N 7 2 - 1 4 7 4 5

NASA TECHNICALM E M O R A N D U M

NASA TM X- 67978

ooI —NO

I

X

<oo<

QUASI-LINEAR THEORY OF ELECTRON DENSITY AND

TEMPERATURE FLUCTUATIONS WITH APPLICATION

TO MHD GENERATORS AND MPD ARC THRUSTERS

by J. Marlin SmithLewis Research CenterCleveland, Ohio

TECHNICAL PAPER proposed for presentation atTenth Aerospace Sciences Meeting sponsored by theAmerican Institute of Aeronautics and AstronauticsSan Diego, California, January 17-19, 1972

https://ntrs.nasa.gov/search.jsp?R=19720007095 2020-02-08T05:27:27+00:00Z

QUASI-LINEAR THEORY OF ELECTRON DENSITY AND TEMPERATURE FLUCTUATIONS WITHAPHjICATION TO MHD GENERATORS AND MPD ARC THRUSTERS

J. Marlin Smith, Aerospace EngineerNational Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio

gCD

Abstract

Fluctuations in electron density and tempera-ture coupled through Ohm's law are studied for anionizable medium. The nonlinear effects are con-sidered in the limit of a third order quasi-lineartreatment. Equations are derived for the ampli-tude of the fluctuation. Conditions under which asteady state can exist in the presence of thefluctuation are examined and effective transportproperties are determined. A comparison is madeto previously considered second order theory. Theeffect of third order terms indicates the possi-bility of fluctuations existing in regions pre-dicted stable by previous analysis.

Introduction

Under conditions appropriate to MHD gener-ators and MPD arc thrusters studies have shownthat fluctuations in electron density and/or tem-perature can be unstable.(1»2) in both of theseanalyses only the linear effect of fluctuationswas considered leading to the usual stabilityanalysis dispersion relation from which the growthor damping of the fluctuation can be ascertained.In this paper we extend these analyses into thenonlinear regime in order to determine the ampli-tude of the fluctuation and its effect upon theproperties of interest.

The oscillations under study primarily arisedue to fluctuations in electron temperature.These temperature fluctuations can be amplifiedthrough increased Ohmic heating arising as a re-sult of increased ionization and/or decreased col-lision frequency. These changes in frequencyoccur as a consequence of the initial electrontemperature perturbation. The dispersion relationdepicting this phenomena was derived in refer-ence (2) by considering the time dependent elec-tron density and temperature equations coupledthrough a generalized Ohm's law.

In this paper the nonlinear effects of thefluctuations are considered on the basis of athird order quasi-linear theory neglecting modecoupling. Even though the theory is strictlyvalid only in the neighborhood of the thresholdof stability, the third order terms are includedin an attempt to increase the applicability of theresults. The analysis is carried out for thosefluctuations which show the maximum rate ofgrowth, i.e., first become unstable. Two limitingcases are considered. In the first limit, theionization is assumed to be frozen. In this limitit is shown that the system can become unstable ifelectron collision frequencies decrease suffi-ciently rapidly with electron temperature. Forthe MHD generator, this mode does not appear to beimportant since operating conditions and workingfluids of interest are such that the instabilitycriterion is seldom satisfied. One notable excep-tion may occur, however, when the gas is stronglypreionized prior to entering the MHD channel. In

this case, the ionization may be sufficient thatCoulomb collisions dominate and the instabilitycriterion is satisfied. This was mentioned inreference (3) as a possible source of the anomo-lous oscillations observed in these experimentswith preionization. In reference (3) the oscilla-tion was referred to as the static instabilitymentioned in reference (1) and is akin to ourfrozen flow limit. In this limit, the medium isassumed to be infinite and homogeneous and the re-sults of the quasi-linear analysis are presentedfor a fluctuation propagating in the direction ofmaximum growth.

For the MPD arc thruster, the fluctuationsare assuned to occur in the region of high currentconcentration in the throat of the device which isdepicted by an annular region between two concen-tric cylinders as shown in figures 1 and 2. Sincedamping due to radiation and heat conduction areleast prominent for long wavelengths, the most un-stable waves are those with the longest wave-lengths. In the case where the interelectrodedistance is small compared to the mean circumfer-ence of the annular region, the longest wavelengthis obviously nearly that of purely rotationalpropagation. The analysis of the MPD arc thrusteris therefore simplified by considering only rota-tional propagation.

In the second limiting case, the ionizationrate is assumed infinite. This case has been ex-tensively analyzed with regard to MHD generatorsand was reviewed in reference (4). In that study,terms were retained to second order in fluctuatingquantities. In our work, third order terms andsome terms ignored in reference (4) as being smallin the range of interest are retained. In therange of interest, i.e., the region in which sta-bility is reached in the presence of fluctuationsof finite amplitude, the effect of these addi-tional terms is minor, at least in so far aspresent experimental data is interpretable. How-ever, an interesting point arises in that theseadditional terms open the possibility of insta-bility occurring in a region where linear theorypredicts stability. In this case, instability canoccur if a fluctuation of sufficient initial am-plitude is present. Such fluctuations may be ex-cited in the preionization region of the MHD duct.

For MPD arc thrusters in the purely rota-tional approximation the mode occurring in the in-finite ionization rate limit is damped. For prop-agation at small angles from the rotational direc-tion the mode can be unstable but requires largevalues of the Hall parameter for instability tooccur. From the analysis of reference (2), thesevalues appear to be much higher than those whichoccur in the high current throat area of presenthigh pressure MPD thrusters. Therefore, it is notexpected that this mode of instability plays animportant role in the rotating spoke phenomenaobserved in these devices.

TM X-67978

Assumptions and Governing Equations

Assumptions

Generalized Ohm's Law

The generalized Ohm's Law is

In those regimes where electrothermal dis-turbances predominate, the following assumptions,simplifications, and restrictions are appropriate:

1. The analysis is restricted to sufficientlyshort wavelengths such that the zeroth order prop-erties do not vary appreciably over this distance,i.e., spatial gradients in zeroth order quantitiesare ignored.

2. Only fluctuations in space and/or time ofthe electron number density and temperature areconsidered. This is reasonable since the rela-tively lighter and more mobile electrons respondto disturbances of greatly different frequenciesthan do the heavier and less mobile neutral atomsand ions.

3. Only the propagation of magnetohydrody-namic disturbances is considered. This restric-tion implies neglecting the displacement currentdensity in Maxwell's equations, ignoring inducedmagnetic field relative to the applied constantmagnetic field, and assuming the plasma to bequasi-neutral.

4. Ion and neutral particle flow velocitiesare equal, i.e., ion slip is ignored. Further-more, all heavy particle flow velocities are as-sumed equal to V-, the gas flow velocity.

** *E = E— —

m ve

n eVp

*

(4)

(5)

where E^ = E. + VQ x K. is the electric field inthe frame of reference moving with the gas flowvelocity VQ in the applied magnetic field BQ,

and (l/nee)Vpe is the force due to electron tem-perature and density gradients.

The electrical conductivity, a, is given by

.2

a = -^— (6)n e

m ve

where the total electron momentum collision fre-quency is

v = v + v +ec es .ei (7)

The species collision frequencies are allowed thefollowing general temperature dependence

A n Tea a e (8)

5. Only propagation in planes perpendicularto the applied magnetic field is considered.

6. Ion and neutral particle temperatures areassumed equal and equal to T., the gas tempera-ture.

where a = c, s, or i corresponding to electroncollisions with carrier gas atoms, seed atoms, orseed ions, respectively. AeQ and aea are arbi-trary constants chosen to fit average collisionfrequency data over the temperature range of in-terest.

For the above limitations the problem is com-pletely determined in terms of the fluctuations ofelectron density and temperature and their cou-pling through Ohm's law and Maxwell's equations.

Maxwell's Equations

Electron Energy Equation

The electron energy equation is

kT

~Te+^ VT

From Maxwell's equations under the assumptionof quasi-charge neutrality and neglecting the in-duced magnetic field,

V x E = 0

7 • £ = 0

Electron Continuity Equation

(1)

(2)

It is assumed that in the region of interest,the ionization is dominated by electron-neutralatom ionizing collisions and by three body recom-bination. The electron density equation thentakes the form

37 ne Vn n n v. - n ve s i e r (3)

It is noted that this equation depends upon thegas flow velocity VQ, rather than the electronflow velocity, V^. This arises as a result ofassumption (4) and the condition of quasi-chargeneutrality.

Ve 2

f-n +v- ' Vn3t e -0 el (9)

where v is the collision frequency for energytransfer given by

m m me e . ev = — v -f — v + — v

m m ec m es m. eic s i(10)

and the VTg contribution to the heat conductionterm and radiation losses are neglected. Theseterms contribute damping terms to the dispersionrelation which vary inversely with the wavelengthrelative to the wavelength independent elasticcollision damping. Since, in quasi-linear theory,one is primarily restricted to the region in theneighborhood of the stability threshold, onlythose waves which are most unstable are of inter-est, i.e., the. longest wavelengths. It is there-fore assumed that at these wavelengths, the wave-length dependent terms are ignorably small and

hence for simplicity are not included in the anal-ysis. The question of the relative importance ofthese terms has been extensively investigated forMHD generator type plasmas in references (5) andand (6).

Quasi-Linear Theory

In the second case we assume the ionizationrate is infinite. In this case the left hand sideof equation (3) can be neglected. The electronnumber density is then specified by

(15)

The nonlinear problem is now treated by con-sidering small departures from the stabilitythreshold so that pertinent quantities can bewritten in the form

f (r,t) = f(r,t) (11)

where f is the oscillatory function for which thetime averaged value defined by the brackets (f) iszero and (f^ is the average value of the quantityf. Furthermore, in keeping with our initial ^assumption (f) is spatially independent. The re-striction of small departures from steady statethen requires

f « (12)

so that a solution may be formed as a power seriesexpansion in f. In the cases considered here, theexpansion is carried out to third order in theproduct of f's. Furthermore, since we are con-sidering small departures, only the fundamentalmode is considered, i.e., mode coupling betweenharmonics or other fundamental modes is ignored.

The perturbation function is then taken to beof the form

f(r_,t) = f (t)sin(i. -£-0)1)

+ f (t)cos(A. • r. - ait) (13)

where i_ and u> are real, i_ • BQ = 0 (only con-sider propagation in plane perpendicular to BQ),and the long term time dependence of the perturba-tion function is contained in the coefficients fsand fc which are assumed to be slowly varyingover a period of the oscillation, t, so that

t+T

f (t')dt' - f (t)S S

fc(t')dt' =

(14)

The problem of interest is then one of determiningunder what circumstances, if any, that the ampli-tude of the fluctuation reaches a steady statevalue.

The general quasi-linear analysis of the setof equations (1) to (10) is complicated by the twotime dependent equations (3) and (9). This allowstwo fundamental modes to propagate simultaneously.Therefore, in order to simplify the analysis, weconsider two limiting cases. In the first case,oscillations in the electron number density areignored. This is justified if either the ionizable

material is completely ionized or the frequency ofthe oscillation is much greater than the ionizationfrequency. The electron number density like thegas dynamic properties is then specifiable by equa-tions peculiar to the problem of interest.

It is further assumed that the electrons are ini-tially in Saha equilibrium so that we are onlyconcerned with small fluctuations from equilibrium.Under this condition, the principle of detail bal-ance^) can be invoked to relate the ionizationcoefficient, v^, to the recombination coefficient,vr, through the equilibrium constant. Then

^ = -T^exPl-r\ \) p f 4nsO r e

(16)

This case has been extensively analyzed with re-gard to MHD generators.(**) In this paper, we ex-tend this second order analysis to third order andalso consider it with regard to high pressure MPDarc thrusters.

Frozen Ionization Rate Limit

This is the case where electron number densityfluctuations are ignored. Before proceeding to thequasi-linear analysis, we consider the resultspreviously obtained(2) from linear stability anal-ysis, for these results extended to include aneutral carrier gas as well as an ionizable speciesthe condition that the system be unstable is

1 + 2

X a. ecO esO + aei

VeiolV0 J

< 0 (17)

Let us first discuss the consequences of thisstability criterion for combustion and nonequilib-rium MHD generators and then consider the MPDthruster case. In the combustion generator in-elastic collisions with the numerous molecularspecies in the combustion gas work fluid make itimpossible to raise the electron temperature muchabove the gas temperature. In this case then, thefactor (1 - To/Te0) in the inequality (17) is ingeneral, too small for the instability criterionto be satisfied.

In the nonequilibrium MHD generator operatingon inert gases seeded with alkali metals the goalis to obtain an elevated electron temperature. Anelectron temperature approximately twice the gastemperature is typical. There, even for the mostunstable direction of propagation, i.e.,

A ' JQ = ^' t'le unstable condition requires the

coefficient + aesOV0

+ aei , -1.

In general, over the temperature range and forneutral gases of interest, the temperature expo-nentials and are positive or at leastonly slightly negative (e.g., see Maxwell-averagedcross sections for electron temperatures near3000° K presented in ref. (8)). Therefore, in

order for the coefficient to be <-l, the plasmamust be Coulomb collision dominated sinceaej - -3/2. Since under normal operating condi-tions of MHD generators the collision frequency isneutral dominated, it appears that the range ofimportance of this mode of instability for thesedevices is small and can be avoided without greatdifficulty. The one exception to this could occurwhen the gas is preionized by strong electricfields prior to entering the MHD channel. In thiscase, high degrees of ionization can be reachedand the resulting instability could propagate intothe generator region. This has been mentioned inreference (3) as a possible source of the anomo-lous fluctuations observed in these experiments.

On the basis of the above discussion and thefact that MFD arc thrusters in general operate inthe Coulomb collision dominated regime, the quasi-linear analysis in the limiting case of frozenionization is developed only for v - vei.

We first consider the equations governingthe average and fluctuating electric currentdensities. Substituting quantities of the formof equation (11) into equation (4) , we obtain

.

5)«E— JL Jk

E ) x £

(18)

where to third order (]J }vector in the direction ofparameter, g, is defined by

(E. }, £ is a unitand the Hall

m ve

(19)

Averaging equation (18) then gives an expressionfor the average current density.

(20)

The fluctuating part of the current densityis then obtained by subtracting equation (20) from(18) , forming the triple cross producte x ( x ) and using equations (1) and (5) ,i.e. ,

J = 0£** = 0, and equation (2), i.e.,to obtain

<E*» - 61 x [i b)]]

(21)

The fluctuating part of the field, E , canbe determined from the fluctuating current densityequation obtained by subtracting equation (20)from (18), forming the dot oroduct i_ • J, usingequation (1) so that E = -V$ , and equation (2) ,i.e. , i, • $ = 0 to obtain

-oi(i. • <E*>) + ejk

+ higher order terms (22)

Equation (20) can now be simplified by using equa-tion (22) to_eliminate \o|**^ and equation (21) toeliminate 6j. We then obtain

<*> = <°><E*> - <B><j> x b

It is convenient at this point to define forlater use the effective values of the conductivityand Hall parameter. These quantities are definedby

<*> = °ef£<I*> - Beff<*> X £

It is obvious from equation (24) that

"eft

Substituting equation (23) into equations (25)and (26)

eff ~ 1 + Z

ZT*eff 1 + Z

where

T = tan = tan

(24)

(25)

(26)

I

(27)

(28)

(29)

(30)

Much of the above manipulation was from hind-sight and for the purpose of obtaining the equa-tions for the average current density, equa-tion (23), and the fluctuating current density,equation (21), solely in terms of the fluctuatingelectrical conductivity, a, and the fluctuatingHall parameter, 6. The above equations are com-pletely general, i.e., not restricted to either ofour limiting cases. Specialization to these lim-iting^cases arises through the dependence of 3and (3 upon ne and fe in these limits.

For the case under consideration

Se = aec ' aes = 0; aei = ~3/2

then from equations (6), (7), and (8)

(31)

SL16

^<Te>2

From equations (19), (7) , (8) , (31) , and (32)

(32)

"0

3

(33)

(34)

where OQ and BQ are defined by equations (6)and (19), respectively, evaluated at the averageelectron temperature, (Te).

We now consider the energy equation (9) byeliminating J • E* in terms of (l/a)s2 by tak-ing the dot product of j with equation (4). Itshould then be obvious from equations (21), (23),(32), and (34) that the energy equation can be ex-panded in a power series in Te/(Te). _Carryingout this expansion to third order in Te/(Te) andaveraging the resulting equation we obtain, afterconsiderable algebraic manipulation, the followingequation for the average electron temperature

(35)

At this point, it is appropriate to considerthe essence of the analysis. First, it is to de-termine under what conditions, if any, the ampli-tude of a fluctuation which is linearly unstablecan reach a steady state. The second purpose isto determine the values of the measured propertiesof the system in the presence of fluctuations.Since most measured properties are obtained withinstruments that cannot resolve oscillations onthe time scale of the fluctuation, they only meas-ure time averaged properties (ideally).

It may be noted that for the third order thetheory considered here, the pertinent equations forthe average values, e.g., equation (31), dependupon the average of the square of the amplitude,(f£>, rather than the amplitude itself. Therefore,it is convenient to cast the equation for theamplitude, Te, in this form. The equation for Tgis obtained by subtracting equation (35) from equa-tion (9), expanding to third order in Te/ Te},multiplying by fe and averaging to obtain

2<> = -2 — n nun T k(/T N - T Je/ m eO 0 2 \ e/ 0

X-I <T21 3 35 3 Ve> <Te>

2 16 2 ^ • ^ f ((T > - T )1 1 15

1 83<f

2 < T

e

e>2]

which closes the set of equations describing theaverage properties of the system in terms of theaverage of the square of the amplitude of thefluctuation,

Before proceeding to analyze the above equa-tions, another property of interest, namely, thefrequency of the fluctuation is determined. Thisquantity is obtained by forming the equation forthe amplitude of the fluctuation, Te, as in theabove derivation, but instead of multiplying_byTe and averaging, we multiply by 1/fc2 i_ • VTeand average. The frequency in the frame of refer-ence moving with the gas flow velocity is then

= (1) + i_ • Vg ' (37)

which was the expression previously determinedfrom linear stability analysis. (2)' In otherwords , the frequency of the oscillation dependsupon the amplitude of the oscillation only through

The Effect of Fluctuations on Ohm's Law. Theconductivity and Hall parameter are often indi-rectly determined in experiments by first measur-ing the electric current density and electromag-netic fields. Then Ohm's Law is applied to deter-mine the conductivity and Hall parameter. In thepresence of fluctuations it is then obvious fromequation (24) that the measured values are theeffective values. Clearly, a discrepancy betweentheory and experiment arises when one calculatesthe conductivity and Hall parameter in their clas-sical form, i.e., equations (6) and (19) evaluatedat the measured electron temperature. It istherefore of interest to compare the ratio of ef-fective to classical values of these parameters.

From equations (27) , (29) , (30) , and (31) weobtain

1 +*<£>'3 Ve

eff8M'

1+7"(1 + O

(38)

In figure 3, this ratio is plotted as a function ofthe amplitude function ( /e) for variousangles of propagation. The interesting point tonote is that the effective conductivity is en-hanced by the presence of the fluctuation forvalues of |T| > /5 but is reduced for values lesothan /5.

From equations (28), (29), (30), (33), and(31), we obtain

eff (39)

Comparison of equations (39) and (38) shows thatin the limiting cases T = 0 ar.d |T| = °°,

(36)

effB0

eff (40)

At intermediate values of T, Beff/B0 dependsupon the value of BQ. Therefore, the dividingpoint between enhancement and degradation of theHall parameter is not simply a function of theangle of propagation as it was for the conduc-tivity ratio. The plot of Beff/BQ for the lim-iting cases is therefore identical to the a

eff/co

plot in figure 3, except that the neutral point1 occurs for values of T given by

(41)

These values of T are plotted in figure 4 forvarious values of BQ. The angles of propagationfor which Beff = BQ are restricted to0 <_ T <_ /5 and -*° <_ T <_ -/5. The asymptoticvalues T = ±/5 occur in the limit BQ * "• Atthe limiting point equality (40) again holds ascan be seen by taking the limit BQ •+ " in equa-tion (39) and comparing the results to equa-tion (38) .

The Effect of Fluctuations for the Case ofMaximum Growth Rate. We are interested in deter-mining whether a steady state can exist in thepresence of an infinitesimal oscillation, and ifso, under what circumstance the oscillation occursand what its amplitude is as a function of experi-mentally measured parameters.

We first consider the amplitude equation (36)for the direction of propagation which results inthe maximum growth rate of the disturbance. Sincein this direction, the fluctuation is first un-stable and grows most rapidly, it is assumed thatfor small displacements from the stability thresh-old this is the observed disturbance. For theCoulomb dominated regime considered here(ae 0, -3/2) this direction, as canbe seen from the linear stability term of equa-tion (17) , occurs for propagation perpendicular tothe average current density, i.e., • ($) = 0.For this case, the amplitude function is given by

2 3t

. 3 29+ 2 8~ 29

(42)

where we have used equation (35) with3/3t <Te) = 0 to eliminate (l/<o»^2 and have

f'ected terms greater than second order in/<Te^

2. We note that the linear term is juststability condition obtained from the linear

analysis and obviously is damping until the in-stability point, TQ/(Te) <_ 2/3, is reached. Be-yond this point the linear term causes a growth inthe amplitude of the oscillation until the non-linear term, which is damping for TQ/<Te) > 2/29,offsets the linear growth term. A steady state isreached when

<Te> 1 29. (2_ T028 \29 ~ (Te

(43)

Obviously in the case in which T0/(Te) < 2/29the nonlinear term contributes to the growth andthe thermal perturbation is unconditionally un-stable. However, it is obvious from equation (43)that as T0/<Te> -* 2/29, <T

2>/<Te>2 * » which vi-

olates the quasi-linear condition of small ampli-tude oscillations, i.e., (T2)/(Te)

2 « 1. There-fore, this point is not considered further.

The amplitude function, T/(Te), as givenby equation (43) , can be calculated by determiningthe temperature ratio TQ/(Te f rom equation (35) .For i. • ($) = 0, the steady state condition3/3t e) = 0, and either the MPD thruster geometryshown in figure 2 or the MHD Hall generator con-figuration, this equation reduces to

<T

(44)

where

3— kT2 kiO

(45)

is the ratio of the ion drift energy to its randomkinetic energy, E = |\E*)| and BQ are the magni-tude of the applied electric and magnetic fields,respectively. In deriving equation (44) , we havealso used equations (23) , (31), and (33).

Equations (43) and (44) can be solved simul-taneously for given values of the Hall parameter(B) and the parameter R to determine the ampli-tude function {T2)/(Te)

2. These values are shownin figure 5, where the amplitude function has beenplotted as a function of the Hall parameter (B) forvarious values of the parameter R. The firstthing to be noted from this figure is that for agiven value of R, there is no instability, i.e.,no physically exceptable solutions exist, until acritical value of (B) is reached corresponding tothe onset of instability. This critical value of(fi) increases with decreasing values of R. Alsoto be noted from figure 5 is that the amplitudefunction reaches an asymptotic value as <[B> •* °°,the magnitude of which depends upon the parameterR.

The Effect of Fluctuations for the Case of aPurely Rotational Disturbance. In the previoussection the medium was taken to be infinite andhomogeneous. As explained in the Introduction inthe case where the disturbance is confined to theannular region between two concentric cylindricalelectrodes, the longest wavelength and hence themost unstable wavelength (due to radiation and heat

conduction damping) is approximately that whichcorresponds to pure rotation. In this section, wetherefore consider the effect of a purely rota-tional fluctuation. This was the case consideredin reference 2.

The amplitude equation (36) to second orderin (T|)/(Te)2 reduces, after considerable alge-braic manipulation, to

3. 29 [2_ _28 [29

2329

53 229 B0

3_29

CO COU + # <r e > 2 J e > 2

(46)

Unfortunately the dependence of equation (46) uponT0/(Te} and B0 ls such tnat it is not Possibleto determine a steady state amplitude of the fluc-tuation for the same assumptions used in the line-arized theory of reference (2). In order to ob-tain the values of the local gas dynamic proper-ties required in the theory from the available ex-perimental data, a number of assumptions had to bemade in the analysis of reference (2). One ofthese was to take the local gas temperature, TQ,to be sufficiently small so that it could be ig-nored in the onset region where the frequency ofthe oscillation was evaluated. If this is donehere, the nonlinear term of equation (46) becomes

3. 29 fc 23 82 _ 3_ 84~|28 [29 29 0 29 Oj (1 +

(47)

which is positive and hence destabilizing even forthe maximum value of BQ for which the linearterm in equation (46) is destabilizing, i.e.,Bg = 2. Therefore, a more complete steady statesolution is required and/or local gas propertiesmust be measured experimentally in order to ascer-tain the validity of this assumption.

However, a second interpretation is possible.Since, in reference (2), the rotational frequencywas calculated at what was assumed to be the up-stream onset point, it could well be that theamplitude of the oscillation is growing at thispoint, reaching its quasi-steady value downstreamof the point in question. In this case, theamplitude may indeed be small at the point atwhich the frequency was calculated so that non-linear effects can be ignored. The effect of non-linearity then manifests itself downstream in thegrowth of the oscillation.

Of further interest is the fact that thelinear and nonlinear coefficients in equation (46),

r - I 29 (~2_ _ T0 .23.2 53 2 TQ 3_ 4I ~ 2 8 29 /T \ 29 P0 29 0 /T\ 29 0

(49)(1 +

respectively, can be positive or negative, depend-ing upon the magnitude of TT and &" In

figure 6, the curves of Cg = 0 and

plotted as functions of and B2..

= 0 are

For bothcoefficients, points (S,, T0/(Te}) above their re-spective curves result in negative values of thecoefficient and hence damp the fluctuation. How-ever, for points (0$, To/(Te)) below their respec-tive curves, the coefficients are positive so thatthe terms contribute to the grcwth of the fluctua-tion. As a result, the TQ/ Te} against 6g planeis divided into four regions having the followingcharacteristics :

1. Region I - Both coefficients are dampingand hence fluctuation is unconditionally damped.

2. Region II - Linear term damping, nonlinearterm growing. In this case the usual linear sta-bility theory would predict a damped disturbance.However, the growth contribution of the nonlinearterm indicates that, for disturbances with initialmagnitude of the amplitude function

iV(Te 2 > Ic0/cll> tne disturbances grow un-stably. Therefore, in this region fluctuationsare damped f or IV e)2 < ICg/Cj and grow un-

stably for <Ti)/<Te>2> |c0/Cl|.

•3. Region III - Both coefficients are positiveand hence any disturbance grows unstably.

4. Region IV - Linear term growing, nonlinearterm damping. Disturbance grows or damps until astable equilibrium point is reached with amplitudefunction <Tl>/<Te>

2 = ICg/Cj.

Whether the above regions exist under actualexperimental conditions only further experimenta-tion can answer. Furthermore, the effect of higherorder terms in those regions in which the theorypredicts the unstable growth of the disturbance isnot known. In these regions, therefore, a fullynonlinear theory must be developed.

Infinite lonization Rate Limit

In the case of infinite ionization rate theelectron number density is related to the electrontemperature by Saha's equation (16). Taking thesquare root of equation (16) and expanding theright hand side in a power series in Te/^Te weobtain for the ratio of 'the fluctuating density toits average value

(48)

/ T T2 T3 '1 |5 . 5 i .. i ,3 xi32 U + 2 <Te) '

5 2 + 2 p 3

T3 1x^T+. . .

^e) J

(50)

The important thing to be noted is that for de-vices of interest in this paper the average elec-tron temperature is in general much less than thetemperature corresponding to the ionizationenergy, i.e.,

Ce)» 1 (51)

so that

Therefore throughout the subsequent analysis termsthe order of Te/^Te are neglected with respectto ne/<ne>.

The equations governing the average currentdensity <^> and the fluctuating current density j.are equations (23) and (21) , respectively. Theseequations are specialized to the present limitingcase by specifying the dependence of o and 6upon Tg/ Te) an^ °e/(ne) •'•n tni-s limit. Ignor-ing Te//Te\ with respect to

ne/(ne) we obtainfrom equations (6), (7), and (8)

' n2 42V_ A ) ,_^_W

/n V\e/,

+ A2(l - A) —— + A(l - A)n2

< n e><n e > 2 + -"( ^^

from equations (19) , (7) , and (8)

+ ,3^JL +<ne> /n "

(54)

where

eiO esO <"e> (55)

and we have ignored the electron number dens? tydependence of the logarithmic factor in theCoulomb cross section.

Expressions for the average electron numberdensity and the average of the square of the fluc-tuating electron number density can be obtained byexpanding the electron energy equation in a powerseries in ne/ igN. This expansion is carried, outto third order^in ne/ ne'), neglecting terms ofthe order of Te/^Te) with respect to He/(ne .Since the method is -analogous to that consideredin the r>revious limiting case to determine Teand \T| only the result is presented here. Forthe amplitude function we obtain

3_

/-2\Cn )\ e/

where

* = -[(1 + A) + 2K] + (1 -L

A)

- 2(1 - A) -5- - 2B I—r- (57)(1 + O ° (1 + T^) J

*2 E [(1 + A) + 2K]Z' (58)

5A

*, = 7 (A - 3)Z - [2K + (A - A )] -7^-J / m 4

+ [(1 + A) + 2K] f(l + A )(A + 2K) - A2]L m mj

+ AH. + j Aj[ ( l + A) + 2K] (59)

A) 2+ <e> 2 ] -^ <"e> ... &>

(60)

/T \i V 1»72K =

A = — esO <ne>

The coefficient *^, of the linear term is propor-tional to the stability term obtained from linearwave analysis. Its consequences are reviewed inreference (4). It is found that the oscillation ismost unstable for propagation at the angle given by

T = -m1 - A

- ^^The linear term is then found to be positive andhence growing for a Hall parameter

> 6c = 2/(l + K)(A + K) (64)

The coefficient *„ i-s tne result obtained

in reference (9) and reviewed in reference (4) .This result was obtained for an expansion of theenergy equation to second order in fie/(ne andunder the assumption that Z was large in theregion of interest so that nonlinear. terms not in-volving Z could be ignored. These previouslyignored terms as well as third order terms arecontained in the coefficient *j.

In figures (7) and (8) the values of'oeff/<o> and 8eff/<B> are plotted, respectively,as a function of <B> for the theory of refer-ence (5), the present third order theory, and forthe experimental data of reference (10) for(i) = 2 amps/cm2. The theoretical curves are cal-culated by the method of reference (9) in which avalue of the critical Hall parameter, 6C, ischosen to best fit the experimental data of fig-ures 7 and 8. This then allows the energy lossfactor K to be calculated from equation (63) .All other parameters are calculated from the ex-perimental data of reference (10). The amplitudefunction, {niV(ne

2> is/th£n calculated fromequation (56) for 8/3t \ne//<ne> = 0. 6eff/<S>and °off/(°) are then obtained from equa-tions (27) and (28) , respectively, using equa-tions (53) and (54).

It is seen from figures 7 and 8 that theadditional terms derived in this paper do not havea major effect upon the interpretation of the ex-perimental data, at least as it is presently in-terpreted. The agreement with effective Hallparameter is slightly better and the agreementwith effective conductivity is slightly worse.The interesting point, however, is that the termscontained in *j can contribute a negative valueto the coefficient of the nonlinear term. Thisopens the possibility of instability occurring ina region where linear theory predicts stability,i.e., *i < 0. In this case, instability can occurfor *3 > *2

if the initial amplitude of thefluctuation is

In figure 9 we have plotted (for the conditions offig. 4 of ref. (5), i.e., 6C = 2, A,,, = A = 1) the

values of (neV(ne/ ^or an unstable solution inthe linear stable region 8 <. Bc = 2. Obviouslyfor these conditions the initial amplitude must belarge in order for the system to be unstable. Onemeans by which an initially large, amplitude fluc-tuation could occur would be propagation of fluc-tuations into the MHD generator from a preionizer.This may also be an explanation of the anomolousresults obtained in the experiments of refer-ence (3) during the application of preionization.

It should be emphasized that the predictionof instability in the linearly stable region bythe present theory is only a possibility since itoccurs as the nonlinear term begins to dominate.Higher order terms may provide sufficient dampingto stabilize this region. This question can onlybe answered by considering the full nonlinearproblem.

In MFD arc thrusters limited by geometry toapproximately rotational propagation, it has beenshown in reference (2) that the linear stability

tenn 4^, requires large values of BQ (the orderof the ratio of the mean circumference of the an-nular region to its width) for instability to occur.Since the analysis indicated that the values of BQin the throat area, where the instability is as-sumed to occur, were of order one it does not ap-pear that this mode of instability is of importancein these devices.

Concluding Remarks

The nonlinear effects of fluctuations in theelectron properties of a partially ionized gas havebeen considered on the basis of a third orderquasi-linear theory neglecting mode coupling. Thetwo limiting cases of frozen flow and infiniteionization rate have been investigated with appli-cation to MHD generator and high pressure MHD arcthrusters operation.

For MHD generators oscillations occurring inthe frozen flow limit are damped for normal operat-ing conditions. However, where strong preioniza-tion is used prior to entering the MHD duct suchoscillations can be excited and this may explainthe anomolous fluctuations observed in the experi-ments of reference (3). The effect of oscillationsin the infinite ionization limit has been previ-ously studied on the basis of a second order quasi-linear theory.(9) A comparison of these resultsto the third order theory developed in this papershow little difference in as far as the interpre-tation of present experimental data is possible.However, the present theory indicates the possi-bility of fluctuations occurring in the regionwhich is linearly stable provided fluctuations of afinite initial amplitude are present in the system.In the MHD generator case these may occur as a re-sult of strong preionization.

In high pressure MPD arcs geometrically con-fined between two concentric cylindrical electrodesit is argued that the most unstable oscillationoccurs for nearly pure rotation propagation. Inthis case oscillations arising in the infiniteionization limit are damped at least for high pres-sures. In the frozen flow limit oscillation canoccur. Based on the assumption of zero gas temper-ature in the onset region, which was used in refer-ence (2) to reduce the existing experimental datafor comparison with linear theory, the quasi-lineartheory predicts unstable growth of the fluctuation.Therefore, as a consequence of this assumption, a"steady state" fluctuation of finite amplitude isnot predicted by quasi-linear theory. The validityof the zero gas temperature assumption requiresfurther experimental information or a more completesteady state theory.

A.A,,, defined by eq. (55) and (62) , respec-tively

A ,a defined by eq. (8)601 601

B_ magnetic field vector

6_ unit vector in magnetic field direction

C ,C defined by eq. (48) and (49), respec-tively

m

n

P

R

r_

T

T

T.i

1' 2 3

Subscripts:

e

eff

i

electric field vector

magnitude of the charge of an electron

electric current density vector

Boltzmann constant

wave vector

particle mass

particle number density

pressure

defined by eq. (45)

position vector

temperature of neutrals and ions

temperature of electrons

temperature equivalent of ionizationpotential

time coordinate

gas flow velocity vector

defined by eq. (29)

Hall parameter

total electron momentum collision fre-quency

ionization coefficient

momentum collision frequency betweenith and jth particles

total electron energy collision fre-quency

recombination coefficient

electrical conductivity

period of oscillation in eq. (14). Inrest of text it is defined by eq. (30)

defined by eq. (57) through (59)

angle between i_ and {j)

wave frequency

heavy particles (neutral atoms and ions)

carrier (nonionizable) atoms

electron

effective value

ions.

component parallel to the vector i_

seed (ionizable) atoms

x,y,z components of x, y, z coordinate system

0 zeroth order value

Superscripts:

defined by eq. (11)

* value in coordinate system moving withgas

Brackets:

^. . .} average value of quantity within brackets

References

1. Kerrebrock, J. L., "Nonequilibrium Ionization Dueto Electron Heating: I. Theory," AIAA Jour-nal, Vol. 2, Ho. 6, June 1964, pp. 1072-1080.

2. Smith, J. M., "Electrothermal Instability - AnExplanation of the MPD Arc Thruster RotatingSpoke Phenomena," Paper 69-231, Mar. 1969,AIAA, New York, N.Y.

3. Louis, J. F., "High Hall Coefficient Experimentsin a Large Disc Generator," Electricity fromMHD, 1968, Vol. II, International AtomicEnergy Agency, Vienna, 1968, pp. 825-849.

4. Solves, A., "Instabilities in Non-equilibriumM.H.D. Plasmas - A Review," Paper 70-40, Jan.1970, AIAA, New York, N.Y.

5. Lutz, M. A., "Radiation and Its Effect on theNonequilibrium Properties of a Seeded Plasma,"AIAA Journal, Vol. 5, No. 8, Aug. 1967, pp.1416-1423.

6. Hiramoto, T., "Critical Condition for the Elec-trothermal Instability in a Partially IonizedPlasma," Physics of Fluids, Vol. 13, No. 6,June 1970, pp. 1492-1498.

7. Fowler, R. H., Statistical Mechanics, 2nd ed.,Cambridge University Press, Cambridge, England,1936.

8. Heighway, J. E. and Nichols, L. D., "BraytonCycle Magnetohydrodynamlc Power Generation WithNonequilibrium Conductivity," TN D-2651, NASA,Cleveland, Ohio, 1965.

9. Solbes, A., "Quasi-Linear Plane Wave Study ofElectrothermal Instabilities," Electricityfrom MHD, 1968, Vol. 1, International AtomicEnergy Agency, Vienna, 1968, pp. 499-518.

10. Brederlow, G., Feneberg, W., and Hodgson, R.,"The Non-Equilibrium Conductivity in an Argon-Potassium Plasma in Crossed Electric and Mag-netic Fields," Electricity from MHD, Vol. II,International Atomic Energy Agency, Vienna,1966, pp. 29-37.

10

o--ov£>xOI

NELECIONI

UNIFOELECTFPARTK

' s / / / / / / / / . ' /

/ / / . ' / / / ' / / '

IRON HEATING ANDZATION REGION

RM ARC REGIONiON PRESSURE » HEA1

:LE PRESSURE

' /

i/Y

/ / / / / , '

/ / / / / /

, , / /

' ' / / /

r rvft

GAS FLOW-

__ SONIC TRANSITIONREGION

SONICPOINT

Figure 1. - Theoretical model of MPD thruster.

INNER ELECTRODEOUTER ELECTRODE

(A) ASSUMED GEOMETRY.

(B) APPROXIMATE GEOMETRY USED FOR ANALYSIS.

Figure 2. -Thruster geometry.

E-6660

2.0

1.5

4.0,—

°eff 1.0

.5

.5 1.02

Figure 3. - Ratio ofeffective to zerothorder electricalconductivity as afunction of ampli-tude function forvarious angles ofpropagation.

tan

V5 2.0

-V5.-2.0

-4.0

-6.0

-8.0

Peff ENHANCED

2.0 4.0 6.0 8.0

Figure 4. - Value of tan <p at which j3eff = 30

as a function of P0.

W

l.Oi--2 d \B

= 2.67

R = 2.00

R = 1.50

R = 1.00

R = .70

10 12 14

Figure 5. - Amplitude function as a function of theHall parameter.

1.00REGION OF (Te) <TQ (NOT OF INTEREST)

REGION OF(Te)^T0

IV \ 2 o 23 2 53 2 o 3

Figure 6. - Damping and growth regions forrotation disturbance.

i-M

1.0

°eff

BREDERLOW ,'EXPERIMENT/REF. 10 -1

s rSOLBES THEORYREF. 9

.4

PRESENT THEORY

Figure 7. - Ratio of effective to average conductivity versusaverage Hall parameter.

1.0

Pfeff .6

PRESENT THEORY-'V \\ rSOLBES THEORY

BREDERLOW EXPERIMENT ->REF. 10

0 1

Figure 8. - Ratio of effective to average Hall parameterversus average Hall parameter.

.05,

.04

.03

.02

.Oil

</ STABLE FORFINITE AM-PLiTUDEFLUCTUATIONS

1.83

Figure 9. - Amplitude functionversus average Hall parameter.

NASA-Lewis-Com'l