Numerical Simulation of Glow Discharge in a Magnetic Field Through the Solution of the Boltzmann Equation

Transcription

1 Numrical Simulation of Glow Discharg in a Magntic Fild Through th Solution of th Boltzmann Equation D. A. Storozhv *1 S.T. Surzhikov 2 1 Moscow Institut of Physics and Tchnology Dolgoprudny Moscow Rgion Russia 2 Institut for Problms in Mchanics Russian Acadmy of Scinc Moscow Russia *1 dmitri.storozhv@gmail.com; 2 surg@ipmnt.ru Abstract- Numrical simulation modl of a Dirct Currnt Discharg (DCD) in xtrnal magntic fild is usd for analysis of bhavior of normal DCD at th initial tim instants aftr switching-on of transvrsal magntic fild. Numrical simulation rsults ar prsntd for two-dimnsional glow discharg at prssur 5 Torr Emf of powr supply of 2 kvolt and magntic fild induction of B = 0.05 T. Th modl is basd on th diffusion-drift thory of gas discharg consisting of continuity and momntum consrvation quations for lctron and ion fluids as wll as th Poisson quation for th slf-consistnt lctric fild. Fatur of our approach is to calculat lctron transport cofficints for diffusion-drift modl by solving th lctron Boltzmann quation. It is shown that th switching-on of transvrsal magntic fild givs ris to th DCD plasma oscillations. Th plasma oscillations ar obsrvd also in moving DCD in prpndicular dirction to applid lctric and magntic fild. Kywords- Glow Discharg in a Magntic Fild; Elctron Kintic in a Magntic Fild; Diffusion-drift Modl of Glow Discharg in a Magntic Fild I. INTRODUCTION Rsarch of fundamntal rgularitis of intraction of gas flows with localizd aras of lctric dischargs is in th cntr of attntion of th modrn arophysics bcaus incrasing of vlocitis and flying hights of aircrafts gradually displac intrsts of classical arodynamics in dirction of physical arodynamics of partially ionizd gas flows [1-10]. Exprimntal and thortical rsarchs of last yars show ral faturs of local control of gas strams for arospac applications [11]. Dvlopmnts and applications of computing physical-chmical modls of wakly ionizd gass ar th componnt parts of th rsarchs. Two groups of such modls ar gnrally usd: th diffusion-drift modl and th simplifid quasi-nutral ambipolar modl. Ths modls hav allowd to ssntially improv undrstanding of physics of gas dischargs in viscous gas strams. Comparison with rsults of xprimntal rsarchs has allowd to dfin aras of applicability of th spcifid modls and to spcify such conditions in gas strams at which xisting modls prdict unsatisfactory rsults [1]. At th sam tim using dvlopd modls for studying physical phnomna at xprimntally studid conditions allows undrstand som nw pculiaritis of th physical procsss which wr not analyzd arlir. Our prsnt papr studis an intraction of a glow discharg with xtrnal magntic fild which is of practical intrst for arophysical applications. With dvlopd thory of glow discharg in magntic fild [12] it is shown that at crtain conditions in gas discharg gap som anomalous bhaviors of glow discharg in xtrnal magntic fild ar obsrvd. Dtaild calculations of th glow discharg at arlir tim stag aftr switching-on of xtrnal transvrsal magntic fild show that at th bginning of th procss cathod layr drifts in th ngativ dirction of th x-axs (s Fig. 1) and only aftr som microsconds th glow discharg bgins to drift in th positiv dirction of x-axs. Fig. 1 Schmatic of a glow discharg with xtrnal transvrsal magntic fild Th trm anomalous is usd hr to strss som pculiaritis of th glow dischargs in xtrnal magntic fild. But of cours it is shown that thr ar no any anomalous phnomna th anomalous bhavior can b xplaind by rgular classical physics of glow dischargs

2 II. STATEMENT OF THE PROBLEM AND GOVERNING EQUATIONS Th problm which is considrd in th papr is formulatd as following: Lt at a som instant of tim an xtrnal magntic fild is switchd-on btwn two plat lctrods with stady stat dirct currnt discharg burning in th normal mod as it is shown in Fig. 1 (about normal mod of DCD s in [13]). As it was stablishd arlir for xampl in [12] if any glow discharg is placd insid transvrs magntic fild thn column of such a discharg shifts in dirction prpndicular to th applid magntic and lctric filds (in th dirction x Fig. 1). Hr w will considr in dtail tmporal volution of th DCD column at initial instants of tim aftr switching-on th xtrnal magntic fild. Data of ionization cofficint and transport proprtis ar calculatd by solving th lctron Boltzmann quation. Dirct currnt discharg is considrd in molcular nitrogn btwn flat lctrods (Fig. 1). Th DCD structur in xtrnal magntic fild is dscribd by continuity quations for concntration of lctrons n and positiv ions n i togthr with th Poisson quations for th lctro-static fild E grad and also by th nrgy consrvation quation for nutral spcis [12]: n D n D n ne x 2 ne y 2 nn t x 1 b x y 1 b y n D n D n x ne ne y 1 nn t x b x y b y n 2 2 ni x y (1) (2) (3) T T T cp qj t x x y y grad nv D n n EVB nv D grad n n EVB i i i i i i 2 2 whr i ar th lctron and ion flux dnsitis; x y ; B is th xtrnal magntic fild induction (its dirction is shown in Fig. 1); qj je ; j i ; E and ar th ionization and rcombination cofficints; i ar th lctron and ion mobilitis; D D i ar th lctron and ion diffusion cofficints; is th part of th Joul hat which is ralizd in th gas hating; Bz Bz b b (7) c c ar th Hall paramtrs for lctrons and ions; is th Larmor frquncy of lctrons n Bz H z mc mc (8) Bz H z (9) mc mc is th Larmor frquncy of ions. Boundary conditions for chargd particls and lctrical potntial ar formulatd as following: ni y 0: 0 i 0 T= Tw; y (4) (5) (6) (10) n V y H : ni 0 0 T= Tw; y E (11)

3 n ni T x 0: 0 0; x x x x (12) n ni T x L: 0 0; x x x x whr T 300 K is th tmpratur of th cathod and anod. w A stady stat solution for glow discharg at givn Emf. of powr supply paramtrs of xtrnal lctric circuit and prssur insid th gas discharg gap is usd as th initial condition at t 0. (13) III. CONSTITUTIVE RELATIONSHIPS It is assumd that transport and thrmo-physic proprtis of nutral particls of th discharg dpnd on th tmpratur thrfor: i p p cm p Vs cm p Vs 293 p p Torr (14) T Di i p T D p T c p 7 1 J g K g M 28 mol M 5 M p g T cm T cpm W (2.2)* M R cm K (2.2)* T T T k k 71.4 K 3.68A J R K mol o whr p is th undisturbd prssur 19 p N is th concntration of th nutral particls. T 7 3 Rcombination cofficint and th part of th Joul hat ar takn as constants: 2 10 cm s =0.5. Tmpratur of lctrons T is prdictd by mpirical corrlation T E 29.96ln T p whr T is th lctron tmpratur K; T is th gas tmpratur K; E/p is th discharg paramtr V(cmTorr). Corrlation (15) is slightly xtrapolatd for cathod rgion bcaus th valu of discharg paramtr is out of rang of th data [14]. (15)

4 Th ionization cofficint is dtrmind as follows (th 1 st Townsnd s cofficint): Bin 1 E p Ain xp (16) E p cm Torr 1 V whr Ain 12 Bin 342. cm Torr cm Torr Valus of th 1st Townsnd cofficints and transport cofficints for lctrons in xtrnal magntic fild ar also calculatd by solving th Boltzmann lctron quation using solvr Bolsig+ [15] whr th ffct of th magntic fild is mimickd by making th lctric fild oscillat at th Larmor frquncy (8). Ths valus ar approximatd by a function (16) at diffrnt rangs of E/N. Constants A and B ar prsntd in Tabl I. TABLE I CONSTANTS IN APPROXIMATION (16) OF IONIZATION COEFFICIENTS WHICH WERE OBTAINED BY SOLVING THE BOLTZMANN ELECTRON EQUATION EN<200Td EN200Td -1 A (cm Torr) B V cm Torr Dpndnt transport cofficints in glow discharg on th magntic fild ar illustratd in Fig. 2. All paramtrs of driftdiffusion modl dcras in a magntic fild. It is shown that th 1 st Townsnd cofficint is th fastst changing function of magntic fild and E/N [13]. So xactly ionization cofficints hav a significant influnc upon lctrodynamic structur of glow discharg. A diffrnc btwn mpirical rlation (16) nd valus obtaind by solving th lctron Boltzmann quation is prsntd in Fig E+22 8E+21 6E+21 4E+21 2E E E E E E E Fig. 2 Valus of th ionization cofficints (multiplid by 1/N) mobility (multiplid by N) and diffusion cofficints (multiplid by N) in a magntic fild for E/N=100Td Fig. 3 Ionization cofficint data (th 1st Townsnd cofficint) for lctrons in magntic fild B=0.05T calculatd by solving th Boltzmann quation (dashd lins) and mpirical rlation (16) (solid lins). Data is prsntd for valus of E/N inhrnt for positiv column (a) and for valus of E/N inhrnt for cathod layr of Glow Discharg (b)

5 Equations (1)(6) ar supplmntd with th quation for an xtrnal circuit which is writtn for a stationary currnt as E V IR (17) whr V is th voltag on th lctrods; I is th discharg currnt; E is th Emf in powr supply and R 0 is th xtrnal rsistanc (this cas corrsponds to on sctiond cathod) 0 IV. NUMERICAL SIMULATION RESULTS Calculations wr prforming for th following initial data: prssur of gas (N 2 ) is p 5 Torr Emf of a powr supply is E 2 kv rsistanc of an xtrnal lctric circuit is R0 300 kohm; th distanc btwn flat surfacs of th cathod and th anod is H 2 cm and lngth of th flat channl is L 6 cm (s Fig. 1). An induction of xtrnal magntic fild guidd along an axis z changd in a rang Bz T. Calculations at Bz 0.1 T showd similar rsults in comparison with th cas of Bz 0.05 T thrfor only rsults corrsponding to B z 0.05 T ar prsntd hr. All calculations wr xcutd counting upon unity of lngth (1 cm) along an axis z. Initial conditions of th calculations wr formd by th following: at an initial instant a quasinutral plasma cloud was situatd abov th cathod at x0 3 cm. Concntration of chargd particls was assumd qual to n0 10 cm 3. Formation of a glow discharg xisting in a mod of normal currnt dnsity was obsrvd aftr procss bginning at s (at absnc of a magntic fild). Obtaind stationary solution for th glow discharg in two-dimnsional flat gomtry was just usd as initial condition for solution of th problm on dynamics of a glow discharg in transvrsal magntic fild. It should b strssd that in usd numrical simulation procdur th Boltzmann quation was intgratd in ach nod of calculation domain dpnding on currnt valus E/N. Th inductivity of th magntic fild xrts influnc on th global structur of th glow discharg. Fig. 4 5 shows lctron and ion contours in th glow discharg at p 5 Torr and E 2 kv at conscutiv instants aftr th magntic fild of Bz 0.05 T is applid. Ths ar transint configurations of th dirct glow discharg. Th spcifid unstady solution is prsntd in Fig. 4. Elctron and ion concntrations in th glow discharg ar shown in Fig. 4. Nar lctrod rgions of th glow discharg (th cathod and anod rgions) and also positiv column ar wll visibl in ths figurs. In all cass prsntd th discharg column movs continuously along lctrod surfacs. First of all it is rvald that a transvrs magntic fild shifts discharg path from th initial position. Comparing numrical simulations for diffrnt magntic filds B on can conclud that vlocity of th discharg drifts prpndicular to applid magntic fild and is proportional to th valu of B. Th avrag vlocity of such drift in th cas of Bz 0.05 T quals to u x cm s. Du to th natur of ambipolar mchanism ths drift vlocitis ar much lss than lctronic drift vlocitis but gratr than th ionic drift vlocitis. Lt us considr rgularity of volution of th glow discharg plasma configuration aftr magntic fild switching-on. Initial configuration of lctrons and ions concntration is shown in Fig. 4 5 (t=0). Bcaus lctrons ar much mor mobil in crossd lctric and magntic fild than ions at th first nanoscond thy bgin to mov in a dirction of an x-axis. Th lctrons clos to anod th larg distanc thy shift at positiv x-dirction. But as th lctrons shift from th initial stadystat configuration a polarization fild incrass. Thrfor from som instant of tim thy bgin to shift in opposit dirction. Aftr that on can obsrv oscillation of lctronic concntration nar to anod which will continu vn aftr bginning of movmnt of th glow discharg column. Intnsity of an lctric fild nar to cathod is much mor strong (on can say that thr is a potntial wll nar to cathod for lctrons) thrfor it is mor difficult for ths lctrons to tak part in oscillation mntiond abov and stimulatd by magntic forc. Not that for considring configuration of lctric and magntic fild th glow discharg bgins to mov in positiv x-dirction in full conformity with classical lctrodynamic lows. This movmnt is distinctly sn at t>1 microscond. Ions ar much mor inrt in comparison with lctrons thrfor thy actually do not fl magntic fild at th initial stag of th procss and thir configuration actually rmains invariabl with th xcption of two spatial rgions (nar to anod and nar to cathod). Nar to anod on can obsrv incrasing of ion concntrations in positiv x-dirction. It can b xplaind by additional ionization of nutral particls by incrasd numbrs of lctrons. Nar to cathod on can obsrv dcrasing of ions concntrations in positiv x-dirction. From th first point of viw it sms that th ions cloud is shiftd in ngativ x-dirction. But actually it is obsrvd only som dcrasing of ionization procsss du to lctrons gon th rgion undr applid magntic fild. Som oscillations in ions concntrations at th bginning of th procss ( t 1 microscond) ar xplaind xcptionally by oscillation of lctronic clouds. Nvrthlss aftr t~1 microscond on can obsrv bginning of movmnt of th glow discharg column in positiv x-dirction

6 t=0s t=12.5s t=25s Fig. 4 Elctron contours (in 10 9 cm -3 ) in th gas discharg gap at tim momnts t = s: with valus of th ionization cofficints calculatd by mpirical rlation (16) (a) and th lctron Boltzmann quation solvr [15] (b) t=0s t=12.5s t=25s Fig. 5 Ion contours (in 10-9 cm -3 ) in th gas discharg gap at tim momnts t = s: with valus of th ionization cofficints calculatd by mpirical rlation (16) (a) and using th lctron Boltzmann quation solvr [15] (b) Usag of th lctron Boltzmann quation to obtain ionization cofficint lads to growth of cathod and anod spots dcras in concntration of lctrons and ions in th cntr of glow discharg. Vlocity of moving discharg in magntic fild also incrass significantly. It is worth to mntion that analogous two- and thr-dimnsion modls of glow discharg in xtrnal magntic fild hav xistd alrady [16-20]. But mpirical rlations for transport cofficints of glow discharg plasma ar usd in ths paprs. Using th lctron transport cofficints obtaind by th solution of th Boltzmann quation will allow us in futur to tak into account hating ffcts in glow dischargs in viw of vibrational xcitation of molcular gas and to considr th thrdimnsional problms. V. CONCLUSIONS A thory and two-dimnsional numrical simulations for modling th lctrodynamic structur of th glow dischargs with a magntic fild ar prsntd. With using dvlopd two-dimnsional modl of glow discharg in a magntic fild is shown that switching-on transvrsal magntic fild rsults in origination of oscillation of lctronic and ions concntration insid th column of glow discharg and in cathod and anod rgions. It is also shown that mntiond oscillations ar kpt during th procss of glow discharg movmnt in xtrnal transvrsal magntic fild

7 All two-dimnsional calculations of th glow discharg structur in nitrogn hav bn prformd undr various kintic paramtrs of drift-diffusion modl. Th computd rsults xhibit good agrmnt with th classic thory of von Engl and Stnbck. Th comparison of numrical simulation rsults obtaind in th papr with thos obtaind by mpirical cofficints of th diffusion-drift modl dmonstrats accptabl agrmnt. But in th both considrd cass th Joul hat was assumd idntical. Using of th lctron nrgy distribution function obtaind by intgration of th Boltzmann quation will allow us in th futur to tak into account a ral part of th Joul nrgy which gos to gas hating. ACKNOWLEDGMENT This study was supportd by th Europan Community s Svnth Framwork Program (FP7/ ) undr grant agrmnt and by th Program of Basic Rsarchs of Russian Acadmy of Scincs. REFERENCES [1] Shang J.S. Surzhikov S.T. Kimml R. Gaitond D. Mnart J. Hays J. Mchanisms of Plasma Actuators for Hyprsonic Flow Control Progrss in Arospac Scincs pp [2] Roth J.R. Shrman D.M. Wilkinson S.P. Elctrodynamic Flow Control with a Glow-Discharg Surfac Plasma AIAA Journal Vol.38 No.7 pp [3] Poggi J. DC Glow Discharg: A Computational Study for Flow Control Applications AIAA pp.118. [4] Kimml R. Hays J. Mnart J. and Shang J.S. Effct of Surfac Plasma Discharg on Boundary Layr at Mach 5 AIAA Rno NV January [5] Enlo C.L. McLaughlin T.E. VanDykn R.D. Kachnr K.D. Jumpr E.R. and Cork T.C. Mchanisms and Rsponss of a Singl Dilctric Barrir Plasma AIAA papr [6] Roy S. Gaitond D. Modling Surfac Discharg Effcts of Atmosphric RF on Gas Flow Control AIAA Papr [7] Shang J.S. Chang C.L. Magnto-Arodynamic Intraction ovr Airfoil AIAA Papr pp.110. [8] Machrt S.O. Shnidr M.N. Mils R.B. Magnto hydrodynamic Control of Hyprsonic Flows and Scramjt Inlts Using Elctron Bam Ionization AIAA Journal 2001 Vol. 40 pp [9] Shnidr M.N. and Machrt S.O. Modling of Plasma Virtual Shap Control of Ram/Scramjt Inlt and Isolator Journal of Propulsion and Powr 2006 Vol.22 pp [10] Nishihara M. Jiang N. Rich J.W. Lmprt W.R. Adamovich I.V. Goginni S. Low-tmpratur suprsonic boundary layr control using rptitivly pulsd magntohydrodynamic forcing Physics of Fluids 2005 Vol.17 P [11] Surzhikov S.T. Shang J.S. "Anomalous" bhavior of Glow Discharg in Extrnal Magntic Fild AIAA papr [12] Surzhikov S.T. and Shang J.S. Two-Componnt Plasma Modl for Two-Dimnsional Glow Discharg in Magntic Fild Journal of Computational Physics pp [13] Raizr Yu. P. Surzhikov S.T. Two-dimnsional structur of th normal glow discharg and th rol of diffusion in forming of cathod and anod currnt spots High Tmp. 26 (3) (1988). [14] Townsnd J.S. Baily V.A. Philos. Mag Vol.42 p.874. [15] BOLSIG CPAT: [16] Bisk N.J. Boyd I.D. Poggi J. Numrical Study of Plasma-Assistd Arodynamc Control for Hyprsonic Vhicls Journal of Spaccraft and Rockts pp [17] Poggi J. Numrical Exploration of Flow Control with Glow Dischargs AIAA papr [18] Gaitond D. Poggi J. An implicit Tchniqu for 3D Turbulnt MGD with th Gnralixd Ohm s Law AIAA papr [19] Poggi J. Numrical Simulation of DC and RF Glow Dischargs AIAA papr [20] Shang J. S. Huang P.G. Yan H. Surzhikov S. T. Computational lctrodynamic simulation of dirct currnt discharg J. Appl. Phys. 105 pp