Uncoupled numerical models
In typical FSI applications, the numerical domain i! contains two main subdomains: the fluid domain il f and the solid domain Qs. The fluid subdomain may contain multiple phases such as air Г2а and water if,,,-with Г2 f = Qw U fiQ-separated by a free surface F^, while the solid domain may contain different types of solids such as rigid bodies and flexible structures. A schematic representation of the different domains described above is presented in Figure 1. The boundary of the numerical domain is denoted as Oil, with partial boundaries denoted as Г such as the interface between solid and fluid Fens-
In the current work, we consider a two-phase fluid domain with air and water, and a solid domain for rigid bodies representing floating structures, while flexible beams are used to represent mooring lines. The proposed approach is implemented using the CFD toolkit Proteus® for simulating the physics in the fluid domain and the multiphysics simulation engine Chrono® for simulating MBD in the solid domain. The theoretical background and numerical implementation for each model are briefly presented below.
Figure 1: Illustration of arbitrarily shaped domain for typical FSI applications with two- phase flow and floating body.
Fluid dynamics and all models using the fluid domain are simulated through Proteus®, a computational toolkit for solving PDEs that is highly specialised for fluid mechanics (https: //proteustoolkit.org). Proteus® uses FEM to numerically solve transport equations within the fluid domain. A brief description of the toolkit, in relation to the methodology used to simulate the case studies in this work, is presented below.
2.1.1 Governing equations
The toolkit solves the Navier-Stokes equations for two-phase incompressible, immiscible flow. The momentum and continuity equations are as follows:
where t is the time, V = (Jj> JLj Jj)> x, У and z are the spatial coordinates, u is the fluid velocity field in vector form, p is the fluid pressure, т is the viscous shear stress tensor and p is the fluid density. The shear stress tensor may additionally include turbulent stresses, but these are not considered in the current work.
When the fluid domain contains two incompressible fluid phases (e.g., air and water), the fluid density p and dynamic viscosity // take different (i.e., discrete) values according to the phase, which depends on the position considered in the fluid domain. The two phases are separated by an interface called the free surface. It is well-known that the free surface within the context of two-phase CFD models can be tracked implicitly by setting up transport equations to capture its evolution. This concept was first used by Hilt and Nichols (1981) with the Volume Of Fluid (VOF) method, according to which a transport equation for the water volume fraction was used to capture the evolution of the free surface. Another approach for implicit tracking is the level set method, which treats the distance of any point from the free surface as a field variable and uses relevant transport equations to track its evolution (Sussman et ah, 1994).
In this work, the free surface is tracked by using a coupled level set/VOF method (Kees et ah, 2011). This approach consists of the following steps:
1. The air volume fraction aa is calculated according to the following transport equation written in conservative form:
2. The distance from the free surface фа<ы is calculated according to the non-conservative form of the level set transport relation:
where 0s(if is a signed distance function taking negative values in the water phase and positive values in the air phase. After being transported by Equation (3), фаdf is not a true signed distance function anymore and must be corrected in order to satisfy the eikonal relation:
which can be recovered by keeping the interface where фасif = 0 fixed and using a pseudo-steady state solution approach (Sussman et ah, 1994).
This approach combines two important elements of the VOF and the level set methods: mass conservation from the VOF equation and smooth and sharp representation of the interface using a signed distance function. The computational cost associated with this approach is nevertheless higher than using a single transport equation for VOF or level set method. Further progress has been made in Quezada de Luna et al. (2010) by developing a faster conservative level set scheme, but it was not implemented in the Proteus toolkit at the time of performing the simulations presented herein.
2.1.2 Boundary conditions and wave generation
Proteus® supports the use of Dirichlet and Neumann boundary conditions within the FEM framework, which are applied to the integral form of the Navier-Stokes equations and other PDE models that use the fluid mesh. The boundary conditions can be classified as Dirichlet, Advective, and Diffusive. The Dirichlet conditions, impose an explicit value of the solution at the boundary based on the weak imposition (Bazilevs and Hughes, 2007). The Advective conditions set the flux through a boundary face. The Diffusive conditions set the gradient flux (equivalent to a Neumann condition). For wall boundaries (e.g., tank walls, or surface of structures), typical free-slip or no-slip conditions are usually imposed. For open boundaries such as the top of a numerical tank, atmospheric conditions are used, imposing a Dirichlet pressure (usually a constant p = 0 for flat top) and Dirichlet VOF (usually the air value, aa = 1 in our case).
Waves are introduced in the domain at the generating boundaries and any reflection or transmission of the waves are absorbed using the relaxation zone method. Relaxation zones (generation or absorption zones) correspond to regions of the domain that introduce a source term in the fluid momentum equation:
where аь is a blending function, <*o is a constant, and Uf is the target velocity (fluid velocity from wave theory in generation zones, and zero velocity in absorption zones).
Specifically, the method implemented here is based on the work presented by Dimakopou- los et al. (2019), with c*o = Щ given the wave period T, with generation zones of length = 1A, where A is the wavelength, absorption zones of length La|)s = 2A, and using the blending function аь introduced by Jacobsen et al. (2012):
where 0 < d < 1 is the scaled distance from the boundary to the end of the relaxation zone. An illustration of relaxation zones in a typical numerical wave tank is presented in Figure 2. This methodology is capable of generating and absorbing waves from various theories, including regular (linear and nonlinear), focused, as well as plane and directional random
Figure 2: Relaxation zones and blending functions in numerical wave tank.
waves. In the current work, regular waves are generated using the Fenton Fourier theory for steady waves (Fenton, 1988), whilst random waves are generated using superimposition of linear wave components from a known spectrum.
2.1.3 Numerical solution
The Proteus® toolkit is based on the FEM, a residual-based solution method that produces an approximation to the true solution. The discretisation of all transport equations for the fluid domain -which includes free surface tracking—follow the weak formulation of the FEM, based on the introduction of test functions in the integral form of the transport equations.
A triangular (in 2D) or tetrahedral (in 3D) mesh was used to discretise the flow variables in space. Pi Lagrange elements with affine linear nodal basis and simplex Gaussian quadrature were used. Piecewise linear basis functions were used to interpolate the field variables within the element, and these were also used as test functions, following the Galerkin method. A first-order accurate implicit backward Euler scheme was employed for advancing the solution algorithm in time, as it is a simple numerically robust scheme. The timestep is restricted according to the flow velocities, following the CFL criterion. More details on the discretisation scheme can be found in de Lataillade (2019).