Limit-cycle oscillations in unsteady flows dominated by intermittent leading-edge vortex shedding

High-frequency limit-cycle oscillations of an airfoil at low Reynolds number are studied numerically. This regime is characterized by large apparent-mass effects and intermittent shedding of leading-edge vortices. Under these conditions, leading-edge vortex shedding has been shown to result in favorable consequences such as high lift and efficiencies in propulsion/power extraction, thus motivating this study. The aerodynamic model used in the aeroelastic framework is a potential-flow-based discrete-vortex method, augmented with intermittent leading-edge vortex shedding based on a leading-edge suction parameter reaching a critical value. This model has been validated extensively in the regime under consideration and is computationally cheap in comparison with Navier–Stokes solvers. The structural model used has degrees of freedom in pitch and plunge, and allows for large amplitudes and cubic stiffening. The aeroelastic framework developed in this paper is employed to undertake parametric studies which evaluate the impact of different types of nonlinearity. Structural configurations with pitch-to-plunge frequency ratios close to unity are considered, where the flutter speeds are lowest (ideal for power generation) and reduced frequencies are highest. The range of reduced frequencies studied is two to three times higher than most airfoil studies, a virtually unexplored regime. Aerodynamic nonlinearity resulting from intermittent leading-edge vortex shedding always causes a supercritical Hopf bifurcation, where limit-cycle oscillations occur at freestream velocities greater than the linear flutter speed. The variations in amplitude and frequency of limit-cycle oscillations as functions of aerodynamic and structural parameters are presented through the parametric studies. The excellent accuracy/cost balance offered by the methodology presented in this paper suggests that it could be successfully employed to investigate optimum setups for power harvesting in the low-Reynoldsnumber regime.

[1]  Earl H. Dowell,et al.  Power extraction from aeroelastic limit cycle oscillations , 2011 .

[2]  Qiang Zhu,et al.  Energy harvesting through flow-induced oscillations of a foil , 2009 .

[3]  T. Kármán.  General aerodynamic theory. Perfect fluids , 1963 .

[4]  Earl H. Dowell,et al.  Nonlinear aeroelasticity in rotorcraft , 1993 .

[5]  Masaki Hamamoto,et al.  Application of fluid–structure interaction analysis to flapping flight of insects with deformable wings , 2007, Adv. Robotics.

[6]  Yuan-Cheng Fung,et al.  An introduction to the theory of aeroelasticity , 1955 .

[7]  M. Triantafyllou,et al.  Oscillating foils of high propulsive efficiency , 1998, Journal of Fluid Mechanics.

[8]  G. Vatistas,et al.  A simpler model for concentrated vortices , 1991 .

[9]  Max F. Platzer,et al.  Time-domain analysis of low-speed airfoil flutter , 1996 .

[10]  T. Theodorsen.  General Theory of Aerodynamic Instability and the Mechanism of Flutter , 1934 .

[11]  Lei-Yong Jiang,et al.  FLUTTER OF AN AIRFOIL WITH A CUBIC RESTORING FORCE , 1999 .

[12]  Philippe Viarouge,et al.  Prototype testing of a hydrokinetic turbine based on oscillating hydrofoils , 2011 .

[13]  Theodore Theodorsen,et al.  Flutter calculations in three degrees of freedom , 1942 .

[14]  J. Edwards,et al.  Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding , 2014, Journal of Fluid Mechanics.

[15]  M. Dickinson,et al.  UNSTEADY AERODYNAMIC PERFORMANCE OF MODEL WINGS AT LOW REYNOLDS NUMBERS , 1993 .

[16]  Ashok Gopalarathnam,et al.  Effect of airfoil shape and Reynolds number on leading edge vortex shedding in unsteady flows , 2012 .

[17]  Adrian L. R. Thomas,et al.  Leading-edge vortices in insect flight , 1996, Nature.

[18]  J. Katz,et al.  Low-Speed Aerodynamics , 1991 .

[19]  W. Durand,et al.  Aerodynamic theory : a general review of progress , 1963 .

[20]  J. G. Leishman,et al.  A Semi-Empirical Model for Dynamic Stall , 1989 .

[21]  T. Kinsey,et al.  Parametric Study of an Oscillating Airfoil in a Power-Extraction Regime , 2008 .

[22]  Stuart J. Price,et al.  THE AEROELASTIC RESPONSE OF A TWO-DIMENSIONAL AIRFOIL WITH BILINEAR AND CUBIC STRUCTURAL NONLINEARITIES , 1995 .

[23]  Hester Bijl,et al.  Nonlinear aeroelastic behavior of an oscillating airfoil during stall-induced vibration , 2008 .

[24]  Carlos E. S. Cesnik,et al.  Computational fluid-structure interaction of a deformable flapping wing for micro air vehicle applications , 2008 .

[25]  J. Edwards,et al.  An unsteady airfoil theory applied to pitching motions validated against experiment and computation , 2013 .

[26]  J. Horáček,et al.  Numerical simulation of flow induced airfoil vibrations with large amplitudes , 2007, Programs and Algorithms of Numerical Mathematics 19.

[27]  O. Bendiksen.  Review of unsteady transonic aerodynamics: Theory and applications , 2011 .

[28]  Max F. Platzer,et al.  A review of progress and challenges in flapping foil power generation , 2014 .

[29]  Toshiyuki Nakata,et al.  Aerodynamic performance of a hovering hawkmoth with flexible wings: a computational approach , 2012, Proceedings of the Royal Society B: Biological Sciences.

[30]  S. J. Price,et al.  NONLINEAR AEROELASTIC ANALYSIS OF AIRFOILS : BIFURCATION AND CHAOS , 1999 .

[31]  Shijun Guo,et al.  Aeroelastic effect on flapping wing performance , 2007 .

[32]  Kai Schneider,et al.  Leading-edge vortex shedding from rotating wings , 2014, 1405.4838.

[33]  J. Gordon Leishman,et al.  Principles of Helicopter Aerodynamics , 2000 .

[34]  J. CARRIERt,et al.  A FAST ADAPTIVE MULTIPOLE ALGORITHM FOR PARTICLE SIMULATIONS * , 2022 .

[35]  A. Leonard.  Vortex methods for flow simulation , 1980 .

[36]  Earl H. Dowell,et al.  Comparison of Theory and Experiment for Nonlinear Flutter and Stall Response of Helicopter Blade , 1991 .

[37]  J. Peiro,et al.  Camber effects in the dynamic aeroelasticity of compliant airfoils , 2009 .

[38]  I. E. Garrick.  Propulsion of a flapping and oscillating airfoil , 1936 .

[39]  Guy Dumas,et al.  Computational aeroelastic simulations of self-sustained pitch oscillations of a NACA0012 at transitional Reynolds numbers , 2011 .

[40]  Ole H. Hald,et al.  Convergence of Vortex methods for Euler's equations, III , 1987 .

[41]  Hao Liu,et al.  Flapping Wings and Aerodynamic Lift: The Role of Leading-Edge Vortices , 2007 .

[42]  Matthew Bryant,et al.  Modeling and Testing of a Novel Aeroelastic Flutter Energy Harvester , 2011 .

[43]  C. Ellington.  The novel aerodynamics of insect flight: applications to micro-air vehicles. , 1999, The Journal of experimental biology.

[44]  C. T. Tran,et al.  Semi-empirical model for the dynamic stall of airfoils in view of the , 1980 .

[45]  Wei Shyy,et al.  Computational aerodynamics of low Reynolds number plunging, pitching and flexible wings for MAV applications , 2008 .

[46]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

[47]  Ramji Kamakoti,et al.  Fluid–structure interaction for aeroelastic applications , 2004 .

[48]  Joseph Katz,et al.  A discrete vortex method for the non-steady separated flow over an airfoil , 1981, Journal of Fluid Mechanics.

[49]  Michael P. Païdoussis,et al.  Cantilevered flexible plates in axial flow: Energy transfer and the concept of flutter-mill , 2009 .