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Insect wings can undergo significant deformation during flapping motion owing to inertial, elastic and aerodynamic forces. Changes in shape then alter aerodynamic forces, resulting in a fully coupled Fluid-Structure Interaction (FSI) problem. Here, we present detailed three-dimensional FSI simulations of deformable blowfly (Calliphora vomitoria) wings in flapping flight. A wing model is proposed using a multi-parameter mass-spring approach, chosen for its implementation simplicity and computational efficiency. We train the model to reproduce static elasticity measurements by optimizing its parameters using a genetic algorithm with covariance matrix adaptation (CMA-ES). Wing models trained with experimental data are then coupled to a high-performance flow solver run on massively parallel supercomputers. Different features of the modeling approach and the intra-species variability of elastic properties are discussed. We found that individuals with different wing stiffness exhibit similar aerodynamic properties characterized by dimensionless forces and power at the same Reynolds number. We further study the influence of wing flexibility by comparing between the flexible wings and their rigid counterparts. Under equal prescribed kinematic conditions for rigid and flexible wings, wing flexibility improves lift-to-drag ratio as well as lift-to-power ratio and reduces peak force observed during wing rotation.

The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we propose a wavelet-based adaptive version of the POD (the wPOD), in order to overcome this limitation. The amount of data to be analyzed is reduced by compressing them using biorthogonal wavelets, yielding a sparse representation while conveniently providing control of the compression error. Numerical analysis shows how the distinct error contributions of wavelet compression and POD truncation can be balanced under certain assumptions, allowing us to efficiently process high-resolution data from three-dimensional simulations of flow problems. Using a synthetic academic test case, we compare our algorithm with the randomized singular value decomposition. Furthermore, we demonstrate the ability of our method analyzing data of a 2D wake flow and a 3D flow generated by a flapping insect computed with direct numerical simulation., Comment: The algorithm can be found as a post processing tool in the open source software package wabbit (https://github.com/adaptive-cfd/WABBIT). Please note, that this paper is a working paper and is not reviewed yet. It was submitted to ACOM Journal at the 10th of November 2020

Wing flexibility plays an essential role in the aerodynamic performance of insects due to the considerable deformation of their wings during flight under the impact of inertial and aerodynamic forces. These forces come from the complex wing kinematics of insects. In this study, both wing structural dynamics and flapping wing motion are taken into account to investigate the effect of wing deformation on the aerodynamic efficiency of a bumblebee in tethered flight. A fluid-structure interaction solver, coupling a mass-spring model for the flexible wing with a pseudo-spectral code solving the incompressible Navier-Stokes equations, is implemented for this purpose. We first consider a tethered bumblebee flying in laminar flow with flexible wings. Compared to the rigid model, flexible wings generate smaller aerodynamic forces but require much less power. Finally, the bumblebee model is put into a turbulent flow to investigate its influence on the force production of flexible wings.

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 fluid-structure 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 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., Comment: PDF file with 18 pages, 22 figures and 3 tables

We present a wavelet-based adaptive method for computing 3D multiscale flows in complex, time-dependent geometries, implemented on massively parallel computers. While our focus is on simulations of flapping insects, it can be used for other flow problems, including turbulence, as well. The incompressible fluid is modeled with an artificial compressibility approach in order to avoid solving elliptical problems. No-slip and in/outflow boundary conditions are imposed using volume penalization. The governing equations are discretized on a locally uniform Cartesian grid with centered finite differences, and integrated in time with a Runge--Kutta scheme, both of 4th order. The domain is partitioned into cubic blocks with equidistant grids with different resolution and, for each block, biorthogonal interpolating wavelets are used as refinement indicators and prediction operators. Thresholding the wavelet coefficients allows to generate dynamically evolving grids, and an adaption strategy tracks the solution in both space and scale. Blocks are distributed among MPI processes and the global topology of the grid is encoded using a tree-like data structure. Analyzing the different physical and numerical parameters allows balancing their individual error contributions and thus ensures optimal convergence while minimizing computational effort. Different validation tests score accuracy and performance of our new open source code, WABBIT (Wavelet Adaptive Block-Based solver for Interactions with Turbulence), on massively parallel computers using fully adaptive grids. Flow simulations of flapping insects demonstrate its applicability to complex, bio-inspired problems.

Flapping insects are remarkably agile fliers, adapted to a highly turbulent environment. We present a series of high resolution numerical simulations of a bumblebee interacting with turbulent inflow. We consider both tethered and free flight, the latter with all six degrees of freedom coupled to the Navier--Stokes equations. To this end we vary the characteristics of the turbulent inflow, either changing the turbulence intensity or the spectral distribution of turbulent kinetic energy. Active control is excluded in order to quantify the passive response real animals exhibit during their reaction time delay, before the wing beat can be adapted. Modifying the turbulence intensity shows no significant impact on the cycle-averaged aerodynamical forces, moments and power, compared to laminar inflow conditions. The fluctuations of aerodynamic observables, however, significantly grow with increasing turbulence intensity. Changing the integral scale of turbulent perturbations, while keeping the turbulence intensity fixed, shows that the fluctuation level of forces and moments is significantly reduced if the integral scale is smaller than the wing length. Our study shows that the scale-dependent energy distribution in the surrounding turbulent flow is a relevant factor conditioning how flying insects control their body orientation.

A numerical approach for solving evolutionary partial differential equations in two and three space dimensions on block-based adaptive grids is presented. The numerical discretization is based on high-order, central finite-differences and explicit time integration. Grid refinement and coarsening are triggered by multiresolution analysis, i.e. thresholding of wavelet coefficients, which allow controlling the precision of the adaptive approximation of the solution with respect to uniform grid computations. The implementation of the scheme is fully parallel using MPI with a hybrid data structure. Load balancing relies on space filling curves techniques. Validation tests for 2D advection equations allow to assess the precision and performance of the developed code. Computations of the compressible Navier-Stokes equations for a temporally developing 2D mixing layer illustrate the properties of the code for nonlinear multi-scale problems. The code is open source.

The aerial environment in the operating domain of small-scale natural and artificial flapping wing fliers is highly complex, unsteady and generally turbulent. Considering flapping flight in an unsteady wind environment with a periodically varying lateral velocity component, we show that body rotations experienced by flapping wing fliers result in the reorientation of the aerodynamic force vector that can render a substantial cumulative deficit in the vertical force. We derive quantitative estimates of the body roll amplitude and the related energetic requirements to maintain the weight support in free flight under such conditions. We conduct force measurements of a miniature hummingbird-inspired robotic flapper and numerical simulations of a bumblebee. In both cases, we demonstrate the loss of weight support due to body roll rotations. Using semi-restrained flight measurements, we demonstrate the increased power requirements to maintain altitude in unsteady winds, achieved by increasing the flapping frequency. Flapping fliers may increase their flapping frequency as well as the stroke amplitude to produce the required increase in aerodynamic force, both of these two types of compensatory control requiring additional energetic cost. We analyze the existing data from experiments on animals flying in von K\'arm\'an streets and find reasonable agreement with the proposed theoretical model.

The natural wind environment that volant insects encounter is unsteady and highly complex, posing significant flight control and stability challenges. Unsteady airflows can range from structured chains of discrete vortices shed in the wake of an object to fully developed chaotic turbulence. It is critical to understand the flight control strategies insects employ to safely navigate in natural environments. We combined experiments on free flying bumblebees with high fidelity numerical simulations and lower order modeling to identify the salient mechanics that mediate insect flight in unsteady winds. We trained bumblebees to fly upwind towards an artificial flower in a wind tunnel under steady wind and in a von Karman street (23Hz) formed in the wake of a cylinder. The bees displayed significantly higher movement in the unsteady vortex street compared to steady winds. Correlation analysis revealed that at lower frequencies, less than 10 Hz, in both steady and unsteady winds the bees mediated lateral movement with body roll, typical casting motion. At higher frequencies in unsteady winds there was a negative correlation between body roll and lateral accelerations. Numerical simulations of a bumblebee in similar conditions permitted the separation of the passive and active components of the flight trajectories. Comparison between the free-flying and numerical bees revealed a novel mechanism that enables bees to passively ride out high frequency perturbations while performing active maneuvers and corrections at lower frequencies. The capacity of maintaining stability by combining passive and active modes at different timescales provides a viable means for volant animals and machines to tackle the control challenges posed by complex airflows.

FluSI, a fully parallel open source software for pseudo-spectral simulations of three-dimensional flapping flight in viscous flows, is presented. It is freely available for non-commercial use under [https://github.com/pseudospectators/FLUSI]. The computational framework runs on high performance computers with distributed memory architectures. The discretization of the three-dimensional incompressible Navier--Stokes equations is based on a Fourier pseudo-spectral method with adaptive time stepping. The complex time varying geometry of insects with rigid flapping wings is handled with the volume penalization method. The modules characterizing the insect geometry, the flight mechanics and the wing kinematics are described. Validation tests for different benchmarks illustrate the efficiency and precision of the approach. Finally, computations of a model insect in the turbulent regime demonstrate the versatility of the software.

Aerodynamic ground effect in flapping-wing insect flight is of importance to comparative morphologies and of interest to the micro-air-vehicle (MAV) community. Recent studies, however, show apparently contradictory results of either some significant extra lift or power savings, or zero ground effect. Here we present a numerical study of fruitfly sized insect takeoff with a specific focus on the significance of leg thrust and wing kinematics. Flapping-wing takeoff is studied using numerical modelling and high performance computing. The aerodynamic forces are calculated using a three-dimensional Navier--Stokes solver based on a pseudo-spectral method with volume penalization. It is coupled with a flight dynamics solver that accounts for the body weight, inertia and the leg thrust, while only having two degrees of freedom: the vertical and the longitudinal horizontal displacement. The natural voluntary takeoff of a fruitfly is considered as reference. The parameters of the model are then varied to explore possible effects of interaction between the flapping-wing model and the ground plane. These modified takeoffs include cases with decreased leg thrust parameter, and/or with periodic wing kinematics, constant body pitch angle. The results show that the ground effect during natural voluntary takeoff is negligible. In the modified takeoffs, when the rate of climb is slow, the difference in the aerodynamic forces due to the interaction with the ground is up to 6%. Surprisingly, depending on the kinematics, the difference is either positive or negative, in contrast to the intuition based on the helicopter theory, which suggests positive excess lift. This effect is attributed to unsteady wing-wake interactions. A similar effect is found during hovering.