Flapping Wing Aerodynamic Model

Rapid prediction of aerodynamic forces and power for rigid flapping wings in hover

Estimate flapping-wing forces and power rapidly. Our quasi-steady aerodynamic model predicts lift, drag and aerodynamic power for rigid wings in hover, helping researchers and engineers investigate aerodynamic trends, compare wing shapes and motion profiles, and guide detailed CFD studies or experiments.

The model accounts for Reynolds number and selected effects of Rossby number, aspect ratio and taper ratio, without requiring case-by-case coefficient retuning within its intended scope. Its low computational cost supports both aerodynamic research and preliminary flapping-wing MAV design.

Parameter ranges investigated

Reynolds number: 100 to inviscid
Rossby number: 1.73 to 7
Aspect ratio: 1 to 7
Taper ratio: 0.25 to 4

These ranges were investigated in separate parameter studies, rather than across all possible combinations.

How it works
The model splits the aerodynamic force into translational, rotational and added-mass components. Its coefficients depend on Reynolds number, with correction factors for Rossby number, aspect ratio and taper ratio.

Accuracy
For the simple harmonic cases, mean power agreed with CFD within 5.5%. For the insect-inspired cases, errors in cycle-averaged power predictions were larger, ranging from approximately 10% to 17%. Cycle-averaged lift and drag coefficients generally agreed within approximately 10%, although mean lift was overpredicted by more than 20% in the hawkmoth case. That discrepancy was associated with the wing’s trailing-edge kink and early shedding of the initial leading-edge vortex.

Limitations
The model is for rigid, flat, thin wings in hover. It does not explicitly resolve vortex growth, vortex shedding or wing–wake interaction, which can add discrepancies, particularly for kinematics with rapid stroke reversal. Significant errors may occur for wings with thicknesses of 5% chord or greater. Forward flight and wing deformation are outside the scope demonstrated. The model can be built in a spreadsheet or as a subroutine in your own code, and detailed CFD or experiments remain valuable for checking promising designs.

Why it matters

The model gives rapid performance estimates for different wings and flow conditions within its scope. It can be built in a spreadsheet or as a subroutine in your own code and coupled with optimisation or control algorithms, which makes it useful for preliminary flapping-wing MAV design. Detailed CFD or experiments remain valuable for checking promising designs.

Read the paper
Lee, Lua, Lim and Yeo (2016), Bioinspiration & Biomimetics, 11(3), 036005. DOI 10.1088/1748-3190/11/3/036005. Use the corrected summary equations in the corrigendum: Bioinspiration & Biomimetics 11, 049501 (2016), corrigendum to 11, 036005. DOI: 10.1088/1748-3190/11/4/049501.

Figure 1: The x-r coordinate system. A single blade element (dr) is shown, where the local chord length c(r) is a function of r. R and b refer to the tip radius and span, respectively.

Figure 2: Instantaneous position of the wing, defined by φ, θ and ψ. The effective angle of attack αₑ is measured relative to the instantaneous direction of motion.

Tested across different wings and motions

We compared the model with CFD and experimental force measurements for several wing shapes and flapping motions, including fruit fly, honey bee and hawkmoth wings. It reproduces the main trends in lift, drag and power through the flapping cycle, with larger deviations in some cases (see the Figures below).

Publication

Y. J. Lee, K. B. Lua, T. T. Lim and K. S. Yeo. A Quasi-Steady Aerodynamic Model for Flapping Flight with Improved Adaptability. Bioinspiration & Biomimetics, Vol. 11, No. 3, April 2016, Article 036005. DOI 10.1088/1748-3190/11/3/036005.

Please refer to the published corrigendum for the correct version of the summary equations. DOI 10.1088/1748-3190/11/4/049501.

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