Competition between two directions of convective rolls in a horizontal porous layer, non-uniformly heated

MECHANICS RESEARCH COMMUNICATIONS Vol.16(l), 19-24, 1989. Printed in the USA. 0093-6413/89 $3.00 + .00 Copyright (c) 1989 Pergamon Press pie

COMPETITION BETWEENTWO DIRECTIONS OF CONVECTIVE ROLLS IN A HORIZONTAL POROUS LAYER, NON-~INIFORMLY HEATED

M.C. Vincourt LACME UFR riath~matiques, Universit( Paul Sabatier, 31062 Toulouse C~dex, France (Received 8 March 1988; accepted for print 5 December 1988)

Introduction The appearance of convection is studied in a saturated porous layer heated from below. When the thermal boundary conditions and the physical properties of the solid matrix are uniform, this phenomenon is described by an evolution equation which for every value of the f i l t r a t i o n Rayleigh number Ra possesses a solution with an identically null f i l t r a t i o n speed. For a c r i t i c a l value Ra of Ra, this configuration loses its s t a b i l i t y , and new asymptotically stable e~uilibria appear. The one towards which the system evolves is determined only by i n i t i a l conditions. Experimenters generally cannot control the homogeneity of data like thermal boundary conditions or the porosity of the solid matrix. These d i f f i c u l t i e s can explain why repeated experiments [!J produce one type of convective rolls , in a box whose aspect ratios should in the purely homogeneous case lead indifferently to two types of rolls. We study what happens when the data is not exactly uniform. This kind of situation generally favors [~-3J one branch of equilibrium points, in a sense which we will specify. Physical problem We consider an homogeneous saturated porous layer in an horizontal parallelepipedic box with height h, with adiabatic walls. The temperature of the bottom is non exactly uniform, but near of TI. The upper boundary is at temperature TO < T 1 •

After non dimensionalization the space variable (x,y,z) belongs to = [O,a] x [O,b] x [ 0 , I ] , whose boundary is @;l. The evolution of the f i l t r a tion speed _V =

C)

and of temperature T are ruled by [4] -t9-

:

20

M.C.

VINCOURT

-V + Ra k T =_Vp

(1.a)

@T

(l.b)

AT- V-VT

v.V = 0 V.n:O on @f~, ~@:T

(l.c)

in C~

(1.d)

0 on @gl,T = 0 on @g2,T= l+ef(x,y) on @g3

where @gl = { ( x ' y ' z ) e g / x = O or a, or y=O or b}, @~2= { ( x ' y ' z ) e g / z = 1 } ' @f @g3 = {(x,y,z) e ~/z=O}. f is a function of x and y, with ~ = 0 on @gl n @R3" ef represents the heterogeneity on the thermic boundary conditions, p is a scalar function of (x,y,z)j k the unit vertical ascendent vector. The f i l t r a t i o n Rayleigh number is [41 Ra : g~(~)f hK(TI-To) with ~ equivalent thermic conductivity of the porous medium, K its permeability, ~, v, (pc)f being the thermic expansion coefficient of the saturating fluid, its kinematic viscosity, and its volumic heat capacity.

Mathematical formulati on System (1) is equivalent to :

I!) Ivl

(!I =

(2 .a)

- ._re

28

~T

DO O = 0 on @g2 u @~3' ~-~=0 on @f~l' V.n=O on @~, T = (1-z)(l+ef) + ~ ,

'

c

_V._V=Oin 2

(2 .b)

c"

Operators L~,and~i are defined in H = {(o)/V_~(HI(~)~/_ v.v=O in ~, V.n=O on ~ , 0 ~.H~(~)/~-~ = 0 on ~21,0 = 0 on ~22 u ~23} by :

.od

l

=

z

a^

~ v] +

(3)

o o~ J ~ ~/ (4)

CONVECTIVE

ROLLS

IN P O R O U S

LAYER

21

is the orthogonal projection on the solenoidal vectors in HI for the scalar product (L2(fl)) 3, where HI : {V ~ (HI(~))3/_V.n -- 0 on ~ } . Contrary to the case cf = 0, this problem does not admit the trivial solutionO . We use an asymptotic constructive method 12-31 to obtain solutions of (2) in a vicinity of (

=O

and uniformly bounded in H for (~,c) in a neighbourhood

of ( ~ c , 0 ) . In such a box, the i d e n t i t y

cf --- 0 would imply t h a t O loses i t s s t a b i l i t y

"~= "~c = 27 , and t h a t a s y m p t o t i c a l l y s t a b l e r o l l s parallel

for

appear [ 5 - 6 ] , with axes

to the s i d e w a l l s . The heterogeneity described by Ef ~ 0 g e n e r a l l y breaks

t h i s symmetry.

Asymptotic method We seek paths (~(u), c(~)) in the plane of parameters (~,c), passing through (~c,0) and on which (2) admits solutions in the form : -

0

= Z i=I

Ui(x,y,Z,T) + 0(N+I)

(5)

With T = ~at and ~ > 0, (2) admits such solutions i f the Ui verify systems Si : Ui = Fi, i=I, .... N with FI = ~ o and where Fi is a function of UI...Ui_ I for i ~ 2. In H, where is the scalar product (L2(~))4 , L~c is selfadjoint, and Si has solutions i f Fi is orthogonal to Ker L~

c

, whose a basis is {XI,X2}, with :

C ~ sin ~0x cOs ~ z ~ XI =

- ~ COST x s i n ~ z / -

cos ~ x s i n ~ z /

When 1611 + 1621 # O, where ~I =

~ and X2 = ~ -

0 z) sin ~ y cos ~ z ~ cos ~ y s i n

~ - cos ~ y s i n ~ z

< G]XI> = - 2 ~ 2

b/o a f(x,y)cos~xdx dy

22

M.C.

and 62 = < GIX 2 >, t h i s c o n d i t i o n

VINCOURT

f o r S I i f ~--Id~ : O. Then ~u UI = AI(~)X I + BI(T)X 2 . S2 admits bounded s o l u t i o n s f o r (J~,E) in a neighbourhood of ( % , 0 ) and e : ± 3 ,

is r e a l i z e d

i f ~ ~ 2 and

Io =

system (2) admits s o l u t i o n s

dA1 = 2 ~AI~+ 6I -

d2

Io : 0. For ~ = 2 , ~ = ~

in the form (5) f o r N = 2 with

~2A1 (A~ + ~ T

+ ~2 :

B~)

(6) E

= 2~BI~+

62 - T

System (6) describes the evolution for great t of formal approximations up to order ~ of solutions of (2) belonging to a neighbourhood ofC ). The ~-independent solutions of (6) furnish approximations of these equilibrium positions of (2). For a and b fixed in ~, (6) depends only on 61 and 62, functions of the heterogeneity in the data, and on ~, r e l a t i n g ~ t o E. Discussion of system (6) For every (61,62) system (6) admits at least one, and at most 9 equilibrium points. Their s t a b i l i t y domains are related to the signs of the second members of (6). When 61 = 62 = 0 [5-6], as C becomes positive, four asymptotically stable equilibria appear, approximating rolls whose axes are parallel to the sidewalls of the box. When 6+ # 0, one can findl a positive real Co such that for ~ < ~o (6) has exactly \ one equilibrium p o i n t | A*I ( ~ ) | , which is stable and depends continuously in C. I t describes a longitudinal roll i f BI BI

~ 0, a slightly modulated one i f

~ AI/I0 for instance. And i t remains stable when ~ grows.

Figures (I) and (2) describe the set of equilibrium points of (6) for every ~ in R, for values of (61,62 ) such that 62 = 0, respectively ~2 = 61/10" Note that 62= 0 occurs as soon as f is y-independent or, more generally, when f(x,y).cos~y is an even function of y - ~ . Such a situation can be produced by the device imposing I

the temperature on the bottom. At least two, at most three bifurcations can occur for ~ > ~o" They produce pairs of unstable equilibria i f they are associated with an unstable branch of solutions recovering s t a b i l i t y . They otherwise produce equilibria whose s t a b i l i t i e s are opposite by,Pairs. But the most important feature is the insulated character of the branch t A*I~ , due to the existence of A such that every i n i t i a l condition with

+

A~ ~ ~ B 1 ] l i e s in the s t a b i l i t y domain of

B C+)

CONVECTIVE ROLLS IN POROUS LAYER

23

A1

~ L1

7

25 0.3 t

1._: .

0.4

± 0.4_

B112 LL . . . . . . .__I~-=.. . ..

1.9 '

r

"

03

04

B_t_=± 0.9 B1

Ftg. 1 T-independent s o l u t i o n s o f (6) f o r 61 = 1, 62 = 0 .....

stable

solutions.

unstable solutions.

B1 = 0.03

AI~ Z B1 = 1.7

B,

11

= 2.3

-' A11

15

BI =-0.02 -0.53

,=I

BI = 0.22 B1 = 0.07

Fig. 2 T-independent solutions of (6) for 61 = 1, ~2 = 0.1.

stable . . . .

unstable

Selection of equilibrium solutions of system (2). When ~I f O, the conduction regime of the t r i v i a l solution of (2) is

V* displaced towards (~,)(~,¢)), whose an approximation up to order IvJ = IE[ I/3 is V* ~[A~(~)X1 + B( ({)X2]. (~.) descrlbesfi]tration speed and temperature distributions which are forced by the non uniformity of the data. Whencrossing over

24

M.C.

VINCOURT

2 the curve Co , of equation C~= ~ c + ~o c~ in the plane (CO~,,~), this solution keeps its stability, and the equilibrium solutions which appear are unstable. Other stable t-independent solutions appear when ~). grows, ~ being fixed. But the bifurcation of 61 = 62 = 0 is replaced by a smooth transition [2] which V* lets the amplitude of (~,) grow, those of the other equilibrium solutions being at most equal. Let us consider an experiment, where successive equilibria are obtained from previous stationnary states, Ra being increased. One can follow the solution (~*) even when other stable independent of t solutions exist, so long as the continuity of (~,) in Ra allows this procedure to furnish i n i t i a l conditions V* belonging to the stability domain of the actual (~,) ( ~ , ¢). Conclusion The introduction in the boundary conditions of a small heterogeneity breaking the symmetries associated to the data of a Bdnard problem destroys the bifurcation describing the appearance of convection. I t displaces the conduction solution towards an asymptotically stable equilibrium solution, which depends continuously on the f i l t r a t i o n Rayleigh number and on the importance of the inhomogenity. The asymptotic method [2] leads to effective calculation of an approximation of this solution, and allows the description of its asymptotic stability and other equilibrium solutionsin a neighbourhood of tbe trivial so lution References I. M. LAYADI. Ecoulements stationnaires et instationnaires de convection naturelle en milieu poreux. Approche exp~rimentale. Th~se I.N.P. Toulouse (1987). 2. J. TAVANTZIS, E.L. REISS, B.J. MATKOWSKY. SIAM, J. Appli. Math. 3_44, p. 322-337 (1978). 3. G. lOSS, D.D. JOSEPH. Elementary s t a b i l i t y and b i f u r c a t i o n theory. Springer Ed. (1980), chap. 111.3, p. 32-44. 4. M.C. COMBARNOUS,S. BORIES. Advances in Hydrosciences, 10, p. 231-308 (1975). 5. P.H. STEEN. J. Fluid Mech. 136, p. 219-241 (1983). 6. H. POLITANO. J. M~canique th~orique et appliqu~e. 5, I , p. 39-54 (1986).