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OCR for page 727
A Numerical Study of Three Dimensional Viscous
Interactions of Vortices with a Free Surface
D. Dommermuth (Science Applications International Corporation, USA)
D. Yue (Massachusetts Institute of Technology, USA)
ABSTRACT
We develop semi-implicit and explicit numerical
methods for the direct simulation of the three-
dimensional Navier-Stokes equations with a free
surface. The efficiency of a novel multigrid flow
solver permits the simulation of three-dimensional
flows with free surfaces at low Reynolds and Froude
numbers. The numerical schemes are used to study
vortex rings and tubes interacting with walls and
free surfaces.
In the case of~vortex rings interacting with a no-
slip wall our observations of the formation of sec-
ondary and tertiary vortex rings agree qualitatively
with experimental measurements. When a free
surface is present, the results are sensitive to the
Froude number. For sufficiently low Froude num-
bers, the free surface behaves like a free-slip wall,
which agrees qualitatively with experimental ob-
servations of vortex rings interacting with clean
free surfaces. At intermediate Froude numbers, the
normal incidence of a vortex ring with a clean free
surface results in the formation of secondary vortex
rings.
Numerical studies of vortex tubes interacting with
free surfaces show two possible mechanisms for the
reconnection of vorticity with a free surface in-
cluding primary and secondary vorticity reconnec-
tions. One type of primary vorticity reconnection
should result in a cusp pattern on the free sur-
face and secondary vorticity reconnections should
manifest themselves as paired dimples on the free
surface. The essential stages of the reconnection
of secondary vorticity with the free surface are as
follows: generation of helical vortex sheets by the
primary vortex tube, stripping of the helical vortex
sheets due to self-induced straining flows, attach-
ment of the helical vortex sheets to the separated
727
free-surface boundary layer, wrapping of U-shaped
vortices around the primary vortex tube, feeding of
boundary-layer vorticity into the U-vortices, and
reconnection of U-vortices with the free surface.
We provide evidence which suggests that the stri-
ations that may be observed on the free-surface
above a pair of trailing tip vortices are caused by
helical vorticity.
~ Introduction
In the field of free-surface hydrodynamics few
things are as complex as the wave and viscous
wakes of ships. Consider the region downstream
of a moving ship where the steady wave pattern of
Kelvin waves is located. This wave pattern is com-
posed of two wave systems which move steadily
with the ship: the transverse and diverging waves.
The transverse waves are normal to the ship track,
and the diverging waves obliquely intersect the ship
track. These two wave systems are bounded by the
cusp line which occurs at an angle of about +19.5°
off the ship track. Most of the steady waves dis-
perse linearly, but if the conditions are right, rays
of the diverging waves that are inside of the cusp
lines may evolve into solitons (Brown, et al, 1988~.
In addition to the Kelvin waves, there is also a
small sector centered near the ship track that con-
tains short unsteady waves. Some possible sources
of these short unsteady waves include breaking
waves and turbulence generation near the ship.
Munik, et al, (1987) hypothesize that Bragg scatter
from these short unsteady waves is responsible for
the narrow V-shaped features that are observed in
Synthetic Aperture Radar (SAR) images of ship
wakes in very calm seas. The viscous turbulent
wake, which is also located near the ship track,
OCR for page 728
has a thrust component that is associated with the
propellerts) and a drag component that is due to
the ship hull. Axial vortices that are generated
in the bow region (Babe, 1969; Mori, 1984; and
Takekuma & Eggers, 1984~) and helical vortices
that are shed by the propeller (Kerwin, 1986) are
embedded in the turbulent wake. Field observa-
tions of ships suggests that bubbles rising through
the turbulent wake transport surfactant material
to the free-surface (Kaiser, et al, 1986~. These bub-
bles appear to be entrained by the bow wave, but
other sources include propeller cavitation. Slicks
consisting of compacted surfactant material form
on each side of the ship track possibly due to the
action of vortices moving surface water away from
the viscous wake's center. These slicks damp cap-
illary and gravity-capillary waves that cross the
ship track (Dorrestein, 1951) and may contribute
to the persistent centerline feature that is observed
in SAR images (Lyden, et al, 1988~. Clearly, ship
wakes are very complex, and there are many mech-
anisms that affect direct and remote observations
of ship wakes.
Faced with such a diversity of phenomena, we have
chosen to focus on a very specialized portion of the
problem. Specifically, we are interested in mod-
eling the viscous interaction of vorticity with a
free surface. Vortices may generate and modu-
late short unsteady surface waves and thus affect
the observable features. Two sources of vertical
flows include turbulent bow-wave breaking (Babe,
1969) and lifting devices (propellers, rudders, &
hydrofoils). In addition, coherent vertical struc-
tures may also occur in the viscous turbulent wake
(Brown & Roshko, 1974, and Roshlco, 1954~. The
three-dimensional interactions of these flow fields
with the free surface are very complicated, but it
is our hope that simple models will provide much
needed insight. Some simple models of the large-
scale vertical flows that occur in ship wakes include
two-dimensional vortex pairs, ring vortices, and
three-dimensional vortex tubes. Even these sim-
ple models are so complicated that they challenge
our basic understanding of vortex interactions with
free surfaces.
As an initial step toward understanding the inter-
actions of ship-scale vortices with a free surface, we
consider the direct simulation of the Navier-Stokes
equations with a free surface at a finite Froude
number. Although the Reynolds numbers that we
are able to simulate so far are considerably be
low those observed in reality, our three-dimensional
method is capable of modeling physical phenom-
ena that are simply not possible using inviscid ap-
proaches. A notable example is the reconnection
of vortex tubes, which is an inherently viscous phe-
nomena. Moreover, extensions to our method may
eventually allow us to study the effects of turbu-
lence, air entrainment, and surfactants.
We can partially classify vertical flows near a free-
surface in terms of Reynolds, Froude, and Weber
numbers. As our velocity scale, we use the cen-
terline velocity of a vortex pair, Uc=2I`/,rs, where
is the circulation and ~ is the span. In corre-
spondence with this velocity, the length scale is the
half span, L=~/2, and for ring vortices the ring ra-
dius is used. Based on these scales, the Reynolds
number is Re=~/~t,, where ~ is the kinematic vis-
cosity; the Froude number is Fr=I`/~g~/2~/2~3/2,
where 9 is gravity; and the Weber number is We =
2pI`2/7r2sT, where p is the fluid density and T is
the surface tension. Although surface contamina-
tion plays a very important role in the interaction
of vorticity with free surfaces, this effect is difficult
to quantify and model, and has not been consid-
ered so far in the present simulations. We note that
the hydrodynamics of ship-scale surfactant films
are discussed by Peltzer, et al, (1990) in this pro-
ceedings. In addition, Hirsa, et alms (1990) exper-
imental and numerical studies of two-dimensional
vortex pairs, which also appears in this proceed-
ings, provides a preliminary model of surfactant
films.
As preparation for our literature review of vortices
interacting with free surfaces, we also discuss the
related problems of vortex interactions with no-
slip walls and the stability and reconnection of
vortex tubes. We conclude our introductory sec-
tion with a review of numerical methods for simu-
lating the incompressible Navier-Stokes equations.
In Sec. 2 we provide the mathematical formula-
tion for describing the viscous interactions of three-
dimensional vortices with free surfaces, and in Sec.
3 we discuss the numerical implementations of our
theory. In Sec. 4 we apply our numerical meth-
ods to three problems: 1) ring vortex interactions
with a wall, 2) ring vortex interactions with a free
surface, and 3) vortex tube interactions with a free
surface. In Sec. 5 we discuss the long-term direc-
tions and promise of this line of research.
728
OCR for page 729
1.1 Interaction of Vortices with WalIs
Harvey & Perry's (1971) studies of a wingtip vor-
tex near a wall confirm observations of aircraft vor-
tices bouncing off the ground. They argue that
the rebound effect may be attributed to unsteady
boundary-layer separation and the formation of
secondary vortices. Their experimental measure-
ments show that motion of a vortex moving paral-
lel to a wall may be stopped or even reversed due
to viscous and inviscid interactions. Inviscid effects
alone are ruled out by Saffman (1978), who proves
that a vortex pair normally incident to a wall in
an inviscid fluid will monotonically approach the
wall and then separate apart due to the velocity
induced by themselves and their images. Wall~er's
(1978) two-dimensional computations show that
explosive boundary-layer growth is possible, but
his boundary-layer theory is not capable of model-
ing the formation of secondary vortices. Peace &
Riley's (1983) two-dimensional Navier-Stokes sim-
ulations show substantial growth of the boundary
layer, but their Reynolds numbers (Re = 0~10~)
are too low for secondary vortex formation to oc-
cur.
WaLker, et ales (1986) studies of vortex rings nor-
mally incident to a wall exhibit many of the same
phenomena as two-dimensional vortices. An im-
portant difference is that ring vortices may stretch.
According to inviscid theory, a ring vortex ap-
proaching a wall will expand radially outward and
the diameter of its core will decrease. However, the
experimental results of Walker, et al, and others
(see Cerra & Smith, 1983) show much more com-
plexity including boundary-layer separation, sec-
ondary and tertiary vortex formation, rebounding,
azimuthal instabilities, and transition to turbu-
lence. For Re < 250, no secondary vortices are
observed and the primary vortex rapidly diffuses
while being held fixed against the wall. Flow sep-
aration and secondary vortex formation occur for
Re > 250. For Re > 3000, Walker, et al, have diffi-
culties producing laminar vortex rings. They also
perform boundary-layer calculations which once
again show explosive growth of the boundary layer,
but fail to predict the formation of secondary and
tertiary vortices.
As a prelude to our study of vortex ring interac-
tions with a free-surface, we have chosen to simu-
late the interactions of a vortex ring with a wall be-
cause Wallcer, et alts (1986) documentation permits
qualitative validation of our Navier-Stokes com-
puter code. In addition, the comparisons between
vortex ring interactions with walls and free sur-
faces are very useful. Our numerical simulations
for the wall case show the formation of secondary
and tertiary vortex rings. In fact, the primary vor-
tex ring is wrapped in a sheath of counter-sign vor-
ticity. Initially, a sharp gradient, which diffuses
with time, exists between the primary vortex and
the shed vortices. Despite the appearance of rapid
dissipation, however, the kinetic energy of these
no-slip wall simulations does not decrease signif-
icantly more rapidly than our simulations with a
free-slip wall.
1.2 Stability and Reconnection of Vor-
tex Tubes
Spreiter & Sacks (1951) show that the trailing
wake shed from an elliptically loaded wing will roll
up into a pair of trailing vortices. The span of
the trailing vortices in terms of the original span
of the foil (L,) is 8 = l/4, and the core ra-
dius, assuming constant strength, is rc = 0.0985s.
This trailing vortex pair is unstable to sinusoidal
disturbances along the length of the tube (Crow,
1970~. The wavelength of the instabilities (Acrou')
in terms of the span are Acrou' ~ 8.6s, and the
e-folding time of the instability is Crow =l.2lTi,
where Ti=2~2/I, is the time it takes the vortices
to move one span length due to their own induc-
tion.
Leonard (1985) indicates that large changes in the
curvature of a vortex tube may lead to the rapid
formation of helical vortex lines and even vortex
breakdown. The exponential growth of helical vor-
ticity is related to unstable perturbations of the
vortex core in the presence of straining fields. The
two-dimensional numerical simulations of Dritschel
(1989) suggest that in addition to causing the ini-
tial instability, the straining flows may also strip
the helical vortices away from the primary vor-
tex. According to Leonard's (1985) review, once
the helical vortices are formed, the axial flows that
are induced by the helical vortices may tend to
straighten the primary vortex tube.
As Crow (1970) notes, trailing vortex tubes that
have gone unstable may eventually join to form
crude vortex rings in the late stages of the in-
stability. This process of reconnection is also ob-
served in the vortex ring experiments of Kambe
729
OCR for page 730
l\kao (1971)1 Pall ~ leaner (19?5), Ostiroa
~ As^ (1977), Schlitz (1987)) and Oslo
Izutsu (1988). The bbserstlons in these expert
meets seems to locate that vortex reconnection
takes place as a rest It of canceH,tlon of vortlc1ty as
two eposltely signed vortex tubes lutersect. Ale-
l~nder ~ Husband (1988) cut-and-connect rmodel
of vortex reconnection (lg88), Stacy is based on
spectra shmmdatlons of ant~parsUel vortex tubes,
describes vortex reconnection in three stages:
1. I~iscid fnducifon. SelElnduced velocities
cause the vortex cores to approach each other
and fern a contact zone. Ibe vortex cores
flatten and stretch to ferry a Pole cross-
sectlon.
2. I. \brt1chy is amUldlated in the con-
tact zone and bridges of cross-Dnked vortex
lines fbrr1 at both ends of the contact zone.
Ibe bridges are orthogonal to the posHlons
the vortex tubes bad pclor to contact.
3. ~. Refits of the orlyln~ vortex
ties Form between the brldees as the vortex
ties pad fit.
NO salutes of vortex rlug reco~ectlons
Facade W~cke~ans ~ Leonardls (1989) vor-
ton ~thod1 Ch~erlaln ~ Lluls fl985' ~te-
~~ence Todd Id Klda1 ~aoka1 ~ Hus-
s~nls (1989) t~ee-~menslon~ spectra methods.
A notable Bathe of Ch~erla~ ~ gluts ~te-
~rence echelon is tbelr use of m~tEpole expand
. - ,. · .. .
Sons to specify a ramatlon conmbon.
According to He~oltz theorems vortex Laments
~ bag ~ ~ -- a Id ~d~ ~
cous) because the vortlcity Held is solenoid. In
ad~tlon Kelv1n~s clrc~atlon theorem states that
vortex Gents move as maters Dues Id coot
be brown ~ ~ lovlscld Had. As a reset vor-
tex reco~ctlon is ~ l~erently viscous phenol
ena. Slggla ~ Parks (1987) an~ysls of vortex
Wants suggests that reco~ectlon shad the
place on viscous tl~ scales' as one fight elect.
Howlers Mehon, e# sZ1 (1989j provide _Hc~
Id theoretlc~ evidence that the reco~ectlon pro-
cess occurs on nearly couvectlve time scales. Thea
Wants we s~ported by Kwouls (1989) emery
Into etudes of vortex rings co~ectlng fib tree
s=~cee ~ch show no time dependence as a ~e
don of Reynolds namer.
The e~er~nts of Saga ~ Henderson (1984)
Id S=p~a (1986) locate that Crow 1nst~1D-
ties can develop ~ the trying awes of deeply s--
~rged by~o~lb. 0= own Navler-Stokes d~a-
dons of a sl~so1d~y perturbed vortex the ~v-
lng pang to a gee sauce show that reco~ecdon
of the parlay vortex the with the bee sauce ~
possible even if the co~utatl~ domain is too
short to Perot Crow lastly to occur. Based
the experl~nt~ evidence we gay speculate
that the preferred ws=kngth of parlay vortex re-
co~ecHon corresponds to a Crow ~st~lUt~ but
shown by on own ~Hc~ signs Crow
st~lUtles me not a necessary conatlon far the
co~ectlon of pray vortlcHy with the tree s=-
~ce.
0~ ~erlc~ d~atlons also 1nacate that the
generation of bedded vort1chy plays ~ essentl~
role in the reco~ect10n of secondly vortlchy filth
the hee-s=~ce. Inst~1Utles in the parlay vor-
tex the Open to be the source of Bach vorucH~
Once the beach voracity ~ grad ~ merges with
the secondly vort1cHy that is separated Mom the
hee~ce bo~d~y layer by the parlay vortex
the. Was Dogleg process eventually leads to the
rmatlon of U-sbeed vortices that me wrapped
goad the parlay vortex the. Ibe D-vortlces
gay connect with the gee sauce at their bases or
tips.
1.3 Iuteractlou of ~rtlces alto ~ee-
Sur~ces
Mama ~ Barnes 'p89) e~ed~nt~ gauges of
jet Rows interacting filth Gee sauces show that
the lnteractlons of vortlchy filth the gee sauce
gay generate short instead awes. They Id
that sauce waves me generated as vortlc~ struc-
t~es in the jet approach the bee sauce.
downstream the sauce Titles me caused by
l~ge-sc~e vortlc~ structures interacting erectly
with gee sauce. They also observe that vortex
lines co~ectlng filth the tree sauce me cow
On ~ downstream. The early development of
the ne=-s=~ce jet UseK is studied experlmen-
t~y by Llep_ (1990~. D=lug the tr~dtlon
phase Mom Lang to tangent A°~1 scourged
Id ne=-s=~ce lets convert az~th~ vortlclty
(vortlc~ rings) luto stre~lse fortlolt~ His e~-
perl~nts show that the presence of a tree sauce
accelerates the Talon of stre~lse vortlolt~
730
OCR for page 731
Sarpkaya, et al, (1988) use experiments and nu-
merical simulations to study the normal incidence
of a two-dimensional vortex pair with the free sur-
face. A novel counter-rotating plate arrangement
is used during the experiments to generate vortex
pairs. For their numerical simulations the vortex
pair is modeled using two point vortices and line
vortices are used to model the free surface. Numer-
ical results and experiments are presented for two
Froude numbers, Fr = 6.4 & 2.8. The Reynolds
numbers are not provided. For low Froude num-
bers (Fr < 0.8), the experiments indicate that the
free-surface acts like a free-slip wall, and the vor-
tex centers spread apart. A scar (a dip in the free-
surface elevation) appears in front of each vortex
and moves with the vortex. At higher Froude num-
bers, both experiments and simulations show that
a hump of fluid rises above the vortex pair. The
vortex pair subsequently spreads apart before dif-
fusing due to turbulence.
Telste's (1989) numerical studies of the interaction
of two point vortices with a free surface suggest
that vortices translating beneath a free surface will
not generate significant wavetrains at low Froude
numbers, Fr < 0.5. For higher Froude numbers
Telste's results indicate that wave breaking may
inhibit the outward expansion of the vortex pair.
For Fr ~ 6.4 the point vortices entrain themselves
in a hump of fluid that is located entirely above
the mean position of free surface. Telste's numeri-
cal simulations indicate that the hump would even-
tually collapse on top of itself. For an intermedi-
ate Froude number (Fr ~ 2.0), steep wave troughs
form on each side of a small hump of fluid. The
curvature of the wave troughs is so high that wave
breaking appears to be imminent.
Willmarth, et al, (1989) also study the normal in-
cidence of a two-dimensional vortex pair with a
free surface. They compare experimental results
to the predictions of an inviscid theory which uses
dipole sheets to represent the free surface and the
vortex pair. They report their Froude number as
Fr = 2.2, and we estimate their Reynolds and We-
ber numbers as Re ~ 5, 600 and We ~ 60. As the
Kelvin oval (the volume of fluid entrained by the
vortex pair) rises upward, the free surface forms an
upwelling that is centered above the vortices. Two
sharp troughs form at the edges of the upwelling.
The vortex pair rises slightly into the upwelling
and then sinks as the upwelling springs back. Yu &
Tryggvason (1990), call this phenomena, which de
pends on the Froude number, an inviscid rebound-
ing effect. Although the flow field induced by the
vortex pair becomes turbulent during the experi-
ment, Willmarth, et al, do not report observing the
formation of secondary vortices.
Ohring & Lugts' (1989) numerical studies of
the viscous interactions of two-dimensional vortex
pairs with a free surface indicate that secondary
vortices form at low Froude numbers. They use the
method of artificial compressibility with a general-
ized coordinate system to solve the Navier-Stokes
equations. They consider two cases with very low
Reynolds numbers (Re < 20) and intermediate
Froude numbers (Fr ~ 0.3 & 1.0~. Surface ten-
sion effects are not modeled, but exact boundary
conditions on the free surface are otherwise speci-
fied. For the high Froude number result, the vortex
pair entrains itself in the upwelling, and a patch of
counter-sign vorticity develops at the base of the
upwelling where there is high curvature. At the low
Froude number, the height of the upwelling is sig-
nificantly reduced and the vortex pair splits apart
upon impact with the free surface. Secondary vor-
tices form and partially wrap themselves around
the primary vortex, but the Reynolds number is
50 low that the vortices lose too much energy for
significant inviscid interactions to take place.
The normal and oblique incidence of vortex rings
with a free surface has been studied experimentally
by Kwon (1989~. Kwon generates laminar vortex
rings with Reynolds numbers, 600 ~ Re < 3200,
and Froude numbers, 0.14 ~ Fr < 0.32. We esti-
mate that his Web er numbers are 0.5 < We < 5.
Kwon documents single and double reconnections
at oblique angles of incidence. According to Kwon
the reconnection time depends on the angle of in-
cidence but not the Reynolds number.
The experiments of Sarpkaya & Henderson (1984)
(see also Sarpkaya, 1986) show that trailing vor-
tices shed by wings may interact with free surfaces
to form striations and scars. The striations oc-
cur during the initial phases of the interaction and
are normal to the wing's track. The scar features
form after the striations, and they are parallel to
the wing's track. The scars appear to be localized
free-surface disturbances which move with the vor-
tices. The early development of the scars is similar
to the free-surface upwellings and the steep wave
troughs that have been already discussed. In the
later stages of the flow, dimples form in the scar
region due to normal vorticity connecting with the
73~
OCR for page 732
free surface. They observe that Crow instabilities
can occur for deeply submerged wings, which of-
ten leads to the connection of vortices with the
free surface and a crescent shaped disturbance on
the free surface. The nondimensional parameters
(in our notation) which characterize their experi-
ments are as follows: 7,000 < Re < 21,000 and
0.3 < Or ~ 1. Based on this data, we also estimate
that 10 < We < 100. In this parameter regime,
Sarpkaya (1986) states that the free-surface signa-
tures are not sensitive to changes in the Reynolds
and Froude numbers. Sarpkaya claims further that
the influence of capillary waves on his free-surface
signatures is very small, although no experimental
evidence is offered.
In contrast to the results of Sarpkaya's experi-
ments, Bernal, et al, (1989) and Kwon (1989) re-
port that the presence of surface active agents on
free surfaces strongly influences the interaction of
ring vortices with free surfaces. They find that
the motions of a vortex ring normally incident to a
contaminated free surface are similar to the mo-
tions caused by a no-slip wall. Secondary and
tertiary vortices are shed, and the primary vor-
tex rebounds. For a cleaner free surface they find
that the shed vortices are weaker and the radial
expansion of the primary vortex is greater. They
report Re ~ 1200 and Fr ~ 0.09, and we esti-
mate We ~ 0.9. These physical parameters are
less than Sarpkaya's (1986) and may explain why
Sarpkaya's observations differ from Bernal, et al,
in regard to the influence of surface contamination
on surface signatures. Bernal, et al, (1989), also
report preliminary experiments with delta wings
towed beneath a free surface. They observe what
appears to be a Reynolds ridge, which is the flow
region beneath a stagnant film on the water sur-
face (see Scott, 1982~. According to them, the
scar features that are observed by Sarpkaya may
in fact be Reynolds ridges. Interestingly, Grosen-
baugh & Yueng (1988) report that surface contam-
ination strongly influences the onset of turbulence
in the bow waves of model ships. They describe
a stagnant zone which causes the free surface to
separate farther upstream than it would in the ab-
sence of surface contamination. This description
fits Scott's definition of a Reynolds ridge.
As one example of the range of scales that are
possible for full-scale ships, consider a lightly-
loaded foil traveling slightly below and parallel
to the free surface. For negative lift, the trail
ing tip vortices will rise upward. We assume a
long span, s=O(lOm) and a low circulation, I, =
0~1m2/~. In our notation we have then relatively
high Reynolds numbers (Re ~ 105), low Froude
numbers (fir ~ 10-2), and high Web er numbers
(We ~ 103~. This parameter regime is far out-
side the numerical simulations and laboratory ex-
periments that we have reviewed in this section.
The impact of this mismatch is not clear. We
note that our own numerical simulations of vortex
interactions with a free surface are near the ap-
propriate Froude numbers, but our Reynolds num-
bers (Re = 0~103~) are too low. Hopefully, the
three-dimensional flow phenomena that we observe
at these low Reynolds numbers capture some of
the important features of flows at much higher
Reynolds numbers. We emphasize that inviscid
approximations are inadequate because the salient
features of vortex interactions with a free surface
such as boundary-layer formation and vortex re-
connections are fundamentally viscous.
Having noted these limitations, we now proceed
to summarize the free-surface phenomena that we
observe in our direct numerical simulations of the
Navier-Stokes equations. Our numerical studies of
vortex rings interacting with a free surface show
that secondary vortex rings may be torn away
from the free-surface boundary layer at intermedi-
ate Froude numbers, Ad at lower Froude numbers
the free surface acts like a free-slip wall. Our nu-
merical simulations at low Froude numbers agree
qualitatively with the vortex ring experiments of
Bernal, et al, (1989) and Kwon (1989~. Our stud-
ies of vortex tubes illustrate two possible mecha-
nisms for the reconnection of vorticity with a free
surface including primary and secondary vortic-
ity reconnections. The free-surface signatures of
these two mechanisms are distinctly different. One
type of primary vorticity reconnection should re-
sult in a cusp pattern on the free surface, and sec-
ondary vorticity reconnection should manifest it-
self as paired dimples on the free surface. These
numerical experiments show that the generation of
helical vortices is an initial stage in the reconnec-
tion of the secondary vorticity with the free surface,
and our simulations also suggest that the striations
that may be observed in the waltes of trailing tip
vortices are caused by the interaction of the helical
vorticity with the free-surface boundary layer.
732
OCR for page 733
1.4 Numerical Methods
A very general method for the direct simulation
of the incompressible Navier-Stokes equations with
free-surface effects is the MAC (Marker-And-Cell)
method (see Hirt, et al, 1975~. MAC methods
are distinguished from other numerical methods by
their technique for solving for the pressure, treat-
ment of the convective terms, formulation on a
staggered grid, and capability for particle tracing.
MAC methods enforce mass conservation by solv-
ing a Poisson equation for the pressure. The MAC
equation for the pressure retains temporal and spa-
tial derivatives of the dilation term (V u) to inhibit
the growth of numerical errors, and the convective
terms in the momentum equations are calculated
using a combination of upwinding and central dif-
ferencing. The MAC time integrator without the
upwinded terms is similar to the explicit FTCS
(E`orward Time Central Space) method (Roache,
1976).
As evidenced by the application of the MAC tech-
nique to a wide range of problems including nonlin-
ear ship wave problems (see Miyata & Nishimura,
1985), the MAC method is useful for determin-
ing qualitative features of very complicated free-
surface flows. For simple geometries the high
accuracy of spectral methods relative to finite-
difference methods has proven to be very de-
sirable. A good example is Orszag & Patter-
son's (1972) direct simulation of homogeneous and
isotropic turbulence using Fourier series. In the-
ory Patera's (1984) spectral element method al-
lows complex geometric modeling with spectral ac-
curacy, but in practice general three-dimensional
problems have not been very accommodating. As
a result, low-order firLite-difference methods with
high resolution appear at this moment to be more
robust than spectral methods.
The MAC and FTCS methods are subject to
stringent stability criteria because of their use of
explicit time integrators, and as a result, sev-
eral semi-implicit schemes have been proposed
for overcoming this deficiency, including SIMPLE
(Patankar & Spalding, 1972), SIMPLER (Pan-
tankar, 1981), PISO (Issa, 1985), and the Finite-
Analytic method (Chen & Chen, 1982~. These four
methods solve the momentum equations for the ve-
locities and the Poisson equation for the pressure
separately, and iterations are used to strongly cou-
ple the velocities with the pressures. An interest
ing feature of Chen & Chen's method is their use
of locally analytic solutions to solve the Navier-
Stokes equations. They claim that this technique
gives superior modeling of the convective terms
without incurring a numerical diffusion penalty.
However, the Finite-Analytic method requires sig-
nificantly more memory without improving the
second-order spatial accuracy of more conventional
finite-difference methods.
In contrast to these iterative solution procedures,
Kim & Moin's (1985) fractional step method uses
only one iteration to couple the velocity field with
the pressure field to maintain a divergence-free
flow. Note that Kim & Main, just like Ghia, et al,
(1977), use Neumann boundary conditions in the
Poisson equation for the pressure, and as a result,
they must satisfy a numerical solvability condition.
Kim & Moin are able to achieve second-order accu-
racy in time by using an explicit Adams-Bashforth
scheme for the convective terms and an implicit
Crank-Nicholson scheme for the viscous terms. A1-
though Kim & Moin's hybrid time-integrator elim-
inates time-step restrictions due to viscosity, their
numerical scheme must still meet a Courant con-
dition because of their explicit treatment of the
convective terms.
A fully-implicit scheme has been devised by
Hartwich & Hsu (1988) who uses Chorin's (1967)
artificial compressibility method as a framework.
Chorin's method inserts a time-dependent pres-
sure term in the mass-conservation equations, and
as the solution nears steady-state, the pressure
term drops out. This artifice permits the use
of advanced compressible flow techniques (e.g.,
Warming & Beam, 1978~. In addition, the tech-
nique can be generalized to time-dependent prob-
lems. Other researchers including Brandt & Di-
nar (1977), Fuchs & Zhao (1984), Vanka (1986),
and Thompson & Ferziger (1989) solve the incom-
pressible equations implicitly. Typically, these im-
plicit primit~ve-variable solution schemes use block
Gauss-Seidel iterations to drive the divergence to
zero, and multigrid methods are used to acceler-
ate the solution procedure. The block in lieu of
point Gauss-Seidel solvers are necessary because
the finite-diRerence approximations to the mass-
conservation equation are not diagonally domi-
nant.
So far we have limited our discussions to the
primitive-variable formulation of the Navier-Stokes
equations. The widespread use of vorticity formu
733
OCR for page 734
rations also warrant a discussion. More compre-
hensive reviews of vortex methods can be found
in Saffman & Baker (1979) and Leonard (1980 &
1985~. There are two main vorticity formulations:
vortex element methods which use line and sheet
singularities and finite-difference approximations
of the vorticity-stream function equations. Vor-
tex element methods are ideally suited for sim-
ulating small regions of rotational flow such as
isolated vortex tubes (Leonard, 1985) and sharp
density interfaces (Dahm, Scheil, & Tryggvason,
1989~. Generally, large core deformations and vis-
cous phenomena are not modeled well by three-
dimensional vortex element methods, but Winck-
elrnans & Leonard (1989) have developed a vorton
(vortex sticks) method to study the initial stages
of vortex reconnection. Finite-difference approxi-
mations to the vorticity-stream function equations
do allow large core deformations with strong vis-
cous interactions, but boundary conditions other
than free-slip boundary conditions are difficult to
implement in three dimensions. We note that vor-
tex methods are continually being improved and
their range of application will certainly increase
with time.
Regardless of what formulation is used to simulate
the incompressible Navier-Stokes equations, im-
plicit schemes are traditionally used for rapidly at-
taining steady-state solutions and explicit schemes
for simulating time-dependent flows. Explicit
schemes are not used to reach steady state because
they require too many time steps, and implicit
schemes are not used for time-dependent problems
because they are computationally more intensive
and the time steps are so small based on accuracy
considerations that stability is often not a prob-
lem for explicit schemes. In some special cases of
time-dependent flows, such as creeping flows and
capillary waves, implicit schemes are used because
the stability limits of explicit schemes are too re-
strictive.
In this paper, we present a second-order explicit
method and a first-order semi-implicit method for
solving the three-dimensional Navier-Stokes equa-
tions in primitive-variable form with and with-
out free surfaces. The unconditional stability of
the semi-implicit scheme is useful for obtaining
steady-state solutions, and the accuracy of the ex-
plicit scheme is desirable for time-domain prob-
lems. In both numerical schemes the effects of
numerical viscosity are reduced to a minimum by
using central differencing. The nonlinear momen-
tum equations in the implicit scheme are solved
using Newton-Raphson linearization and multigrid
iteration. We also use multigrid iteration to solve
the Poisson equations for the pressures in both
time-stepping procedures. The convergence prop-
erties of the numerical schemes are demonstrated
for three test problems: (1) the a~cisymmetric stag-
nation flow against a wall, (2) the viscous attenua-
tion of an axisymmetric standing wave, and (3) the
translation and diffusion of a two-dimensional vor-
tex. The numerical schemes are used to study sev-
eral physical problems including vortex rings and
tubes impacting walls and free surfaces.
2 Mathematical Formulation
2.1 Field Equations
We consider the unsteady incompressible viscous
flow of a Newtonian fluid under a free surface. For
an isotropic and homogeneous fluid, the Navier-
Stokes equations for conservation of momentum
have the form:
- + (u V)U = -Up + R-V2U,
e
(1)
where u = u(:e, y, z, I) = (a, v, w) is a three-
dimensional vector field. Note that the momen-
tum equations have been normalized based on a
length scale L and a velocity scale U. Re = UL/L,
is the Reynolds number where ~ is the kinematic
viscosity. In addition, we have defined p as the
hydrodynamic pressures
P=P-F2Z,
r
(2)
where P is the total pressure which is equal to the
sum of the hydrodynamic plus hydrostatic pres-
sures. The pressure terms are normalized by pU2
where p is the density. F2 = U2/gL is the Froude
number. The vertical coordinate z is positive up-
ward, and the origin is located at the mean free
surface. The dyn~nic pressure may be interpreted
as a Lagrange multiplier which projects the veloc-
ity field onto a divergence-free field as expressed by
the mass-conservation equation:
V · u = 0 .
734
(3)
OCR for page 735
The pressure field itself does not require boundary
conditions or initial conditions, but a well-posed
initial-boundary-value problem does require these
conditions for the velocities. Note, however, that
the field equations and boundary conditions for
the velocity field can be used to deduce the ini-
tial pressure and the behavior of the pressure near
the boundaries. For example, the divergence of
the momentum equations (eqts. 1) used in combi-
nation with the mass-conservation equation (eqt.
3) may be used to derive a useful Poisson equa-
tion for the pressure. This equation expressed in
indicial notation is as follows:
V2p = _ ~Uj UP
ant B~j
Sirrularly, the momentum equations may also be
used to prescribe the normal derivative of the pres-
sure on the boundaries of the fluid. Thus, accord-
ing to the divergence theorem the pressure is sub-
ject to the following solvability condition:
/s an /v {3zi ire ' ~ ~
where V is a volume of fluid, S is the surface
bounding the volume, and n is the unit outward-
pointing normal on that surface.
For the velocities, we are required to specify the
initial velocity field:
u = uO(:z, y, z) at t = 0 . (6)
According to Helmholtz's theorem, we can express
this initial vector velocity field uO in terms of scalar
¢0 and vector JO = A?, ye, ?~Az~o velocity poten-
tials:
Uo=V¢o+VX?po. (7)
The initial velocity field, like the time-dependent
velocity field, is required to be solenoidal. There-
fore, we can show that by taking the divergence of
equation (7) that TO satisfies Laplace's equation:
V2+o = 0 . (8)
Here, TO may represent the effects of currents,
the disturbances of surface-piercing and submerged
bodies, etc. A similar Poisson equation is satisfied
by the vector potential To in terms of the initial
vorticity field wO
V2¢o =-To ~ (9)
where ~ = (wc,~',,wz) may be expressed in terms
of the curl of the velocity field at any instant of
time:
~ = V x u. (10)
Based on this definition, the vorticity field is itself
a solenoidal quantity. In addition to posing ini-
tial conditions for the velocity field, we must also
specify boundary conditions as discussed in the fol-
lowing sections.
2.2 Linearized E ree-Surface Boundary
Conditions
Let the free surface elevation be given by ,1~:r, y, t).
We posit the presence of a viscous free-surface
boundary layer scaled by fS=O(Re i/2~. We ob-
tain a linearization of the free-surface boundary
conditions by assuming small surface slopes, i.e.,
'7~ & t7',=0~) << 1, and consequently ,7 is
much smaller than the other length scales of the
flow including [. Under these assumptions, the
exact (nonlinear) field equations are still valid out-
side the boundary layer, but the nonlinear kine-
matic and dynamic free-surface boundary condi-
tions (Wehausen & Laitone, 1960) reduce to:
uz . O
vz = 0 (11)
1
-P + F2 ~
ql + use + van =
=
(12)
where We = pU2L/T is the Web er number, p is the
water density, and T is the surface tension, and
the conditions are applied on z=O. The first set
of equations (11) state that the shear stresses and
normal stresses are zero on the free surface. Note
that the kinematic free-surface condition (12) is
still nonlinear in the flow variables and is obtained
without otherwise requiring that the velocities be
sit.
73s
OCR for page 736
2.3 E`ree-slip Wall Boundary Conditions
We next consider the case of a free-slip wall. This
type of boundary condition is useful for imposing
symmetry boundary conditions and modeling free-
surfaces at low Froude numbers. By definition the
tangential stresses and the normal flux are zero
on a free-slip boundary. Therefore, the boundary
conditions for the velocity on a free-slip plane at
z = 0 are:
uz=0, vz=0, and w=0, (13)
where the first two equations give zero tangential
stresses and the third equation states that there is
no flux across the plane z = 0.
These boundary conditions for the velocities en-
able us to deduce the behavior on the free-slip wall
of the pressure and vorticity, and also the initial
conditions for the scalar and vector velocity po-
tentials. For example, upon substitution of the
boundary conditions (13) into the z-component of
(1), we can show that pz=0, where the z-derivative
of (3) has been used to eliminate the u~zz=0 term.
Similar arguments may be used to show that the
vorticity vector is always normal to a free-slip wall,
~ = (O,O,wz) on z=0. Finally, in regard to the
initial conditions, we can derive the following rela-
tions for the vector velocity potential on the free-
slip plane z = 0:
(~10 = 0, (~10= 0, B(¢Z)° = 0,
and V21,bo = -~0, 0, (~z)O), (14)
where we have assumed that the scalar velocity
potential is initially zero, .50 = 0.
2.4 No-slip Wall Boundary Conditions
On a no-slip wall the velocity is zero:
u = 0 . (15)
In general, the shear stresses are nonzero and as a
result, boundary-layer formation and flow separa-
tion are possible. Certainly, the flow field near a
no-slip wall is more complex than that for a free-
slip wall, but it is still possible to deduce the behav
ior of certain flow quantities near the wall. For ex-
ample, suppose that a no-slip wall is located on the
boundary x = 0, then the no-slip boundary condi-
tions (15) and the mass-conservation equation (3)
may be used to show u,* = 0. Also, substitution
of the no-slip conditions into the z-component of
momentum (1), gives the normal derivative of the
dynamic pressure:
R ZZ · (16)
e
A similar procedure may be used to show that
the source term is zero in the Poisson equation
for the pressure (4~. Unlike a free-slip wall, the
vorticity vector must be parallel to a no-slip wall,
w = (~v, O) on z = 0, and as a result a vortex
tube cannot terminate normal to a no-slip wall.
For a no-slip wall it is difficult to pose unique ini-
tial conditions and boundary conditions that are
physically meaningful for the vector velocity poten-
tial. One method for eliminating this problem is to
let the velocity be prescribed by free-slip boundary
conditions at time t = 0, and then at time t = 0+
set all velocity components equal to zero. As a re-
sult, a sheet of vorticity is instantaneously formed
on the no-slip wall.
2.5 Conservation of Energy
The vector product of the velocity with the mo-
mentum equations ( 1) integrated over the fluid vol-
ume gives a formula for the conservation of energy,
and the transport theorem in conjunction with di-
vergence theorem may be used to simplify the re-
sulting equations. In indicial notation the equation
is as follows:
d ~ Mini
. =
at JV 2
/s njUit-P6ij + R ~ a0Ui + ~Uj )]
Re /v(8zj + ~Zj)0Zj ' (17)
where the first term represents the change in ki-
netic energy integrated over the material volume of
the fluid (V), the second term represents the work
done by the stresses on the fluid boundaries (S),
and the last term equals the energy that is dissi-
pated. (tij denotes the Kroenecker delta function.)
736
OCR for page 737
Note that the stresses do not do any work on slip
and no-slip walls. Upon substitution of the exact
free-surface boundary conditions into this energy
equation, we may derive the following formula:
d ~ nisi 1 d ~ 2
~ - ~ ~ ~ =
d! Jv 2 ' 2F2 dt Js'
We |S~ ~t( Ri + R2 )
Re JV( Ski + 02'i ) I9~ , (18)
where the first term represents the change in ki-
netic energy, the second term the change in poten-
tial energy, the third term the work done by surface
tension, and the last term the dissipated energy.
Sf is the projection of the free-surface onto the xy-
plane and Rat and R2 are the principal radii of cur-
vature. Note that we have assumed that the work
done by the stresses on all other boundaries besides
the free surface is zero. If the free-surface elevation
and slope are small, then the volume integrals are
evaluated below the mean waterline (z = 0) and
the principal radii of curvature terms simplify to
-(~= + ~vu)
3 Numerical Formulation
In this section we derive semi-implicit and explicit
finite-difference schemes for direct simulation of
the three-dimensional Navier-Stokes equations at
low Reynolds numbers. (See Appendix A for an
outline of the axisymmetric formulation:) We show
how the initial velocity field is set up, and demon-
strate the implementation of periodic, slip, no-slip,
and free-surface boundary conditions. For periodic
and/or wall boundary conditions a solvability con-
dition for the pressure is also implemented. We
then discuss the vectorization of a multigrid solu-
tion technique for solving the field equations. We
conclude the numerical formulation section with
stability analyses and validation studies. The val-
idation studies include numerical simulations of
axisymmetric stagnation flows, attenuation of ax-
isymmetric standing waves, and translation and
diffusion of two-dimensional vortices.
3.1 A Semi-Implicit Time-stepping
Scheme
We derive here a first-order semi-implicit scheme
for solving the Navier-Stokes equations. The basic
concept of our scheme is based on the PISO scheme
(Issa, 1985) which updates the velocities indepen-
dently of the pressure. We use Newton-Raphson
iteration to solve the nonlinear Navier-Stokes equa-
tions, and we assume that the velocities for each
implicit time step are equal to some known esti-
mates plus some small corrections. Let superscript
If" denote the n-th time step, and consider the per-
turbations of the velocities at the In + lath time
step as follows:
Un+l = U + U
vn+! = V + V (19)
wn+! = W + Hi ~
where (u,v,w~n+~ are the unknown velocities at
the next time step, (U,V,W) are the known es-
timates, and (u, v, fur are the unknown errors.
Note that for a fully-implicit scheme the hydro-
dynamic pressure would be expanded in a sim-
ilar way. For our serru-implicit scheme the hy-
drodynamic pressure is always known in terms of
the most recent estimate of (u,v,w~n+~. In (19)
we continuously update (u, v, w~n+i until the de-
sired accuracy has been achieved. First we es-
timate the fluid velocity at the In + Both time
step by letting (U. V, W) = (u, v, win, and then we
find the error terms, (u, v, w). Our improved es-
timate of the velocities is given as (u, v, we+ =
(U. V, W) + (u, v, w) and the new pressure can be
calculated based on these updated velocities. The
Newton-Raphson iterations can stop at this point
if the errors are small enough, or they may con-
tinue by letting (U. V, W) equal our new estimate
of (u, v, we+, etc. Generally, only one to two iter-
ations are necessary because the Newton-Raphson
procedure has quadratic convergence.
Upon substitution of the perturbation expansions
(19) into the momentum equations (1) and elimi-
nation of quadratic error terms, we derive the fol-
lowing set of linearized equations:
us + 2(uU)~ + (uV)v + (Uv),, + (uW)z
+ ( Uw In- -V2u = Ures
Rc
737
OCR for page 778
Figure 23e: 1~1 = 1.25 isosurfaces at t=15.
778
OCR for page 779
Figure 24: The interactions of a vortex tube with
a no-slip wall. The x~ = 0.5 (helical vor-
ticity), ~`,~=0.5 (axial vorticity), and ~ - 0.5
(total vorticity) isosurfaces are plotted at different
instants of time: (a) t=5, (b) t=7.5, (c) t=10, (d)
t=12.5, (e) t-15, (f) t=17.5, & (g) t=20. The view
is from below a no-slip wall, and the primary vor-
tex is moving into and to the left of the page due
to its images across the centerplane and above the
wall. The geometry triad is in a submerged corner
on the centerplane of the computational domain.
Note that the perspective in Part (g) is slightly
different from Parts (a)-(f). The numerical param-
eters for this run are provided in Run 3 of Table
(43.
Figure 24a: x/~ = 0.5 isosurfaces at t=5.
Figure 24a: two = 0.5 isosurfaces at t=5.
\1
779
Figure 24a: 1~',1 - 0.5 isosurfaces at t=5.
-
~1
OCR for page 780
Figure 24b: `/~ = 0.5 isosurfaces at t= 7.5.
_~
Figure 24b: ~-0.5 isosurfaces at t=7.5.
Figure 24b: 1~9,1 = 0.5 isosurfaces at t=7.5.
780
OCR for page 781
Figure 24c: `~ = 0.5 isosurfaces at t=10.
Figure 24c: ~ = 0.5 isosurfaces at t=10.
Figure 24c: levy = 0~5 isosurfaces at t-10.
\
\1
781
OCR for page 782
-
Figure 24d: )~2 + W2 = 0.5 isos~faces at t=12.5.
Figure 24d: ~ ~ _ 0.5 isosurfaces at t=12.5.
Figure 24d: 1~Vl-0.5 isosurfaces at t=12.5.
782
-
OCR for page 783
1
Figure 24e: I+ = 0.5 isosurfaces at t-15
Figure 24e: iw~ = 0.5 isosurfaces at t=15.
Figure 24e: 1~',1 = 0.5 isosurfaces at t=15.
783
OCR for page 784
Figure 24f: ,~ = 0.5 isosurfaces at t_17.5.
Figure 24f: 1wl = 0.5 isosurfaces at t=17.5.
784
Figure 24f: levy = 0~5 isosurfaces at t=17.5.
y
OCR for page 785
Figure 24g: x/~ = 0.5 isosurfaces at t=20.
Figure 24g: ~ = 0.5 isosurfaces at t=20.
/
785
Figure 24g: 1~Vl = 0~5 isosurfaces at t=20.
OCR for page 786
Figure 25: The unwrapping of U-shaped vortex tubes
around the primary Forte:: tube. The two = 0.5 iso-
surface of vorticity is plotted at time t=20. To em-
phasize the U-shaped feature this image has been
reflected about its midspan relative to the images
appearing in Figures (24~. The numerical param-
eters for this run are provided in Run 3 of Table
(4~.
Figure 26: The conservation of energy for a three-
dimensional vortex: tube interacting with a no-slip
wall as a function of time. The kinetic energy E(t)
is compared to the energy that is dissipated D(t)
(the Reynolds number term in eqt. (18) integrated
over time). The results are plotted as follows:
~ 1 ~ denotes E(t)/E(O) and ~-2-~ de
notes (E(O) + D(`t)~/E(O). The numerical param-
eters for this run are provided in Run 3 of Table
(4~.
.00
.95
.90
.85
.80
.75
.69
.65
.60
.55
.50
.45
1 1 1 1 1, 1 ' 1 ' 1 ' 1 ' 1 1 1 ~1
my.
,~
- 1 1 ~1 1
0 2 4
6 8 10
t
Am'
I , 1 , 1 1 1 1
12 14 16 1
786
8 20
J
OCR for page 787
Figure 27: The helical vorticity evaluated on the
wall. The contours of w2 on the wall are plot-
ted for different instants of time: (a) t=10, (b)
t=15, and (c) t=20. Observe that the features are
oriented normal to the axis of the primary vortex
tube which is parallel to the Taxis. The numerical
parameters for this run are provided in Run 3 of
Table (4~.
~9
~ ,
0 .5 1 0 1.5 2.0 2.5 3.0 3 5 4.0
X
787
1
. i
~, .
t . ~ |CONTOUR FROM - 7 TO .2 BY ;OC I
J
0 .s 1 0 1 5 2.0 2 5 3.0 3.5 4.0
X
Figure 27b: t=15.
OCR for page 788
DISCUSSION
P. Ananthakrishnan
University of California at Berkeley, USA (India)
My question/remark is regarding the results corresponding to the
interactions due to a vortex ring that is normally incident on the free
surface. According to Song, et al (in M. Song, N. Kachman, J.T.
Kwon, L.P. Bernal, and G. Tryggvason, Vortex Ring Interaction
with a Free Surface,. preprint, 18th Symp. on Naval Hydro.), a dip
on the free surface right above the vortex core, which moves radially
outward as the vortex ring stretches at the free surface, is observed.
They also observe three-dimensional small scale motions and
propagating waves at higher Froude numbers. But your results (Fig.
17) show a rise in wave elevation at the axis of symmetry that is
comparable in magnitude to the surface depression. Could you
comment on the possible reasons for the difference between the
observed and computed results?
AUTHORS' REPLY
Relative to Song, et al's, (1990) experiments, our numerical
axisymmetric results are valid for much higher Froude numbers.
Even so, we also predict the free-surface depression that occurs above
the vortex core as is evident in our Fig. (16). In regard to the small-
scale three-dimensional features that are observed in experiments, we
believe that some of these features may be explained in terms of U-
vortices. For example, Fig, (2b) of Song, et al's (1990) paper shows
a U-shaped feature wrapped around the primary vortex tube. The
base of the U is on the inside of the ring vortex. Both these features
are consistent with the U-vortex phenomena as explained in our paper
DISCUSSION
Ronald W. Yeung
University of California at Berkeley, USA
In your results corresponding to Fig. (16) and Fig. (17), you have
mentioned the existence of a non-zero mean wave elevation, which
you artificially subtract off to obtain presented results. The non-zero
mean implies that mass is not conserved; I was wondering if the
authors can shed some light on the source of this trouble?
AUTHORS' REPLY
For our linearized free-surface boundary conditions, the free-surface
elevation is much smaller than the boundary-layer thickness (~1 ~ b).
The Froude and Reynolds numbers that are illustrated in Figs. (16 &
17) push the limits of this theory so that an interesting physical
regime could be investigated. Fully-nonlinear free-surface boundary
conditions would eliminate the problem with mass conservation.
DISCUSSION
Fred Stern
The University of Iowa, USA
The authors should not use the terminology Direct simulation. to
refer to their solutions of the unsteady Navier-Stokes equations for
low Reynolds numbers (i.e., laminar flow). Generally, this
terminology has become synonymous with the direct simulation of
turbulence through extremely high-resolution solutions of the
unsteady Navier-Stokes equations for relative high Reynolds numbers
as opposed to the use of the Revnolds-avera~ed Navier-Stokes
equations. Would the authors please specifically point out the novel
aspects of their method since most aspects appear to be familiar. In
fact, this should be the focus of the discussion on the numerical
formulation which is too long and often confusing.
The real value of this work is in the use of such numerical methods
for free-surface flows and, in particular, the applications chosen for
study. The authors appear to have captured many of the observed
phenomena although I have not been able to decipher all that they
have from the figures. The free-surface boundary conditions used
appear to be identical to those used in my paper. Essentially, these
are inviscid approximations in which the viscous-stress conditions are
neglected, or if you like, only satisfied to a very low order. It would
appear then that many of the observed phenomena are pressure-
driven. Also, some of the vorticity in the calculations near the free
surface may be erroneous due to these approximations. Would the
authors please comment on these points?
AUTHORS' REPLY
We agree that ~laminar flow simulation" is more appropriate
terminology than Direct simulations. Both our semi-implicit and
explicit schemes use unique fully-vectorized multi-grid methods to
solve the unsteady three-dimensional Navier-Stokes equations at low
Reynolds numbers with and without free surfaces. Our linearized
free-surface boundary conditions are derived from the exact normal
and tangential stress conditions and the exact kinematic free surface
condition subject to our assumption of small free-surface slopes. The
range of validity of these free-surface boundary conditions are
discussed in our paper and in our answer to Professor Yeung's
question. A curved free surface, just like the afterbody of a bluff
object, may have unfavorable pressure gradients that lead to flow
separation.
788
Representative terms from entire chapter:
primary vortex
