Two-dimensional simulation of the fluttering instability using a pseudospectral method with volume penalization Thomas Engels a,b,⇑ , Dmitry Kolomenskiy c , Kai Schneider a , Jörn Sesterhenn b a Laboratiore de Mécanique, Modélisation et Procédés Propres (M2P2), CNRS et Aix-Marseille Université, France b Institut für Strömungmechanik und Technische Akustik (ISTA), TU Berlin, Germany c McGill University/CRM, Montréal, Canada article info Article history: Received 31 May 2012 Accepted 4 December 2012 Available online xxxx Keywords: Fluid–structure interaction Fluttering instability Volume-penalization method Spectral method abstract We present a new numerical scheme for the simulation of deformable objects immersed in a viscous incompressible fluid. The two-dimensional Navier–Stokes equations are discretized with an efficient Fou- rier pseudo-spectral scheme. Using the volume penalization method arbitrary inflow conditions can be enforced, together with the no-slip conditions at the boundary of the immersed flexible object. With respect to Kolomenskiy and Schneider (2009) [1], where rigid moving obstacles have been considered, the present work extends the volume penalization method to account for moving deformable objects while avoiding numerical oscillations in the hydrodynamic forces. For the solid part, a simple and accu- rate one-dimensional model, the non-linear beam equation, is employed. The coupling between the fluid and solid parts is realized with a fast explicit staggered scheme. The method is applied to the fluttering instability of a slender structure immersed in a free stream. This coupled non-linear system can enter three distinct states: stability of the initial condition or maintenance of an either periodic or chaotic flut- tering motion. We present a detailed parameter study for different Reynolds numbers and reduced free- stream velocities. The dynamics of the transition from a periodic to a chaotic state is investigated. The results are compared with those obtained by an inviscid vortex shedding method [2] and by a viscous linear stability analysis [3], yielding for both satisfactory agreement. New results concerning the transi- tion to chaos are presented. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction In nature inspired fluid dynamics, the complex interaction of some deformable structures with an ambient flow is a common- place problem. Whether it is gliding, swimming or flying, many types of animal locomotion strongly rely on this type of interaction [4]. The archetype problem of fluid–structure interaction is the flapping of a flag in the wind, attracting researchers due to its rich- ness in phenomena. Indeed, a flag exhibits a large variety of possi- ble regimes, depending on its material parameters and the surrounding flow. It can be aligned with the flow in a stable state, or flap dynamically. The latter state can further be subdivided into highly regular and chaotic motion patterns. In some parameter ranges, also bistable behavior has been reported, where the dynamically selected state depends on the initial conditions. Natural swimmers and flapping flyers exploit a combination of active and passive flapping to improve their flight performance [5–7], a source of inspiration for various investigations. Also, aside from locomotion, the fluttering instability occurs in other biologi- cal applications. Huang [8] first pointed out that flutter is encoun- tered in the upper human airways, where the soft palate separates nasal and oral inflow. The instability manifests itself in the occur- rence of snoring or, in severe cases, obstructive sleep apnoea/ hypopnea. In engineering, the flutter phenomenon occurs as a problem in printing machines, where flutter limits the speed for moving sheets [9]. However, it is not only destructive and perturbing if flutter occurs in technical applications. The concept of a flutter- mill, a small scale device for energy harvesting, is based on the fact that if the instability threshold is exceeded, energy is pumped con- tinuously into the structure. This energy, originating from the mean flow, can partly be harvested and used for power generation [10,11]. The past decades have seen an increasing variety of theoretical treatments, providing further insight into the phenomenon. One of the earliest works falling into this category was done by Kornecki et al. [12], where the instability has been analyzed using a potential flow with a linear solid model, see also [13] with improved com- puter accuracy. Eloy et al. [13] also developed a three-dimensional 0045-7949/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2012.12.007 ⇑ Corresponding author at: Laboratiore de Mécanique, Modélisation et Procédés Propres (M2P2), CNRS et Aix-Marseille Université, France. Tel.: +49 17683252194. E-mail addresses: thomas.engels@l3m.univ-mrs.fr, thomas.engels@mailbox. tu-berlin.de (T. Engels). Computers and Structures xxx (2013) xxx–xxx Contents lists available at SciVerse ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc Please cite this article in press as: Engels T et al. Two-dimensional simulation of the fluttering instability using a pseudospectral method with volume penalization. Comput Struct (2013), http://dx.doi.org/10.1016/j.compstruc.2012.12.007