2. Derivation

The general expression of the classical added-mass force, $\boldsymbol {F}$, for any two-dimensional body with a piecewise-continuous outer contour, $S$, undergoing arbitrary motion in an unbounded incompressible fluid is given as (Limacher et al. Reference Limacher, Morton and Wood2018)

(2.1)\begin{equation} \boldsymbol{F} ={-}\rho \frac{\textrm{d}}{\textrm{d} t} ( \boldsymbol{P}_\phi + \boldsymbol{P}_b ), \end{equation}

where $\rho$ is the fluid density, $\textrm {d} / \textrm {d} t$ denotes differentiation with respect to time, and $\boldsymbol{P}_\phi$ and $\boldsymbol{P}_b$ are the potential impulse and the body-volume impulse, respectively. They are defined as

(2.2)\begin{gather} \boldsymbol{P}_\phi ={-} \oint_{S} \boldsymbol{n} \phi^\prime \, \textrm{d} S, \end{gather}

(2.3)\begin{gather}\boldsymbol{P}_b ={-}\boldsymbol{u}_c A, \end{gather}

where $\boldsymbol {n}$ is the outward-facing normal on $S$ (pointing into the fluid), $\boldsymbol {u}_c$ is the instantaneous velocity of the body centroid and $A$ is the cross-sectional area of the body.

The quantity $\phi ^\prime$ is the scalar potential describing the irrotational component of the velocity field according to $\boldsymbol {u}_\phi ^\prime = \boldsymbol {\nabla } \phi ^\prime$, where $\boldsymbol {u}_\phi ^\prime$ is defined in a frame of reference relative to the body centroid (prime superscripts denote the body-fixed frame of reference, consistent with the notation of Limacher et al. (Reference Limacher, Morton and Wood2018)). Although (2.2) only requires $\phi ^\prime$ to be known on $S$, it must be obtained as the solution to the Laplace equation on a two-dimensional domain. When solving for $\nabla ^2 \phi ^\prime = 0$ on the fluid domain outside the body, the solution is subject to the boundary condition $\boldsymbol {\nabla } \phi ^\prime = \boldsymbol {U}_\infty - \boldsymbol {u}_c$ at infinity, where $\boldsymbol {U}_\infty$ is the steady free-stream velocity far from the body, and the impermeable boundary condition $\boldsymbol {n} \boldsymbol {\cdot } \boldsymbol {\nabla } \phi ^\prime = 0$ on $S$.

An equivalent added-mass formulation is given by Saffman (Reference Saffman1992), although he treats the scalar potential describing the velocity in the absolute frame of reference. Of course, the choice of reference frame used in the derivation is immaterial to the final results, but a useful aspect of the approach of Limacher et al. (Reference Limacher, Morton and Wood2018) is in the comparison it facilitates to the seminal work of Wu (Reference Wu1981). Wu's exact and general expression of aerodynamic force in terms of body motion, body geometry and vorticity-field evolution has instigated decades of intervening interest in what we now call vortical impulse theory, for which Wu, Ma & Zhou (Reference Wu, Ma and Zhou2015) is recommended as an authoritative reference.

One further note on boundary conditions is warranted, highlighting definitional differences within the category of classical added-mass analyses. In Limacher et al. (Reference Limacher, Morton and Wood2018), and above, the boundary condition for $\phi ^\prime$ on $S$ does not include the surface-normal velocity component due to body rotation, in contrast to other treatments of the classical added-mass force and moment, e.g. La Mantia & Dabnichki (Reference La Mantia and Dabnichki2012), Fernandez-Feria (Reference Fernandez-Feria2019). The reason for this difference in treatment is explained in the Appendix, where it is also shown to have no effect on the force expressions derived for elliptic airfoils (or for any other rotationally symmetric body).

Two coordinate systems will be employed in the present analysis: an inertial coordinate system $(X,Y)$, with the $X$- and $Y$-directions aligned stream-wise and stream-normal, respectively; and a body-fixed coordinate system $(x,y)$, with the $x$- and $y$-directions aligned with the major and minor axes of the ellipse, respectively (see figure 1). Throughout the paper, uppercase subscripts ($X$ or $Y$) refer to vector components in the inertial frame, and lowercase subscripts ($x$ or $y$) to the body-fixed frame. At any instant in time, the angle from the $x$-axis to the $X$-axis is $\theta$ (positive counterclockwise).

Figure 1. Inertial (X,Y) and body-fixed (x,y) coordinate systems used in the present analysis. Here, c is airfoil chord length (major axis length), b is airfoil thickness (minor axis length), θ is the pitch angle (positive in the sense shown) relative to the free stream, U _∞, and $\boldsymbol {u}_c$ is the velocity of the airfoil centroid (i.e. the mid-chord).

Now, it would be possible to solve for the potential impulse on the ellipse as a function of $\boldsymbol {u}_c$ and $\theta$ using standard potential flow techniques. One could, for example, take the known complex potential around a circle and relate it to the domain around an ellipse by means of a conformal map (Milne-Thomson Reference Milne-Thomson1968). Or, for any body of interest, the potential flow could be solved computationally, thereafter numerically approximating the integral in (2.2). Fortunately, such efforts can be avoided for the ellipse case, as the known added-mass forces for linear acceleration yield sufficient information when combined with (2.1)–(2.3). These known forces are as follows (Brennen Reference Brennen1982):

(2.4)\begin{gather} F_x ={-}\frac{{\rm \pi} \rho b^2}{4} \frac{\textrm{d} u_{cx}}{\textrm{d} t}, \end{gather}

(2.5)\begin{gather}F_y ={-}\frac{{\rm \pi} \rho c^2}{4} \frac{\textrm{d} u_{cy}}{\textrm{d} t}, \end{gather}

where $c$ is the airfoil chord (ellipse major axis length), $b$ is the airfoil thickness (ellipse minor axis length), and $\textrm {d} u_{cx}/ \textrm {d} t$ and $\textrm {d} u_{cy}/ \textrm {d} t$ are the chord-wise and chord-normal components of the mid-chord acceleration, respectively.

Before making use of these expressions, let us further clarify how the components of impulse are related to the components of force. The stream-normal and stream-wise forces are

(2.6)\begin{equation} \frac{1}{\rho} \begin{bmatrix} F_X \\ F_Y \end{bmatrix} ={-}\frac{\textrm{d}}{\textrm{d} t} \begin{bmatrix} P_{\phi X} \\ P_{\phi Y} \end{bmatrix} + \frac{{\rm \pi} b c}{4} \frac{\textrm{d}}{\textrm{d} t} \begin{bmatrix} u_{cX} \\ u_{cY} \end{bmatrix}, \end{equation}

where the area of an ellipse $A = {\rm \pi}bc/4$ has been substituted. It will prove more convenient to treat the chord-wise and chord-normal forces, related to the stream-normal and stream-wise forces by a rotation matrix, ${\boldsymbol{\mathsf{R}}}(\theta )$:

(2.7)\begin{equation} {\boldsymbol{\mathsf{R}}}(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}. \end{equation}

This allows us to write

(2.8)\begin{equation} \frac{1}{\rho} \begin{bmatrix} F_x \\ F_y \end{bmatrix} = {\boldsymbol{\mathsf{R}}}(\theta)\frac{\textrm{d}}{\textrm{d} t} \left\{ {\boldsymbol{\mathsf{R}}}(-\theta) \left( - \begin{bmatrix} P_{\phi x} \\ P_{\phi y} \end{bmatrix} + \frac{{\rm \pi} b c}{4} \begin{bmatrix} u_{cx} \\ u_{cy} \end{bmatrix} \right) \right\}. \end{equation}

If we apply the product rule and note that

(2.9)\begin{equation} {\boldsymbol{\mathsf{R}}}(\theta) \frac{\textrm{d} {\boldsymbol{\mathsf{R}}}(-\theta)}{\textrm{d} t} = \frac{\textrm{d} \theta}{\textrm{d} t} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \end{equation}

then (2.8) becomes

(2.10)\begin{equation} \frac{1}{\rho} \begin{bmatrix} F_x \\ F_y \end{bmatrix} ={-}\frac{\textrm{d} \theta}{\textrm{d} t} \begin{bmatrix} P_{\phi y} \\ -P_{\phi x} \end{bmatrix} - \frac{\textrm{d}}{\textrm{d} t} \begin{bmatrix} P_{\phi x} \\ P_{\phi y} \end{bmatrix} + \frac{{\rm \pi} b c}{4} \left\{ \frac{\textrm{d} \theta}{\textrm{d} t} \begin{bmatrix} u_{cy} \\ -u_{cx} \end{bmatrix} + \frac{\textrm{d}}{\textrm{d} t} \begin{bmatrix} u_{cx} \\ u_{cy} \end{bmatrix} \right\}, \end{equation}

demonstrating how rotation causes the impulse components to contribute perpendicular force components.

For rectilinear motion in still fluid, $\textrm {d} \theta / \textrm {d} t = 0$, and we can solve for the potential impulse by substituting the force expressions in (2.4) and (2.5) into (2.10) and taking the antiderivative with respect to time, yielding

(2.11a,b)\begin{equation} P_{\phi x} = \frac{\rm \pi}{4} b (b+c) u_{cx}; \quad P_{\phi y} = \frac{\rm \pi}{4} c (b+c) u_{cy}. \end{equation}

To generalize these expressions in the presence of a free-stream flow, and to allow for a variable pitch angle relative to that flow, recall that the scalar potential in (2.2) is solved for the boundary condition $\boldsymbol {\nabla } \phi ^\prime = \boldsymbol {U}_\infty - \boldsymbol {u}_c$ at infinity. Accordingly, we replace $u_{cx}$ and $u_{cy}$ with $u_{cx} - U_\infty \cos \theta$ and $u_{cy} - U_\infty \sin \theta$, respectively, yielding

(2.12)\begin{gather} P_{\phi x} = \frac{\rm \pi}{4} b (b+c) ( u_{cx} - U_\infty \cos \theta), \end{gather}

(2.13)\begin{gather}P_{\phi y}=\frac{\rm \pi}{4}c(b+c)(u_{cy}-U_\infty\sin\theta). \end{gather}

Substituting (2.12) and (2.13) back into (2.10), and assuming the free-stream velocity to be steady, i.e. $\textrm {d} U_\infty /\textrm {d} t = 0$, we obtain the following after differentiation and algebraic manipulation:

(2.14)\begin{gather} F_x = \frac{\rm \pi}{4} \rho c^2 \left\{U_\infty \frac{\textrm{d} \theta}{\textrm{d} t}\left(1 - \frac{b^2}{c^2} \right) \sin \theta - u_{cy} \frac{\textrm{d} \theta}{\textrm{d} t} - \frac{b^2}{c^2} \frac{\textrm{d} u_{cx}}{\textrm{d} t} \right\}, \end{gather}

(2.15)\begin{gather}F_y = \frac{\rm \pi}{4} \rho c^2 \left\{ U_\infty \frac{\textrm{d} \theta}{\textrm{d} t} \left(1 - \frac{b^2}{c^2} \right) \cos \theta + u_{cx} \frac{b^2}{c^2} \frac{\textrm{d} \theta}{\textrm{d} t} - \frac{\textrm{d} u_{cy}}{\textrm{d} t} \right\}. \end{gather}

Note that when $\textrm {d} \theta / \textrm {d} t = 0$, these expressions collapse to the original ones given in (2.4) and (2.5), even when $U_\infty \neq 0$. When normalized by $1/2 \rho c U_\infty ^2$, the force coefficients become

(2.16)\begin{gather} C_x = \frac{\rm \pi}{2} \left\{ \frac{\textrm{d} \theta}{\textrm{d} t^*} \left(1-b^{*2} \right) \sin \theta - u_{cy}^* \frac{\textrm{d} \theta}{\textrm{d} t^*} - b^{*2} \frac{\textrm{d} u_{cx}^*}{\textrm{d} t^*} \right\}, \end{gather}

(2.17)\begin{gather}C_y = \frac{\rm \pi}{2} \left\{ \frac{\textrm{d} \theta}{\textrm{d} t^*} \left(1 - b^{*2} \right) \cos \theta + u_{cx}^* b^{*2} \frac{\textrm{d} \theta}{\textrm{d} t^*} - \frac{\textrm{d} u_{cy}^*}{\textrm{d} t^*} \right\}, \end{gather}

where $t^* = t U_\infty /c$, $b^* = b/c$ and the velocities have been normalized by $U_\infty$.

Equations (2.16) and (2.17) provide the promised general expressions for the added-mass force on an elliptic airfoil. One colleague has remarked (and I agree) that it is surprising not to find this result long established. It is of wide applicability and seems unlikely to have escaped the interest of the early pioneers of aerodynamics. One might speculate that it had to await the tidy expression of the added-mass force in terms of a surface integral, as in (2.2) and in the monograph of Saffman (Reference Saffman1992). Earlier derivations of the added-mass force have treated the changing kinetic energy in the unbounded fluid domain, calculating the work done on the fluid to determine the force, e.g. Brennen (Reference Brennen1982) and Lighthill (Reference Lighthill1986). That approach suffers from the non-convergence of the energy integral when the velocity is non-zero in the far field, and thus the general effects of rotation would be difficult to capture in closed form.

In the following section, it will be shown how (2.16) and (2.17) recover Theodorsen's (Reference Theodorsen1935) non-circulatory lift on thin airfoils as a special case.

3. Generalizing the non-circulatory lift of Theodorsen (1935)

Theodorsen (Reference Theodorsen1935) derived the lift on an oscillating thin airfoil using potential flow theory. He assumed small-amplitude motions consisting of pitching about a fixed point on the chord line, located at $x_p$ relative to the mid-chord ($x_p < 0$ for a pivot point ahead of the mid-chord), and plunging of the pivot point transversely to the free-stream direction. (He also considered a third degree of freedom with which we are not herein concerned: a pitching aileron mounted to the trailing edge.) The vertical velocity of the pivot point is denoted by $V_p$, normalized as $V_p^* = V_p/U_\infty$.

Theodorsen's non-circulatory component of the lift coefficient, $C_L$, expressed in the notation of the present work, is

(3.1)\begin{equation} C_L = \frac{\rm \pi}{2} \left\{ \frac{\textrm{d} \theta}{\textrm{d} t^*} - \frac{\textrm{d} V_p^*}{\textrm{d} t^*} - x_p^* \frac{\textrm{d} ^2 \theta}{\textrm{d} t^{*2}} \right\}.\end{equation}

When attempting to generalize for large $\theta$, recent literature has suggested that Theodorsen's expression is rightly applied to the chord-normal rather than stream-normal direction (Polet, Rival & Weymouth Reference Polet, Rival and Weymouth2015; Otomo et al. Reference Ōtomo, Henne, Mulleners, Ramesh and Viola2021), which is now corroborated. Setting $u_{cy}^* = V_p^* \cos \theta + x_p^* \textrm {d} \theta / \textrm {d} t^*$, and $b^* = 0$ in (2.17), we obtain the general expression for chord-normal force coefficient on a thin airfoil:

(3.2)\begin{equation} C_y = \frac{\rm \pi}{2} \left\{\cos\theta \frac{\textrm{d} \theta}{\textrm{d} t^*}+ V_p^* \sin\theta \frac{\textrm{d} \theta}{\textrm{d} t^*} - \frac{\textrm{d} V_p^*}{\textrm{d} t^*} \cos\theta - x_p^* \frac{\textrm{d} ^2 \theta}{\textrm{d} t^{*2}} \right\}.\end{equation}

Invoking the small-angle approximations $\sin \theta \approx \theta$ and $\cos \theta \approx 1$, we recover something similar to (3.1):

(3.3)\begin{equation} C_y \approx \frac{\rm \pi}{2} \left\{ \frac{\textrm{d} \theta}{\textrm{d} t^*} + V_p^* \theta \frac{\textrm{d} \theta}{\textrm{d} t^*} - \frac{\textrm{d} V_p^*}{\textrm{d} t^*} - x_p^* \frac{\textrm{d} ^2 \theta}{\textrm{d} t^{*2}} \right\}. \end{equation}

The second term above is absent in Theodorsen's formulation, but an order-of-magnitude analysis justifies its neglect. For harmonic motions with a pitch amplitude of $\theta _0$, we have the first term of order $O(\textrm {d} \theta / \textrm {d} t^*) \sim O(k \theta _0)$, where $k$ is the reduced frequency. The second term is of order $O(k^2 h_0^* \theta _0^2)$, where $h_0^*$ is the chord-normalized plunge amplitude, such that this term may have a magnitude similar to or greater than that of the first if $O(h_0^*) \gg 1$ or $O(k) \gg 1$. However, the third term is of order $O(k^2 h_0^*)$, which will always be greater than that of the second term so long as $\theta _0 \ll 1$. Thus, for any plunge amplitude and frequency, Theodorsen's expression is a good approximation for small pitch angles.

Remarkably, at least for some cases, Theodorsen remains an adequate approximation of the chord-normal added-mass force even at high pitch angles. As an example, we take the motion of a high-efficiency flapping-wing propulsion experiment from Van Buren, Floryan & Smits (Reference Van Buren, Floryan and Smits2019) and increase the pitch amplitude from $15^\circ$ to $75^\circ$ in increments of $20^\circ$. The combined pitching and plunging motion is characterized by the following five normalized parameters: chord-normalized plunge amplitude, $h_0^*$; pitch amplitude, $\theta _0$; pivot location lying on the chord-line, $x_p^*$; normalized frequency, $f^* = fc/U_\infty$; and the phase difference between the plunging and pitching motions, $\varPhi$, defined such that if the plunge position is

(3.4)\begin{equation} Y_p^* = h_0^* \sin (2{\rm \pi} f^* t^*), \end{equation}

then

(3.5)\begin{equation} \theta = \theta_0 \sin (2{\rm \pi} f^* t^* + \varPhi). \end{equation}

The kinematic parameters for these test cases are given in table 1.

Table 1. Parameters of harmonic motion for thin-airfoil ($b^* = 0$) test cases to be compared in figures 2(a) and 2(b): $h_0^*$ is chord-normalized plunge amplitude, $\theta _0$ is pitch amplitude, $\varPhi$ is the phase difference between the plunge and pitch motions, $x_p^*$ is chord-normalized pivot point location and $f^*$ is normalized frequency.

Figure 2. (a) Chord-normal added-mass force ($C_y$) for thin airfoils from (3.2) and Theodorsen's $C_L$ from (3.1) versus period-normalized time $t/T$ for the pitching and plunging motions characterized in table 1. (b) Stream-normal added-mass force ($C_Y$) from (3.6), and $C_L$ versus $t/T$ for the same airfoil motions.

Figure 2(a) shows that the chord-normal force trend over one period of motion, $T$, is well captured by Theodorsen, although instantaneous deviations are notable; for the extreme case of $\theta _0 = 75^\circ$, the maximum instantaneous difference between $C_y$ and $C_L$ is $23\,\%$ of the root-mean-square value of $C_y$.

Qualitative and quantitative agreement between Theodorsen and the stream-normal added-mass force,

(3.6)\begin{equation} C_Y = C_y \cos \theta - C_x \sin \theta, \end{equation}

is poorer for these same pitch amplitudes, as shown in figure 2(b). This is due to the chord-wise component of the added-mass force, $C_x$, which is often overlooked but does appear in some general treatments, e.g. Sedov (Reference Sedov1965), Limacher et al. (Reference Limacher, Morton and Wood2018) and Van Buren et al. (Reference Van Buren, Floryan and Smits2019). With the same substitutions and simplifications as were made to obtain $C_y$ for thin airfoils, (2.16) yields

(3.7)\begin{equation} C_x = \frac{\rm \pi}{2} \frac{\textrm{d} \theta}{\textrm{d} t^*} \left\{ \sin \theta - V_p^* \cos \theta - x_p^* \frac{\textrm{d} \theta}{\textrm{d} t^*} \right\},\end{equation}

which vanishes as expected for $\textrm {d} \theta / \textrm {d} t^* = 0$.

4. Recovering the circular cylinder case

For the limiting case of $b^* =1$, our general force equations must reduce to the well-known added-mass force on a circular cylinder. Because of its rotational symmetry, the force on a circular cylinder, expressed in the inertial frame, can have no dependence on $\textrm {d} \theta / \textrm {d} t^*$; that is, a rotation of the cylinder about its centre has no effect on the surrounding irrotational flow. Setting $b^*=1$, the first rotation-dependent terms in (2.16) and (2.17) vanish, and we are left with

(4.1)\begin{equation} \begin{bmatrix} C_x \\ C_y \end{bmatrix} ={-}\frac{\rm \pi}{2} \frac{\textrm{d} \theta}{\textrm{d} t^*} \begin{bmatrix} u_{cy}^* \\ - u_{cx}^* \end{bmatrix} + \frac{\rm \pi}{2} \frac{\textrm{d}}{\textrm{d} t^*} \begin{bmatrix} u_{cx} \\ u_{cy} \end{bmatrix}. \end{equation}

The remaining dependence on $\textrm {d} \theta / \textrm {d} t^*$ is due only to the rotation of the reference frame. Reversing the procedure carried out in (2.6)–(2.10), (4.1) becomes

(4.2)\begin{equation} \begin{bmatrix} C_x \\ C_y \end{bmatrix} ={-}\frac{\rm \pi}{2}{\boldsymbol{\mathsf{R}}}(\theta)\frac{\textrm{d}}{\textrm{d} t^*} \left\{ {\boldsymbol{\mathsf{R}}}(-\theta) \begin{bmatrix} u_{cx}^* \\ u_{cy}^* \end{bmatrix} \right\} , \end{equation}

from which, after rotation, we recover the expected force coefficients expressed in the inertial frame:

(4.3)\begin{equation} \begin{bmatrix} C_X \\ C_Y \end{bmatrix} ={-}\frac{\rm \pi}{2} \frac{\textrm{d}}{\textrm{d} t^*} \begin{bmatrix} u_{cX}^* \\ u_{cY}^* \end{bmatrix}. \end{equation}

5. The effect of thickness for pitching and plunging airfoils

The general expressions given in (2.16) and (2.17) allow us to discuss the effect of airfoil thickness on the added-mass force for some canonical harmonic motions within a steady free-stream flow: pure pitching, pure plunging, and combined pitching and plunging.

Pure plunging is merely a subset of rectilinear translation, for which $\textrm {d} \theta / \textrm {d} t^* = 0$ and (2.16) and (2.17) reduce to normalized equivalents of the previously known added-mass forces given in (2.4) and (2.5). Thickness has no effect on the chord-normal added-mass force in this case, and gives rise to a chord-wise component that grows as $b^{*2}$.

For pure pitching about the mid-chord, $x_p^* = 0$ and $u_{cx}^* = u_{cy}^* = 0$, and the force coefficients reduce to the first terms on the right-hand sides of (2.16) and (2.17). This reveals that increasing thickness will reduce both $C_x$ and $C_y$ by the same factor, and the greatest force magnitude for any given $\theta = \theta (t^*)$ will be achieved when $b^* = 0$.

For pure pitching about a point lying on the chord-line at $x_p^* \neq 0$, $u_{cy} \neq 0$ but $u_{cx}* = du_{cx}/dt^* = 0$. Accordingly, we once again find that increasing thickness can only decrease the force magnitude in both directions by modifying the first terms on the right-hand sides of (2.16) and (2.17) .

For the general three-degree-of-freedom case, increasing thickness can either increase or decrease the instantaneous forces, but, for any given motion of the chord-line, the difference in force between a thin and a thick airfoil will always be proportional to $b^{*2}$. That is, if we express the force coefficients for a given motion as a function of $b^*$, i.e. $C_x = C_x(b^*), C_y = C_y(b^*)$, then the instantaneous thickness effects can be stated as

(5.1)\begin{gather} C_x(b^*) - C_x(0) ={-}\frac{\rm \pi}{2} b^{*2} \left( \frac{\textrm{d} \theta}{\textrm{d} t^*}\sin\theta + \frac{\textrm{d} u_{cx}^*}{\textrm{d} t^*} \right), \end{gather}

(5.2)\begin{gather}C_y(b^*) - C_y(0) ={-}\frac{\rm \pi}{2} b^{*2} \frac{\textrm{d} \theta}{\textrm{d} t^*} ( \cos\theta - u_{cx}^* ). \end{gather}

In the static coordinate system, the thickness effects take the form

(5.3)\begin{gather} C_X(b^*) - C_X(0) ={-}b^{*2} f_X \left(\theta, u_{cx}^*, \frac{\textrm{d} \theta}{\textrm{d} t^*}, \frac{\textrm{d} u_{cx}^*}{\textrm{d} t^*}\right), \end{gather}

(5.4)\begin{gather}C_Y(b^*) - C_Y(0) ={-}b^{*2} f_Y \left(\theta, u_{cx}^*,\frac{\textrm{d} \theta}{\textrm{d} t^*}, \frac{\textrm{d} u_{cx}^*}{\textrm{d} t^*}\right), \end{gather}

where the functions $f_X$ and $f_Y$ can yield positive or negative values depending on the instantaneous values of the independent kinematic parameters $u_{cx}^* = u_{cx}^*(t^*)$ and $\theta = \theta (t^*)$ and their derivatives. Interestingly, thickness effects are independent of $u_{cy}^*$ in general.

To give an idea of the possible relative magnitude of the thickness effects, two practical example cases of pitching and plunging airfoils are presented: a high-efficiency case from the propulsion study of Van Buren et al. (Reference Van Buren, Floryan and Smits2019), and the best-performing case from the flapping-wing turbine study of Kinsey & Dumas (Reference Kinsey and Dumas2008). The kinematic parameters of these harmonic motions, as previously introduced in § 3, are listed in table 2. In the design of flapping-wing propulsors and flapping-wing turbines, the stream-wise and stream-normal forces ($C_X$ and $C_Y$) are often the most relevant, and these are plotted in figure 3. The effect of thickness can be characterized by the ratios of peak force magnitudes, defined as

(5.5a,b)\begin{equation} r_X = \frac{\max (|C_X(b^*)|)}{\max (|C_X(b^*=0)|)}, \quad r_Y = \frac{\max (|C_Y(b^*)|)}{\max (|C_Y(b^*=0)|)}. \end{equation}

These ratios are plotted against $b^*$ in figure 4. For the thickest airfoil considered ($b^*=0.3$), the most significant effect was a $21\,\%$ decrease in peak $C_Y$ for the Kinsey & Dumas (Reference Kinsey and Dumas2008) case.

Figure 3. Stream-wise and stream-normal added-mass forces, $C_X$ and $C_Y$, versus $t/T$ for the test cases listed in table 2.

Figure 4. Peak added-mass force magntiude ratios, $r_X$ and $r_Y$, as defined in (5.5a,b), versus $b^*$ for the test cases listed in table 2.

Table 2. Airfoil thicknesses to be considered, and chosen kinematic parameters for pitching and plunging test cases from recent studies to be compared in figures 3(a) and 3(b).

The above results suggest that thickness effects are of secondary importance compared to the effect of airfoil motion on force, and the expressions in (5.1) and (5.2) may serve adequately as approximate corrections for finite-thickness effects for airfoils of other cross-sections. If more exact expressions are desired, the procedure outlined in § 2 can be applied to any rotationally symmetric two-dimensional body; take the known added-mass forces in response to orthogonal components of the body's acceleration, and follow the steps presented in (2.10)–(2.15) to account for body rotation and the presence of a non-zero free-stream flow. To extend this approach to airfoils that do not exhibit rotational symmetry, as is the case with most practical airfoils possessing a sharp trailing edge, definitional differences amongst classical added-mass force analyses must first be addressed, as explained in the Appendix.