Two and three dimensional multi-moment finite volume solver for incompressible Navier–Stokes equations on unstructured grids with arbitrary quadrilateral and hexahedral elements

• A novel solver for incompressible viscous flows on 2 and 3D unstructured grid.
• Simultaneously updating cell average and point value as the computational variables.
• Significant superiority in accuracy and robustness over conventional FVM.
• A practical formulation well-balancing computational complexity and accuracy.

This paper presents an extension of the robust and accurate finite volume method (FVM), so-called VPM (Volume integrated average and Point value based Multi-moment) method, to structured and unstructured grids with arbitrary quadrilateral and hexahedral mesh elements. The VPM method treats two different discretized moments of the physical fields, i.e. the volume integrated average (VIA) and the point values (PV) at the vertices of each cell, as the computational variables, which distinguishes it from conventional FVM. Given the local degrees of freedom in terms of VIA and PVs, we have properly designed the interpolation polynomials of reconstruction for quadrilateral and hexahedral mesh elements, which are then used to build a numerical formulation for incompressible viscous fluid dynamics. Numerical results of benchmark tests in both 2 and 3 dimensions are presented to verify the accuracy and robustness of the proposed method, which shows significant improvement in comparison with conventional FVM. The proposed formulation provides a practical solver that is well-balanced between numerical accuracy and algorithmic complexity.