Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates

A finite-difference method for solving three-dimensional time-dependent incompressible Navier–Stokes equations in arbitrary curvilinear orthogonal coordinates is presented. The method is oriented on turbulent flow simulations and consists of a second-order central difference approximation in space and a third-order semi-implicit Runge–Kutta scheme for time advancement. Spatial discretization retains some important properties of the Navier–Stokes equations, including energy conservation by the nonlinear and pressure-gradient terms. Numerical tests cover Cartesian, cylindrical-polar, spherical, cylindrical elliptic and cylindrical bipolar coordinate systems. Both laminar and turbulent flows are considered demonstrating reasonable accuracy and stability of the method.