In this paper, we study torsional oscillations of a cross section of a thin plate submerged in a quiescent, Newtonian, incompressible, and viscous fluid. The plate is subjected to a prescribed shape-morphing deformation in phase with the rigid oscillation. The problem is completely described by three nondimensional parameters indicating oscillation frequency and amplitude and intensity of the shape-morphing deformation. We conduct a parametric study to investigate the possibility of controlling hydrodynamic moments and power dissipation through an active time-varying shape-morphing strategy. The problem is studied in both the linear and nonlinear flow regimes, by employing the boundary element method and direct numerical simulations via computational fluid dynamics methods, respectively. Investigation of flow physics demonstrates that, similarly to what is observed for the case of flexural oscillations, the shape-morphing strategy is effective in modulating vortex shedding in torsional oscillations. The results show that hydrodynamic power dissipation can be minimized and hydrodynamic moments can be controlled through an optimal imposed shape-morphing deformation. Findings from this study are directly applicable to torsional oscillation-based underwater energy harvesting or sensing and actuation systems, where control of hydrodynamic moments and reduction of hydrodynamic power losses are necessary for optimal device operation.

## I. INTRODUCTION

Understanding the dynamic behavior of flexible structures oscillating in viscous fluids hinges primarily on the accurate estimation of the hydrodynamic loading exerted by the fluid. Such knowledge is important for properly designing, operating, and controlling devices in a myriad of applications including, for example, atomic force microscopy (AFM) in liquids,^{1–4} micromechanical sensors and actuators,^{5–7} piezoelectric fans for heat transfer,^{8–12} bio-inspired and/or robotic propulsion,^{13–24} and smart material-based small scale energy harvesting devices.^{18,25–33} In many of these applications, transverse oscillations of beam-like flexible structures are of primary interest. When small amplitude oscillations are concerned, the fluid is usually modeled as an unsteady Stokes flow. The resulting linear hydrodynamic forces are then typically described in the frequency domain in the form of a complex hydrodynamic function.^{1} For large amplitude oscillations, this approach has been extended via a correction term that captures the effect of nonlinear convective phenomena.^{34–37}

Despite the technical and scientific interest, the problem of torsional oscillations of beam-like structures in viscous fluids has received comparatively less attention from the community. In Ref. 38, the hydrodynamic function approach has been translated to describe hydrodynamic moments during torsional oscillation of submerged cantilever beams. This particular work concentrates on small amplitude oscillations, and the moments are expressed via a linear hydrodynamic function that is exclusively dependent on a frequency parameter, which plays the role of an oscillatory Reynolds number. In Ref. 39, the problem of finite amplitude torsional oscillation of the submerged structure is studied. In this case, the hydrodynamic damping moment is inadequately predicted by the linear hydrodynamic function. A nonlinear correction is thus introduced to describe the resultant hydrodynamic moment while capturing the effect of vortex shedding and convection. The nonlinear form of the hydrodynamic function depends on the frequency parameter and on the (finite) angular amplitude of oscillation. Results demonstrate that, especially in the case of moderately large torsional oscillations, hydrodynamic nonlinearities can give rise to substantial vortex shedding-related damping, which, in turn, can be detrimental to the desired dynamic behavior of the system.

In previous studies, we have observed that damping forces responsible for dissipative effects in the flow can be significantly reduced via the so-called “shape-morphing deformation” strategy applied to beam-like structures undergoing transverse bending. Within this context, superimposed to the gross oscillation of the structure, we consider a prescribed deformation of the solid cross section to an arc of a circle with the desired time-varying curvature in the chord-wise direction with the intent to control fluid–structure interactions. In practical applications of the shape-morphing strategy, the prescribed curvature could be realized by embedding smart materials such as piezoelectric patches,^{40} macro-fiber composite actuators,^{41–44} electroactive polymers,^{45} or shape memory alloys^{46} in the flexible structure. In Ref. 47, shape-morphing is applied to the study of a plate undergoing heaving oscillation in an unbounded fluid, whereas oscillations in the vicinity of a solid wall have been investigated in Ref. 48. We have shown therein that hydrodynamic damping and power dissipation can be considerably reduced and minimized by properly tuning the imposed deformation of the structure. The investigation in the potential of shape-morphing is further extended in Ref. 49, where a complete 3D flow model is used to assess its performance on a cantilever beam undergoing small amplitude flexural oscillations. As compared to alternative flow control strategies, such as those involving permanent geometric modifications,^{50–52} the shape-morphing strategy seems to be superior because of its adaptability to diverse, and potentially changing, flow conditions. Although shape-morphing has been shown to be a promising technique to control hydrodynamic forces and power dissipation in transverse oscillations, its potential has not been studied in the context of torsional oscillations.

To close this knowledge gap, in this work, we investigate the possibility of modulating the hydrodynamic moment and power dissipation associated with finite amplitude torsional oscillations of submerged structures via shape-morphing. To this aim, we study a two-dimensional (2D) fluid problem consisting of a submerged thin one-dimensional line-like solid (henceforth, referred to as “plate”) undergoing finite amplitude torsional oscillations while subjected to a time-varying shape-morphing deformation. Similar to Refs. 38 and 39, the “plate” under study represents a cross section of a beam-like solid that undergoes torsional vibrations and its motion is completely prescribed. Thus, in-plane shape-morphing deformation corresponds to the chord-wise curvature control, similar to Refs. 47–49. We consider both the large amplitude oscillation case and the asymptotic limit of small oscillations. In the latter case, the flow is modeled via the unsteady Stokes equation and the resulting linear problem is solved using the Boundary Element Method (BEM). Conversely, the Navier–Stokes equations are employed for large amplitude oscillations and the resulting nonlinear problem is studied via the Computational Fluid Dynamics (CFD) method. Besides providing numerical solutions, as well as a reference for the asymptotic behavior of the nonlinear problem, the small amplitude oscillation problem allows us to identify scaling laws for the behavior of the hydrodynamic moment and power as a function of the imposed shape-morphing deformation. The problem is studied parametrically by varying three identified governing parameters, stemming from geometry and flow kinematics. Results show that when the plate is subjected to torsional oscillations, hydrodynamic moments can be controlled and damping can be reduced significantly via shape-morphing deformation. In addition, power dissipation can be minimized by properly tuning the “intensity” of shape-morphing, that is, the level of the imposed curvature of the plate. To gather insight into the physics of power dissipation minimization, we consider vortex formation and evolution by observing the vorticity in the flow field. These results shed light in the relatively less explored problem of torsional oscillations and provide insight toward damping control for submerged structures undergoing torsional oscillations. The results from this work are of direct interest in applications related to underwater energy harvesting^{53,54} and in other sensing and actuation applications,^{38,55} where torsional oscillations are relevant.

The rest of the paper is organized as follows. In Sec. II, we state the kinematics and governing equations of the problem in detail. In Sec. III, we discuss the solution methodology adopted for the linear and nonlinear problems and provide analytical insight through a scaling argument. In Sec. IV, we discuss results obtained from the parametric study in terms of hydrodynamic moments, power dissipation, and physics of vorticity. Conclusions are reported in Sec. V. The appendices report implementation details for the CFD analysis and convergence studies.

## II. PROBLEM STATEMENT

### A. Governing equations and nondimensional parameters

In this work, the 2D fluid problem consists of a one-dimensional line-like solid (“plate”) oscillating in an unbounded, otherwise quiescent, incompressible Newtonian viscous fluid while subjected to shape-morphing deformation in its own body reference frame, see Fig. 1. This solid represents an arbitrary cross section of a prototype beam-like solid undergoing torsional oscillations. This 2D simplification is consistent with assumptions and methods in the literature,^{38,39} while neglecting potential end effects^{56} and the development of three-dimensional flow along the axis of the beam^{57,58} that are known to exist for flexural vibrations.

It is important to note that, in this study, the motion of the plate is completely prescribed. Initially, the plate is aligned with the world *X*-axis in its reference configuration, as shown schematically in Fig. 1(b). As the plate rotates, the body frame of the plate with axes *X*′ and *Y*′ forms an angle *ϕ*(*t*) with the world axes *X* and *Y*. In the 2D problem, the rotation matrix that maps the body reference frame of the plate *X*′*Y*′ at time *t* to the world reference frame *XY* can be expressed as

such that {*X*, *Y*} = [Rot]{*X*′, *Y*′}. Here, *ϕ*(*t*) is the time-varying amplitude of torsional oscillation, characterized by the maximum excursion *ϵ*, or angular amplitude, and the radian frequency *ω* as *ϕ*(*t*) = *ϵ* sin(*ωt*), see, for example, Ref. 39.

In the body frame, the plate undergoes a shape-morphing deformation in-phase with the rigid oscillation, see Fig. 1(b). Specifically, each half of the plate deforms to an arc of a circle with the prescribed radius of curvature *R*(*t*). The two deformed halves of the plate are tangential to the *X*′-axis in the origin of the body frame, and they deform in opposite directions with respect to the *X*′-axis. As a result, the shape of the plate is anti-symmetric with respect to the *X*′- and *Y*′-axes, see Fig. 1(b). In the body frame, shape-morphing deformation is thus imposed as

where *X*′_{0} describes the location of a material point of the plate in the reference configuration in the body frame, *R*(*t*) is the radius of curvature of the plate in the *X*′*Y*′-plane at time *t*, defined as $R(t)=1/[\kappa \u0303sin(\omega t)]$, and $\kappa \u0303$ is the maximum curvature of the deformed arcs in Fig. 1(b). While this particular “*S*-shape” embodiment of the shape-morphing deformation is selected as a natural extension from our work on transverse oscillations,^{47–49} this is not the only possible choice. In a practical application, the prescribed chord-wise deformation described in Eq. (2) could be realized via pairs of piezoelectric bimorphs actuated in opposition of the phase. Note, however, that the major developments of this paper and their results are independent of the particular details of this selection.

The fluid flow induced by the shape-morphing torsional motion of the plate is in general described by the 2D Navier–Stokes equations. In the absence of body force, the equations governing the unsteady fluid velocity field ** u**, written in the inertial frame, are

^{59}

where *ρ* is the density of the fluid, D(•)/D*t* denotes the material derivative, ∇ · (•) denotes the divergence operator, ∇*p* is the pressure gradient, *μ* is the dynamic viscosity of the fluid, and ∇^{2}(•) is the Laplacian operator. Velocity boundary conditions are imposed by enforcing null velocity at infinity and no-slip boundary conditions at the moving fluid-solid boundary. The motion of the solid boundary is completely specified. Pressure at infinity is set to zero.

Three nondimensional parameters govern the problem: the nondimensional oscillation amplitude *ϵ* regulating the evolution of *ϕ*(*t*) in Eq. (1), the nondimensional frequency parameter *β*, and the nondimensional curvature *κ*. The last two parameters are defined, respectively, as

In addition to these three parameters, we will also often use a fourth parameter $\xi =\kappa /\u03f5=\kappa \u0303b/\u03f5$ alongside *κ* for convenience of exposition. Note, however, that *ξ*, *κ*, and *ϵ* are not independent.

### B. Hydrodynamic moment and power dissipation

By solving the fluid problem described by Eq. (3), we obtain the time history of the resultant hydrodynamic moment *M*(*t*) per unit length of the plate in the out-of-plane direction. This can be expressed as

where *t*_{(n)} is the traction vector acting on the plate surface at time *t*, denoted by *∂S*(*t*), see Refs. 35 and 47, and ** r** is the position vector of the line element d

*ℓ*in the world frame. The unit normal to the surface is

**, and**

*n*

*e*_{3}is the out-of-plane unit vector.

We assume that the hydrodynamic moment response is harmonic with the torsional oscillation and shares the same radian frequency *ω*, similarly to the assumptions in Ref. 39. Hence, the hydrodynamic moment in the frequency domain can be expressed as

where $M^$ is the phasor of the hydrodynamic moment, that is, $M(t)=ImM^(\omega )ei\omega t$, with $i=\u22121$. Here, Θ(*β*, *ϵ*, *ξ*) represents the complex nonlinear hydrodynamic function which relates the hydrodynamic moment to the torsional oscillation amplitude phasor.^{39} The real and imaginary parts of the hydrodynamic function Θ(*β*, *ϵ*, *ξ*) describe the moment in phase with the angular acceleration and with the angular velocity of the plate, that is, the inertial and damping moment, respectively.^{39} In addition, we posit that the hydrodynamic function Θ(*β*, *ϵ*, *ξ*) can be decomposed into two parts: a linear hydrodynamic function Γ(*β*, *ξ*) and a nonlinear correction term Δ(*β*, *ϵ*, *ξ*). The linear part describes the linear hydrodynamics when the oscillation amplitude is negligible,^{38} that is, in the asymptotic limit *ϵ* → 0, whereas the nonlinear correction accounts for convective nonlinearities because of finite amplitude oscillation.^{39} Therefore, using these hypotheses on its functional form, the complex nonlinear hydrodynamic function Θ(*β*, *ϵ*, *ξ*) can be re-written as

By construction, we further assume that, as *ϵ* → 0, the correction term Δ(*β*, *ϵ*, *ξ*) vanishes so that, in this limit, we have Θ(*β*, 0, *ξ*) = Γ(*β*, *ξ*).

The instantaneous hydrodynamic power dissipation $P(t)$ is evaluated by integrating the dot product of local velocity and the corresponding traction vector over the deformed plate domain as

The average hydrodynamic power dissipation $\u27e8P\u27e9$ can then be obtained by averaging $P(t)$ over one oscillation cycle as

where *T* = 2*π*/*ω* is the period of oscillation and *τ* is the starting time of the averaging window. In steady-state conditions, the exact value of *τ* is immaterial.

## III. SOLUTION

### A. Asymptotic analysis: The unsteady Stokes regime

In this section, we discuss the problem in the linear regime for small amplitude oscillation of the plate and solve for linear hydrodynamic function Γ(*β*, *ξ*) and average hydrodynamic power dissipation in the frequency domain using a variant of the BEM approach developed in Ref. 60. In the limiting case of vanishingly small oscillations, the motion of the plate is geometrically linear; thus, the reference frames introduced in Fig. 1 coincide.^{47} Furthermore, since the hydrodynamic problem is linearized, it is convenient to analyze it in the frequency domain.^{38} With reference to Fig. 2, we thus consider an infinitely thin plate of width *b* and we introduce (in the frequency domain) the stream function $\psi ^(x,y)$ which satisfies the continuity equation, that is, $\xfbx(x,y)=\u2202\psi ^(x,y)/\u2202y$, $\xfby(x,y)=\u2212\u2202\psi ^(x,y)/\u2202x$. For simplicity, we will omit the explicit *ω* dependency for phasor quantities. We denote the vorticity and pressure jumps (scaled with the dynamic viscosity) across the plate width as $Z^(x)=\zeta ^(x,0+)\u2212\zeta ^(x,0\u2212)$ and $\Pi ^(x)=[p^(x,0+)\u2212p^(x,0\u2212)]/\mu $. Additionally, exploiting the anti-symmetry of the problem with respect to the *y*-axis, we have $p^(x,y)=\u2212p^(\u2212x,y)$ and $\zeta ^(x,y)=\zeta ^(\u2212x,y)$. Therefore, by considering the contour in Fig. 2, in the limit of zero thickness, the original boundary integral formulation in Ref. 60 can be written as

where $R=(x\u2212x\xaf)2+(y\u2212\u0233)2$, *α*^{2} = −i*ρω*/*μ*, and *K*_{0}(•) is the zeroth order Bessel function of the third kind.^{61} Comma subscripts *x* and *y* applied to $\Psi ^$ are a shorthand notation for partial derivatives with respect to *x* and *y*, respectively.

Differentiating Eq. (11) with respect to *ȳ* and $x\xaf$, we obtain the velocity components $\xfbx(x\xaf,\u0233)$ and $\xfby(x\xaf,\u0233)$, respectively, on the plate boundary. Since we have no-slip conditions on the plate boundary, we specify the values of $\xfbx(x\xaf,\u0233)$ and $\xfby(x\xaf,\u0233)$ directly from the time derivative of the prescribed displacements of the oscillating plate. In the limiting case of vanishingly small oscillations and geometrically linear motion of the plate, we obtain null displacement in the *X*-direction. The displacement component in the *Y*-direction linearizes as

Neglecting higher order terms, since the reference configuration and the current configuration coincide in the geometrically linearized case, the velocity components are given simply as

Two decoupled integral equations are then obtained for *Ẑ*(*x*) and $\Pi ^(x)$ by applying the boundary conditions for *û*_{x} and *û*_{y}, respectively.^{60} The first result is that *Ẑ*(*x*) = 0 because of the fact that *û*_{x} = 0. Hence, we obtain the integral equation for $\Pi ^(x)$ as

Equation (14) is solved numerically to identify $\Pi ^$ on the plate. The linear hydrodynamic function Γ(*β*, *ξ*) and the average hydrodynamic power dissipation $\u27e8P\u27e9$ are, thus, estimated by integrating $\Pi ^$ over the width of the plate as

Before attempting the solution of Eq. (14), it is illustrative to analyze its qualitative properties. Following Ref. 47 and utilizing the linearity of the integral operator, we decompose the total pressure jump $\Pi ^(x)$ into two contributions to identify the properties of hydrodynamic moment and power dissipation using scaling arguments. Hence, denoting $\Pi ^0(x)$ and $\Pi ^1(x)$ as the pressure jump contributions from rigid torsional oscillation and shape-morphing deformation, respectively, we obtain two integral relations as

This allows us to express the total pressure jump as an affine function of *κ* as

Furthermore, using the boundary condition in Eq. (13) and the pressure jump decomposition in Eq. (17), from Eq. (15), we can express the hydrodynamic function and average hydrodynamic power dissipation as

Therefore, even without the detailed knowledge of the function $\Pi ^(x)$, the overall hydrodynamic function Γ and average power dissipation $\u27e8P\u27e9$ exerted on the shape-morphing plate can be expressed in powers of *κ* within the order $O(\kappa 3)$ as

with an obvious meaning of the symbols. Note that Γ_{0} and $\u27e8P0\u27e9$ here can be interpreted as the hydrodynamic function and average power dissipation, respectively, because of rigid torsional oscillation exclusively. This shows that, in the hydrodynamically linear case, the hydrodynamic moment is an affine function of the curvature parameter *κ* and the average hydrodynamic power dissipation is a parabolic function of *κ*. It is remarkable that these results are entirely consistent with our observations in Ref. 47 for transverse oscillations. These results therefore point to a powerful, and general, underlying concept of vortex shedding modulation via shape-morphing which transcends the specific nature of the vibrations. These analytical results will be confirmed numerically in Sec. IV.

To obtain numerical results for the hydrodynamic function and the hydrodynamic power dissipation, we discretize the plate domain via the boundary element method, as described, for example, in Refs. 38, 47, 51, and 60. Specific details of the numerical procedure are generally standard and can be found, for example, in Ref. 51. We only remark here that the pressure jump $\Pi ^(x)$ is computed for each element (or “panel”) on the plate, where we assume that the pressure is constant over the span of each element and equal to the pressure on the midpoint of that element. Results from the procedure are presented alongside CFD results in Sec. IV.

### B. Two-dimensional nonlinear fluid problem

We obtain the hydrodynamic moment and power dissipation in the nonlinear regime by performing a thorough CFD study via a commercially available code (*ANSYS Fluent 16.1*). We investigate parametrically the behavior of the hydrodynamic moment and power dissipation by varying the three nondimensional parameters over a broad range of values. Specifically, the frequency parameter *β* varies as {100, 200, 500, 1000}, the amplitude parameter *ϵ* varies as {0.001, 0.01, 0.05, 0.1}, and the curvature parameter *ξ* varies from 0 to 7 with the unit increment. The CFD study is performed on a computational model modified and adapted from Ref. 39. The oscillating plate is modeled as a no-slip wall of width *b* = 10 mm at the center of the circular domain. The null velocity and pressure boundary conditions at the far field are specified by defining the outer boundary as the pressure outlet. Details of the CFD model and implementation are further reported in Appendix A. Note that the oscillation frequency and shape-morphing curvature are imposed to the plate in the computational model from the nondimensional parameters as *ω* = (2*πμβ*)/(*ρb*^{2}) and *κ* = *ξϵ*, respectively.

## IV. RESULTS

### A. Hydrodynamic moment and power dissipation

The hydrodynamic moment per unit length *M*(*t*) exerted on the shape-morphing plate is directly extracted from the CFD simulations. The initial transient in the moment time history is preliminarily truncated to obtain a reasonably good representation of its steady-state value. Successively, we identify the real and imaginary parts of the hydrodynamic function in Eq. (6) using an approach similar to the one adopted for hydrodynamic force extraction in Ref. 47. This involves fitting the steady-state moment time history over a selected oscillation cycle to a Fourier series of the form

where *C*_{m} and *S*_{m} are Fourier coefficients. We obtain the coefficients through curve fitting for *m* ≤ 5, along with a possible mean value, that is, *C*_{0} in Eq. (20). Furthermore, we identify the real and imaginary parts of the hydrodynamic function Θ(*β*, *ϵ*, *ξ*) by retaining the first harmonic of Fourier series from the identities $Re\Theta =S1$ and $Im\Theta =C1$, where we take into consideration that low-order harmonics are expected to contribute to most of the moment signal. This procedure, in practice, corresponds to consistently neglecting higher order harmonics in the moment signal. While this representation is only approximately verified for large amplitudes, it is essential for the description of the moments in terms of phasors in the frequency domain as in Eq. (6) and hinges on the popular method of the “describing function,” often used in the analysis of nonlinear systems for the approximate description of amplitude-dependent transfer functions.^{62} In addition, the validity of this approach is supported in part by the data displayed in Table I, where we present the harmonic coefficients of the Fourier series for representative cases. It can be seen that higher harmonic coefficients with *m* ≥ 2 are generally negligible as compared to *C*_{1} and *S*_{1}, except for the coefficients *C*_{3} and *S*_{3} at some instances of more aggressive oscillation cases. Notably, modest second harmonic components are likely due to the plate periodic deformation kinematics along their width direction, as described by Eq. (2a). However, the coefficients associated with the second harmonic *C*_{2} and *S*_{2} are generally observed to be much smaller than the first harmonic coefficients for, virtually, all the cases considered. For all the cases reported in this paper, we select the cycle number *n* = 7 as our curve fitting window to allow adequate time for the transients to be sufficiently decayed.

Fourier coefficients (×10^{6})
. | ϵ = 0.001
. | ϵ = 0.01
. | ϵ = 0.05
. | ϵ = 0.1
. |
---|---|---|---|---|

C_{0} | 1.18 × 10^{−5} | 4.73 × 10^{−4} | 3.33 × 10^{−3} | 7.41 × 10^{−3} |

C_{1} | −2.39 × 10^{−2} | −2.40 × 10^{−1} | −1.23 | −2.75 |

S_{1} | 1.55 × 10^{−1} | 1.55 | 7.77 | 15.5 |

C_{2} | 1.19 × 10^{−5} | 1.73 × 10^{−4} | −4.03 × 10^{−3} | 2.49 × 10^{−2} |

S_{2} | 3.86 × 10^{−6} | −9.37 × 10^{−5} | −1.91 × 10^{−3} | −3.60 × 10^{−4} |

C_{3} | −1.66 × 10^{−3} | −1.67 × 10^{−2} | −8.24 × 10^{−2} | −2.39 × 10^{−1} |

S_{3} | 7.88 × 10^{−2} | 7.85 × 10^{−1} | 3.74 × 10^{−1} | 4.64 × 10^{−1} |

C_{4} | −8.79 × 10^{−6} | −8.47 × 10^{−5} | 1.40 × 10^{−3} | 4.32 × 10^{−3} |

S_{4} | 3.60 × 10^{−5} | 5.28 × 10^{−4} | 2.82 × 10^{−3} | −3.57 × 10^{−3} |

C_{5} | 4.89 × 10^{−4} | 4.90 × 10^{−3} | 2.02 × 10^{−2} | 7.86 × 10^{−2} |

S_{5} | 2.53 × 10^{−3} | 2.54 × 10^{−2} | 1.13 × 10^{−1} | 2.60 × 10^{−3} |

Fourier coefficients (×10^{6})
. | ϵ = 0.001
. | ϵ = 0.01
. | ϵ = 0.05
. | ϵ = 0.1
. |
---|---|---|---|---|

C_{0} | 1.18 × 10^{−5} | 4.73 × 10^{−4} | 3.33 × 10^{−3} | 7.41 × 10^{−3} |

C_{1} | −2.39 × 10^{−2} | −2.40 × 10^{−1} | −1.23 | −2.75 |

S_{1} | 1.55 × 10^{−1} | 1.55 | 7.77 | 15.5 |

C_{2} | 1.19 × 10^{−5} | 1.73 × 10^{−4} | −4.03 × 10^{−3} | 2.49 × 10^{−2} |

S_{2} | 3.86 × 10^{−6} | −9.37 × 10^{−5} | −1.91 × 10^{−3} | −3.60 × 10^{−4} |

C_{3} | −1.66 × 10^{−3} | −1.67 × 10^{−2} | −8.24 × 10^{−2} | −2.39 × 10^{−1} |

S_{3} | 7.88 × 10^{−2} | 7.85 × 10^{−1} | 3.74 × 10^{−1} | 4.64 × 10^{−1} |

C_{4} | −8.79 × 10^{−6} | −8.47 × 10^{−5} | 1.40 × 10^{−3} | 4.32 × 10^{−3} |

S_{4} | 3.60 × 10^{−5} | 5.28 × 10^{−4} | 2.82 × 10^{−3} | −3.57 × 10^{−3} |

C_{5} | 4.89 × 10^{−4} | 4.90 × 10^{−3} | 2.02 × 10^{−2} | 7.86 × 10^{−2} |

S_{5} | 2.53 × 10^{−3} | 2.54 × 10^{−2} | 1.13 × 10^{−1} | 2.60 × 10^{−3} |

In Figs. 3(a) and 3(b), we present the real and imaginary parts of hydrodynamic function Θ(*β*, *ϵ*, *ξ*), respectively, for the representative case of *β* = 500. The hydrodynamic function is shown as a function of the curvature parameter *ξ*. Consistent with the semianalytical argument in Eq. (19), an approximate linear decreasing trend with a slope of −0.0133 is observed in Fig. 3(a). In this figure, CFD results for $Re\Theta $ and BEM results for $Re\Gamma $ for the nonlinear and linear cases, respectively, are depicted with solid and dashed lines, respectively. A striking reduction in the inertial moment (described by the real part of Θ or “added moment of inertia”) is observed, which can be explained by considering the large reduction in the projected area associated with the shape-morphing plate as compared to the flat plate. Similar to that observed in the literature, the real part of the hydrodynamic function is largely independent of the torsional amplitude parameter *ϵ*, see, for example, Ref. 39. Furthermore, the CFD results are partially validated with the BEM results discussed in the asymptotic analysis. The dashed black line superimposed to the CFD results describes the scenario with amplitude *ϵ* → 0. The two results are in good agreement and virtually indistinguishable from each other in the scale of the figure. This strongly supports the conclusion that CFD simulations correctly capture the linear regime of the problem governed by unsteady Stokes hydrodynamics. Additionally, as expected, our BEM results for *ξ* = 0 match closely with the result presented by the fit in Ref. 38, as seen in Fig. 3(a).

Figure 3(b) shows the imaginary part of the hydrodynamic moment or “damping moment,” see Ref. 39, for the representative case with *β* = 500. Results can be roughly categorized into two sets based on the behavior of $Im\Theta $ as a function of curvature parameter *ξ*. For small amplitudes, i.e., *ϵ* ≤ 0.01, the imaginary part of the moment shows an essentially linear decreasing trend with respect to the curvature parameter *ξ*. The linear trend for small amplitude oscillation is consistent with the qualitative observations from Eq. (19). The dashed black line depicting BEM results show good agreement with findings from CFD simulations in the essentially linear regime. Conversely, for large oscillation amplitude, linearity is lost, as shown by the presence of local inflexion points, resulting in a complex dependence of $Im\Theta $ on the nondimensional parameter *ξ*. It is interesting to observe that shape-morphing is comparatively more effective in reducing hydrodynamic damping as the oscillation amplitude increases. We hypothesize that this is due to the particularly effective way in which shape-morphing affects vortex shedding. Indeed, vortex shedding seems to be the dominant effect responsible for nonlinear damping at large oscillation amplitudes.^{39} This observation is further investigated in Sec. IV B.

Similar to the flexural oscillation problem, in Fig. 3(b), we observe zero crossing phenomena for a particular level of imposed shape-morphing deformation for all values of *ϵ*. In this case, the zero crossing occurs for curvature parameter *ξ* ≈ 5. For any increase beyond this level, $Im\Theta $ becomes negative. The zero crossing phenomena and negative value of damping moment physically signify a condition in which hydrodynamic damping moments “favor” the rigid torsional oscillation. These results demonstrate that the shape-morphing strategy can strongly modulate the hydrodynamic moment exerted on the oscillating plate and can significantly contribute in reducing the large damping moment associated with oscillation in the nonlinear regime.^{39} In addition, the appearance of the interesting dynamic phenomenon of negative damping illustrates that shape-morphing can potentially be employed to aid the oscillation of the plate, but, as discussed in what follows, this comes associated with an increased energy cost.

To clarify the last statement, the effect of shape-morphing deformation on the hydrodynamics of the torsionally oscillating submerged plate is further analyzed here by studying the hydrodynamic power dissipation. In Fig. 4, we present the hydrodynamic power dissipation ratio $\u27e8P\u27e9/\u27e8P0\u27e9$ as a function of curvature parameter *ξ*. Here, $\u27e8P\u27e9$ is the hydrodynamic power dissipation associated with the shape-morphing plate, and $\u27e8P0\u27e9$ is the case for *ξ* = 0. Figure 4(a) shows $\u27e8P\u27e9/\u27e8P0\u27e9$ varying as a function of *ξ*, as *β* varies over three orders of magnitude, for amplitude parameter *ϵ* = 0. In Fig. 4(b), we demonstrate the behavior of the ratio as the amplitude parameter *ϵ* varies in *ϵ* ∈ [0.001, 0.1]. The hydrodynamic power dissipation ratio generally shows a nonmonotonic parabolic dependency with the existence of a minimum as the curvature parameter *ξ* varies. Such minimum hydrodynamic power dissipation is observed for the range of curvature parameters between *ξ* ≈ 4–6. The nonmonotonic behavior of the power dissipation ratio and the minimization phenomena are further analyzed in Sec. IV B by studying the vorticity dynamics in the vicinity of the oscillating plate.

From Fig. 4(a), it can be observed that the optimal curvature that results in minimum hydrodynamic power dissipation mildly increases as the frequency parameter *β* increases. Interestingly, the behavior of $\u27e8P\u27e9/\u27e8P0\u27e9$ is not strongly dependent on *β* for the small imposed curvature. In contrast, the effect of *β* is more evident for large values of *ξ*. This observation bears similarity to what is highlighted in flexural oscillation cases.^{47} Moreover, the rapid increase of power dissipation after the minimum is notably steeper for small *β* values as compared to large *β*. This behavior may be due to the larger boundary layer thickness, scaling^{2} as *β*^{−1/2}, associated with smaller *β* cases. In these cases, lighter shape-morphing effort is required to significantly affect the boundary layer as compared to the thin boundary layer associated with a plate oscillating with small *β*. In Fig. 4(b), we present the ratio $\u27e8P\u27e9/\u27e8P0\u27e9$ for a representative case of frequency parameter *β* = 500 and for different oscillation amplitudes, demonstrating that the power dissipation ratio is only mildly dependent on *ϵ*, for any *ξ* value. More specifically, and from a qualitative perspective, deviations in Fig. 4(b) from the parabola pertaining to linearized hydrodynamics is more evident as *ϵ* increases, thus showing that for equal *ξ*, reduction of power dissipation is enhanced for large oscillations.

The overall reduction in hydrodynamic power dissipation is observed to be more than approximately 98% for the shape-morphing parameter varying from *ξ* = 0 to *ξ* = 5 for the representative case presented in Fig. 4(b). As expected, this dramatic reduction occurs in proximity of the zero crossing in the hydrodynamic damping coefficient in Fig. 3(b). This suggests that the shape-morphing strategy can be effective in reducing the dissipative losses during torsional oscillation of plate-like structures submerged in a viscous fluid.

### B. Qualitative observations of vorticity dynamics

In this section, we qualitatively study the vorticity dynamics in the vicinity of the torsionally oscillating submerged shape-morphing plate. Here, we seek to understand and correlate the minimization phenomena observed in Subsection IV A to the vorticity induced by the plate oscillations. This analysis is motivated by the relationship between damping and energy dissipation to the vortex shedding and convection during large amplitude oscillations.^{36,39,47} In Fig. 5, we present three representative cases corresponding to three levels of shape-morphing deformation identified from Fig. 4(b): lower, equal, and higher levels of curvature than the optimal value identified for minimization of hydrodynamic power dissipation. Specifically, in panels (a)–(c), we present the vortex contours corresponding to *ξ* = 3, *ξ* = 5, and *ξ* = 7, respectively, for the same oscillation frequency parameter *β* = 500 and amplitude parameter *ϵ* = 0.1. We calculate the out-of-plane vorticity as *ζ* = (∇ ×** u**) ·

*e*_{3}, where

*e*_{3}is the unit vector normal to the

*XY*-plane. Furthermore, we scale the vorticity with the characteristic vorticity expressed as

*ζ*

_{0}= [(2

*πμ*)/(

*ρb*

^{2})](

*βϵ*) = 3.157 s

^{−1}, see, for example, Refs. 47 and 51.

Each panel of Fig. 5 consists of contour plots at nine instants of time pertaining to the sixth oscillation cycle. On the first snapshot at *t* = (0/8)*T*, the undeformed plate is in its reference configuration, about to begin its counter-clockwise (CCW) motion. The plate reaches its farthest CCW displaced position at *t* = (2/8)*T*, and then, the clockwise (CW) motion initiates. The plate reaches the reference configuration in the fifth frame at *t* = (4/8)*T*. The CW motion of the plate continues until the plate reaches its furthest CW displaced position in the seventh frame at *t* = (6/8)*T*. Finally, the plate moves back to its reference configuration in the ninth frame at *t* = (8/8)*T*.

Qualitative observation of the vorticity patterns demonstrates unique features characteristic of each of the three levels of shape-morphing. In panel (a), we first consider the case of low intensity of shape-morphing. As such, the plate oscillates in a nearly rigid fashion and vortices appear in pairs, one each from the two tips of the plate. As expected and similarly to the case of rigid torsional oscillations studied in Ref. 39, the vortices are CW when the plate moves in the CCW direction and CCW when the plate rotates in the CW direction. Similarly, in the central region of the plate, mild vorticity can be observed to be associated with the bulk rotation. As expected, the sign of this vorticity pattern is the same as the sign of the rotation of the plate, e.g., CW vorticity for CW rotation, and vice versa. Since even in the low intensity case at hand, the shape-morphing motion is opposite to the rigid torsional oscillation, these observations suggest that vortex dynamics is dominated by the rigid torsional oscillation, while the shape-morphing deformation plays a secondary role. This scenario implies that the low pressure regions in the flow field are on the trailing side of each plate half length, similarly to the rigid torsional oscillation case.^{39}

On the other hand, panel (b) presents the vortex dynamics for the plate undergoing (close to) optimal shape-morphing, allowing for minimization of hydrodynamic power dissipation. Corresponding to this minimization, only modest vorticity regions can be observed surrounding the plate. The most striking difference between panel (a) and panel (b) is that the high-vorticity regions that were concentrated in proximity of the tips have now shifted toward the center of the plate and new pockets of vorticity of opposite signs have appeared. For example, for panel (b), at time *t* = (0/8)*T*, while the plate is undergoing gross rotation in the CCW direction, CCW vorticity is observed in the immediate proximity of the tips, while the CW vorticity of panel (a) has moved away from the tips toward the center. This occurs while the bulk motion of the plate has been kept the same between panel (a) and (b). Thus, this is probably due to the shape-morphing deformation which prescribes the tips to move in the opposite direction than the bulk of the plate. As mentioned above, the overall magnitude of the vorticity is significantly reduced. In addition to this, and more specifically, the strong and convected tip vortices observed in panel (a) appear to be much weaker and remain in close proximity to the plate in panel (b). Reduced vortex shedding is thus correlated with reduced hydrodynamic damping as necessarily smaller amounts of energy are injected from the oscillating structure in the fluid.^{36,51} We posit that such development is effectively the overall result of the interaction of bulk motion vs the tip motion and their corresponding tendencies to shed vortices in the fluid. Specifically, optimal shape-morphing tends to minimize the overall velocity distribution on the plate and therefore mitigates overall vortex shedding. This is especially evident for the plate tips, for which optimal shape-morphing tends to minimize tip motion, see also Ref. 47. As a result, vorticity regions surrounding the plate are significantly reduced, and thus, hydrodynamic power dissipation is mitigated as well.

This conclusion is further supported by the investigation of the vorticity field in panel (c), where we present flow fields for the plate undergoing the highest intensity shape-morphing. It appears that for very large shape-morphing deformation, multiple vortices with comparable strength are generated at the tips, almost in a quadrupole fashion. While the vorticity patterns are somewhat more complex than those observed in the previous cases, it is important to note the appearance of a stronger pocket of negative vorticity on the trailing side of the each of the halves of the plate. Because of the intensity of shape-morphing deformation, in this case, the tips are actually moving against the gross motion of the plate and the negative vorticity pockets are now on the leading edge of tips, a phenomenon not observed in panel (a). This is arguably the physical cause of observed negative damping, as now low pressure areas tend to favor the motion of the plate. As compared to the case in panel (b), however, larger regions of vorticity can be observed in the flow, which are therefore associated with increased hydrodynamic power dissipation, as discussed above.

## V. CONCLUSIONS

In this paper, we studied the problem of a plate undergoing steady-state harmonic torsional oscillations in a quiescent viscous fluid while morphing its shape according to a prescribed deformation strategy. Specifically, the time-varying shape-morphing is imposed on the plate as to deform each of its two halves to an arc of the circle, in an antisymmetric fashion, with respect to the plate centroid. The goal of the shape-morphing strategy is to control the hydrodynamic moment and power dissipation during the oscillations. We have studied the problem in two dynamic regimes: the unsteady Stokes flow regime and the nonlinear regime considering the convective phenomena. We have solved the linear problem via a BEM approach and the nonlinear problem by a CFD method. In addition, we have proposed a rigorous analysis of the linear problem to characterize the hydrodynamic moment and power dissipation via scaling arguments. We have estimated moment and power dissipation numerically and analyzed their behavior as a function of the three governing parameters of the problem.

Our results demonstrate that the shape-morphing strategy can be effective in modulating force and power dissipation in plate systems subjected to flexural oscillation. Specifically, because of shape-morphing, the hydrodynamic added inertia decreases linearly as the curvature intensity increases. The damping moment is observed to decrease significantly, with a more complex behavior as a function of the applied curvature. In addition, the hydrodynamic power dissipation shows a nonmonotonic parabolic relationship with the imposed shape-morphing curvature. We also observe that for an optimum level of prescribed shape-morphing deformation, the imaginary part of the hydrodynamic moment, describing the hydrodynamic damping, changes sign and the hydrodynamic power dissipation reaches a minimum. The optimum range of the shape-morphing curvature is found to be *ξ* ≈ 4–6 for the frequencies explored in this study. These findings have been further analyzed from a physical perspective, by qualitatively observing the vorticity dynamics in the vicinity of the oscillating submerged structure for three cases corresponding to the imposed curvature, smaller, equal, and greater than the optimum value. The vorticity dynamics exhibits significantly different behavior in the three cases. Additionally, we hypothesize that multiple vortex generations in the vicinity of the tips, their mutual interactions, and their trajectories play an important role in minimizing the overall extent of the vortex-affected region and, thereby, the average hydrodynamic power dissipation during oscillation. In addition, we have shown how vortex dynamics could explain the occurrence of negative damping from a physics-based perspective.

In conclusion, this work demonstrates that optimal shape-morphing deformation can effectively reduce the hydrodynamic moment and power loss associated with torsional oscillation of submerged sharp-edged structures. This work introduces, for the first time, an effective way to control and reduce the large hydrodynamic damping moment resulting from convective nonlinearities during large amplitude torsional oscillation. Besides their direct applicability to torsional vibration-based systems, the outcomes of this study can be combined with the results for active shape-morphing in flexural oscillations^{47–49} to synthesize techniques to model and control the hydrodynamic response on submerged structures undergoing simultaneous flexural and torsional vibrations.

## ACKNOWLEDGMENTS

This study is based in part on the work supported by the National Science Foundation under Grant No. CMMI-1847513.

### APPENDIX A: CFD IMPLEMENTATION

In our CFD simulation campaign, we model a 2D circular fluid domain of diameter 20*b* with the plate at the center of the domain. The optimal domain size is determined via a sensitivity analysis reported below in Appendix B. The entire domain is then divided into two fluid zones to facilitate the implementation of a dynamic mesh technique to capture the motion of the solid: CV-1 and bulk, see Fig. 6. Zone CV-1 encompasses the most hydrodynamically significant *π*(1 × 1)*b*^{2} area surrounding the plate. This zone is meshed with a minimum cell size of 75 *μ*m. The minimum cell size is determined through a cell size convergence study discussed below in Appendix B. The choice of the triangular unstructured mesh in this zone is necessary to allow for the mesh update in the vicinity of the shape-morphing plate. On the other hand, the bulk zone consists of the quadrilateral dominant mesh. Cell size in the bulk zone varies from 75 *μ*m at the CV-1 boundary to 3 mm at the boundary of the bulk zone. To accommodate for the fine mesh grading, we further subdivide the bulk zone into five sections. Material properties are those of standard liquid water at 20 °C, provided in the software libraries. Specifically, we impose mass density per unit volume *ρ* = 998.2 kg m^{−3} and dynamic viscosity *μ* = 1.003 × 10^{−3} Pa s.

We perform the CFD simulations using the commercial software *ANSYS Fluent 16.1*. We use the dynamic mesh method to provide the prescribed motion of the plate and the corresponding mesh update at the end of each time step. To accommodate mesh deformation corresponding to the plate deformation in the CV-1 zone, the spring based smoothing method is used. In addition, since the deformations in the simulation are large as compared to the cell size, we use the local cell based remeshing method to update the mesh at the end of each time step. In particular, because of remeshing, the cell collapses if the cell size goes below 35 *μ*m, and a new cell is created if the cell size increases beyond 75 *μ*m. A user defined function (UDF) based on the macro DEFINE_GRID_MOTION is used to produce the grid-wise varying shape-morphing motion of the plate. To evaluate the instantaneous hydrodynamic power dissipation $P(t)$ using Eq. (8), an additional UDF based on the macro DEFINE_EXECUTE_AT_END is implemented at the end of each time step. Note that a time step sensitivity analysis, along with the choice of the time steps for each value of the frequency parameter, is reported in Appendix B.

### APPENDIX B: SIMULATION SENSITIVITY ANALYSIS

In this section, we discuss the sensitivity analysis performed to verify simulation convergence. We perform the test to obtain the optimum time step and mesh size by varying each of these parameters, while keeping the remaining parameters fixed. Since we have pressure outlet boundary conditions in the far field surrounding the oscillating plate, it can be presumed that as long as the domain boundaries are sufficiently far removed from the oscillating plate, the exact size of the domain would not affect significantly the results obtained from the simulation. In all our simulations, we use a domain size of area *π*(10*b*)^{2} mm^{2}, consistent with our previous work.

Since we utilize the dynamic mesh method to model the oscillating submerged shape-morphing plate, the choice of the time step plays a vital role in the accuracy and convergence of the simulation. To allow for proper mesh update during each step and satisfactory sampling of the simulation, it is required to reduce the time step so as to result in small changes in the spatial coordinates within a single time step. On the other hand, such a small time step demands large computational cost. Therefore, it is imperative to obtain a compromising value of the time step for each frequency parameter *β*. In Table II, we present the simulation results pertaining to different time steps for a representative case with parameters *β* = 500, *ϵ* = 0.05, and *ξ* = 3. From the analysis, the time step for the simulations is chosen as 2.00 × 10^{−4} for *β* = 500. Note, in fact, that decreasing the time step further to 7.50 × 10^{−4} increases the computational cost by approximately three times with only an approximately 5% difference in the simulation results. Furthermore, in Table III, we list the choice of the time step for the values of each frequency parameter *β*. These selections are in part guided by the Courant–Friedrichs–Lewy condition.^{51,63}

Time step (s) . | Samples per cycle . | $Re\Gamma $ . | %Diff. . | $Im\Gamma $ . | %Diff. . | $\u27e8P\u27e9$ . | %Diff. . |
---|---|---|---|---|---|---|---|

7.50 × 10^{−4} | 266 | 0.0363 | 8.794 | 0.0066 | 4.762 | 1.80 × 10^{−7} | 0 |

5.00 × 10^{−4} | 399 | 0.0363 | 8.794 | 0.0063 | 0 | 1.80 × 10^{−7} | 0 |

2.00 × 10^{−4} | 996 | 0.0398 | … | 0.0063 | … | 1.80 × 10^{−7} | … |

7.50 × 10^{−5} | 2654 | 0.0424 | 6.533 | 0.0060 | 4.762 | 1.80 × 10^{−7} | 0 |

Time step (s) . | Samples per cycle . | $Re\Gamma $ . | %Diff. . | $Im\Gamma $ . | %Diff. . | $\u27e8P\u27e9$ . | %Diff. . |
---|---|---|---|---|---|---|---|

7.50 × 10^{−4} | 266 | 0.0363 | 8.794 | 0.0066 | 4.762 | 1.80 × 10^{−7} | 0 |

5.00 × 10^{−4} | 399 | 0.0363 | 8.794 | 0.0063 | 0 | 1.80 × 10^{−7} | 0 |

2.00 × 10^{−4} | 996 | 0.0398 | … | 0.0063 | … | 1.80 × 10^{−7} | … |

7.50 × 10^{−5} | 2654 | 0.0424 | 6.533 | 0.0060 | 4.762 | 1.80 × 10^{−7} | 0 |

β
. | Time step (s) . | Samples per cycle . | Total samples . |
---|---|---|---|

100 | 5.00 × 10^{−4} | 1991 | 13 937 |

200 | 5.00 × 10^{−4} | 996 | 6 972 |

500 | 2.00 × 10^{−4} | 996 | 6 972 |

1000 | 7.50 × 10^{−4} | 1327 | 9 289 |

β
. | Time step (s) . | Samples per cycle . | Total samples . |
---|---|---|---|

100 | 5.00 × 10^{−4} | 1991 | 13 937 |

200 | 5.00 × 10^{−4} | 996 | 6 972 |

500 | 2.00 × 10^{−4} | 996 | 6 972 |

1000 | 7.50 × 10^{−4} | 1327 | 9 289 |

In Table IV, we present the convergence study performed for the choice of mesh size in the vicinity of the plate in zone CV-1. The choice of optimum mesh size is necessary to perform the simulations with a minimal computational cost and truncation error along with maintaining an acceptable accuracy. This mesh size is also maintained while using the cell based remeshing technique to generate an updated mesh after each five time step. For this analysis, we choose five mesh sizes: 50 *μ*m, 75 *μ*m, 100 *μ*m, 200 *μ*m, and 500 *μ*m. We select the mesh cell size for the simulations to be 75 *μ*m, which ensures a manageable computational cost for less than 5% tolerance.

Min cell size (μm)
. | Plate subdivision . | Total cells . | $Re\Gamma $ . | %Diff. . | $Im\Gamma $ . | %Diff. . | $\u27e8P\u27e9$ . | %Diff. . |
---|---|---|---|---|---|---|---|---|

500 | 25 | 13 929 | 0.0366 | 8.040 | 0.0059 | 6.349 | 1.80 × 10^{−7} | 0 |

200 | 50 | 30 045 | 0.0372 | 6.533 | 0.0061 | 3.175 | 1.80 × 10^{−7} | 0 |

100 | 100 | 110 128 | 0.039 | 2.010 | 0.0063 | 0 | 1.80 × 10^{−7} | 0 |

75 | 150 | 162 992 | 0.0398 | … | 0.0063 | … | 1.80 × 10^{−7} | … |

50 | 200 | 283 224 | 0.0402 | 1.005 | 0.0061 | 3.175 | 1.80 × 10^{−7} | 0 |

Min cell size (μm)
. | Plate subdivision . | Total cells . | $Re\Gamma $ . | %Diff. . | $Im\Gamma $ . | %Diff. . | $\u27e8P\u27e9$ . | %Diff. . |
---|---|---|---|---|---|---|---|---|

500 | 25 | 13 929 | 0.0366 | 8.040 | 0.0059 | 6.349 | 1.80 × 10^{−7} | 0 |

200 | 50 | 30 045 | 0.0372 | 6.533 | 0.0061 | 3.175 | 1.80 × 10^{−7} | 0 |

100 | 100 | 110 128 | 0.039 | 2.010 | 0.0063 | 0 | 1.80 × 10^{−7} | 0 |

75 | 150 | 162 992 | 0.0398 | … | 0.0063 | … | 1.80 × 10^{−7} | … |

50 | 200 | 283 224 | 0.0402 | 1.005 | 0.0061 | 3.175 | 1.80 × 10^{−7} | 0 |

In our analysis, we have observed only a very minor dependence of the simulation parameter requirements on the value of the parameter *ξ*, also consistent with previous results from our group.^{47} However, it seems that values of the parameter *ξ* close to optimality afford less stringent requirements on the simulation parameters for convergence. This is likely to be attributed to the fact that, near optimality, many of the steep gradients in the flow variables are smoothed out (for example, reduction of vorticity), and therefore, the flow is “simpler” in a certain sense. This trend can also be inferred by looking at the results in Table I, which shows for the Fourier coefficients *C*_{1} and *S*_{1}, an approximately linear dependence on the value of *ϵ*. This linear trend is significantly less evident in the absence of shape morphing *ξ* = 0; also see Ref. 39. Further indication of this phenomenon is the relatively small magnitude of the higher harmonics in Table I, which hints to the reduced total harmonic distortion^{25} of the hydrodynamic moment signal in the presence of shape morphing.