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    472

    ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2007, Vol. 47, No. 3, pp. 472486. Pleiades Publishing, Ltd., 2007.Original Russian Text A.S. Arkhipov, A.M. Bishaev, 2007, published in Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 3, pp. 490505.

    Three-Dimensional Numerical Simulation of the PlasmaPlume from a Stationary Plasma Thruster

    A. S. Arkhipov and A. M. Bishaev

    Research Institute of Applied Mechanics and Electrodynamics,Leningradskoe sh. 5, Moscow, 125080 Russia

    e-mail: [email protected]

    Received June 16, 2006

    Abstract

    A numerical method is proposed for simulating the low-density plasma plume exhaustedfrom a stationary plasma thruster in a three-dimensional setting. In contrast to the axisymmetric approx-imation, the problem is formulated so as to determine the effect of the backflow on the upstream regionand the thruster walls, which are of finite size. The numerical method is a generalization of rarefied gasnumerical methods to the case when the force field is not specified analytically. The method takes intoaccount the delta-function character of the boundary ion distribution function and the considerable dif-ference between the velocity scales of ions and neutral atoms, which transform into each other. Numer-ical results are presented that demonstrate the effect of some factors on the plasma plume.

    DOI: 10.1134/S0965542507030116

    Keywords:

    numerical simulation of a plasma plume, model kinetic equations, finite-difference numer-ical method

    INTRODUCTION

    The problem of the plasma flow exhausted from a stationary plasma thruster (SPT) in an axisymmetricsetting was studied in [1]. In [2], the flow was simulated using a special kinetic model that took into accountresonance recharge, i.e., a special interaction between the ions and neutral atoms with the largest interactioncross section. With the appearance of Pentium-type computers, the size of the computational domain wasincreased and numerical results were compared with ion current density distributions measured in experi-ments. A technique for correctly comparing experimental measurements with numerical data and a methodfor improving parameters in the boundary conditions were proposed in [3]. Simulation results revealed thatthere are ion currents in the plume that are directed toward the thruster exit face. To determine the backfloweffect on the thruster and enhance the capabilities of a software package so as to compute flows in generalgeometry, we considered the plume problem in a three-dimensional setting.

    1. STATEMENT OF THE PLUME PROBLEM

    The simulation of a plasma plume is based on a system of model kinetic equations for determining theion distribution functionf

    (

    x

    , x

    ) and the neutral distribution function g

    (

    x

    , w

    )

    . In dimensionless variables, thesystem has the form

    (1.1)

    This expression for the electric field is derived if we use the generalization of the thermalized potential

    hypothesis [1, 2] as set forward in [4, 5]. In (1.1), is the characteristic electron temperature (

    3 eV), U

    0

    is the discharge voltage (300600 V), and k

    is the Boltzmann constant. The right-hand side of (1.1) containsthe ionneutral (

    in

    ), neutralion (

    ni

    ), and neutralneutral (

    nn

    ) interaction frequencies. They are defined inthe same manner as in [2]. The Knudsen numbers involved in the collision frequency expressions are greaterthan or on the order of unity; andf

    0

    , g

    0

    , and g

    M

    model the inverse collisions integrals. Specifically,

    jfxj------- FEk

    fk--------+ in f0 f( ), wj

    gxj------- ni g0 g( ) nn gM g( ), F+

    ekT0e

    2U0----------- , Ek

    xk--------,= = = =

    5/2 ni

    ( )

    2/3

    , j k, 1 2 3., ,= =

    T0e

    f0 niB2/ B1T

    n( )( )3/2

    eB2c

    2/T

    n

    , c x un/ B2,= =

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    Vol. 47

    No. 3

    2007

    THREE-DIMENSIONAL NUMERICAL SIMULATION 473

    (1.2)

    whereB

    1

    = eU

    0

    /(

    k

    ),B

    2

    = eU

    0

    /(

    k

    )

    ; and for l

    = i

    , n are the characteristic temperatures of ions and

    neutral atoms, respectively. The macroscopic parameters in Maxwellian functions (1.2) are determined bythe integrals

    (1.3)

    The factor in the first integral in (1.3) is associated with the introduction of two velocity scales for the

    ion component:

    0

    = (2

    eU

    0

    /

    m

    )

    1/2

    , which is the velocity scale of directed ion motion, and c

    0

    = (2

    k

    /

    m

    )

    1/2

    ,

    which is the velocity scale of ion thermal motion. The scales are related by

    0

    > c0 .

    The geometry of the plume is illustrated in Fig. 1. The boxABCDA1B1C1D1 represents the thruster walls.All the faces are rigid surfaces, except for CC1D1Dwith a circular thruster exit in its center. The dashed lineinsideABCDA1B1C1D1shows the accelerating channelRG. The computations were performed in the exte-rior of this box. The boundary conditions for (1.1) are stated as follows. On CC1D1D(z= 0, 1 x1, 1 y1) for z0 and wz0, we have

    (1.4)

    (1.5)

    Here, , , , and are given functions ofxandy(note thatx=x1,y=x2 , andz=x3). They are defined

    in the same manner as in [3], except for , which can be nonzero in contrast to [3].R1andR2are the inner

    and outer dimensionless radii of the thruster exit, respectively, andBw= /Tw, where Twis the temperature

    of the thruster walls. The parameter nwin the second formula in (1.5) is determined by the balance between

    g0 nn

    B1/ B2Ti( )( )

    3/2e

    B1cw2/B2T

    i( ), cw w B2u

    i,= =

    gM nn Tn( ) 3/2 w u( )2/Tn{ },exp=

    T0i

    T0n

    T0l

    n

    nu

    n3

    2B1---------T u

    2+

    i

    B13/2

    1

    x

    x2

    f x,

    n

    nu

    n 3T/2 u2

    +( )

    n 1

    w

    w2

    g w.d=d=

    B13/2

    T0i

    fn/3/2 B1 x u( )

    2{ }, R2 r R1, r exp x

    2y

    2+ ,=

    0, r R1( );

    =

    gn/3/2 w u( )2{ }, R2 r R1, exp

    nw Bw/( )3/2

    Bww2

    { },exp r R2( ).

    =

    n n u u

    u

    T0n

    C GR

    z

    B

    A1

    DA

    y

    x

    L

    M

    K1

    K

    B1 C1

    D1

    N1

    N

    M1L1

    Fig. 1.

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    ARKHIPOV, BISHAEV

    the incident and reflected fluxes of neutral atoms and ions (see [13]) and is given by the formula

    The boundary conditions on the remaining faces ofABCDA1B1C1D1are set by the second formulas in (1.4)and (1.5) for n= (xn) 0and wn= (wn) 0, where n is the outward normal to the corresponding face.

    In addition to the boundary conditions on the rigid surfaces, we need boundary conditions at infinity.They are given by

    (1.6)

    where 2= n/ andB= /T. In the computations, conditions (1.6) were transferred to the boundaryof the computational domain. The computational domain is shown in Fig. 1 by dashed lines. Specifically,this is the box KLMNK1L1M1N1. An important point in this setting is that the computations are performedupstream and downstream of the plume. As a result, the effect of the backflow on the thruster can be takeninto account. In fact, we consider the problem of the flow over a thruster produced by the thruster itself. Itwas assumed in [13] that the thruster exit face is extended to infinity. Therefore, the boundary Maxwellianfunction of neutral atoms had a zero velocity and the temperature Tw . In the present setting, this boundaryfunction can take arbitrary values uand T. As a result, there appear additional dimensionless parameters

    determining the plasma flow. Due to these parameters, we can simulate the pumping-out of a vacuum cham-ber in which an SPT is usually placed in experimental studies.

    2. GENERAL ISSUES CONCERNING THE CONSTRUCTIONOF THE NUMERICAL SCHEME

    The mathematical problem presented in Section 1 can be stated as follows: Solve Eqs. (1.1)(1.3) for thefunctionsf and gsatisfying boundary conditions (1.4) and (1.5) and the conditions at infinity (1.6). Thisproblem was solved numerically by using a method that generalizes numerical techniques designed for sta-tionary kinetic equations in rarefied gas dynamics (see [6, 7]). Specifically, the method is based on the iter-ation procedure

    (2.1)

    Here, kis the iteration index. All the values in (2.1) indexed by k 1 are known, since they are expressed interms of the ion and neutral macroscopic parameters found at the (k 1)th iteration step.

    The basic method applied to rarefied gas dynamics is the method of characteristics. In most rarefied gasproblems, there is no force field (an exception is [8]). Therefore, the characteristics of the equations' differ-ential parts are straight lines. In the case under study, the equations of characteristics (ion trajectories) at aphase point (x, x) are

    (2.2)

    Given () and (), the solution to the first equation in (2.1) can be represented as

    (2.3)

    Here, tbis the value of when the characteristic intersects the computational domain. This can be the faceABCDA1B1C1D1or KLMNK1L1M1N1 . The parameterfbin (2.3) is the ion distribution function given on thecorresponding boundary.

    A key difference of this problem from rarefied gas dynamics is the presence of an electric field, which isknown only at nodes or cells in physical space. Determined numerically in the course of problem solving,

    nw 2 Bw wz g x y 0 w, , ,( ) wr wdd( )dwz

    0

    B13/21B2

    B1----- z f x y 0 x, , ,( ) r dd( )

    0

    dz+

    .=

    f x x,( ) 0, g x w,( ) 2 B/( )3/2

    B w u( )2

    { } x2 y2 z2+ + ,exp= =

    n0n

    T0n

    jfk

    xj-------- FEj

    k 1 fk

    j--------+ F+ 1f

    k, 1 in

    k 1, F+ 1f0

    k 1, wj

    gk

    xj-------- G+

    1G+

    2 2gk,+= = = =

    2 nik 1 nn

    k 1, G+

    1+ ni

    k 1g0

    k 1, G+

    2 nnk 1

    gMk 1

    .= = =

    dx

    d------ x,

    dx

    d------ E, x 0( ) x, x 0( ) x, E FEk 1 x ( )( ).= = = = =

    x x

    fk

    x x,( ) fb x tb( ) x

    tb( ),( ) 1 x s( )( ) sd0

    tb

    exp F+

    x ( ) x ( ),( ) 1 x s( )( ) sd0

    exp .d0

    tb

    +=

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    THREE-DIMENSIONAL NUMERICAL SIMULATION 475

    the electric field may be such that the ion trajectory never reaches the boundary of the computationaldomain. This situation is fundamentally different from that in [8], where the characteristics were extractedanalytically.

    Formula (2.3) (the solution to the second equation in (2.1) is written likewise, see [7]) can be used toanalyze the features of the solution to Eqs. (2.1). It can be seen thatfkand gkare discontinuous in the velocityspace. Moreover, sinceB1in (1.3) is considerably larger than unity (B1= 20), the function on the circularthruster exit is similar to a delta function. Since the area of the thruster exit is much smaller than that of

    CC1D1D, if no care is taken, then few of the characteristics leaving the thruster exit will arrive at the pointxand most of the information about the plume will be lost. Since the value ofB1is large, the ion distributionfunction has several peaks in the velocity space. Due to the recharge processes, analogous peaks appear inthe neutral distribution function. In the case of a rarefied gas, the distribution function is characterized byseveral peaks in hypersonic flows. The difficulty in the computation of macroscopic parameters in this caseis associated with the fact that, due to the hot particles, the distribution function has a slowly decayingtail. As a result, the infinite limits in integrals (1.3) are modeled by large numbers, while the step size in thevelocity space is chosen according to the narrow delta-function-like peak of the distribution function causedby cold particles. This leads to an unacceptably large CPU time required for solving the problem with somany velocity nodes.

    The following procedure is proposed for overcoming this difficulty. Since Eqs. (2.1) are linear, their solu-tion at the kth iteration can be represented as fk=f1+f2and g

    k= g1+ g2+ g3 , where the functions satisfythe equations

    (2.4)

    Here,f1and g1obey boundary conditions (1.3) and (1.4), whilef2, g2, andg3vanish on the boundary of thecomputational domain. It is assumed that each of these functions is responsible for its own peak in the dis-tribution function. Moreover, these functions, except forf1 , are continuous in the velocity space. Therefore,at each point of physical space, we can introduce a special grid in the velocity space and effectively calculatethe corresponding contributions to the macroscopic parameters at a small number of velocity nodes.

    3. NUMERICAL SCHEME FOR DETERMINING f1

    Equation (2.4) forf1is solved together with boundary condition (1.3). The functionf1vanishes at infinityand on all the boundaries, except for the thruster exit. Therefore, it is the ions exhausted through the thrusterexit that form the ion plume. The influence of these particles is propagated along characteristics (2.2) leav-ing the thruster exit and arriving at the pointxin physical space. SinceB1> 1, the influence of the thruster

    exit is transferred by characteristics along which the ion velocity differs little from u= { (r), (r), (r)}.

    The value of (r)is taken from experiments (see [3]). Specifically, it is on the order of unity over the entire

    thruster exit. The electric field componentEzis of order F. Since Fin the plume is small (0.01), the electricfield has a small effect on the ion trajectories determining the support of the distribution function. Therefore,the thruster exit affects the downstream region, which corresponds toz0 in Fig. 1.

    The characteristics arriving at the pointxcan be represented as

    (3.1)

    For characteristics (3.1) leaving the thruster exit, the velocity x is such thatD= { (tb) 0,R2rR1, r=

    }, where tbis determined by the relation (tb) = 0. The representation offkimplies that =

    if1xi-------- Ei

    f1 i--------+ 1f1, i

    f2xi-------- Ei

    f2 i--------+ 1F

    + 1f2,= =

    wig1xi-------- 2g1, wi

    gsxi-------- sGs 1

    + 2gs, s 2 3, i, 1 2 3., ,= = = =

    ur uz u

    uz

    x s( ) x E x ( )( ) , x s( )d0

    s

    x xs E x ( )( ) d0

    .d0

    s

    += =

    z

    x2

    tb( ) y2

    tb( )+ z nik

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    ARKHIPOV, BISHAEV

    n1+ n2 . Using the notation introduced above, we have

    (3.2)

    Define

    Then, tb=z*/zandD= {zAz0,R2 R2}. In integral (3.2), we switch tothe new integration variables

    (3.3)

    Under transformation (3.3), the domainDbecomes = {Azp,R2rR1, 0 2}. The Jacobianof (3.3) can be written asJ= 2r , where

    After switching to new variables, integral (2.2) is evaluated over the thruster exit area with respect to a singlevelocity variable. This makes it possible to develop a numerical integration scheme taking into account that

    the distribution function is of the delta-function type.The integration intervals in rand are divided into m0and n0subintervals, respectively; i.e.,

    Here, rm= + (m 1)rfor m= 1, 2, , m0and n= (n 1)for n= 1, 2, , n0, where r= (R1R2)/m0and = 2/n0. In each double integral over the domain [rm1, rm] [n 1, n], the values of x, A, (r),

    (r), and are assumed to be constant and are calculated at the points = (rm 1+ rm)/2 and = (n1+

    n)/2. Switching fromptop=Az+ + c , we obtain

    (3.4)

    n1 B1/( )3/2

    n r( ) 3/2 B1 x tb( ) u r( )( )2

    x s( )( ) sd0

    tb

    exp x.d

    D

    =

    x tb( ) x* xtb, x* x E x ( )( ) d0

    , x*d0

    tb

    + x* y* z*, ,{ },= = =

    x tb( ) x A, A E x s( )( ) s, Ad0

    tb

    Ax Ay Az, ,{ }.= = =

    x* xtb( )2

    y* y( )2

    +

    x x* r cos( ), y y* r sin( ), z p, p/z*.= = = =

    D

    J

    J 1 r1 sin x*

    --------- y*

    ---------cos

    x*r

    ---------cos y*r

    ---------sin r1 x* y*,( )

    r ,( )-----------------------+ +=

    +y* r sin

    z*r------------------------- z*

    --------cos r z*

    r--------sin

    x* z*,( ) r ,( )

    -----------------------+ x* r cos

    z*r-------------------------- r z*

    r--------cos z*

    --------sin

    y* z*,( ) r ,( )

    ----------------------+ ,+

    Q G,( ) r ,( )

    -------------------Qr-------

    G-------

    Q-------

    Gr-------.=

    ( ) rdd0

    2

    R1

    1

    ( ) r.ddn 1

    n

    rm 1

    rm

    n 1=

    n0

    m 1=

    m0

    =

    R1

    n

    u J rm n

    uz B11/2

    n1i

    Dmn , Dmnn 1=

    n0

    m 1=

    m0

    n2r J B1e p2

    1

    Azp--------

    3/2

    1

    G,d

    uz B1

    m 1

    m

    rm 1

    rm

    =

    dG e2B1g

    2

    dcddr, g2 r ( )cos qu r r( )+[ ]2

    ( )sin u+[ ]2, + x s( )( ) s,d

    0

    tb

    = = =

    x* qAx( )2

    y* qAy( )2

    + , y* qAy( )/x* qAx( ), qarctan 1.= = =

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    THREE-DIMENSIONAL NUMERICAL SIMULATION 477

    Here, the integral with respect to cis evaluated using Gaussian quadrature formulas of the form

    where Ckand ckare the weights and nodes of the corresponding quadrature formula. When 3, the

    integral is evaluated by the Gaussian quadrature formula for integrals of the form (c) dc. When

    < 3, the integral is divided into two, namely,

    The first integral is evaluated using the Gaussian quadratures for integrals of the form (u)du (the inte-

    gration limits can be made symmetric by applying a linear change of variables). The second integral, inwhich the substitution c2= tis made beforehand, is evaluated by the Gaussian quadrature formulas for inte-

    grals (t)etdt. Introducing the new variables t= r kcos( ) + qkand = sin( ) + qk/kin

    (3.4) and evaluating the double integrals, we obtain

    The difficulty in using the scheme described above is that there are no analytical expressions for x*,Az,J, Az/p, and the derivatives inJ; therefore, they have to be determined numerically. For this purpose, weneed to find characteristics (2.2) that, when leaving the thruster exit with velocities belonging to the supportof the given boundary distribution function, arrive at a given point xin physical space. On such character-istics, we have

    Since 0 on these trajectories, they can be found by solving the problem

    (3.5)

    This problem was solved by the relaxation method according to the scheme q/t=Lq, q(0) = q0, q(z) = q1 .The system of equations was solved numerically using the three-point difference scheme

    where all the nonlinear terms were taken from the previous time step. The difference problem was solvedby usual three-diagonal Gaussian elimination (see [9]). In the case under study, the well-posedness and sta-

    Q h c( )e c2

    cd

    uz B1

    Ckh ck( ),k 1=

    k0

    =

    uz B1

    h

    ec

    2

    uz B1

    Q h c( )e c2

    cd

    uz B1

    0

    2t1/2( )1e

    th t( ) t.d

    0

    +=

    ha

    a

    h0

    ur u

    n1 CkSk/4 qkn/k Qm Rmur( ) nRm+( ), where Skn 1=

    n0

    k 1=

    k0

    m 1=

    m0

    Jk/ 1 Azp--------

    k

    ek 1/2 ,= =

    n n( )Erf kk n( )sgn n 1( )Erf kk n 1( )sgn[ ], n n/, k( ),cos= = =

    Qm B1( )1/2 k

    2um 1

    2( )exp k

    2um

    2( )exp[ ],=

    Rm um( )Erf k um( )sgn um 1( )Erf k um 1( )sgn[ ],=

    ul rl k k( )cos urqk, l+ m 1 , m l, k B1.= = =

    x 0( ) x, z tb( ) 0, x tb( ) rm n, y tb( )cos rm n, xz x( )sin pk ckB1

    1/2uz rm( ) Az.+ += = = = = =

    z

    Lqd

    2q

    dz2

    --------1

    z2

    -----dq

    dz------ Eq

    + 0, q x z( ) y z( ),{ }, Eq Ex Ey,{ }, 0 z z, = = = =

    z2

    pk2

    2 Ez q s( ) s,( )( ) s, q 0( )dz

    z

    rm ncos rm nsin,{ } q0, q z( ) x y,{ } q1.= = = = =

    qil 1+

    qil

    t---------------------

    qi 1+l 1+

    2qil 1+

    qi 1l 1+

    +

    z2----------------------------------------------- Ci

    qi 1+l 1+

    qi 1l 1+

    2z--------------------------- Fi, Ci+

    Ez

    z2

    -----

    i

    l

    , FiEq

    z2

    ------

    i

    l

    ,= = =

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    ARKHIPOV, BISHAEV

    bility condition given in [9] implies that the time step t can be arbitrary if ei= z|Ci|/2 < 1 and is related toz by 2t z2/(ei 1) otherwise. It was assumed that a steady-state regime had been achieved if

    < 0.5p, wherepis the minimum cell size in physical space. When the

    right-hand side of the expression for became negative, the corresponding coefficient Ckwas set equal to

    zero.

    The schemes described above were used in all the computations of the plume problem. Note that mostof the CPU time per iteration was spent on these computations, but they were necessary, since correct allow-ance for the thruster exit made it possible to determine the entire structure of the plume in the downstreamregion.

    Figures 2 and 3 show the lines of equal ion density [1] as produced by different computational methods.Specifically, the results in Fig. 2 were obtained when the distribution function was not partitioned and thecharacteristic from the point xreached the boundary. For Fig. 3, the same computations were performedwith a partitioned distribution function using the above-described procedure for determining the contribu-tion off1 . The resulting density distribution patterns differ significantly. It can be seen that Fig. 2 lacks a

    crossover (i.e., a region of higher ion densities on the axis of symmetry [13]). Thus, although the procedure

    requires considerable CPU time, it correctly reproduces the plasma plume pattern.Asz 0, the integrand for r and in (3.4) becomes similar to a delta function even whenB1 1. The

    exact contribution of f1 is n1= /{2[1 + Erf( )]}. For z 0, the formula derived above for the

    numerical computation of n1yields

    Therefore, the above formulas are at least first-order accurate with respect to r whenz 0. The evalua-tion of the integral with respect to is as accurate as the Gaussian quadrature formula. In nondegeneratecases, the formulas constructed are second-order accurate in r and . This can easily be seen by expand-

    ing all the exponentials and Erf functions in Taylor series.

    When k 0, the values of the density and other macroscopic parameters can increase strongly. Thiseffect is especially pronounced in the axisymmetric case, when the ion density on the axis of symmetry

    becomes on the order of . It is easy to see that the numerical scheme then produces results that are sec-

    ond-order accurate in and r.

    Thus, the computational scheme is uniformly applicable; i.e., it correctly produces the leading asymp-totic term without a specially chosen integration step. Some general ideas behind the construction of suchschemes can be found in [10].

    xl 1+

    xl

    ( )2

    yl 1+

    yl

    ( )2

    +

    0 z z max

    z2

    n uz B1

    n1i

    n rm( ) Ckh pk( ).k 1=

    k0

    B1

    0 1.5 3.0 4.5 6.0z

    1.2

    1.8

    2.4

    3.0 0.01

    0.1

    0.20.50.5

    0.20.2

    0.1

    0.01

    0 1.5 4.53.0 6.0z

    1.2

    1.8

    2.4

    3.00.01 0.02

    0.05

    0.1

    0.2

    0.4

    0.61.01.52.00.2 0.4

    0.6

    Fig. 2. Fig. 3.

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    THREE-DIMENSIONAL NUMERICAL SIMULATION 479

    The function g1(see above), the integrals of which determine the contributions of the boundaries to theneutral macroscopic parameters, can be represented in the kth approximation as

    (3.6)

    where

    The first term in (3.6) describes the effect of the thruster exit. This part of g1is determined using the proce-

    dure constructed for ions. In this case, x* = x,Az= 0, and = 1, and the computation time is reduced con-siderably. In contrast to the ion component, the contribution of this term to the macroscopic parameters ofthe neutral atoms is rather small. This is explained by the fact that the distribution function of neutral atomsexhausted from the thruster exit is not of the delta-function type and the thruster exit is small in area as com-pared with the thruster exit face.

    The second term in (3.6) determines the contributions of the rigid surfaces to the neutral distributionfunction. In the three-dimensional statement of the plume problem, these surfaces are the six faces of theplasma thruster.

    A numerical scheme for determining the contribution from any term in (3.6) to the macroscopic param-eters is constructed as above; i.e., we switch to new integration variables; as a result, the integration is per-formed over the area of the corresponding face with respect to a single velocity variable. The relevant for-mulas are rather simple and are easy to derive by following the method described.

    4. NUMERICAL SCHEME FOR DETERMINING f2

    The functionf2in the above representation of fkobeys the second equation in (2.4). Its solution under

    zero boundary conditions is written as

    (4.1)

    wherefdenotesf2 . The computation off according to (4.1) is based on the following procedure:

    (4.2)

    Here, F+() and1(s) are F+( (), ()) and1( (s)), respectively, and xk= {xk,yl,zm}and xi= {xi, yj,zn} are phase-space nodes, which must be prescribed. When = t, the characteristic enters the cell shownin Fig. 4 (tis known). When = t+1, the characteristic leaves this cell (is to be found). The electric fieldis assumed constant inside the cell and is determined at its center. Specifically, the electric field potentialcalculated at nodes of physical space is numerically differentiated and is then averaged. Due to this assump-

    tion, the expressions for () and () inside the cell can be written as

    (4.3)

    where x1= (t), x1= (t), and is the electric field strength inside the cell. Denote by xb= {xb,yb,z}the cell faces that could be reached by the characteristic if the motion were at the constant velocity x1 . Set-

    ting () = xb in (4.3), we find the value t= t t(= 1, 2, 3) for which an ion moving in the constant

    electric field E would reach the corresponding face ( =xb, =yb, =zb):

    (4.4)

    g1 x w,( ) 3/2

    ew u xw

    0( )[ ]2

    0nw

    lxw

    l( ) Bw/( )3/2

    eBww

    2 l

    ,

    l 1=

    6

    +=

    xw x wtbl, l 2

    k 1x ws( ) s, ld

    0

    tbl

    0 6., ,= = =

    J

    f x x,( ) F+ x ( ) x ( ),( ) 1 x s( )( ) sd0

    exp ,d0

    tb

    =

    f xk

    xi,( ) D, D

    1=

    0

    1 s( ) sd0

    t

    F+ ( ) 1 s( ) sd

    t

    exp .dtt

    t 1+

    exp= =

    x x x

    x x

    x ( ) x1 x1 t( ) E t( )2/2 , x ( )+ x1 E t( ),= =

    x x E

    x

    xb

    x1b

    x2b

    x3b

    t 2 xb

    x1

    ( )/ 1 1

    ( ) Dsgn+( ), D 1( )

    22 xb

    x1

    ( ).= =

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    A positive value ofDin (4.4) is a criterion for the curveABto reach . IfD< 0, then there is a point inside

    the cell at which the corresponding component of changes its sign. The characteristic no longer reaches

    but turns toward = + x (xis the mesh size in the corresponding physical-space coor-

    dinate). In this case, tis determined by the condition that, after turning, an ion reaches the face . Spe-cifically,

    The value t+1is given by the formula t+1= t+ t, where t= min{t1, t2, t3}.

    After t+1has been found, we can calculate D. This is done using a modification of the scheme with

    exponentials. In our case, we have

    (4.5)

    where

    Recall that this scheme is second-order accurate in s. When s 1, (4.5) becomes the usual trapezoidal

    rule. For s> 1, scheme (4.5) gives correct asymptotics up to (s)2. In contrast to [6], varies along the

    characteristic. If the electric field is potential, then varies over a cell in the same manner as the macro-scopic parameters. Therefore, the use of scheme (4.5) is natural.

    In contrast to rarefied gas dynamics, the particle trajectory in problems with a force field may never reachthe boundary of the computational domain. In this case, we set tb= , but the integrals in (4.1) converge,which was used in the computations.

    In this study, as in [6, 7], the distribution function was not stored (only recently computers have appearedthat can store six-dimensional arrays of distribution functions). The contributions off2to the macroscopic

    xb

    x

    xb

    xh

    xb 1

    ( )sgn

    xh

    t 1/E 2 xh

    x

    *

    ( )/E, x*

    + x1 1

    ( )2/ 2E( ).= =

    D 1 q( ) qd0

    t

    F+

    1 es

    ( )( F 1++

    F+

    ( ) e s 1 e s( )/s+( ),+exp=

    s 1 q( ) qdt

    t 1+

    1 1 1++( ) t/2 , Fl+ x tl( ) x tl( ),( ), l 1.+,= =

    x

    x

    At

    B t+ 1

    Fig. 4.

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    THREE-DIMENSIONAL NUMERICAL SIMULATION 481

    parameters were calculated using the quadrature formulas

    (4.6)

    whereAnmare the weights of a quadrature formula and m0n00is the number of velocity nodes. Forfixed xijl, we calculatedffor each xmnby using (4.4) and (4.5), thus, making a contribution to the right-handside of (4.6). Then,f2was computed at the next point in physical space. This procedure was performed with-

    out storing the distribution function at nodes of the six-dimensional phase space. In (4.1), we had F+~

    (B2/B1)3/2exp[B2/ (x )

    2]. In the case of contributions to ion macroscopic parameters, all the

    integrals were multiplied by andB2 1000, so that the integrand in (4.1) was nearly a delta function.

    n

    nu

    n 3/ 2T( ) u2+( )

    n

    k

    Amn

    1

    wmn

    wmn2

    g xij l wmn ,( ), 1=

    0

    n 1=

    n0

    m 1=

    m0

    =

    Tnk 1

    unk/ B2

    B13/2

    3 30.0003 84 840.000084 19 190.00019

    72 720.00072

    SPTPTSPT

    84 840.000084

    14 140.00014 19 190.00019 35 350.00035

    7 70.07

    32 320.032

    1 81 80.108

    1461460.1462612610.261 2992990.299

    2232230.223 1851850.18565650.65

    32 320.032

    1.00

    0.71

    0

    0.71

    1.002.66 2.00 1.33 0.67 0 0 1 2

    Fig. 5.

    Fig. 6.

    7 70.07

    32 320.032

    1 81 80.108

    1461460.146 4524520.452

    2332330.233 1841840.1846436430.643

    1.00

    0.71

    0

    0.71

    1.002.66 2.00 1.33 0.67 0 0 1 2

    22 220.00022

    SPTPTSPT

    34 340.00034 1 10.0001

    16 160.00016

    1 10.0001 43 430.000043

    53 530.00053

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    THREE-DIMENSIONAL NUMERICAL SIMULATION 483

    upstream and downstream of the plume, we were able to determine the influence of ion currents on theseregions.

    First, the thrust was determined with sputtering taken into account. The force exerted by the flow on thevehicle is

    (5.1)

    Generally, the momentum received from the neutral component should be taken into account on the right-hand side of (5.1), but it is considerably lower than the momentum received from the ions. The computationsof Fzshowed that the basic part of this thrust is produced by the ion flow exhausted from the thruster exit.The contribution of the remaining faces does not exceed 0.3% of this value, and all of them produce thrust,except forBB1C1C, which exerts drag.

    Figures 5 and 6 show ion density contour lines in the planesy=xandy= x, respectively, for = 0 in

    the regionz0 (right panels) andlz0 (left panels), where lis the thruster length. The thruster is shownin black. Because of the symmetry of the thruster, the contour line distributions in these planes must be iden-tical. Therefore, the differences they exhibit estimate the accuracy of the numerical method used. A com-

    F mi xnf xd .dS

    =

    u

    35 350.035 35 350.035

    35 350.035

    35 350.035

    66 660.066

    66 660.066

    1921920.192

    1921920.1921291290.129 1291290.129

    98 980.098

    98 980.098 2242240.224

    98 980.098

    1611610.161

    1611610.161

    43 430.043

    43 430.043

    43 430.043

    82 820.082 12120.12

    82 820.082

    82 820.082 1591590.159

    1591590.159

    2372370.237 2672670.267

    12120.12

    1981980.198

    12120.12

    0.5 0 0.51.0

    0.066 0.190 0.310

    0.5 0 0.51.0

    0.054 0.15 0.26

    1.0

    0.5

    0

    0.5

    1.0 0 0.5 1.00.5

    1.0

    0.5

    0

    0.5

    1.0 0 0.5 1.00.5

    Fig. 10.

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    ARKHIPOV, BISHAEV

    parison of the figures shows that the contour lines with values equal to or larger than one-thousandth arereproduced fairly well. Figure 7 displays contour lines in the planeXOZ(y= 0).

    By comparing the plots in Figs. 5 and 6, we can reveal the effect of the problems nonaxisymmetry onthe flow pattern. It can be seen that the plume is virtually axisymmetric in the downstream region. Forz< 0(upstream region), there is a small difference in the distribution of the ion density contour lines caused bythe nonaxisymmetry of the problem.

    An analysis of the plasma flow in the SPT channel shows that the plume exhausted from the thruster must

    be swirled. Estimates show that amounts to 57% of . Since computations could be performed in arbi-

    trary geometry, we were able to determine the swirl effect on the distribution of the plume parameters.We computed the plasma flow in which involved in boundary condition (1.4) was specified as =

    0.1(1 + cos).Figures 8 and 9 show the ion current streamlines for = 0 and 0, respectively, i.e., the integral

    curves of the system dx/dt= qui, x(0) = xw, xwABCDA1B1C1D1 , where q= 1 (light lines) if (xw) 0and q= 1 (dark lines) otherwise. Overall, the streamline pattern with = 0 is similar to that with 0.The difference is that the backflow terminates onABCD andBB1C1Cwhen = 0, while, for the swirling

    plume, it terminates only onBB1C1C, while starting atABCD and terminating on the thruster exit face.

    u uz

    u u

    u u

    uni

    u u

    u

    45 450.045

    45 450.045

    45 450.045 65 650.065

    65 650.065

    65 650.065 1 31 30.103

    1 31 30.103

    84 840.084

    84 840.084

    1421420.142 1221220.122

    1221220.122 26 260.026

    26 260.026

    26 260.026

    26 260.026

    0.5 0 0.51.0

    0.043 0.12 0.02

    0.5 0 0.51.0

    0.038 0.099 0.161.0

    0.5

    0

    0.5

    1.0 0 0.5 1.00.5

    1.0

    0.5

    0

    0.5

    1.0 0 0.5 1.00.5

    53 530.053

    53 530.053

    77 770.077

    77 770.077

    1 11 10.101

    1 11 10.101 1251250.125

    1251250.125

    1491490.149

    28 280.028

    28 280.028 28 280.028

    28 280.028

    1731730.173

    Fig. 11.

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    THREE-DIMENSIONAL NUMERICAL SIMULATION 485

    Figures 1012 display the plots of ni= ni(x,y,z0) (upper panels) and contour lines of these functions in

    the planez=z0(lower panels) for = 0 (left panels) and 0 (right panels). Forz0= 1 (Fig. 10), we cansee that the location of the ion density peak and its value are affected by the plume swirl. For = 0, the

    peak lies on thezaxis (i.e., crossover occurs), while, for 0, the peak is noticeably smaller and, judgingfrom the contour lines, corresponds to the thruster exit.

    Atz0= 1.5 (Fig. 11), the behavior of the ion density is noticeably different. When 0, two peaksbegin to form in the ion density distribution.

    Atz0= 2 (Fig. 12), they finally diverge and the differences in the behavior of the ion density in the plumeswith and without swirl are exhibited most clearly. It can be seen that no crossover occurs in this case, sincethe peaks correspond to the influence of the thruster exit. Thus, the fact that no crossover is sometimes

    observed in experiments can be caused by the swirl of the plume. The dependence of on remainsvague. The presence of two peaks in the density distribution may be caused by the specific character of the

    given dependence of on the azimuthal angle. This issue is to be clarified in experiments.

    u u

    u

    u

    u

    u

    u

    Fig. 12.

    19 190.019

    19 190.019

    19 190.019

    19 190.019

    31 310.031

    31 310.031

    31 310.031

    43 430.043

    43 430.043 56 560.056

    56 560.056

    56 560.056 68 680.068

    68 680.068

    68 680.068

    93 930.093

    93 930.093

    81 810.081

    81 810.081

    81 810.081 43 430.043

    43 430.043

    23 230.023 23 230.023

    23 230.023 23 230.023 38 380.038

    38 380.038

    38 380.038

    54 540.054

    54 540.054

    85 850.085

    85 850.085

    110.1

    69 690.069

    69 690.069 69 690.069

    85 850.085 1161160.116 54 540.054

    0.5 0 0.51.0

    0.032 0.82 0.13

    0.5 0 0.51.0

    0.026 0.066 0.11

    1.0

    0.5

    0

    0.5

    0 0.5 1.00.5

    1.0

    0.5

    0

    0.5

    1.0 0 0.5 1.00.51.0

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    ARKHIPOV, BISHAEV

    To conclude, we note that the possibility of storing multidimensional arrays of distribution functions hasmotivated intensive development of nonstationary numerical methods for rarefied gas dynamics and kinetictheory. Therefore, the next step in the simulation of electric thruster plumes should be concerned with thecreation of nonstationary methods for solving problem (1.1)(1.6). The main difficulty to overcome is totake into account the delta-function character of the boundary ion distribution function and the large differ-ence between the ion and neutral velocity scales. The standard versions of the particle-in-cell method arenot applicable in this case.

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    2. A. M. Bishaev, V. K. Kalashnikov, and V. Kim, A Numerical Study of the Rarefied Plasma Plume of a StationaryAccelerator with Closed-Loop Electron Drift, Fiz. Plazmy 18, 698708 (1992).

    3. A. M. Bishaev, V. K. Kalashnikov, V. Kim, and A. V. Shavykina, Numerical Modeling of the Propagation of aPlasma Plume Produced by a Stationary Plasma Thruster in a Low-Pressure Gas, Fiz. Plazmy 24, 989995 (1998)[Plasma Phys. Rep. 24, 923928 (1998)].

    4. B. I. Volkov, A. I. Morozov, A. G. Sveshnikov, and S. A. Yakunin, Numerical Simulation of Ion Dynamics in Sys-tems with a Closed Drift Circuit, Fiz. Plazmy 7, 245253 (1981).

    5. P. I. Zhevandrov, A. I. Morozov, and S. A. Yakunin, Dynamics of a Plasma Produced by Ionizing a Low-DensityGas, Fiz. Plazmy 10, 353365 (1984).

    6. E. M. Shakhov,Method for Studying Rarefied Gas Flows(Nauka, Moscow, 1974) [in Russian].

    7. I. N. Larina, Drag of a Sphere in a Highly Rarefied Gas, in Numerical Methods in Rarefied Gas Dynamics(Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1973), pp. 2230 [in Russian].

    8. V. I. Zhuk, Solution of a Gas Kinetic Equation under Planets Gravity, Dokl. Akad. Nauk SSSR 233, 325328(1977).

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