INSTABILITIES OF SOME TIME-DEPENDENT FLOWS
by
Rory Thompson
A.B., San Diego State College
(1962)
M.S.,
San Diego State College
(1964)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSA CHUSETTS INSTITUTE OF TECHNOLOGY
September 1968
Signature
of
Author
..
.
......
....
Department or Meteology, 19 Aust
Certified by.
..........................
Accepted by.. .....
Chairman, Depa
Thesis Supervisor
.e
.
men
.
e...
Gr aduate
1968
0Suet
...
Committee on Graduate Students
Lindgren
INSTABILITIES CF SOME TIME-DEPENDENT FLOWS
by
Rory Thompson
Submitted to the Department of Meteorology
on August 19, 1968
in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
ABSTRACT
The
flow of fluid between
considered under
two concentric cylinders is
three time-dependent
forcings as
an
ap-
proach to instabilities of general time-dependent flows. The
first is
by
rotating
about
gravity,
so
geometries,
gravity on
free surfaces
axis slightly
there
is
a
tilted
periodic
the
annulus
with
motion.
is
respect to
For
certain
there are resonances of inertial-gravity modes.
These periodic
flow,
an
a
motions
cause a
including a central vortex
rectified
mean
azimuthal
for a cylinder, according
to analysis of the viscous boundary layers. These mean flows
can
be
unstable to
shear oscillations,
sometimes causing
renarkable commotion which could disrupt geophysical
ing
experiments.
The theory
is found to
model-
accord well with
experiments.
The second type of forcing is periodic torsional motion
of
the inner
wall, with the
outer wall
held still.
causes periodic ring vortices for certain parameter
These
ring
ascertain
vortices
the
problems.
sion time
are
most
studied
practical
by
ways
several
to
This
ranges.
methods to
approach
similar
Oscillations slow with respect to viscous diffuare essentially
quasi-steady
and give
rise
to
essentially ordinary Taylor cells which turn on and off with
each half-cycle.
continuous
Faster
Taylor
oscillations give
cells which
feed
rise
on the
to
more
mean absolute
centrifugal gradient. Critical Reynolds numbers are found by
straight-forward
numerical
integrations
and
by
expansions in timerpurely periodic perturbations
to
narginal
results.
growth
rates.
Experiments
harmonic
correspond
corroborate
Oscillations superposed on a mean rotation
the
always
destabilize.
The
third type
of forcing
is similar
to the second,
except the inner cylinder is suddenly started. This illuminates
what
instability
changing anyway, and when
used.
mean
on
a
flow
that is
a quasi-steady assumption can
be
In this case, one can be used even from small times,
and the results
tensen [R6
fit
experimental
data of
Chen and
quite well.
Thesis Supervisort
Titlet
might
Professor Louis N. Howard
Professor of NMthematics
Chris-
this thesis is dedicated to Luella
MUM
Table of Contents
i
Title Page
2
Abstract
4
Dedication
5
Table of Contents
8
Introduction
9
Literature review for general time dependence
10
Selection of problems
11
11i
Chapter One --
The Slightly Tilted Cylinder
Literature review
11
for inviscid oscillations
11.
rectified boundary drifts
12
shear instabilities with rotation
12
slight tilt
13
Notation and Equations
16
Zero order interior solution
17
eigenvalue expansions
19
effects of finite Froude number
21
resonances
23
Zero order boundary solutions
25
First order mean motion
27
interior and ambiguity
28
integration across the bottom boundary
29
formulation of equations
30
interior v(0+,r)
34
radial flows in boundary
IBM=
35
Shallow water
35
zero order
36
first order mean
37
lagrangian drif t
38
Side-Wall boundary layers
39
Shear instabilities on the mean azimuthal velocity
40
Ekman friction
41
Cartesian analog
42
bound on the shear parameter
43
Experiments
43
description of apparatus and operation
44
description of results
46
verif ication of theory
49
Chapter Two
--
Periodic Centrifugal Instabilities
49
Literature review
53
Notation
53
Steady Taylor problem as example of numerical philosophy
54
Equations and basic f low
57-
Solution of the initial value problem
59
boundary conditions
59
u,v f orms
60
kinetic energy
62
critical Reynolds numbers
64
Quasi-steady and rapid oscillation limits
66
66
Mean Rotation superposed
Inner cylinder mean rotation
68
inertial waves
69
subharmonic resonance
71
Truncated spectral attempt
74
Experiments
74
description of apparatus and procedures
77
results for pure oscillation
77-
results with superposed mean rotation
80
82
Inviscid oscillations
Chapter Three
-
Suddenly Twisted Cylinder
82
Introduction
83
Scaling and estimate for critical
85
Quasi-steady approach
87
Numerical initial value 'solution
91
Proper finite-difference forms
92
'energy preserving'
forms
93
disguised spectral forms
95
asymptotic appraximations
96
Experimental comparison
97
After the critical time
99
approximate
1.00
analog set-up
104
Energy integral bounds
107 Acknow ledgments
109
Bibliography
118
Biographical sketch
Instabilities of Some Time-Dependent Flows
Introduction
All geophysical flows are unsteady: the wind over water,
the potential temperatures around a cumulus, and a long wave
with
a
nascent cyclone
imbedded, all
perturbations grow, though usually
Yet,
of wave
theories
change even
on a longer
growth or
for
mathenatical
separate to
cients
it
leave a
reasons:
means
to
speak of
changing anyway.
They generally do
so
first,
linear problem
in time, and second,
time-scale.
cyclogenesis classically
assume the unperturbed flow to be steady.
this
as the
the
equations
with constant
coeffi-
to avoid difficulties with what
instabilities
on a
f low
which
is
So, how can one start toward understanding
instabilities when the basic solution is not steady?
For certain ranges of parameters (such as the
number,
the
asymptotic
behavior of
Navier-Stokes -equations is
boundary conditions.
not uniquely
of the
determined by
the
always has a
conductive
of no motion, but above a certain Rayleigh number,
there also exist convective
conductive
solution
E.g., the Benard problem of Boussinesq
convection between parallel plates
solution
the
Reynolds
solutions which merge with
the
solution as the Rayleigh number decreases to the
critical parameter.
This
is known
as
"bifurcationI
One
expects
similar
behavior
parameters, even if
have
integral
for
suitable
non-dimensional
some of them measure time-dependence. We
theorems
such
as
Serrin's
(1959a)
which
provide a basis in showing that there exist parameter ranges
for which the basic flow is unique, but where are the limits
of the ranges? Under what circumstances are the limits given
by
the linear theory,
as Howard (1963)
showed is true for
the Benard problem?
The various fluid instabilities are so diverse that
attempt at a general theory would not be appropriate,
an
though
some general approaches to stability such as Serrin (1959a),
Sorger
and
(1966),
interesting
Ito
uniqueness
(1961)
criteria
have
for
yielded
certain
weak
but
classes
of
flows. The classic sources of energy for a fluid instability
are density differences (Benard, Rayleigh-Kelvin, and gravitational
instabilities),
Schlichting),
variously
fields,
and
shear (Kelvin-Helmholtz,
centrifugal
forces
combined and complicated
material
temperature
by
Tollmien-
(Taylor-Gertler),
rotation, magnetic
dendences,
and
various
Of this wide choice, what problems can include
geometries.
time-dependence and yet be tractable?
There has
dependence.
been
some work
done
Most of the relevant
on effects
of
time-
ones such as Lick (1965)
and Currie (1967) which study turning on the heat in
convection,
flow, ignore
and D'Arcy (1951) which studies started concave
the
quasi-steadiness
this may
short
Benard
direct
without
be appropriate,
role of
through
assuming
particular investigation
though it
compared to the time-scale
times this can be a
time,
may hold
over a
of the basic flow.
bit disastrous.
of when
For instance, a
time
Somestick
may
be
balanced upright
vertically at
steady
on a
the right
hinge
frequency,
theory would indicate it
if
is osc 1lated
even though
a
quasi-
to be always unstable.
the other hand, Benjamin and Ursell (1954)
a
it
showed that
On
when
cylinder containing an inviscid fluid with a free surface
vertically with simple-harmonic
condition is shaken
eration,
the fluid can be unstable even if
accel-
the amplitude of
the acceleration is much less than gravity, whereas a quasianalysis
steady
always
predicts
stability.
Yih
considers horizontal oscillation of a plate with a
shallow
fluid on
it
in
the long-wave
shows there exists instability.
(1967)
viscous,
limit, and likewise
This does agree with quasi-
steady theory.
So, what are suitable problems here?
The easiest time-
dependent centrifugal instabiities seem to be open,
though
Conrad and Criminale (1965 ) have done some quasi-steady work
with certain weaknesses,
only)
has completed some.
easiest to handle
steady
and recently Sorger (1968, abstract
Orr-Serrin bounds.
rrthematically are
[Chapter two of this
The functions
simple harmonic
plus
thesis], and impulses [Chapter
three]. Since it is desirable to approach time-dependence in
easy
stages,
observation
it
that
is
worth
weak
investigating
periodic
tide-like
Fultz's
forces
(1965)
on
a
rotating fluid can cause considerable turmoil, unlike what a
quasi-steady theory might suggest.
Chapter One.
This is the subject
of
11
Forcing of the Fluid in a Slightly Tilted
Rotating Cylinder or Annulus
The
most
common
models involve
spheres.
geophysical
annuli
Thus it
rotating cylinder of
as tides
such
or
perhaps
imperfections
of
oscillation modes for a
fluid and described
axisymmetric modes
with an axial
an experiment
to
plunger and disk,
which was finally carried out by Fultz (1960).
theoretically
or
is reasonable to study their responses to
Kelvin (1880) found the
rotation.
dynamic laboratory
(including ~cylinders),
extraneous influences
excite
fluid
Baines (1967)
studied axisynmetric forced oscillations of a
finite rotating cylinder and
found the asymptotic
flow contains
patterns of internal shear for
pseudo-random
forcing slower than the rotation frequency.
experimentally
studied
axisymmetric
periodic
Aldridge (1967)
modes
of
a rotating
sphere excited by a small torsional oscillation.
Aldridge (1967) also studied
in
his sphere
and observed a
the viscous boundary
rectified mean
drift with a
square law dependence on the oscillation amplitude.
reported
roll instabilities on
vortices.
studied
He also
the viscous boundaries with
wavelengths and critical Reynolds
instabilities
layer
number which suggest
the
are essentially time-dependent G'rtler-Taylor
Rectified currents in
boundary layers have
been
for a long time, since at least Rayleigh (1884) and
by many people. There are comprehensive reviews,
Schlichting
(1968).
Longuet-Higgins (1953)
dimensional, non-rotating
gravity
waves in
e.g., by H.
showed how twoshallow
water
cause a forward jet in the bottom boundary layer.
Instability
of
shear motion
has
been reviewed
by Lin
12!
(1949)
and by Betchov and Criminale
dimensional, non-rotating
of stability
study
rotating,
for
The
flows.
wave
has
Busse (1967),
rotating
Johnson (1963)
and gives
numbers
complicated
spheroids
mostly for twogives
of one-dimensional parallel
inviscid fluid,
large
(1967),
in
steady
a
motion
due
considered by
been
and of
a cylinder
shear in a
a stability
cylindrical
to
criterion
shear
layer.
precession
Malkus
of
(1964,1966) and
by Johnson
(1967).
For
annuli, visible effects of the misalignment of the
axis of rotation were reported
McDonald
by Fultz et al.
(1959)
and
and Dicke (1967), but apparently the only previous
studies of the effects
have been by
Fultz found that
(1965).
rectangular box developed
certain
range of water
Fultz
the water in
(1965) and
a powerful central
depths.
resonance
of
inviscid
Crow found
artificial pressure
fluid
to keep
the
in
a
Crow
a rotating, tilted
vortex for
a
the same for a
cylinder and showed that the depth of water corresponded
a
a
cylinder
surface plane.
with
In
to
an
this
thesis, the viscous boundary layers are considered carefully
enough
to
geostrophic
show
how
vortices
ink
by Claes
placed
on
the
intriguing
scallop
turbulence.
A new
that
the
ink will
relieving
the
as well as considering
and improving the correspondence with
(also more thoroughly
paraboloidal surface.
proposed
arise,
ambiguity left by Crow,
all of the resonances,
experiments
can
The results
done) by considering the
also explain- the -puzzle
Rooth (private comnunication)
bottom
shapes
of
which
the
sometimes
observation is also
form
cylinder
often
grow
mde and
concentric rings
as to why
on
has
to form
explained
the bottom.
13
Notation.
The
angular velocity 11- about its
cylinder is rotating at
axis of
symmetry at
Igxo../(Jgj Inli)
angle
C
from vertical
and the mean depth of the water is H.
are taken rotating
the viscosity is
velocity .2.
is
with the cylinder,
chosen
2/
,
The usual cylindrical
so z is
along its axis, r is radial, and e is counterclockwise.
angular
=
).
The outer radius is R, the inner is aR,
coordinates
(so sinE
positive.
The
the. height
So,
around which to linearize is
H+
choosing
(r 2
-
RZ(l+a
)/2)
& counterclockwise from
+ Er cos(e+O t)
the projection of ..
horizontal. The deviation from this depth is denoted
onto
. Let
u be the radial velocity dr/dt, v be the tangential velocity
r d6/dt, and w be the vertical velocity dz/dt.
Let p be the.
deviation
pressure from hydrostatic,
centrifugal force and the
which latter includes
rotating horizontal component
of
gravity as well as the vertical component.
In a frame rotating with the cylinder,
motion for an incompressible,
V VWtW
4)V
homogeneous fluid aret
+LAX'
29
1 'V
-wj)
~
2
t;L
+gH
..Vtq
Y'V
V1
t
the equations of
+1)
it4.W;L
z+
Y(L
IY
scs9+O
t +
wx
= dZ/dt
(r -f21a)/)=0
The boundary conditions aret
U
v=W
V
QUU
v= w=0at z
at r
0
S= Hp + grH
co(+
-Z +r
R1 and
=
ost)+
r
=
O,
aR
R+(r'(l4a2a/
J
[unless a=01,
+
. 2
dz /dt,
W
+
=0.
The last four boundary conditions all hold "atthe surface,
H + f r c os(e Sl t)+2(
z
.
(1+a')/2) +
Surface tension is neglected. Now scale t by Li,
u,v,
and w byefR, p byeRl
Also define E =2)/.(R
.
r and z by R,
and ?by EFR, where
F = f2R/g.
Then the non-dimensional
equations
of motion aret
r -ti
z- A-
+'
DII
_
)2
with boundary conditions of
u = v
V =-W = 0
u
w = -r
at z
= w = 0
at r =
sin(G +t) +F
0,
=
and r =a [# 0],
4F ur +
E[u cos (e+t) -v sin(G+t) +Fug +Fv
=0,
+
+
a.FEi> -
]
with the last f our equalLties holding at
z
F[r
-(1+a
)/2] +dr cos (e+t) + F' I +H/R.
2
'Expand the non-dimensional variables in the snmll parameter
E,
so
u
=
coefficients
u.
+
o(cE ), etc.,
Eu, +
of E to get the
and
separate
zero order (linear) equations
(3) and the f irst order equations (/).
zga
+v.xs
2.
*
Eg =
+
.
E IV
U,
4-,
V* a
O
u,=
at
=
atr=a
2.
1a
(!3
3]
(3)
~
The zero-order inviscid boundary conditions are
w0
u. =O
and p =
and w= -r
atr
at z = 0
=l1and.at r
sin(e+t)
at
the
-+F
a#
+ Fu
z = F (2r' -l
-a )A4 +HI/R.
The zero-order viscous boundary conditions are
u, =v,= o
at z = 0,
[
, = w,= 0 at r = 1 and r = a
and
= 0 at
tFL G-t+
unless a = 0],
4
z = F(2r'-l -
Zero-order Interior Solution
In
the
is negligible
E
interior,
experiments discussed later], so set E
[0(10
= 0 in
) in the
(3) for
the
interior behavior. The equations are linear, so only keeping
the driven components of flow, write
w,= w(r,z) sin(G+t),
v, = v(r,z) cos(G+t), so
p, = p(r,z) cos(e+t), and
u
cos(e+t), u,= u(r,z) sin(G+t),
(r)
%=
+ 2v
-M
=
-v = p/r
-2u
w-W
are the interior equations, with boundary conditions
w = 0
u = 0
p =
and w = -r
Equations (N)
at z = 0,
at r = 1 and r = a [+0],
-FT +Fru
at z = F (2r'-l -ao )/4 + H/R.
imply
w= -
which can
yieldt
u = (9-+
2p/r)/3,
v = (2
+ p/r)/3,
be substituted
into the
continuity equation
to
(S~)
with boundary conditions
= 0
+2p/r =Oat r= land
= r + F p/3 -Fr
/3
z = 0,
at
0 . and
3
)/4 +H/R.
at z = F (2rl-1a
By the usual separation techniques,
Z" (z) +
a [unless a = 0.,
r=
one solves
r R" (r) + rRI' + (Ar~ -1)R = 0.
The solutions are of the form
RZ
[J (\r)
=
-
(Ar),
Y, ( \r) ] c-os
kY'
+a2Y()
with A to solve
[aXJ' (aX)+2 J(aA)][AY (\)+2Y,(X) ] =
(X)+ 2J,(XA) ] [ak YI (ak )42Y,(aX) ]
[,'
a= 0, the solutions.are of the form
F or
RZ = JI (Xr) cos(A.7),
r3
with X to solve
NJt'(X)
< 0,
For
+ 2
'(F
the
there is
+2
no non-zero
for initial estimates,
by
) =0.
At
=
iX
which satisfies
= 0, since I, (x) = J, (ix) is monotonic. Using
Bessel function tables of
found
J (
use of
the
Abramowitz and Stegun (1965)
the first
five roots
of (6
polynomial approximations
to
) were
J,
E.E.Allen (1954) to be
2.7346426, X= 5.691402, X= 8.76658,
X4= 11.87525, and,
14.99735.
For
a
+ 0, there
is also such
an asymptotically
equally
mom
A's,
sequence of
spaced
corresponding to these eigenvalues
a = 0. The motions
iX
also one/=
4
a
for
There is
condition.
surface boundary
through the
gravity
influenced. by
will be
which
inertial waves,
are plainly
nT/(l -a), for
Note this asymptotic relation already holds
(1 -a)/n small.
for
fact, A,
and in
an additional
0, giving
eigen-
function
Z
=
[(gLr)
[a/K' (aA) +2K,(a/) ][/I' (I)
satisfies
where
( a)
. [a I
+21I (a/,A) ]I[/
of the graphs for
+2C]
K', (f)
of
the left side
at aA
always <
1, and very much
.
expand in
If one needs
the most practical
is seen
exponentially
away
Since sinh has but
1.329/a.
Taylor series around/4*
is
to
For
get
for only one given a, probably
(
way to
is
1, but the right
a ->0, /-
so as
is by
find it accurately
( () with tables. E.g. for a
this mode
plainly
Plainly also, the left is
so for a ((
2.113,
>1 for a(
3
as/I increases,
and/
~ 1.329.
)
until a
A-~Tz
K (s). By
(8 ) is positive but the right side
negative
1, one can
and sK'(s) +2
1 (
sII(s) +2
the values
considering
+
)] =
+2I(
can be found approximately by considering the shapes
This
a
(oshz
7)
K(
-
=
to resemble
from the
a Kelvin
walls,
one real zero,
1.81. From (7),
/A
0.9,
using
wave in
as well
decaying
as downward.
there are no
resonances
for this mode.
Each
of the modes
satisfy the homogenous
bottom and side
boundary conditionsy a sum is necessary to satisfy the
non-
nately, as well as being inhomogenous,
some
experiments, so
First
where
can be
used
the case
consider
it
necessary.
expansion is
sort of
Unfortu-
boundary condition.
surface condition
homogenous
as the
a
F
inseparable,
is
snall
expansion
0
=
is
(i.e., a
so
in the
parameter.
cylinder), so
Xsatisf ies
Using a Taylor expansion in F,
yet to
has
The surface. boundary condition
be
satisfied.
this condition is
--
+
+
A
(
at z= H/R. Perhaps such an expansion is questionable for the
higher modes (those f or which nF. > 1),
much,
studied
since they are
but these will not be
more susceptible to friction,
less effected by F, and of less interest anyway.
The coef ficients in the expression
f ound
by
a
satisfying the
Galerkin
technique
of
(
) for p will
making
condition (/D ) orthogonal to
functions J ( 4).
That is,
the
be
error in
each of
the
to make
ONO'
ul
+ F,
multiply each side by rJ (,r)
and integrate over [0,1].
The
20
linear system determines the response [A,
resulting
in which case there is a resonance.
the determinant is zero,
case
The
F
- _sin(X*)
:A
rJ(kr)J(r)dri
equations are
then .the
easy, for
0 is
=
unless
rzJ(a)dr, fpr k=l(l)1
=
Then using equation (6.49) page 89 of Tranter. (1956)
and noting that
t'Jl(ti),
(tJt))
the equations for F
sin(
At-
) J 7-(\)
for k=l(1)PO.
J(),
=
the determinant is zero iff -one of the coefficients
Clearly
of [Ai is zero,
and otherwise
pressure fornally
the
While
o are simply
=
depths (multiples of
H/R
=
the response
has a
1.992,
is
dense set
determinate:
of singular
..957, .625, .458, etc.)
one does not expect to see the higher modes, since viscosity
will damp them more,
-
especially since
1.32, -0.50, 0.30, -0.18, 0.06, etc.
goes rapidly to zero as
n increases,
so the resonances
get
narrower as well as shallower as n increases. The zero-order
solutions for F=O are thust
PO
____
=-_
\L~
c0
abbreviated
pA,
J(
[3J, (,\r
u
v
cos(9+t)
Ico
=[3JO
w
The above are
+J (
r)]cs(f
(Ar) -J( r)]cos(
Ji men
sin(
dimensionlessl
sin(G+t),
cos(e+t),
) sin(9+to.
dime ns iona liz
ing,
one has,
eg.,
24l
) +J
%3J (
U =
One expects the
so the ignitudes depend onIly on(R-*nd H/R.
qualitative behavior of the system for
for
F > 0, at least for strall F's.
determine how
much the
first few
F = 0 to carry
over
So, the main task is to
resonance depths, change
with F.
The
first
coefficient.
error
step
We write p
to solve
is
for
A% cos( X( z) J (
in the surface boundary
the first
just
r)
and rrake
the
condition (I/O) orthogoral to
J, (>r).
()J(r) -F
rJ(Ar)j A,Itan(
1FJ(Xr)
+
(2r -1)J(A,r)
=0.
dr
+FrJ ( r)] -r
system (/1 ) to the first
I.e., we truncate the
element
to
get
A,(
ri'
-F
S' i n(Nitf-) + F
fr
Numerically,
) + IF cos(
cos(
(),r) dr cos (k)
J
+F
cos(
)J
J
\)]I-
(1-i) =J
(
the integral evaluates to 0.130.
This gives a
first approximation to the singularity as
-1.576 tan(1.576H/AR)
-
F(0.459)
=.0,
or for F small,
~ 1.995 -
o.618F.
The coefficients required f or the Galerkin approximation
(i )
were
found
integration schemes,
using the
7-point
and
9-point Gaussian
to yield
A[-.2048s -. O597Fc,]j+A2[.0454Fc]
+A[-e.111F e
+.
A,[.o45Fc] 4A[-. 1898s, +.0 135F c,] +Ai[.0775Fca) +.
A 1-. OllFc1
to
-A.[.0775Fc] +A[-.1859s
3 +.0541Fca]
.3]
3
..
.
=
4805
= -. 1713
=
.0910
22
etcetera,
where s, = sin
First approximations
, etcetera.
to the second and third resonances are found by setting
the
diagonal terms equal to zero. This yields the above estimate
and gives for the second
for the first harmonic,
(H/R)
(.071F)
= Tit tan
or
1.915 + .072F, etc.
.958 + -. 072F,
(H/R)
The first approximation to the third harmonic yields
(H/R)
=
1.621+.291FW,
1.241+.291F, l.862+.291F,
To get a second approximation,
of
ically, to
avoid
difficulty
trigonometric functions at
m7-t
are no
found
the first
longer independent
=
the error
disparate
the
approximation to
the
Note the corrections to
of n.
Carrying out
the
.145 yielded the estimtes
(H/R
1.905
(H/R
.968
(H/R)
given.
combining
to improve the estimate.
computations with F
where
in
above .numer-
minor
One may use the values of the other two
trigonometric terms.
harmonic
one can find the Zeros of the
whole third-order
the
determinant
etc.
.67, 1.25
estimate is about one
These are compared
in the
last place
with experiments later, and
are
to be excellent. certainly within experimental error,
and definite improvements over the F =0
0 estimates.
23
Zero-Order Boundary Solutions
While E
throughout
very
snall allows viscosity
fluid, the high
most of the
-
to
ignored
be
order terms become-
important in regions of. relative height E2 from the top
layers,
) in the usual fashion for boundary
Rescaling (3
bottom.
and
the boundary equations
( z
=
o(E-)] aret
(u-i)
-
0~
0, w,= -r -F
v
u=
=
w
=
0 at
-Fur
at
0 for the top,
=0 for the bottom,
and the solution merges with the interior.
These problems might
problems.
be called
time-dependent Eknan
The top boundary layer requires
layer
and
to
top
change by O(1) across a distance o(Ex), which means the
boundary layer will only negligible change the stresses from
the interior values.
resonance
will
not
In any case,
show
satisfy the free stress
of
the
surface.
vanish,
like the
resonant mode
up,
the strong motions near a
because
they automatically
condition, for the normal
velocity
zero at
the mean
is by
def inition
Then continuity causes the
rest of the stress
-to
f or the nornal derivatives of u, and v, will behave
second derivative
of the
norrral velocity,
which
vanishes at the mean surface.
the bottom boundary layer, u, and v,
Across
change
the stresses,
by O(1),
so must
and hence
be considered
more
explicitly. In- the same fashion as the steady Ekan, problem,
i, one gets
and writing
24
AA
which has characteristic
as- +*"" excludes
Boundedness
parts,
.
roots +(l-i)/C and +(l+i
the roots with positive real
so imposing u = v = 0 at S = 0 gives
-+(
++
3
and imposing continuity and W = 0 at
I= 0 gives
rl 0 J-4
+;
17(fA
W
+
term due to
Note the extra
(modified)
Eknan
non-steadiness soon swamps
convergence,
as
one
proceeds
the
into the
interior.
Restoring the factor e
and taking real parts, one
has in the bottom.boundaryt
Ft
u
0
b
in
o
.
(IfFeeding
this transient Ekn~n suction back into the interior
had only the effect of rotating the pattern by E ,
allowing
the transport of momentum to counteract friction.
Equat ions
(/5) will be used in the first-order forcing in the boundary
layer.
First-Order Mean Motion
The f irst-order equations of motion are
Da
C a-&vdo
-a
+E (
v
,
-
-=
+
+V
o
t
W
+~
W
0
.
+V,--;
-;)U
V2
-
A.,.
w,
v=
u
0
The
motion
mean
3= 0
at
w F+Fu~4
+Fv
wd4 =
and at
r
+1 ,cos (G+t ) +Fu r
r
only
will be
1 and r
=
a [unless 0],
at z
at z1tt)t
considered,
denoted
superbar. Since 8 and t only occur in the combination
a time average is also a 9-average.,
i.e., a steady state has
u
+
--
.V
7)
)(r,z)
77L~r,z)
(e+t),
Write
been reached.
=r,
by a
=
u
These are known
from (r),
though
cumbersome,
except
that
(r,z)
=
0
from sin(G+t)* cos (0+t
in the interior,
0
etc.
The first-order mean equations aret
2V + E (
(rz
-2G
(r,z
-
+ E(7
+ E ew
(r,z
V
u
-
at z = 0
0
w
)(w,
c os (&+t) +F
=-(r
-Fru,) + F
-F
+ucos (t ) -vsine(+t) +F u)
at z
=
H/A
+FV
=
a [unless 0]
+F ru
=
Fru
+F (r 2/2.-suggests splitting
the
interior and boundary layers.
The
Standard boundary-layer theory
into two partst
problem
1 and r
and at r
interior equations aret
(r,z)
+
=
0
-2
u
Ot(r,z)
where
w
=
conditions
other. boundary
to Match the boundary solutions.
In the surface
boundary layer,
by
at
trivial
H/R +-2
and
0
at
it
z =
most O(E").
solutions,
f ound that
was earlier
Thus
to
O(E
equations
),
for
conditions, so the boundary conditi-on
and
N-
(22) below
free
wI
=
Fr
change
have only
stress
surface
effectively
27
holds
at
the
top
of
the
interior.
=
0 from
the
interior,
=0
u
then implies
In
(/7,j),
which with the top
boundary
condition implies
W= 0.
between (fl)
Eliminating p
and (113) gives
but leaves v, ambiguous by any axially symmetric geostrophic
flow.
Since we are particularly interested in such,
inconvenient.
Of course,
ambiguity in the
that
one could also say there is such an
zero-order flow,
any non-driven
this is
flow will
but there
decay from
we can
argue
Ekman friction.
Here that is not evident.
The cases of
principal interest
are resonances,
the nth component of the formulas (/S)
(3 J,(
)+J, (r r)-J,
dominate, so
-
sin"(
f3J, +J, )(J
s in (e+t)
) cos(
+(3J, -J)(-4J, + (3 3,-3JD/Xr) cos' (n
+2j, (3J, +J
when
) sinz (e+t)
) Cosa (e+t)
de,
-J, /Ar) sin(2z/3~) sin4(9+t)
+(3J,-J )(JI / Xr) .sin (2 .,z/f3~) dos'2(9+t)
+J
~
2-sin(2N,pZ/\(3)
f 24(3J, +J )(J
s in
+J /
+2J, (3J,+ J)
Since
the
nth
resonance is
(6+t )
r )
de.,
(3J, --J )(3C J
J
-+4J)
sin (2Nz/r)
stich
that
sin(H/iR)
=
0,
cos(
I,
0H/jR) =
and
the vertical average. of
Thus,
V
(o+,r).
=
Let's go
to
the bottom boundary to get this value.
f or (U,w )-rolls
we look
Physically,
in the boundary,
v, is coupled at the tops
driven by radiation pressure.
of
these rolls by Coriolis force. Suppress writing r, and apply
scaling of the usual boundary variety to get the mean firstorder flow in the boundaryt
(+ =05
0
V
U andw
be found
can
and
Now
+0
at
from the
analytically.
same
the
so one should
linear form,
)
for
boundary layer analysis, and equations
behaved
y=0
result (IS)
of the
(22) are of a
well-
be able
to solve them
Several months of effort were convincing that
answer
never
will
recur,
so
the
easier, and
therefore more reliable, technique of numerical solution was
taken
up.
Unfortunately,
this turned out to require a good
part of the floating point software for the PDP-1
so
also took several
months,
but at
computer,
least the answers are
reproducible.
The system (2)
is a
two boundary value problem on
an
infinite interval. Two methods come to mind, namely shooting
and relaxation. The first is more convenient for determining
which allow solution. Ref ormulate the w--
the eigenvalues
boundary conditions via the continuity equation
0
so
j
=
f'd
=
S/r,
=
i.e., no vertical flux out implies the total horizontal flux
is constant.
For a cylinder, this must be finite (in fact,.
zero) for r
0, so the boundary conditions and equations are
+ =c
a(0)
L
v~(0)
=
/bnd
+ 2T,
-2'~
)$+c=
+
0
=
0
>>
for
,iTdf =0,
is written c
where
and
to emphasize its independence of
conditions on the
five boundary
by the
is determined
,
fourth order equation. This system can be numerically solved
fairly
easily by simultaneously
=&+c -2 y3 , 3=
=y 2 , y
y' =y,-+c
10
= )O
13
0 for
=
~
=
error.]
-+2yy
+2y
e
.
1 to 15 except
infinity
autonatic step-size
=
y , y
IVJ,
n
y
=
yr,y
-2y, I , y=
One integrates this set to
scheme with
y;
+c'-2y, ,y'
y'=yy
where y(0)
computing three solutionst
y&(o)
=
y (a)=
1.
(using a Kutta -Merson-
control to
keep down
the
and checks whether the other boundary conditions -can
be satisfied.
They can be if
there is a linear combination
'U
= W = 0 as
of the three solution vectors which satisf ie
->o,
which is
so if
y
the determinant
y)
y
30,
If
a
bisecting the interval within which
by
yalue for c,
better
search for
the determinant to
the value of
not, use
of
c,. q at the outer edge
Given
the zero is known to lie.
the boundary layer is given by (4 +c)/2, and is plotted for
had already
and }t were found
taking
computer
much
with error
evaluated
matons of E.E.
nential
the
over. 9
were
formula, and the vertical
central
differences.
having to find
hit
necessary while
growing
2x10
of
using
the approxi-
error less
by
were
functions
Hastings
approxinated
expothan 2x
The
(1958).
a
twelve-point
derivatives were approximated
Using
the determinant
the actual initial
the boundary
Bessel
evaluated with
approximations
as
algebra
The trigonometric and
(1952).
functions were
using
integrals
Allen
The
less than
hand
The values
straightforward programs
but
time.
values
asymptotic
little
with as
long
meant
which
possible,
10
their
reached
effectively
4 and )4
Fortuna-tely,
then, so the limitation was not serious.
before
of
( 112.
the integration to
limited
equations
to the
possible solutions
exponentially growing
the undesired
by
above avoided
conditions necessary
conditions at infinity.
shooting, for
a
The
origin in each case.
vortex occurs at the
first-order
that
Note
(2).
in figure
three resonances
the first
to
This was highly
exponentially
solutions rendered the solution highly unstable, as
numerical experiments confirmed.
Thus,
to find the
actual
Figure 2a.
non-dimensional mean azimuthal velocity at the top of the
viscous boundary layer, for the first
I
resonance.
1.5
1.0
0.5
00.2
Flaure 3a.
0.6
0.8
-r5. nondimensional radial mass flux in the bottom boundary
II
Figure 2b.
Second
resonance.
0.5
0
05
8
Figure 3b.
Second
resonance.
4
0
0.5
1.0
Figure 2c.
Third
resonance.
10
Figure 3
Third
resonancE
3*
r-
0.5
1.0
velocities' in
the
boundary
necessary.
Now that
relaxation
technique
another
the eigenvalues c(r,A,)
may be
used.
technique is
are known,
The same
a
programs to
4. and M were used, along with the usual
compute
order
layer,
second-
approximations to the second differences and boundary
(23).
conditions in system
nating
Liepmnn
directions was -used, with
for satisfactory convergence.
guesses,
relaxation in
alter-'
a visual display to check
Starting from random
initial
the convergence was slow due to what appeared to be
a close analog
to slowly-decaying geostrophic
oscillations
with sweep number in the place of time. Over-relaxation just
increased the
frequency.
So,
a
srrall amount
of
damping
(slight increase in the ragnitude of the middle coefficient
in the differencing scheme) was introduced and then
to
zero itself.
This very
effectively killed
relaxed
the oscil-
lations. The resultant non-dimensional radial -mass fluxes in
the boundary are sketched in figure
resonances.
rrass flux
Features of special interest about the depicted
in the
boundary layer
somewhat distorted
occur
(3) for the first three
in n gyres
Ekran
bottom into rings.
spirals and
for the nth
ring vortices, * will
are that .they represent
result
the
resonance.
in sweeping
(closed)
flows
These gyres,
anything
on
or
the
They are sketched in figure (3) for the
first three resonances.
Shallow Water
The above treatmnt has been focused on the resonances,
which occur for depths comparable to the radius. While these
are
the most important cases,
the limit of shallow water is
also of interest, especially since it
is easier, at least if
one takes RF/H = 0.
the sums over Bessel
While
do not converge rapidly, so it
functions still
is easier to start over.
fact, to compute the Lagrangian drift later, it
Proceeding as
coordinates.
use Cartesian
z by H and hence u
scaling
hold, they
is easier to
before,
~~2-VO
with boundary conditions of
0
r = land r = a
at
[+0],
w, =0 at z =0
w0 = -x sin t
-y cos t
at z =1.
The solution is
w=
u
v
=
=
-z (x sin t
(-4xy cos t
except
and v by ES.R/H, the zero-order
interior equations are
u
In
+ y cos t)
+(14y'+lox'
((L4xi + 10 y'-10)cos t
-10)
sin t)/16,
-. 4xy sin t)/16.
Thus,
there are no resonances.
The structure of the
bottom boundary layer is
needed.
The equations of motion there are
__~~.2k~O)-ZL4
with
u,(0)
=
v,(0)
0 u,
=
u,(**), v, +v(o> ) as [+"
It
easiest to solve this by the method of undetermined
cients,
since we know
from the form
seems.
coeffi-
of the solutions (/f)
before that
sin(tAe
v(x,y,t,1v)
-v(x,y,t,oa)
4G e
=
5) + D
Ee sin(t+)
B cos (t-5),
+e.cos(t.-
sin(t- F) + He
cos (t-s),
for coefficients which are functions of x and y only.
Since these
solutions must satisfy the equations of motion (25),
F = A, E = -B, H = -C, G = D.
Four more are given by
u(0)
=
v(0)
=
0,
which imply
16u, = (-4xy) cos t + (14y'
+(10' -12r
j e-;O
s in (t+
=(14x'
+(10-12r2)
+ l0y2 -10)
Cos (t+'.)
sin t
.g)+2
(x _y' )e 'pssi n(t- i
+ 4xyecos (t - (f
16v,
-10)
+ l0x'
cos t
+4xy e
-
,
4xy sin t
s in (t-(O) -2 (x -y' )e
e os (t -()
These give, upon rescaling w, by Ej as before,
16
= -16( au. +
-
)
=-16(y cos t + x sin t) +24e '[y cos (t+,k)+x sin(t+36)]
i'8e
[ y cos (t-rt3 ) +-x sin(t-6v )]',
37
which can be integrated to give w .
be
formed numerically
evaluated
anyway, the integration nay as well
and y derivatives. my be
The x
also be numerical.
Since the stresses must
exactly
by second-order differences, since there are only
quadratic coefficients. Then, one shoots for the eigenvalues
for
the boundary
equations as before,
outer edge of the boundary layer.
,(r,0+)=123 r
This
one
allows
to
determine
velocity
azimuthal
measurable, and
velocities
has been
are so slow
the difference between
integrative
above.
point
tracer
0
is
mean velocity
is independent of 'z.This
be
to
Heyer (1967).
[O( 0')] that one
the Eulerian
experimentally
However,
the
needs to consider
the TlAgrangian mean
and
=
NOf r3
ought
by
Vat the
The result for y
the Eulerian
throughout the interior, since v
mean
and gets
velocity of
an
mean velocity given
The Lagrangian velocity of a particle originally
at
a is v(alt), and is the Eulerian velocity. u at the
current location of the particle, a +Sx.
Expanding in E and
using a Taylor series,
ev.(a9t) +gt v, (a9t) = eu.(a+Jx,t) +2u, (a+x,t)
u,(a+E fdt
+O(E7),t) +u,
=
u.(a,t) + eXu (a,t)
so
we
have the
A
formula
+
given by
(a+O(),t)
u$dtO
Longuet-Higgins (1953)9
, dt oVu .
V = U+
u 0 ) +(
2
The latter increment to the Eulerian mean velocity is easily
found from formulas (241) to be
r'
r- + 5r/8
in a tangential direction, so the mean lagrangian tangential
velocity, after dimensionalizing, is
(+.79I(r/R) - .74o(rAd I(
20
38
Side Wall Boundary layers
far,
So
This is
ignored.
derable
with V here as it
8=
u = 0
zero-order equations
S.
Sr-l), and expand in
given as
are
write
layers,
boundary
side
the
Considering
The continuity equation and
at s = 0 force rescaling u by 2 also.
terms do not enter the equations
interior
consi-
not couple
force does
the Coriolis
been
does on the bottom.
The full viscous
( 3 ).
to a
because they are ignorable
extent, since
have
side-walls
the
of
effects
the
= E , so
until
in
The viscous
the
equations hold outside an E layer, and there is no
need to consider an E or
E layer.
When
g=
E, consider
0-0
A4w,
with
u,= v
0
w
at s
0
and allmerge
withthe
interior.
Since p is independent of s, the v and w equations
simple diffusion equations, and
A cos (9+t),\)=
E sin (G+t) as
the boundary layers)
-
,
are
so writing v=
the speeds just
outside
WO9
The continuity equation may be inte rated to give U
If
now consider
we
zero-order E
we
the
velocity, the
(steady) mean
layer above will reflect in the forcing.
If
subtract the interior forcing from the mean velocity, we
consi-
side-wall problem
the steady
will have effectively
since the E 2 deviation forcing will
dered by Howard (1968),
act as forced mean velocities
(ii,v w) at the outer edge
of
the Ellayer, which can be taken as the inner boundary for an
E
layer.
Howard shows how this E
layer will allow "i to match the interior.
and wi, while an E
0 at
E.g.,
layer will 'balance out u
r
1
in figures
1
is
no
difficulty.
Possible Shear Instability.
The
last
section
tangential velocities
shears.
showed
that
there
will
V, in the interior, and
be
mean
consequently
Thus there is a possibility of shear instabilities.
the cylinder
Since
sinuous flow
and the
is rotating
will not be rapid, the Taylor-Proudan theorem will hold for
the
Thus,
perturbations, even if the mean shear does depend on z.
argument shows that
far
for such a
lateral
dominates
system, the Ekmn
latter
the
so
friction,
and
a scaling
Now,
as two-dimensional.
motion
the
consider
with respect to z
v
is reasonable to average
it
friction
will be
neglected. This gives a barotropic shear problem which seems
to
relevant
more
geophysical
than
dynamics
fluid
the
classical shear problems.
The perturbations are nearly non-divergent:
where the vorticity is defined by
V4
DV
The Ekxran vorticity equation is
where v has
been rescaled
to 0(1) in energy measure,
the
Rossby number
Ro- =
with
A,
A~Vgr
-
amplification factor
the
equations
v2 0+,r) dr
near
and Z the mean vorticity
(12),
resonance from
a
+
.
Since the
perturbation flow is nearly non-divergent, write
u+
Substituting
E u
,+
Ev.
into the vorticity equation
and back into the
continuity equation gives
u=-
+ E'I
I =
V2
+0(E), v
+ 0(E).
+E
+0(E),
Substitute these into ( 0 ) to get
-C
T
+
V2-4.-
-
+ ;
D(),
) suggest
Both experiment and balance of equation (31
R
is 0(E),
t in (31
'= E
=SEL and
so writing R
)and
dropping terms of O(E
- -S
For growth
troughs,
so
of a
(32)
f
e l
one needs
shear instability,
separate
one cannot
6 easily.
r and
rotation has dropped out except in E, one looks for
SV
(
Since
insight
cartesian equation
in the corresponding 'f-plane'
with
tilted
X(32C)
2)
Figure (
periodic in x and y.
at several
of
resonances suggests the
v2
cos x
is an appropriate cartesian form.
can
be found
from any
admissible
Since this is a barotropic shear
requires
'tilted
troughs',
and
An upper bound for inf
q1 whidh gives
problem, a good
one
> 0.
estinate
on physical
expects
grounds that an excellent appraxination shouldc.ome from the
trial formt
sin x
sin x
0
sin(1y -x),
s in(1<y +
x -2e F),g
periodic extension,
For this trial form and v,
'
<
x
<
x
other x.
S
T)
<
2ff
q1+v*f
=
(k'+ + 1)V'1
S -
f'k2
=
_sf V.V
so at nairginal growth,
S= 3(
which has a minimum of S = 3 W
at
k =
=
oo
much shorter
.
The latter
than
x-wavelength imposed
the
by
mean
the
This corresponds to the experimental observation of
shear.
wave-number about thirty around on
the
or G- wavelength
implies a y-
the second mode, and
of the troughs which develop.
Mrked tilt
to
Returning to
the dimensional form, there will be instability if
but not if
the
A,' s
much below,
equations (12)
etc.,
( 3
unless another resonance is active.
by
given
are
>
v(0+,r) r dr (AEz) E
2
infinite
the
inverting
near the nth resonance.
sin(1.65 H/R )I
A = 0.90
sin(3.25 H/R)
A = 0.49
sin(4 .7
to 0 for the coefficients.
of
From equations (12),
A = 2.34
where F has been set
set
-
(3)
H/R)
to .145 for the resonances, but
The error in the coefficients is
only O(F).
The mean square amplitudes for the first three v I s are
.33,
.44,
and .31
,
so the instability bounds are
=
.0591 sin(l.65 H/R)I
using E = l.5xlo~
.132
sin(3.25 H/R)
=
.29
sin( 4 .7
E is
greater
,
H/R),
to match the experiment.
than
any
is predicted.
instability
of
The instability
there
Thus,
increases,
dominant,
if
occur when
'shears may
so
above,
are
of
wedges
However, as the tilt
no one
be instabilities
there may
Note that
sides
right
the
instability reaching down to zero tilt.
even
I
predicted by (34a) are given in figure 4.
bounds
if
=
one component is
for
.Q R
large,
E is not large enough to cause one component to go
unstalble.
When comparing the experimental results given in figure
4 with the instability bounds given above,
experimental v 's are
not
in cylindrical
exactly sinusoidal.
Nonetheless,
remember that the
coordinates,
the
and
are
results compare
well enough to conclude that the observed instabilities
are
due to the vertically averaged barotropic shears, with Ekman
friction.
Eperimenta 1 V erif ication
is time to show that the theory developed so far has
It
some relation to reality. The turntable used was the 1-meter
in the
table
described in
Oceanographic Institution,
the Woods Hole
Iaboratory of
Fluid Dynamics
Turner and Frazel
This turntable is carefully engineered to maintain
(1958).
constant angular velocity, which is continuously
adjustable
over a large range. Tilting was accomplished by a hand winch
from a guard-rail to
the
which
turntable stood, allowing accurate measurement of small
angles.
14.
steel plate upon
the four foot
The cylinder
cm,
5
depth 29
used was clear
cm, and
plexiglass of
radius
circular and
quite accurately
get
A flat clear plastic sheet was used as a lid to
right.
rid of torque from the air.
The
flow *was
mde
television attached to the
waves
visible
with
dye
and
turntable showed the
dust.
A
zero-order
rotating backward on the rotating frame of reference,
but otherwise did not
havve sufficient resolution, and
was
restricted to one view, so was not used further.
A
desired
typical run consisted of filling the cylinder to the
depth
temperature,
with
hot
centering it
and
cold
on the
water
mixed
turntable as
to
closely as
feasible, then speeding the turntable to a fixed voltage
a
dial.
The
angular velocity this
room
on
corresponded to varied
from day to day, so the actual frequency was determined with
a
It
stop-watch.
was not
realized then
that F
would be
important, and setting a particular angular velocity was not
The fluid was allowed to spin up for at least twenty
easy.
pernanganate crystals were
minutes, then potassium
through
dropped
snell holes in the lid to check for completed spin-
up. The permanganate dye was used to trace bottom
motions,
soon as
and flourescent dye was
resonances],
used in the interior.
As
cases
of
completed [except
spin-up was
boundary
the turn-table was slowly
in three
tilted and left for
at least ten minutes, usually thirty. Then observations were
made,
mostly of
the ink
cases were followed
in the
carefully tilted further.
.
on the
bottom though interesting
interior. Then
the table
The results are plotted in figure
If nothing much was observed (besides the
periodic motion), a circled dot is entered.
were observed, a circled
was
R is entered,
zero-order
If rings of ink
with the number
of
rings, counting center dots and ink at the edge as rings. It
my be worth
noting that these
rings were not
due to
the
location of the dye crystals, for they nornally sharpened up
long after the crystals had dissolved and occasionally clear
areas formed over a crystal, except for its thin plume going
either in or out. At higher tilts, the rings became unstable
to wavy disturbances, with wave number
rings and wave numbers
entered
as
2 to 4 for
circled I's
with
30 and up for outer
inner rings.
the number
of
These
rings.
resonances, the instability was violent enough that
rings
did not
have time
to form
are
Near
visible
before powerful vortices
.-c
ANGLE
EAR
cm/sec.
1.19
.69
.59
30
(.86
24
I.66
1.45
1.24
er e'
o
ow
\0--
1.04
#2\
.12
V
\\-1
STABLE
2
M*
LIFT
fig. q
OF EDGE
0
IU
formed, which are entered as circled Vs.
ably
what
Fultz and
Crow observed,
strongly near the f irst resonance.
that
these
vortices
vortex at the
V4,
are not
These are
since they
presumform most
It nay be worth
necessarily
origin derived upon
the first-order
finding the
though their continued existence
noting
theoretical
may depend on similar
causes, because these vortices form from the instability
the
inner ring when this grows
detail.
Occasionally
develop
cusps which may grow
a ring
slowly enough to observe in
of ink along the
wall
will
and spread into the interior.
These are entered as circled
lift
on
E's and were observed at
of the edge at three resonances.
zero
They look as if they
might be caused by an instability on a shear a little way in
from
ink collected
the wall, with the
because it
is .heavier.
G6rtler-Taylor
flow up
However,
centrifugal
and down
over the
they
instability
in the
the corner
ray be
due
of
to a
the zero-order
concave corner.
This idea
is
discussed in the next chapter.
Looking
at the
general agreement
which
get
generally
with
weaker
convergence and
completed diagram,
as
the theory:
n
divergence
increases,
in the
we see
there is a
are
resonances
there
there
are
rings
bottom boundary,
of
with
one ring for the second resonance and two for the
third. The vortices (zero rings) go with the first.
Now let
us look at the quantitative predictions.
The
experiments were mostly
run about 30
rpm, so F =
.145. Thus the zero-order interior theory gave resonances at
H/R
=
1.905,
.968, and .67,1.25- for the first,
second, and
With R = 1.5
third resonances.
cm,
14.05 cm, and 9.72,
tilt
are taken
not
is
center
which
The widths and
being used.
region seem in
depths of the stable
its
like excellent
the experimental error,
to various F's
mostly due
seem
These
well defined.
13.9 cm, and
9.2 cm, though
also a wedge around
confirmations and are within
is
27.5 cm,
at depths
'of
the wedges
experimental tips- of
as the
There is
18.0 cm.
The observed E' s f or zero
18.1 cm.
and occured
instability,
this forecasts H = 27.60
cm,
agreement with the
(a
posteriori) prediction of figure
3 5 .4
radius
annulus with outer
experiment with an
rather crude
A
inner radius 21 cm was also tried on
cm and
The annulus was
similar turntable that could not be tilted.
but the invisible
not deep enough to try the first harmonic,
tilt
in the
automatically
considerable current
singly
-
and edge
to cause
was suff icient
table
a
vortices at
13 cm,
surpri-
close to the predicted second resonance at H =((35.4
21)cm
=
12.4
cm,
considering
the
crudeness
of
the
experiment. These currents and vortices were not observed at
so this was encouraging.
four other random depths,
A later
better
trial
exploratory
built annulus showed a
first resonance but not at
with a
smaller
but
much
few sluggish vortices at the
the others.
Since the
vortices
retained the same size as before (a few cm) they filled over
half of the gap, suggesting they found it
difficult to form,
since incipient vortices were visible to the desirous eye at
the second
interesting
resonance.
is
that a
However,
what
my
be
much
more
relatively strong meriodional circu-
Hg
lation developed
near
the
first
and
second
resonances,
though there was a lid, and spin-up had been complete before
tilting, ink in
corner
the bottom
layer flowed out
and from there, spiraled
to the inner wall renmined
outer
up and forward through the
interior to nearthe upper inner corner.
next
to the
However, the
water
clear while the ink formed
into a central and outer ring (for the second resonance,
H - (R,-RI
)0 /2
at
Such circulations could conceivably exist
in nature or in an experimental model.
..R=
f2 .f
cos at
--ed
Figure 6.
Sketch of basic Taylor aoparatus with oscillating inner cylinder.
Chapter Two --
We
now
consider another
pressible, laminar
involve
Periodic Taylor Problems.
similar
problem of
f luid motion
Mathenatical
periodic,
incom-
in an annulus which
tools, but
mentally different.~ This problem
which
will
is funda-
is a variant of
Taylor's
[1923] classical problem of centrigugal instability when the
inner cylinder
rotates
of
enough
effects if the
expects
a concentric
faster than
(as in
the outer.
rotation speeds depend
centrifugal instabilities
centrifugal potential,
particularly
pair
strong.
or
figure
What-will
on time?
if there
perhaps when
limiting
with
case of
still
the
potential
is
With a view toward Fourier integrals,
chapter considers periodic
cylinders,
be the
is enough mean
the obvious motions to consider are periodic and
This
One
)
6
special
impulsive.
torsional movements of the
attention
to
the
interesting
pure torsional oscillations
of the inner
cylinder while the outer cylinder is fixed.
Literature Review.
The
zero-frequency
(steady)
problem
of
centrifugal
)0
Figure 7.
1
Critical Reynold's number Re =
as a function
of gap width for the steady Taylor problem with
/ 000Cr
I
I
I
I- Ii
i I
I
2000-
500-
100
20(R,-1)/R
.02
L-
.05
I
1
1
.2
1
I
I
.5.7
1
I I I i IL
0.
instability was started and to a. renarkable extent f inished,
by G.I. Taylor (1923)
It
both experimentally and theoretically.
has been pursued extensively sincel thorough reviews nay
be found in Chandrasekhar
(1955) and Coles (1967).
Coster
(1919) considered the two-dimensional flow around a torsionally oscillating cylinder.
experimentally
and f ound
Winny
it
(1932) -tested the
good for
theory
< 600.
Re
Ring vortices were apparently first observed for oscillation
of
the inner
cylinder by Fage
(1935),
theory nor critical parameters.
considered
the special case
(
but
he presents no
Meister and Munzer
(1966)
3 + Esine)t, .
=2 for
=
narrow gap, using Galerkin appraxinmtion, and solved numerically f or
E = 1, and
energy f or f ixed
Carrier
t to be
and
They f ound the kinetic
DiPrim
(1956)
than f or
consider
by expanding
W =0.
the. torsional
in the
oscillation
However, the resultant mean -flow should not be
amplitude.
cylinder,
0 and 10.
less f or W =10
a sphere
oscillations of
considered
=
t
instability
an
like
the
where the generators are
vortices
around
a
parallel to the axis of
rotation. serrin (1959b) used variational techniques to show
that
if
flow in a
gassner
the Reynolds
volume with periodic
(1960)
gives
Rayleigh's principle,
stable for all
does not cover
Crimina le
number is below
t
forcing is stable.
time-dependent
Kirch-
extension
of
that an inviscid flow is centrifugally
0 if 5+
the case of
(1965)
stability f or
a
a certain bound, the
consider
.0 and v(rt)
pure oscillations.
sufficient
axisymmetric vortices
in
O. Note this
Conrad
and
conditions
for
a narrow
gap
f or
torsional
oscillation
of
one
of
the
without a superposed steady mean motion of either
They
use
(19590). variational
Serrin's
f or the critical
several lower bounds
functions
of the
shape of the
with or
cylinders
cylinder.
equations
to give
Reynolds numbers
basic flow,
assuming it
quasi-steady, perhaps relative to a changing amplitude,
always
being
locally in time.
local
a
property
is
but
They seem to
regard stability as
of
flow
a
fluid
rather
than
in time or as a bifurcation of solution f orms as
asymptotic
functionals of the
number
as
parameters.
However,
if the
Reynolds
below the infimum over a cycle
of the basic flow is
of the Reynolds numbers curves they give, then the energy of
a
perturbation must always decrease.
For steady rotation,
their critical Taylor number is about two-thirds of Taylor's
=)0
for
but is a
principle for
cylinder,
0 <
0
definite improvement on Rayleigh' s
.
they get the
For pure oscillation, of one
.
curious
result that
the critical
Reynold's number for the outer cylinder oscillating is lower
than for the inner, though unfortunately they used different
Strouhal numbers (or angle
also
find
of swing.) Conrad and
that superposing
an
oscillation on an outside
rotation lowered their
Reynolds number bound
frequency
Superimposing an
increases.
inner cylinder's mean
large
a lot as
the
oscillation on the.
rotation also lowered
the bound
for
enough amplitude, but actually slightly increased it
for. small amplitude.
outer
Criminale
cylinder
oscillation of
They also
lowered
the inner
the
find that rotation of
Reynold's
cylinder.
number
While
the
bound
for
these are
for
S-3
bounds, Donnelly (1965) found that small oscillations
lower
stabilize
the flow to
meant by this
centrifugal instabilities, though he
that the
torque does
higher mean
until
can
cylinder
the inner
rotation of
superposed on -a mean
not increase
strongly
though periodic rolls appeared
even
at lower mean
Notation and Discussion.
As sketched
in
6
figure
,
we
consider
concentric
cylinders with inner radius R1 and outer R2, with
and d = R2 -R .
R,/R
=
The angular velocities of the cylinders are
(t) and .Q(t).
Def ine non-dimensional parameterst.
dA,
(3
Re=
E
N
where
is the angular frequency of oscillation, and
W
122
S+Q
This
u
As before,
.
thesis
endless pages in
has
a
=
general
of
policy
attempting analytic
equations in which the
inmte
dr/dt, v = rde/dt,
w
=
f
=
dz/dt.
not spending
solutions to approx-
effect of neglecting terms is
unknown, when an approxinate solution to the exact equations
can
be found
more easily
this, consider
fornalism
mining
the
the
numerically.
As
extensive bibliography
an example of
an<d
impressive
devoted by Taylor through Chandrasekhar to detercritical
parameters
for
the
steady
Taylor
3-9
problem, i.e., determining the minimum Re (or Taylor number)
and associated wave number a for which one ny solve
with u
=
V = 0 at
A = (D.-)/(i
I),
where
B = R'
produced only
1923),
were
thrashing
around
the asymptotic
the values
.
-'
(A -A)/( -'f).
classical
The
and r = 1.
r =
for
values
=
While testing a
with
this
as
problem
+ 1
(Taylor,
(Chandrasekhar, 1955), and
program used below, these
reproduced for the special case
that the values for other
has
2.0
values
Upon noticing
had not been done, sufficient
other points were run off (in about half an hour) to produce
figure
for
d/R
7
,
=
critical Reynolds number for .Q
of the
.02 to 0.9.
The
=
0
same program could have worked
out Re for .. +g0.
Initial Value Problem.
The
equations of motion are already given in equations
( I ) where the boundary conditions are now
u
V = w = 0,
V =SIR, at r =R.
u =v =w =0, V =.QR, at r
everything is periodic in z, and the azimuthal velocity
V(r,t) + v(r,e,zt),
with
v d9 dz
=0
defining V.
is
3:5Following
Chandrasekhar (1955),
the equations of
linearize
wave number as k, assume axisymnetry, and
motion, take the
eliminate p and w to get
(-YT
k
(3d)
(,
v
with, boundary conditions as above, and where
D = ,C
and
and
D*
G
the variables are functions of
dimensionalize,
g+ 1/r,
r and t only.
Now non-
so
k =a
/R
, r
+rR
, t +
t
where 0 is the oscillation angular frequencyt
9,
0 at
~AJ-~-~-\
2
~rev
S~,
v = A CD,V) U,
0
)
with u = v =Du = 0 at r =1 and r=
y n(t/lat
r =
Vy=D~)^
(37)
,
at r=
~
that the main f low V
Noting
(since linearized),
independent of u and v
is
worthwhile to solve it
it seeis
first,
so try the problem with scaled boundary conditions
V(1,t)
sint t,
=
V('Pt)
V(r,t) = 0 for 1 < r
and initial conditions
This
turns out to be easy to solve with the aid of a Hankel
transform.
Multiply both sides by
- Y, (pr) J, (p)]
rB, (pr) ~=r[ J, (pr) Y, (p)
where p is a positive
over
= 0,
r,
root of B,
(pr) = 0, and
integrate
solve the resulting ordinary differential equation,
and invert, to get
(32')
As t + oC,
this becomes strictly periodic.
solution
of a diffusion
the interior
1 < r (
denominators
J (p)
equation, so cannot
which
so
J(pI'')Y(p) = J(p)Y(pf^),
of Y(x) = Y,(y) and
graphs
graphs were sketched,
positive
makes
J (P f
-
This V is
ever
zero.
equivalent
if
the
We
have
to
the
J,(x) = J,(y) intersecting.
and were
(x,y) quadrant
is
this
have poles in
one curious
are
found to
the
These
occupy the
whole
in great. wiggles, . but to cleverly
avoid each
other, avoiding
form (
The asymptotic expression for V can be summed to
).
a closed
first
V =
but complicated
order,
cos (t)
e.g. ,
[ -ker,
for
(i)
any difficulties
form in
=
kei, (E-)
in the
Kelvin function
0
(no
outside
+kei (E1)
ker,2(E !) +kei'(Et)f
of
the
cylinder]
ker (9 )
+sin(t) [ker (6:1*ker, (E4) +kei (n 1
a
given
ke
.
(
0
Thus,
for
2V=
0,
=
ker (e ) +kei (/E6-{ker2(E~) +kei (
exp(- E
showing the boundary
-
layer structure
of the
decay due
to
spreading as well as viscous decay.
Numerical Initial Value Problem.
Since
there is no easy analytic method for solving the
non-dimensional equations
methods
for insight.
comparison and
order
( 3'
let
us
try
numerical
We will use three numeric methods for
exploration of
solution
),
to the
the best
initial
methods a
value problem,
a harmonic
system, and a (slightly disguised) spectral method.
initial
value
problem,
we
rephrase
the
For the
non-dimensional
equations so
IIV
(3)
where
=
!
u = D4 u= v =0
.v(1,t) =
(DD,
-a
)u,
at r =1 and r='
sh(t),.
v (y',t)
=At)
second-
The
man f low is
included here because it
easier to expand the
tions a little
turns out to be
than to evaluate a bunch of Kelvin functions,
though the latter are useful for checking.
second-order
difference appraxinations
interior, so using superscripts
out
of
K
equa-
system of ordinary differential
[so
We use the usual
to D and D4 in the
to denote network
rk= 1 +
-1l)
for
2+
k
=
position
O(1)K
and
-2a
k---0
ki~ +((iAA>~
4
AtA @5
where
A,4-
The boundary conditions
all for k = 1(1) K-1.
0
V =Q(t)
u =uK = v =~v =0 and VV =.,(t),
are easy enough, but the boundary condition D u
=
0 is
more
interesting. How can this be imposed when all of the u' s are
now defined?
free
One realizes the values . qA
and provide
(D -a ) =V
the needed
set=0an
two more
and g
are still
conditions, through
Now
impose Du
=
0; the
u
,
so
defines u
=
centered second order apprcximation
Similarly ,
additional constraints
These provide the
If
needed.
one
felt uncomfortable with introducing the artificial u- above,
only defining u
one might think of
then imposing the boundary condition Du
for k = 2 (1)K-2,
u'4,
as u1 = ut/4, u
together, thereby
causing
too
brings the
high
a
0
walls closer
critical
Reynolds
So much for
so one should use the centered scheme.
number,
=
and-
which then define
scheme effectively
this
However,
V def inition
from the
ten hours work and experimentation.
For the numeric solution, all variables were
were set with stall random
zero f or each run, then the
numbers.
Then over-relaxation was used
1(1)K-1 (with u* = u
'
and
1P/ were set as in (14
step could be
taken.
on (
I
) f or k
0]. When the error was small enough,
) and ( 13).
sides of (q04) through (9O,3)
right
with
=
initially
Ifw all of
the
were defined, so a time-
A fourth-order Kutta-Merson
scheme
autonatic error checks and step-size control was used.
The curves u(r),
visu.l
checks,
v(r), V(r),
under sense
that v had the same
similar
and
4(r) were displayed
switch control, so
it was seen
to be 21f), and
a
near the same radii,
A
period as V (forced
shape, with zero crossings
for
typical graph is given as f igure
always had
period
the
same
in which
i
The radial velocity u
.
Sid'n (after a
it
transient
occasionally
kept
a
adjustment
second
appearance f or a while], so had a zero-frequency
It
also grew twice
in each V
period.
component.
Superposed on these
cycles was an exponential growth or decay.
a
mode
In f igure
7 is
typical graph of the kinetic energies of the perturbation
and the
mean
f low.
Two
bumps --correspond to
period, due to the squaring.
kinetic
energy
periodic,
can
basic
Note how well the perturbation
represented
p(t)e
as
for
,
p
we thus 'have a picture of the motionsas the inner
cylinder is past
potential
be
one
in
the
the middle of
V
its swing, the
boundary layer
builds
centrifugal
up 'enough to
encourage u to grow, which advects V to encourage v. Thus, u
will
pulse on swings either way, whereas the sign of v will
change on opposite swings.
enough
If the centrifugal potential
is
to cause more growth than the viscous decay over the
whole cycle,
then
proportional
growth,
each
cycle
once
the
will
result
fastest
in
growing
the
same
mode
is
dominant, and until limiting size is reached.
The whole system was run f or various parameters Re, a
and
N relevant to the experiment with
later.
=
1.0444 described
The initial value problem was run out far. enough
determine
whether
there was
growth
,
or decay.
to.
While the
Figure 8'
Typical display of u, v, and V
for
1.0144, Re
during time integration
8000.
Ur
-
e
#9
I
16
10
Figure 9.
Basic flow and
perturbation kinetic
energies from time
integration for
Re = 83.3,
N
/064 4-V
/0
I
0.1.
0.5,
oscillations
were a handicap, fortunately only a few cycles
were necessary, except near marginal growth.
that
This indicates
the fastest growing (or most slowly decaying) mode was
always quickly
After
dominant.
growth was found. Then a
The
minimized.
over
represent
decay
which gave zero
results are
plotted for
one hundred
.
Re versus
The six points found
of computer
hours
N in
time.. The
seventh is the steady limit from Taylor's narrow gap
or figure
was
was changed until the Re found was
10 and tabulated in table
figure
growth or
until the value
Re was changed
discerned,
the
theory,
7.
For the smaller N's (higher frequency,
smaller angles),
the amplitude of the kinetic energy oscillations were small,
suggesting the pulses were rapid compared'to the decay time,
but not
very efficient.
oscillations
'on-off'
were over
N
( 1, the
kinetic
several decades, which
energy
is a strong
behavior, and suggests the feasibility of a
steady
appraximation,
ever, a
quasi-steady
for
useful
For
for N ((
1, as discussed later.
approximation
N > 1, though
quasi-
then
plainly
will
considerations of
How-
not
be
a mean-
square centrifugal potential nay be.
The critical Re was not sensitive to a
,
explaining the
rather large error estimates for a. The growth rate was much
more sensitive to Re for
closer
to given
small N, so the
finite amplitude
Conrad and Criminale (1965)
and N
=
1.94 if Re i
KE
for small
0 curve
is
N than large.
claim stability for
=
1.060
103 (after translating notation). While
++
UNSTABLE
t
STABLE
rC&C
0.0
1.0
2.0
ANGULAR AMPLITUDE IN RADIANS= N
OSCILLATING. INNER CYLINDER
this
is
the only
f or which they give a value, tfe narrow
gap approximation translates
f or
=
.o444 and
this to a
N = l.
also entered on the graph.
94
claim of
if Re < 163.
stability
This point is
Indeed, there will be
stability
below that Re.
Quasi-Steady and Rapid Oscillation Limits
When
the oscillations are slow enough that the viscous
boundary layer
7TfFi
is thicker
than
the gap
d,
one
realistic.
The
dimensional equations of motion and boundary conditions
are
expects
a
quasi-steady
given by (
).
.
where
u
9
Let us rescale t -S.t,
-~
1[ 0,
- R.,
R.
-
CDV it*
at r =1 and r
. at r = 1,
at r =
so
,
V(r,t) = M(r) sin Nt
steady'
so
= (DD, -2 )u,
V = sinNt
where
be
.2a.''Yi//
-t
v = Du =0
V = 0
assumption to
M and
K are O(1)
insofar as N Re
expansion parameter,
so
+ N Re$
and
S = d/R.
is snall.
Thus,
V is 'quasi-
Let us use this as
=q., + N Re
retain the zero-order equations
Ltt
(%fAE
K(r) cos Nt,
L
+....
an
etc, and
where
U
(DD0 -a:* )uO,
=
=
and r
r= 1
at
=v, =D.u, = 0
1, K = 0 at r = 1, M= K =0 at r
M
r'.
=
These are the quasi-steady equations., so one is
jus tif ie d in
treating stability as depending only on the current time and
is small, i.e.
not the past if N Res
Rejsin Ntl
=
limiting size in
,
which determine
the
rolls will 'switch .ont , reaching their
at which the
times
a ~ 3.16
and
F1705
has
If it is, we have the classical
of time to diffuse.
plenty
if the forcing
much less
than a
period of
oscillation.
They will die just as fast when Isin Ntl approaches its next
zero.
Of
course,
Re (
will
there
at
least
for
for
larger N also, since
with
rotation
same
the
.
it
,
instability
if
This guaranteed
probably
since it
seems clear
that the
be less destabilizing
oscillating forcing will always
steady
no
N ((
figure )0
stability line is entered in
holds
be
maximum
than
angular velocity
Currie (1967) suggest
this
may not be quite true]. But in that case, Taylor (1923)
gave
(though quasi-steady results of
a bound which for
This
this
is
line is asymptotic to
see from figure
boundary
Re < 4710,
independent of
N.
the results presented here.
We
10 that the experiments were conducted
for
layer thicknesses comparable to the gap width, and
merge smoothly
with
the
quasi-steady
prediction
to
the
right, for N (« l.
When
the boundary layer
thickness becomes enough less
than the gap that even the vertical wavelength is
less
than
the dependence of
d,
Then
q
(DD
=
(DD
the parameters on
-
-kl)f =2kOVv/r
(DD* -k')v
=
gives
scale of iou/2,-
4' scale
a
of
u, which
constant, much like the classical
S
Since
whereas
i-v, and
(D*V)u gives gives a v-scale of
combine to force
formula.
P
gives a
d should drop out.
=
.
this
can be
squared
to
yield
Re= cN
for
N large.
The asymptotic region
experiments,, f or the exponent of
at
the
right to about 1.9
N for Re increases from
at N
thickness of the viscous boundary
the
gap, and the wave-number
so the balances above
still
was not reached in the
=
1.5
at the
left.
0
The
here is about a third
of
has just started to increase,
have factors
of two and
three
depending on d as well as
Mean Rotations with Oscillations Superposed
With
the equipment available,
an obvious extension of
the theme of time-dependent centrifugal instabilities was to
superpose mean rotations of the outer and inner cylinders on
the
oscillation
of
the
inner
cylinder.
This
seemed
attractive in connecting to the steady problem.
Donnelly
(1964) experimentally studied
stability of steady
circular Couette f low
effects on the
of a
superposed
oscillation of the inner cylinder. Meister and Munzer (1966)
and Conrad and Criminale (1965)
studied this problem theore-
('-7
4.
.
1.
t 152.
__
0.5
fig. 11
oscillation stabilization vs
0
-
1.0
ga p widt h
tically,
and
agreed
oscillations
can
experimental
points
with
Donnelly
stabilize
of
the
that
mean
onset of
flows.
sharp
Donnelly' s
increase
transmitted torque are replotted in figure
is
small amplitude
1.
-of mean
His abscissa
inverted to show the number of viscous waves in the gap.
This way it seems clearer that the decrease in torque is due
to
the correlation of the phase
outer and inner walls.
has
about
the
same
It
of the viscous wave at the
even seems clear that the
shape as
the
viscous
wave,
effect
and if
Donnelly had tried higher frequencies, he would have found a
weak destabilization.
My
observations of the onset of instability werevisual
rather than by
existence
velocity
of
average torque, and
ring vortices
exceeded
the
indicated the 'periodic
about as
critical
for
soon as
the maximum
steady,
except
for
oscillations too fast to be quasi-steady. These did not show
up much before the instabilities on the mean flow.
The claim here that
rotation
always
destabilize
Donnelly's conclusion,
instability
the
for while
involves
not
the
mean
contradictory
to
Donnelly's definition
of
existence
in the torque, here
of
a
different
solution to the equations of motion and boundary
conditions than the basic
showed
is
involves a sudden increase
definition
asymptotic
oscillations superposed on a
one.
the existence of ring
Donnelly's own
experiments
vortices with a non-zero mean
radial motion below the steady Taylor number.
The periodic ring
vortices only appear
on the
latter
periodic
velocity opposes the mean
above
lations
suggest
happen if their
frequency
natural
modes would grow,
inviscid
possible
oscilresonant
gradient is
stable,
What will
oscillations.
will be 'inertial-elastoid'
there
This and the
frequency
interesting
an
mean centrifugal
If the
instability.
velocity.
the proper
for
possible destabilization
back, when - the
not on the
f orward swing, and
half of the
is driven?
Clearly,
but viscosity would oppose this.
Since the mean flow here is forced by viscosity, it would be
interesting
if
a
could
resonance
an
at
oscillation
inner
a
The possible
instability with viscosity present.
requires
force
twice
harmonic
mechanism
natural
the
frequency, so that the forward pulse may correlate with each
outward
swing of the mode,
the end
of the
swing.
and the backward decay ray match
that there
One suspects
considerable
difficulty in.getting
for the high
Reynolds number
will
be
this mechanism to work,
necessary to
avoid too
much
decay on each cycle also nakes the oscillatory boundary very
thin, so not driving the mode efficiently.
The same
program used
for
the pure
oscillation
was
used, with the boundary. conditions on V changed to include a
dominant rotation.
and
for simple
V = '/) at r =
All runs were for.
rotation of the
= 1/2 and
. The natural frequency was f ound by setting.
oscillated very nicely on the
added.
An
did
= 10,
walls, so V = 1 at r = 1,
the viscosity to zero and integrating in time.
frequency
a'
not
change
inner oscillation
scope.
The solution
The real part of
measurably
when
was superposed
the
viscosity was
on the
inner
cylinder
at that frequency,
twice it,- three
times it, and
They all decayed fairly
several other frequencies.
rapidly
and rather indistinguishably. This suggests that either this
viscous harmonic
mean
used a
mechanism will
state is necessary.
smaller Re
resonance
was
plotted in figure l1.
or a
different
When the forcing of the mean flow
than the
quite
not work,
Re of
discernible.
the perturbations,
Some
the
growth, rates are
This situation nay arise if
the
mean
flow is driven by an azimuthal pressure gradient rather than
'by the viscous drag of the walls.
.003
4'
S
10
-.003
Figure 12.
Growth rate for Re
10000 and
forL= 1 + 0.5 sinNt,
4
=/0*
Harmonic Dec ompos it ion.
Figure
9
and
all of the
other runs
show that the
kinetic energy can be well represented by p(t) e
periodic
with the
twice it.f
or symmetric
Floquet's
frequency of
theorem
the basic
can be expressed X(t)
F periodic in t
=
=F (x,t)
P(t) exp(Rt),
where P is a periodic and R is a constant mtrix.
an
La
norm,
JIXfI
J
= IjP(t)
exp(Rt)J
where
,
positive quadratic form like the kinetic energy.
linear
can
theory, the
be
into
separated
Thus,
dominate.
usual presumption is
modes,
exp(Rt)
,
kinetic energy of any
that of
the
which
where R.
By using
I|X
is a
When using
that the solution
some
will normlly tend
dimesional projector R,,e
the
of
of
p 78-81,
and Levinson,
1955), that the fundamental solution mtrix of
with
p
oscillation, or
This reminds one
oscillations.
(see Coddington
with
,
one
will
toward-a oneof rank 1, and
is
particular solution will tend to
dominant mode,
or
p(t)C
,
as
observed.
Since this form holds so rapidly here, the presumption of a
dominant mode holds.
look
for such
Thus,
modes,
one might find it
especially for
cr'
=
profitable
to
0, which corres-
ponds to the mrginal stability for a steady basic flow. So,
write
y
etc.,
=
qg(r)
+ E
(1(r)
cos nt
+ (iL,(r)
sin nt) ,
substitute these into the equations of motion, and
separate harmonic coefficients to get
-ReNV
ReNVY
-nR eN
DDV
=
=
DDVj
=
(DD, -a)
+
niR e N
=(DD -a
)9
-
+
-nReNV
I
[ (1-8
-sa2
+ Vv
+ (1-
v.V,
+ -Re[-4vi)u
}
,,
+(1+)
[v
-(+
(DD -a' )v
+ Vv
[-V v
]7
v2
+(DV.) u
]
]
-1R e[ (1-6)(D .V)u,-,+(1+3,,)(D,V Ju
nR eNv
=
(DD -a2 )v,,- .R e
+-L Re [
=
Du,
o1,2,3,..., and V
=
lu
+(DV)u
(1+ I) (D V )u
where n = 0,1,2,3,..., i
are u,= vA
((D
+1-
]
]
E)(D,
up]
1,2, and the boundary conditions
=
0 at
r = 1 and
r=
,
for -n
given. Note there are two independent sets
of harmonics. The f irst are the even harmonics f or u and the
odd
for
v. These
experimentally.
component,
u
are the
has
a
ones observed
mean
and
a
whereas v has the fundamental.
numerically and
double
frequency
The other set of
harmonics with u oscillating at the fundamental and v with a
rectified component must require a higher Re.
The
obvious
method
to
equations is to follow Galerkin
the
above system
to n 4
above equations all
and seem
programmed
quite suitable
the
or Lorenz,
constant,
higher subscripts to zero and
The
solve
above
and to
system
of
truncate
setting variables with
solve f or the eigenvalue
Re.
have th.e second-order operator DD
f or relaxation.
This system was
separately by two programmers in quite different
styles.
Overstability
under-relaxing
with
in
a
the relaxation
was
avoided
factor min(l,(2nReNh)
,(a Reh ),).
This was necessary, else there was a rapidly-growing
latory
by
oscil-
numerical instability due to the large coefficients,
in close analogy to
the time-step limitation for
equations such as these arose from.
parabolic
However, even with this
under-relaxation, when the second harmonic was added to
system,
both
programs
gave
kinetic
energies
the
which grew
continually
for
disturbing,
and suggests a new numerical instability, which
any
Re,
though
not
rapidly.
This
is
deserves to be understood.
Let us consider a
simple analog- D y + Re y = 0,
with
y(0) = y(Wf) = 0.
This simple system has non-trivial solu-
tions only if Re
n2 for some integer
=
solution for Re < 1.
trivial
relaxation for various Re's?
n, so has only . the
What happens when one tries a
Experiments were run for
the
usual second-order relaxation scheme,
Ay
For
-2y iyf + y
=
Re
=
sine, but
1, y quickly came
very
slowly
+
Re y ],
to resemble the first arch of a
decayed,
reflecting
the
slightly
different spectral character of the finite-difference operator from
solutions
was.
as
the
differential
decayed, and did
For Re
=
For
so more rapidly
Re
< 1,
the
the smaller Re
1.02, the solution slowly decayed, then
the first mode emerged.
growth.
operator.
grew
For larger Re, there was rapid
So this works nicely, and suggests the
May lie in the boundary conditions for
.
instability
71q
Experiments f or Oscillations.
The
major piece
Karlsson at
S.K.F.
several
places,
elaborate
devices.
Snyder and
H.
is described
such-as Snyder and Karlsson (1964).
and
as well
However,
so
built by
Brown University, and
circulation
temperatures,
used,
of apparatus was
control
devices
as complicated
to
in
It has
maintain
electronic measuring
the latter rarely worked, so could not be
periods
were
measured
with
a
stop-watch, and
existence of the vortices was determined visually.
Both the
inner and outer cylinders could be oscillated independently,
with or without a
fly-wheel
superposed rotation.
allowed very smooth
Installing a
oscillations.
heavy
The radius of
the inner cylinder is 6.0275 + .0003 cm, with length
90 cm.
The radius of the outer cylinder is 6.295 + .0003 cm, so the
gap width is
Thermistors
inner
at various
cylinder
occurred
in the
The voltage
shaft
0.267+ .0006 cm, and
by
0.9575,
locations were
checking
that
output voltages as
from a
=
precision
was displayed against a
no
used to
periodic
'?= 1.0444
center the
variation
the cylinders rotated.
potentiometer on
the
drive
harmonic oscillator at 0.180
cps. The signal was found to be virtually free of harmonics.
The
apparatus included ink outlets,
but these did not
15
The
work.
ring
vortices
were
first
visualized
nacromer, but this was f ound to f loculate.
was
Aluminum
using
powder
tried, but required too large a concentration, settling
glass and
on the
forming slag
inside the
apparatus.
It
should never be used in apparatus that cannot be taken apart
and washed.
tried
The
next.
same is
While
detergent, it
true of
washing
the artificial
this
out
capfuls
in
several
gallons.
Ivory
used from then on.
lever arm
angular
of
Liquid
the oscillation
amplitude of a half
any
seems
more
is cheap, won't
the apparatus,
so
was
The observation procedure was to set the
during operation,
Then
helps to clean
formed
even for just a few
satisfactory than nacromer generally, for it
settle out, and even
Ivory Liquid
with
that beautiful -vortices
was noticed
the inner cylinder was rotating,
while
nacromer
was no
was
counting with a stopwatch.
then to
rotation.
f or there
mean rotation
gear,
measure
the
This did not change
measurable
turned on
-
back-lash.
determined by
and
Then the oscillation
frequency
was counted, using the gear arm crossing a slot in the wheel
to define a cycle.
The
temperature of the bath was
noted.
any existence of vortices at any part of the cycle was
Then
noted.
As
soon
as
vortices were
seen,
the
oscillation
frequency was lowered until they could not be, then up again
until they could. No hysteresis was noted, so the time could
be noted to 0.1 second over a minute or so.
was
also
measured
to , 0.01C.
The temperature
Unfortunately,
the
basic
definition of existence was not so accurate, but varied with
the condition of the nacromer.
It
is also often difficult to
Stabilization for inner oscillation by
outer rotation. 0 = 1.535, N = 0.6
0
090.
x x
L.
EL)
4~D
0
0::
0
5000
Re(inner)
10000
15000
decide if faint, flickering lines are really there. However,
wherever they were definitely observed, a '+'
figure
is entered
When near critical, the rolls only appear near the
.
ends of the swings. The abscissa is angular amplitude (9
,
in
and
the ordinate
is Reynolds
number.
The
=
N
data seems
satisfactory in lying above the zero-growth line nearly on a
constant growth isopleth. The pair of points lying far below
the
line
measuring
are
undoubtedly
the angle
a
blunder,
between the
probably
due
to
end-points of oscillation
the wrong way around. An attempt at measuring wavelength was
made.
While
parallax made
delimitation of the
even
faint on and
harder the
counting and
off bands near
the average at N = 0.7 was about 2.1 waves/cm.
critical,
With R = 6.3
cm, the numerical study result of a ~ 5500 gives 1.9/cm. For
higher
N,
there were
smaller wavelengths
observed.
Some
experiments were run with the outer cylinder rotating
the
inner oscillated.
Just as
one would expect,
while
the outer
rotation stabilized. The results are plotted in figure . The
necessary
oscillating
Reynolds
number
increases approxi-
mately linearly in the outer Reynolds number.
Some data
oscillation
mean rotation
of* the
superposed were
inner cylinder
also gotten
with
before the
a
main bearing
on
wore
out. For these, the oscillation amplitude and frequency were
set
first,
instability
then
was
the
f ound.
forward swing now.
only
were run
mean
in a
Some
angular
The rolls
velocity
only
to
cause
occurred
on the
experiments for inner
much smaller
M.I.T., for a different gap width.
and cruder
oscillation
apparatus at
After grinding down
the
NFA
inner cylinder to be rid of rust spots, the inner radius was
2.08 cm, the outer was 2.57 cm, soj= .807,
the
cylinders was
only be
frequency
limited.
silicone
set* at
for
six
the
However,
oil
seem
magnitude in),
oscillation amplitude could
angles, and
smll
motor
water and
allowed
Reynolds numbers.
results
15 cm. The
and the height of
the
was
range
discrete
mixtures of
several
of
constant
and rather
Dow-Corning
viscosities,
so
further
The results are plotted in figure/I.
reasonably
consistent
over
two
200
orders
The
of
considering that a secretary laughed when she
saw the experimental set-up.
EXPERIMENTAL RESULTS OSCILLATION OF THE INNER CYLINDER
WITH
)= .807
The x's represent oserved ring vortices.
observations of no vortices.
The o 's
UNSTABLE REGfIE
0
00
o
STABLE REGIME
0
-1
0
ANGULAR AMPLITUDE, OR
Ao'~-
N
Coda.
thesis, a very
of preparing this
the final stages
In
relevant paper by Rosenblat (1968) appeared. Rosenblat notes
yet
importance,
the
on
dependence
instabilities
of inviscid,
centrifugal
considers
then
instabilities,
time-
of
effects
the
of
neglect,
between coaxial
periodic flows
He linearizes and assumes axisymmetry, and
cylinders.
also
discusses what instability might mean.
lY
the
is
exponent
for
cos ( -cosLwt),
to be
the disturbances
time-dependence of
where
mean flow, he finds the
rigid oscillations of the
For
the
steady
corresponding
(Rayleigh) problem. If this be complex, and 4) is small, the
will increase exp()-f old during growth, which
disturbance
nmy take it
out of the linear range.
the
phase-differential
in
the
finds
He
He next considers nearly rigid oscillations.
direction, however
radial
This
small, is sufficient to cause instability.
conclusion
certainly does not-extend to viscous flows. It also does not
hold for N ((
1, which is exactly the requirement for nearly
when the
oscillations
rigid
mean flow
is
driven
by wall
oscillations.
Rosenblat shows
steady
mean flow
that small
will be stable,
twice the natural frequency
lation,
though
he
oscillations on
except in
a
stable
a band around
oscil-
of an inertial-elastoid
suppresses
the
dependence
of
this
frequency on vertical wave-number.
f or
all
driving frequencies
This seems to mean
below 2'
,
that
some wave-numbers
will be subharmonically unstable.
When the steady mean f low is. unstable, Rosenblat
a second-order,
which
he
takes
f inds
inviscid decrease in the linear growth rate,
to
explain
the
reduction
in
limiting
amplitude found by Donnelly (19 64). This seems questionable.
Chapter Three - Suddenly Twisted Cylinder.
We
centrifugal
starting
flow on which
now consider - another time dependent
the inner
arise,
may
instabilities
cylinder rotating,
meaning and methods, rather than
that
of
view toward
with a
( j ). As soon
will
on it
thin boundary-layer
starts, a
hand.
just the problem at
The equations of motion are already given in
as the cylinder
suddenly
. When this grows thick enough, the
form, of thickness
radial
to pay
great enough
will be
centrifugal potential
motions. We first are interested in the margin between decay
and growth.
suddenly
consider the
sphere
has
been
no closed expression,
but gets
twisted cylinder,
and does not
started
Mallick (1957) considers the mean flow around a
instabilities.
The
considered
Barrett
by
.impulsively
(1967),
using boundary layer theory,
but
similar problem
of a
suddenly applied temperature in the Benard problem has
been
expanding in
=
instabilities
are
and
A
not involved.
considered by Lick (1965) and repeated by Currie (1967),
using an
assumption
broken-line
quasi-steadiness,
of
appraximate
mean
well
as
temperature
as
but
a
gradient.
Robinson (1965) also considers this Benard problem, using an
erfc
profile, and devotes some attention to when the quasi-
steady approximation
resemblance
to
the
ny
be
numeric
valid.
His
results
initial-value
bear
results
a
of
Foster(1965).
Why should one be
quasi-steadiness
able to make
when the
mean is
such an assumption
as.
much?
The
changing so
usual argument is
that the perturbations have a much shorter
time scale
the
marginal
than
mean,
( cr=
state
and
especially since
is
Consider a
bound.
simple example
the approxination
and growth. However, y(2E)
the bound past which
requiring a finite
However,
requires
it
at
t
=
It
E,
can be nade to
not
initial perturbations grow?
be operationally more
size large enough
of
limiting
This
meaningful,
to be detectable.
is usually inconvenient theoretically,
considerations
a
y = y(0) exp(t-
at y =2E, by large E, so is
sort of definition would
in
t = E as
division between decay
initial perturbation size until y = 2E.
2E
The
y(O), so y cannot grow past its
=
have arbitrarily large
time-scale
ty -Ey.
=
y = y(O) exp(-E/2)
does give the
the
This seems a
0 = ty -Ey or
The actual solution is
tE), which has a minimum of
so
consider
no perturbation
quasi-steady approximation gives
stability
they
) of zero growth rate.
contradiction,
produced.
then
size,
as
since it
well
as
depending on the. particular detection device.
The axisymmetric linearized
equations for the
started
cylinder are
A
-
where
and u = v = w =0 at
all
-
0 as r
+oo,
all
r
=
R,
+
0 as t
V =
at r = R,
+ 0 (r}R).
)
9
We
the margin between
are interested in
= S R and the z scale is L, to
Assume the r scale is
found.
Look at
decay and growth.
0 (quasi-steady Mrgin),
be
i.e., balanee
The scales are
generation and dissipation.
P
SRIC,
L
so the scales in the first equation are
In the experiments, S.Qt
smaller
>
30
>>
1, so the last term is
than the second, giving scale L ~ .t
much
E R, or
R
even R. This is the sort of wave-length one expects first
to
go unstable. Note the sharpening of Robinson's quasi-steady
conclusion of wave-number
unstable.
This small
zero in
of
Another conclusion
generation/dissipation
expects growth
for
t.
just as observed. Thus,
being the
is that
the
so
one
£L
is
such that £t
,
>critical,
ort
balancing generation and dissipation
gives the right forml quantitative considerations will
yield
the
constant c.
problem with
condition
a
First one
V boundary
and one open
most
wavenumber matches the
but non-zero
numerical results better.
scale
scale R
layel?
might remark
with one
is similar enough
convection problem with one rigid
rigid
also
that this
surface
to the classical
and one free boundary
to
suggest
r = R (l+Sx),
Writing
now to
)3 , so
in figure
works beautifully
1100
equation is about
that c in the last
This
.
why.
find out
and using SR for the z-scale for
simplicity, the dimensionless equations of motion are
aV
2j9v
t /S
Note that the coefficient Ta =
the* shape of V.
parameter is Ta, so
8 ,
is small, the only O(l)
critical
it is the
it is natural that
Since it will be independent of
parameter, as argued above.
Sfor sanall
S
We see that when
does
depends on t, as
argument will yield
a boundary layer
the
constant.
Quasi-Steady Approach
The
changes slowly,
then
to
set
so
the
there is
exponential
to
growth-rate
consistent approach is to assume
and
f ind when
pation, so
explicit
assume the basic flow
quasi-steady approach is to
the generation
leaving
dependence
zero
zero.
Perhaps
that one mode is
exactly balances
growth ratr.
on slow
time-dependence,
change
This
for the
a more
dominant
the dissi-
view
has
no
basic state,
-
case
Either
dominating.
the
uses
one
assumption of
assumption under. the
hiding this
(is)
equations
mode
with
0. One can eliminate p and w to get
0 = (DI;,-aI) f-2a ReVv/r,
O = (DDt -a
where .
)v
Ju
R e(DV
-
(DD,-az)u,
u = v= D u = 0
and as r -+O.
at r=
Write
This is solved by 'shooting'.
U
y=
Dy,
y2 =D.u
so
Dy =y3 4a y
y, =t
so
Dy3
=y, -y3 /r,
y
=D
so
Dy
=
= v
so
Dy
=
y4 = Dv
so
Dy4 =
here y ()
is
- y, /r,
y
so
a
=
ay
3
()
+ 2a'Re Vy,/r
y4 -yr /r,
ayg + Re(DV )yY
) = y
= y,()) = yr(I) = 0 = y
two-point boundary
value
similarly to that
of the
solve
sets
for
three
problem.
bottom boundary
of
initial
(i,0,0), (0,1,0 ), and (0,0 ,1).
above
is linear,
satisfied if
solutions
so the
which
in Chapter
conditions can be
of (y, ,y, ,y.)
that the determinant
where
for large enough
lower
Reynolds
intermediate
value
for
One
which
deduces
for the
that
three
One actually
changes sign
Reynolds number, but
number.
Onet
(y 3 'Yq 'y)
The sys tem of equa tions
goes to zero as r goes to infinity.
solves by finding
is solved
conditions
outer boundary
the determinant
. This
Y
=
some-
does not at athere
the. determinant
is an
is asymp-
totically zero from continuity of the solutions with respect
to the initial conditions.
This
numerical
integration
approximation V = erfc((r-l)/).
approximation
figure (jQ),
observed
for
and
instabilities,
method here
gives
a
out
for the
This should be an excellent
small Z .
form a
was carried
The
results are
reasonable
plotted in
lower bound
for
the
suggesting that this 'quasi-steady'
good approximation
to
the
time
of
minimum energy, as in the simple example at the beginning of
this chapter.
One discrepancy of this quasi-steady
the
wave
numbers
growing waves
which
come
found
are presented
for
initial-value
in figure
M .
like the
beginning
this
simple
chapter
--
original perturbation size about
rate as it
than
from the
shaped rather
of
out higher
approach is
These
the fastest
integrations,
curves are even
example considered
they
grow
back
at
the
to their
as long after zero
growth
took to get there, and some of those that dropped
more slowly also grew more slowly and were passed.
a
that
correlation
between
those. wavelengths
which
There is
started
growing first and those which grew most rapidly later, which
explains
some of
the success
of the
quasi-steady method.
Numerical Methods.
Because the equations of motion
are the same for
this
problem as for the oscillating cylinder of the last chapter,
Figure 13.
Predicted stability boundaries and observed instabilities
for the started cylinder.
0.05
ho
011
:2
I
I
I
Re=II53
UNSTABLE
4~.
i.
linear ."N
0
\,jerfc.)
4-
+
4~
Serrin
+
II
'I,
-
-
bound
+
.
10
N
H
STABLE
4.
it
is first
worthwhile to
try this
initial value
problem
with the inner boundary condition changed to
(1,t)
and
the outer boundary changed to r
conditions were
changed
to free
=
4, where the boundary
slip,
U = Dv = DD*u = O,
since this. seemed to work heuristically quite well in figure
I3
R = 4 was far enough out that imposing a virtual
.
from
the
exterior
The most
while
solution
obvious feature
v had the
of
same r-scale
much wider scale.
treatment,
would
difficult a
be
worth
the solutions
as
This suggests
and made
not
did V,
the value
u
mass
while.
was
that
and w had a
of a
boundary
grid representation.
The
number of grid points was increased to 36, which gave smooth
enough
curves on
the screen
to indicate
that both scales
were adequately handled.
Reynolds numbers
50,
of
100,
and
150 were
run
for
various wave-numbers designed to be near nximum growth. The
kinetic energies
non-dimensional.
To compare these
of the
disturbances as
functions of
the
time from starting are given in figure 9
results with the
quasi-steady theory
.
and
the experimental results discussed later, -some level must be
selected at which the
The
disturbances have become
'unstable'.
quasi-steady theory essentially finds the points at the
bottoms of the
zero.
curves, where the
kinetic energy growth
is
Examining these curves, we see that the time to this
point is not very dependent
necessarily
related
rapidly later.
on the wave-number,' and is
to - the wave-number
which
not
grows most
30
3 60
20
-=50,
2=q
10'
/0
-3)
/0
Rs, 2= /0
Figure 14.
Perturbation kinetic energy
versus time .after cylinder starts.
V
The initial
potential,
peaks are
where
declining parts
it
adjustment of
is put
represent
Initially,
viscous decay
the energy
from
to .kinetic.
The
before
the
-main
boundary layer grows thick enough to yield energy. Note that
the higher- wave-numbers
though
they
nay start
decay
more
drawing
during
this
period,
energy earlier.
The wave
number of maximum growth increases with Reynolds number, and
with
a
nearly
linear
in
Re,
which
is
to
say, their
wavelength
remains
thickness.
The growth rates are exponential with log-slopes
nearly
proportional
to
the
boundary
very nearly constant over the period run, and increase
with
Re, perhaps at a 4/3 power rate.
F inite-Difference forms.
So far, the conventional second-order finite-difference
appraximations have
relatively
let us
been
used, largely
because
they
are
easy to program, especially at boundaries.
consider what
f inite-difference f orms
Now
ought to
be
used. Most of the computations essentially involve parabolic
terms, for which
is
the prototypeO
non-zero
absorbed
in interior
might be
accurately computed
eliminating
forcing.
VD
boundary conditions
Let
with
us consider
a special
view
can
be
how this
(v
toward
the purely computation restriction on step size
,
can be
which
very
inconvenient for
fine
layers, such. as in present
necessary
to look at boundary
problem.
Arakawa showed the value of preserving
V
Here we have
integral forms.
let us force a
so
quadratic
f0
finite-difference approximation to this,
Let us
by sums.
where the integrals are approximated
use
for
approximations
f inite-dif ference
three-point
grids
illustration:
and require
these to
hold exactly
V =1
for
and
V
=x,
i.e., to be of first-order accuracy. Now impose the integral
energy condition in the form
or, using the boundary conditions v*
(d +f) v v
+ b+
c') E(v
-2 (a+c)by vk vf
-2ac
vk
imposed four
Above we
= 0
v
= -(a
+ e
1v
=
conditions on
the six
coeffi-
cients, so we may impose the zero and f irst order conditions
here.
a
=0,
There
.
are
b = -i
,
two
solutions
c =
I
1 d =
which is an Euler approximation
these
to
e
=
-
six
equations
and f =
3'
v
_
_
other is
kLV
_k
(x~z
~
-
The
vV'-2v-'VvC-
I
)AX
g
backward Euler,
the
be consistent with this energy equation.
not
0
2ac
f + d
and
actually
2/(X)
=
second order
so
,
this
in the energy
AX.
Note
-V k-)/2A)( would
~(vd
that the centered approximation
k k
-v/
(v
Also note that
is
approximation
and second derivative
approximations.
Thus,
using an Arakawa style approach, the Euler scheme
V
-2V
t+4r).----
is derived as 'energy preserving'.
energy grows
requirement
done with
analog
rapidly.
The
was imposed
61A.
computer,
with
At > (Ax),'2, the
Yet if
trouble
,
-
+iVa
is that
but
energy
the computation is
The
above scheme would
but
only
digital
the
be stable on
computers
an
can handle
systems of the necessary complexity.
Since the
spectral
LXR
by
anything
(3T2. ) is
equation
viewpoint.
of
use
We
linear,
are comitted
computer,
a digital
with finer scale,
to a
so
-if such are
let us
use
discrete grid
cannot
handle
important, a finer
grid must be used. Thus, we can regard V as band-limited,
V
=
~
a~sin px ,
where
a
=--
,)
-- 4
so
=
p a
for arbitrary
V( Et+kt)
at.
=
so
sinp
by the Sampling Theorem, even between the gridpoints.
XV
=pasin
a
Thus,
px,
and a,(t+6t) =a,(t)
(0 )
exp(-pAt)
Combining (n&) and (53L) gives
c(k, J)v(
,t),
(6)
C(k,j)=
where
exp(-p At) sin(t!rj.
[in(A)
(Sq)
P:zg
give the exact solution to.the equations as a finite-
These
difference stepping formula.
once at the beginning.
out
linear forcing is
The matrix C(bt) can be worked
combine this idea with non-
To
(Xx -V )v= F.,
evaluate
followed by the above
with
Runge-
=eV + F, we use it
Kutta or whatever scheme to evaluate
to
using a
instead of
easy enough;
a
time
as
step
( ~3 ) transfornation.
usual,
In fact, there
seems considerable merit in bring ing all of the linear parts
of
the equation
to the
left, on
using everything one knows.
a general
philosophy of
In this case, for the price
of
having n-l interactions to handle instead of 3, we get exact
answers (nth order), which allows arbitrary time-step
This
must be
at least (n-l)/3
However, if a f as t
and
Fourier Transform is
back, the ratio is
smaller for
the
times as large
fine
only
size.
to pay off.
used to find
a.
log2 (n)/3, which can be much
grids visualized.
We
follow
N.A.
Phillips in considering a prototype equation
-(A)
where
terms.
limit
first
U is a
stream velocity
The second
like
At<
imposes
term on
,.
-
___
representing the non-linear
the right
imposes a
time-step
whereas for. conventional 'schemes the
This will be more restrictive than
the first if
(XS
viscous boundary thick-
side represents a
where
the right
ness.
Thus, this new disguised spectral scheme may well
be
for looking at a boundary layer structure, as we are
useful
it
doing here. A.few evaluations of the matrix C(bt) showed
to be near block-diagonal for at snall, and in fact, clearly
the
C(n,k) represents
of
influence
time, only nearby
others,* for smal
the
on
region
each
vorticity can diffuse
We note that
over.
C,(k,m)
expf- (
=
and write .x
(sink )(sin4( )
to see this is
17/n, x = pW/n, O
=
a
partial sum for an integral
Now,
exp[-o(xtj
f
C(k,m)
wise, old
formulas 'are
dx.
interested in o< > 1 (other-
1 and we are
((
exp(-7t)
sin kx sin mn
stable), so
my change
upper
the
limit of integration and use trigonometric addition formulas
to get
C(k, m)
exp -' xj
(2- exp[-(k
- cos (k+m)X - cos 2n-k-m)Xd
[c osk
-exp -(k+m)
et al (19&4I),
from (1.4.11) p 15 of Erdelyi,
Vol.
two terms represent reflection in the walls.
latter
have been
order reflections
for
n
evaluating C
=
,
(properly) suppressed
5,
and
and itconsiderably
nay even be
eases
more accurate
( 6 L 1 ), since there will be no round-off error.
behavior
),(56)
1.
The
Higher
in
the
Numerical experiments show this to hold well-1
appraximation.
even
-exp(2n-k-
shows how far
one needs to
the
work
of
than the sum
The exp[-x2j
evaluate* there is no
use
beyond
going
the
accuracy
of
the
nachine
or data.
While Cartesian coordinates were clearer for exposition,
we are interested in cylindrical coordinates, where
so write
B (r)
Y,(
=
)J(
)Y(
r) -(
is such
r), where
needs to
numerically
invert the natrix B to form C, which hardly pays.
For narrow
that B
0.
For
wide gap, one
gap, one can use the asymptotic formulas (9.2.9) and (9.2.10)
of Abramowitz and Stegun (1965) to get
sinA(("IL-r) + 0(
r)
B~'
Proceeding
now
exactly as
C(k,m) ~14')
(6+
'4~)
for the
Cartesian case,
exp(-o(y )sin(ky)sin(my)
[2 ex p -x( O.e-
ex p- -
we get
dy
e- x
This provides a good approximation for narrow gap which ought
to speed the numeric integration of parabolic. schemes on fine
grids in cylindrical coordinates.
The system of equations (51
This
can be- found in the
DV ~
) requires DyV as well as V.
same disguised spectral fashion as
...
V(r'"')
Experimental Comparison.
Christensen [see Chen
data
and Christensen (1967)]
gathered
on the instability for started cylinder with radius
1/4
inch
in
d istilled water.
reproduced in Table
and
This
N's.
&
Christensen's original
along
,
with the corresponding
data consists
which instability was first
essentially of
observed, as
marked are
satisfying
how well they
in the
the time at
varied, and are
plotted in non-dmensional f orm in f igure /3.
the points
data is
Thus, all of
unstable region,
all fall just
and it
is
above the Marginal
lines given in the last section.
After the critical time.
In the similar
problem of Gortler
cells on a
surfac , Lin (1966) 'neglects the viscous terms.
D'Arcy (1951)
f low
to
concave
His student
used a broken line appraximation to the
compute
the
first and
patterns very close to Gg:rtlers.
second
modes,
basic
and found
Let us look at how
this
might work.
The
axisymmetric
linearized equations
4I$
22W'-
V
of
motion are
Let
us
for an
look
The continuity
equation
momentum
vertical
, and (.3)
P
forces scales
T
time-scale
W
U/o9
,
and
so
(7)
)
becomesThe
growth
appropriate
equation
becomes
perturbation
Now the quasi-steady assumption is that the
time
scale T
Thus,
the viscosity in (S3)
quasi-steady
scale
pressure
a
forces
is much smaller
time scale t0.
is precisely as negligible as
assumption
marginal stability.
than the basic
is
good,
once
just as Lin
Thus.,
past
the
suggests,
the
initial
(and
lamb
before him), the viscosity nay be neglected, so
Substituting
momentum
the
pressure
scale
forces
scales
equation
the viscosity here to
shows
gible.
found
(u
be O(Od
above
jvI o
into
the u-
N(T/t,),
and
T/t,), even more negli-
We get- a curious quasi-centrifugal balance
+ 2Vv/r.
We earlier f ound
that the
f irst o< Is to
evalu8.ted at that time,
so
emerge have
scales
is certainly smll.
However, it happens to have been preserved throughout.
Putting the
u/v scale
into the
v equation
forces
the
growth time scale
T/t
For
this to
=
be sall,
marginal growth time,
-~~~)
either tor else
must be
several times the
o. must be large.
In .fact,
it
is
f ound below
rapidly at this
growth
that the larger
time.
in
the
do grow more
Foster's picture of - the
This fits
disturbances
of
wave numbers
started Benard
analogous
start growing first,
problem. the long disturbances
but
do
not
grow as rapidly as shorter disturbances starting later.
It
also fits the observation that the wavelengths are quite
random when at f inite size.
Again dropping the viscosity as being o(T/t,
V/r)u.
+
Writing
(sin z, cos z), the equations
the variables with e
are
TX-
W
c7W
from which w, p, and v may be eliminated.
=0, u +0
with u()
normalize.
o,
This defines an
).
since V =v(
viscous
as x
It
is
and
aK
that the
(v = w = 0) can
thinner boundary O((T/t,)
care of in a
at x= 0,
to
eigenvalue problem for
worth noting
boundary conditions
1
other
two
both be :taken
without
-affecting
this boundary.
Let us f irst consider the broken line model
V=
{l-
x,
0
as
do
D'Arcy,
Lick and
for 0 <x
1
for x > 1
Currie.
This
gives exponential
/00~
for
solutions
u
the
in
two
must match
which
regions,
1,i must be o at x = Oand as
continuously at x
x
This requires
u
for x < 1,
)
A sin(x
=
for x >,
u= B K (oCx)
which requires [to O(S)]
=
tan
For
tanho(1 -A).
-
8, this determines c,
given c4 and
if
it
We
exists.
see that there are an infinite number of modes for which the
sine
is
just
declining K
so
,
a
enough, past
Jap,
(n+ )' 1,
lowest mode grows fastest.
smoothly
match the
O'~
, so
peak to
and
the
The nth mode has n rolls stacked
radially, with the outer one stretching from inside x =[r1 =
to
]
to D'Arcy's
corresponds
This
00
results and
illuminates the following,
Let
now
us
consider
the
exp(
V = erfc (x/), so
better
To facilitate wave-number
let us
temporarily take
scale, so T/t, = (N
) , and consider
comparisons,
u(0) = 0,
where
x + o0.
constant
=
1, and
approximation
o((,
o.
)
out
of the
is such that u
time-
+
0 as
This is an ordinary differential equation with noncoefficients,
so seemed well suited to solution on
an analog computer.
The function
analog
computer.
multiplying u
We
note
that
must be
the
generated by
erfc
the
satisfies the
differential equation
((b3)
erf c (0) = 2/Vr
with erfc(0) = 1 and
This was first generated from equation (63)
where standard notation is used.
using a circuit
Adjusting the coefficients
a little brought erfc to within .0003 of its exact value
points, and
several
brought erfc
Naturally, x must be restricted,
to zero
usually to
at
as x increased.
5 here
X
and
below.
With erfc well approximated,
be solved.including
/S.
The
whole
circuit,
the whole equation can now
in
one
of
its
forms,
some necessary debiasing, used is shown in figure
This
conception,
circuit
from
considerably
grew
the
initial
the lower right corner represents the equation.
The- solution u(x;,otS ) behaves very like a line a + bx
once
x
>>
1. We want to
f ind when both a
and b match the
exterior solution K, (otr), which requires
a + bx =cK,(c<) + c K'(oc)~ K (o<(l+Sx))
using a Taylor expansion.
A c can be found iff
o< kA('()
For each c<,
and
c,
and 3
ramp a + bx was
,
the equation
also.
(63 ) was
This latter
integrated
was adjusted until
asymptotic to the solution uand the a and b were
recorde.
ILY~
Figure 15 -- An analog circuit used.
TABLE 4
'
Eigenvalues
to satisfy DDyu
with boundary conditions u(O)
wave #
for S
5.0
4.0
3.0
2.0
1.0
0.7
0.5
0.3
0.1
,
=
0.1:
0.4099
0.44965
0.109
0.49514
0.548
0.6079
0.1205
0.6280
0.6426
0.6578
0.6708
0.129
for J = 0.6:
1.0
0.2195
0.0563
=
-
(
0, u(eo)
DV +
y.,
= 0.
ES)u
=
0,
growth rate e--;-
Figure 16.
Contours of
a 0(4
/01
for
£
0.1.
in
Plainly
contour
the
0.8
into play
zero contour, and
a most unstable
causing
check,
digital
this
0<
short waves . and
which will change with time.
,
problem was
computer for a
short
Actually viscosity would come
increases, stabilizing
c<
as
[the
0.1
Thus, long waves grow more slowly than
this quasi-steady range.
For a
for
f or (6q)] is close to the
intersection
(7
/
is recorded and contoured in figure
The ratio
also
integrated
few cases presented
on
the
/
in Table
.
The results are very similar.
Energy Bounds for Stability
A guaranteed
energy
stability bound
equation
for
the
full
can be
the
non-linear, time-dependent
incompressible Navier-Stokes equations,
metry.
gotten from
if we assume axisym-
The formulas follow Serrin's (1959a), though here we
are interested in
growth,
rather
the minimum
than
time at which
the minimum
energy
over
smooth
number
Reynold's
steady basic flow. If we raximize
kinetic
there may
be
for a
of the perturbation
perturbations
(u,v,w)
which
satisfy the boundary conditions, and find the time that this
naximum
becomes positive, no
can grow
stationary
before this
time.
Following
value
the
generation
subject to the
equations
perturbation kinetic energies
of
constraint of
Serrin, we
minus
continuity.
The
find
a
dissipation,
variational-
are similar to the quasi-steady equations in form:
/OS-
Re v A-=0
+2
Reuu-
+2
=
0
with
+
(DD, +
f
-+2
+
+
(DDy +
+
)JV +b
(IjD +
+
,w
0, and u = v
0 at r =1 and as r
w
The interesting featurehere is that these equations give
rigorous
bound, and
do not include
+
a
a vague 'quasi-steady'
assumption. Thus, given the time, and hence the shape V, the
eigenvalue Re of these equations provides an absolute bound.
they
Unfortunately,
If
numerically.
are
very
one relies
perturbations are
difficult
solve,
the observation
on
axisymmetric
to
in
the
even
that the
experiments
under
consideration, the equations can be simplified to
(DD
u =- Re a2
-
-Re (
-
(DD.-a' )v
with
=
u = v = Du = 0 at r
like the
quasi-steady
)v
)u
1 and -as r
".
+
equations already
These are just
integrated,
with
slightly dif f erent V coef f icients. These were integrated f or
several values of
V
so
S
with a 'broken-line'
profile
[-r + (&+1) /r ]
=
DV
for 1 < r <1 +S
,and both 0 for r >1 +S.
The Re's found
are plotted on. figure 13 also. They are not as far below the
actual critical
general
criterion.
tions was that
very
rapidly as
to a2 f or
values
S
as one
might
An interesting
the wave-number a
expect from -such
a
feature of the integrafor minimum Re
increased
S became small, and Re was. quite sensitive
near 1.
Thus, the
picture
seems
fairly
complete: After
the
cylinder starts, a viscous boundary layer forms and thickens
until the Reynolds
30,
number based on
when generation can natch
rolls.
After -this, the
perturbations
decreases,
amplitude, they
angular momentum
growth
from
and shorter
viscosity for
waves nay
of further rolls.
-
1 Z-,
several radii
boundary
for
grow, and
basic
flow,
the transmission of
discouraging
then decays on a scale
the longer
rolls.
intensifies
o< R, which can be
Ink in
the
will be drawn out in disks between the rolls,
as observed.
the
The first mode will grow fastest,
so all of the rolls will have a structure which
out to r
the
As the mixture of rolls reaches
will take over
the
about
dissipation f or certain long
importance of
faster than the long rolls.
finite
its thickness is
inner
just
107
Acknow le dgments
I acknowledge my particular indebtedness tot
Professor L.
Howard, for being an encouraging thesis
N.
advisor.
The Electrical
particular,
to do
Engineering Department,
type
computations,
this thesis.
Landau did some
did
of the
most of the typing,
when the
McKenzie
in
for allowing me to use the PDP-1 computer
the extensive
even
J.
nachine
draw
In this
would not
and
connection Charles
prograrming,
and W.
graphs,
B.
Luella
Thompson
Ackermann helped
cooperate.
Many
other
computer hackers also willingly offered advice.
Prof.
S. K. F. Karlsson of Brown University Engineering
Department
for
allowing
me
access
to
the
Taylor
apparatus, and having patience when accessories ground
down, pipes clogged,
and a
Colin Hackett for
A.M.
flood awakened
helping me with
him at
2
the naze of
pipes and valves.
Prof.
for
C.
Rooth at Woods Hole Oceanographic
the original problem in
Institution
Chapter One, and for use
of the equipment used in those experiments.
Prof. G. Veronis and the National Science Foundation for
providing
two
pleasant
summers
at
the
Woods Hole
NEM
Oceanographic Institution
among some
seminars
summer
Geophysical Fluid
of the
Dynamics
most stimulating
people of the field, and where this thesis was started
and finished. Professors F. Busse and W. V. R. Malkus,
for encouraging discussions there.
Prof. C. S.
Chen for providing the original data for the
started cylinder.
The
Mechanical Engineering Department of M.I.T.
for use
of their analog computer.
Professors E.
rest
Mollo-Christensen,
of the Meteorology
esting things.
M.I.T.,
E.
N.
Department,
for providing
Lorenz,
and the
for doing intera nice
office
overlooking the Charles and other facilities.
The
Hertz and Ford Foundations, for providing a comfortable standard of living.
/09
BIBLICGRA PHY
Abramowitz,
M., and Stegun, I. A.,
Handbook of
(1964),
(Eds),
Mathenatical functions with fLormulas, graiphs, and nath
tables., Wash.,
Aldridge,
K. D.,
U.S. Govt. Printing Office, XiV, l046p.
(1967),
An experimental study of axisymmetric
inertial oscillations of a rotating liquid sphere., Ph.D.
Thesis,
Massachusetts Institute of Technology.
Allen, E. E., (1954),
Comp.,
Analytical approximations.,
Math. Tables Aids
8, 240-241.
Baines, P. G., (1967), Forced oscillations of an enclosed rotating
fluid.,
J. Fluid Mech.,
Barrett, K. E., (1967),
J. Fluid. Mech.,
30, 533-546.
On the impulsively started rotating sphere.,
27, 779-788.
Benjamin, T. B., and Ursell, F., (1954),
The stability of a
plane free surface of a liquid- in vertical periodic motion.,
Proc. Roy. Soc., A 225, 505.
Betchov, R., and Criminale, W. 0., Jr.,
Flows., Academic Press,
Busse, F., (1967),
(1967),
Stability of Parallel
New York, 330p.
Shear flow instabilities in rotating systems.,
Abstract in Geophysical Fluid Dynamics Notes.
11, 221-222.
1/0
Carrier, G. F., and DiPrima, R. C., (1956),
On the torsional
oscillations of a solid sphere in a viscous fluid., J. Appl.
Mech.,
23, 601.
Chandrasekhar,
S., (1958),
rotating cylinders.,
Chandrasekhar, S.,
Proc. Roy. Soc., A, 246, 301-311.
(1961),
Oxford Univ. Press,
The stability of viscous flow between
Hydrodynamic and hydronagnetic stability.,
London.
Chen, C. F., and Christensen, D. K., (1967), Stability of f low
induced by an impulsively started rotating cylinder.
Physics of Fluids, 10, No. 8, 1845-1846.
Coddington, E. A., and Levinson, N.,
(1955),
Theory of ordinary
differential equations., McGraw Hill, New York.
Coles, D.,
(1965), Transition in circular Couette rlow., J. Fluid Mech.
2l, 385-425.
Conrad,
P. W., and Criminale, W. o.,
Jr., (1965), The stability
of time-dependent laminar flow with curved streamlines.
Z. Angew.' Math. Phys.,
16., 569-581.
Coster, D., (1919), The rotational oscillations of. a cylinder
in a viscous fluid., Phil Mag.,
No. 6, 37, 587-594.
Crow, S., (1965),
Geophysical Fluid Dynamics Student Lectures, Woods
Hole Oceanographic Institution.
Currie, I. G., (1967),
The effect of heating rate on stab.ility
J. Fluid Mech.,
of stationary fluids.,
D'Arcy, D.,
29, 33 7 -348.
(1951), Studies in vertical flow., M.S. Thesis,
lvssachusetts Institue of Technology.
Experiments on the stability of
Donnelly, R. J., (1958),
viscous flow between rotating cylinders.
Measurements.,
Donnelly, R.
I. Torque
Proc. Roy. Soc., A 246, 312-325.
J.,
Experiments on the stability of
(1964),
viscous flow between rotating cylinders.
Enhancement of stability by modulation.,
Proc. Roy. Soc.,
A 281, 130-139.
Edmunds,
D. E.,
(1963),
On the uniqueness of viscous flows.,
Arch. Rational Mech. Anal.,
Erdelyi, A.,
(1954),
14,
171-176,
Tables of Integral Transforms.,
McGraw Hill, New York.
Fage, A*,
(1938),
The influence of wall oscillations,
*all
rotation, and entry eddies on the breakdown of laminar
flow in an annular pipe.,
Foster, T. D.,
Proc.
Roy.
Soc., A 165,
Some comments on hydrodynamic
(1964),
501-529.
stability
theory when nonlinear temperature profiles are present.,
unpublished.
Foster, T. D.,
(1965), Stability of a homogeneous fluid cooled
uniformly from above.,
Fultz,
D.,
(1959),
Phys. Fluids,
8,
A note on overstability and the elastoid-
inertia oscillations, of Kelvin, Solberg,
J. Meteor.,
16,
D.,
(1965),
Fultz,
D.,
and Murty,
Lecture given at
T.
S.,
rotating fluids.,
H. P.,
Midwestern Mechanics Conference.
(1968),
law of depth on the instability
Greenspan,
and Bjerknes.,
199-208.
Fultz,
in
1249-1257.
Effects of the radial
of inertial
oscillations
submitted.
and Howard,
L. N.,
(1963),
On the time-dependent
motion of a rotating fluid., J. Fluid Mech., 17, 385-404.
Hammerlin, G.,
(1955),
Uber das eigenivertproblem der
drei-dimensionalen Instabilitat
an konkaven Wanden.,
J. .Rat.
laminarer Grenzschuchten
Mech.
Anal., 4,
279-321.
)13
Hastings, C. B., (1955),
Approximations for digital computers.,
Princeton Univ. Press, Princeton, New Jersey.
Helmholtz, H., (1868), Uber diskontinuierliche Flussigkeitsbewegangen
Monats. konige. preuss. Akad. Wisc. Berlin, 215-228.
Howard,
L. N.,
Geophysical Fluid Dynamics Lecture Series
(1968),
Notes, Woods Hole Oceanographic Instituion.
Howard,
L. N., (1963), Heat transport by turbulent convection.,
J. Fluid Mech.,
Ito, S.,
(1961),
17, 405-432.
The existence and uniqueness of regular
solutions of the non-stationary Navier-Stokes equations,
J. Fac. Sci. Univ. Tokyo, Sect. 1, 9, 103-140.
Johnson, J. A.,
(1963),
The stability of shearing motion in
a rotating fluid., J. Fluid Mech.,
Johnson, L.,
(1967),
17, 337-352.
Geophysical Fluid Dynamics Student Lectures,
Woods Hole Oceanographic Institution.
Kelly, R. E., (1965),
flow.,
The stability of an unsteady Kelvin-Helmholtz
J. Fluid Mech.,
22,
547-560.
Kelly, R. E., (1967),
On the stability of an inviscid shear
layer which is periodic in space and time., J. Fluid Mech.,
27, 657-689.
Kirchgassner, K.,
(1961), Die Instabilitit der Stromung
zwischen zwei rotierenden Zylindern gegenuber Taylor-Wirbeln fur
beliebige Spaltbreiten., Z. Angew.
Lick, Wilbert,
Phys., 12,
14-30.
(1965), The instability of a fluid layer with
time-dependent heating.,
Lin, C. C.,
Math.
(1955),
J. Fluid Mech., 21, Part 3,
565-576.
The theory of hydrodynamic stability.,
Cambridge Univ. Press, Cambridge.
Lin, C.C.,
(1966),
The theory of hydrodynamic stability.,
Cambridge Univ. Press, Cambridge,
Longuet-Higgins,
M. S.,
(1953),
(sec. ed.).
Mass transport in water waves.,
Phil. Trans., A, 24 5 , 535-581.
Longuet-Higgins,
M. S.,
(1960),
mass transport in the boundary
layer at a free oscillating surface.,
McDonald, B. E., and Dicke, R. H. , (1967),
Fluid spin-down., Science,
158,
J. Fluid Mech., 8, 293-306.
Solar Oblateness and
1562-1564.
||6
lllick,
D. D.,
(1957),
Nonunif orm rotation of an infinite
circular cylinder in an infinite viscous f luid., Z.
Math.
Mech., 37, 385-392.
Meister, B., and Munzer, We,
mit modulation., Z. Angew.
Miles,
Angew.
J. W.,
(1964),
(1966),
Math. and Phys.,
17, 537-540.
Free-surface oscillation in a slowly
rotating liquid., J. Fluid Mech.,
Orr, W. McF.,
Das Taylorsche Stabilitatsproblm
(1907),
18,
187-194.
The stability or instability of the
steady motions of a liquid., Proc. Roy. Irish Acad., A27,
Phillips, N. A.,
, No. 2,
Rayleigh,
(1963),
9-138.
Geostrophic Motion., Reviews of Geophysics,
123-176.
Lord, .(1884),
On the circulation of air observed in
Kundt's Tubes, and on some allied acoustical problems.,
Phil. Trans. Roy. Soc., 1-22.
Rayleigh,
Lord,
(1916),
On the dynamics of revolving fluids.,,
Scientific Papers, Vol. 6, 447-453,
Cambridge Univ.
Press, London.
Reynolds, O., (1883), An experimental investigation of the
circumstances which determine whether the motion of water
shall be direct or sinuous, and of the law of resistance in
parallel channels.,
Cambridge Univ.
Scientific Papers, 2,
Press,
London.
51-105,
(1966), Two problems in convection; Thermal
Robinson, J. L.,
instability in an infinite region, and finite amplitude
convection cells., Ph.D. Thesis, Massachusetts Institute of
Technology, Department of lethematics.
Rosenblat, S.,
flows.
(1968),
Part 1.
Centrifugal instability of time-dependent
Inviscid, periodic flows.,
J. Fluid Mech.,
33, 321-336.
Schlichting, H.,
(1968),
Boundary Iayer Theory.,
J. Kestin, Ed.,
McGraw Hill, New York.
Serrin, J., (1959a), On the stability of viscous fluid motions.,
Arch. Rational Mech. Anal., 3, 1-13.
Serrin, J., (1959b),
A note on the Existencie of periodic
solutions of the Navier-Stokes equations.,
Mech. Anal.,
3,
Arch. Rational
120-122.
Snyder, H. A., and Karlsson, S. K. F.,
(1 9 64.), Experiments on the
stability of Couette motion with a radial thermal gradient.,
Phys. Fluids, 7, 1696-1706.
Soberman, R. K.,
(1959),
Onset of convection in liquids
subject to transient heating from below.,
Phys. Fluids, 2,
131-13S
~II7
Sorger, P., (1966),
ber ein Variationsproblem aus der nichlinearen
Stabilit~tstheorie Zaher, inkompressibler Stromungen.,
Z. Angew.
Math. Phys.,
Sparrow, E. M.,
l7, 201-216.
Munro, W. D., and Jonsson, V.
K., (1964),
Instability of the flow between rotating cylinders: the wide-gap
problem.,
J. Fluid Mech.,
Stokes, G. G.,
(1886),
20, 35-46.
On the effect of the rotation of cylinders or
spheres about their own axes in increasing the logarithmic
decrement of the arc of vibration.,
Taylor, G. I.,
(1923),
Math, and Phys. Pap., 5, 207..4
Stability of a viscous liquid contained
between two rotating cylinders., Phil. Trans., A, 223, 289-343.
Tranter, C. J.,
(1956),
John Wiley and Sons,
Integral transforms in rathematical physics.,
New York.
Turner, J. S., and Frazel, R.,
turntable.,
(1958),
Construction of a one-meter
unpublished report.
Urabe, M., (1965),
Galerkin' s procedure f or non-linear periodic
systems., Arch. Rational Mech. Anal.,
Winney, H. F., (1932),
20, 120-152.
Rotary oscilation of a long circular
cylinder in a viscous f luid., Phil. Mg.,
(7),
14,
1026-1032.
BICGRAPHICAL SKETCH
Rory Thompson was born 10 May 1942 in Seattle, Washington
of Alix Saunders and Richard Cuthbert Thompson.
his
high schools, all
junior and two senior
in the San
a
Diego,
honors from
with
and graduated
area,
California
five elementary schools,
then
mother's kindergarten,
He attended
Will C.
Crawford High School there in 1959. He married Luella Hallas
Thomas
in
honors
from San
maths,
with a minor
1960.
Subsequently,
a
San
Diego
in 1962,
State
College
to M.I.T.
he
Earlier,
with high
na joring
received an M.S.
then
post-doctoral to
a
Service.
logical
in physics,
Ford Fellowship
Fellowship and
graduated
Diego State College
mathem~atics from
received
he
the
in
minor scholarships,
four
Dance Federation of
Caradian
Meteoro-
research
hardware
one from
including
He has
California.
computer
programming,
a dishwasher,
the Folk
minor
trainee at the
Navy Electronics Laboratory in San Diego
U.S.
and
well
worked as a
flower salesman, a
installator, a
He
a Hertz
teaching assistantships at San Diego State College, as
as
in
and later
gotten
had
1964.
in
a teacher of
,
and a programmer.
His
publications so far includet
'Goodness
Statist.,
'Bias
Assoc.,
Two-Sample Tests',
of Fit for
Feb.
1966, with C.
B.
Annals of
Bell and J.
of the Cramer -von Mises Test', J. Amer.
Mar.
SEva lua tion
M.
[ath.
Moser.
Statist.
1966.
of I,(b)
Similar Integrals',
'Antipodality of
=
Math.
211
Comput.
Continents and
n (x )/x) cos (bx) dx and
April 1967.
Oceans',
Science,
Sept.
1967.
'Distribution and Power of the Absolute Norrral Scores OneSample Test',
J. Amer.
Govindarajulu and K.A.
Statist. Assoc.
Doksum
Feb.
1968, with Z.