Finite Element Method - Application using Fastflo
Convective cooling of heated plates
Two plates of unit length are centrally placed in a square box. The
plates are maintained at temperature T = 1 and the enclosure is at T = 0.
Initially the air is stationary at T = 0.2. The key non-dimensional
parameter is the Rayleigh number Ra which is a measure of size and
temperature difference. The results shown here are for a Rayleigh number
of 4x106. The pictures show a time sequence of temperature
distributions. An initial plume is set up, but becomes unstable, and goes
into endless oscillations.
Download QuickTime animation
(30.8MB)
Computations
A segregated operator splitting method [2] was used for the
time-marching calculation of pressure and velocity. The temperature was
calculated to second order accuracy using a Crank-Nicolson method with
staggered time stepping. This algorithm is available in the standard
operator splitting module of Fastflo. The boundary fluxes were calculated
by an accurate domain integration method.
The mesh was unstructured triangular, with six-node elements. Quadratic
accuracy was used for velocity, with pressure being restricted to the
corner nodes. The mesh had a total of 11,561 elements with 23,562 nodes,
and was strongly concentrated in the critical thermal boundary layers.
The split operators consist of an advection step, which is unsymmetric,
sandwiched between two symmetric Stokes solutions. The unsymmetric stage
was solved by a BiCGSTAB algorithm, and the Stokes problems by a pressure
equation approach using preconditioned conjugate gradient methods.
The time taken depends on Ra. As Ra increased, the advective stage took
longer to converge, and the time step had to be shortened. At the highest
Ra of 4x106, the time step was 0.1, and the time taken on a 133
MHz DECstation was about 1 minute per step. 60 time units required about
10 hours cpu time
Results
The basic fluid flow is the same for all Ra. The air is
heated as it passes the plates and rises in a well defined plume until it
reaches the top of the enclosure and is forced to recirculate by the outer
walls. The fluid then meets the relatively cool air at the bottom, and so
again rises vertically, before cooling sufficiently and moving along a
meandering path back to the plates.
For Ra £ 105,
the flow remains symmetric and quickly reaches steady state. The
temperature gradients around the plate are steep, and there are two stable
eddies.
At slightly higher Ra, the flow remains symmetric for a
while, but then the plume becomes unsteady, shedding alternating eddies.
As Ra increases further, the plume motion becomes increasingly strong and
larger in amplitude. This generates fine secondary structure on smaller
and smaller scales and with increasing intensity.
The results agree reasonably with experiments (see plot above). The
divergence is most significant in the unsteady regime and is probably due
to three-dimensional effects. It is interesting that the onset of unstable
flow actually diminishes heat transfer. More details of computations and
results can be found in [3].
References
[1] J.G. Symons, K.J. Mahoney and T.C. Bostock, Convective heat
transfer from heated plates in a sealed enclosure: The application to
printed circuit boards, Proc. MECH 88, Brisbane, 8-13 May, (The
Institute of Engineers, Australia, 1988), 84-88.
[2] R. Glowinski, Finite element methods for the numerical
simulation of incompressible viscous flow in Lectures in Applied
Mathematics, 28 (1991).
[3] P.W. Cleary, Transient natural convection at very high Rayleigh
number in CTAC95 (World Scientific, Singapore, 1995).
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