European Journal of Mechanics B/Fluids 55 (2016) 146–156 Contents lists available at ScienceDirect European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu An embedded boundary approach for the simulation of a flexible flapping wing at different density ratio Min Xu, Mingjun Wei ∗ , Tao Yang, Young S. Lee Department of Mechanical and Aerospace Engineering, New Mexico State University, NM, 88001, USA article info Article history: Received 27 October 2014 Received in revised form 2 September 2015 Accepted 15 September 2015 Available online 25 September 2015 Keywords: Embedded boundary Flexible flapping wing Low Reynolds number Fluid–structure interaction Density ratio abstract We have developed a strong-coupling approach based on a uniformly-applied Eulerian description for both fluid and solid and provided a simple monolithic formulation to compute highly flexible structures interacting with surrounding fluid flows. Using a fast-tracking method and a fast solver for the modified pressure equation with variable density, we keep the same low computational cost as in the uniform density case studied previously. The new algorithm is first validated by the simulation of the self-sustained oscillation of a flexible plate. Then, it is applied to study the effect from density ratio on a flexible plate flapping with incoming flow. The simulation shows strong effect of density ratio on the pattern of fluid–structure interaction and the propulsion performance through the change in mass ratio and frequency ratio. © 2015 Elsevier Masson SAS. All rights reserved. 1. Introduction Birds and insects have survived by taking the advantage of flapping flexible wings for high energy efficiency and incredible maneuverability, when fixed rigid wings conventionally used for airplanes fail to meet the need at the same low Reynolds number. The flapping-wing mechanism from the nature has inspired generations of aerial vehicle designs from ancient flight machines to modern unmanned aerial vehicles. Especially, with the recent demand of micro air vehicles (MAVs), flapping-wing design attracts attention by many desirable characteristics (i.e. efficiency, maneuverability, hovering-capability) at low Reynolds number regime [1]. Starting with the pioneer work by Knoller [2] and Betz [3] in thrust generation by a plunging airfoil, numerous research in ex- periments and numerical simulation has been done to understand the propulsion by plunging and pitching foils and has been summa- rized in various places [4–7]. Because of extra complexity brought in by wing flexibility, the majority of earlier works focused only on rigid wings [4,8–10] or prescribed deformable wings [11,12]. However, recently, there were increasing interests and number of works in truly flexible wings with fully coupled fluid–structure in- teraction in both experiments and numerical simulation [7,13,14]. ∗ Corresponding author. E-mail address: mjwei@nmsu.edu (M. Wei). To simulation fluid–structure interaction, the fluid and solid are typically solved by separate equations and algorithms, and they are then coupled at the interface through boundary conditions. There have been many studies for different applications and using different numerical algorithms. Donea et al. [15] applied Arbitrary Lagrangian–Eulerian (ALE) finite element method to solve the fluid and study fluid–structure systems under transient dynamic loading. To understand the flexibility effect of a flapping foil, Zhu [16] used boundary element method to solve the fluid and coupled it with a two-dimensional thin-plate structural model by iterations. Luo et al. [17], in their study of the vocal fold vibration in human phonation, used immersed boundary method to solve the incompressible fluid equation with moving boundary and coupled with a linear viscoelastic solid equation for interaction. Tian et al. [18] applied a similar approach to study the aerodynamics of elastic insect wings. All the above approaches require accurate and explicit representation of boundary conditions (e.g. location, velocity, force) at the fluid–solid interface which link the fluid and solid solvers and play the key role in the convergency between these two solvers. In fact, the convergency is not guaranteed in some cases, and some works resorted to choose weak coupling (i.e. without any interaction) to avoid the convergency problem and save computational time. In our study, we used a strong-coupling approach to simulate a highly flexible wing interacting with surrounding fluid flow in a globally Eulerian framework for both fluid and solid, which avoids entirely the explicit representation and matching of boundary conditions at fluid–solid interface and the associate problem in http://dx.doi.org/10.1016/j.euromechflu.2015.09.006 0997-7546/© 2015 Elsevier Masson SAS. All rights reserved.