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\documentstyle[12pt]{article}\title{ Phase Dynamics
in the Taylor-Couette System}
\author{Mingming Wu, C. David Andereck\\
Department of Physics\\The Ohio State University\\Columbus, Ohio 43210}
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\pagebreak
\section*{\hspace{2.3in}Abstract}
We have studied the phase dynamics of flows in the Taylor-Couette system
by applying
a forced modulation to the upper boundary of a large aspect ratio concentric
cylinders system. The experiments were carried out with three different flow
patterns, Taylor, wavy and turbulent Taylor vortex flow. The
results are consistent with the theoretical model proposed for each flow
pattern. In a one phase variable case, the Taylor vortex flow, the
perturbations diffuse along the axial
direction and the response of the vortex boundary positions to the
modulation is well described by a simple diffusion model.
In a two phase variable case, the wavy vortex flow, the perturbations
propagate as traveling waves parallel with the axis of the system when
there is a strong coupling
between the axial and azimuthal phase variables, and diffusively
for weak coupling. The phase dynamics is
governed by coupled diffusion equations. In the turbulent
Taylor vortex flow, where spatial coherence coexists with the turbulent flow,
the phase dynamics of the coherent structure is
described by a diffusion model with a diffusion coefficient
an order of magnitude larger than for the laminar Taylor vortex flow.
\newline
\vspace{0.1in}
\noindent
PACS numbers: 47.20.-k, 47.30.+s
\pagebreak
\section{Introduction}
Considerable attention has been given to spatially periodic patterns
formed in nonequilibrium fluid dynamical systems such as Taylor-Couette
flow~\cite{re:swinneyb}, Rayleigh-Benard
convection\cite{re:gollub},
viscous fingering~\cite{re:couder},
and mixing layers of two immiscible fluids\cite{re:huerre}.
In general, each system is controlled by an external stress $R$. When $R$
exceeds
a threshold value $R_{c}$, the system undergoes a transition from a
spatially uniform state to a periodic state, thereby breaking the
translational invariance of the system.
A typical example is the Taylor-Couette system, in which the fluid is confined
between two concentric rotating cylinders (see Fig.~\ref{f2}).
The control parameter is the
Reynolds number $R$ ($\propto \Omega$, the inner cylinder rotation frequency).
When R is small, the flow in an infinitely long system
has only an azimuthal velocity component, and it is called simply
circular Couette flow (CCF). When $R$ exceeds
$R_{c}$, a centrifugal instability gives rise to Taylor vortex flow (TVF)
out of the base CCF. In TVF, the velocity field consists of a sequence of
pairs of counter-rotating vortices stacked along the cylinders
(see Fig.~\ref{f1}a).
The
velocity field varies with distance from
the system axis, periodically along the system axis, and is constant
in the azimuthal
direction. The first time-dependent regime, wavy vortex
flow(WVF), in which an azimuthal traveling
wave is superimposed on the TVF, occurs for
slightly greater R in a large radius ratio system (see Fig.~\ref{f1}b).
Upon further increasing
$R$, the flow first becomes quasi-periodic, then
weakly turbulent or chaotic,
and eventually reaches a turbulent Taylor vortex flow (TTVF),
in which
the vortex boundaries are embedded in a turbulent background
(see Fig.~\ref{f1}c).
The vortex sizes are close to those in TVF.(***The size of TTVF is always
a little bit bigger than that of TVF. TTVF has the range of Np 22-28, while
TVF has 30-38***).
The vortex boundaries become difficult to discern when $R/R_{c}$ is increased
to $\sim 700$\cite{re:smith}, which is well above the range of R considered
here.
In principle, all the above phenomena may be understood using
the Navier-Stokes
equation\cite{re:marcus1,re:marcus2,re:davey,re:nagata,re:chandra}.
However, the complexity of the equation has thus far made it
impractical to generate full numerical solutions for comparison with the
potentially huge number of laboratory situations. Furthermore, it is not
always easy to extract the physical insights needed for a deep
understanding
of the flow patterns from such numerical solutions. Therefore, model
equations are of considerable
importance.
For example, amplitude
equations~\cite{re:yahata,re:white,re:philip,re:yahata1}
can describe the dynamics of the states close to the onset of the
first supercritical bifurcation.
Amplitude equations are derived from the Navier-Stokes equation (or other
basic equations) by
the expansion of a small amplitude of the emerging structure.
Many interesting features
of laboratory generated Taylor-Couette flows are contained in this class of
model
equations\cite{re:pfister,re:ahlers}.
In an amplitude equation approach,
the phase and amplitude are independent variables.
For a case where the wavelength has slow time and space variations, the
amplitude is slaved to the phase variable and the amplitude
equation can be simplified to a phase equation. For flow patterns
further above the onset of Taylor vortex flow the amplitude equation is no
longer useful, but the equations for phase variables are presumed still valid.
Pomeau and Manneville~\cite{re:pomeau} and Kuramoto\cite{re:kuramoto}
introduced the method of phase dynamics for treating
nonequilibrium systems. A simple
phase diffusion equation was derived from the Swift-Hohenberg equation.
This
phase equation can be applied to a nonequilibrium system that undergoes a
supercritical bifurcation and forms a spatially periodic pattern.
Pomeau and Manneville
applied their theory to the weakly nonlinear regime (near the onset of the
convection state) of the Rayleigh-Benard (RB) system.
A phase diffusion equation was derived for the RB system and
the diffusion coefficient was found to depend
on the wavevector of
the convective rolls and the distance above the threshold of convection.
Following this,
Wesfreid and Croquette~\cite{re:wesfreid}, and Croquette and Schosseler
~\cite{re:vincent} studied experimentally the
phase dynamics
near the onset of the convection state in the RB system. A modulation,
forced
by periodic injection of fluid, was imposed on
the RB system and the responses of the convective
rolls were studied. Their results partially confirmed the
validity of the phase
diffusion model in the RB system.
Tabeling~\cite{re:tabeling} applied the Pomeau-Manneville formalism to the
Taylor-Couette system and
obtained a similar phase diffusion equation near the onset of TVF.
He showed that the diffusion coefficient depends on the
Taylor vortex wavevector and the distance above threshold (which was true
also for the RB system).
In addition, Tabeling suggested that Snyder's experimental results
from a small aspect ratio Taylor-Couette system showed evidence of phase
diffusive behavior\cite{re:tabeling,re:snyder}. In Snyder's experiment,
the flow state was perturbed in a spatially uniform manner
(a sudden increase of the inner cylinder frequency)
and the adjustments of the cell boundaries
were observed. It was found that the cell sizes exponentially
relaxed to their final
values, which was consistent with Tabeling's calculation.
However, the relaxation time, which is inversely
proportional to the diffusion coefficient, did not seem to depend
on the distance from TVF onset or the average wavevectors of
the vortices, although this was not a primary point of his investigation.
Following Tabeling's work,
numerical simulations of the phase dynamics near the onset
of the TVF have been performed\cite{re:lucke}. In the simulations,
one end boundary of the system
was moved to its final position with various speed
profiles and
the responses of the vortices were studied.
Phase diffusive behavior was found in the region far
from the boundary. An experimental study of the
phase dynamics in TVF was carried out by Gerdts\cite{re:hans,re:gerdts},
in a small aspect
ratio ($\Gamma\sim 20$) system. As in the simulations, the first
vortex was compressed by moving the upper collar of the Taylor-Couette system
and the responses of the vortices in the bulk were studied. The results showed
diffusive behavior and the measured diffusion coefficients were found to
depend on the wavevector.
While a significant amount of work has been done on
phase dynamics for the one phase variable case[17-26],
much less attention has been
paid to the two phase variable situation[27-31]. Based on a symmetry
analysis,
and using a slow time and space variation approximation,
Brand and Cross proposed that
the proper description for wavy vortex flow
involved {\it coupled} diffusion
equations\cite{re:brand,re:brand2,re:brand3}.
They predicted a propagating phase
mode in the strong coupling limit and a diffusive mode in the weak
coupling regime.
The diffusion coefficients and coupling
coefficients can be evaluated
only for states close to the onset of TVF, although the equations themselves
are valid for states far above TVF onset.
Phase dynamics has usually been thought of as only applicable
to well-ordered spatial patterns. However, there are circumstances
throughout fluid dynamics in which relatively coherent structures exist,
embedded in a background of turbulence.
It is reasonable to ask whether
phase dynamics might be relevant for describing the behavior of those coherent
structures, ignoring the small-scale fluctuations. It is in this spirit
that we have studied phase dynamics for turbulent Taylor vortices.
Specifically, we have asked whether phase equations can be used to
describe the slow temporal and spatial changes of the
coherent structures in TTVF (where the coherent structures are the Taylor
vortices themselves)?*** Yes*** Theoretical and experimental work on this
problem has been lacking until now.
An overall experimental study of phase dynamics
in the Taylor-Couette system is presented in this paper. Brief reports of
some of
these experimental results have appeared\cite{re:wu1,re:wu2,re:wu3,re:wu4}.
Section 2 provides the theoretical background. Section 3 describes the
experimental setup and the data acquisition techniques, as well as
the experimental
results on phase dynamics in TVF, wavy vortex flow and turbulent Taylor vortex
flow. Section 4 summarizes our work.
\section{Theoretical Background}
The control parameter for the Taylor-Couette system with the outer cylinder
at rest is the Reynolds number R, defined as
$R=\Omega_{i}r_{i}d/\nu$,
where $\Omega_{i}$ is the inner cylinder rotation rate, $r_{i}$ is the
inner cylinder radius, d is the gap between the inner and outer cylinders
and $\nu$ is the kinematic viscosity. When $R$ exceeds the threshold $R_{c}$
(neglecting Ekman effects near the ends of a finite system),
the flow changes from a uniform circular Couette flow (CCF) to a spatially
periodic Taylor vortex flow (TVF).
In the vicinity of $T_{c}$,
the velocity field can be described as:
\begin{eqnarray}
\vec{V}_{TVF}(\vec{r},t) & = & \vec{V}_{CC}(\vec{r},t)+\vec{v}(\vec{r},t)
\nonumber \\
\vec{v}(\vec{r},t) & = & A(z,t) \vec{v}(r) e^{iq_{c}z}
+A^{*}(z,t)\vec{v}^{*}(r) e^{-iq_{c}z}
\end{eqnarray}
where $\vec{V}_{CC}(\vec{r},t)$ and $\vec{V}_{TVF}(\vec{r},t)$ are
the velocity fields for CCF and TVF, $\vec{v}(\vec{r},t)$ is the
perturbation velocity field, $\vec{v}(r)$ is the eigenfunction of
the velocity field and $q_{c}$ is the critical wavevector corresponding to
the lowest $T_{c}$. A(z,t) is the complex amplitude which
retains only the slow time and space variations. The amplitude equation can be
derived by the general procedure of Newell and Whitehead~\cite{re:white},
it is:
\begin{eqnarray}
\tau_{0} \partial A / \partial t =\epsilon A+
\xi_{0}^{2} \partial^{2}A / \partial z^{2}-g|A|^{2}A
\end{eqnarray}
where $\tau_{0}$ is the perturbation amplitude growth rate, $\xi_{0}$ is
the correlation length, z is the coordinate in the axial direction, g is a
factor relating to the scale of A, and $\epsilon=\frac{R-R_{c}}{R_{c}}$
is the distance
to the onset of TVF.
Due to the periodicity of the TVF in the axial direction, A(z,t) can be
written in the form\cite{re:ahlers}:
\begin{eqnarray}
A(z,t)=A_{0}e^{iqz}e^{i\sigma t}
\end{eqnarray}
where $q=\tilde{q}-q_{c}$, $\tilde{q}$ is the wavevector of the TVF, and $\sigma$
is the growth rate of the perturbation.
A steady state for
$\epsilon > \xi_{0}^{2} q^{2}$ is obtained by substituting Eqn. 3 into
Eqn. 2, with the result being:
\begin{eqnarray}
A_{0}=\sqrt{\frac{\epsilon-\xi_{0}^{2}q^{2}}{g}}
\end{eqnarray}
However, this solution is not stable over the whole range of q. The stability
of $A_{0}$ may be determined by assuming $A(z,t)=\overline{A(z,t)}e^{i\Psi} $
where
\begin{eqnarray}
\overline{A(z,t)}=A_{0}+a(z,t)
\hspace{0.5in}
\Psi=qz+\psi(z,t)
\end{eqnarray}
a(z,t) and $\psi(z,t)$ are small perturbations of the steady solution
$A_{0}e^{iqz}$.
A standard linear stability analysis of Eqn. 2 with the above assumptions
gives
\begin{equation}
\frac{\partial \psi }{ \partial t} =
D_{||}\frac{\partial^{2} \psi}{\partial z^{2}}
\label{eq:diffusion}
\end{equation}
where
\begin{equation}
\label{eq:dform}
D_{||}=\frac{\xi^{2}}{\tau_{0}}
\frac{(\epsilon-3\xi_{0}^{2}q^{2})}{(\epsilon -\xi_{0}^{2}q^{2})}
\end{equation}
The steady state $A_{0}$ is stable when the diffusion coefficient is
positive. Therefore instability occurs in the range
$\xi_{0}^{2}q^{2}<\epsilon<3 \xi_{0}^{2}q^{2}.$
Here $\epsilon<3 \xi_{0}^{2}q^{2} $ produces
the Eckhaus instability\cite{re:ahlers,re:eckhaus,re:cannell,re:ning}.
If we increase R further, the rotational symmetry of the TVF is broken
and a new pattern emerges,
wavy vortex flow (WVF).
Therefore, a second phase variable $\phi$ is introduced
to describe the phase of the azimuthal wave motion. A more precise definition
of $\psi$ and $\phi$ can be seen by examining the form of the
amplitude of the wavy vortex flow:
\begin{equation}
\label{eq:wamp}
A=|B|e^{i\psi}+i|C|e^{i\psi}(e^{i(q_{y}y+\phi)}+c.c.)
\end{equation}
where $q_{y}$ is the azimuthal wavevector, y is the azimuthal position,
B and C are constants and C$\rightarrow$0 at $\epsilon_{w}$, the onset
of WVF. Brand and Cross~\cite{re:brand} proposed coupled diffusion
equations to describe the phase dynamics in WVF according to a symmetry
analysis. The equations for WVF must be invariant under two symmetry operations:
1) z$\rightarrow$ -z, $\psi\rightarrow -\psi$, $\phi\rightarrow\phi$, and
2) y$\rightarrow -y$, $q_{y}\rightarrow -q_{y}$, $\psi\rightarrow\psi$, $\phi\rightarrow -\phi$.
Thus they argue that the appropriate phase equations are:
\begin{eqnarray}
\frac{\partial\psi }{\partial t} & = &
D_{1}\frac{\partial^{2}\psi}{\partial z^{2}}
+C_{1}\frac{\partial\phi}{\partial z}
\nonumber \\
\frac{\partial\phi} {\partial t} & = &
D_{2}\frac{\partial^{2}\phi}{\partial z^{2}}+C_{2}\frac{\partial\psi}
{\partial z}
\label{eq:couple}
\end{eqnarray}
where z is along the system axis, $D_{1}$ and $D_{2}$ are the diffusion
coefficients for the axial and azimuthal directions, and
$C_{1}, C_{2}\propto q_{y}$ represent the coupling strength between axial
and azimuthal motion. The coefficients can be derived following the same
procedure as in the
case of TVF close to onset.
Assume the phase variable $\psi$ (and also $\phi$ through the
coupling of the equations)
is perturbed by a term proportional
to exp[i(Kz-$\omega$t)], and substitute into Eqn.~\ref{eq:couple}.
We then eliminate terms in $\phi$ and find the following relation between K and $\omega$:
\begin{eqnarray}
K^{2}=\frac{i\omega(D_{1}+D_{2})-C_{1}C_{2}+\sqrt{\Delta}}{2D_{1}D_{2}}
\nonumber \\
\Delta=C_{1}^{2}C_{2}^{2}[1-\frac{2i\omega(D_{1}+D_{2})}{C_{1}C_{2}}
-\frac{(D_{1}-D_{2})^{2}\omega^{2}}{C_{1}^{2}C_{2}^{2}}]
\end{eqnarray}
For the strong coupling case, i.e.,
$C_{1}C_{2}\gg 2\omega(D_{1}+D_{2})$,
we have $K=\frac{\omega}{c}$, where $c=\sqrt{C_{1}C_{2}}$.
Eqn.~\ref{eq:couple}
in this case has the characteristic form of a wave equation:
\begin{eqnarray}
\frac{\partial ^{2}\psi}{\partial t^{2}}-
c^{2} \frac{\partial ^{2}\psi}{\partial z^{2}}=0
\end{eqnarray}
For the weak coupling limit, where $C_{1}C_{2}\ll |D_{1}-D_{2}|\omega$,
we have $K^{2}=-\frac{i\omega}{D_{1}}$ or $K^{2}=-\frac{i\omega}{D_{2}}$, a
typical situation for a diffusion process.
In Taylor vortex flow, $C_{1}=C_{2}=0$,
the weak coupling condition is trivially satisfied, and the
diffusion model for
TVF is recovered.
With sufficient increase in $\epsilon$ the flow becomes
chaotic or weakly turbulent. As $\epsilon$ is increased
still further, the flow becomes increasingly turbulent. The
vortices remain as large-scale structures
(similar in
size to the laminar Taylor vortices), periodically stacked
along the axis of the system.
The vortex boundaries fluctuate quasi-regularly with a period of
$\sim 1/\Omega_{i}$. The number of phase variables involved in this case is
uncertain. However, we have found that
the {\it average} vortex boundary position is a well-defined, deterministic
variable.
We argue that a phase equation approach would still be valid
for such a coherent structure of vortices embedded in an otherwise
locally turbulent flow.
A possible analogy for motivating a phase dynamics approach for TTVF is in
the hydrodynamic equations for certain
equilibrium systems\cite{re:wu4}.
For example, consider nematic
liquid crystals\cite{re:gennes}. In nematics the positions of the constituents
show short range order, whereas the molecules align on average
spontaneously parallel to a certain direction characterized by a unit
vector, the director, and thus show
broken orientational symmetry.
The director is a
quantity which is averaged over many molecules in space and over many
collision times. That is, close to equilibrium the
hydrodynamic equations are already averaged over
the shorter time- and length- scales.
We argue that this approach would be useful
for the TTVF and similar flows. That is, we
consider equations for the average location of the vortex boundary
between neighbouring vortices, assuming that the fluctuations of the
vortex boundaries are fast compared to the time scales in which we are
interested.
After the fast fluctuations are averaged out, the coherent structure left
is similar to TVF. Therefore, we predict that the slow perturbations
to the coherent structure will diffuse along the axis of the system in
a manner analogous to the case of TVF.
\section{Experimental Technique and Results}
\subsection{Experimental Setup}
Our experiment was conducted in two concentric cylinders with the outer
one fixed (see Fig.~\ref{f2}). The inner cylinder was made of black Delrin plastic
with radius $r_{i}=5.262cm $ and the outer cylinder was of Plexiglas
with inner radius $r_{o}=5.965cm$, which gave a radius ratio $\eta = \frac{r_{i}}{r_{o}} = 0.882$.
(For further details on the basic
system,
see ref.~\cite{re:john}.)
The inner cylinder rotation frequency was controlled by a
Compumotor stepper motor(model M83-93) which is precise to 0.001 Hz.
A PDP-11/73 was interfaced through the Compumotor indexer to control the
stepper motor. The working fluid region was bounded at both ends
by Teflon rings. The lower end ring fit snugly against the outer cylinder,
with a gap of $0.8mm$ to the inner cylinder.
The upper ring was attached to a traversing mechanism,
described later, which allowed the ring to
oscillate along the axial direction over a maximum amplitude of $\pm 0.5cm$.
There were gaps of $\sim 0.8mm$ between the upper Teflon ring and both
the inner and outer
cylinders. In this way, the fluid could move past the ring when it oscillated.
The distance between the two Teflon rings initially
(before the modulation was added) was 49.5cm, and
therefore the average aspect ratio $\Gamma=\frac{L}{r_{o}-r_{i}}=70.4$.
The working fluid was a solution of double
distilled water and 44$\%$ glycerol by weight, which has a kinematic
viscosity $\nu=4.0cs$. Temperature regulation in the room was within one degree C,
with the fluid temperature itself varying by a few tenths of a degree.
1$\%$ by volume
of Kalliroscope AQ1000 was added for
visualization. The fluid could be used for at least two months before it
deteriorates.
The flow pattern was viewed under room illumination
with a $512 \times 480$ pixel CCD
camera (Javelin Model JE-7242 Newvichip) which was connected to an image
processor (Imaging Technology FG-1024). For a typical situation, this
camera and processor provided a resolution of 34 pixels per vortex.
The image processor board was installed in a PC-AT
(Everex System 1800). The board digitized a picture of the
flow pattern and the PC recorded the light intensity profile along a vertical
line. A C language program was used along with the software package HALO
(Imaging Technology)
for the frame grabbing and basic processing.
The positions of the vortex
boundaries were determined by finding the
local minima of the light intensity profile. In a normal data taking process,
we recorded the intensity profile of a vertical line in the dynamic memory of the
PC,
found the minima and saved their locations into a data file. The data files
were then transferred by ethernet to a VAX 8650
system for data analysis (curve fitting or plotting).
The oscillation of the upper ring was controlled by a traversing mechanism.
A schematic diagram of this device is shown in Fig.~\ref{f3}. It consisted
of two basic parts.
The lower part was a plate connected to the moving Teflon ring.
The connections were made by
three 1/8'' brass rods running through the top cover of the concentric
cylinders system. The plate was supported against gravity by three torsional
springs
that were mounted on the top cover. The upper rotating ring was connected to the
supporting plate by a
ball bearing track. The upper ring was threaded on the inside and a pulley
was mounted on the outside. A timing belt connected the pulley with a Compumotor
M83-93 stepper motor. The translations
of the rotating ring produced the vertical displacement, which was
transmitted to the upper Teflon collar by the supporting plate and the
three brass rods.
Four rotations of the stepper motor produced a 1$mm$ displacement of
the upper collar.
The stepper motor was controlled by a Compumotor indexer,
which was controlled in turn by
a PDP-11/73 computer. A safety stop was designed with the joy stick
input of the indexer to prevent the motor from
overrunning the range, thereby damaging itself and the system.
All the numbers in the following are dimensionless unless
otherwise indicated.
Lengths are scaled by the gap $d$ ($=r_{o}-r_{i}=0.703cm$), and times are
scaled by $d^{2}/\nu$, which is $\sim 12.3 s$ in our system.
\subsection{Experimental results}
\subsubsection{Taylor Vortex Flow}
A TVF pattern (Fig.~\ref{f1}a)
is obtained by increasing the rotation speed slightly above
the critical $R_{c}=112$.
Two types of perturbations, induced by motion
of the upper ring, were applied to
the pattern, either a sinusoidal forcing or a step function
forcing.
The responses of the vortex boundary positions
to the perturbations were then recorded as a function of time.
The axial phase variable $\psi$ is directly proportional to the
vortex boundary position variation. Assuming $z_{n}(z,t)$ is the $n^{th}$
vortex boundary position in the axial direction,
and $z_{n}^{0}=\frac{n\pi}{\tilde{q}}$ is the vortex
boundary position before the upper ring oscillates,
then the phase of the amplitude at the $n^{th}$ vortex boundary
$=(\Psi+q_{c}z)|_{z=z_{n}}=\psi(z_{n},t)+\tilde{q}z_{n}=n \pi$. This gives:
\begin{equation}
\label{eq:zpsi}
z_{n}=z_{n}^{0}-\psi(z_{n},t)/ \tilde{q},
\end{equation}
which shows that the axial phase variable is directly proportional
to the vortex boundary position variation.
In the sinusoidal modulation case, the phase at $z=0$
exactly followed the collar's motion, which follows from the
fact that the period of
the modulation $T\gg d^{2}/ \nu$, the diffusion time through a vortex,
where T is typically several thousand seconds. This
resulted in the following boundary condition:
\begin{eqnarray}
\psi|_{z=0}=\psi_{0} sin(\omega t)
\end{eqnarray}
Here $\psi_{0}/\tilde{q}$ and $\omega$ are the modulation
amplitude and frequency.
Solving Eqn.~\ref{eq:diffusion} with the
above boundary condition, we obtain:
\begin{eqnarray}
\psi(z,t) & = & \psi_{0} e^{-\alpha z} sin(\omega t-\beta z) \\
\alpha & = & \beta=\sqrt{\frac{\omega}{2D_{||}}}
\label{eq:alphabeta}
\end{eqnarray}
Substituting $\psi(z_{n},t)$ into Eqn.~\ref{eq:zpsi} gives:
\begin{eqnarray}
z_{n} & = & z_{n}^{0}-\psi_{0} e^{-\alpha z_{n}}
sin(\omega t-\beta z_{n})/\tilde{q} \nonumber \\
& \simeq & z_{n}^{0}-\psi_{0}
e^{-\alpha z_{n}^{0}}sin(\omega t-
\beta z_{n}^{0})/\tilde{q}
\label{eq:form}
\end{eqnarray}
Eqn.~\ref{eq:form} shows that the vortex boundary position
is a sinusoidal function of time, the amplitude of the motion
decreases exponentially along the axial direction,
and the phase shift between neighboring vortices has a linear dependence on
the axial position. The diffusion
coefficient can be evaluated from the parameters $\alpha$ and $\beta$, and
$\alpha$ and $ \beta$ should have the same value
according to Eqn.~\ref{eq:alphabeta}.
In the second experiment, the upper collar is moved to a final position
at a constant speed, which leads to the approximate boundary condition:
\begin{equation}
\psi|_{z=0} = \psi_{0} H(t)
\label{eq:step}
\end{equation}
where H(t) is the step function, 0 for $t<0 $, 1 for $t\geq 0$. This
is an {\it approximate} boundary
condition owing to the finite time (typically 48 s) for the collar
to reach its final position,
but analysis of our data shows that this approximation has a
negligible effect in the
region beyond a few vortex diameters from the boundary.
However, by imposing a step function, we are measuring the reaction of the
flow pattern
to a modulation with a wide frequency band
instead of just a single frequency. The fact that
phase dynamics should only be valid for slow temporal and spatial changes
indicates
that the step function experiment may not be as accurate as
in the sinusoidal function case.
The advantage of the step function is that it takes about $40min$. to take one
set of data compared with the 6 hours in the sinusoidal case.
This enabled us to collect several data sets ($\sim 10$)
for each flow state and thus
obtain an average value for $D_{||}$.
Substituting Eqn.~\ref{eq:step} into Eqn.~\ref{eq:diffusion}, we obtain
$z_{n}(z,t)$ as:
\begin{eqnarray}
z_{n}(z,t) & = & z_{n}^{0}-
d_{0} {\em erfc} (\frac{z_{n}}{2\sqrt{D_{||}t}}) \nonumber \\
& \simeq & z_{n}^{0}-d_{0} {\em erfc}(\frac{z_{n}^{0}}{2\sqrt{D_{||}t}})
\label{eq:erfc}
\end{eqnarray}
where $d_{0}$ is the distance that the upper collar moves and $\em{erfc}$ is
the complement of the error function\cite{re:mathtable}.
In the periodic modulation case, the sinusoidal motion of the upper collar
had a typical amplitude of $d_{0}/d=0.480$ and a period of 3040 s.
% In order to obtain a TVF pattern with number of vortices $N<70$,
% the inner cylinder rotation frequency is
% increased to $\epsilon\sim 0.8$ or higher (wavy vortex region) to get rid of
% the defects of the flow pattern and decrease $\epsilon$ to the designated
% value. For obtaining a flow pattern with $N>70$, the inner cylinder
% rotation rate changes suddenly from 0 to the designated $\epsilon$ value..
Before commencing the oscillation, the TVF was left at least an hour so that
the flow reached its steady state (this occurs in our system in about half
an hour). Two hours after the oscillation of the upper ring began,
light intensity profiles along a vertical line were recorded once
every $2 min.$ for a period of 5 hours.
Fig.~\ref{f5} is a typical data set.
The response of each
vortex boundary is necessarily a sinusoidal function of time, with an
amplitude and phase shift as predicted by Eqn.~\ref{eq:form}.
By fitting the data with the following equation
\begin{eqnarray}
z_{n}=z_{n}^{0}-a\sin(\omega t-\delta)
\end{eqnarray}
where a and $\delta$ are the amplitude and the phase of
the vortex boundary motion, we found that $\ln a$ and $\delta$
were linearly related
to $z_{n}^{0}$, as shown in Fig.~\ref{f6}.
The slopes of the lines
yield $\alpha$ and $\beta$, from Eqn.~\ref{eq:alphabeta}, and in turn
$D_{||}$.
The variation of $\alpha$ and $\beta$ for different runs was less
than 20$\%$.
For a TVF, the Ekman pumped
end rolls are normally larger than the rolls in the center
of the cylinder.
It has been suggested that, for finite cylinders,
$\alpha$ should always be smaller
than $\beta$ \cite{re:lucke1}, as found by taking the effect of the Ekman
rolls as a perturbation in the derivation of the phase equations from the
amplitude equation. This is consistent with our observations,
as shown in Fig.~\ref{f7}. A detailed calculation is underway to attempt to
verify this assertion. (To compute the diffusion coefficient we have used
only the $\beta$ values, which, coming from the phase shifts, are more
precisely determined. *** Maybe we should do a small derivation ourself
instead of waiting for Lucke to derive them***)
Repeating the above experiment with different modulation frequencies, we
found that the diffusion coefficient is, within our experimental uncertainty,
independent of modulation frequency.
These results are summarized in Table 1.
The diffusion coefficient depends on the Taylor vortex wavevector,
as suggested by the Pomeau-Manneville diffusion model.
From the Eckhaus instability curve, we know there is a limited range of
stable states of TVF for
a fixed $\epsilon$.
The detailed
history of the adjustment of $\Omega_{i}$ selects the particular
$Np$ within the
stable range.
Our procedure for obtaining a stable TVF state with different
$Np$ for a fixed $\epsilon$ is as follows.
For $q=q_{c}$ (or $Np=35$ in our system), we
increase the inner cylinder rotation rate very slowly
(in about 20 min.) from 0 to the desired $\epsilon$. For a larger $Np$
($Np>35$) state, $\Omega_{i}$ is increased abruptly
from 0 to the selected $\epsilon$. For a
smaller $Np$ $(Np<35)$ value, a stable wavy vortex state is obtained first and
any defects that may form are given time to annihilate
or otherwise leave the flow.
Finally, $\Omega_{i}$ is adjusted to the
$\epsilon$ chosen for the experiment.
$D_{||}$ was measured with several different wavevectors and it
was found to decrease when the wavevector q deviated from $q_{c}$.
Fig.~\ref{f8}
shows the typical dependence of $D_{||}$ on the vortex wavevector.
The correlation length $\xi_{0}$ and correlation time $\tau_{0}$ are
obtained by fitting this with Eqn.~\ref{eq:dform}.
These results are summarized in Table~\ref{t4}.
In the step function modulation case, the upper collar moves at
about $0.125mm/s$ to a final position in 48 s, thereby increasing the
aspect ratio by 0.853. The data acquisition
technique used for the oscillatory case was used
here as well. The light intensity profile along a vertical line was recorded
once every 12 s over a period of 40 minutes after the oscillation
was started.
Fig.~\ref{f9} shows a sample result.
From this plot we can see that the response time of
each vortex increases with the distance of the vortex from the upper boundary.
For
instance, the time for the 2$nd$ vortex boundary to reach half way
to its steady state position was 31.1 s after the collar came to rest,
while the 3$rd$ vortex boundary took 71.9 s.
In this case at least, the approximation of Eqn.~\ref{eq:step} is valid for
the vortices beyond the three adjacent to the collar.
Fitting the data with the equation
\begin{eqnarray}
z_{n} & = & z_{n}^{0}-
d_{0} {\em erfc} (\frac{s_{n}}{\sqrt{t}})
\end{eqnarray}
where $s_{n}=z_{n}^{0}/2\sqrt{D_{||}}$ according to Eqn.~\ref{eq:erfc},
we obtain
$s_{n}$ as a linear function of $z_{n}^{0}$, as shown in Fig~\ref{f10}.
The slope of the line in Fig.~\ref{f10} gives the value of $D_{||}$.
Repeating this process for different TVF wavevectors, we obtained a
relation between $D_{||}$ and wavevector similar to that found under periodic
modulation, as shown in Fig.~\ref{f8}b.
The resulting values of $\xi_{0}$ and $\tau_{0}$ are
compared in Table~\ref{t4}
\noindent
with those from the periodic cases and numerical
simulations~\cite{re:dominguez,re:riecke}. Our values for $\xi_{0}$
and $\tau_{0}$ differ from
the values from reference ~\cite{re:dominguez},
while $D_{||}$ from reference ~\cite{re:riecke}
is within our error bars for 3 cases. In both numerical cases, the
system geometry
studied was similar to ours. That we differ with reference
\cite{re:dominguez} suggests
we may have been operating beyond the useful range of
Eqn.~\ref{eq:diffusion},
which is, strictly, only appropriate for $\epsilon\rightarrow 0$.
*** If I remember it right, the calculation of reference 40 requires
$\epsilon\rightarrow 0 $. A more
proper way to say this maybe: The fact that we differ from reference
40 suggests that we may have been operating beyond the range of
$\epsilon\rightarrow 0 $, which is required by the numerical calculation
of reference 40. ***
Further
experiments with $\epsilon$ smaller than 0.06 may yield better agreement.
As shown in Fig.~\ref{f8}, the range of wavevectors accessible to us is
limited due to the large radius ratio of our system. According to
the Eckhaus theory~\cite{re:ahlers,re:eckhaus,re:cannell,re:ning}, the
range of stable wavevectors for TVF
is larger for a smaller radius ratio system, where it is possible to go to
higher $\epsilon$ before reaching the wavy instability. Therefore, it is
expected that
an experiment in such a system will show a greater
dependence of $D_{||}$ on the wavevector\cite{re:hans}.
\subsubsection{Wavy Vortex Flow}
A wavy vortex flow (Fig.~\ref{f1}b)
can be obtained in a large radius ratio system by increasing the Reynolds number
to slightly above the onset of TVF.
The onset of WVF is at $\epsilon_{w}=0.223$ for our system.
The data acquisition process
is complicated by the azimuthal wave motion of the vortex
boundary.
The vortex boundaries move on two
time scales, the slow time $T$ (corresponding to the slow upper boundary
oscillation, with a period of several minutes) and the
fast time $T_{1}$ (corresponding to the azimuthal wave
motion of period $\simeq 1$ s.).
We define $\bar{z}_{n}(\tau)$ to be the average boundary position
variation *** $\bar{z}_{n}(\tau)$ is defined as position not
position variation in TTVF case, to be consistent, it is better to define it
as average vortex boundary position*** of the $n^{th}$ vortex
due to the slow upper boundary modulation, where $\tau$ is real time scaled by $d^2/\nu$.
Thus,
$\psi=q \bar{z}_{n}(\tau)***add - \bar{z}_{n}^{0}$ ***, where q is the axial
wavevector.
The procedure for obtaining $\bar{z}_{n}(\tau)$ is as follows. We
first take about 100 consecutive line profiles (covering approximately 10
azimuthal waves) in $7 s$ and find the intensity minima.
A typical space-time diagram of the wavy vortex boundary positions
is shown in Fig.~\ref{f12}.
The data file records locations of minima of the vertical lines ($z_{n}$,t).
Our interest is in the average position of each vortex boundary. Thus
we need to
rearrange our data file, putting the locations of each vortex boundary
together,
and then averaging the locations for each line.
The following technique is used to sort out the locations of each
vortex boundary.
The original data (locations of vortex boundaries) are put
into an array,
S(z,t), where z is the axial position and t is time.
\begin{eqnarray}
S(z,t) & = & 1 \hspace{0.2in} \mbox{when (z,t) is located at the vortex
boundary} \nonumber\\
& = & 0 \hspace{0.2in} \mbox{otherwise}
\end{eqnarray}
The numbers S(z,t) over the time range (typical
time range is from 1 to 100*****seconds? scaled units?
we must be consistent*****) *** A good question, sorry about the problem.
1-100 means 100 frames, each frame takes 0.07 second, so the time
range here is 0. to 7. sec.. I think it is O.K. to give out the time in
second here to give people a more direct feeling of how long it is,
if you don't
agree, it is O.K. to put dimensionless time in ***
at a fixed z value are then added together.
When the sum equals 0,
it means that particular z value locates the separation region of two vortex
boundaries. Different vortex boundaries are thus identified and the
azimuthal wave
motions are averaged out along each vortex boundary to
obtain $\bar{z}_{n}(\tau)$.
This technique becomes more difficult to apply when the
vortex boundary motions
overlap in the axial direction (for instance, in the case of
modulated wavy vortex flow) since our method can no longer separate
each vortex boundary. In order to find $\bar{z}_{n}(\tau)$ in such a case,
we would need to fit
$S(z,t)$ with a series of sinusoidal functions. Further work will be
necessary to study the phase dynamics in either modulated wavy vortex flow
or large amplitude wavy vortex flow.
We first discuss our measurements of
$\bar{z}_{n}(\tau)$ for flow states with different m.
An m=3 state was obtained by increasing $\epsilon$ slightly above the onset
of TVF, to $\epsilon=0.140$. A typical result is shown in
Fig.~\ref{f13}a.
The amplitude of $\bar{z}_{n}(\tau)$
decreases exponentially with distance from the boundary (Fig.~\ref{f14}a), and
there is a linear phase
shift between oscillations of neighboring vortices(Fig.~\ref{f14}b). These are
characteristics of a diffusive mode, as shown by Eqn.~\ref{eq:form}.
The diffusion coefficient in this case is $\sim 1.4$, comparable to that
for TVF. Increasing $\epsilon$ to 0.813, the system
reaches a new stable state with $m=7$.
In order to obtain $ m=7$ with the same Np,
$\epsilon$ must be changed rapidly from $0.140$ to
$0.813$. With the same modulation period as for the $m=3$ case, $T=53.3$,
there is an axially
propagating wave in the middle region of the cylinders, as shown in the space-time plot of
Fig.~\ref{f13}b, evidently described by $\bar{z}_{n}***add -\bar{z}_{n}^{0}***
\propto sin(kz-\omega t)$,
with small damping\cite{re:brand}.
The complete amplitude
of $\bar{z}_{n}(\tau) $ vs. axial position is shown as the $\Box$ data points
in
Fig.~\ref{f14}a. We identify the behavior in the central region as a propagating mode
because: 1) the amplitude is roughly constant over
many vortices, in contrast to the diffusive case, and 2) the phase shift $\delta$ of
neighboring $\bar{z}_{n}(\tau)$ in this
region
depends linearly on the axial position($\Box$ line of
Fig.~\ref{f14}b),
which is to be expected for a traveling wave, but not a standing wave.
The behavior in the end regions for $T=53.3$ is rather different from that in the bulk.
Diffusive behavior is observed in the vortices
near the {\it upper} (oscillating) collar of the cylinders (0$30$ in Fig.~\ref{f14}a) is evidently a finite
length effect.
The phase shift in this region
is also a linear function of the axial position with a slope close to that
of the phase shift line for the traveling wave in the middle of the system.
***Mingming, what flow state is figure 14 based on?? For T=52 I would expect it
to look much like figure 12. I don't see anything further in your thesis.***
*** Dr. Andereck, it maybe $\epsilon=0.813, m=7, N=27$, look at the lab book
and look at the WVF section, where the data file for phase and amplitude are
named as pha2.dat, pha4.dat... and amp2.dat, amp3.dat ....***.
The effects of modulation frequency on the phase dynamics were also studied.
Fig.~\ref{f14}a and Fig.~\ref{f17} shows that for a fast modulation,
the amplitude drops off more rapidly near the
upper collar than for a slower modulation.
This is consistent with the diffusion model, where a$\propto
exp(-\sqrt{\frac{\omega}{2D_{||}}}z)$. The constant amplitude region where
the propagating mode
exists shortens as the modulation frequency decreases, and it finally
collapses into a straight line for very long periods of oscillation.
For example,
for modulation period $T>101$, ($\ast$ and $\Box$ lines of Fig.~\ref{f17}, and
the $\times$ data points of Fig.~\ref{f14}a),
the amplitude decreases linearly with z, with
a slope $0.0137\sim 1/\Gamma$. The slope here is
scaled by the maximum amplitude of the oscillation of the upper collar.
This is a typical solution of a propagating
wave equation when the wavelength is much longer than the system itself.
The corresponding phase shift between neighboring
$\bar{z}_{n}(\tau)$ ($\ast$ and $\Box$ line of Fig.~\ref{f17a}) is zero,
which means that the modulation is slow enough for each vortex
to respond essentially simultaneously. (The discontinuities in Fig.~\ref{f17a}
come from the fact
that the data were taken separately
from different sections of the flow pattern
in order to obtain higher resolution.
The pattern is sectioned into three parts, $0