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PREDICTION OF RESISTANCE OF FLOATING VESSELS
D Jürgens and M Palm, Voith Turbo Schneider Propulsion GmbH & Co. KG, Germany
M Periç and E Schreck, CD-adapco, Nürnberg Office, Germany
SUMMARY
Optimization of marine vessels requires that the interaction between hull, propulsion and steering devices, and other
appendages is taken into account. In addition, one needs to account for the variable floating position at different
operating conditions.
Due to the complexity of all interactions, experimental optimization is both costly and time-consuming. Computational
methods can help in predicting efficiently the effects of design changes on resistance and other properties of the vessel.
The aim of this paper is to present the results of several validation studies which include both simulations and
experiments, demonstrating the ability of CFD to predict not only the trends but also the quantitative variation in
resistance due to design changes.
1. INTRODUCTION
CFD has successfully been used for the prediction of
resistance of bare ship hulls in a fixed position relative to
undisturbed free surface, as demonstrated at several
recent workshops (Göteborg, 2000; Osaka, 2005).
However, validations of CFD application to predict the
effects of design changes at various operating conditions
when the vessel has several degrees of freedom are less
numerous.
Although Voith manufactures propulsion and steering
devices (Voith-Schneider-Propeller, Voith-Cycloidal-
Rudder etc.), it has proven important in recent years to
perform simulations of flow around the vessel fitted with
all appendages. Often much larger improvements in
performance are obtained by slight modifications to the
hull than would have been possible by optimizing the
propeller alone. On the other hand, optimization of the
hull alone does not necessarily lead to the best solution,
since the behavior of the whole system, when it is freely
floating, can be substantially different.
In this paper results of several validation studies, which
include both simulations and experiments, are presented.
The aim of these studies was to assess the reliability of
prediction of resistance of marine vessels under free-
floating conditions, as well as the possibility to predict
the effects of design changes on resistance. The Flow
around a simple brick-like body is investigated first,
followed by real marine vessels of different shapes.
The next section describes briefly the solution method
used. This is followed by three sections presenting results
of validation studies. The final section summarizes the
findings, with recommendations for further investiga-
tions.
2. COMPUTATIONAL METHOD
All computations reported here are performed using the
CFD software from CD-adapco. It is based on a finite-
volume (FV) method and starts from conservation
equations in integral form. With appropriate initial and
boundary conditions and by means of a number of
discrete approximations, an algebraic equation system
solvable on a computer is obtained. First, the spatial
solution domain is subdivided into a finite number of
contiguous control volumes (CVs) which can be of an
arbitrary polyhedral shape and are typically made smaller
in regions of rapid variation of flow variables. The time
interval of interest is also subdivided into time steps of
appropriate size (not necessarily constant). The
governing equations contain surface and volume
integrals, as well as time and space derivatives. These are
then approximated for each CV and time level using
suitable approximations.
The flow is assumed to be governed by the Reynolds-
averaged Navier-Stokes equations, in which turbulence
effects are included via an eddy-viscosity model (k-ε or
k-ω models are typically used). Thus, the continuity
equation, three momentum component equations, and
two equations for turbulence properties are solved. In
addition, the space-conservation law must be satisfied
because the CVs have to move and change their shape
and location as the vessel or appendages move. These
equations are:
Mass conservation:
Momentum conservation:
Generic transport equation for scalar quantities:
Space-conservation law:
In these equations, ρ stands for fluid density, v is the
fluid velocity vector and vb is the velocity of CV surface;
n is the unit vector normal to CV surface whose area is S
and volume V. T stands for the stress tensor (expressed in
terms of velocity gradients and eddy viscosity), p is the
pressure, I is the unit tensor, φ stands for the scalar
variable (k or ε or ω), Γ is the diffusivity coefficient, b is
the vector of body forces per unit mass and bφ represents
sources or sinks of φ. Since the CV can move arbitrarily,
velocity relative to CV surface appears in the convective
flux terms, and the time derivative expresses the
temporal change along the CV-path.
In order to account for the free surface and allow for its
arbitrary deformation (including fragmentation, trapping
of air bubbles etc.), an additional equation is solved for
the volume fraction c of the gas phase, which can be
treated either as an incompressible fluid or as a
compressible ideal gas:
Liquid and gas are considered as two immiscible
components of a single effective fluid, whose properties
are assumed to vary according to the volume fraction of
each component as follows:
The equations describing the motion of a floating body
are:
Linear motion:
Angular motion:
Here mB is the body mass, IC is its moment of inertia,
vC is the velocity of body's center of mass, ωωωωB is its
angular velocity, FB is the force and MB the moment
acting on the body. The force is made typically of flow-
induced forces (with shear-stress and pressure
contributions) and body weight; the latter does not
contribute to the moment about the center of mass:
Here g stands for gravity acceleration and r for the
position vector relative to a fixed reference frame; index
“B” denotes body and “C” denotes center of body mass.
It is beyond the scope of this paper to go into all the
details of the numerical solution method, so only a brief
description is given here; details can be found in [1,2].
All integrals are approximated by midpoint rule, i.e. the
value of the function to be integrated is first evaluated at
the centre of the integration domain (CV face centres for
surface integrals, CV centre for volume integrals, time
level for time integrals) and then multiplied by the
integration range (face area, cell volume, or time step).
These approximations are of second-order accuracy,
irrespective of the shape of the integration region
(arbitrary polygons for surface integrals, arbitrary
polyhedra for volume integrals). Since variable values
are computed at CV centres, interpolation has to be used
to compute values at face centres and linear interpolation
is predominantly used. However, first-order upwind
interpolation is sometimes blended with linear
interpolation for stability reasons. In order to compute
diffusive fluxes, gradients are also needed at cell faces,
while some source terms in equations for turbulence
quantities require gradients at CV centres. These are also
computed from linear shape functions.
In the equation for volume fraction of the gas phase,
convective fluxes require special treatment. The aim is to
achieve a sharp resolution of the interface between liquid
and gas (one to two cells), which requires special
interpolation of volume fraction. The method used here
represents a blend of upwind, downwind, and central
differencing, depending on the local Courant number, the
profile of volume fraction, and the orientation of
interface relative to cell face; for more details, see [1].
The scheme is adjusted to guarantee that the volume
fraction is always bounded between zero and one, to
avoid non-physical solutions.
The solution of the Navier-Stokes equations is
accomplished in a segregated iterative method, in which
the linearised momentum component equations are
solved first using prevailing pressure and mass fluxes
through cell faces (inner iterations), followed by solving
the pressure-correction equation derived from the
continuity equation (SIMPLE-algorithm; see [2] for more
details). Thereafter equations for volume fraction and
turbulence quantities are solved; the sequence is repeated
(outer iterations) until all non-linear and coupled
equations are satisfied within a prescribed tolerance, after
which the process advances to the next time level.
When the motion of a floating body is also computed, the
outer iteration loop within each time step is extended to
allow for an update of body position. The equations of
body motion are first solved to obtain the velocities using
a predictor-corrector scheme of second order (equivalent
to Crank-Nicolson scheme) and then for displacements
and rotations; the grid within flow domain is adjusted in
every outer iteration of one time step to fit the new body
position. Under-relaxation of body motion is used in a
similar way as when solving the Navier-Stokes
equations; it can be interpreted as adding a virtual mass
to the system [3]. At the end of each time step, the new
body position and the corresponding flow are obtained.
The coupled solution method for flow and body motion
is thus fully implicit. This allows larger time steps and
better stability than explicit schemes in which flow and
body motion are computed one after another.
Grid adaptation to body motion requires special
attention. Three basic approaches are used, depending on
application:
� In the case of moderate motion, as would occur with
ships in small-amplitude waves, one can proceed as
follows: (i) move the grid near body rigidly with the
body, (ii) keep the grid further away from the body
undeformed, (iii) deform the grid in the region
between these two (usually a kind of algebraic
smoothing is applied).
� When a single body in an infinite domain is
considered, one can also move the whole grid with
the body. This can be problematic in the case of large
motions and waves, because the grid needs to be fine
in a larger region in order to capture the free surface
and waves properly than would be the case in the first
approach.
� The third possibility is to use overlapping grids,
where one background grid is adapted to the free
surface (and possible outside boundaries, like shore
or harbour walls), while overlapping grids are
attached to floating bodies and move with them
without deformation. In this case the grid quality is
easier to control and grid motion is easier to handle,
but the solution method needs to account for the
coupling of background and overlapping grid
solutions.
Only the overlapping grid method is applicable to
unlimited motions (including capsizing).
The emphasis of the present study is on determining the
floating position of the vessel and its resistance. In the
next section results are first presented for a simple brick-
like body, followed by some results obtained for real
maritime vessels.
3. FLOATING BRICK
In one validation project, the flow around a brick-like
body in fixed and floating position has been studied at
Voith, both experimentally and numerically. Due to
rectangular body geometry, Cartesian block-structured
grids were used. Two turbulence models with wall
functions were tested: k-ε and k-ω. Computations were
performed on three meshes of different fineness in order
to estimate discretization errors; these were found to be
of the order of 1 % on the finest mesh, which had 1.9
million CVs.
Fig. 1 shows the body in fixed position and the free
surface deformation around it when exposed to water
flow at 1 m/s. Fig. 2 shows the same situation when the
body is free to sink and trim. These pictures show that
there is a high qualitative similarity between predicted
free surface shape and the observed interface
deformation in experiments.
A quantitative comparison of predicted and measured
resistance for fixed body position is shown in Figure 3.
The selected turbulence model makes little difference – a
very good agreement is obtained for all free stream
velocities.
Figure 1: Free surface deformation around fixed brick at
1 m/s: simulation (upper) and experiment (lower)
Figure 2: Free surface deformation around brick free to
sink and trim at 1 m/s: simulation (upper) and experiment
(lower)
Figure 3: Comparison of measured and computed resi-
stance for the fixed brick
Figure 4: Comparison of measured and computed resi-
stance for the floating brick
Figure 5: Comparison of measured and computed pitch
angle for the floating brick
When the body is free to sink and trim, the agreement
between simulation and experiment is less perfect, as can
be seen from Figs. 4 and 5. At free stream velocities
above 1 m/s, simulation under-predicts both resistance
and trim angle. The same mesh was used in both
simulations, but due to high trim angle in the second
case, the angle between streamlines and grid lines
becomes less favourable as the velocity increases. This
might be one reason for discrepancies, while the higher
uncertainty in measured data at higher speed should also
be mentioned.
Since the brick-body with its sharp edges and severe
separation is not highly representative of marine vessels,
no deeper analysis of the observed discrepancies was
undertaken; the obtained results are considered
satisfactory for the intended purpose. A more detailed
analysis is performed for two marine vessels, and the
results are presented in the following sections.
4. VESSEL 1
The first vessel is a typical tug-boat with the following
characteristics: 37 m long, 13.5 m wide, 3.3 m draught.
Both simulations and experiments were performed at
model scale 1:16. Several grids were used to compute the
flow at the speed of 14 kn in calm water, in order to
evaluate grid dependence of the numerical solution. All
grids were block-structured and generated using ICEM-
HEXA mesh generator. The k-ε turbulence model with
wall functions was used. The thickness of near-wall cells
was adjusted so that the dimensionless distance of the
first computational point from wall was around 50 wall
units. Using Richardson extrapolation, it was estimated
that the discretization errors on the grid consisting of 1.8
million CVs were not larger than 3 %, so this grid was
used to compute resistance at nine speeds between 10
and 16 kn.
Figure 6: Variation in wave pattern created by vessel in
calm water at different speeds: 10 kn (top left), 12 kn
(top right), 14.5 kn (bottom left) and 16 kn (bottom right)
Fig. 6 shows how the wave pattern, generated by the
vessel, changes as the speed is increased from 10 to 16
kn. Fig. 7 shows the comparison of computed and
measured resistance for various speeds. Keeping the
vessel fixed in the position corresponding to zero speed
leads to a substantial under-prediction of resistance. Only
when the vessel is allowed to float in the simulation
(with two degrees of freedom – trim and sinkage),
correct forces are predicted; the agreement between
simulation and experiment is then very good over the
whole range of speeds, as evident from Fig. 7. This
shows that it is important to take free surface
deformation and variation of the vessel position into
account when predicting the resistance and the required
propulsion power, since they would otherwise be
substantially under-predicted.
Figure 7: Comparison of predicted and measured
resistance in calm water at various speeds for vessel 1.
5. VESSEL 2
The second vessel is a motor yacht with the following
characteristics: 52.5 m long, 11 m wide, 2.35 m draught.
While the experiments have been performed at model
scale with a scale factor of 1:9, the simulations were
carried out at full scale. The same meshing procedure
and the analysis of grid-dependence of solution as in the
previous case was conducted. The final mesh for which
the discretization errors were estimated to be below 3 %
had approx. 2 million CVs for the half model.
Fig. 8 shows how the wave pattern around the vessel
changes as the speed is increased. The changes are
especially large in the stern region. The wetted hull
surface also increases at the bow, which contributes to
the increase in resistance.
At low speeds (up to 9 kn), there is no significant
difference in predicted resistance for the vessel fixed in
zero-speed position and when it has two degrees of
freedom (to sink and trim), as shown in Fig. 9. At higher
speeds, the difference increases up to about 30 % at 14
kn speed. As in the previous case, the resistance is under-
predicted when the vessel is held fixed. Figure 9 shows
that the agreement between simulation and experiment is
very good when the vessel position is computed as part
of solution. Only at speeds above 13 kn the simulations
predict slightly lower resistance than is found in the
experiment. It is possible that, by adjusting the grid to the
vessel position relative to undisturbed free surface and
local refinement in regions of steep waves a better
agreement could have been obtained; however, this was
not attempted here.
6. REDUCTION OF RESISTANCE
The results presented in the three preceding sections
show that CFD can reliably predict the variation of
resistance as a function of vessel speed and position. It is
thus possible to use CFD in order to evaluate various
ideas for resistance reduction, before experimental
studies – which are more expensive and time consuming
– are attempted. This approach has been used at Voith
with considerable success during the past five years.
Figure 8: Variation in wave pattern created by the vessel
in calm water at different speeds: 9 kn (top left), 11 kn
(top right), 13 kn (bottom left) and 15 kn (bottom right)
Figure 9: Variation in wave pattern created by vessel 2
in calm water at different speeds
As seen in the previous section, the resistance of the
vessel would be substantially lower if it would not sink
and trim (although Figs. 7 and 9 do not contain
experimental data for fixed vessel position, the observed
effects are both qualitatively and quantitatively the same
as in the simulations). One can therefore try to modify
the hull form to reduce the variation in vessel position at
higher speeds. The results of one such study is presented
here, in which the stern of the vessel from the preceding
section is modified by simple add-ons. In one case, a
wedge has been added to the hull bottom at stern, as
shown in Fig. 10. Three more tests were performed with
a plate added to the stern surface which protruded by
different extent below the hull bottom; the largest plate
(plate 3) is also shown in Fig. 10.
Figure 10: Variation of stern geometry: wedge (upper)
and interceptor-plate (lower)
Figure 11: Variation of resistance when the stern
geometry is varied (wedge and plate 3 are shown in Fig.
10; plate 2 is in size about 2/3 and plate 1 about 1/3 of
plate 3)
As one might expect, adding such an obstacle to the
smooth hull surface leads to an increase in resistance
when the vessel is held fixed in the zero-speed position.
This is correctly reproduced in simulations: the dark bars
in Fig. 11 show that both the wedge as well as all plates
result in a higher resistance. Especially for the largest
plate (plate 3) the increase is significant – over 10 %.
However, when the vessel is free to sink and trim, the
resistance with the largest plate is the same as for the
original geometry, while the two smaller plates and the
wedge lead to a reduction of resistance. For the plate 1,
the reduction amounts to about 3.5 %. Thus, a very
simple modification of stern geometry can provide
significant fuel saving.
Figure 12: Variation of trim and sinkage when the stern
geometry is changed
Figure 12 shows how sinkage and trim are affected by
the changes of stern geometry. The sinkage is in all cases
reduced compared to the original geometry, but this
change is relatively moderate (maximum 10 %). The
changes in pitch angle are more significant; for the
wedge and plate 3, the angle changes sign (positive angle
means bow upwards). Maximum saving is achieved
when the pitch angle is minimized in magnitude; a plate
in size between plate 1 and plate 2 would probably
produce zero trim and maximum saving.
7. SYSTEM ANALYSIS
In all validation studies presented in preceding section,
only bare hulls were used (because experiments at model
scale including propeller are difficult and affected by
many uncertainties). The results have demonstrated that
CFD can be trusted for producing reliable predictions of
various effects that influence resistance and the required
power for propulsion.
In another series of studies at Voith, the application of
CFD to predict the performance of Voith-Schneider-
propeller has been analysed. It has been found that CFD
simulations predict the variation of torque on each
propeller blade during its rotation with sufficient
accuracy. Through various measures (modification to the
blade shape, end plates at blade tips, guard plate etc.) that
were developed with the aid of CFD, the efficiency of the
Voith-Schneider-propeller has been significantly
improved over the past five years.
The validation of CFD by a detailed analysis of flow
around components resulted in confidence that it can be
used for predicting the performance of the whole system
(hull, propulsion device and other appendages). The
optimum solution requires that all interactions are taken
into account; optimization of each component alone is
not sufficient. Nowadays, for many Voith-Schneider-
Propellers delivered to a customer, a series of CFD
system simulations (including free surface, rotating
propeller, coupled solution for flow and vessel motion) is
performed. Only in this way it is possible to reliably
predict the required power and the behaviour of the
vessel under operation conditions.
Figure 13: Free surface deformation around vessel 2
without propeller (upper) and with two Voith-Schneider-
propeller at the stern (lower)
The difference in free surface deformation around vessel
2 described in section 5 at 15 kn in calm water with and
without propeller is shown in Fig. 13. The two Voith-
Schneider-propeller at the stern are fitted in cylindrical
blocks that rotate and slide in cylindrical and plane
surfaces along the rest of the grid that is fixed to the hull.
In addition, each blade is fitted into a cylindrical grid
block that moves with the larger cylinder around the
propeller axis, but at the same time partially rotates
around blade axes. The two propeller and the wake are
shown enlarged in Fig. 14. It is obvious that the flow is
substantially different when all interactions are taken into
account, and that such coupled simulations are needed
when the system performance is to be optimized.
8. CONCLUSIONS
The results of several validation studies on prediction of
resistance of marine vessels have been presented. It has
been shown that modern CFD techniques can predict the
resistance and its dependence on speed and geometrical
variations with a sufficient accuracy that allows for
simulations to become an integral part of product design
and optimization. Voith uses CFD not only to optimize
its products but also to optimize the performance of the
whole system ship.
Figure 14: Detail of free surface deformation around
vessel 2 with two Voith-Schneider-propeller at the stern
9. REFERENCES
1. Muzaferija, S., Periç, M.: Computation of free
surface flows using interface-tracking and interface-
capturing methods, chap. 2 in O. Mahrenholtz and M.
Markiewicz (eds.), Nonlinear Water Wave Interaction,
pp. 59-100,WIT Press, Southampton, 1999.
2. Ferziger, J.H., Periç, M.: Computational Methods for
Fluid Dynamics, 3rd ed., Springer, Berlin, 2003
3. Xing-Kaeding, Y.: Unified approach to ship
seakeeping and maneuvering by a RANSE method,
Dissertation, TU Hamburg-Harburg, 2005.
8. AUTHORS’ BIOGRAPHIES
Dirk Jürgens holds the position of the Head of Research
and Development at Voith Turbo Schneider Propulsion.
He is responsible for product development.
Michael Palm is a CFD engineer at Voith Turbo
Schneider Propulsion. His responsibilities include
simulation of flow around Voith-Schneider-Propeller and
coupled simulations of flow around ships propelled by
Voith Turbo devices and their motions.
Milovan Periç holds the current position of the Director
of Technology at CD-adapco. He is responsible for the
development and implementation of new discretisation,
modelling and solution techniques in the CFD software
products of CD-adapco (STAR-CD and STAR-CCM+).
Eberhard Schreck holds the position of a Senior CFD
Development Engineer at CD-adapco. His responsibilit-
ies include development and implementation of various
software modules, in particular those related to coupled
simulation of flow and motion of floating bodies and the
use of moving and overlapping grids.
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