Page 203 - Process Modelling and Simulation With Finite Element Methods

190 Process Modelling and Simulation with Finife Element Methods
'phase', 0,. . .
'geomnum',l, ...
'dl', 1, ...
'intorder',4, ...
' context , ' local ) ;
I
output (j ,1) =Rayleigh (1 ;
)
output (j ,2) =I1;
output (],3) =IZ;
solZ=femeig(fem, 'U', fem.sol.u, 'Eigpar', 10) ;
output (j ,4) =so12 .lambda (1) ;
output (j ,5) =so12. lambda (3) ;
output (j ,6) =solZ. lambda (5) ;
output (j ,7) =so12. lambda (7) ;
output(j,8)=solZ.lambda(9);
end
save bifurc3.mat fern so12;
dlmwrite('bifurc3.dat1,0utput,',');
quit
The m-file script computes fluxes and the first ten eigenvalues for Ra in
[100:100:4000], which shows a crossover between Ra values 1900 and 2000.
Table 5.1 shows the eigenvalues homing in on the critical value of Ra=1956 for
aspect ratio 3:l. Saving the solution and fern structure, as well as the
eigenvalues in a mat-file permits the re-loading of the final solution in the
FEMLAB GUI by importing from MATLAB into FEMLAB. bifurc3.m was
computed as a UNIX background job from the command line:
matlab -nojvm <bifurc3.m >err 2>err &
since it takes a few hours to execute. The save command permitted subsequent
perusal of the solution. The m-file script computes the total convective and
conductive fluxes for each Rayleigh number solution. The critical Rayleigh
number (circa 1956) corresponds to both the zero eigenvalue, but also an abrupt
increase in convective heat transfer.

5.2.3 Agreement with thin layer theory
Recall, the theory of Reid and Harris [S] describes the critical Rayleigh number
for cells with upper and lower rigid boundaries occurs at Rac=1708 with a
wavenumber of 3.1 17 for an infinite layer. Since our layer has aspect ratio 1 :3,
we would not particularly expect agreement. Davis [9] computes the 3-D
solution for finite aspect ratio boxes, and finds substantially higher critical
Rayleigh numbers, approaching the theoretical predictions only at high aspect
ratio. For this reason, we have reproduced our simulations here for aspect ratio
1:lO. Figure 5.5 shows a periodicity of ten in a ten unit periodic layer for the
critical mode, which is in agreement for the theoretical estimate of the
wavenumber. Table 5.2 for the eigenvalues and Figure 5.6 for the Nusselt