A mass-spring fluid-structure interaction solver: Application to flexible revolving wings

Abstract

The secret to the spectacular flight capabilities of flapping insects lies in their wings, which are often approximated as flat, rigid plates. Real wings are however delicate structures, composed of veins and membranes, and can undergo significant deformation. In the present work, we present detailed numerical simulations of such deformable wings. Our results are obtained with a fluidstructure interaction solver, coupling a mass-spring model for the flexible wing with a pseudo-spectral code solving the incompressible Navier-Stokes equations. We impose the no-slip boundary condition through the volume penalization method; the time-dependent complex geometry is then completely described by a mask function. This allows solving the governing equations of the fluid on a regular Cartesian grid. Our implementation for massively parallel computers allows us to perform high resolution computations with Email addresses: dinh-hung.truong@univ-amu.fr (Hung Truong), thomas.engels@ens.fr (Thomas Engels), dkolomenskiy@jamstec.go.jp (Dmitry Kolomenskiy), kai.schneider@univ-amu.fr (Kai Schneider) up to 500 million grid points. The mass-spring model uses a functional approach, thus modeling the different mechanical behaviors of the veins and the membranes of the wing. We perform a series of numerical simulations of a flexible revolving bumblebee wing at a Reynolds number Re = 1800. In order to assess the influence of wing flexibility on the aerodynamics, we vary the elasticity parameters and study rigid, flexible and highly flexible wing models. Code validation is carried out by computing classical benchmarks.

An experimental data-driven mass-spring model of flexible Calliphora wings

“…However, the training can be greatly accelerated by using an initial configuration that takes the distinct properties of veins and membrane into account. This functional approach [20] is chosen here, resulting in the static computational mesh (Fig. 1F).…”

Study of Mosquito Aerodynamics for Imitation as a Small Robot and Flight in a Low-Density Environment

“…Flexible wings help in increasing the L/D ratio for superior performance. Though it generates lower lift and drag than the rigid, the chordwise deformation ceases the increase in effective geometric AoA, thereby changing the total resultant force direction upwards, increasing the L/D [ 107 ]. To better control and play with the dynamic characteristic modes, insects can utilize wing base flexibility [ 108 ].…”

Fluid-structure interaction using volume penalization and mass-spring models with application to flapping bumblebee flight

“…This along with their small wing lengths (from mm to cm) makes it extremely challenging to model the mechanical behavior of insect wings. In our work, a mass-spring system is employed to mimic the dynamics of the complicated membrane-vein network by taking the different mechanical properties between veins and membranes into account [6]. While veins can be considered as rods which resist mainly the torsion and bending deformation, a membrane is fabric-like and behaves like a piece of cloth which resists against the extension deformation.…”

“…However, the scheme is efficient because the fluid and the solid need to be advanced only one time at the current time level. Full details of the fluid-structure interaction (FSI) framework as well as detailed validation of the results can be found in our previous work [6].…”

“…For the mass distribution of the membrane, the same optimization method as in [6] is applied where the objective function is the difference between the mass center of the wing measured in the experiment [26] and the one calculated from the massspring model. For a mass point m i belonging to the membrane at position [x i , y i ], we get: Nominal mass 1 0.0070 0.0209 1 0.0065 0.0180 2 0.0074 0.0237 2 0.0043 0.0071 3 0.0055 0.0076 3 0.0046 0.0024 4 0.0070 0.0063 4 0.0011 0.0001 5 0.0040 0.0031 5 0.0038 0.0043 6 0.0048 0.0094 6 0.0037 0.0005 7 0.0040 0.0019 7 0.0020 0.0012 8 0.0038 0.0009 9 0.0041 0.0023 10 0.0048 0.0064 11 0.0045 0.0017 12 0.0038 0.0018 13 0.0042 0.0010 14 0.0038 0.0020 15 0.0034 0.0008 16 0.0032 0.0005 17 0.0032 0.0004 18 0.0044 0.0009 19 0.0015 0.0001 20 0.0018 0.0001 21 0.0020 0.0009 Table 1 Dimensionless vein diameter d v (adapted from [26]) and their corresponding dimensionless mass m v .…”