Page 192 - Process Modelling and Simulation With Finite Element Methods

Simulation and Nonlinear Dynamics 179

the problem can be found in Drazin and Reid [5]. The gist of the problem is that
a vertically decreasing temperature profile and no flow is identically a solution to
the boundary value problem stated as
au
-+u.VU = ---Vpi-vV2~+-T
1
ag
at P P
v.u = 0

dt
(c.f. equations 3.1) with boundary conditions of

(5.7)

where the bottom temperature is usually greater than the top. The dimensionless
groups that matter are still the Prandtl number and the Rayleigh number:
V
Pr = -
K
(5.8)
ag (ST) h3
Ra =
PVK
where h is the depth of the fluid, 6T is the applied temperature difference, a is
the coefficient of thermal expansion, g is the gravitational acceleration vector, p
is the density, v the kinematic viscosity, and K is the thermal diffusivity.
You can be forgiven for thinking that we have just turned the hot wall-cold
wall problem on its side. In the case of vertical heated walls, motion is
automatically induced even with an infinitesimal temperature difference. Fluid
along a hot vertical wall must rise. For horizontal heated walls, however, a
steady, linear temperature profile with no motion is an exact solution to the
system. If the heating is from above, that makes perfect sense as hot light fluid
will lie over colder dense fluid - gravity supplies a buoyant restoring force to
any fluid element that might be displaced vertically. For heating from below,
however, the stratification has dense fluid over light. Buoyant forces should
overturn this top heavy profile. Yet, if viscosity and thermal conductivity are
strong enough, they resist the motion. Stability theory identifies a critical
Rayleigh number above which convection cells form. Below that Ra,,
dissipation still damps out motion.
So let’s explore these two situations by finite element analysis.