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FLOW LINES IN AN INCOMPRESSIBLE FLUID
L. de Seze, Y. Pomeau
To cite this version:
JOURNAL DE PHYSIQUE Colloque C5, supplément au n° 8, tome 39, août 1978, page C5-95
FLOW LINES IN AN INCOMPRESSIBLE FLUID
L. DE SEZE and Y. POMEAU CEA/DPh., BP n° 2, 91190 Gif sur Yvette, France
Résumé. — On étudie la configuration des lignes de forces d'un champ de vitesse tridimensionnel périodique stationnaire. Les lignes de forces sont presque toutes situées sur une famille de tores qui entourent les lignes de courant fermées. Cette configuration représente expérimentalement la situation physique d'un écoulement convectif stationnaire entre plaques parallèles lorsque le nombre de Rayleigh dépasse le seuil de tridimensionnalité. On peut se représenter le champ de vitesses également comme un réseau plan de texture de Hopf séparées par les variétés tangentes bidimensionnelles pour une double famille de points fixes.
Abstract. — The flow line topology of a tridimensional, periodic and stationary velocity field is studied. Almost all flow lines lie on a family of nested tori which surround the closed flow lines. This configuration represents the experimental situation of a stationary convective flow between horizontal plates for a Rayleigh number just above the tridimensionality threshold. The velocity field can also be viewed as a plane lattice of Hopf defects separated by bidimensional tangent manifolds associated to a double family of fixed points.
In slightly supercritical conditions, the Benard Rayleigh instability yields, in a large horizontal fluid layer a stationary flow made of parallel rolls [1]. This flow is periodic in space : let Oz be the vertical direction, Ox and Oy be the horizontal directions respectively parallel and perpendicular to the axis of the rolls. The fluid velocity in the bulk fluid (that is not too close to the vertical boundaries) depends on space like
dq>
<p being the stream function which is periodic with respect to the variable JC :
00
<P(X, 2) = X S i n ("«*) >/'n(Z)
where ii„(z) must satisfy the boundary conditions :
^
=-dT
= 0 f o r z = ±2 '
z = 0 being the mid horizontal plane of the layer and the unit length being the width of the layer. Further-more ip„(z) has the same parity than zn+1. The precise
form of ij/i and \j/2 may be found, for instance in [1].
As eqs. (1) describe a 2d (two dimensional)
incompres-fdvx dv\
sible flow 1-^—I- -x-J= 0 , the corresponding flow lines are closed : in each vertical plane y = C, they are the level lines of the function <P(x, z) and draw a periodic system of closed curves surrounding the extrema of <P.
For high Prandtl number fluids, when the tempe-rature difference accross the layer increases beyond the onset of convection, a supercritical transition occurs and the flow becomes tridimensional, although it remains stationnary. Its steadiness makes it possible to draw experimentally the flow lines by following the trajectory of small adverted particles. It is thus an interesting question to know the phase portrait (= the topology of this system of flow lines) of this flow. It is known [2] that, in an inviscid steady flow, the flow lines are located on the so called Bernouilli surfaces, which are a continuum of nested tori in the generic case. In our case, the flow is far from being inviscid (typically the Reynolds number is less or of order 1). Thus the general theorem given by Arnol'd does not apply. We have investigated on a desk computer this 3d system of flow lines.
Experimentally the flow is double-periodic in space [1] so that the fluid velocity is such that
( 2n In \
v \x + — , y + -j-, z) = \(x,y,z) a and b being the wavenumbers in the x and y directions respectively. Near the onset of three dimensionality
C5-96 L. DE SEZE A N D Y. POMEAU this flow may be described by adding a small pertur-
bation to the 2d flow defined by the roll system :
a@
v, = q(1
+
E cos 2 ax)-
(y, z)az
0 = -
a ~ ,
-
(x, z) - ~ ( 1+
E COS 2 ax)a@
-
(z, y)ax
aywhere @ takes the same form as in equation (I), E is a
number experimentally of order 0.3, and where
m
@ '(y, z) =
C
sin nby~,(z)n = 1
the function 1, satisfy the same b.c. and have the same symmetry than the functions ll/,(z). In equation (2) we have introduced a parameter q in order to make obvious the fact that the 2d part of the flow (i.e. the amplitude and shape of @) depends smoothly on the temperature difference accross the layer, although its 3d part, that is proportional to q strongly depends on this temperature difference. In particular it vanishes at the onset 3 dimensionality.
Let us consider first the most obvious points about the phase portrait of this flow. The fixed points of the flow (that is the points outside the horizontal boun- daries where the fluid velocity vanishes) are the inter- sections of three surfaces :
and thus form of periodic horizontal lattice of isolated points. Among those fixed points are those with the coordinates
k, k' being any integer. Around each of these points, the flow takes place in a finite parallelepipedic cell. One of these cells is defined as
1
- -
1 7L 7C2 < z < - O d x d - and O < y d b .
2 ' a
As the fluid velocity is tangent to the boundaries of this cell, no flow line leaves it if it starts from inside and no flow line comes into the cell from the outside. Let us consider first the flow on the vertical boun- daries of the cell. For the clarity of the discussion, it is convenient to use explicit formula for the function $,(z) and ~,(z). We have simplified the harmonic structure of the flow by choosing X, = 0 for n 2 2 and (I/, = 0 for n 2 3. This does not simplify the reality too badly, although
+,
is far from being negli- gible. However this simplification has no drastic effect upon the qualitative picture of the flow lines.Furthermore, we have chosen for $,,
$,
and X, very simple polynomial forms :p being a numerical parameter describing the relative amplitude of the second harmonics in x i p 2 . 5 x 10- fits rather well the experimental data. In the vertical planes bounding the elementary cell along they direc- tion, namely the planes x = 0 and x = nla, the flow is bidimensional with the equations
where 6 = 1 at x = 0, and 6 =
-
1 a t x = n/a. Of course the interesting part of this flow is in the sheet -$
d z<
3
and is periodic, of period 2 n / b along the y-direction. In the rectanglethe flow lines join two fixed points, the stable one lying at the corner of the rectangle, the unstable one being on the same vertical edge as the stable point.
In the plane y = 0 and y = n/b the flow is given by
1
V , = ( ~ - ~ z ~ ) - u c o s U X + ~ ~ ( ~ + E C O S ~ u x ) + 16
+
2 apz cos 2 a x.
Again this 2d flow is limited to rectangular cellsbut, contrary to the previous case there is one fixed point inside this rectangle (at least if p is small anough which is the case) and the flow lines spirals away from this points, going toward a stable fixed point just at one of the corner of the rectangle limiting the flow. Let Q and Q' be these two fixed points on the faces of the parallelepipedic cell. This allows one to recon- struct the flow on the vertical boundaries of the cell.
FIAOW LINES IN A N 1NCOMPRESSlBLE FLUID C5-97
difficult to prove this point, although it shows up very clearly in the computations.
The sheet of flow converging toward the fixed point P splits the cell in two parts that are symmetric with respect to P, so that it is just necessary to know the phase portrait of this flow in one of these parts.
In order to make clear this phase portrait, it is convenient to mention first that a simple closed current line makes one turn around the flow line A going through the fixed points P, Q and Q'. Thus, if one looks at the Poincare map in any plane inter- secting this closed line, its intersection with the plane is a fixed point.
Owing to the incompressibility of the flow, the Poincark map close to this line defines an area pre- serving mapping and it happens that the fixed point defined by the intercept of the closed flow line is elliptically stable. Accordingly the flow lines starting in the vicinity of this closed lines fills up a continuum of nested tori expanding, in some sense from this closed line. This picture is, of course, very reminescent of the one given by the famous K.A.M. theorem in Hamiltonian dynamics ([2], p. 405 et sq.). It is also well known in Hamiltonian dynamics that a very complicated fine structure appears around the tori, arising from the possibility of a resonance among the two frequencies of a quasiperiodic motion on a torus. We have not been able to detect neither the existence of these islands, that manifests the reso- nances, for instance in the Henon-Heiles system [3] nor the existence of regions of stochasticity (they should correspond to trajectories filling a region of space with a more complicated structure than a
[I] NORMAND, C., POMEAU, Y., VELARDE, M. G., Rev. Mod. Phys.
49 (1977) 581.
(21 M4thodes Mafh4mofiques de la Mecanique Classique (V. Arnold,
ed. Moscou) (1976).
surface, its Hausdorff dimension could be larger than 2). If this fine structure exists, it seems likely that it will be difficult to put it in evidence by following the trajectory of advected particles.
Let us end with a few remarks made in slightly different points of view. It is difficult, if not impossible, to have any precise idea about the generic phase portrait of 3d divergenceless flow. Despite its peculiar character our study could give some idea in this direction. For that purpose, it is convenient to recall what is a Hopf topological texture (or defect). All the continuous non singular vector fields in R3 (practi- cally an element of S2, or a unit vector, is defined at each point), that take a constant value at
I
r1
+ a, [4] may be classified via the homotopy theory [4], by a discrete index which is an element of the additive groupe Z (= group of positive and negative integers). If this vector field does not correspond to the unit element of the group, one says that is describes a Hopf defect. In order to know the element of Z cor- responding to a given vector field, it is sufficient to look at the way in which three counter-images in R3of three points of S2 are interlaced. [A counter image is a curve in R3 that is the set of points r where n(r) takes a constant value.] In our case, it is easy t o see that in each half of the parallelepipedic cell, the topological index measured in this way is either
+
1 or-
1. Our definition of the index in these cells is not rigorous, since one must avoid to take the counter images of directions defined by the tangent flow near one of the critical points. In this case two counter images could intersect, forbidding to define the topological index.One may thus conjecture that a bounded diver- genceless flow is made of set of fixed points, the invariant 2d manifolds starting from some of these fixed points, define the faces of a polyhedral tiling of space, the edges of the polyhedral being the attracting and repelling Id manifolds of another class of fixed points, which are the vertices of the polyhedral net, 6 edges intersecting accordingly at each vertex. Each polyhedra is thus filled by a Hopf defect. One might imagine that these Hopf defects are generated by instabilities of the boundary layer, then move on the fluid and disappear either by recombining o r are unknotted when two counter images intersect, cor- responding to vectors of the tangent manifold of fixed points. It should be interesting also to know whether these Hopf defects have something to do with the well known thermals or plumes.
[3] H ~ N O N , M., HEILES, C., Astron. J . 69 (1964) 73.
14) GODBILLON, C., Elemmfs de Topologir Algibriqur er sq. (Her-
