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Wavelength selection in axisymmetric cellular structures
Y. Pomeau, P. Manneville
To cite this version:
Y. Pomeau, P. Manneville. Wavelength selection in axisymmetric cellular structures. Journal de Physique, 1981, 42 (8), pp.1067-1074. �10.1051/jphys:019810042080106700�. �jpa-00209092�
1067
Wavelength selection in axisymmetric cellular structures
Y. Pomeau and P. Manneville
Service de Physique Théorique, CEN Saclay, B.P. N° 2, 91190 Gif sur Yvette, France (Reçu le 16 février 1981, accepté le 22 avril 1981 )
Résumé. 2014 Nous considérons des structures cellulaires axisymétriques telles que celles qui se développent en
convection de Rayleigh-Bénard lorsque l’on force un rouleau à la paroi circulaire extérieure. Nous montrons que dans ces circonstances une longueur d’onde unique est sélectionnée. Elle correspond à la condition d’annulation du coefficient de diffusion perpendiculaire (D~ = 0). Cette condition exprime simplement le fait que les rouleaux peuvent être à la fois courbés et stationnaires et ne tendent ni à se courber davantage (D~ 0) ni à se redresser D~ > 0. Nous formulons quelques spéculations sur la sorte de bruit qui apparait lorsque la longueur d’onde
choisie par la structure axisymétrique tombe en dehors de la bande sélectionnée par les parois latérales.
Abstract. 2014 We consider axisymmetric cellular structures, as concentric rolls occurring in Rayleigh-Bénard thermo-
convection when forcing a roll at an outer circular boundary. We show that under these circumstances a unique
wavenumber is selected. It corresponds to the vanishing of the coefficient of perpendicular diffusion (D~ = 0).
This condition expresses simply the fact that the rolls can be both bended and steady and do not tend to become
more curved (D~ 0) or straight (D~ > 0). We make some speculations about the kind of noise occurring when the
wavenumber selected by the axisymmetric structure is outside of the band selected by the lateral boundaries.
LE JOURNAL DE
PHYSIQUE
.-
J. Physique 42 (1981) 1067-1074 AOÛT 1981, 1
Classification
Physics Abstracts
47.25 .
As shown recently, the lateral boundary conditions
are of crucial importance for the determination of the
wavelength of cellular structures, as produced by ther-
moconvective flows [la, b], bluckling phenomena [ 1 c]
or model equations [1d]. However experiments [2]
made in axisymmetric Rayleigh-Bénard thermocon-
vection indicate that a single wavenumber is selected in supercritical conditions, although the lateral boun-
dary conditions restrict [1] this wavenumber to a narrow - but finite - band. Here we show that in
axisymmetric pattern (or concentric rolls) a unique
wavenumber can exist in steady conditions. The wavenumber selected by the axisymmetric pattern
may be outside the band selected by the iateral boun- dary Below this is shown to occur for thermoconvec- tion in porous flows. In this case, no steady axisym-
metric solution exists, evgn near the convection thre- shold. In section 2 below we speculate about a possible
connection (already made in reference [9]) between
this sort of frustration and the occurrence of low
frequency noise near convection threshold.
In what follows we develop first general considera-
tions aiming at proving that a single wavenumber
exists for a steady axisymmetric pattern. It is of interest to note that this proof does not involve any
perturbative calculation with respect to the amplitude
of the fluctuation. It is valid even at a finite distance
from the instability threshold. Then we carry out the
,explicit calculation of the selected wavenumber for three particular problems.
To make a long story short, let us consider the general problem of finding a steady infinite axisymme-
tric pattern at large distance from the centre. In general,
the corresponding equations involve explicitly the
distance to the centre, say r. For instance the radial
Laplacian in 2 dimensions is , Accordingly,
at large distances from the centre one may expand the
non-linear equations for the steady state as
where A is some unknown function of r (A may be
actually a set of functions of r, as in problems to be
considered below). This function may depend eventually on other variables, as the vertical coordi-
nate in the case of thermoconvection in horizontal
layers. Furthermore Lo, LI,,,, are operators with linear and non-linear parts in general.
The interest of the decomposition (1) is the fact
that Lo, L1, ... do not depend explicitly on r, so that
at large distances from the centre (r ~ oo), A can
be taken close to solution of
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019810042080106700
1068
Thus, to find the solution of (1) at large (but not infinite) r, we seek an expansion of A in the form
where Ao, A1, ... are periodic functions of r, the period being the wavelength of the structure for parallel
rolls. At first order in 1/r, one has
where is the operator on the left hand side of equation (2), when linearized around Ao.
In equation (4), the unknown function is A1; but,
in general this equation has no solution, because A
has a non trivial kernel : since Lo is r-independent,
one has
Thus, to make possible the expansion (3), one must verify that the right hand side of (4) is orthogonal to
the kernel of A. This is obviously given by the condi-,
tion
where A + is the kernel of the adjoint operator A’.
The adjointness is of course defined by the scalar
product chosen in equation (5), so that
for any function B(r) in some suitable functional space.
For cellular structures Ao, A,, ... are periodic func-
tions of r, say of period a,, so that the o natural » scalar product is
However, one may adopt, at least in principle,
another definition for this scalar product, introducing
some 03BB-periodic smooth weight 1À(r) in the integrand
on the right hand side of (6). Furthermore, more complicated definitions of this scalar product are
to be found when the functions depend on another
variable (as variable z in cases to be considered below) and/or when A represents actually a finite set of
functions. For the present discussion the correspond- ing generalizations of this scalar product are purely
formal.
It is important to notice that condition (5) is drastic,
because it defines in general (and except for possible
non generic degeneracies) in a unique way the wave-
number at large distances from the centre of the structure : once Ao is fixed, A + is fixed too (up to
trivial multiplicative factor) and, as Li is well
defined, the left hand side of equation (5) is a function
of the wavenumber of the structure only, that is the
single adjustable non trivial parameter for periodic
solutions in supercritical conditions. Accordingly, equation (5) is a non trivial equation to be satisfied
by a single real parameter. In cases to be considered
below, this defines in a unique way the wavenumber in slightly supercritical conditions (this is the domain that is accessible to analytic computations).
We shall present now some remarks about other features of the wavelength selection in axisymmetric
structures.
Remark 1. - The above reasoning could give the impression that an infinite number of solvability
conditions are generated when one follows the expan- sion in r-1. Actually it can be shown that no solva-
bility condition is generated for getting terms even
in r-1 in the expansion (3). But the odd order terms
(r-3, r - 5, .,..) generate formally non trivial solvability
conditions. To satisfy ’them, one uses the freedom implied in the solution of equation (4) : as ~ has a
non trivial kernel (i.e. ôAo/ôr), one may add to any solution of (4) a quantity arbitrary. The
free choice of al (and of similar free parameters at
higher order) is used to satisfy the solvability condi-
tions at order 1/r3. This remark has possible experi-
mental implications. Actually, the large distance expansion of the solution of (Eq. (1)) is now of the form
, and this is the begin- ning of the large r-expansion of
This shows that the phase of the roll structure reaches its asymptotic behaviour as a constant, plus some quantity proportional to 1 /r.
. Remark 2. - It may happen that the non-linear
equations (1) are the Euler-Lagrange equations for
a functional V[A] (as in the model of thermocon- vection between poorly conducting plates to be con-
sidered below). In this case the wavenumber defined
by the solvability condition (5) must give the mini-
mum of V for periodic one-dimensional solutions [3].
As noticed in reference [3], this optimal wavenumber
is at marginal stability for the perpendicular phase
diffusion. In notations of reference [3], this is the wavenumber for which
Thus one may wonder about the general connec-
tion between this condition of marginal stability
and the selection criterium given by equation (5).
It turns out that these two conditions define the same wavenumber in all cases, particularly when equa- tion (1) is not derived from an Euler-Lagrange func-
tional, as it is the case for problems of thermoconvec- tion to be considered below.
The scheme of proof of this result is the following
one : consider a periodic function Ao(x) such that Lo[Ao(x)] = 0 (we are allowed to replace the radius r by a Cartesian coordinate x in Lo, since Lo is r-inde- pendent and does not distinguish between parallel
and concentric rolls). Let qJ(Y) be a slowly varying
function of y, y being the coordinate perpendicular
to x, the problem under consideration being inva- riant, by assumption, under rotation in the (x, y) plane (x, y are the « horizontal » coordinates). Following
the method exposed in reference [3], one gets a diffu- sion equation for ç by expanding the solution of the
equations of motion with respect to the derivatives of ç (that are small by assumption). Let L[A] ] = 0
be the equation for the steady state (this equation is
more general than Lo[A ] = 0, as the function may
depend on more than one « horizontal » variable).
Up to second order in the space derivative of ç,
one has [4] :
where Lo and L, are the same functional as the ones occurring in equations (2) and (4). Furthermore follow-
ing the method exposed in reference [3], one readily
shows that the diffusion coefficient D.1 (in notations
of Ref. [3]) is proportional to (A +, L1[Ao]), so that
the condition D1 = 0 is exactly the first solvability
condition for the 1 /r - expansion, as given in equa- tion (5). All this discussion is valid of course under the condition of strict isotropy in the (x, y) plane.
This mechanism of wavelength selection can be explained as follows : the perpendicular diffusion
process govems the slow motion of smoothly bended
rolls. When the corresponding diffusion coefficient,
i.e. D.1, is positive the roll system relaxes toward
a stable state of straight unbended rolls ; when D1
is negative the rolls tend to become more curved.
This is only in the marginal situation (D.1 = 0) that slightly bended rolls can remain steady.
Remark 3. - The 1/r expansion breaks down in the
vicinity of r = 0. The matching of the outer (r - oo)
and inner solution (r finite, eventually zero) has been
considered by Brown and Stewardson [7]. The final
result is a solution regular and bounded everywhere,
at r = 0 in particular.
In what follows, we have chosen to apply the pre- vious considerations to three problems in the analytic theory of cellular structures : (1) thermoconvection in a fluid between free boundaries ; (2) thermoconvec- tion in a porous médium ; (3) thermoconvection in a
fluid between poorly conducting boundaries. All three cases are considered between parallel hori-
zontal plates.
We develop in some details the calculations in
case 1 ; in case 2, we use some computational tricks
making the computations a little easier. In casé 3
we only give the result, without following the explicit
derivation.
1. Thermoconvection between free boundaries. - This is the classical problem considered in its linear version by Lord Rayleigh. The equations for the axisymmetric steady state are in dimensionless form
where a, R, u, w, 0 and p are the Prandtl number, the Rayleigh number, the radial and vertical velocity, the
temperature and the pressure respectively. Further-
more
Ul UG 1
z being the vertical dimension, so that the boundary
conditions on the lower and upper plates are 0 = u., = w = 0 at z = 0, 1.
In the r -> oo limit, this system of equations reduces
to the equations for parallel rolls. The linear and
weakly non-linear solutions for this case are well known [5] and read
where the small parameter q measures the distance to the curve of marginal linear stability in the (R, k) plane and depends on R and k as
’
with
In equations (8a-c) the subscript 0 in wo, uo and 80
refers to the order zero in the 1/r - expansion.
Let now wl, ul and 01 be the perturbations at first
order in this expansion : that is in the large r-limit
1070
and similar formula for u and 0 ; wo, wl, uo, ul... are
periodic functions of period 2 03C0/k in the variable r.
The first order perturbation in 1/r is given by the
solution of the following linear inhomogeneous sys- tem :
In order to write (9a-d ) in a more compact form, we
have used the notation A r2 + 8;2. Furthermore the boundary conditions at the lower and upper plates
are again w, = Ul,z = 0 = 0 at z = 0,1.
The system of equations (9a-d) is linear with respect
to (wl, ul, 01), it is the explicit form taken in the pre- sent problem by our previous equation (4).
As the coefficients of this linear system depend in a
non trivial manner on r and z, this system cannot be solved in a compact form. However one can obtain its solution by an expansion with respect to the small parameter fi, that is basically the small amplitude of
the zeroth order solution (i.e. wo, uo, 00). The selection
criterion for the wavenumber of this zeroth order solution will appear as a solvability condition for this
expansion. It is the explicit translation of the solvability
condition given in equation (5) in a general form. In
the weakly non-linear case considered here, we deal with finite dimensional function space, so that equa- tion (5) becomes a compatibility condition for a
finite linear algebraic system.
Let us expand the solution of (9a-d ) as
Similar expansions hold for ul and 91.
The lowest order contribution is obtained by neglect- ing in equations (9b-d) terms on the right hand side,
as they are of order 112 at least. Furthermore we keep
on the left hand side the dominant contributions to R,
uo, wo and 0.. The corresponding linear system has a solution if k = kc == n/J2 only : this arises from the fact that, if R = Rc(k), the homogeneous part of
equation (9) has a non trivial solution. Of course, one may add to any solution of this system (if it exists)
an arbitrary solution of the homogeneous equation.
This is equivalent to a global phase change of the solution, which is irrelevant at this order. Thus a
solution of equations (9a-d ) is, at the lowest order
in q :
At next order in fi, one finds again an inhomoge-
neous system for w}l), UB2) and (JB2). The inhomôge-
neous terms have various origins : (1) some of them
come from the right hand side of (9b-d) wherein one
inserts the dominant contribution (in il) to (wl, ul, 01)
and (wo, uo, 90) ; (2) the others arise from the 1-expan- sions of R and k around their zeroth order values :
The q-expansion of k generates terms of order ~2
when one computes for instance ÂUBI) :
Lengthy but straightforward computations show
and
’
The third order calculation with respect to 1 yields,
as it is classical in this field, the sought after solvability
condition. In the present case this solvability condi-
tion gives the value of k2. Again, at third order, one
has to solve inhomogeneous linear equations for (WB3), UB3), 03B81(3) and two sorts of inhomogeneous contri-
butions appear : the one arising from the expansions
of (k, R ) around (kc, Re) and those coming from the right hand sides of (9b-d) wherein one keeps terms as
UO UB2). Of course, one finds a solvability condition
when one considers quantities depending on r as sin kr
or cos kr. Again the calculation is quite heavy, and
rather peculiar algebraic compensations lead to the
. final result k2 = 0. This means that the curve of
selected wavenumber starts « vertically » from the point of coordinates (ke, Re) in the Cartesian (k, R) plane.
2. Thermoconvection in a porous layer. - In this section, we determine near the onset of convection the
wavelength for axisymmetric thermoconvection in a
porous layer [6]. The method differs from the one of
previous section. However, both methods give the
samé result. The present method is closer to the one
used in reference [ 1 b-d ] for studying the wavelength
selection by lateral boundaries. This section is intend-
ed (among other things) to make easier the comparison
between the two situations.
The non-linear equations for steady axis symmetric
convection in an infinite horizontal layer are, in a
dimensionless form [6] :
The boundary conditions on the lower and upper
(heat conducting) plates are w = 0 = 0 at z = 0, 1
and R is the Darcy-Rayleigh number [no confusion
must be made between R, as defined in (12) and in (7)].
All the other symbols have the same meaning as in equation (7). At large distances from the centre
(r - oo) on expands, as explained before, the equations
in powers of 1/r. At zeroth order
Near R = 4 03C02 + e (s 11 0), these equations have
« parallel roll solutions » with a small amplitude :
where X2 ~ e/7r’. Actually a continuum of solutions exists with any horizontal wavenumber in the band
]q-, q+[, q2 being the two roots of Rq2 = (n 2 + q 2)2,
but we limit ourselves to the solution with the wave-
length q = 03C0 that appears at R = 4 03C02.
Consider now the equation ( 12a) in the axisymmetric
case. It reads formally as
where
and
Following the idea of the method explained in
reference [1d], let us multiply equation (15a) by 0,, integrate over z and try to put the result into the form of an exact derivative. This gives for the left
hand side of (15a) :
where
and
At large r, K(r) tends to a constant value, since 0
becomes close to a periodic function of r with a cons- tant amplitude, in the form of the expansion (14).
Near r = 0, 0 must be an even function of r to make finite all terms in L(0) at r = 0, and this makes finite too K(r) at r = 0. Accordingly, integrating dK/dr
from r = 0 to infinity, one gets the finite quantity K(oo) - K(O). The things are different when one
tries to integrate Ml over r from zero to infinity.
Assuming, as before, that 0 tends at r - oo to a
periodic solution in the form
the asymptotic behaviour of M1 at large r is (once the
fast variation over r is averaged) :
To derive this last expression, we have assumed ô - 8 ( = R - 4 03C02). Of course Ml is the dominant term on the right hand side of (16) at r ~ oo, since by integration over r it yields a logarithmic divergence
at r - oo.
It remains now to compute the leading order term in
where N is given in (15c).
For this, in a way very similar to the one followed in reference [1d], we expand the solution of (12) up to order r-1 included and to second order in Bl/2.
From now on (as in previous section), the superscript
i = 1,2 in 03B8(i), w(i) and u(i) refers to the order with
respect to E1/2 (or to X). The starting point of this expansion is
In what follows (i.e. up to order r-’ included), we
shall consider x as constant [7].
Furthermore we shall take as horizontal wave-
number n instead of je + b. As in reference [1d],
this approximation neglects subdominant terms if 03B4 ~ a
1072
From w(’) = AO(1) and from incompressibility :
and
The second order (with respect to Bl/2 or x) contri-
butions are computed from
The intermediate results are
and
This yields
Now U(2) and w (2) are computed from
and
This yields, up to order r-1 :
Let us split now into three parts the dominant contribution to M2, as defined by (18) and (15c) :
where
and where
In the above expressions, the bar stands for the
averaging over the fast variation with respect to r :
Up to order r-l, one has :
From (4a)
Selecting terms of order 1/r in this last equation,
as given in (17) and (21), one gets at the lowest order :
this imposes à c>5 0 at the order e.
If one compares this with the criterium of wave-
length selection by lateral heat-conducting boun-
daries [lb] :
one sees that no system of concentric rolls can be in
equilibrium both with the centre and the outer circular
boundary when the computations of reference [1b]
are valid, that is when ~p > 1, p being the radius of the circular box wherein convection takes place.
One may speculate that this absence of steady solu-
tion due to a « frustration » between the lateral
boundary and the centre will manifest itself by well
defined bursts separated by long time lags of a seeming- ly quiet behaviour. The lateral boundary and the
centre « communicate » by the process of longitudi-
nal phase diffusion, which introduces a long time scale, of order p2 / D Il (D]] Il is the « parallel » diffusion coefficient, as defined in reference [3]), although the
« short » time scale (essentially p-independent) could
be the decay or growth time of a roll near the centre
or the boundary, this decay or growth being triggered by a subcritical instability induced by some wave- length change at large distance from the centre (or boundary).
Furthermore, it is of interest to look at the frequency dependence of this sort of noise. Let us consider a quantity, say Q(t) changing by a finite amount at the generation (or decay) of a roll (Q could be seen, for instance, as the total heat flux through the cell). The
slow time evolution of this quantity should obey an
equation of the form ,
