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OCR for page 254
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Direct numerical simulation of surface tension dominated
and non-dominated breaking waves
A. lafrati, E.F. Campana
(INSEAN - Italian Ship Mode] Basin, Italy)
ABSTRACT
Surface tension effects onto the two-dimensional wave
breaking flow produced by a hydrofoil moving beneath
the free surface are investigated. The study is carried
out numerically by a finite difference approach which
solves the Navier-Stokes equations. The air-water in-
terface is embedded in the computational domain and it
is captured via a Level-Set technique. A heterogeneous
unsteady domain decomposition approach is used, al-
lowing to focus the computational effort onto the free
surface vicinities, while the flow about the body is ap-
proximated by a potential flow model.
Surface tension effects are investigated by progres-
sively reducing the length scale, while keeping Froude
and Reynolds number constant. Different flow regimes
are recovered, ranging from intense plunging jet, even-
tually resulting in large amount of entrapped air, up to
a micro-breaker, in which case air entrapment is sup-
pressed and the jet is replaced by a bulge growing on
the wave crest. At this scale, the surface tension is re-
sponsible for the large curvature at the toe of the bulge,
and when the bulge slides upon the forward face of the
wave, an intense shear layer develops from the toe. In-
stabilities of this shear layer are observed, and, when
increasing the Reynolds number, the shear layer breaks-
up into coherent vortex-structures that interact with the
free surface, eventually leading to the formation of large
surface fluctuations which propagate downstream.
INTRODUCTION
In the present paper, attention is drawn to two as-
pects of the wave breaking process: the role of the sur-
face tension on breakers of progressively smaller length
scale and the effects of viscosity on short waves produc-
ing micro-breakers.
Among the large body of works on breaking waves,
many papers have been studying surface tension effects
theoretically, experimentally or numerically. Tulin (1996)
presented a numerical investigation on the effects of sur-
face tension on the jet development. In the study, car-
ried out with the aid of a potential flow solver, waves
of different length are followed along their evolution to-
ward the breaking. The analyzed length scales range
from ~ = x (corresponding to the case of negligible
surface tension) down to ~ = 25 cm. As the wave-
length is decreased, surface tension forces are relatively
increased and modifications occur to the jet: initially its
tip is rounded and finally the jet is suppressed and re-
placed by bulge on the crest of the wave.
The growing of the bulge was also shown through
some calculations by Longuet-Higgins (19961. Based
on potential flow assumptions (with and without surface
tension) he also showed the appearance of a train of par-
asitic capillary waves upstream the leading edge of the
bulge (referred here as the toe). Within the same po-
tential flow assumptions, surface tension effects on the
bulge-capillary system have been numerically addressed
also by Ceniceros & Hou (1999), where results for a
wide range of surface tension coefficients are reported.
Furthermore, by using a modified boundary integral for-
mulation for water waves to include weak viscous ef-
fects, they show that, depending on the viscous coef-
ficient, the train of these capillary waves may be sup-
pressed.
That these capillaries might be a source of vortic-
ity shedding was initially predicted by Longuet-Higgins
(1992~. Numerical and experimental evidence of the
vorticity field generated by capillary ripples have been
provided by Mui & Dommermuth (1995) and by Lin &
Rockwell (1995), respectively.
In a successive paper, Longuet-Higgins (1994) sug-
gested that the vorticity shed by these capillaries (named
Type 1) may also explain the unexpected appearance of
longer capillary ripples above the toe (named Type 11),
propagating downstream, experimentally observed by Dun-
can et al. (1994~. In his theory, Longuet-Higgins as-
sumed that these Type II downstream ripples might be
primarily caused by instabilities of the shear flow in-
duced by the vorticity shed by the parasitic capillaries.
OCR for page 255
Duncan & Dimas (1996), performing linear stability anal-
ysis of a theoretical velocity profile established that the
downstream ripples are primarily generated at the breaker.
In two more recent papers, Duncan et al. (1999) and
Qiao & Duncan (2001), by performing an experimental
study on gentle spilling breakers induced by the wave fo-
cusing technique, observed that when the bulge becomes
fairly steep the toe began to move, sliding down on the
forward face of the wave. In these papers, it is also ar-
gued that the downstream ripples are very likely induced
by an instability of the shear layer generated between the
downslope flow of the bulge and the underlying upslope
incoming flow.
In the present paper, two-dimensional breaking waves,
produced by the motion of a submerged hydrofoil, are
numerically investigated by solving the unsteady incom-
pressible Navier-Stokes equations. The flow close to
the free surface is described as a two-phase flow and
the Level-Set method is used capture the interface be-
tween air and water. The computational effort is re-
duced by taking advantage of a heterogeneous domain
decomposition approach previously developed (Iafrati et
al. 2000, Iafrati & Campana, 2002~.
A set of Froude-scaled numerical simulations is pro-
duced by varying the speed and the chord of the hydro-
foil, so that breakers of different length scales are ob-
tained. With this mechanism, the relative importance
of viscosity, gravity and surface tension is altered and a
substantial range of variation of some of the character-
istics phenomena of unsteady breakers is modeled and
studied. As to the effects of the surface tension on break-
ers of progressively reduced length scale, starting from
a scale at which the flow is strongly dominated by the
gravity and the perturbed free surface evolve in a plung-
ing breaker, different breaking regimes are recovered,
up to a very short breaking wave. With this set of data,
the modifying action of the surface tension on the jet
formation and on the entrapment of air is discussed and
differences in the vorticity generation mechanisms are
shown.
Finally, for the smallest simulated scale, effects of
the viscosity are also investigated. At this scale, sur-
face tension is highly dominating and it is responsible
for the large curvature at the toe of the bulge. Numerical
results show the development of an intense shear layer
from the toe. At the lowest Reynolds number, weak in-
stabilities of the shear layer are observed, responsible
for extremely small downstream ripples on the free sur-
face. However, being the shear layer still quite stable, it
does not break up into separate vortices. When increas-
ing the Reynolds number, the instabilities grow, and the
shear layer break up into coherent vortex-structures. The
strong interaction of these coherent structures with the
free surface leads to the formation of large free surface
ripples, whose amplitude and wavelength is governed
by the vortex-blobs. Details of the evolution of the co-
herent structures also shown that the combined action
of the vorticity field and of the surface tension produce
a cusp-shaped wave pattern, and secondary separations
from the cusps are also observed.
NUMERICAL MODEL
Domain Decomposition Technique
Wave breaking is usually responsible for a highly com-
plex flow in the free surface vicinities: the air-water in-
terface assumes a complicated topology, eventually lead-
ing to jets, drops, air entrapment, and an intense vortic-
ity field is generated either by the impact processes or
by the action of strong velocity gradients.
In the attempt of accurately describe complex flows,
several numerical models, able to deal with a substan-
tial two-phase flow, have been developed and are now
at hand (a survey is provided by Scardovelli & Zaleski,
1999~. However, the cost of a detailed computation of
the breaking and post-breaking evolution is still very
high.
On the other hand, in absence of breaking, the wavy
flow generated by bodies moving beneath the free sur-
face is rather accurately predictable by much simpler
potential approaches, in spite of the strong assumptions
made about the flow features in the body vicinities (Iafrati
& Campana, 19981. Hence, when attention is mainly
concerned with the flow details in the free surface re-
gion, the use of a domain decomposition approach, that
make use of the most suitable assumptions according to
the flow region, sounds very attractive.
I I r ~~ ~ I T [ I I I I I I I I I I
I I T=T 1 ¢1~t 1 $~ N-S domain
, . . I I I I , , I , I I I I , I
, , I i j I I I I j I I I I j I I I I
Matching surface
~ I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
_ I ~ I T I r 1 1 i I I 1 I T I I 1 1
~ r(c~ :~
-
BEM domain
Figure 1: Sketch of the body and free surface domains
used for the domain decomposition.
On the basis of the above considerations, an unsteady
heterogeneous domain decomposition approach has been
developed to tackle the wave breaking flow induced by
a submerged hydrofoil moving beneath the free surface
(Iafrati & Campana, 20021. In the free surface region,
a viscous flow model with an interface capturing tech-
nique is adopted, while, in the body region, a potential
OCR for page 256
flow approximation is used. A coupling procedure is
developed, allowing an exchange of information with-
out the need of overlapping between the domains.
In the body region the flow is governed by a Laplace
equation for the velocity potential ~
Van = 0 (1)
which satisfies Neumann boundary conditions at the in-
flow ~QB, at the outflow PRO, at the bottom of the
channel ~QB and all along the body contour Arc (see
Fig. 1) while the velocity potential at the matching line
~ is assigned by the integration in time of the unsteady
Bernoulli's equation:
~3'P = 1 (63Q~ ~ (2)
-
~n
Am:
on
—1 (0Qo ) (3)
= 0 (0QB U SAC) (4)
~ = ~ ( 8~) dt (r) (s)
where
'; = ———gx2— 2
In the above equation Q is the fluid density, 9 is the accel-
eration of gravity, x2 is the vertical coordinate oriented
upwards with x2 = 0 at the still water level, u is the
local fluid velocity and pB iS the pressure acting on the
body domain which, by enforcing the continuity of the
normal stresses at the matching line, is given by
p = p —2,ll In ~ (6)
n being the unit vector normal to the matching line, ,u
the local fluid viscosity and pF iS the pressure field at the
matching line provided by the solution of the Navier-
Stokes equations in the free surface domain. Further-
more, a steady Kutta condition is applied at the trailing
edge of the hydrofoil to properly account for the vortex
shedding. More details concerning the coupling strat-
egy and its validation are reported in Iafrati & Campana
(2002~.
Navier-Stokes solver
The two phase flow of air and water is approximated
as that of a single fluid with density and viscosity vary-
ing smoothly through the interface. By assuming the
fluids to be incompressible, the continuity equation in
generalized coordinates simply reads:
Hum 0 (7)
where
,~ Hi i (8)
Is the volume flux normal to the (m iso-surface and J-i
is the inverse of the Jacobian.
According to the smooth variation of fluid proper-
ties through the interface, surface tension effects are in-
cluded by using a continuum model, as suggested by
Brackbill et al. (1992) and recently employed also by
Sussman & Puckett (2000~. As a result, the momentum
equation, in non dimensional form, is:
—(J suit + ~: (Umui) Q 0(m ( taxi )
7-i In _ ~ ~ (J i0(m0(d))
Q Re 0(m ( 06t ~~ )
where hi is the i—th Cartesian velocity component, d
is the Kronecker delta and
Or , Re = UrLrQw We = Ur>/~
low ' ~
(10)
are the Froude, Reynolds and Weber numbers, respec-
tively. Here, Ur and Lr are reference values for veloc-
ity and length, cr is the surface tension coefficient while
Qw, qw are the values of density and dynamic viscosity
in water and are used as reference values. In (9)
Gem J_~ (J(m 05t Bm~ji = J_~ 0fm 6341
Xj ~Xj ~Xj Phi
(1 1)
are metric quantities, ~ is the local curvature and H(d) is
the Heaviside function with the distance function d be-
ing positive in water and negative in air, so that H(d) =
1 in water and H(d) = 0 in air.
The system of the Navier-Stokes equations is solved
by using a finite difference method on a non staggered
grid. The grid stencil and the numerical approach are
similar to those suggested by Zang et al. (19941: carte-
sian velocities and pressure are defined at the cell centers
whereas volume fluxes are defined at the mid point of
the cell faces. A fractional step approach is employed:
the momentum equation is advanced in time by neglect-
ing pressure terms (Predictor step) whose effects are
successively reintroduced by enforcing the continuity of
the velocity field (Corrector step). The diagonal part
of the dominating diffusive terms, i.e. that originated
from Vu, are computed with a Crank-Nicolson scheme,
whereas all the other terms are computed explicitly. With
respect to Zang et al. (1994), a three-steps Runge-Kutta
scheme (Rai & Moin 1991) is adopted here. The grid be-
ing fixed in time, the discretized form of the momentum
equation at the step n is
OCR for page 257
· Step l
· Step 2
· Step 3
(J-1—o~l/\tDI) fur USA=
Y1 [C(uin) + DE(tlin) + Ti(dn)] +
2~1 [—J-lFr2 +DI(0i )
~1 ^1 Ri(¢l)
Hi—Hi =)lQlJ-1
(J 1 _ or2/\tDI) ~ ~ ~~ t) ~
)2 [C(Uil ) + DE (Gil ) + Ti (d1 )] +
(1 [C(uin) + DE (Olin) + Ti (dn)] +
20~2 [—J 1Fr2 + DI (pi)]
Ui2—Ui2 = )2 ~2( -1 + (1 el~-l
(J-1—cx3 /\tDI) (ut up ~ =
)3 [C(Ui ) + DE (ui2) + Ti (d2)] +
(2 [C(Uil ) + DE (Ui1 ) + Ti (d1 )] +
2(X3 L—J 1Fr2 + DI (up)
un+1 _ ui3 = y3 Hi ~ ~ J_ i) + (2 e2 J- l
The coefficients oli, Hi, h are reported in Rai & Moin
(1991) and in literature cited therein. In the above equa-
tions, for the sake of clarity, a compact notation is used
to represent the convective, diffusive and surface tension
contributions:
C(ui) = —d9( (Umui) ~
( ) e Re i9(m ( t96~ )
DE(ui) = R ~: (/lGml 5fi +~Bmkji'~:j)
m 76 | ,
eWe2 Am (J Taxi H(d)) ,
while
0(m ( Act ) (12)
is the gradient operator in generalized coordinates.
In the corrector steps, ~ is the pressure corrector
term which is found by enforcing the continuity of the
velocity field at the end of the substep, (Rai & Moin,
1991, Kim & Moin, 1985~. The procedure is as follows:
once the intermediate velocity field is found, say hi, the
fluxes associated to this velocity field (U~ ~ are com-
puted by (8) at the mid point of the cell faces through
a second order upwind scheme (QUICK). In terms of
fluxes, the corrector step can be written as
Um—Um 'Yt ( pt 0:j )
( Gm~ i90 - 1 )
(13)
so that, by applying the continuity (7) to Ut, the fol-
lowing Poisson equation for the pressure corrector is ob-
tained:
~ ~ Gmi d9¢~l N\ 1 BUT
= _
{3(m ~ Q (j ~ /\t 0(m
(-1 ~ {Gmi 04~-~N
)~ Gym ~ 0~~1 Aim J (14)
When the velocity field is assigned throughout the bound-
ary of the computational domain, (13) provides Neu-
mann boundary conditions for the solution of the Pois-
son equation (14~.
The pressure corrector term is related to the pressure
field by the following equation:
Ri spy ) = (A J 1 _ cot i\tD~) ( pt ,[_1 )
, (15)
but, since the solution of this equation is not straightfor-
ward, usually, an approximate pressure field is obtained
as (Rosenfeld et al. 1991~:
Ripply ~ Ri(~) =' pi = 0' + O(~\t) . (16)
The system of equation discussed above is spatially
discretized by a central finite difference approach, sec-
ond order accurate. As already stated, a second order up-
wind scheme (QUICK) is used to evaluate volume fluxes
at the cell faces. At each substep of the Runge-Kutta
scheme, the momentum equation is solved by using an
approximate factorization approach of the diffusive part,
as suggested by Kim & Moin (1985~. The time step
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Representative terms from entire chapter:
surface tension
is chosen so that the Courant number is always smaller
than ~ and the stability constraint required by surface
tension is satisfied (Brackbill et al. 1992)
1
At / (i ,,, 0~) /\x3
Since not all the viscous contributions are treated implic-
itly, further limitations to the time step can be required.
A multigrid technique is adopted for the solution
of the Poisson equation for the pressure corrector term,
which is the most expensive part of the computational
procedure. A corrector scheme is used for restriction
and prolongation (Brands 1992) and a LSOR method
is employed as high frequency smoother. Since met-
ric quantities and fluid properties appear into the coef-
ficients of the Poisson equation, a simple average of the
metric and of the distance function is used in the restric-
tion phase.
Free surface capturing via Level-Set technique
The air-water interface is captured as the zero level-set
of a signed normal distance from the interface dart)
which, at t = 0, is initialized by assuming d > 0 in
water, d < 0 in air and d = 0 at the interface (Sussman
et al. 1994~. The fluid property f is assumed to vary
from the air to the water values as follows:
~ fa
f (d) = ~ i~ + f 6~ + Em Id sin ( 2~5 )
~ fw
if idl < ~
if dad;
(17)
Here ~ is the half width of the transition region needed
to to evaluate derivatives of fluid properties into the gov-
erning equations (Iafrati et al., 2001~. Accordingly, the
same smoothing is applied to the Heaviside function,
thus obtaining (Sussman & Puckett, 2000)
~ ~ 2 + 2 sin (25) if Ids < ~ (18)
with Had) = 0 if d 5.
During the motion, the distance is transported by the
flow, that is the equation
Mu Vd=0
(19)
is integrated in time to update the distribution of the
function d and then to follow the interface as the level-
set d = 0.
The integration is carried out with the three-steps
Runge-Kutta by using the same discretization scheme
employed for the convective terms, that is
do = dt-i + ~~\tC(
| case | U(cm/s) | L(cm) | Re | Reedy | We | eventually suppressing, the jet formation, which is fi-
A 79.40 20 1000 158720 42 natty replaced by a bulge (Tulin, 1996). Consequently,
B 39.70 5 1000 19840 21 the entrapment of air into the water is also reduced.
C 19.85 1.25 1000 2480 10.5
D 19.85 1.25 2480 2480 10.5
Table 1: Parameters used for numerical simulations. Re
and Reeve denote the Reynolds numbers of the compu-
tations and of the experimental value at corresponding
scale, respectively.
on the wave breaking dynamics have been also inves-
tigated at the smallest length scale (case C), by repeat-
ing the simulation with the corresponding experimental
Reynolds value (case D). A complete description of the
set of parameters used for the computations is reported
in Table 1.
Concerning the set of data employed in the computa-
tions, the fluid domain extends from x = - 15 to x = 15
with the matching surface located at y = - 0.2 and the
top boundary is at y = 0.4. The grid has 768 x 192 cells,
with /\y = 0.0025 and /\x ~ 0.0027 in the breaking
region. The density and viscosity jumps are spread on
a stripe which half thickness is ~ = 0.02. It is worth to
notice that, to reduce the formation of forward propagat-
ing waves generated by an impulsive start, a sinusoidal
ramp is used to accelerate the hydrofoil up to the final
speed which is reached at t = 10.
Effects of surface tension on the wave breaking de- o
velopment
The unsteady process that results into the breaking wave
formation and establishment is numerically computed
for three different length scale at the same Froude num-
ber, thus varying the role of surface tension effects onto
the free surface dynamics, compared to inertial and grav-
ity terms.
To clearly describe how deeply surface tension af-
fects the wave breaking establishment in the three cases
A,B,C, pictures of the resulting free surface shape and
of the vorticity field are shown in Figs. 2,3,4 at three
different stages of the breaking process. Since, due to
the baroclinic contribution, the behavior of the vortic-
ity contours into the transition region can be misleading,
in the figures three different density contours are shown
denoting the air and the water values together with the
mean level. In this way the thickness of the transition
region used in the computations can be also argued.
The comparison among the three different processes,
clearly reveals that, when reducing the length scale, the
dominant effects of surface tension progressively reduces,
net
._
0.1
-0.1
on
C
o.
, -
, .,, .,,, .., .,,,, .,,,,,, .,,,,, . I,, I,
1.2 1.3 1.4 1.5 1.6 1.7 1.8
x
Figure 2: Density (black lines) and vorticity contours
at three different stages of the wave breaking establish-
ment in case A. The three different density contours
denote the air, average and water values, respectively.
Beside a significant change in the wave breaking de-
velopment, a substantially different vorticity production
mechanism is also evident. At the largest scale (case A),
when the jet develops, vorticity contours reveal a weak
shedding of vorticity in the water. In fact, in the under-
side region of the plunging jet, the curvature is rather
small so that the flow is able to follow the free surface
shape. After the first plunging, air is entrapped and, as
a consequence of the momentum exchange due to the
impact, a significant amount of circulation is produced
into the water. Successive stages of the motion show that
vorticity generated in this manner is then convected with
the air bubble. The sequence in Fig. 2 clearly shows
the process which leads to the air entrapment and to the
formation of a highly rotational region that is convected
downstream after wards. Due to the impact, a splash-up
is then formed, thus leading to a new, thinner jet and to
a second, weaker impact.
0.2
0.1
n
o
n
;
1.2 1.3 1.4 1.5 1.6 1.7 1.8
x
In the case B. due to the reduced length scale, the
increased role of surface tension leads to a significant
reduction in the jet intensity and in the entrapment of
air (Fig. 3~. Water-particle velocities in the jet are also
reduced, and the impact is less violent. Furthermore, a
large increase of the curvature at the jet root is found,
which leads to a flow separation and to the development
of a strong shear layer. The mechanisms and the magni-
tude of vorticity production are already sensibly differ-
ent from those of case A.
Figure 3: The same as in Figure 2 but for case B.
In Fig. S. a close up view of the local velocity field
during the formation of the first splash up is shown. At
the beginning, the velocity field suggests that the main
portion of the fluid comes from the jet while, in a later
stage, most of the fluid arrives from the incoming flow
under the forward face of the wave.
0.2 F
0.1
.
no
0 1
o
-0.1
0.2
n
-0 1
Figure 4: The same as in Figure 2 but for case C.
To produce test conditions in which surface tension
has a dominant effect on the evolution of the breaking
wave, a further reduction of the length scale is finally
used (case C). Results shown in Fig. 4 reveal that the
formation of the jet is now completely suppressed and
replaced by a bulge which forms at the crest. In a succes-
sive stage, the toe of the bulge starts its motion sliding
on the forward face of the wave. After an initial accel-
eration, the toe reaches an almost constant speed. As
this motion continues, flow separates due to the sudden
change in the free surface slope at the toe, thus originat-
ing an intense shear layer that propagates downstream,
remaining closely beneath to the free surface. Air en-
trapment is completely suppressed.
0.05 ,
n
-0.05
0.05
o
-0.05
1.4 1.5 1.6
x
Figure 5: Plots of the density contours and of the ve-
locity field (every other vector is shown) at two stages
of the jet plunge (case A).
A careful inspection of the results shown in Fig. 4,
reveals that no capillary waves appear on the free surface
ahead of the bulge, although, for short waves, the large
curvature at the toe is often responsible for the formation
of a train of parasitic capillary waves (Longuet-Higgins,
1992~. The lack of these capillary waves in the numer-
ical results might have several explanations. Actually,
Lin & Rockwell (1995) experimentally show that this
pattern only occurs in a rather narrow range of Froude
numbers, and that a slight variation of speed deeply af-
fects the capillary pattern, reducing the number of crests
up to their disappearance.
I...............
60
50)
40
1n
20
1R
16
14
12
10
30~
,:=
.J ~.........................................................
V . . . . . .
O 0.5 1 1.5 2 2.5 3
X
3.5 4
8
1 1.5 2 2.5
x
3 3.5 4
Figure 6: Time sequences of free surface profiles ob-
tained for case C. Below a close up view about the time
at which the bulge takes its foremost position for the
first time is shown. The At between two profiles is 0.5
in the upper figure, 0.1 in the lower one. The same
quantity is used to shift vertically two successive pro-
files. For the clarity, a vertical scale factor is applied.
Another possible explanation for the lack of capil-
laries upstream the toe can be related to the damping
effect played by viscosity on these small scales (Ceni-
ceros & Hou, 19991. Finally, it has to be remarked that,
due to the use of a continuum model (Brackbill et al.,
1992), the surface tension contribution to the momen-
tum equation is spread on a transition region (18) and
this can play an important role on the accurate descrip-
tion of capillary waves having amplitude comparable or
smaller than 5.
40
35
30
20
10
0 0.5 1 1.5
2
x
2.5
3 3.5 4
Figure 7: Time sequence of the free surface profiles
obtained in case D. Here At = 0.1 is the time delay
and the vertical shift between two successive profiles.
Effects of viscosity on short breaking waves
Wave crest profiles history of a long time simulation is
shown in Fig. 6 for the case C, in a frame of reference
fixed with the hydrofoil. For the sake of clarity, a ver-
tical displacement is applied at each profile. The pro-
file history shows the initial steepening of the crest and
the growing of the bulge. Next, the bulge starts to slide
down the forward face of the wave. After the first large
downslope movement, the toe reduces its forward speed,
reaches its foremost position and begins to retreat. A
damped oscillatory motion is then established with reg-
ular period. The cause of this back and forth movement
can be ascribed to the start from rest, as discussed by
Duncan (1981), whilst the damping is due to the action
of the numerical beaches posed at both sides of the com-
putational domain.
A more refined observation of the profile history re-
veals some interesting details. Indeed, while the damped
motion appears to be quite smooth and the surface pro-
files highly regular, a closer inspection of the crest pro-
files shows the presence of small downstream propagat-
ing surface fluctuations (lower part in Fig. 6~. These
fluctuations are produced each time the toe experiences
its downslope movement and their amplitude is extremely
small.
To get more insights about the nature of these ripples
and the mechanisms responsible for their propagation,
the numerical simulation at the experimental Reynolds
number, Reese = 2480 (case D), is performed and the
resulting profile history is shown in Fig. 7. Several sig-
nificant differences with respect to the case C can be
observed: (i) the first downslope motion appears to be
much faster; (ii) the bulge reaches an upstream fore-
most position; (`iii) much larger downstream propagat-
ing ripples appears, which makes much more evident the
recurrence of their formation each time the bulge is ad-
vancing; (iv) ripples seem to be produced even when the
bulge is at rest; (v ~ the oscillatory motion of the bulge in
more irregular and its amplitude is damped more rapidly.
By carefully looking at the free surface profiles, it
can be noticed that, as soon as they appear, ripples ex-
hibit a growth of their amplitude and wavelength as they
move downstream. Furthermore, initially they propa-
gate downstream with a velocity which is about 0.6 times
the incoming flow speed.
The occurrence of downstream propagating ripples
was experimentally observed by Duncan et al. (1994)
and theoretically investigated by Longuet-Higgins (1994),
where they were denoted as Type II capillaries to distin-
guish them from the parasitic capillaries waves (the Type
I capillaries), located in front of the toe and propagating
upstream.
In Longuet-Higgins (1994) it is speculated that Type
II capillaries are shear layer instability waves and that
their generation is due to the vorticity shed from the
Type I capillaries. Duncan et al. (1994), by using a wave
focusing technique to induce breaking, argued that the
ripples are generated by the instability of the shear layer,
produced between the gravity induced downslope flow
and the incoming upslope flow, which separates at the
toe. Linear stability analysis of measured velocity pro-
files (Duncan & Dimas, 1996) provided a model which
is consistent with the latter assumption.
Present results show the occurrence of Type II rip-
ples although Type I are absent, which agrees with the
Duncan et al. (1994) hypothesis. To better understand
this point, instantaneous distributions of the vorticity are
analyzed in the next section.
-0.25
-0.25
-0.25
-0.25
-0.25
1.~ Z ~.5
-0.25
Figure 8: Sequence of the free surface profiles and of the vorticity contours during the first
upstream motion of the bulge. They refers to the interval t = 14 to t = 16.2 with time step
At = 0.2. The development of shear flow instabilities and the corresponding formation of free
surface ripples is clearly evident in this case.
Vorticity/free surface interaction
The main features of the complex interaction between
the vorticity shed from the toe and the free-surface can
be drawn by the sequence shown in Fig. 8. There, the
later stage of the toe motion between t = 14 to t = 16.2,
is shown. During this time, the bulge front moves from
x = 1.30 up to x = 0.95 and free surface ripples are
established.
The instantaneous vorticity fields illustrate the ini-
tial growth of the instabilities of the shear layer formed
at the toe. Then, instabilities develops into separated co-
herent structures which strongly interacts with the free
surface. Results shown in Fig. 8 thus reveal that ripples
are traces on the free surface of the underlying vortex-
blobs. As soon as ripples are formed, their propaga-
tion speed is the same at which vortex-blobs are con-
vected into the water. Furthermore, the amplitude and
the wavelength of the free surface ripples is essentially
governed by the growth of the size of the vortex-blobs
as they are convected downstream (see the last frames
of Fig. 81.
Free surface ripples display a highly non-symmetric
shape with respect to the horizontal axis: crests are smooth
rounded while, due to surface tension, troughs are cusp-
shaped. Under the action of the flow field induced by the
vortex-blobs, the vertical symmetry is also lost with the
free surface eventually developing an overturning mo-
tion in the downstream direction (Fig. 9~. Due to the
large curvature at the free surface during this stage, a
flow separation from the troughs also takes place, as it is
shown in Fig. lO.
0.06
0.04
0.02
o
-0.02
-0.04
-0.06
-0.08
1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2
x
Figure 9: Free surface profile at t = 15.8 for case D. A
significant asymmetry with respect to the vertical axis
is induced by the interaction with the vortex-blobs.
Shear layer analysis
To characterize the shear layer originating at the toe,
cuts of the velocity field at different stages of the mo-
tion of the bulge are made. In particular four different
vertical cuts of the horizontal velocity component, my),
are performed when: (i) the bulge is still close to the
crest but it is sliding down along the forward face; (ii)
the bulge takes its maximum forward position; (iii) the
bulge is moving backward; (iv) the bulge takes its max-
imum backward position. To make them comparable,
cuts are made at the same horizontal distance Ecus from
the toe. The corresponding plots are reported in Fig. l 1.
To have a rough estimate of the intensity of the shear
layer, the total velocity defect, between the underlying
flow and the flow at the crest, and the corresponding
thickness, along which the shear takes place, are eval-
uated and their ratio reported in Table 2. As expected,
maximum and minimum intensity of the shear corre-
spond to forward and backward motion of the toe, re-
spectively.
Possible correlations among the values of this ratio
and the occurrence of shear layer instabilities is still un-
der investigation. However, a comparison between the
velocity profiles measured during the forward motion of
the bulge at Re = lOOO and at Re = 2480 reveals that,
in the former case, the total velocity defect is smaller
and the thickness of the shear layer is larger (Fig. 12),
thus resulting in a value of the ratio which is 230s-l
and this can explains the rather small size of the surface
ripples observed in case C.
O.1
-0.1
:
1.6 1.8 X 2
Figure Jo: Close up view of the density and vorticity
contours of the last frame of the sequence in Fig. 8. Due
to the strong curvature, a secondary flow separation is
induced at the wave troughs.
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-0.2
-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2
u
Figure ll: Plots of the vertical cuts of the horizon-
tal velocity component taken at a horizontal distance
recut = 0.1 from the toe (case D). The four different
cuts refer to different stages of the bulge movement:
red the bulge is moving forward; green the bulge has
reached its maximum forward position; blue the bulge
is moving backward; pink the bulge has reached its
maximum backward position.
Phase | i\u/b(s-~ ) |
1. Forward motion 350
2. Maximum forward position 300
3. Backward motion 200
4. Maximum backward position 230
Table 2: Ratio between total velocity defect and the
thickness of the shear layer at different stages of the
bulge motion for case D (Fig. l l).
o
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-0.2
-0.4 -0.2 0 0.2
0.4 0.6 0.8 1 1.2
u
Figure 12: Comparison of the vertical cuts for case
C (red) and case D (blue). The significant increase in
the shear layer intensity may explain the much stronger
vorticity production and, as a consequence, ripples for-
mation observed in case D.
CONCLUSIONS
In this paper the effects of surface tension on the
wave breaking dynamics and the effects of viscosity on
micro-breakers have been numerically addressed. By
using a domain decomposition approach, an accurate
description of the breaking and post-breaking evolution
has been obtained by solving the Navier-Stokes equa-
tions only in a narrow region encompassing the free sur-
face.
Results on different length scales are compared, and
effects of the increasing relative importance of the sur-
face tension are fully recovered. The use of a viscous
flow model has also allowed to identify changes in the
vorticity production mechanism. In particular, for low
surface tension, vorticity into the water is mainly intro-
duced by air-entrapment while, for large surface tension,
a intense shear layer develops at the toe of the bulge.
On the smallest analyzed length scale, the evolu-
tion of a micro-breaker has been followed at two differ-
ent Reynolds numbers. Downstream propagating rip-
ples, scarcely visible at the lowest Reynolds number,
have been clearly observed when Re increases. In both
cases, shear layer instabilities, generated between the in-
coming flow and the downsloping flow induced by the
motion of the toe, are responsible for the appearance of
these ripples. For the higher Reynolds number, shear
layer instabilities develops into intense vortex-blobs; the
strong interaction of these coherent structures with the
free surface leads to large ripples that propagates down-
stream.
ACKNOWLEDGMENTS
The work by A.I. was supported by the Ounce of Naval
Research, under grant N.000140010344, through Dr. Pat
Purtell. The work by E.F.C. was supported by the Min-
istero Trasporti e Navigazione in the frame of the IN-
SEAN research plan 2000-02.
REFERENCES
Brackbill, J.U., Kothe, D.B., and Zemach, C., "A con-
tinuum method for modeling surface tension," Journal
Computational Physics, Vol. 100, 1992, pp. 335-354.
Brandt, A., "Guide to multigrid development," Multigrid
Methods, Hackbush W., Trottenberg U. Eds., Springer-
Verlag, Berlin, Germany, 1992.
Ceniceros, H.D., and Hou, T.Y., "Dynamic generation
of capillary waves,", Physics of Fluids, Vol. 11, 1999,
pp. 1042-50.
Duncan, J.H., "An experimental investigation of break-
ing waves produced by a towed hydrofoil," Proceedings
of Royal Society, London, Ser. A, Vol. 377, 1981, pp
331-348.
Duncan, J.H., "The breaking and nonbreaking wave re-
sistance of a two-dimensional hydrofoil," Journal of Flu-
id Mechanics, Vol. 126, 1983, pp. 507-520.
Duncan, J.H., Philomin, V., Behres, M., and Kimmel
J., "The formation of spilling breaking water waves,"
Physics of Fluids, Vol.6, 1994, pp. 2558-2560.
Duncan, J.H., and Dimas, A.A., "Surface ripples due
to steady breaking waves," Journal of Fluid Mechanics,
Vol. 329, 1996, pp. 309-339.
Duncan, J.H., Qiao, H., Philomin, V., and Wenz, A.,
"Gentle spilling breakers: crest profile evolution," Journal
of Fluid Mechanics, Vol. 379, 1999, pp. 191-222.
Iafrati, A., and Campana, E.F., "Unsteady free surface
flow around hydrofoils," Boundary Elements XX, Kassab
A., Brebbia C.A., Chopra M. Eds., Computational Me-
chanics Publications, Southampton, UK, 1998, pp. 451-
460.
Iafrati, A., Olivieri, A., Pistani, F., and Campana, E.F.,
"Numerical and experimental study of the wave break-
ing generated by a submerged hydrofoil," Proceedings
of the 23r~ Symposium on Naval Hydrodynamics, Val de
Reuil, France, 2000.
Iafrati, A., Di Mascio, A., and Campana, E.F., "A level-
set technique applied to unsteady free surface flows,"
International Journal for Numerical Methods in Fluids
Vol. 35, 2001, pp. 281-297, 2001.
Iafrati, A., and Campana, E.F., "A domain decomposi-
tion approach to compute wave breaking," submitted for
publication to the International Journal for Numerical
Methods in Fluids, 2002.
Kim, J., and Moin, P., "Application of a fractional-step
method to incompressible Navier-Stokes equations," Jour-
nalofComputationalPhysics, Vol. 59, 1985, pp. 308-
-
323.
Lin, J.C., and Rockwell, D., "Evolution of a quasi-steady
breaking wave," Journal of Fluid Mechanics, Vol. 302,
1995, pp. 29-44.
Longuet-Higgins, M.S., "Capillary rollers and bores,"
Journal of Fluid Mechanics, Vol. 240, 1992, pp. 659-
679.
Longuet-Higgins, M.S., "Shear instability in spilling break-
ers," Proceedings of Royal Society, London, Ser. A, Vol.
446, 1994, pp. 399-409.
Longuet-Higgins, M.S., "Progress towards understand-
ing how waves break," Proceedings of the 21St Sympo-
sium on Naval Hydrodynamics, Trondheim, Norway, 1996,
pp. 7-28.
Mui, R.C.Y., and Dommermuth, D.G., "The vertical struc-
ture of parasitic capillary waves," Journal of Fluids Engi-
neering, Vol. 117, 1995, pp. 355-361.
Qiao, H., and Duncan, J.H., "Gentle spilling breakers:
crest flow field," Journal of Fluid Mechanics, Vol. 439,
2001, pp. 57-85.
Rai, M.M., and Moin, P., "Direct simulations of tur-
bulent flow using finite-difference schemes," Journal of
Computational Physics, Vol. 96, 1991, pp. 15-53.
Rosenfeld, M., Kwak, D., and Vinokur, M., "A frac-
tional step solution method for the unsteady Incompress-
ible Navier-Stokes equations in generalized coordinate
system," Journal of Computational Physics, Vol. 94,1991,
pp. 102-137.
Russo, G., and Smereka, P., "A remark on computing
distance functions," Journal of Computational Physics, Vol.
163, 2000, pp. 51-67.
Scardovelli, R., and Zaleski, S., "Direct numerical simu-
lation of free surface and interracial flow," Annual Review
of Fluid Mechanics, Vol. 31, 1999, pp. 567-603.
Sethian, J.A., "Level Set Methods and Fast Marching
Methods," Cambridge University Press, 1999.
Sussman, M., and Dommermuth, D.G., "The numerical
simulation of ship waves using Cartesian grid methods,"
Proceedings of the 23rd Symposium on Naval Hydro-
dynamics, Val de Reuil, France, 2000.
Sussman, M., and Fatemi, E., "An efficient, interface-
preserving level-set redistance algorithm and its applica-
tion to interracial incompressible fluid flow," SIAM Jour-
nal Scientific Computation, Vol. 20~4), 1999, pp. 1165-
1191.
Sussman, M., and Puckett, G., "A coupled level-set and
volume-of-fluid method for computing 3D and axisym-
metric incompressible two-phase flows," Journal of Com-
putational Physics, Vol. 162, 2000, pp. 301-337.
Sussman, M., Smereka, P., and Osher, S., "A level set
approach for computing solutions to incompressible two-
phase flow," Journal of Computational Physics, Vol. 114,
1994, pp. 146-159.
Tulin, M.P., "Breaking of ocean waves and downshift-
ing," Waves and Nonlinear Processes in Hydrodynamics,
Grue J., Gjevik B., and Weber J.E. Eds., Kluwer Acad.,
Dordrecht, Netherland, 1996. pp. 170-190.
Zang, Y., Street, R.L., and Koseff, J.R., "A non-staggered
grid, fractional step method for time-dependent incom-
pressible Navier-Stokes equations in curvilinear coor-
dinates," Journal of Computational Physics, Vol. 114,
1994,pp.18-33.
DISCUSSION
D. G. Dommermuth
Science Applications International Corporation,
USA
The authors have made a significant contribution
toward developing a numerical procedure for
simulating and understanding gravity-capillary
waves. For the matching procedure, a
Helmholtz decomposition, whereby the upper
domain is divided into its potential and vertical
components, may give additional insight into the
boundary condition that is imposed by the
authors. Also, could the authors please comment
on what is the next step toward improving our
understanding of small-scale breaking waves?
AUTHORS' REPLY
We thank Dr. Dommermuth for his comments on
the paper. Conceming with the use of a
Helmholtz decomposition, we actually recognize
that it could allow a better understanding of the
mechanisms governing the exchange of
information between the two sub-domains.
Likely, a better way to enforce the matching
condition could be also derived.
As to the second question the authors are
thinking of taking advantage from the two-phase
formulation used to solve this problem and move
toward the analysis of the small-scale wave-wind
interaction, being a very important generation
mechanisms for short surface tension dominated
breaking waves. This point has been studied
experimentally, for instance by Tulin (1996), but
little has been done with numerical tools, which
can help in the understanding of the fundamental
mechanism of this phenomena.
