|
Items in cart [0] |
Questions? Call 888-624-8373 |
| Copyright © 2009. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 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)
ARSTRACT
In She present paper She two-dimem Coal we y flow
generated by m hyd of oil mm leg beneath the he sur-
face is e perimentally observed md m mericclly stud-
led The m mericcl inve tigation is performed by m ms
of cflmite differenceNavier-Stokes solver The fiee sur-
face is embedded in She oompnhtioncl domcm md She
flow either in air md in water are computed The Na vier-
Stok s solver is coupled with c L vel Set tech ique to
ccptmes She mte face location Th presence of She hy-
d of oil is taken mto acco mt either by introducing mit-
ctle body fmces on She g id pomts inside the body con-
tour or by c new domain decomposition approach, de-
veloped to concentrate c mputatiom~l efforts in the flee
surface legion Experimental st dy concerns the wave -
breking domirutedbythe ripples fommation The stages
of the evolution of c breckmg generated after the m-
set of c ccpilk y wave ham are visuali:D:d For c fixed
Froude n ml er md male of attack, She depth of the hy-
d of oil hr. been g Edna lly varied, mtil She condition for
incipient heat mg hr. been reach d D pending on She
condition of the experiment, the wave breaking c m start
from the fo ward face of the second or thi d wave Rests,
hence propagating to the -i t wave, leading to the full
developed event
INTRODUCTION
The knowledge of She mechanisms re ponsible of
She breckmg waves is of g et impo tance for She com-
prehension of m my mutual phffmmerur md th devel-
opment of several engmeermg processes
There are so m my probl ms rel.t ed wish breckmg
waves Nat c complete ii t is hard to compile To be
cord ted to those rented with ships, breaking waves me
producedbyckmo t mymarinevehicle, md are relervnr
m She deflmition of Heir operative conditions Beside of
bemg re portable of She in x ease of the ship's resistmce,
b'ecking use s p By c re levmt r o le m a et ive md pass ive
ship detection problems The hyd odynamic noise pro-
duced by the breakers m I wer to c g eat extent the ef-
flciency the ship's detection equipment, usually located
mside She buk Although She problem may be solved
by inmecsmg She depth of She sonar dome, this not cl-
ways represent c wimmmg hyd odynamic solution On
She of her side, breaking waves me responsible of possi-
ble detection of She ship fiom synthetic cpertme radar
ISAR! images of She see surface
Fmthemm me, breckmg waves are always in close con-
necti m with vo ticity md turbulence production et She
he surface, es well as the generation of cbubblynear
wake of She ship, again c relevmt signature problem, md
c g eat effort is currently devoted toward the indent md-
mg md modelization of These phenomenon (see for :x-
cmpleReference 1)
A long, md far from being complete, list of refer-
ences could be w ite d wn So m my researches have
conh~buted to our basic mderstmding of th breaking
phen memo Nat w have to cord ~ ourself ju t to s me
of previous studies
Theflow structure near She shipwhenb~eckmgevents
occur ht. not l en deeply investigated ~ Miyatc &
h Pi (1984) the problem hits been revi wed md, more
recently, Do g et al (1997) perfommed detailed parti-
cle image velocimet y REV) measurements of the flow
Croat c ship model, carefully am~lysing She wave struc-
tme near th b w md the mechmism of flee surface vor-
ticity production
The -I w structure of ED pilling breakers has been
more extensively st died Hyd of oil generated spilling
breckershavebenexp rimentallyinve tigatedwifhg et
tcc~tc~ by Battles & Sat 3~ (1981), D mc m (1981,1983),
Mcri (1986), Dmcm & Dimas (1996), Lm & Rock-
w 11 (1996) h portions the work of D mc m's research
g c m, Spree nts 9 reference point, e pecislly for She
OCR for page 747
case of the towed hydrofoil.
Theoretical studies suggested that the flow just be-
low the breakers is turbulent, and suggested the exis-
tance of a shear layer beneath the breaking wave, see
Peregrine & Svensen (1978), Longuet - Higgins (1994a),
Longuet - Higgins (1994b), Cointe & Tulin (19944.
The numerical description of the breaking phenom-
ena 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 - break-
ing event. In Sussman et al. (1994) an additional vari-
able, 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 (20004.
The purpose of this paper is to report recent devel-
opments 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 - break-
ing rising from the formation of capillary waves on the
forward face of the gravity wave. The problem has been
treated theoretically by Longuet - Higgins (19924. Pre-
liminary observation of the occurrence of these ripples
and of the breakdown of this type of flow are reported in
the following. ~ J
The numerical approach is here used to study the in- <~ X
ception of the breaking produced by a submerged hydro-
foil, with particular reference to the velocity and pres-
sure 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, to-
gether with a Level Set technique, used to follow the
free surface dynamics, has been developed. This ap-
proach can lead to free surface instabilities in regions
where the grid is highly skewed, unless an high grid re-
finement is used (Iafrati et al. 20004. This in turn implies
that some difficulties may be encountered when study-
ing 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 de-
scription 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 ap-
proach 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 com-
posite material, whose chord and span are respectively
0.4 m and 2 m. The hydrofoil is connected to the car-
riage by two vertical, surface piercing, side struts. Vari-
ation 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 vertical structures leaving the rear part of the hydro-
foil. 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 hydro-
foil has been used to increase the amount of light (Fig. l).
LIGHT
BEHEST
N~ .
; ~ ~— At;
i · ~ ~N
~ N.
i . ~ NO
'. 'I
\ F.
· 1.
. ~-~
x~ EN ~ PLATE
.......................................... N.
..;......
; ,MREQR \W,1~G
Figure 1: Sketch of the experimental apparatus (side
view).
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 devel-
opment.
OCR for page 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 for-
ward 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. Fig-
ure 3 shows the presence of ripples and a wave breaking
on the third crest, while the second is just partially in-
terested. 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 4: Fully developed breaking for the depth 0.12 m
with residualfollowing waves ..
A close - up view of the waves crests) (Fig 5. - top)
shows the appearance of ripples, in particular on the for-
ward 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 suc-
cessive propagation leads to a tranverse instability of the
wave front (Fig 5. - bottom), finally breaking in a three-
dimensional way.
Figure 5: Ripples appearance at depth 0.18 m (top) and
transversal instability (bottom) immediately before the
breaking at the rear crests.
This kind of scenario for breaking waves has been al-
ready experimentally observed by Duncan et al. (1994)
OCR for page 749
OCR for page 751
OCR for page 753
OCR for page 754
OCR for page 755
OCR for page 756
OCR for page 757
OCR for page 758
OCR for page 759
OCR for page 760
OCR for page 761
Representative terms from entire chapter:
capillary waves
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 rel-
atively low Reynolds number and high angle of attack
(Fig. 64. The effects of this separation will be discussed
Figure 6: Vortex shedding from the upper side of the
hydrofoil.
NUMERICAL MODELING
Navier-Stokes solver for the two-phase How
The two-phase flow is modeled as the flow of a sin-
gle fluid whose density and viscosity smoothly changes
across the interface. By assuming both phases to be in-
compressible, in an Eulerian frame of reference the local
fluid properties changes with time only due to the inter-
face motion.
If surface tension and turbulence effects are neglected,
the unsteady non - dimensional Navier - Stokes equa-
tions for an incompressible fluid in generalized coordi-
nates are:
(3(m
(1)
It (I ruin + ~3~ (Um~i) = Q TV (J {3x P)
_l it 1 63 / Gmrl 63'ui i\
Fr2 ReQ0cm k~ i
(2)
where ni is the ~—th Cartesian velocity component and
did is the Kronecker delta. The quantity
U J—~ t36.m <3y
~Xj
is the volume flux normal to the (m iso-surface and J-i
is the inverse of the Jacobian. In Eq. (2) the gravity term
is written in non - dimensional form, being
Fr= v7:
the Froude number and Ur and Lr reference values for
velocity and length, respectively. In the diffusive term
Re = (UrLrQw)/pw
is the reference Reynolds number being Qw, ,uw the val-
ues of density and dynamic viscosity in water that are
also used as reference values. The quantity
Gmrl J-i (3(m (pro <
dxj dx
is the mesh skewness tensor.
The numerical solution of the Navier-Stokes equa-
tions is achieved through a finite difference solver on a
non staggered grid similar to that suggested by Zang et
al. (19941. 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
J-i
always maller th m 0.8 She use of m explicit scheme
for the off-diagom~l diffusive operator m also limit She
maxim m allowable time tep du to She viscous stail-
ity limit when highly skew d g ids me used m legions
where viscous effects me domi mt On th other h Ed
m implicit a o mt of the off-diagom~l diffusive operator
is too expensive from She computational point of view
Ed, how ver, She use of highly skewed g id m viscous
dominated regions should be a voided
Equation (5) is solved th ough ~ fictional step ap-
proach: a auxiliary velocity field Ilk is inh odu ed Ed
She probl m is solved in two teps in ~ -i st step the a -
iliary velocity field is fo Ed by neglecting She p~essme
term from the righthmd side:
( fit 2 TO ever) ~ l;—l4) =
(1+ At ) f + ~ DF,lt)]
_ t ICE-' + ~ DF(l,t-')~
+ I D 'fib>+ .6 + TO
By such acting the a ove expression fi om Es,
mains:
(At -2~eD~)qtb~-tt:)=e~,~i
She auxili By velocity field is fo Ed by solving the pre-
dictor step Eq 6) 6 ough m approximate factorization
of She operator of She discreti:osdmoment m equation
She pressu e field at She new time step is fo Ed by
assummg that the velocity field Ub ~ ~ is rented to ilk, by
She relation:
I (5) it le-
. ') (6
Lk" -l; = eT_, .( id")
(7
where up is the pie.. se conector term By inhodu bra
Eq (7) into Eq (6 the following relation betw en She
pressme field Ed She p~essme conector holds:
n,(ph'') = e (.T~'
- 2T~Dr) ( eJ-~ )) (8)
Once the scaly fu tion up is computed, the above ~eh-
tion could be used to calcite the pressu e field H w-
ever, what working in generalized coordinates its so-
luti m is not shaightforward Ed Instead the following
approximationisused~ose feldetol 1991):
j11 =¢bl t +O,~t )
7 his does not affect the acmay of She m merical scheme
since She pressme itself is never used in th calculation
She pressu e collector up is computed by e forcing
She continuity by Eq (1) in fat, Eq (7) c m be written
as:
,<~_l<+~t(_a ,aqJ;~\
~ -' e: al a,J
md,byEq (3),itfoll _.:
(9)
lb~'=z~-~t (G~"'0'' ) (lo)
Using this expression in the continuity equation, ~ Pois-
son probl m for the pressure con ector is obtained:
a (Guta'~- I a a, 1
a . ~ e 3., J - ~` a<;,, ( )
Whet, She velocity is k own at the boundaries, Eq (10)
provides ~ Neum umboumdary condition for the solution
of Eq (11) The solution of fi is Poisson equation is
pe fommed eifitm by ~ BiCGSIAB ( m der Vor t 1992)
algorithm vifih m ILU preconditioner or by ~ multig id
tech iqm
This Utter has beenfoumd Ether effective, even though
difficulties have been encountered when dealing with
g ids having ~ ve y large ~ pect ratio
Free surface motion via the Level-Set to boi mm
The m meri 91 model described in the previous sec-
tion is used to solve the Navier-8tokes equstiorr m
domain that encloses both air Ed water while the actual
location of She inrerthc must be captured in some way
Although fluid density Ed viscosity are assumed to take
-i ed valet s for each fluid, Hey vary m time du to She
mte Lace motion H _ m m, when using the cone pond-
mg t mspo t equsti ms difficulties may arise du to She
sharp variation of She fluid properties at She interface
h the level-set techmiqu fi is problem is avoided by
assummg fluid properties as fraction of ~ sig ed nor-
mal d. tnce fiom the interface d(x,t) At t = 0 this
faction is mitiali:osdassummgrl > 0 m water, al < Gin
ai Ed al = 0 at the interface (8ussm m et at 1994) The
generic fluid property J is assumed to be J'dl = ,,;, ff
al > {, J(d1 = Jo if al < - 6 Ed
1(~ = (,1~ + 1~/2 + (gin—1~/2 ~i~l(sT~/(254)
of herwise In She clove expression ~ is She he f width
of c transition region inh odu ed to mooch the jump m
She fluid properties The chick ess ~ is chosen so that
She jump cm ~ s et lee t fou cells ~ this way She width
of She jump region, that will be kept constant in time,
decreases when rode ing the cell size Urrserdi & T yg-
g ason 1992)
Durmg the evolution the dist ace is assumed to be
convected by She flow, Thus She equation
t3t+u.v4=o
(12)
is integ Ted to update the di traction of She distance
function At the end of She convective step, since She
ate face is c materiel su face, the fiee su ftce location
is ccptmed es the level rl = 0
In fat, She integ ction of equation (12) does not en-
sme that She chick ess of the jump region is kept con-
st at in pace ad time To avoid She spre thing or con-
cenbation of She t msition zone, She di tance f motion
is re-miri tli7rd et each time tep es the normal distance
from She actual ate face
The problem of the ~einiticlization of She dist ace is
w 11 dis ussed in Sus m m et al (1994) ad Adalsteins-
son & Sethi m (1999) Usually the dist ace fu tion is
reinitialized by iterctmg to steady state the equation:
~ =.S(,~(l—IVAN
where .S(dl is c sig fu tion that is limo on the inter-
fae The main cdvmtag of this cpproah is gnat She
actual ate face location does not need to be computed
et each time step As c d cwbak, She solution of She
al i. e equation needs suitable m merical scheme to pre-
vent o sc il k t ions
How ax, for two-dimensiomd applications, She com-
putatiomd effort needed to locate the fiee su face ad to
recompute She di tance fu tion is not critical For this
reason he interface is reconstructed et each time tep by
explicitly locatmg She position of She level d = 0 ad She
function rl(xjt1 is remiticli:osd by computing, et each
cell center, She sig ed normal distance from She inter-
fae This procedu e is fouled effective in terms of mass
conservation ad in facing complex flows as it is dis-
cussed in lafrcti et at (2000) where several kind of flee
su face flows has e been armiy:osd to validate the proce-
dme
~ order to damp distmba es outgoingfrom She com-
putatiomd domain, c m mericcl beach model is mtro-
du ed m Eq (12) Two beach legions are mtrodu ed
close to She two boundaries of She computatiorul do-
mcin f y = 0 is She still water level, m the beach
legions Eq (12) takes the following form:
tat = u Vr/—v(d + y) (13)
where the coefficient v is :osro et the imp r limits of She
beaches ad g ws quad optically toward the boundaries
of the c mputatiomd domain
Solid boundaries modeled via body forces
~ lafrat i et Ill (2000), it is observedthat in~tib i l n it s
may arise when the mterfa pass f ough legs ms where
She g id is too disto ted unless c r Fly fine g id resolution
(or c Urge valet of b) is employed This is m import at
issu to be solved when th we y flow generated by hy-
d of oil mm rig close to the ate face ht. to be studied
On She other h ad, when attention is mainly focused on
She flee su la flow, m t coo ate description of She flow
clout the body is not sh ictly needed
With the al i. e issu s in mind, She presence of She
solidbodyhcs been modeled th ough cbodyforces cp-
proah, gnat is by mtrodu ing Citable body forces in
g i d ce ll s ms i de the b o dy c ontou The m agnitude of thi s
forces is chosen so that She velocity of the g id pouts m-
side the body contou tend to be equal to She velocity of
She body itself
At t = 0 the flow is assumed to be umifomm with
(it t 1 = (LO) on each g id point of She computtttional
domain Since the frame of reference is cttahed to She
body, for my g id point inside the body co ton She fol-
lowing term is add d to She right h ad side of Eq (2):
.r-'
to being the length of the ramp, is introduced to reduce
the formation of long upstream propagating waves in-
duced by the starting phase. This problem is more ev-
ident 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 approxima-
tion of the body shape. This means that when refining
the grid the solution changes not only due to the accu-
racy 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 re-
sults.
Domain decomposition
For the purposes of the present work, we focus our
attention toward the processes near the free surface. Ac-
cording 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 pre-
dict the flow about the body.
The above consideration suggested to develop a zonal
approach, decomposing the fluid domain in an upper re-
gion, 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 sur-
face (upper) region, the flow is described by the Navier-
Stokes equations. This decomposition allows to concen-
trate 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 74.
~ , ~ ~ , ~ ~ , , ~ ~ , ~ ~ , ~ ~
I T r I T T I I T I I T r I T T I I 1
I ~ ~ I ~ i I ~ ~ I I ~ ~ I I i I I I
-r T ~ I ~ I ~ T kTN I _
I I I I I i I my ~ ~ 'A I ~ ~ ' - ~ ~ i I j ~
I T r I T T I To T I I T r I T T I I 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
r T 1 T r 1 T r T 1 r T 1 T r 1 T r
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
_ _
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 im-
posed 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 sur-
face 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 po-
tential via the unsteady Bernoulli's equation.
In the following, additional details about the poten-
tial 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 out-
let 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 uni-
form distribution of the vorticity density fly, 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 vor-
tex 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
N-S domain body, takes the following form:
Matching surface
Figure 7: The decomposition of the computational do-
main 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 tech-
nique for solving the air-water;J7ow.
Viscous and the inviscid rotational solutions are fully
coupled at each time step, with a simple and effective
D1t Q,I, Fr2 2 + US (~U + ~V)
(16)
where p is the pressure value coming from the Navier-
Stokes solution, ~ is the velocity potential in the abso-
lute frame of reference, nv is the velocity field induced
by the vortex and UP is the velocity of the frame of
reference, that is 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
body contou fl e impermeability con traint is applied:
'tn = (rT~ _u,.) n
where n is fl e umit normcl vector directed mward On
fl e inlet md outlet boumdaries c umiform ff oming md
outgoing fl w is cssi8 ed:
t~t = (_ETP—{A] ~ . n
he solution of the fl w in the potenticl region is
obtaff dbyusingcT roorderpcnelmeflodforfleso-
luti m of the Laplae equation for the velocity potential
he solution of fl e boumd uy valu problem provides fl e
velocity potentiaI m the solid contou s md its T rmal
derivative on the matching su fa his htter is fl en
used, clong with fl e velocity potentia I itself, for the ccl-
coktion of fl e velocity fleld on the matching Iff
VALIDATION
R sults PT sented in fl is section me rektive to fl e
cssessment of some of the charateristics of the mefl -
ods described before he va/idation study hcs been
pe formed clso to show the cpplicability of fl e decom-
position cpproah he study is based on fl e wz y flow
ger mted by c movmgbottom topog cphy md the flow
mdu edby chyd of oil movmgbenethfl e fT e su fa,
bofl in non - breaking md in breckmg c mdition Re-
sultsarecomparedwiththoseobbir dwifl cfullynon-
lff ar boumdary el ments solver md, in fl e case of fl e
hyd of oil, wifl fl e experimenb I date obtamed by Dum-
c m (1983)
Case study: wxry flow ihdueed by a movihg bottom
topogTaphy
he wz y flow generctedby cbottom bump movmg
m c chumel is studied by usmg fl e full Navier-Stokes
solver NS), fl e domain decomposition cpproah DD)
md fl e fully non-lff ar boumd uy elements solwx BE~
he geomeby of thebump, located m x ~ ( - 0.5, 0.S).
is given by fl e foll wmg equation:
:~(1~= - 1+0.1{1 - 4 +4 )
he bump is pla d on c flct bottom at g = - 1 while fl e
still water level is et g = 0 he compubtior~I domcm
extends fi om ~r = - 14 to ~r = 14 in fl e hori ontal di-
rection md fr m the bottom proflle up to y = 0.4 in fl e
ve ticcl direction Numericcl beah models have been
cpplied in the Upstr cm md down TT cm ft e su fa T -
gions, x ~ ( - 14, - 8) md ~r fl (8,14) md fl e maximum
valu u = 2 is cssumed for the dampmg in Eq (13)
h order to perform c fair comparison wifl BEAd T -
sults, c slip b mdary condition is applied on fl e bot-
tom proflle when usmg fl e N8 cpproah 'when us-
mg fl e DD cpproach, the matchmg su fae is located et
y = - 0.2 he dependance of the m merical soluti m on
fl e location of fl e matchmg su fa es ben empiricclly
wxifled, by movmg fl e su fae from very deep up to
1.5 times the depth of the fl st trough he solution hcs
proved to be subst mticlly mdependent fi om fl e location
of the matching
At t = 0 fl e bump is suddenly stsrted et I T~ =
( - 1,0) mdFr = l7/~ = 0.707, Lbeingflehori
zonbl lenght of the bump R mlts obbir d with fle
fl ee dffferent cpproahes et two dffferent time valu s
me sh wn in the flgu es below
01~
10
X
01
^0O ).
x
10
10
Figu e 8: FTT s rfoce pT files gene?ot d by th sudd m
st rtof o bottom bump inochonnel ott = 20 (top)ond
t = 130 ~ottom): FN8 (solid line), DD (dashed Ime),
BEAd (dash-dotted Iff` )
Figu e 8 shows that fl e th ee solutions me in c wxy
good C8 ement et t = 20, i e before wave distmbs
T ah the d wn tream damping mT, while slight dff-
ferer es occur kter in particohr et t = 130, while fl e
BEAd md FN8 solution are still ve y close eah ofl er,
fl e m merica I DD solution is charaterised by m e' es-
sive m mericcl dampmg Among others, two fators c m
be responsible for fl is damping: fl e use of c fl st order
explicit scheme to inte8 cte in time Eq (16) md the use
of Eq (9) rather th m solving Eq (8) to evaluate fl e
Pr ssme fleld in fat, both these fators sugge t that c
small time step has to be used to achieve a good accu-
racy. However, wave phase and wave lenght are quite
well catched from the DD approach.
..
Figure 9: Velocityfield andiree surface profiles gener-
ated 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~pa/pw = 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 im-
pact 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 even-
tually (not shown) its forward face impacts again on the
free surface, leading to another air entrainment and an-
other 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 be-
neath 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 experi-
mental 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 compu-
tational 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.
fIere, for all the grids the values /\y = 0.005, ~ = 0.02
are used for the vertical grid spacing and for the half
width of the jump region (see Fig 10~. A value to = 8
has been assumed for the length of the ramp when using
Eq. ( 154. In Fig. 1 1 one every fourth grid point is shown
for the coarse grid ~ 282 x 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 x
259) and fine grid ~ 672 x 371) are obtained by halving
cells in the body region only. For the fine grid fix =
0.01 and /\y = 0.002 are used in the body region.
1
0.5
n
ens
-1 .R
-2.5
_,9
j jig 3
~ W
~
..
. TO
1
~~.~ ~..~
R9 ma
_
= ~
,
_ ~
it' W~
-15 -10 -5 0 5 10 15 20
Figure 10: Numerical gridfor the FNS solution. One
over fourth grid point is shown of the coarse grid. The
body andfree surface region are clearly recognizable
n
n
>`
1
1
n
1 1
1
1.5
Figure 1 1: Close up view of the grid in the body region,
used in the FNS approach: for clarity, one everyfourth
grid point is shown for the coarse grid
Free surface profiles obtained with the three grids
are shown in Fig. 12 in comparison with the experimen-
tal data obtained by Duncan (19834. The computation
is performed by assuming Qa/Qw = 0.00125, pa/pw =
0.018, and a Reynolds number Re = 10000.
0 1
-0.1
......... .
-2 0
;. —
Figure 12: Comparison among the free surface pro-
f les obtained by the FNS approach and the experimen-
tal data (Duncan 19839: coarse (dash-dotted), medium
(dashed), (ne (solid), Duncan (dot)
\
.
-1 1|
-' 1
-1.1 1
x
~~:~
~::
l ~
Figure 13: Comparison among the u contours about the
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
The comparison put in evidence some features of
the numerical results: (i) good convergence in terms of
wavelength, (ii) poor convergence in terms of wave cm-
plitude, (iii) c phcse shfft simibr for cll the g ids md (iv)
c sp ke in the wa~ profile et x = 6 for the medi m g id
result
Inve tigatmg ctout these points, cttention hcs been
focused m close up vi ws of th velocity field m fhe
lecding edge ~egion, obtamed wifh fhe th ee g ids, de-
picted m Fig 13 he velocity field m fhis ~egion shows
relev mt ch mges when fhe g id is ~efimed ~ fact, durmg
fhis process, due to fhe zero order model used for cssig
body forces, mbst mtial dffferences in fhe computatiorur
shme of th solid bo mdary occurs Neverfheless, cl-
fhough fhe doove limit of the body force cpprocch c m
justffy the poor convergence, th phcse shfft is ckmo t
fhe same for cll the g ids
As c cmefull look et the vorticity shedded from fhe
hailingedgeofthehydofoiHeveals Fig 14),theinter-
actions betw en fhe vo ticity field beneadh the free sur-
face md the wave profile (wifh some dipole rising up
toward the free smface) me ~espons~ble for the sp ke m
fhe wave elevation et x = 6 he fiow separction oc-
curing from the sucti m side, c m sigmfic mtly clter fhe
wave cmplitude md phase
·e qik ~ i~ 3 o-
_s ~
~ ~ o ~3 O~O o
~F_¢ J' _. r
Figure 14: VorEcih con tourr behind the hydnofoil f om
thetopt thebottomt= 13,t= 20,t=22
On fhe basis of fhe ctove considemtion, c fai com-
parison with experiments performed et high Rey olds
n mber carmot be e tablished mless c tmbulence mod-
eling, beside to m improved body fmce formoktion, is
mtroduced
DD nerult
he problems enco mte~ed m the conect p~ediction
of phcse md wave cmplit de, lecd to fhe development
of fhe domcm decomposition mprocch fnat uses fhe m-
viscid rotatiorurl fi w model md the bo mdary element
techmiqme to describe the fiow ctout fhe lifimg body
wheres uses th Navier-Stokes solver coupled wifh fhe
Level-Set techmique to descobe the fiow in the fiee sur-
f a ce ~ egion, where c omp lex mte f a ce t op o logie s m cy de -
velop m breaking condition
N mericcl simoktion have ben carried out by as-
s mingthematchi gsurfacecty= - 0.2 IndheNavier-
Stokes ~egion c 2S6 x 96 g id is employed with g id
pomts s itdoly cluste~ed ctout x = I m the horizontal
di ection md ctout y = 0 m fhe verticcl di ection in
fhis case c mffomm vertical g id spacmg Xy = 0.005 is
usedmy ~ ( - 0.2.0.2),whe~ecs5=0.03isusedasfhe
hcff wid h of fhe j mp
~ Fig 15, the free smface profile obtained by fhe
DD mprocch is compared versus fhe fully non Imear
BE5d re mlt md wifh the experimental data by D mc m
(1983) fcr th same conditions es before
01~
2 o 2 4 d 5 10
x
Figme 15: Compuzuon mong the fnee su~face p?C
fihs obt ined by the DD oppm~h (solid), the full BEM
(dashed) ond the e pezim mt I doto by Dun~n (1983)
(dot)
With respect fhe FNS results, the DD cllows c much
better description of fhe fi st trough, ew~n though m :x-
cessive dampmg of the following waves cppears, md
wave phcse md lenght are m good agreement with :x-
periments too As alrecdy stated, recsons for this :x-
cessive damping are not yet reclly mder tood, cldhough
it is believed to be rehted to the explicit mteg ction of
fhe Bernoulli's equation md to the cpproximati m of fhe
p~essme field given by Eq (9) H wever, it hcs been
wxified that, es long es the matchmg surface is deep
enough with respect to the wave houghs, its position
does not sig ifi mtly affects fhe sohtion
Submerged hydrofoil: breaking regime
In spite of the limits of the numerical solver, simu-
lations of the breaking wave produced by an hydrofoil
have been attempted, aimed at the verification of the ca-
pability of the solver to predict and model the breaking
wave phenomena.
For a non - dimensional submergence of 0.9113, Dun-
can (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. How-
ever, 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 es-
tablishment of the breaker. As a consequence, the dis-
sipative effects played by the breaker on the following
wave are not modeled.
0.1
0.1 ~
0~;
-0.1
of ~ ~ 9; , /\ ' ., '
-0.1 ~ ~
.
................................. .
-2 0 2 4 6 8 10
x
.. , ... ~
·- / ·-.\
·- , ·
, ·
, .
~ ..
:- /
·
i-'
.~:~
x
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 break-
ing 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 - dimen-
sional submergence 0.783, Duncan (1983) observed an
intense wave breaking with a high dissipative effect on
the following wave. The comparison between the nu-
merical solution and the experimental data for this sub-
mergence 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 over-
predicted 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.
n ~ ~
.
~ 0 _
-0.1
~ h I'd: '\
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-2 0 2 4 6 8 10
x
Figure 17: Wave profile behind a hydrofoil at a non - di-
mensional submergence 0.783. An intense breaking was
observed by Duncan (1983) (dot) whereas the poorres-
olution does not allow the numerical approach (solid)
to capture neither the breaker nor the dissipative effects
on the following wave.
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 informa-
tion 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 break-
~ng case.
It is wo th to notice that, de pite the poor resolution,
She dish ibution of the dynamic pressme ch loges sig if-
ic Fitly passing fi om the non - breckmg to the breckmg
stage U fortunately, the poor resolution does not allow
file water to plush down mdtoformfhebreakemegion
CONCLUDING REMARKS
N merical Ed experimental tudies have been car-
ried out et NS AN on the 2D use trQtmg -l w pro-
duced by c tow d hyd of oil N merical simulations of
She two phase flow m air Ed water have l en performed,
Ed validation tudies have l en conducted in the case
of She flow on c bottom b mp, producmg c pi mgmg
breaker he fommation of the jet, the pltsh - up Ed
She post treat mg event me m c qualitative cg cement
with experimental observation Hence, the -I w pest She
submerged hyd of oil ht. been simulated Ed the condi-
tion for She onset of the t ret mg have been inve tigated
his ht. resulted in c detailed arch sis of She computed
velocity Ed p~essme ft id.
A Navier - Stokes solver m generclied coordinates,
together with c L vel Set tech iq~x, used to follow She
flee surface dynamics, ht. I en used Ed two differ-
ent m mericcl codes have been developed, based on She
body force Ed on c domain decomposition approach s
respectively
he bodyfmces approach has been developed in vi w
of delmg with She flow cutout mulUbody co flgmations
es it is She case of c ship with appendages his approach
ht. proved to be useful, although c higher order model
for the cssig ement of the body forces, Ed hence of She
way in which the shape of the body is represented, is
needed in order to gam msight mto the dynamics of She
flee surface, She domain decomposition cpproah hits
proved to be promising, focusing attention Ed compu-
tatiomd efforts in the flee surface region With reference
to She quasi - tecdyb~eckingproducedbythehyd of oil,
m mericcl re mlts discussed h re suggest Nat, Although
She m merical techniques are Cole to detect the incep-
tion of the breckmg, the adopted g id w re too coarse to
resolve She -I w Possible extension of She work is She
inclusion of surface tension effects, allowing c compari-
son with the set of experimental data
he emphasis of She experimental work is on m-
derst Ming the conditions alder which ccpilk y waves
may lorce She breaking on the folowmg wave h cm, sub-
sequently forcing She extension of the t ret mg area to
She forward waves Th work is k gerly alder develop-
ment Ed c new >! tem ht. Greedy been desig ed for
reproducmg She experiments, mckmg also q mtitative
mecsmements
Acknowledgements
This work was supported by th Minut m Wasp oh
e Novigazione in She frame of She NSEAN research pi m
2000-02
REFERENCE S
, Proceedings of the ONR '000 i ee surface turbo
lance and bubbyf It workshop, Cclifomiclostit te of
Techmology, Parader USA), 2000
Adal temsson D Ed Semi m J. A, The fast consh it non
of extension velocities in level set methods, J. Comput
P. a, vol. 148,2-22,1999
A ueb, R. Muzaferijc, S. Peric, M md Yoo, S D,
Computation of flows aro Ed hyd of oils mder the he
s mf a ce, Pmceed ings of He 7th In t Cold on Num h ip
Hydm, ed OffceofNa~lResearch,Nmtes, RANCE,
1999
Bcttjes J. A, Sakai T. Velocity ft Id in c steady breaker,
J. FluidMech ,vol 111, 421-437,1981
Bommarin P. Geomeh ic properties of deep - water h eck-
mg waves J. Fluid Tech, vol. 209, 405-433,1989
Cointe R. Tulin M, A Theo y of steady treaters J.
Fluid Tech, vol. 276,1-20, 1994
Dommermuth D, Immis G. Luth T. Novikov E, Schc-
kgeter E md Talc tt J. 'N mericcl simulation of bow
waves," pro, e dings of 22nd ymp slum on Now I Hy
art dy mics, O floe of Ntsul Research, 1998
Dommermuth D, Mni R. The vo tical tructure of a
wave breaking g avity-capillarity wave, Pn endings of
206h Symp sit m on Now I Hydmdyn m. cs, cd O floe of
Ntsul Ret arch, Washington D C ,1994
D ng RR, Katz J mdH mg I I, i: n Ih ant t ne at
bow waves on a ship model, J. Fhid Tech, vol. 346,
77-115,1997
D mc m, J. H. A experimental investigation of br tk-
mg waves produced by a tow d hyd of oil, Pn c R Soc
LO.rG, vol. A 377, 331-34d, 1981
D mc m, J. H. Th Ine thing md non - bre tkmg wave ~e-
sistmce of a two-dimensiomd hyd of oil, J. Fluid Tech,
vol. 126, 507-520,1983
D mc m J. H. Dimas A A, Smfae ripples due to steady
l he thing waves, J. Fhid Fir ech, vol. 329,309-339,1996
Duncan JH, Philomin V md Qiao H. The transition
to turbulence m a pilling br tker, Pn tcdings of 6hc
206h Symp sit m on Now I Hyde dy mics, cd O he
of Nils al R search, Washington, D C, 1994
L frati A, Camp Ins E F. A level-set tech iq~x applied
to complex he Oaf tee -I _., Pn cc dings of ASME
FEDSU 00, Boston Afar, USA 2000
L frati A, Di Mascio A md Compare E F. A level-set
tech iq~ Plied to msteady I ree turfae -I _ ., Int J.
fw Num Meth in Fhid, in press, 2000
Lcfaurie, B. Nardone, C, Scardovelli, R. Zcleski, S
md Zaoetti, G. Modellmg mergi g md fragmentation
m multiphase fl ws with SUR ER J. Comp t Phys,
vol. 113,134-147,1994
Long ett-Higgins M S. Ccpilbry rollers md bmes, J.
Fluid Mech, vol. 240, 659-679, 1992
Long ett-Higgins M S. The initiction of spillmg breck-
ers, lnt Sym Woves Physicol o d Numenicol Model
ing, U. of British Col mbic, Vancouver, Cam~dc, 1994
Long ett-Higgins M S. Shear instabilities in spillmg brek-
ers,Pnoc R Soc L~nd ,vol A 446, 399-409,1994
LinJ-C,Rockw IlD,Evolutionofqua y-stecdyheck-
mg w~ve, J. Fhid Mech, vol. 302,29-44, 1996
Miyatc H. ~ui T. Nonlinear ship waves, Ad~ Appl
Mech, vol. 24,215-288,1984
Mori, K -h, Subb~ecki g w~ves md criticcl condition
for 6heir cppearance, J. Soc Nov Aeh Jopan, vol. 159,
1-8,1986
M~ferija, S. Peric, M, Scmes, P. md Schellm, T. A
two-fluid Navier-Stokes solver to simokte water~nt y,
Pnoceedings of the 22nd Symp slum on Novol Hydno
dyn mics, cd Of flce of Na~l Research, Wcshmgton,
DC,1998
Pe~eg me D H. Svendson, I A, Spillmg brekers, bores
md hyd mlic j mps, Pnoceed ings of 16th Coost I En
gng Conf Hcmburg, Germ my, 1978
Rose feld M, Kwak D md Vmokur M, A fiactiorul
step solution method for 6he mstecdy incompressible
Navier-Stokes equations in gene~alized coordincte sys-
tems, J. Comp t Phys,vol 94,102-137,1991
Sus m m M, Smerekc P. md Osher S. A level-set cp-
proah for computi g solutions to mcompressible two-
phcseflow, J. Comp t Phys,vol 114,146-159,1994
Unver di S O. md Tryg es on G. A front-tracking medho d
for viscous, incompressible multi-fluid flows, J. Com
p t Phys,vol 100, 25-37,1992
m der Vorst, H. A, Bi-CGSTAB: A fast md smoothly
converging vari mt to Bi-CG for the solution of n mlim
ear sy tem, SlAM JSci Stotut Comp t, vol. 13, 631-
644,1992
Vogt, M md Larsson, L, L vel set medhods for predict-
mg viscous fre-smface fl ws, P~ceedings of 6he 7th
Int Conf on Num Ship Hydno, cd Of flee of Na~l
R search, N mtes, FRANCE, 1999
Z mg, Y. Street, R. L md Koseff, J. R. A non- taggered
g id, fractiorurl tep method for time-depff~dent mcom-
press~ble Navier-Stokes equations in curvilmear coordi-
m~tes, J. Comp t Phys, vol. 114,18-33,1994
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.
(a)
......................
(b)
(C)
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. l9a). When the depth of
the hydrofoil is reduced, three dimensional instabilities
appear (Fig. l9b), eventually leading to wave breaking
(Fig. 1 9c). 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 by the
breaker (Fig. 20) at least beneath the first crest.
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.
