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OCR for page 224
Numerical simulation of two dimensional breaking
waves past a submerged hydrofoil
R. Muscari, A. Di Mascio
(INSEAN, Italian Ship Model Basin, :~taly)
ABSTRACT
A numerical model for the simulation of two-
dimensional spilling breaking waves is described. It
is derived from a previous model which, in turn,
takes the underlying ideas from the Cointe and
Tulin's theory of steady breakers. With respect
to the former model, the present one is local, i.e.
the inception, the extension and the geometry of
the breaker are determined through the local shape
of the water surface. The model has been imple-
mented in a RANSE code, developed for the simu-
lation of ship flows, through a modification in the
boundary conditions. This yields an effective and
simple way to reproduce the breaker influence on
the underlying flow. The resulting code has been
used for the simulation of the flow past a submerged
hydrofoil. The numerical results are compared with
those of the previous model and with the experi-
mental data obtained by Duncan.
INTRODUCTION
The aim of this work is the development and valida-
tion of a numerical model for the simulation of flows
with breaking waves. In particular, we are inter-
ested in flows with steady or quasi-steady spilling
breakers.
Examples of this kind of flows are the waves ap-
proaching to a shore, the wave train generated by
a two-dimensional object in uniform motion near
the free surface or the wave pattern generated by
a ship moving through calm water. Despite the
importance of the effects of the breaking on these
flows has been generally acknowledged (see Ban-
ner and Peregrine (1993~; Melville (1996~; Duncan
(2001) for comprehensive reviews), very little has
been done, from a numerical point of view, in or-
der to include these effects in naval hydrodynamics
codes.
The first attempt, to our knowledge, to practically
include the breaking in a numerical code was done
by Cointe and Tulin (1994), who elaborated a the-
ory describing the shape and effects of the breaker,
and then implemented it in a potential code for
two-dimensional flows. One of the advantages of
Cointe and Tulin's theory is that it yields some
boundary conditions to model the breaker which
are simple but effective and readily applicable to
free surface RANSE codes. This feature has been
exploited by Rhee and Stern (2002) and Muscari
and Di Mascio (2002), who have implemented two
different models in their already existing RANSE
codes, developed for the study of flows past ship
hulls.
In particular, Rhee and Stern (2002) adopt the
same flat-top geometry for the breaker and the
same relationship between its height and that of the
breaking wave as suggested by Cointe and Tulin.
Furthermore, they specify the velocity distribution
along the wave-breaker interface by assuming that,
at the toe of the breaker, the horizontal component
of velocity u undergoes a discontinuity, whose in-
tensity depends on the wave steepness. Beyond the
toe, u is computed by assuming constant the total
head.
In Muscari and Di Mascio (2002) some modifica-
tions in the original theory were introduced, with
the goal of developing a stable and accurate numer-
ical algorithm when coupling the model of break-
ing to their steady state free surface RANSE code.
First of all, it was assumed a different relation be-
tween the height of the breaking region and that of
the following waves. Furthermore, the sharp trian-
gular shape of the breaker adopted by Cointe and
OCR for page 225
Tulin and Rhee and Stern was smoothed out in or-
der to mitigate the abrupt transition in the free
surface dynamic boundary condition and hence en-
hance convergence rate to steady state.
The models cited so far have been implemented only
for two-dimensional flows and share the disadvan-
tage that are non-local. This means that, for ex-
ample, in order to compute the hydrostatic pres-
sure exerted by the breaker on a free surface point
we need to know the height of the following waves
(Cointe and Tulin, 1994; Rhee and Stern, 2002) or
the location of the trough and the crest of the break-
ing wave (Muscari and Di Mascio, 2002~. It is not
evident how these geometrical properties of a two-
dimensional wave train should be interpreted in a
three-dimensional context and, however, the result-
ing algorithm would be inefficient and prone to er-
rors.
With the ultimate objective of simulating spilling
breaking in general naval flows, we develop in this
work a "localized" version of the model described
in Muscari and Di Mascio (2002~. The selection of
the points where breaking occurs, the local height
of the breaker and, hence, the hydrostatic pressure
are all determined by the free surface elevation and
its first derivative. In fact, we think that this is a
necessary step before the model could be general-
ized to three-dimensional flows. We do not consider
viscous effects which deserve further attention and
will be the subject of future work. The model is
described in the next session and, then, its results
are compared to the ones of the "non-local" model
and to the experiments by Duncan (1983~.
A LOCAL MODEL FOR SPILLING BREA-
KING
Following the Cointe and Tulin's idea, we want to
model the effects of the breaker through a suitable
pressure, applied on the patches of free surface in-
volved in the process, and simulating the weight of
the breaker itself. To this purpose, the main prob-
lems to be solved are the detection of the afore-
mentioned patches and the calculation of the local
height of the breaker and, hence, of the pressure to
be applied.
Cointe and Tulin assume that the top of the breaker
coincides with the crest of the wave (see fig. 1) and
compute the vertical distance between the top and
the toe as:
he = 2 - Zl°P
where Fr is the Froude's number. Then, assuming
a flat-top shape for the breaker, they prescribe the
hydrostatic pressure to be applied at the free sur-
face points between Toe and atop:
hi = Stop—zfx) ~ pax) = Fr2
Pb being the density of the air-water mixture in the
breaker, which is taken equal to 0.6.
X~T ~~
I (top
X
Figure 1: Geometry of the breaker in the Cointe
and Tulin's theory.
In Muscari and Di Mascio (2002) some modifica-
tions to the Cointe and Tulin's theory are intro-
duced in order to gain robustness and accuracy for
their RANSE code. Their expression for the verti-
cal distance between the toe and the top, obtained
by Duncan's (1981) experimental data, is:
hm<~ = 0.64 ab
(1)
where ab is the vertical distance between the trough
and the crest of the breaking wave. Furthermore,
the shape of Cointe and Tulin's breaker is smoothed
by an exponential function, in order to enhance con-
vergence to the steady state:
hm~(x) = hC~(x) {1—exp ~—5 ( toe )] }
Stop—Xtoe
Apart from pros and cons of the two formulations,
both of them need the detection of the crest of
the breaking wave and, from this, of the toe of
the breaker. Unfortunately, in a three-dimensional
case these geometrical entities are not easily de-
tected and, moreover, they do not uniquely locate
the breaking zone.
For the development of a local model we need to
define a detector of breaking A(x) in every point of
the free surface. This can be set equal to:
A(x) = N|Z2 + (Z~42
which would yield the wave amplitude in the case
of single sine wave
2
OCR for page 226
z(x) =—A cos ~ x (2)
Evaluating the wave number K as for a plane pro-
gressive wave ~ = F 2 (Newman, 1977), we get:
A(x) = 4~/z + (ax Fr ~ (3)
Figure 2 shows that, for the experimental setup de-
scribed in the following, eq. (2) is a very good ap-
proximation of the actual wave probe.
0.,:
0.05 _
U ~
-0.05
~ 1
I.
-
~C''''"'"""" \
\ I
at\
_ .. ............ \ . ~
, \ ~
. \\
\~.
..........
Figure 2: Actual wave profile (solid line) and
eq. (2) (dashed line) for depth = 18.5 cm.
By eq. (3), we can establish a local criterion for
detecting the inception of breaking, that is:
2 A(x) ~ 0.69 Fr2 (4)
This equation is the natural extension of ab ~
0.69Fr2, proposed in Duncan (1983) and used in
Muscari and Di Mascio (2002), but it is not enough,
by itself, to locate correctly the breaking zone. In
particular, on the forward face of the leading wave
it detects a zone much smaller than the experimen-
tal investigations would suggest and, on the other
side, it includes points of the backward face which
are not directly involved in the breaking process.
However, on the forward face of the wave we have
u ax ~ 0 (5)
where u is the horizontal component of the velocity.
We found that an extremely effective solution is to
activate the model in all free surface points where
eq. (5) is verified and where eq. (4) holds at least
in one point of the considered wave region.
Finally, to evaluate the pressure distribution due to
the breaker, we prescribe the following expression:
hl,oc (X) = /C z2 (X) (Z—Ztoe) (6)
which is obviously equal to zero at the toe and
at the crest of the wave. Ztoe is found through
Ztoe = stop - h* and the height of the breaker is
determined by eq. (1).
In order to close the problem, we assume stop = 2
and resort to the experimental data by Duncan
(1981) to calculate ab:
ab = 0.586 Fr2
The last free parameter, I;, is determined by en-
forcing that the area of the breaker for the present
model and that for the Cointe and Tulin's one be
the same:
[Xtop
/ thct(X)—hi do = 0
Jxtoe
Assembling all elements of the model, we come to
the final expression for the local breaker height
hoc(X) = 14.35 z2 (z + 0.117 Fr2) (7)
which is illustrated in fig. 3.
Figure 3: Geometry of the breaker for the present
model.
From eq. (7) we get the pressure to be applied on
all free surface points between xtOe and Stop to sim-
ulate the presence of the breaker:
p(x)= Fry
This final expression conjugates very good results
with the desired locality of the model. In particu-
lar, to calculate the height of the breaker only the
wave elevation and its first derivative are necessary.
The proposed model will be examined in the fol-
lowing in order to verify its capabilities in a two-
dimensional case. The extension to a more gen-
eral three-dimensional case will be done in a future
work.
3
OCR for page 227
TEST CASE
In order to validate the proposed model we chose to
simulate the flow described in Duncan (1981, 1983)
and illustrated in fig. 4. This flow has been also
used for the validation of the non-local model in
Muscari and Di Mascio (2002), so that we can bene-
fit, together with extensive experimental data, from
other numerical results.
Re = 1.423 x 10
Fr = 0.5672
Figure 4: Experimental setup.
A submerged hydrofoil, a NACA 0012 profile whose
chord is 20.3 cm, is towed in a tank with speed
0.8 m/s and 5° angle of attack. The leading wave of
the train can break or not depending on the depth
of submergence. This latter is varied through the
water level in the tank, whereas the profile is kept
fixed with respect to the bottom.
The numerical solution was computed on a multi-
block fine grid of about 60.000 cells, and on two
coarser grids, each obtained by removing every
other point from the previous finer one. The nu-
merical uncertainty U was evaluated as suggested
in ITTC Quality Manual (2001~; Roache (1997), by
U = NIGH + UI2T
where UH is the contribution from the grid size
| hone _ hmediUm |
(~ = 2 being the theoretical convergence order and
r = 2 the grid refinement ratio for all the compu-
tations reported), whereas UIT is the contribution
from incomplete iterative convergence
UIT= 2h
Ah being the oscillation amplitude of the solution.
Unfortunately, experimental uncertainty was not
available for this test case, and therefore a com-
plete validation as required in ITTC Quality Man-
ual (2001) was not possible. Nevertheless, use-
ful information on the reliability of the model can
be gained when comparing the numerical solutions
with the towing tank data.
RESULTS AND DISCUSSION
First, the depth = 19.3 cm is considered. As re-
ported in Duncan (1983), for this submergence the
steady breaking is not spontaneous but must be
triggered by a disturbance (a surface current cre-
ated by dragging a cloth on the water surface in
front of the hydrofoil).
~o2
s
10.3
1o" t
~ DEPTH = 19.3 CM
~~_~
1 _ _
., . . , . j
~ ~~7,k~ — — - fX
t- I ~~'t;l.`,.
1 ~ it. ~
~ ~?~- -
.................. , ,, - if;
_ 0.38
_M
_ 0.33
_ 028
_ 0.23
0.18
I ~ 1 ~ ..
.08
iter
Figure 5: Residuals and resistance histories.
_ n 1~
In fig. 5, L2-norm of residuals and non-dimensional
resistance histories are shown. The solution was
computed by means of a Full Multigrid - Full Ap-
proximation Scheme (FMG-FAS) with four grid lev-
els, and the solutions on the two finer grid were used
to estimate the uncertainty. It can be seen from the
figure that a stationary regime is reached on each
grid level, and therefore the contribution to uncer-
tainty from incomplete convergence was negligible.
The comparison with the non-local model, fig. 6,
shows that, despite of the different breaker geome-
tries, the computed wave profiles are substantially
equal, with only a very slight phase difference. The
top of the breaker obtained with the local model is
set between the toe and the crest of the wave, not
at the crest itself as it is assumed in the Cointe and
Tulin's theory. This is an obvious consequence of
the locality of the model, but we do not think that
it represents a real drawback.
The wave profile is also in very good agreement
with experiments, fig. 7. The slight difference at the
troughs, which are overpredicted, can be well within
4
OCR for page 228
the experimental uncertainties. As for the numer-
ical uncertainties, their very small value is due to
the fact that convergence to steady state has been
reached (UIT ~ O) and that the wave profile has
almost reached grid independence on the medium
grid (UH ~ O)
O.1 ... . . . ...
0.05
coos
-0.1 _
DEPTH = 19.3: CM
_ ..... . . .
o _ . ~
_. .. . ., .. . . .. . .. . . . .. ..
J :
1 _~. ,,,,, . , ,. ,, . .,, . . 1
1 2 3 4
·f~s
Figure 6: Wave profiles with breaker's geome-
tries. Solid line: local model; dotted line: non-
local model.
our.
O.Oc
o
O.O:
-0.1
1
· . ! . .. .
. . . .
DEPTH = 19.3 CM ~ ~
. ~ ~ ~
~ ~~k ~~ ~ offs
_~: W.
., ~ .
·
. .................................................................................................................................................................................
·...
. .
.
The last case is depth = 15.9 cm. For this submer-
gence the breaking is very intense and it is not clear,
from the experimental wave profile, if a steady or
quasi-steady state can be actually reached, so the
comparison with the numerical data is rather ques-
tionable. The numerical solution too does not at-
tain convergence to steady state and the residuals
oscillates around a small but constant value. Even
so, it is interesting to compare the different data
sets, fig. 9 and fig. 10. With respect to the non-
local model, the local one produces a higher pres-
sure with a consequent reduction of the amplitude
of the wave train. Furthermore, a second breaker
appears on the first following wave. In fact, in the
proposed model there is no need to know a pr~ori
which wave breaks. On the contrary, the inception
of breaking is essentially determined by the local
steepness of the water surface, eq. (3~-~4), and is
propagated to the neighboring points with a sim-
ple, robust but very effective criterion. As could be
expected, comparison with experiments can be only
qualitative. In this sense, the new solution performs
better then the non-local one with a lesser wave
amplitude and a larger extension of the breaking
area.
n ~ _
r DEPTH = 18.5 CM
. . . ~
: ~~!
.O .
T -lo
'- - ~ ''''1"':" ' ''' ' ' '' ' ''' '' ''
. ~ .
/
. . e_f
-
.
..
I
~-
Figure 7: Computed wave profile with numerical
uncertainties vs. experiments.
The test case with depth = 18.5 cm is the the
first experimental setup for which breaking spon-
taneously occurs, and it is mild enough to have
a regular steady following wave train. The gen-
eral considerations done for the previous case still
hold. Here, we report only the comparison with
the experiments, fig. 8, which is again very good.
Although the leading wave is not captured, the fol-
lowing waves show a remarkably good agreement
with the experiments. Even the troughs, that for
the depth = 19.3 cm case were overpredicted, are
well reproduced.
Figure 8: Computed wave profile with numerical
uncertainties vs. experiments.
CONCLUSIONS
A local model for simulating two-dimensional
spilling breaking has been proposed. It is de-
rived from the non-local model described in Mus-
cari and Di Mascio (2002) and represents a nec-
essary premise towards the simulation of three-
dimensional flows with general breaking patterns.
The inception and the shape of the breaker depend
on local characteristics of the free surface, the el-
evation and the gradient, and, as a consequence,
5
OCR for page 229
the extension to three-dimensional flows does not
present any major theoretical difficulties.
The model has been applied to the study of a wave
train created by a hydrofoil towed under the free
surface. For the cases with milder breaking the re-
sults are similar to those obtained by the non-local
model and in very good agreement with the experi-
ments. For the most severe case, the new algorithm
performs better then the old one. Even if it can not
cope with an apparently unsteady flow, nonetheless
the inception of a secondary breaker indicates that
the extension of the phenomenon can not be re-
stricted to the forward face of the leading wave.
01r
nor
O ;
-0.05
~.1
~ , ~ . -., DEPTH = 15.9 CM
.W
_ . I. \ ..
~ ' \
_ . .... it. .. .,\ .
1 ~
. ·.
:
:
. . .. . . . . .
I:
:,. :
~ /
·/
Figure 9: Wave profiles with breaker's geome-
tries. Solid line: local model; dotted line: non-
local model.
0.05
At.
.... . . ~. ........ .. .
'''''I 'my ~. I' /~
:.
. . . .. . . . ...
Figure 10: Computed wave profile vs. experi-
ments.
The natural evolution of this work will be the ap-
plication to a three-dimensional flow and, possibly,
the inclusion of the phenomenology due to the vis-
cosity.
ACKNOWLEDGMENTS
This work was sponsored by the Italian "Ministero
delle Infrastrutture e dei liasporti" through the IN-
SEAN Research Program 2000-2002, and by the
Office of Naval Research contract N00014-00-1-0344
under the administration of Dr. Patrick Purtell.
REFERENCES
Banner, M. L. and Peregrine, D. H. (1993). Wave
breaking in deep water. Arson. Rev. Fluid Mech.,
25:373-397.
Cointe, R. and Tulin, M. P. (1994~. A theory of
steady breakers. J. Fluid Mech., 276: 1-20.
Duncan, J. H. (1981~. An experimental investiga-
tion of breaking waves produced by a towed hy-
drofoil. Proc. R. Soc. Lond. A, 377:331-348.
Duncan, J. H. (1983~. The breaking and non-
breaking wave resistance of a two-dimensional hy-
drofoil. J. Fluid Mech., 126:507-520.
Duncan, J. H. (2001~. Spilling breakers. Aran. Rev.
Fluid Mech., 33:519-547.
ITTC Quality Manual (2001~. Resistance Commit-
tee of 23th ITTC.
Melville, W. K. (1996~. The role of surface-wave
breaking in air-sea interaction. Ann. Rev. Fluid
Mech., 28:279-321.
Muscari, R. and Di Mascio, A. (2002~. A model for
the simulation of spilling breaking waves. Sub-
mitted to J. Ship Research.
Newman, J. N. (1977~. Marine Hydrodynamics.
The MIT Press, Cambridge, MA.
Rhee, S. H. and Stern, F. (2002~. Rans model for
spilling breaking waves. J. Fluid Erlg., 124:1-9.
Roache, P. J. (1997~. Quantification of uncertainty
in computational fluid dynamics. Ar7~rz. Rev. Fluid
Mech., 29:123.
6
OCR for page 230
DISCUSSION
L.J. Doctors
The University of New South Wales, Australia
I would like to thank the authors for a very
interesting paper on the question of wave
breaking, which should have application to
breaking wave system behind a transom in stern
at low speeds.
My question relates to the equation at the top of
column 2 of page 58 of your paper, in which the
height of the wave is given as a constant times
the Froude number squared. Could you kindly
chart the dimensions of this equation?
AUTHORS' REPLY
Thank you very much for the comment.
Regarding the question, all the equations in the
paper are written in non-dimensional form. In
particular, the hydrofoil chord length is the
reference length for the wave height ab
Representative terms from entire chapter:
wave profile
