A novel ghost-cell immersed boundary method (GCIBM) for large eddy simulation (LES) of two dimensional (2D) flows with free surface is presented. 2D incompressible Navier-Stokes equations are numerically solved with finite volume method. Spatial terms are discretized with fourth-order compact scheme on collocated grid. Time stepping is implemented employing a fourth-order explicit Runge-Kutta algorithm. Momentum interpolation method (MIM) is used to deal with velocity-pressure coupling. Volume of fluid (VOF) with piecewise linear interface calculation (PLIC-VOF) is applied to trace free surface. The GCIBM is verified with uniform flows around a circle with Reynolds number 40 and 100, respectively. Investigation on free surface piercing fixed rectangle is carried out for flows with various Reynolds numbers at several different draughts. Velocity vectors, vorticity contours and the corresponding drags are obtained and shown. Tendencies of the flows with respect to draught and Reynolds number are summarized.
Flow around a rectangle is a quest in offshore engineering. Recently, Camarri et el., 2009; Tian et el., 2013; Maiti et el, 2014; Bosch et el., 2015 etc. studied single phase rectangle uniform flow, i.e. uniform flow around a rectangle surrounded by an unbounded fluid without free surface. Such flows are found separate from the rectangle at the leading corner. However, ocean structures, like semi-submersibles, FLNG and FPSO, often pierce free surface, and free surface effect is an important issue. Luo et el., 2015 developed a consistent particle method (CPM) for solving two-phase flows of fluids with large density difference. Later, Luo et el., 2016 applied CPM to simulate water-air flows with compressible air pockets. Flow around free surface piercing fixed rectangle is typical and worth to investigate.
Numerical simulation of solid boundary flows is still a challenge task. Unstructured grid method, such as Verzicco et al, 2000, is traditionally applied to implement rigid boundary condition. But its requirement for much computer resources and the need of grid generation technique greatly limits its application. Recently, the immersed boundary method (IBM) tries to improve the limitation. There are mainly two class of IBM, that is, the (continuous) diffuse interface method and the (discrete) sharp interface method. Comparing with sharp interface method, accuracy of diffuse interface method is lower because solid boundary is blurred to a band of grid points by utilizing a regularized delta function, and smearing occurs across the boundary. Peskin, 1972 proposed IBM employing the diffuse interface method originally to simulate the blood flow near heart valves. The diffuse interface method has advantages in simulation of the flows with flexible boundary, while on rigid boundary, a feedback forcing need be introduced (refer to Lai and Peskin, 2000). After introduction of a feedback forcing, no-slip boundary condition will restrict time step length and spurious oscillations may occur. To release the restriction on time step length and remove the spurious oscillations, Mohd-Yusof 1997 proposed a sharp-interface method, which employs a direct forcing to ensure actual velocity distribution on solid boundary. Later, Fadlun et al, 2000 further developed it by taking finite difference on staggered grid for three dimensional flows. Another sharp interface method to implement the solid boundary condition, like the one by Udaykumar et al, 2001, reshapes the immersed boundary and makes it conform to the solid boundary. This treatment is particularly suitable for two dimensional flows, and rarely used in three dimensions.