A computational model for the evaluation of the spray generation of a Wave Adaptive Modular Vessel

This paper presents part of the work done within the project ‘Advanced Numerical Simulation and Performance Evaluation of WAM-V ® in Spray Generating Conditions’ developed by the International Center for Numerical Methods in Engineering (CIMNE) under Navy Grant N62909-121-7101 issued by the Office of Naval Research Global. One of the primary goals of that project was the development of a computational model for simulation of the Wave Adaptive Modular Vessel (WAM-V®) under spray generating conditions. For this purpose, a Semi-Lagrangian Particle Finite Element Method (SL-PFEM) has been applied. This is the latest development within the framework of the so-called Particle Finite Element Method (PFEM), using the X-IVAS (eXplicit Integration along the Velocity and Acceleration Streamlines) scheme. In this paper we demonstrate the applicability of the SL-PFEM using the X-IVAS scheme for the simulation of the Wave Adaptive Modular Vehicle under spray generating conditions. INTRODUCTION A Wave Adaptive Modular Vessel (WAMV®) is a new class of ship that uses inflatable flexible hulls to conform to the surface of the water. It is similar in design to a catamaran, in that it has a twin hull design and no keel. However, the superstructure is not rigidly attached to the hulls; it uses shock absorbers and ball joints to articulate the vessel, which allows WAM-V to conform to the surface of the water while mitigating the stresses transmitted to the structure. Moreover, the inflatable hulls help to absorb the high frequency wave-loads. These features allow WAM-V to travel efficiently with low wave resistance in rough seas, by surfing on top of the waves rather than cut through them. The objective of the WAM-V is to be a lightweight watercraft capable of moving fast and efficiently on the surface of the sea. WAM-Vs are designed to allow for a variety of applications for either manned or unmanned operations and can be built in different lengths to match specific services. This paper presents part of the work done in the project ‘Advanced Numerical Simulation and Performance Evaluation of WAM-V ® in Spray Generating Conditions’ developed by the International Center for Numerical Methods in Engineering (CIMNE) under Navy Grant N62909-121-7101 issued by the Office of Naval Research Global. The scope of that project included the performance analysis of the WAM-V in waves, taking into account the flexibility of the ship hulls, using fluid-structure interaction computational models (see Figure 1). However, the focus of this paper is one of the primary concerns of that project; the development of a computational model for simulation of the WAM-V under spray generating conditions. In this regards, the final goal was to develop and demonstrate a computational engineering solver that could be used to design strategies to reduce the spray generation of the vessel. Figure 1. Snapshot of a fluid-structure interaction analysis of the WAM-V in irregular sea (colormap shows free surface elevation). When the rest of the components of drag are significantly reduced, the viscous components and any other source of energy dissipation induced by the movement of the vessel, become increasingly important. Because of the inflatable nature of the hulls of the WAM-V, there is little room for hydrodynamic shape optimization. Furthermore, as a consequence of the shape of the hulls, it is likely that they will generate spray when touching the sea surface. Therefore, spray might become an important source of energy dissipation in these little optimized hull shapes. In addition, excessive spray generation can increase the difficulties associated with operating the ship in certain cases (e.g. the possibility of the spray reaching the deck of the craft is a design issue depending on the particular operations the vessel is set to perform). Therefore it becomes obvious the need to characterize and reduce the spray generation, in order to increase the range of operation of this class of vessels. Furthermore, there is a need to understand the dynamics of the vessel and the hulls in different sea states and the generation of spray when sailing in a seaway. The Particle Finite Element Method (PFEM, Idelsohn et al., 2004) is a versatile framework for the analysis of fluid-structure interaction problems. The PFEM combines Lagrangian particle-based techniques with the advantage of the integral formulation of the Finite Element Method (FEM). It has been shown (Idelsohn et al., 2004; Becker, 2015) to successfully simulate a wide variety of complex engineering problems, e.g. freesurface/multi-fluid flows with violent interface motions, multi-fluid mixing and buoyancy-driven segregation problems etc. The latest development within the framework of the PFEM is the X-IVAS (eXplicit Integration along the Velocity and Acceleration Streamlines) scheme (Idelsohn et al., 2012). It is a semi-implicit scheme built over a Semi-Lagrangian (SL) formulation of the PFEM. In this paper we present the application of the SL-PFEM using the X-IVAS scheme for the simulation of the Wave Adaptive Modular Vehicle under spray generating conditions. SEMI-LAGRANGIAN PARTICLE FINITE ELEMENT METHOD Notation: Vectors are written using bold italic font and matrices are written using bold upright font. The independent variables in Lagrangian kinematics are { , }, where represents a label to identify particles and represents the time elapsed after labeling. The primary dependent variable is the fluid particle trajectory denoted as ( , ). The independent variables in Eulerian kinematics are ( , ), where denotes the spatial coordinates. The primary dependent variable is the fluid velocity ( , ). Consider the Eulerian description of the incompressible Navier-Stokes equations. + ( ∙ ) − ∆ + ( / ) =f (1) ∇ ∙ = 0 (2) where is the kinematic viscosity and ( , ), ( , ) are the pressure and the external acceleration fields, respectively. The effective acceleration field ( , ) in the fluid domain is obtained from the momentum balance equation of the flow. = + ( ∙ ) = ∆ − +f (3) Note that the functional dependence on the independent variables is suppressed in equations (1), (2) and (3) for brevity. The fundamental principle of kinematics relates the Eulerian description of the flow with the Lagrangian description as follows. ( , ): = ( , ) = ( ( , ), ) (4) ( , ) = ( , ) = ( ( , ), ) (5) The basic idea of the X-IVAS scheme is to update the fluid particle position and velocity within a time-step ≤ ≤ using ( , ) = ( , ), = ( , ) + (6) ( , ) = ( ( , ), ) = ( , ) + (7) where ( , ) and ( , ) denote spatially continuous piecewise linear approximations of the velocity and acceleration defined on a background simplicial mesh. The matrices , and the vectors , are spatially piecewise constant and depend on the time . The particle trajectory and its velocity computed in this manner are denoted as ( , ) and ( , ), respectively. Nielson and Jung (1999) presented formulas in 2D and 3D to compute the closed-form analytical solution of tangent curves for piecewise linear vector fields defined over simplicial meshes. Thus, the Nielson--Jung formulas can be used to compute the analytical solution of (6). Idelsohn et al. (2012) presented a procedure to compute the analytical solution of (6) and (7) in 2D. However the Nielson--Jung formulas and the calculation procedure described by Idelsohn et al. to compute the analytical solution are not numerically stable; loss of significance occurs due to subtractive cancellations near removable singularities. Recently, Nadukandi (2015) presented numerically stable formulas in 2D and 3D for the closed-form analytical solution of (6) and (7). In the following, we briefly describe the algorithm to implement the SL-PFEM using the XIVAS scheme. First the Lagrangian advection of the particles: ( , ) → ( , ) and ( , ) → ( , ) are done solving the following equations