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1

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: kbel@fluid.mech.ntua.gr(1), iromalefaki@gmail.com(2)

Abstract

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

1 Introduction

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.

2

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

in

n

c

co

on

nj

ju

un

nc

ct

ti

io

on

n

w

wi

it

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h

d

di

ip

po

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e

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ri

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te

ea

ad

dy

y

b

bl

la

ad

de

e

l

lo

oa

ad

d,

,

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

sh

ho

oc

ck

k

w

wa

av

ve

e

g

ge

en

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at

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;

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a

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(

(2

20

00

05

5)

).

.

In the lower frequency band,

t

th

he

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n

no

oi

is

se

e

e

ex

xc

ci

it

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at

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,

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m

ma

am

mm

ma

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ls

s.

.

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.

3

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

source-sink lattice.

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

t

, a vertical oscillatory heaving motion

ht, and thus

4

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

=23deg.

cos sin

sin cos

XUtx z

Yy

Zht x z

(1)

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

Middle wing

section at CL

Tip wing

section

5

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

F

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:

11

, for 1, 2..,

w

Kt

K

ww

lk k l l lk k l

kk

A

UlK

bn , (2)

where k

are the bound vortex ring element strengths and the matrix coefficient is composed by

,1,...,

ij i ij

A

for i KnU and ,1,..,

iiK

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

ww

K

tMNt corresponds to the number of wake panels generated

by the unsteady wing motion up to the time instant t, where

/

w

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

TE TE

W upper lower

. (3)

The first term in the right – hand side of Eq.(2) is defined by:

,1,..,

kkk

kK bun (4)

with k

u denoting the relative flow velocity at the collocation points of the wing

,,

xyz w

ddh

uuu U dt dt

uijrk

, (5)

where ,,ijk

, 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

K

=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:

12 12

,,

ss ss Ux

, (6)

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):

6

12

12

,

uu

s

s , (7)

from which the the total velocity is estimated:

12

uu

12

**

we e

, (8)

where 2

,

1

**

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

2

222

2

1

12

p

pp

CUUUt

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:

2

13 1

2442

L

pc p

FUAHARCkUh c Uhc

cc

, (10)

where /pc

is pitching axis location,

/2kcU

is the reduced frequency

Ck is the Theodorsen

function (lift deficiency factor),

A

is the area of the foil, and

1

2HARAR

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

t

shown for a time interval of 4 periods,

1

,where tan

bb

h

at t a t a t U

(11)

where the foil pitching and heaving oscillatory motions are defined as follows

00

sin 0.5 and sintt htht

. (12)

7

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,

0

=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

2

2

22

1p

p

mdq

ct

, (13)

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

000

'( ,) ( ,) ( ,)

TL

p

tp tp t

xxx

. (14)

8

The loading term is given by

00

ˆ

1

4(,) (1 )

Lfrret

ddpr

p

tdS

cdt r M

n

x

2

0

ˆ

(1 )

frret

dp r dS

rM

n , (15)

where f=0 indicates the moving surfaces n

u the corresponding normal velocity, where dp

denotes the

pressure jump on the blade surface, r

M

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

00 0

ˆ

,/

TT

rtrtr xxxx xx

where

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

023

()

11

,(),

44

PTr PTr

r

Lr

x

xt x xt

dF t

pt Ft

cdt r r

x (16)

where ()

r

Ft denotes the fluctuating unsteady part of the foil force, mainly composed from vertical (lift)

and horizontal (thrust) forces, and /

r

trc denotes the retarded time between the observation point 0

x

and the disturbance generating point T

x.

Similarly for the thickness effect we have

00

(,) ,

4(1)

n

Tfrret

u

p

tdS

trM

x (17)

which is approximated by

2

02

0

() 1

(,) 4()

cr

T

Qr

dQ t

pt dt t

xxx , (18)

where

,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.

9

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

=23deg, using

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.

5. Conclusions

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.

10

Acknowledgements

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

contains.

6. References

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commercial ships, J Acoust Soc Am. , Vol.131, 2012, pp. 92-103.

[3] IMO 2014. Guidelines for the Reduction of Underwater Noise from Commercial Shipping to Address

Adverse Impacts on Marine Life – non mandatory technical advices, International Maritime

Organization (2014) MEPC.1/Circ.833.

[4] IMO 2008. 72nd session of the Marine Environment Protection Committee (MEPC 70) at the

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[5] Belibassakis K., 2018, A velocity-based BEM for modeling generation and propagation of underwater

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[15] Theodorsen, T. 1935, General Theory of Aerodynamic Instability and the Mechanisms of Flutter,

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