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Convective flow patterns in rectangular boxes of finite extent under an external magnetic field
P. Tabeling
To cite this version:
P. Tabeling. Convective flow patterns in rectangular boxes of finite extent under an external magnetic field. Journal de Physique, 1982, 43 (9), pp.1295-1303. �10.1051/jphys:019820043090129500�. �jpa- 00209508�
1295
LE JOURNAL DE
PHYSIQUE
Convective flow patterns in rectangular boxes
of finite extent under an external magnetic field
P. Tabeling
L.G.E.P.-E.S.E., Laboratoire associé au C.N.R.S., Paris VI, Paris XI et E.S.E., Plateau du Moulon, 91190 Gif sur Yvette, France
(Reçu le 16 décembre 1981, accepté le 26 mai 1982)
Résumé. 2014 L’instabilité convective stationnaire dans des boîtes rectangulaires de dimensions finies, en présence
de champ magnétique extérieur, est considérée dans le contexte de la théorie linéaire. Le cas où le champ magnétique
est appliqué verticalement peut se réduire, par une transformation d’échelle, à la configuration hydrodynamique (champ magnétique extérieur nul). Lorsque le champ magnétique est appliqué horizontalement la propriété d’isotropie de l’opérateur régissant l’instabilité est supprimée. Les calculs effectués dans cette configuration mettent
en évidence des transitions d’orientation des rouleaux, et la suppression des dislocations de la structure convective.
Abstract. 2014 Stationary convective instability in rectangular boxes of finite extent, under the action of an external
magnetic field is considered within the framework of linearized theory. In the case where the magnetic field is vertical, by changing the scale accordingly, the situation is shown to be similar to a complete lack of magnetic
field. When the magnetic field is horizontally applied, the isotropic property of the governing operator is broken.
Calculations performed for various situations show transitions from transversal to longitudinal rolls and suppres- sion of dislocations when the horizontal magnetic field is increased.
J. Physique 43 (1982) 1295-1303 SEPTEMBRE 1982,
Classification
Physics Abstracts
47.20
1. Introduction. - In a recent paper, C. Nor- mand [1] has considered the problem of linear sta- tionary convective instability in rectangular boxes
of finite extent. The linear analysis of C. Normand is indeed a preliminary step towards the much more
complicated non linear problem of convective insta-
bility [2], [3]. However the results found in [1] reveal
some very interesting features concerning convective
flows confined in finite boxes. In particular C. Nor-
mand has shown that dislocations of convective struc- tures in the form of parallel rolls can occur in boxes
of large size.
The present paper deals with the action of an exter- nal magnetic field on convective stationary patterns in rectangular boxes of finite extent. The analysis is
carried out within the framework of linear theory. We
shall restrict ourselves to the case of very low values of the magnetic Reynolds number Rm = U * La (where U *, L* are typical velocity and length and #
is the magnetic diffusity of the fluid). In this case the
external magnetic field is virtually unperturbed by the
flow. This assumption is relevant for laboratory expe- riments on convective instability in mercury, such
as those recently performed by Fauve and Libcha- ber [4]. In the case of large magnetic Reynolds num-
bers once convection starts the magnetic flux is expulsed by the convective motions. It follows that
a realistic approach to this problem requires finite amplitude theory. The reader is referred to Kno- bloch et al. [5], Proctor et al. [6] or others for the relevant studies.
A typical effect of an imposed magnetic field on the
convective instability in liquid metals is to inhibit the onset of the rolls whose axes are normal to the
magnetic field. Previous work on the subject has only considered infinite boxes so, naturally, it has
been found that a horizontally applied magnetic field, whatever its strength, lines up the roll axes in
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043090129500
1296
its direction. A more realistic approach must take
the effect of the lateral walls into account since these walls tend to determine the orientation of the convec- tive structure. Hence a balance between magnetic
and finite wall effects must be sought. In this paper,
we shall see that such a balance can be described by
a sort of Langevin parameter i.e., the ratio of some
activation energy over magnetic energy.
The dislocations in convective structures are also affected by the magnetic field. Indeed no conjecture
about the dynamics of the imperfections is given in
the present paper since we have restricted ourselves to linearized theory. However some features about the
influence of a magnetic field on the spatial structure
of the dislocations have been singled out. Such results may provide some guidance in experiments, as to
what sort of imperfections might develop at the early
stages of convection when an external magnetic field
is applied.
2. Equations of flow. - We consider stationary
convective flows of conducting fluids in a rectan-
gular box of finite extent, subjected to an external
constant uniform magnetic induction Bo(Bx, By, Bz)
of arbitrary orientation as shown in figure 1. The
Fig. 1. - The rectangular box, as considered in the paper, with the axes Ox, Oy, Oz.
governing equations for the dimensionless velocity u( Ux’ u, uz), pressure p, temperature 0, current density j
and induced magnetic field h have been previously
stated by many authors (see [7] for instance). The
latter read :
and
where Ra is the Rayleigh number, Q = JB) d2/r¡
is the Chandrasekhar number (in which Q and q
are the conductivity and the viscosity of the fluid), bo is the constant unit vector of components
and ez is a vector of unit length lying in the Oz direc-
tion. In equations (1)-(6),
is the scalar (resp. vector) Laplacian operator. In
contrast with pure hydrodynamic convective flows, equation (2) includes an additional term (j x bo)
which accounts for Lorentz magnetic forces. Once the field orientation is fixed, two control parameters
are sufficient to describe the flow : the Rayleigh
number Ra and the Chandrasekhar number Q.
Other parameters such as the magnetic Reynolds
number are irrelevant so that the above equations
are valid in principle for liquid metal flows (small magnetic Reynolds number) and large scale magne-
tohydrodynamic convection (large magnetic Rey-
nolds number). However, as explained in the intro-
duction, the present paper is relevant only for liquid
metal flows.
The equations (1)-(6) can be rewritten as a single equation of order 6 which governs the vertical velocity
u,. After some calculations we find :
in which A a2 a2. is the bidimensional I Lapla-
in which J =
20132013
+2013
is the bidimensiona! Lapla-In W IC Liz = ðxz + ay2 IS tel Imenslona ap a-
cian operator. We shall now examine the boundary
conditions of the flow. Along the side walls x = + L and y = + M. We have simply :
and 00 = 0 (perfectly insulated side walls) or 0 = 0 (thermally conducting side walls).
Since we are concerned with the horizontal modes of flow and in order to avoid lengthy calculations,
we assume free upper and lower boundary conditions.
Although such conditions are not realistic in view of laboratory experiments, one can estimate that the broader features found under the present assumption
will still remain valid in the rigid-rigid case. Note
that for certain particular orientations of the magnetic’
field, solutions found herein may asymptotically fit
the bulk convective flow of the rigid-rigid case :
this is the case where a strong vertical magnetic field
is applied. Thus we impose on z = 0 and z = 1,
In the general case when the orientation of the external magnetic field is arbitrary, the dependence
on z for the convective velocity has no simple form.
However, in the case when the magnetic field is purely
horizontal or purely vertical, the velocity u,, can be written in the form
in accordance with condition (9). We shall restrict ourselves to these two situations, and hence consider the following case :
(i) vertical magnetic field i.e., bxo = byo = 0 and bzo = 1,
(ii) horizontal magnetic field, parallel to the lateral
wall y = ± M i.e., bxo = 1, and byo = bzo = 0.
In addition, we shall examine only the first mode
n = 1 in (10).
Case (i) : Vertical magnetic field (bxo = byo = 0
and bzo = 1). As C. Normand does [1] we shall intro- duce the operator C defined by
where ao is the critical wave-number for the unbounded
layer. According to previous studies [7], ao is the positive root of the cubic equation
so ao increases with Q and follows the asymptotic law ao - 2-1/6 n2/3 Q 1/6 for large values of Q. Insert- ing (3) and (2) into (1) yields an equation of the form
where
in which Racoo is the critical Rayleigh number for the unbounded layer, given by
Racoo is an increasing function of Q which follows the
asymptotic law Racoo = n2 Q when Q tends to infinity.
Equation (5) is very similar with that obtained by
C. Normand [1] in the hydrodynamic case (zero magnetic field). Such a feature is essentially due to the
fact that a vertical magnetic field does not break
horizontal, rotational and translational invariance.
Since the wave-number ao changes with Q, one can expect some rescaling effects from the magnetic field.
This point will be examined further.
Case (ii) : Horizontal magnetic field (bxo = 1 and byo = bzo = 0). In this case, equation (7) becomes :
in which aõ = n2 /2 is the convective wave-number for the unbounded layer, and A = Ra - Racoo, with Racoo = 27 n4/4 (16). In contrast with case (i), both a 2
and Racoo are independent of the external magnetic
field However the governing equation (15) is comple- tely different from equation (13), since it contains an
additional term
Q 02 (C - a 2 _ n2) W,
OX2 2 ( 0 whichaccounts for the magnetic effects. This term is essen- tially anisotropic. Thus we can expect to find some preferred orientation for the system of convective rolls when Q is increased. Balancing terms in equation (15)
we can expect that, when Q is large, elongated rolls along the Ox direction correspond to the smallest proper values A. This point will be examined further.
3. The approximation l ao. - We shall consider the case when A is small compared with aõ and examine the asymptotic formulation of the problem in cases (i)
and (ii). As in hydrodynamics (without an external magnetic field), one can expect the lowest eigen-
values A to be small when the extent of the box is large.
This results from arguments performed on equa- tion (15) concerning orders of magnitude. Accordingly
Fig. 2. - Sketch of the various regions of flow when A is small compared with ao.
and by using equation (15), it is possible to single out
the following distinct regions of flow :
(I) The interior region, including the major part of the box for which E - À. 1/2 ;
(S) Secondary regions near the lateral walls of the box for which E = 0 (a 2);
one can then use a matching procedure to solve the complete problem. In the interior region, the gover-
ning equations for the vertical convective velocity w
are respectively (to first order) :
Case (i) :
Case (ii) :
1298
The boundary conditions associated to (17), (18) require that we examine the flow in the secondary regions (S). However it is possible to show that,
whatever the thermal boundary conditions on the lateral walls, we have (to first order) :
Condition (19) is justified in Appendix A.
Now we shall examine some features of the flow in
cases (i) and (ii).
Case (i) : Vertical magnetic field - Here we shall
consider the sysiems of rolls with a wave-vector
a(ax, ay) and examine the relation obtained from (17)
for fixed values of A. Using equation (17) yields :
which defines in the wave-number space (ax, ay) a
family of pairs of concentric rings, each one distant 0(À.1/2) from the other. In this case, the isotropy property is not broken by the magnetic field. When
Q is increased but the value of h remains fixed, the
radius ao of the circle increases while the distance 6 between the concentric rings decreases. For large
values of Q, 6 is O(Q - 116 A112). Since b -1 represents the largest characteristic scale for convective flow, it
should correspond to the size of the box and hence we
have the following (crude) estimate
where S is the typical size of the box. Thus, for a given
size of the box, A is an increasing function of Q.
The above features show simple rescaling effect due
to the external magnetic field. By applying trans-
formations x - xaoolao (where ago = n’/2), and
h - Aa 2013(a 2 + g2) to equation (17) we obtain an
identical equation to that given by C. Normand [1]
in the absence of an external magnetic field. When Q
is increased, the wavelength ao increases so that there
are more and more rolls in a given box of finite extent.
Thus we can expect successive transitions from N rolls to N + 1 rolls when Q is increased, while the
convective pattern continues to present the same features as in hydrodynamics.
Case (ii) : Horizontal tnagnetic field. - Equation (18) clearly exhibits anisotropy. The relation associated
to (19) reads :
The curves in figure 3, drawn at a given (small)
value of A, represent the various situations of interest.
When Q is small, in the sense that Q A13 ao,
curves (9) are similar to those found in absence of a
magnetic field; however, the relevant closed lines differ slightly from (perfect) circles since the separation
distance between them is not constant : it is
Fig. 3. - The accessible states of flow in the wave-number space when 2 is given. (a) Q A13 aõ, (b) Q = A13 aõ,
(c) Q > A13 a’, (d ) Q > A13 a’ with an arbitrary orientation of the magnetic field.
À. 1/2/3 - Qa 2/2 À.1/2 in the direction of the magnetic
field. Hence there is a slight pinch of the ring around the point (ao, 0), which should correspond to the brea- king of large scale isotropic invariance. As Q is raised
to the value A13 ao, the two curves coalesce (see Fig. 3b), and further give two symmetric closed loops for larger values of Q (Fig. 3c). Thus when Q
is larger than A13 aõ, the wave-vector a(a ay) is con-
fined into a sector of the plane with semi-angle n/2 - qJ, where :
As Q is further increased, only wave-vectors virtually
normal to the magnetic field satisfy (21). This situation corresponds to quasi longitudinal rolls i.e., rolls with
axes that are quasi parallel to the external magnetic
field. In such a case the flow is strongly anisotropic
since the wave-vector tends to point up normally to
the magnetic field, thus exhibiting a preferred orien-
tation. So when Q is increased from À./3 ao the small
scale isotropic invariance is in turn broken.
The above process can be generalized to any orientation of the horizontal magnetic field, as shown
in figure 3d. The angle cp defined by (22) still represents any slightest angular deviation from the magnetic
field lines for the wave-vector, at a given level of A.
It turns out then that a preferred orientation emerges when the magnetic field is present. This feature can also be singled out by using a variational
approach for problem (18). After integration by parts,
we find :
where the bracket means integration thoughout the
entire box. Note that A has a lower bound 3 Q/2 L2.
Considering orders of magnitude, one can show that elongated convective patterns in the direction of the external magnetic field generally require a small
amount of energy to be excited, which is in accor-
dance with the previous result.
A more quantitative approach to the problem can be attempted for small values of the Chandrasekhar number Q. Returning to equation (18) we consider
the magnetic
term - 2
3 a2w/ax2 as a perturbation sothat the « basic state » is that with no external magne- tic field (Q = 0). Expanding the velocity w and the eigenvalues A as
we find to first order
and
Since A, is positive (with a lower bound 3/2 L2), an increasing amount of energy is required to excite rolls under a weak external magnetic field. Considering
further the order of magnitude shows that the more the rolls are elongated along the magnetic field, the smaller the amount of energy they require.
As a conclusion, one can expect dramatic changes
in the convective structure as Q is gradually raised up from zero. Small scale modifications .outlined above
can be interpreted in some cases as transitions from transversal to longitudinal rolls, while larger scale changes may give rise to changes in the pattern dislo- cations such as those calculated by C. Normand [1]
in the absence of magnetic field. This will be analysed
in detail.
4. Some limiting cases. - Before entering into the
detail of the calculation, one can outline some inte- resting features about the convective pattern when the box is infinite in a given direction. For simplicity we
examine here only the even modes.
(i) Vertical magnetic field. - Let suppose that the box is infinite in the Oy direction so that equa-
tion (17) reduces to
together with
Restricting ourselves to even modes and small values of h, the eigenfunctions w(n) can be written in the form
(to first order) :
where A is a constant, and
When Q is increased, À (n) increases together with the discrepancy À(n+ 1) - À(n) between two successives values of the spectrum. On the other hand, the convec-
tive pattern exhibits successive transitions from N rolls to N + 1 rolls as Q is increased, while the form of their envelope remains unchanged. At large values
of Q, the increase AQ required to get an additional roll in the box can be estimated at Q 1/6 L -1 so that
many transitions occur when the extent of the box is large.
Hence the above results show that the external
magnetic field tends to affect the small scale structure of the convective pattern, leaving its large scale fea-
tures unchanged.
(ii) Horizontal magnetic field. - In this case,
four configurations can be studied, depending on the
orientation of the infinite size of the box with the external field. Figure 4 shows three cases of interest.
In the first case (Fig. 4a), we consider a system of
Fig. 4. - The various cases of interest when the box is
infinitely extended in a given direction.
transverse rolls with wavelength 2 7r/ao confined
between two vertical walls separated by M. Elemen-
tary calculations show that the eigenvalues À. (n) are given by
where 13n are the positive roots of the equation
The above result shows that the eigenvalues increase dramatically with Q, so that a large amount of energy
1300
is required to maintain transverse rolls. In contrast with this situation, longitudinal rolls are much less
sensitive to the magnetic field. Figure (4b) shows the
extreme case where the box is infinite in the field direction. In this case, the discrete eigenspectrum,
as well as the convective flow pattern, is unaffected
by the external magnetic field. In the last case (Fig. 4c),
we consider a system of longitudinal rolls confined between two vertical planes separated by L. Restricting
ourselves to large values of Q, we find the following
estimate for the eigenvalues À. (n) :
The above formula shows that the increasing rate
of À. (n) with Q is small if the box is sufficiently large,
so that the amount of energy required to maintain longitudinal rolls in the external field is small in boxes
elongated in the field direction.
5. Numerical method of solution. - We describe here the calculation method for the eigenvalues A(n)
and eigenfunctions w , restricting ourselves to even modes, i.e., those satisfying the condition
Although the present restriction does not allow for the calculations of all the accessible states of flow,
it should be representative of most of the situations of interest.
Let us consider now the equation (18) together with
the relevant boundary conditions (19). Functions w(x, y) satisfying (19) can be developed in the form
where Apq are constant coefficients,
and
in which PP are the positive roots of the equation
The function Cp(x) form a complete set of orthogonal
functions since we have
for any p and q.
Inserting further (27) and (28) into the governing equation (18) and integrating, leaves a relation of the form
where Fpqmn is defined by
where
and
The system (29) constitutes an eigenvalue problem
which can be treated by standard numerical techni- ques provided that a suitable truncation is performed.
The present Galerkin procedure is different from the numerical technique used by C. Normand in the absence of a magnetic field (the latter used a finite
difference method). The comparison between the two methods at Q = 0 will be described further.
We shall restrict ourselves to the equation (18) i.e.,
to the case when the external magnetic field is hori-
zontally applied along Ox. The other situation (verti-
cal magnetic field) can be reduced to the pure hydro- dynamic case (Q = 0) by using a suitable rescaling,
such as that mentioned in section 3.
6. Numerical results. - Calculations have been
performed in the two following cases :
a) box with its larger side parallel to the external
magnetic field :
b) box with its larger side normal to the external
magnetic field
Case a : box extended in the magnetic field direction.
- A good convergence of our results was obtained
once the extreme values of p and q are greater than 4 and 14 respectively. All calculations presented here
have been performed with 80 terms for series (29), i.e.
with p and q running in the range [1, 16] and [1, 5]
respectively.
