Keywords: unsteady vortex lattice method, flapping wings, B-splines, shape optimisation.

In this paper, shape optimisation of flapping wings in forward flight is considered. This analysis is performed by combining a local gradient-based optimizer with the unsteady vortex lattice method (UVLM). Although the UVLM applies only to incompressible, inviscid flows where the separation lines are known a priori, Persson et al. [1] showed through a detailed comparison between UVLM and higher-fidelity computational fluid dynamics methods for flapping flight that the UVLM schemes produce accurate results for attached flow cases and even remain trend-relevant in the presence of flow separation. As such, they recommended the use of an aerodynamic model based on UVLM to perform preliminary design studies of flapping wing vehicles Unlike standard computational fluid dynamics schemes, this method requires meshing of the wing surface only and not of the whole flow domain [2]. From the design or optimisation perspective taken in our work, it is fairly common (and sometimes entirely necessary, as a result of the excessive computational cost of the highest fidelity tools such as Navier-Stokes solvers) to rely upon such a moderate level of modelling fidelity to traverse the design space in an economical manner. The objective of the work, described in this paper, is to identify a set of optimised shapes that maximise the propulsive efficiency, defined as the ratio of the propulsive power over the aerodynamic power, under lift, thrust, and area constraints.

The shape of the wings is modelled using B-splines, a technology used in the computer-aided design (CAD) field for decades. This basis can be used to smoothly discretize wing shapes with few degrees of freedom, referred to as control points. The locations of the control points constitute the design variables.

The results suggest that changing the shape yields significant improvement in the performance of the flapping wings. The optimisation pushes the design to "bird-like" shapes with substantial increase in the time-averaged thrust, while the average aerodynamic power is increased. Furthermore, increasing the number of variables (i.e., providing the wing shape with greater degrees of spatial freedom) is observed to enable superior designs. To gain a better understanding of the reasons for which the obtained optimised shapes produce efficient flapping flights, the wake pattern and its vorticity strength are examined. This work described in this paper should facilitate better guidance for shape design of engineered flying systems.

1

P. Persson, D. Willis, J. Peraire, "Numerical Simulation of Flapping Wings Using a Panel Method and High-Order Navier-Stokes Solver", International Journal for Numerical Methods in Engineering, 89, 1296-1316, 2012.

2

J. Katz, A. Plotkin, "Low-Speed Aerodynamics", Cambridge University Press, MA, 2001.

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