SYMBOLIC COMPUTATION AND EQUATION ON ... - Science Direct

Computers & FluidsVol. 16, No. 3, pp. 229-238, 1988 Printed in Great Britain. All rights reserved

0045-7930/88 $3.00+ 0.00 Copyright © 1988 Pergamon Press pie

SYMBOLIC COMPUTATION AND EQUATION ON THE CENTER MANIFOLD: APPLICATION TO THE COUETTE-TAYLOR PROBLEM P. LAURE and Y. DEMAY Laboratoire de Mathematiques, Parc Valrose, 06034 Nice Cedex, France

(Received 11 December 1986; in revised form 7 October 1987) Abstraet--A particular case of the transition from the Couette flow to the Taylor cells is mathematically studied using the Center Manifold theorem. In this degenerate case, the stability of the Taylor vortex flow is calculated using fifth order terms in amplitude. We describe a method for computing these high order terms. Moreover, the numerical values are obtained with a Fortran program generated by a Macsyma program. 1. I N T R O D U C T I O N

In this paper, a particular case of the transition from the Couette flow to the Taylor vortices flow in the Couette-Taylor problem is studied. The amplitude of the bifurcated solution is classically obtained as an expansion in a power series of the parameter of bifurcation. The first coefficient of this expansion has been computed numerically in Refs [1] or [2] and can vanish for some values of the parameters. In this degenerate case, we compute the following term. Due to the next term in the expansion, the saturation is reached and it is proved that the Taylor vortices are still stable in a neighborhood of this degenerate case. This can be done by solving a system of recurrent equations. These computations are performed using both numerical analysis and the symbolic computation. Usually, the computer is used after analytical or algebraic manual studies and the program realizes numerical calculations or logical tests. Systems such as Macsyma or Reduce are able to manipulate Taylor series or polynomials (i.e. factorization, expansion, simplification of expressions without any arithmetic values of the variables occurring in these computations). This method has been previously used in various fields of Mathematics and Mechanics (see for example Ref. [3]). The mathematical framework used is the theory of bifurcation, which predicts the dynamical behavior when the basic flow becomes unstable. The process is to reduce a system of partial differential equations to its form on the center manifold (of finite dimension). These equations, predicting all the dynamic of the solutions, are obtained by Taylor expansion in the neighborhood of the bifurcation point. As a consequence, the amplitude of the bifurcated solution is obtained as an expansion in bifurcation parameter. The transition is then studied by computing the first coefficients of this expansion. This method has already been used to study the Couette-Taylor Problem (Refs [1] and [4] in the general case and Ref. [2] in the small gap approximation) and the convection in a differentially heated box [5]. By using recent developments of the bifurcation theory in the presence of symmetry and by writing the equations on a normal form [6], it is possible to minimize the number of coefficients to compute. Demay and Iooss [1] have analytically obtained the intermediate equations for the first coefficient. Macsyma and the recurrent structure of these intermediate equations are used to get to a higher order. Moreover, a Fortran program has been automatically generated using the existing and checked basic routines. 2. T H E C O U E T T E - T A Y L O R

PROBLEM

2. I. The apparatus and the basic flow Let us consider the motion of a viscous incompressible fluid between two infinite coaxial cylinders of radii R~ and R 2 (R2 > R~), rotating about their axis with angular velocities f~ and f~2 (Fig. 1). CAE

~6/~B

229

230

P. LAUREand Y. DEMA¥ A~

|

• •

. . . ' ,

•

..

•

. . . . , |

..:.

.. -

• .

. ".

e

Fig. 1. Geometry of the problem• The dimensionless velocity field V and pressure p are obtained by using scales RI, f~- ', RI f~ for length, time and velocity respectively. They satisfy the Navier-Stokes equations:

+(vvv) t

= -Vp+

'av

(1)

V.V=O

where ~ = ( ~ R~)/v is the Reynolds number and v the kinematic viscosity. The Couette flow, observed for small Reynolds number is azimuthal. It can be written in cylindrical coordinates (r, 0, z): p°=C+A2

+ 2B log(r) - ~rir2

(2)

where we note R2

f~2

a2o9 -- 1

A-a2_l

and

B-

a2(1 - 0 9 ) a2~o _ 1 ,

and

1 ~"X ulrDu2o + alo~-U2o + ~- ) + iotulzu2o im 2

UlrhU2z + - - UloU2z r

(34)

+ i~UlzU2z

APPENDIX

II

We describe in this part how we get the Fortran program using a Macsyma program.

(1) Algebraic Equations We describe how to obtain the recurrent algebraic equations with Macsyma. The following algorithm allows to compute all the coet~cients %, such as p + r = s. It can be easily translated in Macsyma which has an Algol programming language. (*CONDITION (26)*) f o r p ' = 0 thru s do for s ' = 1 thru 2*(s - p ' ) for r ' = 0 thru s' do

do

q' = s' -- r ' if q' > / r ' then write the equation (19) for 3-uple (p', q', r') else write that ~p,q,,, is conjugate to ~p'r.q, end ( * C O M P U T A T I O N OF C O E F F I C I E N T S 0t~,*) for p' = 0 thru s do write the equation (19) for 3-uple (p', s - p ' + I, s - p ' ) The statement "write the equation (19) for the 3-uple (p, q, r)" is performed by the function " w r i t e eq" which mainly use the Macsyma function sum: write e q ( p , q, r): = block([dte,gche,pl,ql,rl,L0,M,L1],

238

P. LAURE and Y. DEMAY dte:sum(sum(sum(funbili(M,phi[p 1,q 1,r 1],phi[p -- p l,q - q l,r - r 1]),r 1,0,r),q 1,0,q),p 1,0,p), dte:sum(sum(phi[p - p l , q - q l , r - ql]*((q - q l ) * a l p h a [ p l , q l ] + (r - q l ) * c a l p h a [ p l , q l ] ) , q l , ~c0,r),p 1,0,p) - dte, if p > = 1 then dte:dte-funmake(Ll,[phi[p - 1,q,r,]]), gche:funmake(L0,[phi[p,q,r]])~ print(" "),print(" "), print("- . . . . . . . . . . . ordre ",mu Ap*z0^q*cz0^r), print(" "),print(" "),print(gche = dte),print(" "));

(2) The Fortran Program It is obtained by c h a n g i n g the call of M a c s y m a function sum by loops of the following form: "for i : - - t h r u ~ o

call(Name of F o r t r a n subroutine, list of p a r a m e t e r s ) "

For example the F o r t r a n s t a t e m e n t s allowing to c o m p u t e quadratic terms of Fpqr are obtained by M a c s y m a p r o g r a m " t e r m e s _ bili(p,q,r)': termes bili(p,q,r): = block([som,va,pl ,q l,r l,p2,q2,r2], for p2:0 t h r u p do for q2:0 thru q do for r2:0 thru r do ifq2+r2>0then if testbili([p - p2,q - q2,r - r2],[p2,q2,r2] then if [p - p2,q - q2,r - r2] = [p2, q2, r2] then c a l l ( B I L I l , [ q 2 - r2,0,co(U,[p2,q2,r2]),F,N]) else call(BILl,[q - q2 - r + r2,0,co(U,[p - p2,q - q2,r - r2]),q2 - r2,0,co(U,[p2,q2,r2]),F,N]));

--*

testbili(I 1,12): = if I1 = [ ] then true else if first(12) > lirst(ll) then false else if first(12) < first(ll) then true else testbili(rest(l l),rest(12)); call(fun,lis): = block([va], lis:concatl(lis), va:string(apply(fun,lis)), print(concat(" call",va))); concatl(list): = apply(append,maplist(lambda([x],if atom(x) then [x] else concatl(x)),list)); co(U,list): = if list = [ ] then U else co(concat (U,first(list)),rest(list)); Finally, the part o f F o r t r a n p r o g r a m resolving eqn (21) for 3-uple(1,2, 1) is the form C C * R E S O L U T I O N O R D R E M U * * I Z0**2 C Z 0 * * I C

.................................

C CALL ZERO(N,F) C C C ---+

SECOND CALL CALL CALL CALL

MEMBRE

BILINEAIRE

B I L I ( 2 , 0 , U I 2 0 , - 1,0,U001,F,N) BILI(0,0,UI 11,1,0,U010,F,N) BILI(1,0,U110,0,0,U011,F,N) B I L I ( - 1,0,UI01,2,0,U020,F,N)

SECOND

MEMBRE

AVEC COEFFICIENT

C O E F = I* C01 + 0 * C O N J G ( C 0 1 ) CALL MULT(COEF,U110,X,N) CALL SOM(X,F,N) C O E F = 2* C10 + I * C O N J G ( C I 0 ) CALL MULT(COEF,U021,X,N) CALL SOM(X,F,N) CALL LI(U021,0,F021,X,N) CALL SOM(X,F,N) P R I N T 1121 1121 F O R M A T ( 5 X , / , 6 X , ' * U121 *') C C A L L L 0 ( I , 0 , 0 , F , U I 2 1 , C I I,N) PRINT III,Cll 111 F O R M A T ( 5 X , / , / , ' * CI 1 = (',E14.7,',',E14.7,')') where the F o r t r a n s u b r o u t i n e s L0, L1 a n d BILl c o r r e s p o n d to operators

0

.2a,t ,, and M~lkt~Z~2 respectively.