Modern CFD applications require the treatment of general complex domains to accurately model the emerging flow patterns. In the present work, a new finite difference scheme is employed and tested for the numerical solution of the incompressible Navier-Stokes equations in a complex domain described in curvilinear coordinates. A staggered grid discretization is used on both the physical and computational domains. A subgrid based computation of the Jacobian and the metric coefficients of the transformation is used. The discretization methods employed with the current methodology include, low (1st, 2nd) and higher order (4th) compact schemes for the temporal, advection and diffusion terms of the N-S equations. The temporal discretization is carried out by either a 1st Order Explicit Scheme, a 2nd Order Predictor--Corrector Method, or a 4th Order Explicit Runge-Kutta Method. Since the algorithm is tested over a variety of complex domains, the effective boundary conditions treatment is very important, especially in curvilinear coordinates, where the shape and the diversity of the boundary regions (slip / no-slip walls, inlet, outlet, symmetry, periodic, free surface, etc.) deviates from its simpler cartesian counterpart. The incompressibility condition, properly transformed in curvilinear coordinates, is enforced by an iterative procedure employing either a modified local pressure correction technique or the globally defined numerical solution of a general elliptic BVP (a Poisson-type equation). The enforcement of the incompressibility condition to the numerical solution, at each time step, produced by a high order numerical scheme is commonly accepted to be the most computationally demanding part of the overall algorithm. To expand the computational applications of the current methodology, the governing equations can be depth-averaged to produce the well-known shallow water equations, which include bathymerty forces and friction. Coupled with the Exner equation for describing the morphological evolution, a single system of equations is numerically solved by a high-resolution finite volume scheme of the relaxation type. This numerical scheme is based on classical relaxation models previously developed, where neither approximate Riemann solvers nor characteristic decompositions are required. Bed-load sediment transport simulations are presented, targeted to describe the morphodynamics in coastal areas. Different forms of the bed-load transport flux are considered in the Exner equation. The results obtained by the proposed Navier-Stokes solution algorithm, exhibit very good agreement with other experimental and numerical calculations for a variety of flow domains and grid configurations. The overall numerical solver effectively treats the general complex domains, for different types of boundary conditions. To test the validity of the results, obtained by the shallow water equations coupled with the Exner equation, comparisons are made for benchmark cases in the bibliography, as well as with commercial software, in an coastal area on the island of Crete.
