ISBN
82-553-
D325-1
Applied
Mathematics
No
2
-.September
12
NONLINEAR
THERMAL
CONVECTION
IN
ANISOTROPIC
POROUS
MEDIA
by
~77
Oddmund
Kvernvold
and
Peder
A.
Tyvand
Oslo
PREPRINT
SERIES
-
Matematisk
institutt,
Universitetet
i Oslo
c
NONLINEAR
THERMAL
CONVECTION
IN
ANISOTROPIC
POROUS
MEDIA
Oddmund
Kvernvold
and
Peder
A. Tyvand
Department
of
Mechanics
University
of
Oslo
Oslo,
Norway
Abstract
In
this
paper
a
theoretical
investigation
of
convection
currents
in
anisotropic
porous
media
is
performed.
The
porous
layer
is
homogeneous
and
bounded
by
two
infinite,
perfectly
heatconducting
horizontal
planes.
The
criterion
for
the
onset
of
convection
is
derived.
The
supercritical,
steady
two-dimen-
sional
motion,
the
heat
transport
and
the
s:tability
of
the
Ill.Otion
are
investigated.
The
results
may
be
applied
in
insulation
technique.
-
--
-~----
- -
----
----
--~
- - -
~----
--
---
-
---
--
--
--
--
--
-
------------
--~-------
-----~----·---
- . - ---·-·------
--·------
-
-----
~
..
- 2 -
NOMENCLATURE
cp'
specific
heat
at
constant
pressure;
[fl),
dimensionless
tensor
of
effective
thermal
diffusivity;
v'
=
c:x'
a
a
) •
ay'
az
,
V12,
a
2 .
a2
=
+
--
.
ax2
ay2
'
g ,
acceleration
due
to
gravity;
H ,
heat
fluL;
h ,
thickness
of
porous
layer;
...............
i,j,k,
unit
vectors;
X ,
dimensionless
permeability
tensor;
K ,
permeability;
N ,
truncation
parameter;
Nu
, "Nusselt number;
p ,
dimensionless
pressure;
R ,
Rayleigh
number K
3
gytiTh/icm
3
v;
R
0
,
Rayleigh
number
for
the
onset
of
convection;
T ,
dimensionless
temperature;
T
0
,
reference
temperature;
tiT,
temperature
difference
between
lower
and
upper
plane;
.....
v =
(u,v,w),
dimensionless
velocity
vector;
.
x,-Y
,z_,
d1m~nsionJ~s-~
_CSJ:'tesian _
coordir1ates
!
~-
--~---------------------···-~---~-----
---~---
----------------------------
--
- -
Greek
letters
Y,
coefficient
of
thermal
volume
expansion;
n
1
2
,thermal
anisotropy
parameters
defined
by
(2.8);
,
e,
dimensionless
temperature;
~,
thermal
diffusivity;
A,
thermal
conductivity;
- 3 -
~
1
,
2
,
permeability
anisotropy
parameterndefined
by
(2.8);
¥ ,
kinematic
viscosity;
p ,
density;
p
0
,
standard
density;
Subscripts
c ,
critical;
f ,
fluid;
m ,
solid-fluid
mixture;
x,y,z,
partial
derivatives;
1,2,3,
tensor
components
in
x-,
y-
and
z-direction,
respectively;
I,II,
longitudinal
and
transverse
tensor
components
for
a
transversely
isotropic
medium.
Superscripts
T"l"
,
horizontal
mean;
(*)
,
transformation
given
by
(4.1)
and
(4.2);
( ) (
n)
,order.
OS'
eer1ea
e%pana1on;
(
)',
small
disturbance
of
the
two-dimensional
steady
solution.
- 4 -
1
INTRODUCTION
Free
thermal
convection
in
porous
media
has
received
con-
siderable
interest
due
to
its
technical
and
geophysical
applica-
tions.
So
far,
theoretical
and
experimental
investigations
have
usually
been
concerned
with
isotropic
porous
media.
However,
in
many
problems
the
porous
materials
are
of
anisotropic
nature.
This
is
the
case
for
fibrous
insulation
materials,
where
ccnvection
currents
may
occur.
Another
important
example
is
groundwater
motion
in
sediments
and
other
anisotropic
rocks,
especially
in
areas
with
geothermal
activity.
The
papers
on
convection
in
anisotropic
media
are
rather
new
and
not
numerous.
Castinel
and
Combarnous [
1]
derived
the
stabi-
lity
criterion
for
porous
media
with
anisotropic
permeability,
and
made
experiments
concerning
the
supercritical
heat
transport
and
temperature
field.
Ephere
[2]
extended
the
stability
analysis
to
media
with
anisotropJ
also
in
thermal
diffusivity,
and Tyvand
[3]
took
into
account
the
effect
of
hydrodynamic
dispersion
caused
by
a
uniform
basic
flow.
Burns,
Chow
and
Tien
[4]
incorporated
anisotropic
permeability
in
their
study
of
convection
in
vertical
slots.
Their
study
is
relevant
to
insulation
between
walls,
while
our
present
study
is
relevant
to
insulation
between
floors
and
ceilings
in
buildings.
Nonlinear
convect1on
-1n
1sotrop1c
porous
media
-was-treate-d-
numerically
by
Elder
[SJ,
Straus
[6]
and
Kvernvold
[7],
and
ana-
lytically
by
Palm,
Weber
and
Kvernvold
{8].
In
this
paper
the
onset
of
convection
is
analysed
for
a more
general
type
of
anisotropic
rnedia
than
in
[1,2].
Moreover,
the
effects
of
anisotropy
on
the
supercritical
motion
and
the
heat
