Noise generation and propagation
by biomimetic dynamic-foil thruster
Kostas Belibassakis1 and Iro Malefaki2
School of Naval Architecture & Marine Engineering,
National Technical University of Athens, Greece
Email: firstname.lastname@example.org(1), email@example.com(2)
Biomimetic flapping-foil thrusters are able to operate efficiently while offering desirable levels of thrust
required for the propulsion of a small vessel or an Autonomous Underwater Vehicle (AUV). Extended
review of hydrodynamic scaling laws in aquatic locomotion and fishlike swimming can be found in
Triantafyllou et al (2005). Flapping-foil configurations have been investigated both as main propulsion
devices and for augmenting ship propulsion in waves; see also the review by Wu et al (2020). In this work
biomimetic systems are studied with application to small vessel or AUV propulsion and their comparative
performance with standard marine propellers concerning the reduction of noise. A three-dimensional model
of the lifting flow around the dynamic foil is presented and its application is discussed as regards the
prediction of the hydrodynamically generated noise, in conjunction with methods allowing for the
calculation of acoustic propagation and spatial evolution of the spectrum, based on data concerning the noise
sources on the dynamic foil, coupled with the solution of the hydroacoustic problem.
Keywords: Biomimetic flapping-foil thrusters, small vessel / AUV propulsion, hydrodynamic noise
The seas become substantially noisier the last decades and anthropogenic sources contribute substantially in
this degradation trend with detrimental effects on sea life and particularly on marine mammals; see, e.g.,
Duarte et al (2021). Shipping, resource exploration, and infrastructure development have increased the
anthrophony (sounds generated by human activities), whereas the biophony (sounds of biological origin) has
been reduced by hunting, fishing, and habitat degradation. In particular, shipping noise has a significant
impact on the marine environment as demonstrated by the fact that at low frequencies below 300 Hz,
ambient noise levels have been increased by 15-20dB over the last century (McKenna et al 2012).
Many recent studies have shown that underwater-radiated noise from commercial ships may have both short
and long-term negative consequences on marine life, especially marine mammals. The issue of underwater
noise and impact on marine mammals was first raised at IMO in 2004. It was noted that continuous
anthropogenic noise in the ocean was primarily generated by shipping. Since ships routinely cross
international boundaries, management of such noise required a coordinated international response.
Moreover, in 2008, the IMO Marine Environment Protection Committee (MEPC) agreed to develop non-
mandatory technical guidelines to minimize the introduction of incidental noise from commercial shipping
operations into the marine environment to reduce potential adverse impacts on marine life.
As far as the radiated noise is concerned, it has been found that different components are dominant at
different speeds. In particular, hydrodynamic noise due to propeller operation in the wake of the ship and
machinery is dominant at low speeds, whereas propeller noise is dominant at higher speed especially when
cavitation takes place; see also Belibassakis (2018). Marine propellers are the standard devices used for ship
propulsion and are characterized by increased load distribution on the disc while operating at high rotational
speed conditions. The increased flow speed is the main reason leading to the appearance in almost all cases
of partial cavitation manifested near the tip region and occasionally also at the hub of marine propeller blades
and the trailing vortex sheets. The fast variation of the generated bubble cavitation volume on the propeller
blades, acting as acoustic monopole terms, i
leads to the generation of intensive noise, especially at the blade frequency and the first harmonics,
while at higher frequencies noise is caused by sheet cavity collapse and s
In the lower frequency band,
On the other hand, flapping-foil thrusters are systems operating
at substantial lower frequency as compared with marine propellers and are characterized by much smaller
power concentration. The latter biomimetic devices are able to operate very efficiently while offering
desirable levels of thrust required for the propulsion a small vessel or an Autonomous Underwater Vehicle
(AUV); see, e.g., Triantafyllou et al (2000), Rozhdestvensky & Ryzhov (2003). Extended review of
hydrodynamic scaling laws in aquatic locomotion and fishlike swimming can be found in Triantafyllou et al
(2005). Moreover, flapping-foil configurations have been investigated both as main propulsion devices and
for augmenting ship propulsion in waves, substantially improving the performance by exploitation of
renewable wave energy. More details can be found in Belibassakis & Politis (2013), Belibassakis & Filippas
(2015); see also the review Wu et al (2020). In the framework of Seatech H2020 project entitled “Next
generation short-sea ship dual-fuel engine and propulsion retrofit technologies” (https://seatech2020.eu/) a
concept of symbiotic ship engine and propulsion innovations is studied, that when combined, are expected to
lead to significant increase in fuel efficiency and emission reductions. The proposed renewable-energy-
based propulsion innovation is based on the bio-mimetic dynamic wing, mounted at the ship bow to augment
ship propulsion in moderate and higher sea states, capturing wave energy and producing extra thrust while
damping ship motions.
In this work biomimetic flapping thrusters are considered with application to the propulsion of small vessel
and AUV and approximate models are presented in order to evaluate their comparative performance with
standard marine propellers concerning the reduction of noise. More specifically, a three-dimensional model
of the lifting flow around the dynamic foil operating as an unsteady flapping thruster is described. The
method is based on vortex-ring elements and its application is subsequently presented concerning the
prediction of the dynamical behavior of the system and the hydrodynamically generated noise. Finally results
are presented showing the effectiveness of the model to be used, in conjunction with methods allowing for
the calculation of acoustic propagation and spatial evolution of the acoustic spectrum, based on data
concerning the noise sources on the dynamic foil, coupled with the solvers of the hydroacoustic problem.
2 The vortex ring element method for flapping thruster performance
A vortex ring element method based on quadrilateral elements will be used to discretize the wing and its
trailing vortex wake and the singularity strengths are calculated to satisfy directly the no-entrance boundary
condition on the surface of the foil, along with the Kutta condition. A general foil geometry is modelled
including camber, thickness and various planform shapes and aspect ratio AR=s2/A, where s is the span of
the wing, c the midchord length and A the planform area. A main difference with the lifting surface Vortex-
Lattice model (Katz & Plotkin 1990), is that the exact boundary condition is satisfied on the actual wing
surface, in contrast with lifting surface models where the boundary condition is satisfied on the mean camber
surface and the thickness effects are taken into account by linearization procedure and a corresponding
Figure 1: Discretization of a flapping wing and its trailing vortex sheet by means of quadrilateral
elements carrying constant dipole strength (left), which is equivalent with vortex ring elements
(right). Only the half symmetric part with respect to the centerplane of wing of AR=8 is shown.
The method is based on the discretization of the wing sections into number of chordwise elements for a
number of spanwise sections as shown in the Fig.1 above. A scanning procedure is applied in order to define
the 4 nodal points of the vortex ring elements, which then are used to calculate the influence coefficients on
the collocation points (defined as the centroids of the ring elements on the body surface).
The wing undergoes an oscillatory heaving and pitching motion, with same frequency and phase difference
around 90deg, while traveling at constant speed, in order to operate in a flapping mode; see Triantafyllou et
al (2000, 2005). The most important parameters are the Strouhal number
0/Str h U
, the heaving
motion amplitude h0/c and the pitching amplitude 0
, where ω is the angular frequency, and Udenotes the
incident parallel inflow due to the steady forward speed of the flapping thruster.
In treating time – dependent motion of bodies, the selection of the coordinate systems becomes important. It
is useful to describe the unsteady motion of the wing on which the flow – tangency condition is applied in a
body – fixed coordinate system (x,y,z); see Katz & Plotkin (1990). The motion of the origin is prescribed in
an inertial frame of reference (X,Y,Z). In the present work, the flapping wing starts from rest, and we also
consider the wing to perform a pitching angle
, a vertical oscillatory heaving motion
ht, and thus
Figure 2. Vortex wake development of flapping thruster operating at a Strouhal number Str=0.23, with
heaving amplitude with h0/c=0.75 and pitching amplitude 0
=23deg, during 4 periods of oscillation
(left), Morino – type Kutta condition (right).
Figure 3. Pressure distribution of every wing section in the case of flapping thruster at a Strouhal
number Str=0.23, with heaving amplitude with h0/c=0.75 and pitching amplitude 0
Zht x z
The solution is based on a time – stepping technique, and at the beginning of the motion only the bound
vortex ring elements on the unsteady thruster exist. The closing segment of the trailing – edge vortex
section at CL
elements represent sthe starting vortex. At the first time step, there will be no wake panels. During the
second time step, the wing is moved along its flight path and each trailing – edge vortex panel sheds a wake
panel with a vortex strength equal to its circulation in the previous time step. This time step methodology can
be continued for any type of foil path and at each time step the vortex wake corner points can be moved by
the local velocity, so that the wake rollup can be simulated.
The problem is solved by calculating the influence coefficients of the induced potential and velocity
, , , , 1,... .
ij ij ij ij
UV forij KU by each vortex ring element on each collocation point on the wing,
which is selected as the centroid of each quadrilateral element. The latter quantities are used to set up a linear
system of equations by constructing the coefficient matrix. To this respect, the flow – tangency condition is
implemented on the wing surface, requiring zero normal velocity. Consequently, in the present case, the
discrete system of equations expressing the flow tangency condition at the collocation points on the wing is:
, for 1, 2..,
lk k l l lk k l
bn , (2)
are the bound vortex ring element strengths and the matrix coefficient is composed by
ij i ij
for i KnU and ,1,..,
n the unit normal vector directed to the exterior of the
body. In Eq.(2) k is an one – dimensional counter for each collocation point, and l for each vortex
ring element. The index
tMNt corresponds to the number of wake panels generated
by the unsteady wing motion up to the time instant t, where
Nt t t
, where t
is the time
step. The system is supplemented by a Morino-type Kutta condition used to determine the vortex
ring intensity in the wake element adjacent to the trailing edge is in this case connected with the
ring intensities of the first element in the lower wing side and the last element in the upper wing
side (see Fig. 2) as follows
W upper lower
The first term in the right – hand side of Eq.(2) is defined by:
kK bun (4)
u denoting the relative flow velocity at the collocation points of the wing
uuu U dt dt
, are the unit vectors along the axes x,y,z respectively and w
r denotes the position vector on
the wing. In the right hand side of Eq.(2) the influence of the vortex-ring elements modelling the shed wake
vorticity on the wing wake is included. The summation is over the ww
=M×N vortex elements of the
wake (M in the spanwise direction and w
N n the downstream direction) which are generated from the
motion of the wing, after discetization to equal time steps
t. The total potential on the surface S of the
unsteady wing is approximately:
ss ss Ux
and it can be used for the calculation of the velocity on the wing by covariant differentiation of the potential
in curvilinear coordinates on the wing (s1-cordwise and s2 -spanwise):
s , (7)
from which the the total velocity is estimated:
ee denotes the physical components of the surface contravariant base on the wing. After obtaining
the velocity, the pressure distribution is calculated by applying the unsteady Bernoulli’s equation providing
the instantaneous distribution of the pressure coefficient
w . (9)
Finally, time dependent hydrodynamic responses concerning flapping thruster forces and moments are
calculated by pressure integration over the wing surface. Indicative results are presented in Figs 2 and 3 as
obtained by the present method.
3 Comparison with unsteady hydrofoil theory
In this section results from the present 3D unsteady panel method are compared against unsteady hydrofoil
theory by Theodorsen (1935) and experimental data from Schouveiler et al (2005).
The case of flapping wing of large aspect ratio of Fig.2 is studied for verification. Using Theodorsen theory
(see Katz & Plotkin 1990) in the case of wings of finite aspect ratio the lift force L
F can be estimated as:
FUAHARCkUh c Uhc
is pitching axis location,
is the reduced frequency
Ck is the Theodorsen
function (lift deficiency factor),
is the area of the foil, and
is a 3D correction from
lifting – line theory (elliptic wing). In Fig.4 at the top subplot the time-history of foil angle of attack
shown for a time interval of 4 periods,
at t a t a t U
where the foil pitching and heaving oscillatory motions are defined as follows
sin 0.5 and sintt htht
Figure 4. Comparison of present method prediction with Theodorsen’s unsteady hydrofoil theory and
experimental data by by Schouveiler et al (2005) in the case of foil of AR=6, at Str=0.23, h0/c=0.75,
=23deg. Top subplot: angle of attack α(t). Lower subplot: Lift nd thrust coefficients as calculated by
the present method and compared with theory (red line) and measured data (symbols).
In the last subplot the vertical (lift) force coefficient is shown by using black line and the horizontal (thrust)
force by using cyan line, as calculated by the present method, and are compared against the measured data
shown by using symbols and the unsteady hydrofoil theory results by using red lines. It is observed that the
present method provides compatible predictions concerning the integrated quantities with unsteady hydrofoil
theory and the experiment approximating satisfactorily the maxima of both the lift and thrust forces.
4. Flapping thruster noise prediction
As the flapping thruster operates, it is subjected mainly to unsteady pressure loads. Low frequency noise is
caused by the fluctuations of foil pressure and volume flow disturbance due to oscillatory motion. The
usual formulation for the acoustic pressure p
generated from rotating machinery is based on the Ffowcs
Williams and Hawkings (1969) equation as follows
where c is the speed of sound in the medium (c=1500-1550m/s for water) and the various terms in the
right-hand side correspond to the acoustic monopole, dipole and quadrupole source terms (Farassat &
Myers 1988). The quadrupole term becomes important for strongly transonic flow phenomena at higher
frequencies. Taking into account that the speed of sound in water is much greater than the flow velocities,
and focusing on the low-frequency part of the generated noise spectrum the contributions by the latter term
are neglected in the present work. Farassat (2001) formulation is employed offering an integral
representation of the solution of Eq.(13) forced by the monopole and dipole terms. The acoustic pressure
field is accordingly given by thickness and loading components, as follows
'( ,) ( ,) ( ,)
tp tp t
The loading term is given by
4(,) (1 )
cdt r M
dp r dS
n , (15)
where f=0 indicates the moving surfaces n
u the corresponding normal velocity, where dp
pressure jump on the blade surface, r
denotes Mach number in the r-direction and the integrand is
calculated at retarded time. For relatively large distances (of the order of several propeller diameters) of
the observation point from the propeller, we use the approximation
rtrtr xxxx xx
Ttx denote the center of lift and thrust force on the flapping wing. Using the fact that the Mach
number is very small, Eq.(18) leads to the following simplification
xt x xt
cdt r r
Ft denotes the fluctuating unsteady part of the foil force, mainly composed from vertical (lift)
and horizontal (thrust) forces, and /
trc denotes the retarded time between the observation point 0
and the disturbance generating point T
Similarly for the thickness effect we have
which is approximated by
pt dt t
xxx , (18)
,Qk tx denotes the center of volume c
Q displaced by the foil. In the case of an unsteady cavitating
foil thruster the latter term will include also the bubble cavitation volume.
Indicative results obtained by the above simplified model are presented in Fig.5 in the vicinity of the
flapping thruster and at large distances. The acoustic field generated by the flapping thruster of middle
chord c=1m and AR=6 operating in the same as before conditions in water (c=1500m/s) is presented in Fig. 5
as calculated by the present method. Results are presented at a time instant after 3.7 periods of oscillation,
starting from rest. The contribution of the monopole and the dipole term, which is dominant in the examined
case, are clearly observed. In the examined case since the foil flow is not cavitating and the intensity of the
acoustic field is very small. Future work will be directed to the incorporation of unsteady cavitation effects
which are expected to be important in the case of flapping thrusters operating near the free surface.
Figure 5. Calculation of acoustic field generated by the flapping thruster operating in water
(c=1500m/s), in the case of foil of Fig.4 of AR=6, flapping at Str=0.23, h0/c=0.75, 0
the calculated hydrodynamic loads by the present method. Top subplot: near field from dipole and
monopole term and total acoustic field in the vicinity of the flapping thruster. Lower subplot:
calculated field at large distances.
In the present work a 3D vortex – ring element method has been presented for calculating the flow over
wings in unsteady conditions with application to the performance of flapping thrusters operating at low
Strouhal numbers. The method is shown to provide compatible predictions with unsteady hydrofoil theory
and experimental data. Next, a simplified model is presented for the prediction of the hydrodynamically
generated noise, based on data concerning the noise sources on the dynamic foil, coupled with the solution of
the hydroacoustic problem. The present model will be used, in conjunction with methods allowing for the
calculation of acoustic propagation for calculating spatial evolution of the noise spectrum, for comparative
studies with the noise from standard marine propellers. An important fact is that the utilization of the
flapping thruster to augment ship propulsion could enhance the combined ship/AUV propulsive performance
dropping at the same time the power feed of marine propeller and reducing the overall generated noise level.
Future work will be directed to the incorporation of cavitation effects which are expected to be important in
the case of flapping thrusters operating at low submergence depths, including biomimetic flapping thrusters
that are currently studied for augmenting ship propulsion in waves. Also, the reflection and scattering effects
by the vessel or AUV hull surface and the refraction effects due to variable sound speed profile on longer-
distance acoustic propagation characteristics will be considered.
The present work has been supported by Seatech H2020 project received funding from the European
Union’s Horizon 2020 research and innovation program under the grant agreement No 857840. The opinions
expressed in this document reflect only the author’s view and in no way reflect the European Commission’s
opinions. The European Commission is not responsible for any use that may be made of the information it
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