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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 746
Numerical and Experimental Study of the Wave Breaking
Generated by a Submerged Hydrofoil
A.Iafrati, A.Olivieri, F.Pistani, E.Campana (CETENA S.P.A., Italy)
ABSTRACT
In the present paper the two-dimensional wavy flow generated by an hydrofoil moving beneath the free surface is
experimentally observed and numerically studied. The numerical investigation is performed by means of a finite
difference Navier-Stokes solver. The free surface is embedded in the computational domain and the flow either in air and
in water are computed. The Navier-Stokes solver is coupled with a Level Set technique to captures the interface location.
The presence of the hydrofoil is taken into account either by introducing suitable body forces on the grid points inside the
body contour or by a new domain decomposition approach, developed to concentrate computational efforts in the free
surface region. Experimental study concerns the wave-breaking dominated by the ripples formation. The stages of the
evolution of a breaking generated after the onset of a capillary wave train are visualized. For a fixed Froude number and
angle of attack, the depth of the hydrofoil has been gradually varied, until the condition for incipient breaking has been
reached. Depending on the condition of the experiment, the wave breaking can start from the forward face of the second
or third wave crests, hence propagating to the first wave, leading to the full developed event.
INTRODUCTION
The knowledge of the mechanisms responsible of the breaking waves is of great importance for the comprehension
of many natural phenomena and the development of several engineering processes.
There are so many problems related with breaking waves that a complete list is hard to compile. To be confined to
those related with ships, breaking waves are produced by almost any marine vehicle, and are relevant in the definition of
their operative conditions. Beside of being responsible of the increase of the ship's resistance, breaking waves play a
relevant role in active and passive ship detection problems. The hydrodynamic noise produced by the breakers can lower
to a great extent the efficiency the ship's detection equipment, usually located inside the bulb. Although the problem may
be solved by increasing the depth of the sonar dome, this not always represent a winning hydrodynamic solution. On the
other side, breaking waves are responsible of possible detection of the ship from synthetic aperture radar (SAR) images of
the sea surface.
Furthermore, breaking waves are always in close connection with vorticity and turbulence production at the free
surface, as well as the generation of a bubbly near wake of the ship, again a relevant signature problem, and a great effort
is currently devoted toward the understanding and modelization of these phenomenon (see for example Reference 1).
A long, and far from being complete, list of references could be write down. So many researches have contributed to
our basic understanding of the breaking phenomena that we have to confine ourself just to some of previous studies.
The flow structure near the ship when breaking events occur has not been deeply investigated. In Miyata & Inui
(1984) the problem has been reviewed and, more recently, Dong et al. (1997) performed detailed particle image
velocimetry (PIV) measurements of the flow about a ship model, carefully analysing the wave structure near the bow and
the mechanism of free surface vorticity production.
The flow structure of 2D spilling breakers has been more extensively studied. Hydrofoil generated spilling breakers
have been experimentally investigated with great accuracy by Battjes & Sakay (1981), Duncan (1981, 1983), Mori
(1986), Duncan & Dimas (1996), Lin & Rockwell (1996). In particular the work of Duncan's research group, represents a
reference point, especially for the
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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 747
case of the towed hydrofoil.
Theoretical studies suggested that the flow just below the breakers is turbulent, and suggested the existance of a
shear layer beneath the breaking wave, see Peregrine & Svensen (1978), Longuet-Higgins (1994a), Longuet-Higgins
(1994b), Cointe & Tulin (1994).
The numerical description of the breaking phenomena via moving grid approaches is not straightforward. Recently,
new techniques describing the two-phase flow of both air and water have been developed, allowing for a complete
description of the breaking and post-breaking event. In Sussman et al. (1994) an additional variable, i.e. the signed normal
distance from the interface, is introduced and the free surface location is identified as the zero level set (LS) of this
quantity. The distance is a continuous function across the interface, reinitialized at each time step. The capability of this
approach to deal with complex flows in which topological changes of the interface occur has been proved by some recent
papers by Azcueta et al. (1999), Vogt & Larsson (1999), Iafrati et al. (2000), Iafrati & Campana (2000).
The purpose of this paper is to report recent developments at INSEAN in the numerical and experimental
investigation on breaking waves, in the framework of a cooperative project involving ONR, IIHR and DTMB.
Present experimental study concerns the wave-breaking rising from the formation of capillary waves on the forward
face of the gravity wave. The problem has been treated theoretically by Longuet-Higgins (1992). Preliminary observation
of the occurrence of these ripples and of the breakdown of this type of flow are reported in the following.
The numerical approach is here used to study the inception of the breaking produced by a submerged hydrofoil, with
particular reference to the velocity and pressure fields. Several numerical schemes have been adopted, ranging from a
simple inviscid rotational formulation (mainly used for verification purposes), to the solution of the Navier-Stokes
equations in the full domain. A Navier-Stokes solver in generalized coordinates, together with a Level Set technique, used
to follow the free surface dynamics, has been developed. This approach can lead to free surface instabilities in regions
where the grid is highly skewed, unless an high grid refinement is used (Iafrati et al. 2000). This in turn implies that some
difficulties may be encountered when studying the free surface flow induced by bodies moving close to the interface, due
to the distortion of the body fitted grid. Nevertheless, when attention is mainly devoted to the free surface dynamics rather
than to a detailed description of the flow about the body, the above problem has been overcome, either by using an
orthogonal grid and introducing suitable “body forces” that mimic the presence of the solid boundary, or developing a
new approach based on a domain decomposition technique.
EXPERIMENTAL INVESTIGATION
Experimental system and techniques
The experiments have been carried out at INSEAN basin n.2 (220 m long, 9 m wide and 3.5 m deep). The towed
hydrofoil is a NACA 0012 profile made of composite material, whose chord and span are respectively 0.4 m and 2 m. The
hydrofoil is connected to the carriage by two vertical, surface piercing, side struts. Variation of the angle of attack and
rotation along the z axis are allowed. Images of the generated wave pattern have been taken using a video camera and
pictures have been subsequently extracted. Moreover, a submerged video camera has been applied to visualize the flow
around the hydrofoil. A fluorescent substance, introduced upstream the hydrofoil by a thin duct, has been used to enhance
the vortical structures leaving the rear part of the hydrofoil. The light source needed for the underwater images has been
provided by an 800 watt photo-floodlight with Fresnel lens placed close to the surface on the rear part of the hydrofoil. A
flat mirror mounted below the hydrofoil has been used to increase the amount of light (Fig. 1). The tests have been carried
out at a constant Froude number of 0.177, while the hydrofoil has a 10 angle of attack (nose up). To detect the onset of the
wave breaking, the hydrofoil depth has been slowly decreased during the carriage run, following the evolution of the
breaking from the initial stage up to its complete development.
Figure 1: Sketch of the experimental apparatus (side view).
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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 748
Wave-pattern and vortex-shedding visualization
The different phases of the wave breaking process have been filmed by a video camera placed on the forward part of
the apparatus, just above the free surface.
Figures 2 to 4 show the main phases of the wave breaking process. Figure 2 refers to a hydrofoil depth of 0.2 m, and
the wave formation is scarcely perceivable. By gradually decreasing the depth of the hydrofoil the wave pattern
developed, and ripples are observed. Figure 3 shows the presence of ripples and a wave breaking on the third crest, while
the second is just partially interested. Finally, figure 4 shows well developed wave breaking, along with the presence of
residual waves.
Figure 2: The wave pattern produced by the hydrofoil at depth of 0.2 m.
Figure 3: Same as before but with depth 0.16 m. The
breaking develops at the rear crests before extending in
the forward direction.
Figure 5: Ripples appearance at depth 0.18 m (top) and
transversal instability (bottom) immediately before the
breaking at the rear crests.
Figure 4: Fully developed breaking for the depth 0.12 m
with residual following waves..
A close-up view of the waves crestsi (Fig 5.—top) shows the appearance of ripples, in particular on the forward face
of the second and third wave crests, before the breaking region reaches the first crests and the breaking fully develops.
The formation of ripples and their successive propagation leads to a tranverse instability of the wave front (Fig 5.—
bottom), finally breaking in a three-dimensional way.
This kind of scenario for breaking waves has been already experimentally observed by Duncan et al. (1994)
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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 749
for waves of wavelength of about 20 cm, as in the present study.
Pictures acquired by the underwater camera show the presence of vortex shedding from the rear part of the hydrofoil.
This is due to the combined effect of relatively low Reynolds number and high angle of attack (Fig. 6). The effects of this
separation will be discussed later.
Figure 6: Vortex shedding from the upper side of the hydrofoil.
NUMERICAL MODELING
Navier-Stokes solver for the two-phase flow
The two-phase flow is modeled as the flow of a single fluid whose density and viscosity smoothly changes across the
interface. By assuming both phases to be incompressible, in an Eulerian frame of reference the local fluid properties
changes with time only due to the interface motion.
If surface tension and turbulence effects are neglected, the unsteady non-dimensional Navier-Stokes equations for an
incompressible fluid in generalized coordinates are:
(1)
(2)
where ui is the i-th cartesian velocity component and δij is the Kronecker delta. The quantity
(3)
is the volume flux normal to the ξm iso-surface and J−1 is the inverse of the Jacobian. In Eq. (2) the gravity term is
written in non-dimensional form, being
the Froude number and Ur and Lr reference values for velocity and length, respectively. In the diffusive term
is the reference Reynolds number being µ w the values of density and dynamic viscosity in water that are also
used as reference values. The quantity
(4)
is the mesh skewness tensor.
The numerical solution of the Navier-Stokes equations is achieved through a finite difference solver on a non
staggered grid similar to that suggested by Zang et al. (1994). Cartesian velocities and pressure are defined at the cell
centers whereas volume fluxes are defined at the mid point of the cell faces. For the computation of the convective terms
and to enforce the continuity, fluxes at cell faces are evaluated by using a quadratic upwind scheme (QUICK) to
interpolate cartesian velocities.
The momentum equation is integrated in time with a semi-implicit scheme: explicit terms are computed with a
variable time step Adam-Bashfort scheme while a Crank-Nicolson discretization is employed for the implicit terms. Since
the grid is time independent, the discretized form of Eq. (2) is
(5)
where ∆t=tk+1−tk and represents the convective terms at the step k, Ri is the gradient operator in
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curvilinear coordinates, DI and DE are the diagonal and off-diagonal diffusive operator. The use of an explicit scheme for
the convective terms limits the time step: this is chosen so that the maximum Courant number all along the computational
domain is

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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 750
always smaller than 0.8. The use of an explicit scheme for the off-diagonal diffusive operator can also limit the maximum
allowable time step due to the viscous stability limit when highly skewed grids are used in regions where viscous effects
are dominant. On the other hand an implicit account of the off-diagonal diffusive operator is too expensive from the
computational point of view and, however, the use of highly skewed grids in viscous dominated regions should be avoided.
Equation (5) is solved through a fractional step approach: an auxiliary velocity field is introduced and the problem
is solved in two steps. In a first step the auxiliary velocity field is found by neglecting the pressure term from the right
hand side:
By subtracting the above expression from Eq. (5) it remains:
(6)
The auxiliary velocity field is found by solving the predictor step (Eq. 6) through an approximate factorization of the
operator of the discretized momentum equation.
The pressure field at the new time step is found by assuming that the velocity field is related to by the
relation:
(7)
where is the pressure corrector term. By introducing Eq. (7) into Eq. (6) the following relation between the
pressure field and the pressure corrector holds:
(8)
Once the scalar function is computed, the above relation could be used to calculate the pressure field. However,
when working in generalized coordinates its solution is not straightforward and instead the following approximation is
used (Rosenfeld et al. 1991):
(9)
This does not affect the accuracy of the numerical scheme since the pressure itself is never used in the calculation.
The pressure corrector is computed by enforcing the continuity by Eq (1). In fact, Eq. (7) can be written as:
and, by Eq. (3), it follows:
(10)
Using this expression in the continuity equation, a Poisson problem for the pressure corrector is obtained:
(11)
When the velocity is known at the boundaries, Eq. (10) provides a Neumann boundary condition for the solution of
Eq. (11). The solution of this Poisson equation is performed either by a BiCGSTAB (van der Vorst 1992) algorithm with
an ILU preconditions or by a multigrid technique.
This latter has been found rather effective, even though difficulties have been encountered when dealing with grids
having a very large aspect ratio.
Free surface motion via the Level-Set technique
The numerical model described in the previous section is used to solve the Navier-Stokes equations in a domain that
encloses both air and water while the actual location of the interface must be captured in some way. Although fluid
density and viscosity are assumed to take fixed values for each fluid, they vary in time due to the interface motion.
However, when using the corresponding transport equations difficulties may arise due to the sharp variation of the fluid
properties at the interface.
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In the level-set technique this problem is avoided by assuming fluid properties as functions of a signed normal
distance from the interface d(x, t). At t=0 this function is initialized assuming d>0 in water, d<0 in air and d=0 at the
interface (Sussman et al. 1994). The generic fluid property ∫ is assumed to be ∫(d)=∫w if d>δ,∫ (d)=∫a if d<−δ and

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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 751
otherwise. In the above expression δ is the half width of a transition region introduced to smooth the jump in the
fluid properties. The thickness δ is chosen so that the jump covers at least four cells. In this way the width of the jump
region, that will be kept constant in time, decreases when reducing the cell size (Unverdi & Tryggvason 1992).
During the evolution the distance is assumed to be convected by the flow, thus the equation
(12)
is integrated to update the distribution of the distance function. At the end of the convective step, since the interface
is a material surface, the free surface location is captured as the level d=0.
In fact, the integration of equation (12) does not ensure that the thickness of the jump region is kept constant in space
and time. To avoid the spreading or concentration of the transition zone, the distance function is re-initialized at each time
step as the normal distance from the actual interface.
The problem of the reinitialization of the distance is well discussed in Sussman et al. (1994) and Adalsteinsson &
Sethian (1999). Usually the distance function is reinitialized by iterating to steady state the equation:
where S(d) is a sign function that is zero on the interface. The main advantage of this approach is that the actual
interface location does not need to be computed at each time step. As a drawback, the solution of the above equation
needs suitable numerical scheme to prevent oscillations.
However, for two-dimensional applications, the computational effort needed to locate the free surface and to
recompute the distance function is not critical. For this reason the interface is reconstructed at each time step by explicitly
locating the position of the level d=0 and the function d(x, t) is reinitialized by computing, at each cell center, the signed
normal distance from the interface. This procedure is found effective in terms of mass conservation and in facing complex
flows as it is discussed in Iafrati et al. (2000) where several kind of free surface flows have been analyzed to validate the
procedure.
In order to damp disturbances outgoing from the computational domain, a numerical beach model is introduced in
Eq. (12). Two beach regions are introduced close to the two boundaries of the computational domain. If y=0 is the still
water level, in the beach regions Eq. (12) takes the following form:
(13)
where the coefficient v is zero at the inner limits of the beaches and grows quadratically toward the boundaries of the
computational domain.
Solid boundaries modeled via body forces
In Iafrati et al. (2000), it is observed that instabilities may arise when the interface pass through regions where the
grid is too distorted unless a very fine grid resolution (or a large value of δ) is employed. This is an important issue to be
solved when the wavy flow generated by hydrofoil moving close to the interface has to be studied. On the other hand,
when attention is mainly focused on the free surface flow, an accurate description of the flow about the body is not
strictly needed.
With the above issues in mind, the presence of the solid body has been modeled through a body forces approach, that
is by introducing suitable body forces in grid cells inside the body contour. The magnitude of this forces is chosen so that
the velocity of the grid points inside the body contour tend to be equal to the velocity of the body itself.
At t=0 the flow is assumed to be uniform with (u, v)=(1, 0) on each grid point of the computational domain. Since
the frame of reference is attached to the body, for any grid point inside the body contour the following term is added to
the right hand side of Eq. (2):
(14)
where C∫ is a friction coefficient whose effect will asymptotically reduces the velocity of points inside the body up to
the rest (Dommermuth et al. 1998). It is worth to remark that this kind of transient, in which the velocity field inside the
body progressively frozen up to the rest, is rather unphysical but it is acceptable when steady body velocity have to be
considered.
The function S(t) is a smooth function
(15)
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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 752
t0 being the length of the ramp, is introduced to reduce the formation of long upstream propagating waves induced by
the starting phase. This problem is more evident in two-dimensional problems when these waves propagate keeping their
amplitude constant.
The use of Eq. (14) for grid points inside the body and qi=0 elsewhere is a only a zero order approximation of the
body shape. This means that when refining the grid the solution changes not only due to the accuracy in the description of
the fluid flow but also due to the changes in the body shape. This question will be discussed more deeply when analysing
the numerical results.
Domain decomposition
For the purposes of the present work, we focus our attention toward the processes near the free surface. According to
this, an accurate description of the flow field about the body is not really needed. Furthermore, when considering high
Reynolds number flows, the full Navier-Stokes approach becomes very expensive and, moreover, a turbulence model
should be included to accurately predict the flow about the body.
The above consideration suggested to develop a zonal approach, decomposing the fluid domain in an upper region,
near the free surface, and a lower region, where the body is located.
In the body (lower) region, an inviscid flow model is assumed. The flow is described via a velocity potential and a
suitable Kutta condition can be used to describe the rotational flow about the hydrofoil. In the free surface (upper) region,
the flow is described by the Navier-Stokes equations. This decomposition allows to concentrate the computational effort,
to a great extent devoted to the solution of the Navier-Stokes equations, in a small domain enclosing the free surface (see
Fig 7).
Figure 7: The decomposition of the computational domain in a lower (body) region, computed via a BEM solver, and an
upper (free surface) region, where the Navier-Stokes solver is coupled with a Level Set technique for solving the air-water
flow.
Viscous and the inviscid rotational solutions are fully coupled at each time step, with a simple and effective
procedure. The potential flow region is resolved first. Neumann boundary conditions are applied on the two sides of the
computational domain (inflow and outflow of the body region), on the bottom and on the body contour, while a Dirichlet
boundary condition is applied onto the matching surface and a Kutta condition is imposed at the trailing edge of the
hydrofoil.
The solution of the flow in the body region provides the normal and tangential velocity components at the matching
surface. This velocity is used as a boundary condition for the Navier-Stokes solver in the free surface region. At the end
of the advancement in time the Navier-Stokes solver provides the pressure field on the matching surface that is used to
update the velocity potential via the unsteady Bernoulli's equation.
In the following, additional details about the potential solution and the coupling procedure are discussed. Although
the coupling procedure here suggested could work even in the three-dimensional case, details below refers to the two-
dimensional case.
The potential domain is limited on the top by the matching surface, on the two sides by the inlet and outlet vertical
sections, and by the solid boundaries, that is by the body and/or the bottom. When the flow about an hydrofoil is
investigated, a Kutta condition is enforced at the trailing edge. To this aim, a vortex line, with a uniform distribution of
the vorticity density γ, is introduced within the hydrofoil, ranging from the leading edge to the trailing edge. The vorticity
density is fixed, so that the average of the velocities at the midpoint of the two panels at the trailing edge is parallel to the
vortex line. A further simplification is introduced, in that the vortex shedding, characterizing the initial transient, is not
accounted for in this model. This simplified model is acceptable when attention is mainly focused in the final quasi-steady
solution.
As stated above, the velocity potential is assigned on the matching surface by integrating the unsteady Bernoulli's
equation that, in a frame of reference attached to the body, takes the following form:
(16)
where p is the pressure value coming from the Navier-Stokes solution, is the velocity potential in the absolute
frame of reference, uv is the velocity field induced by the vortex and UB is the velocity of the frame of reference, that is
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attached to the moving body or to the moving bottom.
All along the other boundaries, the normal derivative of is assigned. On the moving bottom and/or on the

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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 753
body contour the impermeability constraint is applied:
where n is the unit normal vector directed inward. On the inlet and outlet boundaries a uniform incoming and
outgoing flow is assigned:
The solution of the flow in the potential region is obtained by using a zero order panel method for the solution of the
Laplace equation for the velocity potential. The solution of the boundary value problem provides the velocity potential on
the solid contours and its normal derivative on the matching surface. This latter is then used, along with the velocity
potential itself, for the calculation of the velocity field on the matching line.
VALIDATION
Results presented in this section are relative to the assessment of some of the characteristics of the methods
described before. The validation study has been performed also to show the applicability of the decomposition approach.
The study is based on the wavy flow generated by a moving bottom topography and the flow induced by a hydrofoil
moving beneath the free surface, both in non-breaking and in breaking condition. Results are compared with those
obtained with a fully nonlinear boundary elements solver and, in the case of the hydrofoil, with the experimental data
obtained by Duncan (1983).
Case study: wavy flow induced by a moving bottom topography
The wavy flow generated by a bottom bump moving in a channel is studied by using the full Navier-Stokes solver
(FNS), the domain decomposition approach (DD) and the fully non-linear boundary elements solver (BEM).
The geometry of the bump, located in x ∈ (−0.5, 0.5), is given by the following equation:
The bump is placed on a flat bottom at y=−1 while the still water level is at y=0. The computational domain extends
from x=−14 to x=14 in the horizontal direction and from the bottom profile up to y=0.4 in the vertical direction.
Numerical beach models have been applied in the upstream and downstream free surface regions, x ∈ (−14, −8) and x ∈
(8, 14) and the maximum value v=2 is assumed for the damping in Eq. (13).
In order to perform a fair comparison with BEM results, a slip boundary condition is applied on the bottom profile
when using the FNS approach. When using the DD approach, the matching surface is located at y=−0.2. The dependance
of the numerical solution on the location of the matching surface as been empirically verified, by moving the surface from
very deep up to 1.5 times the depth of the first trough. The solution has proved to be substantially independent from the
location of the matching.
At t=0 the bump is suddenly started at UB=(−1, 0) and L being the horizzontal lenght of the bump.
Results obtained with the three different approaches at two different time values are shown in the figures below.
Figure 8: Free surface profiles generated by the sudden start of a bottom bump in a channel at t=20 (top) and t=130
(bottom): FNS (solid line), DD (dashed line), BEM (dash-dotted line)
Figure 8 shows that the three solutions are in a very good agreement at t=20, i.e. before wave disturbs reach the
downstream damping zone, while slight differences occur later. In particular at t=130, while the BEM and FNS solution
are still very close each other, the numerical DD solution is characterised by an excessive numerical damping. Among
others, two factors can be responsible for this damping: the use of a first order explicit scheme to integrate in time Eq.
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(16) and the use of Eq. (9) rather than solving Eq. (8) to evaluate the pressure field. In fact, both these factors suggest that a

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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 754
small time step has to be used to achieve a good accuracy. However, wave phase and wave lenght are quite well catched
from the DD approach.
Figure 9: Velocity field and free surface profiles generated by the sudden start of a bottom bump in a channel: t=6.4 (top) and
t=6.8 (bottom)
In order to show the effectiveness of the Level Set approach in the prediction of wave breaking, the flow over an
high bump, leading to the breaking of the free surface, has been also carried out. The height/lenght ratio of the bump is H/
L=0.4, and simulation has been performed with Re=104, µ a/µw=0.018. In Fig. 9, two frames of the time history of the
impact and successive phases of the breaking are shown, together with the corresponding velocity field in water and air.
More detailed results for this type of flow can be found in Iafrati et al. (2000). After t=6, the jet is sufficiently developed
so that it impacts the free surface. The impact of the jet on the free surface lead to air entrainment and also to a splash-up
just ahead of the impact point onto the free surface. The splash-up evolves and eventually (not shown) its forward face
impacts again on the free surface, leading to another air entrainment and another splash-up process. The phenomena
proceeds as described, even though gradually decreasing the splash-up intensity. This behaviour is qualitatively consistent
with that described, for istance in Bonmarin (1989).
Submerged hydrofoil: non breaking regime
The wavy flow generated by a hydrofoil moving beneath the free surface is studied by using the FNS and the DD
approaches. When using the FNS approach, body forces are introduced to model the presence of the solid boundary. As a
first application the non-breaking wavy flow is analyzed. To this aim, following experimental data obtained by Duncan
(1983), a NACA 0012 profile, 5° angle of attack, moving at Fr=0.567 at a non-dimensional depth 1.034, is considered. In
all cases, the computational domain extends from x=−20 to x=20 in the horizontal direction and from y=−3 to y=1 in the
vertical direction, y=0 being the still water level. As to the boundary condition, u= (1, 0) is applied all along the boundary
of the computational domain in the FNS solution. In order to damp disturbances outgoing from the computational domain
numerical beach models are introduced in the regions x ∈ (−20, −12) and x ∈ (12, 20).
FNS results
In order to check the convergence properties of the body force approach three different grids are used. In all cases a
uniform horizontal spacing is used in x ∈ (−1, 3) and a constant growth factor is used to fill the domain. In the vertical
direction uniform spacing is used both in the body region and in the free surface region. Here, for all the grids the values
∆y=0.005, δ=0.02 are used for the vertical grid spacing and for the half width of the jump region (see Fig 10). A value
t0=8 has been assumed for the length of the ramp when using Eq. (15). In Fig. 11 one every fourth grid point is shown for
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the coarse grid (282×199). This lead to a mesh with sufficient grid points per wavelenght in the free surface region
between x=−1. and x=3., (about 50 points for the case study). According to this, medium (426×

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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 755
259) and fine grid (672×371) are obtained by halving cells in the body region only. For the fine grid ∆x= 0.01 and
∆y=0.002 are used in the body region.
Figure 10: Numerical grid for the FNS solution. One over
fourth grid point is shown of the coarse grid. The body
and free surface region are clearly recognisable
Figure 11: Close up view of the grid in the body region,
used in the FNS approach: for clarity, one every fourth
grid point is shown for the coarse grid
Figure 12: Comparison among the free surface profiles
obtained by the FNS approach and the experimental data
(Duncan 1983): coarse (dash-dotted), medium (dashed), Figure 13: Comparison among the u contours about the
fine (solid), Duncan (dot) leading edge of the hydrofoil: from the top to the botton
coarse, medium and fine grid. The dashed line represent
the section of the hydrofoil
Free surface profiles obtained with the three grids are shown in Fig. 12 in comparison with the experimental data
obtained by Duncan (1983). The computation is performed by assuming µ a/µw= 0.018, and a
Reynolds number Re=10000.
The comparison put in evidence some features of the numerical results: (i) good convergence in terms of
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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 756
wavelength, (ii) poor convergence in terms of wave amplitude, (iii) a phase shift similar for all the grids and (iv) a spike
in the wave profile at x=6 for the medium grid result.
Investigating about these points, attention has been focused on close up views of the velocity field in the leading
edge region, obtained with the three grids, depicted in Fig. 13. The velocity field in this region shows relevant changes
when the grid is refined. In fact, during this process, due to the zero order model used for assign body forces, substantial
differences in the computational shape of the solid boundary occurs. Nevertheless, although the above limit of the body
force approach can justify the poor convergence, the phase shift is almost the same for all the grids.
As a carefull look at the vorticity shedded from the trailing edge of the hydrofoil reveals (Fig. 14), the interactions
between the vorticity field beneath the free surface and the wave profile (with some dipole rising up toward the free
surface) are responsible for the spike in the wave elevation at x=6. The flow separation occuring from the suction side,
can significantly alter the wave amplitude and phase.
Figure 14: Vorticity contours behind the hydrofoil: from the top to the bottom t=18, t=20, t=22
On the basis of the above consideration, a fair comparison with experiments performed at high Reynolds number
cannot be established unless a turbulence modeling, beside to an improved body force formulation, is introduced.
DD results
The problems encountered in the correct prediction of phase and wave amplitude, lead to the development of the
domain decomposition approach that uses the inviscid rotational flow model and the boundary element technique to
describe the flow about the lifting body whereas uses the Navier-Stokes solver coupled with the Level-Set technique to
describe the flow in the free surface region, where complex interface topologies may develop in breaking condition.
Numerical simulation have been carried out by assuming the matching surface at y=−0.2. In the Navier-Stokes
region a 256x96 grid is employed with grid points suitably clustered about x=1 in the horizontal direction and about y=0
in the vertical direction. In this case a uniform vertical grid spacing ∆y=0.005 is used in y ∈ (−0.2, 0.2), whereas δ=0.03 is
used as the half width of the jump.
In Fig. 15, the free surface profile obtained by the DD approach is compared versus the fully non linear BEM result
and with the experimental data by Duncan (1983) for the same conditions as before.
Figure 15: Comparison among the free surface profiles obtained by the DD approach (solid), the full BEM (dashed) and the
experimental data by Duncan (1983) (dot)
With respect the FNS results, the DD allows a much better description of the first trough, even though an excessive
damping of the following waves appears, and wave phase and lenght are in good agreement with experiments too. As
already stated, reasons for this excessive damping are not yet really understood, although it is believed to be related to the
explicit integration of the Bernoulli's equation and to the approximation of the pressure field given by Eq. (9). However, it
has been verified that, as long as the matching surface is deep enough with respect to the wave troughs, its position does
not significantly affects the solution.
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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 757
Submerged hydrofoil: breaking regime
In spite of the limits of the numerical solver, simulations of the breaking wave produced by an hydrofoil have been
attempted, aimed at the verification of the capability of the solver to predict and model the breaking wave phenomena.
For a non-dimensional submergence of 0.9113, Duncan (1983) observed that a weak spilling breaker is present,
reducing the following wave system. The comparison between the numerical solution and the experimental data is
reported in Fig. 16. It shows that the amplitude of the first crest is overpredicted and also the following wave system is
not damped, but for numerical effects. However, the close up view shows that a good agreement is achieved in terms of
the slopes of the front and back of the first wave, meaning that something is occurring. Differences are due to the low
resolution adopted in the numerical simulations, that is the wave tries to break but the resolution does not allow to
correctly capture the establishment of the breaker. As a consequence, the dissipative effects played by the breaker on the
following wave are not modeled.
Figure 16: Top: Wave profile behind a hydrofoil at a non-dimensional submergence 0.9113. Bottom: Close up view of the
first wave. In this condition, a breaking condition was observed by Duncan (1983) (dots). The adopted grid resolution allow
numerical solutions to (solid) capture the asymmetry but is not able to fully resolve the breaker region.
If the hydrofoil submergence is further reduced, the intensity of the wave breaking grows. For a non-dimensional
submergence 0.783, Duncan (1983) observed an intense wave breaking with a high dissipative effect on the following
wave. The comparison between the numerical solution and the experimental data for this submergence is reported in
Fig. 17: in this computation, due to the more pronunced wave trough, the matching surface has been located at y=−0.36.
As for the previous case the first crest is largely overpredicted and no dissipative effects on the following wave are
predicted. Also in this case, however, the strong asymmetry of the first wave can be noted.
Figure 17: Wave profile behind a hydrofoil at a non-dimensional submergence 0.783. An intense breaking was observed by
Duncan (1983) (dot) whereas the poor resolution does not allow the numerical approach (solid) to capture neither the
breaker nor the dissipative effects on the following wave.
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Figure 18: Dynamic pressure distribution in non-breaking (top) and breaking condition: 0.911 (center), 0.783 (bottom).
Even though the free surface elevation is the most obvious quantity to check, additional important information can
be provided by looking at the dynamic pressure field. In Fig. 18 the contours of the dynamic pressure are shown for the
non breaking case and for the two breaking case.

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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 758
It is worth to notice that, despite the poor resolution, the distribution of the dynamic pressure changes significantly
passing from the non-breaking to the breaking stage. Unfortunately, the poor resolution does not allow the water to splash
down and to form the breaker region.
CONCLUDING REMARKS
Numerical and experimental studies have been carried out at INSEAN on the 2D wave breaking flow produced by a
towed hydrofoil. Numerical simulations of the two phase flow in air and water have been performed, and validation
studies have been conducted in the case of the flow on a bottom bump, producing a plunging breaker. The formation of
the jet, the splash-up and the post breaking event are in a qualitative agreement with experimental observation. Hence, the
flow past the submerged hydrofoil has been simulated and the condition for the onset of the breaking have been
investigated. This has resulted in a detailed analysis of the computed velocity and pressure fields.
A Navier-Stokes solver in generalized coordinates, together with a Level Set technique, used to follow the free
surface dynamics, has been used and two different numerical codes have been developed, based on the body force and on
a domain decomposition approaches respectively.
The body forces approach has been developed in view of dealing with the flow about multibody configurations as it
is the case of a ship with appendages. This approach has proved to be useful, although a higher order model for the
assignement of the body forces, and hence of the way in which the shape of the body is represented, is needed. In order to
gain insight into the dynamics of the free surface, the domain decomposition approach has proved to be promising,
focusing attention and computational efforts in the free surface region. With reference to the quasi-steady breaking
produced by the hydrofoil, numerical results discussed here suggest that, although the numerical techniques are able to
detect the inception of the breaking, the adopted grid were too coarse to resolve the flow. Possible extension of the work
is the inclusion of surface tension effects, allowing a comparison with the set of experimental data.
The emphasis of the experimental work is on understanding the conditions under which capillary waves may force
the breaking on the folowing wave train, subsequently forcing the extension of the breaking area to the forward waves.
The work is largerly under development and a new system has already been designed for reproducing the experiments,
making also quantitative measurements.
ACKNOWLEDGEMENTS
This work was supported by the Ministero Trasporti e Navigazione in the frame of the INSEAN research plan 2000–
02.
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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 760
DISCUSSION
J.H.Duncan
Naval Surface Warfare Center, Carderock
USA
I find the photographs in Figure 5 particularly interesting. In my experiments with surface-tension-dominated
unsteady breakers with wavelengths of about 1 meter, a bulge forms at the crest with capillary waves upstream of the
leading edge (toe) of the bulge. The transition to turbulent flow seems to be initiated by flow separation at the toe. In your
photographs, it appears that the transition to turbulent flow is initiated in the region of capillary waves upstream of the
toe. In particular, it appears that the capillary waves are breaking first. Would the authors please comment further on this
phenomenon. It would also be very interesting if they can provide a qualitative description of the temporal evolution of
the free surface after the first appearance of these breaking capillary waves.
In the numerical calculations, the authors have attacked an exceedingly difficult problem. As they pointed out,
increased resolution will be needed to accurately compute the flow. However, given the present resolution, the authors
have compared the behavior of the following wavetrain under breaking conditions to experimental measurements. Have
they also examined the vertical distribution of horizontal velocity in the following wavetrain to look for evidence of the
wake found near the free surface in the experiments?
AUTHOR'S REPLY
We thank Prof. Duncan for the questions and for calling our attention to some flow details that deserve some more
comments.
About the different breaking mechanisms of the capillary waves, a possible explanation lay in the different water
quality. Indeed, in the experiment carried out by Prof. Duncan, the quality of the water is frequently cleaned through
filtering and the value of the surface tension is assessed with great accuracy. On the other side, being the present
experiment carried out in a large towing tank (220 m long), the presence of dust on the free surface cannot be avoided, as
it may be seen in Fig. 5. This can cause the growing of instabilities of the capillary wave front, eventually leading to the
difference in the breaking event.
A qualitative description of the observed temporal evolution of the free surface, after the appearance of capillary
waves, is sketched in Fig. 19 below.
Fig. 19—Sketch of the 3-dimensional instabilities (top view). The arrow shows the velocity of the hydrofoil (represented with
a thick black ribbon), the solid (dashed) lines represent the gravity (capillary) waves.
If the depth of the hydrofoil is large enough, some capillary waves appear on the forward face of the second and
third crests (Fig. 19a). When the depth of the hydrofoil is reduced, three dimensional instabilities appear (Fig. 19b),
eventually leading to wave breaking (Fig. 19c). Finally, depending on the depth, the breaking may also propagate to the
first crest.
Concerning the last question raised by Prof. Duncan, due to the poor resolution used in the calculation here
presented, the computed wake past the breaking cannot be seen. However, a calculation for the bump case with a more
refined grid (640x256) at low Reynolds number (Re=1000) has been carried out. Results show that a slackness is operated
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NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 761
Fig. 20—Vorticity contours for a spilling breaking condition. The black ribbon represents the free surface location and the
distribution of the velocity is shown along some vertical lines.
Fig. 20 also shows the intense counter-clockwise vorticity originating close to the toe of the bulge. Due to the low
Reynolds number, vorticity is rapidly diffused into the fluid domain.
DISCUSSION
D.Dommermuth
Science Applications International Corp., USA
The authors have developed a unique procedure for modeling breaking waves. Could they please compare the
domain decomposition method that is described in their paper to the Schwarz alternating method
(Schwarz, 1890)?
AUTHOR'S REPLY
We thank Prof. Dommermuth for the interesting question. The Domain Decomposition (DD) approach we have used
in the paper does not need an overlapping region and is assigned on the matching surface. To apply the Schwarz
alternating method an overlapping is needed and the normal component of the velocity must be exchanged between the
subdomains. In contrast with the former approach, the latter algorithm does not require an explicit time integration for the
exchanged variable. Nevertheless, some subiterations are necessary, whose number depend on the extension of the
overlapping region. As a consequence, an a priori comparison of the two different approaches in terms of computational
efficiency do not permit to establish which is the best choice. The development of the Schwarz method, in order to
compare the two approaches in terms of CPU time and accuracy, is a part of ongoing activity.
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