# Oscillation and collective behaviour in convective flows

A. Gergely  Physics Department, Babeş-Bolyai University, Cluj-Napoca, Romania
Cs. Paizs  Faculty of Chemistry and Chemical Engineering, Babeş-Bolyai
University, Cluj-Napoca, Romania  R. Tötös  Faculty of Chemistry and Chemical
Engineering, Babeş-Bolyai University, Cluj-Napoca, Romania  Z. Néda
<EMAIL_ADDRESS>Physics Department, Babeş-Bolyai University, Cluj-
Napoca, Romania

###### Abstract

Oscillation and collective behavior in convection-driven fluid columns are
investigated and discussed in analogy with similar phenomenon observed for the
flickering flames of candle bundles. It is shown experimentally that an
ascending circular Helium gas column performs an oscillation which is similar
in several aspects to the oscillation of diffusion flames. Increasing the
nozzle diameter leads to a decrease in the oscillation frequency, while
increasing the flow rate results in an increase in this frequency. For helium
columns oscillating at nearby frequency and placed close to each other anti-
phase synchronization and beating phenomena is observed. A simple toy-model
based on elementary hydrodynamics describes the observed oscillations and
leads to oscillation frequencies in the right order of magnitude.

††preprint: AIP/123-QED

##  I Introduction

Oscillation and collective behavior of diffusive flames is an intriguing and
well studied problem Chamberlin and Rose (1948); Durox  _et al._ (1995, 1997);
Huang  _et al._ (1999); Kitahata  _et al._ (2009); Ghosh  _et al._ (2010);
Okamoto  _et al._ (2016); Chen  _et al._ (2019); Gergely  _et al._ (2020).
Experimental results suggests however Yuan  _et al._ (1994), that similar
oscillations are present in rising gas columns, and therefore the two
phenomenon could be more strongly related than it is nowadays believed. More
specifically one could ask, whether the existence of the flame or the chemical
reactions inside of it, captured in the currently used models is a necessary
ingredient to understand the oscillations in diffusive flames, or
hydrodynamics by itself is enough to understand this interesting phenomena.
Some carefully conducted experiments and computer simulations could reveal
more analogies between the two phenomenon and therefore could help in a better
understanding for both of them. The present work intends to contribute in such
directions.

A candle is a simple system consisting of a wick and the surrounding
combustible material (usually paraffin). The fuel of the candle does not burn
in solid form, for the combustion to take place we must first evaporate the
fuel. The combustion reaction takes place in the boundary layer of the fuel
vapor and air, based on which the candle flames are associated with diffusion
flames. It has long been known Chamberlin and Rose (1948) that under certain
conditions the volume of diffusion flames changes periodically over time, this
phenomenon is called the oscillation of diffusion flames.

At normal atmospheric oxygen concentration (21% $O_{2}$), the flame of a
single candle burns in a stable manner. In order to obtain oscillations the
candles must be arranged in a bundle. Different physical parameters affects
the oscillation frequency of the candle bundle. In our previous work Gergely
_et al._ (2020), we investigated experimentally how the oscillation frequency
changes as a function of the number of candles in the bundle and how the
oxygen concentration around the candle bundle affects the oscillation. We have
shown that for the compact and hollow arrangements of the candles inside the
bundle the oscillation frequency decreases as the number of candles are
increased in the bundle. We also proved that as the oxygen concentration
increases, the oscillation frequency decreased. We observed that a high oxygen
concentration can cause oscillation in cases where this would not occur at a
normal oxygen concentration. Interestingly, high oxygen concentration can also
stop the oscillations in cases where this would occur at a normal oxygen
concentration.

If the flame of two candle bundles are placed nearby each other, collective
behavior in form of synchronization appears as a result of the interaction
between their flickering Kitahata  _et al._ (2009). We thoroughly examined
this collective behaviour as a function of the distance between bundles by a
properly defined synchronisation order parameter Boda  _et al._ (2012);
Gergely  _et al._ (2020). It was found that for small flame distances, in-
phase synchronisation develops. At larger distances this turns sharply in
counter-phase synchronization and by further increasing the distance between
the bundles one observes a graduate decrease in the synchronization level.

In the seminal work of Kitahata et. al Kitahata  _et al._ (2009), the authors
conclude that the coupling mechanism is originated in the thermal radiation
emitted by the candles. The phenomena of flame oscillation and the
synchronization of nearby flames is modeled by two coupled evolution equations
in which the temperature and oxygen concentration inside the flame are the
relevant dynamical variables. Collective behavior is accounted by considering
similar evolution equations for the interacting flames and a coupling
mechanism between them through thermal radiation. The experimental results
presented in Gergely  _et al._ (2020) contradicted however the existence of
such a coupling mechanism and lead to an improved dynamical system model both
for the oscillation phenomena and for the observed synchronization. The model
proposed in Gergely  _et al._ (2020) is similar with the original model of
Kitahata et. al and it is still based on the chemical reactions that takes
place inside the flame. In this improved model the coupling mechanism is
realized via the oxygen flow induced by the periodic changes in the flame
size. Interestingly, this improved and oversimplified model described
excellently all the carefully gathered experimental results.

Intriguing similarities with some purely hydrodynamical phenomena could
seriously question however whether the existence of a flame or a chemical
reaction inside of it is needed in order to understand the oscillation and
synchronization phenomena. For example, in Yuan  _et al._ (1994) , the authors
observed that a Helium column flowing laterally into the air performs similar
oscillations to the one in diffusion flames. This raises the possibility that
the role of flame in our candle experiments is only to create the convective
flow in which hydrodynamic instabilities occur and this flow is causing all
the interesting physics connected to the oscillation of diffusive flames. In
the followings we will approach this question both experimentally and
theoretically. We present experimental results pointing to quantitative
analogies. On the theoretical side, we consider a simple analytical approach
based on elementary hydrodynamics which is intended to estimate the
oscillation frequencies as a function of the relevant parameters of the flow.
Finally, we discuss all the qualitative and quantitative analogies revealed
between the dynamical behavior of diffusion flames and the ones observed in
convective flows.

##  II Experimental studies

###  II.1 Experimental setup

With the experiments detailed below, we are looking to answer whether
hydrodynamic instabilities in convective flows are able to explain by itself
the oscillations and collective behavior observed for diffusive flames.

The flow must be produced without any chemical reaction, therefore a
controlled flow of Helium gas in air was considered to be a good solution.
Since the flow of Helium in air does not emit visible light, the experimental
methods used for candle flames Gergely  _et al._ (2020) had to be modified.
The Schlieren technique Settles (2001); Leptuch and Agrawal (2006) proved to
be a simple and efficient method to visualize the flow in rising Helium
columns.

Schlieren imaging is a method for visualising and qualitatively characterising
the refractive index distribution in a transparent medium. There are several
types of Schlieren equipments, we have built one of the cheapest and simplest
off-axis mirrored version presented in Settles (2001) page 47. A schematic
diagram of our equipment is shown in Figure 1.

Figure 1: Schematic drawing of the used Schlieren setup. The following
elements are indicated: (1) -fast camera, (2) -razor blade, (3) -parabolic
mirror, (4) -circular gap, (5) -light-emitting diode.

The operation principle is relatively simple. With the help of a light-
emitting diode, a circular gap is illuminated, creating a nearly point-like
light source. This light source is imaged using a parabolic mirror, and a
blade is placed at the location of the image so that it obscures half of it.
The uncovered light enters the camera lens used for recording. If there is
anisotropy in the refractive index in the space between the light source and
the mirror, a refraction will occur, resulting a change in the amount of light
obscured by the blade. The change in brightness results in dark and light
areas in the camera image, making the refractive index gradients visible. The
refractive index of Helium used in our experiments is lower than the
refractive index of air at room temperature, so their flow can be observed
using this technique. The largest change in brightness are in those places
where the greatest amount of light refraction due to optical anisotropy
occurs. The anisotropy results in an increase or decrease in brightness
depending on the direction of light refraction and the location of the blade
in the equipment. For example Figure 3 shows frames made by the Schlieren
technique on a rising Helium column. In these cases an increase in brightness
relative to the background occurs in those parts of the images where the
thickness of the optically less dense Helium decreases from left to right. In
the case of the projection seen by the camera, the largest change in the layer
thickness of the Helium is at the edges of the column, so the largest increase
in brightness will occur there. Due to the high-frequency oscillations
observed in the gas columns (above 10 Hz), for recording we used the Sony
Cyber-Shot DSC-RX100 VI photo camera which allowed HD format recording at 250
fps.

In our experiments we monitored the movement of these edges at a given height.
For the proper image processing algorithm the Otsu method for the gray-scale
images was selected Otsu (1979); FreddyCree (2020). This method consists of
choosing a critical intensity level below which the value of the recorded
pixel is assigned zero, and otherwise it is assigned the logical one. If the
Otsu threshold is selected properly, the pixels near the boundary layer will
be 1 and the other pixels will become zero. After applying the cutoff, going
from left to right we identify the first pixel whose value is one,
approximating the location of the boundary layer at a given height. Movies
with original recordings and the ones processed with the Otsu method can be
consulted on our YouTube channel Gergely (2020).

The nozzles used to initiate the Helium column were realized by 3D printing.
They consist of 3 parts: an inlet part, a hexagonal structure that ensures the
laminar nature of the initial flow, and an outlet part through where the gas
leaves. The modular design was needed so that the hexagonal structure and the
inlet parts did not have to be manufactured separately for the outlets with
different diameter, saving by this time and plastic. Several different
plastics (PC, PLA, CPE) and nozzle diameters for the 3D printer were tested
(0.25, 0.4, 0.6, 0.8 mm) and the best results were obtained with a combination
of 0.4 mm nozzle diameter and PLA plastic. The 3D printed elements are shown
in Figure 2.

Figure 2: Elements of the nozzles realised with a 3D printer. Element (a) is
the outlet of the nozzle, element (b) is the hexagonal structure that ensures
the laminar nature of the flow, and element (c) is the inlet through which
Helium is introduced into the nozzle.

Helium was introduced into the nozzle through a $1/4$ inch tube. One defining
parameter of the Helium’s flow is the debit (yield) of the gas flowing through
the nozzle. This was controlled by a needle valve and measured with a precise
rotameter.

Time-lapse images obtained with the Schlieren technique are illustrated in
Figure 3, and some recorded movies can be consulted in Gergely (2021a).

Figure 3: Time-lapse images taken using the Schlieren technique for a column
of Helium gas flowing out in vertical direction from the nozzle.

###  II.2 Results

####  II.2.1 Oscillations

For studying the oscillation of the ascending Helium column, the jet was
produced with circular cross-section nozzles of various diameters. Our
experiments were performed at room temperature and normal atmospheric
pressure, the purity of the used Helium was 99%. Each experiment was repeated
5 times, and the standard deviation for the obtained results was visualized
with error bars on the plots. The video recordings obtained from the
experiments can be viewed on the Gergely (2021b, a) youtube playlists.

At a constant Helium yield of $\Phi=$46 $\pm$ 2.3 $l/min$, it was examined the
oscillation frequency variation with the nozzle diameter. As shown in Figure
4, it was observed that the increasing nozzle diameter determined the decrease
of the oscillation frequency, which can be well approximated by a power law
function with exponent $-1.64$. These results are somehow in accordance with
those obtained with candle bundles, for which a power-law like trend was also
observed for the oscillation frequency as a function of the number of candles
in the bundle Gergely  _et al._ (2020).

Figure 4: Oscillation frequency of the Helium column as a function of the
nozzle diameter for a yield of $\Phi=$46 $\pm$ 2.3 $l/min$. The fitted power
function is shown with a continuous line. Please note the double logarithmic
scale. (In the case of nozzle diameter, the high margin of error is due to the
eccentricity of the nozzle and the error was considered as the difference
between the largest and the smallest measured diameter.)

The effect of the flow rate was examined using a 2 cm diameter circular
nozzle. Our results are shown in Figure 5, where one will observe that as the
yield increases, the oscillation frequency increases. This increase is well
approximated by a linear trend.

Figure 5: Oscillation frequency of the helium column as a function of the
yield (flow debit) obtained for nozzle with a diameter of 2 cm. With
increasing flow yield, the frequency of the oscillation increases in an almost
linear manner. (For the yield, the error was considered to be equal to the
flow debit corresponding to one division on the scale of the rotameter.)

####  II.2.2 Collective behaviour

For studying the collective behavior of the flow oscillations, experiments
with two Helium columns were performed.

If two diffusion flames are placed nearby each other collective behaviour in
form of synchronization of the flickering can appear. Kitahata  _et al._
(2009); Forrester (2015); Okamoto  _et al._ (2016); Manoj  _et al._ (2018);
Yang  _et al._ (2019); Dange  _et al._ (2019); Fujisawa  _et al._ (2020);
Gergely  _et al._ (2020). Similarly with the studies performed on candle
flames, first we will examine the collective oscillation frequency and the
synchronization order parameter $z$ for two Helium columns with the same flow
parameters (yield and nozzle diameter) as a function of the distance between
the nozzles. The experimental apparatus is pretty much the same as the one
used for one Helium column, the only difference is that now two nozzles are
used, with a fine control on the distance between their close edges. Our
results are summarized in Figure 6. On Figure 6a we plot the measured
oscillation frequency as a function of the distance between the nozzles and in
Figure 6b we show the value of the synchronization order-parameter as a
function of the nozzles distance. Results on both graphs are for nozzles of 2
cm diameter and Helium flow rate of 46 $\pm$ 2.3 l / min. The synchronization
order-parameter $z$ is a number in the interval $[-1,1]$, an it is defined and
determined from the Otsu processed images in the same manner as it was done in
Gergely  _et al._ (2020). The value $z=1$ means complete in-phase synchrony
while $z=-1$ indicates a perfect counter-phase synchronization.

From these results and the ones plotted in Figures 4 and 5 one can conclude
that for short distances the frequency is significantly higher than the one
observed for non-interacting Helium columns with the same parameters (flow
rate and nozzle diameter). It can also be observed that for the entire
distance range that we have examined anti-phase oscillation dominates. This is
different from the case of the collective behavior observed for candle flames,
where at short separation distances in-phase synchronization is also observed.
More on such similarities and differences will be discussed in the concluding
section.

Figure 6: Figure (a) shows the common oscillation frequency of the Helium
columns flowing out of two identical 2 cm diameter nozzles with a flow rate of
$\Phi$=46 $\pm$ 2.3 l/min as a function of the distance between the nozzles.
Figure (b) plots the synchronization order parameter $z$ (defined in Gergely
_et al._ (2020)) obtained from the oscillating time series of the Helium
flows.

As previously observed, another interesting collective behavior for
interacting diffusion flames with slightly different frequencies also occurs
Chen  _et al._ (2019). In such case a phenomenon similar to the "beating"
known in acoustics is observable. For flickering candle flames, this means
that the amplitude of oscillation for one of the flames performs a long period
modulation.

In order to test whether such beating is observable for Helium columns as well
we use two approaches to produce slightly different oscillating frequencies.
In the first approach we kept the Helium yield constant and varied the nozzle
diameter, while in the second approach we modified the Helium yield for a
constant nozzle diameter.

Figure 7: Beating-like phenomenon observed for two interacting Helium gas
columns. For Figures a, b, c, the distance between the nozzles is 1, 1.5, 2
cm, respectively, and the yield of Helium flow is $\Phi=34.5\pm$ 2.3 l / min
and $\Phi=46\pm$ 2.3 l / min, respectively. The nozzle diameter is 2 cm and
the plotted time series are for the oscillation of the Helium column with the
lower debit. For Figures d, e, f, the nozzle distances are 1, 1.5, 2 cm,
respectively, the Helium flow debit for both nozzles is $\Phi=46\pm 2.3$ l /
min and the nozzle diameters are 2.25 cm and 2.5 cm. The plotted time-series
are for the Helium column initiated from the 2.5 cm diameter nozzle.

As the graphs in Figure 7 illustrates we were able to obtain the beating-like
phenomenon with both methods. Figure 7a,b,c shows the beating realized with
different yields. In these experiments the nozzle diameters are fixed for 2 cm
and the outflow yields are $\Phi=$34.5 $\pm$ 2.3 l/min and $\Phi=$46 $\pm$ 2.3
l/min. In these figures the time series of the Helium column with the lower
yield is plotted for the distance between nozzles fixed at 1, 1.5 and 2 cm,
respectively.

For the beating phenomena observed with nozzles of different diameters and the
same flow yields the time-series are shown in Figure 7d,e,f. In these
experiments the flow yield is $\Phi=46\pm$ 2.3 l / min, and the nozzle
diameters are 2.25 and 2.5 cm. The plotted time series for the oscillation of
the Helium column are for the flow from the larger diameter nozzle. In Figures
7d, e, f the distance between the nozzles are again 1, 1.5 and 2 cm,
respectively.

##  III Analytical approach for the oscillation frequency

We present here a toy-model for understanding the oscillation frequency of the
rising Helium gas column as a function of the flow-rate and nozzle diameter.
This oversimplified model is unfortunately not appropriate to tackle the
problems related to collective behavior.

Our basic assumption is that the Helium column accelerated by gravity becomes
unstable at a given Reynolds number. Accepting this, we assume that for a
given volume element the time from the outflow up to the formation of
instability will approximate the oscillation period of the Helium gas column.
Definitely, this is an oversimplified approach, since the dynamical feedback
of this instability on the outflow from the nozzle is neglected and therefore
the periodicity for the formation of these instabilities are also lost.
Nevertheless, we hope that this approach will estimate a correct time-scale
for the formation of the first instability and assume that this time-scale
drives the periodicity of the observed oscillations.

The Reynolds number for a cylindrical fluid column can be given by the
following equation

$\begin{split}R_{e}(t)=\frac{2\cdot v(t)\cdot r(t)}{\nu},\end{split}$ (1)

where $v(t)$ denotes the velocity of the considered fluid element, $r(t)$
denotes its radius and $\nu$ denotes the dynamic viscosity of the fluid.

According to our assumption the oscillation period $\tau$ will be estimated as
the time necessary for the considered cylindrical fluid element’s Reynolds
number to reach a critical value $R_{e}^{c}$:

$\begin{split}R_{e}(\tau)=R_{e}^{c}\end{split}$ (2)

In the following we examine the above model in two cases where simple
analytical results can be drawn. In the first limit we neglect viscosity,
while in the second approximation friction effects are taken into
consideration to describe the rising dynamics of the Helium gas. In the later
approach we discuss again two cases. First we assume no slip condition for the
air-Helium boundary layer, fixing the velocity of Helium on this interface at
0. Then we discuss a much more realistic case, where we allow the movement of
the air-Helium boundary layer implementing a semi-slip boundary condition.

###  III.1 Dynamics with no friction

The buoyancy force acting on a Helium element with volume $V$ can be written
as:

$\begin{split}F_{V}=V\cdot g\cdot(\rho_{Air}-\rho_{He})\end{split}$ (3)

Here $g$ denotes the gravitational acceleration, $\rho_{He}$ and $\rho_{Air}$
denote the density of Helium and air, respectively. If one neglects the
friction in the air-Helium boundary layer, based on the Newtonian equation of
motion, the velocity of a Helium gas element with an initial velocity $v_{0}$
will be:

$\begin{split}v(t)=v_{0}+\frac{g\cdot(\rho_{Air}-\rho_{He})}{\rho_{He}}\cdot
t\end{split}$ (4)

To calculate the Reynolds number, we also need the radius $r(t)$ of the
cylindrical Helium column element at time-moment $t$. This is determined from
the continuity equation as follows:

$\begin{split}r(t)=r_{0}\cdot\sqrt{\frac{v_{0}}{v(t)}}\end{split}$ (5)

We denoted by $r_{0}$ and $v_{0}$ the radius and the velocity of the Helium
gas column at the moment of outflow from the nozzle, respectively. From
equations (5), (4) and (1) one gets the Reynolds number for the flow at time
$t$:

$\begin{split}R_{e}(t)=\frac{2\cdot
r_{0}\cdot\sqrt{\left(v_{0}+\frac{g\cdot(\rho_{Air}-\rho_{He})}{\rho_{He}}\cdot
t\right)\cdot v_{0}}}{\nu}\end{split}$ (6)

Using this result and our approximation (2) for the estimation of the
$f=1/\tau$ oscillation frequency we get:

$\begin{split}f=\frac{g\cdot(\rho_{Air}-\rho_{He})\cdot v_{0}\cdot
r_{0}^{2}}{\left({\left(\frac{R_{e}^{c}\cdot\nu}{2}\right)}^{2}-{(v_{0}\cdot
r_{0})}^{2}\right)\cdot\rho_{He}}\end{split}$ (7)

In the above equation, the values of all parameters are known from the
experiments except for the critical Reynolds number $R_{e}^{c}$. Using a
realistic estimate for the critical Reynolds number: one between a laminar and
turbulent flow, the model gives a correct order of magnitude for the
oscillation frequency and also correctly reproduces the experimentally
obtained trends for oscillation frequency as a function of nozzle diameter and
outflow yield. Reconsidering now equation (7) to include that flow yield
$\Phi$ instead of flow velocity $v$ we get:

$\begin{split}f=\frac{g\cdot\left(\rho_{Air}-\rho_{He}\right)\cdot\frac{\Phi}{\pi}}{\left({\left(\frac{R_{e}^{c}\cdot\nu}{2}\right)}^{2}-{\left(\frac{\Phi}{r_{0}\cdot\pi}\right)}^{2}\right)\cdot\rho_{He}}\end{split}$
(8)

Equations (7) and (8) allow now to plot the trends for the estimated
oscillation frequency as a function of nozzle radius ($r_{0}$) and flow debit
($\Phi$). Considering some realistic critical Reynolds number $R_{e}^{c}$
values in Figures 8a and 8b we compare the theoretical trends with the results
of the experiments. The trends offered by our simple model are in good
agreement with the observations and also the predicted oscillation frequencies
are of the right orders of magnitude.

Figure 8:  The Helium column’s oscillation frequency predicted by
approximation (8) and the results observed in the experiments. Figure (a)
presents the results for different nozzle diameters and Figure (b) is the
result for different outflow rate. We illustrate the theoretically predicted
trend for several $R_{e}^{c}$ values as indicated in the legend. For the
theoretical calculations we used the parameters for our experiments with
Helium: $g=10\ m/s^{2},\ \rho_{Air}=1.2\ kg/m^{3},\ \rho_{He}=0.17\ kg/m^{3},\
\nu=11.52\cdot 10^{-5}\ m^{2}/s$. We have chosen $\Phi=46\pm 2.3\ l/min$ for
Figure (a) and $r_{0}=0.01\ m$ for Figure (b). The model correctly predicts
the experimentally obtained trends and the magnitude of the obtained
frequencies.

###  III.2 Dynamics with friction

We reconsider now the previous approach by introducing viscosity. First we
will use no-slip condition and then semi-slip boundary condition for the
Helium-air interface.

####  III.2.1 No-slip boundary condition

We consider now the effect of friction with no-slip boundary condition at the
interface of Helium and air. In deriving the average velocity of a fluid
element as a function of time, we assume that the velocity of the Helium
column has a cylindrical symmetry and the velocity $v_{z}(r,t)$ as a function
of radius has a parabolic profile.

The equation of motion is written for the average velocity of a cylindrical
volume element of height $\langle v_{z}(t)\rangle\cdot da$. The mass of the
volume element can be given as

$\begin{split}dm=\rho_{He}\cdot\langle
v_{z}\rangle\cdot\pi\cdot{r_{z}}^{2}\cdot da,\end{split}$ (9)

where $r_{z}$ denotes the radius of the considered element at a height $z$.
The volume element is affected by the friction force and the buoyancy force.
The buoyancy force is approximated as:

$\begin{split}dF_{b}=(\rho_{Air}-\rho_{He})\cdot\langle
v_{z}\rangle\cdot\pi\cdot{r_{z}}^{2}\cdot g\cdot da\end{split}$ (10)

The friction force is derived from the shear stress $\kappa$, which for a
$\mu$ kinematic viscosity can be written as:

$\begin{split}\kappa=\frac{dv_{z}}{dr}\cdot\mu\end{split}$ (11)

The friction force between the air and the Helium is then:

$dF_{f}=\mu\cdot 2\cdot\pi\cdot r_{z}\cdot\langle v_{z}\rangle\cdot
da\cdot\left.\frac{dv_{z}}{dr}\right|_{r=r_{z}}$ (12)

For the no-slip boundary condition the velocity $v_{z}$ as a function of
radius is described by the parabolic profile:

$\begin{split}v_{z}(t,r)=2\cdot\langle
v_{z}(t)\rangle\cdot\left(1-\frac{r^{2}}{{r_{z}}^{2}}\right)\end{split}$ (13)

Using the above equation the friction force from (12) can be estimated as:

$\begin{split}dF_{f}=-8\cdot\pi\cdot\mu\cdot{\langle v_{z}\rangle}^{2}\cdot
da\end{split}$ (14)

Assuming that the parabolic profile is maintained throughout the acceleration,
the equation of motion for the chosen cylindrical element can be written using
the average velocity :

$\begin{split}dm\frac{d\langle v_{z}\rangle}{dt}=dF_{f}+dF_{b}\end{split}$
(15)

Using now the formula for $dm$, buoyancy and friction forces given by
equations (9), (10) and (14), respectively, we get:

$\begin{split}\rho_{He}\langle v_{z}\rangle\pi\cdot{r_{z}}^{2}\frac{d\langle
v_{z}\rangle}{dt}=-8\pi\cdot\mu{\langle
v_{z}\rangle}^{2}+(\rho_{Air}-\rho_{He})\langle
v_{z}\rangle\pi{r_{z}}^{2}g\end{split}$ (16)

The $r_{z}$ radius can be estimated from the continuity equation:

$\begin{split}\pi\cdot{r_{z}}^{2}\cdot\langle v_{z}\rangle=\Phi\end{split}$
(17)

Rearranging now equation (16), we are lead to

$\begin{split}\frac{d\langle
v_{z}\rangle}{dt}=\frac{-8\cdot\pi\cdot\mu\cdot{\langle
v_{z}\rangle}^{2}}{\rho_{He}\cdot\Phi}+\frac{(\rho_{Air}-\rho_{He})\cdot
g}{\rho_{He}},\end{split}$ (18)

where we introduced the notations:

$\begin{split}A=\frac{8\cdot\pi\cdot\mu}{\rho_{He}\cdot\Phi}\\\
B=\frac{(\rho_{Air}-\rho_{He})\cdot g}{\rho_{He}}\end{split}$ (19)

Separating the variables $t$ and $\langle v_{z}\rangle$ and integrating
equation (18)

$\begin{split}\int_{v_{z}(0)}^{v_{z}(t)}\frac{d\langle
v_{z}\rangle}{B-A\cdot{\langle v_{z}\rangle}^{2}}=\int_{0}^{t}dt,\end{split}$
(20)

finally leads to:

$\begin{split}\frac{\tanh^{-1}\left(\sqrt{\frac{A}{B}}\cdot\langle
v_{z}(t)\rangle\right)-\tanh^{-1}\left(\sqrt{\frac{A}{B}}\cdot\langle
v_{z}(0)\rangle\right)}{\sqrt{A\cdot B}}=t\end{split}$ (21)

In order to estimate the oscillation frequency the average velocity $\langle
v_{z}(t)\rangle$ is expressed from the Reynolds number and $\langle
v_{z}(0)\rangle$ is expressed from the continuity equation as a function of
flow yield $\Phi$ and nozzle radius $r_{0}$. Using equations (1) and (17), the
velocity can be given as:

$\begin{split}\langle
v_{z}\rangle=\frac{\pi\cdot{R_{e}}^{2}\cdot\nu^{2}}{4\cdot\Phi}\end{split}$
(22)

From equations (17), (20) and (21) the oscillation frequency as a function of
the critical Reynolds number is derived:

$\begin{split}f=\frac{\sqrt{A\cdot
B}}{\tanh^{-1}\left(\sqrt{\frac{A}{B}}\cdot\frac{\pi\cdot{R_{e}^{c}}^{2}\cdot\nu^{2}}{4\cdot\Phi}\right)-\tanh^{-1}\left(\sqrt{\frac{A}{B}}\cdot\frac{\Phi}{\pi\cdot{r_{0}}^{2}}\right)}\end{split}$
(23)

Let us examine now the predictions of the model with the implemented no-slip
boundary conditions and friction. First we conclude that it is not possible to
find a critical Reynolds number for which the above model gives a positive
oscillation frequency for the entire experimentally studied flow yield range.
The reason for this is the unrealistic no-slip boundary condition. In reality
one would expect that the Helium column induces a strong flow in the
surrounding air. This airflow depends on the Helium yield, therefore a correct
frequency formula would require a correction term that depends on the flow
yield. Since we do not have experimental data on the dependence of the airflow
rate on the Helium flow yield, we cannot estimate this term. Assuming however
that this correction term depends only on the yield, equation (23) could give
a good approximation for a constant yield. In the experimental study where the
effect of the nozzle diameter was investigated, the Helium flow rate was
fixed, therefore accepting the above argument, our toy-model could approximate
the experimental results well in this case.

In Figure 9, the experimentally measured oscillation frequency as a function
of the nozzle diameter is compared with the theoretical prediction for no-slip
boundary condition using different $R_{e}^{c}$ values, as indicated in the
legend. For the model we used the following parameters : $g=10\ m/s^{2},\
\rho_{Air}=1.2\ kg/m^{3},\ \rho_{He}=0.17\ kg/m^{3},\ \nu=11.52\cdot 10^{-5}\
m^{2}/s,\ \mu=1.96\cdot 10^{-5}\ Pa\cdot s,\ \Phi=46\pm 2.3\ l/min,\
R_{e}^{c}=519$.

Figure 9: Oscillation frequencies predicted by approximation (23) (continuous
line) with different $R_{e}^{c}$ choices in comparison with the experimentally
observed oscillation frequency as a function of the nozzle diameter. The
following parameters were used in the calculations: $g=10\ m/s^{2},\
\rho_{Air}=1.2\ kg/m^{3},\ \rho_{He}=0.17\ kg/m^{3},\ \nu=11.52\cdot 10^{-5}\
m^{2}/s,\ \mu=1.96\cdot 10^{-5}\ Pa\cdot s,\ \Phi=46\pm 2.3\ l/min$.

####  III.2.2 Semi-slip boundary condition

Let us briefly discuss how the application of a non-zero velocity for the
Helium-air boundary layer would improve the predictions of the model. Since
the derivation steps are almost identical to those presented in the previous
subsection, only the differences are outlined in the followings. We assume
again that the velocity profile of the Helium column can be described by a
parabolic function as a function of the radius $r$, the only difference from
the no-slip boundary condition is that now the velocity at $r_{z}$ should not
be set to 0. Based on the above consideration, the radial profile of the speed
can be approximated in the form

$v_{z}(t,r)=\frac{2\cdot\langle
v_{z}(t)\rangle}{\left(2-\beta\right)}\cdot\left(1-\frac{r^{2}}{{r_{z}}^{2}}\cdot\beta\right),$
(24)

where $\beta$ can take a value between $0$ and $1$. This constant governs the
difference of the velocity at $r=r_{z}$ relative to the value $0$, considered
for no-slip boundary conditions. For $\beta=0$ we get the frictionless case
while for $\beta=1$ we get the no-slip boundary condition. In between this
extreme values is reality.

The shape of the velocity profile is contained only in the equation of the
friction force. Using this form in equation (12), the friction force becomes:

$\begin{split}dF_{f}=-8\cdot\pi\cdot\mu\cdot{\langle
v_{z}\rangle}^{2}\cdot\frac{\beta}{2-\beta}\cdot da\end{split}$ (25)

From this point on we continue with the same straightforward steps that are
used for the no-slip boundary condition leading to a difference in the value
of $A$:

$\begin{split}A=\frac{8\cdot\pi\cdot\mu\cdot\beta}{\rho_{He}\cdot\Phi\cdot(2-\beta)}\end{split}$
(26)

Using the $A$ given above we are lead to a model with two free parameters,
$\beta$ and $R_{e}^{c}$. Considering now the following experimental
parameters: $g=10\ m/s^{2},\ \rho_{Air}=1.2\ kg/m^{3},\ \rho_{He}=0.17\
kg/m^{3},\ \nu=11.52\cdot 10^{-5}\ m^{2}/s,\ \mu=1.96\cdot 10^{-5}\ Pa\cdot s$
and $\beta=0.18$ we find an acceptable trend for the oscillation frequencies
as a function of the nozzle diameter and flow yield. Results for some
reasonable $R_{e}^{c}$ values are summarized in Figure 10.

Figure 10: Experimentally observed oscillation frequencies for the Helium
column oscillations in comparison with the frequency values predicted by the
equation (23) particularised for semi-slip boundary conditions using
$\beta=0.18$ and several reasonable $R_{e}^{c}$ values as indicated in the
legend. Figure (a) is for the oscillation frequency as a function of the flow
yield (nozzle diameter is $r_{0}=2cm$) and Figure (b) shows the frequency as a
function of the nozzle diameter for a yield of $\Phi=46$ l/min. In agreement
with our experimental setup the following parameters were used: $g=10\
m/s^{2},\ \rho_{Air}=1.2\ kg/m^{3},\ \rho_{He}=0.17\ kg/m^{3},\ \nu=11.52\cdot
10^{-5}\ m^{2}/s,\ \mu=1.96\cdot 10^{-5}\ Pa\cdot s$

##  IV Discussion and Conclusions

Oscillation of diffusion flames is a complex phenomenon in which both,
physical and chemical processes concomitantly takes place. Consequently, it is
difficult to conclude which are the relevant processes responsible for this
behavior. In our previous work Gergely  _et al._ (2020), we explained the
oscillation of candle flames and their synchronization using a dynamical model
that incorporates the effects of the chemical reaction taking place during the
combustion. Although the proposed model described the collective behavior and
the experimentally obtained trends for the oscillation frequency well, there
are experimental results suggesting that similar oscillations can be explained
solely by hydrodynamic instabilities Yuan  _et al._ (1994). In the present
work we used both experimental and theoretical approaches to investigate
whether the hydrodynamic instabilities that occur in rising gas columns can
explain the relevant features of the oscillations observed for the diffusion
flames.

First, we have shown experimentally that for certain experimental parameters
rising gas columns can induce oscillations that are similar to the one
observed in diffusion flames. The quantitative analysis of the oscillation in
convective flow was performed for a rising Helium gas column. For such a
Helium flow, we have shown that for a constant flow-rate the frequency of the
oscillation decreases with increasing nozzle diameter, similarly with the
flickering frequency of candle bundles as a function of the number of candles
in the pack. The decreasing trend observed for the Helium flow can be well
approximated by a power function with $-1.64$ power exponent.

For a constant nozzle diameter, the oscillation frequency of Helium column
increases roughly linearly with the floe debit of Helium. For two interacting
Helium columns a collective oscillation similar with the one observed for the
diffusion flames were observed.

The main difference however relative to diffusion flames is that for the
Helium columns only counter-phase synchronization is observable. It is thus
conceivable that the mechanism leading to synchronization is fundamentally
different but it is also possible that the interaction between the Helium
columns was not strong enough to find the in-phase synchronization at short
separation distances. Another possible explanation could be that for diffusion
flames whenever one observes the in-phase synchrony the involved hydrodynamic
flows are not really separated but visually we detect the hot part of the
flames as separate regions. When collective behavior was found, we examined
the dependence of the oscillation frequency as a function of the separation
distance. Here we observed a similar decreasing trend as the one observed for
the counter-phase synchronization in diffusion flames.

The observed oscillation with slightly different frequencies for two
interacting Helium columns is similar to the phenomenon of beating, known in
acoustics. This was observed in the case of diffusion flames as well and it
was reported by Chen et. al Chen  _et al._ (2019). In our experiments beating
was observed both by using different nozzle diameters at the same flow yield
and with different yields and the same nozzle diameter of the two Helium
columns.

In order to approach the observed oscillations theoretically we proposed a
simplified but analytically treatable model. Our main assumption is built on
the observation that the Reynolds number of the Helium column that is
accelerating under the effect buoyancy increases with time until reaching a
critical Reynolds number where the flow becomes unstable. The oscillation
frequency was approximated as the time needed to reach this situation. The
theoretical model was discussed in two cases: a frictionless approach and
another approach with taking into account friction as well. The simple
frictionless case offered already an unexpectedly good trend for the
oscillation frequency as a function of the nozzle diameter and the outflow
yield. The model discussed by incorporating friction with no-slip boundary
condition at the Helium-air interface gives a better agreement for the
dependence of the experimentally measured oscillation frequency as a function
of nozzle size but it fails in describing the experimental results observed
for the oscillation frequency as a function of the flow rate. Using instead of
the no-slip boundary condition semi-slip boundary conditions the model results
are improved at the cost of an extra fitting parameter. We are confident that
by taking into account also the extra resistance of the flow due to the
developed instability, the predictions of such an approach can be
significantly improved. Unfortunately our theoretical approach was not
suitable to account for the collective behavior of the oscillating flows.

In conclusion, we can state that our study proves our hypothesis according to
which the instabilities in a Helium column ascending from a circular nozzle
into the air behaves in a similar manner to the oscillation of diffusion
flames. Seemingly the hydrodynamic processes by their own are able to explain
the oscillations observed for the diffusion flames. The extremely simple
analytical approach considered by us for this complex phenomenon, leads to
qualitatively good trends and right orders of magnitude for the oscillation
frequency. This promising agreement suggests once again the power of simple,
analytically solvable models in approaching universalities in complex systems.

Acknowledgment Work supported by the UEFISCDI PN-III-P4-ID-PCE-2020-0647
research grant. The work of A. Gergely is also supported by the Collegium
Talentum Programme of Hungary

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