N O T I C E
THIS DOCUMENT HAS BEEN REPRODUCED FROM
MICROFICHE. ALTHOUGH IT IS RECOGNIZED THAT
CERTAIN PORTIONS ARE ILLEGIBLE, IT IS BEING RELEASED
IN THE INTEREST OF MAKING AVAILABLE AS MUCH
INFORMATION AS POSSIBLE
'"
024
CORNELL
UNIVERSITY
Center
for
Radiophysics
and
Space
Research
0
(NASA-CR-162251)
E
MODAL
REPRESENFFECTS
CF
TRUNCATION
IN
(Cornell
u·
TATIONS
OF
THERMAL
CONVECTION
HC
A03/MF
~~~.'
Ithaca,
N.
Y.)
35
P
CSCL 20D
G3/34
EFFECTS
OF
TRUNCATION
IN
ITHACA,
N.
y~
,
CRSR
730
N80-10457
fJnclas
39668
MODAL
REPRESENTATIONS
OF
THERMAL
CONVECTION
Philip
S. Marcus
~l~
(
I
EFFECTS
OF
TRUNCATION
IN
MODAL
REPRESENTATIONS
OF
THERMAL
CONVECTION'"
Philip
S. Marcus
Center
for
Radiophysics
and Space
Research
Cornell
University
November 1979
off
•
Supported
~n
part
by
National
Science
Foundation
Grants
ATM
76-10424 and
AST
78-20708 and
NASA
Grant
NGR-33-010-1SS.
,
t-
-2-
ABSTRACT
We
examine
the
Galerkin
(including
single-mode
and
Lorenz)
equations
for
convection
in
a
sphere
to
determine
which
physi-
cal
processes
are
neglected
when
tht
e~uations
of
motion
are
truncated
too
severely.
We
test
Jur
conclusions
by
calculating
solutions
to
the
equations
of
motion
for
different
values
of
the
Rayleigh
number
and
for
different
values
of
the
limit
of
the
horizontal
spatial
resolution.
We
show
that
the
transitions
from
steady-state
to
periodic,
then
to
aperiodic
convection
depend
not
only
on
the
Rayleigh
number
but
also
very
strongly
on
the
horizontal
resolution.
All
of
our
models
are
well-
resolved
in
the
vertical
direction,
so
the
transitions
do
not
appear
to
be due
to
poorly
resolved
boundary-layers.
One
of
the
effects
of
truncation
is
to
enhance
the
high
wavenumber end
of
the
kinetic
energy
and
thermal
variance
spectra.
Our
numerical
examples
indicate
that
as
long
as
the
kinetic
energy
spectrum
decreases
with
wavenumber, a
truncation
gives
a
qualitatively
correct
solution.
--~--,,-.--.~""""""'------"~-'-
-~-.-----.---
•.
--
_-&....
--
1
-3-
REPRODUCIBILITY
OF
THE
I.
INTRODUCTION
ORIGINAL
PAGE
IS
Poor?
In
Rayleigh-Bernard
convection,
discrete
transitions
from
steady-state
to
periodic
to
aperiodic
convection
have
been
experi-
mentally
observed.
(See
the
recent
reviews
by
Fenstermacher
et
al.
1978
and
Busse
1978.)
As
the
Rayleigh
number
is
increased
and
the
fluid
becomes more
"turbulent",the
Fourier
spectrum
(in
time)
of
the
velocity
develops
a
single
spike
(and
its
overtones)
and
shows a
gradual
increase
of
the
broad
band
background
noise
that
eventually
overwhelms
the
spikes.
Although
the
transitions
depend
not
only
on
the
Rayleigh
numbe~
but
also
on
the
Prandtl
number
and
initial
conditions,
there
has
recently
been
much
interest
in
trying
to
compute
these
transitions
from
the
actual
equations
of
motion.
In
attempting
to
compute
time-dependent
numerical
solutions
to
the
three-dimensional
Navier-Stokes
equation,one
is
forced
to
make
severe
approximations.
When
simplifying
the
equations
of
motion
to
make them
numerically
tractable,one
hopes
to
esta-
blish
a compromise
so
that
the
modified
equations
are
uncompli-
cated
enough
to
be
easily
solved,
yet
complete
enough
that
the
underlying
physics
of
the
fluid
dynamics
is
not
lost.
The
crudest
approximation
is
the
Lorenz (1963)
model.
The
Lorenz
model
predicts
not
only
the
transitions
to
steady-state
and
time-dependent
convection,
but
also
a
sequence
of
bifurca-
tions
that
eventually
leads
to
chaotic
(aperiodic)
behavior.
For
low
Rayleigh
numbers
near
the
onset
of
convection
the
heat
flux
(Nusselt
number)
predicted
by
the
Lorenz reodel
is
in
fair
•
,-