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.