H. Kamkar and H.K. Moffatt- A dynamic runaway effect associated with flux expulsion in...

8/3/2019 H. Kamkar and H.K. Moffatt- A dynamic runaway effect associated with flux expulsion in magnetohydrodynamic ch

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J.Fluid Mech. (1982) , vol. 1 2 1 , p p . 107-122Printed in Great Britain

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A dynamic runaway effect associated with flux expulsionin magnetohydrodynam ic channel flowBy H. K A M K A R t

A N D H. K. M O F F A T TSchool of Mathematics , University of Bristol

Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge(Received 3 October 1981)

Pressure-driven flow along a channel in the presence of an applied magnetic fieldwhich is periodic in the streamwise direction is considered. The configuration is suchthat th e transverse component of field B, is non-zero on the centreline y = 0, but itsstreamwise average (B,) is zero. In this situation, flux expulsion due to reconnectionof field lines occurs when the pressure gradient is sufficiently large. This leads to adecrease in the Lorentz forces, hence t o an acceleration of the flow, and hence tostronger flux expulsion. When viscous effects are weak (i.e. a t high Hartmann number)this creates a runaway effect, which appears a t a critical value of th e pressure gradient.This critical value is determined in the inviscid limit, and numerical and analyticalmethods are used to explore th e associated cusp-catastrophe behaviour when effectsof weak viscosity are taken into account.1. Introduction

The phenomenon of flux expulsion in magnetohydrodynamics occurs a t highmagnetic Reynolds number, whenever a flow with closed streamlines acts upon amagnetic field transverse to t he flow, The purely kinematic aspects of this phenomenonhave been widely studied (Zeldovich 1957; E. N. Parker 1963; R . L. Parker 1966;Weiss 1966) and are well understood. It is also known (Galloway, Proctor & Weiss1978; Proctor & Galloway 1979) that flux expulsion can persist even in circumstancesin which the magnetic field has a strong dynamic influence.

I n this paper, we study an aspect of the dynamic behaviour associated with fluxexpulsion, which has been investigated in the simpler context of solid conductingrotors by Gimblett & Peckover (1979); this is that, as the forces which drive themotion are increased, the ohmic resistance to motion can decreuse, leading to a run-away situation in which th e kinetic energy of the motion increases very rapidly, beingultimately controlled by any other weak dissipative processes th at may be present.This behaviour was recognized by Gimblett & Peckover as a typical cusp-catastrophebehaviour, with associated hysteresis effects. The same type of behaviour in a fluidcontext (rotation of a fluid cylinder in the presence of a rotating transverse field) hasbeen identified by Moffatt (1980).If the flow does not have closed streamlines, flux expulsion can still occur provided

t Present address: Berkeley Nuclear Laboratories, Berkeley, Gloucester, England.

www.moffatt.tc


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108 H . Kamkur and H . K . Mofjatt

7 w + l J17FEBYFFlow_

( U ) (6 1FIGURE. Sketch of configuration considered: (a)magnetic field with no flow; ( b ) magnetic fieldshowing flux expulsion due to flow under spplied pressme gradient.that the applied field is non-uniform, and a particle trajectory crosses a region orregions in which the transverse component of applied field changes sign. The con-figuration of figure 1, which we study in this paper, falls within this general category.Specifically we consider the plane two-dimensional flow of an incompressible fluid ofdensity p and electrical conductivity v, long a channel IyI c b , driven by an appliedpressure gradient -pG, and in the presence of a magnetic field B whose normalcomponent B, is prescribed? on the boundaries in the form

B , = B,coskx on y = f b . (1.1)Figure 1(U) hows a distribution of electromagnets which could (in idealized circum-stances) produce such a field, and the resulting flux lines when the fluid is a t rest (i.e.when G = 0 ) . This type of configuration is of potential importance in the context ofliquid-metal cooling circuits of breeder reactors; here, magnetic fields may be used tocontrol the flow structure, and it is important to understand the influence of non-uniform fields, among which the field of figure 1(a) ay be regarded as a particularprototype.

Consider first, in a qualitative manner, what happens when G increases throughpositive values. Assuming th at the flow is laminar, the field B and velocity U will beperiodic in x with period 2n/k Let U and B represent average values (with respectto x)of U , and lByl on the cedtreline y = 0. Band U are related through the inductionequation and the equation of motion. The induction equation describes the distortionof flux lines in the downstream direction, and the elimination of strong field gradientsby ohmic diffusion. When U is large, this process will evidently result in expulsion offlux from the core region of the duct, and the formation of magnetic boundary layerson y = i (figure 1b ) ;we may therefore infer a relationship

B = P ( U ) , (1 .2 )where F( ) s monotonic decreasing, and exponentially small as U -f m. (The fact tha tthe asymptotic decrease is exponential is confirmed by the analysis of $4.2 below.)This is represented by the solid curve of figure 2(a) ,which may be described as theflux equilibrium curve.

t A referee has pointed out t ha t, i f the exciting current consists of a current sheet J, sin kxon y = f , the regions lyI > b being filled with highly permeable materia l, then it is B , ratherthan E , that is prescribed on y = f . This leads to a rather different boundary-value problem,but with similar qualitative behaviour.


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Dyn am ic ru naw ay effect in M H D channelJow 109

Dynamical equilibrium curves

\,Fl ux eauilibrium curve

A3 A * A ,FIGURE. Flux equilibrium curve (solid)and dynamic equilibrium curves (dashed). ntersectionsrepresent possible steady states which are of Hartmann type ( H )or Poiseuille type ( P ) stateslabelled J are unstable. (a) nviscid situation, showing runaway behaviour if G > G,; ( b )weaklyviscous s ituation, showing the possibility of three solutions (H2,, , PJ .

Consider now the dominant force balance in the equation of motion. The inducedcurrents in the core are of orderand the Lorentz force is of order j N CUB, ( 1 . 3 )

F - j B N aUB2 . (1 .4)If viscous forces are negligible, then a steady st at e is possible only if this force is inequilibrium with the applied pressure gradient, i.e.

vUB2N pG. (1.5)This simple relationship between U and B is represented (for different values of G )by the dashed curves of figure 2 ( a ) ,which may be described as dynamic equilibriumcurves.

Both the induction equation and the equation of motion can be satisfied only atpoints of intersection of a dashed curve with t he solid curve, e.g. a t th e points H and


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110 H . Kamkar and H . K . MoflattJ in figure 2 ( a ) . The point H represents a stable equilibrium, whereas the point Jrepresents an unstable equilibrium; this may be seen by considering the behaviour ofthe system a t points such as R or S. Suppose that G is given and tha t we st art a t thepoint R on the corresponding dynamic equilibrium curve. A t R , B is greater than isrequired by flux equilibrium; consequently flux is expelled and B decreases. TheLorentz force then decreases and so the flow accelerates, i.e. U increases. We thusfollow a pa th in the (B ,U)-plane from R towards H as indicated in the figure. Similarlyif we s tar t a t the point S , B increases and U decreases until again we arrive at H .Hence H is stable. Similarly J is unstable.

If G exceeds a critical value G, for which the dashed curve touches the flux equi-librium curve, then there is no intersection and therefore no possibility of an inviscidsteady-state solution. We have then a runaway situation: as the flow acceleratesunder the applied pressure gradient the flux-expulsion effect is so strong that there isa dramatic decrease in th e resisting Lorentz force; in this situation the velocity canonly be controlled by viscosity (no matter how weak this may be).

The effect of weak viscosity may be crudely represented by modifying the estimate( 1 . 5 ) to the formThe dynamic equilibrium curves are then shifted downwards slightly (the dot tedcurves of figure 2b ) , each curve intersecting the axis B = 0 a t the Poiseuille valueU, N Gb2 / v .Three possible curves are shown in figure 2 ( b ) abelled (i) , (ii),(iii).Curve(i) s a runaway curve which now intersects the flux equilibrium curve a t th e pointPl,ery near to the Poiseuille value (represented by A l) . Curve (ii)shows three inter-sections H,, J, and P,, f which H , and Pz re stable and J, is unstable. As v increasesfur ther, we lose the intersections J and P, nd (curve (iii)) we have again a singleintersection at H,; this flow is characterized by strong field and weak velocity, and ismore akin to Hartmann flow th an to Poiseuille flow.

The above description is of course purely qualitative, but the more detailed analysisof the subsequent sections corroborates this description, and provides a quanti tat ivedetermination of the critical pressure gradient a t which runaway occurs.

aUB2+ p vU / b 2 N pG. (1.6)

2. Mean-flow equationsI n order to provide this quanti tat ive analysis, we need to replace ( 1 . 2 )and ( 1 . 6 )by

the appropriate governing differential equations for mean fields (averaged over x),and solve these either analytically or numerically. We shall, for th e moment, retaintime-dependence in these equations, which can then also des ibe the approach to

Let A = A(x , , t ) , be the vector potential of the field B (satisfying V . A = 0), i.e.equilibrium (as represented for example by the trajectory R-t2n figure 2a).B, = aA/ay, B, = -aA/ax. ( 2 . 1 )

The current distribution in th e fluid is thenj = p u , l V ~ B = - p ; l V 2 A i , ,F = j A B = -po1(V2A) VA.and the Lorentz force is

The electric field E is given byE = - , aA/at -Vg5, (2.4)


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Dyna mic runaw ay effect in M H D channel f low 1 1 1and is related to j by Ohms law,

j = V ( E + U A ), (2.5)where U ( X , y , ) is the velocity field, satisfying V . U = 0. Since V . = 0 and the flow istwo-dimensional, the divergence of (2.5) ives

v2q4 = 0 , (2.6)so that the electrostatic contribution -Vq4 to E is determined by the remote boundaryconditions. In conventional Hartmann flow, with a uniform applied field, this con-tribution provides the means of closing the current circuit, either within the fluid orwithin the duct boundary (see e.g. Hunt & Shercliff 1971). n the present context,however, the periodicity in the x-direction implies that the net flux of j in the z-directionis zero, and the problem of closing the current circuit does not arise. We shall supposethat there is no externally applied electric field, and in particular that there is noexternally maintained potential difference between the remote boundaries z = f o(where zo $ b ) . There will of course be boundary layers on z = f o which will dependon their electrical properties, but outside these layers the assumption of a two-dimensional flow and field is justified, and the relevant solution of (2.6) s $ = cst;(2.5)hen gives, in the usual way,

aA/at +U . V A = h V 2 A ,where h = (p0g)-1.The Navier-Stokes equation is

& / a t + U , V U = -p-VP - pop)- (VA) V A + vVU,where ( a p i a x ) = -PG (2.9)

(2.8)

is the applied pressure gradient, which we suppose t o be constant. Here, and sub-sequently, the angular brackets { ) denote averaging over a wavelength 2nlk n thex-direction.

Now let U = U ( y , t ) i , + u @ , y , t ) , (2.10)where U= (U ,v, ), (U) = 0. (2.11)It is obvious from the periodicity of the applied field that

( A ) = 0.The average of (2.7)hen gives

(2.12)

a(vA) /ay = 0, i.e. {vA)= l ( t ) ,say, (2.13)and the average of the two components of (2.8) ives

the latter integrates, in conjunction with (2.9),o give(U)+ ( 2 p O p ) - l ( ( V A ) )= C x + f 2 ( t ) . (2.16)

The equations for the fluctuating fields A , U and P = P - P ) are now simplyobtained; the fluctuating part of (2.7)s

8A aA-t + U ax+ V . (uA- u A ) ) = hV2A, (2.17)


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112and the fluctuating part of ( 2 . 8 ) is

H . Kamkar and H . K . Moffatt

(V2AVA-(V2AVA))+~V2~.2 . 1 8 )ur au I 1-+U-+V.(uu-(uu)) = - - V P - -at 8X P POPThe underlined terms involve interaction of fluctuating quantities, and the treatmentof such terms is in general exceedingly awkward. If, however, conditions are such that

IuI < U , (2 .19)then these underlined terms will be negligible compared with U8AIax (in ( 2 . 1 7 ) )andUaulax (in (2.18)).Let us suppose th at this is the case; then, provided we are not toofar from equilibrium, the dominant balance in ( 2 . 1 7 ) is between mean convection anddiffusion, i.e.in order of magnitude (cf. ( 1 . 3 ) ) ,and, assuming viscous effects are weak, the dominantbalance in ( 2 . 1 4 ) is(cf. ( 1 . 4 ) ) .Likewise, the dominant balance of the x-component of ( 2 . 1 8 ) is

UaAIax N hV2A (2 .20)

G N ( , L L ~ ~ ) - ~ ( ~ ~ A / ~ z V ~ A )(flOph)-l (Bt) ( 2 . 2 1 )

uauyax - poph)-1U ( B ;- B;)) , (2 .22)or, again in order of magnitude,

U kl u ( N (poph)- U(Bt)N G. (2 .23)Hence ( 2 . 2 4 )Now (B;) Q &Bi with equality only at the boundaries y = _+ b ) ,and so ( 2 . 2 4 )becomes

-


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D y n a m i c ru n a w a y ef fect in M N D channel Jlow 113Let us now introduce dimensionless variables

and let7A

Then ( 2 . 2 8 ) and (2 .29 ) become

= y / b , 7 - b2kGh-l t ,= B o k - 1 9 [ i f ( 7 ,7) e i k 2 ] ,

U = ( A / b 2 k )8.

where-8_ -a7

( 2 . 3 1 )( 2 . 3 2a )( 2 . 3 2b )

( 2 . 3 3 a )(2 .33b )

( 2 . 3 4 )the four independent dimensionless parameters that characterize the problem. Interms of these parameters, the condition ( 2 . 2 6 )becomes simply/3 < Q 2 . ( 2 . 3 5 )Note that the Hartmann number M = B,b/( ,uophv): and the magnetic Prandtlnumber P, = v/h are related to these parameters by

M = (&E ) - * , P, = B E . ( 2 . 3 6 )We shall be particularly concerned with the situation when K = O ( l ) ,Q = O ( 1 )and

( 2 . 3 7 )< 1 , so thatThe la tter condition is realistic in the context of liquid metals, for whichP, is generallyin th e range 10-5-10-7.

419 1 , P, < 1 .The boundary conditions associated with ( 2 . 3 3 )are (from ( 2 . 3 0 ) )

f (& 1 , 7 ) = 1 , o( 1 , ~ ) 0. ( 2 . 3 8 )However, it is clear that both f and U are even functions of 7 ; we may thereforerestrict attent ion to the interval 0 < 7 < 1 , and adopt boundary conditions

hf 7 = 0 , % = o on 7 = 0 ,f = 1 , U = O on ~ = 1 .

h

( 2 . 3 9 a )(2 .39b )If /3 < 1 , ( 2 . 3 3 a , b ) may be fur ther simplified. I n this situation, the adjustment t o

flux equilibrium is instantaneous, and the term paf1a.r in ( 2 . 3 3 a ) can be dropped.( 2 . 4 0 a )e then have

aO/& = l - ( Z Q ) - O / f ( 2 - + ~ 8 2 8 / 8 7 2 , ( 2 . 4 0b )equations which, in effect, describe evolution along the flux equilibrium curve offigure 2 ( a )or 2 ( b ) .

I n the computational work described in the following sections, the parameter /3 in( 2 . 3 3 a , b ) was set equal to unity; this must merely be regarded as a computationallyconvenient choice: any conclusions concerning steady-state solutions of ( 2 . 3 3 a , b ) areindependent of the value of p chosen, and would be the same if, instead, a small valuesatisfying ( 2 . 3 5 )were chosen.

i8f= a 2 / a y - K 2 ) f ,


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114 H . Ka m k a r a nd H . K . Moffatt3. Conditions for the existence of a steady inviscid solution

We may expect, from the introductory discussion, that in the inviscid limit ( e = 0)a steady solution of ( 2 . 3 3 )and ( 2 . 3 8 )will exist? only if Q is not too large, i.e. only forQ < Q c ( ~ ) , here Q c ( ~ )s a function to be determined. The critical 'runaway' pressuregradient (from ( 2 . 3 4 b ) )will then be given by

Suppose first th at a steady solution {f(r),o(7))oes exist. Letf(7) P ( 7 )exp [%(7)1,p ' ( 0 )= q ' (0 ) = 0,P ( 1 )= 1,

p " - p q ' 2 - K P -- 0,

1q ( 1 ) = 0.J

where, from ( 2 . 3 8 ) ,

Then with (e = 0), the steady form of (2 .33)becomes

2p'q'+pq" = o p ,Opz = 2Q.

( 3 . 4 a )( 3 . 4 b )( 3 . 4 c )

Note that , sincep ( ) = 1 , and 0must be finite, we must havep ( 7 )> 0 (0 < 7 < 1 ) . ( 3 . 5 )

We may now eliminate l and q from (3 .4) to obtain a single equation for p ; for2Q = O pz = p(2p 'q ' +par ' )= ( p 2 q ' ) ' ,

p2q' = 2Q7.nd so, since q ' ( 0 ) = 0,p " = K21) -Q 2V

P 3 'Hence ( 3 . 4 ~ )ecomesand we have the boundary conditionsp ' ( 0 ) = 0 , p ( 1 )= 1 . ( 3 . 9 )

This defines an intriguing nonlinear two-point boundary-value problem. The solution,if it exists, must evidently satisfy0 < p ( r )< 1 , p " ( 7 )> 0 (0 < 7 < 1 ) . ( 3 . 1 0 )

An alternative statement of the problem involves the Green function G(6 ,q )defined( 3 . 1 1 )

(andG(7 , ) = G(6,7 ) ) .It may easily be shown thatp(7)satisfies the nonlinear integralequation (3 .12)c-t The no-slip condition U (1) = 0 is of course dropped in the inviscid limit.


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Dy na mi c runaway effect in M H D channel flow 115We may deduce a necessary condition for the existence of a solution; for the condition

1cosh K '

p ( 0 )> 0 implies thatHence, sincep([) < 1 ,

t 2G ( t , )

H. Kamkar and H.K. Moffatt- A dynamic runaway effect associated with flux expulsion in...

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