CFD MODELLING OF PLANING HULLS WITH PARTIALLY VENTILATED BOTTOM

RINA

International Conference

The William Froude ConFerenCe:advanCes in TheoreTiCal and applied

hydrodynamiCs – pasT and FuTure24 - 25 November 2010

portsmouth, uK

The royal institution of naval architects

CFD MODELLING OF PLANING HULLS WITH PARTIALLY VENTILATED BOTTOM

Stefano Brizzolara

, Alessandro Federici§, Marine CFD Group,

University of Genoa, Italy;

§ University Pole of La Spezia, Italy

SUMMARY

Paper intends to present the main results achieved from a CFD investigation about planing hulls with partially ventilated

bottom, with the aim to characterise the main feature and behaviour of these important class of planing hulls, of which

very few hydrodynamic results and design data are known. After a brief introduction about the level accuracies obtained

in the simulation of conventional planing hulls with a state of the art RANSE VOF solver, with specific adaptation

introduced by the Marine CFD Group, mainly about mesh type generation, the paper will focus on stepped hulls, dealing

with studies aimed to the correct mesh definition, to the correct problem definition and numerical schemes. The

presented application cases regard two quite different hulls types: the first is a deep-V prismatic hull form, with a

cathedral type of spray rail and transverse step whose numerical results are compared with experiments in terms of total

resistance and visual comparison of the two phase flow below the bottom. The second example is a lower deadrise

planing hull with mixed convex/concave ordinates, warped and tapered at stern for which a rather comprehensive set of

towing tank tests results is presented and used for validation purposes of the CFD models.

NOMENCLATURE

BOA Breadth over all (m) BWL Waterline breadth in static condition

CRT Vol Volumetric Total Resistance Coefficient

defined as (2 RT -1 V-2 -2/3)

Fn Volumetric Froude number,

calculated as (V g -1/2 -1/6)

V0 Velocity of undisturbed flow (m s-1)

Displaced volume (m3)

g Gravity acceleration (9.8066 m s-2)

FnL Length Froude number IAV Draft on fwd perp. (mm)

IAD Draft on aft perp. (mm) LPP Length between perpendicular (m)

LWL Waterline length in static condition (m)

LCG Longitudinal Centre of Gravity, zero XG point is at Stern (m)

RO Wave resistance (N)

RA Shear resistance (N) RT Total advance resistance (N)

VS Ship velocity (m s-1)

VM Model velocity (m s-1)

S Trim angle in static condition, positive

means Stern sinking (deg)

D Trim angle in dynamic condition, positive is Stern sinking (deg)

Specific mass of water (kg m-3)

1. INTRODUCTION

The simulation of the hydrodynamics characteristics of

planing hulls is rather complex and involve different

fluid phenomena, such as large free surface deformation,

production of jet flows and sprays, wave breaking,

viscous boundary layer development in pressure gradient

and flow separation from sharp edges. The numerical

solution of the free surface RANSE seems the most

promising and comprehensive method to take into

account of the shape variability of hull forms, as other

faster methods based on potential flow solution [1] can

do, with the capability of considering also the effects of

different types of appendages. In fact, the most recent

examples of numerical studies in the field are using these

kind of flow solvers [2] [3]. The Marine CFD Group of

Genoa also brought its contribution in this field.

This paper, in fact, continues the series of studies

dedicated to RANSE simulations of planing hulls, the

first studies being published in [4] and [5] about an

original 2DOF method developed into a commercial

RANSE solver in order to more quickly find the running

attitude of planing hull at speed; more recent application

on different types of contemporary fast crafts, complete

of details and propulsion system, such as spray rails and

waterjet inlets in [6]; and finally systematic studies about

hydrodynamic forces on stern appendages for trim setting

and control in [7].

Figure 1 – Conventional (top) and stepped planing hull

(bottom) . Wetted surface at rest is evidenced in red.

Present paper, instead, is focusing the attention on the

specific problem of stepped hulls (see figure 1) with

partially ventilated bottom. To the authors knowledge

few studies have been published on this argument,

although planing hulls with transverse steps have been

used since around 1930 (Italy with Baglietto shipyard

being one of the innovators) and are still widely used as

primary mean of drag reduction in very fast planing hulls

from racing to fast attack crafts and motor yachts.

Among the few authors that published studies about the

argument, for sure there is Eugene Clement who after

several model test campaign devised a simplified design

method for planing hulls partially ventilated bottom [8].

The general lack of reference data about the achievable

drag reduction and the portion of ventilated bottom

obtained with a certain design of transverse steps make

the design of these kind of hulls rather arbitrary and in

general always require a set of model tests to confirm (if

not optimize) the intended effect. In this respect, this

paper also intends to introduce a series of reference

experimental tests conducted in the years 1930 in the first

Italian towing tank, located in La Spezia naval base. The

towing tank (146x6x3 meters) built on 1887 and

destroyed during WW2, was at the time the fourth in

Europe, and types of studies made there had many

analogies with those of William Froude and his son

Edmund in Torquay, although aside the tests on warship

models, also smaller fast units, like the planing attack

crafts for Xa MAS, taken as second example of hull for

the validation of CFD results in this paper.

The first type of hull form assumed in the study and

presented in the next chapter, instead, is a rather

conventional deep-V planing hull, conceived for a super

high speed ferry design that was modified by the

designer with a cathedral type of spray rail and a

transverse step with forward rake. The level of

confidence of RANSE simulations for this kind of hull in

its original and stepped version will be presented in the

following two paragraphs. Finally the last paragraph will

be dedicated to a systematic numerical analysis of the

influence of the step height on the resistance

characteristics of a different type of hull.

2. RANSE SOLVER

A state of art of RANSE solver with VoF method to

represent the free surface has been selected to simulate

the Flap and the Interceptor behind a prismatic planing

hull. The software suite [9] has the capability to solve the

turbulent viscous flow around a body in stationary

conditions, with the VoF method to predict the free

surface around it.

The solver is applied to the following group of equations

which express the mass and momentum balance with an

Eulerian approach and Reynolds time-Average approach

with the appropriate boundary conditions valid for the

specific type of problem. The RANS equations can be

expressed for an incompressible flow as follows:

MSTUPU

U

Re

0

(1)

where U is the average velocity vector field, P is the

average pressure field, μ is the dynamic viscosity, TRe is

the tensor of Reynolds stresses and SM is the vector of

momentum sources. The component of TRe are computed

in agreement with the k-ε turbulence model selected for

this application:

kC

kCgraddivdiv

t

kgraddivkdivt

k

kkxx

ijijt

k

t

ijij

k

t

ijijtij

i

j

j

itij

2

21

Re

2

2

3

22

3

2

DDU

DDU

DDUU

(2)

Where μt is the turbulent viscosity, k is the turbulent

kinetic energy and ε is the dissipation term of turbulent

kinetic energy. The realizable k-ε turbulence model was

selected to close the hydrodynamic problem together

with a two layer wall function applied for the cell near

the wall.

The wall function is an analytical treatment for the first

cell near the wall where the velocity vector and all other

scalar quantities are extrapolated from the known

quantities on the wall boundary surface. The two layer

wall function model, is a model that impose a first thin

linear layer near the wall, and a second logarithmic layer

over the first, this model assume that the first cell

centroid near the wall is lies within the logarithmic

region of the boundary layer. The wall treatment is

optimized to compute a mesh with a y+<100. All the

available models with relative analytical formulations are

listed in [9].

The RANS solver is based on a Finite-Volume method to

discretise the physical domain. The equation for an

incompressible multi-phase fluid has been used in the

simulation, with one more transport equation for the

VoF, that represent the fraction of water present inside

each cell.

0

UVoFdiv

t

VoF (3)

This new equation guarantees to find the correct shape of

free surface defined by the point of transition between

water and air, this method is powerful for the problem

when occurred the wave breaking, or when the air effects

are important on the free surface shape.

The simulation has been built to reach a stationary

solution but it can reached its only across a time-

marching solution to guarantee to update the solution of

VoF quantity.

To solve the time-marching equations, it’s used an

implicit solver to find the field of all hydrodynamic

unknown quantities for each time step, in junction with a

iterator solver for each time step. The software uses a

SIMPLE method to conjugate pressure field and velocity

field, and a AMG (Algebraic Multi-Grid) solver to

accelerate the convergence of the solution.

3. ON THE SET-UP AND ACCURACY FOR A

RANSE MODEL OF A THE DEEP-V BASIC HULL

We are used to refer to pure planing condition when the

hull is running at an advance volumetric Froude number

of Fn > 2.5-3.0, in that case the weight of the boat is

just almost completely sustained by dynamic lift force

due to the dynamic pressure distribution acting on the

hull bottom: the wave resistance practically becomes

negligible and main drag components are the shear drag,

followed by induced resistance (from lift), both

proportional to the square of the velocity. In this

condition, hydrostatic component of lift, proportional to

the submerged volume is small compared to the total

displaced volume at rest. The dynamic trim is important

since it changes the wetted surface and its length as well

as . So dynamic trim induce an important non linearity

into the resistance component, depending on the center of

gravity position and to the hull form, at pre-planing

speeds, augmenting their value with respect to a pure

quadratic behavior, while in pure planing regime the

dynamic trim generally decreases, the dynamic rise

generally increases, with the result of a reduction of

induced drag and lower frictional resistance. Using this

effect we can highly increase the speed, without having a

prohibitive increasing of drag. Variable deadrise angles

of the ordinates along length (warped hulls), bended

longitudinal sections in the aft part of the hull (rocker or

hook) as well as tapered beam and spray rails can all

have an influence in the hydrodynamic behavior of the

hull in pre-planing and planing regimes, since they can

have a direct influence on the dynamic trim and rise

experienced by the vessel at high speed.

Another way of playing on the running attitude of the

vessel and of reducing its wetted surface at high speeds is

to ventilate part of the bottom surface, by realizing a

transverse step at a proper longitudinal position aft of the

spray root line and by opening a to create and maintain

an air pocket on the hull bottom aft of the step (see

Figure 1). Eventually this the water flow separating from

the lower step edge, forming a free surface, will reattach

to the bottom after a certain longitudinal distance to

create another wetted portion of the hull forward of the

transom, from which will separated again to form the

wave trough in the hull wake. The length and volume of

the air pocket beneath the hull and is of course very

much influenced by longitudinal position, the height and

the sweeping angle of the step.

3.1 CHARACTERISTICS OF FIRST HULL

The first deep-V planing hull (about 25 degree of

deadrise) was given to the authors as a reference hull for

the CFD validation study and was already tested with a

systematic campaign of model tests in towing tank by the

designer, with and without steps. The hull devised for a

super-fast ferry of about 70m in length, has been created

without deck and superstructure, and it is cut with a

simple flat deck also in the CFD model and the

aerodynamic forces developing on it are neglected in the

calculation of the total drag.

The deep-V planing hull has a monohedric shape aft of

midship, ending in a vertical transom and has a rather

long slender entrance body to ensure good seakeeping. It

has a single hard chine with a flat spray rail to better

diverge the separated spray flow from the side at high

speeds, while augmenting the lift force. Figure 9 presents

the body plan of the stepped version of this hull, whose

simulations are described later.

As already discussed in previous studies in case of the

some hulls of series 62 [5], the correct prediction of the

jet spray separation from chines is the most critical issue

to obtain valid results from the RANSE solvers. This is

critically discussed in the next paragraph with regards to

the mesh type to be used.

3.2 MESH SPECIFICATIONS

The key for a correct solution of the flow in the jet spray

region forward of spray root line and the separated flow

from chine aft of the spray root line is a correct mesh

topology and resolution. In parallel to the mesh density,

to obtain a valid courant number also at high speeds, the

time-step must be reduced. These two requirements

together imply a rather high computing time needed to

reach the steady state solution.

As regards the mesh, a very high refinement of cells is

preferable around the free surface in the spray region to

minimize a non physical inclusion of air under the keel.

Figure 2 present the VoF distribution obtained with two

different mesh resolution: colour scale represent the

percentage of fraction of water (volume of fluid)

calculated in any cell. So the cells with 100% of water

are coloured in red, while in blue those with 100% of air.

For the less dense mesh, a marked numerical diffusion of

air under the hull bottom is noted; this error artificially

decreases the frictional resistance, because, as from the

volume of fluid theory, cells with only a fraction of

water, will experience a proportionally less dense and

viscous fluid. On the other hand the wetted surface tends

to increase, with this numerical air diffusion, because the

spray root line becomes less sharply defined, and the

pressure area increases its extension and so the pressure

resistance. The result of the above described numerical

diffusion has not much influence on the predicted

resistance. In fact, the decrease of shear force is balanced

by a virtual increase of pressure resistance, so that the

total drag does not vary much between different mesh

resolutions. So the mere looking at the resistance will not

give a good indication about the proper mesh to use. One

should better analyse the VOF distribution under the hull

bottom to judge if the mesh type and resolution are valid.

In fact, in the high refined mesh case, it is noted that the

spray root line results very well defined and it has a more

realistic swept back angle as well.

This is because the VoF interface capturing method, fill

those mesh cells cut by the free surface, with a fraction

of water (between 0% and 100%) and the complement of

air, so that those cells have a hybrid fluid obtained by a

relative mixture of the two fluids, air and water, with

material proprieties, density and viscosity, that

correspond to the weighted average of filling ratio of

each fluid in the cell. The transport equation, then,

diffuse the mixture from partially filled cells to the

contiguous ones, bringing the flow mixture also below

the hull bottom. Naturally by refining the mesh (compare

figure 2), the same effect is still experienced, but on a set

of smaller cells, occupying a smaller overall volume. In

this way, the effect on the results is less pronounced,

although, not completely eliminated.

Graph 1 –Shear and pressure resistance ratio on total predicted

resistance for different mesh resolutions, obtained by CFD on

the first hull model without step

Graph 1 presents the variation of relative weight of

shear/pressure resistance on total resistance predicted

with CFD models by systematically varying the mesh

resolution from 250k cells to 5.7M cells, confirming the

above mentioned trend.

For practical purposes, if a good trade off between

computational costs and solution accuracy, a number of

about 1M cells with a nested refinement as that presented

in figure 3 is sufficient to obtain accuracies in the range

of 2% as verified in many different cases[11]. The

comparison of the predicted resistance by the present

method and model test total resistance volumetric

coefficient curve is given in Grahp 2 at the end of the

paper.

4. STEPPED HULL FORMS

A stepped hull is generally obtained from a conventional

planing hull by cutting, or inversely extruding, one or

more transverse steps on the bottom surface (figure 1).

As already discussed earlier in the paper, the

hydrodynamic function of the step is to create a negative

pressure zone immediately behind itself, in order to

naturally (sometimes also forcedly) create a separated

free surface flow downstream of its lower edge, with an

air pocket of a certain extent attached to the hull. Its

action must be complemented, in the simplest case, by a

way for the air to be connected with and flow into the

depressurized air pocket. Otherwise the step would only

increase the resistance, directly because the negative

pressure pulls behind the step back face, indirectly

because the reduced pressure below the hull bottom tends

to sink the hull and increasing the necessary trim angle to

obtain the same lift force. It is clear that the position and

sweep angle of the step line and the step height as well as

the way of air ventilation, are all important parameters to

be considered to optimize the resistance reduction of a

stepped hull. In general the beneficial effect is

experienced only above a certain speed, around the hump

speed of the hull. Below that the resistance

characteristics of the stepped version will be poorer than

the original hull form.

Incidentally it is worth to mention that beside the

advance resistance, another important effect to be

investigated to design stepped planing hulls is the

dynamic stability of the hull, i.e. porpoising and chine

walking phenomena, which can be effectively studied

with the same kind of RANSE solvers presented in this

paper.

Two rather different hulls forms are presented in this

paper: a deep-V, straight section, prismatic hull and

mixed convex-concave sections with much lower

deadrise angle. They will be referenced in the paper as

first and second model and their body plans are given in

figure 9 and figure 10, respectively.

4.1 GENERAL DESIGN

The Step divide the hull in two portions, one forward of

it, namely the forward-body, the other aft of it: the after-

body. The reduction of shear resistance obtained with the

stepped hulls starts in general to be significant for around

Fn=3.0 and rather important for Fn>5.0. So in general,

it is a good solution for very high speed crafts. For the

specific case of model-1, the predicted resistance

reduction obtained by ventilation of the bottom with the

given step design at different speed is presented in Graph

3, in which a rather limited reduction of about 5% is

noted at the top speed (Fn4.7).

For a proper design of the step it should be considered to

use only the Fwd-body to carry out the major part of the

lift (90%) to sustain the craft, while using the Aft-body

to give a small portion of lift force (10%) only to adjust

the trim angle to an optimum value; so attention is to be

paid for the proper definition of the geometry of the

After-body that must be consistent with the geometry of

the step.

4.2 DESIGN OF 1

st STEPPED HULL MODEL

The model, as originally designed, had a quite small step

(in height) with a negative sweep-back angle. The flat

spray rail at the chine, still present in the forward body

has been substituted by a very sharp-V type chine, as

visible on the vertical of the body plan of figure 9. The

0 20 40 60 80

100 120

5.700.000 Cells 2.100.000 Cells 1.000.000 Cells 465.000 Cells 250.000 Cells

Shear Pressione

original idea of the designer, was probably that to

contain the trapped air pocket below the hull bottom, in

similitude to what it is done in SES crafts. Model was

also equipped, in fact, with an air feeding system

composed by rectangular conduits connecting the deck

with the bottom. Tests and simulations were all

performed for the naked hull version, without

appendages.

4.3 DESIGN OF 2nd

STEPPED HULL MODEL

The Vessel model was made in scale 1:10 according to

Italian Navy's towing tank tests made in La Spezia on

1951. The model is named b112, as tested in series III

and series IV reported in [10].

The parent hull was a MAS hull, of the type still widely

used on hull form for small and fast planing crafts, also

for pleasure purposes. The step is rectilinear, without any

sweep-back angle, and it is obtained with a reduction on

the vertical plane of the step, namely "Bassofondo", to

bring the air from the side to the keel just downstream of

the step, as visible in the body plan of Figure 11. All The

sections are not straight and typical for this class of

boats; there is no chine plate, and an upright side with a

small negative angle. The width is reduced at stern with a

flat transom inclined of a small backward angle. The bow

is round with a deep-V sections.

The deck in the CFD model, has been realized with a

transversally flat surface that connect both sides, but, as

in the first model, the aerodynamic forces arising on it

are not considered in the calculation of total resistance.

Among all the other tests reported in [10], this model was

chosen because it was one of the few tested without

appendages and propellers.

4.4 MESH SPECIFICATION

If the problem of refining mesh around the free surface is

important for conventional planing hulls, this is even

more true for stepped hulls. In fact, in the latter case,

there are not only the problems of virtual air inclusion at

the bow (spray root line) and the flow separation from

the chine (water on side), but there is also that of the

correct prediction of the separated flow from step that

reattaches on the bottom of the after-body keel with the

same above problems. In fact, if the flow arriving at the

step already has a numerical inclusion of air coming from

the first spray root line, this gets worse when it reattach

on after-body after having mixed, also in the reality with

air in the air-cushion downstream the step.

So to accurately predict this complicated flow mixture

the mesh has to be highly refined in the afterbody,

between step and transom, as shown in the example of

Figure 3, that refers to the CFD model-1. In the same

figure, the ventilation pipes (in white) and the hole on the

chine necessary to bring that air behind the step.

Figure 4, instead, presents the comparison between

After-body and Forward-body spray root line solution. In

the Fwd-body the inclusion of air is limited, because the

free surface of the incoming is quite sharp and interests

only one cell; in the Aft-body, instead, the inflow is more

complex and many more cells are interested by a mixed

flow (air/water) that is caused by the wrongly predicted

separation behind the step, as visible in the Figure 5. A

method to predict a sharp separation also aft of the step is

needed to improve the solution, as will be presented in

the next paragraph.

4.5 INTERFACE SOLUTION

The proposed solution is to build a virtual interface

surface behind the step aligned with the local flow, in

order to refine the mesh in the proximity of the interface

with the mesher routine. The interface has only the role

to topologically divide the mesh along the guessed free

surface to align the cells sides with the free surface,

reducing the number of cells with a mixture of fluids.

The effect of this different mesh can be seen in Figure 5,

showing the flow capture without interface (top picture),

with interface and double prism layer around (bottom

picture) the surface. In the second case the flow is very

well captured. The combination of this virtual interface

with nested mesh refinement, previously discussed, can

ensure good results as presented in the comparison of top

and bottom pictures in figure 6.

The problem associated with this virtual interface

method, though, is of course that the free surface is not

known a priori, so a guess must be done and refined after

the first obtained solution. In our cases we notice that the

free surface is initially aligned with the Fwd-body keel

and after about 10 heights it turns in the direction of the

undisturbed flow. This behaviour can be approximated

with two planes at a certain angle between them.

Secondly, at present, the solver [9] does not accept the

interface model with the 6 DoF solver, so the trim and

rise must be first found with a more approximated model

and then imposed by trial and error in this more accurate

model. This procedure usually works with few

adjustments, but it is very time consuming. In fact, the

use of interface modifies the pressure distribution in the

after-body, so the trim changes from one interface shape

and the other and needs an iterative search of the right

trim (and sinkage). The following table summarises the

differences found between the simulation without

interface and the simulation with it. The drag values of

the two simulations have been corrected to correspond to

exactly the same lift force taken from the experimental

model.

Table 1 – Model1: calculated CFD resistance components

and correlation error with experimental results, Fn= 4.1

Drag Exp. Drag CFD Error % Drag Shear Drag Prex N N N N

Without 187.6 169.6 -10 90.0 79.6 With Interf. 187.6 176.1 -6 106.1 70.0

As expected the model with specially refined mesh get

closer to the experimental results, and as commented

before shear drag is increases and pressure drag reduces

with respect to the coarse mesh model. This is consistent,

since in the first case (w/o interface) the wetted surface is

higher, but is in contact with a fluid with less density and

viscosity than in the second case.

The underestimation error on shear force noted on the

model with a coarse mesh implies an corresponding

underestimation of the resistance reduction obtained with

a certain step design with respect to the original hull

form. In fact this underestimation error on shear is

expected also for full scale CFD simulations, though,

obviously it will be smaller in relative sense, being lower

at full scale the relative weight of shear to pressure

resistance.

4.6 COMPUTATIONAL ACCURACY

From a qualitative point of view, the predicted air/water

mixed flow pattern obtained with the above mentioned

CFD models is very similar to the observations done in

towing tank; we can see, for instance figure 7 and 8,

which present a photograph from below of the model

towed at Fn= 4.1 and the same view obtained from the

CFD model, in the same conditions. The two cases are

very close, in terms of compared the stagnation line,

spray area and wetted and ventilated portion of the

bottom: the spray extent in the far field, instead, is not so

accurately captured due to the increasing mesh cells size

toward the boundaries.

From a quantitative point of view, the level of accuracy

that can be obtained on the CFD simulation of stepped

hull can be better discussed on a more consistent and

systematic series of experimental tests made in case of

model 2. The tests on this hull model (figure 10) has

been repeated with two different longitudinal center of

gravity values. The complete result of the tests are

reported in Table 3 and Table 5, respectively, referred to

as Series III and Series IV. For each series of tests, not

only the resistance components, calculated with

ATTC’47 method used at that time, but also dynamic

trim and sinkage are reported for each run speed.

4.6 (a) Series III

Series III are a 27 runs testes with Fn going from 0.627

to 6.269. The model is a 1.55 LPP m long and 0.39 BOA m

large; with a weight of 15.107 kg and a LCG of 0.8 m.

The craft has a static angle of trim of S= -0.81°, positive

bow down, as reported in table 2.

The model geometry scale is 1:10.

Table 3 presents the measured resistance values in model

scale, and a tentative of interpretation of the resistance

components in model scale, performed on the basis of the

mean wetted length and surface taken from visual

observation during each run and ATTC’47 frictional

correlation line, following the practice used at the time of

tests.

In pure planning condition, the error on total resistance

between towing tank tests and CFD simulations, reported

in table 4 is very small; never exceeding 3.4 %. For the

case at 20 kn (Fn=2), the error is higher, but the flow

regime there is not purely planning and another mesh

typology and domain size should be used. For improving

the results, this case was studied with a more refined

mesh obtained with the virtual interface technique and

the respective result is reported at bottom of table 4; as

visible the error in the predicted resistance reduces, but it

is still high (10%) due to the limited domain size, valid

for pure planing conditions and not for semi-planing

regimes.

Trim predictions, differently to drag ones, present some

noticeable uncertainty, varying around ±10%. The

precision of the measurement of trim values at those

time, though is also uncertain, rending the discussion a

little difficult. Moreover, due to the ventilation of a large

portion of the aft body (at some speed the whole aft

body) the dynamic trim angle is not stationary but it

oscillates around a mean value, as noted from CFD

simulations. The same it is expected to happen also in the

model tests, though a single (averaged?) value was

reported. Again also in this case, the virtual interface

solution approach can minimize this error (from 12% to

5%) being the pressure distribution on after-body part,

that has a large influence on the trim angle, more

correctly predicted .

Graph 4 represents the trends of the predicted total drag

volumetric coefficient against volumetric Froude number

obtained by CFD simulations with the towing tank

results. As clear the accuracy is very good for Fn>2.5.

4.6 (b) Series IV

Series IV present similar trends with respect to Series III,

the mainly difference being the LCG set at 7.5m from the

transom. The weight and the wetted length are the same;

the static trim angle is S= -0.07°, as reported in table 2.

In table 5 we have all the experimental results from

towing tank about this condition.

This series of tests was investigated in more details with

a larger number of CFD runs, to derive a better defined

resistance curve. For this type of test, a very good

correlation is again found for the runs at high speeds, for

which a maximum error of about 2% is noted. At slow

speed, in pre-planning regime, the error rapidly increases

due to low Froude number, but in any case remains lower

than 6%.

In Graph 5, it can be noted also graphically how the total

drag coefficient trend over speed is close to that obtained

from towing tank measurements, at high speed. A final

notice can be done on dynamic trim angle D that

presents an average error of about 8%. Also in this case

the error on trim is increasing with speed, highlighting a

possible problem casued by the unsteady and fluctuating

nature of this signal over time. In fact particularly in this

series, at very highest speeds, the particular shape of the

free surface calculated aft of the step edge, keeps the

after-body floating in air although at a very close

proximity to the water. This interesting and rather

unusual condition is also confirmed by mode test

sketches, reporting wetted part of the after-body.

4.7 INFLUENCE OF GEOMETRIC

PARAMETERS

The influence of geometric parameters were studied with

regard to the last case, Series IV: the original step height

of the model tested in towing tank was taken as reference

height, while two additional ones have been added,

namely +30% and -30%. The CFD simulations covered

the complete planing regime range, with three calculation

speeds: 20, 30 and 40 kn in ship scale.

The geometry modification was restricted to the after-

body of the hull. In this respect, to increase the step

height the after-body bottom surface was parallelly

shifted up of a normal distance corresponding to the

height reduction; inversely to decrease the step height the

bottom surface is shifted downward.

4.7 (a) Step Height

Table 6 presents the CFD results obtained for the model

with the original step height, while Table 7 reports the

same Lift and Drag force components subdivided

between after-body and forward-body.

Complementally, Table 8 to Table 11, report the same

kind of results for two modifications of the step height: a

variation of ±30%. Graph 6 presents an extract of the

obtained CFD results in terms of friction and pressure

resistance components for the three different step heights

as a function of speed. As visible, the pressure resistance

of the forward body decreases when the step height

reduces, ad it is partly compensated by a contemporary

increase of the frictional resistance: this changes are

primarily due to the difference in dynamic attitude

assumed by the hull with different steps height (trim

increases and sinkage decreases with step height).

The shear resistance of the after-body reaches almost null

values at high speeds (Fn>3.0) for 0% and +30% step

heights since the wetted area of the bottom is nearly zero,

while for the lowest step the shear resistance maintains

on a considerable value. As regards the pressure

resistance, for high speeds, the best step variant seems to

be that of the original design: the lowest height being

better at the lowest speed as expected.

These conflicting trends noted for the shear and pressure

resistance make the final results in terms of total

resistance not very much dependent from the step height,

at least in the investigated range of variation. Total

resistance seems anyhow to be slightly decreased in the

highest speed range (Fn>3.0) for the smaller step.

Some more definite trends could be found with a more

significant step height variation. For this reason new

CFD calculations are currently planned and will give

definitive guidelines about step influence on drag

reduction. Naturally, if the step height is reduced too

much, there is the possibility to largely reduce or even to

loose the ventilated portion of the after-body, with the

result of having a significant increase of drag, due to

separated recirculating flow induced aft of the step

irregularity in the fully wetted flow under the hull

bottom.

5. CONCLUSIONS

Paper outlined the current techniques devised and applied

by the Marine CFD Group to increase the accuracy in the

numerical prediction of fast planing hull resistance in

calm water, in both cases for conventional hull forms as

well as for hulls with transverse steps. The techniques

regards mainly the generation of a proper topology for

the mesh and a refined mesh cells, aligned locally with

the predicted free surface shape in the stagnation line

region as well as in the separation region aft of the step.

In particular the presented CFD models are able to

accurately define the main physical aspects of the flow

under a planing hull with steps at high speeds: the

position and shape of the spray root line and the spray

area are correctly predicted; so is correctly predicted the

free surface separation from the step lower edge and the

ventilated portion of the bottom downstream of it, also

for complicated hull, step and chine shapes like those of

the presented model-1.

The level of accuracy on the numerical results achievable

with the described techniques are generally satisfactory

for engineering purposes, in the pure planing regime,

being in the average error within a 3% with respect to

reliable experimental results.

The hydrodynamic effect of a transverse step has been

discussed on the basis of the detailed CFD results, on two

very different hull shapes, and analysing the results

obtained for model-2 from a systematic variation of the

step height and test speed. In general, for the examined

cases, the effect of the step results beneficial, due to an

important reduction of the shear drag component with

respect to the original hull without step.

When the step height is reduced but still sufficient to

induce ventilation on the aft-body bottom, the CFD

simulations seem to indicate a reduction in total

resistance, primarily justified by a lowest value of the

running trim angle which in general means higher

induced drag (pressure) resistance on the forward body.

CFD models are useful in this respect not much to

evaluate the drag reduction difference between one step

height and another, but rather to find the lower step

height to ensure a proper after-body ventilation, the final

key for drag reduction for stepped hulls.

Eventually, once the minimum step height to induce

ventilation is set, a final CFD study on the most

appropriate shape of the after-body to ensure the lowest

trim angle and pressure resistance on the forward and

after-body is suggested and can be effectively performed

by the proposed CFD models.

6. AKNOWLEDGEMENTS

Authors wish to acknowledge the support of this work by

Promostudi University Pole of La Spezia within a

broader research program dedicated to the numerical

hydrodynamic simulations for the design of motor/sailing

yacht. A special thanks to Prof. Marco Ferrando for his

expert hint in using model tests of the towing tank of La

Spezia as reference data for CFD validations.

7. REFERENCES

1. Savander B.R., Scorpio S. M., Taylor R.K., “Steady

hydrodynamic analysis of planing surfaces”, Journal of

Ship Research, vol. 46, no.4, pp. 248-279, 2002

2. Caponnetto, M. “Practical CFD Simulations for

Planing Hulls”, 2nd

High Performance Marine Vehicles

Conference (HIPER), pp. 128-138. Hamburg, 2001.

3. Thornill E., Bose N., Veitch B., Liu P. “ Planing Hull

Performance Evaluation Using a General Purpose CFD

Code”, Proceedings of Twenty-Fourth Symposium on

Naval Hydrodynamics, NAP, 2003.

4. Brizzolara S., Serra F. “Accuracy of CFD Codes in the

Prediction of Planing Surfaces Hydrodynamic

Characteristics”. 2nd

Int. Conference on marine Research

and Transportation, ICMRT”07. ISCHIA. 28-30 June,

2007. (vol. 1, pp. A-1-A-12). ISBN: 88-901174-3-5

5. Villa D., Vatteroni G., Brizzolara S.“CFD Calculations

of Planing Hulls Hydrodynamics”, Star European

Conference, London, March 2009

6. Brizzolara S., Villa D., “CFD Simulation of Planing

Hulls”, Seventh International Conference On High-

Performance Marine Vehicles Melbourne, Florida, USA

13-15 Oct. 2010.

7. Brizzolara S., Villa D. “A systematic CFD Analysis of

Flaps / Interceptors Hydrodynamic Performance”, Fast

2009 Int. Conference on High Speed Fast Ship Design,

Athens, 2009.

8. Clement E.P., Pope J.D. “Stepless and Stepped

Planing Hulls – Graphs for Performance Predictions and

Design” DTMB Report 1490, Jan 1961.

9. CD-ADAPCO “Star-CCM+ User and Theory

Manual”, version 4.04.011, 2009

10. Permanent Commission for War Material and

Experiments, “Annale n°124, Fascicolo IV” series of

model tests results in the towing tank of La Spezia, 1951.

11. Federici A., “Prediction by CFD Methods about

Hydrodynamic Behavior of Planing Hulls with

Cambered Step and Stern Vee Hydrofoil”, Master

Degree Thesis with Press Dignity in Yacht and Power

Craft Engineering at University of La Spezia, December

2009.

8. AUTHORS BIOGRAPHY

Stefano Brizzolara, Naval Architect and Marine

Engineer, MRINA, MSNAME, PhD in numerical

hydrodynamics. is researcher at the University of Genoa,

where he holds the two courses of “computational

hydrodynamics for ship design” for the MSc degree in

naval architecture and yacht design.

Head of the Marine CFD Group, besides specialist

consultancy work for industries, he is currently guiding

different research projects for ONR, Italian Ministry of

Defence and EU all dealing with non conventional high

efficiency hull form and propulsors design, devised and

optimised by CFD methods. His previous experience

includes navy ships and propellers design in the

hydrodynamic design office of Fincantieri Naval

Business Unit and experimental research at the cavitation

tunnel of the Italian Navy in Rome.

Alessandro Federici, MSc in yacht engineering, holds a

research grant in La Spezia University Pole for

specializing, inside the Marine CFD Group, on the

application of RANSE solvers to typical yacht design

problems; in particular fast and non conventional hulls

simulations for resistance in calm water and motions in

waves.

Figure 2 – Model 1: VOF distribution on the hull surface. Mesh with 1M (top) and 5.7M cells (bottom)

Figure 3 – Model-1: generated mesh typology for the stepped version

Figure 4 – Model 1: VoF distribution in the spray root line region in the fore-body (top) and aft-body (bottom)

Figure 5 – Predicted Air/Water (VoF) distribution behind the Step with standard (top) and refined mesh typology (bottom)

Figure 6 – Comparison of VOF distribution between the case without interface (top) and with it (bottom)

Figure 7 - Underwater Photo from model tests– Flow pattern and bottom ventilation at Fn= 4.1

Figure 8 - RANSE Simulation results in the same case of figure 7

Figure 9 - Body Plan of First Stepped Hull

Figure 10 - Body Plan of Second Stepped Hull

TABLES OF MODEL TESTS-CFD RESULTS for Model-2

LPP 15.500 dm

LWL 14.959 dm

BOA 3.900 dm

BWL 3.890 dm

IAV 72.7 mm

IAD 50.7 mm

XG 0.800 m

S -0.81 deg

LPP 15.500 dm

LWL 14.912 dm

BOA 3.900 dm

BWL 3.890 dm

IAV 62.0 mm

IAD 60.0 mm

XG 0.750 m

S -0.07 deg Table 2 - Main Geometrical and Load Characteristics of Model-2, for Test Series III (left) and IV (right)

Table 3 - Series III - Towing Tank Results

Table 4 - Series III - CFD Results

V S V S FnV FnL V M R O R A (15°C) R O R A (15°C) R T C RT Vol I AV I AD I AV I AD [Kn] [m/s] [m/s] [Kg] [Kg] [N] [N] [N] *1000 [dm] [dm] [mm] [mm] [deg]

6 3.087 0.627 0.255 0.976 0.104 0.106 1.020 1.040 2.060 70.78 0.03 -0.02 75.7 48.7 -1.00 8 4.116 0.836 0.340 1.301 0.302 0.184 2.963 1.805 4.768 92.19 0.09 -0.04 81.7 46.7 -1.29 10 5.144 1.045 0.425 1.627 0.812 0.285 7.966 2.796 10.762 133.05 0.13 -0.02 85.7 48.7 -1.37 12 6.173 1.254 0.510 1.952 1.258 0.390 12.341 3.826 16.167 138.86 -0.10 0.20 62.7 70.7 0.30 14 7.202 1.463 0.595 2.278 1.407 0.494 13.803 4.846 18.649 117.61 -0.26 0.31 46.7 81.7 1.29 16 8.231 1.672 0.680 2.603 1.574 0.628 15.441 6.161 21.602 104.34 -0.34 0.32 38.7 82.7 1.63 18 9.260 1.881 0.765 2.928 1.959 0.778 19.218 7.632 26.850 102.50 -0.37 0.31 35.7 81.7 1.70 20 10.289 2.090 0.849 3.254 1.953 0.872 19.159 8.554 27.713 85.66 -0.40 0.30 32.7 80.7 1.77 22 11.318 2.299 0.934 3.579 1.806 0.923 17.717 9.055 26.771 68.40 -0.52 0.31 20.7 81.7 2.25 24 12.347 2.507 1.019 3.904 1.689 0.958 16.569 9.398 25.967 55.76 -0.67 0.32 5.7 82.7 2.84 26 13.376 2.717 1.104 4.230 1.558 1.005 15.284 9.859 25.143 45.99 -0.79 0.34 -6.3 84.7 3.36 28 14.404 2.925 1.189 4.555 1.466 1.041 14.381 10.212 24.594 38.79 -0.89 0.33 -16.3 83.7 3.69 30 15.433 3.134 1.274 4.880 1.389 1.022 13.626 10.026 23.652 32.50 -0.96 0.31 -23.3 81.7 3.88 32 16.462 3.344 1.359 5.206 1.323 1.003 12.979 9.839 22.818 27.55 -1.00 0.29 -27.3 79.7 3.95 34 17.491 3.552 1.444 5.531 1.266 0.989 12.419 9.702 22.122 23.67 -1.04 0.27 -31.3 77.7 4.02 36 18.520 3.761 1.529 5.856 1.219 0.985 11.958 9.663 21.621 20.63 -1.08 0.26 -35.3 76.7 4.13 38 19.549 3.970 1.614 6.182 1.188 0.987 11.654 9.682 21.337 18.27 -1.12 0.24 -39.3 74.7 4.21 40 20.578 4.179 1.699 6.507 1.111 1.003 10.899 9.839 20.738 16.03 -1.14 0.21 -41.3 71.7 4.17 42 21.607 4.389 1.784 6.833 1.040 1.028 10.202 10.085 20.287 14.22 -1.15 0.19 -42.3 69.7 4.13 44 22.636 4.597 1.869 7.158 0.981 1.064 9.624 10.438 20.061 12.81 -1.16 0.17 -43.3 67.7 4.10 46 23.664 4.806 1.954 7.483 0.924 1.111 9.064 10.899 19.963 11.67 -1.16 0.14 -43.3 64.7 3.99 48 24.693 5.015 2.039 7.809 0.869 1.171 8.525 11.488 20.012 10.74 -1.14 0.11 -41.3 61.7 3.80 50 25.722 5.224 2.124 8.134 0.817 1.243 8.015 12.194 20.209 10.00 -1.12 0.08 -39.3 58.7 3.62 52 26.751 5.433 2.209 8.459 0.766 1.330 7.514 13.047 20.562 9.40 -1.11 0.06 -38.3 56.7 3.51 54 27.780 5.642 2.294 8.785 0.727 1.428 7.132 14.009 21.141 8.96 -1.09 0.04 -36.3 54.7 3.36 56 28.809 5.851 2.379 9.110 0.624 1.537 6.121 15.078 21.199 8.36 -1.08 0.03 -35.3 53.7 3.29 58 29.838 6.060 2.464 9.435 0.533 1.655 5.229 16.236 21.464 7.89 -1.08 0.02 -35.3 52.7 3.25 60 30.867 6.269 2.548 9.761 0.462 1.790 4.532 17.560 22.092 7.59 -1.09 0.01 -36.3 51.7 3.25

Trim Ship Model

V S V S FnV FnL V M R PREX R SHEAR R T Error C RT Vol I AD D Errore [Kn] [m/s] [m/s] [N] [N] [N] [%] *1000 [mm] [deg] [%]

20 10.289 2.090 0.849 3.254 16.953 6.745 23.698 -14.49 73.25 78.76 1.99 12.03 30 15.433 3.134 1.274 4.880 15.603 7.250 22.853 -3.38 31.41 82.03 4.26 9.92 40 20.578 4.179 1.699 6.507 15.100 5.922 21.022 1.37 16.25 72.13 4.65 11.47 60 30.867 6.269 2.548 9.761 11.419 11.042 22.461 1.67 7.72 47.00 2.91 -10.37

20 10.289 2.090 0.849 3.254 17.766 6.957 24.723 -10.79 76.42 82.10 1.87 5.43

Ship Trim Model

Table 5 - Series IV - Towing Tank Results

Table 6 - Series IV - CFD Results

Table 7- Series IV - Aft & Fwd body's resistance component

Table 8 - Series IV - Step Height -30% - CFD Results

Table 9 - Series IV - Step Height -30% - Aft & Fwd body's resistance component

V S FnV Lift Overall R PREX R SHEAR R T Overall Lift Overall R PREX R SHEAR R T Overall [Kn] [N] [%] [N] [N] [N] [%] [N] [%] [N] [N] [N] [%]

20 2.090 62.221 41.98 2.495 1.857 4.351 19.72 86.005 58.02 12.945 4.769 17.714 80.28 30 3.134 32.639 22.02 2.487 1.409 3.897 17.81 115.617 77.98 12.121 5.859 17.980 82.19 40 4.179 11.781 8.01 1.448 0.299 1.747 8.72 135.374 91.99 12.191 6.109 18.300 91.28

After-Body Forward-Body Ship

V S V S FnV FnL V M R O R A (15°C) R O R A (15°C) R T C RT Vol I AV I AD I AV I AD D [Kn] [m/s] [m/s] [Kg] [Kg] [N] [N] [N] *1000 [dm] [dm] [mm] [mm] [deg]

6 3.087 0.627 0.255 0.976 0.108 0.102 1.059 1.001 2.060 70.78 0.04 0.03 66 63 -0.11 8 4.116 0.836 0.340 1.301 0.315 0.175 3.090 1.717 4.807 92.94 0.11 0.03 73 63 -0.37 10 5.144 1.045 0.425 1.627 0.784 0.266 7.691 2.609 10.301 127.35 0.04 0.10 66 70 0.15 12 6.173 1.254 0.510 1.952 1.171 0.369 11.488 3.620 15.107 129.76 -0.18 0.26 44 86 1.55 14 7.202 1.463 0.596 2.278 1.414 0.481 13.871 4.719 18.590 117.24 -0.32 0.33 30 93 2.33 16 8.231 1.672 0.681 2.603 1.609 0.606 15.784 5.945 21.729 104.96 -0.39 0.32 23 92 2.55 18 9.260 1.881 0.766 2.928 1.775 0.725 17.413 7.112 24.525 93.62 -0.46 0.29 16 89 2.70 20 10.289 2.090 0.851 3.254 1.666 0.784 16.343 7.691 24.035 74.29 -0.55 0.28 7 88 2.99 22 11.318 2.299 0.936 3.579 1.640 0.800 16.088 7.848 23.936 61.16 -0.67 0.29 -5 89 3.47 24 12.347 2.507 1.021 3.904 1.575 0.830 15.451 8.142 23.593 50.66 -0.77 0.28 -15 88 3.80 26 13.376 2.717 1.106 4.230 1.507 0.893 14.784 8.760 23.544 43.06 -0.85 0.27 -23 87 4.06 28 14.404 2.925 1.191 4.555 1.418 0.912 13.911 8.947 22.857 36.05 -0.91 0.25 -29 85 4.21 30 15.433 3.134 1.276 4.880 1.340 0.928 13.145 9.104 22.249 30.58 -0.94 0.23 -32 83 4.24 32 16.462 3.344 1.361 5.206 1.277 0.939 12.527 9.212 21.739 26.25 -0.97 0.21 -34.5 81 4.26 34 17.491 3.552 1.446 5.531 1.224 0.946 12.007 9.280 21.288 22.77 -0.99 0.19 -37 79 4.28 36 18.520 3.761 1.531 5.856 1.186 0.951 11.635 9.329 20.964 20.01 -1.02 0.16 -39.5 76 4.26 38 19.549 3.970 1.617 6.182 1.163 0.957 11.409 9.388 20.797 17.81 -1.04 0.14 -41.5 74 4.26 40 20.578 4.179 1.702 6.507 1.122 0.970 11.007 9.516 20.523 15.86 -1.06 0.11 -43.5 71 4.22 42 21.607 4.389 1.787 6.833 1.091 0.993 10.703 9.741 20.444 14.33 -1.06 0.08 -44 68 4.13 44 22.636 4.597 1.872 7.158 1.055 1.025 10.350 10.055 20.405 13.03 -1.07 0.06 -44.5 65.5 4.06 46 23.664 4.806 1.957 7.483 1.013 1.067 9.938 10.467 20.405 11.93 -1.07 0.03 -45 63 3.99 48 24.693 5.015 2.042 7.809 0.961 1.121 9.427 10.997 20.424 10.96 -1.07 0.01 -45 61 3.91 50 25.722 5.224 2.127 8.134 0.896 1.191 8.790 11.684 20.473 10.13 -1.07 0.00 -45 60 3.88 52 26.751 5.433 2.212 8.459 0.814 1.278 7.985 12.537 20.523 9.39 -1.08 0.00 -46 60 3.91 54 27.780 5.642 2.297 8.785 0.728 1.372 7.142 13.459 20.601 8.74 -1.11 0.01 -49 61 4.06

Trim Ship Model

V S V S FnV FnL V M R PREX R SHEAR R T Errore C RT Vol I AD D Errore [Kn] [m/s] [m/s] [N] [N] [N] [%] *1000 [mm] [deg] [%]

20 10.289 2.090 0.851 3.254 16.323 6.303 22.626 -5.86 69.93 90.66 3.19 6.69 24 12.347 2.507 1.021 3.904 16.133 6.844 22.977 -2.61 49.34 91.23 4.13 8.65 30 15.433 3.134 1.276 4.880 15.436 5.753 21.190 -4.76 29.12 71.74 4.66 9.82 34 17.491 3.552 1.446 5.531 15.432 6.094 21.526 1.12 23.03 76.65 4.61 7.83 40 20.578 4.179 1.702 6.507 15.075 6.031 21.106 2.84 16.31 70.20 4.56 7.90 44 22.636 4.597 1.872 7.158 15.229 5.077 20.305 -0.49 12.97 69.26 4.74 16.80 50 25.722 5.224 2.127 8.134 14.762 5.674 20.435 -0.19 10.11 64.50 4.46 15.15

Ship Model Trim

V S FnV Lift Overall R PREX R SHEAR R T Overall Lift Overall R PREX R SHEAR R T Overall [Kn] [N] [%] [N] [N] [N] [%] [N] [%] [N] [N] [N] [%]

20 2.090 59.671 40.24 2.990 1.720 4.710 20.82 88.599 59.76 13.333 4.583 17.916 79.18 30 3.134 9.623 6.37 1.422 0.339 1.761 8.31 141.386 93.63 14.015 5.414 19.429 91.69 40 4.179 9.591 6.47 1.565 0.590 2.155 10.21 138.646 93.53 13.509 5.442 18.951 89.79

Forward-Body After-Body Ship

V S V S FnV FnL V M Lift Error R PREX R SHEAR R T Var on 0% C RT Vol Rise D Var on 0% [Kn] [m/s] [m/s] [N] [%] [N] [N] [N] [%] *1000 [mm] [deg] [%]

20 10.289 2.090 0.851 3.254 148.226 0.02 15.440 6.626 22.065 -2.48 68.20 0.39 2.77 -13.17 30 15.433 3.134 1.276 4.880 148.256 0.04 14.608 7.269 21.877 3.24 30.06 3.78 4.11 -11.85 40 20.578 4.179 1.702 6.507 147.155 -0.71 13.639 6.408 20.047 -5.02 15.50 4.15 4.07 -10.81

Trim Ship Model

Table 10 - Series IV - Step Height +30% - CFD Results

Table 11 - Series IV - Step Height -30% - Aft & Fwd body's resistance component

Graph 2 – Model-1 (without steps): comparison of the predicted total resistance by CFD results and towing tank measurements

V S FnV Lift Overall R PREX R SHEAR R T Overall Lift Overall R PREX R SHEAR R T Overall [Kn] [N] [%] [N] [N] [N] [%] [N] [%] [N] [N] [N] [%]

20 2.090 57.351 38.71 3.252 1.399 4.650 20.31 90.807 61.29 13.706 4.540 18.246 79.69 30 3.134 21.959 14.81 2.460 0.518 2.979 13.25 126.320 85.19 14.906 4.598 19.504 86.75 40 4.179 3.876 2.62 0.928 0.069 0.997 4.72 144.105 97.38 16.159 3.963 20.122 95.28

After-Body Forward-Body Ship

V S V S FnV FnL V M Lift Error R PREX R SHEAR R T Var su 0% C RT Vol Rise D Var su 0% [Kn] [m/s] [m/s] [N] [%] [N] [N] [N] [%] *1000 [mm] [deg] [%]

20 10.289 2.090 0.851 3.254 148.159 -0.03 16.958 5.938 22.896 1.19 70.77 -0.50 3.49 9.20 30 15.433 3.134 1.276 4.880 148.278 0.05 17.366 5.116 22.482 6.10 30.90 7.70 5.28 13.21 40 20.578 4.179 1.702 6.507 147.981 -0.15 17.087 4.032 21.119 0.06 16.32 8.70 5.50 20.64

Ship Trim Model

15

20

25

30

35

40

2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8

103

CRT

Fn

Without Step

With Step

Graph 3 – Model-1: comparison of the hull resistance in the conventional and modified with step versions (numerically predicted)

0

10

20

30

40

50

60

70

80

90

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

103CRT

Fn

Towing Tank

CFD

Graph 4 - Model2, Series III: Total Resistance Coefficient Vs Volumetric Froude Number

0

10

20

30

40

50

60

70

80

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

103CRT

Fn

Towing Tank

CFD

Graph 5 – Model2, Series IV: Total Resistance Coefficient Vs Volumetric Froude Number

Graph 6 - Model2, Serie IV: Shear and pressure resistance components dependence on step height at different speeds, evaluated on

the after-body and forward body parts of the hull