AN ADVANCED FINITE ELEMENT METHOD FOR FLUID DYNAMIC ANALYSIS OF AMERICAS CUP and IMS BOATS J....

AN ADVANCED FINITE AN ADVANCED FINITE ELEMENT METHOD FOR ELEMENT METHOD FOR

FLUID DYNAMIC ANALYSIS OF FLUID DYNAMIC ANALYSIS OF AMERICA’S CUP and IMS AMERICA’S CUP and IMS

BOATSBOATS

J. García-Espinosa, [email protected]. García-Espinosa, [email protected]. Luco-Salman, M. Salas2, R. Luco-Salman, M. Salas2,

[email protected]@uach.clM. López-Rodríguez, M. López-Rodríguez,

[email protected]@nautatec.comE. Oñate, [email protected]. Oñate, [email protected]

High Performance Yacht Design Conference 2002

ContentsContents

• RANSE solver. Motivation.RANSE solver. Motivation.– Finite Increment Calculus (FIC) formulationFinite Increment Calculus (FIC) formulation– Implicit Fractional Step schemeImplicit Fractional Step scheme– Monolithic Predictor-Corrector schemeMonolithic Predictor-Corrector scheme

• Free surface solverFree surface solver• Sink and Trim treatmentSink and Trim treatment• Numerical and Implementation AspectsNumerical and Implementation Aspects• ExamplesExamples

– IMS AMC-CRCIMS AMC-CRC– Rioja de EspañaRioja de España

• ConclusionsConclusions

RANSE solver. Motivation.RANSE solver. Motivation.

• Implicit scheme: In most of the cases of interest for the Implicit scheme: In most of the cases of interest for the naval architecture the time step imposed by the stability naval architecture the time step imposed by the stability criteria (smallest elements) may be orders of magnitude criteria (smallest elements) may be orders of magnitude smaller than the time step required to obtain time-smaller than the time step required to obtain time-accurate results (physical time step), rendering explicit accurate results (physical time step), rendering explicit schemes impractical.schemes impractical.

• Pressure-Velocity segregation: Monolithic schemes treat Pressure-Velocity segregation: Monolithic schemes treat advection term in an implicit manner, which avoids the advection term in an implicit manner, which avoids the mentioned disadvantages. Nevertheless, these methods mentioned disadvantages. Nevertheless, these methods are very expensive from a computational point of view are very expensive from a computational point of view (velocity and pressure discrete equations are coupled).(velocity and pressure discrete equations are coupled).

• Finite Increment Calculus: Stability of the numerical Finite Increment Calculus: Stability of the numerical algorithm is still today an open issue. There is a need for algorithm is still today an open issue. There is a need for an accurate, pressure and advection stable algorithm, an accurate, pressure and advection stable algorithm, based on the physics of the problem.based on the physics of the problem.

Finite Increment Calculus (FIC) Finite Increment Calculus (FIC) FoundationsFoundations

0A Bq q A B

C

d1 d2

d

x

qA qB

We consider a convection-diffusion problem in a 1D domain of length L. The equation of balance of fluxes in a sub-domain of size d is

where qA and qB are the incoming fluxes at points A and B. The flux q includes both convective and diffusive terms.

Let us express now the fluxes qA and qB in terms of the flux at an arbitrary point C within the balance domain. Expanding qA and qB in Taylor series around point C up to second order terms gives

2 2 2 2

3 31 21 1 2 22 2

0 02 2A C B C

C CC C

dq d d q dq d d qq q d d q q d d

dx dx dx dx

Substituting above eqs. Into balance equation gives

2

20

2

dq h d q

dx dx 1 2h d d with

Finite Increment Calculus (FIC) Finite Increment Calculus (FIC) FormulationFormulation

10 , 1,2,3

2

10 1,2,3

2

10 1,2

2

i

i

j

mm j

j

dd j

j

j

rr h on i j

x

rr h on j

x

rr h on j

x

1,2,3

1,2,3

i

ijim i j

j i j

id

i

ii

u pr u u

t x x x

ur i

x

r u it x

We consider the motion around a body of a viscous incompressible fluid (RANSE-NS) including a free surface (FS). The stabilized finite calculus (FIC) form of the governing differential equations for the three dimensional (3D) problem can be written as:

These eqs. are the starting point for deriving a variety of stabilized numerical methods for solving the incompressible NS-RANSE equations. It can be shown, that a number of stabilized methods allowing equal order interpolations for velocity and pressure fields and stable and accurate advection terms integration can be derived from this formulation

Implicit Fractional Step FormulationImplicit Fractional Step Formulation

1 1

02

i

nnn n nmijn ni i

i j jj i j j

ru u pu u h

t x x x x

1 1

02

i

nnn n nmijn ni i

i j jj i j j

ru u pu u h

t x x x x

Discretization in time of the stabilized momentum equation using the trapezoidal rule (or θ method) as

An implicit fractional step method can be simply derived by splitting above equation as follows

1 1n n ni i

i

u u t p px

This allow us to uncouple pressure and velocity calculations without loss of accuracy ( splitting errors of order 0(t2) )

2

1 1

2n n i d

ji i i j

u rt p p hx x x x

Implicit Fractional Step formulationImplicit Fractional Step formulation

1 1

02

i

nnn n nmijn ni i

i j jj i j j

ru u pu u h

t x x x x

1 1n n ni i

i

u u t p px

Characteristics of the scheme:

• Stable (convective terms - equal order velocity-pressure interpolations)

• Implicit (higher time step)• Second order accuracy (theta scheme)• All the problems to be solved are scalar (less CPU and memory

requirements)

2

1 1

2n n i d

ji i i j

u rt p p hx x x x

Monolithic Predictor-Corrector schemeMonolithic Predictor-Corrector scheme

,, 11, 1 ,

, 1 , 1 10

2i

n mn mn m n n mmijn m n mi i

i j jj i j j

ru u pu u h

t x x x x

Characteristics of the scheme:

• Stable (convective terms - equal order velocity-pressure interpolations)

• Implicit and monolithic (higher time step)• Second order accuracy (theta scheme)• All the problems to be solved are scalar (less CPU and memory

requirements)

, 1 ,2

, 1 , 1

2i

n m n mn m n m d

ji i i j

u rt p p hx x x x

All the arguments to derive Fractional Step scheme are still valid if we replace pressure term pn by any other pressure evaluation. In particular we may write (m iteration counter)

Free surface solverFree surface solver

10 1,2

2

1,2,3

j

j

ii

rr h on j

x

r u it x

FIC method can be directly applied to the free surface equation.

33

1j ij i i j jn n p n p n

R n R

Transpiration technique is used to couple free surface condition with RANSE solver. This technique is based on imposing pressure at free surface obtained by stress continuity as (neglecting tangential components):

Being the surface tension coefficient and R the average curvature radius. Above condition is applied on a reference surface. FS eq. is also solved on this reference surface not necessary being the exact free surface. In order to improve accuracy of the solver a mesh updating procedure is applied.

Sink and Trim treatmentSink and Trim treatment

, yz

wp y

MFz gA gI

Dynamic sinkage and trim angle are calculated by

where z is a correction of the sinkage at the center of gravity, is a trim angle correction, Fz and My are a net heave force and a trim moment. Awp is the water plane area, and Iy is the corresponding moment of inertia about the y axis. In order to take these changes into account a mesh updating process is carried out automatically several times during the calculation process. This process is based on a strategy of nodal displacement diffusion through the mesh. Nodal displacements are due to free surface deformation and sink and trim effects. However only sink and trim effects are taken into account close to the ship body.

Final geometry updating and mesh Final geometry updating and mesh regenerationregeneration

Finally a new calculation is performed with the real geometry. This is done in four steps:1.New free surface NURBS definition, taking the resulting deformation into account, is generated:

• NURBS Cartesian support grid of MxN points is created.

• Z coordinate of the points, representing the wave elevation, is interpolated into the grid.

• Finally, the NURBS surface based on the support grid is generated.

2.Geometry of the vessel if moved according to calculated sinkage and trim angle. 3.New control volume and mesh are automatically generated4.New calculation is carried out with fixed mesh

Numerical and Implementation Numerical and Implementation AspectsAspects

• RANSE and FS equations are integrated by RANSE and FS equations are integrated by standard Finite Element Method standard Finite Element Method (linear/quadratic tetrahedra, hexahedra, prisms, (linear/quadratic tetrahedra, hexahedra, prisms, …)…)

• RANSE and FS solvers have been optimized for RANSE and FS solvers have been optimized for working with unstructured meshed of linear working with unstructured meshed of linear tetrahedra/trianglestetrahedra/triangles

• Implicit fractional step algorithm is used to go Implicit fractional step algorithm is used to go faster to steady statefaster to steady state

• Boundary conditions may be defined by Boundary conditions may be defined by analytical functions allowing to run different analytical functions allowing to run different drifts angles with the same geometry/meshdrifts angles with the same geometry/mesh

Numerical and Implementation Numerical and Implementation AspectsAspects• RANSE-FS solver has been RANSE-FS solver has been

implemented within the CFD implemented within the CFD system Tdynsystem Tdyn

• Tdyn includes a fully integrated Tdyn includes a fully integrated pre/postprocessor based on GiD pre/postprocessor based on GiD system, incorporating advanced system, incorporating advanced CAD tools (NURBS importation, CAD tools (NURBS importation, creation and edition)creation and edition)

• Data insertion (control volume Data insertion (control volume generation, physical properties, generation, physical properties, boundary conditions, etc) is guided boundary conditions, etc) is guided by the use of wizard toolsby the use of wizard tools

• Mesh can be automatically Mesh can be automatically generated from the CAD generated from the CAD information within the system. It information within the system. It also allows elements size also allows elements size assignment and quality check of assignment and quality check of the resulting meshthe resulting mesh

• System also includes a set of System also includes a set of postprocessing options and tools postprocessing options and tools for report generationfor report generation

Application: IMS AMC-CRCApplication: IMS AMC-CRC

• The first analyzed example is The first analyzed example is an IMS 37’ boatan IMS 37’ boat

• Simulations have been Simulations have been carried out at full scale, using carried out at full scale, using a a two layer k turbulence two layer k turbulence model, in combination with model, in combination with an extended law of the wallan extended law of the wall

• Results are compared to Results are compared to experimental data experimental data extrapolations performed at extrapolations performed at Ship Hydrodynamics Centre -Ship Hydrodynamics Centre -Australian Maritime College Australian Maritime College (AMC) of the Maritime (AMC) of the Maritime Engineering Cooperative Engineering Cooperative Research Centre (CRC)Research Centre (CRC)

ApplicationApplication: : IMS AMC-CRCIMS AMC-CRC

Acknowledgements: This work has partially funded by Universidad Austral de Chile.


TestTest Heel Heel angleangle

Drift Drift angleangle

Rudder Rudder angleangle

E10D3E10D3 10º10º 3º3º 3º3º

E20D5E20D5 20º20º 5º5º 5º5º

Case E10D3 was towed at equivalent velocities of 5, 6, 7 and 8 knotsCase E20D5 was towed at equivalent velocities of 6, 7, 7.5 and 8 knotsSimulations were performed with fixed model. The position of the model in these calculations were obtained by static equilibrium of sails and hydrostatic forces.Meshes have been generated with the same quality criteria (in terms of size transition and minimum angle), obtaining about 350 000 tetrahedral elements in every case.

ApplicationApplication: : IMS AMC-CRCIMS AMC-CRCPressure and velocity contoursStreamlines around hull and appendages

Application: Rioja de EspañaApplication: Rioja de España• The second analyzed example is The second analyzed example is

the Spanish America’s Cup boat the Spanish America’s Cup boat Rioja de España, participant in Rioja de España, participant in the edition of 1995the edition of 1995

• Simulations have been carried Simulations have been carried out at full scale, using a out at full scale, using a two two layer k-e turbulence model, in layer k-e turbulence model, in combination with an extended combination with an extended law of the walllaw of the wall

• Results are compared to Results are compared to experimental data experimental data extrapolations performed at El extrapolations performed at El Pardo towing tank with aPardo towing tank with a model model at scale 1/3.5at scale 1/3.5

ApplicationApplication: Rioja de España: Rioja de España

Acknowledgements: Authors thank National Institute of Aerospace Technique (INTA) for permitting the publication of the towing tank tests of Rioja de España, and the model basin El Pardo (CEHIPAR) for sending the full documentation of the tests. Special gratefulness to IZAR shipbuilders for allowing the publication of the Rioja de España hull forms.


TestTest GeometryGeometry Heel Heel angleangle

Drift Drift angleangle

C0D0C0D0 Hull, no Hull, no appendagesappendages 0º0º 0º0º

E0D0E0D0 Hull, bulb and keelHull, bulb and keel 0º0º 0º0º

E15D2E15D2 Hull, bulb, keel and Hull, bulb, keel and rudderrudder 15º15º 2º2º



Every case was towed at equivalent velocities of 10, 9, 8.5, 8.0, 7.5 and 7.0 knots


TestTest SymmetrySymmetry # # ElementsElements

# Nodes# Nodes

C0D0C0D0 YesYes 300 000300 000 75 00075 000

E0D0E0D0 YesYes 700 000700 000 175 000175 000

E15D2E15D2 NoNo 1 500 0001 500 000 380 000380 000

E15D4E15D4 NoNo 1 500 0001 500 000 380 000380 000

E25D2E25D2 NoNo 1 500 0001 500 000 380 000380 000All grids used have been generated with the same quality criteria (in terms of size transition and minimum angle) and using element sizes from 5mm to 2000 mm.

All meshes have been generated with the same quality criteria (in terms of size transition and minimum angle) and using element sizes from 5mm to 2000 mm.

Application: Rioja de EspañaApplication: Rioja de España

E15D2 keel-bulb union detail

Final meshE0D0

E25D2 Final Mesh

E15D2 Initial mesh

Application: Rioja de EspañaApplication: Rioja de EspañaE25D2 Streamlines and E25D2 Streamlines and velocity modulus contours velocity modulus contours (V = 9 Kn)(V = 9 Kn)

Application: Rioja de EspañaApplication: Rioja de España

E25D2 Pressure, velocity and eddy viscosity contours

Application: Rioja de EspañaApplication: Rioja de EspañaE25D2 Streamlines on hull and appendages

Application: Rioja de EspañaApplication: Rioja de EspañaE25D2 Wave profile 9Kn

Experimental Simulation

ApplicationApplication: Rioja de España: Rioja de EspañaE25D2 Wave profiles and pressure contours 9Kn

ApplicationApplication: Rioja de España: Rioja de EspañaE0D0 Pressure contours 10Kn

ApplicationApplication: Rioja de España: Rioja de EspañaE25D2 Pressure contours 9Kn


E0D0 E15D2

E15D4

E25D2

ConclusionsConclusions• FIC technique applied to RANSE-FS equations allows to derive a number FIC technique applied to RANSE-FS equations allows to derive a number

of stabilized schemes allowing equal-order velocity-pressure interpolation of stabilized schemes allowing equal-order velocity-pressure interpolation and adequate treatment of advection terms.and adequate treatment of advection terms.

• An implicit second-order accurate monolithic scheme, based on the FIC An implicit second-order accurate monolithic scheme, based on the FIC formulation has been derived to solve incompressible free surface flow formulation has been derived to solve incompressible free surface flow problems. The final system of equations resulting from the time and problems. The final system of equations resulting from the time and space discretization is solved in each time step in an uncoupled manner.space discretization is solved in each time step in an uncoupled manner.

• The numerical experience indicates that the formulation is very efficient The numerical experience indicates that the formulation is very efficient for free surfaces flows, when the critical time step of the problem is some for free surfaces flows, when the critical time step of the problem is some orders of magnitude smaller than the time step required to obtain time-orders of magnitude smaller than the time step required to obtain time-accurate results (physical time step), and that its time accuracy is accurate results (physical time step), and that its time accuracy is excellent (i.e. 4 CPU h / 1.5Mtetras standard PC single processor, excellent (i.e. 4 CPU h / 1.5Mtetras standard PC single processor, including s&t final calculation).including s&t final calculation).

• Sink and trim effects as well as free surface deformations are taking into Sink and trim effects as well as free surface deformations are taking into account in the solution process by automatic mesh updating and a final account in the solution process by automatic mesh updating and a final geometry and mesh regeneration.geometry and mesh regeneration.

• The solver has been optimized for using unstructured meshes of linear The solver has been optimized for using unstructured meshes of linear tetrahedra, allowing a simple and automatic mesh generation from tetrahedra, allowing a simple and automatic mesh generation from complex NURBS based geometry, and the best mesh refinement control.complex NURBS based geometry, and the best mesh refinement control.

ConclusionsConclusions• RANSE-FS solver has been integrated within the RANSE-FS solver has been integrated within the

pre/postprocessing environment (GiD)pre/postprocessing environment (GiD)• Graphical environment has been adapted to naval Graphical environment has been adapted to naval

architecture needs by developing wizard tools architecture needs by developing wizard tools • Numerical results obtained in the analysis of America’s Cup Numerical results obtained in the analysis of America’s Cup

Rioja de EspañaRioja de España and IMS AMC-CRC boats indicate that the and IMS AMC-CRC boats indicate that the proposed method can be used with confidence for practical proposed method can be used with confidence for practical design purposes.design purposes.– Evaluation of total resistance gives less that 5% difference with Evaluation of total resistance gives less that 5% difference with

towing tank extrapolations in most part of the analysis range.towing tank extrapolations in most part of the analysis range.– Evaluation of induced (lift) forces in non-symmetric cases gives Evaluation of induced (lift) forces in non-symmetric cases gives

even less differences.even less differences.– Obtained wave profiles are also very close to those measured Obtained wave profiles are also very close to those measured

in towing tank.in towing tank.– Dynamic sinkage and trim angles show similar behavior in both Dynamic sinkage and trim angles show similar behavior in both

experimental and numerical results. However, scale effects experimental and numerical results. However, scale effects don’t allow further conclusions. don’t allow further conclusions.

– Qualitative results including: wave maps, streamlines, pressure Qualitative results including: wave maps, streamlines, pressure and velocity contours and turbulence distribution, show also and velocity contours and turbulence distribution, show also reasonable patternsreasonable patterns

For further information ...For further information ...

http://www.compassis.com email:[email protected]://www.compassis.com/productos/en/tdyn/descargar

AN ADVANCED FINITE ELEMENT METHOD FOR FLUID DYNAMIC ANALYSIS OF AMERICAS CUP and IMS BOATS J....

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