The wake of an oscillating cylinder at low Reynolds numbers is a nonlinear system in which a limit cycle due to natural vortex shedding is modulated, generating in phase space a flow on a torus. We experimentally show that the system displays Arnol'd tongues for rational frequency ratios, and approximates the devil's staircase along the critical line. The "singularity spectrum" as well as spectral peaks at various Fibonacci sequences accompanying quasiperiodic transition to chaos show's that the system belongs to the same universality class as the sine circle map.PACS numbers: 47.15. Gf, 47.25.Gk In low-dimensional dynamical systems, detailed predictions have been made for the "universal" features of transition to chaos by period-doubling 1,2 and quasiperiodic 3 " 6 routes. Experiments in small-aspect-ratio closed-flow systems 7 " 9 have gone a long way in establishing the validity of these predictions to fluid flows. However, experiments in open-flow systems (those with imposed unidirectional main flow) with little or no confinement have paid heed to these predictions only rarely. 10 " 13 The best case for showing some conformity with features of nonlinear dynamics is the flow behind circular cylinders. 10,13 Here, we study at low Reynolds numbers the flow behind a circular cylinder oscillating transverse to an oncoming stream, and show that it exhibits some quantitative features of universality.Briefly, with increasing Reynolds number, the flow behind a stationary cylinder first undergoes a Hopf bifurcation 14 from the steady state to a periodic state characterized by the vortex-shedding mode at a frequency /o, say. We have shown in Ref. 14 that the post-critical state can be modeled by the Landau equation, and determined the Landau constants. For cylinder aspect ratio (that is, the length to diameter ratio) exceeding about 60, details of this bifurcation are independent of the aspect ratio, and the critical Reynolds number (based on the cylinder diameter D and the oncoming velocity) is about 46. For the present measurements, the working fluid was air, and the Reynolds number about 55. A modulation was imposed by our causing the cylinder to oscillate transverse to the main flow at a frequency f e , the amplitude of oscillation being then a measure of the nonlinear coupling between the two modes. The system has two competing frequencies (/o and f e ) yielding two control parameters, f e /fo and the nondimensional amplitude of oscillation, a/D. Once the external modulation is imposed, we expect /o to shift to fb, say. This is similar in spirit to the convection experiments of Refs. 7 and 8, and the well-studied sine circle map.

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The dynamic motion of floating wind turbines is studied using numerical simulations. Floating wind turbines in the deep ocean avoid many of the concerns with land-based wind turbines while allowing access to strong stable winds. The full three-dimensional Navier-Stokes equations are solved on a regular structured grid, using a level set method for the free surface and an immersed boundary method for the turbine platform. The tethers, the tower, the nacelle and the rotor weight are included using reduced order dynamic models, resulting in an efficient numerical approach which can handle nearly all the nonlinear wave forces on the platform, while imposing no limitation on the platform motion. Wind is modeled as a constant thrust force and rotor gyroscopic effects are accounted for. Other aerodynamic loadings and aero-elastic effects are not considered. Several tests, including comparison with other numerical, experimental and grid study tests, have been done to validate and verify the numerical approach. Also for further validation, a 100 : 1 scale model Tension Leg Platform (TLP) floating wind turbine has been simulated and the results are compared with water flume experiments conducted by our research group. The model has been extended to full scale systems and the response of the tension leg and spar buoy floating wind turbines has been studied. The tension leg platform response to different amplitude waves is examined and for large waves a nonlinear trend is seen. The nonlinearity limits the motion and shows that the linear assumption will lead to over prediction of the TLP response. Studying the flow field behind the TLP for moderate amplitude waves shows vortices during the transient response of the platform but not at the steady state, probably due to the small Keulegan-Carpenter number. The effects of changing the platform shape are considered and finally the nonlinear response of the platform to a large amplitude wave leading to slacking of the tethers is simulated. For the spar buoy floating wind turbine, the response to regular periodic waves is studied first. Then, the model is extended to irregular waves to study the interaction of the buoy with more realistic sea state. The results are presented for a harsh condition, in which waves over 17 m are generated, and linear models might not be accurate enough. The results are studied in both time and frequency domain without relying on any experimental data or linear assumption. Finally a design study has been conducted on the spar buoy platform to study the effects of tethers position, tethers stiffness, and platform aspect ratio, on the response of the floating wind turbine. It is shown that higher aspect ratio platforms generally lead to lower mean pitch and surge responses, but it may also lead to nonlinear trend in standard deviation in pitch and heave, and that the tether attachment points design near the platform center of gravity generally leads to a more stable platform in comparison with attachment points near the tank top or bottom of the platfor...

Articles you may be interested inModel test and simulation of modified spar type floating offshore wind turbine with three catenary mooring lines Validation of a FAST semi-submersible floating wind turbine numerical model with DeepCwind test data J. Renewable Sustainable Energy 5, 023116 (2013); 10.1063/1.4796197 CFD-based design load analysis of 5MW offshore wind turbine AIP Conf.A nonlinear computational model, based on solving the Navier-Stokes equation, is used to study the motion of a 5 MW spar buoy floating wind turbine in moderate and extreme sea states with irregular waves. The main advantages of using the current model are that there is no limitation on the platform motion, the hydrodynamic loads do not rely on experimental data, and nonlinear hydrodynamic loads can be predicted. The current work extends a previously developed Navier-Stokes model for regular periodic waves on a tension leg platform floating wind turbine. Free decay tests are performed, and pitch, heave, and surge natural frequencies are determined. The responses of the spar buoy to operating conditions with significant wave height of 8 m and mean period of 10 s, and an extreme sea states including waves over 17 m height are studied. For the extreme sea state, a nonlinear model is required, since the platform response amplitudes are not small with respect to the spar buoy diameter. Effects not included in linear models, such as platform pitch angles higher than 10 , complete submergence of the platform tank and tether slacking are captured. Finally, a design study on spar buoy aspect ratio is performed for one sea state and it is shown that higher aspect ratio spars generally lead to lower mean pitch and surge responses as expected, but also may lead to a nonlinear trend in the standard deviations in pitch and heave, probably due to the increase in wind and wave moments on the spar. V C 2014 AIP Publishing LLC.

A forced Landau-Stuart equation is studied in order to derive a low-dimensional model describing the temporal behavior of a paradigm open flow, the two-dimensional forced cylinder wake. Numerical results from the model exhibit several characteristics of circle maps, and compare qualitatively to previous experimental results for an oscillating cylinder wake. The low-dimensional model is also shown to reduce to a circle map in the limit of small forcing amplitudes. Observation of circle map dynamics in the forced Landau-Stuart equation strengthens the conjecture that globally unstable fluid flows are amenable to a dynamical systems approach focusing on the study of low-dimensional iterative maps. The established connection between the Landau-Stuart equation and the circle map unifies certain aspects of spatiotemporal stability and low-dimensional chaos theory.