###### Abstract

Three-dimensional wake of an oscillating foil with combined heaving and
pitching motion is numerically evaluated at a range of chord-based Strouhal
number (0.32 $\leq St_{c}\leq$ 0.56) and phase offset (90∘ $\leq\phi\leq$
270∘) at $Re=$ 8000\. The changes in $\phi$ and $St_{c}$ reflect a unique
route of transition in mechanisms that govern the origin of spanwise
instabilities and growth of secondary wake structures. At lower $St_{c}$,
heave dominated kinematics demonstrates a strong secondary leading edge vortex
($LEV$) as the source of growing spanwise instability on the primary $LEV$,
followed by an outflux of streamwise vorticity filaments from the secondary
$LEV$. With increasing heave domination, the origin of stronger spanwise
instability is governed by a counter-rotating trailing edge vortex ($TEV$) and
$LEV$ that leads to growth of streamwise secondary structures. A decreasing
heave domination ultimately coincides with an absence of strong $LEV$
undulations and secondary structures. The consistent transition routes are
represented on a phase-space map, where a progression of spanwise instability
and growth of secondary structures becomes evident within regimes of decreased
heave domination. The increasing strength of circulation for the primary
$LEV$, with increasing $St_{c}$, provides a crucial reasoning for this newly
identified progression.

On the Association of Kinematics, Spanwise Instability and Growth of Secondary
Vortex Structures in the Wake of Oscillating Foils

Suyash Verma1, Muhammad Saif Ullah Khalid1,2, and Arman Hemmati1⋆

1Department of Mechanical Engineering, University of Alberta, Edmonton,
Alberta T6G 2R3, Canada

2Department of Mechanical and Mechatronics Engineering, Lakehead University,
Thunder Bay, ON P7B 5E1, Canada

⋆Corresponding Author, Email<EMAIL_ADDRESS>
##  1 INTRODUCTION

Achieving fast decay of vortical structures is crucial in modern high-speed
aircraft to reduce the risk of incidents during take-off or landing, while
also improving stealth for tactical submarines through reduced noise
propagation [19, 25, 18]. Such aspects are also useful in technologies that
improve the propulsive lift and thrust generation of bio-inspired robotic
swimmers during marine surveillance [36], operations of conventional
helicopter rotors [52], and flight performance of micro aerial vehicles
(MAV$s$) [39]. In the past, vortex decay processes required fundamental
understanding of their formation and advection through experiments. These
studies conducted assessments on equal or unequal strength counter-rotating
vortical pairs [19, 25, 6, 22]. The observations suggested novel interaction
mechanisms of paired structures that eventually developed wavy undulations in
spanwise direction, which were then characterized by a particular wavelength
and periodicity [25, 22]. Detailed theoretical and evolution aspects of vortex
instabilities had been considered, which helped out understanding
disintegration of vortical structures in wakes [9]. On a similar note, origins
of wake instabilities garnered sufficient attention besides characterizing
their spatio-temporal development stages during the evolution process [30,
53]. This further revealed intricate physical connection to the formation of
secondary vortical structures as well [30], whose role in promoting wake
three-dimensionality was immense. Primarily, such secondary vortical
structures had been largely characterized in wakes of stationary and
oscillating bluff bodies including cylinders (with varying cross-sectional
shapes) and hydrofoils or airfoils [30, 4, 31, 32, 10, 34]. Currently,
association of secondary wake structures with prevailing spanwise instability
on the primary vortex still require more attention, specifically for foils
oscillating in complex multi-degree of freedom motion [42]. This study extends
our knowledge by characterizing the origins of secondary wake structures,
including their inherent association with the governing spanwise instability
and prescribed coupled kinematics of an oscillating foil.

Elliptic instability had been primarily associated with counter-rotating pairs
of spanwise coherent structures (or rollers), either with equal or unequal
strength [19, 25, 22]. Experiments and numerical studies that modeled such
instabilities focused on isolated pair of vortices, wherein the temporal
evolution of spatial dislocations on rollers was quite vivid [19, 25, 6, 22].
Crow’s linear stability model effectively described the nature of equal
strength counter-rotating vortex pair [35], where the early growth phase of
instability coincided with vortex pair merger at specific spanwise intervals.
This subsequently led to vortex ring-like arrangement [35]. This model was
further extended to a system of multiple rollers, including pairs that
constituted unequal strength counter-rotating spanwise coherent structures
[25]. Klien et al. [17] assessed the wrapping mechanism for a weak vortex
filament around the stronger paired roller, which eventually yielded a vortex
loop. Stability analysis for paired vortices was conducted by Ortega and Savas
[16], wherein, the qualitative observations supported the previous analysis of
Klien et al. [17]. A follow-up study also provided a quantitative assessment
in terms of circulation strength ($\Gamma$), internal strain-field
distribution, and evolution mechanisms of sinusoidal instability. This
revealed spatial wavelength of lower magnitude [25], compared to Crow’s
estimation for equal strength vortex pair [35]. Bristol et al. [6] extended
the analysis to a co-rotating vortex pair that depicted formation of vorticity
bridges on account of the elliptic instability that subsequently contributed
to vortex merger in the wake.

Besides the mentioned fundamental studies on spanwise instabilities of
isolated vortex pairs, their dominant contribution to wake three-
dimensionality had been well observed for stationary bluff bodies, such as a
circular cylinder [53, 3, 4] or the blunt trailing edge airfoil [32, 13].
Experimental visualizations suggested that the growth of streamwise coherent
structures, called ribs, were also associated with spanwise instability modes
that possessed different spatio-temporal characteristics [53]. Mode A, in case
of stationary cylinder, was characterized by core vorticity outflux from the
rollers that consequently led to tongue like dislocations in the wake [53,
30]. These later elongated into pairs of counter-rotating streamwise filaments
with alternating periodicity and a spanwise wavelength ($\lambda_{z}$) equal
to four times the cylinder diameter [53]. Mode B, in contrast, featured
streamwise filaments characterized by $\lambda_{z}$ on the order of cylinder
diameter [53]. These thin strands were associated with the straining of
streamwise vorticity that existed in the braid region between two consecutive
rollers [53, 3, 4]. Mittal and Balachander [30] highlighted the formation of
secondary hairpin-like structures that also appeared as spatial dislocation on
spanwise rollers, shed in the wake of stationary circular cylinder. Upon
evolution, these dislocated formations elongated and formed rib pairs [30].
These studies [53, 30] therefore confirmed that the intermediary structures
played a pivotal role in governing three-dimensionality of the wake. Extension
of this work on cylinders with varying cross-sections (i.e.square) also
exhibited other instability modes [31], while characterizing their association
with secondary vortex pairs. Even for the case of a blunt trailing edge
($BTE$) airfoil, experiments [13] and computational [32] studies utilizing
Floquet stability analysis, highlighted the existence of instability modes
(Mode S), whose spatio-temporal features of secondary wake structures were
different compared to previously identified modes [53]. Extensive knowledge
and understanding obtained in cases of stationary bluff bodies motivated fluid
dynamicists to expand the instability assessments onto rigid bodies that
perform prescribed oscillations [10, 34, 42, 40, 24, 37, 27]. These outlined
different aspects of secondary wake structures and how they influenced the
wake three-dimensional features.

Kinematics prescribed on foils generally included single degree of freedom
oscillations in the form of either pitching or heaving [10, 34, 40, 11, 12,
41, 8]. The aspects of three-dimensional wakes for such oscillating bodies
were initially evaluated for a pitching foil with low aspect ratios [11, 12,
41]. Visbal [40] initially expanded on the secondary vorticity outflux
observed near the leading edge of a heaving foil at high Reynolds numbers
($Re\approx$ 104). Existence of long and short wavelength instability modes
and the associated secondary vortex arrangement were also examined [10], where
the spatio-temporal features of modes differed in terms of spanwise wavelength
and streamwise periodicity. Deng et al. [10] performed Floquet analysis for
the case of pitching foil that highlighted a direct correspondence of
secondary vortex pair formation, in conjunction with asymmetric arrangement of
primary rollers shed behind the foil. Sun et al. [34] discussed features of
Mode A, Mode B, and Mode S for a heaving foil that were characterized by
$\lambda_{z}\approx$ 1.05, 0.2, and 0.39, respectively. Some studies also
presented limited assessments on foils with combined pitching and heaving
oscillations [42, 21]. Floquet stability analysis of Moriche et al. [21] on a
foil with combined heaving and pitching motion provided details on the three-
dimensional wake transition, along with associated impacts on aerodynamic
force generation. Although the observations showed minimal effects on forces,
the simulations revealed that the onset of three-dimensionality on the
infinite span oscillating wing was linked to the bending of trailing edge
vortex ($TEV$) [21]. Cheireghin et al. [8] observed the existence of
sinusoidal undulation on the shed $LEV$ filament in the wake of a high aspect
ratio heaving swept wing. The origins, however, remained unclear and they were
speculated to be either an instability of oscillating shear flow, mixing
layer, or the vortex filament itself [8]. It was also noted by Cheireghin et
al. [8] that increasing circulation of $LEV$ at high reduced frequencies ($k$)
led to stronger deformations, which also coincided with significant effects on
lift and bending moments. The presence of secondary structures and their
association with the $LEV$ dynamics was, however, not evident in the study of
Cheireghin et al.[8]. Verma and Hemmati [42] investigated the three-
dimensional wake of an infinite span foil with combined heaving and pitching
oscillations, at $Re_{c}=$ 8000, which confirmed that despite changes in
spatial arrangement of rollers in the wake, spanwise instability on the
leading and trailing edge vortical structures closely resembled cooperative
instability [19, 25, 6, 22] of the counter-rotating or co-rotating vortex
pairs [42]. The wavelength ($\lambda_{z}$), estimated for $LEV$ filaments,
were in strong agreement with previous experimental studies [19, 25, 22].
Verma $\&$ Hemmati [42] further reported that tongue-like vortex dislocations
and conjoint horseshoe-hairpin secondary structures had an apparent
association with the cooperative instability [19, 25, 6, 22], in terms of
similar magnitude for the estimated $\lambda_{z}$. While the presence of
secondary wake structures provided few unique insights into wakes of an
oscillating foil, discussion with respect to onset mechanisms of instability
and their association to prescribed kinematics was fairly limited [42]. Son et
al. [37] recently provided comparative assessments of $LEV$ dynamics on an
oscillating finite aspect ratio wing, and an airfoil of infinite span, with
prescribed heaving motion at $Re=$ 104. Their experimental and computational
observations were in agreement with the hypothesis of Verma $\&$ Hemmati [42],
in terms of the counter-rotating rollers being responsible for the resulting
spanwise undulations of $LEV$ filament. Their study [37] also found a
dependence of $LEV$ spanwise undulations on the reduced frequency ($k$), which
contributed to differences in circulation ratio of $LEV$ and $TEV$ within one
oscillation cycle. This coincided with the occurrence of Crow’s instability
mechanism [35] triggered by either a $LEV-TEV$ pair (at $k=$ 2 and 3), or an
$LEV$ paired with its image $LEV$ structure that generated on account of
vortex-foil interaction [40] (at $k=$ 1) [37]. A similar qualitative
assessment of changes in $LEV$ instability features that were associated with
strong vortex-foil interaction, with an increase in the aspect ratio ($A.R$)
of the heaving foil, was recently completed by Hammer et al. [27] at $Re=$
$2$x$10^{5}$. Despite this higher $Re$, the qualitative and quantitative
observations were in agreement with instability aspects reported at lower $Re$
[42, 40, 8, 37]. The effects were dominant as $A.R$ increased from 4 to 8, but
above 8, the changes were not noticeable. Although an association of $LEV$
instability with $k$ and $A.R$ of oscillating foil were apparent in the study
of Son et al.[37] and Hammer et al.[27], respectively, they were limited to
heaving and a single motion or geometrical parameter (i.e. $k$ or $A.R$). This
hints at the need for an adequate extension of this particular association to
more complex prescribed kinematics, and its relationship to eventual growth of
secondary wake structures.

Here, we provide detailed insights on specific mechanisms that govern the
onset of a spanwise instability on primary rollers, and its association with
the growth of secondary wake structures. The work particularly expands on the
results of Verma $\&$ Hemmati [42, 47], by looking at a wider kinematic
parameter space (i.e chord based Strouhal number, $St_{c}$, and phase offset
between heave and pitch, $\phi$) for an oscillating foil in combined heaving
and pitching motion. As identified in our previous study [42], spanwise
undulations observed on rollers near the leading edge held key details for the
induced spanwise instability, that consequently promoted the growth of
secondary wake structures. Thus, to ensure that a strong $LEV$ generation is
captured in our numerical study, we specifically focus on the oscillations
that govern heave dominated coupled kinematics [48]. A brief preliminary
discussion was recently presented by Verma $\&$ Hemmati [49], which outlined
the association of spanwise wake instability and the growth of secondary wake
structures at $St_{c}=$ 0.32 and $\phi=$ 90∘. This paper elaborates their
findings [49], while also revealing the effects on spanwise instability and
secondary wake structures, as kinematics slowly undergoes transition and
enters a pitch-dominated regime of foil motion. The methodology and problem
description in Section 2 provide further details on the prescribed kinematics,
along with the numerical procedure employed in the current study. This is
followed by a comprehensive discussion of novel findings in the Section 3, and
a summary of the current ongoing work in Section 4.

##  2 Problem Description

The flow around an infinitely span foil with a maximum thickness ($D$) to
chord length ($c$) ratio of $D/c=0.1$ is examined numerically for a range of
chord based ($St_{c}=fc/U_{\infty}=0.32-0.56$) and amplitude based Strouhal
numbers (0.05 $\leq St_{A}\leq$ 0.4). The foil cross-section shown in zoomed
snippet of Figure 1 resembles a teardrop hydrofoil shape, which was used in
recent experimental investigations [50, 51]. These studies focused on
propulsive performance and scaling relations of underwater propulsors, and
suggested that the teardrop foil geometry represented a simplified model for
fish tail fins [50]. The Reynolds number is $Re=U_{\infty}c/\nu=$ 8000, which
is consistent with previous studies [42] and agrees closely with the
biological characteristics of swimming fish and bio-inspired flights [1, 36].
Here, $U_{\infty}$ and $\nu$ represent the freestream velocity and fluid’s
kinematic viscosity, respectively.

Figure 1: Schematic of the foil geometry and motion.

The prescribed foil kinematics is characterized by a coupled heaving and
pitching motion, where the location of pitch axis is approximately 0.05$c$
from the leading edge. Figure 1 marks the heave and pitch amplitudes as
$h_{o}$ and $\theta_{o}$, respectively. The resultant trailing edge amplitude
is also shown as $A_{T}$. The motion profiles of heave ($h$) and pitch
($\theta$), where pitching has a phase advancement (or offset) of $\phi$
relative to heaving, are represented using:

$h(t)=h_{o}\sin(2\pi ft)$ (1) $\theta(t)=\theta_{o}\sin(2\pi ft+\phi)$ (2)

In order to present a broader association of spanwise instability, secondary
wake structures and kinematics, we vary the phase offset ($\phi$) between
heaving and pitching motion in the range of 90∘ and 270∘. This leads to
changes in $A_{T}$ relative to a fixed $h_{o}/c$ ( = 0.25) and $\theta$ =
(10∘) [48]. $\phi$ below 225∘ are suggestive of a heave-dominated kinematics,
based on the relatively higher $h_{L}$ compared to $A_{T}$ [44]. At $\phi\geq$
225∘, the oscillating foil kinematics starts transitioning towards pitch
dominated motion settings [44]. On account of variation in $\phi$ and $A_{T}$,
Strouhal number ($St_{A}=2fA_{T}/U_{\infty}$) is varied in the range specified
earlier, where the maximum $St_{A}$ corresponds to both $\phi=$ 90∘ and 270∘,
while the least $St_{A}$ is observed at $\phi=$ 180∘ [48]. Anderson et al.[1]
indicated that significant transitions in the wake of flapping foils were
observable at 0.2 $<St_{A}<$ 0.4. This also coincides with the range
corresponding to optimal propulsive efficiency in swimming mammals [36, 38].
In addition to above considerations, the most propulsively efficient phase
offset is 90∘ according to Anderson et al.[2], which was also employed by
Zheng et al.[55] and Lagopoulos et al.[20]. This phase offset corresponds to
$\phi=$ 270∘ following the reference coordinate system employed by Van Buren
et al.[50].

###  2.1 Numerical details

The continuity and Navier-Stokes equations were solved directly using
OpenFOAM, which is a numerical package based on the finite-volume method. It
is extensively used for simulating wake dynamics of oscillating foils and
panels [42, 43, 44, 45, 46, 48, 33, 14, 55, 5]. The oscillatory foil dynamics
was modeled using Overset Grid Assembly (OGA) method, based on a stationary
background grid and a moving overset grid that were merged for the simulation
[28]. Extensive details of the method can be found in Verma $\&$ Hemmati [42,
46].

The computational domain is also presented in Figure 1, which highlights the
C-type overset boundary containing the foil. The boundary conditions at the
inlet are prescribed a uniform fixed velocity (Dirichlet) and a zero normal
gradient (Neumann) for pressure. At the outlet, a zero-gradient outflow
boundary condition is implied [10]. The top and bottom walls are further
prescribed a slip boundary condition that effectively model open-channel or
free-surface flows, and closely resemble the experimental and computational
conditions of Van Buren et al.[50] and Hemmati et al.[14], respectively. At
the foil boundary, a no-slip condition for velocity and a zero-gradient
condition for pressure is ensured. The periodic boundary condition is further
implemented on the side boundaries, coinciding with the spanwise extent of the
foil. This provides an effective way to model an infinitely span case without
the end or tip effects. The spanwise length of the foil ($\approx$ $\pi c$)
follows the assessments conducted in recent studies [54, 49], which showed
imminent presence of three-dimensionality with a similar range of kinematics
assumed here. A spanwise domain length analysis was further conducted here to
establish the insensitivity of the wavelength for spanwise instability on the
spanwise length of the computational domain. Figure 2 further exhibits non-
homogeneous hexahedral grid, comprising of a C-type grid to model the foil
geometry and motion. This region overlaps a rectangular background grid.
Higher grid refinements in critical overlapping regions ensured accurate
simulation of the near body vortex shedding and wake interactions. The grid
further expands towards the domain boundaries with an expansion factor of less
than $1.02$. The flow is solved using overPimpleDyMFoam solver, which
integrates the functionality of OGA and PIMPLE algorithms. Second order
accurate backward and central difference schemes are employed for temporal and
spatial discretizations, respectively [33]. The momentum equations are solved
using Preconditioned Biconjugate Gradient (PbiCGSTAB) method, whereas
Preconditioned Conjugate Gradient (PCG) method is employed for the Poisson
equation [10, 55]. The simulations are completed using the Cedar and Narval
high performance clusters, operated by Digital Research Alliance of Canada.
The parallel decomposition and assignment of computational domain utilizes 96
CPUs with a total of 190 GB memory and 1440 simulation hours per case.

Figure 2: Schematics of the grid and Overset Grid Assembly method. Table 1: Grid refinement details for the current study. $N_{total}$ represents the sum of hexahedral elements in background grid and overset grid. Study | $N_{total}$ | $\overline{C_{T}}$ | $C_{L}^{rms}$ | $\epsilon_{T}$ | $\epsilon_{L}^{rms}$ | $x=$ | $1c$ | $2.5c$ | $5c$
---|---|---|---|---|---|---|---|---|---
Grid1 | $8.4\times 10^{6}$ | 0.64 | 2.86 | 0.084 | 0.010 | $\delta^{*}=$ | 7.3 | 16.9 | 55.2
Grid2 | $1.7\times 10^{7}$ | 0.60 | 2.84 | 0.017 | 0.003 | $\delta^{*}=$ | 3.7 | 8.5 | 15.8
Grid3 | $3.1\times 10^{7}$ | 0.59 | 2.83 | 0.001 | - | $\delta^{*}=$ | 1.8 | 4.2 | 7.9
Exp. | - | 0.59 | - | - | - | $\delta^{*}=$ | - | - | -

A spatial convergence analysis is completed at $Re=$8000, $h_{o}/c=$0.25,
$\theta_{o}=$15∘, $\phi=$270∘ and $St_{c}=$0.67. This enables comparative
evaluation of the numerical results with respect to experiments of Van Buren
et al.[50]. Table 1 and Figure 3 summarize the grid convergence results
involving three grids, Grid1, Grid2 and Grid3. The ratio ($\delta^{*}$) of
minimum grid size element ($\Delta x$) to Kolmogorov scale ($\eta$) is kept
approximately below 10, within the critical region near the foil ($x<$
2.5$c$), specifically for Grid2 and Grid3 (see Table 1), where the origin of
spanwise instability and secondary structures are speculated to emerge and
grow as the wake evolves [42, 37]. Here, Kolmogorov scale is calculated based
on kinematic viscosity ($\nu$) and the dissipation rate of turbulence kinetic
energy ($\epsilon$), computed by $\eta\approx(\nu^{3}/\epsilon)^{0.25}$ [29].
The coefficient of thrust ($\overline{C_{T}}=\bar{T}/0.5\rho
scU_{\infty}^{2}$) and root-mean-square ($rms$) of lift coefficient
($C_{L}=L/0.5\rho scU_{\infty}^{2}$), which is averaged over the final 10
oscillating cycles following the statistical convergence, are used as a
quantitative estimate for spatial grid convergence. The relative error in
prediction of $\overline{C_{T}}$
($\epsilon_{T}=|\overline{C_{T}}_{,exp}-\overline{C_{T}}|/\overline{C_{T}}_{,exp}$),
calculated with respect to the experimental results [50], was below 5$\%$ for
Grid2. Similarly,
$\epsilon_{L}^{rms}(=|C_{L,Grid3}^{rms}-C_{L}^{rms}|/C_{L,Grid3}^{rms})$,
calculated with respect to the finest grid (Grid3) was below 0.1$\%$. The
corresponding experimental results for $C_{L}$ are not yet available.

(a) (b)

Figure 3: Comparison of (a) unsteady variation of $C_{T}$, and (b) cross-
stream velocity profiles at increasing streamwise distance ($x^{+}$) for three
grids.

Figure 3(a) shows the instantaneous $C_{T}$ for the three grids, which
demonstrates the insensitivity of propulsive performance to the spatial grid
resolution. Similar agreement also holds in terms of wake characteristics,
where the cross-stream distribution of mean streamwise velocity
($\overline{U_{x}^{+}}$) at increasing streamwise distance $(X^{+})$ (Figure
3b) are relatively similar for successively refined grids. This provides
sufficient confidence to use Grid2 for our analysis. The time-step size
($\delta t$) is also set in agreement with Verma $\&$ Hemmati[42]. The ratio
of eddy turnover time ($\tau_{\eta}$) to the set $\delta t$ for the smallest
dissipative eddy [29] yield at least 100 time-steps, which thus ensures that
our simulations have adequate resolution to capture essential turbulent
characteristics [23]. Extensive details for verification and validation of the
numerical solver with respect to the domain size, spatial and temporal grid,
OGA solver and boundary conditions can be found in Hemmati et al.[14] and
Verma $\&$ Hemmati[42, 46].

##  3 RESULTS AND DISCUSSION

We begin with elucidating the onset of spanwise instability on the $LEV$s and
associated growth of secondary wake structures at a low chord-based Strouhal
number ($St_{c}=$ 0.32), while $\phi$ varies from 90∘ to 270∘. The transition
in growth mechanisms that leads to the onset of secondary wake structures is
described, which coincide with the changes of kinematics from heave to pitch
domination. Moreover, the progression of secondary structure growth at
kinematics governed by onset of pitch domination ($\approx$270∘) is further
evaluated at an increasing $St_{c}$. Quantitative assessments of circulation
strength of primary $LEV$s reveals physical reasoning behind the observed
progression.

###  3.1 Transition in growth mechanisms of secondary wake structures

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

Figure 4: Wakes corresponding to (a-d) $\phi=$ 90∘, (e-h) $\phi=$ 180∘, (i)
$\phi=$ 225∘ and (j) $\phi=$ 270∘, at $St_{c}=$ 0.32. The oscillation time
varies from $t^{+}=$ 0 (a,e) to 0.75 (d,h) in increments of 0.25 for the cases
of $\phi=$ 90∘ and 180∘. $t^{+}$ corresponds to 0.75 for $\phi=$ 225∘ and
270∘. Each stage is represented using iso-surfaces of $\lambda_{2}(=-0.05)$,
which are colored based on $|\omega_{z}^{+}|=$ 5.

We first describe the growth mechanisms that primarily govern the onset of
secondary wake structures at $St_{c}=$ 0.32. Figures 4(a-d) and 4(e-h) present
the three-dimensional instantaneous wake structures for the cases
corresponding to $\phi=$ 90∘ and 180∘, respectively. The normalized time
instants ($t^{+}$= $t/T$) in Figure 4 vary from $t^{+}=$ 0 to 0.75, in
increments of 0.25. The primary vortex (or rollers marked as $LEV1_{ac}$ and
$LEV2_{ac}$) and coherent streamwise secondary structures in the form of
elongated hairpin-like formations (marked by $HP$) are clearly identifiable
for both cases. These wake structures are represented using iso-surfaces of
$\lambda_{2}$ criterion [15], which is normalized with respect to $U_{\infty}$
and $c$, i.e. $\lambda_{2}^{+}$ =$\lambda c^{2}/U_{\infty}^{2}$. This method
had been proven effective and accurate to extract vortex cores [15].

At $\phi=$ 90∘ (see Figure 4(a)), as the primary $LEV_{1}^{ac}$ gains strength
and grows in size, a secondary roller ($LEV1_{s}^{c}$) begins to form in close
vicinity of $LEV_{1}^{ac}$. The outflux of counter-rotating vorticity via
vortex-foil interaction is likely the cause of this secondary $LEV$ formation,
as was also suggested by Visbal [40] and Son et al.[37]. However, as
$LEV1_{s}^{c}$ further evolve downstream, the cooperative elliptic instability
[19, 25] starts developing on these counter-rotating vortex structures of
unequal strength, which then leads to undulations of both $LEV$ filaments (see
Figure 4(b)). This is followed by a secondary streamwise vorticity outflux via
$LEV1_{s}^{c}$, owing to its relatively weak strength compared to
$LEV_{1}^{ac}$, which subsequently reveals weak hairpin-like structures
(marked as $HP_{1}$ in Figure 4(c)). The legs of these hairpin-like structures
elongate to form ribs, downstream of the foil trailing edge due to the braid
vorticity straining, similar to the mechanism outlined by Mittal $\&$
Balachander [30] in the wake of a stationary circular cylinder. An interaction
of primary and secondary $LEV$ rollers is the governing mechanism for the
growth of secondary structures at $\phi=$ 90∘, which coincides with heave
dominated kinematics. We identify this mechanism by label $``1"$ in this
study. The spanwise hairpin-like arrangement that originate via mechanism
$``1"$ will therefore be referred as $HPn^{1}$, where $n$ is used for vortex
numbering in the wake, corresponding to different settings of kinematics.

The stages of the wake evolution at $\phi=$ 180∘ are now discussed to
understand the distinct changes in characteristics of spanwise instability and
secondary structures, as kinematics change compared to $\phi=$ 90∘. Verma $\&$
Hemmati [48] reported that there is a clear decrease in trailing edge
amplitude relative to the leading edge, which coincided with a decrease in
peak effective angle of attack ($\alpha_{o}$) within an oscillation cycle.
However, both cases still represent a heave-dominated kinematic regime for
oscillating foils in the coupled motion [44, 48].

Figures 4(e-h) depicts three-dimensional iso-surface ($\lambda_{2}=$ -0.05)
plots at every quarter of the oscillation cycle. At $t^{+}=$ 0 (Figure 4(e)),
development of a primary roller ($LEV_{2}^{ac}$) is observed near the leading
edge while hairpin-like structures (marked $HP2^{\prime}$) are evident
downstream of the trailing edge. These secondary structures are formed through
a similar process of core vorticity outflux and subsequent shear straining of
streamwise filaments that emanate from rollers during the oscillation cycle
[53, 30]. However, it is still important to investigate if these streamwise
vorticity filaments remain associated with a secondary $LEV$, as observed in
the case of $\phi=$ 90∘. At $t^{+}=$ 0.25 (Figure 4(f)), a thin neighboring
filament ($LEV_{2,s}^{c}$) forms besides $LEV_{2}^{ac}$, although in a
partially diffused form. Verma $\&$ Hemmati [49] recently attributed this
effect to the reduction in normalized circulation,
$\Gamma^{+}=\Gamma/(U_{\infty}c)$, of the primary $LEV$ ($LEV_{2}^{ac}$) at
$\phi=$ 180∘, compared to $LEV_{1}^{ac}$ observed at $\phi=$ 90∘. This
contributes to a reduction in intensity of vortex-foil interaction that
subsequently decrease the outflux of secondary spanwise vorticity. Spanwise
undulation of primary roller ($LEV_{2}^{ac}$) emerge at $t^{+}=$ 0.25 (Figure
4(f)), owing to the cooperative instability induced by the neighboring paired
vortex $LEV_{2,s}^{c}$. As the primary and secondary $LEV$ pair approach the
trailing edge at approximately $t^{+}=$ 0.5 (see Figure 4(g)), the outgrowth
of thin streamwise hairpin-like filaments ($HP_{2}^{1}$) is evident. However
at $t^{+}=$ 0.75 (Figure 4(h)), the coincident growth of $TEV_{2}^{c}$ from
the bottom part of the foil inhibits the continued elongation and straining
process of $HP_{2}^{1}$ filaments identified in Figure 4(g). Thus, it is
important to note here that although the growth of streamwise filaments from a
secondary $LEV$ is consistent with the observations at $\phi=$ 90∘, the
suppression mechanism by a growing TEV actually leads to distinct changes in
spatio-temporal evolution of secondary hairpin-like structures in the wake.
This is evident in Figure 4(g), where only fragmented $HP2^{1}$ filaments are
visible. At the same instant, the cooperative elliptic instability still
retains its dominance due to the presence of counter-rotating vortex rollers
($LEV_{2}^{ac}$ and $TEV_{2}^{c}$). This subsequently promotes core vorticity
outflux from $TEV_{2}^{c}$, which then demonstrates valley and bulge like
features [30] along the spanwise direction. Such dislocations eventually
evolve into dominant secondary hairpin-like structures ($HP2^{2}$), as
identified at the beginning of oscillation cycle (see Figure 4(e)), which
appears to be stronger in the wake compared to $HP2^{1}$. From this reasoning,
we infer that the presence of spanwise cooperative instability is consistent
during distinct evolution mechanisms discussed within the heave dominated
kinematics at $\phi=$ 90∘ and 180∘, respectively. However, the origins of
dominant secondary wake structures does not necessarily associate with the
presence of secondary $LEV$ roller (mechanism $``1"$) alone, despite the heave
dominated kinematics. At $\phi=$ 180∘, a growing $TEV$ and its paired
interaction with primary $LEV$ provides a second mechanism that can govern the
growth of secondary wake structures. This will now be referred as mechanism
$``2"$.

Interestingly, increasing $\phi$ further towards 225∘ and 270∘ reveals the
absence of secondary hairpin-like structures, as shown in Figures 4(i) and
4(j), respectively. This also coincides with the onset of pitch domination for
a foil in coupled heaving and pitching oscillation. However, the presence of
spanwise undulations on the primary rollers ($LEV_{3}^{ac}$ and
$LEV_{4}^{ac}$) in the wake of foils is still identifiable. These small-
amplitude undulations are attributed to the weak foil-vortex interactions that
occur during the $LEV$ advection over the foil boundary. The much weaker
secondary $LEV$ is observed to either diffuse or merge with the shear layer
separating from the bottom part of the foil.The entire temporal evolution of
vortex structures within one oscillation cycle are not presented here for
brevity (see supplementary videos for reference), since the observations that
concerned growth and advection of the primary $LEVs$ and $TEVs$ are mostly
similar to $\phi=$ 180∘. Disappearance of secondary wake structures also
coincides with decreased circulation strength of the primary rollers as
discussed by Verma $\&$ Hemmati [49]. Cheireghin et al.[8] further hinted in
their study that a larger deformation of the $LEV$ filament could be
attributed to an increased circulation strength. However, no plausible
association with the growth of secondary wake structures was discussed in
their study [8].

We now quantitatively assess the variation of pressure gradient over the foil
surface as the primary $LEV$ approaches the trailing edge, under kinematics
characterized by a transition from heave- to pitch-dominated motion. This
provides further explanation in addition to the decreasing circulation
strength of primary $LEV$ [49] for the transition from mechanism $``1"$ to
$``2"$ detailed above. Particularly, streamwise pressure gradient along the
chordwise extent of the foil demonstrate an association between the growth of
secondary vortex, and the subsequent core vorticity outflux observed in
mechanism $``1"$ (which leads to formation of streamwise hairpin-like
filaments). To proceed, we utilize the methodology explained by Obabco and
Cassel [26] to identify the mechanism of secondary recirculation zone
formation in the vicinity of a larger vortex core. In brief, Obabco and Cassel
[26] highlighted that large scale vortical interactions were characterized by
a rapid change in the streamwise pressure gradient, which coincided with a
growth of secondary re-circulation zone [26]. Particularly, the region
characterized by the primary re-circulation zone experiences a sudden
streamwise compression due to a rapid change in pressure gradient, that favors
and accelerates the growth of a new secondary re-circulation zone [26]. We
assess this correspondence of streamwise pressure gradient and growth of
secondary re-circulation region, which in our study will represent the
secondary vortex formation observed at $\phi=$ 90∘ and 180∘ in Figure 4(b) and
4(f), respectively. The coincident association of surface pressure
distribution and secondary hairpin-like growth through mechanism $``1"$ will
also establish a novel finding for the wake evolution of oscillating foils.

(a) (b)

(c)

(d) (e)

(f)

(g) (h)

(i)

(j) (k)

(l)

Figure 5: Span-averaged $dp_{w}/dx$ over the foil surface in streamwise
direction at first three quarter phases of an oscillation cycle. The cases
correspond to $\phi=$ 90∘ (a,b,c), $\phi=$ 180∘ (d,e,f), $\phi=$ 225∘ (g,h,i)
and $\phi=$ 270∘ (j,k,l).

Figure 5 depicts span-averaged profiles of pressure gradient on foil in the
chordwise direction ($dp_{w}/dx$), for different $\phi$. Three time-instants
corresponding to the first three quarter phases of an oscillation cycle, i.e.
$t^{+}=$ 0 (leftmost column), 0.25 (middle column) and 0.5 (rightmost column),
are examined here to evaluate the streamwise compression of vortical flow
[26], and its potential contribution to the growth of the secondary $LEV$. A
subsequent connection to the streamwise hairpin-like evolution (through
mechanism $``1"$) will also be established. The distribution of $dp_{w}/dx$
for $\phi=$ 90∘ at $t^{+}=$ 0 (see Figure 5(a)), indicates that the local
minima coincides with the advecting $LEV1_{ac}$ (Figure 4(a)). A rapid
increase in $dp_{w}/dx$ towards the upstream direction features a streamwise
flow compression, very similar to the observations of Obabco and Casell [26].
This yields a favorable pressure gradient (indicated by an overshoot towards a
positive value) for the onset of $LEV1_{c}^{s}$. This onset is marked as a
sharp drop at $X^{+}\approx 0.16$. As $LEV1_{ac}$ and $LEV1_{c}^{s}$ advect
downstream, Figure 5(b) shows a local minima of relatively low magnitude,
which corresponds to $LEV1_{c}^{s}$. This minima eventually approached 0 at
$t^{+}=$ 0.5 (see Figure 5(c)). Drop in the magnitude of pressure gradient
minima in the region featuring secondary $LEV$ coincides with the eventual
streamwise hairpin-like evolution (through mechanism $``1"$).

The profiles of $dp_{w}/dx$ at $\phi=$ 180∘ (see Figure 5(d-f)) provide
another crucial quantitative evidence for the lack of secondary $LEV$
transformation to dominant secondary hairpin-like structures via mechanism
$``1"$. This is consistent with the circulation strength assessment evaluated
recently by Verma $\&$ Hemmati [49]. A local minima in $dp_{w}/dx$ is observed
in Figure 5(d), which corresponds to $LEV2_{ac}$. This is relatively smaller
in magnitude compared to the local minima at $\phi=$ 90∘. A sharp rise towards
a positive $dp_{w}/dx$ in the upstream chord direction is further evident
although a sharp (peak) drop is not noticeable. A plateaued $dp_{w}/dx$ region
eventually becomes evident at $t^{+}=$ 0.25, which coincides with the location
of diffused $LEV2_{c}^{s}$ in Figure 4(f). The smoother drop in $dp_{w}/dx$
indicates that the streamwise flow compression is weaker, which hence leads to
a weak secondary $LEV$ growth. This reflects an important association between
kinematics and the evolution of secondary structures, where the reduced
capacity of mechanism $``1"$ becomes clear with an increase in $\phi$ to 180∘.
On a similar note, a transition from mechanism $``1"$ to $``2"$ coincides with
the discussed changes in streamwise distribution of $dp_{w}/dx$. With
transition of kinematics towards an onset of pitch domination at $\phi=$ 225∘
and 270∘, a similar trend is persistent with respect to the growth of
secondary $LEV$. No substantial drop is seen upstream of the primary $LEVs$
($LEV3_{ac}$ and $LEV4_{ac}$) in Figures 5(g), 5(h), 5(j) and 5(k),
respectively. Coincidentally, absence of the secondary hairpin-like
arrangement through mechanism $``1"$ is evident in these kinematic regime (see
Figure 4(i,h)). It is important to note that there are several other features
that can be extracted and discussed from the results in Figure 5. Such
discussions, however, fall outside the scope of the current study.

Based on the observations at $St_{c}=$ 0.32, a transition in mechanisms that
govern the growth of secondary structures is clear with respect to the change
in heave-dominated kinematics, towards an onset of pitch domination. This
explanation provides us with another important aspect of vortex dynamics
around oscillating foils (representing flapping wings and tail-fins), which
relates to the potentially similar role of these mechanisms for increasing
$St_{c}$. It holds importance, because a plausible answer to this question
expands the applicability of these mechanisms to a broader set of kinematics.

###  3.2 Influence of increasing $St_{c}$

####  3.2.1 $St_{c}=$ 0.40

Flow observations at $St_{c}=$ 0.4 and $\phi=$ 90∘ demonstrates mechanism
$``1"$ of the secondary structure formation, similar to $St_{c}=$ 0.32. The
wake topology is not shown or discussed here for brevity (see supplementary
videos). However, in order to qualitatively reveal a progression of secondary
structure growth towards kinematics dominated by decreasing heave domination,
wake topologies corresponding to $\phi=$ 180∘ and 225∘ are shown in Figure
6(a-d) and 6(e-h), respectively. The wake at $\phi=$ 180∘ reflects growth of
the primary $LEV5_{ac}$, in a paired configuration with a weaker secondary
vortex $LEV5_{c}^{s}$ (see Figures 6(a) and 6(b)). At the same instant, we
also observe formation of a strong secondary hairpin-like vortex
($HP3^{2^{\prime}}$) downstream of the trailing edge, that evolved in the
previous oscillation cycle via mechanism $``2"$ (i.e. deformation and
dislocation growth on $TEV$). At $t^{+}=$ 0.5 (see Figure 6(c)) thin strands
of coherent streamwise structures evolve via core vorticity outflux from
$LEV5_{c}^{s}$, which then form weak hairpin-like structures. These are marked
as $HP3^{1}$ at $t^{+}=$ 0.75 (see Figures 6(d)). The size of $HP3^{1}$
appears much smaller (visually) when compared with $HP3^{2\prime}$. This is
also consistent with flow observations at $\phi=$ 180∘ at $St_{c}=$ 0.32.

Wake topology at $\phi=$ 225∘, however, demonstrates some contrasting features
with reference to the growth of secondary structures at $St_{c}=$ 0.32. This
is evident in Figures 6(e-h). Formation of thin secondary hairpin-like
filaments are now evident at $\phi=$ 225∘. These streamwise pairs are marked
as $HP4^{2\prime}$ in Figure 6(e), which were formed in the previous shedding
cycle. Despite the advection of paired primary and secondary rollers
($LEV6_{ac}$ and $LEV6_{c}^{s}$), an outflux of streamwise vorticity does not
appear to be imminent at later stages of the oscillation cycle. However, at
$t^{+}=$ 0.75 (see Figure 6(h)), a paired configuration of $TEV6_{c}$ and
$LEV6_{ac}$ is noticeable, which leads to an elliptic instability between the
counter-rotating rollers. This consequently promotes the formation of
secondary hairpin-like structures marked as $HP4^{2^{\prime}}$ (Figure 6(e))
at $t^{+}=$ 0.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 6: Wakes corresponding to (a-d) $\phi=$ 180∘ and (e-h) $\phi=$ 225∘, at
$St_{c}=$ 0.40. The oscillation time varies from $t^{+}=$ (a,b) 0 to (d,h)
0.75. Each stage is represented using iso-surfaces of $\lambda_{2}(=-0.05)$,
which are colored based on $|\omega_{z}^{+}|=$ 5.

Based on the observations discussed at $St_{c}=$ 0.40, it is evident that the
wake topology at a decreased heave domination (i.e. at $\phi=$ 225∘) is now
characterized by growth of secondary hairpin-like structures through mechanism
$``2"$. The wake at $\phi=$ 270∘, however, remains devoid of any secondary
hairpin-like formation (wake evolution not shown here for brevity). On
comparing with the results discussed at $St_{c}=$ 0.32, we also observe a
consistent transition of growth mechanisms which concern evolution of
secondary hairpin-like structures. The transition follows a route
characterized by mechanism $``1"$ at the beginning of heave domination (i.e.
at $\phi=$ 90∘). This is followed by mechanism $``2"$ during peak heave
domination at $\phi=$ 180∘. As the heave domination begins to decrease
($\approx$ 225∘), mechanism $``2"$ is observed to be persistent which is then
succeeded by an absence of secondary structure at the onset of pitch
domination (at $\phi=$ 270∘). We will next evaluate the applicability of this
characteristic transition route at higher $St_{c}$.

####  3.2.2 $St_{c}=$ 0.48

(a) (b)

(c) (d)

(e) (f)

Figure 7: Wakes corresponding to (a,b) $\phi=$ 180∘, (c,d) $\phi=$ 180∘, and
(e,f) $\phi=$ 270∘, at $St_{c}=$ 0.48. The oscillation time corresponds to
$t^{+}=$ 0.5 (a,c,e) and 0.75 (b,d,f), respectively. Each stage is represented
using iso-surfaces of $\lambda_{2}(=-0.05)$, which are colored based on
$|\omega_{z}^{+}|=$ 5.

An increase in $St_{c}$ to 0.48 coincides with the secondary hairpin-like
formation within the entire $\phi$ range (90∘ to 270∘), characterized by a
transition from heave domination to an onset of pitch-dominated kinematics. At
$\phi=$ 90∘, mechanism $``1"$ contributes towards the secondary hairpin-like
formation (see supplementary videos), similar to the observations at a lower
$St_{c}$. Figure 7 provides instantaneous three-dimensional wake topologies at
an increasing $\phi$ from 180∘ to 270∘. The time instants were chosen at
$t^{+}=$ 0.5 and 0.75, respectively. At $\phi=$ 180∘ (see Figures 7(a) and
7(b)), a paired secondary ($LEV8_{c}^{s}$) and primary ($LEV8_{ac}$) roller
configuration is evident near the trailing edge of the foil (at $t^{+}=$ 0.5
in Figure 7(a)). A subsequent evolution of hairpin arrangement ($HP5^{1}$) is
observable at $t^{+}=$ 0.75 in Figure 7(b), which is now governed by mechanism
$``1"$. Therefore, mechanism $``2"$, which governed the growth of secondary
hairpin-like structures at lower $St_{c}(=$ 0.32 and 0.40), has now
transitioned to mechanism $``1"$, characterized by an elliptic instability and
subsequent outflux of streamwise vorticity through a secondary $LEV$. This
also reflects that mechanism $``1"$ remains sustained at peak heave domination
(i.e. at $\phi=$ 90∘ and 180∘), as $St_{c}$ increases to 0.48.

In order to identify if mechanism $``1"$ transitions to $``2"$, with
decreasing heave domination ($\phi>$ 180∘), we now investigate the wake at
$\phi=$ 225∘. This will also help establish if the consistent route of
transition concerning the growth mechanisms of secondary structures, remains
evident with increasing $St_{c}$, while kinematics change from heave to
domination. Figures 7(c) and 7(d) provides another indication of qualitatively
stronger secondary hairpin-like formation at $\phi=$ 225∘, which were either
absent (at $St_{c}=$ 0.32), or remained sufficiently small in the wake (at
$St_{c}=$ 0.40). The presence and growth of hairpin arrangement, marked by
$HP6^{2\prime}$ in Figures 7(c) and 7(d) is attributed to mechanism $``2"$.
$LEV-TEV$ pair is formed by structures $TEV9_{c}$ and $LEV9_{ac}$, where the
elliptic instability features are well developed on the former roller. These
features or dislocations outgrow to form secondary structures, i.e.
$HP6^{2\prime}$. Indeed, the growth mechanism transitions to $``2"$ as heave
domination decreases at $\phi=$ 225∘, which was suggested earlier at $St_{c}<$
0.48.

Although there exists a consistent transition mechanism from $``1"$ to $``2"$,
we notice that the wake at $\phi=$ 270∘ (Figure 7(e) and 7(f)) now depicts
spanwise hairpin arrangement of substantially smaller spatial scales, while
dominant secondary structure growth is still absent. In order to track the
formation mechanism of these hairpin-like structures, Figure 7(e) shows the
presence of primary ($LEV10_{ac}$) and secondary ($LEV10_{c}^{s}$) rollers in
a paired configuration near the trailing edge. The coinciding elliptic
instability induce the core vorticity outflux, which contributes to the
process of mechanism $``1"$ in onset of secondary hairpins (marked as
$HP7^{1}$) evident in Figure 7(f).

Hence, it is interesting to note that with the onset of pitch-dominated
kinematics at $St_{c}=$ 0.48, mechanism $``1"$ starts to dominate the wake
evolution rather than $``2"$. A possible reasoning is increasing strength and
timing for development of $TEV$ structures at $\phi=$ 270∘, since the
kinematics coincide with a relative increase in trailing edge amplitude
compared to the leading edge [48]. During this prolonged time for development
of $TEVs$, the primary $LEV$ and small scale hairpin-like structures (through
mechanism $``1"$) advect past the trailing edge, thereby failing to experience
any significant interaction with $TEVs$.

####  3.2.3 $St_{c}=$ 0.56

(a) (b)

(c)

Figure 8: Wakes corresponding to (a) $\phi=$ 90∘, (b) $\phi=$ 180∘ and (c)
$\phi=$ 270∘ at $St_{c}=$ 0.56. The oscillation time corresponds to $t^{+}=$
0.75. Each stage is represented using iso-surfaces of $\lambda_{2}(=-0.05)$,
which are colored based on $|\omega_{z}^{+}|=$ 5.

As $St_{c}$ further increases to 0.56, wake topologies corresponding to
$\phi=$ 180∘-270∘ now demonstrate dominant hairpin-like growth (see Figure 8).
Besides $\phi=$ 90∘, the transition from heave- to the onset of pitch-
dominated kinematics coincides with the presence of secondary hairpin-like
structures along the spanwise direction (marked as $HP8^{1^{\prime}}$,
$HP9^{1^{\prime}}$ and $HP10^{1^{\prime}}$ for $\phi=$ 180∘, 225∘, and 270∘,
respectively). The origin of these secondary hairpin-like structures is also
entirely attributed to mechanism $``1"$.

As we compare the findings at higher $St_{c}$ (i.e. at 0.48 and 0.56), it is
clear that while the route to transition from mechanism $``1"$ to $``2"$
remains persistent, a simultaneous progression of the secondary structure
growth also occurs for $\phi$. This coincide with a weaker heave domination.
This characteristic was not observable at $St_{c}<$ 0.48. Also, mechanism
$``1"$ completely dominates the range of $\phi$ at $St_{c}>$ 0.48. This
suggests that the dependence of mechanisms $``1"$ and $``2"$ on transitioning
nature of kinematics is lost at $St_{c}=$ 0.56.

###  3.3 Representative transition models on $\phi-St_{c}$ phase-space

The increase in $\phi$ and $St_{c}$ depicted transition of mechanisms that
contributed to the growth of secondary hairpin-like structures. It is useful
to map these growth mechanisms on a $\phi-St_{c}$ phase-space diagram such
that a consistent pattern in their apparent transition can be established.
Figure 9 shows a schematic that lays out the mechanisms observed at varying
$\phi$ and $St_{c}$ in the previous section. Two patterns are evident. The
first demonstrates that the transition from mechanism $``1"$ to an absence of
the secondary wake structures occurs via an intermediate stage that is
governed by mechanism $``2"$. We term this transition model as Series $A$,
which either coincides with a change in kinematics from heave domination (at
$\phi=$ 90∘) to the onset of pitch dominated motion (at $\phi=$ 270∘), or a
decrease in $St_{c}$. Note that the absence of secondary wake structures for
Series A model, corresponding to $\phi=$ 180∘, is not captured within the
range of decreasing $St_{c}$ considered here, and may be expected at $St_{c}<$
0.32.

Regarding the second pattern, we see that at the onset of pitch domination
(i.e. $\phi=$ 270∘), there is a vivid transition from mechanism $``1"$ to an
absence of secondary wake structures. However, in contrast to Series A model,
this characterizes a lack of intermediate stage or mechanism $``2"$ within the
decreasing range of $St_{c}$. Hence, we define this transition model as Series
$B$.

Figure 9: Phase-map representing dominance of growth mechanisms for secondary
hairpin-like structures at increasing $\phi-St_{c}$.

####  3.3.1 Insights from Circulation

The $\phi-St_{c}$ map in Figure 9 reveals two major transition models that
characterizes the change in growth of secondary wake structures from mechanism
$``1"$ to either $``2"$, or directly to their absence. These patterns demand
quantitative justification based on changes in the physical features of
vortical structures that primarily govern the onset of secondary hairpins in
the wake. The vortical structure that is pertinent to both mechanisms $``1"$
and $``2"$ is the primary $LEV$. We therefore discuss the quantitative
circulation ($\Gamma$) variation of the primary $LEV$ within one oscillation
cycle. The recent study by Verma $\&$ Hemmati [49] confirmed that as
kinematics transition from heave domination to an onset of pitch dominated
regime (i.e. from $\phi=$ 90∘ to 270∘), a decrease in $\Gamma^{+}$ for the
primary $LEV$ was evident. This substantially reduces the intensity of core
vorticity outflux through the interaction of foil boundary and primary $LEV$,
thus leading to an absence of secondary wake structures. This justification
supports the Series $A$ model represented at $St_{c}=$ 0.32 and 0.40,
respectively. Here, we present supplementary $\Gamma^{+}$ assessments in order
to provide physical reasoning behind Series $A$ and $B$ transition models
observed with respect to a decreasing $St_{c}$ from 0.56 to 0.32 (see Figure
9), and for transitioning kinematics from heave to pitch domination.

(a) (b)

(c)

Figure 10: Variation of $\Gamma^{+}$ at increasing $St_{c}$ within 1
oscillation cycle for $\phi$ corresponding to (a) 180∘, (b) 225∘ and (c) 270∘.

Figure 10(a-c) presents the instantaneous $\Gamma^{+}$ variation within one
oscillation cycle corresponding to $\phi=$ 180 (Figure 10(a)), 225∘(Figure
10(b)) and 270∘(Figure 10(c)). The variation of $\Gamma^{+}$ for $\phi=$ 90∘
is not shown since it depicts formation of secondary hairpin-like structures
via mechanism $``1"$, across the entire $St_{c}$ range (see Figure 9). A
consistent increase in peak $\Gamma^{+}$ magnitude of primary $LEV$ with
increasing $St_{c}$ is noticeable despite the transition of kinematics from
heave to pitch domination. A stronger primary $LEV$ can therefore lead to a
higher outflux of streamwise vorticity in two ways. The first way involves
coherent streamwise vorticity outflux from secondary $LEV$ (i.e. mechanism
$``1"$), while the second way present a strong enough interaction with the
developing $TEV$ near the trailing edge. The latter involves secondary
hairpin’s evolution via mechanism $``2"$. This observation also remains
consistent with recent literature [8, 37] that suggested a greater $LEV$
deformation for a higher $St_{c}$.

##  4 CONCLUSIONS

The association between coupled kinematics and the wake three-dimensionality
behind an oscillating foil is numerically investigated in this study. The foil
kinematics are modified based on changing phase offset ($\phi$) between
coupled heave and pitch motion, and an increasing reduced frequency ($St_{c}$)
at $Re=$ 8000\. The considered range of $\phi$ is largely representative of a
heave-dominated kinematics, which also enables a dominant leading edge vortex
formation that undergoes deformation on account of the spanwise instability.
It concurrently plays a role in the evolution of secondary wake structures.

At lower $St_{c}$, spanwise undulations of the primary roller remain evident
due to the developed instability. The wake corresponding to $\phi=$ 90∘ and
180∘, where the relative leading to trailing edge amplitude is greatest within
the heave-dominated kinematics considered here[48], depicts a dominant growth
of secondary hairpin-like structures. These are essentially an outcome of core
vorticity outflux from either the formation of a secondary $LEV$, that
neighbors a primary $LEV$ ($\phi=$ 90∘), or a $TEV$ interacting with a primary
$LEV$ shed behind the foil ($\phi=$ 180∘). Contrary to these cases, it is
observed that at $\phi=$ 225∘ and 270∘, the wake coincides with the absence of
hairpin-like structures. This range of $\phi$ also coincides with a decreasing
leading edge amplitude relative to the trailing edge, thus presenting the
onset of pitch domination at $\phi=$ 270∘. This transition in mechanisms that
govern the growth of secondary hairpin-like structures is consistent with
increasing $St_{c}$.

The transition in growth mechanisms is characterized by two major routes
(Series $A$ and $B$) that reflect either a change in kinematics from heave to
pitch domination, or a decrease in $St_{c}$. These routes confirm that a
transition from mechanism $``1"$ to an absence of secondary hairpin-like
structure either occurs directly or via an intermediate stage (i.e. mechanism
$``2"$). Quantitative assessment on circulation of primary $LEVs$ indicates
that the reduction in their strength coincides with a lower possibility of
core vorticity outflux, as $\phi$ increases from 90∘ to 270∘. This explains
the disappearance of secondary wake structures for $\phi=$ 225∘ and 270∘ at
lower $St_{c}$. However, as $St_{c}$ increases, the higher $\Gamma^{+}$
associated with primary $LEV$ promotes the formation of secondary hairpin-like
structures within the entire range of $\phi$. Overall, a close association
between the kinematics, evolving vortex instability, and growth of secondary
wake structures is observed for the first time in literature for oscillating
foils in couple heaving and pitching kinematics.

##  5 ACKNOWLEDGMENTS

This study has received support from the Future Energy Systems, and Compute
Canada clusters were implemented for the required computational analysis.

##  6 DATA AVAILABILITY

The data that supports the findings of this research are available within the
article.

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