Soap bubbles in a spin

In the February issue we reported how soap bubbles can be used to model the behaviour of twisting winds – Hamid Kellay picks up the story to give us a deeper insight into just how he goes about this amazing technique…

Put some soapy water in a dish, take a straw, and try blowing some air into this water. The result is intuitive: soap bubbles or half soap bubbles will form and float on the surface of the water. If the water is hot enough (50°C say) and you peer at the surface of the bubbles, conveniently lit by the ceiling lights, you will notice that there is movement on the surface of the bubble. The experiments we do in my lab are just more controlled versions of this simple observation that one can make in the kitchen sink.

What do we control and how do we make our observations? Instead of a dish, we use a large and heavy cylindrical copper block with a groove drilled all around. This copper block can be heated to the desired temperature with a water circulation thermostat that pumps heated water from a reservoir, where the temperature is set, through the copper block where a hollow ring was drilled for the water circulation (Figure 1). Soap water is then poured in the groove and left to heat up to the right temperature. A pipette attached to a flexible tube is used to blow the bubble shown.

[caption id="attachment_37712" align="alignright" width="200"] Figure 1: The bubble set up. Left: a brass block with a groove (1) for soap solution and a circular hole (2) for water circulation from a thermostat. Right: a brass disk (1) with a circular groove (3) can be rotated using a continuous motor (6) connected to it by a shaft (5). This disk is heated by the proximity of a hollow annulus (2) connected to a water circulation bath. The bubble is blown using the soap solution in the groove (3). The inner side of the brass disk is covered by a Teflon coating (2mm thick) to minimise the heating of the air inside the bubble. The temperature at the equator of the bubble is set by the temperature of the water bath.[/caption]

As mentioned, the heated water promotes movement on the surface of this bubble. This agitation can be more or less intense depending on the temperature of the copper block or the soapy solution out of which the bubble was made.

To view what is happening on the surface, all is needed is a colour camera and a relatively intense white light source with a diffuser screen to shed light on the bubble surface and bring out the interference colours with which the agitation can be viewed. Figure 2 shows the thermal plumes coming out of the equator of the bubble and rising up to the pole. These plumes are the essential bricks of how a fluid transports heat from hot to cold regions. This thermal convection is widespread in a variety of industrial and natural settings.

Now, on our bubble surface, the plumes are not the only interesting characters. Once in a while, large single vortices emerge out of this agitation. These vortices usually form when a large plume, which is the result of the merging of a few smaller ones, rises to near the top of the bubble and forms a swirl as shown in Figure 2. While this is one possible way to form such vortices, there are probably other possible ways leading to the formation of such vortices. Once formed, the large vortices can be followed using video imaging.  Note that the vortex has a well-defined centre or 'eye' and develops as a spiral looking structure of dimensions between 1 and 2 cm, which is considerable considering the bubble diameter is about 10cm and that its thickness is only a few micrometres. These single vortices can also be obtained in numerical simulations of thermal convection on the surface of a sphere so they are not particular to soap bubbles.

[caption id="attachment_37713" align="alignleft" width="200"] Figure 2: Images showing the detachment of thermal plumes from near the equator and rising towards the pole as well as an image of the full bubble with a vortex and a zoom on the vortex. The colours are interference colours of white light being reflected by the thin water layer constituting the bubble.[/caption]

Single vortices such as the ones seen here are not very common in fluid flows. In general, vortices come in pairs as in the wake behind an obstacle where a von Karman Street (swirling vortices caused by the unsteady separation of flow of a fluid around a blunt body) is produced with pairs of vortices detaching periodically behind the obstacle. Single vortices can be observed on the surface of some planets and much debate about their life time (think of Jupiter’s red spot) or their structure and their location is still ongoing. The half bubble system therefore allows investigating the properties of these creatures in detail and in a laboratory setting where curvature is naturally present (the bubble is spherical) and rotation can be added. For this purpose the set up was modified conveniently so rotation rates up to 2Hz can be imposed. The effect of rotation turns out to be highly non-trivial as it suppresses long lifetime vortices and confines them near the pole.

An important question which can be addressed in detail is how fast these vortices rotate and does this rotation speed have any particular features. Measurements of the rotation velocity of the vortex can be carried out over long periods of time spanning several turnover times of the vortex. A typical turnover time of these vortices is a fraction of a second (0.1s) which is roughly the time needed for a particle to make a full turn around the centre of the vortex. An example of the long-time dynamics of this rotation velocity is displayed in Figure 3. This long time variation, from several vortices, shows periods of constant low velocity followed by an intensification period with higher velocities and vorticities. The vortex followed here ends up reducing its velocity in the end and practically disappears in the background flow.

Another curious aspect we have observed is that during the intensification period, the trajectory of the centre of the vortex shows a trochoidal motion as seen in Figure 3 (insets): the vortex centre wobbles a few times around its mean position. The period of oscillation of the centre is roughly one turnover time. This is seen for different vortices with the period of trochoidal motion being roughly the same and close to 0.1s. Such trochoidal motion of the centre of vortices has been observed and documented for some tropical cyclones and has given rise to much debate about the origin of this motion and its link to the intensity of these vortices. The link between such trochoidal motion and the intensity of the vortex in our case is not clear since this motion was observed both for the increasing intensity phase as well as during the decreasing intensity one. Different reasons have been proposed to explain this peculiar feature observed during the motion of some tropical cyclones including instability of the core of the vortex or the existence of a double vortex structure giving rise to a periodic displacement of the vortex core with respect to its periphery. Our velocity measurements seem to exclude the double vortex hypothesis and visualisations of the vortex do not seem to indicate large deformations of the core during this phase.

[caption id="attachment_37714" align="alignright" width="200"] Figure 3: a) Long time dynamics of the rotation velocity of a single vortex, the upper left inset shows the trajectory with visible trochoidal motion during the intensification phase marked by the red and black dots. The two components of the centre position are shown in the lower inset.
b) Superposition of intensification events from the bubble vortices and from the numerics: vortex intensity from five different intensification events at different frequencies (0, 0.2 and 0.6 Hz) versus time. Three intensification events from the numerics are also shown. The velocity axis has been normalised by the maximum velocity and the time has been normalised by a characteristic time ? and shifted so that the position of the maximum velocity is at zero. Inset: characteristic time versus maximum velocity for the bubble vortices used in this graph.
c) Intensification for several hurricanes in the North Eastern pacific. Here again the time and velocity axis have been normalised as in (a). The time constants are given along with the name of the TC in the figure.
d) superposition of hurricane intensity (from five different hurricanes in the Atlantic as well as two compilations for the Atlantic and Pacific oceans ), vortex intensity (five different intensification events at a frequency of 0, 0.2 and 0.6 Hz) versus time as well as three intensification events from the numerics.[/caption]

Let me come back to the intensification of vortices. In Figure 3, several intensification events from the soap bubble as well as from our numerical simulations are shown. Note that both data sets show an intensification period where the velocity increases up to a maximum value we note V_{max} followed by a decline in intensity. The time scale in this graph has been shifted, so that V_{max} occurs at zero time, and normalised by a characteristic time ? while the velocity was normalised by V_{max}. A surprising feature of this graph is that through a simple rescaling of the velocity and time axes all the data collapse onto a single curve indicating that the intensification shows similar features for different vortices at different bubble rotation rates as well as for vortices from the numerical simulations. The time constant ? used to rescale this data is shown in the inset of this figure and turns out to be roughly 0.07s for the bubble vortices and for the vortices from the numerical simulations. This time constant seems comparable to the turnover time of these vortices which is roughly 0.1s.

The vortices studied here are quasi two dimensional and therefore very different from natural giant vortices such as tropical cyclones; nonetheless, some of the properties observed (trochoidal motion and intensification) are also characteristic of hurricanes and typhoons. For the sake of comparison and keeping in mind the differences in mechanisms as well as in the energetics of the vortices and tropical cyclones, the question of whether such qualitative similarities can be made more quantitative has been examined. In a previous study we had shown that the trajectories of our vortices and that of tropical cyclones do share some quantitative features.

One of the particular features examined concerns the relation between the intensification of the bubble vortices and that of tropical cyclones which are the only known vortices for which intensification, as observed in our experiments, is documented. In Figure 3 we show the variation of the wind velocity V(t) versus time t for a few hurricanes . Note that the normalisation used above is used here also to superimpose intensification data from different hurricanes. This figure shows that such normalisation works reasonably well for this set of data with time constants which vary but hover around six hours. We went a step further and superimposed data from other hurricanes alongside our data (shown in Figure 3). The curves from different tropical cyclones, from our vortices, and from our numerics are superimposed in this representation suggesting that the variation of the velocity versus time is similar for such very different vortices. In itself this result is perhaps not surprising since many different vortices may show an increase and a decrease in intensity. However, a notable feature is that the characteristic time ? needed to normalise the data turns out to be roughly constant and of order six hours for TCs and 0.07s for our vortices as shown in the inset. For the numerics the value of ? is about 0.07s for the three vortices shown in very good agreement with the experiments. Each system therefore seems to be characterised by a single time constant. While the exact meaning of this time constant is not clear at present, its order of magnitude points to roughly one turnover time for the experiments and the numerics, and roughly one turnover time for the tropical cyclones if the radius of hurricane force winds is considered.

In order to test whether for tropical cyclones the time constant is robust and not just a coincidence, we have performed a similar analysis on a large ensemble of events. A time constant was obtained from superimposing all the data onto the same graph. The histogram of ? values obtained from an analysis of 171 TCs in the Atlantic and the Pacific basins is well defined suggesting that the value of ? has a mean of six hours with a standard deviation of about two hours. Furthermore, and in Figure 3, we have added data from a compilation of tropical cyclone intensity variation with time (over 56 storms in the Atlantic and 73 storms in the Pacific). The normalisation as above of this data set using a time constant of six hours works reasonably well. The collapse of the data proposed here indicates that the dynamics of the intensification of vortices may have generic features which may be useful in the understanding of such a difficult problem: the intensity variation of intense events such as hurricanes or typhoons.

Further reading

Intensity of vortices: from soap bubbles to hurricanes T. Meuel, Y. L. Xiong, P. Fischer, C. H. Bruneau, M. Bessafi, and H. Kellay,  Scientific Reports 3, Article number: 3455 doi:10.1038/srep03455, 2013. http://www.nature.com/srep/2013/131213/srep03455/full/srep03455.html

Hurricane Track Forecast Cones from fluctuations T. Meuel, G. Prado, F. Seychelles, M. Besaffi, and H. Kellay, Scientific Reports 2, Article number: 446 doi:10.1038/srep00446, 2012. http://www.nature.com/srep/2012/120614/srep00446/full/srep00446.html

Author Hamid Kellay is Professor of Physics at Université de Bordeaux and Institut Universitaire de France Laboratoire Ondes et Matière d' Aquitaine (CNRS and U. Bordeaux)

http://youtu.be/rnn7WktwyMo

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