Content uploaded by Vaibhav Joshi

Author content

All content in this area was uploaded by Vaibhav Joshi on May 27, 2022

Content may be subject to copyright.

Eﬀect of combined heaving and pitching on propulsion of single and

tandem ﬂapping foils

Amit S. Hegde1, Pardha S. Gurugubelli2, and Vaibhav Joshi ∗3

1Department of Mechanical Engineering, Birla Institute of Technology & Science Pilani,

Vidya Vihar, Pilani, Rajasthan 333031, India

2Computing Lab, Department of Mechanical Engineering, Birla Institute of Technology &

Science Pilani, Hyderabad Campus, Hyderabad 500078, India

3Department of Mechanical Engineering, Birla Institute of Technology & Science Pilani, K K

Birla Goa Campus, NH 17B Bypass Road, Zuarinagar, Sancoale, Goa 403726, India

Abstract

In this study, we present a two and three-dimensional numerical investigation to understand the combined

eﬀects of the non-dimensional heave amplitude varying from 0 to 1 and the pitch amplitude ranging from 0◦

to 30◦on the propulsive performance for a single and tandem foil system. Flow dynamics across single and

tandem ﬂapping foils has been considered at Reynolds number of Re = 1100 and reduced frequency of f∗= 0.2.

The numerical framework consists of arbitrary Lagrangian-Eulerian moving mesh based algorithm coupled with

the variational modeling of the incompressible ﬂow equations. We initially present a systematic analysis on

the thrust generation due to the kinematic parameters for a single foil with the aid of eﬀective angle of attack,

projected area of the foil to the ﬂow direction, time-averaged pressure and streamwise velocity in the wake & wake

signature. The signiﬁcance of eﬀective angle of attack and the projected area of the foil has been emphasized

in comprehending the dynamics of lift and drag forces and their relationship with the propulsion. We next

investigate the relation between the streamwise gap and kinematic parameters on propulsion for the tandem

foil system. We show that the propulsive performance strongly depends on the upstream wake interacting

with the downstream foil, and the timing of the interaction due to the gap and phase between the kinematic

motion of the foils. Through a control volume analysis for both single and tandem foils, the time-averaged

pressure and streamwise velocity have been investigated to explain the eﬀect of kinematic parameters on the

hydrodynamic forces. Typically in the literature, the formation of jet in the wake has been attributed to thrust

generation. However, in this study, we emphasize and show the signiﬁcance of the time-averaged pressure in

the wake apart from the streamwise velocity (jet) for predicting the thrust forces. The study is concluded with

a three-dimensional demonstration of the tandem foils to understand the possible three-dimensional eﬀects due

to the large amplitude ﬂapping and wake-foil interaction.

Keywords: Flapping, Tandem, Biomimetics, Pitching, Heaving, Propulsion

1 Introduction

Bio-inspiration has been a key factor in the design of underwater vehicles and robots. Recently, there has

been an increased interest in learning from nature to improve underwater propulsive performance. The natural

propulsion systems utilized by aquatic creatures can out-perform the conventional propulsive devices by as much

as 40%, as mentioned in Mannam et al. (2020); Yu & Wang (2005). Such propulsion systems have many beneﬁts

over screw propellers and other traditional propulsors including the absence of cavitation (Huang et al., 2013;

Arndt, 2012; Mengjie et al., 2020; Mannam et al., 2020), low acoustic signature (Wagenhoﬀer et al., 2021) and

excellent manoeuvring performance (Roper et al., 2011), among others.

Bio-inspired underwater propulsion can be broadly classiﬁed into two categories (Chu et al., 2012), viz.,

ﬁnned propulsion and jet propulsion. Finned propulsion systems employ ﬂapping foils to generate thrust while

jet propulsion utilize ﬂexible membranes and surfaces to squeeze water through a small space (creating a jet).

Several species of ﬁsh use multiple ﬁns to generate thrust and consequently, some bio-mimetic marine vehicles

(Mannam & Krishnankutty, 2018; Licht et al., 2004a,b) also use multiple foils for propulsion. Recently, ﬂapping

∗Corresponding author: vaibhavj@goa.bits-pilani.ac.in

1

foils have been employed in marine engineering to harvest wave energy in Ji & Huang (2017), power generation

by turbines in Kinsey & Dumas (2012), ship propulsion in Belibassakis & Politis (2013) and in propulsion of

unmanned underwater vehicles (UUVs) (Ramamurti et al., 2010; Zhang et al., 2021). The ﬂapping foil system

has very high potential to enhance propulsion and stability of ships and UUVs (Triantafyllou et al., 2004). If

the ﬁns/foils are arranged in-line, the conﬁguration is known as a tandem conﬁguration. On the other hand, if

the foils are arranged such that they are parallel to one another, the conﬁguration is said to be side-by-side.

In addition to making use of ﬂapping ﬁns/foils, ﬁsh often swim together in formations known as “schools”.

It has been shown that this collective swimming behavior oﬀers a hydrodynamic advantage to a ﬁsh within the

formation as a result of the wake created by ﬁsh leading the school (Marras & Porﬁri, 2012). The enhanced

hydrodynamic performance corresponds to the tandem conﬁguration of ﬁns/foils and could be beneﬁcial for

underwater vehicles. To create robust and reliable tandem ﬂapping foil propulsors, it is essential that the

physics behind the relevant ﬂow phenomena is well understood. Therefore, in this study we focus on the

ﬂapping dynamics of a tandem foil conﬁguration.

Literature pertaining to ﬂapping foils highlight several governing parameters such as Reynolds number of

the ﬂow, foil geometry (chord length and thickness), and foil kinematics (Wu et al., 2020). In addition, the

frequency of ﬂapping is also of great signiﬁcance. Based on the combinations of above-mentioned parameters,

a ﬂapping foil can either produce thrust or extract energy from the surrounding ﬂow, as noted by Kinsey &

Dumas (2006). Several studies have been carried out in the energy extraction regime for single as well as tandem

ﬂapping foils by Xu et al. (2019); Karbasian et al. (2015); Ma et al. (2021); Ribeiro et al. (2021). In the case

of tandem foils, the phase diﬀerence in the ﬂapping and the streamwise gap between the foils are also crucial

parameters (Wu et al., 2020).

Several works have investigated the relationship between the phase diﬀerence of tandem ﬂapping foils and

the propulsive performance. As studied by Cong et al. (2020), the performance of the downstream foil is

aﬀected signiﬁcantly by the phase diﬀerence between the upstream and the downstream foil for streamwise gap

distances of 0.25-0.75 chord lengths at Re = 200. Studies carried out by Sampath et al. (2020) at Re = 36500

have revealed that the downstream foil generates more thrust than the upstream foil when it lags by a quarter

cycle, but performs worse if it leads by the same amount. It has also been found that the downstream foil

generates maximum thrust when the ﬂapping of both the foils is in-phase at Re = 5000 by Lua et al. (2016).

Propulsive performance of in-line multiple foils with ﬁxed spacing of g/c = 0.25 at Reynolds numbers of 500

and 1000 were considered by Han et al. (2022) where the eﬀects of phase diﬀerence and Reynolds number were

investigated.

Apart from the phase diﬀerence, the gap between the tandem foils has signiﬁcant eﬀect on the ﬂow dynamics

and propulsive performance. For synchronized plunging foils, Chen et al. (2022) found that at Re = 5000, thrust

enhancement is maximum when the streamwise gap between the foils is between 1.5 and 2 chord lengths. Studies

on tandem self-propelled ﬂexible ﬂapping plates have also been carried out by Ryu et al. (2020); Peng et al.

(2018). At Reynolds number of 100, signiﬁcant improvements in thrust generation was obtained by reducing

the streamwise gap by Ryu et al. (2020). At Re = 200, the propulsive eﬃciency was found to be larger when the

upstream plate was longer than the downstream plate (Peng et al., 2018). A numerical study by Pan & Dong

(2020) at Re = 1000 identiﬁed that an increase in the streamwise spacing in a school of ﬂapping foils reduced

the inﬂuence of the lateral neighbors on the performance of the ﬂapping foils. Recently, Joshi & Mysa (2021a,b)

studied the eﬀect of gap and the chord sizes for the tandem foils on the propulsion at Re = 1100. The mechanism

of wake interaction with the downstream foil was identiﬁed and studied in detail. A periodic variation in the

thrust performance for the downstream foil was observed with the gap between the foils, indicating the crucial

eﬀect of the wake interaction. Furthermore, the combined eﬀects of phase diﬀerence and the gap between the

tandem foils was studied experimentally by Boschitsch et al. (2014). The gap between the tandem foils were

considered from 0.5 to 5 times the chord length along with varying Strouhal numbers and phase diﬀerence by

Muscutt et al. (2017) at Reynolds number of 7000. It was found that the downstream foil produced from null

to almost twice the thrust of a single foil, depending on the gap and phase diﬀerence. The vortex interaction

between the foils were also discussed in detail.

Works by Mandujano & M´alaga (2018); Alam & Muhammad (2020); Chao et al. (2021a,b); Thakor et al.

(2020); Das et al. (2016) dealt with pure pitching motion of the foil. Non-dimensionalizing with respect to

the foil thickness d, Alam & Muhammad (2020) studied pitching foil at 0.21 ≥Std=f d/U∞≥0.33, 1.1≥

A∗=A/d ≥1.6, and Chao et al. (2021a,b) considered Std= 0.1−0.3 and A∗= 0.5−2, where Adenotes

the peak-to-peak amplitude of the trailing edge. Furthermore, the parameters utilized by Thakor et al. (2020)

consisted of 2◦−6◦of pitch amplitude and reduced frequency (πfc/U∞) in the range 3 −9, cbeing the chord

of the foil. Studies by Deng et al. (2016); Floryan et al. (2017) focused on pure pitching and heaving motions

separately. Experiments conducted by Floryan et al. (2019) involved combined heaving and pitching motion

where h0/c = 0.1−0.75, pitch amplitude θ0= 5◦−40◦with frequency of 0.1 Hz and chord-based Reynolds

number of 8000. Computations by Yu et al. (2017) considered frequency range 1 −30 Hz, pitch amplitude

3◦−19◦,h0/c = 0.1−0.9 and Reynolds number 1000 −1600000. As per the knowledge of the authors, studies

dealing with the combined motion of heaving and pitching of the single ﬂapping foil are scarce and no such

2

detailed study concerning the kinematic motion parameters exists for the tandem foil conﬁguration.

Most of the works in the literature talk about the transition from the drag-producing von-K´arm´an vortex

street to the thrust-producing inverted von-K´arm´an street in the wake due to change of kinematic parameters

in the ﬂapping motion of the foil. It has also been noted that while investigating the thrust-producing regime,

researchers have mostly relied on the time-averaged streamwise velocity in the wake of the foil to comprehend the

thrust generation. It was pointed out in Alam & Muhammad (2020) that the jet formation and wake signature

are attributes of the thrust production and do not give an insight about its origin. The trend of mean thrust

generation still remains unanswered for the scenarios where ﬂapping foils with positive thrust are compared. Is

the comparison of time-averaged streamwise velocity in the wake for these cases enough to predict the trend in

the mean thrust force? We try to answer this question in the current work by considering the inﬂuence of the

kinematic motion parameters (heave and pitch amplitudes) on propulsion for single as well as tandem foils.

Majority of the computational research conducted on ﬂapping foils has been two-dimensional (Wu et al.,

2020) and the three-dimensional spanwise as well as end eﬀects have not been taken into consideration. Recent

works by Lagopoulos et al. (2021); Arranz et al. (2020); Jurado et al. (2022) have studied the eﬀect of aspect

ratio of the foil on propulsive performance. However, the conditions under which three-dimensional (3D) ﬂow

eﬀects become important are yet to be studied in detail for tandem foils.

In the present study, we numerically investigate the ﬂow dynamics of a single and tandem ﬂapping foil

system at low Reynolds number of 1100. This Reynolds number falls in the range 103−104which has been

found to be relevant for bird ﬂight and swimming of ﬁshes (Taylor et al., 2003; Triantafyllou et al., 1993). We

employ a moving mesh arbitrary Lagrangian-Eulerian framework for the ﬂapping motion of the foil. The ﬂuid

dynamics is modeled with the help of variational ﬁnite element method applied to incompressible Navier-Stokes

equations.

We try to shed some light on the following questions from the present work:

•How does the kinematic parameters such as heave and pitch amplitudes aﬀect the propulsive performance

for combined heaving and pitching of single and tandem foils?

•Can the trend in mean thrust force be predicted solely by observing the time-averaged streamwise velocity

in the wake of the ﬂapping foil (single and tandem) system?

•What is the inﬂuence of the streamwise gap between the tandem foils on the thrust generation capability

of the downstream foil?

•How does the three-dimensional ﬂow behave during the wake-foil interaction for large amplitude ﬂapping

foils in tandem conﬁguration?

The article is organized in the following manner. First, we brieﬂy discuss the numerical framework in

section 2. The next section 3 discusses the deﬁnition of the various parameters utilized in the study. Flapping

dynamics of a single foil along with the eﬀects of the kinematic parameters are studied in section 4. The

tandem arrangement of ﬂapping foils is examined in section 5. This is followed by the demonstration of the

three-dimensional simulation for tandem foils in Section 6. Finally, the key ﬁndings are summarized and the

study is concluded in section 7.

2 Numerical framework

In the current study, the ﬂapping dynamics of the foils is modeled using the moving mesh arbitrary Lagrangian-

Eulerian (ALE) framework. Discretization of the ﬂow equations is performed using a stabilized Petrov-Galerkin

variational formulation, while the foil motion is speciﬁed by satisfying the kinematic equilibrium condition or

the velocity continuity at the interface between the ﬂuid and the foil. Here, we brieﬂy review the governing

equations of the formulation for the sake of completeness.

The ﬂow is modeled with the help of incompressible Navier-Stokes equations written in the ALE framework

as

ρf∂vf

∂t χ

+ρf(vf−w)· ∇vf=∇ · σf+ρfbf,(1)

∇ · vf= 0,(2)

where the ﬂuid velocity is given by vf= (vf

x, vf

y) with its X- and Y- components, and the mesh velocity is

denoted as w. The body force acting on a ﬂuid element is written as bfand the ﬂuid density is given as ρf. We

consider a Newtonian ﬂuid for which the Cauchy stress tensor can be written as σf=−pI+µf(∇vf+ (∇vf)T)

in which the ﬂuid pressure is denoted by p, ﬂuid dynamic viscosity by µfand the identity matrix by I. In

Eq. (1), χdenotes the ALE referential coordinate system pertaining to the moving mesh coordinates. The

governing equations are temporally discretized in the time interval t∈[tn, tn+1] with the help of the Generalized-

αmethod by Chung & Hulbert (1993) while the spatial discretization is carried out by stabilized ﬁnite element

approximations. Detailed description of the present formulation can be found in the works by Joshi & Jaiman

(2018, 2019); Jaiman & Joshi (2022).

3

θ(t)

h(t)

c

(a)

θu(t)

hu(t)

cu

θd(t)

hd(t)

cd

g

(b)

Figure 1: Flapping kinematic motion of the foil consisting of heaving and pitching motion for (a) single foil,

and (b) tandem foils.

The ﬂapping kinematic motion of the foil is illustrated in Fig. 1. For a single foil as shown in Fig. 1(a),

the kinematics consists of a heave component h(t) = h0sin(2πft +φh) along with a pitch component θ(t) =

θ0sin(2πf t) about a pitching axis located at the leading edge of the foil. Here, h0,θ0,fand φhdenote the

heave amplitude, pitch amplitude, ﬂapping frequency and the phase diﬀerence between the heaving and pitching

motion, respectively. We also consider the tandem conﬁguration of ﬂapping foils with a gap of gbetween the foils

(Fig. 1(b)). The chord length of the upstream and the downstream foils are denoted by cuand cd, respectively.

The upstream foil’s motion is given as

θu(t) = θu

0sin(2πf ut),(3)

hu(t) = hu

0sin(2πf ut+φu

h),(4)

where hu

0,θu

0,fuand φu

hrepresent the heave amplitude, pitch amplitude, ﬂapping frequency and phase diﬀerence

between the heaving and pitching motion, respectively, for the upstream foil. Similarly, the motion of the

downstream foil is given by

θd(t) = θd

0sin(2πf dt+ϕ),(5)

hd(t) = hd

0sin(2πf dt+φd

h+ϕ),(6)

where ϕis the phase diﬀerence between the kinematic motions of the upstream and downstream foils and the

other symbols have their usual meanings. In the current work, we consider National Advisory Committee for

Aeronautics (NACA) 0015 foils in tandem with cu/cd= 1, ϕ= 0◦and φu

h=φd

h= 90◦.

The current problem involves an interaction between the ﬂuid and structure in which the structural displace-

ments are imposed on the surface of the foil. This means that there is a one-way coupling that is facilitated by

matching the structural and ﬂuid velocities at the boundary between the foil and the ﬂuid (kinematic equilibrium

condition). The mathematical equation associated with this boundary condition is given as

vf(e

ϕ(X, t), t) = vs(X, t),∀X∈Γfs,(7)

where e

ϕrepresents a one-to-one mapping between the structural position Xat time t= 0 and its corresponding

position at time t > 0. Γfs is the ﬂuid-structure interface at t= 0 and vsdenotes the structural velocity.

The nonlinear Navier-Stokes equations are solved by the Newton-Raphson iterative technique. The coupling

between the ﬂow equations and the moving mesh framework is carried out in a partitioned iterative manner,

the details of which can be found in the work by Joshi & Mysa (2021b); Jaiman & Joshi (2022). The above

formulation has been veriﬁed and validated, consisting of mesh convergence and time convergence studies in the

earlier work by Joshi & Mysa (2021b) for the single and tandem ﬂapping foils and will not be discussed in the

present work for brevity.

3 Parameters of interest

The non-dimensional parameters pertaining to the single ﬂapping foil are the Reynolds number Re = (ρfU∞c)/µf,

non-dimensional heave amplitude h0/c and non-dimensional ﬂapping frequency f∗= (fc)/U∞, where U∞is

4

the freestream velocity of the ﬂow. Similarly, for the tandem foils, we consider the characteristic length as the

chord of the downstream foil cd. Thus, Reynolds number Re = (ρfU∞cd)/µf, non-dimensional heave ampli-

tude of upstream and downstream foils are denoted by hu

0/cdand hd

0/cd, non-dimensional ﬂapping frequency of

upstream and downstream foils are given by f∗

u= (fucd)/U∞and f∗

d= (fdcd)/U∞, respectively, and the gap

ratio between the foils is g/cd. Note that we consider cu/cd= 1 in the present study.

The propulsive performance of the ﬂapping foils is determined by evaluating the integrated values of the

ﬂuid forces on the foil surface. The instantaneous coeﬃcients are given as follows for a single foil:

CY=FY

1

2ρfU2

∞cl =1

1

2ρfU2

∞cl ZΓfs(t)

(σf·n)·nydΓ,(8)

CX=FX

1

2ρfU2

∞cl =1

1

2ρfU2

∞cl ZΓfs(t)

(σf·n)·nxdΓ,(9)

CT=−FX

1

2ρfU2

∞cl =−CX,(10)

CP=P

1

2ρfU3

∞cl =−FYvs

y,heave −MZω

1

2ρfU3

∞cl .(11)

In the set of equations given above, CYand CXare the force coeﬃcients in the transverse and inline directions to

the freestream velocity U∞respectively, CTis the thrust coeﬃcient and CPdenotes the power coeﬃcient. FX,

FYand Pare the X-component of the force, Y-component of the force and the power supplied to the structure

respectively. The moment of the forces acting about the pitching axis of the foil is denoted by MZwhile the

heave translational velocity of the foil is written as vs

y,heave = 2πf h0cos(2πf t +φh). The angular velocity of the

foil in pitching motion is denoted by ω. Here, cand l= 1 are the chord and the span of the foil respectively.

The variable Xrepresents the time averaged mean value of Xover a time period Tof the oscillation. The

propulsive eﬃciency of the foil can thus be written as η=CT/CP. Similarly, one can extend the coeﬃcients

for the tandem foils. The mean thrust coeﬃcient and propulsive eﬃciency of the combined tandem foil system

can be evaluated by considering the combined mean thrust and power coeﬃcients of the foils.

Next, we discuss the propulsive performance of a single and tandem foil conﬁgurations subjected to a

freestream ﬂow under the variation of the heave amplitude and pitch amplitude. We investigate the diﬀerent

mechanisms of thrust generation to comprehensively understand the ﬂow dynamics of ﬂapping foils for such

scenarios.

4 Flapping of a single foil

The single ﬂapping foil system has received signiﬁcant attention in the context of both numerical and experi-

mental studies. In particular, studies on the eﬀect of kinematic parameters on the propulsive performance of

the foil have been performed by Das et al. (2016); Deng et al. (2016); Yu et al. (2017); Mandujano & M´alaga

(2018); Alam & Muhammad (2020); Chao et al. (2021a,b); Thakor et al. (2020); Floryan et al. (2017) for a

range of Reynolds number, ﬂapping frequency, heave and pitch amplitudes. However, the combined motion

of heaving and pitching for a single foil and the inﬂuence of the kinematic parameters on propulsion have not

been studied comprehensively and no such detailed study exists for the tandem foils. Prior to understanding

the eﬀects of heave and pitch amplitudes on the performance of the tandem foil conﬁguration, we discuss their

eﬀects for a single isolated foil in this section.

We perform two-dimensional computations and give insights about the inﬂuence of varying heave and pitch

amplitudes and explain the trends with the help of wake signatures. Furthermore, we comprehend the generation

of thrust with the help of a control volume analysis which gives a complete picture of the time-averaged propulsive

performance for the foil.

A ﬂapping foil generates thrust through the development of a leading edge vortex (LEV) in the propulsion

regime. During the downstroke, this LEV is responsible for suction pressure (negative pressure) on the upper

surface of the foil, whereas a positive pressure exists at the lower surface. This pressure diﬀerential along with

the orientation of the ﬂapping foil during the ﬂapping motion leads to a net thrust force. This is depicted in

Fig. 2(a). Therefore, the favorable conditions for generation of thrust during the downstroke of a ﬂapping foil

are (Joshi & Mysa, 2021b): (i) suction pressure on the upper surface, and (ii) positive pressure on the lower

surface.

The various components of the net resultant force Ron the foil are shown in Fig. 2(b). As a result of

the incoming freestream velocity and the heave velocity of the foil, the incoming ﬂow eﬀective velocity Ueﬀ is

inclined to the horizontal direction by an angle tan−1(−vs

y,heave(t)/U∞). The eﬀective angle of attack for the

foil can thus be deﬁned as

αeﬀ (t) = tan−1−vs

y,heave(t)

U∞−θ(t),(12)

5

High pressure on

LEV

Thrust

U∞

Low pressure on

upper surface

lower surface

(a)

U∞

vs

y,heave

θ(t)

αeﬀ (t)

FL

FD

FY

FT=−FX

R

Ueﬀ

Aproj (t)

(b)

Figure 2: Flapping single foil: (a) generation of thrust as a result of LEV during the downstroke, and (b)

instantaneous force components during ﬂapping.

which represents the inclination of Ueﬀ with the chord of the ﬂapping foil. The components of the force along

the eﬀective velocity and perpendicular to it are known as the drag (FD) and lift (FL) forces, respectively. The

resultant force can also be decomposed in the direction of freestream velocity as FXand perpendicular to it as

FY. A relationship exists between these two decompositions and can be written as

FD=FXcos(αeﬀ +θ) + FYsin(αeﬀ +θ),(13)

FL=−FXsin(αeﬀ +θ) + FYcos(αeﬀ +θ).(14)

The drag (CD) and lift (CL) coeﬃcients can also be deﬁned by non-dimensionalizing these forces with (1/2)ρfU2

∞cl.

We also quantify the projected area of the foil as seen from the eﬀective ﬂow direction Ueﬀ . For this scenario,

the foil is assumed to be a straight line connecting the leading edge to the trailing edge. Based on the prescribed

ﬂapping motion, the projected area is evaluated as

Aproj (t) = |csin(αeﬀ (t))|l, (15)

where l= 1 is the span of the foil.

4.1 Eﬀect of heave amplitude (h0)on propulsion

The temporal variation of the force coeﬃcients in the X and Y directions in a ﬂapping cycle considering f∗= 0.2,

θ0= 30◦and Re = 1100 is shown for diﬀerent heave amplitudes in Fig. 3. It can be observed that an increase

in the heave amplitude leads to higher thrust generation for the single foil with maximum thrust coeﬃcient

noted for h0/c = 1. It can be deduced that the mean thrust coeﬃcient CTover a ﬂapping cycle increases with

increase in heave amplitude of the foil, where CTis negative for h0/c = 0 and transitions to a positive thrust

as heave amplitude increases. To understand this average variation of the thrust with heave amplitude, we

consider a control volume surrounding the ﬂapping foil, as shown in Fig. 4. We will apply the conservation of

linear momentum principle in the streamwise direction. Let ep1Aand ep2Adenote the pressure forces on the left

and right boundaries of the control volume, eu1and eu2represent the velocities of the ﬂuid at the boundaries

and FTis the reaction to the thrust force on the control volume. Following Newton’s second law,

XFx= ˙m2eu2−˙m1eu1(16)

ep1A−ep2A+FT=ρfA(eu2

2−eu2

1) (17)

FT=ρfA(eu2

2−eu2

1)+(ep2−ep1)A. (18)

As can be observed, the mean thrust force on the foil is a consequence of not just the change in velocity but

also the change in the pressure across the control volume. Note here that the qualtities ep1,ep2,eu1and eu2are

6

(a) (b)

Figure 3: Temporal variation of the force coeﬃcients in a ﬂapping cycle for a single foil with varying h0/c (ﬁxed

θ0= 30◦): (a) CT, and (b) CY.

ep1Aep2A

eu1eu2

FT

Control volume

l0

xt

Figure 4: Control volume analysis of the ﬂapping foil.

7

(a) (b)

(c) (d) (e)

Figure 5: Control volume analysis of the ﬂapping foil for the eﬀect of heave amplitude: (a) time-averaged

pressure at the left boundary (p1), (b) time-averaged streamwise velocity at the left boundary (u1), (c) time-

averaged pressure at the right boundary (p2), (d) time-averaged streamwise velocity at the right boundary (u2),

and (e) evaluated quantities based on Eq. (18) for varying h0/c.

8

(a) (b)

Figure 6: Temporal variation of the force coeﬃcients in a ﬂapping cycle for a single foil with varying h0/c (ﬁxed

θ0= 30◦): (a) CDand (b) CL.

evaluated in an averaged sense and vary uniformly over the two boundaries. They are deﬁned as follows:

ep1A=Zl0

0

p1l dy, (19)

eu2

1A=Zl0

0

u2

1l dy, (20)

where l0and lare the width of the control volume and the span (in the third dimension, l= 1 for two-dimensional

study), respectively. Here, p1and u1are the time-averaged variation of the quantities across the Y-direction,

as shown in Figs. 5(a) and (b), respectively. Similar expressions can be deﬁned for ep2Aand eu2

2Afor which the

time-averaged quantities across Y-direction are also depicted in Figs. 5(c) and (d).

For the current scenario of varying heave amplitude, it can be noticed from Figs. 5(a) and (b) that the

variation in u1and p1across diﬀerent h0/c is negligible as they represent freestream quantities. The diﬀerence

between the cases is observed for u2and p2. The peak of the time-averaged velocity u2is negative for h0/c = 0

depicting a velocity-deﬁcit region in the wake, while it transitions to a jet-like velocity proﬁle for h0/c = 1

(see Fig. 5(d)). On the other hand, the negative pressure is larger for h0/c = 0 compared to h0/c = 0.4

(Fig. 5(c)). For h0/c = 0, the velocity is small, while the pressure has a large negative value, resulting in a

larger contribution of the (ep2−ep1)Aterm in Eq. (18) giving negative thrust, which is clearly visible in Fig.

5(e). As the heave amplitude is increased to h0/c = 1, the velocity attains large values in the wake making

the contribution of ρfA(eu2

2−eu2

1) term larger resulting in higher thrust. Thus, the increasing trend of the mean

thrust coeﬃcient can be explained by observing the variations in both velocity and pressure at the downstream

wake of the ﬂapping foil as summarized in Fig. 5(e).

The temporal variation of the drag and lift coeﬃcients in a ﬂapping cycle is shown for diﬀerent heave

amplitudes in Fig. 6. The peak value of the drag coeﬃcient in a ﬂapping cycle ﬁrst decreases from h0/c ∈[0,0.4]

and then increases with h0/c wherein the maximum drag coeﬃcient is observed for h0/c = 1. The variation in

CLis quite similar to CY. However, the amplitude of CDis smaller compared to CLdepicting lift as the main

contributor to the resultant force on the foil. During the downstroke (ﬁrst-half of the ﬂapping cycle), increasing

the heave amplitude increases the heave velocity leading to an increase in the change in the amplitude and

direction of the eﬀective velocity Ueﬀ (see the schematic in Fig. 2(b)). As the direction of the eﬀective velocity

changes, so does the projected area of the foil to Ueﬀ . This area has been plotted in Fig. 7(a). The variation in

the area directly correlates to the drag coeﬃcient as with increasing heave amplitude, the projected area also

decreases till h0/c = 0.4 and then increases. On the other hand, the lift coeﬃcient variation is explained by the

changing eﬀective angle of attack, depicted in Fig. 7(b). During the downstroke, the eﬀective angle of attack

at a time instant increases with increasing heave amplitude which conﬁrms the increase in CLat a time instant

in the downstroke.

9

(a) (b)

Figure 7: Temporal variation of the following quantities in a ﬂapping cycle for a single foil with varying h0/c

(ﬁxed θ0= 30◦): (a) Apro j, and (b) αeﬀ .

Next, we look into the wake signature of the ﬂapping foil to conﬁrm our observations regarding the force

coeﬃcients and their correlation with the eﬀective angle of attack. The Z-vorticity contour plots (Left) of the

wake of the ﬂapping foil along with the pressure distribution (Middle) and forces (Right) are depicted in Fig.

8. The transition from drag-producing to thrust-producing wake can be noticed in the wake signature of the

ﬂapping foil. At h0/c = 0 (Fig. 8(a)), the clockwise (CW) and the counter-clockwise (CCW) vortices (blue and

red in color respectively) are aligned such that they produce a von-K´arm´an (vK) vortex street, which is drag

producing. As heave amplitude is increased to h0/c = 0.4 (Fig. 8(b)), the vortex street resembles a scenario

close to the transition to thrust-producing inverted von-K´arm´an (IvK) vortex street (this transition has been

referred as “feathering limit” in the literature). At h0/c = 1 (Fig. 8(c)), we observe the thrust-producing IvK

vortex street. The pressure arrow plots for the same instances of the ﬂapping foil depict the increased suction

and positive pressure on the upper and lower surfaces of the foil, respectively with heave amplitude. This results

in thrust-favoring conditions as the heave amplitude increases. The forces on the foil at the same instances are

also depicted in the ﬁgure. With the increase in h0/c, the eﬀective angle of attack increases. This results in the

resultant force transitioning such that its X-component gives a more positive thrust with the increase in h0/c.

Moreover, the change of the projected area Apro j(t) can also be noticed in Fig. 8(Right) which decreases from

h0/c = 0 to h0/c = 0.4 and then increases for h0/c = 1. As mentioned earlier, this translates directly to the

trend in the drag coeﬃcient in Fig. 6(a).

The main contributor to the resultant force in this case is the lift force. Thus, for the parameters considered,

an increase in the thrust coeﬃcient with heave amplitude is noticed as a consequence of increase in the eﬀective

angle of attack. This dependence of the thrust coeﬃcient on the eﬀective angle of attack for heaving foil

corroborates the observations by Van Buren et al. (2019), where it was pointed out that the thrust is entirely

a consequence of lift forces. At ﬁxed pitch amplitude and f∗, the mean thrust coeﬃcient was found to increase

with heave amplitude. The study conducted by Yu et al. (2017) observed a similar eﬀect of changing the heave

amplitude for ﬁxed ﬂapping frequency and pitch amplitude.

4.2 Eﬀect of pitch amplitude (θ0)on propulsion

For observing the variation of the propulsive performance with the pitch amplitude, the heave amplitude is

ﬁxed at h0/c = 1 along with the Reynolds number Re = 1100 and the ﬂapping frequency f∗= 0.2. The pitch

amplitude is varied between 0◦and 30◦with increments of 5◦. The time history of the thrust coeﬃcient is

shown in Fig. 9(a). It is seen that the maximum thrust coeﬃcient in a ﬂapping cycle increases with an increase

in the pitch amplitude, similar to the observations by Yu et al. (2017) for low frequencies. Furthermore, the

instant of the peak thrust coeﬃcient is delayed as the pitch amplitude increases.

It can also be deduced that the mean thrust coeﬃcient is positive for all θ0values, however it increases

with θ0. A control volume analysis similar to the previous subsection (Fig. 4) can be performed here, with

10

FL=FY

R

Ueﬀ =U∞

Aproj(t)

FD=FX

(a)

U∞

vs

y,heave

FL

FD

FY

FX

R

Ueﬀ

Aproj(t)

(b)

U∞

vs

y,heave

FL

FD

FY

FT=−FX

R

Ueﬀ

Aproj(t)

(c)

Figure 8: Flapping single foil at t/T = 0.25, θ0= 30◦and (a) h0/c = 0, (b) h0/c = 0.4, and (c) h0/c = 1. The

Z-vorticity contours, pressure distribution along the surface of the downstream foil and the instantaneous forces

are depicted in the Left, Middle and Right columns, respectively.

11

(a) (b)

Figure 9: Temporal variation of the force coeﬃcients in a ﬂapping cycle for a single foil with varying θ0(ﬁxed

h0/c = 1): (a) CT, and (b) CY.

variations in p2and u2at the right boundary of the control volume shown in Fig. 10. Here, we observe a

peculiar behavior. Although the velocity eu2is higher for θ0= 0◦, it has the lowest mean thrust among the cases

considered. Reviewing the conservation of linear momentum derived in Eq. (18), it can be observed that in this

scenario, the second term (ep2−ep1)Ahas signiﬁcant eﬀect as higher negative pressures are observed for θ0= 0◦

case, in comparison to others (depicted in Fig. 10(c)). Therefore, it is important to consider the pressure

distribution at the wake of the foil to get a clear understanding of the thrust generation. The time-averaged

streamwise velocity pattern solely does not paint a clear picture of the average thrust. As can be observed from

Fig. 10(c), there are competing eﬀects of the two terms, viz., (ep2−ep1)Aand ρfA(eu2

2−eu2

1), which result in the

trend for the generated thrust.

The interplay of the drag and lift forces can be observed in Fig. 11. In contrast to the previous subsection,

here the drag force is noted to have signiﬁcant contribution to the resultant force, especially for smaller θ0values.

The peak value of the drag coeﬃcient decreases with increasing pitch amplitude. As the heave amplitude is

constant for the cases considered, the eﬀective velocity Ueﬀ remains at the same amplitude and direction across

the various pitch amplitudes. However, as the pitch amplitude changes, so does the projected area of the foil to

the ﬂow direction, as shown in Fig. 12(a). The variation in the projected area translates to the drag coeﬃcient

plot. The lift coeﬃcient (Fig. 11(b)) has a similar variation to CY. Minor changes are observed across the

diﬀerent pitch amplitudes, with highest lift noted for θ0= 0◦during the downstroke. The lift coeﬃcient can be

correlated with the changing eﬀective angle of attack across pitch amplitudes, depicted in Fig. 12(b).

The wake signature of the ﬂapping foil is visualized by Z-vorticity contours in Fig. 13 along with pressure

distribution and the forces on the foil. As all the pitch amplitudes produce net thrust, an IvK vortex street

is observed for all the cases. Furthermore, with increase in pitch amplitude, the width of the wake increases

which can be conﬁrmed by the time-averaged streamwise velocity plot in Fig. 10(b). The pressure distribution

across the various pitch amplitudes more or less remain the same. However, it is the inclination of the foil

with the horizontal which changes, leading to a decrease in the projected area of the foil to the incoming ﬂow.

The inclination due to the increasing pitch amplitude also inclines the resultant force towards the negative X

direction, increasing the thrust component of the force. Another perspective could be the increase in the frontal

area of the foil to the freestream velocity U∞through which the pressure diﬀerential across the upper and lower

surfaces of the foil tends to give more thrust component as pitch amplitude increases.

The key takeaways from the analysis of the single ﬂapping foil are:

•Eﬀective angle of attack (αeﬀ(t)) relates to the lift force and the projected area to the incoming ﬂow

(Aproj (t)) relates to the drag force.

•Average thrust increases with the heave amplitude with majority contribution from the lift force which

increases with increasing eﬀective angle of attack.

•The mean thrust increases with the pitch amplitude, while decreasing the projected area of the foil to the

incoming ﬂow Ueﬀ . On the other hand, the projected area to the freestream velocity U∞increases resulting in

12

(a) (b) (c)

Figure 10: Control volume analysis of the ﬂapping foil for the eﬀect of pitch amplitude: (a) time-averaged

pressure at the right boundary (p2), (b) time-averaged streamwise velocity at the right boundary (u2), and (c)

evaluated quantities based on Eq. (18) for varying θ0.

(a) (b)

Figure 11: Temporal variation of the force coeﬃcients in a ﬂapping cycle for a single foil with varying θ0(ﬁxed

h0/c = 1): (a) CD, and (b) CL.

13

(a) (b)

Figure 12: Temporal variation of the following quantities in a ﬂapping cycle for a single foil with varying θ0

(ﬁxed h0/c = 1): (a) Aproj , and (b) αeﬀ .

higher component of thrust force.

•The behavior of both the time-averaged pressure and streamwise velocity at the wake of the foil are

necessary to quantify the mean thrust.

With this background context about the inﬂuence of the kinematic parameters on the propulsive character-

istics of a single isolated ﬂapping foil, we next consider the more complex tandem conﬁguration.

5 Flapping foils in tandem conﬁguration

In the tandem conﬁguration of ﬂapping foils, the downstream foil interacts with the wake of the upstream foil

resulting in a very complex interaction. Unlike the single isolated foil discussed in the previous section, the

ﬂow physics is not that straightforward to understand in this scenario. It is very diﬃcult to deﬁne the basic

parameters such as eﬀective angle of attack and the projected area for the downstream foil, as the direction

of incoming ﬂow is disturbed by the upstream foil. Therefore, we focus on the vortex interaction mechanisms

to comprehend the behavior of the downstream foil under the inﬂuence of the upstream foil’s wake. We also

consider the projected frontal area of the ﬂapping foil to the freestream velocity U∞in our discussion. The eﬀect

of the gap between the ﬂapping foils on the propulsive performance of the downstream foil was investigated

in detail by Joshi & Mysa (2021b) for gap ratios of g/cd∈[1,14] for foils of identical chord length, i.e.,

cu/cd= 1. Various mechanisms of vortex interactions were proposed and categorized into constructive and

destructive interactions, which have been noticed by Broering et al. (2012); Gopalkrishnan et al. (1994); Lewin

& Haj-Hariri (2003); Muscutt et al. (2017).

A constructive interaction occurs when same-signed vortices interact, leading to supply of vorticity on

the surface of the downstream foil. On the other hand, a destructive interaction involves the interaction of

opposite-signed vortices which drives away the vorticity on the surface of the foil. It is the combination of

above interactions on the upper and lower surfaces of the foil which determines the favorable and unfavorable

conditions for generation of thrust. Major interactions have been shown in Fig. 14, the details of which can

be found in the work by Joshi & Mysa (2021b). To summarize, a constructive interaction (same-signed) on the

upper surface (Interaction-1 or I-1) and a destructive one (opposite-signed) on the lower surface (Interaction-

4 or I-4) of the foil favors the generation of thrust during the downstroke. On the contrary, a destructive

interaction on the upper surface (Interaction-2 or I-2) and a constructive one on the lower surface (Interaction-4

or I-4) leads to unfavorable thrust generating conditions. Furthermore, when an opposite-signed vortex travels

in proximity to the upper surface of the foil, it tends to pull the shear layer leading to an enlargement of LEV,

which is favorable (Interaction-3 or I-3) (not shown in Fig. 14). The earlier (later) the favorable (unfavorable)

condition occurs during the downstroke, the better is the thrust generation for the downstream foil in the

tandem arrangement.

14

U∞

vs

y,heave

FLFD

FY

FT=−FX

R

Ueﬀ

Aproj(t)

(a)

U∞

vs

y,heave

FL

FD

FY

FT=−FX

R

Ueﬀ

θ(t)

Aproj(t)

(b)

U∞

vs

y,heave

FL

FD

FY

FT=−FX

R

Ueﬀ

θ(t)

Aproj(t)

(c)

Figure 13: Flapping single foil at t/T = 0.25, h0/c = 1 and (a) θ0= 0◦, (b) θ0= 15◦, and (c) θ0= 25◦. The

Z-vorticity contours, pressure distribution along the surface of the downstream foil and the instantaneous forces

are depicted in the Left, Middle and Right columns, respectively.

vortex interaction

vortex interaction

on upper surface (I-1)

on lower surface (I-4)

Opposite signed

Same signed

(a)

vortex interaction

vortex interaction

on upper surface (I-2)

on lower surface (I-5)

Same signed

Opposite signed

(b)

Figure 14: Major types of interactions of the downstream foil with the upstream foil’s wake for (a) favorable,

and (b) unfavorable, thrust generating conditions in the work by Joshi & Mysa (2021b).

15

(a) (b) (c)

Figure 15: Variation of the mean thrust coeﬃcient for the tandem ﬂapping foils at hd

0/cd= 1, θu

0=θd

0= 30◦

with varying upstream foil heave amplitude hu

0/cdfor (a) upstream foil, (b) downstream foil, and (c) tandem

foil system.

Based on the gap ratios and the considered parameters in Joshi & Mysa (2021b) (Re = 1100, cu/cd= 1,

θu

0=θd

0= 30◦,f∗

u=f∗

d= 0.2, hu

0/cd=hd

0/cd= 1, φu

h=φd

h= 90◦), a periodic variation in the thrust and

propulsive eﬃciency was noticed with the increase in gap ratio, similar to the observations in the literature.

The gap ratio of g/cd= 4 manifested lowest mean thrust coeﬃcient (most unfavorable condition) and g/cd= 7

depicted the highest thrust coeﬃcient (most favorable condition). Therefore, in the current investigation to study

the eﬀects of heave and pitch amplitudes on the propulsive performance, we consider these two representative

gap ratios. We perform two-dimensional computations of the tandem ﬂapping foils to investigate the eﬀects of

kinematic motion on propulsion.

5.1 Eﬀect of heave amplitude of upstream foil (hu

0)on propulsion

Here, we discuss the inﬂuence of the heave amplitude of the upstream foil on the thrust and propulsive eﬃciency

of the tandem foils. As mentioned earlier, two gap ratios of g/cd= 4,7 are considered. The downstream heave

amplitude is ﬁxed (hd

0/cd= 1) and both the upstream and downstream pitch amplitudes are held constant

(θu

0=θd

0= 30◦). As expected, the mean thrust coeﬃcient of the upstream foil follows the trend of the single

foil, indicating null interference of the downstream foil on the upstream foil characteristics, as shown in Fig.

15(a). An increase in CTis observed with hu

0/cdwhich has been discussed in detail in the previous section

pertaining to a single ﬂapping foil. For the downstream foil, the variation in mean thrust is shown in Fig.

15(b). The mean thrust decreases and increases with hu

0/cdfor g/cd= 4 and g/cd= 7, respectively. For the

combined tandem system, the mean thrust coeﬃcient is depicted in Fig. 15(c). For the gap ratio of 7, the

combined CTis observed to increase with hu

0/cd, reaching a maximum value of 2.34 at hu

0/cd= 1. In contrast,

at g/cd= 4, the thrust decreases slightly at hu

0/cd= 0.4 and then increases marginally. The maximum mean

thrust for g/cd= 4 is 0.84 at hu

0/cd= 0. The propulsive eﬃciency is depicted in Fig. 16. The eﬃciency for the

upstream foil is identical to the isolated single foil. It increases sharply as the dimensionless heave amplitude

grows from 0.6 to 0.8 and continues to increase at higher amplitudes albeit slowly. For hu

0/cd= [0,0.2,0.4],

negative thrust or drag is observed for the upstream foil and therefore, the eﬃciency has not been plotted. For

the downstream foil, a decrease in eﬃciency with increasing hu

0/cdis observed for g/cd= 4, while it increases

gradually for g/cd= 7. Note that in Figs. 15(b) and 16(b), the “single” foil depicts the results for an isolated

foil with h0/c = 1 as the heave amplitude of the downstream foil is also unchanged, i.e., hd

0/cd= 1. For the

combined tandem foils shown in Fig. 16(c), the propulsive eﬃciency follows a similar trend as that of the mean

thrust coeﬃcient. The maximum eﬃciency of 41% is observed for g/cd= 7 at hu

0/cd= 1 and about 32% for

g/cd= 4 at hu

0/cd= 0.2.

The average thrust coeﬃcient variation with hu

0/cdcan be realized by revisiting the control volume analysis

discussed in Section 4.1 for a single foil. Here, we will apply the conservation of linear momentum in the

streamwise direction on a control volume enclosing the tandem foil conﬁguration. The mean thrust force is

given by FT=ρfA(eu2

2−eu2

1) + (ep2−ep1)A. We have observed that the variation in the freestream quantities

such as ep1and eu1on the left boundary of the control volume for various kinematic parameters is negligible.

On the other hand, the quantities ep2and eu2on the right boundary of the control volume depict the averaged

wake of the tandem system and are of importance to the study. These quantities are evaluated by integrating

the time-averaged pressure (p2) and velocity (u2) across the Y-direction similar to Eqs. (19) and (20).

16

(a) (b) (c)

Figure 16: Variation of the propulsive eﬃciency ηfor the tandem ﬂapping foils at hd

0/cd= 1, θu

0=θd

0= 30◦

with varying upstream foil heave amplitude hu

0/cdfor (a) upstream foil, (b) downstream foil, and (c) tandem

foil system.

(a) (b) (c)

Figure 17: Control volume analysis of the tandem system of foils: (a) time-averaged pressure p2, (b) time-

averaged streamwise velocity u2, at the right boundary of the control volume; and (c) evaluated quantities

based on Eq. (18) for representative hu

0/cd; for the gap ratio of g/cd= 4.

The variation in p2and u2at a distance of xt= 6cdfrom the leading edge of the downstream foil is shown

in Figs. 17(a-b) and 18(a-b) for gap ratio of 4 and 7, respectively. The average pressure distribution across

the right boundary of the control volume is such that the quantity (ep2−ep1)Ais almost identical across the

diﬀerent heave amplitudes, which is observed in Figs. 17(c) and 18(c). Therefore, the average streamwise

velocity distribution u2dictates the mean thrust of the tandem system in this scenario which is clear from the

evaluated quantity ρfA(eu2

2−eu2

1) and the trend in FTin the ﬁgure.

The variation of the thrust coeﬃcient of the downstream foil for the two gap ratios under various hu

0/cdis

shown in Fig. 19. It can be observed that the thrust generation capability of the downstream foil decreases

with increasing hu

0/cdfor g/cd= 4, while a contrasting eﬀect is observed for high gap ratio of 7. These transient

variations in thrust are a result of favorable or unfavorable interactions of the wake of the upstream foil with

the downstream foil. This is shown in the wake signature diagram by visualizing the Z-vorticity contours at

diﬀerent instances for the gap ratios g/cd= 4 and g/cd= 7 in Figs. 20 and 21, respectively. For g/cd= 4, at

hu

0/cd= 0.2, the CCW vortex just passes the upper surface of the foil in close proximity at t/T = 0.28 (I-3)

which leads to the LEV getting pulled on the upper surface during the downstroke (see Fig. 20(a)). This gives

the favorable condition for thrust generation near the mid-stroke. The negative and positive pressure on the

upper and lower surface of the downstream foil respectively is represented by the arrow diagram in the ﬁgure.

As the heave amplitude is increased to hu

0/cd= 0.6−1, the interaction becomes unfavorable as the CCW vortex

interacts with the upper and lower surface of the foil fairly early in the downstroke (see Figs. 20(b) and (c)).

This early interaction takes away the CW vorticity from the upper surface eventually leading to premature

shedding of the LEV. Furthermore, the interaction also supplies the CCW vorticity to the lower surface of the

foil strengthening the shear layer resulting in suction pressure on the lower surface (Figs. 20(b) and (c)).

17

(a) (b) (c)

Figure 18: Control volume analysis of the tandem system of foils: (a) time-averaged pressure p2, (b) time-

averaged streamwise velocity u2, at the right boundary of the control volume; and (c) evaluated quantities

based on Eq. (18) for representative hu

0/cd; for the gap ratio of g/cd= 7.

(a) (b)

Figure 19: Temporal variation of the thrust coeﬃcient for the downstream foil for varying hu

0/cdat g/cd: (a) 4,

and (b) 7.

18

(a)

(b)

(c)

Figure 20: Flapping foils at diﬀerent time instances of a ﬂapping cycle at g/cd= 4, hd

0/cd= 1, θu

0=θd

0= 30◦

and hu

0/cd: (a) 0.2, (b) 0.6, and (c) 1. The Z-vorticity contours and pressure distribution along the surface of

the downstream foil are depicted on the Left and Right column at the instant.

19

At g/cd= 7, the variation of the mean thrust coeﬃcient is opposite to that of g/cd= 4. Although all the

hu

0/cdcases generate positive thrust or are favorable conditions; with increase in hu

0/cd, the interaction of the

CW vorticity with the downstream foil occurs earlier (I-1 and I-4) during the downstroke, as can be seen in Fig.

21. For example, this interaction occurs at t/T = 0.42 (almost the end of downstroke) for hu

0/cd= 0.2 (Fig.

21(a)) but occurs at the start of the downstroke t/T = 0.03 for hu

0/cd= 1 (Fig. 21(c)). This earlier interaction

generates a larger LEV (as vorticity is supplied due to constructive interaction on the upper surface) during

the prime conﬁguration of the downstroke where the projected area of the foil to the freestream direction is the

largest, thus leading to higher thrust.

5.2 Eﬀect of heave amplitude of downstream foil (hd

0)on propulsion

Next, we understand the eﬀect of varying the heave amplitude of the downstream foil on its propulsive perfor-

mance. To accomplish this, we ﬁx the heave amplitude of the upstream foil hu

0/cd= 1, the pitch amplitudes as

θu

0=θd

0= 30◦and the non-dimensional ﬂapping frequency is selected as f∗= 0.2. The heave amplitude of the

downstream foil is varied in the range hd

0/cd∈[0,1].

The variation in the mean thrust coeﬃcient and the propulsive eﬃciency for the tandem foils are shown in

Figs. 22 and 23, respectively. Note that as the upstream heave amplitude is ﬁxed, the propulsive performance

of the upstream foil will be identical to an isolated single foil at h0/c = 1, θ0= 30◦and f∗= 0.2. It is observed

that the mean thrust for the downstream foil decreases with decrease in the heave amplitude of the downstream

foil for both the gap ratios, as shown in Fig. 22(a), similar to the variation for a single foil with decreasing h0/c.

At g/cd= 4, we do not observe a positive thrust coeﬃcient for the downstream foil for the heave amplitudes

considered. On the other hand, for g/cd= 7, the thrust coeﬃcient is negative for hd

0/cd= [0,0.2]. For the

combined system (Fig. 22(b)), the mean thrust increases for both the gap ratios with increasing hd

0/cd, similar

to the behavior of the downstream foil as the thrust of the upstream foil (constant) gets added to that of the

downstream foil in the combined system. The propulsive eﬃciency for the downstream foil, shown in Fig. 23(a),

ﬁrst increases slightly when hd

0/cdis decreased from 1 to 0.8, and then decreases monotonically for g/cd= 7.

As the average thrust of the combined system is always positive, the propulsive eﬃciency is deﬁned for all

the cases, as depicted in Fig. 23(b). For g/cd= 4, the eﬃciency is observed to increase linearly with hd

0/cd,

reaching a maximum value of 31.4% at hd

0/cd= 1. The behavior of the eﬃciency is nonlinear in nature and has

a maximum value of 42% at hd

0/cd= 0.8 for g/cd= 7.

Carrying out the control volume analysis similar to previous sections, the variation in the mean thrust force

can be explained. The quantities ep2and eu2at the right boundary of the control volume are evaluated using

Eq. (18) with the help of time-averaged pressure (p2) and velocity u2which are plotted in Figs. 24 and 25

for g/cd= 4 and 7, respectively. For both the gap ratios, the pressure distribution is such that the evaluated

quantity (ep2−ep1)Adoes not change much for hd

0/cd∈[0.4,0.8] indicating the streamwise velocity distribution

to be the dominant term representing the generated thrust. On the other hand, for hd

0/cd∈[0,0.4], the change

in the velocity term ρfA(eu2

2−eu2

1) is observed to be very small indicating the pressure term to be the dominant

term and dictating the thrust trend. This observation is similar to that of the single isolated foil in Section 4.1.

The temporal change in the thrust coeﬃcient for the downstream foil is shown in Fig. 26 for the two gap

ratios. For both the gap ratios, the peak of the thrust coeﬃcient in a ﬂapping cycle increases with the heave

amplitude hd

0/cd. However, this increase is signiﬁcant for g/cd= 7. We focus now on the vortex interaction

of the upstream foil’s wake with the downstream foil to get an insight about the instantaneous change in the

thrust. The vortex interaction can be visualized in Figs. 27 and 28 for g/cd= 4 and g/cd= 7, respectively. In

Section 4.1, it was observed that for a single foil, at lower h0/c, the wake is drag producing vK vortex street and

as the heave amplitude increases, the propulsive performance increases with the wake resembling IvK vortex

street. A similar observation can be inferred for the tandem conﬁguration when the heave amplitude of the

downstream foil is increased. For gap ratio of g/cd= 4, the individual wake of the downstream foil is inherently

vK vortex street at hd

0/cd= 0 (Fig. 27(a)). The incoming IvK wake of the upstream foil interacts with the

downstream foil in such a manner that the opposite-signed vortices of the vK and IvK vortex streets pair up.

This pairing can also be observed for hd

0/cd= 0.4 (Fig. 27(b)). With increase in hd

0/cd, the CCW vortex of the

upstream foil’s wake interacts with the downstream foil at the early of the downstroke resulting in I-2 where

the CW vortex shear layer on the upper surface of the foil is prematurely shed (seen at t/T = 0.28 in Fig.

27(c)). Thus, for all the cases at g/cd= 4, the thrust is negative (unfavorable condition) for the downstream

foil and it is inherently unfavorable as a result of vK street for lower hd

0/cd. The negative thrust (or drag)

for the downstream foil can also be noticed from the pressure arrow plots where most of the foil has suction

pressure during the downstroke (unfavorable).

For higher gap ratio of g/cd= 7, the vK vortex street for the downstream foil is observed at hd

0/cd= 0

(Fig. 28(a)) which leads to the negative thrust. With increase in the heave amplitude of the downstream

foil, favorable condition for thrust generation occurs (Figs. 28(b) and (c)). The CW vortex from the wake of

the upstream foil interacts with the upper and lower surface of the downstream foil. This results in supply

20

(a)

(b)

(c)

Figure 21: Flapping foils at diﬀerent time instances of a ﬂapping cycle at g/cd= 7, hd

0/cd= 1, θu

0=θd

0= 30◦

and hu

0/cd: (a) 0.2, (b) 0.6, and (c) 1. The Z-vorticity contours and pressure distribution along the surface of

the downstream foil are depicted on the Left and Right column at the instant.

21

(a) (b)

Figure 22: Variation of the mean thrust coeﬃcient for the (a) downstream foil, and (b) tandem foil system, at

hu

0/cd= 1, θu

0=θd

0= 30◦with varying downstream foil heave amplitude hd

0/cd.

(a) (b)

Figure 23: Variation of the propulsive eﬃciency for the (a) downstream foil, and (b) tandem foil system, at

hu

0/cd= 1, θu

0=θd

0= 30◦with varying downstream foil heave amplitude hd

0/cd.

(a) (b) (c)

Figure 24: Control volume analysis of the tandem system of foils: (a) time-averaged pressure p2, (b) time-

averaged streamwise velocity u2, at the right boundary of the control volume; and (c) evaluated quantities

based on Eq. (18) for representative hd

0/cd; for the gap ratio of g/cd= 4.

22

(a) (b) (c)

Figure 25: Control volume analysis of the tandem system of foils: (a) time-averaged pressure p2, (b) time-

based on Eq. (18) for representative hd

0/cd; for the gap ratio of g/cd= 7.

(a) (b)

Figure 26: Temporal variation of the thrust coeﬃcient for the downstream foil for varying hd

0/cdat g/cd: (a) 4,

and (b) 7.

23

(a)

(b)

(c)

Figure 27: Flapping foils at diﬀerent time instances of a ﬂapping cycle at g/cd= 4, hu

0/cd= 1, θu

0=θd

0= 30◦

and hd

0/cd: (a) 0, (b) 0.4, and (c) 0.8. The Z-vorticity contours and pressure distribution along the surface of

the downstream foil are depicted on the Left and Right column at the instant.

24

(a)

(b)

(c)

Figure 28: Flapping foils at diﬀerent time instances of a ﬂapping cycle at g/cd= 7, hu

0/cd= 1, θu

0=θd

0= 30◦

and hd

0/cd: (a) 0, (b) 0.4, and (c) 0.8. The Z-vorticity contours and pressure distribution along the surface of

the downstream foil are depicted on the Left and Right column at the instant.

25

(a) (b) (c)

Figure 29: Variation of the mean thrust coeﬃcient for the tandem ﬂapping foils at hu

0/cd=hd

0/cd= 1, θd

0= 30◦

with varying upstream foil pitch amplitude θu

0for (a) upstream foil, (b) downstream foil, and (c) tandem foil

system.

of vorticity to the upper surface (I-1) and the vorticity on the lower surface is driven away (I-4). A pressure

diﬀerential is created based on the suction pressure on the upper surface and high pressure on the lower surface

of the foil during downstroke, leading to thrust. These interactions (I-1 and I-4) occur at the early stages of the

downstroke as hd

0/cdincreases, thus extending the favorable condition in the duration where projected area of

the foil to the freestream direction is maximum, giving higher average thrust.

5.3 Eﬀect of pitch amplitude of upstream foil (θu

0)on propulsion

In this subsection, we examine the eﬀect of the pitch amplitude of the upstream foil on the thrust and propulsive

eﬃciency of the tandem conﬁguration. Once again, two gap ratios (g/cd= 4,7) are considered while the

downstream pitch amplitude is ﬁxed at θd

0= 30◦. Here, the heave amplitudes of both the upstream and the

downstream foil are held constant (hu

0/cd=hd

0/cd= 1).

The mean thrust coeﬃcient of the upstream foil for both the gap ratios (Fig. 29(a)) exhibits a similar

behavior to that of the single foil indicating that the downstream foil does not inﬂuence the ﬂow around the

upstream foil. As the pitch amplitude (θu

0) is increased to 30◦from zero, the mean thrust coeﬃcient grows larger,

peaking around θu

0= 25◦before dropping slightly at θu

0= 30◦. The trend in CTwith θu

0can be explained in

terms of the behavior of the single foil, explained in the previous section. The variation in the mean thrust

coeﬃcient for the downstream foil with θu

0is shown in Fig. 29(b). For g/cd= 4, the thrust generated by the

downstream foil increases slightly with pitch amplitude. On the other hand, at g/cd= 7, the thrust generated

by the downstream foil exhibits a more complex behavior; it decreases gradually until the region θu

0∈[20◦,25◦]

before starting to rise again. The thrust coeﬃcient for a single foil at identical parameters (h0/c = 1 and

θ0= 30◦) is also shown in the ﬁgure for comparison. Owing to the small variation in the mean thrust coeﬃcient

in this case, we can infer that the pitch amplitude of the upstream foil has a minor eﬀect on the propulsive

performance of the downstream foil. Therefore, for the combined tandem system, the mean thrust slightly

increases as a result of the increasing trend for the upstream foil, as shown in Fig. 29(c), reaching a maximum

of 2.34 and 0.76 for g/cd= 7 and 4, respectively, both at θu

0= 30◦.

The propulsive eﬃciency of the upstream foil increases almost linearly with upstream pitch amplitude and

negligible variation is observed as the gap ratio is changed, as shown in Fig. 30(a). The trends for both

g/cd= 4 and g/cd= 7 are identical to that of the single foil, as expected. On the other hand, the variation in

the eﬃciency of the downstream foil is irregular for g/cd= 7 (Fig. 30(b)) as a slight variation between 0.36 and

0.415 is observed. Note that as the average thrust coeﬃcient is negative for all the g/cd= 4 cases, they are not

shown in the eﬃciency plot. The propulsive eﬃciency for the combined tandem foils increases with θu

0for both

the gap ratios reaching a maximum of 41% and 31.4% for g/cd= 7 and g/cd= 4, respectively, as observed in

Fig. 30(c).

The control volume analysis provides an insight into the dominant term in determining the mean thrust

generation. The pressure and streamwise velocity distribution on the right boundary of the control volume are

shown in Figs. 31 and 32 for the two gap ratios. From the plots, it can be inferred that for g/cd= 4, the term

(ep2−ep1)Ahas a signiﬁcant contribution for θu

0∈[5◦,15◦] and its variation translates to the thrust generation.

On the other hand, both the velocity and pressure terms tend to reach an asymptotic value as θu

0reaches 30◦

leading to a minor increase in the mean thrust. At the gap ratio of 7, a signiﬁcant diﬀerence is observed for the

26

(a) (b) (c)

Figure 30: Variation of the propulsive eﬃciency for the tandem ﬂapping foils at hu

0/cd=hd

0/cd= 1, θd

0= 30◦

with varying upstream foil pitch amplitude θu

0for (a) upstream foil, (b) downstream foil, and (c) tandem foil

system.

(a) (b) (c)

Figure 31: Control volume analysis of the tandem system of foils: (a) time-averaged pressure p2, (b) time-

based on Eq. (18) for representative θu

0; for the gap ratio of g/cd= 4.

pressure distribution in Fig. 32(a) compared to the velocity (Fig. 32(b)) for various θu

0values. This gives an

indication of the pressure term (ep2−ep1)Abeing a dominant factor in determining the thrust of the tandem foil

system. Therefore, the mean thrust follows the trend of the pressure term and increases with θu

0, albeit slowly.

The instantaneous change in the thrust coeﬃcient for the downstream foil at various θu

0is depicted in Fig.

33. Minor changes are observed in the variation with the thrust coeﬃcient at the mid-downstroke (t/T = 0.25)

in a ﬂapping cycle; increasing and decreasing with increase in θu

0for g/cd= 4 and 7, respectively. To elucidate

the variation of the thrust of the downstream foil, we turn our attention to the Z-vorticity contours presented

in Figs. 34 and 35 for g/cd= 4 and 7, respectively. It can be observed that as θu

0increases, the wake of the

upstream foil is wider, identical to the observation for a single foil in Section 4.2. For g/cd= 4, the thrust

coeﬃcient is negative for all the cases as a consequence of unfavorable condition. However, there is a slight

increase in the mean thrust with θu

0. This can be explained by observing the increase in the delay of the shedding

of the premature LEV by the CCW vortex as θu

0increases (as a consequence of the wider wake). The shedding

occurs around t/T = 0.15, 0.18 and 0.22 for θu

0= 5◦, 15◦and 30◦, respectively (Fig. 34). As the interaction gets

delayed in the downstroke, the average thrust coeﬃcient increases, although slightly. For higher gap ratio of

g/cd= 7, we observe a slight dip in CTtill θu

0= 20◦and then an increase till 30◦. At θu

0= 5◦(Fig. 35(a)), the

stronger CCW vorticity from the upstream foil’s wake is in close proximity to the downstream foil at t/T = 0.28

(around mid-downstroke). This leads to the favorable condition (I-3) which pulls the shear layer on the leading

edge of the downstream foil, forming LEV. As the angle is increased to θu

0= 15◦in Fig. 35(b), the interaction

I-3 is weaker (due to the weaker CCW vortex) and delayed to t/T = 0.34. Furthermore, the stronger CCW

vortex is far away from the upper surface due to the wider wake of the upstream foil at higher θu

0. This delay in

the interaction leads to a slightly lower mean thrust compared to 5◦and is the reason for the decreasing trend

27

(a) (b) (c)

Figure 32: Control volume analysis of the tandem system of foils: (a) time-averaged pressure p2, (b) time-

based on Eq. (18) for representative θu

0; for the gap ratio of g/cd= 7.

(a) (b)

Figure 33: Temporal variation of the thrust coeﬃcient for the downstream foil for varying θu

0at g/cd: (a) 4,

and (b) 7.

28

in CT. For θu

0= 25◦, the favorable condition for thrust generation occurs via I-1 and I-4 during the start of the

downstroke (Fig. 35(c)) which results in increasing trend for the average thrust.

5.4 Eﬀect of pitch amplitude of downstream foil (θd

0)on propulsion

Having studied the eﬀect of the pitch amplitude of the upstream foil, we now focus on the pitch amplitude of

the downstream foil. The heave amplitudes of the upstream and downstream foils are ﬁxed (hu

0/cd=hd

0/cd= 1)

and the upstream pitch amplitude is θu

0= 30◦in this situation.

The inﬂuence of θd

0on the mean thrust coeﬃcient of the downstream foil is shown in Fig. 36(a). For the

reference case of an isolated (single) foil, the mean thrust coeﬃcient increases until θd

0= 25◦after which it

gradually settles at CT≈0.8. In comparison, for g/cd= 4, CTincreases and then decreases with θd

0. The peak

value of CToccurs at θd

0= 10◦and it is noted that CTfor all the cases is very close to zero. On the other

hand, for g/cd= 7, the mean thrust coeﬃcient increases consistently for θd

0∈[0◦,30◦] with a maximum value

close to 1.6. At low values of θd

0, the behavior of the mean thrust is observed to be similar to that of a single

foil. The combined tandem system behaves similarly to the downstream foil characteristics as the thrust of the

upstream foil (constant) gets added to that of the downstream foil.

The variation in propulsive eﬃciency is shown in Fig. 37. Here, the isolated foil exhibits an almost linear

increase in ηwith θd

0. For g/cd= 4, the propulsive eﬃciency of the downstream foil increases with downstream

pitch amplitude before peaking at θd

0= 20◦and then decreasing again. When the gap ratio is increased to

g/cd= 7, the behavior of ηclosely matches that of the isolated foil case. On the other hand, it is interesting

to note that the eﬃciency of the combined system increases monotonically for both the gap ratios, as shown in

Fig. 37(b).

The control volume analysis shows a similar variation as that of a single foil presented in Section 4.2. The

time averaged pressure and velocity for the two gaps are provided in Figs. 38 and 39. For g/cd= 4, the combined

eﬀects of the velocity and pressure distributions drive the thrust variation with θd

0. The thrust increases slightly

till θd

0= 15◦and then decreases with further increase in the pitch amplitude. The competing eﬀects of the two

terms leads to a minor variation of the thrust with θd

0. At high gap ratio of 7, the contribution of the velocity

term ρfA(eu2

2−eu2

1) seems to be small across θd

0(see Fig. 39(c)). Thus, the pressure term dictates the trend in

the thrust coeﬃcient, resulting in an increase in CTwith θd

0.

The time history of the thrust coeﬃcient for the downstream foil is shown in Fig. 40 for the two gap

ratios. A negligible variation for various θd

0for g/cd= 4 is observed. However, the maximum thrust in a

cycle increases with θd

0for the higher gap ratio of 7. Let us further analyse the ﬂow dynamics to comprehend

the variation in the propulsive performance. As observed for a single isolated foil, the mean thrust coeﬃcient

increases with θ0for f∗= 0.2 as the frontal area to the freestream velocity U∞increases. Therefore, if the

upstream foil would not be present, one would observe an increase in CT. In the case of tandem conﬁguration,

the behavior of the downstream foil is inﬂuenced by the interaction of it with the wake of the upstream foil,

leading to either favorable or unfavorable conditions for propulsion. For g/cd= 4, there seems to be a competing

eﬀect between the increasing frontal area to freestream velocity of the downstream foil and the eﬀect of the

unfavorable condition. For all the cases of θd

0, there is an unfavorable condition caused due to the interaction

of CCW vortex of the upstream foil’s wake with the leading edge of the downstream foil (I-2 and I-5) resulting

in premature LEV shedding. For θd

0= 0◦(Fig. 41(a)), a secondary LEV is formed during the downstroke.

As θd

0increases, the projected frontal area of the downstream foil also increases leading to an increase in CT

at θd

0= 15◦(Fig. 41(b)). However with further increase in θd

0, there is no generation of secondary LEV and

premature shedding of LEV is more dominant leading to decrease in average thrust (Fig. 41(c)). Therefore, at

lower θd

0, the downstream foil behaves similar to an isolated foil (increase in mean CTwith θd

0: although slightly

due to already existing unfavorable condition), while for higher θd

0, the unfavorable condition is more prominent

and reduces the propulsive performance of the downstream foil.

Contrary to low gap ratio, we observe an increase in mean thrust for g/cd= 7. The favorable thrust

generating condition as a consequence of I-1 and I-4 can be observed for θd

0= 5◦in Fig. 42(a). A comparison of

the vorticity contours at the quarter time period for θd

0= 15◦and 25◦is depicted in Fig. 42(b-c). The favorable

condition occurs for all the θd

0cases as indicated by the pressure arrow plots showing negative and positive

pressures on the upper and lower surfaces of the foil, respectively. However, the projected frontal area to the

freestream velocity increases with θd

0leading to an increase in the component of the net force in the freestream

direction, i.e., thrust.

Therefore, the type of vortex interaction from the wake of the upstream foil with the downstream foil;

and the time and duration of interaction in the ﬂapping cycle determines the propulsive performance of the

downstream foil. From an operational perspective, the tandem foil system generates maximum thrust where

interactions leading to favorable conditions occur at the downstream foil. Moreover, the propulsion eﬃciency is

determined by the amount of thrust generated and the power input to the system. Optimal regimes where the

thrust is higher and the power input is lower leads to higher eﬃciency of the propulsive system.

29

(a)

(b)

(c)

Figure 34: Flapping foils at diﬀerent time instances of a ﬂapping cycle at g/cd= 4, hu

0/cd=hd

0/cd= 1, θd

0= 30◦

and θu

0: (a) 5◦, (b) 15◦, and (c) 30◦. The Z-vorticity contours and pressure distribution along the surface of the

downstream foil are depicted on the Left and Right column at the instant.

30

(a)

(b)

(c)

Figure 35: Flapping foils at diﬀerent time instances of a ﬂapping cycle at g/cd= 7, hu

0/cd=hd

0/cd= 1, θd

0= 30◦

and θu

0: (a) 5◦, (b) 15◦, and (c) 25◦. The Z-vorticity contours and pressure distribution along the surface of the

downstream foil are depicted on the Left and Right column at the instant.

31

(a) (b)

Figure 36: Variation of the mean thrust coeﬃcient for the (a) downstream foil, and (b) tandem foil system, at

hu

0/cd=hd

0/cd= 1, θu

0= 30◦with varying downstream foil pitch amplitude θd

0.

(a) (b)

Figure 37: Variation of the propulsive eﬃciency for the (a) downstream foil, and (b) tandem foil system, at

hu

0/cd=hd

0/cd= 1, θu

0= 30◦with varying downstream foil pitch amplitude θd

0.

(a) (b) (c)

Figure 38: Control volume analysis of the tandem system of foils: (a) time-averaged pressure p2, (b) time-

based on Eq. (18) for representative θd

0; for the gap ratio of g/cd= 4.

32

(a) (b) (c)

Figure 39: Control volume analysis of the tandem system of foils: (a) time-averaged pressure p2, (b) time-

based on Eq. (18) for representative θd

0; for the gap ratio of g/cd= 7.

(a) (b)

Figure 40: Temporal variation of the thrust coeﬃcient for the downstream foil for varying θd

0at g/cd: (a) 4,

and (b) 7.

33

(a)

(b)

(c)

Figure 41: Flapping foils at diﬀerent time instances of a ﬂapping cycle at g/cd= 4, hu

0/cd=hd

0/cd= 1, θu

0= 30◦

and θd

0: (a) 0◦, (b) 15◦, and (c) 25◦. The Z-vorticity contours and pressure distribution along the surface of the

downstream foil are depicted on the Left and Right column at the instant.

34

(a)

(b)

(c)

Figure 42: Flapping foils at diﬀerent time instances of a ﬂapping cycle at g/cd= 7, hu

0/cd=hd

0/cd= 1, θu

0= 30◦

and θd

0: (a) 5◦, (b) 15◦, and (c) 25◦. The Z-vorticity contours and pressure distribution along the surface of the

downstream foil are depicted on the Left and Right column at the instant.

35

cd

20cd

15cd

cug

5cd

40cd

U∞

Top (Slip condition)

Bottom (Slip condition)

Inlet Outlet

No-slip condition

X

Y

Z

Figure 43: Schematic of the three-dimensional computational domain for a uniform ﬂow across tandem ﬂapping

foils at Re = 1100.

6 Three-dimensional tandem ﬂapping foils

In order to understand if the wake-foil interaction in tandem ﬂapping exhibits any three-dimensionality, we

investigate the tandem foil system considering the parameters Re = 1100, g/cd= 7, hu

0/cd=hd

0/cd= 1,

θu

0=θd

0= 30◦,φu

h=φd

h= 90◦and ϕ= 0◦. The computational domain with the boundary conditions

is depicted in Fig. 43. A freestream velocity in the X-direction is imposed on the inlet boundary which is

15cdfrom the upstream foil, while a stress-free condition is satisﬁed at the outlet boundary (20cdfrom the

downstream foil). The slip or no-penetration condition is satisﬁed at the top and bottom boundaries which are

equidistant (20cd) from the center of the foils. A no-slip condition is satisﬁed at the surface of the two foils.

The two-dimensional mesh created for the computations presented in the previous sections is extruded in the

third dimension with a span size of 5cdconsidering ∆z/cd= 0.125 as the resolution in the Z-direction. The

boundaries perpendicular to the Z-axis representing the total span of the foils are considered to be periodic

boundaries. Note that this conﬁguration models the foil to have an inﬁnite span, eliminating the end-eﬀects.

The discretization consists of around 4.1 million grid points consisting of approximately 4 million eight-node

hexahedral elements.

The time history of the thrust coeﬃcient for the upstream and the downstream foils is shown in Fig. 44(a).

The temporal variation of the thrust coeﬃcient for the two-dimensional case has also been plotted. It can

be observed that for the parameters considered, there is no diﬀerence between the propulsive response of the

tandem conﬁguration of ﬂapping foils in three-dimensions as compared to the two-dimensional results. This

is further corroborated by the visualization of the vortex structures in Fig. 44(b-d). The three-dimensional

vortex structures are depicted by the iso-surfaces of Q-criterion at 0.25 colored by the freestream velocity. The

two-dimensional Z-vorticity is also shown at three diﬀerent spans of z/cd∈[0,2.5,5]. In spite of the large

amplitude ﬂapping of the foils and the wake-foil interaction in the tandem conﬁguration, the ﬂow structures are

observed to be two-dimensional at Re = 1100. This suggests the adoption of the two-dimensional simulation

can be employed for understanding the ﬂow dynamics at this Reynolds number. Further study needs to be

carried out at high Reynolds numbers including turbulence eﬀects to understand the three-dimensionality of

the ﬂow patterns.

7 Conclusions

The present study numerically investigated the inﬂuence of heave (hu

0/cd∈[0,1] and hd

0/cd∈[0,1]) and pitch

amplitudes (θu

0∈[0◦,30◦] and θd

0∈[0◦,30◦]) on the propulsion of a single and tandem foil system. The

combination of these kinematic parameters was explored and their eﬀects on the performance of NACA 0015

foils in tandem were studied.

For a single isolated foil, the lift and the drag forces were observed to be directly proportional with the

eﬀective angle of attack and the projected area of the foil to the incoming ﬂow direction, respectively. Through

36

(a) (b)

(c) (d)

Figure 44: Three-dimensional ﬂapping at g/cd= 7, hu

0/cd=hd

0/cd= 1, θu

0=θd

0= 30◦,f∗

u=f∗

d= 0.2 and

Re = 1100: (a) comparison of the thrust coeﬃcient in a ﬂapping cycle for 2D and 3D computations, and iso-

surfaces of Q-criterion colored by streamwise velocity (and Z-vorticity at various layers along the span) at t/T:

(b) 0, (c) 0.2, and (d) 0.4.

37

a control volume analysis, it was shown that both the time-averaged pressure and the streamwise velocity

at the wake are crucial indicators to quantify the mean thrust force. Corroborating the ﬁndings from the

literature (Van Buren et al., 2019; Yu et al., 2017), the propulsive performance of the single foil increased with

heave amplitude pertaining to the increase in the eﬀective angle of attack resulting in larger lift force, which

has a major contribution to the thrust. Increase in pitch amplitude also increased the propulsive force as a

consequence of increasing projected area of the foil to the freestream velocity resulting in higher component of

the thrust force.

The salient ﬁndings from the investigation of the tandem foil system at two gap ratios are summarized as

follows:

•The upstream foil behaved as the single isolated foil with no interference from the downstream foil for the

considered streamwise gaps between the foils.

•The time-averaged pressure and streamwise velocity variation in the wake of the tandem system collectively

determined the mean propulsive force for the system, as shown by the control volume analysis. The trend of

the mean thrust for the foil system is similar to that of single foil when considering the eﬀects of heave (hd

0)

and pitch (θd

0) amplitudes of the downstream foil.

•The transient behavior of the thrust for the tandem system can be interpreted by examining the vortex

interaction patterns and linking it with favorable and unfavorable conditions for thrust generation provided in

the literature (Joshi & Mysa, 2021b; Broering et al., 2012; Gopalkrishnan et al., 1994; Lewin & Haj-Hariri,

2003; Muscutt et al., 2017).

•At the low Reynolds number of 1100, the wake structures due to the wake-foil interaction are observed to

be two-dimensional for large amplitude combined heave and pitch motions.

Acknowledgments

The computational work for this article was performed on the High Performance Computing resource at BITS

Pilani, K K Birla Goa Campus and the resource at Computing Lab, Department of Mechanical Engineering,

BITS Pilani, Hyderabad Campus.

Declaration of Interests

The authors report no conﬂict of interest.

References

Alam, Md Mahbub & Muhammad, Zaka 2020 Dynamics of ﬂow around a pitching hydrofoil. Journal of

Fluids and Structures 99, 103151.

Arndt, Roger E.A. 2012 Some remarks on hydrofoil cavitation. Journal of Hydrodynamics 2012 24:3 24,

305–314.

Arranz, G., Flores, O. & Garc

´

ıa-Villalba, M. 2020 Three-dimensional eﬀects on the aerodynamic

performance of ﬂapping wings in tandem conﬁguration. Journal of Fluids and Structures 94, 102893.

Belibassakis, Kostas A. & Politis, Gerasimos K. 2013 Hydrodynamic performance of ﬂapping wings for

augmenting ship propulsion in waves. Ocean Engineering 72, 227–240.

Boschitsch, Birgitt M., Dewey, Peter A. & Smits, Alexander J. 2014 Propulsive performance

of unsteady tandem hydrofoils in an in-line conﬁguration. Physics of Fluids 26 (5), 051901, arXiv:

https://doi.org/10.1063/1.4872308.

Broering, Timothy M., Lian, Yongsheng & Henshaw, William 2012 Numerical investigation of

energy extraction in a tandem ﬂapping wing conﬁguration. AIAA Journal 50 (11), 2295–2307, arXiv:

https://doi.org/10.2514/1.J051104.

Chao, Li Ming, Alam, Md Mahbub & Ji, Chunning 2021aDrag–thrust transition and wake structures of

a pitching foil undergoing asymmetric oscillation. Journal of Fluids and Structures 103, 103289.

Chao, Li Ming, Alam, Md Mahbub, Ji, Chunning & Wang, Hanfeng 2021bFlow map of foil undergoing

combined fast and slow pitching. Physics of Fluids 33, 101902.

Chen, Zhe, Li, Xiong & Chen, Long 2022 Enhanced performance of tandem plunging airfoils with an

asymmetric pitching motion. Physics of Fluids 34, 011910.

38

Chu, Won Shik, Lee, Kyung Tae, Song, Sung Hyuk, Han, Min Woo, Lee, Jang Yeob, Kim,

Hyung Soo, Kim, Min Soo, Park, Yong Jai, Cho, Kyu Jin & Ahn, Sung Hoon 2012 Review of

biomimetic underwater robots using smart actuators. International Journal of Precision Engineering and

Manufacturing 2012 13:7 13, 1281–1292.

Chung, J. & Hulbert, G. M. 1993 A time integration algorithm for structural dynamics with improved

numerical dissipation: The generalized-αmethod. Journal of Applied Mechanics 60 (2), 371–375.

Cong, Longfei, Teng, Bin & Cheng, Liang 2020 Hydrodynamic behavior of two-dimensional tandem-

arranged ﬂapping ﬂexible foils in uniform ﬂow. Physics of Fluids 32 (2), 021903.

Das, Anil, Shukla, Ratnesh K. & Govardhan, Raghuraman N. 2016 Existence of a sharp transition in

the peak propulsive eﬃciency of a low- pitching foil. Journal of Fluid Mechanics 800, 307–326.

Deng, Jian, Sun, Liping, Teng, Lubao, Pan, Dingyi & Shao, Xueming 2016 The correlation between

wake transition and propulsive eﬃciency of a ﬂapping foil: A numerical study. Physics of Fluids 28 (9),

094101.

Floryan, Daniel, Van Buren, Tyler, Rowley, Clarence W. & Smits, Alexander J. 2017 Scaling

the propulsive performance of heaving and pitching foils. Journal of Fluid Mechanics 822, 386–397.

Floryan, Daniel, Van Buren, Tyler & Smits, Alexander J. 2019 Large-amplitude oscillations of foils

for eﬃcient propulsion. Phys. Rev. Fluids 4, 093102.

Gopalkrishnan, R., Triantafyllou, M. S., Triantafyllou, G. S. & Barrett, D. 1994 Active vorticity

control in a shear ﬂow using a ﬂapping foil. Journal of Fluid Mechanics 274, 1–21.

Han, Pan, Pan, Yu, Liu, Geng & Dong, Haibo 2022 Propulsive performance and vortex wakes of multiple

tandem foils pitching in-line. Journal of Fluids and Structures 108, 103422.

Huang, Biao, Ducoin, Antoine & Young, Yin Lu 2013 Physical and numerical investigation of cavitating

ﬂows around a pitching hydrofoil. Physics of Fluids 25, 102109.

Jaiman, R. K. & Joshi, V. 2022 Computational Mechanics of Fluid-Structure Interaction, 1st edn. Springer.

Ji, Fang & Huang, Diangui 2017 Eﬀects of reynolds number on energy extraction performance of a two

dimensional undulatory ﬂexible body. Ocean Engineering 142, 185–193.

Joshi, V. & Jaiman, R. K. 2018 A positivity preserving and conservative variational scheme for phase-ﬁeld

modeling of two-phase ﬂows. Journal of Computational Physics 360, 137–166.

Joshi, V. & Jaiman, R. K. 2019 A hybrid variational allen-cahn/ale scheme for the coupled analysis of

two-phase ﬂuid-structure interaction. International Journal for Numerical Methods in Engineering 117 (4),

405–429, arXiv: https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.5961.

Joshi, V. & Mysa, R. C. 2021aHydrodynamic analysis of tandem ﬂapping hydrofoils. In ASME 40th Inter-

national Conference on Ocean, Oﬀshore and Arctic Engineering (OMAE). Virtual, Online.

Joshi, Vaibhav & Mysa, Ravi Chaithanya 2021bMechanism of wake-induced ﬂow dynamics in tandem

ﬂapping foils: Eﬀect of the chord and gap ratios on propulsion. Physics of Fluids 33, 087104.

Jurado, R., Arranz, G., Flores, O. & Garc

´

ıa-Villalba, M. 2022 Numerical simulation of ﬂow over

ﬂapping wings in tandem: Wingspan eﬀects. Physics of Fluids 34, 017114.

Karbasian, H. R., Esfahani, J. A. & Barati, E. 2015 Simulation of power extraction from tidal currents

by ﬂapping foil hydrokinetic turbines in tandem formation. Renewable Energy 81, 816–824.

Kinsey, T. & Dumas, G. 2006 Parametric study of an oscillating airfoil in power extraction regime. 24th

AIAA Applied Aerodynamics Conference .

Kinsey, T. & Dumas, G. 2012 Optimal tandem conﬁguration for oscillating-foils hydrokinetic turbine. Journal

of Fluids Engineering, Transactions of the ASME 134 (3), cited By 71.

Lagopoulos, N. S., Weymouth, G. D. & Ganapathisubramani, B. 2021 Eﬀect of aspect ratio on the

propulsive performance of tandem ﬂapping foils. arXiv .

Lewin, G. C. & Haj-Hariri, H. 2003 Modelling thrust generation of a two-dimensional heaving foil in a

viscous ﬂow. Journal of Fluid Mechanics 492, 339–362.

39

Licht, Stephen, Hover, Franz & Triantafyllou, Michael S. 2004aDesign of a ﬂapping foil underwater

vehicle. 2004 International Symposium on Underwater Technology, UT’04 - Proceedings pp. 311–316.

Licht, Stephen, Polidoro, Victor, Flores, Melissa, Hover, Franz S. & Triantafyllou,

Michael S. 2004bDesign and projected performance of a ﬂapping foil auv. IEEE Journal of Oceanic Engi-

neering 29, 786–794.

Lua, K. B., Lu, H., Zhang, X. H., Lim, T. T. & Yeo, K. S. 2016 Aerodynamics of two-dimensional

ﬂapping wings in tandem conﬁguration. Physics of Fluids 28 (12), 121901.

Ma, Penglei, Wang, Yong, Xie, Yudong & Liu, Guijie 2021 Behaviors of two semi-passive oscillating

hydrofoils with a tandem conﬁguration. Energy 214, 118908.

Mandujano, F. & M´

alaga, C. 2018 On the forced ﬂow around a rigid ﬂapping foil. Physics of Fluids 30,

061901.

Mannam, Naga Praveen Babu, Alam, Md Mahbub & Krishnankutty, P. 2020 Review of biomimetic

ﬂexible ﬂapping foil propulsion systems on diﬀerent planetary bodies. Results in Engineering 8, 100183.

Mannam, Naga Praveen Babu & Krishnankutty, Parameswaran 2018 Hydrodynamic study of ﬂapping

foil propulsion system ﬁtted to surface and underwater vehicles. Ships and Oﬀshore Structures 13 (6), 575–

583.

Marras, Stefano & Porfiri, Maurizio 2012 Fish and robots swimming together: attraction towards the

robot demands biomimetic locomotion. Journal of The Royal Society Interface 9, 1856–1868.

Mengjie, Zhang, Biao, Huang, Zhongdong, Qian, Taotao, Liu, Qin, Wu, Hanzhe, Zhang & Guoyu,

Wang 2020 Cavitating ﬂow structures and corresponding hydrodynamics of a transient pitching hydrofoil in

diﬀerent cavitation regimes. International Journal of Multiphase Flow 132, 103408.

Muscutt, L. E., Weymouth, G. D. & Ganapathisubramani, B. 2017 Performance augmentation mech-

anism of in-line tandem ﬂapping foils. Journal of Fluid Mechanics 827, 484–505.

Pan, Yu & Dong, Haibo 2020 Computational analysis of hydrodynamic interactions in a high-density ﬁsh

school. Physics of Fluids 32, 121901.

Peng, Ze-Rui, Huang, Haibo & Lu, Xi-Yun 2018 Collective locomotion of two self-propelled ﬂapping plates

with diﬀerent propulsive capacities. Physics of Fluids 30, 111901.

Ramamurti, Ravi, Geder, Jason, Palmisano, John, Ratna, Banahalli & Sandberg, William C.

2010 Computations of ﬂapping ﬂow propulsion for unmanned underwater vehicle design. AIAA Journal 48 (1),

188–201, arXiv: https://doi.org/10.2514/1.43389.

Ribeiro, Bernardo Luiz R., Su, Yunxing, Guillaumin, Quentin, Breuer, Kenneth S. & Franck,

Jennifer A. 2021 Wake-foil interactions and energy harvesting eﬃciency in tandem oscillating foils. Phys.

Rev. Fluids 6, 074703.

Roper, D. T., Sharma, S., Sutton, R. & Culverhouse, P. 2011 A review of developments towards

biologically inspired propulsion systems for autonomous underwater vehicles:. Proceedings of the Institution

of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment 225, 77–96.

Ryu, Jaeha, Yang, Jongmin, Park, Sung Goon & Sung, Hyung Jin 2020 Phase-mediated locomotion

of two self-propelled ﬂexible plates in a tandem arrangement. Physics of Fluids 32, 041901.

Sampath, Kaushik, Geder, Jason D., Ramamurti, Ravi, Pruessner, Marius D. & Koehler, Ray-

mond 2020 Hydrodynamics of tandem ﬂapping pectoral ﬁns with varying stroke phase oﬀsets. Phys. Rev.

Fluids 5, 094101.

Taylor, G., Nudds, R. & Thomas, A. 2003 Flying and swimming animals cruise at a Strouhal number

tuned for high power eﬃciency. Nature 425, 707–711.

Thakor, Mitesh, Kumar, Gaurav, Das, Debopam & De, Ashoke 2020 Investigation of asymmetrically

pitching airfoil at high reduced frequency. Physics of Fluids 32, 053607.

Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. 1993 Optimal thrust development

in oscillating foils with application to ﬁsh propulsion. Journal of Fluids and Structures 7(2), 205 – 224.

40

Triantafyllou, M.S., Techet, A.H. & Hover, F.S. 2004 Review of experimental work in biomimetic

foils. IEEE Journal of Oceanic Engineering 29 (3), 585–594.

Van Buren, Tyler, Floryan, Daniel & Smits, Alexander J. 2019 Scaling and performance of simultane-

ously heaving and pitching foils. AIAA Journal 57 (9), 3666–3677, arXiv: https://doi.org/10.2514/1.J056635.

Wagenhoffer, Nathan, Moored, Keith W. & Jaworski, Justin W. 2021 Unsteady propulsion and the

acoustic signature of undulatory swimmers in and out of ground eﬀect. Physical Review Fluids 6, 033101.

Wu, Xia, Zhang, Xiantao, Tian, Xinliang, Li, Xin & Lu, Wenyue 2020 A review on ﬂuid dynamics of

ﬂapping foils. Ocean Engineering 195, 106712.

Xu, Wenhua, Xu, Guodong, Duan, Wenyang, Song, Zhijie & Lei, Jie 2019 Experimental and numerical

study of a hydrokinetic turbine based on tandem ﬂapping hydrofoils. Energy 174, 375–385.

Yu, Dengtao, Sun, Xiaojing, Bian, Xiutao, Huang, Diangui & Zheng, Zhongquan 2017 Numerical

study of the eﬀect of motion parameters on propulsive eﬃciency for an oscillating airfoil. Journal of Fluids

and Structures 68, 245–263.

Yu, Junzhi & Wang, Long 2005 Parameter optimization of simpliﬁed propulsive model for biomimetic robot

ﬁsh. Proceedings - IEEE International Conference on Robotics and Automation 2005, 3306–3311.

Zhang, Yue, Yang, Fuchun, Wang, Dianrui & Jiang, Xiaofeng 2021 Numerical investigation of a

new three-degree-of-freedom motion trajectory on propulsion performance of ﬂapping foils for uuvs. Ocean

Engineering 224, 108763.

41