## Appendix A

The Theodorsen theory is highly linearized, using small angle assumptions which sometimes set cosine equal to one and sine equal to zero. This linearity is desirable for the applications of this study because it simplifies solving for the forces and position of the airfoil over time. In Equation (

12), we showed that the time-averaged lateral force from Theodorsen does not depend on the flapping motion of the airfoil. The goal of this appendix is to show that the time averaged lateral force with a pitch bias is still independent of the flapping motion, even when non-linearities are considered.

Theodorsen splits the lateral forces into an added mass and a circulatory component (Equation (6)). Over one flapping cycle, the circulatory component averaged to a constant value which did not depend on the flapping motion and the added mass component averaged to zero (Equation (

12)). A non-linear version of the circulatory component is not readily available, and so the lift forces will be left untouched. However, we can check a non-linear version of the added mass forces and see if it averages to zero even when no small angle assumptions are made.

Sedov ([

38]) derives the following equations for the 2D added mass forces on an object translating and rotating through a fluid:

where

${U}_{\Vert}$ is the velocity of the object through the fluid in tangential direction (

${\overrightarrow{\mathrm{e}}}_{\Vert}$),

${U}_{\perp}$ is the velocity of the object through the fluid in the perpendicular direction (

${\overrightarrow{\mathrm{e}}}_{\perp}$), and

$\dot{\theta}$ is the rotational velocity of the object.

${\lambda}_{ij}$ represents added mass coefficients, which depend on the shape of the object, where ‖ refers to the tangential direction, ⊥ refers to the normal direction, and

$\theta $ refers to the rotational direction. This coordinate system is shown in

Figure A1 for reference.

**Figure A1.**
Coordinate system used by Sedov. The plate is moving through the water to the left at a speed u. The y direction represents the lateral direction in the water channel reference frame, and the x direction represents the streamwise direction in the water channel reference frame. A negative streamwise force ${F}_{x}$ would represent thrust. The directions ${\overrightarrow{\mathrm{e}}}_{\Vert}$ and ${\overrightarrow{\mathrm{e}}}_{\perp}$ are fixed to the plate and represent the tangential and normal directions, respectively.

**Figure A1.**
Coordinate system used by Sedov. The plate is moving through the water to the left at a speed u. The y direction represents the lateral direction in the water channel reference frame, and the x direction represents the streamwise direction in the water channel reference frame. A negative streamwise force ${F}_{x}$ would represent thrust. The directions ${\overrightarrow{\mathrm{e}}}_{\Vert}$ and ${\overrightarrow{\mathrm{e}}}_{\perp}$ are fixed to the plate and represent the tangential and normal directions, respectively.

We will be assuming the flapping hydrofoil is thin and flat enough to be considered a flat plate. Sedov derives the added mass coefficients for a flat plate, listed below:

where

b is the semi-chord of the flat plate, equal to

$c/2$. All of the added mass coefficients in the tangential direction (‖) are zero because a flat plate has no thickness. Substituting in the added mass coefficients gives the tangential and normal forces on the flat plate:

Now we will consider a reference frame aligned with the global water channel reference frame. The

x direction points in the same direction as incoming flow in the water channel, and the

y direction points in the lateral direction. The hydrofoil is moving through the fluid at a velocity

$-u$ in the

x direction, which corresponds to moving to the left in

Figure A1. The hydrofoil may also be moving laterally with a velocity

$\dot{y}$ in the

y direction. The hydrofoil is rotated by

$\theta $ relative to the water channel reference frame. Transforming the velocities of the plate into the reference frame of the airfoil gives the tangential and normal velocities of the plate through the fluid:

Substituting these equations into the relations for the normal and tangential forces for a flat plate gives

To obtain the final lateral and streamwise forces in the water channel reference frame, we rotate the normal and tangential forces (

${F}_{\Vert}$ and

F) to the water channel reference frame by the angle of the airfoil

$\theta $. This produces the lateral and streamwise added mass forces

${F}_{y}$ and

${F}_{x}$:

For a static airfoil (

$\dot{y}=0$), the lateral force

${F}_{y}$ and streamwise force

${F}_{x}$ are defined by

Here we are interested in

${F}_{y}$. If small angle assumptions are applied here (

$cos\left(\theta \right)\approx 1$ and

$sin\left(\theta \right)\approx 0$), the above equation for the lateral added mass force matches the added mass component of the Theodorsen model (Equation (

5)).

The angle of the airfoil

$\theta \left(t\right)$ is defined in Equation (

1) as the pitch bias plus a sinusoidal flapping component (

$\theta ={\theta}_{\mathrm{bias}}+\alpha sin\left(2\mathsf{\pi}ft\right)$). Substituting

$\theta $ into Equation (A13) for

${F}_{y}$ and integrating over flapping one cycle gives the time-averaged lateral force. This integration is lengthy to do by hand, but can be accomplished with extensive application of sine and cosine addition formulas and by employing the following identities for the Bessel function of the first kind

${J}_{n}$ applied to

$\alpha $:

${J}_{n}\left(\alpha \right)=\frac{1}{\mathsf{\pi}}{\int}_{0}^{\mathsf{\pi}}cos(\alpha sin\left(x\right)-nx)dx$,

${J}_{n+1}\left(\alpha \right)=\frac{2n}{\alpha}{J}_{n}\left(\alpha \right)-{J}_{n-1}\left(\alpha \right)$. Below is integration of

${F}_{y}$ term by term:

The first two terms integrate directly to zero, and the last two terms cancel out. Therefore, the time-averaged lateral force due to added mass of a flapping airfoil with a pitch bias is zero. Using a non-linear version of the added mass forces that makes no small angle assumptions does not change the result of Equation (

12).

Surprisingly, the time-averaged added mass forces in the streamwise direction happens to also average to zero. Substituting in

$\theta $ into Equation (

A12) and integrating term by term using the same process as above:

The first term averages to zero and the last two cancel each other out. This result is surprising because the Garrick model predicts a net thrust due to time-averaged added mass terms. Added mass terms in Garrick arise from the term

${C}_{L,Theo}\theta $ in the equation for the thrust (Equation (

13)), where

${C}_{L,Theo}$ is the lift from Theodorsen which includes added mass forces. Theodorsen only considers perpendicular added mass forces, while the Sedov model considers both the perpendicular and tangential added mass forces (Equation (

A10)). Time averaging the term

${C}_{L,\mathrm{Theo}}\theta $ in the Garrick model corresponds to time-averaging

$-{F}_{\perp}sin\theta $ in the Sedov model. Integrating only this term in the Sedov model produces thrust (result shown in Equation (A20)). However, the perpendicular added mass term is canceled out in the Sedov model because the Sedov model also time averages the tangential added mass force in the term

${F}_{\Vert}cos\theta $, where the result is shown in Equation (

A18).

Without thrust from added mass forces, the Garrick model predicts a net drag from lift forces [

19]. Presumably, an underlying assumption of the Garrick model results in thrust being generated from added mass forces rather than circulatory forces. It is possible that the flat wake assumption prevents the vorticity distribution in the Garrick model from producing thrust.

## Appendix B

The Theodorsen model is described in Equation (6), which uses the auxiliary functions

${\theta}^{*}=\alpha {\mathrm{e}}^{2\mathsf{\pi}ft\mathrm{i}}$ and

${y}^{*}={h}_{0}{\mathrm{e}}^{(2\mathsf{\pi}ft+\varphi )\mathrm{i}}$ along with complex multiplication to represent the amplitude and phase shift caused by the Theodorsen function

${C}^{*}\left(k\right)$. The high frequency component of the trajectory can be derived by solving for

${h}_{0}$ and

$\varphi $ given that

$\alpha $ and

f are prescribed. Below is the algebra involved in solving for

${h}_{0}$ and

$\varphi $. Equation (6) will be left in its complex form, because taking the imaginary component of the equation is less convenient and not necessary for solving for

${h}_{0}$ and

$\varphi $.

It is convenient to separate the terms that involve

$\theta $, which are prescribed, from the terms that involve

y, which are unknown:

Substituting

${\theta}^{*}$ and

${y}^{*}$ into the above expression and simplifying terms gives

Substituting

$k=\pi c/u$, multiplying both sides by −1, and further simplifying leads to

Then, substituting

${C}^{*}\left(k\right)=F+G\mathrm{i}$ and grouping imaginary and real terms, we can then isolate

${h}_{0}$ and

$\varphi $:

Using the placeholder terms

${A}_{4},{A}_{5},{A}_{6},$ and

${A}_{7}$ defined in Equations (

26)–(29), and rearranging terms,

The heaving amplitude

${h}_{0}$ can be found by taking the magnitude of both sides:

The phase

$\varphi $ can be extracted by taking the complex argument of both sides: