9 -!" 95 46 76 <; :7956(4 MHD/ $-$ %&' PSB @9(;=>?= $ '$ $ )' ( &#&" $ '!"# $ % $ ( / - )'!"# $ ' $ ( / - )' $ ' ' ' @ :> :(4( A B (? 56" 4 56" &7 9 /6 5# :7 ;% < # =>- 9? @ &% G# # K &4 &#I J4 < G H &#)' ( I dis C mɺ )& / BC / DE @ 6)/ Q) ( "% 4 G Q) N O ' ( P" & % E 6" 9 /6 9 H7 Q#" ( &' &7 /6 / S6B 9 $? @ &/ HE ( & "% = U WB N #< V D &7 9 4 56" G 4 U IV &7 Q) E 7 &G &7 XH % &YMUSC @ V "% OMUSC &N 5 / ka BC / 6 g s& / 7 / 6 cm W PSB #)'N )' &7 ( 5 / 67 ' & V # b cy N 9 U / < PN I E &7 a V / N% / /-B &N ( (dg# Y %6 9 &%N V V G 5 )' &7&N' & G 9 < # =>- G H /:$/$ Dvlopmnt and Validation of an MHD Cod fo Study of Agon plasma low Caactistics in t PSB ust M Aanga Dpatmnt of Aospac Engining aculty of Nw cnologis and Engining Said Bsti Univsity an Ian Ebaimi aculty of Aospac Engining K N oosi Univsity of cnology an Ian M Sams aculty of Aospac Engining K N oosi Univsity of cnology an Ian Abstact Applying o s scm to solv t MHD uations lads to duction of t numical viscosity and gowt of t accuacy By incasing I mɺ in MPD tust tat accompanis stong xpansion na t lctods tips o s scm as faild In t dis ons w nd mo numical viscosity ybidiing o s scm wit HE mtod povids a stabl solution and limits t numical viscosity in t ot ons o aciv a ig-solution mtod nw modification of MUSC tcniu as bn mployd is mtod is calld OMUSC tcniu wic as low dispsion and dissipation os o validation of dvlopd algoitm t Pinctonfull-scal bncmak tust wit catod lngt of 76cm mass flow at of 6g s and total discag cunt of 5kA as bn simulatd compaison of t pdictd sults of magntic fild and nclosd cunt wit t masud data sows good ualitativ and uantitativ agmnt calculatd tust is 567 N wic sowsabout6% diffnc compad wit masud valu utmo tis simulation poply pdicts t xpimntally xaust plum stuctu Kywods: Plasma flow Magntoydo dynamic (MHD) uations imann solv ust Numical Modl ing ( ) [-] "% 4 E -Y # N & &N )' ( ( & 99 W [] #& 6 k ("% &- W ( - D% = W =>- D 5#l - m4 =>- "% = < E- )' 5 ' G H )'N / )' &#)' J ("- N PSB S 5 gy #)'N ( MPD) < G H Mico-instabilitis Pincton ull-scal Bncmak (PSB) ust Magnto-Plasma-Dynamic m_aanga@sbuaci :
[]@9(?( (G PSB @9(-- 6 < # =>- 9 v9 5#l X# PSB )' G H / &N (MHD) /6 G 9 N O - 5 u" U ( & U &7 gu Y G#? O G @ N O &- 7 [7] - - 5 u" - S% % &N [] H & )' G [9] "v ( x6/ 7-5 W - - Y [9] P/ Y G# BC / V &N - &G H / &N @ % &Y 56" E ka N ' @ [] &- 5#l ' ( P" & &7 )'? @ (6)/ 9 HE O G u" 5 W G# HE ka BC / U H & @ - S% &N - 5 & HE @ [] "% 4 O - 5 &G &7 /6? @ &' &7 /6 9 5"-t> @ &7 &#& $/ U &7 /6 G x H @? @ N m [] P/ &7 /6 &? @ & 9 @ ( &7 &#& 9 $ % = G @ ); '" &#G V O E @ v9 5#l %? @ (6)/ HE P/ "% - 5 W G# n ' W/ U O [] n 4 9 & ; @ U XH BC / V & o - ) H % % / 5 kan 5 V &N )' ( 5 W [4] 5 G# ( &N = < ka BC / BC - & =>- 5#l 9 & "% U QE9 m4 %N (dg# O #C' O @ N =>- (G! G/ U @ &% 9 % N% 5 " &N & # - 5 W su =>- #& 6 k H- "% "% - = & )' MACH N O [5] W ( &N &- = ( & D #& 6 9 W- S7 =k 5#l ( / )' ( 4W [6] 5 G# )# ' "% 4 &N - 5 u" 5"-t> @ G U N S7 5"-t> @ ;% &#@ &' )G# 7 & l &# V :7 ( #C' O 5? @ #@ V 6 @ ( &7 /6 5 " #l B ' / $ - 5 W N ") [] )' G 9 ( &#C' -Y & MHD =>- 9 E 7-6 Magntoydodynamic (MHD) Euations 7 Madan-admo Villani 9 Hatn ax van and Einfldt Slf-Consistnc Saa s Euilibium Ioniation Modl Multi-block Abitay Coodinat Hydo-magntic (MACH) Simulation ool 4 ax idics Mtod 5 o s Scm
tg N$ G# v )# &$ N = n k ν ω χ s B AN i i i i i= { N () (7) =>- G W =7 G7 4 N ( E ) V (9) # Q$ #' 6 S S =7 U 4 Q# G &#=7 # # # % # ( &y W Q$ ) m 6 (9) - N% 5 " N 9 &y Q$ #& 6 [9] P/ $ U }U &y - 9 S6B & (- ε ( ε p ) u ( ε p ) w t p p = N 5 kb ωi χi i i = p p k n ( ε p ) u u w η ( ) ρ ν i i k k M B [9] () ~ 4 N O ( ) / ); () - U G 6/ WV - G G H # ); ( n u) ( n u) ( n w) n i i i i = ɺ ωi ; i = 6 t ( ωɺ i ) G 6/ X (- () =7 U }U [] P/ mv dg# V &# mv U & SN> o # # % G H V % N% &G H 9 - l % Y (dg# x [9] P/ ' - )- &7 &N }N9 #l # ' PSB )' &#@ N [4] P/ N #5#l ( N Sd# - &#%l N O #C' O 9 =O  C' / MHD =>- #C'G N% =>- W =>- %N U &!" &#)' ( I dis 6" (4 V N mɺ )BG7 V P (l 56" V) 9 #& / U &# N MPD 6 5#l ( SE W ' ( [] # ' &7 &y Q$ 9H Idis - mɺ 56" [7] W )' H H H H PSB PSB % $ - )-K) I dis ( ka) 5 6 5 / &7 @?? HE HE 5"-t> HE-? 5 W - - - - - - P/ [7] [9] [] [] [6] 5#l v9 K- " < # =>- &U V =C # MPD )' G H 7- &#$/ S ) B =>- &- =>- &y YU $/ < B =>- # ) &y G U = S D S t [9] # 5G 4 N & V &#< U = ρ ρu ρw Bθ ε () - # 7- $/ // &# ' () N = Bθ Bθ = ρu ρu p ρuw ubθ u ε µ p µ () Bθ Bθ = ρw ρwu ρw p wbθ w ε µ p µ N Y N &y ); () Y p B γ µ θ ε = ρ ( u w ) =C =>- 7- =7 S () % # &U V Bθ Bθ S = ρ ρ ρ ε (4) u u wu u µ p µ (5) =>- m4 m 6 &no &# E E D = S = s E B k k θ = µ E B k k θ = µ (6) (5) (6) # (7) ()
max ( λ ) min ( λ ) b = max max V C b = min min V C min @ N # &l V (G (' λ &#O m G C C V V (5) max λ? (- 7 &#O / 7 HE @ / U WB N ˆ; (G (' $ #C' O &N /6 &? @ ( G G7 l V &' &7 O N &NE &#< V [] P/ U o Uvit = U t S D S b U o Uvit = U t S D S % o b (6)/ HE % (6) &7 9 Q) &#< V W 4 &#@ # V WB N W = ρ u w Bθ p p ni 9 4 56" &#@ N "% WB 6 V [5]MUSC @ N O &7 E &#WB #< V m9 &N &#< XH % &Y &V E S @ ( &% =H' P" D &#Y U & &7 $ MUSC @ N v9 5#l () O % x [6] o OMUSC S N ˆ; G &#< V ( = ɶ φ ( ) = ɶ φ ɶ φ φ ɶ φ φ % U N = U ( ( φ )) ( ( φ )) W = max min = 75 75 = max min = 75 75 = = l l l l WB (7) # () &#O N # V Š" o ; 7 l N% / ) & N 56" () Y < ou # K E 9 &4 I c4 56" 9 ( '/ &y V D % BC ƒ &y Q$ '" V o ( U ( % O &y N '/ <? @ C' O &# 9 &#/ G7 # Q &7 /6 & &#@ 6/ % ( K) 6" 9 E V Y ( V I E #& G 56" I dis mɺ &7 /6 &U V I % U &#WB "% D% #l "% &7 &#@ - G &#@ V N m [ &' &7 /6 & 6] P/ W GO P/ O HE 5"-t> &#@ &#@ V &' &7 /6 & [ ] #l H7 #? @ #C' O &#& G B$ & [5 ] P/ "% = O 7 /6 N &7 v9 5#l () I ( X $/ @ U? @ &% // -? o G V 9 V O9 (G! XH &#C' O x N = // U? =>- = U U A ( U ) U o A = M Λ Λ M n n n n x [7] P/ Idis mɺ Λ n n M [] () () &#t I o N V &N &#/? @ ' ( P" &#@ N 6" 9 E &7 /6 &? @ m N O &[] ( $ / o @ v9 5#l ( ( &' ( x MHD =>- (6)/ N &' &7 /6 9 HE U N = @ ( %? @ b U = b - b b - b b - ( ) b ( U ) HE - b ( U U ) [4] (4) &#C' -Y & MHD =>- 9 E 7-4
tg N$ G# v )# &$ Y ( BC / V ( ' & 9 % WG7 / ); & N E &#WB ( % U < =V' N E &7 9 & &#% &y Q$ ( H7 % U '" V N &# () Y < ( P" & % O & N E U &#WB 6 N # BC / V ' I dis $ V E 56" & 7 gy G7 O V ( / ma Ž'" =Ž7 N (Ž Ž O # =7 V ( p = p B θ / µ ) Ž ) m N N & ) '" O // &# u" 5#l ( [9] 9 D G7 W & G# & (- [5] "% O N & G # % WV - 9 S6B # gy # 9 N %d 56" :7 Jv ( mb G# v9 5#l ( N b/ $/ # gy & V V 5C n #l "% # bn & N G V V U & & ' E N ( p ) Ž'" =7 N OŽ 7-Ž 7 O 6 7- // =7 V N 9 N &6 o & '" u7 % WV 7- % / S7 ) & QV Q G# V o%g = N ( u" ( O 6 < U & N Y ~ - 4 m N O 4Bθ / = = µ W / =G &G7 % [4] &U / / &#N () "% O < G# @6< S7 I N 7 < & c7 } N & N ( N &7 &% 5 V (dg# O N ( & Ž n SN> "% O [4] k (ψ = B θ )Ž/ PŽ - &7 Q) I = U m 5G G &6 XH @ &no &N % & B g @ N &' $/ (dg# su S N S% (- & O < N S% N &#V W MHD =>- & 9 & ( / O N S% # N V (; τ W = ħ ϑ λ [7] N 7 N &#V "% max : (- N V τ = µ ħ η :< no N V MD τ = n k ħ k HC B ( 7 ϑ { } t = min τ τ τ W MD HC :9 no N V { } ħ = min o ( N = N S% < no N V W MHD =>- & -9 - N V V < k (9) N 9 }N ) N V N %6 >G- (- - - / 5 ϑ &N ( % 4 k "% k ' / MPD &#)' B # &!" ; c N o U ' & N N &B ; B" $ % c6 E N G H / 6 D G Qy J N G H / KN 9 ( # N 7 MHD =>- I =v" 4 MHD =>- D 9 W / ( u" KN > ( N " [4] &N N 9 9 &G H / "% N " 9 & & SN> / % W-" 5 " ( 9 ) 7 ( Ma = w a = ) o U C' in in in γ k w = Z m [9] U N = & in B Zff in in ff in in in C' (dg# kœ 5 gy Z ff () - N O & / ); & / V mɺ ρin = w π ( ) in a c C' J- m (- ) c a () 6/ x6/ '" $ - G G H 6/ ); & '" # PG/9 N U N & < ~ 4 &% [4] B θ in µ I = π dis W9 4 N Y () 5
-Y & MHD =>- 9 E 7 - ([7] N?( 5?4) 56 @9( :-- ) =>- &# G4 N =< ' &y < &U 7- S )G# 9 " 9H Y G# -9 #U @ " N & # G4 BC / U ( ( 7 / 65 5 4 k t~ 5 / 4 ka V &#C' - - siduals - -4-5 -6-7 - -9 Dnsity adial momntum Axial momntum Magntic fild Engy E6 E6 im stp ' - (@: )-4- ' 5 k < IY PN IY V cy a 9H Y G# # ' &%N 5# < = V J- W 56" o U / &#N U K Š H / V &N IY #)'N o 9 ( 6 % / Y?% > /% M ) 5( (-- %&' '?%-4 5N )' " - U E 7 OU - N &N [7] o ' # J- W N G# )' O - N 5#l ( 5C U &# N J ( &#U / = [9 6-4] B4 #l G# K C G4 N (dg# N mv gy ;~ = ( " / )' ( "% N% ' & / 5 / 7 / 6 W ka& BC 6 g s% 6
B" N ) Q V 4 / ); V A-II O E =- Y [7] " % = S ) 4 W B 5C )' / / m B 5C (9 ) " " ' % ' 6 A-II 5 )# & $ G# v tg N $ J - 5 6" 9 $ &N 5# / 9 % ' 6 G / o G O Q# N / B 4 6 & 7 &N E O = #' ( 5 7 5 ( ) S ;( P6Q N? -5 - ); P N G U A-II a " ' = #' c Y A-II = E / B 9 ); E V 9 ' & / K 5# V J - 5 6" N / o N O / B 9 6E ) [7] (4) Y N I nclosd = =?( ( ;( O P6Q N?-6 - U / I Y P N ' 7 π Bθ (4) µ E a Q( ( W " ( : 4 A-II ; [7]@9 ( W S V & -9 - )Q ; Q) Out flow 6 CG# N 9 / I Y " # K / = O K x g v H 4 Y G# C N ( S N m V gy Y # K Y G# " # y " N v 9 5# l = E y " k g v [9] P/ Q$ & # = < $ K # Stamlin Inn flow 4E E4 6E5 E6 E7 5E E9 4E9 7 5 A-II ; N? - - ( ka ) 7 ( 7 & U O = < 7 " ' V U B /5 9/5 9 N K N & B " XH K N & 5 B " < k & 7 a ( N - G E & 7 V ( ' 5 & %N V N ' B7 K N & 9 B " XH C K v 9 &N ( Jv ( & 5( ; P6Q N? -7 - K 7 [7] =?( ( ; P6Q N?- -
( 5 / 67 U ' & V V (dg# XH %6 / [7] ( 56 / E V N Y N l v Isp = (6) mg ɺ 96 &%N V V k 95 Y %6 / 7) [7] k 4N-5 -O) G @ ( & 5#l ( - &G H / &7 &N $/ (O U & ( "% MPD )' 4 O HE? &#@ MHD =>- // PSB )' - Q) N "% / < IY PN V &N b cy N 9 E a U O =#' &7 &N # ' " &QV E A-II E =-Y N 9 / B 9 / N% / V U &U 7 " G 5 / 5 / 5 9 N K N & 5 B" E a CG# < k B / 7- =< V ( H7 ( # O G &%N N 9 &%N V 5# / 9 I J- 56" < m l v ' & ( H7 V V k 95 ( 5 / 67 XH %7 N G &%N :7 Jv ( "% X gy = ( 7 V c4 (- Y (& ZQ) ;E (&( Y' -)-- / 5 ()( PE) [7] N 4 V&) B" / 7- =< 5G ( ) N N & / 9 / N J- 56" 9H Y G# ( J4 5# / 55 ( kg m s) / 47 9 N / 9 G 9 J- 56" ( ); V N I @C / / / 6 N ); 5# 6 & 5# = E " ( $ ' I Y 7 56" / 5# &#C' -Y & MHD =>- 9 E 7 - / t ( ) < = &no ) G H P/ ) ( Vm // ( N ) '& 9 / ( m s ) n/ b ( ka) BC / ( s) l v ( Am ) / ); - - / ( JK ) (6 k V &#< WV t/ ( m ) G 6/ =n - ); ( Pa) '" -9 / 6 ( C) 9 l &# t :-6 A B D E g I dis I sp k B M n p 4 V&) (& ZQ) 9' /(-- )4 \6?( & & / 5" ()( PE) [] N ' & / N & (5) W) U > B θ = v( ρv da) p µ da [9] U 4 (5)
tg N$ G# v )# &$ MHD =>- 9" N$ tg v G# &$ )# [7] #C' O &> @ N O $ '#l GB7 BE "< G H )' &N 9 O 4- UO 4 G [] Villani D D Engy oss Mcanisms in a Magnto plasma dynamic Act PD sis Pincton Univsity Pincton Nw Jsy 9 [9] Aanga M Ebaimi Sams M Numical simulation of non-uilibium plasma flow in a cylindical MPD tust using a ig-od flux-diffnc splitting mtod Acta Astonautica Vol No pp 9-4 4 # =>- 9 N$ tg v G# &$ )# [] G H )' &N $/ HE @ N O < 9!"# 5 ' ' "D< / &7 &N N$ tg v G# &$ )# [] '#l GB7 BE < G H )' - &G H 9-9 UO G 4 # #N [] Janunn P A positiv consvativ mtod fo magntoydodynamics basd on H and o mtods Jounal of Computational Pysics Vol 6 pp 649-66 [] any C B Computational gasdynamics Cambigd Univsity Pss Nw Yok 99 [4] Einfldt t al On godunov-typ mtods na low dnsitis Jounal of Computational Pysics Vol 9 pp 7-95 99 [5] Van B owads t ultimat consvativ diffnc scm II monoticity and consvation combind in a scond-od scm Jounal of Computational Pysics Vol 4 pp 6 7 974 [6] Yan t al Optimiation of t MUSC scm by dispsion and dissipation Scinc Cina Pysics Mcanics and Astonomy Vol 55 Iss 5 pp 44-5 [7] Boyl M J Acclation pocsss in t uasi-stady magnto plasma dynamic discag PD sis Pincton Univsity Pincton Nw Jsy 974 [] Coy J S Mass Momntum and Engy low fom an MPD Acclato PD sis Pincton Univsity Pincton Nw Jsy 97 =7 ( K ) ) 7 7- O V &#< ) 7 ) 7 &U O &#< ( Jm )&y ); ( Om m) G H k V ) l V O l V &Y4 t -7 4 π ( N A l % ) < &DnO ( s ) #& 6 5% t " ( kg m ) ); ( J) 6 &y ) G 5 =7 (#Q #) () =n # & N U WB t V 7- $/ &U $/ YU $/ Λ S u U v w W ε η λ Λ γ µ ν AN i ρ χ ωɺ B?-7 i in n θ B4- U WB N ˆ; G U WB N G N(-9 [] ow Jaskowsky W Clak K and Jan Eosion masumnts on uasi-stady magnto plasma dynamic tusts J Spaccaft Vol 9 No 4 pp 49-5 9 [] Uibai Onst voltag as and anod spots in uasistady magnto plasma dynamic tusts PD sis Pincton Univsity [] Caldo G and Couii E Y Numical fluid simulation of an MPD tust wit al gomtyin d Intnational Elctic Populsion Confnc Sattl WA USA 99 [4] Sankaan K Couii E Y and Jadin S C Compaison of simulatd magnto plasma dynamic tust flow filds to xpimntal masumnts Jounal of Populsion and Pow Vol No pp 9-5 [5] Pama B J -dimnsional modling and analysis of magnto plasma dynamic acclation MSc sis Aiona Stat Univsity / &7 &N N$ tg v G# &$ )# [6] t O (!" {' < U G H W ( / - )' $ t< 9 &9- $ 9