Wave Profile for Proforce Current Bearing Waves

Volume 50

Article 13

1996

Wave Profile for Proforce Current Bearing Waves

Mostafa Hemmati

Arkansas Tech University

Steven Young

Arkansas Tech University

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Recommended Citation

Hemmati, Mostafa and Young, Steven (1996) "Wave Profile for Proforce Current Bearing Waves,"Journal of the Arkansas Academy of Science: Vol. 50 , Article 13.

I

Wave Profile

for Proforce

Current

Bearing

Waves

Mostafa Ilemma!iand Steven Young

Physical Science Department Arkansas TechUniversity

Russellville, AR72801

Abstract

Acomplete waveprofile for proforce style of breakdown waves witha current _behind the shock front is discussed.

The solution oftheelectron fluid dynamical equations in the sheath region forproforce current bearing waves conforms

withtheexpected conditins forthe values of the dynamical variables at the trailing edge ofthe wave.The waveprofilefor

electric field and electron velocity, temperature, and number density arepresented.

Introduction

The model is that ofan infiniteplane wavetraveling

in thepositive xdirection with aspeed V.Assuming that

the neutral particles are at restin thelaboratory frame,in

the wave frame they are being swept into the wavefront withaspeed -V.Considering the wave fronttobe the

ori-ginofthe waveframe, x=0, _the wavewillextend to x=

-°°. The wave front at x =0 is a strong discontinuity (a

shock front) which separates the neutral gas in front of

the wave from the three component fluid composed of

neutral particles, ions,and electrons. The structure of the

waveconsists oftwodistinct regions. First,a thin

dynami-cal region at the wave front which is somewhat thicker than aDebye length and willbe referred toas the shock

layer or the sheath region. Here the electric field,

elec-tron velocity, and the electron gas temperature firstattain maximum values, and then the electric field reduces to

zero as the electrons slow down to the same velocity as theheavy particles. Second, abroad quasi-neutral region,

where at _{the beginning} ofthis region the electrons _posses a high temperature, and further ioni/.ation takes place until the electrons cool below the ionizing temperature.

Breakdown waves for which electron mobility is in the

direction of wavepropagation willbe referred toas

pro-force waves.

Aone-dimensional, _{steady-profile, multifluid}model

isemployed to describe the breakdown wavepropagation intoaneutral medium subjected toa strongelectric field. In order to _describe the waveinterms of electron vari-ables _{only, electron-fluid equations} are decoupled from

the rest ofthe equations. Inthe one-dimensional

approxi-mation, _the set _{of electron fluid dynamical equations,}

composed of the equations ofconservation of mass,

momentum, and _{energy coupled} with thePosisson's

equa-tion,have been applied to a wave ofsteady profile. The

set _{of equations} has been discussed in_detail intwo

previ-ouspapers: (a)Fowler etal. (1984), and (b) Hemmati and

Young (1995). Interms of the dimensionless variables

the set ofequations are

d(«/v) _(D

4

_{

l',.(r

-

|)₄

„//«}

- -

,;, >.H,¦

-

_{!). (2)}

$

{

„»•(* I)2 _ImH>(.k 2)(nn +,„/»

"'^

||| } -

-'.«,{

:i,,fl₄(-.

-

I)2

}.

(3)

8-«<"

"•

(4)

Equations (1) through (3) are the equations of

conserva-tion ofmass, momentum, _{and energy, respectively.} With notime variationin the wave frame, Maxwell'sequations

in one dimension reduce topoisson's equation, equation

(4), alone. The variables v,_T

e

,

n,E, x,are electron veloci-ty, electron temperature, electron number density, elec-tric field(applied fieldplusspace charge field),and posi-tioninside the sheath. In dimensionless form they are

represented by i),6, v, J],and

_£.

Thesymbol 0,K,_fi,and u are ioni/.ationpotential, elastic collision_frequency, ioni/a-tionfrequency, and ioni/.ation rate respectively, with a

and krepresenting the waveparameters.

Solution of theEquations

The waveisconsidered tohave astrong discontinuity

atit's front,and interms ofnondimensional _{variables the}

shock front boundary conditions which have proven tobe

sucessful (Fowleret al. 1984) are

//,_/I): _,,,=|; _{0,= 1,(1}

-

_tf.,)/tt;

5(1+V)

-{

IB

"

+

1)

)

l

(5)

where vh()x

,

ij)iare electron number density, electron

temperature, and electron velocity, respectively; T]\ is the

electric field,_and

_0\

is the electron temperature

the trailing edge of the sheath are

iy_a

=

o: i-.₂= i: ,,'₂=_0. (6)

To achieve solution for the set ofelectron fluid dynamical equations, the equations areintegrated bytrail and error through the sheath region, as described below. To make the integration of the set _{of equations through}

the sheath region possible, themomentum balance

equa-tionis expanded and other equations from the set are

used tosolve for eletron velocity derivative

(It/-

_

KU'( I

-

_I'MI+/()

-

_<\Onfl

-

(>(¦«'

-

i)i.< "

r2

-

nO

(7)

This allows forthe singularity inherent in_the set _of

equa-tions toappear in the denominator ofequation (7). The movable singularity present between ij)]and 1 is used to integrate the set ofequations by trialand error. For0

<

ij)

<

_{1 when ij)}2

-

_{a0 approaches} zero, _{the electron velocity}

derivative willapproach infinity.This represents the pres-ence of a shock within the sheath. A shock inside the

sheath is not allowed; therefore, the numerator in

equa-tion(7) has to approach zero at the same time as the

denominator approaches zero. For agiven wave speed a and electron number denisty V1( this allows choice of

ini-tial value of by trail and error for agiven K.After

reaching the singularity, forapproximately ten

integra-tionsteps the denominator and the numerator in

equa-tion (7) areheld constant _{until both change signs} after

passing through the singularity. The integration of the set

ofequations is continued until i)approaches 1. At the

trailing edge ofthe wave, _{the conditions} (6), however,

have to_{be satisfied} in theirentirety. Ifconditions (6) are

not satisfied, _{the process} has tobegin witha newvalue of

/cand i){

.

The ionizationrate, u,iscalculated by usingan

equa-tionderived by Fowler (1983). His derivation is based on

free trajectory theory, where ionization due toboth

directed and random motion ofelectrons is included.

(8)

Analysis and Results

Forbreakdown waves withalarge current behind the

shock front,_{Poisson's equation and the initial boundary}

condition on electron temperature have tobe modified

(Hemmati and Young 1995). In dimensionless variables themodified Poisson's equation and theboundary condi-tionon electron temperature respectively are

S

=_{S(e- '>}

₊

*•'•. (9)

0

=«"'<!

-*''¦>

,

K

t (10)

'.

where, I=_rnK₀K

,

in equations (9) and (10) is the dimen-sionless current, and I( is the current behind the shock

front. Uman(1971), considering long laboratory sparks as

miniature lightning, calculates the current values from

electric fieldmeasurements. He reports peak currents in

the order of 103_{A forleader steps, and peak currents in}

the range of10 kA ~ _{100 kA}for the lightning return

stroke. These current values correspond to a

dimension-less current, I,value inthe range of 0.005 and 0.5. For

larger current values, the computer integration of the set

of electron fluid dynamical equations becomes very diffi-cult and time consuming. For larger current values in

order to make the passage through thesingularity

possi-ble, the number ofintegration steps near the singularity

where the numerator and the denominator inequation (7) are kept constant must be increased up to twenty steps. This is the cause of the kink in the enclosed graphs. Earlier, Hemmati and Young(1995) have

report-ed the variation ofelectric field and electron velocity

inside the sheath for several current _{values. Their results} meet _{the expected boundary conditions} at _{the trailing}

edge of the sheath. Acomplete waveprofile, for alarger

current value, i=0.25,isthe subject ofthe present paper. Earlierattempts inintegration ofthe set ofequations

for this value of the current resulted inonly the passage through the singularity. In our recent attempts, notonly

have webeen able to pass through the singularity, but

also our best result is satisfactory inmeeting the condi-tions at the trailing edge ofthe sheath. For asatisfactory

solution at the end of the sheath, _r\has to approach zero

while electrons acquire the same speed as those ofheavy particles (i)

—

>

1).Atthe end of the sheath, _{the electron}

temperature, 6,has toremain positive.Apositive electron

temperature indicates electron capacity for further ioniza-tion.

Figure 1 is agraph of electric field,77, as afunction

of electron velocity, ty._{As expected, the beginning} value

of the electric fieldisequal to_{that of the applied field (77}

=_1),

and itapproches zero at_the end of the sheath.

Figure 2is a graph of electric field,77, as a function of position, £,inside the sheath. Comparing the sheath

thickness for1=0.25 with those of lowercurrent values,

Hemmati and Young(1995) shows that as the current

value behind the shock frontincreases, the sheath becomes thicker.

Figures 3 and 4 are graphs ofelectron temperature,

0, and electron number density, V, as a function of posi-tion inside the sheath. Our results arefor a =_{0.01, which}

represents a wave speed value of 3 x 107m/s, and K=

0.8033678. Forlarge current _values the passage through

the singularity requires alarger number ofsignificant

fig-ures in variable values such as K.The other variable

Fig. 1. Net electric field, 7],as a function ofelectron velocity, ij),inside the sheath.

-Fig. 3. Electron temperature, 9,as afunction of position,

£,, insidethe sheath.

Fig. 2. Net electric field T), as a function ofposition, £,

inside the sheath.

ues at the shock front for our successful solution are v,=

0.029 and =

0.267. The singularity appears at

_£

=0.34.

r Conclusions

For current value ofi=0.5, the integration of the set

ofequations become practically impossible. This

indi-cates the existence ofa cut-offpointforcurrent values for

which the solution of the electron fluid dynamical equa-tions ispossible. The existence of a cut-offpoint for

cur-rent agrees with thereported experimental current _values

(Uman 1971) and is a validation of the fluidmodel and the set of electron fluiddynamical equations.

Acknowledgments.

—

_{The authors would like to}

Fig.4. Election number density, v, as a function of

posi-tion, £,_{inside the sheath.}

express their gratitude to _the Scientific Information Liaison Office and Arkansas Space Grant consortium for their financialsupport forthisproject.

Literature Cited

Fowler, R.G.,M.Hemmati, R.P. Scott and S.

Parsenajadh. 1984.Electric breakdown waves: Exact numerical solutions. PartI.Phys. Fluids. 27:1521-1526.

Fowler,R.G. 1988. Atrajectory theory ofioni/.ation in

strongelectric fields.

_J.

Phys. B: At. Mol.Phys.

16:4495-4510.

effect ofcurrent _{behind the shock front}on wave

structure. Proceedings of Arkansas Acad. ofSci.

Uman, M.A.1971. Comparison oflightning andlong

lab-oratoryspark. Proceedings of theIEEE.59:457-466.