Corrigendum: An analytical model and scaling of chordwise flexible flapping wings in forward flight (2016 Bioinspir. Biomim. 12 016006)

In the main paper, we have used the following expression for Theodorsen's lift equation [1]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L=&~\frac{\pi {{\rho }_{f}}{{c}^{2}}}{4}\left\{-\ddot{h}+U{{{\dot{\alpha }}}_{a}}-c\left(\frac{1}{2}-\frac{a}{c} \right){{{\ddot{\alpha }}}_{a}} \right\}\nonumber \\ & +\pi {{\rho }_{f}}UcC\left(k \right)\left\{-\dot{h}+U{{\alpha }_{a}}+c\left(\frac{3}{4}-\frac{a}{c} \right){{{\dot{\alpha }}}_{a}} \right\}\nonumber \end{align} \tag{ c1 }$$

as shown in section 2.3, equation (3). However, the original form of Theodorsen's lift equation is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{L}_{{\rm Theodorsen}}}=&~\frac{\pi {{\rho }_{f}}{{c}^{2}}}{4}\left\{-\ddot{h}+U{{{\dot{\alpha }}}_{a}}+c\left(\frac{1}{2}-\frac{a}{c} \right){{{\ddot{\alpha }}}_{a}} \right\}\nonumber \\ & +\pi {{\rho }_{f}}UcC\left(k \right)\left\{-\dot{h}+U{{\alpha }_{a}}+c\left(\frac{3}{4}-\frac{a}{c} \right){{{\dot{\alpha }}}_{a}} \right\},\nonumber \end{align} \tag{ c2 }$$

where the coefficient of the pitch acceleration term ${{\ddot{\alpha }}_{a}}$ has a reversed sign. This ${{\ddot{\alpha }}_{a}}$ term plays an important role in this study as the influence of the wing deformation on the resulting lift is accounted for by considering the passive pitch due to wing deformation, such that ${{\alpha }_{a}}\approx -{{w}_{{\rm TE}}}~/c$, where w_TE is the trailing-edge displacement relative to the leading-edge and c is considered as the length of the elastic flat plate in the present study. The pivot location a is considered at the leading edge of the flat plate which is a  =  0.

However, the use of the original form of Theodorsen's lift equation (equation (c2)) results in a normalized relative trailing-edge wing deformation amplitude w_a/h_a as shown in figure C1(a). The correlation shown in figure C1(a) is much worse compared to the agreement shown in the main paper (figure 1(a) in the main paper; also figure C1(b)). The solution in figure C1(b) was obtained with equation (c1).

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In this corrigendum, we show that the modified Theodorsen's equation (equation (c1)), which we have used in the main paper, needs to be used in place of the original Theodorsen's lift model to accurately capture the considered fluid-structure interaction.

Theodorsen's theory [3] assumes small amplitude motions and linearized potential flow solutions. Therefore, the flow oscillations that are induced by the wing motion and the wake and their effects on the pressure variations on the airfoil are neglected in the original Theodorsen's model. However, the high-fidelity computations [2] show that the induced flow oscillations are not small. In fact, the flow velocity in the transverse direction ${{U}_{y}}$ scales with the relative trailing edge displacement ${{w}_{{\rm TE}}}$ as shown in figure C2. The flow acceleration amplitude is estimated by calculating $\left(\max \left({{{\dot{U}}}_{y}} \right)-\min \left({{{\dot{U}}}_{y}} \right) \right)/2$ from the high-fidelity numerical results [2] and taking the average within the plunge range at the midchord of the flat plate, which is the center of pressure of the added mass force. The phase of flow acceleration is obtained by considering the first order harmonic representation of ${{\dot{U}}_{y}}$.

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To account for the combined effect of flow induced and body oscillations, we consider the added mass F_am as derived by Brennen [4] as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{F}_{{\rm am}}}=-{{M}_{ij}}{{\dot{W}}_{j}}+\left({{M}_{ij}}+{{\rho }_{f}}{{V}_{d}}{{\delta }_{ij}} \right){{\dot{U}}_{j}},\nonumber \end{align} \tag{ c3 }$$

where M_ij is the added mass tensor that represents the inertia of the potential flow, W_j is the velocity of the body, ρ_f is the density of the fluid, V_d is the volume of the fluid displaced by the body, and U_j is the fluid velocity.

In this study, the volume of an ideal flat plate is zero and, hence, the volume of the fluid displaced by the flat plate is zero (${{V}_{d}}=0$) [5]. The apparent mass of the flat plate M_ij in the transverse direction is $\pi {{\rho }_{f}}{{c}^{2}}/4$ and the vertical acceleration of the body is ${{\dot{W}}_{j}}=\ddot{h}+c\left(\frac{1}{2}-\frac{a}{c} \right){{\ddot{\alpha }}_{a}}$. The negative sign for the acceleration of the body is due to the direction considered where plunge h is negative in downward direction and ${{\alpha }_{a}}\approx -{{w}_{{\rm TE}}}/c$. Then, evaluating equation (c3) for the considered motions and including the circulatory effects with pivot location a  =  0 yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{L}_{{\rm full}}}=&~\frac{\pi {{\rho }_{f}}{{c}^{2}}}{4}\left\{-\ddot{h}+U{{{\dot{\alpha }}}_{a}}+\frac{c}{2}{{{\ddot{\alpha }}}_{a}}+{{{\dot{U}}}_{y}} \right\}\nonumber \\ & +\pi {{\rho }_{f}}U~cC\left(k \right)\left\{-\dot{h}+U{{\alpha }_{a}}+\frac{3c}{4}{{{\dot{\alpha }}}_{a}} \right\}.\nonumber \end{align} \tag{ c4 }$$

Compared to the original form of the Theodorsen (equation (c2)), there is an additional term with ${{\dot{U}}_{y}}$ which represents the induced flow oscillation effects. The strong agreement between the transverse flow acceleration ${{\dot{U}}_{y}}$ and relative trailing edge acceleration ${{\ddot{w}}_{{\rm TE}}}$ as shown in figure C2 suggests that we can model the flow acceleration as ${{\dot{U}}_{y}}\approx {{\ddot{w}}_{{\rm TE}}}=-c{{\ddot{\alpha }}_{a}}$. Then, the lift due to both the body and flow oscillations, given by equations (c3) and (c4) becomes equation (c1), which is what we have used in the main paper.

This discussion suggests that the considered fluid-structure interaction is also affected by the flow oscillations induced by the body motion and the wake. This higher order effect may not be neglected for motions with relatively large deformations. The application of Brennen's added mass (equation (c3)) leads to the lift model used in main paper in equation (3) in section 2.3. The rest of the results, discussion and conclusion remain unaltered in the main paper.