# Bubble drag reduction requires large bubbles

## Abstract

In the maritime industry, the injection of air bubbles into the turbulent boundary layer under the ship hull is seen as one of the most promising techniques to reduce the overall fuel consumption. However, the exact mechanism behind bubble drag reduction is unknown. Here we show that bubble drag reduction in turbulent flow dramatically depends on the bubble size. By adding minute concentrations (6 ppm) of the surfactant Triton X-100 into otherwise completely unchanged strongly turbulent Taylor-Couette flow containing bubbles, we dramatically reduce the drag reduction from more than 40% to about 4%, corresponding to the trivial effect of the bubbles on the density and viscosity of the liquid. The reason for this striking behavior is that the addition of surfactants prevents bubble coalescence, leading to much smaller bubbles. Our result demonstrates that bubble deformability is crucial for bubble drag reduction in turbulent flow and opens the door for an optimization of the process.

Theoretical, numerical and experimental studies on drag reduction (DR) of a solid body moving in a turbulent flow have been performed for more than three decades cec10 (); mad84 (); mad85 (); lvov2005 (); murai2014 (); kumagai2015 (). A few volume percent () of bubbles can reduce the overall drag up to 40% and beyond san06 (); deu04 (); sug04 (); ber05 (); ber07 (); elbing2008 (); gil13 (); elbing2013 (). However, the exact physics behind this drag reduction mechanism is unknown, thus hindering further progress and optimization, and even the dependence of the effect on the bubble size is controversial mer89 (); fer04 (); lu05 (), though it is believed to be independent of the bubble size cec10 ().

In this Letter, we experimentally investigated the mechanism behind bubble drag reduction in a Taylor-Couette (TC) system, i.e. the flow between two independently rotating coaxial cylinders. The TC system can be seen as “drosophila” of physics of fluids, with many concepts in fluid dynamics being tested therewith, ranging from instabilities, to pattern formation, to turbulence, see the reviews far14 (); gro16 (). Here we inject bubbles into the system, which due to the density difference to water experience a centripetal force towards the inner cylinder, mimicking the upwards gravitational force acting on bubbles under a ship hull.

The experiments are performed in the Twente Turbulent Taylor-Couette facility (TC) gil11a (), with the inner one strongly rotating, corresponding to very large Reynolds number of . The setup has an inner cylinder with a radius of and an outer cylinder with a radius of , resulting in a radius ratio of . The inner cylinder rotates with a frequency up to , resulting in Reynolds numbers up to , in which is kinematic viscosity of water-bubble mixture. The outer cylinder is at rest. The cylinders have a height of , resulting in an aspect ratio of . The flow is cooled through both endplates to prevent viscous heating through the viscous dissipation. The torque is measured with a co-axial torque transducer (Honeywell Hollow Reaction Torque Sensor 2404-1K, maximum capacity of \unit115\newton\meter), mounted inside the middle section of the inner cylinder, to avoid measurement uncertainties due to seals- and bearing friction and endplate effects. Details are described in ref. gil11a (). The gap between the cylinders is either fully filled with water () or, when measuring with bubbles, partly filled with water (). The effective viscosity and density of a bubbly liquid can be approximated using and the Einstein relation gil13 (); ein1906 (): , in which and are the density and the viscosity of the single phase liquid, and is the global volume fraction of air. Air bubbles form over the entire cylinder height because of the large turbulent fluctuations and the high centripetal forces.

The main result is seen in figure 1a,b, where we show the drag coefficient at as function of time for four different bubble concentrations. It is calculated as (see figure 1a) from the measured required torque to keep the inner cylinder rotating at the fixed angular velocity . While with bubble volume concentration between 2% and 4% the drag is remarkably reduced between 18% - 43% as compared to the single phase flow case without bubble gil13 () – here the percentage of drag reduction is expressed as – adding the surfactant Triton X-100 at at a concentration of only 6 ppm reduces the drag reduction within (the time needed for Triton X to mix over the whole system) to the value corresponding to the volumetric gas concentration of 2% - 4%. The same holds for weaker turbulence – here we tested down to (see figure 1c) – though for weaker turbulence the original drag reduction effect through the bubbles is less pronounced.

Figure 2 shows snapshots of the bubbly turbulence at three different lengthscales (reflecting the multiscale character of bubbly turbulence) without (upper row) and with (lower row) the addition of Triton X-100. It is seen that the addition of the surfactant dramatically changes the structure of the turbulent dispersed bubbly flow, resulting in much smaller bubbles (with the same total volume concentration) in the case with Triton X-100. The reason is that the surfactant suppresses bubble coalescence takagi2008 (); takagi2011 (). Earlier studies noticed the role of the bubble Weber number in bubble drag reduction gil13 (); ber05 (); murai2014 (). The Weber numbers before and after addition of Tritox X-100 are estimated as follows: From fig. 2, we estimate that the equivalent bubble diameters are of order mm) for clean water, and mm) for water with Triton X-100, respectively. The surface tension between water and air is known for clean water, i.e. 73 mN/m at room temperatures. After the addition of 6ppm Triton X-100 (equivalent to mol/L), the surface tension lowers to 40 mN/m gob97 (). The velocity fluctuations are impossible to measure after the addition of the surfactant, the flow is too dense to be optically accessible. We know that without bubbles, gil13 () in the bulk of the flow, and that this ratio is constant over a large range of Reynolds numbers, as long as the flow is fully turbulent hui12 (). Furthermore, it has been shown that this ratio does not change much after adding a few percent of mm-sized bubbles gil13 (). For a rotation rate of 20 Hz, we calculate that 0.76 m/s. We assume that this is a reasonable measure for the fluctuations in our bubbly flow. For lower Reynolds numbers, the velocity fluctuations become smaller, resulting in lower Weber numbers.

From the figures we estimate the corresponding Weber numbers in the two cases as and , implying that prior to injection of the surfactant the bubbles can deform (as indeed seen from the figures 2b,c), whereas this is not possible after Triton X-100 was added (which is consistent with figures 2e,f). As shown in figure 1c, drag reduction is less pronounced at lower Reynolds numbers. The physical reason for this trend is that the Weber number of the bubbles decreases when reducing the Reynolds number.

Our findings give strong evidence that the bubble deformability is crucial in the drag reduction mechanism, as already speculated in refs. lu05 (); ber07 (); gil13 (), but disputed by other authors. We note that both the shape change of the bubble and the bubble coating by the surfactant will also modify the lift force coefficient of the lift acting on the bubble mag00 (); takagi2008 (); takagi2011 (); elbing2013 (); muradoglu2014 () and thus the bubble distribution in the flow. Apparently, the large and deforming bubbles, which accumulate close to the inner cylinder gil13 (), hinder the angular momentum exchange between boundary layer and bulk by partly blocking the emission of coherent structures from the boundary layer towards the bulk and reducing the Reynolds stress, thus leading to drag reduction lu05 (); gil13 (); muradoglu2014 (); kit05 ().

Our result have strong bearing on the projected bubble drag reduction in the navel industry. Not only surfactants, but also ions of the various dissolved salts have a strong effect on coalescence properties of bubbles, either enhancing or suppressing coalescence craig2004 (). As seen from our experiments, tests of bubbly drag reduction in fresh water facilities will therefore lead to very different results as in the salty ocean water.

Our results however also offer opportunities to enhance drag reduction in pipelines transporting liquified natural gases (LNGs) close to the boiling point by adding appropriate surfactants helping coalescence ike14 (). Going beyond bubbly multiphase flow towards emulsions of e.g. oil in water wong2015 (), also here the global drag will be strongly affected by the local coalescence behavior of the droplets, thus opening opportunities to influenced the overall drag by the use of surfactants.

###### Acknowledgements.

We would like to thank Gert-Wim Bruggert and Martin Bos for their continuous technical support over the years. We acknowledge stimulation discussions with Dennis Bakhuis, and Rodrigo Ezeta Aparicio and Michiel van Limbeek. The work was supported by the Dutch Foundation for Fundamental Research on Matter (FOM) and the Dutch Technology Foundation STW.### References

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