Low-order face-based approaches for incompressible computational fluid dynamics

The fast and accurate simulation of laminar and turbulent incompressible flows is crucial
in science and engineering. Low-order numerical strategies, particularly finite-volume (FV)
methods based on cell-centred and vertex-centred approaches, remain essential in CFD, especially
in industrial settings, due to their favourable trade-off between computational cost and
accuracy. However, these methods face challenges such as stabilising convective-dominated
flows, handling velocity-pressure coupling, and managing numerical flux reconstruction on
distorted and stretched grids. The face-centred finite volume (FCFV) method emerges as an
alternative to standard FV methods. Derived from a mixed formulation of the discontinuous
Galerkin method, FCFV avoids flux reconstruction at cell faces, making its accuracy and
convergence nearly insensitive to grid quality. It also satisfies the Ladyzhenskaya-Babuˇska-
Brezzi (LBB) condition without special treatment for velocity-pressure coupling. This work
showcases the FCFV method for simulating laminar and turbulent incompressible flows for
the first time. The formulation is based on the Reynolds-averaged Navier-Stokes (RANS)
equations coupled with the negative Spalart-Allmaras (SA) model. Three new convective
stabilisations inspired by Riemann solvers are proposed, along with monolithic and staggered
solution strategies for the RANS-SA system and two relaxation strategies for pseudo-time
marching. A new hybrid pressure FCFV formulation is also introduced to improve the
FCFV accuracy in laminar flow simulations by enriching the pressure space through relaxed
compressibility conditions. Both FCFV and hybrid pressure FCFV achieve first-order convergence
of velocity, velocity gradient tensor, and pressure, accurately predicting engineering
quantities such as drag and lift on structured and unstructured meshes. By avoiding gradient
reconstruction, these methods are less sensitive to mesh quality, even on highly distorted
grids. Numerical benchmarks for laminar and turbulent, steady and transient cases assess
the performance, accuracy, and robustness of the proposed methodologies. Implemented in
Fortran 90, these methods lay the foundation for the integration of FCFV techniques within
the CFD community. The work concludes with a detailed discussion of the Fortran 90
FCFV code implementation.