Page 134 - Process Modelling and Simulation With Finite Element Methods

Multiphysics 121

This model m-file can now be read directly into the FEMLAB GUI from the file
menu, the Open model m-file option. waterdensity.m computes the Gr=O and
Gr=l solutions. You can now migrate to higher Gr by hand with parametric
contiunuation. I wrote a second variation of the model m-file highgrav.m which
migrates to higher Gr using automated parametric continuation and a doubly
refined mesh. It arrives to Gr=105 after about a day on my fastest linux PC
workstation as a background job. Expert criticism suggests that we should use a
pressure datum introduced in point mode to improve the convergence rate.
Chapter five discusses how to do this in detail, in the vicinity of equation (5.9).
The only new trick in this MATLAB program is saving the fem structure as a
mat-file for later interrogation in MATLAB. See the line just before the end:
save highgrav.mat fem
The temperature isotherms tell an interesting story in Figure 3.5 that the fluid is
starting to stratify, with cold fluid under hot:

isotherms (T)
1
368 2
0.9 363 5
358 7
0.8 354
349 2
07 344 4
339 7
0.6 334 9
330 1
325 4
0.5
320 6
315 9
0.4 311 1
306 3
0.3 301 6
296 8
02 292
287 3
0.1 282 5
277 a
0
0 02 0.4 0.6 0.8 1
Figure 3.5 Isotherms for water density model with Gr=100000.
Similarly, the core recirculation warps as the flow in the cold region slows.
Figure 3.6 gives the streamlines, which with the given scale, show a substantial
strengthening.