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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 (1𝑚−100𝑚) and can reach velocities of 0.5𝑚/𝑠−2𝑚/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=

𝜌𝑈∞C/𝜇=8500 where 𝜌is the water density, 𝑈∞is 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)=ℎ0sin(2𝑓 𝜋𝑡)

| {z }

𝑦ℎ(𝑡)

+ (1− P ) C sin[𝜃(𝑡)]

| {z }

𝑦𝜃(𝑡)

(1)

𝑤𝑖𝑡 ℎ 𝜃 (𝑡)=𝜃0sin(2𝑓 𝜋𝑡 +𝜓)(2)

where subscripts 𝑓,ℎand 𝜃denote the front (or single foil), heaving and pitching components

respectively.

Here, the instantaneous pitching angle is expressed as the 𝜃(𝑡), while ℎ0and 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 ∼ ℎ0=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 𝑑𝑦ℎ/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)=𝑦ℎ(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 ℎ(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 𝑆C≥1(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 𝑈∞is 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=0◦and SC=2. Simulation

points are characterised by ◦while the best ﬁt curve is depicted via a dashed line.

Here we should note that, similar ˜

𝐶𝑇−𝐴𝑅 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=0◦and 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 ∼ [4−6]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 ∼ [2−8]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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Cite this article: N. S. Lagopoulos, G. D. Weymouth and B. Ganapathisubramani (2021). Eﬀect of aspect ratio on the propulsive performance of

tandem ﬂapping foils. Flow,This draft was prepared using the LaTeX style ﬁle belonging to FLOW: Applications of Fluid Mechanics, E1. doi: