Simulation of unsteady ship maneuvering using free-surface RANS solver

7/28/2019 Simulation of unsteady ship maneuvering using free-surface RANS solver

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26th

Symposium on Naval HydrodynamicsRome, Italy, 17-22 September 2006

Simulation of unsteady ship maneuveringusing free-surface RANS solver

E. JACQUIN, P.-E. GUILLERM, A. DROUET, P. PERDON,

(Bassin d'essais des carnes (DGA), France)

B. ALESSANDRINI, (Ecole Centrale de Nantes, France)

INTRODUCTION

Applications of RANS with free surface solvers fornaval hydrodynamics started about ten or fifteenyears ago. Those solvers show great interest sincethey notably increase computations accuracy which is

mainly due to the use of non-linear free surfaceconditions and viscosity in the equations. Nowadaysthey are commonly used in ship-resistanceapplications by almost all research institutes and fewindustrial parties. More recently their field ofapplication extends to hull form optimization (Tahara

et al. [1], Jacquin et al. [2], Campana et al. [3]) andfirst computations of ship maneuver based on Navier-stokes simulations were also carried out on the Serie60 ship for steady cases (Alessandrini [4]) and forced

motions (Di Mascio [5]). In the same way, in orderto extend applications of RANS solvers, Wilson and

Stern [6] [7] performed computations of forced rolland roll decay with very accurate results compared to

experiments.

Thus, the field of application of these tools isincredibly wide, specifically for unsteady simulations

where the model can move in all directions of amaneuvering ship. The interest of such fully unsteadynumerical simulations for maneuvering simulations isto naturally take into account the viscous flow around

the hull or appendages, the wake in the propellerplane, and then non-linear effects of hull / propeller /

rudder interactions.

This paper presents the recent developmentsperformed at Bassin d'essais des carenes and Ecole

Centrale Nantes in order to simulate unsteady selfpropelled maneuvering ship with the RANS codeICARE.

In a first part, we briefly describe the theory andequations used in the ICARE solver, and adaptationsperformed for steady maneuvering simulations.Validation of the solver on three different ships insteady maneuvers is then presented : pure drift, puregyrating, and a combination of drift and gyrating.

Then an example of forced oscillatory unsteadymaneuvering simulation (forced sway or yaw motions

with forward speed) show the ability of the solver to

compute unsteady applications.

The second part of the paper is dedicated to the

description of the development performed for freeunsteady simulations of a self propelled maneuveringship : six degrees of motion capabilities, actuator diskand moving appendages. The development andvalidation of the solver is composed of successivesteps, and an application on a simplified case is

presented at this stage.

HYDRODYNAMICS: RANS WITH FREESURFACE SOLVER ICARE

ICARE [9][10]is a RANS with free-surface solverdeveloped by Ecole Centrale Nantes through FrenchNavy (DGA / Bassin d'essais des carenes) support.

Governing equations, turbulence model

The convective form of Reynolds Averaged Navier-Stokes Equations is written through partial

transformation from cartesian space (x1

,x2

,x3

) tocurvilinear space ),,( 321 fitted to the hull and

the free surface at each time-step. The dependantunknowns of the system are the free-surfaceelevation, the three cartesian velocity components(ui), the pressure (p) including the gravitational

effects ( 3gx ) and the turbulent kinetic energy

( k3

2 ).

Mean momentum transport equations are written inthe moving referential attached to the hull:


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01

))((

,,,,

,,,

=++

+

quaaugpa

uaafuuau

k

jjit

i

kij

ij

effk

k

j

j

kit

i

k

i

eff

i

g

ij

it

(1)

Where ai is the contravariant basis, gij thecontravariant metric tensor, fi the control grid

functions and igu the grid velocity which lead the

displacement of the mesh. Inertia forces due to nonGalilean referential (gyration, accelerated translation)are taken into account in the qi terms. In translation

case (with or without drift angle) inertia forces areexpressed as follows where Ua is the hull velocity:

( )

( )

==

==

==

0

sin

cos

33

,

22

,

11

qq

Uqq

Uqq

d

tad

tad

(2)

In gyration case, inertia forces must include Coriolisand centrifugal forces:

( ) ( )

( ) ( )

==

++==

++==

0

2

2

33

1222,

1122

2112,

2211

qq

uxxxxqq

uxxxxqq

r

rtrr

rtrr

(3)

Where is the hull rotation velocity and R(x1r, x2

r)

the rotation center location.

Mass conservation is expressed as the classicalcontinuity equation:

0, =ij

ji ua (4)

Finally to close the equations set we used a classical

k turbulence model proposed by Wilcox [11],introducing a specific dissipation rate without lowReynolds formulation requirement. Transport

equation of turbulent kinetic energy and dissipation

rate are written as follows:

=++

++

=++

++

0Pr/)(

))()((

0*Pr)*(

)*)*()((

2

,

,,,

,

,,,

kg

aafuua

kkg

kaafuuak

ij

ij

t

j

j

kit

i

k

j

t

i

g

ij

it

ij

ij

t

j

j

kit

i

k

j

t

i

g

ij

it

(5)

with:

kt *= (6)

and:

===

===

1*;5.0*;09.0*

9

5;5.0;

40

3

(7)

Free surface conditions

Free surface boundary conditions are the kinematics

condition, the two tangential dynamic conditions and

the normal dynamic condition. Kinematics condition

coming from the continuity hypothesis expresses that

the fluid particles of free surface stay on it:

{ } 0))((3

2,1),(,, =+ uhuubh jiji

gij

it (8)

where ib is the bi-dimensional contravariant basis

based on the discretization of free surface only.

Dynamic conditions of the free surface are given by

the continuity of strains at the free surface. If thepressure is assumed to be constant above free surface,

normal dynamic condition is:

02 ,33

23

=r

uaaa

a

ghp ikkjji

eff (9)

where is the surface tension coefficient (that is a

physical way to smooth free surface near the hull)and r the free surface medium curvature radius.

Tangential dynamic conditions are simply given by alinear combination of first order velocitiesderivatives:

0,3 =ij

ji uga (10)

Discretization

General schemes are based on second order (in spaceand time) implicit finite differences. Discrete

unknowns are distributed on a structured curvilineargrid fitted to the hull and the free surface. VelocityCartesian components, kinetic turbulent energy andspecific dissipation rate are located on the grid nodes.Pressure is located at the center of each volume andfree surface elevation is located on the center of free

surface interfaces.

Convective terms are computed using an upwindsecond order scheme that needs a 13 nodes cell.

Diffusive terms need 7 nodes for second order


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derivatives and 12 nodes to express second-ordercross derivatives while pressure gradient requires 8nodes for each component.

Concerning the free surface, it has been shown thatthe classical way using normal dynamic condition asa Dirichlet condition on the pressure and uncoupledkinematics equation as transport equation to computefree surface elevation leads to problems connected to

mass conservation under free surface. An efficientsolution consists in using a fully coupled algorithm(Alessandrini and Al.) that requires at each time thelinear solution of mean momentum equations,continuity equation and all boundary conditionsincluding the free surface condition. In order to invert

this system with iterative solvers the linear system ismodified using free surface boundary conditions to

express the flux through free surface. In this case,conditioning number decreases dramatically and fullycoupled system can be inverted by iterativealgorithms.

Resulting linear system for velocity (U) and pseudo

velocity (U~

) components, pressure (P) and free

surface elevation (H) is written as follows

=

h

p

u

u

f

f

f

f

H

P

U

U

MM

MM

MM

MMM

~

1111

1111

1111

111111~

00

00

00

0

(11)

Thus, we obtain at each time pressure forces (normalcomponent) and friction forces (tangentialcomponent) summing the whole efforts calculated onfacets of the hull or of appendages.

Gyration case

From a boundary condition point of view,computations past a ship with pure drift angle andcomputations with gyration motion are very similar:working in relative referential moving with the hull,Coriolis and centrifugal forces have to be added as

source terms in momentum equation. This sectionshows that gyration case is quite more difficult andrequires some numerical cautions.

In this relative referential, velocity field traducingflow at rest (without hull effect) is written as followsfor the drift case:

( )

( )

=

=

=

0

sin

cos

3

2

1

u

Uu

Uu

r

ar

ar

(12)

and:

( )( )

=

=

=

03

112

221

u

xxu

xxu

r

rr

rr

(13)

in the gyration case.

In both cases these velocity fields have to verifymomentum and continuity equations (1) and (4) withpressure gradient and turbulent viscosity equal tozero. For uniform velocity field (12) it is very easy tocheck since convective and diffusive terms cancel.

We obtain for momentum and continuity equations,respectively:

0;3

1

=

=

=ii

iri

r

ir

x

uq

t

u (14)

For velocity field coming from gyration case it is

easy to see that the continuity equation is verified andmomentum equations give:

( )( )

( )

( )( )

( )( )

( )

( )( )

( )( )

( )( )

( )( )

+

==

+

==

xxxxxx

qxxxx

xxxxxx

qxxxx

r

III

r

II

tr

rr

I

tr

r

III

r

II

tr

rr

I

tr

222222,

11

2222,

11

112112,

22

1112,

22

2

2

(15)

where Coriolis (III) terms are exactly balanced bycentrifugal terms (II) for half part and convectiveterms (I) for the other part.

Problems appear when these equations arediscretized. In the drift case, due to the uniformity ofvelocity field, discrete momentum and continuityequations are verified exactly. Unfortunately in thegyration case two problems appear due to

linearization and discretization.

First issue is shown considering that discretederivatives of velocity field are unable to give exactly

rotation velocity:

+

x

u

x

u rr2

1

1

1

;0 (16)

0;2

2

1

2

x

u

x

u rr (17)

Where is a general discretization operator used inthe present formulation.


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A solution consists in adding source terms in order tobalance convection residuals. These consistent terms(they converge to zero when discretization step goesto zero) are computed as follows:

=

+

+=

+

+=

qq

ux

uu

x

uuqq

ux

uu

x

uuqq

rr

rr

rr

rrr

rr

rr

rrr

33

1

2

22

1

2222

2

2

12

1

1111

(18)

Second issue is due to convective terms linearization:convective velocities are usually expressed using

previous non-linear iteration velocities andconvective terms are of course (because it is

impossible for non-linear terms) not completelyimplicit. Then, solving momentum equations requiresgood convergence rates on linear and non-linearprocesses.

STEADY MANEUVERING VALIDATION

Even if the final aim of the present work is theunsteady maneuvering simulation, a large effort hasbeen made in order to evaluate abilities of Navier-

Stokes with free-surface computations to accuratelypredict average forces and moments applied on hullfor drift and gyration cases. We present in this sectionexamples of validation with pure drift (KVLCC2Mand HTC) and gyrating with and without drift angle(Tanker).

All forces and moments coefficients are given withrespect to the reference axes , defined with origine atship mid-ship, x-axis directed forward, the y-axis to

starboard and the z-axis downward (all according toconventions for maneuvering analysis).

KVLCC2M test case

The KVLCC2M model is 4.970m long, and wastested at NMRI for a speed of 0.994 m/s and driftangles from 0 to 12 by 3 steps.

Figure 1: HVLCC2M hull

Drag, sway and yaw moment coefficients for thosetests conditions are given in tables and Figures belowand are compared with experiments.

() Cx Cy Cn Cx Cy Cn

0 -0.0170 -0.0002 0.0000 -0.0176 0.0000 -0.0001

3 -0.0172 0.0122 0.0076 -0.0178 0.0126 0.0061

6 -0.0171 0.0272 0.0147 -0.0177 0.0256 0.0139

9 -0.0172 0.0481 0.0206 -0.0173 0.0455 0.0194

12 -0.0170 0.0732 0.0257 -0.0175 0.0708 0.0254

Computations Experiments MNRI

Table 1: KVLCC2M, drift cases, double flow model

() Cx Cy Cn

0 -2.9% - -

3 -3.3% -3.1% 23.4%

6 -3.7% 6.4% 5.9%

9 -0.7% 5.7% 6.1%

12 -2.7% 3.3% 1.2%

Differences (%)

Table 2: Differences between computations and model test

Tables 1 and 2 show that drag and lift coefficientscalculated with ICARE are closed to experimentalresults with a gap of about 3% on drag and 5% on lift.

Moment coefficients are also relatively wellpredicted, except for the smallest drift anglecorresponding to the lower values and the largest

measurement uncertainty.

Results are plotted on the Figures below.

Figure 2: CX, KVLCC2M


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Figures below compare velocity components U, V, Wmeasured in the propeller plane with results of thecalculation for the 6 drift angle case.

Y/Lpp

u/U0,v/U0,w/U0

-0.05 0 0.05 0.1

-0.2

0

0.2

0.4

0.6

0.8

Experiments6 UExperiments6 VExperiments6 WComputations UComputations V

Computations W

Figure 8: Velocity components U, V, W in the propellerplane, =6

In the same way, following figures show pressure

coefficients on the hull for several longitudinalsections.

Figure 9: Pressure coefficient on hull at X = -0.4, =6

Figure 10: Pressure coefficient on hull at X = 0.4, =6

Comparisons show a good match between computedand measured pressure coefficients. Nevertheless,

magnitudes of pressure minima are slightlyunderestimate.

Hamburgh test case (HTC)

The HTC model is 6.4033 long and was tested at

HSVA in the Virtue WP3 Eurpean Project. Theresults presented in this sections are composed of

pure drift cases, with two different Froude numbersof 0.132 and 0.238. The results composed ofresistance, side force and momentum coefficients aresummed up in the tables and charts below.


0 -0.0155 0.0000 0.0000 -0.0142 0.0000 0.0000

5 -0.0160 0.0208 0.0130 -0.0153 0.0198 0.0119

10 -0.0182 0.0520 0.0270 -0.0165 0.0476 0.0233

20 -0.0151 0.1471 0.0557 -0.0147 0.1335 0.0481

30 -0.0090 0.2581 0.0813 -0.0081 0.2439 0.0732

Computation Experiments

Table 3: HTC, Fn=0.132

() Cx Cy Cn

0 9.5 - -

5 4.8 4.9 9.6

10 9.9 9.3 16.0

20 2.8 10.2 15.7

30 11.0 5.8 11.0

Differences (%)

Table 4: Differences between computations andexperiments, Fn = 0.132

Figure 11: CX, HTC, Fn=0.132


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Figure 12: CY, HTC, Fn=0.132

Figure 13: CN, HTC, Fn=0.132

The results obtained for the second Froude number

are presented bellow.


0 -0.0169 0.0000 0.0000 -0.0141 0.0000 0.0000

5 -0.0176 0.0199 0.0134 -0.0153 0.0210 0.0124

10 -0.0192 0.0492 0.0278 -0.0168 0.0479 0.0245

20 -0.0175 0.1426 0.0572 -0.0185 0.1289 0.046830 -0.0210 0.2623 0.0838 -0.0175 0.2403 0.0704

Computation Experiments

Table 5: HTC, Fn=0.238

() Cx Cy Cn

0 20.0 - -

5 15.2 5.2 7.6

10 14.3 2.9 13.6

20 5.5 10.6 22.1

30 20.2 9.2 19.0

Differences (%)

Table 6: Differences between computations and

experiments, Fn = 0.238

Figure 14: CX, HTC, Fn=0.238

Figure 15: CY, HTC, Fn=0.238

Figure 16: CN, HTC, Fn=0.238

The comparisons of results of the computations with

experiments show a relatively good agreement.Nevertheless, differences are more important on thiscase than on the KVLCC2M case. This is perhaps


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due to free surface effects that could be neglected inthe KVLCC2M model, but not the HTC due to higherFroude number.

Figure 17: HTC streamlines and free surface, =0



Tanker

Experiments on a tanker have been carried out in the

gyration tank of Bassin d'essais des carenes in Paris.The experimental devices allowed to performgyration rates from L/R = 0.2 to 1., and for each caseseveral drift angles from 0 to 30.

Figures below present and compare computationswith experimental data (Cx, Cy, Cn and Cn/Cy) forFn=0.2 and gyration rates of 0.2, 0.5 and 1.

Drift angle ()

Cx

0 5 10 15 20 25 30

ExperimentsCalculations

L/R = 0.2

L/R = 0.5

L/R = 1

Figure 20: Cx, Fn=0.2, L/R = 0.2, 0.5 et 1

Drift angle ()

Cy

0 5 10 15 20 25 30


L/R = 0.2

L/R = 0.5

L/R = 1

Figure 21: Cy, Fn=0.2, L/R = 0.2, 0.5 et 1

Drift angle ()

Cn

0 5 10 15 20 25 30


L/R = 0.2

L/R = 0.5

L/R = 1

Figure 22: Cn, Fn=0.2, L/R = 0.2, 0.5 et 1


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Drift angle ()

Cn/Cy

0 5 10 15 20 25 30

Experiments

Calculations

L/R = 0.2

L/R = 0.5

L/R = 1

Figure 23: Cn/Cy, Fn=0.2, L/R = 0.2, 0.5 et 1

Results are in relatively good agreement withexperiments. Even if coefficients are underestimatedby the computations, the trends due to gyrating rate

variations are very well predicted. Larger errors areobtained for the most severe condition (L/R=1) wherethe ship turns on his own length which creates manylarge flow separations along the whole hull (seefigures hereafter).

Figure 24: Free surface and streamlines, L/R=0.5, drift angle 0, 10 and 20 from top to bottom


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UNSTEADY FORCED MOTION

Prior to six degrees of freedom free running

maneuvering simulations, computations with forced

ship motion were carried out. Experiments will becarried out in the VIRTUE European project, and arenot yet available. Nevertheless, the computationsallow to check the ability of the solver to computeunsteady computations.

Forced sway and yaw oscillatory motions werecomputed, and ship trajectory are presented in thefollowing figures.

Figure 25 : Forced oscillatory sway motion

Figure 26 : Forced oscillatory yaw motion

For the forced oscillatory sway motions, amplitude isequal to 0.25 Lpp, and period is equal to 10 seconds.For the forced oscillatory yaw motion, amplitude ofmotion is 15, and period is equal to 32 seconds.

Forces coefficients associated to ship motion are

presented on figures 27 to 32, using solid line forforces, and dot lines for ship motion.

Forced oscillatory sway motion

T

Cx

TRy

60 70 80 90

-0.2

-0.1

0

0.1

0.2

Figure 27: Cx, Oscillatory sway motion, T=10s

T

Cy

TRy

60 70 80 90

-0.2

-0.1

0

0.1

0.2

Figure 28: Cy, Oscillatory sway motion, T=10s

T

Cy

TRy

60 70 80 90

-0.2

-0.1

0

0.1

0.2

Figure 29: Cn, Oscillatory sway motion, T=10s


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Forced oscillatory sway motion

T

Cx

ROz

50 100-15

-10

-5

0

5

10

15

Figure 30: Cx, Oscillatory yaw motion, T=32s

T

Cy

ROz

50 100-15

-10

-5

0

5

10

15

Figure 31: Cy, Oscillatory yaw motion, T=32s

T

Cn

ROz

50 100-15

-10

-5

0

5

10

15

Figure 32: Cn, Oscillatory yaw motion, T=32s

Forces curves present irregular shapes due to wakeeffects or flow separations. Further analysis will beperformed with experimental results. The next figuresshow iso-velocity and free surface elevation for threeship positions during forced sway motion.

Figure 33: Iso-velocity, forced sway motions

Figure 34: Free surface elevations, forced sway motions


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DEVELOPMENT OF FREE MANOEUVRING

CAPABILITIES

The first part of this paper presented ICARE solver

and its validation on steady and unsteady forcedmaneuvers of ship. In order to extend the capabilities

of the solver for free running unsteady maneuvers,three modules where added and are shortly presented:six degrees of motion capabilities, movingappendages, and propulsion with actuator disk. Theend of this part presents an application of freemaneuvering ship on a simplified case, performed to

evaluate the feasibility of such computations.

SIX DEGREES OF FREEDOM SHIP MOTION

General presentation

In order to compute unsteady maneuvers of ship, thesolver must be able to compute the motions of theship under forces computed at each time step. This isdone by solving the standard Eulers law in the body

fixed coordinate frame centered on G. The sixcomponents of ship velocity and position are thenobtained, and allow to move the ship by moving thegrid, whereas the velocity are used for boundaryconditions on the hull.

Coordinate frame system

Navier-Stokes equations are solved in the fixedgeneral axis center on 0 (R0). This coordinate frame isGalilean, so acceleration terms of the fluid in theNavier-Stokes equations do not have to be taken intoaccount, and then reduce the complexity of theproblem to solve.

The Eulers laws are solved in the body fixedcoordinate frame center on G (RG).

Figure 35 : Fixed reference and body fixed axis

Figure 36 : Decomposition of the rotation of the ship

The rotation matrix used to transform coordinatesfrom the fixed coordinate frame to the body fixed

coordinate frame is defined by:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

++

+

cccssscsscsc

scssscccsssc

scscc

P RR3

Motion equations

The ship motion equations are written using theEulers law:

// /

G RR G R R

dVm V F

dt

+ =

rr rr

( )( ) /

RR R R R G R

dI I M

dt

+ =

rrr r

where

m is the mass of the ship

RI is the inertial matrix of the ship

RGV /r

is the velocity vector of the center of gravity

Rr

is the angular velocity vector

RF/r

is the total forces acting the ship

(hydrodynamic, gravitational and external forces)

RGM /r

is the total momentum acting on the ship

Assuming that the integration scheme has thefollowing form (general multi-step methods) :


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( )( )p

1 1

0

,n n i n ii

v v C f t v+ + =

= + ,

Where :

C are coefficients of implicit or explicit multi-step

methods

is the time step

1nv + is the unknown velocity components at time

step n+1

nv is the known velocity components at time step n

Euler laws can then be written in the following form :

MV F=r r

With :

m

m

mM

Ixx Ixy Ixz

Iyx Iyy Iyz

Izx Izy Izz

=

,

Vectors components are defined by :

1n ni i iV v v

+=

( )( )( )0

.p

j j ji j i

j

F C f V i =

=

rrr

Where i from 1 to 3 refers to x, y and z forces

coordinates, and I from 4 to 6 refers to momentumcoordinates.

Velocity components Vr

are calculated by solving

the linear system. Positions of the ship are then

directly integration from ship velocities.

Choice of integration scheme

Even if ship motion integration seems to be an easy

task, important numerical difficulties arrive inapplication for unsteady RANS simulation.

In the field of hydrodynamic, ship motion integrationis used for example in sea-keeping for time domainsimulations. Majority of solvers use a 4

thorder

explicit Runge-kutta scheme, associated withrelatively small time steps (at least 50 iterations perperiod of motion). A second point is that the forces

are decomposed in different terms, with an explicitknowledge of added mass terms. Those terms are

then transferred to the left hand side of the equationof motion, and integrated with the mass of the ship.

The transposition to RANS code leads to two

important difficulties :

Time step is highly time consuming and cannot be reduce drastically for practicalapplications.

Added mass terms are part of the forcescomputed by the solver, and can not be

explicitly known. Those terms are thenintegrated in the right hand side of the

equation of motion, and have a very badimpact on the stability of integrationscheme.

Several integration schemes have then been tested ona simplified equation of ship heave motion, with a

added mass term in the right hand side.

1 aHyd

mkF z z

m m m

= = +

&& &&

With initial condition : 0 0( )t z=

Where :

m mass of the ship

k hydrostatic stiffness

am added mass terms

Euler implicit and explicit schemes, implicit andexplicit multi-step methods, and predictor/correctormethods were tested. Example of resolution is shownfor two selected integration schemes.


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The first is the classical Euler scheme, explicit forvelocity integration, and implicit for positionintegration.

1

1 1

n n n

n n n

z z h f

z z h z

+

+ +

= +

= +

& &

&

The second scheme is a predictor corrector scheme,based on explicit and implicit second order multi-stepmethods (Adams Baschfrd and Adams Moultonmethods)

Predictor step :

1 1

1 1

3 1

2 2

1 12 2

n n n n

n n n n

z z h f f

z h z z

+

+ +

= +

= + +

& &

& &

Corrector step :

1 1

1 1

1 1

2 2

1 1

2 2

n n n n

n n n n

z z h f f

z z h z z

+ +

+ +

= + +

= + +

& &

& &

The results show that with the same number of timestep per period (50 time step), both integrationsschemes give an accurate prediction of the motion.

But if added mass is added in the right hand side ofthe equation of motion, only the predictor/correctorintegration scheme is stable. Unfortunately, thisstability is not obtained with added mass greater than1.5 time the ship mass.

t

Z

0 0.1 0.2 0.3 0.4 0.5

-0.001

-0.0005

0

0.0005

0.001

0.0015Euler Ma = 0 x MEuler Ma = 1 x MAnalytic solution

Figure 37 : Stability of Euler integration scheme

t

Z

0 0.2 0.4 0.6 0.8 1

-0.001

-0.0005

0

0.0005

0.001

0.0015Predictor/corrector 2nd orderMa = 0 x MPredictor/corrector 2nd orderMa = 1 x MAnalytic solution

Figure 38 : Stability of second order predictorcorrector integration scheme

MOVING APPENDAGES CAPABILITIES

In order to simulate practical ship maneuvering(zigzag for example), rotating appendages must be

taken into account during the simulations.

Two different techniques have been developed:spring network analogy to deform the mesh aroundthe appendages and mesh interpolation of pre-computed meshes at several rudder angles.

Whereas grid interpolation is relatively easy, the use

of spring networks is more complicated and is shortlypresented in this section.

Spring network method

The principle of this method is to grid segments as aspring network, so that fixed displacement of the

boundaries (appendages) propagate to the whole grid,and avoid mesh overlapping within reasonable limits.

First methods were only based on compression

springs (Batina [12]) and have been improved withthe addition of torsion springs (Farhat [13] [14])

allowing the increase of method abilities and then themagnitude of feasible deformations.

The domain is divided into three parts according to

conditions applied on it:

0m p = + + (22)


15/19

Figure 39: Boundaries definition

Nodes displacements are imposed on the moving

boundaries (m). IfiD

uuris the displacement of point I

we obtain:

{ },

i i md D i=

uur uur(23)

On the domain boundary nodes displacement is nullwhich means:

{ }00, id i= uur r

(24)

Finally we define boundary projection where nodeshave to stay on surface. For instance this law is

applied on mesh nodes of the hull located nearmoving appendages. This condition is written:

{ }. 0, pi p

d n i = uur r

(25)

Figure 40: Compression springs

Considering two neighboring points Mi and Mj beingpart of mesh, given je

uurvector carried by the MiMj

segment, the segment MiMj length is defined with

ij ij ij ijl e e e= = uur uur uur

.

Thus we are able to determine the associated unitary

vector ijij

ij

ei

l=

uurur

.

Writing duur

and duur

as the respective displacement

of Mi and Mj. So the compression or extension of a

segment is defined with ( )j i ijd d i uur uur ur

. The following

formula gives us the force at Mi point exerted by Mj:

( )( )lin

ij ij j i ij ijf k d d i i=

uuur uur uur ur ur

Spring stiffness is classically chosen as the inverseratio of the segment length, so that the shorter nodesdistances give the stiffer spring:

1ij

ij

kl

=

Static balance of efforts at Mi point is written:

Nb voisins

1

0linijj

f=

=uuur r

Or with previous relations:

Nb voisins Nb voisins

1 1

. .t t

ij ij ij i ij ij ij j

j j

k i i d k i i d = =

=

ur ur uur ur ur uur

Nodes displacements are computed by solving thelinear system either with an iterative method that

propagates displacements or with a direct method thatinverts the matrix constituted with previous terms.

The use of compression springs doesnt avoid nodes

to go on opposite faces, and are limited in practice to

rotations of about 10 degrees. A cure is to introducein the previous system torsion springs between twoneighboring segments.

Figure 41: Torsion springs

The spring stiffness is then chosen so that it increases

when the angle between two segments tends to zero

or pi, with for example)(sin

12 ijk

i

ijk

iC

= .

Writing that displacements of points i, j, k makes

angles variations in the triangle ijk which allow todetermine moment created by the torsion spring at i.Linking angles variations to points i, j, kdisplacements we obtain a set of equations similar

with the one obtain previously, but more complex,

and solve with the same method.

Mi

Mj

Mk

jkC

Mk

Mj

Mi

Mi

Mj

u

m

p

0


16/19

The method sketch applied on appendages rotation isdescribed below. Some projection steps are requiredin order to guarantee the hull geometry andintersections quality between hull and appendages.

Figure 42 : Sketch of the spring method applied on the

appendage rotation

Figures below show an example of the spring method

applied on a rudder rotation. This method is appliedto a 3D mesh around a rudder. Amplitudes arerespectively 0 and 15 degrees and the method usedconserve the mesh quality required for the flow

solver.

Figure 43: Example of 3D unsteady deformation of themesh for rotating rudder

SIMULATION OF AN SELF-PROPELLED SHIP

WITH AN ACTUATOR DISC

Using Navier-Stokes computations to predict the realmaneuvering ship behavior is fully achieved only ifthe propeller is taken into account in the simulations.We have to consider here the interactions of the

propeller with the hull and appendages. The hull dragusually increases because of the pressure fieldmodification and also appendages drag since they arelocated in the propeller flow.

Several approaches can then be considered. The first

one consists in directly modeling the propeller in thecomputation with its own mesh. The propeller

rotation is taken into account with the rotation of themesh in a cylinder. These computations are one of the

aim of the numerical naval hydrodynamics but needsa huge mesh density and so very long CPU time. The

second approach, which has been developed in thepresent work, consists in simulating the propellereffect on the flow with an external forces in meanmomentum equations. This solution called actuator

disc method allows to get, in a way more or lesscomplex, suction and wake effects induced by the

propeller and with a reduced CPU time.

Actuator disc method

The method presented here is based on an explicitdistribution of forces, i.e. the force distribution isimposed without coupling the up-flow [15]. For givenvalue of RPM, and then KT and KQ coming fromopen water curves, we obtain a force distribution inthe propeller disk with axial and tangential

repartitions respectively defined by:

** 1 rrAf rr = and( )

bb rrr

rrAf

+

=

*

**

1

1

The non-dimensional radius of the propeller r* and of

the boss rb are respectively defined by:

r

rrr

b

b

=

1

* and

p

bb

R

Rr =

At that step the force distribution is only defined as afunction of the propeller radius. Amplitudes of radialArand tangential A forces are then computed writingthat the integrations of forces and of moments on thepropeller disc are respectively equal to thrust andpropeller torque.

Calculation of stiffness for

compression and torsion

Constitution of the matrix KX=B

Computation of the linear system

Rotation of appendage mesh points

Projection of points on hull

Projection of points on hull


17/19

2

2 20

0

p

m

R x dx

r

R x

T L U f dxrd dr

+

=

and:2

3 20

0

p

m

R x dx

R x

Q L U rf dxrd dr

+

=

The force distribution is computed at each node of thepropeller disc mesh. The volume used is then the

volume of a fictive cell around the considered nodeand bounded by the center of neighboring cells.

The figure bellow present an example of forces

distribution in the propeller disc.

Figure 44: Forces distribution in the propeller disc

The effect of this actuator disc in the flow around a ship

combatant is presented in the next figure.

Figure 45: Self-propelled ship, streamlinesand iso-velocity

UNSTEADY MANEUVERING APPLICATIONS

The first unsteady applications are presented in this

section and concern the turning of a series 60.

Experiments were carried out in the oceanic tank atKrilov Institute in 1994 [16] for Bassin d'essais descarnes. Future computations carried out in the sameconditions will allow to evaluate the accuracy ofnumerical results obtained.

At first, unsteady simulations have been performedby prescribing a force equivalent to the force on therudder when turned. That force is assumed constant.

Then, in a second time, computations were performedwith a real rudder at a specific angle directly meshed.

The six degrees of freedom of the ship are free, whichmeans that the ship finds its own balance undereffects of inertia, hydrodynamics and external forces.

The boat speed is introduced by a constant forceapplied at propeller location so that ship speed is thesame as the one obtained in experiments. Future

simulations will integrate ship propulsion using anactuator disk in order to get more realistic propellereffects and speed loss during turning.

Figures below show the rudder turning influence onflow. We can see in particular at rudder end thevortex created between intrados and extrados.

Figure 46: Flow around rudder at 0 and 10


18/19

Figures below show free surface elevation andstreamlines around hull during turning.

Figure 47: Free surface and streamlines around series 60

Under effect of rudder incidence and moment createdaround vertical axis the ship has then a circular

trajectory that is shown on the next figure whereseveral ship positions have been drawn. These firstcomputations show a similar behavior withmaneuvering ship, with the prediction of snap-rollphase when rudder is turned, and speed decreasewhen ship is turning.

Figure 48:Unsteady RANS computations of Series 60gyration circle obtained with force at rudder location

CONCLUSIONS

Initially developed to predict drag resistance, Navier-

Stokes with free-surface solvers extend nowadays

their applications to seakeeping and maneuverability.This paper shows an example applications in the fieldof maneuverability for steady cases, forced motionsand attempt of free maneuvering ship. The ultimateaim of those developments is to directly compute the

whole unsteady movement including interactionsbetween propellers, hull and appendages. Morespecifically, three points have been studied. The firstone was the ability to predict accurately the shipdynamics with six degrees of freedom only subject toforces and moments computed by the solver. The

second point was to simulate the rudder turningduring ship maneuvering. And the last point was the

ability to simulate self propulsion of ship, taking intoaccount propeller effects in the flow.

First unsteady calculations have been made on a

Series 60 ship with a five degrees turned rudder. Shipis then driven by a constant force imposing the initialspeed before turning. Next we will focus oncomparison between numerical results and

experiments, taking into account the propeller.

Even if the first results presented in this paper havebeen obtained for a simplified case, they show the

abilities of such approachs, able to take into accountcomplex interactions between hull, appendages and

propeller when hull is in incidence or in turning.

ACKNOWLEDGMENTS

The work presented in this paper was sponsored byDGA/SPN under PEA 1999 and 2004 projects.

Applications of forced motions and free running shiphave been supported by the European ProjectVIRTUE grant 516201.


19/19

REFERENCES

[1] Y. Tahara, E. Paterson, F. Stern, Y.

Himeno,, Flow- and Wave-Field Optimisation of

Surface Combatants Using CFD-Based OptimizationMethods , 23th

Symposium on NavalHydrodynamics, Val de Reuil, France, Septembre2000.

[2] E. Jacquin, Q. Derbanne, D. Bellevre, S.Cordier, B. Alessandrini, Y. Roux, Hull FormOptimization Using A Free Surface RANS Solver,25th Symposium on Naval Hydrodynamics, St Johns,

Newfoundland and Labrador, Canada, Aot 2004.

[3] E. F. Campana, D. Peri, Y. Tahara, F. Stern,Comparison and Validation of CFD Based Local

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25

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[4] B. Alessandrini, G. Delhommeau, Viscous freesurface flow past a ship in drift and in rotating

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[5] A. Di Mascio, R. Broglia, R. Muscari,

Unsteady RANS Simulation of a Manoeuvring ShipHull, 25th Symposium on Naval Hydrodynamics, StJohns, Newfoundland and Labrador, Canada, Aot2004.

[6] R. Wilson, F. Stern, Unsteady RANS

Simulation of a Surface Combatant with RollMotion, 24th Symposium on Naval Hydrodynamics,Fukuoka, Japon, Juillet 2002.

[7] R. Wilson, P. Carrica, F. Stern, UnsteadyRANS method for ship motions with application toroll for a surfacecombatant, Computers and Fluids2006, Vol. 35, pp. 501-524.

[9] B. Alessandrini, G. Delhommeau, Simulationof three-dimensional unsteady viscous free surfaceflow around a ship model,International Journal for

Numerical Methods in Fluids, vol 19, pp 321-342,1994

[10] B. Alessandrini, G. Delhommeau, A fullycoupled Navier-Stokes solver for calculations ofturbulent incompressible free surface flow past a ship

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[11] D. C. Wilcox, Multiscale model for turbulentflows, AIAA Journal, Vol 26, pp. 1211-1320,November 1988.

[12] J.T. Batina, Unsteady Euler airfoil solutionsusing unstructured dynamic meshes, AIAA Paper n89-0150, AIAA 27th Aerospace Sciences Meeting,Reno, NV, USA (1989)

[13] C. Farhat, C. Degand, B. Koobu, M.Lesoinne, Torsional springs for two-dimensionaldynamic unstructured fluid meshes , ComputationalMethods Appl. Mech. Engrg, 1998 (231-245)

[14] C. Degand, C. Farhat, A three-dimensionaltorsional spring analogy method for unstructureddynamic meshes , Computers and Structures 2002(305-316)

[15] F. Stern, H.T. Kim, V.C Patel, H.V. Chen, Aviscous flox approach to the computation of propellerhull interaction, Journal of Ship research, Vol 32,

n4, 1988, pp. 246-262

[16] Free running tests in calm water and in waveson Ship Model of Series 60, Krylov ShipbuildingResearch Institue, St Petersburg 1994.

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Simulation of unsteady ship maneuvering using free-surface RANS solver

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