Abstract
A novel lifting line formulation is presented for the quasi-steady aerodynamic evaluation of insect-like wings in hovering flight. The approach allows accurate estimation of aerodynamic forces from geometry and kinematic information alone and provides for the first time quantitative information on the relative contribution of induced and profile drag associated with lift production for insect-like wings in hover. The main adaptation to the existing lifting line theory is the use of an equivalent angle of attack, which enables capture of the steady non-linear aerodynamics at high angles of attack. A simple methodology to include non-ideal induced effects due to wake periodicity and effective actuator disc area within the lifting line theory is included in the model. Low Reynolds number effects as well as the edge velocity correction required to account for different wing planform shapes are incorporated through appropriate modification of the wing section lift curve slope. The model has been successfully validated against measurements from revolving wing experiments and high order computational fluid dynamics simulations. Model predicted mean lift to weight ratio results have an average error of 4% compared to values from computational fluid dynamics for eight different insect cases. Application of an unmodified linear lifting line approach leads on average to a 60% overestimation in the mean lift force required for weight support, with most of the discrepancy due to use of linear aerodynamics. It is shown that on average for the eight insects considered, the induced drag contributes 22% of the total drag based on the mean cycle values and 29% of the total drag based on the mid half-stroke values.
Highlights
The classical lifting line theory (LLT), developed by Prandtl a century ago provided the first satisfactory analytical treatment for the evaluation of the aerodynamics of a finite wing [1,2,3,4,5,6]
The solutions delivered by the LLT are closed form and they are many orders of magnitude faster to evaluate compared to higher order computational methods; they are able to provide deep insight into how different wing parameters affect the aerodynamic performance [6]
The developed modelling capability provides a framework to adapt the original LLT for hovering flight and opens the door for simplified yet accurate modelling of 3d lifting surfaces at different operating conditions
Summary
The classical lifting line theory (LLT), developed by Prandtl a century ago provided the first satisfactory analytical treatment for the evaluation of the aerodynamics of a finite wing [1,2,3,4,5,6]. The Kutta-Joukowski theorem can be applied at each wing section, which is assumed to behave as a 2d wing at a modified angle of attack referred to as the effective angle of attack. This concept led Prandtl to his well-known linear equation governing the circulation on a finite lifting surface, which will be formally introduced later in this work in section ‘LLT fundamental equations’. The most well-known solution methodology is that presented by Glauert [8] who provided a solution in the form of an infinite Fourier sine series with the series coefficients obtained from the collocation method
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