Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators
This paper reports the results of the numerical studies of the axisymmetric flows in the plasma accelerators with the impenetrable equipotential electrodes of the various geometries. The calculations were performed using the two-dimensional two-fluid magnetohydrodynamic model taking into account the...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2010
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| Cite this: | Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators / A.N. Kozlov // Вопросы атомной науки и техники. — 2010. — № 6. — С. 97-99. — Бібліогр.: 9 назв. — англ. |
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| citation_txt | Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators / A.N. Kozlov // Вопросы атомной науки и техники. — 2010. — № 6. — С. 97-99. — Бібліогр.: 9 назв. — англ. |
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| description | This paper reports the results of the numerical studies of the axisymmetric flows in the plasma accelerators with the impenetrable equipotential electrodes of the various geometries. The calculations were performed using the two-dimensional two-fluid magnetohydrodynamic model taking into account the Hall effect and the conductivity tensor of the medium. The numerical experiments have allowed to reveal the influence of the electrode form on effect of occurrence of the current crisis.
Представлены результаты численных исследований осесимметричных потоков в плазменном ускорителе с непроницаемыми эквипотенциальными электродами различной геометрии. Расчеты выполнены в рамках двумерной двухжидкостной МГД-модели с учетом эффекта Холла и тензора проводимости среды. Численные эксперименты позволили выявить влияние формы электродов на эффект возникновения кризиса тока.
Представлено результати чисельних досліджень вісесиметричних потоків у плазмовому прискорювачі з непроникними еквіпотенціальними електродами різної геометрії. Розрахунки виконано в рамках двовимірної дворідинної МГД-моделі з урахуванням ефекту Холу і тензора провідності середовища. Чисельні експерименти дозволили виявити вплив форми електродів на ефект виникнення кризи струму.
The work has been executed at the financial support of Russian Foundation of Basic Research (grant 09-01-12056) and Russian Academy of Science (the program No. 14 (1) of Presidium RAS).
|
| first_indexed | 2025-12-07T16:45:57Z |
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INFLUENCE OF GEOMETRY OF THE IMPENETRABLE ELECTRODES
ON PROCESS OF FORMATION OF THE CURRENT CRISIS
IN THE PLASMA ACCELERATORS
A.N. Kozlov
Keldysh Institute for Applied Mathematics, RAS, Moscow, Russia
E-mail: ankoz@keldysh.ru
This paper reports the results of the numerical studies of the axisymmetric flows in the plasma accelerators with the
impenetrable equipotential electrodes of the various geometries. The calculations were performed using the two-
dimensional two-fluid magnetohydrodynamic model taking into account the Hall effect and the conductivity tensor of
the medium. The numerical experiments have allowed to reveal the influence of the electrode form on effect of
occurrence of the current crisis.
PACS: 52.30.Cv, 52.59.Dk, 52.65.-y
1. INTRODUCTION
Now the various modifications of the quasi-steady
plasma accelerators (QSPA) are tested (see, for example,
[1-6]), which allow partially to solve a problem of
interaction of the plasma streams with electrodes. In
practice the continuous impenetrable electrodes continue
to be used in a lot of cases.
Earlier the simplified numerical models which did not
include the dependence of coefficients in the equations
and boundary conditions from parameter ee τω were
used. The present researches are executed on the basis of
the full MHD model according to [7]. The various
modifications of the two-fluid MHD model answer the
statement of the various boundary conditions and have
been used earlier for the comparison of the two-
dimensional numerical and analytical models [8], and also
for the analysis of the ion current transport regime in
QSPA with the penetrated electrodes including the
additional longitudinal magnetic field [9]. In this case it is
a question of numerical researches in QSPA with
continuous electrodes of the various geometries at the
presence of the unique azimuthal magnetic field.
2. FORMULATION OF PROBLEM
Following [13] we consider that plasma is
quasineutral and ignore the inertia of
electrons ( ). We will restrict ourselves to the
dynamics of a hydrogen plasma (
nnn ei ==
ie mm <<
pi mmmZ === ,1 )
under condition of . TTT ei =≅
The initial transport equations using the above
assumptions lead to the following system:
0=+ Vρ
∂
ρ∂
div
t
; [ ]Hj
V
,1
c
P
td
d
=∇+ρ ;
( ) ( )
n
div
e
P
T
e
kdivQdivP
td
d e jjqV +∇
−
+−=+ ,
1γ
ερ ;
E
H
rot
tc
−=
∂
∂1 ; [ ] RHVE
ne
P
nec ee
11,1
+∇−−= ;
( )einerotc VVHj −==
π4
; ( )∇+
∂
∂
= ,V
ttd
d ;
( ) TccPPP vpei ρ−=+= 2 ; Tcv2=ε , where
; iVV = P is the total pressure, q is the heat flux;
nm=ρ is the heavy-particle density, and j is the
electric current. The force of friction is the
sum of the friction force due to the presence of the
relative velocity
TRRR j +=
jR
neie /jVVu −=−= and the thermal
force dependent on the temperature gradient. The
transport coefficients in magnetic field depend on
TR
ee τω .
As the normalizing quantities we use the following
dimensional constants: the channel length , the
characteristic values of the concentration
L
( )000 nmn =ρ , temperature , and azimuthal
magnetic field component at the channel inlet
. Here is the radius of the
outer electrode; is the discharge current in the
system. By means of these values the units of the
characteristic velocity
0T
0
0
0 /2 RcJHH d== ϕ 0R
dJ
000 4/ ρπHV = and the
current density LHcj π4/00 = are formed for
example. Four dimensionless parameters participate in
model:
04 n
m
Le
c
π
ξ = is the parameter characterizing
the role of the Hall effect; is the ratio
of the gas (
2
00 /8 HPπβ =
000 TnkP = ) and magnetic pressures;
ρνξτω /Hee = ; and
is the magnetic viscosity which is inverse proportional to
the magnetic Reynolds number corresponding to Spitzer
conductivity .
σπν 0
2 4/Re/1 VLcm ==
2/3
0Re Tm σ=
The statement of boundary conditions assumes that at
the channel inlet 0=z there is the subsonic plasma
flowing with the known distributions of density and
temperature.
Fig. 1. Geometry of electrodes
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. № 6. 97
Series: Plasma Physics (16), p. 97-99.
Further we believe as 1=ρ and at the inlet.
Without taking into account the equation of an electric
circuit we consider that the current has a constant and
1=T
constrHr == 0ϕ at where . The
boundary conditions assumes that the electrodes are
equipotential ( ) and impenetrable (
0=z LRr /00 =
0=τE 0=nV ). The
polarity of electrodes specified Fig. 1 answers the
standard experimental researches.
The form of the internal cathode is certain according
to the analytical model [8] for the external electrode
(line II on Fig. 1). The forms of anode
I and III are set by pieces of a parabola at
( ) constrzr a == 0
5.00 => zz .
3. CALCULATIONS OF PLASMA FLOWS
We choose as the characteristic units of a problem for
calculation of the base variant, for example, the following
values: ; ; 316
0 cm102 −⋅=n eVT 30 = kAJ d 500= ;
; . In this case the values of the
dimensionless parameters of the problem are
mL 6.0= mR 25.00 =
15.0=β ;
0027.0=ξ ; 8.3340 =σ . If and 1=T 1=ρ we have
9.0=ee τω and 003.0=ν .
The theoretical analysis [1] of the plasma dynamics
across a magnetic field in a vicinities of the
equipotential impenetrable electrode ( ,
ϕHH =
VE ⊥ 0=nV ,
) has been spent on the basis of the generalized
Ohm’s law. As the account of the Hall effect
and parameter
0=τE
ie VV ≠
ee τω leads to the occurrence of the
longitudinal Hall’s component of a current
⊥≅ jj ee τω|| and to the pushing of plasma from the
anode. In turn the concentration decrease in a vicinity of
an electrode increases the parameter ee τω and increases
a current along the anode and the pushing of plasma from
an electrode even more. As a result under certain
conditions there can be a full reorganization of the flow
structure and the large-amplitude oscillations occur. In
experiments this phenomenon exerts the greatest effect on
the volt-ampere characteristics when the discharge current
in the system cannot exceed some critical value . crJ
In model the decrease leads to the qualitative
reorganization of flow and to formation of the obviously
expressed layer in the vicinity of anode adequating to the
more high-velocity stream of the rarefied plasma.
0n
The distributions of a) longitudinal components of a
current and b) parameter zj ee τω along the anode for
three values , other parameters specified above and
are presented in Fig. 2. The
distribution of a radial current does not vary practically.
At the same time the decrease of parameter leads to
the essential growth of value
0n
( ) constrzra == 0
0n
zj . The essential growth
of a variable ee τω is simultaneously observed. The
given distributions answer the stationary flows calculated
by means of the relaxation method.
Even greater decrease of the characteristic
concentration of particles up to value
conducts to the qualitative
reorganization of processes. The flow is not become
stationary. The fast increase of values
316
0 cm1069.0 −⋅=n
zj and ee τω in
the vicinity of the outlet part of the anode is observed in
the full conformity with the theory. Accordingly for the
given magnitude of a discharge current the
value can be considered as the
critical value of the characteristic concentration . If
we have the laminar stationary flows. In case of
kAJ d 500=
316
0 cm107.0 −⋅=n
crn
crnn >0
crnn <0 the quickly increasing instability is observed.
As a result of a series of calculations for the various
values of the discharge current the critical values of
the characteristic plasma concentration at the inlet in
the accelerator channel presented in Fig. 3 have been
certain accordingly for three different profiles of anode I,
II and III represented on Fig. 1. It has appeared that in a
plane of variables ( , ) the boundary between the
laminar and unstable modes is the linear function. We can
see that values decrease at transition from a profile
of electrode I to geometry II and accordingly from the
form of an electrode II to III. In other words the
narrowing of the accelerating channel expands the area of
the parameters adequating to the laminar flows and
interferes with development of the current crisis.
dJ
crn
dJ crn
crn
-7
-6
-5
-4
-3
-2
-1
0
0,0 0,2 0,4 0,6 0,8 1,0
jz
3
2
Z
1
a)
a
0,0 0,2 0,4 0,6 0,8 1,0
0
1
2
3
4
5
6
7 ω τ
2
3
b)
1
Z
b
Fig. 2. Distributions of a) longitudinal current
component; b) parameter ee τω along anode:
1- n0=2×1016 cm-3, 2- n0 = 1016 cm-3, 3- n0=0.7×1016 cm-3
98
4. CONCLUSIONS
0 2 4 6 8 10
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8
III
III
instability
flows
steady-state
Jp 10 - 2(KA)
ncr10 - 16(cm - 3)
Researches have confirmed the theory of phenomenon
of the current crisis in the vicinity of anode in the plasma
accelerator channel with the continuous electrodes. The
formation of a layer near to the anode and the occurrence
of the processes which proceed to the current crisis are
revealed. The comparison to the experimental data
defining the presence of the critical modes has been
executed in the terms of the values and . The
transition from the traditional geometry of the external
electrode
dJ mJ
( ) constrzr a == 0 to the extending anode
(profile I on Fig. 1) promotes the occurrence of the
phenomenon of the current crisis.
10-2 (кА)
The work has been executed at the financial support of
Russian Foundation of Basic Research (grant 09-01-
12056) and Russian Academy of Science (the program
No. 14 (1) of Presidium RAS).
Fig. 3. The critical values of the characteristic plasma
concentration at the inlet in the accelerator channel
for different profiles of the anode (Fig. 1)
Earlier the experiments [1] have led to a following
approximate relation in which the critical
value of a discharge current participates, the
constant
KJJ mcr ≈/2
crJ
K depends basically on the geometry of the
channel, and is the mass flux (g/s)
expressed in the current units. This flux is easy for
defining in calculations.
im mmeJ /&& = m&
REFERENCES
1. A.I. Morozov. Introduction in Plasmadynamics.
Moscow: “Fizmatlit”, 2nd issue. 2008 (in Russian).
2. V.I. Tereshin, A.N. Bandura, O.V. Byrka,
V.V. Chebotarev, I.E. Garkusha, I. Landman,
V.A. Makhlaj, I.M. Neklyudov, D.G. Solyakov,
A.V. Tsarenko // Plasma Phys. Contr. Fusion. 2007,
v. 49, p. А231.
To each point of the graph represented in Fig. 3 there
corresponds the magnitude , and . It has
appeared that the corresponding graph in the plane of
variables ( , ) defines also the linear
dependences. In this plane of variables the straight lines
adequating to three various forms of the anode are in
essence parallel each other. On an inclination of these
straight lines it is easy to calculate the coefficient
m& mJ mJln
dJln mJln
α in
relations KJJ dm lnlnln −⋅= α or
which define the critical values of the mass flux at the
certain discharge current or on the contrary the critical
values of the discharge current at the known mass flux.
Calculations lead to value
KJJ md ≈/α
38.1≅α for profile I on Fig. 1,
42.1≅α for form of electrode II and 47.1≅α for
geometry III. The results of the numerical experiments
allow to speak about the comprehensible qualitative
conformity with available approximate experimental data.
3. V.G. Belan, S.P. Zolotarev, V.F. Levashov,
V.S. Mainashev, A.I. Morozov, V.L. Podkoviirov,
Iu.V. Skvortsov // Fiz. Plasmy. 1990, v.16, N 2, p.96.
4. S.I. Ananin, V.M. Astashinskii, E.A. Kostyukevich,
A.A. Man’kovskii, L.Ya. Min’ko // Plasma Physics
Reports. 1998, v. 24, p. 936.
5. G.A. Dyakonov, V.B. Tikhonov // Fiz. Plasmy. 1994,
v. 20, N 6, p. 533 (in Russian).
6. A.N. Kozlov, S.P. Drukarenko, N.S. Klimov,
A.A. Moskacheva, V.L. Podkovyrov // Problems of
Atomic Science and Technology. Series “Plasma
Physics”(15). 2009, N 1, p. 92-94.
7. S.I. Braginskii. Transport phenomena in plasma //
Reviews of Plasma Physics /ed. V.A. Leontovich. New
York: “Consultants Bureau”. 1966, v. 1, p. 253.
8. A.N. Kozlov // J. Plasma Physics. 2008, v. 74, N 2,
p. 261-286.
9. A.N. Kozlov // J. of Applied Mechanics and Technical
Physics. 2009, v. 50, N 3, p. 396-405.
Article received 13.09.10
ВЛИЯНИЕ ГЕОМЕТРИИ НЕПРОНИЦАЕМЫХ ЭЛЕКТРОДОВ НА ПРОЦЕСС ФОРМИРОВАНИЯ
КРИЗИСА ТОКА В ПЛАЗМЕННЫХ УСКОРИТЕЛЯХ
А.Н. Козлов
Представлены результаты численных исследований осесимметричных потоков в плазменном ускорителе с
непроницаемыми эквипотенциальными электродами различной геометрии. Расчеты выполнены в рамках
двумерной двухжидкостной МГД-модели с учетом эффекта Холла и тензора проводимости среды. Численные
эксперименты позволили выявить влияние формы электродов на эффект возникновения кризиса тока.
ВПЛИВ ГЕОМЕТРІЇ НЕПРОНИКНИХ ЕЛЕКТРОДІВ НА ПРОЦЕС ФОРМУВАННЯ КРИЗИ ТОКУ
В ПЛАЗМОВИХ ПРИСКОРЮВАЧАХ
А.М. Козлов
Представлено результати чисельних досліджень вісесиметричних потоків у плазмовому прискорювачі з
непроникними еквіпотенціальними електродами різної геометрії. Розрахунки виконано в рамках двовимірної
дворідинної МГД-моделі з урахуванням ефекту Холу і тензора провідності середовища. Чисельні експерименти
дозволили виявити вплив форми електродів на ефект виникнення кризи струму.
99
INFLUENCE OF GEOMETRY OF THE IMPENETRABLE ELECTRODES ON PROCESS OF FORMATION OF THE CURRENT CRISIS IN THE PLASMA ACCELERATORS
A.N. Kozlov
1. INTRODUCTION
ВЛИЯНИЕ ГЕОМЕТРИИ НЕПРОНИЦАЕМЫХ ЭЛЕКТРОДОВ НА ПРОЦЕСС ФОРМИРОВАНИЯ КРИЗИСА ТОКА В ПЛАЗМЕННЫХ УСКОРИТЕЛЯХ
ВПЛИВ ГЕОМЕТРІЇ НЕПРОНИКНИХ ЕЛЕКТРОДІВ НА ПРОЦЕС ФОРМУВАННЯ КРИЗИ ТОКУ В ПЛАЗМОВИХ ПРИСКОРЮВАЧАХ
|
| id | nasplib_isofts_kiev_ua-123456789-17470 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:45:57Z |
| publishDate | 2010 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kozlov, A.N. 2011-02-26T21:40:42Z 2011-02-26T21:40:42Z 2010 Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators / A.N. Kozlov // Вопросы атомной науки и техники. — 2010. — № 6. — С. 97-99. — Бібліогр.: 9 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17470 This paper reports the results of the numerical studies of the axisymmetric flows in the plasma accelerators with the impenetrable equipotential electrodes of the various geometries. The calculations were performed using the two-dimensional two-fluid magnetohydrodynamic model taking into account the Hall effect and the conductivity tensor of the medium. The numerical experiments have allowed to reveal the influence of the electrode form on effect of occurrence of the current crisis. Представлены результаты численных исследований осесимметричных потоков в плазменном ускорителе с непроницаемыми эквипотенциальными электродами различной геометрии. Расчеты выполнены в рамках двумерной двухжидкостной МГД-модели с учетом эффекта Холла и тензора проводимости среды. Численные эксперименты позволили выявить влияние формы электродов на эффект возникновения кризиса тока. Представлено результати чисельних досліджень вісесиметричних потоків у плазмовому прискорювачі з непроникними еквіпотенціальними електродами різної геометрії. Розрахунки виконано в рамках двовимірної дворідинної МГД-моделі з урахуванням ефекту Холу і тензора провідності середовища. Чисельні експерименти дозволили виявити вплив форми електродів на ефект виникнення кризи струму. The work has been executed at the financial support of Russian Foundation of Basic Research (grant 09-01-12056) and Russian Academy of Science (the program No. 14 (1) of Presidium RAS). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Динамика плазмы и взаимодействие плазма – стенка Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators Влияние геометрии непроницаемых электродов на процесс формирования кризиса тока в плазменных ускорителях Вплив геометрії непроникних електродів на процес формування кризи току в плазмових прискорювачах Article published earlier |
| spellingShingle | Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators Kozlov, A.N. Динамика плазмы и взаимодействие плазма – стенка |
| title | Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators |
| title_alt | Влияние геометрии непроницаемых электродов на процесс формирования кризиса тока в плазменных ускорителях Вплив геометрії непроникних електродів на процес формування кризи току в плазмових прискорювачах |
| title_full | Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators |
| title_fullStr | Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators |
| title_full_unstemmed | Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators |
| title_short | Influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators |
| title_sort | influence of geometry of the impenetrable electrodes on process of formation of the current crisis in the plasma accelerators |
| topic | Динамика плазмы и взаимодействие плазма – стенка |
| topic_facet | Динамика плазмы и взаимодействие плазма – стенка |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/17470 |
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