# Wake Interference Effects on Flapping Dynamics of Elastic Inverted Foil

Aarshana R. Parekh1<EMAIL_ADDRESS>Rajeev. K. Jaiman1  Mechanical
Engineering Department, The University of British Columbia, Vancouver, BC,
Canada V6T 1Z4

###### Abstract

Using high-fidelity simulations, we study the self-induced flapping dynamics
of an inverted elastic foil when it is placed in tandem with a stationary
circular cylinder. The effect of wake interference on the inverted foil’s
coupled dynamics is systematically examined at a fixed Reynolds number ($Re$)
as a function of non-dimensional bending rigidity ($K_{B}$) and the structure
to fluid mass ratio ($m^{*}$). Our results show that the inverted foil
experiences a relatively higher transverse fluid loading under unsteady wake
flow. Consequently, flapping instability is induced at a higher $K_{B}$ value
as compared to an isolated inverted elastic foil. There exists a critical
$K_{B,Cr}=0.25$, above which the downstream foil is synchronized with the
unsteady wake, and the cylinder controls the flapping response and the wake
vortex dynamics. During synchronization, two additional flapping modes are
observed as a function of decreasing $K_{B}$, namely small amplitude flapping
and moderate amplitude flapping modes. Below $K_{B,Cr}$, the downstream foil
undergoes self-induced large-amplitude flapping similar to an isolated foil
counterpart. Interestingly, unlike an isolated inverted foil, in the tandem
setup, the flapping dynamics are strongly influenced by both $K_{B}$ and
$m^{*}.$ The findings from this study aim to generalize our understanding of
the self-induced flapping dynamics of inverted foils and have relevance to the
development of inverted foil-based renewable energy harvesters.

###### keywords:

##  1 Introduction

When submerged in fluid flow, an elastic cantilevered foil experiences self-
induced, self-sustained periodic oscillations, commonly known as the self-
induced flapping phenomenon (Shelley and Zhang, 2011). When the foil is
cantilevered at the trailing edge (TE), with its leading edge (LE) left free
to oscillate, i.e., the inverted configuration, its flapping response presents
an intriguing fluid-structure interaction problem that is commonly observed in
nature and various engineering applications. In the past few decades, the
study of inverted foil flapping dynamics has observed an increase in
popularity owing not only to its prevalence but also to the rich non-linear
dynamics associated with complex wake vortex interactions and large
deformation of the flexible structure. Particularly, there is a need for a
generalized understanding of the underlying flapping mechanism for
optimization and control strategies to achieve the desired flapping response.

Studies have shown that for an inverted foil of a fixed aspect ratio ($AR$)
under steady uniform flow, the dynamics are characterized by three critical
non-dimensional parameters, namely Reynolds number ($Re$), the structure-to-
fluid mass ratio ($m^{*}$), and bending rigidity ($K_{B}$). At a given a $Re$,
there exists a critical $K_{B,Cr}$ below which the inverted foil loses its
stability and undergoes self-sustained flapping motion (Kim et al., 2013;
Gurugubelli and Jaiman, 2015; Sader et al., 2016). For a range of $K_{B}$,
known as the large amplitude flapping (LAF) regime, the foil undergoes limit
cycle oscillations with large peak-to-peak amplitudes. These large amplitude
oscillations are driven by the combined effect of periodic vortex shedding
from the leading edge and the foil’s inherent fluid elastic instability
(Gurugubelli and Jaiman, 2019). Compared to $K_{B}$, during LAF, the dynamics
are only weakly influenced by $m^{*}$ and relatively insensitive to changes in
$Re$ (Gurugubelli and Jaiman, 2015; Tang et al., 2015; Shoele and Mittal,
2016; Gurugubelli and Jaiman, 2019).

While extensive research has been conducted on inverted foil dynamics in
uniform flow, recent studies have demonstrated intriguing phenomena when the
foil is subject to unsteady flow, particularly the coupled dynamics of an
inverted foil when placed in tandem with a rigid bluff body. Studies like (Kim
et al., 2017) and (Ojo et al., 2022) have revealed that in some scenarios, due
to wake interference, an additional small amplitude flapping mode is observed
along with the modes observed for an isolated inverted foil. The occurrence of
these different modes is determined by the balance between fluid forces
arising from the foil’s interaction with the unsteady wake and the structure’s
elastic and inertial forces. The upstream unsteady wake, in turn, is dependent
on cylinder geometry and proximity.

While these findings shed light upon the inverted foil’s flapping response in
wake flow, its underlying physics is not completely understood and requires
further investigation. Such knowledge is especially relevant in the field of
small-scale renewable energy harvesting. Inverted foil has demonstrated
superior energy harvesting potential. Many energy harvesting models have been
proposed based on this approach wherein the fluid kinetic energy is harnessed
in the form of the foil’s internal strain energy during flapping and via
piezoelectric material, converting it to electric power (Allen and Smits,
2001; Akcabay and Young, 2012; Michelin and Doaré, 2013; Orrego et al., 2017).
While the electric power output for these devices typically relies on the
flapping amplitude and frequency of the foil, it is strongly dependent on
upstream wind and ocean environments. To achieve robust energy harvesting
systems with a consistent power output, it is critical to expand our knowledge
of the flapping response of an inverted foil in unsteady and perturbed flow
conditions, thus motivating our study toward the wake interference effect on
inverted foils.

This study examines the coupled dynamics of a flexible inverted foil subject
to the wake flow of an upstream rigid circular cylinder through a series of
high-fidelity numerical simulations. Through a comparative study of the
inverted foil response in isolated and tandem setups, we aim to investigate
the following key questions: (i) How does the wake interference influence the
foil’s flapping response? (ii) what is the intrinsic relationship between the
wake vortices and different flapping modes? and (iii) how does mass ratio
influence the flapping dynamics in the tandem setup? A systematic study with
the considered parameter range that addresses these questions has not yet been
studied to the best of the authors’ knowledge.

##  2 Problem setup and numerical methodology

We study the self-induced flapping dynamics of an inverted foil by considering
a Quasi-3D computational setup. Figure 1 displays the problem setup and the
computational domain with information on the boundary conditions. We consider
an inverted foil $\Omega_{s}$ of length $L$, width $W=0.75L$, and thickness
$h=0.01L$. For the tandem arrangement, the foil is positioned in the
downstream region of a rigid circular cylinder of diameter $D$ at a streamwise
distance $g^{*}$ along the center line. Both setups are subjected to uniform
incompressible axial flow $\Omega_{f}(t)$, with velocity magnitude
$U_{\infty}$ as shown in figure 1. The foil is placed with its TE at
$(0,0,0)$. A uniform velocity $\mathbf{u}^{f}=(0,U_{\infty},0)$ condition is
applied at the inlet $\Gamma^{f}_{in}$ boundary, and traction-free condition
$(\boldsymbol{\sigma}^{f}\cdot\mathbf{n}^{f}=0)$ is considered for the outlet
$\Gamma^{f}_{out}$ boundary. Here, $\Gamma^{f}_{top}$ and
$\Gamma^{f}_{bottom}$ represent the sides of the domain where the slip
condition is implemented. No-slip condition is applied for the cylinder and
foil surfaces and periodic boundary condition is applied for the domain sides.

Figure 1: Schematic of the computational domain of an inverted foil placed downstream of a stationary circular cylinder | $M1$ | $M2$ | $M3$
---|---|---|---
No. of nodes | 36670 | 73194 | 149212
No. of Elements | 114227 | 223481 | 451083
rms of transverse tip displacement $A^{rms}_{z}/L$ |  000.1039 (59.414$\%$) |  000.245 (4.29$\%$) | 0.256
Mean drag coefficient $\overline{C_{d}}$ | 0.0829 (70.22$\%$) | 0.2864 (2.8735$\%$) | 0.2784
rms lift coefficient $C^{rms}_{l}$ | 0.5059 (47.32$\%$) | 0.9562 (0.447$\%$) | 0.9605
Table 1: Grid convergence study results for the flexible inverted foil
interacting with wake flow of a stationary circular cylinder at $Re=1000$,
$m^{*}=0.1$, and $K_{B}=0.2$

The incompressible fluid flow dynamics are governed by unsteady three-
dimensional Naiver-Stokes equations in the arbitrary Lagrangian-Eulerian (ALE)
reference frame, which are defined as:

$\rho^{f}\frac{\partial\mathbf{u}^{f}}{\partial
t}+\rho^{f}\left(\mathbf{u}^{f}-\mathbf{w}\right)\cdot\nabla\mathbf{u}^{f}=\nabla\cdot\boldsymbol{\sigma}^{f}+\mathbf{f}^{f}\quad\text{in}\;\Omega^{f}(t),$
(1) $\nabla\cdot\mathbf{u}^{f}=0\quad\text{in}\;\Omega^{f}(t),$ (2)
$\boldsymbol{\sigma}^{f}=-p\mathbf{I}+\mathbf{T},\quad\mathbf{T}=2\mu^{f}\boldsymbol{\epsilon}^{f}(\mathbf{u}^{f}),\quad\boldsymbol{\epsilon}^{f}(\mathbf{u}^{f})=\frac{1}{2}\left[\nabla\mathbf{u}^{f}+(\nabla\mathbf{u}^{f})^{T}\right]$
(3)

where $\mathbf{u}^{f}=\mathbf{u}^{f}(\mathbf{x},t)$ and
$\mathbf{w}=\mathbf{w}(\mathbf{x},t)$ represent the fluid and mesh velocities
defined at each spatial point $\mathbf{x}\in\Omega^{f}(t)$, respectively.
$\rho^{f}$ and $\boldsymbol{\sigma}^{f}$ represent the density and Cauchy
stress tensor for a Newtonian fluid as defined in equation (3), and
$\mathbf{f}^{f}$ denotes the body force applied to the fluid. $p$ represents
the fluid pressure, $\mathbf{I}$ denotes the second-order identity tensor, and
$\mathbf{T}$ is the viscous stress tensor.

The elastic foil’s deformation is governed by the non-linear structural
equation, defined as:

$\rho^{s}\frac{\partial\mathbf{u}^{s}}{\partial
t}=\nabla\cdot\boldsymbol{\sigma}^{s}+\mathbf{f}^{s}\quad\text{in}\;\Omega^{s},$
(4)

where $\mathbf{u}^{s}=\mathbf{u}^{s}(\mathbf{z},t)$ is the structural velocity
defined at each point on the structure $\mathbf{z}\in\Omega^{s}$,
$\mathbf{f}^{s}$ represents the fluid forces applied on the solid, and
$\boldsymbol{\sigma}^{s}$ denotes the first Piola-Kirchoff stress tensor. In
this study, the structure is modeled as a St. Venant-Kirchoff hyperelastic
material. In this coupled fluid-structure system, the velocity and traction
continuity conditions are satisfied along the interface as follows:

$\int_{\boldsymbol{\varphi}^{s}(\gamma,t)}\boldsymbol{\sigma}^{f}(\boldsymbol{\varphi}^{s}(\mathbf{z},t),t)\cdot\mathbf{n}^{f}d\Gamma+\int_{\gamma}\boldsymbol{\sigma}^{s}(\mathbf{z},t)\cdot\mathbf{n}^{s}d\Gamma=0\quad\forall\gamma\subset\Gamma_{0},$
(5)

$\mathbf{u}^{f}(\boldsymbol{\varphi}^{s}(\mathbf{z},t),t)=\mathbf{u}^{f}(\mathbf{z},t)\quad\forall\mathbf{z}\in\Gamma_{0},$
(6)

where $\mathbf{n}^{f}$ and $\mathbf{n}^{s}$ are the outward normals to the
deformed fluid and the undeformed solid interface boundaries, respectively.
$\Gamma_{0}$ denotes the fluid-structure interface $\Gamma(t)$ at time $t=0$.
$\boldsymbol{\varphi}^{s}$ is the displacement function that maps each
Lagrangian point $\mathbf{z}\in\Omega^{s}$ to its deformed position at time
$t$. $\gamma$ is any part of the interface $\Gamma_{0}$ and
$\boldsymbol{\varphi}^{s}(\gamma,t)$ represents the corresponding fluid part
over the interface $\Gamma(t)$ at $t$.

A higher-order variational method based on fully coupled three-dimensional
Navier Stokes and nonlinear structural dynamic equations has been utilized to
model the coupled FSI problem. This methodology is based on the combined field
explicit interface (CFEI) formulation proposed by (Liu et al., 2014). The
solver’s numerical stability, validity, and ability to accurately simulate
flapping dynamics have been demonstrated in our prior work (Liu et al., 2014;
Gurugubelli and Jaiman, 2015). For a detailed description of the numerical
algorithm and implementation, the readers are referred to (Liu et al., 2014).
To determine the adequacy of the computation mesh, a convergence study is
performed for three different meshes $M1$, $M2$, and $M3$. The flexible foil
in the tandem arrangement has been considered at $Re=1000$, $m^{*}=0.1$, and
$K_{B}=0.2$. Table 1 lists the r.m.s of the normalized transverse tip
displacement $A^{rms}_{z}/L$ and lift coefficient, as well as the mean drag
coefficient for the different meshes. The bracketed values denote the
percentage difference in numerical simulations with respect to mesh $M3$. It
is seen that the relative errors using mesh $M2$ are less than $5\%$;
therefore, all the numerical simulations are carried out using mesh $M2$.

The dynamics of the flexible foil are strongly influenced by three key non-
dimensional parameters, i.e., $Re$, $K_{B}$ and $m^{*}$, defined as:

$Re=\frac{\rho^{f}U_{\infty}L}{\mu^{f}},\>K_{B}=\frac{B}{\rho^{f}U_{\infty}^{2}L^{3}},\;m^{*}=\frac{\rho^{s}h}{\rho^{f}L},$
(7)

where $E$, $\nu$, and $\rho^{s}$ denote Young’s modulus, Poisson’s ratio, and
density, respectively. $B$ represents the flexural rigidity, which is defined
as $B=Eh^{3}/12\left(1-v^{2}\right)$, and $K_{B}$ determines the relative
importance of the foil’s bending force to the fluid’s inertial force. The
derived quantities that we study include the lift and drag force coefficients.
These force coefficients are computed by integrating the element-wise
contributions of pressure and viscous stresses along the foil surface. The
drag coefficient $C_{d}$ and lift coefficient $C_{l}$ are defined as

$C_{d}=\frac{1}{\frac{1}{2}\rho^{f}U^{2}L}\int_{\Gamma}(\rho^{f}\cdot\mathbf{n})\cdot\mathbf{n}_{y}d\Gamma,\quad
C_{l}=\frac{1}{\frac{1}{2}\rho^{f}U^{2}L}\int_{\Gamma}(\rho^{f}\cdot\mathbf{n})\cdot\mathbf{n}_{z}d\Gamma,$
(8)

where $\mathbf{n}_{y}$ and $\mathbf{n}_{z}$ are the Cartesian components of
normal vector $\mathbf{n}$. Our present study considers a fixed $g^{*}=2.8L$
and $D=0.4L$. The considered $g^{*}$, is greater than the critical value $4D$
beyond which the shear layers roll up alternatively to form Kármán vortices
between the cylinder and the foil only experiences wake interference effects
(Zdravkovich, 1987; Sharman et al., 2005).

A fixed $Re=1000$ and $\nu=0.3$ is considered, and the foil’s dynamics in both
isolated and tandem setups are analyzed for $K_{B}\in[1\times 10^{-3},1]$ and
$m^{*}\in[0.1,10]$. The non-dimensional parameters $K_{B}$ and $m^{*}$ are
varied by changing Young’s modulus and density of the structure, respectively.

##  3 Results and discussion

Figure 2: Response characteristics of the inverted foil at $Re=1000$ and
$m^{*}=0.1$: (a) variation of maximum transverse tip displacement for the
isolated and downstream foil as a function of $K_{B}$, (b) variation of fluid
loading the isolated and downstream foil as a function of $K_{B}$, (c-d) power
spectra of the transverse vibration amplitudes and lift forces for the
downstream foil at $K_{B}=[0.15,0.2,0.25]$, (e) variation of flapping and lift
frequencies on bending stiffness $K_{B}$.

###  3.1 Response characteristics

We first investigate how the flapping response changes for an inverted foil
when placed in tandem with an upstream bluff body compared to the isolated
case. The foil’s coupled dynamics are characterized as a function of
decreasing $K_{B}$ at $m^{*}=0.1$. Figure 2a shows the variation of maximum
transverse tip displacements of the foil with respect to $K_{B}$. The maximum
transverse tip displacement is defined as
$A^{max}_{z}/L=\sqrt{2}A^{rms}_{z}/L$. Here $A^{rms}_{z}/L$, calculated using
$A^{rms}_{z}/L=\left(\sqrt{\frac{1}{n}\Sigma_{t}(A_{z}-\overline{A_{z}}})^{2}\right)/L$,
where $A_{z}$ and $\overline{A_{z}}$ represent the transverse tip displacement
and mean transverse displacement respectively. Compared to the isolated foil,
for which flapping instability is observed for $K_{B}<0.35$, in the tandem
arrangement, the foil performs sustained oscillations for $K_{B}\leq 1$. Thus,
the range of $K_{B}$ where flapping instability is observed is wider for the
inverted foil in the tandem arrangement than an isolated foil. Two additional
flapping regimes are identified before the downstream foil transitions into
the LAF regime based on the foil’s maximum transverse tip displacement, namely
the (i) small amplitude flapping regime (SAF) when $A^{max}_{z}/L<0.2$ and
(ii) moderate amplitude flapping regime (MAF) when $0.2<A^{max}_{z}/L\leq
0.5$. The typical full-body response of the downstream inverted foil in SAF,
MAF, and LAF regimes is shown in figure 2a. The SAF and MAF regimes are
observed for $0.6\leq K_{B}\leq 1$ and $0.2\leq K_{B}<0.6$, respectively. The
maximum transverse tip displacement for both regimes gradually increases as
$K_{B}$ decreases. Consequently, the transition to LAF is smoother as compared
to the isolated case. The downstream foil experiences the large amplitude
flapping for a narrower range of $K_{B}\in[0.1,0.15]$ compared to the isolated
case of $K_{B}\in[0.1,0.2]$. For $K_{B}<0.1$, as $K_{B}$ is decreased,
downstream foil transitions to deflected flapping mode and finally flipped
flapping mode for very small $K_{B}\in[0.01,0.001]$ similar to the isolated
foil.

Figure 2b shows the variation of the mean drag coefficient $\overline{C_{d}}$
and the root mean square of the lift coefficient $C^{rms}_{l}$ with respect to
$K_{B}$ for the inverted foil in both setups. Qualitatively, the drag force
exerted on the inverted foil in both setups follows similar trends as a
function $K_{B}$. Wherein the drag force increases as $K_{B}$ decreases,
instability is induced, and deformation increases. In contrast, an overall
higher lift force is exerted on the downstream foil as compared to the
isolated case. For $K_{B}>0.35$, a lift coefficient for the downstream foil is
around $O(10)$ times higher than the drag force and up to $O(10^{3})$ times
greater than the lift coefficient for the isolated foil thereby resulting in
the early onset of flapping instability. These results demonstrate that for
$K_{B}>0.35$, the oscillations of the downstream foil in the tandem setup are
driven by the fluid loading due to the unsteady wake.

Figures 2c and 2d show the power spectra of the coefficient of lift and
transverse tip displacement about the mean deformed position
$(A_{z}-\overline{A_{z}})/L$ for $K_{B}\in[0.15,0.25]$. Here, $f_{vs}$ denotes
the vortex shedding frequency of the upstream rigid circular cylinder. One
must note that at the considered separation distance $g^{*}=2.8L$, the
downstream inverted foil, does not influence the cylinder dynamics. Therefore,
a constant $f_{vs}=1.9342$ is observed for all values of $K_{B}$ based on the
Strouhal number of $0.21$. Multiple frequencies exist in both the temporal
response of $C_{l}$ and $(A_{z}-\overline{A_{z}})/L$ that are demonstrated by
the two dominant peaks observed in the frequency spectra till $K_{B}>0.2$,
below which a single dominant frequency is recorded.

Figure 2e demonstrates the variation of the dominant frequency ratio of the
transverse force $f_{C_{l}}/f_{vs}$ and tip displacement $f_{A_{z}}/f_{vs}$ of
the downstream foil with $K_{B}$. A frequency match exists between the
transverse tip displacement and lift force for all values of $K_{B}$. We
observe that for $K_{B}\geq 0.25$, the dominant frequency $f_{C_{l}}\approx
f_{A_{z}}\approx f_{vs}$ and is unaffected by the change in foil flexibility.
A sharp decline in the dominant frequency ratio is observed for $K_{B}<0.25$
with $f_{A_{z}}\approx f_{C_{l}}\approx 0.2f_{vs}$. These results demonstrate
that for $K_{B}>0.25$, the cylinder wake strongly influences the inverted
foil’ response, wherein the foil’s motion is synchronized with the cylinder
vortex shedding frequency. During synchronization, depending on foil
flexibility, the inverted foil undergoes high-frequency oscillations with
either small or moderate amplitudes. Below $K_{B}=0.25$, the influence of the
wake reduces, and the foil is no longer synchronized with the cylinder wake.
As a result, the downstream foil rapidly starts oscillating in LAF mode,
similar to an isolated foil, as $K_{B}$ is reduced.

###  3.2 Influence of wake interference on flow field characteristics

Figure 3: $x$-vorticity contours at $Re=1000$, $m^{*}=0.1$ for: (a-b) isolated
foil at $K_{B}=0.35$ and $K_{B}=0.15$, (c-d) downstream foil at $K_{B}=0.35$,
and $K_{B}=0.15$. The dotted and solid lines represent negative and positive
vorticity contours, respectively.

We next investigate the vortex organization and formation for the inverted
foil for $K_{B}\in[0.1,1]$ in both configurations. Figures 3(a-b) and 3(c-d)
represent the vorticity contours for the isolated foil and downstream inverted
foil at the maximum tip displacement above $Z/L=0$ at
$K_{B}=0.35\;\text{and}\;0.15$, respectively. As demonstrated in these
figures, for the tandem arrangement, a steady Kármán wake is generated in the
cylinder wake in the 2S shedding mode as described by (Willamson and Roshko,
1988). The wake vortices interfere with the shear layer developed on the
downstream foil surface and induce flapping instability for a higher $K_{B}$
value at which the isolated foil is still in the stable state, as shown in
figures 3a and 3c. Consequently, two single opposite signed vortices are
generated on the foil and shed into the wake per one time period, which
coalesce with the same signed vortices in the foil’s immediate vicinity,
resulting in a C(2S) vortex shedding pattern observed in the near wake of foil
(Willamson and Roshko, 1988). The C(2S) pattern quickly develops into the 2S
pattern further downstream. Our investigation revealed a similar wake topology
in the tandem arrangement for $K_{B}>0.25$.

At $K_{B}=0.15$, the downstream foil oscillates in the LAF regime similar to
an isolated foil as demonstrated by figures 3b and 3d. The wake vortices flow
above and below the foil surface and interact with the foil’s shed vortices in
the near wake. As a result, two pairs of vortices are shed over one cycle,
resulting in a 2P vortex pattern observed in the wakes like the isolated foil.
This 2P pattern is observed for the downstream foil for $K_{B}\in[0.1,0.2]$.
Based on these results, for $K_{B}\in[0.1,1]$ primarily, two vortex-shedding
patterns are observed in the wake of the downstream foil. The 2S mode is
observed for $K_{B}\geq 0.25$, below which 2P mode is observed. Thus, we can
deduce that when the inverted foil is synchronized with the vortex shedding of
the upstream cylinder, it undergoes 2S vortex shedding like the cylinder.
Below $K_{B}=0.25$, the foil starts behaving like an isolated foil and no
longer oscillates in sync with the cylinder vortex shedding. The consequent
change in flapping frequency leads to a change in the vortex shedding pattern
from 2S to 2P mode.

###  3.3 Effect of mass ratio $m^{*}$

Figure 4: Response characteristics of the downstream foil when
$m^{*}\in[0.1,10]$ at $K_{B}=[0.15,0.2,0.4]$: (a) variation of maximum
transverse tip displacement as a function of $m^{*}$, and (b) variation of
oscillation and lift frequency as a function of $m^{*}$

We next investigate the influence of mass ratio on the inverted foil’s
response in the tandem setup. Figures 4a and 4b show the maximum transverse
tip amplitude and the oscillation frequency as a function of the mass ratio at
$K_{B}=0.15,0.2$ and $0.4$. It is worth noting that the flapping amplitude is
synchronized with the lift force acting on the foil for all cases. These
figures show three distinct trends observed in the flapping response for
varying $K_{B}$ and $m^{*}$. At $K_{B}=0.4$, the flapping amplitude
monotonically reduces as $m^{*}$ increases. Moreover, synchronization between
the foil and the cylinder wake is observed for all values of $m^{*}$. For
moderately flexible foil at $K_{B}=0.2$, when $m^{*}$ is increased, the
flapping amplitude steadily grows and saturates at $A^{max}_{z}/L\approx
0.78$. Meanwhile, a marginal decrease is observed in the flapping frequency.
At $K_{B}=0.15$, the flapping amplitude is mostly unaffected by the variation
of $m^{*}$. Similar to $K_{B}=0.2$, an increase in $m^{*}$ results in a
decrease in the flapping frequency; however, the influence of $m^{*}$ is
comparatively stronger. As shown in figure 4b, synchronization between the
cylinder vortex shedding and the foil is not observed for any value of $m^{*}$
at $K_{B}=0.15$ and $0.2$.

As presented in $\S$3.1, for $m^{*}=0.1$ and $K_{B}=0.4$, the inverted foil is
relatively stiff, and it undergoes a flapping response induced by the unsteady
wake through the synchronization process. Owing to the increase in the foil’s
inertia force as $m^{*}$ increases, the unsteady loading from the wake is
insufficient in balancing the inertia forces and maintaining the flapping
amplitude. When stiffness is reduced, changes in $m^{*}$ have a drastically
different effect on the flapping amplitudes, as shown in figure 4a.

At $m^{*}=0.1$ and $K_{B}=0.2$, the downstream foil, due to decreased
stiffness, the foil is no longer synchronized with the wake. However, due to
wake interference, the downstream foil undergoes flapping in the MAF regime,
as shown in figure 4b. When the mass ratio is increased, the influence of wake
flow on the inverted foil dynamics reduces owing to the increase in the
density of the foil. As a result, the downstream foil now undergoes flapping
in the LAF-like mode observed in an isolated foil counterpart.

At $K_{B}=0.15$, the inverted foil is highly flexible, and the cylinder wake
has limited influence on the downstream foil flapping dynamics. It undergoes
LAF with maximum peak-to-peak amplitudes of $A^{max}_{z}/L\approx 0.85$ at
$m^{*}=0.1$ as observed in figure 4a. The flapping motion is driven
predominantly by the combined effect of periodic vortex shedding from the foil
and its inherent fluid-elastic instability, akin to an isolated foil (Kim et
al., 2013). As a result, when $m^{*}$ increases, the dampening effect on the
foil’s inertia forces by the wake is reduced, leading to a marginal increase
in the flapping amplitude and decrease in the frequency. Based on these
results, it is evident that in the tandem arrangement of foils with higher
stiffness values, the effect of the upstream cylinder is more predominant for
lighter foils as compared to heavier foils.

##  4 Conclusion

In this work, we have investigated the response of an inverted foil when
placed in tandem with a stationary circular cylinder using high-fidelity
numerical simulations. The dynamics are explored systematically as a function
of the non-dimensional bending stiffness and mass ratio at a fixed Reynolds
number and the gap ratio. Our analysis demonstrates that due to interaction
with the cylinder’s unsteady wake, the inverted foil experiences a larger
transverse fluid loading than the isolated case. Consequently, the downstream
becomes unstable for a higher stiffness value and undergoes sustained
oscillations for a wider range of the non-dimensional bending stiffness. As a
function of increasing flexibility, two additional flapping modes are observed
for the downstream foil before it transitions to large amplitude flapping
mode: (i) small amplitude flapping and (ii) moderate amplitude flapping modes.

There exists a critical non-dimensional bending stiffness value of
$K_{B,Cr}=0.25$, above which synchronization between the downstream foil and
cylinder vortex shedding is observed, which drives the foil’s flapping
response. During synchronization, the inverted foil oscillates with small to
moderate amplitudes as stiffness is reduced. Also, a C(2S) shedding pattern
wake is observed, which quickly develops to a 2S pattern in the near wake of
the foil. Below this threshold stiffness, the cylinder loses control over the
foil, and the downstream foil undergoes sustained flapping like an isolated
foil. Upon investigating the downstream inverted foil dynamics at different
mass ratios, our results indicated that, compared to uniform flow, when
subject to the unsteady wake of a bluff body, the wake interference effects on
the foil, along with the occurrence of various flapping modes, are highly
influenced by the mass ratio. These results can be applied towards the
development of novel optimization and control concepts that maximize the
energy-harvesting potential of inverted foils as well as ensure a consistent
energy output. Eventually, these control strategies can be leveraged for the
design of efficient and robust inverted foil-based energy-harvesting array
systems.

[Acknowledgements]The research was enabled in part through computational
resources and services provided by The Digital Research Alliance of Canada
(https://ccdb.alliancecan.ca/) and the Advanced Research Computing facility at
the University of British Columbia (https://arc.ubc.ca/).

[Funding]The authors would like to acknowledge the Natural Sciences and
Engineering Research Council of Canada (NSERC) for funding the project.

[Author ORCIDs]A. Parekh, https://orcid.org/0009-0007-3454-7871

[Declaration of Interests]The authors report no conflict of interest.

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