This research entails the numerical simulation of physical flow instabilities that can occur in two-phase pipe flows with a new efficient algorithm. The fluids are assumed to be immiscible, and the flow is incompressible and isothermal in a straight circular pipe section with a certain inclination. The numerical algorithm that was developed consists of a flow solver and a sharp interface model that solve the Navier-Stokes equations in cylindrical coordinates. The simulation results obtained with the new method are validated through comparison with other models and with experiments. The focus lies on obtaining an accurate and efficient algorithm that can ultimately be used for Direct Numerical Simulations or Large Eddy Simulations of turbulent two-phase pipe flows.

We present a finite difference discretization of the incompressible Navier–Stokes equations in cylindrical coordinates. This currently is, to the authors' knowledge, the only scheme available that is demonstrably capable of conserving mass, momentum and kinetic energy (in the absence of viscosity) on both uniform and non-uniform grids. Simultaneously, we treat the inherent discretization issues that arise due to the presence of the coordinate singularity at the polar axis. We demonstrate the validity of the conservation claims by performing a number of numerical experiments with the proposed scheme, and we show that it is second order accurate in space using the Method of Manufactured Solutions.