234 NUMERICAL SOLUTION OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATION
fact that von Neumann’s stability analysis is most commonly used, each method
has its own advantages as well as limitations. A comparison and evaluation of
these three methods can be found in Chapter 3, Section A-5, of Roache (1972).
By using a model equation (4.2.1), we have shown that the substantial deriva-
tive may be approximated by the upwind differencing scheme, which is compu-
tationally stable if the ratio τ/h is chosen appropriate for the local fluid velocity.
Furthermore, the upwind differencing scheme introduces, in addition to the trun-
cation errors, an artificial diffusion whose effect is to smear out perturbations in
the numerical solution. For thorough referencing and a more detailed discussion
on the subject and its related topics, Sections A-8 to A-11 in Chapter 3 of Roache
(1972) are recommended.
4.3 B
´
ENARD AND TAYLOR INSTABILITIES
When a horizontal layer of fluid is heated from below in a gravitational field, the
density of the fluid at any location becomes smaller than the fluid just above it.
If a fluid parcel is displaced slightly upward into the region of higher density, a
buoyant force will assist it to move further upward. Similarly, if the fluid parcel
is displaced downward into a region of smaller density, it will keep moving
in the same direction. Without a sufficiently large, viscous, retarding force, this
situation is said to be unstable, and the instability appears in the form of a net
of hexagonal convection cells.
Lord Rayleigh (1916) made the first theoretical analysis of the so-called
B´enard problem concerning the stability of a fluid layer in the presence of a tem-
perature gradient parallel to the gravitational force. An extensive treatment on this
subject based on linearized theories can be found in the book by Chandrasekhar
(1961). An example of a solution procedure for the linearized equations for his
problem was presented in Section 3.8 of this book. As we have observed there, the
linearized theory predicts only the onset of instabilities. Once the flow becomes
unstable, the initially small disturbances will grow with time, and the subsequent
fluid motion will be governed by the nonlinear Navier-Stokes equation. Now that
we already have a successful upwind-difference numerical scheme for approxi-
mating the nonlinear terms in that equation, the method used in Section 4.2 will
be adopted here to study the B
´
enard problem numerically on the computer.
The governing equations for the fluid motion are (3.1.1) to (3.1.3), the conti-
nuity, the Navier-Stokes (adding the buoyant force), the energy equations, and,
also, the equation of state. For the last equation we use, instead of (3.1.5), the
following form:
ρ = ρ
0
1 − α(T −T
0
)
(4.3.1)
where α is the coefficient of volume expansion and T
0
is the temperature at which
the fluid density is ρ
0
. For ordinary gases or liquids, α is of the order of 10
−3
or 10
−4
. Based on this fact, considerable simplifications can be made by using
the Boussinesq approximation that, if temperature variations are not too large, ρ