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J. García-Espinosa, E. Oñate

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Motivation To develop a general environment for analysis of ship aero+hydro-dynamics. Main topics: Two fluids NS solver (stability + robustness + accuracy) Free surface (accuracy) Fluid-Rigid solid interaction Optimization for only hydrodynamics analysis Different types of problems (still water, seakeeping, manouvering, …)

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Introduction The main problem of the free surface capturing algorithms is the treatment of the pressure (gradient) discontinuity at the interface due to the variation of the properties there. In most of these algorithms the sharp transition of the properties at the interface is smoothed (extended to a band of several elements width) and therefore the accuracy in capturing the free surface is reduced. Ω 1 : Fluid 1 Ω 2 : Fluid 2

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Introduction A new free surface capturing approach able to overcome/improve most of the difficulties/ shortcomings of the existing algorithms is presented. This approach is based on: Stabilisation of governing equations (incompressible two- fluids Navier-Stokes equations) by means of the FIC method. Application of ALE techniques. Free surface movement is solved using a level set approach. Monolithic Fractional-Step type Navier-Stokes integration scheme. Application of domain decomposition techniques to improve the accuracy in the solution of governing equations in the interface between both fluids.

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Problem Statement: Two fluids Navier Stokes equations Two (incompressible) fluids (non homogeneous) Navier Stokes equations: With: And the necessary initial and boundary conditions

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Problem statement: Level set equation Let be Ψ a function (level set), defined as follows: Therefore we can re-write the density field as: And finally, obtain an equivalent equation for the mass conservation in terms of Ψ (level set):

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Problem Statement: Level set – Navier Stokes equations Two (incompressible) fluids level-set type Navier Stokes equations: With: And the necessary initial and boundary conditions

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Problem Statement: ALE* formulation We may easily re-write the previous equations in an Arbitrary Lagrangian-Eulerian frame: Where u m is the (mesh) deformation velocity of the moving domain: * The algorithm is Lagrangian for the movement of the ship but the free surface problem (level set) is solved in an Eulerian way

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Overlapping Domain Decomposition Tecnique Let K be a finite element partition of domain Ω, and consider a domain decomposition of Ω into three disjoint sub domains Ω 3 (t), Ω 3 (t) and Ω 5 (t): Ω 1 (Fluid 1) Ω 2 (Fluid 2)

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Overlapping Domain Decomposition Tecnique From this partition let us define two overlapping domains Ω* 1, Ω* 2 in such a way that Ω*1Ω*1Ω*1Ω*1 Ω*2Ω*2Ω*2Ω*2 * 1 * 1 * 2 * 2

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Overlapping Domain Decomposition Tecnique We can write an equivalent problem, using a standard Dirichlet- Neumann domain decomposition technique (using Ω* 1, Ω* 2 decomposition). The resulting variational problem is (inluding FIC stabilisation terms): Ω*1Ω*1Ω*1Ω*1 Ω*2Ω*2Ω*2Ω*2

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Problem statement: Boundary condition at the interface Dirichlet conditions (compatibility of velocities at the interface) are applied on Γ* 1 (boundary of And Neumann condition are applied on Γ* 2 (jump condition) t* is evaluated from the resulting velocity and pressure field on Ω* 1. Pressure evaluation must take into account the jump condition given by: Where γ is the coefficient of surface tension, and p 1, p 2 are the pressure values evaluated on the real free surface interface Γ. These are extrapolated to Γ * to impose the conditions on the defined interfaces.

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Monolithic Fractional-Step type scheme Integration of Navier-Stokes equations in every domain is done by means of a Monolithic Fractional-Step type scheme. This scheme is based on the iterative solution of the momentum equation, where the pressure is updated by using the solution of a velocity divergence free correction (m iteration counter): This scheme only requires to solve scalar problems, with the subsequent savings on CPU time and memory. The discretization of the problem is done using standard FEM technique

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Adaptation for solving Monophase Flow In many cases of interest in naval/marine applications, the aerodynamics effects can be neglected (density and viscosity ratio are about 1000). It is important for these cases to adapt the ODDLS technique to solve monophase problems, reducing the computational cost and capturing the free surface with the necessary accuracy and maintaining the advantages of the proposed method. In these cases, the computational domain is reduced to the nodes in the water plus those in the air being connected to the interface (Ω* 1 ). The later nodes are used to impose the pressure and velocity boundary conditions on the interface.

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Monophase flow algorithm 1. Solve momentum conservation equation on Ω* 1 using the following free surface boundary condition for the velocity: 2. Solve the mass continuity equation on Ω* 1, applying the following free surface boundary condition: 3. Solve level set equation on Ω (update free surface position) and reinitiate distance field 4. Evaluate forces on ship boundary and move it (update the mesh accordingly) 5. Go to 1 until convergence.

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NT-130 We will show different applications of the presented technique to the analysis of a semi-planning hull. The general characteristics of this boat are shown next. Main Characteristics LOA14.0 m Moulded Draft1.05 m Moulded Beam3.54 m Design Speed14 Kn (Fn = 0.65)

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Applications

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NT-130 Still Water Analyses The first example consist of the towing of the hull at different speeds (still water). Characteristics of the analyses: 2 sets of analysis were done: fixed ship and free to sink and trim. 4 different speeds were run. All the cases were run twice, using two different meshes of 2.0 and 3.2 million linear tetrahedra. Additional cases with different meshes ranging from 1.5 to 16 million linear tetrahedra were used to study the influence of the mesh density in the results. All the cases were run using an ILES-type turbulence model.

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NT-130 Still Water Analyses: Dynamic Sinkage and Trim effect Fn=0.65 Fn=0.55

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NT-130 Still water Drag / Towing force

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NT-130 Still Water Analyses Snapshot of the results (V = 12kn): Mesh of 16 million tetrahedra Left: (dynamic) pressure Down: velocity modulus

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NT-130 Results / Mesh dependency

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Applications

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NT-130 Head Waves Analyses The second example consist of the towing of the hull at different speeds with head waves. Characteristics of the analyses: The analyses were carried out with the ship free to sink and trim. 4 different speeds were run for every set of analyses with an unstructured 2.5 million linear tetrahedra mesh. The analysed wave length range from 1.0 to 1.5 x LOA (about critical values for slamming). All the cases were run using an ILES-type turbulence model.

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NT-130 Wave Generator The waves are generated by defining an oscillating velocity boundary condition at the inlet of the basin. The velocity is obtained from the equivalent movement of a wall given by x=d·sin(ωt), where ω is the wave generator frequency and d is the amplitude of the movement (stroke). The resulting boundary condition is as follows (k is the wave number and V o is the ship speed): For this problem linear wave theory states that the relation between wave number, channel depth and wave-maker frequency is ω 2 =g·k·tanh(kh). Where k is the wave number, h is the water depth and g the gravity acceleration The waves are dumped at the right hand side of the tank by disposing a (numerical) beach.

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NT-130 Head Waves Fn = 0.65 (right) Fn = 0.55 (down)

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NT-130 Head Waves analyses View of the mesh (longitudinal cut, free surface and ship) for a specific time step.

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NT-130 Head waves analysis Left: Pressure forces (OX) Down: Trim angle

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NT-130 Other Other type of practical analyses can be carried out with this model, including stability analysis (i.e. roll motion decay) and different manouvering analysis.

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Conclusions The present work describes a new methodology for the analysis of naval problems (ODDLS). ODDLS method is based on the domain decomposition technique combined with the Level Set technique and a FIC stabilized FEM. The ODDLS approximation increases the accuracy of the free surface capturing (level set equation) as well as the solution of the governing equations in the interface between two fluids. The greater accuracy in the solution of the interface between the fluids allows the use of non- structured meshes, as well as larger elements in the free surface. The method can be simplified by solving only one of the two fluids, which increases the efficiency in most of the naval/marine applications where the effect of one of the fluids can be neglected. The proposed ODDLS methodology has also been integrated with an ALE algorithm for the treatment of the dynamic movement of the ship. The presented application examples show that the method can be used for the analysis of a wide range of naval studies.

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