Biomedical Engineering Reference
In-Depth Information
14.2.3
Numerical Results
14.2.3.1
Linear Stability Analysis
It can be shown that the principle of exchange of stability is satisfied in the
present problem as well as in the case of Benard convection. We then look for
solutions of the form
ψ
1
(
x, y, t
)=
φ
(
y
)exp
γt
exp
ikx
where
γ
and
k
are real numbers, representing the growth (or damping) rate
and the wavenumber of the perturbations. The onset of convection corre-
sponds to
γ
= 0. The stability boundary,
Ra(k,Pe),
is therefore determined
by solving the eigenvalue problem of the system (14.21-14.24) for marginally
stable perturbations with
∂N
1
∂t
=
γ
= 0. It should be noted that the results
presented here apply to a rectangular cavity of aspect ratio
F
as well as to
an infinite horizontal porous layer for a wavenumber
k
=
nπ/F
with
n
being
the number of the convection cells in the considered domain. As the problem
is governed by two parameters,
Ra
and
Pe
, the Rayleigh number at the onset
of convection is a function of
k
and
Pe
.
Figure 14.3(a) shows a family of stability curves
Ra
versus
k
for values of
Pe
varying from 0.1 to 20. It divides the parameter space
(Ra, k)
into two
regions: the region above the stability curve is unstable and the region below
the curve is stable. For each value of
Pe
we get one stability curve. Each curve
has a minimum at
Ra
=
Ra
m
,k
=
k
m
. This is referred to as the critical point
for onset of convection. Figure 14.3(d) for the critical conditions as functions
of
Pe
shows that
k
m
is an increasing function of
Pe
, that is, the plow patterns
change from elongated to slender shape as the swimming velocity is increased.
For
Pe
smaller than 1, the stability curves almost coincide as can be seen
in Figure 14.3(b) and the critical points shown in Figure 14.3(e) may be
approximated by
k
c
=0
.
9
Pe
1
/
2
Ra
c
=12+2
Pe
and
(14.28)
For
Pe
greater than 1, the stability curves are strongly dependent on
Pe
(Figure 14.3a). However, by setting the length scale to
h
=
D
c
/V
c
instead
of
H
, they coalesce into a
unique stability curve with a critical point Ra
c
=
10
.
2
,k
c
=0
.
7 for all
Pe
as shown in Figures 14.3(c) and 14.3(f). It should
be noted that this renormalization leads to a renormalized Rayleigh number,
Ra
∗
=
Ra/P e
, and a wavenumber,
k
∗
=
k/Pe
, based on the length scale,
h
=
D
c
/V
c
.
On the basis of this result, the values of
Pe
chosen for the numerical
simulations are varying between 0.1 and 5 while the Rayleigh numbers
Ra
may take the value several times as large as the critical one.
Also of interest is the growth rate of convection as predicted by the linear
stability theory. For a given Rayleigh number above the critical value
Ra
m
,
there exists a range of wavenumber for which the system is unstable. As shown
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