Downloaded 13 Jan 2006 to 131.215.225.172. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
The
spatial
stability
of
a
class
of
similarity
solutions
L.
Durlofsky
and
J.
F.
Brady
Department
of
Chemical
Engineering,
Massachusetts
Institute
of
Technology,
Cambridge,
Massachusetts
02139
(Received
31
May
1983;
accepted
4 November
1983)
The
spatial
stability
of
a class
of
exact
similarity
solutions
of
the
Navier-Stokes
equations
whose
longitudinal
velocity
is
of
the
form
xf'(y),
where
xis
the
stream
wise
coordinate
andf'(y)
is a
function
of
the
transverse,
cross-stream
wise,
coordinate
y
only,
is determined.
These
similarity
solutions
correspond
to
the
flow
in
an
infinitely
long
channel
or
tube
whose
surface
is either
uniformly
porous
or
moves
with
a velocity
linear
in
x.
Small
perturbations
to
the
streamwise
velocity
of
the
form
x'
g'(y)
are
assumed,
resulting
in
an
eigenvalue
problem
for
A,
which
is solved
numerically.
For
the
porous
wall
problem,
it is shown
that
similarity
solutions
in
whichf'(y)
is a
monotonic
function
of
y
are
spatially
stable,
while
those
that
are
not
monotonic
are
spatially
unstable.
For
the
accelerating-wall
problem,
the
interpretation
of
the
stability
results
is
not
unambiguous
and
two
interpretations
are
offered.
In
one
interpretation
the
conclusions
are
the
same
as
for
the
porous
problem-monotonic
solutions
are
stable;
the
second
interpretation
is
more
restrictive
in
that
some
of
the
monotonic
as
well
as
the
nonmonotonic
solutions
are
unstable.
I.
INTRODUCTION
Similarity
solutions
to
the
equations
of
motion
have
been
in
use
at
least
since
Blasius
1
assumed
a self-similar
ve-
locity
profile
for
the
boundary-layer
flow
over
a flat
plate.
Many
of
the
exact
solutions
to
the
Navier-Stokes
equations
are
based
on
the
assumption
of
self-similarity;
thus
reducing
the
equations
of
motion
to
one
or
more
ordinary
differential
equations
and
greatly
simplifying
the
analysis.
Important
as
similarity
solutions
are
in
helping
us
understand
the
behav-
ior
of
fluids,
there
is
no
assurance
that
these
solutions
repre-
sent
a physically
realizable
flow.
The
flow
domains
are
often
unbounded,
and
the
similarity
solutions
possess
singularities
either
at
infinity
or
at
the
origin;
thus
casting
some
doubt
as
to
their
uniform
validity.
It
has
generally
been
accepted,
however,
that
similarity
solutions
can
and
do
provide
at
least
a local
description
of
some
flow.
Several
recent
studies
have
examined
the
question
of
the
applicability
of
similarity
solu-
tions
and
have shown
that
even
a local
validity
may
not
be
possible.
2
-6
In
this
paper
we
wish
to
investigate
the
validity
of
a
particular
class
of
similarity
solutions.
These
solutions
cor-
respond
to
the
flow
in
a uniformly
porous
channel
or
tube
and
to
the
flow
in
a channel
or
tube
with
an
accelerating
surface
velocity
(cf.
Fig.
1
).
The
similarity
solutions
for
the
porous
wall
problem
have
been
studied by
Terrill
and
oth-
ers,7-12
and
those
for
the
accelerating
surface
problem
by
Brady
and
AcrivosY
We
are
interested
in
these
problems
because
the
similarity
solutions
possess
some
rather
unusual,
and
perhaps
unphysical,
features
such
as
the
existence
of
multiple
solutions
for
some
range
of
Reynolds
number.
Even
more
disturbing,
however,
is
the
discrepancy
which
exists
between
the
axisymmetric
(tube)
and
two-dimensional
(channel)
flows
for
positive
Reynolds
numbers
for
both
the
porous
wall
and
accelerating
surface
problems:
a range
of
Reynolds
number
exists
within
which
there
is
no
solution
to
the
axisymmetric
similarity
equation,
while
solutions
to
the
two-dimensional
case
exist
for
all
values
of
the
Reynolds
number.
This
inconsistency
and
lack
of
similarity
solutions
for
the
axisymmetric
problems
calls
into
question
their
va-
lidity.
Brady
and
Acrivos
5
addressed
this
paradox
of
nonexis-
tence
in
their
study
of
the
flow
in
a channel
or
tube
with
an
accelerating
surface
velocity.
Its
cause
was
traced
to
the
as-
sumption
that
the
tube
and
channel
were
infinite
in
extent.
By
an
appropriate,
boundary-layer-like,
numerical
analysis
of
the
equations
of
motion,
they
showed
that
above
a critical
value
of
the
Reynolds
number
Rc,
the
flow
throughout
the
entire
domain
was
influenced
by
the
end
condition.
Thus,
the
similarity
solution
no
longer
represented
any
real
flow.
Recently,
Brady
6
has
shown
that
the
same
phenomenon
oc-
curs
in
the
porous
wall
problem.
It
should
be
noted,
how-
ever,
that
although
the
domain
is finite,
the
actual
length
of
the
channel
or
tube
can
be
arbitrarily
large.
The
end
condition
may
be
viewed
as
an
0
(
1
),
mass-con-
serving,
numerical
perturbation
to
the
similarity
velocity
profile.
At
low
values
of
R,
this
perturbation
decayed
as
it
was
convected
longitudinally
away
from
the
end,
and
the
similarity
solution
was
valid
over
a region
far
removed
from
the
end.
At
Rc,
however,
the
perturbation
was
convected
all
the
way
to
the
origin,
and
the
region
of
validity
of
the
similar-
ity
solution
shrunk
to
zero.
Further,
Brady
and
Acrivos
5
found
that
this
critical
value
of
R
was
insensitive
to
the
de-
tailed
nature
of
the
end
condition.
For
flow
in
a
tube
with
an
accelerating
surface
velocity,
Rc
coincided
exactly
with
the
value
ofR,
10.25,
beyond
which
no
solution
to
the
similarity
equation
existed.
A
critical
R
was
also
shown
to
exist
for
flow
in
a
channel
with
an
accelerating
surface
velocity
(Rc
~57),
even
though
solutions
to
the
similarity
equation
exist
for
all
R.
For
the
porous
wall
problem
(Ref.
5),
critical
Reynolds
numbers
of
2.3
and
6,
respectively,
were
also
found.
It
was
shown,
however,
that
the
similarity
solutions
for
the
porous
wall
problem
may
regain
validity
at
larger
R
if
the
end
condition
is sufficiently
"close"
to
a similarity
solu-
tion.
Thus,
we
see
that
beyond
Rc
the
similarity
solutions
for
this class
of
flows
may
or
may
not
describe
the
motion
in
a
1068
Phys.
Fluids
27
(5),
May
1984
0031-9171/84/051068-09$01.90
©
1984
American
Institute
of
Physics
1068
Downloaded 13 Jan 2006 to 131.215.225.172. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
finite
channel
or
tube.
The
important
influence
of
the
end
condition
and
the
presence
of
critical
Reynolds
numbers
suggests
that
a spatial
stability
analysis
of
the
similarity
equations
may
aid
in
un-
derstanding
how
and
why
these
solutions
lose
validity.
In
particular,
it
would
be
of
interest
to
see
if
these
critical
Reyn-
olds
numbers
could
be
predicted
by
such
an
analysis.
Be-
cause
these
similarity
equations
are
very
much
like
the
clas-
sical
boundary-layer
equations,
it is instructive
to
first
con-
sider
the
spatial
stability
of
the
Falkner-Skan
equation
for
the
boundary-layer
flow
over
a wedge.
Serrin
14
proved
that
for
an
arbitrary
initial
velocity
profile,
the
magnitude
of
the
deviation
of
the
actual
flow
from
that
predicted
by
the
simi-
larity
solution
is
o[(l
+
m
log
x)lxm]
for
m>O
as
X---+oo
downstream.
[Here,
m
is
the
exponent
of
the
free
stream
velocity
U
(x)
=
Cxm,
where
Cis
a constant.
His
analysis
is
restricted
to
the
case
m
>0.]
Thus,
for
these
flows
an
0
(
1)
perturbation
to
the
similarity
solution,
regardless
of
its
form,
will
decay
as
it is convected
downstream;
i.e.,
the
similarity
solution
will
be
valid
asymptotically
far
downstream
from
the
leading
edge.
Chen
and
Libby
15
also
investigated
the
validity
of
solu-
tions
to
the
Falkner-Skan
equation
by
means
of
a stability
analysis.
They
assumed
small
perturbations
to
the
stream-
wise
velocity
of
the
formxt
¢
(77),
wherex
is
the
distance
from
the
leading
edge,
77
is
the
similarity
variable,
and¢
(77)
is
the
perturbation
function.
Here
A
is
the
unknown
eigenvalue:
A<
0
represents
stability
in
the
sense
that
small
perturba-
tions
will
decay
as
X---+oo
(downstream),
and
A>
0
would
therefore
indicate
spatial
instability.
16
For
m>O,
only
nega-
tive
eigenvalues
were
found,
in
agreement
with
Serrin's
anal-
ysis.
Form
<
0,
the
solutions
to
the
Falkner-Skan
equation
are
no
longer
unique,
there
being
at
least
two
different
solu-
tions.
One
branch
of
solutions
has
reverse
flow
adjacent
to
the
surface-fluid
moving
towards
the
leading
edge,
and
it
connects
with
the
unidirectional
flow
branch
at
the
point
where
the
shear
stress
at
the
surface
vanishes,
m
= -
0.0904.
For
the
reverse
flow
branch,
they
found
both
positive
and
negative
A
'sand
attributed
the
presence
of
the
positive
eigenvalues
as
an
indication
of
instability
(see
Sec.
II).
The
undirectional
flow
branch
gave
only
negative
A's.
The
agreement
between
the
analyses
of
Chen
and
Lib-
by15
and
Serrin
16
for
m>O
and
the
conclusions
concerning
the
stability
of
the
branches
form
<
0
indicate
the
utility
of
a
linear
stability
analysis
for
determining
solution
validity.
Our
approach
will
be
quite
similar
to
that
of
Chen
and
Libby.
15
In
Sec.
II
we
consider
the
spatial
stability
of
the
two-dimensional
porous
wall
and
accelerating
surface
prob-
lems.
We
show
that
the
stability
results
may
be
interpreted
in
two
ways.
One
interpretation
applies
to
both
the
porous
and
accelerating
surface
problems
and
maintains
that
the
simi-
larity
solutions
which
are
monotonic
functions
of
the
cross-
streamwise
coordinate
are
stable,
while
those
that
are
not
are
unstable.
The
other
interpretation
applies
only
to
the
accel-
erating
surface
problem
and
further
limits
the
spatially
sta-
ble
flows.
It
implies
that
all
of
the
nonmonotonic
solutions
are
unstable
and
that
the
monotonic
solutions,
which
exist
for
-
oo
<
R
<
oo,
are
stable
only
for
0
<
R
<
11.0.
In
neither
case,
then,
are
we
able
to
determine
the
critical
Reyn-
1069
Phys.
Fluids,
Vol.
27,
No.5,
May
1984
olds
numbers
reported
in
Refs.
5
and
6.
Section
III
is devoted
to
the
axisymmetric
tube
problems.
Here,
a critical
Reyn-
olds
number
is found,
and
it
agrees
precisely
with
the
Reyn-
olds
number
at
which
similarity
solutions
cease
to
exist.
The
conclusions
we
reach
regarding
stability
or
instability
of
the
similarity
solutions
are
with
regard
to
a particular
class
of
spatial
perturbations.
Questions
of
temporal
stability
and
stability
to
a
broader
class
of
spatial
perturbations
are
not
addressed
in
this
study.
II.
THE
TWO-DIMENSIONAL
CHANNEL
PROBLEMS
A.
Porous
channel
flow
Referring
to
Fig.
1,
we
denote
the
channel
half-width
by
h,
the
fluid
viscosity
by
f.l,
its
density
by
p,
and
the
normal
velocity
through
the
porous
wall
by
V.
For
an
infinitely
long
channel,
the
Navier-Stokes
equations
admit
an
exact
simi-
larity
solution
of
the
form
u
=
-xf'(y),
v
=f(y),
P
=PolY)+
!flx
2
,
where
u
and
v
are
the
x
andy
components
ofvelocity,p
is the
pressure,
and
fJ
is
a constant.
For
flows
symmetric
about
y
=
0,
the
similarity
function/satisfies
f"'
+
fJ
=
-
R[(f')2
-.If"],
f(O)
=f"(O)
=
0,
/(1)
=
1,
/'(1)
=
0.
(1)
(2)
Here,
R
=
p
Vh
I
J.l
is
the
Reynolds
number;
R
>
0
corre-
sponds
to
suction,
and
R
<
0
to
injection.
The
unknown
pres-
sure
coefficient
fJ
is determined
by
the
fourth
boundary
con-
dition.
The
solutions
to
Eqs.
(1)
and
(2)
have
been
studied
ex-
tensively,?-11
and
we
shall
comment
only
briefly
on
their
structure.
There
exists
a single
continuous
solution
extend-
ing
from
-
oo
to
+
oo
in
R.
The
longitudinal
velocity
pro-
files
are
monotonic
functions
of
y,
with
the
fluid
coming
in
from
infinity
for
suction
(R
>
0)
and
going
out
towards
infin-
ity
for
injection
(R
<
0).
The
suction
solutions
develop
a
boundary
layer
near
the
wall
as
R---+
oo;
the
injection
solu-
tions
do
not.
There
are
two
additional
suction
solutions
which
appear
atR
=
12.165
and
extend
toR---+
+
oo.
These
solutions
have
velocity
profiles
in
which
the
longitudinal
ve-
locity
is no
longer
a monotonic function
of
y,
and
for
some
solutions
there
is a region
of
reverse
flow
at
the
centerline
with
the
fluid
there
now
moving
out
towards
infinity.
In
Fig.
2 we
show
a plot
of
fJ
vs
R where
we
have
labeled
the
different
c·:r·:'·,·"'
-ro
\ÂŁ.!!
•••
0
oy
FIG.
I.
Schematic
diagram
for
the
two-dimensional
channel
similarity
so-
lutions.
Uniformly
porous:
u
=
0,
v
=
V.
Accelerating
surface:
u
=Ex,
v=O.
L.
Durlofsky
and
J.
F.
Brady
1069
Downloaded 13 Jan 2006 to 131.215.225.172. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
0
{3
-20
rr
III
-40
0
20
40
R
FIG.
2.
The
pressure
coefficient
/3
of
the
similarity
solutions
for
the
two-
dimensional
flow
in
a uniformly
porous
channel
as
a function
of
the
Reyn-
olds
number.
The
three
solution
families
are
labeled
I, II,
and
III.
solutions
I, II,
and
III.
The
point
X
marks
the
juncture
of
branches
II
and
III
and
is
the
point
where
the
centerline
velocity
is
zero.
To
determine
the
spatial
stability
of
these
similarity
so-
lutions,
we
write for
the
velocity
and
pressure
fields
u
= -
xf'(y)
-
.0
g'(y),
v
=
f(y)
+Ax"-
1
g(y),
(3)
P=PolY)+
!/3x
2
+[11(l+A)]/3,tx
1
+",
where
g'
is
the
perturbation
to
the
longitudinal
velocity
and
13.-t
is
the
perturbation
to
the
pressure.
Substituting
into
the
x-momentum
equation
and
neglecting
terms
of
second
order
in
the
small
perturbation
g,
the
linearized
stability
equation
for
g
is
g'"
+
13.-t
=
R[/g"-
(1
+A
)f'g'
+
Aj"g],
(4)
with
g(O)
=
g"(O)
=
g(1)
=
g'(1)
=
0,
g'(O)
=
1,
(5)
where
without
loss
of
generality
we
have
set
g'(O)
equal
to
unity.
Both
A
and
13.-t
are
unknown
eigenvalues.
Given
a
si-
milarity
solution/at
a particular
R,
the
two-point
boundary-
value
problem
[(4)
and
(5)]
was
solved
using
Newton's
meth-
od
to
determine
g,
13.-t,
and
A.
17
Whether
the
eigenvalue
A
is
greater
than
or
less
than
unity
will
determine
the
spatial
stability
of
the
similarity
so-
lutions.
Some
care
is
needed,
however,
when
considering
in
which
direction,
increasing
or
decreasing
x,
the
perturba-
tions propagate.
For
the
group
I
solutions,
a negative
Reyn-
olds
number
corresponds
to
injection
with
the
fluid
moving
out
towards
infinity.
Here,
we
wish
to
consider
a perturba-
tion
occurring
at
some
finite
value
of
x
and
ask
whether
this
perturbation
decays
relative
to
the
similarity
solution
as
it
is
convected
with
the
flow
as
X---+oo.
Thus,
A<
1 represents
stability.
For
positive
Reynolds
numbers
corresponding
to
suction,
the
situation
is
reversed.
The
flow
is
now
moving
inward
from
infinity
to
the
origin
x
=
0.
Here,
we
imagine
a
perturbation
at
some
finite
x
and
ask
whether
it decays
rela-
tive
to
the
similarity
solution
as
x---+0.
18
Thus,
A
>
1 repre-
1070
Phys.
Fluids,
Vol.
27,
No.5,
May
1984
sents
stability.
For
the
group
III
solutions
in
which
there
is
fluid
moving
in
both
the
positive
and
negative
x
directions,
the
direction
of
propagation
of
the
perturbation
and
hence
the
stability
criterion
with
A
are
no
longer
as
unambiguous.
This
difficulty
becomes
even
more
apparent
in
the
accelerat-
ing
surface
problems
because
fluid
always
moves
in
both
directions.
When
we
consider
the
accelerating-surface
chan-
nel
problem
later
in
this
section,
we
shall
offer
an
interpreta-
tion
of
A
with
regard
to
the
direction
of
flow.
For
the
mo-
ment,
we
shall
simply
consider
the
group
I
porous
channel
results.
The
minimum
magnitude
eigenvalues
A
for
the
group
I
porous
channel
solutions
are
plotted
versus
the
Reynolds
number
in
Fig.
3.
No
negative
A's
were
found
for R
>
0
(suc-
tion),
nor
were
there
positive
A
's for
R
<
0
(injection).
For
R
>
0,
A
asymptotically
approaches
1 as
R---+oo;
thus,
suc-
tion
flow
is
always
stable,
approaching
marginal
stability
as
R
tends
to
infinity.
For
R
<
0,
A
asymptotically
approaches
- 3
as
R---+
-
oo;
injection
flow
is
always
stable,
even
as-
ymptotically
in
R.
The
asymmetry
in
the
asymptotic
forms
of
A
vs
R
is
not
surprising
because
the
similarity
flows
are
themselves
quite
different
at
large
I
R
I:
a boundary
layer
ad-
jacent
to the
wall
forms
in
suction
flow,
but
not
for
injection.
The
similarity
solutions
are,
however,
symmetric
about
R
=
0,
and
this
symmetry
is
present
in
the
stability
analysis.
As
R---+0,
from
above
or
below,
A~AofR
with
A
0
=
18.81.
There
are
of
course
additional
eignevalues,
and
they
show
the
same
behavior
as
the
minimum
magnitude
eigenvalues,
with
differentA
0
's
and
asymptotic
values
for
large
IRI.
The
spatial
stability
analysis
of
the
group
I
solutions
has
shown
these
solutions
to
be
stable
for
all
R.
We
have
not
found
(as
was
initially
hoped)
a critical
Reynolds
number
for
loss
of
stability.
The
predictions
of
the
stability
analysis
for
R
>
0
do
seem,
however,
to
agree
well
with
the
numerical
results
of
Ref.
6.
There
it was
shown
that
for
R
~50,
certain
0
(
1)
numerical
perturbations
to
the
inlet
condition
result
in
velocity
and
pressure
profiles
as
x---+0
that
are
superpositions
10
5
;-------
-
A
0
-5
-10
f--
-15
-50
A-~
R
-25
0
R
A-~
R
-
-
25
50
FIG.
3.
The
minimum
magnitude
eigenvalues
A
for
the
group
I porous
channel
solutions.
Here
A.~
1
as
R--->
oo,
A.~
-
3 as
R--->
-
oo,
and
A.~
18.81/R
as
R---+0.
The
eigenvalues
indicate
the
group
I solutions
are
spatially
stable.
The
dashed
lines
in
this
and
all
subsequent
figures
of
stabil-
ity
results
correspond
to
A.
=
1.
L.
Durlofsky
and
J.
F.
Brady
1070