Simulation of Green Water Loading Using the Navier-stokes Equations

Simulating viscous flows with a free surface causes special difficulties, since its position will change continuously. Therefore, besides solving the Navier-Stokes equations, the position of the free surface must be determined every time step. In the present method, the Navier-Stokes equations are solved on a three-dimensional Cartesian grid. A Volume-of-Fluid function is used for the position of the fluid. Since the method is able to handle arbitrary forms of the geometry, many types of industrial flow problems can be simulated. In this paper the problem of green water loading on the foredeck of a ship is discussed and a comparison is made with experimental results. Waterheights, pressures and water contours are produced and compared with model tests. Also forces on different structures placed on the deck are compared and analyzed. INTRODUCTION When a ship at sea is sailing or moving in the waves, it may get water on the foredeck. This water, which flows on the deck in high waves when the relative wave motion around the bow is exceeding the deck level, is called green water. As a result of this green water loading, damage to superstructures on the deck is still a common occurrence. The Maritime Research Institute Netherlands (MARIN) has done extensive model test research to this phenomenon during the last few years [1], [2]. In the paper a simulation method will be described with which this phenomenon can be investigated numerically. The simulation of green water flow on the foredeck of a ship is a complex problem, since the water will behave wildly when it flows on the deck, causing effects like air bubble entrapment. The tests also show complex high velocity flow patterns on the deck. Besides the model test research MARIN has done, it also investigated the non linear relative wave motions around the bow with a boundary integral method, modeling the flow with a potential function [3]. However, fluid flow is best described by the complete Navier-Stokes equations. In 1995, at the University of Groningen (RuG), the development of a computer program called ComFlo has been started which can solve fluid flow with free surfaces in 3D-complex geometries. Here the Navier-Stokes equations are solved on a Cartesian grid. No motion of the geometry has been implemented yet, so this will cause some differences between the tests and the simulation. The inflow conditions at the boundaries of the domain will be determined by the data of the model tests instead of simulating an incoming wave. MATHEMATICAL MODEL The motion of water, and in general the motion of a viscous, incompressible fluid can be described by the incompressible Navier-Stokes equations, consisting of conservation of mass and conservation of momentum: r u = 0 (1) @u @t + (u r)u = rp + u + F (2) where u = (u; v; w) is the velocity, is the density, p is the pressure, r is the gradient operator, r is the divergence operator, and is the Laplace operator. Further is the kinematic viscosity and F = (Fx; Fy; Fz) is an external body force, e.g. gravity. Further, boundary conditions are required for the solid boundary, the free surface and eventually inand outflow boundaries. At the solid boundary a no-slip condition is used: u = 0. Free-slip walls are also possible, resulting in the conditions un = 0 and = 0. Here un = u n is the component of the velocity perpendicular to the wall, = @ut @n is the tangential stress, where ut is the velocity component in the tangential direction. At the free surface the boundary conditions consist of two components: p+ 2 @un @n = p0 + 2 H (3) @un @t + @ut @n = 0 (4) where is the dynamic viscosity, p0 is the atmospheric pressure, is the surface tension and 2H is the total curvature of the surface. These boundary conditions describe the continuity of normal and tangential stresses at the free-surface. Further, for the free surface displacement an equation is required: Suppose the position of the free surface is described by s(x; t) = 0, then the movement of the free surface becomes Ds Dt = @s @t + u rs = 0: (5) At inflow boundaries the velocity u is prescribed, and at outflow boundaries the homogeneous Neumann condition @u @n is used. This is better than prescribing the normal component of the velocity, since then a boundary layer could easily be created. Further, at outflow boundaries the pressure p is set equal to the atmospheric value p0. NUMERICAL MODEL In this section the mathematical model will be discretized to obtain a numerical model. Description of geometry and free surface First a Cartesian grid is laid over the three dimensional domain. The discretization is done on a totally staggered grid, which means that the pressure will be set in the cell centers and the velocity components in the middle of the cell faces between two cells (figure 1). Like all figures of the discretization of geometries in this paper, this is a 2-dimensional example. Extension to 3D is straightforward. Since complex geometries are used, the grid cells will run through the boundaries in several ways. This is also the case for the free surface, with an extra complexity since the free surface is time-dependent. Also the main reasons why Cartesian grids are used, can be inferred from this. In the first place each cell has the same orientation which is an advantage with respect v p p