B
´
ENARD AND TAYLOR INSTABILITIES 243
After T = 1.0 the changes in stream pattern are not drastic. At T = 5.0the
left boundary of the middle cell has shifted to the right from its position shown
in the plot for T = 1.0. The displacement of the lower portion of that boundary
is a little longer than that of the upper portion, resulting in a curved interface.
Although a steady state has not been reached yet at T = 5.0, the flow does not
seem to have any further significant changes.
Temperature profiles are not plotted in the output because of their slow varia-
tion with respect to time. In general, the isothermal lines, except those for θ = 0
and 1, have the shape of the cutaway view of an opened umbrella.
Since the pictures shown in the output describe only the behavior of half of
the flow, the total number of convective cells to appear in the channel under the
present arrangement is two at the beginning, changes to four later, and finally
becomes six as the flow continuously evolves. Thus, we start from an unstable
situation of the fluid system having a density inversion; the numerical solution of
the governing equations leads us to a final state, at which the system may be said
to be most stable under the imposed temperature boundary conditions. Once the
computer program has been written, we can easily change the dimensions of the
channel, try different fluid media by varying the Prandtl number, or examine the
effect of temperature gradient by assigning various values to the Grashof number,
and we observe the results of our numerical experiments on the computer.
There are many advantages to using numerical methods to examine the stability
of a fluid flow. If a linearized analytic method, as in Section 3.8, were used to
study the problem considered in this section, a certain number of cells might be
predicted for the onset of instability. However, this prediction may not be valid
at later stages, when the ignored inertial force becomes important, as revealed
by our numerical result that the number of cells is changing with time. It seems
there is no guarantee that the most unstable condition predicted by a linearized
theory will show up in the final state. Furthermore, the inequality in cell size and
the curved cell boundaries are all nonlinear phenomena and cannot be predicted
using linearized theories. Higher-order analyses are generally tedious and are
usually formulated based on the linearized result. The validity of the result so
obtained is also uncertain.
The numerical solution in Program 4.2 is obtained under the assumption of
a two-dimensional flow. The result is unrealistic by virtue of the fact that the
observed convection cells are hexagonal when looking from the top, so that the
actual fluid motion is three dimensional.
Project for Further Study: Perform a numerical experiment on the fluid sys-
tem shown in Fig. 4.3.1, but with some of the boundary conditions changed. It
is now assumed that at the initial instant, θ = 0 everywhere, except θ = 1on
the upper surface. Thereafter, the temperatures at all bounding surfaces will be
kept at their initial values. The upper surface is considered to be a rigid free
surface where both vorticity and shear stress vanish. Assume that there is no ini-
tial motion in the water layer. The situation stated in this problem is somewhat