Effect of combined heaving and pitching on propulsion of single and tandem flapping foils

Abstract

In this study, we present 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 at Reynolds number 1100 and reduced frequency 0.2. We initially present a systematic analysis on the thrust generation due to the kinematic parameters for a single foil. The significance of effective angle of attack and the projected area of the foil has been emphasized in explaining 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 between the foils. Through a control volume analysis, 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.

Full text

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