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24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Efficient Methods to Compute Steady Ship Viscous Flow with
Free Surface
Hoyte C. Raven, Bram Starke,
Maritime Research Institute Netheriands (MARIN), NetherIands.
Abstract
The computation of the steady viscous flow with free
surface around a ship hull is addressed. To improve its
practical applicability, two different alternative meth-
ods are proposed:
· a 'composite approach' in which the wave sur-
face is computed beforehand using an inviscid-
flow approximation;
· a new formulation which permits to solve the vis-
cous free-surface problem by iteration, without
time integration.
The former method is practical, provides accurate pre-
dictions of many flow features, but neglects viscous ef-
fects on the wave pattern. Applications are shown to the
KCS test case, and to a case with wave-induced flow
separation.
The latter method solves the complete free-
surface RANS problem by an unconventional approach
which removes several difficulties and promises signif-
icant reductions of computation time and complexity.
Results are shown for a 2D bottom bump, a 3D pres-
sure patch and a Series 60 case.
1 INTRODUCTION
The practical application of Computational Fluid Dy-
namics methods for predicting the steady flow around a
ship hull has made much progress over the last decade.
Today, several of the CFD tools play an important role
in the ship hull form design. In particular nonlinear
free-surface potential flow codes for predicting a ship's
wave pattern are routinely used at institutes and ship-
yards since several years. These codes are generally ro-
bust, efficient and versatile, and give a good prediction
of the quality of a design from the viewpoint of wave
making (Raven, 1998~.
Complementary to that, in recent years the use
of RANS solvers for predicting the viscous flow around
the hull has increased. This provides important new
possibilities such as prediction of flow separation, wake
fields, thrust deduction, local flow directions etc. Sub-
stantial hull form improvements can thus be achieved at
least for fuller hull forms.
The complementary use of inviscid wave pat-
tern predictions and RANS predictions for viscous flow
generally implies a separate consideration of these as-
pects. Often the viscous flow computations suppose
symmetry boundary conditions at the undisturbed wa-
ter surface, the so-called 'double-body' approximation.
Thus neither an effect of the free water surface upon
the viscous flow, nor an effect of the boundary layer
and wake upon the wave pattern is incorporated. Ob-
viously, the same approximation underlies most towing
tank work, in particular resistance extrapolation meth-
ods; but taking into account these interaction effects is
a desired next step.
This objective has given rise to perhaps the
most active field of numerical ship hydrodynamics to-
day, the development of solution methods for the prob-
lem of the steady viscous flow around a ship hull with
free-surface boundary conditions. A variety of methods
has been proposed and appreciable progress has been
made, as testified by the results presented at the recent
Gothenburg Workshop (Larsson et al, 2000~.
However, at present such RANS/FS
(Reynolds-Averaged Navier-Stokes equations with
Free Surface) methods are not widely applied in
practical ship design yet. Besides still required further
improvement in accuracy, the computation times and
memory requirements of these methods are a draw-
back. Within the short time frame of a usual merchant
ship design project, one often resorts to the simpler
double-body approximation to be able to complete the
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Representative terms from entire chapter:
free surface
study in time.
Therefore, to have practical benefit of
RANS/FS solvers, certain further improvements are
desired. The present paper addresses two questions in
this regard:
· Can more efficient approaches be devised to
solve the full RANS/FS problem?
· Is a full RANS/FS method always necessary or
are there suitable intermediate levels that include
the relevant physics but require less computa-
tional effort?
In Section 2, we discuss the state of the art in
RANS/FS methods, and indicate the aspects we believe
should be improved. Section 3 describes a practical ap-
proach that combines a free-surface potential flow code
and a RANS solver. This appears to provide a solution
that is equivalent to that of a full RANS/FS solver in
the largest part of the domain, but at a fraction of the
cost. In Section 4 we derive a new formulation of a
full RANS/FS algorithm that solves the steady prob-
lem by iteration, discarding any time-dependence; and
we show the results obtained so far, which indicate fast
convergence for the test cases tried. Prospects are dis-
cussed and conclusions drawn in Section 5.
2 STATE OF THE ART IN RANS/FS
METHODS
Methods used
While there is a variety of RANS/FS methods for ship
hydrodynamic applications, they have several aspects in
common, which we shall describe now to provide some
background to further discussions.
Almost all RANS/FS methods solve the
RANS problem for the flow around the ship hull sub-
ject to the usual boundary conditions on the hull (i.e.
no-slip, sometimes wall functions); uniform-flow con-
ditions at a remote outer boundary; and fully nonlinear
free-surface boundary conditions (FSBC's) imposed at
the actual water surface. The velocity components (in a
coordinate system fixed to the ship, with x positive aft)
are u,v,w, ((x,y) is the wave height, and all quantities
are nondimensionalised using ship speed U. a reference
length L, and gravity acceleration g. The general form
of the FSBC's then is:
· a kinematic condition that the free surface moves
with the flow;
~t+U
dieted the wave profile along the hull. However, the
wave pattern away from the hull was much more vari-
able, and generally subject to appreciable numerical
damping. In particular for the 5415 test case with its
more diverging waves, several predictions had already
lost half of the bow wave amplitude in the wave cut at
y/L= 0.172. Also the predicted stern wave heights in
the wave cuts presented at the workshop were generally
poor and most variable for RANS/FS methods: The el-
evation of the first stern wave crest was underestimated
by 20 to 85 %. The only inviscid prediction presented,
from RAPID, had just the right stern wave elevation at
the position of the wave cut (but a too large amplitude
further aft and right aft of the transom). Since solving
the full RANS/FS problem would be expected to pro-
vide a more accurate prediction due to incorporation of
viscous effects on the stern wave system, there is still
some work to do.
To improve the predicted wave amplitudes at a
distance from the hull, denser meshes should probably
be used; but it still is difficult today to reach sufficient
resolution to accurately predict the wave pattern even
at some distance from the hull, and at the same time
have sufficient resolution in the boundary layer; in par-
ticular at high (full-scale) Reynolds numbers and low
Froude numbers. Computation times are very substan-
tial already, e.g. often of the order of 100 hours CPU on
a fast workstation, for a grid of 1.5 2 million cells.
Nevertheless, clear progress is evident com-
pared to results published earlier, and the methods
clearly have good prospects to produce accurate solu-
tions in a near future.
Desired improvements
For a sensible study of ship wave making and to deduce
desired modifications to the hull shape in order to re-
duce dominant wave components, it is not sufficient to
have only an accurate hull wave profile, since that may
lead to wrong conclusions; and it is important to have
a prediction of the wave pattern up to e.g. 0.3L off the
ship's centreplane. Also short, diverging waves should
be represented. Therefore, to improve the accuracy of
the predictions, minimising the numerical wave damp-
ing is indispensable in the first place.
Secondly, we believe there is room for re-
ducing computation time and complexity of RANS/FS
computations; and a possible approach is presented in
Section 4.
In the third place, virtually all RANS/FS meth-
ods proposed aim at solving the complete problem
of the RANS equations subject to the nonlinear free-
surface boundary conditions imposed at the real wave
surface over the entire domain. It may be useful to
have available a hierarchy of simpler approaches, from
which the best compromise between completeness and
complexity can be selected dependent on the strength
of physical phenomena in an application considered. A
possible hierarchy of this kind will be proposed in Sec-
tion 5.
Finally, all RANS/FS solutions we have seen
so far have been for model scale. To really exploit
the additional capabilities of RANS/FS methods, they
must be able to give accurate predictions of the full-
scale wave pattern as well. The particular advantage of
RANS/FS methods, the incorporation of viscous effects
on the wave pattern, is of little practical interest if it
only applies to model scale; for the model-scale resis-
tance prediction thus obtained we do not even see a rea-
sonable way of extrapolating it to full scale, unless by
techniques used for model tests which again disregard
those same effects.
The present paper addresses the second and
third item mentioned above, and provides some indirect
information on the 4th item.
COMPUTING VISCOUS FLOW
UNDER A PRECOMPUTED WAVE
SURFACE
Full RANS/FS incorporates the interaction between
wave generation and viscous flow; both wave erects on
the viscous flow and viscous effects on the wave gener-
ation and propagation. Wave effects on the viscous flow
around a ship hull may be significant for all cases with
substantial wavemaking, the wavy upper boundary of
the viscous flow domain having an effect on the devel-
opment of the boundary layer all along the hull. On the
other hand, viscous effects on the wave pattern are gen-
erally insignificant along most of the hull (except those
connected with wave breaking) and only substantial in
the stern area, as is confirmed by many validations of
nonlinear free-surface potential-flow methods (Raven,
1998~. Therefore, along most of the hull there is no in-
teraction between the viscous flow and the free surface,
but just a one-way action of the wave pattern on the vis-
cous flow.
This can be incorporated completely by first
computing the wave pattern using a panel code, and
next to compute the viscous flow under that wave sur-
face, imposing free-slip boundary conditions at that sur-
face. Surprisingly, this possibility seems to have been
largely disregarded so far, except for the similar work at
NTUA (Garofallidis, 1996) in which viscous flow was
computed under a measured wave surface.
Compared with a full RANS/FS solution we
have to disregard one of the boundary conditions, the
normal component of the dynamic boundary condition
at the wave surface. The degree to which that condi-
tion appears to be violated by the solution, i.e. the re-
sulting deviation from atmospheric pressure at the wave
surface, indicates the validity of the approximation, as
discussed below.
This 'composite approach' has been briefly
proposed in (Winds & Raven, 2000) and (Hoekstra et
al, 2000), and in the meantime has been refined and ap-
plied. It uses two essentially standard tools: an inviscid
wave pattern prediction, and a viscous flow prediction
without free surface. The methods we use will be briefly
discussed next.
Inviscid wave pattern computation
The calculation of the wave pattern is made using a non-
linear free-surface potential flow method. We use the
code RAPID (Raven, 1996,1998), an iterative 'raised-
panel' method. The hull surface and a surface at a spec-
ified distance above the wave surface are covered with
source panels that, together with the incoming undis-
turbed flow, determine the velocity field. The strengths
of the source panels are solved from the boundary con-
ditions. On the hull, zero normal velocity is imposed;
on the wave surface, the kinematic condition (eq. 1,
in steady form) and the dynamic condition (2) are im-
posed. Therefore, the free-surface boundary conditions
correspond with those imposed in a viscous how, except
for the tangential dynamic conditions (31.
The boundary conditions being nonlinear and
to be imposed on an unknown wave surface, the solu-
tion needs to be found by an iterative process in which
the free surface and the coefficients in the FSBC's are
updated and the trim and sinkage of the hull are ad-
justed. In practical cases, ~10) iterations are needed.
The method is known to predict the wave pat-
tern accurately, with as principal exception an overesti-
mation of the stern wave system due to the neglect of
viscous effects. Computation times are modest, of the
order of 15 minutes on a usual PC for a practical case.
Computation of the viscous flow
All viscous-flow computations are carried out with the
computer code PARNASSOS (Hoekstra, 1999; Hoek-
stra & E,ca, 1998), which solves the steady Reynolds-
averaged Navier-Stokes equations on a boundary-fitted
mesh. The momentum and mass-conservation equa-
tions are fully coupled and solved in primitive-variable
form, without resorting to pressure correction meth-
ods or the artificial compressibility approach. Addi-
tional transport equations associated with the turbu-
lence model are treated as uncoupled from the mass and
momentum equations. Menter's one-equation model
(Menter, 1997) is used, extended with the correction by
Dacles-Mariani et al. (19951.
The governing equations are integrated down
to the wall (no wall-functions are used, not even for the
full-scale computations). Mesh points are strongly clus-
tered towards the hull to capture the gradients in the
boundary layer. The resulting very high aspect ratio of
the cells near the hull puts high demands on the solver
for the linear systems, which is one of the motivations
to maintain the coupling between the equations in the
iterative solution.
All terms in the momentum and mass-
conservation equations are discretised at least second-
order accurately. To avoid negative turbulence quanti-
ties, only in the transport equations in the turbulence
models we use a first-order upwind scheme for convec-
tion. The resulting set of non-linear algebraic equations
is quasi-Newton linearised.
In order to reduce the size of the discrete equa-
tion system we use a marching solution scheme. The
velocities and pressure of a number of transverse grid
planes are solved simultaneously. The grid planes are
visited in downstream order, while the elliptic character
of the RANS-equations is numerically recovered by it-
eration; contrary to earlier versions no 'parabolisation'
is applied. Each step of this iteration scheme includes
not only the downstream sweep through the computa-
tional domain, in which the eddy viscosity, the veloci-
ties and the pressure are updated, but also an additional
upwind sweep in which only the pressure is updated. A
computation is continued until the maximum variation
of the static pressure coefficient between successive it-
erations drops below 1 x 10-4.
In usual ship computations, the computational
domain extends from 25% of the ship's length in front
of the bow to 25 to 50% of the ship's length beyond the
stern. The width and the depth of the mesh are taken
approximately equal to twice the breadth and twice the
draught of the ship, respectively. At the outer bound-
ary of the viscous-flow domain, boundary conditions
are imposed that are derived from a potential-flow cal-
culation.
The PARNASSOS code is applied on a regu-
lar basis in practical ship design projects at MARIN,
now in some 70 computations per year; a number that
is quickly increasing.
The composite approach
The composite approach consists of first computing the
wave pattern using the free-surface potential flow code;
next, to generate a 3D grid in the domain under the re-
sulting wave surface; and then to solve the RANS prob-
lem on that grid. Mainly the last step has some particu-
lar aspects, to be discussed now.
The boundary conditions imposed to the vis-
cous flow are the usual no-slip condition on the hull;
tangential velocities and pressure at the outer boundary,
taken from the inviscid wave pattern calculation; and an
outflow condition for the pressure derivative also taken
from the inviscid flow.
For the free surface, there is now one degree
of freedom less compared to the original free-surface
problem, since the wave elevation is now fixed. Conse-
quently, like in any fixed-domain RANS problem only
three boundary conditions can be imposed for veloci-
ties and pressure at this surface, and one has to be sacri-
ficed. In our implementation the imposed set of bound-
ary conditions models a free-slip surface: the kinematic
FSBC (in steady form) and the tangential components
of the dynamic condition are retained, giving the fol-
lowing boundary conditions for the velocity compo-
nents:
AX + viny—w = 0, or V.n = 0
t.~.n = 0
(4)
(5)
Zero normal derivatives are imposed for all turbulence
quantities.
The pressure is evidently coupled to the en-
tire velocity field, and is primarily evaluated from the
discretised momentum balance normal to the wave sur-
face. We thus ignore the normal component of the dy-
namic FSBC, eq. (2). The error in this condition, i.e.
the deviation from the atmospheric pressure at the wave
surface, gives an indication of the validity of the given
wave surface for the viscous flow. From eq. (2) this er-
ror can be expressed in a local wave-height difference
according to
A; = Fn2/`yr
where ibid = a =~/(PU21-
(6)
I r rig/ kI- - ~
If the pressure deviation turns out to be zero
everywhere, our solution is identical to one of a full
RANS/FS solution. However, in practice at least some
local pressure deviations will be found. The meaning of
a solution having some local pressure deviations may
not be immediately apparent but can be understood as
follows. Suppose that the viscous flow under the pre-
scribed wave surface is found to have a pressure devia-
tion /~r~x,y). Then this prescribed wave pattern would
be correct for an imaginary case in which the same pres-
sure field llyr would act as an external pressure on the
wave surface. The required change of the prescribed
wave surface for our actual case is then the change of
the wave pattern that would result from a removal of
that external pressure field. In the spirit of a linearisa-
tion we may assume this to be qualitatively equal to
minus the wave pattern of such an external pressure dis-
tribution, i.e. a local increase of the wave elevation ap-
proximately in agreement with eq. (6), plus a trailing
~ ,---~00005~
(
Figure 1: Sten, wave pattern for Series 60 case, model scale;
bottom: wave height from inviscid flow computation; top:
wave height difference from pressure residual in RANS com-
putation;
wave pattern within a 'Kelvin wedge' downstream of
it. Upstream of the location of the pressure residual, its
effect decays quickly with distance. In other words: if
our result has only a local pressure deviation at an iso-
lated spot, we will have very nearly the full RANS/FS
solution everywhere upstream of it, but not in a Kelvin
wedge area downstream of it.
The fact that the change of the trailing wave
system is not observed in the pressure residual, is un-
derstandable by considering the fact that such a change
is a consequence of the generation and propagation of
wave energy. By the elimination of the dynamic bound-
ary condition from the system we solve, this compu-
tation does not display any wave physics by itself. A
local pressure error thus will not cause a trailing wave
system.
Besides the pressure deviations resulting from
the modelling, there may be some due to numerical er-
rors as well. Both the inviscid-flow and the viscous-
flow computations will involve slight discretisation and
other numerical errors. Since the numerical methods
are entirely different, the numerical errors will have
a different pattern. This again may cause some pres-
sure residuals at the prescribed wave surface, which
have no physical meaning. This kind of pressure resid-
uals should vanish upon grid refinement, and Windt &
Raven (2000) show an example of this.
Results and applications
Series 60
Figure 1 shows the pressure residual, expressed as a
wave-height difference, for the Series 60 Cb = 0.60
model at Froude number En = 0.316 and model-scale
Reynolds number Rn = 3.4 x 106, as reported by Windt
and Raven (2000). Compared with the wave height
itself, the wave-height difference is only substantial
around the stern, due to viscous effects on the stern
wave system. Besides, very close to the bow and the
bow wave crest some small wave height differences oc-
cur due to numerical errors (not shown in the figure).
Everywhere else the difference is negligible (less than
5 x 10-4L or 2.7 % of the maximum wave elevation).
KCS container ship
The composite approach has also been applied to the
'KRISO Container Ship' (KCS) at Froude number
Fn = 0.26, both for model and full-scale Reynolds
numbers (Rn = 1.4 x 107 and 2.5 x 109, respectively).
For the computation on model scale the mesh consists
of 345 points in main stream direction, 101 points in
wall-normal direction and 53 points in girthwise direc-
tion. For the computation on full scale the number of
points in wall-normal direction is increased to 151. The
maximum distance of the grid points adjacent to the hull
is below y+ = 0.6.
The availability of extensive experimental
wave-height data allows a detailed comparison of the
inviscid wave height prediction with reality. Fig. 2 com-
pares the wave pattern from RAPID with the data and
shows good agreement, with differences that are essen-
tially confined to the region aft of the transom and the
stern wave system.
Figure 2: Wave pattern of KCS; top: RAPID prediction; bot-
tom: experiment
-0.006 ~.0045 ~.003 ~.00150.0005 0.002 0.0035 0.005 0.0065 0.008
Figure 3: KCS, model scale; top: wave height difference from
pressure residual in composite solution; bottom: wave height
difference between experiment and inviscid solution.
Fig. 3 attempts to correlate these differences
with the pressure residuals in the composite solution.
The bottom half shows the predicted wave elevation
Figure 4: As Fig. 3, close-up of stern area
subtracted from the experimental wave pattern. Some
minor deviations occur along the crests of the bow wave
system. These can be due to small phase errors or insuf-
ficient resolution; of course, subtracting the two wave
signals is a severe test of the accuracy. However, the
principal deviations occur aft of the transom. It seems
rational to attribute these to the neglect of viscous ef-
fects on the stern wave system, and this is essentially
confirmed by the top half of Fig. 3, which shows the
wave-height difference from the pressure residual in
the viscous-flow computation at model scale. Again the
principal deviations occur directly aft of the transom.
Fig. 4 is a close-up of the stern area of the same figure.
It illustrates that the differences between the inviscid
prediction and the experimental data have the form of
a persistent error in the entire stern wave system, while
the pressure differences in the viscous solution are just
local. This is a clear demonstration of the statement
made above on the meaning of a pressure residual.
~ c
~ c
o~_~
~ i'' '"\\.
~1 ~ ~
experimen ~1
prescribed wave surface from Rapid
- - corrected wave surface from Parnassos, model scale
~ ---;--- corrected wave surface from Parnassos, full scale
JO
° 0.5 0.6 0.7 0.8
,==
x/L
1
0.9
Figure 5: Stern wave of KCS, at the centreline
1.0
For a more detailed comparison between the
experimental and predicted stern wave systems, wave
cuts have been made near the centreline aft of the tran-
som, see Fig. 5. Substantial differences between the ex-
periments and the inviscid predictions are found, e.g. a
30 % overestimation of the first wave crest at the centre-
line. However, the first experimental marker lies higher
than the transom edge (which is at the start of the full
line in the graph), indicating the presence of a thin dead-
water zone; the transom was not entirely cleared in the
experiment. This, and the fact that a boundary layer is
shed from the hull, leads to the reduced wave amplitude
and the forward shift of the stern wave peak.
If we now add the computed wave-height dif-
ferences (eq. (6~) to the imposed wave elevation, we
get the dashed line shown in Fig. 5, for the model-scale
Reynolds number. This of course neglects the effect on
the trailing wave system but gives an impression of the
magnitude of the correction. It indicates a reduction of
the stern wave height, of approximately the right size,
confirming again that the neglect of viscous effects on
the stern wave system is the main cause of errors in the
inviscid wave pattern computation.
The same can be done for full scale. Under
the same wave surface we have generated an appro-
priately refined grid and have run the RANS computa-
tion for the full-scale Reynolds number of 2.5 x 109. Of
course viscous effects decrease for increasing Reynolds
number, most notably resulting in a decrease of the
boundary-layer thickness and the width of the wake.
This is e.g. reflected in the predicted wake fraction,
which decreases from 0.29 to 0.18 between model and
full scale in the present computations. The full-scale
composite solution shows significantly smaller pressure
deviations at the stern; and applying these to the im-
posed wave elevations we obtain the dotted line in the
same figure. The conclusion that at full scale the vis-
cous effects on the wave pattern will be one half to 2/3
of those at model scale here seems justified.
The effect of the wavy surface on the wake
field is depicted in Fig. 6. In this figure a comparison
is made between the double-body solution, the present
solution beneath the precomputed wavy surface and ex-
perimental data at the line z/L = - 0.03 in the propeller
plane of the KCS container vessel. A moderate im-
provementis foundin the axial velocity for y/L > 0.01,
while further towards the centerline, that is, in the inner
part of the wake, the predicted axial velocity is hardly
affected by the shape of the surface. A more substan-
tial improvement is found for the vertical velocity com-
ponent (w/U), which shows an average increase of ap-
proximately dw/U = 0.02 throughout the entire cross
section. Similar results were reported at the Gothenburg
Workshop (Larsson et al, 2000) for full RANS/FS com-
1
0.8
0.7
0.6
~0.5
0.4
0.3
0.2
0.1
o
; . 1 ' ' ' ' I ' ' ' ' I
UIU
0.9 o
/
/ o
o
A
_
o
-O.1
0 0.005 0.01
y/L
V I U
0.015 0.02
Figure 6: Wake field of KCS. Dahed line: without wavy sur-
face; solid line, win wavy surface incorporated; symbols: ex-
penment.
putations.
In general, the results of the composite ap-
proach for the KCS case were competitive with those
of RANS/FS codes presented at the Gothenburg Work-
shop; and for the wave pattern, were even superior to
most, with as single exception the area just aft of the
transom. A considerable advantage from the practical
point of view, however, is the large reduction in calcula-
tion time. In the present computations both the conver-
gence behaviour and the calculation time are compara-
ble to double-body computations. An indication of the
required computational effort (on a single R12000 pro-
cessor of an SGI Octane workstation) is given in Table
1. The inviscid computation of the wave pattern preced-
ing it took just a few minutes of CPU time.
Table 1: Computation data for the KCS case
Scale model full
number of grid nodes 1.85M 2.75M
number of global iterations 189 151
tolerance ACp~r 10-4 10-4
CPU time [h] 10 15.75
Tanker stern
A second example to indicate the practical applicabil-
ity of the composite approach is given in Fig. 7. Here,
the result is shown of a tuft test performed in one of
MARIN's model basins for a full-block ship (Cb=
0.89) at a Froude number Fn = 0.159 and a model-
scale Reynolds number Rn = 1.3 x 107. This was part
of a research project by the CRS-Pods working group
of MARIN's Cooperative Research Ships.
In the photograph from the experiment a sub-
Figure 7: Flow directions along the afterbody of a full tanker
form. Top: experiment; middle: RANS without wavy surface;
bottom: RANS with wavy surface.
stantial region with flow reversal can be seen close to
the waterline near the stern. In the tests it was observed
that the extent of this flow separation depended on ship
speed, and that the onset of separation coincided with
the onset of wave breaking at the same location. There-
fore this was an evident case of viscous / wave interac-
tion.
The experimentally determined flow pattern
near the hull is compared with the viscous flow pat-
tern predicted by a double-body computation as well
as by the composite approach. As can be seen in the
middle figure, the flow separation is not present in the
double-body computation. The computation under a
precomputed wave surface, however, does indeed pre-
dict the experimentally observed area of flow reversal,
albeit slightly smaller in size. The correct representa-
tion of the stronger adverse pressure gradient in the
free-surface flow appears to be important. Of course,
some differences can be expected near the free surface,
since the present approach does not take into account
the interaction between the flow reversal and the free
surface and does not model wave breaking. Neverthe-
less, it can be seen that it is capable of capturing some
relevant physical phenomena in the flow, and in the
present case indicates the wish for a hull form modi-
fication to suppress this wave-induced flow separation.
4 A STEADY ITERATIVE
SOLUTION METHOD FOR FREE-
SURFACE VISCOUS FLOWS
Motivation
While the composite approach described in the previ-
ous section is a step forward and works efficiently, the
fact that the wave pattern is determined by an inviscid
approximation is a restriction. The absence of viscous
effects on the wave pattern causes deviations from the
experimental data at the stern, as was observed for the
Series 60 and KCS cases considered. While these de-
viations are localised, they make the prediction of the
stern wave amplitudes, and thereby of the wave resis-
tance, inaccurate; and in principle also the viscous flow
then will be locally less accurate. Therefore, the method
is limited and a complete, fully interactive RANS/FS
method is still desired; albeit possibly confined to just
the stern area.
In setting up such a method, we have cho-
sen a quite unconventional approach, first published in
(Raven & Van Brummelen, 1999) and (Van Brumme-
len & Raven, 20001. The motivation was that we want
to avoid some drawbacks of most RANS/FS methods:
the long computation times, the need to pass through
all temporary stages to find the steady state solution,
the persistent time dependence, etc. Therefore, we shall
first consider more closely the basis of these problems.
Time dependence and integration times
As discussed in Section 2, virtually all methods pro-
posed so far follow a time-dependent procedure until a
steady result is obtained. The wave surface is updated
using the kinematic free-surface boundary condition, or
a similar algorithm is applied for the convection of the
wave surface in fixed-grid methods.
However, this procedure is the origin of vari-
ous difficulties:
· The ship has to be accelerated smoothly to its de-
sired speed, an impulsive start is usually not per-
mitted. This takes time, extending the total inte-
gration time to be covered by the computation to
come to a steady result.
· Once the ship speed has become constant, it still
takes substantial time before a steady wave sys-
tem has established in an area around a ship.
Since in deep water the group velocity is just half
the wave phase speed, the wave energy spreads
rather slowly, and it takes time before a wave ele-
vation at a distance has taken its steady value. It is
impossible to find steady state earlier than that if
time accuracy is conserved. To give an example:
for a point at a distance of 1 ship length from the
ship's centreline, a wave component with diver-
gence angle of 60 degrees can only begin to settle
to its steady elevation once the ship has travelled
over 4.6 ship lengths at constant speed.
· The initial acceleration of the ship generates tran-
sient waves. Before a steady result can be ob-
tained, these have to leave the domain or be ab-
sorbed by damping regions. If artificial bound-
aries are imperfectly transparent, persistent time-
dependence may occur in calculations due to
sloshing between the boundaries.
· As is analysed in (Van Brummelen et al, 2001),
the asymptotic decay of the unsteady effects is
determined by one particular wave component,
the transient wave that propagates in the same di-
rection as the vessel and has phase speed twice
the ship speed. Its group velocity equals the ship
speed, such that its wave energy stays with the
ship and only decays by dispersion. In 3D cases,
the asymptotic decay of this wave is only ~(1/~.
The effect of this wave is evident in many pub-
lished convergence histories for wave resistance,
as persistent, slowly decaying oscillations with
nondimensional period AT.V/L= 8~Fn2. This
only further prolongs the required integration,
sometimes causing users to just take an aver-
age over the last few periods instead of a desired
steady solution.
· Since the algorithm essentially uncouples the dy-
namic conditions (imposed to the RANS solu-
tion) and the kinematic free-surface conditions
(used to update the wave surface) in each time
step, small time steps must be made and a CFL
condition usually needs to be respected.
· If we refine the discretisation, the time step needs
to be reduced. However, in order to have bene-
fit of the better accuracy, also the unsteadiness
must be reduced. As derived in (Van Brummelen
et al, 2001), for a 3D method with second-order
discretisation, this means that the number of time
steps must be of ~(h-3) halving the grid spac-
ing could ask for an 8-fold increase of the number
of time steps!
· The no-slip boundary condition on the hull, to-
gether with the time-dependent kinematic free-
surface condition, produces the 'contact line
problem': the waterline location cannot move
during the time integration. Generally this prob-
lem is avoided rather than solved.
Of course, there are ways to alleviate some
of the difficulties mentioned, and not all methods suf-
fer equally. Nevertheless, the number of required time
steps generally ranges from 3000 to 30000 for pub-
lished 3D ship cases. At each step the grid needs to be
adjusted in a surface-fitting method, or the free surface
reconstructed in a VOF method; and the effort is ambi-
tious to say the least.
The obvious solution for these difficulties is
to avoid all unsteadiness and to solve a strictly steady
form of the problem directly by iteration. The problem
then only admits wave solutions that satisfy the steady
dispersion relation, excludes any transients at startup,
reduces reflection problems at artificial boundaries, and
alleviates contact line problems. Exactly this has been
done in free-surface potential flow solution methods
since a long time. For these it is evident that for solving
steady problems, steady solution methods are far more
efficient than unsteady ones. Solving this steady prob-
lem for RANS equations seems just as desirable the
only question is: how?
Derivation of the method
The free-surface boundary conditions to be satisfied, in
a steady form, are:
u~x+v~y-w=o atz=;
Fn2~r—(=0 at z=;
t.~.n = 0
(7)
(8)
(9)
The dynamics of the waves are essentially
governed by the normal component of the dynamic con-
dition, eq. (8), and the kinematic condition (7); the tan-
gential components of the dynamic condition cause the
appearance of weak free-surface boundary layers with
little effect on the wave pattern.
Iterative formulations for the steady RANS/FS
problem could be based on alternatingly imposing the
normal dynamic and the kinematic boundary condi-
tion; like in time stepping approaches. One might im-
pose to the RANS equations the kinematic condition
on a guessed wave surface, together with the tangen-
tial dynamic conditions; and next update the wave sur-
face from the normal dynamic condition, i.e. from the
pressure difference at the wave surface. This would be
a straightforward extension of the composite approach
described in Section 3.
However, such a scheme would suffer from a
drawback it shares with the usual time-stepping meth-
ods: it uncouples the kinematic and normal dynamic
condition. None of these two conditions has any wave-
like character by itself: wave solutions, a dispersion re-
lation, group velocity and all other properties of free-
surface waves only arise from the combination of both
conditions. Uncoupling leaves the task of generating a
wave pattern to the iterative algorithm; and we believe
we can do better by already building it into our RANS
solution at each iteration.
Here again, we look at what is being done in
inviscid methods. All steady nonlinear inviscid meth-
ods apply an iterative procedure, imposing a combined
kinematic / dynamic FSBC in each iteration. Since our
method is inspired by this, we now will pay some atten-
tion to the derivation of the FSBC's in those.
Free-surface potential flows
For inviscid flows, a velocity potential ~ is usually in-
troduced, the velocity v= V¢, and Bernoulli's equa-
tion is invoked for replacing the pressure yt. In potential
flow the kinematic and dynamic FSBC thus become:
~x; + by - 0z = 0 at z = ~
iFn2~1—¢2_~2_~2~_~=0
(11)
The next step then made is to combine both
equations, by eliminating the wave elevation ~ from the
kinematic condition:
2 Fn2 Fax ,33 + (p), ,,jG] + at ~ ~ (~2 + dp2 + (p2) + ¢ = 0
(12)
at z = if.
This condition forms the basis of iterative
schemes for the nonlinear free-surface potential flow
problem. After linearization it is imposed in each itera-
tion, and the resulting velocity field is then substituted
in the dynamic condition to obtain an updated wave sur-
face. Unlike the separate FSBC's, the combined condi-
tion already incorporates the essence of gravity waves
and ship wave making. Even after linearization relative
to a uniform flow and an undisturbed water surface, the
resulting condition, the familiar Kelvin condition
Fn2~+¢z=0 at z=0, (13)
provides a qualitatively correct wave pattern in a single
step without any iteration. It is this degree of 'implicit-
ness' that we aim for in our RANS/FS method.
Free-surface viscous flows
Returning now to the viscous problem, we may not
use Bernoulli's equation or a velocity potential, but the
combination of kinematic and normal dynamic condi-
tion can be made in exactly the same way. Substituting
the wave elevation from the dynamic condition into the
kinematic FSBC we get:
Fn2(u~\U+v,~lv+w<~~ w=0 at z=~ (14)
The set of the combined FSBC and the dy-
namic conditions corresponds exactly with the original
conditions, and ensures that the pressure variation, the
normal velocity and the shear stress vanish at the wave
surface. The advantage of the combined FSBC comes
once we use it in the context of an iterative procedure.
Our iterative process consists of solving the
RANS equations under a guessed wave surface, on
which the free-surface boundary condition (14) is im-
posed, together with the tangential dynamic conditions;
followed by a free-surface and grid update based on the
normal dynamic condition (8~. The combined FSBC in
itself demands neither a zero normal velocity nor a con-
stant pressure at the estimated wave surface; but im-
poses a relation between the normal velocity and the
(10) pressure field, such that the velocity at z= ~ is par-
allel to an isobar surface. Once the solution has been
obtained, the dynamic FSBC updates the wave surface
to that isobar surface.
Van Brummelen et al (2001) give a more pre-
cise derivation of the free-surface boundary condition
(14) and derive that the theoretical asymptotic conver-
gence rate of this iterative process is independent of the
grid spacing. Again demanding that the residual at the
free surface is reduced to the level of the discretisation
errors, the number of iterations now is Clog 1/h), as
opposed to the ~(h-3) number of time steps for a time-
dependent approach.
0.1 i 1
vo.6.
0.4
0.2
o
Figure 8: Example of a grid used in the numerical exper~-
ments. The grid is coarsened for illustration purposes. -o. 05
Implementation
Compared with the free-slip conditions imposed in Sec-
tion 3, for a precomputed wave surface, the normal ve-
locity component is not set to zero as in (4) but the com-
bined FSBC (14) is imposed. This is nonlinear in the
unknown pressure and velocities, and is Newton lin-
earised. The explicit contributions are taken from the
previous inner iteration (in the RANS solution).
Second-order differences are used for the
FSBC. An upstream difference is required for the
gyr/9x term, contrary to its downstream difference in
the ~—momentum equation.
Results
2D Bottom bump
An initial, and so far the most complete, study has been
done for a 2D test case. Use has been made of the code
PARNAX (Hoekstra, 1999), which is a 2D finite-volume
code very similar to PARNASSOS . To the solution oro-
cess described before, an outer iteration loop was added
in which the free surface and grid are adjusted using the
normal dynamic condition (8) after a sufficient degree
of convergence of the inner loop that solves the RANS
equations.
The test case is a bottom bump in a shallow
2D flow, at a depth Froude number of 0.43 and Re =
1.5 x lO5. The geometry of the obstacle is
y= - 1+ 4 `?3x~(x~—t),
O ~ x~ ~ ~ , (15)
with hb and e the height and length of the obstacle,
non-dimensionalized with the undisturbed water depth.
Choosing e = 2 and hb = 0.2, the setup is in accordance
with (Cahouet, 1984). In this case rather strongly non-
linear waves are generated (up to 70 % of the stagnation
height). At inflow a boundary-layer velocity profile was
specified in agreement with the data. Near the outflow
additional wave damping has been applied to prevent
possible problems there. Grids of 400 * 70 and 800 *
70 have been used. Figure 8 illustrates the geometry of
the bump and the grid obtained after convergence.
The main interest of the test is the convergence
properties of the iteration for the free surface. Fig. 9
n no
\
\
-0.1- O i
Figure 9: Example of convergence of Me free-surface shape
in consecutive iterations; 400*70 grid.
shows the wave profile in consecutive free-surface it-
erations, for the hb = 0.20 case. The calculations were
started from a flat water surface. Notably, already the
first iteration (i.e. a RANS solution under a flat water
surface, subject to the FSBC's (14) and (9) ~ produces
a wave pattern (labelled 'b') which is qualitatively cor-
rect but has reduced amplitude due to the linearization.
This entire free-surface adjustment is then applied, the
grid updated, and a next iteration performed. The im-
plicitness of the combined FSBC permits large free-
surface adjustments in a single step and produces a very
fast convergence of the wave profile. Notwithstanding
the substantial nonlinearity, 9 iterations suffice. In (Van
Brummelen et al, 2001) the convergence for this test
case is further analysed, and the theoretical result of a
convergence rate independent of the grid density is es-
sentially confirmed. The convergence ratio is estimated
as ~ = 0.45 to 0.52. In an additional test for a reduced
bump height of hb = 0.15, generating less steep waves,
convergence of the wave profile was obtained in just 3
iterations and the convergence ratio was co = 0.15.
Compared to calculations using other methods
for the same case, which e.g. required time integration
up to a nondimensional time T = 60 (Vogt, 1998) or
~1000) free surface iterations (Tzabiras, 1997), we
believe that the behaviour of our method is most en-
couraging. Admittedly, from the point of view of the
computation time one iteration in our method cannot
be compared with one time step in other methods; the
many time steps in time-dependent methods also repre-
sent the iterative solution of the RANS equations them-
selves. Nevertheless, we believe that the strong reduc-
tion of the number of free-surface adjustments in itself
is an important improvement. For the present case, the
solution of the nonlinear free-surface problem takes just
2 3 times the computation time of a case with sym-
metry boundary conditions.
Figure 10 compares the result with the ex-
of
o.os
art
-o~os
1 2 3 4 ~
x
Figure 10: Computed wave elevation (solid line) and mea-
surements from (Cahouet, 1984) (symbols), for hb = 0.20.
The obstacle is located in the interval x ~ to, 2~. 800 * 70 and.
perimental data from Cahouet. Agreement is generally
good for the wave length, but the wave amplitude is
overestimated and exceeds the experimental uncertainty
quoted. Comparable levels of agreement for this test
case have been shown in other publications.
3D Pressure patch
Next we considere the wave pattern generated by a 3D
pressure patch travelling over the free surface. The pa-
rameters of the pressure distribution were taken equal
to those of Wyatt (2000~. It has a Gaussian distribution,
petty) =A.e-(r/B) ; r= ~/(x—xO)2+y2 (16)
in which we took A = 0.05, B= 0.5, x0 = -6.5. The
Froude number based on the unit length was 0.6. The
flow is essentially inviscid since no other perturbation
is present.
The domain considered extends fromx = - 7.5
to x = - 0.5. A 160 * 40 * 40 grid was used, which was
essentially uniform at the water surface. The computa-
tion was started from an undisturbed water surface and
uniform flow.
For not too large pressure amplitudes, no free-
surface update and grid adjustment have been done. The
RANS solution then predicts a wave patteIn shown in
Fig. 11. To check the result, the same case has been
computed using the inviscid-flow code RAPID, using a
panel size comparable to the free-surface cell size. The
agreement is fairly good, as shown by the two longitu-
dinal wave cuts in Fig. 12. There is evidence of a some-
what larger numerical damping and a slightly larger
wave length in the RANS solution.
Since these results were obtained without any
free-surface update, they confirm that also in 3D the
x z
Yet
Figure 11: Wave pattern of pressure patch at Fn = 0.6, from
RANS computation.
--- Rapid, nonlinear
— Parnassos, linear FSBC
:,
7 ~
~\ ~ \ , ~ By.
. .
~ j ,.
~ , I . 1 , I . 1 . I . I . I
-8.5 -7.5 -6.5 -5.5 4.5 -3.5 -2.5 -1.5 -( ).5
x/L
Figure 12: Wave cut at centreline (top) and at y/L = 1.05
(bottom) for pressure patch at Fn = 0.6, comparison of RAPID
and PARNASSOS .
physics of wave making are embodied in the RANS
problem subject to the combined FSBC; and that for
mild, essentially linear waves no free-surface update is
needed at all. The CPU time required to get this result
amounted to 12.5 minutes CPU on a single SG R14000
processor at 500 MHz.
Series 60 case
The last application we show is the viscous flow and
wave pattern around the Series 60 CB = 0.60 model at
Fn = 0.316, Rn = 3.4* 106. The domain extends from
Figure 13: Wave pattern for Series 60 CB = 0.60 at Fn =
0.316, found from RANS with lineansed FSBC.
0.02 _
nn1 _
-0.01
-0.02
0.02
nn
-0.0
~-
-0.5 o 0.5
X/L
o- ~
_ ·\ ·7
2
~ - ~ 1 1 ~ 1 1 1 , , , ~ 1 , , ~ , 1 ~ ~
-0.5 o 0.5 1
X/L
7.~
. · .
al, - - ~
1
Figure 14: Senes 60 CB = 0.60 at En = 0.316, longitudinal
wave cuts at y/L = 0.0755 and 0.2067, compared with exper-
imental data
0.25L upstream of the bow to 0.5L downstream of the
stern, and had a half width of 0.5L. A grid of 320 * 121
* 44 = 1.7 M cells was used, under an undisturbed wa-
ter surface. The computation was started from scratch
using the free-surface boundary conditions and contin-
ued until pressure changes between iterations were be-
low 5 x 10-4 everywhere. At present the convergence
of the computation with free surface is still slower than
usual, and this needs to be addressed.
Fig. 13 shows the wave pattern obtained. One
should note that no free-surface adjustment has been
made in this computation, and the wave pattern thus is
a partly linearised result. Fig. 14 compares longitudinal
cuts at y/L = 0.0755 and 0.2067
with experimental data from Toda et al(1991)
and shows quite reasonable agreement. Further away
from the hull the computed wave amplitude is some-
what too small, partly due to the linearisation of the
FSBC, partly due to numerical damping.
This development is underway and progress is
still made. However, the purpose of this computation
was not to show any final agreement with the data, but
to illustrate
· that the mechanism of free-surface flow predic-
tion using the combined FSBC works as in the
other test cases;
that the result, although provisional, shows the
right wave physics;
· that, without any grid update and by just mod-
ifying a boundary condition, one already can
get fairly close to the experimental wave pattern
here;
· that no further precautions at all were necessary
at the bow or along the waterline: since there is
no contradiction at the waterline between the no-
slip condition and steady FSBC's, no problems
were experienced.
Of course, much more needs to be done.
Specifically, stability and convergence rate of the com-
putation with free-surface boundary conditions needs
to be addressed, iteration for the free surface and grid
needs to be performed, and efficiency improved. Even
so we believe that the cases done so far hold promise
for an efficient solution of the full RANS/FS problem
in a limited number of iterations.
5 DISCUSSION AND CONCLUSIONS
In search for a further extension of the already impor-
tant role of CFD in practical ship design, we have con-
sidered possibilities to make the combined computation
of viscous flow and wave pattern more readily applica-
ble in practical ship design. The present paper describes
two different methods:
· a composite approach, in which the viscous flow
is computed under a wave surface determined be-
forehand from a potential flow calculation;
· a full RANS/FS method that solves the problem
by iteration rather than by time stepping.
The composite approach is a combination of
tools that are already extensively used in design, i.e.
a nonlinear free-surface panel code and a RANS code
without free surface. This makes it directly applica-
ble in practice. Inspection of the results for Series 60
and KCS has shown the composite solution to be es-
sentially equivalent to a full RANS/FS solution, except
in a Kelvin wedge starting at the stern. Compared to
a double-body RANS solution the composite approach
uses no more computation time, but adds the wave ef-
fects upon the viscous flow. This was found to be a rel-
evant step forward: e.g. it leads to a small improvement
in the wake field for the KCS case, but, more impor-
tantly, permitted to predict the wave-induced flow sep-
aration observed experimentally at the stern of a tanker.
On the other hand, viscous effects on the wave pattern
are neglected; and there seems to be no obvious way to
extend the method to a more complete solution.
The other method, iterative solution of the
RANS/FS problem, is meant to reduce the long com-
putation times needed by most RANS/FS methods. We
have shown that by a particular formulation of the free-
surface boundary conditions in steady form, all time-
dependence can be omitted. This eliminates the long
integration times, large number of time steps, resecting
transient waves and contact line problems encountered
in several other methods. Results for a 2D bottom bump
show convergence of the wave pattern in 9 iterations.
For a 3D pressure patch, a single iteration appears al-
ready to provide an accurate wave pattern. For a Series
60 case, our initial results show that again the first iter-
ation, a RANS solution on a grid under an undisturbed
wave surface, provides a qualitatively correct wave pat-
tern. While there is still much development needed, our
present results seem to offer good prospects for a quick
convergence to a final solution.
These two methods can be regarded as mem-
bers of a hierarchy of computational methods consisting
of the following levels:
· an inviscid wave pattern calculation, and a
double-body RANS solution; as two separate
components without any mutual influence. This
is the traditional level that is widely used now in
ship design.
· the composite approach proposed here. Basically
.
the same components are used, but we impose
free-slip conditions in the RANS solution. This
adds the wave effects upon the viscous flow but
lacks interaction near the stern.
· RANS with linearised FSBCs. The RANS equa-
tions are now solved on a fixed grid under an esti-
mated wave surface (either undisturbed, or from
an inviscid flow code) subject to the combined
FSBC proposed in this paper. This adds viscous
erects on the wave pattern and viscous/wave in-
teraction. If the resulting wave elevation update
is small, the linearisation is accurate and no grid
update is needed.
Finally, the fully nonlinear RANS/FS, by the iter-
ative formulation proposed. This does need grid
updates and iteration. Like the previous level, it
adds the full viscous/wave interaction, but now
without any assumption of the magnitude of the
adjustments.
The methods we propose in this paper illustrate and
anticipate the various levels. We believe that efficient
methods to compute steady ship viscous flow with free
surface can be designed by choosing the right level from
this hierarchy in the right case and in the right part of
the flow domain; and we hope this will bring a large-
scale and fruitful application of such methods in practi-
cal ship design closer.
Acknowledgement
We thank Mervyn Lewis, Centre for Mathematics and
Computer Science (CWI) in Amsterdam, for providing
the Series 60 result in Figs 13-14. He cames out parallel
work on the method of Section 4, in a project supported
by the Netherlands Technology Foundation STW. We
acknowledge that support and appreciate the benefit we
have from our cooperation.
The permission from the CRS-Pods working
group of MARIN's Cooperative Research Ships pro-
gram to publish Fig. 7 is gratefully acknowledged.
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Garofallidis, D.A., "Experimental and numerical
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Larsson. L.. Stern. F.. and Bertram, V. (2000),
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resistance.html
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DISCUSSION
H. Soding
Technical University of Hamburg, Germany
My colleagues in Hamburg and I stopped similar
developments because we were not successful in
handling breaking waves by solving Euler or Rans
equations with grids fitted to the free surface.
Neglecting breaking waves often leads to non-
convergence. Thus I think one must either learn to
handle breaking waves with surface-fitted grids, or
abandon such grids for steady ship flow
computations.
AUTHORS' REPLY
Prof. Soeding addresses the question whether one
should use surface-fitting or surface-capturing
methods. There does not seem to be a single answer
to that question at present, and both classes of
methods are being proposed in the literature.
In any case, the composite approach will not be
affected by wave breaking, as long as we are able to
compute the wave surface beforehand by a panel
method; which is no problem for the great majority
of cases (except those having severe wave breaking
in reality).
As a matter of fact, our second method, the steady
RANS/FS solution, probably will have problems
for cases having substantial wave breaking in the
steady flow, like other surface-fitting methods; but,
as opposed to the time-dependent approaches, will
be insensitive to any wave breaking in an
intermediate stage. We expect that ad-hoc local
models will be sufficient to remove these problems.
DISCUSSION
Chi Yang
George Mason University, USA
Authors have presented a very interesting and smart
approach for computing the steady viscous flow
with free surface. This approach is based on the
assumption that the viscous effects on wave
elevation can be neglected. It therefore leads to a
very efficient method and also removes the
difficulty associated with moving RAN S grids in
the viscosity of the hull during the simulation. I
would like to know how authors reconstruct the
deformed free surface from the inviscid solution to
generate RAN S grids. What type of RANS grids is
used? The mesh movement will be relative small if
the RAN S simulation starts from the inviscid
solution. Have authors tried to update the wave
elevation during RAN S simulation and compare the
results with these obtained without the viscous
effect on wave elevation? In order to save
computing time, the inviscid solution can also be
used as initial solution for RAN S computation to
speed up the convergence. I would like to have
your comment on that.
AUTHORS' REPLY
Our "composite approach" is based on the
observation that viscous effects on the wave
elevation can be neglected in the majority of the
domain, except, most notably, the stern region. The
free-surface for the viscous-flow computation is
obtained by fitting a B-spline surface through the
collocation points on the free surface in the inviscid
solution. An orthogonal grid generator is then used
to define the mesh on the free surface, which forms
one of the boundary planes of the single-block, HO-
mesh we used in these computations.
With respect to the update of the wave elevation in
the RAN S computations the answer is twofold. In
the case of the composite approach, further grid
updates based on the pressure residual at the free
surface would correspond to a decoupling of the
kinematic and dynamic FSBC's. This is what we
have avoided using the combined conditions in our
steady iterative method for free-surface flows; as
motivated in our paper. Hence we do not intend to
perform grid updates in the composite approach. In
the case of,the steady iterative method, grid updates
have been performed for the 2D bottom bump and
the 3D pressure patch, but not yet for the Series 60
hull. Nevertheless, we expect to perform these
updates in the near future and undoubtedly will
compare the results with inviscid-flow predictions.
The last question addresses the possible speed-up of
the convergence of the viscous-flow computations
using the inviscid-flow solution as an initial
estimation. It is recalled that in our viscous-flow
solver (PARNASSOS) we impose the pressure and
the tangential velocities obtained from an inviscid-
flow computation at the outer boundary of the
domain. Rather than evaluating the inviscid
velocities at every node in the interior of the grid as
well, including the ones in the boundary layer, we
basically start with a uniform pressure and velocity
field. The initial sweeps on the coarser meshes in
.
our gee -sequencing procedure then essentially
generate the initial estimation of the pressure and
velocity field for the finer meshes. This procedure
has been found to be quite stable and robust and
leads to the computation times listed in our paper.
