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275847 Single Image Turbulence Analysis for Drag Reduction Affected by Flexibility of Polymers

The addition of very small amounts of flexible polymer to a flow reduces drag in turbulent flow by almost 60% at the maximum. The phenomenon is known well as Toms’ effect or drag reduction (DR), and is used in many industries to improve the energy efficiency. Lumley emphasized that visco-elastic effects due to polymers can occur only at higher hydrodynamic frequencies in turbulent flow. This notion of elastic behavior at high frequencies is historically for molten and entangled polymer chain. He also mentioned the strongly enhanced viscosity in the regions of flow, while the viscosity in the laminar sublayer near the wall should remain low. This suggestion is related to shear thinning near the wall, that Lumley explained the reduction of turbulent losses. That is the reason why shear viscosity or velocity profile near the wall has been got much attention in turbulent researches. On the other hand, there were experiments with polymer injection at the center of a pipe. In the case, the drag reduction was also founded where wall effects are not involved. By this experiment, de Gennes claimed that not only the viscosity enhancements near the wall but also the elastic modulus of polymers were important. In the explanation of viscoelasticity, the relaxation time of polymer is important to see the Deborah number corresponding to a “coil-stretch transition”. The evidence of flexible polymer stretch in extensional flows was shown experimentally by Chu.

*et al*at a first time. For rigid polymer, its orientation was observed by a birefringence technique in rather concentrated solution. Indeed, such an extension of polymers is expected to increase the extensional viscosity of flows. If we compare an intrinsic viscosity and an extensional viscosity of polymer solution, the extensional viscosity might be higher than the intrinsic viscosity.

In our previous research, we proposed that, since the polymer stretching is important for the increase of extensional viscosity, naturally stretched rigid polymers also contribute to the increase. In order to observe the effects of polymers on turbulence, two-dimensional turbulence made by flowing soap films was used. Flowing soap films are known as a 2D flow where dynamics of water layer is easily visualized. Since the soap films reflect illumination light at the front and the back of the surface, optical path differences make the interference patterns of the soap film. Thus, the interference patterns are affected by the thickness of water layer, which has information of dynamics of water layer as 2D turbulence. The experimental setup was originally developed by Rutgers. Our original technique is a single image analysis system of the turbulence, which is called Film Interference Flow Imaging (FIFI). When a comb is perpendicularly inserted to the film, the 2D turbulence is generated under the comb. At the same time, extensional stress is generated at the comb which leads to the extension of polymers and increase of extensional viscosity. Thus, the fluctuation of water layer is affected by polymers, which is observed by interference patterns. The patterns were recorded with a digital video camera at the data acquisition area which was 15cm behind the comb.

In this study, the phenomena have been precisely studied by FIFI focusing on turbulence analysis. Soap solutions which were used to make 2D turbulence contain sodium dodecylbenzenesulfonate (SDBS) as a surfactant at the concentration of 2wt%. To observe the polymer effects, polyethyleneoxide (PEO, molecular weight of 3.5x10^{6}) was used as a flexible polymer at the concentration of 0.25, 0.5, 0.75, 1.0, 1.5 and 2.0x10^{-3}wt%. As a rigid polymer, hydroxypropyl cellulose (HPC, molecular weight is greater than 1.0x10^{6}) was used at the concentration of 0.01, 0.02 and 0.05wt%.

The interference pattern of 2D turbulence without polymer contained many fully-developed vortices. The vortices in turbulence became streamwisely long and spanwisely thin with addition of little amounts of PEO. It indicated that the inhibition of vortex mergers which is characteristic of 2D turbulence as an inverse transfer of energy, that is, the 2D turbulence become laminar flow with addition of PEO. In addition, such an effect is suddenly occurred at a critical concentration, which was 0.75wt% of PEO in this study. In the case of HPC, the effect was hardly seen at a low concentration, and gradually observed at much higher concentration compared to that of PEO. While the vortices become streamwisely long and spanwisely thin at the 0.05wt% of HPC, the shape of vortices are very different from those in the case of PEO. It was recognized by interference patterns that not only the shape of vortices but also the fluctuations at the small scales seem to be affected by PEO.

These interference patterns were analyzed by single image analysis. First, the power spectrum of the interference images of the 2D turbulence, <*I*^{2}(*k*_{x}, *k*_{y})>, was calculated by the pixel intensity G with the hamming window. The *k*_{x} and *k*_{y} are the spatial frequencies that are perpendicular and parallel to the flow direction respectively. <*I*^{2}(*k*_{x}, *k*_{y})> of the pixel intensity G characterizes the strength of the thickness fluctuation of water layer on spatial frequencies of *k*_{x} and *k*_{y}. The power spectrum <* I*^{2}(*k*_{x}, 0)> of the interference pattern for SDBS 2wt% solution showed scaling behavior as the power component is -1.56. This value almost fits with the -5/3 that is predicted by theory for 2D turbulence. The power components increase from -5/3 to -1 with increase of PEO and HPC concentrations. This variation can be explained by theory of fluid dynamics which suggests that polymers prohibit the inverse energy in 2D turbulence. While the phenomena are especially known in PEO so far, the variation of the power components also gradually occurred with the HPC additives. The power spectrum <* I*^{2}(0, *k*_{y})> was also calculated for each images. <* I*^{2}(0, *k*_{y})> has the information of flow direction. By combining the information of <* I*^{2}(*k*_{x}, 0)> and <* I*^{2}(0, *k*_{y})>, how the turbulence become to laminar flow should be detected.

Second, single image analysis called Curvature analysis was performed to calculate the curvature of the interference patterns. The interference patterns have the information of the thickness fluctuation of the soap film. Thickness is known to be related to the vorticity of the turbulence. Furthermore, since the thickness field is related to the vorticity, the thickness is also related to the instantaneous streamlines. Streamlines is the family of curves that are instantaneously tangent to the velocity vector of the flow. That is the reason why, it can be reasonable that the curvatures of the interference pattern is related to the variations of the velocity vector, that is, velocity fluctuations. The characteristic patterns of the interference image were obtained after the Weighted Median Filtering operation. The contour of the interference pattern was traced, and the curvature of the contour was calculated. Mathematically, the curvature of a line on a plane was calculated by a differentiation of tangential line. The curvature histogram which is related to the distribution of velocity fluctuation was calculated for each polymer concentration. In addition, the curvature histogram was fitted by a distribution function proposed as velocity fluctuation. Indeed, the histogram was fitted well by the functions, which indicate the distribution becomes sharp with addition of polymers. This indicated flow laminarization occurs. PEO might be efficient for DR compared to HPC. The curvature analysis detected the subtle difference of variations of the interference patterns between PEO and HPC.

The methods we proposed and used were efficient tools for turbulent analysis. We now consider the extensional viscosity is important for DR, which will be simply analyzed by FIFI systems.

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