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AN ADVANCED FINITE ELEMENT METHOD FOR FLUID DYNAMIC ANALYSIS OF AMERICAS CUP and IMS BOATS J. García-Espinosa, R. Luco-Salman, M. Salas2, M. López-Rodríguez, E. Oñate, High Performance Yacht Design Conference 2002

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Contents RANSE solver. Motivation. RANSE solver. Motivation. –Finite Increment Calculus (FIC) formulation –Implicit Fractional Step scheme –Monolithic Predictor-Corrector scheme Free surface solver Free surface solver Sink and Trim treatment Sink and Trim treatment Numerical and Implementation Aspects Numerical and Implementation Aspects Examples Examples –IMS AMC-CRC –Rioja de España Conclusions Conclusions

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RANSE solver. Motivation. Implicit scheme: In most of the cases of interest for the naval architecture the time step imposed by the stability criteria (smallest elements) may be orders of magnitude smaller than the time step required to obtain time-accurate results (physical time step), rendering explicit schemes impractical. Implicit scheme: In most of the cases of interest for the naval architecture the time step imposed by the stability criteria (smallest elements) may be orders of magnitude smaller than the time step required to obtain time-accurate results (physical time step), rendering explicit schemes impractical. Pressure-Velocity segregation: Monolithic schemes treat advection term in an implicit manner, which avoids the mentioned disadvantages. Nevertheless, these methods are very expensive from a computational point of view (velocity and pressure discrete equations are coupled). Pressure-Velocity segregation: Monolithic schemes treat advection term in an implicit manner, which avoids the mentioned disadvantages. Nevertheless, these methods are very expensive from a computational point of view (velocity and pressure discrete equations are coupled). Finite Increment Calculus: Stability of the numerical algorithm is still today an open issue. There is a need for an accurate, pressure and advection stable algorithm, based on the physics of the problem. Finite Increment Calculus: Stability of the numerical algorithm is still today an open issue. There is a need for an accurate, pressure and advection stable algorithm, based on the physics of the problem.

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Finite Increment Calculus (FIC) Foundations AB C d1d1 d2d2 d x qAqA qBqB 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 q A and q B are the incoming fluxes at points A and B. The flux q includes both convective and diffusive terms. Let us express now the fluxes q A and q B in terms of the flux at an arbitrary point C within the balance domain. Expanding q A and q B in Taylor series around point C up to second order terms gives Substituting above eqs. Into balance equation gives with

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Finite Increment Calculus (FIC) Formulation 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

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Implicit Fractional Step Formulation 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 This allow us to uncouple pressure and velocity calculations without loss of accuracy ( splitting errors of order 0(t 2 ) )

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Implicit Fractional Step formulation 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)

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

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

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Sink and Trim treatment 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, F z and M y are a net heave force and a trim moment. A wp is the water plane area, and I y 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.

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Final geometry updating and mesh regeneration 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 generated 4.New calculation is carried out with fixed mesh

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Numerical and Implementation Aspects RANSE and FS equations are integrated by standard Finite Element Method (linear/quadratic tetrahedra, hexahedra, prisms, …) RANSE and FS equations are integrated by standard Finite Element Method (linear/quadratic tetrahedra, hexahedra, prisms, …) RANSE and FS solvers have been optimized for working with unstructured meshed of linear tetrahedra/triangles RANSE and FS solvers have been optimized for working with unstructured meshed of linear tetrahedra/triangles Implicit fractional step algorithm is used to go faster to steady state Implicit fractional step algorithm is used to go faster to steady state Boundary conditions may be defined by analytical functions allowing to run different drifts angles with the same geometry/mesh Boundary conditions may be defined by analytical functions allowing to run different drifts angles with the same geometry/mesh

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Numerical and Implementation Aspects RANSE-FS solver has been implemented within the CFD system Tdyn RANSE-FS solver has been implemented within the CFD system Tdyn Tdyn includes a fully integrated pre/postprocessor based on GiD system, incorporating advanced CAD tools (NURBS importation, creation and edition) Tdyn includes a fully integrated pre/postprocessor based on GiD system, incorporating advanced CAD tools (NURBS importation, creation and edition) Data insertion (control volume generation, physical properties, boundary conditions, etc) is guided by the use of wizard tools Data insertion (control volume generation, physical properties, boundary conditions, etc) is guided by the use of wizard tools Mesh can be automatically generated from the CAD information within the system. It also allows elements size assignment and quality check of the resulting mesh Mesh can be automatically generated from the CAD information within the system. It also allows elements size assignment and quality check of the resulting mesh System also includes a set of postprocessing options and tools for report generation System also includes a set of postprocessing options and tools for report generation

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Application: IMS AMC-CRC The first analyzed example is an IMS 37 boat The first analyzed example is an IMS 37 boat Simulations have been carried out at full scale, using a two layer k turbulence model, in combination with an extended law of the wall Simulations have been carried out at full scale, using a two layer k turbulence model, in combination with an extended law of the wall Results are compared to experimental data extrapolations performed at Ship Hydrodynamics Centre - Australian Maritime College (AMC) of the Maritime Engineering Cooperative Research Centre (CRC) Results are compared to experimental data extrapolations performed at Ship Hydrodynamics Centre - Australian Maritime College (AMC) of the Maritime Engineering Cooperative Research Centre (CRC)

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Application: IMS AMC-CRC Acknowledgements: This work has partially funded by Universidad Austral de Chile.

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Application: IMS AMC-CRC Test Heel angle Drift angle Rudder angle E10D310º3º3º E20D520º5º5º Case E10D3 was towed at equivalent velocities of 5, 6, 7 and 8 knots Case E20D5 was towed at equivalent velocities of 6, 7, 7.5 and 8 knots Simulations 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 tetrahedral elements in every case.

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Application: IMS AMC-CRC Pressure and velocity contours Streamlines around hull and appendages

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Application: IMS AMC-CRC

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Application: Rioja de España The second analyzed example is the Spanish Americas Cup boat Rioja de España, participant in the edition of 1995 The second analyzed example is the Spanish Americas Cup boat Rioja de España, participant in the edition of 1995 Simulations have been carried out at full scale, using a two layer k-e turbulence model, in combination with an extended law of the wall Simulations have been carried out at full scale, using a two layer k-e turbulence model, in combination with an extended law of the wall Results are compared to experimental data extrapolations performed at El Pardo towing tank with a model at scale 1/3.5 Results are compared to experimental data extrapolations performed at El Pardo towing tank with a model at scale 1/3.5

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Application: 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.

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Application: Rioja de España TestGeometry Heel angle Drift angle C0D0 Hull, no appendages 0º0º E0D0 Hull, bulb and keel 0º0º E15D2 Hull, bulb, keel and rudder 15º2º E15D4 15º4º E25D2 25º2º Every case was towed at equivalent velocities of 10, 9, 8.5, 8.0, 7.5 and 7.0 knots

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Application: Rioja de España TestSymmetry # Elements # Nodes C0D0Yes E0D0Yes E15D2No E15D4No E25D2No All 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.

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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 España E15D2 keel-bulb union detail Final mesh E0D0 E25D2 Final Mesh E15D2 Initial mesh

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

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Application: Rioja de España E25D2 Pressure, velocity and eddy viscosity contours

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Application: Rioja de España E25D2 Streamlines on hull and appendages

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Application: Rioja de España E25D2 Wave profile 9Kn Experimental Simulation

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Application: Rioja de España E25D2 Wave profiles and pressure contours 9Kn

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Application: Rioja de España E0D0 Pressure contours 10Kn

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Application: Rioja de España E25D2 Pressure contours 9Kn

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Application: Rioja de España E0D0E15D2 E15D4E25D2

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Conclusions FIC technique applied to RANSE-FS equations allows to derive a number of stabilized schemes allowing equal-order velocity-pressure interpolation and adequate treatment of advection terms. FIC technique applied to RANSE-FS equations allows to derive a number of stabilized schemes allowing equal-order velocity-pressure interpolation and adequate treatment of advection terms. An implicit second-order accurate monolithic scheme, based on the FIC formulation has been derived to solve incompressible free surface flow problems. The final system of equations resulting from the time and space discretization is solved in each time step in an uncoupled manner. An implicit second-order accurate monolithic scheme, based on the FIC formulation has been derived to solve incompressible free surface flow problems. The final system of equations resulting from the time and space discretization is solved in each time step in an uncoupled manner. The numerical experience indicates that the formulation is very efficient 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-accurate results (physical time step), and that its time accuracy is excellent (i.e. 4 CPU h / 1.5Mtetras standard PC single processor, including s&t final calculation). The numerical experience indicates that the formulation is very efficient 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-accurate results (physical time step), and that its time accuracy is excellent (i.e. 4 CPU h / 1.5Mtetras standard PC single processor, including s&t final calculation). 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 geometry and mesh regeneration. 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 geometry and mesh regeneration. The solver has been optimized for using unstructured meshes of linear tetrahedra, allowing a simple and automatic mesh generation from complex NURBS based geometry, and the best mesh refinement control. The solver has been optimized for using unstructured meshes of linear tetrahedra, allowing a simple and automatic mesh generation from complex NURBS based geometry, and the best mesh refinement control.

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Conclusions RANSE-FS solver has been integrated within the pre/postprocessing environment (GiD) RANSE-FS solver has been integrated within the pre/postprocessing environment (GiD) Graphical environment has been adapted to naval architecture needs by developing wizard tools Graphical environment has been adapted to naval architecture needs by developing wizard tools Numerical results obtained in the analysis of Americas Cup Rioja de España and IMS AMC-CRC boats indicate that the proposed method can be used with confidence for practical design purposes. Numerical results obtained in the analysis of Americas Cup Rioja de España and IMS AMC-CRC boats indicate that the proposed method can be used with confidence for practical design purposes. –Evaluation of total resistance gives less that 5% difference with towing tank extrapolations in most part of the analysis range. –Evaluation of induced (lift) forces in non-symmetric cases gives even less differences. –Obtained wave profiles are also very close to those measured in towing tank. –Dynamic sinkage and trim angles show similar behavior in both experimental and numerical results. However, scale effects dont allow further conclusions. –Qualitative results including: wave maps, streamlines, pressure and velocity contours and turbulence distribution, show also reasonable patterns

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