Ionization and recombination rates in boundary layer of magnetohydrodynamic channel electrod

Reshotko, E.

Sherman, A.

Yeh, H.

English

Boundary layer over segmented electrode wall in MHD channel when nonequilibrium plasma flow

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First Quarterly Report
M ID BOUNDARY LAYERS WITH
NON-EQUILIBRIUM IONIZATION AND
Flh1r-7
 „ ATE S
(March 1, 1969 - June 1, 1969)
by
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Arthur Sherman
Hsuan Yeh
Eli Reshotko (consultant) k^%	 s
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` Prepared under Contract No. NASw - 1866 by
1 -	 u ^ t
Computer and Applied Sciences, Inc.
Phi ladelphic, Pennsylvania
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for
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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
•
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TABLE OF CONTENTS
Section	 Page
. I nt roduct i or.	 2
II. Analysis of governing equations 	 3
11L. Programming	 12
IV. Future efforts
	 13
I1. INTRODUCTION
The present report is the first quarterly report under Contract NASw-
` 1866 and covers the period March 1, 1969 to June 1, 1969, The problem being
considered is the boundary layer that will develop over the segmented electrode
wall in an MHD Channel when a non-equilibrium piasma is flowing.
As a result of work completed under NASw-1586 we have developed
Ia theoretical analysis for a non-equilibrium uoundary layer explicitely taking into
account the electrr n temperature differential equation and the plasma sheath. This
analysis was developed using a finite difference format so that finite electrodes along
a wall could be treated in a non-similar manner.
The present program seeks to extend this earlier study to inc IL-de
finiie ionization and recombination rates in order to make the analysis apply in a
more realistic way to physically important situations.
-2-
II. ANALYSIS OF GOVERNING EQUATIONS
In order to pursue our boundary layer analysis now we must abandon
the convenient assumption that the electron density can be calculated simply
from the Saha Law at the electron temperature. Rather, we now need an additional
differential equation governing the behaviour of this variable. This relationship is
obtained from the equation governing conservation of electron number density and takes
the following form
n	 n 	 2	 n 2
Pu ax ( P ) + P v ( p ) a rec . ne 
(ne )eq. 
i 777F2e eq
_	 a_ 	 ( R	 ) + a ( Pi jx )
ax	 a	 by	 e
+	 a	 Pi K	 dn Tby
	 a IBS	 by ( e e)
However, in the present study we have elected to consider
	
Pi = 0
(i.,o. neglect ion slip) so that the last three terms can be omitted.
The rational for such an assumption is that 	 a, is only significanti
when	 Pe is large and in that case other phenomena (i.e. plasma instabilities)
that ire not being considered become important. Since the earlier analysis included
there is clearly no difriculty in doing s^ again if we choose to.i
6
- 3 -
P-
6
The overall system of equations can then be written as
Overall Continuity
ax ( P u ) + —	 ( pv) =0Y
Overa I I Momentum
pu - ax + pv 
a 
u	 P.0 u00Y
dui_ + _a (N ^U )	 (2)d x	 r, y	 a y
W
Overa I I Energy
2
pu a h + oil a h _ - p-0 u,0 u dui + P( a u )ax	 by	 dx	 by
	+ a (F	 K	 aTi +-5	 kT e , J e )
	
by .1
	 .
	 by	 2	 e	 Y
+	 Ex + j -y-j P ( ne )	 pv	 ( p )y
_ by
(3)
-4 -
0Electron Energy
3 kun	 ^ Te + 3 k u T Une + 3 kvn dTe + 3 k v T	 6ne
e Rax
	
2	 e d x	 2	 e a y	 2	 e 6y
5 n k T ( ^u + dv ) _	 a (K	 a T e + 5	 k Te	 )
2	 e	 e	 a x	 t)+	 y	 ay	 e	 ay	 2	 e Je y
Ye s
Jex Ex + je y (Ey - U B  ) + 3p e k ( T - T ON ) 	m s
nn
_ I p j a X( p ) + p „ a_ (_pey
Electron Continuit y
n	 n	 2	 2Z	 e 	 e _	 n e
p 
u a x ( p )+ pv d y ( p) Q rec a (n a )eq . 1- ne^
(4)
(5)
- 5 -
1
i
Generalizes{ Ohms Law
ix- —01 +_ 	 E x 	(3e	 a Y	 e e(EY
- u Bz % 	 k Pe —? (n T)
i ae	 e n e
,1y - i as 2 (5y - u B ) + ae E x + en
	
a	 (n e e )De 	 e	 Y
(6)
Where it should be noted that we have included ;n the generalized
Ohms Law terms involving gradients of the electron pressure. We have also assumed
in the right-hand-side of Equation (5) that the dominant process of recombination
is due to three-body impact.
Next, Equation's (6) cre used to replace terms involving andje in
the p receeding equations, and the resulting set are transformed to the Levy-Lees
variables:
Transformation
X
(x ) _	 ( P P )r ux dx
0
Y
^ (x, Y)_^^ ^	 P dY
V ^ fo
I
OP
-6-
1then
a 
x = ( p u)r a + ^x -a
a	 pux	 a
ay	 N,72—k a^
Introducing
V =
	
2 k
	
( f l < + ` /pv )(PP u^) r 	 x	 v
after transformation the resulting equations are:
Continuity
2k 
a 
f ^ + av + f' = 0
Momentum
2 k f,	 a f' + V o f = 2 k	 du=	 g- f , 2
ak	 a<	 Um	 d
+ a	 ^ a f'
	
a^	 aZ
(7)
(8)
-7-
Energy
•
2
	
+ v d- i 2	 d 	 = _ 2 k ux duio r'9
	
a t	 h ,^	 d	 h 10	 d
2 ,^ ( a f' ) 2 + a	 t 
.^ I	 + T e ou a	 e_
ha,	 a^	 at	 Pr ail
	 T^ aT	 a^
a
+ 5 f2 k Tee y^ ae +_ -Lk- Bzjy.f.^
2	 (PP)r u,,,C PTY, e i3< 
	
(PN)r C P T,r, P.,
^	 1
+	 2T	 u 2 [CE x? + !+M— ( 1 + P ) 9
-P	 (P P) r Pa,	 a	 e
2	 I P2	 °^. a a 2 _ 2^
	
Cp T^ (PN) r ^ m	 eq.
	
e	 g 	 a
(9)
wkere
a rec . = 5.6 x 10 27 ( k Te )- 2 cm 6/ sec
e
with ( k Te ) in eV.
e
-8-
Electron Energy
3	
^P^
k
	r TC P U 2	 2^ f , a e + v a 6 + 2 9 V e d Tecom2	 g	 2^ me	 S	 ea, d
N
+-'^ k Tew9+I
	
`^rec m^ 3 01 ( °e2- X21 1c9	 I
+	 2gmc g	 a	 .3S	 (0
	
00
	
d
U 2	 (e' A rTeov ^a ( ^ a0 ) __.t,_U.^ 5 kTe^	 7B
2 9	 at	 0^ \12^g 2	 e	 JY^ a
v Cr E 2 +	 00 ( 1+ 2	 k "YOo ^oo U^o Tejo d t3 +x	 C7-re )	 e	 V2f g
+	 9 ( a Ir7 a _ d	 )
a5	 a^
+ 3 —°t - -me k 10,, T  w g 'T^ - e ) :Ecs	 (10)g	 c	 e oo	 s	 ' 3
Electron Continuity
2reC	 0	 1	 22 ; f • d + V	 _ 2 ('^	 _r	 a ., o^	 (11)d	 07	 r V`,,^ m e	 9	 5
•
-9-
f' =	 u --
u
D = e-
T e ,s^'
_ T
g	
Tc,
n
O= e
n
6
Now the dependent variables have been defined to be
Also, the boundary conditions on the four variables in the four
partial differential equations are
--7	 f' > 1	 e —'^ 1
n m
e^ cg —^ 1
	 a —> a cc =	
PCV
 ---) 0	 f' = 0
g = g , ( x) or g ' = g 'W (x)
e= e w
-10-
•Where 9 w is determined from the plasma s:.eath analysis as had
been done in our previous work .
At this point then, all the necessary equaticns are available for
solutions that will yield u, T, T e , and ne as functions of x and y.
111. PROGRAMMING
In addition to the formulation of a ^-v set of equations including
finite rate effects, we have been reprogramming the computer program that had
been developed under NASw-1586 to convert it for use on the University of
Pennsylvania's IBM 360,475. The original program had been run on a General
Electric 635.
At this time a new set of punched cards has been prepared, the
program has been compiled, and a number of minor difficulties are being worked out.
Programming of the new problem with finite rates will commence
as soon as ' t-
 , finite difference format has been completed.
- 12 -
IV. FUTURE EFFORTS
During the first quarter major emphasis was placed on a proper
formulation of the governing equations including finite rates, and on the conversion
of our computer programs for use on the IBM 360/75.
In the immediate future, we plan to complete formulation of the
new finite difference formulation of our equations. At the same time we expect to
complete conversion of our computer prog rams and run a number of aeditionai cal-
culations with infinite rates.
Subsequently, we shall program the new system and run a new series
of calculations. In addition, we plan to make a more careful review of the sheath
rrodel that has been used to establish our boundary condition on T
e 
at the wall.
- 13 -


MHD boundary layers with non-equilibrium ionization and finite rates Quarterly report, 1 Jun. - 1 Sep. 1969

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