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S. CHABLES. CHABLEaa F. ROGIER F. ROGIERbb

a - a - ONERA, 2 av. Ed. Belin, 31055 TOULOUSE Cedex 4 ONERA, 2 av. Ed. Belin, 31055 TOULOUSE Cedex 4

b - b - MIP, Université Paul Sabatier, 118, route de Narbonne, 31062 MIP, Université Paul Sabatier, 118, route de Narbonne, 31062 TOULOUSE CedexTOULOUSE Cedex

Numerical Investigation and Numerical Investigation and Modeling of Stationary Modeling of Stationary Plasma Thruster Low Plasma Thruster Low Frequency OscillationsFrequency Oscillations

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Scheme Scheme of a of a SPTSPT

Magnetic field Electrons confined in the channel Electric field Ions accelerated

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Advantages : tAdvantages : thrustershrusters lower and lower and more more accurate accurate than chemical conventional systemthan chemical conventional systemss (thrust of (thrust of some mN).some mN).

Development since three decades by the Russians.Development since three decades by the Russians. Development nowadays in U.SA. (M.I.T.), in Development nowadays in U.SA. (M.I.T.), in

Russia, in Japan and in France (LPMI, CPAT, Russia, in Japan and in France (LPMI, CPAT, CNES, ONERA, LPGP, SNECMA, Astrium ...).CNES, ONERA, LPGP, SNECMA, Astrium ...).

Research axes :Research axes : - - clarifications of the mechanisms clarifications of the mechanisms responsible for the plasma conductivityresponsible for the plasma conductivity

- reduction of the divergence of the plumereduction of the divergence of the plume- reduction of the low frequency oscillationsreduction of the low frequency oscillations Numerical simulationsNumerical simulations

ContextContext

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1.1. IntroductionIntroduction

2.2. Physical modelPhysical model

3.3. Study of the linear instability modes Study of the linear instability modes

4.4. A simplified modelA simplified model

5.5. Control of the instabilitiesControl of the instabilities

6.6. ConclusionConclusion

PlanPlan

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Physical model : Physical model : ModeliModelizzationation

Assumptions :Assumptions : Ions transport scale time .Ions transport scale time . Strong electron-neutral elastic collision rate : Strong electron-neutral elastic collision rate :

Maxwellian fluid for the electrons.Maxwellian fluid for the electrons. BallisBallisttic neutrals and ions.ic neutrals and ions. Magnetic field not sensitive to the plasma.Magnetic field not sensitive to the plasma. Ions not sensitive to the magnetic field.Ions not sensitive to the magnetic field. Monokinetic neutral distributionMonokinetic neutral distribution QuasineutralityQuasineutrality Inclusion of an electron-wall collision frequency Inclusion of an electron-wall collision frequency

(Bœuf-Garrigues model)(Bœuf-Garrigues model)

00

mme

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Physical modelPhysical model

e ei

iss

eie

eiivixi

eix

nndxdUnknk

dxd

dxd

nnkdxdn

dxd

nnkfmefv

tf

nnkfvtf

100

0

00

0000

)))(/exp()(35)((

35

)()(

)(..

)(.

3R

vdfn

3R

vdfvvn

vdfvmn

2

2

1

Avec et

1D model

202

0

))(()/()(

nkmeB

nkme

ee

e

e))1(16exp()( 2max

LxBxB

and neutral and ion distribution functions, , and are the neutral, ion and electron densities. is the electron mean energy. is the electric potential, the magnetic field. and the ionization rate, the elastic collision rate and the inelastic collision rate. the electron-wall collision frequency and the electron mobility.

0f if 0n

in enB ik

ek k

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LinLinearizationearization

dx

dAv

vA

f

fF

vxF

FKv

FA

x

FA

t

F

vx

i

vx

0

1

10

0

00

0

0

0),,0(

..

solution. statesteady the, 00110

0

1

0

1

0

1 ),,( ,),,( ,),,( : terms thesecontainsK dwwxtfdyvytfdyvytfLx

Remark : the non local terms don’t allow us to conclude to the dissipation of the perturbed solution.

Friedrichs system

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Link linear – non linearLink linear – non linear

Discretization : the problem is unstable if the positive real part of the eigenvalues is positive

Discretized system :

Observation : linear instability The link between the ion Vlasov The link between the ion Vlasov equation and the elequation and the eleectric potential ctric potential is the mechanism responsible for is the mechanism responsible for the instabilities.the instabilities.

0. avec r rii

tjij tftft

tCtt

))(),(()(

)()(

1,

1,0

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LiLink linear – non linearnk linear – non linear

Link between the growth rate and the amplitude of Link between the growth rate and the amplitude of the non linear oscillationsthe non linear oscillations..

Amplitude of the non linear oscillations obtained from the transient model

Growth rate obtained from the linear model.

Variation function of . , , 0max0 vB

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Simplified modelSimplified model

Assumptions :Assumptions : Electron mean energy given and constant.Electron mean energy given and constant. Neutral density given and constantNeutral density given and constant.. Source term neglected.Source term neglected. Linear model :Linear model :

0, )(

0

0

0

tx

x

j

xn

x

x

f

xx

fv

t

f

R

with :22

0

00

)/()( ee

e

e meBnk

nk

m

m

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Theorical instabilityTheorical instability

Theorem :Theorem : The problem is bivalued. If is The problem is bivalued. If is solution so is solution too.solution so is solution too.

Unstable problemUnstable problem Relationship with Buneman instabilityRelationship with Buneman instability Control of the instabilities : topology of B, Control of the instabilities : topology of B,

electron-wall collision frequency.electron-wall collision frequency.

*

ii 24

112

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ConclusionConclusion

Treated points :Treated points : Spectral study of the linear instability with a Spectral study of the linear instability with a

stationary quasineutral hybrid model stationary quasineutral hybrid model relationship relationship between the growth rate and the amplitude of the between the growth rate and the amplitude of the non linear model.non linear model.

Simplified model to explain the instabilities Simplified model to explain the instabilities relationship with the Buneman instabilities.relationship with the Buneman instabilities.

Control of the instabilities by modifying the magnetic Control of the instabilities by modifying the magnetic field of the electron-wall collision frequency.field of the electron-wall collision frequency.

Predictive modelPredictive model

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