Starting from a specific industrial problem related to the propellant flow within a liquid propellant rocket engine, a numerical method for simulating three-dimensional, generic barotropic flows in rotating frames is developed. A novel finite volume compressible approach for unstructured grids is proposed, suitably preconditioned for accurately dealing with nearly-incompressible flows. The time-advancing is performed by a novel, generic, linearized implicit scheme. A constructive procedure for solving the one-dimensional Riemann problem associated with a generic convex barotropic state law is presented as well. All the proposed numerical ingredients are validated against one-dimensional exact solutions or three-dimensional experimental data related to complex, industrial flows also involving cavitation phenomena.