the net ponderomotive force, since negative ionsimpart momentumto the air in the opposite direction to posi-

tive ions. Gadri37

has shown that an atmospheric glow dis-

charge is characterized by the same phenomenology as low-

pressure dc glow discharge.

Research on high-speed jet control is progressing fast in

a very competitive areae.g., Refs.38and39, but it is so farbeyond our reach.

The aim of this paper is to present a self-consistent two-

dimensional modeling of the temporal and spatial develop-

ment of asymmetric DBD plasma actuator using an electro-

aElectronic mail: [email protected].

bElectronic mail: [email protected].

JOURNAL OF APPLIED PHYSICS 107, 12010

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0021-8979/2010/1078/1/0/$30.00 2010 American Institute of Physics107, 1-1
http://dx.doi.org/10.1063/1.3383056http://dx.doi.org/10.1063/1.3383056http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://dx.doi.org/10.1063/1.3383056http://dx.doi.org/10.1063/1.3383056http://-/?-http://-/?-http://dx.doi.org/10.1063/1.3383056http://dx.doi.org/10.1063/1.3383056


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hydrodynamicsEHDcodeCODEHDdeveloped by us. Thecomputational model solves the governing equations in the

drift-diffusion approximation and uses a plasma kinetic

model. Figure1 shows the rather simple configuration of the

plasma actuator in coplanar configuration.

II. NUMERICAL MODEL

A. Description

To subdue numerical complexity no detailed plasma

chemistry with neutral heavy species is presently addressed.

At this stage, it is only considered the kinetics involving

electrically charged species supposedly playing a determi-

nant role at atmospheric pressure, N2+, N4

+, O2+, O2

, and

electrons. From the charged species populations balanceequations and as well the electric field controlling their dy-namicsPoisson equation, it can be studied the EHD effectswith interest to plasma actuators, like the body forces acting

on the plasma horizontallyfor neutral flow controland per-pendicularly for boundary-layer control to the energizedelectrode and also the induced neutral particles average

speed using Bernoulli equation.

The applied voltage has a sinusoidal wave form Vat= Vdc + Vrmssint /2, where the root mean square voltage,Vrms, in this case study is 5 kV and the applied frequency is

f= 5 kHz. Therefore, the dynamical time isT=200 s.

The simulations were done for a two-dimensional flat

staggered geometry, while assuming the plasma homoge-neous along the OZ-axissee Fig.1. The computational do-main is a two-dimensional area with the total length along

the OX-axis Lx=4 mm and height Ly =4 mm. The grid has

Cartesian coordinates and the time stepping was chosen

typically of the order of 1 ns. This is essentially a surface

discharge arrangement with asymmetric electrodes. As

shown in Fig. 1, the simulated physical domain consists of

conductive copper stripswith negligible thickness of widthw =1 mm, separated by a d = 0.065 cm thick dielectric with

width equal to 3 mm and relative dielectric permittivity

r=5. The electrical capacity of the reactor is assumed to be

given by the conventional formula C=roS/d.

B. Transport parameters and rate coefficients

The working gas is an airlike mixture of a fixed

fraction of nitrogen N2 =N2 /N=0.78 and oxygen O2=O2 /N=0.22, as normally present at sea level at p=1 atm. The electron homogeneous Boltzmann equation

EHBE is solved with the two-term expansion in sphericalharmonics.

4042The gas temperature is altogether assumed

constant, both spatially and in time frame, with Tg =300 K,and the same applies to the vibrational temperature of nitro-

gen TvN2=2000 K and oxygen TvO2=2000 K, which

are consistent with Ref.43. This assumption avoids the need

to include a complex vibrational kinetic model. The output is

the electrons energy distribution functionEEDF, which fortypical conditions does not depend on the electron concen-

trations. Electron-electron collisions make the EEDF Max-

wellian but they are not important in molecular gases likenitrogen at low degree of ionization in our case, typicallythe degree of ionization is ne /N10

5.Using the set of cross sections of excitation by electron

impact taken from Ref. 44, rates coefficients and transport

parameters needed for the electronic kinetics are obtained.

So far, the species included in the present model are the

following: N4+, N2

+, O2+, O2

, and electrons. At atmospheric

pressure N4+ ions need to be introduced since their concen-

trations can possibly be of the same order of magnitude

or even higher than N2+,

45partially due to the process of

ion conversion,46

N2+ + N2 + N2N4

+ + N2, which occurs at a

higher rate than the direct ionization, with constant kic1 = 5

1029 cm6 s1. Ion diffusion were obtained using

Einstein Smoluchowski relation and mobility coefficients

were taken from,47,48

O2

N=6.851021 V1 m1 s1 on

the range of E /N with interest here, O2

+N=6.91

1021 V1 m1 s1, and N2

+N=5.371021 V1 m1 s1.

The gas density at p = 1 atm and assuming Tg =300 K is N

=2.4471025 m3.

At atmospheric pressure the local equilibrium assump-

tion holds and the transport coefficients ionN2, ion

O2, e,

p,De, andDpdepend on space and time r , tonly throughthe local value of the electric field Er , t; this is the so calledhydrodynamic regime, thoroughly assumed in the present

model.

The motion of the gas has an appreciable effect on the

motion of ions for gas flow velocity above 10 3 104 m /s,

since this is comparable to the drift velocity of ions in the

electric field. Although the gas flow has no direct effect on

the motion of electrons, the coupling between electrons and

ions through the ambipolar electric field does affect electrons

motion. To avoid the use of NavierStokes equations and to

obtain a faster numerical solution of the present hydrody-

namic problem, it is here assumed that the gas flow does not

alter the plasma characteristics and is much smaller than the

charged particle drift velocity.

With the above assumptions, charged species can be de-

scribed by continuity equations and momentum transport

equations in the drift-diffusion approximation. This last ap-

proximation is valid if their drift energy is negligible with

FIG. 1. Schematic of the asymmetric plasma actuator.

1-2 M. J. Pinheiro and A. A. Martins J. Appl. Phys. 107, 1 2010

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of the different charged particles. The applied voltage has a

sinusoidal wave form

Vat=Vdc +V0 sint , 7

where Vdc is the dc bias voltage considered here as fixed toground,Vdc = 0and is the applied angular frequency. V0 isthe maximum amplitude, and the root mean square voltage in

our case of study is Vrms = 5 kV, were the applied frequency

is f=5 kHz.The total current convective plus displacement current

was determined using the following equation given by Mor-

row and Sato:53

Idt=e

V

V

npvpnevennvnDpnpz

+Dene

z

+Dnnn

z ELdv+ 0

V

V

ELt

ELdv, 8whereVdv is the volume occupied by the discharge, EL isthe space-charge free component of the electric field. The

last integral when applied to our geometry gives the dis-

placement current component

Idispt=0

d2V

t

V

dv. 9

The flux density of secondary electrons onto the cathode is

given by

jset= jpt , 10

withjpdenoting the flux density of positive ions. We assume

throughout the calculations that Auger electrons are pro-

duced by impact of positive ions on the cathode with effi-

ciency = 5102.

Secondary electron emission plays a fundamental role on

the working of the asymmetric DBD plasma actuator. The

progressive accumulation of electric charges over the dielec-

tric surface develops a so-called memory voltage, whose

expression is given by49

Vmt=1

Cds

t0

t

Idtdt+Vmt0. 11

Here, Cds is the equivalent capacitance of the discharge.

As charged particles are generated in the plasma volume,

the space-charge electric field is determined by solving the

Poissons equation coupled to the particles governing equa-

tions

V= e

0npnenn. 12

Here, np, ne, and n n denote, respectively, positive ion, nega-

tive ion, and electron number densities. The following

boundary conditions were assumed at the electrode and di-

electric surfaces:

over the electrode Dirichlet boundary condition:Vx,y = 0 , t= V Vm,

over the insulatorNeumann boundary condition: En=E n=/ 20.

The flux of electric charges impinging on the dielectric

surface builds up a surface charge density which was cal-

culated by balancing the flux to the dielectric and it is gov-

erned by

t=ep,ne,n . 13

Here, p,n and e,n represent the normal component of the

flux of positive and negative ions and electrons to the dielec-

tric surface. It is assumed that ions and electrons recombine

instantaneously on the perfectly absorbing surface. As it will

be discussed later, this simplified assumption constitute a

drawback of the present model, but at our knowledge this

important issue is not yet resolved in the literature.

The entire set of equations were solved together self-

consistently at each 1 ns time step, as illustrated with the

program flow chart shown in Fig. 2.

III. RESULTS

The electrohydrodynamic area of research has grown to

a large extent lately, but it still remains to achieve a better

understanding on how the charged particles transfer momen-

tum to neutrals, and what physical limitations restrain the

applicability of the device herein discussed for boundary

control and neutral flow propulsion.

A. Electrical characteristics

Figure3 shows the evolution along a full period of the

calculated electric currentconvective plus displacement cur-rents, applied voltage, gas voltage, and memory voltage.

FIG. 2. Program flow chart of the EHD codeCODEHD.


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At the same time and in the reversed direction, a

streamer of positive ions N2+ is driven to the cathode, as

shown at the time series illustrated in Fig. 7. During these

events the streamers ponderomotive forces attain their high-

est magnitude. The event occurring at the instant of time

t=6.75 s portrays the arrival of N 2+ ions to the virtualcathode.

The N2+ ions are mainly created in volume and are im-

mediately driven to the virtual cathode, while an average

number of them are retained nearby the anode, decreasing

almost linearly with distance from the energized electrode

surface. Thus, it is clear that the propagation of the electron

avalanche near and above the dielectric surface is of con-siderable importance, dictating the strength of the body

forces and gas speed. We hope in a next paper to treat this

issue.

Conditions for maximizing ion-driven gas flows were

obtained by Rickard et al.,55

and they concluded that, irre-

spective of geometry, ponderomotive forces on the gas are

maximized by increasing current density and by decreasing

mobilityi.e., charge carriers which exercise highest drag onthe neutral gas. Therefore, N2

+ ions seem to be the best

candidate for this purpose.

Our numerical model shows that when the present con-

ditions prevail, heavy species such as N4+ move more slowly

with the varying external electric field. In fact, the most ac-

tive species in the process of momentum transfer are the

electrons and N2+ ions, although the molecular ions, due to

their mass, contribute in majority to control the boundary

layer and propulsive force.

From Fig. 7 it is noticeable the multiplication of N2+

ions when flowing from the electrode edge to the dielectric

surface, flowing along the reverse way as electrons did. The

ions feeding along the dielectric surface are due to a relative

biggerdielectric width which favors the increase of the ion

swarm,56

increasing therefore the gas speed due to the mo-

mentum exchange onset from charged particles to neutrals.

Notice that at t= 6 s nitrogen ions leave the region at the

boundary between the electrode and the dielectric, which

corresponds to a region of maximum electric field seeFig.4.

Figure 8 shows contours of constant potential at t

= 5 s. We can see the decrease of the potential above the

energized electrode and a field reversal region toward the

dielectric sidewith the negative electrode below. The nega-tive glow remains in the proximity of this region, while a

second region of field reversal is also momentarily observed.

The region of negative electric potential that appears at 5 s

in the first half-cycle is due to the presence of an excess of

negative charge due to O2 ions, while the region of maxi-

mum electric potential has as a field source all other positive

ions.

Numerical simulations have shown that while negative

ions in the air do not contribute significantly to the pondero-motive forces, they can play a role in the discharge working

processes.33

Hence, the phenomenology and typical struc-

tures developed by normal glow discharge are also displayed

by the OAUGDP one atmosphere uniform glow discharge

plasma, a plasma actuator device used for plasma depositionor etching, and aerodynamic boundary layer and flow con-

trol. These aspects were also shown in previous publications,

like the one-dimensional numerical simulations of the

OAUGDP

done by Gadri37,57

and fast photography ob-

tained by Massines et al.58

The EHD force acting on the charged particles is given

by26,30

F=enineEnikTi+nekTe+meuemiuiS,

14

where ni and n e are, respectively, the ion and electron num-

ber density; ue and ui are the electron and ion mean veloci-

ties, and S is the charged particle production rate. The last

term of Eq.14 is here neglected, however, since its impor-tance confines to other phenomena such as electrophoresis

and cathophoresis,30

and its contribution to the induced ve-

locity is smaller due to the very small electron/neutral atoms

mass ratio.59

Although we had assumed a constant ion tem-

perature in equilibrium with the gas temperature, we calcu-

0.08

0.16

0.240.32

0.40

0.08

0.16

0.24

0.32

0.4010

8

109

1010

t=6 s

N2 +

Density(c

m -3)

Y

(cm)

X(cm)

0.08

0.16

0.24

0.32

0.40

0.08

0.16

0.24

0.32

0.40

108

109

1010

1011

1012

t=6.75 s

N2 +

Density(c

m -3

)

Y(cm)

X(cm)

(b)(a)

FIG. 7. Two-dimensional distribution of the N2+ ion density predicted by the

numerical code for typical conditions, during an avalanche occurring at the

first half-cycle.

9E2

9E29E2

1.8E3

1.8E3

2.7E3

0.08 0.16 0.24 0.32 0.40

0.050

0.075

0.100

t=5.13 s

Y(cm)

X (cm)

FIG. 8. Contours of constant potential at t= 5 s. Same conditions as inFig. 3.


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lated the electron temperature by solving independently the

EHBE and verified that the order of magnitude of the second

term is about 1% of the coulombian force term, which con-

stitute the main term of the theory of paraelectric gas flow

control developed by Roth.27

In addition, we calculated the

electrostriction force termnot included in Eq.14and con-cluded that it is not significant, contributing at maximum

only 1% of the total ponderomotive force. Subsequently, the

ponderomotive forces were averaged over the area of calcu-

lation.

It is found that the calculated space averaged pondero-

motive forces per unit volume increase when the electrode

width increases.6

On average, during the second half-cycle

the ponderomotive force magnitude decreases with a magni-

tude of a few Newton per meter as shown in Fig. 9,a result

consistent with experimental results as those presented in

Ref.60.This happens when the voltage polarity is reversed

and the energized electrode plays the role of the cathode.This is due to the potential gradient reduction on the edge of

the expanding plasmasee also Ref.14. This numerical re-sult is contrary to the experimental study presented in Ref.

61,showing that the forces decay exponentially with increas-

ing electrode diameter. This is due to the role of the dielec-

tric, which in our model was assumed to absorb electric

charged particles instead of feeding the swarms and strength-

ening the body forces see also Ref. 6. However, there isstill no consensus on the ponderomotive force dependency.

For example, Singh and Subrata62

obtained the magnitude of

approximated force and have shown that it increases with the

fourth power of the amplitude of the rf potential, implying

that the induced fluid velocity also increases. This is cer-tainly an aspect that must be dealt with more caution.

In fact, electrons are faster than ions. After the first

breakdown, they start to charge negatively the dielectric dur-

ing the first half-cycle; positive ions gain more energy and

the electrons are also increasingly accelerated, due to a

higher growing potential. Therefore, the positive ions density

is bigger during this half-cycle, resulting in stronger pon-

deromotive forces.

During the second half-cycle, positive ions which aremainly formed in volume tend toward both the electrodeand the dielectric surface; the charge surface density on the

dielectric start to become less negative and the gas voltage

decreases, generating less ions and electrons, resulting in

smaller body forces; this is the mechanism of an asymmetric

flat panel device. These findings are consistent with other

models50

and experimental work.43,60

Otherwise, if the discharge is entirely symmetrical in

both half-cycles, it is expected that the average gas speed

equals zero. In fact, it is the asymmetry in the streamers that

gives an overall positive gas speed along the axis.

In our present model, calculations of EHD ponderomo-

tive force have shown that its maximum intensity is attained

during electron avalanches, with typical values on the order

of 5109 N / m3. Fxpoints along OXpropelling direction,while Fy points downwards boundary layer control. Ourcalculations show that the resulting average gas speed is

about 10 m/s and the net EHD body forceswith the presentconditions have comparable values in the first and second-

half cycle, although slightly bigger during the first half-cycle, as shown in Figs. 9 and 10. Although the Navier

Stokes are not solved, we estimate the gas speed by using

Bernoullis equation6,59

v0=E0

. 15

Here, is the mass density of the neutral working gas. It is

clear that the successive streamers that charge the dielectric

surface are responsible for pulling the flow upstream, unidi-

rectionally, as recent experiments have shown.60

Therefore,

we can envisage methods to significantly improve their

strength by using modes of plasma-assisted electron emis-sion from ferroelectric ceramics of other high-k materials.63

IV. CONCLUSION

A two-dimensional fluid model of an asymmetric plasma

actuator displays the behavior of charged species during both

half-cycles when electrodes are subject to a sinusoidal ap-

plied voltage. The actuator is strongly dominated by N2+

dynamics, charged species form preferentially at the edge of

the electrode with the insulator, and their subsequent behav-

ior and ability to provide an unidirectional gas speed results

from the interaction of the charged species with the dielec-

0 50 100 150 200

-0.64

-0.32

0.00

0.32

0.64

Fx:

Fy:

Bodyforces(N/m)

time (s)

FIG. 9. Electrohydrodynamic forcesin N/m as a function of time. Sameconditions as in Fig. 3.

0 50 100 150 200

-10

-5

0

5

10

15

20

G

asspeed(m/s)

time (s)

FIG. 10. Color online Gas speed in m/s as a function of time. Sameconditions as in Fig.9.


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tric, in particular, the effect of the electric field above the

insulator and the propensity of the dielectric surface to ad-

sorb or not charged species, and thus controlling the plasma

density of the streamers. An appropriate model to describe a

realistic interaction of charged species with the dielectric, for

plasma density enhancement, remains to be done, lacking in

the literature a more careful study of this important issue.

ACKNOWLEDGMENTSThe authors gratefully acknowledge partial financial sup-

port by the Reitoria da Universidade Tcnica de Lisboa and

the Fundao Calouste Gulbenkian. We would also like to

thank important financial support to one of the authors

A.A.M. in the form of a PhD Scholarship from FCTFundao para a Cincia e a Tecnologia.

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