A Vortex Lattice Method for the Hydrodynamic Solution of Lifting

Bodies Traveling Close and Across a Free Surface

RAFFAELE SOLARI1, PATRIZIA BAGNERINI2, GIULIANO VERNENGO1

1Department of Electric, Electronic, Telecommunication Engineering and Naval Architecture (DITEN)

Polytechnic School of the University of Genoa,

Via Montallegro 1, 16145, GE, ITALY

2Department of Mechanical, Energy, Management and Transportation Engineering (DIME)

Polytechnic School of the University of Genoa,

Via all'Opera Pia 15, 16145, GE, ITALY

Abstract: The hydrodynamics performance of submerged and surface-piercing lifting bodies is analyzed by a

potential flow model based on a Vortex Lattice Method (VLM). Such a numerical scheme, widely applied in

aerodynamics, is particularly suitable to model the lifting effects thanks to the vortex distribution used to dis-

cretize the boundaries of the lifting bodies. The method has been developed with specific boundary conditions

to account for the development of steady free surface wave patterns. Both submerged bodies, such as flat plates

and hydrofoils, as well as planing hulls can be studied. The method is validated by comparison against available

experimental data and other Computational Fluid Dynamic (CFD) results from Reynolds Averaged Navier Stokes

(RANS) approaches. In all the analyzed cases, namely 2D and 3D flat plates, a NACA hydrofoil, planning flat

plates and prismatic planing hulls, results have been found to be consistent with those taken as reference. The

obtained hydrodynamic predictionsare discussed highlighting the advantages and the possible improvements of

the developed approach.

Key-Words: Vortex Lattice Method (VLM); Boundary Element Method (BEM); Free Surface; Hydrofoils;

Planing Surfaces.

Received: April 21, 2021. Revised: January 15, 2022. Accepted: January 27, 2022. Published: March 8, 2022.

1 Introduction

Design high performance floating and submerged

vessels has always been a great challenge in hydro-

dynamics. This is mainly related to the effects of

the dynamic pressures on the lifting body that induce

changes in their running attitudes and, particularly,

to the interactions that arise with the water free sur-

face. Such interactions on a side alter the pressure

distribution on the body surface and, on the other side,

produce waves that propagates in the far field down-

stream.

This hydrodynamic problem has been deeply stud-

ied both numerically and experimentally. Consider-

ing the latter approach the two components of the

wave resistance, related to the breaking and the non-

breaking waves, have been analyzed for a submerged

hydrofoil [1, 2]. The pressure distribution at the wet

surface of a planing plate has been measured for sev-

eral configurations and conditions at the NASA re-

search center [3]. Considering the numerical solution

of the problem, a variety of methods have been pro-

posed. Boundary Element Methods (BEM) relying

on potential flow theory have been developed to solve

related problems such as the performance of cavitat-

ing and supercavitating hydrofoils [4, 5, 6] or sub-

cavitating hydrofoils interacting with a free surface

[7] and are widely applied in design by optimization

processes thanks to their inherent computational ef-

ficiency [8, 9]. Among potential flow based meth-

ods, there are also vortex based approaches relying

on the so called Vortex Lattice Method (VLM), ini-

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tially developed to solve aerodynamics related prob-

lems [10]. Despite it is particularly suitable to pre-

dict lifting effects of a body, relatively few authors

have developed VLM based approaches to predict the

performance of lifting bodies such as planing hulls

[11, 12, 13, 14]. Other hydrodynamic solutions of

planing hulls have been presented based on high-

fidelity Reynolds Averaged Navier Stokes (RANS)

methods [15, 16, 17, 18, 19, 20, 21].

A VLM for the solution of lifting bodies, both sub-

merged and surface piercing, is proposed. The math-

ematical formulation of the problem, i.e. the devel-

opment of the suitable set of boundary conditions, is

presented and the numerical scheme is described. An

extensive validation of the developed VLM is then

presented proving the accuracy of the solution. In

particular the VLM is applied to the hydrodynamic

analysis of submerged and surface piercing flat plates,

both 2D and 3D, to the solution of submerged, finite,

NACA hydrofoils and to the performance prediction

of a prismatic planing hull. The obtained results are

compared with available experimental measurements

and against results obtained from other computational

methods.

2 Free Surface Hydrodynamic

Problem Formulation

The fluid domain changes based on the position of

the lifting body under analysis, according to the

schematic representation shown in Fig. 1. Consid-

ering the fully submerged lifting body SB, e.g. a hy-

drofoil, the fluid wake detached by its trailing edge

SWneeds to be included to fulfill the so-called Kutta

condition and the free surface SFSis a continuous in-

terface where stationary waves (both transverse and

divergent) could develop. On the other hand, as the

lifting body crosses the free surface, as e.g. in the

case of a planing hull, a more complex wave pattern

is generated since the fluid interface is forced to be

attached to the bottom of the lifting body. Both cases

have been developed for deep water condition, mean-

ing that the sea bottom surface S∞is far enough not

be included into the model.

The solution is formulated in the framework of a so-

called potential flow theory, i.e. assuming the flow

to be ideal, irrotational and incompressible. Introduc-

ing the Cartesian reference system shown in Figure 2,

consistent with the body, a velocity potential, scalar,

function Φcan be defined so that ∇Φ = −→

v, being

−→

v= [U+vx, vy, vz]the global flow velocity vec-

tor. In particular, such a potential function can be

seen as the superimposition of a free stream poten-

tial ϕ0=x·Uwith a potential ϕconsequence of the

disturbance to the flow generated by the presence of

the body, resulting in Φ(x, y, z) = x·U+ϕ(x, y, z).

According to the mentioned hypothesis, the Laplace

Eq. (1) hold in the fluid domain:

∇2Φ = 0, in D(1)

A Neumann-type boundary condition on the body wet

surface is imposed to ensure the non-penetration of

the fluid particles. Being −→

n= (nx, ny, nz)the nor-

mal vector to the body surface, the normal velocity is

null at the body boundaries according to Eq. (2):

∇Φ·−→

n=∂ϕ

∂n +Unx= 0, at SB(2)

The perturbation needs to decay at a large distance

−→

r= (xr, yr, zr)to the body. This radiation condition

is expressed by Eq. 3:

lim

−→

r−→∞ ∇Φ = 0 (3)

Considering the free surface as a function z=

η(x, y), two boundary conditions hold, namely a kine-

matic and a dynamic condition. The first, according

to Eq. 4, guarantees that the fluid particles belong-

ing to the free surface will remain on that interface.

The second is a direct consequence of the Bernoulli's

theorem, written in Eq. 5, keeping constant the atmo-

spheric pressure patm at the interface.

∂Φ

∂z − ∇Φ· ∇η= 0, on z =η(x, y)(4)

2(∇Φ)2−1

2U2+gη = 0, on z =η(x, y)(5)

An additional boundary condition is required for a

submerged lifting body, ensuring the uniqueness of

the solution. From a physical perspective the Kutta

condition imposes that the TE of the hydrofoil has to

be a stagnation point, then ∆pT E = 0. The free sur-

face boundary conditions can be combined consider-

ing the gradient of the dynamic boundary condition,

obtaining the following non-linear condition:

2∇Φ· ∇(∇Φ)2+g∂Φ

∂z = 0, on z =η(x, y)(6)

Despite the advantage of being independent from the

free surface elevation η(x, y), which is part of the so-

lution, Eq. (6) is still very complex to be fulfilled.

For this reason, it can be further linearized by consid-

ering that it can be imposed at the undisturbed free

surface level z= 0, instead of the actual free surface

z=eta(x, y)[22, 23], resulting in the following Eq.

(7):

U2∂ϕ2

∂x2+g∂ϕ

∂z = 0, on z = 0 (7)

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(a) HS

b= 1.42% (b) HS

b= 2.85%

Figure 1: Schematic representation of the fluid domain for both the submerged (left) and surface-piercing (right)

cases.

Figure 2: Reference system used for the formulation

of the hydrodynamic problem.

According to the presented formulation, the Bound-

ary Value Problem (BVP) for lifting bodies travel-

ing close or across a free surface, in deep, water can

be formulated in terms of two boundary conditions,

namely Eq. (2) for the body and Eq. (7) for the lin-

earized free surface.

3 Solution trough the Vortex Lattice

Method

The potential function in a generic point of the con-

sidered domain can be written according to the Green

equation, Eq. (8), based on the distribution of singular

elements over the boundaries, as:

Φ(x, y, z) = 1

4π∫S

µ−→

n· ∇(1

r)dS +U∞x(8)

being S=SB+η(x, y)and µthe strength of a

constant distribution of dipoles. The numerical so-

lution of the BVP is achieved by using a Vortex Lat-

tice Method (VLM). According to this approach, each

material surface (the body and the free surface) is dis-

cretized by using a lattice of vortex built by the so-

called vortex rings, as shown in Fig. 3. Each of these

vortex rings is made of four straight vortex filaments,

each carrying a constant circulation Γistrength, be-

ing circulation around the hydrofoil is defined as the

Figure 3: Generic representation of the VLM panel

mesh discretization for a wing and particular of one

of its vortex rings.

vorticity flux over a surface:

Γ = ∫S

−→

ζ·−→

n dS (9)

The vorticity is defined as the curl of the velocity −→

ζ=

∇ × −→

v. A collocation point Pcoll is defined for each

vortex ring as the center of the mean plane defined

by the ring. The boundary conditions are enforced

at those collocation points. The velocity induced by

a vortex ring, that is equivalent to that induced by a

constant distribution of dipoles, is found according to

the Biot-Savart law, defined in Eq. (10) as:

−→

vind =Γ

4πIC

d−→

l×−→

r3(10)

being −→

land −→

rthe vector length of a vortex fila-

ment and the vector distance of the collocation point

from the filament itself, respectively. Considering

Eq. (10), the circulation strength Γof each vortex ring

is the unknown to be found as solution of a linear sys-

tem. The rest, called influence coefficient, only de-

pends on the relative position of the collocation point

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with respect to the vortex ring and can be directly

found.

Once the BVP has been solved, the total, linearized,

pressure ptat a point P, made of a static and a dynamic

contribution, psand pd, respectively, can be found as:

pt=ps+pd=−ρg(zp+η(xp, yp)) −ρU ∂ϕ

∂x (11)

The dynamic pressure in Eq. (11) is computed ac-

counting for the tangential velocity jump ∆−→

vtan at

the lifting surface due to the presence of the vortex

lattice. This contribution is computed considering the

mean distribution of circulation −→

γij over a vortex

ring of area Aij :

∆−→

vtan =−→

nij ×−→

γij =1

Aij

∑

k=1

σΓk·−→

lij (12)

being σ= 0.5everywhere but at the leading edge vor-

tex filament for which σ= 1. Then the total induced

velocity at each collocation point is computed as:

−→

v±

ind =−→

vind ±1

2∆−→

vtan (13)

Considering a discretization made of N×Mvortex

rings, the lift and the induced drag are computed from

the knowledge of the pressure as follows:

∑

i=1

∑

j=1

Lij =

∑

i=1

∑

j=1 −pT ij ·Aij ·nZij (14)

∑

i=1

∑

j=1

Dij =

∑

i=1

∑

j=1 −pT ij ·Aij ·nXij

(15)

The Kutta condition, used in the case of a submerged

hydrofoil, imposes that the strength of the vortexes at

the TE is null, that is the circulation at the TE is re-

leased in the wake behind it. Such a further boundary

condition is used to ensure the presence of a stagna-

tion point at the TE (valid for attached flows with-

out separation) and it is simply enforced by imposing

ΓT E = ΓW ake.

As regards the free surface boundary condition, to

prevent numerical damping of the free surface eleva-

tion, a four point derivation scheme is used for the

term ∂ϕ/∂x [22]. Moreover, the collocation points

of the free surface are placed at a minimum distance

dh/c= 0.1÷0.2with respect to the undisturbed free

surface level (z=0) to let the generation of the com-

plete wave pattern.

4 Validation and verification of the

VLM

The method has been validate against available exper-

imental data and other numerical results on several

cases, namely a submerged flat plate, a submerged

hydrofoil and a planing prismatic hull. In the follow-

ing sections results of this validation process are pre-

sented and discussed, highlighting the accuracy of the

obtained predictions.

4.1 Submerged Finite Flat Plate

The hydrodynamic performance of a deeply sub-

merged flat plate, with AR = 3 and t/c= 2.3%,

have been studied by using a CFD [24] at Re = 8·104.

The comparison of the results in terms of lift and drag

coefficients are displayed in Fig. 4 and Fig. 5, re-

spectively. The obtained predictions have also been

compared against theoretical results obtained by ap-

plying the 2D thin profile theory, corrected to include

the three dimensional effects. In particular, the theo-

retical lift coefficient for an infinitely AR wing, i.e. a

2D case, is defined as:

C2D

L=∂CL

∂α ·sin(α) = 2π·sin(α)(16)

being αand ∂CL/∂α the angle of attack and the slope

of the lift curve, respectively. Among the possible

formulations, two have been selected for the AR cor-

rection of the lift curve slope, reported in Eq. (17) and

Eq. (18), and one formula for the drag coefficient, re-

ported in Eq. (19), respectively.

∂C3D

∂α =f·

∂C2D

∂α

1 + (57.3·

∂C2D

∂α

πAR P

l)(17)

∂C3D

∂α =∂C2D

∂α ·AR

AR + 2 (18)

C3D

D=2C3D

πeAR (19)

In the previous equations, Pand lare the half perime-

ter and the span of the wing, respectively, f=

[0.98; 1] is a coefficient given as a function of the AR,

and e= 1.78(1 −0.045AR0.68)−0.64 is the Oswald

coefficient, which accounts for a correction for non-

elliptic wings.

The lifting coefficient predicted by the proposed

method is in very good agreement with the CFD re-

sults up to about α= 12deg, that is over all the lin-

ear trend of the curve. This is consistent with the as-

sumptions of the potential flow theory used to develop

the method. For higher angles of attack, predominant

viscous effects rise, highlighting a larger separation

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among the viscous and inviscid predictions. Consid-

ering the drag coefficient, a relatively good agreement

is found. Considering that only the induced drag is

predicted by the VLM, a correction through Blausius

formulation for the friction contribution Cf, valid for

laminar flow regimes (Re < 2.5·105), is applied. As

for the lift coefficient, the drag coefficient predicted

by the simplified formula is consistent with the other

results only for small angles of attack, in the order of

α < 6deg.

Figure 4: Comparison of the computed lift coeffi-

cients for the AR=3, deeply submerged, flat plate.

Figure 5: Comparison of the computed drag coeffi-

cients for the AR=3, deeply submerged, flat plate.

4.2 Submerged Infinite Flat Plate

A 2D flat plate (AR → ∞)with t/c= 1% has been

analyzed at Re = 104and results have been compared

against available wind tunnel experiments [25], then

neglecting the effect of the free surface. The compar-

isons of the lift coefficient is shown in Fig. 6. Again,

the results are consistent within the linear trend of the

lift curve. Being this case referred to a 2D flat plate,

far from the free surface, the induced drag related to

the three dimensional effect is null. The predicted

pressure coefficient at the center line of the flat plate is

compared against experimental measurements in Fig.

7. The pressure distribution is computed according to

the velocity jump induced by the vortex distribution,

as in Eq. (12). The relatively slight difference found

on the face of the plate is ascribed to the effect of

the thickness, that is not modeled in the present VLM

(e.g. by using a source distribution superimposed to

the vortex lattice).

Figure 6: Comparison of the computed lift coeffi-

cients for the AR → ∞, deeply submerged, flat plate.

Figure 7: Comparison of the computed pressure dis-

tribution at the center line for the AR → ∞, deeply

submerged, flat plate.

4.3 Submerged Finite NACA hydrofoil

The predicted free surface elevation at the center line

of a NACA profile, with a mean camber line a=0.8, is

compared against available numerical prediction [26]

in Fig. 8. The hydrofoil, AR = 1.5and t/c= 0%,

is analyzed at a draught over chord ratio T/c= 1,

α= 5deg and at a relatively high speed, correspond-

ing to F n = 0.58. The two methods, relying on

the same theoretical approach, provide very similar

results. The complete three dimensional, stationary,

wave pattern generated by the hydrofoil is shown in

Fig. 9. It is worth noticing that to avoid reflections of

the waves a relatively large free surface domain has

to be discretized. The dimensions of the free surface,

especially its length behind the profile location, very

much depends on the speed of the hydrofoil and on the

submergence ratio since they affect the wave length

and, consequently, the wave celerity. In general, it is

in the order of several chord lengths (Laft/c > 10).

The method has been used to analyze the effect of the

angle of attack variation on the free surface elevation

and on the overall performance of the hydrofoil. The

predicted free surface profiles at the center line are

shown in Fig. 10. As αincreases a higher pressure

disturbance is generated, resulting in a larger wave

amplitude of the first wave hollow. Such a maximum,

negative, wave amplitude is also anticipated due to the

relative shift forward of the center of pressure of the

hydrofoil. These characteristics are maintained over

the whole development of the wave profile. Consis-

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tently, the lift and drag coefficients, referred in Fig.

11 to the 2D case, increase as αincreases. While the

CL/C2D

Lratio increment is in the extent of about 5%

from α= 0deg to α= 10deg, the CD/C2D

Dratio

increases of about 14% over the same range of α.

Figure 8: Wave profile at the symmetry plane of the

free surface for the NACA a=0.8 mean camber line at

T/c=1. Blue line with markers: VLM computation.

Red circles: computation from [26]

Figure 9: Global view of the computed free surface

for the NACA a=0.8 mean camber line at T/c=1.

Figure 10: Wave profiles at the symmetry plane of the

free surface for the NACA a=0.8 mean camber line at

T/c=1 for several angle of attacks α.

4.4 Surface-Piercing Lifting Bodies

The method has been conceived to account for sur-

face piercing lifting bodies, as in the case of planing

hulls. This is a very interesting application in the field

of naval architecture since there are very few poten-

tial flow based methods developed to analyze it. The

pressure coefficients at the half-span line of two plan-

ing flat plates experimentally tested at NASA [3] are

Figure 11: Variation of the CLand CDwith respect to

the 2D limit case for the NACA a=0.8 mean camber

line for several angles of attack α.

Table 1: Planing surfaces particulars and lift and drag

coefficients comparison.

Case 1 Case 2

AR 2.87 1.07

F nB10.11 12.14

τ6.0 6.0

CV LM

L6.10E-02 8.80E-02

CSavitsky

L5.90E-02 8.50E-02

ϵ(CL)3.4% 3.4%

CV LM

D6.40E-03 9.30E-03

CSavitsky

D6.40E-03 9.00E-03

ϵ(CD)3.2% 3.3%

compared to the numerical prediction carried out by

using the proposed VLM and to those obtained by us-

ing another BEM [27], in Fig 12 and in Fig. 13, re-

spectively. In Table 1, the main particulars of the two

cases and the comparison of the corresponding lift and

drag coefficients computed by the VLM and by the

well known Savitsky's method [28] are reported. The

Froude number is related to the beam (the span) of

the surface, F nB=V/√gB. The trim angle τis the

equivalent of the angle of attack αfor an hydrofoil.

The relative percentage errors of a quantity ϵ(x)are

computed as:

ϵ(x) = 100 ·|xref −x|

xref

(20)

In both cases, the CPdistributions are in very well

agreement with the experimental measurements. In

particular they better fit the experimental distribution

especially close to the forward stagnation line, called

spray root line, where the peak of the pressure occurs.

This is a key feature for a reliable prediction of the

overall planing hull performance. When compared to

the Savitsky's predictions, used as term of reference,

the percentage errors on both the lift and the drag co-

efficients are slightly beyond the 3%, confirming that

the global hydrodynamic performance are well pre-

dicted.

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Figure 12: Pressure coefficient at the center line of

a planing flat plate, L/b= 2.87,F r = 10.11,τ=

6deg. Blue circles: experimental measurements. Red

solid line with circles: VLM computations. Black

dashed line with squares: BEM computations from

[27].

Figure 13: Pressure coefficient at the center line of

a planing flat plate, L/b= 1.07,F r = 12.14,τ=

6deg. Blue circles: experimental measurements. Red

solid line with circles: VLM computations. Black

dashed line with squares: BEM computations from

[27].

The last analysis deals with prismatic planing hull.

In this case a so called deadrise angle βis introduced.

This is similar to the dihedral angle typically used in

wing design. The hull have been designed accord-

ing to the Savitsky's method [28]. Its main partic-

ulars are listed in Table 2, being LK,LCand LM

the wet lengths at keel and chine and the mean wet

length, respectively, d the draught at transom and

λ= (LK+LC)/2Bthe mean wet length ratio. The

predicted lift and drag coefficients are compared to

those obtained by Savitsky's method in Fig. 14 and in

Fig. 15, respectively. The main differences between

the two predictions are in the pre-planing regime, i.e.

for F n < 1. In this phase the hull is still supported

by a relatively large hydrostatic force, as shown in

Fig. ??. At higher speeds, F n > 1, the dynamic

pressure becomes prevalent compared to the hydro-

static component, supporting the hull that runs in fully

planning regime. Since the VLM method has been

developed particularly for this kind of flow regime,

the obtained agreement is satisfactory. The steady

wave patterns corresponding to the two Froude num-

bers F n = [0.77,1.80] are shown in Fig. 16 and Fig.

Table 2: Planing surfaces particulars and lift and drag

coefficients comparison.

Particular Units Hull

L [m] 15

B [m] 4

d [m] 1

τ[deg] 5.0

β[deg] 10.0

λ[-] 2.55

LK[m] 11.47

LC[m] 8.91

LM[m] 10.2

17. The free surface elevation is colored by using a

color map from blue to yellow. Consistently with the

known evidences on ship wave patter formation, in

the pre-planing regime (F n = 0.77) there are both di-

vergent and transverse waves (the latter being almost

disappearing due to the relatively high speed). The so

called Kelvin angle ψ≈19deg from the bow apex,

defining the existence domain of the generated waves

in the plane of the free surface, is present. As the crit-

ical Froude number is overcome F n > 1, the hull

starts planing and the Kelvin sector narrows while

the transverse waves completely disappear revealing

a wave pattern made of divergent waves only.

Figure 14: Lift coefficient vs Froude number for a

prismatic planing hull, λ= 2.55,τ= 5deg,β=

10deg. Red: VLM computations. Blue: Savitsky's

method.

5 Conclusion

The hydrodynamic problem of either submerged and

surface-piercing lifting bodies has been solved in the

framework of a potential flow theory. In particular,

a Vortex Lattice Method has been developed to pre-

dict both the pressure distribution over these lifting

surfaces and the steady free surface elevation gen-

erated by the presence of the body. The boundary

value problem has been linearized in order to achieve

a fast and reliable numerical solution. The method

has been validated by comparison against available

experimental data and against computational results

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Figure 15: Drag coefficient vs Froude number for a

prismatic planing hull, λ= 2.55,τ= 5deg,β=

10deg. Red: VLM computations. Blue: Savitsky's

method.

Figure 16: Complete wave pattern predicted for a

prismatic planing hull, λ= 2.55,τ= 5deg,β=

10deg, at Fr=0.77. The wave elevation is displayed

by using a colormap from blue (z=-1m) to yellow

(z=2m).

from other sources, namely a conventional Boundary

Element Method, a viscous RANS based solver, Sav-

itsky's semi-empirical method for planing hulls and

theoretical results from thin foil theory, including cor-

rections for finite Aspect Ratios.

The validation process assesses the accuracy of the

method, that results to be in good agreement with the

majority of the analyzed cases in terms of pressure, lift

and drag coefficients. The method has been further

applied to analyzed the free surface formation of both

a submerged hydrofoil, i.e. a NACA a=0.8 mean cam-

ber line, and a prismatic planing hull. The obtained

trends of the free surface, despite not directly vali-

dated against experimental measurements, are consis-

tent with the theoretical knowledge on these phenom-

ena, then confirming the VLM predictions.

Figure 17: Complete wave pattern predicted for a

prismatic planing hull, λ= 2.55,τ= 5deg,β=

10deg, at Fr=1.80. The wave elevation is displayed

by using a colormap from blue (z=-1m) to yellow

(z=2m).

References

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breaking waves produced by a towed hydro-

foil. Proceedings of the Royal Society of Lon-

don. A. Mathematical and Physical Sciences,

377(1770):331--348, 1981.

[2] James H Duncan. The breaking and non-

breaking wave resistance of a two-dimensional

hydrofoil. Journal of fluid mechanics, 126:507-

-520, 1983.

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Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Raffaele Solari: Methodology, software, investiga-

tion, validation, writing - review and editing.

Patrizia Bagnerini: Methodology, supervision, writ-

ing - review and editing.

Giuliano Vernengo: Conceptualization, methodol-

ogy, visualization, supervision, writing - original

draft, review and editing.

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International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_US

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DOI: 10.37394/232013.2022.17.4

Raffaele Solari, Patrizia Bagnerini, Giuliano Vernengo

E-ISSN: 2224-347X

Volume 17, 2022