Implicit preconditioned numerical schemes for the simulation of three-dimensional barotropic flows

A numerical method for simulating three-dimensional, generic barotropic flows on unstructured grids is developed. Space and time discretizations are separately considered. A finite volume compressible approach, based on a suitable Roe numerical flux function, is proposed and the accuracy of the resulting semi-discrete formulation for nearly-incompressible flows is ensured by ad hoc preconditioning. Moreover, a linearized implicit time-advancing technique is proposed, only relying on the algebraic properties of the Roe flux function and therefore applicable to a variety of problems. This implicit strategy is extended so as to incorporate the aforementioned preconditioning. The considered numerical ingredients are firstly defined in a one-dimensional context; after validation, they are extended to three-dimensional non-rotating as well as rotating frames. Finally, the resulting numerical method is validated by considering complex industrial flows, namely the water flow around a hydrofoil (for which specific experimental data are available) and the water flow around a rotating turbo-pump inducer. By starting from a particular industrial problem (namely the numerical simulation of propellant flows around an axial inducer belonging to the feed turbo-pump system of a liquid propellant rocket engine), a numerical method which can be applied to generic barotropic flows is defined. Along the way, a constructive procedure for solving the 1D Riemann problem associated with a generic convex barotropic state law is proposed. This solution, also exploited for defining a Godunov numerical flux suitable for incorporation into finite volume schemes, is systematically used in order to define exact benchmarks for the quantitative validation of the proposed one-dimensional numerical methods.