Unsteady 2D and 3D Navier-Stokes Solver with Application of Multigrid Scheme to Pressure Poisson Fractional Step on Arbitrary Unstructured Grids in Various Applications with Emphasis on Ship Motion

A 3D unsteady computer solver is presented to compute incompressible Navier-Stokes equations combined with the volume of fraction (VOF) method on an arbitrary unstructured domain. This is done to simulate fluid flows in various applications, especially around a marine vessel. The Navier-Stokes solver is based on the fractional steps method coupled with a finite volume scheme and collocated grids by which velocity components and pressure fields are defined at the center of the control volume. However, the fluxes are defined at the midpoint on their corresponding cell faces. On the other hand, the CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes) scheme is applied to capture the free surface. In the presented fractional step method, the pressure Poisson equation suffers from poor convergence rate by simple iterative methods like Successive Overrelaxation (SOR), especially in simulating complex geometrics like a ship with appendages. Therefore, to accelerate the convergence rate, an agglomeration multigrid method is applied on arbitrary moving mesh for solving pressure Poisson equation with two well-known cycles, V and W. In order to maintain accuracy, the geometry details should not change in grid coarsening procedure. Therefore, the boundary faces are assumed to be fixed in all grids level. This assumption requires nonstandard cells in coarsening procedures. To investigate the performance of the applied algorithm, various flows including one and two-phase flows are studied in two and three dimensions. It is found that the multigrid method can speed up the convergence rate of fractional step twofold. In most cases (not all), W cycle displays better performance. It is also concluded that the efficiency of the cycle depends on the number of meshes and complexity of the problem and this is mainly due to the data transferring between grids. Therefore, the type of cycle should be selected judiciously and carefully, while considering the mesh size and flow properties.