A CFD-informed quasi-steady model of flapping wing aerodynamics

Abstract

Aerodynamic performance and agility during flapping flight are determined by the combination of wing shape and kinematics. The degree of morphological and kinematic optimisation is unknown and depends upon a large parameter space. Aimed at providing an accurate and computationally inexpensive modelling tool for flapping-wing aerodynamics, we propose a novel CFD (computational fluid dynamics)-informed quasi-steady model (CIQSM), which assumes that the aerodynamic forces on a flapping wing can be decomposed into the quasi-steady forces and parameterised based on CFD results. Using least-squares fitting, we determine a set of proportional coefficients for the quasi-steady model relating wing kinematics to instantaneous aerodynamic force and torque; we calculate power with the product of quasi-steady torques and angular velocity. With the quasi-steady model fully and independently parameterised on the basis of high-fidelity CFD modelling, it is capable of predicting flapping-wing aerodynamic forces and power more accurately than the conventional blade element model (BEM) does. The improvement can be attributed to, for instance, taking into account the effects of the induced downwash and the wing tip vortex on the force generation and power consumption. Our model is validated by comparing the aerodynamics of a CFD model and the present quasi-steady model using the example case of a hovering hawkmoth. It demonstrates that the CIQSM outperforms the conventional BEM while remaining computationally cheap, and hence can be an effective tool for revealing the mechanisms of optimization and control of kinematics and morphology in flapping-wing flight for both bio-flyers and unmanned air systems.

Keywords: Biological Fluid Dynamics; Computational methods; Swimming/flying.

Figure 1

(a) Definition of global coordinate system, stroke plane angle, body angle and flapping angles: positional, feathering and elevation angle. (b, c) Definition of wing-fixed coordinate system viewed with (b) wing planform and (c) cross section.

Figure 2

Parameterised lift-force coefficients as a function of geometric angle of attack.

Figure 3

Definition of translational and rotational torque.

Figure 4

Computational model of a hovering hawkmoth. (a) Wing-body morphological model for CFD and wing models for BEM with chordwise and spanwise blades, and (b) kinematic models of a hovering hawkmoth. (c) Wing tip trajectories and wing attitude of a realistic (black) and a modified wing kinematics (purple). (d -h) Modified flapping angles.

Figure 5

Aerodynamic forces and power predicted by CFD with realistic wing kinematics. (a-c) Time-courses of aerodynamic force with respect to (a) global coordinate system and (b) wing fixed coordinate system, and (c) aerodynamic power. The shaded area corresponds to the downstroke.

Figure 6

Comparisons of (a) mean vertical forces and (b) mean aerodynamic powers predicted by CFD (horizontal axis) and BEM (vertical axis).

Figure 7

Comparison of (a) aerodynamic vertical force and (b) aerodynamic power simulated by CFD (dashed black) and CIQSM (solid black). Coloured components sum to solid black lines.

Figure 8

Time-courses of (i) aerodynamic forces and (ii) flow structure at several time instants A-E by (a) realistic kinematics and (b) 10 % delayed rotation. While the LEV highlighted by red line is kept attached on the wing with realistic wing kinematics through the downstroke, the LEV is detached at early downstroke (B) by 10 % delayed rotation. Vortex wake identified by iso-surfaces at Q=0.5, coloured by spanwise vorticity.

Figure 9

Error estimation of the quasi-steady predictions illustrated by the probability distributions of the estimated error and PCC of (a) mean vertical aerodynamic force and (b) mean aerodynamic power by the CIQSM. (i) Mean error of the CIQSM predictions compared with high-fidelity CFD simulations. (ii) PCC of mean aerodynamic vertical forces or power between CFD and quasi-steady predictions. Vertical black lines show the prediction of a BEM, which is significantly less accurate in all cases.

Figure 10

Comparisons of (a) mean vertical forces and (b) mean aerodynamic powers predicted by CFD (horizontal axis) and CIQSM (vertical axis) constructed using 23 input cases. Mean standard deviation of mean vertical forces and powers are 0.0769 mN and 0.28 mW, respectively, which are smaller than the size of the plotted points. A grey circle, a triangle and a square show the predictions by realistic, 10 % delayed and 10 % advanced rotation, respectively.

Figure 11

Coefficients for CIQSM calculated using 23 input cases. (a-c) The lift, drag and normal force coefficients for CIQSM. The experimental values by Usherwood & Ellington (2002) are shown for comparisons. (d-e) Rotational lift and drag coefficients. The C_RL by Zheng et al. (2013) and maximum drag coefficients by Usherwood & Ellington (2002) are shown for comparisons. (f -h) Ratio of the shape dependent coefficients for added mass forces (f) I_am1, (g) I_am2 and (h) I_am3 by CIQSM and BEM. Shaded region around lines and error bars display the standard deviation.

Figure 12

Optimized wing kinematics and its aerodynamic performances. (a) Angular motion of the cross section during down and upstroke. Circles show the leading edges of the cross sections. (b) Time-courses of the optimized flapping angles. (c-d) Time-courses of the aerodynamic vertical force and aerodynamic power with optimized wing kinematics calculated by the CIQSM and CFD. The mean values are shown by the horizontal lines in each panel.