AD-AGOG 215 STD RESEARCH CORP ARCADIA CALIF FIG 9043cOWPUTER STUDY OF HIGH1 MAGNETIC REYNOLDS NUMBER MH CHANNEL F--E,C (U)

.OV G0 D A OLIVER. T F SWEAN, C 0 BANGERTER N0001.-77-C-O?#

UNCLASSIFIED ITDR-80-4I NL

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MICROCOPY RESOLUTION TEST CHART

STD RESEARCH.CORPORATIONPOST OFFICE BOX 'C', ARCADIA, CALIFORNIA 91006

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/ - STDR-80-41

A COMPUTER STUDY OF

HIGH MAGNETIC REYNOLDS

o NUMBER MHD CHANNEL FLOW

Final Report

Contract No. N00014-77-C-0574

Submitted to D IUnited States Department of the Navy SNOV 28 19802

Office of Naval Research

Arlington, Virginia 22217 A

October 31 1980

Approved for Public Release; distribution unlimited

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A COMPUTER STUDY OF

HIGH MAGNETIC REYNOLDS

4NUMBER MHD CHANNEL FLOW

by

D. A. Oliver, T. F. Swean, Jr., C. D. Bangerter,

IC. D. Maxwell, S. T. Demetriades

IJFinal Report

Contract No. N00014-77-C-0574

I Work Unit NR 099-415

Research Sponsored by the

IOffice of Naval Research

IReproduction in whole or in part is permitted for

Iany purpose of the United States government.

I Ia.. P &-: ,

REPORT DOCUMkENTATION PAGE mE? U4TUCIN

12. GOVT ACcas5somNo MO. acciptlivs CATALOG 1,1,014111

IA Computer Study of High Magnetic 1 Aug 77 *= 31 Oc80Reynold Nube MI- Channe FlowNMUIg STDR- 80-41

D-f Mha-rkham, C. D. /Banterter ; 5 Ni91-7C97jaw-S. T.IDetriades C.-L /AAa____________

Ia. PROGRAM ELEMENT. PROJECT, TASK

STD Research Corporation L/ AREA 46 WORK UNT NUuegs

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IS. SUPPLEMENTARY NOTES

j IS. Key WOROS (C.~enti. on free side 0I Reeo we 10mifr W No"~ number)

Mtagnetohydrodynamnic Flow, High Magnetic Reynolds Number ]Flow,g Pulsed MHD ]Flows.

j20. A"\ACT (CenUau m Pe6,V side. It 0448000 M 1~1?' &Is 8141111116W)

IMA summary of progress made in the prediction of magnetogasdynamicphenomena in explosion MUD flows. Theoretical and computational resultsare presented for quasi- one -dimensional, two-dimensional, quasi-three-dimensional and high Reynolds number flows in both weak and strong MUD

DO 1473 goV TIOM OF I NSov 111 is OSSOLEIET

ISgCumr~v CLAINuP9 Art Ior 11ASEMnDt

A COMPUTER STUDY OF HIGH MAGNETIC

REYNOLDS NUMBER MHD CHANNEL FLOW

1.0 INTRODUCTIONIThis is a final report describing theoretical and

*computational studies of the fully coupled motions of a highspeed, high temperature plasma flow interacting with an applied

magnetid field and an external circuit.

The flows of interest may be created by chemical

detonations and the ensuing process may be either inherentlyunsteady [1] or quasi-steady [2]. In the unsteady process

1 (Fig. 1) the flow consists of a hot plasma ("plasmoid") formedbetween a driven ionizing shock wave and its following contactsurface. The plasmoid is created by a sudden release of energy

into a driver section which is in contact with a test gas in

which the plasmoid propagates. The conducting plasmoid entersa region in which an externally imposed magnetic field B(0 ) and

electrodes coupled to an external circuit exist (Fig. 1). The

plasma conducts current to this external circuit and is subjectto Lorentz forces and Joule heating as it propagates through

the magnetic field. If the explosion drive is a chemicalsource, such a plasmoid will be of the order of 5-20 cm in

length in traversing a magnetic field region of the order of

100 cm at velocities of the order of 104 ms -1 . The plasmoid

Imay exist at pressures up to 1 k bar and energies of up to5 eV.

IIn a steady flow device, the physical schematic is the

same as in Fig. 1 except that a stagnation chamber exists ahead

Iof the MHD channel in which the explosion driven gases are

processed. This configuration yields a hypersonic channel flow

which is quasi-steady over the flow time scale in the channel.

If Cr0 , poUoo are the characteristic electrical

1

conductivity, mass density, velocity, and applied magnetic

field, the flow may be specified by an interaction number i and

magnetic Reynolds number rm (in addition to the gasdynamic Mach

number):1=00 R = ooUo(1)

~p 0 u0

For an interaction region of length L, the nondimensional

Jnumbers are defined asL L

I S f idx fr dx (2)0 0

When Rm >> 1, the appropriate measure of the interaction is the

parameter S defined as

S a(B1-/ROL)/(p 0U2) (3)

where the spatial average <> is over the electrically conduct-

ing portion of the region L. For a uniform plasmoid of length

a, these numbers become I - OB2a/ PoUo, Rm - uoUoa ,

S - I/Rm. In the case of a steady flow situation, the

characteristic length, a, may be selected as the height of the

duct.

]It should also be noted that the square of the Alfven2Mach number, MA, is equal to the reciprocal of the interaction

]parameter S (see Appendix A). Thus, electromagnetic effects

can nominally propagate upstream ("upstream wakes") for R. >> 1

only for S 1 or MA < 1 (see [31). By strong interaction we

mean flows for which S > 1 and by high magnetic Reynolds number

we mean Rm > 1.

The primary parameters describing the motion are thus

the gasdynamic Mach number M, the interaction parameter S (or

I), and the magnetic Reynolds number R,. In addition, a number

[ of secondary parameters are required to specify the motion.

2

These include the ratio of electrode length to duct height, andwhen current flows to the external circuit, the external

current parameter £JROIc/B0 where Ic is the current per unit

depth flowing in the external circuit.

CONTACTSURFACE

ELECTRODES PLASIOiCSO FRONT

_ADR IVER LOAD

ISOURCE -

-- MAGNET

~FIELD COIL

ORDRIVER GAS RI VEN GAS

*19-2974

Fig. 1. Schematic of explosion-driven plasma flow.

1I

13

1 2.0 REVIEW OF PROGRESS

IIn these studies the authors at STD Research Corporation

have presented solutions to a succession of progressively more

Jcomplex initial value and boundary value problems building upto the strong-interaction, high magnetic Reynolds number

motion. Thus in [4], the strong interaction, high magnetic

Reynolds number motions were studied in the quasi-one-

dimensional approximation. In that work several important

phenomena were revealed for the first time. It was shown that

under conditions of strong interaction and moderate-to-high

magnetic Reynolds numbers the interaction zone progressively

decouples from the shock front and evolves into a magneto-

hydrodynamic discontinuity (suitably dispersed depending upon

the magnetic Reynolds number) which moves at the particle speed

of the fluid rather than at the shock front speed. It was

further shown in [4] that upstream running shock waves will in

general be created in the interaction zone under conditions of

large interaction parameters.

In [5] the authors considered the two-dimensional motion

under conditions of low to high magnetic Reynolds numbers but

under conditions of weak interaction. In this case, the

electrical motion is that which results from a two-dimensional,

high-speed, turbulent flow proceeding in a constant area duct;

since the interaction parameter is vanishingly small the fluid

motion is not influenced by the electrical motion. Since there

are no geometrical nonuniformities or magnetohydrodynamic

forces operating on the fluid, there are no shock wave

interactions set up. The principal fluid nonuniformities are

the turbulent boundary layers generated on the duct walls. In

these studies the detailed eddy-current motions in the non-

uniform regions of the applied magnetic field were revealed;

but more importantly, the current distribution within the

conducting fluid resulting from the electrode system and the

external circuit current were also revealed. In particular it4[

I was shown that for magnetic Reynolds numbers in the range 10 <

Rm < 20 the current flowing between the electrodes is convectedfully downstream of the electrode gap. As a result, the

Lorentz forces which would act on the fluid exist downstream of

(and not within) the interelectrode gap. It was further shown

that even for electrode systems which are maintained such that

the electrode edges do not extend into the fringe field of the

applied magnetic field, there is still considerable coupling of

the fringe field induced eddy-cells and the electrode system,

particularly when the external circuit is open.

In (5] the authors studied a quasi-three-dimensional

description of the strong interaction-high magnetic Reynolds

number regime. There it was shown that the Lorentz forces

concentrated in the magnetic boundary layers under strong

interaction conditions would stall these boundary layers and

establish a recirculation cell system in the duct with a

central jet of plasma which would be relatively free of Lorentz

forces. Because of the restrictions of the quasi-three-

dimensional model, the reflected-upstream moving shock waves as

well as the details of the recirculation cells cannot be

described. Only the onset of stall and recirculation can be

predicted.

15k

3. RESEARCH PUBLICATIONS

The following publications have originated from thepresent research:

D. A. Oliver, T. F. Swean, Jr., and D. M. Markham,"Magnetohydrodynamics of Hypervelocity Pulsed Flows" to appearin the AIAA Journal, (1980)

D. A. Oliver, T. F. Swean, Jr., D. M. Markham, C. D.Maxwell, and S. T. Demetriades, "High Magnetic Reynolds Numberand Strong Interaction Phenomena in MHD Channel Flows," Proc.,7th Int'l. Conf. on MHD Electrical Power Generation, MIT,Cambridge, MA, June 1980

D. A. Oliver, T. F. Swean, Jr., D. M. Markham and S. T.Demetriades, "Strong Interaction Magnetogasdynamics of Shock-Generated Plasma," AIAA 18th Aerospace Sciences Meeting,Pasadena, CA, Paper No. AIAA-80-0027, January 1980

D. A. Oliver, T. F. Swean, Jr., D. M. Markham, C. D.Bangerter and S. T. Demetriades, "Magnetogasdynamic Phenomenain Pulsed MHD Flows," STD Research Corporation Interim SummaryReport No. STDR-79-8 prepared for Office of Naval Research,October 1979

S. T. Demetriades, D. M. Markham, C. D. Maxwell, D. A.Oliver and T. F. Swean, Jr., "Magnetogasdynamic Phenomena inPulsed MHD Flows," STD Research Corporation Interim SummaryReport No. STDR-79-2 prepared for Office of Naval Research,January 1979

S. T. Demetriades, D. M. Markham, C. D. Maxwell, D. A.Oliver and T. F. Swean, Jr., "Magnetogasdynamic Phenomena inPulsed MHD Flows," Paper presented at ONR Nonideal PlasmaWorkshop, Pasadena, CA, November 1978

Lk

I

REFERENCES

[1] C. D. Bangerter, B. D. Hopkins and T. R. Brogan, Proc.,Sixth Int'l Conf. on Magnetohydrodynamic ElectricalPower Generation, Washington, D.C., Vol. IV, p. 155,June 1975

(2] S. P. Gill, D. W. Baum and H. Calvin, "Explosive MHDResearch," Artec Assoc. Annual Report No. 119AR to ONR,April 1975 - April 1976

[3] W. R. Sears and E. L. Resler, Jr., "Sub- and Super-Alfv~nic Flows Past Bodies," Advances in AeronauticalScience, Vols. 3 and 4, Pergamon Press, London, p. 657,1962

[4] T. F. Swean, Jr., C. D. Bangerter and D. Markham,"Strong Interaction Magnetogasdynamics ofShock-Generated Plasma," AIAA 18th Aerospace SciencesMeetin Pasadena, CA, Paper No. AIAA-80-0027, January

(5] D. A. Oliver, T. F. Swean, Jr., D. M. Markham, C. D.Maxwell, and S. T. Demetriades, "High Magnetic ReynoldsNumber and Strong Interaction Phenomena in MHD ChannelFlows," Proc., 7th Int'l. Conf. on MHD Electrical PowerGeneration, MIT, Cambridge, MA, June 1980

i

APPENDIX A

THE MATHEMATICAL DESCRIPTION OF MAGNETOGASDYNAMIC FLOWS'

AT STRONG INTERACTION AND HIGH MAGNETIC REYNOLDS NUMBER

U.

i

I

2. THE MATHEMATICAL DESCRIPTION OF MAGNETOGASDYNAMIC

FLOWS AT STRONG INTERACTION AND HIGH MAGNETIC REYNOLDS NUMBER

2. 1 Fluid Conservation Laws

We may describe the fluid in terms of its mass density, p,

velocity U, and internal energy e. We describe the electromagnetic

effects in terms of the electric field E and magnetic field B. These

variables are considered to be general functions of space x and time t.

The conservation laws for mass, momentum, and energy are

at

- (pu) + 17- (pUU) -T + J XB (2)

a P( U u2/2 )] + +U(E +u/Z) Pul = V (F- U~) -V 7 * E (3)

In the conservation laws, ; is the total pressure tensor and

q is the heat flux vector. This system of conservation laws is completed

in the limit of infinitely fast kinetics by the kinetic and caloric equations

of state

p - p(p, -) (4)

C (p, T) (5)

where p is the isotropic part of the stress tensor iT and T is the

temperature. For a general fluid the state equations, Eqs. (4), (5) cannot be

- explicitly given but are embedded in the general statistical mechanical

description of the equilibrium thermochemistry of the system.

_

In the case of a perfect gas with particular gas constant R

and specific heat ratio y explicit formulae may be given:

p= pRT (6)

-1= (-y-l) RT (7)

p =P(- 1) (8)

2.2 The Electromagnetic Contributions

The electrical equations (consisting of the Maxwell equations

and the generalized Ohm's law) govern the electric and magnetic fields

E,B and the conduction current density ". In the hydromagnetic limit

these are

xE =(at

V x B = (10)

V.B =0 (11)

7 = r(E+ U FB + (K2

where is the thermal diffusion flux ector,

"TK K .

andK" is given by Ii]

a -[0")VT +9)V~T x F+ 0) (VT x B) XB

NSI[ (1)vp+P(2) VPaX F+ 0(3) (7P B")XB] (13)

3

I

The ", *"andP " are transport coefficients defined in

[11, T is the electron temperature and the 'subscript a denotes aplasma component.

The contributions to the momentum and energy of the systemby the electromagnetic field are contained within the Lorentz force,

7 X N and the Lorentz power E. The Lorentz force may be

represented in terms of the Maxwell stress tensor T. as

T XW = T (14)

wher'e the Maxwell stress tensor is defined as

Correspondingly the Lorentz power may be represented in terms of the

Poynting flux S and the electromagnetic energy density e

S m (16)at

The Poynting flux S is defined as

= (17)

while the electromagnetic energy density is

em I 'o t B 2/Z (18)

Let us expand the Poynting flux in terms of 7, F through the use of

the Ohm's law

* 7=

, We have

4

.1os'~( ) ^ ( x^"g -(Aoo), X -90' (U ) X ^" o-

The term 3 X B is simply V - T The term (U X B) XB is readily

shown to be

which can be rearranged to

( X F) X U (F F - T)-"(B 2 lz)

The Poynting flux is therefore represented as

S " em -V- + r1T7-Kx (19)

where rl a (90(r) is the magnetic diffusivity. We note that the Poynting

flux can be decomposed into four constituent parts: (a) a purely convected

flux of electromagnetic energy carried by the motion of the medium

(emt6); (b) a power flow represented by work done per unit time by theMaxwell stresses acting on the moving medium -U .T ; (c) a diffusive

flux of electromagnetic energy driven by gradients of the Maxwell stress

tensor (+r. . ); and (d) a power flow represented by work done per unit

time by the thermal diffusion flux (-nJ K X B).

The electric and magnetic fields and currents may be expressed

in terms of vecter potentials A and scalar potential * as

B X A (20a)

(ZOb)

aI

d

2.3 Fluid- Electrical System

The mass, momentum, and energy equations for the general,

viscous, hydromagnetic system may now be expressed as

PA + 'M 0(2!)at

+7.r=7.r (22)at

+ 7 (T - ) - . 7- + 6 ( (23)at

In the above pT = p is the momentum density and r is the total fluid

and electromagnetic momentum flux

P " + ;I F(24)

The total energy density is

e = p(e + U2 /2) + e m (25)

and H is the total enthalpy flux vector

H = [(e+p)T- • U (26)

The fluid stress tensor 7 has been decomposed into a pressure p and

viscous stress tensor -r where

P 3 . Trace (i') (27)

andV 7 -- T - + (28)

r We may define the electromagnetic pressure pm as the mean

normal compressive Maxwell stress:

6

Pm Trace( ) (29)

Expressed in terms of the magnetic field intensity B, the pressure is

m 3(30)

or expressed in terms of the electromagnetic energy-density em

o 1 (31)

The Maxwell stress tensor may be decomposed into a mag-

netic pressure p. and a Maxwell stress deviator from isotropy T, as

" T , Pm (32)

Equation (32) may be thought of as the defining equation for the Maxwell

stress deviators T*. The mass, momentum, and energy equations may

now be rewritten in alternative form as

2 + .V 0 o(33)at

+ V. . = 7. F + V. - T (34)

a.wt +V - U- -T* + T "T B) (35)

In the above, G is the total momentum flux with only the magnetic pressure

included as the magnetic contribution

G UU + (p + pm) (36)

while 9 is the total enthalpy flux consisting of both fluid and electro-

magnetic pressure and energy contributions.

e= [ + (p + Pm )] (37)

i

I

In terms of the total energy density e, the momentum density

g, and the electromagnetic energy density am, the state equations (6) -

(8) become

p (Y-)[e - *m - 2/ 2P] (38)

T = p/pB = ' am Z/Z] (39)

Let us now consider the transformation of the electrical

equations (9)-(12) into more useful forms. Combining Eqs. (9)-(12) we

obtain the governing equation for the magnetic induction M'.

if - V x ("x) =- x (n x 1) + 7 x (4o)at

We note that given the magnetic induction F(x',t) governed by Eq. (40)

one immediately has specified the Maxwell stress tensor 7 and the

electromagnetic energy density em. Further, the current density T is

determined from B as

7 1 x I

and the electric field E as

"=-" x F + rV x B-

2.4 Viscous and Heat Conduction Effects

Let us now make some observations about the viscous stress

tensor 7 and the heat flux vector q. The Navier-Stokes moments of

the Boltzmann equation yield kinetic theory forms of these quantities:

T L a21&(VU) 0 (41)

q L = -T (42)

!In the above, g and X are the coefficients of viscosity and thermal

conductivity and (7U) is the synumetrized, traceless velocity gradient

tensor:

a0j u. au- 1 auji X2 ax 38

We denote the stress tensor and heat flux vector with a subscript L to

denote that these are laminar quantities. If, on the other hand, we

interpret the fluid variables p,U,T,""" as turbulent mean quantities,

then T and ; contain turbulent contributions due to the turbulent velocity

and enthalpy correlations. Hence, the complete stress and heat flux

fields for a turbulent hydromagnetic medium are

PU ) + L

where U',h' are the turbulently fluctuating velocity and enthalpy and ()denotes an ensemble average. A detailed higher order closure theory for

7 the turbulent contributions (pU U ), (pU h') is given by Demetriades,

Argyropoulos . and Lackner [2]. The radiative heat flux is qR" Since

the optical depths in dense, explosion generated plasma are so small,

the radiative heat flux is only important in layers near the plasma surface

of the order of the radiation free path.

2.5 Nondimensional Forms

Let us consider the nondimensionalization of the fluid equations

(33) - (35) and the electrical equation (4 ). For this purpose let us

specify characteristic values of the variables as p0,U0ee0,P0,T aswell as magnetic field Boo We define a characteristic length L and

characteristic time t0 L/U 0 . We indicate nondimensional variables

with (.I

9

I,

The fluid and electrical conservation laws then become

-[;

ae LI ]+[ 1")M2Rp 1at L

- m • (45)

atM

-7 - O

at

In the form in which the Maxwell stress tensor is decomposed the left handsides of the momentum and energy equations take the forms

SV- -T*

atI

1a +7.H-sV.(U •T*) - +O t 8t

JThe nondimensional variables are defined as

It *U 0 t/L V LV

U TUlaP •P/Po

1 N W /P0 U0

to

T T/T 0

PE LYXiMJ] + U+ Se

* (B /*

E F /U 0 B 0

B / B 0

Az A/(B0 /L)

* /(U 0B 0L)

J/(^OUBO)

-- m ) --Z -- -- I

r *p uu+ (ym 2 1 p I- sr

H [+ (yM 2 )p + S] 6

r r/(B /40 L)

where

* X(p0,To). go0 a (p0.TO). no in(PO. TO)

The state equations (38). (30) in nondimensional form are

401

P=Y(Y-)Mz[e -Sem -~i

p

j It can be seen that the general viscous hydromagnetic equations

contain six fundamental nondimensional parameters. These are

M Mach number U0/( P0/P0

S Interaction parameter (B,/Z',L0 )/(P 0 Uz)

RE Magnetic Reynolds number (UoL/Ai0 )

R Viscous Reynolds number P0 UoL/IL0

PR Viscous Prandtl number (CpILO/x O )

We note that the Alfvdn speed CA is defined as

1/2CA -- (BZ/'O)

and the Alvin Mach number is MA = U/CA. Hence the interaction param-

eter S is also twice the reciprocal of the square of the Alfven Mach number.

For flows in the absence of viscous and diffusion effects

Rm a*, R e -a* we obtain the inviscid hydromagnetic equations:

I (47)at~

- 10 (.48)

12

+ H T (49)at

- VX(UX B=O (50)

at

The jump equations across hydromagnetic shocks immediately follow

from Eqs. (47)-(50).

G - S • } = 0 (52)

{H U T t 0 (53)

{ x (x = 0 (54)

where { } denotes the difference in the quantity across the shock surface

and n is the normal to the shock surface.

Since the electrical conductivity achievable in nonideal plasma

is large but finite, the magnetic Reynolds numbers are not infinite but

perhaps vary in the range 1 R m :5 20.

In this range, the appropriate system is the nviscid, finite

conductivity hydromagnetic system:

+=0 (55)ata + .r -o (56)

I1

I- - _

+ H -SR M '7 7T) (57)a " m

at

or in the alternative form, the momentum and energy equations are

--- + S.G : 'ST (56a)a t

a_e+ 7.H = S 7'" (U SR M .(l.T ) (56b)at"• ')- m

Z.6 Applied and Induced Fields

Let us separate the magnetic field B into an applied portion9(o) sustained by currents external to the plasma and plasma induced

portion 9(i) which results from currents flowing within the plasma:

= -(o) + M(i) (57)

The induction equation, Eq. (58), then becomes

-{ ~ i } = =i R1 --

--(o) -(o)-i Vx(rVxB ) - B (58)

where o is denoted (0)

at

Ii1

1 1

APPENDIX B

AIAA PAPER AIAA-80-0027

STRONG INTERACTION MAGNETOGASDYNAMI CS

OF SHOCK-GENERATED PLASMA

D. A. Oliver, T. F. Swean, Jr., D. M. Markhamand S. T. Demetriades

I

I AIAA-80-0027Strong Interaction Magnetogasdynamicsof Shock-Generated PlasmaD. A. Oliver, T. F. Swean, Jr.,D. M. Markham and S. T. Demetriades,STD Research Corp., Arcadia, Ca.

AIAA 18thi AEROSPACE SCIENCES MEETING

January 14-16, 1980/Pasadena, California

FI per uisin to cpy r repubish, eoW the Ameelon Intlutlof Aoremutlu and Atrwa ,lt.1290 Avenue of the Amelsa, New Yrk. N.Y. 1001

STRONG INTERACTION MAGNETOGASOYNAMICS OF SHOCK-GENERATED PLASMA

D. A. Oliver, T. F. Swean. D. M. Markham. and S. T. DemetriadesSTD Research Corporation

Arcadia. CaliforniaIAbstract

The magnetohydrodynamic behavior of shock- When R > > 1. the appropriate measure of thegenerated pulses of plasma ("plasmoids") in intereAton is the parameter S defined asencounter with an applied magnetic field isstudied. Reflection and transmission of theplasmoid shock front is revealed as Well as the Sa (3)existence of Joule heated zones which are BO/

convectively unstable. It is shown that stronginteraction plasmoids are not delimited by shockfront and following contact surface; rather, they where the spatial average () is over the elect-develop their own internal, evolving structure rically conducting portion of the region L. For a

under the mutual Influence of self Joule heatlng, uniform plasmoid of length a. these numbers becomeand induced magnetic field. As a result, the Zinteraction zone behind an ionizing shock wave will I a ,OB 0 a/p 0 Uof Rm = qo9oU0a, S z I/Reprogressively decouple from the shock front andfollow a trajectory given by the particle motion of Pulsed magnetohydrodynamic flows have beenthe gas rather than the wave motion of the shock examined in the Case of low magnetic Reynoldsfront. number (Rm << 1) and weak interaction (I < 1)

Because of the low interaction, these studiesI. Introduction revealed simple current flow through the plasmoid

and weak magnetohydrodynamic deceleration.

In the present work we study the behavior of The transverse ionizing shock wave which formsshock-generated magnetohydrodynamic flow over the the front of the plasmoid has been extensivelyrange of weak and strong magnetohydrodynamic" studied in the limit of infinitely large magneticinteraction and low to high magnetic Reynolds Reynolds numberS'', In addition to the expositionnumbers. Such a flow consists of a hot plasma of the general Rankine-Hugoniot conditions for* 5plasmoid5 formed between a driven ionizing shock these shocks

$ it has also been demonstrated the

wave and its following contact surface. The such shock waves can be reflected as well as trans-

plasMOid is created by a sudden release of energy mitted upon encounter with an externally imposedin a driver section which is in contact with a test magnetic field. These studies also showed that thegas in which the plasmoid propagates. Such a flow electric field in front of the shock must be self-may be driven, for example, by the use of focused consistently determined with the dynamical state

chemical explosivesl,2. behind the shock and the electrical boundary

conditions imposed upon the gas.The conducting plasmoid enters a region in

w hich an externally imposed magnetic field T and In the present study, we examine theelectrodes coupled to an external circuit %xist magnetohydrodynamics of the whole plasmoid In its(Fig. 1). The plasma conducts current to this encounter with, and transit through an externallyexternal circuit and is subject to Lorentz forces imposed magnetic field. We show that under con-and Joule heating as it propagates through the ditions of strong interaction. hypervelocitymagnetic field. If the explosion drive is a plasmoIds can possess a rich variety of magneto-chemical source, such a plasmoid will be of the hydrodynamic phenomena including magneticallyorder of 5-20 cm in length in traversing a magnetic reflected shock waves, embedded MHD disconti-field region of thl ordir of 100 cm at velocities nuities, and significant periods of transonic flowof the order of 10 ms . The plasmoid may exist within the plasmold. In particular, we reveal theat pressures up to 1 k bar and energies of S eV. dynamics of reflected and transmitted waves through

the plasmoid in both the low and high magneticI f .. Pv U are the characteristic electrical Reynolds number regime. We reveal the behavior of

conducti4

ty' ;&ss density and velocity within the electrothermally unstable plasmoids. We show that.plasmold. the flow may be specified by an inter- in general, the plasmoid is not delimited by theaction number per unit length i and magnetic region between the shock front and contact surface.Reynolds number per unit length r (in addition to Instead. the plasmoid develops its own internal.the goadynamic Mach number), a evolving structure governed by the mutual inter-

action of self-heating and induced self-fields.

(1) Shock-generated hypervelocity flows of this0 m a 0 0 OU 0 kind are subject to a variety of nonideal

0~ phenomena. These include wall Interaction effects(viscous losses. gas leakage, and ohtic voltage

For an Interaction region of length L. the non- drops in boundary layers), thermal radiationdimensional numbers are defined as losses, and kinetic/ionization relaxation effects

L L behind shock waves. In the present study we ignoreL these effects and examine those phenomena which

f I dK Rm f r m dx (2) arise specifically from the magnetohydrodynamic

0

0

1

m m am , |lm, .

interaction. of the plasmoid is then given by Ohm's law and theMaxwell equation in the MHD approximation:

In Part II of what follows we present a mathe-matical description of the flow of quasi-one-dimensional plasmolds. In Part III we examine the J a (E US - U 0 (7)dynamics of strong Interaction plasmoids with anapplied magnetic field but at low magnetic Reynoldsnumber. In Part ZV we similarly consider stronginteraction pIasmoids but at large magnetic Rey-nolds number. In Part V we Illustrate the behaviora (of "transitional" plasaoids. These are flows In (E - U -U (-)which the plasmoid enters the magnetic field atrelatively low values of interaction parameter andmagnetic Reynolds number. As a result of self- where rj a (oar) " is the magnetic diffusivity.Joule heating, however. the plasmoid conductance iselevated as it progresses through the field External Interaction conditions with ancarrying it into the strong interaction, high external circuit including inductive coupling withmagnetic Reynolds number regime, the applied magnetic field coil are required to

complete the description of actual flow situations.11. Mathematical Description Rather than Include such circuit detail in these

illustrations we assume that the external circuitA. Conservation Laws for Fluid and Electricity is configured so that an electric field r .

t(O0.E.0) is maintained within the interelectrodeWe consider a quasi-one-dimonsional descrip- region whose magnitude is uniform in space and

tion of the gas moving over the spatial coordinate given byx In time t. Average properties (over the cross-section) of the duct describing the flow are its E a K(UB) (9)density P . internal energy C , velocity U. andpressure p. The magnetic f eld Ir we separate Intoan applied field E. and a field V. induced by the where K is a "load" parameter (0 S K 5: 1). For aplasma currents. %ram these we Aeleot the total passive external circuit, the value X z 0mass. momentum, and energy densities corresponds to a shorted external circuit; for K

z r 1 the external circuit Is open circuited./ + B I The fluid variables P p.T. w.U are nondimen-

as the state W(x.t) specifying the flow at any sionalized by the values Po.p 0 To. ir0U 0point in space and time: characterizing the Interior of the iniLial pla-old

before encounter with the magnetic field. Nor

r 1dimensional space and time 7.t are defined in termst 1 (14) of x. nondimensionalized by the plasmoid length a.

M(x, n and t by a/U0 . The nondlmensional parametersgoverning the interaction are the Mach number M,the gas heat ratio Y . either of the interaction

The fluid conservation laWs are parameters I or S. and the magnetic Reynolds numberRm

aW 8L B. Boundary and Initial Conditions

X In the limit of Rm -4 . the system consisting

of Eqs. (4) and (7) is fully hyperbolic. ForIn the above. F represents the convected fluxes of finite. R , the system is mixed hyperbolic/parabolicmass, momentum, and energy whle €ontains the with emledded regions where resistive effectsLmrent u frce and pwer assoilated Zcth the applied occur. The boundary and initial conditions whichL orgnti field and te Joule disspetion. e specify the Interaction problem for an explosionare epresied in terms of current density 1 a generated plamoid encounter with a magnetic fieldaneic f id are as follows. As an initial condition we take anm 80. idealized explosion driven flow in which the

r rn 0 plasmoid of liven breadth a occupies the hot zone. between contact surface and shock front' (Fig. 31).

L n/ p B AS .=o (6 At time t a 0 the shock front is located at theL 0 edge of the magnetic field. Over the time scale

Bt/e+p 522po) L 'T -cTX No) for the dynamics of Interest the shock front of theplaamold will run continuously into the quiescentdriver gas while the backward running rarefaction

In this illustration study we assume that the continuously runs into the explosive Source. Hencekinetic effects are confined to the relaxation the boundary conditions for the fluid equations arelayer at the shook, and further, that the those of specified explosion and quiescent statesrelaxation layer is thin sh c opared to the overall at the boundaries a a L1 . x a ..L respectively.thickness of the plassold. The boundary condition for the induced magnetic

thiened of trom Eqp() ha f ymetydcrsFor the samle geometry (xy,z) of Fig. 1, the field , from Eq. (8) is that of symeetry acrossmagnetic field T is given by I0,Ol). the electric the overall plasmoid so that at x % L . I a -a.geld by I R 000), aind the current by i * which lie outside the region of any ourrjnt flow 2

710.J.0). The description for the near fields J.9I

2

--- --

aI(-L2at) -Si(LIt) reestablishes high velocity flow through the

plasmoid and reencounters the transmitted shock

The applied magnetic field B0 is uniform in both front which has been decelerated. During thisspace and time. period of strong wave dynamics the current

distribution within *the plasmoid is stronglyC. Numerical Procedures affected (Fig. 5). The current is diminished to

very small values during the period of plasmoidThe solutions to the initial-value problems deceleration behind the reflected shock I. and then

formulated above and to be discussed in Parts III. returns back to enhanced levels after passage ofIV. and V are computationally generated with secord the fast reflected shock II. The low magneticorder accurate explicit finite difference opera- Reynolds number of the plasmoid allows the currenttars. The hyperbolic system is treated with the to diffuse nearly uniformly throughout its breadth.MacCormack version of the Lax-Wendroff-Richtmyeroperator 10 . For the space-time grid utilized. IV. Interaction at High Magnetic Reynolds Numbercomparisons were made with the analyticallyavailable solutions for the zero and infinite R . We next consider the behavior of a uniform con-zero interact!on limits. At the extreme pressure ductivity plasmoid in the high magnetic Reynoldsratios of 10 between driver and driven gas for number regime. For this case the incident plamoidthese explosion generated plasmoids, the maximum has an interaction parameter I x 20 as in the pre-variations between the computationally generated vious illustration but a magnetic Reynolds numberand analytical solutions within the plasmoid R z 5. Correspondingly. the interaction number S(expressed as a fraction of the analytical has the reduced value S a 4. ,The full conditionssolution) are 0.025 in velocity, 0.04 in pressure, for this flow are given in Table II. The encounterand 0.08 in temperature. of this plaseoid with the magnetic field is similar

to that of Part III. The features peculiar to theI1. Interaction at Low Magnetic Reynolds Number higher Reynolds number are best perceived in the

current distribution of Fig. 7 which is moreWe now proceed to the first of several nonuniform compared to that of Fig. 8. When the

illustrations of the foregoing description. We current levels rise behind the reflected shock 1I.consider first the strong interaction of a plasmoid they do so by directly following the shock until itwith the magnetic field but at low magnetic merges with the transmitted front. The current max-Reynolds number. In Fig. 2 the kinematics of this imum then remains at the shock front of the plas-situation are shown. When the incident plasmoid moid. As a result of decelerating Lorentz forcesencounters the magnetic field, the leading shock concentrated immediately behind the shock front.front may be both transmitted and reflected. In the shock front is slowed and the overall breadththe case of reflected fronts, the rear (contact of the plasmoid is decreased as it progressessurface) of the plasmoid subsequently interacts through the magnetic field. This is in contrast towith the reflected shock front. This colliding the plasmoid dynamics of Parts III and V.disturbance then radiates a fast and slow reflectedshock (denoted shock I) back through the plammoid V. Transitional Plasmoids(which consists of subsonic flow behind thereflected shock front) where it then collides with We now turn to consideration of plasmoid behaviorthe now strongly decelerated shock front, with a coupled electrical conductivity model. In

contrast to the previous illustrations in which theFor the illustration shown here, we s3 lect a conductivity is spatially uniform within the high

plasmoid with Mach number M x 1.64. interaction temperature plasmoid and vanishes outside, weparameters I z 20. S s 200. and Reynolds number R consider a conductivity which is appropriatelya 0.1. just before encountering the magnetic field, coupled to the thermodynamic state of the gas. AsThe full conditions for the flow are given in Table a result, local regions within the plasmoid can beI. We impose the condition that the electrical rendered more conductive by the self-Joule heatingconductivity is spatially uniform within the high of the plasmoid. With the electrical conductivitytemperature plasmoid (we consider electrical strongly coupled to the Joule dissipation, aconductivity functions which are consistently plasmoid in the low I. low R range can evolve intocoupled to the gas thermodynamic state in Part V). the large I. large R ra ge as it progressesThis uniform conductivity distribution is achieved through the magnetic flld and experiences furtherIn the computations with the model conductivity Joule heating. We term such flows "transitional"function plasmoids.

0 T/T 0 < 1/3 We consider a conductivity function of the form

0 T/To t t/3 .1 +-1 (10)

which effectively switches on a constant conduc- In the above @, is an electrical conductivity of ativity ff within the plasmoid and switches the neutral specieAn background and ae is the Coulombconductiviy off outside the zone in which the conductivity. We use as a suamwy representationplasmold exists. The dynamics of the interaction of these two contributions the formsare exhibited in Figs. 4. 5, and 6. When theplasmoid shock front encounters the magnetic field, ..it is both reflected and transmitted. The (.L (11)reflected shock I collides with the contact surface Cn 0 Toand initiates a fast reflected shock II and a slowreflected shock. The fast reflected shock 1I

3

..- T (12) state, the plasmoid is electrothermally unstable

tw 2 o ff lA and creates Its own. evolving region of enhancedelectrical conductance which carries most of the

where Z ff s the average effective ionic charge current and is convected at the fluid speed withinand ' 0.' InA. n > 0 are parameters for a given the plasmoid. The shock front becomes increasinglygas. free of current and runs progressively farther

ahead of the unstable current structure embedded inThe conductivity function Eqs. (10)-(12) is the plasmoid interior.

dominated by the partially ionized conductivity ff

a t low temperature and goes over to the CouloMB Nonideal phenomena such as viscous well layers.(fully-ionized) conductivity at high temperature. kinetic-relazation effects behind the shock front.This conductivity function has the property 8a/8 T- and thermal radiation loses can play important0 over the entire range of temperature. Since roles in these strong interaction flows. The basicthere are no thermal energy loss mechanisms (Which structure of the ma gnetohydrodynamic interactionwould be principally radiative) Included in the itself, however. is a prerequisite to the

* model, the plasmoid is unconditionally eleetro- description and understanding of these additional* thermally unstableS' U . This convective modifying effects.

instability is simply a growth of temperature non-uniformities Within the plsamoid due to intensi y- VII. AcknowledgementsIng Joule hesting resulting from growing electricalconductivity. This work Was sponsored by the U.S. Office of

Naval Research, Contract go. MOOO14-77-C-O574.The interaction of a representative transi-

tional plasmoid is shown in Figs. 8-10. This ReAerencesplasmoid has interaction parameters I s 1. 3 aand magnetic Reynolds number R a 1 just before it 1. M. Jones. "Explosion Driven Linear MHDenters the magnetic field. The complete conditions Generators.* Proc. Conf. on Me0agaussfor this flow are given in Table III. It should be Mainetic Field Generation by Exelosives.noted that the Interaction at entry into the mag- Frescati. Italy. Sept. 1965netic field is considerably smaller than the inter-action described in Parts III and IV. The plasmoid 2. 1. 1. Glass. S. K. Chan. and H.L. Brode.progresses into the field where it begins to self- *Strong Planar Shock Waves Generated byheat and decelerate. The modest Interaction at Explosively-Driven Spherical INplosions."plasmold entry to the magnetic field does not AIAA J.. Vol. 12. No. 3. March 1974. p. 367create distinct reflected waves, but rather astrong and continuous deceleration. Magnetic 3. J. Roscieszewski and W. Gallaher. *Shock TubeReynolds number and interaction parameter grow Flow Interaction with en Electromagneticsignificantly has the plasma is heated. At time t Field.* Proc. Seventh Int'l. Shock Tubez 2.6. the magnetic Reynolds number is in excess of Symp, ed. I. I. Glass. Univ. of Toronto. p.10 and the current has progressively become sheet- 575-489 (1970)like. It should be noted that the current maximano longer follow imediately behind the transmitted 4. J. J. Roscesze*wski and T. T. Yeh. "Shock Tubeshock front. Rather, the current concentrates in Flow Passing Through a Section of a Linearthe electrothermally heated zone which is then MHD Generator.' AZAAJ.. Vol. 11. No. 12.convected at the local fluid speed rather than p. 1756 (1973)radiated at the shock front %peed. Thisdecelerating electrothermal instability is then 5. C. K. Chu and Robert A. Gross. "Shock Waves inswept up by collision with the waves initiated by Plans Physics." Adv. P1. Phys. ad. A.the arrival of the contact surface at the magnetic Simon and W.B. Thompson. Val.2.p.139.(1969)field lIlet. A feature of note is the developmentof reversed current flow in the upstream portion of 6. C. K. Chu. 'Dynamics of Ionizing Shock Waves:the current sheet due to the sharply diminished Shocks in Transverse Magnetic Fields,'magnetic field behind the current sheet at large Phys. Fluids. Vol. 7. No. S. p. 1349.aa. August 1965

VI. Summary Remarks 7. mitri A. Bout and Robert A. Gross. Inter-action of an Ionizing Shock Wave with a

In this study we have illustrated significant Transverse Magnetic Field.' Phys. Fluids,purely magnetogadynamic phenomena which occur when Vol. 13. No. 6. p. 1473. June 1970a hypervelocity pulse of plasma ("plasmoid') en-counters an applied magnetic field under strong 8. Ditri A. Bout. Richard S. Post. and Hermaninteraction condition. With uniform electrical Presby. "Ioni ing Shocks Incident Upon aconductivity within the plasmoid (and vanishing Transverse Magnetic Field.' Phys. fluids.electrical conductivity outside), reflection and Vol. 13, No. 5, p. 1399. May 1970transmission of the plasmoid shock front arepossible coupled with strongly nonuniform current 9. Ta. B. Zel'dovieh and Yu. P. Raizer. Physics ofevolution in time. Such plasmods with large Shook Waves and High Tenoerature Hydro-magnetic Reynolds numbers have current distri- dinami Phenma, Vol. 1. Academic Press.butons (and decelerating Lorentz forces) scens- ly. 1966, p. 23-238trated Imediately behind the shook front. As aresult of shock front deceleration, these plasoids 10. R. F. Warming. et al.. "Second- and Third-Orderdiminish In breadth as they proceed through the Nonentered Difference Schemes for Non-magnetic field. With electrical conductivity linear Hyperbolic Equations,' AZAA J.. Vol.within the plasmoid coupled to its thermodynamic 11, Mo. 2. February 1973

g11. D. A. Oliver. "A Constricted Discharge in C....

Nagntohydrodynamic Plaisa" Proc. 15thSLWEFEt MILuypsiu on Engiearing Aspects of L' KLCh MC

Wsntohydrodynamics. Univ. Pennsylvania, ,

Philadelphia, p. IX.'I. May 1976 TRAECTRIE if

12. 3. T. Dmetriadets. C. D. Maxwell. G. S. /AGSINCE OF IRMSTCKAI

Argyooulos. and G. Fbilds..Bonardi. on/ , avEno T '

on Enaineertnc Aspects tj Manthdro/ *--dyai G altech. Pasadena. CA. p. 64,Hrn 1970 COLLISION Of AULECTto

FRONT AND *- C

TABLE r CONTACT SURFACt FSTK RELCE

Conditions for Interaction at Low Magnetic 119FLECTED TRASMITTEDReynolds Number SMACK FRONT I , FRONT

- REFLECTION ZONE Of

do Quiescent driven gas teperature ftASNOIO SMACK FRONT

Po Quiescent driven gas pressurerEAGIONS OF MAGNET'IC FIELD LAM -

N u 1.64 Va 1.5 ftIOIITEETOR

0 232 Fig. 2. Space-time diagram of atroag intilerSON,TD/Ta 4 K a0.5hyperveLocity, plasmoida. Planetoid tra jectory IniTo'ta 4 KS0.5the limit of vanishin interaction ia delioited by

the shock front trajectory U'5" and coattact mar-TABUE 11 face trajectory C ' *C Plamid shock front re-Conditions for Intersation at High Magnetic flection S, and toanamiision 89810 are initiatedReynolds Number at encounter with magnetic field. Contat surface

encountars reflected front at C, which generatesM s 1.64 Y a 1.5 fast reflected abock 11 (CIC"). slow reflected

PDPa 232 Shock ZX (CC''), and Convected wave CC'C").

40ABLE IN IANO1

Conditions for Tansitional Plasmoid

4, WEI FELDM s 1. 64 0 a 1060 1 DRIVER GAS rW

PO/pc I 232 n a 3.11 Iiz-5 z f/nz01 -LI X-Li

Y z 15 K ~.*na 0.16X2 5I 0 P

Po)

CONTACT FI1LO COIL

T. b)

OIVE

.4 10RI19 GAS - OAIVN GA Fig. 3. initial condition for plamid latereties

for the Wessue to), temperature Ib) * and velocity(ac). Shock front of plammoid of given breadth a

ig. 1. schemtic of explosion-driven plasmoid flow loeated at magnetic field edge at time t *0

Aft .616410 W.4

Fi. ~ Vlcif"iel of0WT4 UM.~i intasttrufa pled i il ne srn neato

CI a 20), low aa.~etic Reynolds number (Rn u P0. T condito6 Shoc frn4fpes, sbt elce

from (ateti fieldK Oa) an"rnnttditulgei ild()elce W a olie ihcntac srfce(d an iitats fst efecedshck () ndslw eflctd hoiL(a) Rflctd av

0.16 S1u) MUM Usao Mm.?__

0.0 W

0.5.11

Fig. 4. Cueltenity edisiutin foracnitin hoa apied mntic iealdo cundenite(acdion(Ia20e. shl masneticsmeidoldlocitier dimai0h1)ehindireflense shock front (b).l Af t resa flce

tcsufc(danintaeafatreflected shock 11 (a)cretlvsrieC) and so refleine ufomtruplsock. (gth uniformcon-adu1ti(it mdel, wihornutiited isuiomehock front (f)) anContact surface (h) Atarte lonucin frmondutc

iengs nber Note tr0.1),cyrregnt is difsedanaly niforml overiiswt h plasmoiddrn h

peio ofrfeto6n eelct ftesokfot

b) PUUIS 411411hL&V KATO AT LOW CUMUNI

Fi.6 Teatr distrbutio foM condition oT Fig. v9hevOeunfomteaueoVlsodiaffected~~~~~~~~~~~~~~~KNW byLCTL wave MyETic *ados iifintybyJuehtngWekItacrenlelsdeo

reflected hock frontICE (a). as hw in i.l edt ekeeain ntmeauedet ol et

ing~~~~~~~~~~~~~~~~~~d (b).MI ote tAT heate rgowihnpsmiattttisscneteatowvlcity.Trsmteshock1. front isM mhr~dclrtd otc raei uhlg eeeaed during9 AThi peCod After

2~ el. . C. a I mgaa

Fig. 6. Temprtur distribution for tongdineationso pFg. tris uniformynampcsratre similatoi ths

eflebited inoc fios. I4 h(aever thew n .leynod towe petiore Inonteiforatureneto(n Jdueateing fi.oe ditribation reio wurren pls intate bya etcmer cofvce patow shocirot (ansgieshoc ton Is shacrrelt eelsaind.lowntalct raegin bechn eflecelted du ringthi peid Abfureteleesrs n hf eato lsod()C)atrpassage of reflected shock fro) ol etngIcesssgiianl hrb lvtn l. Noe hacunto conepratue() iFoitllobin asg ffetd shock frn1i.nrstt that of Finteio iwer convente isnarlunliiform ran igewe that of Fi.fweecroent isd neatl abeont)c behinde shoc feriodetott.so

'111,--. .... .. ~- , ,- - ~ - - --

WSt wet. lo

Fi.8 eoct itibto o trnstin# MlITid P~mid ntr aeicfed ihI 2.N ,Su2. Mds ntrcina etytoagei il Aoes not gnateditnc rfeced ve, u

ratherstrongsad cotinuou deceeratio. Magntic Renolds umbea. d ineltonprmtestodu 4gtecu.7ftsi.Sokfotpoed truhmgei il it otnosdclrtoCs.mc h enod ubrrahe ausR 0,tecreths eoesetlkewt elgraton-ieieniigdioniut wiinalresboizoeoth pas idCoatsrfe

enc.8. elctditiuinfrat toa lsad aso etersvt magnetic field wibh inIae aontrs runn1,e () d C)

ratersrnI n otnosdclrto.Mgei enlsnme an ntrcto parameter growW~ W~AILtdurig th couse o tmansit Shc frn poeesthog mgetcfil wt oniuosdeeerto(a). Once5IIt tIhe Reyold nubrrahsvle m51,tecrethsbcm he-iewt el

36.10

WUM too.M

(4)~f 11411111111"TS "wagh.91M ,6? i (UM lS

Fi. .uren dstibtin ortmnitocl ~suod f 118.Curntcocetrtin ehndshckfrn

at ntr tome~eti fild ays(a)wit coresoningheaingandconuctvit enancmmn (rfleteIn tempe atur fielof g 10).~ ShocT frn rpgts ha bt eoe nrasnllreo

Fi.urednete fieldriun upstranspotiona of smi curen nheet CurnConetai).eidshc rnat nty o agetc ied gow () it crrspndig eain ad onuciviy nhncmet relete

S.""

6.2 (4) MRNSW Am INSTASaU"

S.'.,

46

I..[i

M&W .

P s T 9 Z 4

III'

APPENDIX C

HIGH MAGNETIC REYNOLDS NUMBER EFFECTS

AND STRONG INTERACTIVE PHENOMENA IN MHD CHANNEL FLOWS

D. A. Oliver, T. F. Swean, Jr., D. M. Markham, C. D. Maxwelland S. T. Demetriades

HSIGN HNPI3T1C UYNOLOS W ZER AM STRONG IMPACTZON PHIPOW)MEA IN 142D CMWU I FLOWS

D. A. Oliver, T. T. Swean. Jr., 0. M. MarkhamC. 0. Ma ell and S. 1. DometriadesSTO Research Corporation

Arcadia, California 91006

1. INT24ICOtN In Part IV we consider the three-dimenional stronginteraction dynamics of a steady hydromsagnetic channel

Zn this paper we mamine mom* phenomena of interest flow at high magnetic Reynolds numbers. The quasi-in sagnetohydrodynamic (Her ) channel flay a high one-dimensional. strong-interaction, high R unsteadymagnetic Reynolds numer. Such flows may be created by problem is treated in fef. 3.

chemical explosion* and the esutig MW process may beeither inherently unsteady A or quasi-steady.

1 it. Unsteady Nigh haumetic Reynolds Number

Electrical Conducti on

Zn the unsteedy process (Mig. 1) the flow consists

of a hot plasma (plasoide) formed between a driven A. Governing Ilectrical Eq1aioftioazinag shock wave and its following contact surface.The plammoid is created by a sudden release of energy Let us now consider the form of the electricalinto a driver section which is in contact with a test equations. The Maxwell equations and OLO's law governgas in which the plammoid propagates. The conducting the electric field i, magnetic field 0. and currentpLamid entera a ion i which an externally imposed density T. Let pa the axis aligned with an appliedmagnetic field S and electrodes coupled to an magnetic field and let %.y be the coordinatesexternal circuit mist (Fig. 1). The plasma conducts defining the plane of conduction. The current vectorcurret to this external circuit and is aubject to - - -Lorentz forces and 4oule heating as it propagates then becomes J - J(Jx.J .01. the magnetic vectorthrough the magnetic field. If the explosion drive is - - -y -

a chemical source, mh plsesoid will be of the order becomes a - 3(OO..), and the velocity U - U(XU 0 0)of 5-20 an in length in traversing a magnetic field -

Rin of im 0fdar of 100 cm at velocities of the re .7 in l current rector noodineasiomaLiaed asogder of 1f as . The plasaoid may xist at pressures ms.0'i_ th aximm 'aloe of the ap. lNdup to k br a eergiee of up to S eV.

magnetic field. 0 is the velocity nondlmn~lonalisedIn a steady flow device. the physical schematic is -

the me as in Pig. I except that a stagnation chamber on the duct inlet velocity V . is the magnetic fieldexists ahead of the HO channel in which the explosion nondimerALoneliaed on the maximsum applied magneticdriven gas*s are processed. This configuration yields field as. We d4, le the magnetic into thea hyperonlic channuel flow whtich is quasi-st*dy over appie~ld mportion 9 4" an portion a; which is

the flow time scale in the channel . induced by the currents flowing in the plama. Thea hperoni chnne flw wichis uas-stadyove aple porations3 and a portin 5 whichbne isff are the characteristic electrical axwoll equations and Ohm's law then may be combined in

condctiil; ensty d veociy *~* fo...ay the induction equation form asconductivity. 01450 density and velocity, the flow way

be specified by n interaction number i and magneticReynolds nmber re (in addition to the geedynamic Mach in() + i 4 .Z~m 5. , _._

ebone numbers are defined

diffusivity. I is t~he mantcI odsnme uead m* 0 m upon ( 0 ro. and a and

Fhrn i ntr, the apropriae measure of the iter- a(0) m ...ation numbeth parameter 3 defined so at

S (I q0 )./ (pU) ) The current densty como nen armWhenR P,1 theta aerpae mea suvrte ofecterl (0) x

osamcting prtioen of tO region L. For a uniformb

plams _ of longth a, these numers becme The beundary conditions appropriate to Eq. (4) are1 s 0 SO/pU 0 , ra a SrpoUOa G - I/% , that an insulating boudaries.

In Part It of what follows we consider (t) , constant 16)

two-dme seioal electrical conduction under conditionsof high i fot the umetedy flow of a plasid and itsemcem ves with the applied magnetic field And the wals on conductor*electrode System. in Part Itt we consider the

- two-dimensional steady. hypersonic flow of plasme t 0high manet&c Reynolds number througm a constant ares dn

if reemagnem dus. We desribe the current distribution were n is the iorel coordinate to the cedacterin the platn of Pte Lz and III under the conditions urfce.

of high at but for vanishing inceraction I.

-i & 5

...I . . . . .. -, .. . , .. dr

S. The unsteady Flow Process downstream from the duct entry from the driver. At thestation 10 cmupstream from the generator inlet (x

I. the unsteady flow process, a plasmoid of initial 0.50 a) the conditions of the flow arebreadt a enters the magnetic field and electroderegion atI.t-0. The front of the plasoid mes~ Velocity a: duct centotcline 9612 mlsec

behind the shock front and beyond the contact surface M4ach number 3. 14move at the plasmod velocity Ct . Within the plainaid Boundary layer thickness 5. S NoI(damarkad by the shock-front anw contact surface),* theelectrical conductivity is W~ while outside the At the center of the generator section (x - 0.60 a)conductivity has the value 4,c0a' hs ale r

The applied magnetic field along the duct axis is Velocity at duct centerline 9S20 to/secassumed to be given by the following model profile Temperature at duct centerline 36,930 Xfunction2 Mach number 3.11

Boundry layer thickness 6.5 Me

0 x xIZ At the exit of the duct, (x - 0.70 a) these values are

Velocity at duct centerline 9430 m/secCS) tempe)(8)rature at duct centerline 36,970 K

Poach number 3.07Dx( L2 oundary layer thickness 7.5 so

- expo~x-/Z)IThis distribution of nonuniform velocity and

(1-exwe-L/Z) J temperature in x and y (with corresponding density and* conductivity nonuniformity) resulting from the boundary

whr (0) j.temxmmapidlayers Is used as the basi* for the electricalwher 6 s th maimumappiedfield. The values calculations utilizing Eq. (4).* The nondimensionalized

selectsP are L. - 4h and a - 2/h -here h is the duct values are based upon the velocity and conductivity atheight. the dut Inlet (Z 0).* The Reynolds number is based

upon the channel height, h and hes the value R * 7.C. ResultinegElectrical Distributions The velocity and temperature profiles at varicus

Stations along the duct are shown in Fig. 4. SinceOiven an external load condition, and a given these calculations decouple the electricity from the

plasomid geadynamic condition we seek to describa the fluid behavior (weak or vanishing interaction) only theresulting current and induced magnetic field distri- Magnetic Reynolds number, R is a relevant electricalbutions as the plasoid progresses through the mw6 duct parameter.mand generator section. we shall consider conditionsfor U U - 4/3 (corresponding to strong shockso in noble a. Resulting glectrical Distributionsgases and initial plasmoid breadths a 0 h A Themaximum nondimeonsi. al current flowing in the load The appropriate electrical equations to be solvedcircuit is a rl I/ 144jf) where z is the maxims current are Bqs. (4) and (5) under steady state conditions.per unit length flowinT to the external load. Boundary conditions are as given in Eqs. (6) and ( 7).

The velocity and temperature profiles (and theThe current streamline distributions land corresponding corresponding electrical conductivity profile) will beinduced magnetic field distribution I for a magnetic similar to those shown in rig. 31 however we shallReynolds number R - 10 are shown in Fig. 2. The con- reveal the effects of different absolute electricalplete dynamics o? interaction between the generator oonductiviity levels (or velocity or size levels) bycurrent and the eddy-currents induced by the magnetic varying the magnetic Reynolds number for the variousfield distribution are revealed in rig. 2. computed results.

111. Steady IUC Channel Flow at High 3. In Fig. 4 we exhibit the open circuit and loadedcondition for a magnetic Reynolds number ft - 1. *it

A. Channel. Applied Magnetic Field. and Plow can be seen that there is an onset of eddy cellsConfiguration induced in the regrions of magnetic field gradients and

a convection of the current pettern downstream of theWe now consider the electrical conduction resulting generator. The principal generator current, however.

from the steady flow of plasma at high magnetic is still confined to the region between the electrodes.Reynolds number in a constant ares duct. The flownexist within the duct geometry and applied magnetic In Fig. S we exhibit the same situation but for afield distribution shown in Fig. iIin which the magnetic Reynolds number R - 7. We note that consid-generator electrode length is equa to the duct height. erable current flow induced by the magnetic field?he applied magnetic field is given by Eq.. (6). gradients exists under open circuit conditions. Rate

how the upstream eddy cell actually couples into theThe flow field has been calculated as that generator electrodes. Under load, we see that theJ resulting from a reservoir which feeds the duct with generator Current exists c-ofletelv downstream from the

arge. at an inlet velocity of 10 hoo/sec. an Internal electrodes.eneorgy of 37 NJ/kg. a pressure of 10 It bar, end anminmal electrical Conductivity or 2S.000 O/n at the tn Fig. 6 the conduction field for a magneticduct inlet. The resulting turbulent boundary layer Reynolds number No a 17.5 is shown.interaction to considerable as the flow proceeds downthe duct. These flow distributions have been Again, oons~derabl* eddy cwrent activity existscasulated with the STD Research Corporation Q30 family beth at open circuit and under load. The generatorof ides". current is again driven out of the interelectrode gapI and exists downstream of the electrodes.

The generator mlectrodas are located at 0.6 m

IV. Three-Dimensional Strong Interaction Dynamics 1, R > 1, it is possible to show that the transverseat High Magnetic Reynolds Number Loretz force is of order I/R tines the axial Lorentz

force. Hence, cross plane momentum is weakly generated

A. Flow Structure Prediction in Hydromagntic Channel by Lorentz forces compared to axial generation. TheFlow primary source of cross plane velocity nonuniformities

is the redistribution of axial mass flow due to theIn short burst MUD generator channel flow nonuniform deceleration.

(timeecale 100 s - 1000 p s) utilizing explosivelygenerated and compressed plasma as a stagnation Let us now consider the electrical description.source, the flow is very nearly a steady channel flow. With I - (U .0,0). the three-dimensional inductionAt the plasma conditions of electrical conductivity, equation can b e expressed aslength scale, and velocities achievable in such flowsboth the interaction parameter I (ratio of Loreints ai uforce to change of momentum) and magnetic Reynolds Ua : "T -- x + X( ' V) U -V x:(W7 x F) (9)number R are large. For the hydromagnetic flows

typical mof the generators characterized by theoperating conditions of Table I we find the magnetic where xI is the unit vector in the x-direction end n -Reynolds number R to be about 20 and the interaction (P0 0) is the magnetic diffusivity. The boundaryparameter I to bi about 15/mater. These conditions conditions for Y are as follows. As can be seen fromindicate that the deceleration time for the fluid is of Pig. 7 , specifications of the boundary data for U onthe order of the diffusion time for the external the channel perimeter requires solution of the externalmagnetic field (consisting of applied and possible (outside the channel) as well as internal problem forself-excited combinations) to penetrate through the . As an alternative to solution of the externalplama. One may therefore expect very significant problem, we approximate the boundary conditions forinteraction between the velocity field of the flow and a Y 5 asthe nonuniform Lorentz forces associated with the y Znonuniform magnetic field diffusion. B A• ( +AB)exp(-x) y > 0

In Fig. 7 we indicate schematically the interaction w A Y

of the magnetic field with the high 3 channel flow. on ZAwey from the front and throughout e channel. thelines are more sharply bent through the magnetic B :

I-ex p (*

Xx )

boundary layers and resemble the cross plane structure Z eshown in Fig. 7. At high magnetic Reynolds number the ('O)magnetic boundary layers are thin and transverse r B a 0gradients in y,z are much stronger than axial gradients ) yin x. Correspondingly, the generator current is on y Z -Concentrated in the magnetic boundary layers. At the - z : +channel inlet the magnetic field is most stronglyexcluded from the flowl at the exit the field haspenetrated most deeply. We now ask how a hydromagnotic 41, U -

hflow will respond in Its velocity, current. orentz where --od in the decay term for the fundamentalforce, and power generation distributions to thi difusion bode in a square duct, a is the far-fieldfundamental nonuniformity situation in high magnetic external value of the magnetic fiel4 (totally in the zReynolds Nmbar, strong interaction flow. direction), end AS is the maximum compression of the

magnetic field at the generator inlet. It can be seenthat the approximate model (Sq. (10) possesses the

TABLc 1 correct limiting values at x - ca. For R >> 1, thevalues at x - 0 are also close to them

full-fieldConditions for Illustrative High Magnetic exclusion solutions. The model is least accurate for

Reynolds Number Channel Flow conditions of R Z 1.S

For 2 the boundary conditions are a I 1 /2 onGenerator Inlet Conditions a - w where r

is the total current-per unft depth<"> 7 x 103 /s flowing in the generator in the transverse direction.T 2. x 10-elX

P -. bjr C. Magnetic Diffusion and Flow Interaction

5 SxlO mho/m(0) We now consider the predictions for flow in the

External agnetic Field (B ) 10 T generator channel of Fig. 7 under the conditions ofGenerator Cross Section (h) 5 cm Table 1. For these conditions we set the generatorR 22 current at the nominal value1m 15 "

r a (i-K) w(wuB)

B. The Mathematical Model with K = 1/2 and w = the channel width. We assume thatno field has penetrated the plama at the electrode

The fluid mechanical evolution is calculated with edge x - 0. (In general. one will expect a startingthe $TO Research Corporation Q30 code family magnetic boundary layer thickness at the channeldescribed an mand 9 • In the version of the Q30 family entrance dependent upon the plasma nozzle geometry and

" utilized in these calculations we center attention on the fringe field location.)the axial velocity field U (x,y,s) and neglect theeffects of the cross flow oA secondary flow velocity The distribution of the transverse current J andfield. Since 1 3> 1, the flow is dominated by the magnetic field component a for various 1xialelectramagnetic rather than fluid mechanical viscous stations in the channel are shownein Fig. a . (Thereeffects, although turbulent viscous effects are is an axially directed eddy current J associated withincluded in the calculation. Under conditions of I P) the fields S a-i U which is ,.ot shown here. 1 k x *

1.4 Cm the magnetic boundary layers aze exceedingly channel 1S cm long, only 10% of the ideal power isthin ( 3 Ims). They have grown to 1.4 cm at x = 14 cm available and for a 28 cm channel only 27% isa" 2.9 as at x * 2 . The Inward diffusing magnetic available. These results must be tempered by thefield component a z is initially concentrated along the observation that beyond 12 cm the boundary layers havewells parallel bo the external magnetic field (elee- begun to stall; it is not likely that power extractiontrode wallas see Fig. 1). but the z component on the will be enhanced by realistic inclusion of the recir-

side wall builds up as the flow proceeds down the culation zones under stalled conditions. Hence, thechannel. The transverse current J is initially power extraction levels beyond x - 12 cm will likelycncentrated lonq the sidewalls and Ydiffuses inward not be reached.from these wills.

V. AcknowledgementIn rig. 9 we observe the impact of these

distributions of current and magnetic field on Lorentz The authors would like to acknowledge manyforce and axial velocity. in the early stages of the interesting discussions of this problem with S. T.flow evolution the maximum axial Lorentz force is an Demetriades and C. D. Bangerter. This work wasthe corners. This is because the maximum field B supported by the U.S. Office of Naval Research underexists on the electrode walls parallel to the z axis Contract ONR-M-00013-77-C-0574.and maxinum current J exists on the sidewalls parallelto the y axis. The laxinue product of the two there- Referencesfore exists in the corners. Throughout the entireevolution, the maximum Lorentz force is concentrated inthe magnetic boundary layers along the sidewalls. 1. C. D. Bangerter and B. 0. Hopkins, Hercules, Inc.,

and T. R. Bogan, )WPPSCO, Inc., Proc. 6thThe impact on the velocity distribution is strong Int'l. Conf. on Magnetohydrodynamic Electrical

and significant. The initially flat velocity profile Power Generation. Vol. IV. p. 1SS.Washinqton,is rapidly decelerated in the sidewall layers. by 12 D.C.. June 1975am the flow is actually reversed in the corners and theregions of reverse flow spread. Because of the 2. S. P. Gill, 0. V. Sam, and H. Calvin, "cptoelvedeceleration of the wall regions by the inward P= Research, = Artec Assoc., Annual Ppt. No.diffusing Lorents force the core flow is accelerated. 119AR to 01, April 1975 - April 1976From an initial value of 7 km/s the core flow isaccelerated to nearly 10 ka/s by x - 28 cm. Since 3. T. F. Swean. C. D. Bangerter and 0. M. Markham,there is little Lorentz force in the core region, this Dyuamics of Strong Interaction Shock-high velocity region is not effectively coupled by the Generated Plasma, Paper No. AIAA-80-0027,generator. AIM 18th Aerospace Sciences Heting,

Pasadena, Calif., January 1980It should be noted that once reverse flow occurs

(stalled MO boundary layers) the present prediction of 4. S. T. Dametriadee, G. S. Arqyropoulos. C. 0.the actual velocity field in the wall layer region Maxwell, "Progress in Analytical Modeling ofcannot be considered correct since upstream feedback is MW Power Generators." Proc., 12th Symposiumprohibited by the method of calculation. The present on Engineering Aspects of Magnetohydrodyna-calculation is only definitive in its prediction that mics. Argonne Nat'l. Lab., Argonne, IL, pp.stall will Indeed occur. in actuality we conjecture 1.5.1-1.5.12, March 1972that the flow under these conditions will appear asshown in Fig. 10 in which recirculation eddies are S. S1 Research Corporation. ' TD/MHD Codestrapped within the boundary layer region of intense Introductory User's Guide: Code: QUE3DEE,Lorentz force while the core flow jets through outside Family: Q30, pt. 4o. STDR-102-79, July 1979the region of the eddies. In such a stalled flow,there is little Lorentz force coupling to the highmomentum jet in the central region of the channel.

As a demonstration that stall Is not a peculiarity

of the corner region, we show the Lorentz force and

velocity profiles at 21 cm for a higher interactionsituation (a - 2 x 1 0 mho/m) but otherwise conditionsidentical to Table I (Fig.ll). In rig. 12 we see thatthe entire sidewall has reversed flow under theseconditions.

In rig. 13 we show the distribution of power CONTACTproduction -( .7 F) over the cross section. Power is ELECTRODES ,PLAsAOIConly produced in the mid-well layer regions and is SsGCK RONT

actually consumned in the corner and sidewall layerswhere reversed flow takes place. The overall powerextraction per unit length, dP/dx. is defined as theintegral of the power density over the channel crosssections EXPLOSIVEDRIVER LOAD

1E,.f ( -F.i :dx - A

The ideal power atraction for uniform flow is

- J ' 'AGN(T

(1) KO( I f BK A fIELO COIL.dxideal DR 0IVE& W - t-'O iVIN GAS

i In Fig. 14 de exhibit the ratio of tr.ese quantities for 9-2974

the conditions of Table 1. It ian be seen that for a Fiq. 1. Schemati.. ,f explosion Arive. plasmoi f'low.

1o

I

I i

USOK_ _-_1.1,2.5 ()

-0CONTOURSINTERVAL

0.05

USHOCK

o -. 0.5O.S CONTOUR

INTERVALo.5 0.05

USHOCK0.5 2.5 ( )

. 0.5

CONTOUR0.5 INTERVAL

-0 0.1

-3.00.25 1 -. 0.

0.2 CONTOUR

INTERVAL0 0. 0.05-0 '

i-3.5 USOK (e)

CONTOUR

:.So-INTERVAL

L 0.050-3509

riq. 2. Induced magnetic field isolevels (current density streamlines) at R - 10 in unsteady passage ofexplosion-driven plasmoid. At time t - 0, the shock front is at the edqe of the applied maqgnetic field.Upper fiqure in each time sequence indicates plasmoid position. LoWer fiqure indicates magnetic fieldisolevels. Note intense eddy-current cells in (a) and (e) when plasmoid is in region of applied maqnetic

field gradient and out of ranqe of electrodes. tn (bl, the generator current is concentrated in a thin

zone behind the shock front. In fc), the plasmoid has progressed lownstream so that the qenerator current

is stronqly convected lownstream. Ir (d), the qenerator current an" lownstream eddy cell interact

toqether.

8 00

MIMMI __ __ _ __ __ _ __ __ _

I .INTERVA

.2.1 0.I .O 0.5

0-3592. Fig. S. induced magnetic field leolevels -a( for

I ap steady winttohydrohyna flow with thq I . * profile of i q. 3 and R - 7. I I-(4), the

generator is at open cLrLi, (J 0) a4d In (b)

fdthe generator i(sn- 1).

I I

0-3S90Fig. 3. Velocity and temperature profiles between

electrodes in turbulent supersonic plasma flowin an HO generator duct.

I(0 )

4.s) 0

I.$ COUOUI NTEIVAL

0 0.$ 1______________________ .

.4.05 0.c (0) 2.•0

L I I_ CoNTouR 0-3593

0.02 Fig. 6. Induced magnetic field isolevels a L) forsteady magneotohydrodynamic flow with profilessimilar to those of Fig. 3 and R - 17.54 in(a).a. (Ol (aI . the generator is at open c itcut ( . _ O 1

-.O cotouR and in (b) the generator is under load (1" 1).INTERVAL0.05

0-3591

Fig. 4. Induced mangetic field isolevels a"' forsteady magnetohydrodynamic flow with profilessimilar to those of Fig. 3 and t -_ 1. In (a),the generator is at open ci cui t-(,- 0) and in(hi the generator is under load (.r 1).

"-....ruct r, Oft * i-kt.I :qI

8-2970

Fig. 7. Magnet c field nonumifomities Ln xqh Magneticftynolds Mumber channel flow.

6

0

0*0~ -ti-150,0-* z tq q

200*1~~~~~@ a t lVI. mASV

I, Ir

IBM

0 LPx~~ ~ ~ t" o+*9*

1043ssz~~~~s 1032* aUwqc10+3zoo~t

£c

I-

; O .3SSl ' 2 aT $ X W # *a X V+O 3 ? 0 0' - a T60-P3 66*9

PA6NETIC WAUnOM

IJIWII I(- III.- -1. -

LAVER KVELWIMENT\ STALL POINT RECIRCILATION CELL

X A

FL9.10. ConjeCtured Velocity streamlines in stalledhigh m4agnetic Reynolds Neumber channel flow

- I -

showi4Jng recirlation catllstr aped (0-- boundar y~g1,lol'etl'.Ln4ntyaot o o a

layer region.oe e~o8 onued_ _ _ _ __ _ _ 4 ;- K*1..c

tfi

8-2874

,A0 s d X.is~ .

Fig. 1. necntu odlistrob sta xe 21 mc rralled-hi Mdnticer Interaction case ( f0

10 ho/) bu othrwise ondtons as I Table 1 k. c4ame o ee ercln e nt eih

shevin rrigs.laio cel andge in Poerisprduadrya

layer wallrioon.

Q.4

0.2 -

ihPOINT

8-285A ITArig. 12. Vloceity oc distribution at x 21 m corre-ng 0-2877

Ingnd to higher Interaction case ( a 2 x10a/8 but Otherwise conditions as Tabl 1. rig. u. Rai fower attraction pit evoltiongrth cas

Note tat enire sdew~llflow s revrsed ideaopoee exrc is Warodlcers inaid

this case. oner mall regionelngh

I