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Effect of aspect ratio on the propulsive performance of tandem flapping foils

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In this work, we describe the impact of aspect ratio (AR) on the performance of optimally phased, identical flapping flippers in a tandem configuration. Three-dimensional simulations are performed for seven sets of single and tandem finite foils at a moderate Reynolds number, with thrust producing, heave-to-pitch coupled kinematics. Increasing slenderness (or aspect ratio - AR) is found to improve thrust coefficients and thrust augmentation but the benefits level off towards higher values of AR. On the other hand, the propulsive efficiency shows no significant change with increasing AR, while the hind foil outperforms the single by a small margin. Further analysis of the spanwise development and propagation of vortical structures allows us to gain some insights on the mechanisms of these wake interactions and provide valuable information for the design of novel biomimetic propulsion systems.
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RESEARCH ARTICLE
Effect of aspect ratio on the propulsive performance of
tandem flapping foils
N. S. Lagopoulos 1,3, G. D. Weymouth 2,4, B. Ganapathisubramani 1
1Aerodynamics and Flight Mechanics Group, University of Southampton, UK
2Southampton Marine and Maritime Institute, University of Southampton, UK
3Dolprop Industries AB, Ekerö, Sweden
4Alan Turing Institute, London, UK
*Corresponding authors. E-mails: nikolaos@dolprop.se and g.d.weymouth@soton.ac.uk
Received: XX 2021; Revised: XX XX 2021; Accepted: XX XX 2021
Keywords: Vortex dynamics; Swimming/flying; Autonomous underwater vehicles
Abstract
In this work, we describe the impact of aspect ratio (AR) on the performance of optimally phased, identical flapping
flippers in a tandem configuration. Three-dimensional simulations are performed for seven sets of single and tandem
finite foils at a moderate Reynolds number, with thrust producing, heave-to-pitch coupled kinematics. Increasing
slenderness (or aspect ratio - AR) is found to improve thrust coefficients and thrust augmentation but the benefits
level off towards higher values of AR. On the other hand, the propulsive efficiency shows no significant change
with increasing AR, while the hind foil outperforms the single by a small margin. Further analysis of the spanwise
development and propagation of vortical structures allows us to gain some insights on the mechanisms of these
wake interactions and provide valuable information for the design of novel biomimetic propulsion systems.
Impact Statement Tandem flapping foils has the potential to be used for propulsion, especially among bio-
inspired AUV designers, due to their superior performance over single flippers. In this study, we evaluate the
importance of aspect ratio on the thrust-augmenting effect of in-line flapping, known as wake recapture. It is
shown that flipper elongation impacts the interaction between the hind flipper (or follower) and its incoming
flow, as it strengthens the trailing edge vortices, shed in the wake of the front flipper. This affects both
the thrust generating capacity and the optimal phasing of the flippers, allowing the engineer to determine
the vehicle´s suitability towards certain missions, simply based on foil slenderness. An in-depth analysis of
the wake dynamics enables us to distinguish the limitations as well as ways to optimize this approach by
monitoring the transition towards a quasi two-dimensional flow.
1. Introduction
Flapping foil mechanisms are the basic means of propulsion and control within the avian and aquatic
fauna. These systems are often more agile, durable and efficient compared to conventional man-made
propulsors (Weymouth,2016). Thus, many studies have focused on the analysis of these biological con-
figurations in terms of kinematics (Khalid et al.,2021;Cimarelli et al.,2021) fluid-structure interaction
(Kim et al.,2013;Zurman-Nasution et al.,2020) as well as the effects of planform geometry (Dagenais
and Aegerter,2020;Zurman-Nasution et al.,2021b) and flexibility (Shi et al.,2020;Fernandez-Feria
and Alaminos-Quesada,2021).
©The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative
Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in
any medium, provided the original work is properly cited.
arXiv:2112.11538v1 [physics.flu-dyn] 21 Dec 2021
E1-2
Figure 1. Two examples of bio-inspired AUVs that combine front and back flipper oscillation (tandem
arrangement) as a means of propulsion: (a) performs a pure pitching motion (Long et al.,2006) while
(b) uses a combination of rolling and pitching (Weymouth et al.,2017). A simplified version of the latter
kinematics is utilized by this study.
Tandem flapping configurations e.g. insect wings (Alexander,1984;Thomas et al.,2004), plesiosaur
flippers (Robinson and JA,1975;Hawthorne et al.,2019) etc. are shown to outperform single flappers
due to certain foil-wake interactions commonly referred to as 𝑤𝑎𝑘𝑒 𝑟𝑒𝑐𝑎 𝑝𝑡𝑢𝑟𝑒 (Broering and Lian,
2012;Muscutt et al.,2017b). This has inspired researchers to experiment on quadruple foil systems
to propel autonomous underwater vehicles (AUVs), utilizing a variety of harmonic kinematics e.g
pitch, roll, coupled motion etc. Most of these tetrapodal swimmers (see figure 1) are electric-powered,
designed for a wide range of depths (1𝑚100𝑚) and can reach velocities of 0.5𝑚/𝑠2𝑚/𝑠(Licht et al.,
2004;Long et al.,2006;Weymouth et al.,2017), which are comparable to modern propeller-driven,
ocean-going AUVs of a similar size and weight (Yuh,2000).
Of particular interest, towards the design of these systems, is the 𝑎𝑠𝑝𝑒𝑐𝑡 𝑟𝑎𝑡𝑖𝑜 AR (for rectangular
wings AR =𝑊/C where 𝑊is the wingspan and Cis the chord length) due to its impact on the system´s
thrust generation capacity. Slender flippers for example, are widely considered beneficial to both thrust
and efficiency (Green and Smits,2008;Shao et al.,2010;Dewey et al.,2013) which is further implied
by the predominance of high AR wings among birds and insects (Ellington,1984;Azuma,1992;
Usherwood and Ellington,2002). As a result, early research was driven towards two dimensional or
quasi two dimensional approaches both experimentally (Koochesfahani,1989;Triantafyllou et al.,1993)
and numerically (Pedro et al.,2003;Mittal,2004;Guglielmini and Blondeaux,2004).
Unlike avian organisms however, aquatic animals demonstrate a great variety of AR with non
migratory fish utilizing mostly low AR (Walker and Westneat,2002;Combes and Daniel,2001) as they
are considered more suitable for their drag-based paddling motion. Furthermore, comparisons among
various species suggest that high AR‘s benefit cruising efficiency while low AR‘s promotes thrust
generation in short bursts (Flammang and Lauder,2009;Domenici,2010) which is also supported
by recent experiments (Lee et al.,2017). A preference towards lower AR in aquatic propulsion can
be additionally attributed to the much higher density of water, which leads to greater added-mass
associated bending moments (Dong et al.,2006). This can significantly constrain the design of an
AUV by determining manufacturing costs, durability, mission envelope etc. and thus demonstrates the
necessity of finite flipper analysis.
Contemporary literature on finite wings of varying AR often focuses on single flapping configurations
(Zurman-Nasution et al.,2021b;Hammer et al.,2021;Zhong et al.,2021) with only a few studies related
to tandem arrangements (Arranz et al.,2020). A key feature of the above is the presence of tip vortices
that transform the two dimensional wake into a complex chain of ring-like formations (Shao et al.,2010;
Li et al.,2018). Moreover, the majority of these studies utilizes insect and small fish kinematics and/or
Flow E1-3
Figure 2. Structural details of an AR=4 hydrofoil, where the (a) frontal,(b) upper and (c) side view
are presented. A detailed model of the flipper in the form of an IGS file can be found online, within the
supplementary material of this study.
geometries which, although quite suitable for special applications, are less relevant to open water designs.
On the other hand, certain heave-to-pitch combinations are considered dominant in cetacean locomotion
(Ayancik et al.,2020;Han et al.,2020) (where spanwise flexibility of the caudal fin is comparatively
low (Gough et al.,2018;Adams and Fish,2019)). Furthermore, heave-to-pitch coupling is considered
sufficient to represent the mid-chord kinematics of flipper-based, AUV´s and/or aquatic animals using
roll-to-pitch combinations (Muscutt et al.,2017a) such as sea turtles, penguins and most notably the
tetrapodal plesiosaurs (Carpenter et al.,2010). Besides, the effect of flipper AR on the wake recapture
remains unknown, despite its aforementioned importance within tandem flapping AUV consepts.
The present study attempts to address these issues via the numerical analysis of rectangular flippers
with elliptical tip, undergoing heave-to-pitch coupling for a chord based Reynolds number, 𝑅𝑒 C=
𝜌𝑈C/𝜇=8500 where 𝜌is the water density, 𝑈is the freestream velocity and 𝜇is the dynamic
viscosity. Seven AR are tested in both single and tandem configurations of identical AR for an amplitude
based Strouhal, 𝑆𝑡 𝐴=𝑓·2A/𝑈=0.4where 𝑓is the frequency of oscillation and 2Ais the peak-
to-peak amplitude of the trailing edge (TE) (Triantafyllou et al.,1991). In addition, the phase lag and
distance between consecutive flippers are kept constant, selected for maximum thrust augmentation at
the given 𝑆𝑡 𝐴in 2D (Muscutt et al.,2017b). Here, the choice of 𝑆𝑡 𝐴is based on the observed range
of Strouhals utilized by swimming and flying organisms (Triantafyllou et al.,1993). Furthermore, the
test cases are evaluated in terms of thrust coefficient, relative thrust augmentation and hydrodynamic
efficiency. To this end, we compare the single/tandem flipper sensitivity to AR and attempt to shed light
on the three dimensional aspect of the wake to wake interaction.
2. Methodology
2.1. Flipper Geometry and kinematics
We consider a rigid NACA0016 with a thickness 𝐷=0.16 C, a rectangular planform section where
the width is equal to 1Cand a tapered elliptical tip as shown in figure 2. Here the elliptical section has
a span of 1Cwhile 𝑊is the total span of the flipper. Thus, for the sake of simplicity, we use the AR
definition of rectangular flippers (explained in section 1) and we set our baseline test case at AR =2
proceeding towards AR =8in increments of AR =1.
E1-4
Figure 3. The kinematic parameters and coordinate system of an oscillating foil undergoing (a) heave
(b) pitch and (c) coupled motion. Redrawn from (Lagopoulos et al.,2019).
The kinematics parameters of the hydrofoil can be seen in figure 3. As stated above, the flippers
utilize heave to pitch coupling which is achieved by the superposition of the two harmonic components.
More specifically, pitch refers to the sinusoidal rotation about the pivot point P=0.25 (normalised by
C) while heave is a sinusoidal, vertical translation with respect to the centreline:
𝑦𝑓(𝑡)=0sin(2𝑓 𝜋𝑡)
| {z }
𝑦(𝑡)
+ (1− P ) C sin[𝜃(𝑡)]
| {z }
𝑦𝜃(𝑡)
(1)
𝑤𝑖𝑡 ℎ 𝜃 (𝑡)=𝜃0sin(2𝑓 𝜋𝑡 +𝜓)(2)
where subscripts 𝑓,and 𝜃denote the front (or single foil), heaving and pitching components
respectively.
Here, the instantaneous pitching angle is expressed as the 𝜃(𝑡), while 0and 𝜃0represent the
amplitudes of the two motions. Note that although the total peak-to-peak amplitude is a combination
of these TE displacements, the chosen kinematic parameters result in A ∼ 0=1C. Furthermore, the
heave to pitch phase difference is set to 𝜓=90, which is shown to maximize the propulsive efficiency
within the frequency range of interest (Platzer and Jones,2008). Lastly, the combined (or effective)
angle of attack 𝛼(𝑡)equals to the summation of 𝜃(𝑡)and the heave-induced angle of attack so that its
amplitude is expressed as:
𝛼0=arctan 2𝜋 𝑓 ℎ0
𝑈
𝜃0(3)
where 2𝜋 𝑓 ℎ0is the amplitude of 𝑑𝑦/𝑑𝑡. Within this study, all simulations are conducted for 𝛼0=20
due to its dominance within modern cetaceans (Fish and Rohr,1999) and relevant studies in tetrapodal
swimming (Muscutt et al.,2017a).
The complete tandem arrangement is depicted in figure 4. To distinguish parameters referring to the
front or the back foil, we utilize the subscripts 𝑓(already mentioned above) and 𝑏respectively, for the
remaining of this study. Furthermore, to describe the foil-to-foil interaction, we introduce two more
parameters: the phase lag and the inter foil spacing.
The phase lag between the two foils is expressed as 𝜑and will be referred to as simply the 𝑝ℎ𝑎 𝑠𝑒.
Thus the back flipper´s motion is described as:
𝑦𝑏(𝑡)=𝑦(𝑡+𝜑) + 𝑦𝜃(𝑡+𝜙)(4)
Spacing Sis the distance between the TE of the front foil and the leading edge (LE) of the back foil
towards the streamwise direction. Therefore the chord normalised spacing S𝐶is defined as:
Flow E1-5
Figure 4. Details of a tandem foil configuration, undergoing synchronous (𝜙=0), heave-to-pitch
coupling.
SC=S
C(5)
Here S𝐶=2to allow comparison with relevant studies of the same 𝑆𝑡 𝐴(Muscutt et al.,2017b).
Moreover, 𝜙=0𝑜as preliminary simulations found that it maximizes thrust augmentation for the chosen
SCand 𝑆𝑡 𝐴in 2D, the details of which can be found in Appendix A.
2.2. Performance Metrics
Within a flapping cycle, the flipper experiences the time-dependent thrust 𝐹𝑋(𝑡), the side force
(lift/downforce) 𝐹𝑌(𝑡)and moment 𝑀𝑍(𝑡)around P. In this study we focus on the thrust generation
capacity of the fore and hind flipper, characterised by the thrust coefficient:
𝐶𝑇=𝐹𝑋
1
2𝜌𝑈2
𝐺(6)
where 𝐺=C · (𝑊− C) + 0.25𝜋C2is the planform area. Cycle averaged quantities are presented with
a tilde (e) to distinguish them from their instantaneous counterparts. Furthermore, the two dimensional
thrust coefficient (where 𝐺is replaced by C) is distinguished by the use of the subscript 𝑡instead of 𝑇.
Another important parameter is the propulsive (hydrodynamic) efficiency (𝜂) of each flipper. This is
simply the ratio between the power of the generated thrust and the power imparted to the flipper so that
it overcomes the loads imposed by the fluid:
𝜂=𝑇𝑈
𝑃(7)
where 𝑇is the thrust and 𝑃the input power defined as:
𝑃(𝑡)=𝐹𝑌(𝑡)𝑑𝑦 (𝑡)
𝑑𝑡 +𝑀𝑍(𝑡)𝑑𝜃(𝑡)
𝑑𝑡 (8)
To compare the performance of the single and tandem configurations we normalize the above values
by the equivalent parameters of the single (front) flipper:
𝐶
𝑇 ,𝑏 =
˜
𝐶𝑇 ,𝑏
˜
𝐶𝑇 , 𝑓
, 𝜂
𝑏=𝜂𝑏
𝜂𝑓
(9)
E1-6
Figure 5. The impact of AR on the (a) thrust coefficient and (b) efficiency of the single flipper, undergoing
heave-to-pitch coupling. Simulation points are characterised by while the best fit curve is depicted via
a dashed line.
where denotes relative terms. Previous studies suggest little to no alteration of the front foil´s loads
and efficiency by the presence of the hind for 𝑆C1(Muscutt et al.,2017b). Thus, any normalised
parameters presented here are associated with the back flipper.
2.3. Computational Method
The CFD solver selected for this work can simulate complex geometries and moving boundaries for a
variety of Reynolds numbers in 2D and 3D domains, by utilizing the boundary data immersion method
(BDIM). BDIM solves a combined set of analytic meta-equations for an immersed solid and its ambient
flow, achieving a smoothed interface via an integral kernel function (Weymouth and Yue,2011). The
technique shows a quadratic convergence and has been validated for flapping foil applications at a wide
range of kinematics (Maertens and Weymouth,2015;Polet et al.,2015).
The mesh configuration is formed by a rectangular Cartesian grid with a dense uniform domain
around the body and near wake while exponential grid stretching is used for the far-field. The boundary
conditions consist of a uniform inflow, zero-gradient out flow and free-slip conditions on the upper
and lower boundaries. Furthermore no slip boundary conditions are used on the oscillating foil and
symmetric conditions are enforced towards the spanwise direction. Mesh density is expressed in terms
of grid points per chord. A uniform grid of Δ𝑥= Δ𝑦= Δ 𝑧=𝐶/64 is used for this study, yielding
relatively fast results while the standard deviation of the estimated thrust is 8% of the simulations
with four times the resolution in both 2D and 3D (Zurman-Nasution et al.,2021a).
3. Results and Discussion
3.1. AR effect on the single flipper
The performance of the single flipper at varying AR can be seen in figure 5. Elongation leads to a sharp
increase of the thrust coefficient until 𝐴𝑅 4where the curve starts to asymtote for higher AR, where
𝛿˜
𝐶𝑇
𝛿 𝐴𝑅 3% (beyond AR > 6). This is still far away from the two dimensional value of ˜
𝐶𝑡, 𝑓 =0.675 (see
Appendix A). Unlike ˜
𝐶𝑇, the propulsive efficiency seems almost insensitive to slenderness possibly due
to the use of optimal kinematic parameters, with a negligible decline observed for 𝐴𝑅 between 2 and 4.
A qualitative comparison of the flow field around different AR´s can be seen in figure 6. The flapping
motion generates a cylindrical vortex across the leading edge (LEV) of the flipper which appears almost
undisturbed by the tip effects. Turbulent structures can be seen only downstream of the elliptical edge,
Flow E1-7
Figure 6. Snapshots of normalised vorticity at 𝑡/𝑇=1where 𝑇=1/𝑓, for single flappers of (a)
𝐴𝑅 =2, (b) 𝐴𝑅 =4, (c) 𝐴𝑅 =6and (d) 𝐴 𝑅 =8. The direction of the free-stream flow 𝑈is right to
left. Areas of undisturbed, two dimensional wake are characterised by rectangles of red dashed lines.
Figure 7. (a) Spanwise averaged vorticity for single flippers at 𝑡/𝑇=0.5where the TEV is enclosed by
a box of yellow dashed line for 𝐴𝑅 =2. (b) The resultant circulation over kinematic viscosity (𝜈) of the
TEV, calculated at this instance for all AR´s of this study.
indicating local breakdown of the trailing edge vortex (TEV). This breakdown propagates towards the
root covering a distance of 2.5Cwhich remains constant, regardless of flipper AR. Consequently, an
increasing AR allows the formation of elongated, undisturbed TEVs leading to quasi-two-dimensional
wake.
In order to quantify the above observations, we examine changes in the strength of the downstream
wake, for varying levels of planform slenderness. This can be achieved by calculating the circulation (Γ)
of a TEV at a chosen instant of the flapping cycle, from a spanwise-averaged flow field. Variations along
the span will be absorbed in this spanwise-averaging process leading to increased circulation for only
the most coherent TEVs. Details of the procedure can be found in Appendix B. As shown in figure 7a,
the TEV appears to become more compact for higher AR, causing Γto saturate at a constant value (see
figure 7b). This in turn, leads to a constant velocity surplus across the flipper span which is reflected in
behaviour of the thrust in figure 5.
E1-8
Figure 8. The impact of AR in terms of (a) thrust coefficient and (b) efficiency, on the fore and hind
flippers of a tandem configuration, undergoing heave-to-pitch coupling at 𝜙=0and SC=2. Simulation
points are characterised by while the best fit curve is depicted via a dashed line.
Here we should note that, similar ˜
𝐶𝑇𝐴𝑅 relationships have been reported by Shao et al. (2010)
despite the latter´s different planform geometry (no wingtip) and significantly lower 𝑅𝑒 C. As both studies
utilize heave-to-pitch coupling at 𝜓=90𝑜, a similar dynamic separation should be expected at least in
2D. Moreover, the chosen kinematics are essentially two dimensional ( 𝛿 𝑦 𝑓
𝛿𝑧 =𝛿 𝑦 𝑓
𝛿𝑧 =0), further reducing
the importance of the spanwise geometry, although they still cause minor discrepancies between the two
studies e.g. in ˜
𝐶𝑇. This, in turn, can support the predominance of medium slenderness (𝐴𝑅 ∼ [4,6]
using our paper´s definition) caudal fins observed in cetaceans (Ayancik et al.,2020) where the same
kinematics are used. Indeed, fins of too low AR would reduce the propulsive capacity of the species,
yet much larger ones would be structurally demanding without offering any significant hydrodynamic
advantage. It should be noted, however, that these animals demonstrate a plethora of wingtip geometries,
combined with at least some level of flexibility (Fish and Rohr,1999). Thus, although a deeper analysis
in the area is required, this topic is beyond the focus of the present work.
3.2. AR effect on the tandem configuration
The addition of another upstream oscillating body alters the flow field and determines the propulsive
characteristics of the downstream foil. This is made clear in figure 8a where the rear flipper demonstrates
significantly higher ˜
𝐶𝑇than the front, due to wake recapture. Once again, elongation has a greater impact
on low AR´s but ˜
𝐶𝑇appears to stabilize at a rate that is marginally slower than that of a single foil
(or that of the front foil). This suggest that the wake recapture tends to benefit over a larger range of
ARs compared to performance improvement of a single foil. On the other hand, the efficiency shows a
negligible improvement while following the same trend as for the front foil (see figure 8b). This is not
surprising since the optimal 𝜙for thrust augmentation often differs from the one of improved efficiency
(Muscutt et al.,2017b).
In order to investigate the relative augmentation of thrust for the back foil in more detail, we examine
˜
𝐶𝑇 ,𝑏
(Which is the ratio of thrust of the back foil to that of the front foil) in figure 9a. It can be seen that
there is a sharp increase in this ratio for 𝐴 𝑅 ∼ [2,4](from ˜
𝐶𝑇 ,𝑏
=1.3to ˜
𝐶𝑇 ,𝑏
=1.45) and the ratio
seems to level out around AR = 4 (at ˜
𝐶𝑇 ,𝑏
1.42) which remains approximately constant beyond this
aspect ratio. This suggests that the rate of increase in thrust for the front and back foils essentially follow
each other proportionately. Consequently, there are no further benefits beyond AR = 4 in terms of relative
augmentation, although there is still a benefit in the overall thrust produced by the pair of flippers.
Flow E1-9
Figure 9. The impact of AR, in terms of relative (a) thrust and (b) efficiency augmentation, on the
hind flipper of the tandem configuration, undergoing heave-to-pitch coupling at 𝜙=0and SC=2.
Simulation points are depicted as while the dashed lines represent the best fit curves.
Interestingly, figure 9b shows that relative efficiency 𝜂𝑏(which is the ratio of efficiency of back foil
to the front foil) remains practically unchanged, showing very minor gains of about 2.4% throughout
the entire range of AR (see figure 9b). This is probably related to the flipper kinematics being already
optimized for maximum efficiency, in conjunction with the, previously mentioned, use of a thrust-
specific 𝜙. Thus, it may be possible to alter the kinematics and/or the planform of the flippers to make
further gains in this area. However, this is beyond the scope of the current study.
The above observations are obviously related to the flow field development between the two foils. This
is evident in figure 10, where the wakes of tandem arrangements for 𝐴 𝑅 =2and 𝐴𝑅 =8are compared
(animations of these test cases can be found in the supplementary material). As mentioned previously,
wingtip effects are proportionally higher in the wake of 𝐴𝑅 =2compared to 𝐴𝑅 =8(figures 10a and
10b) resulting in the break-up of TEV shed from the front and the back foils (figures 10c and 10d).
Specifically, the break-up of the TEV from the front foil means that the back foil does not experience
a coherent wake across its span. Therefore, any additional benefit that could come from wake-vortex
capture is not significant. This also limits paired effects of vortices in the wake of the rear-foil. The figure
shows that the distance between shed-vortices of the two foils appears to decrease with increasing aspect
ratio and this is consistent observations in previous studies (Dong et al.,2006;Shao et al.,2010). This
also suggests that low AR foils enable a greater range of phase-difference in the kinematics between
the two foils since it is easier for the back foil to weave between the incoming vortices, which is a key
characteristic of wake recapture (Muscutt et al.,2017b). Finally, as AR increases, the incoming wake
for the back foil takes a quasi-2D form (except at around the tip) and relative augmentation begins to
stagnate since no additional benefit is extracted.
These findings can be further quantified by computing the spanwise-averaged circulation of the
TEV or the back flipper for different aspect ratios. As shown in figure 11a, wake recapture allows the
formation of a noticeably larger and stronger TEV compared with those shed by the front foil. However,
its compactness/coherence is more dependent on flipper elongation, which alters the TEV circulation
with AR (see figure 11b). Note that the circulation is computed based on a given box size and further
information on the effect of box size on the computation of this circulation is in Appendix B. For low
AR, the vortex appears to be diffused due to interactions between the main TEV and the tip. As AR
increases, the spanwise-averaged TEV becomes more coherent and its circulation rises (for a given
box size). However, this value starts to level off at higher AR, following the same trend as the thrust
E1-10
Figure 10. Snapshots of normalised vorticity at 𝑡/𝑇=1for tandem configurations. A top view compar-
ison shows that the wake of 𝐴𝑅 =2(a) suffers significantly from vortex breakdown while the wake of
𝐴𝑅 =8(c) remains mostly unaffected. This is more evident at a side view although the back foil (yellow
dashed line) of both 𝐴𝑅 =2(c) and 𝐴𝑅 =8(d) manages to weave through the incoming vortex pair
(boxes of red dashed line) of the front flipper, due to proper 𝜙adjustment.
coefficient. It can also be seen that the circulation values in the back foil are significantly higher than
that of the front foil, which is also consistent with the thrust results.
From an application point of view, the above findings suggest that a set of high AR flippers would
be preferable for steady cruising, while lower AR permit a more agile behaviour at the cost of speed.
Moreover, tandem arrangements of 𝐴 𝑅 ∼ [46]may be a prudent compromise between augmentation
benefits and mechanical behaviour, since the relative thrust enhancement has reached a saturated state
while the size is still small enough to withstand the large unsteady loads. Finally,we speculate that it
might be beneficial for the rear foil to be at slightly lower AR than the front in order to avoid the tip
vortex effect that could reduce the relative augmentation.
4. Conclusions
The propulsive characteristics of single and tandem flapping foils were examined numerically under
a heave-to-pitch coupling motion, for seven flipper sets of 𝐴𝑅 ∼ [28]of rectangular flippers with
elliptical tip at 𝑅𝑒C=8500. Each set had the same AR and the test were conducted for the fixed
combination 𝑆𝑡 𝐴=0.4− S𝐶=2at 𝜙=0𝑜which was found to optimise wake recapture in 2D.
Our analysis shows that flipper elongation has a positive impact on the thrust coefficients of both
single and tandem configurations at low AR´s but this effect weakens as we move towards higher AR.
Flow E1-11
Figure 11. (a) Spanwise averaged vorticity for back flippers of a tandem configuration, at 𝑡/𝑇=0.5
where the TEV is enclosed in a box of yellow dashed line for 𝐴𝑅 =2. (b) The resultant Γ/𝜈calculated
at this instance, for the TEV of both front and back foils with 𝐴𝑅 ∼ [2,8].
More specifically, an increasing AR benefits the wake recapture of the tandem configuration, which
results in a slower convergence of the back flipper´s thrust coefficient. On the other hand the efficiency
remains virtually unaffected due to the flippers´optimal kinematics and the thrust-targeting 𝜙.
Fundamentally, the above are related to the enhanced strength and cohesion of the TEV´s pair shed
at each stroke. In particular, snapshots of instantaneous vorticity show that 3D effects have a localized
behaviour around the wing tip of the front foil which remains constant throughout the range of AR´s. This
affects spanwise-averaged TEV circulation, which increases with elongation but eventually saturates,
so that the benefits diminish towards two dimensional concepts (𝐴𝑅 ∼ ∞). Low AR´s, however, lead to
weaker TEV´s which move away from the centerline and decay faster in the streamwise direction. This is
vital for the hind flipper as a stronger incoming wake will induce higher acceleration of the surrounding
flow while a wider, weaker vortex pair can enable a greater range of 𝜙since weaving within the vortices
becomes easier. Consequently, wake recapture leads to an augmented Γfor the TEV of the rear foil,
combined with a higher sensitivity towards flipper slenderness. This results in a comparatively sharper
circulation increase and a slower convergence, in a similar fashion to the ˜
𝐶𝑇results.
This study provides evidence of the AR impact on wake recapture under kinematics commonly
used in the natural world. In addition, our findings provide some design rules for more versatile bio-
inspired systems by revealing the hydrodynamic benefits and limitations of the single and tandem flipper
arrangements. It should be noted that, the singular effect of aspect ratio has been addressed here for a
simple planform. However, natural systems can achieve a wide range of planforms for the same aspect
ratio and it is possible to design even more through engineering tools. Therefore, the effects of planform
shape can be further explored for a given aspect ratio that is most suitable for tandem foils.
APPENDIX
A. Phase Optimisation
To set our reference test case in terms of maximum thrust augmentation, a preliminary study was
conducted in 2D, evaluating the phasing 𝜙of the tandem configuration for the chosen spacing, kinematic
parameters and ambient conditions. Tandem foil simulations were performed, starting from 𝜙=0𝑜and
progressing at increments of Δ𝜙=45𝑜until 𝜙=315𝑜, while the single foil was found to produce
˜
𝐶𝑡, 𝑓 0.675. Figure 12 shows that the modification of the hind foil´s thrust due to interacting with the
incoming wake, follows a cosine-like curve with respect to the phase lag, as shown in similar studies
E1-12
Figure 12. Impact of 𝜙on the two dimensional wake recapture, expressed via the relative thrust
augmentation of the two foils. Here 𝐶
𝑇 , 𝑓 =1since the front foil experiences no flow field changes,
which coincides with ˜
𝐶𝑡, 𝑓 0.675. Simulation points are depicted as while the dashed lines represent
the best fit curves.
Figure 13. Sensitivity analysis of the integration area, used to calculate the circulation of the front
foil´s TEV. Here the vorticity is first spanwise averaged, at 𝑡/𝑇=0.5as shown in (a) for 𝐴𝑅 =2. Then
it is integrated within boxes of increasing size until circulation values begin to drop (b).
(Muscutt et al.,2017a). Clearly, optimal 𝐶
𝑇 ,𝑏 is found for 𝜙=0𝑜and therefore it is chosen for all the
simulations presented in the current study.
B. Sensitivity Analysis of the TEV Circulation
In this study, circulation is calculated via the integration of vorticity over a rectangular cell (see figure
13a). As we focus on the TEV analysis, the size and location of this area should be optimized to enclose
the exact size of the vortex while minimizing ambient interference. Therefore, after visually choosing an
initial location, we gradually increase the area of integration until the overall circulation begins to drop
(see figure 13b). Having finalized the size of the box, we re-evaluate its location by moving its center
towards the y and x axis until a position of maximum circulation is identified. Due to the shape/size of
Flow E1-13
the TEV for various ARs, this procedure was conducted individually for all single and tandem test cases
such that the “highest” circulation is obtained for each case.
Acknowledgements. We would like to thank A.N. Zurman-Nasution and M. Lauber for our fruitful discussions throughout the
duration of this project. Furthermore, we would like to thank the IRIDIS High Performance Computing Facility, with its associated
support services at the University of Southampton, for their aid towards the completion of our study.
Funding Statement. This research was supported financially by the Office of Naval Research award N62909-18-1-2091 and the
Engineering and Physical Sciences Research Council doctoral training award [1789955].
Declaration of Interests. The authors declare no conflict of interest.
Author Contributions. Conceptualization: N.S.L, G.W. and B.G. Investigation: N.S.L. Writing original draft: N.S.L. Writing
review and editing: G.W. and B.G.
Data Availability Statement. All data supporting this study, including useful supplementary material, is openly available via
the University of Southampton repository at (AVAILABLE UPON ACCEPTANCE).
Ethical Standards. The research meets all ethical guidelines, including adherence to the legal requirements of the study country.
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