Environmental Engineering Reference
In-Depth Information
the corresponding condition for an external force directed parallel to the bound-
ary layer. We take the
z
-axistobenormaltotheboundarylayerandtheexternal
forcetobedirectedalongthe
x
-axis. Then, to the second equation of the set of
equations (5.3), we add the term
m
(
w
r
)
w
and compare the
x
components of this
equation. This comparison yields
mw
x
L
F
(
T
1
T
2
)
N
w
x
δ
.
(5.17)
T
2
Equation (5.17) gives the estimate
1/4
2
TL
N
2
Fm
3
(
T
1
η
δ
T
2
)
for the thickness of the boundary layer. Hence, the ratio of the heat flux
q
due to
convection and the flux
q
cond
due to thermal conduction is
q
q
cond
L
δ
Gr
1/4
.
(5.18)
The ratio of the heat fluxes in this case is seen to be different from that when
the external force is perpendicular to the boundary layer (see (5.15)). However, the
convective heat flux in this case is still considerably larger than the heat flux due to
thermal conduction in a motionless gas.
5.1.5
Instability of Convective Motion
We find that the convective motion of an ionized gas (as of any gas or liquid) oc-
curs as a Rayleigh-Taylor instability that results in convective gas movement along
simple closed trajectories for a simple geometry of its boundaries, and the space is
divided in Benard cells, so gas trajectories (or Taylor vortices) do not intersect the
boundaries of these cells. But this motion becomes more complicated with increas-
ing Rayleigh number [10-12], and new types of convective motion develop when
the Rayleigh and Grashof numbers become sufficiently large. The orderly convec-
tive motion becomes disturbed, and this disturbance increases until the stability of
the convective motion of a gas is entirely disrupted, giving rise to disordered and
turbulent flow of the gas. This will happen even if the gas is contained in a station-
ary enclosure. To analyze the development of turbulent gas flow we consider once
again the Rayleigh problem: a gas at rest between two parallel and infinite planes
maintained at different constant temperatures is subjected to an external force. We
shall analyze the convective motion of a gas described by (5.11), and corresponding
to sufficiently high Rayleigh numbers with
n
2. In this case there can develop
simultaneously at least two different types of convection.
Figure 5.2 shows two types of convective motion for the Rayleigh number Ra
D
4
corresponding to the wave number
k
1
D
9.4/
L
for
n
D
1andto
k
2
D
4.7/
L
108
π
for
n
D
2. To analyze this example using these parameters, we combine the solu-
tions so that the gas flows corresponding to
n
D
1andto
n
D
2travelinthesame
