The videos were processed using MATLAB codes which were written and .... [1] J. Niu and D. L. Hu, âDrag reduction of a hairy disk,â Phys. Fluids, vol. 23, no.
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Experiments on vertical flexible filaments in crossflow Jorge Silvaa,b, Andrea Cioncolinia, Antonio Filipponea a
School of Mechanical, Aerospace and Civil Engineering University of Manchester Manchester, United Kingdom b Facultad de Ingeniería en Mecánica y Ciencias de la Producción Escuela Superior Politécnica del Litoral, ESPOL Campus Gustavo Galindo Km 30.5 Vía Perimetral, P.O. Box 09-01-5863 Guayaquil, Ecuador Abstract—In this paper we introduce an experimental framework we have employed to study the mechanics of flexible filaments in air crossflow. First, we briefly introduce the route of the experimental work focusing on the basic steps towards obtaining data, starting from filament manufacturing up to its testing in a wind tunnel. Second, the data analysis methodology is illustrated with some examples. The methodology will prove useful for future experimental studies related to filament mechanics, for example in their use for flow control or energy harvesting.
I.
INTRODUCTION
The study of reconfiguration and kinematics of flexible filaments is key to understanding the behaviour of filaments in fluid flow, as this can aid explaining differences observed between studies in which different materials or geometries were employed. Characterisation of the mechanical properties of filaments and their further employment in kinematical tests is nonexistent. Additionally, the three-dimensional motion of filaments has not been experimentally explored, with experimental studies being carried on two-dimensional setups such as soap films. Furthermore, there is no experimental evidence of fluid force magnitudes on inclined (i.e. except the cross-flow configuration) cylindrical structures at Reynolds numbers below 2000 (based on filament diameter). This study provides data to fill in these gaps in knowledge and offers a framework for future research related to the use of flexible filaments for various applications; for example, to study the potential use of filaments for drag reduction [1] or energy harvesting [2]. Moreover, these experimental data can provide valuable information for calibration of numerical codes as well as having an understanding on the filament dynamics when interacting with a fluid flow. II.
EXPERIMENTAL METHODS
The filaments used for these experiments were manufactured from commercial silicone on a 3-D printer (model: 3D-Bioplotter; make: Envisiontec GMBH, Germany). This was found to be a reliable method for producing filaments which properties were homogeneous and reproducible. It provided monofilaments of uniform geometry, homogeneous mechanical properties and smooth surface finishing. Other commercially available fibres were either too stiff, exhibited
permanent deformations, consisted of multiple threads or had highly irregular surface finishing. The silicone filaments have been characterised and properties of interest were measured using various techniques. The natural frequency of the filaments was determined from forced vibration tests. Fig. 1 shows the first three modes of motion obtained via shaker tests. The flexural rigidity can be inferred from static deflection experiments. The elasticity modulus can be determined from tensile tests, the diameter from Scanning Electron Microscopy measurements, and the density can be obtained via volumetric methods.
Figure 1. Filaments first three modes of motion from left to right.
Filaments of aspect ratio (length/diameter) ranging from 30 to 190 were then tested in hanging vertical cantilever configuration inside a commercial wind tunnel (make: Armfield) in order to have a crossflow configuration. The tests consisted of subjecting each filament to a steady fluid load and simultaneously record it via two high speed cameras located perpendicular between each other in order to capture its motion in three dimensions. Each filament was tested at increasing wind speeds (1 to 15 m/s in steps of 0.5 m/s) and then decreasing. The videos were processed using MATLAB codes which were written and tuned for this application. The coordinates from the filaments were then used for static and dynamic analyses. At low wind speeds, the filaments were reconfigured and remain practically static, while at sufficiently large flow speeds they exhibited vibration-like motion before progressing into large-amplitude flapping at higher wind speeds. Fig. 2 shows a typical snapshot obtained from the filament’s motion.
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(a)
(b)
Figure 2. Snapshot obtained from filament’s motion (X-Y plane, flow from left to right).
III.
ANALYSIS METHODOLOGY AND RESULTS
From the static reconfiguration of the filament it was seen that it remains practically rectilinear on most of its length. A modelling of the filament as an Euler-Bernoulli beam under steady fluid loading shows that it behaves well according to that beam theory as shown in Fig. 3(a). Moreover, from the reconfigured shape it is possible to estimate the normal forces on the rectilinear part of the filament. For the dynamic analyses, the post-processing methodology includes standard time-series analysis tools (histograms, power spectra, root-mean-squared values, autocorrelation function) as well as nonlinear time-series analysis techniques (phase-space reconstruction and Lyapunov exponents). The use of several time-series diagnostics tools can provide complementary evidence to contrast with the conclusions obtained from standard time-series methods alone [3], [4]. The information coming from modes and trajectories of motion also aid on the interpretation of the behavior of the filaments. Furthermore, the energy content on the filament can be estimated from the experimental data. Together, these techniques provide information on the system’s mechanics which otherwise would be experimentally challenging to measure. However, for the sake of brevity, not all of the analyses will be covered here. Filaments of small aspect ratio (L/d < 38) were found to be too stiff and did not incur in large amplitude motion. On the contrary, more slender filaments exhibited an abrupt increase of amplitude of motion at a certain critical speed. Fig. 3(b) shows an example of the envelopes of motion for a filament of L/d=76, consisting of the shapes of the filament obtained during 1 minute of video recording at a sampling frequency of 200 Hertz. This shows a clear three-dimensional motion just at the onset of flutter. The critical speed of flutter was found to asymptotically decrease with increasing aspect ratio, confirming the trends observed in other publications [5]. One example of the results for the analysis of the free end’s motion at this velocity is shown in Fig. 4. At this speed the motion of the filament has become large in amplitude. The power spectrum shows one dominant peak, with harmonics of the dominant frequency and also a subharmonic. The autocorrelation function is periodic and decays slowly. The displacement histogram is bridge-shaped while the reconstructed attractor is limit-cycle ring-like. The maximal Lyapunov exponent at this wind speed was estimated to be around 0.01, which theoretically means that the motion is chaotic. However, all the other evidence suggests that the motion is clearly a periodic flutter. Since experimental systems are always contaminated with noise and there is inevitably some variability in parameters, it is expected to have a chaotic component which is reflected in the slow decay of the autocorrelation function as well as a positive and very small Lyapunov exponent.
Figure 3. (a) Reconfiguration comparison between experiment and theoretical model. Filament data: L=75 mm, d=0.40 mm, E=0.3 MPa, ρs=1000 kg/m3; Flow data: U=3.5 m/s, ρf=1.2 kg/m3. (b) Envelopes of motion (axes in mm) for a filament of L/d=76, side and top views, 6.5 m/s.
Notably, at this speed the motion in the top view (see Fig. 5) is not the same as in the side view: motion is still periodic but more complex, characterized by two dominant frequencies; a bell-shaped histogram and a reconstructed attractor with a different topology. This is noteworthy because it shows that the filament dynamics is not necessarily qualitatively the same when observed in different planes. At even higher wind speeds the periodicity is less evident, as can be appreciated from the short-decay autocorrelation plot and a flat histogram in Fig. 6. However, the reconstructed attractor still shows periodicity and the frequency spectra still has dominant peaks. Therefore, the presence of dominant chaotic dynamics cannot be assured even though the Lyapunov exponent at this speed is greater than 0.1. IV.
CONCLUSIONS
The methodology here introduced provides a framework for current and future experimental studies related to filament mechanics. Ranging from filament manufacturing and characterisation, their testing, data post-processing up to the analysis methodology; this set of tools will allow studying the filament mechanics and understand better their behaviour in fluid flow before proceeding to more complex cases of interest (e.g. filaments attached to bluff bodies). This study provides evidence on filament motion in an experimental setup where both top and side views have been resolved and analysed. It was found that the filament dynamics is not necessarily qualitatively the same when observed in different planes and that the motion is three-dimensional even at the onset of flutter. In general, short filaments tend to have a simpler dynamics (L/d < 38), while longer filaments exhibit a richer behaviour. Experimental data is subject to disturbances and noise and thus standard time series analysis techniques might give a partial picture of the dynamics of the system being studied. The recourse to several time series analysis tools was proven to be useful when studying dynamic systems experimentally. This is particularly evident at high wind velocities, where the reconstructed attractor may highlight periodicity better that other more classic time series analysis techniques.
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Figure 4. Side view, filament L/d=76, U=6.5 m/s, free end. (a) Time series of displacement, (b) Power spectrum density, (c) Autocorrelation function of displacement, (d) Histogram of displacement, (e) Reconstructed phase-space.
Figure 5. Top view, filament L/d=76, U=6.5 m/s, free end. (a) Time series of displacement, (b) Power spectrum density, (c) Autocorrelation function of displacement, (d) Histogram of displacement, (e) Reconstructed phase-space.
Figure 6. Side view, filament L/d=76, U=14.5 m/s, free end. (a) Time series of displacement, (b) Power spectrum density, (c) Autocorrelation function of displacement, (d) Histogram of displacement, (e) Reconstructed phase-space. [3]
ACKNOWLEDGEMENTS Jorge Silva wishes to thank the National Government of the Republic of Ecuador for funding provided through a scholarship by SENESCYT under the program Convocatoria Abierta 2012-I. REFERENCES [1] [2]
J. Niu and D. L. Hu, “Drag reduction of a hairy disk,” Phys. Fluids, vol. 23, no. 10, p. 101701, 2011. C. Grouthier, S. Michelin, R. Bourguet, Y. Modarres-Sadeghi, and E. de Langre, “On the efficiency of energy harvesting using vortex-induced vibrations of cables,” J. Fluids Struct., vol. 49, pp. 427–440, 2014.
[4] [5]
F. C. Moon, Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA, 2004. M. P. Paidoussis, Fluid-Structure Interactions: Slender Structures and Axial Flow, Volume 1, Second. Academic Press, 2014. L. Schouveiler, C. Eloy, and P. Le Gal, “Flow-induced vibrations of high mass ratio flexible filaments freely hanging in a flow,” Phys. Fluids, vol. 17, no. 4, p. 47104, Apr. 2005.
