AN INVESTIGATION OF TRANSITION AND TURBULENCE
IN OSCILLATORY STOKES LAYERS
by
Rayhaneh Akhavan-Alizadeh
B.S., California Institute of Technology (1980)
S.M., Massachusetts Institute of Technology (1982)
Submitted to ths Department of Mechanical Engineering
in Partial Fulfillment of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE, 1987
c
Massachusetts Institute of Technology, 1987
Signature of Author
Dep ttment
of Mechanical Engineering
May 1987
Certified
by ,
Professor loger D. Kamm
Thesis Supervisor
Certified by
_. \.....
_
Profe sor Ascher H. Shapiro
Pfs Thesis Supervisor
Accepted by
Professor Ain A. Sonin ARCHIVES
Chairman, Department Committee on Graduate Studie§:~TE:,NLr.G;
JUL 0 2 1987
LIBRARIES
- _-
A] INVESTIGATION OF TRANSITION AND TURBULENCE
IN OSCILLATORY STOKES LAYERS
by
Rayhaneh Akhavan
Submitted to the Department of Mechanical Engineering
in May, 1987 in partial fulfillment of the requirements
for the Degree of Doctor of Philcsophy in
Mechanical Engineering
ABSTRACT
Transition to turbulence in bounded oscillatory Stokes
layers is investigated experimentally by LDV measurements
for flow in a pipe, and numerically by direct simulations
of the Navier-Stokes equations for flow in a channel. The
experiments indicate a transitional Reynolds number based on
Stokes boundary layer thickness of Re-500. The resulting
turbulent flow is characterized by the sudden appearance of
turbulence with the start of the deceleration phase of the
flow during the
"laminar"
to
reversion
cycle, and
acceleration phase. It is proposed that generation of
turbulence in this flow is due to turbulent "bursts" similar
in nature to those that occur for steady wall-bounded shear
flows. Furthermore relaminarization of the flow during the
acceleration phase is attributed to cessation of "bursting"
because of the favorable pressure gradients that are present
during this period.
Investigation of the stability of oscillatory channel
flow to various infinitesimal and finite-amplitude two- and
that;
(a) An
reveals
disturbances
three-dimensional
infinitesimal perturbation introduced in the boundary layer
would, over the course of a few cycles migrate to the center
of the channel where it would decay with viscous time scales
as predicted by a Floquet analysis. However, as long as the
disturbance is still localized within the boundary layer, it
would amplify with inviscid growth rates as predicted by
finite-amplitude twoNo
(b)
theories.
quasi-steady
dimensional equilibrium states were found; however, quasiequilibria (with viscous decay rates) were observed at
transitional Reynolds numbers. (c) The secondary instability
of these two-dimensional finite-amplitude quasi-equilibrium
states to infinitesimal three-dimensional perturbations can
successfully explain the critical Reynolds numbers and
turbulent flow structures that were observed in experiments.
Thesis Supervisors: Prof.
Prof.
Thesis Committee : Prof.
Prof.
Prof.
R.D.
A.H.
D.J.
C.F.
A.T.
KAMM
SHAPIRO
BENNEY
DEWEY
PATERA
-3-
ACKNOWLIEDGENTS
I would
like
to
extend
my
sincere
thesis supervisors Professors Roger
for their guidance
, patience
period of my stay at MIT.
group
could
have
I
given
independent growth as
Kamm and Ascher Shapiro
and
support throughout the
have always felt that no other
me
was
gratitude to my
as
provided
much
by
like to acknowledge
the
invaluable
Patera without whom
the
computational
opportunity
for
them. I would also
help of Professor Tony
aspects of his work
would not have been possible. Thanks also to Professor Steve
Orszag for helpful discussions in the earlier stages of this
work.
The friendship and assistance of all the members of the
Fluid Mechanics Lab. has provided a wonderful environment to
work in. In particular,
I
would
like to thank Dick Fenner
for all his help in constructing the experimental apparatus.
Finally, the love and support
of my family and my good
friend, Martin made this all possible.
-4-
TABLE OF CONTENTS
PAGE
1
.............................................
TITLE PAGE
2
...........................................
ABSTRACT
3
ACKNOWLEDGMENTS........................................
TABLE OF CO
4
NTENTS
......................................
I.
INTRODUCTION ......................................
II.
EXPERIMENTS.......................................
5
16
2.1
2.2
Experimental Methods .........................
Experimental Results .........................
2.2.1 Calibration Run ........................
2.2.2 Turbulent Flow Regime ..................
2.2.3 Effect of Re and A.....................
16
24
24
25
48
2.3
Discussion ...................................
78
III. NUMERICAL SIMULATIONS.
.
...........................
84
3.1
3.2
Problem Formulation and Numerical Methods.... 84
92
Small Amplitude Disturbances (Linear Theory)
3.2.1 Two- and Three-Dimensional Disturbances 94
3.2.2 The Linearized Problem ................. 96
3.2.3 Effect of Initial Conditions........... 98
3.2.4 Results ................................104
3.3 Finite Amplitude Disturbances................ 121
3.3.1 Two-Dimensional Disturbances........... 122
3.3.2 Three-Dimensional Instability......... 134
3.3.3 Secondary Instability and Transition... 141
IV.
SUMMARY AND CONCLUSIONS ........................... 147
REFERENCES............................................. 152
-5-
I.
INTRODUCTION
The flow induced in a
semi-infinite body of fluid by a
plate undergoing sinusoidal
oscillations
was originally investigated by
moves with velocity
Stokes
Upcoswt,
parallel to the plate, in
boundary
layer
then the unidirectional flow,
the surrounding fluid is given by
thickness.
The
representative boundary layer in
involving periodic flows within
Our concern is with
the
its own plane
[1851). If the plate
u(z,t) - Upexp(-z/6)cos(wt-z/6), where
of
in
a
6v2/w;
Stokes
is a measure
layer
is
the
large class of problems
or around rigid boundaries.
stability
and turbulent motion of
the bounded oscillatory Stokes layer.
Previous work on transition in purely oscillatory flows
is rather
scarce.
Here
"purely" oscillatory flows
modulated about some
we
make
(about
non-zero
the
a
mean.
distinction between
zero
The
mean) and flows
two problems are
quite different, as the instability of flows modulated about
a non-zero mean is usually associated with the (steady) mean
flow. In this case, the modulations can usually be accounted
for
by
perturbation
contrasts with the
methods
situation
about
for
where the basic flow is unsteady.
the
mean
flow. This
purely oscillatory flows
-6-
A
number
literature
of
on
oscillatory
experiments
the
flows.
problem
new
of
these
As
performed in bounded
channels), a
have
reported
transition
experiments
geometries
parameter
been
(such
as
describing
are
in
the
in
in
purely
generally
pipes or in
ratio
of the
physical dimensions of the flow (for example, pipe radius or
channel half-width) to
is introduced
problem; A-d/2vw/2v . Alternatively, A
- the
can be thought of as
of the flow to the
is related
r=d/2v
to
, by the
ratio of the diffusive time scale
the
period
the
Stokes boundary layer thickness
the
of oscillation. Note also that A
known
parameter
relation
A-/
.
that
Womersley number,
All investigators agree
A
(2£A) transition is governed
is
based on the Stokes boundary
that for sufficiently large
by a Reynolds number
as
layer thickness. Furthermore, a value of Re 6 on the order of
500-550 is more
often
cited
number. Here, Re =-U
0 6/,
as
the transitional Reynolds
where U o is the amplitude of cross-
sectional mean velocity and
layer thickness. For example,
Sergeev [1966] found the
6-vFi/w
for
is the Stokes boundary
flow
in a straight pipe
transitional Reynoldz number to be
while Hino, et al.
Re -500 for the range of 2.8<A<28;
[1976]
and Ohmi, et al. [1982] each found the transitional Reynolds
number to be Re
and 2<A<29
the
range of 2A<6
(Hino, et al.)
al.),
respectively.
In a different
550 for
(Ohmi,
et
geometry, Li [1954] observed turbulence for Re6>565 for flow
in a channel with
for
the
an
transitional
oscillating
Reynolds
bottom. The value of 500
number,
however,
is
not
-7-
that the flow in a
found the flow due
plate to become
pipe became turbulent at a Re6
resonant
while
cf 280 (30<A<50);
and Thomann [1975] observed
Merkli
universal; for example,
[1957] and Collins
Vincent
to
at
turbulent
waves on a smooth
gravity
periodic
[1963]
numbers of 110 and
Reynolds
160, respectively.
studies, the works of Hino
Of the various experimental
most
detailed. They distinguish four
their
experiments: (I) laminar flows,
and coworkers are the
in
classes of flows
(II) distorted laminar or
characterized
by
weakly turbulent flows, which are
appearance
the
small
of
amplitude
perturbations in the early stages of the acceleration phase,
(III)
turbulent
explosive appearance of
deceleration
during the
which
flows,
acceleration
which
recover
phase,
(IV)
observed
by
Ohmi, et al. observed
bursts occurred
in
any
that
the
the
to
by
the
start of the
"laminar" flow
turbulent flows that
cycle. Flow regime (IV) has
remain turbulent throughout the
not been
with
turbulence
and
phase
characterized
are
the investigators, although
of
was increased, turbulent
as Re
later
of the acceleration
stages
phase as well as in the deceleration phase.
In the experiments
of
Hino,
the transitional Reynolds number of
flow regime (III)
observed by Hino
independent
and
coworkers
al. and Ohmi, et al.,
et
of
550 marked the onset of
A.
over
Flow regime (II) was
a
range of Reynolds
-8-
numbers
6
Re mitRe
6
550,
where
dependent on A; for example,
observed for 70Re
was observed for
hand,
found
the
value
for A6.2
6
of
Re min
was
flow regime (II) was
(550 while for A-1.9 the same flow regime
550.
460<Re
onset
that
Ohmi,
of
flow
et
al.
regime
on the other
(II)
in
their
experiments agreed well with the critical Reynolds number of
Re -280 which was proposed
words, flow regime (II)
280<Re <550
Thomann
between
by
was
independent
of
[1975] themselves
flows of type
Merkli and Thomann. In other
observed
the
do
(II)
by
value
not
and
Ohmi, et al. for
of
make
A.
any
Merkli and
distinction
(III), and therefore
it is
not possible to determine in giving the value of 280 for the
transitional Reynolds number
which
type
of flow they were
referring to.
In
general,
oscillatory
when
flows,
reference
the
intended. The details of
was studied by
Hino
rectangular duct at
mean
velocity,
the
and
regime
the
is
(III)
to
described
turbulent
above is
flow structure in this regime
coworkers
Re =876
made
and
[1983]
A12.8
for
. The profiles of
intensities,
turbulence
flow in a
the Reynolds
stresses and the turbulent energy production rate were found
to be
completely
different
between
the
accelerating and
decelerating phases, and in either case quite different from
turbulence
in
flows
which
Nevertheless, they concluded
are
that
steady
in
the
mean.
the basic processes such
as ejection, sweep and interaction towards and away from the
wall are the same as those of 'steady' wall turbulence.
-9-
of
the
stability of Stokes
new.
The
mathematics
of the
case
quite
Theoretical investigation
layers
is
also
relatively
even
problem,
the
for
'linearized'
arises
complicated. The complexity
disturbances
because in the presence
flow, the linearized equations for
of a time-periodic basic
the
is
are
that
coefficients
have
periodic
the time dependence of the
functions of time; consequently,
perturbation is not separable and the method of normal modes
suggested
have been
analysis
by
Two classes of solutions
applicable.
is no more (formally)
in
the
theory
Floquet
for this problem;
literature
and
solution
based
on
the
assumption of quasi-steadiness. Von Kerczek and Davis [1974]
noted that since the
linearized equations have coefficients
time, the results of Floquet
that are periodic functions of
theory [Coddington and Levinson
employed to predict that
disturbance
(x,z,r)=e
stream
function
time normalized to the
the
net
within each cycle. The
from one cycle to the
the
form
r=wt is the
wavenumber (with x normalized to
is the Floquet exponent and
A
(and
cycle
periodic function of time with
flow and represents the
of
oscillation of the basic
of
growth
disturbance field from one
be
conjugate. Here,
the Stokes layer thickness),
represents
should
period
streamwise
Yih (1968)] can be
periodic steady state the
the
in
x+Ar(z,)+complex
flow, a is the
(1955),
phase
to
the
change)
next, and
of
the
is a
the same period as the basic
variation
of the disturbance field
disturbances experience a net growth
next
if
Real(A)>O . Von
erczek and
-10-
Davis used a Galerkin technique to numerically integrate the
resulting system of equations for the geometry consisting of
an oscillating plate with a second stationary plate placed a
distance of eight boundary
range
of
wavenumbers
O<Re <800,
they
thicknesses away. For the
0.3<a<1.3
found
negative, indicating a
layer
A
to
net
and
be
real
decay
Reynolds
and
in
numbers
all cases
of disturbances from one
cycle to the next. The periodic part of the stream function,
O(z,r), was also found
within a
cycle;
to experience no considerable growth
thereby
ruling
within a cycle the disturbances
that the flow
might
von Kercz.k and
absolutely
range
of
may
the possibility that
grow to such an extent
become nonlinearly unstable. Therefore
Davis
stable
out
concluded
to
parameters
that
infinitesimal
investigated
the Stokes layer is
disturbances
and
possibly
for the
for
all
Reynolds numbers.
As a corollary
to
their
point out that since the
work,
principal Floquet exponent A found
in their numerical results was
in
conclude that the
for
time
von Kerczek and Davis
scale
all cases 0(1), one must
the
growth or decay of
linear disturbances is of the same order of magnitude as the
time scale of
oscillation
of
the
basic
flow. This would
serve as a warning against the use of quasi-steady theories;
as the quasi-steady assumption
would
be
disturbances varied much faster than 1/w.
valid only if the
-l-
Hall [1978] investigated
infinite Stokes layer using
the
linear
Floquet
stability of the
theory, and found this
flow to be also stable for all Reynolds numbers investigated
(O0Re <300). In addition
to
a
the Orr-Sommerfeld equation
was found to also have
for all
Reynolds
associated
for
this
considered the effect
The
of
and
only a discrete set of
'finite'
a
Hall
also
second boundary a
as was done in the problem
Davis. For the latter problem
eigenvalues exists. Hall showed that
disturbance
of
the 'finite' Stokes layer
the least stable discrete eigenvalue of
Stokes
Stokes
spectrum.
introducing
addressed by von Kerczek
the infinite
dangerous modes were
continuous
layer,
was not the same as
infinite Stokes layer
most
long way from the Stokes
the least stable
the
set of eigenvalues,
a continuous spectrum of eigenvalues
numbers.
with
discrete
layer;
layer
continuous spectrum
of
were
the
separation between the two
rather
the
found
to
infinite
eigenmodes of the
'merge'
Stokes
into
the
layer as the
plates
tended to infinity. Thus
he concluded that the disturbances
given by von Kerczek and
Davis have no connection
with
those of the infinite Stokes
layer.
Despite the objections raised
against the use
stability
of
of
quasi-steady
Stokes
steadiness is a fair
layers,
by von Kerozek and Davis
theories
one
assumption
inflectional instabilities of the
if
can
in studying the
show
that
quasi-
one is considering the
Stokes layer. First, note
-12-
that at all time during
infinite
number
of
the
cycle
inflection
Fjortoft's criterion
for
flat plate oscillating
plate
progresses within a cycle.
these
of
oscillation
flow
Upcoswt
scale
may
for
become
by
inviscidly
the growth or decay of
is
much shorter than the
(11/w); (6/Up)-/Re
equation for a succession
there is an
away from the wall as time
points
neglect the time dependence of
stability of the
example, for a
(y/-0=O) at wt=O (as well as
inflectional instabilities (6/Up)
period
satisfying
At asymptotically large Reynolds
inflection
unstable. Since the time
each
For
velocity
other ones at y/6=nr) which move
numbers,
points,
instability.
with
inflection point on the
the Stokes layer has an
(l11w), one can
the
base flow and study the
solving
the (inviscid) Rayleigh
of quasi-steady velocity profiles
for different times during the cycle. This approach has been
adopted
by
Cowley
1985]
and
[1985]. One problem with this
normal
modes,
which
are
as
they
move
type
Monkewitz
character
away
difficult to compare these
from
with
from
the
results
and Bunster
of analysis is that the
associated
points, completely change
next
by
wall.
the inflection
one cycle to the
This
makes it
to those of the Floquet
theory (which assumes the flow has reached a periodic steady
state). Monkewitz and
mode associated with
Bunster
an
inflection
the wall would pass its energy
inflection point closer to
out
in
the
free
exponentially weak.
suggest
stream
the
a
scenario where a
point moving away from
to a mode associated with an
wall, before it gets damped
where
inflection
points
are
-13-
Cowley
analysis
[1985]
carried
the
stability
of
out
of
asymptotically large Reynolds
interaction between
analysis.
His
growth rates were observed
before the start
of
the
modes
showed
wavelengths comparable to 6
the
inviscid
numbers
different
results
an
can
quasi--steady
Stokes
layer
for
where the effects of
was
that
included
in the
disturbances
with
grow. The most significant
between
r/4 and r/2; i.e,
acceleration
ust
phase of the cycle.
None of these modes were found to evolve in such a way as to
return to their
original
factor) after one
theory.
cycle,
Therefore,
as
(multiplied
would
although
necessarily incompatible
theory, no direct
state
with
be required by Floquet
these
the
comparison
by a constant
results
results
between
the
are
not
of the Floquet
two theories is
possible either.
In short, the present
to the stability of flat
status of knowledge with regards
Stokes layers can be summarized as
follows. On the one hand, experimental results show that the
flow undergoes
start of the
violent
transition
deceleration
numbers on the
order
other hand, have
so
phase
to
of
turbulence with the
the cycle at Reynolds
of
500.
Theoretical studies, on the
far
been
restricted to infinitesimal
disturbances and either
predict
the
stable in the
steady
state
periodic
flow to be absolutely
(Floquet theory), or
consist of analyses which are valid for asymptotically large
Reynolds numbers and
predict
transient
instability but at
-14-
phases during
the
cycle
opposite
to
where turbulence is
observed in experiments (inviscid quasi-steady theories). In
either case no
for
critical
comparison
to
transitional Reynolds
Reynolds
the
number can be identified
experimental
number.
Clearly
value
a
for
the
large gap exists
between theory and experiment in the problem of stability of
flat Stokes layers.
In the present work
this
gap
by
we
performing
numerical studies
of
have
tried
simultaneous
bounded
to bridge some of
experimental
oscillatory
and
Stokes flows at
transitional and turbulent Reynolds numbers. The experiments
were performed in circular
pipes. We felt these experiments
were necessary; because even
have
been
reported
in
though a number of experiments
literature
on
the
stability
of
oscillatory Stokes layers, the only detailed work is that of
Hino and coworkers
flow condition,
1983]
Re =876,
experimental results
with, and serve as
are
a
Hino and coworkers. We
which
is
A=12.8
for
. As
the
confirmation
believe
restricted to a single
most
of
we
shall see, our
part in agreement
the earlier work of
this confirmation by itself
is quite important. Furthermore, comparisons between the two
sets of experiments has allowed
us to draw conclusions upon
key features common to both sets that were neglected (or may
have been rejected as noise) by Hino and coworkers.
-15-
To
identify
the
nature
explain the experimentally
Reynolds
numbers,
and
of
disturbances
observed
to
gain
that could
values of transitional
an
understanding
of the
physics of the transition back and forth between the laminar
and turbulent
states
that
values of Reynolds number
direct numerical
three-dimensional
is
of
Navier-Stokes
ease of computation
channel
and
in experiments at
above Retrans ; we have performed
simulations
flow in a channel. The
observed
in
the
full, time-dependent
equation
for
oscillatory
geometry was chosen for the
light
of
the close agreement
between our experimental results for flow in a pipe, and the
experiments of
Hino,
et
al.
which
were
performed
in a
channel. The channel flow is subjected to initial conditions
that consist
of
various
dimensional infinitesimal
combinations
of
two- and three-
and finite-amplitude disturbances
and the time evolution of the disturbances are followed. The
disturbances that most
observed
features
are
closely reproduce the experimentally
identified,
mechanisms of transition is discussed.
and
the
dynamics and
-16-
II.
EXPERIMENTS
Experimental Methods
2.1
were
The experiments
pipe, 2 cm in
Figure
1. The
such that under all
chosen
was
pipe
length of the quartz
in
schematically
shown
experimental setup
8 meters long using the
and
diameter
inner
a quartz circular
in
performed
flow conditions the tube length was at least three times the
stroke length
of
the
flow
in
quartz
the
pipe; thereby
ensuring that the flow in the middle section of the pipe was
any
not suffering from
tube geometry was
A
effects.
end
chosen for the experiments over a two-dimensional channel or
the great care required to
an oscillating plate, because of
channel and the complex flow
ensure two-dimensionality in a
systems that would be required for an oscillating plate. The
choice of a
critical,
particular
as
correspondence
earlier
between
studies
experimental
the
seem to be too
not
does
geometry
structure
of
show
the
a close
turbulent
oscillatory flows in all three geometries.
The sinusoidal
generated by a
flow
piston
electrohydraulic
whose
servovalve
pipe was
into
the
quartz
motion
was
controlled by an
rate
(
MOOG
model
76-102
) with
feedback control on the velocity. The sinusoidal output of a
function generator
served
as
control circuitry. The quality
the
of
input
command
for the
the sinusoidal flow rate
-17-
RA
if
j
es
I-
tb
Cd
.
p,
Cd
t2o ·,
Q
Q3
E
0)
)
000
TO
TO
aO
00
.,1
-i
0
Ca
I
I
I
-1
I
I
I
I
3
v
o
0
v
-- -
-18-
was checked by monitoring the
generated by this flow system
motion of
the
piston
and
6L3-VTZ)
Schaevitz model
a
with
transducer (LVDT,
velocity
computing the flow
by
also
rates based on LDA measurements of the velocity profiles. In
an
each case
results of
FFT
these
was
fit
two
measurements
the output. The
on
performed
for
the
flow rate of
poorest sinusoidal quality is shown in Figures 2a and 2b. As
seen in these
deviation from a pure
maximum
the
figures,
sinusoid occurs near the point
zero flow. At this point
of
the O-ring seals on the piston,
because of some backlash in
the motion of the piston could not follow the input command.
on
Despite considerable effort
our
to overcome this
part
eliminated. There was no mean
backlash, the problem was not
flow.
Velocity measurements were
in
color laser doppler anemometer
and green (514 nm)
lines
simultaneously
measure
of
an
components. The LDA was equipped
backscatter
operated in the
signal
The
processing.
20x20x100 microns
front lense in
and
the
LDA
fringes
system
laser were used to
Argon
and
was
velocity
were used for
Counters
volume
were
radial
a Bragg cell and was
with
mode.
probe
the blue (488 nm)
which
axial
the
a DISA two
using
obtained
2
had
dimensions
of
microns apart. The
mounted on a two-axis
linear translation stage with micrometer control of position
to within
position
.0001".
the
probe
This
traversing
volume
at
diameter within the quartz pipe.
mechanism
selected
was used to
points
along
a
-19-
.6
.4
.2
-. e
E
0
-. 2
-. 4
-.
7r/2
37r/2
wt
Figure
2.
Cross-sectional mean velocity calculated from the
output of the LVDT that was used to measure the
velocity of the piston.
-20-
.U
.8
.6
.4
0
-
-. 2
-. 4
-. 6
-.8
-1
n
-
2
7/2
wt
3r/2
Figure 2ab Cross-sectional mean velocity calculated by
integrating the velocity profiles that were
obtained by the LDA system.
-2 1-
arrival of each doppler
accomplished by a
The computer was
was
burst
in
flow
basic
motion of the
flow
basic
the
Because
into
triggered
to
system,
the
monitored the motion of
few
time
the
cycles,
the
measured to evaluate the
This period was then
1/240th
register
of
a
cycle
within
the
trigger
point
was
the velocity transducer that
of
determined from the output
computer clock was
the
The
counting.
start
and A/D converters. In
corresponding to (positive-going)
point
rate
clock, a DMA
internal
(DRV11-J),
parallel interface board
zero flow
an
with
equipped
a
be determined. This was
PDP11/23 data acquisition computer.
MINC
every cycle, at
time of
relation to the periodic
to
had
the
unsteady,
piston. For the first
driving
triggers
was
between
successive
precise
period of the basic flow.
used
to
set
it
would
overflow
computer.
indicate time in fractions
The register was reset at
the clock so that every
This
and
register
increment a
would then
of
1/240th of the cycle period.
the
beginning of each cycle. The
data acquisition protocol can then be summarized as follows.
Once both LDA counters had
15
se
of each
other),
verified a doppler burst (within
the
computer would be interrupted
via the parallel interface board.
1/240tb
f the period of
the
The time in increments of
cycle
would be read from the
appropriate register, and the doppler information consisting
of the doppler frequencies, the number of fringes associated
with each doppler burst and the
computer for
later
processing.
time would be stored in the
In
addition
to the above
_9--
information, the
quartz
pipe
transducer)
instantaneous
(measured
and
the
with
flow
measured by the velocity
pressure
a
rate
drop
through the
Microswitch
through
the
pressure
system
(as
transducer attached to the piston)
were sampled through the A/D
converters 240 times per cycle
and stored.
Typically on the order
of
were processed in this manner.
the amount of seeding
that
2000 doppler bursts per sec
The rate limiting factor was
could
be
used before the flow
became opaque.
The
optical
distortion
of
eliminated by index-matching
quartz (nd - 1.458)
using
the
an
thiocyanate. The solution was
of refraction was
achieved
ammonium thiocyanate salt
refractometer (Extech,
range
measure the index of
working
pipe
fluid to that of
aqueous solution of ammonium
distilled water).A hand held
K502192)
+0.2%
with
accuracy
0-90% sucrose
was
used
, v-1.32 centistokes) .
windows
immersed
in
to
refraction of the solution. Otherwise,
the solution had properties very close to water (pl.1
pipe was
was
mixed until the correct index
and
and
quartz
(approximately equal weights of
model
(nd-1.333-1.512)
the
At
a
g/cm3
the measurement site the quartz
rectangular
chamber
with quartz
which was filled with the same index-matched fluid.
-23-
A
variety
of
particles
seeding
latex, silicon carbide
and
microns) was chosen
signal to noise
over
ratio
scales of turbulence.
the
and
of
diameter
others
ability
alumina,
dioxde were examined.
titanium
Titanium dioxide (with particles
including
because
to
from .5 to 1
of its good
follow the smaller
-24-
2.2 Experimental Results
2.2.1 Calibration Run
To check the
the
velocity
of ouz,experimental methods,
reliability
profiles
for
an
(Re -233, A-3.6) were measured.
equations of motion have a
1955] and
Womersley
check on the
rate, Q=Qocoswt,
the
In
of
laminar
the laminar regime, the
and
[1956],
results.
instantaneous
For
provide a direct
a sinusoidal flow
velocity
profiles are
given by,
u(r,wt)
B cos(wt+6) + (l-A) sin(wt+6)
+ CD/)(1)
I (1-C/)
U0
where,
v
- A/V2
(note that xc
ber(c) ber(rlv)
Womersley number)
+ bei(c) bei(rvW)
2
A
ber
()
bei(x) ber(rvV/)
B
+ bei
2
()
- ber(c) bei(rvlw)
ber2 (n) + bei 2 (c)
ber(x) bei'(x) - bei(c) ber'(c)
C
2
ber2(i)
+ bei2(I)
ber(c) ber'(c) + bei(x) bei'(n)
D
6
ber2 (c) + bei2(c)
tan-1 {
1-C/v
D/v
flow
closed form solution as given by
Uchida
experimental
example
-25-
experiments and the theoretical
The comparison between
solution of Uchida is
3 and 4. Figure 3
Figures
in
shown
shows the time evolution of axial velocity at various radial
The
locations.
experimental velocity profiles for various
In Figure 4, the
cycle
the
phases during
curve. The agreement
theory
backlash in
the
from
deviated
this
phase
O-ring
seals,
the
pure
sinusoid
as
a
experiment is in
'turnaround' in the motion
to
At
piston.
driving
and
maximum deviation from theory occurs
at wt-r/2, which corresponds
the
to the theoretical
compared
are
between
general very good. The
of
with respect to the core.
lead
boundary layer have a phase
in the
points
the
however
location,
each
at
variation
sinusoidal
predicts
solution
theoretical
because
motion
already
of some
of the piston
discussed in
section 2.1 .
2.2.2
Turbulent Flow Regime
Hino et al. [1976] have fairly extensively investigated
regime
the transitional
already
discussed
observed
two
experiments.
in
classes
One
class
an
in
detail
of
of
small
in
can
be
and
amplitude
superimposed on the laminar
the
pipe flow. As
introduction,
non-laminar
laminar" or "weakly turbulent"
appearance
oscillatory
flows
identified
in
they
their
as "distorted
is characterized by the
disturbances
that
are
profile during the acceleration
-26-
1%
* Z-
.1
0
C-
u
_
E
1::
-/
2
ir/2
37r/2
wt
Figure
3.
Phase variation of axial velocity at fixed radial
locations compared to theory in the laminar flow
regime. (Re 6 -233, A-3.6)
-27-
An
..
.1
.I
'C
-. 1
-. 1
-. 2
.i
.05
0
-. 05
-. 10
-. 15
I
-l
r/R
Figure
4.
-1.0
-. 8
-. 6
-. 4
-. 2
0
.2
.4
.6
r/R
Experimental velocity profiles at various phases
compared to theory for laminar flow. (Re -233,
A-3.6)
.8
1.0
-28-
phase of the
cycle.
of
type
other
The
flow
is what is
generally considered "turbulent" and is characterized by the
sudden explosive
of
appearance
turbulent
bursts with the
start of the deceleration phase and recovers to laminar flow
during the acceleration phase. Our main concern in this work
of the latter "turbulent" type
is with the characterization
of flow.
"turbulent"
for
The stability boundary
flows (of the
second kind described abcve) is very well defined in our, as
as
well
other
these flows were
(3<A<10)
investigators
experiments.
in
observed
experiments for Re >500
our
observations
with
consistent
In particular,
of
earlier
investigators.
the
In contrast,
laminar"
flows
seems
region
be
to
dominance
of
strongly
at
observed these flows for 70<Re <550
460<Re
<550 at a A of
1.9;
while
dependent
on
the
example, Hino, et al.
For
specific experimental apparatus.
of "distorted
a A of 6.2, and for
the same flow regime
was
observed by Ohmi, et al.
for 280<Re '550 independent of the
not
made any attempt to identify these
value of A. We have
flows in our experiments.
In
the
turbulent
measurements of the
A-10.6
Re -530,
flow
1080,
regime
we
field
for
1720;
and
have
made
detailed
four flow conditions;
A-5.?
Re -957.
The
-29-
turbulent flow will first
characteristics of the
resulting
be discussed in detail
for the flow condition corresponding
to Re6-1080, A-10.6 .
of the features demonstrated by
Most
this flow are common to all the flow conditions studied, and
considered as a prototype
be
may
point
as such this data
flow. The effect of different values for Re
and A will then
be discussed.
Figure 5
phase
the
shows
averaged axial velocity
variation
of the ensemble
different radial positions. The
at
ensemble-averaged velocity is defined as,
1
N
u(r,wt) - -
(2)
u (rwt±+t)
N J-1
order
where N is typically on the
cycle is resolved
to
2t=1/240th
within
particles through
transit
finite
the
the
of the period of
corrected for the bias errors
oscillation. The averages are
associated with
of 100, and phase of the
volume
probe
time
of
Buchave,
the seeding
1979]. The
solid lines are the theoretical laminar curve, and are given
for reference.
The most striking feature
of
these velocity traces is
the explosive appearance of turbulence with the start of the
deceleration
phase
of
the
cycle,
which
transfers
the
momentum of the high
speed
fluid
towards the wall and the
momentum of the
speed
fluid
towards the center. This
low
momentum exchange appears in
the
velocity traces of Figure
-30-
2
1
0
Q
"
U
r/2
wt
Figure-fa.
For caption see Figure 5c.
3r/ 2
-31-
2
1
0
U
0
U
a
1E1
0
7r/2
wt
Figure
5b.
For caption see Figure 5c.
37r/2
-32-
A
I
O
-
4)D
-X/2
Figure
r/2
wt
5c.
37r/2
Phase variation of the ensemble averaged
turbulent axial velocity compared to the
laminar theoretical curve for various radial
positions. (Re6-1080, A-10.6)
-33-
5, as a dip at wt-O for 0.7<r0.9
and a simultaneous hump in
the velocity trace for 0.9<r<1.0 .
A direct consequence of this enhanced momentum exchange
is that
the
phase
difference
boundary layer and the core
end of the deceleration
in
the
motion between the
is significantly reduced by the
phase.
This
point is best seen in
plots of instantaneous average velocity profiles (Figure 6).
Once again the solid
line
represents the laminar solution.
Note, for example, that at the time wt-r/2 (corresponding to
zero flow rate)
there
is
practically
no phase difference
between the boundary layer and the core.
In
general,
the
during the acceleration
r/2<wtr
) are similar
result in an
ensemble-averaged
phase
to
oscillating
of
the
velocity profiles
cycle ( -/2<wt,0
,
the
laminar profiles that would
pipe
flow, starting from initial
conditions u(r,O)-O . These profiles, however, are different
from Uchida's solution because the
flow has not yet reached
a periodic steady state. As confirmation of this point, note
that the thickness of the boundary layer in the acceleration
phase
grows
approximately
in
proportion
to
v,
as in
Stokes' first problem.
During the deceleration phase of the cycle, ( Owt(r/2,
r<wt<3r/2 ) the
Just
outside
average
the
velocity
boundary
profiles are inflexional
layers.
The
shape
of
these
-34-
Wt
-
-r/3
r/6
0
r/3
r/2
1.0
.a
.a
.37
.4
.3
.2
.$
0
U (m/sec)
27/3
5/6
r
7r/6
4r/3
1.0
.0
.
I
-
.
. 4
.
.2
.1
0
-
U (m/sec)
Instantaneous ensemble averaged velocity profiles
in the turbulent flow regime compared to the
laminar theoretical curve. (Re -1080, A-10.6)
-35-
wt - - /3
-7r/6
r/3
7r/6
0
7/2
o00
a
o
ga
a
10-I
a
a
a 'O
a
aa C.
a -0
a
o
o
o
a
a
o
0
a
a
0
4
o
o
o
I1
I
a
4
a
aaa
0
a
o
a
44
0
o0-
a
o
C10
II
o
a
a
0
a
I
o
o
o
a
o
o
o
a
a
a
'
o
o
0
0o
a
o
0
o
o
o
oa
a
a
a
a
a
I.
a
a
a
o
C.
a
1
2r/ 3 5r/6
U (m/sec)
7r/6
47r/3
o10
o
o
0
o°
o
o
o
0a
o
a*OOC
oaO
o
0
0
0-t
o
aOC
o
o
*
ao
o
o
o
o
I-
o
o
o
o
o
0
o
o
o
C.
o
0
a
0
% o)o
0
o
aO
I 0-1
a
'
I 0
_
-2 I
L
-
^_
·
_
._
-
-
-
1
·
U (m/sec)
Figure 7.
Semi-logarithmic plots of ensemble averaged
axial velocity in the turbulent flow regime.
(Re -1080, A-10.6)
-36-
profiles is
very
similar
to
obtain in steady wall flows
turbulent burst, and
instantaneous
during
profiles that
the ejection phase of a
suggests
that the fundamental process
of generation of turbulence in
oscillatory pipe flow may be
the same as that of
steady
wall
flows. This point will be
discussed in more detail later.
The contrast between
the
acceleration phase and those
best shown in Figure 7,
velocity profiles during the
for
where these profiles are plotted on
semi-logarithmic scale. As seen
acceleration
phase
the
in
regions
logarithmic law are very
phase the
the deceleration phase is
this figure, during the
dominated
by
the
semi-
narrow, while for the deceleration
semi-logarithmic
region
extends
across
a much
larger portion of the tube.
Instantaneous turbulent velocities
difference
between
the
instantaneous
are
defined as the
and
the
ensemble-
averaged velocities,
u'(r,wt) - u(r,wt) - u(r,wt)
(3)
v'(r,wt) - v(r,wt) - v(r,wt)
(4)
The turbulent intensities (u
stress (u'v')
2
' 2)
and <v'2 >) and the Reynolds
can then be defined as,
u'2 (r,wt)
1
- N
N
2
u1 2 (r,wt+±t)
j-1
(5)
-37-
2 (r,wt)>
,v'
1
N
N
J-1
1
N
-
(u'v'(r,wt), -
d)2
, <v
shown in Figures 8
intensities
J-1
of
variation
intensities (<u'2
>)
axial
and
radial turbulence
various radial locations is
at
10. Throughout the pipe, turbulence
and
remain
(7)
uj(rwt+±t) v(r,wt+t)
N
The phase
(6)
v-2 (r,wt+6t)
at
a
very
acceleration portion of the
cycle.
low
level
during
the
However, with the start
of deceleration, turbulence appears abruptly in the boundary
layer ( 0.93
r ( 0.99 ) and propagates towards the center.
The abrupt generation of
turbulence during the deceleration
r - 0.94
phase seems to first occur at
turbulence levels rapidly
wt--r/8. Thereafter
intensify throughout the boundary
layer; with the axial turbulence intensity reaching its peak
value around r-0.98, wt-ir/10.During the later stages of the
intensities gradually drop to
deceleration phase turbulence
low levels.
Spatially,
substantially
turbulence
axial
reduced
as
the
approached. The distribution
across the pipe
at
different
intensities
center
of
of
the
pipe
are
is
axial turbulence intensity
phases
during
the cycle is
shown in Figure 9, and confirms the above arguments.
-38-
.1I
~~~~~~~~I
I
I
1-
I
~ -
r-O
_
s
r -. 1
0-^"c4
r-
~--~------
C1i- __
-.
_
_
2
r- . 2
__
U
4
0
r- .3
I.
S
I--
of,-%__l_
_
,~L
__C
----
I·
U-IDr
r-.4
N
I·u~~hC~
.
1
Al__
,,,^__ A . , ;·>
.
, ," ns~
r -.5
-0
__ W
-'I-~CI~·._~
r.6_
. .d_.:= CB's~,'·*PC~e'O
r
wo
I---
6
·
7r/2
wt
Figure 8a. For caption see Figure 8c.
37r/2
-39-
.1
I
I
r-.7
-.
I
K
-. 7
,,
I
r-.8
o0
C)
a,
r.86
N
*.,. r-. 89
~
5-p~
*·~2·
-
as'= *r.92
L*_ _X*
,,*q.
I
's .
·
.
I
I
/2
wt
Figuze
·
d'
b
as~ gsA
r~~~~~~~
-7r2r/2
.
r~·~LI~ks,,
8b- For caption see Figure 8c.
I
I
3/2
-40-
__
.1
·
I
s.
.
,r·
e
-ft.
-%
a
I
0
%ft-.
.1
I
S4.
N
. -140'
*.
.sr-.93
i
·· ·
._ no
A-
~~k·C~~ChlP
~
~
c ~AVd~~'
0*~~~~~~~~~~~~'
*%dsdPu1.,.*.*pAJ
..
.
.
*
_
-%'
~~,qL
_'S
-,P
.
· ·
,%t16·
0P
C
0
0P
*
0
;... o
*
· 0.
;dla
'
-
E
·
0
r - . 98
%'
00.s
.
;
, - - e
+r
'?
0
,
::J
r-. 97
r:Wa· I -
r
.0
-
0
04111
X
o
-
.f.
P·Bq.~
, A. .
,., ,
w 00
*6%,·
-?·
0.
C4
R
0,%e
* 000.
0-...
n
1 p~muAin
0
r0*-.- . 96
96
~
"4S~,0*
·
.
*
..-. .......
1.
,
.
.
~b~b~sC~'NOWE
v
'
_
0~
.
.-
I
rys-
_
cL
0
W" -
0
I
7r/2
2 ~~~~~~.0
r-.99
. _c~rc~pe.. .
I
37r/2
wt
Phase variation of axial turbulence intensities
-10.6)
at various radial positions. (Re -1080,
-41-
wt
-r/3
-7r/6
0
7r/6
7r/3
7r/2
.0O
. a
.8
.7-
.
.2
.o
0
u'
27r/3
57r/6
7r
77r/6
,*t
(m/scc )
47r/3
L.0
.
.
.3
I
.
.4
.3
.Z
.1
O
u'
Figure
9.
(m/sec)4
Distribution of axial turbulence intensity across
the pipe radius for various phases. (Re -1080,
A-10.6)
-42-
Radial turbulence intensities
magnitudes on the order
and 11). In addition,
pipe
is
more
<v' 2 , is very similar
1/5th
the
uniform
Notwithstanding these
of
The Reynolds
stress
cross-section
at
with
gradually
begin
the
to
u'2 ) (see Figures 10
of
a
given
differences,
phase
the
(Figure
11).
phase variation of
u' 2 > .
to that of
of
Reynolds stresses (<u'v'>)
for selected phases during the cycle.
is
almost
throughout
cycle, but
in general have
distribution of <v'2> across the
The radial distribution
is shown in Figure 12
(<v'2>),
the
start
grow
negligible
acceleration
of
within
the
over the whole
phase
of
the
deceleration phase it
the
boundary
layer and
spreads upwards towards the center as time progresses within
the cycle. By the end of the deceleration phase the Reynolds
stresses once again reduce to very low levels.
-43-
I I
.u-
I
.
I
I
r=O.
0
I
r=.l 1
I
r=.2
14)
o
. r=.3
.-*.
c e-
r= .4
r=. 5
.0
.0
0~e~cs'
·
.
I
- r2
1
*
.
..
·
_-Idbl
.
*
··
*.
.
r- .6
w0C%
I
ir/2
wt
Figure la. For caption see figure 10c.
-
..
cr·o emL
·
I
*
-
0
~Waq
4
I
37r/2
-44-
.02
_
I
·I
·
I
_...·
-%-
.- r=.7
.'
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rs·~~u~hw~
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0
0-~
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For caption see figure 10c.
*
92
e~
--
37r/2
-45-
_
.02
%
,.p?
._
~Bb~d.
f...00~%
0
**
0.
·Ul1
_~__~L_~,s
~_iyM%
1
1
I
I
0
0.
:,'N,,~~~~~~~*
0
e .·
o
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qr
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.
r=94
·
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,
·
*
.
·
..
0
.j,~.,P-
r=.93
0
.'.
i!
.
~,~.~.,
F
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..*
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.
L
..
0
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.
r=.96
.
.*.:%
%'6.
,--a~
%-'·
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.
r=.97
-amp'...
N
4)
0
-
v
.
-- %. "OF w
a_
-
,_
....
r·
aa
ee
------IILL
.
I:
0
-r
M ~P;C
er b
0.
L
k=C··
.
·
o
/.
r=.98
-
r=.99
0
-
s
r~-
L~e-F
--
1W
.· i·~~·
r= .992
I-
I-
r/2
-
37r/2
wt
Figure 10c, Phase variation of radial turbulence intensities
at various radial positions. (Re -1080, A-10.6)
-46-
wt - -X71/3
7r/6
0
7r/3
7r/2
.0o
.8
.
..S
.4
.,
.2
.1
0
v' 2 (m/sec) 2
27r/3
57r/6
7-
77r/6
4r/3
.0
.a
.
.s
..
.2
.1
0
Rae
c.
v' 2 (m/sec)
Figure 11. Distribution of radial turbulence intensity
across the pipe radius for various phases.
(Re -1080, A-10.6)
-47-
wt
-/3
-r/6
0
7r/6
?r/3
7r/2
1.0
.
..
..- 4
.
.2
.
ao
r
2r/3
5r//6
u'v' (m/sec)"
7/6
4X/3
.a
.6
..
.a
1-
.s
.4
.a
.2
o1
a
11
u'v' (m/sec)'
;Figure
La
Distribution of Reynolds stress across the pipe
radius for various phases. (Re -1080, A-10.6)
-48-
Effect
2.2.3
of Re 6 and A
The effect of Re
and
A
on the characteristics of the
turbulent oscillatory pipe flow
has been briefly considered
by studying two other flows; one
with the same A (-10.6) as
the case discussed above, but
at
a higher Re s (-1720), and
the other at roughly the same
Re
(~1000) but at a lower A
for
the
(A-5.7) .
The experimental results
are shown in figures 13
through
the case discussed in the
the lower value of
are set up during
not as
severe.
A,
the
adverse pressure gradients that
Accordingly,
wt--r/8, r=.93)
section in that, due to
deceleration
during the later stages
at
20. This flow differs from
previous
the
phase of the cycle are
turbulence
when
adverse
pressure
the
not as violent in the
layer
boundary
the
pipe
seen in the velocity traces
to the backlash in the
observed only
gradients
are
observed turbulence is
but tends to be more
cross section. The "glitches"
of
motion
discussed in section 2.1 .
rate for this flow
is
of the deceleration phase (starting
sufficiently high. Furthermore,
spread out through
case Re -957, A-5.7
Figure 13 at wt-r/2 are due
of the driving piston as was
The curves of instantaneous flow
condition
were
2b, and manifest the same "glitch".
shown in Figures 2a and
-49-
1.0
.5
U
V
on
la
IP
ir/2
-7r/2
wt
Figure
13a.
For caption see Figure 13c.
37r/2
-50-
1.0
.5
TO
6)
IS
7r/2
wt
Figure
13b. For caption see Figure 13c.
37r/2
-51-
. U
.5
0
O
ID
-7r/ 2r2
37/2
t
Figure 13cL Phase variation
turbulent
of
the
axial velocity
ensemble
averaged
compared to the
laminar theoretical curve for various radial
positions. (Re6-957, A-s.7)
-52-
wt - -X/3
0
7r/6
ir/3
7r/2
1.o0
.7
-
.
. 4
.3a
.2
.
0
U (m/sec)
2X/3 5r/6
X
7r/6
4r/3
.0
.a
.I
.
-
1.4
.3
1-
.
. 2
.
-1.0
-.
S
0
U (m/sec)
Instantaneous ensemble averaged velocity profiles
in the turbulent flow regime compared to the
laminar theoretical curve. (Re -957, A-5. 7)
-53-
0
wt - -7/3
o10
o
aoo
-
-
--
a
a
a
a
0
4
C
31
3.
II
C
a
o
oa
o
o
-a
-
o
1 0
10'
oa
oa
a
0
to0-
.
0
o
a
a
a
oa
a
a
a
ao
r/2
c)
a
a
o
o
-o
a
a
a
a
-
aaa
a
o
a
0
r/3
a
_-
I_.4
7/6
C
a
a
o
o
'C
C
C
0
c
,C
C
2
o
.6
27r/3
a
:)
0
CI
o)
1.0
U (m/sec)
5Sr/6
7r
7r/6
41r/3
I
I
4
0
I
U (m/sec)
Figure
15.
Semi-logarithmic plots of ensemble averaged
axial velocity in the turbulent flow regime.
(Re -957, A-5.7)
-54-
.05
_
__
I
I
Pd'.
0
Wp
-I
r=.l
0%
0..
0~~~~~--
r-.
-
b~Y~·
l~bWr/nL~L~L~
~Q~l~ia .,^f W%,%-·C
~~
w ---
V
=b .2
-
-r=. 2
~~~~~~~~~~~-A
%.r=.
3 .:a
N
_'i%
lu
0
- .
00
-1
S
r=.4
(4
=-I-Lc-~-~-~
Ld
P-6-
-
IPBbs
1U
0
-0
'
r=.5
.r= .6
b`·.~
*Sw;,s
-__A·l~s-r
~4~~.ch~hu~~~·r~kJ:I._r·
I
I
7r/2
wt
For caption see Figure 16c.
37r/2
-55-
. 05
__
_
r=.7
TO
r=.76
r= .8
r=.84
M
0
4a·-,
11
11
~
·--- · ~~~=:.
;
i~~~
~~gbg%
r=.86
9
bc"-lb
p-
--
---r
.
1Re
-
-'
0
·
0'
.v
......
-' TV,.
I
--- ~a*%
.n r
-%WdV%#M-a.
0
·
1
*~o
0 %,I.
h0, E P.A U~ E % %>
4
r-.89
V.
*
16./
r=.92
%
OI
011
1-P
I
7r/2
wt
Figure
16b. For caption see Figure 16c.
37r/2
-56-
.05
I1
r
_
ms~
0
, .w·.9_e,_g _
~~~~~Ib
./
.
_
w
~C.
' Cc'~.B'-~
%
r93.
C
0
-A10*
%
0
' ~.·- .. k~r=.96$··
-
·- "--I'---a
.,
ct
-
.4
0
0
AI~ h4_q___S,^.b*S
__
I%*
.
C4
I
*_"
4
d~~ci~c
r=.97-
r=.98:~~~~~~~~~~~~~~~~~~~~=
98r= .99_
,_c--
-r--as-
--
-- hCJ
r=.99;
I
II
7r/2
wt
Figure
- - - - --
37r/2
10c. Phase variation of axial turbulence intensities
at various radial positions. (Re -957, 5A5.7)
-57-
wt - -7r/3
0
r/ 6
r/3
7r/2
L.0
.o
.a
I
.6
11
.
.4
.5
.2
.1
0
27r/3
57r/6
Xir
?~/6
47r/3
/3
(m/sec ) 2
u'
(m/sec)2
L.O
oO
.8
.7
.6a
.5
.3
.2
· I
0
C:
A
Distribution of axial turbulence intensity across
the pipe radius for various phases. (Re -957,
A-5.?)
-58-
--.UU5
-
I
I
I
r=O.
A
:
r=.l
.. r=.2
7
*
go
A_
r=.
~~~~~r-
I
-X7r/2
7T/2
18a. For caption see figure
,~~~i '=.h
I
i
37/2
wt
Figure
A
18c.
-59-
.005
_
·
r=. 7
0
' r=.76
*:,...
-
·
l'e~~~=
,
-
0'~
N
t)
.
~
.
_,
*
qmV'
_________·--~c~
e~-~
C4
-
/.
r=
ck-~b4I~;5~4IELH'.
m
ft
;l
.
r=.84
. ..
f*
0
.
· ~~c
~
1% ·
..
;I,
'
-
.
dll
Am~w
-
^ur
_'f"
b
-
h
idP
.. ~~CL·/·
,:/ ,
-% ~
-r.89· CC/P
: :
.
r=.92.2
- -e
---~-----4U~L~I-t~l·F· -- A 0O--CM · ·---
I
!
7r/2
wt
For caption see figure 18c.
--
-
3r/2
-60-
.005
_
_
I·
_
_
_
r=.93
0
-- Y
13
rLvrrhr
--
I~C~
_*
A,1 e
v~--r
ww-"
-
m
-c-r
--
I-- -
-CT---C
----
--
I
--
1
---
RAN
-
--
-
·-
r=. 94
.V-WOc~te",
: r=.96
,.. ~·..
--
----
e' m*.*
ardh
- -
---
-
-
v
C;
r=.98
-r=.99
r=.992
7r/2
37r/2
Figure 18c. Phase variation of radial turbulence intensities
6
at various radial positions. (Re -957, A-.?7)
-61-
wt - -/3
0
7r/6
7r/3
7r/2
L.O
.O
.
.
1.4
- .* 4
.2
.1
0
2r/3
Sr /6
7r/6
4r/3
v'
,'
(m/sec)4
1.0
.
.a
.7
.
I
-
.5
.4
.5
.2
.I
0
O
.00
V,2
(m/sec)2
Distribution of radial turbulence intensity
across the pipe radius for various phases.
(Re -957 ,-5.7)
-62-
wt - -"r/3
0
r/6
7r/3
7r/2
1.0
.
.
.9
I
q,-4
.
.
.2
.1
0
rr
u'v' (m/sec)'
21r/3
57r/6
X
77r/6
4ir/3
1.0
.
.
.7
.
..
.
.2
.1
0
-
Figure
01
0
u'v' (m/sec)2
20. Distribution of Reynolds stress across the pipe
6
radius for various phases. (Re -957, A-5.7)
-63-
of
In contrast, the characteristics
same A but at a
Re 6
higher
the
28. In this case,
For
an
always be a region
instantaneous
mild favorable pressure gradient),
for a substantial portion of
turbulent
the flow may remain
flow,
oscillatory
near
the
phase
number
Reynolds
is
has
to
oscillation is large compared
of turbulence so that
no
stop.
to
however, there will
of zero flow where the
low
and
the
flow
is
gradient, and as a result
experiencing a favorable pressure
the bursting process
phase ( when the flow
Reynolds number is sufficiently high
suggesting that if the
the cycle.
of turbulent flow extends to
acceleration
a
is still experiencing
flow at the
is shown in figures 21 through
region
the later stages of the
the
If
the
time scale of
time scale of dissipation
residual
turbulence is left, the
flow has to relaminarize during this period. This point will
be discussed in more detail later.
-64-
q
2
0
a
m
7r/2
wt
Figure 21a.
For caption see Figure 21c.
37r/2
-65-
2
0
ar
1-1
0
r/2
wt
For caption see Figure 21c.
37r/2
-66-
q
2
0
O
I.:
-X/2
r/2
37/2
wt
Figure 21c. Phase variation of the ensemble averaged
turbulent axial velocity compared to the
laminar theoretical curve for various radial
positions. (Re6-1720, A-10.6)
-67-
wt - -/3
7r/6
0
7r/3
7r/2
1.0
.
.a
54
.
.4
.2
.2
.I
0
U (m/sec)
27r/3
57r/6
7r1
7 r/6
4ir/3
L.
e-Id
-
4-
!-
o
e
U (m/sec)
Instantaneous ensemble averaged velocity profiles
in the turbulent flow regime compared to the
laminar theoretical curve. (Re6 -1720, A-10.6)
-68-
wt - -7r/3
,r/6
0
-7r/6
r/3
7r/2
10
o
o
o
o
o
o
-
0
o
0
0o
ooo
0141
LO'-
:
-
a
-4
-
9
o
o
o
I
o
o
o
o
ooo
o
C
o
C
o
o
o
o
oo
o0
-40
a
o
o
o
o
o
o
o
0
o
o
a
oa
o
,C
,CI
a
O
Oa
o
(
·-
C
10-
·. ~mm
.a
2
i
27r/3
U (m/sec)
8
4
o4
o
ao
o
oo
o
o
o
o
oo
I
o
aa
O-I
o
oo
ao
a0
laM
o
o
o
0o
o
a0
0o
o
oa
o
o
8I%
a
o
o
47r/3
77r/6
7r
57r/ 6
100
o
oa0'o
oa
o
a o
o
oo
oa
a
a
a
0 -0'
C,'
o~~~
o~~~
o~~~
o°
o
10
~ ~
o-0
aO
I*C
o
O0C
Co
aO
9
1
0-1
I10--
.
-2
.
*
-
.
.,
.
I
.
.
.
._
o
U (m/sec)
Semi-logarithmic plots of ensemble averaged
axial velocity in the turbulent flow regime.
(Re 6-1720,
-10.6)
-69-
2
0
To
N
p
go
El
1-
-7r/2
7r/2
wt
For caption see Figure 24o.
37r/2
-70-
.2
0
o
o
0
,=
E!
0
(4
-7r/2
r/2
wt
Fiwure24b.For caption see Figure 24c.
37;/2
-71-
.2
UI ·
- _ ,~?~,
_
0
~
. ~. r=.93
sPr
.
r=.94
r=.96
.97_
c)
'-S
oil,
~~~~~~~~d
.0
I
-7r/2
I
lr/2
wt
He
.0~~,
. %'"-,"'.._r=. 99
_~~'~~~,
I
37r/2
Figure 24c. Phase variation of axial turbulence intensities
at various radial positions. (Re6-1720, A-10.6)
-72-
wt
- -7/3
0
ir/6
7r/3
77r/6
47r/3
7r/2
L.0
.o
.a
.7r
.
.
.
2r/3
5ir/6
u'
(m/sec)'
.O
.O
.a
.I
I_
.
4
.3
.2
.o
0
u,2 (m/sec)
Distribution of axial turbulence intensity across
the pipe radius for various phases. (Re -1720,
-10. 6)
-73-
. 04
-
I
I-
.02
r=O.
'S.
.-.
0
...
·
~
, .
--lE
i-
r
.AwW4.- . fto
v
.-
e'/be.S4h
:~.*...
....
' ~-.
u.
- r=.1
~
.
/...
i--
%O
O.rcr
ftA
,V..O~
%:r
r·
r~e
4-ks =rb:B
r= q
0
--
**
*%:.
0%
fwok
N.
1-1
-'. r
-A
r=.4
WM
0
**~A'I*.~~·i%
-
,
-
-
.
Jrr
-
-
%- . .01
~
~
~
~
I_
-r=
.9 ._14
*,.0
r
-t,--Jb~A
C
_
*O b-
· _·CI
· P~~~~~__
_ _~~~_
~~·_·(~~~.
_· itP.,
.,O.:~ ""
I
-J
"--: 'A.-O.m.
*
%W*J
·
0
r=. 6
e*
a
7r/2
wt
For caption see figure 26c.
-
-
-
-
S
37r/2
-74-
.02
0
%',
*,
-~~~~
.·
, L:.._____
__-_·.-____
hmm..m.mmh'"
¶-
.
*,,~,.
*
-0
I0
··,,.,..*
N
(4
0
0
ao
a
A
.~.
· S -,-,~"'.-*.,,,~,,~
~,
,
-P··k.
r;Y;
'Yl~ru,
r=.8
. -0:
0f
~
r.
u'*
W
;S'~~~~~~0
I
-
r= . 7
e.
....
_
_
__
_
O
0
0
*
· 0 o.
S
· 500
a
eel
f~b01.,
V'r=.
-
-
8
4-
.. : r=86
...
d='
% .C-· .
W
-
-r0r~h~·
:
s.s-
89.1I.r=.
*
lvvlv
0m-p
_--
q.V".hc
*
p
*~*~e
*4
I
-r/2
I
7r/2
wt
26b.
For caption see figure 26c.
r=.92
* I,
~~hIEI..%lPU.~l~VdIP'r
"ee
6w"
00
Figure
0
V
I
37r/2
-75-
~~ ~~~~~I
.04
_
'
.
_
=
I
_
I
.02
0
'_
., · .%
_
~~~~%ft
.%16-,V,
.
-_VIP
~ ~ +.+~
I
r
U
%
% 1%.0
__
_
-el
AO
C
1
-
-
__
-
~~~r
.94
'~_96
~
P)Erg~anoCI
^6md'O**040%,O%
*
r=.97
.. !-%
~ ~ ~~~~_
C
-
I
-
=
*4.
q
4
a~
.7:WI~;I(L~~b~L)
,-
%'-%,.%.
-
-
ir/2
-
-
1~~~~~
-
-
-
37r/2
wt
Fiure
26o,
Phase variation of radial turbulence intensities
at various radial positions. (Re -1720, A-10.6)
-76-
wt - -r/3
rr/6
0
7r/3
7r/2
1.0
.a
.O
.
.7
..
..I
..
.2
.1
0
1%
27r/3 Sr /6
v' 1 (m/sec)'
7r/6
47r/3
1.0
.a
.6
.O
.6
-
.5
.2
.1
0
o
v' 2 (m/sec)22
Figure 27. Distribution of radial turbulence intensity
across the pipe radius for various phases.
(Re -1720,A-10.6)
-77-
wt - -/3
-7r/6
0
r/6
r/3
Ir/2
1.0
.
.7-
i1>%
.-
.
.4
.2
.1
0
u'v'
27r/3
51r/6
Wr
77r/6
A
(m/sec)4
4r/3
1.0
.8
.8
.7
.8
.
I
.5
-
.
.2
.1
0
u'v' (/sec)
Figure 28. Distribution of Reynolds stress across the pipe
radius for various phases. (Re -1720, A-10.6)
2
-78-
2.3 Discussion
One of the major objectives
gain
some
processes
understanding
that
of
underlie
between the "laminar"
of
the
and
the present work is to
the
fundamental
transition
turbulent
back
physical
and
forth
states of oscillatory
pipe flow.
We have already
mentioned
that
the ensemble averaged
velocity profiles during the deceleration phase of the cycle
(when
the
flow
is
turbulent)
look
very
similar
to
instantaneous profiles that obtain during the ejection phase
of
turbulent
bursts
in
steady
[Kline,et.al (1967), Kovasznay,
profiles are inflectional
a result of momentum
near the wall and
the
and
flows
(1962)], in that the
ust outside the boundary layer as
exchange
between
higher
contrast, the profiles
cycle are full,
et.al
wall-bounded
for
speed fluid further away. In
the
suggest
the low speed fluid
acceleration
that
phase of the
the relaminarization of
flow during this period may be due to cessation of turbulent
bursts.
Let us start
steady wall
by
flows.
reviewing
We
can
the bursting phenomenon in
then
consider
the
effect of
-79-
of
acceleration or deceleration
process
and
decide
relaminarization
bursting is
of
the
flow
assumption
if
the
the
flo w
because
on the bursting
with
of
regard
cessation
to
of
ustifiable.
Consider the (inviscid) vorticity field associated with
a
uniform
shear
flow.
The
undisturbed flow consists
vorticity
of
component the
vortex
which
spanwise direction are deformed
and
these disturbances). Due
vortex a positive u3
tip of the
loop
velocity lifts up the
at
T)
U-shaped loops in the
[1960]
component
while
points
tip
originally in the
for the origin of
self-induction of the U-shaped
velocity
component is induced
29). If this flow is
were
into
Lin
to
(points
this
which has a streamwise vorticity
lines
xl-x2 plane (see Benney
for
spanwise vortex lines forming
sheets of transverse vorticity (Figure
perturbed by a disturbance
field
of
is induced at the
a negative u 3 velocity
V
(valleys). This induced
the
vortex bringing it into
regions of higher axial velocity, and pushes down the vortex
filament near the valleys into regions of low velocity. As a
result, the tip of the
vortex
higher axial velocity relative
would have to move
faster
of
but
to the valleys and therefore
the
to the valleys, giving
vortex filament and the
-shaped filament. At the same time, the the
original vortex filament is
direction
ends up in a region of
compared
rise to further stretching
formation of an
loop
acquires
a
no
more purely in the spanwise
strong
streamwise
component
-80-
Figure 29. Conceptual model of the bursting process in
steady wall-bounded shear flows (from Hinze,
Turbulence).
-81-
especially
around
the
legs
of
the
vortex
loop.
This
streamwise vorticity pumps the low speed fluid away from the
wall and results in a defect in instantaneous axial velocity
profiles which is accompanied
by
very intense shear layers
in the velocity profiles. A turbulent burst is the result of
local
inflectional
instability
of
these
intense
shear
layers.
when the flow is accelerating, because of the favorable
pressure gradients that
are
superimposed
profiles at the "peaks" cannot
result, the
streamwise
stretch as effectively
the flow the
become as inflectional. As a
vortex
and
on
filaments
the
bursting
cannot
lift and
process does not
occur.
Deceleration of the
fluid,
on
the
other hand, would
tend to enhance the "defect" in the velocity profiles at the
"peaks", and would result in more violent bursting.
For
the
oscillatory
favorable) pressure
gradients
with time. This variation,
individual
this
point,
oscillatory pipe flow
are
not
because
the
recall
convective time scales on
flow,
the
(adverse
or
constant, but vary
however, is immaterial during an
bursting process
burst is much shorter that
To see
pipe
the
varies
the time scale of the
time variation of the flow.
that
the
order
on
a
turbulent burst has
of (d/U), whereas the
time
scale (1/w). The
-82-
ratio of these two time scales is the Strouhal number - U/wd
which
is of the same order as Re /A ,, 1.0 .
The
relaminarization
acceleration phase
of
requires,
in
bursting during this period,
left in the flow from
the
eddies
the
of
that
limiting step in
and
the
the
to cessation of
decay of turbulence. In this
dissipation is a very passive
it
inviscid
(Tennekes
addition
during
that no residual turbulence is
regards, one must recall that
dictated by
flow
deceleration phase. Thus we need
to estimate the time scale
process in the sense
the
proceeds
inertial
Lumley).
which large eddies can transfer
a rate that is
behavior
In
dissipation
at
other
process
of the large
words
is
the time
the rate at
their energy to the smaller
scales. This time scale is given by u'/l, where 1 represents
the size of the largest
hand, maintain their
with
the
mean
eddies.
vorticity
shear
flow
vorticities of the same
This reasoning is
the
steady flows, and can
order
basis
be
because d/U,,l/w. From the
These eddies, on the other
as
and
a result of interaction
thus
have characteristic
as the mean shear; u'/lU/d.
of
mixing length theories in
extended to our nonsteady problem
above
arguments, the time scale
for decay of turbulence can be estimated as d/U, which again
is much shorter than 1/w
if Re /A>>l. Therefore, as long as
the Strouhal number=U/wd-Re /A ,>1, the turbulence generated
during the deceleration phase
does
the acceleration
in
phase,
and
not get carried over to
the
absence
during this period the flow has to relaminarize.
of bursting
-83-
In summary, it
appears
that
the
transition back and
forth between the laminar and turbulent states is mainly due
to
cessation
acceleration
of
phase
"bursting"
of the
cycle,
pressure gradients superimposed
experiments present indirect
portions
during
as
on
of
the
a result of favorable
the boundary layer. Our
evidence
that the fundamental
bursting process is the same as that for steady wall-bounded
flows. Further experimentation,
of
Lagrangian
flow
(in particular in the form
visualization)
can
provide
better
evidence for this claim.
We will instead
proceed
to investigate the transition
process by direct numerical simulations of the Navier-Stokes
equations. These
section.
results
will
be
discussed
in
the next
-84-
III.
NUMERICAL SIMULATIONS
3.1
Problem Formulation and Numerical Methods
In this
section
investigation of
channel
which
infinitesimal
we
the
present
stability
is
of
subjected
and
results
The
to
various
pseudospectral
two-
flow in a
classes
and
of
three-
full three-dimensional time-
dependent Navier-Stokes equations
using the
a numerical
oscillatory
finite-amplitude
dimensional disturbances.
of
are
solved in each case,
formulation
suggested
by Orszag
(1971].
The flow geometry
is
a
has rigid walls at z -
+h
and extends to infinity in the x
and y
all
directions.
In
two-dimensional channel which
that
follows
the channel half
width, h, is taken to be ten times the Stokes boundary layer
thickness ( h
10v2v/w ). The flow rate per unit width into
this channel is imposed as Q -Qocos(wt )
The choice of
a
study versus a pipe
channel
for
geometry for the theoretical
experiments
convenience. The characteristics of
pipe and a channel
flow
structures
are
are
curvature of the pipe.
is mainly a matter of
the turbulent flow in a
very similar, because the turbulent
in
For
general
steady
much
smaller
than
the
flows the large pool of
-85-
experimental data is in support of this fact. For the Stokes
problem, we
base
this
argument
on
between our experimental results for
results of
Hino
and
coworkers
the
close similarity
flow in a pipe and the
which
were
obtained in a
channel.
We
solve
the
*
Navier-Stokes
rotation form,
a*
*
where
| )
p
*
Vxv
-
*
expressed
in
2
xW -V( --+
-V
at
equations
+ fx
2
is the vorticity and (
p*
-
+
P
I2
1v1 2
) is the
2
pressure head. The mean pressure gradient is not included in
p
*
but is represented
by
the
external
*
forcing f ,
(f
*
-
VP /p). Here, * denotes dimensional quantities.
If
one
further
introduces
quantities,
t
wt
-
--- , where U -
v
Uo
2h
*
x-
where 6 V2iv
,
p*
*
Iv I
P
2
11
2
0
f
-
2
the
non-dimensional
-86-
the
Navier-Stokes
equations
can
be
represented
in
the
nondimensional form,
av
Re
-At
Re
(--)(xw)
2
-
1
(
2
This equation is solved
boundary conditions
at
Re
+ - Vv
2
)
+
(--)f
2
numerically subject to no slip
z-+10,
and
with periodic boundary
conditions in the horizontal directions;
27rn
v(
for all
x
27rm
+ -
, y +
integers
n
-
and
, z
m,
, t
and
) - v
( x,y,z,t
)
specified streamwise and
spanwise wavenumbers a and P.
The numerical scheme is that used in earlier studies of
transition
in
steady
wall-bounded
coworkers. Various details of the
we
by
Orszag
and
the numerical code may be
found in the literature [Orszag and
Patera(1983)]. Here
flows
summarize
ells (1980), Orszag and
the basic discretization
and time-stepping procedures.
The velocity field is
represented using Fourier series
in x and y and Chebyshev polynomials in z,
P
N
v l
Mianx
E
eip
vnmp(t) eia
my
p-0 n--N m--M
N
1m
E
E
Vnm(Z,t) ei
n--N m--M
nx
eiImy
T(z)
-87-
The time-stepping procedure uses
splitting technique),
where
a fractional step (or
incompressibility
and viscous
boundary conditions are imposed in different fractional time
steps. The time-stepping procedure is given by,
'n+1 vn
6
( i)
3
---
1-1
xw-(v -(vxw)
-
2
Re
(-
n-
+ f
2
At)
2
-n+l -
V
(ii)
rn+l
--VH
-
Re
-At)
2
n
+
V.v l
I I w+l1o
at z-+10
L 1
VA
(iii)
E
-M.& 1
-, V pyrL
At
I Iv
n +
_ _1~r 2_n+1
_-
v
2
= 0 at
z=+10
The first step incorporates nonlinear effects. The time
stepping procedure
is
an
Adams-Bashforth explicit scheme,
which incurs errors of O(At2 ).
The second step
imposes
equations are reduced to a
incompressibility. The vector
Poisson
equation for w which is
-88-
solved subject to the inviscid boundary conditions (i.e, w-O
at
z-+10).
At
the
end
incompressible but the
of
this
viscous
step
the
flow
is
boundary conditions are not
satisfied.
The last
step
incorporates
imposes the viscous
boundary
the
viscous
effects and
conditions. The time-stepping
scheme is only first order but implicit.
In the present
problem
than the external forcing
This
can
be
the
(pressure gradient) is specified.
implemented
by
divergence-free, it merely
has
and vn+l
noting
an
that
since
f
is
additive effect on vn+l
in steps (i) and (ii). In other word the effect of
the external forcing is to
n+
in step (iii) given by v
v n+ l
1 where,
Re 6
v
lv
introduce a contribution to vn +
1
At
V2vn+l
+ (
2
) f
2
'n+
0
at z
One could thus
solve
+10
for
v
forcing in a pre-processing stage,
v
1
2
- Vv
At
I
flow rate (Q=cost) rather
n+l
+
2
v
-
0
at
z
+10
corresponding to a unit
-89-
At each time step, equations
without the effect
of
(i) thru (iii) are then solved
external
to give v " n+l The
forcing
correct solution to vn + l is then,
n+l - v
where fn+
n+l
+ (Re /2) fn+
is determined
by
the
v
imposed flow rate at time
(n+1), i.e;
f v
n+ l
dA + (Re 6 /2) fn+l f v
dA
n+
or,
Re 6
Qn+l - f v
This completes
the
dA
dA
fv
2
n+ l
description
of
the
main
part of the
program.
Initial conditions for the
generally consist of
are
superposed
an
the
unperturbed
various
finite-amplitude two-
laminar flow on which
combinations
and
(time-independent)
runs reported in this study
of
infinitesimal or
three-dimensional
Orr-Sommerfeld
eigenmodes of
equation
for
the
"frozen" initial profile. Note
that even though the initial
disturbance is strictly
only
valid
independent, the results obtained
the
full
time-dependent
if
the flow was time-
in the steady state using
simulation
and
the
initial
-90-
disturbances described above should accurately represent the
full time-dependent
A number of
of the flow.
behavior
have
tests
performed to verify the
been
accuracy of the direct numerical simulation. As a start, the
of
evolution
the
least
stable
Sommerfeld equation is followed
the direct simulation and the
predicted by the
eigenmode
for
of
the
Orr-
a frozen profile using
results are compared to those
Orr-Sommerfeld
equation.
A
few of these
results are summarized in table 1. In other tests results of
the full
simulation
of
small
linearized
with those of the
simulations
for
two-
and
disturbances were compared
equations; and finally direct
three-dimensional
infinitesimal
disturbances were compared to verify that they correspond to
each other according
to
Squires'
theorem. These results
will be discussed in later sections.
Typical parameter
P-64(or
128),
values
2N-32,
Convergence in all
these
2M-2,
for
direct
Courant
parameters
has
either halving or doubling the resolution.
simulations are
number
-
0.1
.
been verified by
-91-
frozen profile of wt=0
2D
2D
3D
disturbance disturbance
Re 6
frozen profile of wt-7/2
3D
disturbance disturbance
1000
1000V2
1000
1 000\/
0.5
0. 5/X/
0.5
0.5 //
0.5 /A/i
3
0o.5/v
-. 06858357 -. 04849591
Real(w)
.54467293
.38514192
Im(w)
.00175319
.00121291
.02015786
8x65
8x65
8x65
8x65
Courant
.1
.1
.1
.1
Final
Time
r/10
7r/10
7r/10
r/10
1.309
1.309
23.722
23.722
1.346
1.341
23.406
23.446
-85.557
-85. 557
10.773
10.773
-85.813
10.772
10.772
Resolution
2N x (P+1)
predicted
amp. chni
.01425378
e(Re/2)Im(w)t
computed
amp. chng
predicted
phase chng
e-i(Re/2)Real(w)t
computed
phase chng
-85.813
Table 1. Comparison of evolution of small-amplitude
disturbances for frozen profiles to prediction
of Orr-Sommerfeld equation.
-92-
3.2 Small Amplitude Dsturbances
The linear stability
of
(Linear Theory)
oscillatory Stokes layers has
been the subJect of a number of investigations. The simplest
and most popular
approach
is
where the time dependence
and the
stability
studied
using .the
of
the quasi-static assumption,
of
the
basic profile is ignored
instantaneous
the
classical
Sommerfeld eigenvalue equation.
of which frozen profile
to
of oscillatory flow in a
"frozen"
profiles is
(time-independent) Orr-
One
question is the choice
use. For the particular example
channel
with an imposed flow rate
of Q - cost, two such quasi-steady stability maps, one for a
time of t-0 and the
other
for
t/2,
are shown in Figures
30a and 30b. As one would expect, the two stability maps are
quite different. The
minimum
critical Reynolds number, for
example, has a value Re crit.
80 for the profile at t=r/2;
but a significantly larger value
The most dangerous
streamwise
(Re
rit. - 500 ) for tO.
wavenumbers, however, are in
both cases close to a of 0.5. If the quasi-steady problem is
at all relevant to
the
Stokes layers, the
stability
search
for
unstable
problem of stability of oscillatory
modes
maps
(in
of
the
problem) in the range of wavenumbers a -
Figure 30 suggest a
full
time-dependent
0.2 to 0.7.
-93-
wt-7r/2
"o,
;U
10-'
10-2
Rev
wt-O
lno
Au
i
10-'
i
%J
i
i
10-'
1
10
.
lo
I
jI
03
10
,
'0
Re
Figure 50. Iso-growth curves for the frozen profiles at
wt-r/2
and wt-O.
-94-
3.2.1
Two and Three-Dimensional
Squire's theorem provides
for the linear stability
Reynol.dsnumber
equations
von
the
that
Drazin
transformation can
be
and
&
that the evolution of
any
related to a 2-D
one
at
confirmed in the
linear stability problem of
of
2-D disturbance at a lower
Reid,1981].
used
erczek
a very useful transformation
problem of "steady" unidirectional
shear flows, which reduces
any 3-D disturbance to
1Disturbar&es
in
the
Exactly the same
"unsteady" linearized
Davis,1974]
with the conclusion
infinitesimal 3-D disturbance is
a
lower Reynolds number. This is
results
of
the numerical simulation, one
example of which is shown
in
Figure 31. In this figure the
evolution
of
a
3-D
disturbance
disturbance
are
compared.
dimensional
disturbance
Here
energy
and
its
En2
is
(in
all
equivalent 2-D
the
the
total
streamwise
fourier modes l<n<N ) and is given by,
N
10
(2)
*(
En2- eat J(Vn
vn
)
( ))dz
f (Vn(2)
n-l -10
where * denotes complex conjugate.
Similarly En3 is the total three-dimensional energy,
M
En3 - £
N
m--M n--N
10
*
(3)
-10
z
(vi d(
two-
II,
U
-95-
-i- T-r=-
- I - I-
I . .. .
...
. .
. .
. .
. .
-2
-4
-6
-8
M
0
;.10
-10
-12
-4
-14
-16
-18
-20
-22
--
-_"m
-
I
U
A ,I
X
Ir/2
o
.
.
I
I
I
I
.
.
.
.
.
.
I
I
I
1
I
-1
3r/2
I
I
I
.
I
.
2x
I
I
I
I
I
-
I
I
I
I
.
I
I
-2
--t
-6
-8
e
-10
-12
U-4
-14
-16
-1is
-LJ
-22
..,,..;.,,,.,..,,,,,.,...,
....,...,,..,,..,
-V
a
7/2
7
33r/2
2r
F gure B1. Equivalence of two- and three-dimensional infinitesimal
disturbances by Squires Theorem. For 2D disturbances
Re -1000., a-0.5 . For 3D disturbances Re -1000\ ,
a-I-0.5/Vr. Initial conditions are the least stable
eigenmodes of O-S equation for the frozen initial profile.
-96-
The
comparison
also
serves
as
an
extra
check
on
the
integrity of the numerical code.
Consequently,
only
two-dimensional
infinitesimal
disturbances will be considered in this section.
3.2.2 The Linearized Problem
For infinitesimal
disturbances,
in computation time is
achieved
a substantial savings
by solving, instead of the
w-Oill
nonlinear problem, the linearized disturbance equations
given by:
a
Re
-(u',v',w')=
~2
--
(Uu' +w'U
~t
Re
x)
Uv'x ,Uw'
6
-(
2
Here v' is the
z,
disturbance
1
1, H
X
3
7 ) + -V
Z
2
velocity,
2
(u',v',w')
and only one fourier
mode is kept in the streamwise direction;
v' =
v'(t) einx
T (z)
n-l,+l
P
v'(z,t) eia nx
n--l,+l
Except for the
implementation
of
the nonlinear terms, the
method of solution is the same as that described earlier.
-97-
0
-2
-4
-6
-8
CN -10
-12
-14
-I i
-16
-18
-20
-22
-;)4
Figure
0
32.
W/2
Ir
3r/2
2r
5r/2
3T ?
x/2
4
9x/2
5r
Comparison of the results of the full threedimensional simulation to the linearized problem
for the evolution of two-dimensional disturbances
at Re -1000.0 a0.5 .
-98-
For
infinitesimal
disturbances,
the
results
of the
linearized equations should, of course, be in agreement with
the full nonlinear problem. A comparison of the two is shown
in Figure 32 for Re =1000 and
a-0.5 . The slight wiggles at
large times in the full simulation are due to the excitation
of higher order modes by roundoff errors.
The results to be discussed in the rest of this section
were obtained using the linearized equations.
3.2.3Effect of nitiat Conditions
It
was
mentioned
earlier
that
the
Navier-Stokes
equations are solved subject
to initial conditions that are
the least stable
of the Orr-Sommerfeld equation
eigenmodes
for the initial "frozen" profile. It was also mentioned that
for large
times
the
solution
initial condition. This
point
becomes
is
shown in Figures 33(a-d), where
different
times
(
t=O,
and
subjected to the
least
"frozen" profile.
In addition
disturbance,
the
across the channel width
is
by the results
flow is started at two
t=w/2
)
and
disturbance
to
distribution
verified
the
stable
independent of the
in
each case
of the initial
the overall energy
of
the
of the
disturbance energy
also shown for selected times.
Comparison of the growth rates in the two cases (as shown in
Figures 33a,b),
or
the
energy for large times
distributions
(as
shown
in
of
the disturbance
Figures 33c and 33d)
-99-
U
-2
-6
-8
C
-10
-16
-18
-20
-22
-24
0
1/2
r
3/2
21
51/2
3I
7r,'2
41
9 /2
5
Figure 33(a.b). Evolution of two-dimensional disturbances at
Re - 1000.0, a - 0.5 for two distinct initial
conditions. Note that at large times the growth
rates ( Figures 33a , b ) and distributions of the
disturbance
energy
across
the channel width
( Figures 33c , d ) become independent
of
the
initial condition.
-t0-
Distribution
I
r-
t
r
r -
--- r
of E2 at time 0
--7----------
-rr
I
-
--.
III-r
I
I
.I
I
Distribution
._
I
of E2 at time r/2
-
18
18
16
16
14
14
12
12
10
- 10
8
8
6
4
4
2
2
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n I
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0
c
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a
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t
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Distribution
.
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00
rYx
co
Y7
.
.
I
.
0
a1O
0
a,
0)
K
C
-K?
X
of E2 at time
.1n
IC
Distribution
_t
of E2 at time 3r/2
LJ
18
18
16
14
14
12
12
10
>- 10
0
8
6
6
4
4
2
2
0
0 0
w w
N
x
r
W
XK
O
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0
w
0 M0 0 0
N
K
0
0
0
¢
r
x
u
x
S
N
eVN
x
K
cr
A
9
---
.Elg==,re
33
For caption see Figure 33a.
I,
-101-
Distribution
of E2 at time 2r
Distribution
"N
of E2 at time 57r/2
20
CU
18
16
16
I
14
14
.1
12
12
>- 10
>-
8
.
6
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r7
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en
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Distribution
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(14
20
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M
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a)
a
x
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10
o
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e
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of E2 at time 3r
I
r r..
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18
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For
Fig.re
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o
p
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w
caption
330. For caption
C3o
C
C
o
C3
o
C
0
x
r
_
O
0D
S
see
Figure
33a.
ee Figure 33a.
Tq
_0
(_
_
N
-102-
OISTRIBUTION OF E2
?r/2
T TIME=
20
18
16
14
12
> 10
8
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2
0
DISTRIBUTION OF E2
T TIME=
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r
^^
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DISTRIBUTION OF E2
T TIME= 3
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Lu
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an
atn
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Eigure 33d. For caption see Figure 33a.
;
-
=
l
i
W
-
-103-
OISTRIBUTION OF E2 AT TIME=
27
OISTRIBUTION OF E2 AT TIME- 57r/2
20
20
18
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12
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T
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to
Figure 33d. For caption see Figure 33a.
. . . .
. | .
. T
. E s. s U
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Tl / r w
f. I. s w WWI W
o
a
x
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0
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x
x
LI
0
VI
U,
-104-
proves that the final solution is independent of the form of
the initial disturbance.
3.2.4 Results
In Figures
34
and
35
we
show
how
a
typical two-
dimensional disturbance which starts off as the least stable
eigenmode of the
(time-independent) Orr-Sommerfeld equation
for the initial "frozen" profile,
evolves to a steady state
solution which is that
by
predicted
a Floquet analysis of
the time-dependent Orr-Sommerfeld equation.
In addition to
(shown in Figure
34),
distribution of the
width at
various
the
overall
we
find
times
so
it
disturbance
35).
why
different
useful to consider the
energy
(Figure
immediately becomes obvious
results that are
energy of the disturbance
across the channel
From
Figure
35 it
quasi-steady theories give
from
those
of the Floquet
analysis of the Orr-Sommerfeld equation.
The
specified
initial
as
disturbance
the
least
stable
Scmmerfeld equation for
the
plot of distribution
energy
of
peak at the "critical layer",
at successive
inflexion
field
(at time
eigenmode
"frozen"
for
of
r/2)
the
is
Orr-
initial profile. The
this
time has a major
with two less prominent peaks
points.
Between
r/2
and
r, the
-105-
-2
-1
-4
-6
-8
CCq
a
-10
a -12
-14
I
-18
1
-20
1
-22
1
-P4
I
O
Figure 34
I
T
I
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2W
I
3W
I
4V
I
SW
6r
It
Evolution of the energy of two-dimensional
disturbances at Re 1000.0, a=0.5 . The initial
conditions correspond to the least stable eigen
modes of the Orr-Sommerfeld equation for the
frozen initial profile.
-106-
=
r... aT TIME
c-..
, o
T
Et=
r
2
2a
16
t2
a
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21
20
18
It6
1;
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CN
r o 6o
x
C* Ir~
6*
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(4C
-10 7-
DISTRIBUTIONOF E2
T TIME: 57r/2
- ---- - - -- - - -- - - - - - - . . . . . . . . . . ,.. .I.,,.. - .
20
18
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DISTRIBUTION OF E2 T TIME: 37
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-108-
DISTRIBUTION OF E2 AT TIME: 9r/2
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DISTRIBUTION OF E2
DISTRIBUTION OF E2 AT TIME= 11r/2
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L
D
Fiure 35. Evolution of two-dimensional disturbances at
Re -1000.0, a-0.5 .
-110-
disturbance evolves
in
ways
not
too
different from what
would be predicted by quasi-steady theories; with the growth
rates of the same
order
of
magnitude and distributions of
disturbance energy across the channel similar to the "least"
stable eigenmode of the
36 for a
Orr-Sommerfeld equation (see Figure
comparison).
intrinsic nature of
During
the
disturbance has moved
the
away
center. In the next half
the
Instead, it excites
from
the
wall.
period, because of the
basic
velocity profiles, the
from
period
3r/2), the disturbance is too
able to excite
this
the
wall and towards the
(for
example at a time of
far
away from the wall to be
same
modes
that were excited earlier.
modes
that
are localized further away
This
"mode-hopping"
disturbance has travelled all the
channel, at which
state. It
is
point
this
the
way
flow
steady-state
continues
until the
to the center of the
settles
behavior
into a steadythat
a Floquet
analysis of the time-dependent Orr-Sommerfeld equation would
predict.
In the steady-state,
with
the
basic
the
flow,i.e;
differs from that at
time
disturbances are synchronous
the
t
disturbance
at
time t+27
only by an exponential factor
(with no phase change). This means the Floquet exponents are
real, in agreement with the
Davis. This is to be
results
expected,
the Orr-Sommerfeld equation that
is its complex conjugate,
since
if
found by von
erczek
it can be shown from
A is an eigenvalue, so
. Thus the solution should either
-111-
^^^
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.024
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.016
0o
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.012
.010
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.006
.004
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0
-_- fn?
r/2
3r/2
wt
Figere 36. Comparison of the growth rates predicted by the
time-dependent simulation (solid line), to the
least stable eigenmode of the Orr-Sommerfeld
equation (dots) for r/2<t,3r/2. (Two-dimensional
disturbances a-0.5, Re -1000)
.
-11247
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. a1-o
The disturbance field at Re 6 -1000.0, a-0.5 as
seen in a frame of reference moving with the
centerline velocity.
.
-115-
simultaneous disturbance waves
consist of pairs of
and
·
At+iax
e
et
propagate
which
in
e At+iax
opposite directions, or
form of a standing wave
the
it should be of
At+iax
is real. Furthermore, the .umerical
V(z,t)e t +
, where A
else
simulations show that in the steady state,the phase speed of
the disturbance
is
the
same
as
the
instantaneous axial
velocity in the core. In other words the disturbances merely
with the mean flow. This point
get convected back and forth
is shown in
Fgure
37,
moving
frame of reference
the
where
with
disturbance field in a
the centerline velocity is
shown.
other combinations of Reynolds
The results for various
number and streamwise wavenumber are shown in Figures 38 and
39. In each case
the
similar to the case
off
in
the
of the disturbance is very
evolution
discussed above; the disturbance starts
boundary
and
layer
reaches
a
steady-state
solution with eigenmodes near the center of the channel. The
are
resulting Floquet exponents
plotted
in Figure 40. The
general trends in the behavior of A as a function of Re
and
a, are identical to those obtained by von Kerczek and Davis,
but the absolute
magnitudes
expected, since as we
seen,
have
localized within the boundary
consists of eigenmodes that
This is to be
the disturbances are not
layers but migrate away until
they settle into a steady-state
boundary conditions. For a
are smaller.
of A
that is compatible with the
channel flow,
are
this steady-state
clustered about the center
-116-
-
~~~~-
0
- -
- - -
I
,
i
-i
J
-5
-10
-15
).3
Ct
.5
-20
-25
-30
-35
).7_
I
Figure
I
2
0
38.
I
I
2r
I
31
i
41
5r
I
I
61
71
Evolution of two-dimensional disturbances with
streamwise wavenumber a=0.3, 0.5, 0.7 at
Re61000.0 .
-117-
,,
U
I
.I
.
.
.I .
.'l
I
I
'
.
.
II
.
T
Il
I
I
]
I
1
'
-
I ji
-5
-10
-15
L
Ca
rut, -20
-25
-30
Re-200.
-35
-dn
-- 'U
I
o
I
f
_
I
I
2s
I
3t
.
4r
I
5r
I
I
64
I
71
Figure U. Evolution of two-dimensional disturbances with
a0.5 at Re 6 - 200.0, 500.0, 1000.0
-118-
a,0.0
0
-.1
a-0.3
O
-. 2
0o
O
a
a-0.5
-. 3
.4
5
a-0.7
-. 6
-,7
.8
-. 9
I
-1.0
500
0
I~~~~~~~~~~~~~~~~~~~~
1000
Re
Figure 40.
The principal Floquet exponent as function of
Re
and a .
-119-
of
the
channel.
disturbances
For
an
migrate
away
from
"merge" with the continuous
layer, as was shown by
on the other hand,
consists of
an
infinite
Stokes
the
layer,
boundary
the
layer
and
spectrum of the infinite Stokes
Hall
[1978]. Von Kerczek and Davis,
consider
a
oscillating
bounded Stokes layer, which
plate
with another stationary
plate a few boundary layer thicknesses away. The requirement
that the disturbances go
to
zero on this second stationary
plate, results in a problem that is distinct from either the
infinite Stokes layer or the channel.
Nevertheless, our results agree
with those obtained by
other investigators for other examples of oscillatory Stokes
layers, in that
the
flow
is
steady-state) to infinitesimal
the
principal
studied,
but
Floquet
for
a
found
be stable (in the
disturbances,.
exponents
given
to
negative
streamwise
Not only are
in
all
cases
wavenumber,
the
oscillatory channel flow is
more stable than the motionless
(Re =0)
Floquet
state;
with
the
exponent
a
monotonic
decreasing function of Re6.
Finally, one
should
revealed by the results
namely, that if
a
layers near the
wall,
not
overlook
of the direct numerical simulation;
disturbance
it
the acceleration portion
another phenomenon
is
present in the boundary
can substantially amplify during
of
the
cycle. In an experimental
setting, this result is probably more relevant than those of
the Floquet analysis,
since
a
certain
level
of noise is
-120-
always present throughout the flow field. In this situation,
are small, they would amplify
and provided the disturbances
of
phase
during the acceleration
during the deceleration phase.
reporting
observed
"weak"
that
two
transitional
describe
one
appear
during
that
disturbances
cycle and be damped
Indeed, Hino et al.[1978] in
of
classes
experimentally,
the
as
flows
consisting
are
of
the acceleration
portion of the cycle.
As a
result
might be amplified
of
to
process,very small disturbances
this
such
an
fails. In that case, and for tose
extent
that linear theory
disturbances that are not
sufficiently small to be treated by a linearized theory, one
must study the
stability
of
the
flow to finite-amplitude
disturbances. This case is considered next.
-121-
3.3 Finite Amplitude Disturbances
When the perturbation is
results. Such
a
theory
infinitesimal a linear theory
is
very
attractive
resulting instability is independent
amplitude of
the
disturbance.
linear theory cannot explain
is observed in experiments.
in
that the
of the form or initial
However,
as
we have seen,
the transition phenomenon that
We
therefore, need to consider
more general (finite-amplitude) disturbances.
We
start
disturbances
by
of
a
finite
dimensional
theory
transition
(in
understanding
amplitude.
cannot
been
shown
the
basic laminar profile
(or
of
two-dimensional
Even
all
though
the
a
details
twoof
three-dimensionality),
behavior
is
important
[
Herbert and Morkovin(1980) ,
that
the transition process for a
large class of "steady" shear
equilibria
its
two-dimensional
Orszag and Patera(1983)
instability of
explain
particular
the
because it has
consideration
flows can be explained as the
intermediate
state
consisting
of the
plus two-dimensional finite amplitude
quasi-equilibria)
dimensional disturbances.
to
secondary
three-
-122-
3.3.1 Two-DimensionalDisturbances
The nonlinear theory
not yet
critical
been
developed.
Reynolds
Nevertheless,
for
Stokes
One
problem
number
according
subcritical
nonlinear
destabilizing
terms
for
in
is
the
to
linear
finite-amplitude
(travelling waves with constant
the
flow instability has
the
lack
of a
theory.
equilibria
energy) may be possible, if
Navier-Stokes
disturbances
with
equations are
energies
above some
threshold value.
There are two
finite amplitude
routes
to
equilibria.
equilibrium solutions
techniques. In the
are
second
determining
In
the
directly
first
approach, the
obtained by eigenvalue
approach,
disturbances are followed by
two-dimensional
evolution of specific
direct numerical simulation of
the Navier-Stokes equations. The eigenvalue approach is more
efficient
and
would
solutions within the
directly
(E,R,a)
locate
space
the
equilibrium
(although
it does not
give the time scales of evolution to the equilibrium state).
However, it requires some assumption about the specific form
of the equilibrium solution.
the equilibrium solutions are
wave with constant phase
In
of
speed,
steady flows, for example,
the
and
form of a travelling
can be represented by
-123-
ian(x-ct)
00
solutions are steady
in
a
(real) wavespeed c and are
Galilean
nonlinear
frame moving with the
periodic in x. Truncation of the
fourier series representing
coupled
. These
V) (z) e
n-- o n
-
the stream function,
would
(but
result
in a set of N+1
time-independent)
differential equations for An
which
can
ordinary
be solved to give
the equilibrium solutions [Herbert(1977)].
For
the
periodic.
Stokes
One
problem,
might
the
therefore
basic
flow
anticipate
is
time
equilibrium
solutions to be of the general form,
ian(x-cr)
00
=
0
(z£r
e
n=-oo n
where c is real (and
time normalized to the
mean flow. In
other
possibly
basic
words,
they exist) are expected to
zero)
and r is the physical
period
of oscillation of the
the
be
equilibrium solutions (if
time periodic with the same
period of oscillation as the basic flow.
The
above
formulation
would
result
in
a nonlinear
eigenvalue problem for the time periodic functions
can be solved in
the
(E,R,a)
n', which
space to locate any possible
equilibria.
In the present work,
follow
the
we
evolution
adopt the second approach and
of
disturbances by direct numerical
Stokes equations. Because of
specific
two-dimensional
simulations of the Navier-
the relatively large amount of
-124-
computation time involved in
each
simulation, we limit our
search to three wavenumbers ( a=0.3,
single Reynolds number of
1000.
0.5 and 0.7 ) all at a
In each case the evolution
of the least stable eigenmode of the (time-independent) OrrSommerfeld equation
followed
by
for
direct
the
numerical
disturbance is normalized to
The
simulations
initial
were
(frozen)
simulation.
have
obtained
a
profile is
The
initial
total energy of 0.04.
with
a
resolution P128,
2N-32.
The results of the
simulation
for
the case a=0.5 are
shown in Figures 41 through
44. The most important question
is whether the disturbances
evolve to an equilibrium state,
or do they decay;
and
if
a
plot
so
what
are the time scales of
decay.
Figure 41
is
disturbance field as a
of
the
function
of
overall
energy of the
time. After an initial
transient period, the disturbances are seen to settle into a
state of slow
decay,
suggesting
amplitude disturbance would
that
the initial finite-
eventually
die away. The decay
rates, however, are 0(1) (actually,
which represents the oscillation
Recalling that Stokes flows
A--.38) on a time scale
period
of the basic flow.
result from an inherent balance
between the viscous terms and temporal acceleration ( and as
such the period of
oscillation
time scale;
1/w
62/v
is
the same as the viscous
), the decay of the two-dimensional
-125-
U
r
a-. 5
-10
-20
l
)
Figure
7r/2
l
37r/2
·
2r
_
l
·
57r/2
41. Evolution of two-dimensional finite-amplitude
disturbances for a-O.5, Re -1000.0 .
-126-
finite amplitude state
phenomena, on
the
is
seen
other
to
hand,
be viscous. Transition
happen
on convective time
scales; therefore, if the mechanism that triggers transition
were to involve
the
above two-dimensional finite amplitude
state, the viscous decay
field would be
process. This
of
of the two-dimensional disturbance
little
point
importance during the transition
will
be
of
the
discussed
further
in later
field
from its
sections.
The
development
disturbance
initial "prescribed" form as
the
the Orr-Sommerfeld equation,
to the final quasi-equilibrium
state can be seen
in
distribution of the
more
least stable eigenmode of
detail
energy
of
in
the
Figure 42 where the
disturbance across the
channel is plotted at various times. The initial disturbance
field is localized near the
walls
at the (linear) critical layer
with a major energy peak
and two less prominent peaks
at neighboring inflection points. Within a very short period
(with
convective
migrate to the
time
center
scales)
of
the
quasi-equilibrium state where
back and forth with the
though,
channel
they
the
and
disturbances
settle into a
basically get convected
mean flow, maintaining their energy
to O(1/R).
Both the disturbance
attenuation rate A, are
theory results of the
field
not
and
the
magnitude of the
too different from the Floquet
linear theory. Nonlinear effects tend
-127-
20
---
.,--
----
---
---
__,..
-I
20
.
18
18
16
16
14
14
12
12
.. ?
10
>- 10
8
8
6
6
4
4
2
2
....
n
U
..I......
--
Ln 0
a
:FPF·-·
888888888 88
o '0 a
of
0
a
-,I~.T,,.
.....I..
..~
Distribution
20
18
16
16
14
14
12
12
10
>- 10
8
8
6
6
4
4
2
2
0
.0005 .0010 .0015 .0020 .0025 .0030 .0035 .0040
C)
Lu
0
8
8
8
8
o
8
8
In
C0
Wi
0
ln
C
8
8
8
8
8
8
of E2 at time 3zr/2
20
18
3
In
Distribution
2 at time 7r
. . . .. . . .. . . . . ..
0
of E2 at time r/2+lr/10
Distribution
of E2 at time r/2
Distribution
0
f igure 42.
.0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008
-12 8-
Distribution
of E2 at time 2r
Distribution
20
2
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
o
o
"
J ITc~ (
w
..
.
.
.
Figure 42.
.
.
c
.
co
.
0
IN
.
INrI
O0 N
.
CD
t
.
,
,q,
0
i
L
C
.
,
I
_
ooO
,
,
.
,
I
-
CD
oo
of E2 at time 57r/2
-
,
,
I
0
O
o
,
I
.
....
I
0
O
O
C
I
-
I
O
O
o
C
v
.
.
,
,
.
r
I
(
r-
c
0
O
O
o
C
Evolution of two-dimensioanl finite-amplitude
disturbances for a-05, Re -1000.0 . The initial
disturbance is the least stable eigenmode of the
Orr-Sommerfeld equation for the frozen initial
profile.
O
C
o
-129-
to spread out the perturbation
over
a wider section
of the
cross-section, and to result in slightly larger values for A
( A--.38 for the finite amplitude disturbance as compared to
A--.25 for the linear case ).
The effect of the nonlinear
profiles is
shown
in
Figure
appears near the center
boundary layers'
as
of
this
terms on the mean velocity
43.
the
is
The
biggest correction
channel
where
and away from the
the
disturbances are
localized.
Finally, the contribution of various streamwise Fourier
modes to the
overall
shown in Figure
44.
energy
Even
of
the
disturbance field is
though
16 fourier modes (2N=32)
only
the first five are shown
were used in the simulation,
in Figure 44, as higher modes have very small amplitudes. At
the time wt=S5/2, for
example,
the
various modes have the
following energy contents:
n
En
1
2
3
4
0.112E-02
0.102E-05
0.630E-7
0.858E-08
n
En
5
It should be
fourier
10
0.171E-08
modes
noted
however,
become
insufficient number of
affect the initial
mainly because
15
0.621E-12
that
even
unimportant
streamwise
transient
small
0.707E-16
round
at
though the higher
large
times,
an
fourier components would
behavior of the disturbances;
off
errors
introduced in the
boundary layers could result in new instabilities because of
-130-
wt-7r/ 2
Wtlr
n
.
1.
1.2
1.0
1.0
.8
.8
.6
.6
.4
.4
-.2
-
3
I
0
I I
I
I
I
I
I
I
I
.2
I
- -..2
-. 6
-. 8
! IC)
& .6
C1
2
4
6
8
10
z
12
14
16
18
20
0
2
4
6
8
wt-37r/2
10
z
12
14
16
18
20
12
14
16
18
20
wt -27r
1.2I
. I
1.c I
1.C
I .i
.8I
.6i
-.E
.4
-. E
I
I
I
-I .
_.
-.-
I
-. 8
-.
-1.0
-I .
-1.2
0
2
4
6
8
10
12
14
16
18
20
Figure 43
0
2
4
6
8
10
z
-131-
wt-Sir/2
1.2
1.0
.8
.6
.4
-. 2
-
0
-.4
-. 6
-. 8
-1.0
-1.2
0
2
4
6
8
Figure 43.
10
z
12
14
16
18
20
Average velocity profiles in the presence of twodimensional finitite-amplitude disturbances
( a-0.5, Re 1000.0 ) compared to the laminar
profiles.
-132-
0
ILY·\
n-I
-1
-0
n- 2
n-3
n-4
n-5
-20
n-6
r
3/2
27r
57r/2
0
7r/2
Figure 44
Development of higher order modes for an initial
finite-amplitude disturbance with a-0.5,
Re -1000.0
-133-
the highly unstable nature
of
above simulation convergence has
the boundary layers. For the
been verified by repeating
the simulation with 2N-16, P-65, which gave nearly identical
results.
-13 4-
3.3.2 Three-Dimensional
Instability
When only two-dimensional
flow, as we have seen,
disturbances are allowed the
evolves to a quasi-equilibrium state
of travelling waves that decay
no further interactions with
this flow remains well
over viscous time scales. If
other disturbances take place,
ordered
at
all
times and shows no
sign of the chaotic
small-scale structure of real turbulent
flows. However, the
same
two-dimensional state may undergo
further instabilities if
subjected
dimensional disturbances
[ Herbert
to infinitesimal threeand
Morkovin (1980)
Orszag & Patera (1983) ].
Let A be the amplitude
B that of
the
of the two-dimensional wave and
infinitesimal three-dimensional disturbance.
With A>>B, the
development
of
the two-dimensional wave is
not influenced by the three-dimensional disturbance, but the
amplitude
of B depends
on A according
d(lnB)
to
Stuart(1962)
i,
2
= bo + bAI
+
dt
With bO,
there are
the presence
of
the
two
possibilities; if b
two-dimensional
is negative,
wave causes stronger
damping of the three-dimensional
disturbance. However if bl
is positive, the two-dimensional
wave feeds energy into the
three-dimensional
disturbance;
leading
to
an exponential
growth of the three-dimensional wave. The exponential growth
-135-
of
the
three-dimensional
indefinitely,
disturbance
as in a short
comparable to A
and
the
states
disturbances seems to
problem
flows
as
therefore, seem
the
of B becomes
to be reformulated.
to infinitesimal three-dimensional
explain
shown
by
universally many features of
in
a wide class of "steady"
Orszag
ustified in
two-dimensional
has
continue
instability of two-dimensional
the early stages of transition
shear
not
time the amplitude
Nevertheless, this secondary
finite amplitude
does
&
Patera
[1983]. We
investigating the stability of
finite-amplitude
states
of
the
oscillatory channel flow (discussed in the previous section)
to secondary three-dimensional disturbances.
Once again, we
the full
employ
time-dependent
investigation.
In
direct numerical simulations of
Navier
particular
Stokes
we
equations
will
study
for our
the
time
evolution of flows resulting from initial conditions,
v(x,y,z,t=o) - U
+
Here U is the basic laminar
2D +
V3D
flow, v2D is the initial finite
amplitude two-dimensional disturbance with wave vector (a,O)
which is specified to
eigenmode of
the
be
in
the
Orr-Sommerfeld
form of the least stable
equation
for
the frozen
initial profile and is normalized to have a total energy E2 D
of
0.04;
and
V3 D
is
the
initial
dimensional disturbance which has
vortex with wave vector (0,)
and
the
has
infinitesimal
three-
form of a streamwise
a total energy
of
-136-
10
8- .
Only one mode is
kept in the spanwise direction owing
to linearity and separability.
N
modes (2N-32) are kept in
the streamwise direction because of nonlinear effects.
As
the
instability
persistence of
the
depends
two-dimensional
down to transitional Reynolds
of
the
will choose for a, the
on
the
finite amplitude state
numbers (since in the absence
two-dimensional
wave
disturbance has to die away
critically
the
three-dimensional
according to linear theory), we
most stable of the three wavenumbers
studied in the previous section, namely a-0.5 .
The stability of this flow to disturbances with various
spanwise wavenumbers
,
simulated numerically at Re 61000,
is illustrated in Figure 45.
verify that
instability.
there
The
is,
indeed,
channel
a strong three-dimensional
three-dimensional
energy by a factor of 10
travel 5
To begin with, the simulations
in
widths
corresponds to a convective
at
disturbance
grows
in
the time it takes the fluid to
maximum
(and
bulk
velocity. This
not viscous) growth rate,
and suggests that the instability mechanism is inviscid.
Secondly, the instability is
most effective at P>0(1),
emphasizing the three-dimensional nature of the instability.
There is a weak maximum in
at P
the magnitude of the growth rate
1, but beyond that the preference in
is very weak.
- 1 37-
ar .5
,.1.0
-2 .0
X -10
-20
ir/2
Fiure
3r /4r
45
Spanwise wavenumber dependence of the threedimensional growth rate at Re 1000.0, ca-0.5
-138-
We may
characterize
the
transition
process by three
parameters; the transitional Reynolds number, the time scale
of the instability
and
the
spatial
dmensionality
of the
perturbation responsible for the instability. We have so far
shown the instability
described
and grows on convective time
above is three-dimensional
scales. It still remains to be
shown that the instability cuts
at Re6
off
500, which is
the value observed experimentally for transition. This point
is demonstrated'in Figure 46 where the evolution of the twodimensional
quasi-equilibrium
state
and
the
three-
dimensional perturbation are followed at Reynolds numbers of
200, 500 and 1000 for a=0.5 and P-1.0 . Note that a Reynolds
number on the order of
Reynolds number below
decay.
This
considering
the
The
singled out as the critical
may
require
observation
positive
point
disturbance needs the
to continue
is
which three-dimensional perturbations
statement
disturbance shows
numbers.
500
its
that
growth
is,
some
the
at
that
clarification
three-dimensional
all
the
three Reynolds
three-dimensional
two-dimensional finite-amplitude wave
growth.
For
Reynolds
numbers
below 500
before the three-dimensional disturbance has grown to finite
amplitudes,
the
dropped
such
to
level
of
small
maintain the instability
dimensional wave never
the
two-dimensional
amplitudes
process.
grows
instability is turned off.
to
This
As
that
a
it
can
wave
has
no more
result the three-
finite amplitudes and the
argument has been verified
by continuing the simulation at Re -200 to longer times, and
-139-
U
X
-10
-20
3r/4
r/2
Figure
46-
Reynolds-number dependence of the three=1.0
dimensional growth rate at a0.5,
Observe that disturbances decay for Re <500.
7r
-140-
the
ultimately decay
as
a
result
dimensional wave. Note also
three Reynolds numbers,
to
perturbation.
viscous
of
that
the two-dimensional base state
mainly
is
disturbance
three-dimensional
the
is
indicating
damping
the
of
indeed
decay
seen
to
of the two-
the rate of decay of
roughly
the same in all
that
the cutoff is due
the
three-dimensional
-14 1-
3.3.3
Secondary Instability and Transition
We would like to determine to what extent the secondary
instability mechanism discussed
the
transition
process
this section relates to
in
that
been
has
observed
in
experiments.
With respect to
parameter is
the
transition,
Reynolds
number predicted by
been shown to be
in
the
the single most important
number.
instability has already
secondary
good
critical Reynolds
The
with the value of 500
agreement
that is observed in experiments.
The next question is
whether the secondary instability
presented here saturates in an ordered state when it reaches
finite amplitudes or whether it results in chaotic behavior;
and in
general
to
resemble turbulent
what
flow
degree
does
structures
experiments. To answer this
question
large numerical
at
simulation
a
the
that
Re
initial
development
of
an
are
observed in
we have carried out a
of
initial disturbances with wavenumbers a=0.5,
The
resulting flow
1000.
and with
-1.0 .
infinitesimal three-
dimensional disturbance ( with wave vector (0,6) and initial
-14 2-
energy-10- 8 ) in
dimensional
energy-4xl0
the
wave
2
(
)
with
was
resolution P128,
the time t9r/10
presence
of
wave
first
a finite-amplitude two-
vector
simulated
(a,0)
using
and initial
a
run
with
2N-32, 2M-2 and starting at time t-r/2. At
when
the three-dimensional disturbance had
grown to an energy
of
.015
.0475) the run was
restarted
(and
with
En2 still equal to
with resolution P-128, 2M-32,
2N=16.
The results of the full simulation are shown in Figures
47 and 49. In Figure
47,
the
overall energy of the three-
dimensional disturbance field
(integrated
cross-section) is plotted as a
over the channel
function of the phase of the
cycle. For comparison the overall energy of the disturbances
(integrated
over
the
pipe
cross-section)
from
the
experimental run at Re =1080, A=10.6 is also shown in Figure
48. Note the excellent agreement
between the results of the
simulation for 3/2<wt<5r/2
experimental results. (The
and
results of the simulation for r/2wt3r/2
are not in as good
an agreement because of the effect of the artificial initial
conditions). In particular the
decay
during
the
disturbances
acceleration
explosively reappear
with
the
phase
start
(wt-2r) in accord
with
of
average
velocity
striking
similarity
instantaneous
simulations show
of
can be seen to
(3r/2<wt<2r)
and
the deceleration
experiments. Furthermore, the plots
profiles (Figure 6), in that
from
to
the
numerical
the experimental
for the acceleration phase the
-143-
A
.
. 1
.16
.14
.12
.10
.08
.06
.04
.02
0
r/2
Figure
3r/ 2
wt
57r/2
47Z Phase variation of the overall energy of threedimensional disturbances from the numerical
simulation (Re6-1000. H-10.0)
-144-
,,
.10
.16
"
l-
.14
el
C.4
.12
+
.10
C%1
(.
'
.06
.04
.02
n
3r/2/2
-/7r/2
wt
Figure.8.
Phase variation of the ove'rall energy of
disturbances from experiments (Re -1000.,A-10.0).
Here the value of w' is estimated to be equal to
V' .
-145-
wt - 3/2
4
r/8
+7r/4
+3r /8
10
0-1
a
"I
1P.
4
2
2
a
2r'
-,0
+7r/8
+r/4
+32r/8
5r/2
U/U 0
a
S
2
0
U/U 0
Figure 49.
Average velocity profiles for various phases
from the numerical simulation at Re -1000.,
A-10.0 (curve B) compared to the laminar
solution (curve A).
-146-
profiles are
"full",
while
during
the deceleration phase
they are inflectional outside the boundary layer.
-147-
IV.
SUMMARY AND CONCLUSIONS
In this work we
have
gaps between theory and
attempted
to bridge some of the
experiment regarding transition and
turbulence in bounded oscillatory Stokes flows. Earlier work
on this subject
consist
of,
on
the one hand, experiments
that show the flow undergoes transition at a Re6500;
the other hand, theoretical
form of Floquet analyses
of
studies
that are either in the
the "linearized" equations and
predict all disturbances would decay,
various
quasi-steady
treatments
equations which in turn
are
and on
or are in the form of
of
the
"linearized"
subjects of dispute because of
the assumption of quasi-steadiness.
Our experiments for oscillatory flow in a pipe, for the
range of
parameters
agreement with
230<Re <1710,
earlier
investigations
undergoes transition at a
of A. The resulting
3<A<10
Re'-500
, are
in good
show
the flow
and
independent of the value
turbulent
flow is characterized by the
sudden explosive appearance of
turbulence with the start of
the
deceleration
phase
"laminar" flow during
profiles
of
of
the
the
and
reversion to
the acceleration phase. Instantaneous
ensemble-averaged
strong 'clue' to
cycle,
mechanism
acceleration phase, these
axial
velocity
provide
a
of instability. During the
profiles
are "full" and resemble
-148-
the laminar profiles that
flow starting from
would
initial
be obtained for sinusoidal
conditions u(r,O)=O. During the
deceleration
phase,
inflectional
ust outside the boundary layers as a result of
however,
the
momentum exchange between the low
velocity
profiles
are
speed fluid near the wall
and the higher speed fluid further towards the center. These
profiles
have
a
striking
profiles that have been
during
the
"peaks");
ejection
and
similarity
observed
phase
suggest
of
that
generation of turbulence in
"instantaneous"
in steady boundary layers
the
the
the
to
turbulent
fundamental
bursts (at
process
of
oscillatory flow may be in
the form of bursts similar in
nature to those that occur in
steady boundary
relaminarization
during the
layers.
The
acceleration
cessation of "bursting"
phase
can
because
of
then
of the flow
be attributed to
the favorable pressure
gradients that are present during this period.
The starting point for
is the development of
nonlinear
a
burst in steady shear flows,
streamwise
interaction
of
vorticity
two-
and
as a result of
three-dimensional
disturbance waves. Furthermore the development of the threedimensional waves
infinitesimal
can
be
traced
three-dimensional
presence of finite-amplitude
quasi-equilibria. Both
development
of
disturbances, are
the
back
to
instability of
perturbations
in
the
two-dimensional equilibria or
nonlinear
infinitesimal
independent
of
the
interaction
and the
three-dimensional
shape
of
the mean
-149-
profiles; and are
the
result
three-dimensional
waves
of
interaction
a
in
general
of two- and
shear
flow.
It
therefore, seems reasonable to
expect the same mechanism to
prevail for oscillatory flows,
provided the two- and three-
dimensional waves can
flow, in this
case,
Re 6 /A,
ratio
oscillation
develop.
would
representing
to
the
be
The
time
variation of the
unimportant
the
ratio
(convective)
of
as long as the
time
time
scale
scale
of
of
the
studied
the
instability, is large.
To
substantiate
these
ideas,
we
have
stability of the oscillatory flow in a channel (for A=10) to
various classes of
and
infinitesimal
three-dimensional
and finite-amplitude two-
disturbances.
The
results
of this
study can be summarized as follows,
- For infinitesimal
be
extended
conclusion
to
the
that
oscillatory
only
disturbances need to
disturbances, Squires' theorem can
be
flow
problem,
two-dimensional
considered.
two-dimensional disturbance which
The
initially
with the
infinitesimal
development of a
is in the form
of the least stable eigenmode of the Orr-Sommerfeld equation
for the "frozen"
simulations of the
initial
profile
Navier-Stokes
is followed by direction
equations. The results of
the simulation immediately explain the discrepancies between
quasi-steady theories and
Floquet
theory treatments of the
linearized equations. For the first cycle the disturbance is
-150-
seen to develop in ways
not too different from quasi-steady
predictions. During this period
inflection points or
critical
profiles. Because of the
the disturbance follows the
layers
intrinsic
at the end of the cycle
of the instantaneous
nature of the profiles,
the disturbance ends up in a radial
location too far away from
the
wall
to be able to perturb
the same modes that were excited earlier. Instead it excites
modes that are located further away from the wall. This mode
hopping continues until
the
way to the center of the
into a
periodic
Floquet
theory.
disturbance
channel, at which point it settles
steady-state
and
decays
It
be
noted
should
disturbance is still localized
does
amplify
has moved all the
with
inviscid
growth, however, can
that is observed in
not
as predicted by
that
while
the
near the boundary layers, it
growth
rates.
This inviscid
explain the transition phenomenon
experiments; because maximum growth for
this type of
instability
(i.e, at the
start
of
is
acceleration)
phase to where turbulence
disturbances, however,
flow regime that was
observed around flow reversal
which is opposite in
is observed in experiments. These
may
explain
observed
the "weakly turbulent"
in Hino's experiments during
the acceleration phase of the cycle.
- No
finite-amplitude
two-dimensional equilibria were
found, however quasi-equilibria
were
observed
for
(with
viscous decay rates)
two-dimensional
finite-amplitude
disturbance at transitional Reynolds numbers.
-151-
states
equilibrium
is
three-dimensional disturbance
(on
convective
to
scales)
time
seen
is
qu,asi-
the transitional Re6
explain
The infinitesimal
in experiments.
observed
above
three-dimensional
infinitesimal
to
disturbances can successfully
of 500 that
the
of
instability
secondary
-The
to grow explosively
finite
amplitudes.
The
mechanism of instability is the result of interaction of the
and
two-and three-dimensional waves
is
independent of the
shape (and inflectional nature) of the profiles. Here, again
the
time
variation
of
the
basic
flow
is
not
of much
consequence as long as the ratio Re /A>>,,1.
-
Nonlinear
interaction
of
the
dimensional disturbance waves, shows
the
explosively
grow
average
instantaneous
during
the
acceleration
with
the
three-
structures that are in
phase
start
velocity
deceleration
acceleration.
and
experiments. Disturbances are seen
very good agreement with
to decay during
two-
phase
of
profiles
and
are
of the cycle, and
deceleration.
The
are inflectional
full
during the
-152-
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